fun x y => pos x + neg y = pos y + neg x. Global Instance equiv_is_equiv_rel@{} : EquivRel . Proof. split. - hnf. reflexivity. - hnf. unfold . intros ??;apply symmetry. - hnf. unfold . intros a b c E1 E2. apply (left_cancellation (+) (neg b)). rewrite (plus_assoc (neg b) (pos a)). rewrite (plus_comm (neg b) (pos a)), E1. rewrite (plus_comm (pos b)). rewrite <-plus_assoc. rewrite E2. rewrite (plus_comm (pos c) (neg b)). rewrite plus_assoc. rewrite (plus_comm (neg a)). rewrite <-plus_assoc. rewrite (plus_comm (neg a)). reflexivity. Qed.
Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.