(********************************************************************)

Inductive is_balanced_avl_left_shift (l r : avl_tree) : bal -> Prop :=
  | Is_Left_Balanced_Avl_Left_Shift :
      height_avl l = S (S (height_avl r)) ->
      is_balanced_avl_left_shift l r Left_Balanced
  | Is_Fully_Balanced_Avl_Left_Shift :
      height_avl l = S (height_avl r) ->
      is_balanced_avl_left_shift l r Balanced
  | Is_Right_Balanced_Avl_Left_Shift :
      height_avl l = height_avl r ->
      is_balanced_avl_left_shift l r Right_Balanced.(********************************************************************)

Inductive is_balanced_avl_left_shift (l r : avl_tree) : bal -> Prop :=
  | Is_Left_Balanced_Avl_Left_Shift :
      height_avl l = S (S (height_avl r)) ->
      is_balanced_avl_left_shift l r Left_Balanced
  | Is_Fully_Balanced_Avl_Left_Shift :
      height_avl l = S (height_avl r) ->
      is_balanced_avl_left_shift l r Balanced
  | Is_Right_Balanced_Avl_Left_Shift :
      height_avl l = height_avl r ->
      is_balanced_avl_left_shift l r Right_Balanced.(********************************************************************)

Inductive is_balanced_avl_left_shift (l r : avl_tree) : bal -> Prop :=
  | Is_Left_Balanced_Avl_Left_Shift :
      height_avl l = S (S (height_avl r)) ->
      is_balanced_avl_left_shift l r Left_Balanced
  | Is_Fully_Balanced_Avl_Left_Shift :
      height_avl l = S (height_avl r) ->
      is_balanced_avl_left_shift l r Balanced
  | Is_Right_Balanced_Avl_Left_Shift :
      height_avl l = height_avl r ->
      is_balanced_avl_left_shift l r Right_Balanced.(********************************************************************)

Inductive is_balanced_avl_left_shift (l r : avl_tree) : bal -> Prop :=
  | Is_Left_Balanced_Avl_Left_Shift :
      height_avl l = S (S (height_avl r)) ->
      is_balanced_avl_left_shift l r Left_Balanced
  | Is_Fully_Balanced_Avl_Left_Shift :
      height_avl l = S (height_avl r) ->
      is_balanced_avl_left_shift l r Balanced
  | Is_Right_Balanced_Avl_Left_Shift :
      height_avl l = height_avl r ->
      is_balanced_avl_left_shift l r Right_Balanced.

(* Specifications: *)
Lemma is_left_balanced_is_left_balanced_left_shift : forall (l0 l r0 : avl_tree) (b0 : bal), is_balanced_avl l0 r0 b0 -> height_avl l = S (height_avl l0) -> is_balanced_avl_left_shift l r0 b0.
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.