Module mathcomp.analysis.showcase.summability

From HB Require Import structures.
From mathcomp Require Import boot order ssralg ssrint ssrnum finmap matrix.
From mathcomp Require Import interval zmodp.
From mathcomp Require Import boolp classical_sets.
From mathcomp Require Import ereal reals topology normedtype.

This file proposes a replacement for the definition `summable` (file `realsum.v`).

Unset SsrOldRewriteGoalsOrder.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Import GRing.Theory Num.Def Num.Theory.

Local Open Scope classical_set_scope.

From mathcomp Require fintype bigop finmap.

Section totally.

Import fintype bigop finmap.
Local Open Scope fset_scope.
Definition { : choiceType} : set_system {fset I} :=
filter_from setT (fun => [set | A `<=` B]).

Instance { : choiceType} : ProperFilter (@totally I).

Proof.

eapply filter_from_proper; last by move=> A _; exists A; rewrite /= fsubset_refl.
apply: filter_fromT_filter; first by exists fset0.
by move=> A B /=; exists (A `|` B) => P /=; rewrite fsubUset => /andP[].
Qed.

Definition { : choiceType} { : zmodType}
  ( : I -> R) ( : {fset I}) : R := \sum_( : A) x (val i).

Definition ( : choiceType) { : numDomainType} { : normedModType K}
   ( : I -> R) : R := lim (partial_sum x @ totally).

Definition ( : choiceType) { : realType} { : normedModType K}
   ( : I -> R) :=
   \forall \near +oo%R, \forall \near totally,
   (partial_sum (fun => `|x i|) J <= M)%R.

End totally.

Files