Lemma istriangle_hr_abs' : ∏ x y : hr_ConstructiveField, hr_abs x - hr_abs y <= hr_abs (x + y)%ring. Proof. intros x y. apply_pr2 (plusNonnegativeReals_lecompat_l (hr_abs y)). rewrite <- maxNonnegativeReals_minus_plus. apply maxNonnegativeReals_le. - assert (Hx : x = ((x + y) + (- y))%ring). { now rewrite ringassoc1, ringrinvax1, ringrunax1. } pattern x at 1 ; rewrite Hx. rewrite <- (hr_abs_opp y). apply istriangle_hr_abs. - apply plusNonnegativeReals_le_r. 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.