Module CleanupLabelsproof


Correctness proof for clean-up of labels

Require Import Coqlib.
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 Smallstep.
Require Import Op.
Require Import Locations.
Require Import Linear.
Require Import CleanupLabels.
Require Import Annotations.

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 public_preserved:
  forall (s: ident), Genv.public_symbol tge s = Genv.public_symbol ge s.
Proof.
  intros; unfold ge, tge, tprog, transf_program.
  apply Genv.public_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 l; 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 l; 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 remove_unused_labels_cons:
  forall bto i c,
  remove_unused_labels bto (i :: c) =
  match i with
  | Llabel lbl =>
      if Labelset.mem lbl bto then i :: remove_unused_labels bto c else remove_unused_labels bto c
  | _ =>
      i :: remove_unused_labels bto c
  end.
Proof.
  unfold remove_unused_labels; intros. rewrite list_fold_right_eq. auto.
Qed.


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.
  rewrite remove_unused_labels_cons.
  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 f sp ls c,
      incl c f.(fn_code) ->
      match_stackframes
        (Stackframe f sp ls c)
        (Stackframe (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.

Lemma match_parent_locset:
  forall s ts,
  list_forall2 match_stackframes s ts ->
  parent_locset ts = parent_locset s.
Proof.
  induction 1; simpl. auto. inv H; auto.
Qed.

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; try rewrite remove_unused_labels_cons.
 Lgetstack *)  left; econstructor; split.
  econstructor; eauto.
  econstructor; eauto with coqlib.
 Lsetstack *)  left; econstructor; split.
  econstructor; eauto.
  econstructor; eauto with coqlib.
 Lop *)  left; econstructor; split.
  econstructor; eauto. instantiate (1 := v). rewrite <- H.
  apply eval_operation_preserved. exact symbols_preserved.
  econstructor; eauto with coqlib.
 Lload *)  assert (eval_addressing tge sp addr (LTL.reglist rs args) = Some a).
    rewrite <- H. apply eval_addressing_preserved. exact symbols_preserved.
  left; econstructor; split.
  econstructor; eauto.
  econstructor; eauto with coqlib.
 Lstore *)  assert (eval_addressing tge sp addr (LTL.reglist rs args) = Some a).
    rewrite <- H. apply eval_addressing_preserved. exact symbols_preserved.
  left; econstructor; split.
  econstructor; eauto.
  econstructor; eauto with coqlib.
 Lcall *)  left; econstructor; split.
  econstructor. eapply find_function_translated; eauto.
  symmetry; apply sig_function_translated.
  econstructor; eauto. constructor; auto. constructor; eauto with coqlib.
 Ltailcall *)  left; econstructor; split.
  econstructor. erewrite match_parent_locset; eauto. eapply find_function_translated; eauto.
  symmetry; apply sig_function_translated.
  simpl. eauto.
  econstructor; eauto.
 Lbuiltin *)  left; econstructor; split.
  econstructor.
  eapply eval_builtin_args_preserved with (ge1 := ge); eauto. exact symbols_preserved.
  eapply external_call_symbols_preserved; eauto.
  exact symbols_preserved. exact public_preserved. exact varinfo_preserved.
  eauto.
  econstructor; eauto with coqlib.
 Llabel *)  case_eq (Labelset.mem lbl (labels_branched_to (fn_code f))); intros.
 not eliminated *)  left; econstructor; split.
  constructor.
  econstructor; eauto with coqlib.
 eliminated *)  right. split. simpl. omega. split. auto. econstructor; eauto with coqlib.
 Lgoto *)  left; econstructor; split.
  econstructor. eapply find_label_translated; eauto. red; auto.
  econstructor; eauto. eapply find_label_incl; eauto.
 Lcond taken *)  left; econstructor; split.
  econstructor. auto. eauto. eapply find_label_translated; eauto. red; auto.
  econstructor; eauto. eapply find_label_incl; eauto.
 Lcond not taken *)  left; econstructor; split.
  eapply exec_Lcond_false; eauto.
  econstructor; eauto with coqlib.
 Ljumptable *)  left; econstructor; split.
  econstructor. eauto. eauto. eapply find_label_translated; eauto.
  red. eapply list_nth_z_in; eauto. eauto.
  econstructor; eauto. eapply find_label_incl; eauto.
 Lreturn *)  left; econstructor; split.
  econstructor; eauto.
  erewrite <- match_parent_locset; eauto.
  econstructor; eauto with coqlib.
 internal function *)  left; econstructor; split.
  econstructor; simpl; eauto.
  econstructor; eauto with coqlib.
 external function *)  left; econstructor; split.
  econstructor; eauto. eapply external_call_symbols_preserved'; eauto.
  exact symbols_preserved. exact public_preserved. exact varinfo_preserved.
  econstructor; eauto with coqlib.
 return *)  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 H6. econstructor; eauto.
Qed.

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

Lemma match_stackframes_same_sps:
  forall s ts,
    list_forall2 match_stackframes s ts ->
    map (fun s => match s with | Stackframe _ sp _ _ => sp end) s = map (fun s => match s with | Stackframe _ sp _ _ => sp end) ts.
Proof.
  induction 1; intros.
  - reflexivity.
  - inv H; simpl; rewrite IHlist_forall2; reflexivity.
Qed.

Theorem transf_step_correct':
  forall s1 t s2, step_safe ge s1 t s2 ->
  forall s1' (MS: match_states s1 s1'),
  (exists s2', step_safe tge s1' t s2' /\ match_states s2 s2')
  \/ (measure s2 < measure s1 /\ t = E0 /\ match_states s2 s1')%nat.
Proof.
  intros until 1; destruct H as [HA WA].
  induction HA; intros; inv MS; try rewrite remove_unused_labels_cons.
 Lgetstack *)  left; econstructor; split.
  split; auto. econstructor; eauto.
  econstructor; eauto with coqlib.
 Lsetstack *)  left; econstructor; split.
  split; auto. econstructor; eauto.
  econstructor; eauto with coqlib.
 Lop *)  left; econstructor; split.
  split; auto. econstructor; eauto. instantiate (1 := v). rewrite <- H.
  apply eval_operation_preserved. exact symbols_preserved.
  econstructor; eauto with coqlib.
 Lload *)  assert (eval_addressing tge sp addr (LTL.reglist rs args) = Some a).
    rewrite <- H. apply eval_addressing_preserved. exact symbols_preserved.
  left; econstructor; split.
  split; auto. econstructor; eauto.
  intros; erewrite <- match_stackframes_same_sps; eauto.
  eapply annot_sem_preserved. eapply symbols_preserved. eapply WA. rewrite H1 in H2; inv H2; auto.
  econstructor; eauto with coqlib.
 Lstore *)  assert (eval_addressing tge sp addr (LTL.reglist rs args) = Some a).
    rewrite <- H. apply eval_addressing_preserved. exact symbols_preserved.
  left; econstructor; split.
  split; auto. econstructor; eauto.
  intros; erewrite <- match_stackframes_same_sps; eauto.
  eapply annot_sem_preserved. eapply symbols_preserved. eapply WA. rewrite H1 in H2; inv H2; auto.
  econstructor; eauto with coqlib.
 Lcall *)  left; econstructor; split.
  split; auto. econstructor. eapply find_function_translated; eauto.
  symmetry; apply sig_function_translated.
  econstructor; eauto. constructor; auto. constructor; eauto with coqlib.
 Ltailcall *)  left; econstructor; split.
  split; auto. econstructor. erewrite match_parent_locset; eauto. eapply find_function_translated; eauto.
  symmetry; apply sig_function_translated.
  simpl. eauto.
  econstructor; eauto.
 Lbuiltin *)  left; econstructor; split.
  split; auto. econstructor.
  eapply eval_builtin_args_preserved with (ge1 := ge); eauto. exact symbols_preserved.
  eapply external_call_symbols_preserved; eauto.
  exact symbols_preserved. exact public_preserved. exact varinfo_preserved.
  eauto.
  econstructor; eauto with coqlib.
 Llabel *)  case_eq (Labelset.mem lbl (labels_branched_to (fn_code f))); intros.
 not eliminated *)  left; econstructor; split.
  split; auto. constructor.
  econstructor; eauto with coqlib.
 eliminated *)  right. split. simpl. omega. split. auto. econstructor; eauto with coqlib.
 Lgoto *)  left; econstructor; split.
  split; auto. econstructor. eapply find_label_translated; eauto. red; auto.
  econstructor; eauto. eapply find_label_incl; eauto.
 Lcond taken *)  left; econstructor; split.
  split; auto. econstructor. auto. eauto. eapply find_label_translated; eauto. red; auto.
  econstructor; eauto. eapply find_label_incl; eauto.
 Lcond not taken *)  left; econstructor; split.
  split; auto. eapply exec_Lcond_false; eauto.
  econstructor; eauto with coqlib.
 Ljumptable *)  left; econstructor; split.
  split; auto. econstructor. eauto. eauto. eapply find_label_translated; eauto.
  red. eapply list_nth_z_in; eauto. eauto.
  econstructor; eauto. eapply find_label_incl; eauto.
 Lreturn *)  left; econstructor; split.
  split; auto. econstructor; eauto.
  erewrite <- match_parent_locset; eauto.
  econstructor; eauto with coqlib.
 internal function *)  left; econstructor; split.
  split; auto. econstructor; simpl; eauto.
  econstructor; eauto with coqlib.
 external function *)  left; econstructor; split.
  split; auto. econstructor; eauto. eapply external_call_symbols_preserved'; eauto.
  exact symbols_preserved. exact public_preserved. exact varinfo_preserved.
  econstructor; eauto with coqlib.
 return *)  inv H3. inv H1. left; econstructor; split.
  split; auto. econstructor; eauto.
  econstructor; eauto.
Qed.

Theorem transf_program_correct':
  forward_simulation (Linear.semantics_safe prog) (Linear.semantics_safe tprog).
Proof.
  eapply forward_simulation_opt.
  eexact public_preserved.
  eexact transf_initial_states.
  eexact transf_final_states.
  eexact transf_step_correct'.
Qed.

End CLEANUP.

Generated by coq2html