(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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(** Normalisation functions for rational numbers. *)
Require Export QArith_base.
Notation Z2P := Z.to_pos (only parsing).
Notation Z2P_correct := Z2Pos.id (only parsing).
(** Simplification of fractions using [Z.gcd]. This version can compute within Coq. *)
Definition Qred (q:Q) :=
let (q1,q2) := q in
let (r1,r2) := snd (Z.ggcd q1 (Zpos q2))
in r1#(Z.to_pos r2).
Add Morphism Qred with signature (Qeq ==> Qeq) as Qred_comp.
Definition Qplus' (p q : Q) := Qred (Qplus p q).
Definition Qmult' (p q : Q) := Qred (Qmult p q).
Definition Qminus' x y := Qred (Qminus x y).
Add Morphism Qplus' with signature (Qeq ==> Qeq ==> Qeq) as Qplus'_comp.
Add Morphism Qmult' with signature (Qeq ==> Qeq ==> Qeq) as Qmult'_comp.
Add Morphism Qminus' with signature (Qeq ==> Qeq ==> Qeq) as Qminus'_comp.

Lemma Qred_opp: forall q, Qred (-q) = - (Qred q).