intros t0 t k key above_t0 equiv_del_t0.
inversion_clear equiv_del_t0.
apply lookup_greater_above.
intros k0 d0 lookup_k0.
apply lookup_above_greater with t0 d0.
elim (equal_dec k0 key).
intros equal_k0.
elimtype False.
apply H with k0 d0; assumption.
intros not_equal_k0.
apply H1; assumption.
assumption.
Qed.
(**********************************************************************)
Inductive hasnot_shrunk_left (t : avl_tree) (k : Int)
(d : B) (l r : avl_tree) : bal -> Prop :=
| Hasnot_Shrunk_Left_Bal :
forall b : bal,
is_balanced_avl l r b ->
height_avl t = height_avl (Avl_Node k d l r b) ->
hasnot_shrunk_left t k d l r b
| Hasnot_Shrunk_Left_Shift_Left :
is_balanced_avl_left_shift l r Left_Balanced ->
height_avl t = S (height_avl l) ->
hasnot_shrunk_left t k d l r Left_Balanced
| Hasnot_Shrunk_Left_Shift_Balanced :
is_balanced_avl_left_shift l r Balanced ->
height_avl t = S (height_avl l) ->
hasnot_shrunk_left t k d l r Balanced.
Inductive has_shrunk_left (t : avl_tree) (k : Int)
(d : B) (l r : avl_tree) : bal -> Prop :=
| Has_Shrunk_Left_Shift_Left :
is_balanced_avl_left_shift l r Left_Balanced ->
height_avl t = height_avl l -> has_shrunk_left t k d l r Left_Balanced
| Has_Shrunk_Left_Shift_Right :
is_balanced_avl_left_shift l r Right_Balanced ->
height_avl t = S (height_avl l) ->
has_shrunk_left t k d l r Right_Balanced.
Inductive bal_shrunk_left_spec (k : Int) (d : B) (l r : avl_tree)
(b : bal) : Set :=
Balance_Shrunk_Left_Spec_Intro :
forall t : avl_tree,
is_avl t ->
lin_avl t = lin_avl (Avl_Node k d l r b) ->
equiv t (Avl_Node k d l r b) ->
{hasnot_shrunk_left t k d l r b} + {has_shrunk_left t k d l r b} ->
bal_shrunk_left_spec k d l r b.

Lemma balance_shrunk_left :
forall (k : Int) (d : B) (l r : avl_tree) (b : bal),
is_avl l ->
is_below_avl l k ->
is_avl r ->
is_above_avl r k ->
{is_balanced_avl l r b} + {is_balanced_avl_left_shift l r b} ->
bal_shrunk_left_spec k d l r b.