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(*                 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.

Lemma Inverses_equiv : Equivalence Inverses_eq.
Canonical Structure Inverses_setoid : Setoid := Inverses_equiv.