proposition: Lemma Q_abs_le_pr@{} : forall x y : Q, abs x <= y <-> - y <= x /\ x <= y.
proof: Proof. intros x y;split. - intros E. split. + apply flip_le_negate. rewrite involutive. transitivity (abs x);trivial. apply Qabs_le_neg_raw. + transitivity (abs x);trivial. apply Qabs_le_raw. - intros [E1 E2]. destruct (total_abs_either x) as [[E3 E4]|[E3 E4]];rewrite E4. + trivial. + apply flip_le_negate;rewrite involutive;trivial. 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