Fusion des modifications faites sur les branches "tailcalls" et "smallstep".

En particulier:
- Semantiques small-step depuis RTL jusqu'a PPC
- Cminor independant du processeur
- Ajout passes Selection et Reload
- Ajout des langages intermediaires CminorSel et LTLin correspondants
- Ajout des tailcalls depuis Cminor jusqu'a PPC


git-svn-id: https://yquem.inria.fr/compcert/svn/compcert/trunk@384 fca1b0fc-160b-0410-b1d3-a4f43f01ea2e

1 files changed, 0 insertions, 1207 deletions

diff --git a/backend/Cmconstrproof.v b/backend/Cmconstrproof.v
deleted file mode 100644
index 35b3d8a0..00000000
--- a/backend/Cmconstrproof.v
+++ /dev/null
@@ -1,1207 +0,0 @@
-(** Correctness of the Cminor smart constructors.  This file states
-  evaluation rules for the smart constructors, for instance that [add
-  a b] evaluates to [Vint(Int.add i j)] if [a] evaluates to [Vint i]
-  and [b] to [Vint j].  It then proves that these rules are
-  admissible, that is, satisfied for all possible choices of [a] and
-  [b].  The Cminor producer can then use these evaluation rules
-  (theorems) to reason about the execution of terms produced by the
-  smart constructors.
-*)
-
-Require Import Coqlib.
-Require Import Compare_dec.
-Require Import Maps.
-Require Import AST.
-Require Import Integers.
-Require Import Floats.
-Require Import Values.
-Require Import Mem.
-Require Import Events.
-Require Import Op.
-Require Import Globalenvs.
-Require Import Cminor.
-Require Import Cmconstr.
-
-Section CMCONSTR.
-
-Variable ge: Cminor.genv.
-
-(** * Lifting of let-bound variables *)
-
-Inductive insert_lenv: letenv -> nat -> val -> letenv -> Prop :=
-  | insert_lenv_0:
-      forall le v,
-      insert_lenv le O v (v :: le)
-  | insert_lenv_S:
-      forall le p w le' v,
-      insert_lenv le p w le' ->
-      insert_lenv (v :: le) (S p) w (v :: le').
-
-Lemma insert_lenv_lookup1:
-  forall le p w le',
-  insert_lenv le p w le' ->
-  forall n v,
-  nth_error le n = Some v -> (p > n)%nat ->
-  nth_error le' n = Some v.
-Proof.
-  induction 1; intros.
-  omegaContradiction.
-  destruct n; simpl; simpl in H0. auto. 
-  apply IHinsert_lenv. auto. omega.
-Qed.
-
-Lemma insert_lenv_lookup2:
-  forall le p w le',
-  insert_lenv le p w le' ->
-  forall n v,
-  nth_error le n = Some v -> (p <= n)%nat ->
-  nth_error le' (S n) = Some v.
-Proof.
-  induction 1; intros.
-  simpl. assumption.
-  simpl. destruct n. omegaContradiction. 
-  apply IHinsert_lenv. exact H0. omega.
-Qed.
-
-Scheme eval_expr_ind_3 := Minimality for eval_expr Sort Prop
-  with eval_condexpr_ind_3 := Minimality for eval_condexpr Sort Prop
-  with eval_exprlist_ind_3 := Minimality for eval_exprlist Sort Prop.
-
-Hint Resolve eval_Evar eval_Eop eval_Eload eval_Estore
-             eval_Ecall eval_Econdition eval_Ealloc
-             eval_Elet eval_Eletvar 
-             eval_CEtrue eval_CEfalse eval_CEcond
-             eval_CEcondition eval_Enil eval_Econs: evalexpr.
-
-Lemma eval_list_one:
-  forall sp le e m1 a t m2 v,
-  eval_expr ge sp le e m1 a t m2 v ->
-  eval_exprlist ge sp le e m1 (a ::: Enil) t m2 (v :: nil).
-Proof.
-  intros. econstructor. eauto. constructor. traceEq.
-Qed.
-
-Lemma eval_list_two:
-  forall sp le e m1 a1 t1 m2 v1 a2 t2 m3 v2 t,
-  eval_expr ge sp le e m1 a1 t1 m2 v1 ->
-  eval_expr ge sp le e m2 a2 t2 m3 v2 ->
-  t = t1 ** t2 ->
-  eval_exprlist ge sp le e m1 (a1 ::: a2 ::: Enil) t m3 (v1 :: v2 :: nil).
-Proof.
-  intros. econstructor. eauto. econstructor. eauto. constructor. 
-  reflexivity. traceEq.
-Qed.
-
-Lemma eval_list_three:
-  forall sp le e m1 a1 t1 m2 v1 a2 t2 m3 v2 a3 t3 m4 v3 t,
-  eval_expr ge sp le e m1 a1 t1 m2 v1 ->
-  eval_expr ge sp le e m2 a2 t2 m3 v2 ->
-  eval_expr ge sp le e m3 a3 t3 m4 v3 ->
-  t = t1 ** t2 ** t3 ->
-  eval_exprlist ge sp le e m1 (a1 ::: a2 ::: a3 ::: Enil) t m4 (v1 :: v2 :: v3 :: nil).
-Proof.
-  intros. econstructor. eauto. econstructor. eauto. econstructor. eauto. constructor. 
-  reflexivity. reflexivity. traceEq.
-Qed.
-
-Hint Resolve eval_list_one eval_list_two eval_list_three: evalexpr.
-
-Lemma eval_lift_expr:
-  forall w sp le e m1 a t m2 v,
-  eval_expr ge sp le e m1 a t m2 v ->
-  forall p le', insert_lenv le p w le' ->
-  eval_expr ge sp le' e m1 (lift_expr p a) t m2 v.
-Proof.
-  intros w.
-  apply (eval_expr_ind_3 ge
-    (fun sp le e m1 a t m2 v =>
-      forall p le', insert_lenv le p w le' ->
-      eval_expr ge sp le' e m1 (lift_expr p a) t m2 v)
-    (fun sp le e m1 a t m2 vb =>
-      forall p le', insert_lenv le p w le' ->
-      eval_condexpr ge sp le' e m1 (lift_condexpr p a) t m2 vb)
-    (fun sp le e m1 al t m2 vl =>
-      forall p le', insert_lenv le p w le' ->
-      eval_exprlist ge sp le' e m1 (lift_exprlist p al) t m2 vl));
-  simpl; intros; eauto with evalexpr.
-
-  destruct v1; eapply eval_Econdition;
-  eauto with evalexpr; simpl; eauto with evalexpr.
-
-  eapply eval_Elet. eauto. apply H2. apply insert_lenv_S; auto. auto.
-
-  case (le_gt_dec p n); intro. 
-  apply eval_Eletvar. eapply insert_lenv_lookup2; eauto.
-  apply eval_Eletvar. eapply insert_lenv_lookup1; eauto.
-
-  destruct vb1; eapply eval_CEcondition;
-  eauto with evalexpr; simpl; eauto with evalexpr.
-Qed.
-
-Lemma eval_lift:
-  forall sp le e m1 a t m2 v w,
-  eval_expr ge sp le e m1 a t m2 v ->
-  eval_expr ge sp (w::le) e m1 (lift a) t m2 v.
-Proof.
-  intros. unfold lift. eapply eval_lift_expr.
-  eexact H. apply insert_lenv_0. 
-Qed.
-Hint Resolve eval_lift: evalexpr.
-
-(** * Useful lemmas and tactics *)
-
-Ltac EvalOp := eapply eval_Eop; eauto with evalexpr.
-
-Ltac TrivialOp cstr :=
-  unfold cstr; intros; EvalOp.
-
-(** The following are trivial lemmas and custom tactics that help
-  perform backward (inversion) and forward reasoning over the evaluation
-  of operator applications. *)  
-
-Lemma inv_eval_Eop_0:
-  forall sp le e m1 op t m2 v,
-  eval_expr ge sp le e m1 (Eop op Enil) t m2 v ->
-  t = E0 /\ m2 = m1 /\ eval_operation ge sp op nil = Some v.
-Proof.
-  intros. inversion H. inversion H6. 
-  intuition. congruence.
-Qed.
-  
-Lemma inv_eval_Eop_1:
-  forall sp le e m1 op t a1 m2 v,
-  eval_expr ge sp le e m1 (Eop op (a1 ::: Enil)) t m2 v ->
-  exists v1,
-  eval_expr ge sp le e m1 a1 t m2 v1 /\
-  eval_operation ge sp op (v1 :: nil) = Some v.
-Proof.
-  intros. 
-  inversion H. inversion H6. inversion H18. 
-  subst. exists v1; intuition. rewrite E0_right. auto.
-Qed.
-
-Lemma inv_eval_Eop_2:
-  forall sp le e m1 op a1 a2 t3 m3 v,
-  eval_expr ge sp le e m1 (Eop op (a1 ::: a2 ::: Enil)) t3 m3 v ->
-  exists t1, exists t2, exists m2, exists v1, exists v2,
-  eval_expr ge sp le e m1 a1 t1 m2 v1 /\
-  eval_expr ge sp le e m2 a2 t2 m3 v2 /\
-  t3 = t1 ** t2 /\
-  eval_operation ge sp op (v1 :: v2 :: nil) = Some v.
-Proof.
-  intros. 
-  inversion H. subst. inversion H6. subst. inversion H8. subst.
-  inversion H11. subst. 
-  exists t1; exists t0; exists m0; exists v0; exists v1.
-  intuition. traceEq.
-Qed.
-
-Ltac SimplEval :=
-  match goal with
-  | [ |- (eval_expr _ ?sp ?le ?e ?m1 (Eop ?op Enil) ?t ?m2 ?v) -> _] =>
-      intro XX1;
-      generalize (inv_eval_Eop_0 sp le e m1 op t m2 v XX1);
-      clear XX1;
-      intros [XX1 [XX2 XX3]];
-      subst t m2; simpl in XX3; 
-      try (simplify_eq XX3; clear XX3;
-      let EQ := fresh "EQ" in (intro EQ; rewrite EQ))
-  | [ |- (eval_expr _ ?sp ?le ?e ?m1 (Eop ?op (?a1 ::: Enil)) ?t ?m2 ?v) -> _] =>
-      intro XX1;
-      generalize (inv_eval_Eop_1 sp le e m1 op t a1 m2 v XX1);
-      clear XX1;
-      let v1 := fresh "v" in let EV := fresh "EV" in
-      let EQ := fresh "EQ" in
-      (intros [v1 [EV EQ]]; simpl in EQ)
-  | [ |- (eval_expr _ ?sp ?le ?e ?m1 (Eop ?op (?a1 ::: ?a2 ::: Enil)) ?t ?m2 ?v) -> _] =>
-      intro XX1;
-      generalize (inv_eval_Eop_2 sp le e m1 op a1 a2 t m2 v XX1);
-      clear XX1;
-      let t1 := fresh "t" in let t2 := fresh "t" in
-      let m := fresh "m" in
-      let v1 := fresh "v" in let v2 := fresh "v" in
-      let EV1 := fresh "EV" in let EV2 := fresh "EV" in
-      let EQ := fresh "EQ" in let TR := fresh "TR" in
-      (intros [t1 [t2 [m [v1 [v2 [EV1 [EV2 [TR EQ]]]]]]]]; simpl in EQ)
-  | _ => idtac
-  end.
-
-Ltac InvEval H :=
-  generalize H; SimplEval; clear H.
-
-(** ** Admissible evaluation rules for the smart constructors *)
-
-(** 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.
-*)
-
-Theorem eval_negint:
-  forall sp le e m1 a t m2 x,
-  eval_expr ge sp le e m1 a t m2 (Vint x) ->
-  eval_expr ge sp le e m1 (negint a) t m2 (Vint (Int.neg x)).
-Proof.
-  TrivialOp negint. 
-Qed.
-
-Theorem eval_negfloat:
-  forall sp le e m1 a t m2 x,
-  eval_expr ge sp le e m1 a t m2 (Vfloat x) ->
-  eval_expr ge sp le e m1 (negfloat a) t m2 (Vfloat (Float.neg x)).
-Proof.
-  TrivialOp negfloat.
-Qed.
-
-Theorem eval_absfloat:
-  forall sp le e m1 a t m2 x,
-  eval_expr ge sp le e m1 a t m2 (Vfloat x) ->
-  eval_expr ge sp le e m1 (absfloat a) t m2 (Vfloat (Float.abs x)).
-Proof.
-  TrivialOp absfloat.
-Qed.
-
-Theorem eval_intoffloat:
-  forall sp le e m1 a t m2 x,
-  eval_expr ge sp le e m1 a t m2 (Vfloat x) ->
-  eval_expr ge sp le e m1 (intoffloat a) t m2 (Vint (Float.intoffloat x)).
-Proof.
-  TrivialOp intoffloat.
-Qed.
-
-Theorem eval_floatofint:
-  forall sp le e m1 a t m2 x,
-  eval_expr ge sp le e m1 a t m2 (Vint x) ->
-  eval_expr ge sp le e m1 (floatofint a) t m2 (Vfloat (Float.floatofint x)).
-Proof.
-  TrivialOp floatofint.
-Qed.
-
-Theorem eval_floatofintu:
-  forall sp le e m1 a t m2 x,
-  eval_expr ge sp le e m1 a t m2 (Vint x) ->
-  eval_expr ge sp le e m1 (floatofintu a) t m2 (Vfloat (Float.floatofintu x)).
-Proof.
-  TrivialOp floatofintu.
-Qed.
-
-Theorem eval_notint:
-  forall sp le e m1 a t m2 x,
-  eval_expr ge sp le e m1 a t m2 (Vint x) ->
-  eval_expr ge sp le e m1 (notint a) t m2 (Vint (Int.not x)).
-Proof.
-  unfold notint; intros until x; case (notint_match a); intros.
-  InvEval H. FuncInv. EvalOp. simpl. congruence. 
-  InvEval H. FuncInv. EvalOp. simpl. congruence. 
-  InvEval H. FuncInv. EvalOp. simpl. congruence. 
-  eapply eval_Elet. eexact H. 
-  eapply eval_Eop.
-  eapply eval_Econs. apply eval_Eletvar. simpl. reflexivity.
-  eapply eval_Econs. apply eval_Eletvar. simpl. reflexivity.
-  apply eval_Enil. reflexivity. reflexivity. 
-  simpl. rewrite Int.or_idem. auto. traceEq.
-Qed.
-
-Lemma eval_notbool_base:
-  forall sp le e m1 a t m2 v b,
-  eval_expr ge sp le e m1 a t m2 v ->
-  Val.bool_of_val v b ->
-  eval_expr ge sp le e m1 (notbool_base a) t m2 (Val.of_bool (negb b)).
-Proof. 
-  TrivialOp notbool_base. simpl. 
-  inversion H0. 
-  rewrite Int.eq_false; auto.
-  rewrite Int.eq_true; auto.
-  reflexivity.
-Qed.
-
-Hint Resolve Val.bool_of_true_val Val.bool_of_false_val
-             Val.bool_of_true_val_inv Val.bool_of_false_val_inv: valboolof.
-
-Theorem eval_notbool:
-  forall a sp le e m1 t m2 v b,
-  eval_expr ge sp le e m1 a t m2 v ->
-  Val.bool_of_val v b ->
-  eval_expr ge sp le e m1 (notbool a) t m2 (Val.of_bool (negb b)).
-Proof.
-  assert (N1: forall v b, Val.is_false v -> Val.bool_of_val v b -> Val.is_true (Val.of_bool (negb b))).
-    intros. inversion H0; simpl; auto; subst v; simpl in H.
-    congruence. apply Int.one_not_zero. contradiction.
-  assert (N2: forall v b, Val.is_true v -> Val.bool_of_val v b -> Val.is_false (Val.of_bool (negb b))).
-    intros. inversion H0; simpl; auto; subst v; simpl in H.
-    congruence. 
-
-  induction a; simpl; intros; try (eapply eval_notbool_base; eauto).
-  destruct o; try (eapply eval_notbool_base; eauto).
-
-  destruct e. InvEval H. injection XX3; clear XX3; intro; subst v.
-  inversion H0. rewrite Int.eq_false; auto. 
-  simpl; eauto with evalexpr.
-  rewrite Int.eq_true; simpl; eauto with evalexpr.
-  eapply eval_notbool_base; eauto.
-
-  inversion H. subst. 
-  simpl in H11. eapply eval_Eop; eauto.
-  simpl. caseEq (eval_condition c vl); intros.
-  rewrite H1 in H11. 
-  assert (b0 = b). 
-  destruct b0; inversion H11; subst v; inversion H0; auto.
-  subst b0. rewrite (Op.eval_negate_condition _ _ H1). 
-  destruct b; reflexivity.
-  rewrite H1 in H11; discriminate.
-
-  inversion H; eauto 10 with evalexpr valboolof.
-  inversion H; eauto 10 with evalexpr valboolof.
-
-  inversion H. subst. eapply eval_Econdition with (t2 := t8). eexact H34.
-  destruct v4; eauto. auto.
-Qed.
-
-Theorem eval_addimm:
-  forall sp le e m1 n a t m2 x,
-  eval_expr ge sp le e m1 a t m2 (Vint x) ->
-  eval_expr ge sp le e m1 (addimm n a) t m2 (Vint (Int.add x n)).
-Proof.
-  unfold addimm; intros until x.
-  generalize (Int.eq_spec n Int.zero). case (Int.eq n Int.zero); intro.
-  subst n. rewrite Int.add_zero. auto.
-  case (addimm_match a); intros.
-  InvEval H0. EvalOp. simpl. rewrite Int.add_commut. auto.
-  InvEval H0. destruct (Genv.find_symbol ge s); discriminate.
-  InvEval H0. 
-  destruct sp; simpl in XX3; discriminate.
-  InvEval H0. FuncInv. EvalOp. simpl. subst x. 
-  rewrite Int.add_assoc. decEq; decEq; decEq. apply Int.add_commut.
-  EvalOp. 
-Qed. 
-
-Theorem eval_addimm_ptr:
-  forall sp le e m1 n t a m2 b ofs,
-  eval_expr ge sp le e m1 a t m2 (Vptr b ofs) ->
-  eval_expr ge sp le e m1 (addimm n a) t m2 (Vptr b (Int.add ofs n)).
-Proof.
-  unfold addimm; intros until ofs.
-  generalize (Int.eq_spec n Int.zero). case (Int.eq n Int.zero); intro.
-  subst n. rewrite Int.add_zero. auto.
-  case (addimm_match a); intros.
-  InvEval H0. 
-  InvEval H0. EvalOp. simpl. 
-    destruct (Genv.find_symbol ge s). 
-    rewrite Int.add_commut. congruence.
-    discriminate.
-  InvEval H0. destruct sp; simpl in XX3; try discriminate.
-  inversion XX3. EvalOp. simpl. decEq. decEq. 
-  rewrite Int.add_assoc. decEq. apply Int.add_commut.
-  InvEval H0. FuncInv. subst b0; subst ofs. EvalOp. simpl. 
-    rewrite (Int.add_commut n m). rewrite Int.add_assoc. auto.
-  EvalOp. 
-Qed.
-
-Theorem eval_add:
-  forall sp le e m1 a t1 m2 x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vint x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vint y) ->
-  eval_expr ge sp le e m1 (add a b) (t1**t2) m3 (Vint (Int.add x y)).
-Proof.
-  intros until y. unfold add; case (add_match a b); intros.
-  InvEval H. rewrite Int.add_commut. apply eval_addimm. 
-  rewrite E0_left; assumption.
-  InvEval H. FuncInv. InvEval H0. FuncInv. 
-    replace (Int.add x y) with (Int.add (Int.add i i0) (Int.add n1 n2)).
-    apply eval_addimm. EvalOp. 
-    subst x; subst y. 
-    repeat rewrite Int.add_assoc. decEq. apply Int.add_permut. 
-  InvEval H. FuncInv. 
-    replace (Int.add x y) with (Int.add (Int.add i y) n1).
-    apply eval_addimm. EvalOp.
-    subst x. repeat rewrite Int.add_assoc. decEq. apply Int.add_commut.
-  InvEval H0. FuncInv.
-    apply eval_addimm. rewrite E0_right. auto.
-  InvEval H0. FuncInv. 
-    replace (Int.add x y) with (Int.add (Int.add x i) n2).
-    apply eval_addimm. EvalOp.
-    subst y. rewrite Int.add_assoc. auto.
-  EvalOp.
-Qed.
-
-Theorem eval_add_ptr:
-  forall sp le e m1 a t1 m2 p x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vptr p x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vint y) ->
-  eval_expr ge sp le e m1 (add a b) (t1**t2) m3 (Vptr p (Int.add x y)).
-Proof.
-  intros until y. unfold add; case (add_match a b); intros.
-  InvEval H. 
-  InvEval H. FuncInv. InvEval H0. FuncInv. 
-    replace (Int.add x y) with (Int.add (Int.add i i0) (Int.add n1 n2)).
-    apply eval_addimm_ptr. subst b0. EvalOp. 
-    subst x; subst y.
-    repeat rewrite Int.add_assoc. decEq. apply Int.add_permut. 
-  InvEval H. FuncInv. 
-    replace (Int.add x y) with (Int.add (Int.add i y) n1).
-    apply eval_addimm_ptr. subst b0. EvalOp.
-    subst x. repeat rewrite Int.add_assoc. decEq. apply Int.add_commut.
-  InvEval H0. apply eval_addimm_ptr. rewrite E0_right. auto.
-  InvEval H0. FuncInv. 
-    replace (Int.add x y) with (Int.add (Int.add x i) n2).
-    apply eval_addimm_ptr. EvalOp.
-    subst y. rewrite Int.add_assoc. auto.
-  EvalOp.
-Qed.
-
-Theorem eval_add_ptr_2:
-  forall sp le e m1 a t1 m2 p x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vint x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vptr p y) ->
-  eval_expr ge sp le e m1 (add a b) (t1**t2) m3 (Vptr p (Int.add y x)).
-Proof.
-  intros until y. unfold add; case (add_match a b); intros.
-  InvEval H. 
-    apply eval_addimm_ptr. rewrite E0_left. auto.
-  InvEval H. FuncInv. InvEval H0. FuncInv. 
-    replace (Int.add y x) with (Int.add (Int.add i0 i) (Int.add n1 n2)).
-    apply eval_addimm_ptr. subst b0. EvalOp. 
-    subst x; subst y.
-    repeat rewrite Int.add_assoc. decEq. 
-    rewrite (Int.add_commut n1 n2). apply Int.add_permut. 
-  InvEval H. FuncInv. 
-    replace (Int.add y x) with (Int.add (Int.add y i) n1).
-    apply eval_addimm_ptr. EvalOp. 
-    subst x. repeat rewrite Int.add_assoc. auto.
-  InvEval H0. 
-  InvEval H0. FuncInv. 
-    replace (Int.add y x) with (Int.add (Int.add i x) n2).
-    apply eval_addimm_ptr. EvalOp. subst b0; reflexivity.
-    subst y. repeat rewrite Int.add_assoc. decEq. apply Int.add_commut.
-  EvalOp.
-Qed.
-
-Theorem eval_sub:
-  forall sp le e m1 a t1 m2 x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vint x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vint y) ->
-  eval_expr ge sp le e m1 (sub a b) (t1**t2) m3 (Vint (Int.sub x y)).
-Proof.
-  intros until y.
-  unfold sub; case (sub_match a b); intros.
-  InvEval H0. rewrite Int.sub_add_opp. 
-    apply eval_addimm. rewrite E0_right. assumption.
-  InvEval H. FuncInv. InvEval H0. FuncInv.
-    replace (Int.sub x y) with (Int.add (Int.sub i i0) (Int.sub n1 n2)).
-    apply eval_addimm. EvalOp.
-    subst x; subst y.
-    repeat rewrite Int.sub_add_opp.
-    repeat rewrite Int.add_assoc. decEq. 
-    rewrite Int.add_permut. decEq. symmetry. apply Int.neg_add_distr.
-  InvEval H. FuncInv. 
-    replace (Int.sub x y) with (Int.add (Int.sub i y) n1).
-    apply eval_addimm. EvalOp.
-    subst x. rewrite Int.sub_add_l. auto.
-  InvEval H0. FuncInv. 
-    replace (Int.sub x y) with (Int.add (Int.sub x i) (Int.neg n2)).
-    apply eval_addimm. EvalOp.
-    subst y. rewrite (Int.add_commut i n2). symmetry. apply Int.sub_add_r. 
-  EvalOp.
-Qed.
-
-Theorem eval_sub_ptr_int:
-  forall sp le e m1 a t1 m2 p x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vptr p x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vint y) ->
-  eval_expr ge sp le e m1 (sub a b) (t1**t2) m3 (Vptr p (Int.sub x y)).
-Proof.
-  intros until y.
-  unfold sub; case (sub_match a b); intros.
-  InvEval H0. rewrite Int.sub_add_opp. 
-    apply eval_addimm_ptr. rewrite E0_right. assumption.
-  InvEval H. FuncInv. InvEval H0. FuncInv.
-    subst b0.
-    replace (Int.sub x y) with (Int.add (Int.sub i i0) (Int.sub n1 n2)).
-    apply eval_addimm_ptr. EvalOp.
-    subst x; subst y.
-    repeat rewrite Int.sub_add_opp.
-    repeat rewrite Int.add_assoc. decEq. 
-    rewrite Int.add_permut. decEq. symmetry. apply Int.neg_add_distr.
-  InvEval H. FuncInv. subst b0.
-    replace (Int.sub x y) with (Int.add (Int.sub i y) n1).
-    apply eval_addimm_ptr. EvalOp.
-    subst x. rewrite Int.sub_add_l. auto.
-  InvEval H0. FuncInv. 
-    replace (Int.sub x y) with (Int.add (Int.sub x i) (Int.neg n2)).
-    apply eval_addimm_ptr. EvalOp.
-    subst y. rewrite (Int.add_commut i n2). symmetry. apply Int.sub_add_r. 
-  EvalOp.
-Qed.
-
-Theorem eval_sub_ptr_ptr:
-  forall sp le e m1 a t1 m2 p x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vptr p x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vptr p y) ->
-  eval_expr ge sp le e m1 (sub a b) (t1**t2) m3 (Vint (Int.sub x y)).
-Proof.
-  intros until y.
-  unfold sub; case (sub_match a b); intros.
-  InvEval H0. 
-  InvEval H. FuncInv. InvEval H0. FuncInv.
-    replace (Int.sub x y) with (Int.add (Int.sub i i0) (Int.sub n1 n2)).
-    apply eval_addimm. EvalOp. 
-    simpl; unfold eq_block. subst b0; subst b1; rewrite zeq_true. auto.
-    subst x; subst y.
-    repeat rewrite Int.sub_add_opp.
-    repeat rewrite Int.add_assoc. decEq. 
-    rewrite Int.add_permut. decEq. symmetry. apply Int.neg_add_distr.
-  InvEval H. FuncInv. subst b0.
-    replace (Int.sub x y) with (Int.add (Int.sub i y) n1).
-    apply eval_addimm. EvalOp.
-    simpl. unfold eq_block. rewrite zeq_true. auto.
-    subst x. rewrite Int.sub_add_l. auto.
-  InvEval H0. FuncInv. subst b0.
-    replace (Int.sub x y) with (Int.add (Int.sub x i) (Int.neg n2)).
-    apply eval_addimm. EvalOp.
-    simpl. unfold eq_block. rewrite zeq_true. auto.
-    subst y. rewrite (Int.add_commut i n2). symmetry. apply Int.sub_add_r. 
-  EvalOp. simpl. unfold eq_block. rewrite zeq_true. auto.
-Qed.
-
-Lemma eval_rolm:
-  forall sp le e m1 a amount mask t m2 x,
-  eval_expr ge sp le e m1 a t m2 (Vint x) ->
-  eval_expr ge sp le e m1 (rolm a amount mask) t m2 (Vint (Int.rolm x amount mask)).
-Proof.
-  intros until x. unfold rolm; case (rolm_match a); intros.
-  InvEval H. eauto with evalexpr. 
-  case (Int.is_rlw_mask (Int.and (Int.rol mask1 amount) mask)).
-  InvEval H. FuncInv. EvalOp. simpl. subst x. 
-  decEq. decEq. 
-  replace (Int.and (Int.add amount1 amount) (Int.repr 31))
-     with (Int.modu (Int.add amount1 amount) (Int.repr 32)).
-  symmetry. apply Int.rolm_rolm. 
-  change (Int.repr 31) with (Int.sub (Int.repr 32) Int.one).
-  apply Int.modu_and with (Int.repr 5). reflexivity.
-  EvalOp. 
-  EvalOp.
-Qed.
-
-Theorem eval_shlimm:
-  forall sp le e m1 a n t m2 x,
-  eval_expr ge sp le e m1 a t m2 (Vint x) ->
-  Int.ltu n (Int.repr 32) = true ->
-  eval_expr ge sp le e m1 (shlimm a n) t m2 (Vint (Int.shl x n)).
-Proof.
-  intros.  unfold shlimm.
-  generalize (Int.eq_spec n Int.zero); case (Int.eq n Int.zero); intro.
-  subst n. rewrite Int.shl_zero. auto.
-  rewrite H0.
-  replace (Int.shl x n) with (Int.rolm x n (Int.shl Int.mone n)).
-  apply eval_rolm. auto. symmetry. apply Int.shl_rolm. exact H0.
-Qed.
-
-Theorem eval_shruimm:
-  forall sp le e m1 a n t m2 x,
-  eval_expr ge sp le e m1 a t m2 (Vint x) ->
-  Int.ltu n (Int.repr 32) = true ->
-  eval_expr ge sp le e m1 (shruimm a n) t m2 (Vint (Int.shru x n)).
-Proof.
-  intros.  unfold shruimm.
-  generalize (Int.eq_spec n Int.zero); case (Int.eq n Int.zero); intro.
-  subst n. rewrite Int.shru_zero. auto.
-  rewrite H0.
-  replace (Int.shru x n) with (Int.rolm x (Int.sub (Int.repr 32) n) (Int.shru Int.mone n)).
-  apply eval_rolm. auto. symmetry. apply Int.shru_rolm. exact H0.
-Qed.
-
-Lemma eval_mulimm_base:
-  forall sp le e m1 a t n m2 x,
-  eval_expr ge sp le e m1 a t m2 (Vint x) ->
-  eval_expr ge sp le e m1 (mulimm_base n a) t m2 (Vint (Int.mul x n)).
-Proof.
-  intros; unfold mulimm_base. 
-  generalize (Int.one_bits_decomp n). 
-  generalize (Int.one_bits_range n).
-  change (Z_of_nat wordsize) with 32.
-  destruct (Int.one_bits n).
-  intros. EvalOp. 
-  destruct l.
-  intros. rewrite H1. simpl. 
-  rewrite Int.add_zero. rewrite <- Int.shl_mul.
-  apply eval_shlimm. auto. auto with coqlib. 
-  destruct l.
-  intros. apply eval_Elet with t m2 (Vint x) E0. auto.
-  rewrite H1. simpl. rewrite Int.add_zero. 
-  rewrite Int.mul_add_distr_r.
-  rewrite <- Int.shl_mul.
-  rewrite <- Int.shl_mul.
-  EvalOp. eapply eval_Econs. 
-  apply eval_shlimm. apply eval_Eletvar. simpl. reflexivity. 
-  auto with coqlib.
-  eapply eval_Econs.
-  apply eval_shlimm. apply eval_Eletvar. simpl. reflexivity.
-  auto with coqlib.
-  auto with evalexpr.
-  reflexivity. traceEq. reflexivity. traceEq. 
-  intros. EvalOp. 
-Qed.
-
-Theorem eval_mulimm:
-  forall sp le e m1 a n t m2 x,
-  eval_expr ge sp le e m1 a t m2 (Vint x) ->
-  eval_expr ge sp le e m1 (mulimm n a) t m2 (Vint (Int.mul x n)).
-Proof.
-  intros until x; unfold mulimm.
-  generalize (Int.eq_spec n Int.zero); case (Int.eq n Int.zero); intro.
-  subst n. rewrite Int.mul_zero. 
-  intro. eapply eval_Elet; eauto with evalexpr. traceEq.
-  generalize (Int.eq_spec n Int.one); case (Int.eq n Int.one); intro.
-  subst n. rewrite Int.mul_one. auto.
-  case (mulimm_match a); intros.
-  InvEval H1. EvalOp. rewrite Int.mul_commut. reflexivity.
-  InvEval H1. FuncInv. 
-  replace (Int.mul x n) with (Int.add (Int.mul i n) (Int.mul n n2)).
-  apply eval_addimm. apply eval_mulimm_base. auto.
-  subst x. rewrite Int.mul_add_distr_l. decEq. apply Int.mul_commut.
-  apply eval_mulimm_base. assumption.
-Qed.
-
-Theorem eval_mul:
-  forall sp le e m1 a t1 m2 x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vint x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vint y) ->
-  eval_expr ge sp le e m1 (mul a b) (t1**t2) m3 (Vint (Int.mul x y)).
-Proof.
-  intros until y.
-  unfold mul; case (mul_match a b); intros.
-  InvEval H. rewrite Int.mul_commut. apply eval_mulimm. 
-  rewrite E0_left; auto.
-  InvEval H0. rewrite E0_right. apply eval_mulimm. auto.
-  EvalOp.
-Qed.
-
-Theorem eval_divs:
-  forall sp le e m1 a t1 m2 x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vint x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vint y) ->
-  y <> Int.zero ->
-  eval_expr ge sp le e m1 (divs a b) (t1**t2) m3 (Vint (Int.divs x y)).
-Proof.
-  TrivialOp divs. simpl. 
-  predSpec Int.eq Int.eq_spec y Int.zero. contradiction. auto.
-Qed.
-
-Lemma eval_mod_aux:
-  forall divop semdivop,
-  (forall sp x y,
-   y <> Int.zero ->
-   eval_operation ge sp divop (Vint x :: Vint y :: nil) =
-   Some (Vint (semdivop x y))) ->
-  forall sp le e m1 a t1 m2 x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vint x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vint y) ->
-  y <> Int.zero ->
-  eval_expr ge sp le e m1 (mod_aux divop a b) (t1**t2) m3
-   (Vint (Int.sub x (Int.mul (semdivop x y) 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. reflexivity. reflexivity. 
-  apply H. assumption.
-  eapply eval_Econs. apply eval_Eletvar. simpl; reflexivity.
-  apply eval_Enil. reflexivity. reflexivity. 
-  simpl; reflexivity. apply eval_Enil. 
-  reflexivity. reflexivity. reflexivity.
-  reflexivity. traceEq.
-Qed.
-
-Theorem eval_mods:
-  forall sp le e m1 a t1 m2 x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vint x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vint y) ->
-  y <> Int.zero ->
-  eval_expr ge sp le e m1 (mods a b) (t1**t2) m3 (Vint (Int.mods x y)).
-Proof.
-  intros; unfold mods. 
-  rewrite Int.mods_divs. 
-  eapply eval_mod_aux; eauto. 
-  intros. simpl. predSpec Int.eq Int.eq_spec y0 Int.zero. 
-  contradiction. auto.
-Qed.
-
-Lemma eval_divu_base:
-  forall sp le e m1 a t1 m2 x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vint x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vint y) ->
-  y <> Int.zero ->
-  eval_expr ge sp le e m1 (Eop Odivu (a ::: b ::: Enil)) (t1**t2) m3 (Vint (Int.divu x y)).
-Proof.
-  intros. EvalOp. simpl. 
-  predSpec Int.eq Int.eq_spec y Int.zero. contradiction. auto.
-Qed.
-
-Theorem eval_divu:
-  forall sp le e m1 a t1 m2 x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vint x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vint y) ->
-  y <> Int.zero ->
-  eval_expr ge sp le e m1 (divu a b) (t1**t2) m3 (Vint (Int.divu x y)).
-Proof.
-  intros until y.
-  unfold divu; case (divu_match b); intros.
-  InvEval H0. caseEq (Int.is_power2 y). 
-  intros. rewrite (Int.divu_pow2 x y i H0).
-  apply eval_shruimm. rewrite E0_right. auto.
-  apply Int.is_power2_range with y. auto.
-  intros. subst n2. eapply eval_divu_base. eexact H. EvalOp. auto.
-  eapply eval_divu_base; eauto.
-Qed.
-
-Theorem eval_modu:
-  forall sp le e m1 a t1 m2 x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vint x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vint y) ->
-  y <> Int.zero ->
-  eval_expr ge sp le e m1 (modu a b) (t1**t2) m3 (Vint (Int.modu x y)).
-Proof.
-  intros until y; unfold modu; case (divu_match b); intros.
-  InvEval H0. caseEq (Int.is_power2 y). 
-  intros. rewrite (Int.modu_and x y i H0).
-  rewrite <- Int.rolm_zero. apply eval_rolm. rewrite E0_right; auto.
-  intro. rewrite Int.modu_divu. eapply eval_mod_aux. 
-  intros. simpl. predSpec Int.eq Int.eq_spec y0 Int.zero.
-  contradiction. auto.
-  eexact H. EvalOp. auto. auto.
-  rewrite Int.modu_divu. eapply eval_mod_aux. 
-  intros. simpl. predSpec Int.eq Int.eq_spec y0 Int.zero.
-  contradiction. auto.
-  eexact H. eexact H0. auto. auto.
-Qed.
-
-Theorem eval_andimm:
-  forall sp le e m1 n a t m2 x,
-  eval_expr ge sp le e m1 a t m2 (Vint x) ->
-  eval_expr ge sp le e m1 (andimm n a) t m2 (Vint (Int.and x n)).
-Proof.
-  intros.  unfold andimm. case (Int.is_rlw_mask n).
-  rewrite <- Int.rolm_zero. apply eval_rolm; auto.
-  EvalOp. 
-Qed.
-
-Theorem eval_and:
-  forall sp le e m1 a t1 m2 x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vint x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vint y) ->
-  eval_expr ge sp le e m1 (and a b) (t1**t2) m3 (Vint (Int.and x y)).
-Proof.
-  intros until y; unfold and; case (mul_match a b); intros.
-  InvEval H. rewrite Int.and_commut. 
-  rewrite E0_left; apply eval_andimm; auto.
-  InvEval H0. rewrite E0_right; apply eval_andimm; auto.
-  EvalOp.
-Qed.
-
-Remark eval_same_expr_pure:
-  forall a1 a2 sp le e m1 t1 m2 v1 t2 m3 v2,
-  same_expr_pure a1 a2 = true ->
-  eval_expr ge sp le e m1 a1 t1 m2 v1 ->
-  eval_expr ge sp le e m2 a2 t2 m3 v2 ->
-  t1 = E0 /\ t2 = E0 /\ a2 = a1 /\ v2 = v1 /\ m2 = m1.
-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. 
-  assert (v2 = v1). congruence. tauto.
-  discriminate.
-Qed.
-
-Lemma eval_or:
-  forall sp le e m1 a t1 m2 x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vint x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vint y) ->
-  eval_expr ge sp le e m1 (or a b) (t1**t2) m3 (Vint (Int.or x y)).
-Proof.
-  intros until y; unfold or; case (or_match a b); intros.
-  generalize (Int.eq_spec amount1 amount2); case (Int.eq amount1 amount2); intro.
-  case (Int.is_rlw_mask (Int.or mask1 mask2)).
-  caseEq (same_expr_pure t0 t3); intro.
-  simpl. InvEval H. FuncInv. InvEval H0. FuncInv. 
-  generalize (eval_same_expr_pure _ _ _ _ _ _ _ _ _ _ _ _ H2 EV EV0).
-  intros [EQ1 [EQ2 [EQ3 [EQ4 EQ5]]]]. 
-  injection EQ4; intro EQ7. subst.
-  EvalOp. simpl. rewrite Int.or_rolm. auto.
-  simpl. EvalOp. 
-  simpl. EvalOp. 
-  simpl. EvalOp. 
-  EvalOp.
-Qed.
-
-Theorem eval_xor:
-  forall sp le e m1 a t1 m2 x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vint x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vint y) ->
-  eval_expr ge sp le e m1 (xor a b) (t1**t2) m3 (Vint (Int.xor x y)).
-Proof. TrivialOp xor. Qed.
-
-Theorem eval_shl:
-  forall sp le e m1 a t1 m2 x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vint x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vint y) ->
-  Int.ltu y (Int.repr 32) = true ->
-  eval_expr ge sp le e m1 (shl a b) (t1**t2) m3 (Vint (Int.shl x y)).
-Proof.
-  intros until y; unfold shl; case (shift_match b); intros.
-  InvEval H0. rewrite E0_right. apply eval_shlimm; auto.
-  EvalOp. simpl. rewrite H1. auto.
-Qed.
-
-Theorem eval_shr:
-  forall sp le e m1 a t1 m2 x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vint x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vint y) ->
-  Int.ltu y (Int.repr 32) = true ->
-  eval_expr ge sp le e m1 (shr a b) (t1**t2) m3 (Vint (Int.shr x y)).
-Proof.
-  TrivialOp shr. simpl. rewrite H1. auto.
-Qed.
-
-Theorem eval_shru:
-  forall sp le e m1 a t1 m2 x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vint x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vint y) ->
-  Int.ltu y (Int.repr 32) = true ->
-  eval_expr ge sp le e m1 (shru a b) (t1**t2) m3 (Vint (Int.shru x y)).
-Proof.
-  intros until y; unfold shru; case (shift_match b); intros.
-  InvEval H0. rewrite E0_right; apply eval_shruimm; auto.
-  EvalOp. simpl. rewrite H1. auto.
-Qed.
-
-Theorem eval_addf:
-  forall sp le e m1 a t1 m2 x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vfloat x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vfloat y) ->
-  eval_expr ge sp le e m1 (addf a b) (t1**t2) m3 (Vfloat (Float.add x y)).
-Proof.
-  intros until y; unfold addf; case (addf_match a b); intros.
-  InvEval H. FuncInv. EvalOp. 
-  econstructor; eauto. econstructor; eauto. econstructor; eauto. constructor.
-  traceEq. simpl. subst x. reflexivity.
-  InvEval H0. FuncInv. eapply eval_Elet. eexact H. EvalOp. 
-  econstructor; eauto with evalexpr. 
-  econstructor; eauto with evalexpr. 
-  econstructor. apply eval_Eletvar. simpl; reflexivity.
-  constructor. reflexivity. traceEq.
-  subst y. rewrite Float.addf_commut. reflexivity. auto.
-  EvalOp.
-Qed.
- 
-Theorem eval_subf:
-  forall sp le e m1 a t1 m2 x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vfloat x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vfloat y) ->
-  eval_expr ge sp le e m1 (subf a b) (t1**t2) m3 (Vfloat (Float.sub x y)).
-Proof.
-  intros until y; unfold subf; case (subf_match a b); intros.
-  InvEval H. FuncInv. EvalOp. 
-  econstructor; eauto. econstructor; eauto. econstructor; eauto. constructor.
-  traceEq. subst x. reflexivity.
-  EvalOp.
-Qed.
-
-Theorem eval_mulf:
-  forall sp le e m1 a t1 m2 x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vfloat x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vfloat y) ->
-  eval_expr ge sp le e m1 (mulf a b) (t1**t2) m3 (Vfloat (Float.mul x y)).
-Proof. TrivialOp mulf. Qed.
-
-Theorem eval_divf:
-  forall sp le e m1 a t1 m2 x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vfloat x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vfloat y) ->
-  eval_expr ge sp le e m1 (divf a b) (t1**t2) m3 (Vfloat (Float.div x y)).
-Proof. TrivialOp divf. Qed.
-
-Theorem eval_cast8signed:
-  forall sp le e m1 a t m2 v,
-  eval_expr ge sp le e m1 a t m2 v ->
-  eval_expr ge sp le e m1 (cast8signed a) t m2 (Val.cast8signed v).
-Proof. 
-  intros until v; unfold cast8signed; case (cast8signed_match a); intros.
-  replace (Val.cast8signed v) with v. auto. 
-  InvEval H. inversion EQ. destruct v0; simpl; auto. rewrite Int.cast8_signed_idem. reflexivity.
-  EvalOp.
-Qed.
-
-Theorem eval_cast8unsigned:
-  forall sp le e m1 a t m2 v,
-  eval_expr ge sp le e m1 a t m2 v ->
-  eval_expr ge sp le e m1 (cast8unsigned a) t m2 (Val.cast8unsigned v).
-Proof. 
-  intros until v; unfold cast8unsigned; case (cast8unsigned_match a); intros.
-  replace (Val.cast8unsigned v) with v. auto. 
-  InvEval H. inversion EQ. destruct v0; simpl; auto. rewrite Int.cast8_unsigned_idem. reflexivity.
-  EvalOp.
-Qed.
-
-Theorem eval_cast16signed:
-  forall sp le e m1 a t m2 v,
-  eval_expr ge sp le e m1 a t m2 v ->
-  eval_expr ge sp le e m1 (cast16signed a) t m2 (Val.cast16signed v).
-Proof. 
-  intros until v; unfold cast16signed; case (cast16signed_match a); intros.
-  replace (Val.cast16signed v) with v. auto. 
-  InvEval H. inversion EQ. destruct v0; simpl; auto. rewrite Int.cast16_signed_idem. reflexivity.
-  EvalOp.
-Qed.
-
-Theorem eval_cast16unsigned:
-  forall sp le e m1 a t m2 v,
-  eval_expr ge sp le e m1 a t m2 v ->
-  eval_expr ge sp le e m1 (cast16unsigned a) t m2 (Val.cast16unsigned v).
-Proof. 
-  intros until v; unfold cast16unsigned; case (cast16unsigned_match a); intros.
-  replace (Val.cast16unsigned v) with v. auto. 
-  InvEval H. inversion EQ. destruct v0; simpl; auto. rewrite Int.cast16_unsigned_idem. reflexivity.
-  EvalOp.
-Qed.
-
-Theorem eval_singleoffloat:
-  forall sp le e m1 a t m2 v,
-  eval_expr ge sp le e m1 a t m2 v ->
-  eval_expr ge sp le e m1 (singleoffloat a) t m2 (Val.singleoffloat v).
-Proof. 
-  intros until v; unfold singleoffloat; case (singleoffloat_match a); intros.
-  replace (Val.singleoffloat v) with v. auto. 
-  InvEval H. inversion EQ. destruct v0; simpl; auto. rewrite Float.singleoffloat_idem. reflexivity.
-  EvalOp.
-Qed.
-
-Theorem eval_cmp:
-  forall sp le c e m1 a t1 m2 x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vint x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vint y) ->
-  eval_expr ge sp le e m1 (cmp c a b) (t1**t2) m3 (Val.of_bool (Int.cmp c x y)).
-Proof. 
-  TrivialOp cmp. 
-  simpl. case (Int.cmp c x y); auto.
-Qed.
-
-Theorem eval_cmp_null_r:
-  forall sp le c e m1 a t1 m2 p x b t2 m3 v,
-  eval_expr ge sp le e m1 a t1 m2 (Vptr p x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vint Int.zero) ->
-  (c = Ceq /\ v = Vfalse) \/ (c = Cne /\ v = Vtrue) ->
-  eval_expr ge sp le e m1 (cmp c a b) (t1**t2) m3 v.
-Proof. 
-  TrivialOp cmp. 
-  simpl. elim H1; intros [EQ1 EQ2]; subst c; subst v; reflexivity.
-Qed.
-
-Theorem eval_cmp_null_l:
-  forall sp le c e m1 a t1 m2 p x b t2 m3 v,
-  eval_expr ge sp le e m1 a t1 m2 (Vint Int.zero) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vptr p x) ->
-  (c = Ceq /\ v = Vfalse) \/ (c = Cne /\ v = Vtrue) ->
-  eval_expr ge sp le e m1 (cmp c a b) (t1**t2) m3 v.
-Proof. 
-  TrivialOp cmp. 
-  simpl. elim H1; intros [EQ1 EQ2]; subst c; subst v; reflexivity.
-Qed.
-
-Theorem eval_cmp_ptr:
-  forall sp le c e m1 a t1 m2 p x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vptr p x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vptr p y) ->
-  eval_expr ge sp le e m1 (cmp c a b) (t1**t2) m3 (Val.of_bool (Int.cmp c x y)).
-Proof. 
-  TrivialOp cmp. 
-  simpl. unfold eq_block. rewrite zeq_true. 
-  case (Int.cmp c x y); auto.
-Qed.
-
-Theorem eval_cmpu:
-  forall sp le c e m1 a t1 m2 x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vint x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vint y) ->
-  eval_expr ge sp le e m1 (cmpu c a b) (t1**t2) m3 (Val.of_bool (Int.cmpu c x y)).
-Proof. 
-  TrivialOp cmpu. 
-  simpl. case (Int.cmpu c x y); auto.
-Qed.
-
-Theorem eval_cmpf:
-  forall sp le c e m1 a t1 m2 x b t2 m3 y,
-  eval_expr ge sp le e m1 a t1 m2 (Vfloat x) ->
-  eval_expr ge sp le e m2 b t2 m3 (Vfloat y) ->
-  eval_expr ge sp le e m1 (cmpf c a b) (t1**t2) m3 (Val.of_bool (Float.cmp c x y)).
-Proof. 
-  TrivialOp cmpf. 
-  simpl. case (Float.cmp c x y); auto.
-Qed.
-
-Lemma eval_base_condition_of_expr:
-  forall sp le a e m1 t m2 v (b: bool),
-  eval_expr ge sp le e m1 a t m2 v ->
-  Val.bool_of_val v b ->
-  eval_condexpr ge sp le e m1 
-                (CEcond (Ccompuimm Cne Int.zero) (a ::: Enil))
-                t m2 b.
-Proof.
-  intros. 
-  eapply eval_CEcond. eauto with evalexpr. 
-  inversion H0; simpl. rewrite Int.eq_false; auto. auto. auto.
-Qed.
-
-Lemma eval_condition_of_expr:
-  forall a sp le e m1 t m2 v (b: bool),
-  eval_expr ge sp le e m1 a t m2 v ->
-  Val.bool_of_val v b ->
-  eval_condexpr ge sp le e m1 (condexpr_of_expr a) t m2 b.
-Proof.
-  induction a; simpl; intros;
-    try (eapply eval_base_condition_of_expr; eauto; fail).
-  destruct o; try (eapply eval_base_condition_of_expr; eauto; fail).
-
-  destruct e. InvEval H. inversion XX3; subst v.
-  inversion H0. 
-  rewrite Int.eq_false; auto. constructor.
-  subst i; rewrite Int.eq_true. constructor.
-  eapply eval_base_condition_of_expr; eauto.
-
-  inversion H. subst. eapply eval_CEcond; eauto. simpl in H11.
-  destruct (eval_condition c vl); try discriminate.
-  destruct b0; inversion H11; subst; inversion H0; congruence.
-
-  inversion H. subst.
-  destruct v1; eauto with evalexpr.
-Qed.
-
-Theorem eval_conditionalexpr_true:
-  forall sp le e m1 a1 t1 m2 v1 t2 a2 m3 v2 a3,
-  eval_expr ge sp le e m1 a1 t1 m2 v1 ->
-  Val.is_true v1 ->
-  eval_expr ge sp le e m2 a2 t2 m3 v2 ->
-  eval_expr ge sp le e m1 (conditionalexpr a1 a2 a3) (t1**t2) m3 v2.
-Proof.
-  intros; unfold conditionalexpr.
-  apply eval_Econdition with t1 m2 true t2; auto.
-  eapply eval_condition_of_expr; eauto with valboolof.
-Qed.
-
-Theorem eval_conditionalexpr_false:
-  forall sp le e m1 a1 t1 m2 v1 a2 t2 m3 v2 a3,
-  eval_expr ge sp le e m1 a1 t1 m2 v1 ->
-  Val.is_false v1 ->
-  eval_expr ge sp le e m2 a3 t2 m3 v2 ->
-  eval_expr ge sp le e m1 (conditionalexpr a1 a2 a3) (t1**t2) m3 v2.
-Proof.
-  intros; unfold conditionalexpr.
-  apply eval_Econdition with t1 m2 false t2; auto.
-  eapply eval_condition_of_expr; eauto with valboolof.
-Qed.
-
-Lemma eval_addressing:
-  forall sp le e m1 a t m2 v b ofs,
-  eval_expr ge sp le e m1 a t m2 v ->
-  v = Vptr b ofs ->
-  match addressing a with (mode, args) =>
-    exists vl,
-    eval_exprlist ge sp le e m1 args t m2 vl /\ 
-    eval_addressing ge sp mode vl = Some v
-  end.
-Proof.
-  intros until v. unfold addressing; case (addressing_match a); intros.
-  InvEval H. exists (@nil val). split. eauto with evalexpr. 
-  simpl. auto.
-  InvEval H. exists (@nil val). split. eauto with evalexpr. 
-  simpl. auto.
-  InvEval H. InvEval EV. rewrite E0_left in TR. subst t1. FuncInv. 
-    congruence.
-    destruct (Genv.find_symbol ge s); congruence.
-    exists (Vint i0 :: nil). split. eauto with evalexpr. 
-    simpl. subst v. destruct (Genv.find_symbol ge s). congruence.
-    discriminate.
-  InvEval H. FuncInv. 
-    congruence.
-    exists (Vptr b0 i :: nil). split. eauto with evalexpr. 
-    simpl. congruence.
-  InvEval H. FuncInv. 
-    congruence.
-    exists (Vint i :: Vptr b0 i0 :: nil).
-    split. eauto with evalexpr. simpl. 
-    rewrite Int.add_commut. congruence.
-    exists (Vptr b0 i :: Vint i0 :: nil).
-    split. eauto with evalexpr. simpl. congruence.
-  exists (v :: nil). split. eauto with evalexpr. 
-    subst v. simpl. rewrite Int.add_zero. auto.
-Qed.
-
-Theorem eval_load:
-  forall sp le e m1 a t m2 v chunk v',
-  eval_expr ge sp le e m1 a t m2 v ->
-  Mem.loadv chunk m2 v = Some v' ->
-  eval_expr ge sp le e m1 (load chunk a) t m2 v'.
-Proof.
-  intros. generalize H0; destruct v; simpl; intro; try discriminate.
-  unfold load. 
-  generalize (eval_addressing _ _ _ _ _ _ _ _ _ _ H (refl_equal _)).
-  destruct (addressing a). intros [vl [EV EQ]]. 
-  eapply eval_Eload; eauto. 
-Qed.
-
-Theorem eval_store:
-  forall sp le e m1 a1 t1 m2 v1 a2 t2 m3 v2 chunk m4,
-  eval_expr ge sp le e m1 a1 t1 m2 v1 ->
-  eval_expr ge sp le e m2 a2 t2 m3 v2 ->
-  Mem.storev chunk m3 v1 v2 = Some m4 ->
-  eval_expr ge sp le e m1 (store chunk a1 a2) (t1**t2) m4 v2.
-Proof.
-  intros. generalize H1; destruct v1; simpl; intro; try discriminate.
-  unfold store.
-  generalize (eval_addressing _ _ _ _ _ _ _ _ _ _ H (refl_equal _)).
-  destruct (addressing a1). intros [vl [EV EQ]]. 
-  eapply eval_Estore; eauto. 
-Qed.
-
-Theorem exec_ifthenelse_true:
-  forall sp e m1 a t1 m2 v ifso ifnot t2 e3 m3 out,
-  eval_expr ge sp nil e m1 a t1 m2 v ->
-  Val.is_true v ->
-  exec_stmt ge sp e m2 ifso t2 e3 m3 out ->
-  exec_stmt ge sp e m1 (ifthenelse a ifso ifnot) (t1**t2) e3 m3 out.
-Proof.
-  intros. unfold ifthenelse.
-  apply exec_Sifthenelse with t1 m2 true t2.
-  eapply eval_condition_of_expr; eauto with valboolof.
-  auto. auto.
-Qed.
-
-Theorem exec_ifthenelse_false:
-  forall sp e m1 a t1 m2 v ifso ifnot t2 e3 m3 out,
-  eval_expr ge sp nil e m1 a t1 m2 v ->
-  Val.is_false v ->
-  exec_stmt ge sp e m2 ifnot t2 e3 m3 out ->
-  exec_stmt ge sp e m1 (ifthenelse a ifso ifnot) (t1**t2) e3 m3 out.
-Proof.
-  intros. unfold ifthenelse.
-  apply exec_Sifthenelse with t1 m2 false t2.
-  eapply eval_condition_of_expr; eauto with valboolof.
-  auto. auto.
-Qed.
-
-End CMCONSTR.
-