proposition: Lemma CRabs_pos : forall x:CR, 0 <= x -> CRabs x == x.
proof: Proof. intros x H. apply CRle_antisym. split. - apply CRabs_AbsSmall. split. 2: apply CRle_refl. apply (@CRle_trans _ 0). 2: exact H. rewrite <- CRopp_0. apply CRopp_le_compat, H. - intro e. simpl. unfold Cap_raw; simpl. apply (Qle_trans _ 0). apply (Qopp_le_compat 0), Qpos_nonneg. rewrite <- Qle_minus_iff. apply Qle_Qabs. Qed.
imports: CoRN.algebra.RSetoid CoRN.metric2.Metric CoRN.metric2.UniformContinuity CoRN.model.totalorder.QposMinMax CoRN.reals.fast.CRArith CoRN.model.metric2.Qmetric Coq.QArith.Qabs CoRN.model.totalorder.QMinMax CoRN.logic.Stability