Lemma pathssminus ( n m : nat ) ( p : natlth m ( S n ) ) : S ( sub n m ) = sub ( S n ) m. Proof. revert m p. induction n. - intros m p. destruct m. + auto. + apply fromempty. apply nopathstruetofalse. apply pathsinv0. assumption. - intros m p. destruct m. + auto. + apply IHn. apply p. 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.