induction w; intros.
elimtype False.
rewrite plus_0_r in H0.
apply (lt_not_le x b H0)...
simpl.
destruct (le_lt_eq_dec _ _ H); [right | left]...
apply IHw...
omega.
Qed.

(* Specifications: *)
Lemma In_nats_inv (w x b: nat): In x (nats b w) -> b <= x < b + w.
Proof with auto.