intros l l0 r0 b0 Balanced_l0 height_r.
inversion_clear Balanced_l0.
apply Is_Left_Balanced_Avl_Right_Shift.
rewrite H in height_r.
symmetry  in |- *.
injection height_r; trivial.
apply Is_Fully_Balanced_Avl_Right_Shift.
rewrite <- height_r; assumption.
apply Is_Right_Balanced_Avl_Right_Shift.
rewrite <- H.
rewrite height_r; trivial.
Qed.

Lemma equiv_del_trans_left :
forall (k k0 : Int) (d0 : B) (l0 r0 l : avl_tree) (b0 : bal),
Less k k0 ->
is_above r0 k0 ->
equiv_del k l0 l ->
equiv_del k (Avl_Node k0 d0 l0 r0 b0) (Avl_Node k0 d0 l r0 b0).