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.
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 RInt_gen.

Context {V : NormedModule R_AbsRing}.

Definition is_RInt_gen (f : RV) (Fa Fb : (RProp) → Prop) (l : V) :=
   (If : R × RV),
   filter_prod Fa Fb (fun abis_RInt f (fst ab) (snd ab) (If ab))
     filterlim If (filter_prod Fa Fb) (locally l).
Definition ex_RInt_gen (f : RV) (Fa Fb : (RProp) → 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.

End RInt_gen.

Basic properties of integrals


Section Property.

Context {V : NormedModule R_AbsRing}.

Lemma is_RInt_gen_ext {Fa Fb : (RProp) → Prop}
  {FFa : Filter Fa} {FFb : Filter Fb} (f g : RV) (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 lis_RInt_gen g Fa Fb l.

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

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

Lemma is_RInt_gen_Chasles {Fa Fc : (RProp) → Prop}
  {FFa : Filter Fa} {FFc : Filter Fc}
  (f : RV) (b : R) (l1 l2 : V) :
  is_RInt_gen f Fa (at_point b) l1is_RInt_gen f (at_point b) Fc l2
  → is_RInt_gen f Fa Fc (plus l1 l2).

End Property.

Composition


Section Compose.

Context {V : NormedModule R_AbsRing}.

Lemma is_RInt_gen_comp_opp {Fa Fb : (RProp) → Prop}
  {FFa : Filter Fa} {FFb : Filter Fb} (f : RV) (l : V) :
  is_RInt_gen f (filtermap opp Fa) (filtermap opp Fb) l
  is_RInt_gen (fun yopp (f (- y))) Fa Fb l.

Lemma is_RInt_gen_comp_lin {Fa Fb : (RProp) → Prop}
  {FFa : Filter Fa} {FFb : Filter Fb} (f : RV) (u v : R) (l : V) :
  is_RInt_gen f (filtermap (fun au × a + v) Fa) (filtermap (fun bu × b + v) Fb) l
    → is_RInt_gen (fun yscal u (f (u × y + v))) Fa Fb l.

End Compose.

Operations


Section Operations.

Context {V : NormedModule R_AbsRing}.

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

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

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

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

End Operations.

Lemma RInt_gen_norm {V : NormedModule R_AbsRing} {Fa Fb : (RProp) → Prop}
  {FFa : ProperFilter Fa} {FFb : ProperFilter Fb} (f : RV) (g : RR) (lf : V) (lg : R) :
  filter_prod Fa Fb (fun abfst ab snd ab)
  → filter_prod Fa Fb (fun ab x, fst ab x snd abnorm (f x) g x)
  → is_RInt_gen f Fa Fb lfis_RInt_gen g Fa Fb lg
    → norm lf lg.

Lemma is_RInt_gen_Derive {Fa Fb : (RProp) → Prop} {FFa : Filter Fa} {FFb : Filter Fb}
  (f : RR) (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).