SemiRingOrder (≤). Proof. repeat (split; try apply _). - intros x y E. exists (- x + y). rewrite simple_associativity, plus_negate_r, plus_0_l. reflexivity. - intros x y E. rewrite <-(plus_0_l x), <-(plus_0_l y), <-!(plus_negate_l z), <-!simple_associativity. apply (order_preserving _). trivial. Qed.
Require Import