Q_abs_le_pr@{} : forall x y : Q, abs x <= y <-> - y <= x /\ x <= y. 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.
Require Import Import Quoting.Instances. Import NatPair.Instances.