(** Counted sequences of transitions *)
Inductive starN (ge: genv): nat -> state -> trace -> state -> Prop :=
| starN_refl: forall s,
starN ge O s E0 s
| starN_step: forall n s t t1 s' t2 s'',
step ge s t1 s' -> starN ge n s' t2 s'' -> t = t1 ** t2 ->
starN ge (S n) s t s''.
Remark starN_star:
forall ge n s t s', starN ge n s t s' -> star ge s t s'.
Remark star_starN:
forall ge s t s', star ge s t s' -> exists n, starN ge n s t s'.
(** Infinitely many transitions *)
CoInductive forever (ge: genv): state -> traceinf -> Prop :=
| forever_intro: forall s1 t s2 T,
step ge s1 t s2 -> forever ge s2 T ->
forever ge s1 (t *** T).

(* Specifications: *)
Lemma star_forever:
forall ge s1 t s2, star ge s1 t s2 ->
forall T, forever ge s2 T ->
forever ge s1 (t *** T).