Module NUsualOrderedType <: UsualOrderedType.
Definition t:=N.
Definition eq:=@eq N.
Definition eq_refl := @refl_equal t.
Definition eq_sym := @sym_eq t.
Definition eq_trans := @trans_eq t.
Definition lt p q:= Nless p q = true.
Definition lt_trans := Nless_trans.

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