proposition: Lemma nonneg_minus_compat (x y z : R) : 0 ≤ z -> x ≤ y -> x - z ≤ y.
proof: Proof. intros E1 E2. rewrite plus_comm, (plus_conjugate_alt y (- z)), involutive. apply (order_preserving (-(z) +)). transitivity y; trivial. pattern y at 1;apply (transport _ (plus_0_r y)). apply (order_preserving (y +)). trivial. Qed.
imports: HoTT.Classes.interfaces.abstract_algebra HoTT.Classes.interfaces.orders HoTT.Classes.theory.groups HoTT.Classes.theory.rings HoTT.Classes.orders.semirings