induction w...
intros.
rewrite nats_Sw.
rewrite IHw.
simpl.
rewrite plus_n_Sm...
Qed.

Lemma split_pow2_range n:
nats 1 (pow 2 n) = 1 :: concat (map (fun x => nats (pow 2 x + 1) (pow 2 x)) (nats 0 n)).