text: //===- 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/DenseSet.h"
  #include "llvm/ADT/FoldingSet.h"
  #include "llvm/ADT/SetVector.h"
  #include "llvm/Analysis/LoopInfo.h"
  #include "llvm/IR/ConstantRange.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/DataTypes.h"
  
  namespace llvm {
  class APInt;
  class AssumptionCache;
  class Constant;
  class ConstantInt;
  class DominatorTree;
  class Type;
  class ScalarEvolution;
  class DataLayout;
  class TargetLibraryInfo;
  class LLVMContext;
  class Operator;
  class SCEV;
  class SCEVAddRecExpr;
  class SCEVConstant;
  class SCEVExpander;
  class SCEVPredicate;
  class SCEVUnknown;
  class Function;
  
  template <> struct FoldingSetTrait<SCEV>;
  template <> struct FoldingSetTrait<SCEVPredicate>;
  
  /// 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;
  
  private:
    SCEV(const SCEV &) = delete;
    void operator=(const SCEV &) = delete;
  
  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), SubclassData(0) {}
  
    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:
    typedef DenseMap<const SCEV *, SmallVector<const SCEVPredicate *, 4>>
        PredicateMap;
  
    /// 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;
    }
  };
  
  /// 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);
    }
  
  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 typedef for HasRecMap.
    ///
    typedef DenseMap<const SCEV *, bool> HasRecMapType;
  
    /// This is a cache to record whether a SCEV contains any scAddRecExpr.
    HasRecMapType HasRecMap;
  
    /// The typedef for ExprValueMap.
    ///
    typedef std::pair<Value *, ConstantInt *> ValueOffsetPair;
    typedef DenseMap<const SCEV *, SetVector<ValueOffsetPair>> ExprValueMapType;
  
    /// 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 typedef for ValueExprMap.
    ///
    typedef DenseMap<SCEVCallbackVH, const SCEV *, DenseMapInfo<Value *>>
        ValueExprMapType;
  
    /// 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;
  
    /// Set to true by isLoopBackedgeGuardedByCond when we're walking the set of
    /// conditions dominating the backedge of a loop.
    bool WalkingBEDominatingConds;
  
    /// Set to true by isKnownPredicateViaSplitting when we're trying to prove a
    /// predicate by splitting it into a set of independent predicates.
    bool ProvingSplitPredicate;
  
    /// Memoized values for the GetMinTrailingZeros
    DenseMap<const SCEV *, uint32_t> MinTrailingZerosCache;
  
    /// 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
      bool MaxOrZero; // Not taken either exactly MaxNotTaken or zero times
  
      /// 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);
      }
  
      /// 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;
      bool hasAlwaysTruePredicate() const {
        return !Predicate || Predicate->isAlwaysTrue();
      }
  
      explicit ExitNotTakenInfo(PoisoningVH<BasicBlock> ExitingBlock,
                                const SCEV *ExactNotTaken,
                                std::unique_ptr<SCEVUnionPredicate> Predicate)
          : ExitingBlock(ExitingBlock), ExactNotTaken(ExactNotTaken),
            Predicate(std::move(Predicate)) {}
    };
  
    /// 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;
  
      /// \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), MaxOrZero(false) {}
  
      BackedgeTakenInfo(BackedgeTakenInfo &&) = default;
      BackedgeTakenInfo &operator=(BackedgeTakenInfo &&) = default;
  
      typedef std::pair<BasicBlock *, ExitLimit> EdgeExitInfo;
  
      /// 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(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,
    /// TBB, and FBB.
    ///
    /// \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,
                                       BasicBlock *TBB, BasicBlock *FBB,
                                       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, TBB, FBB, 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;
      BasicBlock *TBB;
      BasicBlock *FBB;
      bool AllowPredicates;
  
    public:
      ExitLimitCache(const Loop *L, BasicBlock *TBB, BasicBlock *FBB,
                     bool AllowPredicates)
          : L(L), TBB(TBB), FBB(FBB), AllowPredicates(AllowPredicates) {}
  
      Optional<ExitLimit> find(const Loop *L, Value *ExitCond, BasicBlock *TBB,
                               BasicBlock *FBB, bool ControlsExit,
                               bool AllowPredicates);
  
      void insert(const Loop *L, Value *ExitCond, BasicBlock *TBB,
                  BasicBlock *FBB, bool ControlsExit, bool AllowPredicates,
                  const ExitLimit &EL);
    };
  
    typedef ExitLimitCache ExitLimitCacheTy;
    ExitLimit computeExitLimitFromCondCached(ExitLimitCacheTy &Cache,
                                             const Loop *L, Value *ExitCond,
                                             BasicBlock *TBB, BasicBlock *FBB,
                                             bool ControlsExit,
                                             bool AllowPredicates);
    ExitLimit computeExitLimitFromCondImpl(ExitLimitCacheTy &Cache, const Loop *L,
                                           Value *ExitCond, BasicBlock *TBB,
                                           BasicBlock *FBB, 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, TBB, and FBB. 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,
                                       BasicBlock *TBB, BasicBlock *FBB,
                                       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 isKnownViaSimpleReasoning(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);
  
  public:
    ScalarEvolution(Function &F, TargetLibraryInfo &TLI, AssumptionCache &AC,
                    DominatorTree &DT, LoopInfo &LI);
    ~ScalarEvolution();
    ScalarEvolution(ScalarEvolution &&Arg);
  
    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);
  
    /// Return the Value set from which the SCEV expr is generated.
    SetVector<ValueOffsetPair> *getSCEVValues(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 *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 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);
  
    /// 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);
  
    /// 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:
    /// 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);
  
    /// 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);
  
  private:
    FoldingSet<SCEV> UniqueSCEVs;
    FoldingSet<SCEVPredicate> UniquePreds;
    BumpPtrAllocator SCEVAllocator;
  
    /// 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;
  };
  
  /// Analysis pass that exposes the \c ScalarEvolution for a function.
  class ScalarEvolutionAnalysis
      : public AnalysisInfoMixin<ScalarEvolutionAnalysis> {
    friend AnalysisInfoMixin<ScalarEvolutionAnalysis>;
    static AnalysisKey Key;
  
  public:
    typedef ScalarEvolution Result;
  
    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;
  
  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.
    typedef std::pair<unsigned, const SCEV *> RewriteEntry;
  
    /// 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;
  
    /// The backedge taken count.
    const SCEV *BackedgeCount;
  };
  }
  
  #endif
  