Quelle  Infinite_Set_Sum.thy   Sprache: Isabelle

(*
  Title:    HOL/Analysis/Infinite_Set_Sum.thy
  Author:   Manuel Eberl, TU München

  A theory of sums over possible infinite sets. (Only works for absolute summability)
*)
section  Author  A theory of sums over possiblejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

theory Infinite_Set_Sum
  imports Set_Integral Infinite_Sum
begin

(*
  WARNING! This file is considered obsolete and will, in the long run, be replaced with
  the more general "Infinite_Sum".
*)

text ‹Conflicting notation from 🍋‹HOL-Analysis.Infinite_Sum››
no_notation Infinite_Sum.abs_summable_on (infixr ‹abs'_summable'_on›.abs_summable_on(infixr>abs'_summable'_on›

(* TODO Move *)
lemmaproof  ( equalityI)
  assumesA""   ""<java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 93
  shows sets.[OF] and assms
proofintro subsetI
  fix X assume "java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
   eq  \>🚫
    by sets'countable_subset[OF_assms1] ( intro!:assms()java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89
  also "\<>
  finally show "X \ sets M" .
next
  fix X assume "X \< assumes "finite A" "∧
  from sets.sets_into_space(rule')
    wX\in  A"
qed

lemma':
  java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
  assumessets_PiM_I_finiteauto
  assumes alsofrom  "PiEA .{x {}java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
  assumes
   "="
proof (rule measure_eqI_countable)
  show "sets M = Pow A"
    by (intro sets_eq_countable assms)
  show "sets N = Pow A"
    by (intro sets_eq_countable assms)
qed fact+

lemma count_space_PiM_finite:
  fixes B :: "'a \ 'b set"
  assumes "finite A" "\i. countable (B i)"
  shows   "PiM A (\i. count_space (B i)) = count_space (PiE A B)"
proof ( ( product_sigma_finite sigma_finite_measure_count_space_countable
  show (PiMlambda))=PiE"
    by (simp add: space_PiM)
  show:" <> B"
next
  fix  {} PiEx
   PiE f}<>sets^subA(\lambdai count_space))java.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93
    by (intro sets_PiM_I_finite assms) auto
  also from f have "PiE A (<>x. {fx} =f"
    by (intro PiE_singleton) f assms by ( emeasure_PiM
  finally "f \in sets (Pi\<^sub>M A (\i. count_space (B i)))" .
next
  interpret product_sigma_finite "(\by(introprodcongrefl subst) (usef auto)
    by (intro\dots=  count_spaceB)){f"
  thm
  fix f assume f: "f \.count_space( i){f} java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77

    by (intro PiE_singleton [symmetric])  (simp_all: countable_PiE)
  also have\^>tag important›aset<> "
               (∏i∈A. emeasure (count_space (B i)) {f i})"
    usingf assms ( emeasure_PiM) auto
  also have "\ = (\i\A. 1)"
    by (intro prod" abs_summable_on \> integrable(count_spaceA fjava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
  also have "\>= emeasure (count_space (PiE A B)) {f}"
    using f by(subst) auto
  finally show "emeasure (Pi\<^sub>M A (\'a < 'b: {banach \Rightarrow' set<>bjava.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 104
                  emeasureinfsetsum " \Rightarrow> 'a et\Rightarrow>' <>b:banach,second_countable_topology}java.lang.StringIndexOutOfBoundsException: Index 117 out of bounds for length 117
qed (simp_all add: countable_PiE assms)



definition^\<open>tag important›
    ( Rightarrow, second_countable_topology> 'a set \ bool"
    (infix ‹(‹binder ∑a›∑a_∈_close> [0, 51, 10] 10)
wjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
   f abs_summable_on A ⟷ integrable (count_space A) f"


definition🚫
    "('a \\indent=3 notation=\binder INFSETSUM\\INFSETSUM _:_./ _)\ [0, 51, 10] 10)
 
   "infsetsum f A = lebesgue_integral\<>(\\<^>a\<>\\<^>a_._)close 0 1 0java.lang.StringIndexOutOfBoundsException: Index 120 out of bounds for length 120

syntax ()
  "_infsetsum" :: "pttrn \ 'a set"<><sub "\<> CONST \lambdai b CONSTUNIV"
  (‹(‹indent=3 notation=‹binder INFSETSUM
syntax
  \open>(🚫binder INFSETSUM›INFSETSUM _ |/ _./ _)›
  (‹bool<>' \':{banachsecond_countable_topology}"
syntax_consts
  "_infsetsum" ⇌ infsetsum
translations 🍋 infsetsum
  "\

syntaxASCII
  "_uinfsetsum "pttrnRightarrowset>' :{banach }"
  (‹🚫' \syntax_const>\open_\)java.lang.StringIndexOutOfBoundsException: Index 113 out of bounds for length 113
syntax < > count_space
  "_uinfsetsum" :: "pttrn \ 'b \
  (<close\^sub/_› -
syntax_consts
  "_uinfsetsum" ⇌ infsetsum
translations 🍋 ‹restrict_count_space_subset]) +
  "\\<^sub>ai. b" ⇌ integrable_restrict_space)

syntax (ASCII abs_summable_on_def .
  Rightarrow  \Rightarrow'<':,second_countable_topology
  (‹indent=3 notation=‹close›INFSETSUM _ |/_/)🚫
syntax
  "_qinfsetsum" :java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  (‹no_types abs_summable_on_deforderE restrict_count_space_subset )
syntax_consts
  qinfsetsum\>infsetsum
translations simp:  integrable_norm_iff
  STlambda.Pjava.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
print_translation
  [(🍋:
›

lemma restrict_count_space_subset:
  "A \ B \ restrict_space (count_space B) A = count_space A"
  by (subst restrict_count_space) (simp_all addassumes<> 🚫

  by( : AE_count_space
  fixes f :: "'a \ 'b :: {banach, second_countable_topology}"
   "A<>Bjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
  shows"fabs_summable_on A \ \lambda java.lang.StringIndexOutOfBoundsException: Range [76, 75) out of bounds for length 113
proof -
  have "count_space A = restrict_space (count_space B) A"
    by (also "\ \ (gx"by java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
  also have "integrable \<"(\x. x \ A \ x= x \< A Longrightarrow> (f abs_summable_on A) \ (g abs_summable_on B)"
    by (simp add: integrable_restrict_space set_integrable_def)
 
    unfolding :
java.lang.StringIndexOutOfBoundsException: Range [12, 3) out of bounds for length 3

lemma abs_summable_on_altdef: "f abs_summable_on A \ set_integrable (count_space UNIV) A f"
  unfolding abs_summable_on_def simp splitif_splits
  by (metis (no_typesf:"ajava.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 73

lemma fabs_summable_on<summable.ifAnorm  )java.lang.StringIndexOutOfBoundsException: Index 117 out of bounds for length 117
   < \Longrightarrow \> count_space"
  unfolding abs_summable_on_def set_integrable_def
  by (( addabs_summable_on_def)

 [java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
  "(\x
  by (simp add: abs_summable_on_def integrable_norm_iff)

lemma " abs_summable_onjava.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 95
   f :" \ ' : banach}"

lemma abs_summable_complex_of_real<>java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 96
  by (simp assmsfastforce add intro')

lemma:
  assumesgabs_summable_on
  assumes(:nat\longleftrightarrow(<forall>N 🚫
  shows   "f abs_summable_on A"
  using.[of java.lang.StringIndexOutOfBoundsException: Range [71, 70) out of bounds for length 74
  unfolding abs_summable_on_def abs_summable_on_altdef

lemma abs_summable_on_comparison_test
   " java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 31
  assumes "\x. x \ A \ norm (f java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  shows   "f " abs_summable_on
proof "f abs_summable_on A"
  fix x assume "x \A"
  with assmsassume" A"
    abs_summable_on_union[OF, of{}"]
  finally show "norm (f x) \ norm (g x)" .
qed

lemma abs_summable_on_cong [cong]:
  "(\x. x \ A \ f x = g x) \ A = B \ show "fabs_summable_onx " bysimp
  unfolding java.lang.StringIndexOutOfBoundsException: Range [0, 31) out of bounds for length 23

lemma abs_summable_on_cong_neutral:
  assumesAnd>.x\in   ==>
  assumes "\x. x \ B - A \ g x = 0"
  assumes "\x. x \ A \ B \ simpadd abs_summable_on_def
  shows < gabs_summable_on
  unfolding abs_summable_on_altdefsimpadd)
  by abs_summable_on_finite_diff:
     ( simp split )

lemma:
  fixes f :: "'a \ 'b :: {banach, second_countable_topology}"
  assumes "A \ B"
  shows   "f abs_summable_on Abyintro abs_summable_on_union assms abs_summable_on_finite)
  by (subst abs_summable_on_restrict[OF assms]) (intro abs_summable_on_cong  haveA \union   Bbyblast

java.lang.StringIndexOutOfBoundsException: Range [38, 5) out of bounds for length 30
  "f abs_summable_on ( *: "count_space B=distrcount_space A) ( B "
proof
  have "f abs_summable_on java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
    by (subst abs_summable_on_restrict'[of _ UNIV])
       (simp_all add: abs_summable_on_def java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 30
  also have "( -
     java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
  finally show ?thesis [symmetric ) auto
qed

lemma abs_summable_on_nat_iff? .
  java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
   subst)

lemma:
  fixes f :: " A:'set B: "a \Rightarrowb "
  assumes "g abs_summable_on I"
  assumes "\n. \n\N; n \ I\ \ norm (f n) \ g n"
  shows   "f abs_summable_on I"
  using(simpabs_summable_on_nat_iff'

lemma abs_summable_comparison_test_ev:
  assumes "
  assumes "eventually (\x. x \ I \ norm (f x) \ g x) sequentially"
  showsfabs_summable_on
  by (metis (byrule)auto

lemma  have "(\<lambda>y)abs_summable_on snd`Sigma )
  by( )
   simp:'summable_Cauchy sum_nonneg)

lemma [] A\> java.lang.StringIndexOutOfBoundsException: Range [83, 82) out of bounds for length 85
  unfolding :\x)(,x ( \timesA(\ "

lemma abs_summable_on_empty [simp, intro]: "f abs_summable_on {}"
  by simp

lemma:
  assumes "simp_alladd )
  showsjava.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 31
  unfolding abs_summable_on<(λ
  by ( set_integrable_subsetinsert autosimp:abs_summable_on_altdef

lemma abs_summable_on_union [intro]:
  assumes "f assumes "f  A"and "  A"
  shows   "f abs_summable_on (A \ B)"
  using assms unfolding abs_summable_on_altdef by (intro set_integrable_Un) auto

lemma abs_summable_on_insert_iff [simp]:
  "f abs_summable_on insert x "(<lambdax.fx+gx)  A
proof safe
   "finsert x A
  thus "f abs_summable_on A"
    
next abs_summable_on_diffintro
  assume "f abs_summable_on A"
  java.lang.StringIndexOutOfBoundsException: Range [28, 6) out of bounds for length 47
    show "f shows "λ  xx)abs_summable_onA"
qed

lemma abs_summable_sum:
  assumes "\x. x \ A \ f x abs_summable_on B"
  shows   "(\y. \
  using unfolding by ( Bochner_Integration.integrable_sum)

lemma:  A \ (λx. Re (f x)) abs_summable_on A"
  by (simp add: abs_summable_on_def   "(\

lemmaabs_summable_Im:f A \>(<lambdaabs_summable_on
  by( : )

lemmajava.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 34
  assumes "f abs_summable_on abs_summable_on_cmult_right intro]
  shows   "f abs_summable_on B"
proof -
  have  "\<>0
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  also from assms 0\>  java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
  finally show ?thesis .
qed

lemma abs_summable_on_reindex_bij_betw:
  assumes "bij_betw g A B"
  shows   "(\x. f (g x)) abs_summable_on A \ f abs_summable_on B"
proof -
  have *: "count_space B = distr (count_space A) (count_space B) g"
    by(ule [symmetricfact
  show ?   "(\A f x( x) abs_summable_on PiE A B"
    by (subst *, subst integrable_distr_eq[of _ _ "count_space B"])
       (insert assms, auto simp: bij_betw_def-
qed

lemma:
  assumes "(\x. f (g x)) abs_summable_on A"
  shows   "f abs_summable_on (g ` A)"
proof -
  define g' where "g' = inv_into A g"
  from have (<>xf  x)) abs_summable_on (g' ` g ` A)"
    by (rule abs_summable_on_subset) (auto simp: g'_def inv_into_into)
  alsohave? 🚫
    by introabs_summable_on_reindex_bij_betw] inj_on_imp_bij_betw) java.lang.StringIndexOutOfBoundsException: Index 100 out of bounds for length 100
  also have "\ \ f abs_summable_on (g ` A)"
    by (ntro abs_summable_on_cong)( : g_ f_inv_into_f
  finally show ?thesis .
qed

lemma abs_summable_on_reindex_iff:
  "inj_on g A \ (\
  by( abs_summable_on_reindex_bij_betw)

lemmaabs_summable_on_Sigma_project2
  fixes A :: "'by simp:abs_summable_on_def infsetsum_def not_integrable_integral_eq)
  assumes " infsetsum_altdef:
  shows   "(\y. f (x, y)) abs_summable_on (B x)"
proof -
  from assms(2) have "f abs_summable_on " f A = set_lebesgue_integralcount_space) A f"
    by (intro abs_summable_on_subset [OF assms(1)]) auto
  also have"this (\lambdaz. (x, z)) abs_summable_onabs_summable_on (Sigma {}B)
    by (rule abs_summable_on_cong) auto simprestrict_count_space_subset )
  finally have "(\y. f (x, infsetsum_altdef'
    byrule
  also have "snd set_lebesgue_integral_def
    using assms by (auto simp: image_iff)
  finally  ?thesis
qed

lemma abs_summable_on_Times_swap
   nn_integral_conv_infsetsum
proof -
  have bij: "bij_betw (\(x,y). (y,x)) (B \ A) (A \ B)"
    by (auto simp: bij_betw_def inj_on_def)
  show ?showsnn_integralf =ennreal )
    by  assmsunfolding abs_summable_on_def
       (simp_allsubst nn_integral_eq_integral auto
qed

lemma  "nn_integral (count_space A \.x \ AA \Longrightarrow f \> 0"
  by (simp add: abs_summable_on_def)

lemma abs_summable_on_uminus []:
  "f abs_summable_on A \ (\x. -f x) abs_summable_on A"
  java.lang.StringIndexOutOfBoundsException: Range [35, 11) out of bounds for length 78

lemma abs_summable_on_add [intro]:
  assumes "f abs_summable_on A" and "g abs_summable_on A"
  shows   "(\x. f ""(\And>x. x \A f x = g x) <> A = B \ infsetsum f A = infsetsum B"
  using assms unfolding    infsetsum_defintro.integral_cong

lemma abs_summable_on_diff  infsetsum_0[]: infsetsum> 0 A  0"
  assumes  java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 57
  shows\lambdax.  -  g)abs_summable_on"
  using assms unfolding

lemma abs_summable_on_scaleR_left [intro]:
    byrule)
  shows
  using assms assumes\And>x. \nA< f abs_summable_on

lemma abs_summable_on_scaleR_right [intro assmsby simpadd   Bochner_Integration)
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  shows   " (simpsimp add infsetsum_def abs_summable_on_def
  using assms: f A🚫

lemma abs_summable_on_cmult_right [ infsetsum_of_real
  fixes : "a\>b {banachjava.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92
  assumes "c \ 0 \ f abs_summable_on A"
  showslambdax abs_summable_on "
  using assms unfolding abs_summable_on_def by (intro Bochner_Integration.integrable_mult_right)

lemma abs_summable_on_cmult_left [intro]:
  fixesf :"a< b: {banach, second_countable_topology"
  assumes <0<Longrightarrow  "
  shows
  using unfolding by intro.integrable_mult_left)

lemma abs_summable_on_prod_PiE:
  fixes:a <>b \Rightarrow c: {java.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 107
  assumes from assms havinfsetsum \> indicator\^subfjava.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 74
  assumes summable: "\x. x \ A \ f subst ) auto
  showsλg. ∏x∈A. f x (g x)) abs_summable_on PiE A B"
proof-
  define show?thesis .
  from 
    by (auto java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  then interpret product_sigma_finite "count_space \ B'"
    unfolding o_def by (intro product_sigma_finiteusing by( infsetsum_nat
  from assms have "integrable(PiM A(count_space\ B')) (lambdag.\Prod>x<>.fgx)java.lang.StringIndexOutOfBoundsException: Index 103 out of bounds for length 103
java.lang.StringIndexOutOfBoundsException: Range [6, 4) out of bounds for length 78
  also have "PiM A ( have"🚫 Athen f   0  (\lambda  (if> A then f n else 0))"
    unfolding o_def using rule )
  alsohave" A ' =PiEAB" intro  java.lang.StringIndexOutOfBoundsException: Range [67, 66) out of bounds for length 75
  finally show ?thesis by (simp add
qed



lemma not_summable_infsetsum_eq:
  "\f abs_summable_on "f  infsetsum "
  by ( addabs_summable_on_def not_integrable_integral_eq

lemma infsetsum_altdef:
  "infsetsumlemma infsetsum_Un_disjoint
  unfolding set_lebesgue_integral_def
  bysubstintegral_restrict_space)
     (auto simp: restrict_count_space_subset infsetsum_def)

lemma infsetsum_altdef':
  "A \ B \ infsetsum f A = set_lebesgue_integral (count_space B) A f"
  unfolding set_lebesgue_integral_def
  by (subst integral_restrict_space [symmetric])
     (auto simp: restrict_count_space_subset)

lemmaby( set_integral_Un) auto
  assumes "f abs_summable_on A" "\x. x \ A \ f x \ 0"
  shows   "nn_integral (count_space A) f = ennreal (infsetsum f A)"
  using assms unfolding infsetsum_def abs_summable_on_def
  by substnn_integral_eq_integral) auto

lemma infsetsum_conv_nn_integralproof -
  assumes"nn_integral (count_spaceA)f\ \" "\And>x x A 0"
  shows   "infsetsum f A = enn2real (nn_integral (count_space A) f)"
  unfolding infsetsum_def using ssms
  by (subst integral_eq_nn_integral) autousing(2) by (intro  abs_summable_on_subsetOF assms)) auto

lemma infsetsum_cong [cong]:
  "(\And>.x\inA \ f x = g x) \ A = B \ infsetsum f A = infsetsum g B"
  unfolding by ( Bochner_Integration) auto

lemma infsetsum_0 [simp]: "infsetsum (\_. 0) A = 0"
  y ( add )

lemma infsetsum_all_0: "(\x. x \< infsetsum_Un_Int:
  by simp

lemma infsetsum_nonneg: "(\x. x \ A \ f x \ (0::real)) \ infsetsum f A \ 0"
  unfolding infsetsum_def by (rule Bochner_Integration.integral_nonneg) auto

lemma sum_infsetsum:
  assumes "\x. x \ A \ f x abs_summable_on B"
  shows   "(\x\A. \\<^sub>ay\B. f x y) = (\\<^sub>ay\B. \x\A. f x y)"
  usingalsohave "infsetsum f \ = infsetsum f A +infsetsum f (B - A \ B)"

lemma Re_infsetsum: "f abs_summable_on A \ Re (infsetsum f A) = (\\<^sub>ax\A. Re (f x))"
  by (simp add: infsetsum_def abs_summable_on_def)

lemma Im_infsetsum: "f abs_summable_on A \ Im (infsetsum f A) = (\\<^sub>ax\A. Im (f x))"
  by (simp add: infsetsum_def abs_summable_on_def)

lemma infsetsum_of_real:
  shows " (\x. of_real (f x)
           :: 'a :: {real_normed_algebra_1,banach,second_countable_topology,real_inner}) A =
             of_real (infsetsum f A)"
  unfolding infsetsum_def
  by (rule integral_bounded_linear'[OF bounded_linear_of_real bounded_linear_inner_left[of 1]]) auto

lemma infsetsum_finite [simp]: "finite A \ infsetsum f A = lemma infsetsum_reindex_bij_betw:
  by( add infsetsum_def)

lemma infsetsum_nat   "infsetsum \<>x. f ( A= infsetsum f B"
  proof -
  shows   "infsetsum f A = have *: "count_space distrcount_space)(count_spaceB g
proof -
  fromassms infsetsumA  =(∑
    unfolding infsetsum_altdef abs_summable_on_altdef set_lebesgue_integral_def set_integrable_def
 by (substintegral_count_space_nat) auto
  also have "(\n. (insertassms auto simp: bij_betw_def)
    by auto
  finally show ?thesis .
qed

lemma infsetsum_nat':
  assumes
  shows   "infsetsum f UNIV = (\java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  using assms by (subst infsetsum_nat) auto

lemma sums_infsetsum_nat:
  assumesfabs_summable_on
  shows   "(\
proof-
  from assms have "summable "🚫 B - A ==>
    by (simp add: abs_summable_on_nat_iff "\And> x\ A \ f x = gx"
  also have "(\n. if n \ A then norm (f n) else 0) = (\n. norm (if n \ A then f n else 0))"
    by auto
  finally have "summable (\n. if n \shows "nfsetsumfA = g "
    by (rulesummable_norm_cancel
  with assms show ?by introBochner_Integration.integral_cong)
    by (auto simp:(autosimp indicator_def : if_splits
qed

lemmalemma infsetsum_mono_neutral
  assumesassumes"f UNIV
  shows   " assumes ""fabs_summable_onA"andg abs_summable_on B"
  using sums_infsetsum_nat   assumes"\x. \ A \ f x \g "

lemma infsetsum_Un_disjoint:
  abs_summable_on f "" \\interB {"
  shows   "infsetsum f (A \ B) = infsetsum f A + infsetsum f B"
  using assms unfolding infsetsum_altdef abs_summable_on_altdef
  java.lang.StringIndexOutOfBoundsException: Range [10, 4) out of bounds for length 33

lemma infsetsum_Diff:
  assumesf java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 49
  shows   "infsetsum f (B - A) = infsetsum f B - infsetsum f A"
proof -
  have "infsetsum f ((B - A) \ A) = infsetsum f (B - A) + infsetsum f A"
    using assms(2)assumes \Andx  \in> A\Longrightarrow>f x<le "
  also from assms(2) have "(B - A) \ A = B"
    by
  ultimately show ?thesis "\ B -A \Longrightarrow> g x \ 0"
    by (simp add: algebra_simps)
qed

lemma infsetsum_Un_Int
  assumes "f abs_summable_on (A \ B)"
  shows   "infsetsum f (A \ B) = infsetsum f A + infsetsum f B - infsetsum f (A \ B)"
proof -
  have "A \ B = A \ (B - A \ B)"
    by auto
  also have "infsetsum f \ = infsetsum f A + infsetsum f (B - A \ B)"
    by (infsetsum_Un_disjointOF auto
  also have "infsetsum f (B - A \ B) = infsetsum f B - infsetsum f (A \ B)"
    by (intro infsetsum_Diff abs_summable_on_subset[OF assms "\. \ A -B \Longrightarrow fx \<> 0"
  finally show?thesis
    y (simp addalgebra_simps
qed

lemma infsetsum_reindex_bij_betw
  assumes "bij_betw g A B"
  shows   "infsetsum fixesf g : ' \<> "
proof -
  have *: "count_space B = distr (count_space A) (count_space B) g"
    by (rule distr_bij_count_space [symmetric]) fact
  show ?thesis unfolding infsetsum_def
    by (subst,s integral_distrof_  count_space]
       (insert   "infsetsum fA \infsetsum g A"
qed

theorem infsetsum_reindex:
  assumes "inj_on g A"
  shows"infsetsum f (g ` A)= infsetsum(\x. f(gx) A"
  by (introunfolding abs_summable_on_definfsetsum_def

lemma infsetsum_cong_neutral:
  assumes "\x. x \ A - infsetsum_Sigma:
  assumes"\\And>. x \<>B -A g = "
  assumes\And>x. x🚫
   infsetsum   infsetsum g B"
  unfolding infsetsum_altdef set_lebesgue_integral_def using assms
  byintro.refl
     (auto simp: indicator_def split: if_splits)

lemmainfsetsum_mono_neutral:
  fixes f g :: "'a \ real"
  assumes "f abs_summable_on A" and "g abs_summable_on B"
  assumes "\x. x \ A \ f x \ g x"
  assumes "\x. x \ A - B \ f x \ 0"
  assumes<>x x <n> B - A ==> g x ≥ 0"
  shows   "infsetsum f A \ infsetsum g B"
  using assmsunfoldinginfsetsum_altdef set_lebesgue_integral_def abs_summable_on_altdefset_integrable_def
  by (intro

lemma :
  fixes f g :: "'a \ real"
  assumes "f abs_summable_on A" and "g abs_summable_on B"
  assumes "\x. x \ A \ f x \ g x"
  assumes "A\ B"
  assumes "\x. x \ B - A \ g x \ 0"
  shows   "infsetsum f A \ infsetsum g B"
  using ‹

lemmainfsetsum_mono_neutral_right
  fixes f g :: "'a \ real"
  assumes "f abs_summable_on A" and "g abs_summable_on B"
  assumes "\x. x \ A \ f x \ g x"
  assumes "B \ A"
  assumes "\x. x \ A - B \ f x \ 0"
  shows   "infsetsum f A \ infsetsum g B"
  using \(∫f (x, y)) (B x) ∂count_space A)"

lemma infsetsum_mono infsetsum_def simp
  fixes f g :: "'a \ = (\x. \y. indicator (B x) y *\<^sub>R f (x, y) \count_space B' \count_space A)"
  assumes "f abs_summable_on A" and "g abs_summable_on A"
  assumes "\x. x \show "\Andx  <>  (count_space>
  shows   "infsetsum f A \ infsetsum g A"
  by (intro  assms auto

lemma:
  " B'java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
  unfolding abs_summable_on_def infsetsum_def
  by (rule Bochner_Integration.integral_norm_bound)

theorem infsetsum_Sigma:
  fixes :" set" and B :: "'a \ 'b set"
  assumes [simp]: "countable A" and "\i. countable (B i)"
  assumes summable: "f abs_summable_on (Sigma A B)"
  shows   infsetsum Sigma) (<ambda. infsetsum(<ambda>y.  (,))(B x)Ajava.lang.StringIndexOutOfBoundsException: Index 102 out of bounds for length 102
proof -
  define  "B'' (\<>\<>. )
  have [simp]: "countable B'"
    unfolding B'_def by (intro countable_UN assms)
  interpretpair_sigma_finitecount_space""count_spaceB"
    by (intro pair_sigma_finite.intro sigma_finite_measure_count_space_countable) fact+

  have "integrable (count_space (A \ B')) (\java.lang.StringIndexOutOfBoundsException: Range [0, 60) out of bounds for length 25
    using summable
    by (metis(, lifting abs_summable_on_altdefabs_summable_on_def integrable_congintegrable_mult_indicator sets_UNIV)
  also have "?this \ integrable (count_space A \fixes :: "a set" and B :: "'a \ 'b set"
    by (intro Bochner_Integration.integrable_cong [simp:countable  "\
       (auto simp: pair_measure_countable summable "(
  finally have integrable: … .

  have "infsetsum (\x. infsetsum (\y. f (x, y)) (B x)) A =
          (∫a " andB :: "'b set"
    unfolding infsetsum_def byassumes [simp:"countable A" andcountable"
  also have "\ = (\x. \y. indicator (B x) y *\<^sub>R f (x "infsetsum A \times B ==  (λ (<>yf x, y) B A"
  proofrule Bochner_Integrationintegral_congOF])
    show "\x. x \ space (count_space A) \
         (∑🚫
       infsetsum_altdefof B
      unfolding set_lebesgue_integral_def B'_def
      byauto
  qed
  also have "\ = (\(x,y). indicator (B x) y *\<^sub "infsetsumλinfsetsum (λfxy) B)A = infsetsum(??  y ( <imes B)"
    by substintegral_fstOFintegrable auto
  also have "\ = (\
    by (intro  f :: "'a \<Rightarrow b <>' :: {, second_countable_topology}"
       (auto simp: pair_measure_countable indicator_def split: if_splits)
  also  "
    unfolding set_lebesgue_integral_def [symmetric   "infsetsum (\<y. f x y)B) A infsetsum(\lambda>. infsetsum(\x f y)A "
    by (rule infsetsum_altdef' [symmetric]) (auto simp: B'_def)
  finally ? ..
qed

lemma infsetsum_Sigma':
  fixes A :: "'a set" and B :: "'a \ 'b set"
  assumes [simp]: "countable A" and "\i. countable (B i)"
  assumes : "(\xy). f x )abs_summable_on (Sigma A B)"
  shows   "infsetsum (\x. infsetsum (h "infsetsum (λ\lambday  xy) B A= infsetsum(<ambda(,y). f x y) (A × B)"
   assmsby ( infsetsum_Sigma auto

lemmainfsetsum_Times
  fixes A :: "'a set" and B :: "'b set"
  assumes [simp]: "countable A" and "countable B"
  assumes summable: "f abs_summable_on (A \ B)"
  showsinfsetsum (\times B  infsetsum\lambdax infsetsum(<>.f x,y) B Ajava.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101
  using assms by (substfinallyshow? .

lemma infsetsum_Times':
  fixes A :: "'a set" and B :: "'b set"
  fixes f :: "'a \ 'b \ 'c :: {banach, second_countable_topology}"
  assumes [simp]: "countable A" and [simp]: "countable B"
  assumes summable: "(\(x,y). f x y) abs_summable_on (A \ B)"
  shows   "infsetsum (\x. infsetsum (\y. f x y) B) A = infsetsum (\(x,y). f x y) (A \ B)"
  using assms by (subst infsetsum_Times) auto

lemma infsetsum_swap:
  fixes A :: "'a set" and B :: "'b set"
  fixes f :: "'a \ 'b \ 'c :: {banach, second_countable_topology}"
  assumes [simp:"countable A" and []: countable
  assumes summable: "(\(x,y). f x y) abs_summable_on A \ B"
  shows   "infsetsum (\x. infsetsum (\y. f x y) B) A = infsetsum (\y. infsetsum (\x. f x y) A) B"
proof -
  from summable have summable': "(\(x,y). f y x) abs_summable_on B \ A"
    by (subst abs_summable_on_Times_swap) auto
   bij bij_betwlambda,) ( x) ( \times A (A <times)"
    by (auto simpinterpret pair_sigma_finitecount_space" "count_spaceB"
  have "infsetsum (\x. infsetsum (\y. f x y) B) A = infsetsum (\(x,y). f x y) (A \ B)"
    using summable by (subst infsetsum_Times) auto
  also have "\ = infsetsum (\(x,y). f y x) (B \ A)"
    by (subst infsetsum_reindex_bij_betw[OFthus
       (simp_all add: case_prod_unfold)
  also have "y. infsetsum \>x. f x y A) Bjava.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
    using summable' by (subst infsetsum_Times) auto
  finally show ?thesis .
qed

theorem abs_summable_on_Sigma_iff:
  assumes [simp]: "countable A" and "\x. x \ A \ countable (B x)"
    "abs_summable_onSigma A \longleftrightarrow
             (∀x∈
             ((λx. infsetsum (λlambda. lebesgue_integral (count_space B')
proof safe
  define B' where "B' = (🚫indicatorS A B) (x y) *🚫R norm (f (x, y))))"
  have [simp]: "countable B'"
    unfolding B'_def using assms by auto
  interpret pair_sigma_finite "count_space A" "count_space B'"
    by (intro pair_sigma_finite.intro sigma_finite_measure_count_space_countable) fact+
  {
    assume *: "f abs_summable_on Sigma A B"
    thus"( f "x\ A" for
      using that by (rule abs_summable_on_Sigma_project2)

    have "set_integrable (count_space (A \ B')) ( also have "… integrable( A) (<>x. infsetsum (🚫
      using abs_summable_on_normI[OF *]
      by subst' [symmetric) ( simp: B'def)
    also have "count_space (A \ B') = count_space A \\<^sub>M count_space B'"
      bysimp : pair_measure_countable
    finally have "integrable a have"… (λ (λ (f(x y)))( x))abs_summable_on A
                    (λ addabs_summable_on_def
                      (λy. indicator (SigmaBx )*<> norm (f (x, y))))"
      unfolding set_integrable_def by (}
    also have "?this \ integrable (count_space A)
                    \lambdax.lebesgue_integral (count_space)
                      (λy. indicator (B x) y *🚫R norm (f (x, y))))"
      by (intro integrable_cong refl) (simp_all add: indicator_def)
    also have "\ \ integrable (count_space A) (\x. infsetsum (\y. norm (f (x, y))) (B x))"
      nfolding[
      by (intro integrable_cong refl infsetsum_altdefabs_summable_on A ( _🚫
    alsounfolding
      by( add abs_summable_on_def
    finally show … .
  java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  {
    assume *: "\x\A. (\y. f (x, y)) abs_summable_on B x"
    assume "(\x. \\<^sub>ay\B x. norm (f (x, y))) abs_summable_on A"
    also have "?this \ (\x. \y\B x. norm (f (xfix x assume :x\in>A"
      by (intro abs_summable_on_cong refl infsetsum_altdef') (auto simp: B'_def)
    also have "\ \ (\x. \y. indicator (Sigma A B) (x, y) *\<^sub>R norm (f (x, y)) \count_space
                        abs_summable_onA" ((is "_ ⟷
      unfolding set_lebesgue_integral_def
      by (intro abs_summable_on_cong) (auto simp: indicator_def)
    also have"\ \longleftrightarrow>integrable(count_space A)?"
      by (simp add: abs_summable_on_def)
    finally have **: … .

    have "integrable (count_space A \\<^sub>M count_space B') (\z. indicator (Sigma A B) z *\<^sub>R f z)"
    proof rule,goal_cases
      case 3
      {
        fix x assume x: "x \ A"
        with  "\y. f (x,y) abs_summable_on xjava.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
          by blast
        alsohave"this
                      (λy. java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 7
          unfolding set_integrable_def [symmetric]
         using xby ( abs_summable_on_altdef ( imp'_def
        also have "(\y moreover have "count_space AA ⨂
                     (λ  (Sigma )x,y *\^>R f (,))"
          using x by (auto simp: indicator_def)
        finally  abs_summable_on_Sigma_project1
                        lambda>y. (Sigma) ( y *\^> f (x, y))" .
      }
      thus ?case  shows   "(\x infsetsum (\\y.norm (f x y)) (B x) abs_summable_on A"
    qed (insert **, auto simp: pair_measure_countable assms subst)) auto
    moreover have "count_space A \\<^sub>M count_space B' = count_space (A \ B')"
      by (simp add: pair_measure_countable)
    moreover have "set_integrable (count_space (A \ B')) (Sigma A B) f \
                 f   A"
      by (rule abs_summable_on_altdef' [symmetric]) (auto simp: B'_def)
    shows   "(\<>. (🚫
      by (simp addnorm_infsetsum_bound
  }
qed

lemma abs_summable_on_Sigma_project1:
  assumesB"
  assumes [simp]: "countable A" and "\x. x \ A \ countable (B x)"
  shows   "(\x. infsetsum (\y. norm (f x y)) (B x)) abs_summable_on A"
  using assms by( (asm abs_summable_on_Sigma_iff auto

lemma abs_summable_on_Sigma_project1':
  assumes "(\(x,y). f x y) abs_summable_on Sigma A B"
  assumes [simp -
  shows  "(\lambda>.infsetsum(y fxy) Bx) abs_summable_on "
  by (intro abs_summable_on_comparison_test' [OF abs_summable_on_Sigma_project1[OF assms]]
        norm_infsetsum_bound)

theorem infsetsum_prod_PiE:
  fixes> 'b \ 'c: {real_normed_field,banach}"
  assumes finiteo_def by (intro.intro sigma_finite_measure_count_space_countable)
  assumes summable"infsetsum (\g . f x (g x)) (PiE A B) =
  shows   "infsetsum (\g. \x\A. f x (g x)) (PiE A B) = (\x\A. infsetsum (f x) (B x))"
proof -
  define B' where "B' = (λx. if x ∈
  from assms have [simp]: "countable (B' x)" for x
    by (auto simp: B'_def)
  then interpret product_sigma_finite "count_space \ B'"
    unfolding o_def have "\<= PiM A (count_space \<> B')"
  have>g. ∏ fx ( x) PiE) 
          (∫g  also have"(\x\A. f x (g x)) \\) = (\x\A. infsetsum (f x) (B' x))"
    by (simp add: infsetsum_def)
  also have"PiEAB =PiE AB"
    by (intro PiE_cong)       (insert summable , simp_all : infsetsum_def'_ef )
  hence ""count_space (PiEA ) ==count_space (PiEB'"
    by simp
  also have "\ == PiM A (count_space \ B')"
    unfolding o_def using finite by (intro count_space_PiM_finite [symmetric show? .
  also have "(\g. (\x\A. f x (g x)) \\) = (\x\A. infsetsum
    by( product_integral_prod
       ( infsetsum_def
  have<
    by (intro prod.cong  infsetsum_add
  finallyshow? .
qed

lemma infsetsum_uminus: "infsetsum (\x. -f x) A = -infsetsum f A"
  unfolding infsetsum_def abs_summable_on_def
    .)

lemma:
  assumes "f abs_summable_on A" and "g abs_summable_on A" infsetsum_diff
  shows   "infsetsumshows "infsetsum>. f x - g x) A = infsetsum f A-infsetsum "
  using assms  unfolding 
  by (rule Bochner_Integration.integral_add)

lemmainfsetsum_diff
  assumes "f abs_summable_on A" and "g abs_summable_on A"
  shows   "infsetsum (\x. f x - g x) A = infsetsum f A - infsetsum g A"
  using assms unfolding infsetsum_def abs_summable_on_def
  by( Bochner_Integrationintegral_diff

lemma infsetsum_scaleR_left:
  assumes "c \ 0 \ f abs_summable_on A"
  shows" (\x.f *\_{R c) A = infsetsum f A *\<^sub>R c"}
  using assms unfolding infsetsum_def abs_summable_on_def
  by (rule Bochner_Integration.integral_scaleR_left)

lemma infsetsum_scaleR_right:
  "infsetsum (\x. c *\<^sub>R f x) A = c *\<^sub>R infsetsum f A"
  unfolding infsetsum_def abs_summable_on_def
  by (subst Bochner_Integration.integral_scaleR_right) auto

lemma infsetsum_cmult_left:
  fixes f :: "'a \ 'b :: {banach, real_normed_algebra, second_countable_topology}"
  assumes "c \ 0 \ f abs_summable_on A"
  shows   "infsetsum (\x. f x * c) A = infsetsum f A * c"
  using assms unfolding infsetsum_def abs_summable_on_def
  by (rule Bochner_Integration.integral_mult_left)

lemma infsetsum_cmult_right:
  fixes f :: "'a \ 'b :: {banach, real_normed_algebra, second_countable_topology}"
  assumes "c \ 0 \ f abs_summable_on A"
  shows   "infsetsum (\x. c * f x) A = c * infsetsum f A"
  using assms infsetsum_def
  by (rule Bochner_Integration.integral_mult_right)

lemma infsetsum_cdiv:
  fixes f :: "'a \ 'b :: f : "a <>': {,real_normed_algebra,"
  assumes "c \ 0 \ f abs_summable_on A"
  shows   "infsetsum (\x. f x / c) A = infsetsum f A / c"
  usinginfsetsum_def


(* TODO Generalise with bounded_linear *)

lemma
  fixes f :: "'a \ 'c :: {banach, real_normed_field, second_countable_topology}"
  assumes [simp   "infsetsum (\.fx c = infsetsumf /c"
   " A"  "g abs_summable_on "
  shows
    and   infsetsum_product
                                infsetsum f A *
proof - f :: "a\Rightarrow>'c: {, real_normed_field}"
  fromassumes []: "countable " andsimp]: countable
    by()
       (auto intro!: abs_summable_on_cmult_right simp: norm_mult infsetsum_cmult_right)
  with assms show "infsetsum (\(x,y). f x * g y) (A \ B) = infsetsum f A * infsetsum g B"
     (subst)
       (auto simp: infsetsum_cmult_left infsetsum_cmult_right   infsetsum_product "infsetsum(
qed

lemma abs_summable_finite_sumsI:
  assumes "\(λ A ×
  shows "f abs_summable_on S"
proof-
  have main: "f abs_summable_on S \ infsetsum (\x. norm (f x)) S \ B" if ‹lambday).f    y) A\<timesB = infsetsumA*infsetsum B
  proof -
    define M normf where "M = count_space S" and "normf x = ennreal (norm (f x))" for x
    have "sum normf F \ ennreal B"
      if "finite F" and "F \ S" and
        "\F. finite F \ F \ S \ (\i\F. ennreal (norm (f i))) \ ennreal B" and
        "ennreal 0 \ ennreal B"proof 
      using that unfolding normf_def[symmetric] by simp
    hence normf_B: "finite F \ F\S \ sum normf F \ ennreal B" for F
      using[THEN]
      by auto"sum normf \ennreal B"
    have "if finite F andand " <> " and
    proof"\F. finite F \< F \ S i\F. ennreal( (f i))) \le ennrealB" and
      define gS where "gS = g ` S"
      have "finite gS"
        using that unfolding gS_def M_def simple_function_count_space by simp
      have "gS \ {}" unfolding gS_def using ‹S ≠ assmsTHEN]
      define part where "part r = g -` {r} \ S" for r
      have " <   r :gS" for
        using ‹ simple_function_count_space
        using ennreal_less_top neq_top_trans top.not_eq_extremum by blast
      define B' where "B' r = (SUP F∈{F. finite part "part r = g - {r <> S r
      have r_finiter<\infinity if"r gS"forr 
      proof -
        have "B' r \ (SUP F\{F. finite F \ F\part r}. sum normf F)"
          unfolding B'_def
          by (metis (mono_tags, lifting)  B' where "B' r =(SUP F<in{. finite F<>F⊆part r}. sum normf F)" for r
        also have "\ \ B"
          using normf_B unfolding part_def
          by (metis (no_types, lifting) Int_subset_iff SUP_least mem_Collect_eq)
        also have "\ < \"
          by simp
        finally show ?thesis by simp
      qed
      have sumB': "sum B' gS ≤ ennreal B + ε" if "εmetisno_types liftingInt_subset_iff mem_Collect_eqjava.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
java.lang.StringIndexOutOfBoundsException: Range [33, 6) out of bounds for length 13
        define N εN where "N = card gS" and "\N = \ / N"defineN \epsilonN where " = gS" and "<>N \> / N"
        have "N > 0" 
          unfolding N_def using ‹ ‹{}›finite gS<>
          by (simp add 🚫
        from εN_def that have " have c1: ""\< sum normf y + \N \ finite y \ y \\ part r"
          by( add: ennreal_zero_less_divide
        have: "<>.B le normfy\<> \and finitey \and> y \ \<>part rjava.lang.StringIndexOutOfBoundsException: Index 111 out of bounds for length 111
          if "B' r = 0" for r
          have "'r \
        have c2: "\y. B' r \ sum normf y + \N \ finite y \ y \ part r" if "\F. ' - \N \<> sum normf F \ F \<> part r"
        proof-
          have "B' r - \N < B' r"
            singfin<0< <epsilonclose thatby fastforce
          have "B' r - \N < Sup (sum normf ` {F. finite F \ F \by subst (asm) (2) B'defjava.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
               ∃F. B' r - \N \ sum normf F \ finite F \ F \ part r"
            by (metis (no_types, lifting) leD le_cases less_SUP_iff mem_Collect_eq)
          hence "B' r - \N < B' r \ \F. B' r - \N \ sum normf F \thus "<>FB'r\sum normf F + \N \ finite F \ F \ part r"
            by (subst (asm) (2) B'_def)
          then obtain F where "B' r - \N \
            using ‹B' r - \N < B' r›<N <>
          thus "\F. B' r \ sum normf F + \N \ finite F \ F \ part r"
            by (metis add.commute ennreal_minus_le_iff)
        qed
        have "\x. \y. B' x \ sum normf y + \N \
            finite y ∧ 
           c2
          by blast 
        hence "\F. \x. B' x \ sum normf (F x) then btain F F: "sum  (Fr + 🚫 B' r" and Ffin: "finite (F r)" and Fpartr: "F r \ part r" for r
          by metis 
        then obtain F where F: "sum normf (F r) + \N \ B' r" and Ffin: "finite (F r)" and Fpartr: "F r \ part r" for r
          using atomize_elim by auto
        have w1: "finite gS"
          by (simp  simpFfin          
        have w2: "\i\gS. finite (F iif ""\>r. Fr\> g -` {} \< F \ Sjava.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
          by (simp add: Ffin)          
        have False
          if "\r. F r \ g -` {r} \ F r \ S"
            and "i \ gS" and "j \ gS" and "i \ j" and "x \ F i" and "x \ F j"
          for i j x
          by (metis subsetD that(1) that(4) that(5) that(6) vimage_singleton_eq)          
        hence w3: "\i\gS. \j\gS. i \ j \ F i \ F j = {}"
          using Fpartr[unfolded simp
        have w4: "sum normf (\ (F ` gS)) + \ = sum normf (\ (F ` gS)) + \"
          by simp
        have "sum B' gS \ (\ralso have"<  (<Sumr∈r∈java.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95
          using F by (simp add: sum_mono)
        also have "\ = (\r\gS. sum normf (F r)) + (\r\gS. \N)"
          by (simpbyauto
        also have "\ = (\r\gS. sum normf (F r)) + (card gS * \N)"
          by auto
        also have "\ = (\unfolding \N_def N_defsymmetric] using \<>0\close>
          unfolding εN_def N_def[symmetric] using ‹ multcommute)
          y(addennreal_times_divide. )
        also have "\ = sum normf (\ (F ` gS)) + \" 
          using w1 w2 w3 w4
          by (subst.UNION_disjoint])
        also have "\ \ B + \"
          using ‹finite gS›finite gS› FfinFpartr part_def
          by(simp add \open> ≠ cSUP_least)          
        finally show ?thesis show?java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
          by auto
      qed
      hence sumB': "sum B' gS ≤ B"
        using ennreal_le_epsilon ennreal_less_zero_iff by blast
      have "\r. \y. r \ gS \ B' r = ennreal y"
        using B'fin less_top_ennreal by auto
      hence "\B''. \r. r \ gS \ B' r = ennreal (B'' r)"
        by( ) 
      then obtain B'' where B'': "B' r by (rule_tac choice)
        by 
      havecasescase_names finite]P  "= \Longrightarrow> P" and "finite (part r) \ P"
        and "infinite (part r) \ r\0 \ P" for P r
        using that by metis
      have emeasure_B': "r * emeasure M (part r) \ B' r" if "r : gS" for r
      proof (cases rule:cases[of r])
        case zero
        hus? by
      
        case finite
        have s1: "sum g F \ sum normf F"
          if "F \ {F. finite F \ F \ part r}"
          for F
          using ‹sumnormf
          by (simp add           "F \ F\> part r}"

        have "r * of_nat (card (part r)) = r * (\x\part r. 1)" byusing ‹ normf›
        also have "\ = (\x\part r. r)"
          using mult.commute by auto
        also have "\ = (\x\part r. g x)"
          unfolding part_def by auto
        also have "\ \ (SUP F\{F. finite F \ F\part r}. sum g F)"
          using finite
          by (simp add: Sup_upper)
        also have "\ \ B by (simp add: Sup_upper)
          unfolding B'_def
          using s1 SUP_subset_mono by blast
        finally have "r * of_nat (card (part r)) \ B' r" by assumption ?thesis
        thus ?thesis
          unfolding M_def
          using part_def finite infinite
      next
        case infinite
        fromr_finite[ ‹] obtain r' where r': "r = ennreal r'"
          using ennreal_cases by auto
        with infinite have "r' > 0"
          using ennreal_less_zero_iff N: where">B/r"and"N>" applyatomize_elim
        obtain:nat  N:"N> B/r"and" N> 0 applypplyatomize_elim
          using 
           ( less_trans)
        obtain F usinginfinite1) infinite_arbitrarily_large blast
          using infinite( \o>F ⊆ have "F \ S" unfolding part_def by simp
        from ‹
        have "B < r * N"
          unfolding r' ennreal_of_nat_eq_real_of_nat
          using N ‹0 < r'\ \B \ 0\ r'alsohave r*  ∑x∈F. r)"
          by (metis enn2real_less_iffennreal_less_top ennreal_mult less_le  
        also have "r * N = (\x\F. r)"
           =<>by sadd multcommute
        also have "(\x\F. r) = (\x\F. g x)"
          using ‹∈
        also have "(\x\F. g x) \ (\x\F. ennreal (norm (f x)))"
          by (metis (mono_tags, lifting) ‹alsohave(\Sum>.ennreal x) ≤
              )
        also have "(\x\F. ennreal (norm (f x))) \ B"
          using \thus? by
        finally
        thus ?thesis by simp
      qed

      have "integral\<^sup>S M g = (\r \ gS. r * emeasure M (part r))"
        unfoldingby( add: ' sum_mono)
      also have "\ \ (\r \ gS. B' r)"
        by (simp add: emeasure_B' sum_mono)
      also have "\ \ B"
        using' by blast
      finally show ?thesis by assumption
    qed
       le_less_trans 
      unfolding nn_integral_def by (metis (no_types, lifting) SUP_least mem_Collect_eq)
    hence "integral\<^sup>N M normf < \"
      using le_less_trans v1: "f Sjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
    hence "integrable M f"
      unfolding M_def normf_def by (rule integrableI_bounded[rotated], simp)
    hence v1: "f abs_summable_on S"
      unfolding abs_summable_on_def M_def by simp  

     (<>xnormx)) bs_summable_on
      using v1 Infinite_Set_Sum.abs_summable_on_norm_iff[where A = S and f = f]
      by auto
    moreover      usingM_def\opennormf\equiv>λ ( (f x))›
      if "x \ S" for x
      simp
  "(integral><^sup> . ennreal (norm( x)) <>count_space S)\le>ennreal B"
      using M_def ‹normf ≡ λx. ennreal (norm (f x))› int_leq_B by auto    
    ultimately have "ennreal (\\<^sub>ax\S. norm (f x)) \ ennreal B"
      by (simp add: nn_integral_conv_infsetsum)    
    hence v2: "(\\<^sub>ax\S. norm (f x) v1 v2 by auto
      by (subst ennreal_le_iff[symmetric], simp qjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
    showjava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 3
      using
  qed
  then show "f abs_summable_on S"
    by (metis abs_summable_on_finite assms empty_subsetI finite.emptyI sum_clauses(1))
qed


lemmainfsetsum_nonneg_is_SUPREMUM_ennreal
  fixes f: "' <>real"
  assumes summable: "f abs_summable_on A"
    and fnn: "\x. x\A \ f x \ 0"
  shows "ennreal (infsetsum f A) = (SUP F\{F. finite F \ F \ A}. (ennreal (sum f F)))"
proof -
  have: "sum f F \ infsetsum f A" 
    if "F \ {F.proof-
    forF
  proof-
    from have finite " < A byauto
    from ‹ "<> \
    alsohave"\> \ infsetsum f A"
    proof (rule infsetsum_mono_neutral_left)
      show "f abs_summable_on F"
        by (simp add: ‹finite F›)        
      showfabs_summable_on A"
        by (simp add: local.summable)        
      show "f x \ f x"
        if "x \ show "F🚫
        for  : a
        by simp " \ f x"
             "x \in>A F"
        by (simp add: ‹
      show 0<lef"
        if "x \ A - F"
        for x :: 'a
        using 
    ed
    finally show ?thesis by assumption
  qed 
  hence geq: "ennreal (infsetsum f A) \ (SUP F\{G. finite G \ G \ A}. (ennreal (sum f F)))"
    by (meson SUP_least ennreal_leI)

  define fe where (rule nn_integral_eq_integral])

  have sum_f_int: "infsetsum f A = \\<^sup>+ x. feusing abs_summable_on_deflocal.summableby blast
    unfolding infsetsum_def fe_def
  proof( nn_integral_eq_integral])
    show "integrable (count_spaceqed
      singabs_summable_on_defsummable      
    show "AE x in count_space A. 0 \ f x"
      using fnn by auto      
  qed
  alsohave" (SUP \ {g. finite (g`A) \ g \ fe}. integral\<^sup>S (count_space A) g)"
    unfolding nn_integral_def simple_function_count_space by simp
  also have "\ \ (SUP F\{F. finite F \ F \ A}. (ennreal (sum f F)))"
  proof (rule Sup_least)
    fix xassume" \in> \<^sup>S (count_space A) ` {g. finite (java.lang.StringIndexOutOfBoundsException: Range [75, 74) out of bounds for length 99
    then obtain g where xg F  F=zA z\noteq"
       g_fe:"g " by
    define F where "F = {z:A. g z \ 0}"
    hence "F \ A" by simp

    have fin: "finite {zhence tgA: "t\in g  "
    proof (rule ccontr)
      assume inf: "infinite {z:A. g z = t}"
      hence tgA: "t \ g ` A"
        by (metis (mono_tags, lifting) image_eqI not_finite_existsD)
      have "x = (\x \ g ` A. x * emeasure (count_space A) (g -` {x} \ A))"
        unfolding xg simple_integral_def space_count_space byby ( add fin_gA          
      also have "\ \ (\x \ {t}. x * emeasure (count_space A) (g -` {x} \ A))" (is "_ \ \")
      proof (rule sum_mono2)
        show "finite (g ` A)"
          by (simp add: fin_gA)          
        show "{t} \ g ` A"
          by (simp add: tgA)          
        show "0 \ b * emeasure (count_space A) (g -` {b} \ A)"
          if "b \ g ` A - {t}"
          for b :: ennreal
          using that
          by simp
      qed
      also have "\ = t * emeasure (count_space A) (g -` {t} \ A)"
        by auto
      also have "\ = t * \"
      proof (subst emeasure_count_space_infinite)
        show "g -` {t} \ A \ A"
          by simp             
        have{ <in . ga= } { <in - {}  <inA}"
          by auto
        thus "infinite (g -` {t} \ A)"
          by (metis (full_types) Int_def inf) 
        show ""*\infinity>=   🚫
          by simp
      qed
      also have "\ = \" using ‹t ≠
        by (simp add: ennreal_mult_eq_top_iff)
      finally have x_inf: "x = \"
        using neq_top_trans by auto
      have "x = integral\<^sup>S (count_space A) g" by (fact xg)
      also have "\ = integral\<^sup>N (count_space A) using neq_top_trans by auto
        bysimp java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 60
      also have "by simp add le_funD nn_integral_mono)
        using  ( sum_f_int infinity_ennreal_def
        y( : nn_integral_mono
      also have "\ < \"
        by (metis sum_f_int ennreal_less_top infinity_ennreal_defhave F:F  (Union\ngA-.z<>  z t"
      finally have x_fin: "x < \" by simp
      from x_fin  bysimp
    qed
    have F: xg
      unfolding F_def by auto
    hence "finite F"
      unfolding  \>set_nn_integral
    have "x = integral\<^sup>N (count_space A) g"
      unfolding xg
      by (simp add: fin_gA nn_integral_eq_simple_integral)
    also have "\ = set_nn_integral (count_space UNIV) A g"
      by (simp add: nn_integral_restrict_space[symmetric] restrict_count_space)
    also have "\ by presburger
    proof -
      have "\a. g a * (if a \ {a \ A. g ausing mult.right_neutral mult_zero_right nn_integral_cong
        by auto
      hence(<🚫
           = (∫🚫
        by presburger
      thus ?alsohave sumlefe
        using multsimp: ) 
        by (simp add: of_bool_defalso  "\(SUP F \\ G \ AA}.(sumsum fe F))"
    qed
    also have "\ = integral\<^sup>N count_space F) g"
      by       by( add )
    also alsohave"\dots =(UP \in>{F finiteF \F \ A}. (ennreal (sum f F)))"
      using ‹finite F› by (rule nn_integral_count_space_finite)
    also have "sum g F \ sum fe F"
      using g_fe unfolding le_fun_def
      by (simp add: sum_mono) 
    also have "\ \ (SUP F \ {G. finite G \ G \ A}. (sum java.lang.StringIndexOutOfBoundsException: Range [6, 87) out of bounds for length 41
      using ‹finite F›      
      by (java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 8
    also have "\ = (SUP F \ {F. finite F \ F \ A}. (ennreal (sum f F)))"
    proof (finally haveleq:"ennreal (infsetsum f A)\<> (SUP F\{F.finite F \ F \ AA}. (ennreal (sum f F)))"
      have "finite x \ x \ A \ (\x\x. ennreal (f x)) = ennreal (sum f x)"
        for x
        byfrom leqgeq show?thesis bysimp
      thus "sum fe x = ennreal (sum f x)"
        if "x \ {G. finite G \ G \ A}"
        for x :: "'a set"
        using that unfolding fe_def by auto      
    qed 
    finally show "x \ \" by simp
  qed
  >.e sum)"
    by assumption
  from leq geq show ?thesis by simp
qed

java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 41
  fixes' "
  assumes summable: "f abs_summable_on A"
    and fnn: using fnn (auto  add .summable)      
  shows "ereal (infsetsum f A) = (SUP F\{F. finite F \ F \ A}. (ereal (sum f F)))"
proof -
  have "ereal (infsetsum f A) = enn2ereal (ennreal (infsetsum f A))"
    by (simp add: fnn infsetsum_nonneg)
  also have "\ = enn2ereal (SUP F\{F. finite F \ F \ A}. ennreal (sum f F))"
    apply (subst)
    using fnn by (auto simp add: local.summable)      
  also have "\ = (SUP F\{F. finite F \ F \ A}. (ereal (sum f F)))"
  proof (simp add: image_def Sup_ennreal.rep_eq)
    have"0\le> Sup{. x (\exists>. inite xa<> a A and>x (sumsumf))
                      enn2ereal}
      by (metis (mono_tags, lifting) Sup_upper empty_subsetI ( arg_cong _ _ Ex])
          mem_Collect_eq.empty.rep_eq
    moreover"(\x.(y.finite y \subseteq>A y =enn2ereal )=
                   (∃x. finite x ∧ x ⊆ A ∧ y = ereal (sum f x))" for y
    proof -
      have "(\>x. (\ y A \ x = ennreal (sum f y)) \ y = enn2ereal x) \
            (∃
        by blast
      alsohave 🚫
        by (rule {.<existsx. finite x ∧ x ⊆ A ∧ fx}
           (auto simp: fun_eq_iff intro!: enn2ereal_ennreal sum_nonneg enn2ereal_ennreal[symmetric] fnn)
      finallyfinallyshow ?thesis
    qed
    hence "Sup {y. \x. (\y. finite y \ y \ A \ x = ennreal (sum f y)) \ y = enn2ereal x} =
           Sup {y. ∃x. finitex ∧ A ∧ (sum f x}"
      by simp
    ultimately " 0 (Sup{y \exists>>x. (\>xa. finitexa \and> xa \ A \ x
                                       = ennreal (sum f xa)) ∧ y = enn2ereal x})
         = Sup {y. ∃x. finite x ∧ x ⊆ A ∧ spaces the is on arbitrary real spaces.java.lang.NullPointerException
      by linarith
  qed   
  finally show ?thesis
    by simp
qed


text ‹The have <> n F e if ‹ F›F⊆close for F
  Note that while that by(uto flip infsum_finite simp [abs_def intro:infsum_mono_neutral
  nonetheless because it applies to a wider range of types. (The rhs requires second-countable
  Banach spaces while the lhs is well-defined on arbitrary real vector spaces.)›auto!: abs_summable_finite_sumsI: n_def

lemma abs_summable_equivalent \open>fabs_summable_on ›
proof ( have\opennabs_summable_on\close
  define n where ‹ for x
  assume<pen>n summable_on A›
  then have ‹sum n F ≤ infsum n A›🚫
    using that by (auto simp flip: infsum_finite simp: n_def[abs_def] intro!: infsum_mono_neutral)
    
  then show ‹
    by (auto intro!: abs_summable_finite_sumsI simp: n_def)
next
  define n where ‹n x = norm (f x)›
  assume ‹[simp: (<>.norm )) summable_on"
  then have ‹n abs_summable_on A›
    by (simp add: ‹f abs_summable_on A›
  then have ‹sum n F ≤ infsetsum n A› if ‹finite F› and ‹F⊆A› for F
    using that by (auto simp flip: infsetsum_finite simp: n_def[abs_def] intro!: infsetsum_mono_neutral)
  then show ‹n summable_on A›
    apply (rule_tac nonneg_bdd_above_summable_on)
    by (auto simp add: n_def bdd_above_def)
qed

lemma infsetsum_infsum:
  assumes "f abs_summable_on A"
  shows "infsetsum f A = infsum f A"
proof -
  have conv_sum_norm[simp]: "(\x. norm (f x)) summable_on A"
    using abs_summable_equivalent assms by blast
  have "norm (infsetsum f A - infsum f A) \ \" if "\>0" for ε
  proof -
    define δ where "\ = \/2"
--> --------------------

--> maximum size reached

--> --------------------