proposition: Lemma Qabs_is_join@{} : forall q : Q, abs q = join (- q) q.
proof: Proof. intros q. symmetry. destruct (total_abs_either q) as [[E1 E2]|[E1 E2]];rewrite E2. - apply join_r. transitivity (0:Q);trivial. apply flip_nonneg_negate;trivial. - apply join_l. transitivity (0:Q);trivial. apply flip_nonpos_negate;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