Proof with reflexivity.
induction n; intros.
rewrite (eq_nil v)...
rewrite (eq_cons v).
simpl.
rewrite IHn...
Qed.

Lemma nb_nats_plus x n y: nb_nats n (x + y) = app (nb_nats n x) (nb_nats (n + x) y).
Proof with auto.