Section avl_trees.
Variable B : Set.
(*********************************************************)
(*		Definition bal and avl_tree	  	 *)
Inductive bal : Set :=
| Left_Balanced : bal
| Balanced : bal
| Right_Balanced : bal.
Inductive avl_tree : Set :=
| Avl_Nil : avl_tree
| Avl_Node : Int -> B -> avl_tree -> avl_tree -> bal -> avl_tree.
(*********************************************************)
(*		Definition is_below		  	 *)
Inductive is_below : avl_tree -> Int -> Prop :=
| Below_Nil : forall k0 : Int, is_below Avl_Nil k0
| Below_Node :
forall (k : Int) (d : B) (l r : avl_tree) (b : bal) (k0 : Int),
Less k k0 ->
is_below l k0 -> is_below r k0 -> is_below (Avl_Node k d l r b) k0.
intros.
inversion H; assumption.
Qed.
intros.
inversion H; assumption.
Qed.
intros.
inversion H; assumption.
Qed.

Lemma below_trans :
forall (t : avl_tree) (k0 k1 : Int),
is_below t k0 -> Less k0 k1 -> is_below t k1.