revert k n; induction s; intros; cbn.
- decide (n = k)... subst. unfold bound in H. decide (k < n0). omega. congruence.
- cbn in H. eapply Bool.andb_true_iff in H as []. erewrite IHs1, IHs2...
- erewrite IHs... omega.
Qed.

(* Specifications: *)
Lemma closed_subst s k u : closed s -> subst s k u = s.
Lemma closed_app s t : closed (s t) -> closed s /\ closed t.
Lemma app_closed s t : closed s -> closed t -> closed (s t).
Definition I : term := .\"x"; "x".