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.

Definition always_diverges (p : stmt) : Prop :=
  forall s p' s',
    step_star s p s' p' ->
    p' <> Snop.

Definition WTN : stmt :=
  Swhile (Eval (Vbool true)) Snop.

Lemma while_t_diverges :
    always_diverges WTN.
Proof.
  unfold always_diverges, WTN; intros.
  prep_induction H. induction H; intros; subst.
  - congruence.
  - inv H.

Abort.

Lemma while_t_sstar_cases :
  forall s s' p',
    step_star
      s WTN
      s' p' ->
    p' = Sseq Snop WTN \/
    p' = WTN.
Proof.
  unfold WTN; intros.
  prep_induction H. induction H; intros; subst.
  - right. reflexivity.
  - inv H.

Abort.

Definition WTN' : stmt :=
  Sseq Snop (Swhile (Eval (Vbool true)) Snop).

Lemma while_t_sstar_cases :
  forall s p s' p',
    step_star
      s p
      s' p' ->
    (p = WTN \/ p = WTN') ->
    p' = WTN \/ p' = WTN'.
Proof.
  unfold WTN, WTN'; intros.
  prep_induction H. induction H; intros; subst.
  - inv H0.
    + left. reflexivity.
    + right. reflexivity.
  - apply IHstep_star. inv H1.
    + inv H.
      * auto.
      * inv H7.
    + inv H.
      * auto.
      * inv H7.
Qed.

Lemma while_t_diverges :
    always_diverges WTN.
Proof.
  unfold always_diverges, WTN; intros.
  apply while_t_sstar_cases in H.
  - unfold WTN, WTN' in *.
    inv H; congruence.
  - unfold WTN, WTN' in *; auto.
Qed.