(** a [UsualOrderedType] is in particular an [OrderedType]. *)

Module UOT_to_OT (U:UsualOrderedType) <: OrderedType := U.Definition t := nat.Definition eq := @eq nat.Definition eq_refl := @eq_refl t.Definition eq_sym := @eq_sym t.Definition eq_trans := @eq_trans t.Definition lt := lt.

Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y.