forall e (q r : Q), close e q r <-> abs (q - r) < ' e. Proof. intros e q r;split. - intros [E1 E2]. destruct (total le 0 (q - r)) as [E|E]. + rewrite (Qabs_of_nonneg _ E). trivial. + rewrite (Qabs_of_nonpos _ E). apply flip_lt_negate. rewrite involutive. trivial. - intros E. split;[apply flip_lt_negate;rewrite involutive|]; apply le_lt_trans with (abs (q - r));trivial. + apply Qabs_le_neg_raw. + apply Qabs_le_raw. Qed.
Require Import Import NatPair.Instances. Import Quoting.Instances.