Proof with auto.
induction n...
simpl pow.
rewrite plus_0_r.
rewrite nats_plus.
rewrite IHn.
rewrite nats_Sw'.
rewrite map_app.
simpl.
rewrite concat_app.
rewrite plus_0_r.
simpl.
rewrite app_nil_r...
Qed.

Lemma nats_Sb w b: nats (S b) w = map S (nats b w).
Proof with auto.