Lemma istriangle_hr_abs : ∏ x y : hr_ConstructiveField, hr_abs (x + y)%ring <= hr_abs x + hr_abs y. Proof. intros x y. pattern x at 1 ; rewrite <- (hr_to_NR_bij x) ; pattern y at 1 ; rewrite <- (hr_to_NR_bij y). rewrite NR_to_hr_plus. eapply istrans_leNonnegativeReals. { apply NR_to_hr_abs. } apply maxNonnegativeReals_le ; apply plusNonnegativeReals_lecompat. - apply maxNonnegativeReals_le_l. - apply maxNonnegativeReals_le_l. - apply maxNonnegativeReals_le_r. - apply maxNonnegativeReals_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.