intros l r b Balanced_l.
inversion_clear Balanced_l; intros Balanced_shift_l;
inversion_clear Balanced_shift_l.
apply n_Sn_false with (S (height_avl r)).
rewrite <- H0.
symmetry  in |- *; assumption.
apply n_Sn_false with (height_avl r).
rewrite <- H0.
symmetry  in |- *; assumption.
apply n_Sn_false with (height_avl l).
rewrite H.
assumption.
Qed.
(********************************************************************)
Inductive is_balanced_avl_right_shift (l r : avl_tree) : bal -> Prop :=
| Is_Left_Balanced_Avl_Right_Shift :
height_avl l = height_avl r ->
is_balanced_avl_right_shift l r Left_Balanced
| Is_Fully_Balanced_Avl_Right_Shift :
S (height_avl l) = height_avl r ->
is_balanced_avl_right_shift l r Balanced
| Is_Right_Balanced_Avl_Right_Shift :
S (S (height_avl l)) = height_avl r ->
is_balanced_avl_right_shift l r Right_Balanced.

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