proposition: Lemma kth_S_comm : forall delta gamma n, kth_S delta (kth_S gamma n) = kth_S gamma (kth_S delta n).
proof: Proof. induction delta. intros. induction gamma. compute. reflexivity. unfold kth_S. fold kth_S. unfold kth_S in IHgamma. fold kth_S in IHgamma. rewrite IHgamma. reflexivity. intros. unfold kth_S. fold kth_S. rewrite IHdelta. induction gamma. compute. reflexivity. unfold kth_S. fold kth_S. rewrite IHgamma. reflexivity. Qed.
imports: PeanoNat Compare