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.
Definition compare : forall x y : t, Compare lt eq x y.
intros (x1,x2) (y1,y2).
destruct (O1.compare x1 y1).
apply LT; unfold lt; auto.
destruct (O2.compare x2 y2).
apply LT; unfold lt; auto.
apply EQ; unfold eq; auto.
apply GT; unfold lt; auto.
apply GT; unfold lt; auto.
Defined.
Definition eq_dec : forall x y : t, {eq x y} + {~ eq x y}.

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