Module CleanupLabelsproof

Require Import FSets.
Require Import Coqlib Ordered Integers.
Require Import AST Linking.
Require Import Values Memory Events Globalenvs Smallstep.
Require Import Op Locations Linear.
Require Import CleanupLabels.

Module LabelsetFacts := FSetFacts.Facts(Labelset).

Definition match_prog (p tp: Linear.program) :=
  match_program (fun ctx f tf => tf = transf_fundef f) eq p tp.

Lemma transf_program_match:
  forall p, match_prog p (transf_program p).

Section CLEANUP.
Context `{external_calls_prf: ExternalCalls}.

Variable fn_stack_requirements: ident -> Z.

Variables prog tprog: program.
Hypothesis TRANSL: match_prog prog tprog.
Let ge := Genv.globalenv prog.
Let tge := Genv.globalenv tprog.

Lemma symbols_preserved:
  forall id,
  Genv.find_symbol tge id = Genv.find_symbol ge id.
Proof (Genv.find_symbol_transf TRANSL).

Lemma senv_preserved:
  Senv.equiv ge tge.
Proof (Genv.senv_transf TRANSL).

Lemma genv_next_preserved:
  Genv.genv_next tge = Genv.genv_next ge.

Lemma functions_translated:
  forall v f,
  Genv.find_funct ge v = Some f ->
  Genv.find_funct tge v = Some (transf_fundef f).
Proof (Genv.find_funct_transf TRANSL).

Lemma function_ptr_translated:
  forall v f,
  Genv.find_funct_ptr ge v = Some f ->
  Genv.find_funct_ptr tge v = Some (transf_fundef f).
Proof (Genv.find_funct_ptr_transf TRANSL).

Lemma sig_function_translated:
  forall f,
  funsig (transf_fundef f) = funsig f.

Lemma find_function_translated:
  forall ros ls f,
  find_function ge ros ls = Some f ->
  find_function tge ros ls = Some (transf_fundef f).


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).

Remark add_label_branched_to_contains:
  forall ls i lbl,
  instr_branches_to i lbl ->
  Labelset.In lbl (add_label_branched_to ls i).

Lemma labels_branched_to_correct:
  forall c i lbl,
  In i c -> instr_branches_to i lbl -> Labelset.In lbl (labels_branched_to c).


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.


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').

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).

Lemma find_label_incl:
  forall lbl c c', find_label lbl c = Some c' -> incl c' c.


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 sz,
      list_forall2 match_stackframes s ts ->
      match_states (Callstate s f ls m sz)
                   (Callstate ts (transf_fundef f) ls m sz)
  | 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 init_ls,
  forall s ts,
  list_forall2 match_stackframes s ts ->
  parent_locset init_ls ts = parent_locset init_ls s.

Section WITHINITLS.

Variable init_ls: locset.

Lemma ros_is_function_translated:
  forall ros rs i,
    ros_is_function ge ros rs i ->
    ros_is_function tge ros rs i.


Theorem transf_step_correct:
  forall s1 t s2, step fn_stack_requirements init_ls ge s1 t s2 ->
  forall s1' (MS: match_states s1 s1'),
  (exists s2', step fn_stack_requirements init_ls tge s1' t s2' /\ match_states s2 s2')
  \/ (measure s2 < measure s1 /\ t = E0 /\ match_states s2 s1')%nat.

End WITHINITLS.

Lemma transf_initial_states:
  forall st1, initial_state fn_stack_requirements prog st1 ->
  exists st2, initial_state fn_stack_requirements tprog st2 /\ match_states st1 st2.

Lemma transf_final_states:
  forall st1 st2 r,
  match_states st1 st2 -> final_state st1 r -> final_state st2 r.

Theorem transf_program_correct:
  forward_simulation (Linear.semantics fn_stack_requirements prog) (Linear.semantics fn_stack_requirements tprog).

End CLEANUP.