(** This file is part of the Coquelicot formalization of real analysis in Coq: http://coquelicot.saclay.inria.fr/  Copyright (C) 2011-2015 Sylvie Boldo #<br /># Copyright (C) 2011-2015 Catherine Lelay #<br /># Copyright (C) 2011-2015 Guillaume Melquiond  This library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version.  This library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the COPYING file for more details. *)
(** This file contains the definition and properties of the Riemann integral, defined on a normed module on [R]. For real functions, a total function [RInt] is available. *)
Section is_RInt.
(** * Definition of Riemann integral *)
Context {V : NormedModule R_AbsRing}.
Definition is_RInt (f : R -> V) (a b : R) (If : V) :=
filterlim (fun ptd => scal (sign (b-a)) (Riemann_sum f ptd)) (Riemann_fine a b) (locally If).
Definition ex_RInt (f : R -> V) (a b : R) :=
exists If : V, is_RInt f a b If.
(** ** Usual properties *)
(** The integral between a and a is null *)

(* Specifications: *)
Lemma is_RInt_point :
forall (f : R -> V) (a : R),
is_RInt f a a zero.
Lemma ex_RInt_point :
forall (f : R -> V) a,
ex_RInt f a a.