intros k t0 t below_t0 equiv_t.
inversion_clear equiv_t.
apply lookup_less_below.
intros k0 d0 lookup_k0.
apply lookup_below_less with t0 d0.
apply H; assumption.
assumption.
Qed.

Lemma Balanced_shrunk_left_balanced_shift :
forall (l0 r r0 : avl_tree) (b0 : bal),
is_balanced_avl l0 r0 b0 ->
height_avl r0 = S (height_avl r) -> is_balanced_avl_left_shift l0 r b0.