(* *********************************************************************)
(*                                                                     *)
(*              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.     *)
(*                                                                     *)
(* *********************************************************************)

Require Import Coqlib Errors.
Require Import Integers Floats AST Linking.
Require Import Values Memory Events Globalenvs Smallstep.
Require Import Op Locations Machblock Conventions Asmblock.
Require Import Asmblockgenproof0.
Require Import PostpassScheduling.
Require Import Asmblockgenproof.
Require Import Axioms.

Local Open Scope error_monad_scope.

Definition match_prog (p tp: Asmblock.program) :=
  match_program (fun _ f tf => transf_fundef f = OK tf) eq p tp.

Lemma transf_program_match:
  forall p tp, transf_program p = OK tp -> match_prog p tp.
Proof.
  intros. eapply match_transform_partial_program; eauto.
Qed.

Inductive bblock_equiv (ge: Genv.t fundef unit) (f: function) (bb bb': bblock) :=
  | bblock_equiv_intro:
      (forall rs m,
      exec_bblock ge f bb rs m = exec_bblock ge f bb' rs m) ->
      bblock_equiv ge f bb bb'.

Lemma app_nonil {A: Type} (l l': list A) : l <> nil -> l ++ l' <> nil.
Proof.
  intros. destruct l; simpl.
  - contradiction.
  - discriminate.
Qed.

Lemma app_nonil2 {A: Type} : forall (l l': list A), l' <> nil -> l ++ l' <> nil.
Proof.
  destruct l.
  - intros. simpl; auto.
  - intros. rewrite <- app_comm_cons. discriminate.
Qed.

Program Definition concat2 (bb bb': bblock) : res bblock :=
  match (exit bb) with
  | None => 
      match (header bb') with
      | nil => 
          match (exit bb') with 
          | Some (PExpand (Pbuiltin _ _ _)) => Error (msg "PostpassSchedulingproof.concat2: builtin not alone")
          | _ => OK {| header := header bb; body := body bb ++ body bb'; exit := exit bb' |}
          end
      | _ => Error (msg "PostpassSchedulingproof.concat2")
      end
  | _ => Error (msg "PostpassSchedulingproof.concat2")
  end.
Next Obligation.
  apply wf_bblock_refl. constructor.
  - destruct bb' as [hd' bdy' ex' WF']. destruct bb as [hd bdy ex WF]. simpl in *.
    apply wf_bblock_refl in WF'. apply wf_bblock_refl in WF.
    inversion_clear WF'. inversion_clear WF. clear H1 H3.
    inversion H2; inversion H0.
    + left. apply app_nonil. auto.
    + right. auto.
    + left. apply app_nonil2. auto.
    + right. auto.
  - unfold builtin_alone. intros. rewrite H0 in H.
    assert (Some (PExpand (Pbuiltin ef args res)) <> Some (PExpand (Pbuiltin ef args res))).
    apply (H ef args res). contradict H1. auto.
Qed.

Fixpoint concat_all (lbb: list bblock) : res bblock :=
  match lbb with
  | nil => Error (msg "PostpassSchedulingproof.concatenate: empty list")
  | bb::nil => OK bb
  | bb::lbb =>
      do bb' <- concat_all lbb;
      concat2 bb bb'
  end.

Axiom verified_schedule_correct:
  forall ge f bb lbb,
  verified_schedule bb = OK lbb ->
  exists tbb, 
     concat_all lbb = OK tbb
  /\ bblock_equiv ge f bb tbb.

Axiom verified_schedule_single_inst: forall bb, size bb = 1 -> verified_schedule bb = OK (bb::nil).

Remark builtin_body_nil:
  forall bb ef args res, exit bb = Some (PExpand (Pbuiltin ef args res)) -> body bb = nil.
Proof.
  intros. destruct bb as [hd bdy ex WF]. simpl in *.
  apply wf_bblock_refl in WF. inv WF. unfold builtin_alone in H1.
  eapply H1; eauto.
Qed.

Lemma verified_schedule_builtin_idem:
  forall bb ef args res lbb,
  exit bb = Some (PExpand (Pbuiltin ef args res)) ->
  verified_schedule bb = OK lbb ->
  lbb = bb :: nil.
Proof.
  intros. exploit builtin_body_nil; eauto. intros.
  rewrite verified_schedule_single_inst in H0.
  - inv H0. auto.
  - unfold size. rewrite H. rewrite H1. simpl. auto.
Qed.

Lemma concat2_noexit:
  forall a b bb,
  concat2 a b = OK bb ->
  exit a = None.
Proof.
  intros. destruct a as [hd bdy ex WF]; simpl in *.
  destruct ex as [e|]; simpl in *; auto.
  unfold concat2 in H. simpl in H. discriminate.
Qed.

Lemma concat2_decomp:
  forall a b bb,
  concat2 a b = OK bb ->
     body bb = body a ++ body b
  /\ exit bb = exit b.
Proof.
  intros. exploit concat2_noexit; eauto. intros.
  destruct a as [hda bda exa WFa]; destruct b as [hdb bdb exb WFb]; destruct bb as [hd bd ex WF]; simpl in *.
  subst exa.
  unfold concat2 in H; simpl in H.
  destruct hdb.
  - destruct exb.
    + destruct c.
      * destruct i. discriminate.
      * monadInv H. split; auto.
    + monadInv H. split; auto.
  - discriminate.
Qed.

Lemma exec_body_app:
  forall l l' ge rs m rs'' m'',
  exec_body ge (l ++ l') rs m = Next rs'' m'' ->
  exists rs' m',
       exec_body ge l rs m = Next rs' m'
    /\ exec_body ge l' rs' m' = Next rs'' m''.
Proof.
  induction l.
  - intros. simpl in H. repeat eexists. auto.
  - intros. rewrite <- app_comm_cons in H. simpl in H.
    destruct (exec_basic_instr ge a rs m) eqn:EXEBI.
    + apply IHl in H. destruct H as (rs1 & m1 & EXEB1 & EXEB2).
      repeat eexists. simpl. rewrite EXEBI. eauto. auto.
    + discriminate.
Qed.

Lemma exec_body_pc:
  forall l ge rs1 m1 rs2 m2,
  exec_body ge l rs1 m1 = Next rs2 m2 ->
  rs2 PC = rs1 PC.
Proof.
  induction l.
  - intros. inv H. auto.
  - intros until m2. intro EXEB.
    inv EXEB. destruct (exec_basic_instr _ _ _) eqn:EBI; try discriminate.
    eapply IHl in H0. rewrite H0.
    erewrite exec_basic_instr_pc; eauto.
Qed.

Lemma next_eq {A: Type}:
  forall (rs rs':A) m m',
  rs = rs' -> m = m' -> Next rs m = Next rs' m'.
Proof.
  intros. congruence.
Qed.

Lemma regset_double_set:
  forall r1 r2 (rs: regset) v1 v2,
  r1 <> r2 ->
  (rs # r1 <- v1 # r2 <- v2) = (rs # r2 <- v2 # r1 <- v1).
Proof.
  intros. apply functional_extensionality. intros r. destruct (preg_eq r r1).
  - subst. rewrite Pregmap.gso; auto. repeat (rewrite Pregmap.gss). auto.
  - destruct (preg_eq r r2).
    + subst. rewrite Pregmap.gss. rewrite Pregmap.gso; auto. rewrite Pregmap.gss. auto.
    + repeat (rewrite Pregmap.gso; auto).
Qed.

Lemma regset_double_set_id:
  forall r (rs: regset) v1 v2,
  (rs # r <- v1 # r <- v2) = (rs # r <- v2).
Proof.
  intros. apply functional_extensionality. intros. destruct (preg_eq r x).
  - subst r. repeat (rewrite Pregmap.gss; auto).
  - repeat (rewrite Pregmap.gso); auto.
Qed.

Lemma exec_load_pc_var:
  forall ge t rs m rd ra ofs rs' m' v,
  exec_load ge t rs m rd ra ofs = Next rs' m' ->
  exec_load ge t rs # PC <- v m rd ra ofs = Next rs' # PC <- v m'.
Proof.
  intros. unfold exec_load in *. rewrite Pregmap.gso; try discriminate.
  destruct (Mem.loadv _ _ _).
  - inv H. apply next_eq; auto. apply functional_extensionality. intros. rewrite regset_double_set; auto. discriminate.
  - discriminate.
Qed.

Lemma exec_store_pc_var:
  forall ge t rs m rd ra ofs rs' m' v,
  exec_store ge t rs m rd ra ofs = Next rs' m' ->
  exec_store ge t rs # PC <- v m rd ra ofs = Next rs' # PC <- v m'.
Proof.
  intros. unfold exec_store in *. rewrite Pregmap.gso; try discriminate.
  destruct (Mem.storev _ _ _).
  - inv H. apply next_eq; auto.
  - discriminate.
Qed.

Lemma exec_basic_instr_pc_var:
  forall ge i rs m rs' m' v,
  exec_basic_instr ge i rs m = Next rs' m' ->
  exec_basic_instr ge i (rs # PC <- v) m = Next (rs' # PC <- v) m'.
Proof.
  intros. unfold exec_basic_instr in *. destruct i.
  - unfold exec_arith_instr in *. exploreInst.
      all: try (inv H; apply next_eq; auto;
      apply functional_extensionality; intros; rewrite regset_double_set; auto; discriminate).
  - exploreInst; apply exec_load_pc_var; auto.
  - exploreInst; apply exec_store_pc_var; auto.
  - destruct (Mem.alloc _ _ _) as (m1 & stk). repeat (rewrite Pregmap.gso; try discriminate).
    destruct (Mem.storev _ _ _ _); try discriminate.
    inv H. apply next_eq; auto. apply functional_extensionality. intros.
    rewrite (regset_double_set GPR32 PC); try discriminate.
    rewrite (regset_double_set GPR12 PC); try discriminate.
    rewrite (regset_double_set GPR14 PC); try discriminate. reflexivity.
  - repeat (rewrite Pregmap.gso; try discriminate).
    destruct (Mem.loadv _ _ _); try discriminate.
    destruct (rs GPR12); try discriminate.
    destruct (Mem.free _ _ _ _); try discriminate.
    inv H. apply next_eq; auto.
    rewrite (regset_double_set GPR32 PC).
    rewrite (regset_double_set GPR12 PC). reflexivity.
    all: discriminate.
  - destruct rs0; try discriminate. inv H. apply next_eq; auto.
    repeat (rewrite Pregmap.gso; try discriminate). apply regset_double_set; discriminate.
  - destruct rd; try discriminate. inv H. apply next_eq; auto.
    repeat (rewrite Pregmap.gso; try discriminate). apply regset_double_set; discriminate.
  - inv H. apply next_eq; auto.
Qed.

Lemma exec_body_pc_var:
  forall l ge rs m rs' m' v,
  exec_body ge l rs m = Next rs' m' ->
  exec_body ge l (rs # PC <- v) m = Next (rs' # PC <- v) m'.
Proof.
  induction l.
  - intros. simpl. simpl in H. inv H. auto.
  - intros. simpl in *.
    destruct (exec_basic_instr ge a rs m) eqn:EXEBI; try discriminate.
    erewrite exec_basic_instr_pc_var; eauto.
Qed.

Lemma pc_set_add:
  forall rs v r x y,
  0 <= x <= Ptrofs.max_unsigned ->
  0 <= y <= Ptrofs.max_unsigned ->
  rs # r <- (Val.offset_ptr v (Ptrofs.repr (x + y))) = rs # r <- (Val.offset_ptr (rs # r <- (Val.offset_ptr v (Ptrofs.repr x)) r) (Ptrofs.repr y)).
Proof.
  intros. apply functional_extensionality. intros r0. destruct (preg_eq r r0).
  - subst. repeat (rewrite Pregmap.gss); auto.
    destruct v; simpl; auto.
    rewrite Ptrofs.add_assoc.
    cutrewrite (Ptrofs.repr (x + y) = Ptrofs.add (Ptrofs.repr x) (Ptrofs.repr y)); auto.
    unfold Ptrofs.add.
    cutrewrite (x + y = Ptrofs.unsigned (Ptrofs.repr x) + Ptrofs.unsigned (Ptrofs.repr y)); auto.
    repeat (rewrite Ptrofs.unsigned_repr); auto.
  - repeat (rewrite Pregmap.gso; auto).
Qed.

Lemma concat2_straight:
  forall a b bb rs m rs'' m'' f ge,
  size a <= Ptrofs.max_unsigned -> size b <= Ptrofs.max_unsigned ->
  concat2 a b = OK bb ->
  exec_bblock ge f bb rs m = Next rs'' m'' ->
  exists rs' m',
       exec_bblock ge f a rs m = Next rs' m'
    /\ rs' PC = Val.offset_ptr (rs PC) (Ptrofs.repr (size a))
    /\ exec_bblock ge f b rs' m' = Next rs'' m''.
Proof.
  intros until ge. intros LTA LTB CONC2 EXEB.
  exploit concat2_noexit; eauto. intros EXA.
  exploit concat2_decomp; eauto. intros. inv H.
  unfold exec_bblock in EXEB. destruct (exec_body ge (body bb) rs m) eqn:EXEB'; try discriminate.
  rewrite H0 in EXEB'. apply exec_body_app in EXEB'. destruct EXEB' as (rs1 & m1 & EXEB1 & EXEB2).
  repeat eexists.
    unfold exec_bblock. rewrite EXEB1. rewrite EXA. simpl. eauto.
    exploit exec_body_pc. eapply EXEB1. intros. rewrite <- H. auto.
    unfold exec_bblock. unfold nextblock. erewrite exec_body_pc_var; eauto.
    rewrite <- H1. unfold nextblock in EXEB. rewrite regset_double_set_id.
    assert (size bb = size a + size b).
    { unfold size. rewrite H0. rewrite H1. rewrite app_length. rewrite EXA. simpl. rewrite Nat.add_0_r.
      repeat (rewrite Nat2Z.inj_add). omega. }
    clear EXA H0 H1. rewrite H in EXEB.
    assert (rs1 PC = rs0 PC). { apply exec_body_pc in EXEB2. auto. }
    rewrite H0. rewrite <- pc_set_add; auto.
    exploit AB.size_positive. instantiate (1 := a). intro. omega.
    exploit AB.size_positive. instantiate (1 := b). intro. omega.
Qed.

Axiom TODO: False.

Lemma concat_all_exec_bblock (ge: Genv.t fundef unit) (f: function) :
  forall a bb rs m lbb rs'' m'', 
  lbb <> nil ->
  concat_all (a :: lbb) = OK bb ->
  exec_bblock ge f bb rs m = Next rs'' m'' ->
  exists bb' rs' m',
       concat_all lbb = OK bb'
    /\ exec_bblock ge f a rs m = Next rs' m'
    /\ rs' PC = Val.offset_ptr (rs PC) (Ptrofs.repr (size a))
    /\ exec_bblock ge f bb' rs' m' = Next rs'' m''.
Proof.
  intros until m''. intros Hnonil CONC EXEB.
  simpl in CONC.
  destruct lbb as [|b lbb]; try contradiction. clear Hnonil.
  monadInv CONC. exploit concat2_straight; eauto. 1-2: destruct TODO. intros (rs' & m' & EXEB1 & PCeq & EXEB2).
  exists x. repeat econstructor. all: eauto.
Admitted.

Lemma concat2_size:
  forall a b bb, concat2 a b = OK bb -> size bb = size a + size b.
Proof.
  intros. unfold concat2 in H.
  destruct a as [hda bda exa WFa]; destruct b as [hdb bdb exb WFb]; destruct bb as [hd bdy ex WF]; simpl in *.
  destruct exa; try discriminate. destruct hdb; try discriminate. destruct exb; try discriminate.
  - destruct c.
    + destruct i; try discriminate.
    + inv H. unfold size; simpl. rewrite app_length. rewrite Nat.add_0_r. rewrite <- Nat2Z.inj_add. rewrite Nat.add_assoc. reflexivity.
  - inv H. unfold size; simpl. rewrite app_length. repeat (rewrite Nat.add_0_r). rewrite <- Nat2Z.inj_add. reflexivity.
Qed.

Lemma concat_all_size :
  forall lbb a bb bb',
  concat_all (a :: lbb) = OK bb ->
  concat_all lbb = OK bb' ->
  size bb = size a + size bb'.
Proof.
  intros. unfold concat_all in H. fold concat_all in H.
  destruct lbb; try discriminate.
  monadInv H. rewrite H0 in EQ. inv EQ.
  apply concat2_size. assumption.
Qed.

Lemma ptrofs_add_repr :
  forall a b,
  Ptrofs.unsigned (Ptrofs.add (Ptrofs.repr a) (Ptrofs.repr b)) = Ptrofs.unsigned (Ptrofs.repr (a + b)).
Proof.
  intros a b.
  rewrite Ptrofs.add_unsigned. repeat (rewrite Ptrofs.unsigned_repr_eq).
  rewrite <- Zplus_mod. auto.
Qed.

Section PRESERVATION.

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

Lemma transf_function_no_overflow:
  forall f tf,
  transf_function f = OK tf -> size_blocks tf.(fn_blocks) <= Ptrofs.max_unsigned.
Proof.
  intros. monadInv H. destruct (zlt Ptrofs.max_unsigned (size_blocks x.(fn_blocks))); inv EQ0.
  omega.
Qed.

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

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

Lemma functions_translated:
  forall v f,
  Genv.find_funct ge v = Some f ->
  exists tf,
  Genv.find_funct tge v = Some tf /\ transf_fundef f = OK tf.
Proof (Genv.find_funct_transf_partial TRANSL).

Lemma function_ptr_translated:
  forall v f,
  Genv.find_funct_ptr ge v = Some f ->
  exists tf,
  Genv.find_funct_ptr tge v = Some tf /\ transf_fundef f = OK tf.
Proof (Genv.find_funct_ptr_transf_partial TRANSL).

Lemma functions_transl:
  forall fb f tf,
  Genv.find_funct_ptr ge fb = Some (Internal f) ->
  transf_function f = OK tf ->
  Genv.find_funct_ptr tge fb = Some (Internal tf).
Proof.
  intros. exploit function_ptr_translated; eauto.
  intros (tf' & A & B). monadInv B. rewrite H0 in EQ. inv EQ. auto.
Qed.

Inductive match_states: state -> state -> Prop :=
  | match_states_intro:
      forall s1 s2, s1 = s2 -> match_states s1 s2.

Lemma prog_main_preserved:
  prog_main tprog = prog_main prog.
Proof (match_program_main TRANSL).

Lemma prog_main_address_preserved:
  (Genv.symbol_address (Genv.globalenv prog) (prog_main prog) Ptrofs.zero) =
  (Genv.symbol_address (Genv.globalenv tprog) (prog_main tprog) Ptrofs.zero).
Proof.
  unfold Genv.symbol_address. rewrite symbols_preserved.
  rewrite prog_main_preserved. auto.
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.
    eapply (Genv.init_mem_transf_partial TRANSL); eauto.
  - econstructor; eauto. subst ge0. subst rs0. rewrite prog_main_address_preserved. auto.
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. econstructor; eauto.
Qed.

Lemma transf_find_bblock:
  forall ofs f bb tf,
  find_bblock (Ptrofs.unsigned ofs) (fn_blocks f) = Some bb ->
  transf_function f = OK tf ->
  exists lbb,
     verified_schedule bb = OK lbb
  /\ exists c, code_tail (Ptrofs.unsigned ofs) (fn_blocks tf) (lbb ++ c).
Proof.
Admitted.

Lemma transf_exec_bblock:
  forall f tf bb rs m,
  transf_function f = OK tf ->
  exec_bblock ge f bb rs m = exec_bblock tge tf bb rs m.
Proof.
Admitted.

Lemma transf_step_simu:
  forall tf b lbb ofs c tbb rs m rs' m',
  Genv.find_funct_ptr tge b = Some (Internal tf) ->
  size_blocks (fn_blocks tf) <= Ptrofs.max_unsigned ->
  rs PC = Vptr b ofs ->
  code_tail (Ptrofs.unsigned ofs) (fn_blocks tf) (lbb ++ c) ->
  concat_all lbb = OK tbb ->
  exec_bblock tge tf tbb rs m = Next rs' m' ->
  plus step tge (State rs m) E0 (State rs' m').
Proof.
  induction lbb.
  - intros until m'. simpl. intros. discriminate.
  - intros until m'. intros GFIND SIZE PCeq TAIL CONC EXEB.
    destruct lbb.
    + simpl in *. clear IHlbb. inv CONC. eapply plus_one. econstructor; eauto. eapply find_bblock_tail; eauto.
    + exploit concat_all_exec_bblock; eauto; try discriminate.
    intros (tbb0 & rs0 & m0 & CONC0 & EXEB0 & PCeq' & EXEB1).
    eapply plus_left.
    econstructor.
      3: eapply find_bblock_tail. rewrite <- app_comm_cons in TAIL. 3: eauto.
      all: eauto.
    eapply plus_star. eapply IHlbb; eauto. rewrite PCeq in PCeq'. simpl in PCeq'. all: eauto.
    eapply code_tail_next_int; eauto.
Qed.

Theorem transf_step_correct:
  forall s1 t s2, step ge s1 t s2 ->
  forall s1' (MS: match_states s1 s1'),
  (exists s2', plus step tge s1' t s2' /\ match_states s2 s2').
Proof.
  induction 1; intros; inv MS.
  - exploit function_ptr_translated; eauto. intros (tf & FFP & TRANSF). monadInv TRANSF.
    exploit transf_find_bblock; eauto. intros (lbb & VES & c & TAIL).
    exploit verified_schedule_correct; eauto. intros (tbb & CONC & BBEQ).
    assert (NOOV: size_blocks x.(fn_blocks) <= Ptrofs.max_unsigned).
      eapply transf_function_no_overflow; eauto. 

    erewrite transf_exec_bblock in H2; eauto.
    inv BBEQ. rewrite H3 in H2.
    exists (State rs' m'). split; try (constructor; auto).
    eapply transf_step_simu; eauto.

  - exploit function_ptr_translated; eauto. intros (tf & FFP & TRANSF). monadInv TRANSF.
    exploit transf_find_bblock; eauto. intros (lbb & VES & c & TAIL).
    exploit verified_schedule_builtin_idem; eauto. intros. subst lbb.

    remember (State (nextblock _ _) _) as s'. exists s'.
    split; try constructor; auto.
    eapply plus_one. subst s'.
    eapply exec_step_builtin.
      3: eapply find_bblock_tail. simpl in TAIL. 3: eauto.
      all: eauto.
    eapply eval_builtin_args_preserved with (ge1 := ge). exact symbols_preserved. eauto.
    eapply external_call_symbols_preserved; eauto. apply senv_preserved.

  - exploit function_ptr_translated; eauto. intros (tf & FFP & TRANSF). monadInv TRANSF.
    remember (State _ m') as s'. exists s'. split; try constructor; auto.
    subst s'. eapply plus_one. eapply exec_step_external; eauto.
    eapply external_call_symbols_preserved; eauto. apply senv_preserved.
Qed.

Theorem transf_program_correct:
  forward_simulation (Asmblock.semantics prog) (Asmblock.semantics tprog).
Proof.
  eapply forward_simulation_plus.
  - apply senv_preserved.
  - apply transf_initial_states.
  - apply transf_final_states.
  - apply transf_step_correct.
Qed.

End PRESERVATION.