rounded_le' `{Rounded A} : forall e u v, close e u v -> forall d, ' e <= ' d -> close d u v. Proof. intros e u v xi d E. apply le_equiv_lt in E. destruct E as [E|E]. - apply pos_eq in E. rewrite <-E;trivial. - pose proof (pos_eq _ (_ + _) (Qpos_diff_pr _ _ E)) as E'. rewrite E'. apply rounded_plus. trivial. Qed.
Require Import Import NatPair.Instances. Import Quoting.Instances.