(* 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                                                     *)
Set Automatic Coercions Import.
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Integral_domain_facts.
Require Export Cfield_cat.
Require Export Abelian_group_facts.
Require Export Ring_util.
(** Title "The field of fractions of an integral domain" "with decidable equality." *)
Section Def.
Variable R : INTEGRAL_DOMAIN.
Variable diff10 : ~ Equal (ring_unit R) (monoid_unit R).
Set Strict Implicit.
Unset Implicit Arguments.
Record fraction : Type :=
{num : R; den : R; den_prf : ~ Equal den (monoid_unit R)}.
Set Implicit Arguments.
Unset Strict Implicit.
Hint Resolve den_prf: algebra.
Definition eqfraction (x y : fraction) :=
Equal (ring_mult (num x) (den y)) (ring_mult (num y) (den x)).

Lemma eqfraction_refl : reflexive eqfraction.
red in |- *.