proposition: Lemma Qabs_triangle_alt_aux : forall x y : Q, abs x - abs y <= abs (x - y).
proof: Proof. intros q r. apply (order_reflecting (+ (abs r))). assert (Hrw : abs q - abs r + abs r = abs q) by ring_tac.ring_with_integers (NatPair.Z nat); rewrite Hrw;clear Hrw. etransitivity;[|apply Q_triangle_le]. assert (Hrw : q - r + r = q) by ring_tac.ring_with_integers (NatPair.Z nat); rewrite Hrw;clear Hrw. reflexivity. Qed.
imports: HoTT.Classes.implementations.peano_naturals HoTT.Classes.implementations.natpair_integers HoTT.Classes.interfaces.abstract_algebra HoTT.Classes.interfaces.naturals HoTT.Classes.interfaces.rationals HoTT.Classes.interfaces.orders HoTT.Classes.theory.groups HoTT.Classes.theory.integers HoTT.Classes.theory.dec_fields HoTT.Classes.orders.sum HoTT.Classes.orders.dec_fields HoTT.Classes.orders.lattices HoTT.Classes.theory.additional_operations HoTT.Classes.tactics.ring_quote HoTT.Classes.tactics.ring_tac