intros t; elim t; clear t.
intros.
apply Below_Nil.
intros k d l ih_l r ih_r b k0 H.
apply Below_Node.
apply (H k d).
apply Lookup_Equal.
apply ih_l.
intros k1 d0 lookup_l.
apply (H k1 d0).
apply Lookup_Left; assumption.
apply ih_r.
intros k1 d0 lookup_r.
apply (H k1 d0).
apply Lookup_Right; assumption.
Qed.

Lemma lookup_greater_above :
forall (t : avl_tree) (k0 : Int),
(forall (k : Int) (d : B), lookup k t d -> Less k0 k) -> is_above t k0.