intros l l0 r0 b0 Balanced_r0 height_l.
inversion_clear Balanced_r0.
apply Is_Left_Balanced_Avl_Right_Shift.
rewrite H in height_l.
injection height_l; trivial.
apply Is_Fully_Balanced_Avl_Right_Shift.
rewrite <- H; assumption.
apply Is_Right_Balanced_Avl_Right_Shift.
rewrite height_l.
assumption.
Qed.

Lemma is_balanced_is_balanced_right_shift :
forall (l l0 r0 : avl_tree) (b0 : bal),
is_balanced_avl l0 r0 b0 ->
height_avl l0 = S (height_avl l) -> is_balanced_avl_right_shift l r0 b0.