proposition: Lemma cut_minus_plus_r x y z : 0 ≤ z → x ∸ (y + z) = (x ∸ y) ∸ z.
proof: Proof. intros E. case (total (≤) x y); intros Exy. rewrite (cut_minus_0 x y) by easy. rewrite cut_minus_0_l, cut_minus_0; try easy. now apply semirings.plus_le_compat_r. rewrite (cut_minus_plus_rev_r y (x ∸ y) z). now rewrite cut_minus_le, commutativity. Qed.
imports: Coq.setoid_ring.Ring MathClasses.interfaces.abstract_algebra MathClasses.interfaces.additional_operations MathClasses.interfaces.orders MathClasses.orders.minmax