(** Infinitely many silent transitions *)

CoInductive forever_silent (ge: genv): state -> Prop :=
  | forever_silent_intro: forall s1 s2,
      step ge s1 E0 s2 -> forever_silent ge s2 ->
      forever_silent ge s1.(** Infinitely many silent transitions *)

CoInductive forever_silent (ge: genv): state -> Prop :=
  | forever_silent_intro: forall s1 s2,
      step ge s1 E0 s2 -> forever_silent ge s2 ->
      forever_silent ge s1.(** An alternate definition. *)

CoInductive forever_silent_N (ge: genv) : A -> state -> Prop :=
  | forever_silent_N_star: forall s1 s2 a1 a2,
      star ge s1 E0 s2 ->
      order a2 a1 ->
      forever_silent_N ge a2 s2 ->
      forever_silent_N ge a1 s1
  | forever_silent_N_plus: forall s1 s2 a1 a2,
      plus ge s1 E0 s2 ->
      forever_silent_N ge a2 s2 ->
      forever_silent_N ge a1 s1.(** An alternate definition. *)

CoInductive forever_silent_N (ge: genv) : A -> state -> Prop :=
  | forever_silent_N_star: forall s1 s2 a1 a2,
      star ge s1 E0 s2 ->
      order a2 a1 ->
      forever_silent_N ge a2 s2 ->
      forever_silent_N ge a1 s1
  | forever_silent_N_plus: forall s1 s2 a1 a2,
      plus ge s1 E0 s2 ->
      forever_silent_N ge a2 s2 ->
      forever_silent_N ge a1 s1.(** An alternate definition. *)

CoInductive forever_silent_N (ge: genv) : A -> state -> Prop :=
  | forever_silent_N_star: forall s1 s2 a1 a2,
      star ge s1 E0 s2 ->
      order a2 a1 ->
      forever_silent_N ge a2 s2 ->
      forever_silent_N ge a1 s1
  | forever_silent_N_plus: forall s1 s2 a1 a2,
      plus ge s1 E0 s2 ->
      forever_silent_N ge a2 s2 ->
      forever_silent_N ge a1 s1.

Lemma forever_silent_N_inv: forall ge a s, forever_silent_N ge a s -> exists s', exists a', step ge s E0 s' /\ forever_silent_N ge a' s'.