induction w; simpl; intros.
apply NoDup_nil.
apply NoDup_cons...
intro.
destruct (In_nats_inv _ _ _ H).
apply (le_Sn_n _ H0).
Qed.

(* Specifications: *)
Lemma nats_plus y x z: nats x (y + z) = nats x y ++ nats (y + x) z.
Proof with auto.