Require Import BoolEqual.
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.


Inductive condition : Type :=
  | Ccomp (c: comparison) (* signed integer comparison *)
  | Ccompu (c: comparison) (* unsigned integer comparison *)
  | Ccompimm (c: comparison) (n: int) (* signed integer comparison with a constant *)
  | Ccompuimm (c: comparison) (n: int) (* unsigned integer comparison with a constant *)
  | Ccompl (c: comparison) (* signed 64-bit integer comparison *)
  | Ccomplu (c: comparison) (* unsigned 64-bit integer comparison *)
  | Ccomplimm (c: comparison) (n: int64) (* signed 64-bit integer comparison with a constant *)
  | Ccompluimm (c: comparison) (n: int64) (* unsigned 64-bit integer comparison with a constant *)
  | Ccompf (c: comparison) (* 64-bit floating-point comparison *)
  | Cnotcompf (c: comparison) (* negation of a floating-point comparison *)
  | Ccompfs (c: comparison) (* 32-bit floating-point comparison *)
  | Cnotcompfs (c: comparison) (* negation of a floating-point comparison *)
  | Cmaskzero (n: int) (* test (arg & constant) == 0 *)
  | Cmasknotzero (n: int). (* test (arg & constant) != 0 *)


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


Inductive operation : Type :=
  | Omove (* rd = r1 *)
  | Ointconst (n: int) (* rd is set to the given integer constant *)
  | Olongconst (n: int64) (* rd is set to the given integer constant *)
  | Ofloatconst (n: float) (* rd is set to the given float constant *)
  | Osingleconst (n: float32)(* rd is set to the given float constant *)
  | Oindirectsymbol (id: ident) (* rd is set to the address of the symbol *)
  | Ocast8signed (* rd is 8-bit sign extension of r1 *)
  | Ocast8unsigned (* rd is 8-bit zero extension of r1 *)
  | Ocast16signed (* rd is 16-bit sign extension of r1 *)
  | Ocast16unsigned (* rd is 16-bit zero extension of r1 *)
  | Oneg (* rd = - r1 *)
  | Osub (* rd = r1 - r2 *)
  | Omul (* rd = r1 * r2 *)
  | Omulimm (n: int) (* rd = r1 * n *)
  | Omulhs (* rd = high part of r1 * r2, signed *)
  | Omulhu (* rd = high part of r1 * r2, unsigned *)
  | Odiv (* rd = r1 / r2 (signed) *)
  | Odivu (* rd = r1 / r2 (unsigned) *)
  | Omod (* rd = r1 % r2 (signed) *)
  | Omodu (* rd = r1 % r2 (unsigned) *)
  | Oand (* rd = r1 & r2 *)
  | Oandimm (n: int) (* rd = r1 & n *)
  | Oor (* rd = r1 | r2 *)
  | Oorimm (n: int) (* rd = r1 | n *)
  | Oxor (* rd = r1 ^ r2 *)
  | Oxorimm (n: int) (* rd = r1 ^ n *)
  | Onot (* rd = ~r1 *)
  | Oshl (* rd = r1 << r2 *)
  | Oshlimm (n: int) (* rd = r1 << n *)
  | Oshr (* rd = r1 >> r2 (signed) *)
  | Oshrimm (n: int) (* rd = r1 >> n (signed) *)
  | Oshrximm (n: int) (* rd = r1 / 2^n (signed) *)
  | Oshru (* rd = r1 >> r2 (unsigned) *)
  | Oshruimm (n: int) (* rd = r1 >> n (unsigned) *)
  | Ororimm (n: int) (* rotate right immediate *)
  | Oshldimm (n: int) (* rd = r1 << n | r2 >> (32-n) *)
  | Olea (a: addressing) (* effective address *)
  | Omakelong (* rd = r1 << 32 | r2 *)
  | Olowlong (* rd = low-word(r1) *)
  | Ohighlong (* rd = high-word(r1) *)
  | Ocast32signed (* rd is 32-bit sign extension of r1 *)
  | Ocast32unsigned (* rd is 32-bit zero extension of r1 *)
  | Onegl (* rd = - r1 *)
  | Oaddlimm (n: int64) (* rd = r1 + n *)
  | Osubl (* rd = r1 - r2 *)
  | Omull (* rd = r1 * r2 *)
  | Omullimm (n: int64) (* rd = r1 * n *)
  | Omullhs (* rd = high part of r1 * r2, signed *)
  | Omullhu (* rd = high part of r1 * r2, unsigned *)
  | Odivl (* rd = r1 / r2 (signed) *)
  | Odivlu (* rd = r1 / r2 (unsigned) *)
  | Omodl (* rd = r1 % r2 (signed) *)
  | Omodlu (* rd = r1 % r2 (unsigned) *)
  | Oandl (* rd = r1 & r2 *)
  | Oandlimm (n: int64) (* rd = r1 & n *)
  | Oorl (* rd = r1 | r2 *)
  | Oorlimm (n: int64) (* rd = r1 | n *)
  | Oxorl (* rd = r1 ^ r2 *)
  | Oxorlimm (n: int64) (* rd = r1 ^ n *)
  | Onotl (* rd = ~r1 *)
  | Oshll (* rd = r1 << r2 *)
  | Oshllimm (n: int) (* rd = r1 << n *)
  | Oshrl (* rd = r1 >> r2 (signed) *)
  | Oshrlimm (n: int) (* rd = r1 >> n (signed) *)
  | Oshrxlimm (n: int) (* rd = r1 / 2^n (signed) *)
  | Oshrlu (* rd = r1 >> r2 (unsigned) *)
  | Oshrluimm (n: int) (* rd = r1 >> n (unsigned) *)
  | Ororlimm (n: int) (* rotate right immediate *)
  | Oleal (a: addressing) (* effective address *)
  | Onegf (* rd = - r1 *)
  | Oabsf (* rd = abs(r1) *)
  | Oaddf (* rd = r1 + r2 *)
  | Osubf (* rd = r1 - r2 *)
  | Omulf (* rd = r1 * r2 *)
  | Odivf (* rd = r1 / r2 *)
  | Onegfs (* rd = - r1 *)
  | Oabsfs (* rd = abs(r1) *)
  | Oaddfs (* rd = r1 + r2 *)
  | Osubfs (* rd = r1 - r2 *)
  | Omulfs (* rd = r1 * r2 *)
  | Odivfs (* rd = r1 / r2 *)
  | Osingleoffloat (* rd is r1 truncated to single-precision float *)
  | Ofloatofsingle (* rd is r1 extended to double-precision float *)
  | Ointoffloat (* rd = signed_int_of_float64(r1) *)
  | Ofloatofint (* rd = float64_of_signed_int(r1) *)
  | Ointofsingle (* rd = signed_int_of_float32(r1) *)
  | Osingleofint (* rd = float32_of_signed_int(r1) *)
  | Olongoffloat (* rd = signed_long_of_float64(r1) *)
  | Ofloatoflong (* rd = float64_of_signed_long(r1) *)
  | Olongofsingle (* rd = signed_long_of_float32(r1) *)
  | Osingleoflong (* rd = float32_of_signed_long(r1) *)
  | Ocmp (cond: condition). (* rd = 1 if condition holds, rd = 0 otherwise. *)


Definition eq_condition (x y: condition) : {x=y} + {x<>y}.

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

Definition beq_operation: forall (x y: operation), bool.

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

Global Opaque eq_condition eq_addressing eq_operation.


Definition offset_in_range (n: Z) : bool := zle Int.min_signed n && zle n Int.max_signed.

Definition addressing_valid (a: addressing) : bool :=
  match a with
  | Aindexed n => offset_in_range n
  | Aindexed2 n => offset_in_range n
  | Ascaled sc ofs => offset_in_range ofs
  | Aindexed2scaled sc ofs => offset_in_range ofs
  | Aglobal s ofs => true
  | Abased s ofs => true
  | Abasedscaled sc s ofs => true
  | Ainstack ofs => offset_in_range (Ptrofs.signed ofs)
  end.

Evaluation functions

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)
  | Ccompl c, v1 :: v2 :: nil => Val.cmpl_bool c v1 v2
  | Ccomplu c, v1 :: v2 :: nil => Val.cmplu_bool (Mem.valid_pointer m) c v1 v2
  | Ccomplimm c n, v1 :: nil => Val.cmpl_bool c v1 (Vlong n)
  | Ccompluimm c n, v1 :: nil => Val.cmplu_bool (Mem.valid_pointer m) c v1 (Vlong 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)
  | Ccompfs c, v1 :: v2 :: nil => Val.cmpfs_bool c v1 v2
  | Cnotcompfs c, v1 :: v2 :: nil => option_map negb (Val.cmpfs_bool c v1 v2)
  | Cmaskzero n, v1 :: nil => Val.maskzero_bool v1 n
  | Cmasknotzero n, v1 :: nil => option_map negb (Val.maskzero_bool v1 n)
  | _, _ => None
  end.

Definition eval_addressing32
    (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 (Int.repr n)))
  | Aindexed2 n, v1::v2::nil =>
      Some (Val.add (Val.add v1 v2) (Vint (Int.repr n)))
  | Ascaled sc ofs, v1::nil =>
      Some (Val.add (Val.mul v1 (Vint (Int.repr sc))) (Vint (Int.repr ofs)))
  | Aindexed2scaled sc ofs, v1::v2::nil =>
      Some(Val.add v1 (Val.add (Val.mul v2 (Vint (Int.repr sc))) (Vint (Int.repr ofs))))
  | Aglobal s ofs, nil =>
      if Archi.ptr64 then None else Some (Genv.symbol_address genv s ofs)
  | Abased s ofs, v1::nil =>
      if Archi.ptr64 then None else Some (Val.add (Genv.symbol_address genv s ofs) v1)
  | Abasedscaled sc s ofs, v1::nil =>
      if Archi.ptr64 then None else Some (Val.add (Genv.symbol_address genv s ofs) (Val.mul v1 (Vint (Int.repr sc))))
  | Ainstack ofs, nil =>
      if Archi.ptr64 then None else Some(Val.offset_ptr sp ofs)
  | _, _ => None
  end.

Definition eval_addressing64
    (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.addl v1 (Vlong (Int64.repr n)))
  | Aindexed2 n, v1::v2::nil =>
      Some (Val.addl (Val.addl v1 v2) (Vlong (Int64.repr n)))
  | Ascaled sc ofs, v1::nil =>
      Some (Val.addl (Val.mull v1 (Vlong (Int64.repr sc))) (Vlong (Int64.repr ofs)))
  | Aindexed2scaled sc ofs, v1::v2::nil =>
      Some(Val.addl v1 (Val.addl (Val.mull v2 (Vlong (Int64.repr sc))) (Vlong (Int64.repr ofs))))
  | Aglobal s ofs, nil =>
      if Archi.ptr64 then Some (Genv.symbol_address genv s ofs) else None
  | Ainstack ofs, nil =>
      if Archi.ptr64 then Some(Val.offset_ptr sp ofs) else None
  | _, _ => None
  end.

Definition eval_addressing
    (F V: Type) (genv: Genv.t F V) (sp: val)
    (addr: addressing) (vl: list val) : option val :=
  if Archi.ptr64
  then eval_addressing64 genv sp addr vl
  else eval_addressing32 genv sp addr vl.

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)
  | Olongconst n, nil => Some (Vlong n)
  | Ofloatconst n, nil => Some (Vfloat n)
  | Osingleconst n, nil => Some (Vsingle n)
  | Oindirectsymbol id, nil => Some (Genv.symbol_address genv id Ptrofs.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))
  | Omulhs, v1::v2::nil => Some (Val.mulhs v1 v2)
  | Omulhu, v1::v2::nil => Some (Val.mulhu v1 v2)
  | 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))
  | Onot, v1::nil => Some(Val.notint v1)
  | 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_addressing32 genv sp addr vl
  | Omakelong, v1::v2::nil => Some(Val.longofwords v1 v2)
  | Olowlong, v1::nil => Some(Val.loword v1)
  | Ohighlong, v1::nil => Some(Val.hiword v1)
  | Ocast32signed, v1 :: nil => Some (Val.longofint v1)
  | Ocast32unsigned, v1 :: nil => Some (Val.longofintu v1)
  | Onegl, v1::nil => Some (Val.negl v1)
  | Oaddlimm n, v1::nil => Some (Val.addl v1 (Vlong n))
  | Osubl, v1::v2::nil => Some (Val.subl v1 v2)
  | Omull, v1::v2::nil => Some (Val.mull v1 v2)
  | Omullimm n, v1::nil => Some (Val.mull v1 (Vlong n))
  | Omullhs, v1::v2::nil => Some (Val.mullhs v1 v2)
  | Omullhu, v1::v2::nil => Some (Val.mullhu v1 v2)
  | Odivl, v1::v2::nil => Val.divls v1 v2
  | Odivlu, v1::v2::nil => Val.divlu v1 v2
  | Omodl, v1::v2::nil => Val.modls v1 v2
  | Omodlu, v1::v2::nil => Val.modlu v1 v2
  | Oandl, v1::v2::nil => Some(Val.andl v1 v2)
  | Oandlimm n, v1::nil => Some (Val.andl v1 (Vlong n))
  | Oorl, v1::v2::nil => Some(Val.orl v1 v2)
  | Oorlimm n, v1::nil => Some (Val.orl v1 (Vlong n))
  | Oxorl, v1::v2::nil => Some(Val.xorl v1 v2)
  | Oxorlimm n, v1::nil => Some (Val.xorl v1 (Vlong n))
  | Onotl, v1::nil => Some(Val.notl v1)
  | Oshll, v1::v2::nil => Some (Val.shll v1 v2)
  | Oshllimm n, v1::nil => Some (Val.shll v1 (Vint n))
  | Oshrl, v1::v2::nil => Some (Val.shrl v1 v2)
  | Oshrlimm n, v1::nil => Some (Val.shrl v1 (Vint n))
  | Oshrxlimm n, v1::nil => Val.shrxl v1 (Vint n)
  | Oshrlu, v1::v2::nil => Some (Val.shrlu v1 v2)
  | Oshrluimm n, v1::nil => Some (Val.shrlu v1 (Vint n))
  | Ororlimm n, v1::nil => Some (Val.rorl v1 (Vint n))
  | Oleal addr, _ => eval_addressing64 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)
  | Onegfs, v1::nil => Some(Val.negfs v1)
  | Oabsfs, v1::nil => Some(Val.absfs v1)
  | Oaddfs, v1::v2::nil => Some(Val.addfs v1 v2)
  | Osubfs, v1::v2::nil => Some(Val.subfs v1 v2)
  | Omulfs, v1::v2::nil => Some(Val.mulfs v1 v2)
  | Odivfs, v1::v2::nil => Some(Val.divfs v1 v2)
  | Osingleoffloat, v1::nil => Some(Val.singleoffloat v1)
  | Ofloatofsingle, v1::nil => Some(Val.floatofsingle v1)
  | Ointoffloat, v1::nil => Val.intoffloat v1
  | Ofloatofint, v1::nil => Val.floatofint v1
  | Ointofsingle, v1::nil => Val.intofsingle v1
  | Osingleofint, v1::nil => Val.singleofint v1
  | Olongoffloat, v1::nil => Val.longoffloat v1
  | Ofloatoflong, v1::nil => Val.floatoflong v1
  | Olongofsingle, v1::nil => Val.longofsingle v1
  | Osingleoflong, v1::nil => Val.singleoflong v1
  | Ocmp c, _ => option_map Val.of_bool (eval_condition c vl m)
  | _, _ => None
  end.

Remark eval_addressing_Aglobal:
  forall (F V: Type) (genv: Genv.t F V) sp id ofs,
  eval_addressing genv sp (Aglobal id ofs) nil = Some (Genv.symbol_address genv id ofs).

Remark eval_addressing_Ainstack:
  forall (F V: Type) (genv: Genv.t F V) sp ofs,
  eval_addressing genv sp (Ainstack ofs) nil = Some (Val.offset_ptr sp ofs).

Remark eval_addressing_Ainstack_inv:
  forall (F V: Type) (genv: Genv.t F V) sp ofs vl v,
  eval_addressing genv sp (Ainstack ofs) vl = Some v -> vl = nil /\ v = Val.offset_ptr sp ofs.

Ltac FuncInv :=
  match goal with
  | H: (match ?x with nil => _ | _ :: _ => _ end = Some _) |- _ =>
      destruct x; simpl in H; FuncInv
  | H: (match ?v with Vundef => _ | Vint _ => _ | Vfloat _ => _ | Vptr _ _ => _ end = Some _) |- _ =>
      destruct v; simpl in H; FuncInv
  | H: (if Archi.ptr64 then _ else _) = Some _ |- _ =>
      destruct Archi.ptr64 eqn:?; FuncInv
  | H: (Some _ = Some _) |- _ =>
      injection H; intros; clear H; FuncInv
  | H: (None = Some _) |- _ =>
      discriminate H
  | _ =>
      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
  | Ccompl _ => Tlong :: Tlong :: nil
  | Ccomplu _ => Tlong :: Tlong :: nil
  | Ccomplimm _ _ => Tlong :: nil
  | Ccompluimm _ _ => Tlong :: nil
  | Ccompf _ => Tfloat :: Tfloat :: nil
  | Cnotcompf _ => Tfloat :: Tfloat :: nil
  | Ccompfs _ => Tsingle :: Tsingle :: nil
  | Cnotcompfs _ => Tsingle :: Tsingle :: nil
  | Cmaskzero _ => Tint :: nil
  | Cmasknotzero _ => Tint :: nil
  end.

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

Definition type_of_addressing := type_of_addressing_gen Tptr.
Definition type_of_addressing32 := type_of_addressing_gen Tint.
Definition type_of_addressing64 := type_of_addressing_gen Tlong.

Definition type_of_operation (op: operation) : list typ * typ :=
  match op with
  | Omove => (nil, Tint)
  | Ointconst _ => (nil, Tint)
  | Olongconst _ => (nil, Tlong)
  | Ofloatconst f => (nil, Tfloat)
  | Osingleconst f => (nil, Tsingle)
  | Oindirectsymbol _ => (nil, Tptr)
  | 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)
  | Omulhs => (Tint :: Tint :: nil, Tint)
  | Omulhu => (Tint :: 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)
  | Onot => (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_addressing32 addr, Tint)
  | Omakelong => (Tint :: Tint :: nil, Tlong)
  | Olowlong => (Tlong :: nil, Tint)
  | Ohighlong => (Tlong :: nil, Tint)
  | Ocast32signed => (Tint :: nil, Tlong)
  | Ocast32unsigned => (Tint :: nil, Tlong)
  | Onegl => (Tlong :: nil, Tlong)
  | Oaddlimm _ => (Tlong :: nil, Tlong)
  | Osubl => (Tlong :: Tlong :: nil, Tlong)
  | Omull => (Tlong :: Tlong :: nil, Tlong)
  | Omullimm _ => (Tlong :: nil, Tlong)
  | Omullhs => (Tlong :: Tlong :: nil, Tlong)
  | Omullhu => (Tlong :: Tlong :: nil, Tlong)
  | Odivl => (Tlong :: Tlong :: nil, Tlong)
  | Odivlu => (Tlong :: Tlong :: nil, Tlong)
  | Omodl => (Tlong :: Tlong :: nil, Tlong)
  | Omodlu => (Tlong :: Tlong :: nil, Tlong)
  | Oandl => (Tlong :: Tlong :: nil, Tlong)
  | Oandlimm _ => (Tlong :: nil, Tlong)
  | Oorl => (Tlong :: Tlong :: nil, Tlong)
  | Oorlimm _ => (Tlong :: nil, Tlong)
  | Oxorl => (Tlong :: Tlong :: nil, Tlong)
  | Oxorlimm _ => (Tlong :: nil, Tlong)
  | Onotl => (Tlong :: nil, Tlong)
  | Oshll => (Tlong :: Tint :: nil, Tlong)
  | Oshllimm _ => (Tlong :: nil, Tlong)
  | Oshrl => (Tlong :: Tint :: nil, Tlong)
  | Oshrlimm _ => (Tlong :: nil, Tlong)
  | Oshrxlimm _ => (Tlong :: nil, Tlong)
  | Oshrlu => (Tlong :: Tint :: nil, Tlong)
  | Oshrluimm _ => (Tlong :: nil, Tlong)
  | Ororlimm _ => (Tlong :: nil, Tlong)
  | Oleal addr => (type_of_addressing64 addr, Tlong)
  | 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)
  | Onegfs => (Tsingle :: nil, Tsingle)
  | Oabsfs => (Tsingle :: nil, Tsingle)
  | Oaddfs => (Tsingle :: Tsingle :: nil, Tsingle)
  | Osubfs => (Tsingle :: Tsingle :: nil, Tsingle)
  | Omulfs => (Tsingle :: Tsingle :: nil, Tsingle)
  | Odivfs => (Tsingle :: Tsingle :: nil, Tsingle)
  | Osingleoffloat => (Tfloat :: nil, Tsingle)
  | Ofloatofsingle => (Tsingle :: nil, Tfloat)
  | Ointoffloat => (Tfloat :: nil, Tint)
  | Ofloatofint => (Tint :: nil, Tfloat)
  | Ointofsingle => (Tsingle :: nil, Tint)
  | Osingleofint => (Tint :: nil, Tsingle)
  | Olongoffloat => (Tfloat :: nil, Tlong)
  | Ofloatoflong => (Tlong :: nil, Tfloat)
  | Olongofsingle => (Tsingle :: nil, Tlong)
  | Osingleoflong => (Tlong :: nil, Tsingle)
  | Ocmp c => (type_of_condition c, Tint)
  end.


Section SOUNDNESS.

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

Remark type_add:
  forall v1 v2, Val.has_type (Val.add v1 v2) Tint.

Remark type_addl:
  forall v1 v2, Val.has_type (Val.addl v1 v2) Tlong.

Lemma type_of_addressing64_sound:
  forall addr vl sp v,
  eval_addressing64 genv sp addr vl = Some v ->
  Val.has_type v Tlong.

Lemma type_of_addressing32_sound:
  forall addr vl sp v,
  eval_addressing32 genv sp addr vl = Some v ->
  Val.has_type v Tint.

Corollary type_of_addressing_sound:
  forall addr vl sp v,
  eval_addressing genv sp addr vl = Some v ->
  Val.has_type v Tptr.

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)).

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
  | Ccompl c => Ccompl(negate_comparison c)
  | Ccomplu c => Ccomplu(negate_comparison c)
  | Ccomplimm c n => Ccomplimm (negate_comparison c) n
  | Ccompluimm c n => Ccompluimm (negate_comparison c) n
  | Ccompf c => Cnotcompf c
  | Cnotcompf c => Ccompf c
  | Ccompfs c => Cnotcompfs c
  | Cnotcompfs c => Ccompfs 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: Z) (addr: addressing) :=
  match addr with
  | Ainstack ofs => Ainstack (Ptrofs.add ofs (Ptrofs.repr delta))
  | _ => addr
  end.

Definition shift_stack_operation (delta: Z) (op: operation) :=
  match op with
  | Olea addr => Olea (shift_stack_addressing delta addr)
  | Oleal addr => Oleal (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_addressing32:
  forall F V (ge: Genv.t F V) sp addr vl delta,
  eval_addressing32 ge (Vptr sp Ptrofs.zero) (shift_stack_addressing delta addr) vl =
  eval_addressing32 ge (Vptr sp (Ptrofs.repr delta)) addr vl.

Lemma eval_shift_stack_addressing64:
  forall F V (ge: Genv.t F V) sp addr vl delta,
  eval_addressing64 ge (Vptr sp Ptrofs.zero) (shift_stack_addressing delta addr) vl =
  eval_addressing64 ge (Vptr sp (Ptrofs.repr delta)) addr vl.

Lemma eval_shift_stack_addressing:
  forall F V (ge: Genv.t F V) sp addr vl delta,
  eval_addressing ge (Vptr sp Ptrofs.zero) (shift_stack_addressing delta addr) vl =
  eval_addressing ge (Vptr sp (Ptrofs.repr delta)) addr vl.

Lemma eval_shift_stack_operation:
  forall F V (ge: Genv.t F V) sp op vl m delta,
  eval_operation ge (Vptr sp Ptrofs.zero) (shift_stack_operation delta op) vl m =
  eval_operation ge (Vptr sp (Ptrofs.repr delta)) op vl m.


Definition offset_addressing_total (addr: addressing) (delta: Z) : addressing :=
  match addr with
  | Aindexed n => Aindexed (n + delta)
  | Aindexed2 n => Aindexed2 (n + delta)
  | Ascaled sc n => Ascaled sc (n + delta)
  | Aindexed2scaled sc n => Aindexed2scaled sc (n + delta)
  | Aglobal s n => Aglobal s (Ptrofs.add n (Ptrofs.repr delta))
  | Abased s n => Abased s (Ptrofs.add n (Ptrofs.repr delta))
  | Abasedscaled sc s n => Abasedscaled sc s (Ptrofs.add n (Ptrofs.repr delta))
  | Ainstack n => Ainstack (Ptrofs.add n (Ptrofs.repr delta))
  end.

Definition offset_addressing (addr: addressing) (delta: Z) : option addressing :=
  let addr' := offset_addressing_total addr delta in
  if addressing_valid addr' then Some addr' else None.

Lemma eval_offset_addressing_total_32:
  forall (F V: Type) (ge: Genv.t F V) sp addr args delta v,
  eval_addressing32 ge sp addr args = Some v ->
  eval_addressing32 ge sp (offset_addressing_total addr delta) args = Some(Val.add v (Vint (Int.repr delta))).

Lemma eval_offset_addressing_total_64:
  forall (F V: Type) (ge: Genv.t F V) sp addr args delta v,
  eval_addressing64 ge sp addr args = Some v ->
  eval_addressing64 ge sp (offset_addressing_total addr delta) args = Some(Val.addl v (Vlong (Int64.repr delta))).


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 ->
  Archi.ptr64 = false ->
  eval_addressing ge sp addr' args = Some(Val.add v (Vint (Int.repr delta))).


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


Definition op_depends_on_memory (op: operation) : bool :=
  match op with
  | Ocmp (Ccompu _) => negb Archi.ptr64
  | Ocmp (Ccompuimm _ _) => negb Archi.ptr64
  | Ocmp (Ccomplu _) => Archi.ptr64
  | Ocmp (Ccompluimm _ _) => Archi.ptr64
  | _ => 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 globals_addressing (addr: addressing) : list ident :=
  match addr with
  | Aglobal s n => s :: nil
  | Abased s n => s :: nil
  | Abasedscaled sc s n => s :: nil
  | _ => nil
  end.

Definition globals_operation (op: operation) : list ident :=
  match op with
  | Oindirectsymbol s => s :: nil
  | Olea addr => globals_addressing addr
  | Oleal addr => globals_addressing addr
  | _ => nil
  end.

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_addressing32_preserved:
  forall sp addr vl,
  eval_addressing32 ge2 sp addr vl = eval_addressing32 ge1 sp addr vl.

Lemma eval_addressing64_preserved:
  forall sp addr vl,
  eval_addressing64 ge2 sp addr vl = eval_addressing64 ge1 sp addr vl.

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 F1 F2 V1 V2: Type.
Variable ge1: Genv.t F1 V1.
Variable ge2: Genv.t F2 V2.
Variable f: meminj.

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 (Ptrofs.unsigned ofs) = true ->
  Mem.valid_pointer m2 b2 (Ptrofs.unsigned (Ptrofs.add ofs (Ptrofs.repr delta))) = true.

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

Hypothesis valid_different_pointers_inj:
  forall b1 ofs1 b2 ofs2 b1' delta1 b2' delta2,
  b1 <> b2 ->
  Mem.valid_pointer m1 b1 (Ptrofs.unsigned ofs1) = true ->
  Mem.valid_pointer m1 b2 (Ptrofs.unsigned ofs2) = true ->
  f b1 = Some (b1', delta1) ->
  f b2 = Some (b2', delta2) ->
  b1' <> b2' \/
  Ptrofs.unsigned (Ptrofs.add ofs1 (Ptrofs.repr delta1)) <> Ptrofs.unsigned (Ptrofs.add ofs2 (Ptrofs.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.inject_list _ nil _ |- _ ] =>
      inv H; InvInject
  | [ H: Val.inject_list _ (_ :: _) _ |- _ ] =>
      inv H; InvInject
  | _ => idtac
  end.

Lemma eval_condition_inj:
  forall cond vl1 vl2 b,
  Val.inject_list 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_addressing32_inj:
  forall addr sp1 vl1 sp2 vl2 v1,
  (forall id ofs,
      In id (globals_addressing addr) ->
      Val.inject f (Genv.symbol_address ge1 id ofs) (Genv.symbol_address ge2 id ofs)) ->
  Val.inject f sp1 sp2 ->
  Val.inject_list f vl1 vl2 ->
  eval_addressing32 ge1 sp1 addr vl1 = Some v1 ->
  exists v2, eval_addressing32 ge2 sp2 addr vl2 = Some v2 /\ Val.inject f v1 v2.

Lemma eval_addressing64_inj:
  forall addr sp1 vl1 sp2 vl2 v1,
  (forall id ofs,
      In id (globals_addressing addr) ->
      Val.inject f (Genv.symbol_address ge1 id ofs) (Genv.symbol_address ge2 id ofs)) ->
  Val.inject f sp1 sp2 ->
  Val.inject_list f vl1 vl2 ->
  eval_addressing64 ge1 sp1 addr vl1 = Some v1 ->
  exists v2, eval_addressing64 ge2 sp2 addr vl2 = Some v2 /\ Val.inject f v1 v2.

Lemma eval_addressing_inj:
  forall addr sp1 vl1 sp2 vl2 v1,
  (forall id ofs,
      In id (globals_addressing addr) ->
      Val.inject f (Genv.symbol_address ge1 id ofs) (Genv.symbol_address ge2 id ofs)) ->
  Val.inject f sp1 sp2 ->
  Val.inject_list f vl1 vl2 ->
  eval_addressing ge1 sp1 addr vl1 = Some v1 ->
  exists v2, eval_addressing ge2 sp2 addr vl2 = Some v2 /\ Val.inject f v1 v2.

Lemma eval_operation_inj:
  forall op sp1 vl1 sp2 vl2 v1,
  (forall id ofs,
      In id (globals_operation op) ->
      Val.inject f (Genv.symbol_address ge1 id ofs) (Genv.symbol_address ge2 id ofs)) ->
  Val.inject f sp1 sp2 ->
  Val.inject_list f vl1 vl2 ->
  eval_operation ge1 sp1 op vl1 m1 = Some v1 ->
  exists v2, eval_operation ge2 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 (Ptrofs.unsigned ofs) = true ->
  Mem.valid_pointer m2 b2 (Ptrofs.unsigned (Ptrofs.add ofs (Ptrofs.repr delta))) = true.

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 (Ptrofs.unsigned ofs) = true ->
  Mem.weak_valid_pointer m2 b2 (Ptrofs.unsigned (Ptrofs.add ofs (Ptrofs.repr delta))) = true.

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 (Ptrofs.unsigned ofs) = true ->
  0 <= Ptrofs.unsigned ofs + Ptrofs.unsigned (Ptrofs.repr delta) <= Ptrofs.max_unsigned.

Remark valid_different_pointers_extends:
  forall m1 b1 ofs1 b2 ofs2 b1' delta1 b2' delta2,
  b1 <> b2 ->
  Mem.valid_pointer m1 b1 (Ptrofs.unsigned ofs1) = true ->
  Mem.valid_pointer m1 b2 (Ptrofs.unsigned ofs2) = true ->
  Some(b1, 0) = Some (b1', delta1) ->
  Some(b2, 0) = Some (b2', delta2) ->
  b1' <> b2' \/
  Ptrofs.unsigned(Ptrofs.add ofs1 (Ptrofs.repr delta1)) <> Ptrofs.unsigned(Ptrofs.add ofs2 (Ptrofs.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 (Genv.symbol_address genv id ofs) (Genv.symbol_address genv id ofs).

Lemma eval_condition_inject:
  forall cond vl1 vl2 b m1 m2,
  Val.inject_list 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.inject_list f vl1 vl2 ->
  eval_addressing genv (Vptr sp1 Ptrofs.zero) addr vl1 = Some v1 ->
  exists v2,
     eval_addressing genv (Vptr sp2 Ptrofs.zero) (shift_stack_addressing delta addr) vl2 = Some v2
  /\ Val.inject f v1 v2.

Lemma eval_operation_inject:
  forall op vl1 vl2 v1 m1 m2,
  Val.inject_list f vl1 vl2 ->
  Mem.inject f m1 m2 ->
  eval_operation genv (Vptr sp1 Ptrofs.zero) op vl1 m1 = Some v1 ->
  exists v2,
     eval_operation genv (Vptr sp2 Ptrofs.zero) (shift_stack_operation delta op) vl2 m2 = Some v2
  /\ Val.inject f v1 v2.

End EVAL_INJECT.