proposition: Lemma hr_to_NR_opp : ∏ x : hr_commring, hr_to_NR (- x)%ring = (hr_to_NRneg x ,, hr_to_NRpos x).
proof: 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.
imports: UniMath.Algebra.Groups UniMath.Foundations.Preamble UniMath.MoreFoundations.Orders UniMath.MoreFoundations.Tactics UniMath.OrderTheory.Lattice.Lattice UniMath.RealNumbers.Prelim UniMath.RealNumbers.Sets UniMath.RealNumbers.NonnegativeRationals UniMath.RealNumbers.NonnegativeReals