(* -*- coding: utf-8 -*- *)
(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)
Local Open Scope R_scope.
Set Implicit Arguments.
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(**           Riemann's Integral            *)
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Definition Riemann_integrable (f:R -> R) (a b:R) : Type :=
forall eps:posreal,
{ phi:StepFun a b &
{ psi:StepFun a b |
(forall t:R,
Rmin a b <= t <= Rmax a b -> Rabs (f t - phi t) <= psi t) /\
Rabs (RiemannInt_SF psi) < eps } }.
Definition phi_sequence (un:nat -> posreal) (f:R -> R)
(a b:R) (pr:Riemann_integrable f a b) (n:nat) :=
projT1 (pr (un n)).

Lemma RiemannInt_P3 :
forall (f:R -> R) (a b:R) (un:nat -> posreal) (vn wn:nat -> StepFun a b),
Un_cv un 0 ->
(forall n:nat,
(forall t:R, Rmin a b <= t <= Rmax a b -> Rabs (f t - vn n t) <= wn n t) /\
Rabs (RiemannInt_SF (wn n)) < un n) ->
{ l:R | Un_cv (fun N:nat => RiemannInt_SF (vn N)) l }.