(** And we now check the induction principle on omega: *)

Theorem omega_ind
:
forall P : PRED,
Compat P ->
P unit ZERO id ->
(forall (X : Typ1) (A : Rel X) (a : X),
ELT X A a (opt nat) OMEGA (none nat) ->
P X A a -> P (opt X) (SUCC X A a) (none X)) ->
forall (X : Typ1) (A : Rel X) (a : X),
ELT X A a (opt nat) OMEGA (none nat) -> P X A a.