Section omega_compat.
Local Ltac omega ::= lia.
intuition.
Qed.
End omega_compat.
unfold N.lt; intros ?? H.
rewrite nat_of_Ncompare in H.
apply nat_compare_Lt_lt; assumption.
Qed.

Theorem Nlt_in : forall n m, (nat_of_N n < nat_of_N m)%nat
-> n < m.