intros m1 n1 H1.
rewrite <- (Nat2N.id (S n1)); rewrite <- (Nat2N.id (S m1)).
simpl; apply nat_of_P_gt_Gt_compare_morphism; auto.
Qed.

Theorem Ngt_gt_rev: forall n  m, n > m -> (N.to_nat n > N.to_nat m)%nat.
intros; apply gt_Ngt; repeat rewrite N2Nat.id; auto.