(* *********************************************************************)
(*                                                                     *)
(*              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 AST Integers Floats.
Require Import Values Memory Builtins Globalenvs.
Require Import Cminor Op CminorSel.
Require Import SelectOp.
Require Import OpHelpers OpHelpersproof.

Local Open Scope cminorsel_scope.
Local Transparent Archi.ptr64.

(** * 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 *)

Section CMCONSTR.
Variable prog: program.
Variable hf: helper_functions.
Hypothesis HELPERS: helper_functions_declared prog hf.
Let ge := Genv.globalenv prog.
Variable sp: val.
Variable e: env.
Variable m: mem.

(** 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 (Genv.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.offset_ptr sp 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.
  TrivialExists.
  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.
  subst x. rewrite Val.not_xor. rewrite Val.xor_assoc. TrivialExists.
  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 Ptrofs.add_zero. auto.
  case (addimm_match a); intros; InvEval; simpl; TrivialExists; simpl.
  rewrite Int.add_commut. auto.
  unfold Genv.symbol_address. destruct (Genv.find_symbol ge s); simpl; auto. rewrite Ptrofs.add_commut; auto.
  destruct sp; simpl; auto. rewrite Ptrofs.add_assoc. do 3 f_equal. apply Ptrofs.add_commut.
  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.
  subst. TrivialExists.
  subst. rewrite Val.add_commut. TrivialExists.
  TrivialExists.
Qed.

Theorem eval_rsubimm: forall n, unary_constructor_sound (rsubimm n) (fun v => Val.sub (Vint n) v).
Proof.
  red; intros until x. unfold rsubimm; case (rsubimm_match a); intros.
  InvEval. TrivialExists.
  InvEval. subst x. econstructor; split. EvalOp. unfold eval_operation; eauto.
  destruct v1; simpl; auto. rewrite Int.sub_add_r. rewrite <- Int.sub_add_opp.
  auto.
  InvEval. subst x. econstructor; split. EvalOp. simpl; eauto.
  fold (Val.sub (Vint m0) v1). destruct v1; simpl; auto.
  rewrite ! Int.sub_add_opp.  rewrite Int.neg_add_distr. rewrite Int.neg_involutive.
  rewrite (Int.add_permut i). rewrite (Int.add_commut i). auto.
  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.
  apply eval_rsubimm; auto.
  subst. TrivialExists.
  subst. TrivialExists.
  TrivialExists.
Qed.

Theorem eval_negint: unary_constructor_sound negint (fun v => Val.sub Vzero v).
Proof.
  red; intros. unfold negint. apply eval_rsubimm; auto.
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) eqn:?; simpl.
  destruct (shlimm_match a); intros.
  InvEval. simpl; rewrite Heqb. TrivialExists.
  destruct (Int.ltu (Int.add n n1) Int.iwordsize) 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) eqn:?; simpl.
  destruct (shrimm_match a); intros.
  InvEval. simpl; rewrite Heqb. TrivialExists.
  destruct (Int.ltu (Int.add n n1) Int.iwordsize) 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) eqn:?; simpl.
  destruct (shruimm_match a); intros.
  InvEval. simpl; rewrite Heqb. TrivialExists.
  destruct (Int.ltu (Int.add n n1) Int.iwordsize) 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) 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_mulhs: binary_constructor_sound mulhs Val.mulhs.
Proof.
  unfold mulhs; red; intros; TrivialExists.
Qed.
  
Theorem eval_mulhu: binary_constructor_sound mulhu Val.mulhu.
Proof.
  unfold mulhu; red; intros; 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.
  predSpec Int.eq Int.eq_spec n Int.mone.
  intros. exists x; split; auto.
  subst. destruct x; simpl; auto. rewrite Int.and_mone; 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.
  predSpec Int.eq Int.eq_spec n Int.mone.
  intros. exists (Vint Int.mone); split. EvalOp.
  destruct x; simpl; auto. subst n. rewrite Int.or_mone. 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) 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) eqn:?; auto.
  destruct (Int.ltu n2 Int.iwordsize) 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) 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) eqn:?; auto.
  destruct (Int.ltu n2 Int.iwordsize) 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.
  predSpec Int.eq Int.eq_spec n Int.mone.
  intros. subst n. rewrite <- Val.not_xor. apply eval_notint; auto.
  intros. destruct (xorimm_match a); intros; InvEval.
  TrivialExists. simpl. rewrite Int.xor_commut; auto.
  subst. rewrite Val.xor_assoc. simpl. rewrite Int.xor_commut. TrivialExists.
  subst x. rewrite Val.not_xor. rewrite Val.xor_assoc.
  rewrite (Val.xor_commut (Vint Int.mone)). 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_base:
  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_base a b) v /\ Val.lessdef z v.
Proof.
  intros. unfold divs_base. exists z; split. EvalOp. auto.
Qed.

Theorem eval_mods_base:
  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_base a b) v /\ Val.lessdef z v.
Proof.
  intros; unfold mods_base.
  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_divu_base:
  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_base a b) v /\ Val.lessdef z v.
Proof.
  intros. unfold divu_base. exists z; split. EvalOp. auto.
Qed.

Theorem eval_modu_base:
  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_base a b) v /\ Val.lessdef z v.
Proof.
  intros; unfold modu_base.
  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_shrximm:
  forall le a n x z,
  eval_expr ge sp e m le a x ->
  Val.shrx x (Vint n) = Some z ->
  exists v, eval_expr ge sp e m le (shrximm a n) v /\ Val.lessdef z v.
Proof.
  intros. unfold shrximm.
  predSpec Int.eq Int.eq_spec n Int.zero.
  subst n. exists x; split; auto.
  destruct x; simpl in H0; try discriminate.
  destruct (Int.ltu Int.zero (Int.repr 31)); inv H0.
  replace (Int.shrx i Int.zero) with i. auto.
  unfold Int.shrx, Int.divs. rewrite Int.shl_zero.
  change (Int.signed Int.one) with 1. rewrite Z.quot_1_r. rewrite Int.repr_signed; auto.
  econstructor; split. EvalOp. 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_negfs: unary_constructor_sound negfs Val.negfs.
Proof.
  red; intros. TrivialExists.
Qed.

Theorem eval_absfs: unary_constructor_sound absfs Val.absfs.
Proof.
  red; intros. TrivialExists.
Qed.

Theorem eval_addfs: binary_constructor_sound addfs Val.addfs.
Proof.
  red; intros; TrivialExists.
Qed.

Theorem eval_subfs: binary_constructor_sound subfs Val.subfs.
Proof.
  red; intros; TrivialExists.
Qed.

Theorem eval_mulfs: binary_constructor_sound mulfs Val.mulfs.
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_compfs:
  forall c, binary_constructor_sound (compfs c) (Val.cmpfs c).
Proof.
  intros; red; intros. unfold compfs. TrivialExists.
Qed.

Theorem eval_select:
  forall le ty cond al vl a1 v1 a2 v2 a b,
  select ty cond al a1 a2 = Some a ->
  eval_exprlist ge sp e m le al vl ->
  eval_expr ge sp e m le a1 v1 ->
  eval_expr ge sp e m le a2 v2 ->
  eval_condition cond vl m = Some b ->
  exists v, 
     eval_expr ge sp e m le a v
  /\ Val.lessdef (Val.select (Some b) v1 v2 ty) v.
Proof.
  unfold select; intros.
  destruct (match ty with Tint | Tfloat | Tsingle => true | _ => false end); inv H.
  rewrite <- H3; TrivialExists. 
Qed.

Theorem eval_cast8signed: unary_constructor_sound cast8signed (Val.sign_ext 8).
Proof.
  red; intros until x. unfold cast8signed; case (cast8signed_match a); intros.
  InvEval. TrivialExists.
  TrivialExists.
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. lia.
Qed.

Theorem eval_cast16signed: unary_constructor_sound cast16signed (Val.sign_ext 16).
Proof.
  red; intros until x. unfold cast16signed; case (cast16signed_match a); intros.
  InvEval. TrivialExists.
  TrivialExists.
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. lia.
Qed.

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

Theorem eval_floatofsingle: unary_constructor_sound floatofsingle Val.floatofsingle.
Proof.
  red; intros. unfold floatofsingle. 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 until y; unfold floatofint. case (floatofint_match a); intros.
  InvEval. simpl in H0. TrivialExists.
  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 until y; unfold floatofintu. case (floatofintu_match a); intros.
  InvEval. simpl in H0. TrivialExists.
  TrivialExists.
Qed.

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

Theorem eval_singleofint:
  forall le a x y,
  eval_expr ge sp e m le a x ->
  Val.singleofint x = Some y ->
  exists v, eval_expr ge sp e m le (singleofint a) v /\ Val.lessdef y v.
Proof.
  intros until y; unfold singleofint. case (singleofint_match a); intros.
  InvEval. simpl in H0. TrivialExists.
  TrivialExists.
Qed.

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

Theorem eval_singleofintu:
  forall le a x y,
  eval_expr ge sp e m le a x ->
  Val.singleofintu x = Some y ->
  exists v, eval_expr ge sp e m le (singleofintu a) v /\ Val.lessdef y v.
Proof.
  intros until y; unfold singleofintu. case (singleofintu_match a); intros.
  InvEval. simpl in H0. TrivialExists.
  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 Ptrofs.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 Ptrofs.add_zero; auto.
  exists (v :: nil); split. eauto with evalexpr. subst. simpl. rewrite Ptrofs.add_zero; auto.
Qed.

Theorem eval_builtin_arg:
  forall a v,
  eval_expr ge sp e m nil a v ->
  CminorSel.eval_builtin_arg ge sp e m (builtin_arg a) v.
Proof.
  intros until v. unfold builtin_arg; case (builtin_arg_match a); intros; InvEval.
- constructor.
- constructor.
- constructor.
- simpl in H5. inv H5. constructor.
- subst v. constructor; auto.
- inv H. InvEval. simpl in H6; inv H6. constructor; auto.
- inv H. InvEval. simpl in H6. rewrite <- Genv.shift_symbol_address_32 in H6 by auto.
  inv H6. constructor; auto.
- inv H. repeat constructor; auto.
- constructor; auto.
Qed.

(* floating-point division without HELPERS *)
Theorem eval_divf_base:
  forall le a b x 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 (divf_base a b) v /\ Val.lessdef (Val.divf x y) v.
Proof.
  intros; unfold divf_base.
  TrivialExists.
Qed.

Theorem eval_divfs_base:
  forall le a b x 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 (divfs_base a b) v /\ Val.lessdef (Val.divfs x y) v.
Proof.
  intros; unfold divfs_base.
  TrivialExists.
Qed.

(** Platform-specific known builtins *)

Theorem eval_platform_builtin:
  forall bf al a vl v le,
  platform_builtin bf al = Some a ->
  eval_exprlist ge sp e m le al vl ->
  platform_builtin_sem bf vl = Some v ->
  exists v', eval_expr ge sp e m le a v' /\ Val.lessdef v v'.
Proof.
  intros. discriminate.
Qed.

End CMCONSTR.