intros p; elim p using Pind; simpl.
intros H1; inversion H1.
intros n1 _; rewrite nat_of_P_succ_morphism.
intros H1; inversion H1.
intros p1 p2 H1 H2; absurd (nat_of_P p2 > nat_of_P p1)%nat; auto with arith.
apply nat_of_P_gt_Gt_compare_morphism; auto.
Qed.

(* Specifications: *)
Theorem le_Nle: forall n m, N.of_nat n <= N.of_nat m -> (n <= m)%nat.
intros n m; case n; case m; unfold N.le; simpl; auto with arith.