intros key k d l r b data avl_t less lookup_t.
elim (inv_lookup key k d l r b data lookup_t); intro u0.
elimtype False.
apply (less_irrefl key).
elim u0; clear u0; intros u0 u1.
rewrite u0 in less.
assumption.
elim u0; clear u0; intro u0.
elimtype False.
apply (lookup_below_false key l data u0).
apply (below_trans l k key).
apply is_below_avl_is_below.
apply (inv_is_avl_left k d l r b avl_t).
apply (inv_is_avl_is_is_below_avl k d l r b avl_t).
assumption.
assumption.
Qed.

Lemma lookup_avl_equal :
forall (k1 k : Int) (t : avl_tree) (d1 d : B),
is_avl t -> lookup k1 t d1 -> lookup k t d -> Equal k1 k -> d1 = d.