(** 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_ind2:
forall (P: state -> trace -> state -> Prop),
(forall s1 t s2, step s1 t s2 -> P s1 t s2) ->
(forall s1 t1 s2 t2 s3 t,
step s1 t1 s2 -> plus s2 t2 s3 -> P s2 t2 s3 -> t = t1 ** t2 ->
P s1 t s3) ->
forall s1 t s2, plus s1 t s2 -> P s1 t s2.