apply equiv_del_intro.
intros k equal d lookup_nil.
apply (inv_lookup_nil k d lookup_nil).
intros k d notequal lookup_nil.
apply lookup_nil.
intros k d notequal lookup_nil.
apply lookup_nil.
Qed.

(* Specifications: *)
Lemma equiv_del_right_semi_leave :
forall (k : Int) (d : B) (r : avl_tree) (b : bal),
is_avl (Avl_Node k d Avl_Nil r b) ->
equiv_del k (Avl_Node k d Avl_Nil r b) r.