proposition: Lemma istriangle_hr_abs' : ∏ x y : hr_ConstructiveField, hr_abs x - hr_abs y <= hr_abs (x + y)%ring.
proof: 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.
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