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

Set Implicit Arguments.


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


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


Inductive operation : Type :=
  | Omove: operation (* rd = r1 *)
  | Ointconst: int -> operation (* rd is set to the given integer constant *)
  | Ofloatconst: float -> operation (* rd is set to the given float constant *)
  | Ocast8signed: operation (* rd is 8-bit sign extension of r1 *)
  | Ocast8unsigned: operation (* rd is 8-bit zero extension of r1 *)
  | Ocast16signed: operation (* rd is 16-bit sign extension of r1 *)
  | Ocast16unsigned: operation (* rd is 16-bit zero extension of r1 *)
  | Oneg: operation (* rd = - r1 *)
  | Osub: operation (* rd = r1 - r2 *)
  | Omul: operation (* rd = r1 * r2 *)
  | Omulimm: int -> operation (* rd = r1 * n *)
  | Odiv: operation (* rd = r1 / r2 (signed) *)
  | Odivu: operation (* rd = r1 / r2 (unsigned) *)
  | Omod: operation (* rd = r1 % r2 (signed) *)
  | Omodu: operation (* rd = r1 % r2 (unsigned) *)
  | Oand: operation (* rd = r1 & r2 *)
  | Oandimm: int -> operation (* rd = r1 & n *)
  | Oor: operation (* rd = r1 | r2 *)
  | Oorimm: int -> operation (* rd = r1 | n *)
  | Oxor: operation (* rd = r1 ^ r2 *)
  | Oxorimm: int -> operation (* rd = r1 ^ n *)
  | Oshl: operation (* rd = r1 << r2 *)
  | Oshlimm: int -> operation (* rd = r1 << n *)
  | Oshr: operation (* rd = r1 >> r2 (signed) *)
  | Oshrimm: int -> operation (* rd = r1 >> n (signed) *)
  | Oshrximm: int -> operation (* rd = r1 / 2^n (signed) *)
  | Oshru: operation (* rd = r1 >> r2 (unsigned) *)
  | Oshruimm: int -> operation (* rd = r1 >> n (unsigned) *)
  | Ororimm: int -> operation (* rotate right immediate *)
  | Olea: addressing -> operation (* effective address *)
  | Onegf: operation (* rd = - r1 *)
  | Oabsf: operation (* rd = abs(r1) *)
  | Oaddf: operation (* rd = r1 + r2 *)
  | Osubf: operation (* rd = r1 - r2 *)
  | Omulf: operation (* rd = r1 * r2 *)
  | Odivf: operation (* rd = r1 / r2 *)
  | Osingleoffloat: operation (* rd is r1 truncated to single-precision float *)
  | Ointoffloat: operation (* rd = signed_int_of_float(r1) *)
  | Ofloatofint: operation (* rd = float_of_signed_int(r1) *)
  | Ocmp: condition -> operation. (* rd = 1 if condition holds, rd = 0 otherwise. *)


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).


Definition eq_addressing (x y: addressing) : {x=y} + {x<>y}.

Definition eq_operation (x y: operation): {x=y} + {x<>y}.

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.


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)
  | 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))
  | 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
  | 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)
  | Ointconst _ => (nil, Tint)
  | Ofloatconst _ => (nil, Tfloat)
  | 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)
  | 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, Tfloat)
  | Ointoffloat => (Tfloat :: nil, Tint)
  | Ofloatofint => (Tint :: nil, Tfloat)
  | Ocmp c => (type_of_condition c, Tint)
  end.


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.

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)).

Lemma type_of_chunk_correct:
  forall chunk m addr v,
  Mem.loadv chunk m addr = Some v ->
  Val.has_type v (type_of_chunk chunk).

End SOUNDNESS.

Manipulating and transforming 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.


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).


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.

Lemma type_shift_stack_operation:
  forall delta op, type_of_operation (shift_stack_operation delta op) = type_of_operation op.

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.

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.


Definition op_for_binary_addressing (addr: addressing) : operation := Olea addr.

Lemma eval_op_for_binary_addressing:
  forall (F V: Type) (ge: Genv.t F V) sp addr args v m,
  (length args >= 2)%nat ->
  eval_addressing ge sp addr args = Some v ->
  eval_operation ge sp (op_for_binary_addressing addr) args m = Some v.

Lemma type_op_for_binary_addressing:
  forall addr,
  (length (type_of_addressing addr) >= 2)%nat ->
  type_of_operation (op_for_binary_addressing addr) = (type_of_addressing addr, Tint).



Definition two_address_op (op: operation) : bool :=
  match op with
  | Omove => false
  | Ointconst _ => false
  | Ofloatconst _ => false
  | Ocast8signed => false
  | Ocast8unsigned => false
  | Ocast16signed => false
  | Ocast16unsigned => false
  | Oneg => true
  | Osub => true
  | Omul => true
  | Omulimm _ => true
  | Odiv => true
  | Odivu => true
  | Omod => true
  | Omodu => true
  | Oand => true
  | Oandimm _ => true
  | Oor => true
  | Oorimm _ => true
  | Oxor => true
  | Oxorimm _ => true
  | Oshl => true
  | Oshlimm _ => true
  | Oshr => true
  | Oshrimm _ => true
  | Oshrximm _ => true
  | Oshru => true
  | Oshruimm _ => true
  | Ororimm _ => true
  | Olea addr => false
  | Onegf => true
  | Oabsf => true
  | Oaddf => true
  | Osubf => true
  | Omulf => true
  | Odivf => true
  | Osingleoffloat => false
  | Ointoffloat => false
  | Ofloatofint => false
  | Ocmp c => false
  end.


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


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.


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.

Invariance and compatibility properties.

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.

Lemma eval_operation_preserved:
  forall sp op vl m,
  eval_operation ge2 sp op vl m = eval_operation ge1 sp op vl m.

End GENV_TRANSF.


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 valid_pointer_no_overflow:
  forall b1 ofs b2 delta,
  f b1 = Some(b2, delta) ->
  Mem.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.

Remark val_add_inj:
  forall v1 v1' v2 v2',
  val_inject f v1 v1' -> val_inject f v2 v2' -> val_inject f (Val.add v1 v2) (Val.add v1' v2').

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.

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.

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.

End EVAL_COMPAT.


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.

Remark valid_pointer_no_overflow_extends:
  forall m1 b1 ofs b2 delta,
  Some(b1, 0) = Some(b2, delta) ->
  Mem.valid_pointer m1 b1 (Int.unsigned ofs) = true ->
  0 <= Int.unsigned ofs + Int.unsigned (Int.repr delta) <= Int.max_unsigned.

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)).

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.

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.

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.

End EVAL_LESSDEF.


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).

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.

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.

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.

End EVAL_INJECT.