intros l0 l r0 b0 Balanced_l0 height_l.
inversion_clear Balanced_l0.
apply Is_Left_Balanced_Avl_Left_Shift.
rewrite <- H; assumption.
apply Is_Fully_Balanced_Avl_Left_Shift.
rewrite <- H; assumption.
apply Is_Right_Balanced_Avl_Left_Shift.
rewrite <- H; assumption.
Qed.

Lemma is_balanced_is_balanced_left_shift_false :
forall (l r : avl_tree) (b : bal),
is_balanced_avl l r b -> is_balanced_avl_left_shift l r b -> False.