Lemma hzabsvalandlth ( n m : hz ) ( p : hzleh 0 n ) ( p' : hzlth n m ) : natlth ( hzabsval n ) ( hzabsval m ). Proof. intros. destruct ( natlthorgeh ( hzabsval n ) ( hzabsval m ) ) as [ h | k ]. - assumption. - apply fromempty. apply ( isirreflhzlth m ). apply ( hzlehlthtrans _ n _ ). + rewrite <- ( hzabsvalgeh0 ). * rewrite <- ( hzabsvalgeh0 p ). apply nattohzandleh. apply k. * apply hzgthtogeh. apply ( hzgthgehtrans _ n _ ); assumption. + assumption. Defined.
Require Import UniMath.Foundations.PartA. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.NaturalNumbers. Require Import UniMath.Algebra.RigsAndRings. Require Import UniMath.Algebra.Domains_and_Fields. Require Import UniMath.NumberSystems.Integers. Require Import UniMath.Algebra.Monoids.