(** 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 plus_three:
forall s1 t1 s2 t2 s3 t3 s4 t,
step s1 t1 s2 -> step s2 t2 s3 -> step s3 t3 s4 -> t = t1 ** t2 ** t3 ->
plus s1 t s4.