intros.
inversion H; assumption.
Qed.
(************************************************************************)
(* lookup                                                               *)
Inductive lookup (key : Int) : avl_tree -> B -> Prop :=
| Lookup_Equal :
forall (d : B) (l r : avl_tree) (b : bal),
lookup key (Avl_Node key d l r b) d
| Lookup_Left :
forall (k : Int) (d : B) (l r : avl_tree) (b : bal) (data : B),
lookup key l data -> lookup key (Avl_Node k d l r b) data
| Lookup_Right :
forall (k : Int) (d : B) (l r : avl_tree) (b : bal) (data : B),
lookup key r data -> lookup key (Avl_Node k d l r b) data.

(* Specifications: *)
Lemma inv_lookup_nil :
forall (key : Int) (data : B), lookup key Avl_Nil data -> False.