Proof using.
intros n ln t l H; elim H; clear H n ln t l.
intros n t l ln1 ln2 a; case ln1; simpl in |- *; auto.
intros n ln1 ln2 t1 t2 l1 l2 H H0 H1 H2 ln0 ln3 a H3 H4 H5.
case app_inv_app with (1 := H3).
intros (ln4, H7); auto.
cut (ln3 = ln4 ++ ln2);
[ intros E1
| apply app_inv_head with (l1 := ln0 ++ a :: nil); repeat rewrite app_ass;
simpl in |- *; rewrite <- H3; rewrite H7; rewrite app_ass;
auto ].
case H0 with (1 := H7); auto; clear H0 H2.
intros n1 H8; apply H5; rewrite E1; auto with datatypes.
intros (ln5, HH); left; exists (ln5 ++ ln2).
apply trans_equal with (1 := E1); rewrite HH; auto.
intros (HH1, HH2).
cut (ln0 = nil /\ ln4 = nil /\ a = S n);
[ intros (HH3, (HH4, HH5))
| generalize HH1; rewrite H7; case ln0; simpl in |- *;
[ case ln4; try (intros; discriminate); (intros HH6; injection HH6; auto)
| intros n0 l; case l; simpl in |- *; intros; discriminate ] ].
generalize E1 H1; case ln2; simpl in |- *; auto; clear E1 H1.
intros E1 H1; case height_pred_not_nil2 with (1 := H1); auto.
generalize (height_pred_length _ _ _ _ H1); case l2; simpl in |- *; auto;
intros; discriminate.
intros n0 ln5 E1 H1; case height_pred_larger_strict with (n1 := n0) (1 := H1);
simpl in |- *; auto with datatypes.
intros HH6; Contradict HH6; apply le_not_lt; rewrite <- HH5; apply H5;
rewrite E1; auto with datatypes.
intros (H8, H9); left; exists (nil (A:=nat)); injection H8.
intros HH7 HH8; rewrite HH5; rewrite <- HH8; rewrite <- HH7; rewrite E1;
rewrite HH4; auto.
intros (ln4, H7); auto.
cut (ln0 = ln1 ++ ln4);
[ intros E1
| apply app_inv_tail with (l1 := a :: ln3); rewrite <- H3; rewrite H7;
rewrite app_ass; auto ].
case H2 with (1 := H7); auto.
intros n1 H6; apply H4; rewrite E1; auto with datatypes.
intros (HH1, HH2).
cut (ln3 = nil /\ ln4 = nil /\ a = S n);
[ intros (HH3, (HH4, HH5))
| generalize HH1; rewrite H7; case ln4; simpl in |- *;
[ case ln3; try (intros; discriminate); (intros HH6; injection HH6; auto)
| intros n0 l; case l; simpl in |- *; intros; discriminate ] ].
case height_pred_larger_ex with (1 := H); auto.
intros n1; rewrite <- HH5; intros (HH6, HH7).
Contradict HH7; apply lt_not_le; apply H4; rewrite E1; auto with datatypes.
Qed.

(* The first maximum height in the list is immediately repeated once *)
Theorem height_pred_disj_larger :
forall (n a : nat) (ln1 ln2 : list nat) (t : btree A) (l : list (btree A)),
height_pred n (ln1 ++ a :: ln2) l t ->
(forall n1 : nat, In n1 ln1 -> n1 < a) ->
(forall n1 : nat, In n1 ln2 -> n1 <= a) ->
(exists ln3 : _, ln2 = a :: ln3) \/
(ln1 = nil /\ a = n /\ ln2 = nil) /\ l = t :: nil.