(*********************************************************)
(*		Definition is_balanced		  	 *)


Inductive is_balanced_avl (l r : avl_tree) : bal -> Prop :=
  | Is_Left_Balanced_Avl :
      height_avl l = S (height_avl r) -> is_balanced_avl l r Left_Balanced
  | Is_Fully_Balanced_Avl :
      height_avl l = height_avl r -> is_balanced_avl l r Balanced
  | Is_Right_Balanced_Avl :
      S (height_avl l) = height_avl r -> is_balanced_avl l r Right_Balanced.(*********************************************************)
(*		Definition is_balanced		  	 *)


Inductive is_balanced_avl (l r : avl_tree) : bal -> Prop :=
  | Is_Left_Balanced_Avl :
      height_avl l = S (height_avl r) -> is_balanced_avl l r Left_Balanced
  | Is_Fully_Balanced_Avl :
      height_avl l = height_avl r -> is_balanced_avl l r Balanced
  | Is_Right_Balanced_Avl :
      S (height_avl l) = height_avl r -> is_balanced_avl l r Right_Balanced.(*********************************************************)
(*		Definition is_balanced		  	 *)


Inductive is_balanced_avl (l r : avl_tree) : bal -> Prop :=
  | Is_Left_Balanced_Avl :
      height_avl l = S (height_avl r) -> is_balanced_avl l r Left_Balanced
  | Is_Fully_Balanced_Avl :
      height_avl l = height_avl r -> is_balanced_avl l r Balanced
  | Is_Right_Balanced_Avl :
      S (height_avl l) = height_avl r -> is_balanced_avl l r Right_Balanced.(*********************************************************)
(*		Definition is_balanced		  	 *)


Inductive is_balanced_avl (l r : avl_tree) : bal -> Prop :=
  | Is_Left_Balanced_Avl :
      height_avl l = S (height_avl r) -> is_balanced_avl l r Left_Balanced
  | Is_Fully_Balanced_Avl :
      height_avl l = height_avl r -> is_balanced_avl l r Balanced
  | Is_Right_Balanced_Avl :
      S (height_avl l) = height_avl r -> is_balanced_avl l r Right_Balanced.

Lemma inv_left_balanced_avl : forall l r : avl_tree, is_balanced_avl l r Left_Balanced -> height_avl l = S (height_avl r).