(* *********************************************************************)
(*                                                                     *)
(*              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 proof for expression simplification. *)

Require Import Coq.Program.Equality.
Require Import Axioms.
Require Import Coqlib.
Require Import Maps.
Require Import AST.
Require Import Errors.
Require Import Integers.
Require Import Values.
Require Import Memory.
Require Import Events.
Require Import Smallstep.
Require Import Globalenvs.
Require Import Csyntax.
Require Import Csem.
Require Import Cstrategy.
Require Import Clight.
Require Import SimplExpr.
Require Import SimplExprspec.

Section PRESERVATION.

Variable prog: C.program.
Variable tprog: Clight.program.
Hypothesis TRANSL: transl_program prog = OK tprog.

Let ge := Genv.globalenv prog.
Let tge := Genv.globalenv tprog.

(** Invariance properties. *)

Lemma symbols_preserved:
  forall (s: ident), Genv.find_symbol tge s = Genv.find_symbol ge s.
Proof
  (Genv.find_symbol_transf_partial transl_fundef _ TRANSL).

Lemma function_ptr_translated:
  forall b f,
  Genv.find_funct_ptr ge b = Some f ->
  exists tf,
  Genv.find_funct_ptr tge b = Some tf /\ transl_fundef f = OK tf.
Proof
  (Genv.find_funct_ptr_transf_partial transl_fundef _ TRANSL).

Lemma functions_translated:
  forall v f,
  Genv.find_funct ge v = Some f ->
  exists tf,
  Genv.find_funct tge v = Some tf /\ transl_fundef f = OK tf.
Proof
  (Genv.find_funct_transf_partial transl_fundef _ TRANSL).

Lemma varinfo_preserved:
  forall b, Genv.find_var_info tge b = Genv.find_var_info ge b.
Proof
  (Genv.find_var_info_transf_partial transl_fundef _ TRANSL).

Lemma block_is_volatile_preserved:
  forall b, block_is_volatile tge b = block_is_volatile ge b.
Proof.
  intros. unfold block_is_volatile. rewrite varinfo_preserved. auto.
Qed.

Lemma type_of_fundef_preserved:
  forall f tf, transl_fundef f = OK tf ->
  type_of_fundef tf = C.type_of_fundef f.
Proof.
  intros. destruct f; monadInv H.
  exploit transl_function_spec; eauto. intros [A [B [C D]]].
  simpl. unfold type_of_function, C.type_of_function. congruence.
  auto.
Qed.

Lemma function_return_preserved:
  forall f tf, transl_function f = OK tf ->
  fn_return tf = C.fn_return f.
Proof.
  intros. unfold transl_function in H.
  destruct (transl_stmt (C.fn_body f) initial_generator); inv H.
  auto.
Qed.

Lemma type_of_global_preserved:
  forall b ty,
  Csem.type_of_global ge b = Some ty ->
  type_of_global tge b = Some ty.
Proof.
  intros until ty. unfold Csem.type_of_global, type_of_global.
  rewrite varinfo_preserved. destruct (Genv.find_var_info ge b). auto. 
  case_eq (Genv.find_funct_ptr ge b); intros. 
  inv H0. exploit function_ptr_translated; eauto. intros [tf [A B]]. 
  rewrite A. decEq. apply type_of_fundef_preserved; auto.
  congruence.
Qed.

(** Translation of simple expressions. *)

Lemma tr_simple_nil:
  (forall le dst r sl a tmps, tr_expr le dst r sl a tmps ->
   dst = For_val \/ dst = For_effects -> simple r = true -> sl = nil)
/\(forall le rl sl al tmps, tr_exprlist le rl sl al tmps ->
   simplelist rl = true -> sl = nil).
Proof.
  assert (A: forall dst a, dst = For_val \/ dst = For_effects -> final dst a = nil).
    intros. destruct H; subst dst; auto.
  apply tr_expr_exprlist; intros; simpl in *; try discriminate; auto.
  rewrite H0; auto. simpl; auto.
  rewrite H0; auto. simpl; auto.
  destruct H1; congruence.
  destruct (andb_prop _ _ H6). inv H1.
  rewrite H0; auto. simpl; auto.
  rewrite H9 in H8; discriminate.
  rewrite H0; auto. simpl; auto.
  rewrite H0; auto. simpl; auto.
  destruct (andb_prop _ _ H7). rewrite H0; auto. rewrite H2; auto. simpl; auto.
  rewrite H0; auto. simpl; auto.
  destruct (andb_prop _ _ H13). destruct (andb_prop _ _ H14).
  rewrite H2; auto. simpl; auto.
  destruct (andb_prop _ _ H14). destruct (andb_prop _ _ H15). intuition congruence.
  destruct (andb_prop _ _ H13). destruct (andb_prop _ _ H14). intuition congruence.
  destruct (andb_prop _ _ H6). rewrite H0; auto.
Qed.

Lemma tr_simple_expr_nil:
  forall le dst r sl a tmps, tr_expr le dst r sl a tmps ->
  dst = For_val \/ dst = For_effects -> simple r = true -> sl = nil.
Proof (proj1 tr_simple_nil).

Lemma tr_simple_exprlist_nil:
  forall le rl sl al tmps, tr_exprlist le rl sl al tmps ->
  simplelist rl = true -> sl = nil.
Proof (proj2 tr_simple_nil).

(** Evaluation of simple expressions and of their translation *)

Remark deref_loc_preserved:
  forall ty m b ofs t v,
  deref_loc ge ty m b ofs t v -> deref_loc tge ty m b ofs t v.
Proof.
  intros. inv H. 
  eapply deref_loc_value; eauto.
  eapply deref_loc_volatile; eauto.
  eapply volatile_load_preserved with (ge1 := ge); auto.
  exact symbols_preserved. exact block_is_volatile_preserved.
  eapply deref_loc_reference; eauto.
  eapply deref_loc_copy; eauto.
Qed.

Remark assign_loc_preserved:
  forall ty m b ofs v t m',
  assign_loc ge ty m b ofs v t m' -> assign_loc tge ty m b ofs v t m'.
Proof.
  intros. inv H. 
  eapply assign_loc_value; eauto.
  eapply assign_loc_volatile; eauto.
  eapply volatile_store_preserved with (ge1 := ge); auto.
  exact symbols_preserved. exact block_is_volatile_preserved.
  eapply assign_loc_copy; eauto.
Qed.

Lemma tr_simple:
 forall e m,
 (forall r v,
  eval_simple_rvalue ge e m r v ->
  forall le dst sl a tmps,
  tr_expr le dst r sl a tmps ->
  match dst with
  | For_val => sl = nil /\ C.typeof r = typeof a /\ eval_expr tge e le m a v
  | For_effects => sl = nil
  | For_test tyl s1 s2 =>
      exists b, sl = makeif (fold_left Ecast tyl b) s1 s2 :: nil
             /\ C.typeof r = typeof b
             /\ eval_expr tge e le m b v
  end)
/\
 (forall l b ofs,
  eval_simple_lvalue ge e m l b ofs ->
  forall le sl a tmps,
  tr_expr le For_val l sl a tmps ->
  sl = nil /\ C.typeof l = typeof a /\ eval_lvalue tge e le m a b ofs).
Proof.
Opaque makeif.
  intros e m.
  apply (eval_simple_rvalue_lvalue_ind ge e m); intros until tmps; intros TR; inv TR.
(* value *)
  auto.
  auto.
  exists a0; auto.
(* rvalof *)
  inv H7; try congruence. 
  exploit H0; eauto. intros [A [B C]].
  subst sl1; simpl.
  assert (eval_expr tge e le m a v).
    eapply eval_Elvalue. eauto. congruence. rewrite <- B. eapply deref_loc_preserved; eauto. 
  destruct dst; auto.
  econstructor. split. simpl; eauto. auto. 
(* addrof *)
  exploit H0; eauto. intros [A [B C]].
  subst sl1; simpl.
  assert (eval_expr tge e le m (Eaddrof a1 ty) (Vptr b ofs)). econstructor; eauto.
  destruct dst; auto. simpl; econstructor; eauto. 
(* unop *)
  exploit H0; eauto. intros [A [B C]].
  subst sl1; simpl.
  assert (eval_expr tge e le m (Eunop op a1 ty) v). econstructor; eauto. congruence.
  destruct dst; auto. simpl; econstructor; eauto.
(* binop *)
  exploit H0; eauto. intros [A [B C]].
  exploit H2; eauto. intros [D [E F]].
  subst sl1 sl2; simpl.
  assert (eval_expr tge e le m (Ebinop op a1 a2 ty) v). econstructor; eauto. congruence.
  destruct dst; auto. simpl; econstructor; eauto.
(* cast *)
  exploit H0; eauto. intros [A [B C]].
  subst sl1; simpl.
  assert (eval_expr tge e le m (Ecast a1 ty) v). econstructor; eauto. congruence.
  destruct dst; auto. simpl; econstructor; eauto.
(* condition *)
  exploit H2; eauto. intros [A [B C]]. subst sl1. 
  assert (tr_expr le For_val (if b then r2 else r3) (if b then sl2 else sl3) (if b then a2 else a3) (if b then tmp2 else tmp3)).
    destruct b; auto.
  exploit H5; eauto. intros [D [E F]]. 
  assert (eval_expr tge e le m (Econdition a1 a2 a3 ty) v).
    econstructor; eauto. congruence. rewrite <- E. auto. 
  destruct dst; auto. simpl; econstructor; eauto. 
  intuition congruence.
  intuition congruence.
(* sizeof *)
  destruct dst.
  split; auto. split; auto. constructor.
  auto.
  exists (Esizeof ty1 ty). split. auto. split. auto. constructor.
(* alignof *)
  destruct dst.
  split; auto. split; auto. constructor.
  auto.
  exists (Ealignof ty1 ty). split. auto. split. auto. constructor.
(* var local *)
  split; auto. split; auto. apply eval_Evar_local; auto. 
(* var global *)
  split; auto. split; auto. apply eval_Evar_global; auto.
    rewrite symbols_preserved; auto.
    eapply type_of_global_preserved; eauto. 
(* deref *)
  exploit H0; eauto. intros [A [B C]]. subst sl1.
  split; auto. split; auto. constructor; auto.
(* field struct *)
  exploit H0; eauto. intros [A [B C]]. subst sl1.
  split; auto. split; auto. rewrite B in H1. eapply eval_Efield_struct; eauto.
(* field union *)
  exploit H0; eauto. intros [A [B C]]. subst sl1.
  split; auto. split; auto. rewrite B in H1. eapply eval_Efield_union; eauto.
Qed.

Lemma tr_simple_rvalue:
  forall e m r v,
  eval_simple_rvalue ge e m r v ->
  forall le dst sl a tmps,
  tr_expr le dst r sl a tmps ->
  match dst with
  | For_val => sl = nil /\ C.typeof r = typeof a /\ eval_expr tge e le m a v
  | For_effects => sl = nil
  | For_test tyl s1 s2 =>
      exists b, sl = makeif (fold_left Ecast tyl b) s1 s2 :: nil /\ C.typeof r = typeof b /\ eval_expr tge e le m b v
  end.
Proof.
  intros e m. exact (proj1 (tr_simple e m)).
Qed.

Lemma tr_simple_lvalue: 
  forall e m l b ofs,
  eval_simple_lvalue ge e m l b ofs ->
  forall le sl a tmps,
  tr_expr le For_val l sl a tmps ->
  sl = nil /\ C.typeof l = typeof a /\ eval_lvalue tge e le m a b ofs.
Proof.
  intros e m. exact (proj2 (tr_simple e m)).
Qed.

Lemma tr_simple_exprlist:
  forall le rl sl al tmps,
  tr_exprlist le rl sl al tmps ->
  forall e m tyl vl,
  eval_simple_list ge e m rl tyl vl ->
  sl = nil /\ eval_exprlist tge e le m al tyl vl.
Proof.
  induction 1; intros. 
  inv H. split. auto. constructor.
  inv H4.
  exploit tr_simple_rvalue; eauto. intros [A [B C]].
  exploit IHtr_exprlist; eauto. intros [D E].
  split. subst; auto. econstructor; eauto. congruence.
Qed.

(** Commutation between the translation of expressions and left contexts. *)

Lemma typeof_context:
  forall k1 k2 C, leftcontext k1 k2 C ->
  forall e1 e2, C.typeof e1 = C.typeof e2 ->
  C.typeof (C e1) = C.typeof (C e2).
Proof.
  induction 1; intros; auto. 
Qed.

Inductive compat_dest: (C.expr -> C.expr) -> destination -> destination -> list statement -> Prop :=
  | compat_dest_base: forall dst,
      compat_dest (fun x => x) dst dst nil
  | compat_dest_val: forall C dst sl,
      compat_dest C For_val dst sl
  | compat_dest_effects: forall C dst sl,
      compat_dest C For_effects dst sl
  | compat_dest_paren: forall C ty dst' dst sl,
      compat_dest C dst' (cast_destination ty dst) sl ->
      compat_dest (fun x => C.Eparen (C x) ty) dst' dst sl.

Lemma compat_dest_not_test:
  forall  C dst' dst sl,
  compat_dest C dst' dst sl ->
  dst = For_val \/ dst = For_effects ->
  dst' = For_val \/ dst' = For_effects.
Proof.
  induction 1; intros; auto.
  apply IHcompat_dest. destruct H0; subst; auto.
Qed.

Lemma compat_dest_change:
  forall C1 dst' dst1 sl1 C2 dst2 sl2,
  compat_dest C1 dst' dst1 sl1 ->
  dst1 = For_val \/ dst1 = For_effects ->
  compat_dest C2 dst' dst2 sl2.
Proof.
  intros. exploit compat_dest_not_test; eauto. intros [A | A]; subst dst'; constructor.
Qed.

Lemma compat_dest_test:
  forall C tyl s1 s2 dst sl,
  compat_dest C (For_test tyl s1 s2) dst sl ->
  exists tyl', exists tyl'', dst = For_test tyl'' s1 s2 /\ tyl = tyl' ++ tyl''.
Proof.
  intros. dependent induction H. 
  exists (@nil type); exists tyl; auto.
  exploit IHcompat_dest; eauto. intros [l1 [l2 [A B]]]. 
  destruct dst; simpl in A; inv A. 
  exists (l1 ++ ty :: nil); exists tyl0; split; auto. rewrite app_ass. auto. 
Qed.

Scheme leftcontext_ind2 := Minimality for leftcontext Sort Prop
  with leftcontextlist_ind2 := Minimality for leftcontextlist Sort Prop.
Combined Scheme leftcontext_leftcontextlist_ind from leftcontext_ind2, leftcontextlist_ind2.

Lemma tr_expr_leftcontext_rec:
 (
  forall from to C, leftcontext from to C ->
  forall le e dst sl a tmps,
  tr_expr le dst (C e) sl a tmps ->
  exists dst', exists sl1, exists sl2, exists a', exists tmp',
  tr_expr le dst' e sl1 a' tmp'
  /\ sl = sl1 ++ sl2
  /\ compat_dest C dst' dst sl2
  /\ incl tmp' tmps
  /\ (forall le' e' sl3,
        tr_expr le' dst' e' sl3 a' tmp' ->
        (forall id, ~In id tmp' -> le'!id = le!id) ->
        C.typeof e' = C.typeof e ->
        tr_expr le' dst (C e') (sl3 ++ sl2) a tmps)
 ) /\ (
  forall from C, leftcontextlist from C ->
  forall le e sl a tmps,
  tr_exprlist le (C e) sl a tmps ->
  exists dst', exists sl1, exists sl2, exists a', exists tmp',
  tr_expr le dst' e sl1 a' tmp'
  /\ sl = sl1 ++ sl2
  /\ match dst' with For_test _ _ _ => False | _ => True end
  /\ incl tmp' tmps
  /\ (forall le' e' sl3,
        tr_expr le' dst' e' sl3 a' tmp' ->
        (forall id, ~In id tmp' -> le'!id = le!id) ->
        C.typeof e' = C.typeof e ->
        tr_exprlist le' (C e') (sl3 ++ sl2) a tmps)
).
Proof.

Ltac TR :=
  econstructor; econstructor; econstructor; econstructor; econstructor;
  split; [eauto | split; [idtac | split; [eauto | split]]].

Ltac NOTIN :=
  match goal with
  | [ H1: In ?x ?l, H2: list_disjoint ?l _ |- ~In ?x _ ] =>
        red; intro; elim (H2 x x); auto; fail
  | [ H1: In ?x ?l, H2: list_disjoint _ ?l |- ~In ?x _ ] =>
        red; intro; elim (H2 x x); auto; fail
  end.

Ltac UNCHANGED :=
  match goal with
  | [ H: (forall (id: ident), ~In id _ -> ?le' ! id = ?le ! id) |-
         (forall (id: ident), In id _ -> ?le' ! id = ?le ! id) ] =>
      intros; apply H; NOTIN
  end.

  generalize compat_dest_change; intro CDC.
  apply leftcontext_leftcontextlist_ind; intros.

(* base *)
  TR. rewrite <- app_nil_end; auto. constructor. red; auto.
  intros. rewrite <- app_nil_end; auto. 
(* deref *)
  inv H1. 
  exploit H0; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. subst sl1; rewrite app_ass; eauto. auto. 
  intros. rewrite <- app_ass. econstructor; eauto. 
(* field *)
  inv H1.
  exploit H0. eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. subst sl1; rewrite app_ass; eauto. auto. 
  intros. rewrite <- app_ass. econstructor; eauto.
(* rvalof *)
  inv H1.
  exploit H0; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. subst sl1; rewrite app_ass; eauto. red; eauto. 
  intros. rewrite <- app_ass; econstructor; eauto. 
  exploit typeof_context; eauto. congruence.
(* addrof *)
  inv H1. 
  exploit H0; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. subst sl1; rewrite app_ass; eauto. auto. 
  intros. rewrite <- app_ass. econstructor; eauto.
(* unop *) 
  inv H1. 
  exploit H0; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. subst sl1; rewrite app_ass; eauto. auto. 
  intros. rewrite <- app_ass. econstructor; eauto. 
(* binop left *)
  inv H1.
  exploit H0; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. subst sl1. rewrite app_ass. eauto. 
  red; auto. 
  intros. rewrite <- app_ass. econstructor; eauto.
  eapply tr_expr_invariant; eauto. UNCHANGED. 
(* binop right *)
  inv H2. 
  assert (sl1 = nil) by (eapply tr_simple_expr_nil; eauto). subst sl1; simpl.
  exploit H1; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. subst sl2. rewrite app_ass. eauto. 
  red; auto. 
  intros. rewrite <- app_ass. change (sl3 ++ sl2') with (nil ++ sl3 ++ sl2'). rewrite app_ass. econstructor; eauto.
  eapply tr_expr_invariant; eauto. UNCHANGED. 
(* cast *)
  inv H1.
  exploit H0; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. subst sl1; rewrite app_ass; eauto. auto. 
  intros. rewrite <- app_ass. econstructor; eauto.
(* condition *)
  inv H1.
  (* simple *)
  exploit H0; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. subst sl1. rewrite app_ass. eauto. 
  red; auto. 
  intros. rewrite <- app_ass. econstructor. auto. auto. eauto.
  eapply tr_expr_invariant; eauto. UNCHANGED.
  eapply tr_expr_invariant; eauto. UNCHANGED.
  auto. auto. auto. auto. auto. 
  (* for val *)
  exploit H0; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. 
  rewrite Q. rewrite app_ass. eauto. 
  red; auto.
  intros. rewrite <- app_ass. econstructor. auto. apply S; auto. 
  eapply tr_expr_invariant; eauto. UNCHANGED.
  eapply tr_expr_invariant; eauto. UNCHANGED.
  auto. auto. auto. auto. auto. auto. auto. 
  (* for effects *)
  exploit H0; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. 
  rewrite Q. rewrite app_ass. eauto. 
  red; auto.
  intros. rewrite <- app_ass. econstructor. auto. auto. apply S; auto. 
  eapply tr_expr_invariant; eauto. UNCHANGED.
  eapply tr_expr_invariant; eauto. UNCHANGED.
  auto. auto. auto. auto. auto.
(* assign left *)
  inv H1.
  (* for effects *)
  exploit H0; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. subst sl1. rewrite app_ass. eauto. 
  red; auto. 
  intros. rewrite <- app_ass. econstructor. apply S; auto.
  eapply tr_expr_invariant; eauto. UNCHANGED.
  auto. auto. auto.
  (* for val *)
  exploit H0; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. subst sl1. rewrite app_ass. eauto. 
  red; auto. 
  intros. rewrite <- app_ass. econstructor. apply S; auto.
  eapply tr_expr_invariant; eauto. UNCHANGED.
  auto. auto. auto. auto. auto. auto.
  eapply typeof_context; eauto.
  auto. 
(* assign right *)
  inv H2. 
  (* for effects *)
  assert (sl1 = nil) by (eapply tr_simple_expr_nil; eauto). subst sl1; simpl.
  exploit H1; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. subst sl2. rewrite app_ass. eauto. 
  red; auto. 
  intros. rewrite <- app_ass. change (sl3 ++ sl2') with (nil ++ (sl3 ++ sl2')). rewrite app_ass. 
  econstructor. 
  eapply tr_expr_invariant; eauto. UNCHANGED.
  apply S; auto. auto. auto. auto.
  (* for val *)
  assert (sl1 = nil) by (eapply tr_simple_expr_nil; eauto). subst sl1; simpl.
  exploit H1; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. subst sl2. rewrite app_ass. eauto. 
  red; auto. 
  intros. rewrite <- app_ass. change (sl3 ++ sl2') with (nil ++ (sl3 ++ sl2')). rewrite app_ass. 
  econstructor. 
  eapply tr_expr_invariant; eauto. UNCHANGED.
  apply S; auto. auto. auto. auto. auto. auto. auto. auto. 
  eapply typeof_context; eauto.
(* assignop left *)
  inv H1.
  (* for effects *)
  exploit H0; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. subst sl1. rewrite app_ass. eauto. 
  red; auto. 
  intros. rewrite <- app_ass. econstructor. apply S; auto.
  eapply tr_expr_invariant; eauto. UNCHANGED. 
  symmetry; eapply typeof_context; eauto. eauto.  
  auto. auto. auto. auto. auto. auto.
  (* for val *)
  exploit H0; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. subst sl1. rewrite app_ass. eauto. 
  red; auto. 
  intros. rewrite <- app_ass. econstructor. apply S; auto.
  eapply tr_expr_invariant; eauto. UNCHANGED. 
  eauto. auto. auto. auto. auto. auto. auto. auto. auto. auto. auto. 
  eapply typeof_context; eauto.
(* assignop right *)
  inv H2.
  (* for effects *)
  assert (sl1 = nil) by (eapply tr_simple_expr_nil; eauto). subst sl1; simpl.
  exploit H1; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. subst sl2. rewrite app_ass. eauto. 
  red; auto.
  intros. rewrite <- app_ass. change (sl0 ++ sl2') with (nil ++ sl0 ++ sl2'). rewrite app_ass. econstructor. 
  eapply tr_expr_invariant; eauto. UNCHANGED.
  apply S; auto. auto. eauto. auto. auto. auto. auto. auto. auto. 
  (* for val *)
  assert (sl1 = nil) by (eapply tr_simple_expr_nil; eauto). subst sl1; simpl.
  exploit H1; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. subst sl2. rewrite app_ass. eauto. 
  red; auto. 
  intros. rewrite <- app_ass. change (sl0 ++ sl2') with (nil ++ sl0 ++ sl2'). rewrite app_ass. econstructor. 
  eapply tr_expr_invariant; eauto. UNCHANGED.
  apply S; auto. eauto. auto. auto. auto. auto. auto. auto. auto. auto. auto. auto. auto. 
(* postincr *)
  inv H1.
  (* for effects *)
  exploit H0; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. rewrite Q; rewrite app_ass; eauto. red; auto. 
  intros. rewrite <- app_ass. econstructor; eauto. 
  symmetry; eapply typeof_context; eauto. 
  (* for val *)
  exploit H0; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. rewrite Q; rewrite app_ass; eauto. red; auto. 
  intros. rewrite <- app_ass. econstructor; eauto. 
  eapply typeof_context; eauto.
(* call left *)
  inv H1.
  (* for effects *)
  exploit H0; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. rewrite Q; rewrite app_ass; eauto. red; auto. 
  intros. rewrite <- app_ass. econstructor. apply S; auto.
  eapply tr_exprlist_invariant; eauto. UNCHANGED.
  auto. auto. auto.
  (* for val *)
  exploit H0; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. rewrite Q; rewrite app_ass; eauto. red; auto. 
  intros. rewrite <- app_ass. econstructor. auto. apply S; auto.
  eapply tr_exprlist_invariant; eauto. UNCHANGED.
  auto. auto. auto. auto. 
(* call right *)
  inv H2.
  (* for effects *)
  assert (sl1 = nil) by (eapply tr_simple_expr_nil; eauto). subst sl1; simpl. 
  exploit H1; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. rewrite Q; rewrite app_ass; eauto. destruct dst'; contradiction || constructor.
  red; auto. 
  intros. rewrite <- app_ass. change (sl3++sl2') with (nil ++ sl3 ++ sl2'). rewrite app_ass. econstructor.
  eapply tr_expr_invariant; eauto. UNCHANGED.
  apply S; auto. auto. auto. auto.
  (* for val *)
  assert (sl1 = nil) by (eapply tr_simple_expr_nil; eauto). subst sl1; simpl. 
  exploit H1; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. rewrite Q; rewrite app_ass; eauto. destruct dst'; contradiction || constructor.
  red; auto. 
  intros. rewrite <- app_ass. change (sl3++sl2') with (nil ++ sl3 ++ sl2'). rewrite app_ass. econstructor.
  auto. eapply tr_expr_invariant; eauto. UNCHANGED.
  apply S; auto.
  auto. auto. auto. auto.
(* comma *)
  inv H1.
  exploit H0; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. rewrite Q; rewrite app_ass; eauto. red; auto. 
  intros. rewrite <- app_ass. econstructor. apply S; auto.
  eapply tr_expr_invariant; eauto. UNCHANGED.
  auto. auto. auto.
(* paren *)
  inv H1. 
  (* for val *)
  exploit H0; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. rewrite Q. rewrite app_ass. eauto. red; auto. 
  intros. rewrite <- app_ass. econstructor; eauto. 
  (* for effects *)
  exploit H0; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. rewrite Q. eauto. constructor; auto. auto.
  intros. econstructor; eauto.
(* cons left *)
  inv H1.
  exploit H0; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. subst sl1. rewrite app_ass. eauto. 
  exploit compat_dest_not_test; eauto. intros [A|A]; subst dst'; auto.
  red; auto. 
  intros. rewrite <- app_ass. econstructor. apply S; auto.
  eapply tr_exprlist_invariant; eauto.  UNCHANGED.
  auto. auto. auto.
(* cons right *)
  inv H2.
  assert (sl1 = nil) by (eapply tr_simple_expr_nil; eauto). subst sl1; simpl. 
  exploit H1; eauto. intros [dst' [sl1' [sl2' [a' [tmp' [P [Q [U [R S]]]]]]]]].
  TR. subst sl2. eauto. 
  red; auto. 
  intros. change sl3 with (nil ++ sl3). rewrite app_ass. econstructor.
  eapply tr_expr_invariant; eauto. UNCHANGED.
  apply S; auto.
  auto. auto. auto.
Qed.

Theorem tr_expr_leftcontext:
  forall C le r dst sl a tmps,
  leftcontext RV RV C ->
  tr_expr le dst (C r) sl a tmps ->
  exists dst', exists sl1, exists sl2, exists a', exists tmp',
  tr_expr le dst' r sl1 a' tmp'
  /\ sl = sl1 ++ sl2
  /\ compat_dest C dst' dst sl2
  /\ incl tmp' tmps
  /\ (forall le' r' sl3,
        tr_expr le' dst' r' sl3 a' tmp' ->
        (forall id, ~In id tmp' -> le'!id = le!id) ->
        C.typeof r' = C.typeof r ->
        tr_expr le' dst (C r') (sl3 ++ sl2) a tmps).
Proof.
  intros. eapply (proj1 tr_expr_leftcontext_rec); eauto.
Qed.

Theorem tr_top_leftcontext:
  forall e le m dst rtop sl a tmps,
  tr_top tge e le m dst rtop sl a tmps ->
  forall r C,
  rtop = C r ->
  leftcontext RV RV C ->
  exists dst', exists sl1, exists sl2, exists a', exists tmp',
  tr_top tge e le m dst' r sl1 a' tmp'
  /\ sl = sl1 ++ sl2
  /\ compat_dest C dst' dst sl2
  /\ incl tmp' tmps
  /\ (forall le' m' r' sl3,
        tr_expr le' dst' r' sl3 a' tmp' ->
        (forall id, ~In id tmp' -> le'!id = le!id) ->
        C.typeof r' = C.typeof r ->
        tr_top tge e le' m' dst (C r') (sl3 ++ sl2) a tmps).
Proof.
  induction 1; intros.
(* val for val *)
  inv H2; inv H1.
  exists For_val; econstructor; econstructor; econstructor; econstructor.
  split. apply tr_top_val_val; eauto.
  split. instantiate (1 := nil); auto.
  split. constructor.
  split. apply incl_refl.
  intros. rewrite <- app_nil_end. constructor; auto.
(* val for test *)
  inv H2; inv H1.
  exists (For_test tyl s1 s2); econstructor; econstructor; econstructor; econstructor.
  split. apply tr_top_val_test; eauto.
  split. instantiate (1 := nil); auto.
  split. constructor.
  split. apply incl_refl.
  intros. rewrite <- app_nil_end. constructor; eauto.
(* base *)
  subst r. exploit tr_expr_leftcontext; eauto.
  intros [dst' [sl1 [sl2 [a' [tmp' [P [Q [R [S T]]]]]]]]].
  exists dst'; exists sl1; exists sl2; exists a'; exists tmp'.
  split. apply tr_top_base; auto.
  split. auto. split. auto. split. auto.
  intros. apply tr_top_base. apply T; auto. 
(* paren *)
  inv H1; inv H0.
  (* at top *)
  exists (For_test tyl s1 s2); econstructor; econstructor; econstructor; econstructor.
  split. apply tr_top_paren_test; eauto.
  split. instantiate (1 := nil). rewrite <- app_nil_end; auto.
  split. constructor.
  split. apply incl_refl.
  intros. rewrite <- app_nil_end. constructor; eauto.
  (* below *)
  exploit (IHtr_top r0 C0); auto. 
  intros [dst' [sl1 [sl2 [a' [tmp' [P [Q [T [R S]]]]]]]]].
  exists dst'; exists sl1; exists sl2; exists a'; exists tmp'.
  split. auto.
  split. auto.
  split. constructor; auto.
  split. auto.
  intros. apply tr_top_paren_test. apply S; auto.
Qed.

Theorem tr_top_testcontext:
  forall C tyl s1 s2 dst sl2 r sl1 a tmps e le m,
  compat_dest C (For_test tyl s1 s2) dst sl2 ->
  tr_top tge e le m (For_test tyl s1 s2) r sl1 a tmps ->
  tr_top tge e le m dst (C r) (sl1 ++ sl2) a tmps.
Proof.
  intros. dependent induction H.
  rewrite <- app_nil_end. auto.
  exploit compat_dest_test; eauto. intros [l1 [l2 [A B]]].
  destruct dst; simpl in A; inv A. 
  apply tr_top_paren_test. eapply IHcompat_dest; eauto. 
Qed.

(** Semantics of smart constructors *)

Lemma eval_simpl_expr_sound:
  forall e le m a v, eval_expr tge e le m a v ->
  match eval_simpl_expr a with Some v' => v' = v | None => True end.
Proof.
  induction 1; simpl; auto. 
  destruct (eval_simpl_expr a); auto. subst. rewrite H0. auto. 
  inv H; simpl; auto.
Qed.

Lemma step_makeif:
  forall f a s1 s2 k e le m v1 b,
  eval_expr tge e le m a v1 ->
  bool_val v1 (typeof a) = Some b ->
  star step tge (State f (makeif a s1 s2) k e le m)
             E0 (State f (if b then s1 else s2) k e le m).
Proof.
  intros. functional induction (makeif a s1 s2).
  exploit eval_simpl_expr_sound; eauto. rewrite e0. intro EQ; subst v. 
  rewrite e1 in H0. inv H0. constructor.
  exploit eval_simpl_expr_sound; eauto. rewrite e0. intro EQ; subst v. 
  rewrite e1 in H0. inv H0. constructor.
  apply star_one. eapply step_ifthenelse; eauto. 
  apply star_one. eapply step_ifthenelse; eauto. 
Qed.

Lemma step_make_set:
  forall id a ty m b ofs t v e le f k,
  deref_loc ge ty m b ofs t v ->
  eval_lvalue tge e le m a b ofs ->
  typeof a = ty ->
  step tge (State f (make_set id a) k e le m)
         t (State f Sskip k e (PTree.set id v le) m).
Proof.
  intros. exploit deref_loc_preserved; eauto. rewrite <- H1. intros DEREF.
  unfold make_set. destruct (type_is_volatile (typeof a)) as []_eqn.
  econstructor; eauto.
  assert (t = E0). inv H; congruence. subst t.
  constructor. eapply eval_Elvalue; eauto.
Qed. 

Fixpoint Kseqlist (sl: list statement) (k: cont) :=
  match sl with
  | nil => k
  | s :: l => Kseq s (Kseqlist l k)
  end.

Remark Kseqlist_app:
  forall sl1 sl2 k,
  Kseqlist (sl1 ++ sl2) k = Kseqlist sl1 (Kseqlist sl2 k).
Proof.
  induction sl1; simpl; congruence.
Qed.

(*
Axiom only_scalar_types_volatile:
  forall ty, type_is_volatile ty = true -> exists chunk, access_mode ty = By_value chunk.
*)

Lemma step_tr_rvalof:
  forall ty m b ofs t v e le a sl a' tmp f k,
  deref_loc ge ty m b ofs t v ->
  eval_lvalue tge e le m a b ofs ->
  tr_rvalof ty a sl a' tmp ->
  typeof a = ty ->
  exists le',
    star step tge (State f Sskip (Kseqlist sl k) e le m)
               t (State f Sskip k e le' m)
  /\ eval_expr tge e le' m a' v
  /\ typeof a' = typeof a
  /\ forall x, ~In x tmp -> le'!x = le!x.
Proof.
  intros. inv H1. 
  (* non volatile *)
  assert (t = E0). inv H; auto. congruence. subst t.  
  exists le; intuition. apply star_refl.
  eapply eval_Elvalue; eauto. eapply deref_loc_preserved; eauto.
  (* volatile *)
  exists (PTree.set t0 v le); intuition.
  simpl. eapply star_two. econstructor. eapply step_vol_read; eauto. 
  eapply deref_loc_preserved; eauto. traceEq.
  constructor. apply PTree.gss.
  apply PTree.gso. congruence. 
Qed.

(** Matching between continuations *)


Inductive match_cont : Csem.cont -> cont -> Prop :=
  | match_Kstop:
      match_cont Csem.Kstop Kstop
  | match_Kseq: forall s k ts tk,
      tr_stmt s ts ->
      match_cont k tk ->
      match_cont (Csem.Kseq s k) (Kseq ts tk)
  | match_Kwhile2: forall r s k s' ts tk,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s ts ->
      match_cont k tk ->
      match_cont (Csem.Kwhile2 r s k)
                 (Kwhile expr_true (Ssequence s' ts) tk)
  | match_Kdowhile1: forall r s k s' ts tk,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s ts ->
      match_cont k tk ->
      match_cont (Csem.Kdowhile1 r s k)
                 (Kfor2 expr_true s' ts tk)
  | match_Kfor3: forall r s3 s k ts3 s' ts tk,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s3 ts3 ->
      tr_stmt s ts ->
      match_cont k tk ->
      match_cont (Csem.Kfor3 r s3 s k)
                 (Kfor2 expr_true ts3 (Ssequence s' ts) tk)
  | match_Kfor4: forall r s3 s k ts3 s' ts tk,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s3 ts3 ->
      tr_stmt s ts ->
      match_cont k tk ->
      match_cont (Csem.Kfor4 r s3 s k)
                 (Kfor3 expr_true ts3 (Ssequence s' ts) tk)
  | match_Kswitch2: forall k tk,
      match_cont k tk ->
      match_cont (Csem.Kswitch2 k) (Kswitch tk)
  | match_Kcall_none: forall f e C ty k tf le sl tk a dest tmps,
      transl_function f = Errors.OK tf ->
      leftcontext RV RV C ->
      (forall v m, tr_top tge e le m dest (C (C.Eval v ty)) sl a tmps) ->
      match_cont_exp dest a k tk ->
      match_cont (Csem.Kcall f e C ty k)
                 (Kcall None tf e le (Kseqlist sl tk))
  | match_Kcall_some: forall f e C ty k dst tf le sl tk a dest tmps,
      transl_function f = Errors.OK tf ->
      leftcontext RV RV C ->
      (forall v m, tr_top tge e (PTree.set dst v le) m dest (C (C.Eval v ty)) sl a tmps) ->
      match_cont_exp dest a k tk ->
      match_cont (Csem.Kcall f e C ty k)
                 (Kcall (Some dst) tf e le (Kseqlist sl tk))

with match_cont_exp : destination -> expr -> Csem.cont -> cont -> Prop :=
  | match_Kdo: forall k a tk,
      match_cont k tk ->
      match_cont_exp For_effects a (Csem.Kdo k) tk
  | match_Kifthenelse_1: forall a s1 s2 k ts1 ts2 tk,
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      match_cont k tk ->
      match_cont_exp For_val a (Csem.Kifthenelse s1 s2 k) (Kseq (Sifthenelse a ts1 ts2) tk)
  | match_Kifthenelse_2: forall a s1 s2 k ts1 ts2 tk,
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      match_cont k tk ->
      match_cont_exp (For_test nil ts1 ts2) a (Csem.Kifthenelse s1 s2 k) tk
  | match_Kwhile1: forall r s k s' a ts tk,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s ts ->
      match_cont k tk ->
      match_cont_exp (For_test nil Sskip Sbreak) a
         (Csem.Kwhile1 r s k)
         (Kseq ts (Kwhile expr_true (Ssequence s' ts) tk))
  | match_Kdowhile2: forall r s k s' a ts tk,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s ts ->
      match_cont k tk ->
      match_cont_exp (For_test nil Sskip Sbreak) a
        (Csem.Kdowhile2 r s k)
        (Kfor3 expr_true s' ts tk)
  | match_Kfor2: forall r s3 s k s' a ts3 ts tk,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s3 ts3 ->
      tr_stmt s ts ->
      match_cont k tk ->
      match_cont_exp (For_test nil Sskip Sbreak) a
        (Csem.Kfor2 r s3 s k)
        (Kseq ts (Kfor2 expr_true ts3 (Ssequence s' ts) tk))
  | match_Kswitch1: forall ls k a tls tk,
      tr_lblstmts ls tls ->
      match_cont k tk ->
      match_cont_exp For_val a (Csem.Kswitch1 ls k) (Kseq (Sswitch a tls) tk)
  | match_Kreturn: forall k a tk,
      match_cont k tk ->
      match_cont_exp For_val a (Csem.Kreturn k) (Kseq (Sreturn (Some a)) tk).

Lemma match_cont_call:
  forall k tk,
  match_cont k tk ->
  match_cont (Csem.call_cont k) (call_cont tk).
Proof.
  induction 1; simpl; auto. constructor. econstructor; eauto. econstructor; eauto.
Qed.

Lemma match_cont_exp_for_test_inv:
  forall tyl s1 s2 a a' k tk,
  match_cont_exp (For_test tyl s1 s2) a k tk ->
  match_cont_exp (For_test tyl s1 s2) a' k tk.
Proof.
  intros. inv H; econstructor; eauto.
Qed.

(** Matching between states *)

Inductive match_states: Csem.state -> state -> Prop :=
  | match_exprstates: forall f r k e m tf sl tk le dest a tmps,
      transl_function f = Errors.OK tf ->
      tr_top tge e le m dest r sl a tmps ->
      match_cont_exp dest a k tk ->
      match_states (Csem.ExprState f r k e m)
                   (State tf Sskip (Kseqlist sl tk) e le m)
  | match_regularstates: forall f s k e m tf ts tk le,
      transl_function f = Errors.OK tf ->
      tr_stmt s ts ->
      match_cont k tk ->
      match_states (Csem.State f s k e m)
                   (State tf ts tk e le m)
  | match_callstates: forall fd args k m tfd tk,
      transl_fundef fd = Errors.OK tfd ->
      match_cont k tk ->
      match_states (Csem.Callstate fd args k m)
                   (Callstate tfd args tk m)
  | match_returnstates: forall res k m tk,
      match_cont k tk ->
      match_states (Csem.Returnstate res k m)
                   (Returnstate res tk m)
  | match_stuckstate: forall S,
      match_states Csem.Stuckstate S.

Lemma push_seq:
  forall f sl k e le m,
  star step tge (State f (makeseq sl) k e le m)
             E0 (State f Sskip (Kseqlist sl k) e le m).
Proof.
  intros. unfold makeseq. generalize Sskip. revert sl k. 
  induction sl; simpl; intros.
  apply star_refl.
  eapply star_right. apply IHsl. constructor. traceEq.
Qed.

(** Additional results on translation of statements *)

Lemma tr_select_switch:
  forall n ls tls,
  tr_lblstmts ls tls ->
  tr_lblstmts (Csem.select_switch n ls) (select_switch n tls).
Proof.
  induction 1; simpl.
  constructor; auto.
  destruct (Int.eq n0 n). constructor; auto. auto.
Qed.

Lemma tr_seq_of_labeled_statement:
  forall ls tls,
  tr_lblstmts ls tls ->
  tr_stmt (Csem.seq_of_labeled_statement ls) (seq_of_labeled_statement tls).
Proof.
  induction 1; simpl. auto. constructor; auto.
Qed.

(** Commutation between translation and the "find label" operation. *)

Section FIND_LABEL.

Variable lbl: label.

Definition nolabel (s: statement) : Prop :=
  forall k, find_label lbl s k = None.

Fixpoint nolabel_list (sl: list statement) : Prop :=
  match sl with
  | nil => True
  | s1 :: sl' => nolabel s1 /\ nolabel_list sl'
  end.

Lemma nolabel_list_app:
  forall sl2 sl1, nolabel_list sl1 -> nolabel_list sl2 -> nolabel_list (sl1 ++ sl2).
Proof.
  induction sl1; simpl; intros. auto. tauto. 
Qed.

Lemma makeseq_nolabel:
  forall sl, nolabel_list sl -> nolabel (makeseq sl).
Proof.
  assert (forall sl s, nolabel s -> nolabel_list sl -> nolabel (makeseq_rec s sl)).
  induction sl; simpl; intros. auto. destruct H0. apply IHsl; auto. 
  red. intros; simpl. rewrite H. apply H0.
  intros. unfold makeseq. apply H; auto. red. auto. 
Qed.

Lemma small_stmt_nolabel:
  forall s, small_stmt s = true -> nolabel s.
Proof.
  induction s; simpl; intros; congruence || (red; auto).
  destruct (andb_prop _ _ H). intros; simpl. rewrite IHs1; auto. apply IHs2; auto. 
Qed.

Lemma makeif_nolabel:
  forall a s1 s2, nolabel s1 -> nolabel s2 -> nolabel (makeif a s1 s2).
Proof.
  intros. functional induction (makeif a s1 s2); auto. 
  red; simpl; intros. rewrite H; auto.
  red; simpl; intros. rewrite H; auto.
Qed.

Lemma tr_rvalof_nolabel:
  forall ty a sl a' tmp, tr_rvalof ty a sl a' tmp -> nolabel_list sl.
Proof.
  destruct 1; simpl; intuition. red; simpl; auto.
Qed.

Definition nolabel_dest (dst: destination) : Prop :=
  match dst with
  | For_val => True
  | For_effects => True
  | For_test tyl s1 s2 => nolabel s1 /\ nolabel s2
  end.

Lemma nolabel_final:
  forall dst a, nolabel_dest dst -> nolabel_list (final dst a).
Proof.
  destruct dst; simpl; intros. auto. auto. 
  split; auto. destruct H. apply makeif_nolabel; auto.
Qed. 

Lemma nolabel_dest_cast:
  forall ty dst, nolabel_dest dst -> nolabel_dest (cast_destination ty dst).
Proof.
  unfold nolabel_dest; intros; destruct dst; auto.
Qed.

Ltac NoLabelTac :=
  match goal with
  | [ |- nolabel_list nil ] => exact I
  | [ |- nolabel_list (final _ _) ] => apply nolabel_final; NoLabelTac
  | [ |- nolabel_list (_ :: _) ] => simpl; split; NoLabelTac
  | [ |- nolabel_list (_ ++ _) ] => apply nolabel_list_app; NoLabelTac
  | [ |- nolabel_dest For_val ] => exact I
  | [ |- nolabel_dest For_effects ] => exact I
  | [ H: _ -> nolabel_list ?x |- nolabel_list ?x ] => apply H; NoLabelTac
  | [ |- nolabel _ ] => red; intros; simpl; auto
  | [ |- _ /\ _ ] => split; NoLabelTac
  | _ => auto
  end.

Lemma tr_find_label_expr:
  (forall le dst r sl a tmps, tr_expr le dst r sl a tmps -> nolabel_dest dst -> nolabel_list sl)
/\(forall le rl sl al tmps, tr_exprlist le rl sl al tmps -> nolabel_list sl).
Proof.
  apply tr_expr_exprlist; intros; NoLabelTac.
  destruct H1. apply makeif_nolabel; auto.
  eapply tr_rvalof_nolabel; eauto.
  apply makeif_nolabel; NoLabelTac.
  rewrite (makeseq_nolabel sl2); auto.
  rewrite (makeseq_nolabel sl3); auto.
  apply makeif_nolabel; NoLabelTac.
  rewrite (makeseq_nolabel sl2). auto. apply H4. apply nolabel_dest_cast; auto. 
  rewrite (makeseq_nolabel sl3). auto. apply H6. apply nolabel_dest_cast; auto.
  eapply tr_rvalof_nolabel; eauto.
  eapply tr_rvalof_nolabel; eauto.
  eapply tr_rvalof_nolabel; eauto.
  unfold make_set. destruct (type_is_volatile (typeof a1)); auto.
  apply nolabel_dest_cast; auto.
Qed.

Lemma tr_find_label_top:
  forall e le m dst r sl a tmps,
  tr_top tge e le m dst r sl a tmps -> nolabel_dest dst -> nolabel_list sl.
Proof.
  induction 1; intros; NoLabelTac.
  destruct H1. apply makeif_nolabel; auto. 
  eapply (proj1 tr_find_label_expr); eauto.
Qed.

Lemma tr_find_label_expression:
  forall r s a, tr_expression r s a -> forall k, find_label lbl s k = None.
Proof.
  intros. inv H.
  assert (nolabel (makeseq sl)). apply makeseq_nolabel.
  eapply tr_find_label_top with (e := empty_env) (le := PTree.empty val) (m := Mem.empty).
  eauto. exact I. 
  apply H.
Qed.

Lemma tr_find_label_expr_stmt:
  forall r s, tr_expr_stmt r s -> forall k, find_label lbl s k = None.
Proof.
  intros. inv H.
  assert (nolabel (makeseq sl)). apply makeseq_nolabel.
  eapply tr_find_label_top with (e := empty_env) (le := PTree.empty val) (m := Mem.empty).
  eauto. exact I. 
  apply H.
Qed.

Lemma tr_find_label_if:
  forall r s1 s2 s,
  tr_if r s1 s2 s ->
  small_stmt s1 = true -> small_stmt s2 = true ->
  forall k, find_label lbl s k = None.
Proof.
  intros. inv H. 
  assert (nolabel (makeseq sl)). apply makeseq_nolabel.
  eapply tr_find_label_top with (e := empty_env) (le := PTree.empty val) (m := Mem.empty).
  eauto. split; apply small_stmt_nolabel; auto.
  apply H.
Qed.

Lemma tr_find_label:
  forall s k ts tk
    (TR: tr_stmt s ts)
    (MC: match_cont k tk),
  match Csem.find_label lbl s k with
  | None =>
      find_label lbl ts tk = None
  | Some (s', k') =>
      exists ts', exists tk',
          find_label lbl ts tk = Some (ts', tk')
       /\ tr_stmt s' ts'
       /\ match_cont k' tk'
  end
with tr_find_label_ls:
  forall s k ts tk
    (TR: tr_lblstmts s ts)
    (MC: match_cont k tk),
  match Csem.find_label_ls lbl s k with
  | None =>
      find_label_ls lbl ts tk = None
  | Some (s', k') =>
      exists ts', exists tk',
          find_label_ls lbl ts tk = Some (ts', tk')
       /\ tr_stmt s' ts'
       /\ match_cont k' tk'
  end.
Proof.
  induction s; intros; inversion TR; subst; clear TR; simpl.
  auto.
  eapply tr_find_label_expr_stmt; eauto.
(* seq *)
  exploit (IHs1 (Csem.Kseq s2 k)); eauto. constructor; eauto.
  destruct (Csem.find_label lbl s1 (Csem.Kseq s2 k)) as [[s' k'] | ].
  intros [ts' [tk' [A [B C]]]]. rewrite A. exists ts'; exists tk'; auto.
  intro EQ. rewrite EQ. eapply IHs2; eauto. 
(* if no-opt *)
  rename s' into sr. 
  rewrite (tr_find_label_expression _ _ _ H2).
  exploit (IHs1 k); eauto.
  destruct (Csem.find_label lbl s1 k) as [[s' k'] | ].
  intros [ts' [tk' [A [B C]]]]. rewrite A. exists ts'; exists tk'; intuition.
  intro EQ. rewrite EQ. eapply IHs2; eauto.
(* if opt *)
  rewrite (tr_find_label_if _ _ _ _ H7); auto.
  exploit (IHs1 k); eauto.
  destruct (Csem.find_label lbl s1 k) as [[s' k'] | ].
  intros [ts' [tk' [A [B C]]]]. 
  exploit small_stmt_nolabel. eexact H4. instantiate (1 := tk). congruence.
  intros.
  exploit (IHs2 k); eauto.
  destruct (Csem.find_label lbl s2 k) as [[s' k'] | ].
  intros [ts' [tk' [A [B C]]]]. 
  exploit small_stmt_nolabel. eexact H6. instantiate (1 := tk). congruence.
  auto.
(* while *)
  rename s' into sr. 
  rewrite (tr_find_label_if _ _ _ _ H1); auto.
  eapply IHs; eauto. econstructor; eauto.
(* dowhile *)
  rename s' into sr.
  rewrite (tr_find_label_if _ _ _ _ H1); auto.
  exploit (IHs (Kdowhile1 e s k)); eauto. econstructor; eauto.
  destruct (Csem.find_label lbl s (Kdowhile1 e s k)) as [[s' k'] | ].
  intros [ts' [tk' [A [B C]]]]. rewrite A. exists ts'; exists tk'; intuition.
  intro EQ. rewrite EQ. auto. 
(* for skip *)
  rename s' into sr.
  rewrite (tr_find_label_if _ _ _ _ H4); auto.
  exploit (IHs3 (Csem.Kfor3 e s2 s3 k)); eauto. econstructor; eauto.
  destruct (Csem.find_label lbl s3 (Csem.Kfor3 e s2 s3 k)) as [[s' k'] | ].
  intros [ts' [tk' [A [B C]]]]. rewrite A. exists ts'; exists tk'; intuition.
  intro EQ. rewrite EQ.
  exploit (IHs2 (Csem.Kfor4 e s2 s3 k)); eauto. econstructor; eauto.
(* for not skip *)
  rename s' into sr.
  rewrite (tr_find_label_if _ _ _ _ H3); auto.
  exploit (IHs1 (Csem.Kseq (C.Sfor C.Sskip e s2 s3) k)); eauto. 
    econstructor; eauto. econstructor; eauto. 
  destruct (Csem.find_label lbl s1
               (Csem.Kseq (C.Sfor C.Sskip e s2 s3) k)) as [[s' k'] | ].
  intros [ts' [tk' [A [B C]]]]. rewrite A. exists ts'; exists tk'; intuition.
  intro EQ; rewrite EQ.
  exploit (IHs3 (Csem.Kfor3 e s2 s3 k)); eauto. econstructor; eauto.
  destruct (Csem.find_label lbl s3 (Csem.Kfor3 e s2 s3 k)) as [[s'' k''] | ].
  intros [ts' [tk' [A [B C]]]]. rewrite A. exists ts'; exists tk'; intuition.
  intro EQ'. rewrite EQ'. 
  exploit (IHs2 (Csem.Kfor4 e s2 s3 k)); eauto. econstructor; eauto.
(* break, continue, return 0 *)
  auto. auto. auto.
(* return 1 *)
  rewrite (tr_find_label_expression _ _ _ H0). auto.
(* switch *)
  rewrite (tr_find_label_expression _ _ _ H1). apply tr_find_label_ls. auto. constructor; auto.
(* labeled stmt *)
  destruct (ident_eq lbl l). exists ts0; exists tk; auto. apply IHs; auto. 
(* goto *)
  auto.

  induction s; intros; inversion TR; subst; clear TR; simpl.
(* default *)
  apply tr_find_label; auto. 
(* case *)
  exploit (tr_find_label s (Csem.Kseq (Csem.seq_of_labeled_statement s0) k)); eauto.
  econstructor; eauto. apply tr_seq_of_labeled_statement; eauto.
  destruct (Csem.find_label lbl s
    (Csem.Kseq (Csem.seq_of_labeled_statement s0) k)) as [[s' k'] | ].
  intros [ts' [tk' [A [B C]]]]. rewrite A. exists ts'; exists tk'; auto.
  intro EQ. rewrite EQ. eapply IHs; eauto.
Qed.
 
End FIND_LABEL.

(** Anti-stuttering measure *)

(** There are some stuttering steps in the translation:
- The execution of [Sdo a] where [a] is side-effect free,
  which is three transitions in the source:
<<
    Sdo a, k  --->  a, Kdo k ---> rval v, Kdo k ---> Sskip, k
>>
  but the translation, which is [Sskip], makes no transitions.
- The reduction [C.Ecomma (C.Eval v) r2 --> r2].
- The reduction [C.Eparen (C.Eval v) --> C.Eval v] in a [For_effects] context.

The following measure decreases for these stuttering steps. *)

Fixpoint esize (a: C.expr) : nat :=
  match a with
  | C.Eloc _ _ _ => 1%nat
  | C.Evar _ _ => 1%nat
  | C.Ederef r1 _ => S(esize r1)
  | C.Efield l1 _ _ => S(esize l1)
  | C.Eval _ _ => O
  | C.Evalof l1 _ => S(esize l1)
  | C.Eaddrof l1 _ => S(esize l1)
  | C.Eunop _ r1 _ => S(esize r1)
  | C.Ebinop _ r1 r2 _ => S(esize r1 + esize r2)%nat
  | C.Ecast r1 _ => S(esize r1)
  | C.Econdition r1 _ _ _ => S(esize r1)
  | C.Esizeof _ _ => 1%nat
  | C.Ealignof _ _ => 1%nat
  | C.Eassign l1 r2 _ => S(esize l1 + esize r2)%nat
  | C.Eassignop _ l1 r2 _ _ => S(esize l1 + esize r2)%nat
  | C.Epostincr _ l1 _ => S(esize l1)
  | C.Ecomma r1 r2 _ => S(esize r1 + esize r2)%nat
  | C.Ecall r1 rl2 _ => S(esize r1 + esizelist rl2)%nat
  | C.Eparen r1 _ => S(esize r1)
  end

with esizelist (el: C.exprlist) : nat :=
  match el with
  | C.Enil => O
  | C.Econs r1 rl2 => (esize r1 + esizelist rl2)%nat
  end.

Definition measure (st: Csem.state) : nat :=
  match st with
  | Csem.ExprState _ r _ _ _ => (esize r + 1)%nat
  | Csem.State _ C.Sskip _ _ _ => 0%nat
  | Csem.State _ (C.Sdo r) _ _ _ => (esize r + 2)%nat
  | Csem.State _ (C.Sifthenelse r _ _) _ _ _ => (esize r + 2)%nat
  | _ => 0%nat
  end.

Lemma leftcontext_size:
  forall from to C,
  leftcontext from to C ->
  forall e1 e2,
  (esize e1 < esize e2)%nat ->
  (esize (C e1) < esize (C e2))%nat
with leftcontextlist_size:
  forall from C,
  leftcontextlist from C ->
  forall e1 e2,
  (esize e1 < esize e2)%nat ->
  (esizelist (C e1) < esizelist (C e2))%nat.
Proof.
  induction 1; intros; simpl; auto with arith. exploit leftcontextlist_size; eauto. auto with arith.
  induction 1; intros; simpl; auto with arith. exploit leftcontext_size; eauto. auto with arith.
Qed.

(** Forward simulation for expressions. *)

Lemma tr_val_gen:
  forall le dst v ty a tmp,
  typeof a = ty ->
  (forall tge e le' m,
      (forall id, In id tmp -> le'!id = le!id) ->
      eval_expr tge e le' m a v) ->
  tr_expr le dst (C.Eval v ty) (final dst a) a tmp.
Proof.
  intros. destruct dst; simpl; econstructor; auto.
Qed.

Lemma estep_simulation:
  forall S1 t S2, Cstrategy.estep ge S1 t S2 ->
  forall S1' (MS: match_states S1 S1'),
  exists S2',
     (plus step tge S1' t S2' \/
       (star step tge S1' t S2' /\ measure S2 < measure S1)%nat)
  /\ match_states S2 S2'.
Proof.
  induction 1; intros; inv MS.
(* expr *)
  assert (tr_expr le dest r sl a tmps).
    inv H9. contradiction. contradiction. auto. inv H. 
  econstructor; split.
  right; split. apply star_refl. destruct r; simpl; (contradiction || omega).
  econstructor; eauto.
  instantiate (1 := tmps).
  exploit tr_simple_rvalue; eauto. destruct dest.
  intros [A [B C]]. subst sl. apply tr_top_val_val; auto.
  intros A. subst sl. apply tr_top_base. constructor. 
  intros [b [A [B C]]]. subst sl. apply tr_top_val_test; auto.
(* rval volatile *)
  exploit tr_top_leftcontext; eauto. clear H11.
  intros [dst' [sl1 [sl2 [a' [tmp' [P [Q [T [R S]]]]]]]]].
  inv P. inv H2. inv H7; try congruence. 
  exploit tr_simple_lvalue; eauto. intros [SL [TY EV]]. subst sl0; simpl.
  econstructor; split.
  left. eapply plus_two. constructor. eapply step_vol_read; eauto. 
  rewrite <- TY. eapply deref_loc_preserved; eauto. congruence. auto.
  econstructor; eauto. change (final dst' (Etempvar t0 (C.typeof l)) ++ sl2) with (nil ++ (final dst' (Etempvar t0 (C.typeof l)) ++ sl2)).
  apply S. apply tr_val_gen. auto. intros. constructor. rewrite H5; auto. apply PTree.gss.
  intros. apply PTree.gso. red; intros; subst; elim H5; auto. 
  auto.
(* condition *)
  exploit tr_top_leftcontext; eauto. clear H10. 
  intros [dst' [sl1 [sl2 [a' [tmp' [P [Q [T [R S]]]]]]]]].
  inv P. inv H3. 
  (* simple *)
  intuition congruence.
  (* for value *)
  exploit tr_simple_rvalue; eauto. intros [SL [TY EV]].
  subst sl0; simpl Kseqlist. destruct b.
  econstructor; split.
  left. eapply plus_left. constructor.
  eapply star_trans. apply step_makeif with (b := true) (v1 := v); auto. congruence.
  eapply star_left. constructor. apply push_seq. reflexivity. reflexivity. traceEq. 
  replace (Kseqlist sl3 (Kseq (Sset t (Ecast a2 ty)) (Kseqlist sl2 tk)))
     with (Kseqlist (sl3 ++ Sset t (Ecast a2 ty) :: sl2) tk).
  eapply match_exprstates; eauto. 
  change (Sset t (Ecast a2 ty) :: sl2) with ((Sset t (Ecast a2 ty) :: nil) ++ sl2). rewrite <- app_ass. 
  apply S. econstructor; eauto.  auto. auto. 
  rewrite Kseqlist_app. auto.
  econstructor; split.
  left. eapply plus_left. constructor.
  eapply star_trans. apply step_makeif with (b := false) (v1 := v); auto. congruence.
  eapply star_left. constructor. apply push_seq. reflexivity. reflexivity. traceEq. 
  replace (Kseqlist sl4 (Kseq (Sset t (Ecast a3 ty)) (Kseqlist sl2 tk)))
     with (Kseqlist (sl4 ++ Sset t (Ecast a3 ty) :: sl2) tk).
  eapply match_exprstates; eauto. 
  change (Sset t (Ecast a3 ty) :: sl2) with ((Sset t (Ecast a3 ty) :: nil) ++ sl2). rewrite <- app_ass. 
  apply S. econstructor; eauto.  auto. auto. 
  rewrite Kseqlist_app. auto.
   (* for effects *)
  exploit tr_simple_rvalue; eauto. intros [SL [TY EV]].
  subst sl0; simpl Kseqlist. destruct b.
  econstructor; split.
  left. eapply plus_left. constructor.
  eapply star_trans. apply step_makeif with (b := true) (v1 := v); auto. congruence.
  apply push_seq. 
  reflexivity. traceEq.
  rewrite <- Kseqlist_app.
  econstructor. eauto. apply S.
    econstructor; eauto. apply tr_expr_monotone with tmp2; eauto. 
    econstructor; eauto. 
  auto. auto.
  econstructor; split.
  left. eapply plus_left. constructor.
  eapply star_trans. apply step_makeif with (b := false) (v1 := v); auto. congruence.
  apply push_seq. 
  reflexivity. traceEq.
  rewrite <- Kseqlist_app.
  econstructor. eauto. apply S.
    econstructor; eauto. apply tr_expr_monotone with tmp3; eauto. 
    econstructor; eauto. 
  auto. auto.
(* assign *)
  exploit tr_top_leftcontext; eauto. clear H12. 
  intros [dst' [sl1 [sl2 [a' [tmp' [P [Q [T [R S]]]]]]]]].
  inv P. inv H4.
  (* for effects *)
  exploit tr_simple_rvalue; eauto. intros [SL2 [TY2 EV2]].
  exploit tr_simple_lvalue; eauto. intros [SL1 [TY1 EV1]].
  subst; simpl Kseqlist.
  econstructor; split.
  left. eapply plus_left. constructor.
  apply star_one. econstructor; eauto.
  rewrite <- TY1; rewrite <- TY2; eauto.
  rewrite <- TY1. eapply assign_loc_preserved; eauto.
  traceEq.
  econstructor. auto. change sl2 with (nil ++ sl2). apply S.
  constructor. auto. auto. auto.
  (* for value *)
  exploit tr_simple_rvalue; eauto. intros [SL2 [TY2 EV2]].
  exploit tr_simple_lvalue. eauto.
    eapply tr_expr_invariant with (le' := PTree.set t0 v le). eauto. 
    intros. apply PTree.gso. intuition congruence.
  intros [SL1 [TY1 EV1]].
  subst; simpl Kseqlist.
  econstructor; split.
  left. eapply plus_left. constructor. 
  eapply star_left. constructor. eauto. 
  eapply star_left. constructor.
  apply star_one. econstructor; eauto. constructor. apply PTree.gss.
  simpl. rewrite <- TY1; eauto.
  rewrite <- TY1. eapply assign_loc_preserved; eauto.
  reflexivity. reflexivity. traceEq.
  econstructor. auto. apply S.
  apply tr_val_gen. auto. intros. econstructor; eauto. constructor. 
  rewrite H4; auto. apply PTree.gss. 
  intros. apply PTree.gso. intuition congruence.
  auto. auto.
(* assignop *)
  exploit tr_top_leftcontext; eauto. clear H15. 
  intros [dst' [sl1 [sl2 [a' [tmp' [P [Q [T [R S]]]]]]]]].
  inv P. inv H6.
  (* for effects *)
  exploit tr_simple_lvalue; eauto. intros [SL1 [TY1 EV1]].
  exploit step_tr_rvalof; eauto. intros [le' [EXEC [EV3 [TY3 INV]]]].
  exploit tr_simple_lvalue. eauto. eapply tr_expr_invariant with (le := le) (le' := le'). eauto. 
  intros. apply INV. NOTIN. intros [? [? EV1']].
  exploit tr_simple_rvalue. eauto. eapply tr_expr_invariant with (le := le) (le' := le'). eauto. 
  intros. apply INV. NOTIN. simpl. intros [SL2 [TY2 EV2]].
  subst; simpl Kseqlist. 
  econstructor; split.
  left. eapply star_plus_trans. rewrite app_ass. rewrite Kseqlist_app. eexact EXEC.
  eapply plus_two. simpl. econstructor. econstructor. eexact EV1'. 
    econstructor. eexact EV3. eexact EV2. 
    rewrite TY3; rewrite <- TY1; rewrite <- TY2; eauto.
    rewrite <- TY1; eauto.
    rewrite <- TY1. eapply assign_loc_preserved; eauto.
  reflexivity. traceEq.
  econstructor. auto. change sl2 with (nil ++ sl2). apply S.
  constructor. auto. auto. auto.
  (* for value *)
  exploit tr_simple_lvalue; eauto. intros [SL1 [TY1 EV1]].
  exploit step_tr_rvalof; eauto. intros [le' [EXEC [EV3 [TY3 INV]]]].
  exploit tr_simple_lvalue. eauto. eapply tr_expr_invariant with (le := le) (le' := le'). eauto. 
  intros. apply INV. NOTIN. intros [? [? EV1']].
  exploit tr_simple_rvalue. eauto. eapply tr_expr_invariant with (le := le) (le' := le'). eauto. 
  intros. apply INV. NOTIN. simpl. intros [SL2 [TY2 EV2]].
  exploit tr_simple_lvalue. eauto. 
    eapply tr_expr_invariant with (le := le) (le' := PTree.set t v3 le'). eauto. 
    intros. rewrite PTree.gso. apply INV. NOTIN. intuition congruence.
  intros [? [? EV1'']].
  subst; simpl Kseqlist.
  econstructor; split.
  left. rewrite app_ass. rewrite Kseqlist_app. 
  eapply star_plus_trans. eexact EXEC.
  simpl. eapply plus_four. econstructor. econstructor.
    econstructor. eexact EV3. eexact EV2. 
    rewrite TY3; rewrite <- TY1; rewrite <- TY2. eauto.
  econstructor. econstructor.
    eexact EV1''. constructor. apply PTree.gss. 
    simpl. rewrite <- TY1; eauto.
    rewrite <- TY1. eapply assign_loc_preserved; eauto. 
  reflexivity. traceEq.
  econstructor. auto. apply S.
  apply tr_val_gen. auto. intros. econstructor; eauto. constructor. 
  rewrite H10; auto. apply PTree.gss. 
  intros. rewrite PTree.gso. apply INV. 
  red; intros; elim H10; auto. 
  intuition congruence.
  auto. auto.
(* assignop stuck *)
  exploit tr_top_leftcontext; eauto. clear H12. 
  intros [dst' [sl1 [sl2 [a' [tmp' [P [Q [T [R S]]]]]]]]].
  inv P. inv H4.
  (* for effects *)
  exploit tr_simple_lvalue; eauto. intros [SL1 [TY1 EV1]].
  exploit tr_simple_rvalue; eauto. intros [SL2 [TY2 EV2]].
  exploit step_tr_rvalof; eauto. intros [le' [EXEC [EV3 [TY3 INV]]]].
  subst; simpl Kseqlist. 
  econstructor; split.
  right; split. rewrite app_ass. rewrite Kseqlist_app. eexact EXEC. 
  simpl. omega.
  constructor.
  (* for value *)
  exploit tr_simple_lvalue; eauto. intros [SL1 [TY1 EV1]].
  exploit tr_simple_rvalue; eauto. intros [SL2 [TY2 EV2]].
  exploit step_tr_rvalof; eauto. intros [le' [EXEC [EV3 [TY3 INV]]]].
  subst; simpl Kseqlist. 
  econstructor; split.
  right; split. rewrite app_ass. rewrite Kseqlist_app. eexact EXEC. 
  simpl. omega.
  constructor.
(* postincr *)
  exploit tr_top_leftcontext; eauto. clear H14. 
  intros [dst' [sl1 [sl2 [a' [tmp' [P [Q [T [R S]]]]]]]]].
  inv P. inv H5.
  (* for effects *)
  exploit tr_simple_lvalue; eauto. intros [SL1 [TY1 EV1]].
  exploit step_tr_rvalof; eauto. intros [le' [EXEC [EV3 [TY3 INV]]]].
  exploit tr_simple_lvalue. eauto. eapply tr_expr_invariant with (le := le) (le' := le'). eauto. 
  intros. apply INV. NOTIN. intros [? [? EV1']].
  subst; simpl Kseqlist.
  econstructor; split.
  left. rewrite app_ass; rewrite Kseqlist_app. 
  eapply star_plus_trans. eexact EXEC. 
  eapply plus_two. simpl. constructor.
  econstructor; eauto.
  unfold transl_incrdecr. destruct id; simpl in H2. 
  econstructor. eauto. constructor. simpl. rewrite TY3; rewrite <- TY1. eauto.
  econstructor. eauto. constructor. simpl. rewrite TY3; rewrite <- TY1. eauto.
  rewrite <- TY1. instantiate (1 := v3). destruct id; auto. 
  rewrite <- TY1. eapply assign_loc_preserved; eauto. 
  reflexivity. traceEq.
  econstructor. auto. change sl2 with (nil ++ sl2). apply S.
  constructor. auto. auto. auto.
  (* for value *)
  exploit tr_simple_lvalue; eauto. intros [SL1 [TY1 EV1]].
  exploit tr_simple_lvalue. eauto.
    eapply tr_expr_invariant with (le' := PTree.set t v1 le). eauto. 
    intros. apply PTree.gso. intuition congruence.
  intros [SL2 [TY2 EV2]].
  subst; simpl Kseqlist.
  econstructor; split.
  left. eapply plus_four. constructor.
  eapply step_make_set; eauto. 
  constructor.
  econstructor. eauto. 
  unfold transl_incrdecr. destruct id; simpl in H2. 
  econstructor. constructor. apply PTree.gss. constructor. simpl. eauto.
  econstructor. constructor. apply PTree.gss. constructor. simpl. eauto.
  rewrite <- TY1. instantiate (1 := v3). destruct id; auto. 
  rewrite <- TY1. eapply assign_loc_preserved; eauto. 
  traceEq.
  econstructor. auto. apply S.
  apply tr_val_gen. auto. intros. econstructor; eauto.
  rewrite H5; auto. apply PTree.gss. 
  intros. apply PTree.gso. intuition congruence.
  auto. auto.
(* postincr stuck *)
  exploit tr_top_leftcontext; eauto. clear H11. 
  intros [dst' [sl1 [sl2 [a' [tmp' [P [Q [T [R S]]]]]]]]].
  inv P. inv H3.
  (* for effects *)
  exploit tr_simple_lvalue; eauto. intros [SL1 [TY1 EV1]].
  exploit step_tr_rvalof; eauto. intros [le' [EXEC [EV3 [TY3 INV]]]].
  subst. simpl Kseqlist. 
  econstructor; split.
  right; split. rewrite app_ass; rewrite Kseqlist_app. eexact EXEC.
  simpl; omega.
  constructor.
  (* for value *)
  exploit tr_simple_lvalue; eauto. intros [SL1 [TY1 EV1]].
  subst. simpl Kseqlist. 
  econstructor; split.
  left. eapply plus_two. constructor. eapply step_make_set; eauto. 
  traceEq.
  constructor.
(* comma *)
  exploit tr_top_leftcontext; eauto. clear H9. 
  intros [dst' [sl1 [sl2 [a' [tmp' [P [Q [T [R S]]]]]]]]].
  inv P. inv H1.
  exploit tr_simple_rvalue; eauto. simpl; intro SL1.
  subst sl0; simpl Kseqlist.
  econstructor; split.
  right; split. apply star_refl. simpl. apply plus_lt_compat_r. 
  apply (leftcontext_size _ _ _ H). simpl. omega. 
  econstructor; eauto. apply S. 
  eapply tr_expr_monotone; eauto. 
  auto. auto. 
(* paren *)
  exploit tr_top_leftcontext; eauto. clear H9. 
  intros [dst' [sl1 [sl2 [a' [tmp' [P [Q [T [R S]]]]]]]]].
  inv P. inv H2.
  (* for value *)
  exploit tr_simple_rvalue; eauto. intros [SL1 [TY1 EV1]].
  subst sl0; simpl Kseqlist.
  econstructor; split.
  left. eapply plus_left. constructor. apply star_one.
  econstructor. econstructor; eauto. rewrite <- TY1; eauto. traceEq.
  econstructor; eauto. 
  change sl2 with (final For_val (Etempvar t (C.typeof r)) ++ sl2). apply S.
  constructor. auto. intros. constructor. rewrite H2; auto. apply PTree.gss.
  intros. apply PTree.gso. intuition congruence.
  auto. 
  (* for effects *)
  econstructor; split.
  right; split. apply star_refl. simpl. apply plus_lt_compat_r.
  apply (leftcontext_size _ _ _ H). simpl. omega.
  econstructor; eauto.
  exploit tr_simple_rvalue; eauto. destruct dst'; simpl.
  (* dst' = For_val: impossible *)
  congruence.
  (* dst' = For_effects: easy *)
  intros A. subst sl1. apply S. constructor; auto. auto. auto. 
  (* dst' = For_test: then dest is For_test as well and C is a string of C.Eparen,
     so we can apply tr_top_paren. *)
  intros [b [A [B D]]].
  eapply tr_top_testcontext; eauto. 
  subst sl1. apply tr_top_val_test; auto. econstructor; eauto. rewrite <- B; auto. 
  (* already reduced *)
  econstructor; split.
  right; split. apply star_refl. simpl. apply plus_lt_compat_r.
  apply (leftcontext_size _ _ _ H). simpl. omega.
  econstructor; eauto. instantiate (1 := @nil ident).
  inv H9. 
    inv H0. eapply tr_top_testcontext; eauto. simpl. constructor. auto. econstructor; eauto.
    exploit tr_simple_rvalue; eauto. simpl. intros [b [A [B D]]]. 
    eapply tr_top_testcontext; eauto. subst sl1. apply tr_top_val_test. auto. econstructor; eauto. congruence. 
    inv H0.
(* call *)
  exploit tr_top_leftcontext; eauto. clear H12. 
  intros [dst' [sl1 [sl2 [a' [tmp' [P [Q [U [R S]]]]]]]]].
  inv P. inv H5.
  (* for effects *)
  exploit tr_simple_rvalue; eauto. intros [SL1 [TY1 EV1]].
  exploit tr_simple_exprlist; eauto. intros [SL2 EV2].
  subst. simpl Kseqlist.
  exploit functions_translated; eauto. intros [tfd [J K]].
  econstructor; split. 
  left. eapply plus_left. constructor.  apply star_one.
  econstructor; eauto. rewrite <- TY1; eauto.
  exploit type_of_fundef_preserved; eauto. congruence.
  traceEq.
  constructor; auto. econstructor; eauto.
  intros. change sl2 with (nil ++ sl2). apply S.
  constructor. auto. auto. 
  (* for value *)
  exploit tr_simple_rvalue; eauto. intros [SL1 [TY1 EV1]].
  exploit tr_simple_exprlist; eauto. intros [SL2 EV2].
  subst. simpl Kseqlist.
  exploit functions_translated; eauto. intros [tfd [J K]].
  econstructor; split. 
  left. eapply plus_left. constructor.  apply star_one.
  econstructor; eauto. rewrite <- TY1; eauto.
  exploit type_of_fundef_preserved; eauto. congruence.
  traceEq.
  constructor; auto. econstructor; eauto.
  intros. apply S.
  destruct dst'; constructor.
  auto. intros. constructor. rewrite H5; auto. apply PTree.gss. 
  auto. intros. constructor. rewrite H5; auto. apply PTree.gss.  
  intros. apply PTree.gso. intuition congruence.
  auto. 
Qed.

(** Forward simulation for statements. *)

Lemma tr_top_val_for_val_inv:
  forall e le m v ty sl a tmps,
  tr_top tge e le m For_val (C.Eval v ty) sl a tmps ->
  sl = nil /\ typeof a = ty /\ eval_expr tge e le m a v.
Proof.
  intros. inv H. auto. inv H0. auto. 
Qed.

Lemma tr_top_val_for_test_inv:
  forall s1 s2 e le m v ty sl a tmps,
  tr_top tge e le m (For_test nil s1 s2) (C.Eval v ty) sl a tmps ->
  exists b, sl = makeif b s1 s2 :: nil /\ typeof b = ty /\ eval_expr tge e le m b v.
Proof.
  intros. inv H. exists a0; auto. 
  inv H0. exists a0; auto.
Qed.

Lemma bind_parameters_preserved:
  forall e m params args m',
  bind_parameters ge e m params args m' ->
  bind_parameters tge e m params args m'.
Proof.
  induction 1; econstructor; eauto. eapply assign_loc_preserved; eauto.
Qed.

Lemma sstep_simulation:
  forall S1 t S2, Csem.sstep ge S1 t S2 ->
  forall S1' (MS: match_states S1 S1'),
  exists S2',
     (plus step tge S1' t S2' \/
       (star step tge S1' t S2' /\ measure S2 < measure S1)%nat)
  /\ match_states S2 S2'.
Proof.
  induction 1; intros; inv MS.
(* do 1 *)
  inv H6. inv H0. 
  econstructor; split.
  right; split. apply push_seq. 
  simpl. omega.
  econstructor; eauto. constructor. auto.
(* do 2 *)
  inv H7. inv H6. inv H.  
  econstructor; split. 
  right; split. apply star_refl. simpl. omega.
  econstructor; eauto. constructor.

(* seq *)
  inv H6. econstructor; split. left. apply plus_one. constructor.
  econstructor; eauto. constructor; auto.
(* skip seq *)
  inv H6; inv H7. econstructor; split.
  left. apply plus_one; constructor.
  econstructor; eauto.
(* continue seq *)
  inv H6; inv H7. econstructor; split.
  left. apply plus_one; constructor.
  econstructor; eauto. constructor.
(* break seq *)
  inv H6; inv H7. econstructor; split.
  left. apply plus_one; constructor.
  econstructor; eauto. constructor.

(* ifthenelse *)
  inv H6.
  (* not optimized *)
  inv H2. econstructor; split.
  left. eapply plus_left. constructor. apply push_seq. traceEq.
  econstructor; eauto. econstructor; eauto.
  (* optimized *)
  inv H10. econstructor; split.
  right; split. apply push_seq. simpl. omega.
  econstructor; eauto. constructor; auto. 
(* ifthenelse *)
  inv H8. 
  (* not optimized *)
  exploit tr_top_val_for_val_inv; eauto. intros [A [B C]]. subst. 
  econstructor; split.
  left. eapply plus_two. constructor.
  apply step_ifthenelse with (v1 := v) (b := b); auto. traceEq.
  destruct b; econstructor; eauto.
  (* optimized *)
  exploit tr_top_val_for_test_inv; eauto. intros [c [A [B C]]]. subst. 
  econstructor; split.
  left. simpl. eapply plus_left. constructor.
  apply step_makeif with (v1 := v) (b := b); auto. traceEq.
  econstructor; eauto.
  destruct b; auto.

(* while *)
  inv H6. inv H1. econstructor; split.
  left. eapply plus_left. eapply step_while_true. constructor.
  auto. eapply star_left. constructor.
  apply push_seq.
  reflexivity. traceEq.
  econstructor; eauto. econstructor; eauto. econstructor; eauto.
(* while false *)
  inv H8. 
  exploit tr_top_val_for_test_inv; eauto. intros [b [A [B C]]]. subst. 
  econstructor; split.
  left. simpl. eapply plus_left. constructor.
  eapply star_trans. apply step_makeif with (v1 := v) (b := false); auto.
  eapply star_two. constructor. apply step_break_while. 
  reflexivity. reflexivity. traceEq.
  constructor; auto. constructor.
(* while true *)
  inv H8. 
  exploit tr_top_val_for_test_inv; eauto. intros [b [A [B C]]]. subst. 
  econstructor; split.
  left. simpl. eapply plus_left. constructor.
  eapply star_right. apply step_makeif with (v1 := v) (b := true); auto.
  constructor.
  reflexivity. traceEq.
  constructor; auto. constructor; auto. 
(* skip-or-continue while *)
  assert (ts = Sskip \/ ts = Scontinue). destruct H; subst s0; inv H7; auto.
  inv H8.
  econstructor; split.
  left. apply plus_one. apply step_skip_or_continue_while; auto. 
  constructor; auto. constructor; auto. 
(* break while *)
  inv H6. inv H7. 
  econstructor; split.
  left. apply plus_one. apply step_break_while. 
  constructor; auto. constructor.

(* dowhile *)
  inv H6. 
  econstructor; split.
  left. apply plus_one.
  apply step_for_true with (Vint Int.one). constructor. auto. 
  constructor; auto. constructor; auto.
(* skip_or_continue dowhile *)
  assert (ts = Sskip \/ ts = Scontinue). destruct H; subst s0; inv H7; auto.
  inv H8. inv H4.
  econstructor; split.
  left. eapply plus_left. apply step_skip_or_continue_for2. auto.
  apply push_seq. 
  reflexivity. traceEq.
  econstructor; eauto. econstructor; auto. econstructor; eauto. 
(* dowhile false *)
  inv H8. 
  exploit tr_top_val_for_test_inv; eauto. intros [b [A [B C]]]. subst. 
  econstructor; split.
  left. simpl. eapply plus_left. constructor. 
  eapply star_right. apply step_makeif with (v1 := v) (b := false); auto. 
  constructor. 
  reflexivity. traceEq.
  constructor; auto. constructor.
(* dowhile true *)
  inv H8. 
  exploit tr_top_val_for_test_inv; eauto. intros [b [A [B C]]]. subst. 
  econstructor; split.
  left. simpl. eapply plus_left. constructor. 
  eapply star_right. apply step_makeif with (v1 := v) (b := true); auto. 
  constructor. 
  reflexivity. traceEq.
  constructor; auto. constructor; auto. 
(* break dowhile *)
  inv H6. inv H7.
  econstructor; split.
  left. apply plus_one. apply step_break_for2. 
  constructor; auto. constructor.

(* for start *)
  inv H7. congruence. 
  econstructor; split. 
  left; apply plus_one. constructor.
  econstructor; eauto. constructor; auto. econstructor; eauto. 
(* for *)
  inv H6; try congruence. inv H2. 
  econstructor; split.
  left. eapply plus_left. apply step_for_true with (Vint Int.one). 
    constructor. auto.
  eapply star_left. constructor. apply push_seq.
  reflexivity. traceEq.
  econstructor; eauto. constructor; auto. econstructor; eauto.
(* for false *)
  inv H8. exploit tr_top_val_for_test_inv; eauto. intros [b [A [B C]]]. subst.
  econstructor; split.
  left. simpl. eapply plus_left. constructor.
  eapply star_trans. apply step_makeif with (v1 := v) (b := false); auto.
  eapply star_two. constructor. apply step_break_for2. 
  reflexivity. reflexivity. traceEq.
  constructor; auto. constructor.
(* for true *)
  inv H8. exploit tr_top_val_for_test_inv; eauto. intros [b [A [B C]]]. subst.
  econstructor; split.
  left. simpl. eapply plus_left. constructor. 
  eapply star_right. apply step_makeif with (v1 := v) (b := true); auto.
  constructor.
  reflexivity. traceEq.
  constructor; auto. constructor; auto. 
(* skip_or_continue for3 *)
  assert (ts = Sskip \/ ts = Scontinue). destruct H; subst x; inv H7; auto.
  inv H8.
  econstructor; split.
  left. apply plus_one. apply step_skip_or_continue_for2. auto.
  econstructor; eauto. econstructor; auto.
(* break for3 *)
  inv H6. inv H7. 
  econstructor; split.
  left. apply plus_one. apply step_break_for2.
  econstructor; eauto. constructor.
(* skip for4 *)
  inv H6. inv H7. 
  econstructor; split.
  left. apply plus_one. constructor.
  econstructor; eauto. constructor; auto. 

(* return none *)
  inv H7. econstructor; split.
  left. apply plus_one. econstructor; eauto. 
  constructor. apply match_cont_call; auto. 
(* return some 1 *)
  inv H6. inv H0. econstructor; split.
  left; eapply plus_left. constructor. apply push_seq. traceEq.
  econstructor; eauto. constructor. auto.
(* return some 2 *)
  inv H9. exploit tr_top_val_for_val_inv; eauto. intros [A [B C]]. subst.
  econstructor; split.
  left. eapply plus_two. constructor. econstructor. eauto.
  replace (fn_return tf) with (C.fn_return f). eauto.
  exploit transl_function_spec; eauto. intuition congruence. 
  eauto. traceEq.
  constructor. apply match_cont_call; auto.
(* skip return *)
  inv H9. 
  assert (is_call_cont tk). inv H10; simpl in *; auto.
  econstructor; split.
  left. apply plus_one. apply step_skip_call; eauto.
  rewrite <- H0. apply function_return_preserved; auto.
  constructor. auto.

(* switch *)
  inv H6. inv H1. 
  econstructor; split. 
  left; eapply plus_left. constructor. apply push_seq. traceEq.
  econstructor; eauto. constructor; auto. 
(* expr switch *)
  inv H7. exploit tr_top_val_for_val_inv; eauto. intros [A [B C]]. subst.
  econstructor; split.
  left; eapply plus_two. constructor. econstructor; eauto. traceEq.
  econstructor; eauto.
  apply tr_seq_of_labeled_statement. apply tr_select_switch. auto. 
  constructor; auto.

(* skip-or-break switch *)
  assert (ts = Sskip \/ ts = Sbreak). destruct H; subst x; inv H7; auto.
  inv H8.
  econstructor; split.
  left; apply plus_one. apply step_skip_break_switch. auto. 
  constructor; auto. constructor.

(* continue switch *)
  inv H6. inv H7.
  econstructor; split.
  left; apply plus_one. apply step_continue_switch.
  constructor; auto. constructor.

(* label *)
  inv H6. econstructor; split.
  left; apply plus_one. constructor.
  constructor; auto.

(* goto *)
  inv H7.
  exploit transl_function_spec; eauto. intros [A [B [C D]]].
  exploit tr_find_label. eexact A. apply match_cont_call. eauto. 
  instantiate (1 := lbl). rewrite H. 
  intros [ts' [tk' [P [Q R]]]]. 
  econstructor; split. 
  left. apply plus_one. econstructor; eauto.
  econstructor; eauto. 

(* internal function *)
  monadInv H7. 
  exploit transl_function_spec; eauto. intros [A [B [C D]]].
  econstructor; split.
  left; apply plus_one. eapply step_internal_function. 
  rewrite C; rewrite D; auto.
  rewrite C; rewrite D; eauto.
  rewrite C. eapply bind_parameters_preserved; eauto.
  constructor; auto. 

(* external function *)
  monadInv H5. 
  econstructor; split.
  left; apply plus_one. econstructor; eauto.
  eapply external_call_symbols_preserved; eauto. 
  exact symbols_preserved. exact varinfo_preserved. 
  constructor; auto.

(* return *)
  inv H3.
  (* none *)
  econstructor; split.
  left; apply plus_one. constructor.
  econstructor; eauto.
  (* some *)
  econstructor; split.
  left; apply plus_one. constructor.
  econstructor; eauto.
Qed.

(** Semantic preservation *)

Theorem simulation:
  forall S1 t S2, Cstrategy.step ge S1 t S2 ->
  forall S1' (MS: match_states S1 S1'),
  exists S2',
     (plus step tge S1' t S2' \/
       (star step tge S1' t S2' /\ measure S2 < measure S1)%nat)
  /\ match_states S2 S2'.
Proof.
  intros S1 t S2 STEP. destruct STEP. 
  apply estep_simulation; auto.
  apply sstep_simulation; auto.
Qed.

Lemma transl_initial_states:
  forall S,
  Csem.initial_state prog S ->
  exists S', Clight.initial_state tprog S' /\ match_states S S'.
Proof.
  intros. inv H. 
  exploit function_ptr_translated; eauto. intros [tf [FIND TR]].
  econstructor; split.
  econstructor.
  apply (Genv.init_mem_transf_partial _ _ TRANSL). eauto. 
  simpl. fold tge. rewrite symbols_preserved.
  replace (prog_main tprog) with (prog_main prog). eexact H1.
  symmetry. unfold transl_program in TRANSL.
  eapply transform_partial_program_main; eauto.
  eexact FIND.
  rewrite <- H3. apply type_of_fundef_preserved. auto.
  constructor. auto. constructor.
Qed.

Lemma transl_final_states:
  forall S S' r,
  match_states S S' -> Csem.final_state S r -> Clight.final_state S' r.
Proof.
  intros. inv H0. inv H. inv H4. constructor.
Qed.

Theorem transl_program_correct:
  forward_simulation (Cstrategy.semantics prog) (Clight.semantics tprog).
Proof.
  eapply forward_simulation_star_wf with (order := ltof _ measure).
  eexact symbols_preserved.
  eexact transl_initial_states.
  eexact transl_final_states.
  apply well_founded_ltof.
  exact simulation.
Qed.

End PRESERVATION.