{x y} : x < y -> exists z, 0 < z /\ y = x + z. Proof. intros E. destruct (strict_srorder_partial_minus x y E) as [z Ez]. exists z. split; [| trivial]. apply (strictly_order_reflecting (x+)). rewrite <-Ez, rings.plus_0_r. trivial. Qed.
Require Import