path: root/backend/ForwardMovesproof.v

Require Import FunInd.
Require Import Coqlib Maps Errors Integers Floats Lattice Kildall.
Require Import AST Linking.
Require Import Values Memory Globalenvs Events Smallstep.
Require Import Registers Op RTL.
Require Import ForwardMoves.


Definition match_prog (p tp: RTL.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).
Proof.
  intros. eapply match_transform_program; eauto.
Qed.

Section PRESERVATION.

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

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 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 sig_preserved:
  forall f, funsig (transf_fundef f) = funsig f.
Proof.
  destruct f; trivial.
Qed.

Lemma find_function_translated:
  forall ros rs fd,
  find_function ge ros rs = Some fd ->
  find_function tge ros rs = Some (transf_fundef fd).
Proof.
  unfold find_function; intros. destruct ros as [r|id].
  eapply functions_translated; eauto.
  rewrite symbols_preserved. destruct (Genv.find_symbol ge id); try congruence.
  eapply function_ptr_translated; eauto.
Qed.

Lemma transf_function_at:
  forall f pc i,
  f.(fn_code)!pc = Some i ->
  (transf_function f).(fn_code)!pc =
    Some(transf_instr (forward_map f) pc i).
Proof.
  intros until i. intro CODE.
  unfold transf_function; simpl.
  rewrite PTree.gmap.
  unfold option_map.
  rewrite CODE.
  reflexivity.
Qed.

(*
Definition fmap_sem (fmap : option (PMap.t RB.t)) (pc : node) (rs : regset) :=
  forall x : reg,
    (rs # (subst_arg fmap pc x)) = (rs # x).
 *)

Lemma apply_instr'_bot :
  forall code,
  forall pc,
    RB.eq (apply_instr' code pc RB.bot) RB.bot.
Proof.
  reflexivity.
Qed.

Definition get_rb_sem (rb : RB.t) (rs : regset) :=
  match rb with
  | None => False
  | Some rel =>
    forall x : reg,
      (rs # (get_r rel x)) = (rs # x)
  end.

Lemma get_rb_sem_ge:
  forall rb1 rb2 : RB.t,
    (RB.ge rb1 rb2) ->
    forall rs : regset,
      (get_rb_sem rb2 rs) -> (get_rb_sem rb1 rs).
Proof.
  destruct rb1 as [r1 | ];
    destruct rb2 as [r2 | ];
    unfold get_rb_sem;
    simpl;
    intros GE rs RB2RS;
    try contradiction.
  unfold RELATION.ge in GE.
  unfold get_r in *.
  intro x.
  pose proof (GE x) as GEx.
  pose proof (RB2RS x) as RB2RSx.
  destruct (r1 ! x) as [r1x | ] in *;
    destruct (r2 ! x) as [r2x | ] in *;
    congruence.
Qed.

Definition fmap_sem (fmap : option (PMap.t RB.t))
  (pc : node) (rs : regset) :=
  match fmap with
  | None => True
  | Some m => get_rb_sem (PMap.get pc m) rs
  end.

Lemma subst_arg_ok:
  forall f,
  forall pc,
  forall rs,
  forall arg,
    fmap_sem (forward_map f) pc rs ->
    rs # (subst_arg (forward_map f) pc arg) = rs # arg.
Proof.
  intros until arg.
  intro SEM.
  unfold fmap_sem in SEM.
  destruct (forward_map f) as [map |]in *; trivial.
  simpl.
  unfold get_rb_sem in *.
  destruct (map # pc).
  2: contradiction.
  apply SEM.
Qed.

Lemma subst_args_ok:
  forall f,
  forall pc,
  forall rs,
  fmap_sem (forward_map f) pc rs ->
  forall args,
    rs ## (subst_args (forward_map f) pc args) = rs ## args.
Proof.
  induction args; trivial.
  simpl.
  f_equal.
  apply subst_arg_ok; assumption.
  assumption.
Qed.

Lemma kill_ok:
  forall dst,
  forall mpc,
  forall rs,
  forall v,
    get_rb_sem (Some mpc) rs ->
    get_rb_sem (Some (kill dst mpc)) rs # dst <- v.
Proof.
  unfold get_rb_sem.
  intros until v.
  intros SEM x.
  destruct (Pos.eq_dec x dst) as [EQ | NEQ].
  {
    subst dst.
    rewrite Regmap.gss.
    unfold kill, get_r.
    rewrite PTree.gfilter1.
    rewrite PTree.grs.
    apply Regmap.gss.
  }
  rewrite (Regmap.gso v rs NEQ).
  unfold kill, get_r in *.
  rewrite PTree.gfilter1.
  rewrite PTree.gro by assumption.
  pose proof (SEM x) as SEMx.
  destruct (mpc ! x).
  {
    destruct (Pos.eq_dec dst r).
    {
      subst dst.
      rewrite Regmap.gso by assumption.
      reflexivity.
    }
    rewrite Regmap.gso by congruence.
    assumption.
  }
  rewrite Regmap.gso by assumption.
  reflexivity.
Qed.

Ltac TR_AT :=
  match goal with
  | [ A: (fn_code _)!_ = Some _ |- _ ] =>
        generalize (transf_function_at _ _ _ A); intros
  end.

Inductive match_frames: RTL.stackframe -> RTL.stackframe -> Prop :=
| match_frames_intro: forall res f sp pc rs,
    (fmap_sem (forward_map f) pc rs) ->
    match_frames (Stackframe res f sp pc rs)
                 (Stackframe res (transf_function f) sp pc rs).

Inductive match_states: RTL.state -> RTL.state -> Prop :=
  | match_regular_states: forall stk f sp pc rs m stk'
                                 (STACKS: list_forall2 match_frames stk stk'),
      (fmap_sem (forward_map f) pc rs) ->
      match_states (State stk f sp pc rs m)
                   (State stk' (transf_function f) sp pc rs m)
  | match_callstates: forall stk f args m stk'
        (STACKS: list_forall2 match_frames stk stk'),
      match_states (Callstate stk f args m)
                   (Callstate stk' (transf_fundef f) args m)
  | match_returnstates: forall stk v m stk'
        (STACKS: list_forall2 match_frames stk stk'),
      match_states (Returnstate stk v m)
                   (Returnstate stk' v m).

Lemma step_simulation:
  forall S1 t S2, RTL.step ge S1 t S2 ->
  forall S1', match_states S1 S1' ->
              exists S2', RTL.step tge S1' t S2' /\ match_states S2 S2'.
Proof.
  induction 1; intros S1' MS; inv MS; try TR_AT.
- (* nop *)
  econstructor; split. eapply exec_Inop; eauto.
  constructor; auto.
  
  simpl in *.
  unfold fmap_sem in *.
  destruct (forward_map _) as [map |] eqn:MAP in *; trivial.
  apply get_rb_sem_ge with (rb2 := map # pc); trivial.
  replace (map # pc) with (apply_instr' (fn_code f) pc (map # pc)).
  {
    eapply DS.fixpoint_solution with (code := fn_code f) (successors := successors_instr); try eassumption.
    2: apply apply_instr'_bot.
    simpl. tauto.
  }
  unfold apply_instr'.
  unfold get_rb_sem in *.
  destruct (map # pc) in *; try contradiction.
  rewrite H.
  reflexivity.
- (* op *)
  econstructor; split.
  eapply exec_Iop with (v := v); eauto.
  rewrite <- H0.
  rewrite subst_args_ok by assumption.
  apply eval_operation_preserved. exact symbols_preserved.
  constructor; auto.
  
  admit.
  
(* load *)
- econstructor; split.
  assert (eval_addressing tge sp addr rs ## args = Some a).
  rewrite <- H0.
  apply eval_addressing_preserved. exact symbols_preserved.
  eapply exec_Iload; eauto.
  rewrite subst_args_ok; assumption.
  constructor; auto.

  simpl in *.
  unfold fmap_sem in *.
  destruct (forward_map _) as [map |] eqn:MAP in *; trivial.
  destruct (map # pc) as [mpc |] eqn:MPC in *; try contradiction.
  apply get_rb_sem_ge with (rb2 := Some (kill dst mpc)).
  {
    replace (Some (kill dst mpc)) with (apply_instr' (fn_code f) pc (map # pc)).
    {
      eapply DS.fixpoint_solution with (code := fn_code f) (successors := successors_instr); try eassumption.
      2: apply apply_instr'_bot.
      simpl. tauto.
    }
    unfold apply_instr'.
    rewrite H.
    rewrite MPC.
    reflexivity.
  }
  apply kill_ok.
  assumption.
  
- (* load notrap1 *)
  econstructor; split.
  assert (eval_addressing tge sp addr rs ## args = None).
  rewrite <- H0. apply eval_addressing_preserved. exact symbols_preserved.
  eapply exec_Iload_notrap1; eauto.
  rewrite subst_args_ok; assumption.
  constructor; auto.

  simpl in *.
  unfold fmap_sem in *.
  destruct (forward_map _) as [map |] eqn:MAP in *; trivial.
  destruct (map # pc) as [mpc |] eqn:MPC in *; try contradiction.
  apply get_rb_sem_ge with (rb2 := Some (kill dst mpc)).
  {
    replace (Some (kill dst mpc)) with (apply_instr' (fn_code f) pc (map # pc)).
    {
      eapply DS.fixpoint_solution with (code := fn_code f) (successors := successors_instr); try eassumption.
      2: apply apply_instr'_bot.
      simpl. tauto.
    }
    unfold apply_instr'.
    rewrite H.
    rewrite MPC.
    reflexivity.
  }
  apply kill_ok.
  assumption.
  
- (* load notrap2 *)
  econstructor; split.
  assert (eval_addressing tge sp addr rs ## args = Some a).
  rewrite <- H0. apply eval_addressing_preserved. exact symbols_preserved.
  eapply exec_Iload_notrap2; eauto.
  rewrite subst_args_ok; assumption.
  constructor; auto.

  simpl in *.
  unfold fmap_sem in *.
  destruct (forward_map _) as [map |] eqn:MAP in *; trivial.
  destruct (map # pc) as [mpc |] eqn:MPC in *; try contradiction.
  apply get_rb_sem_ge with (rb2 := Some (kill dst mpc)).
  {
    replace (Some (kill dst mpc)) with (apply_instr' (fn_code f) pc (map # pc)).
    {
      eapply DS.fixpoint_solution with (code := fn_code f) (successors := successors_instr); try eassumption.
      2: apply apply_instr'_bot.
      simpl. tauto.
    }
    unfold apply_instr'.
    rewrite H.
    rewrite MPC.
    reflexivity.
  }
  apply kill_ok.
  assumption.
  
- (* store *)
  econstructor; split.
  assert (eval_addressing tge sp addr rs ## args = Some a).
  rewrite <- H0. apply eval_addressing_preserved. exact symbols_preserved.
  eapply exec_Istore; eauto.
  rewrite subst_args_ok; assumption.
  constructor; auto.

  simpl in *.
  unfold fmap_sem in *.
  destruct (forward_map _) as [map |] eqn:MAP in *; trivial.
  apply get_rb_sem_ge with (rb2 := map # pc); trivial.
  replace (map # pc) with (apply_instr' (fn_code f) pc (map # pc)).
  {
    eapply DS.fixpoint_solution with (code := fn_code f) (successors := successors_instr); try eassumption.
    2: apply apply_instr'_bot.
    simpl. tauto.
  }
  unfold apply_instr'.
  unfold get_rb_sem in *.
  destruct (map # pc) in *; try contradiction.
  rewrite H.
  reflexivity.
  
(* call *)
- econstructor; split.
  eapply exec_Icall with (fd := transf_fundef fd); eauto.
    eapply find_function_translated; eauto.
    apply sig_preserved.
  rewrite subst_args_ok by assumption.
  constructor. constructor; auto. constructor.

  admit.
  
(* tailcall *)
- econstructor; split.
  eapply exec_Itailcall with (fd := transf_fundef fd); eauto.
    eapply find_function_translated; eauto.
    apply sig_preserved.
  rewrite subst_args_ok by assumption.
  constructor. auto.
  
(* builtin *)
- econstructor; split.
  eapply exec_Ibuiltin; eauto.
    eapply eval_builtin_args_preserved with (ge1 := ge); eauto. exact symbols_preserved.
    eapply external_call_symbols_preserved; eauto. apply senv_preserved.
  constructor; auto.

  admit.
  
(* cond *)
- econstructor; split.
  eapply exec_Icond; eauto.
  rewrite subst_args_ok; eassumption.
  constructor; auto.

  admit.
  
(* jumptbl *)
- econstructor; split.
  eapply exec_Ijumptable; eauto.
  rewrite subst_arg_ok; eassumption.
  constructor; auto.

  admit.
  
(* return *)
- destruct or as [arg | ].
  {
    econstructor; split.
    eapply exec_Ireturn; eauto.
    unfold regmap_optget.
    rewrite subst_arg_ok by eassumption.
    constructor; auto.
  }
    econstructor; split.
    eapply exec_Ireturn; eauto.
    constructor; auto.
  
  
(* internal function *)
-  simpl. econstructor; split.
  eapply exec_function_internal; eauto.
  constructor; auto.

  admit.
  
(* external function *)
- econstructor; split.
  eapply exec_function_external; eauto.
    eapply external_call_symbols_preserved; eauto. apply senv_preserved.
    constructor; auto.

(* return *)
- inv STACKS. inv H1.
  econstructor; split.
  eapply exec_return; eauto.
  constructor; auto.

  admit.
Admitted.


Lemma transf_initial_states:
  forall S1, RTL.initial_state prog S1 ->
  exists S2, RTL.initial_state tprog S2 /\ match_states S1 S2.
Proof.
  intros. inv H. econstructor; split.
  econstructor.
    eapply (Genv.init_mem_transf TRANSL); eauto.
    rewrite symbols_preserved. rewrite (match_program_main TRANSL). eauto.
    eapply function_ptr_translated; eauto.
    rewrite <- H3; apply sig_preserved.
  constructor. constructor.
Qed.

Lemma transf_final_states:
  forall S1 S2 r, match_states S1 S2 -> RTL.final_state S1 r -> RTL.final_state S2 r.
Proof.
  intros. inv H0. inv H. inv STACKS. constructor.
Qed.

Theorem transf_program_correct:
  forward_simulation (RTL.semantics prog) (RTL.semantics tprog).
Proof.
  eapply forward_simulation_step.
  apply senv_preserved.
  eexact transf_initial_states.
  eexact transf_final_states.
  exact step_simulation.
Qed.

End PRESERVATION.