Module CleanupLabelsproof


Correctness proof for clean-up of labels

Require Import Coqlib.
Require Import Maps.
Require Import Ordered.
Require Import FSets.
Require Import AST.
Require Import Integers.
Require Import Values.
Require Import Memory.
Require Import Events.
Require Import Globalenvs.
Require Import Errors.
Require Import Smallstep.
Require Import Op.
Require Import Locations.
Require Import LTLin.
Require Import CleanupLabels.

Module LabelsetFacts := FSetFacts.Facts(Labelset).

Section CLEANUP.

Variable prog: program.
Let tprog := transf_program prog.
Let ge := Genv.globalenv prog.
Let tge := Genv.globalenv tprog.

Lemma symbols_preserved:
  forall (s: ident), Genv.find_symbol tge s = Genv.find_symbol ge s.
Proof.
  intros; unfold ge, tge, tprog, transf_program.
  apply Genv.find_symbol_transf.
Qed.

Lemma varinfo_preserved:
  forall b, Genv.find_var_info tge b = Genv.find_var_info ge b.
Proof.
  intros; unfold ge, tge, tprog, transf_program.
  apply Genv.find_var_info_transf.
Qed.

Lemma functions_translated:
  forall (v: val) (f: fundef),
  Genv.find_funct ge v = Some f ->
  Genv.find_funct tge v = Some (transf_fundef f).
Proof.
  intros.
  exact (Genv.find_funct_transf transf_fundef _ _ H).
Qed.

Lemma function_ptr_translated:
  forall (b: block) (f: fundef),
  Genv.find_funct_ptr ge b = Some f ->
  Genv.find_funct_ptr tge b = Some (transf_fundef f).
Proof.
  intros.
  exact (Genv.find_funct_ptr_transf transf_fundef _ _ H).
Qed.

Lemma sig_function_translated:
  forall f,
  funsig (transf_fundef f) = funsig f.
Proof.
  intros. destruct f; reflexivity.
Qed.

Lemma find_function_translated:
  forall ros ls f,
  find_function ge ros ls = Some f ->
  find_function tge ros ls = Some (transf_fundef f).
Proof.
  unfold find_function; intros; destruct ros; simpl.
  apply functions_translated; auto.
  rewrite symbols_preserved. destruct (Genv.find_symbol ge i).
  apply function_ptr_translated; auto.
  congruence.
Qed.

Correctness of labels_branched_to.

Definition instr_branches_to (i: instruction) (lbl: label) : Prop :=
  match i with
  | Lgoto lbl' => lbl = lbl'
  | Lcond cond args lbl' => lbl = lbl'
  | Ljumptable arg tbl => In lbl tbl
  | _ => False
  end.

Remark add_label_branched_to_incr:
  forall ls i, Labelset.Subset ls (add_label_branched_to ls i).
Proof.
  intros; red; intros; destruct i; simpl; auto.
  apply Labelset.add_2; auto.
  apply Labelset.add_2; auto.
  revert H; induction l0; simpl. auto. intros; apply Labelset.add_2; auto.
Qed.

Remark add_label_branched_to_contains:
  forall ls i lbl,
  instr_branches_to i lbl ->
  Labelset.In lbl (add_label_branched_to ls i).
Proof.
  destruct i; simpl; intros; try contradiction.
  apply Labelset.add_1; auto.
  apply Labelset.add_1; auto.
  revert H. induction l0; simpl; intros.
  contradiction.
  destruct H. apply Labelset.add_1; auto. apply Labelset.add_2; auto.
Qed.

Lemma labels_branched_to_correct:
  forall c i lbl,
  In i c -> instr_branches_to i lbl -> Labelset.In lbl (labels_branched_to c).
Proof.
  intros.
  assert (forall c' bto,
             Labelset.Subset bto (fold_left add_label_branched_to c' bto)).
  induction c'; intros; simpl; red; intros.
  auto.
  apply IHc'. apply add_label_branched_to_incr; auto.

  assert (forall c' bto,
             In i c' -> Labelset.In lbl (fold_left add_label_branched_to c' bto)).
  induction c'; simpl; intros.
  contradiction.
  destruct H2.
  subst a. apply H1. apply add_label_branched_to_contains; auto.
  apply IHc'; auto.

  unfold labels_branched_to. auto.
Qed.

Commutation with find_label.

Lemma find_label_commut:
  forall lbl bto,
  Labelset.In lbl bto ->
  forall c c',
  find_label lbl c = Some c' ->
  find_label lbl (remove_unused_labels bto c) = Some (remove_unused_labels bto c').
Proof.
  induction c; simpl; intros.
  congruence.
  unfold is_label in H0. destruct a; simpl; auto.
  destruct (peq lbl l). subst l. inv H0.
  rewrite Labelset.mem_1; auto.
  simpl. rewrite peq_true. auto.
  destruct (Labelset.mem l bto); auto. simpl. rewrite peq_false; auto.
Qed.

Corollary find_label_translated:
  forall f i c' lbl c,
  incl (i :: c') (fn_code f) ->
  find_label lbl (fn_code f) = Some c ->
  instr_branches_to i lbl ->
  find_label lbl (fn_code (transf_function f)) =
     Some (remove_unused_labels (labels_branched_to (fn_code f)) c).
Proof.
  intros. unfold transf_function; unfold cleanup_labels; simpl.
  apply find_label_commut. eapply labels_branched_to_correct; eauto.
  apply H; auto with coqlib.
  auto.
Qed.

Lemma find_label_incl:
  forall lbl c c', find_label lbl c = Some c' -> incl c' c.
Proof.
  induction c; simpl; intros.
  discriminate.
  destruct (is_label lbl a). inv H; auto with coqlib. auto with coqlib.
Qed.

Correctness of clean-up

Inductive match_stackframes: stackframe -> stackframe -> Prop :=
  | match_stackframe_intro:
      forall res f sp ls c,
      incl c f.(fn_code) ->
      match_stackframes
        (Stackframe res f sp ls c)
        (Stackframe res (transf_function f) sp ls
          (remove_unused_labels (labels_branched_to f.(fn_code)) c)).

Inductive match_states: state -> state -> Prop :=
  | match_states_intro:
      forall s f sp c ls m ts
        (STACKS: list_forall2 match_stackframes s ts)
        (INCL: incl c f.(fn_code)),
      match_states (State s f sp c ls m)
                   (State ts (transf_function f) sp (remove_unused_labels (labels_branched_to f.(fn_code)) c) ls m)
  | match_states_call:
      forall s f ls m ts,
      list_forall2 match_stackframes s ts ->
      match_states (Callstate s f ls m)
                   (Callstate ts (transf_fundef f) ls m)
  | match_states_return:
      forall s ls m ts,
      list_forall2 match_stackframes s ts ->
      match_states (Returnstate s ls m)
                   (Returnstate ts ls m).

Definition measure (st: state) : nat :=
  match st with
  | State s f sp c ls m => List.length c
  | _ => O
  end.

Theorem transf_step_correct:
  forall s1 t s2, step ge s1 t s2 ->
  forall s1' (MS: match_states s1 s1'),
  (exists s2', step tge s1' t s2' /\ match_states s2 s2')
  \/ (measure s2 < measure s1 /\ t = E0 /\ match_states s2 s1')%nat.
Proof.
  induction 1; intros; inv MS; simpl.
  left; econstructor; split.
  econstructor; eauto. instantiate (1 := v). rewrite <- H.
  apply eval_operation_preserved. exact symbols_preserved.
  econstructor; eauto with coqlib.
  assert (eval_addressing tge sp addr (map rs args) = Some a).
    rewrite <- H. apply eval_addressing_preserved. exact symbols_preserved.
  left; econstructor; split.
  econstructor; eauto.
  econstructor; eauto with coqlib.
  assert (eval_addressing tge sp addr (map rs args) = Some a).
    rewrite <- H. apply eval_addressing_preserved. exact symbols_preserved.
  left; econstructor; split.
  econstructor; eauto.
  econstructor; eauto with coqlib.
  left; econstructor; split.
  econstructor. eapply find_function_translated; eauto.
  symmetry; apply sig_function_translated.
  econstructor; eauto. constructor; auto. constructor; eauto with coqlib.
  left; econstructor; split.
  econstructor. eapply find_function_translated; eauto.
  symmetry; apply sig_function_translated.
  simpl. eauto.
  econstructor; eauto.
  left; econstructor; split.
  econstructor; eauto. eapply external_call_symbols_preserved; eauto.
  exact symbols_preserved. exact varinfo_preserved.
  econstructor; eauto with coqlib.
  case_eq (Labelset.mem lbl (labels_branched_to (fn_code f))); intros.
  left; econstructor; split.
  constructor.
  econstructor; eauto with coqlib.
  right. split. omega. split. auto. econstructor; eauto with coqlib.
  left; econstructor; split.
  econstructor. eapply find_label_translated; eauto. red; auto.
  econstructor; eauto. eapply find_label_incl; eauto.
  left; econstructor; split.
  econstructor. auto. eapply find_label_translated; eauto. red; auto.
  econstructor; eauto. eapply find_label_incl; eauto.
  left; econstructor; split.
  eapply exec_Lcond_false; eauto.
  econstructor; eauto with coqlib.
  left; econstructor; split.
  econstructor. eauto. eauto. eapply find_label_translated; eauto.
  red. eapply list_nth_z_in; eauto.
  econstructor; eauto. eapply find_label_incl; eauto.
  left; econstructor; split.
  econstructor; eauto.
  econstructor; eauto with coqlib.
  left; econstructor; split.
  econstructor; simpl; eauto.
  econstructor; eauto with coqlib.
  left; econstructor; split.
  econstructor; eauto. eapply external_call_symbols_preserved; eauto.
  exact symbols_preserved. exact varinfo_preserved.
  econstructor; eauto with coqlib.
  inv H3. inv H1. left; econstructor; split.
  econstructor; eauto.
  econstructor; eauto.
Qed.

Lemma transf_initial_states:
  forall st1, initial_state prog st1 ->
  exists st2, initial_state tprog st2 /\ match_states st1 st2.
Proof.
  intros. inv H.
  econstructor; split.
  eapply initial_state_intro with (f := transf_fundef f).
  eapply Genv.init_mem_transf; eauto.
  rewrite symbols_preserved; eauto.
  apply function_ptr_translated; auto.
  rewrite sig_function_translated. auto.
  constructor; auto. constructor.
Qed.

Lemma transf_final_states:
  forall st1 st2 r,
  match_states st1 st2 -> final_state st1 r -> final_state st2 r.
Proof.
  intros. inv H0. inv H. inv H4. constructor.
Qed.

Theorem transf_program_correct:
  forward_simulation (LTLin.semantics prog) (LTLin.semantics tprog).
Proof.
  eapply forward_simulation_opt.
  eexact symbols_preserved.
  eexact transf_initial_states.
  eexact transf_final_states.
  eexact transf_step_correct.
Qed.

End CLEANUP.

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