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(* 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                          *)
(* original version using nat instead of Z                                  *)
Record Mat2 : Set := mat2 {M11 : nat; M12 : nat; M21 : nat; M22 : nat}.
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 _ _ _).