intros H1; inversion H1.
intros p H1; inversion H1.
intros; apply nat_of_P_lt_Lt_compare_complement_morphism; auto.
Qed.

Theorem lt_Nlt: forall n m, N.of_nat n < N.of_nat m -> (n < m)%nat.
intros n m; case n; case m; unfold N.lt; simpl; try (intros; discriminate); auto with arith.