Reorganized the development, modularizing away machine-dependent parts.

Started to merge the ARM code generator.
Started to add support for PowerPC/EABI.
Use ocamlbuild to construct executable from Caml files.


git-svn-id: https://yquem.inria.fr/compcert/svn/compcert/trunk@930 fca1b0fc-160b-0410-b1d3-a4f43f01ea2e

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+(* *********************************************************************)
+(*                                                                     *)
+(*              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.     *)
+(*                                                                     *)
+(* *********************************************************************)
+
+(** Corollaries of the main semantic preservation theorem. *)
+
+Require Import Classical.
+Require Import Coqlib.
+Require Import AST.
+Require Import Integers.
+Require Import Values.
+Require Import Events.
+Require Import Globalenvs.
+Require Import Smallstep.
+Require Import Csyntax.
+Require Import Csem.
+Require Import Asm.
+Require Import Compiler.
+Require Import Errors.
+
+(** * Determinism of Asm semantics *)
+
+(** In this section, we show that the semantics for the Asm language
+  are deterministic, in a sense to be made precise later.
+  There are two sources of apparent non-determinism:
+- The semantics leaves unspecified the results of calls to external 
+  functions.  Different results to e.g. a "read" operation can of
+  course lead to different behaviors for the program.
+  We address this issue by modeling a notion of deterministic
+  external world that uniquely determines the results of external calls.
+- For diverging executions, the trace of I/O events is not uniquely
+  determined: it can contain events that will never be performed
+  because the program diverges earlier.  We address this issue
+  by showing the existence of a minimal trace for diverging executions.
+
+*)
+
+(** ** Deterministic worlds *)
+
+(** An external world is a function that, given the name of an
+  external call and its arguments, returns either [None], meaning
+  that this external call gets stuck, or [Some(r,w)], meaning
+  that this external call succeeds, has result [r], and changes
+  the world to [w]. *)
+
+Inductive world: Set :=
+  World: (ident -> list eventval -> option (eventval * world)) -> world.
+
+Definition nextworld (w: world) (evname: ident) (evargs: list eventval) :
+                     option (eventval * world) :=
+  match w with World f => f evname evargs end.
+
+(** A trace is possible in a given world if all events correspond
+  to non-stuck external calls according to the given world.
+  Two predicates are defined, for finite and infinite traces respectively:
+- [possible_trace w t w'], where [w] is the initial state of the
+  world, [t] the finite trace of interest, and [w'] the state of the
+  world after performing trace [t].
+- [possible_traceinf w T], where [w] is the initial state of the
+  world and [T] the possibly infinite trace of interest.
+*)
+
+Inductive possible_trace: world -> trace -> world -> Prop :=
+  | possible_trace_nil: forall w,
+      possible_trace w E0 w
+  | possible_trace_cons: forall w0 evname evargs evres w1 t w2,
+      nextworld w0 evname evargs = Some (evres, w1) ->
+      possible_trace w1 t w2 ->
+      possible_trace w0 (mkevent evname evargs evres :: t) w2.
+
+Lemma possible_trace_app:
+  forall t2 w2 w0 t1 w1,
+  possible_trace w0 t1 w1 -> possible_trace w1 t2 w2 ->
+  possible_trace w0 (t1 ** t2) w2.
+Proof.
+  induction 1; simpl; intros.
+  auto.
+  econstructor; eauto.
+Qed.
+
+Lemma possible_trace_app_inv:
+  forall t2 w2 t1 w0,
+  possible_trace w0 (t1 ** t2) w2 ->
+  exists w1, possible_trace w0 t1 w1 /\ possible_trace w1 t2 w2.
+Proof.
+  induction t1; simpl; intros.
+  exists w0; split. constructor. auto.
+  inv H. exploit IHt1; eauto. intros [w1 [A B]]. 
+  exists w1; split. econstructor; eauto. auto.
+Qed.
+
+CoInductive possible_traceinf: world -> traceinf -> Prop :=
+  | possible_traceinf_nil: forall w0,
+      possible_traceinf w0 Enilinf
+  | possible_traceinf_cons: forall w0 evname evargs evres w1 T,
+      nextworld w0 evname evargs = Some (evres, w1) ->
+      possible_traceinf w1 T ->
+      possible_traceinf w0 (Econsinf (mkevent evname evargs evres) T).
+
+Lemma possible_traceinf_app:
+  forall t2 w0 t1 w1,
+  possible_trace w0 t1 w1 -> possible_traceinf w1 t2 ->
+  possible_traceinf w0 (t1 *** t2).
+Proof.
+  induction 1; simpl; intros.
+  auto.
+  econstructor; eauto.
+Qed.
+
+Lemma possible_traceinf_app_inv:
+  forall t2 t1 w0,
+  possible_traceinf w0 (t1 *** t2) ->
+  exists w1, possible_trace w0 t1 w1 /\ possible_traceinf w1 t2.
+Proof.
+  induction t1; simpl; intros.
+  exists w0; split. constructor. auto.
+  inv H. exploit IHt1; eauto. intros [w1 [A B]]. 
+  exists w1; split. econstructor; eauto. auto.
+Qed.
+
+Ltac possibleTraceInv :=
+  match goal with
+  | [H: possible_trace _ (_ ** _) _ |- _] =>
+      let P1 := fresh "P" in
+      let w := fresh "w" in
+      let P2 := fresh "P" in
+      elim (possible_trace_app_inv _ _ _ _ H); clear H;
+      intros w [P1 P2];
+      possibleTraceInv
+  | [H: possible_traceinf _ (_ *** _) |- _] =>
+      let P1 := fresh "P" in
+      let w := fresh "w" in
+      let P2 := fresh "P" in
+      elim (possible_traceinf_app_inv _ _ _ H); clear H;
+      intros w [P1 P2];
+      possibleTraceInv
+  | _ => idtac
+  end.
+
+(** Determinism properties of [event_match]. *)
+
+Remark eventval_match_deterministic:
+  forall ev1 ev2 ty v1 v2,
+  eventval_match ev1 ty v1 -> eventval_match ev2 ty v2 ->
+  (ev1 = ev2 <-> v1 = v2).
+Proof.
+  intros. inv H; inv H0; intuition congruence.
+Qed.
+
+Remark eventval_list_match_deterministic:
+  forall ev1 ty v, eventval_list_match ev1 ty v ->
+  forall ev2, eventval_list_match ev2 ty v -> ev1 = ev2.
+Proof.
+  induction 1; intros.
+  inv H. auto.
+  inv H1. decEq.
+  rewrite (eventval_match_deterministic _ _ _ _ _ H H6). auto.
+  eauto.
+Qed.
+
+Lemma event_match_deterministic:
+  forall w0 t1 w1 t2 w2 ef vargs vres1 vres2,
+  possible_trace w0 t1 w1 ->
+  possible_trace w0 t2 w2 ->
+  event_match ef vargs t1 vres1 ->
+  event_match ef vargs t2 vres2 ->
+  vres1 = vres2 /\ t1 = t2 /\ w1 = w2.
+Proof.
+  intros. inv H1. inv H2. 
+  assert (eargs = eargs0). eapply eventval_list_match_deterministic; eauto. subst eargs0.
+  inv H. inv H12. inv H0. inv H12.
+  rewrite H11 in H10. inv H10. intuition. 
+  rewrite <- (eventval_match_deterministic _ _ _ _ _ H4 H5). auto.
+Qed.
+
+(** ** Determinism of Asm transitions. *)
+
+Remark extcall_arguments_deterministic:
+  forall rs m sg args args',
+  extcall_arguments rs m sg args ->
+  extcall_arguments rs m sg args' -> args = args'.
+Proof.
+  assert (
+    forall rs m ll args,
+    extcall_args rs m ll args ->
+    forall args', extcall_args rs m ll args' -> args = args').
+  induction 1; intros.
+  inv H. auto.
+  inv H1. decEq; eauto.
+  inv H; inv H4; congruence.
+  unfold extcall_arguments; intros; eauto.
+Qed.
+
+Lemma step_deterministic:
+  forall ge s0 t1 s1 t2 s2 w0 w1 w2,
+  step ge s0 t1 s1 -> step ge s0 t2 s2 ->
+  possible_trace w0 t1 w1 -> possible_trace w0 t2 w2 ->
+  s1 = s2 /\ t1 = t2 /\ w1 = w2.
+Proof.
+  intros. inv H; inv H0.
+  assert (c0 = c) by congruence. subst c0.
+  assert (i0 = i) by congruence. subst i0.
+  split. congruence. split. auto. inv H1; inv H2; auto.
+  congruence.
+  congruence.
+  assert (ef0 = ef) by congruence. subst ef0.
+  assert (args0 = args). eapply extcall_arguments_deterministic; eauto. subst args0.
+  exploit event_match_deterministic. eexact H1. eexact H2. eauto. eauto.
+  intros [A [B C]]. intuition congruence.
+Qed.
+
+Lemma initial_state_deterministic:
+  forall p s1 s2,
+  initial_state p s1 -> initial_state p s2 -> s1 = s2.
+Proof.
+  intros. inv H; inv H0. reflexivity. 
+Qed.
+
+Lemma final_state_not_step:
+  forall ge st r t st', final_state st r -> step ge st t st' -> False.
+Proof.
+  intros. inv H. inv H0.
+  unfold Vzero in H1. congruence.
+  unfold Vzero in H1. congruence. 
+Qed.
+
+Lemma final_state_deterministic:
+  forall st r1 r2, final_state st r1 -> final_state st r2 -> r1 = r2.
+Proof.
+  intros. inv H; inv H0. congruence.
+Qed.
+
+(** ** Determinism for terminating executions. *)
+
+(*
+Lemma star_star_inv:
+  forall ge s t1 s1, star step ge s t1 s1 ->
+  forall t2 s2 w w1 w2, star step ge s t2 s2 ->
+  possible_trace w t1 w1 -> possible_trace w t2 w2 ->
+  exists t, (star step ge s1 t s2 /\ t2 = t1 ** t)
+        \/  (star step ge s2 t s1 /\ t1 = t2 ** t).
+Proof.
+  induction 1; intros.
+  exists t2; left; split; auto.
+  inv H2. exists (t1 ** t2); right; split. econstructor; eauto. auto.
+  possibleTraceInv. 
+  exploit step_deterministic. eexact H. eexact H5. eauto. eauto. 
+  intros [U [V W]]. subst s5 t3 w3. 
+  exploit IHstar; eauto. intros [t [ [Q R] | [Q R] ]].
+  subst t4. exists t; left; split. auto. traceEq.
+  subst t2. exists t; right; split. auto. traceEq.
+Qed.
+*)
+
+Lemma steps_deterministic:
+  forall ge s0 t1 s1, star step ge s0 t1 s1 -> 
+  forall r1 r2 t2 s2 w0 w1 w2, star step ge s0 t2 s2 -> 
+  final_state s1 r1 -> final_state s2 r2 ->
+  possible_trace w0 t1 w1 -> possible_trace w0 t2 w2 ->
+  t1 = t2 /\ r1 = r2.
+Proof.
+  induction 1; intros. 
+  inv H. split. auto. eapply final_state_deterministic; eauto.
+  byContradiction. eapply final_state_not_step with (st := s); eauto.
+  inv H2. byContradiction. eapply final_state_not_step with (st := s0); eauto.
+  possibleTraceInv.
+  exploit step_deterministic. eexact H. eexact H7. eauto. eauto. 
+  intros [A [B C]]. subst s5 t3 w3.
+  exploit IHstar. eexact H8. eauto. eauto. eauto. eauto. 
+  intros [A B]. subst t4 r2. 
+  auto.
+Qed.
+
+(** ** Determinism for infinite transition sequences. *)
+
+Lemma forever_star_inv:
+  forall ge s t s', star step ge s t s' ->
+  forall T w w', forever step ge s T ->
+  possible_trace w t w' -> possible_traceinf w T ->
+  exists T',
+  forever step ge s' T' /\ possible_traceinf w' T' /\ T = t *** T'.
+Proof.
+  induction 1; intros.
+  inv H0. exists T; auto.
+  subst t. possibleTraceInv. 
+  inv H2. possibleTraceInv. 
+  exploit step_deterministic.
+    eexact H. eexact H1. eauto. eauto. intros [A [B C]]; subst s4 t1 w1.
+  exploit IHstar; eauto. intros [T' [A [B C]]]. 
+  exists T'; split. auto.
+  split. auto.
+  rewrite C. rewrite Eappinf_assoc; auto.
+Qed.
+
+Lemma star_final_not_forever:
+  forall ge s1 t s2 r T w1 w2,
+  star step ge s1 t s2 ->
+  final_state s2 r -> forever step ge s1 T ->
+  possible_trace w1 t w2 -> possible_traceinf w1 T ->
+  False.
+Proof.
+  intros. exploit forever_star_inv; eauto. intros [T' [A [B C]]]. 
+  inv A. eapply final_state_not_step; eauto.
+Qed.
+
+(** ** Minimal traces for divergence. *)
+
+(** There are two mutually exclusive way in which a program can diverge.
+  It can diverge in a reactive fashion: it performs infinitely many
+  external calls, and the internal computations between two external
+  calls are always finite.  Or it can diverge silently: after a finite
+  number of external calls, it enters an infinite sequence of internal
+  computations. *)
+
+Definition reactive (ge: genv) (s: state) (w: world) :=
+  forall t s1 w1,
+  star step ge s t s1 -> possible_trace w t w1 ->
+  exists s2, exists t', exists s3, exists w2,
+  star step ge s1 E0 s2 
+  /\ step ge s2 t' s3
+  /\ possible_trace w1 t' w2
+  /\ t' <> E0.
+
+Definition diverges_silently (ge: genv) (s: state) :=
+  forall s2, star step ge s E0 s2 -> exists s3, step ge s2 E0 s3.
+
+Definition diverges_eventually (ge: genv) (s: state) (w: world) :=
+  exists t, exists s1, exists w1,
+  star step ge s t s1 /\ possible_trace w t w1 /\ diverges_silently ge s1.
+
+(** Using classical logic, we show that any infinite sequence of transitions
+  that is possible in a deterministic world is of one of the two forms 
+  described above. *)
+
+Lemma reactive_or_diverges:
+  forall ge s T w,
+  forever step ge s T -> possible_traceinf w T ->
+  reactive ge s w \/ diverges_eventually ge s w.
+Proof.
+  intros. elim (classic (diverges_eventually ge s w)); intro.
+  right; auto.
+  left. red; intros.
+  generalize (not_ex_all_not trace _ H1 t).
+  intro. clear H1.
+  generalize (not_ex_all_not state _ H4 s1).
+  intro. clear H4.
+  generalize (not_ex_all_not world _ H1 w1).
+  intro. clear H1.
+  elim (not_and_or _ _ H4); clear H4; intro.
+  contradiction.
+  elim (not_and_or _ _ H1); clear H1; intro.
+  contradiction.
+  generalize (not_all_ex_not state _ H1). intros [s2 A]. clear H1.
+  destruct (imply_to_and _ _ A). clear A.
+  exploit forever_star_inv.
+    eapply star_trans. eexact H2. eexact H1. reflexivity.
+    eauto. rewrite E0_right. eauto. eauto. 
+  intros [T' [A [B C]]]. 
+  inv A. possibleTraceInv. 
+  exists s2; exists t0; exists s3; exists w4. intuition.
+  subst t0. apply H4. exists s3; auto.
+Qed.
+
+(** Moreover, a program cannot be both reactive and silently diverging. *)
+
+Lemma reactive_not_diverges:
+  forall ge s w,
+  reactive ge s w -> diverges_eventually ge s w -> False.
+Proof.
+  intros. destruct H0 as [t [s1 [w1 [A [B C]]]]]. 
+  destruct (H _ _ _ A B) as [s2 [t' [s3 [w2 [P [Q [R S]]]]]]].
+  destruct (C _ P) as [s4 T].
+  assert (s3 = s4 /\ t' = E0 /\ w2 = w1).
+    eapply step_deterministic; eauto. constructor.
+  intuition congruence.
+Qed.
+
+(** A program that silently diverges can be given any finite or
+  infinite trace of events.  In particular, taking [T' = Enilinf],
+  it can be given the empty trace of events. *)
+
+Lemma diverges_forever:
+  forall ge s1 T w T',
+  diverges_silently ge s1 ->
+  forever step ge s1 T ->
+  possible_traceinf w T ->
+  forever step ge s1 T'.
+Proof.
+  cofix COINDHYP; intros. inv H0. possibleTraceInv. 
+  assert (exists s3, step ge s1 E0 s3). apply H. constructor.
+  destruct H0 as [s3 C]. 
+  assert (s2 = s3 /\ t = E0 /\ w0 = w). eapply step_deterministic; eauto. constructor.
+  destruct H0 as [Q [R S]]. subst s3 t w0. 
+  change T' with (E0 *** T'). econstructor. eassumption. 
+  eapply COINDHYP; eauto. 
+  red; intros. apply H. eapply star_left; eauto. 
+Qed.
+
+(** The trace of I/O events generated by a reactive diverging program
+  is uniquely determined up to bisimilarity. *)
+
+Lemma reactive_sim:
+  forall ge s w T1 T2,
+  reactive ge s w ->
+  forever step ge s T1 ->
+  forever step ge s T2 ->
+  possible_traceinf w T1 ->
+  possible_traceinf w T2 ->
+  traceinf_sim T1 T2.
+Proof.
+  cofix COINDHYP; intros.
+  elim (H E0 s w); try constructor. 
+  intros s2 [t' [s3 [w2 [A [B [C D]]]]]].
+  assert (star step ge s t' s3). eapply star_right; eauto. 
+  destruct (forever_star_inv _ _ _ _ H4 _ _ _ H0 C H2)
+  as [T1' [P [Q R]]].
+  destruct (forever_star_inv _ _ _ _ H4 _ _ _ H1 C H3)
+  as [T2' [S [T U]]].
+  destruct t'. unfold E0 in D. congruence.
+  assert (t' = nil). inversion B. inversion H7. auto. subst t'.
+  subst T1 T2. simpl. constructor.
+  apply COINDHYP with ge s3 w2; auto.
+  red; intros. eapply H. eapply star_trans; eauto.
+  eapply possible_trace_app; eauto. 
+Qed.
+
+(** A trace is minimal for a program if it is a prefix of all possible
+  traces. *)
+
+Definition minimal_trace (ge: genv) (s: state) (w: world) (T: traceinf) :=
+  forall T',
+  forever step ge s T' -> possible_traceinf w T' ->
+  traceinf_prefix T T'.
+
+(** For any program that diverges with some possible trace [T1],
+  the set of possible traces admits a minimal element [T].
+  If the program is reactive, this trace is [T1]. 
+  If the program silently diverges, this trace is the finite trace
+  of events performed prior to silent divergence. *)
+
+Lemma forever_minimal_trace:
+  forall ge s T1 w,
+  forever step ge s T1 -> possible_traceinf w T1 ->
+  exists T,
+     forever step ge s T
+  /\ possible_traceinf w T
+  /\ minimal_trace ge s w T.
+Proof.
+  intros. 
+  destruct (reactive_or_diverges _ _ _ _ H H0).
+  (* reactive *)
+  exists T1; split. auto. split. auto. red; intros.
+  elim (reactive_or_diverges _ _ _ _ H2 H3); intro.
+  apply traceinf_sim_prefix. eapply reactive_sim; eauto. 
+  elimtype False. eapply reactive_not_diverges; eauto.
+  (* diverges *) 
+  elim H1. intros t [s1 [w1 [A [B C]]]].
+  exists (t *** Enilinf); split.
+  exploit forever_star_inv; eauto. intros [T' [P [Q R]]]. 
+  eapply star_forever. eauto. 
+  eapply diverges_forever; eauto.
+  split. eapply possible_traceinf_app. eauto. constructor. 
+  red; intros. exploit forever_star_inv. eauto. eexact H2. eauto. eauto.
+  intros [T2 [P [Q R]]].
+  subst T'. apply traceinf_prefix_app. constructor.
+Qed.
+
+(** ** Refined semantics for program executions. *)
+
+(** We now define the following variant [exec_program'] of the
+  [exec_program] predicate defined in module [Smallstep].
+  In the diverging case [Diverges T], the new predicate imposes that the
+  finite or infinite trace [T] is minimal. *)
+
+Inductive exec_program' (p: program) (w: world): program_behavior -> Prop :=
+  | program_terminates': forall s t s' w' r,
+      initial_state p s ->
+      star step (Genv.globalenv p) s t s' ->
+      possible_trace w t w' ->
+      final_state s' r ->
+      exec_program' p w (Terminates t r)
+  | program_diverges': forall s T,
+      initial_state p s ->
+      forever step (Genv.globalenv p) s T ->
+      possible_traceinf w T ->
+      minimal_trace (Genv.globalenv p) s w T ->
+      exec_program' p w (Diverges T).
+
+(** We show that any [exec_program] execution corresponds to
+  an [exec_program'] execution. *)
+
+Definition possible_behavior (w: world) (b: program_behavior) : Prop :=
+  match b with
+  | Terminates t r => exists w', possible_trace w t w'
+  | Diverges T => possible_traceinf w T
+  end.
+
+Inductive matching_behaviors: program_behavior -> program_behavior -> Prop :=
+  | matching_behaviors_terminates: forall t r,
+      matching_behaviors (Terminates t r) (Terminates t r)
+  | matching_behaviors_diverges: forall T1 T2,
+      traceinf_prefix T2 T1 ->
+      matching_behaviors (Diverges T1) (Diverges T2).
+
+Theorem exec_program_program':
+  forall p b w,
+  exec_program p b -> possible_behavior w b ->
+  exists b', exec_program' p w b' /\ matching_behaviors b b'.
+Proof.
+  intros. inv H; simpl in H0.
+  (* termination *)
+  destruct H0 as [w' A].
+  exists (Terminates t r).
+  split. econstructor; eauto. constructor.
+  (* divergence *)
+  exploit forever_minimal_trace; eauto. intros [T0 [A [B C]]].
+  exists (Diverges T0); split.
+  econstructor; eauto.
+  constructor. apply C; auto.
+Qed.
+
+(** Moreover, [exec_program'] is deterministic, in that the behavior
+  associated with a given program and external world is uniquely
+  defined up to bisimilarity between infinite traces. *)
+
+Inductive same_behaviors: program_behavior -> program_behavior -> Prop :=
+  | same_behaviors_terminates: forall t r,
+      same_behaviors (Terminates t r) (Terminates t r)
+  | same_behaviors_diverges: forall T1 T2,
+      traceinf_sim T2 T1 ->
+      same_behaviors (Diverges T1) (Diverges T2).
+
+Theorem exec_program'_deterministic:
+  forall p b1 b2 w,
+  exec_program' p w b1 -> exec_program' p w b2 ->
+  same_behaviors b1 b2.
+Proof.
+  intros. inv H; inv H0;
+  assert (s0 = s) by (eapply initial_state_deterministic; eauto); subst s0.
+  (* terminates, terminates *)
+  exploit steps_deterministic. eexact H2. eexact H5. eauto. eauto. eauto. eauto.
+  intros [A B]. subst. constructor.
+  (* terminates, diverges *)
+  byContradiction. eapply star_final_not_forever; eauto.
+  (* diverges, terminates *)
+  byContradiction. eapply star_final_not_forever; eauto.
+  (* diverges, diverges *)
+  constructor. apply traceinf_prefix_2_sim; auto.
+Qed.
+
+Lemma matching_behaviors_same:
+  forall b b1' b2',
+  matching_behaviors b b1' -> same_behaviors b1' b2' ->
+  matching_behaviors b b2'.
+Proof.
+  intros. inv H; inv H0. 
+  constructor.
+  constructor. apply traceinf_prefix_compat with T2 T1.
+  auto. apply traceinf_sim_sym; auto. apply traceinf_sim_refl.
+Qed.
+
+(** * Additional semantic preservation property *)
+
+(** Combining the semantic preservation theorem from module [Main]
+  with the determinism of Asm executions, we easily obtain
+  additional, stronger semantic preservation properties.
+  The first property states that, when compiling a Clight
+  program that has well-defined semantics, all possible executions
+  of the resulting Asm code correspond to an execution of
+  the source Clight program, in the sense of the [matching_behaviors]
+  predicate. *)
+
+Theorem transf_c_program_correct_strong:
+  forall p tp b w,
+  transf_c_program p = OK tp ->
+  Csem.exec_program p b ->
+  possible_behavior w b ->
+  (exists b', exec_program' tp w b')
+/\(forall b', exec_program' tp w b' ->
+   exists b0, Csem.exec_program p b0 /\ matching_behaviors b0 b').
+Proof.
+  intros.
+  assert (Asm.exec_program tp b).
+    eapply transf_c_program_correct; eauto.
+  exploit exec_program_program'; eauto. 
+  intros [b' [A B]].
+  split. exists b'; auto.
+  intros. exists b. split. auto.
+  apply matching_behaviors_same with b'. auto.
+  eapply exec_program'_deterministic; eauto.
+Qed.
+
+Section SPECS_PRESERVED.
+
+(** The second additional results shows that if one execution
+  of the source Clight program satisfies a given specification
+  (a predicate on the observable behavior of the program),
+  then all executions of the produced Asm program satisfy
+  this specification as well.  *) 
+
+Variable spec: program_behavior -> Prop.
+
+(* Since the execution trace for a diverging Clight program
+   is not uniquely defined (the trace can contain events that
+   the program will never perform because it loops earlier),
+   this result holds only if the specification is closed under
+   prefixes in the case of diverging executions.  This is the
+   case for all safety properties (some undesirable event never
+   occurs), but not for liveness properties (some desirable event
+   always occurs). *)
+
+Hypothesis spec_safety:
+  forall T T', traceinf_prefix T T' -> spec (Diverges T') -> spec (Diverges T).
+
+Theorem transf_c_program_preserves_spec:
+  forall p tp b w,
+  transf_c_program p = OK tp ->
+  Csem.exec_program p b ->
+  possible_behavior w b ->
+  spec b ->
+  (exists b', exec_program' tp w b')
+/\(forall b', exec_program' tp w b' -> spec b').
+Proof.
+  intros.
+  assert (Asm.exec_program tp b).
+    eapply transf_c_program_correct; eauto.
+  exploit exec_program_program'; eauto. 
+  intros [b' [A B]].
+  split. exists b'; auto.
+  intros b'' EX.
+  assert (same_behaviors b' b''). eapply exec_program'_deterministic; eauto.
+  inv B; inv H4. 
+  auto.
+  apply spec_safety with T1. 
+  eapply traceinf_prefix_compat with T2 T1. 
+  auto. apply traceinf_sim_sym; auto. apply traceinf_sim_refl.
+  auto.
+Qed.
+
+End SPECS_PRESERVED.