intros t; elim t; clear t.
intros.
apply Below_Nil.
intros k d l ih_l r ih_r b k0 k1 H0 H1.
apply Below_Node.
apply (less_trans k k0 k1).
apply (inv_below_key k d l r b k0 H0).
assumption.
apply (ih_l k0 k1 (inv_below_left k d l r b k0 H0) H1).
apply (ih_r k0 k1 (inv_below_right k d l r b k0 H0) H1).
Qed.
(*********************************************************)
(*		Definition is_above		  	 *)
Inductive is_above : avl_tree -> Int -> Prop :=
| Above_Nil : forall k0 : Int, is_above Avl_Nil k0
| Above_Node :
forall (k : Int) (d : B) (l r : avl_tree) (b : bal) (k0 : Int),
Less k0 k ->
is_above l k0 -> is_above r k0 -> is_above (Avl_Node k d l r b) k0.

Lemma inv_above_key :
forall (k : Int) (d : B) (l r : avl_tree) (b : bal) (k0 : Int),
is_above (Avl_Node k d l r b) k0 -> Less k0 k.