(** The pointed graph ((opt nat), OMEGA, (some nat O)) is bisimilar to the pointed graph (unit ZERO id) that represents the empty set. This is done by extensionality. *)

(* Specifications: *)
Lemma OMEGA_ZERO : EQV (opt nat) OMEGA (some nat O) unit ZERO id.
(** Inductive case: for any n:nat such that (wf_nat n), the pointed graph ((opt nat), OMEGA, (some nat (S n))) is bisimilar to the successor of the pointed graph ((opt nat), OMEGA, (some nat n)). Again, this is done by extensionality. *)
Lemma OMEGA_SUCC :
forall n : nat,
wf_nat n ->
EQV (opt nat) OMEGA (some nat (S n)) (opt (opt nat))
(SUCC (opt nat) OMEGA (some nat n)) (none (opt nat)).