Require Import List.
Require Import String.
Require Import ZArith.

Open Scope list_scope.
Open Scope string_scope.
Open Scope Z_scope.

Require Import StructTactics.
Require Import ImpSyntax.
Require Import ImpCommon.
Require Import ImpEval.
Require Import ImpStep.
Require Import ImpSemanticsFacts.
Require Import ImpInterp.

Lemma interp_op1_eval_op1 :
  forall op v v',
    interp_op1 op v = Some v' ->
    eval_unop op v v'.
Proof.
  unfold interp_op1; intros.
  repeat break_match; subst;
    discriminate || solve_by_inversion' ee.
Qed.

Lemma eval_op1_interp_op1 :
  forall op v v',
    eval_unop op v v' ->
    interp_op1 op v = Some v'.
Proof.
  inversion 1; auto.
Qed.

Lemma interp_op2_eval_op2 :
  forall op v1 v2 v',
    interp_op2 op v1 v2 = Some v' ->
    eval_binop op v1 v2 v'.
Proof.
  unfold interp_op2; intros.
  repeat break_match; subst;
    try discriminate;
    find_inversion; ee.
Qed.

Lemma eval_op2_interp_op2 :
  forall op v1 v2 v',
    eval_binop op v1 v2 v' ->
    interp_op2 op v1 v2 = Some v'.
Proof.
  inversion 1; auto.
  - simpl. break_match; [congruence | auto].
  - simpl. break_match; [congruence | auto].
Qed.

Lemma interp_e_eval_e :
  forall s h e v,
    interp_e s h e = Some v ->
    eval_e s h e v.
Proof.
  induction e; simpl; intros.
  - inv H; ee.
  - ee.
  - repeat break_match; try discriminate.
    ee. apply interp_op1_eval_op1; auto.
  - repeat break_match; try discriminate.
    ee. apply interp_op2_eval_op2; auto.
  - repeat break_match; try discriminate.
    + find_inversion. eapply eval_len_s; eauto.
    + find_inversion. eapply eval_len_a; eauto.
  - repeat break_match; try discriminate.
    + find_inversion. eapply eval_idx_s; eauto.
    + eapply eval_idx_a; eauto.
Qed.

Lemma eval_e_interp_e :
  forall s h e v,
    eval_e s h e v ->
    interp_e s h e = Some v.
Proof.
  induction 1; simpl; auto.
  - repeat break_match; try discriminate.
    find_inversion. apply eval_op1_interp_op1; auto.
  - repeat break_match; try discriminate.
    repeat find_inversion. apply eval_op2_interp_op2; auto.
  - break_match; try discriminate. find_inversion.
    break_match; try discriminate. find_inversion.
    reflexivity.
  - break_match; try discriminate.
    find_inversion. reflexivity.
  - break_match; try discriminate.
    find_inversion. repeat find_rewrite.
    do 2 (break_match; try omega). auto.
  - break_match; try discriminate. find_inversion.
    break_match; try discriminate. find_inversion.
    break_match; try omega.
    break_match; try discriminate.
    find_inversion; auto.
Qed.

Lemma interps_e_evals_e :
  forall s h es vs,
    interps_e s h es = Some vs ->
    evals_e s h es vs.
Proof.
  induction es; simpl; intros.
  - find_inversion. ee.
  - repeat break_match; try discriminate.
    find_inversion. ee.
    apply interp_e_eval_e; auto.
Qed.

Lemma evals_e_interps_e :
  forall s h es vs,
    evals_e s h es vs ->
    interps_e s h es = Some vs.
Proof.
  induction 1; simpl; intros; auto.
  find_apply_lem_hyp eval_e_interp_e.
  repeat find_rewrite. auto.
Qed.

Lemma interp_s_step :
  forall s h p s' h' p',
    interp_s s h p = Some (s', h', p') ->
    step s h p s' h' p'.
Proof.
  induction p; simpl; intros.
  - discriminate.
  - repeat break_match; try discriminate.
    find_inversion. ee; apply interp_e_eval_e; auto.
  - repeat break_match; try discriminate.
    find_inversion. ee; apply interp_e_eval_e; auto.
  - repeat break_match; try discriminate.
    find_inversion. ee; apply interp_e_eval_e; auto.
  - repeat break_match; try discriminate.
    + invc H; ee. apply interp_e_eval_e; auto.
    + invc H; ee. apply interp_e_eval_e; auto.
  - repeat break_match; try discriminate.
    + invc H; ee. apply interp_e_eval_e; auto.
    + invc H; ee. apply interp_e_eval_e; auto.
  - break_if.
    + find_inversion; ee.
    + repeat break_match; try discriminate.
      find_inversion; ee.
Qed.

Lemma interp_s_nostep :
  forall s h p s' h' p',
    interp_s s h p = None ->
    ~ step s h p s' h' p'.
Proof.
  unfold not; intros. prep_induction H0.
  induction H0; simpl; intros; subst.
  - find_apply_lem_hyp eval_e_interp_e.
    find_rewrite; discriminate.
  - repeat (find_apply_lem_hyp eval_e_interp_e).
    repeat find_rewrite.
    break_if. discriminate. omega.
  - repeat (find_apply_lem_hyp eval_e_interp_e).
    repeat find_rewrite.
    repeat break_if; try discriminate; try omega.
  - find_apply_lem_hyp eval_e_interp_e.
    find_rewrite; discriminate.
  - find_apply_lem_hyp eval_e_interp_e.
    find_rewrite; discriminate.
  - find_apply_lem_hyp eval_e_interp_e.
    find_rewrite; discriminate.
  - find_apply_lem_hyp eval_e_interp_e.
    find_rewrite; discriminate.
  - break_if; subst. discriminate. congruence.
  - break_if; subst. discriminate.
    repeat break_match; subst. discriminate. auto.
Qed.

Inductive result_ok :
  store -> heap -> stmt -> expr -> result -> Prop :=
| result_ok_timeout :
    forall s1 h1 p1 s2 h2 p2 ret,
      step_star
        s1 h1 p1
        s2 h2 p2 ->
      result_ok
        s1 h1 p1 ret
        (Timeout s2 h2 p2 ret)
| result_ok_done :
    forall s1 h1 p1 s2 h2 ret v,
      step_star
        s1 h1 p1
        s2 h2 Snop ->
      eval_e s2 h2 ret v ->
      result_ok
        s1 h1 p1 ret
        (Done h2 v)
| result_ok_stuck_prog :
    forall s1 h1 p1 s2 h2 p2 ret,
      step_star
        s1 h1 p1
        s2 h2 p2 ->
      p2 <> Snop ->
      (forall s3 h3 p3,
        ~ step
            s2 h2 p2
            s3 h3 p3) ->
      result_ok
        s1 h1 p1 ret
        (Stuck s2 h2 p2 ret)
| result_ok_stuck_ret :
    forall s1 h1 p1 s2 h2 ret,
      step_star
        s1 h1 p1
        s2 h2 Snop ->
      (forall v, ~ eval_e s2 h2 ret v) ->
      result_ok
        s1 h1 p1 ret
        (Stuck s2 h2 Snop ret).

Lemma interp_s_interps_p :
  forall n s1 h1 p1 s2 h2 p2 ret res,
    interp_s s1 h1 p1 = Some (s2, h2, p2) ->
    interps_p n s2 h2 p2 ret = res ->
    interps_p (S n) s1 h1 p1 ret = res.
Proof.
  simpl; intros. break_match; subst.
  - discriminate.
  - find_rewrite; auto.
Qed.

Lemma interps_p_inv :
  forall n s h p ret res,
    interps_p n s h p ret = res ->
    result_ok s h p ret res.
Proof.
  induction n; simpl; intros; subst.
  - repeat ee.
  - repeat break_match; subst.
    + eapply result_ok_done; eauto.
      repeat ee. eapply interp_e_eval_e; eauto.
    + eapply result_ok_stuck_ret; eauto.
      repeat ee. unfold not; intros.
      find_apply_lem_hyp eval_e_interp_e.
      congruence.
    + remember (interps_p n s1 h0 s0 ret).
      symmetry in Heqr. find_copy_apply_hyp_hyp.
      on (result_ok _ _ _ _ _), inv.
      * repeat ee. eapply interp_s_step; eauto.
      * repeat ee. eapply interp_s_step; eauto.
      * repeat ee. eapply interp_s_step; eauto.
      * eapply result_ok_stuck_ret; eauto.
        repeat ee. eapply interp_s_step; eauto.
    + repeat ee. intros.
      apply interp_s_nostep; auto.
Qed.