Lemma hr_to_NRposneg_zero : ∏ x : hr_commring, 0 < hr_to_NRpos x -> hr_to_NRneg x = 0. Proof. intros x. rewrite <- (hr_to_NR_bij x). generalize (hr_to_NR x) ; clear x ; intros x. rewrite hr_to_NRpos_NR_to_hr, hr_to_NRneg_NR_to_hr. intros Hx. apply minusNonnegativeReals_eq_zero. apply lt_leNonnegativeReals. apply_pr2 ispositive_minusNonnegativeReals. exact Hx. 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.