intros key k0 d0 l0 r0 b0 r b avl_t0 less equiv_r0.
inversion_clear equiv_r0.
apply equiv_del_intro.
intros k equal data lookup_t.
elim (inv_lookup k k0 d0 l0 r b data lookup_t); intro u0.
elim u0; clear u0; intros u0 u1.
rewrite u0 in less.
rewrite (equal_eq k key equal) in less.
apply less_irrefl with key; assumption.
elim u0; clear u0; intro u0.
rewrite (equal_eq k key equal) in u0.
apply lookup_below_false with key l0 data.
assumption.
apply below_trans with k0.
inversion_clear avl_t0.
apply is_below_avl_is_below; assumption.
assumption.
apply H with k data; assumption.
intros k data notequal lookup_t0.
inversion_clear lookup_t0.
apply Lookup_Equal.
apply Lookup_Left; assumption.
apply Lookup_Right.
apply H0; assumption.
intros k data notequal lookup_t.
inversion_clear lookup_t.
apply Lookup_Equal.
apply Lookup_Left; assumption.
apply Lookup_Right.
apply H1; assumption.
Qed.
(**********************************************************************)

Lemma below_equiv_below :
forall (k : Int) (t0 t : avl_tree),
is_below t0 k -> equiv t t0 -> is_below t k.