path: root/arm/SelectOpproof.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.     *)
(*                                                                     *)
(* *********************************************************************)

(** Correctness of instruction selection for operators *)

Require Import Coqlib.
Require Import Maps.
Require Import AST.
Require Import Integers.
Require Import Floats.
Require Import Values.
Require Import Memory.
Require Import Events.
Require Import Globalenvs.
Require Import Smallstep.
Require Import Cminor.
Require Import Op.
Require Import CminorSel.
Require Import SelectOp.

Open Local Scope cminorsel_scope.

Section CMCONSTR.

Variable ge: genv.
Variable sp: val.
Variable e: env.
Variable m: mem.

(** * Useful lemmas and tactics *)

(** The following are trivial lemmas and custom tactics that help
  perform backward (inversion) and forward reasoning over the evaluation
  of operator applications. *)  

Ltac EvalOp := eapply eval_Eop; eauto with evalexpr.

Ltac InvEval1 :=
  match goal with
  | [ H: (eval_expr _ _ _ _ _ (Eop _ Enil) _) |- _ ] =>
      inv H; InvEval1
  | [ H: (eval_expr _ _ _ _ _ (Eop _ (_ ::: Enil)) _) |- _ ] =>
      inv H; InvEval1
  | [ H: (eval_expr _ _ _ _ _ (Eop _ (_ ::: _ ::: Enil)) _) |- _ ] =>
      inv H; InvEval1
  | [ H: (eval_exprlist _ _ _ _ _ Enil _) |- _ ] =>
      inv H; InvEval1
  | [ H: (eval_exprlist _ _ _ _ _ (_ ::: _) _) |- _ ] =>
      inv H; InvEval1
  | _ =>
      idtac
  end.

Ltac InvEval2 :=
  match goal with
  | [ H: (eval_operation _ _ _ nil _ = Some _) |- _ ] =>
      simpl in H; inv H
  | [ H: (eval_operation _ _ _ (_ :: nil) _ = Some _) |- _ ] =>
      simpl in H; FuncInv
  | [ H: (eval_operation _ _ _ (_ :: _ :: nil) _ = Some _) |- _ ] =>
      simpl in H; FuncInv
  | [ H: (eval_operation _ _ _ (_ :: _ :: _ :: nil) _ = Some _) |- _ ] =>
      simpl in H; FuncInv
  | _ =>
      idtac
  end.

Ltac InvEval := InvEval1; InvEval2; InvEval2.

Ltac TrivialExists :=
  match goal with
  | [ |- exists v, _ /\ Val.lessdef ?a v ] => exists a; split; [EvalOp | auto]
  end.

(** * Correctness of the smart constructors *)

(** We now show that the code generated by "smart constructor" functions
  such as [Selection.notint] behaves as expected.  Continuing the
  [notint] example, we show that if the expression [e]
  evaluates to some integer value [Vint n], then [Selection.notint e]
  evaluates to a value [Vint (Int.not n)] which is indeed the integer
  negation of the value of [e].

  All proofs follow a common pattern:
- Reasoning by case over the result of the classification functions
  (such as [add_match] for integer addition), gathering additional
  information on the shape of the argument expressions in the non-default
  cases.
- Inversion of the evaluations of the arguments, exploiting the additional
  information thus gathered.
- Equational reasoning over the arithmetic operations performed,
  using the lemmas from the [Int] and [Float] modules.
- Construction of an evaluation derivation for the expression returned
  by the smart constructor.
*)

Definition unary_constructor_sound (cstr: expr -> expr) (sem: val -> val) : Prop :=
  forall le a x,
  eval_expr ge sp e m le a x ->
  exists v, eval_expr ge sp e m le (cstr a) v /\ Val.lessdef (sem x) v.

Definition binary_constructor_sound (cstr: expr -> expr -> expr) (sem: val -> val -> val) : Prop :=
  forall le a x b y,
  eval_expr ge sp e m le a x ->
  eval_expr ge sp e m le b y ->
  exists v, eval_expr ge sp e m le (cstr a b) v /\ Val.lessdef (sem x y) v.

Theorem eval_addrsymbol:
  forall le id ofs,
  exists v, eval_expr ge sp e m le (addrsymbol id ofs) v /\ Val.lessdef (symbol_address ge id ofs) v.
Proof.
  intros. unfold addrsymbol. econstructor; split. 
  EvalOp. simpl; eauto. 
  auto.
Qed.

Theorem eval_addrstack:
  forall le ofs,
  exists v, eval_expr ge sp e m le (addrstack ofs) v /\ Val.lessdef (Val.add sp (Vint ofs)) v.
Proof.
  intros. unfold addrstack. econstructor; split.
  EvalOp. simpl; eauto. 
  auto.
Qed.

Theorem eval_notint: unary_constructor_sound notint Val.notint.
Proof.
  unfold notint; red; intros until x; case (notint_match a); intros; InvEval.
  subst x. TrivialExists.
  exists v1; split; auto. subst. destruct v1; simpl; auto. rewrite Int.not_involutive; auto.
  exists (eval_shift s v1); split. EvalOp. subst x. destruct (eval_shift s v1); simpl; auto. rewrite Int.not_involutive; auto.
  TrivialExists.
Qed.

Theorem eval_addimm:
  forall n, unary_constructor_sound (addimm n) (fun x => Val.add x (Vint n)).
Proof.
  red; unfold addimm; intros until x.
  predSpec Int.eq Int.eq_spec n Int.zero.
  subst n. intros. exists x; split; auto. 
  destruct x; simpl; auto. rewrite Int.add_zero. auto. rewrite Int.add_zero. auto.
  case (addimm_match a); intros; InvEval; simpl; TrivialExists; simpl.
  rewrite Int.add_commut. auto.
  unfold symbol_address. destruct (Genv.find_symbol ge s); simpl; auto. rewrite Int.add_commut; auto.
  rewrite Val.add_assoc. rewrite Int.add_commut. auto.
  subst x. rewrite Val.add_assoc. rewrite Int.add_commut. auto.
Qed. 

Theorem eval_add: binary_constructor_sound add Val.add.
Proof.
  red; intros until y.
  unfold add; case (add_match a b); intros; InvEval.
  rewrite Val.add_commut. apply eval_addimm; auto.
  apply eval_addimm; auto.
  subst. 
  replace (Val.add (Val.add v1 (Vint n1)) (Val.add v0 (Vint n2)))
     with (Val.add (Val.add v1 v0) (Val.add (Vint n1) (Vint n2))).
  apply eval_addimm. EvalOp.
  repeat rewrite Val.add_assoc. decEq. apply Val.add_permut.
  subst. 
  replace (Val.add (Val.add v1 (Vint n1)) y)
     with (Val.add (Val.add v1 y) (Vint n1)).
  apply eval_addimm. EvalOp.
  repeat rewrite Val.add_assoc. decEq. apply Val.add_commut.
  subst. rewrite <- Val.add_assoc. apply eval_addimm. EvalOp.
  subst. rewrite Val.add_commut. TrivialExists.
  subst. TrivialExists.
  TrivialExists.
Qed.

Theorem eval_sub: binary_constructor_sound sub Val.sub.
Proof.
  red; intros until y.
  unfold sub; case (sub_match a b); intros; InvEval.
  rewrite Val.sub_add_opp. apply eval_addimm; auto.
  subst. rewrite Val.sub_add_l. rewrite Val.sub_add_r. 
    rewrite Val.add_assoc. simpl. rewrite Int.add_commut. rewrite <- Int.sub_add_opp.
    apply eval_addimm; EvalOp.
  subst. rewrite Val.sub_add_l. apply eval_addimm; EvalOp.
  subst. rewrite Val.sub_add_r. apply eval_addimm; EvalOp.
  TrivialExists.
  subst. TrivialExists.
  subst. TrivialExists.
  TrivialExists.
Qed.

Theorem eval_negint: unary_constructor_sound negint (fun v => Val.sub Vzero v).
Proof.
  red; intros. unfold negint. TrivialExists.
Qed.

Theorem eval_shlimm:
  forall n, unary_constructor_sound (fun a => shlimm a n)
                                    (fun x => Val.shl x (Vint n)).
Proof.
Opaque mk_shift_amount.
  red; intros until x.  unfold shlimm.
  predSpec Int.eq Int.eq_spec n Int.zero.
  intros; subst. exists x; split; auto. destruct x; simpl; auto. rewrite Int.shl_zero; auto.
  destruct (Int.ltu n Int.iwordsize) as []_eqn; simpl.
  destruct (shlimm_match a); intros.
  InvEval. simpl; rewrite Heqb. TrivialExists.
  destruct (Int.ltu (Int.add n n1) Int.iwordsize) as []_eqn.
  InvEval. subst x. exists (Val.shl v1 (Vint (Int.add n n1))); split. EvalOp.
  simpl. rewrite mk_shift_amount_eq; auto. 
  destruct v1; simpl; auto. rewrite s_range. simpl. rewrite Heqb. rewrite Heqb0.
  rewrite Int.add_commut. rewrite Int.shl_shl; auto. apply s_range. rewrite Int.add_commut; auto.
  TrivialExists. simpl. rewrite mk_shift_amount_eq; auto. 
  TrivialExists. simpl. rewrite mk_shift_amount_eq; auto. 
  intros; TrivialExists. simpl. constructor. eauto. constructor. EvalOp. simpl; eauto. constructor. auto.
Qed.

 Theorem eval_shrimm:
  forall n, unary_constructor_sound (fun a => shrimm a n)
                                    (fun x => Val.shr x (Vint n)).
Proof.
  red; intros until x.  unfold shrimm.
  predSpec Int.eq Int.eq_spec n Int.zero.
  intros; subst. exists x; split; auto. destruct x; simpl; auto. rewrite Int.shr_zero; auto.
  destruct (Int.ltu n Int.iwordsize) as []_eqn; simpl.
  destruct (shrimm_match a); intros.
  InvEval. simpl; rewrite Heqb. TrivialExists.
  destruct (Int.ltu (Int.add n n1) Int.iwordsize) as []_eqn.
  InvEval. subst x. exists (Val.shr v1 (Vint (Int.add n n1))); split. EvalOp.
  simpl. rewrite mk_shift_amount_eq; auto. 
  destruct v1; simpl; auto. rewrite s_range. simpl. rewrite Heqb. rewrite Heqb0.
  rewrite Int.add_commut. rewrite Int.shr_shr; auto. apply s_range. rewrite Int.add_commut; auto.
  TrivialExists. simpl. rewrite mk_shift_amount_eq; auto.  
  TrivialExists. simpl. rewrite mk_shift_amount_eq; auto. 
  intros; TrivialExists. simpl. constructor. eauto. constructor. EvalOp. simpl; eauto. constructor. auto.
Qed.

Theorem eval_shruimm:
  forall n, unary_constructor_sound (fun a => shruimm a n)
                                    (fun x => Val.shru x (Vint n)).
Proof.
  red; intros until x.  unfold shruimm.
  predSpec Int.eq Int.eq_spec n Int.zero.
  intros; subst. exists x; split; auto. destruct x; simpl; auto. rewrite Int.shru_zero; auto.
  destruct (Int.ltu n Int.iwordsize) as []_eqn; simpl.
  destruct (shruimm_match a); intros.
  InvEval. simpl; rewrite Heqb. TrivialExists.
  destruct (Int.ltu (Int.add n n1) Int.iwordsize) as []_eqn.
  InvEval. subst x. exists (Val.shru v1 (Vint (Int.add n n1))); split. EvalOp.
  simpl. rewrite mk_shift_amount_eq; auto. 
  destruct v1; simpl; auto. destruct (Int.ltu n1 Int.iwordsize) as []_eqn; simpl; auto.
  rewrite Heqb; rewrite Heqb0. rewrite Int.add_commut. rewrite Int.shru_shru; auto. rewrite Int.add_commut; auto.
  TrivialExists. simpl. rewrite mk_shift_amount_eq; auto. 
  TrivialExists. simpl. rewrite mk_shift_amount_eq; auto. 
  intros; TrivialExists. simpl. constructor. eauto. constructor. EvalOp. simpl; eauto. constructor. auto.
Qed.

Lemma eval_mulimm_base:
  forall n, unary_constructor_sound (mulimm_base n) (fun x => Val.mul x (Vint n)).
Proof.
  intros; red; intros; unfold mulimm_base.
  assert (DFL: exists v, eval_expr ge sp e m le (Eop Omul (Eop (Ointconst n) Enil ::: a ::: Enil)) v /\ Val.lessdef (Val.mul x (Vint n)) v).
  TrivialExists. econstructor. EvalOp. simpl; eauto. econstructor. eauto. constructor.
  rewrite Val.mul_commut. auto.
  generalize (Int.one_bits_decomp n). 
  generalize (Int.one_bits_range n).
  destruct (Int.one_bits n).
  intros. auto.
  destruct l.
  intros. rewrite H1. simpl. 
  rewrite Int.add_zero.
  replace (Vint (Int.shl Int.one i)) with (Val.shl Vone (Vint i)). rewrite Val.shl_mul.
  apply eval_shlimm. auto. simpl. rewrite H0; auto with coqlib.
  destruct l.
  intros. rewrite H1. simpl.
  exploit (eval_shlimm i (x :: le) (Eletvar 0) x). constructor; auto. intros [v1 [A1 B1]].
  exploit (eval_shlimm i0 (x :: le) (Eletvar 0) x). constructor; auto. intros [v2 [A2 B2]].
  exploit (eval_add (x :: le)). eexact A1. eexact A2. intros [v [A B]].
  exists v; split. econstructor; eauto. 
  rewrite Int.add_zero.
  replace (Vint (Int.add (Int.shl Int.one i) (Int.shl Int.one i0)))
     with (Val.add (Val.shl Vone (Vint i)) (Val.shl Vone (Vint i0))).
  rewrite Val.mul_add_distr_r.
  repeat rewrite Val.shl_mul. eapply Val.lessdef_trans. 2: eauto. apply Val.add_lessdef; auto.
  simpl. repeat rewrite H0; auto with coqlib. 
  intros. auto.
Qed.


Theorem eval_mulimm:
  forall n, unary_constructor_sound (mulimm n) (fun x => Val.mul x (Vint n)).
Proof.
  intros; red; intros until x; unfold mulimm.
  predSpec Int.eq Int.eq_spec n Int.zero. 
  intros. exists (Vint Int.zero); split. EvalOp. 
  destruct x; simpl; auto. subst n. rewrite Int.mul_zero. auto.
  predSpec Int.eq Int.eq_spec n Int.one.
  intros. exists x; split; auto.
  destruct x; simpl; auto. subst n. rewrite Int.mul_one. auto.
  case (mulimm_match a); intros; InvEval.
  TrivialExists. simpl. rewrite Int.mul_commut; auto.
  subst. rewrite Val.mul_add_distr_l. 
  exploit eval_mulimm_base; eauto. instantiate (1 := n). intros [v' [A1 B1]].
  exploit (eval_addimm (Int.mul n n2) le (mulimm_base n t2) v'). auto. intros [v'' [A2 B2]].
  exists v''; split; auto. eapply Val.lessdef_trans. eapply Val.add_lessdef; eauto. 
  rewrite Val.mul_commut; auto.
  apply eval_mulimm_base; auto.
Qed.

Theorem eval_mul: binary_constructor_sound mul Val.mul.
Proof.
  red; intros until y.
  unfold mul; case (mul_match a b); intros; InvEval.
  rewrite Val.mul_commut. apply eval_mulimm. auto. 
  apply eval_mulimm. auto.
  TrivialExists.
Qed.

Theorem eval_andimm:
  forall n, unary_constructor_sound (andimm n) (fun x => Val.and x (Vint n)).
Proof.
  intros; red; intros until x. unfold andimm.
  predSpec Int.eq Int.eq_spec n Int.zero. 
  intros. exists (Vint Int.zero); split. EvalOp. 
  destruct x; simpl; auto. subst n. rewrite Int.and_zero. auto.
  case (andimm_match a); intros.
  InvEval. TrivialExists. simpl. rewrite Int.and_commut; auto.
  InvEval. subst. rewrite Val.and_assoc. simpl. rewrite Int.and_commut. TrivialExists. 
  TrivialExists.
Qed.

Theorem eval_and: binary_constructor_sound and Val.and.
Proof.
  red; intros until y; unfold and; case (and_match a b); intros; InvEval.
  rewrite Val.and_commut. apply eval_andimm; auto.
  apply eval_andimm; auto.
  subst. rewrite Val.and_commut. TrivialExists.
  subst. TrivialExists.
  subst. rewrite Val.and_commut. TrivialExists.
  subst. TrivialExists.
  subst. rewrite Val.and_commut. TrivialExists.
  subst. TrivialExists.
  TrivialExists.
Qed.

Theorem eval_orimm:
  forall n, unary_constructor_sound (orimm n) (fun x => Val.or x (Vint n)).
Proof.
  intros; red; intros until x. unfold orimm.
  predSpec Int.eq Int.eq_spec n Int.zero.
  intros. subst. exists x; split; auto.
  destruct x; simpl; auto. rewrite Int.or_zero; auto.
  destruct (orimm_match a); intros; InvEval.
  TrivialExists. simpl. rewrite Int.or_commut; auto.
  subst. rewrite Val.or_assoc. simpl. rewrite Int.or_commut. TrivialExists. 
  TrivialExists.
Qed.

Remark eval_same_expr:
  forall a1 a2 le v1 v2,
  same_expr_pure a1 a2 = true ->
  eval_expr ge sp e m le a1 v1 ->
  eval_expr ge sp e m le a2 v2 ->
  a1 = a2 /\ v1 = v2.
Proof.
  intros until v2.
  destruct a1; simpl; try (intros; discriminate). 
  destruct a2; simpl; try (intros; discriminate).
  case (ident_eq i i0); intros.
  subst i0. inversion H0. inversion H1. split. auto. congruence. 
  discriminate.
Qed.

Theorem eval_or: binary_constructor_sound or Val.or.
Proof.
  red; intros until y; unfold or; case (or_match a b); intros; InvEval.
  rewrite Val.or_commut. apply eval_orimm; auto.
  apply eval_orimm; auto.
(* shl - shru *)
  destruct (Int.eq (Int.add n1 n2) Int.iwordsize && same_expr_pure t1 t2) as []_eqn.
  destruct (andb_prop _ _ Heqb0).
  generalize (Int.eq_spec (Int.add n1 n2) Int.iwordsize); rewrite H1; intros.
  exploit eval_same_expr; eauto. intros [EQ1 EQ2]. subst. 
  exists (Val.ror v0 (Vint n2)); split. EvalOp.
  destruct v0; simpl; auto. 
  destruct (Int.ltu n1 Int.iwordsize) as []_eqn; auto.
  destruct (Int.ltu n2 Int.iwordsize) as []_eqn; auto.
  simpl. rewrite <- Int.or_ror; auto.
  subst. TrivialExists.
  econstructor. EvalOp. simpl; eauto. econstructor; eauto. constructor.
  simpl. auto.
(* shru - shr *)
  destruct (Int.eq (Int.add n2 n1) Int.iwordsize && same_expr_pure t1 t2) as []_eqn.
  destruct (andb_prop _ _ Heqb0).
  generalize (Int.eq_spec (Int.add n2 n1) Int.iwordsize); rewrite H1; intros.
  exploit eval_same_expr; eauto. intros [EQ1 EQ2]. subst. 
  exists (Val.ror v0 (Vint n1)); split. EvalOp.
  destruct v0; simpl; auto. 
  destruct (Int.ltu n1 Int.iwordsize) as []_eqn; auto.
  destruct (Int.ltu n2 Int.iwordsize) as []_eqn; auto.
  simpl. rewrite Int.or_commut. rewrite <- Int.or_ror; auto.
  subst. TrivialExists.
  econstructor. EvalOp. simpl; eauto. econstructor; eauto. constructor.
  simpl. auto.
(* orshift *)
  subst. rewrite Val.or_commut. TrivialExists.
  subst. TrivialExists.
(* default *)
  TrivialExists.
Qed.

Theorem eval_xorimm:
  forall n, unary_constructor_sound (xorimm n) (fun x => Val.xor x (Vint n)).
Proof.
  intros; red; intros until x. unfold xorimm.
  predSpec Int.eq Int.eq_spec n Int.zero. 
  intros. exists x; split. auto.  
  destruct x; simpl; auto. subst n. rewrite Int.xor_zero. auto.
  destruct (xorimm_match a); intros; InvEval.
  TrivialExists. simpl. rewrite Int.xor_commut; auto.
  subst. rewrite Val.xor_assoc. simpl. rewrite Int.xor_commut. TrivialExists. 
  TrivialExists.
Qed.

Theorem eval_xor: binary_constructor_sound xor Val.xor.
Proof.
  red; intros until y; unfold xor; case (xor_match a b); intros; InvEval.
  rewrite Val.xor_commut. apply eval_xorimm; auto.
  apply eval_xorimm; auto.
  subst. rewrite Val.xor_commut. TrivialExists.
  subst. TrivialExists.
  TrivialExists.
Qed.

Lemma eval_mod_aux:
  forall divop semdivop,
  (forall sp x y m, eval_operation ge sp divop (x :: y :: nil) m = semdivop x y) ->
  forall le a b x y z,
  eval_expr ge sp e m le a x ->
  eval_expr ge sp e m le b y ->
  semdivop x y = Some z ->
  eval_expr ge sp e m le (mod_aux divop a b) (Val.sub x (Val.mul z y)).
Proof.
  intros; unfold mod_aux.
  eapply eval_Elet. eexact H0. eapply eval_Elet. 
  apply eval_lift. eexact H1.
  eapply eval_Eop. eapply eval_Econs. 
  eapply eval_Eletvar. simpl; reflexivity.
  eapply eval_Econs. eapply eval_Eop. 
  eapply eval_Econs. eapply eval_Eop.
  eapply eval_Econs. apply eval_Eletvar. simpl; reflexivity.
  eapply eval_Econs. apply eval_Eletvar. simpl; reflexivity.
  apply eval_Enil.  
  rewrite H. eauto.
  eapply eval_Econs. apply eval_Eletvar. simpl; reflexivity.
  apply eval_Enil.  
  simpl; reflexivity. apply eval_Enil. 
  reflexivity.
Qed.

Theorem eval_divs:
  forall le a b x y z,
  eval_expr ge sp e m le a x ->
  eval_expr ge sp e m le b y ->
  Val.divs x y = Some z ->
  exists v, eval_expr ge sp e m le (divs a b) v /\ Val.lessdef z v.
Proof.
  intros. unfold divs. exists z; split. EvalOp. auto.
Qed.

Theorem eval_mods:
  forall le a b x y z,
  eval_expr ge sp e m le a x ->
  eval_expr ge sp e m le b y ->
  Val.mods x y = Some z ->
  exists v, eval_expr ge sp e m le (mods a b) v /\ Val.lessdef z v.
Proof.
  intros; unfold mods. 
  exploit Val.mods_divs; eauto. intros [v [A B]].
  subst. econstructor; split; eauto.
  apply eval_mod_aux with (semdivop := Val.divs); auto.
Qed.

Theorem eval_divuimm:
  forall le n a x z,
  eval_expr ge sp e m le a x ->
  Val.divu x (Vint n) = Some z ->
  exists v, eval_expr ge sp e m le (divuimm a n) v /\ Val.lessdef z v.
Proof.
  intros; unfold divuimm. 
  destruct (Int.is_power2 n) as []_eqn. 
  replace z with (Val.shru x (Vint i)). apply eval_shruimm; auto.
  eapply Val.divu_pow2; eauto.
  TrivialExists. 
  econstructor. eauto. econstructor. EvalOp. simpl; eauto. constructor. auto.
Qed.

Theorem eval_divu:
  forall le a x b y z,
  eval_expr ge sp e m le a x ->
  eval_expr ge sp e m le b y ->
  Val.divu x y = Some z ->
  exists v, eval_expr ge sp e m le (divu a b) v /\ Val.lessdef z v.
Proof.
  intros until z. unfold divu; destruct (divu_match b); intros; InvEval.
  eapply eval_divuimm; eauto.
  TrivialExists. 
Qed.

Theorem eval_moduimm:
  forall le n a x z,
  eval_expr ge sp e m le a x ->
  Val.modu x (Vint n) = Some z ->
  exists v, eval_expr ge sp e m le (moduimm a n) v /\ Val.lessdef z v.
Proof.
  intros; unfold moduimm. 
  destruct (Int.is_power2 n) as []_eqn.
  replace z with (Val.and x (Vint (Int.sub n Int.one))). TrivialExists.
  eapply Val.modu_pow2; eauto.
  exploit Val.modu_divu; eauto. intros [v [A B]].
  subst. econstructor; split; eauto.
  apply eval_mod_aux with (semdivop := Val.divu); auto.
  EvalOp.
Qed.

Theorem eval_modu:
  forall le a x b y z,
  eval_expr ge sp e m le a x ->
  eval_expr ge sp e m le b y ->
  Val.modu x y = Some z ->
  exists v, eval_expr ge sp e m le (modu a b) v /\ Val.lessdef z v.
Proof.
  intros until y; unfold modu; case (modu_match b); intros; InvEval.
  eapply eval_moduimm; eauto.
  exploit Val.modu_divu; eauto. intros [v [A B]].
  subst. econstructor; split; eauto.
  apply eval_mod_aux with (semdivop := Val.divu); auto.
Qed.

Theorem eval_shl: binary_constructor_sound shl Val.shl.
Proof.
  red; intros until y; unfold shl; case (shl_match b); intros.
  InvEval. apply eval_shlimm; auto.
  TrivialExists. 
Qed.

Theorem eval_shr: binary_constructor_sound shr Val.shr.
Proof.
  red; intros until y; unfold shr; case (shr_match b); intros.
  InvEval. apply eval_shrimm; auto.
  TrivialExists. 
Qed.

Theorem eval_shru: binary_constructor_sound shru Val.shru.
Proof.
  red; intros until y; unfold shru; case (shru_match b); intros.
  InvEval. apply eval_shruimm; auto.
  TrivialExists. 
Qed.


Theorem eval_negf: unary_constructor_sound negf Val.negf.
Proof.
  red; intros. TrivialExists. 
Qed.

Theorem eval_absf: unary_constructor_sound absf Val.absf.
Proof.
  red; intros. TrivialExists. 
Qed.

Theorem eval_addf: binary_constructor_sound addf Val.addf.
Proof.
  red; intros; TrivialExists.
Qed.
 
Theorem eval_subf: binary_constructor_sound subf Val.subf.
Proof.
  red; intros; TrivialExists.
Qed.

Theorem eval_mulf: binary_constructor_sound mulf Val.mulf.
Proof.
  red; intros; TrivialExists.
Qed.

Theorem eval_divf: binary_constructor_sound divf Val.divf.
Proof.
  red; intros; TrivialExists.
Qed.

Section COMP_IMM.

Variable default: comparison -> int -> condition.
Variable intsem: comparison -> int -> int -> bool.
Variable sem: comparison -> val -> val -> val.

Hypothesis sem_int: forall c x y, sem c (Vint x) (Vint y) = Val.of_bool (intsem c x y).
Hypothesis sem_undef: forall c v, sem c Vundef v = Vundef.
Hypothesis sem_eq: forall x y, sem Ceq (Vint x) (Vint y) = Val.of_bool (Int.eq x y).
Hypothesis sem_ne: forall x y, sem Cne (Vint x) (Vint y) = Val.of_bool (negb (Int.eq x y)).
Hypothesis sem_default: forall c v n, sem c v (Vint n) = Val.of_optbool (eval_condition (default c n) (v :: nil) m).

Lemma eval_compimm:
  forall le c a n2 x,
  eval_expr ge sp e m le a x ->
  exists v, eval_expr ge sp e m le (compimm default intsem c a n2) v
         /\ Val.lessdef (sem c x (Vint n2)) v.
Proof.
  intros until x.
  unfold compimm; case (compimm_match c a); intros.
(* constant *)
  InvEval. rewrite sem_int. TrivialExists. simpl. destruct (intsem c0 n1 n2); auto.
(* eq cmp *)
  InvEval. inv H. simpl in H5. inv H5. 
  destruct (Int.eq_dec n2 Int.zero). subst n2. TrivialExists. 
  simpl. rewrite eval_negate_condition.
  destruct (eval_condition c0 vl m); simpl.
  unfold Vtrue, Vfalse. destruct b; simpl; rewrite sem_eq; auto.
  rewrite sem_undef; auto.
  destruct (Int.eq_dec n2 Int.one). subst n2. TrivialExists.
  simpl. destruct (eval_condition c0 vl m); simpl.
  unfold Vtrue, Vfalse. destruct b; simpl; rewrite sem_eq; auto.
  rewrite sem_undef; auto.
  exists (Vint Int.zero); split. EvalOp. 
  destruct (eval_condition c0 vl m); simpl.
  unfold Vtrue, Vfalse. destruct b; rewrite sem_eq; rewrite Int.eq_false; auto.
  rewrite sem_undef; auto.
(* ne cmp *)
  InvEval. inv H. simpl in H5. inv H5. 
  destruct (Int.eq_dec n2 Int.zero). subst n2. TrivialExists. 
  simpl. destruct (eval_condition c0 vl m); simpl.
  unfold Vtrue, Vfalse. destruct b; simpl; rewrite sem_ne; auto.
  rewrite sem_undef; auto.
  destruct (Int.eq_dec n2 Int.one). subst n2. TrivialExists.
  simpl. rewrite eval_negate_condition. destruct (eval_condition c0 vl m); simpl.
  unfold Vtrue, Vfalse. destruct b; simpl; rewrite sem_ne; auto.
  rewrite sem_undef; auto.
  exists (Vint Int.one); split. EvalOp. 
  destruct (eval_condition c0 vl m); simpl.
  unfold Vtrue, Vfalse. destruct b; rewrite sem_ne; rewrite Int.eq_false; auto.
  rewrite sem_undef; auto.
(* default *)
  TrivialExists. simpl. rewrite sem_default. auto.
Qed.

Hypothesis sem_swap:
  forall c x y, sem (swap_comparison c) x y = sem c y x.

Lemma eval_compimm_swap:
  forall le c a n2 x,
  eval_expr ge sp e m le a x ->
  exists v, eval_expr ge sp e m le (compimm default intsem (swap_comparison c) a n2) v
         /\ Val.lessdef (sem c (Vint n2) x) v.
Proof.
  intros. rewrite <- sem_swap. eapply eval_compimm; eauto. 
Qed.

End COMP_IMM.

Theorem eval_comp:
  forall c, binary_constructor_sound (comp c) (Val.cmp c).
Proof.
  intros; red; intros until y. unfold comp; case (comp_match a b); intros; InvEval.
  eapply eval_compimm_swap; eauto. 
  intros. unfold Val.cmp. rewrite Val.swap_cmp_bool; auto.
  eapply eval_compimm; eauto. 
  TrivialExists.
Qed.

Theorem eval_compu:
  forall c, binary_constructor_sound (compu c) (Val.cmpu (Mem.valid_pointer m) c).
Proof.
  intros; red; intros until y. unfold compu; case (compu_match a b); intros; InvEval.
  eapply eval_compimm_swap; eauto. 
  intros. unfold Val.cmpu. rewrite Val.swap_cmpu_bool; auto.
  eapply eval_compimm; eauto. 
  TrivialExists.
Qed.

Theorem eval_compf:
  forall c, binary_constructor_sound (compf c) (Val.cmpf c).
Proof.
  intros; red; intros. unfold compf. TrivialExists.
Qed.

Theorem eval_cast8signed: unary_constructor_sound cast8signed (Val.sign_ext 8).
Proof.
  red; intros. unfold cast8signed.
  exploit (eval_shlimm (Int.repr 24)); eauto. intros [v1 [A1 B1]].
  exploit (eval_shrimm (Int.repr 24)). eexact A1. intros [v2 [A2 B2]].
  exists v2; split; auto. 
  destruct x; simpl; auto. simpl in *. inv B1. simpl in *. inv B2. 
  rewrite Int.sign_ext_shr_shl. auto. compute; auto.
Qed.

Theorem eval_cast8unsigned: unary_constructor_sound cast8unsigned (Val.zero_ext 8).
Proof.
  red; intros until x. unfold cast8unsigned.
  rewrite Val.zero_ext_and. apply eval_andimm. compute; auto.
Qed.

Theorem eval_cast16signed: unary_constructor_sound cast16signed (Val.sign_ext 16).
Proof.
  red; intros. unfold cast16signed.
  exploit (eval_shlimm (Int.repr 16)); eauto. intros [v1 [A1 B1]].
  exploit (eval_shrimm (Int.repr 16)). eexact A1. intros [v2 [A2 B2]].
  exists v2; split; auto. 
  destruct x; simpl; auto. simpl in *. inv B1. simpl in *. inv B2. 
  rewrite Int.sign_ext_shr_shl. auto. compute; auto.
Qed.

Theorem eval_cast16unsigned: unary_constructor_sound cast16unsigned (Val.zero_ext 16).
Proof.
  red; intros until x. unfold cast8unsigned.
  rewrite Val.zero_ext_and. apply eval_andimm. compute; auto.
Qed.

Theorem eval_singleoffloat: unary_constructor_sound singleoffloat Val.singleoffloat.
Proof.
  red; intros. unfold singleoffloat. TrivialExists.
Qed.

Theorem eval_intoffloat:
  forall le a x y,
  eval_expr ge sp e m le a x ->
  Val.intoffloat x = Some y ->
  exists v, eval_expr ge sp e m le (intoffloat a) v /\ Val.lessdef y v.
Proof.
  intros; unfold intoffloat. TrivialExists. 
Qed.

Theorem eval_floatofint:
  forall le a x y,
  eval_expr ge sp e m le a x ->
  Val.floatofint x = Some y ->
  exists v, eval_expr ge sp e m le (floatofint a) v /\ Val.lessdef y v.
Proof.
  intros; unfold floatofint. TrivialExists. 
Qed.

Theorem eval_intuoffloat:
  forall le a x y,
  eval_expr ge sp e m le a x ->
  Val.intuoffloat x = Some y ->
  exists v, eval_expr ge sp e m le (intuoffloat a) v /\ Val.lessdef y v.
Proof.
  intros; unfold intuoffloat. TrivialExists. 
Qed.

Theorem eval_floatofintu:
  forall le a x y,
  eval_expr ge sp e m le a x ->
  Val.floatofintu x = Some y ->
  exists v, eval_expr ge sp e m le (floatofintu a) v /\ Val.lessdef y v.
Proof.
  intros; unfold floatofintu. TrivialExists. 
Qed.

Theorem eval_addressing:
  forall le chunk a v b ofs,
  eval_expr ge sp e m le a v ->
  v = Vptr b ofs ->
  match addressing chunk a with (mode, args) =>
    exists vl,
    eval_exprlist ge sp e m le args vl /\ 
    eval_addressing ge sp mode vl = Some v
  end.
Proof.
  intros until v. unfold addressing; case (addressing_match a); intros; InvEval.
  exists (@nil val). split. eauto with evalexpr. simpl. auto.
  exists (v1 :: nil); split. eauto with evalexpr. simpl. congruence.
  destruct (can_use_Aindexed2shift chunk); simpl.
  exists (v1 :: v0 :: nil); split. eauto with evalexpr. congruence.
  exists (Vptr b ofs :: nil); split. constructor. EvalOp. simpl. congruence. constructor.
  simpl. rewrite Int.add_zero; auto.
  destruct (can_use_Aindexed2 chunk); simpl.
  exists (v1 :: v0 :: nil); split. eauto with evalexpr. congruence.
  exists (Vptr b ofs :: nil); split. constructor. EvalOp. simpl. congruence. constructor.
  simpl. rewrite Int.add_zero; auto.
  exists (v :: nil); split. eauto with evalexpr. subst. simpl. rewrite Int.add_zero; auto.
Qed.

End CMCONSTR.