proposition: Lemma nonneg_plus_le_compat_r x z : 0 ≤ z ↔ x ≤ x + z.
proof: Proof. rewrite <-(plus_0_r x) at 1. split; intros. now apply (order_preserving _). now apply (order_reflecting (x+)). Qed.
imports: Coq.setoid_ring.Ring MathClasses.interfaces.abstract_algebra MathClasses.interfaces.orders MathClasses.theory.rings MathClasses.orders.orders MathClasses.orders.maps