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.
unfold N.lt; intros.
rewrite nat_of_Ncompare.
apply (proj1 (nat_compare_lt _ _)); assumption.
Qed.

Theorem Nge_out : forall n m, n >= m
-> (nat_of_N n >= nat_of_N m)%nat.