(* Now prove a few other utility lemmas about union and intersection. *)

(* Specifications: *)
Lemma union_idem : forall x,
x = union_tree x x.
Lemma intersect_idem : forall x,
x = intersect_tree x x.
Lemma union_absorb : forall x y,
x = union_tree x (intersect_tree x y).
Lemma intersect_absorb : forall x y,
x = intersect_tree x (union_tree x y).