(* This program is free software; you can redistribute it and/or      *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1   *)
(* of the License, or (at your option) any later version.             *)
(*                                                                    *)
(* This program is distributed in the hope that it will be useful,    *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of     *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
(*                                                                    *)
(* You should have received a copy of the GNU Lesser General Public   *)
(* License along with this program; if not, write to the Free         *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA                                                     *)
(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
(* ---                        matrix.v                                  --- *)
(*                                                                          *)
(* Author: Pierre Casteran.                                                 *)
(*    LABRI, URA CNRS 1304,                                                 *)
(*    Departement d'Informatique, Universite Bordeaux I,                    *)
(*    33405 Talence CEDEX,                                                  *)
(*    e-mail:  casteran@labri.u-bordeaux.fr                                 *)
(*          2-2 matrices with natural coefficients                          *)
(* Ported to ZArith by P.Letouzey (nov 2001)                                *)
Record Mat2 : Set := mat2 {M11 : Z; M12 : Z; M21 : Z; M22 : Z}.
Definition Id2 := mat2 1 0 0 1.
Definition Mat_mult (M M' : Mat2) :=
mat2 (M11 M * M11 M' + M12 M * M21 M') (M11 M * M12 M' + M12 M * M22 M')
(M21 M * M11 M' + M22 M * M21 M') (M21 M * M12 M' + M22 M * M22 M').
Axiom
Mat_assoc :
forall M M' M'' : Mat2,
Mat_mult M (Mat_mult M' M'') = Mat_mult (Mat_mult M M') M''.

Lemma matrix : monoid Mat2.
refine (mkmonoid Mat2 Id2 Mat_mult _ _ _).