(** ** Define [includes] *)

Definition includes (x y:RegExp) : Prop := (x || y) =R= y.Notation "x <= y" := (includes x y) (at level 70).Definition prop_eq (p q:Prop) : Prop := (p <-> q).

Lemma Cat_least_upper_bound : forall x y, x <= (x || y) /\ y <= (x || y) /\ (forall z, x <= z -> y <= z -> (x || y) <= z).