intros.
inversion_clear H.
apply H2; assumption.
Qed.

Lemma inv_equiv_ins_equal1 :
forall (key : Int) (update : B -> B) (init : B) (t0 t : avl_tree) (k : Int),
equiv_ins key update init t0 t ->
Equal k key ->
(forall data : B, lookup k t0 data -> False) -> lookup k t (update init).