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