path: root/ia32/Op.v

(* *********************************************************************)
(*                                                                     *)
(*              The Compcert verified compiler                         *)
(*                                                                     *)
(*          Xavier Leroy, INRIA Paris-Rocquencourt                     *)
(*                                                                     *)
(*  Copyright Institut National de Recherche en Informatique et en     *)
(*  Automatique.  All rights reserved.  This file is distributed       *)
(*  under the terms of the INRIA Non-Commercial License Agreement.     *)
(*                                                                     *)
(* *********************************************************************)

(** Operators and addressing modes.  The abstract syntax and dynamic
  semantics for the CminorSel, RTL, LTL and Mach languages depend on the
  following types, defined in this library:
- [condition]:  boolean conditions for conditional branches;
- [operation]: arithmetic and logical operations;
- [addressing]: addressing modes for load and store operations.

  These types are IA32-specific and correspond roughly to what the 
  processor can compute in one instruction.  In other terms, these
  types reflect the state of the program after instruction selection.
  For a processor-independent set of operations, see the abstract
  syntax and dynamic semantics of the Cminor language.
*)

Require Import Coqlib.
Require Import AST.
Require Import Integers.
Require Import Floats.
Require Import Values.
Require Import Memory.
Require Import Globalenvs.
Require Import Events.

Set Implicit Arguments.

(** Conditions (boolean-valued operators). *)

Inductive condition : Type :=
  | Ccomp: comparison -> condition      (**r signed integer comparison *)
  | Ccompu: comparison -> condition     (**r unsigned integer comparison *)
  | Ccompimm: comparison -> int -> condition (**r signed integer comparison with a constant *)
  | Ccompuimm: comparison -> int -> condition  (**r unsigned integer comparison with a constant *)
  | Ccompf: comparison -> condition     (**r floating-point comparison *)
  | Cnotcompf: comparison -> condition  (**r negation of a floating-point comparison *)
  | Cmaskzero: int -> condition         (**r test [(arg & constant) == 0] *)
  | Cmasknotzero: int -> condition.     (**r test [(arg & constant) != 0] *)

(** Addressing modes.  [r1], [r2], etc, are the arguments to the 
  addressing. *)

Inductive addressing: Type :=
  | Aindexed: int -> addressing         (**r Address is [r1 + offset] *)
  | Aindexed2: int -> addressing        (**r Address is [r1 + r2 + offset] *)
  | Ascaled: int -> int -> addressing   (**r Address is [r1 * scale + offset] *)
  | Aindexed2scaled: int -> int -> addressing
                                   (**r Address is [r1 + r2 * scale + offset] *)
  | Aglobal: ident -> int -> addressing (**r Address is [symbol + offset] *)
  | Abased: ident -> int -> addressing  (**r Address is [symbol + offset + r1] *)
  | Abasedscaled: int -> ident -> int -> addressing  (**r Address is [symbol + offset + r1 * scale] *)
  | Ainstack: int -> addressing.        (**r Address is [stack_pointer + offset] *)

(** Arithmetic and logical operations.  In the descriptions, [rd] is the
  result of the operation and [r1], [r2], etc, are the arguments. *)

Inductive operation : Type :=
  | Omove: operation                    (**r [rd = r1] *)
  | Ointconst: int -> operation         (**r [rd] is set to the given integer constant *)
  | Ofloatconst: float -> operation     (**r [rd] is set to the given float constant *)
  | Oindirectsymbol: ident -> operation (**r [rd] is set to the address of the symbol *)
(*c Integer arithmetic: *)
  | Ocast8signed: operation             (**r [rd] is 8-bit sign extension of [r1] *)
  | Ocast8unsigned: operation           (**r [rd] is 8-bit zero extension of [r1] *)
  | Ocast16signed: operation            (**r [rd] is 16-bit sign extension of [r1] *)
  | Ocast16unsigned: operation          (**r [rd] is 16-bit zero extension of [r1] *)
  | Oneg: operation                     (**r [rd = - r1] *)
  | Osub: operation                     (**r [rd = r1 - r2] *)
  | Omul: operation                     (**r [rd = r1 * r2] *)
  | Omulimm: int -> operation           (**r [rd = r1 * n] *)
  | Odiv: operation                     (**r [rd = r1 / r2] (signed) *)
  | Odivu: operation                    (**r [rd = r1 / r2] (unsigned) *)
  | Omod: operation                     (**r [rd = r1 % r2] (signed) *)
  | Omodu: operation                    (**r [rd = r1 % r2] (unsigned) *)
  | Oand: operation                     (**r [rd = r1 & r2] *)
  | Oandimm: int -> operation           (**r [rd = r1 & n] *)
  | Oor: operation                      (**r [rd = r1 | r2] *)
  | Oorimm: int -> operation            (**r [rd = r1 | n] *)
  | Oxor: operation                     (**r [rd = r1 ^ r2] *)
  | Oxorimm: int -> operation           (**r [rd = r1 ^ n] *)
  | Oshl: operation                     (**r [rd = r1 << r2] *)
  | Oshlimm: int -> operation           (**r [rd = r1 << n] *)
  | Oshr: operation                     (**r [rd = r1 >> r2] (signed) *)
  | Oshrimm: int -> operation           (**r [rd = r1 >> n] (signed) *)
  | Oshrximm: int -> operation          (**r [rd = r1 / 2^n] (signed) *)
  | Oshru: operation                    (**r [rd = r1 >> r2] (unsigned) *)
  | Oshruimm: int -> operation          (**r [rd = r1 >> n] (unsigned) *)
  | Ororimm: int -> operation           (**r rotate right immediate *)
  | Oshldimm: int -> operation          (**r [rd = r1 << n | r2 >> (32-n)] *)
  | Olea: addressing -> operation       (**r effective address *)
(*c Floating-point arithmetic: *)
  | Onegf: operation                    (**r [rd = - r1] *)
  | Oabsf: operation                    (**r [rd = abs(r1)] *)
  | Oaddf: operation                    (**r [rd = r1 + r2] *)
  | Osubf: operation                    (**r [rd = r1 - r2] *)
  | Omulf: operation                    (**r [rd = r1 * r2] *)
  | Odivf: operation                    (**r [rd = r1 / r2] *)
  | Osingleoffloat: operation           (**r [rd] is [r1] truncated to single-precision float *)
(*c Conversions between int and float: *)
  | Ointoffloat: operation              (**r [rd = signed_int_of_float(r1)] *)
  | Ofloatofint: operation              (**r [rd = float_of_signed_int(r1)] *)
(*c Manipulating 64-bit integers: *)
  | Omakelong: operation                (**r [rd = r1 << 32 | r2] *)
  | Olowlong: operation                 (**r [rd = low-word(r1)] *)
  | Ohighlong: operation                (**r [rd = high-word(r1)] *)
(*c Boolean tests: *)
  | Ocmp: condition -> operation.       (**r [rd = 1] if condition holds, [rd = 0] otherwise. *)

(** Derived operators. *)

Definition Oaddrsymbol (id: ident) (ofs: int) : operation := Olea (Aglobal id ofs).
Definition Oaddrstack (ofs: int) : operation := Olea (Ainstack ofs).
Definition Oaddimm (n: int) : operation := Olea (Aindexed n).

(** Comparison functions (used in modules [CSE] and [Allocation]). *)

Definition eq_condition (x y: condition) : {x=y} + {x<>y}.
Proof.
  generalize Int.eq_dec; intro.
  assert (forall (x y: comparison), {x=y}+{x<>y}). decide equality.
  decide equality.
Defined.

Definition eq_addressing (x y: addressing) : {x=y} + {x<>y}.
Proof.
  generalize Int.eq_dec; intro.
  assert (forall (x y: ident), {x=y}+{x<>y}). exact peq.
  decide equality.
Defined.

Definition eq_operation (x y: operation): {x=y} + {x<>y}.
Proof.
  generalize Int.eq_dec; intro.
  generalize Float.eq_dec; intro.
  generalize Int64.eq_dec; intro.
  decide equality.
  apply peq.
  apply eq_addressing.
  apply eq_condition.
Defined.

Global Opaque eq_condition eq_addressing eq_operation.

(** * Evaluation functions *)

Definition symbol_address (F V: Type) (genv: Genv.t F V) (id: ident) (ofs: int) : val :=
  match Genv.find_symbol genv id with
  | Some b => Vptr b ofs
  | None => Vundef
  end.

(** Evaluation of conditions, operators and addressing modes applied
  to lists of values.  Return [None] when the computation can trigger an
  error, e.g. integer division by zero.  [eval_condition] returns a boolean,
  [eval_operation] and [eval_addressing] return a value. *)

Definition eval_condition (cond: condition) (vl: list val) (m: mem): option bool :=
  match cond, vl with
  | Ccomp c, v1 :: v2 :: nil => Val.cmp_bool c v1 v2
  | Ccompu c, v1 :: v2 :: nil => Val.cmpu_bool (Mem.valid_pointer m) c v1 v2
  | Ccompimm c n, v1 :: nil => Val.cmp_bool c v1 (Vint n)
  | Ccompuimm c n, v1 :: nil => Val.cmpu_bool (Mem.valid_pointer m) c v1 (Vint n)
  | Ccompf c, v1 :: v2 :: nil => Val.cmpf_bool c v1 v2
  | Cnotcompf c, v1 :: v2 :: nil => option_map negb (Val.cmpf_bool c v1 v2)
  | Cmaskzero n, Vint n1 :: nil => Some (Int.eq (Int.and n1 n) Int.zero)
  | Cmasknotzero n, Vint n1 :: nil => Some (negb (Int.eq (Int.and n1 n) Int.zero))
  | _, _ => None
  end.

Definition eval_addressing
    (F V: Type) (genv: Genv.t F V) (sp: val)
    (addr: addressing) (vl: list val) : option val :=
  match addr, vl with
  | Aindexed n, v1::nil =>
      Some (Val.add v1 (Vint n))
  | Aindexed2 n, v1::v2::nil =>
      Some (Val.add (Val.add v1 v2) (Vint n))
  | Ascaled sc ofs, v1::nil =>
      Some (Val.add (Val.mul v1 (Vint sc)) (Vint ofs))
  | Aindexed2scaled sc ofs, v1::v2::nil =>
      Some(Val.add v1 (Val.add (Val.mul v2 (Vint sc)) (Vint ofs)))
  | Aglobal s ofs, nil =>
      Some (symbol_address genv s ofs)
  | Abased s ofs, v1::nil =>
      Some (Val.add (symbol_address genv s ofs) v1)
  | Abasedscaled sc s ofs, v1::nil =>
      Some (Val.add (symbol_address genv s ofs) (Val.mul v1 (Vint sc)))
  | Ainstack ofs, nil =>
      Some(Val.add sp (Vint ofs))
  | _, _ => None
  end.

Definition eval_operation
    (F V: Type) (genv: Genv.t F V) (sp: val)
    (op: operation) (vl: list val) (m: mem): option val :=
  match op, vl with
  | Omove, v1::nil => Some v1
  | Ointconst n, nil => Some (Vint n)
  | Ofloatconst n, nil => Some (Vfloat n)
  | Oindirectsymbol id, nil => Some (symbol_address genv id Int.zero)
  | Ocast8signed, v1 :: nil => Some (Val.sign_ext 8 v1)
  | Ocast8unsigned, v1 :: nil => Some (Val.zero_ext 8 v1)
  | Ocast16signed, v1 :: nil => Some (Val.sign_ext 16 v1)
  | Ocast16unsigned, v1 :: nil => Some (Val.zero_ext 16 v1)
  | Oneg, v1::nil => Some (Val.neg v1)
  | Osub, v1::v2::nil => Some (Val.sub v1 v2)
  | Omul, v1::v2::nil => Some (Val.mul v1 v2)
  | Omulimm n, v1::nil => Some (Val.mul v1 (Vint n))
  | Odiv, v1::v2::nil => Val.divs v1 v2
  | Odivu, v1::v2::nil => Val.divu v1 v2
  | Omod, v1::v2::nil => Val.mods v1 v2
  | Omodu, v1::v2::nil => Val.modu v1 v2
  | Oand, v1::v2::nil => Some(Val.and v1 v2)
  | Oandimm n, v1::nil => Some (Val.and v1 (Vint n))
  | Oor, v1::v2::nil => Some(Val.or v1 v2)
  | Oorimm n, v1::nil => Some (Val.or v1 (Vint n))
  | Oxor, v1::v2::nil => Some(Val.xor v1 v2)
  | Oxorimm n, v1::nil => Some (Val.xor v1 (Vint n))
  | Oshl, v1::v2::nil => Some (Val.shl v1 v2)
  | Oshlimm n, v1::nil => Some (Val.shl v1 (Vint n))
  | Oshr, v1::v2::nil => Some (Val.shr v1 v2)
  | Oshrimm n, v1::nil => Some (Val.shr v1 (Vint n))
  | Oshrximm n, v1::nil => Val.shrx v1 (Vint n)
  | Oshru, v1::v2::nil => Some (Val.shru v1 v2)
  | Oshruimm n, v1::nil => Some (Val.shru v1 (Vint n))
  | Ororimm n, v1::nil => Some (Val.ror v1 (Vint n))
  | Oshldimm n, v1::v2::nil => Some (Val.or (Val.shl v1 (Vint n))
                                            (Val.shru v2 (Vint (Int.sub Int.iwordsize n))))
  | Olea addr, _ => eval_addressing genv sp addr vl
  | Onegf, v1::nil => Some(Val.negf v1)
  | Oabsf, v1::nil => Some(Val.absf v1)
  | Oaddf, v1::v2::nil => Some(Val.addf v1 v2)
  | Osubf, v1::v2::nil => Some(Val.subf v1 v2)
  | Omulf, v1::v2::nil => Some(Val.mulf v1 v2)
  | Odivf, v1::v2::nil => Some(Val.divf v1 v2)
  | Osingleoffloat, v1::nil => Some(Val.singleoffloat v1)
  | Ointoffloat, v1::nil => Val.intoffloat v1
  | Ofloatofint, v1::nil => Val.floatofint v1
  | Omakelong, v1::v2::nil => Some(Val.longofwords v1 v2)
  | Olowlong, v1::nil => Some(Val.loword v1)
  | Ohighlong, v1::nil => Some(Val.hiword v1)
  | Ocmp c, _ => Some(Val.of_optbool (eval_condition c vl m))
  | _, _ => None
  end.

Ltac FuncInv :=
  match goal with
  | H: (match ?x with nil => _ | _ :: _ => _ end = Some _) |- _ =>
      destruct x; simpl in H; try discriminate; FuncInv
  | H: (match ?v with Vundef => _ | Vint _ => _ | Vfloat _ => _ | Vptr _ _ => _ end = Some _) |- _ =>
      destruct v; simpl in H; try discriminate; FuncInv
  | H: (Some _ = Some _) |- _ =>
      injection H; intros; clear H; FuncInv
  | _ =>
      idtac
  end.

(** * Static typing of conditions, operators and addressing modes. *)

Definition type_of_condition (c: condition) : list typ :=
  match c with
  | Ccomp _ => Tint :: Tint :: nil
  | Ccompu _ => Tint :: Tint :: nil
  | Ccompimm _ _ => Tint :: nil
  | Ccompuimm _ _ => Tint :: nil
  | Ccompf _ => Tfloat :: Tfloat :: nil
  | Cnotcompf _ => Tfloat :: Tfloat :: nil
  | Cmaskzero _ => Tint :: nil
  | Cmasknotzero _ => Tint :: nil
  end.

Definition type_of_addressing (addr: addressing) : list typ :=
  match addr with
  | Aindexed _ => Tint :: nil
  | Aindexed2 _ => Tint :: Tint :: nil
  | Ascaled _ _ => Tint :: nil
  | Aindexed2scaled _ _ => Tint :: Tint :: nil
  | Aglobal _ _ => nil
  | Abased _ _ => Tint :: nil
  | Abasedscaled _ _ _ => Tint :: nil
  | Ainstack _ => nil
  end.

Definition type_of_operation (op: operation) : list typ * typ :=
  match op with
  | Omove => (nil, Tint)   (* treated specially *)
  | Ointconst _ => (nil, Tint)
  | Ofloatconst f => (nil, if Float.is_single_dec f then Tsingle else Tfloat)
  | Oindirectsymbol _ => (nil, Tint)
  | Ocast8signed => (Tint :: nil, Tint)
  | Ocast8unsigned => (Tint :: nil, Tint)
  | Ocast16signed => (Tint :: nil, Tint)
  | Ocast16unsigned => (Tint :: nil, Tint)
  | Oneg => (Tint :: nil, Tint)
  | Osub => (Tint :: Tint :: nil, Tint)
  | Omul => (Tint :: Tint :: nil, Tint)
  | Omulimm _ => (Tint :: nil, Tint)
  | Odiv => (Tint :: Tint :: nil, Tint)
  | Odivu => (Tint :: Tint :: nil, Tint)
  | Omod => (Tint :: Tint :: nil, Tint)
  | Omodu => (Tint :: Tint :: nil, Tint)
  | Oand => (Tint :: Tint :: nil, Tint)
  | Oandimm _ => (Tint :: nil, Tint)
  | Oor => (Tint :: Tint :: nil, Tint)
  | Oorimm _ => (Tint :: nil, Tint)
  | Oxor => (Tint :: Tint :: nil, Tint)
  | Oxorimm _ => (Tint :: nil, Tint)
  | Oshl => (Tint :: Tint :: nil, Tint)
  | Oshlimm _ => (Tint :: nil, Tint)
  | Oshr => (Tint :: Tint :: nil, Tint)
  | Oshrimm _ => (Tint :: nil, Tint)
  | Oshrximm _ => (Tint :: nil, Tint)
  | Oshru => (Tint :: Tint :: nil, Tint)
  | Oshruimm _ => (Tint :: nil, Tint)
  | Ororimm _ => (Tint :: nil, Tint)
  | Oshldimm _ => (Tint :: Tint :: nil, Tint)
  | Olea addr => (type_of_addressing addr, Tint)
  | Onegf => (Tfloat :: nil, Tfloat)
  | Oabsf => (Tfloat :: nil, Tfloat)
  | Oaddf => (Tfloat :: Tfloat :: nil, Tfloat)
  | Osubf => (Tfloat :: Tfloat :: nil, Tfloat)
  | Omulf => (Tfloat :: Tfloat :: nil, Tfloat)
  | Odivf => (Tfloat :: Tfloat :: nil, Tfloat)
  | Osingleoffloat => (Tfloat :: nil, Tsingle)
  | Ointoffloat => (Tfloat :: nil, Tint)
  | Ofloatofint => (Tint :: nil, Tfloat)
  | Omakelong => (Tint :: Tint :: nil, Tlong)
  | Olowlong => (Tlong :: nil, Tint)
  | Ohighlong => (Tlong :: nil, Tint)
  | Ocmp c => (type_of_condition c, Tint)
  end.

(** Weak type soundness results for [eval_operation]:
  the result values, when defined, are always of the type predicted
  by [type_of_operation]. *)

Section SOUNDNESS.

Variable A V: Type.
Variable genv: Genv.t A V.

Lemma type_of_addressing_sound:
  forall addr vl sp v,
  eval_addressing genv sp addr vl = Some v ->
  Val.has_type v Tint.
Proof with (try exact I).
  intros. destruct addr; simpl in H; FuncInv; subst; simpl.
  destruct v0...
  destruct v0... destruct v1... destruct v1...
  destruct v0...
  destruct v0... destruct v1... destruct v1...
  unfold symbol_address; destruct (Genv.find_symbol genv i)...
  unfold symbol_address; destruct (Genv.find_symbol genv i)...
  unfold symbol_address; destruct (Genv.find_symbol genv i)... destruct v0...
  destruct v0...
  unfold symbol_address; destruct (Genv.find_symbol genv i0)... destruct v0...
  destruct sp...
Qed.

Lemma type_of_operation_sound:
  forall op vl sp v m,
  op <> Omove ->
  eval_operation genv sp op vl m = Some v ->
  Val.has_type v (snd (type_of_operation op)).
Proof with (try exact I).
  intros.
  destruct op; simpl in H0; FuncInv; subst; simpl.
  congruence.
  exact I.
  destruct (Float.is_single_dec f); auto. 
  unfold symbol_address; destruct (Genv.find_symbol genv i)...
  destruct v0...
  destruct v0...
  destruct v0...
  destruct v0...
  destruct v0...
  destruct v0; destruct v1... simpl. destruct (eq_block b b0)...
  destruct v0; destruct v1...
  destruct v0...
  destruct v0; destruct v1; simpl in *; inv H0.
    destruct (Int.eq i0 Int.zero || Int.eq i (Int.repr Int.min_signed) && Int.eq i0 Int.mone); inv H2...
  destruct v0; destruct v1; simpl in *; inv H0. destruct (Int.eq i0 Int.zero); inv H2...
  destruct v0; destruct v1; simpl in *; inv H0.
    destruct (Int.eq i0 Int.zero || Int.eq i (Int.repr Int.min_signed) && Int.eq i0 Int.mone); inv H2...
  destruct v0; destruct v1; simpl in *; inv H0. destruct (Int.eq i0 Int.zero); inv H2...
  destruct v0; destruct v1...
  destruct v0...
  destruct v0; destruct v1...
  destruct v0...
  destruct v0; destruct v1...
  destruct v0...
  destruct v0; destruct v1; simpl... destruct (Int.ltu i0 Int.iwordsize)...
  destruct v0; simpl... destruct (Int.ltu i Int.iwordsize)...
  destruct v0; destruct v1; simpl... destruct (Int.ltu i0 Int.iwordsize)...
  destruct v0; simpl... destruct (Int.ltu i Int.iwordsize)...
  destruct v0; simpl in H0; try discriminate. destruct (Int.ltu i (Int.repr 31)); inv H0...
  destruct v0; destruct v1; simpl... destruct (Int.ltu i0 Int.iwordsize)...
  destruct v0; simpl... destruct (Int.ltu i Int.iwordsize)...
  destruct v0; simpl... destruct (Int.ltu i Int.iwordsize)...
  destruct v0; simpl... destruct (Int.ltu i Int.iwordsize)...
  destruct v1; simpl... destruct (Int.ltu (Int.sub Int.iwordsize i) Int.iwordsize)...
  eapply type_of_addressing_sound; eauto.
  destruct v0...
  destruct v0...
  destruct v0; destruct v1...
  destruct v0; destruct v1...
  destruct v0; destruct v1...
  destruct v0; destruct v1...
  destruct v0... apply Float.singleoffloat_is_single.
  destruct v0; simpl in H0; inv H0. destruct (Float.intoffloat f); inv H2...
  destruct v0; simpl in H0; inv H0...
  destruct v0; destruct v1...
  destruct v0...
  destruct v0...
  destruct (eval_condition c vl m); simpl... destruct b... 
Qed.

End SOUNDNESS.

(** * Manipulating and transforming operations *)

(** Recognition of move operations. *)

Definition is_move_operation
    (A: Type) (op: operation) (args: list A) : option A :=
  match op, args with
  | Omove, arg :: nil => Some arg
  | _, _ => None
  end.

Lemma is_move_operation_correct:
  forall (A: Type) (op: operation) (args: list A) (a: A),
  is_move_operation op args = Some a ->
  op = Omove /\ args = a :: nil.
Proof.
  intros until a. unfold is_move_operation; destruct op;
  try (intros; discriminate).
  destruct args. intros; discriminate.
  destruct args. intros. intuition congruence. 
  intros; discriminate.
Qed.

(** [negate_condition cond] returns a condition that is logically
  equivalent to the negation of [cond]. *)

Definition negate_condition (cond: condition): condition :=
  match cond with
  | Ccomp c => Ccomp(negate_comparison c)
  | Ccompu c => Ccompu(negate_comparison c)
  | Ccompimm c n => Ccompimm (negate_comparison c) n
  | Ccompuimm c n => Ccompuimm (negate_comparison c) n
  | Ccompf c => Cnotcompf c
  | Cnotcompf c => Ccompf c
  | Cmaskzero n => Cmasknotzero n
  | Cmasknotzero n => Cmaskzero n
  end.

Lemma eval_negate_condition:
  forall cond vl m,
  eval_condition (negate_condition cond) vl m = option_map negb (eval_condition cond vl m).
Proof.
  intros. destruct cond; simpl.
  repeat (destruct vl; auto). apply Val.negate_cmp_bool.
  repeat (destruct vl; auto). apply Val.negate_cmpu_bool.
  repeat (destruct vl; auto). apply Val.negate_cmp_bool.
  repeat (destruct vl; auto). apply Val.negate_cmpu_bool.
  repeat (destruct vl; auto). 
  repeat (destruct vl; auto). destruct (Val.cmpf_bool c v v0); auto. destruct b; auto.
  destruct vl; auto. destruct v; auto. destruct vl; auto. 
  destruct vl; auto. destruct v; auto. destruct vl; auto. simpl. rewrite negb_involutive. auto.
Qed.

(** Shifting stack-relative references.  This is used in [Stacking]. *)

Definition shift_stack_addressing (delta: int) (addr: addressing) :=
  match addr with
  | Ainstack ofs => Ainstack (Int.add delta ofs)
  | _ => addr
  end.

Definition shift_stack_operation (delta: int) (op: operation) :=
  match op with
  | Olea addr => Olea (shift_stack_addressing delta addr)
  | _ => op
  end.

Lemma type_shift_stack_addressing:
  forall delta addr, type_of_addressing (shift_stack_addressing delta addr) = type_of_addressing addr.
Proof.
  intros. destruct addr; auto. 
Qed.

Lemma type_shift_stack_operation:
  forall delta op, type_of_operation (shift_stack_operation delta op) = type_of_operation op.
Proof.
  intros. destruct op; auto. simpl. decEq. apply type_shift_stack_addressing. 
Qed.

Lemma eval_shift_stack_addressing:
  forall F V (ge: Genv.t F V) sp addr vl delta,
  eval_addressing ge sp (shift_stack_addressing delta addr) vl =
  eval_addressing ge (Val.add sp (Vint delta)) addr vl.
Proof.
  intros. destruct addr; simpl; auto.
  rewrite Val.add_assoc. simpl. auto.
Qed.

Lemma eval_shift_stack_operation:
  forall F V (ge: Genv.t F V) sp op vl m delta,
  eval_operation ge sp (shift_stack_operation delta op) vl m =
  eval_operation ge (Val.add sp (Vint delta)) op vl m.
Proof.
  intros. destruct op; simpl; auto.
  apply eval_shift_stack_addressing. 
Qed.

(** Offset an addressing mode [addr] by a quantity [delta], so that
  it designates the pointer [delta] bytes past the pointer designated
  by [addr].  May be undefined, in which case [None] is returned. *)

Definition offset_addressing (addr: addressing) (delta: int) : option addressing :=
  match addr with
  | Aindexed n => Some(Aindexed (Int.add n delta))
  | Aindexed2 n => Some(Aindexed2 (Int.add n delta))
  | Ascaled sc n => Some(Ascaled sc (Int.add n delta))
  | Aindexed2scaled sc n => Some(Aindexed2scaled sc (Int.add n delta))
  | Aglobal s n => Some(Aglobal s (Int.add n delta))
  | Abased s n => Some(Abased s (Int.add n delta))
  | Abasedscaled sc s n => Some(Abasedscaled sc s (Int.add n delta))
  | Ainstack n => Some(Ainstack (Int.add n delta))
  end.

Lemma eval_offset_addressing:
  forall (F V: Type) (ge: Genv.t F V) sp addr args delta addr' v,
  offset_addressing addr delta = Some addr' ->
  eval_addressing ge sp addr args = Some v ->
  eval_addressing ge sp addr' args = Some(Val.add v (Vint delta)).
Proof.
  intros. destruct addr; simpl in H; inv H; simpl in *; FuncInv; inv H.
  rewrite Val.add_assoc; auto.
  rewrite !Val.add_assoc; auto.
  rewrite !Val.add_assoc; auto.
  rewrite !Val.add_assoc; auto.
  unfold symbol_address. destruct (Genv.find_symbol ge i); auto. 
  unfold symbol_address. destruct (Genv.find_symbol ge i); auto.
  rewrite Val.add_assoc. rewrite Val.add_permut. rewrite Val.add_commut. auto. 
  unfold symbol_address. destruct (Genv.find_symbol ge i0); auto.
  rewrite Val.add_assoc. rewrite Val.add_permut. rewrite Val.add_commut. auto. 
  rewrite Val.add_assoc. auto. 
Qed.

(** Operations that are so cheap to recompute that CSE should not factor them out. *)

Definition is_trivial_op (op: operation) : bool :=
  match op with
  | Omove => true
  | Ointconst _ => true
  | Olea (Aglobal _ _) => true
  | Olea (Ainstack _) => true
  | _ => false
  end.

(** Operations that depend on the memory state. *)

Definition op_depends_on_memory (op: operation) : bool :=
  match op with
  | Ocmp (Ccompu _) => true
  | _ => false
  end.

Lemma op_depends_on_memory_correct:
  forall (F V: Type) (ge: Genv.t F V) sp op args m1 m2,
  op_depends_on_memory op = false ->
  eval_operation ge sp op args m1 = eval_operation ge sp op args m2.
Proof.
  intros until m2. destruct op; simpl; try congruence.
  destruct c; simpl; try congruence. reflexivity. 
Qed.

(** Checking whether two addressings, applied to the same arguments, produce
  separated memory addresses.  Used in [CSE]. *)

Definition addressing_separated (chunk1: memory_chunk) (addr1: addressing)
                               (chunk2: memory_chunk) (addr2: addressing) : bool :=
  match addr1, addr2 with
  | Aindexed ofs1, Aindexed ofs2 => 
      Int.no_overlap ofs1 (size_chunk chunk1) ofs2 (size_chunk chunk2)
  | Aglobal s1 ofs1, Aglobal s2 ofs2 => 
      if ident_eq s1 s2 then Int.no_overlap ofs1 (size_chunk chunk1) ofs2 (size_chunk chunk2) else true
  | Abased s1 ofs1, Abased s2 ofs2 => 
      if ident_eq s1 s2 then Int.no_overlap ofs1 (size_chunk chunk1) ofs2 (size_chunk chunk2) else true
  | Ainstack ofs1, Ainstack ofs2 =>
      Int.no_overlap ofs1 (size_chunk chunk1) ofs2 (size_chunk chunk2)
  | _, _ => false
  end.

Lemma addressing_separated_sound:
  forall (F V: Type) (ge: Genv.t F V) sp chunk1 addr1 chunk2 addr2 vl b1 n1 b2 n2,
  addressing_separated chunk1 addr1 chunk2 addr2 = true ->
  eval_addressing ge sp addr1 vl = Some(Vptr b1 n1) ->
  eval_addressing ge sp addr2 vl = Some(Vptr b2 n2) ->
  b1 <> b2 \/ Int.unsigned n1 + size_chunk chunk1 <= Int.unsigned n2 \/ Int.unsigned n2 + size_chunk chunk2 <= Int.unsigned n1.
Proof.
  unfold addressing_separated; intros.
  generalize (size_chunk_pos chunk1) (size_chunk_pos chunk2); intros SZ1 SZ2.
  destruct addr1; destruct addr2; try discriminate; simpl in *; FuncInv.
(* Aindexed *)
  destruct v; simpl in *; inv H1; inv H2.
  right. apply Int.no_overlap_sound; auto. 
(* Aglobal *)
  unfold symbol_address in *. 
  destruct (Genv.find_symbol ge i1) eqn:?; inv H2.
  destruct (Genv.find_symbol ge i) eqn:?; inv H1.
  destruct (ident_eq i i1). subst.
  replace (Int.unsigned n1) with (Int.unsigned (Int.add Int.zero n1)).
  replace (Int.unsigned n2) with (Int.unsigned (Int.add Int.zero n2)).
  right. apply Int.no_overlap_sound; auto. 
  rewrite Int.add_commut; rewrite Int.add_zero; auto.
  rewrite Int.add_commut; rewrite Int.add_zero; auto.
  left. red; intros; elim n. subst. eapply Genv.genv_vars_inj; eauto.
(* Abased *)
  unfold symbol_address in *. 
  destruct (Genv.find_symbol ge i1) eqn:?; simpl in *; try discriminate.
  destruct v; inv H2.
  destruct (Genv.find_symbol ge i) eqn:?; inv H1.
  destruct (ident_eq i i1). subst.
  rewrite (Int.add_commut i0 i3). rewrite (Int.add_commut i2 i3).
  right. apply Int.no_overlap_sound; auto. 
  left. red; intros; elim n. subst. eapply Genv.genv_vars_inj; eauto.
(* Ainstack *)
  destruct sp; simpl in *; inv H1; inv H2.
  right. apply Int.no_overlap_sound; auto. 
Qed.

(** * Invariance and compatibility properties. *)

(** [eval_operation] and [eval_addressing] depend on a global environment
  for resolving references to global symbols.  We show that they give
  the same results if a global environment is replaced by another that
  assigns the same addresses to the same symbols. *)

Section GENV_TRANSF.

Variable F1 F2 V1 V2: Type.
Variable ge1: Genv.t F1 V1.
Variable ge2: Genv.t F2 V2.
Hypothesis agree_on_symbols:
  forall (s: ident), Genv.find_symbol ge2 s = Genv.find_symbol ge1 s.

Lemma eval_addressing_preserved:
  forall sp addr vl,
  eval_addressing ge2 sp addr vl = eval_addressing ge1 sp addr vl.
Proof.
  intros.
  unfold eval_addressing, symbol_address; destruct addr; try rewrite agree_on_symbols;
  reflexivity.
Qed.

Lemma eval_operation_preserved:
  forall sp op vl m,
  eval_operation ge2 sp op vl m = eval_operation ge1 sp op vl m.
Proof.
  intros.
  unfold eval_operation; destruct op; auto.
  unfold symbol_address. rewrite agree_on_symbols. auto.
  apply eval_addressing_preserved.
Qed.

End GENV_TRANSF.

(** Compatibility of the evaluation functions with value injections. *)

Section EVAL_COMPAT.

Variable F V: Type.
Variable genv: Genv.t F V.
Variable f: meminj.

Hypothesis symbol_address_inj: 
  forall id ofs,
  val_inject f (symbol_address genv id ofs) (symbol_address genv id ofs).

Variable m1: mem.
Variable m2: mem.

Hypothesis valid_pointer_inj:
  forall b1 ofs b2 delta,
  f b1 = Some(b2, delta) ->
  Mem.valid_pointer m1 b1 (Int.unsigned ofs) = true ->
  Mem.valid_pointer m2 b2 (Int.unsigned (Int.add ofs (Int.repr delta))) = true.

Hypothesis weak_valid_pointer_inj:
  forall b1 ofs b2 delta,
  f b1 = Some(b2, delta) ->
  Mem.weak_valid_pointer m1 b1 (Int.unsigned ofs) = true ->
  Mem.weak_valid_pointer m2 b2 (Int.unsigned (Int.add ofs (Int.repr delta))) = true.

Hypothesis weak_valid_pointer_no_overflow:
  forall b1 ofs b2 delta,
  f b1 = Some(b2, delta) ->
  Mem.weak_valid_pointer m1 b1 (Int.unsigned ofs) = true ->
  0 <= Int.unsigned ofs + Int.unsigned (Int.repr delta) <= Int.max_unsigned.

Hypothesis valid_different_pointers_inj:
  forall b1 ofs1 b2 ofs2 b1' delta1 b2' delta2,
  b1 <> b2 ->
  Mem.valid_pointer m1 b1 (Int.unsigned ofs1) = true ->
  Mem.valid_pointer m1 b2 (Int.unsigned ofs2) = true ->
  f b1 = Some (b1', delta1) ->
  f b2 = Some (b2', delta2) ->
  b1' <> b2' \/
  Int.unsigned (Int.add ofs1 (Int.repr delta1)) <> Int.unsigned (Int.add ofs2 (Int.repr delta2)).

Ltac InvInject :=
  match goal with
  | [ H: val_inject _ (Vint _) _ |- _ ] =>
      inv H; InvInject
  | [ H: val_inject _ (Vfloat _) _ |- _ ] =>
      inv H; InvInject
  | [ H: val_inject _ (Vptr _ _) _ |- _ ] =>
      inv H; InvInject
  | [ H: val_list_inject _ nil _ |- _ ] =>
      inv H; InvInject
  | [ H: val_list_inject _ (_ :: _) _ |- _ ] =>
      inv H; InvInject
  | _ => idtac
  end.

Lemma eval_condition_inj:
  forall cond vl1 vl2 b,
  val_list_inject f vl1 vl2 ->
  eval_condition cond vl1 m1 = Some b ->
  eval_condition cond vl2 m2 = Some b.
Proof.
  intros. destruct cond; simpl in H0; FuncInv; InvInject; simpl; auto.
  inv H3; inv H2; simpl in H0; inv H0; auto.
  eauto 3 using val_cmpu_bool_inject, Mem.valid_pointer_implies.
  inv H3; simpl in H0; inv H0; auto.
  eauto 3 using val_cmpu_bool_inject, Mem.valid_pointer_implies.
  inv H3; inv H2; simpl in H0; inv H0; auto.
  inv H3; inv H2; simpl in H0; inv H0; auto.
  inv H3; try discriminate; inv H5; auto.
  inv H3; try discriminate; inv H5; auto.
Qed.

Ltac TrivialExists :=
  match goal with
  | [ |- exists v2, Some ?v1 = Some v2 /\ val_inject _ _ v2 ] =>
      exists v1; split; auto
  | _ => idtac
  end.

Lemma eval_addressing_inj:
  forall addr sp1 vl1 sp2 vl2 v1,
  val_inject f sp1 sp2 ->
  val_list_inject f vl1 vl2 ->
  eval_addressing genv sp1 addr vl1 = Some v1 ->
  exists v2, eval_addressing genv sp2 addr vl2 = Some v2 /\ val_inject f v1 v2.
Proof.
  intros. destruct addr; simpl in H1; simpl; FuncInv; InvInject; TrivialExists.
  apply Values.val_add_inject; auto.
  apply Values.val_add_inject; auto. apply Values.val_add_inject; auto.
  apply Values.val_add_inject; auto. inv H4; simpl; auto.
  apply Values.val_add_inject; auto. apply Values.val_add_inject; auto. inv H2; simpl; auto.
  apply Values.val_add_inject; auto.
  apply Values.val_add_inject; auto. inv H4; simpl; auto.
  apply Values.val_add_inject; auto.
Qed.

Lemma eval_operation_inj:
  forall op sp1 vl1 sp2 vl2 v1,
  val_inject f sp1 sp2 ->
  val_list_inject f vl1 vl2 ->
  eval_operation genv sp1 op vl1 m1 = Some v1 ->
  exists v2, eval_operation genv sp2 op vl2 m2 = Some v2 /\ val_inject f v1 v2.
Proof.
  intros. destruct op; simpl in H1; simpl; FuncInv; InvInject; TrivialExists.
  inv H4; simpl; auto.
  inv H4; simpl; auto.
  inv H4; simpl; auto.
  inv H4; simpl; auto.
  inv H4; simpl; auto.
  inv H4; inv H2; simpl; auto. econstructor; eauto. 
    rewrite Int.sub_add_l. auto.
    destruct (eq_block b1 b0); auto. subst. rewrite H1 in H0. inv H0. rewrite dec_eq_true. 
    rewrite Int.sub_shifted. auto.
  inv H4; inv H2; simpl; auto.
  inv H4; simpl; auto.
  inv H4; inv H3; simpl in H1; inv H1. simpl.
    destruct (Int.eq i0 Int.zero || Int.eq i (Int.repr Int.min_signed) && Int.eq i0 Int.mone); inv H2. TrivialExists.
  inv H4; inv H3; simpl in H1; inv H1. simpl. 
    destruct (Int.eq i0 Int.zero); inv H2. TrivialExists.
  inv H4; inv H3; simpl in H1; inv H1. simpl.
    destruct (Int.eq i0 Int.zero || Int.eq i (Int.repr Int.min_signed) && Int.eq i0 Int.mone); inv H2. TrivialExists.
  inv H4; inv H3; simpl in H1; inv H1. simpl. 
    destruct (Int.eq i0 Int.zero); inv H2. TrivialExists.
  inv H4; inv H2; simpl; auto.
  inv H4; simpl; auto.
  inv H4; inv H2; simpl; auto.
  inv H4; simpl; auto.
  inv H4; inv H2; simpl; auto.
  inv H4; simpl; auto.
  inv H4; inv H2; simpl; auto. destruct (Int.ltu i0 Int.iwordsize); auto.
  inv H4; simpl; auto. destruct (Int.ltu i Int.iwordsize); auto.
  inv H4; inv H2; simpl; auto. destruct (Int.ltu i0 Int.iwordsize); auto.
  inv H4; simpl; auto. destruct (Int.ltu i Int.iwordsize); auto.
  inv H4; simpl in H1; try discriminate. simpl. 
  destruct (Int.ltu i (Int.repr 31)); inv H1. TrivialExists.
  inv H4; inv H2; simpl; auto. destruct (Int.ltu i0 Int.iwordsize); auto.
  inv H4; simpl; auto. destruct (Int.ltu i Int.iwordsize); auto.
  inv H4; simpl; auto. destruct (Int.ltu i Int.iwordsize); auto.
  inv H4; simpl; auto. destruct (Int.ltu i Int.iwordsize); auto.
  inv H2; simpl; auto. destruct (Int.ltu (Int.sub Int.iwordsize i) Int.iwordsize); auto.
  eapply eval_addressing_inj; eauto. 
  inv H4; simpl; auto.
  inv H4; simpl; auto.
  inv H4; inv H2; simpl; auto.
  inv H4; inv H2; simpl; auto.
  inv H4; inv H2; simpl; auto.
  inv H4; inv H2; simpl; auto.
  inv H4; simpl; auto.
  inv H4; simpl in H1; inv H1. simpl. destruct (Float.intoffloat f0); simpl in H2; inv H2.
  exists (Vint i); auto.
  inv H4; simpl in H1; inv H1. simpl. TrivialExists.
  inv H4; inv H2; simpl; auto.
  inv H4; simpl; auto.
  inv H4; simpl; auto.
  subst v1. destruct (eval_condition c vl1 m1) eqn:?.
  exploit eval_condition_inj; eauto. intros EQ; rewrite EQ.
  destruct b; simpl; constructor.
  simpl; constructor.
Qed.

End EVAL_COMPAT.

(** Compatibility of the evaluation functions with the ``is less defined'' relation over values. *)

Section EVAL_LESSDEF.

Variable F V: Type.
Variable genv: Genv.t F V.

Remark valid_pointer_extends:
  forall m1 m2, Mem.extends m1 m2 ->
  forall b1 ofs b2 delta,
  Some(b1, 0) = Some(b2, delta) ->
  Mem.valid_pointer m1 b1 (Int.unsigned ofs) = true ->
  Mem.valid_pointer m2 b2 (Int.unsigned (Int.add ofs (Int.repr delta))) = true.
Proof.
  intros. inv H0. rewrite Int.add_zero. eapply Mem.valid_pointer_extends; eauto. 
Qed.

Remark weak_valid_pointer_extends:
  forall m1 m2, Mem.extends m1 m2 ->
  forall b1 ofs b2 delta,
  Some(b1, 0) = Some(b2, delta) ->
  Mem.weak_valid_pointer m1 b1 (Int.unsigned ofs) = true ->
  Mem.weak_valid_pointer m2 b2 (Int.unsigned (Int.add ofs (Int.repr delta))) = true.
Proof.
  intros. inv H0. rewrite Int.add_zero. eapply Mem.weak_valid_pointer_extends; eauto. 
Qed.

Remark weak_valid_pointer_no_overflow_extends:
  forall m1 b1 ofs b2 delta,
  Some(b1, 0) = Some(b2, delta) ->
  Mem.weak_valid_pointer m1 b1 (Int.unsigned ofs) = true ->
  0 <= Int.unsigned ofs + Int.unsigned (Int.repr delta) <= Int.max_unsigned.
Proof.
  intros. inv H. rewrite Zplus_0_r. apply Int.unsigned_range_2.
Qed.

Remark valid_different_pointers_extends:
  forall m1 b1 ofs1 b2 ofs2 b1' delta1 b2' delta2,
  b1 <> b2 ->
  Mem.valid_pointer m1 b1 (Int.unsigned ofs1) = true ->
  Mem.valid_pointer m1 b2 (Int.unsigned ofs2) = true ->
  Some(b1, 0) = Some (b1', delta1) ->
  Some(b2, 0) = Some (b2', delta2) ->
  b1' <> b2' \/
  Int.unsigned(Int.add ofs1 (Int.repr delta1)) <> Int.unsigned(Int.add ofs2 (Int.repr delta2)).
Proof.
  intros. inv H2; inv H3. auto.
Qed.

Lemma eval_condition_lessdef:
  forall cond vl1 vl2 b m1 m2,
  Val.lessdef_list vl1 vl2 ->
  Mem.extends m1 m2 ->
  eval_condition cond vl1 m1 = Some b ->
  eval_condition cond vl2 m2 = Some b.
Proof.
  intros. eapply eval_condition_inj with (f := fun b => Some(b, 0)) (m1 := m1).
  apply valid_pointer_extends; auto.
  apply weak_valid_pointer_extends; auto.
  apply weak_valid_pointer_no_overflow_extends.
  apply valid_different_pointers_extends; auto.
  rewrite <- val_list_inject_lessdef. eauto. auto.
Qed.

Lemma eval_operation_lessdef:
  forall sp op vl1 vl2 v1 m1 m2,
  Val.lessdef_list vl1 vl2 ->
  Mem.extends m1 m2 ->
  eval_operation genv sp op vl1 m1 = Some v1 ->
  exists v2, eval_operation genv sp op vl2 m2 = Some v2 /\ Val.lessdef v1 v2.
Proof.
  intros. rewrite val_list_inject_lessdef in H.
  assert (exists v2 : val,
          eval_operation genv sp op vl2 m2 = Some v2
          /\ val_inject (fun b => Some(b, 0)) v1 v2).
  eapply eval_operation_inj with (m1 := m1) (sp1 := sp).
  intros. rewrite <- val_inject_lessdef; auto.
  apply valid_pointer_extends; auto.
  apply weak_valid_pointer_extends; auto.
  apply weak_valid_pointer_no_overflow_extends.
  apply valid_different_pointers_extends; auto.
  rewrite <- val_inject_lessdef; auto.
  eauto. auto. 
  destruct H2 as [v2 [A B]]. exists v2; split; auto. rewrite val_inject_lessdef; auto. 
Qed.

Lemma eval_addressing_lessdef:
  forall sp addr vl1 vl2 v1,
  Val.lessdef_list vl1 vl2 ->
  eval_addressing genv sp addr vl1 = Some v1 ->
  exists v2, eval_addressing genv sp addr vl2 = Some v2 /\ Val.lessdef v1 v2.
Proof.
  intros. rewrite val_list_inject_lessdef in H.
  assert (exists v2 : val,
          eval_addressing genv sp addr vl2 = Some v2
          /\ val_inject (fun b => Some(b, 0)) v1 v2).
  eapply eval_addressing_inj with (sp1 := sp).
  intros. rewrite <- val_inject_lessdef; auto.
  rewrite <- val_inject_lessdef; auto.
  eauto. auto. 
  destruct H1 as [v2 [A B]]. exists v2; split; auto. rewrite val_inject_lessdef; auto. 
Qed.

End EVAL_LESSDEF.

(** Compatibility of the evaluation functions with memory injections. *)

Section EVAL_INJECT.

Variable F V: Type.
Variable genv: Genv.t F V.
Variable f: meminj.
Hypothesis globals: meminj_preserves_globals genv f.
Variable sp1: block.
Variable sp2: block.
Variable delta: Z.
Hypothesis sp_inj: f sp1 = Some(sp2, delta).

Remark symbol_address_inject:
  forall id ofs, val_inject f (symbol_address genv id ofs) (symbol_address genv id ofs).
Proof.
  intros. unfold symbol_address. destruct (Genv.find_symbol genv id) eqn:?; auto.
  exploit (proj1 globals); eauto. intros. 
  econstructor; eauto. rewrite Int.add_zero; auto.
Qed.

Lemma eval_condition_inject:
  forall cond vl1 vl2 b m1 m2,
  val_list_inject f vl1 vl2 ->
  Mem.inject f m1 m2 ->
  eval_condition cond vl1 m1 = Some b ->
  eval_condition cond vl2 m2 = Some b.
Proof.
  intros. eapply eval_condition_inj with (f := f) (m1 := m1); eauto.
  intros; eapply Mem.valid_pointer_inject_val; eauto.
  intros; eapply Mem.weak_valid_pointer_inject_val; eauto.
  intros; eapply Mem.weak_valid_pointer_inject_no_overflow; eauto.
  intros; eapply Mem.different_pointers_inject; eauto.
Qed.

Lemma eval_addressing_inject:
  forall addr vl1 vl2 v1,
  val_list_inject f vl1 vl2 ->
  eval_addressing genv (Vptr sp1 Int.zero) addr vl1 = Some v1 ->
  exists v2, 
     eval_addressing genv (Vptr sp2 Int.zero) (shift_stack_addressing (Int.repr delta) addr) vl2 = Some v2
  /\ val_inject f v1 v2.
Proof.
  intros. 
  rewrite eval_shift_stack_addressing. simpl.
  eapply eval_addressing_inj with (sp1 := Vptr sp1 Int.zero); eauto.
  exact symbol_address_inject.
Qed.

Lemma eval_operation_inject:
  forall op vl1 vl2 v1 m1 m2,
  val_list_inject f vl1 vl2 ->
  Mem.inject f m1 m2 ->
  eval_operation genv (Vptr sp1 Int.zero) op vl1 m1 = Some v1 ->
  exists v2,
     eval_operation genv (Vptr sp2 Int.zero) (shift_stack_operation (Int.repr delta) op) vl2 m2 = Some v2
  /\ val_inject f v1 v2.
Proof.
  intros. 
  rewrite eval_shift_stack_operation. simpl.
  eapply eval_operation_inj with (sp1 := Vptr sp1 Int.zero) (m1 := m1); eauto.
  exact symbol_address_inject.
  intros; eapply Mem.valid_pointer_inject_val; eauto.
  intros; eapply Mem.weak_valid_pointer_inject_val; eauto.
  intros; eapply Mem.weak_valid_pointer_inject_no_overflow; eauto.
  intros; eapply Mem.different_pointers_inject; eauto.
Qed.

End EVAL_INJECT.