proposition: Lemma union_app: forall h k, union h k ≡ h ++ k.
proof: Proof. induction h as [|x h IHh]; simpl union. reflexivity. induction k as [|y k IHk]. lattice. case cmp_spec. intros ->. fold_cons. rewrite IHh. lattice. intros _. fold_cons. rewrite IHh. lattice. intros _. fold_cons. rewrite IHk. lattice. Qed.
imports: lattice comparisons List