Lemma hr_to_NR_opp : ∏ x : hr_commring, hr_to_NR (- x)%ring = (hr_to_NRneg x ,, hr_to_NRpos x). Proof. intros x. rewrite <- (hr_to_NR_bij x), NR_to_hr_opp. unfold hr_to_NRneg, hr_to_NRpos. generalize (hr_to_NR x) ; clear x ; intros x. unfold hr_to_NR, NR_to_hr. rewrite !setquotunivcomm. reflexivity. Qed.
Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.OrderTheory.Lattice.Lattice. Require Import UniMath.RealNumbers.Prelim. Require Import UniMath.RealNumbers.Sets. Require Import UniMath.RealNumbers.NonnegativeRationals.