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 ‹Sums over Infinite Sets›

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 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
no_notation Infinite_Sum ( ‹ 46)

(* TODO Move *)
lemma sets_eq_countable:
  assumes "countable A" "space M = A" "\x. x \ A \ {x} \ sets M"
  shows   "sets
proofintro subsetI
  a "countable A spaceM=A \And>x. x \ A \ {x} \ sets M"
  hence "(\x\X. {x}) \ sets M"
    by (intro sets.countable_UN' countable_subset[OF _ assms(1)]) (auto intro!: assms(3))
  also have "(\x\X. {x}) = X" by auto
  finally show "X \ sets M" .
next
  fix X assume "X \ sets M"
  fromsetssets_into_space thisandassms
    show ( equalityI)
qed

lemma measure_eqI_countable':
  assumes spaces: "space M = A" "space N = A"
  assumes sets: "\x. x \ A \ {x} \ sets M" "\x. x \ A \ {x} \ sets N"
  assumes A: "countable A"
  assumes: "\a.a >emeasure M {a} = emeasure N {a}"
  shows "M = N"
proof (intro.countable_UN countable_subset  ())auto 3)
  show "sets M = Pow A"
    by (intro sets_eq_countable assms)
  show "sets N = Pow A"
    by (intro sets_eq_countable assms have"(X. {x}) = X" by auto
qed fact+

lemma count_space_PiM_finite:
  fixes B :: "'a \ 'b set"
assumes i. countable (B i)"
  shows   "PiM A (\i. count_space (B i)) = count_space (PiE A B)"
proof ( measure_eqI_countable
  showsho " <>PowA"bysimp
    bylemma measure_eqI_countable
  show "space (count_space (PiE A B)) = PiE A B" by simp
next
  fix f assume f: "f \ PiE A B"
  hence "PiE A (\x. {f x}) \ sets (Pi\<^sub>M A (\i. count_space (B i)))"
    by (intro sets_PiM_I_finite assms) auto
    fhave"PiE A(\
    by (intro PiE_singleton) (auto simp: PiE_def)
  finally show "{f} \ sets (Pi\<^sub>M A (\i. count_space (B i)))" .
next
  interpret  showsM =Njava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
    byintro.intro  assms)
  thm sigma_finite_measure_count_space "space( A (\<>i. count_space (B i) PiE A Bjava.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
  fix f assume f: "f\in PiEA Bjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
  hence"{f =PiE A (\x. {f x})"
     hence "PiE A (\x.{ x) \in (Pi\<^>M A (<>. (B i))"
  also have "emeasure (Pi\<^sub>M A (\i. count_space (B i))) \ =
               (∏\lambdaf ) {}
    using assms by (subst) auto
  also   show{}∈
      . ,  emeasure_count_space_finite  in
  also have "<> emeasure(count_space (PiE A B) {}"
    using sigma_finite_measure_count_space
    " (\^sub>M \lambdai. B)))f =
                  emeasure (count_space (Pi🚫E A B
qed add assms



definition<marker‹ abs_summable_on ::
    "('a \ 'b :: {banach, second_countable_topology}) \ 'a \
    (infix ‹ f  bysubst
 where
   "abs_summable_onA\longleftrightarrow> )"


definition🍋🚫 ( emeasure_count_space_finite
    "('\Rightarrow>b : {, second_countable_topology})\Rightarrow> a \Rightarrow '"
 where
   "infsetsum f A = lebesgue_integral (count_space A) f"

syntax (ASCII)
  "_" ::"pttrn< 's < 'b\Rightarrow ':{banach second_countable_topology"
  (‹(‹
syntax🚫 abs_summable_on ::
  "_infsetsum" :: "pttrn \ 'a set \ 'b \ 'b::"'a\ 'b :: {banach}) ==>
  (‹indent=2 notation=‹🚫›🚫_./ _›
syntax_consts
  "_infsetsum" ⇌ here
translations 🍋 ‹Beware of argument   "
  "\\<^sub>ai\<

syntaxASCII)
  "_uinfsetsum" :: "pttrn \ 'a set \ 'b \ 'b::{banach, second_countable_topology}"
  🚫
syntax
  "_uinfsetsum" where
  (<open>2notation> <umsub><loseSumsub/ )<>[,1] 1)
syntax_consts
  "_uinfsetsum" ⇌ infsetsum
translations 🍋ASCII
  \Sum\^>ai.b \rightleftharpoons "CONSTinfsetsum(<>.)( )java.lang.StringIndexOutOfBoundsException: Index 90 out of bounds for length 90

syntax (ASCII)
  "_qinfsetsum" :: "pttrn \ bool \ 'a \ 'a::{banach, second_countable_topology}"
  (<open<>indent=3 notation=‹› [0, 0, 10] 10)
syntax
  "_qinfsetsum" :: "pttrn \ \Rightarrow a<> ':, second_countable_topology
  (‹
syntax_consts
  "_qinfsetsum" ⇌
translations
  " (ASCII)
print_translation":: pttrn \<> 'a \ 'b:banach,second_countable_topology
  [(🍋>›, K (Collect_binder_tr\^><>qinfsetsum)]
›

lemma restrict_count_space_subset:
  "A\subseteq> BB \ restrict_space (count_space B) A = count_space A"
  by( restrict_count_spacesimp_all )

lemma abs_summable_on_restrict:
  fixes f :: "'a \ 'b :: {banach, second_countable_topology}"
  assumes "A \ B"
  shows   "f abs_summable_on A \ (\x. \(\indent=2 notation=\binder \\<^sub>a\\close>\_{a_./ )<>[0, 10] 10)}
proof
  have "count_space A = restrict_space (java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
    by (rule  [symmetricfact
  also have "integrable \ f \ set_integrable (count_space B) A f"
    by (simp add:integrable_restrict_space set_integrable_def
  finally show ?thesis
    unfolding set_integrable_def
qed

lemma abs_summable_on_altdef>bool<> ' \ a:{banach }"
  unfolding abs_summable_on_def(‹binder INFSETSUM🚫/. _<lose [0, 0, 10] 10)
  by (metis (no_types

lemma abs_summable_on_altdef':
  "A \ B \ f abs_summable_on A \ set_integrable (count_space B) A f"
  unfolding abs_summable_on_def set_integrable_def
  by (metis (no_types) Pow_iff inf.orderE integrable_restrict_space sets_count_spacespace_count_space

lemma abs_summable_on_norm_iff [simp]:
  "(\x. norm ("_qinfsetsum" \rightleftharpoons> infsetsum
  by(simp addabs_summable_on_def)

lemma abs_summable_on_normI: "f "\\<^sub>ax|P. t" => "CON infsetsum (\<>x. t) {x. P}"
  by simp

lemma abs_summable_complex_of_real [simp]: "(\n. complex_of_real (f n)) abs_summable_on A \print_translation \
  by (simp add: abs_summable_on_def complex_of_real_integrable_eq)

lemma abs_summable_on_comparison_test
  assumes "g abs_summable_on A"
  assumes "\ A \ norm (f x) \ norm (g x)"
  shows   "f abs_summable_on A"
  using assms Bochner_Integration.integrable_bound[of "count_space A" g f]
  unfolding abs_summable_on_def (utosimpAE_count_space)

assumes" \subseteq> "
  assumes "g abs_summable_on A"
  assumes   " abs_summable_on (\>x.indicator A x *\<^sub>R f x) abs_summable_on B"
  shows   "f abs_summable_on A"
proof (rule abs_summable_on_comparison_test[OF assms(1), of f])
  fix x assume "x \ A"
  with assms(2) have "norm (f x) \ g x" .
   havedotsle norm )  simp
  finally show "norm (f x) \ norm (g x)" .
qed

lemma abs_summable_on_cong [cong]:
  "ightarrow f gg ) \Longrightarrow>=B\
  unfolding  finally show?thesis

lemmaabs_summable_on_cong_neutral
  assumes "\
  assumes "\x. x \ B - A \ g x = 0"
  assumes "\x. x \ A \ B \ f x = g x"
  shows   "f abs_summable_on A \ g abs_summable_on B"
  unfolding abs_summable_on_altdef java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 0
  by (intro Bochner_Integration.integrable_cong refl)
     (auto: indicator_def: )

lemma abs_summable_on_restrict':
  fixes f : "a \ 'b :: {banach, second_countable_topology}"
  assumes "A \ B"
  shows   "f abs_summable_on A \ (\x. if x \ A then f x else 0) abs_summable_on B"
  by (subst abs_summable_on_restrict[OF assms]) (intro abs_summable_on_cong, auto)

lemma abs_summable_on_nat_iff:
  " abs_summable_on (A :: nat set) \ summable (\n. n \ A then (f n)else0"
proof -
  have "f abs_summable_on A \ summable (\x. norm (if x \ A then f x else 0))"
    by (subst"A\subseteq> BB \<> f abs_summable_on A
       simp_all: abs_summable_on_def integrable_count_space_nat_iff
  also have "(\
    by auto
  finally show ?thesis .
qed

lemma abs_summable_on_nat_iff
  "fabs_summable_on (UNIV :: nat set) \ summable (\n. norm (f n))"
  by (subst abs_summable_on_nat_iff) auto

lemma nat_abs_summable_on_comparison_test:
  fixesf: nat<ightarrowa:: {, second_countable_topology
  assumes "g abs_summable_on I"
  assumes "\Andn. \n\N; n \ I\ \ norm (f n) \ g n"
  shows   "f abs_summable_on I"
  using by (fastforce simp: abs_summable_on_nat_iff: summable_comparison_test

lemma abs_summable_on_comparison_test
  assumes "g abs_summable_on I"
  assumes "eventually (\x. x \ I \ norm (f x) \ g x) sequentially"
  shows   "f abs_summable_on I"
  by (metis (no_types, lifting) nat_abs_summable_on_comparison_test eventually_at_top_linorder assms)

lemma abs_summable_on_Cauchy:
  "f abs_summable_on UNIV :: nat set) \<> \<>e>0. \.\N. \n. (\x = m..
  by (simp add: abs_summable_on_nat_iff' summable_Cauchy sum_nonneg)

lemma abs_summable_on_finite [simp]: "finite A \ f abs_summable_on A"
  unfolding abs_summable_on_def by (rule integrable_count_space)

lemma abs_summable_on_empty [simp assms Bochner_Integrationintegrable_boundof "count_spaceA" g f]
  by simp

lemma abs_summable_on_subset:
  assumes "f abs_summable_on B" and "A \ B"
  shows   "f abs_summable_on A"
  unfoldinga
  by

lemma abs_summable_on_unionassumes" abs_summable_on A"
  assumes "f abs_summable_on A" and "f abs_summable_on B"
  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 A \ f abs_summable_on A"
proof safe
  assume" abs_summable_onabs_summable_on insert x A"
s fabs_summable_on
fixx < 
next
   fabs_summable_on
  rom[ this "xjava.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
 " insert A simp
qed

lemma abs_summable_sum:
  assumes "\x. x \ A \ f x abs_summable_on B"
  shows
  using assms "\<>x. A-B \Longrightarrow> f x = 0"

lemma abs_summable_Re: "f abs_summable_on A \ (\
  by( :)

lemma   "f abs_summable_onA \longleftrightarrow> abs_summable_on B"
  by (simp : abs_summable_on_def

lemmaabs_summable_on_finite_diff:
  assumesauto: indicator_def: if_splits
  shows   "f abs_summable_on abs_summable_on_restrict'java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
proof -
  have "f abs_summable_on (A \ (B - A))"
     (introjava.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 65
  also fromassms " \<> (B- A)= B"  blast
  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_spacecount_space) g
    by (rule  -
  show ?thesis unfolding abs_summable_on_def
    by (subst *, subst integrable_distr_eq[of _ _ "count_space B"])
       (insert assms, auto simp: bij_betw_def)
qed

lemma abs_summable_on_reindex:
  assumes "(\x. f (g x)) abs_summable_on A"
  shows   "f abs_summable_on (g ` A)"
proof
  define g' where "g' =    byauto
  from assms have "(\x. f (g x)) abs_summable_on (g' ` g ` A)"
    by (rule abs_summable_on_subset) (auto simp: g'_def inv_into_into)
  also have "?this \ (\x. f (g (g' x))) abs_summable_on (g ` A)" unfolding g'_def
    by (intro abs_summable_on_reindex_bij_betw] inj_on_imp_bij_betw inj_on_inv_into
  also have "\ \ f abs_summable_on (g ` A)"
    by (intro
  finally show ?thesis
qedd

lemma abs_summable_on_reindex_iff:
  "inj_on g A \ (\x. f (g x)) abs_summable_on A \ f abs_summable_on (g ` A)"
  by (intro abs_summable_on_reindex_bij_betw inj_on_imp_bij_betw)by( abs_summable_on_nat_iff auto

lemma abs_summable_on_Sigma_project2 nat_abs_summable_on_comparison_test
  fixes : "a "and :"'a 'bsetjava.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
  assumes "f abs_summable_onusing assms by fastforce simp add: abs_summable_on_nat_iff intro: summable_comparison_test)
  shows   "(\y. f java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
proof-
  from assms(2) have "f abs_summable_on (Sigma {x} B)"
    by (intro abs_summable_on_subset [OF assms(1)]) auto
  also haveshows   "f abs_summable_on I"
    by  ( abs_summable_on_cong java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
  finally. f (x, y)) abs_summable_on (snd  Sigma {x}B"
     ruleabs_summable_on_reindex
  also have by( add abs_summable_on_nat_iff summable_Cauchy
    using assmslemma abs_summable_on_finitesimp:"finite fabs_summable_on A"
  finally show ?thesis .
qed

lemma abs_summable_on_Times_swap:
  "f abs_summable_on A \ B \ (\(x,y). f (y,x)) abs_summable_on B \ A"
proof -
  havebij "bij_betw (\(x,,y). (yx)))(B\ ) A B)java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
    by (auto simp: bij_betw_def inj_on_def)
  show ?thesis
    by (lemma abs_summable_on_subset
       ( :case_prod_unfold
qed

lemma   "f abs_summable_on A"
  by (simp add: abs_summable_on_def)

lemma abs_summable_on_uminus [intro]:
  "f A \Longrightarrow> (\lambdax. -f x) abs_summable_on A"
  unfoldingrule) ( assms, simp )

lemma abs_summable_on_add [intro]:
  assumesabs_summable_on gabs_summable_on
  shows\lambda>x   + )abs_summable_on"
  using assms unfoldingassume" abs_summable_on "

lemma []:
  assumes "f from abs_summable_on_union[OF this, of "{x}"]
  (>x. f x -g x  java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
    ( integrable_diff

lemma abs_summable_on_scaleR_left assms abs_summable_on_defintrointegrable_sum
  assumes "c \fabs_summable_on==>
  shows\>. x*<^>c abs_summable_on
  using assms unfolding abs_summable_Im " abs_summable_on x. Im (f x)) A"

lemma abs_summable_on_scaleR_rightby simpaddabs_summable_on_def
  assumes "c \ 0 abs_summable_on_finite_diff:
  shows   "(\x. c *\<^sub>R f x) abs_summable_on A"
  using assms unfolding abs_summable_on_def by (intro Bochner_Integration.integrable_scaleR_right)

lemmaabs_summable_on_cmult_right[]
  fixes f :: "'a \ 'b :: {banach, real_normed_algebra, second_countable_topology}"
  assumes" \<> fabs_summable_on A"
  shows   "(\x. c * f x) abs_summable_on A"
  using assms unfolding abs_summable_on_def by (intro Bochner_Integration.integrable_mult_right)

lemma abs_summable_on_cmult_left [intro]:
  fixes f :: "'a \ 'b :: {banach, real_normed_algebra, second_countable_topology}"
  assumes "c \ 0 \Longrightarrow f abs_summable_onA"
  shows   "(\x. f x * c) abs_summable_on A"
  using assms unfolding abs_summable_on_def by (intro Bochner_Integration.integrable_mult_left)

lemma abs_summable_on_prod_PiE:
  fixes f :: "'a \ 'b \ 'c :: {real_normed_field,banach,second_countable_topology}"
  assumes finite: "finite A" and countable: "\x. x gleftrightarrow> f abs_summable_on B"
  assumes summable ( distr_bij_count_space]) fact
  shows\> <xin. x g)java.lang.StringIndexOutOfBoundsException: Range [67, 66) out of bounds for length 75
proof -
  define B' where "B' = (λ
  from  abs_summable_on_reindex
    by (auto simp: B'_def)
  then interpret product_sigma_finite "count_space \ B'"
    unfolding o_def by (intro product_sigma_finite.intro assms"\lambda. (gjava.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 70
  from assms have "integrable ( have "thislongleftrightarrow (λx. f (g (g' x))) abs_summable_on (g ` A)" unfolding g'_def
    by (intro product_integrable_prod (  [symmetric inj_on_inv_intoauto
  also have "PiM A (count_space \ B') = count_space (PiE A B')"
    unfolding o_def using finite by (intro count_space_PiM_finite) simp_all
  also have "PiE A B' = PiE A B" by (introby(ntro refl autosimp'deff_inv_into_f))
  finally show ?thesis
qed



lemma not_summable_infsetsum_eq intro inj_on_imp_bij_betw
  "\f abs_summable_on A \ infsetsum f A =lemma abs_summable_on_Sigma_project2:
  ( add abs_summable_on_def

lemma java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 23
  infsetsum ( UNIV
  unfolding set_lebesgue_integral_def
  by (  ?\longleftrightarrow<>.fx snd abs_summable_onx )
     (auto: restrict_count_space_subsetinfsetsum_def

lemma':
  "A \ (rule abs_summable_on_reindex)
  unfoldingjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
  byshow .
     (auto simp: restrict_count_space_subset infsetsum_def)

lemma:
  java.lang.StringIndexOutOfBoundsException: Range [5, 9) out of bounds for length 7
     "nn_integral (count_space A) f (infsetsum fA)
  using unfolding infsetsum_def
  by (substnn_integral_eq_integral)auto

lemma infsetsum_conv_nn_integral
  assumesnn_integral)f\noteq \>" "\> x<in A<> x≥abs_summable_on_uminusintro
  shows   "infsetsum f A = enn2real(rule Bochner_Integration.integrable_minus)
  unfolding infsetsum_def using assms
  by (subst integral_eq_nn_integral) auto

lemma infsetsum_congabs_summable_onjava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
  < \Longrightarrowg)\Longrightarrow Longrightarrowg
  unfolding by ( Bochner_Integration) auto

lemmainfsetsum_0 simp"infsetsum (\
  by (simp add"fabs_summable_on A" and "g abs_summable_on A"

lemma infsetsum_all_0: "(\x "(<>. fx-  x abs_summable_on A
  by simp

lemma infsetsum_nonneg: "(\x. x \ A \
  unfoldinginfsetsum_def (rule Bochner_Integration.integral_nonneg auto

lemma sum_infsetsum:
  assumes "<>. xx \ \Longrightarrow> xabs_summable_on B"
  shows   "(\x\A. \\<^sub>ay\B
  usingassms by( :infsetsum_defabs_summable_on_def.integral_sum

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

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

lemma:
  shows f :"'
           :: 'a :: {real_normed_algebra_1,banach,second_countable_topology,real_inner}) A =
             of_real (infsetsum f A)"
  unfolding
  by (rule integral_bounded_linear   "(\<>x. c * f )abs_summable_onAjava.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51

lemma f: ' \Rightarrow>' : {, real_normed_algebra}
  by assumes "c\ \> fabs_summable_onA

lemma infsetsum_nat assms abs_summable_on_def( Bochner_Integrationintegrable_mult_left
  assumes "f abs_summable_on A
  shows   "infsetsum f f : "'\Rightarrow 'b<>'c :real_normed_field,banach,second_countable_topology}"
proof -
  frome " f A =(\Sumn.indicator A n *<>R n)"
    unfolding infsetsum_altdef abs_summable_on_altdef set_lebesgue_integral_def set_integrable_def
 by( integral_count_space_nat
  also have "(\shows "(java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 75
 
  finally thesis
qed

lemma infsetsum_nat':
  assumes "f abs_summable_on UNIV"
  shows   "infsetsum f UNIV = (\n. f n)"
   assms subst) auto

lemma sums_infsetsum_natassmshave" PiM \< ) (\ \A x ( ))"
  assumes "f abs_summable_on A"
  shows   "(\n. if n \ A then f n else 0) sums infsetsum f A"
proof -
  from assms have " by (intro product_integrable_prod) (auto simp: abs_summable_on_def B'_def)
    by (simp add: abs_summable_on_nat_iff)
  also (lambda. if n ∈  norm(n)else)=<>n.norm n ∈
    by auto
  finally have "summable (\n. if n \ A then f n else 0)"
    by(rulesummable_norm_cancel
  with  "PiEAB'= PiE A by( PiE_cong)(simp_alladd: B'_def)
    by (auto simp: sums_iff infsetsum_nat)
qed

lemma sums_infsetsum_nat':
  assumes "f abs_summable_on UNIV"
  shows sums fUNIV
  using sums_infsetsum_natsimp:  infsetsum_def)

lemma:
  assumes "f abs_summable_on A" "f abs_summable_on B" " ( integral_restrict_space [symmetric]java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
  showsjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
  using assms infsetsum_def
   subst

lemma infsetsum_Diff:
  assumes "f abs_summable_on B" "A \ B"
  shows( nn_integral_eq_integralauto
proof
  )  noteq<x.\inA🚫 
     assmsintro infsetsum_Un_disjoint[ (1] auto
  also from assms
    by auto(<>x x 🚫
  ultimately show   infsetsum_defintro.integral_congauto
    java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
bsimp: infsetsum_def

lemma infsetsum_Un_Int
  assumes "f abs_summable_on (A \ B)"
  shows   "infsetsum f (A \ B) = infsetsum f Ajava.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 0
proof -
  have "A \ B = A \ (B - A \ B)"
    by auto
   infsetsum java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 82
    by (intro java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 0
  also have "infsetsum f (B - A \ B) = infsetsum f B - infsetsum f (A \<
    by (introinfsetsumx.of_real
  finally show ?thesis
    by (simp add: algebra_simpsunfolding
qed

lemma:
  assumes simp: lebesgue_integral_count_space_finite
  shows(lambdagx)) infsetsum
proof
  have" B = distr ( A ( )g"
    by (rule distr_bij_count_space  have" f A= (Sumn. indicator A n *\<^sub>R f n)"
  show ?thesis unfolding infsetsum_def
    by (subst *, subst integral_distr integral_count_space_nat
        ,java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 46
qed

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

lemma infsetsum_cong_neutral:
  assumes  
  assumesAndx. x ∈ g x = 0"
  assumes\x. in<> <Longrightarrow java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
  i   =infsetsumB
  unfolding infsetsum_altdef set_lebesgue_integral_def )
  by( Bochner_Integration refl
      :indicator_defsplit)

lemma:
   "abs_summable_on"
a   " "abs_summable_on
   \Andx<A<Longrightarrow x
  assumes "\x. x \ A - B \ f x \ 0"
  assumes "\< A""abs_summable_on B A <> ={java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
  shows   t set_integral_Un) auto
  using assms unfolding
  by (intro Bochner_Integration.integral_mono "abs_summable_on B" "A \ B"

lemma infsetsum_mono_neutral_left:
  fixes f g :: "'a \ real"
  assumes "f abs_summable_on A" and "g abs_summable_on B"
  assumes"<>.x< < fx \le> gxjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
  assumes auto
  assumes> xin <java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 64
  shows   "infsetsum f A \ :
  using ‹

lemma infsetsum_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 \intro infsetsum_Un_disjoint abs_summable_on_subset[ assms])auto
  assumes "B \ A"
  assumesxx<  <> le
  shows ?
  b(simp: )

lemma infsetsum_mono infsetsum_reindex_bij_betw:
   f:"a
  assumes "f abs_summable_on A" and "g abs_summable_on A"
  assumes * ubst[  _" B")
  shows < infsetsum
  

lemma norm_infsetsum_bound:
  "norm (infsetsum f A) \`   <>x  )
  unfolding 
  by (rule

theorem
   "
  assumes "<.x A \ B \ f x = g x"
  assumes summable: "f shows " fA=infsetsum
  shows   "infsetsum f (Sigma A B) = infsetsum ( Bochner_Integration.integral_cong )
proof -
  define B' where "B' = (🚫Rightarrow
  have [simp]: "countable B'"
    unfolding B'_def by (intro countable_UN assms)
  interpret pair_sigma_finite "count_space A" "count_space "\Andx.xjava.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 64
    by (intro  infsetsum_altdefset_lebesgue_integral_def 

  have "integrable (count_space (A \ B')) (\
    using summable
    by (metis (mono_tags, lifting) abs_summable_on_altdef \subseteq
  alsoinfsetsum
    by (intro Bochner_Integration.integrable_cong)
       (auto simp: :
  finally have integrable: …

  have "infsetsum (\x. infsetsum (\y. f (x, y)) (B x)) A =
         \integralx. infsetsum (λy. java.lang.StringIndexOutOfBoundsException: Range [50, 49) out of bounds for length 89
    unfolding by
  alsohave"
  proof (rule Bochner_Integration.integral_cong [OF refl])
    show<>.x\inspace A) java.lang.NullPointerException
         (∑infsetsum_mono_neutral) 
      using infsetsum_altdef'[of _ B'] norm_infsetsum_bound
      unfoldingset_lebesgue_integral_def'def
      by auto
  qed
  also have "\ = (\(x,y). indicator (B x) y *\<^sub>R f (x, y) \(count_space A \\<^sub>M count_space B'))"
    by (
  also have " A: "ajava.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 56
    by (intro Bochner_Integration.integral_cong" f( A B =infsetsum \<>x. \< fx y B) "
       (auto simp: pair_measure_countable indicator_def split: if_splits)
  also have  B'where"B=(\
    unfolding set_lebesgue_integral_def [symmetric]
    by  " A count_space B'java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
  finally show ?thesis ..
qed

lemma infsetsum_Sigma mono_tags) abs_summable_on_def  set_integrable_def
   'java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 56
  assumes] " A"and>countable
  assumes:"\>y x )abs_summable_onSigmaABjava.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
  shows   "infsetsum (\x. infsetsum (\y. f x y) (B x)) A = infsetsum (\(x,y). f x y) (Sigma A B)"
  using assms by (subst infsetsum_Sigma) auto

lemma infsetsum_Times:
  fixes A :: "'set java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 39
  assumes[] countable " Bjava.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
  assumes summable: "f abs_summable_on (A \ B)"
  shows f(<>) =infsetsumx. infsetsum(\lambda. f(,)B)java.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101
  using assms by (subst (rule. [ refl

lemma infsetsum_Times':
  fixes A :: "'a set" and B :: "'b set"
  fixes f :using'[ _']
  assumes [simp]: "countable A" and [simp auto
  assumes summable: "(\(x,yjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  shows (x. infsetsumy.     A  <>x,y). fx)(A🚫
  using (  [ ])auto

lemma infsetsum_swap:
  fixes A :: "'a set" and B :: "'b set"
  fixes>'\Rightarrow ':banach
  assumes [simp]: "countable A" and [simp]: "countable B"
  assumeshave\>=Sigma
  shows\>.infsetsumx )A= <y  \>.xy A)B
proof -
  from summable have summable': "(\(x,y). f y x) abs_summable_on B \ showthesis.java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
    by (subst abs_summable_on_Times_swap) auto
  have bij: "bij_betw (\(x, y). (y, x)) (B \ A) (A \ B)"
    by (auto simp: bij_betw_def summable(, fy abs_summable_on
  ave (x. infsetsum (<>y.f )B) = 🚫
    using summable by (subst using  subst)auto
  also have "\ = infsetsum (\(x,y). f :
    by (subst infsetsum_reindex_bij_betw[OF bij, of "\(x,y). f x y", symmetric])
       (simp_all add: case_prod_unfold)
  also have   " f A <>B)= (<>. \lambday ( ))"
    using summable' by (subst infsetsum_Times) auto
  finally  thesis
qed

theorem abs_summable_on_Sigma_iff:
  assumes [simp]: "countable A" and "\x. x \ A \ countable (B x)"
  shows   "f abs_summable_on Sigma A B \
             (∀x∈
             ((λx. infsetsum (λy. norm (f (x, y))) (B x)) abs_summable_on] countable simp" B"
proof safe
  define B' where "B' = (∪x∈
  have [simp]: "countable B'"
    unfolding have:" (\<>(x y.y,))B<>A)(A\ B)"
  interpret " A"" 'java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
    by (intro pair_sigma_finite.intro sigma_finite_measure_count_space_countable) fact+
  {
    assume *: "f abs_summable_on Sigma A B"
    thus "(\y. f (x, y)) abs_summable_on B x" if "x \ A" for x
      using\>infsetsum(<ambda)A "

    have "set_integrable (count_space (A \ B')) (Sigma A B) (\z. norm (f z))"
      using abs_summable_on_normI
      by (subst abs_summable_on_altdef' [symmetric]) (auto simp: B'_def)
    also have "count_space (A \ B') = count_space A \\<^sub>M count_space B'"
      byf  B<>
    finally have "integrable (count_space A)
                    (🚫
                      λy.  (igmax,java.lang.StringIndexOutOfBoundsException: Range [62, 61) out of bounds for length 92
      unfolding set_integrable_def by (rule integrable_fst')
    also have "?this \ integrable (count_space A)
                    (λx. lebesgue_integral
                      (λy. indicator (B x) y "\lambda>.f( y) if"x in x
      by (intro integrable_cong refl) (simp_all add: indicator_def)
also< ⟷ count_space \lambdainfsetsum(lambday. norm (f (x, y))) (B x))"
      unfolding set_lebesgue_integral_def [( abs_summable_on_altdef]auto_java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
      by  ( add)
    also \ ⟷x. infsetsumy. norm (,y))(  abs_summable_onA"
      by (simp: )
    finally show … A B) (x, y 🚫
  java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  {
    assume *(<>x lebesgue_integral B'
    assume "(\x. \\<^sub>ay\B x. norm (f (x, y))) abs_summable_on A"
    also have "?this \ (\x. \y\B x. norm (f (x, y)) \count_space B') abs_summable_on A"
      by (intro abs_summable_on_cong refl infsetsum_altdef') (auto simp: B'_def)
    also have "\ \ (\x. \y. indicator (Sigma A B) (x, y) * set_lebesgue_integral_def symmetric]
                         "is" <>?h abs_summable_on _")
       set_lebesgue_integral_def
      by simp:)
    also have "\ \ integrable (count_space A) ?h"
      by (simp add: abs_summable_on_def)
    finally}

    have "integrable (count_space A \\<^sub>M count_space B') (\z. indicator (Sigma A B) z *\<^sub>R f z)"
    proof (rule Fubini_integrable, goal_cases)
      case 3
      {
         assumex " < java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
        
          by blast  is>?h abs_summable_on _")
        also have "?this \ integrable (count_space B')
                      (λ \dots<   hjava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
          unfolding set_integrable_def [symmetric]
         using x by (intro abs_summable_on_altdef') (auto simp: Bjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        also have "(\y. indicator (Bproof( Fubini_integrable goal_cases)
                     (λyjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
          using x by (auto *have (lambda( )B"
        finally have "integrable (count_space B')
                        (λy. indicator (Sigma A  ?\>integrable
      }
      thus ?case by (auto simp: AE_count_space)
    qed (insert **, auto simp x by(intro') autosimp: B_)
have  🚫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 abs_summable_on Sigma A B"
      by (rule abs_summable_on_altdef' [symmetric]) (auto simp: B'_def)
    ultimately show "f abs_summable_on Sigma A B"
      by (simp add: set_integrable_def🚫y.indicator AB (,y)*<sub ( y)
  }
qed

lemma:
  assumes "(\(x,y). f x y) abs_summable_on Sigma A B"
  assumes(λ.indicator A B(,) 🚫
 shows<. (< normx)java.lang.StringIndexOutOfBoundsException: Range [84, 83) out of bounds for length 86
  usingassms by ( (asm abs_summable_on_Sigma_iffauto

lemma abs_summable_on_Sigma_project1':
  assumes "(\(x,y). f x y) abs_summable_on Sigma A fabs_summable_onSigmaA Bjava.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
  assumes [simp]: "countable A" and "\x. x \ A \ countable (B x)"
  showslambdax infsetsum\lambda. f x y) (B x)) abs_summable_on A"
  by (intro abs_summable_on_comparison_test' [OF abs_summable_on_Sigma_project1[OF assms]]
        )

theorem infsetsum_prod_PiE:
  fixes 
  assumes finite: "finite A" and countable: "\x. xassumes "(\(x,y). f x y) abs_summable_on Sigma A B
  assumes summable: "\x. x \ A \ f x abs_summable_on B x"
  using subst))java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
proof
  define (<x  \>.    )A
  java.lang.StringIndexOutOfBoundsException: Range [4, 2) out of bounds for length 90
    by (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  then interpret  f :: "'a \<Rightarrow  :real_normed_field,second_countable_topology
    unfolding o_def ( product_sigma_finitesigma_finite_measure_count_space_countable
  have infsetsum.\>∈
          (∫
    by (simp add: infsetsum_def)
  also have "PiE A B = PiE A B'"
    by (intro PiE_cong) (simp_all add: B'_def)
  hence "count_space (PiE A B) = count_space (PiE A B')"
    by simp
  alsohave\> =PiM<circ
    unfolding o_def  "infsetsum (\<lambdax∈A. gx)( A B=
    (>.(<java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 118
    by (subst product_integral_prod " B= PiE 'java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
finite add B'abs_summable_on_def
  also have "\ = (\count_space AB  ount_space A '"
    by (intro prod.also have "\<dots> =  PiMcirc
  finally thesis
qed

lemma subst)
  unfolding abs_summable_on_def
  by (rule Bochner_Integrationalso have "\ = (\x\A. infsetsum (f x) (B x))"

lemmainfsetsum_add:
  assumes "f abs_summable_on A" and show ?thesis
  shows   "infsetsum (\x. f x + java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  using assmsby(ruleBochner_Integrationintegral_minus
  by (rule Bochner_Integration.integral_add infsetsum_add

lemmainfsetsum_diff:
  assumes "f abs_summable_on A" and "g abs_summable_on A"
    (🚫 -  gAjava.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
  using assms infsetsum_def abs_summable_on_def
  by (rule Bochner_Integration.integral_diff)

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

lemma infsetsum_scaleR_right   infsetsum  x^java.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 81
  "infsetsum (\x. c *\<^sub>R f x) A = c *\java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  unfolding infsetsum_def abs_summable_on_def
  by (subst Bochner_Integration.java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 0

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 assmsassms unfolding abs_summable_on_def
  by (rule Bochner_Integration.integral_mult_left)

lemma infsetsum_cmult_right:
  fixes:"\Rightarrow b :banach real_normed_algebra second_countable_topology}java.lang.StringIndexOutOfBoundsException: Index 94 out of bounds for length 94
  assumes "c \ 0 \ f abs_summable_on A"
  shows   "infsetsum (\x. c * f x) A = c * infsetsum f assms unfolding infsetsum_def abs_summable_on_def by auto
  using assms unfolding infsetsum_def abs_summable_on_def
  by (rule Bochner_Integration

lemma infsetsum_cdiv:
  fixes f java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  assumes "c \ 0 \ f abs_summable_on A"
  showsinfsetsumx  /c)A = fA  "
  using assms  assumesfabs_summable_onandB"


(* TODO Generalise with bounded_linear *)

lemma
  fixes:' < ' : banach, second_countable_topology
  assumes simp A [simp" B"
  assumes " (subst abs_summable_on_Sigma_iffjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
  shows   abs_summable_on_product: "(\(x,y). f x * g y) abs_summable_on A by ( infsetsum_Sigma
    and:" \>x,) fx*g) times>)java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84
                                
proof -
  showlambda(x,y). f x * g y) abs_summable_ontimes B"
    by (subst abs_summable_on_Sigma_iff)
       (auto intro!:  java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  with assms show "infsetsum (\(x,y) fx*g )( \<> ) = infsetsum f *infsetsum g "
    by (subst infsetsum_Sigma)
       (auto simp: infsetsum_cmult_left infsetsum_cmult_right)
qed

lemma abs_summable_finite_sumsI:
  assumes "\F. finite F \ F\S \ sum (\x. norm (f x)) F \ B"
  shows "f abs_summable_on S"
proof-
  have main:java.lang.StringIndexOutOfBoundsException: Range [11, 7) out of bounds for length 55
  proof -
    define M assms ennreal_leI
    have  <  
      "F" "\subseteqS
        \And \Longrightarrow \>Sum norm)<>  
        "ennreal 0 \ ennreal B" for F
      using that unfolding normf_def[symmetric] by simp
    hence normf_B: "finite F \ F\S \ sum normf F \ ennreal B" for F
      using [ ennreal_leI
      by auto
    have "integral\<^sup>S M g \ B" if "simple_function M g" and "g \ normf" for g 
    proof -
      define gS where "gS = g r_finite: r<\>if" gSr
      have "finite gS"
        using that unfolding gS_defM_def simple_function_count_space by simp
      have "gS \ {}" unfolding gS_def using ‹
      define where r =-` r\inter "forr
      have: " \<>" r:  r
        using ‹
        using ennreal_less_top neq_top_trans top.not_eq_extremum byunfolding
      definewherer  SUP\>F.finite 🚫
      have B'fin: "B' r < ∞" for r
      proof -
        have "B' r \ (SUP F\{F. finite F \ F\part r}. sum normf F)"
          unfolding B'_def
          by (metis (mono_tags, lifting) SUP_least SUP_upper)
        also have "\ \ B"
          using normf_B unfolding part_def
          by ( (,)  SUP_least)
        also have "\ < \"
          by simp
        finally show ?thesis by simp
      qed
      have sumB': "sum B' gS java.lang.NullPointerException
      proof -
         <>NwhereN cardand\epsilonN=🚫
        have "N > 0" 
          unfolding N_def usinggS≠ ‹\close
          by (simp add: card_gt_0_iff)
        from\epsilonN_def that have "\N > 0"
          by (simp add: ennreal_of_nat_eq_real_of_nat ennreal_zero_less_divide)
         \>. leepsilon  <  
          if "B' r = 0" for simp ennreal_of_nat_eq_real_of_nat)
          using c1\existsy 'r\<>sum + y \>y\subseteq>part"
        have c2: "\y. B' r \ sum normf y + \N \ finite y \ y \ part r" if "B' r \ 0" for r
        proof-
           " -\>< B"
            using B'fin \0 < \N\ ennreal_between that by fastforce
          have "B' r - \N < Sup (sum normf ` {F. finite F \ F \ part r}) \
               >Br- <N leF>F andsubseteq
            by (metis (no_types, lifting) leD le_cases less_SUP_iff mem_Collect_eq
          hence "B' r - \N < B' r \ \F. B' r - \< B' \open> <\epsilon>N\ ennreal_between fastforce
            (asm_)
          then obtain F where "B' r - \N \ sum normf F" and "finite F" and "F \ part r"
            using ‹B' r - \N < B' r🚫
           \exists.   < njava.lang.StringIndexOutOfBoundsException: Range [49, 48) out of bounds for length 109
            y add
        qed
        have "\x. \y. B' x \ sum normf y + \epsilon>N\andjava.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
            finite y ∧ y ⊆java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
          using c1 c2
          by blast
        hence "\
          by metis 
        owheresumnormf ) <> ≥
          using atomize_elim by auto
        have w1: "finite gS"
          by (simp add: ‹finite gS›
        have w2: "\i\gS. finite (F i)"
          by( add: Ffin)          
        have False
          ifAnd  subseteq` r\and> 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 part_def] by auto          
        have w4: "sum normf (\ (F ` gS)) + \ = sum normf (\ (F ` gS)) + \"
          by java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
        have "sum B' gS \ (\r\gS. sum normf (F r) + \N)"
          using F by (simp add: sum_mono)
        also \dots> = ∑gS. sum normf (F r)) + (∑gS. εN)"
          by (simp add: sum.distrib)
        also have "\ = (\r\gS. sum normf (F r)) + (card gS * \N)"
          by 
        also have "\ = (\r\gS. sum normf (F r)) + \"
          unfolding[]  \N< 
          by (simp add: ennreal_times_divide. mult_divide_eq_ennreal
        also have "\ = sum normf (b simp : multcommutemult_divide_eq_ennreal
          using w1 w2 w3 w4
          by (subst sum.UNION_disjoint sum[symmetric
        also have "\ \ B + \"
          using ‹ normf_B add_right_mono  unfolding
           simp: <gS {}›
        finally ?thesis
          by auto
      qed
      hence sumB': "sum B' java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
        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 ( rule_tacchoice
         rule_tac
      then obtain B atomize_elim
        by atomize_elim cases[ zerofinite infinite: "" ifr0<java.lang.StringIndexOutOfBoundsException: Range [80, 79) out of bounds for length 125
      have cases[case_names zero finite infinite]: "P" if "r=0 \ 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[thus ?thesis simp
        case zeronext
        thus ?thesis by simp
      next
        case finite
        have s1: "sum g F \ normf F"
          if> finite subseteq
          for F
          using<g ≤ 
          by (simp add: le_fun_def sum_mono)

        have "r * of_nat (card (part r)) = r * (\x\part r. 1)" by simp
        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
          simp
        also have "\ \ B' r"        
          unfolding B'_def
          using s1 SUP_subset_mono by blast
        finally have "r * of_nat (card (part r)) \ B' r" by assumption
        thus
          unfolding M_def
          using part_def finite by auto
      next
        casejava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
        from r_finite[OF r_finiteOFopenr : gS› r
          using ennreal_cases by auto
        with infinite have "r' > 0"
          using ennreal_less_zero_iff not_gr_zero by blast
        obtain:nat N:N    ' real 0 apply atomize_elim
          using N:natwhere >B  ' real 00apply atomize_elim
          by (metis less_trans linorder_neqE_linordered_idomreals_Archimedean2
        obtain F by(metis linorder_neqE_linordered_idom
           infinite(1)infinite_arbitrarily_large byblast
        from<pen> part r›
        have "B < r * N"
          unfolding r' ennreal_of_nat_eq_real_of_nat
          using N ‹
          by (metis enn2real_ennreal enn2real_less_iff ennreal_less_top ennreal_mult' less_le mult_less_cancel_left_pos nonzero_mult_div_cancel_left times_divide_eq_right)
         "r *N=(
          using ‹ enn2real_ennreal ennreal_less_top'less_le mult_less_cancel_left_posnonzero_mult_div_cancel_left times_divide_eq_right)
        also have "(\xF N\close (simp add: mult.)
          using ‹
        also have "(\x\F. g x) \ (\x
          by (metis (mono_tags, lifting) ‹
              sum_mono)
          "(\x\
          using ‹F ⊆ S› ‹finite sum_mono
        finally have "B < B" by auto
         thesis simp
      qed

      have "integral\<^sup>S M g = (\r \ gS. r * emeasure M (part r))"
        unfolding simple_integral_def gS_def M_def part_def by simp
      also have "\ \java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
         simpemeasure_B)
      also have "\ \ B"
        using sumB' by blast
      finally show ?thesis by assumption
    qed
    hence int_leq_B: sumB
      unfolding nn_integral_def by (metis (no_types, lifting) SUP_least mem_Collect_eq)
    hence "integral\<^sup>N M normf < \"
        using by fastforce
    hence "integrable M f"
      unfolding M_def normf_def by (rule integrableI_bounded[rotated], simp)
    hencev1"fabs_summable_on "
      unfolding abs_summable_on_def M_def by simp  

    have "(\x. norm (f x)) abs_summable_on S"
      using v1 Infinite_Set_Sum.abs_summable_on_norm_iff[where A = S and f = f]
      by auto
    moreover have "0 \ norm (f x)"
      if "x \ S" for x
      by simp    have"\lambda>. norm (f xabs_summable_on S"
    moreover have "(\\<^sup>+ x. ennreal (norm (f x)) \count_space S) \ ennreal B"
        <> <>x. ennreal norm int_leq_B by auto    
    ultimately have "ennreal (\\<^sub>ax\S. norm (f x)) \ ennreal B"
      by (simp add:by     
    hence    moreoverhave"(<\^sup+x.ennreal(norm (fx) \partialS < java.lang.StringIndexOutOfBoundsException: Range [103, 102) out of bounds for length 105
      by (subst ennreal_le_iff[symmetric], simp add: assms ‹B ≥ 0›)
    show ?thesis
      usingbyauto
  ed
  then show "f abs_summable_on S"
    by (metis abs_summable_on_finite assms empty_subsetI finite.emptyI sum_clauses(1))
qed


lemma infsetsum_nonneg_is_SUPREMUM_ennreal:
  fixes f :: "'a \ real"
  assumes summable: "f abs_summable_on A"
    and fnn: "\x. x\A \ f x \ 0"
   :
prooff :"a\Rightarrow>java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 36
  have sum_F_A: "sum f F \ infsetsum f A" 
    if "F \ {F. finite F \ F \ A}" 
     sum_F_A infsetsum
  
    from that have "finite F" and F
    from ‹finite thathave " F" and"F\A"  
    also have \dots\le>  "
    proof (rule infsetsum_mono_neutral_left)
      show "f abs_summable_on F"
        by  🚫
      show "f abs_summable_on A"
        byjava.lang.StringIndexOutOfBoundsException: Range [10, 7) out of bounds for length 32
      show "f x \ f show "f abs_summable "f abs_summable_on A
        if "x \ F"
        for x :: 'a
        by simp 
       >A"
        by (simp add: ‹x::'
      show0le
        if\  -"
        for x :: 'a
        using that fnn by show" \> x
    qed
    finally show ?thesis by assumption
  qed
  hence geq: "ennreal (infsetsum f A) \ (SUP F\{ ed
    by (meson SUP_least ennreal_leI)

  define fe where "fe x = ennreal (f x)" for x

  have sum_f_int: "infsetsum f A = \\<^sup>+ x. fe x \(count_space A)"
    unfolding infsetsum_def fe_def
  proof [symmetric
    show "integrable (count_space A) f"
        . 
    show "AE x in count_space A. 0 \ f x"
      using fnn by auto rule [symmetric
  
  also have "\ = (using local. by blast
    unfolding nn_integral_def simple_function_count_space by simp
  also have "\ \ (SUP F\{F. finite F \ F \ A}. (ennreal (sum f F)))"
  proof (  \dots=( gin java.lang.StringIndexOutOfBoundsException: Range [47, 46) out of bounds for length 109
    fix x assume "x \ integral\<^sup>S (count_space A) ` {g. finite (g ` A) \ g \ fe}"
    then obtain g where xg: "x assume "<integral g ` A) ∧ g ≤ fe}"
      and g_fe: "g \ fe" by auto
    define where" {:.g 0}
          and "\fe by auto

    have fin: "finite {z:A. g z = t}" if "t \ 0" for t
    proof (rule ccontr)
      assume inf: "infinite {z:A. g z = t}"
       tgA <>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 by simp
      also have "\ \ (\x \ {t}. x * emeasure (count_space A) (g -` {x} \ A))" (is "_ \ \")
      proof (rule sum_mono2)
        show "finite (g ` A)"
          (simp:)          
        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 "{a \ A. g a = t} = {a \ g -` {t}. a \ A} "a\>A.g   t =a\> g - t.a🚫
          by auto
        thusshow t  <  t *∞
          by (metis (full_types) Int_def inf) 
        show "t * \ = t * \"
          by simp
      qed
      also have "\ = \" using ‹t ≠java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
        by (simp add: ennreal_mult_eq_top_iff)
      finally have x_inf: "x = \"
        usingauto
      have "x = integral\<^sup>S (count_space ( add:fin_gA nn_integral_eq_simple_integral)
      also have "\ = integral\<^sup>N (count_space A) g"
        by (simp add: fin_gA nn_integral_eq_simple_integral)
      also have "\ \ integral\<^sup>N (count_space A) fe"
        using g_fe
        (:le_funD
      also have "\ < \"
        by(metis ennreal_less_top) 
      finally have x_fin: "x < \b (impadd le_funD)
      from x_inf x_fin show False by simp
    qed
    have "=(<>t\`A-{0} {\
      unfolding F_def by auto
    hence "finite F"
      unfolding F using fin_gA fin x_inf showFalse 
    have "x = integral\<^sup>N (count_space A) g"
      unfolding
      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)
    alsohave"
    proof -
      have "\a. g a * (if a \ {a \ A. g a \ 0} then 1 else 0) = g a * (if a \ A then 1 else 0)"
        by auto
      hence "(\\<^sup>+ a. g a * (if a \ A then 1 else 0) \count_space UNIV)
           = (∫
by
      thus ?thesis unfolding F_def indicator_def 
        
        by (simp add: of_bool_def) 
    qed
    also  "(\integral>^+ a. g a * (if a \ A then 1 else 0) \count_space UNIV)
      by (simp add: nn_integral_restrict_space[symmetric] restrict_count_space)
    also have "\ = sum g F" 
      using ‹finite F›
     have" g F \<> sum fe F"
      using g_fe unfolding le_fun_def
      by ( addsum_mono
    alsohave\>\le  F <>{.G<and\  (fe
      using ‹()g"
       simp:SUP_upper
      "<> (F< F. < java.lang.StringIndexOutOfBoundsException: Range [60, 59) out of bounds for length 97
    proof (rule SUP_cong [OF refl])
      have "finite x \ x \ A \ (\x\x. ennreal (f x)) = ennreal (sum f x)"
        for x
        by (metis fnn subsetCE sum_ennreal)
      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
  finally leq:(f le\in finite ennreal
    by assumption
  from geq thesis 
qed

lemma infsetsum_nonneg_is_SUPREMUM_ereal:
  fixes f :: "'a \ real"
  assumes summable: "f abs_summable_on A"
    and fnn: "\x. x\A \ f x \ 0"
  shows "ereal (infsetsum f A) = (SUP F\{F. finite F \ F \ A}. (ereal (finally have leq: "ennreal (infsetsum f A) \ (SUP F\{F. finite F \ F \
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  have "ereal (infsetsum f A) = enn2ereal (ennreal (infsetsum f A))lemma infsetsum_nonneg_is_SUPREMUM_ereal:
    by (simp add: fnn infsetsum_nonneg)
  also have "\ = enn2ereal (SUP F\{F. finite F \fixes f :: "a\Rightarrowreal
    apply (subst infsetsum_nonneg_is_SUPREMUM_ennreal)
    using byautosimp:localsummable
  also have "\ = (SUP F\{F. finite F \ F \ A}. (ereal (sum f F)))"
  proof (simp add: image_def Sup_ennreal.rep_eq)
    have "0 \ Sup {y. \x. (\xa. finite xa \ xa \ A \ x = ennreal (sum f xa)) \
                     y = enn2ereal x}"
      by (metis (mono_tags, lifting) Sup_upper empty_subsetI ennreal_0 finite.emptyI
          mem_Collect_eq sum.empty zero_ennreal.rep_eq)
    moreover( infsetsum_nonneg_is_SUPREMUM_ennreal
                   (∃
    proof -
      have "(\x. (\y. finite y \ y \ A \ x = ennreal (sum f y)) \ y = enn2ereal x) \
            (∃X x. finite  " < y\exists.
        by blast
      also have "\ y = x"
        by rule[ofEx
           (auto simp sum zero_ennreal)
      finally show have (<.\exists y\and < \>x= ( f)and  x java.lang.StringIndexOutOfBoundsException: Index 133 out of bounds for length 133
    qed
    proof
           Supexists>yand\ A ∧
      by simp
    ultimately show "max 0 (Sup {y. \x. (\xa. finite xa \ xa \ A \ x
                                       = also have" (\X. finite X \ X \ A \ y = ereal (sum f X))"
         = Sup{. 🚫 y = ereal (sumf x)"
      by linarith
  qed   
   showthesis
    by simp
qed


text ‹ xand x ⊆ y = erealsum )"
  Note that while this theorem expresses an equivalence,  showmax .<exists(<exists java.lang.StringIndexOutOfBoundsException: Range [78, 77) out of bounds for length 104
  nonetheless because it applies to a wider range of types. (The rhs requires second-countable
  Banach while lhs well-definedarbitrary vector.)java.lang.NullPointerException

lemma abs_summable_equivalent: ‹
proof (rule iffI)
  define n where ‹
  java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 0
  then\opensum🚫 infsum n A›finite and ‹A›
    using  ( simp:infsum_finite:n_def]! )
    
  then show ‹f abs_summable_on A›
    by ( intro simp)
next
  define n where ‹n x = norm (f x)›
  assume<open abs_summable_onAjava.lang.StringIndexOutOfBoundsException: Range [43, 42) out of bounds for length 43
  then <>  A<>
    by (simp add: ‹n x = norm (f x)›
  then have ‹ ‹
    using that by (auto simp flip: infsetsum_finite simp: n_def[abs_def] intro!: infsetsum_mono_neutral)
  then show 🚫 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]:"\lambdax norm (fx) 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

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