proposition: Lemma minusNonnegativeRationals_lecompat_l : ∏ k x y : NonnegativeRationals, x <= y -> x - k <= y - k.
proof: Proof. intros k x y Hxy. generalize (isdecrel_leNonnegativeRationals k x) ; apply sumofmaps ; intros Hkx. - apply_pr2 (plusNonnegativeRationals_lecompat_r k). rewrite !minusNonnegativeRationals_plus_r. + exact Hxy. + now apply istrans_leNonnegativeRationals with (2 := Hxy). + exact Hkx. - rewrite minusNonnegativeRationals_eq_zero. + now apply isnonnegative_NonnegativeRationals. + now apply lt_leNonnegativeRationals, notge_ltNonnegativeRationals. Qed.
imports: UniMath.MoreFoundations.Tactics UniMath.MoreFoundations.PartA UniMath.MoreFoundations.Sets UniMath.MoreFoundations.Orders UniMath.RealNumbers.Sets UniMath.RealNumbers.Fields UniMath.Algebra.DivisionRig UniMath.OrderTheory.Lattice.Lattice UniMath.RealNumbers.Prelim