intros t key; elim t; clear t.
intros.
apply Above_Avl_Nil.
intros k d l ih_l r ih_r b above_t.
apply Above_Avl_Node.
apply (inv_above_key k d l r b key above_t).
apply ih_l.
apply (inv_above_left k d l r b key above_t).
Qed.
(*****************************************************************)
(*           height_avl                                          *)
Fixpoint height_avl (t : avl_tree) : nat :=
match t with
| Avl_Nil => 0
| Avl_Node _ _ l _ Left_Balanced => S (height_avl l)
| Avl_Node _ _ l _ Balanced => S (height_avl l)
| Avl_Node _ _ _ r Right_Balanced => S (height_avl r)
end.

Lemma height_avl_O_nil : forall t : avl_tree, height_avl t = 0 -> t = Avl_Nil.
intros t; elim t; clear t.