intros key k d l r b data lookup_t less below_l.
elim (inv_lookup key k d l r b data lookup_t); clear lookup_t; intro u0.
elimtype False.
elim u0; clear u0; intros u0 u1.
apply less_irrefl with key.
rewrite u0 in less.
assumption.
elim u0; clear u0; intro u0.
elimtype False.
apply less_irrefl with key.
apply lookup_below_less with l data; assumption.
assumption.
Qed.

Lemma lookup_below_false :
forall (key : Int) (t : avl_tree) (data : B),
lookup key t data -> is_below t key -> False.