proposition: Lemma StepFun_P34 : forall (a b:R) (f:StepFun a b), a <= b -> Rabs (RiemannInt_SF f) <= RiemannInt_SF (mkStepFun (StepFun_P32 f)).
proof: Proof. intros; unfold RiemannInt_SF; decide (Rle_dec a b) with H. replace (Int_SF (subdivision_val (mkStepFun (StepFun_P32 f))) (subdivision (mkStepFun (StepFun_P32 f)))) with (Int_SF (map Rabs (subdivision_val f)) (subdivision f)). - apply StepFun_P33; assert (H0 := StepFun_P29 f); unfold is_subdivision in H0; elim H0; intros; unfold adapted_couple in p; decompose [and] p; assumption. - apply StepFun_P17 with (fun x:R => Rabs (f x)) a b; [ apply StepFun_P31; apply StepFun_P1 | apply (StepFun_P1 (mkStepFun (StepFun_P32 f))) ]. Qed.
imports: Rbase Rfunctions Ranalysis_reg Classical_Prop List RList Lia Lra