Library Coquelicot.RInt_gen

This file is part of the Coquelicot formalization of real analysis in Coq: http://coquelicot.saclay.inria.fr/

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.

From Coq Require Import Reals ssreflect.

Require Import Rbar Hierarchy RInt Lim_seq Continuity Derive Rcomplements RInt_analysis.

This file describes improper integrals, such as integrals with an infinity endpoint or integrals of a function with a singularity. A few properties are given: Chasles, operations, composition, derivation.

Open Scope R_scope.

Improper Riemann integral

Section is_RInt_gen.

Context {V : NormedModule R_AbsRing}.

Definition is_RInt_gen (f : R → V) (Fa Fb : (R → Prop) → Prop) (l : V) :=
  filterlimi (fun ab ⇒ is_RInt f (fst ab) (snd ab)) (filter_prod Fa Fb) (locally l).
Definition ex_RInt_gen (f : R → V) (Fa Fb : (R → Prop) → Prop) :=
  ∃ l : V , is_RInt_gen f Fa Fb l.

Definition is_RInt_gen_at_point f a b l :
  is_RInt_gen f (at_point a) (at_point b) l
    ↔ is_RInt f a b l.

Basic properties of integrals

Lemma is_RInt_gen_ext {Fa Fb : (R → Prop) → Prop}
  {FFa : Filter Fa} {FFb : Filter Fb} (f g : R → V) (l : V) :
  filter_prod Fa Fb (fun ab ⇒ ∀ x, Rmin (fst ab) (snd ab) < x < Rmax (fst ab) (snd ab)
                               → f x = g x) →
  is_RInt_gen f Fa Fb l → is_RInt_gen g Fa Fb l.

Lemma ex_RInt_gen_ext {Fa Fb : (R → Prop) → Prop}
  {FFa : Filter Fa} {FFb : Filter Fb} (f g : R → V) :
  filter_prod Fa Fb (fun ab ⇒ ∀ x, Rmin (fst ab) (snd ab) < x < Rmax (fst ab) (snd ab)
                               → f x = g x) →
  ex_RInt_gen f Fa Fb → ex_RInt_gen g Fa Fb.

Lemma ex_RInt_gen_ext_eq {Fa Fb : (R → Prop) → Prop}
  {FFa : Filter Fa} {FFb : Filter Fb} (f g : R → V) :
  (∀ x, f x = g x ) → ex_RInt_gen f Fa Fb → ex_RInt_gen g Fa Fb.

Lemma is_RInt_gen_point (f : R → V) (a : R) :
  is_RInt_gen f (at_point a) (at_point a) zero.

Lemma is_RInt_gen_swap {Fa Fb : (R → Prop) → Prop}
  {FFa : Filter Fa} {FFb : Filter Fb} (f : R → V) (l : V) :
  is_RInt_gen f Fb Fa l → is_RInt_gen f Fa Fb (opp l).

Lemma is_RInt_gen_Chasles {Fa Fc : (R → Prop) → Prop}
  {FFa : Filter Fa} {FFc : Filter Fc}
  (f : R → V) (b : R) (l1 l2 : V) :
  is_RInt_gen f Fa (at_point b) l1 → is_RInt_gen f (at_point b) Fc l2
  → is_RInt_gen f Fa Fc (plus l1 l2).

Lemma ex_RInt_gen_Chasles :
  ∀ {Fa Fc : (R → Prop) → Prop},
  ∀ {FFa : Filter Fa} {FFc : Filter Fc},
  ∀ (f : R → V) (b : R),
  ex_RInt_gen f Fa (at_point b) →
  ex_RInt_gen f (at_point b) Fc →
  ex_RInt_gen f Fa Fc.

Composition

Operations