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.