intros key t; elim t; clear t.
intros data k0 lookup_t above_t.
elimtype False.
apply (inv_lookup_nil key data lookup_t).
intros k d l ih_l r ih_r b data k0 lookup_t above_t.
elim (inv_lookup key k d l r b data lookup_t); clear lookup_t; intro u0.
elim u0; clear u0; intros u0 u1.
rewrite <- u0.
apply (inv_above_key k d l r b k0 above_t).
elim u0; clear u0; intro u0.
apply (ih_l data).
assumption.
apply (inv_above_left k d l r b k0 above_t).
apply (ih_r data).
assumption.
apply (inv_above_right k d l r b k0 above_t).
Qed.

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