intros key update init t0 t k0 equiv_t0 greater below_t0.
apply lookup_greater_above.
intros k d lookup_t.
elim (equal_dec k key).
intro equal.
apply (less_equal_less k0 key k).
assumption.
apply equal_sym; assumption.
intro notequal.
generalize
(inv_equiv_ins_notequal1 key update init t0 t k d equiv_t0 notequal lookup_t).
intro lookup_t0.
apply (lookup_above_greater k t0 d); assumption.
Qed.
(***************************************************************************)

Lemma leave_is_avl :
forall (k : Int) (d : B), is_avl (Avl_Node k d Avl_Nil Avl_Nil Balanced).