Module Nat_as_OT <: UsualOrderedType.
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.
Definition compare x y : Compare lt eq x y.

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