intros l l0 r0 b0 Balanced_l0 height_l.
inversion_clear Balanced_l0.
apply Is_Left_Balanced_Avl.
rewrite height_l; assumption.
apply Is_Fully_Balanced_Avl.
rewrite height_l; assumption.
apply Is_Right_Balanced_Avl.
rewrite height_l; assumption.
Qed.

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