Require Import Coq.FSets.FMapPositive.
Require Import PArith.
Require Import Rustre.Common.
Require Import Rustre.Minimp.Syntax.
Require Import Rustre.Minimp.Semantics.

Require Import Relations.
Require Import Morphisms.
Require Import Setoid.

Definition stmt_eval_eq s1 s2: Prop :=
  ∀ prog menv env menv´ env´,
    stmt_eval prog menv env s1 (menv´, env´)
    ↔
    stmt_eval prog menv env s2 (menv´, env´).

Lemma stmt_eval_eq_refl:
  reflexive stmt stmt_eval_eq.
Proof. now apply iff_refl. Qed.

Lemma stmt_eval_eq_sym:
  symmetric stmt stmt_eval_eq.
Proof.
  intros s1 s2 Heq prog menv env menv´ env´.
  split; apply Heq.
Qed.

Lemma stmt_eval_eq_trans:
  transitive stmt stmt_eval_eq.
Proof.
  intros s1 s2 s3 Heq1 Heq2 prog menv env menv´ env´.
  split; intro HH; [apply Heq2, Heq1|apply Heq1, Heq2]; exact HH.
Qed.

Add Relation stmt (stmt_eval_eq)
    reflexivity proved by stmt_eval_eq_refl
    symmetry proved by stmt_eval_eq_sym
    transitivity proved by stmt_eval_eq_trans
  as stmt_eval_equiv.

Instance stmt_eval_eq_Proper:
  Proper (eq ==> eq ==> eq ==> stmt_eval_eq ==> eq ==> iff) stmt_eval.
Proof.
  intros prog´ prog HR1 menv´ menv HR2 env´ env HR3 s1 s2 Heq r´ r HR4;
  subst; destruct r as [menv´ env´].
  now apply Heq.
Qed.

Instance stmt_eval_eq_Comp_Proper:
  Proper (stmt_eval_eq ==> stmt_eval_eq ==> stmt_eval_eq) Comp.
Proof.
  intros s s´ Hseq t t´ Hteq prog menv env menv´ env´.
  split; inversion_clear 1;
  [rewrite Hseq, Hteq in *; econstructor; eassumption
  |rewrite <-Hseq, <-Hteq in *; econstructor; eassumption].
Qed.

Lemma Comp_assoc:
  ∀ s1 s2 s3,
    stmt_eval_eq (Comp s1 (Comp s2 s3)) (Comp (Comp s1 s2) s3).
Proof.
  intros prog s1 s2 s3 menv env menv´ env´.
  split;
    intro HH;
    repeat progress
           match goal with
           | H:stmt_eval _ _ _ (Comp _ _) _ |- _ ⇒ inversion H; subst; clear H
           | |- _ ⇒ repeat econstructor; now eassumption
           end.
Qed.

Lemma stmt_eval_eq_Comp_Skip1:
  ∀ s, stmt_eval_eq (Comp Skip s) s.
Proof.
  intros s prog menv env menv´ env´.
  split.
  - inversion_clear 1;
    try match goal with
        | H:stmt_eval _ _ _ Skip _ |- _ ⇒ inversion H; subst; assumption
        end.
  - intro HH; econstructor; [now econstructor|eassumption].
Qed.

Lemma stmt_eval_eq_Comp_Skip2:
  ∀ s, stmt_eval_eq (Comp s Skip) s.
Proof.
  intros s prog menv env menv´ env´.
  split.
  - inversion_clear 1;
    try match goal with
        | H:stmt_eval _ _ _ Skip _ |- _ ⇒ inversion H; subst; assumption
        end.
  - intro HH; econstructor; [eassumption|now constructor].
Qed.

Instance stmt_eval_eq_Ifte_Proper:
  Proper (eq ==> stmt_eval_eq ==> stmt_eval_eq ==> stmt_eval_eq) Ifte.
Proof.
  intros e e´ Heeq s s´ Hseq t t´ Hteq prog menv env menv´ env´.
  rewrite <-Heeq.
  split; inversion_clear 1;
  first [apply Iifte_true;[assumption|]
        |apply Iifte_false;[assumption|]];
  first [rewrite Hseq in ×
        |rewrite Hteq in *]; assumption.
Qed.