(** No transitions: stuck state *)

Definition nostep (s: state) : Prop :=
  forall t s', ~(step s t s').

(* Specifications: *)
Lemma star_one: forall s1 t s2, step s1 t s2 -> star s1 t s2.
Lemma star_E0_ind: forall (P: state -> state -> Prop), (forall s, P s s) -> (forall s1 s2 s3, step s1 E0 s2 -> P s2 s3 -> P s1 s3) -> forall s1 s2, star s1 E0 s2 -> P s1 s2.
Lemma plus_one: forall s1 t s2, step s1 t s2 -> plus s1 t s2.
Lemma plus_star: forall s1 t s2, plus s1 t s2 -> star s1 t s2.
Lemma star_inv: forall s1 t s2, star s1 t s2 -> (s2 = s1 /\ t = E0) \/ plus s1 t s2.
Lemma plus_E0_ind: forall (P: state -> state -> Prop), (forall s1 s2 s3, step s1 E0 s2 -> star s2 E0 s3 -> P s1 s3) -> forall s1 s2, plus s1 E0 s2 -> P s1 s2.