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

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