Proof using.
intros n n1 ln t l H; generalize n1; elim H; clear H n ln t l n1; auto.
intros n ln1 ln2 t1 t2 l1 l2 H H0 H1 H2 n1 H3; left;
apply lt_le_trans with (S n); auto.
case in_app_or with (1 := H3).
intros H4; apply height_pred_larger with (1 := H); auto.
intros H4; apply height_pred_larger with (1 := H1); auto.
Qed.

(* Specifications: *)
(* There always a larger element that the initial height in the heigh list *)
Theorem height_pred_larger_ex :
forall (n : nat) (ln : list nat) (t : btree A) (l : list (btree A)),
height_pred n ln l t -> exists n1 : _, In n1 ln /\ n <= n1.