proposition: Lemma distribComplete_prf : forall (x:StepF (msp_as_RSetoid CR)), is_RegularFunction (@ball LinfStepQ) (distribComplete_raw x).
proof: Proof. unfold distribComplete_raw. intros x a b. induction x. apply (@regFun_prf _ Qball). simpl (ball (m:=LinfStepQ)). set (f:=(fun z : RegularFunction (@ball Q_as_MetricSpace) => approximate z a)) in *. set (g:=(fun z : RegularFunction (@ball Q_as_MetricSpace) => approximate z b)) in *. change (Map f (glue o x1 x2)) with (glue o (Map f x1:StepQ) (Map f x2)). change (Map g (glue o x1 x2)) with (glue o (Map g x1:StepQ) (Map g x2)). rewrite -> (StepFSupBallGlueGlue _ _ o (Map f x1 : StepQ) (Map f x2) (Map g x1 : StepQ) (Map g x2)). auto. Qed.
imports: CoRN.algebra.RSetoid CoRN.metric2.Metric CoRN.metric2.UniformContinuity CoRN.model.metric2.IntegrableFunction CoRN.model.metric2.BoundedFunction CoRN.model.metric2.CRmetric CoRN.reals.fast.CRArith CoRN.reals.fast.CRIR CoRN.ftc.Integral CoRN.model.structures.StepQsec CoRN.model.structures.OpenUnit CoRN.model.totalorder.QMinMax CoRN.model.totalorder.QposMinMax Coq.QArith.Qabs CoRN.tactics.Qauto Coq.QArith.Qround CoRN.model.metric2.L1metric CoRN.model.metric2.LinfMetric CoRN.model.ordfields.Qordfield CoRN.algebra.COrdFields2 CoRN.model.metric2.Qmetric CoRN.reals.fast.ContinuousCorrect CoRN.ftc.IntegrationRules CoRN.ftc.MoreIntegrals CoRN.tactics.CornTac CoRN.model.metric2.LinfDistMonad