Lemma dualelement_lt_comp {n : nat} (i j : ⟦ n ⟧%stn) : i < j -> (dualelement i) > (dualelement j). Proof. intros lt. do 2 rewrite dualelement_defs_eq; unfold dualelement'. apply minusgth0inv; simpl. rewrite natminusminusassoc, natpluscomm, <- natminusminusassoc, minusminusmmn. 2: {apply (natgthtogehm1 _ _ (pr2 j)). } apply (minusgth0 _ _ lt). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Combinatorics.Maybe. Require Import UniMath.Combinatorics.StandardFiniteSets. Require Import UniMath.Algebra.Domains_and_Fields. Require Import UniMath.Algebra.RigsAndRings.