(** * 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. *)
(** ** [Eps] *)
(** [Eps] corresponds to 1 in Kleene algebra. *)
(** ** [Cat] is associative. *)

Lemma matches_Cat_Cat_left : forall s s' s'' r r' r'',
r ~== s -> r' ~== s' -> r'' ~== s'' ->
((r ++ r') ++ r'') ~== ((s ++ s') ++ s'').