//===- llvm/Analysis/ScalarEvolution.h - Scalar Evolution -------*- C++ -*-===//
//
//                     The LLVM Compiler Infrastructure
//
// This file is distributed under the University of Illinois Open Source
// License. See LICENSE.TXT for details.
//
//===----------------------------------------------------------------------===//
//
// The ScalarEvolution class is an LLVM pass which can be used to analyze and
// categorize scalar expressions in loops.  It specializes in recognizing
// general induction variables, representing them with the abstract and opaque
// SCEV class.  Given this analysis, trip counts of loops and other important
// properties can be obtained.
//
// This analysis is primarily useful for induction variable substitution and
// strength reduction.
//
//===----------------------------------------------------------------------===//

#ifndef LLVM_ANALYSIS_SCALAREVOLUTION_H
#define LLVM_ANALYSIS_SCALAREVOLUTION_H

#include "llvm/ADT/APInt.h"
#include "llvm/ADT/ArrayRef.h"
#include "llvm/ADT/DenseMap.h"
#include "llvm/ADT/DenseMapInfo.h"
#include "llvm/ADT/FoldingSet.h"
#include "llvm/ADT/Hashing.h"
#include "llvm/ADT/Optional.h"
#include "llvm/ADT/PointerIntPair.h"
#include "llvm/ADT/SetVector.h"
#include "llvm/ADT/SmallPtrSet.h"
#include "llvm/ADT/SmallVector.h"
#include "llvm/Analysis/LoopInfo.h"
#include "llvm/IR/ConstantRange.h"
#include "llvm/IR/Function.h"
#include "llvm/IR/InstrTypes.h"
#include "llvm/IR/Instructions.h"
#include "llvm/IR/Operator.h"
#include "llvm/IR/PassManager.h"
#include "llvm/IR/ValueHandle.h"
#include "llvm/IR/ValueMap.h"
#include "llvm/Pass.h"
#include "llvm/Support/Allocator.h"
#include "llvm/Support/Casting.h"
#include "llvm/Support/Compiler.h"
#include <algorithm>
#include <cassert>
#include <cstdint>
#include <memory>
#include <utility>

namespace llvm {

class AssumptionCache;
class BasicBlock;
class Constant;
class ConstantInt;
class DataLayout;
class DominatorTree;
class GEPOperator;
class Instruction;
class LLVMContext;
class raw_ostream;
class ScalarEvolution;
class SCEVAddRecExpr;
class SCEVUnknown;
class StructType;
class TargetLibraryInfo;
class Type;
class Value;

/// This class represents an analyzed expression in the program.  These are
/// opaque objects that the client is not allowed to do much with directly.
///
class SCEV : public FoldingSetNode {
  friend struct FoldingSetTrait<SCEV>;

  /// A reference to an Interned FoldingSetNodeID for this node.  The
  /// ScalarEvolution's BumpPtrAllocator holds the data.
  FoldingSetNodeIDRef FastID;

  // The SCEV baseclass this node corresponds to
  const unsigned short SCEVType;

protected:
  /// This field is initialized to zero and may be used in subclasses to store
  /// miscellaneous information.
  unsigned short SubclassData = 0;

public:
  /// NoWrapFlags are bitfield indices into SubclassData.
  ///
  /// Add and Mul expressions may have no-unsigned-wrap <NUW> or
  /// no-signed-wrap <NSW> properties, which are derived from the IR
  /// operator. NSW is a misnomer that we use to mean no signed overflow or
  /// underflow.
  ///
  /// AddRec expressions may have a no-self-wraparound <NW> property if, in
  /// the integer domain, abs(step) * max-iteration(loop) <=
  /// unsigned-max(bitwidth).  This means that the recurrence will never reach
  /// its start value if the step is non-zero.  Computing the same value on
  /// each iteration is not considered wrapping, and recurrences with step = 0
  /// are trivially <NW>.  <NW> is independent of the sign of step and the
  /// value the add recurrence starts with.
  ///
  /// Note that NUW and NSW are also valid properties of a recurrence, and
  /// either implies NW. For convenience, NW will be set for a recurrence
  /// whenever either NUW or NSW are set.
  enum NoWrapFlags {
    FlagAnyWrap = 0,    // No guarantee.
    FlagNW = (1 << 0),  // No self-wrap.
    FlagNUW = (1 << 1), // No unsigned wrap.
    FlagNSW = (1 << 2), // No signed wrap.
    NoWrapMask = (1 << 3) - 1
  };

  explicit SCEV(const FoldingSetNodeIDRef ID, unsigned SCEVTy)
      : FastID(ID), SCEVType(SCEVTy) {}
  SCEV(const SCEV &) = delete;
  SCEV &operator=(const SCEV &) = delete;

  unsigned getSCEVType() const { return SCEVType; }

  /// Return the LLVM type of this SCEV expression.
  Type *getType() const;

  /// Return true if the expression is a constant zero.
  bool isZero() const;

  /// Return true if the expression is a constant one.
  bool isOne() const;

  /// Return true if the expression is a constant all-ones value.
  bool isAllOnesValue() const;

  /// Return true if the specified scev is negated, but not a constant.
  bool isNonConstantNegative() const;

  /// Print out the internal representation of this scalar to the specified
  /// stream.  This should really only be used for debugging purposes.
  void print(raw_ostream &OS) const;

  /// This method is used for debugging.
  void dump() const;
};

// Specialize FoldingSetTrait for SCEV to avoid needing to compute
// temporary FoldingSetNodeID values.
template <> struct FoldingSetTrait<SCEV> : DefaultFoldingSetTrait<SCEV> {
  static void Profile(const SCEV &X, FoldingSetNodeID &ID) { ID = X.FastID; }

  static bool Equals(const SCEV &X, const FoldingSetNodeID &ID, unsigned IDHash,
                     FoldingSetNodeID &TempID) {
    return ID == X.FastID;
  }

  static unsigned ComputeHash(const SCEV &X, FoldingSetNodeID &TempID) {
    return X.FastID.ComputeHash();
  }
};

inline raw_ostream &operator<<(raw_ostream &OS, const SCEV &S) {
  S.print(OS);
  return OS;
}

/// An object of this class is returned by queries that could not be answered.
/// For example, if you ask for the number of iterations of a linked-list
/// traversal loop, you will get one of these.  None of the standard SCEV
/// operations are valid on this class, it is just a marker.
struct SCEVCouldNotCompute : public SCEV {
  SCEVCouldNotCompute();

  /// Methods for support type inquiry through isa, cast, and dyn_cast:
  static bool classof(const SCEV *S);
};

/// This class represents an assumption made using SCEV expressions which can
/// be checked at run-time.
class SCEVPredicate : public FoldingSetNode {
  friend struct FoldingSetTrait<SCEVPredicate>;

  /// A reference to an Interned FoldingSetNodeID for this node.  The
  /// ScalarEvolution's BumpPtrAllocator holds the data.
  FoldingSetNodeIDRef FastID;

public:
  enum SCEVPredicateKind { P_Union, P_Equal, P_Wrap };

protected:
  SCEVPredicateKind Kind;
  ~SCEVPredicate() = default;
  SCEVPredicate(const SCEVPredicate &) = default;
  SCEVPredicate &operator=(const SCEVPredicate &) = default;

public:
  SCEVPredicate(const FoldingSetNodeIDRef ID, SCEVPredicateKind Kind);

  SCEVPredicateKind getKind() const { return Kind; }

  /// Returns the estimated complexity of this predicate.  This is roughly
  /// measured in the number of run-time checks required.
  virtual unsigned getComplexity() const { return 1; }

  /// Returns true if the predicate is always true. This means that no
  /// assumptions were made and nothing needs to be checked at run-time.
  virtual bool isAlwaysTrue() const = 0;

  /// Returns true if this predicate implies \p N.
  virtual bool implies(const SCEVPredicate *N) const = 0;

  /// Prints a textual representation of this predicate with an indentation of
  /// \p Depth.
  virtual void print(raw_ostream &OS, unsigned Depth = 0) const = 0;

  /// Returns the SCEV to which this predicate applies, or nullptr if this is
  /// a SCEVUnionPredicate.
  virtual const SCEV *getExpr() const = 0;
};

inline raw_ostream &operator<<(raw_ostream &OS, const SCEVPredicate &P) {
  P.print(OS);
  return OS;
}

// Specialize FoldingSetTrait for SCEVPredicate to avoid needing to compute
// temporary FoldingSetNodeID values.
template <>
struct FoldingSetTrait<SCEVPredicate> : DefaultFoldingSetTrait<SCEVPredicate> {
  static void Profile(const SCEVPredicate &X, FoldingSetNodeID &ID) {
    ID = X.FastID;
  }

  static bool Equals(const SCEVPredicate &X, const FoldingSetNodeID &ID,
                     unsigned IDHash, FoldingSetNodeID &TempID) {
    return ID == X.FastID;
  }

  static unsigned ComputeHash(const SCEVPredicate &X,
                              FoldingSetNodeID &TempID) {
    return X.FastID.ComputeHash();
  }
};

/// This class represents an assumption that two SCEV expressions are equal,
/// and this can be checked at run-time.
class SCEVEqualPredicate final : public SCEVPredicate {
  /// We assume that LHS == RHS.
  const SCEV *LHS;
  const SCEV *RHS;

public:
  SCEVEqualPredicate(const FoldingSetNodeIDRef ID, const SCEV *LHS,
                     const SCEV *RHS);

  /// Implementation of the SCEVPredicate interface
  bool implies(const SCEVPredicate *N) const override;
  void print(raw_ostream &OS, unsigned Depth = 0) const override;
  bool isAlwaysTrue() const override;
  const SCEV *getExpr() const override;

  /// Returns the left hand side of the equality.
  const SCEV *getLHS() const { return LHS; }

  /// Returns the right hand side of the equality.
  const SCEV *getRHS() const { return RHS; }

  /// Methods for support type inquiry through isa, cast, and dyn_cast:
  static bool classof(const SCEVPredicate *P) {
    return P->getKind() == P_Equal;
  }
};

/// This class represents an assumption made on an AddRec expression. Given an
/// affine AddRec expression {a,+,b}, we assume that it has the nssw or nusw
/// flags (defined below) in the first X iterations of the loop, where X is a
/// SCEV expression returned by getPredicatedBackedgeTakenCount).
///
/// Note that this does not imply that X is equal to the backedge taken
/// count. This means that if we have a nusw predicate for i32 {0,+,1} with a
/// predicated backedge taken count of X, we only guarantee that {0,+,1} has
/// nusw in the first X iterations. {0,+,1} may still wrap in the loop if we
/// have more than X iterations.
class SCEVWrapPredicate final : public SCEVPredicate {
public:
  /// Similar to SCEV::NoWrapFlags, but with slightly different semantics
  /// for FlagNUSW. The increment is considered to be signed, and a + b
  /// (where b is the increment) is considered to wrap if:
  ///    zext(a + b) != zext(a) + sext(b)
  ///
  /// If Signed is a function that takes an n-bit tuple and maps to the
  /// integer domain as the tuples value interpreted as twos complement,
  /// and Unsigned a function that takes an n-bit tuple and maps to the
  /// integer domain as as the base two value of input tuple, then a + b
  /// has IncrementNUSW iff:
  ///
  /// 0 <= Unsigned(a) + Signed(b) < 2^n
  ///
  /// The IncrementNSSW flag has identical semantics with SCEV::FlagNSW.
  ///
  /// Note that the IncrementNUSW flag is not commutative: if base + inc
  /// has IncrementNUSW, then inc + base doesn't neccessarily have this
  /// property. The reason for this is that this is used for sign/zero
  /// extending affine AddRec SCEV expressions when a SCEVWrapPredicate is
  /// assumed. A {base,+,inc} expression is already non-commutative with
  /// regards to base and inc, since it is interpreted as:
  ///     (((base + inc) + inc) + inc) ...
  enum IncrementWrapFlags {
    IncrementAnyWrap = 0,     // No guarantee.
    IncrementNUSW = (1 << 0), // No unsigned with signed increment wrap.
    IncrementNSSW = (1 << 1), // No signed with signed increment wrap
                              // (equivalent with SCEV::NSW)
    IncrementNoWrapMask = (1 << 2) - 1
  };

  /// Convenient IncrementWrapFlags manipulation methods.
  LLVM_NODISCARD static SCEVWrapPredicate::IncrementWrapFlags
  clearFlags(SCEVWrapPredicate::IncrementWrapFlags Flags,
             SCEVWrapPredicate::IncrementWrapFlags OffFlags) {
    assert((Flags & IncrementNoWrapMask) == Flags && "Invalid flags value!");
    assert((OffFlags & IncrementNoWrapMask) == OffFlags &&
           "Invalid flags value!");
    return (SCEVWrapPredicate::IncrementWrapFlags)(Flags & ~OffFlags);
  }

  LLVM_NODISCARD static SCEVWrapPredicate::IncrementWrapFlags
  maskFlags(SCEVWrapPredicate::IncrementWrapFlags Flags, int Mask) {
    assert((Flags & IncrementNoWrapMask) == Flags && "Invalid flags value!");
    assert((Mask & IncrementNoWrapMask) == Mask && "Invalid mask value!");

    return (SCEVWrapPredicate::IncrementWrapFlags)(Flags & Mask);
  }

  LLVM_NODISCARD static SCEVWrapPredicate::IncrementWrapFlags
  setFlags(SCEVWrapPredicate::IncrementWrapFlags Flags,
           SCEVWrapPredicate::IncrementWrapFlags OnFlags) {
    assert((Flags & IncrementNoWrapMask) == Flags && "Invalid flags value!");
    assert((OnFlags & IncrementNoWrapMask) == OnFlags &&
           "Invalid flags value!");

    return (SCEVWrapPredicate::IncrementWrapFlags)(Flags | OnFlags);
  }

  /// Returns the set of SCEVWrapPredicate no wrap flags implied by a
  /// SCEVAddRecExpr.
  LLVM_NODISCARD static SCEVWrapPredicate::IncrementWrapFlags
  getImpliedFlags(const SCEVAddRecExpr *AR, ScalarEvolution &SE);

private:
  const SCEVAddRecExpr *AR;
  IncrementWrapFlags Flags;

public:
  explicit SCEVWrapPredicate(const FoldingSetNodeIDRef ID,
                             const SCEVAddRecExpr *AR,
                             IncrementWrapFlags Flags);

  /// Returns the set assumed no overflow flags.
  IncrementWrapFlags getFlags() const { return Flags; }

  /// Implementation of the SCEVPredicate interface
  const SCEV *getExpr() const override;
  bool implies(const SCEVPredicate *N) const override;
  void print(raw_ostream &OS, unsigned Depth = 0) const override;
  bool isAlwaysTrue() const override;

  /// Methods for support type inquiry through isa, cast, and dyn_cast:
  static bool classof(const SCEVPredicate *P) {
    return P->getKind() == P_Wrap;
  }
};

/// This class represents a composition of other SCEV predicates, and is the
/// class that most clients will interact with.  This is equivalent to a
/// logical "AND" of all the predicates in the union.
///
/// NB! Unlike other SCEVPredicate sub-classes this class does not live in the
/// ScalarEvolution::Preds folding set.  This is why the \c add function is sound.
class SCEVUnionPredicate final : public SCEVPredicate {
private:
  using PredicateMap =
      DenseMap<const SCEV *, SmallVector<const SCEVPredicate *, 4>>;

  /// Vector with references to all predicates in this union.
  SmallVector<const SCEVPredicate *, 16> Preds;

  /// Maps SCEVs to predicates for quick look-ups.
  PredicateMap SCEVToPreds;

public:
  SCEVUnionPredicate();

  const SmallVectorImpl<const SCEVPredicate *> &getPredicates() const {
    return Preds;
  }

  /// Adds a predicate to this union.
  void add(const SCEVPredicate *N);

  /// Returns a reference to a vector containing all predicates which apply to
  /// \p Expr.
  ArrayRef<const SCEVPredicate *> getPredicatesForExpr(const SCEV *Expr);

  /// Implementation of the SCEVPredicate interface
  bool isAlwaysTrue() const override;
  bool implies(const SCEVPredicate *N) const override;
  void print(raw_ostream &OS, unsigned Depth) const override;
  const SCEV *getExpr() const override;

  /// We estimate the complexity of a union predicate as the size number of
  /// predicates in the union.
  unsigned getComplexity() const override { return Preds.size(); }

  /// Methods for support type inquiry through isa, cast, and dyn_cast:
  static bool classof(const SCEVPredicate *P) {
    return P->getKind() == P_Union;
  }
};

struct ExitLimitQuery {
  ExitLimitQuery(const Loop *L, BasicBlock *ExitingBlock, bool AllowPredicates)
      : L(L), ExitingBlock(ExitingBlock), AllowPredicates(AllowPredicates) {}

  const Loop *L;
  BasicBlock *ExitingBlock;
  bool AllowPredicates;
};

template <> struct DenseMapInfo<ExitLimitQuery> {
  static inline ExitLimitQuery getEmptyKey() {
    return ExitLimitQuery(nullptr, nullptr, true);
  }

  static inline ExitLimitQuery getTombstoneKey() {
    return ExitLimitQuery(nullptr, nullptr, false);
  }

  static unsigned getHashValue(ExitLimitQuery Val) {
    return hash_combine(hash_combine(Val.L, Val.ExitingBlock),
                        Val.AllowPredicates);
  }

  static bool isEqual(ExitLimitQuery LHS, ExitLimitQuery RHS) {
    return LHS.L == RHS.L && LHS.ExitingBlock == RHS.ExitingBlock &&
           LHS.AllowPredicates == RHS.AllowPredicates;
  }
};

/// The main scalar evolution driver. Because client code (intentionally)
/// can't do much with the SCEV objects directly, they must ask this class
/// for services.
class ScalarEvolution {
public:
  /// An enum describing the relationship between a SCEV and a loop.
  enum LoopDisposition {
    LoopVariant,   ///< The SCEV is loop-variant (unknown).
    LoopInvariant, ///< The SCEV is loop-invariant.
    LoopComputable ///< The SCEV varies predictably with the loop.
  };

  /// An enum describing the relationship between a SCEV and a basic block.
  enum BlockDisposition {
    DoesNotDominateBlock,  ///< The SCEV does not dominate the block.
    DominatesBlock,        ///< The SCEV dominates the block.
    ProperlyDominatesBlock ///< The SCEV properly dominates the block.
  };

  /// Convenient NoWrapFlags manipulation that hides enum casts and is
  /// visible in the ScalarEvolution name space.
  LLVM_NODISCARD static SCEV::NoWrapFlags maskFlags(SCEV::NoWrapFlags Flags,
                                                    int Mask) {
    return (SCEV::NoWrapFlags)(Flags & Mask);
  }
  LLVM_NODISCARD static SCEV::NoWrapFlags setFlags(SCEV::NoWrapFlags Flags,
                                                   SCEV::NoWrapFlags OnFlags) {
    return (SCEV::NoWrapFlags)(Flags | OnFlags);
  }
  LLVM_NODISCARD static SCEV::NoWrapFlags
  clearFlags(SCEV::NoWrapFlags Flags, SCEV::NoWrapFlags OffFlags) {
    return (SCEV::NoWrapFlags)(Flags & ~OffFlags);
  }

  ScalarEvolution(Function &F, TargetLibraryInfo &TLI, AssumptionCache &AC,
                  DominatorTree &DT, LoopInfo &LI);
  ScalarEvolution(ScalarEvolution &&Arg);
  ~ScalarEvolution();

  LLVMContext &getContext() const { return F.getContext(); }

  /// 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 isSCEVable(Type *Ty) const;

  /// Return the size in bits of the specified type, for which isSCEVable must
  /// return true.
  uint64_t getTypeSizeInBits(Type *Ty) const;

  /// 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-sized integer type.
  Type *getEffectiveSCEVType(Type *Ty) const;

  // Returns a wider type among {Ty1, Ty2}.
  Type *getWiderType(Type *Ty1, Type *Ty2) const;

  /// Return true if the SCEV is a scAddRecExpr or it contains
  /// scAddRecExpr. The result will be cached in HasRecMap.
  bool containsAddRecurrence(const SCEV *S);

  /// Erase Value from ValueExprMap and ExprValueMap.
  void eraseValueFromMap(Value *V);

  /// Return a SCEV expression for the full generality of the specified
  /// expression.
  const SCEV *getSCEV(Value *V);

  const SCEV *getConstant(ConstantInt *V);
  const SCEV *getConstant(const APInt &Val);
  const SCEV *getConstant(Type *Ty, uint64_t V, bool isSigned = false);
  const SCEV *getTruncateExpr(const SCEV *Op, Type *Ty);
  const SCEV *getZeroExtendExpr(const SCEV *Op, Type *Ty, unsigned Depth = 0);
  const SCEV *getSignExtendExpr(const SCEV *Op, Type *Ty, unsigned Depth = 0);
  const SCEV *getAnyExtendExpr(const SCEV *Op, Type *Ty);
  const SCEV *getAddExpr(SmallVectorImpl<const SCEV *> &Ops,
                         SCEV::NoWrapFlags Flags = SCEV::FlagAnyWrap,
                         unsigned Depth = 0);
  const SCEV *getAddExpr(const SCEV *LHS, const SCEV *RHS,
                         SCEV::NoWrapFlags Flags = SCEV::FlagAnyWrap,
                         unsigned Depth = 0) {
    SmallVector<const SCEV *, 2> Ops = {LHS, RHS};
    return getAddExpr(Ops, Flags, Depth);
  }
  const SCEV *getAddExpr(const SCEV *Op0, const SCEV *Op1, const SCEV *Op2,
                         SCEV::NoWrapFlags Flags = SCEV::FlagAnyWrap,
                         unsigned Depth = 0) {
    SmallVector<const SCEV *, 3> Ops = {Op0, Op1, Op2};
    return getAddExpr(Ops, Flags, Depth);
  }
  const SCEV *getMulExpr(SmallVectorImpl<const SCEV *> &Ops,
                         SCEV::NoWrapFlags Flags = SCEV::FlagAnyWrap,
                         unsigned Depth = 0);
  const SCEV *getMulExpr(const SCEV *LHS, const SCEV *RHS,
                         SCEV::NoWrapFlags Flags = SCEV::FlagAnyWrap,
                         unsigned Depth = 0) {
    SmallVector<const SCEV *, 2> Ops = {LHS, RHS};
    return getMulExpr(Ops, Flags, Depth);
  }
  const SCEV *getMulExpr(const SCEV *Op0, const SCEV *Op1, const SCEV *Op2,
                         SCEV::NoWrapFlags Flags = SCEV::FlagAnyWrap,
                         unsigned Depth = 0) {
    SmallVector<const SCEV *, 3> Ops = {Op0, Op1, Op2};
    return getMulExpr(Ops, Flags, Depth);
  }
  const SCEV *getUDivExpr(const SCEV *LHS, const SCEV *RHS);
  const SCEV *getUDivExactExpr(const SCEV *LHS, const SCEV *RHS);
  const SCEV *getURemExpr(const SCEV *LHS, const SCEV *RHS);
  const SCEV *getAddRecExpr(const SCEV *Start, const SCEV *Step, const Loop *L,
                            SCEV::NoWrapFlags Flags);
  const SCEV *getAddRecExpr(SmallVectorImpl<const SCEV *> &Operands,
                            const Loop *L, SCEV::NoWrapFlags Flags);
  const SCEV *getAddRecExpr(const SmallVectorImpl<const SCEV *> &Operands,
                            const Loop *L, SCEV::NoWrapFlags Flags) {
    SmallVector<const SCEV *, 4> NewOp(Operands.begin(), Operands.end());
    return getAddRecExpr(NewOp, L, Flags);
  }

  /// Checks if \p SymbolicPHI can be rewritten as an AddRecExpr under some
  /// Predicates. If successful return these <AddRecExpr, Predicates>;
  /// The function is intended to be called from PSCEV (the caller will decide
  /// whether to actually add the predicates and carry out the rewrites).
  Optional<std::pair<const SCEV *, SmallVector<const SCEVPredicate *, 3>>>
  createAddRecFromPHIWithCasts(const SCEVUnknown *SymbolicPHI);

  /// Returns an expression for a GEP
  ///
  /// \p GEP The GEP. The indices contained in the GEP itself are ignored,
  /// instead we use IndexExprs.
  /// \p IndexExprs The expressions for the indices.
  const SCEV *getGEPExpr(GEPOperator *GEP,
                         const SmallVectorImpl<const SCEV *> &IndexExprs);
  const SCEV *getSMaxExpr(const SCEV *LHS, const SCEV *RHS);
  const SCEV *getSMaxExpr(SmallVectorImpl<const SCEV *> &Operands);
  const SCEV *getUMaxExpr(const SCEV *LHS, const SCEV *RHS);
  const SCEV *getUMaxExpr(SmallVectorImpl<const SCEV *> &Operands);
  const SCEV *getSMinExpr(const SCEV *LHS, const SCEV *RHS);
  const SCEV *getUMinExpr(const SCEV *LHS, const SCEV *RHS);
  const SCEV *getUnknown(Value *V);
  const SCEV *getCouldNotCompute();

  /// Return a SCEV for the constant 0 of a specific type.
  const SCEV *getZero(Type *Ty) { return getConstant(Ty, 0); }

  /// Return a SCEV for the constant 1 of a specific type.
  const SCEV *getOne(Type *Ty) { return getConstant(Ty, 1); }

  /// Return an expression for sizeof AllocTy that is type IntTy
  const SCEV *getSizeOfExpr(Type *IntTy, Type *AllocTy);

  /// Return an expression for offsetof on the given field with type IntTy
  const SCEV *getOffsetOfExpr(Type *IntTy, StructType *STy, unsigned FieldNo);

  /// Return the SCEV object corresponding to -V.
  const SCEV *getNegativeSCEV(const SCEV *V,
                              SCEV::NoWrapFlags Flags = SCEV::FlagAnyWrap);

  /// Return the SCEV object corresponding to ~V.
  const SCEV *getNotSCEV(const SCEV *V);

  /// Return LHS-RHS.  Minus is represented in SCEV as A+B*-1.
  const SCEV *getMinusSCEV(const SCEV *LHS, const SCEV *RHS,
                           SCEV::NoWrapFlags Flags = SCEV::FlagAnyWrap,
                           unsigned Depth = 0);

  /// Return a SCEV corresponding to a conversion of the input value to the
  /// specified type.  If the type must be extended, it is zero extended.
  const SCEV *getTruncateOrZeroExtend(const SCEV *V, Type *Ty);

  /// Return a SCEV corresponding to a conversion of the input value to the
  /// specified type.  If the type must be extended, it is sign extended.
  const SCEV *getTruncateOrSignExtend(const SCEV *V, Type *Ty);

  /// Return a SCEV corresponding to a conversion of the input value to the
  /// specified type.  If the type must be extended, it is zero extended.  The
  /// conversion must not be narrowing.
  const SCEV *getNoopOrZeroExtend(const SCEV *V, Type *Ty);

  /// Return a SCEV corresponding to a conversion of the input value to the
  /// specified type.  If the type must be extended, it is sign extended.  The
  /// conversion must not be narrowing.
  const SCEV *getNoopOrSignExtend(const SCEV *V, Type *Ty);

  /// Return a SCEV corresponding to a conversion of the input value to the
  /// specified type. If the type must be extended, it is extended with
  /// unspecified bits. The conversion must not be narrowing.
  const SCEV *getNoopOrAnyExtend(const SCEV *V, Type *Ty);

  /// Return a SCEV corresponding to a conversion of the input value to the
  /// specified type.  The conversion must not be widening.
  const SCEV *getTruncateOrNoop(const SCEV *V, Type *Ty);

  /// Promote the operands to the wider of the types using zero-extension, and
  /// then perform a umax operation with them.
  const SCEV *getUMaxFromMismatchedTypes(const SCEV *LHS, const SCEV *RHS);

  /// Promote the operands to the wider of the types using zero-extension, and
  /// then perform a umin operation with them.
  const SCEV *getUMinFromMismatchedTypes(const SCEV *LHS, const SCEV *RHS);

  /// Transitively follow the chain of pointer-type operands until reaching a
  /// SCEV that does not have a single pointer operand. This returns a
  /// SCEVUnknown pointer for well-formed pointer-type expressions, but corner
  /// cases do exist.
  const SCEV *getPointerBase(const SCEV *V);

  /// Return a SCEV expression for the specified value at the specified scope
  /// in the program.  The L value specifies a loop nest to evaluate the
  /// expression at, where null is the top-level or a specified loop is
  /// immediately inside of the loop.
  ///
  /// This method can be used to compute the exit value for a variable defined
  /// in a loop by querying what the value will hold in the parent loop.
  ///
  /// In the case that a relevant loop exit value cannot be computed, the
  /// original value V is returned.
  const SCEV *getSCEVAtScope(const SCEV *S, const Loop *L);

  /// This is a convenience function which does getSCEVAtScope(getSCEV(V), L).
  const SCEV *getSCEVAtScope(Value *V, const Loop *L);

  /// Test whether entry to the loop is protected by a conditional between LHS
  /// and RHS.  This is used to help avoid max expressions in loop trip
  /// counts, and to eliminate casts.
  bool isLoopEntryGuardedByCond(const Loop *L, ICmpInst::Predicate Pred,
                                const SCEV *LHS, const SCEV *RHS);

  /// Test whether the backedge of the loop is protected by a conditional
  /// between LHS and RHS.  This is used to eliminate casts.
  bool isLoopBackedgeGuardedByCond(const Loop *L, ICmpInst::Predicate Pred,
                                   const SCEV *LHS, const SCEV *RHS);

  /// Returns the maximum trip count of the loop if it is a single-exit
  /// loop and we can compute a small maximum for that loop.
  ///
  /// Implemented in terms of the \c getSmallConstantTripCount overload with
  /// the single exiting block passed to it. See that routine for details.
  unsigned getSmallConstantTripCount(const Loop *L);

  /// Returns the maximum trip count of this loop as a normal unsigned
  /// value. Returns 0 if the trip count is unknown or not constant. This
  /// "trip count" assumes that control exits via ExitingBlock. More
  /// precisely, it is the number of times that control may reach ExitingBlock
  /// before taking the branch. For loops with multiple exits, it may not be
  /// the number times that the loop header executes if the loop exits
  /// prematurely via another branch.
  unsigned getSmallConstantTripCount(const Loop *L, BasicBlock *ExitingBlock);

  /// Returns the upper bound of the loop trip count as a normal unsigned
  /// value.
  /// Returns 0 if the trip count is unknown or not constant.
  unsigned getSmallConstantMaxTripCount(const Loop *L);

  /// Returns the largest constant divisor of the trip count of the
  /// loop if it is a single-exit loop and we can compute a small maximum for
  /// that loop.
  ///
  /// Implemented in terms of the \c getSmallConstantTripMultiple overload with
  /// the single exiting block passed to it. See that routine for details.
  unsigned getSmallConstantTripMultiple(const Loop *L);

  /// 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!). As explained in the comments
  /// for getSmallConstantTripCount, this assumes that control exits the loop
  /// via ExitingBlock.
  unsigned getSmallConstantTripMultiple(const Loop *L,
                                        BasicBlock *ExitingBlock);

  /// Get the expression for the number of loop iterations for which this loop
  /// is guaranteed not to exit via ExitingBlock. Otherwise return
  /// SCEVCouldNotCompute.
  const SCEV *getExitCount(const Loop *L, BasicBlock *ExitingBlock);

  /// If the specified loop has a predictable backedge-taken count, return it,
  /// otherwise return a SCEVCouldNotCompute object. The backedge-taken count is
  /// the number of times the loop header will be branched to from within the
  /// loop, assuming there are no abnormal exists like exception throws. This is
  /// one less than the trip count of the loop, since it doesn't count the first
  /// iteration, when the header is branched to from outside the loop.
  ///
  /// Note that it is not valid to call this method on a loop without a
  /// loop-invariant backedge-taken count (see
  /// hasLoopInvariantBackedgeTakenCount).
  const SCEV *getBackedgeTakenCount(const Loop *L);

  /// Similar to getBackedgeTakenCount, except it will add a set of
  /// SCEV predicates to Predicates that are required to be true in order for
  /// the answer to be correct. Predicates can be checked with run-time
  /// checks and can be used to perform loop versioning.
  const SCEV *getPredicatedBackedgeTakenCount(const Loop *L,
                                              SCEVUnionPredicate &Predicates);

  /// When successful, this returns a SCEVConstant that is greater than or equal
  /// to (i.e. a "conservative over-approximation") of the value returend by
  /// getBackedgeTakenCount.  If such a value cannot be computed, it returns the
  /// SCEVCouldNotCompute object.
  const SCEV *getMaxBackedgeTakenCount(const Loop *L);

  /// Return true if the backedge taken count is either the value returned by
  /// getMaxBackedgeTakenCount or zero.
  bool isBackedgeTakenCountMaxOrZero(const Loop *L);

  /// Return true if the specified loop has an analyzable loop-invariant
  /// backedge-taken count.
  bool hasLoopInvariantBackedgeTakenCount(const Loop *L);

  /// This method should be called by the client when it has changed a loop in
  /// a way that may effect ScalarEvolution's ability to compute a trip count,
  /// or if the loop is deleted.  This call is potentially expensive for large
  /// loop bodies.
  void forgetLoop(const Loop *L);

  /// This method should be called by the client when it has changed a value
  /// in a way that may effect its value, or which may disconnect it from a
  /// def-use chain linking it to a loop.
  void forgetValue(Value *V);

  /// Called when the client has changed the disposition of values in
  /// this loop.
  ///
  /// We don't have a way to invalidate per-loop dispositions. Clear and
  /// recompute is simpler.
  void forgetLoopDispositions(const Loop *L) { LoopDispositions.clear(); }

  /// Determine the minimum number of zero bits that S is guaranteed to end in
  /// (at every loop iteration).  It is, at the same time, the minimum number
  /// of times S is divisible by 2.  For example, given {4,+,8} it returns 2.
  /// If S is guaranteed to be 0, it returns the bitwidth of S.
  uint32_t GetMinTrailingZeros(const SCEV *S);

  /// Determine the unsigned range for a particular SCEV.
  /// NOTE: This returns a copy of the reference returned by getRangeRef.
  ConstantRange getUnsignedRange(const SCEV *S) {
    return getRangeRef(S, HINT_RANGE_UNSIGNED);
  }

  /// Determine the min of the unsigned range for a particular SCEV.
  APInt getUnsignedRangeMin(const SCEV *S) {
    return getRangeRef(S, HINT_RANGE_UNSIGNED).getUnsignedMin();
  }

  /// Determine the max of the unsigned range for a particular SCEV.
  APInt getUnsignedRangeMax(const SCEV *S) {
    return getRangeRef(S, HINT_RANGE_UNSIGNED).getUnsignedMax();
  }

  /// Determine the signed range for a particular SCEV.
  /// NOTE: This returns a copy of the reference returned by getRangeRef.
  ConstantRange getSignedRange(const SCEV *S) {
    return getRangeRef(S, HINT_RANGE_SIGNED);
  }

  /// Determine the min of the signed range for a particular SCEV.
  APInt getSignedRangeMin(const SCEV *S) {
    return getRangeRef(S, HINT_RANGE_SIGNED).getSignedMin();
  }

  /// Determine the max of the signed range for a particular SCEV.
  APInt getSignedRangeMax(const SCEV *S) {
    return getRangeRef(S, HINT_RANGE_SIGNED).getSignedMax();
  }

  /// Test if the given expression is known to be negative.
  bool isKnownNegative(const SCEV *S);

  /// Test if the given expression is known to be positive.
  bool isKnownPositive(const SCEV *S);

  /// Test if the given expression is known to be non-negative.
  bool isKnownNonNegative(const SCEV *S);

  /// Test if the given expression is known to be non-positive.
  bool isKnownNonPositive(const SCEV *S);

  /// Test if the given expression is known to be non-zero.
  bool isKnownNonZero(const SCEV *S);

  /// Splits SCEV expression \p S into two SCEVs. One of them is obtained from
  /// \p S by substitution of all AddRec sub-expression related to loop \p L
  /// with initial value of that SCEV. The second is obtained from \p S by
  /// substitution of all AddRec sub-expressions related to loop \p L with post
  /// increment of this AddRec in the loop \p L. In both cases all other AddRec
  /// sub-expressions (not related to \p L) remain the same.
  /// If the \p S contains non-invariant unknown SCEV the function returns
  /// CouldNotCompute SCEV in both values of std::pair.
  /// For example, for SCEV S={0, +, 1}<L1> + {0, +, 1}<L2> and loop L=L1
  /// the function returns pair:
  /// first = {0, +, 1}<L2>
  /// second = {1, +, 1}<L1> + {0, +, 1}<L2>
  /// We can see that for the first AddRec sub-expression it was replaced with
  /// 0 (initial value) for the first element and to {1, +, 1}<L1> (post
  /// increment value) for the second one. In both cases AddRec expression
  /// related to L2 remains the same.
  std::pair<const SCEV *, const SCEV *> SplitIntoInitAndPostInc(const Loop *L,
                                                                const SCEV *S);

  /// We'd like to check the predicate on every iteration of the most dominated
  /// loop between loops used in LHS and RHS.
  /// To do this we use the following list of steps:
  /// 1. Collect set S all loops on which either LHS or RHS depend.
  /// 2. If S is non-empty
  /// a. Let PD be the element of S which is dominated by all other elements.
  /// b. Let E(LHS) be value of LHS on entry of PD.
  ///    To get E(LHS), we should just take LHS and replace all AddRecs that are
  ///    attached to PD on with their entry values.
  ///    Define E(RHS) in the same way.
  /// c. Let B(LHS) be value of L on backedge of PD.
  ///    To get B(LHS), we should just take LHS and replace all AddRecs that are
  ///    attached to PD on with their backedge values.
  ///    Define B(RHS) in the same way.
  /// d. Note that E(LHS) and E(RHS) are automatically available on entry of PD,
  ///    so we can assert on that.
  /// e. Return true if isLoopEntryGuardedByCond(Pred, E(LHS), E(RHS)) &&
  ///                   isLoopBackedgeGuardedByCond(Pred, B(LHS), B(RHS))
  bool isKnownViaInduction(ICmpInst::Predicate Pred, const SCEV *LHS,
                           const SCEV *RHS);

  /// Test if the given expression is known to satisfy the condition described
  /// by Pred, LHS, and RHS.
  bool isKnownPredicate(ICmpInst::Predicate Pred, const SCEV *LHS,
                        const SCEV *RHS);

  /// Test if the condition described by Pred, LHS, RHS is known to be true on
  /// every iteration of the loop of the recurrency LHS.
  bool isKnownOnEveryIteration(ICmpInst::Predicate Pred,
                               const SCEVAddRecExpr *LHS, const SCEV *RHS);

  /// Return true if, for all loop invariant X, the predicate "LHS `Pred` X"
  /// is monotonically increasing or decreasing.  In the former case set
  /// `Increasing` to true and in the latter case set `Increasing` to false.
  ///
  /// A predicate is said to be monotonically increasing if may go from being
  /// false to being true as the loop iterates, but never the other way
  /// around.  A predicate is said to be monotonically decreasing if may go
  /// from being true to being false as the loop iterates, but never the other
  /// way around.
  bool isMonotonicPredicate(const SCEVAddRecExpr *LHS, ICmpInst::Predicate Pred,
                            bool &Increasing);

  /// Return true if the result of the predicate LHS `Pred` RHS is loop
  /// invariant with respect to L.  Set InvariantPred, InvariantLHS and
  /// InvariantLHS so that InvariantLHS `InvariantPred` InvariantRHS is the
  /// loop invariant form of LHS `Pred` RHS.
  bool isLoopInvariantPredicate(ICmpInst::Predicate Pred, const SCEV *LHS,
                                const SCEV *RHS, const Loop *L,
                                ICmpInst::Predicate &InvariantPred,
                                const SCEV *&InvariantLHS,
                                const SCEV *&InvariantRHS);

  /// Simplify LHS and RHS in a comparison with predicate Pred. Return true
  /// iff any changes were made. If the operands are provably equal or
  /// unequal, LHS and RHS are set to the same value and Pred is set to either
  /// ICMP_EQ or ICMP_NE.
  bool SimplifyICmpOperands(ICmpInst::Predicate &Pred, const SCEV *&LHS,
                            const SCEV *&RHS, unsigned Depth = 0);

  /// Return the "disposition" of the given SCEV with respect to the given
  /// loop.
  LoopDisposition getLoopDisposition(const SCEV *S, const Loop *L);

  /// Return true if the value of the given SCEV is unchanging in the
  /// specified loop.
  bool isLoopInvariant(const SCEV *S, const Loop *L);

  /// Determine if the SCEV can be evaluated at loop's entry. It is true if it
  /// doesn't depend on a SCEVUnknown of an instruction which is dominated by
  /// the header of loop L.
  bool isAvailableAtLoopEntry(const SCEV *S, const Loop *L);

  /// Return true if the given SCEV changes value in a known way in the
  /// specified loop.  This property being true implies that the value is
  /// variant in the loop AND that we can emit an expression to compute the
  /// value of the expression at any particular loop iteration.
  bool hasComputableLoopEvolution(const SCEV *S, const Loop *L);

  /// Return the "disposition" of the given SCEV with respect to the given
  /// block.
  BlockDisposition getBlockDisposition(const SCEV *S, const BasicBlock *BB);

  /// Return true if elements that makes up the given SCEV dominate the
  /// specified basic block.
  bool dominates(const SCEV *S, const BasicBlock *BB);

  /// Return true if elements that makes up the given SCEV properly dominate
  /// the specified basic block.
  bool properlyDominates(const SCEV *S, const BasicBlock *BB);

  /// Test whether the given SCEV has Op as a direct or indirect operand.
  bool hasOperand(const SCEV *S, const SCEV *Op) const;

  /// Return the size of an element read or written by Inst.
  const SCEV *getElementSize(Instruction *Inst);

  /// Compute the array dimensions Sizes from the set of Terms extracted from
  /// the memory access function of this SCEVAddRecExpr (second step of
  /// delinearization).
  void findArrayDimensions(SmallVectorImpl<const SCEV *> &Terms,
                           SmallVectorImpl<const SCEV *> &Sizes,
                           const SCEV *ElementSize);

  void print(raw_ostream &OS) const;
  void verify() const;
  bool invalidate(Function &F, const PreservedAnalyses &PA,
                  FunctionAnalysisManager::Invalidator &Inv);

  /// Collect parametric terms occurring in step expressions (first step of
  /// delinearization).
  void collectParametricTerms(const SCEV *Expr,
                              SmallVectorImpl<const SCEV *> &Terms);

  /// Return in Subscripts the access functions for each dimension in Sizes
  /// (third step of delinearization).
  void computeAccessFunctions(const SCEV *Expr,
                              SmallVectorImpl<const SCEV *> &Subscripts,
                              SmallVectorImpl<const SCEV *> &Sizes);

  /// Split this SCEVAddRecExpr into two vectors of SCEVs representing the
  /// subscripts and sizes of an array access.
  ///
  /// The delinearization is a 3 step process: the first two steps compute the
  /// sizes of each subscript and the third step computes the access functions
  /// for the delinearized array:
  ///
  /// 1. Find the terms in the step functions
  /// 2. Compute the array size
  /// 3. Compute the access function: divide the SCEV by the array size
  ///    starting with the innermost dimensions found in step 2. The Quotient
  ///    is the SCEV to be divided in the next step of the recursion. The
  ///    Remainder is the subscript of the innermost dimension. Loop over all
  ///    array dimensions computed in step 2.
  ///
  /// To compute a uniform array size for several memory accesses to the same
  /// object, one can collect in step 1 all the step terms for all the memory
  /// accesses, and compute in step 2 a unique array shape. This guarantees
  /// that the array shape will be the same across all memory accesses.
  ///
  /// FIXME: We could derive the result of steps 1 and 2 from a description of
  /// the array shape given in metadata.
  ///
  /// Example:
  ///
  /// A[][n][m]
  ///
  /// for i
  ///   for j
  ///     for k
  ///       A[j+k][2i][5i] =
  ///
  /// The initial SCEV:
  ///
  /// A[{{{0,+,2*m+5}_i, +, n*m}_j, +, n*m}_k]
  ///
  /// 1. Find the different terms in the step functions:
  /// -> [2*m, 5, n*m, n*m]
  ///
  /// 2. Compute the array size: sort and unique them
  /// -> [n*m, 2*m, 5]
  /// find the GCD of all the terms = 1
  /// divide by the GCD and erase constant terms
  /// -> [n*m, 2*m]
  /// GCD = m
  /// divide by GCD -> [n, 2]
  /// remove constant terms
  /// -> [n]
  /// size of the array is A[unknown][n][m]
  ///
  /// 3. Compute the access function
  /// a. Divide {{{0,+,2*m+5}_i, +, n*m}_j, +, n*m}_k by the innermost size m
  /// Quotient: {{{0,+,2}_i, +, n}_j, +, n}_k
  /// Remainder: {{{0,+,5}_i, +, 0}_j, +, 0}_k
  /// The remainder is the subscript of the innermost array dimension: [5i].
  ///
  /// b. Divide Quotient: {{{0,+,2}_i, +, n}_j, +, n}_k by next outer size n
  /// Quotient: {{{0,+,0}_i, +, 1}_j, +, 1}_k
  /// Remainder: {{{0,+,2}_i, +, 0}_j, +, 0}_k
  /// The Remainder is the subscript of the next array dimension: [2i].
  ///
  /// The subscript of the outermost dimension is the Quotient: [j+k].
  ///
  /// Overall, we have: A[][n][m], and the access function: A[j+k][2i][5i].
  void delinearize(const SCEV *Expr, SmallVectorImpl<const SCEV *> &Subscripts,
                   SmallVectorImpl<const SCEV *> &Sizes,
                   const SCEV *ElementSize);

  /// Return the DataLayout associated with the module this SCEV instance is
  /// operating on.
  const DataLayout &getDataLayout() const {
    return F.getParent()->getDataLayout();
  }

  const SCEVPredicate *getEqualPredicate(const SCEV *LHS, const SCEV *RHS);

  const SCEVPredicate *
  getWrapPredicate(const SCEVAddRecExpr *AR,
                   SCEVWrapPredicate::IncrementWrapFlags AddedFlags);

  /// Re-writes the SCEV according to the Predicates in \p A.
  const SCEV *rewriteUsingPredicate(const SCEV *S, const Loop *L,
                                    SCEVUnionPredicate &A);
  /// Tries to convert the \p S expression to an AddRec expression,
  /// adding additional predicates to \p Preds as required.
  const SCEVAddRecExpr *convertSCEVToAddRecWithPredicates(
      const SCEV *S, const Loop *L,
      SmallPtrSetImpl<const SCEVPredicate *> &Preds);

private:
  /// A CallbackVH to arrange for ScalarEvolution to be notified whenever a
  /// Value is deleted.
  class SCEVCallbackVH final : public CallbackVH {
    ScalarEvolution *SE;

    void deleted() override;
    void allUsesReplacedWith(Value *New) override;

  public:
    SCEVCallbackVH(Value *V, ScalarEvolution *SE = nullptr);
  };

  friend class SCEVCallbackVH;
  friend class SCEVExpander;
  friend class SCEVUnknown;

  /// The function we are analyzing.
  Function &F;

  /// Does the module have any calls to the llvm.experimental.guard intrinsic
  /// at all?  If this is false, we avoid doing work that will only help if
  /// thare are guards present in the IR.
  bool HasGuards;

  /// The target library information for the target we are targeting.
  TargetLibraryInfo &TLI;

  /// The tracker for @llvm.assume intrinsics in this function.
  AssumptionCache &AC;

  /// The dominator tree.
  DominatorTree &DT;

  /// The loop information for the function we are currently analyzing.
  LoopInfo &LI;

  /// This SCEV is used to represent unknown trip counts and things.
  std::unique_ptr<SCEVCouldNotCompute> CouldNotCompute;

  /// The type for HasRecMap.
  using HasRecMapType = DenseMap<const SCEV *, bool>;

  /// This is a cache to record whether a SCEV contains any scAddRecExpr.
  HasRecMapType HasRecMap;

  /// The type for ExprValueMap.
  using ValueOffsetPair = std::pair<Value *, ConstantInt *>;
  using ExprValueMapType = DenseMap<const SCEV *, SetVector<ValueOffsetPair>>;

  /// ExprValueMap -- This map records the original values from which
  /// the SCEV expr is generated from.
  ///
  /// We want to represent the mapping as SCEV -> ValueOffsetPair instead
  /// of SCEV -> Value:
  /// Suppose we know S1 expands to V1, and
  ///  S1 = S2 + C_a
  ///  S3 = S2 + C_b
  /// where C_a and C_b are different SCEVConstants. Then we'd like to
  /// expand S3 as V1 - C_a + C_b instead of expanding S2 literally.
  /// It is helpful when S2 is a complex SCEV expr.
  ///
  /// In order to do that, we represent ExprValueMap as a mapping from
  /// SCEV to ValueOffsetPair. We will save both S1->{V1, 0} and
  /// S2->{V1, C_a} into the map when we create SCEV for V1. When S3
  /// is expanded, it will first expand S2 to V1 - C_a because of
  /// S2->{V1, C_a} in the map, then expand S3 to V1 - C_a + C_b.
  ///
  /// Note: S->{V, Offset} in the ExprValueMap means S can be expanded
  /// to V - Offset.
  ExprValueMapType ExprValueMap;

  /// The type for ValueExprMap.
  using ValueExprMapType =
      DenseMap<SCEVCallbackVH, const SCEV *, DenseMapInfo<Value *>>;

  /// This is a cache of the values we have analyzed so far.
  ValueExprMapType ValueExprMap;

  /// Mark predicate values currently being processed by isImpliedCond.
  SmallPtrSet<Value *, 6> PendingLoopPredicates;

  /// Mark SCEVUnknown Phis currently being processed by getRangeRef.
  SmallPtrSet<const PHINode *, 6> PendingPhiRanges;

  /// Set to true by isLoopBackedgeGuardedByCond when we're walking the set of
  /// conditions dominating the backedge of a loop.
  bool WalkingBEDominatingConds = false;

  /// Set to true by isKnownPredicateViaSplitting when we're trying to prove a
  /// predicate by splitting it into a set of independent predicates.
  bool ProvingSplitPredicate = false;

  /// Memoized values for the GetMinTrailingZeros
  DenseMap<const SCEV *, uint32_t> MinTrailingZerosCache;

  /// Return the Value set from which the SCEV expr is generated.
  SetVector<ValueOffsetPair> *getSCEVValues(const SCEV *S);

  /// Private helper method for the GetMinTrailingZeros method
  uint32_t GetMinTrailingZerosImpl(const SCEV *S);

  /// Information about the number of loop iterations for which a loop exit's
  /// branch condition evaluates to the not-taken path.  This is a temporary
  /// pair of exact and max expressions that are eventually summarized in
  /// ExitNotTakenInfo and BackedgeTakenInfo.
  struct ExitLimit {
    const SCEV *ExactNotTaken; // The exit is not taken exactly this many times
    const SCEV *MaxNotTaken; // The exit is not taken at most this many times

    // Not taken either exactly MaxNotTaken or zero times
    bool MaxOrZero = false;

    /// A set of predicate guards for this ExitLimit. The result is only valid
    /// if all of the predicates in \c Predicates evaluate to 'true' at
    /// run-time.
    SmallPtrSet<const SCEVPredicate *, 4> Predicates;

    void addPredicate(const SCEVPredicate *P) {
      assert(!isa<SCEVUnionPredicate>(P) && "Only add leaf predicates here!");
      Predicates.insert(P);
    }

    /*implicit*/ ExitLimit(const SCEV *E);

    ExitLimit(
        const SCEV *E, const SCEV *M, bool MaxOrZero,
        ArrayRef<const SmallPtrSetImpl<const SCEVPredicate *> *> PredSetList);

    ExitLimit(const SCEV *E, const SCEV *M, bool MaxOrZero,
              const SmallPtrSetImpl<const SCEVPredicate *> &PredSet);

    ExitLimit(const SCEV *E, const SCEV *M, bool MaxOrZero);

    /// Test whether this ExitLimit contains any computed information, or
    /// whether it's all SCEVCouldNotCompute values.
    bool hasAnyInfo() const {
      return !isa<SCEVCouldNotCompute>(ExactNotTaken) ||
             !isa<SCEVCouldNotCompute>(MaxNotTaken);
    }

    bool hasOperand(const SCEV *S) const;

    /// Test whether this ExitLimit contains all information.
    bool hasFullInfo() const {
      return !isa<SCEVCouldNotCompute>(ExactNotTaken);
    }
  };

  /// Information about the number of times a particular loop exit may be
  /// reached before exiting the loop.
  struct ExitNotTakenInfo {
    PoisoningVH<BasicBlock> ExitingBlock;
    const SCEV *ExactNotTaken;
    std::unique_ptr<SCEVUnionPredicate> Predicate;

    explicit ExitNotTakenInfo(PoisoningVH<BasicBlock> ExitingBlock,
                              const SCEV *ExactNotTaken,
                              std::unique_ptr<SCEVUnionPredicate> Predicate)
        : ExitingBlock(ExitingBlock), ExactNotTaken(ExactNotTaken),
          Predicate(std::move(Predicate)) {}

    bool hasAlwaysTruePredicate() const {
      return !Predicate || Predicate->isAlwaysTrue();
    }
  };

  /// Information about the backedge-taken count of a loop. This currently
  /// includes an exact count and a maximum count.
  ///
  class BackedgeTakenInfo {
    /// A list of computable exits and their not-taken counts.  Loops almost
    /// never have more than one computable exit.
    SmallVector<ExitNotTakenInfo, 1> ExitNotTaken;

    /// The pointer part of \c MaxAndComplete is an expression indicating the
    /// least maximum backedge-taken count of the loop that is known, or a
    /// SCEVCouldNotCompute. This expression is only valid if the predicates
    /// associated with all loop exits are true.
    ///
    /// The integer part of \c MaxAndComplete is a boolean indicating if \c
    /// ExitNotTaken has an element for every exiting block in the loop.
    PointerIntPair<const SCEV *, 1> MaxAndComplete;

    /// True iff the backedge is taken either exactly Max or zero times.
    bool MaxOrZero = false;

    /// \name Helper projection functions on \c MaxAndComplete.
    /// @{
    bool isComplete() const { return MaxAndComplete.getInt(); }
    const SCEV *getMax() const { return MaxAndComplete.getPointer(); }
    /// @}

  public:
    BackedgeTakenInfo() : MaxAndComplete(nullptr, 0) {}
    BackedgeTakenInfo(BackedgeTakenInfo &&) = default;
    BackedgeTakenInfo &operator=(BackedgeTakenInfo &&) = default;

    using EdgeExitInfo = std::pair<BasicBlock *, ExitLimit>;

    /// Initialize BackedgeTakenInfo from a list of exact exit counts.
    BackedgeTakenInfo(SmallVectorImpl<EdgeExitInfo> &&ExitCounts, bool Complete,
                      const SCEV *MaxCount, bool MaxOrZero);

    /// Test whether this BackedgeTakenInfo contains any computed information,
    /// or whether it's all SCEVCouldNotCompute values.
    bool hasAnyInfo() const {
      return !ExitNotTaken.empty() || !isa<SCEVCouldNotCompute>(getMax());
    }

    /// Test whether this BackedgeTakenInfo contains complete information.
    bool hasFullInfo() const { return isComplete(); }

    /// Return an expression indicating the exact *backedge-taken*
    /// count of the loop if it is known or SCEVCouldNotCompute
    /// otherwise.  If execution makes it to the backedge on every
    /// iteration (i.e. there are no abnormal exists like exception
    /// throws and thread exits) then this is the number of times the
    /// loop header will execute minus one.
    ///
    /// If the SCEV predicate associated with the answer can be different
    /// from AlwaysTrue, we must add a (non null) Predicates argument.
    /// The SCEV predicate associated with the answer will be added to
    /// Predicates. A run-time check needs to be emitted for the SCEV
    /// predicate in order for the answer to be valid.
    ///
    /// Note that we should always know if we need to pass a predicate
    /// argument or not from the way the ExitCounts vector was computed.
    /// If we allowed SCEV predicates to be generated when populating this
    /// vector, this information can contain them and therefore a
    /// SCEVPredicate argument should be added to getExact.
    const SCEV *getExact(const Loop *L, ScalarEvolution *SE,
                         SCEVUnionPredicate *Predicates = nullptr) const;

    /// Return the number of times this loop exit may fall through to the back
    /// edge, or SCEVCouldNotCompute. The loop is guaranteed not to exit via
    /// this block before this number of iterations, but may exit via another
    /// block.
    const SCEV *getExact(BasicBlock *ExitingBlock, ScalarEvolution *SE) const;

    /// Get the max backedge taken count for the loop.
    const SCEV *getMax(ScalarEvolution *SE) const;

    /// Return true if the number of times this backedge is taken is either the
    /// value returned by getMax or zero.
    bool isMaxOrZero(ScalarEvolution *SE) const;

    /// Return true if any backedge taken count expressions refer to the given
    /// subexpression.
    bool hasOperand(const SCEV *S, ScalarEvolution *SE) const;

    /// Invalidate this result and free associated memory.
    void clear();
  };

  /// Cache the backedge-taken count of the loops for this function as they
  /// are computed.
  DenseMap<const Loop *, BackedgeTakenInfo> BackedgeTakenCounts;

  /// Cache the predicated backedge-taken count of the loops for this
  /// function as they are computed.
  DenseMap<const Loop *, BackedgeTakenInfo> PredicatedBackedgeTakenCounts;

  /// This map contains entries for all of the PHI instructions that we
  /// attempt to compute constant evolutions for.  This allows us to avoid
  /// potentially expensive recomputation of these properties.  An instruction
  /// maps to null if we are unable to compute its exit value.
  DenseMap<PHINode *, Constant *> ConstantEvolutionLoopExitValue;

  /// This map contains entries for all the expressions that we attempt to
  /// compute getSCEVAtScope information for, which can be expensive in
  /// extreme cases.
  DenseMap<const SCEV *, SmallVector<std::pair<const Loop *, const SCEV *>, 2>>
      ValuesAtScopes;

  /// Memoized computeLoopDisposition results.
  DenseMap<const SCEV *,
           SmallVector<PointerIntPair<const Loop *, 2, LoopDisposition>, 2>>
      LoopDispositions;

  struct LoopProperties {
    /// Set to true if the loop contains no instruction that can have side
    /// effects (i.e. via throwing an exception, volatile or atomic access).
    bool HasNoAbnormalExits;

    /// Set to true if the loop contains no instruction that can abnormally exit
    /// the loop (i.e. via throwing an exception, by terminating the thread
    /// cleanly or by infinite looping in a called function).  Strictly
    /// speaking, the last one is not leaving the loop, but is identical to
    /// leaving the loop for reasoning about undefined behavior.
    bool HasNoSideEffects;
  };

  /// Cache for \c getLoopProperties.
  DenseMap<const Loop *, LoopProperties> LoopPropertiesCache;

  /// Return a \c LoopProperties instance for \p L, creating one if necessary.
  LoopProperties getLoopProperties(const Loop *L);

  bool loopHasNoSideEffects(const Loop *L) {
    return getLoopProperties(L).HasNoSideEffects;
  }

  bool loopHasNoAbnormalExits(const Loop *L) {
    return getLoopProperties(L).HasNoAbnormalExits;
  }

  /// Compute a LoopDisposition value.
  LoopDisposition computeLoopDisposition(const SCEV *S, const Loop *L);

  /// Memoized computeBlockDisposition results.
  DenseMap<
      const SCEV *,
      SmallVector<PointerIntPair<const BasicBlock *, 2, BlockDisposition>, 2>>
      BlockDispositions;

  /// Compute a BlockDisposition value.
  BlockDisposition computeBlockDisposition(const SCEV *S, const BasicBlock *BB);

  /// Memoized results from getRange
  DenseMap<const SCEV *, ConstantRange> UnsignedRanges;

  /// Memoized results from getRange
  DenseMap<const SCEV *, ConstantRange> SignedRanges;

  /// Used to parameterize getRange
  enum RangeSignHint { HINT_RANGE_UNSIGNED, HINT_RANGE_SIGNED };

  /// Set the memoized range for the given SCEV.
  const ConstantRange &setRange(const SCEV *S, RangeSignHint Hint,
                                ConstantRange CR) {
    DenseMap<const SCEV *, ConstantRange> &Cache =
        Hint == HINT_RANGE_UNSIGNED ? UnsignedRanges : SignedRanges;

    auto Pair = Cache.try_emplace(S, std::move(CR));
    if (!Pair.second)
      Pair.first->second = std::move(CR);
    return Pair.first->second;
  }

  /// Determine the range for a particular SCEV.
  /// NOTE: This returns a reference to an entry in a cache. It must be
  /// copied if its needed for longer.
  const ConstantRange &getRangeRef(const SCEV *S, RangeSignHint Hint);

  /// Determines the range for the affine SCEVAddRecExpr {\p Start,+,\p Stop}.
  /// Helper for \c getRange.
  ConstantRange getRangeForAffineAR(const SCEV *Start, const SCEV *Stop,
                                    const SCEV *MaxBECount, unsigned BitWidth);

  /// Try to compute a range for the affine SCEVAddRecExpr {\p Start,+,\p
  /// Stop} by "factoring out" a ternary expression from the add recurrence.
  /// Helper called by \c getRange.
  ConstantRange getRangeViaFactoring(const SCEV *Start, const SCEV *Stop,
                                     const SCEV *MaxBECount, unsigned BitWidth);

  /// We know that there is no SCEV for the specified value.  Analyze the
  /// expression.
  const SCEV *createSCEV(Value *V);

  /// Provide the special handling we need to analyze PHI SCEVs.
  const SCEV *createNodeForPHI(PHINode *PN);

  /// Helper function called from createNodeForPHI.
  const SCEV *createAddRecFromPHI(PHINode *PN);

  /// A helper function for createAddRecFromPHI to handle simple cases.
  const SCEV *createSimpleAffineAddRec(PHINode *PN, Value *BEValueV,
                                            Value *StartValueV);

  /// Helper function called from createNodeForPHI.
  const SCEV *createNodeFromSelectLikePHI(PHINode *PN);

  /// Provide special handling for a select-like instruction (currently this
  /// is either a select instruction or a phi node).  \p I is the instruction
  /// being processed, and it is assumed equivalent to "Cond ? TrueVal :
  /// FalseVal".
  const SCEV *createNodeForSelectOrPHI(Instruction *I, Value *Cond,
                                       Value *TrueVal, Value *FalseVal);

  /// Provide the special handling we need to analyze GEP SCEVs.
  const SCEV *createNodeForGEP(GEPOperator *GEP);

  /// Implementation code for getSCEVAtScope; called at most once for each
  /// SCEV+Loop pair.
  const SCEV *computeSCEVAtScope(const SCEV *S, const Loop *L);

  /// This looks up computed SCEV values for all instructions that depend on
  /// the given instruction and removes them from the ValueExprMap map if they
  /// reference SymName. This is used during PHI resolution.
  void forgetSymbolicName(Instruction *I, const SCEV *SymName);

  /// Return the BackedgeTakenInfo for the given loop, lazily computing new
  /// values if the loop hasn't been analyzed yet. The returned result is
  /// guaranteed not to be predicated.
  const BackedgeTakenInfo &getBackedgeTakenInfo(const Loop *L);

  /// Similar to getBackedgeTakenInfo, but will add predicates as required
  /// with the purpose of returning complete information.
  const BackedgeTakenInfo &getPredicatedBackedgeTakenInfo(const Loop *L);

  /// Compute the number of times the specified loop will iterate.
  /// If AllowPredicates is set, we will create new SCEV predicates as
  /// necessary in order to return an exact answer.
  BackedgeTakenInfo computeBackedgeTakenCount(const Loop *L,
                                              bool AllowPredicates = false);

  /// Compute the number of times the backedge of the specified loop will
  /// execute if it exits via the specified block. If AllowPredicates is set,
  /// this call will try to use a minimal set of SCEV predicates in order to
  /// return an exact answer.
  ExitLimit computeExitLimit(const Loop *L, BasicBlock *ExitingBlock,
                             bool AllowPredicates = false);

  /// Compute the number of times the backedge of the specified loop will
  /// execute if its exit condition were a conditional branch of ExitCond.
  ///
  /// \p ControlsExit is true if ExitCond directly controls the exit
  /// branch. In this case, we can assume that the loop exits only if the
  /// condition is true and can infer that failing to meet the condition prior
  /// to integer wraparound results in undefined behavior.
  ///
  /// If \p AllowPredicates is set, this call will try to use a minimal set of
  /// SCEV predicates in order to return an exact answer.
  ExitLimit computeExitLimitFromCond(const Loop *L, Value *ExitCond,
                                     bool ExitIfTrue, bool ControlsExit,
                                     bool AllowPredicates = false);

  // Helper functions for computeExitLimitFromCond to avoid exponential time
  // complexity.

  class ExitLimitCache {
    // It may look like we need key on the whole (L, ExitIfTrue, ControlsExit,
    // AllowPredicates) tuple, but recursive calls to
    // computeExitLimitFromCondCached from computeExitLimitFromCondImpl only
    // vary the in \c ExitCond and \c ControlsExit parameters.  We remember the
    // initial values of the other values to assert our assumption.
    SmallDenseMap<PointerIntPair<Value *, 1>, ExitLimit> TripCountMap;

    const Loop *L;
    bool ExitIfTrue;
    bool AllowPredicates;

  public:
    ExitLimitCache(const Loop *L, bool ExitIfTrue, bool AllowPredicates)
        : L(L), ExitIfTrue(ExitIfTrue), AllowPredicates(AllowPredicates) {}

    Optional<ExitLimit> find(const Loop *L, Value *ExitCond, bool ExitIfTrue,
                             bool ControlsExit, bool AllowPredicates);

    void insert(const Loop *L, Value *ExitCond, bool ExitIfTrue,
                bool ControlsExit, bool AllowPredicates, const ExitLimit &EL);
  };

  using ExitLimitCacheTy = ExitLimitCache;

  ExitLimit computeExitLimitFromCondCached(ExitLimitCacheTy &Cache,
                                           const Loop *L, Value *ExitCond,
                                           bool ExitIfTrue,
                                           bool ControlsExit,
                                           bool AllowPredicates);
  ExitLimit computeExitLimitFromCondImpl(ExitLimitCacheTy &Cache, const Loop *L,
                                         Value *ExitCond, bool ExitIfTrue,
                                         bool ControlsExit,
                                         bool AllowPredicates);

  /// Compute the number of times the backedge of the specified loop will
  /// execute if its exit condition were a conditional branch of the ICmpInst
  /// ExitCond and ExitIfTrue. If AllowPredicates is set, this call will try
  /// to use a minimal set of SCEV predicates in order to return an exact
  /// answer.
  ExitLimit computeExitLimitFromICmp(const Loop *L, ICmpInst *ExitCond,
                                     bool ExitIfTrue,
                                     bool IsSubExpr,
                                     bool AllowPredicates = false);

  /// Compute the number of times the backedge of the specified loop will
  /// execute if its exit condition were a switch with a single exiting case
  /// to ExitingBB.
  ExitLimit computeExitLimitFromSingleExitSwitch(const Loop *L,
                                                 SwitchInst *Switch,
                                                 BasicBlock *ExitingBB,
                                                 bool IsSubExpr);

  /// Given an exit condition of 'icmp op load X, cst', try to see if we can
  /// compute the backedge-taken count.
  ExitLimit computeLoadConstantCompareExitLimit(LoadInst *LI, Constant *RHS,
                                                const Loop *L,
                                                ICmpInst::Predicate p);

  /// Compute the exit limit of a loop that is controlled by a
  /// "(IV >> 1) != 0" type comparison.  We cannot compute the exact trip
  /// count in these cases (since SCEV has no way of expressing them), but we
  /// can still sometimes compute an upper bound.
  ///
  /// Return an ExitLimit for a loop whose backedge is guarded by `LHS Pred
  /// RHS`.
  ExitLimit computeShiftCompareExitLimit(Value *LHS, Value *RHS, const Loop *L,
                                         ICmpInst::Predicate Pred);

  /// If the loop is known to execute a constant number of times (the
  /// condition evolves only from constants), try to evaluate a few iterations
  /// of the loop until we get the exit condition gets a value of ExitWhen
  /// (true or false).  If we cannot evaluate the exit count of the loop,
  /// return CouldNotCompute.
  const SCEV *computeExitCountExhaustively(const Loop *L, Value *Cond,
                                           bool ExitWhen);

  /// Return the number of times an exit condition comparing the specified
  /// value to zero will execute.  If not computable, return CouldNotCompute.
  /// If AllowPredicates is set, this call will try to use a minimal set of
  /// SCEV predicates in order to return an exact answer.
  ExitLimit howFarToZero(const SCEV *V, const Loop *L, bool IsSubExpr,
                         bool AllowPredicates = false);

  /// Return the number of times an exit condition checking the specified
  /// value for nonzero will execute.  If not computable, return
  /// CouldNotCompute.
  ExitLimit howFarToNonZero(const SCEV *V, const Loop *L);

  /// Return the number of times an exit condition containing the specified
  /// less-than comparison will execute.  If not computable, return
  /// CouldNotCompute.
  ///
  /// \p isSigned specifies whether the less-than is signed.
  ///
  /// \p ControlsExit is true when the LHS < RHS condition directly controls
  /// the branch (loops exits only if condition is true). In this case, we can
  /// use NoWrapFlags to skip overflow checks.
  ///
  /// If \p AllowPredicates is set, this call will try to use a minimal set of
  /// SCEV predicates in order to return an exact answer.
  ExitLimit howManyLessThans(const SCEV *LHS, const SCEV *RHS, const Loop *L,
                             bool isSigned, bool ControlsExit,
                             bool AllowPredicates = false);

  ExitLimit howManyGreaterThans(const SCEV *LHS, const SCEV *RHS, const Loop *L,
                                bool isSigned, bool IsSubExpr,
                                bool AllowPredicates = false);

  /// Return a predecessor of BB (which may not be an immediate predecessor)
  /// which has exactly one successor from which BB is reachable, or null if
  /// no such block is found.
  std::pair<BasicBlock *, BasicBlock *>
  getPredecessorWithUniqueSuccessorForBB(BasicBlock *BB);

  /// Test whether the condition described by Pred, LHS, and RHS is true
  /// whenever the given FoundCondValue value evaluates to true.
  bool isImpliedCond(ICmpInst::Predicate Pred, const SCEV *LHS, const SCEV *RHS,
                     Value *FoundCondValue, bool Inverse);

  /// Test whether the condition described by Pred, LHS, and RHS is true
  /// whenever the condition described by FoundPred, FoundLHS, FoundRHS is
  /// true.
  bool isImpliedCond(ICmpInst::Predicate Pred, const SCEV *LHS, const SCEV *RHS,
                     ICmpInst::Predicate FoundPred, const SCEV *FoundLHS,
                     const SCEV *FoundRHS);

  /// Test whether the condition described by Pred, LHS, and RHS is true
  /// whenever the condition described by Pred, FoundLHS, and FoundRHS is
  /// true.
  bool isImpliedCondOperands(ICmpInst::Predicate Pred, const SCEV *LHS,
                             const SCEV *RHS, const SCEV *FoundLHS,
                             const SCEV *FoundRHS);

  /// Test whether the condition described by Pred, LHS, and RHS is true
  /// whenever the condition described by Pred, FoundLHS, and FoundRHS is
  /// true. Here LHS is an operation that includes FoundLHS as one of its
  /// arguments.
  bool isImpliedViaOperations(ICmpInst::Predicate Pred,
                              const SCEV *LHS, const SCEV *RHS,
                              const SCEV *FoundLHS, const SCEV *FoundRHS,
                              unsigned Depth = 0);

  /// Test whether the condition described by Pred, LHS, and RHS is true.
  /// Use only simple non-recursive types of checks, such as range analysis etc.
  bool isKnownViaNonRecursiveReasoning(ICmpInst::Predicate Pred,
                                       const SCEV *LHS, const SCEV *RHS);

  /// Test whether the condition described by Pred, LHS, and RHS is true
  /// whenever the condition described by Pred, FoundLHS, and FoundRHS is
  /// true.
  bool isImpliedCondOperandsHelper(ICmpInst::Predicate Pred, const SCEV *LHS,
                                   const SCEV *RHS, const SCEV *FoundLHS,
                                   const SCEV *FoundRHS);

  /// Test whether the condition described by Pred, LHS, and RHS is true
  /// whenever the condition described by Pred, FoundLHS, and FoundRHS is
  /// true.  Utility function used by isImpliedCondOperands.  Tries to get
  /// cases like "X `sgt` 0 => X - 1 `sgt` -1".
  bool isImpliedCondOperandsViaRanges(ICmpInst::Predicate Pred, const SCEV *LHS,
                                      const SCEV *RHS, const SCEV *FoundLHS,
                                      const SCEV *FoundRHS);

  /// Return true if the condition denoted by \p LHS \p Pred \p RHS is implied
  /// by a call to \c @llvm.experimental.guard in \p BB.
  bool isImpliedViaGuard(BasicBlock *BB, ICmpInst::Predicate Pred,
                         const SCEV *LHS, const SCEV *RHS);

  /// Test whether the condition described by Pred, LHS, and RHS is true
  /// whenever the condition described by Pred, FoundLHS, and FoundRHS is
  /// true.
  ///
  /// This routine tries to rule out certain kinds of integer overflow, and
  /// then tries to reason about arithmetic properties of the predicates.
  bool isImpliedCondOperandsViaNoOverflow(ICmpInst::Predicate Pred,
                                          const SCEV *LHS, const SCEV *RHS,
                                          const SCEV *FoundLHS,
                                          const SCEV *FoundRHS);

  /// 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 *getConstantEvolutionLoopExitValue(PHINode *PN, const APInt &BEs,
                                              const Loop *L);

  /// Test if the given expression is known to satisfy the condition described
  /// by Pred and the known constant ranges of LHS and RHS.
  bool isKnownPredicateViaConstantRanges(ICmpInst::Predicate Pred,
                                         const SCEV *LHS, const SCEV *RHS);

  /// Try to prove the condition described by "LHS Pred RHS" by ruling out
  /// integer overflow.
  ///
  /// For instance, this will return true for "A s< (A + C)<nsw>" if C is
  /// positive.
  bool isKnownPredicateViaNoOverflow(ICmpInst::Predicate Pred, const SCEV *LHS,
                                     const SCEV *RHS);

  /// Try to split Pred LHS RHS into logical conjunctions (and's) and try to
  /// prove them individually.
  bool isKnownPredicateViaSplitting(ICmpInst::Predicate Pred, const SCEV *LHS,
                                    const SCEV *RHS);

  /// Try to match the Expr as "(L + R)<Flags>".
  bool splitBinaryAdd(const SCEV *Expr, const SCEV *&L, const SCEV *&R,
                      SCEV::NoWrapFlags &Flags);

  /// Compute \p LHS - \p RHS and returns the result as an APInt if it is a
  /// constant, and None if it isn't.
  ///
  /// This is intended to be a cheaper version of getMinusSCEV.  We can be
  /// frugal here since we just bail out of actually constructing and
  /// canonicalizing an expression in the cases where the result isn't going
  /// to be a constant.
  Optional<APInt> computeConstantDifference(const SCEV *LHS, const SCEV *RHS);

  /// Drop memoized information computed for S.
  void forgetMemoizedResults(const SCEV *S);

  /// Return an existing SCEV for V if there is one, otherwise return nullptr.
  const SCEV *getExistingSCEV(Value *V);

  /// Return false iff given SCEV contains a SCEVUnknown with NULL value-
  /// pointer.
  bool checkValidity(const SCEV *S) const;

  /// Return true if `ExtendOpTy`({`Start`,+,`Step`}) can be proved to be
  /// equal to {`ExtendOpTy`(`Start`),+,`ExtendOpTy`(`Step`)}.  This is
  /// equivalent to proving no signed (resp. unsigned) wrap in
  /// {`Start`,+,`Step`} if `ExtendOpTy` is `SCEVSignExtendExpr`
  /// (resp. `SCEVZeroExtendExpr`).
  template <typename ExtendOpTy>
  bool proveNoWrapByVaryingStart(const SCEV *Start, const SCEV *Step,
                                 const Loop *L);

  /// Try to prove NSW or NUW on \p AR relying on ConstantRange manipulation.
  SCEV::NoWrapFlags proveNoWrapViaConstantRanges(const SCEVAddRecExpr *AR);

  bool isMonotonicPredicateImpl(const SCEVAddRecExpr *LHS,
                                ICmpInst::Predicate Pred, bool &Increasing);

  /// Return SCEV no-wrap flags that can be proven based on reasoning about
  /// how poison produced from no-wrap flags on this value (e.g. a nuw add)
  /// would trigger undefined behavior on overflow.
  SCEV::NoWrapFlags getNoWrapFlagsFromUB(const Value *V);

  /// Return true if the SCEV corresponding to \p I is never poison.  Proving
  /// this is more complex than proving that just \p I is never poison, since
  /// SCEV commons expressions across control flow, and you can have cases
  /// like:
  ///
  ///   idx0 = a + b;
  ///   ptr[idx0] = 100;
  ///   if (<condition>) {
  ///     idx1 = a +nsw b;
  ///     ptr[idx1] = 200;
  ///   }
  ///
  /// where the SCEV expression (+ a b) is guaranteed to not be poison (and
  /// hence not sign-overflow) only if "<condition>" is true.  Since both
  /// `idx0` and `idx1` will be mapped to the same SCEV expression, (+ a b),
  /// it is not okay to annotate (+ a b) with <nsw> in the above example.
  bool isSCEVExprNeverPoison(const Instruction *I);

  /// This is like \c isSCEVExprNeverPoison but it specifically works for
  /// instructions that will get mapped to SCEV add recurrences.  Return true
  /// if \p I will never generate poison under the assumption that \p I is an
  /// add recurrence on the loop \p L.
  bool isAddRecNeverPoison(const Instruction *I, const Loop *L);

  /// Similar to createAddRecFromPHI, but with the additional flexibility of
  /// suggesting runtime overflow checks in case casts are encountered.
  /// If successful, the analysis records that for this loop, \p SymbolicPHI,
  /// which is the UnknownSCEV currently representing the PHI, can be rewritten
  /// into an AddRec, assuming some predicates; The function then returns the
  /// AddRec and the predicates as a pair, and caches this pair in
  /// PredicatedSCEVRewrites.
  /// If the analysis is not successful, a mapping from the \p SymbolicPHI to
  /// itself (with no predicates) is recorded, and a nullptr with an empty
  /// predicates vector is returned as a pair.
  Optional<std::pair<const SCEV *, SmallVector<const SCEVPredicate *, 3>>>
  createAddRecFromPHIWithCastsImpl(const SCEVUnknown *SymbolicPHI);

  /// Compute the backedge taken count knowing the interval difference, the
  /// stride and presence of the equality in the comparison.
  const SCEV *computeBECount(const SCEV *Delta, const SCEV *Stride,
                             bool Equality);

  /// Compute the maximum backedge count based on the range of values
  /// permitted by Start, End, and Stride. This is for loops of the form
  /// {Start, +, Stride} LT End.
  ///
  /// Precondition: the induction variable is known to be positive.  We *don't*
  /// assert these preconditions so please be careful.
  const SCEV *computeMaxBECountForLT(const SCEV *Start, const SCEV *Stride,
                                     const SCEV *End, unsigned BitWidth,
                                     bool IsSigned);

  /// Verify if an linear IV with positive stride can overflow when in a
  /// less-than comparison, knowing the invariant term of the comparison,
  /// the stride and the knowledge of NSW/NUW flags on the recurrence.
  bool doesIVOverflowOnLT(const SCEV *RHS, const SCEV *Stride, bool IsSigned,
                          bool NoWrap);

  /// Verify if an linear IV with negative stride can overflow when in a
  /// greater-than comparison, knowing the invariant term of the comparison,
  /// the stride and the knowledge of NSW/NUW flags on the recurrence.
  bool doesIVOverflowOnGT(const SCEV *RHS, const SCEV *Stride, bool IsSigned,
                          bool NoWrap);

  /// Get add expr already created or create a new one.
  const SCEV *getOrCreateAddExpr(SmallVectorImpl<const SCEV *> &Ops,
                                 SCEV::NoWrapFlags Flags);

  /// Get mul expr already created or create a new one.
  const SCEV *getOrCreateMulExpr(SmallVectorImpl<const SCEV *> &Ops,
                                 SCEV::NoWrapFlags Flags);

  /// Find all of the loops transitively used in \p S, and fill \p LoopsUsed.
  /// A loop is considered "used" by an expression if it contains
  /// an add rec on said loop.
  void getUsedLoops(const SCEV *S, SmallPtrSetImpl<const Loop *> &LoopsUsed);

  /// Find all of the loops transitively used in \p S, and update \c LoopUsers
  /// accordingly.
  void addToLoopUseLists(const SCEV *S);

  FoldingSet<SCEV> UniqueSCEVs;
  FoldingSet<SCEVPredicate> UniquePreds;
  BumpPtrAllocator SCEVAllocator;

  /// This maps loops to a list of SCEV expressions that (transitively) use said
  /// loop.
  DenseMap<const Loop *, SmallVector<const SCEV *, 4>> LoopUsers;

  /// Cache tentative mappings from UnknownSCEVs in a Loop, to a SCEV expression
  /// they can be rewritten into under certain predicates.
  DenseMap<std::pair<const SCEVUnknown *, const Loop *>,
           std::pair<const SCEV *, SmallVector<const SCEVPredicate *, 3>>>
      PredicatedSCEVRewrites;

  /// The head of a linked list of all SCEVUnknown values that have been
  /// allocated. This is used by releaseMemory to locate them all and call
  /// their destructors.
  SCEVUnknown *FirstUnknown = nullptr;
};

/// Analysis pass that exposes the \c ScalarEvolution for a function.
class ScalarEvolutionAnalysis
    : public AnalysisInfoMixin<ScalarEvolutionAnalysis> {
  friend AnalysisInfoMixin<ScalarEvolutionAnalysis>;

  static AnalysisKey Key;

public:
  using Result = ScalarEvolution;

  ScalarEvolution run(Function &F, FunctionAnalysisManager &AM);
};

/// Printer pass for the \c ScalarEvolutionAnalysis results.
class ScalarEvolutionPrinterPass
    : public PassInfoMixin<ScalarEvolutionPrinterPass> {
  raw_ostream &OS;

public:
  explicit ScalarEvolutionPrinterPass(raw_ostream &OS) : OS(OS) {}

  PreservedAnalyses run(Function &F, FunctionAnalysisManager &AM);
};

class ScalarEvolutionWrapperPass : public FunctionPass {
  std::unique_ptr<ScalarEvolution> SE;

public:
  static char ID;

  ScalarEvolutionWrapperPass();

  ScalarEvolution &getSE() { return *SE; }
  const ScalarEvolution &getSE() const { return *SE; }

  bool runOnFunction(Function &F) override;
  void releaseMemory() override;
  void getAnalysisUsage(AnalysisUsage &AU) const override;
  void print(raw_ostream &OS, const Module * = nullptr) const override;
  void verifyAnalysis() const override;
};

/// An interface layer with SCEV used to manage how we see SCEV expressions
/// for values in the context of existing predicates. We can add new
/// predicates, but we cannot remove them.
///
/// This layer has multiple purposes:
///   - provides a simple interface for SCEV versioning.
///   - guarantees that the order of transformations applied on a SCEV
///     expression for a single Value is consistent across two different
///     getSCEV calls. This means that, for example, once we've obtained
///     an AddRec expression for a certain value through expression
///     rewriting, we will continue to get an AddRec expression for that
///     Value.
///   - lowers the number of expression rewrites.
class PredicatedScalarEvolution {
public:
  PredicatedScalarEvolution(ScalarEvolution &SE, Loop &L);

  const SCEVUnionPredicate &getUnionPredicate() const;

  /// Returns the SCEV expression of V, in the context of the current SCEV
  /// predicate.  The order of transformations applied on the expression of V
  /// returned by ScalarEvolution is guaranteed to be preserved, even when
  /// adding new predicates.
  const SCEV *getSCEV(Value *V);

  /// Get the (predicated) backedge count for the analyzed loop.
  const SCEV *getBackedgeTakenCount();

  /// Adds a new predicate.
  void addPredicate(const SCEVPredicate &Pred);

  /// Attempts to produce an AddRecExpr for V by adding additional SCEV
  /// predicates. If we can't transform the expression into an AddRecExpr we
  /// return nullptr and not add additional SCEV predicates to the current
  /// context.
  const SCEVAddRecExpr *getAsAddRec(Value *V);

  /// Proves that V doesn't overflow by adding SCEV predicate.
  void setNoOverflow(Value *V, SCEVWrapPredicate::IncrementWrapFlags Flags);

  /// Returns true if we've proved that V doesn't wrap by means of a SCEV
  /// predicate.
  bool hasNoOverflow(Value *V, SCEVWrapPredicate::IncrementWrapFlags Flags);

  /// Returns the ScalarEvolution analysis used.
  ScalarEvolution *getSE() const { return &SE; }

  /// We need to explicitly define the copy constructor because of FlagsMap.
  PredicatedScalarEvolution(const PredicatedScalarEvolution &);

  /// Print the SCEV mappings done by the Predicated Scalar Evolution.
  /// The printed text is indented by \p Depth.
  void print(raw_ostream &OS, unsigned Depth) const;

  /// Check if \p AR1 and \p AR2 are equal, while taking into account
  /// Equal predicates in Preds.
  bool areAddRecsEqualWithPreds(const SCEVAddRecExpr *AR1,
                                const SCEVAddRecExpr *AR2) const;

private:
  /// Increments the version number of the predicate.  This needs to be called
  /// every time the SCEV predicate changes.
  void updateGeneration();

  /// Holds a SCEV and the version number of the SCEV predicate used to
  /// perform the rewrite of the expression.
  using RewriteEntry = std::pair<unsigned, const SCEV *>;

  /// Maps a SCEV to the rewrite result of that SCEV at a certain version
  /// number. If this number doesn't match the current Generation, we will
  /// need to do a rewrite. To preserve the transformation order of previous
  /// rewrites, we will rewrite the previous result instead of the original
  /// SCEV.
  DenseMap<const SCEV *, RewriteEntry> RewriteMap;

  /// Records what NoWrap flags we've added to a Value *.
  ValueMap<Value *, SCEVWrapPredicate::IncrementWrapFlags> FlagsMap;

  /// The ScalarEvolution analysis.
  ScalarEvolution &SE;

  /// The analyzed Loop.
  const Loop &L;

  /// The SCEVPredicate that forms our context. We will rewrite all
  /// expressions assuming that this predicate true.
  SCEVUnionPredicate Preds;

  /// Marks the version of the SCEV predicate used. When rewriting a SCEV
  /// expression we mark it with the version of the predicate. We use this to
  /// figure out if the predicate has changed from the last rewrite of the
  /// SCEV. If so, we need to perform a new rewrite.
  unsigned Generation = 0;

  /// The backedge taken count.
  const SCEV *BackedgeCount = nullptr;
};

} // end namespace llvm

#endif // LLVM_ANALYSIS_SCALAREVOLUTION_H