text: //===- ScalarEvolution.cpp - Scalar Evolution Analysis --------------------===//
  //
  // Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
  // See https://llvm.org/LICENSE.txt for license information.
  // SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
  //
  //===----------------------------------------------------------------------===//
  //
  // This file contains the implementation of the scalar evolution analysis
  // engine, which is used primarily to analyze expressions involving induction
  // variables in loops.
  //
  // There are several aspects to this library.  First is the representation of
  // scalar expressions, which are represented as subclasses of the SCEV class.
  // These classes are used to represent certain types of subexpressions that we
  // can handle. We only create one SCEV of a particular shape, so
  // pointer-comparisons for equality are legal.
  //
  // One important aspect of the SCEV objects is that they are never cyclic, even
  // if there is a cycle in the dataflow for an expression (ie, a PHI node).  If
  // the PHI node is one of the idioms that we can represent (e.g., a polynomial
  // recurrence) then we represent it directly as a recurrence node, otherwise we
  // represent it as a SCEVUnknown node.
  //
  // In addition to being able to represent expressions of various types, we also
  // have folders that are used to build the *canonical* representation for a
  // particular expression.  These folders are capable of using a variety of
  // rewrite rules to simplify the expressions.
  //
  // Once the folders are defined, we can implement the more interesting
  // higher-level code, such as the code that recognizes PHI nodes of various
  // types, computes the execution count of a loop, etc.
  //
  // TODO: We should use these routines and value representations to implement
  // dependence analysis!
  //
  //===----------------------------------------------------------------------===//
  //
  // There are several good references for the techniques used in this analysis.
  //
  //  Chains of recurrences -- a method to expedite the evaluation
  //  of closed-form functions
  //  Olaf Bachmann, Paul S. Wang, Eugene V. Zima
  //
  //  On computational properties of chains of recurrences
  //  Eugene V. Zima
  //
  //  Symbolic Evaluation of Chains of Recurrences for Loop Optimization
  //  Robert A. van Engelen
  //
  //  Efficient Symbolic Analysis for Optimizing Compilers
  //  Robert A. van Engelen
  //
  //  Using the chains of recurrences algebra for data dependence testing and
  //  induction variable substitution
  //  MS Thesis, Johnie Birch
  //
  //===----------------------------------------------------------------------===//
  
  #include "llvm/Analysis/ScalarEvolution.h"
  #include "llvm/ADT/APInt.h"
  #include "llvm/ADT/ArrayRef.h"
  #include "llvm/ADT/DenseMap.h"
  #include "llvm/ADT/DepthFirstIterator.h"
  #include "llvm/ADT/EquivalenceClasses.h"
  #include "llvm/ADT/FoldingSet.h"
  #include "llvm/ADT/None.h"
  #include "llvm/ADT/Optional.h"
  #include "llvm/ADT/STLExtras.h"
  #include "llvm/ADT/ScopeExit.h"
  #include "llvm/ADT/Sequence.h"
  #include "llvm/ADT/SetVector.h"
  #include "llvm/ADT/SmallPtrSet.h"
  #include "llvm/ADT/SmallSet.h"
  #include "llvm/ADT/SmallVector.h"
  #include "llvm/ADT/Statistic.h"
  #include "llvm/ADT/StringRef.h"
  #include "llvm/Analysis/AssumptionCache.h"
  #include "llvm/Analysis/ConstantFolding.h"
  #include "llvm/Analysis/InstructionSimplify.h"
  #include "llvm/Analysis/LoopInfo.h"
  #include "llvm/Analysis/ScalarEvolutionDivision.h"
  #include "llvm/Analysis/ScalarEvolutionExpressions.h"
  #include "llvm/Analysis/TargetLibraryInfo.h"
  #include "llvm/Analysis/ValueTracking.h"
  #include "llvm/Config/llvm-config.h"
  #include "llvm/IR/Argument.h"
  #include "llvm/IR/BasicBlock.h"
  #include "llvm/IR/CFG.h"
  #include "llvm/IR/Constant.h"
  #include "llvm/IR/ConstantRange.h"
  #include "llvm/IR/Constants.h"
  #include "llvm/IR/DataLayout.h"
  #include "llvm/IR/DerivedTypes.h"
  #include "llvm/IR/Dominators.h"
  #include "llvm/IR/Function.h"
  #include "llvm/IR/GlobalAlias.h"
  #include "llvm/IR/GlobalValue.h"
  #include "llvm/IR/GlobalVariable.h"
  #include "llvm/IR/InstIterator.h"
  #include "llvm/IR/InstrTypes.h"
  #include "llvm/IR/Instruction.h"
  #include "llvm/IR/Instructions.h"
  #include "llvm/IR/IntrinsicInst.h"
  #include "llvm/IR/Intrinsics.h"
  #include "llvm/IR/LLVMContext.h"
  #include "llvm/IR/Metadata.h"
  #include "llvm/IR/Operator.h"
  #include "llvm/IR/PatternMatch.h"
  #include "llvm/IR/Type.h"
  #include "llvm/IR/Use.h"
  #include "llvm/IR/User.h"
  #include "llvm/IR/Value.h"
  #include "llvm/IR/Verifier.h"
  #include "llvm/InitializePasses.h"
  #include "llvm/Pass.h"
  #include "llvm/Support/Casting.h"
  #include "llvm/Support/CommandLine.h"
  #include "llvm/Support/Compiler.h"
  #include "llvm/Support/Debug.h"
  #include "llvm/Support/ErrorHandling.h"
  #include "llvm/Support/KnownBits.h"
  #include "llvm/Support/SaveAndRestore.h"
  #include "llvm/Support/raw_ostream.h"
  #include <algorithm>
  #include <cassert>
  #include <climits>
  #include <cstddef>
  #include <cstdint>
  #include <cstdlib>
  #include <map>
  #include <memory>
  #include <tuple>
  #include <utility>
  #include <vector>
  
  using namespace llvm;
  using namespace PatternMatch;
  
  #define DEBUG_TYPE "scalar-evolution"
  
  STATISTIC(NumArrayLenItCounts,
            "Number of trip counts computed with array length");
  STATISTIC(NumTripCountsComputed,
            "Number of loops with predictable loop counts");
  STATISTIC(NumTripCountsNotComputed,
            "Number of loops without predictable loop counts");
  STATISTIC(NumBruteForceTripCountsComputed,
            "Number of loops with trip counts computed by force");
  
  static cl::opt<unsigned>
  MaxBruteForceIterations("scalar-evolution-max-iterations", cl::ReallyHidden,
                          cl::ZeroOrMore,
                          cl::desc("Maximum number of iterations SCEV will "
                                   "symbolically execute a constant "
                                   "derived loop"),
                          cl::init(100));
  
  // FIXME: Enable this with EXPENSIVE_CHECKS when the test suite is clean.
  static cl::opt<bool> VerifySCEV(
      "verify-scev", cl::Hidden,
      cl::desc("Verify ScalarEvolution's backedge taken counts (slow)"));
  static cl::opt<bool> VerifySCEVStrict(
      "verify-scev-strict", cl::Hidden,
      cl::desc("Enable stricter verification with -verify-scev is passed"));
  static cl::opt<bool>
      VerifySCEVMap("verify-scev-maps", cl::Hidden,
                    cl::desc("Verify no dangling value in ScalarEvolution's "
                             "ExprValueMap (slow)"));
  
  static cl::opt<bool> VerifyIR(
      "scev-verify-ir", cl::Hidden,
      cl::desc("Verify IR correctness when making sensitive SCEV queries (slow)"),
      cl::init(false));
  
  static cl::opt<unsigned> MulOpsInlineThreshold(
      "scev-mulops-inline-threshold", cl::Hidden,
      cl::desc("Threshold for inlining multiplication operands into a SCEV"),
      cl::init(32));
  
  static cl::opt<unsigned> AddOpsInlineThreshold(
      "scev-addops-inline-threshold", cl::Hidden,
      cl::desc("Threshold for inlining addition operands into a SCEV"),
      cl::init(500));
  
  static cl::opt<unsigned> MaxSCEVCompareDepth(
      "scalar-evolution-max-scev-compare-depth", cl::Hidden,
      cl::desc("Maximum depth of recursive SCEV complexity comparisons"),
      cl::init(32));
  
  static cl::opt<unsigned> MaxSCEVOperationsImplicationDepth(
      "scalar-evolution-max-scev-operations-implication-depth", cl::Hidden,
      cl::desc("Maximum depth of recursive SCEV operations implication analysis"),
      cl::init(2));
  
  static cl::opt<unsigned> MaxValueCompareDepth(
      "scalar-evolution-max-value-compare-depth", cl::Hidden,
      cl::desc("Maximum depth of recursive value complexity comparisons"),
      cl::init(2));
  
  static cl::opt<unsigned>
      MaxArithDepth("scalar-evolution-max-arith-depth", cl::Hidden,
                    cl::desc("Maximum depth of recursive arithmetics"),
                    cl::init(32));
  
  static cl::opt<unsigned> MaxConstantEvolvingDepth(
      "scalar-evolution-max-constant-evolving-depth", cl::Hidden,
      cl::desc("Maximum depth of recursive constant evolving"), cl::init(32));
  
  static cl::opt<unsigned>
      MaxCastDepth("scalar-evolution-max-cast-depth", cl::Hidden,
                   cl::desc("Maximum depth of recursive SExt/ZExt/Trunc"),
                   cl::init(8));
  
  static cl::opt<unsigned>
      MaxAddRecSize("scalar-evolution-max-add-rec-size", cl::Hidden,
                    cl::desc("Max coefficients in AddRec during evolving"),
                    cl::init(8));
  
  static cl::opt<unsigned>
      HugeExprThreshold("scalar-evolution-huge-expr-threshold", cl::Hidden,
                    cl::desc("Size of the expression which is considered huge"),
                    cl::init(4096));
  
  static cl::opt<bool>
  ClassifyExpressions("scalar-evolution-classify-expressions",
      cl::Hidden, cl::init(true),
      cl::desc("When printing analysis, include information on every instruction"));
  
  static cl::opt<bool> UseExpensiveRangeSharpening(
      "scalar-evolution-use-expensive-range-sharpening", cl::Hidden,
      cl::init(false),
      cl::desc("Use more powerful methods of sharpening expression ranges. May "
               "be costly in terms of compile time"));
  
  //===----------------------------------------------------------------------===//
  //                           SCEV class definitions
  //===----------------------------------------------------------------------===//
  
  //===----------------------------------------------------------------------===//
  // Implementation of the SCEV class.
  //
  
  #if !defined(NDEBUG) || defined(LLVM_ENABLE_DUMP)
  LLVM_DUMP_METHOD void SCEV::dump() const {
    print(dbgs());
    dbgs() << '\n';
  }
  #endif
  
  void SCEV::print(raw_ostream &OS) const {
    switch (getSCEVType()) {
    case scConstant:
      cast<SCEVConstant>(this)->getValue()->printAsOperand(OS, false);
      return;
    case scPtrToInt: {
      const SCEVPtrToIntExpr *PtrToInt = cast<SCEVPtrToIntExpr>(this);
      const SCEV *Op = PtrToInt->getOperand();
      OS << "(ptrtoint " << *Op->getType() << " " << *Op << " to "
         << *PtrToInt->getType() << ")";
      return;
    }
    case scTruncate: {
      const SCEVTruncateExpr *Trunc = cast<SCEVTruncateExpr>(this);
      const SCEV *Op = Trunc->getOperand();
      OS << "(trunc " << *Op->getType() << " " << *Op << " to "
         << *Trunc->getType() << ")";
      return;
    }
    case scZeroExtend: {
      const SCEVZeroExtendExpr *ZExt = cast<SCEVZeroExtendExpr>(this);
      const SCEV *Op = ZExt->getOperand();
      OS << "(zext " << *Op->getType() << " " << *Op << " to "
         << *ZExt->getType() << ")";
      return;
    }
    case scSignExtend: {
      const SCEVSignExtendExpr *SExt = cast<SCEVSignExtendExpr>(this);
      const SCEV *Op = SExt->getOperand();
      OS << "(sext " << *Op->getType() << " " << *Op << " to "
         << *SExt->getType() << ")";
      return;
    }
    case scAddRecExpr: {
      const SCEVAddRecExpr *AR = cast<SCEVAddRecExpr>(this);
      OS << "{" << *AR->getOperand(0);
      for (unsigned i = 1, e = AR->getNumOperands(); i != e; ++i)
        OS << ",+," << *AR->getOperand(i);
      OS << "}<";
      if (AR->hasNoUnsignedWrap())
        OS << "nuw><";
      if (AR->hasNoSignedWrap())
        OS << "nsw><";
      if (AR->hasNoSelfWrap() &&
          !AR->getNoWrapFlags((NoWrapFlags)(FlagNUW | FlagNSW)))
        OS << "nw><";
      AR->getLoop()->getHeader()->printAsOperand(OS, /*PrintType=*/false);
      OS << ">";
      return;
    }
    case scAddExpr:
    case scMulExpr:
    case scUMaxExpr:
    case scSMaxExpr:
    case scUMinExpr:
    case scSMinExpr: {
      const SCEVNAryExpr *NAry = cast<SCEVNAryExpr>(this);
      const char *OpStr = nullptr;
      switch (NAry->getSCEVType()) {
      case scAddExpr: OpStr = " + "; break;
      case scMulExpr: OpStr = " * "; break;
      case scUMaxExpr: OpStr = " umax "; break;
      case scSMaxExpr: OpStr = " smax "; break;
      case scUMinExpr:
        OpStr = " umin ";
        break;
      case scSMinExpr:
        OpStr = " smin ";
        break;
      default:
        llvm_unreachable("There are no other nary expression types.");
      }
      OS << "(";
      ListSeparator LS(OpStr);
      for (const SCEV *Op : NAry->operands())
        OS << LS << *Op;
      OS << ")";
      switch (NAry->getSCEVType()) {
      case scAddExpr:
      case scMulExpr:
        if (NAry->hasNoUnsignedWrap())
          OS << "<nuw>";
        if (NAry->hasNoSignedWrap())
          OS << "<nsw>";
        break;
      default:
        // Nothing to print for other nary expressions.
        break;
      }
      return;
    }
    case scUDivExpr: {
      const SCEVUDivExpr *UDiv = cast<SCEVUDivExpr>(this);
      OS << "(" << *UDiv->getLHS() << " /u " << *UDiv->getRHS() << ")";
      return;
    }
    case scUnknown: {
      const SCEVUnknown *U = cast<SCEVUnknown>(this);
      Type *AllocTy;
      if (U->isSizeOf(AllocTy)) {
        OS << "sizeof(" << *AllocTy << ")";
        return;
      }
      if (U->isAlignOf(AllocTy)) {
        OS << "alignof(" << *AllocTy << ")";
        return;
      }
  
      Type *CTy;
      Constant *FieldNo;
      if (U->isOffsetOf(CTy, FieldNo)) {
        OS << "offsetof(" << *CTy << ", ";
        FieldNo->printAsOperand(OS, false);
        OS << ")";
        return;
      }
  
      // Otherwise just print it normally.
      U->getValue()->printAsOperand(OS, false);
      return;
    }
    case scCouldNotCompute:
      OS << "***COULDNOTCOMPUTE***";
      return;
    }
    llvm_unreachable("Unknown SCEV kind!");
  }
  
  Type *SCEV::getType() const {
    switch (getSCEVType()) {
    case scConstant:
      return cast<SCEVConstant>(this)->getType();
    case scPtrToInt:
    case scTruncate:
    case scZeroExtend:
    case scSignExtend:
      return cast<SCEVCastExpr>(this)->getType();
    case scAddRecExpr:
      return cast<SCEVAddRecExpr>(this)->getType();
    case scMulExpr:
      return cast<SCEVMulExpr>(this)->getType();
    case scUMaxExpr:
    case scSMaxExpr:
    case scUMinExpr:
    case scSMinExpr:
      return cast<SCEVMinMaxExpr>(this)->getType();
    case scAddExpr:
      return cast<SCEVAddExpr>(this)->getType();
    case scUDivExpr:
      return cast<SCEVUDivExpr>(this)->getType();
    case scUnknown:
      return cast<SCEVUnknown>(this)->getType();
    case scCouldNotCompute:
      llvm_unreachable("Attempt to use a SCEVCouldNotCompute object!");
    }
    llvm_unreachable("Unknown SCEV kind!");
  }
  
  bool SCEV::isZero() const {
    if (const SCEVConstant *SC = dyn_cast<SCEVConstant>(this))
      return SC->getValue()->isZero();
    return false;
  }
  
  bool SCEV::isOne() const {
    if (const SCEVConstant *SC = dyn_cast<SCEVConstant>(this))
      return SC->getValue()->isOne();
    return false;
  }
  
  bool SCEV::isAllOnesValue() const {
    if (const SCEVConstant *SC = dyn_cast<SCEVConstant>(this))
      return SC->getValue()->isMinusOne();
    return false;
  }
  
  bool SCEV::isNonConstantNegative() const {
    const SCEVMulExpr *Mul = dyn_cast<SCEVMulExpr>(this);
    if (!Mul) return false;
  
    // If there is a constant factor, it will be first.
    const SCEVConstant *SC = dyn_cast<SCEVConstant>(Mul->getOperand(0));
    if (!SC) return false;
  
    // Return true if the value is negative, this matches things like (-42 * V).
    return SC->getAPInt().isNegative();
  }
  
  SCEVCouldNotCompute::SCEVCouldNotCompute() :
    SCEV(FoldingSetNodeIDRef(), scCouldNotCompute, 0) {}
  
  bool SCEVCouldNotCompute::classof(const SCEV *S) {
    return S->getSCEVType() == scCouldNotCompute;
  }
  
  const SCEV *ScalarEvolution::getConstant(ConstantInt *V) {
    FoldingSetNodeID ID;
    ID.AddInteger(scConstant);
    ID.AddPointer(V);
    void *IP = nullptr;
    if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;
    SCEV *S = new (SCEVAllocator) SCEVConstant(ID.Intern(SCEVAllocator), V);
    UniqueSCEVs.InsertNode(S, IP);
    return S;
  }
  
  const SCEV *ScalarEvolution::getConstant(const APInt &Val) {
    return getConstant(ConstantInt::get(getContext(), Val));
  }
  
  const SCEV *
  ScalarEvolution::getConstant(Type *Ty, uint64_t V, bool isSigned) {
    IntegerType *ITy = cast<IntegerType>(getEffectiveSCEVType(Ty));
    return getConstant(ConstantInt::get(ITy, V, isSigned));
  }
  
  SCEVCastExpr::SCEVCastExpr(const FoldingSetNodeIDRef ID, SCEVTypes SCEVTy,
                             const SCEV *op, Type *ty)
      : SCEV(ID, SCEVTy, computeExpressionSize(op)), Ty(ty) {
    Operands[0] = op;
  }
  
  SCEVPtrToIntExpr::SCEVPtrToIntExpr(const FoldingSetNodeIDRef ID, const SCEV *Op,
                                     Type *ITy)
      : SCEVCastExpr(ID, scPtrToInt, Op, ITy) {
    assert(getOperand()->getType()->isPointerTy() && Ty->isIntegerTy() &&
           "Must be a non-bit-width-changing pointer-to-integer cast!");
  }
  
  SCEVIntegralCastExpr::SCEVIntegralCastExpr(const FoldingSetNodeIDRef ID,
                                             SCEVTypes SCEVTy, const SCEV *op,
                                             Type *ty)
      : SCEVCastExpr(ID, SCEVTy, op, ty) {}
  
  SCEVTruncateExpr::SCEVTruncateExpr(const FoldingSetNodeIDRef ID, const SCEV *op,
                                     Type *ty)
      : SCEVIntegralCastExpr(ID, scTruncate, op, ty) {
    assert(getOperand()->getType()->isIntOrPtrTy() && Ty->isIntOrPtrTy() &&
           "Cannot truncate non-integer value!");
  }
  
  SCEVZeroExtendExpr::SCEVZeroExtendExpr(const FoldingSetNodeIDRef ID,
                                         const SCEV *op, Type *ty)
      : SCEVIntegralCastExpr(ID, scZeroExtend, op, ty) {
    assert(getOperand()->getType()->isIntOrPtrTy() && Ty->isIntOrPtrTy() &&
           "Cannot zero extend non-integer value!");
  }
  
  SCEVSignExtendExpr::SCEVSignExtendExpr(const FoldingSetNodeIDRef ID,
                                         const SCEV *op, Type *ty)
      : SCEVIntegralCastExpr(ID, scSignExtend, op, ty) {
    assert(getOperand()->getType()->isIntOrPtrTy() && Ty->isIntOrPtrTy() &&
           "Cannot sign extend non-integer value!");
  }
  
  void SCEVUnknown::deleted() {
    // Clear this SCEVUnknown from various maps.
    SE->forgetMemoizedResults(this);
  
    // Remove this SCEVUnknown from the uniquing map.
    SE->UniqueSCEVs.RemoveNode(this);
  
    // Release the value.
    setValPtr(nullptr);
  }
  
  void SCEVUnknown::allUsesReplacedWith(Value *New) {
    // Remove this SCEVUnknown from the uniquing map.
    SE->UniqueSCEVs.RemoveNode(this);
  
    // Update this SCEVUnknown to point to the new value. This is needed
    // because there may still be outstanding SCEVs which still point to
    // this SCEVUnknown.
    setValPtr(New);
  }
  
  bool SCEVUnknown::isSizeOf(Type *&AllocTy) const {
    if (ConstantExpr *VCE = dyn_cast<ConstantExpr>(getValue()))
      if (VCE->getOpcode() == Instruction::PtrToInt)
        if (ConstantExpr *CE = dyn_cast<ConstantExpr>(VCE->getOperand(0)))
          if (CE->getOpcode() == Instruction::GetElementPtr &&
              CE->getOperand(0)->isNullValue() &&
              CE->getNumOperands() == 2)
            if (ConstantInt *CI = dyn_cast<ConstantInt>(CE->getOperand(1)))
              if (CI->isOne()) {
                AllocTy = cast<GEPOperator>(CE)->getSourceElementType();
                return true;
              }
  
    return false;
  }
  
  bool SCEVUnknown::isAlignOf(Type *&AllocTy) const {
    if (ConstantExpr *VCE = dyn_cast<ConstantExpr>(getValue()))
      if (VCE->getOpcode() == Instruction::PtrToInt)
        if (ConstantExpr *CE = dyn_cast<ConstantExpr>(VCE->getOperand(0)))
          if (CE->getOpcode() == Instruction::GetElementPtr &&
              CE->getOperand(0)->isNullValue()) {
            Type *Ty = cast<GEPOperator>(CE)->getSourceElementType();
            if (StructType *STy = dyn_cast<StructType>(Ty))
              if (!STy->isPacked() &&
                  CE->getNumOperands() == 3 &&
                  CE->getOperand(1)->isNullValue()) {
                if (ConstantInt *CI = dyn_cast<ConstantInt>(CE->getOperand(2)))
                  if (CI->isOne() &&
                      STy->getNumElements() == 2 &&
                      STy->getElementType(0)->isIntegerTy(1)) {
                    AllocTy = STy->getElementType(1);
                    return true;
                  }
              }
          }
  
    return false;
  }
  
  bool SCEVUnknown::isOffsetOf(Type *&CTy, Constant *&FieldNo) const {
    if (ConstantExpr *VCE = dyn_cast<ConstantExpr>(getValue()))
      if (VCE->getOpcode() == Instruction::PtrToInt)
        if (ConstantExpr *CE = dyn_cast<ConstantExpr>(VCE->getOperand(0)))
          if (CE->getOpcode() == Instruction::GetElementPtr &&
              CE->getNumOperands() == 3 &&
              CE->getOperand(0)->isNullValue() &&
              CE->getOperand(1)->isNullValue()) {
            Type *Ty = cast<GEPOperator>(CE)->getSourceElementType();
            // Ignore vector types here so that ScalarEvolutionExpander doesn't
            // emit getelementptrs that index into vectors.
            if (Ty->isStructTy() || Ty->isArrayTy()) {
              CTy = Ty;
              FieldNo = CE->getOperand(2);
              return true;
            }
          }
  
    return false;
  }
  
  //===----------------------------------------------------------------------===//
  //                               SCEV Utilities
  //===----------------------------------------------------------------------===//
  
  /// Compare the two values \p LV and \p RV in terms of their "complexity" where
  /// "complexity" is a partial (and somewhat ad-hoc) relation used to order
  /// operands in SCEV expressions.  \p EqCache is a set of pairs of values that
  /// have been previously deemed to be "equally complex" by this routine.  It is
  /// intended to avoid exponential time complexity in cases like:
  ///
  ///   %a = f(%x, %y)
  ///   %b = f(%a, %a)
  ///   %c = f(%b, %b)
  ///
  ///   %d = f(%x, %y)
  ///   %e = f(%d, %d)
  ///   %f = f(%e, %e)
  ///
  ///   CompareValueComplexity(%f, %c)
  ///
  /// Since we do not continue running this routine on expression trees once we
  /// have seen unequal values, there is no need to track them in the cache.
  static int
  CompareValueComplexity(EquivalenceClasses<const Value *> &EqCacheValue,
                         const LoopInfo *const LI, Value *LV, Value *RV,
                         unsigned Depth) {
    if (Depth > MaxValueCompareDepth || EqCacheValue.isEquivalent(LV, RV))
      return 0;
  
    // Order pointer values after integer values. This helps SCEVExpander form
    // GEPs.
    bool LIsPointer = LV->getType()->isPointerTy(),
         RIsPointer = RV->getType()->isPointerTy();
    if (LIsPointer != RIsPointer)
      return (int)LIsPointer - (int)RIsPointer;
  
    // Compare getValueID values.
    unsigned LID = LV->getValueID(), RID = RV->getValueID();
    if (LID != RID)
      return (int)LID - (int)RID;
  
    // Sort arguments by their position.
    if (const auto *LA = dyn_cast<Argument>(LV)) {
      const auto *RA = cast<Argument>(RV);
      unsigned LArgNo = LA->getArgNo(), RArgNo = RA->getArgNo();
      return (int)LArgNo - (int)RArgNo;
    }
  
    if (const auto *LGV = dyn_cast<GlobalValue>(LV)) {
      const auto *RGV = cast<GlobalValue>(RV);
  
      const auto IsGVNameSemantic = [&](const GlobalValue *GV) {
        auto LT = GV->getLinkage();
        return !(GlobalValue::isPrivateLinkage(LT) ||
                 GlobalValue::isInternalLinkage(LT));
      };
  
      // Use the names to distinguish the two values, but only if the
      // names are semantically important.
      if (IsGVNameSemantic(LGV) && IsGVNameSemantic(RGV))
        return LGV->getName().compare(RGV->getName());
    }
  
    // For instructions, compare their loop depth, and their operand count.  This
    // is pretty loose.
    if (const auto *LInst = dyn_cast<Instruction>(LV)) {
      const auto *RInst = cast<Instruction>(RV);
  
      // Compare loop depths.
      const BasicBlock *LParent = LInst->getParent(),
                       *RParent = RInst->getParent();
      if (LParent != RParent) {
        unsigned LDepth = LI->getLoopDepth(LParent),
                 RDepth = LI->getLoopDepth(RParent);
        if (LDepth != RDepth)
          return (int)LDepth - (int)RDepth;
      }
  
      // Compare the number of operands.
      unsigned LNumOps = LInst->getNumOperands(),
               RNumOps = RInst->getNumOperands();
      if (LNumOps != RNumOps)
        return (int)LNumOps - (int)RNumOps;
  
      for (unsigned Idx : seq(0u, LNumOps)) {
        int Result =
            CompareValueComplexity(EqCacheValue, LI, LInst->getOperand(Idx),
                                   RInst->getOperand(Idx), Depth + 1);
        if (Result != 0)
          return Result;
      }
    }
  
    EqCacheValue.unionSets(LV, RV);
    return 0;
  }
  
  // Return negative, zero, or positive, if LHS is less than, equal to, or greater
  // than RHS, respectively. A three-way result allows recursive comparisons to be
  // more efficient.
  // If the max analysis depth was reached, return None, assuming we do not know
  // if they are equivalent for sure.
  static Optional<int>
  CompareSCEVComplexity(EquivalenceClasses<const SCEV *> &EqCacheSCEV,
                        EquivalenceClasses<const Value *> &EqCacheValue,
                        const LoopInfo *const LI, const SCEV *LHS,
                        const SCEV *RHS, DominatorTree &DT, unsigned Depth = 0) {
    // Fast-path: SCEVs are uniqued so we can do a quick equality check.
    if (LHS == RHS)
      return 0;
  
    // Primarily, sort the SCEVs by their getSCEVType().
    SCEVTypes LType = LHS->getSCEVType(), RType = RHS->getSCEVType();
    if (LType != RType)
      return (int)LType - (int)RType;
  
    if (EqCacheSCEV.isEquivalent(LHS, RHS))
      return 0;
  
    if (Depth > MaxSCEVCompareDepth)
      return None;
  
    // Aside from the getSCEVType() ordering, the particular ordering
    // isn't very important except that it's beneficial to be consistent,
    // so that (a + b) and (b + a) don't end up as different expressions.
    switch (LType) {
    case scUnknown: {
      const SCEVUnknown *LU = cast<SCEVUnknown>(LHS);
      const SCEVUnknown *RU = cast<SCEVUnknown>(RHS);
  
      int X = CompareValueComplexity(EqCacheValue, LI, LU->getValue(),
                                     RU->getValue(), Depth + 1);
      if (X == 0)
        EqCacheSCEV.unionSets(LHS, RHS);
      return X;
    }
  
    case scConstant: {
      const SCEVConstant *LC = cast<SCEVConstant>(LHS);
      const SCEVConstant *RC = cast<SCEVConstant>(RHS);
  
      // Compare constant values.
      const APInt &LA = LC->getAPInt();
      const APInt &RA = RC->getAPInt();
      unsigned LBitWidth = LA.getBitWidth(), RBitWidth = RA.getBitWidth();
      if (LBitWidth != RBitWidth)
        return (int)LBitWidth - (int)RBitWidth;
      return LA.ult(RA) ? -1 : 1;
    }
  
    case scAddRecExpr: {
      const SCEVAddRecExpr *LA = cast<SCEVAddRecExpr>(LHS);
      const SCEVAddRecExpr *RA = cast<SCEVAddRecExpr>(RHS);
  
      // There is always a dominance between two recs that are used by one SCEV,
      // so we can safely sort recs by loop header dominance. We require such
      // order in getAddExpr.
      const Loop *LLoop = LA->getLoop(), *RLoop = RA->getLoop();
      if (LLoop != RLoop) {
        const BasicBlock *LHead = LLoop->getHeader(), *RHead = RLoop->getHeader();
        assert(LHead != RHead && "Two loops share the same header?");
        if (DT.dominates(LHead, RHead))
          return 1;
        else
          assert(DT.dominates(RHead, LHead) &&
                 "No dominance between recurrences used by one SCEV?");
        return -1;
      }
  
      // Addrec complexity grows with operand count.
      unsigned LNumOps = LA->getNumOperands(), RNumOps = RA->getNumOperands();
      if (LNumOps != RNumOps)
        return (int)LNumOps - (int)RNumOps;
  
      // Lexicographically compare.
      for (unsigned i = 0; i != LNumOps; ++i) {
        auto X = CompareSCEVComplexity(EqCacheSCEV, EqCacheValue, LI,
                                       LA->getOperand(i), RA->getOperand(i), DT,
                                       Depth + 1);
        if (X != 0)
          return X;
      }
      EqCacheSCEV.unionSets(LHS, RHS);
      return 0;
    }
  
    case scAddExpr:
    case scMulExpr:
    case scSMaxExpr:
    case scUMaxExpr:
    case scSMinExpr:
    case scUMinExpr: {
      const SCEVNAryExpr *LC = cast<SCEVNAryExpr>(LHS);
      const SCEVNAryExpr *RC = cast<SCEVNAryExpr>(RHS);
  
      // Lexicographically compare n-ary expressions.
      unsigned LNumOps = LC->getNumOperands(), RNumOps = RC->getNumOperands();
      if (LNumOps != RNumOps)
        return (int)LNumOps - (int)RNumOps;
  
      for (unsigned i = 0; i != LNumOps; ++i) {
        auto X = CompareSCEVComplexity(EqCacheSCEV, EqCacheValue, LI,
                                       LC->getOperand(i), RC->getOperand(i), DT,
                                       Depth + 1);
        if (X != 0)
          return X;
      }
      EqCacheSCEV.unionSets(LHS, RHS);
      return 0;
    }
  
    case scUDivExpr: {
      const SCEVUDivExpr *LC = cast<SCEVUDivExpr>(LHS);
      const SCEVUDivExpr *RC = cast<SCEVUDivExpr>(RHS);
  
      // Lexicographically compare udiv expressions.
      auto X = CompareSCEVComplexity(EqCacheSCEV, EqCacheValue, LI, LC->getLHS(),
                                     RC->getLHS(), DT, Depth + 1);
      if (X != 0)
        return X;
      X = CompareSCEVComplexity(EqCacheSCEV, EqCacheValue, LI, LC->getRHS(),
                                RC->getRHS(), DT, Depth + 1);
      if (X == 0)
        EqCacheSCEV.unionSets(LHS, RHS);
      return X;
    }
  
    case scPtrToInt:
    case scTruncate:
    case scZeroExtend:
    case scSignExtend: {
      const SCEVCastExpr *LC = cast<SCEVCastExpr>(LHS);
      const SCEVCastExpr *RC = cast<SCEVCastExpr>(RHS);
  
      // Compare cast expressions by operand.
      auto X =
          CompareSCEVComplexity(EqCacheSCEV, EqCacheValue, LI, LC->getOperand(),
                                RC->getOperand(), DT, Depth + 1);
      if (X == 0)
        EqCacheSCEV.unionSets(LHS, RHS);
      return X;
    }
  
    case scCouldNotCompute:
      llvm_unreachable("Attempt to use a SCEVCouldNotCompute object!");
    }
    llvm_unreachable("Unknown SCEV kind!");
  }
  
  /// Given a list of SCEV objects, order them by their complexity, and group
  /// objects of the same complexity together by value.  When this routine is
  /// finished, we know that any duplicates in the vector are consecutive and that
  /// complexity is monotonically increasing.
  ///
  /// Note that we go take special precautions to ensure that we get deterministic
  /// results from this routine.  In other words, we don't want the results of
  /// this to depend on where the addresses of various SCEV objects happened to
  /// land in memory.
  static void GroupByComplexity(SmallVectorImpl<const SCEV *> &Ops,
                                LoopInfo *LI, DominatorTree &DT) {
    if (Ops.size() < 2) return;  // Noop
  
    EquivalenceClasses<const SCEV *> EqCacheSCEV;
    EquivalenceClasses<const Value *> EqCacheValue;
  
    // Whether LHS has provably less complexity than RHS.
    auto IsLessComplex = [&](const SCEV *LHS, const SCEV *RHS) {
      auto Complexity =
          CompareSCEVComplexity(EqCacheSCEV, EqCacheValue, LI, LHS, RHS, DT);
      return Complexity && *Complexity < 0;
    };
    if (Ops.size() == 2) {
      // This is the common case, which also happens to be trivially simple.
      // Special case it.
      const SCEV *&LHS = Ops[0], *&RHS = Ops[1];
      if (IsLessComplex(RHS, LHS))
        std::swap(LHS, RHS);
      return;
    }
  
    // Do the rough sort by complexity.
    llvm::stable_sort(Ops, [&](const SCEV *LHS, const SCEV *RHS) {
      return IsLessComplex(LHS, RHS);
    });
  
    // Now that we are sorted by complexity, group elements of the same
    // complexity.  Note that this is, at worst, N^2, but the vector is likely to
    // be extremely short in practice.  Note that we take this approach because we
    // do not want to depend on the addresses of the objects we are grouping.
    for (unsigned i = 0, e = Ops.size(); i != e-2; ++i) {
      const SCEV *S = Ops[i];
      unsigned Complexity = S->getSCEVType();
  
      // If there are any objects of the same complexity and same value as this
      // one, group them.
      for (unsigned j = i+1; j != e && Ops[j]->getSCEVType() == Complexity; ++j) {
        if (Ops[j] == S) { // Found a duplicate.
          // Move it to immediately after i'th element.
          std::swap(Ops[i+1], Ops[j]);
          ++i;   // no need to rescan it.
          if (i == e-2) return;  // Done!
        }
      }
    }
  }
  
  /// Returns true if \p Ops contains a huge SCEV (the subtree of S contains at
  /// least HugeExprThreshold nodes).
  static bool hasHugeExpression(ArrayRef<const SCEV *> Ops) {
    return any_of(Ops, [](const SCEV *S) {
      return S->getExpressionSize() >= HugeExprThreshold;
    });
  }
  
  //===----------------------------------------------------------------------===//
  //                      Simple SCEV method implementations
  //===----------------------------------------------------------------------===//
  
  /// Compute BC(It, K).  The result has width W.  Assume, K > 0.
  static const SCEV *BinomialCoefficient(const SCEV *It, unsigned K,
                                         ScalarEvolution &SE,
                                         Type *ResultTy) {
    // Handle the simplest case efficiently.
    if (K == 1)
      return SE.getTruncateOrZeroExtend(It, ResultTy);
  
    // We are using the following formula for BC(It, K):
    //
    //   BC(It, K) = (It * (It - 1) * ... * (It - K + 1)) / K!
    //
    // Suppose, W is the bitwidth of the return value.  We must be prepared for
    // overflow.  Hence, we must assure that the result of our computation is
    // equal to the accurate one modulo 2^W.  Unfortunately, division isn't
    // safe in modular arithmetic.
    //
    // However, this code doesn't use exactly that formula; the formula it uses
    // is something like the following, where T is the number of factors of 2 in
    // K! (i.e. trailing zeros in the binary representation of K!), and ^ is
    // exponentiation:
    //
    //   BC(It, K) = (It * (It - 1) * ... * (It - K + 1)) / 2^T / (K! / 2^T)
    //
    // This formula is trivially equivalent to the previous formula.  However,
    // this formula can be implemented much more efficiently.  The trick is that
    // K! / 2^T is odd, and exact division by an odd number *is* safe in modular
    // arithmetic.  To do exact division in modular arithmetic, all we have
    // to do is multiply by the inverse.  Therefore, this step can be done at
    // width W.
    //
    // The next issue is how to safely do the division by 2^T.  The way this
    // is done is by doing the multiplication step at a width of at least W + T
    // bits.  This way, the bottom W+T bits of the product are accurate. Then,
    // when we perform the division by 2^T (which is equivalent to a right shift
    // by T), the bottom W bits are accurate.  Extra bits are okay; they'll get
    // truncated out after the division by 2^T.
    //
    // In comparison to just directly using the first formula, this technique
    // is much more efficient; using the first formula requires W * K bits,
    // but this formula less than W + K bits. Also, the first formula requires
    // a division step, whereas this formula only requires multiplies and shifts.
    //
    // It doesn't matter whether the subtraction step is done in the calculation
    // width or the input iteration count's width; if the subtraction overflows,
    // the result must be zero anyway.  We prefer here to do it in the width of
    // the induction variable because it helps a lot for certain cases; CodeGen
    // isn't smart enough to ignore the overflow, which leads to much less
    // efficient code if the width of the subtraction is wider than the native
    // register width.
    //
    // (It's possible to not widen at all by pulling out factors of 2 before
    // the multiplication; for example, K=2 can be calculated as
    // It/2*(It+(It*INT_MIN/INT_MIN)+-1). However, it requires
    // extra arithmetic, so it's not an obvious win, and it gets
    // much more complicated for K > 3.)
  
    // Protection from insane SCEVs; this bound is conservative,
    // but it probably doesn't matter.
    if (K > 1000)
      return SE.getCouldNotCompute();
  
    unsigned W = SE.getTypeSizeInBits(ResultTy);
  
    // Calculate K! / 2^T and T; we divide out the factors of two before
    // multiplying for calculating K! / 2^T to avoid overflow.
    // Other overflow doesn't matter because we only care about the bottom
    // W bits of the result.
    APInt OddFactorial(W, 1);
    unsigned T = 1;
    for (unsigned i = 3; i <= K; ++i) {
      APInt Mult(W, i);
      unsigned TwoFactors = Mult.countTrailingZeros();
      T += TwoFactors;
      Mult.lshrInPlace(TwoFactors);
      OddFactorial *= Mult;
    }
  
    // We need at least W + T bits for the multiplication step
    unsigned CalculationBits = W + T;
  
    // Calculate 2^T, at width T+W.
    APInt DivFactor = APInt::getOneBitSet(CalculationBits, T);
  
    // Calculate the multiplicative inverse of K! / 2^T;
    // this multiplication factor will perform the exact division by
    // K! / 2^T.
    APInt Mod = APInt::getSignedMinValue(W+1);
    APInt MultiplyFactor = OddFactorial.zext(W+1);
    MultiplyFactor = MultiplyFactor.multiplicativeInverse(Mod);
    MultiplyFactor = MultiplyFactor.trunc(W);
  
    // Calculate the product, at width T+W
    IntegerType *CalculationTy = IntegerType::get(SE.getContext(),
                                                        CalculationBits);
    const SCEV *Dividend = SE.getTruncateOrZeroExtend(It, CalculationTy);
    for (unsigned i = 1; i != K; ++i) {
      const SCEV *S = SE.getMinusSCEV(It, SE.getConstant(It->getType(), i));
      Dividend = SE.getMulExpr(Dividend,
                               SE.getTruncateOrZeroExtend(S, CalculationTy));
    }
  
    // Divide by 2^T
    const SCEV *DivResult = SE.getUDivExpr(Dividend, SE.getConstant(DivFactor));
  
    // Truncate the result, and divide by K! / 2^T.
  
    return SE.getMulExpr(SE.getConstant(MultiplyFactor),
                         SE.getTruncateOrZeroExtend(DivResult, ResultTy));
  }
  
  /// Return the value of this chain of recurrences at the specified iteration
  /// number.  We can evaluate this recurrence by multiplying each element in the
  /// chain by the binomial coefficient corresponding to it.  In other words, we
  /// can evaluate {A,+,B,+,C,+,D} as:
  ///
  ///   A*BC(It, 0) + B*BC(It, 1) + C*BC(It, 2) + D*BC(It, 3)
  ///
  /// where BC(It, k) stands for binomial coefficient.
  const SCEV *SCEVAddRecExpr::evaluateAtIteration(const SCEV *It,
                                                  ScalarEvolution &SE) const {
    return evaluateAtIteration(makeArrayRef(op_begin(), op_end()), It, SE);
  }
  
  const SCEV *
  SCEVAddRecExpr::evaluateAtIteration(ArrayRef<const SCEV *> Operands,
                                      const SCEV *It, ScalarEvolution &SE) {
    assert(Operands.size() > 0);
    const SCEV *Result = Operands[0];
    for (unsigned i = 1, e = Operands.size(); i != e; ++i) {
      // The computation is correct in the face of overflow provided that the
      // multiplication is performed _after_ the evaluation of the binomial
      // coefficient.
      const SCEV *Coeff = BinomialCoefficient(It, i, SE, Result->getType());
      if (isa<SCEVCouldNotCompute>(Coeff))
        return Coeff;
  
      Result = SE.getAddExpr(Result, SE.getMulExpr(Operands[i], Coeff));
    }
    return Result;
  }
  
  //===----------------------------------------------------------------------===//
  //                    SCEV Expression folder implementations
  //===----------------------------------------------------------------------===//
  
  const SCEV *ScalarEvolution::getLosslessPtrToIntExpr(const SCEV *Op,
                                                       unsigned Depth) {
    assert(Depth <= 1 &&
           "getLosslessPtrToIntExpr() should self-recurse at most once.");
  
    // We could be called with an integer-typed operands during SCEV rewrites.
    // Since the operand is an integer already, just perform zext/trunc/self cast.
    if (!Op->getType()->isPointerTy())
      return Op;
  
    // What would be an ID for such a SCEV cast expression?
    FoldingSetNodeID ID;
    ID.AddInteger(scPtrToInt);
    ID.AddPointer(Op);
  
    void *IP = nullptr;
  
    // Is there already an expression for such a cast?
    if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP))
      return S;
  
    // It isn't legal for optimizations to construct new ptrtoint expressions
    // for non-integral pointers.
    if (getDataLayout().isNonIntegralPointerType(Op->getType()))
      return getCouldNotCompute();
  
    Type *IntPtrTy = getDataLayout().getIntPtrType(Op->getType());
  
    // We can only trivially model ptrtoint if SCEV's effective (integer) type
    // is sufficiently wide to represent all possible pointer values.
    // We could theoretically teach SCEV to truncate wider pointers, but
    // that isn't implemented for now.
    if (getDataLayout().getTypeSizeInBits(getEffectiveSCEVType(Op->getType())) !=
        getDataLayout().getTypeSizeInBits(IntPtrTy))
      return getCouldNotCompute();
  
    // If not, is this expression something we can't reduce any further?
    if (auto *U = dyn_cast<SCEVUnknown>(Op)) {
      // Perform some basic constant folding. If the operand of the ptr2int cast
      // is a null pointer, don't create a ptr2int SCEV expression (that will be
      // left as-is), but produce a zero constant.
      // NOTE: We could handle a more general case, but lack motivational cases.
      if (isa<ConstantPointerNull>(U->getValue()))
        return getZero(IntPtrTy);
  
      // Create an explicit cast node.
      // We can reuse the existing insert position since if we get here,
      // we won't have made any changes which would invalidate it.
      SCEV *S = new (SCEVAllocator)
          SCEVPtrToIntExpr(ID.Intern(SCEVAllocator), Op, IntPtrTy);
      UniqueSCEVs.InsertNode(S, IP);
      addToLoopUseLists(S);
      return S;
    }
  
    assert(Depth == 0 && "getLosslessPtrToIntExpr() should not self-recurse for "
                         "non-SCEVUnknown's.");
  
    // Otherwise, we've got some expression that is more complex than just a
    // single SCEVUnknown. But we don't want to have a SCEVPtrToIntExpr of an
    // arbitrary expression, we want to have SCEVPtrToIntExpr of an SCEVUnknown
    // only, and the expressions must otherwise be integer-typed.
    // So sink the cast down to the SCEVUnknown's.
  
    /// The SCEVPtrToIntSinkingRewriter takes a scalar evolution expression,
    /// which computes a pointer-typed value, and rewrites the whole expression
    /// tree so that *all* the computations are done on integers, and the only
    /// pointer-typed operands in the expression are SCEVUnknown.
    class SCEVPtrToIntSinkingRewriter
        : public SCEVRewriteVisitor<SCEVPtrToIntSinkingRewriter> {
      using Base = SCEVRewriteVisitor<SCEVPtrToIntSinkingRewriter>;
  
    public:
      SCEVPtrToIntSinkingRewriter(ScalarEvolution &SE) : SCEVRewriteVisitor(SE) {}
  
      static const SCEV *rewrite(const SCEV *Scev, ScalarEvolution &SE) {
        SCEVPtrToIntSinkingRewriter Rewriter(SE);
        return Rewriter.visit(Scev);
      }
  
      const SCEV *visit(const SCEV *S) {
        Type *STy = S->getType();
        // If the expression is not pointer-typed, just keep it as-is.
        if (!STy->isPointerTy())
          return S;
        // Else, recursively sink the cast down into it.
        return Base::visit(S);
      }
  
      const SCEV *visitAddExpr(const SCEVAddExpr *Expr) {
        SmallVector<const SCEV *, 2> Operands;
        bool Changed = false;
        for (auto *Op : Expr->operands()) {
          Operands.push_back(visit(Op));
          Changed |= Op != Operands.back();
        }
        return !Changed ? Expr : SE.getAddExpr(Operands, Expr->getNoWrapFlags());
      }
  
      const SCEV *visitMulExpr(const SCEVMulExpr *Expr) {
        SmallVector<const SCEV *, 2> Operands;
        bool Changed = false;
        for (auto *Op : Expr->operands()) {
          Operands.push_back(visit(Op));
          Changed |= Op != Operands.back();
        }
        return !Changed ? Expr : SE.getMulExpr(Operands, Expr->getNoWrapFlags());
      }
  
      const SCEV *visitUnknown(const SCEVUnknown *Expr) {
        assert(Expr->getType()->isPointerTy() &&
               "Should only reach pointer-typed SCEVUnknown's.");
        return SE.getLosslessPtrToIntExpr(Expr, /*Depth=*/1);
      }
    };
  
    // And actually perform the cast sinking.
    const SCEV *IntOp = SCEVPtrToIntSinkingRewriter::rewrite(Op, *this);
    assert(IntOp->getType()->isIntegerTy() &&
           "We must have succeeded in sinking the cast, "
           "and ending up with an integer-typed expression!");
    return IntOp;
  }
  
  const SCEV *ScalarEvolution::getPtrToIntExpr(const SCEV *Op, Type *Ty) {
    assert(Ty->isIntegerTy() && "Target type must be an integer type!");
  
    const SCEV *IntOp = getLosslessPtrToIntExpr(Op);
    if (isa<SCEVCouldNotCompute>(IntOp))
      return IntOp;
  
    return getTruncateOrZeroExtend(IntOp, Ty);
  }
  
  const SCEV *ScalarEvolution::getTruncateExpr(const SCEV *Op, Type *Ty,
                                               unsigned Depth) {
    assert(getTypeSizeInBits(Op->getType()) > getTypeSizeInBits(Ty) &&
           "This is not a truncating conversion!");
    assert(isSCEVable(Ty) &&
           "This is not a conversion to a SCEVable type!");
    assert(!Op->getType()->isPointerTy() && "Can't truncate pointer!");
    Ty = getEffectiveSCEVType(Ty);
  
    FoldingSetNodeID ID;
    ID.AddInteger(scTruncate);
    ID.AddPointer(Op);
    ID.AddPointer(Ty);
    void *IP = nullptr;
    if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;
  
    // Fold if the operand is constant.
    if (const SCEVConstant *SC = dyn_cast<SCEVConstant>(Op))
      return getConstant(
        cast<ConstantInt>(ConstantExpr::getTrunc(SC->getValue(), Ty)));
  
    // trunc(trunc(x)) --> trunc(x)
    if (const SCEVTruncateExpr *ST = dyn_cast<SCEVTruncateExpr>(Op))
      return getTruncateExpr(ST->getOperand(), Ty, Depth + 1);
  
    // trunc(sext(x)) --> sext(x) if widening or trunc(x) if narrowing
    if (const SCEVSignExtendExpr *SS = dyn_cast<SCEVSignExtendExpr>(Op))
      return getTruncateOrSignExtend(SS->getOperand(), Ty, Depth + 1);
  
    // trunc(zext(x)) --> zext(x) if widening or trunc(x) if narrowing
    if (const SCEVZeroExtendExpr *SZ = dyn_cast<SCEVZeroExtendExpr>(Op))
      return getTruncateOrZeroExtend(SZ->getOperand(), Ty, Depth + 1);
  
    if (Depth > MaxCastDepth) {
      SCEV *S =
          new (SCEVAllocator) SCEVTruncateExpr(ID.Intern(SCEVAllocator), Op, Ty);
      UniqueSCEVs.InsertNode(S, IP);
      addToLoopUseLists(S);
      return S;
    }
  
    // trunc(x1 + ... + xN) --> trunc(x1) + ... + trunc(xN) and
    // trunc(x1 * ... * xN) --> trunc(x1) * ... * trunc(xN),
    // if after transforming we have at most one truncate, not counting truncates
    // that replace other casts.
    if (isa<SCEVAddExpr>(Op) || isa<SCEVMulExpr>(Op)) {
      auto *CommOp = cast<SCEVCommutativeExpr>(Op);
      SmallVector<const SCEV *, 4> Operands;
      unsigned numTruncs = 0;
      for (unsigned i = 0, e = CommOp->getNumOperands(); i != e && numTruncs < 2;
           ++i) {
        const SCEV *S = getTruncateExpr(CommOp->getOperand(i), Ty, Depth + 1);
        if (!isa<SCEVIntegralCastExpr>(CommOp->getOperand(i)) &&
            isa<SCEVTruncateExpr>(S))
          numTruncs++;
        Operands.push_back(S);
      }
      if (numTruncs < 2) {
        if (isa<SCEVAddExpr>(Op))
          return getAddExpr(Operands);
        else if (isa<SCEVMulExpr>(Op))
          return getMulExpr(Operands);
        else
          llvm_unreachable("Unexpected SCEV type for Op.");
      }
      // Although we checked in the beginning that ID is not in the cache, it is
      // possible that during recursion and different modification ID was inserted
      // into the cache. So if we find it, just return it.
      if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP))
        return S;
    }
  
    // If the input value is a chrec scev, truncate the chrec's operands.
    if (const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(Op)) {
      SmallVector<const SCEV *, 4> Operands;
      for (const SCEV *Op : AddRec->operands())
        Operands.push_back(getTruncateExpr(Op, Ty, Depth + 1));
      return getAddRecExpr(Operands, AddRec->getLoop(), SCEV::FlagAnyWrap);
    }
  
    // Return zero if truncating to known zeros.
    uint32_t MinTrailingZeros = GetMinTrailingZeros(Op);
    if (MinTrailingZeros >= getTypeSizeInBits(Ty))
      return getZero(Ty);
  
    // The cast wasn't folded; create an explicit cast node. We can reuse
    // the existing insert position since if we get here, we won't have
    // made any changes which would invalidate it.
    SCEV *S = new (SCEVAllocator) SCEVTruncateExpr(ID.Intern(SCEVAllocator),
                                                   Op, Ty);
    UniqueSCEVs.InsertNode(S, IP);
    addToLoopUseLists(S);
    return S;
  }
  
  // Get the limit of a recurrence such that incrementing by Step cannot cause
  // signed overflow as long as the value of the recurrence within the
  // loop does not exceed this limit before incrementing.
  static const SCEV *getSignedOverflowLimitForStep(const SCEV *Step,
                                                   ICmpInst::Predicate *Pred,
                                                   ScalarEvolution *SE) {
    unsigned BitWidth = SE->getTypeSizeInBits(Step->getType());
    if (SE->isKnownPositive(Step)) {
      *Pred = ICmpInst::ICMP_SLT;
      return SE->getConstant(APInt::getSignedMinValue(BitWidth) -
                             SE->getSignedRangeMax(Step));
    }
    if (SE->isKnownNegative(Step)) {
      *Pred = ICmpInst::ICMP_SGT;
      return SE->getConstant(APInt::getSignedMaxValue(BitWidth) -
                             SE->getSignedRangeMin(Step));
    }
    return nullptr;
  }
  
  // Get the limit of a recurrence such that incrementing by Step cannot cause
  // unsigned overflow as long as the value of the recurrence within the loop does
  // not exceed this limit before incrementing.
  static const SCEV *getUnsignedOverflowLimitForStep(const SCEV *Step,
                                                     ICmpInst::Predicate *Pred,
                                                     ScalarEvolution *SE) {
    unsigned BitWidth = SE->getTypeSizeInBits(Step->getType());
    *Pred = ICmpInst::ICMP_ULT;
  
    return SE->getConstant(APInt::getMinValue(BitWidth) -
                           SE->getUnsignedRangeMax(Step));
  }
  
  namespace {
  
  struct ExtendOpTraitsBase {
    typedef const SCEV *(ScalarEvolution::*GetExtendExprTy)(const SCEV *, Type *,
                                                            unsigned);
  };
  
  // Used to make code generic over signed and unsigned overflow.
  template <typename ExtendOp> struct ExtendOpTraits {
    // Members present:
    //
    // static const SCEV::NoWrapFlags WrapType;
    //
    // static const ExtendOpTraitsBase::GetExtendExprTy GetExtendExpr;
    //
    // static const SCEV *getOverflowLimitForStep(const SCEV *Step,
    //                                           ICmpInst::Predicate *Pred,
    //                                           ScalarEvolution *SE);
  };
  
  template <>
  struct ExtendOpTraits<SCEVSignExtendExpr> : public ExtendOpTraitsBase {
    static const SCEV::NoWrapFlags WrapType = SCEV::FlagNSW;
  
    static const GetExtendExprTy GetExtendExpr;
  
    static const SCEV *getOverflowLimitForStep(const SCEV *Step,
                                               ICmpInst::Predicate *Pred,
                                               ScalarEvolution *SE) {
      return getSignedOverflowLimitForStep(Step, Pred, SE);
    }
  };
  
  const ExtendOpTraitsBase::GetExtendExprTy ExtendOpTraits<
      SCEVSignExtendExpr>::GetExtendExpr = &ScalarEvolution::getSignExtendExpr;
  
  template <>
  struct ExtendOpTraits<SCEVZeroExtendExpr> : public ExtendOpTraitsBase {
    static const SCEV::NoWrapFlags WrapType = SCEV::FlagNUW;
  
    static const GetExtendExprTy GetExtendExpr;
  
    static const SCEV *getOverflowLimitForStep(const SCEV *Step,
                                               ICmpInst::Predicate *Pred,
                                               ScalarEvolution *SE) {
      return getUnsignedOverflowLimitForStep(Step, Pred, SE);
    }
  };
  
  const ExtendOpTraitsBase::GetExtendExprTy ExtendOpTraits<
      SCEVZeroExtendExpr>::GetExtendExpr = &ScalarEvolution::getZeroExtendExpr;
  
  } // end anonymous namespace
  
  // The recurrence AR has been shown to have no signed/unsigned wrap or something
  // close to it. Typically, if we can prove NSW/NUW for AR, then we can just as
  // easily prove NSW/NUW for its preincrement or postincrement sibling. This
  // allows normalizing a sign/zero extended AddRec as such: {sext/zext(Step +
  // Start),+,Step} => {(Step + sext/zext(Start),+,Step} As a result, the
  // expression "Step + sext/zext(PreIncAR)" is congruent with
  // "sext/zext(PostIncAR)"
  template <typename ExtendOpTy>
  static const SCEV *getPreStartForExtend(const SCEVAddRecExpr *AR, Type *Ty,
                                          ScalarEvolution *SE, unsigned Depth) {
    auto WrapType = ExtendOpTraits<ExtendOpTy>::WrapType;
    auto GetExtendExpr = ExtendOpTraits<ExtendOpTy>::GetExtendExpr;
  
    const Loop *L = AR->getLoop();
    const SCEV *Start = AR->getStart();
    const SCEV *Step = AR->getStepRecurrence(*SE);
  
    // Check for a simple looking step prior to loop entry.
    const SCEVAddExpr *SA = dyn_cast<SCEVAddExpr>(Start);
    if (!SA)
      return nullptr;
  
    // Create an AddExpr for "PreStart" after subtracting Step. Full SCEV
    // subtraction is expensive. For this purpose, perform a quick and dirty
    // difference, by checking for Step in the operand list.
    SmallVector<const SCEV *, 4> DiffOps;
    for (const SCEV *Op : SA->operands())
      if (Op != Step)
        DiffOps.push_back(Op);
  
    if (DiffOps.size() == SA->getNumOperands())
      return nullptr;
  
    // Try to prove `WrapType` (SCEV::FlagNSW or SCEV::FlagNUW) on `PreStart` +
    // `Step`:
  
    // 1. NSW/NUW flags on the step increment.
    auto PreStartFlags =
      ScalarEvolution::maskFlags(SA->getNoWrapFlags(), SCEV::FlagNUW);
    const SCEV *PreStart = SE->getAddExpr(DiffOps, PreStartFlags);
    const SCEVAddRecExpr *PreAR = dyn_cast<SCEVAddRecExpr>(
        SE->getAddRecExpr(PreStart, Step, L, SCEV::FlagAnyWrap));
  
    // "{S,+,X} is <nsw>/<nuw>" and "the backedge is taken at least once" implies
    // "S+X does not sign/unsign-overflow".
    //
  
    const SCEV *BECount = SE->getBackedgeTakenCount(L);
    if (PreAR && PreAR->getNoWrapFlags(WrapType) &&
        !isa<SCEVCouldNotCompute>(BECount) && SE->isKnownPositive(BECount))
      return PreStart;
  
    // 2. Direct overflow check on the step operation's expression.
    unsigned BitWidth = SE->getTypeSizeInBits(AR->getType());
    Type *WideTy = IntegerType::get(SE->getContext(), BitWidth * 2);
    const SCEV *OperandExtendedStart =
        SE->getAddExpr((SE->*GetExtendExpr)(PreStart, WideTy, Depth),
                       (SE->*GetExtendExpr)(Step, WideTy, Depth));
    if ((SE->*GetExtendExpr)(Start, WideTy, Depth) == OperandExtendedStart) {
      if (PreAR && AR->getNoWrapFlags(WrapType)) {
        // If we know `AR` == {`PreStart`+`Step`,+,`Step`} is `WrapType` (FlagNSW
        // or FlagNUW) and that `PreStart` + `Step` is `WrapType` too, then
        // `PreAR` == {`PreStart`,+,`Step`} is also `WrapType`.  Cache this fact.
        SE->setNoWrapFlags(const_cast<SCEVAddRecExpr *>(PreAR), WrapType);
      }
      return PreStart;
    }
  
    // 3. Loop precondition.
    ICmpInst::Predicate Pred;
    const SCEV *OverflowLimit =
        ExtendOpTraits<ExtendOpTy>::getOverflowLimitForStep(Step, &Pred, SE);
  
    if (OverflowLimit &&
        SE->isLoopEntryGuardedByCond(L, Pred, PreStart, OverflowLimit))
      return PreStart;
  
    return nullptr;
  }
  
  // Get the normalized zero or sign extended expression for this AddRec's Start.
  template <typename ExtendOpTy>
  static const SCEV *getExtendAddRecStart(const SCEVAddRecExpr *AR, Type *Ty,
                                          ScalarEvolution *SE,
                                          unsigned Depth) {
    auto GetExtendExpr = ExtendOpTraits<ExtendOpTy>::GetExtendExpr;
  
    const SCEV *PreStart = getPreStartForExtend<ExtendOpTy>(AR, Ty, SE, Depth);
    if (!PreStart)
      return (SE->*GetExtendExpr)(AR->getStart(), Ty, Depth);
  
    return SE->getAddExpr((SE->*GetExtendExpr)(AR->getStepRecurrence(*SE), Ty,
                                               Depth),
                          (SE->*GetExtendExpr)(PreStart, Ty, Depth));
  }
  
  // Try to prove away overflow by looking at "nearby" add recurrences.  A
  // motivating example for this rule: if we know `{0,+,4}` is `ult` `-1` and it
  // does not itself wrap then we can conclude that `{1,+,4}` is `nuw`.
  //
  // Formally:
  //
  //     {S,+,X} == {S-T,+,X} + T
  //  => Ext({S,+,X}) == Ext({S-T,+,X} + T)
  //
  // If ({S-T,+,X} + T) does not overflow  ... (1)
  //
  //  RHS == Ext({S-T,+,X} + T) == Ext({S-T,+,X}) + Ext(T)
  //
  // If {S-T,+,X} does not overflow  ... (2)
  //
  //  RHS == Ext({S-T,+,X}) + Ext(T) == {Ext(S-T),+,Ext(X)} + Ext(T)
  //      == {Ext(S-T)+Ext(T),+,Ext(X)}
  //
  // If (S-T)+T does not overflow  ... (3)
  //
  //  RHS == {Ext(S-T)+Ext(T),+,Ext(X)} == {Ext(S-T+T),+,Ext(X)}
  //      == {Ext(S),+,Ext(X)} == LHS
  //
  // Thus, if (1), (2) and (3) are true for some T, then
  //   Ext({S,+,X}) == {Ext(S),+,Ext(X)}
  //
  // (3) is implied by (1) -- "(S-T)+T does not overflow" is simply "({S-T,+,X}+T)
  // does not overflow" restricted to the 0th iteration.  Therefore we only need
  // to check for (1) and (2).
  //
  // In the current context, S is `Start`, X is `Step`, Ext is `ExtendOpTy` and T
  // is `Delta` (defined below).
  template <typename ExtendOpTy>
  bool ScalarEvolution::proveNoWrapByVaryingStart(const SCEV *Start,
                                                  const SCEV *Step,
                                                  const Loop *L) {
    auto WrapType = ExtendOpTraits<ExtendOpTy>::WrapType;
  
    // We restrict `Start` to a constant to prevent SCEV from spending too much
    // time here.  It is correct (but more expensive) to continue with a
    // non-constant `Start` and do a general SCEV subtraction to compute
    // `PreStart` below.
    const SCEVConstant *StartC = dyn_cast<SCEVConstant>(Start);
    if (!StartC)
      return false;
  
    APInt StartAI = StartC->getAPInt();
  
    for (unsigned Delta : {-2, -1, 1, 2}) {
      const SCEV *PreStart = getConstant(StartAI - Delta);
  
      FoldingSetNodeID ID;
      ID.AddInteger(scAddRecExpr);
      ID.AddPointer(PreStart);
      ID.AddPointer(Step);
      ID.AddPointer(L);
      void *IP = nullptr;
      const auto *PreAR =
        static_cast<SCEVAddRecExpr *>(UniqueSCEVs.FindNodeOrInsertPos(ID, IP));
  
      // Give up if we don't already have the add recurrence we need because
      // actually constructing an add recurrence is relatively expensive.
      if (PreAR && PreAR->getNoWrapFlags(WrapType)) {  // proves (2)
        const SCEV *DeltaS = getConstant(StartC->getType(), Delta);
        ICmpInst::Predicate Pred = ICmpInst::BAD_ICMP_PREDICATE;
        const SCEV *Limit = ExtendOpTraits<ExtendOpTy>::getOverflowLimitForStep(
            DeltaS, &Pred, this);
        if (Limit && isKnownPredicate(Pred, PreAR, Limit))  // proves (1)
          return true;
      }
    }
  
    return false;
  }
  
  // Finds an integer D for an expression (C + x + y + ...) such that the top
  // level addition in (D + (C - D + x + y + ...)) would not wrap (signed or
  // unsigned) and the number of trailing zeros of (C - D + x + y + ...) is
  // maximized, where C is the \p ConstantTerm, x, y, ... are arbitrary SCEVs, and
  // the (C + x + y + ...) expression is \p WholeAddExpr.
  static APInt extractConstantWithoutWrapping(ScalarEvolution &SE,
                                              const SCEVConstant *ConstantTerm,
                                              const SCEVAddExpr *WholeAddExpr) {
    const APInt &C = ConstantTerm->getAPInt();
    const unsigned BitWidth = C.getBitWidth();
    // Find number of trailing zeros of (x + y + ...) w/o the C first:
    uint32_t TZ = BitWidth;
    for (unsigned I = 1, E = WholeAddExpr->getNumOperands(); I < E && TZ; ++I)
      TZ = std::min(TZ, SE.GetMinTrailingZeros(WholeAddExpr->getOperand(I)));
    if (TZ) {
      // Set D to be as many least significant bits of C as possible while still
      // guaranteeing that adding D to (C - D + x + y + ...) won't cause a wrap:
      return TZ < BitWidth ? C.trunc(TZ).zext(BitWidth) : C;
    }
    return APInt(BitWidth, 0);
  }
  
  // Finds an integer D for an affine AddRec expression {C,+,x} such that the top
  // level addition in (D + {C-D,+,x}) would not wrap (signed or unsigned) and the
  // number of trailing zeros of (C - D + x * n) is maximized, where C is the \p
  // ConstantStart, x is an arbitrary \p Step, and n is the loop trip count.
  static APInt extractConstantWithoutWrapping(ScalarEvolution &SE,
                                              const APInt &ConstantStart,
                                              const SCEV *Step) {
    const unsigned BitWidth = ConstantStart.getBitWidth();
    const uint32_t TZ = SE.GetMinTrailingZeros(Step);
    if (TZ)
      return TZ < BitWidth ? ConstantStart.trunc(TZ).zext(BitWidth)
                           : ConstantStart;
    return APInt(BitWidth, 0);
  }
  
  const SCEV *
  ScalarEvolution::getZeroExtendExpr(const SCEV *Op, Type *Ty, unsigned Depth) {
    assert(getTypeSizeInBits(Op->getType()) < getTypeSizeInBits(Ty) &&
           "This is not an extending conversion!");
    assert(isSCEVable(Ty) &&
           "This is not a conversion to a SCEVable type!");
    assert(!Op->getType()->isPointerTy() && "Can't extend pointer!");
    Ty = getEffectiveSCEVType(Ty);
  
    // Fold if the operand is constant.
    if (const SCEVConstant *SC = dyn_cast<SCEVConstant>(Op))
      return getConstant(
        cast<ConstantInt>(ConstantExpr::getZExt(SC->getValue(), Ty)));
  
    // zext(zext(x)) --> zext(x)
    if (const SCEVZeroExtendExpr *SZ = dyn_cast<SCEVZeroExtendExpr>(Op))
      return getZeroExtendExpr(SZ->getOperand(), Ty, Depth + 1);
  
    // Before doing any expensive analysis, check to see if we've already
    // computed a SCEV for this Op and Ty.
    FoldingSetNodeID ID;
    ID.AddInteger(scZeroExtend);
    ID.AddPointer(Op);
    ID.AddPointer(Ty);
    void *IP = nullptr;
    if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;
    if (Depth > MaxCastDepth) {
      SCEV *S = new (SCEVAllocator) SCEVZeroExtendExpr(ID.Intern(SCEVAllocator),
                                                       Op, Ty);
      UniqueSCEVs.InsertNode(S, IP);
      addToLoopUseLists(S);
      return S;
    }
  
    // zext(trunc(x)) --> zext(x) or x or trunc(x)
    if (const SCEVTruncateExpr *ST = dyn_cast<SCEVTruncateExpr>(Op)) {
      // It's possible the bits taken off by the truncate were all zero bits. If
      // so, we should be able to simplify this further.
      const SCEV *X = ST->getOperand();
      ConstantRange CR = getUnsignedRange(X);
      unsigned TruncBits = getTypeSizeInBits(ST->getType());
      unsigned NewBits = getTypeSizeInBits(Ty);
      if (CR.truncate(TruncBits).zeroExtend(NewBits).contains(
              CR.zextOrTrunc(NewBits)))
        return getTruncateOrZeroExtend(X, Ty, Depth);
    }
  
    // If the input value is a chrec scev, and we can prove that the value
    // did not overflow the old, smaller, value, we can zero extend all of the
    // operands (often constants).  This allows analysis of something like
    // this:  for (unsigned char X = 0; X < 100; ++X) { int Y = X; }
    if (const SCEVAddRecExpr *AR = dyn_cast<SCEVAddRecExpr>(Op))
      if (AR->isAffine()) {
        const SCEV *Start = AR->getStart();
        const SCEV *Step = AR->getStepRecurrence(*this);
        unsigned BitWidth = getTypeSizeInBits(AR->getType());
        const Loop *L = AR->getLoop();
  
        if (!AR->hasNoUnsignedWrap()) {
          auto NewFlags = proveNoWrapViaConstantRanges(AR);
          setNoWrapFlags(const_cast<SCEVAddRecExpr *>(AR), NewFlags);
        }
  
        // If we have special knowledge that this addrec won't overflow,
        // we don't need to do any further analysis.
        if (AR->hasNoUnsignedWrap())
          return getAddRecExpr(
              getExtendAddRecStart<SCEVZeroExtendExpr>(AR, Ty, this, Depth + 1),
              getZeroExtendExpr(Step, Ty, Depth + 1), L, AR->getNoWrapFlags());
  
        // Check whether the backedge-taken count is SCEVCouldNotCompute.
        // Note that this serves two purposes: It filters out loops that are
        // simply not analyzable, and it covers the case where this code is
        // being called from within backedge-taken count analysis, such that
        // attempting to ask for the backedge-taken count would likely result
        // in infinite recursion. In the later case, the analysis code will
        // cope with a conservative value, and it will take care to purge
        // that value once it has finished.
        const SCEV *MaxBECount = getConstantMaxBackedgeTakenCount(L);
        if (!isa<SCEVCouldNotCompute>(MaxBECount)) {
          // Manually compute the final value for AR, checking for overflow.
  
          // Check whether the backedge-taken count can be losslessly casted to
          // the addrec's type. The count is always unsigned.
          const SCEV *CastedMaxBECount =
              getTruncateOrZeroExtend(MaxBECount, Start->getType(), Depth);
          const SCEV *RecastedMaxBECount = getTruncateOrZeroExtend(
              CastedMaxBECount, MaxBECount->getType(), Depth);
          if (MaxBECount == RecastedMaxBECount) {
            Type *WideTy = IntegerType::get(getContext(), BitWidth * 2);
            // Check whether Start+Step*MaxBECount has no unsigned overflow.
            const SCEV *ZMul = getMulExpr(CastedMaxBECount, Step,
                                          SCEV::FlagAnyWrap, Depth + 1);
            const SCEV *ZAdd = getZeroExtendExpr(getAddExpr(Start, ZMul,
                                                            SCEV::FlagAnyWrap,
                                                            Depth + 1),
                                                 WideTy, Depth + 1);
            const SCEV *WideStart = getZeroExtendExpr(Start, WideTy, Depth + 1);
            const SCEV *WideMaxBECount =
              getZeroExtendExpr(CastedMaxBECount, WideTy, Depth + 1);
            const SCEV *OperandExtendedAdd =
              getAddExpr(WideStart,
                         getMulExpr(WideMaxBECount,
                                    getZeroExtendExpr(Step, WideTy, Depth + 1),
                                    SCEV::FlagAnyWrap, Depth + 1),
                         SCEV::FlagAnyWrap, Depth + 1);
            if (ZAdd == OperandExtendedAdd) {
              // Cache knowledge of AR NUW, which is propagated to this AddRec.
              setNoWrapFlags(const_cast<SCEVAddRecExpr *>(AR), SCEV::FlagNUW);
              // Return the expression with the addrec on the outside.
              return getAddRecExpr(
                  getExtendAddRecStart<SCEVZeroExtendExpr>(AR, Ty, this,
                                                           Depth + 1),
                  getZeroExtendExpr(Step, Ty, Depth + 1), L,
                  AR->getNoWrapFlags());
            }
            // Similar to above, only this time treat the step value as signed.
            // This covers loops that count down.
            OperandExtendedAdd =
              getAddExpr(WideStart,
                         getMulExpr(WideMaxBECount,
                                    getSignExtendExpr(Step, WideTy, Depth + 1),
                                    SCEV::FlagAnyWrap, Depth + 1),
                         SCEV::FlagAnyWrap, Depth + 1);
            if (ZAdd == OperandExtendedAdd) {
              // Cache knowledge of AR NW, which is propagated to this AddRec.
              // Negative step causes unsigned wrap, but it still can't self-wrap.
              setNoWrapFlags(const_cast<SCEVAddRecExpr *>(AR), SCEV::FlagNW);
              // Return the expression with the addrec on the outside.
              return getAddRecExpr(
                  getExtendAddRecStart<SCEVZeroExtendExpr>(AR, Ty, this,
                                                           Depth + 1),
                  getSignExtendExpr(Step, Ty, Depth + 1), L,
                  AR->getNoWrapFlags());
            }
          }
        }
  
        // Normally, in the cases we can prove no-overflow via a
        // backedge guarding condition, we can also compute a backedge
        // taken count for the loop.  The exceptions are assumptions and
        // guards present in the loop -- SCEV is not great at exploiting
        // these to compute max backedge taken counts, but can still use
        // these to prove lack of overflow.  Use this fact to avoid
        // doing extra work that may not pay off.
        if (!isa<SCEVCouldNotCompute>(MaxBECount) || HasGuards ||
            !AC.assumptions().empty()) {
  
          auto NewFlags = proveNoUnsignedWrapViaInduction(AR);
          setNoWrapFlags(const_cast<SCEVAddRecExpr *>(AR), NewFlags);
          if (AR->hasNoUnsignedWrap()) {
            // Same as nuw case above - duplicated here to avoid a compile time
            // issue.  It's not clear that the order of checks does matter, but
            // it's one of two issue possible causes for a change which was
            // reverted.  Be conservative for the moment.
            return getAddRecExpr(
                  getExtendAddRecStart<SCEVZeroExtendExpr>(AR, Ty, this,
                                                           Depth + 1),
                  getZeroExtendExpr(Step, Ty, Depth + 1), L,
                  AR->getNoWrapFlags());
          }
          
          // For a negative step, we can extend the operands iff doing so only
          // traverses values in the range zext([0,UINT_MAX]). 
          if (isKnownNegative(Step)) {
            const SCEV *N = getConstant(APInt::getMaxValue(BitWidth) -
                                        getSignedRangeMin(Step));
            if (isLoopBackedgeGuardedByCond(L, ICmpInst::ICMP_UGT, AR, N) ||
                isKnownOnEveryIteration(ICmpInst::ICMP_UGT, AR, N)) {
              // Cache knowledge of AR NW, which is propagated to this
              // AddRec.  Negative step causes unsigned wrap, but it
              // still can't self-wrap.
              setNoWrapFlags(const_cast<SCEVAddRecExpr *>(AR), SCEV::FlagNW);
              // Return the expression with the addrec on the outside.
              return getAddRecExpr(
                  getExtendAddRecStart<SCEVZeroExtendExpr>(AR, Ty, this,
                                                           Depth + 1),
                  getSignExtendExpr(Step, Ty, Depth + 1), L,
                  AR->getNoWrapFlags());
            }
          }
        }
  
        // zext({C,+,Step}) --> (zext(D) + zext({C-D,+,Step}))<nuw><nsw>
        // if D + (C - D + Step * n) could be proven to not unsigned wrap
        // where D maximizes the number of trailing zeros of (C - D + Step * n)
        if (const auto *SC = dyn_cast<SCEVConstant>(Start)) {
          const APInt &C = SC->getAPInt();
          const APInt &D = extractConstantWithoutWrapping(*this, C, Step);
          if (D != 0) {
            const SCEV *SZExtD = getZeroExtendExpr(getConstant(D), Ty, Depth);
            const SCEV *SResidual =
                getAddRecExpr(getConstant(C - D), Step, L, AR->getNoWrapFlags());
            const SCEV *SZExtR = getZeroExtendExpr(SResidual, Ty, Depth + 1);
            return getAddExpr(SZExtD, SZExtR,
                              (SCEV::NoWrapFlags)(SCEV::FlagNSW | SCEV::FlagNUW),
                              Depth + 1);
          }
        }
  
        if (proveNoWrapByVaryingStart<SCEVZeroExtendExpr>(Start, Step, L)) {
          setNoWrapFlags(const_cast<SCEVAddRecExpr *>(AR), SCEV::FlagNUW);
          return getAddRecExpr(
              getExtendAddRecStart<SCEVZeroExtendExpr>(AR, Ty, this, Depth + 1),
              getZeroExtendExpr(Step, Ty, Depth + 1), L, AR->getNoWrapFlags());
        }
      }
  
    // zext(A % B) --> zext(A) % zext(B)
    {
      const SCEV *LHS;
      const SCEV *RHS;
      if (matchURem(Op, LHS, RHS))
        return getURemExpr(getZeroExtendExpr(LHS, Ty, Depth + 1),
                           getZeroExtendExpr(RHS, Ty, Depth + 1));
    }
  
    // zext(A / B) --> zext(A) / zext(B).
    if (auto *Div = dyn_cast<SCEVUDivExpr>(Op))
      return getUDivExpr(getZeroExtendExpr(Div->getLHS(), Ty, Depth + 1),
                         getZeroExtendExpr(Div->getRHS(), Ty, Depth + 1));
  
    if (auto *SA = dyn_cast<SCEVAddExpr>(Op)) {
      // zext((A + B + ...)<nuw>) --> (zext(A) + zext(B) + ...)<nuw>
      if (SA->hasNoUnsignedWrap()) {
        // If the addition does not unsign overflow then we can, by definition,
        // commute the zero extension with the addition operation.
        SmallVector<const SCEV *, 4> Ops;
        for (const auto *Op : SA->operands())
          Ops.push_back(getZeroExtendExpr(Op, Ty, Depth + 1));
        return getAddExpr(Ops, SCEV::FlagNUW, Depth + 1);
      }
  
      // zext(C + x + y + ...) --> (zext(D) + zext((C - D) + x + y + ...))
      // if D + (C - D + x + y + ...) could be proven to not unsigned wrap
      // where D maximizes the number of trailing zeros of (C - D + x + y + ...)
      //
      // Often address arithmetics contain expressions like
      // (zext (add (shl X, C1), C2)), for instance, (zext (5 + (4 * X))).
      // This transformation is useful while proving that such expressions are
      // equal or differ by a small constant amount, see LoadStoreVectorizer pass.
      if (const auto *SC = dyn_cast<SCEVConstant>(SA->getOperand(0))) {
        const APInt &D = extractConstantWithoutWrapping(*this, SC, SA);
        if (D != 0) {
          const SCEV *SZExtD = getZeroExtendExpr(getConstant(D), Ty, Depth);
          const SCEV *SResidual =
              getAddExpr(getConstant(-D), SA, SCEV::FlagAnyWrap, Depth);
          const SCEV *SZExtR = getZeroExtendExpr(SResidual, Ty, Depth + 1);
          return getAddExpr(SZExtD, SZExtR,
                            (SCEV::NoWrapFlags)(SCEV::FlagNSW | SCEV::FlagNUW),
                            Depth + 1);
        }
      }
    }
  
    if (auto *SM = dyn_cast<SCEVMulExpr>(Op)) {
      // zext((A * B * ...)<nuw>) --> (zext(A) * zext(B) * ...)<nuw>
      if (SM->hasNoUnsignedWrap()) {
        // If the multiply does not unsign overflow then we can, by definition,
        // commute the zero extension with the multiply operation.
        SmallVector<const SCEV *, 4> Ops;
        for (const auto *Op : SM->operands())
          Ops.push_back(getZeroExtendExpr(Op, Ty, Depth + 1));
        return getMulExpr(Ops, SCEV::FlagNUW, Depth + 1);
      }
  
      // zext(2^K * (trunc X to iN)) to iM ->
      // 2^K * (zext(trunc X to i{N-K}) to iM)<nuw>
      //
      // Proof:
      //
      //     zext(2^K * (trunc X to iN)) to iM
      //   = zext((trunc X to iN) << K) to iM
      //   = zext((trunc X to i{N-K}) << K)<nuw> to iM
      //     (because shl removes the top K bits)
      //   = zext((2^K * (trunc X to i{N-K}))<nuw>) to iM
      //   = (2^K * (zext(trunc X to i{N-K}) to iM))<nuw>.
      //
      if (SM->getNumOperands() == 2)
        if (auto *MulLHS = dyn_cast<SCEVConstant>(SM->getOperand(0)))
          if (MulLHS->getAPInt().isPowerOf2())
            if (auto *TruncRHS = dyn_cast<SCEVTruncateExpr>(SM->getOperand(1))) {
              int NewTruncBits = getTypeSizeInBits(TruncRHS->getType()) -
                                 MulLHS->getAPInt().logBase2();
              Type *NewTruncTy = IntegerType::get(getContext(), NewTruncBits);
              return getMulExpr(
                  getZeroExtendExpr(MulLHS, Ty),
                  getZeroExtendExpr(
                      getTruncateExpr(TruncRHS->getOperand(), NewTruncTy), Ty),
                  SCEV::FlagNUW, Depth + 1);
            }
    }
  
    // The cast wasn't folded; create an explicit cast node.
    // Recompute the insert position, as it may have been invalidated.
    if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;
    SCEV *S = new (SCEVAllocator) SCEVZeroExtendExpr(ID.Intern(SCEVAllocator),
                                                     Op, Ty);
    UniqueSCEVs.InsertNode(S, IP);
    addToLoopUseLists(S);
    return S;
  }
  
  const SCEV *
  ScalarEvolution::getSignExtendExpr(const SCEV *Op, Type *Ty, unsigned Depth) {
    assert(getTypeSizeInBits(Op->getType()) < getTypeSizeInBits(Ty) &&
           "This is not an extending conversion!");
    assert(isSCEVable(Ty) &&
           "This is not a conversion to a SCEVable type!");
    assert(!Op->getType()->isPointerTy() && "Can't extend pointer!");
    Ty = getEffectiveSCEVType(Ty);
  
    // Fold if the operand is constant.
    if (const SCEVConstant *SC = dyn_cast<SCEVConstant>(Op))
      return getConstant(
        cast<ConstantInt>(ConstantExpr::getSExt(SC->getValue(), Ty)));
  
    // sext(sext(x)) --> sext(x)
    if (const SCEVSignExtendExpr *SS = dyn_cast<SCEVSignExtendExpr>(Op))
      return getSignExtendExpr(SS->getOperand(), Ty, Depth + 1);
  
    // sext(zext(x)) --> zext(x)
    if (const SCEVZeroExtendExpr *SZ = dyn_cast<SCEVZeroExtendExpr>(Op))
      return getZeroExtendExpr(SZ->getOperand(), Ty, Depth + 1);
  
    // Before doing any expensive analysis, check to see if we've already
    // computed a SCEV for this Op and Ty.
    FoldingSetNodeID ID;
    ID.AddInteger(scSignExtend);
    ID.AddPointer(Op);
    ID.AddPointer(Ty);
    void *IP = nullptr;
    if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;
    // Limit recursion depth.
    if (Depth > MaxCastDepth) {
      SCEV *S = new (SCEVAllocator) SCEVSignExtendExpr(ID.Intern(SCEVAllocator),
                                                       Op, Ty);
      UniqueSCEVs.InsertNode(S, IP);
      addToLoopUseLists(S);
      return S;
    }
  
    // sext(trunc(x)) --> sext(x) or x or trunc(x)
    if (const SCEVTruncateExpr *ST = dyn_cast<SCEVTruncateExpr>(Op)) {
      // It's possible the bits taken off by the truncate were all sign bits. If
      // so, we should be able to simplify this further.
      const SCEV *X = ST->getOperand();
      ConstantRange CR = getSignedRange(X);
      unsigned TruncBits = getTypeSizeInBits(ST->getType());
      unsigned NewBits = getTypeSizeInBits(Ty);
      if (CR.truncate(TruncBits).signExtend(NewBits).contains(
              CR.sextOrTrunc(NewBits)))
        return getTruncateOrSignExtend(X, Ty, Depth);
    }
  
    if (auto *SA = dyn_cast<SCEVAddExpr>(Op)) {
      // sext((A + B + ...)<nsw>) --> (sext(A) + sext(B) + ...)<nsw>
      if (SA->hasNoSignedWrap()) {
        // If the addition does not sign overflow then we can, by definition,
        // commute the sign extension with the addition operation.
        SmallVector<const SCEV *, 4> Ops;
        for (const auto *Op : SA->operands())
          Ops.push_back(getSignExtendExpr(Op, Ty, Depth + 1));
        return getAddExpr(Ops, SCEV::FlagNSW, Depth + 1);
      }
  
      // sext(C + x + y + ...) --> (sext(D) + sext((C - D) + x + y + ...))
      // if D + (C - D + x + y + ...) could be proven to not signed wrap
      // where D maximizes the number of trailing zeros of (C - D + x + y + ...)
      //
      // For instance, this will bring two seemingly different expressions:
      //     1 + sext(5 + 20 * %x + 24 * %y)  and
      //         sext(6 + 20 * %x + 24 * %y)
      // to the same form:
      //     2 + sext(4 + 20 * %x + 24 * %y)
      if (const auto *SC = dyn_cast<SCEVConstant>(SA->getOperand(0))) {
        const APInt &D = extractConstantWithoutWrapping(*this, SC, SA);
        if (D != 0) {
          const SCEV *SSExtD = getSignExtendExpr(getConstant(D), Ty, Depth);
          const SCEV *SResidual =
              getAddExpr(getConstant(-D), SA, SCEV::FlagAnyWrap, Depth);
          const SCEV *SSExtR = getSignExtendExpr(SResidual, Ty, Depth + 1);
          return getAddExpr(SSExtD, SSExtR,
                            (SCEV::NoWrapFlags)(SCEV::FlagNSW | SCEV::FlagNUW),
                            Depth + 1);
        }
      }
    }
    // If the input value is a chrec scev, and we can prove that the value
    // did not overflow the old, smaller, value, we can sign extend all of the
    // operands (often constants).  This allows analysis of something like
    // this:  for (signed char X = 0; X < 100; ++X) { int Y = X; }
    if (const SCEVAddRecExpr *AR = dyn_cast<SCEVAddRecExpr>(Op))
      if (AR->isAffine()) {
        const SCEV *Start = AR->getStart();
        const SCEV *Step = AR->getStepRecurrence(*this);
        unsigned BitWidth = getTypeSizeInBits(AR->getType());
        const Loop *L = AR->getLoop();
  
        if (!AR->hasNoSignedWrap()) {
          auto NewFlags = proveNoWrapViaConstantRanges(AR);
          setNoWrapFlags(const_cast<SCEVAddRecExpr *>(AR), NewFlags);
        }
  
        // If we have special knowledge that this addrec won't overflow,
        // we don't need to do any further analysis.
        if (AR->hasNoSignedWrap())
          return getAddRecExpr(
              getExtendAddRecStart<SCEVSignExtendExpr>(AR, Ty, this, Depth + 1),
              getSignExtendExpr(Step, Ty, Depth + 1), L, SCEV::FlagNSW);
  
        // Check whether the backedge-taken count is SCEVCouldNotCompute.
        // Note that this serves two purposes: It filters out loops that are
        // simply not analyzable, and it covers the case where this code is
        // being called from within backedge-taken count analysis, such that
        // attempting to ask for the backedge-taken count would likely result
        // in infinite recursion. In the later case, the analysis code will
        // cope with a conservative value, and it will take care to purge
        // that value once it has finished.
        const SCEV *MaxBECount = getConstantMaxBackedgeTakenCount(L);
        if (!isa<SCEVCouldNotCompute>(MaxBECount)) {
          // Manually compute the final value for AR, checking for
          // overflow.
  
          // Check whether the backedge-taken count can be losslessly casted to
          // the addrec's type. The count is always unsigned.
          const SCEV *CastedMaxBECount =
              getTruncateOrZeroExtend(MaxBECount, Start->getType(), Depth);
          const SCEV *RecastedMaxBECount = getTruncateOrZeroExtend(
              CastedMaxBECount, MaxBECount->getType(), Depth);
          if (MaxBECount == RecastedMaxBECount) {
            Type *WideTy = IntegerType::get(getContext(), BitWidth * 2);
            // Check whether Start+Step*MaxBECount has no signed overflow.
            const SCEV *SMul = getMulExpr(CastedMaxBECount, Step,
                                          SCEV::FlagAnyWrap, Depth + 1);
            const SCEV *SAdd = getSignExtendExpr(getAddExpr(Start, SMul,
                                                            SCEV::FlagAnyWrap,
                                                            Depth + 1),
                                                 WideTy, Depth + 1);
            const SCEV *WideStart = getSignExtendExpr(Start, WideTy, Depth + 1);
            const SCEV *WideMaxBECount =
              getZeroExtendExpr(CastedMaxBECount, WideTy, Depth + 1);
            const SCEV *OperandExtendedAdd =
              getAddExpr(WideStart,
                         getMulExpr(WideMaxBECount,
                                    getSignExtendExpr(Step, WideTy, Depth + 1),
                                    SCEV::FlagAnyWrap, Depth + 1),
                         SCEV::FlagAnyWrap, Depth + 1);
            if (SAdd == OperandExtendedAdd) {
              // Cache knowledge of AR NSW, which is propagated to this AddRec.
              setNoWrapFlags(const_cast<SCEVAddRecExpr *>(AR), SCEV::FlagNSW);
              // Return the expression with the addrec on the outside.
              return getAddRecExpr(
                  getExtendAddRecStart<SCEVSignExtendExpr>(AR, Ty, this,
                                                           Depth + 1),
                  getSignExtendExpr(Step, Ty, Depth + 1), L,
                  AR->getNoWrapFlags());
            }
            // Similar to above, only this time treat the step value as unsigned.
            // This covers loops that count up with an unsigned step.
            OperandExtendedAdd =
              getAddExpr(WideStart,
                         getMulExpr(WideMaxBECount,
                                    getZeroExtendExpr(Step, WideTy, Depth + 1),
                                    SCEV::FlagAnyWrap, Depth + 1),
                         SCEV::FlagAnyWrap, Depth + 1);
            if (SAdd == OperandExtendedAdd) {
              // If AR wraps around then
              //
              //    abs(Step) * MaxBECount > unsigned-max(AR->getType())
              // => SAdd != OperandExtendedAdd
              //
              // Thus (AR is not NW => SAdd != OperandExtendedAdd) <=>
              // (SAdd == OperandExtendedAdd => AR is NW)
  
              setNoWrapFlags(const_cast<SCEVAddRecExpr *>(AR), SCEV::FlagNW);
  
              // Return the expression with the addrec on the outside.
              return getAddRecExpr(
                  getExtendAddRecStart<SCEVSignExtendExpr>(AR, Ty, this,
                                                           Depth + 1),
                  getZeroExtendExpr(Step, Ty, Depth + 1), L,
                  AR->getNoWrapFlags());
            }
          }
        }
  
        auto NewFlags = proveNoSignedWrapViaInduction(AR);
        setNoWrapFlags(const_cast<SCEVAddRecExpr *>(AR), NewFlags);
        if (AR->hasNoSignedWrap()) {
          // Same as nsw case above - duplicated here to avoid a compile time
          // issue.  It's not clear that the order of checks does matter, but
          // it's one of two issue possible causes for a change which was
          // reverted.  Be conservative for the moment.
          return getAddRecExpr(
              getExtendAddRecStart<SCEVSignExtendExpr>(AR, Ty, this, Depth + 1),
              getSignExtendExpr(Step, Ty, Depth + 1), L, AR->getNoWrapFlags());
        }
  
        // sext({C,+,Step}) --> (sext(D) + sext({C-D,+,Step}))<nuw><nsw>
        // if D + (C - D + Step * n) could be proven to not signed wrap
        // where D maximizes the number of trailing zeros of (C - D + Step * n)
        if (const auto *SC = dyn_cast<SCEVConstant>(Start)) {
          const APInt &C = SC->getAPInt();
          const APInt &D = extractConstantWithoutWrapping(*this, C, Step);
          if (D != 0) {
            const SCEV *SSExtD = getSignExtendExpr(getConstant(D), Ty, Depth);
            const SCEV *SResidual =
                getAddRecExpr(getConstant(C - D), Step, L, AR->getNoWrapFlags());
            const SCEV *SSExtR = getSignExtendExpr(SResidual, Ty, Depth + 1);
            return getAddExpr(SSExtD, SSExtR,
                              (SCEV::NoWrapFlags)(SCEV::FlagNSW | SCEV::FlagNUW),
                              Depth + 1);
          }
        }
  
        if (proveNoWrapByVaryingStart<SCEVSignExtendExpr>(Start, Step, L)) {
          setNoWrapFlags(const_cast<SCEVAddRecExpr *>(AR), SCEV::FlagNSW);
          return getAddRecExpr(
              getExtendAddRecStart<SCEVSignExtendExpr>(AR, Ty, this, Depth + 1),
              getSignExtendExpr(Step, Ty, Depth + 1), L, AR->getNoWrapFlags());
        }
      }
  
    // If the input value is provably positive and we could not simplify
    // away the sext build a zext instead.
    if (isKnownNonNegative(Op))
      return getZeroExtendExpr(Op, Ty, Depth + 1);
  
    // The cast wasn't folded; create an explicit cast node.
    // Recompute the insert position, as it may have been invalidated.
    if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;
    SCEV *S = new (SCEVAllocator) SCEVSignExtendExpr(ID.Intern(SCEVAllocator),
                                                     Op, Ty);
    UniqueSCEVs.InsertNode(S, IP);
    addToLoopUseLists(S);
    return S;
  }
  
  /// getAnyExtendExpr - Return a SCEV for the given operand extended with
  /// unspecified bits out to the given type.
  const SCEV *ScalarEvolution::getAnyExtendExpr(const SCEV *Op,
                                                Type *Ty) {
    assert(getTypeSizeInBits(Op->getType()) < getTypeSizeInBits(Ty) &&
           "This is not an extending conversion!");
    assert(isSCEVable(Ty) &&
           "This is not a conversion to a SCEVable type!");
    Ty = getEffectiveSCEVType(Ty);
  
    // Sign-extend negative constants.
    if (const SCEVConstant *SC = dyn_cast<SCEVConstant>(Op))
      if (SC->getAPInt().isNegative())
        return getSignExtendExpr(Op, Ty);
  
    // Peel off a truncate cast.
    if (const SCEVTruncateExpr *T = dyn_cast<SCEVTruncateExpr>(Op)) {
      const SCEV *NewOp = T->getOperand();
      if (getTypeSizeInBits(NewOp->getType()) < getTypeSizeInBits(Ty))
        return getAnyExtendExpr(NewOp, Ty);
      return getTruncateOrNoop(NewOp, Ty);
    }
  
    // Next try a zext cast. If the cast is folded, use it.
    const SCEV *ZExt = getZeroExtendExpr(Op, Ty);
    if (!isa<SCEVZeroExtendExpr>(ZExt))
      return ZExt;
  
    // Next try a sext cast. If the cast is folded, use it.
    const SCEV *SExt = getSignExtendExpr(Op, Ty);
    if (!isa<SCEVSignExtendExpr>(SExt))
      return SExt;
  
    // Force the cast to be folded into the operands of an addrec.
    if (const SCEVAddRecExpr *AR = dyn_cast<SCEVAddRecExpr>(Op)) {
      SmallVector<const SCEV *, 4> Ops;
      for (const SCEV *Op : AR->operands())
        Ops.push_back(getAnyExtendExpr(Op, Ty));
      return getAddRecExpr(Ops, AR->getLoop(), SCEV::FlagNW);
    }
  
    // If the expression is obviously signed, use the sext cast value.
    if (isa<SCEVSMaxExpr>(Op))
      return SExt;
  
    // Absent any other information, use the zext cast value.
    return ZExt;
  }
  
  /// Process the given Ops list, which is a list of operands to be added under
  /// the given scale, update the given map. This is a helper function for
  /// getAddRecExpr. As an example of what it does, given a sequence of operands
  /// that would form an add expression like this:
  ///
  ///    m + n + 13 + (A * (o + p + (B * (q + m + 29)))) + r + (-1 * r)
  ///
  /// where A and B are constants, update the map with these values:
  ///
  ///    (m, 1+A*B), (n, 1), (o, A), (p, A), (q, A*B), (r, 0)
  ///
  /// and add 13 + A*B*29 to AccumulatedConstant.
  /// This will allow getAddRecExpr to produce this:
  ///
  ///    13+A*B*29 + n + (m * (1+A*B)) + ((o + p) * A) + (q * A*B)
  ///
  /// This form often exposes folding opportunities that are hidden in
  /// the original operand list.
  ///
  /// Return true iff it appears that any interesting folding opportunities
  /// may be exposed. This helps getAddRecExpr short-circuit extra work in
  /// the common case where no interesting opportunities are present, and
  /// is also used as a check to avoid infinite recursion.
  static bool
  CollectAddOperandsWithScales(DenseMap<const SCEV *, APInt> &M,
                               SmallVectorImpl<const SCEV *> &NewOps,
                               APInt &AccumulatedConstant,
                               const SCEV *const *Ops, size_t NumOperands,
                               const APInt &Scale,
                               ScalarEvolution &SE) {
    bool Interesting = false;
  
    // Iterate over the add operands. They are sorted, with constants first.
    unsigned i = 0;
    while (const SCEVConstant *C = dyn_cast<SCEVConstant>(Ops[i])) {
      ++i;
      // Pull a buried constant out to the outside.
      if (Scale != 1 || AccumulatedConstant != 0 || C->getValue()->isZero())
        Interesting = true;
      AccumulatedConstant += Scale * C->getAPInt();
    }
  
    // Next comes everything else. We're especially interested in multiplies
    // here, but they're in the middle, so just visit the rest with one loop.
    for (; i != NumOperands; ++i) {
      const SCEVMulExpr *Mul = dyn_cast<SCEVMulExpr>(Ops[i]);
      if (Mul && isa<SCEVConstant>(Mul->getOperand(0))) {
        APInt NewScale =
            Scale * cast<SCEVConstant>(Mul->getOperand(0))->getAPInt();
        if (Mul->getNumOperands() == 2 && isa<SCEVAddExpr>(Mul->getOperand(1))) {
          // A multiplication of a constant with another add; recurse.
          const SCEVAddExpr *Add = cast<SCEVAddExpr>(Mul->getOperand(1));
          Interesting |=
            CollectAddOperandsWithScales(M, NewOps, AccumulatedConstant,
                                         Add->op_begin(), Add->getNumOperands(),
                                         NewScale, SE);
        } else {
          // A multiplication of a constant with some other value. Update
          // the map.
          SmallVector<const SCEV *, 4> MulOps(drop_begin(Mul->operands()));
          const SCEV *Key = SE.getMulExpr(MulOps);
          auto Pair = M.insert({Key, NewScale});
          if (Pair.second) {
            NewOps.push_back(Pair.first->first);
          } else {
            Pair.first->second += NewScale;
            // The map already had an entry for this value, which may indicate
            // a folding opportunity.
            Interesting = true;
          }
        }
      } else {
        // An ordinary operand. Update the map.
        std::pair<DenseMap<const SCEV *, APInt>::iterator, bool> Pair =
            M.insert({Ops[i], Scale});
        if (Pair.second) {
          NewOps.push_back(Pair.first->first);
        } else {
          Pair.first->second += Scale;
          // The map already had an entry for this value, which may indicate
          // a folding opportunity.
          Interesting = true;
        }
      }
    }
  
    return Interesting;
  }
  
  bool ScalarEvolution::willNotOverflow(Instruction::BinaryOps BinOp, bool Signed,
                                        const SCEV *LHS, const SCEV *RHS) {
    const SCEV *(ScalarEvolution::*Operation)(const SCEV *, const SCEV *,
                                              SCEV::NoWrapFlags, unsigned);
    switch (BinOp) {
    default:
      llvm_unreachable("Unsupported binary op");
    case Instruction::Add:
      Operation = &ScalarEvolution::getAddExpr;
      break;
    case Instruction::Sub:
      Operation = &ScalarEvolution::getMinusSCEV;
      break;
    case Instruction::Mul:
      Operation = &ScalarEvolution::getMulExpr;
      break;
    }
  
    const SCEV *(ScalarEvolution::*Extension)(const SCEV *, Type *, unsigned) =
        Signed ? &ScalarEvolution::getSignExtendExpr
               : &ScalarEvolution::getZeroExtendExpr;
  
    // Check ext(LHS op RHS) == ext(LHS) op ext(RHS)
    auto *NarrowTy = cast<IntegerType>(LHS->getType());
    auto *WideTy =
        IntegerType::get(NarrowTy->getContext(), NarrowTy->getBitWidth() * 2);
  
    const SCEV *A = (this->*Extension)(
        (this->*Operation)(LHS, RHS, SCEV::FlagAnyWrap, 0), WideTy, 0);
    const SCEV *B = (this->*Operation)((this->*Extension)(LHS, WideTy, 0),
                                       (this->*Extension)(RHS, WideTy, 0),
                                       SCEV::FlagAnyWrap, 0);
    return A == B;
  }
  
  std::pair<SCEV::NoWrapFlags, bool /*Deduced*/>
  ScalarEvolution::getStrengthenedNoWrapFlagsFromBinOp(
      const OverflowingBinaryOperator *OBO) {
    SCEV::NoWrapFlags Flags = SCEV::NoWrapFlags::FlagAnyWrap;
  
    if (OBO->hasNoUnsignedWrap())
      Flags = ScalarEvolution::setFlags(Flags, SCEV::FlagNUW);
    if (OBO->hasNoSignedWrap())
      Flags = ScalarEvolution::setFlags(Flags, SCEV::FlagNSW);
  
    bool Deduced = false;
  
    if (OBO->hasNoUnsignedWrap() && OBO->hasNoSignedWrap())
      return {Flags, Deduced};
  
    if (OBO->getOpcode() != Instruction::Add &&
        OBO->getOpcode() != Instruction::Sub &&
        OBO->getOpcode() != Instruction::Mul)
      return {Flags, Deduced};
  
    const SCEV *LHS = getSCEV(OBO->getOperand(0));
    const SCEV *RHS = getSCEV(OBO->getOperand(1));
  
    if (!OBO->hasNoUnsignedWrap() &&
        willNotOverflow((Instruction::BinaryOps)OBO->getOpcode(),
                        /* Signed */ false, LHS, RHS)) {
      Flags = ScalarEvolution::setFlags(Flags, SCEV::FlagNUW);
      Deduced = true;
    }
  
    if (!OBO->hasNoSignedWrap() &&
        willNotOverflow((Instruction::BinaryOps)OBO->getOpcode(),
                        /* Signed */ true, LHS, RHS)) {
      Flags = ScalarEvolution::setFlags(Flags, SCEV::FlagNSW);
      Deduced = true;
    }
  
    return {Flags, Deduced};
  }
  
  // We're trying to construct a SCEV of type `Type' with `Ops' as operands and
  // `OldFlags' as can't-wrap behavior.  Infer a more aggressive set of
  // can't-overflow flags for the operation if possible.
  static SCEV::NoWrapFlags
  StrengthenNoWrapFlags(ScalarEvolution *SE, SCEVTypes Type,
                        const ArrayRef<const SCEV *> Ops,
                        SCEV::NoWrapFlags Flags) {
    using namespace std::placeholders;
  
    using OBO = OverflowingBinaryOperator;
  
    bool CanAnalyze =
        Type == scAddExpr || Type == scAddRecExpr || Type == scMulExpr;
    (void)CanAnalyze;
    assert(CanAnalyze && "don't call from other places!");
  
    int SignOrUnsignMask = SCEV::FlagNUW | SCEV::FlagNSW;
    SCEV::NoWrapFlags SignOrUnsignWrap =
        ScalarEvolution::maskFlags(Flags, SignOrUnsignMask);
  
    // If FlagNSW is true and all the operands are non-negative, infer FlagNUW.
    auto IsKnownNonNegative = [&](const SCEV *S) {
      return SE->isKnownNonNegative(S);
    };
  
    if (SignOrUnsignWrap == SCEV::FlagNSW && all_of(Ops, IsKnownNonNegative))
      Flags =
          ScalarEvolution::setFlags(Flags, (SCEV::NoWrapFlags)SignOrUnsignMask);
  
    SignOrUnsignWrap = ScalarEvolution::maskFlags(Flags, SignOrUnsignMask);
  
    if (SignOrUnsignWrap != SignOrUnsignMask &&
        (Type == scAddExpr || Type == scMulExpr) && Ops.size() == 2 &&
        isa<SCEVConstant>(Ops[0])) {
  
      auto Opcode = [&] {
        switch (Type) {
        case scAddExpr:
          return Instruction::Add;
        case scMulExpr:
          return Instruction::Mul;
        default:
          llvm_unreachable("Unexpected SCEV op.");
        }
      }();
  
      const APInt &C = cast<SCEVConstant>(Ops[0])->getAPInt();
  
      // (A <opcode> C) --> (A <opcode> C)<nsw> if the op doesn't sign overflow.
      if (!(SignOrUnsignWrap & SCEV::FlagNSW)) {
        auto NSWRegion = ConstantRange::makeGuaranteedNoWrapRegion(
            Opcode, C, OBO::NoSignedWrap);
        if (NSWRegion.contains(SE->getSignedRange(Ops[1])))
          Flags = ScalarEvolution::setFlags(Flags, SCEV::FlagNSW);
      }
  
      // (A <opcode> C) --> (A <opcode> C)<nuw> if the op doesn't unsign overflow.
      if (!(SignOrUnsignWrap & SCEV::FlagNUW)) {
        auto NUWRegion = ConstantRange::makeGuaranteedNoWrapRegion(
            Opcode, C, OBO::NoUnsignedWrap);
        if (NUWRegion.contains(SE->getUnsignedRange(Ops[1])))
          Flags = ScalarEvolution::setFlags(Flags, SCEV::FlagNUW);
      }
    }
  
    return Flags;
  }
  
  bool ScalarEvolution::isAvailableAtLoopEntry(const SCEV *S, const Loop *L) {
    return isLoopInvariant(S, L) && properlyDominates(S, L->getHeader());
  }
  
  /// Get a canonical add expression, or something simpler if possible.
  const SCEV *ScalarEvolution::getAddExpr(SmallVectorImpl<const SCEV *> &Ops,
                                          SCEV::NoWrapFlags OrigFlags,
                                          unsigned Depth) {
    assert(!(OrigFlags & ~(SCEV::FlagNUW | SCEV::FlagNSW)) &&
           "only nuw or nsw allowed");
    assert(!Ops.empty() && "Cannot get empty add!");
    if (Ops.size() == 1) return Ops[0];
  #ifndef NDEBUG
    Type *ETy = getEffectiveSCEVType(Ops[0]->getType());
    for (unsigned i = 1, e = Ops.size(); i != e; ++i)
      assert(getEffectiveSCEVType(Ops[i]->getType()) == ETy &&
             "SCEVAddExpr operand types don't match!");
    unsigned NumPtrs = count_if(
        Ops, [](const SCEV *Op) { return Op->getType()->isPointerTy(); });
    assert(NumPtrs <= 1 && "add has at most one pointer operand");
  #endif
  
    // Sort by complexity, this groups all similar expression types together.
    GroupByComplexity(Ops, &LI, DT);
  
    // If there are any constants, fold them together.
    unsigned Idx = 0;
    if (const SCEVConstant *LHSC = dyn_cast<SCEVConstant>(Ops[0])) {
      ++Idx;
      assert(Idx < Ops.size());
      while (const SCEVConstant *RHSC = dyn_cast<SCEVConstant>(Ops[Idx])) {
        // We found two constants, fold them together!
        Ops[0] = getConstant(LHSC->getAPInt() + RHSC->getAPInt());
        if (Ops.size() == 2) return Ops[0];
        Ops.erase(Ops.begin()+1);  // Erase the folded element
        LHSC = cast<SCEVConstant>(Ops[0]);
      }
  
      // If we are left with a constant zero being added, strip it off.
      if (LHSC->getValue()->isZero()) {
        Ops.erase(Ops.begin());
        --Idx;
      }
  
      if (Ops.size() == 1) return Ops[0];
    }
  
    // Delay expensive flag strengthening until necessary.
    auto ComputeFlags = [this, OrigFlags](const ArrayRef<const SCEV *> Ops) {
      return StrengthenNoWrapFlags(this, scAddExpr, Ops, OrigFlags);
    };
  
    // Limit recursion calls depth.
    if (Depth > MaxArithDepth || hasHugeExpression(Ops))
      return getOrCreateAddExpr(Ops, ComputeFlags(Ops));
  
    if (SCEV *S = std::get<0>(findExistingSCEVInCache(scAddExpr, Ops))) {
      // Don't strengthen flags if we have no new information.
      SCEVAddExpr *Add = static_cast<SCEVAddExpr *>(S);
      if (Add->getNoWrapFlags(OrigFlags) != OrigFlags)
        Add->setNoWrapFlags(ComputeFlags(Ops));
      return S;
    }
  
    // Okay, check to see if the same value occurs in the operand list more than
    // once.  If so, merge them together into an multiply expression.  Since we
    // sorted the list, these values are required to be adjacent.
    Type *Ty = Ops[0]->getType();
    bool FoundMatch = false;
    for (unsigned i = 0, e = Ops.size(); i != e-1; ++i)
      if (Ops[i] == Ops[i+1]) {      //  X + Y + Y  -->  X + Y*2
        // Scan ahead to count how many equal operands there are.
        unsigned Count = 2;
        while (i+Count != e && Ops[i+Count] == Ops[i])
          ++Count;
        // Merge the values into a multiply.
        const SCEV *Scale = getConstant(Ty, Count);
        const SCEV *Mul = getMulExpr(Scale, Ops[i], SCEV::FlagAnyWrap, Depth + 1);
        if (Ops.size() == Count)
          return Mul;
        Ops[i] = Mul;
        Ops.erase(Ops.begin()+i+1, Ops.begin()+i+Count);
        --i; e -= Count - 1;
        FoundMatch = true;
      }
    if (FoundMatch)
      return getAddExpr(Ops, OrigFlags, Depth + 1);
  
    // Check for truncates. If all the operands are truncated from the same
    // type, see if factoring out the truncate would permit the result to be
    // folded. eg., n*trunc(x) + m*trunc(y) --> trunc(trunc(m)*x + trunc(n)*y)
    // if the contents of the resulting outer trunc fold to something simple.
    auto FindTruncSrcType = [&]() -> Type * {
      // We're ultimately looking to fold an addrec of truncs and muls of only
      // constants and truncs, so if we find any other types of SCEV
      // as operands of the addrec then we bail and return nullptr here.
      // Otherwise, we return the type of the operand of a trunc that we find.
      if (auto *T = dyn_cast<SCEVTruncateExpr>(Ops[Idx]))
        return T->getOperand()->getType();
      if (const auto *Mul = dyn_cast<SCEVMulExpr>(Ops[Idx])) {
        const auto *LastOp = Mul->getOperand(Mul->getNumOperands() - 1);
        if (const auto *T = dyn_cast<SCEVTruncateExpr>(LastOp))
          return T->getOperand()->getType();
      }
      return nullptr;
    };
    if (auto *SrcType = FindTruncSrcType()) {
      SmallVector<const SCEV *, 8> LargeOps;
      bool Ok = true;
      // Check all the operands to see if they can be represented in the
      // source type of the truncate.
      for (unsigned i = 0, e = Ops.size(); i != e; ++i) {
        if (const SCEVTruncateExpr *T = dyn_cast<SCEVTruncateExpr>(Ops[i])) {
          if (T->getOperand()->getType() != SrcType) {
            Ok = false;
            break;
          }
          LargeOps.push_back(T->getOperand());
        } else if (const SCEVConstant *C = dyn_cast<SCEVConstant>(Ops[i])) {
          LargeOps.push_back(getAnyExtendExpr(C, SrcType));
        } else if (const SCEVMulExpr *M = dyn_cast<SCEVMulExpr>(Ops[i])) {
          SmallVector<const SCEV *, 8> LargeMulOps;
          for (unsigned j = 0, f = M->getNumOperands(); j != f && Ok; ++j) {
            if (const SCEVTruncateExpr *T =
                  dyn_cast<SCEVTruncateExpr>(M->getOperand(j))) {
              if (T->getOperand()->getType() != SrcType) {
                Ok = false;
                break;
              }
              LargeMulOps.push_back(T->getOperand());
            } else if (const auto *C = dyn_cast<SCEVConstant>(M->getOperand(j))) {
              LargeMulOps.push_back(getAnyExtendExpr(C, SrcType));
            } else {
              Ok = false;
              break;
            }
          }
          if (Ok)
            LargeOps.push_back(getMulExpr(LargeMulOps, SCEV::FlagAnyWrap, Depth + 1));
        } else {
          Ok = false;
          break;
        }
      }
      if (Ok) {
        // Evaluate the expression in the larger type.
        const SCEV *Fold = getAddExpr(LargeOps, SCEV::FlagAnyWrap, Depth + 1);
        // If it folds to something simple, use it. Otherwise, don't.
        if (isa<SCEVConstant>(Fold) || isa<SCEVUnknown>(Fold))
          return getTruncateExpr(Fold, Ty);
      }
    }
  
    if (Ops.size() == 2) {
      // Check if we have an expression of the form ((X + C1) - C2), where C1 and
      // C2 can be folded in a way that allows retaining wrapping flags of (X +
      // C1).
      const SCEV *A = Ops[0];
      const SCEV *B = Ops[1];
      auto *AddExpr = dyn_cast<SCEVAddExpr>(B);
      auto *C = dyn_cast<SCEVConstant>(A);
      if (AddExpr && C && isa<SCEVConstant>(AddExpr->getOperand(0))) {
        auto C1 = cast<SCEVConstant>(AddExpr->getOperand(0))->getAPInt();
        auto C2 = C->getAPInt();
        SCEV::NoWrapFlags PreservedFlags = SCEV::FlagAnyWrap;
  
        APInt ConstAdd = C1 + C2;
        auto AddFlags = AddExpr->getNoWrapFlags();
        // Adding a smaller constant is NUW if the original AddExpr was NUW.
        if (ScalarEvolution::maskFlags(AddFlags, SCEV::FlagNUW) ==
                SCEV::FlagNUW &&
            ConstAdd.ule(C1)) {
          PreservedFlags =
              ScalarEvolution::setFlags(PreservedFlags, SCEV::FlagNUW);
        }
  
        // Adding a constant with the same sign and small magnitude is NSW, if the
        // original AddExpr was NSW.
        if (ScalarEvolution::maskFlags(AddFlags, SCEV::FlagNSW) ==
                SCEV::FlagNSW &&
            C1.isSignBitSet() == ConstAdd.isSignBitSet() &&
            ConstAdd.abs().ule(C1.abs())) {
          PreservedFlags =
              ScalarEvolution::setFlags(PreservedFlags, SCEV::FlagNSW);
        }
  
        if (PreservedFlags != SCEV::FlagAnyWrap) {
          SmallVector<const SCEV *, 4> NewOps(AddExpr->op_begin(),
                                              AddExpr->op_end());
          NewOps[0] = getConstant(ConstAdd);
          return getAddExpr(NewOps, PreservedFlags);
        }
      }
    }
  
    // Skip past any other cast SCEVs.
    while (Idx < Ops.size() && Ops[Idx]->getSCEVType() < scAddExpr)
      ++Idx;
  
    // If there are add operands they would be next.
    if (Idx < Ops.size()) {
      bool DeletedAdd = false;
      // If the original flags and all inlined SCEVAddExprs are NUW, use the
      // common NUW flag for expression after inlining. Other flags cannot be
      // preserved, because they may depend on the original order of operations.
      SCEV::NoWrapFlags CommonFlags = maskFlags(OrigFlags, SCEV::FlagNUW);
      while (const SCEVAddExpr *Add = dyn_cast<SCEVAddExpr>(Ops[Idx])) {
        if (Ops.size() > AddOpsInlineThreshold ||
            Add->getNumOperands() > AddOpsInlineThreshold)
          break;
        // If we have an add, expand the add operands onto the end of the operands
        // list.
        Ops.erase(Ops.begin()+Idx);
        Ops.append(Add->op_begin(), Add->op_end());
        DeletedAdd = true;
        CommonFlags = maskFlags(CommonFlags, Add->getNoWrapFlags());
      }
  
      // If we deleted at least one add, we added operands to the end of the list,
      // and they are not necessarily sorted.  Recurse to resort and resimplify
      // any operands we just acquired.
      if (DeletedAdd)
        return getAddExpr(Ops, CommonFlags, Depth + 1);
    }
  
    // Skip over the add expression until we get to a multiply.
    while (Idx < Ops.size() && Ops[Idx]->getSCEVType() < scMulExpr)
      ++Idx;
  
    // Check to see if there are any folding opportunities present with
    // operands multiplied by constant values.
    if (Idx < Ops.size() && isa<SCEVMulExpr>(Ops[Idx])) {
      uint64_t BitWidth = getTypeSizeInBits(Ty);
      DenseMap<const SCEV *, APInt> M;
      SmallVector<const SCEV *, 8> NewOps;
      APInt AccumulatedConstant(BitWidth, 0);
      if (CollectAddOperandsWithScales(M, NewOps, AccumulatedConstant,
                                       Ops.data(), Ops.size(),
                                       APInt(BitWidth, 1), *this)) {
        struct APIntCompare {
          bool operator()(const APInt &LHS, const APInt &RHS) const {
            return LHS.ult(RHS);
          }
        };
  
        // Some interesting folding opportunity is present, so its worthwhile to
        // re-generate the operands list. Group the operands by constant scale,
        // to avoid multiplying by the same constant scale multiple times.
        std::map<APInt, SmallVector<const SCEV *, 4>, APIntCompare> MulOpLists;
        for (const SCEV *NewOp : NewOps)
          MulOpLists[M.find(NewOp)->second].push_back(NewOp);
        // Re-generate the operands list.
        Ops.clear();
        if (AccumulatedConstant != 0)
          Ops.push_back(getConstant(AccumulatedConstant));
        for (auto &MulOp : MulOpLists) {
          if (MulOp.first == 1) {
            Ops.push_back(getAddExpr(MulOp.second, SCEV::FlagAnyWrap, Depth + 1));
          } else if (MulOp.first != 0) {
            Ops.push_back(getMulExpr(
                getConstant(MulOp.first),
                getAddExpr(MulOp.second, SCEV::FlagAnyWrap, Depth + 1),
                SCEV::FlagAnyWrap, Depth + 1));
          }
        }
        if (Ops.empty())
          return getZero(Ty);
        if (Ops.size() == 1)
          return Ops[0];
        return getAddExpr(Ops, SCEV::FlagAnyWrap, Depth + 1);
      }
    }
  
    // If we are adding something to a multiply expression, make sure the
    // something is not already an operand of the multiply.  If so, merge it into
    // the multiply.
    for (; Idx < Ops.size() && isa<SCEVMulExpr>(Ops[Idx]); ++Idx) {
      const SCEVMulExpr *Mul = cast<SCEVMulExpr>(Ops[Idx]);
      for (unsigned MulOp = 0, e = Mul->getNumOperands(); MulOp != e; ++MulOp) {
        const SCEV *MulOpSCEV = Mul->getOperand(MulOp);
        if (isa<SCEVConstant>(MulOpSCEV))
          continue;
        for (unsigned AddOp = 0, e = Ops.size(); AddOp != e; ++AddOp)
          if (MulOpSCEV == Ops[AddOp]) {
            // Fold W + X + (X * Y * Z)  -->  W + (X * ((Y*Z)+1))
            const SCEV *InnerMul = Mul->getOperand(MulOp == 0);
            if (Mul->getNumOperands() != 2) {
              // If the multiply has more than two operands, we must get the
              // Y*Z term.
              SmallVector<const SCEV *, 4> MulOps(Mul->op_begin(),
                                                  Mul->op_begin()+MulOp);
              MulOps.append(Mul->op_begin()+MulOp+1, Mul->op_end());
              InnerMul = getMulExpr(MulOps, SCEV::FlagAnyWrap, Depth + 1);
            }
            SmallVector<const SCEV *, 2> TwoOps = {getOne(Ty), InnerMul};
            const SCEV *AddOne = getAddExpr(TwoOps, SCEV::FlagAnyWrap, Depth + 1);
            const SCEV *OuterMul = getMulExpr(AddOne, MulOpSCEV,
                                              SCEV::FlagAnyWrap, Depth + 1);
            if (Ops.size() == 2) return OuterMul;
            if (AddOp < Idx) {
              Ops.erase(Ops.begin()+AddOp);
              Ops.erase(Ops.begin()+Idx-1);
            } else {
              Ops.erase(Ops.begin()+Idx);
              Ops.erase(Ops.begin()+AddOp-1);
            }
            Ops.push_back(OuterMul);
            return getAddExpr(Ops, SCEV::FlagAnyWrap, Depth + 1);
          }
  
        // Check this multiply against other multiplies being added together.
        for (unsigned OtherMulIdx = Idx+1;
             OtherMulIdx < Ops.size() && isa<SCEVMulExpr>(Ops[OtherMulIdx]);
             ++OtherMulIdx) {
          const SCEVMulExpr *OtherMul = cast<SCEVMulExpr>(Ops[OtherMulIdx]);
          // If MulOp occurs in OtherMul, we can fold the two multiplies
          // together.
          for (unsigned OMulOp = 0, e = OtherMul->getNumOperands();
               OMulOp != e; ++OMulOp)
            if (OtherMul->getOperand(OMulOp) == MulOpSCEV) {
              // Fold X + (A*B*C) + (A*D*E) --> X + (A*(B*C+D*E))
              const SCEV *InnerMul1 = Mul->getOperand(MulOp == 0);
              if (Mul->getNumOperands() != 2) {
                SmallVector<const SCEV *, 4> MulOps(Mul->op_begin(),
                                                    Mul->op_begin()+MulOp);
                MulOps.append(Mul->op_begin()+MulOp+1, Mul->op_end());
                InnerMul1 = getMulExpr(MulOps, SCEV::FlagAnyWrap, Depth + 1);
              }
              const SCEV *InnerMul2 = OtherMul->getOperand(OMulOp == 0);
              if (OtherMul->getNumOperands() != 2) {
                SmallVector<const SCEV *, 4> MulOps(OtherMul->op_begin(),
                                                    OtherMul->op_begin()+OMulOp);
                MulOps.append(OtherMul->op_begin()+OMulOp+1, OtherMul->op_end());
                InnerMul2 = getMulExpr(MulOps, SCEV::FlagAnyWrap, Depth + 1);
              }
              SmallVector<const SCEV *, 2> TwoOps = {InnerMul1, InnerMul2};
              const SCEV *InnerMulSum =
                  getAddExpr(TwoOps, SCEV::FlagAnyWrap, Depth + 1);
              const SCEV *OuterMul = getMulExpr(MulOpSCEV, InnerMulSum,
                                                SCEV::FlagAnyWrap, Depth + 1);
              if (Ops.size() == 2) return OuterMul;
              Ops.erase(Ops.begin()+Idx);
              Ops.erase(Ops.begin()+OtherMulIdx-1);
              Ops.push_back(OuterMul);
              return getAddExpr(Ops, SCEV::FlagAnyWrap, Depth + 1);
            }
        }
      }
    }
  
    // If there are any add recurrences in the operands list, see if any other
    // added values are loop invariant.  If so, we can fold them into the
    // recurrence.
    while (Idx < Ops.size() && Ops[Idx]->getSCEVType() < scAddRecExpr)
      ++Idx;
  
    // Scan over all recurrences, trying to fold loop invariants into them.
    for (; Idx < Ops.size() && isa<SCEVAddRecExpr>(Ops[Idx]); ++Idx) {
      // Scan all of the other operands to this add and add them to the vector if
      // they are loop invariant w.r.t. the recurrence.
      SmallVector<const SCEV *, 8> LIOps;
      const SCEVAddRecExpr *AddRec = cast<SCEVAddRecExpr>(Ops[Idx]);
      const Loop *AddRecLoop = AddRec->getLoop();
      for (unsigned i = 0, e = Ops.size(); i != e; ++i)
        if (isAvailableAtLoopEntry(Ops[i], AddRecLoop)) {
          LIOps.push_back(Ops[i]);
          Ops.erase(Ops.begin()+i);
          --i; --e;
        }
  
      // If we found some loop invariants, fold them into the recurrence.
      if (!LIOps.empty()) {
        // Compute nowrap flags for the addition of the loop-invariant ops and
        // the addrec. Temporarily push it as an operand for that purpose.
        LIOps.push_back(AddRec);
        SCEV::NoWrapFlags Flags = ComputeFlags(LIOps);
        LIOps.pop_back();
  
        //  NLI + LI + {Start,+,Step}  -->  NLI + {LI+Start,+,Step}
        LIOps.push_back(AddRec->getStart());
  
        SmallVector<const SCEV *, 4> AddRecOps(AddRec->operands());
        // This follows from the fact that the no-wrap flags on the outer add
        // expression are applicable on the 0th iteration, when the add recurrence
        // will be equal to its start value.
        AddRecOps[0] = getAddExpr(LIOps, Flags, Depth + 1);
  
        // Build the new addrec. Propagate the NUW and NSW flags if both the
        // outer add and the inner addrec are guaranteed to have no overflow.
        // Always propagate NW.
        Flags = AddRec->getNoWrapFlags(setFlags(Flags, SCEV::FlagNW));
        const SCEV *NewRec = getAddRecExpr(AddRecOps, AddRecLoop, Flags);
  
        // If all of the other operands were loop invariant, we are done.
        if (Ops.size() == 1) return NewRec;
  
        // Otherwise, add the folded AddRec by the non-invariant parts.
        for (unsigned i = 0;; ++i)
          if (Ops[i] == AddRec) {
            Ops[i] = NewRec;
            break;
          }
        return getAddExpr(Ops, SCEV::FlagAnyWrap, Depth + 1);
      }
  
      // Okay, if there weren't any loop invariants to be folded, check to see if
      // there are multiple AddRec's with the same loop induction variable being
      // added together.  If so, we can fold them.
      for (unsigned OtherIdx = Idx+1;
           OtherIdx < Ops.size() && isa<SCEVAddRecExpr>(Ops[OtherIdx]);
           ++OtherIdx) {
        // We expect the AddRecExpr's to be sorted in reverse dominance order,
        // so that the 1st found AddRecExpr is dominated by all others.
        assert(DT.dominates(
             cast<SCEVAddRecExpr>(Ops[OtherIdx])->getLoop()->getHeader(),
             AddRec->getLoop()->getHeader()) &&
          "AddRecExprs are not sorted in reverse dominance order?");
        if (AddRecLoop == cast<SCEVAddRecExpr>(Ops[OtherIdx])->getLoop()) {
          // Other + {A,+,B}<L> + {C,+,D}<L>  -->  Other + {A+C,+,B+D}<L>
          SmallVector<const SCEV *, 4> AddRecOps(AddRec->operands());
          for (; OtherIdx != Ops.size() && isa<SCEVAddRecExpr>(Ops[OtherIdx]);
               ++OtherIdx) {
            const auto *OtherAddRec = cast<SCEVAddRecExpr>(Ops[OtherIdx]);
            if (OtherAddRec->getLoop() == AddRecLoop) {
              for (unsigned i = 0, e = OtherAddRec->getNumOperands();
                   i != e; ++i) {
                if (i >= AddRecOps.size()) {
                  AddRecOps.append(OtherAddRec->op_begin()+i,
                                   OtherAddRec->op_end());
                  break;
                }
                SmallVector<const SCEV *, 2> TwoOps = {
                    AddRecOps[i], OtherAddRec->getOperand(i)};
                AddRecOps[i] = getAddExpr(TwoOps, SCEV::FlagAnyWrap, Depth + 1);
              }
              Ops.erase(Ops.begin() + OtherIdx); --OtherIdx;
            }
          }
          // Step size has changed, so we cannot guarantee no self-wraparound.
          Ops[Idx] = getAddRecExpr(AddRecOps, AddRecLoop, SCEV::FlagAnyWrap);
          return getAddExpr(Ops, SCEV::FlagAnyWrap, Depth + 1);
        }
      }
  
      // Otherwise couldn't fold anything into this recurrence.  Move onto the
      // next one.
    }
  
    // Okay, it looks like we really DO need an add expr.  Check to see if we
    // already have one, otherwise create a new one.
    return getOrCreateAddExpr(Ops, ComputeFlags(Ops));
  }
  
  const SCEV *
  ScalarEvolution::getOrCreateAddExpr(ArrayRef<const SCEV *> Ops,
                                      SCEV::NoWrapFlags Flags) {
    FoldingSetNodeID ID;
    ID.AddInteger(scAddExpr);
    for (const SCEV *Op : Ops)
      ID.AddPointer(Op);
    void *IP = nullptr;
    SCEVAddExpr *S =
        static_cast<SCEVAddExpr *>(UniqueSCEVs.FindNodeOrInsertPos(ID, IP));
    if (!S) {
      const SCEV **O = SCEVAllocator.Allocate<const SCEV *>(Ops.size());
      std::uninitialized_copy(Ops.begin(), Ops.end(), O);
      S = new (SCEVAllocator)
          SCEVAddExpr(ID.Intern(SCEVAllocator), O, Ops.size());
      UniqueSCEVs.InsertNode(S, IP);
      addToLoopUseLists(S);
    }
    S->setNoWrapFlags(Flags);
    return S;
  }
  
  const SCEV *
  ScalarEvolution::getOrCreateAddRecExpr(ArrayRef<const SCEV *> Ops,
                                         const Loop *L, SCEV::NoWrapFlags Flags) {
    FoldingSetNodeID ID;
    ID.AddInteger(scAddRecExpr);
    for (unsigned i = 0, e = Ops.size(); i != e; ++i)
      ID.AddPointer(Ops[i]);
    ID.AddPointer(L);
    void *IP = nullptr;
    SCEVAddRecExpr *S =
        static_cast<SCEVAddRecExpr *>(UniqueSCEVs.FindNodeOrInsertPos(ID, IP));
    if (!S) {
      const SCEV **O = SCEVAllocator.Allocate<const SCEV *>(Ops.size());
      std::uninitialized_copy(Ops.begin(), Ops.end(), O);
      S = new (SCEVAllocator)
          SCEVAddRecExpr(ID.Intern(SCEVAllocator), O, Ops.size(), L);
      UniqueSCEVs.InsertNode(S, IP);
      addToLoopUseLists(S);
    }
    setNoWrapFlags(S, Flags);
    return S;
  }
  
  const SCEV *
  ScalarEvolution::getOrCreateMulExpr(ArrayRef<const SCEV *> Ops,
                                      SCEV::NoWrapFlags Flags) {
    FoldingSetNodeID ID;
    ID.AddInteger(scMulExpr);
    for (unsigned i = 0, e = Ops.size(); i != e; ++i)
      ID.AddPointer(Ops[i]);
    void *IP = nullptr;
    SCEVMulExpr *S =
      static_cast<SCEVMulExpr *>(UniqueSCEVs.FindNodeOrInsertPos(ID, IP));
    if (!S) {
      const SCEV **O = SCEVAllocator.Allocate<const SCEV *>(Ops.size());
      std::uninitialized_copy(Ops.begin(), Ops.end(), O);
      S = new (SCEVAllocator) SCEVMulExpr(ID.Intern(SCEVAllocator),
                                          O, Ops.size());
      UniqueSCEVs.InsertNode(S, IP);
      addToLoopUseLists(S);
    }
    S->setNoWrapFlags(Flags);
    return S;
  }
  
  static uint64_t umul_ov(uint64_t i, uint64_t j, bool &Overflow) {
    uint64_t k = i*j;
    if (j > 1 && k / j != i) Overflow = true;
    return k;
  }
  
  /// Compute the result of "n choose k", the binomial coefficient.  If an
  /// intermediate computation overflows, Overflow will be set and the return will
  /// be garbage. Overflow is not cleared on absence of overflow.
  static uint64_t Choose(uint64_t n, uint64_t k, bool &Overflow) {
    // We use the multiplicative formula:
    //     n(n-1)(n-2)...(n-(k-1)) / k(k-1)(k-2)...1 .
    // At each iteration, we take the n-th term of the numeral and divide by the
    // (k-n)th term of the denominator.  This division will always produce an
    // integral result, and helps reduce the chance of overflow in the
    // intermediate computations. However, we can still overflow even when the
    // final result would fit.
  
    if (n == 0 || n == k) return 1;
    if (k > n) return 0;
  
    if (k > n/2)
      k = n-k;
  
    uint64_t r = 1;
    for (uint64_t i = 1; i <= k; ++i) {
      r = umul_ov(r, n-(i-1), Overflow);
      r /= i;
    }
    return r;
  }
  
  /// Determine if any of the operands in this SCEV are a constant or if
  /// any of the add or multiply expressions in this SCEV contain a constant.
  static bool containsConstantInAddMulChain(const SCEV *StartExpr) {
    struct FindConstantInAddMulChain {
      bool FoundConstant = false;
  
      bool follow(const SCEV *S) {
        FoundConstant |= isa<SCEVConstant>(S);
        return isa<SCEVAddExpr>(S) || isa<SCEVMulExpr>(S);
      }
  
      bool isDone() const {
        return FoundConstant;
      }
    };
  
    FindConstantInAddMulChain F;
    SCEVTraversal<FindConstantInAddMulChain> ST(F);
    ST.visitAll(StartExpr);
    return F.FoundConstant;
  }
  
  /// Get a canonical multiply expression, or something simpler if possible.
  const SCEV *ScalarEvolution::getMulExpr(SmallVectorImpl<const SCEV *> &Ops,
                                          SCEV::NoWrapFlags OrigFlags,
                                          unsigned Depth) {
    assert(OrigFlags == maskFlags(OrigFlags, SCEV::FlagNUW | SCEV::FlagNSW) &&
           "only nuw or nsw allowed");
    assert(!Ops.empty() && "Cannot get empty mul!");
    if (Ops.size() == 1) return Ops[0];
  #ifndef NDEBUG
    Type *ETy = Ops[0]->getType();
    assert(!ETy->isPointerTy());
    for (unsigned i = 1, e = Ops.size(); i != e; ++i)
      assert(Ops[i]->getType() == ETy &&
             "SCEVMulExpr operand types don't match!");
  #endif
  
    // Sort by complexity, this groups all similar expression types together.
    GroupByComplexity(Ops, &LI, DT);
  
    // If there are any constants, fold them together.
    unsigned Idx = 0;
    if (const SCEVConstant *LHSC = dyn_cast<SCEVConstant>(Ops[0])) {
      ++Idx;
      assert(Idx < Ops.size());
      while (const SCEVConstant *RHSC = dyn_cast<SCEVConstant>(Ops[Idx])) {
        // We found two constants, fold them together!
        Ops[0] = getConstant(LHSC->getAPInt() * RHSC->getAPInt());
        if (Ops.size() == 2) return Ops[0];
        Ops.erase(Ops.begin()+1);  // Erase the folded element
        LHSC = cast<SCEVConstant>(Ops[0]);
      }
  
      // If we have a multiply of zero, it will always be zero.
      if (LHSC->getValue()->isZero())
        return LHSC;
  
      // If we are left with a constant one being multiplied, strip it off.
      if (LHSC->getValue()->isOne()) {
        Ops.erase(Ops.begin());
        --Idx;
      }
  
      if (Ops.size() == 1)
        return Ops[0];
    }
  
    // Delay expensive flag strengthening until necessary.
    auto ComputeFlags = [this, OrigFlags](const ArrayRef<const SCEV *> Ops) {
      return StrengthenNoWrapFlags(this, scMulExpr, Ops, OrigFlags);
    };
  
    // Limit recursion calls depth.
    if (Depth > MaxArithDepth || hasHugeExpression(Ops))
      return getOrCreateMulExpr(Ops, ComputeFlags(Ops));
  
    if (SCEV *S = std::get<0>(findExistingSCEVInCache(scMulExpr, Ops))) {
      // Don't strengthen flags if we have no new information.
      SCEVMulExpr *Mul = static_cast<SCEVMulExpr *>(S);
      if (Mul->getNoWrapFlags(OrigFlags) != OrigFlags)
        Mul->setNoWrapFlags(ComputeFlags(Ops));
      return S;
    }
  
    if (const SCEVConstant *LHSC = dyn_cast<SCEVConstant>(Ops[0])) {
      if (Ops.size() == 2) {
        // C1*(C2+V) -> C1*C2 + C1*V
        if (const SCEVAddExpr *Add = dyn_cast<SCEVAddExpr>(Ops[1]))
          // If any of Add's ops are Adds or Muls with a constant, apply this
          // transformation as well.
          //
          // TODO: There are some cases where this transformation is not
          // profitable; for example, Add = (C0 + X) * Y + Z.  Maybe the scope of
          // this transformation should be narrowed down.
          if (Add->getNumOperands() == 2 && containsConstantInAddMulChain(Add))
            return getAddExpr(getMulExpr(LHSC, Add->getOperand(0),
                                         SCEV::FlagAnyWrap, Depth + 1),
                              getMulExpr(LHSC, Add->getOperand(1),
                                         SCEV::FlagAnyWrap, Depth + 1),
                              SCEV::FlagAnyWrap, Depth + 1);
  
        if (Ops[0]->isAllOnesValue()) {
          // If we have a mul by -1 of an add, try distributing the -1 among the
          // add operands.
          if (const SCEVAddExpr *Add = dyn_cast<SCEVAddExpr>(Ops[1])) {
            SmallVector<const SCEV *, 4> NewOps;
            bool AnyFolded = false;
            for (const SCEV *AddOp : Add->operands()) {
              const SCEV *Mul = getMulExpr(Ops[0], AddOp, SCEV::FlagAnyWrap,
                                           Depth + 1);
              if (!isa<SCEVMulExpr>(Mul)) AnyFolded = true;
              NewOps.push_back(Mul);
            }
            if (AnyFolded)
              return getAddExpr(NewOps, SCEV::FlagAnyWrap, Depth + 1);
          } else if (const auto *AddRec = dyn_cast<SCEVAddRecExpr>(Ops[1])) {
            // Negation preserves a recurrence's no self-wrap property.
            SmallVector<const SCEV *, 4> Operands;
            for (const SCEV *AddRecOp : AddRec->operands())
              Operands.push_back(getMulExpr(Ops[0], AddRecOp, SCEV::FlagAnyWrap,
                                            Depth + 1));
  
            return getAddRecExpr(Operands, AddRec->getLoop(),
                                 AddRec->getNoWrapFlags(SCEV::FlagNW));
          }
        }
      }
    }
  
    // Skip over the add expression until we get to a multiply.
    while (Idx < Ops.size() && Ops[Idx]->getSCEVType() < scMulExpr)
      ++Idx;
  
    // If there are mul operands inline them all into this expression.
    if (Idx < Ops.size()) {
      bool DeletedMul = false;
      while (const SCEVMulExpr *Mul = dyn_cast<SCEVMulExpr>(Ops[Idx])) {
        if (Ops.size() > MulOpsInlineThreshold)
          break;
        // If we have an mul, expand the mul operands onto the end of the
        // operands list.
        Ops.erase(Ops.begin()+Idx);
        Ops.append(Mul->op_begin(), Mul->op_end());
        DeletedMul = true;
      }
  
      // If we deleted at least one mul, we added operands to the end of the
      // list, and they are not necessarily sorted.  Recurse to resort and
      // resimplify any operands we just acquired.
      if (DeletedMul)
        return getMulExpr(Ops, SCEV::FlagAnyWrap, Depth + 1);
    }
  
    // If there are any add recurrences in the operands list, see if any other
    // added values are loop invariant.  If so, we can fold them into the
    // recurrence.
    while (Idx < Ops.size() && Ops[Idx]->getSCEVType() < scAddRecExpr)
      ++Idx;
  
    // Scan over all recurrences, trying to fold loop invariants into them.
    for (; Idx < Ops.size() && isa<SCEVAddRecExpr>(Ops[Idx]); ++Idx) {
      // Scan all of the other operands to this mul and add them to the vector
      // if they are loop invariant w.r.t. the recurrence.
      SmallVector<const SCEV *, 8> LIOps;
      const SCEVAddRecExpr *AddRec = cast<SCEVAddRecExpr>(Ops[Idx]);
      const Loop *AddRecLoop = AddRec->getLoop();
      for (unsigned i = 0, e = Ops.size(); i != e; ++i)
        if (isAvailableAtLoopEntry(Ops[i], AddRecLoop)) {
          LIOps.push_back(Ops[i]);
          Ops.erase(Ops.begin()+i);
          --i; --e;
        }
  
      // If we found some loop invariants, fold them into the recurrence.
      if (!LIOps.empty()) {
        //  NLI * LI * {Start,+,Step}  -->  NLI * {LI*Start,+,LI*Step}
        SmallVector<const SCEV *, 4> NewOps;
        NewOps.reserve(AddRec->getNumOperands());
        const SCEV *Scale = getMulExpr(LIOps, SCEV::FlagAnyWrap, Depth + 1);
        for (unsigned i = 0, e = AddRec->getNumOperands(); i != e; ++i)
          NewOps.push_back(getMulExpr(Scale, AddRec->getOperand(i),
                                      SCEV::FlagAnyWrap, Depth + 1));
  
        // Build the new addrec. Propagate the NUW and NSW flags if both the
        // outer mul and the inner addrec are guaranteed to have no overflow.
        //
        // No self-wrap cannot be guaranteed after changing the step size, but
        // will be inferred if either NUW or NSW is true.
        SCEV::NoWrapFlags Flags = ComputeFlags({Scale, AddRec});
        const SCEV *NewRec = getAddRecExpr(
            NewOps, AddRecLoop, AddRec->getNoWrapFlags(Flags));
  
        // If all of the other operands were loop invariant, we are done.
        if (Ops.size() == 1) return NewRec;
  
        // Otherwise, multiply the folded AddRec by the non-invariant parts.
        for (unsigned i = 0;; ++i)
          if (Ops[i] == AddRec) {
            Ops[i] = NewRec;
            break;
          }
        return getMulExpr(Ops, SCEV::FlagAnyWrap, Depth + 1);
      }
  
      // Okay, if there weren't any loop invariants to be folded, check to see
      // if there are multiple AddRec's with the same loop induction variable
      // being multiplied together.  If so, we can fold them.
  
      // {A1,+,A2,+,...,+,An}<L> * {B1,+,B2,+,...,+,Bn}<L>
      // = {x=1 in [ sum y=x..2x [ sum z=max(y-x, y-n)..min(x,n) [
      //       choose(x, 2x)*choose(2x-y, x-z)*A_{y-z}*B_z
      //   ]]],+,...up to x=2n}.
      // Note that the arguments to choose() are always integers with values
      // known at compile time, never SCEV objects.
      //
      // The implementation avoids pointless extra computations when the two
      // addrec's are of different length (mathematically, it's equivalent to
      // an infinite stream of zeros on the right).
      bool OpsModified = false;
      for (unsigned OtherIdx = Idx+1;
           OtherIdx != Ops.size() && isa<SCEVAddRecExpr>(Ops[OtherIdx]);
           ++OtherIdx) {
        const SCEVAddRecExpr *OtherAddRec =
          dyn_cast<SCEVAddRecExpr>(Ops[OtherIdx]);
        if (!OtherAddRec || OtherAddRec->getLoop() != AddRecLoop)
          continue;
  
        // Limit max number of arguments to avoid creation of unreasonably big
        // SCEVAddRecs with very complex operands.
        if (AddRec->getNumOperands() + OtherAddRec->getNumOperands() - 1 >
            MaxAddRecSize || hasHugeExpression({AddRec, OtherAddRec}))
          continue;
  
        bool Overflow = false;
        Type *Ty = AddRec->getType();
        bool LargerThan64Bits = getTypeSizeInBits(Ty) > 64;
        SmallVector<const SCEV*, 7> AddRecOps;
        for (int x = 0, xe = AddRec->getNumOperands() +
               OtherAddRec->getNumOperands() - 1; x != xe && !Overflow; ++x) {
          SmallVector <const SCEV *, 7> SumOps;
          for (int y = x, ye = 2*x+1; y != ye && !Overflow; ++y) {
            uint64_t Coeff1 = Choose(x, 2*x - y, Overflow);
            for (int z = std::max(y-x, y-(int)AddRec->getNumOperands()+1),
                   ze = std::min(x+1, (int)OtherAddRec->getNumOperands());
                 z < ze && !Overflow; ++z) {
              uint64_t Coeff2 = Choose(2*x - y, x-z, Overflow);
              uint64_t Coeff;
              if (LargerThan64Bits)
                Coeff = umul_ov(Coeff1, Coeff2, Overflow);
              else
                Coeff = Coeff1*Coeff2;
              const SCEV *CoeffTerm = getConstant(Ty, Coeff);
              const SCEV *Term1 = AddRec->getOperand(y-z);
              const SCEV *Term2 = OtherAddRec->getOperand(z);
              SumOps.push_back(getMulExpr(CoeffTerm, Term1, Term2,
                                          SCEV::FlagAnyWrap, Depth + 1));
            }
          }
          if (SumOps.empty())
            SumOps.push_back(getZero(Ty));
          AddRecOps.push_back(getAddExpr(SumOps, SCEV::FlagAnyWrap, Depth + 1));
        }
        if (!Overflow) {
          const SCEV *NewAddRec = getAddRecExpr(AddRecOps, AddRecLoop,
                                                SCEV::FlagAnyWrap);
          if (Ops.size() == 2) return NewAddRec;
          Ops[Idx] = NewAddRec;
          Ops.erase(Ops.begin() + OtherIdx); --OtherIdx;
          OpsModified = true;
          AddRec = dyn_cast<SCEVAddRecExpr>(NewAddRec);
          if (!AddRec)
            break;
        }
      }
      if (OpsModified)
        return getMulExpr(Ops, SCEV::FlagAnyWrap, Depth + 1);
  
      // Otherwise couldn't fold anything into this recurrence.  Move onto the
      // next one.
    }
  
    // Okay, it looks like we really DO need an mul expr.  Check to see if we
    // already have one, otherwise create a new one.
    return getOrCreateMulExpr(Ops, ComputeFlags(Ops));
  }
  
  /// Represents an unsigned remainder expression based on unsigned division.
  const SCEV *ScalarEvolution::getURemExpr(const SCEV *LHS,
                                           const SCEV *RHS) {
    assert(getEffectiveSCEVType(LHS->getType()) ==
           getEffectiveSCEVType(RHS->getType()) &&
           "SCEVURemExpr operand types don't match!");
  
    // Short-circuit easy cases
    if (const SCEVConstant *RHSC = dyn_cast<SCEVConstant>(RHS)) {
      // If constant is one, the result is trivial
      if (RHSC->getValue()->isOne())
        return getZero(LHS->getType()); // X urem 1 --> 0
  
      // If constant is a power of two, fold into a zext(trunc(LHS)).
      if (RHSC->getAPInt().isPowerOf2()) {
        Type *FullTy = LHS->getType();
        Type *TruncTy =
            IntegerType::get(getContext(), RHSC->getAPInt().logBase2());
        return getZeroExtendExpr(getTruncateExpr(LHS, TruncTy), FullTy);
      }
    }
  
    // Fallback to %a == %x urem %y == %x -<nuw> ((%x udiv %y) *<nuw> %y)
    const SCEV *UDiv = getUDivExpr(LHS, RHS);
    const SCEV *Mult = getMulExpr(UDiv, RHS, SCEV::FlagNUW);
    return getMinusSCEV(LHS, Mult, SCEV::FlagNUW);
  }
  
  /// Get a canonical unsigned division expression, or something simpler if
  /// possible.
  const SCEV *ScalarEvolution::getUDivExpr(const SCEV *LHS,
                                           const SCEV *RHS) {
    assert(!LHS->getType()->isPointerTy() &&
           "SCEVUDivExpr operand can't be pointer!");
    assert(LHS->getType() == RHS->getType() &&
           "SCEVUDivExpr operand types don't match!");
  
    FoldingSetNodeID ID;
    ID.AddInteger(scUDivExpr);
    ID.AddPointer(LHS);
    ID.AddPointer(RHS);
    void *IP = nullptr;
    if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP))
      return S;
  
    // 0 udiv Y == 0
    if (const SCEVConstant *LHSC = dyn_cast<SCEVConstant>(LHS))
      if (LHSC->getValue()->isZero())
        return LHS;
  
    if (const SCEVConstant *RHSC = dyn_cast<SCEVConstant>(RHS)) {
      if (RHSC->getValue()->isOne())
        return LHS;                               // X udiv 1 --> x
      // If the denominator is zero, the result of the udiv is undefined. Don't
      // try to analyze it, because the resolution chosen here may differ from
      // the resolution chosen in other parts of the compiler.
      if (!RHSC->getValue()->isZero()) {
        // Determine if the division can be folded into the operands of
        // its operands.
        // TODO: Generalize this to non-constants by using known-bits information.
        Type *Ty = LHS->getType();
        unsigned LZ = RHSC->getAPInt().countLeadingZeros();
        unsigned MaxShiftAmt = getTypeSizeInBits(Ty) - LZ - 1;
        // For non-power-of-two values, effectively round the value up to the
        // nearest power of two.
        if (!RHSC->getAPInt().isPowerOf2())
          ++MaxShiftAmt;
        IntegerType *ExtTy =
          IntegerType::get(getContext(), getTypeSizeInBits(Ty) + MaxShiftAmt);
        if (const SCEVAddRecExpr *AR = dyn_cast<SCEVAddRecExpr>(LHS))
          if (const SCEVConstant *Step =
              dyn_cast<SCEVConstant>(AR->getStepRecurrence(*this))) {
            // {X,+,N}/C --> {X/C,+,N/C} if safe and N/C can be folded.
            const APInt &StepInt = Step->getAPInt();
            const APInt &DivInt = RHSC->getAPInt();
            if (!StepInt.urem(DivInt) &&
                getZeroExtendExpr(AR, ExtTy) ==
                getAddRecExpr(getZeroExtendExpr(AR->getStart(), ExtTy),
                              getZeroExtendExpr(Step, ExtTy),
                              AR->getLoop(), SCEV::FlagAnyWrap)) {
              SmallVector<const SCEV *, 4> Operands;
              for (const SCEV *Op : AR->operands())
                Operands.push_back(getUDivExpr(Op, RHS));
              return getAddRecExpr(Operands, AR->getLoop(), SCEV::FlagNW);
            }
            /// Get a canonical UDivExpr for a recurrence.
            /// {X,+,N}/C => {Y,+,N}/C where Y=X-(X%N). Safe when C%N=0.
            // We can currently only fold X%N if X is constant.
            const SCEVConstant *StartC = dyn_cast<SCEVConstant>(AR->getStart());
            if (StartC && !DivInt.urem(StepInt) &&
                getZeroExtendExpr(AR, ExtTy) ==
                getAddRecExpr(getZeroExtendExpr(AR->getStart(), ExtTy),
                              getZeroExtendExpr(Step, ExtTy),
                              AR->getLoop(), SCEV::FlagAnyWrap)) {
              const APInt &StartInt = StartC->getAPInt();
              const APInt &StartRem = StartInt.urem(StepInt);
              if (StartRem != 0) {
                const SCEV *NewLHS =
                    getAddRecExpr(getConstant(StartInt - StartRem), Step,
                                  AR->getLoop(), SCEV::FlagNW);
                if (LHS != NewLHS) {
                  LHS = NewLHS;
  
                  // Reset the ID to include the new LHS, and check if it is
                  // already cached.
                  ID.clear();
                  ID.AddInteger(scUDivExpr);
                  ID.AddPointer(LHS);
                  ID.AddPointer(RHS);
                  IP = nullptr;
                  if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP))
                    return S;
                }
              }
            }
          }
        // (A*B)/C --> A*(B/C) if safe and B/C can be folded.
        if (const SCEVMulExpr *M = dyn_cast<SCEVMulExpr>(LHS)) {
          SmallVector<const SCEV *, 4> Operands;
          for (const SCEV *Op : M->operands())
            Operands.push_back(getZeroExtendExpr(Op, ExtTy));
          if (getZeroExtendExpr(M, ExtTy) == getMulExpr(Operands))
            // Find an operand that's safely divisible.
            for (unsigned i = 0, e = M->getNumOperands(); i != e; ++i) {
              const SCEV *Op = M->getOperand(i);
              const SCEV *Div = getUDivExpr(Op, RHSC);
              if (!isa<SCEVUDivExpr>(Div) && getMulExpr(Div, RHSC) == Op) {
                Operands = SmallVector<const SCEV *, 4>(M->operands());
                Operands[i] = Div;
                return getMulExpr(Operands);
              }
            }
        }
  
        // (A/B)/C --> A/(B*C) if safe and B*C can be folded.
        if (const SCEVUDivExpr *OtherDiv = dyn_cast<SCEVUDivExpr>(LHS)) {
          if (auto *DivisorConstant =
                  dyn_cast<SCEVConstant>(OtherDiv->getRHS())) {
            bool Overflow = false;
            APInt NewRHS =
                DivisorConstant->getAPInt().umul_ov(RHSC->getAPInt(), Overflow);
            if (Overflow) {
              return getConstant(RHSC->getType(), 0, false);
            }
            return getUDivExpr(OtherDiv->getLHS(), getConstant(NewRHS));
          }
        }
  
        // (A+B)/C --> (A/C + B/C) if safe and A/C and B/C can be folded.
        if (const SCEVAddExpr *A = dyn_cast<SCEVAddExpr>(LHS)) {
          SmallVector<const SCEV *, 4> Operands;
          for (const SCEV *Op : A->operands())
            Operands.push_back(getZeroExtendExpr(Op, ExtTy));
          if (getZeroExtendExpr(A, ExtTy) == getAddExpr(Operands)) {
            Operands.clear();
            for (unsigned i = 0, e = A->getNumOperands(); i != e; ++i) {
              const SCEV *Op = getUDivExpr(A->getOperand(i), RHS);
              if (isa<SCEVUDivExpr>(Op) ||
                  getMulExpr(Op, RHS) != A->getOperand(i))
                break;
              Operands.push_back(Op);
            }
            if (Operands.size() == A->getNumOperands())
              return getAddExpr(Operands);
          }
        }
  
        // Fold if both operands are constant.
        if (const SCEVConstant *LHSC = dyn_cast<SCEVConstant>(LHS)) {
          Constant *LHSCV = LHSC->getValue();
          Constant *RHSCV = RHSC->getValue();
          return getConstant(cast<ConstantInt>(ConstantExpr::getUDiv(LHSCV,
                                                                     RHSCV)));
        }
      }
    }
  
    // The Insertion Point (IP) might be invalid by now (due to UniqueSCEVs
    // changes). Make sure we get a new one.
    IP = nullptr;
    if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;
    SCEV *S = new (SCEVAllocator) SCEVUDivExpr(ID.Intern(SCEVAllocator),
                                               LHS, RHS);
    UniqueSCEVs.InsertNode(S, IP);
    addToLoopUseLists(S);
    return S;
  }
  
  static const APInt gcd(const SCEVConstant *C1, const SCEVConstant *C2) {
    APInt A = C1->getAPInt().abs();
    APInt B = C2->getAPInt().abs();
    uint32_t ABW = A.getBitWidth();
    uint32_t BBW = B.getBitWidth();
  
    if (ABW > BBW)
      B = B.zext(ABW);
    else if (ABW < BBW)
      A = A.zext(BBW);
  
    return APIntOps::GreatestCommonDivisor(std::move(A), std::move(B));
  }
  
  /// Get a canonical unsigned division expression, or something simpler if
  /// possible. There is no representation for an exact udiv in SCEV IR, but we
  /// can attempt to remove factors from the LHS and RHS.  We can't do this when
  /// it's not exact because the udiv may be clearing bits.
  const SCEV *ScalarEvolution::getUDivExactExpr(const SCEV *LHS,
                                                const SCEV *RHS) {
    // TODO: we could try to find factors in all sorts of things, but for now we
    // just deal with u/exact (multiply, constant). See SCEVDivision towards the
    // end of this file for inspiration.
  
    const SCEVMulExpr *Mul = dyn_cast<SCEVMulExpr>(LHS);
    if (!Mul || !Mul->hasNoUnsignedWrap())
      return getUDivExpr(LHS, RHS);
  
    if (const SCEVConstant *RHSCst = dyn_cast<SCEVConstant>(RHS)) {
      // If the mulexpr multiplies by a constant, then that constant must be the
      // first element of the mulexpr.
      if (const auto *LHSCst = dyn_cast<SCEVConstant>(Mul->getOperand(0))) {
        if (LHSCst == RHSCst) {
          SmallVector<const SCEV *, 2> Operands(drop_begin(Mul->operands()));
          return getMulExpr(Operands);
        }
  
        // We can't just assume that LHSCst divides RHSCst cleanly, it could be
        // that there's a factor provided by one of the other terms. We need to
        // check.
        APInt Factor = gcd(LHSCst, RHSCst);
        if (!Factor.isIntN(1)) {
          LHSCst =
              cast<SCEVConstant>(getConstant(LHSCst->getAPInt().udiv(Factor)));
          RHSCst =
              cast<SCEVConstant>(getConstant(RHSCst->getAPInt().udiv(Factor)));
          SmallVector<const SCEV *, 2> Operands;
          Operands.push_back(LHSCst);
          Operands.append(Mul->op_begin() + 1, Mul->op_end());
          LHS = getMulExpr(Operands);
          RHS = RHSCst;
          Mul = dyn_cast<SCEVMulExpr>(LHS);
          if (!Mul)
            return getUDivExactExpr(LHS, RHS);
        }
      }
    }
  
    for (int i = 0, e = Mul->getNumOperands(); i != e; ++i) {
      if (Mul->getOperand(i) == RHS) {
        SmallVector<const SCEV *, 2> Operands;
        Operands.append(Mul->op_begin(), Mul->op_begin() + i);
        Operands.append(Mul->op_begin() + i + 1, Mul->op_end());
        return getMulExpr(Operands);
      }
    }
  
    return getUDivExpr(LHS, RHS);
  }
  
  /// Get an add recurrence expression for the specified loop.  Simplify the
  /// expression as much as possible.
  const SCEV *ScalarEvolution::getAddRecExpr(const SCEV *Start, const SCEV *Step,
                                             const Loop *L,
                                             SCEV::NoWrapFlags Flags) {
    SmallVector<const SCEV *, 4> Operands;
    Operands.push_back(Start);
    if (const SCEVAddRecExpr *StepChrec = dyn_cast<SCEVAddRecExpr>(Step))
      if (StepChrec->getLoop() == L) {
        Operands.append(StepChrec->op_begin(), StepChrec->op_end());
        return getAddRecExpr(Operands, L, maskFlags(Flags, SCEV::FlagNW));
      }
  
    Operands.push_back(Step);
    return getAddRecExpr(Operands, L, Flags);
  }
  
  /// Get an add recurrence expression for the specified loop.  Simplify the
  /// expression as much as possible.
  const SCEV *
  ScalarEvolution::getAddRecExpr(SmallVectorImpl<const SCEV *> &Operands,
                                 const Loop *L, SCEV::NoWrapFlags Flags) {
    if (Operands.size() == 1) return Operands[0];
  #ifndef NDEBUG
    Type *ETy = getEffectiveSCEVType(Operands[0]->getType());
    for (unsigned i = 1, e = Operands.size(); i != e; ++i) {
      assert(getEffectiveSCEVType(Operands[i]->getType()) == ETy &&
             "SCEVAddRecExpr operand types don't match!");
      assert(!Operands[i]->getType()->isPointerTy() && "Step must be integer");
    }
    for (unsigned i = 0, e = Operands.size(); i != e; ++i)
      assert(isLoopInvariant(Operands[i], L) &&
             "SCEVAddRecExpr operand is not loop-invariant!");
  #endif
  
    if (Operands.back()->isZero()) {
      Operands.pop_back();
      return getAddRecExpr(Operands, L, SCEV::FlagAnyWrap); // {X,+,0}  -->  X
    }
  
    // It's tempting to want to call getConstantMaxBackedgeTakenCount count here and
    // use that information to infer NUW and NSW flags. However, computing a
    // BE count requires calling getAddRecExpr, so we may not yet have a
    // meaningful BE count at this point (and if we don't, we'd be stuck
    // with a SCEVCouldNotCompute as the cached BE count).
  
    Flags = StrengthenNoWrapFlags(this, scAddRecExpr, Operands, Flags);
  
    // Canonicalize nested AddRecs in by nesting them in order of loop depth.
    if (const SCEVAddRecExpr *NestedAR = dyn_cast<SCEVAddRecExpr>(Operands[0])) {
      const Loop *NestedLoop = NestedAR->getLoop();
      if (L->contains(NestedLoop)
              ? (L->getLoopDepth() < NestedLoop->getLoopDepth())
              : (!NestedLoop->contains(L) &&
                 DT.dominates(L->getHeader(), NestedLoop->getHeader()))) {
        SmallVector<const SCEV *, 4> NestedOperands(NestedAR->operands());
        Operands[0] = NestedAR->getStart();
        // AddRecs require their operands be loop-invariant with respect to their
        // loops. Don't perform this transformation if it would break this
        // requirement.
        bool AllInvariant = all_of(
            Operands, [&](const SCEV *Op) { return isLoopInvariant(Op, L); });
  
        if (AllInvariant) {
          // Create a recurrence for the outer loop with the same step size.
          //
          // The outer recurrence keeps its NW flag but only keeps NUW/NSW if the
          // inner recurrence has the same property.
          SCEV::NoWrapFlags OuterFlags =
            maskFlags(Flags, SCEV::FlagNW | NestedAR->getNoWrapFlags());
  
          NestedOperands[0] = getAddRecExpr(Operands, L, OuterFlags);
          AllInvariant = all_of(NestedOperands, [&](const SCEV *Op) {
            return isLoopInvariant(Op, NestedLoop);
          });
  
          if (AllInvariant) {
            // Ok, both add recurrences are valid after the transformation.
            //
            // The inner recurrence keeps its NW flag but only keeps NUW/NSW if
            // the outer recurrence has the same property.
            SCEV::NoWrapFlags InnerFlags =
              maskFlags(NestedAR->getNoWrapFlags(), SCEV::FlagNW | Flags);
            return getAddRecExpr(NestedOperands, NestedLoop, InnerFlags);
          }
        }
        // Reset Operands to its original state.
        Operands[0] = NestedAR;
      }
    }
  
    // Okay, it looks like we really DO need an addrec expr.  Check to see if we
    // already have one, otherwise create a new one.
    return getOrCreateAddRecExpr(Operands, L, Flags);
  }
  
  const SCEV *
  ScalarEvolution::getGEPExpr(GEPOperator *GEP,
                              const SmallVectorImpl<const SCEV *> &IndexExprs) {
    const SCEV *BaseExpr = getSCEV(GEP->getPointerOperand());
    // getSCEV(Base)->getType() has the same address space as Base->getType()
    // because SCEV::getType() preserves the address space.
    Type *IntIdxTy = getEffectiveSCEVType(BaseExpr->getType());
    // FIXME(PR23527): Don't blindly transfer the inbounds flag from the GEP
    // instruction to its SCEV, because the Instruction may be guarded by control
    // flow and the no-overflow bits may not be valid for the expression in any
    // context. This can be fixed similarly to how these flags are handled for
    // adds.
    SCEV::NoWrapFlags OffsetWrap =
        GEP->isInBounds() ? SCEV::FlagNSW : SCEV::FlagAnyWrap;
  
    Type *CurTy = GEP->getType();
    bool FirstIter = true;
    SmallVector<const SCEV *, 4> Offsets;
    for (const SCEV *IndexExpr : IndexExprs) {
      // Compute the (potentially symbolic) offset in bytes for this index.
      if (StructType *STy = dyn_cast<StructType>(CurTy)) {
        // For a struct, add the member offset.
        ConstantInt *Index = cast<SCEVConstant>(IndexExpr)->getValue();
        unsigned FieldNo = Index->getZExtValue();
        const SCEV *FieldOffset = getOffsetOfExpr(IntIdxTy, STy, FieldNo);
        Offsets.push_back(FieldOffset);
  
        // Update CurTy to the type of the field at Index.
        CurTy = STy->getTypeAtIndex(Index);
      } else {
        // Update CurTy to its element type.
        if (FirstIter) {
          assert(isa<PointerType>(CurTy) &&
                 "The first index of a GEP indexes a pointer");
          CurTy = GEP->getSourceElementType();
          FirstIter = false;
        } else {
          CurTy = GetElementPtrInst::getTypeAtIndex(CurTy, (uint64_t)0);
        }
        // For an array, add the element offset, explicitly scaled.
        const SCEV *ElementSize = getSizeOfExpr(IntIdxTy, CurTy);
        // Getelementptr indices are signed.
        IndexExpr = getTruncateOrSignExtend(IndexExpr, IntIdxTy);
  
        // Multiply the index by the element size to compute the element offset.
        const SCEV *LocalOffset = getMulExpr(IndexExpr, ElementSize, OffsetWrap);
        Offsets.push_back(LocalOffset);
      }
    }
  
    // Handle degenerate case of GEP without offsets.
    if (Offsets.empty())
      return BaseExpr;
  
    // Add the offsets together, assuming nsw if inbounds.
    const SCEV *Offset = getAddExpr(Offsets, OffsetWrap);
    // Add the base address and the offset. We cannot use the nsw flag, as the
    // base address is unsigned. However, if we know that the offset is
    // non-negative, we can use nuw.
    SCEV::NoWrapFlags BaseWrap = GEP->isInBounds() && isKnownNonNegative(Offset)
                                     ? SCEV::FlagNUW : SCEV::FlagAnyWrap;
    return getAddExpr(BaseExpr, Offset, BaseWrap);
  }
  
  std::tuple<SCEV *, FoldingSetNodeID, void *>
  ScalarEvolution::findExistingSCEVInCache(SCEVTypes SCEVType,
                                           ArrayRef<const SCEV *> Ops) {
    FoldingSetNodeID ID;
    void *IP = nullptr;
    ID.AddInteger(SCEVType);
    for (unsigned i = 0, e = Ops.size(); i != e; ++i)
      ID.AddPointer(Ops[i]);
    return std::tuple<SCEV *, FoldingSetNodeID, void *>(
        UniqueSCEVs.FindNodeOrInsertPos(ID, IP), std::move(ID), IP);
  }
  
  const SCEV *ScalarEvolution::getAbsExpr(const SCEV *Op, bool IsNSW) {
    SCEV::NoWrapFlags Flags = IsNSW ? SCEV::FlagNSW : SCEV::FlagAnyWrap;
    return getSMaxExpr(Op, getNegativeSCEV(Op, Flags));
  }
  
  const SCEV *ScalarEvolution::getMinMaxExpr(SCEVTypes Kind,
                                             SmallVectorImpl<const SCEV *> &Ops) {
    assert(!Ops.empty() && "Cannot get empty (u|s)(min|max)!");
    if (Ops.size() == 1) return Ops[0];
  #ifndef NDEBUG
    Type *ETy = getEffectiveSCEVType(Ops[0]->getType());
    for (unsigned i = 1, e = Ops.size(); i != e; ++i) {
      assert(getEffectiveSCEVType(Ops[i]->getType()) == ETy &&
             "Operand types don't match!");
      assert(Ops[0]->getType()->isPointerTy() ==
                 Ops[i]->getType()->isPointerTy() &&
             "min/max should be consistently pointerish");
    }
  #endif
  
    bool IsSigned = Kind == scSMaxExpr || Kind == scSMinExpr;
    bool IsMax = Kind == scSMaxExpr || Kind == scUMaxExpr;
  
    // Sort by complexity, this groups all similar expression types together.
    GroupByComplexity(Ops, &LI, DT);
  
    // Check if we have created the same expression before.
    if (const SCEV *S = std::get<0>(findExistingSCEVInCache(Kind, Ops))) {
      return S;
    }
  
    // If there are any constants, fold them together.
    unsigned Idx = 0;
    if (const SCEVConstant *LHSC = dyn_cast<SCEVConstant>(Ops[0])) {
      ++Idx;
      assert(Idx < Ops.size());
      auto FoldOp = [&](const APInt &LHS, const APInt &RHS) {
        if (Kind == scSMaxExpr)
          return APIntOps::smax(LHS, RHS);
        else if (Kind == scSMinExpr)
          return APIntOps::smin(LHS, RHS);
        else if (Kind == scUMaxExpr)
          return APIntOps::umax(LHS, RHS);
        else if (Kind == scUMinExpr)
          return APIntOps::umin(LHS, RHS);
        llvm_unreachable("Unknown SCEV min/max opcode");
      };
  
      while (const SCEVConstant *RHSC = dyn_cast<SCEVConstant>(Ops[Idx])) {
        // We found two constants, fold them together!
        ConstantInt *Fold = ConstantInt::get(
            getContext(), FoldOp(LHSC->getAPInt(), RHSC->getAPInt()));
        Ops[0] = getConstant(Fold);
        Ops.erase(Ops.begin()+1);  // Erase the folded element
        if (Ops.size() == 1) return Ops[0];
        LHSC = cast<SCEVConstant>(Ops[0]);
      }
  
      bool IsMinV = LHSC->getValue()->isMinValue(IsSigned);
      bool IsMaxV = LHSC->getValue()->isMaxValue(IsSigned);
  
      if (IsMax ? IsMinV : IsMaxV) {
        // If we are left with a constant minimum(/maximum)-int, strip it off.
        Ops.erase(Ops.begin());
        --Idx;
      } else if (IsMax ? IsMaxV : IsMinV) {
        // If we have a max(/min) with a constant maximum(/minimum)-int,
        // it will always be the extremum.
        return LHSC;
      }
  
      if (Ops.size() == 1) return Ops[0];
    }
  
    // Find the first operation of the same kind
    while (Idx < Ops.size() && Ops[Idx]->getSCEVType() < Kind)
      ++Idx;
  
    // Check to see if one of the operands is of the same kind. If so, expand its
    // operands onto our operand list, and recurse to simplify.
    if (Idx < Ops.size()) {
      bool DeletedAny = false;
      while (Ops[Idx]->getSCEVType() == Kind) {
        const SCEVMinMaxExpr *SMME = cast<SCEVMinMaxExpr>(Ops[Idx]);
        Ops.erase(Ops.begin()+Idx);
        Ops.append(SMME->op_begin(), SMME->op_end());
        DeletedAny = true;
      }
  
      if (DeletedAny)
        return getMinMaxExpr(Kind, Ops);
    }
  
    // Okay, check to see if the same value occurs in the operand list twice.  If
    // so, delete one.  Since we sorted the list, these values are required to
    // be adjacent.
    llvm::CmpInst::Predicate GEPred =
        IsSigned ? ICmpInst::ICMP_SGE : ICmpInst::ICMP_UGE;
    llvm::CmpInst::Predicate LEPred =
        IsSigned ? ICmpInst::ICMP_SLE : ICmpInst::ICMP_ULE;
    llvm::CmpInst::Predicate FirstPred = IsMax ? GEPred : LEPred;
    llvm::CmpInst::Predicate SecondPred = IsMax ? LEPred : GEPred;
    for (unsigned i = 0, e = Ops.size() - 1; i != e; ++i) {
      if (Ops[i] == Ops[i + 1] ||
          isKnownViaNonRecursiveReasoning(FirstPred, Ops[i], Ops[i + 1])) {
        //  X op Y op Y  -->  X op Y
        //  X op Y       -->  X, if we know X, Y are ordered appropriately
        Ops.erase(Ops.begin() + i + 1, Ops.begin() + i + 2);
        --i;
        --e;
      } else if (isKnownViaNonRecursiveReasoning(SecondPred, Ops[i],
                                                 Ops[i + 1])) {
        //  X op Y       -->  Y, if we know X, Y are ordered appropriately
        Ops.erase(Ops.begin() + i, Ops.begin() + i + 1);
        --i;
        --e;
      }
    }
  
    if (Ops.size() == 1) return Ops[0];
  
    assert(!Ops.empty() && "Reduced smax down to nothing!");
  
    // Okay, it looks like we really DO need an expr.  Check to see if we
    // already have one, otherwise create a new one.
    const SCEV *ExistingSCEV;
    FoldingSetNodeID ID;
    void *IP;
    std::tie(ExistingSCEV, ID, IP) = findExistingSCEVInCache(Kind, Ops);
    if (ExistingSCEV)
      return ExistingSCEV;
    const SCEV **O = SCEVAllocator.Allocate<const SCEV *>(Ops.size());
    std::uninitialized_copy(Ops.begin(), Ops.end(), O);
    SCEV *S = new (SCEVAllocator)
        SCEVMinMaxExpr(ID.Intern(SCEVAllocator), Kind, O, Ops.size());
  
    UniqueSCEVs.InsertNode(S, IP);
    addToLoopUseLists(S);
    return S;
  }
  
  const SCEV *ScalarEvolution::getSMaxExpr(const SCEV *LHS, const SCEV *RHS) {
    SmallVector<const SCEV *, 2> Ops = {LHS, RHS};
    return getSMaxExpr(Ops);
  }
  
  const SCEV *ScalarEvolution::getSMaxExpr(SmallVectorImpl<const SCEV *> &Ops) {
    return getMinMaxExpr(scSMaxExpr, Ops);
  }
  
  const SCEV *ScalarEvolution::getUMaxExpr(const SCEV *LHS, const SCEV *RHS) {
    SmallVector<const SCEV *, 2> Ops = {LHS, RHS};
    return getUMaxExpr(Ops);
  }
  
  const SCEV *ScalarEvolution::getUMaxExpr(SmallVectorImpl<const SCEV *> &Ops) {
    return getMinMaxExpr(scUMaxExpr, Ops);
  }
  
  const SCEV *ScalarEvolution::getSMinExpr(const SCEV *LHS,
                                           const SCEV *RHS) {
    SmallVector<const SCEV *, 2> Ops = { LHS, RHS };
    return getSMinExpr(Ops);
  }
  
  const SCEV *ScalarEvolution::getSMinExpr(SmallVectorImpl<const SCEV *> &Ops) {
    return getMinMaxExpr(scSMinExpr, Ops);
  }
  
  const SCEV *ScalarEvolution::getUMinExpr(const SCEV *LHS,
                                           const SCEV *RHS) {
    SmallVector<const SCEV *, 2> Ops = { LHS, RHS };
    return getUMinExpr(Ops);
  }
  
  const SCEV *ScalarEvolution::getUMinExpr(SmallVectorImpl<const SCEV *> &Ops) {
    return getMinMaxExpr(scUMinExpr, Ops);
  }
  
  const SCEV *
  ScalarEvolution::getSizeOfScalableVectorExpr(Type *IntTy,
                                               ScalableVectorType *ScalableTy) {
    Constant *NullPtr = Constant::getNullValue(ScalableTy->getPointerTo());
    Constant *One = ConstantInt::get(IntTy, 1);
    Constant *GEP = ConstantExpr::getGetElementPtr(ScalableTy, NullPtr, One);
    // Note that the expression we created is the final expression, we don't
    // want to simplify it any further Also, if we call a normal getSCEV(),
    // we'll end up in an endless recursion. So just create an SCEVUnknown.
    return getUnknown(ConstantExpr::getPtrToInt(GEP, IntTy));
  }
  
  const SCEV *ScalarEvolution::getSizeOfExpr(Type *IntTy, Type *AllocTy) {
    if (auto *ScalableAllocTy = dyn_cast<ScalableVectorType>(AllocTy))
      return getSizeOfScalableVectorExpr(IntTy, ScalableAllocTy);
    // We can bypass creating a target-independent constant expression and then
    // folding it back into a ConstantInt. This is just a compile-time
    // optimization.
    return getConstant(IntTy, getDataLayout().getTypeAllocSize(AllocTy));
  }
  
  const SCEV *ScalarEvolution::getStoreSizeOfExpr(Type *IntTy, Type *StoreTy) {
    if (auto *ScalableStoreTy = dyn_cast<ScalableVectorType>(StoreTy))
      return getSizeOfScalableVectorExpr(IntTy, ScalableStoreTy);
    // We can bypass creating a target-independent constant expression and then
    // folding it back into a ConstantInt. This is just a compile-time
    // optimization.
    return getConstant(IntTy, getDataLayout().getTypeStoreSize(StoreTy));
  }
  
  const SCEV *ScalarEvolution::getOffsetOfExpr(Type *IntTy,
                                               StructType *STy,
                                               unsigned FieldNo) {
    // We can bypass creating a target-independent constant expression and then
    // folding it back into a ConstantInt. This is just a compile-time
    // optimization.
    return getConstant(
        IntTy, getDataLayout().getStructLayout(STy)->getElementOffset(FieldNo));
  }
  
  const SCEV *ScalarEvolution::getUnknown(Value *V) {
    // Don't attempt to do anything other than create a SCEVUnknown object
    // here.  createSCEV only calls getUnknown after checking for all other
    // interesting possibilities, and any other code that calls getUnknown
    // is doing so in order to hide a value from SCEV canonicalization.
  
    FoldingSetNodeID ID;
    ID.AddInteger(scUnknown);
    ID.AddPointer(V);
    void *IP = nullptr;
    if (SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) {
      assert(cast<SCEVUnknown>(S)->getValue() == V &&
             "Stale SCEVUnknown in uniquing map!");
      return S;
    }
    SCEV *S = new (SCEVAllocator) SCEVUnknown(ID.Intern(SCEVAllocator), V, this,
                                              FirstUnknown);
    FirstUnknown = cast<SCEVUnknown>(S);
    UniqueSCEVs.InsertNode(S, IP);
    return S;
  }
  
  //===----------------------------------------------------------------------===//
  //            Basic SCEV Analysis and PHI Idiom Recognition Code
  //
  
  /// Test if values of the given type are analyzable within the SCEV
  /// framework. This primarily includes integer types, and it can optionally
  /// include pointer types if the ScalarEvolution class has access to
  /// target-specific information.
  bool ScalarEvolution::isSCEVable(Type *Ty) const {
    // Integers and pointers are always SCEVable.
    return Ty->isIntOrPtrTy();
  }
  
  /// Return the size in bits of the specified type, for which isSCEVable must
  /// return true.
  uint64_t ScalarEvolution::getTypeSizeInBits(Type *Ty) const {
    assert(isSCEVable(Ty) && "Type is not SCEVable!");
    if (Ty->isPointerTy())
      return getDataLayout().getIndexTypeSizeInBits(Ty);
    return getDataLayout().getTypeSizeInBits(Ty);
  }
  
  /// Return a type with the same bitwidth as the given type and which represents
  /// how SCEV will treat the given type, for which isSCEVable must return
  /// true. For pointer types, this is the pointer index sized integer type.
  Type *ScalarEvolution::getEffectiveSCEVType(Type *Ty) const {
    assert(isSCEVable(Ty) && "Type is not SCEVable!");
  
    if (Ty->isIntegerTy())
      return Ty;
  
    // The only other support type is pointer.
    assert(Ty->isPointerTy() && "Unexpected non-pointer non-integer type!");
    return getDataLayout().getIndexType(Ty);
  }
  
  Type *ScalarEvolution::getWiderType(Type *T1, Type *T2) const {
    return  getTypeSizeInBits(T1) >= getTypeSizeInBits(T2) ? T1 : T2;
  }
  
  const SCEV *ScalarEvolution::getCouldNotCompute() {
    return CouldNotCompute.get();
  }
  
  bool ScalarEvolution::checkValidity(const SCEV *S) const {
    bool ContainsNulls = SCEVExprContains(S, [](const SCEV *S) {
      auto *SU = dyn_cast<SCEVUnknown>(S);
      return SU && SU->getValue() == nullptr;
    });
  
    return !ContainsNulls;
  }
  
  bool ScalarEvolution::containsAddRecurrence(const SCEV *S) {
    HasRecMapType::iterator I = HasRecMap.find(S);
    if (I != HasRecMap.end())
      return I->second;
  
    bool FoundAddRec =
        SCEVExprContains(S, [](const SCEV *S) { return isa<SCEVAddRecExpr>(S); });
    HasRecMap.insert({S, FoundAddRec});
    return FoundAddRec;
  }
  
  /// Try to split a SCEVAddExpr into a pair of {SCEV, ConstantInt}.
  /// If \p S is a SCEVAddExpr and is composed of a sub SCEV S' and an
  /// offset I, then return {S', I}, else return {\p S, nullptr}.
  static std::pair<const SCEV *, ConstantInt *> splitAddExpr(const SCEV *S) {
    const auto *Add = dyn_cast<SCEVAddExpr>(S);
    if (!Add)
      return {S, nullptr};
  
    if (Add->getNumOperands() != 2)
      return {S, nullptr};
  
    auto *ConstOp = dyn_cast<SCEVConstant>(Add->getOperand(0));
    if (!ConstOp)
      return {S, nullptr};
  
    return {Add->getOperand(1), ConstOp->getValue()};
  }
  
  /// Return the ValueOffsetPair set for \p S. \p S can be represented
  /// by the value and offset from any ValueOffsetPair in the set.
  ScalarEvolution::ValueOffsetPairSetVector *
  ScalarEvolution::getSCEVValues(const SCEV *S) {
    ExprValueMapType::iterator SI = ExprValueMap.find_as(S);
    if (SI == ExprValueMap.end())
      return nullptr;
  #ifndef NDEBUG
    if (VerifySCEVMap) {
      // Check there is no dangling Value in the set returned.
      for (const auto &VE : SI->second)
        assert(ValueExprMap.count(VE.first));
    }
  #endif
    return &SI->second;
  }
  
  /// Erase Value from ValueExprMap and ExprValueMap. ValueExprMap.erase(V)
  /// cannot be used separately. eraseValueFromMap should be used to remove
  /// V from ValueExprMap and ExprValueMap at the same time.
  void ScalarEvolution::eraseValueFromMap(Value *V) {
    ValueExprMapType::iterator I = ValueExprMap.find_as(V);
    if (I != ValueExprMap.end()) {
      const SCEV *S = I->second;
      // Remove {V, 0} from the set of ExprValueMap[S]
      if (auto *SV = getSCEVValues(S))
        SV->remove({V, nullptr});
  
      // Remove {V, Offset} from the set of ExprValueMap[Stripped]
      const SCEV *Stripped;
      ConstantInt *Offset;
      std::tie(Stripped, Offset) = splitAddExpr(S);
      if (Offset != nullptr) {
        if (auto *SV = getSCEVValues(Stripped))
          SV->remove({V, Offset});
      }
      ValueExprMap.erase(V);
    }
  }
  
  /// Check whether value has nuw/nsw/exact set but SCEV does not.
  /// TODO: In reality it is better to check the poison recursively
  /// but this is better than nothing.
  static bool SCEVLostPoisonFlags(const SCEV *S, const Value *V) {
    if (auto *I = dyn_cast<Instruction>(V)) {
      if (isa<OverflowingBinaryOperator>(I)) {
        if (auto *NS = dyn_cast<SCEVNAryExpr>(S)) {
          if (I->hasNoSignedWrap() && !NS->hasNoSignedWrap())
            return true;
          if (I->hasNoUnsignedWrap() && !NS->hasNoUnsignedWrap())
            return true;
        }
      } else if (isa<PossiblyExactOperator>(I) && I->isExact())
        return true;
    }
    return false;
  }
  
  /// Return an existing SCEV if it exists, otherwise analyze the expression and
  /// create a new one.
  const SCEV *ScalarEvolution::getSCEV(Value *V) {
    assert(isSCEVable(V->getType()) && "Value is not SCEVable!");
  
    const SCEV *S = getExistingSCEV(V);
    if (S == nullptr) {
      S = createSCEV(V);
      // During PHI resolution, it is possible to create two SCEVs for the same
      // V, so it is needed to double check whether V->S is inserted into
      // ValueExprMap before insert S->{V, 0} into ExprValueMap.
      std::pair<ValueExprMapType::iterator, bool> Pair =
          ValueExprMap.insert({SCEVCallbackVH(V, this), S});
      if (Pair.second && !SCEVLostPoisonFlags(S, V)) {
        ExprValueMap[S].insert({V, nullptr});
  
        // If S == Stripped + Offset, add Stripped -> {V, Offset} into
        // ExprValueMap.
        const SCEV *Stripped = S;
        ConstantInt *Offset = nullptr;
        std::tie(Stripped, Offset) = splitAddExpr(S);
        // If stripped is SCEVUnknown, don't bother to save
        // Stripped -> {V, offset}. It doesn't simplify and sometimes even
        // increase the complexity of the expansion code.
        // If V is GetElementPtrInst, don't save Stripped -> {V, offset}
        // because it may generate add/sub instead of GEP in SCEV expansion.
        if (Offset != nullptr && !isa<SCEVUnknown>(Stripped) &&
            !isa<GetElementPtrInst>(V))
          ExprValueMap[Stripped].insert({V, Offset});
      }
    }
    return S;
  }
  
  const SCEV *ScalarEvolution::getExistingSCEV(Value *V) {
    assert(isSCEVable(V->getType()) && "Value is not SCEVable!");
  
    ValueExprMapType::iterator I = ValueExprMap.find_as(V);
    if (I != ValueExprMap.end()) {
      const SCEV *S = I->second;
      if (checkValidity(S))
        return S;
      eraseValueFromMap(V);
      forgetMemoizedResults(S);
    }
    return nullptr;
  }
  
  /// Return a SCEV corresponding to -V = -1*V
  const SCEV *ScalarEvolution::getNegativeSCEV(const SCEV *V,
                                               SCEV::NoWrapFlags Flags) {
    if (const SCEVConstant *VC = dyn_cast<SCEVConstant>(V))
      return getConstant(
                 cast<ConstantInt>(ConstantExpr::getNeg(VC->getValue())));
  
    Type *Ty = V->getType();
    Ty = getEffectiveSCEVType(Ty);
    return getMulExpr(V, getMinusOne(Ty), Flags);
  }
  
  /// If Expr computes ~A, return A else return nullptr
  static const SCEV *MatchNotExpr(const SCEV *Expr) {
    const SCEVAddExpr *Add = dyn_cast<SCEVAddExpr>(Expr);
    if (!Add || Add->getNumOperands() != 2 ||
        !Add->getOperand(0)->isAllOnesValue())
      return nullptr;
  
    const SCEVMulExpr *AddRHS = dyn_cast<SCEVMulExpr>(Add->getOperand(1));
    if (!AddRHS || AddRHS->getNumOperands() != 2 ||
        !AddRHS->getOperand(0)->isAllOnesValue())
      return nullptr;
  
    return AddRHS->getOperand(1);
  }
  
  /// Return a SCEV corresponding to ~V = -1-V
  const SCEV *ScalarEvolution::getNotSCEV(const SCEV *V) {
    if (const SCEVConstant *VC = dyn_cast<SCEVConstant>(V))
      return getConstant(
                  cast<ConstantInt>(ConstantExpr::getNot(VC->getValue())));
  
    // Fold ~(u|s)(min|max)(~x, ~y) to (u|s)(max|min)(x, y)
    if (const SCEVMinMaxExpr *MME = dyn_cast<SCEVMinMaxExpr>(V)) {
      auto MatchMinMaxNegation = [&](const SCEVMinMaxExpr *MME) {
        SmallVector<const SCEV *, 2> MatchedOperands;
        for (const SCEV *Operand : MME->operands()) {
          const SCEV *Matched = MatchNotExpr(Operand);
          if (!Matched)
            return (const SCEV *)nullptr;
          MatchedOperands.push_back(Matched);
        }
        return getMinMaxExpr(SCEVMinMaxExpr::negate(MME->getSCEVType()),
                             MatchedOperands);
      };
      if (const SCEV *Replaced = MatchMinMaxNegation(MME))
        return Replaced;
    }
  
    Type *Ty = V->getType();
    Ty = getEffectiveSCEVType(Ty);
    return getMinusSCEV(getMinusOne(Ty), V);
  }
  
  /// Compute an expression equivalent to S - getPointerBase(S).
  static const SCEV *removePointerBase(ScalarEvolution *SE, const SCEV *P) {
    assert(P->getType()->isPointerTy());
  
    if (auto *AddRec = dyn_cast<SCEVAddRecExpr>(P)) {
      // The base of an AddRec is the first operand.
      SmallVector<const SCEV *> Ops{AddRec->operands()};
      Ops[0] = removePointerBase(SE, Ops[0]);
      // Don't try to transfer nowrap flags for now. We could in some cases
      // (for example, if pointer operand of the AddRec is a SCEVUnknown).
      return SE->getAddRecExpr(Ops, AddRec->getLoop(), SCEV::FlagAnyWrap);
    }
    if (auto *Add = dyn_cast<SCEVAddExpr>(P)) {
      // The base of an Add is the pointer operand.
      SmallVector<const SCEV *> Ops{Add->operands()};
      const SCEV **PtrOp = nullptr;
      for (const SCEV *&AddOp : Ops) {
        if (AddOp->getType()->isPointerTy()) {
          // If we find an Add with multiple pointer operands, treat it as a
          // pointer base to be consistent with getPointerBase.  Eventually
          // we should be able to assert this is impossible.
          if (PtrOp)
            return SE->getZero(P->getType());
          PtrOp = &AddOp;
        }
      }
      *PtrOp = removePointerBase(SE, *PtrOp);
      // Don't try to transfer nowrap flags for now. We could in some cases
      // (for example, if the pointer operand of the Add is a SCEVUnknown).
      return SE->getAddExpr(Ops);
    }
    // Any other expression must be a pointer base.
    return SE->getZero(P->getType());
  }
  
  const SCEV *ScalarEvolution::getMinusSCEV(const SCEV *LHS, const SCEV *RHS,
                                            SCEV::NoWrapFlags Flags,
                                            unsigned Depth) {
    // Fast path: X - X --> 0.
    if (LHS == RHS)
      return getZero(LHS->getType());
  
    // If we subtract two pointers with different pointer bases, bail.
    // Eventually, we're going to add an assertion to getMulExpr that we
    // can't multiply by a pointer.
    if (RHS->getType()->isPointerTy()) {
      if (!LHS->getType()->isPointerTy() ||
          getPointerBase(LHS) != getPointerBase(RHS))
        return getCouldNotCompute();
      LHS = removePointerBase(this, LHS);
      RHS = removePointerBase(this, RHS);
    }
  
    // We represent LHS - RHS as LHS + (-1)*RHS. This transformation
    // makes it so that we cannot make much use of NUW.
    auto AddFlags = SCEV::FlagAnyWrap;
    const bool RHSIsNotMinSigned =
        !getSignedRangeMin(RHS).isMinSignedValue();
    if (maskFlags(Flags, SCEV::FlagNSW) == SCEV::FlagNSW) {
      // Let M be the minimum representable signed value. Then (-1)*RHS
      // signed-wraps if and only if RHS is M. That can happen even for
      // a NSW subtraction because e.g. (-1)*M signed-wraps even though
      // -1 - M does not. So to transfer NSW from LHS - RHS to LHS +
      // (-1)*RHS, we need to prove that RHS != M.
      //
      // If LHS is non-negative and we know that LHS - RHS does not
      // signed-wrap, then RHS cannot be M. So we can rule out signed-wrap
      // either by proving that RHS > M or that LHS >= 0.
      if (RHSIsNotMinSigned || isKnownNonNegative(LHS)) {
        AddFlags = SCEV::FlagNSW;
      }
    }
  
    // FIXME: Find a correct way to transfer NSW to (-1)*M when LHS -
    // RHS is NSW and LHS >= 0.
    //
    // The difficulty here is that the NSW flag may have been proven
    // relative to a loop that is to be found in a recurrence in LHS and
    // not in RHS. Applying NSW to (-1)*M may then let the NSW have a
    // larger scope than intended.
    auto NegFlags = RHSIsNotMinSigned ? SCEV::FlagNSW : SCEV::FlagAnyWrap;
  
    return getAddExpr(LHS, getNegativeSCEV(RHS, NegFlags), AddFlags, Depth);
  }
  
  const SCEV *ScalarEvolution::getTruncateOrZeroExtend(const SCEV *V, Type *Ty,
                                                       unsigned Depth) {
    Type *SrcTy = V->getType();
    assert(SrcTy->isIntOrPtrTy() && Ty->isIntOrPtrTy() &&
           "Cannot truncate or zero extend with non-integer arguments!");
    if (getTypeSizeInBits(SrcTy) == getTypeSizeInBits(Ty))
      return V;  // No conversion
    if (getTypeSizeInBits(SrcTy) > getTypeSizeInBits(Ty))
      return getTruncateExpr(V, Ty, Depth);
    return getZeroExtendExpr(V, Ty, Depth);
  }
  
  const SCEV *ScalarEvolution::getTruncateOrSignExtend(const SCEV *V, Type *Ty,
                                                       unsigned Depth) {
    Type *SrcTy = V->getType();
    assert(SrcTy->isIntOrPtrTy() && Ty->isIntOrPtrTy() &&
           "Cannot truncate or zero extend with non-integer arguments!");
    if (getTypeSizeInBits(SrcTy) == getTypeSizeInBits(Ty))
      return V;  // No conversion
    if (getTypeSizeInBits(SrcTy) > getTypeSizeInBits(Ty))
      return getTruncateExpr(V, Ty, Depth);
    return getSignExtendExpr(V, Ty, Depth);
  }
  
  const SCEV *
  ScalarEvolution::getNoopOrZeroExtend(const SCEV *V, Type *Ty) {
    Type *SrcTy = V->getType();
    assert(SrcTy->isIntOrPtrTy() && Ty->isIntOrPtrTy() &&
           "Cannot noop or zero extend with non-integer arguments!");
    assert(getTypeSizeInBits(SrcTy) <= getTypeSizeInBits(Ty) &&
           "getNoopOrZeroExtend cannot truncate!");
    if (getTypeSizeInBits(SrcTy) == getTypeSizeInBits(Ty))
      return V;  // No conversion
    return getZeroExtendExpr(V, Ty);
  }
  
  const SCEV *
  ScalarEvolution::getNoopOrSignExtend(const SCEV *V, Type *Ty) {
    Type *SrcTy = V->getType();
    assert(SrcTy->isIntOrPtrTy() && Ty->isIntOrPtrTy() &&
           "Cannot noop or sign extend with non-integer arguments!");
    assert(getTypeSizeInBits(SrcTy) <= getTypeSizeInBits(Ty) &&
           "getNoopOrSignExtend cannot truncate!");
    if (getTypeSizeInBits(SrcTy) == getTypeSizeInBits(Ty))
      return V;  // No conversion
    return getSignExtendExpr(V, Ty);
  }
  
  const SCEV *
  ScalarEvolution::getNoopOrAnyExtend(const SCEV *V, Type *Ty) {
    Type *SrcTy = V->getType();
    assert(SrcTy->isIntOrPtrTy() && Ty->isIntOrPtrTy() &&
           "Cannot noop or any extend with non-integer arguments!");
    assert(getTypeSizeInBits(SrcTy) <= getTypeSizeInBits(Ty) &&
           "getNoopOrAnyExtend cannot truncate!");
    if (getTypeSizeInBits(SrcTy) == getTypeSizeInBits(Ty))
      return V;  // No conversion
    return getAnyExtendExpr(V, Ty);
  }
  
  const SCEV *
  ScalarEvolution::getTruncateOrNoop(const SCEV *V, Type *Ty) {
    Type *SrcTy = V->getType();
    assert(SrcTy->isIntOrPtrTy() && Ty->isIntOrPtrTy() &&
           "Cannot truncate or noop with non-integer arguments!");
    assert(getTypeSizeInBits(SrcTy) >= getTypeSizeInBits(Ty) &&
           "getTruncateOrNoop cannot extend!");
    if (getTypeSizeInBits(SrcTy) == getTypeSizeInBits(Ty))
      return V;  // No conversion
    return getTruncateExpr(V, Ty);
  }
  
  const SCEV *ScalarEvolution::getUMaxFromMismatchedTypes(const SCEV *LHS,
                                                          const SCEV *RHS) {
    const SCEV *PromotedLHS = LHS;
    const SCEV *PromotedRHS = RHS;
  
    if (getTypeSizeInBits(LHS->getType()) > getTypeSizeInBits(RHS->getType()))
      PromotedRHS = getZeroExtendExpr(RHS, LHS->getType());
    else
      PromotedLHS = getNoopOrZeroExtend(LHS, RHS->getType());
  
    return getUMaxExpr(PromotedLHS, PromotedRHS);
  }
  
  const SCEV *ScalarEvolution::getUMinFromMismatchedTypes(const SCEV *LHS,
                                                          const SCEV *RHS) {
    SmallVector<const SCEV *, 2> Ops = { LHS, RHS };
    return getUMinFromMismatchedTypes(Ops);
  }
  
  const SCEV *ScalarEvolution::getUMinFromMismatchedTypes(
      SmallVectorImpl<const SCEV *> &Ops) {
    assert(!Ops.empty() && "At least one operand must be!");
    // Trivial case.
    if (Ops.size() == 1)
      return Ops[0];
  
    // Find the max type first.
    Type *MaxType = nullptr;
    for (auto *S : Ops)
      if (MaxType)
        MaxType = getWiderType(MaxType, S->getType());
      else
        MaxType = S->getType();
    assert(MaxType && "Failed to find maximum type!");
  
    // Extend all ops to max type.
    SmallVector<const SCEV *, 2> PromotedOps;
    for (auto *S : Ops)
      PromotedOps.push_back(getNoopOrZeroExtend(S, MaxType));
  
    // Generate umin.
    return getUMinExpr(PromotedOps);
  }
  
  const SCEV *ScalarEvolution::getPointerBase(const SCEV *V) {
    // A pointer operand may evaluate to a nonpointer expression, such as null.
    if (!V->getType()->isPointerTy())
      return V;
  
    while (true) {
      if (auto *AddRec = dyn_cast<SCEVAddRecExpr>(V)) {
        V = AddRec->getStart();
      } else if (auto *Add = dyn_cast<SCEVAddExpr>(V)) {
        const SCEV *PtrOp = nullptr;
        for (const SCEV *AddOp : Add->operands()) {
          if (AddOp->getType()->isPointerTy()) {
            // Cannot find the base of an expression with multiple pointer ops.
            if (PtrOp)
              return V;
            PtrOp = AddOp;
          }
        }
        if (!PtrOp) // All operands were non-pointer.
          return V;
        V = PtrOp;
      } else // Not something we can look further into.
        return V;
    }
  }
  
  /// Push users of the given Instruction onto the given Worklist.
  static void
  PushDefUseChildren(Instruction *I,
                     SmallVectorImpl<Instruction *> &Worklist) {
    // Push the def-use children onto the Worklist stack.
    for (User *U : I->users())
      Worklist.push_back(cast<Instruction>(U));
  }
  
  void ScalarEvolution::forgetSymbolicName(Instruction *PN, const SCEV *SymName) {
    SmallVector<Instruction *, 16> Worklist;
    PushDefUseChildren(PN, Worklist);
  
    SmallPtrSet<Instruction *, 8> Visited;
    Visited.insert(PN);
    while (!Worklist.empty()) {
      Instruction *I = Worklist.pop_back_val();
      if (!Visited.insert(I).second)
        continue;
  
      auto It = ValueExprMap.find_as(static_cast<Value *>(I));
      if (It != ValueExprMap.end()) {
        const SCEV *Old = It->second;
  
        // Short-circuit the def-use traversal if the symbolic name
        // ceases to appear in expressions.
        if (Old != SymName && !hasOperand(Old, SymName))
          continue;
  
        // SCEVUnknown for a PHI either means that it has an unrecognized
        // structure, it's a PHI that's in the progress of being computed
        // by createNodeForPHI, or it's a single-value PHI. In the first case,
        // additional loop trip count information isn't going to change anything.
        // In the second case, createNodeForPHI will perform the necessary
        // updates on its own when it gets to that point. In the third, we do
        // want to forget the SCEVUnknown.
        if (!isa<PHINode>(I) ||
            !isa<SCEVUnknown>(Old) ||
            (I != PN && Old == SymName)) {
          eraseValueFromMap(It->first);
          forgetMemoizedResults(Old);
        }
      }
  
      PushDefUseChildren(I, Worklist);
    }
  }
  
  namespace {
  
  /// Takes SCEV S and Loop L. For each AddRec sub-expression, use its start
  /// expression in case its Loop is L. If it is not L then
  /// if IgnoreOtherLoops is true then use AddRec itself
  /// otherwise rewrite cannot be done.
  /// If SCEV contains non-invariant unknown SCEV rewrite cannot be done.
  class SCEVInitRewriter : public SCEVRewriteVisitor<SCEVInitRewriter> {
  public:
    static const SCEV *rewrite(const SCEV *S, const Loop *L, ScalarEvolution &SE,
                               bool IgnoreOtherLoops = true) {
      SCEVInitRewriter Rewriter(L, SE);
      const SCEV *Result = Rewriter.visit(S);
      if (Rewriter.hasSeenLoopVariantSCEVUnknown())
        return SE.getCouldNotCompute();
      return Rewriter.hasSeenOtherLoops() && !IgnoreOtherLoops
                 ? SE.getCouldNotCompute()
                 : Result;
    }
  
    const SCEV *visitUnknown(const SCEVUnknown *Expr) {
      if (!SE.isLoopInvariant(Expr, L))
        SeenLoopVariantSCEVUnknown = true;
      return Expr;
    }
  
    const SCEV *visitAddRecExpr(const SCEVAddRecExpr *Expr) {
      // Only re-write AddRecExprs for this loop.
      if (Expr->getLoop() == L)
        return Expr->getStart();
      SeenOtherLoops = true;
      return Expr;
    }
  
    bool hasSeenLoopVariantSCEVUnknown() { return SeenLoopVariantSCEVUnknown; }
  
    bool hasSeenOtherLoops() { return SeenOtherLoops; }
  
  private:
    explicit SCEVInitRewriter(const Loop *L, ScalarEvolution &SE)
        : SCEVRewriteVisitor(SE), L(L) {}
  
    const Loop *L;
    bool SeenLoopVariantSCEVUnknown = false;
    bool SeenOtherLoops = false;
  };
  
  /// Takes SCEV S and Loop L. For each AddRec sub-expression, use its post
  /// increment expression in case its Loop is L. If it is not L then
  /// use AddRec itself.
  /// If SCEV contains non-invariant unknown SCEV rewrite cannot be done.
  class SCEVPostIncRewriter : public SCEVRewriteVisitor<SCEVPostIncRewriter> {
  public:
    static const SCEV *rewrite(const SCEV *S, const Loop *L, ScalarEvolution &SE) {
      SCEVPostIncRewriter Rewriter(L, SE);
      const SCEV *Result = Rewriter.visit(S);
      return Rewriter.hasSeenLoopVariantSCEVUnknown()
          ? SE.getCouldNotCompute()
          : Result;
    }
  
    const SCEV *visitUnknown(const SCEVUnknown *Expr) {
      if (!SE.isLoopInvariant(Expr, L))
        SeenLoopVariantSCEVUnknown = true;
      return Expr;
    }
  
    const SCEV *visitAddRecExpr(const SCEVAddRecExpr *Expr) {
      // Only re-write AddRecExprs for this loop.
      if (Expr->getLoop() == L)
        return Expr->getPostIncExpr(SE);
      SeenOtherLoops = true;
      return Expr;
    }
  
    bool hasSeenLoopVariantSCEVUnknown() { return SeenLoopVariantSCEVUnknown; }
  
    bool hasSeenOtherLoops() { return SeenOtherLoops; }
  
  private:
    explicit SCEVPostIncRewriter(const Loop *L, ScalarEvolution &SE)
        : SCEVRewriteVisitor(SE), L(L) {}
  
    const Loop *L;
    bool SeenLoopVariantSCEVUnknown = false;
    bool SeenOtherLoops = false;
  };
  
  /// This class evaluates the compare condition by matching it against the
  /// condition of loop latch. If there is a match we assume a true value
  /// for the condition while building SCEV nodes.
  class SCEVBackedgeConditionFolder
      : public SCEVRewriteVisitor<SCEVBackedgeConditionFolder> {
  public:
    static const SCEV *rewrite(const SCEV *S, const Loop *L,
                               ScalarEvolution &SE) {
      bool IsPosBECond = false;
      Value *BECond = nullptr;
      if (BasicBlock *Latch = L->getLoopLatch()) {
        BranchInst *BI = dyn_cast<BranchInst>(Latch->getTerminator());
        if (BI && BI->isConditional()) {
          assert(BI->getSuccessor(0) != BI->getSuccessor(1) &&
                 "Both outgoing branches should not target same header!");
          BECond = BI->getCondition();
          IsPosBECond = BI->getSuccessor(0) == L->getHeader();
        } else {
          return S;
        }
      }
      SCEVBackedgeConditionFolder Rewriter(L, BECond, IsPosBECond, SE);
      return Rewriter.visit(S);
    }
  
    const SCEV *visitUnknown(const SCEVUnknown *Expr) {
      const SCEV *Result = Expr;
      bool InvariantF = SE.isLoopInvariant(Expr, L);
  
      if (!InvariantF) {
        Instruction *I = cast<Instruction>(Expr->getValue());
        switch (I->getOpcode()) {
        case Instruction::Select: {
          SelectInst *SI = cast<SelectInst>(I);
          Optional<const SCEV *> Res =
              compareWithBackedgeCondition(SI->getCondition());
          if (Res.hasValue()) {
            bool IsOne = cast<SCEVConstant>(Res.getValue())->getValue()->isOne();
            Result = SE.getSCEV(IsOne ? SI->getTrueValue() : SI->getFalseValue());
          }
          break;
        }
        default: {
          Optional<const SCEV *> Res = compareWithBackedgeCondition(I);
          if (Res.hasValue())
            Result = Res.getValue();
          break;
        }
        }
      }
      return Result;
    }
  
  private:
    explicit SCEVBackedgeConditionFolder(const Loop *L, Value *BECond,
                                         bool IsPosBECond, ScalarEvolution &SE)
        : SCEVRewriteVisitor(SE), L(L), BackedgeCond(BECond),
          IsPositiveBECond(IsPosBECond) {}
  
    Optional<const SCEV *> compareWithBackedgeCondition(Value *IC);
  
    const Loop *L;
    /// Loop back condition.
    Value *BackedgeCond = nullptr;
    /// Set to true if loop back is on positive branch condition.
    bool IsPositiveBECond;
  };
  
  Optional<const SCEV *>
  SCEVBackedgeConditionFolder::compareWithBackedgeCondition(Value *IC) {
  
    // If value matches the backedge condition for loop latch,
    // then return a constant evolution node based on loopback
    // branch taken.
    if (BackedgeCond == IC)
      return IsPositiveBECond ? SE.getOne(Type::getInt1Ty(SE.getContext()))
                              : SE.getZero(Type::getInt1Ty(SE.getContext()));
    return None;
  }
  
  class SCEVShiftRewriter : public SCEVRewriteVisitor<SCEVShiftRewriter> {
  public:
    static const SCEV *rewrite(const SCEV *S, const Loop *L,
                               ScalarEvolution &SE) {
      SCEVShiftRewriter Rewriter(L, SE);
      const SCEV *Result = Rewriter.visit(S);
      return Rewriter.isValid() ? Result : SE.getCouldNotCompute();
    }
  
    const SCEV *visitUnknown(const SCEVUnknown *Expr) {
      // Only allow AddRecExprs for this loop.
      if (!SE.isLoopInvariant(Expr, L))
        Valid = false;
      return Expr;
    }
  
    const SCEV *visitAddRecExpr(const SCEVAddRecExpr *Expr) {
      if (Expr->getLoop() == L && Expr->isAffine())
        return SE.getMinusSCEV(Expr, Expr->getStepRecurrence(SE));
      Valid = false;
      return Expr;
    }
  
    bool isValid() { return Valid; }
  
  private:
    explicit SCEVShiftRewriter(const Loop *L, ScalarEvolution &SE)
        : SCEVRewriteVisitor(SE), L(L) {}
  
    const Loop *L;
    bool Valid = true;
  };
  
  } // end anonymous namespace
  
  SCEV::NoWrapFlags
  ScalarEvolution::proveNoWrapViaConstantRanges(const SCEVAddRecExpr *AR) {
    if (!AR->isAffine())
      return SCEV::FlagAnyWrap;
  
    using OBO = OverflowingBinaryOperator;
  
    SCEV::NoWrapFlags Result = SCEV::FlagAnyWrap;
  
    if (!AR->hasNoSignedWrap()) {
      ConstantRange AddRecRange = getSignedRange(AR);
      ConstantRange IncRange = getSignedRange(AR->getStepRecurrence(*this));
  
      auto NSWRegion = ConstantRange::makeGuaranteedNoWrapRegion(
          Instruction::Add, IncRange, OBO::NoSignedWrap);
      if (NSWRegion.contains(AddRecRange))
        Result = ScalarEvolution::setFlags(Result, SCEV::FlagNSW);
    }
  
    if (!AR->hasNoUnsignedWrap()) {
      ConstantRange AddRecRange = getUnsignedRange(AR);
      ConstantRange IncRange = getUnsignedRange(AR->getStepRecurrence(*this));
  
      auto NUWRegion = ConstantRange::makeGuaranteedNoWrapRegion(
          Instruction::Add, IncRange, OBO::NoUnsignedWrap);
      if (NUWRegion.contains(AddRecRange))
        Result = ScalarEvolution::setFlags(Result, SCEV::FlagNUW);
    }
  
    return Result;
  }
  
  SCEV::NoWrapFlags
  ScalarEvolution::proveNoSignedWrapViaInduction(const SCEVAddRecExpr *AR) {
    SCEV::NoWrapFlags Result = AR->getNoWrapFlags();
  
    if (AR->hasNoSignedWrap())
      return Result;
  
    if (!AR->isAffine())
      return Result;
  
    const SCEV *Step = AR->getStepRecurrence(*this);
    const Loop *L = AR->getLoop();
  
    // Check whether the backedge-taken count is SCEVCouldNotCompute.
    // Note that this serves two purposes: It filters out loops that are
    // simply not analyzable, and it covers the case where this code is
    // being called from within backedge-taken count analysis, such that
    // attempting to ask for the backedge-taken count would likely result
    // in infinite recursion. In the later case, the analysis code will
    // cope with a conservative value, and it will take care to purge
    // that value once it has finished.
    const SCEV *MaxBECount = getConstantMaxBackedgeTakenCount(L);
  
    // Normally, in the cases we can prove no-overflow via a
    // backedge guarding condition, we can also compute a backedge
    // taken count for the loop.  The exceptions are assumptions and
    // guards present in the loop -- SCEV is not great at exploiting
    // these to compute max backedge taken counts, but can still use
    // these to prove lack of overflow.  Use this fact to avoid
    // doing extra work that may not pay off.
  
    if (isa<SCEVCouldNotCompute>(MaxBECount) && !HasGuards &&
        AC.assumptions().empty())
      return Result;
  
    // If the backedge is guarded by a comparison with the pre-inc  value the
    // addrec is safe. Also, if the entry is guarded by a comparison with the
    // start value and the backedge is guarded by a comparison with the post-inc
    // value, the addrec is safe.
    ICmpInst::Predicate Pred;
    const SCEV *OverflowLimit =
      getSignedOverflowLimitForStep(Step, &Pred, this);
    if (OverflowLimit &&
        (isLoopBackedgeGuardedByCond(L, Pred, AR, OverflowLimit) ||
         isKnownOnEveryIteration(Pred, AR, OverflowLimit))) {
      Result = setFlags(Result, SCEV::FlagNSW);
    }
    return Result;
  }
  SCEV::NoWrapFlags
  ScalarEvolution::proveNoUnsignedWrapViaInduction(const SCEVAddRecExpr *AR) {
    SCEV::NoWrapFlags Result = AR->getNoWrapFlags();
  
    if (AR->hasNoUnsignedWrap())
      return Result;
  
    if (!AR->isAffine())
      return Result;
  
    const SCEV *Step = AR->getStepRecurrence(*this);
    unsigned BitWidth = getTypeSizeInBits(AR->getType());
    const Loop *L = AR->getLoop();
  
    // Check whether the backedge-taken count is SCEVCouldNotCompute.
    // Note that this serves two purposes: It filters out loops that are
    // simply not analyzable, and it covers the case where this code is
    // being called from within backedge-taken count analysis, such that
    // attempting to ask for the backedge-taken count would likely result
    // in infinite recursion. In the later case, the analysis code will
    // cope with a conservative value, and it will take care to purge
    // that value once it has finished.
    const SCEV *MaxBECount = getConstantMaxBackedgeTakenCount(L);
  
    // Normally, in the cases we can prove no-overflow via a
    // backedge guarding condition, we can also compute a backedge
    // taken count for the loop.  The exceptions are assumptions and
    // guards present in the loop -- SCEV is not great at exploiting
    // these to compute max backedge taken counts, but can still use
    // these to prove lack of overflow.  Use this fact to avoid
    // doing extra work that may not pay off.
  
    if (isa<SCEVCouldNotCompute>(MaxBECount) && !HasGuards &&
        AC.assumptions().empty())
      return Result;
  
    // If the backedge is guarded by a comparison with the pre-inc  value the
    // addrec is safe. Also, if the entry is guarded by a comparison with the
    // start value and the backedge is guarded by a comparison with the post-inc
    // value, the addrec is safe.
    if (isKnownPositive(Step)) {
      const SCEV *N = getConstant(APInt::getMinValue(BitWidth) -
                                  getUnsignedRangeMax(Step));
      if (isLoopBackedgeGuardedByCond(L, ICmpInst::ICMP_ULT, AR, N) ||
          isKnownOnEveryIteration(ICmpInst::ICMP_ULT, AR, N)) {
        Result = setFlags(Result, SCEV::FlagNUW);
      }
    }
  
    return Result;
  }
  
  namespace {
  
  /// Represents an abstract binary operation.  This may exist as a
  /// normal instruction or constant expression, or may have been
  /// derived from an expression tree.
  struct BinaryOp {
    unsigned Opcode;
    Value *LHS;
    Value *RHS;
    bool IsNSW = false;
    bool IsNUW = false;
  
    /// Op is set if this BinaryOp corresponds to a concrete LLVM instruction or
    /// constant expression.
    Operator *Op = nullptr;
  
    explicit BinaryOp(Operator *Op)
        : Opcode(Op->getOpcode()), LHS(Op->getOperand(0)), RHS(Op->getOperand(1)),
          Op(Op) {
      if (auto *OBO = dyn_cast<OverflowingBinaryOperator>(Op)) {
        IsNSW = OBO->hasNoSignedWrap();
        IsNUW = OBO->hasNoUnsignedWrap();
      }
    }
  
    explicit BinaryOp(unsigned Opcode, Value *LHS, Value *RHS, bool IsNSW = false,
                      bool IsNUW = false)
        : Opcode(Opcode), LHS(LHS), RHS(RHS), IsNSW(IsNSW), IsNUW(IsNUW) {}
  };
  
  } // end anonymous namespace
  
  /// Try to map \p V into a BinaryOp, and return \c None on failure.
  static Optional<BinaryOp> MatchBinaryOp(Value *V, DominatorTree &DT) {
    auto *Op = dyn_cast<Operator>(V);
    if (!Op)
      return None;
  
    // Implementation detail: all the cleverness here should happen without
    // creating new SCEV expressions -- our caller knowns tricks to avoid creating
    // SCEV expressions when possible, and we should not break that.
  
    switch (Op->getOpcode()) {
    case Instruction::Add:
    case Instruction::Sub:
    case Instruction::Mul:
    case Instruction::UDiv:
    case Instruction::URem:
    case Instruction::And:
    case Instruction::Or:
    case Instruction::AShr:
    case Instruction::Shl:
      return BinaryOp(Op);
  
    case Instruction::Xor:
      if (auto *RHSC = dyn_cast<ConstantInt>(Op->getOperand(1)))
        // If the RHS of the xor is a signmask, then this is just an add.
        // Instcombine turns add of signmask into xor as a strength reduction step.
        if (RHSC->getValue().isSignMask())
          return BinaryOp(Instruction::Add, Op->getOperand(0), Op->getOperand(1));
      return BinaryOp(Op);
  
    case Instruction::LShr:
      // Turn logical shift right of a constant into a unsigned divide.
      if (ConstantInt *SA = dyn_cast<ConstantInt>(Op->getOperand(1))) {
        uint32_t BitWidth = cast<IntegerType>(Op->getType())->getBitWidth();
  
        // If the shift count is not less than the bitwidth, the result of
        // the shift is undefined. Don't try to analyze it, because the
        // resolution chosen here may differ from the resolution chosen in
        // other parts of the compiler.
        if (SA->getValue().ult(BitWidth)) {
          Constant *X =
              ConstantInt::get(SA->getContext(),
                               APInt::getOneBitSet(BitWidth, SA->getZExtValue()));
          return BinaryOp(Instruction::UDiv, Op->getOperand(0), X);
        }
      }
      return BinaryOp(Op);
  
    case Instruction::ExtractValue: {
      auto *EVI = cast<ExtractValueInst>(Op);
      if (EVI->getNumIndices() != 1 || EVI->getIndices()[0] != 0)
        break;
  
      auto *WO = dyn_cast<WithOverflowInst>(EVI->getAggregateOperand());
      if (!WO)
        break;
  
      Instruction::BinaryOps BinOp = WO->getBinaryOp();
      bool Signed = WO->isSigned();
      // TODO: Should add nuw/nsw flags for mul as well.
      if (BinOp == Instruction::Mul || !isOverflowIntrinsicNoWrap(WO, DT))
        return BinaryOp(BinOp, WO->getLHS(), WO->getRHS());
  
      // Now that we know that all uses of the arithmetic-result component of
      // CI are guarded by the overflow check, we can go ahead and pretend
      // that the arithmetic is non-overflowing.
      return BinaryOp(BinOp, WO->getLHS(), WO->getRHS(),
                      /* IsNSW = */ Signed, /* IsNUW = */ !Signed);
    }
  
    default:
      break;
    }
  
    // Recognise intrinsic loop.decrement.reg, and as this has exactly the same
    // semantics as a Sub, return a binary sub expression.
    if (auto *II = dyn_cast<IntrinsicInst>(V))
      if (II->getIntrinsicID() == Intrinsic::loop_decrement_reg)
        return BinaryOp(Instruction::Sub, II->getOperand(0), II->getOperand(1));
  
    return None;
  }
  
  /// Helper function to createAddRecFromPHIWithCasts. We have a phi
  /// node whose symbolic (unknown) SCEV is \p SymbolicPHI, which is updated via
  /// the loop backedge by a SCEVAddExpr, possibly also with a few casts on the
  /// way. This function checks if \p Op, an operand of this SCEVAddExpr,
  /// follows one of the following patterns:
  /// Op == (SExt ix (Trunc iy (%SymbolicPHI) to ix) to iy)
  /// Op == (ZExt ix (Trunc iy (%SymbolicPHI) to ix) to iy)
  /// If the SCEV expression of \p Op conforms with one of the expected patterns
  /// we return the type of the truncation operation, and indicate whether the
  /// truncated type should be treated as signed/unsigned by setting
  /// \p Signed to true/false, respectively.
  static Type *isSimpleCastedPHI(const SCEV *Op, const SCEVUnknown *SymbolicPHI,
                                 bool &Signed, ScalarEvolution &SE) {
    // The case where Op == SymbolicPHI (that is, with no type conversions on
    // the way) is handled by the regular add recurrence creating logic and
    // would have already been triggered in createAddRecForPHI. Reaching it here
    // means that createAddRecFromPHI had failed for this PHI before (e.g.,
    // because one of the other operands of the SCEVAddExpr updating this PHI is
    // not invariant).
    //
    // Here we look for the case where Op = (ext(trunc(SymbolicPHI))), and in
    // this case predicates that allow us to prove that Op == SymbolicPHI will
    // be added.
    if (Op == SymbolicPHI)
      return nullptr;
  
    unsigned SourceBits = SE.getTypeSizeInBits(SymbolicPHI->getType());
    unsigned NewBits = SE.getTypeSizeInBits(Op->getType());
    if (SourceBits != NewBits)
      return nullptr;
  
    const SCEVSignExtendExpr *SExt = dyn_cast<SCEVSignExtendExpr>(Op);
    const SCEVZeroExtendExpr *ZExt = dyn_cast<SCEVZeroExtendExpr>(Op);
    if (!SExt && !ZExt)
      return nullptr;
    const SCEVTruncateExpr *Trunc =
        SExt ? dyn_cast<SCEVTruncateExpr>(SExt->getOperand())
             : dyn_cast<SCEVTruncateExpr>(ZExt->getOperand());
    if (!Trunc)
      return nullptr;
    const SCEV *X = Trunc->getOperand();
    if (X != SymbolicPHI)
      return nullptr;
    Signed = SExt != nullptr;
    return Trunc->getType();
  }
  
  static const Loop *isIntegerLoopHeaderPHI(const PHINode *PN, LoopInfo &LI) {
    if (!PN->getType()->isIntegerTy())
      return nullptr;
    const Loop *L = LI.getLoopFor(PN->getParent());
    if (!L || L->getHeader() != PN->getParent())
      return nullptr;
    return L;
  }
  
  // Analyze \p SymbolicPHI, a SCEV expression of a phi node, and check if the
  // computation that updates the phi follows the following pattern:
  //   (SExt/ZExt ix (Trunc iy (%SymbolicPHI) to ix) to iy) + InvariantAccum
  // which correspond to a phi->trunc->sext/zext->add->phi update chain.
  // If so, try to see if it can be rewritten as an AddRecExpr under some
  // Predicates. If successful, return them as a pair. Also cache the results
  // of the analysis.
  //
  // Example usage scenario:
  //    Say the Rewriter is called for the following SCEV:
  //         8 * ((sext i32 (trunc i64 %X to i32) to i64) + %Step)
  //    where:
  //         %X = phi i64 (%Start, %BEValue)
  //    It will visitMul->visitAdd->visitSExt->visitTrunc->visitUnknown(%X),
  //    and call this function with %SymbolicPHI = %X.
  //
  //    The analysis will find that the value coming around the backedge has
  //    the following SCEV:
  //         BEValue = ((sext i32 (trunc i64 %X to i32) to i64) + %Step)
  //    Upon concluding that this matches the desired pattern, the function
  //    will return the pair {NewAddRec, SmallPredsVec} where:
  //         NewAddRec = {%Start,+,%Step}
  //         SmallPredsVec = {P1, P2, P3} as follows:
  //           P1(WrapPred): AR: {trunc(%Start),+,(trunc %Step)}<nsw> Flags: <nssw>
  //           P2(EqualPred): %Start == (sext i32 (trunc i64 %Start to i32) to i64)
  //           P3(EqualPred): %Step == (sext i32 (trunc i64 %Step to i32) to i64)
  //    The returned pair means that SymbolicPHI can be rewritten into NewAddRec
  //    under the predicates {P1,P2,P3}.
  //    This predicated rewrite will be cached in PredicatedSCEVRewrites:
  //         PredicatedSCEVRewrites[{%X,L}] = {NewAddRec, {P1,P2,P3)}
  //
  // TODO's:
  //
  // 1) Extend the Induction descriptor to also support inductions that involve
  //    casts: When needed (namely, when we are called in the context of the
  //    vectorizer induction analysis), a Set of cast instructions will be
  //    populated by this method, and provided back to isInductionPHI. This is
  //    needed to allow the vectorizer to properly record them to be ignored by
  //    the cost model and to avoid vectorizing them (otherwise these casts,
  //    which are redundant under the runtime overflow checks, will be
  //    vectorized, which can be costly).
  //
  // 2) Support additional induction/PHISCEV patterns: We also want to support
  //    inductions where the sext-trunc / zext-trunc operations (partly) occur
  //    after the induction update operation (the induction increment):
  //
  //      (Trunc iy (SExt/ZExt ix (%SymbolicPHI + InvariantAccum) to iy) to ix)
  //    which correspond to a phi->add->trunc->sext/zext->phi update chain.
  //
  //      (Trunc iy ((SExt/ZExt ix (%SymbolicPhi) to iy) + InvariantAccum) to ix)
  //    which correspond to a phi->trunc->add->sext/zext->phi update chain.
  //
  // 3) Outline common code with createAddRecFromPHI to avoid duplication.
  Optional<std::pair<const SCEV *, SmallVector<const SCEVPredicate *, 3>>>
  ScalarEvolution::createAddRecFromPHIWithCastsImpl(const SCEVUnknown *SymbolicPHI) {
    SmallVector<const SCEVPredicate *, 3> Predicates;
  
    // *** Part1: Analyze if we have a phi-with-cast pattern for which we can
    // return an AddRec expression under some predicate.
  
    auto *PN = cast<PHINode>(SymbolicPHI->getValue());
    const Loop *L = isIntegerLoopHeaderPHI(PN, LI);
    assert(L && "Expecting an integer loop header phi");
  
    // The loop may have multiple entrances or multiple exits; we can analyze
    // this phi as an addrec if it has a unique entry value and a unique
    // backedge value.
    Value *BEValueV = nullptr, *StartValueV = nullptr;
    for (unsigned i = 0, e = PN->getNumIncomingValues(); i != e; ++i) {
      Value *V = PN->getIncomingValue(i);
      if (L->contains(PN->getIncomingBlock(i))) {
        if (!BEValueV) {
          BEValueV = V;
        } else if (BEValueV != V) {
          BEValueV = nullptr;
          break;
        }
      } else if (!StartValueV) {
        StartValueV = V;
      } else if (StartValueV != V) {
        StartValueV = nullptr;
        break;
      }
    }
    if (!BEValueV || !StartValueV)
      return None;
  
    const SCEV *BEValue = getSCEV(BEValueV);
  
    // If the value coming around the backedge is an add with the symbolic
    // value we just inserted, possibly with casts that we can ignore under
    // an appropriate runtime guard, then we found a simple induction variable!
    const auto *Add = dyn_cast<SCEVAddExpr>(BEValue);
    if (!Add)
      return None;
  
    // If there is a single occurrence of the symbolic value, possibly
    // casted, replace it with a recurrence.
    unsigned FoundIndex = Add->getNumOperands();
    Type *TruncTy = nullptr;
    bool Signed;
    for (unsigned i = 0, e = Add->getNumOperands(); i != e; ++i)
      if ((TruncTy =
               isSimpleCastedPHI(Add->getOperand(i), SymbolicPHI, Signed, *this)))
        if (FoundIndex == e) {
          FoundIndex = i;
          break;
        }
  
    if (FoundIndex == Add->getNumOperands())
      return None;
  
    // Create an add with everything but the specified operand.
    SmallVector<const SCEV *, 8> Ops;
    for (unsigned i = 0, e = Add->getNumOperands(); i != e; ++i)
      if (i != FoundIndex)
        Ops.push_back(Add->getOperand(i));
    const SCEV *Accum = getAddExpr(Ops);
  
    // The runtime checks will not be valid if the step amount is
    // varying inside the loop.
    if (!isLoopInvariant(Accum, L))
      return None;
  
    // *** Part2: Create the predicates
  
    // Analysis was successful: we have a phi-with-cast pattern for which we
    // can return an AddRec expression under the following predicates:
    //
    // P1: A Wrap predicate that guarantees that Trunc(Start) + i*Trunc(Accum)
    //     fits within the truncated type (does not overflow) for i = 0 to n-1.
    // P2: An Equal predicate that guarantees that
    //     Start = (Ext ix (Trunc iy (Start) to ix) to iy)
    // P3: An Equal predicate that guarantees that
    //     Accum = (Ext ix (Trunc iy (Accum) to ix) to iy)
    //
    // As we next prove, the above predicates guarantee that:
    //     Start + i*Accum = (Ext ix (Trunc iy ( Start + i*Accum ) to ix) to iy)
    //
    //
    // More formally, we want to prove that:
    //     Expr(i+1) = Start + (i+1) * Accum
    //               = (Ext ix (Trunc iy (Expr(i)) to ix) to iy) + Accum
    //
    // Given that:
    // 1) Expr(0) = Start
    // 2) Expr(1) = Start + Accum
    //            = (Ext ix (Trunc iy (Start) to ix) to iy) + Accum :: from P2
    // 3) Induction hypothesis (step i):
    //    Expr(i) = (Ext ix (Trunc iy (Expr(i-1)) to ix) to iy) + Accum
    //
    // Proof:
    //  Expr(i+1) =
    //   = Start + (i+1)*Accum
    //   = (Start + i*Accum) + Accum
    //   = Expr(i) + Accum
    //   = (Ext ix (Trunc iy (Expr(i-1)) to ix) to iy) + Accum + Accum
    //                                                             :: from step i
    //
    //   = (Ext ix (Trunc iy (Start + (i-1)*Accum) to ix) to iy) + Accum + Accum
    //
    //   = (Ext ix (Trunc iy (Start + (i-1)*Accum) to ix) to iy)
    //     + (Ext ix (Trunc iy (Accum) to ix) to iy)
    //     + Accum                                                     :: from P3
    //
    //   = (Ext ix (Trunc iy ((Start + (i-1)*Accum) + Accum) to ix) to iy)
    //     + Accum                            :: from P1: Ext(x)+Ext(y)=>Ext(x+y)
    //
    //   = (Ext ix (Trunc iy (Start + i*Accum) to ix) to iy) + Accum
    //   = (Ext ix (Trunc iy (Expr(i)) to ix) to iy) + Accum
    //
    // By induction, the same applies to all iterations 1<=i<n:
    //
  
    // Create a truncated addrec for which we will add a no overflow check (P1).
    const SCEV *StartVal = getSCEV(StartValueV);
    const SCEV *PHISCEV =
        getAddRecExpr(getTruncateExpr(StartVal, TruncTy),
                      getTruncateExpr(Accum, TruncTy), L, SCEV::FlagAnyWrap);
  
    // PHISCEV can be either a SCEVConstant or a SCEVAddRecExpr.
    // ex: If truncated Accum is 0 and StartVal is a constant, then PHISCEV
    // will be constant.
    //
    //  If PHISCEV is a constant, then P1 degenerates into P2 or P3, so we don't
    // add P1.
    if (const auto *AR = dyn_cast<SCEVAddRecExpr>(PHISCEV)) {
      SCEVWrapPredicate::IncrementWrapFlags AddedFlags =
          Signed ? SCEVWrapPredicate::IncrementNSSW
                 : SCEVWrapPredicate::IncrementNUSW;
      const SCEVPredicate *AddRecPred = getWrapPredicate(AR, AddedFlags);
      Predicates.push_back(AddRecPred);
    }
  
    // Create the Equal Predicates P2,P3:
  
    // It is possible that the predicates P2 and/or P3 are computable at
    // compile time due to StartVal and/or Accum being constants.
    // If either one is, then we can check that now and escape if either P2
    // or P3 is false.
  
    // Construct the extended SCEV: (Ext ix (Trunc iy (Expr) to ix) to iy)
    // for each of StartVal and Accum
    auto getExtendedExpr = [&](const SCEV *Expr,
                               bool CreateSignExtend) -> const SCEV * {
      assert(isLoopInvariant(Expr, L) && "Expr is expected to be invariant");
      const SCEV *TruncatedExpr = getTruncateExpr(Expr, TruncTy);
      const SCEV *ExtendedExpr =
          CreateSignExtend ? getSignExtendExpr(TruncatedExpr, Expr->getType())
                           : getZeroExtendExpr(TruncatedExpr, Expr->getType());
      return ExtendedExpr;
    };
  
    // Given:
    //  ExtendedExpr = (Ext ix (Trunc iy (Expr) to ix) to iy
    //               = getExtendedExpr(Expr)
    // Determine whether the predicate P: Expr == ExtendedExpr
    // is known to be false at compile time
    auto PredIsKnownFalse = [&](const SCEV *Expr,
                                const SCEV *ExtendedExpr) -> bool {
      return Expr != ExtendedExpr &&
             isKnownPredicate(ICmpInst::ICMP_NE, Expr, ExtendedExpr);
    };
  
    const SCEV *StartExtended = getExtendedExpr(StartVal, Signed);
    if (PredIsKnownFalse(StartVal, StartExtended)) {
      LLVM_DEBUG(dbgs() << "P2 is compile-time false\n";);
      return None;
    }
  
    // The Step is always Signed (because the overflow checks are either
    // NSSW or NUSW)
    const SCEV *AccumExtended = getExtendedExpr(Accum, /*CreateSignExtend=*/true);
    if (PredIsKnownFalse(Accum, AccumExtended)) {
      LLVM_DEBUG(dbgs() << "P3 is compile-time false\n";);
      return None;
    }
  
    auto AppendPredicate = [&](const SCEV *Expr,
                               const SCEV *ExtendedExpr) -> void {
      if (Expr != ExtendedExpr &&
          !isKnownPredicate(ICmpInst::ICMP_EQ, Expr, ExtendedExpr)) {
        const SCEVPredicate *Pred = getEqualPredicate(Expr, ExtendedExpr);
        LLVM_DEBUG(dbgs() << "Added Predicate: " << *Pred);
        Predicates.push_back(Pred);
      }
    };
  
    AppendPredicate(StartVal, StartExtended);
    AppendPredicate(Accum, AccumExtended);
  
    // *** Part3: Predicates are ready. Now go ahead and create the new addrec in
    // which the casts had been folded away. The caller can rewrite SymbolicPHI
    // into NewAR if it will also add the runtime overflow checks specified in
    // Predicates.
    auto *NewAR = getAddRecExpr(StartVal, Accum, L, SCEV::FlagAnyWrap);
  
    std::pair<const SCEV *, SmallVector<const SCEVPredicate *, 3>> PredRewrite =
        std::make_pair(NewAR, Predicates);
    // Remember the result of the analysis for this SCEV at this locayyytion.
    PredicatedSCEVRewrites[{SymbolicPHI, L}] = PredRewrite;
    return PredRewrite;
  }
  
  Optional<std::pair<const SCEV *, SmallVector<const SCEVPredicate *, 3>>>
  ScalarEvolution::createAddRecFromPHIWithCasts(const SCEVUnknown *SymbolicPHI) {
    auto *PN = cast<PHINode>(SymbolicPHI->getValue());
    const Loop *L = isIntegerLoopHeaderPHI(PN, LI);
    if (!L)
      return None;
  
    // Check to see if we already analyzed this PHI.
    auto I = PredicatedSCEVRewrites.find({SymbolicPHI, L});
    if (I != PredicatedSCEVRewrites.end()) {
      std::pair<const SCEV *, SmallVector<const SCEVPredicate *, 3>> Rewrite =
          I->second;
      // Analysis was done before and failed to create an AddRec:
      if (Rewrite.first == SymbolicPHI)
        return None;
      // Analysis was done before and succeeded to create an AddRec under
      // a predicate:
      assert(isa<SCEVAddRecExpr>(Rewrite.first) && "Expected an AddRec");
      assert(!(Rewrite.second).empty() && "Expected to find Predicates");
      return Rewrite;
    }
  
    Optional<std::pair<const SCEV *, SmallVector<const SCEVPredicate *, 3>>>
      Rewrite = createAddRecFromPHIWithCastsImpl(SymbolicPHI);
  
    // Record in the cache that the analysis failed
    if (!Rewrite) {
      SmallVector<const SCEVPredicate *, 3> Predicates;
      PredicatedSCEVRewrites[{SymbolicPHI, L}] = {SymbolicPHI, Predicates};
      return None;
    }
  
    return Rewrite;
  }
  
  // FIXME: This utility is currently required because the Rewriter currently
  // does not rewrite this expression:
  // {0, +, (sext ix (trunc iy to ix) to iy)}
  // into {0, +, %step},
  // even when the following Equal predicate exists:
  // "%step == (sext ix (trunc iy to ix) to iy)".
  bool PredicatedScalarEvolution::areAddRecsEqualWithPreds(
      const SCEVAddRecExpr *AR1, const SCEVAddRecExpr *AR2) const {
    if (AR1 == AR2)
      return true;
  
    auto areExprsEqual = [&](const SCEV *Expr1, const SCEV *Expr2) -> bool {
      if (Expr1 != Expr2 && !Preds.implies(SE.getEqualPredicate(Expr1, Expr2)) &&
          !Preds.implies(SE.getEqualPredicate(Expr2, Expr1)))
        return false;
      return true;
    };
  
    if (!areExprsEqual(AR1->getStart(), AR2->getStart()) ||
        !areExprsEqual(AR1->getStepRecurrence(SE), AR2->getStepRecurrence(SE)))
      return false;
    return true;
  }
  
  /// A helper function for createAddRecFromPHI to handle simple cases.
  ///
  /// This function tries to find an AddRec expression for the simplest (yet most
  /// common) cases: PN = PHI(Start, OP(Self, LoopInvariant)).
  /// If it fails, createAddRecFromPHI will use a more general, but slow,
  /// technique for finding the AddRec expression.
  const SCEV *ScalarEvolution::createSimpleAffineAddRec(PHINode *PN,
                                                        Value *BEValueV,
                                                        Value *StartValueV) {
    const Loop *L = LI.getLoopFor(PN->getParent());
    assert(L && L->getHeader() == PN->getParent());
    assert(BEValueV && StartValueV);
  
    auto BO = MatchBinaryOp(BEValueV, DT);
    if (!BO)
      return nullptr;
  
    if (BO->Opcode != Instruction::Add)
      return nullptr;
  
    const SCEV *Accum = nullptr;
    if (BO->LHS == PN && L->isLoopInvariant(BO->RHS))
      Accum = getSCEV(BO->RHS);
    else if (BO->RHS == PN && L->isLoopInvariant(BO->LHS))
      Accum = getSCEV(BO->LHS);
  
    if (!Accum)
      return nullptr;
  
    SCEV::NoWrapFlags Flags = SCEV::FlagAnyWrap;
    if (BO->IsNUW)
      Flags = setFlags(Flags, SCEV::FlagNUW);
    if (BO->IsNSW)
      Flags = setFlags(Flags, SCEV::FlagNSW);
  
    const SCEV *StartVal = getSCEV(StartValueV);
    const SCEV *PHISCEV = getAddRecExpr(StartVal, Accum, L, Flags);
  
    ValueExprMap[SCEVCallbackVH(PN, this)] = PHISCEV;
  
    // We can add Flags to the post-inc expression only if we
    // know that it is *undefined behavior* for BEValueV to
    // overflow.
    if (auto *BEInst = dyn_cast<Instruction>(BEValueV))
      if (isLoopInvariant(Accum, L) && isAddRecNeverPoison(BEInst, L))
        (void)getAddRecExpr(getAddExpr(StartVal, Accum), Accum, L, Flags);
  
    return PHISCEV;
  }
  
  const SCEV *ScalarEvolution::createAddRecFromPHI(PHINode *PN) {
    const Loop *L = LI.getLoopFor(PN->getParent());
    if (!L || L->getHeader() != PN->getParent())
      return nullptr;
  
    // The loop may have multiple entrances or multiple exits; we can analyze
    // this phi as an addrec if it has a unique entry value and a unique
    // backedge value.
    Value *BEValueV = nullptr, *StartValueV = nullptr;
    for (unsigned i = 0, e = PN->getNumIncomingValues(); i != e; ++i) {
      Value *V = PN->getIncomingValue(i);
      if (L->contains(PN->getIncomingBlock(i))) {
        if (!BEValueV) {
          BEValueV = V;
        } else if (BEValueV != V) {
          BEValueV = nullptr;
          break;
        }
      } else if (!StartValueV) {
        StartValueV = V;
      } else if (StartValueV != V) {
        StartValueV = nullptr;
        break;
      }
    }
    if (!BEValueV || !StartValueV)
      return nullptr;
  
    assert(ValueExprMap.find_as(PN) == ValueExprMap.end() &&
           "PHI node already processed?");
  
    // First, try to find AddRec expression without creating a fictituos symbolic
    // value for PN.
    if (auto *S = createSimpleAffineAddRec(PN, BEValueV, StartValueV))
      return S;
  
    // Handle PHI node value symbolically.
    const SCEV *SymbolicName = getUnknown(PN);
    ValueExprMap.insert({SCEVCallbackVH(PN, this), SymbolicName});
  
    // Using this symbolic name for the PHI, analyze the value coming around
    // the back-edge.
    const SCEV *BEValue = getSCEV(BEValueV);
  
    // NOTE: If BEValue is loop invariant, we know that the PHI node just
    // has a special value for the first iteration of the loop.
  
    // If the value coming around the backedge is an add with the symbolic
    // value we just inserted, then we found a simple induction variable!
    if (const SCEVAddExpr *Add = dyn_cast<SCEVAddExpr>(BEValue)) {
      // If there is a single occurrence of the symbolic value, replace it
      // with a recurrence.
      unsigned FoundIndex = Add->getNumOperands();
      for (unsigned i = 0, e = Add->getNumOperands(); i != e; ++i)
        if (Add->getOperand(i) == SymbolicName)
          if (FoundIndex == e) {
            FoundIndex = i;
            break;
          }
  
      if (FoundIndex != Add->getNumOperands()) {
        // Create an add with everything but the specified operand.
        SmallVector<const SCEV *, 8> Ops;
        for (unsigned i = 0, e = Add->getNumOperands(); i != e; ++i)
          if (i != FoundIndex)
            Ops.push_back(SCEVBackedgeConditionFolder::rewrite(Add->getOperand(i),
                                                               L, *this));
        const SCEV *Accum = getAddExpr(Ops);
  
        // This is not a valid addrec if the step amount is varying each
        // loop iteration, but is not itself an addrec in this loop.
        if (isLoopInvariant(Accum, L) ||
            (isa<SCEVAddRecExpr>(Accum) &&
             cast<SCEVAddRecExpr>(Accum)->getLoop() == L)) {
          SCEV::NoWrapFlags Flags = SCEV::FlagAnyWrap;
  
          if (auto BO = MatchBinaryOp(BEValueV, DT)) {
            if (BO->Opcode == Instruction::Add && BO->LHS == PN) {
              if (BO->IsNUW)
                Flags = setFlags(Flags, SCEV::FlagNUW);
              if (BO->IsNSW)
                Flags = setFlags(Flags, SCEV::FlagNSW);
            }
          } else if (GEPOperator *GEP = dyn_cast<GEPOperator>(BEValueV)) {
            // If the increment is an inbounds GEP, then we know the address
            // space cannot be wrapped around. We cannot make any guarantee
            // about signed or unsigned overflow because pointers are
            // unsigned but we may have a negative index from the base
            // pointer. We can guarantee that no unsigned wrap occurs if the
            // indices form a positive value.
            if (GEP->isInBounds() && GEP->getOperand(0) == PN) {
              Flags = setFlags(Flags, SCEV::FlagNW);
  
              const SCEV *Ptr = getSCEV(GEP->getPointerOperand());
              if (isKnownPositive(getMinusSCEV(getSCEV(GEP), Ptr)))
                Flags = setFlags(Flags, SCEV::FlagNUW);
            }
  
            // We cannot transfer nuw and nsw flags from subtraction
            // operations -- sub nuw X, Y is not the same as add nuw X, -Y
            // for instance.
          }
  
          const SCEV *StartVal = getSCEV(StartValueV);
          const SCEV *PHISCEV = getAddRecExpr(StartVal, Accum, L, Flags);
  
          // Okay, for the entire analysis of this edge we assumed the PHI
          // to be symbolic.  We now need to go back and purge all of the
          // entries for the scalars that use the symbolic expression.
          forgetSymbolicName(PN, SymbolicName);
          ValueExprMap[SCEVCallbackVH(PN, this)] = PHISCEV;
  
          // We can add Flags to the post-inc expression only if we
          // know that it is *undefined behavior* for BEValueV to
          // overflow.
          if (auto *BEInst = dyn_cast<Instruction>(BEValueV))
            if (isLoopInvariant(Accum, L) && isAddRecNeverPoison(BEInst, L))
              (void)getAddRecExpr(getAddExpr(StartVal, Accum), Accum, L, Flags);
  
          return PHISCEV;
        }
      }
    } else {
      // Otherwise, this could be a loop like this:
      //     i = 0;  for (j = 1; ..; ++j) { ....  i = j; }
      // In this case, j = {1,+,1}  and BEValue is j.
      // Because the other in-value of i (0) fits the evolution of BEValue
      // i really is an addrec evolution.
      //
      // We can generalize this saying that i is the shifted value of BEValue
      // by one iteration:
      //   PHI(f(0), f({1,+,1})) --> f({0,+,1})
      const SCEV *Shifted = SCEVShiftRewriter::rewrite(BEValue, L, *this);
      const SCEV *Start = SCEVInitRewriter::rewrite(Shifted, L, *this, false);
      if (Shifted != getCouldNotCompute() &&
          Start != getCouldNotCompute()) {
        const SCEV *StartVal = getSCEV(StartValueV);
        if (Start == StartVal) {
          // Okay, for the entire analysis of this edge we assumed the PHI
          // to be symbolic.  We now need to go back and purge all of the
          // entries for the scalars that use the symbolic expression.
          forgetSymbolicName(PN, SymbolicName);
          ValueExprMap[SCEVCallbackVH(PN, this)] = Shifted;
          return Shifted;
        }
      }
    }
  
    // Remove the temporary PHI node SCEV that has been inserted while intending
    // to create an AddRecExpr for this PHI node. We can not keep this temporary
    // as it will prevent later (possibly simpler) SCEV expressions to be added
    // to the ValueExprMap.
    eraseValueFromMap(PN);
  
    return nullptr;
  }
  
  // Checks if the SCEV S is available at BB.  S is considered available at BB
  // if S can be materialized at BB without introducing a fault.
  static bool IsAvailableOnEntry(const Loop *L, DominatorTree &DT, const SCEV *S,
                                 BasicBlock *BB) {
    struct CheckAvailable {
      bool TraversalDone = false;
      bool Available = true;
  
      const Loop *L = nullptr;  // The loop BB is in (can be nullptr)
      BasicBlock *BB = nullptr;
      DominatorTree &DT;
  
      CheckAvailable(const Loop *L, BasicBlock *BB, DominatorTree &DT)
        : L(L), BB(BB), DT(DT) {}
  
      bool setUnavailable() {
        TraversalDone = true;
        Available = false;
        return false;
      }
  
      bool follow(const SCEV *S) {
        switch (S->getSCEVType()) {
        case scConstant:
        case scPtrToInt:
        case scTruncate:
        case scZeroExtend:
        case scSignExtend:
        case scAddExpr:
        case scMulExpr:
        case scUMaxExpr:
        case scSMaxExpr:
        case scUMinExpr:
        case scSMinExpr:
          // These expressions are available if their operand(s) is/are.
          return true;
  
        case scAddRecExpr: {
          // We allow add recurrences that are on the loop BB is in, or some
          // outer loop.  This guarantees availability because the value of the
          // add recurrence at BB is simply the "current" value of the induction
          // variable.  We can relax this in the future; for instance an add
          // recurrence on a sibling dominating loop is also available at BB.
          const auto *ARLoop = cast<SCEVAddRecExpr>(S)->getLoop();
          if (L && (ARLoop == L || ARLoop->contains(L)))
            return true;
  
          return setUnavailable();
        }
  
        case scUnknown: {
          // For SCEVUnknown, we check for simple dominance.
          const auto *SU = cast<SCEVUnknown>(S);
          Value *V = SU->getValue();
  
          if (isa<Argument>(V))
            return false;
  
          if (isa<Instruction>(V) && DT.dominates(cast<Instruction>(V), BB))
            return false;
  
          return setUnavailable();
        }
  
        case scUDivExpr:
        case scCouldNotCompute:
          // We do not try to smart about these at all.
          return setUnavailable();
        }
        llvm_unreachable("Unknown SCEV kind!");
      }
  
      bool isDone() { return TraversalDone; }
    };
  
    CheckAvailable CA(L, BB, DT);
    SCEVTraversal<CheckAvailable> ST(CA);
  
    ST.visitAll(S);
    return CA.Available;
  }
  
  // Try to match a control flow sequence that branches out at BI and merges back
  // at Merge into a "C ? LHS : RHS" select pattern.  Return true on a successful
  // match.
  static bool BrPHIToSelect(DominatorTree &DT, BranchInst *BI, PHINode *Merge,
                            Value *&C, Value *&LHS, Value *&RHS) {
    C = BI->getCondition();
  
    BasicBlockEdge LeftEdge(BI->getParent(), BI->getSuccessor(0));
    BasicBlockEdge RightEdge(BI->getParent(), BI->getSuccessor(1));
  
    if (!LeftEdge.isSingleEdge())
      return false;
  
    assert(RightEdge.isSingleEdge() && "Follows from LeftEdge.isSingleEdge()");
  
    Use &LeftUse = Merge->getOperandUse(0);
    Use &RightUse = Merge->getOperandUse(1);
  
    if (DT.dominates(LeftEdge, LeftUse) && DT.dominates(RightEdge, RightUse)) {
      LHS = LeftUse;
      RHS = RightUse;
      return true;
    }
  
    if (DT.dominates(LeftEdge, RightUse) && DT.dominates(RightEdge, LeftUse)) {
      LHS = RightUse;
      RHS = LeftUse;
      return true;
    }
  
    return false;
  }
  
  const SCEV *ScalarEvolution::createNodeFromSelectLikePHI(PHINode *PN) {
    auto IsReachable =
        [&](BasicBlock *BB) { return DT.isReachableFromEntry(BB); };
    if (PN->getNumIncomingValues() == 2 && all_of(PN->blocks(), IsReachable)) {
      const Loop *L = LI.getLoopFor(PN->getParent());
  
      // We don't want to break LCSSA, even in a SCEV expression tree.
      for (unsigned i = 0, e = PN->getNumIncomingValues(); i != e; ++i)
        if (LI.getLoopFor(PN->getIncomingBlock(i)) != L)
          return nullptr;
  
      // Try to match
      //
      //  br %cond, label %left, label %right
      // left:
      //  br label %merge
      // right:
      //  br label %merge
      // merge:
      //  V = phi [ %x, %left ], [ %y, %right ]
      //
      // as "select %cond, %x, %y"
  
      BasicBlock *IDom = DT[PN->getParent()]->getIDom()->getBlock();
      assert(IDom && "At least the entry block should dominate PN");
  
      auto *BI = dyn_cast<BranchInst>(IDom->getTerminator());
      Value *Cond = nullptr, *LHS = nullptr, *RHS = nullptr;
  
      if (BI && BI->isConditional() &&
          BrPHIToSelect(DT, BI, PN, Cond, LHS, RHS) &&
          IsAvailableOnEntry(L, DT, getSCEV(LHS), PN->getParent()) &&
          IsAvailableOnEntry(L, DT, getSCEV(RHS), PN->getParent()))
        return createNodeForSelectOrPHI(PN, Cond, LHS, RHS);
    }
  
    return nullptr;
  }
  
  const SCEV *ScalarEvolution::createNodeForPHI(PHINode *PN) {
    if (const SCEV *S = createAddRecFromPHI(PN))
      return S;
  
    if (const SCEV *S = createNodeFromSelectLikePHI(PN))
      return S;
  
    // If the PHI has a single incoming value, follow that value, unless the
    // PHI's incoming blocks are in a different loop, in which case doing so
    // risks breaking LCSSA form. Instcombine would normally zap these, but
    // it doesn't have DominatorTree information, so it may miss cases.
    if (Value *V = SimplifyInstruction(PN, {getDataLayout(), &TLI, &DT, &AC}))
      if (LI.replacementPreservesLCSSAForm(PN, V))
        return getSCEV(V);
  
    // If it's not a loop phi, we can't handle it yet.
    return getUnknown(PN);
  }
  
  const SCEV *ScalarEvolution::createNodeForSelectOrPHI(Instruction *I,
                                                        Value *Cond,
                                                        Value *TrueVal,
                                                        Value *FalseVal) {
    // Handle "constant" branch or select. This can occur for instance when a
    // loop pass transforms an inner loop and moves on to process the outer loop.
    if (auto *CI = dyn_cast<ConstantInt>(Cond))
      return getSCEV(CI->isOne() ? TrueVal : FalseVal);
  
    // Try to match some simple smax or umax patterns.
    auto *ICI = dyn_cast<ICmpInst>(Cond);
    if (!ICI)
      return getUnknown(I);
  
    Value *LHS = ICI->getOperand(0);
    Value *RHS = ICI->getOperand(1);
  
    switch (ICI->getPredicate()) {
    case ICmpInst::ICMP_SLT:
    case ICmpInst::ICMP_SLE:
    case ICmpInst::ICMP_ULT:
    case ICmpInst::ICMP_ULE:
      std::swap(LHS, RHS);
      LLVM_FALLTHROUGH;
    case ICmpInst::ICMP_SGT:
    case ICmpInst::ICMP_SGE:
    case ICmpInst::ICMP_UGT:
    case ICmpInst::ICMP_UGE:
      // a > b ? a+x : b+x  ->  max(a, b)+x
      // a > b ? b+x : a+x  ->  min(a, b)+x
      if (getTypeSizeInBits(LHS->getType()) <= getTypeSizeInBits(I->getType())) {
        bool Signed = ICI->isSigned();
        const SCEV *LA = getSCEV(TrueVal);
        const SCEV *RA = getSCEV(FalseVal);
        const SCEV *LS = getSCEV(LHS);
        const SCEV *RS = getSCEV(RHS);
        if (LA->getType()->isPointerTy()) {
          // FIXME: Handle cases where LS/RS are pointers not equal to LA/RA.
          // Need to make sure we can't produce weird expressions involving
          // negated pointers.
          if (LA == LS && RA == RS)
            return Signed ? getSMaxExpr(LS, RS) : getUMaxExpr(LS, RS);
          if (LA == RS && RA == LS)
            return Signed ? getSMinExpr(LS, RS) : getUMinExpr(LS, RS);
        }
        auto CoerceOperand = [&](const SCEV *Op) -> const SCEV * {
          if (Op->getType()->isPointerTy()) {
            Op = getLosslessPtrToIntExpr(Op);
            if (isa<SCEVCouldNotCompute>(Op))
              return Op;
          }
          if (Signed)
            Op = getNoopOrSignExtend(Op, I->getType());
          else
            Op = getNoopOrZeroExtend(Op, I->getType());
          return Op;
        };
        LS = CoerceOperand(LS);
        RS = CoerceOperand(RS);
        if (isa<SCEVCouldNotCompute>(LS) || isa<SCEVCouldNotCompute>(RS))
          break;
        const SCEV *LDiff = getMinusSCEV(LA, LS);
        const SCEV *RDiff = getMinusSCEV(RA, RS);
        if (LDiff == RDiff)
          return getAddExpr(Signed ? getSMaxExpr(LS, RS) : getUMaxExpr(LS, RS),
                            LDiff);
        LDiff = getMinusSCEV(LA, RS);
        RDiff = getMinusSCEV(RA, LS);
        if (LDiff == RDiff)
          return getAddExpr(Signed ? getSMinExpr(LS, RS) : getUMinExpr(LS, RS),
                            LDiff);
      }
      break;
    case ICmpInst::ICMP_NE:
      // n != 0 ? n+x : 1+x  ->  umax(n, 1)+x
      if (getTypeSizeInBits(LHS->getType()) <= getTypeSizeInBits(I->getType()) &&
          isa<ConstantInt>(RHS) && cast<ConstantInt>(RHS)->isZero()) {
        const SCEV *One = getOne(I->getType());
        const SCEV *LS = getNoopOrZeroExtend(getSCEV(LHS), I->getType());
        const SCEV *LA = getSCEV(TrueVal);
        const SCEV *RA = getSCEV(FalseVal);
        const SCEV *LDiff = getMinusSCEV(LA, LS);
        const SCEV *RDiff = getMinusSCEV(RA, One);
        if (LDiff == RDiff)
          return getAddExpr(getUMaxExpr(One, LS), LDiff);
      }
      break;
    case ICmpInst::ICMP_EQ:
      // n == 0 ? 1+x : n+x  ->  umax(n, 1)+x
      if (getTypeSizeInBits(LHS->getType()) <= getTypeSizeInBits(I->getType()) &&
          isa<ConstantInt>(RHS) && cast<ConstantInt>(RHS)->isZero()) {
        const SCEV *One = getOne(I->getType());
        const SCEV *LS = getNoopOrZeroExtend(getSCEV(LHS), I->getType());
        const SCEV *LA = getSCEV(TrueVal);
        const SCEV *RA = getSCEV(FalseVal);
        const SCEV *LDiff = getMinusSCEV(LA, One);
        const SCEV *RDiff = getMinusSCEV(RA, LS);
        if (LDiff == RDiff)
          return getAddExpr(getUMaxExpr(One, LS), LDiff);
      }
      break;
    default:
      break;
    }
  
    return getUnknown(I);
  }
  
  /// Expand GEP instructions into add and multiply operations. This allows them
  /// to be analyzed by regular SCEV code.
  const SCEV *ScalarEvolution::createNodeForGEP(GEPOperator *GEP) {
    // Don't attempt to analyze GEPs over unsized objects.
    if (!GEP->getSourceElementType()->isSized())
      return getUnknown(GEP);
  
    SmallVector<const SCEV *, 4> IndexExprs;
    for (Value *Index : GEP->indices())
      IndexExprs.push_back(getSCEV(Index));
    return getGEPExpr(GEP, IndexExprs);
  }
  
  uint32_t ScalarEvolution::GetMinTrailingZerosImpl(const SCEV *S) {
    if (const SCEVConstant *C = dyn_cast<SCEVConstant>(S))
      return C->getAPInt().countTrailingZeros();
  
    if (const SCEVPtrToIntExpr *I = dyn_cast<SCEVPtrToIntExpr>(S))
      return GetMinTrailingZeros(I->getOperand());
  
    if (const SCEVTruncateExpr *T = dyn_cast<SCEVTruncateExpr>(S))
      return std::min(GetMinTrailingZeros(T->getOperand()),
                      (uint32_t)getTypeSizeInBits(T->getType()));
  
    if (const SCEVZeroExtendExpr *E = dyn_cast<SCEVZeroExtendExpr>(S)) {
      uint32_t OpRes = GetMinTrailingZeros(E->getOperand());
      return OpRes == getTypeSizeInBits(E->getOperand()->getType())
                 ? getTypeSizeInBits(E->getType())
                 : OpRes;
    }
  
    if (const SCEVSignExtendExpr *E = dyn_cast<SCEVSignExtendExpr>(S)) {
      uint32_t OpRes = GetMinTrailingZeros(E->getOperand());
      return OpRes == getTypeSizeInBits(E->getOperand()->getType())
                 ? getTypeSizeInBits(E->getType())
                 : OpRes;
    }
  
    if (const SCEVAddExpr *A = dyn_cast<SCEVAddExpr>(S)) {
      // The result is the min of all operands results.
      uint32_t MinOpRes = GetMinTrailingZeros(A->getOperand(0));
      for (unsigned i = 1, e = A->getNumOperands(); MinOpRes && i != e; ++i)
        MinOpRes = std::min(MinOpRes, GetMinTrailingZeros(A->getOperand(i)));
      return MinOpRes;
    }
  
    if (const SCEVMulExpr *M = dyn_cast<SCEVMulExpr>(S)) {
      // The result is the sum of all operands results.
      uint32_t SumOpRes = GetMinTrailingZeros(M->getOperand(0));
      uint32_t BitWidth = getTypeSizeInBits(M->getType());
      for (unsigned i = 1, e = M->getNumOperands();
           SumOpRes != BitWidth && i != e; ++i)
        SumOpRes =
            std::min(SumOpRes + GetMinTrailingZeros(M->getOperand(i)), BitWidth);
      return SumOpRes;
    }
  
    if (const SCEVAddRecExpr *A = dyn_cast<SCEVAddRecExpr>(S)) {
      // The result is the min of all operands results.
      uint32_t MinOpRes = GetMinTrailingZeros(A->getOperand(0));
      for (unsigned i = 1, e = A->getNumOperands(); MinOpRes && i != e; ++i)
        MinOpRes = std::min(MinOpRes, GetMinTrailingZeros(A->getOperand(i)));
      return MinOpRes;
    }
  
    if (const SCEVSMaxExpr *M = dyn_cast<SCEVSMaxExpr>(S)) {
      // The result is the min of all operands results.
      uint32_t MinOpRes = GetMinTrailingZeros(M->getOperand(0));
      for (unsigned i = 1, e = M->getNumOperands(); MinOpRes && i != e; ++i)
        MinOpRes = std::min(MinOpRes, GetMinTrailingZeros(M->getOperand(i)));
      return MinOpRes;
    }
  
    if (const SCEVUMaxExpr *M = dyn_cast<SCEVUMaxExpr>(S)) {
      // The result is the min of all operands results.
      uint32_t MinOpRes = GetMinTrailingZeros(M->getOperand(0));
      for (unsigned i = 1, e = M->getNumOperands(); MinOpRes && i != e; ++i)
        MinOpRes = std::min(MinOpRes, GetMinTrailingZeros(M->getOperand(i)));
      return MinOpRes;
    }
  
    if (const SCEVUnknown *U = dyn_cast<SCEVUnknown>(S)) {
      // For a SCEVUnknown, ask ValueTracking.
      KnownBits Known = computeKnownBits(U->getValue(), getDataLayout(), 0, &AC, nullptr, &DT);
      return Known.countMinTrailingZeros();
    }
  
    // SCEVUDivExpr
    return 0;
  }
  
  uint32_t ScalarEvolution::GetMinTrailingZeros(const SCEV *S) {
    auto I = MinTrailingZerosCache.find(S);
    if (I != MinTrailingZerosCache.end())
      return I->second;
  
    uint32_t Result = GetMinTrailingZerosImpl(S);
    auto InsertPair = MinTrailingZerosCache.insert({S, Result});
    assert(InsertPair.second && "Should insert a new key");
    return InsertPair.first->second;
  }
  
  /// Helper method to assign a range to V from metadata present in the IR.
  static Optional<ConstantRange> GetRangeFromMetadata(Value *V) {
    if (Instruction *I = dyn_cast<Instruction>(V))
      if (MDNode *MD = I->getMetadata(LLVMContext::MD_range))
        return getConstantRangeFromMetadata(*MD);
  
    return None;
  }
  
  void ScalarEvolution::setNoWrapFlags(SCEVAddRecExpr *AddRec,
                                       SCEV::NoWrapFlags Flags) {
    if (AddRec->getNoWrapFlags(Flags) != Flags) {
      AddRec->setNoWrapFlags(Flags);
      UnsignedRanges.erase(AddRec);
      SignedRanges.erase(AddRec);
    }
  }
  
  ConstantRange ScalarEvolution::
  getRangeForUnknownRecurrence(const SCEVUnknown *U) {
    const DataLayout &DL = getDataLayout();
  
    unsigned BitWidth = getTypeSizeInBits(U->getType());
    const ConstantRange FullSet(BitWidth, /*isFullSet=*/true);
  
    // Match a simple recurrence of the form: <start, ShiftOp, Step>, and then
    // use information about the trip count to improve our available range.  Note
    // that the trip count independent cases are already handled by known bits.
    // WARNING: The definition of recurrence used here is subtly different than
    // the one used by AddRec (and thus most of this file).  Step is allowed to
    // be arbitrarily loop varying here, where AddRec allows only loop invariant
    // and other addrecs in the same loop (for non-affine addrecs).  The code
    // below intentionally handles the case where step is not loop invariant.
    auto *P = dyn_cast<PHINode>(U->getValue());
    if (!P)
      return FullSet;
  
    // Make sure that no Phi input comes from an unreachable block. Otherwise,
    // even the values that are not available in these blocks may come from them,
    // and this leads to false-positive recurrence test.
    for (auto *Pred : predecessors(P->getParent()))
      if (!DT.isReachableFromEntry(Pred))
        return FullSet;
  
    BinaryOperator *BO;
    Value *Start, *Step;
    if (!matchSimpleRecurrence(P, BO, Start, Step))
      return FullSet;
  
    // If we found a recurrence in reachable code, we must be in a loop. Note
    // that BO might be in some subloop of L, and that's completely okay.
    auto *L = LI.getLoopFor(P->getParent());
    assert(L && L->getHeader() == P->getParent());
    if (!L->contains(BO->getParent()))
      // NOTE: This bailout should be an assert instead.  However, asserting
      // the condition here exposes a case where LoopFusion is querying SCEV
      // with malformed loop information during the midst of the transform.
      // There doesn't appear to be an obvious fix, so for the moment bailout
      // until the caller issue can be fixed.  PR49566 tracks the bug.
      return FullSet;
  
    // TODO: Extend to other opcodes such as mul, and div
    switch (BO->getOpcode()) {
    default:
      return FullSet;
    case Instruction::AShr:
    case Instruction::LShr:
    case Instruction::Shl:
      break;
    };
  
    if (BO->getOperand(0) != P)
      // TODO: Handle the power function forms some day.
      return FullSet;
  
    unsigned TC = getSmallConstantMaxTripCount(L);
    if (!TC || TC >= BitWidth)
      return FullSet;
  
    auto KnownStart = computeKnownBits(Start, DL, 0, &AC, nullptr, &DT);
    auto KnownStep = computeKnownBits(Step, DL, 0, &AC, nullptr, &DT);
    assert(KnownStart.getBitWidth() == BitWidth &&
           KnownStep.getBitWidth() == BitWidth);
  
    // Compute total shift amount, being careful of overflow and bitwidths.
    auto MaxShiftAmt = KnownStep.getMaxValue();
    APInt TCAP(BitWidth, TC-1);
    bool Overflow = false;
    auto TotalShift = MaxShiftAmt.umul_ov(TCAP, Overflow);
    if (Overflow)
      return FullSet;
  
    switch (BO->getOpcode()) {
    default:
      llvm_unreachable("filtered out above");
    case Instruction::AShr: {
      // For each ashr, three cases:
      //   shift = 0 => unchanged value
      //   saturation => 0 or -1
      //   other => a value closer to zero (of the same sign)
      // Thus, the end value is closer to zero than the start.
      auto KnownEnd = KnownBits::ashr(KnownStart,
                                      KnownBits::makeConstant(TotalShift));
      if (KnownStart.isNonNegative())
        // Analogous to lshr (simply not yet canonicalized)
        return ConstantRange::getNonEmpty(KnownEnd.getMinValue(),
                                          KnownStart.getMaxValue() + 1);
      if (KnownStart.isNegative())
        // End >=u Start && End <=s Start
        return ConstantRange::getNonEmpty(KnownStart.getMinValue(),
                                          KnownEnd.getMaxValue() + 1);
      break;
    }
    case Instruction::LShr: {
      // For each lshr, three cases:
      //   shift = 0 => unchanged value
      //   saturation => 0
      //   other => a smaller positive number
      // Thus, the low end of the unsigned range is the last value produced.
      auto KnownEnd = KnownBits::lshr(KnownStart,
                                      KnownBits::makeConstant(TotalShift));
      return ConstantRange::getNonEmpty(KnownEnd.getMinValue(),
                                        KnownStart.getMaxValue() + 1);
    }
    case Instruction::Shl: {
      // Iff no bits are shifted out, value increases on every shift.
      auto KnownEnd = KnownBits::shl(KnownStart,
                                     KnownBits::makeConstant(TotalShift));
      if (TotalShift.ult(KnownStart.countMinLeadingZeros()))
        return ConstantRange(KnownStart.getMinValue(),
                             KnownEnd.getMaxValue() + 1);
      break;
    }
    };
    return FullSet;
  }
  
  /// Determine the range for a particular SCEV.  If SignHint is
  /// HINT_RANGE_UNSIGNED (resp. HINT_RANGE_SIGNED) then getRange prefers ranges
  /// with a "cleaner" unsigned (resp. signed) representation.
  const ConstantRange &
  ScalarEvolution::getRangeRef(const SCEV *S,
                               ScalarEvolution::RangeSignHint SignHint) {
    DenseMap<const SCEV *, ConstantRange> &Cache =
        SignHint == ScalarEvolution::HINT_RANGE_UNSIGNED ? UnsignedRanges
                                                         : SignedRanges;
    ConstantRange::PreferredRangeType RangeType =
        SignHint == ScalarEvolution::HINT_RANGE_UNSIGNED
            ? ConstantRange::Unsigned : ConstantRange::Signed;
  
    // See if we've computed this range already.
    DenseMap<const SCEV *, ConstantRange>::iterator I = Cache.find(S);
    if (I != Cache.end())
      return I->second;
  
    if (const SCEVConstant *C = dyn_cast<SCEVConstant>(S))
      return setRange(C, SignHint, ConstantRange(C->getAPInt()));
  
    unsigned BitWidth = getTypeSizeInBits(S->getType());
    ConstantRange ConservativeResult(BitWidth, /*isFullSet=*/true);
    using OBO = OverflowingBinaryOperator;
  
    // If the value has known zeros, the maximum value will have those known zeros
    // as well.
    uint32_t TZ = GetMinTrailingZeros(S);
    if (TZ != 0) {
      if (SignHint == ScalarEvolution::HINT_RANGE_UNSIGNED)
        ConservativeResult =
            ConstantRange(APInt::getMinValue(BitWidth),
                          APInt::getMaxValue(BitWidth).lshr(TZ).shl(TZ) + 1);
      else
        ConservativeResult = ConstantRange(
            APInt::getSignedMinValue(BitWidth),
            APInt::getSignedMaxValue(BitWidth).ashr(TZ).shl(TZ) + 1);
    }
  
    if (const SCEVAddExpr *Add = dyn_cast<SCEVAddExpr>(S)) {
      ConstantRange X = getRangeRef(Add->getOperand(0), SignHint);
      unsigned WrapType = OBO::AnyWrap;
      if (Add->hasNoSignedWrap())
        WrapType |= OBO::NoSignedWrap;
      if (Add->hasNoUnsignedWrap())
        WrapType |= OBO::NoUnsignedWrap;
      for (unsigned i = 1, e = Add->getNumOperands(); i != e; ++i)
        X = X.addWithNoWrap(getRangeRef(Add->getOperand(i), SignHint),
                            WrapType, RangeType);
      return setRange(Add, SignHint,
                      ConservativeResult.intersectWith(X, RangeType));
    }
  
    if (const SCEVMulExpr *Mul = dyn_cast<SCEVMulExpr>(S)) {
      ConstantRange X = getRangeRef(Mul->getOperand(0), SignHint);
      for (unsigned i = 1, e = Mul->getNumOperands(); i != e; ++i)
        X = X.multiply(getRangeRef(Mul->getOperand(i), SignHint));
      return setRange(Mul, SignHint,
                      ConservativeResult.intersectWith(X, RangeType));
    }
  
    if (const SCEVSMaxExpr *SMax = dyn_cast<SCEVSMaxExpr>(S)) {
      ConstantRange X = getRangeRef(SMax->getOperand(0), SignHint);
      for (unsigned i = 1, e = SMax->getNumOperands(); i != e; ++i)
        X = X.smax(getRangeRef(SMax->getOperand(i), SignHint));
      return setRange(SMax, SignHint,
                      ConservativeResult.intersectWith(X, RangeType));
    }
  
    if (const SCEVUMaxExpr *UMax = dyn_cast<SCEVUMaxExpr>(S)) {
      ConstantRange X = getRangeRef(UMax->getOperand(0), SignHint);
      for (unsigned i = 1, e = UMax->getNumOperands(); i != e; ++i)
        X = X.umax(getRangeRef(UMax->getOperand(i), SignHint));
      return setRange(UMax, SignHint,
                      ConservativeResult.intersectWith(X, RangeType));
    }
  
    if (const SCEVSMinExpr *SMin = dyn_cast<SCEVSMinExpr>(S)) {
      ConstantRange X = getRangeRef(SMin->getOperand(0), SignHint);
      for (unsigned i = 1, e = SMin->getNumOperands(); i != e; ++i)
        X = X.smin(getRangeRef(SMin->getOperand(i), SignHint));
      return setRange(SMin, SignHint,
                      ConservativeResult.intersectWith(X, RangeType));
    }
  
    if (const SCEVUMinExpr *UMin = dyn_cast<SCEVUMinExpr>(S)) {
      ConstantRange X = getRangeRef(UMin->getOperand(0), SignHint);
      for (unsigned i = 1, e = UMin->getNumOperands(); i != e; ++i)
        X = X.umin(getRangeRef(UMin->getOperand(i), SignHint));
      return setRange(UMin, SignHint,
                      ConservativeResult.intersectWith(X, RangeType));
    }
  
    if (const SCEVUDivExpr *UDiv = dyn_cast<SCEVUDivExpr>(S)) {
      ConstantRange X = getRangeRef(UDiv->getLHS(), SignHint);
      ConstantRange Y = getRangeRef(UDiv->getRHS(), SignHint);
      return setRange(UDiv, SignHint,
                      ConservativeResult.intersectWith(X.udiv(Y), RangeType));
    }
  
    if (const SCEVZeroExtendExpr *ZExt = dyn_cast<SCEVZeroExtendExpr>(S)) {
      ConstantRange X = getRangeRef(ZExt->getOperand(), SignHint);
      return setRange(ZExt, SignHint,
                      ConservativeResult.intersectWith(X.zeroExtend(BitWidth),
                                                       RangeType));
    }
  
    if (const SCEVSignExtendExpr *SExt = dyn_cast<SCEVSignExtendExpr>(S)) {
      ConstantRange X = getRangeRef(SExt->getOperand(), SignHint);
      return setRange(SExt, SignHint,
                      ConservativeResult.intersectWith(X.signExtend(BitWidth),
                                                       RangeType));
    }
  
    if (const SCEVPtrToIntExpr *PtrToInt = dyn_cast<SCEVPtrToIntExpr>(S)) {
      ConstantRange X = getRangeRef(PtrToInt->getOperand(), SignHint);
      return setRange(PtrToInt, SignHint, X);
    }
  
    if (const SCEVTruncateExpr *Trunc = dyn_cast<SCEVTruncateExpr>(S)) {
      ConstantRange X = getRangeRef(Trunc->getOperand(), SignHint);
      return setRange(Trunc, SignHint,
                      ConservativeResult.intersectWith(X.truncate(BitWidth),
                                                       RangeType));
    }
  
    if (const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(S)) {
      // If there's no unsigned wrap, the value will never be less than its
      // initial value.
      if (AddRec->hasNoUnsignedWrap()) {
        APInt UnsignedMinValue = getUnsignedRangeMin(AddRec->getStart());
        if (!UnsignedMinValue.isNullValue())
          ConservativeResult = ConservativeResult.intersectWith(
              ConstantRange(UnsignedMinValue, APInt(BitWidth, 0)), RangeType);
      }
  
      // If there's no signed wrap, and all the operands except initial value have
      // the same sign or zero, the value won't ever be:
      // 1: smaller than initial value if operands are non negative,
      // 2: bigger than initial value if operands are non positive.
      // For both cases, value can not cross signed min/max boundary.
      if (AddRec->hasNoSignedWrap()) {
        bool AllNonNeg = true;
        bool AllNonPos = true;
        for (unsigned i = 1, e = AddRec->getNumOperands(); i != e; ++i) {
          if (!isKnownNonNegative(AddRec->getOperand(i)))
            AllNonNeg = false;
          if (!isKnownNonPositive(AddRec->getOperand(i)))
            AllNonPos = false;
        }
        if (AllNonNeg)
          ConservativeResult = ConservativeResult.intersectWith(
              ConstantRange::getNonEmpty(getSignedRangeMin(AddRec->getStart()),
                                         APInt::getSignedMinValue(BitWidth)),
              RangeType);
        else if (AllNonPos)
          ConservativeResult = ConservativeResult.intersectWith(
              ConstantRange::getNonEmpty(
                  APInt::getSignedMinValue(BitWidth),
                  getSignedRangeMax(AddRec->getStart()) + 1),
              RangeType);
      }
  
      // TODO: non-affine addrec
      if (AddRec->isAffine()) {
        const SCEV *MaxBECount = getConstantMaxBackedgeTakenCount(AddRec->getLoop());
        if (!isa<SCEVCouldNotCompute>(MaxBECount) &&
            getTypeSizeInBits(MaxBECount->getType()) <= BitWidth) {
          auto RangeFromAffine = getRangeForAffineAR(
              AddRec->getStart(), AddRec->getStepRecurrence(*this), MaxBECount,
              BitWidth);
          ConservativeResult =
              ConservativeResult.intersectWith(RangeFromAffine, RangeType);
  
          auto RangeFromFactoring = getRangeViaFactoring(
              AddRec->getStart(), AddRec->getStepRecurrence(*this), MaxBECount,
              BitWidth);
          ConservativeResult =
              ConservativeResult.intersectWith(RangeFromFactoring, RangeType);
        }
  
        // Now try symbolic BE count and more powerful methods.
        if (UseExpensiveRangeSharpening) {
          const SCEV *SymbolicMaxBECount =
              getSymbolicMaxBackedgeTakenCount(AddRec->getLoop());
          if (!isa<SCEVCouldNotCompute>(SymbolicMaxBECount) &&
              getTypeSizeInBits(MaxBECount->getType()) <= BitWidth &&
              AddRec->hasNoSelfWrap()) {
            auto RangeFromAffineNew = getRangeForAffineNoSelfWrappingAR(
                AddRec, SymbolicMaxBECount, BitWidth, SignHint);
            ConservativeResult =
                ConservativeResult.intersectWith(RangeFromAffineNew, RangeType);
          }
        }
      }
  
      return setRange(AddRec, SignHint, std::move(ConservativeResult));
    }
  
    if (const SCEVUnknown *U = dyn_cast<SCEVUnknown>(S)) {
  
      // Check if the IR explicitly contains !range metadata.
      Optional<ConstantRange> MDRange = GetRangeFromMetadata(U->getValue());
      if (MDRange.hasValue())
        ConservativeResult = ConservativeResult.intersectWith(MDRange.getValue(),
                                                              RangeType);
  
      // Use facts about recurrences in the underlying IR.  Note that add
      // recurrences are AddRecExprs and thus don't hit this path.  This
      // primarily handles shift recurrences.
      auto CR = getRangeForUnknownRecurrence(U);
      ConservativeResult = ConservativeResult.intersectWith(CR);
  
      // See if ValueTracking can give us a useful range.
      const DataLayout &DL = getDataLayout();
      KnownBits Known = computeKnownBits(U->getValue(), DL, 0, &AC, nullptr, &DT);
      if (Known.getBitWidth() != BitWidth)
        Known = Known.zextOrTrunc(BitWidth);
  
      // ValueTracking may be able to compute a tighter result for the number of
      // sign bits than for the value of those sign bits.
      unsigned NS = ComputeNumSignBits(U->getValue(), DL, 0, &AC, nullptr, &DT);
      if (U->getType()->isPointerTy()) {
        // If the pointer size is larger than the index size type, this can cause
        // NS to be larger than BitWidth. So compensate for this.
        unsigned ptrSize = DL.getPointerTypeSizeInBits(U->getType());
        int ptrIdxDiff = ptrSize - BitWidth;
        if (ptrIdxDiff > 0 && ptrSize > BitWidth && NS > (unsigned)ptrIdxDiff)
          NS -= ptrIdxDiff;
      }
  
      if (NS > 1) {
        // If we know any of the sign bits, we know all of the sign bits.
        if (!Known.Zero.getHiBits(NS).isNullValue())
          Known.Zero.setHighBits(NS);
        if (!Known.One.getHiBits(NS).isNullValue())
          Known.One.setHighBits(NS);
      }
  
      if (Known.getMinValue() != Known.getMaxValue() + 1)
        ConservativeResult = ConservativeResult.intersectWith(
            ConstantRange(Known.getMinValue(), Known.getMaxValue() + 1),
            RangeType);
      if (NS > 1)
        ConservativeResult = ConservativeResult.intersectWith(
            ConstantRange(APInt::getSignedMinValue(BitWidth).ashr(NS - 1),
                          APInt::getSignedMaxValue(BitWidth).ashr(NS - 1) + 1),
            RangeType);
  
      // A range of Phi is a subset of union of all ranges of its input.
      if (const PHINode *Phi = dyn_cast<PHINode>(U->getValue())) {
        // Make sure that we do not run over cycled Phis.
        if (PendingPhiRanges.insert(Phi).second) {
          ConstantRange RangeFromOps(BitWidth, /*isFullSet=*/false);
          for (auto &Op : Phi->operands()) {
            auto OpRange = getRangeRef(getSCEV(Op), SignHint);
            RangeFromOps = RangeFromOps.unionWith(OpRange);
            // No point to continue if we already have a full set.
            if (RangeFromOps.isFullSet())
              break;
          }
          ConservativeResult =
              ConservativeResult.intersectWith(RangeFromOps, RangeType);
          bool Erased = PendingPhiRanges.erase(Phi);
          assert(Erased && "Failed to erase Phi properly?");
          (void) Erased;
        }
      }
  
      return setRange(U, SignHint, std::move(ConservativeResult));
    }
  
    return setRange(S, SignHint, std::move(ConservativeResult));
  }
  
  // Given a StartRange, Step and MaxBECount for an expression compute a range of
  // values that the expression can take. Initially, the expression has a value
  // from StartRange and then is changed by Step up to MaxBECount times. Signed
  // argument defines if we treat Step as signed or unsigned.
  static ConstantRange getRangeForAffineARHelper(APInt Step,
                                                 const ConstantRange &StartRange,
                                                 const APInt &MaxBECount,
                                                 unsigned BitWidth, bool Signed) {
    // If either Step or MaxBECount is 0, then the expression won't change, and we
    // just need to return the initial range.
    if (Step == 0 || MaxBECount == 0)
      return StartRange;
  
    // If we don't know anything about the initial value (i.e. StartRange is
    // FullRange), then we don't know anything about the final range either.
    // Return FullRange.
    if (StartRange.isFullSet())
      return ConstantRange::getFull(BitWidth);
  
    // If Step is signed and negative, then we use its absolute value, but we also
    // note that we're moving in the opposite direction.
    bool Descending = Signed && Step.isNegative();
  
    if (Signed)
      // This is correct even for INT_SMIN. Let's look at i8 to illustrate this:
      // abs(INT_SMIN) = abs(-128) = abs(0x80) = -0x80 = 0x80 = 128.
      // This equations hold true due to the well-defined wrap-around behavior of
      // APInt.
      Step = Step.abs();
  
    // Check if Offset is more than full span of BitWidth. If it is, the
    // expression is guaranteed to overflow.
    if (APInt::getMaxValue(StartRange.getBitWidth()).udiv(Step).ult(MaxBECount))
      return ConstantRange::getFull(BitWidth);
  
    // Offset is by how much the expression can change. Checks above guarantee no
    // overflow here.
    APInt Offset = Step * MaxBECount;
  
    // Minimum value of the final range will match the minimal value of StartRange
    // if the expression is increasing and will be decreased by Offset otherwise.
    // Maximum value of the final range will match the maximal value of StartRange
    // if the expression is decreasing and will be increased by Offset otherwise.
    APInt StartLower = StartRange.getLower();
    APInt StartUpper = StartRange.getUpper() - 1;
    APInt MovedBoundary = Descending ? (StartLower - std::move(Offset))
                                     : (StartUpper + std::move(Offset));
  
    // It's possible that the new minimum/maximum value will fall into the initial
    // range (due to wrap around). This means that the expression can take any
    // value in this bitwidth, and we have to return full range.
    if (StartRange.contains(MovedBoundary))
      return ConstantRange::getFull(BitWidth);
  
    APInt NewLower =
        Descending ? std::move(MovedBoundary) : std::move(StartLower);
    APInt NewUpper =
        Descending ? std::move(StartUpper) : std::move(MovedBoundary);
    NewUpper += 1;
  
    // No overflow detected, return [StartLower, StartUpper + Offset + 1) range.
    return ConstantRange::getNonEmpty(std::move(NewLower), std::move(NewUpper));
  }
  
  ConstantRange ScalarEvolution::getRangeForAffineAR(const SCEV *Start,
                                                     const SCEV *Step,
                                                     const SCEV *MaxBECount,
                                                     unsigned BitWidth) {
    assert(!isa<SCEVCouldNotCompute>(MaxBECount) &&
           getTypeSizeInBits(MaxBECount->getType()) <= BitWidth &&
           "Precondition!");
  
    MaxBECount = getNoopOrZeroExtend(MaxBECount, Start->getType());
    APInt MaxBECountValue = getUnsignedRangeMax(MaxBECount);
  
    // First, consider step signed.
    ConstantRange StartSRange = getSignedRange(Start);
    ConstantRange StepSRange = getSignedRange(Step);
  
    // If Step can be both positive and negative, we need to find ranges for the
    // maximum absolute step values in both directions and union them.
    ConstantRange SR =
        getRangeForAffineARHelper(StepSRange.getSignedMin(), StartSRange,
                                  MaxBECountValue, BitWidth, /* Signed = */ true);
    SR = SR.unionWith(getRangeForAffineARHelper(StepSRange.getSignedMax(),
                                                StartSRange, MaxBECountValue,
                                                BitWidth, /* Signed = */ true));
  
    // Next, consider step unsigned.
    ConstantRange UR = getRangeForAffineARHelper(
        getUnsignedRangeMax(Step), getUnsignedRange(Start),
        MaxBECountValue, BitWidth, /* Signed = */ false);
  
    // Finally, intersect signed and unsigned ranges.
    return SR.intersectWith(UR, ConstantRange::Smallest);
  }
  
  ConstantRange ScalarEvolution::getRangeForAffineNoSelfWrappingAR(
      const SCEVAddRecExpr *AddRec, const SCEV *MaxBECount, unsigned BitWidth,
      ScalarEvolution::RangeSignHint SignHint) {
    assert(AddRec->isAffine() && "Non-affine AddRecs are not suppored!\n");
    assert(AddRec->hasNoSelfWrap() &&
           "This only works for non-self-wrapping AddRecs!");
    const bool IsSigned = SignHint == HINT_RANGE_SIGNED;
    const SCEV *Step = AddRec->getStepRecurrence(*this);
    // Only deal with constant step to save compile time.
    if (!isa<SCEVConstant>(Step))
      return ConstantRange::getFull(BitWidth);
    // Let's make sure that we can prove that we do not self-wrap during
    // MaxBECount iterations. We need this because MaxBECount is a maximum
    // iteration count estimate, and we might infer nw from some exit for which we
    // do not know max exit count (or any other side reasoning).
    // TODO: Turn into assert at some point.
    if (getTypeSizeInBits(MaxBECount->getType()) >
        getTypeSizeInBits(AddRec->getType()))
      return ConstantRange::getFull(BitWidth);
    MaxBECount = getNoopOrZeroExtend(MaxBECount, AddRec->getType());
    const SCEV *RangeWidth = getMinusOne(AddRec->getType());
    const SCEV *StepAbs = getUMinExpr(Step, getNegativeSCEV(Step));
    const SCEV *MaxItersWithoutWrap = getUDivExpr(RangeWidth, StepAbs);
    if (!isKnownPredicateViaConstantRanges(ICmpInst::ICMP_ULE, MaxBECount,
                                           MaxItersWithoutWrap))
      return ConstantRange::getFull(BitWidth);
  
    ICmpInst::Predicate LEPred =
        IsSigned ? ICmpInst::ICMP_SLE : ICmpInst::ICMP_ULE;
    ICmpInst::Predicate GEPred =
        IsSigned ? ICmpInst::ICMP_SGE : ICmpInst::ICMP_UGE;
    const SCEV *End = AddRec->evaluateAtIteration(MaxBECount, *this);
  
    // We know that there is no self-wrap. Let's take Start and End values and
    // look at all intermediate values V1, V2, ..., Vn that IndVar takes during
    // the iteration. They either lie inside the range [Min(Start, End),
    // Max(Start, End)] or outside it:
    //
    // Case 1:   RangeMin    ...    Start V1 ... VN End ...           RangeMax;
    // Case 2:   RangeMin Vk ... V1 Start    ...    End Vn ... Vk + 1 RangeMax;
    //
    // No self wrap flag guarantees that the intermediate values cannot be BOTH
    // outside and inside the range [Min(Start, End), Max(Start, End)]. Using that
    // knowledge, let's try to prove that we are dealing with Case 1. It is so if
    // Start <= End and step is positive, or Start >= End and step is negative.
    const SCEV *Start = AddRec->getStart();
    ConstantRange StartRange = getRangeRef(Start, SignHint);
    ConstantRange EndRange = getRangeRef(End, SignHint);
    ConstantRange RangeBetween = StartRange.unionWith(EndRange);
    // If they already cover full iteration space, we will know nothing useful
    // even if we prove what we want to prove.
    if (RangeBetween.isFullSet())
      return RangeBetween;
    // Only deal with ranges that do not wrap (i.e. RangeMin < RangeMax).
    bool IsWrappedSet = IsSigned ? RangeBetween.isSignWrappedSet()
                                 : RangeBetween.isWrappedSet();
    if (IsWrappedSet)
      return ConstantRange::getFull(BitWidth);
  
    if (isKnownPositive(Step) &&
        isKnownPredicateViaConstantRanges(LEPred, Start, End))
      return RangeBetween;
    else if (isKnownNegative(Step) &&
             isKnownPredicateViaConstantRanges(GEPred, Start, End))
      return RangeBetween;
    return ConstantRange::getFull(BitWidth);
  }
  
  ConstantRange ScalarEvolution::getRangeViaFactoring(const SCEV *Start,
                                                      const SCEV *Step,
                                                      const SCEV *MaxBECount,
                                                      unsigned BitWidth) {
    //    RangeOf({C?A:B,+,C?P:Q}) == RangeOf(C?{A,+,P}:{B,+,Q})
    // == RangeOf({A,+,P}) union RangeOf({B,+,Q})
  
    struct SelectPattern {
      Value *Condition = nullptr;
      APInt TrueValue;
      APInt FalseValue;
  
      explicit SelectPattern(ScalarEvolution &SE, unsigned BitWidth,
                             const SCEV *S) {
        Optional<unsigned> CastOp;
        APInt Offset(BitWidth, 0);
  
        assert(SE.getTypeSizeInBits(S->getType()) == BitWidth &&
               "Should be!");
  
        // Peel off a constant offset:
        if (auto *SA = dyn_cast<SCEVAddExpr>(S)) {
          // In the future we could consider being smarter here and handle
          // {Start+Step,+,Step} too.
          if (SA->getNumOperands() != 2 || !isa<SCEVConstant>(SA->getOperand(0)))
            return;
  
          Offset = cast<SCEVConstant>(SA->getOperand(0))->getAPInt();
          S = SA->getOperand(1);
        }
  
        // Peel off a cast operation
        if (auto *SCast = dyn_cast<SCEVIntegralCastExpr>(S)) {
          CastOp = SCast->getSCEVType();
          S = SCast->getOperand();
        }
  
        using namespace llvm::PatternMatch;
  
        auto *SU = dyn_cast<SCEVUnknown>(S);
        const APInt *TrueVal, *FalseVal;
        if (!SU ||
            !match(SU->getValue(), m_Select(m_Value(Condition), m_APInt(TrueVal),
                                            m_APInt(FalseVal)))) {
          Condition = nullptr;
          return;
        }
  
        TrueValue = *TrueVal;
        FalseValue = *FalseVal;
  
        // Re-apply the cast we peeled off earlier
        if (CastOp.hasValue())
          switch (*CastOp) {
          default:
            llvm_unreachable("Unknown SCEV cast type!");
  
          case scTruncate:
            TrueValue = TrueValue.trunc(BitWidth);
            FalseValue = FalseValue.trunc(BitWidth);
            break;
          case scZeroExtend:
            TrueValue = TrueValue.zext(BitWidth);
            FalseValue = FalseValue.zext(BitWidth);
            break;
          case scSignExtend:
            TrueValue = TrueValue.sext(BitWidth);
            FalseValue = FalseValue.sext(BitWidth);
            break;
          }
  
        // Re-apply the constant offset we peeled off earlier
        TrueValue += Offset;
        FalseValue += Offset;
      }
  
      bool isRecognized() { return Condition != nullptr; }
    };
  
    SelectPattern StartPattern(*this, BitWidth, Start);
    if (!StartPattern.isRecognized())
      return ConstantRange::getFull(BitWidth);
  
    SelectPattern StepPattern(*this, BitWidth, Step);
    if (!StepPattern.isRecognized())
      return ConstantRange::getFull(BitWidth);
  
    if (StartPattern.Condition != StepPattern.Condition) {
      // We don't handle this case today; but we could, by considering four
      // possibilities below instead of two. I'm not sure if there are cases where
      // that will help over what getRange already does, though.
      return ConstantRange::getFull(BitWidth);
    }
  
    // NB! Calling ScalarEvolution::getConstant is fine, but we should not try to
    // construct arbitrary general SCEV expressions here.  This function is called
    // from deep in the call stack, and calling getSCEV (on a sext instruction,
    // say) can end up caching a suboptimal value.
  
    // FIXME: without the explicit `this` receiver below, MSVC errors out with
    // C2352 and C2512 (otherwise it isn't needed).
  
    const SCEV *TrueStart = this->getConstant(StartPattern.TrueValue);
    const SCEV *TrueStep = this->getConstant(StepPattern.TrueValue);
    const SCEV *FalseStart = this->getConstant(StartPattern.FalseValue);
    const SCEV *FalseStep = this->getConstant(StepPattern.FalseValue);
  
    ConstantRange TrueRange =
        this->getRangeForAffineAR(TrueStart, TrueStep, MaxBECount, BitWidth);
    ConstantRange FalseRange =
        this->getRangeForAffineAR(FalseStart, FalseStep, MaxBECount, BitWidth);
  
    return TrueRange.unionWith(FalseRange);
  }
  
  SCEV::NoWrapFlags ScalarEvolution::getNoWrapFlagsFromUB(const Value *V) {
    if (isa<ConstantExpr>(V)) return SCEV::FlagAnyWrap;
    const BinaryOperator *BinOp = cast<BinaryOperator>(V);
  
    // Return early if there are no flags to propagate to the SCEV.
    SCEV::NoWrapFlags Flags = SCEV::FlagAnyWrap;
    if (BinOp->hasNoUnsignedWrap())
      Flags = ScalarEvolution::setFlags(Flags, SCEV::FlagNUW);
    if (BinOp->hasNoSignedWrap())
      Flags = ScalarEvolution::setFlags(Flags, SCEV::FlagNSW);
    if (Flags == SCEV::FlagAnyWrap)
      return SCEV::FlagAnyWrap;
  
    return isSCEVExprNeverPoison(BinOp) ? Flags : SCEV::FlagAnyWrap;
  }
  
  bool ScalarEvolution::isSCEVExprNeverPoison(const Instruction *I) {
    // Here we check that I is in the header of the innermost loop containing I,
    // since we only deal with instructions in the loop header. The actual loop we
    // need to check later will come from an add recurrence, but getting that
    // requires computing the SCEV of the operands, which can be expensive. This
    // check we can do cheaply to rule out some cases early.
    Loop *InnermostContainingLoop = LI.getLoopFor(I->getParent());
    if (InnermostContainingLoop == nullptr ||
        InnermostContainingLoop->getHeader() != I->getParent())
      return false;
  
    // Only proceed if we can prove that I does not yield poison.
    if (!programUndefinedIfPoison(I))
      return false;
  
    // At this point we know that if I is executed, then it does not wrap
    // according to at least one of NSW or NUW. If I is not executed, then we do
    // not know if the calculation that I represents would wrap. Multiple
    // instructions can map to the same SCEV. If we apply NSW or NUW from I to
    // the SCEV, we must guarantee no wrapping for that SCEV also when it is
    // derived from other instructions that map to the same SCEV. We cannot make
    // that guarantee for cases where I is not executed. So we need to find the
    // loop that I is considered in relation to and prove that I is executed for
    // every iteration of that loop. That implies that the value that I
    // calculates does not wrap anywhere in the loop, so then we can apply the
    // flags to the SCEV.
    //
    // We check isLoopInvariant to disambiguate in case we are adding recurrences
    // from different loops, so that we know which loop to prove that I is
    // executed in.
    for (unsigned OpIndex = 0; OpIndex < I->getNumOperands(); ++OpIndex) {
      // I could be an extractvalue from a call to an overflow intrinsic.
      // TODO: We can do better here in some cases.
      if (!isSCEVable(I->getOperand(OpIndex)->getType()))
        return false;
      const SCEV *Op = getSCEV(I->getOperand(OpIndex));
      if (auto *AddRec = dyn_cast<SCEVAddRecExpr>(Op)) {
        bool AllOtherOpsLoopInvariant = true;
        for (unsigned OtherOpIndex = 0; OtherOpIndex < I->getNumOperands();
             ++OtherOpIndex) {
          if (OtherOpIndex != OpIndex) {
            const SCEV *OtherOp = getSCEV(I->getOperand(OtherOpIndex));
            if (!isLoopInvariant(OtherOp, AddRec->getLoop())) {
              AllOtherOpsLoopInvariant = false;
              break;
            }
          }
        }
        if (AllOtherOpsLoopInvariant &&
            isGuaranteedToExecuteForEveryIteration(I, AddRec->getLoop()))
          return true;
      }
    }
    return false;
  }
  
  bool ScalarEvolution::isAddRecNeverPoison(const Instruction *I, const Loop *L) {
    // If we know that \c I can never be poison period, then that's enough.
    if (isSCEVExprNeverPoison(I))
      return true;
  
    // For an add recurrence specifically, we assume that infinite loops without
    // side effects are undefined behavior, and then reason as follows:
    //
    // If the add recurrence is poison in any iteration, it is poison on all
    // future iterations (since incrementing poison yields poison). If the result
    // of the add recurrence is fed into the loop latch condition and the loop
    // does not contain any throws or exiting blocks other than the latch, we now
    // have the ability to "choose" whether the backedge is taken or not (by
    // choosing a sufficiently evil value for the poison feeding into the branch)
    // for every iteration including and after the one in which \p I first became
    // poison.  There are two possibilities (let's call the iteration in which \p
    // I first became poison as K):
    //
    //  1. In the set of iterations including and after K, the loop body executes
    //     no side effects.  In this case executing the backege an infinte number
    //     of times will yield undefined behavior.
    //
    //  2. In the set of iterations including and after K, the loop body executes
    //     at least one side effect.  In this case, that specific instance of side
    //     effect is control dependent on poison, which also yields undefined
    //     behavior.
  
    auto *ExitingBB = L->getExitingBlock();
    auto *LatchBB = L->getLoopLatch();
    if (!ExitingBB || !LatchBB || ExitingBB != LatchBB)
      return false;
  
    SmallPtrSet<const Instruction *, 16> Pushed;
    SmallVector<const Instruction *, 8> PoisonStack;
  
    // We start by assuming \c I, the post-inc add recurrence, is poison.  Only
    // things that are known to be poison under that assumption go on the
    // PoisonStack.
    Pushed.insert(I);
    PoisonStack.push_back(I);
  
    bool LatchControlDependentOnPoison = false;
    while (!PoisonStack.empty() && !LatchControlDependentOnPoison) {
      const Instruction *Poison = PoisonStack.pop_back_val();
  
      for (auto *PoisonUser : Poison->users()) {
        if (propagatesPoison(cast<Operator>(PoisonUser))) {
          if (Pushed.insert(cast<Instruction>(PoisonUser)).second)
            PoisonStack.push_back(cast<Instruction>(PoisonUser));
        } else if (auto *BI = dyn_cast<BranchInst>(PoisonUser)) {
          assert(BI->isConditional() && "Only possibility!");
          if (BI->getParent() == LatchBB) {
            LatchControlDependentOnPoison = true;
            break;
          }
        }
      }
    }
  
    return LatchControlDependentOnPoison && loopHasNoAbnormalExits(L);
  }
  
  ScalarEvolution::LoopProperties
  ScalarEvolution::getLoopProperties(const Loop *L) {
    using LoopProperties = ScalarEvolution::LoopProperties;
  
    auto Itr = LoopPropertiesCache.find(L);
    if (Itr == LoopPropertiesCache.end()) {
      auto HasSideEffects = [](Instruction *I) {
        if (auto *SI = dyn_cast<StoreInst>(I))
          return !SI->isSimple();
  
        return I->mayHaveSideEffects();
      };
  
      LoopProperties LP = {/* HasNoAbnormalExits */ true,
                           /*HasNoSideEffects*/ true};
  
      for (auto *BB : L->getBlocks())
        for (auto &I : *BB) {
          if (!isGuaranteedToTransferExecutionToSuccessor(&I))
            LP.HasNoAbnormalExits = false;
          if (HasSideEffects(&I))
            LP.HasNoSideEffects = false;
          if (!LP.HasNoAbnormalExits && !LP.HasNoSideEffects)
            break; // We're already as pessimistic as we can get.
        }
  
      auto InsertPair = LoopPropertiesCache.insert({L, LP});
      assert(InsertPair.second && "We just checked!");
      Itr = InsertPair.first;
    }
  
    return Itr->second;
  }
  
  bool ScalarEvolution::loopIsFiniteByAssumption(const Loop *L) {
    // A mustprogress loop without side effects must be finite.
    // TODO: The check used here is very conservative.  It's only *specific*
    // side effects which are well defined in infinite loops.
    return isMustProgress(L) && loopHasNoSideEffects(L);
  }
  
  const SCEV *ScalarEvolution::createSCEV(Value *V) {
    if (!isSCEVable(V->getType()))
      return getUnknown(V);
  
    if (Instruction *I = dyn_cast<Instruction>(V)) {
      // Don't attempt to analyze instructions in blocks that aren't
      // reachable. Such instructions don't matter, and they aren't required
      // to obey basic rules for definitions dominating uses which this
      // analysis depends on.
      if (!DT.isReachableFromEntry(I->getParent()))
        return getUnknown(UndefValue::get(V->getType()));
    } else if (ConstantInt *CI = dyn_cast<ConstantInt>(V))
      return getConstant(CI);
    else if (GlobalAlias *GA = dyn_cast<GlobalAlias>(V))
      return GA->isInterposable() ? getUnknown(V) : getSCEV(GA->getAliasee());
    else if (!isa<ConstantExpr>(V))
      return getUnknown(V);
  
    Operator *U = cast<Operator>(V);
    if (auto BO = MatchBinaryOp(U, DT)) {
      switch (BO->Opcode) {
      case Instruction::Add: {
        // The simple thing to do would be to just call getSCEV on both operands
        // and call getAddExpr with the result. However if we're looking at a
        // bunch of things all added together, this can be quite inefficient,
        // because it leads to N-1 getAddExpr calls for N ultimate operands.
        // Instead, gather up all the operands and make a single getAddExpr call.
        // LLVM IR canonical form means we need only traverse the left operands.
        SmallVector<const SCEV *, 4> AddOps;
        do {
          if (BO->Op) {
            if (auto *OpSCEV = getExistingSCEV(BO->Op)) {
              AddOps.push_back(OpSCEV);
              break;
            }
  
            // If a NUW or NSW flag can be applied to the SCEV for this
            // addition, then compute the SCEV for this addition by itself
            // with a separate call to getAddExpr. We need to do that
            // instead of pushing the operands of the addition onto AddOps,
            // since the flags are only known to apply to this particular
            // addition - they may not apply to other additions that can be
            // formed with operands from AddOps.
            const SCEV *RHS = getSCEV(BO->RHS);
            SCEV::NoWrapFlags Flags = getNoWrapFlagsFromUB(BO->Op);
            if (Flags != SCEV::FlagAnyWrap) {
              const SCEV *LHS = getSCEV(BO->LHS);
              if (BO->Opcode == Instruction::Sub)
                AddOps.push_back(getMinusSCEV(LHS, RHS, Flags));
              else
                AddOps.push_back(getAddExpr(LHS, RHS, Flags));
              break;
            }
          }
  
          if (BO->Opcode == Instruction::Sub)
            AddOps.push_back(getNegativeSCEV(getSCEV(BO->RHS)));
          else
            AddOps.push_back(getSCEV(BO->RHS));
  
          auto NewBO = MatchBinaryOp(BO->LHS, DT);
          if (!NewBO || (NewBO->Opcode != Instruction::Add &&
                         NewBO->Opcode != Instruction::Sub)) {
            AddOps.push_back(getSCEV(BO->LHS));
            break;
          }
          BO = NewBO;
        } while (true);
  
        return getAddExpr(AddOps);
      }
  
      case Instruction::Mul: {
        SmallVector<const SCEV *, 4> MulOps;
        do {
          if (BO->Op) {
            if (auto *OpSCEV = getExistingSCEV(BO->Op)) {
              MulOps.push_back(OpSCEV);
              break;
            }
  
            SCEV::NoWrapFlags Flags = getNoWrapFlagsFromUB(BO->Op);
            if (Flags != SCEV::FlagAnyWrap) {
              MulOps.push_back(
                  getMulExpr(getSCEV(BO->LHS), getSCEV(BO->RHS), Flags));
              break;
            }
          }
  
          MulOps.push_back(getSCEV(BO->RHS));
          auto NewBO = MatchBinaryOp(BO->LHS, DT);
          if (!NewBO || NewBO->Opcode != Instruction::Mul) {
            MulOps.push_back(getSCEV(BO->LHS));
            break;
          }
          BO = NewBO;
        } while (true);
  
        return getMulExpr(MulOps);
      }
      case Instruction::UDiv:
        return getUDivExpr(getSCEV(BO->LHS), getSCEV(BO->RHS));
      case Instruction::URem:
        return getURemExpr(getSCEV(BO->LHS), getSCEV(BO->RHS));
      case Instruction::Sub: {
        SCEV::NoWrapFlags Flags = SCEV::FlagAnyWrap;
        if (BO->Op)
          Flags = getNoWrapFlagsFromUB(BO->Op);
        return getMinusSCEV(getSCEV(BO->LHS), getSCEV(BO->RHS), Flags);
      }
      case Instruction::And:
        // For an expression like x&255 that merely masks off the high bits,
        // use zext(trunc(x)) as the SCEV expression.
        if (ConstantInt *CI = dyn_cast<ConstantInt>(BO->RHS)) {
          if (CI->isZero())
            return getSCEV(BO->RHS);
          if (CI->isMinusOne())
            return getSCEV(BO->LHS);
          const APInt &A = CI->getValue();
  
          // Instcombine's ShrinkDemandedConstant may strip bits out of
          // constants, obscuring what would otherwise be a low-bits mask.
          // Use computeKnownBits to compute what ShrinkDemandedConstant
          // knew about to reconstruct a low-bits mask value.
          unsigned LZ = A.countLeadingZeros();
          unsigned TZ = A.countTrailingZeros();
          unsigned BitWidth = A.getBitWidth();
          KnownBits Known(BitWidth);
          computeKnownBits(BO->LHS, Known, getDataLayout(),
                           0, &AC, nullptr, &DT);
  
          APInt EffectiveMask =
              APInt::getLowBitsSet(BitWidth, BitWidth - LZ - TZ).shl(TZ);
          if ((LZ != 0 || TZ != 0) && !((~A & ~Known.Zero) & EffectiveMask)) {
            const SCEV *MulCount = getConstant(APInt::getOneBitSet(BitWidth, TZ));
            const SCEV *LHS = getSCEV(BO->LHS);
            const SCEV *ShiftedLHS = nullptr;
            if (auto *LHSMul = dyn_cast<SCEVMulExpr>(LHS)) {
              if (auto *OpC = dyn_cast<SCEVConstant>(LHSMul->getOperand(0))) {
                // For an expression like (x * 8) & 8, simplify the multiply.
                unsigned MulZeros = OpC->getAPInt().countTrailingZeros();
                unsigned GCD = std::min(MulZeros, TZ);
                APInt DivAmt = APInt::getOneBitSet(BitWidth, TZ - GCD);
                SmallVector<const SCEV*, 4> MulOps;
                MulOps.push_back(getConstant(OpC->getAPInt().lshr(GCD)));
                MulOps.append(LHSMul->op_begin() + 1, LHSMul->op_end());
                auto *NewMul = getMulExpr(MulOps, LHSMul->getNoWrapFlags());
                ShiftedLHS = getUDivExpr(NewMul, getConstant(DivAmt));
              }
            }
            if (!ShiftedLHS)
              ShiftedLHS = getUDivExpr(LHS, MulCount);
            return getMulExpr(
                getZeroExtendExpr(
                    getTruncateExpr(ShiftedLHS,
                        IntegerType::get(getContext(), BitWidth - LZ - TZ)),
                    BO->LHS->getType()),
                MulCount);
          }
        }
        break;
  
      case Instruction::Or:
        // If the RHS of the Or is a constant, we may have something like:
        // X*4+1 which got turned into X*4|1.  Handle this as an Add so loop
        // optimizations will transparently handle this case.
        //
        // In order for this transformation to be safe, the LHS must be of the
        // form X*(2^n) and the Or constant must be less than 2^n.
        if (ConstantInt *CI = dyn_cast<ConstantInt>(BO->RHS)) {
          const SCEV *LHS = getSCEV(BO->LHS);
          const APInt &CIVal = CI->getValue();
          if (GetMinTrailingZeros(LHS) >=
              (CIVal.getBitWidth() - CIVal.countLeadingZeros())) {
            // Build a plain add SCEV.
            return getAddExpr(LHS, getSCEV(CI),
                              (SCEV::NoWrapFlags)(SCEV::FlagNUW | SCEV::FlagNSW));
          }
        }
        break;
  
      case Instruction::Xor:
        if (ConstantInt *CI = dyn_cast<ConstantInt>(BO->RHS)) {
          // If the RHS of xor is -1, then this is a not operation.
          if (CI->isMinusOne())
            return getNotSCEV(getSCEV(BO->LHS));
  
          // Model xor(and(x, C), C) as and(~x, C), if C is a low-bits mask.
          // This is a variant of the check for xor with -1, and it handles
          // the case where instcombine has trimmed non-demanded bits out
          // of an xor with -1.
          if (auto *LBO = dyn_cast<BinaryOperator>(BO->LHS))
            if (ConstantInt *LCI = dyn_cast<ConstantInt>(LBO->getOperand(1)))
              if (LBO->getOpcode() == Instruction::And &&
                  LCI->getValue() == CI->getValue())
                if (const SCEVZeroExtendExpr *Z =
                        dyn_cast<SCEVZeroExtendExpr>(getSCEV(BO->LHS))) {
                  Type *UTy = BO->LHS->getType();
                  const SCEV *Z0 = Z->getOperand();
                  Type *Z0Ty = Z0->getType();
                  unsigned Z0TySize = getTypeSizeInBits(Z0Ty);
  
                  // If C is a low-bits mask, the zero extend is serving to
                  // mask off the high bits. Complement the operand and
                  // re-apply the zext.
                  if (CI->getValue().isMask(Z0TySize))
                    return getZeroExtendExpr(getNotSCEV(Z0), UTy);
  
                  // If C is a single bit, it may be in the sign-bit position
                  // before the zero-extend. In this case, represent the xor
                  // using an add, which is equivalent, and re-apply the zext.
                  APInt Trunc = CI->getValue().trunc(Z0TySize);
                  if (Trunc.zext(getTypeSizeInBits(UTy)) == CI->getValue() &&
                      Trunc.isSignMask())
                    return getZeroExtendExpr(getAddExpr(Z0, getConstant(Trunc)),
                                             UTy);
                }
        }
        break;
  
      case Instruction::Shl:
        // Turn shift left of a constant amount into a multiply.
        if (ConstantInt *SA = dyn_cast<ConstantInt>(BO->RHS)) {
          uint32_t BitWidth = cast<IntegerType>(SA->getType())->getBitWidth();
  
          // If the shift count is not less than the bitwidth, the result of
          // the shift is undefined. Don't try to analyze it, because the
          // resolution chosen here may differ from the resolution chosen in
          // other parts of the compiler.
          if (SA->getValue().uge(BitWidth))
            break;
  
          // We can safely preserve the nuw flag in all cases. It's also safe to
          // turn a nuw nsw shl into a nuw nsw mul. However, nsw in isolation
          // requires special handling. It can be preserved as long as we're not
          // left shifting by bitwidth - 1.
          auto Flags = SCEV::FlagAnyWrap;
          if (BO->Op) {
            auto MulFlags = getNoWrapFlagsFromUB(BO->Op);
            if ((MulFlags & SCEV::FlagNSW) &&
                ((MulFlags & SCEV::FlagNUW) || SA->getValue().ult(BitWidth - 1)))
              Flags = (SCEV::NoWrapFlags)(Flags | SCEV::FlagNSW);
            if (MulFlags & SCEV::FlagNUW)
              Flags = (SCEV::NoWrapFlags)(Flags | SCEV::FlagNUW);
          }
  
          Constant *X = ConstantInt::get(
              getContext(), APInt::getOneBitSet(BitWidth, SA->getZExtValue()));
          return getMulExpr(getSCEV(BO->LHS), getSCEV(X), Flags);
        }
        break;
  
      case Instruction::AShr: {
        // AShr X, C, where C is a constant.
        ConstantInt *CI = dyn_cast<ConstantInt>(BO->RHS);
        if (!CI)
          break;
  
        Type *OuterTy = BO->LHS->getType();
        uint64_t BitWidth = getTypeSizeInBits(OuterTy);
        // If the shift count is not less than the bitwidth, the result of
        // the shift is undefined. Don't try to analyze it, because the
        // resolution chosen here may differ from the resolution chosen in
        // other parts of the compiler.
        if (CI->getValue().uge(BitWidth))
          break;
  
        if (CI->isZero())
          return getSCEV(BO->LHS); // shift by zero --> noop
  
        uint64_t AShrAmt = CI->getZExtValue();
        Type *TruncTy = IntegerType::get(getContext(), BitWidth - AShrAmt);
  
        Operator *L = dyn_cast<Operator>(BO->LHS);
        if (L && L->getOpcode() == Instruction::Shl) {
          // X = Shl A, n
          // Y = AShr X, m
          // Both n and m are constant.
  
          const SCEV *ShlOp0SCEV = getSCEV(L->getOperand(0));
          if (L->getOperand(1) == BO->RHS)
            // For a two-shift sext-inreg, i.e. n = m,
            // use sext(trunc(x)) as the SCEV expression.
            return getSignExtendExpr(
                getTruncateExpr(ShlOp0SCEV, TruncTy), OuterTy);
  
          ConstantInt *ShlAmtCI = dyn_cast<ConstantInt>(L->getOperand(1));
          if (ShlAmtCI && ShlAmtCI->getValue().ult(BitWidth)) {
            uint64_t ShlAmt = ShlAmtCI->getZExtValue();
            if (ShlAmt > AShrAmt) {
              // When n > m, use sext(mul(trunc(x), 2^(n-m)))) as the SCEV
              // expression. We already checked that ShlAmt < BitWidth, so
              // the multiplier, 1 << (ShlAmt - AShrAmt), fits into TruncTy as
              // ShlAmt - AShrAmt < Amt.
              APInt Mul = APInt::getOneBitSet(BitWidth - AShrAmt,
                                              ShlAmt - AShrAmt);
              return getSignExtendExpr(
                  getMulExpr(getTruncateExpr(ShlOp0SCEV, TruncTy),
                  getConstant(Mul)), OuterTy);
            }
          }
        }
        break;
      }
      }
    }
  
    switch (U->getOpcode()) {
    case Instruction::Trunc:
      return getTruncateExpr(getSCEV(U->getOperand(0)), U->getType());
  
    case Instruction::ZExt:
      return getZeroExtendExpr(getSCEV(U->getOperand(0)), U->getType());
  
    case Instruction::SExt:
      if (auto BO = MatchBinaryOp(U->getOperand(0), DT)) {
        // The NSW flag of a subtract does not always survive the conversion to
        // A + (-1)*B.  By pushing sign extension onto its operands we are much
        // more likely to preserve NSW and allow later AddRec optimisations.
        //
        // NOTE: This is effectively duplicating this logic from getSignExtend:
        //   sext((A + B + ...)<nsw>) --> (sext(A) + sext(B) + ...)<nsw>
        // but by that point the NSW information has potentially been lost.
        if (BO->Opcode == Instruction::Sub && BO->IsNSW) {
          Type *Ty = U->getType();
          auto *V1 = getSignExtendExpr(getSCEV(BO->LHS), Ty);
          auto *V2 = getSignExtendExpr(getSCEV(BO->RHS), Ty);
          return getMinusSCEV(V1, V2, SCEV::FlagNSW);
        }
      }
      return getSignExtendExpr(getSCEV(U->getOperand(0)), U->getType());
  
    case Instruction::BitCast:
      // BitCasts are no-op casts so we just eliminate the cast.
      if (isSCEVable(U->getType()) && isSCEVable(U->getOperand(0)->getType()))
        return getSCEV(U->getOperand(0));
      break;
  
    case Instruction::PtrToInt: {
      // Pointer to integer cast is straight-forward, so do model it.
      const SCEV *Op = getSCEV(U->getOperand(0));
      Type *DstIntTy = U->getType();
      // But only if effective SCEV (integer) type is wide enough to represent
      // all possible pointer values.
      const SCEV *IntOp = getPtrToIntExpr(Op, DstIntTy);
      if (isa<SCEVCouldNotCompute>(IntOp))
        return getUnknown(V);
      return IntOp;
    }
    case Instruction::IntToPtr:
      // Just don't deal with inttoptr casts.
      return getUnknown(V);
  
    case Instruction::SDiv:
      // If both operands are non-negative, this is just an udiv.
      if (isKnownNonNegative(getSCEV(U->getOperand(0))) &&
          isKnownNonNegative(getSCEV(U->getOperand(1))))
        return getUDivExpr(getSCEV(U->getOperand(0)), getSCEV(U->getOperand(1)));
      break;
  
    case Instruction::SRem:
      // If both operands are non-negative, this is just an urem.
      if (isKnownNonNegative(getSCEV(U->getOperand(0))) &&
          isKnownNonNegative(getSCEV(U->getOperand(1))))
        return getURemExpr(getSCEV(U->getOperand(0)), getSCEV(U->getOperand(1)));
      break;
  
    case Instruction::GetElementPtr:
      return createNodeForGEP(cast<GEPOperator>(U));
  
    case Instruction::PHI:
      return createNodeForPHI(cast<PHINode>(U));
  
    case Instruction::Select:
      // U can also be a select constant expr, which let fall through.  Since
      // createNodeForSelect only works for a condition that is an `ICmpInst`, and
      // constant expressions cannot have instructions as operands, we'd have
      // returned getUnknown for a select constant expressions anyway.
      if (isa<Instruction>(U))
        return createNodeForSelectOrPHI(cast<Instruction>(U), U->getOperand(0),
                                        U->getOperand(1), U->getOperand(2));
      break;
  
    case Instruction::Call:
    case Instruction::Invoke:
      if (Value *RV = cast<CallBase>(U)->getReturnedArgOperand())
        return getSCEV(RV);
  
      if (auto *II = dyn_cast<IntrinsicInst>(U)) {
        switch (II->getIntrinsicID()) {
        case Intrinsic::abs:
          return getAbsExpr(
              getSCEV(II->getArgOperand(0)),
              /*IsNSW=*/cast<ConstantInt>(II->getArgOperand(1))->isOne());
        case Intrinsic::umax:
          return getUMaxExpr(getSCEV(II->getArgOperand(0)),
                             getSCEV(II->getArgOperand(1)));
        case Intrinsic::umin:
          return getUMinExpr(getSCEV(II->getArgOperand(0)),
                             getSCEV(II->getArgOperand(1)));
        case Intrinsic::smax:
          return getSMaxExpr(getSCEV(II->getArgOperand(0)),
                             getSCEV(II->getArgOperand(1)));
        case Intrinsic::smin:
          return getSMinExpr(getSCEV(II->getArgOperand(0)),
                             getSCEV(II->getArgOperand(1)));
        case Intrinsic::usub_sat: {
          const SCEV *X = getSCEV(II->getArgOperand(0));
          const SCEV *Y = getSCEV(II->getArgOperand(1));
          const SCEV *ClampedY = getUMinExpr(X, Y);
          return getMinusSCEV(X, ClampedY, SCEV::FlagNUW);
        }
        case Intrinsic::uadd_sat: {
          const SCEV *X = getSCEV(II->getArgOperand(0));
          const SCEV *Y = getSCEV(II->getArgOperand(1));
          const SCEV *ClampedX = getUMinExpr(X, getNotSCEV(Y));
          return getAddExpr(ClampedX, Y, SCEV::FlagNUW);
        }
        case Intrinsic::start_loop_iterations:
          // A start_loop_iterations is just equivalent to the first operand for
          // SCEV purposes.
          return getSCEV(II->getArgOperand(0));
        default:
          break;
        }
      }
      break;
    }
  
    return getUnknown(V);
  }
  
  //===----------------------------------------------------------------------===//
  //                   Iteration Count Computation Code
  //
  
  const SCEV *ScalarEvolution::getTripCountFromExitCount(const SCEV *ExitCount) {
    // Get the trip count from the BE count by adding 1.  Overflow, results
    // in zero which means "unknown".
    return getAddExpr(ExitCount, getOne(ExitCount->getType()));
  }
  
  static unsigned getConstantTripCount(const SCEVConstant *ExitCount) {
    if (!ExitCount)
      return 0;
  
    ConstantInt *ExitConst = ExitCount->getValue();
  
    // Guard against huge trip counts.
    if (ExitConst->getValue().getActiveBits() > 32)
      return 0;
  
    // In case of integer overflow, this returns 0, which is correct.
    return ((unsigned)ExitConst->getZExtValue()) + 1;
  }
  
  unsigned ScalarEvolution::getSmallConstantTripCount(const Loop *L) {
    auto *ExitCount = dyn_cast<SCEVConstant>(getBackedgeTakenCount(L, Exact));
    return getConstantTripCount(ExitCount);
  }
  
  unsigned
  ScalarEvolution::getSmallConstantTripCount(const Loop *L,
                                             const BasicBlock *ExitingBlock) {
    assert(ExitingBlock && "Must pass a non-null exiting block!");
    assert(L->isLoopExiting(ExitingBlock) &&
           "Exiting block must actually branch out of the loop!");
    const SCEVConstant *ExitCount =
        dyn_cast<SCEVConstant>(getExitCount(L, ExitingBlock));
    return getConstantTripCount(ExitCount);
  }
  
  unsigned ScalarEvolution::getSmallConstantMaxTripCount(const Loop *L) {
    const auto *MaxExitCount =
        dyn_cast<SCEVConstant>(getConstantMaxBackedgeTakenCount(L));
    return getConstantTripCount(MaxExitCount);
  }
  
  unsigned ScalarEvolution::getSmallConstantTripMultiple(const Loop *L) {
    SmallVector<BasicBlock *, 8> ExitingBlocks;
    L->getExitingBlocks(ExitingBlocks);
  
    Optional<unsigned> Res = None;
    for (auto *ExitingBB : ExitingBlocks) {
      unsigned Multiple = getSmallConstantTripMultiple(L, ExitingBB);
      if (!Res)
        Res = Multiple;
      Res = (unsigned)GreatestCommonDivisor64(*Res, Multiple);
    }
    return Res.getValueOr(1);
  }
  
  unsigned ScalarEvolution::getSmallConstantTripMultiple(const Loop *L,
                                                         const SCEV *ExitCount) {
    if (ExitCount == getCouldNotCompute())
      return 1;
  
    // Get the trip count
    const SCEV *TCExpr = getTripCountFromExitCount(ExitCount);
  
    const SCEVConstant *TC = dyn_cast<SCEVConstant>(TCExpr);
    if (!TC)
      // Attempt to factor more general cases. Returns the greatest power of
      // two divisor. If overflow happens, the trip count expression is still
      // divisible by the greatest power of 2 divisor returned.
      return 1U << std::min((uint32_t)31,
                            GetMinTrailingZeros(applyLoopGuards(TCExpr, L)));
  
    ConstantInt *Result = TC->getValue();
  
    // Guard against huge trip counts (this requires checking
    // for zero to handle the case where the trip count == -1 and the
    // addition wraps).
    if (!Result || Result->getValue().getActiveBits() > 32 ||
        Result->getValue().getActiveBits() == 0)
      return 1;
  
    return (unsigned)Result->getZExtValue();
  }
  
  /// Returns the largest constant divisor of the trip count of this loop as a
  /// normal unsigned value, if possible. This means that the actual trip count is
  /// always a multiple of the returned value (don't forget the trip count could
  /// very well be zero as well!).
  ///
  /// Returns 1 if the trip count is unknown or not guaranteed to be the
  /// multiple of a constant (which is also the case if the trip count is simply
  /// constant, use getSmallConstantTripCount for that case), Will also return 1
  /// if the trip count is very large (>= 2^32).
  ///
  /// As explained in the comments for getSmallConstantTripCount, this assumes
  /// that control exits the loop via ExitingBlock.
  unsigned
  ScalarEvolution::getSmallConstantTripMultiple(const Loop *L,
                                                const BasicBlock *ExitingBlock) {
    assert(ExitingBlock && "Must pass a non-null exiting block!");
    assert(L->isLoopExiting(ExitingBlock) &&
           "Exiting block must actually branch out of the loop!");
    const SCEV *ExitCount = getExitCount(L, ExitingBlock);
    return getSmallConstantTripMultiple(L, ExitCount);
  }
  
  const SCEV *ScalarEvolution::getExitCount(const Loop *L,
                                            const BasicBlock *ExitingBlock,
                                            ExitCountKind Kind) {
    switch (Kind) {
    case Exact:
    case SymbolicMaximum:
      return getBackedgeTakenInfo(L).getExact(ExitingBlock, this);
    case ConstantMaximum:
      return getBackedgeTakenInfo(L).getConstantMax(ExitingBlock, this);
    };
    llvm_unreachable("Invalid ExitCountKind!");
  }
  
  const SCEV *
  ScalarEvolution::getPredicatedBackedgeTakenCount(const Loop *L,
                                                   SCEVUnionPredicate &Preds) {
    return getPredicatedBackedgeTakenInfo(L).getExact(L, this, &Preds);
  }
  
  const SCEV *ScalarEvolution::getBackedgeTakenCount(const Loop *L,
                                                     ExitCountKind Kind) {
    switch (Kind) {
    case Exact:
      return getBackedgeTakenInfo(L).getExact(L, this);
    case ConstantMaximum:
      return getBackedgeTakenInfo(L).getConstantMax(this);
    case SymbolicMaximum:
      return getBackedgeTakenInfo(L).getSymbolicMax(L, this);
    };
    llvm_unreachable("Invalid ExitCountKind!");
  }
  
  bool ScalarEvolution::isBackedgeTakenCountMaxOrZero(const Loop *L) {
    return getBackedgeTakenInfo(L).isConstantMaxOrZero(this);
  }
  
  /// Push PHI nodes in the header of the given loop onto the given Worklist.
  static void
  PushLoopPHIs(const Loop *L, SmallVectorImpl<Instruction *> &Worklist) {
    BasicBlock *Header = L->getHeader();
  
    // Push all Loop-header PHIs onto the Worklist stack.
    for (PHINode &PN : Header->phis())
      Worklist.push_back(&PN);
  }
  
  const ScalarEvolution::BackedgeTakenInfo &
  ScalarEvolution::getPredicatedBackedgeTakenInfo(const Loop *L) {
    auto &BTI = getBackedgeTakenInfo(L);
    if (BTI.hasFullInfo())
      return BTI;
  
    auto Pair = PredicatedBackedgeTakenCounts.insert({L, BackedgeTakenInfo()});
  
    if (!Pair.second)
      return Pair.first->second;
  
    BackedgeTakenInfo Result =
        computeBackedgeTakenCount(L, /*AllowPredicates=*/true);
  
    return PredicatedBackedgeTakenCounts.find(L)->second = std::move(Result);
  }
  
  ScalarEvolution::BackedgeTakenInfo &
  ScalarEvolution::getBackedgeTakenInfo(const Loop *L) {
    // Initially insert an invalid entry for this loop. If the insertion
    // succeeds, proceed to actually compute a backedge-taken count and
    // update the value. The temporary CouldNotCompute value tells SCEV
    // code elsewhere that it shouldn't attempt to request a new
    // backedge-taken count, which could result in infinite recursion.
    std::pair<DenseMap<const Loop *, BackedgeTakenInfo>::iterator, bool> Pair =
        BackedgeTakenCounts.insert({L, BackedgeTakenInfo()});
    if (!Pair.second)
      return Pair.first->second;
  
    // computeBackedgeTakenCount may allocate memory for its result. Inserting it
    // into the BackedgeTakenCounts map transfers ownership. Otherwise, the result
    // must be cleared in this scope.
    BackedgeTakenInfo Result = computeBackedgeTakenCount(L);
  
    // In product build, there are no usage of statistic.
    (void)NumTripCountsComputed;
    (void)NumTripCountsNotComputed;
  #if LLVM_ENABLE_STATS || !defined(NDEBUG)
    const SCEV *BEExact = Result.getExact(L, this);
    if (BEExact != getCouldNotCompute()) {
      assert(isLoopInvariant(BEExact, L) &&
             isLoopInvariant(Result.getConstantMax(this), L) &&
             "Computed backedge-taken count isn't loop invariant for loop!");
      ++NumTripCountsComputed;
    } else if (Result.getConstantMax(this) == getCouldNotCompute() &&
               isa<PHINode>(L->getHeader()->begin())) {
      // Only count loops that have phi nodes as not being computable.
      ++NumTripCountsNotComputed;
    }
  #endif // LLVM_ENABLE_STATS || !defined(NDEBUG)
  
    // Now that we know more about the trip count for this loop, forget any
    // existing SCEV values for PHI nodes in this loop since they are only
    // conservative estimates made without the benefit of trip count
    // information. This is similar to the code in forgetLoop, except that
    // it handles SCEVUnknown PHI nodes specially.
    if (Result.hasAnyInfo()) {
      SmallVector<Instruction *, 16> Worklist;
      PushLoopPHIs(L, Worklist);
  
      SmallPtrSet<Instruction *, 8> Discovered;
      while (!Worklist.empty()) {
        Instruction *I = Worklist.pop_back_val();
  
        ValueExprMapType::iterator It =
          ValueExprMap.find_as(static_cast<Value *>(I));
        if (It != ValueExprMap.end()) {
          const SCEV *Old = It->second;
  
          // SCEVUnknown for a PHI either means that it has an unrecognized
          // structure, or it's a PHI that's in the progress of being computed
          // by createNodeForPHI.  In the former case, additional loop trip
          // count information isn't going to change anything. In the later
          // case, createNodeForPHI will perform the necessary updates on its
          // own when it gets to that point.
          if (!isa<PHINode>(I) || !isa<SCEVUnknown>(Old)) {
            eraseValueFromMap(It->first);
            forgetMemoizedResults(Old);
          }
          if (PHINode *PN = dyn_cast<PHINode>(I))
            ConstantEvolutionLoopExitValue.erase(PN);
        }
  
        // Since we don't need to invalidate anything for correctness and we're
        // only invalidating to make SCEV's results more precise, we get to stop
        // early to avoid invalidating too much.  This is especially important in
        // cases like:
        //
        //   %v = f(pn0, pn1) // pn0 and pn1 used through some other phi node
        // loop0:
        //   %pn0 = phi
        //   ...
        // loop1:
        //   %pn1 = phi
        //   ...
        //
        // where both loop0 and loop1's backedge taken count uses the SCEV
        // expression for %v.  If we don't have the early stop below then in cases
        // like the above, getBackedgeTakenInfo(loop1) will clear out the trip
        // count for loop0 and getBackedgeTakenInfo(loop0) will clear out the trip
        // count for loop1, effectively nullifying SCEV's trip count cache.
        for (auto *U : I->users())
          if (auto *I = dyn_cast<Instruction>(U)) {
            auto *LoopForUser = LI.getLoopFor(I->getParent());
            if (LoopForUser && L->contains(LoopForUser) &&
                Discovered.insert(I).second)
              Worklist.push_back(I);
          }
      }
    }
  
    // Re-lookup the insert position, since the call to
    // computeBackedgeTakenCount above could result in a
    // recusive call to getBackedgeTakenInfo (on a different
    // loop), which would invalidate the iterator computed
    // earlier.
    return BackedgeTakenCounts.find(L)->second = std::move(Result);
  }
  
  void ScalarEvolution::forgetAllLoops() {
    // This method is intended to forget all info about loops. It should
    // invalidate caches as if the following happened:
    // - The trip counts of all loops have changed arbitrarily
    // - Every llvm::Value has been updated in place to produce a different
    // result.
    BackedgeTakenCounts.clear();
    PredicatedBackedgeTakenCounts.clear();
    LoopPropertiesCache.clear();
    ConstantEvolutionLoopExitValue.clear();
    ValueExprMap.clear();
    ValuesAtScopes.clear();
    LoopDispositions.clear();
    BlockDispositions.clear();
    UnsignedRanges.clear();
    SignedRanges.clear();
    ExprValueMap.clear();
    HasRecMap.clear();
    MinTrailingZerosCache.clear();
    PredicatedSCEVRewrites.clear();
  }
  
  void ScalarEvolution::forgetLoop(const Loop *L) {
    SmallVector<const Loop *, 16> LoopWorklist(1, L);
    SmallVector<Instruction *, 32> Worklist;
    SmallPtrSet<Instruction *, 16> Visited;
  
    // Iterate over all the loops and sub-loops to drop SCEV information.
    while (!LoopWorklist.empty()) {
      auto *CurrL = LoopWorklist.pop_back_val();
  
      // Drop any stored trip count value.
      BackedgeTakenCounts.erase(CurrL);
      PredicatedBackedgeTakenCounts.erase(CurrL);
  
      // Drop information about predicated SCEV rewrites for this loop.
      for (auto I = PredicatedSCEVRewrites.begin();
           I != PredicatedSCEVRewrites.end();) {
        std::pair<const SCEV *, const Loop *> Entry = I->first;
        if (Entry.second == CurrL)
          PredicatedSCEVRewrites.erase(I++);
        else
          ++I;
      }
  
      auto LoopUsersItr = LoopUsers.find(CurrL);
      if (LoopUsersItr != LoopUsers.end()) {
        for (auto *S : LoopUsersItr->second)
          forgetMemoizedResults(S);
        LoopUsers.erase(LoopUsersItr);
      }
  
      // Drop information about expressions based on loop-header PHIs.
      PushLoopPHIs(CurrL, Worklist);
  
      while (!Worklist.empty()) {
        Instruction *I = Worklist.pop_back_val();
        if (!Visited.insert(I).second)
          continue;
  
        ValueExprMapType::iterator It =
            ValueExprMap.find_as(static_cast<Value *>(I));
        if (It != ValueExprMap.end()) {
          eraseValueFromMap(It->first);
          forgetMemoizedResults(It->second);
          if (PHINode *PN = dyn_cast<PHINode>(I))
            ConstantEvolutionLoopExitValue.erase(PN);
        }
  
        PushDefUseChildren(I, Worklist);
      }
  
      LoopPropertiesCache.erase(CurrL);
      // Forget all contained loops too, to avoid dangling entries in the
      // ValuesAtScopes map.
      LoopWorklist.append(CurrL->begin(), CurrL->end());
    }
  }
  
  void ScalarEvolution::forgetTopmostLoop(const Loop *L) {
    while (Loop *Parent = L->getParentLoop())
      L = Parent;
    forgetLoop(L);
  }
  
  void ScalarEvolution::forgetValue(Value *V) {
    Instruction *I = dyn_cast<Instruction>(V);
    if (!I) return;
  
    // Drop information about expressions based on loop-header PHIs.
    SmallVector<Instruction *, 16> Worklist;
    Worklist.push_back(I);
  
    SmallPtrSet<Instruction *, 8> Visited;
    while (!Worklist.empty()) {
      I = Worklist.pop_back_val();
      if (!Visited.insert(I).second)
        continue;
  
      ValueExprMapType::iterator It =
        ValueExprMap.find_as(static_cast<Value *>(I));
      if (It != ValueExprMap.end()) {
        eraseValueFromMap(It->first);
        forgetMemoizedResults(It->second);
        if (PHINode *PN = dyn_cast<PHINode>(I))
          ConstantEvolutionLoopExitValue.erase(PN);
      }
  
      PushDefUseChildren(I, Worklist);
    }
  }
  
  void ScalarEvolution::forgetLoopDispositions(const Loop *L) {
    LoopDispositions.clear();
  }
  
  /// Get the exact loop backedge taken count considering all loop exits. A
  /// computable result can only be returned for loops with all exiting blocks
  /// dominating the latch. howFarToZero assumes that the limit of each loop test
  /// is never skipped. This is a valid assumption as long as the loop exits via
  /// that test. For precise results, it is the caller's responsibility to specify
  /// the relevant loop exiting block using getExact(ExitingBlock, SE).
  const SCEV *
  ScalarEvolution::BackedgeTakenInfo::getExact(const Loop *L, ScalarEvolution *SE,
                                               SCEVUnionPredicate *Preds) const {
    // If any exits were not computable, the loop is not computable.
    if (!isComplete() || ExitNotTaken.empty())
      return SE->getCouldNotCompute();
  
    const BasicBlock *Latch = L->getLoopLatch();
    // All exiting blocks we have collected must dominate the only backedge.
    if (!Latch)
      return SE->getCouldNotCompute();
  
    // All exiting blocks we have gathered dominate loop's latch, so exact trip
    // count is simply a minimum out of all these calculated exit counts.
    SmallVector<const SCEV *, 2> Ops;
    for (auto &ENT : ExitNotTaken) {
      const SCEV *BECount = ENT.ExactNotTaken;
      assert(BECount != SE->getCouldNotCompute() && "Bad exit SCEV!");
      assert(SE->DT.dominates(ENT.ExitingBlock, Latch) &&
             "We should only have known counts for exiting blocks that dominate "
             "latch!");
  
      Ops.push_back(BECount);
  
      if (Preds && !ENT.hasAlwaysTruePredicate())
        Preds->add(ENT.Predicate.get());
  
      assert((Preds || ENT.hasAlwaysTruePredicate()) &&
             "Predicate should be always true!");
    }
  
    return SE->getUMinFromMismatchedTypes(Ops);
  }
  
  /// Get the exact not taken count for this loop exit.
  const SCEV *
  ScalarEvolution::BackedgeTakenInfo::getExact(const BasicBlock *ExitingBlock,
                                               ScalarEvolution *SE) const {
    for (auto &ENT : ExitNotTaken)
      if (ENT.ExitingBlock == ExitingBlock && ENT.hasAlwaysTruePredicate())
        return ENT.ExactNotTaken;
  
    return SE->getCouldNotCompute();
  }
  
  const SCEV *ScalarEvolution::BackedgeTakenInfo::getConstantMax(
      const BasicBlock *ExitingBlock, ScalarEvolution *SE) const {
    for (auto &ENT : ExitNotTaken)
      if (ENT.ExitingBlock == ExitingBlock && ENT.hasAlwaysTruePredicate())
        return ENT.MaxNotTaken;
  
    return SE->getCouldNotCompute();
  }
  
  /// getConstantMax - Get the constant max backedge taken count for the loop.
  const SCEV *
  ScalarEvolution::BackedgeTakenInfo::getConstantMax(ScalarEvolution *SE) const {
    auto PredicateNotAlwaysTrue = [](const ExitNotTakenInfo &ENT) {
      return !ENT.hasAlwaysTruePredicate();
    };
  
    if (any_of(ExitNotTaken, PredicateNotAlwaysTrue) || !getConstantMax())
      return SE->getCouldNotCompute();
  
    assert((isa<SCEVCouldNotCompute>(getConstantMax()) ||
            isa<SCEVConstant>(getConstantMax())) &&
           "No point in having a non-constant max backedge taken count!");
    return getConstantMax();
  }
  
  const SCEV *
  ScalarEvolution::BackedgeTakenInfo::getSymbolicMax(const Loop *L,
                                                     ScalarEvolution *SE) {
    if (!SymbolicMax)
      SymbolicMax = SE->computeSymbolicMaxBackedgeTakenCount(L);
    return SymbolicMax;
  }
  
  bool ScalarEvolution::BackedgeTakenInfo::isConstantMaxOrZero(
      ScalarEvolution *SE) const {
    auto PredicateNotAlwaysTrue = [](const ExitNotTakenInfo &ENT) {
      return !ENT.hasAlwaysTruePredicate();
    };
    return MaxOrZero && !any_of(ExitNotTaken, PredicateNotAlwaysTrue);
  }
  
  bool ScalarEvolution::BackedgeTakenInfo::hasOperand(const SCEV *S) const {
    return Operands.contains(S);
  }
  
  ScalarEvolution::ExitLimit::ExitLimit(const SCEV *E)
      : ExitLimit(E, E, false, None) {
  }
  
  ScalarEvolution::ExitLimit::ExitLimit(
      const SCEV *E, const SCEV *M, bool MaxOrZero,
      ArrayRef<const SmallPtrSetImpl<const SCEVPredicate *> *> PredSetList)
      : ExactNotTaken(E), MaxNotTaken(M), MaxOrZero(MaxOrZero) {
    assert((isa<SCEVCouldNotCompute>(ExactNotTaken) ||
            !isa<SCEVCouldNotCompute>(MaxNotTaken)) &&
           "Exact is not allowed to be less precise than Max");
    assert((isa<SCEVCouldNotCompute>(MaxNotTaken) ||
            isa<SCEVConstant>(MaxNotTaken)) &&
           "No point in having a non-constant max backedge taken count!");
    for (auto *PredSet : PredSetList)
      for (auto *P : *PredSet)
        addPredicate(P);
    assert((isa<SCEVCouldNotCompute>(E) || !E->getType()->isPointerTy()) &&
           "Backedge count should be int");
    assert((isa<SCEVCouldNotCompute>(M) || !M->getType()->isPointerTy()) &&
           "Max backedge count should be int");
  }
  
  ScalarEvolution::ExitLimit::ExitLimit(
      const SCEV *E, const SCEV *M, bool MaxOrZero,
      const SmallPtrSetImpl<const SCEVPredicate *> &PredSet)
      : ExitLimit(E, M, MaxOrZero, {&PredSet}) {
  }
  
  ScalarEvolution::ExitLimit::ExitLimit(const SCEV *E, const SCEV *M,
                                        bool MaxOrZero)
      : ExitLimit(E, M, MaxOrZero, None) {
  }
  
  class SCEVRecordOperands {
    SmallPtrSetImpl<const SCEV *> &Operands;
  
  public:
    SCEVRecordOperands(SmallPtrSetImpl<const SCEV *> &Operands)
      : Operands(Operands) {}
    bool follow(const SCEV *S) {
      Operands.insert(S);
      return true;
    }
    bool isDone() { return false; }
  };
  
  /// Allocate memory for BackedgeTakenInfo and copy the not-taken count of each
  /// computable exit into a persistent ExitNotTakenInfo array.
  ScalarEvolution::BackedgeTakenInfo::BackedgeTakenInfo(
      ArrayRef<ScalarEvolution::BackedgeTakenInfo::EdgeExitInfo> ExitCounts,
      bool IsComplete, const SCEV *ConstantMax, bool MaxOrZero)
      : ConstantMax(ConstantMax), IsComplete(IsComplete), MaxOrZero(MaxOrZero) {
    using EdgeExitInfo = ScalarEvolution::BackedgeTakenInfo::EdgeExitInfo;
  
    ExitNotTaken.reserve(ExitCounts.size());
    std::transform(
        ExitCounts.begin(), ExitCounts.end(), std::back_inserter(ExitNotTaken),
        [&](const EdgeExitInfo &EEI) {
          BasicBlock *ExitBB = EEI.first;
          const ExitLimit &EL = EEI.second;
          if (EL.Predicates.empty())
            return ExitNotTakenInfo(ExitBB, EL.ExactNotTaken, EL.MaxNotTaken,
                                    nullptr);
  
          std::unique_ptr<SCEVUnionPredicate> Predicate(new SCEVUnionPredicate);
          for (auto *Pred : EL.Predicates)
            Predicate->add(Pred);
  
          return ExitNotTakenInfo(ExitBB, EL.ExactNotTaken, EL.MaxNotTaken,
                                  std::move(Predicate));
        });
    assert((isa<SCEVCouldNotCompute>(ConstantMax) ||
            isa<SCEVConstant>(ConstantMax)) &&
           "No point in having a non-constant max backedge taken count!");
  
    SCEVRecordOperands RecordOperands(Operands);
    SCEVTraversal<SCEVRecordOperands> ST(RecordOperands);
    if (!isa<SCEVCouldNotCompute>(ConstantMax))
      ST.visitAll(ConstantMax);
    for (auto &ENT : ExitNotTaken)
      if (!isa<SCEVCouldNotCompute>(ENT.ExactNotTaken))
        ST.visitAll(ENT.ExactNotTaken);
  }
  
  /// Compute the number of times the backedge of the specified loop will execute.
  ScalarEvolution::BackedgeTakenInfo
  ScalarEvolution::computeBackedgeTakenCount(const Loop *L,
                                             bool AllowPredicates) {
    SmallVector<BasicBlock *, 8> ExitingBlocks;
    L->getExitingBlocks(ExitingBlocks);
  
    using EdgeExitInfo = ScalarEvolution::BackedgeTakenInfo::EdgeExitInfo;
  
    SmallVector<EdgeExitInfo, 4> ExitCounts;
    bool CouldComputeBECount = true;
    BasicBlock *Latch = L->getLoopLatch(); // may be NULL.
    const SCEV *MustExitMaxBECount = nullptr;
    const SCEV *MayExitMaxBECount = nullptr;
    bool MustExitMaxOrZero = false;
  
    // Compute the ExitLimit for each loop exit. Use this to populate ExitCounts
    // and compute maxBECount.
    // Do a union of all the predicates here.
    for (unsigned i = 0, e = ExitingBlocks.size(); i != e; ++i) {
      BasicBlock *ExitBB = ExitingBlocks[i];
  
      // We canonicalize untaken exits to br (constant), ignore them so that
      // proving an exit untaken doesn't negatively impact our ability to reason
      // about the loop as whole.
      if (auto *BI = dyn_cast<BranchInst>(ExitBB->getTerminator()))
        if (auto *CI = dyn_cast<ConstantInt>(BI->getCondition())) {
          bool ExitIfTrue = !L->contains(BI->getSuccessor(0));
          if ((ExitIfTrue && CI->isZero()) || (!ExitIfTrue && CI->isOne()))
            continue;
        }
  
      ExitLimit EL = computeExitLimit(L, ExitBB, AllowPredicates);
  
      assert((AllowPredicates || EL.Predicates.empty()) &&
             "Predicated exit limit when predicates are not allowed!");
  
      // 1. For each exit that can be computed, add an entry to ExitCounts.
      // CouldComputeBECount is true only if all exits can be computed.
      if (EL.ExactNotTaken == getCouldNotCompute())
        // We couldn't compute an exact value for this exit, so
        // we won't be able to compute an exact value for the loop.
        CouldComputeBECount = false;
      else
        ExitCounts.emplace_back(ExitBB, EL);
  
      // 2. Derive the loop's MaxBECount from each exit's max number of
      // non-exiting iterations. Partition the loop exits into two kinds:
      // LoopMustExits and LoopMayExits.
      //
      // If the exit dominates the loop latch, it is a LoopMustExit otherwise it
      // is a LoopMayExit.  If any computable LoopMustExit is found, then
      // MaxBECount is the minimum EL.MaxNotTaken of computable
      // LoopMustExits. Otherwise, MaxBECount is conservatively the maximum
      // EL.MaxNotTaken, where CouldNotCompute is considered greater than any
      // computable EL.MaxNotTaken.
      if (EL.MaxNotTaken != getCouldNotCompute() && Latch &&
          DT.dominates(ExitBB, Latch)) {
        if (!MustExitMaxBECount) {
          MustExitMaxBECount = EL.MaxNotTaken;
          MustExitMaxOrZero = EL.MaxOrZero;
        } else {
          MustExitMaxBECount =
              getUMinFromMismatchedTypes(MustExitMaxBECount, EL.MaxNotTaken);
        }
      } else if (MayExitMaxBECount != getCouldNotCompute()) {
        if (!MayExitMaxBECount || EL.MaxNotTaken == getCouldNotCompute())
          MayExitMaxBECount = EL.MaxNotTaken;
        else {
          MayExitMaxBECount =
              getUMaxFromMismatchedTypes(MayExitMaxBECount, EL.MaxNotTaken);
        }
      }
    }
    const SCEV *MaxBECount = MustExitMaxBECount ? MustExitMaxBECount :
      (MayExitMaxBECount ? MayExitMaxBECount : getCouldNotCompute());
    // The loop backedge will be taken the maximum or zero times if there's
    // a single exit that must be taken the maximum or zero times.
    bool MaxOrZero = (MustExitMaxOrZero && ExitingBlocks.size() == 1);
    return BackedgeTakenInfo(std::move(ExitCounts), CouldComputeBECount,
                             MaxBECount, MaxOrZero);
  }
  
  ScalarEvolution::ExitLimit
  ScalarEvolution::computeExitLimit(const Loop *L, BasicBlock *ExitingBlock,
                                        bool AllowPredicates) {
    assert(L->contains(ExitingBlock) && "Exit count for non-loop block?");
    // If our exiting block does not dominate the latch, then its connection with
    // loop's exit limit may be far from trivial.
    const BasicBlock *Latch = L->getLoopLatch();
    if (!Latch || !DT.dominates(ExitingBlock, Latch))
      return getCouldNotCompute();
  
    bool IsOnlyExit = (L->getExitingBlock() != nullptr);
    Instruction *Term = ExitingBlock->getTerminator();
    if (BranchInst *BI = dyn_cast<BranchInst>(Term)) {
      assert(BI->isConditional() && "If unconditional, it can't be in loop!");
      bool ExitIfTrue = !L->contains(BI->getSuccessor(0));
      assert(ExitIfTrue == L->contains(BI->getSuccessor(1)) &&
             "It should have one successor in loop and one exit block!");
      // Proceed to the next level to examine the exit condition expression.
      return computeExitLimitFromCond(
          L, BI->getCondition(), ExitIfTrue,
          /*ControlsExit=*/IsOnlyExit, AllowPredicates);
    }
  
    if (SwitchInst *SI = dyn_cast<SwitchInst>(Term)) {
      // For switch, make sure that there is a single exit from the loop.
      BasicBlock *Exit = nullptr;
      for (auto *SBB : successors(ExitingBlock))
        if (!L->contains(SBB)) {
          if (Exit) // Multiple exit successors.
            return getCouldNotCompute();
          Exit = SBB;
        }
      assert(Exit && "Exiting block must have at least one exit");
      return computeExitLimitFromSingleExitSwitch(L, SI, Exit,
                                                  /*ControlsExit=*/IsOnlyExit);
    }
  
    return getCouldNotCompute();
  }
  
  ScalarEvolution::ExitLimit ScalarEvolution::computeExitLimitFromCond(
      const Loop *L, Value *ExitCond, bool ExitIfTrue,
      bool ControlsExit, bool AllowPredicates) {
    ScalarEvolution::ExitLimitCacheTy Cache(L, ExitIfTrue, AllowPredicates);
    return computeExitLimitFromCondCached(Cache, L, ExitCond, ExitIfTrue,
                                          ControlsExit, AllowPredicates);
  }
  
  Optional<ScalarEvolution::ExitLimit>
  ScalarEvolution::ExitLimitCache::find(const Loop *L, Value *ExitCond,
                                        bool ExitIfTrue, bool ControlsExit,
                                        bool AllowPredicates) {
    (void)this->L;
    (void)this->ExitIfTrue;
    (void)this->AllowPredicates;
  
    assert(this->L == L && this->ExitIfTrue == ExitIfTrue &&
           this->AllowPredicates == AllowPredicates &&
           "Variance in assumed invariant key components!");
    auto Itr = TripCountMap.find({ExitCond, ControlsExit});
    if (Itr == TripCountMap.end())
      return None;
    return Itr->second;
  }
  
  void ScalarEvolution::ExitLimitCache::insert(const Loop *L, Value *ExitCond,
                                               bool ExitIfTrue,
                                               bool ControlsExit,
                                               bool AllowPredicates,
                                               const ExitLimit &EL) {
    assert(this->L == L && this->ExitIfTrue == ExitIfTrue &&
           this->AllowPredicates == AllowPredicates &&
           "Variance in assumed invariant key components!");
  
    auto InsertResult = TripCountMap.insert({{ExitCond, ControlsExit}, EL});
    assert(InsertResult.second && "Expected successful insertion!");
    (void)InsertResult;
    (void)ExitIfTrue;
  }
  
  ScalarEvolution::ExitLimit ScalarEvolution::computeExitLimitFromCondCached(
      ExitLimitCacheTy &Cache, const Loop *L, Value *ExitCond, bool ExitIfTrue,
      bool ControlsExit, bool AllowPredicates) {
  
    if (auto MaybeEL =
            Cache.find(L, ExitCond, ExitIfTrue, ControlsExit, AllowPredicates))
      return *MaybeEL;
  
    ExitLimit EL = computeExitLimitFromCondImpl(Cache, L, ExitCond, ExitIfTrue,
                                                ControlsExit, AllowPredicates);
    Cache.insert(L, ExitCond, ExitIfTrue, ControlsExit, AllowPredicates, EL);
    return EL;
  }
  
  ScalarEvolution::ExitLimit ScalarEvolution::computeExitLimitFromCondImpl(
      ExitLimitCacheTy &Cache, const Loop *L, Value *ExitCond, bool ExitIfTrue,
      bool ControlsExit, bool AllowPredicates) {
    // Handle BinOp conditions (And, Or).
    if (auto LimitFromBinOp = computeExitLimitFromCondFromBinOp(
            Cache, L, ExitCond, ExitIfTrue, ControlsExit, AllowPredicates))
      return *LimitFromBinOp;
  
    // With an icmp, it may be feasible to compute an exact backedge-taken count.
    // Proceed to the next level to examine the icmp.
    if (ICmpInst *ExitCondICmp = dyn_cast<ICmpInst>(ExitCond)) {
      ExitLimit EL =
          computeExitLimitFromICmp(L, ExitCondICmp, ExitIfTrue, ControlsExit);
      if (EL.hasFullInfo() || !AllowPredicates)
        return EL;
  
      // Try again, but use SCEV predicates this time.
      return computeExitLimitFromICmp(L, ExitCondICmp, ExitIfTrue, ControlsExit,
                                      /*AllowPredicates=*/true);
    }
  
    // Check for a constant condition. These are normally stripped out by
    // SimplifyCFG, but ScalarEvolution may be used by a pass which wishes to
    // preserve the CFG and is temporarily leaving constant conditions
    // in place.
    if (ConstantInt *CI = dyn_cast<ConstantInt>(ExitCond)) {
      if (ExitIfTrue == !CI->getZExtValue())
        // The backedge is always taken.
        return getCouldNotCompute();
      else
        // The backedge is never taken.
        return getZero(CI->getType());
    }
  
    // If it's not an integer or pointer comparison then compute it the hard way.
    return computeExitCountExhaustively(L, ExitCond, ExitIfTrue);
  }
  
  Optional<ScalarEvolution::ExitLimit>
  ScalarEvolution::computeExitLimitFromCondFromBinOp(
      ExitLimitCacheTy &Cache, const Loop *L, Value *ExitCond, bool ExitIfTrue,
      bool ControlsExit, bool AllowPredicates) {
    // Check if the controlling expression for this loop is an And or Or.
    Value *Op0, *Op1;
    bool IsAnd = false;
    if (match(ExitCond, m_LogicalAnd(m_Value(Op0), m_Value(Op1))))
      IsAnd = true;
    else if (match(ExitCond, m_LogicalOr(m_Value(Op0), m_Value(Op1))))
      IsAnd = false;
    else
      return None;
  
    // EitherMayExit is true in these two cases:
    //   br (and Op0 Op1), loop, exit
    //   br (or  Op0 Op1), exit, loop
    bool EitherMayExit = IsAnd ^ ExitIfTrue;
    ExitLimit EL0 = computeExitLimitFromCondCached(Cache, L, Op0, ExitIfTrue,
                                                   ControlsExit && !EitherMayExit,
                                                   AllowPredicates);
    ExitLimit EL1 = computeExitLimitFromCondCached(Cache, L, Op1, ExitIfTrue,
                                                   ControlsExit && !EitherMayExit,
                                                   AllowPredicates);
  
    // Be robust against unsimplified IR for the form "op i1 X, NeutralElement"
    const Constant *NeutralElement = ConstantInt::get(ExitCond->getType(), IsAnd);
    if (isa<ConstantInt>(Op1))
      return Op1 == NeutralElement ? EL0 : EL1;
    if (isa<ConstantInt>(Op0))
      return Op0 == NeutralElement ? EL1 : EL0;
  
    const SCEV *BECount = getCouldNotCompute();
    const SCEV *MaxBECount = getCouldNotCompute();
    if (EitherMayExit) {
      // Both conditions must be same for the loop to continue executing.
      // Choose the less conservative count.
      // If ExitCond is a short-circuit form (select), using
      // umin(EL0.ExactNotTaken, EL1.ExactNotTaken) is unsafe in general.
      // To see the detailed examples, please see
      // test/Analysis/ScalarEvolution/exit-count-select.ll
      bool PoisonSafe = isa<BinaryOperator>(ExitCond);
      if (!PoisonSafe)
        // Even if ExitCond is select, we can safely derive BECount using both
        // EL0 and EL1 in these cases:
        // (1) EL0.ExactNotTaken is non-zero
        // (2) EL1.ExactNotTaken is non-poison
        // (3) EL0.ExactNotTaken is zero (BECount should be simply zero and
        //     it cannot be umin(0, ..))
        // The PoisonSafe assignment below is simplified and the assertion after
        // BECount calculation fully guarantees the condition (3).
        PoisonSafe = isa<SCEVConstant>(EL0.ExactNotTaken) ||
                     isa<SCEVConstant>(EL1.ExactNotTaken);
      if (EL0.ExactNotTaken != getCouldNotCompute() &&
          EL1.ExactNotTaken != getCouldNotCompute() && PoisonSafe) {
        BECount =
            getUMinFromMismatchedTypes(EL0.ExactNotTaken, EL1.ExactNotTaken);
  
        // If EL0.ExactNotTaken was zero and ExitCond was a short-circuit form,
        // it should have been simplified to zero (see the condition (3) above)
        assert(!isa<BinaryOperator>(ExitCond) || !EL0.ExactNotTaken->isZero() ||
               BECount->isZero());
      }
      if (EL0.MaxNotTaken == getCouldNotCompute())
        MaxBECount = EL1.MaxNotTaken;
      else if (EL1.MaxNotTaken == getCouldNotCompute())
        MaxBECount = EL0.MaxNotTaken;
      else
        MaxBECount = getUMinFromMismatchedTypes(EL0.MaxNotTaken, EL1.MaxNotTaken);
    } else {
      // Both conditions must be same at the same time for the loop to exit.
      // For now, be conservative.
      if (EL0.ExactNotTaken == EL1.ExactNotTaken)
        BECount = EL0.ExactNotTaken;
    }
  
    // There are cases (e.g. PR26207) where computeExitLimitFromCond is able
    // to be more aggressive when computing BECount than when computing
    // MaxBECount.  In these cases it is possible for EL0.ExactNotTaken and
    // EL1.ExactNotTaken to match, but for EL0.MaxNotTaken and EL1.MaxNotTaken
    // to not.
    if (isa<SCEVCouldNotCompute>(MaxBECount) &&
        !isa<SCEVCouldNotCompute>(BECount))
      MaxBECount = getConstant(getUnsignedRangeMax(BECount));
  
    return ExitLimit(BECount, MaxBECount, false,
                     { &EL0.Predicates, &EL1.Predicates });
  }
  
  ScalarEvolution::ExitLimit
  ScalarEvolution::computeExitLimitFromICmp(const Loop *L,
                                            ICmpInst *ExitCond,
                                            bool ExitIfTrue,
                                            bool ControlsExit,
                                            bool AllowPredicates) {
    // If the condition was exit on true, convert the condition to exit on false
    ICmpInst::Predicate Pred;
    if (!ExitIfTrue)
      Pred = ExitCond->getPredicate();
    else
      Pred = ExitCond->getInversePredicate();
    const ICmpInst::Predicate OriginalPred = Pred;
  
    // Handle common loops like: for (X = "string"; *X; ++X)
    if (LoadInst *LI = dyn_cast<LoadInst>(ExitCond->getOperand(0)))
      if (Constant *RHS = dyn_cast<Constant>(ExitCond->getOperand(1))) {
        ExitLimit ItCnt =
          computeLoadConstantCompareExitLimit(LI, RHS, L, Pred);
        if (ItCnt.hasAnyInfo())
          return ItCnt;
      }
  
    const SCEV *LHS = getSCEV(ExitCond->getOperand(0));
    const SCEV *RHS = getSCEV(ExitCond->getOperand(1));
  
    // Try to evaluate any dependencies out of the loop.
    LHS = getSCEVAtScope(LHS, L);
    RHS = getSCEVAtScope(RHS, L);
  
    // At this point, we would like to compute how many iterations of the
    // loop the predicate will return true for these inputs.
    if (isLoopInvariant(LHS, L) && !isLoopInvariant(RHS, L)) {
      // If there is a loop-invariant, force it into the RHS.
      std::swap(LHS, RHS);
      Pred = ICmpInst::getSwappedPredicate(Pred);
    }
  
    // Simplify the operands before analyzing them.
    (void)SimplifyICmpOperands(Pred, LHS, RHS);
  
    // If we have a comparison of a chrec against a constant, try to use value
    // ranges to answer this query.
    if (const SCEVConstant *RHSC = dyn_cast<SCEVConstant>(RHS))
      if (const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(LHS))
        if (AddRec->getLoop() == L) {
          // Form the constant range.
          ConstantRange CompRange =
              ConstantRange::makeExactICmpRegion(Pred, RHSC->getAPInt());
  
          const SCEV *Ret = AddRec->getNumIterationsInRange(CompRange, *this);
          if (!isa<SCEVCouldNotCompute>(Ret)) return Ret;
        }
  
    switch (Pred) {
    case ICmpInst::ICMP_NE: {                     // while (X != Y)
      // Convert to: while (X-Y != 0)
      if (LHS->getType()->isPointerTy()) {
        LHS = getLosslessPtrToIntExpr(LHS);
        if (isa<SCEVCouldNotCompute>(LHS))
          return LHS;
      }
      if (RHS->getType()->isPointerTy()) {
        RHS = getLosslessPtrToIntExpr(RHS);
        if (isa<SCEVCouldNotCompute>(RHS))
          return RHS;
      }
      ExitLimit EL = howFarToZero(getMinusSCEV(LHS, RHS), L, ControlsExit,
                                  AllowPredicates);
      if (EL.hasAnyInfo()) return EL;
      break;
    }
    case ICmpInst::ICMP_EQ: {                     // while (X == Y)
      // Convert to: while (X-Y == 0)
      if (LHS->getType()->isPointerTy()) {
        LHS = getLosslessPtrToIntExpr(LHS);
        if (isa<SCEVCouldNotCompute>(LHS))
          return LHS;
      }
      if (RHS->getType()->isPointerTy()) {
        RHS = getLosslessPtrToIntExpr(RHS);
        if (isa<SCEVCouldNotCompute>(RHS))
          return RHS;
      }
      ExitLimit EL = howFarToNonZero(getMinusSCEV(LHS, RHS), L);
      if (EL.hasAnyInfo()) return EL;
      break;
    }
    case ICmpInst::ICMP_SLT:
    case ICmpInst::ICMP_ULT: {                    // while (X < Y)
      bool IsSigned = Pred == ICmpInst::ICMP_SLT;
      ExitLimit EL = howManyLessThans(LHS, RHS, L, IsSigned, ControlsExit,
                                      AllowPredicates);
      if (EL.hasAnyInfo()) return EL;
      break;
    }
    case ICmpInst::ICMP_SGT:
    case ICmpInst::ICMP_UGT: {                    // while (X > Y)
      bool IsSigned = Pred == ICmpInst::ICMP_SGT;
      ExitLimit EL =
          howManyGreaterThans(LHS, RHS, L, IsSigned, ControlsExit,
                              AllowPredicates);
      if (EL.hasAnyInfo()) return EL;
      break;
    }
    default:
      break;
    }
  
    auto *ExhaustiveCount =
        computeExitCountExhaustively(L, ExitCond, ExitIfTrue);
  
    if (!isa<SCEVCouldNotCompute>(ExhaustiveCount))
      return ExhaustiveCount;
  
    return computeShiftCompareExitLimit(ExitCond->getOperand(0),
                                        ExitCond->getOperand(1), L, OriginalPred);
  }
  
  ScalarEvolution::ExitLimit
  ScalarEvolution::computeExitLimitFromSingleExitSwitch(const Loop *L,
                                                        SwitchInst *Switch,
                                                        BasicBlock *ExitingBlock,
                                                        bool ControlsExit) {
    assert(!L->contains(ExitingBlock) && "Not an exiting block!");
  
    // Give up if the exit is the default dest of a switch.
    if (Switch->getDefaultDest() == ExitingBlock)
      return getCouldNotCompute();
  
    assert(L->contains(Switch->getDefaultDest()) &&
           "Default case must not exit the loop!");
    const SCEV *LHS = getSCEVAtScope(Switch->getCondition(), L);
    const SCEV *RHS = getConstant(Switch->findCaseDest(ExitingBlock));
  
    // while (X != Y) --> while (X-Y != 0)
    ExitLimit EL = howFarToZero(getMinusSCEV(LHS, RHS), L, ControlsExit);
    if (EL.hasAnyInfo())
      return EL;
  
    return getCouldNotCompute();
  }
  
  static ConstantInt *
  EvaluateConstantChrecAtConstant(const SCEVAddRecExpr *AddRec, ConstantInt *C,
                                  ScalarEvolution &SE) {
    const SCEV *InVal = SE.getConstant(C);
    const SCEV *Val = AddRec->evaluateAtIteration(InVal, SE);
    assert(isa<SCEVConstant>(Val) &&
           "Evaluation of SCEV at constant didn't fold correctly?");
    return cast<SCEVConstant>(Val)->getValue();
  }
  
  /// Given an exit condition of 'icmp op load X, cst', try to see if we can
  /// compute the backedge execution count.
  ScalarEvolution::ExitLimit
  ScalarEvolution::computeLoadConstantCompareExitLimit(
    LoadInst *LI,
    Constant *RHS,
    const Loop *L,
    ICmpInst::Predicate predicate) {
    if (LI->isVolatile()) return getCouldNotCompute();
  
    // Check to see if the loaded pointer is a getelementptr of a global.
    // TODO: Use SCEV instead of manually grubbing with GEPs.
    GetElementPtrInst *GEP = dyn_cast<GetElementPtrInst>(LI->getOperand(0));
    if (!GEP) return getCouldNotCompute();
  
    // Make sure that it is really a constant global we are gepping, with an
    // initializer, and make sure the first IDX is really 0.
    GlobalVariable *GV = dyn_cast<GlobalVariable>(GEP->getOperand(0));
    if (!GV || !GV->isConstant() || !GV->hasDefinitiveInitializer() ||
        GEP->getNumOperands() < 3 || !isa<Constant>(GEP->getOperand(1)) ||
        !cast<Constant>(GEP->getOperand(1))->isNullValue())
      return getCouldNotCompute();
  
    // Okay, we allow one non-constant index into the GEP instruction.
    Value *VarIdx = nullptr;
    std::vector<Constant*> Indexes;
    unsigned VarIdxNum = 0;
    for (unsigned i = 2, e = GEP->getNumOperands(); i != e; ++i)
      if (ConstantInt *CI = dyn_cast<ConstantInt>(GEP->getOperand(i))) {
        Indexes.push_back(CI);
      } else if (!isa<ConstantInt>(GEP->getOperand(i))) {
        if (VarIdx) return getCouldNotCompute();  // Multiple non-constant idx's.
        VarIdx = GEP->getOperand(i);
        VarIdxNum = i-2;
        Indexes.push_back(nullptr);
      }
  
    // Loop-invariant loads may be a byproduct of loop optimization. Skip them.
    if (!VarIdx)
      return getCouldNotCompute();
  
    // Okay, we know we have a (load (gep GV, 0, X)) comparison with a constant.
    // Check to see if X is a loop variant variable value now.
    const SCEV *Idx = getSCEV(VarIdx);
    Idx = getSCEVAtScope(Idx, L);
  
    // We can only recognize very limited forms of loop index expressions, in
    // particular, only affine AddRec's like {C1,+,C2}<L>.
    const SCEVAddRecExpr *IdxExpr = dyn_cast<SCEVAddRecExpr>(Idx);
    if (!IdxExpr || IdxExpr->getLoop() != L || !IdxExpr->isAffine() ||
        isLoopInvariant(IdxExpr, L) ||
        !isa<SCEVConstant>(IdxExpr->getOperand(0)) ||
        !isa<SCEVConstant>(IdxExpr->getOperand(1)))
      return getCouldNotCompute();
  
    unsigned MaxSteps = MaxBruteForceIterations;
    for (unsigned IterationNum = 0; IterationNum != MaxSteps; ++IterationNum) {
      ConstantInt *ItCst = ConstantInt::get(
                             cast<IntegerType>(IdxExpr->getType()), IterationNum);
      ConstantInt *Val = EvaluateConstantChrecAtConstant(IdxExpr, ItCst, *this);
  
      // Form the GEP offset.
      Indexes[VarIdxNum] = Val;
  
      Constant *Result = ConstantFoldLoadThroughGEPIndices(GV->getInitializer(),
                                                           Indexes);
      if (!Result) break;  // Cannot compute!
  
      // Evaluate the condition for this iteration.
      Result = ConstantExpr::getICmp(predicate, Result, RHS);
      if (!isa<ConstantInt>(Result)) break;  // Couldn't decide for sure
      if (cast<ConstantInt>(Result)->getValue().isMinValue()) {
        ++NumArrayLenItCounts;
        return getConstant(ItCst);   // Found terminating iteration!
      }
    }
    return getCouldNotCompute();
  }
  
  ScalarEvolution::ExitLimit ScalarEvolution::computeShiftCompareExitLimit(
      Value *LHS, Value *RHSV, const Loop *L, ICmpInst::Predicate Pred) {
    ConstantInt *RHS = dyn_cast<ConstantInt>(RHSV);
    if (!RHS)
      return getCouldNotCompute();
  
    const BasicBlock *Latch = L->getLoopLatch();
    if (!Latch)
      return getCouldNotCompute();
  
    const BasicBlock *Predecessor = L->getLoopPredecessor();
    if (!Predecessor)
      return getCouldNotCompute();
  
    // Return true if V is of the form "LHS `shift_op` <positive constant>".
    // Return LHS in OutLHS and shift_opt in OutOpCode.
    auto MatchPositiveShift =
        [](Value *V, Value *&OutLHS, Instruction::BinaryOps &OutOpCode) {
  
      using namespace PatternMatch;
  
      ConstantInt *ShiftAmt;
      if (match(V, m_LShr(m_Value(OutLHS), m_ConstantInt(ShiftAmt))))
        OutOpCode = Instruction::LShr;
      else if (match(V, m_AShr(m_Value(OutLHS), m_ConstantInt(ShiftAmt))))
        OutOpCode = Instruction::AShr;
      else if (match(V, m_Shl(m_Value(OutLHS), m_ConstantInt(ShiftAmt))))
        OutOpCode = Instruction::Shl;
      else
        return false;
  
      return ShiftAmt->getValue().isStrictlyPositive();
    };
  
    // Recognize a "shift recurrence" either of the form %iv or of %iv.shifted in
    //
    // loop:
    //   %iv = phi i32 [ %iv.shifted, %loop ], [ %val, %preheader ]
    //   %iv.shifted = lshr i32 %iv, <positive constant>
    //
    // Return true on a successful match.  Return the corresponding PHI node (%iv
    // above) in PNOut and the opcode of the shift operation in OpCodeOut.
    auto MatchShiftRecurrence =
        [&](Value *V, PHINode *&PNOut, Instruction::BinaryOps &OpCodeOut) {
      Optional<Instruction::BinaryOps> PostShiftOpCode;
  
      {
        Instruction::BinaryOps OpC;
        Value *V;
  
        // If we encounter a shift instruction, "peel off" the shift operation,
        // and remember that we did so.  Later when we inspect %iv's backedge
        // value, we will make sure that the backedge value uses the same
        // operation.
        //
        // Note: the peeled shift operation does not have to be the same
        // instruction as the one feeding into the PHI's backedge value.  We only
        // really care about it being the same *kind* of shift instruction --
        // that's all that is required for our later inferences to hold.
        if (MatchPositiveShift(LHS, V, OpC)) {
          PostShiftOpCode = OpC;
          LHS = V;
        }
      }
  
      PNOut = dyn_cast<PHINode>(LHS);
      if (!PNOut || PNOut->getParent() != L->getHeader())
        return false;
  
      Value *BEValue = PNOut->getIncomingValueForBlock(Latch);
      Value *OpLHS;
  
      return
          // The backedge value for the PHI node must be a shift by a positive
          // amount
          MatchPositiveShift(BEValue, OpLHS, OpCodeOut) &&
  
          // of the PHI node itself
          OpLHS == PNOut &&
  
          // and the kind of shift should be match the kind of shift we peeled
          // off, if any.
          (!PostShiftOpCode.hasValue() || *PostShiftOpCode == OpCodeOut);
    };
  
    PHINode *PN;
    Instruction::BinaryOps OpCode;
    if (!MatchShiftRecurrence(LHS, PN, OpCode))
      return getCouldNotCompute();
  
    const DataLayout &DL = getDataLayout();
  
    // The key rationale for this optimization is that for some kinds of shift
    // recurrences, the value of the recurrence "stabilizes" to either 0 or -1
    // within a finite number of iterations.  If the condition guarding the
    // backedge (in the sense that the backedge is taken if the condition is true)
    // is false for the value the shift recurrence stabilizes to, then we know
    // that the backedge is taken only a finite number of times.
  
    ConstantInt *StableValue = nullptr;
    switch (OpCode) {
    default:
      llvm_unreachable("Impossible case!");
  
    case Instruction::AShr: {
      // {K,ashr,<positive-constant>} stabilizes to signum(K) in at most
      // bitwidth(K) iterations.
      Value *FirstValue = PN->getIncomingValueForBlock(Predecessor);
      KnownBits Known = computeKnownBits(FirstValue, DL, 0, &AC,
                                         Predecessor->getTerminator(), &DT);
      auto *Ty = cast<IntegerType>(RHS->getType());
      if (Known.isNonNegative())
        StableValue = ConstantInt::get(Ty, 0);
      else if (Known.isNegative())
        StableValue = ConstantInt::get(Ty, -1, true);
      else
        return getCouldNotCompute();
  
      break;
    }
    case Instruction::LShr:
    case Instruction::Shl:
      // Both {K,lshr,<positive-constant>} and {K,shl,<positive-constant>}
      // stabilize to 0 in at most bitwidth(K) iterations.
      StableValue = ConstantInt::get(cast<IntegerType>(RHS->getType()), 0);
      break;
    }
  
    auto *Result =
        ConstantFoldCompareInstOperands(Pred, StableValue, RHS, DL, &TLI);
    assert(Result->getType()->isIntegerTy(1) &&
           "Otherwise cannot be an operand to a branch instruction");
  
    if (Result->isZeroValue()) {
      unsigned BitWidth = getTypeSizeInBits(RHS->getType());
      const SCEV *UpperBound =
          getConstant(getEffectiveSCEVType(RHS->getType()), BitWidth);
      return ExitLimit(getCouldNotCompute(), UpperBound, false);
    }
  
    return getCouldNotCompute();
  }
  
  /// Return true if we can constant fold an instruction of the specified type,
  /// assuming that all operands were constants.
  static bool CanConstantFold(const Instruction *I) {
    if (isa<BinaryOperator>(I) || isa<CmpInst>(I) ||
        isa<SelectInst>(I) || isa<CastInst>(I) || isa<GetElementPtrInst>(I) ||
        isa<LoadInst>(I) || isa<ExtractValueInst>(I))
      return true;
  
    if (const CallInst *CI = dyn_cast<CallInst>(I))
      if (const Function *F = CI->getCalledFunction())
        return canConstantFoldCallTo(CI, F);
    return false;
  }
  
  /// Determine whether this instruction can constant evolve within this loop
  /// assuming its operands can all constant evolve.
  static bool canConstantEvolve(Instruction *I, const Loop *L) {
    // An instruction outside of the loop can't be derived from a loop PHI.
    if (!L->contains(I)) return false;
  
    if (isa<PHINode>(I)) {
      // We don't currently keep track of the control flow needed to evaluate
      // PHIs, so we cannot handle PHIs inside of loops.
      return L->getHeader() == I->getParent();
    }
  
    // If we won't be able to constant fold this expression even if the operands
    // are constants, bail early.
    return CanConstantFold(I);
  }
  
  /// getConstantEvolvingPHIOperands - Implement getConstantEvolvingPHI by
  /// recursing through each instruction operand until reaching a loop header phi.
  static PHINode *
  getConstantEvolvingPHIOperands(Instruction *UseInst, const Loop *L,
                                 DenseMap<Instruction *, PHINode *> &PHIMap,
                                 unsigned Depth) {
    if (Depth > MaxConstantEvolvingDepth)
      return nullptr;
  
    // Otherwise, we can evaluate this instruction if all of its operands are
    // constant or derived from a PHI node themselves.
    PHINode *PHI = nullptr;
    for (Value *Op : UseInst->operands()) {
      if (isa<Constant>(Op)) continue;
  
      Instruction *OpInst = dyn_cast<Instruction>(Op);
      if (!OpInst || !canConstantEvolve(OpInst, L)) return nullptr;
  
      PHINode *P = dyn_cast<PHINode>(OpInst);
      if (!P)
        // If this operand is already visited, reuse the prior result.
        // We may have P != PHI if this is the deepest point at which the
        // inconsistent paths meet.
        P = PHIMap.lookup(OpInst);
      if (!P) {
        // Recurse and memoize the results, whether a phi is found or not.
        // This recursive call invalidates pointers into PHIMap.
        P = getConstantEvolvingPHIOperands(OpInst, L, PHIMap, Depth + 1);
        PHIMap[OpInst] = P;
      }
      if (!P)
        return nullptr;  // Not evolving from PHI
      if (PHI && PHI != P)
        return nullptr;  // Evolving from multiple different PHIs.
      PHI = P;
    }
    // This is a expression evolving from a constant PHI!
    return PHI;
  }
  
  /// getConstantEvolvingPHI - Given an LLVM value and a loop, return a PHI node
  /// in the loop that V is derived from.  We allow arbitrary operations along the
  /// way, but the operands of an operation must either be constants or a value
  /// derived from a constant PHI.  If this expression does not fit with these
  /// constraints, return null.
  static PHINode *getConstantEvolvingPHI(Value *V, const Loop *L) {
    Instruction *I = dyn_cast<Instruction>(V);
    if (!I || !canConstantEvolve(I, L)) return nullptr;
  
    if (PHINode *PN = dyn_cast<PHINode>(I))
      return PN;
  
    // Record non-constant instructions contained by the loop.
    DenseMap<Instruction *, PHINode *> PHIMap;
    return getConstantEvolvingPHIOperands(I, L, PHIMap, 0);
  }
  
  /// EvaluateExpression - Given an expression that passes the
  /// getConstantEvolvingPHI predicate, evaluate its value assuming the PHI node
  /// in the loop has the value PHIVal.  If we can't fold this expression for some
  /// reason, return null.
  static Constant *EvaluateExpression(Value *V, const Loop *L,
                                      DenseMap<Instruction *, Constant *> &Vals,
                                      const DataLayout &DL,
                                      const TargetLibraryInfo *TLI) {
    // Convenient constant check, but redundant for recursive calls.
    if (Constant *C = dyn_cast<Constant>(V)) return C;
    Instruction *I = dyn_cast<Instruction>(V);
    if (!I) return nullptr;
  
    if (Constant *C = Vals.lookup(I)) return C;
  
    // An instruction inside the loop depends on a value outside the loop that we
    // weren't given a mapping for, or a value such as a call inside the loop.
    if (!canConstantEvolve(I, L)) return nullptr;
  
    // An unmapped PHI can be due to a branch or another loop inside this loop,
    // or due to this not being the initial iteration through a loop where we
    // couldn't compute the evolution of this particular PHI last time.
    if (isa<PHINode>(I)) return nullptr;
  
    std::vector<Constant*> Operands(I->getNumOperands());
  
    for (unsigned i = 0, e = I->getNumOperands(); i != e; ++i) {
      Instruction *Operand = dyn_cast<Instruction>(I->getOperand(i));
      if (!Operand) {
        Operands[i] = dyn_cast<Constant>(I->getOperand(i));
        if (!Operands[i]) return nullptr;
        continue;
      }
      Constant *C = EvaluateExpression(Operand, L, Vals, DL, TLI);
      Vals[Operand] = C;
      if (!C) return nullptr;
      Operands[i] = C;
    }
  
    if (CmpInst *CI = dyn_cast<CmpInst>(I))
      return ConstantFoldCompareInstOperands(CI->getPredicate(), Operands[0],
                                             Operands[1], DL, TLI);
    if (LoadInst *LI = dyn_cast<LoadInst>(I)) {
      if (!LI->isVolatile())
        return ConstantFoldLoadFromConstPtr(Operands[0], LI->getType(), DL);
    }
    return ConstantFoldInstOperands(I, Operands, DL, TLI);
  }
  
  
  // If every incoming value to PN except the one for BB is a specific Constant,
  // return that, else return nullptr.
  static Constant *getOtherIncomingValue(PHINode *PN, BasicBlock *BB) {
    Constant *IncomingVal = nullptr;
  
    for (unsigned i = 0, e = PN->getNumIncomingValues(); i != e; ++i) {
      if (PN->getIncomingBlock(i) == BB)
        continue;
  
      auto *CurrentVal = dyn_cast<Constant>(PN->getIncomingValue(i));
      if (!CurrentVal)
        return nullptr;
  
      if (IncomingVal != CurrentVal) {
        if (IncomingVal)
          return nullptr;
        IncomingVal = CurrentVal;
      }
    }
  
    return IncomingVal;
  }
  
  /// getConstantEvolutionLoopExitValue - If we know that the specified Phi is
  /// in the header of its containing loop, we know the loop executes a
  /// constant number of times, and the PHI node is just a recurrence
  /// involving constants, fold it.
  Constant *
  ScalarEvolution::getConstantEvolutionLoopExitValue(PHINode *PN,
                                                     const APInt &BEs,
                                                     const Loop *L) {
    auto I = ConstantEvolutionLoopExitValue.find(PN);
    if (I != ConstantEvolutionLoopExitValue.end())
      return I->second;
  
    if (BEs.ugt(MaxBruteForceIterations))
      return ConstantEvolutionLoopExitValue[PN] = nullptr;  // Not going to evaluate it.
  
    Constant *&RetVal = ConstantEvolutionLoopExitValue[PN];
  
    DenseMap<Instruction *, Constant *> CurrentIterVals;
    BasicBlock *Header = L->getHeader();
    assert(PN->getParent() == Header && "Can't evaluate PHI not in loop header!");
  
    BasicBlock *Latch = L->getLoopLatch();
    if (!Latch)
      return nullptr;
  
    for (PHINode &PHI : Header->phis()) {
      if (auto *StartCST = getOtherIncomingValue(&PHI, Latch))
        CurrentIterVals[&PHI] = StartCST;
    }
    if (!CurrentIterVals.count(PN))
      return RetVal = nullptr;
  
    Value *BEValue = PN->getIncomingValueForBlock(Latch);
  
    // Execute the loop symbolically to determine the exit value.
    assert(BEs.getActiveBits() < CHAR_BIT * sizeof(unsigned) &&
           "BEs is <= MaxBruteForceIterations which is an 'unsigned'!");
  
    unsigned NumIterations = BEs.getZExtValue(); // must be in range
    unsigned IterationNum = 0;
    const DataLayout &DL = getDataLayout();
    for (; ; ++IterationNum) {
      if (IterationNum == NumIterations)
        return RetVal = CurrentIterVals[PN];  // Got exit value!
  
      // Compute the value of the PHIs for the next iteration.
      // EvaluateExpression adds non-phi values to the CurrentIterVals map.
      DenseMap<Instruction *, Constant *> NextIterVals;
      Constant *NextPHI =
          EvaluateExpression(BEValue, L, CurrentIterVals, DL, &TLI);
      if (!NextPHI)
        return nullptr;        // Couldn't evaluate!
      NextIterVals[PN] = NextPHI;
  
      bool StoppedEvolving = NextPHI == CurrentIterVals[PN];
  
      // Also evaluate the other PHI nodes.  However, we don't get to stop if we
      // cease to be able to evaluate one of them or if they stop evolving,
      // because that doesn't necessarily prevent us from computing PN.
      SmallVector<std::pair<PHINode *, Constant *>, 8> PHIsToCompute;
      for (const auto &I : CurrentIterVals) {
        PHINode *PHI = dyn_cast<PHINode>(I.first);
        if (!PHI || PHI == PN || PHI->getParent() != Header) continue;
        PHIsToCompute.emplace_back(PHI, I.second);
      }
      // We use two distinct loops because EvaluateExpression may invalidate any
      // iterators into CurrentIterVals.
      for (const auto &I : PHIsToCompute) {
        PHINode *PHI = I.first;
        Constant *&NextPHI = NextIterVals[PHI];
        if (!NextPHI) {   // Not already computed.
          Value *BEValue = PHI->getIncomingValueForBlock(Latch);
          NextPHI = EvaluateExpression(BEValue, L, CurrentIterVals, DL, &TLI);
        }
        if (NextPHI != I.second)
          StoppedEvolving = false;
      }
  
      // If all entries in CurrentIterVals == NextIterVals then we can stop
      // iterating, the loop can't continue to change.
      if (StoppedEvolving)
        return RetVal = CurrentIterVals[PN];
  
      CurrentIterVals.swap(NextIterVals);
    }
  }
  
  const SCEV *ScalarEvolution::computeExitCountExhaustively(const Loop *L,
                                                            Value *Cond,
                                                            bool ExitWhen) {
    PHINode *PN = getConstantEvolvingPHI(Cond, L);
    if (!PN) return getCouldNotCompute();
  
    // If the loop is canonicalized, the PHI will have exactly two entries.
    // That's the only form we support here.
    if (PN->getNumIncomingValues() != 2) return getCouldNotCompute();
  
    DenseMap<Instruction *, Constant *> CurrentIterVals;
    BasicBlock *Header = L->getHeader();
    assert(PN->getParent() == Header && "Can't evaluate PHI not in loop header!");
  
    BasicBlock *Latch = L->getLoopLatch();
    assert(Latch && "Should follow from NumIncomingValues == 2!");
  
    for (PHINode &PHI : Header->phis()) {
      if (auto *StartCST = getOtherIncomingValue(&PHI, Latch))
        CurrentIterVals[&PHI] = StartCST;
    }
    if (!CurrentIterVals.count(PN))
      return getCouldNotCompute();
  
    // Okay, we find a PHI node that defines the trip count of this loop.  Execute
    // the loop symbolically to determine when the condition gets a value of
    // "ExitWhen".
    unsigned MaxIterations = MaxBruteForceIterations;   // Limit analysis.
    const DataLayout &DL = getDataLayout();
    for (unsigned IterationNum = 0; IterationNum != MaxIterations;++IterationNum){
      auto *CondVal = dyn_cast_or_null<ConstantInt>(
          EvaluateExpression(Cond, L, CurrentIterVals, DL, &TLI));
  
      // Couldn't symbolically evaluate.
      if (!CondVal) return getCouldNotCompute();
  
      if (CondVal->getValue() == uint64_t(ExitWhen)) {
        ++NumBruteForceTripCountsComputed;
        return getConstant(Type::getInt32Ty(getContext()), IterationNum);
      }
  
      // Update all the PHI nodes for the next iteration.
      DenseMap<Instruction *, Constant *> NextIterVals;
  
      // Create a list of which PHIs we need to compute. We want to do this before
      // calling EvaluateExpression on them because that may invalidate iterators
      // into CurrentIterVals.
      SmallVector<PHINode *, 8> PHIsToCompute;
      for (const auto &I : CurrentIterVals) {
        PHINode *PHI = dyn_cast<PHINode>(I.first);
        if (!PHI || PHI->getParent() != Header) continue;
        PHIsToCompute.push_back(PHI);
      }
      for (PHINode *PHI : PHIsToCompute) {
        Constant *&NextPHI = NextIterVals[PHI];
        if (NextPHI) continue;    // Already computed!
  
        Value *BEValue = PHI->getIncomingValueForBlock(Latch);
        NextPHI = EvaluateExpression(BEValue, L, CurrentIterVals, DL, &TLI);
      }
      CurrentIterVals.swap(NextIterVals);
    }
  
    // Too many iterations were needed to evaluate.
    return getCouldNotCompute();
  }
  
  const SCEV *ScalarEvolution::getSCEVAtScope(const SCEV *V, const Loop *L) {
    SmallVector<std::pair<const Loop *, const SCEV *>, 2> &Values =
        ValuesAtScopes[V];
    // Check to see if we've folded this expression at this loop before.
    for (auto &LS : Values)
      if (LS.first == L)
        return LS.second ? LS.second : V;
  
    Values.emplace_back(L, nullptr);
  
    // Otherwise compute it.
    const SCEV *C = computeSCEVAtScope(V, L);
    for (auto &LS : reverse(ValuesAtScopes[V]))
      if (LS.first == L) {
        LS.second = C;
        break;
      }
    return C;
  }
  
  /// This builds up a Constant using the ConstantExpr interface.  That way, we
  /// will return Constants for objects which aren't represented by a
  /// SCEVConstant, because SCEVConstant is restricted to ConstantInt.
  /// Returns NULL if the SCEV isn't representable as a Constant.
  static Constant *BuildConstantFromSCEV(const SCEV *V) {
    switch (V->getSCEVType()) {
    case scCouldNotCompute:
    case scAddRecExpr:
      return nullptr;
    case scConstant:
      return cast<SCEVConstant>(V)->getValue();
    case scUnknown:
      return dyn_cast<Constant>(cast<SCEVUnknown>(V)->getValue());
    case scSignExtend: {
      const SCEVSignExtendExpr *SS = cast<SCEVSignExtendExpr>(V);
      if (Constant *CastOp = BuildConstantFromSCEV(SS->getOperand()))
        return ConstantExpr::getSExt(CastOp, SS->getType());
      return nullptr;
    }
    case scZeroExtend: {
      const SCEVZeroExtendExpr *SZ = cast<SCEVZeroExtendExpr>(V);
      if (Constant *CastOp = BuildConstantFromSCEV(SZ->getOperand()))
        return ConstantExpr::getZExt(CastOp, SZ->getType());
      return nullptr;
    }
    case scPtrToInt: {
      const SCEVPtrToIntExpr *P2I = cast<SCEVPtrToIntExpr>(V);
      if (Constant *CastOp = BuildConstantFromSCEV(P2I->getOperand()))
        return ConstantExpr::getPtrToInt(CastOp, P2I->getType());
  
      return nullptr;
    }
    case scTruncate: {
      const SCEVTruncateExpr *ST = cast<SCEVTruncateExpr>(V);
      if (Constant *CastOp = BuildConstantFromSCEV(ST->getOperand()))
        return ConstantExpr::getTrunc(CastOp, ST->getType());
      return nullptr;
    }
    case scAddExpr: {
      const SCEVAddExpr *SA = cast<SCEVAddExpr>(V);
      if (Constant *C = BuildConstantFromSCEV(SA->getOperand(0))) {
        if (PointerType *PTy = dyn_cast<PointerType>(C->getType())) {
          unsigned AS = PTy->getAddressSpace();
          Type *DestPtrTy = Type::getInt8PtrTy(C->getContext(), AS);
          C = ConstantExpr::getBitCast(C, DestPtrTy);
        }
        for (unsigned i = 1, e = SA->getNumOperands(); i != e; ++i) {
          Constant *C2 = BuildConstantFromSCEV(SA->getOperand(i));
          if (!C2)
            return nullptr;
  
          // First pointer!
          if (!C->getType()->isPointerTy() && C2->getType()->isPointerTy()) {
            unsigned AS = C2->getType()->getPointerAddressSpace();
            std::swap(C, C2);
            Type *DestPtrTy = Type::getInt8PtrTy(C->getContext(), AS);
            // The offsets have been converted to bytes.  We can add bytes to an
            // i8* by GEP with the byte count in the first index.
            C = ConstantExpr::getBitCast(C, DestPtrTy);
          }
  
          // Don't bother trying to sum two pointers. We probably can't
          // statically compute a load that results from it anyway.
          if (C2->getType()->isPointerTy())
            return nullptr;
  
          if (C->getType()->isPointerTy()) {
            C = ConstantExpr::getGetElementPtr(Type::getInt8Ty(C->getContext()),
                                               C, C2);
          } else {
            C = ConstantExpr::getAdd(C, C2);
          }
        }
        return C;
      }
      return nullptr;
    }
    case scMulExpr: {
      const SCEVMulExpr *SM = cast<SCEVMulExpr>(V);
      if (Constant *C = BuildConstantFromSCEV(SM->getOperand(0))) {
        // Don't bother with pointers at all.
        if (C->getType()->isPointerTy())
          return nullptr;
        for (unsigned i = 1, e = SM->getNumOperands(); i != e; ++i) {
          Constant *C2 = BuildConstantFromSCEV(SM->getOperand(i));
          if (!C2 || C2->getType()->isPointerTy())
            return nullptr;
          C = ConstantExpr::getMul(C, C2);
        }
        return C;
      }
      return nullptr;
    }
    case scUDivExpr: {
      const SCEVUDivExpr *SU = cast<SCEVUDivExpr>(V);
      if (Constant *LHS = BuildConstantFromSCEV(SU->getLHS()))
        if (Constant *RHS = BuildConstantFromSCEV(SU->getRHS()))
          if (LHS->getType() == RHS->getType())
            return ConstantExpr::getUDiv(LHS, RHS);
      return nullptr;
    }
    case scSMaxExpr:
    case scUMaxExpr:
    case scSMinExpr:
    case scUMinExpr:
      return nullptr; // TODO: smax, umax, smin, umax.
    }
    llvm_unreachable("Unknown SCEV kind!");
  }
  
  const SCEV *ScalarEvolution::computeSCEVAtScope(const SCEV *V, const Loop *L) {
    if (isa<SCEVConstant>(V)) return V;
  
    // If this instruction is evolved from a constant-evolving PHI, compute the
    // exit value from the loop without using SCEVs.
    if (const SCEVUnknown *SU = dyn_cast<SCEVUnknown>(V)) {
      if (Instruction *I = dyn_cast<Instruction>(SU->getValue())) {
        if (PHINode *PN = dyn_cast<PHINode>(I)) {
          const Loop *CurrLoop = this->LI[I->getParent()];
          // Looking for loop exit value.
          if (CurrLoop && CurrLoop->getParentLoop() == L &&
              PN->getParent() == CurrLoop->getHeader()) {
            // Okay, there is no closed form solution for the PHI node.  Check
            // to see if the loop that contains it has a known backedge-taken
            // count.  If so, we may be able to force computation of the exit
            // value.
            const SCEV *BackedgeTakenCount = getBackedgeTakenCount(CurrLoop);
            // This trivial case can show up in some degenerate cases where
            // the incoming IR has not yet been fully simplified.
            if (BackedgeTakenCount->isZero()) {
              Value *InitValue = nullptr;
              bool MultipleInitValues = false;
              for (unsigned i = 0; i < PN->getNumIncomingValues(); i++) {
                if (!CurrLoop->contains(PN->getIncomingBlock(i))) {
                  if (!InitValue)
                    InitValue = PN->getIncomingValue(i);
                  else if (InitValue != PN->getIncomingValue(i)) {
                    MultipleInitValues = true;
                    break;
                  }
                }
              }
              if (!MultipleInitValues && InitValue)
                return getSCEV(InitValue);
            }
            // Do we have a loop invariant value flowing around the backedge
            // for a loop which must execute the backedge?
            if (!isa<SCEVCouldNotCompute>(BackedgeTakenCount) &&
                isKnownPositive(BackedgeTakenCount) &&
                PN->getNumIncomingValues() == 2) {
  
              unsigned InLoopPred =
                  CurrLoop->contains(PN->getIncomingBlock(0)) ? 0 : 1;
              Value *BackedgeVal = PN->getIncomingValue(InLoopPred);
              if (CurrLoop->isLoopInvariant(BackedgeVal))
                return getSCEV(BackedgeVal);
            }
            if (auto *BTCC = dyn_cast<SCEVConstant>(BackedgeTakenCount)) {
              // Okay, we know how many times the containing loop executes.  If
              // this is a constant evolving PHI node, get the final value at
              // the specified iteration number.
              Constant *RV = getConstantEvolutionLoopExitValue(
                  PN, BTCC->getAPInt(), CurrLoop);
              if (RV) return getSCEV(RV);
            }
          }
  
          // If there is a single-input Phi, evaluate it at our scope. If we can
          // prove that this replacement does not break LCSSA form, use new value.
          if (PN->getNumOperands() == 1) {
            const SCEV *Input = getSCEV(PN->getOperand(0));
            const SCEV *InputAtScope = getSCEVAtScope(Input, L);
            // TODO: We can generalize it using LI.replacementPreservesLCSSAForm,
            // for the simplest case just support constants.
            if (isa<SCEVConstant>(InputAtScope)) return InputAtScope;
          }
        }
  
        // Okay, this is an expression that we cannot symbolically evaluate
        // into a SCEV.  Check to see if it's possible to symbolically evaluate
        // the arguments into constants, and if so, try to constant propagate the
        // result.  This is particularly useful for computing loop exit values.
        if (CanConstantFold(I)) {
          SmallVector<Constant *, 4> Operands;
          bool MadeImprovement = false;
          for (Value *Op : I->operands()) {
            if (Constant *C = dyn_cast<Constant>(Op)) {
              Operands.push_back(C);
              continue;
            }
  
            // If any of the operands is non-constant and if they are
            // non-integer and non-pointer, don't even try to analyze them
            // with scev techniques.
            if (!isSCEVable(Op->getType()))
              return V;
  
            const SCEV *OrigV = getSCEV(Op);
            const SCEV *OpV = getSCEVAtScope(OrigV, L);
            MadeImprovement |= OrigV != OpV;
  
            Constant *C = BuildConstantFromSCEV(OpV);
            if (!C) return V;
            if (C->getType() != Op->getType())
              C = ConstantExpr::getCast(CastInst::getCastOpcode(C, false,
                                                                Op->getType(),
                                                                false),
                                        C, Op->getType());
            Operands.push_back(C);
          }
  
          // Check to see if getSCEVAtScope actually made an improvement.
          if (MadeImprovement) {
            Constant *C = nullptr;
            const DataLayout &DL = getDataLayout();
            if (const CmpInst *CI = dyn_cast<CmpInst>(I))
              C = ConstantFoldCompareInstOperands(CI->getPredicate(), Operands[0],
                                                  Operands[1], DL, &TLI);
            else if (const LoadInst *Load = dyn_cast<LoadInst>(I)) {
              if (!Load->isVolatile())
                C = ConstantFoldLoadFromConstPtr(Operands[0], Load->getType(),
                                                 DL);
            } else
              C = ConstantFoldInstOperands(I, Operands, DL, &TLI);
            if (!C) return V;
            return getSCEV(C);
          }
        }
      }
  
      // This is some other type of SCEVUnknown, just return it.
      return V;
    }
  
    if (const SCEVCommutativeExpr *Comm = dyn_cast<SCEVCommutativeExpr>(V)) {
      // Avoid performing the look-up in the common case where the specified
      // expression has no loop-variant portions.
      for (unsigned i = 0, e = Comm->getNumOperands(); i != e; ++i) {
        const SCEV *OpAtScope = getSCEVAtScope(Comm->getOperand(i), L);
        if (OpAtScope != Comm->getOperand(i)) {
          // Okay, at least one of these operands is loop variant but might be
          // foldable.  Build a new instance of the folded commutative expression.
          SmallVector<const SCEV *, 8> NewOps(Comm->op_begin(),
                                              Comm->op_begin()+i);
          NewOps.push_back(OpAtScope);
  
          for (++i; i != e; ++i) {
            OpAtScope = getSCEVAtScope(Comm->getOperand(i), L);
            NewOps.push_back(OpAtScope);
          }
          if (isa<SCEVAddExpr>(Comm))
            return getAddExpr(NewOps, Comm->getNoWrapFlags());
          if (isa<SCEVMulExpr>(Comm))
            return getMulExpr(NewOps, Comm->getNoWrapFlags());
          if (isa<SCEVMinMaxExpr>(Comm))
            return getMinMaxExpr(Comm->getSCEVType(), NewOps);
          llvm_unreachable("Unknown commutative SCEV type!");
        }
      }
      // If we got here, all operands are loop invariant.
      return Comm;
    }
  
    if (const SCEVUDivExpr *Div = dyn_cast<SCEVUDivExpr>(V)) {
      const SCEV *LHS = getSCEVAtScope(Div->getLHS(), L);
      const SCEV *RHS = getSCEVAtScope(Div->getRHS(), L);
      if (LHS == Div->getLHS() && RHS == Div->getRHS())
        return Div;   // must be loop invariant
      return getUDivExpr(LHS, RHS);
    }
  
    // If this is a loop recurrence for a loop that does not contain L, then we
    // are dealing with the final value computed by the loop.
    if (const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(V)) {
      // First, attempt to evaluate each operand.
      // Avoid performing the look-up in the common case where the specified
      // expression has no loop-variant portions.
      for (unsigned i = 0, e = AddRec->getNumOperands(); i != e; ++i) {
        const SCEV *OpAtScope = getSCEVAtScope(AddRec->getOperand(i), L);
        if (OpAtScope == AddRec->getOperand(i))
          continue;
  
        // Okay, at least one of these operands is loop variant but might be
        // foldable.  Build a new instance of the folded commutative expression.
        SmallVector<const SCEV *, 8> NewOps(AddRec->op_begin(),
                                            AddRec->op_begin()+i);
        NewOps.push_back(OpAtScope);
        for (++i; i != e; ++i)
          NewOps.push_back(getSCEVAtScope(AddRec->getOperand(i), L));
  
        const SCEV *FoldedRec =
          getAddRecExpr(NewOps, AddRec->getLoop(),
                        AddRec->getNoWrapFlags(SCEV::FlagNW));
        AddRec = dyn_cast<SCEVAddRecExpr>(FoldedRec);
        // The addrec may be folded to a nonrecurrence, for example, if the
        // induction variable is multiplied by zero after constant folding. Go
        // ahead and return the folded value.
        if (!AddRec)
          return FoldedRec;
        break;
      }
  
      // If the scope is outside the addrec's loop, evaluate it by using the
      // loop exit value of the addrec.
      if (!AddRec->getLoop()->contains(L)) {
        // To evaluate this recurrence, we need to know how many times the AddRec
        // loop iterates.  Compute this now.
        const SCEV *BackedgeTakenCount = getBackedgeTakenCount(AddRec->getLoop());
        if (BackedgeTakenCount == getCouldNotCompute()) return AddRec;
  
        // Then, evaluate the AddRec.
        return AddRec->evaluateAtIteration(BackedgeTakenCount, *this);
      }
  
      return AddRec;
    }
  
    if (const SCEVZeroExtendExpr *Cast = dyn_cast<SCEVZeroExtendExpr>(V)) {
      const SCEV *Op = getSCEVAtScope(Cast->getOperand(), L);
      if (Op == Cast->getOperand())
        return Cast;  // must be loop invariant
      return getZeroExtendExpr(Op, Cast->getType());
    }
  
    if (const SCEVSignExtendExpr *Cast = dyn_cast<SCEVSignExtendExpr>(V)) {
      const SCEV *Op = getSCEVAtScope(Cast->getOperand(), L);
      if (Op == Cast->getOperand())
        return Cast;  // must be loop invariant
      return getSignExtendExpr(Op, Cast->getType());
    }
  
    if (const SCEVTruncateExpr *Cast = dyn_cast<SCEVTruncateExpr>(V)) {
      const SCEV *Op = getSCEVAtScope(Cast->getOperand(), L);
      if (Op == Cast->getOperand())
        return Cast;  // must be loop invariant
      return getTruncateExpr(Op, Cast->getType());
    }
  
    if (const SCEVPtrToIntExpr *Cast = dyn_cast<SCEVPtrToIntExpr>(V)) {
      const SCEV *Op = getSCEVAtScope(Cast->getOperand(), L);
      if (Op == Cast->getOperand())
        return Cast; // must be loop invariant
      return getPtrToIntExpr(Op, Cast->getType());
    }
  
    llvm_unreachable("Unknown SCEV type!");
  }
  
  const SCEV *ScalarEvolution::getSCEVAtScope(Value *V, const Loop *L) {
    return getSCEVAtScope(getSCEV(V), L);
  }
  
  const SCEV *ScalarEvolution::stripInjectiveFunctions(const SCEV *S) const {
    if (const SCEVZeroExtendExpr *ZExt = dyn_cast<SCEVZeroExtendExpr>(S))
      return stripInjectiveFunctions(ZExt->getOperand());
    if (const SCEVSignExtendExpr *SExt = dyn_cast<SCEVSignExtendExpr>(S))
      return stripInjectiveFunctions(SExt->getOperand());
    return S;
  }
  
  /// Finds the minimum unsigned root of the following equation:
  ///
  ///     A * X = B (mod N)
  ///
  /// where N = 2^BW and BW is the common bit width of A and B. The signedness of
  /// A and B isn't important.
  ///
  /// If the equation does not have a solution, SCEVCouldNotCompute is returned.
  static const SCEV *SolveLinEquationWithOverflow(const APInt &A, const SCEV *B,
                                                 ScalarEvolution &SE) {
    uint32_t BW = A.getBitWidth();
    assert(BW == SE.getTypeSizeInBits(B->getType()));
    assert(A != 0 && "A must be non-zero.");
  
    // 1. D = gcd(A, N)
    //
    // The gcd of A and N may have only one prime factor: 2. The number of
    // trailing zeros in A is its multiplicity
    uint32_t Mult2 = A.countTrailingZeros();
    // D = 2^Mult2
  
    // 2. Check if B is divisible by D.
    //
    // B is divisible by D if and only if the multiplicity of prime factor 2 for B
    // is not less than multiplicity of this prime factor for D.
    if (SE.GetMinTrailingZeros(B) < Mult2)
      return SE.getCouldNotCompute();
  
    // 3. Compute I: the multiplicative inverse of (A / D) in arithmetic
    // modulo (N / D).
    //
    // If D == 1, (N / D) == N == 2^BW, so we need one extra bit to represent
    // (N / D) in general. The inverse itself always fits into BW bits, though,
    // so we immediately truncate it.
    APInt AD = A.lshr(Mult2).zext(BW + 1);  // AD = A / D
    APInt Mod(BW + 1, 0);
    Mod.setBit(BW - Mult2);  // Mod = N / D
    APInt I = AD.multiplicativeInverse(Mod).trunc(BW);
  
    // 4. Compute the minimum unsigned root of the equation:
    // I * (B / D) mod (N / D)
    // To simplify the computation, we factor out the divide by D:
    // (I * B mod N) / D
    const SCEV *D = SE.getConstant(APInt::getOneBitSet(BW, Mult2));
    return SE.getUDivExactExpr(SE.getMulExpr(B, SE.getConstant(I)), D);
  }
  
  /// For a given quadratic addrec, generate coefficients of the corresponding
  /// quadratic equation, multiplied by a common value to ensure that they are
  /// integers.
  /// The returned value is a tuple { A, B, C, M, BitWidth }, where
  /// Ax^2 + Bx + C is the quadratic function, M is the value that A, B and C
  /// were multiplied by, and BitWidth is the bit width of the original addrec
  /// coefficients.
  /// This function returns None if the addrec coefficients are not compile-
  /// time constants.
  static Optional<std::tuple<APInt, APInt, APInt, APInt, unsigned>>
  GetQuadraticEquation(const SCEVAddRecExpr *AddRec) {
    assert(AddRec->getNumOperands() == 3 && "This is not a quadratic chrec!");
    const SCEVConstant *LC = dyn_cast<SCEVConstant>(AddRec->getOperand(0));
    const SCEVConstant *MC = dyn_cast<SCEVConstant>(AddRec->getOperand(1));
    const SCEVConstant *NC = dyn_cast<SCEVConstant>(AddRec->getOperand(2));
    LLVM_DEBUG(dbgs() << __func__ << ": analyzing quadratic addrec: "
                      << *AddRec << '\n');
  
    // We currently can only solve this if the coefficients are constants.
    if (!LC || !MC || !NC) {
      LLVM_DEBUG(dbgs() << __func__ << ": coefficients are not constant\n");
      return None;
    }
  
    APInt L = LC->getAPInt();
    APInt M = MC->getAPInt();
    APInt N = NC->getAPInt();
    assert(!N.isNullValue() && "This is not a quadratic addrec");
  
    unsigned BitWidth = LC->getAPInt().getBitWidth();
    unsigned NewWidth = BitWidth + 1;
    LLVM_DEBUG(dbgs() << __func__ << ": addrec coeff bw: "
                      << BitWidth << '\n');
    // The sign-extension (as opposed to a zero-extension) here matches the
    // extension used in SolveQuadraticEquationWrap (with the same motivation).
    N = N.sext(NewWidth);
    M = M.sext(NewWidth);
    L = L.sext(NewWidth);
  
    // The increments are M, M+N, M+2N, ..., so the accumulated values are
    //   L+M, (L+M)+(M+N), (L+M)+(M+N)+(M+2N), ..., that is,
    //   L+M, L+2M+N, L+3M+3N, ...
    // After n iterations the accumulated value Acc is L + nM + n(n-1)/2 N.
    //
    // The equation Acc = 0 is then
    //   L + nM + n(n-1)/2 N = 0,  or  2L + 2M n + n(n-1) N = 0.
    // In a quadratic form it becomes:
    //   N n^2 + (2M-N) n + 2L = 0.
  
    APInt A = N;
    APInt B = 2 * M - A;
    APInt C = 2 * L;
    APInt T = APInt(NewWidth, 2);
    LLVM_DEBUG(dbgs() << __func__ << ": equation " << A << "x^2 + " << B
                      << "x + " << C << ", coeff bw: " << NewWidth
                      << ", multiplied by " << T << '\n');
    return std::make_tuple(A, B, C, T, BitWidth);
  }
  
  /// Helper function to compare optional APInts:
  /// (a) if X and Y both exist, return min(X, Y),
  /// (b) if neither X nor Y exist, return None,
  /// (c) if exactly one of X and Y exists, return that value.
  static Optional<APInt> MinOptional(Optional<APInt> X, Optional<APInt> Y) {
    if (X.hasValue() && Y.hasValue()) {
      unsigned W = std::max(X->getBitWidth(), Y->getBitWidth());
      APInt XW = X->sextOrSelf(W);
      APInt YW = Y->sextOrSelf(W);
      return XW.slt(YW) ? *X : *Y;
    }
    if (!X.hasValue() && !Y.hasValue())
      return None;
    return X.hasValue() ? *X : *Y;
  }
  
  /// Helper function to truncate an optional APInt to a given BitWidth.
  /// When solving addrec-related equations, it is preferable to return a value
  /// that has the same bit width as the original addrec's coefficients. If the
  /// solution fits in the original bit width, truncate it (except for i1).
  /// Returning a value of a different bit width may inhibit some optimizations.
  ///
  /// In general, a solution to a quadratic equation generated from an addrec
  /// may require BW+1 bits, where BW is the bit width of the addrec's
  /// coefficients. The reason is that the coefficients of the quadratic
  /// equation are BW+1 bits wide (to avoid truncation when converting from
  /// the addrec to the equation).
  static Optional<APInt> TruncIfPossible(Optional<APInt> X, unsigned BitWidth) {
    if (!X.hasValue())
      return None;
    unsigned W = X->getBitWidth();
    if (BitWidth > 1 && BitWidth < W && X->isIntN(BitWidth))
      return X->trunc(BitWidth);
    return X;
  }
  
  /// Let c(n) be the value of the quadratic chrec {L,+,M,+,N} after n
  /// iterations. The values L, M, N are assumed to be signed, and they
  /// should all have the same bit widths.
  /// Find the least n >= 0 such that c(n) = 0 in the arithmetic modulo 2^BW,
  /// where BW is the bit width of the addrec's coefficients.
  /// If the calculated value is a BW-bit integer (for BW > 1), it will be
  /// returned as such, otherwise the bit width of the returned value may
  /// be greater than BW.
  ///
  /// This function returns None if
  /// (a) the addrec coefficients are not constant, or
  /// (b) SolveQuadraticEquationWrap was unable to find a solution. For cases
  ///     like x^2 = 5, no integer solutions exist, in other cases an integer
  ///     solution may exist, but SolveQuadraticEquationWrap may fail to find it.
  static Optional<APInt>
  SolveQuadraticAddRecExact(const SCEVAddRecExpr *AddRec, ScalarEvolution &SE) {
    APInt A, B, C, M;
    unsigned BitWidth;
    auto T = GetQuadraticEquation(AddRec);
    if (!T.hasValue())
      return None;
  
    std::tie(A, B, C, M, BitWidth) = *T;
    LLVM_DEBUG(dbgs() << __func__ << ": solving for unsigned overflow\n");
    Optional<APInt> X = APIntOps::SolveQuadraticEquationWrap(A, B, C, BitWidth+1);
    if (!X.hasValue())
      return None;
  
    ConstantInt *CX = ConstantInt::get(SE.getContext(), *X);
    ConstantInt *V = EvaluateConstantChrecAtConstant(AddRec, CX, SE);
    if (!V->isZero())
      return None;
  
    return TruncIfPossible(X, BitWidth);
  }
  
  /// Let c(n) be the value of the quadratic chrec {0,+,M,+,N} after n
  /// iterations. The values M, N are assumed to be signed, and they
  /// should all have the same bit widths.
  /// Find the least n such that c(n) does not belong to the given range,
  /// while c(n-1) does.
  ///
  /// This function returns None if
  /// (a) the addrec coefficients are not constant, or
  /// (b) SolveQuadraticEquationWrap was unable to find a solution for the
  ///     bounds of the range.
  static Optional<APInt>
  SolveQuadraticAddRecRange(const SCEVAddRecExpr *AddRec,
                            const ConstantRange &Range, ScalarEvolution &SE) {
    assert(AddRec->getOperand(0)->isZero() &&
           "Starting value of addrec should be 0");
    LLVM_DEBUG(dbgs() << __func__ << ": solving boundary crossing for range "
                      << Range << ", addrec " << *AddRec << '\n');
    // This case is handled in getNumIterationsInRange. Here we can assume that
    // we start in the range.
    assert(Range.contains(APInt(SE.getTypeSizeInBits(AddRec->getType()), 0)) &&
           "Addrec's initial value should be in range");
  
    APInt A, B, C, M;
    unsigned BitWidth;
    auto T = GetQuadraticEquation(AddRec);
    if (!T.hasValue())
      return None;
  
    // Be careful about the return value: there can be two reasons for not
    // returning an actual number. First, if no solutions to the equations
    // were found, and second, if the solutions don't leave the given range.
    // The first case means that the actual solution is "unknown", the second
    // means that it's known, but not valid. If the solution is unknown, we
    // cannot make any conclusions.
    // Return a pair: the optional solution and a flag indicating if the
    // solution was found.
    auto SolveForBoundary = [&](APInt Bound) -> std::pair<Optional<APInt>,bool> {
      // Solve for signed overflow and unsigned overflow, pick the lower
      // solution.
      LLVM_DEBUG(dbgs() << "SolveQuadraticAddRecRange: checking boundary "
                        << Bound << " (before multiplying by " << M << ")\n");
      Bound *= M; // The quadratic equation multiplier.
  
      Optional<APInt> SO = None;
      if (BitWidth > 1) {
        LLVM_DEBUG(dbgs() << "SolveQuadraticAddRecRange: solving for "
                             "signed overflow\n");
        SO = APIntOps::SolveQuadraticEquationWrap(A, B, -Bound, BitWidth);
      }
      LLVM_DEBUG(dbgs() << "SolveQuadraticAddRecRange: solving for "
                           "unsigned overflow\n");
      Optional<APInt> UO = APIntOps::SolveQuadraticEquationWrap(A, B, -Bound,
                                                                BitWidth+1);
  
      auto LeavesRange = [&] (const APInt &X) {
        ConstantInt *C0 = ConstantInt::get(SE.getContext(), X);
        ConstantInt *V0 = EvaluateConstantChrecAtConstant(AddRec, C0, SE);
        if (Range.contains(V0->getValue()))
          return false;
        // X should be at least 1, so X-1 is non-negative.
        ConstantInt *C1 = ConstantInt::get(SE.getContext(), X-1);
        ConstantInt *V1 = EvaluateConstantChrecAtConstant(AddRec, C1, SE);
        if (Range.contains(V1->getValue()))
          return true;
        return false;
      };
  
      // If SolveQuadraticEquationWrap returns None, it means that there can
      // be a solution, but the function failed to find it. We cannot treat it
      // as "no solution".
      if (!SO.hasValue() || !UO.hasValue())
        return { None, false };
  
      // Check the smaller value first to see if it leaves the range.
      // At this point, both SO and UO must have values.
      Optional<APInt> Min = MinOptional(SO, UO);
      if (LeavesRange(*Min))
        return { Min, true };
      Optional<APInt> Max = Min == SO ? UO : SO;
      if (LeavesRange(*Max))
        return { Max, true };
  
      // Solutions were found, but were eliminated, hence the "true".
      return { None, true };
    };
  
    std::tie(A, B, C, M, BitWidth) = *T;
    // Lower bound is inclusive, subtract 1 to represent the exiting value.
    APInt Lower = Range.getLower().sextOrSelf(A.getBitWidth()) - 1;
    APInt Upper = Range.getUpper().sextOrSelf(A.getBitWidth());
    auto SL = SolveForBoundary(Lower);
    auto SU = SolveForBoundary(Upper);
    // If any of the solutions was unknown, no meaninigful conclusions can
    // be made.
    if (!SL.second || !SU.second)
      return None;
  
    // Claim: The correct solution is not some value between Min and Max.
    //
    // Justification: Assuming that Min and Max are different values, one of
    // them is when the first signed overflow happens, the other is when the
    // first unsigned overflow happens. Crossing the range boundary is only
    // possible via an overflow (treating 0 as a special case of it, modeling
    // an overflow as crossing k*2^W for some k).
    //
    // The interesting case here is when Min was eliminated as an invalid
    // solution, but Max was not. The argument is that if there was another
    // overflow between Min and Max, it would also have been eliminated if
    // it was considered.
    //
    // For a given boundary, it is possible to have two overflows of the same
    // type (signed/unsigned) without having the other type in between: this
    // can happen when the vertex of the parabola is between the iterations
    // corresponding to the overflows. This is only possible when the two
    // overflows cross k*2^W for the same k. In such case, if the second one
    // left the range (and was the first one to do so), the first overflow
    // would have to enter the range, which would mean that either we had left
    // the range before or that we started outside of it. Both of these cases
    // are contradictions.
    //
    // Claim: In the case where SolveForBoundary returns None, the correct
    // solution is not some value between the Max for this boundary and the
    // Min of the other boundary.
    //
    // Justification: Assume that we had such Max_A and Min_B corresponding
    // to range boundaries A and B and such that Max_A < Min_B. If there was
    // a solution between Max_A and Min_B, it would have to be caused by an
    // overflow corresponding to either A or B. It cannot correspond to B,
    // since Min_B is the first occurrence of such an overflow. If it
    // corresponded to A, it would have to be either a signed or an unsigned
    // overflow that is larger than both eliminated overflows for A. But
    // between the eliminated overflows and this overflow, the values would
    // cover the entire value space, thus crossing the other boundary, which
    // is a contradiction.
  
    return TruncIfPossible(MinOptional(SL.first, SU.first), BitWidth);
  }
  
  ScalarEvolution::ExitLimit
  ScalarEvolution::howFarToZero(const SCEV *V, const Loop *L, bool ControlsExit,
                                bool AllowPredicates) {
  
    // This is only used for loops with a "x != y" exit test. The exit condition
    // is now expressed as a single expression, V = x-y. So the exit test is
    // effectively V != 0.  We know and take advantage of the fact that this
    // expression only being used in a comparison by zero context.
  
    SmallPtrSet<const SCEVPredicate *, 4> Predicates;
    // If the value is a constant
    if (const SCEVConstant *C = dyn_cast<SCEVConstant>(V)) {
      // If the value is already zero, the branch will execute zero times.
      if (C->getValue()->isZero()) return C;
      return getCouldNotCompute();  // Otherwise it will loop infinitely.
    }
  
    const SCEVAddRecExpr *AddRec =
        dyn_cast<SCEVAddRecExpr>(stripInjectiveFunctions(V));
  
    if (!AddRec && AllowPredicates)
      // Try to make this an AddRec using runtime tests, in the first X
      // iterations of this loop, where X is the SCEV expression found by the
      // algorithm below.
      AddRec = convertSCEVToAddRecWithPredicates(V, L, Predicates);
  
    if (!AddRec || AddRec->getLoop() != L)
      return getCouldNotCompute();
  
    // If this is a quadratic (3-term) AddRec {L,+,M,+,N}, find the roots of
    // the quadratic equation to solve it.
    if (AddRec->isQuadratic() && AddRec->getType()->isIntegerTy()) {
      // We can only use this value if the chrec ends up with an exact zero
      // value at this index.  When solving for "X*X != 5", for example, we
      // should not accept a root of 2.
      if (auto S = SolveQuadraticAddRecExact(AddRec, *this)) {
        const auto *R = cast<SCEVConstant>(getConstant(S.getValue()));
        return ExitLimit(R, R, false, Predicates);
      }
      return getCouldNotCompute();
    }
  
    // Otherwise we can only handle this if it is affine.
    if (!AddRec->isAffine())
      return getCouldNotCompute();
  
    // If this is an affine expression, the execution count of this branch is
    // the minimum unsigned root of the following equation:
    //
    //     Start + Step*N = 0 (mod 2^BW)
    //
    // equivalent to:
    //
    //             Step*N = -Start (mod 2^BW)
    //
    // where BW is the common bit width of Start and Step.
  
    // Get the initial value for the loop.
    const SCEV *Start = getSCEVAtScope(AddRec->getStart(), L->getParentLoop());
    const SCEV *Step = getSCEVAtScope(AddRec->getOperand(1), L->getParentLoop());
  
    // For now we handle only constant steps.
    //
    // TODO: Handle a nonconstant Step given AddRec<NUW>. If the
    // AddRec is NUW, then (in an unsigned sense) it cannot be counting up to wrap
    // to 0, it must be counting down to equal 0. Consequently, N = Start / -Step.
    // We have not yet seen any such cases.
    const SCEVConstant *StepC = dyn_cast<SCEVConstant>(Step);
    if (!StepC || StepC->getValue()->isZero())
      return getCouldNotCompute();
  
    // For positive steps (counting up until unsigned overflow):
    //   N = -Start/Step (as unsigned)
    // For negative steps (counting down to zero):
    //   N = Start/-Step
    // First compute the unsigned distance from zero in the direction of Step.
    bool CountDown = StepC->getAPInt().isNegative();
    const SCEV *Distance = CountDown ? Start : getNegativeSCEV(Start);
  
    // Handle unitary steps, which cannot wraparound.
    // 1*N = -Start; -1*N = Start (mod 2^BW), so:
    //   N = Distance (as unsigned)
    if (StepC->getValue()->isOne() || StepC->getValue()->isMinusOne()) {
      APInt MaxBECount = getUnsignedRangeMax(applyLoopGuards(Distance, L));
      APInt MaxBECountBase = getUnsignedRangeMax(Distance);
      if (MaxBECountBase.ult(MaxBECount))
        MaxBECount = MaxBECountBase;
  
      // When a loop like "for (int i = 0; i != n; ++i) { /* body */ }" is rotated,
      // we end up with a loop whose backedge-taken count is n - 1.  Detect this
      // case, and see if we can improve the bound.
      //
      // Explicitly handling this here is necessary because getUnsignedRange
      // isn't context-sensitive; it doesn't know that we only care about the
      // range inside the loop.
      const SCEV *Zero = getZero(Distance->getType());
      const SCEV *One = getOne(Distance->getType());
      const SCEV *DistancePlusOne = getAddExpr(Distance, One);
      if (isLoopEntryGuardedByCond(L, ICmpInst::ICMP_NE, DistancePlusOne, Zero)) {
        // If Distance + 1 doesn't overflow, we can compute the maximum distance
        // as "unsigned_max(Distance + 1) - 1".
        ConstantRange CR = getUnsignedRange(DistancePlusOne);
        MaxBECount = APIntOps::umin(MaxBECount, CR.getUnsignedMax() - 1);
      }
      return ExitLimit(Distance, getConstant(MaxBECount), false, Predicates);
    }
  
    // If the condition controls loop exit (the loop exits only if the expression
    // is true) and the addition is no-wrap we can use unsigned divide to
    // compute the backedge count.  In this case, the step may not divide the
    // distance, but we don't care because if the condition is "missed" the loop
    // will have undefined behavior due to wrapping.
    if (ControlsExit && AddRec->hasNoSelfWrap() &&
        loopHasNoAbnormalExits(AddRec->getLoop())) {
      const SCEV *Exact =
          getUDivExpr(Distance, CountDown ? getNegativeSCEV(Step) : Step);
      const SCEV *Max = getCouldNotCompute();
      if (Exact != getCouldNotCompute()) {
        APInt MaxInt = getUnsignedRangeMax(applyLoopGuards(Exact, L));
        APInt BaseMaxInt = getUnsignedRangeMax(Exact);
        if (BaseMaxInt.ult(MaxInt))
          Max = getConstant(BaseMaxInt);
        else
          Max = getConstant(MaxInt);
      }
      return ExitLimit(Exact, Max, false, Predicates);
    }
  
    // Solve the general equation.
    const SCEV *E = SolveLinEquationWithOverflow(StepC->getAPInt(),
                                                 getNegativeSCEV(Start), *this);
    const SCEV *M = E == getCouldNotCompute()
                        ? E
                        : getConstant(getUnsignedRangeMax(E));
    return ExitLimit(E, M, false, Predicates);
  }
  
  ScalarEvolution::ExitLimit
  ScalarEvolution::howFarToNonZero(const SCEV *V, const Loop *L) {
    // Loops that look like: while (X == 0) are very strange indeed.  We don't
    // handle them yet except for the trivial case.  This could be expanded in the
    // future as needed.
  
    // If the value is a constant, check to see if it is known to be non-zero
    // already.  If so, the backedge will execute zero times.
    if (const SCEVConstant *C = dyn_cast<SCEVConstant>(V)) {
      if (!C->getValue()->isZero())
        return getZero(C->getType());
      return getCouldNotCompute();  // Otherwise it will loop infinitely.
    }
  
    // We could implement others, but I really doubt anyone writes loops like
    // this, and if they did, they would already be constant folded.
    return getCouldNotCompute();
  }
  
  std::pair<const BasicBlock *, const BasicBlock *>
  ScalarEvolution::getPredecessorWithUniqueSuccessorForBB(const BasicBlock *BB)
      const {
    // If the block has a unique predecessor, then there is no path from the
    // predecessor to the block that does not go through the direct edge
    // from the predecessor to the block.
    if (const BasicBlock *Pred = BB->getSinglePredecessor())
      return {Pred, BB};
  
    // A loop's header is defined to be a block that dominates the loop.
    // If the header has a unique predecessor outside the loop, it must be
    // a block that has exactly one successor that can reach the loop.
    if (const Loop *L = LI.getLoopFor(BB))
      return {L->getLoopPredecessor(), L->getHeader()};
  
    return {nullptr, nullptr};
  }
  
  /// SCEV structural equivalence is usually sufficient for testing whether two
  /// expressions are equal, however for the purposes of looking for a condition
  /// guarding a loop, it can be useful to be a little more general, since a
  /// front-end may have replicated the controlling expression.
  static bool HasSameValue(const SCEV *A, const SCEV *B) {
    // Quick check to see if they are the same SCEV.
    if (A == B) return true;
  
    auto ComputesEqualValues = [](const Instruction *A, const Instruction *B) {
      // Not all instructions that are "identical" compute the same value.  For
      // instance, two distinct alloca instructions allocating the same type are
      // identical and do not read memory; but compute distinct values.
      return A->isIdenticalTo(B) && (isa<BinaryOperator>(A) || isa<GetElementPtrInst>(A));
    };
  
    // Otherwise, if they're both SCEVUnknown, it's possible that they hold
    // two different instructions with the same value. Check for this case.
    if (const SCEVUnknown *AU = dyn_cast<SCEVUnknown>(A))
      if (const SCEVUnknown *BU = dyn_cast<SCEVUnknown>(B))
        if (const Instruction *AI = dyn_cast<Instruction>(AU->getValue()))
          if (const Instruction *BI = dyn_cast<Instruction>(BU->getValue()))
            if (ComputesEqualValues(AI, BI))
              return true;
  
    // Otherwise assume they may have a different value.
    return false;
  }
  
  bool ScalarEvolution::SimplifyICmpOperands(ICmpInst::Predicate &Pred,
                                             const SCEV *&LHS, const SCEV *&RHS,
                                             unsigned Depth) {
    bool Changed = false;
    // Simplifies ICMP to trivial true or false by turning it into '0 == 0' or
    // '0 != 0'.
    auto TrivialCase = [&](bool TriviallyTrue) {
      LHS = RHS = getConstant(ConstantInt::getFalse(getContext()));
      Pred = TriviallyTrue ? ICmpInst::ICMP_EQ : ICmpInst::ICMP_NE;
      return true;
    };
    // If we hit the max recursion limit bail out.
    if (Depth >= 3)
      return false;
  
    // Canonicalize a constant to the right side.
    if (const SCEVConstant *LHSC = dyn_cast<SCEVConstant>(LHS)) {
      // Check for both operands constant.
      if (const SCEVConstant *RHSC = dyn_cast<SCEVConstant>(RHS)) {
        if (ConstantExpr::getICmp(Pred,
                                  LHSC->getValue(),
                                  RHSC->getValue())->isNullValue())
          return TrivialCase(false);
        else
          return TrivialCase(true);
      }
      // Otherwise swap the operands to put the constant on the right.
      std::swap(LHS, RHS);
      Pred = ICmpInst::getSwappedPredicate(Pred);
      Changed = true;
    }
  
    // If we're comparing an addrec with a value which is loop-invariant in the
    // addrec's loop, put the addrec on the left. Also make a dominance check,
    // as both operands could be addrecs loop-invariant in each other's loop.
    if (const SCEVAddRecExpr *AR = dyn_cast<SCEVAddRecExpr>(RHS)) {
      const Loop *L = AR->getLoop();
      if (isLoopInvariant(LHS, L) && properlyDominates(LHS, L->getHeader())) {
        std::swap(LHS, RHS);
        Pred = ICmpInst::getSwappedPredicate(Pred);
        Changed = true;
      }
    }
  
    // If there's a constant operand, canonicalize comparisons with boundary
    // cases, and canonicalize *-or-equal comparisons to regular comparisons.
    if (const SCEVConstant *RC = dyn_cast<SCEVConstant>(RHS)) {
      const APInt &RA = RC->getAPInt();
  
      bool SimplifiedByConstantRange = false;
  
      if (!ICmpInst::isEquality(Pred)) {
        ConstantRange ExactCR = ConstantRange::makeExactICmpRegion(Pred, RA);
        if (ExactCR.isFullSet())
          return TrivialCase(true);
        else if (ExactCR.isEmptySet())
          return TrivialCase(false);
  
        APInt NewRHS;
        CmpInst::Predicate NewPred;
        if (ExactCR.getEquivalentICmp(NewPred, NewRHS) &&
            ICmpInst::isEquality(NewPred)) {
          // We were able to convert an inequality to an equality.
          Pred = NewPred;
          RHS = getConstant(NewRHS);
          Changed = SimplifiedByConstantRange = true;
        }
      }
  
      if (!SimplifiedByConstantRange) {
        switch (Pred) {
        default:
          break;
        case ICmpInst::ICMP_EQ:
        case ICmpInst::ICMP_NE:
          // Fold ((-1) * %a) + %b == 0 (equivalent to %b-%a == 0) into %a == %b.
          if (!RA)
            if (const SCEVAddExpr *AE = dyn_cast<SCEVAddExpr>(LHS))
              if (const SCEVMulExpr *ME =
                      dyn_cast<SCEVMulExpr>(AE->getOperand(0)))
                if (AE->getNumOperands() == 2 && ME->getNumOperands() == 2 &&
                    ME->getOperand(0)->isAllOnesValue()) {
                  RHS = AE->getOperand(1);
                  LHS = ME->getOperand(1);
                  Changed = true;
                }
          break;
  
  
          // The "Should have been caught earlier!" messages refer to the fact
          // that the ExactCR.isFullSet() or ExactCR.isEmptySet() check above
          // should have fired on the corresponding cases, and canonicalized the
          // check to trivial case.
  
        case ICmpInst::ICMP_UGE:
          assert(!RA.isMinValue() && "Should have been caught earlier!");
          Pred = ICmpInst::ICMP_UGT;
          RHS = getConstant(RA - 1);
          Changed = true;
          break;
        case ICmpInst::ICMP_ULE:
          assert(!RA.isMaxValue() && "Should have been caught earlier!");
          Pred = ICmpInst::ICMP_ULT;
          RHS = getConstant(RA + 1);
          Changed = true;
          break;
        case ICmpInst::ICMP_SGE:
          assert(!RA.isMinSignedValue() && "Should have been caught earlier!");
          Pred = ICmpInst::ICMP_SGT;
          RHS = getConstant(RA - 1);
          Changed = true;
          break;
        case ICmpInst::ICMP_SLE:
          assert(!RA.isMaxSignedValue() && "Should have been caught earlier!");
          Pred = ICmpInst::ICMP_SLT;
          RHS = getConstant(RA + 1);
          Changed = true;
          break;
        }
      }
    }
  
    // Check for obvious equality.
    if (HasSameValue(LHS, RHS)) {
      if (ICmpInst::isTrueWhenEqual(Pred))
        return TrivialCase(true);
      if (ICmpInst::isFalseWhenEqual(Pred))
        return TrivialCase(false);
    }
  
    // If possible, canonicalize GE/LE comparisons to GT/LT comparisons, by
    // adding or subtracting 1 from one of the operands.
    switch (Pred) {
    case ICmpInst::ICMP_SLE:
      if (!getSignedRangeMax(RHS).isMaxSignedValue()) {
        RHS = getAddExpr(getConstant(RHS->getType(), 1, true), RHS,
                         SCEV::FlagNSW);
        Pred = ICmpInst::ICMP_SLT;
        Changed = true;
      } else if (!getSignedRangeMin(LHS).isMinSignedValue()) {
        LHS = getAddExpr(getConstant(RHS->getType(), (uint64_t)-1, true), LHS,
                         SCEV::FlagNSW);
        Pred = ICmpInst::ICMP_SLT;
        Changed = true;
      }
      break;
    case ICmpInst::ICMP_SGE:
      if (!getSignedRangeMin(RHS).isMinSignedValue()) {
        RHS = getAddExpr(getConstant(RHS->getType(), (uint64_t)-1, true), RHS,
                         SCEV::FlagNSW);
        Pred = ICmpInst::ICMP_SGT;
        Changed = true;
      } else if (!getSignedRangeMax(LHS).isMaxSignedValue()) {
        LHS = getAddExpr(getConstant(RHS->getType(), 1, true), LHS,
                         SCEV::FlagNSW);
        Pred = ICmpInst::ICMP_SGT;
        Changed = true;
      }
      break;
    case ICmpInst::ICMP_ULE:
      if (!getUnsignedRangeMax(RHS).isMaxValue()) {
        RHS = getAddExpr(getConstant(RHS->getType(), 1, true), RHS,
                         SCEV::FlagNUW);
        Pred = ICmpInst::ICMP_ULT;
        Changed = true;
      } else if (!getUnsignedRangeMin(LHS).isMinValue()) {
        LHS = getAddExpr(getConstant(RHS->getType(), (uint64_t)-1, true), LHS);
        Pred = ICmpInst::ICMP_ULT;
        Changed = true;
      }
      break;
    case ICmpInst::ICMP_UGE:
      if (!getUnsignedRangeMin(RHS).isMinValue()) {
        RHS = getAddExpr(getConstant(RHS->getType(), (uint64_t)-1, true), RHS);
        Pred = ICmpInst::ICMP_UGT;
        Changed = true;
      } else if (!getUnsignedRangeMax(LHS).isMaxValue()) {
        LHS = getAddExpr(getConstant(RHS->getType(), 1, true), LHS,
                         SCEV::FlagNUW);
        Pred = ICmpInst::ICMP_UGT;
        Changed = true;
      }
      break;
    default:
      break;
    }
  
    // TODO: More simplifications are possible here.
  
    // Recursively simplify until we either hit a recursion limit or nothing
    // changes.
    if (Changed)
      return SimplifyICmpOperands(Pred, LHS, RHS, Depth+1);
  
    return Changed;
  }
  
  bool ScalarEvolution::isKnownNegative(const SCEV *S) {
    return getSignedRangeMax(S).isNegative();
  }
  
  bool ScalarEvolution::isKnownPositive(const SCEV *S) {
    return getSignedRangeMin(S).isStrictlyPositive();
  }
  
  bool ScalarEvolution::isKnownNonNegative(const SCEV *S) {
    return !getSignedRangeMin(S).isNegative();
  }
  
  bool ScalarEvolution::isKnownNonPositive(const SCEV *S) {
    return !getSignedRangeMax(S).isStrictlyPositive();
  }
  
  bool ScalarEvolution::isKnownNonZero(const SCEV *S) {
    return getUnsignedRangeMin(S) != 0;
  }
  
  std::pair<const SCEV *, const SCEV *>
  ScalarEvolution::SplitIntoInitAndPostInc(const Loop *L, const SCEV *S) {
    // Compute SCEV on entry of loop L.
    const SCEV *Start = SCEVInitRewriter::rewrite(S, L, *this);
    if (Start == getCouldNotCompute())
      return { Start, Start };
    // Compute post increment SCEV for loop L.
    const SCEV *PostInc = SCEVPostIncRewriter::rewrite(S, L, *this);
    assert(PostInc != getCouldNotCompute() && "Unexpected could not compute");
    return { Start, PostInc };
  }
  
  bool ScalarEvolution::isKnownViaInduction(ICmpInst::Predicate Pred,
                                            const SCEV *LHS, const SCEV *RHS) {
    // First collect all loops.
    SmallPtrSet<const Loop *, 8> LoopsUsed;
    getUsedLoops(LHS, LoopsUsed);
    getUsedLoops(RHS, LoopsUsed);
  
    if (LoopsUsed.empty())
      return false;
  
    // Domination relationship must be a linear order on collected loops.
  #ifndef NDEBUG
    for (auto *L1 : LoopsUsed)
      for (auto *L2 : LoopsUsed)
        assert((DT.dominates(L1->getHeader(), L2->getHeader()) ||
                DT.dominates(L2->getHeader(), L1->getHeader())) &&
               "Domination relationship is not a linear order");
  #endif
  
    const Loop *MDL =
        *std::max_element(LoopsUsed.begin(), LoopsUsed.end(),
                          [&](const Loop *L1, const Loop *L2) {
           return DT.properlyDominates(L1->getHeader(), L2->getHeader());
         });
  
    // Get init and post increment value for LHS.
    auto SplitLHS = SplitIntoInitAndPostInc(MDL, LHS);
    // if LHS contains unknown non-invariant SCEV then bail out.
    if (SplitLHS.first == getCouldNotCompute())
      return false;
    assert (SplitLHS.second != getCouldNotCompute() && "Unexpected CNC");
    // Get init and post increment value for RHS.
    auto SplitRHS = SplitIntoInitAndPostInc(MDL, RHS);
    // if RHS contains unknown non-invariant SCEV then bail out.
    if (SplitRHS.first == getCouldNotCompute())
      return false;
    assert (SplitRHS.second != getCouldNotCompute() && "Unexpected CNC");
    // It is possible that init SCEV contains an invariant load but it does
    // not dominate MDL and is not available at MDL loop entry, so we should
    // check it here.
    if (!isAvailableAtLoopEntry(SplitLHS.first, MDL) ||
        !isAvailableAtLoopEntry(SplitRHS.first, MDL))
      return false;
  
    // It seems backedge guard check is faster than entry one so in some cases
    // it can speed up whole estimation by short circuit
    return isLoopBackedgeGuardedByCond(MDL, Pred, SplitLHS.second,
                                       SplitRHS.second) &&
           isLoopEntryGuardedByCond(MDL, Pred, SplitLHS.first, SplitRHS.first);
  }
  
  bool ScalarEvolution::isKnownPredicate(ICmpInst::Predicate Pred,
                                         const SCEV *LHS, const SCEV *RHS) {
    // Canonicalize the inputs first.
    (void)SimplifyICmpOperands(Pred, LHS, RHS);
  
    if (isKnownViaInduction(Pred, LHS, RHS))
      return true;
  
    if (isKnownPredicateViaSplitting(Pred, LHS, RHS))
      return true;
  
    // Otherwise see what can be done with some simple reasoning.
    return isKnownViaNonRecursiveReasoning(Pred, LHS, RHS);
  }
  
  Optional<bool> ScalarEvolution::evaluatePredicate(ICmpInst::Predicate Pred,
                                                    const SCEV *LHS,
                                                    const SCEV *RHS) {
    if (isKnownPredicate(Pred, LHS, RHS))
      return true;
    else if (isKnownPredicate(ICmpInst::getInversePredicate(Pred), LHS, RHS))
      return false;
    return None;
  }
  
  bool ScalarEvolution::isKnownPredicateAt(ICmpInst::Predicate Pred,
                                           const SCEV *LHS, const SCEV *RHS,
                                           const Instruction *Context) {
    // TODO: Analyze guards and assumes from Context's block.
    return isKnownPredicate(Pred, LHS, RHS) ||
           isBasicBlockEntryGuardedByCond(Context->getParent(), Pred, LHS, RHS);
  }
  
  Optional<bool>
  ScalarEvolution::evaluatePredicateAt(ICmpInst::Predicate Pred, const SCEV *LHS,
                                       const SCEV *RHS,
                                       const Instruction *Context) {
    Optional<bool> KnownWithoutContext = evaluatePredicate(Pred, LHS, RHS);
    if (KnownWithoutContext)
      return KnownWithoutContext;
  
    if (isBasicBlockEntryGuardedByCond(Context->getParent(), Pred, LHS, RHS))
      return true;
    else if (isBasicBlockEntryGuardedByCond(Context->getParent(),
                                            ICmpInst::getInversePredicate(Pred),
                                            LHS, RHS))
      return false;
    return None;
  }
  
  bool ScalarEvolution::isKnownOnEveryIteration(ICmpInst::Predicate Pred,
                                                const SCEVAddRecExpr *LHS,
                                                const SCEV *RHS) {
    const Loop *L = LHS->getLoop();
    return isLoopEntryGuardedByCond(L, Pred, LHS->getStart(), RHS) &&
           isLoopBackedgeGuardedByCond(L, Pred, LHS->getPostIncExpr(*this), RHS);
  }
  
  Optional<ScalarEvolution::MonotonicPredicateType>
  ScalarEvolution::getMonotonicPredicateType(const SCEVAddRecExpr *LHS,
                                             ICmpInst::Predicate Pred) {
    auto Result = getMonotonicPredicateTypeImpl(LHS, Pred);
  
  #ifndef NDEBUG
    // Verify an invariant: inverting the predicate should turn a monotonically
    // increasing change to a monotonically decreasing one, and vice versa.
    if (Result) {
      auto ResultSwapped =
          getMonotonicPredicateTypeImpl(LHS, ICmpInst::getSwappedPredicate(Pred));
  
      assert(ResultSwapped.hasValue() && "should be able to analyze both!");
      assert(ResultSwapped.getValue() != Result.getValue() &&
             "monotonicity should flip as we flip the predicate");
    }
  #endif
  
    return Result;
  }
  
  Optional<ScalarEvolution::MonotonicPredicateType>
  ScalarEvolution::getMonotonicPredicateTypeImpl(const SCEVAddRecExpr *LHS,
                                                 ICmpInst::Predicate Pred) {
    // A zero step value for LHS means the induction variable is essentially a
    // loop invariant value. We don't really depend on the predicate actually
    // flipping from false to true (for increasing predicates, and the other way
    // around for decreasing predicates), all we care about is that *if* the
    // predicate changes then it only changes from false to true.
    //
    // A zero step value in itself is not very useful, but there may be places
    // where SCEV can prove X >= 0 but not prove X > 0, so it is helpful to be
    // as general as possible.
  
    // Only handle LE/LT/GE/GT predicates.
    if (!ICmpInst::isRelational(Pred))
      return None;
  
    bool IsGreater = ICmpInst::isGE(Pred) || ICmpInst::isGT(Pred);
    assert((IsGreater || ICmpInst::isLE(Pred) || ICmpInst::isLT(Pred)) &&
           "Should be greater or less!");
  
    // Check that AR does not wrap.
    if (ICmpInst::isUnsigned(Pred)) {
      if (!LHS->hasNoUnsignedWrap())
        return None;
      return IsGreater ? MonotonicallyIncreasing : MonotonicallyDecreasing;
    } else {
      assert(ICmpInst::isSigned(Pred) &&
             "Relational predicate is either signed or unsigned!");
      if (!LHS->hasNoSignedWrap())
        return None;
  
      const SCEV *Step = LHS->getStepRecurrence(*this);
  
      if (isKnownNonNegative(Step))
        return IsGreater ? MonotonicallyIncreasing : MonotonicallyDecreasing;
  
      if (isKnownNonPositive(Step))
        return !IsGreater ? MonotonicallyIncreasing : MonotonicallyDecreasing;
  
      return None;
    }
  }
  
  Optional<ScalarEvolution::LoopInvariantPredicate>
  ScalarEvolution::getLoopInvariantPredicate(ICmpInst::Predicate Pred,
                                             const SCEV *LHS, const SCEV *RHS,
                                             const Loop *L) {
  
    // If there is a loop-invariant, force it into the RHS, otherwise bail out.
    if (!isLoopInvariant(RHS, L)) {
      if (!isLoopInvariant(LHS, L))
        return None;
  
      std::swap(LHS, RHS);
      Pred = ICmpInst::getSwappedPredicate(Pred);
    }
  
    const SCEVAddRecExpr *ArLHS = dyn_cast<SCEVAddRecExpr>(LHS);
    if (!ArLHS || ArLHS->getLoop() != L)
      return None;
  
    auto MonotonicType = getMonotonicPredicateType(ArLHS, Pred);
    if (!MonotonicType)
      return None;
    // If the predicate "ArLHS `Pred` RHS" monotonically increases from false to
    // true as the loop iterates, and the backedge is control dependent on
    // "ArLHS `Pred` RHS" == true then we can reason as follows:
    //
    //   * if the predicate was false in the first iteration then the predicate
    //     is never evaluated again, since the loop exits without taking the
    //     backedge.
    //   * if the predicate was true in the first iteration then it will
    //     continue to be true for all future iterations since it is
    //     monotonically increasing.
    //
    // For both the above possibilities, we can replace the loop varying
    // predicate with its value on the first iteration of the loop (which is
    // loop invariant).
    //
    // A similar reasoning applies for a monotonically decreasing predicate, by
    // replacing true with false and false with true in the above two bullets.
    bool Increasing = *MonotonicType == ScalarEvolution::MonotonicallyIncreasing;
    auto P = Increasing ? Pred : ICmpInst::getInversePredicate(Pred);
  
    if (!isLoopBackedgeGuardedByCond(L, P, LHS, RHS))
      return None;
  
    return ScalarEvolution::LoopInvariantPredicate(Pred, ArLHS->getStart(), RHS);
  }
  
  Optional<ScalarEvolution::LoopInvariantPredicate>
  ScalarEvolution::getLoopInvariantExitCondDuringFirstIterations(
      ICmpInst::Predicate Pred, const SCEV *LHS, const SCEV *RHS, const Loop *L,
      const Instruction *Context, const SCEV *MaxIter) {
    // Try to prove the following set of facts:
    // - The predicate is monotonic in the iteration space.
    // - If the check does not fail on the 1st iteration:
    //   - No overflow will happen during first MaxIter iterations;
    //   - It will not fail on the MaxIter'th iteration.
    // If the check does fail on the 1st iteration, we leave the loop and no
    // other checks matter.
  
    // If there is a loop-invariant, force it into the RHS, otherwise bail out.
    if (!isLoopInvariant(RHS, L)) {
      if (!isLoopInvariant(LHS, L))
        return None;
  
      std::swap(LHS, RHS);
      Pred = ICmpInst::getSwappedPredicate(Pred);
    }
  
    auto *AR = dyn_cast<SCEVAddRecExpr>(LHS);
    if (!AR || AR->getLoop() != L)
      return None;
  
    // The predicate must be relational (i.e. <, <=, >=, >).
    if (!ICmpInst::isRelational(Pred))
      return None;
  
    // TODO: Support steps other than +/- 1.
    const SCEV *Step = AR->getStepRecurrence(*this);
    auto *One = getOne(Step->getType());
    auto *MinusOne = getNegativeSCEV(One);
    if (Step != One && Step != MinusOne)
      return None;
  
    // Type mismatch here means that MaxIter is potentially larger than max
    // unsigned value in start type, which mean we cannot prove no wrap for the
    // indvar.
    if (AR->getType() != MaxIter->getType())
      return None;
  
    // Value of IV on suggested last iteration.
    const SCEV *Last = AR->evaluateAtIteration(MaxIter, *this);
    // Does it still meet the requirement?
    if (!isLoopBackedgeGuardedByCond(L, Pred, Last, RHS))
      return None;
    // Because step is +/- 1 and MaxIter has same type as Start (i.e. it does
    // not exceed max unsigned value of this type), this effectively proves
    // that there is no wrap during the iteration. To prove that there is no
    // signed/unsigned wrap, we need to check that
    // Start <= Last for step = 1 or Start >= Last for step = -1.
    ICmpInst::Predicate NoOverflowPred =
        CmpInst::isSigned(Pred) ? ICmpInst::ICMP_SLE : ICmpInst::ICMP_ULE;
    if (Step == MinusOne)
      NoOverflowPred = CmpInst::getSwappedPredicate(NoOverflowPred);
    const SCEV *Start = AR->getStart();
    if (!isKnownPredicateAt(NoOverflowPred, Start, Last, Context))
      return None;
  
    // Everything is fine.
    return ScalarEvolution::LoopInvariantPredicate(Pred, Start, RHS);
  }
  
  bool ScalarEvolution::isKnownPredicateViaConstantRanges(
      ICmpInst::Predicate Pred, const SCEV *LHS, const SCEV *RHS) {
    if (HasSameValue(LHS, RHS))
      return ICmpInst::isTrueWhenEqual(Pred);
  
    // This code is split out from isKnownPredicate because it is called from
    // within isLoopEntryGuardedByCond.
  
    auto CheckRanges = [&](const ConstantRange &RangeLHS,
                           const ConstantRange &RangeRHS) {
      return RangeLHS.icmp(Pred, RangeRHS);
    };
  
    // The check at the top of the function catches the case where the values are
    // known to be equal.
    if (Pred == CmpInst::ICMP_EQ)
      return false;
  
    if (Pred == CmpInst::ICMP_NE) {
      if (CheckRanges(getSignedRange(LHS), getSignedRange(RHS)) ||
          CheckRanges(getUnsignedRange(LHS), getUnsignedRange(RHS)))
        return true;
      auto *Diff = getMinusSCEV(LHS, RHS);
      return !isa<SCEVCouldNotCompute>(Diff) && isKnownNonZero(Diff);
    }
  
    if (CmpInst::isSigned(Pred))
      return CheckRanges(getSignedRange(LHS), getSignedRange(RHS));
  
    return CheckRanges(getUnsignedRange(LHS), getUnsignedRange(RHS));
  }
  
  bool ScalarEvolution::isKnownPredicateViaNoOverflow(ICmpInst::Predicate Pred,
                                                      const SCEV *LHS,
                                                      const SCEV *RHS) {
    // Match X to (A + C1)<ExpectedFlags> and Y to (A + C2)<ExpectedFlags>, where
    // C1 and C2 are constant integers. If either X or Y are not add expressions,
    // consider them as X + 0 and Y + 0 respectively. C1 and C2 are returned via
    // OutC1 and OutC2.
    auto MatchBinaryAddToConst = [this](const SCEV *X, const SCEV *Y,
                                        APInt &OutC1, APInt &OutC2,
                                        SCEV::NoWrapFlags ExpectedFlags) {
      const SCEV *XNonConstOp, *XConstOp;
      const SCEV *YNonConstOp, *YConstOp;
      SCEV::NoWrapFlags XFlagsPresent;
      SCEV::NoWrapFlags YFlagsPresent;
  
      if (!splitBinaryAdd(X, XConstOp, XNonConstOp, XFlagsPresent)) {
        XConstOp = getZero(X->getType());
        XNonConstOp = X;
        XFlagsPresent = ExpectedFlags;
      }
      if (!isa<SCEVConstant>(XConstOp) ||
          (XFlagsPresent & ExpectedFlags) != ExpectedFlags)
        return false;
  
      if (!splitBinaryAdd(Y, YConstOp, YNonConstOp, YFlagsPresent)) {
        YConstOp = getZero(Y->getType());
        YNonConstOp = Y;
        YFlagsPresent = ExpectedFlags;
      }
  
      if (!isa<SCEVConstant>(YConstOp) ||
          (YFlagsPresent & ExpectedFlags) != ExpectedFlags)
        return false;
  
      if (YNonConstOp != XNonConstOp)
        return false;
  
      OutC1 = cast<SCEVConstant>(XConstOp)->getAPInt();
      OutC2 = cast<SCEVConstant>(YConstOp)->getAPInt();
  
      return true;
    };
  
    APInt C1;
    APInt C2;
  
    switch (Pred) {
    default:
      break;
  
    case ICmpInst::ICMP_SGE:
      std::swap(LHS, RHS);
      LLVM_FALLTHROUGH;
    case ICmpInst::ICMP_SLE:
      // (X + C1)<nsw> s<= (X + C2)<nsw> if C1 s<= C2.
      if (MatchBinaryAddToConst(LHS, RHS, C1, C2, SCEV::FlagNSW) && C1.sle(C2))
        return true;
  
      break;
  
    case ICmpInst::ICMP_SGT:
      std::swap(LHS, RHS);
      LLVM_FALLTHROUGH;
    case ICmpInst::ICMP_SLT:
      // (X + C1)<nsw> s< (X + C2)<nsw> if C1 s< C2.
      if (MatchBinaryAddToConst(LHS, RHS, C1, C2, SCEV::FlagNSW) && C1.slt(C2))
        return true;
  
      break;
  
    case ICmpInst::ICMP_UGE:
      std::swap(LHS, RHS);
      LLVM_FALLTHROUGH;
    case ICmpInst::ICMP_ULE:
      // (X + C1)<nuw> u<= (X + C2)<nuw> for C1 u<= C2.
      if (MatchBinaryAddToConst(RHS, LHS, C2, C1, SCEV::FlagNUW) && C1.ule(C2))
        return true;
  
      break;
  
    case ICmpInst::ICMP_UGT:
      std::swap(LHS, RHS);
      LLVM_FALLTHROUGH;
    case ICmpInst::ICMP_ULT:
      // (X + C1)<nuw> u< (X + C2)<nuw> if C1 u< C2.
      if (MatchBinaryAddToConst(RHS, LHS, C2, C1, SCEV::FlagNUW) && C1.ult(C2))
        return true;
      break;
    }
  
    return false;
  }
  
  bool ScalarEvolution::isKnownPredicateViaSplitting(ICmpInst::Predicate Pred,
                                                     const SCEV *LHS,
                                                     const SCEV *RHS) {
    if (Pred != ICmpInst::ICMP_ULT || ProvingSplitPredicate)
      return false;
  
    // Allowing arbitrary number of activations of isKnownPredicateViaSplitting on
    // the stack can result in exponential time complexity.
    SaveAndRestore<bool> Restore(ProvingSplitPredicate, true);
  
    // If L >= 0 then I `ult` L <=> I >= 0 && I `slt` L
    //
    // To prove L >= 0 we use isKnownNonNegative whereas to prove I >= 0 we use
    // isKnownPredicate.  isKnownPredicate is more powerful, but also more
    // expensive; and using isKnownNonNegative(RHS) is sufficient for most of the
    // interesting cases seen in practice.  We can consider "upgrading" L >= 0 to
    // use isKnownPredicate later if needed.
    return isKnownNonNegative(RHS) &&
           isKnownPredicate(CmpInst::ICMP_SGE, LHS, getZero(LHS->getType())) &&
           isKnownPredicate(CmpInst::ICMP_SLT, LHS, RHS);
  }
  
  bool ScalarEvolution::isImpliedViaGuard(const BasicBlock *BB,
                                          ICmpInst::Predicate Pred,
                                          const SCEV *LHS, const SCEV *RHS) {
    // No need to even try if we know the module has no guards.
    if (!HasGuards)
      return false;
  
    return any_of(*BB, [&](const Instruction &I) {
      using namespace llvm::PatternMatch;
  
      Value *Condition;
      return match(&I, m_Intrinsic<Intrinsic::experimental_guard>(
                           m_Value(Condition))) &&
             isImpliedCond(Pred, LHS, RHS, Condition, false);
    });
  }
  
  /// isLoopBackedgeGuardedByCond - Test whether the backedge of the loop is
  /// protected by a conditional between LHS and RHS.  This is used to
  /// to eliminate casts.
  bool
  ScalarEvolution::isLoopBackedgeGuardedByCond(const Loop *L,
                                               ICmpInst::Predicate Pred,
                                               const SCEV *LHS, const SCEV *RHS) {
    // Interpret a null as meaning no loop, where there is obviously no guard
    // (interprocedural conditions notwithstanding).
    if (!L) return true;
  
    if (VerifyIR)
      assert(!verifyFunction(*L->getHeader()->getParent(), &dbgs()) &&
             "This cannot be done on broken IR!");
  
  
    if (isKnownViaNonRecursiveReasoning(Pred, LHS, RHS))
      return true;
  
    BasicBlock *Latch = L->getLoopLatch();
    if (!Latch)
      return false;
  
    BranchInst *LoopContinuePredicate =
      dyn_cast<BranchInst>(Latch->getTerminator());
    if (LoopContinuePredicate && LoopContinuePredicate->isConditional() &&
        isImpliedCond(Pred, LHS, RHS,
                      LoopContinuePredicate->getCondition(),
                      LoopContinuePredicate->getSuccessor(0) != L->getHeader()))
      return true;
  
    // We don't want more than one activation of the following loops on the stack
    // -- that can lead to O(n!) time complexity.
    if (WalkingBEDominatingConds)
      return false;
  
    SaveAndRestore<bool> ClearOnExit(WalkingBEDominatingConds, true);
  
    // See if we can exploit a trip count to prove the predicate.
    const auto &BETakenInfo = getBackedgeTakenInfo(L);
    const SCEV *LatchBECount = BETakenInfo.getExact(Latch, this);
    if (LatchBECount != getCouldNotCompute()) {
      // We know that Latch branches back to the loop header exactly
      // LatchBECount times.  This means the backdege condition at Latch is
      // equivalent to  "{0,+,1} u< LatchBECount".
      Type *Ty = LatchBECount->getType();
      auto NoWrapFlags = SCEV::NoWrapFlags(SCEV::FlagNUW | SCEV::FlagNW);
      const SCEV *LoopCounter =
        getAddRecExpr(getZero(Ty), getOne(Ty), L, NoWrapFlags);
      if (isImpliedCond(Pred, LHS, RHS, ICmpInst::ICMP_ULT, LoopCounter,
                        LatchBECount))
        return true;
    }
  
    // Check conditions due to any @llvm.assume intrinsics.
    for (auto &AssumeVH : AC.assumptions()) {
      if (!AssumeVH)
        continue;
      auto *CI = cast<CallInst>(AssumeVH);
      if (!DT.dominates(CI, Latch->getTerminator()))
        continue;
  
      if (isImpliedCond(Pred, LHS, RHS, CI->getArgOperand(0), false))
        return true;
    }
  
    // If the loop is not reachable from the entry block, we risk running into an
    // infinite loop as we walk up into the dom tree.  These loops do not matter
    // anyway, so we just return a conservative answer when we see them.
    if (!DT.isReachableFromEntry(L->getHeader()))
      return false;
  
    if (isImpliedViaGuard(Latch, Pred, LHS, RHS))
      return true;
  
    for (DomTreeNode *DTN = DT[Latch], *HeaderDTN = DT[L->getHeader()];
         DTN != HeaderDTN; DTN = DTN->getIDom()) {
      assert(DTN && "should reach the loop header before reaching the root!");
  
      BasicBlock *BB = DTN->getBlock();
      if (isImpliedViaGuard(BB, Pred, LHS, RHS))
        return true;
  
      BasicBlock *PBB = BB->getSinglePredecessor();
      if (!PBB)
        continue;
  
      BranchInst *ContinuePredicate = dyn_cast<BranchInst>(PBB->getTerminator());
      if (!ContinuePredicate || !ContinuePredicate->isConditional())
        continue;
  
      Value *Condition = ContinuePredicate->getCondition();
  
      // If we have an edge `E` within the loop body that dominates the only
      // latch, the condition guarding `E` also guards the backedge.  This
      // reasoning works only for loops with a single latch.
  
      BasicBlockEdge DominatingEdge(PBB, BB);
      if (DominatingEdge.isSingleEdge()) {
        // We're constructively (and conservatively) enumerating edges within the
        // loop body that dominate the latch.  The dominator tree better agree
        // with us on this:
        assert(DT.dominates(DominatingEdge, Latch) && "should be!");
  
        if (isImpliedCond(Pred, LHS, RHS, Condition,
                          BB != ContinuePredicate->getSuccessor(0)))
          return true;
      }
    }
  
    return false;
  }
  
  bool ScalarEvolution::isBasicBlockEntryGuardedByCond(const BasicBlock *BB,
                                                       ICmpInst::Predicate Pred,
                                                       const SCEV *LHS,
                                                       const SCEV *RHS) {
    if (VerifyIR)
      assert(!verifyFunction(*BB->getParent(), &dbgs()) &&
             "This cannot be done on broken IR!");
  
    // If we cannot prove strict comparison (e.g. a > b), maybe we can prove
    // the facts (a >= b && a != b) separately. A typical situation is when the
    // non-strict comparison is known from ranges and non-equality is known from
    // dominating predicates. If we are proving strict comparison, we always try
    // to prove non-equality and non-strict comparison separately.
    auto NonStrictPredicate = ICmpInst::getNonStrictPredicate(Pred);
    const bool ProvingStrictComparison = (Pred != NonStrictPredicate);
    bool ProvedNonStrictComparison = false;
    bool ProvedNonEquality = false;
  
    auto SplitAndProve =
      [&](std::function<bool(ICmpInst::Predicate)> Fn) -> bool {
      if (!ProvedNonStrictComparison)
        ProvedNonStrictComparison = Fn(NonStrictPredicate);
      if (!ProvedNonEquality)
        ProvedNonEquality = Fn(ICmpInst::ICMP_NE);
      if (ProvedNonStrictComparison && ProvedNonEquality)
        return true;
      return false;
    };
  
    if (ProvingStrictComparison) {
      auto ProofFn = [&](ICmpInst::Predicate P) {
        return isKnownViaNonRecursiveReasoning(P, LHS, RHS);
      };
      if (SplitAndProve(ProofFn))
        return true;
    }
  
    // Try to prove (Pred, LHS, RHS) using isImpliedViaGuard.
    auto ProveViaGuard = [&](const BasicBlock *Block) {
      if (isImpliedViaGuard(Block, Pred, LHS, RHS))
        return true;
      if (ProvingStrictComparison) {
        auto ProofFn = [&](ICmpInst::Predicate P) {
          return isImpliedViaGuard(Block, P, LHS, RHS);
        };
        if (SplitAndProve(ProofFn))
          return true;
      }
      return false;
    };
  
    // Try to prove (Pred, LHS, RHS) using isImpliedCond.
    auto ProveViaCond = [&](const Value *Condition, bool Inverse) {
      const Instruction *Context = &BB->front();
      if (isImpliedCond(Pred, LHS, RHS, Condition, Inverse, Context))
        return true;
      if (ProvingStrictComparison) {
        auto ProofFn = [&](ICmpInst::Predicate P) {
          return isImpliedCond(P, LHS, RHS, Condition, Inverse, Context);
        };
        if (SplitAndProve(ProofFn))
          return true;
      }
      return false;
    };
  
    // Starting at the block's predecessor, climb up the predecessor chain, as long
    // as there are predecessors that can be found that have unique successors
    // leading to the original block.
    const Loop *ContainingLoop = LI.getLoopFor(BB);
    const BasicBlock *PredBB;
    if (ContainingLoop && ContainingLoop->getHeader() == BB)
      PredBB = ContainingLoop->getLoopPredecessor();
    else
      PredBB = BB->getSinglePredecessor();
    for (std::pair<const BasicBlock *, const BasicBlock *> Pair(PredBB, BB);
         Pair.first; Pair = getPredecessorWithUniqueSuccessorForBB(Pair.first)) {
      if (ProveViaGuard(Pair.first))
        return true;
  
      const BranchInst *LoopEntryPredicate =
          dyn_cast<BranchInst>(Pair.first->getTerminator());
      if (!LoopEntryPredicate ||
          LoopEntryPredicate->isUnconditional())
        continue;
  
      if (ProveViaCond(LoopEntryPredicate->getCondition(),
                       LoopEntryPredicate->getSuccessor(0) != Pair.second))
        return true;
    }
  
    // Check conditions due to any @llvm.assume intrinsics.
    for (auto &AssumeVH : AC.assumptions()) {
      if (!AssumeVH)
        continue;
      auto *CI = cast<CallInst>(AssumeVH);
      if (!DT.dominates(CI, BB))
        continue;
  
      if (ProveViaCond(CI->getArgOperand(0), false))
        return true;
    }
  
    return false;
  }
  
  bool ScalarEvolution::isLoopEntryGuardedByCond(const Loop *L,
                                                 ICmpInst::Predicate Pred,
                                                 const SCEV *LHS,
                                                 const SCEV *RHS) {
    // Interpret a null as meaning no loop, where there is obviously no guard
    // (interprocedural conditions notwithstanding).
    if (!L)
      return false;
  
    // Both LHS and RHS must be available at loop entry.
    assert(isAvailableAtLoopEntry(LHS, L) &&
           "LHS is not available at Loop Entry");
    assert(isAvailableAtLoopEntry(RHS, L) &&
           "RHS is not available at Loop Entry");
  
    if (isKnownViaNonRecursiveReasoning(Pred, LHS, RHS))
      return true;
  
    return isBasicBlockEntryGuardedByCond(L->getHeader(), Pred, LHS, RHS);
  }
  
  bool ScalarEvolution::isImpliedCond(ICmpInst::Predicate Pred, const SCEV *LHS,
                                      const SCEV *RHS,
                                      const Value *FoundCondValue, bool Inverse,
                                      const Instruction *Context) {
    // False conditions implies anything. Do not bother analyzing it further.
    if (FoundCondValue ==
        ConstantInt::getBool(FoundCondValue->getContext(), Inverse))
      return true;
  
    if (!PendingLoopPredicates.insert(FoundCondValue).second)
      return false;
  
    auto ClearOnExit =
        make_scope_exit([&]() { PendingLoopPredicates.erase(FoundCondValue); });
  
    // Recursively handle And and Or conditions.
    const Value *Op0, *Op1;
    if (match(FoundCondValue, m_LogicalAnd(m_Value(Op0), m_Value(Op1)))) {
      if (!Inverse)
        return isImpliedCond(Pred, LHS, RHS, Op0, Inverse, Context) ||
                isImpliedCond(Pred, LHS, RHS, Op1, Inverse, Context);
    } else if (match(FoundCondValue, m_LogicalOr(m_Value(Op0), m_Value(Op1)))) {
      if (Inverse)
        return isImpliedCond(Pred, LHS, RHS, Op0, Inverse, Context) ||
                isImpliedCond(Pred, LHS, RHS, Op1, Inverse, Context);
    }
  
    const ICmpInst *ICI = dyn_cast<ICmpInst>(FoundCondValue);
    if (!ICI) return false;
  
    // Now that we found a conditional branch that dominates the loop or controls
    // the loop latch. Check to see if it is the comparison we are looking for.
    ICmpInst::Predicate FoundPred;
    if (Inverse)
      FoundPred = ICI->getInversePredicate();
    else
      FoundPred = ICI->getPredicate();
  
    const SCEV *FoundLHS = getSCEV(ICI->getOperand(0));
    const SCEV *FoundRHS = getSCEV(ICI->getOperand(1));
  
    return isImpliedCond(Pred, LHS, RHS, FoundPred, FoundLHS, FoundRHS, Context);
  }
  
  bool ScalarEvolution::isImpliedCond(ICmpInst::Predicate Pred, const SCEV *LHS,
                                      const SCEV *RHS,
                                      ICmpInst::Predicate FoundPred,
                                      const SCEV *FoundLHS, const SCEV *FoundRHS,
                                      const Instruction *Context) {
    // Balance the types.
    if (getTypeSizeInBits(LHS->getType()) <
        getTypeSizeInBits(FoundLHS->getType())) {
      // For unsigned and equality predicates, try to prove that both found
      // operands fit into narrow unsigned range. If so, try to prove facts in
      // narrow types.
      if (!CmpInst::isSigned(FoundPred) && !FoundLHS->getType()->isPointerTy()) {
        auto *NarrowType = LHS->getType();
        auto *WideType = FoundLHS->getType();
        auto BitWidth = getTypeSizeInBits(NarrowType);
        const SCEV *MaxValue = getZeroExtendExpr(
            getConstant(APInt::getMaxValue(BitWidth)), WideType);
        if (isKnownPredicate(ICmpInst::ICMP_ULE, FoundLHS, MaxValue) &&
            isKnownPredicate(ICmpInst::ICMP_ULE, FoundRHS, MaxValue)) {
          const SCEV *TruncFoundLHS = getTruncateExpr(FoundLHS, NarrowType);
          const SCEV *TruncFoundRHS = getTruncateExpr(FoundRHS, NarrowType);
          if (isImpliedCondBalancedTypes(Pred, LHS, RHS, FoundPred, TruncFoundLHS,
                                         TruncFoundRHS, Context))
            return true;
        }
      }
  
      if (LHS->getType()->isPointerTy())
        return false;
      if (CmpInst::isSigned(Pred)) {
        LHS = getSignExtendExpr(LHS, FoundLHS->getType());
        RHS = getSignExtendExpr(RHS, FoundLHS->getType());
      } else {
        LHS = getZeroExtendExpr(LHS, FoundLHS->getType());
        RHS = getZeroExtendExpr(RHS, FoundLHS->getType());
      }
    } else if (getTypeSizeInBits(LHS->getType()) >
        getTypeSizeInBits(FoundLHS->getType())) {
      if (FoundLHS->getType()->isPointerTy())
        return false;
      if (CmpInst::isSigned(FoundPred)) {
        FoundLHS = getSignExtendExpr(FoundLHS, LHS->getType());
        FoundRHS = getSignExtendExpr(FoundRHS, LHS->getType());
      } else {
        FoundLHS = getZeroExtendExpr(FoundLHS, LHS->getType());
        FoundRHS = getZeroExtendExpr(FoundRHS, LHS->getType());
      }
    }
    return isImpliedCondBalancedTypes(Pred, LHS, RHS, FoundPred, FoundLHS,
                                      FoundRHS, Context);
  }
  
  bool ScalarEvolution::isImpliedCondBalancedTypes(
      ICmpInst::Predicate Pred, const SCEV *LHS, const SCEV *RHS,
      ICmpInst::Predicate FoundPred, const SCEV *FoundLHS, const SCEV *FoundRHS,
      const Instruction *Context) {
    assert(getTypeSizeInBits(LHS->getType()) ==
               getTypeSizeInBits(FoundLHS->getType()) &&
           "Types should be balanced!");
    // Canonicalize the query to match the way instcombine will have
    // canonicalized the comparison.
    if (SimplifyICmpOperands(Pred, LHS, RHS))
      if (LHS == RHS)
        return CmpInst::isTrueWhenEqual(Pred);
    if (SimplifyICmpOperands(FoundPred, FoundLHS, FoundRHS))
      if (FoundLHS == FoundRHS)
        return CmpInst::isFalseWhenEqual(FoundPred);
  
    // Check to see if we can make the LHS or RHS match.
    if (LHS == FoundRHS || RHS == FoundLHS) {
      if (isa<SCEVConstant>(RHS)) {
        std::swap(FoundLHS, FoundRHS);
        FoundPred = ICmpInst::getSwappedPredicate(FoundPred);
      } else {
        std::swap(LHS, RHS);
        Pred = ICmpInst::getSwappedPredicate(Pred);
      }
    }
  
    // Check whether the found predicate is the same as the desired predicate.
    if (FoundPred == Pred)
      return isImpliedCondOperands(Pred, LHS, RHS, FoundLHS, FoundRHS, Context);
  
    // Check whether swapping the found predicate makes it the same as the
    // desired predicate.
    if (ICmpInst::getSwappedPredicate(FoundPred) == Pred) {
      // We can write the implication
      // 0.  LHS Pred      RHS  <-   FoundLHS SwapPred  FoundRHS
      // using one of the following ways:
      // 1.  LHS Pred      RHS  <-   FoundRHS Pred      FoundLHS
      // 2.  RHS SwapPred  LHS  <-   FoundLHS SwapPred  FoundRHS
      // 3.  LHS Pred      RHS  <-  ~FoundLHS Pred     ~FoundRHS
      // 4. ~LHS SwapPred ~RHS  <-   FoundLHS SwapPred  FoundRHS
      // Forms 1. and 2. require swapping the operands of one condition. Don't
      // do this if it would break canonical constant/addrec ordering.
      if (!isa<SCEVConstant>(RHS) && !isa<SCEVAddRecExpr>(LHS))
        return isImpliedCondOperands(FoundPred, RHS, LHS, FoundLHS, FoundRHS,
                                     Context);
      if (!isa<SCEVConstant>(FoundRHS) && !isa<SCEVAddRecExpr>(FoundLHS))
        return isImpliedCondOperands(Pred, LHS, RHS, FoundRHS, FoundLHS, Context);
  
      // Don't try to getNotSCEV pointers.
      if (LHS->getType()->isPointerTy() || FoundLHS->getType()->isPointerTy())
        return false;
  
      // There's no clear preference between forms 3. and 4., try both.
      return isImpliedCondOperands(FoundPred, getNotSCEV(LHS), getNotSCEV(RHS),
                                   FoundLHS, FoundRHS, Context) ||
             isImpliedCondOperands(Pred, LHS, RHS, getNotSCEV(FoundLHS),
                                   getNotSCEV(FoundRHS), Context);
    }
  
    // Unsigned comparison is the same as signed comparison when both the operands
    // are non-negative.
    if (CmpInst::isUnsigned(FoundPred) &&
        CmpInst::getSignedPredicate(FoundPred) == Pred &&
        isKnownNonNegative(FoundLHS) && isKnownNonNegative(FoundRHS))
      return isImpliedCondOperands(Pred, LHS, RHS, FoundLHS, FoundRHS, Context);
  
    // Check if we can make progress by sharpening ranges.
    if (FoundPred == ICmpInst::ICMP_NE &&
        (isa<SCEVConstant>(FoundLHS) || isa<SCEVConstant>(FoundRHS))) {
  
      const SCEVConstant *C = nullptr;
      const SCEV *V = nullptr;
  
      if (isa<SCEVConstant>(FoundLHS)) {
        C = cast<SCEVConstant>(FoundLHS);
        V = FoundRHS;
      } else {
        C = cast<SCEVConstant>(FoundRHS);
        V = FoundLHS;
      }
  
      // The guarding predicate tells us that C != V. If the known range
      // of V is [C, t), we can sharpen the range to [C + 1, t).  The
      // range we consider has to correspond to same signedness as the
      // predicate we're interested in folding.
  
      APInt Min = ICmpInst::isSigned(Pred) ?
          getSignedRangeMin(V) : getUnsignedRangeMin(V);
  
      if (Min == C->getAPInt()) {
        // Given (V >= Min && V != Min) we conclude V >= (Min + 1).
        // This is true even if (Min + 1) wraps around -- in case of
        // wraparound, (Min + 1) < Min, so (V >= Min => V >= (Min + 1)).
  
        APInt SharperMin = Min + 1;
  
        switch (Pred) {
          case ICmpInst::ICMP_SGE:
          case ICmpInst::ICMP_UGE:
            // We know V `Pred` SharperMin.  If this implies LHS `Pred`
            // RHS, we're done.
            if (isImpliedCondOperands(Pred, LHS, RHS, V, getConstant(SharperMin),
                                      Context))
              return true;
            LLVM_FALLTHROUGH;
  
          case ICmpInst::ICMP_SGT:
          case ICmpInst::ICMP_UGT:
            // We know from the range information that (V `Pred` Min ||
            // V == Min).  We know from the guarding condition that !(V
            // == Min).  This gives us
            //
            //       V `Pred` Min || V == Min && !(V == Min)
            //   =>  V `Pred` Min
            //
            // If V `Pred` Min implies LHS `Pred` RHS, we're done.
  
            if (isImpliedCondOperands(Pred, LHS, RHS, V, getConstant(Min),
                                      Context))
              return true;
            break;
  
          // `LHS < RHS` and `LHS <= RHS` are handled in the same way as `RHS > LHS` and `RHS >= LHS` respectively.
          case ICmpInst::ICMP_SLE:
          case ICmpInst::ICMP_ULE:
            if (isImpliedCondOperands(CmpInst::getSwappedPredicate(Pred), RHS,
                                      LHS, V, getConstant(SharperMin), Context))
              return true;
            LLVM_FALLTHROUGH;
  
          case ICmpInst::ICMP_SLT:
          case ICmpInst::ICMP_ULT:
            if (isImpliedCondOperands(CmpInst::getSwappedPredicate(Pred), RHS,
                                      LHS, V, getConstant(Min), Context))
              return true;
            break;
  
          default:
            // No change
            break;
        }
      }
    }
  
    // Check whether the actual condition is beyond sufficient.
    if (FoundPred == ICmpInst::ICMP_EQ)
      if (ICmpInst::isTrueWhenEqual(Pred))
        if (isImpliedCondOperands(Pred, LHS, RHS, FoundLHS, FoundRHS, Context))
          return true;
    if (Pred == ICmpInst::ICMP_NE)
      if (!ICmpInst::isTrueWhenEqual(FoundPred))
        if (isImpliedCondOperands(FoundPred, LHS, RHS, FoundLHS, FoundRHS,
                                  Context))
          return true;
  
    // Otherwise assume the worst.
    return false;
  }
  
  bool ScalarEvolution::splitBinaryAdd(const SCEV *Expr,
                                       const SCEV *&L, const SCEV *&R,
                                       SCEV::NoWrapFlags &Flags) {
    const auto *AE = dyn_cast<SCEVAddExpr>(Expr);
    if (!AE || AE->getNumOperands() != 2)
      return false;
  
    L = AE->getOperand(0);
    R = AE->getOperand(1);
    Flags = AE->getNoWrapFlags();
    return true;
  }
  
  Optional<APInt> ScalarEvolution::computeConstantDifference(const SCEV *More,
                                                             const SCEV *Less) {
    // We avoid subtracting expressions here because this function is usually
    // fairly deep in the call stack (i.e. is called many times).
  
    // X - X = 0.
    if (More == Less)
      return APInt(getTypeSizeInBits(More->getType()), 0);
  
    if (isa<SCEVAddRecExpr>(Less) && isa<SCEVAddRecExpr>(More)) {
      const auto *LAR = cast<SCEVAddRecExpr>(Less);
      const auto *MAR = cast<SCEVAddRecExpr>(More);
  
      if (LAR->getLoop() != MAR->getLoop())
        return None;
  
      // We look at affine expressions only; not for correctness but to keep
      // getStepRecurrence cheap.
      if (!LAR->isAffine() || !MAR->isAffine())
        return None;
  
      if (LAR->getStepRecurrence(*this) != MAR->getStepRecurrence(*this))
        return None;
  
      Less = LAR->getStart();
      More = MAR->getStart();
  
      // fall through
    }
  
    if (isa<SCEVConstant>(Less) && isa<SCEVConstant>(More)) {
      const auto &M = cast<SCEVConstant>(More)->getAPInt();
      const auto &L = cast<SCEVConstant>(Less)->getAPInt();
      return M - L;
    }
  
    SCEV::NoWrapFlags Flags;
    const SCEV *LLess = nullptr, *RLess = nullptr;
    const SCEV *LMore = nullptr, *RMore = nullptr;
    const SCEVConstant *C1 = nullptr, *C2 = nullptr;
    // Compare (X + C1) vs X.
    if (splitBinaryAdd(Less, LLess, RLess, Flags))
      if ((C1 = dyn_cast<SCEVConstant>(LLess)))
        if (RLess == More)
          return -(C1->getAPInt());
  
    // Compare X vs (X + C2).
    if (splitBinaryAdd(More, LMore, RMore, Flags))
      if ((C2 = dyn_cast<SCEVConstant>(LMore)))
        if (RMore == Less)
          return C2->getAPInt();
  
    // Compare (X + C1) vs (X + C2).
    if (C1 && C2 && RLess == RMore)
      return C2->getAPInt() - C1->getAPInt();
  
    return None;
  }
  
  bool ScalarEvolution::isImpliedCondOperandsViaAddRecStart(
      ICmpInst::Predicate Pred, const SCEV *LHS, const SCEV *RHS,
      const SCEV *FoundLHS, const SCEV *FoundRHS, const Instruction *Context) {
    // Try to recognize the following pattern:
    //
    //   FoundRHS = ...
    // ...
    // loop:
    //   FoundLHS = {Start,+,W}
    // context_bb: // Basic block from the same loop
    //   known(Pred, FoundLHS, FoundRHS)
    //
    // If some predicate is known in the context of a loop, it is also known on
    // each iteration of this loop, including the first iteration. Therefore, in
    // this case, `FoundLHS Pred FoundRHS` implies `Start Pred FoundRHS`. Try to
    // prove the original pred using this fact.
    if (!Context)
      return false;
    const BasicBlock *ContextBB = Context->getParent();
    // Make sure AR varies in the context block.
    if (auto *AR = dyn_cast<SCEVAddRecExpr>(FoundLHS)) {
      const Loop *L = AR->getLoop();
      // Make sure that context belongs to the loop and executes on 1st iteration
      // (if it ever executes at all).
      if (!L->contains(ContextBB) || !DT.dominates(ContextBB, L->getLoopLatch()))
        return false;
      if (!isAvailableAtLoopEntry(FoundRHS, AR->getLoop()))
        return false;
      return isImpliedCondOperands(Pred, LHS, RHS, AR->getStart(), FoundRHS);
    }
  
    if (auto *AR = dyn_cast<SCEVAddRecExpr>(FoundRHS)) {
      const Loop *L = AR->getLoop();
      // Make sure that context belongs to the loop and executes on 1st iteration
      // (if it ever executes at all).
      if (!L->contains(ContextBB) || !DT.dominates(ContextBB, L->getLoopLatch()))
        return false;
      if (!isAvailableAtLoopEntry(FoundLHS, AR->getLoop()))
        return false;
      return isImpliedCondOperands(Pred, LHS, RHS, FoundLHS, AR->getStart());
    }
  
    return false;
  }
  
  bool ScalarEvolution::isImpliedCondOperandsViaNoOverflow(
      ICmpInst::Predicate Pred, const SCEV *LHS, const SCEV *RHS,
      const SCEV *FoundLHS, const SCEV *FoundRHS) {
    if (Pred != CmpInst::ICMP_SLT && Pred != CmpInst::ICMP_ULT)
      return false;
  
    const auto *AddRecLHS = dyn_cast<SCEVAddRecExpr>(LHS);
    if (!AddRecLHS)
      return false;
  
    const auto *AddRecFoundLHS = dyn_cast<SCEVAddRecExpr>(FoundLHS);
    if (!AddRecFoundLHS)
      return false;
  
    // We'd like to let SCEV reason about control dependencies, so we constrain
    // both the inequalities to be about add recurrences on the same loop.  This
    // way we can use isLoopEntryGuardedByCond later.
  
    const Loop *L = AddRecFoundLHS->getLoop();
    if (L != AddRecLHS->getLoop())
      return false;
  
    //  FoundLHS u< FoundRHS u< -C =>  (FoundLHS + C) u< (FoundRHS + C) ... (1)
    //
    //  FoundLHS s< FoundRHS s< INT_MIN - C => (FoundLHS + C) s< (FoundRHS + C)
    //                                                                  ... (2)
    //
    // Informal proof for (2), assuming (1) [*]:
    //
    // We'll also assume (A s< B) <=> ((A + INT_MIN) u< (B + INT_MIN)) ... (3)[**]
    //
    // Then
    //
    //       FoundLHS s< FoundRHS s< INT_MIN - C
    // <=>  (FoundLHS + INT_MIN) u< (FoundRHS + INT_MIN) u< -C   [ using (3) ]
    // <=>  (FoundLHS + INT_MIN + C) u< (FoundRHS + INT_MIN + C) [ using (1) ]
    // <=>  (FoundLHS + INT_MIN + C + INT_MIN) s<
    //                        (FoundRHS + INT_MIN + C + INT_MIN) [ using (3) ]
    // <=>  FoundLHS + C s< FoundRHS + C
    //
    // [*]: (1) can be proved by ruling out overflow.
    //
    // [**]: This can be proved by analyzing all the four possibilities:
    //    (A s< 0, B s< 0), (A s< 0, B s>= 0), (A s>= 0, B s< 0) and
    //    (A s>= 0, B s>= 0).
    //
    // Note:
    // Despite (2), "FoundRHS s< INT_MIN - C" does not mean that "FoundRHS + C"
    // will not sign underflow.  For instance, say FoundLHS = (i8 -128), FoundRHS
    // = (i8 -127) and C = (i8 -100).  Then INT_MIN - C = (i8 -28), and FoundRHS
    // s< (INT_MIN - C).  Lack of sign overflow / underflow in "FoundRHS + C" is
    // neither necessary nor sufficient to prove "(FoundLHS + C) s< (FoundRHS +
    // C)".
  
    Optional<APInt> LDiff = computeConstantDifference(LHS, FoundLHS);
    Optional<APInt> RDiff = computeConstantDifference(RHS, FoundRHS);
    if (!LDiff || !RDiff || *LDiff != *RDiff)
      return false;
  
    if (LDiff->isMinValue())
      return true;
  
    APInt FoundRHSLimit;
  
    if (Pred == CmpInst::ICMP_ULT) {
      FoundRHSLimit = -(*RDiff);
    } else {
      assert(Pred == CmpInst::ICMP_SLT && "Checked above!");
      FoundRHSLimit = APInt::getSignedMinValue(getTypeSizeInBits(RHS->getType())) - *RDiff;
    }
  
    // Try to prove (1) or (2), as needed.
    return isAvailableAtLoopEntry(FoundRHS, L) &&
           isLoopEntryGuardedByCond(L, Pred, FoundRHS,
                                    getConstant(FoundRHSLimit));
  }
  
  bool ScalarEvolution::isImpliedViaMerge(ICmpInst::Predicate Pred,
                                          const SCEV *LHS, const SCEV *RHS,
                                          const SCEV *FoundLHS,
                                          const SCEV *FoundRHS, unsigned Depth) {
    const PHINode *LPhi = nullptr, *RPhi = nullptr;
  
    auto ClearOnExit = make_scope_exit([&]() {
      if (LPhi) {
        bool Erased = PendingMerges.erase(LPhi);
        assert(Erased && "Failed to erase LPhi!");
        (void)Erased;
      }
      if (RPhi) {
        bool Erased = PendingMerges.erase(RPhi);
        assert(Erased && "Failed to erase RPhi!");
        (void)Erased;
      }
    });
  
    // Find respective Phis and check that they are not being pending.
    if (const SCEVUnknown *LU = dyn_cast<SCEVUnknown>(LHS))
      if (auto *Phi = dyn_cast<PHINode>(LU->getValue())) {
        if (!PendingMerges.insert(Phi).second)
          return false;
        LPhi = Phi;
      }
    if (const SCEVUnknown *RU = dyn_cast<SCEVUnknown>(RHS))
      if (auto *Phi = dyn_cast<PHINode>(RU->getValue())) {
        // If we detect a loop of Phi nodes being processed by this method, for
        // example:
        //
        //   %a = phi i32 [ %some1, %preheader ], [ %b, %latch ]
        //   %b = phi i32 [ %some2, %preheader ], [ %a, %latch ]
        //
        // we don't want to deal with a case that complex, so return conservative
        // answer false.
        if (!PendingMerges.insert(Phi).second)
          return false;
        RPhi = Phi;
      }
  
    // If none of LHS, RHS is a Phi, nothing to do here.
    if (!LPhi && !RPhi)
      return false;
  
    // If there is a SCEVUnknown Phi we are interested in, make it left.
    if (!LPhi) {
      std::swap(LHS, RHS);
      std::swap(FoundLHS, FoundRHS);
      std::swap(LPhi, RPhi);
      Pred = ICmpInst::getSwappedPredicate(Pred);
    }
  
    assert(LPhi && "LPhi should definitely be a SCEVUnknown Phi!");
    const BasicBlock *LBB = LPhi->getParent();
    const SCEVAddRecExpr *RAR = dyn_cast<SCEVAddRecExpr>(RHS);
  
    auto ProvedEasily = [&](const SCEV *S1, const SCEV *S2) {
      return isKnownViaNonRecursiveReasoning(Pred, S1, S2) ||
             isImpliedCondOperandsViaRanges(Pred, S1, S2, FoundLHS, FoundRHS) ||
             isImpliedViaOperations(Pred, S1, S2, FoundLHS, FoundRHS, Depth);
    };
  
    if (RPhi && RPhi->getParent() == LBB) {
      // Case one: RHS is also a SCEVUnknown Phi from the same basic block.
      // If we compare two Phis from the same block, and for each entry block
      // the predicate is true for incoming values from this block, then the
      // predicate is also true for the Phis.
      for (const BasicBlock *IncBB : predecessors(LBB)) {
        const SCEV *L = getSCEV(LPhi->getIncomingValueForBlock(IncBB));
        const SCEV *R = getSCEV(RPhi->getIncomingValueForBlock(IncBB));
        if (!ProvedEasily(L, R))
          return false;
      }
    } else if (RAR && RAR->getLoop()->getHeader() == LBB) {
      // Case two: RHS is also a Phi from the same basic block, and it is an
      // AddRec. It means that there is a loop which has both AddRec and Unknown
      // PHIs, for it we can compare incoming values of AddRec from above the loop
      // and latch with their respective incoming values of LPhi.
      // TODO: Generalize to handle loops with many inputs in a header.
      if (LPhi->getNumIncomingValues() != 2) return false;
  
      auto *RLoop = RAR->getLoop();
      auto *Predecessor = RLoop->getLoopPredecessor();
      assert(Predecessor && "Loop with AddRec with no predecessor?");
      const SCEV *L1 = getSCEV(LPhi->getIncomingValueForBlock(Predecessor));
      if (!ProvedEasily(L1, RAR->getStart()))
        return false;
      auto *Latch = RLoop->getLoopLatch();
      assert(Latch && "Loop with AddRec with no latch?");
      const SCEV *L2 = getSCEV(LPhi->getIncomingValueForBlock(Latch));
      if (!ProvedEasily(L2, RAR->getPostIncExpr(*this)))
        return false;
    } else {
      // In all other cases go over inputs of LHS and compare each of them to RHS,
      // the predicate is true for (LHS, RHS) if it is true for all such pairs.
      // At this point RHS is either a non-Phi, or it is a Phi from some block
      // different from LBB.
      for (const BasicBlock *IncBB : predecessors(LBB)) {
        // Check that RHS is available in this block.
        if (!dominates(RHS, IncBB))
          return false;
        const SCEV *L = getSCEV(LPhi->getIncomingValueForBlock(IncBB));
        // Make sure L does not refer to a value from a potentially previous
        // iteration of a loop.
        if (!properlyDominates(L, IncBB))
          return false;
        if (!ProvedEasily(L, RHS))
          return false;
      }
    }
    return true;
  }
  
  bool ScalarEvolution::isImpliedCondOperands(ICmpInst::Predicate Pred,
                                              const SCEV *LHS, const SCEV *RHS,
                                              const SCEV *FoundLHS,
                                              const SCEV *FoundRHS,
                                              const Instruction *Context) {
    if (isImpliedCondOperandsViaRanges(Pred, LHS, RHS, FoundLHS, FoundRHS))
      return true;
  
    if (isImpliedCondOperandsViaNoOverflow(Pred, LHS, RHS, FoundLHS, FoundRHS))
      return true;
  
    if (isImpliedCondOperandsViaAddRecStart(Pred, LHS, RHS, FoundLHS, FoundRHS,
                                            Context))
      return true;
  
    return isImpliedCondOperandsHelper(Pred, LHS, RHS,
                                       FoundLHS, FoundRHS);
  }
  
  /// Is MaybeMinMaxExpr an (U|S)(Min|Max) of Candidate and some other values?
  template <typename MinMaxExprType>
  static bool IsMinMaxConsistingOf(const SCEV *MaybeMinMaxExpr,
                                   const SCEV *Candidate) {
    const MinMaxExprType *MinMaxExpr = dyn_cast<MinMaxExprType>(MaybeMinMaxExpr);
    if (!MinMaxExpr)
      return false;
  
    return is_contained(MinMaxExpr->operands(), Candidate);
  }
  
  static bool IsKnownPredicateViaAddRecStart(ScalarEvolution &SE,
                                             ICmpInst::Predicate Pred,
                                             const SCEV *LHS, const SCEV *RHS) {
    // If both sides are affine addrecs for the same loop, with equal
    // steps, and we know the recurrences don't wrap, then we only
    // need to check the predicate on the starting values.
  
    if (!ICmpInst::isRelational(Pred))
      return false;
  
    const SCEVAddRecExpr *LAR = dyn_cast<SCEVAddRecExpr>(LHS);
    if (!LAR)
      return false;
    const SCEVAddRecExpr *RAR = dyn_cast<SCEVAddRecExpr>(RHS);
    if (!RAR)
      return false;
    if (LAR->getLoop() != RAR->getLoop())
      return false;
    if (!LAR->isAffine() || !RAR->isAffine())
      return false;
  
    if (LAR->getStepRecurrence(SE) != RAR->getStepRecurrence(SE))
      return false;
  
    SCEV::NoWrapFlags NW = ICmpInst::isSigned(Pred) ?
                           SCEV::FlagNSW : SCEV::FlagNUW;
    if (!LAR->getNoWrapFlags(NW) || !RAR->getNoWrapFlags(NW))
      return false;
  
    return SE.isKnownPredicate(Pred, LAR->getStart(), RAR->getStart());
  }
  
  /// Is LHS `Pred` RHS true on the virtue of LHS or RHS being a Min or Max
  /// expression?
  static bool IsKnownPredicateViaMinOrMax(ScalarEvolution &SE,
                                          ICmpInst::Predicate Pred,
                                          const SCEV *LHS, const SCEV *RHS) {
    switch (Pred) {
    default:
      return false;
  
    case ICmpInst::ICMP_SGE:
      std::swap(LHS, RHS);
      LLVM_FALLTHROUGH;
    case ICmpInst::ICMP_SLE:
      return
          // min(A, ...) <= A
          IsMinMaxConsistingOf<SCEVSMinExpr>(LHS, RHS) ||
          // A <= max(A, ...)
          IsMinMaxConsistingOf<SCEVSMaxExpr>(RHS, LHS);
  
    case ICmpInst::ICMP_UGE:
      std::swap(LHS, RHS);
      LLVM_FALLTHROUGH;
    case ICmpInst::ICMP_ULE:
      return
          // min(A, ...) <= A
          IsMinMaxConsistingOf<SCEVUMinExpr>(LHS, RHS) ||
          // A <= max(A, ...)
          IsMinMaxConsistingOf<SCEVUMaxExpr>(RHS, LHS);
    }
  
    llvm_unreachable("covered switch fell through?!");
  }
  
  bool ScalarEvolution::isImpliedViaOperations(ICmpInst::Predicate Pred,
                                               const SCEV *LHS, const SCEV *RHS,
                                               const SCEV *FoundLHS,
                                               const SCEV *FoundRHS,
                                               unsigned Depth) {
    assert(getTypeSizeInBits(LHS->getType()) ==
               getTypeSizeInBits(RHS->getType()) &&
           "LHS and RHS have different sizes?");
    assert(getTypeSizeInBits(FoundLHS->getType()) ==
               getTypeSizeInBits(FoundRHS->getType()) &&
           "FoundLHS and FoundRHS have different sizes?");
    // We want to avoid hurting the compile time with analysis of too big trees.
    if (Depth > MaxSCEVOperationsImplicationDepth)
      return false;
  
    // We only want to work with GT comparison so far.
    if (Pred == ICmpInst::ICMP_ULT || Pred == ICmpInst::ICMP_SLT) {
      Pred = CmpInst::getSwappedPredicate(Pred);
      std::swap(LHS, RHS);
      std::swap(FoundLHS, FoundRHS);
    }
  
    // For unsigned, try to reduce it to corresponding signed comparison.
    if (Pred == ICmpInst::ICMP_UGT)
      // We can replace unsigned predicate with its signed counterpart if all
      // involved values are non-negative.
      // TODO: We could have better support for unsigned.
      if (isKnownNonNegative(FoundLHS) && isKnownNonNegative(FoundRHS)) {
        // Knowing that both FoundLHS and FoundRHS are non-negative, and knowing
        // FoundLHS >u FoundRHS, we also know that FoundLHS >s FoundRHS. Let us
        // use this fact to prove that LHS and RHS are non-negative.
        const SCEV *MinusOne = getMinusOne(LHS->getType());
        if (isImpliedCondOperands(ICmpInst::ICMP_SGT, LHS, MinusOne, FoundLHS,
                                  FoundRHS) &&
            isImpliedCondOperands(ICmpInst::ICMP_SGT, RHS, MinusOne, FoundLHS,
                                  FoundRHS))
          Pred = ICmpInst::ICMP_SGT;
      }
  
    if (Pred != ICmpInst::ICMP_SGT)
      return false;
  
    auto GetOpFromSExt = [&](const SCEV *S) {
      if (auto *Ext = dyn_cast<SCEVSignExtendExpr>(S))
        return Ext->getOperand();
      // TODO: If S is a SCEVConstant then you can cheaply "strip" the sext off
      // the constant in some cases.
      return S;
    };
  
    // Acquire values from extensions.
    auto *OrigLHS = LHS;
    auto *OrigFoundLHS = FoundLHS;
    LHS = GetOpFromSExt(LHS);
    FoundLHS = GetOpFromSExt(FoundLHS);
  
    // Is the SGT predicate can be proved trivially or using the found context.
    auto IsSGTViaContext = [&](const SCEV *S1, const SCEV *S2) {
      return isKnownViaNonRecursiveReasoning(ICmpInst::ICMP_SGT, S1, S2) ||
             isImpliedViaOperations(ICmpInst::ICMP_SGT, S1, S2, OrigFoundLHS,
                                    FoundRHS, Depth + 1);
    };
  
    if (auto *LHSAddExpr = dyn_cast<SCEVAddExpr>(LHS)) {
      // We want to avoid creation of any new non-constant SCEV. Since we are
      // going to compare the operands to RHS, we should be certain that we don't
      // need any size extensions for this. So let's decline all cases when the
      // sizes of types of LHS and RHS do not match.
      // TODO: Maybe try to get RHS from sext to catch more cases?
      if (getTypeSizeInBits(LHS->getType()) != getTypeSizeInBits(RHS->getType()))
        return false;
  
      // Should not overflow.
      if (!LHSAddExpr->hasNoSignedWrap())
        return false;
  
      auto *LL = LHSAddExpr->getOperand(0);
      auto *LR = LHSAddExpr->getOperand(1);
      auto *MinusOne = getMinusOne(RHS->getType());
  
      // Checks that S1 >= 0 && S2 > RHS, trivially or using the found context.
      auto IsSumGreaterThanRHS = [&](const SCEV *S1, const SCEV *S2) {
        return IsSGTViaContext(S1, MinusOne) && IsSGTViaContext(S2, RHS);
      };
      // Try to prove the following rule:
      // (LHS = LL + LR) && (LL >= 0) && (LR > RHS) => (LHS > RHS).
      // (LHS = LL + LR) && (LR >= 0) && (LL > RHS) => (LHS > RHS).
      if (IsSumGreaterThanRHS(LL, LR) || IsSumGreaterThanRHS(LR, LL))
        return true;
    } else if (auto *LHSUnknownExpr = dyn_cast<SCEVUnknown>(LHS)) {
      Value *LL, *LR;
      // FIXME: Once we have SDiv implemented, we can get rid of this matching.
  
      using namespace llvm::PatternMatch;
  
      if (match(LHSUnknownExpr->getValue(), m_SDiv(m_Value(LL), m_Value(LR)))) {
        // Rules for division.
        // We are going to perform some comparisons with Denominator and its
        // derivative expressions. In general case, creating a SCEV for it may
        // lead to a complex analysis of the entire graph, and in particular it
        // can request trip count recalculation for the same loop. This would
        // cache as SCEVCouldNotCompute to avoid the infinite recursion. To avoid
        // this, we only want to create SCEVs that are constants in this section.
        // So we bail if Denominator is not a constant.
        if (!isa<ConstantInt>(LR))
          return false;
  
        auto *Denominator = cast<SCEVConstant>(getSCEV(LR));
  
        // We want to make sure that LHS = FoundLHS / Denominator. If it is so,
        // then a SCEV for the numerator already exists and matches with FoundLHS.
        auto *Numerator = getExistingSCEV(LL);
        if (!Numerator || Numerator->getType() != FoundLHS->getType())
          return false;
  
        // Make sure that the numerator matches with FoundLHS and the denominator
        // is positive.
        if (!HasSameValue(Numerator, FoundLHS) || !isKnownPositive(Denominator))
          return false;
  
        auto *DTy = Denominator->getType();
        auto *FRHSTy = FoundRHS->getType();
        if (DTy->isPointerTy() != FRHSTy->isPointerTy())
          // One of types is a pointer and another one is not. We cannot extend
          // them properly to a wider type, so let us just reject this case.
          // TODO: Usage of getEffectiveSCEVType for DTy, FRHSTy etc should help
          // to avoid this check.
          return false;
  
        // Given that:
        // FoundLHS > FoundRHS, LHS = FoundLHS / Denominator, Denominator > 0.
        auto *WTy = getWiderType(DTy, FRHSTy);
        auto *DenominatorExt = getNoopOrSignExtend(Denominator, WTy);
        auto *FoundRHSExt = getNoopOrSignExtend(FoundRHS, WTy);
  
        // Try to prove the following rule:
        // (FoundRHS > Denominator - 2) && (RHS <= 0) => (LHS > RHS).
        // For example, given that FoundLHS > 2. It means that FoundLHS is at
        // least 3. If we divide it by Denominator < 4, we will have at least 1.
        auto *DenomMinusTwo = getMinusSCEV(DenominatorExt, getConstant(WTy, 2));
        if (isKnownNonPositive(RHS) &&
            IsSGTViaContext(FoundRHSExt, DenomMinusTwo))
          return true;
  
        // Try to prove the following rule:
        // (FoundRHS > -1 - Denominator) && (RHS < 0) => (LHS > RHS).
        // For example, given that FoundLHS > -3. Then FoundLHS is at least -2.
        // If we divide it by Denominator > 2, then:
        // 1. If FoundLHS is negative, then the result is 0.
        // 2. If FoundLHS is non-negative, then the result is non-negative.
        // Anyways, the result is non-negative.
        auto *MinusOne = getMinusOne(WTy);
        auto *NegDenomMinusOne = getMinusSCEV(MinusOne, DenominatorExt);
        if (isKnownNegative(RHS) &&
            IsSGTViaContext(FoundRHSExt, NegDenomMinusOne))
          return true;
      }
    }
  
    // If our expression contained SCEVUnknown Phis, and we split it down and now
    // need to prove something for them, try to prove the predicate for every
    // possible incoming values of those Phis.
    if (isImpliedViaMerge(Pred, OrigLHS, RHS, OrigFoundLHS, FoundRHS, Depth + 1))
      return true;
  
    return false;
  }
  
  static bool isKnownPredicateExtendIdiom(ICmpInst::Predicate Pred,
                                          const SCEV *LHS, const SCEV *RHS) {
    // zext x u<= sext x, sext x s<= zext x
    switch (Pred) {
    case ICmpInst::ICMP_SGE:
      std::swap(LHS, RHS);
      LLVM_FALLTHROUGH;
    case ICmpInst::ICMP_SLE: {
      // If operand >=s 0 then ZExt == SExt.  If operand <s 0 then SExt <s ZExt.
      const SCEVSignExtendExpr *SExt = dyn_cast<SCEVSignExtendExpr>(LHS);
      const SCEVZeroExtendExpr *ZExt = dyn_cast<SCEVZeroExtendExpr>(RHS);
      if (SExt && ZExt && SExt->getOperand() == ZExt->getOperand())
        return true;
      break;
    }
    case ICmpInst::ICMP_UGE:
      std::swap(LHS, RHS);
      LLVM_FALLTHROUGH;
    case ICmpInst::ICMP_ULE: {
      // If operand >=s 0 then ZExt == SExt.  If operand <s 0 then ZExt <u SExt.
      const SCEVZeroExtendExpr *ZExt = dyn_cast<SCEVZeroExtendExpr>(LHS);
      const SCEVSignExtendExpr *SExt = dyn_cast<SCEVSignExtendExpr>(RHS);
      if (SExt && ZExt && SExt->getOperand() == ZExt->getOperand())
        return true;
      break;
    }
    default:
      break;
    };
    return false;
  }
  
  bool
  ScalarEvolution::isKnownViaNonRecursiveReasoning(ICmpInst::Predicate Pred,
                                             const SCEV *LHS, const SCEV *RHS) {
    return isKnownPredicateExtendIdiom(Pred, LHS, RHS) ||
           isKnownPredicateViaConstantRanges(Pred, LHS, RHS) ||
           IsKnownPredicateViaMinOrMax(*this, Pred, LHS, RHS) ||
           IsKnownPredicateViaAddRecStart(*this, Pred, LHS, RHS) ||
           isKnownPredicateViaNoOverflow(Pred, LHS, RHS);
  }
  
  bool
  ScalarEvolution::isImpliedCondOperandsHelper(ICmpInst::Predicate Pred,
                                               const SCEV *LHS, const SCEV *RHS,
                                               const SCEV *FoundLHS,
                                               const SCEV *FoundRHS) {
    switch (Pred) {
    default: llvm_unreachable("Unexpected ICmpInst::Predicate value!");
    case ICmpInst::ICMP_EQ:
    case ICmpInst::ICMP_NE:
      if (HasSameValue(LHS, FoundLHS) && HasSameValue(RHS, FoundRHS))
        return true;
      break;
    case ICmpInst::ICMP_SLT:
    case ICmpInst::ICMP_SLE:
      if (isKnownViaNonRecursiveReasoning(ICmpInst::ICMP_SLE, LHS, FoundLHS) &&
          isKnownViaNonRecursiveReasoning(ICmpInst::ICMP_SGE, RHS, FoundRHS))
        return true;
      break;
    case ICmpInst::ICMP_SGT:
    case ICmpInst::ICMP_SGE:
      if (isKnownViaNonRecursiveReasoning(ICmpInst::ICMP_SGE, LHS, FoundLHS) &&
          isKnownViaNonRecursiveReasoning(ICmpInst::ICMP_SLE, RHS, FoundRHS))
        return true;
      break;
    case ICmpInst::ICMP_ULT:
    case ICmpInst::ICMP_ULE:
      if (isKnownViaNonRecursiveReasoning(ICmpInst::ICMP_ULE, LHS, FoundLHS) &&
          isKnownViaNonRecursiveReasoning(ICmpInst::ICMP_UGE, RHS, FoundRHS))
        return true;
      break;
    case ICmpInst::ICMP_UGT:
    case ICmpInst::ICMP_UGE:
      if (isKnownViaNonRecursiveReasoning(ICmpInst::ICMP_UGE, LHS, FoundLHS) &&
          isKnownViaNonRecursiveReasoning(ICmpInst::ICMP_ULE, RHS, FoundRHS))
        return true;
      break;
    }
  
    // Maybe it can be proved via operations?
    if (isImpliedViaOperations(Pred, LHS, RHS, FoundLHS, FoundRHS))
      return true;
  
    return false;
  }
  
  bool ScalarEvolution::isImpliedCondOperandsViaRanges(ICmpInst::Predicate Pred,
                                                       const SCEV *LHS,
                                                       const SCEV *RHS,
                                                       const SCEV *FoundLHS,
                                                       const SCEV *FoundRHS) {
    if (!isa<SCEVConstant>(RHS) || !isa<SCEVConstant>(FoundRHS))
      // The restriction on `FoundRHS` be lifted easily -- it exists only to
      // reduce the compile time impact of this optimization.
      return false;
  
    Optional<APInt> Addend = computeConstantDifference(LHS, FoundLHS);
    if (!Addend)
      return false;
  
    const APInt &ConstFoundRHS = cast<SCEVConstant>(FoundRHS)->getAPInt();
  
    // `FoundLHSRange` is the range we know `FoundLHS` to be in by virtue of the
    // antecedent "`FoundLHS` `Pred` `FoundRHS`".
    ConstantRange FoundLHSRange =
        ConstantRange::makeExactICmpRegion(Pred, ConstFoundRHS);
  
    // Since `LHS` is `FoundLHS` + `Addend`, we can compute a range for `LHS`:
    ConstantRange LHSRange = FoundLHSRange.add(ConstantRange(*Addend));
  
    // We can also compute the range of values for `LHS` that satisfy the
    // consequent, "`LHS` `Pred` `RHS`":
    const APInt &ConstRHS = cast<SCEVConstant>(RHS)->getAPInt();
    // The antecedent implies the consequent if every value of `LHS` that
    // satisfies the antecedent also satisfies the consequent.
    return LHSRange.icmp(Pred, ConstRHS);
  }
  
  bool ScalarEvolution::canIVOverflowOnLT(const SCEV *RHS, const SCEV *Stride,
                                          bool IsSigned) {
    assert(isKnownPositive(Stride) && "Positive stride expected!");
  
    unsigned BitWidth = getTypeSizeInBits(RHS->getType());
    const SCEV *One = getOne(Stride->getType());
  
    if (IsSigned) {
      APInt MaxRHS = getSignedRangeMax(RHS);
      APInt MaxValue = APInt::getSignedMaxValue(BitWidth);
      APInt MaxStrideMinusOne = getSignedRangeMax(getMinusSCEV(Stride, One));
  
      // SMaxRHS + SMaxStrideMinusOne > SMaxValue => overflow!
      return (std::move(MaxValue) - MaxStrideMinusOne).slt(MaxRHS);
    }
  
    APInt MaxRHS = getUnsignedRangeMax(RHS);
    APInt MaxValue = APInt::getMaxValue(BitWidth);
    APInt MaxStrideMinusOne = getUnsignedRangeMax(getMinusSCEV(Stride, One));
  
    // UMaxRHS + UMaxStrideMinusOne > UMaxValue => overflow!
    return (std::move(MaxValue) - MaxStrideMinusOne).ult(MaxRHS);
  }
  
  bool ScalarEvolution::canIVOverflowOnGT(const SCEV *RHS, const SCEV *Stride,
                                          bool IsSigned) {
    
    unsigned BitWidth = getTypeSizeInBits(RHS->getType());
    const SCEV *One = getOne(Stride->getType());
  
    if (IsSigned) {
      APInt MinRHS = getSignedRangeMin(RHS);
      APInt MinValue = APInt::getSignedMinValue(BitWidth);
      APInt MaxStrideMinusOne = getSignedRangeMax(getMinusSCEV(Stride, One));
  
      // SMinRHS - SMaxStrideMinusOne < SMinValue => overflow!
      return (std::move(MinValue) + MaxStrideMinusOne).sgt(MinRHS);
    }
  
    APInt MinRHS = getUnsignedRangeMin(RHS);
    APInt MinValue = APInt::getMinValue(BitWidth);
    APInt MaxStrideMinusOne = getUnsignedRangeMax(getMinusSCEV(Stride, One));
  
    // UMinRHS - UMaxStrideMinusOne < UMinValue => overflow!
    return (std::move(MinValue) + MaxStrideMinusOne).ugt(MinRHS);
  }
  
  const SCEV *ScalarEvolution::getUDivCeilSCEV(const SCEV *N, const SCEV *D) {
    // umin(N, 1) + floor((N - umin(N, 1)) / D)
    // This is equivalent to "1 + floor((N - 1) / D)" for N != 0. The umin
    // expression fixes the case of N=0.
    const SCEV *MinNOne = getUMinExpr(N, getOne(N->getType()));
    const SCEV *NMinusOne = getMinusSCEV(N, MinNOne);
    return getAddExpr(MinNOne, getUDivExpr(NMinusOne, D));
  }
  
  const SCEV *ScalarEvolution::computeMaxBECountForLT(const SCEV *Start,
                                                      const SCEV *Stride,
                                                      const SCEV *End,
                                                      unsigned BitWidth,
                                                      bool IsSigned) {
    // The logic in this function assumes we can represent a positive stride.
    // If we can't, the backedge-taken count must be zero.
    if (IsSigned && BitWidth == 1)
      return getZero(Stride->getType());
  
    // Calculate the maximum backedge count based on the range of values
    // permitted by Start, End, and Stride.
    const SCEV *MaxBECount;
    APInt MinStart =
        IsSigned ? getSignedRangeMin(Start) : getUnsignedRangeMin(Start);
  
    APInt StrideForMaxBECount =
        IsSigned ? getSignedRangeMin(Stride) : getUnsignedRangeMin(Stride);
  
    // We assume either the stride is positive, or the backedge-taken count
    // is zero. So force StrideForMaxBECount to be at least one.
    APInt One(BitWidth, 1);
    StrideForMaxBECount = IsSigned ? APIntOps::smax(One, StrideForMaxBECount)
                                   : APIntOps::umax(One, StrideForMaxBECount);
  
    APInt MaxValue = IsSigned ? APInt::getSignedMaxValue(BitWidth)
                              : APInt::getMaxValue(BitWidth);
    APInt Limit = MaxValue - (StrideForMaxBECount - 1);
  
    // Although End can be a MAX expression we estimate MaxEnd considering only
    // the case End = RHS of the loop termination condition. This is safe because
    // in the other case (End - Start) is zero, leading to a zero maximum backedge
    // taken count.
    APInt MaxEnd = IsSigned ? APIntOps::smin(getSignedRangeMax(End), Limit)
                            : APIntOps::umin(getUnsignedRangeMax(End), Limit);
  
    // MaxBECount = ceil((max(MaxEnd, MinStart) - MinStart) / Stride)
    MaxEnd = IsSigned ? APIntOps::smax(MaxEnd, MinStart)
                      : APIntOps::umax(MaxEnd, MinStart);
  
    MaxBECount = getUDivCeilSCEV(getConstant(MaxEnd - MinStart) /* Delta */,
                                 getConstant(StrideForMaxBECount) /* Step */);
  
    return MaxBECount;
  }
  
  ScalarEvolution::ExitLimit
  ScalarEvolution::howManyLessThans(const SCEV *LHS, const SCEV *RHS,
                                    const Loop *L, bool IsSigned,
                                    bool ControlsExit, bool AllowPredicates) {
    SmallPtrSet<const SCEVPredicate *, 4> Predicates;
  
    const SCEVAddRecExpr *IV = dyn_cast<SCEVAddRecExpr>(LHS);
    bool PredicatedIV = false;
  
    if (!IV && AllowPredicates) {
      // Try to make this an AddRec using runtime tests, in the first X
      // iterations of this loop, where X is the SCEV expression found by the
      // algorithm below.
      IV = convertSCEVToAddRecWithPredicates(LHS, L, Predicates);
      PredicatedIV = true;
    }
  
    // Avoid weird loops
    if (!IV || IV->getLoop() != L || !IV->isAffine())
      return getCouldNotCompute();
  
    // A precondition of this method is that the condition being analyzed
    // reaches an exiting branch which dominates the latch.  Given that, we can
    // assume that an increment which violates the nowrap specification and
    // produces poison must cause undefined behavior when the resulting poison
    // value is branched upon and thus we can conclude that the backedge is
    // taken no more often than would be required to produce that poison value.
    // Note that a well defined loop can exit on the iteration which violates
    // the nowrap specification if there is another exit (either explicit or
    // implicit/exceptional) which causes the loop to execute before the
    // exiting instruction we're analyzing would trigger UB.
    auto WrapType = IsSigned ? SCEV::FlagNSW : SCEV::FlagNUW;
    bool NoWrap = ControlsExit && IV->getNoWrapFlags(WrapType);
    ICmpInst::Predicate Cond = IsSigned ? ICmpInst::ICMP_SLT : ICmpInst::ICMP_ULT;
  
    const SCEV *Stride = IV->getStepRecurrence(*this);
  
    bool PositiveStride = isKnownPositive(Stride);
  
    // Avoid negative or zero stride values.
    if (!PositiveStride) {
      // We can compute the correct backedge taken count for loops with unknown
      // strides if we can prove that the loop is not an infinite loop with side
      // effects. Here's the loop structure we are trying to handle -
      //
      // i = start
      // do {
      //   A[i] = i;
      //   i += s;
      // } while (i < end);
      //
      // The backedge taken count for such loops is evaluated as -
      // (max(end, start + stride) - start - 1) /u stride
      //
      // The additional preconditions that we need to check to prove correctness
      // of the above formula is as follows -
      //
      // a) IV is either nuw or nsw depending upon signedness (indicated by the
      //    NoWrap flag).
      // b) loop is single exit with no side effects.
      //
      //
      // Precondition a) implies that if the stride is negative, this is a single
      // trip loop. The backedge taken count formula reduces to zero in this case.
      //
      // Precondition b) implies that the unknown stride cannot be zero otherwise
      // we have UB.
      //
      // The positive stride case is the same as isKnownPositive(Stride) returning
      // true (original behavior of the function).
      //
      // We want to make sure that the stride is truly unknown as there are edge
      // cases where ScalarEvolution propagates no wrap flags to the
      // post-increment/decrement IV even though the increment/decrement operation
      // itself is wrapping. The computed backedge taken count may be wrong in
      // such cases. This is prevented by checking that the stride is not known to
      // be either positive or non-positive. For example, no wrap flags are
      // propagated to the post-increment IV of this loop with a trip count of 2 -
      //
      // unsigned char i;
      // for(i=127; i<128; i+=129)
      //   A[i] = i;
      //
      if (PredicatedIV || !NoWrap || isKnownNonPositive(Stride) ||
          !loopIsFiniteByAssumption(L))
        return getCouldNotCompute();
  
      // We allow a potentially zero stride, but we need to divide by stride
      // below.  Since the loop can't be infinite and this check must control
      // the sole exit, we can infer the exit must be taken on the first
      // iteration (e.g. backedge count = 0) if the stride is zero.  Given that,
      // we know the numerator in the divides below must be zero, so we can
      // pick an arbitrary non-zero value for the denominator (e.g. stride)
      // and produce the right result.
      // FIXME: Handle the case where Stride is poison?
      auto wouldZeroStrideBeUB = [&]() {
        // If RHS isn't loop invariant, bail out for now. This isn't necessary
        // for the proof, but isLoopEntryGuardedByCond only works on
        // loop-invariant values.
        if (!isLoopInvariant(RHS, L))
          return false;
  
        // Proof by contradiction.  Suppose the stride were zero.  If we can
        // prove that the backedge *is* taken on the first iteration, then since
        // we know this condition controls the sole exit, we must have an
        // infinite loop.  We can't have a (well defined) infinite loop per
        // check just above.
        // Note: The (Start - Stride) term is used to get the start' term from
        // (start' + stride,+,stride). Remember that we only care about the
        // result of this expression when stride == 0 at runtime.
        auto *StartIfZero = getMinusSCEV(IV->getStart(), Stride);
        return isLoopEntryGuardedByCond(L, Cond, StartIfZero, RHS);
      };
      if (!isKnownNonZero(Stride) && !wouldZeroStrideBeUB()) {
        Stride = getUMaxExpr(Stride, getOne(Stride->getType()));
      }
    } else if (!Stride->isOne() && !NoWrap) {
      auto isUBOnWrap = [&]() {
        // Can we prove this loop *must* be UB if overflow of IV occurs?
        // Reasoning goes as follows:
        // * Suppose the IV did self wrap.
        // * If Stride evenly divides the iteration space, then once wrap
        //   occurs, the loop must revisit the same values.
        // * We know that RHS is invariant, and that none of those values
        //   caused this exit to be taken previously.  Thus, this exit is
        //   dynamically dead.
        // * If this is the sole exit, then a dead exit implies the loop
        //   must be infinite if there are no abnormal exits.
        // * If the loop were infinite, then it must either not be mustprogress
        //   or have side effects. Otherwise, it must be UB.
        // * It can't (by assumption), be UB so we have contradicted our
        //   premise and can conclude the IV did not in fact self-wrap.
        // From no-self-wrap, we need to then prove no-(un)signed-wrap.  This
        // follows trivially from the fact that every (un)signed-wrapped, but
        // not self-wrapped value must be LT than the last value before
        // (un)signed wrap.  Since we know that last value didn't exit, nor
        // will any smaller one.
  
        if (!isLoopInvariant(RHS, L))
          return false;
  
        auto *StrideC = dyn_cast<SCEVConstant>(Stride);
        if (!StrideC || !StrideC->getAPInt().isPowerOf2())
          return false;
  
        if (!ControlsExit || !loopHasNoAbnormalExits(L))
          return false;
  
        return loopIsFiniteByAssumption(L);
      };
  
      // Avoid proven overflow cases: this will ensure that the backedge taken
      // count will not generate any unsigned overflow. Relaxed no-overflow
      // conditions exploit NoWrapFlags, allowing to optimize in presence of
      // undefined behaviors like the case of C language.
      if (canIVOverflowOnLT(RHS, Stride, IsSigned) && !isUBOnWrap())
        return getCouldNotCompute();
    }
  
    const SCEV *Start = IV->getStart();
  
    // Preserve pointer-typed Start/RHS to pass to isLoopEntryGuardedByCond.
    // Use integer-typed versions for actual computation.
    const SCEV *OrigStart = Start;
    const SCEV *OrigRHS = RHS;
    if (Start->getType()->isPointerTy()) {
      Start = getLosslessPtrToIntExpr(Start);
      if (isa<SCEVCouldNotCompute>(Start))
        return Start;
    }
    if (RHS->getType()->isPointerTy()) {
      RHS = getLosslessPtrToIntExpr(RHS);
      if (isa<SCEVCouldNotCompute>(RHS))
        return RHS;
    }
  
    // When the RHS is not invariant, we do not know the end bound of the loop and
    // cannot calculate the ExactBECount needed by ExitLimit. However, we can
    // calculate the MaxBECount, given the start, stride and max value for the end
    // bound of the loop (RHS), and the fact that IV does not overflow (which is
    // checked above).
    if (!isLoopInvariant(RHS, L)) {
      const SCEV *MaxBECount = computeMaxBECountForLT(
          Start, Stride, RHS, getTypeSizeInBits(LHS->getType()), IsSigned);
      return ExitLimit(getCouldNotCompute() /* ExactNotTaken */, MaxBECount,
                       false /*MaxOrZero*/, Predicates);
    }
  
    // We use the expression (max(End,Start)-Start)/Stride to describe the
    // backedge count, as if the backedge is taken at least once max(End,Start)
    // is End and so the result is as above, and if not max(End,Start) is Start
    // so we get a backedge count of zero.
    const SCEV *BECount = nullptr;
    auto *StartMinusStride = getMinusSCEV(OrigStart, Stride);
    // Can we prove (max(RHS,Start) > Start - Stride?
    if (isLoopEntryGuardedByCond(L, Cond, StartMinusStride, Start) &&
        isLoopEntryGuardedByCond(L, Cond, StartMinusStride, RHS)) {
      // In this case, we can use a refined formula for computing backedge taken
      // count.  The general formula remains:
      //   "End-Start /uceiling Stride" where "End = max(RHS,Start)"
      // We want to use the alternate formula:
      //   "((End - 1) - (Start - Stride)) /u Stride"
      // Let's do a quick case analysis to show these are equivalent under
      // our precondition that max(RHS,Start) > Start - Stride.
      // * For RHS <= Start, the backedge-taken count must be zero.
      //   "((End - 1) - (Start - Stride)) /u Stride" reduces to
      //   "((Start - 1) - (Start - Stride)) /u Stride" which simplies to
      //   "Stride - 1 /u Stride" which is indeed zero for all non-zero values
      //     of Stride.  For 0 stride, we've use umin(1,Stride) above, reducing
      //     this to the stride of 1 case.
      // * For RHS >= Start, the backedge count must be "RHS-Start /uceil Stride".
      //   "((End - 1) - (Start - Stride)) /u Stride" reduces to
      //   "((RHS - 1) - (Start - Stride)) /u Stride" reassociates to
      //   "((RHS - (Start - Stride) - 1) /u Stride".
      //   Our preconditions trivially imply no overflow in that form.
      const SCEV *MinusOne = getMinusOne(Stride->getType());
      const SCEV *Numerator =
          getMinusSCEV(getAddExpr(RHS, MinusOne), StartMinusStride);
      if (!isa<SCEVCouldNotCompute>(Numerator)) {
        BECount = getUDivExpr(Numerator, Stride);
      }
    }
  
    const SCEV *BECountIfBackedgeTaken = nullptr;
    if (!BECount) {
      auto canProveRHSGreaterThanEqualStart = [&]() {
        auto CondGE = IsSigned ? ICmpInst::ICMP_SGE : ICmpInst::ICMP_UGE;
        if (isLoopEntryGuardedByCond(L, CondGE, OrigRHS, OrigStart))
          return true;
  
        // (RHS > Start - 1) implies RHS >= Start.
        // * "RHS >= Start" is trivially equivalent to "RHS > Start - 1" if
        //   "Start - 1" doesn't overflow.
        // * For signed comparison, if Start - 1 does overflow, it's equal
        //   to INT_MAX, and "RHS >s INT_MAX" is trivially false.
        // * For unsigned comparison, if Start - 1 does overflow, it's equal
        //   to UINT_MAX, and "RHS >u UINT_MAX" is trivially false.
        //
        // FIXME: Should isLoopEntryGuardedByCond do this for us?
        auto CondGT = IsSigned ? ICmpInst::ICMP_SGT : ICmpInst::ICMP_UGT;
        auto *StartMinusOne = getAddExpr(OrigStart,
                                         getMinusOne(OrigStart->getType()));
        return isLoopEntryGuardedByCond(L, CondGT, OrigRHS, StartMinusOne);
      };
  
      // If we know that RHS >= Start in the context of loop, then we know that
      // max(RHS, Start) = RHS at this point.
      const SCEV *End;
      if (canProveRHSGreaterThanEqualStart()) {
        End = RHS;
      } else {
        // If RHS < Start, the backedge will be taken zero times.  So in
        // general, we can write the backedge-taken count as:
        //
        //     RHS >= Start ? ceil(RHS - Start) / Stride : 0
        //
        // We convert it to the following to make it more convenient for SCEV:
        //
        //     ceil(max(RHS, Start) - Start) / Stride
        End = IsSigned ? getSMaxExpr(RHS, Start) : getUMaxExpr(RHS, Start);
  
        // See what would happen if we assume the backedge is taken. This is
        // used to compute MaxBECount.
        BECountIfBackedgeTaken = getUDivCeilSCEV(getMinusSCEV(RHS, Start), Stride);
      }
  
      // At this point, we know:
      //
      // 1. If IsSigned, Start <=s End; otherwise, Start <=u End
      // 2. The index variable doesn't overflow.
      //
      // Therefore, we know N exists such that
      // (Start + Stride * N) >= End, and computing "(Start + Stride * N)"
      // doesn't overflow.
      //
      // Using this information, try to prove whether the addition in
      // "(Start - End) + (Stride - 1)" has unsigned overflow.
      const SCEV *One = getOne(Stride->getType());
      bool MayAddOverflow = [&] {
        if (auto *StrideC = dyn_cast<SCEVConstant>(Stride)) {
          if (StrideC->getAPInt().isPowerOf2()) {
            // Suppose Stride is a power of two, and Start/End are unsigned
            // integers.  Let UMAX be the largest representable unsigned
            // integer.
            //
            // By the preconditions of this function, we know
            // "(Start + Stride * N) >= End", and this doesn't overflow.
            // As a formula:
            //
            //   End <= (Start + Stride * N) <= UMAX
            //
            // Subtracting Start from all the terms:
            //
            //   End - Start <= Stride * N <= UMAX - Start
            //
            // Since Start is unsigned, UMAX - Start <= UMAX.  Therefore:
            //
            //   End - Start <= Stride * N <= UMAX
            //
            // Stride * N is a multiple of Stride. Therefore,
            //
            //   End - Start <= Stride * N <= UMAX - (UMAX mod Stride)
            //
            // Since Stride is a power of two, UMAX + 1 is divisible by Stride.
            // Therefore, UMAX mod Stride == Stride - 1.  So we can write:
            //
            //   End - Start <= Stride * N <= UMAX - Stride - 1
            //
            // Dropping the middle term:
            //
            //   End - Start <= UMAX - Stride - 1
            //
            // Adding Stride - 1 to both sides:
            //
            //   (End - Start) + (Stride - 1) <= UMAX
            //
            // In other words, the addition doesn't have unsigned overflow.
            //
            // A similar proof works if we treat Start/End as signed values.
            // Just rewrite steps before "End - Start <= Stride * N <= UMAX" to
            // use signed max instead of unsigned max. Note that we're trying
            // to prove a lack of unsigned overflow in either case.
            return false;
          }
        }
        if (Start == Stride || Start == getMinusSCEV(Stride, One)) {
          // If Start is equal to Stride, (End - Start) + (Stride - 1) == End - 1.
          // If !IsSigned, 0 <u Stride == Start <=u End; so 0 <u End - 1 <u End.
          // If IsSigned, 0 <s Stride == Start <=s End; so 0 <s End - 1 <s End.
          //
          // If Start is equal to Stride - 1, (End - Start) + Stride - 1 == End.
          return false;
        }
        return true;
      }();
  
      const SCEV *Delta = getMinusSCEV(End, Start);
      if (!MayAddOverflow) {
        // floor((D + (S - 1)) / S)
        // We prefer this formulation if it's legal because it's fewer operations.
        BECount =
            getUDivExpr(getAddExpr(Delta, getMinusSCEV(Stride, One)), Stride);
      } else {
        BECount = getUDivCeilSCEV(Delta, Stride);
      }
    }
  
    const SCEV *MaxBECount;
    bool MaxOrZero = false;
    if (isa<SCEVConstant>(BECount)) {
      MaxBECount = BECount;
    } else if (BECountIfBackedgeTaken &&
               isa<SCEVConstant>(BECountIfBackedgeTaken)) {
      // If we know exactly how many times the backedge will be taken if it's
      // taken at least once, then the backedge count will either be that or
      // zero.
      MaxBECount = BECountIfBackedgeTaken;
      MaxOrZero = true;
    } else {
      MaxBECount = computeMaxBECountForLT(
          Start, Stride, RHS, getTypeSizeInBits(LHS->getType()), IsSigned);
    }
  
    if (isa<SCEVCouldNotCompute>(MaxBECount) &&
        !isa<SCEVCouldNotCompute>(BECount))
      MaxBECount = getConstant(getUnsignedRangeMax(BECount));
  
    return ExitLimit(BECount, MaxBECount, MaxOrZero, Predicates);
  }
  
  ScalarEvolution::ExitLimit
  ScalarEvolution::howManyGreaterThans(const SCEV *LHS, const SCEV *RHS,
                                       const Loop *L, bool IsSigned,
                                       bool ControlsExit, bool AllowPredicates) {
    SmallPtrSet<const SCEVPredicate *, 4> Predicates;
    // We handle only IV > Invariant
    if (!isLoopInvariant(RHS, L))
      return getCouldNotCompute();
  
    const SCEVAddRecExpr *IV = dyn_cast<SCEVAddRecExpr>(LHS);
    if (!IV && AllowPredicates)
      // Try to make this an AddRec using runtime tests, in the first X
      // iterations of this loop, where X is the SCEV expression found by the
      // algorithm below.
      IV = convertSCEVToAddRecWithPredicates(LHS, L, Predicates);
  
    // Avoid weird loops
    if (!IV || IV->getLoop() != L || !IV->isAffine())
      return getCouldNotCompute();
  
    auto WrapType = IsSigned ? SCEV::FlagNSW : SCEV::FlagNUW;
    bool NoWrap = ControlsExit && IV->getNoWrapFlags(WrapType);
    ICmpInst::Predicate Cond = IsSigned ? ICmpInst::ICMP_SGT : ICmpInst::ICMP_UGT;
  
    const SCEV *Stride = getNegativeSCEV(IV->getStepRecurrence(*this));
  
    // Avoid negative or zero stride values
    if (!isKnownPositive(Stride))
      return getCouldNotCompute();
  
    // Avoid proven overflow cases: this will ensure that the backedge taken count
    // will not generate any unsigned overflow. Relaxed no-overflow conditions
    // exploit NoWrapFlags, allowing to optimize in presence of undefined
    // behaviors like the case of C language.
    if (!Stride->isOne() && !NoWrap)
      if (canIVOverflowOnGT(RHS, Stride, IsSigned))
        return getCouldNotCompute();
  
    const SCEV *Start = IV->getStart();
    const SCEV *End = RHS;
    if (!isLoopEntryGuardedByCond(L, Cond, getAddExpr(Start, Stride), RHS)) {
      // If we know that Start >= RHS in the context of loop, then we know that
      // min(RHS, Start) = RHS at this point.
      if (isLoopEntryGuardedByCond(
              L, IsSigned ? ICmpInst::ICMP_SGE : ICmpInst::ICMP_UGE, Start, RHS))
        End = RHS;
      else
        End = IsSigned ? getSMinExpr(RHS, Start) : getUMinExpr(RHS, Start);
    }
  
    if (Start->getType()->isPointerTy()) {
      Start = getLosslessPtrToIntExpr(Start);
      if (isa<SCEVCouldNotCompute>(Start))
        return Start;
    }
    if (End->getType()->isPointerTy()) {
      End = getLosslessPtrToIntExpr(End);
      if (isa<SCEVCouldNotCompute>(End))
        return End;
    }
  
    // Compute ((Start - End) + (Stride - 1)) / Stride.
    // FIXME: This can overflow. Holding off on fixing this for now;
    // howManyGreaterThans will hopefully be gone soon.
    const SCEV *One = getOne(Stride->getType());
    const SCEV *BECount = getUDivExpr(
        getAddExpr(getMinusSCEV(Start, End), getMinusSCEV(Stride, One)), Stride);
  
    APInt MaxStart = IsSigned ? getSignedRangeMax(Start)
                              : getUnsignedRangeMax(Start);
  
    APInt MinStride = IsSigned ? getSignedRangeMin(Stride)
                               : getUnsignedRangeMin(Stride);
  
    unsigned BitWidth = getTypeSizeInBits(LHS->getType());
    APInt Limit = IsSigned ? APInt::getSignedMinValue(BitWidth) + (MinStride - 1)
                           : APInt::getMinValue(BitWidth) + (MinStride - 1);
  
    // Although End can be a MIN expression we estimate MinEnd considering only
    // the case End = RHS. This is safe because in the other case (Start - End)
    // is zero, leading to a zero maximum backedge taken count.
    APInt MinEnd =
      IsSigned ? APIntOps::smax(getSignedRangeMin(RHS), Limit)
               : APIntOps::umax(getUnsignedRangeMin(RHS), Limit);
  
    const SCEV *MaxBECount = isa<SCEVConstant>(BECount)
                                 ? BECount
                                 : getUDivCeilSCEV(getConstant(MaxStart - MinEnd),
                                                   getConstant(MinStride));
  
    if (isa<SCEVCouldNotCompute>(MaxBECount))
      MaxBECount = BECount;
  
    return ExitLimit(BECount, MaxBECount, false, Predicates);
  }
  
  const SCEV *SCEVAddRecExpr::getNumIterationsInRange(const ConstantRange &Range,
                                                      ScalarEvolution &SE) const {
    if (Range.isFullSet())  // Infinite loop.
      return SE.getCouldNotCompute();
  
    // If the start is a non-zero constant, shift the range to simplify things.
    if (const SCEVConstant *SC = dyn_cast<SCEVConstant>(getStart()))
      if (!SC->getValue()->isZero()) {
        SmallVector<const SCEV *, 4> Operands(operands());
        Operands[0] = SE.getZero(SC->getType());
        const SCEV *Shifted = SE.getAddRecExpr(Operands, getLoop(),
                                               getNoWrapFlags(FlagNW));
        if (const auto *ShiftedAddRec = dyn_cast<SCEVAddRecExpr>(Shifted))
          return ShiftedAddRec->getNumIterationsInRange(
              Range.subtract(SC->getAPInt()), SE);
        // This is strange and shouldn't happen.
        return SE.getCouldNotCompute();
      }
  
    // The only time we can solve this is when we have all constant indices.
    // Otherwise, we cannot determine the overflow conditions.
    if (any_of(operands(), [](const SCEV *Op) { return !isa<SCEVConstant>(Op); }))
      return SE.getCouldNotCompute();
  
    // Okay at this point we know that all elements of the chrec are constants and
    // that the start element is zero.
  
    // First check to see if the range contains zero.  If not, the first
    // iteration exits.
    unsigned BitWidth = SE.getTypeSizeInBits(getType());
    if (!Range.contains(APInt(BitWidth, 0)))
      return SE.getZero(getType());
  
    if (isAffine()) {
      // If this is an affine expression then we have this situation:
      //   Solve {0,+,A} in Range  ===  Ax in Range
  
      // We know that zero is in the range.  If A is positive then we know that
      // the upper value of the range must be the first possible exit value.
      // If A is negative then the lower of the range is the last possible loop
      // value.  Also note that we already checked for a full range.
      APInt A = cast<SCEVConstant>(getOperand(1))->getAPInt();
      APInt End = A.sge(1) ? (Range.getUpper() - 1) : Range.getLower();
  
      // The exit value should be (End+A)/A.
      APInt ExitVal = (End + A).udiv(A);
      ConstantInt *ExitValue = ConstantInt::get(SE.getContext(), ExitVal);
  
      // Evaluate at the exit value.  If we really did fall out of the valid
      // range, then we computed our trip count, otherwise wrap around or other
      // things must have happened.
      ConstantInt *Val = EvaluateConstantChrecAtConstant(this, ExitValue, SE);
      if (Range.contains(Val->getValue()))
        return SE.getCouldNotCompute();  // Something strange happened
  
      // Ensure that the previous value is in the range.  This is a sanity check.
      assert(Range.contains(
             EvaluateConstantChrecAtConstant(this,
             ConstantInt::get(SE.getContext(), ExitVal - 1), SE)->getValue()) &&
             "Linear scev computation is off in a bad way!");
      return SE.getConstant(ExitValue);
    }
  
    if (isQuadratic()) {
      if (auto S = SolveQuadraticAddRecRange(this, Range, SE))
        return SE.getConstant(S.getValue());
    }
  
    return SE.getCouldNotCompute();
  }
  
  const SCEVAddRecExpr *
  SCEVAddRecExpr::getPostIncExpr(ScalarEvolution &SE) const {
    assert(getNumOperands() > 1 && "AddRec with zero step?");
    // There is a temptation to just call getAddExpr(this, getStepRecurrence(SE)),
    // but in this case we cannot guarantee that the value returned will be an
    // AddRec because SCEV does not have a fixed point where it stops
    // simplification: it is legal to return ({rec1} + {rec2}). For example, it
    // may happen if we reach arithmetic depth limit while simplifying. So we
    // construct the returned value explicitly.
    SmallVector<const SCEV *, 3> Ops;
    // If this is {A,+,B,+,C,...,+,N}, then its step is {B,+,C,+,...,+,N}, and
    // (this + Step) is {A+B,+,B+C,+...,+,N}.
    for (unsigned i = 0, e = getNumOperands() - 1; i < e; ++i)
      Ops.push_back(SE.getAddExpr(getOperand(i), getOperand(i + 1)));
    // We know that the last operand is not a constant zero (otherwise it would
    // have been popped out earlier). This guarantees us that if the result has
    // the same last operand, then it will also not be popped out, meaning that
    // the returned value will be an AddRec.
    const SCEV *Last = getOperand(getNumOperands() - 1);
    assert(!Last->isZero() && "Recurrency with zero step?");
    Ops.push_back(Last);
    return cast<SCEVAddRecExpr>(SE.getAddRecExpr(Ops, getLoop(),
                                                 SCEV::FlagAnyWrap));
  }
  
  // Return true when S contains at least an undef value.
  static inline bool containsUndefs(const SCEV *S) {
    return SCEVExprContains(S, [](const SCEV *S) {
      if (const auto *SU = dyn_cast<SCEVUnknown>(S))
        return isa<UndefValue>(SU->getValue());
      return false;
    });
  }
  
  namespace {
  
  // Collect all steps of SCEV expressions.
  struct SCEVCollectStrides {
    ScalarEvolution &SE;
    SmallVectorImpl<const SCEV *> &Strides;
  
    SCEVCollectStrides(ScalarEvolution &SE, SmallVectorImpl<const SCEV *> &S)
        : SE(SE), Strides(S) {}
  
    bool follow(const SCEV *S) {
      if (const SCEVAddRecExpr *AR = dyn_cast<SCEVAddRecExpr>(S))
        Strides.push_back(AR->getStepRecurrence(SE));
      return true;
    }
  
    bool isDone() const { return false; }
  };
  
  // Collect all SCEVUnknown and SCEVMulExpr expressions.
  struct SCEVCollectTerms {
    SmallVectorImpl<const SCEV *> &Terms;
  
    SCEVCollectTerms(SmallVectorImpl<const SCEV *> &T) : Terms(T) {}
  
    bool follow(const SCEV *S) {
      if (isa<SCEVUnknown>(S) || isa<SCEVMulExpr>(S) ||
          isa<SCEVSignExtendExpr>(S)) {
        if (!containsUndefs(S))
          Terms.push_back(S);
  
        // Stop recursion: once we collected a term, do not walk its operands.
        return false;
      }
  
      // Keep looking.
      return true;
    }
  
    bool isDone() const { return false; }
  };
  
  // Check if a SCEV contains an AddRecExpr.
  struct SCEVHasAddRec {
    bool &ContainsAddRec;
  
    SCEVHasAddRec(bool &ContainsAddRec) : ContainsAddRec(ContainsAddRec) {
      ContainsAddRec = false;
    }
  
    bool follow(const SCEV *S) {
      if (isa<SCEVAddRecExpr>(S)) {
        ContainsAddRec = true;
  
        // Stop recursion: once we collected a term, do not walk its operands.
        return false;
      }
  
      // Keep looking.
      return true;
    }
  
    bool isDone() const { return false; }
  };
  
  // Find factors that are multiplied with an expression that (possibly as a
  // subexpression) contains an AddRecExpr. In the expression:
  //
  //  8 * (100 +  %p * %q * (%a + {0, +, 1}_loop))
  //
  // "%p * %q" are factors multiplied by the expression "(%a + {0, +, 1}_loop)"
  // that contains the AddRec {0, +, 1}_loop. %p * %q are likely to be array size
  // parameters as they form a product with an induction variable.
  //
  // This collector expects all array size parameters to be in the same MulExpr.
  // It might be necessary to later add support for collecting parameters that are
  // spread over different nested MulExpr.
  struct SCEVCollectAddRecMultiplies {
    SmallVectorImpl<const SCEV *> &Terms;
    ScalarEvolution &SE;
  
    SCEVCollectAddRecMultiplies(SmallVectorImpl<const SCEV *> &T, ScalarEvolution &SE)
        : Terms(T), SE(SE) {}
  
    bool follow(const SCEV *S) {
      if (auto *Mul = dyn_cast<SCEVMulExpr>(S)) {
        bool HasAddRec = false;
        SmallVector<const SCEV *, 0> Operands;
        for (auto Op : Mul->operands()) {
          const SCEVUnknown *Unknown = dyn_cast<SCEVUnknown>(Op);
          if (Unknown && !isa<CallInst>(Unknown->getValue())) {
            Operands.push_back(Op);
          } else if (Unknown) {
            HasAddRec = true;
          } else {
            bool ContainsAddRec = false;
            SCEVHasAddRec ContiansAddRec(ContainsAddRec);
            visitAll(Op, ContiansAddRec);
            HasAddRec |= ContainsAddRec;
          }
        }
        if (Operands.size() == 0)
          return true;
  
        if (!HasAddRec)
          return false;
  
        Terms.push_back(SE.getMulExpr(Operands));
        // Stop recursion: once we collected a term, do not walk its operands.
        return false;
      }
  
      // Keep looking.
      return true;
    }
  
    bool isDone() const { return false; }
  };
  
  } // end anonymous namespace
  
  /// Find parametric terms in this SCEVAddRecExpr. We first for parameters in
  /// two places:
  ///   1) The strides of AddRec expressions.
  ///   2) Unknowns that are multiplied with AddRec expressions.
  void ScalarEvolution::collectParametricTerms(const SCEV *Expr,
      SmallVectorImpl<const SCEV *> &Terms) {
    SmallVector<const SCEV *, 4> Strides;
    SCEVCollectStrides StrideCollector(*this, Strides);
    visitAll(Expr, StrideCollector);
  
    LLVM_DEBUG({
      dbgs() << "Strides:\n";
      for (const SCEV *S : Strides)
        dbgs() << *S << "\n";
    });
  
    for (const SCEV *S : Strides) {
      SCEVCollectTerms TermCollector(Terms);
      visitAll(S, TermCollector);
    }
  
    LLVM_DEBUG({
      dbgs() << "Terms:\n";
      for (const SCEV *T : Terms)
        dbgs() << *T << "\n";
    });
  
    SCEVCollectAddRecMultiplies MulCollector(Terms, *this);
    visitAll(Expr, MulCollector);
  }
  
  static bool findArrayDimensionsRec(ScalarEvolution &SE,
                                     SmallVectorImpl<const SCEV *> &Terms,
                                     SmallVectorImpl<const SCEV *> &Sizes) {
    int Last = Terms.size() - 1;
    const SCEV *Step = Terms[Last];
  
    // End of recursion.
    if (Last == 0) {
      if (const SCEVMulExpr *M = dyn_cast<SCEVMulExpr>(Step)) {
        SmallVector<const SCEV *, 2> Qs;
        for (const SCEV *Op : M->operands())
          if (!isa<SCEVConstant>(Op))
            Qs.push_back(Op);
  
        Step = SE.getMulExpr(Qs);
      }
  
      Sizes.push_back(Step);
      return true;
    }
  
    for (const SCEV *&Term : Terms) {
      // Normalize the terms before the next call to findArrayDimensionsRec.
      const SCEV *Q, *R;
      SCEVDivision::divide(SE, Term, Step, &Q, &R);
  
      // Bail out when GCD does not evenly divide one of the terms.
      if (!R->isZero())
        return false;
  
      Term = Q;
    }
  
    // Remove all SCEVConstants.
    erase_if(Terms, [](const SCEV *E) { return isa<SCEVConstant>(E); });
  
    if (Terms.size() > 0)
      if (!findArrayDimensionsRec(SE, Terms, Sizes))
        return false;
  
    Sizes.push_back(Step);
    return true;
  }
  
  // Returns true when one of the SCEVs of Terms contains a SCEVUnknown parameter.
  static inline bool containsParameters(SmallVectorImpl<const SCEV *> &Terms) {
    for (const SCEV *T : Terms)
      if (SCEVExprContains(T, [](const SCEV *S) { return isa<SCEVUnknown>(S); }))
        return true;
  
    return false;
  }
  
  // Return the number of product terms in S.
  static inline int numberOfTerms(const SCEV *S) {
    if (const SCEVMulExpr *Expr = dyn_cast<SCEVMulExpr>(S))
      return Expr->getNumOperands();
    return 1;
  }
  
  static const SCEV *removeConstantFactors(ScalarEvolution &SE, const SCEV *T) {
    if (isa<SCEVConstant>(T))
      return nullptr;
  
    if (isa<SCEVUnknown>(T))
      return T;
  
    if (const SCEVMulExpr *M = dyn_cast<SCEVMulExpr>(T)) {
      SmallVector<const SCEV *, 2> Factors;
      for (const SCEV *Op : M->operands())
        if (!isa<SCEVConstant>(Op))
          Factors.push_back(Op);
  
      return SE.getMulExpr(Factors);
    }
  
    return T;
  }
  
  /// Return the size of an element read or written by Inst.
  const SCEV *ScalarEvolution::getElementSize(Instruction *Inst) {
    Type *Ty;
    if (StoreInst *Store = dyn_cast<StoreInst>(Inst))
      Ty = Store->getValueOperand()->getType();
    else if (LoadInst *Load = dyn_cast<LoadInst>(Inst))
      Ty = Load->getType();
    else
      return nullptr;
  
    Type *ETy = getEffectiveSCEVType(PointerType::getUnqual(Ty));
    return getSizeOfExpr(ETy, Ty);
  }
  
  void ScalarEvolution::findArrayDimensions(SmallVectorImpl<const SCEV *> &Terms,
                                            SmallVectorImpl<const SCEV *> &Sizes,
                                            const SCEV *ElementSize) {
    if (Terms.size() < 1 || !ElementSize)
      return;
  
    // Early return when Terms do not contain parameters: we do not delinearize
    // non parametric SCEVs.
    if (!containsParameters(Terms))
      return;
  
    LLVM_DEBUG({
      dbgs() << "Terms:\n";
      for (const SCEV *T : Terms)
        dbgs() << *T << "\n";
    });
  
    // Remove duplicates.
    array_pod_sort(Terms.begin(), Terms.end());
    Terms.erase(std::unique(Terms.begin(), Terms.end()), Terms.end());
  
    // Put larger terms first.
    llvm::sort(Terms, [](const SCEV *LHS, const SCEV *RHS) {
      return numberOfTerms(LHS) > numberOfTerms(RHS);
    });
  
    // Try to divide all terms by the element size. If term is not divisible by
    // element size, proceed with the original term.
    for (const SCEV *&Term : Terms) {
      const SCEV *Q, *R;
      SCEVDivision::divide(*this, Term, ElementSize, &Q, &R);
      if (!Q->isZero())
        Term = Q;
    }
  
    SmallVector<const SCEV *, 4> NewTerms;
  
    // Remove constant factors.
    for (const SCEV *T : Terms)
      if (const SCEV *NewT = removeConstantFactors(*this, T))
        NewTerms.push_back(NewT);
  
    LLVM_DEBUG({
      dbgs() << "Terms after sorting:\n";
      for (const SCEV *T : NewTerms)
        dbgs() << *T << "\n";
    });
  
    if (NewTerms.empty() || !findArrayDimensionsRec(*this, NewTerms, Sizes)) {
      Sizes.clear();
      return;
    }
  
    // The last element to be pushed into Sizes is the size of an element.
    Sizes.push_back(ElementSize);
  
    LLVM_DEBUG({
      dbgs() << "Sizes:\n";
      for (const SCEV *S : Sizes)
        dbgs() << *S << "\n";
    });
  }
  
  void ScalarEvolution::computeAccessFunctions(
      const SCEV *Expr, SmallVectorImpl<const SCEV *> &Subscripts,
      SmallVectorImpl<const SCEV *> &Sizes) {
    // Early exit in case this SCEV is not an affine multivariate function.
    if (Sizes.empty())
      return;
  
    if (auto *AR = dyn_cast<SCEVAddRecExpr>(Expr))
      if (!AR->isAffine())
        return;
  
    const SCEV *Res = Expr;
    int Last = Sizes.size() - 1;
    for (int i = Last; i >= 0; i--) {
      const SCEV *Q, *R;
      SCEVDivision::divide(*this, Res, Sizes[i], &Q, &R);
  
      LLVM_DEBUG({
        dbgs() << "Res: " << *Res << "\n";
        dbgs() << "Sizes[i]: " << *Sizes[i] << "\n";
        dbgs() << "Res divided by Sizes[i]:\n";
        dbgs() << "Quotient: " << *Q << "\n";
        dbgs() << "Remainder: " << *R << "\n";
      });
  
      Res = Q;
  
      // Do not record the last subscript corresponding to the size of elements in
      // the array.
      if (i == Last) {
  
        // Bail out if the remainder is too complex.
        if (isa<SCEVAddRecExpr>(R)) {
          Subscripts.clear();
          Sizes.clear();
          return;
        }
  
        continue;
      }
  
      // Record the access function for the current subscript.
      Subscripts.push_back(R);
    }
  
    // Also push in last position the remainder of the last division: it will be
    // the access function of the innermost dimension.
    Subscripts.push_back(Res);
  
    std::reverse(Subscripts.begin(), Subscripts.end());
  
    LLVM_DEBUG({
      dbgs() << "Subscripts:\n";
      for (const SCEV *S : Subscripts)
        dbgs() << *S << "\n";
    });
  }
  
  /// Splits the SCEV into two vectors of SCEVs representing the subscripts and
  /// sizes of an array access. Returns the remainder of the delinearization that
  /// is the offset start of the array.  The SCEV->delinearize algorithm computes
  /// the multiples of SCEV coefficients: that is a pattern matching of sub
  /// expressions in the stride and base of a SCEV corresponding to the
  /// computation of a GCD (greatest common divisor) of base and stride.  When
  /// SCEV->delinearize fails, it returns the SCEV unchanged.
  ///
  /// For example: when analyzing the memory access A[i][j][k] in this loop nest
  ///
  ///  void foo(long n, long m, long o, double A[n][m][o]) {
  ///
  ///    for (long i = 0; i < n; i++)
  ///      for (long j = 0; j < m; j++)
  ///        for (long k = 0; k < o; k++)
  ///          A[i][j][k] = 1.0;
  ///  }
  ///
  /// the delinearization input is the following AddRec SCEV:
  ///
  ///  AddRec: {{{%A,+,(8 * %m * %o)}<%for.i>,+,(8 * %o)}<%for.j>,+,8}<%for.k>
  ///
  /// From this SCEV, we are able to say that the base offset of the access is %A
  /// because it appears as an offset that does not divide any of the strides in
  /// the loops:
  ///
  ///  CHECK: Base offset: %A
  ///
  /// and then SCEV->delinearize determines the size of some of the dimensions of
  /// the array as these are the multiples by which the strides are happening:
  ///
  ///  CHECK: ArrayDecl[UnknownSize][%m][%o] with elements of sizeof(double) bytes.
  ///
  /// Note that the outermost dimension remains of UnknownSize because there are
  /// no strides that would help identifying the size of the last dimension: when
  /// the array has been statically allocated, one could compute the size of that
  /// dimension by dividing the overall size of the array by the size of the known
  /// dimensions: %m * %o * 8.
  ///
  /// Finally delinearize provides the access functions for the array reference
  /// that does correspond to A[i][j][k] of the above C testcase:
  ///
  ///  CHECK: ArrayRef[{0,+,1}<%for.i>][{0,+,1}<%for.j>][{0,+,1}<%for.k>]
  ///
  /// The testcases are checking the output of a function pass:
  /// DelinearizationPass that walks through all loads and stores of a function
  /// asking for the SCEV of the memory access with respect to all enclosing
  /// loops, calling SCEV->delinearize on that and printing the results.
  void ScalarEvolution::delinearize(const SCEV *Expr,
                                   SmallVectorImpl<const SCEV *> &Subscripts,
                                   SmallVectorImpl<const SCEV *> &Sizes,
                                   const SCEV *ElementSize) {
    // First step: collect parametric terms.
    SmallVector<const SCEV *, 4> Terms;
    collectParametricTerms(Expr, Terms);
  
    if (Terms.empty())
      return;
  
    // Second step: find subscript sizes.
    findArrayDimensions(Terms, Sizes, ElementSize);
  
    if (Sizes.empty())
      return;
  
    // Third step: compute the access functions for each subscript.
    computeAccessFunctions(Expr, Subscripts, Sizes);
  
    if (Subscripts.empty())
      return;
  
    LLVM_DEBUG({
      dbgs() << "succeeded to delinearize " << *Expr << "\n";
      dbgs() << "ArrayDecl[UnknownSize]";
      for (const SCEV *S : Sizes)
        dbgs() << "[" << *S << "]";
  
      dbgs() << "\nArrayRef";
      for (const SCEV *S : Subscripts)
        dbgs() << "[" << *S << "]";
      dbgs() << "\n";
    });
  }
  
  bool ScalarEvolution::getIndexExpressionsFromGEP(
      const GetElementPtrInst *GEP, SmallVectorImpl<const SCEV *> &Subscripts,
      SmallVectorImpl<int> &Sizes) {
    assert(Subscripts.empty() && Sizes.empty() &&
           "Expected output lists to be empty on entry to this function.");
    assert(GEP && "getIndexExpressionsFromGEP called with a null GEP");
    Type *Ty = nullptr;
    bool DroppedFirstDim = false;
    for (unsigned i = 1; i < GEP->getNumOperands(); i++) {
      const SCEV *Expr = getSCEV(GEP->getOperand(i));
      if (i == 1) {
        Ty = GEP->getSourceElementType();
        if (auto *Const = dyn_cast<SCEVConstant>(Expr))
          if (Const->getValue()->isZero()) {
            DroppedFirstDim = true;
            continue;
          }
        Subscripts.push_back(Expr);
        continue;
      }
  
      auto *ArrayTy = dyn_cast<ArrayType>(Ty);
      if (!ArrayTy) {
        Subscripts.clear();
        Sizes.clear();
        return false;
      }
  
      Subscripts.push_back(Expr);
      if (!(DroppedFirstDim && i == 2))
        Sizes.push_back(ArrayTy->getNumElements());
  
      Ty = ArrayTy->getElementType();
    }
    return !Subscripts.empty();
  }
  
  //===----------------------------------------------------------------------===//
  //                   SCEVCallbackVH Class Implementation
  //===----------------------------------------------------------------------===//
  
  void ScalarEvolution::SCEVCallbackVH::deleted() {
    assert(SE && "SCEVCallbackVH called with a null ScalarEvolution!");
    if (PHINode *PN = dyn_cast<PHINode>(getValPtr()))
      SE->ConstantEvolutionLoopExitValue.erase(PN);
    SE->eraseValueFromMap(getValPtr());
    // this now dangles!
  }
  
  void ScalarEvolution::SCEVCallbackVH::allUsesReplacedWith(Value *V) {
    assert(SE && "SCEVCallbackVH called with a null ScalarEvolution!");
  
    // Forget all the expressions associated with users of the old value,
    // so that future queries will recompute the expressions using the new
    // value.
    Value *Old = getValPtr();
    SmallVector<User *, 16> Worklist(Old->users());
    SmallPtrSet<User *, 8> Visited;
    while (!Worklist.empty()) {
      User *U = Worklist.pop_back_val();
      // Deleting the Old value will cause this to dangle. Postpone
      // that until everything else is done.
      if (U == Old)
        continue;
      if (!Visited.insert(U).second)
        continue;
      if (PHINode *PN = dyn_cast<PHINode>(U))
        SE->ConstantEvolutionLoopExitValue.erase(PN);
      SE->eraseValueFromMap(U);
      llvm::append_range(Worklist, U->users());
    }
    // Delete the Old value.
    if (PHINode *PN = dyn_cast<PHINode>(Old))
      SE->ConstantEvolutionLoopExitValue.erase(PN);
    SE->eraseValueFromMap(Old);
    // this now dangles!
  }
  
  ScalarEvolution::SCEVCallbackVH::SCEVCallbackVH(Value *V, ScalarEvolution *se)
    : CallbackVH(V), SE(se) {}
  
  //===----------------------------------------------------------------------===//
  //                   ScalarEvolution Class Implementation
  //===----------------------------------------------------------------------===//
  
  ScalarEvolution::ScalarEvolution(Function &F, TargetLibraryInfo &TLI,
                                   AssumptionCache &AC, DominatorTree &DT,
                                   LoopInfo &LI)
      : F(F), TLI(TLI), AC(AC), DT(DT), LI(LI),
        CouldNotCompute(new SCEVCouldNotCompute()), ValuesAtScopes(64),
        LoopDispositions(64), BlockDispositions(64) {
    // To use guards for proving predicates, we need to scan every instruction in
    // relevant basic blocks, and not just terminators.  Doing this is a waste of
    // time if the IR does not actually contain any calls to
    // @llvm.experimental.guard, so do a quick check and remember this beforehand.
    //
    // This pessimizes the case where a pass that preserves ScalarEvolution wants
    // to _add_ guards to the module when there weren't any before, and wants
    // ScalarEvolution to optimize based on those guards.  For now we prefer to be
    // efficient in lieu of being smart in that rather obscure case.
  
    auto *GuardDecl = F.getParent()->getFunction(
        Intrinsic::getName(Intrinsic::experimental_guard));
    HasGuards = GuardDecl && !GuardDecl->use_empty();
  }
  
  ScalarEvolution::ScalarEvolution(ScalarEvolution &&Arg)
      : F(Arg.F), HasGuards(Arg.HasGuards), TLI(Arg.TLI), AC(Arg.AC), DT(Arg.DT),
        LI(Arg.LI), CouldNotCompute(std::move(Arg.CouldNotCompute)),
        ValueExprMap(std::move(Arg.ValueExprMap)),
        PendingLoopPredicates(std::move(Arg.PendingLoopPredicates)),
        PendingPhiRanges(std::move(Arg.PendingPhiRanges)),
        PendingMerges(std::move(Arg.PendingMerges)),
        MinTrailingZerosCache(std::move(Arg.MinTrailingZerosCache)),
        BackedgeTakenCounts(std::move(Arg.BackedgeTakenCounts)),
        PredicatedBackedgeTakenCounts(
            std::move(Arg.PredicatedBackedgeTakenCounts)),
        ConstantEvolutionLoopExitValue(
            std::move(Arg.ConstantEvolutionLoopExitValue)),
        ValuesAtScopes(std::move(Arg.ValuesAtScopes)),
        LoopDispositions(std::move(Arg.LoopDispositions)),
        LoopPropertiesCache(std::move(Arg.LoopPropertiesCache)),
        BlockDispositions(std::move(Arg.BlockDispositions)),
        UnsignedRanges(std::move(Arg.UnsignedRanges)),
        SignedRanges(std::move(Arg.SignedRanges)),
        UniqueSCEVs(std::move(Arg.UniqueSCEVs)),
        UniquePreds(std::move(Arg.UniquePreds)),
        SCEVAllocator(std::move(Arg.SCEVAllocator)),
        LoopUsers(std::move(Arg.LoopUsers)),
        PredicatedSCEVRewrites(std::move(Arg.PredicatedSCEVRewrites)),
        FirstUnknown(Arg.FirstUnknown) {
    Arg.FirstUnknown = nullptr;
  }
  
  ScalarEvolution::~ScalarEvolution() {
    // Iterate through all the SCEVUnknown instances and call their
    // destructors, so that they release their references to their values.
    for (SCEVUnknown *U = FirstUnknown; U;) {
      SCEVUnknown *Tmp = U;
      U = U->Next;
      Tmp->~SCEVUnknown();
    }
    FirstUnknown = nullptr;
  
    ExprValueMap.clear();
    ValueExprMap.clear();
    HasRecMap.clear();
    BackedgeTakenCounts.clear();
    PredicatedBackedgeTakenCounts.clear();
  
    assert(PendingLoopPredicates.empty() && "isImpliedCond garbage");
    assert(PendingPhiRanges.empty() && "getRangeRef garbage");
    assert(PendingMerges.empty() && "isImpliedViaMerge garbage");
    assert(!WalkingBEDominatingConds && "isLoopBackedgeGuardedByCond garbage!");
    assert(!ProvingSplitPredicate && "ProvingSplitPredicate garbage!");
  }
  
  bool ScalarEvolution::hasLoopInvariantBackedgeTakenCount(const Loop *L) {
    return !isa<SCEVCouldNotCompute>(getBackedgeTakenCount(L));
  }
  
  static void PrintLoopInfo(raw_ostream &OS, ScalarEvolution *SE,
                            const Loop *L) {
    // Print all inner loops first
    for (Loop *I : *L)
      PrintLoopInfo(OS, SE, I);
  
    OS << "Loop ";
    L->getHeader()->printAsOperand(OS, /*PrintType=*/false);
    OS << ": ";
  
    SmallVector<BasicBlock *, 8> ExitingBlocks;
    L->getExitingBlocks(ExitingBlocks);
    if (ExitingBlocks.size() != 1)
      OS << "<multiple exits> ";
  
    if (SE->hasLoopInvariantBackedgeTakenCount(L))
      OS << "backedge-taken count is " << *SE->getBackedgeTakenCount(L) << "\n";
    else
      OS << "Unpredictable backedge-taken count.\n";
  
    if (ExitingBlocks.size() > 1)
      for (BasicBlock *ExitingBlock : ExitingBlocks) {
        OS << "  exit count for " << ExitingBlock->getName() << ": "
           << *SE->getExitCount(L, ExitingBlock) << "\n";
      }
  
    OS << "Loop ";
    L->getHeader()->printAsOperand(OS, /*PrintType=*/false);
    OS << ": ";
  
    if (!isa<SCEVCouldNotCompute>(SE->getConstantMaxBackedgeTakenCount(L))) {
      OS << "max backedge-taken count is " << *SE->getConstantMaxBackedgeTakenCount(L);
      if (SE->isBackedgeTakenCountMaxOrZero(L))
        OS << ", actual taken count either this or zero.";
    } else {
      OS << "Unpredictable max backedge-taken count. ";
    }
  
    OS << "\n"
          "Loop ";
    L->getHeader()->printAsOperand(OS, /*PrintType=*/false);
    OS << ": ";
  
    SCEVUnionPredicate Pred;
    auto PBT = SE->getPredicatedBackedgeTakenCount(L, Pred);
    if (!isa<SCEVCouldNotCompute>(PBT)) {
      OS << "Predicated backedge-taken count is " << *PBT << "\n";
      OS << " Predicates:\n";
      Pred.print(OS, 4);
    } else {
      OS << "Unpredictable predicated backedge-taken count. ";
    }
    OS << "\n";
  
    if (SE->hasLoopInvariantBackedgeTakenCount(L)) {
      OS << "Loop ";
      L->getHeader()->printAsOperand(OS, /*PrintType=*/false);
      OS << ": ";
      OS << "Trip multiple is " << SE->getSmallConstantTripMultiple(L) << "\n";
    }
  }
  
  static StringRef loopDispositionToStr(ScalarEvolution::LoopDisposition LD) {
    switch (LD) {
    case ScalarEvolution::LoopVariant:
      return "Variant";
    case ScalarEvolution::LoopInvariant:
      return "Invariant";
    case ScalarEvolution::LoopComputable:
      return "Computable";
    }
    llvm_unreachable("Unknown ScalarEvolution::LoopDisposition kind!");
  }
  
  void ScalarEvolution::print(raw_ostream &OS) const {
    // ScalarEvolution's implementation of the print method is to print
    // out SCEV values of all instructions that are interesting. Doing
    // this potentially causes it to create new SCEV objects though,
    // which technically conflicts with the const qualifier. This isn't
    // observable from outside the class though, so casting away the
    // const isn't dangerous.
    ScalarEvolution &SE = *const_cast<ScalarEvolution *>(this);
  
    if (ClassifyExpressions) {
      OS << "Classifying expressions for: ";
      F.printAsOperand(OS, /*PrintType=*/false);
      OS << "\n";
      for (Instruction &I : instructions(F))
        if (isSCEVable(I.getType()) && !isa<CmpInst>(I)) {
          OS << I << '\n';
          OS << "  -->  ";
          const SCEV *SV = SE.getSCEV(&I);
          SV->print(OS);
          if (!isa<SCEVCouldNotCompute>(SV)) {
            OS << " U: ";
            SE.getUnsignedRange(SV).print(OS);
            OS << " S: ";
            SE.getSignedRange(SV).print(OS);
          }
  
          const Loop *L = LI.getLoopFor(I.getParent());
  
          const SCEV *AtUse = SE.getSCEVAtScope(SV, L);
          if (AtUse != SV) {
            OS << "  -->  ";
            AtUse->print(OS);
            if (!isa<SCEVCouldNotCompute>(AtUse)) {
              OS << " U: ";
              SE.getUnsignedRange(AtUse).print(OS);
              OS << " S: ";
              SE.getSignedRange(AtUse).print(OS);
            }
          }
  
          if (L) {
            OS << "\t\t" "Exits: ";
            const SCEV *ExitValue = SE.getSCEVAtScope(SV, L->getParentLoop());
            if (!SE.isLoopInvariant(ExitValue, L)) {
              OS << "<<Unknown>>";
            } else {
              OS << *ExitValue;
            }
  
            bool First = true;
            for (auto *Iter = L; Iter; Iter = Iter->getParentLoop()) {
              if (First) {
                OS << "\t\t" "LoopDispositions: { ";
                First = false;
              } else {
                OS << ", ";
              }
  
              Iter->getHeader()->printAsOperand(OS, /*PrintType=*/false);
              OS << ": " << loopDispositionToStr(SE.getLoopDisposition(SV, Iter));
            }
  
            for (auto *InnerL : depth_first(L)) {
              if (InnerL == L)
                continue;
              if (First) {
                OS << "\t\t" "LoopDispositions: { ";
                First = false;
              } else {
                OS << ", ";
              }
  
              InnerL->getHeader()->printAsOperand(OS, /*PrintType=*/false);
              OS << ": " << loopDispositionToStr(SE.getLoopDisposition(SV, InnerL));
            }
  
            OS << " }";
          }
  
          OS << "\n";
        }
    }
  
    OS << "Determining loop execution counts for: ";
    F.printAsOperand(OS, /*PrintType=*/false);
    OS << "\n";
    for (Loop *I : LI)
      PrintLoopInfo(OS, &SE, I);
  }
  
  ScalarEvolution::LoopDisposition
  ScalarEvolution::getLoopDisposition(const SCEV *S, const Loop *L) {
    auto &Values = LoopDispositions[S];
    for (auto &V : Values) {
      if (V.getPointer() == L)
        return V.getInt();
    }
    Values.emplace_back(L, LoopVariant);
    LoopDisposition D = computeLoopDisposition(S, L);
    auto &Values2 = LoopDispositions[S];
    for (auto &V : make_range(Values2.rbegin(), Values2.rend())) {
      if (V.getPointer() == L) {
        V.setInt(D);
        break;
      }
    }
    return D;
  }
  
  ScalarEvolution::LoopDisposition
  ScalarEvolution::computeLoopDisposition(const SCEV *S, const Loop *L) {
    switch (S->getSCEVType()) {
    case scConstant:
      return LoopInvariant;
    case scPtrToInt:
    case scTruncate:
    case scZeroExtend:
    case scSignExtend:
      return getLoopDisposition(cast<SCEVCastExpr>(S)->getOperand(), L);
    case scAddRecExpr: {
      const SCEVAddRecExpr *AR = cast<SCEVAddRecExpr>(S);
  
      // If L is the addrec's loop, it's computable.
      if (AR->getLoop() == L)
        return LoopComputable;
  
      // Add recurrences are never invariant in the function-body (null loop).
      if (!L)
        return LoopVariant;
  
      // Everything that is not defined at loop entry is variant.
      if (DT.dominates(L->getHeader(), AR->getLoop()->getHeader()))
        return LoopVariant;
      assert(!L->contains(AR->getLoop()) && "Containing loop's header does not"
             " dominate the contained loop's header?");
  
      // This recurrence is invariant w.r.t. L if AR's loop contains L.
      if (AR->getLoop()->contains(L))
        return LoopInvariant;
  
      // This recurrence is variant w.r.t. L if any of its operands
      // are variant.
      for (auto *Op : AR->operands())
        if (!isLoopInvariant(Op, L))
          return LoopVariant;
  
      // Otherwise it's loop-invariant.
      return LoopInvariant;
    }
    case scAddExpr:
    case scMulExpr:
    case scUMaxExpr:
    case scSMaxExpr:
    case scUMinExpr:
    case scSMinExpr: {
      bool HasVarying = false;
      for (auto *Op : cast<SCEVNAryExpr>(S)->operands()) {
        LoopDisposition D = getLoopDisposition(Op, L);
        if (D == LoopVariant)
          return LoopVariant;
        if (D == LoopComputable)
          HasVarying = true;
      }
      return HasVarying ? LoopComputable : LoopInvariant;
    }
    case scUDivExpr: {
      const SCEVUDivExpr *UDiv = cast<SCEVUDivExpr>(S);
      LoopDisposition LD = getLoopDisposition(UDiv->getLHS(), L);
      if (LD == LoopVariant)
        return LoopVariant;
      LoopDisposition RD = getLoopDisposition(UDiv->getRHS(), L);
      if (RD == LoopVariant)
        return LoopVariant;
      return (LD == LoopInvariant && RD == LoopInvariant) ?
             LoopInvariant : LoopComputable;
    }
    case scUnknown:
      // All non-instruction values are loop invariant.  All instructions are loop
      // invariant if they are not contained in the specified loop.
      // Instructions are never considered invariant in the function body
      // (null loop) because they are defined within the "loop".
      if (auto *I = dyn_cast<Instruction>(cast<SCEVUnknown>(S)->getValue()))
        return (L && !L->contains(I)) ? LoopInvariant : LoopVariant;
      return LoopInvariant;
    case scCouldNotCompute:
      llvm_unreachable("Attempt to use a SCEVCouldNotCompute object!");
    }
    llvm_unreachable("Unknown SCEV kind!");
  }
  
  bool ScalarEvolution::isLoopInvariant(const SCEV *S, const Loop *L) {
    return getLoopDisposition(S, L) == LoopInvariant;
  }
  
  bool ScalarEvolution::hasComputableLoopEvolution(const SCEV *S, const Loop *L) {
    return getLoopDisposition(S, L) == LoopComputable;
  }
  
  ScalarEvolution::BlockDisposition
  ScalarEvolution::getBlockDisposition(const SCEV *S, const BasicBlock *BB) {
    auto &Values = BlockDispositions[S];
    for (auto &V : Values) {
      if (V.getPointer() == BB)
        return V.getInt();
    }
    Values.emplace_back(BB, DoesNotDominateBlock);
    BlockDisposition D = computeBlockDisposition(S, BB);
    auto &Values2 = BlockDispositions[S];
    for (auto &V : make_range(Values2.rbegin(), Values2.rend())) {
      if (V.getPointer() == BB) {
        V.setInt(D);
        break;
      }
    }
    return D;
  }
  
  ScalarEvolution::BlockDisposition
  ScalarEvolution::computeBlockDisposition(const SCEV *S, const BasicBlock *BB) {
    switch (S->getSCEVType()) {
    case scConstant:
      return ProperlyDominatesBlock;
    case scPtrToInt:
    case scTruncate:
    case scZeroExtend:
    case scSignExtend:
      return getBlockDisposition(cast<SCEVCastExpr>(S)->getOperand(), BB);
    case scAddRecExpr: {
      // This uses a "dominates" query instead of "properly dominates" query
      // to test for proper dominance too, because the instruction which
      // produces the addrec's value is a PHI, and a PHI effectively properly
      // dominates its entire containing block.
      const SCEVAddRecExpr *AR = cast<SCEVAddRecExpr>(S);
      if (!DT.dominates(AR->getLoop()->getHeader(), BB))
        return DoesNotDominateBlock;
  
      // Fall through into SCEVNAryExpr handling.
      LLVM_FALLTHROUGH;
    }
    case scAddExpr:
    case scMulExpr:
    case scUMaxExpr:
    case scSMaxExpr:
    case scUMinExpr:
    case scSMinExpr: {
      const SCEVNAryExpr *NAry = cast<SCEVNAryExpr>(S);
      bool Proper = true;
      for (const SCEV *NAryOp : NAry->operands()) {
        BlockDisposition D = getBlockDisposition(NAryOp, BB);
        if (D == DoesNotDominateBlock)
          return DoesNotDominateBlock;
        if (D == DominatesBlock)
          Proper = false;
      }
      return Proper ? ProperlyDominatesBlock : DominatesBlock;
    }
    case scUDivExpr: {
      const SCEVUDivExpr *UDiv = cast<SCEVUDivExpr>(S);
      const SCEV *LHS = UDiv->getLHS(), *RHS = UDiv->getRHS();
      BlockDisposition LD = getBlockDisposition(LHS, BB);
      if (LD == DoesNotDominateBlock)
        return DoesNotDominateBlock;
      BlockDisposition RD = getBlockDisposition(RHS, BB);
      if (RD == DoesNotDominateBlock)
        return DoesNotDominateBlock;
      return (LD == ProperlyDominatesBlock && RD == ProperlyDominatesBlock) ?
        ProperlyDominatesBlock : DominatesBlock;
    }
    case scUnknown:
      if (Instruction *I =
            dyn_cast<Instruction>(cast<SCEVUnknown>(S)->getValue())) {
        if (I->getParent() == BB)
          return DominatesBlock;
        if (DT.properlyDominates(I->getParent(), BB))
          return ProperlyDominatesBlock;
        return DoesNotDominateBlock;
      }
      return ProperlyDominatesBlock;
    case scCouldNotCompute:
      llvm_unreachable("Attempt to use a SCEVCouldNotCompute object!");
    }
    llvm_unreachable("Unknown SCEV kind!");
  }
  
  bool ScalarEvolution::dominates(const SCEV *S, const BasicBlock *BB) {
    return getBlockDisposition(S, BB) >= DominatesBlock;
  }
  
  bool ScalarEvolution::properlyDominates(const SCEV *S, const BasicBlock *BB) {
    return getBlockDisposition(S, BB) == ProperlyDominatesBlock;
  }
  
  bool ScalarEvolution::hasOperand(const SCEV *S, const SCEV *Op) const {
    return SCEVExprContains(S, [&](const SCEV *Expr) { return Expr == Op; });
  }
  
  void
  ScalarEvolution::forgetMemoizedResults(const SCEV *S) {
    ValuesAtScopes.erase(S);
    LoopDispositions.erase(S);
    BlockDispositions.erase(S);
    UnsignedRanges.erase(S);
    SignedRanges.erase(S);
    ExprValueMap.erase(S);
    HasRecMap.erase(S);
    MinTrailingZerosCache.erase(S);
  
    for (auto I = PredicatedSCEVRewrites.begin();
         I != PredicatedSCEVRewrites.end();) {
      std::pair<const SCEV *, const Loop *> Entry = I->first;
      if (Entry.first == S)
        PredicatedSCEVRewrites.erase(I++);
      else
        ++I;
    }
  
    auto RemoveSCEVFromBackedgeMap =
        [S](DenseMap<const Loop *, BackedgeTakenInfo> &Map) {
          for (auto I = Map.begin(), E = Map.end(); I != E;) {
            BackedgeTakenInfo &BEInfo = I->second;
            if (BEInfo.hasOperand(S))
              Map.erase(I++);
            else
              ++I;
          }
        };
  
    RemoveSCEVFromBackedgeMap(BackedgeTakenCounts);
    RemoveSCEVFromBackedgeMap(PredicatedBackedgeTakenCounts);
  }
  
  void
  ScalarEvolution::getUsedLoops(const SCEV *S,
                                SmallPtrSetImpl<const Loop *> &LoopsUsed) {
    struct FindUsedLoops {
      FindUsedLoops(SmallPtrSetImpl<const Loop *> &LoopsUsed)
          : LoopsUsed(LoopsUsed) {}
      SmallPtrSetImpl<const Loop *> &LoopsUsed;
      bool follow(const SCEV *S) {
        if (auto *AR = dyn_cast<SCEVAddRecExpr>(S))
          LoopsUsed.insert(AR->getLoop());
        return true;
      }
  
      bool isDone() const { return false; }
    };
  
    FindUsedLoops F(LoopsUsed);
    SCEVTraversal<FindUsedLoops>(F).visitAll(S);
  }
  
  void ScalarEvolution::addToLoopUseLists(const SCEV *S) {
    SmallPtrSet<const Loop *, 8> LoopsUsed;
    getUsedLoops(S, LoopsUsed);
    for (auto *L : LoopsUsed)
      LoopUsers[L].push_back(S);
  }
  
  void ScalarEvolution::verify() const {
    ScalarEvolution &SE = *const_cast<ScalarEvolution *>(this);
    ScalarEvolution SE2(F, TLI, AC, DT, LI);
  
    SmallVector<Loop *, 8> LoopStack(LI.begin(), LI.end());
  
    // Map's SCEV expressions from one ScalarEvolution "universe" to another.
    struct SCEVMapper : public SCEVRewriteVisitor<SCEVMapper> {
      SCEVMapper(ScalarEvolution &SE) : SCEVRewriteVisitor<SCEVMapper>(SE) {}
  
      const SCEV *visitConstant(const SCEVConstant *Constant) {
        return SE.getConstant(Constant->getAPInt());
      }
  
      const SCEV *visitUnknown(const SCEVUnknown *Expr) {
        return SE.getUnknown(Expr->getValue());
      }
  
      const SCEV *visitCouldNotCompute(const SCEVCouldNotCompute *Expr) {
        return SE.getCouldNotCompute();
      }
    };
  
    SCEVMapper SCM(SE2);
  
    while (!LoopStack.empty()) {
      auto *L = LoopStack.pop_back_val();
      llvm::append_range(LoopStack, *L);
  
      auto *CurBECount = SCM.visit(
          const_cast<ScalarEvolution *>(this)->getBackedgeTakenCount(L));
      auto *NewBECount = SE2.getBackedgeTakenCount(L);
  
      if (CurBECount == SE2.getCouldNotCompute() ||
          NewBECount == SE2.getCouldNotCompute()) {
        // NB! This situation is legal, but is very suspicious -- whatever pass
        // change the loop to make a trip count go from could not compute to
        // computable or vice-versa *should have* invalidated SCEV.  However, we
        // choose not to assert here (for now) since we don't want false
        // positives.
        continue;
      }
  
      if (containsUndefs(CurBECount) || containsUndefs(NewBECount)) {
        // SCEV treats "undef" as an unknown but consistent value (i.e. it does
        // not propagate undef aggressively).  This means we can (and do) fail
        // verification in cases where a transform makes the trip count of a loop
        // go from "undef" to "undef+1" (say).  The transform is fine, since in
        // both cases the loop iterates "undef" times, but SCEV thinks we
        // increased the trip count of the loop by 1 incorrectly.
        continue;
      }
  
      if (SE.getTypeSizeInBits(CurBECount->getType()) >
          SE.getTypeSizeInBits(NewBECount->getType()))
        NewBECount = SE2.getZeroExtendExpr(NewBECount, CurBECount->getType());
      else if (SE.getTypeSizeInBits(CurBECount->getType()) <
               SE.getTypeSizeInBits(NewBECount->getType()))
        CurBECount = SE2.getZeroExtendExpr(CurBECount, NewBECount->getType());
  
      const SCEV *Delta = SE2.getMinusSCEV(CurBECount, NewBECount);
  
      // Unless VerifySCEVStrict is set, we only compare constant deltas.
      if ((VerifySCEVStrict || isa<SCEVConstant>(Delta)) && !Delta->isZero()) {
        dbgs() << "Trip Count for " << *L << " Changed!\n";
        dbgs() << "Old: " << *CurBECount << "\n";
        dbgs() << "New: " << *NewBECount << "\n";
        dbgs() << "Delta: " << *Delta << "\n";
        std::abort();
      }
    }
  
    // Collect all valid loops currently in LoopInfo.
    SmallPtrSet<Loop *, 32> ValidLoops;
    SmallVector<Loop *, 32> Worklist(LI.begin(), LI.end());
    while (!Worklist.empty()) {
      Loop *L = Worklist.pop_back_val();
      if (ValidLoops.contains(L))
        continue;
      ValidLoops.insert(L);
      Worklist.append(L->begin(), L->end());
    }
    // Check for SCEV expressions referencing invalid/deleted loops.
    for (auto &KV : ValueExprMap) {
      auto *AR = dyn_cast<SCEVAddRecExpr>(KV.second);
      if (!AR)
        continue;
      assert(ValidLoops.contains(AR->getLoop()) &&
             "AddRec references invalid loop");
    }
  }
  
  bool ScalarEvolution::invalidate(
      Function &F, const PreservedAnalyses &PA,
      FunctionAnalysisManager::Invalidator &Inv) {
    // Invalidate the ScalarEvolution object whenever it isn't preserved or one
    // of its dependencies is invalidated.
    auto PAC = PA.getChecker<ScalarEvolutionAnalysis>();
    return !(PAC.preserved() || PAC.preservedSet<AllAnalysesOn<Function>>()) ||
           Inv.invalidate<AssumptionAnalysis>(F, PA) ||
           Inv.invalidate<DominatorTreeAnalysis>(F, PA) ||
           Inv.invalidate<LoopAnalysis>(F, PA);
  }
  
  AnalysisKey ScalarEvolutionAnalysis::Key;
  
  ScalarEvolution ScalarEvolutionAnalysis::run(Function &F,
                                               FunctionAnalysisManager &AM) {
    return ScalarEvolution(F, AM.getResult<TargetLibraryAnalysis>(F),
                           AM.getResult<AssumptionAnalysis>(F),
                           AM.getResult<DominatorTreeAnalysis>(F),
                           AM.getResult<LoopAnalysis>(F));
  }
  
  PreservedAnalyses
  ScalarEvolutionVerifierPass::run(Function &F, FunctionAnalysisManager &AM) {
    AM.getResult<ScalarEvolutionAnalysis>(F).verify();
    return PreservedAnalyses::all();
  }
  
  PreservedAnalyses
  ScalarEvolutionPrinterPass::run(Function &F, FunctionAnalysisManager &AM) {
    // For compatibility with opt's -analyze feature under legacy pass manager
    // which was not ported to NPM. This keeps tests using
    // update_analyze_test_checks.py working.
    OS << "Printing analysis 'Scalar Evolution Analysis' for function '"
       << F.getName() << "':\n";
    AM.getResult<ScalarEvolutionAnalysis>(F).print(OS);
    return PreservedAnalyses::all();
  }
  
  INITIALIZE_PASS_BEGIN(ScalarEvolutionWrapperPass, "scalar-evolution",
                        "Scalar Evolution Analysis", false, true)
  INITIALIZE_PASS_DEPENDENCY(AssumptionCacheTracker)
  INITIALIZE_PASS_DEPENDENCY(LoopInfoWrapperPass)
  INITIALIZE_PASS_DEPENDENCY(DominatorTreeWrapperPass)
  INITIALIZE_PASS_DEPENDENCY(TargetLibraryInfoWrapperPass)
  INITIALIZE_PASS_END(ScalarEvolutionWrapperPass, "scalar-evolution",
                      "Scalar Evolution Analysis", false, true)
  
  char ScalarEvolutionWrapperPass::ID = 0;
  
  ScalarEvolutionWrapperPass::ScalarEvolutionWrapperPass() : FunctionPass(ID) {
    initializeScalarEvolutionWrapperPassPass(*PassRegistry::getPassRegistry());
  }
  
  bool ScalarEvolutionWrapperPass::runOnFunction(Function &F) {
    SE.reset(new ScalarEvolution(
        F, getAnalysis<TargetLibraryInfoWrapperPass>().getTLI(F),
        getAnalysis<AssumptionCacheTracker>().getAssumptionCache(F),
        getAnalysis<DominatorTreeWrapperPass>().getDomTree(),
        getAnalysis<LoopInfoWrapperPass>().getLoopInfo()));
    return false;
  }
  
  void ScalarEvolutionWrapperPass::releaseMemory() { SE.reset(); }
  
  void ScalarEvolutionWrapperPass::print(raw_ostream &OS, const Module *) const {
    SE->print(OS);
  }
  
  void ScalarEvolutionWrapperPass::verifyAnalysis() const {
    if (!VerifySCEV)
      return;
  
    SE->verify();
  }
  
  void ScalarEvolutionWrapperPass::getAnalysisUsage(AnalysisUsage &AU) const {
    AU.setPreservesAll();
    AU.addRequiredTransitive<AssumptionCacheTracker>();
    AU.addRequiredTransitive<LoopInfoWrapperPass>();
    AU.addRequiredTransitive<DominatorTreeWrapperPass>();
    AU.addRequiredTransitive<TargetLibraryInfoWrapperPass>();
  }
  
  const SCEVPredicate *ScalarEvolution::getEqualPredicate(const SCEV *LHS,
                                                          const SCEV *RHS) {
    FoldingSetNodeID ID;
    assert(LHS->getType() == RHS->getType() &&
           "Type mismatch between LHS and RHS");
    // Unique this node based on the arguments
    ID.AddInteger(SCEVPredicate::P_Equal);
    ID.AddPointer(LHS);
    ID.AddPointer(RHS);
    void *IP = nullptr;
    if (const auto *S = UniquePreds.FindNodeOrInsertPos(ID, IP))
      return S;
    SCEVEqualPredicate *Eq = new (SCEVAllocator)
        SCEVEqualPredicate(ID.Intern(SCEVAllocator), LHS, RHS);
    UniquePreds.InsertNode(Eq, IP);
    return Eq;
  }
  
  const SCEVPredicate *ScalarEvolution::getWrapPredicate(
      const SCEVAddRecExpr *AR,
      SCEVWrapPredicate::IncrementWrapFlags AddedFlags) {
    FoldingSetNodeID ID;
    // Unique this node based on the arguments
    ID.AddInteger(SCEVPredicate::P_Wrap);
    ID.AddPointer(AR);
    ID.AddInteger(AddedFlags);
    void *IP = nullptr;
    if (const auto *S = UniquePreds.FindNodeOrInsertPos(ID, IP))
      return S;
    auto *OF = new (SCEVAllocator)
        SCEVWrapPredicate(ID.Intern(SCEVAllocator), AR, AddedFlags);
    UniquePreds.InsertNode(OF, IP);
    return OF;
  }
  
  namespace {
  
  class SCEVPredicateRewriter : public SCEVRewriteVisitor<SCEVPredicateRewriter> {
  public:
  
    /// Rewrites \p S in the context of a loop L and the SCEV predication
    /// infrastructure.
    ///
    /// If \p Pred is non-null, the SCEV expression is rewritten to respect the
    /// equivalences present in \p Pred.
    ///
    /// If \p NewPreds is non-null, rewrite is free to add further predicates to
    /// \p NewPreds such that the result will be an AddRecExpr.
    static const SCEV *rewrite(const SCEV *S, const Loop *L, ScalarEvolution &SE,
                               SmallPtrSetImpl<const SCEVPredicate *> *NewPreds,
                               SCEVUnionPredicate *Pred) {
      SCEVPredicateRewriter Rewriter(L, SE, NewPreds, Pred);
      return Rewriter.visit(S);
    }
  
    const SCEV *visitUnknown(const SCEVUnknown *Expr) {
      if (Pred) {
        auto ExprPreds = Pred->getPredicatesForExpr(Expr);
        for (auto *Pred : ExprPreds)
          if (const auto *IPred = dyn_cast<SCEVEqualPredicate>(Pred))
            if (IPred->getLHS() == Expr)
              return IPred->getRHS();
      }
      return convertToAddRecWithPreds(Expr);
    }
  
    const SCEV *visitZeroExtendExpr(const SCEVZeroExtendExpr *Expr) {
      const SCEV *Operand = visit(Expr->getOperand());
      const SCEVAddRecExpr *AR = dyn_cast<SCEVAddRecExpr>(Operand);
      if (AR && AR->getLoop() == L && AR->isAffine()) {
        // This couldn't be folded because the operand didn't have the nuw
        // flag. Add the nusw flag as an assumption that we could make.
        const SCEV *Step = AR->getStepRecurrence(SE);
        Type *Ty = Expr->getType();
        if (addOverflowAssumption(AR, SCEVWrapPredicate::IncrementNUSW))
          return SE.getAddRecExpr(SE.getZeroExtendExpr(AR->getStart(), Ty),
                                  SE.getSignExtendExpr(Step, Ty), L,
                                  AR->getNoWrapFlags());
      }
      return SE.getZeroExtendExpr(Operand, Expr->getType());
    }
  
    const SCEV *visitSignExtendExpr(const SCEVSignExtendExpr *Expr) {
      const SCEV *Operand = visit(Expr->getOperand());
      const SCEVAddRecExpr *AR = dyn_cast<SCEVAddRecExpr>(Operand);
      if (AR && AR->getLoop() == L && AR->isAffine()) {
        // This couldn't be folded because the operand didn't have the nsw
        // flag. Add the nssw flag as an assumption that we could make.
        const SCEV *Step = AR->getStepRecurrence(SE);
        Type *Ty = Expr->getType();
        if (addOverflowAssumption(AR, SCEVWrapPredicate::IncrementNSSW))
          return SE.getAddRecExpr(SE.getSignExtendExpr(AR->getStart(), Ty),
                                  SE.getSignExtendExpr(Step, Ty), L,
                                  AR->getNoWrapFlags());
      }
      return SE.getSignExtendExpr(Operand, Expr->getType());
    }
  
  private:
    explicit SCEVPredicateRewriter(const Loop *L, ScalarEvolution &SE,
                          SmallPtrSetImpl<const SCEVPredicate *> *NewPreds,
                          SCEVUnionPredicate *Pred)
        : SCEVRewriteVisitor(SE), NewPreds(NewPreds), Pred(Pred), L(L) {}
  
    bool addOverflowAssumption(const SCEVPredicate *P) {
      if (!NewPreds) {
        // Check if we've already made this assumption.
        return Pred && Pred->implies(P);
      }
      NewPreds->insert(P);
      return true;
    }
  
    bool addOverflowAssumption(const SCEVAddRecExpr *AR,
                               SCEVWrapPredicate::IncrementWrapFlags AddedFlags) {
      auto *A = SE.getWrapPredicate(AR, AddedFlags);
      return addOverflowAssumption(A);
    }
  
    // If \p Expr represents a PHINode, we try to see if it can be represented
    // as an AddRec, possibly under a predicate (PHISCEVPred). If it is possible
    // to add this predicate as a runtime overflow check, we return the AddRec.
    // If \p Expr does not meet these conditions (is not a PHI node, or we
    // couldn't create an AddRec for it, or couldn't add the predicate), we just
    // return \p Expr.
    const SCEV *convertToAddRecWithPreds(const SCEVUnknown *Expr) {
      if (!isa<PHINode>(Expr->getValue()))
        return Expr;
      Optional<std::pair<const SCEV *, SmallVector<const SCEVPredicate *, 3>>>
      PredicatedRewrite = SE.createAddRecFromPHIWithCasts(Expr);
      if (!PredicatedRewrite)
        return Expr;
      for (auto *P : PredicatedRewrite->second){
        // Wrap predicates from outer loops are not supported.
        if (auto *WP = dyn_cast<const SCEVWrapPredicate>(P)) {
          auto *AR = cast<const SCEVAddRecExpr>(WP->getExpr());
          if (L != AR->getLoop())
            return Expr;
        }
        if (!addOverflowAssumption(P))
          return Expr;
      }
      return PredicatedRewrite->first;
    }
  
    SmallPtrSetImpl<const SCEVPredicate *> *NewPreds;
    SCEVUnionPredicate *Pred;
    const Loop *L;
  };
  
  } // end anonymous namespace
  
  const SCEV *ScalarEvolution::rewriteUsingPredicate(const SCEV *S, const Loop *L,
                                                     SCEVUnionPredicate &Preds) {
    return SCEVPredicateRewriter::rewrite(S, L, *this, nullptr, &Preds);
  }
  
  const SCEVAddRecExpr *ScalarEvolution::convertSCEVToAddRecWithPredicates(
      const SCEV *S, const Loop *L,
      SmallPtrSetImpl<const SCEVPredicate *> &Preds) {
    SmallPtrSet<const SCEVPredicate *, 4> TransformPreds;
    S = SCEVPredicateRewriter::rewrite(S, L, *this, &TransformPreds, nullptr);
    auto *AddRec = dyn_cast<SCEVAddRecExpr>(S);
  
    if (!AddRec)
      return nullptr;
  
    // Since the transformation was successful, we can now transfer the SCEV
    // predicates.
    for (auto *P : TransformPreds)
      Preds.insert(P);
  
    return AddRec;
  }
  
  /// SCEV predicates
  SCEVPredicate::SCEVPredicate(const FoldingSetNodeIDRef ID,
                               SCEVPredicateKind Kind)
      : FastID(ID), Kind(Kind) {}
  
  SCEVEqualPredicate::SCEVEqualPredicate(const FoldingSetNodeIDRef ID,
                                         const SCEV *LHS, const SCEV *RHS)
      : SCEVPredicate(ID, P_Equal), LHS(LHS), RHS(RHS) {
    assert(LHS->getType() == RHS->getType() && "LHS and RHS types don't match");
    assert(LHS != RHS && "LHS and RHS are the same SCEV");
  }
  
  bool SCEVEqualPredicate::implies(const SCEVPredicate *N) const {
    const auto *Op = dyn_cast<SCEVEqualPredicate>(N);
  
    if (!Op)
      return false;
  
    return Op->LHS == LHS && Op->RHS == RHS;
  }
  
  bool SCEVEqualPredicate::isAlwaysTrue() const { return false; }
  
  const SCEV *SCEVEqualPredicate::getExpr() const { return LHS; }
  
  void SCEVEqualPredicate::print(raw_ostream &OS, unsigned Depth) const {
    OS.indent(Depth) << "Equal predicate: " << *LHS << " == " << *RHS << "\n";
  }
  
  SCEVWrapPredicate::SCEVWrapPredicate(const FoldingSetNodeIDRef ID,
                                       const SCEVAddRecExpr *AR,
                                       IncrementWrapFlags Flags)
      : SCEVPredicate(ID, P_Wrap), AR(AR), Flags(Flags) {}
  
  const SCEV *SCEVWrapPredicate::getExpr() const { return AR; }
  
  bool SCEVWrapPredicate::implies(const SCEVPredicate *N) const {
    const auto *Op = dyn_cast<SCEVWrapPredicate>(N);
  
    return Op && Op->AR == AR && setFlags(Flags, Op->Flags) == Flags;
  }
  
  bool SCEVWrapPredicate::isAlwaysTrue() const {
    SCEV::NoWrapFlags ScevFlags = AR->getNoWrapFlags();
    IncrementWrapFlags IFlags = Flags;
  
    if (ScalarEvolution::setFlags(ScevFlags, SCEV::FlagNSW) == ScevFlags)
      IFlags = clearFlags(IFlags, IncrementNSSW);
  
    return IFlags == IncrementAnyWrap;
  }
  
  void SCEVWrapPredicate::print(raw_ostream &OS, unsigned Depth) const {
    OS.indent(Depth) << *getExpr() << " Added Flags: ";
    if (SCEVWrapPredicate::IncrementNUSW & getFlags())
      OS << "<nusw>";
    if (SCEVWrapPredicate::IncrementNSSW & getFlags())
      OS << "<nssw>";
    OS << "\n";
  }
  
  SCEVWrapPredicate::IncrementWrapFlags
  SCEVWrapPredicate::getImpliedFlags(const SCEVAddRecExpr *AR,
                                     ScalarEvolution &SE) {
    IncrementWrapFlags ImpliedFlags = IncrementAnyWrap;
    SCEV::NoWrapFlags StaticFlags = AR->getNoWrapFlags();
  
    // We can safely transfer the NSW flag as NSSW.
    if (ScalarEvolution::setFlags(StaticFlags, SCEV::FlagNSW) == StaticFlags)
      ImpliedFlags = IncrementNSSW;
  
    if (ScalarEvolution::setFlags(StaticFlags, SCEV::FlagNUW) == StaticFlags) {
      // If the increment is positive, the SCEV NUW flag will also imply the
      // WrapPredicate NUSW flag.
      if (const auto *Step = dyn_cast<SCEVConstant>(AR->getStepRecurrence(SE)))
        if (Step->getValue()->getValue().isNonNegative())
          ImpliedFlags = setFlags(ImpliedFlags, IncrementNUSW);
    }
  
    return ImpliedFlags;
  }
  
  /// Union predicates don't get cached so create a dummy set ID for it.
  SCEVUnionPredicate::SCEVUnionPredicate()
      : SCEVPredicate(FoldingSetNodeIDRef(nullptr, 0), P_Union) {}
  
  bool SCEVUnionPredicate::isAlwaysTrue() const {
    return all_of(Preds,
                  [](const SCEVPredicate *I) { return I->isAlwaysTrue(); });
  }
  
  ArrayRef<const SCEVPredicate *>
  SCEVUnionPredicate::getPredicatesForExpr(const SCEV *Expr) {
    auto I = SCEVToPreds.find(Expr);
    if (I == SCEVToPreds.end())
      return ArrayRef<const SCEVPredicate *>();
    return I->second;
  }
  
  bool SCEVUnionPredicate::implies(const SCEVPredicate *N) const {
    if (const auto *Set = dyn_cast<SCEVUnionPredicate>(N))
      return all_of(Set->Preds,
                    [this](const SCEVPredicate *I) { return this->implies(I); });
  
    auto ScevPredsIt = SCEVToPreds.find(N->getExpr());
    if (ScevPredsIt == SCEVToPreds.end())
      return false;
    auto &SCEVPreds = ScevPredsIt->second;
  
    return any_of(SCEVPreds,
                  [N](const SCEVPredicate *I) { return I->implies(N); });
  }
  
  const SCEV *SCEVUnionPredicate::getExpr() const { return nullptr; }
  
  void SCEVUnionPredicate::print(raw_ostream &OS, unsigned Depth) const {
    for (auto Pred : Preds)
      Pred->print(OS, Depth);
  }
  
  void SCEVUnionPredicate::add(const SCEVPredicate *N) {
    if (const auto *Set = dyn_cast<SCEVUnionPredicate>(N)) {
      for (auto Pred : Set->Preds)
        add(Pred);
      return;
    }
  
    if (implies(N))
      return;
  
    const SCEV *Key = N->getExpr();
    assert(Key && "Only SCEVUnionPredicate doesn't have an "
                  " associated expression!");
  
    SCEVToPreds[Key].push_back(N);
    Preds.push_back(N);
  }
  
  PredicatedScalarEvolution::PredicatedScalarEvolution(ScalarEvolution &SE,
                                                       Loop &L)
      : SE(SE), L(L) {}
  
  const SCEV *PredicatedScalarEvolution::getSCEV(Value *V) {
    const SCEV *Expr = SE.getSCEV(V);
    RewriteEntry &Entry = RewriteMap[Expr];
  
    // If we already have an entry and the version matches, return it.
    if (Entry.second && Generation == Entry.first)
      return Entry.second;
  
    // We found an entry but it's stale. Rewrite the stale entry
    // according to the current predicate.
    if (Entry.second)
      Expr = Entry.second;
  
    const SCEV *NewSCEV = SE.rewriteUsingPredicate(Expr, &L, Preds);
    Entry = {Generation, NewSCEV};
  
    return NewSCEV;
  }
  
  const SCEV *PredicatedScalarEvolution::getBackedgeTakenCount() {
    if (!BackedgeCount) {
      SCEVUnionPredicate BackedgePred;
      BackedgeCount = SE.getPredicatedBackedgeTakenCount(&L, BackedgePred);
      addPredicate(BackedgePred);
    }
    return BackedgeCount;
  }
  
  void PredicatedScalarEvolution::addPredicate(const SCEVPredicate &Pred) {
    if (Preds.implies(&Pred))
      return;
    Preds.add(&Pred);
    updateGeneration();
  }
  
  const SCEVUnionPredicate &PredicatedScalarEvolution::getUnionPredicate() const {
    return Preds;
  }
  
  void PredicatedScalarEvolution::updateGeneration() {
    // If the generation number wrapped recompute everything.
    if (++Generation == 0) {
      for (auto &II : RewriteMap) {
        const SCEV *Rewritten = II.second.second;
        II.second = {Generation, SE.rewriteUsingPredicate(Rewritten, &L, Preds)};
      }
    }
  }
  
  void PredicatedScalarEvolution::setNoOverflow(
      Value *V, SCEVWrapPredicate::IncrementWrapFlags Flags) {
    const SCEV *Expr = getSCEV(V);
    const auto *AR = cast<SCEVAddRecExpr>(Expr);
  
    auto ImpliedFlags = SCEVWrapPredicate::getImpliedFlags(AR, SE);
  
    // Clear the statically implied flags.
    Flags = SCEVWrapPredicate::clearFlags(Flags, ImpliedFlags);
    addPredicate(*SE.getWrapPredicate(AR, Flags));
  
    auto II = FlagsMap.insert({V, Flags});
    if (!II.second)
      II.first->second = SCEVWrapPredicate::setFlags(Flags, II.first->second);
  }
  
  bool PredicatedScalarEvolution::hasNoOverflow(
      Value *V, SCEVWrapPredicate::IncrementWrapFlags Flags) {
    const SCEV *Expr = getSCEV(V);
    const auto *AR = cast<SCEVAddRecExpr>(Expr);
  
    Flags = SCEVWrapPredicate::clearFlags(
        Flags, SCEVWrapPredicate::getImpliedFlags(AR, SE));
  
    auto II = FlagsMap.find(V);
  
    if (II != FlagsMap.end())
      Flags = SCEVWrapPredicate::clearFlags(Flags, II->second);
  
    return Flags == SCEVWrapPredicate::IncrementAnyWrap;
  }
  
  const SCEVAddRecExpr *PredicatedScalarEvolution::getAsAddRec(Value *V) {
    const SCEV *Expr = this->getSCEV(V);
    SmallPtrSet<const SCEVPredicate *, 4> NewPreds;
    auto *New = SE.convertSCEVToAddRecWithPredicates(Expr, &L, NewPreds);
  
    if (!New)
      return nullptr;
  
    for (auto *P : NewPreds)
      Preds.add(P);
  
    updateGeneration();
    RewriteMap[SE.getSCEV(V)] = {Generation, New};
    return New;
  }
  
  PredicatedScalarEvolution::PredicatedScalarEvolution(
      const PredicatedScalarEvolution &Init)
      : RewriteMap(Init.RewriteMap), SE(Init.SE), L(Init.L), Preds(Init.Preds),
        Generation(Init.Generation), BackedgeCount(Init.BackedgeCount) {
    for (auto I : Init.FlagsMap)
      FlagsMap.insert(I);
  }
  
  void PredicatedScalarEvolution::print(raw_ostream &OS, unsigned Depth) const {
    // For each block.
    for (auto *BB : L.getBlocks())
      for (auto &I : *BB) {
        if (!SE.isSCEVable(I.getType()))
          continue;
  
        auto *Expr = SE.getSCEV(&I);
        auto II = RewriteMap.find(Expr);
  
        if (II == RewriteMap.end())
          continue;
  
        // Don't print things that are not interesting.
        if (II->second.second == Expr)
          continue;
  
        OS.indent(Depth) << "[PSE]" << I << ":\n";
        OS.indent(Depth + 2) << *Expr << "\n";
        OS.indent(Depth + 2) << "--> " << *II->second.second << "\n";
      }
  }
  
  // Match the mathematical pattern A - (A / B) * B, where A and B can be
  // arbitrary expressions. Also match zext (trunc A to iB) to iY, which is used
  // for URem with constant power-of-2 second operands.
  // It's not always easy, as A and B can be folded (imagine A is X / 2, and B is
  // 4, A / B becomes X / 8).
  bool ScalarEvolution::matchURem(const SCEV *Expr, const SCEV *&LHS,
                                  const SCEV *&RHS) {
    // Try to match 'zext (trunc A to iB) to iY', which is used
    // for URem with constant power-of-2 second operands. Make sure the size of
    // the operand A matches the size of the whole expressions.
    if (const auto *ZExt = dyn_cast<SCEVZeroExtendExpr>(Expr))
      if (const auto *Trunc = dyn_cast<SCEVTruncateExpr>(ZExt->getOperand(0))) {
        LHS = Trunc->getOperand();
        // Bail out if the type of the LHS is larger than the type of the
        // expression for now.
        if (getTypeSizeInBits(LHS->getType()) >
            getTypeSizeInBits(Expr->getType()))
          return false;
        if (LHS->getType() != Expr->getType())
          LHS = getZeroExtendExpr(LHS, Expr->getType());
        RHS = getConstant(APInt(getTypeSizeInBits(Expr->getType()), 1)
                          << getTypeSizeInBits(Trunc->getType()));
        return true;
      }
    const auto *Add = dyn_cast<SCEVAddExpr>(Expr);
    if (Add == nullptr || Add->getNumOperands() != 2)
      return false;
  
    const SCEV *A = Add->getOperand(1);
    const auto *Mul = dyn_cast<SCEVMulExpr>(Add->getOperand(0));
  
    if (Mul == nullptr)
      return false;
  
    const auto MatchURemWithDivisor = [&](const SCEV *B) {
      // (SomeExpr + (-(SomeExpr / B) * B)).
      if (Expr == getURemExpr(A, B)) {
        LHS = A;
        RHS = B;
        return true;
      }
      return false;
    };
  
    // (SomeExpr + (-1 * (SomeExpr / B) * B)).
    if (Mul->getNumOperands() == 3 && isa<SCEVConstant>(Mul->getOperand(0)))
      return MatchURemWithDivisor(Mul->getOperand(1)) ||
             MatchURemWithDivisor(Mul->getOperand(2));
  
    // (SomeExpr + ((-SomeExpr / B) * B)) or (SomeExpr + ((SomeExpr / B) * -B)).
    if (Mul->getNumOperands() == 2)
      return MatchURemWithDivisor(Mul->getOperand(1)) ||
             MatchURemWithDivisor(Mul->getOperand(0)) ||
             MatchURemWithDivisor(getNegativeSCEV(Mul->getOperand(1))) ||
             MatchURemWithDivisor(getNegativeSCEV(Mul->getOperand(0)));
    return false;
  }
  
  const SCEV *
  ScalarEvolution::computeSymbolicMaxBackedgeTakenCount(const Loop *L) {
    SmallVector<BasicBlock*, 16> ExitingBlocks;
    L->getExitingBlocks(ExitingBlocks);
  
    // Form an expression for the maximum exit count possible for this loop. We
    // merge the max and exact information to approximate a version of
    // getConstantMaxBackedgeTakenCount which isn't restricted to just constants.
    SmallVector<const SCEV*, 4> ExitCounts;
    for (BasicBlock *ExitingBB : ExitingBlocks) {
      const SCEV *ExitCount = getExitCount(L, ExitingBB);
      if (isa<SCEVCouldNotCompute>(ExitCount))
        ExitCount = getExitCount(L, ExitingBB,
                                    ScalarEvolution::ConstantMaximum);
      if (!isa<SCEVCouldNotCompute>(ExitCount)) {
        assert(DT.dominates(ExitingBB, L->getLoopLatch()) &&
               "We should only have known counts for exiting blocks that "
               "dominate latch!");
        ExitCounts.push_back(ExitCount);
      }
    }
    if (ExitCounts.empty())
      return getCouldNotCompute();
    return getUMinFromMismatchedTypes(ExitCounts);
  }
  
  /// This rewriter is similar to SCEVParameterRewriter (it replaces SCEVUnknown
  /// components following the Map (Value -> SCEV)), but skips AddRecExpr because
  /// we cannot guarantee that the replacement is loop invariant in the loop of
  /// the AddRec.
  class SCEVLoopGuardRewriter : public SCEVRewriteVisitor<SCEVLoopGuardRewriter> {
    ValueToSCEVMapTy &Map;
  
  public:
    SCEVLoopGuardRewriter(ScalarEvolution &SE, ValueToSCEVMapTy &M)
        : SCEVRewriteVisitor(SE), Map(M) {}
  
    const SCEV *visitAddRecExpr(const SCEVAddRecExpr *Expr) { return Expr; }
  
    const SCEV *visitUnknown(const SCEVUnknown *Expr) {
      auto I = Map.find(Expr->getValue());
      if (I == Map.end())
        return Expr;
      return I->second;
    }
  };
  
  const SCEV *ScalarEvolution::applyLoopGuards(const SCEV *Expr, const Loop *L) {
    auto CollectCondition = [&](ICmpInst::Predicate Predicate, const SCEV *LHS,
                                const SCEV *RHS, ValueToSCEVMapTy &RewriteMap) {
      // If we have LHS == 0, check if LHS is computing a property of some unknown
      // SCEV %v which we can rewrite %v to express explicitly.
      const SCEVConstant *RHSC = dyn_cast<SCEVConstant>(RHS);
      if (Predicate == CmpInst::ICMP_EQ && RHSC &&
          RHSC->getValue()->isNullValue()) {
        // If LHS is A % B, i.e. A % B == 0, rewrite A to (A /u B) * B to
        // explicitly express that.
        const SCEV *URemLHS = nullptr;
        const SCEV *URemRHS = nullptr;
        if (matchURem(LHS, URemLHS, URemRHS)) {
          if (const SCEVUnknown *LHSUnknown = dyn_cast<SCEVUnknown>(URemLHS)) {
            Value *V = LHSUnknown->getValue();
            auto Multiple =
                getMulExpr(getUDivExpr(URemLHS, URemRHS), URemRHS,
                           (SCEV::NoWrapFlags)(SCEV::FlagNUW | SCEV::FlagNSW));
            RewriteMap[V] = Multiple;
            return;
          }
        }
      }
  
      if (!isa<SCEVUnknown>(LHS) && isa<SCEVUnknown>(RHS)) {
        std::swap(LHS, RHS);
        Predicate = CmpInst::getSwappedPredicate(Predicate);
      }
  
      // Check for a condition of the form (-C1 + X < C2).  InstCombine will
      // create this form when combining two checks of the form (X u< C2 + C1) and
      // (X >=u C1).
      auto MatchRangeCheckIdiom = [this, Predicate, LHS, RHS, &RewriteMap]() {
        auto *AddExpr = dyn_cast<SCEVAddExpr>(LHS);
        if (!AddExpr || AddExpr->getNumOperands() != 2)
          return false;
  
        auto *C1 = dyn_cast<SCEVConstant>(AddExpr->getOperand(0));
        auto *LHSUnknown = dyn_cast<SCEVUnknown>(AddExpr->getOperand(1));
        auto *C2 = dyn_cast<SCEVConstant>(RHS);
        if (!C1 || !C2 || !LHSUnknown)
          return false;
  
        auto ExactRegion =
            ConstantRange::makeExactICmpRegion(Predicate, C2->getAPInt())
                .sub(C1->getAPInt());
  
        // Bail out, unless we have a non-wrapping, monotonic range.
        if (ExactRegion.isWrappedSet() || ExactRegion.isFullSet())
          return false;
        auto I = RewriteMap.find(LHSUnknown->getValue());
        const SCEV *RewrittenLHS = I != RewriteMap.end() ? I->second : LHS;
        RewriteMap[LHSUnknown->getValue()] = getUMaxExpr(
            getConstant(ExactRegion.getUnsignedMin()),
            getUMinExpr(RewrittenLHS, getConstant(ExactRegion.getUnsignedMax())));
        return true;
      };
      if (MatchRangeCheckIdiom())
        return;
  
      // For now, limit to conditions that provide information about unknown
      // expressions. RHS also cannot contain add recurrences.
      auto *LHSUnknown = dyn_cast<SCEVUnknown>(LHS);
      if (!LHSUnknown || containsAddRecurrence(RHS))
        return;
  
      // Check whether LHS has already been rewritten. In that case we want to
      // chain further rewrites onto the already rewritten value.
      auto I = RewriteMap.find(LHSUnknown->getValue());
      const SCEV *RewrittenLHS = I != RewriteMap.end() ? I->second : LHS;
      const SCEV *RewrittenRHS = nullptr;
      switch (Predicate) {
      case CmpInst::ICMP_ULT:
        RewrittenRHS =
            getUMinExpr(RewrittenLHS, getMinusSCEV(RHS, getOne(RHS->getType())));
        break;
      case CmpInst::ICMP_SLT:
        RewrittenRHS =
            getSMinExpr(RewrittenLHS, getMinusSCEV(RHS, getOne(RHS->getType())));
        break;
      case CmpInst::ICMP_ULE:
        RewrittenRHS = getUMinExpr(RewrittenLHS, RHS);
        break;
      case CmpInst::ICMP_SLE:
        RewrittenRHS = getSMinExpr(RewrittenLHS, RHS);
        break;
      case CmpInst::ICMP_UGT:
        RewrittenRHS =
            getUMaxExpr(RewrittenLHS, getAddExpr(RHS, getOne(RHS->getType())));
        break;
      case CmpInst::ICMP_SGT:
        RewrittenRHS =
            getSMaxExpr(RewrittenLHS, getAddExpr(RHS, getOne(RHS->getType())));
        break;
      case CmpInst::ICMP_UGE:
        RewrittenRHS = getUMaxExpr(RewrittenLHS, RHS);
        break;
      case CmpInst::ICMP_SGE:
        RewrittenRHS = getSMaxExpr(RewrittenLHS, RHS);
        break;
      case CmpInst::ICMP_EQ:
        if (isa<SCEVConstant>(RHS))
          RewrittenRHS = RHS;
        break;
      case CmpInst::ICMP_NE:
        if (isa<SCEVConstant>(RHS) &&
            cast<SCEVConstant>(RHS)->getValue()->isNullValue())
          RewrittenRHS = getUMaxExpr(RewrittenLHS, getOne(RHS->getType()));
        break;
      default:
        break;
      }
  
      if (RewrittenRHS)
        RewriteMap[LHSUnknown->getValue()] = RewrittenRHS;
    };
    // Starting at the loop predecessor, climb up the predecessor chain, as long
    // as there are predecessors that can be found that have unique successors
    // leading to the original header.
    // TODO: share this logic with isLoopEntryGuardedByCond.
    ValueToSCEVMapTy RewriteMap;
    for (std::pair<const BasicBlock *, const BasicBlock *> Pair(
             L->getLoopPredecessor(), L->getHeader());
         Pair.first; Pair = getPredecessorWithUniqueSuccessorForBB(Pair.first)) {
  
      const BranchInst *LoopEntryPredicate =
          dyn_cast<BranchInst>(Pair.first->getTerminator());
      if (!LoopEntryPredicate || LoopEntryPredicate->isUnconditional())
        continue;
  
      bool EnterIfTrue = LoopEntryPredicate->getSuccessor(0) == Pair.second;
      SmallVector<Value *, 8> Worklist;
      SmallPtrSet<Value *, 8> Visited;
      Worklist.push_back(LoopEntryPredicate->getCondition());
      while (!Worklist.empty()) {
        Value *Cond = Worklist.pop_back_val();
        if (!Visited.insert(Cond).second)
          continue;
  
        if (auto *Cmp = dyn_cast<ICmpInst>(Cond)) {
          auto Predicate =
              EnterIfTrue ? Cmp->getPredicate() : Cmp->getInversePredicate();
          CollectCondition(Predicate, getSCEV(Cmp->getOperand(0)),
                           getSCEV(Cmp->getOperand(1)), RewriteMap);
          continue;
        }
  
        Value *L, *R;
        if (EnterIfTrue ? match(Cond, m_LogicalAnd(m_Value(L), m_Value(R)))
                        : match(Cond, m_LogicalOr(m_Value(L), m_Value(R)))) {
          Worklist.push_back(L);
          Worklist.push_back(R);
        }
      }
    }
  
    // Also collect information from assumptions dominating the loop.
    for (auto &AssumeVH : AC.assumptions()) {
      if (!AssumeVH)
        continue;
      auto *AssumeI = cast<CallInst>(AssumeVH);
      auto *Cmp = dyn_cast<ICmpInst>(AssumeI->getOperand(0));
      if (!Cmp || !DT.dominates(AssumeI, L->getHeader()))
        continue;
      CollectCondition(Cmp->getPredicate(), getSCEV(Cmp->getOperand(0)),
                       getSCEV(Cmp->getOperand(1)), RewriteMap);
    }
  
    if (RewriteMap.empty())
      return Expr;
    SCEVLoopGuardRewriter Rewriter(*this, RewriteMap);
    return Rewriter.visit(Expr);
  }
  