proposition: Lemma CRabs_neg: forall x, x <= 0 -> CRabs x == - x.
proof: Proof. intros x H. apply CRle_antisym. split. - apply CRabs_AbsSmall. split. apply (CRplus_le_l _ _ (-x)). ring_simplify. apply CRle_refl. apply (@CRle_trans _ 0). 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, <- Qabs_opp. 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