proposition: Lemma CR_lt_as_Cauchy_IR_lt_1 : forall (x y:CR), (x < y)%CR -> (CRasCauchy_IR x)[<](CRasCauchy_IR y).
proof: Proof. intros x y [e He]. apply shift_zero_less_minus'. apply (less_leEq_trans _ [0] (Cauchy_CReals.inject_Q _ (proj1_sig e))). eapply ing_lt. apply Qpos_ispos. unfold cg_minus. stepr (CRasCauchy_IR (y-x))%CR. stepl (CRasCauchy_IR (' proj1_sig e)%CR). rewrite <- Cauchy_IR_le_as_CR_le. do 2 rewrite -> CRasCR. assumption. eapply CR_inject_Q_as_Cauchy_IR_inject_Q. stepl (CRasCauchy_IR y[+]CRasCauchy_IR(- x)%CR). apply plus_resp_eq. eapply CR_opp_as_Cauchy_IR_opp. eapply CR_plus_as_Cauchy_IR_plus. Qed.
imports: CoRN.algebra.RSetoid CoRN.metric2.Metric CoRN.metric2.UniformContinuity CoRN.model.totalorder.QposMinMax CoRN.model.reals.Cauchy_IR CoRN.tactics.CornTac Coq(Lia) CoRN.model.metric2.Qmetric CoRN.model.totalorder.QMinMax Coq.QArith.Qabs CoRN.reals.fast.CRFieldOps