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

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

Require Import ImpSyntax.
Require Import ImpCommon.
Require Import ImpEval.

Inductive step :
  store -> heap -> stmt ->
  store -> heap -> stmt -> Prop :=
| step_set :
    forall s h x e v,
      eval_e s h e v ->
      step
        s h (Sset x e)
        (update s x v) h Snop

| step_ifelse_t :
    forall s h e p1 p2,
      eval_e s h e (Vbool true) ->
      step
        s h (Sifelse e p1 p2)
        s h p1
| step_ifelse_f :
    forall s h e p1 p2,
      eval_e s h e (Vbool false) ->
      step
        s h (Sifelse e p1 p2)
        s h p2
| step_while_t :
    forall s h e p,
      eval_e s h e (Vbool true) ->
      step
        s h (Swhile e p)
        s h (Sseq p (Swhile e p))
| step_while_f :
    forall s h e p,
      eval_e s h e (Vbool false) ->
      step
        s h (Swhile e p)
        s h Snop
| step_seq_nop :
    forall s h p2,
      step
        s h (Sseq Snop p2)
        s h p2
| step_seq :
    forall s h p1 p2 s' h' p1',
      step
        s h p1
        s' h' p1' ->
      step
        s h (Sseq p1 p2)
        s' h' (Sseq p1' p2).

Inductive step_star :
  store -> heap -> stmt ->
  store -> heap -> stmt -> Prop :=
| step_star_refl :
    forall s h p,
      step_star
        s h p
        s h p
| step_star_l :
    forall s1 h1 p1 s2 h2 p2 s3 h3 p3,
      step
        s1 h1 p1
        s2 h2 p2 ->
      step_star
        s2 h2 p2
        s3 h3 p3 ->
      step_star
        s1 h1 p1
        s3 h3 p3.