Zle_plus_preserving_l' : forall z : Z, OrderPreserving ((+) z). Proof. red. apply (Z_ind3 (fun _ _ _ => _ -> _)). intros [pc nc] [pa na] [pb nb]. rewrite !Zle_def;unfold PairT.Tle;simpl. intros E. assert (Hrw : pc + pa + (nc + nb) = (pc + nc) + (pa + nb)) by ring_with_nat. rewrite Hrw;clear Hrw. assert (Hrw : pc + pb + (nc + na) = (pc + nc) + (pb + na)) by ring_with_nat. rewrite Hrw;clear Hrw. apply (order_preserving _),E. Qed.
Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.