proposition: Lemma ARabs_neg : forall x:AR, ARle x 0 -> ARabs x = -x.
proof: Proof. intros x xneg. apply (injective ARtoCR). pose proof (ARtoCR_preserves_abs x). unfold cast in H5. rewrite H5. clear H5. rewrite CRabs_neg. symmetry. apply ARtoCR_preserves_opp. rewrite <- ARtoCR_preserves_0. apply (ARtoCR_preserves_le x AR0), xneg. 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