(** Natural numbers form a preorder with le (less than or equal relation). *)
(** The pre-order category ω. *)
Definition OmegaCat : Category := PreOrder_Cat OmegaPreOrder.
Notation "'ω'" := (OmegaCat) : omegacat_scope.

Lemma le_Tle_trans (n m k : nat) (H : le n m) (H' : le m k) :
le_Tle (le_trans _ _ _ H H') = Tle_trans _ _ _ (le_Tle H) (le_Tle H').