(** Infinitely many silent transitions *)

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

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

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

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

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

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