intros.
inversion_clear H.
apply H3; assumption.
Qed.

Lemma inv_equiv_ins_notequal0 :
forall (key : Int) (update : B -> B) (init : B) (t0 t : avl_tree)
(k : Int) (data : B),
equiv_ins key update init t0 t ->
~ Equal k key -> lookup k t0 data -> lookup k t data.