(** * Coq codes *)
(** ** Dependencies *)
Require Export RegExp.Utility.
Require Export RegExp.Definitions.
Require Export RegExp.Boolean.
(** ** [Cat] is morphism *)
Add Parametric Morphism : Cat with
signature re_eq ==> re_eq ==> re_eq as Cat_mor.
(** ** [Empty] *)
(** [Empty] corresponds to 0 in Kleene algebra. *)
(** ** [Cat] matches concatination of strings. *)
(** If [Cat] matches a string, the string can be divided into two strings which match the two [RegExp] respectively. *)

Lemma divide_Cat : forall s r' r'', (r' ++ r'') ~== s ->
{s':string & {s'':string | s = (s' ++ s'')%string /\ r' ~== s' /\ r'' ~== s'' }}.