proposition: Lemma ARabs_pos : forall x:AR, ARle 0 x -> ARabs x = x.
proof: Proof. intros x xpos. apply (injective ARtoCR). pose proof (ARtoCR_preserves_abs x). unfold cast in H5. rewrite H5. clear H5. apply CRabs_pos. rewrite <- ARtoCR_preserves_0. apply (ARtoCR_preserves_le AR0 x), xpos. Qed.
imports: CoRN.algebra.RSetoid CoRN.metric2.Metric CoRN.metric2.UniformContinuity Coq.QArith.Qabs CoRN.model.metric2.Qmetric CoRN.model.totalorder.QposMinMax CoRN.model.totalorder.QMinMax CoRN.reals.fast.CRabs CoRN.reals.faster.ARArith