forall x y : Q, abs (abs x - abs y) <= abs (x - y). Proof. intros q r. rewrite (Qabs_is_join (abs q - abs r)). apply join_le. - rewrite <-(Qabs_neg (q - r)),<-!negate_swap_r. apply Qabs_triangle_alt_aux. - apply Qabs_triangle_alt_aux. Qed.
Require Import Import Quoting.Instances. Import NatPair.Instances.