Library Coquelicot.RInt_gen

This file is part of the Coquelicot formalization of real analysis in Coq: http://coquelicot.saclay.inria.fr/
Copyright (C) 2011-2015 Sylvie Boldo
Copyright (C) 2011-2015 Catherine Lelay
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.

Require Import Reals mathcomp.ssreflect.ssreflect.
Require Import Rbar Hierarchy RInt Lim_seq Continuity Derive.
Require Import 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).

Composition



Operations


Lemma is_RInt_gen_scal {Fa Fb : (R → Prop) → Prop}
  {FFa : Filter Fa} {FFb : Filter Fb} (f : R → V) (k : R) (l : V) :
  is_RInt_gen f Fa Fb l →
  is_RInt_gen (fun y ⇒ scal k (f y)) Fa Fb (scal k l).

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

Lemma is_RInt_gen_plus {Fa Fb : (R → Prop) → Prop}
  {FFa : Filter Fa} {FFb : Filter Fb} (f g : R → V) (lf lg : V) :
  is_RInt_gen f Fa Fb lf →
  is_RInt_gen g Fa Fb lg →
  is_RInt_gen (fun y ⇒ plus (f y) (g y)) Fa Fb (plus lf lg).

Lemma is_RInt_gen_minus {Fa Fb : (R → Prop) → Prop}
  {FFa : Filter Fa} {FFb : Filter Fb} (f g : R → V) (lf lg : V) :
  is_RInt_gen f Fa Fb lf →
  is_RInt_gen g Fa Fb lg →
  is_RInt_gen (fun y ⇒ minus (f y) (g y)) Fa Fb (minus lf lg).

End is_RInt_gen.

Section RInt_gen.

Context {V : CompleteNormedModule R_AbsRing}.

Definition RInt_gen (f : R → V) (a b : (R → Prop) → Prop) :=
  iota (is_RInt_gen f a b).

Lemma is_RInt_gen_unique {Fa Fb : (R → Prop) → Prop}
  {FFa : ProperFilter' Fa} {FFb : ProperFilter' Fb} (f : R → V) (l : V) :
  is_RInt_gen f Fa Fb l → RInt_gen f Fa Fb = l.

Lemma RInt_gen_correct {Fa Fb : (R → Prop) → Prop}
  {FFa : ProperFilter' Fa} {FFb : ProperFilter' Fb} (f : R → V) :
  ex_RInt_gen f Fa Fb → is_RInt_gen f Fa Fb (RInt_gen f Fa Fb).

Lemma RInt_gen_norm {Fa Fb : (R → Prop) → Prop}
  {FFa : ProperFilter Fa} {FFb : ProperFilter Fb} (f : R → V) (g : R → R) (lf : V) (lg : R) :
  filter_prod Fa Fb (fun ab ⇒ fst ab ≤ snd ab) →
  filter_prod Fa Fb (fun ab ⇒ ∀ x, fst ab ≤ x ≤ snd ab → norm (f x) ≤ g x) →
  is_RInt_gen f Fa Fb lf → is_RInt_gen g Fa Fb lg →
  norm lf ≤ lg.

Lemma RInt_gen_ext {Fa Fb : (R → Prop) → Prop}
  {FFa : ProperFilter Fa} {FFb : ProperFilter 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 → RInt_gen f Fa Fb = RInt_gen g Fa Fb.

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

End RInt_gen.

Lemma is_RInt_gen_Derive {Fa Fb : (R → Prop) → Prop} {FFa : Filter Fa} {FFb : Filter Fb}
  (f : R → R) (la lb : R) :
  filter_prod Fa Fb
    (fun ab ⇒ ∀ x : R, Rmin (fst ab) (snd ab) ≤ x ≤ Rmax (fst ab) (snd ab) → ex_derive f x)
  → filter_prod Fa Fb
    (fun ab ⇒ ∀ x : R, Rmin (fst ab) (snd ab) ≤ x ≤ Rmax (fst ab) (snd ab) → continuous (Derive f) x)
  → filterlim f Fa (locally la) → filterlim f Fb (locally lb)
  → is_RInt_gen (Derive f) Fa Fb (lb - la).

Section Complements_RInt_gen.

Context {V : CompleteNormedModule R_AbsRing}.

Lemma ex_RInt_gen_at_point f a b : @ex_RInt_gen V f (at_point a) (at_point b) ↔ ex_RInt f a b.

Lemma RInt_gen_at_point f a b :
  ex_RInt f a b → @RInt_gen V f (at_point a) (at_point b) = RInt f a b.

End Complements_RInt_gen.