Proof using.
intros n t l H; generalize n; elim H; clear n t l H.
intros t l n; exists (n :: nil); auto.
intros t1 t2 l1 l2 l3 H H0 H1 H2 n.
case (H0 (S n)); intros ln1 HH1.
case (H2 (S n)); intros ln2 HH2.
exists (ln1 ++ ln2); auto.
Qed.

(* Specifications: *)
(* Elements in the height list are always larger than the initial height *)
Theorem height_pred_larger :
forall (n n1 : nat) (ln : list nat) (t : btree A) (l : list (btree A)),
height_pred n ln l t -> In n1 ln -> n <= n1.