Proof using.
intros t1 l1 l2 H; generalize l2; elim H; clear t1 l1 l2 H.
intros t l l2; case l2.
simpl in |- *; intros; discriminate.
intros b l0; case l0; simpl in |- *; auto.
intros H; exists b; auto.
intros; discriminate.
intros t1 t2 l1 l2 l3 H H0 H1 H2 l0 H3.
case (H0 (first_n l0 (length l1))); auto.
rewrite first_n_length; auto;
(rewrite <- H3; rewrite length_app; auto with arith).
intros t4 HH1.
case (H2 (skip_n l0 (length l1))); auto.
rewrite skip_n_length; auto;
(rewrite <- H3; rewrite length_app; rewrite minus_plus; auto with arith).
intros t5 HH2.
exists (node t4 t5); auto.
rewrite <- (first_n_skip_n_app _ (length l1) l0); auto.
Qed.

(* A height predicate can be completed in a substitution *)
Theorem height_pred_subst_pred :
forall (n : nat) (ln : list nat) (t1 : btree A) (l1 l2 : list (btree A)),
height_pred n ln l1 t1 ->
length l1 = length l2 ->
exists t2 : btree A, height_pred n ln l2 t2 /\ subst_pred l1 l2 t1 t2.