Qabs_is_join@{} : forall q : Q, abs q = join (- q) q. 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.
Require Import Import Quoting.Instances. Import NatPair.Instances.