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_lt_Lt_compare_morphism; auto.
Qed.

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