(* 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                                                     *)
(****************************************************************************)
(*                 The Calculus of Inductive Constructions                  *)
(*                                                                          *)
(*                                Projet Coq                                *)
(*                                                                          *)
(*                     INRIA                        ENS-CNRS                *)
(*              Rocquencourt                        Lyon                    *)
(*                                                                          *)
(*                                 Coq V6.1                                 *)
(*                               Oct 1st 1996                               *)
(*                                                                          *)
(****************************************************************************)
(*                              Inverses_Group.v                            *)
(****************************************************************************)
Require Export Group.
Set Implicit Arguments.
Unset Strict Implicit.
(* The Monoid of inverse pairs on a Monoid *)
Section inv_def.
Variable A : Monoid.
(* The set Inverses_set of inverse pairs on A *)
Structure Inverses : Type :=
{Inv_elt_l : A;
Inv_elt_r : A;
Prf_invs1 : Inverses_rel Inv_elt_l Inv_elt_r;
Prf_invs2 : Inverses_rel Inv_elt_r Inv_elt_l}.
Definition Inverses_eq (x y : Inverses) := Inv_elt_l x =_S Inv_elt_l y.

(* Specifications: *)
Lemma Inverses_equiv : Equivalence Inverses_eq.
Canonical Structure Inverses_setoid : Setoid := Inverses_equiv.