intros t; elim t; clear t.
intros.
apply Above_Nil.
intros k d l ih_l r ih_r b k0 k1 H0 H1.
apply Above_Node.
apply (less_trans k1 k0 k H1).
apply (inv_above_key k d l r b k0 H0).
apply (ih_l k0 k1 (inv_above_left k d l r b k0 H0) H1).
apply (ih_r k0 k1 (inv_above_right k d l r b k0 H0) H1).
Qed.
(*********************************************************)
(*		Definition height         	  	 *)
Fixpoint height (t : avl_tree) : nat :=
match t with
| Avl_Nil => 0
| Avl_Node _ _ l r _ => S (max (height l) (height r))
end.

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