(** One or several transitions.  Also known as the transitive closure. *)
Inductive plus : state -> trace -> state -> Prop :=
| plus_left: forall s1 t1 s2 t2 s3 t,
step s1 t1 s2 -> star s2 t2 s3 -> t = t1 ** t2 ->
plus s1 t s3.

Lemma star_inv:
forall s1 t s2,
star s1 t s2 ->
(s2 = s1 /\ t = E0) \/ plus s1 t s2.