intros t key; elim t; clear t.
intros.
apply Below_Avl_Nil.
intros k d l ih_l r ih_r b below_t.
apply Below_Avl_Node.
apply (inv_below_key k d l r b key below_t).
apply ih_r.
apply (inv_below_right k d l r b key below_t).
Qed.
(*********************************************************)
(*		Definition is_above_avl		  	 *)
Inductive is_above_avl : avl_tree -> Int -> Prop :=
| Above_Avl_Nil : forall k0 : Int, is_above_avl Avl_Nil k0
| Above_Avl_Node :
forall (k : Int) (d : B) (l r : avl_tree) (b : bal) (k0 : Int),
Less k0 k -> is_above_avl l k0 -> is_above_avl (Avl_Node k d l r b) k0.

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