diff --git a/src/HOL/Analysis/Infinite_Sum.thy b/src/HOL/Analysis/Infinite_Sum.thy
--- a/src/HOL/Analysis/Infinite_Sum.thy
+++ b/src/HOL/Analysis/Infinite_Sum.thy
@@ -1,2433 +1,2434 @@
 (*
   Title:    HOL/Analysis/Infinite_Sum.thy
   Author:   Dominique Unruh, University of Tartu
             Manuel Eberl, University of Innsbruck
 
   A theory of sums over possibly infinite sets.
 *)
 
 section \<open>Infinite sums\<close>
 \<^latex>\<open>\label{section:Infinite_Sum}\<close>
 
+theory Infinite_Sum
+  imports
+    Elementary_Topology
+    "HOL-Library.Extended_Nonnegative_Real"
+    "HOL-Library.Complex_Order"
+begin
+
 text \<open>In this theory, we introduce the definition of infinite sums, i.e., sums ranging over an
 infinite, potentially uncountable index set with no particular ordering.
 (This is different from series. Those are sums indexed by natural numbers,
 and the order of the index set matters.)
 
 Our definition is quite standard: $s:=\sum_{x\in A} f(x)$ is the limit of finite sums $s_F:=\sum_{x\in F} f(x)$ for increasing $F$.
 That is, $s$ is the limit of the net $s_F$ where $F$ are finite subsets of $A$ ordered by inclusion.
 We believe that this is the standard definition for such sums.
 See, e.g., Definition 4.11 in \<^cite>\<open>"conway2013course"\<close>.
 This definition is quite general: it is well-defined whenever $f$ takes values in some
 commutative monoid endowed with a Hausdorff topology.
 (Examples are reals, complex numbers, normed vector spaces, and more.)\<close>
 
-theory Infinite_Sum
-  imports
-    Elementary_Topology
-    "HOL-Library.Extended_Nonnegative_Real"
-    "HOL-Library.Complex_Order"
-begin
 
 subsection \<open>Definition and syntax\<close>
 
 definition has_sum :: \<open>('a \<Rightarrow> 'b :: {comm_monoid_add, topological_space}) \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool\<close> where
   \<open>has_sum f A x \<longleftrightarrow> (sum f \<longlongrightarrow> x) (finite_subsets_at_top A)\<close>
 
 definition summable_on :: "('a \<Rightarrow> 'b::{comm_monoid_add, topological_space}) \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "summable'_on" 46) where
   "f summable_on A \<longleftrightarrow> (\<exists>x. has_sum f A x)"
 
 definition infsum :: "('a \<Rightarrow> 'b::{comm_monoid_add,t2_space}) \<Rightarrow> 'a set \<Rightarrow> 'b" where
   "infsum f A = (if f summable_on A then Lim (finite_subsets_at_top A) (sum f) else 0)"
 
 abbreviation abs_summable_on :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "abs'_summable'_on" 46) where
   "f abs_summable_on A \<equiv> (\<lambda>x. norm (f x)) summable_on A"
 
 syntax (ASCII)
   "_infsum" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b::topological_comm_monoid_add"  ("(3INFSUM (_/:_)./ _)" [0, 51, 10] 10)
 syntax
   "_infsum" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b::topological_comm_monoid_add"  ("(2\<Sum>\<^sub>\<infinity>(_/\<in>_)./ _)" [0, 51, 10] 10)
 translations \<comment> \<open>Beware of argument permutation!\<close>
   "\<Sum>\<^sub>\<infinity>i\<in>A. b" \<rightleftharpoons> "CONST infsum (\<lambda>i. b) A"
 
 syntax (ASCII)
   "_univinfsum" :: "pttrn \<Rightarrow> 'a \<Rightarrow> 'a"  ("(3INFSUM _./ _)" [0, 10] 10)
 syntax
   "_univinfsum" :: "pttrn \<Rightarrow> 'a \<Rightarrow> 'a"  ("(2\<Sum>\<^sub>\<infinity>_./ _)" [0, 10] 10)
 translations
   "\<Sum>\<^sub>\<infinity>x. t" \<rightleftharpoons> "CONST infsum (\<lambda>x. t) (CONST UNIV)"
 
 syntax (ASCII)
   "_qinfsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a"  ("(3INFSUM _ |/ _./ _)" [0, 0, 10] 10)
 syntax
   "_qinfsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a"  ("(2\<Sum>\<^sub>\<infinity>_ | (_)./ _)" [0, 0, 10] 10)
 translations
   "\<Sum>\<^sub>\<infinity>x|P. t" => "CONST infsum (\<lambda>x. t) {x. P}"
 
 print_translation \<open>
 let
   fun sum_tr' [Abs (x, Tx, t), Const (@{const_syntax Collect}, _) $ Abs (y, Ty, P)] =
         if x <> y then raise Match
         else
           let
             val x' = Syntax_Trans.mark_bound_body (x, Tx);
             val t' = subst_bound (x', t);
             val P' = subst_bound (x', P);
           in
             Syntax.const @{syntax_const "_qinfsum"} $ Syntax_Trans.mark_bound_abs (x, Tx) $ P' $ t'
           end
     | sum_tr' _ = raise Match;
 in [(@{const_syntax infsum}, K sum_tr')] end
 \<close>
 
 subsection \<open>General properties\<close>
 
 lemma infsumI:
   fixes f g :: \<open>'a \<Rightarrow> 'b::{comm_monoid_add, t2_space}\<close>
   assumes \<open>has_sum f A x\<close>
   shows \<open>infsum f A = x\<close>
   by (metis assms finite_subsets_at_top_neq_bot infsum_def summable_on_def has_sum_def tendsto_Lim)
 
 lemma infsum_eqI:
   fixes f g :: \<open>'a \<Rightarrow> 'b::{comm_monoid_add, t2_space}\<close>
   assumes \<open>x = y\<close>
   assumes \<open>has_sum f A x\<close>
   assumes \<open>has_sum g B y\<close>
   shows \<open>infsum f A = infsum g B\<close>
   by (metis assms(1) assms(2) assms(3) finite_subsets_at_top_neq_bot infsum_def summable_on_def has_sum_def tendsto_Lim)
 
 lemma infsum_eqI':
   fixes f g :: \<open>'a \<Rightarrow> 'b::{comm_monoid_add, t2_space}\<close>
   assumes \<open>\<And>x. has_sum f A x \<longleftrightarrow> has_sum g B x\<close>
   shows \<open>infsum f A = infsum g B\<close>
   by (metis assms infsum_def infsum_eqI summable_on_def)
 
 lemma infsum_not_exists:
   fixes f :: \<open>'a \<Rightarrow> 'b::{comm_monoid_add, t2_space}\<close>
   assumes \<open>\<not> f summable_on A\<close>
   shows \<open>infsum f A = 0\<close>
   by (simp add: assms infsum_def)
 
 lemma summable_iff_has_sum_infsum: "f summable_on A \<longleftrightarrow> has_sum f A (infsum f A)"
   using infsumI summable_on_def by blast
 
 lemma has_sum_infsum[simp]:
   assumes \<open>f summable_on S\<close>
   shows \<open>has_sum f S (infsum f S)\<close>
   using assms by (auto simp: summable_on_def infsum_def has_sum_def tendsto_Lim)
 
 lemma has_sum_cong_neutral:
   fixes f g :: \<open>'a \<Rightarrow> 'b::{comm_monoid_add, topological_space}\<close>
   assumes \<open>\<And>x. x\<in>T-S \<Longrightarrow> g x = 0\<close>
   assumes \<open>\<And>x. x\<in>S-T \<Longrightarrow> f x = 0\<close>
   assumes \<open>\<And>x. x\<in>S\<inter>T \<Longrightarrow> f x = g x\<close>
   shows "has_sum f S x \<longleftrightarrow> has_sum g T x"
 proof -
   have \<open>eventually P (filtermap (sum f) (finite_subsets_at_top S))
       = eventually P (filtermap (sum g) (finite_subsets_at_top T))\<close> for P
   proof 
     assume \<open>eventually P (filtermap (sum f) (finite_subsets_at_top S))\<close>
     then obtain F0 where \<open>finite F0\<close> and \<open>F0 \<subseteq> S\<close> and F0_P: \<open>\<And>F. finite F \<Longrightarrow> F \<subseteq> S \<Longrightarrow> F \<supseteq> F0 \<Longrightarrow> P (sum f F)\<close>
       by (metis (no_types, lifting) eventually_filtermap eventually_finite_subsets_at_top)
     define F0' where \<open>F0' = F0 \<inter> T\<close>
     have [simp]: \<open>finite F0'\<close> \<open>F0' \<subseteq> T\<close>
       by (simp_all add: F0'_def \<open>finite F0\<close>)
     have \<open>P (sum g F)\<close> if \<open>finite F\<close> \<open>F \<subseteq> T\<close> \<open>F \<supseteq> F0'\<close> for F
     proof -
       have \<open>P (sum f ((F\<inter>S) \<union> (F0\<inter>S)))\<close>
         by (intro F0_P) (use \<open>F0 \<subseteq> S\<close> \<open>finite F0\<close> that in auto)
       also have \<open>sum f ((F\<inter>S) \<union> (F0\<inter>S)) = sum g F\<close>
         by (intro sum.mono_neutral_cong) (use that \<open>finite F0\<close> F0'_def assms in auto)
       finally show ?thesis .
     qed
     with \<open>F0' \<subseteq> T\<close> \<open>finite F0'\<close> show \<open>eventually P (filtermap (sum g) (finite_subsets_at_top T))\<close>
       by (metis (no_types, lifting) eventually_filtermap eventually_finite_subsets_at_top)
   next
     assume \<open>eventually P (filtermap (sum g) (finite_subsets_at_top T))\<close>
     then obtain F0 where \<open>finite F0\<close> and \<open>F0 \<subseteq> T\<close> and F0_P: \<open>\<And>F. finite F \<Longrightarrow> F \<subseteq> T \<Longrightarrow> F \<supseteq> F0 \<Longrightarrow> P (sum g F)\<close>
       by (metis (no_types, lifting) eventually_filtermap eventually_finite_subsets_at_top)
     define F0' where \<open>F0' = F0 \<inter> S\<close>
     have [simp]: \<open>finite F0'\<close> \<open>F0' \<subseteq> S\<close>
       by (simp_all add: F0'_def \<open>finite F0\<close>)
     have \<open>P (sum f F)\<close> if \<open>finite F\<close> \<open>F \<subseteq> S\<close> \<open>F \<supseteq> F0'\<close> for F
     proof -
       have \<open>P (sum g ((F\<inter>T) \<union> (F0\<inter>T)))\<close>
         by (intro F0_P) (use \<open>F0 \<subseteq> T\<close> \<open>finite F0\<close> that in auto)
       also have \<open>sum g ((F\<inter>T) \<union> (F0\<inter>T)) = sum f F\<close>
         by (intro sum.mono_neutral_cong) (use that \<open>finite F0\<close> F0'_def assms in auto)
       finally show ?thesis .
     qed
     with \<open>F0' \<subseteq> S\<close> \<open>finite F0'\<close> show \<open>eventually P (filtermap (sum f) (finite_subsets_at_top S))\<close>
       by (metis (no_types, lifting) eventually_filtermap eventually_finite_subsets_at_top)
   qed
 
   then have tendsto_x: "(sum f \<longlongrightarrow> x) (finite_subsets_at_top S) \<longleftrightarrow> (sum g \<longlongrightarrow> x) (finite_subsets_at_top T)" for x
     by (simp add: le_filter_def filterlim_def)
 
   then show ?thesis
     by (simp add: has_sum_def)
 qed
 
 lemma summable_on_cong_neutral: 
   fixes f g :: \<open>'a \<Rightarrow> 'b::{comm_monoid_add, topological_space}\<close>
   assumes \<open>\<And>x. x\<in>T-S \<Longrightarrow> g x = 0\<close>
   assumes \<open>\<And>x. x\<in>S-T \<Longrightarrow> f x = 0\<close>
   assumes \<open>\<And>x. x\<in>S\<inter>T \<Longrightarrow> f x = g x\<close>
   shows "f summable_on S \<longleftrightarrow> g summable_on T"
   using has_sum_cong_neutral[of T S g f, OF assms]
   by (simp add: summable_on_def)
 
 lemma infsum_cong_neutral: 
   fixes f g :: \<open>'a \<Rightarrow> 'b::{comm_monoid_add, t2_space}\<close>
   assumes \<open>\<And>x. x\<in>T-S \<Longrightarrow> g x = 0\<close>
   assumes \<open>\<And>x. x\<in>S-T \<Longrightarrow> f x = 0\<close>
   assumes \<open>\<And>x. x\<in>S\<inter>T \<Longrightarrow> f x = g x\<close>
   shows \<open>infsum f S = infsum g T\<close>
   by (smt (verit, best) assms has_sum_cong_neutral infsum_eqI')
 
 lemma has_sum_cong: 
   assumes "\<And>x. x\<in>A \<Longrightarrow> f x = g x"
   shows "has_sum f A x \<longleftrightarrow> has_sum g A x"
   using assms by (intro has_sum_cong_neutral) auto
 
 lemma summable_on_cong:
   assumes "\<And>x. x\<in>A \<Longrightarrow> f x = g x"
   shows "f summable_on A \<longleftrightarrow> g summable_on A"
   by (metis assms summable_on_def has_sum_cong)
 
 lemma infsum_cong:
   assumes "\<And>x. x\<in>A \<Longrightarrow> f x = g x"
   shows "infsum f A = infsum g A"
   using assms infsum_eqI' has_sum_cong by blast
 
 lemma summable_on_cofin_subset:
   fixes f :: "'a \<Rightarrow> 'b::topological_ab_group_add"
   assumes "f summable_on A" and [simp]: "finite F"
   shows "f summable_on (A - F)"
 proof -
   from assms(1) obtain x where lim_f: "(sum f \<longlongrightarrow> x) (finite_subsets_at_top A)"
     unfolding summable_on_def has_sum_def by auto
   define F' where "F' = F\<inter>A"
   with assms have "finite F'" and "A-F = A-F'"
     by auto
   have "filtermap ((\<union>)F') (finite_subsets_at_top (A-F))
       \<le> finite_subsets_at_top A"
   proof (rule filter_leI)
     fix P assume "eventually P (finite_subsets_at_top A)"
     then obtain X where [simp]: "finite X" and XA: "X \<subseteq> A" 
       and P: "\<forall>Y. finite Y \<and> X \<subseteq> Y \<and> Y \<subseteq> A \<longrightarrow> P Y"
       unfolding eventually_finite_subsets_at_top by auto
     define X' where "X' = X-F"
     hence [simp]: "finite X'" and [simp]: "X' \<subseteq> A-F"
       using XA by auto
     hence "finite Y \<and> X' \<subseteq> Y \<and> Y \<subseteq> A - F \<longrightarrow> P (F' \<union> Y)" for Y
       using P XA unfolding X'_def using F'_def \<open>finite F'\<close> by blast
     thus "eventually P (filtermap ((\<union>) F') (finite_subsets_at_top (A - F)))"
       unfolding eventually_filtermap eventually_finite_subsets_at_top
       by (rule_tac x=X' in exI, simp)
   qed
   with lim_f have "(sum f \<longlongrightarrow> x) (filtermap ((\<union>)F') (finite_subsets_at_top (A-F)))"
     using tendsto_mono by blast
   have "((\<lambda>G. sum f (F' \<union> G)) \<longlongrightarrow> x) (finite_subsets_at_top (A - F))"
     if "((sum f \<circ> (\<union>) F') \<longlongrightarrow> x) (finite_subsets_at_top (A - F))"
     using that unfolding o_def by auto
   hence "((\<lambda>G. sum f (F' \<union> G)) \<longlongrightarrow> x) (finite_subsets_at_top (A-F))"
     using tendsto_compose_filtermap [symmetric]
     by (simp add: \<open>(sum f \<longlongrightarrow> x) (filtermap ((\<union>) F') (finite_subsets_at_top (A - F)))\<close> 
         tendsto_compose_filtermap)
   have "\<forall>Y. finite Y \<and> Y \<subseteq> A - F \<longrightarrow> sum f (F' \<union> Y) = sum f F' + sum f Y"
     by (metis Diff_disjoint Int_Diff \<open>A - F = A - F'\<close> \<open>finite F'\<close> inf.orderE sum.union_disjoint)
   hence "\<forall>\<^sub>F x in finite_subsets_at_top (A - F). sum f (F' \<union> x) = sum f F' + sum f x"
     unfolding eventually_finite_subsets_at_top
     using exI [where x = "{}"]
     by (simp add: \<open>\<And>P. P {} \<Longrightarrow> \<exists>x. P x\<close>) 
   hence "((\<lambda>G. sum f F' + sum f G) \<longlongrightarrow> x) (finite_subsets_at_top (A-F))"
     using tendsto_cong [THEN iffD1 , rotated]
       \<open>((\<lambda>G. sum f (F' \<union> G)) \<longlongrightarrow> x) (finite_subsets_at_top (A - F))\<close> by fastforce
   hence "((\<lambda>G. sum f F' + sum f G) \<longlongrightarrow> sum f F' + (x-sum f F')) (finite_subsets_at_top (A-F))"
     by simp
   hence "(sum f \<longlongrightarrow> x - sum f F') (finite_subsets_at_top (A-F))"
     using tendsto_add_const_iff by blast    
   thus "f summable_on (A - F)"
     unfolding summable_on_def has_sum_def by auto
 qed
 
 lemma
   fixes f :: "'a \<Rightarrow> 'b::{topological_ab_group_add}"
   assumes \<open>has_sum f B b\<close> and \<open>has_sum f A a\<close> and AB: "A \<subseteq> B"
   shows has_sum_Diff: "has_sum f (B - A) (b - a)"
 proof -
   have finite_subsets1:
     "finite_subsets_at_top (B - A) \<le> filtermap (\<lambda>F. F - A) (finite_subsets_at_top B)"
   proof (rule filter_leI)
     fix P assume "eventually P (filtermap (\<lambda>F. F - A) (finite_subsets_at_top B))"
     then obtain X where "finite X" and "X \<subseteq> B" 
       and P: "finite Y \<and> X \<subseteq> Y \<and> Y \<subseteq> B \<longrightarrow> P (Y - A)" for Y
       unfolding eventually_filtermap eventually_finite_subsets_at_top by auto
 
     hence "finite (X-A)" and "X-A \<subseteq> B - A"
       by auto
     moreover have "finite Y \<and> X-A \<subseteq> Y \<and> Y \<subseteq> B - A \<longrightarrow> P Y" for Y
       using P[where Y="Y\<union>X"] \<open>finite X\<close> \<open>X \<subseteq> B\<close>
       by (metis Diff_subset Int_Diff Un_Diff finite_Un inf.orderE le_sup_iff sup.orderE sup_ge2)
     ultimately show "eventually P (finite_subsets_at_top (B - A))"
       unfolding eventually_finite_subsets_at_top by meson
   qed
   have finite_subsets2: 
     "filtermap (\<lambda>F. F \<inter> A) (finite_subsets_at_top B) \<le> finite_subsets_at_top A"
     apply (rule filter_leI)
       using assms unfolding eventually_filtermap eventually_finite_subsets_at_top
       by (metis Int_subset_iff finite_Int inf_le2 subset_trans)
 
   from assms(1) have limB: "(sum f \<longlongrightarrow> b) (finite_subsets_at_top B)"
     using has_sum_def by auto
   from assms(2) have limA: "(sum f \<longlongrightarrow> a) (finite_subsets_at_top A)"
     using has_sum_def by blast
   have "((\<lambda>F. sum f (F\<inter>A)) \<longlongrightarrow> a) (finite_subsets_at_top B)"
   proof (subst asm_rl [of "(\<lambda>F. sum f (F\<inter>A)) = sum f \<circ> (\<lambda>F. F\<inter>A)"])
     show "(\<lambda>F. sum f (F \<inter> A)) = sum f \<circ> (\<lambda>F. F \<inter> A)"
       unfolding o_def by auto
     show "((sum f \<circ> (\<lambda>F. F \<inter> A)) \<longlongrightarrow> a) (finite_subsets_at_top B)"
       unfolding o_def 
       using tendsto_compose_filtermap finite_subsets2 limA tendsto_mono
         \<open>(\<lambda>F. sum f (F \<inter> A)) = sum f \<circ> (\<lambda>F. F \<inter> A)\<close> by fastforce
   qed
 
   with limB have "((\<lambda>F. sum f F - sum f (F\<inter>A)) \<longlongrightarrow> b - a) (finite_subsets_at_top B)"
     using tendsto_diff by blast
   have "sum f X - sum f (X \<inter> A) = sum f (X - A)" if "finite X" and "X \<subseteq> B" for X :: "'a set"
     using that by (metis add_diff_cancel_left' sum.Int_Diff)
   hence "\<forall>\<^sub>F x in finite_subsets_at_top B. sum f x - sum f (x \<inter> A) = sum f (x - A)"
     by (rule eventually_finite_subsets_at_top_weakI)  
   hence "((\<lambda>F. sum f (F-A)) \<longlongrightarrow> b - a) (finite_subsets_at_top B)"
     using tendsto_cong [THEN iffD1 , rotated]
       \<open>((\<lambda>F. sum f F - sum f (F \<inter> A)) \<longlongrightarrow> b - a) (finite_subsets_at_top B)\<close> by fastforce
   hence "(sum f \<longlongrightarrow> b - a) (filtermap (\<lambda>F. F-A) (finite_subsets_at_top B))"
     by (subst tendsto_compose_filtermap[symmetric], simp add: o_def)
   thus ?thesis
     using finite_subsets1 has_sum_def tendsto_mono by blast
 qed
 
 
 lemma
   fixes f :: "'a \<Rightarrow> 'b::{topological_ab_group_add}"
   assumes "f summable_on B" and "f summable_on A" and "A \<subseteq> B"
   shows summable_on_Diff: "f summable_on (B-A)"
   by (meson assms summable_on_def has_sum_Diff)
 
 lemma
   fixes f :: "'a \<Rightarrow> 'b::{topological_ab_group_add,t2_space}"
   assumes "f summable_on B" and "f summable_on A" and AB: "A \<subseteq> B"
   shows infsum_Diff: "infsum f (B - A) = infsum f B - infsum f A"
   by (metis AB assms has_sum_Diff infsumI summable_on_def)
 
 lemma has_sum_mono_neutral:
   fixes f :: "'a\<Rightarrow>'b::{ordered_comm_monoid_add,linorder_topology}"
   (* Does this really require a linorder topology? (Instead of order topology.) *)
   assumes \<open>has_sum f A a\<close> and "has_sum g B b"
   assumes \<open>\<And>x. x \<in> A\<inter>B \<Longrightarrow> f x \<le> g x\<close>
   assumes \<open>\<And>x. x \<in> A-B \<Longrightarrow> f x \<le> 0\<close>
   assumes \<open>\<And>x. x \<in> B-A \<Longrightarrow> g x \<ge> 0\<close>
   shows "a \<le> b"
 proof -
   define f' g' where \<open>f' x = (if x \<in> A then f x else 0)\<close> and \<open>g' x = (if x \<in> B then g x else 0)\<close> for x
   have [simp]: \<open>f summable_on A\<close> \<open>g summable_on B\<close>
     using assms(1,2) summable_on_def by auto
   have \<open>has_sum f' (A\<union>B) a\<close>
     by (smt (verit, best) DiffE IntE Un_iff f'_def assms(1) has_sum_cong_neutral)
   then have f'_lim: \<open>(sum f' \<longlongrightarrow> a) (finite_subsets_at_top (A\<union>B))\<close>
     by (meson has_sum_def)
   have \<open>has_sum g' (A\<union>B) b\<close>
   by (smt (verit, best) DiffD1 DiffD2 IntE UnCI g'_def assms(2) has_sum_cong_neutral)
   then have g'_lim: \<open>(sum g' \<longlongrightarrow> b) (finite_subsets_at_top (A\<union>B))\<close>
     using has_sum_def by blast
 
   have "\<And>X i. \<lbrakk>X \<subseteq> A \<union> B; i \<in> X\<rbrakk> \<Longrightarrow> f' i \<le> g' i"
     using assms by (auto simp: f'_def g'_def)
   then have \<open>\<forall>\<^sub>F x in finite_subsets_at_top (A \<union> B). sum f' x \<le> sum g' x\<close>
     by (intro eventually_finite_subsets_at_top_weakI sum_mono)
   then show ?thesis
     using f'_lim finite_subsets_at_top_neq_bot g'_lim tendsto_le by blast
 qed
 
 lemma infsum_mono_neutral:
   fixes f :: "'a\<Rightarrow>'b::{ordered_comm_monoid_add,linorder_topology}"
   assumes "f summable_on A" and "g summable_on B"
   assumes \<open>\<And>x. x \<in> A\<inter>B \<Longrightarrow> f x \<le> g x\<close>
   assumes \<open>\<And>x. x \<in> A-B \<Longrightarrow> f x \<le> 0\<close>
   assumes \<open>\<And>x. x \<in> B-A \<Longrightarrow> g x \<ge> 0\<close>
   shows "infsum f A \<le> infsum g B"
   by (rule has_sum_mono_neutral[of f A _ g B _]) (use assms in \<open>auto intro: has_sum_infsum\<close>)
 
 lemma has_sum_mono:
   fixes f :: "'a\<Rightarrow>'b::{ordered_comm_monoid_add,linorder_topology}"
   assumes "has_sum f A x" and "has_sum g A y"
   assumes \<open>\<And>x. x \<in> A \<Longrightarrow> f x \<le> g x\<close>
   shows "x \<le> y"
   using assms has_sum_mono_neutral by force
 
 lemma infsum_mono:
   fixes f :: "'a\<Rightarrow>'b::{ordered_comm_monoid_add,linorder_topology}"
   assumes "f summable_on A" and "g summable_on A"
   assumes \<open>\<And>x. x \<in> A \<Longrightarrow> f x \<le> g x\<close>
   shows "infsum f A \<le> infsum g A"
   by (meson assms has_sum_infsum has_sum_mono)
 
 lemma has_sum_finite[simp]:
   assumes "finite F"
   shows "has_sum f F (sum f F)"
   using assms
   by (auto intro: tendsto_Lim simp: finite_subsets_at_top_finite infsum_def has_sum_def principal_eq_bot_iff)
 
 lemma summable_on_finite[simp]:
   fixes f :: \<open>'a \<Rightarrow> 'b::{comm_monoid_add,topological_space}\<close>
   assumes "finite F"
   shows "f summable_on F"
   using assms summable_on_def has_sum_finite by blast
 
 lemma infsum_finite[simp]:
   assumes "finite F"
   shows "infsum f F = sum f F"
   using assms by (auto intro: tendsto_Lim simp: finite_subsets_at_top_finite infsum_def principal_eq_bot_iff)
 
 lemma has_sum_finite_approximation:
   fixes f :: "'a \<Rightarrow> 'b::{comm_monoid_add,metric_space}"
   assumes "has_sum f A x" and "\<epsilon> > 0"
   shows "\<exists>F. finite F \<and> F \<subseteq> A \<and> dist (sum f F) x \<le> \<epsilon>"
 proof -
   have "(sum f \<longlongrightarrow> x) (finite_subsets_at_top A)"
     by (meson assms(1) has_sum_def)
   hence *: "\<forall>\<^sub>F F in (finite_subsets_at_top A). dist (sum f F) x < \<epsilon>"
     using assms(2) by (rule tendstoD)
   thus ?thesis
     unfolding eventually_finite_subsets_at_top by fastforce
 qed
 
 lemma infsum_finite_approximation:
   fixes f :: "'a \<Rightarrow> 'b::{comm_monoid_add,metric_space}"
   assumes "f summable_on A" and "\<epsilon> > 0"
   shows "\<exists>F. finite F \<and> F \<subseteq> A \<and> dist (sum f F) (infsum f A) \<le> \<epsilon>"
 proof -
   from assms have "has_sum f A (infsum f A)"
     by (simp add: summable_iff_has_sum_infsum)
   from this and \<open>\<epsilon> > 0\<close> show ?thesis
     by (rule has_sum_finite_approximation)
 qed
 
 lemma abs_summable_summable:
   fixes f :: \<open>'a \<Rightarrow> 'b :: banach\<close>
   assumes \<open>f abs_summable_on A\<close>
   shows \<open>f summable_on A\<close>
 proof -
   from assms obtain L where lim: \<open>(sum (\<lambda>x. norm (f x)) \<longlongrightarrow> L) (finite_subsets_at_top A)\<close>
     unfolding has_sum_def summable_on_def by blast
   then have *: \<open>cauchy_filter (filtermap (sum (\<lambda>x. norm (f x))) (finite_subsets_at_top A))\<close>
     by (auto intro!: nhds_imp_cauchy_filter simp: filterlim_def)
   have \<open>\<exists>P. eventually P (finite_subsets_at_top A) \<and>
               (\<forall>F F'. P F \<and> P F' \<longrightarrow> dist (sum f F) (sum f F') < e)\<close> if \<open>e>0\<close> for e
   proof -
     define d P where \<open>d = e/4\<close> and \<open>P F \<longleftrightarrow> finite F \<and> F \<subseteq> A \<and> dist (sum (\<lambda>x. norm (f x)) F) L < d\<close> for F
     then have \<open>d > 0\<close>
       by (simp add: d_def that)
     have ev_P: \<open>eventually P (finite_subsets_at_top A)\<close>
       using lim
       by (auto simp add: P_def[abs_def] \<open>0 < d\<close> eventually_conj_iff eventually_finite_subsets_at_top_weakI tendsto_iff)
     
     moreover have \<open>dist (sum f F1) (sum f F2) < e\<close> if \<open>P F1\<close> and \<open>P F2\<close> for F1 F2
     proof -
       from ev_P
       obtain F' where \<open>finite F'\<close> and \<open>F' \<subseteq> A\<close> and P_sup_F': \<open>finite F \<and> F \<supseteq> F' \<and> F \<subseteq> A \<Longrightarrow> P F\<close> for F
         by atomize_elim (simp add: eventually_finite_subsets_at_top)
       define F where \<open>F = F' \<union> F1 \<union> F2\<close>
       have \<open>finite F\<close> and \<open>F \<subseteq> A\<close>
         using F_def P_def[abs_def] that \<open>finite F'\<close> \<open>F' \<subseteq> A\<close> by auto
       have dist_F: \<open>dist (sum (\<lambda>x. norm (f x)) F) L < d\<close>
         by (metis F_def \<open>F \<subseteq> A\<close> P_def P_sup_F' \<open>finite F\<close> le_supE order_refl)
 
       have dist_F_subset: \<open>dist (sum f F) (sum f F') < 2*d\<close> if F': \<open>F' \<subseteq> F\<close> \<open>P F'\<close> for F'
       proof -
         have \<open>dist (sum f F) (sum f F') = norm (sum f (F-F'))\<close>
           unfolding dist_norm using \<open>finite F\<close> F' by (subst sum_diff) auto
         also have \<open>\<dots> \<le> norm (\<Sum>x\<in>F-F'. norm (f x))\<close>
           by (rule order.trans[OF sum_norm_le[OF order.refl]]) auto
         also have \<open>\<dots> = dist (\<Sum>x\<in>F. norm (f x)) (\<Sum>x\<in>F'. norm (f x))\<close>
           unfolding dist_norm using \<open>finite F\<close> F' by (subst sum_diff) auto
         also have \<open>\<dots> < 2 * d\<close>
           using dist_F F' unfolding P_def dist_norm real_norm_def by linarith
         finally show \<open>dist (sum f F) (sum f F') < 2*d\<close> .
       qed
 
       have \<open>dist (sum f F1) (sum f F2) \<le> dist (sum f F) (sum f F1) + dist (sum f F) (sum f F2)\<close>
         by (rule dist_triangle3)
       also have \<open>\<dots> < 2 * d + 2 * d\<close>
         by (intro add_strict_mono dist_F_subset that) (auto simp: F_def)
       also have \<open>\<dots> \<le> e\<close>
         by (auto simp: d_def)
       finally show \<open>dist (sum f F1) (sum f F2) < e\<close> .
     qed
     then show ?thesis
       using ev_P by blast
   qed
   then have \<open>cauchy_filter (filtermap (sum f) (finite_subsets_at_top A))\<close>
     by (simp add: cauchy_filter_metric_filtermap)
   moreover have "complete (UNIV::'b set)"
     by (meson Cauchy_convergent UNIV_I complete_def convergent_def)
   ultimately obtain L' where \<open>(sum f \<longlongrightarrow> L') (finite_subsets_at_top A)\<close>
     using complete_uniform[where S=UNIV] by (force simp add: filterlim_def)
   then show ?thesis
     using summable_on_def has_sum_def by blast
 qed
 
 text \<open>The converse of @{thm [source] abs_summable_summable} does not hold:
   Consider the Hilbert space of square-summable sequences.
   Let $e_i$ denote the sequence with 1 in the $i$th position and 0 elsewhere.
   Let $f(i) := e_i/i$ for $i\geq1$. We have \<^term>\<open>\<not> f abs_summable_on UNIV\<close> because $\lVert f(i)\rVert=1/i$
   and thus the sum over $\lVert f(i)\rVert$ diverges. On the other hand, we have \<^term>\<open>f summable_on UNIV\<close>;
   the limit is the sequence with $1/i$ in the $i$th position.
 
   (We have not formalized this separating example here because to the best of our knowledge,
   this Hilbert space has not been formalized in Isabelle/HOL yet.)\<close>
 
 lemma norm_has_sum_bound:
   fixes f :: "'b \<Rightarrow> 'a::real_normed_vector"
     and A :: "'b set"
   assumes "has_sum (\<lambda>x. norm (f x)) A n"
   assumes "has_sum f A a"
   shows "norm a \<le> n"
 proof -
   have "norm a \<le> n + \<epsilon>" if "\<epsilon>>0" for \<epsilon>
   proof-
     have "\<exists>F. norm (a - sum f F) \<le> \<epsilon> \<and> finite F \<and> F \<subseteq> A"
       using has_sum_finite_approximation[where A=A and f=f and \<epsilon>="\<epsilon>"] assms \<open>0 < \<epsilon>\<close>
       by (metis dist_commute dist_norm)
     then obtain F where "norm (a - sum f F) \<le> \<epsilon>"
       and "finite F" and "F \<subseteq> A"
       by (simp add: atomize_elim)
     hence "norm a \<le> norm (sum f F) + \<epsilon>"
       by (metis add.commute diff_add_cancel dual_order.refl norm_triangle_mono)
     also have "\<dots> \<le> sum (\<lambda>x. norm (f x)) F + \<epsilon>"
       using norm_sum by auto
     also have "\<dots> \<le> n + \<epsilon>"
     proof (intro add_right_mono [OF has_sum_mono_neutral])
       show "has_sum (\<lambda>x. norm (f x)) F (\<Sum>x\<in>F. norm (f x))"
         by (simp add: \<open>finite F\<close>)
     qed (use \<open>F \<subseteq> A\<close> assms in auto)
     finally show ?thesis 
       by assumption
   qed
   thus ?thesis
     using linordered_field_class.field_le_epsilon by blast
 qed
 
 lemma norm_infsum_bound:
   fixes f :: "'b \<Rightarrow> 'a::real_normed_vector"
     and A :: "'b set"
   assumes "f abs_summable_on A"
   shows "norm (infsum f A) \<le> infsum (\<lambda>x. norm (f x)) A"
 proof (cases "f summable_on A")
   case True
   have "has_sum (\<lambda>x. norm (f x)) A (\<Sum>\<^sub>\<infinity>x\<in>A. norm (f x))"
     by (simp add: assms)
   then show ?thesis
     by (metis True has_sum_infsum norm_has_sum_bound)
 next
   case False
   obtain t where t_def: "(sum (\<lambda>x. norm (f x)) \<longlongrightarrow> t) (finite_subsets_at_top A)"
     using assms unfolding summable_on_def has_sum_def by blast
   have sumpos: "sum (\<lambda>x. norm (f x)) X \<ge> 0"
     for X
     by (simp add: sum_nonneg)
   have tgeq0:"t \<ge> 0"
   proof(rule ccontr)
     define S::"real set" where "S = {s. s < 0}"
     assume "\<not> 0 \<le> t"
     hence "t < 0" by simp
     hence "t \<in> S"
       unfolding S_def by blast
     moreover have "open S"
     proof-
       have "closed {s::real. s \<ge> 0}"
         using Elementary_Topology.closed_sequential_limits[where S = "{s::real. s \<ge> 0}"]
         by (metis Lim_bounded2 mem_Collect_eq)
       moreover have "{s::real. s \<ge> 0} = UNIV - S"
         unfolding S_def by auto
       ultimately have "closed (UNIV - S)"
         by simp
       thus ?thesis
         by (simp add: Compl_eq_Diff_UNIV open_closed) 
     qed
     ultimately have "\<forall>\<^sub>F X in finite_subsets_at_top A. (\<Sum>x\<in>X. norm (f x)) \<in> S"
       using t_def unfolding tendsto_def by blast
     hence "\<exists>X. (\<Sum>x\<in>X. norm (f x)) \<in> S"
       by (metis (no_types, lifting) eventually_mono filterlim_iff finite_subsets_at_top_neq_bot tendsto_Lim)
     then obtain X where "(\<Sum>x\<in>X. norm (f x)) \<in> S"
       by blast
     hence "(\<Sum>x\<in>X. norm (f x)) < 0"
       unfolding S_def by auto      
     thus False by (simp add: leD sumpos)
   qed
   have "\<exists>!h. (sum (\<lambda>x. norm (f x)) \<longlongrightarrow> h) (finite_subsets_at_top A)"
     using t_def finite_subsets_at_top_neq_bot tendsto_unique by blast
   hence "t = (Topological_Spaces.Lim (finite_subsets_at_top A) (sum (\<lambda>x. norm (f x))))"
     using t_def unfolding Topological_Spaces.Lim_def
     by (metis the_equality)     
   hence "Lim (finite_subsets_at_top A) (sum (\<lambda>x. norm (f x))) \<ge> 0"
     using tgeq0 by blast
   thus ?thesis unfolding infsum_def 
     using False by auto
 qed
 
 lemma infsum_tendsto:
   assumes \<open>f summable_on S\<close>
   shows \<open>((\<lambda>F. sum f F) \<longlongrightarrow> infsum f S) (finite_subsets_at_top S)\<close>
   using assms by (simp flip: has_sum_def)
 
 lemma has_sum_0: 
   assumes \<open>\<And>x. x\<in>M \<Longrightarrow> f x = 0\<close>
   shows \<open>has_sum f M 0\<close>
   by (metis assms finite.intros(1) has_sum_cong has_sum_cong_neutral has_sum_finite sum.neutral_const)
 
 lemma summable_on_0:
   assumes \<open>\<And>x. x\<in>M \<Longrightarrow> f x = 0\<close>
   shows \<open>f summable_on M\<close>
   using assms summable_on_def has_sum_0 by blast
 
 lemma infsum_0:
   assumes \<open>\<And>x. x\<in>M \<Longrightarrow> f x = 0\<close>
   shows \<open>infsum f M = 0\<close>
   by (metis assms finite_subsets_at_top_neq_bot infsum_def has_sum_0 has_sum_def tendsto_Lim)
 
 text \<open>Variants of @{thm [source] infsum_0} etc. suitable as simp-rules\<close>
 lemma infsum_0_simp[simp]: \<open>infsum (\<lambda>_. 0) M = 0\<close>
   by (simp_all add: infsum_0)
 lemma summable_on_0_simp[simp]: \<open>(\<lambda>_. 0) summable_on M\<close>
   by (simp_all add: summable_on_0)
 lemma has_sum_0_simp[simp]: \<open>has_sum (\<lambda>_. 0) M 0\<close>
   by (simp_all add: has_sum_0)
 
 
 lemma has_sum_add:
   fixes f g :: "'a \<Rightarrow> 'b::{topological_comm_monoid_add}"
   assumes \<open>has_sum f A a\<close>
   assumes \<open>has_sum g A b\<close>
   shows \<open>has_sum (\<lambda>x. f x + g x) A (a + b)\<close>
 proof -
   from assms have lim_f: \<open>(sum f \<longlongrightarrow> a)  (finite_subsets_at_top A)\<close>
     and lim_g: \<open>(sum g \<longlongrightarrow> b)  (finite_subsets_at_top A)\<close>
     by (simp_all add: has_sum_def)
   then have lim: \<open>(sum (\<lambda>x. f x + g x) \<longlongrightarrow> a + b) (finite_subsets_at_top A)\<close>
     unfolding sum.distrib by (rule tendsto_add)
   then show ?thesis
     by (simp_all add: has_sum_def)
 qed
 
 lemma summable_on_add:
   fixes f g :: "'a \<Rightarrow> 'b::{topological_comm_monoid_add}"
   assumes \<open>f summable_on A\<close>
   assumes \<open>g summable_on A\<close>
   shows \<open>(\<lambda>x. f x + g x) summable_on A\<close>
   by (metis (full_types) assms(1) assms(2) summable_on_def has_sum_add)
 
 lemma infsum_add:
   fixes f g :: "'a \<Rightarrow> 'b::{topological_comm_monoid_add, t2_space}"
   assumes \<open>f summable_on A\<close>
   assumes \<open>g summable_on A\<close>
   shows \<open>infsum (\<lambda>x. f x + g x) A = infsum f A + infsum g A\<close>
 proof -
   have \<open>has_sum (\<lambda>x. f x + g x) A (infsum f A + infsum g A)\<close>
     by (simp add: assms(1) assms(2) has_sum_add)
   then show ?thesis
     using infsumI by blast
 qed
 
 
 lemma has_sum_Un_disjoint:
   fixes f :: "'a \<Rightarrow> 'b::topological_comm_monoid_add"
   assumes "has_sum f A a"
   assumes "has_sum f B b"
   assumes disj: "A \<inter> B = {}"
   shows \<open>has_sum f (A \<union> B) (a + b)\<close>
 proof -
   define fA fB where \<open>fA x = (if x \<in> A then f x else 0)\<close>
     and \<open>fB x = (if x \<notin> A then f x else 0)\<close> for x
   have fA: \<open>has_sum fA (A \<union> B) a\<close>
     by (smt (verit, ccfv_SIG) DiffD1 DiffD2 UnCI fA_def assms(1) has_sum_cong_neutral inf_sup_absorb)
   have fB: \<open>has_sum fB (A \<union> B) b\<close>
     by (smt (verit, best) DiffD1 DiffD2 IntE Un_iff fB_def assms(2) disj disjoint_iff has_sum_cong_neutral)
   have fAB: \<open>f x = fA x + fB x\<close> for x
     unfolding fA_def fB_def by simp
   show ?thesis
     unfolding fAB
     using fA fB by (rule has_sum_add)
 qed
 
 lemma summable_on_Un_disjoint:
   fixes f :: "'a \<Rightarrow> 'b::topological_comm_monoid_add"
   assumes "f summable_on A"
   assumes "f summable_on B"
   assumes disj: "A \<inter> B = {}"
   shows \<open>f summable_on (A \<union> B)\<close>
   by (meson assms(1) assms(2) disj summable_on_def has_sum_Un_disjoint)
 
 lemma infsum_Un_disjoint:
   fixes f :: "'a \<Rightarrow> 'b::{topological_comm_monoid_add, t2_space}"
   assumes "f summable_on A"
   assumes "f summable_on B"
   assumes disj: "A \<inter> B = {}"
   shows \<open>infsum f (A \<union> B) = infsum f A + infsum f B\<close>
   by (intro infsumI has_sum_Un_disjoint has_sum_infsum assms)  
 
 lemma norm_summable_imp_has_sum:
   fixes f :: "nat \<Rightarrow> 'a :: banach"
   assumes "summable (\<lambda>n. norm (f n))" and "f sums S"
   shows   "has_sum f (UNIV :: nat set) S"
   unfolding has_sum_def tendsto_iff eventually_finite_subsets_at_top
 proof (safe, goal_cases)
   case (1 \<epsilon>)
   from assms(1) obtain S' where S': "(\<lambda>n. norm (f n)) sums S'"
     by (auto simp: summable_def)
   with 1 obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> \<bar>S' - (\<Sum>i<n. norm (f i))\<bar> < \<epsilon>"
     by (auto simp: tendsto_iff eventually_at_top_linorder sums_def dist_norm abs_minus_commute)
   
   show ?case
   proof (rule exI[of _ "{..<N}"], safe, goal_cases)
     case (2 Y)
     from 2 have "(\<lambda>n. if n \<in> Y then 0 else f n) sums (S - sum f Y)"
       by (intro sums_If_finite_set'[OF \<open>f sums S\<close>]) (auto simp: sum_negf)
     hence "S - sum f Y = (\<Sum>n. if n \<in> Y then 0 else f n)"
       by (simp add: sums_iff)
     also have "norm \<dots> \<le> (\<Sum>n. norm (if n \<in> Y then 0 else f n))"
       by (rule summable_norm[OF summable_comparison_test'[OF assms(1)]]) auto
     also have "\<dots> \<le> (\<Sum>n. if n < N then 0 else norm (f n))"
       using 2 by (intro suminf_le summable_comparison_test'[OF assms(1)]) auto
     also have "(\<lambda>n. if n \<in> {..<N} then 0 else norm (f n)) sums (S' - (\<Sum>i<N. norm (f i)))" 
       by (intro sums_If_finite_set'[OF S']) (auto simp: sum_negf)
     hence "(\<Sum>n. if n < N then 0 else norm (f n)) = S' - (\<Sum>i<N. norm (f i))"
       by (simp add: sums_iff)
     also have "S' - (\<Sum>i<N. norm (f i)) \<le> \<bar>S' - (\<Sum>i<N. norm (f i))\<bar>" by simp
     also have "\<dots> < \<epsilon>" by (rule N) auto
     finally show ?case by (simp add: dist_norm norm_minus_commute)
   qed auto
 qed
 
 lemma norm_summable_imp_summable_on:
   fixes f :: "nat \<Rightarrow> 'a :: banach"
   assumes "summable (\<lambda>n. norm (f n))"
   shows   "f summable_on UNIV"
   using norm_summable_imp_has_sum[OF assms, of "suminf f"] assms
   by (auto simp: sums_iff summable_on_def dest: summable_norm_cancel)
 
 text \<open>The following lemma indeed needs a complete space (as formalized by the premise \<^term>\<open>complete UNIV\<close>).
   The following two counterexamples show this:
   \begin{itemize}
   \item Consider the real vector space $V$ of sequences with finite support, and with the $\ell_2$-norm (sum of squares).
       Let $e_i$ denote the sequence with a $1$ at position $i$.
       Let $f : \mathbb Z \to V$ be defined as $f(n) := e_{\lvert n\rvert} / n$ (with $f(0) := 0$).
       We have that $\sum_{n\in\mathbb Z} f(n) = 0$ (it even converges absolutely). 
       But $\sum_{n\in\mathbb N} f(n)$ does not exist (it would converge against a sequence with infinite support).
   
   \item Let $f$ be a positive rational valued function such that $\sum_{x\in B} f(x)$ is $\sqrt 2$ and $\sum_{x\in A} f(x)$ is 1 (over the reals, with $A\subseteq B$).
       Then $\sum_{x\in B} f(x)$ does not exist over the rationals. But $\sum_{x\in A} f(x)$ exists.
   \end{itemize}
 
   The lemma also requires uniform continuity of the addition. And example of a topological group with continuous 
   but not uniformly continuous addition would be the positive reals with the usual multiplication as the addition.
   We do not know whether the lemma would also hold for such topological groups.\<close>
 
 lemma summable_on_subset:
   fixes A B and f :: \<open>'a \<Rightarrow> 'b::{ab_group_add, uniform_space}\<close>
   assumes \<open>complete (UNIV :: 'b set)\<close>
   assumes plus_cont: \<open>uniformly_continuous_on UNIV (\<lambda>(x::'b,y). x+y)\<close>
   assumes \<open>f summable_on A\<close>
   assumes \<open>B \<subseteq> A\<close>
   shows \<open>f summable_on B\<close>
 proof -
   let ?filter_fB = \<open>filtermap (sum f) (finite_subsets_at_top B)\<close>
   from \<open>f summable_on A\<close>
   obtain S where \<open>(sum f \<longlongrightarrow> S) (finite_subsets_at_top A)\<close> (is \<open>(sum f \<longlongrightarrow> S) ?filter_A\<close>)
     using summable_on_def has_sum_def by blast
   then have cauchy_fA: \<open>cauchy_filter (filtermap (sum f) (finite_subsets_at_top A))\<close> (is \<open>cauchy_filter ?filter_fA\<close>)
     by (auto intro!: nhds_imp_cauchy_filter simp: filterlim_def)
 
   have \<open>cauchy_filter (filtermap (sum f) (finite_subsets_at_top B))\<close>
   proof (unfold cauchy_filter_def, rule filter_leI)
     fix E :: \<open>('b\<times>'b) \<Rightarrow> bool\<close> assume \<open>eventually E uniformity\<close>
     then obtain E' where \<open>eventually E' uniformity\<close> and E'E'E: \<open>E' (x, y) \<longrightarrow> E' (y, z) \<longrightarrow> E (x, z)\<close> for x y z
       using uniformity_trans by blast
     obtain D where \<open>eventually D uniformity\<close> and DE: \<open>D (x, y) \<Longrightarrow> E' (x+c, y+c)\<close> for x y c
       using plus_cont \<open>eventually E' uniformity\<close>
       unfolding uniformly_continuous_on_uniformity filterlim_def le_filter_def uniformity_prod_def
       by (auto simp: case_prod_beta eventually_filtermap eventually_prod_same uniformity_refl)
     have DE': "E' (x, y)" if "D (x + c, y + c)" for x y c
       using DE[of "x + c" "y + c" "-c"] that by simp
 
     from \<open>eventually D uniformity\<close> and cauchy_fA have \<open>eventually D (?filter_fA \<times>\<^sub>F ?filter_fA)\<close>
       unfolding cauchy_filter_def le_filter_def by simp
     then obtain P1 P2
       where ev_P1: \<open>eventually (\<lambda>F. P1 (sum f F)) ?filter_A\<close> 
         and ev_P2: \<open>eventually (\<lambda>F. P2 (sum f F)) ?filter_A\<close>
         and P1P2E: \<open>P1 x \<Longrightarrow> P2 y \<Longrightarrow> D (x, y)\<close> for x y
       unfolding eventually_prod_filter eventually_filtermap
       by auto
     from ev_P1 obtain F1 where F1: \<open>finite F1\<close> \<open>F1 \<subseteq> A\<close> \<open>\<And>F. F\<supseteq>F1 \<Longrightarrow> finite F \<Longrightarrow> F\<subseteq>A \<Longrightarrow> P1 (sum f F)\<close>
       by (metis eventually_finite_subsets_at_top)
     from ev_P2 obtain F2 where F2: \<open>finite F2\<close> \<open>F2 \<subseteq> A\<close> \<open>\<And>F. F\<supseteq>F2 \<Longrightarrow> finite F \<Longrightarrow> F\<subseteq>A \<Longrightarrow> P2 (sum f F)\<close>
       by (metis eventually_finite_subsets_at_top)
     define F0 F0A F0B where \<open>F0 \<equiv> F1 \<union> F2\<close> and \<open>F0A \<equiv> F0 - B\<close> and \<open>F0B \<equiv> F0 \<inter> B\<close>
     have [simp]: \<open>finite F0\<close>  \<open>F0 \<subseteq> A\<close>
       using \<open>F1 \<subseteq> A\<close> \<open>F2 \<subseteq> A\<close> \<open>finite F1\<close> \<open>finite F2\<close> unfolding F0_def by blast+
  
     have *: "E' (sum f F1', sum f F2')"
       if "F1'\<supseteq>F0B" "F2'\<supseteq>F0B" "finite F1'" "finite F2'" "F1'\<subseteq>B" "F2'\<subseteq>B" for F1' F2'
     proof (intro DE'[where c = "sum f F0A"] P1P2E)
       have "P1 (sum f (F1' \<union> F0A))"
         using that assms F1(1,2) F2(1,2) by (intro F1) (auto simp: F0A_def F0B_def F0_def)
       thus "P1 (sum f F1' + sum f F0A)"
         by (subst (asm) sum.union_disjoint) (use that in \<open>auto simp: F0A_def\<close>)
     next
       have "P2 (sum f (F2' \<union> F0A))"
         using that assms F1(1,2) F2(1,2) by (intro F2) (auto simp: F0A_def F0B_def F0_def)
       thus "P2 (sum f F2' + sum f F0A)"
         by (subst (asm) sum.union_disjoint) (use that in \<open>auto simp: F0A_def\<close>)      
     qed
 
     show \<open>eventually E (?filter_fB \<times>\<^sub>F ?filter_fB)\<close>
       unfolding eventually_prod_filter
     proof (safe intro!: exI)
       show "eventually (\<lambda>x. E' (x, sum f F0B)) (filtermap (sum f) (finite_subsets_at_top B))"
        and "eventually (\<lambda>x. E' (sum f F0B, x)) (filtermap (sum f) (finite_subsets_at_top B))"
         unfolding eventually_filtermap eventually_finite_subsets_at_top
         by (rule exI[of _ F0B]; use * in \<open>force simp: F0B_def\<close>)+
     next
       show "E (x, y)" if "E' (x, sum f F0B)" and "E' (sum f F0B, y)" for x y
         using E'E'E that by blast
     qed
   qed
 
   then obtain x where \<open>?filter_fB \<le> nhds x\<close>
     using cauchy_filter_complete_converges[of ?filter_fB UNIV] \<open>complete (UNIV :: _)\<close>
     by (auto simp: filtermap_bot_iff)
   then have \<open>(sum f \<longlongrightarrow> x) (finite_subsets_at_top B)\<close>
     by (auto simp: filterlim_def)
   then show ?thesis
     by (auto simp: summable_on_def has_sum_def)
 qed
 
 text \<open>A special case of @{thm [source] summable_on_subset} for Banach spaces with less premises.\<close>
 
 lemma summable_on_subset_banach:
   fixes A B and f :: \<open>'a \<Rightarrow> 'b::banach\<close>
   assumes \<open>f summable_on A\<close>
   assumes \<open>B \<subseteq> A\<close>
   shows \<open>f summable_on B\<close>
   by (rule summable_on_subset[OF _ _ assms])
      (auto simp: complete_def convergent_def dest!: Cauchy_convergent)
 
 lemma has_sum_empty[simp]: \<open>has_sum f {} 0\<close>
   by (meson ex_in_conv has_sum_0)
 
 lemma summable_on_empty[simp]: \<open>f summable_on {}\<close>
   by auto
 
 lemma infsum_empty[simp]: \<open>infsum f {} = 0\<close>
   by simp
 
 lemma sum_has_sum:
   fixes f :: "'a \<Rightarrow> 'b::topological_comm_monoid_add"
   assumes finite: \<open>finite A\<close>
   assumes conv: \<open>\<And>a. a \<in> A \<Longrightarrow> has_sum f (B a) (s a)\<close>
   assumes disj: \<open>\<And>a a'. a\<in>A \<Longrightarrow> a'\<in>A \<Longrightarrow> a\<noteq>a' \<Longrightarrow> B a \<inter> B a' = {}\<close>
   shows \<open>has_sum f (\<Union>a\<in>A. B a) (sum s A)\<close>
   using assms finite conv 
 proof (induction)
   case empty
   then show ?case 
     by simp
 next
   case (insert x A)
   have \<open>has_sum f (B x) (s x)\<close>
     by (simp add: insert.prems)
   moreover have IH: \<open>has_sum f (\<Union>a\<in>A. B a) (sum s A)\<close>
     using insert by simp
   ultimately have \<open>has_sum f (B x \<union> (\<Union>a\<in>A. B a)) (s x + sum s A)\<close>
     using insert by (intro has_sum_Un_disjoint) auto
   then show ?case
     using insert.hyps by auto
 qed
 
 
 lemma summable_on_finite_union_disjoint:
   fixes f :: "'a \<Rightarrow> 'b::topological_comm_monoid_add"
   assumes finite: \<open>finite A\<close>
   assumes conv: \<open>\<And>a. a \<in> A \<Longrightarrow> f summable_on (B a)\<close>
   assumes disj: \<open>\<And>a a'. a\<in>A \<Longrightarrow> a'\<in>A \<Longrightarrow> a\<noteq>a' \<Longrightarrow> B a \<inter> B a' = {}\<close>
   shows \<open>f summable_on (\<Union>a\<in>A. B a)\<close>
   using finite conv disj by induction (auto intro!: summable_on_Un_disjoint)
 
 lemma sum_infsum:
   fixes f :: "'a \<Rightarrow> 'b::{topological_comm_monoid_add, t2_space}"
   assumes finite: \<open>finite A\<close>
   assumes conv: \<open>\<And>a. a \<in> A \<Longrightarrow> f summable_on (B a)\<close>
   assumes disj: \<open>\<And>a a'. a\<in>A \<Longrightarrow> a'\<in>A \<Longrightarrow> a\<noteq>a' \<Longrightarrow> B a \<inter> B a' = {}\<close>
   shows \<open>sum (\<lambda>a. infsum f (B a)) A = infsum f (\<Union>a\<in>A. B a)\<close>
   by (rule sym, rule infsumI)
      (use sum_has_sum[of A f B \<open>\<lambda>a. infsum f (B a)\<close>] assms in auto)
 
 text \<open>The lemmas \<open>infsum_comm_additive_general\<close> and \<open>infsum_comm_additive\<close> (and variants) below both state that the infinite sum commutes with
   a continuous additive function. \<open>infsum_comm_additive_general\<close> is stated more for more general type classes
   at the expense of a somewhat less compact formulation of the premises.
   E.g., by avoiding the constant \<^const>\<open>additive\<close> which introduces an additional sort constraint
   (group instead of monoid). For example, extended reals (\<^typ>\<open>ereal\<close>, \<^typ>\<open>ennreal\<close>) are not covered
   by \<open>infsum_comm_additive\<close>.\<close>
 
 
 lemma has_sum_comm_additive_general: 
   fixes f :: \<open>'b :: {comm_monoid_add,topological_space} \<Rightarrow> 'c :: {comm_monoid_add,topological_space}\<close>
   assumes f_sum: \<open>\<And>F. finite F \<Longrightarrow> F \<subseteq> S \<Longrightarrow> sum (f \<circ> g) F = f (sum g F)\<close>
       \<comment> \<open>Not using \<^const>\<open>additive\<close> because it would add sort constraint \<^class>\<open>ab_group_add\<close>\<close>
   assumes cont: \<open>f \<midarrow>x\<rightarrow> f x\<close>
     \<comment> \<open>For \<^class>\<open>t2_space\<close>, this is equivalent to \<open>isCont f x\<close> by @{thm [source] isCont_def}.\<close>
   assumes infsum: \<open>has_sum g S x\<close>
   shows \<open>has_sum (f \<circ> g) S (f x)\<close> 
 proof -
   have \<open>(sum g \<longlongrightarrow> x) (finite_subsets_at_top S)\<close>
     using infsum has_sum_def by blast
   then have \<open>((f \<circ> sum g) \<longlongrightarrow> f x) (finite_subsets_at_top S)\<close>
     by (meson cont filterlim_def tendsto_at_iff_tendsto_nhds tendsto_compose_filtermap tendsto_mono)
   then have \<open>(sum (f \<circ> g) \<longlongrightarrow> f x) (finite_subsets_at_top S)\<close>
     using tendsto_cong f_sum
     by (simp add: Lim_transform_eventually eventually_finite_subsets_at_top_weakI)
   then show \<open>has_sum (f \<circ> g) S (f x)\<close>
     using has_sum_def by blast 
 qed
 
 lemma summable_on_comm_additive_general:
   fixes f :: \<open>'b :: {comm_monoid_add,topological_space} \<Rightarrow> 'c :: {comm_monoid_add,topological_space}\<close>
   assumes \<open>\<And>F. finite F \<Longrightarrow> F \<subseteq> S \<Longrightarrow> sum (f \<circ> g) F = f (sum g F)\<close>
     \<comment> \<open>Not using \<^const>\<open>additive\<close> because it would add sort constraint \<^class>\<open>ab_group_add\<close>\<close>
   assumes \<open>\<And>x. has_sum g S x \<Longrightarrow> f \<midarrow>x\<rightarrow> f x\<close>
     \<comment> \<open>For \<^class>\<open>t2_space\<close>, this is equivalent to \<open>isCont f x\<close> by @{thm [source] isCont_def}.\<close>
   assumes \<open>g summable_on S\<close>
   shows \<open>(f \<circ> g) summable_on S\<close>
   by (meson assms summable_on_def has_sum_comm_additive_general has_sum_def infsum_tendsto)
 
 lemma infsum_comm_additive_general:
   fixes f :: \<open>'b :: {comm_monoid_add,t2_space} \<Rightarrow> 'c :: {comm_monoid_add,t2_space}\<close>
   assumes f_sum: \<open>\<And>F. finite F \<Longrightarrow> F \<subseteq> S \<Longrightarrow> sum (f \<circ> g) F = f (sum g F)\<close>
       \<comment> \<open>Not using \<^const>\<open>additive\<close> because it would add sort constraint \<^class>\<open>ab_group_add\<close>\<close>
   assumes \<open>isCont f (infsum g S)\<close>
   assumes \<open>g summable_on S\<close>
   shows \<open>infsum (f \<circ> g) S = f (infsum g S)\<close>
   using assms
   by (intro infsumI has_sum_comm_additive_general has_sum_infsum) (auto simp: isCont_def)
 
 lemma has_sum_comm_additive: 
   fixes f :: \<open>'b :: {ab_group_add,topological_space} \<Rightarrow> 'c :: {ab_group_add,topological_space}\<close>
   assumes \<open>additive f\<close>
   assumes \<open>f \<midarrow>x\<rightarrow> f x\<close>
     \<comment> \<open>For \<^class>\<open>t2_space\<close>, this is equivalent to \<open>isCont f x\<close> by @{thm [source] isCont_def}.\<close>
   assumes infsum: \<open>has_sum g S x\<close>
   shows \<open>has_sum (f \<circ> g) S (f x)\<close>
   using assms
   by (intro has_sum_comm_additive_general has_sum_infsum) (auto simp: isCont_def additive.sum) 
 
 lemma summable_on_comm_additive:
   fixes f :: \<open>'b :: {ab_group_add,t2_space} \<Rightarrow> 'c :: {ab_group_add,topological_space}\<close>
   assumes \<open>additive f\<close>
   assumes \<open>isCont f (infsum g S)\<close>
   assumes \<open>g summable_on S\<close>
   shows \<open>(f \<circ> g) summable_on S\<close>
   by (meson assms(1) assms(2) assms(3) summable_on_def has_sum_comm_additive has_sum_infsum isContD)
 
 lemma infsum_comm_additive:
   fixes f :: \<open>'b :: {ab_group_add,t2_space} \<Rightarrow> 'c :: {ab_group_add,t2_space}\<close>
   assumes \<open>additive f\<close>
   assumes \<open>isCont f (infsum g S)\<close>
   assumes \<open>g summable_on S\<close>
   shows \<open>infsum (f \<circ> g) S = f (infsum g S)\<close>
   by (rule infsum_comm_additive_general; auto simp: assms additive.sum)
 
 lemma nonneg_bdd_above_has_sum:
   fixes f :: \<open>'a \<Rightarrow> 'b :: {conditionally_complete_linorder, ordered_comm_monoid_add, linorder_topology}\<close>
   assumes \<open>\<And>x. x\<in>A \<Longrightarrow> f x \<ge> 0\<close>
   assumes \<open>bdd_above (sum f ` {F. F\<subseteq>A \<and> finite F})\<close>
   shows \<open>has_sum f A (SUP F\<in>{F. finite F \<and> F\<subseteq>A}. sum f F)\<close>
 proof -
   have \<open>(sum f \<longlongrightarrow> (SUP F\<in>{F. finite F \<and> F\<subseteq>A}. sum f F)) (finite_subsets_at_top A)\<close>
   proof (rule order_tendstoI)
     fix a assume \<open>a < (SUP F\<in>{F. finite F \<and> F\<subseteq>A}. sum f F)\<close>
     then obtain F where \<open>a < sum f F\<close> and \<open>finite F\<close> and \<open>F \<subseteq> A\<close>
       by (metis (mono_tags, lifting) Collect_cong Collect_empty_eq assms(2) empty_subsetI finite.emptyI less_cSUP_iff mem_Collect_eq)
     show \<open>\<forall>\<^sub>F x in finite_subsets_at_top A. a < sum f x\<close>
       unfolding eventually_finite_subsets_at_top
     proof (rule exI[of _ F], safe)
       fix Y assume Y: "finite Y" "F \<subseteq> Y" "Y \<subseteq> A"
       have "a < sum f F"
         by fact
       also have "\<dots> \<le> sum f Y"
         using assms Y by (intro sum_mono2) auto
       finally show "a < sum f Y" .
     qed (use \<open>finite F\<close> \<open>F \<subseteq> A\<close> in auto)
   next
     fix a assume *: \<open>(SUP F\<in>{F. finite F \<and> F\<subseteq>A}. sum f F) < a\<close>
     have \<open>sum f F < a\<close> if \<open>F\<subseteq>A\<close> and \<open>finite F\<close> for F
     proof -
       have "sum f F \<le> (SUP F\<in>{F. finite F \<and> F\<subseteq>A}. sum f F)"
         by (rule cSUP_upper) (use that assms(2) in \<open>auto simp: conj_commute\<close>)
       also have "\<dots> < a"
         by fact
       finally show ?thesis .
     qed
     then show \<open>\<forall>\<^sub>F x in finite_subsets_at_top A. sum f x < a\<close>
       by (rule eventually_finite_subsets_at_top_weakI)
   qed
   then show ?thesis
     using has_sum_def by blast
 qed
 
 lemma nonneg_bdd_above_summable_on:
   fixes f :: \<open>'a \<Rightarrow> 'b :: {conditionally_complete_linorder, ordered_comm_monoid_add, linorder_topology}\<close>
   assumes \<open>\<And>x. x\<in>A \<Longrightarrow> f x \<ge> 0\<close>
   assumes \<open>bdd_above (sum f ` {F. F\<subseteq>A \<and> finite F})\<close>
   shows \<open>f summable_on A\<close>
   using assms(1) assms(2) summable_on_def nonneg_bdd_above_has_sum by blast
 
 lemma nonneg_bdd_above_infsum:
   fixes f :: \<open>'a \<Rightarrow> 'b :: {conditionally_complete_linorder, ordered_comm_monoid_add, linorder_topology}\<close>
   assumes \<open>\<And>x. x\<in>A \<Longrightarrow> f x \<ge> 0\<close>
   assumes \<open>bdd_above (sum f ` {F. F\<subseteq>A \<and> finite F})\<close>
   shows \<open>infsum f A = (SUP F\<in>{F. finite F \<and> F\<subseteq>A}. sum f F)\<close>
   using assms by (auto intro!: infsumI nonneg_bdd_above_has_sum)
 
 lemma nonneg_has_sum_complete:
   fixes f :: \<open>'a \<Rightarrow> 'b :: {complete_linorder, ordered_comm_monoid_add, linorder_topology}\<close>
   assumes \<open>\<And>x. x\<in>A \<Longrightarrow> f x \<ge> 0\<close>
   shows \<open>has_sum f A (SUP F\<in>{F. finite F \<and> F\<subseteq>A}. sum f F)\<close>
   using assms nonneg_bdd_above_has_sum by blast
 
 lemma nonneg_summable_on_complete:
   fixes f :: \<open>'a \<Rightarrow> 'b :: {complete_linorder, ordered_comm_monoid_add, linorder_topology}\<close>
   assumes \<open>\<And>x. x\<in>A \<Longrightarrow> f x \<ge> 0\<close>
   shows \<open>f summable_on A\<close>
   using assms nonneg_bdd_above_summable_on by blast
 
 lemma nonneg_infsum_complete:
   fixes f :: \<open>'a \<Rightarrow> 'b :: {complete_linorder, ordered_comm_monoid_add, linorder_topology}\<close>
   assumes \<open>\<And>x. x\<in>A \<Longrightarrow> f x \<ge> 0\<close>
   shows \<open>infsum f A = (SUP F\<in>{F. finite F \<and> F\<subseteq>A}. sum f F)\<close>
   using assms nonneg_bdd_above_infsum by blast
 
 lemma has_sum_nonneg:
   fixes f :: "'a \<Rightarrow> 'b::{ordered_comm_monoid_add,linorder_topology}"
   assumes "has_sum f M a"
     and "\<And>x. x \<in> M \<Longrightarrow> 0 \<le> f x"
   shows "a \<ge> 0"
   by (metis (no_types, lifting) DiffD1 assms(1) assms(2) empty_iff has_sum_0 has_sum_mono_neutral order_refl)
 
 lemma infsum_nonneg:
   fixes f :: "'a \<Rightarrow> 'b::{ordered_comm_monoid_add,linorder_topology}"
   assumes "\<And>x. x \<in> M \<Longrightarrow> 0 \<le> f x"
   shows "infsum f M \<ge> 0" (is "?lhs \<ge> _")
   by (metis assms has_sum_infsum has_sum_nonneg infsum_not_exists linorder_linear)
 
 lemma has_sum_mono2:
   fixes f :: "'a \<Rightarrow> 'b::{topological_ab_group_add, ordered_comm_monoid_add,linorder_topology}"
   assumes "has_sum f A S" "has_sum f B S'" "A \<subseteq> B"
   assumes "\<And>x. x \<in> B - A \<Longrightarrow> f x \<ge> 0"
   shows   "S \<le> S'"
 proof -
   have "has_sum f (B - A) (S' - S)"
     by (rule has_sum_Diff) fact+
   hence "S' - S \<ge> 0"
     by (rule has_sum_nonneg) (use assms(4) in auto)
   thus ?thesis
     by (metis add_0 add_mono_thms_linordered_semiring(3) diff_add_cancel)
 qed
 
 lemma infsum_mono2:
   fixes f :: "'a \<Rightarrow> 'b::{topological_ab_group_add, ordered_comm_monoid_add,linorder_topology}"
   assumes "f summable_on A" "f summable_on B" "A \<subseteq> B"
   assumes "\<And>x. x \<in> B - A \<Longrightarrow> f x \<ge> 0"
   shows   "infsum f A \<le> infsum f B"
   by (rule has_sum_mono2[OF has_sum_infsum has_sum_infsum]) (use assms in auto)
 
 lemma finite_sum_le_has_sum:
   fixes f :: "'a \<Rightarrow> 'b::{topological_ab_group_add, ordered_comm_monoid_add,linorder_topology}"
   assumes "has_sum f A S" "finite B" "B \<subseteq> A"
   assumes "\<And>x. x \<in> A - B \<Longrightarrow> f x \<ge> 0"
   shows   "sum f B \<le> S"
 proof (rule has_sum_mono2)
   show "has_sum f A S"
     by fact
   show "has_sum f B (sum f B)"
     by (rule has_sum_finite) fact+
 qed (use assms in auto)
 
 lemma finite_sum_le_infsum:
   fixes f :: "'a \<Rightarrow> 'b::{topological_ab_group_add, ordered_comm_monoid_add,linorder_topology}"
   assumes "f summable_on A" "finite B" "B \<subseteq> A"
   assumes "\<And>x. x \<in> A - B \<Longrightarrow> f x \<ge> 0"
   shows   "sum f B \<le> infsum f A"
   by (rule finite_sum_le_has_sum[OF has_sum_infsum]) (use assms in auto)
 
 lemma has_sum_reindex:
   assumes \<open>inj_on h A\<close>
   shows \<open>has_sum g (h ` A) x \<longleftrightarrow> has_sum (g \<circ> h) A x\<close>
 proof -
   have \<open>has_sum g (h ` A) x \<longleftrightarrow> (sum g \<longlongrightarrow> x) (finite_subsets_at_top (h ` A))\<close>
     by (simp add: has_sum_def)
   also have \<open>\<dots> \<longleftrightarrow> ((\<lambda>F. sum g (h ` F)) \<longlongrightarrow> x) (finite_subsets_at_top A)\<close>
     by (metis assms filterlim_filtermap filtermap_image_finite_subsets_at_top)
   also have \<open>\<dots> \<longleftrightarrow> (sum (g \<circ> h) \<longlongrightarrow> x) (finite_subsets_at_top A)\<close>
   proof (intro tendsto_cong eventually_finite_subsets_at_top_weakI sum.reindex)
     show "\<And>X. \<lbrakk>finite X; X \<subseteq> A\<rbrakk> \<Longrightarrow> inj_on h X"
       using assms subset_inj_on by blast
   qed
   also have \<open>\<dots> \<longleftrightarrow> has_sum (g \<circ> h) A x\<close>
     by (simp add: has_sum_def)
   finally show ?thesis .
 qed
 
 lemma summable_on_reindex:
   assumes \<open>inj_on h A\<close>
   shows \<open>g summable_on (h ` A) \<longleftrightarrow> (g \<circ> h) summable_on A\<close>
   by (simp add: assms summable_on_def has_sum_reindex)
 
 lemma infsum_reindex:
   assumes \<open>inj_on h A\<close>
   shows \<open>infsum g (h ` A) = infsum (g \<circ> h) A\<close>
   by (metis (no_types, opaque_lifting) assms finite_subsets_at_top_neq_bot infsum_def 
         summable_on_reindex has_sum_def has_sum_infsum has_sum_reindex tendsto_Lim)
 
 lemma summable_on_reindex_bij_betw:
   assumes "bij_betw g A B"
   shows   "(\<lambda>x. f (g x)) summable_on A \<longleftrightarrow> f summable_on B"
 proof -
   have "inj_on g A"
     using assms bij_betw_imp_inj_on by blast
   then have \<open>(\<lambda>x. f (g x)) summable_on A \<longleftrightarrow> f summable_on g ` A\<close>
     by (metis (mono_tags, lifting) comp_apply summable_on_cong summable_on_reindex)
   also have \<open>\<dots> \<longleftrightarrow> f summable_on B\<close>
     using assms bij_betw_imp_surj_on by blast
   finally show ?thesis .
 qed
 
 lemma infsum_reindex_bij_betw:
   assumes "bij_betw g A B"
   shows   "infsum (\<lambda>x. f (g x)) A = infsum f B"
 proof -
   have \<open>infsum (\<lambda>x. f (g x)) A = infsum f (g ` A)\<close>
     by (metis (mono_tags, lifting) assms bij_betw_imp_inj_on infsum_cong infsum_reindex o_def)
   also have \<open>\<dots> = infsum f B\<close>
     using assms bij_betw_imp_surj_on by blast
   finally show ?thesis .
 qed
 
 lemma sum_uniformity:
   assumes plus_cont: \<open>uniformly_continuous_on UNIV (\<lambda>(x::'b::{uniform_space,comm_monoid_add},y). x+y)\<close>
   assumes \<open>eventually E uniformity\<close>
   obtains D where \<open>eventually D uniformity\<close> 
     and \<open>\<And>M::'a set. \<And>f f' :: 'a \<Rightarrow> 'b. card M \<le> n \<and> (\<forall>m\<in>M. D (f m, f' m)) \<Longrightarrow> E (sum f M, sum f' M)\<close>
 proof (atomize_elim, insert \<open>eventually E uniformity\<close>, induction n arbitrary: E rule:nat_induct)
   case 0
   then show ?case
     by (metis card_eq_0_iff equals0D le_zero_eq sum.infinite sum.not_neutral_contains_not_neutral uniformity_refl)
 next
   case (Suc n)
   from plus_cont[unfolded uniformly_continuous_on_uniformity filterlim_def le_filter_def, rule_format, OF Suc.prems]
   obtain D1 D2 where \<open>eventually D1 uniformity\<close> and \<open>eventually D2 uniformity\<close> 
     and D1D2E: \<open>D1 (x, y) \<Longrightarrow> D2 (x', y') \<Longrightarrow> E (x + x', y + y')\<close> for x y x' y'
     apply atomize_elim
     by (auto simp: eventually_prod_filter case_prod_beta uniformity_prod_def eventually_filtermap)
 
   from Suc.IH[OF \<open>eventually D2 uniformity\<close>]
   obtain D3 where \<open>eventually D3 uniformity\<close> and D3: \<open>card M \<le> n \<Longrightarrow> (\<forall>m\<in>M. D3 (f m, f' m)) \<Longrightarrow> D2 (sum f M, sum f' M)\<close> 
     for M :: \<open>'a set\<close> and f f'
     by metis
 
   define D where \<open>D x \<equiv> D1 x \<and> D3 x\<close> for x
   have \<open>eventually D uniformity\<close>
     using D_def \<open>eventually D1 uniformity\<close> \<open>eventually D3 uniformity\<close> eventually_elim2 by blast
 
   have \<open>E (sum f M, sum f' M)\<close> 
     if \<open>card M \<le> Suc n\<close> and DM: \<open>\<forall>m\<in>M. D (f m, f' m)\<close>
     for M :: \<open>'a set\<close> and f f'
   proof (cases \<open>card M = 0\<close>)
     case True
     then show ?thesis
       by (metis Suc.prems card_eq_0_iff sum.empty sum.infinite uniformity_refl) 
   next
     case False
     with \<open>card M \<le> Suc n\<close> obtain N x where \<open>card N \<le> n\<close> and \<open>x \<notin> N\<close> and \<open>M = insert x N\<close>
       by (metis card_Suc_eq less_Suc_eq_0_disj less_Suc_eq_le)
 
     from DM have \<open>\<And>m. m\<in>N \<Longrightarrow> D (f m, f' m)\<close>
       using \<open>M = insert x N\<close> by blast
     with D3[OF \<open>card N \<le> n\<close>]
     have D2_N: \<open>D2 (sum f N, sum f' N)\<close>
       using D_def by blast
 
     from DM 
     have \<open>D (f x, f' x)\<close>
       using \<open>M = insert x N\<close> by blast
     then have \<open>D1 (f x, f' x)\<close>
       by (simp add: D_def)
 
     with D2_N
     have \<open>E (f x + sum f N, f' x + sum f' N)\<close>
       using D1D2E by presburger
 
     then show \<open>E (sum f M, sum f' M)\<close>
       by (metis False \<open>M = insert x N\<close> \<open>x \<notin> N\<close> card.infinite finite_insert sum.insert)
   qed
   with \<open>eventually D uniformity\<close>
   show ?case 
     by auto
 qed
 
 lemma has_sum_Sigma:
   fixes A :: "'a set" and B :: "'a \<Rightarrow> 'b set"
     and f :: \<open>'a \<times> 'b \<Rightarrow> 'c::{comm_monoid_add,uniform_space}\<close>
   assumes plus_cont: \<open>uniformly_continuous_on UNIV (\<lambda>(x::'c,y). x+y)\<close>
   assumes summableAB: "has_sum f (Sigma A B) a"
   assumes summableB: \<open>\<And>x. x\<in>A \<Longrightarrow> has_sum (\<lambda>y. f (x, y)) (B x) (b x)\<close>
   shows "has_sum b A a"
 proof -
   define F FB FA where \<open>F = finite_subsets_at_top (Sigma A B)\<close> and \<open>FB x = finite_subsets_at_top (B x)\<close>
     and \<open>FA = finite_subsets_at_top A\<close> for x
 
   from summableB
   have sum_b: \<open>(sum (\<lambda>y. f (x, y)) \<longlongrightarrow> b x) (FB x)\<close> if \<open>x \<in> A\<close> for x
     using FB_def[abs_def] has_sum_def that by auto
   from summableAB
   have sum_S: \<open>(sum f \<longlongrightarrow> a) F\<close>
     using F_def has_sum_def by blast
 
   have finite_proj: \<open>finite {b| b. (a,b) \<in> H}\<close> if \<open>finite H\<close> for H :: \<open>('a\<times>'b) set\<close> and a
     by (metis (no_types, lifting) finite_imageI finite_subset image_eqI mem_Collect_eq snd_conv subsetI that)
 
   have \<open>(sum b \<longlongrightarrow> a) FA\<close>
   proof (rule tendsto_iff_uniformity[THEN iffD2, rule_format])
     fix E :: \<open>('c \<times> 'c) \<Rightarrow> bool\<close>
     assume \<open>eventually E uniformity\<close>
     then obtain D where D_uni: \<open>eventually D uniformity\<close> and DDE': \<open>\<And>x y z. D (x, y) \<Longrightarrow> D (y, z) \<Longrightarrow> E (x, z)\<close>
       by (metis (no_types, lifting) \<open>eventually E uniformity\<close> uniformity_transE)
     from sum_S obtain G where \<open>finite G\<close> and \<open>G \<subseteq> Sigma A B\<close>
       and G_sum: \<open>G \<subseteq> H \<Longrightarrow> H \<subseteq> Sigma A B \<Longrightarrow> finite H \<Longrightarrow> D (sum f H, a)\<close> for H
       unfolding tendsto_iff_uniformity
       by (metis (mono_tags, lifting) D_uni F_def eventually_finite_subsets_at_top)
     have \<open>finite (fst ` G)\<close> and \<open>fst ` G \<subseteq> A\<close>
       using \<open>finite G\<close> \<open>G \<subseteq> Sigma A B\<close> by auto
     thm uniformity_prod_def
     define Ga where \<open>Ga a = {b. (a,b) \<in> G}\<close> for a
     have Ga_fin: \<open>finite (Ga a)\<close> and Ga_B: \<open>Ga a \<subseteq> B a\<close> for a
       using \<open>finite G\<close> \<open>G \<subseteq> Sigma A B\<close> finite_proj by (auto simp: Ga_def finite_proj)
 
     have \<open>E (sum b M, a)\<close> if \<open>M \<supseteq> fst ` G\<close> and \<open>finite M\<close> and \<open>M \<subseteq> A\<close> for M
     proof -
       define FMB where \<open>FMB = finite_subsets_at_top (Sigma M B)\<close>
       have \<open>eventually (\<lambda>H. D (\<Sum>a\<in>M. b a, \<Sum>(a,b)\<in>H. f (a,b))) FMB\<close>
       proof -
         obtain D' where D'_uni: \<open>eventually D' uniformity\<close> 
           and \<open>card M' \<le> card M \<and> (\<forall>m\<in>M'. D' (g m, g' m)) \<Longrightarrow> D (sum g M', sum g' M')\<close>
         for M' :: \<open>'a set\<close> and g g'
           using sum_uniformity[OF plus_cont \<open>eventually D uniformity\<close>] by blast
         then have D'_sum_D: \<open>(\<forall>m\<in>M. D' (g m, g' m)) \<Longrightarrow> D (sum g M, sum g' M)\<close> for g g'
           by auto
 
         obtain Ha where \<open>Ha a \<supseteq> Ga a\<close> and Ha_fin: \<open>finite (Ha a)\<close> and Ha_B: \<open>Ha a \<subseteq> B a\<close>
           and D'_sum_Ha: \<open>Ha a \<subseteq> L \<Longrightarrow> L \<subseteq> B a \<Longrightarrow> finite L \<Longrightarrow> D' (b a, sum (\<lambda>b. f (a,b)) L)\<close> if \<open>a \<in> A\<close> for a L
         proof -
           from sum_b[unfolded tendsto_iff_uniformity, rule_format, OF _ D'_uni[THEN uniformity_sym]]
           obtain Ha0 where \<open>finite (Ha0 a)\<close> and \<open>Ha0 a \<subseteq> B a\<close>
             and \<open>Ha0 a \<subseteq> L \<Longrightarrow> L \<subseteq> B a \<Longrightarrow> finite L \<Longrightarrow> D' (b a, sum (\<lambda>b. f (a,b)) L)\<close> if \<open>a \<in> A\<close> for a L
             unfolding FB_def eventually_finite_subsets_at_top unfolding prod.case by metis
           moreover define Ha where \<open>Ha a = Ha0 a \<union> Ga a\<close> for a
           ultimately show ?thesis
             using that[where Ha=Ha]
             using Ga_fin Ga_B by auto
         qed
 
         have \<open>D (\<Sum>a\<in>M. b a, \<Sum>(a,b)\<in>H. f (a,b))\<close> if \<open>finite H\<close> and \<open>H \<subseteq> Sigma M B\<close> and \<open>H \<supseteq> Sigma M Ha\<close> for H
         proof -
           define Ha' where \<open>Ha' a = {b| b. (a,b) \<in> H}\<close> for a
           have [simp]: \<open>finite (Ha' a)\<close> and [simp]: \<open>Ha' a \<supseteq> Ha a\<close> and [simp]: \<open>Ha' a \<subseteq> B a\<close> if \<open>a \<in> M\<close> for a
             unfolding Ha'_def using \<open>finite H\<close> \<open>H \<subseteq> Sigma M B\<close> \<open>Sigma M Ha \<subseteq> H\<close> that finite_proj by auto
           have \<open>Sigma M Ha' = H\<close>
             using that by (auto simp: Ha'_def)
           then have *: \<open>(\<Sum>(a,b)\<in>H. f (a,b)) = (\<Sum>a\<in>M. \<Sum>b\<in>Ha' a. f (a,b))\<close>
             by (simp add: \<open>finite M\<close> sum.Sigma)
           have \<open>D' (b a, sum (\<lambda>b. f (a,b)) (Ha' a))\<close> if \<open>a \<in> M\<close> for a
             using D'_sum_Ha \<open>M \<subseteq> A\<close> that by auto
           then have \<open>D (\<Sum>a\<in>M. b a, \<Sum>a\<in>M. sum (\<lambda>b. f (a,b)) (Ha' a))\<close>
             by (rule_tac D'_sum_D, auto)
           with * show ?thesis
             by auto
         qed
         moreover have \<open>Sigma M Ha \<subseteq> Sigma M B\<close>
           using Ha_B \<open>M \<subseteq> A\<close> by auto
         ultimately show ?thesis
           unfolding FMB_def eventually_finite_subsets_at_top
           by (intro exI[of _ "Sigma M Ha"])
              (use Ha_fin that(2,3) in \<open>fastforce intro!: finite_SigmaI\<close>)
       qed
       moreover have \<open>eventually (\<lambda>H. D (\<Sum>(a,b)\<in>H. f (a,b), a)) FMB\<close>
         unfolding FMB_def eventually_finite_subsets_at_top
       proof (rule exI[of _ G], safe)
         fix Y assume Y: "finite Y" "G \<subseteq> Y" "Y \<subseteq> Sigma M B"
         have "Y \<subseteq> Sigma A B"
           using Y \<open>M \<subseteq> A\<close> by blast
         thus "D (\<Sum>(a,b)\<in>Y. f (a, b), a)"
           using G_sum[of Y] Y by auto
       qed (use \<open>finite G\<close> \<open>G \<subseteq> Sigma A B\<close> that in auto)
       ultimately have \<open>\<forall>\<^sub>F x in FMB. E (sum b M, a)\<close>
         by eventually_elim (use DDE' in auto)
       then show \<open>E (sum b M, a)\<close>
         by (rule eventually_const[THEN iffD1, rotated]) (force simp: FMB_def)
     qed
     then show \<open>\<forall>\<^sub>F x in FA. E (sum b x, a)\<close>
       using \<open>finite (fst ` G)\<close> and \<open>fst ` G \<subseteq> A\<close>
       by (auto intro!: exI[of _ \<open>fst ` G\<close>] simp add: FA_def eventually_finite_subsets_at_top)
   qed
   then show ?thesis
     by (simp add: FA_def has_sum_def)
 qed
 
 lemma summable_on_Sigma:
   fixes A :: "'a set" and B :: "'a \<Rightarrow> 'b set"
     and f :: \<open>'a \<Rightarrow> 'b \<Rightarrow> 'c::{comm_monoid_add, t2_space, uniform_space}\<close>
   assumes plus_cont: \<open>uniformly_continuous_on UNIV (\<lambda>(x::'c,y). x+y)\<close>
   assumes summableAB: "(\<lambda>(x,y). f x y) summable_on (Sigma A B)"
   assumes summableB: \<open>\<And>x. x\<in>A \<Longrightarrow> (f x) summable_on (B x)\<close>
   shows \<open>(\<lambda>x. infsum (f x) (B x)) summable_on A\<close>
 proof -
   from summableAB obtain a where a: \<open>has_sum (\<lambda>(x,y). f x y) (Sigma A B) a\<close>
     using has_sum_infsum by blast
   from summableB have b: \<open>\<And>x. x\<in>A \<Longrightarrow> has_sum (f x) (B x) (infsum (f x) (B x))\<close>
     by (auto intro!: has_sum_infsum)
   show ?thesis
     using plus_cont a b 
     by (auto intro: has_sum_Sigma[where f=\<open>\<lambda>(x,y). f x y\<close>, simplified] simp: summable_on_def)
 qed
 
 lemma infsum_Sigma:
   fixes A :: "'a set" and B :: "'a \<Rightarrow> 'b set"
     and f :: \<open>'a \<times> 'b \<Rightarrow> 'c::{comm_monoid_add, t2_space, uniform_space}\<close>
   assumes plus_cont: \<open>uniformly_continuous_on UNIV (\<lambda>(x::'c,y). x+y)\<close>
   assumes summableAB: "f summable_on (Sigma A B)"
   assumes summableB: \<open>\<And>x. x\<in>A \<Longrightarrow> (\<lambda>y. f (x, y)) summable_on (B x)\<close>
   shows "infsum f (Sigma A B) = infsum (\<lambda>x. infsum (\<lambda>y. f (x, y)) (B x)) A"
 proof -
   from summableAB have a: \<open>has_sum f (Sigma A B) (infsum f (Sigma A B))\<close>
     using has_sum_infsum by blast
   from summableB have b: \<open>\<And>x. x\<in>A \<Longrightarrow> has_sum (\<lambda>y. f (x, y)) (B x) (infsum (\<lambda>y. f (x, y)) (B x))\<close>
     by (auto intro!: has_sum_infsum)
   show ?thesis
     using plus_cont a b by (auto intro: infsumI[symmetric] has_sum_Sigma simp: summable_on_def)
 qed
 
 lemma infsum_Sigma':
   fixes A :: "'a set" and B :: "'a \<Rightarrow> 'b set"
     and f :: \<open>'a \<Rightarrow> 'b \<Rightarrow> 'c::{comm_monoid_add, t2_space, uniform_space}\<close>
   assumes plus_cont: \<open>uniformly_continuous_on UNIV (\<lambda>(x::'c,y). x+y)\<close>
   assumes summableAB: "(\<lambda>(x,y). f x y) summable_on (Sigma A B)"
   assumes summableB: \<open>\<And>x. x\<in>A \<Longrightarrow> (f x) summable_on (B x)\<close>
   shows \<open>infsum (\<lambda>x. infsum (f x) (B x)) A = infsum (\<lambda>(x,y). f x y) (Sigma A B)\<close>
   using infsum_Sigma[of \<open>\<lambda>(x,y). f x y\<close> A B]
   using assms by auto
 
 text \<open>A special case of @{thm [source] infsum_Sigma} etc. for Banach spaces. It has less premises.\<close>
 lemma
   fixes A :: "'a set" and B :: "'a \<Rightarrow> 'b set"
     and f :: \<open>'a \<Rightarrow> 'b \<Rightarrow> 'c::banach\<close>
   assumes [simp]: "(\<lambda>(x,y). f x y) summable_on (Sigma A B)"
   shows infsum_Sigma'_banach: \<open>infsum (\<lambda>x. infsum (f x) (B x)) A = infsum (\<lambda>(x,y). f x y) (Sigma A B)\<close> (is ?thesis1)
     and summable_on_Sigma_banach: \<open>(\<lambda>x. infsum (f x) (B x)) summable_on A\<close> (is ?thesis2)
 proof -
   have fsum: \<open>(f x) summable_on (B x)\<close> if \<open>x \<in> A\<close> for x
   proof -
     from assms
     have \<open>(\<lambda>(x,y). f x y) summable_on (Pair x ` B x)\<close>
       by (meson image_subset_iff summable_on_subset_banach mem_Sigma_iff that)
     then have \<open>((\<lambda>(x,y). f x y) \<circ> Pair x) summable_on (B x)\<close>
       by (metis summable_on_reindex inj_on_def prod.inject)
     then show ?thesis
       by (auto simp: o_def)
   qed
   show ?thesis1
     using fsum assms infsum_Sigma' isUCont_plus by blast
   show ?thesis2
     using fsum assms isUCont_plus summable_on_Sigma by blast
 qed
 
 lemma infsum_Sigma_banach:
   fixes A :: "'a set" and B :: "'a \<Rightarrow> 'b set"
     and f :: \<open>'a \<times> 'b \<Rightarrow> 'c::banach\<close>
   assumes [simp]: "f summable_on (Sigma A B)"
   shows \<open>infsum (\<lambda>x. infsum (\<lambda>y. f (x,y)) (B x)) A = infsum f (Sigma A B)\<close>
   using assms
   by (subst infsum_Sigma'_banach) auto
 
 lemma infsum_swap:
   fixes A :: "'a set" and B :: "'b set"
   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{comm_monoid_add,t2_space,uniform_space}"
   assumes plus_cont: \<open>uniformly_continuous_on UNIV (\<lambda>(x::'c,y). x+y)\<close>
   assumes \<open>(\<lambda>(x, y). f x y) summable_on (A \<times> B)\<close>
   assumes \<open>\<And>a. a\<in>A \<Longrightarrow> (f a) summable_on B\<close>
   assumes \<open>\<And>b. b\<in>B \<Longrightarrow> (\<lambda>a. f a b) summable_on A\<close>
   shows \<open>infsum (\<lambda>x. infsum (\<lambda>y. f x y) B) A = infsum (\<lambda>y. infsum (\<lambda>x. f x y) A) B\<close>
 proof -
   have "(\<lambda>(x, y). f y x) \<circ> prod.swap summable_on A \<times> B"
     by (simp add: assms(2) summable_on_cong)
   then have fyx: \<open>(\<lambda>(x, y). f y x) summable_on (B \<times> A)\<close>
     by (metis has_sum_reindex infsum_reindex inj_swap product_swap summable_iff_has_sum_infsum)
   have \<open>infsum (\<lambda>x. infsum (\<lambda>y. f x y) B) A = infsum (\<lambda>(x,y). f x y) (A \<times> B)\<close>
     using assms infsum_Sigma' by blast
   also have \<open>\<dots> = infsum (\<lambda>(x,y). f y x) (B \<times> A)\<close>
     apply (subst product_swap[symmetric])
     apply (subst infsum_reindex)
     using assms by (auto simp: o_def)
   also have \<open>\<dots> = infsum (\<lambda>y. infsum (\<lambda>x. f x y) A) B\<close>
     by (smt (verit) fyx assms(1) assms(4) infsum_Sigma' infsum_cong)
   finally show ?thesis .
 qed
 
 lemma infsum_swap_banach:
   fixes A :: "'a set" and B :: "'b set"
   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::banach"
   assumes \<open>(\<lambda>(x, y). f x y) summable_on (A \<times> B)\<close>
   shows "infsum (\<lambda>x. infsum (\<lambda>y. f x y) B) A = infsum (\<lambda>y. infsum (\<lambda>x. f x y) A) B"
 proof -
   have \<section>: \<open>(\<lambda>(x, y). f y x) summable_on (B \<times> A)\<close>
     by (metis (mono_tags, lifting) assms case_swap inj_swap o_apply product_swap summable_on_cong summable_on_reindex)
   have \<open>infsum (\<lambda>x. infsum (\<lambda>y. f x y) B) A = infsum (\<lambda>(x,y). f x y) (A \<times> B)\<close>
     using assms infsum_Sigma'_banach by blast
   also have \<open>\<dots> = infsum (\<lambda>(x,y). f y x) (B \<times> A)\<close>
     apply (subst product_swap[symmetric])
     apply (subst infsum_reindex)
     using assms by (auto simp: o_def)
   also have \<open>\<dots> = infsum (\<lambda>y. infsum (\<lambda>x. f x y) A) B\<close>
     by (metis (mono_tags, lifting) \<section> infsum_Sigma'_banach infsum_cong)
   finally show ?thesis .
 qed
 
 lemma nonneg_infsum_le_0D:
   fixes f :: "'a \<Rightarrow> 'b::{topological_ab_group_add,ordered_ab_group_add,linorder_topology}"
   assumes "infsum f A \<le> 0"
     and abs_sum: "f summable_on A"
     and nneg: "\<And>x. x \<in> A \<Longrightarrow> f x \<ge> 0"
     and "x \<in> A"
   shows "f x = 0"
 proof (rule ccontr)
   assume \<open>f x \<noteq> 0\<close>
   have ex: \<open>f summable_on (A-{x})\<close>
     by (rule summable_on_cofin_subset) (use assms in auto)
   have pos: \<open>infsum f (A - {x}) \<ge> 0\<close>
     by (rule infsum_nonneg) (use nneg in auto)
 
   have [trans]: \<open>x \<ge> y \<Longrightarrow> y > z \<Longrightarrow> x > z\<close> for x y z :: 'b by auto
 
   have \<open>infsum f A = infsum f (A-{x}) + infsum f {x}\<close>
     by (subst infsum_Un_disjoint[symmetric]) (use assms ex in \<open>auto simp: insert_absorb\<close>)
   also have \<open>\<dots> \<ge> infsum f {x}\<close> (is \<open>_ \<ge> \<dots>\<close>)
     using pos by (rule add_increasing) simp
   also have \<open>\<dots> = f x\<close> (is \<open>_ = \<dots>\<close>)
     by (subst infsum_finite) auto
   also have \<open>\<dots> > 0\<close>
     using \<open>f x \<noteq> 0\<close> assms(4) nneg by fastforce
   finally show False
     using assms by auto
 qed
 
 lemma nonneg_has_sum_le_0D:
   fixes f :: "'a \<Rightarrow> 'b::{topological_ab_group_add,ordered_ab_group_add,linorder_topology}"
   assumes "has_sum f A a" \<open>a \<le> 0\<close>
     and nneg: "\<And>x. x \<in> A \<Longrightarrow> f x \<ge> 0"
     and "x \<in> A"
   shows "f x = 0"
   by (metis assms(1) assms(2) assms(4) infsumI nonneg_infsum_le_0D summable_on_def nneg)
 
 lemma has_sum_cmult_left:
   fixes f :: "'a \<Rightarrow> 'b :: {topological_semigroup_mult, semiring_0}"
   assumes \<open>has_sum f A a\<close>
   shows "has_sum (\<lambda>x. f x * c) A (a * c)"
 proof -
   from assms have \<open>(sum f \<longlongrightarrow> a) (finite_subsets_at_top A)\<close>
     using has_sum_def by blast
   then have \<open>((\<lambda>F. sum f F * c) \<longlongrightarrow> a * c) (finite_subsets_at_top A)\<close>
     by (simp add: tendsto_mult_right)
   then have \<open>(sum (\<lambda>x. f x * c) \<longlongrightarrow> a * c) (finite_subsets_at_top A)\<close>
     by (metis (mono_tags) tendsto_cong eventually_finite_subsets_at_top_weakI sum_distrib_right)
   then show ?thesis
     using infsumI has_sum_def by blast
 qed
 
 lemma infsum_cmult_left:
   fixes f :: "'a \<Rightarrow> 'b :: {t2_space, topological_semigroup_mult, semiring_0}"
   assumes \<open>c \<noteq> 0 \<Longrightarrow> f summable_on A\<close>
   shows "infsum (\<lambda>x. f x * c) A = infsum f A * c"
 proof (cases \<open>c=0\<close>)
   case True
   then show ?thesis by auto
 next
   case False
   then have \<open>has_sum f A (infsum f A)\<close>
     by (simp add: assms)
   then show ?thesis
     by (auto intro!: infsumI has_sum_cmult_left)
 qed
 
 lemma summable_on_cmult_left:
   fixes f :: "'a \<Rightarrow> 'b :: {t2_space, topological_semigroup_mult, semiring_0}"
   assumes \<open>f summable_on A\<close>
   shows "(\<lambda>x. f x * c) summable_on A"
   using assms summable_on_def has_sum_cmult_left by blast
 
 lemma has_sum_cmult_right:
   fixes f :: "'a \<Rightarrow> 'b :: {topological_semigroup_mult, semiring_0}"
   assumes \<open>has_sum f A a\<close>
   shows "has_sum (\<lambda>x. c * f x) A (c * a)"
 proof -
   from assms have \<open>(sum f \<longlongrightarrow> a) (finite_subsets_at_top A)\<close>
     using has_sum_def by blast
   then have \<open>((\<lambda>F. c * sum f F) \<longlongrightarrow> c * a) (finite_subsets_at_top A)\<close>
     by (simp add: tendsto_mult_left)
   then have \<open>(sum (\<lambda>x. c * f x) \<longlongrightarrow> c * a) (finite_subsets_at_top A)\<close>
     by (metis (mono_tags, lifting) tendsto_cong eventually_finite_subsets_at_top_weakI sum_distrib_left)
   then show ?thesis
     using infsumI has_sum_def by blast
 qed
 
 lemma infsum_cmult_right:
   fixes f :: "'a \<Rightarrow> 'b :: {t2_space, topological_semigroup_mult, semiring_0}"
   assumes \<open>c \<noteq> 0 \<Longrightarrow> f summable_on A\<close>
   shows \<open>infsum (\<lambda>x. c * f x) A = c * infsum f A\<close>
   using assms has_sum_cmult_right infsumI summable_iff_has_sum_infsum by fastforce
 
 lemma summable_on_cmult_right:
   fixes f :: "'a \<Rightarrow> 'b :: {t2_space, topological_semigroup_mult, semiring_0}"
   assumes \<open>f summable_on A\<close>
   shows "(\<lambda>x. c * f x) summable_on A"
   using assms summable_on_def has_sum_cmult_right by blast
 
 lemma summable_on_cmult_left':
   fixes f :: "'a \<Rightarrow> 'b :: {t2_space, topological_semigroup_mult, division_ring}"
   assumes \<open>c \<noteq> 0\<close>
   shows "(\<lambda>x. f x * c) summable_on A \<longleftrightarrow> f summable_on A"
 proof
   assume \<open>f summable_on A\<close>
   then show \<open>(\<lambda>x. f x * c) summable_on A\<close>
     by (rule summable_on_cmult_left)
 next
   assume \<open>(\<lambda>x. f x * c) summable_on A\<close>
   then have \<open>(\<lambda>x. f x * c * inverse c) summable_on A\<close>
     by (rule summable_on_cmult_left)
   then show \<open>f summable_on A\<close>
     by (metis (no_types, lifting) assms summable_on_cong mult.assoc mult.right_neutral right_inverse)
 qed
 
 lemma summable_on_cmult_right':
   fixes f :: "'a \<Rightarrow> 'b :: {t2_space, topological_semigroup_mult, division_ring}"
   assumes \<open>c \<noteq> 0\<close>
   shows "(\<lambda>x. c * f x) summable_on A \<longleftrightarrow> f summable_on A"
 proof
   assume \<open>f summable_on A\<close>
   then show \<open>(\<lambda>x. c * f x) summable_on A\<close>
     by (rule summable_on_cmult_right)
 next
   assume \<open>(\<lambda>x. c * f x) summable_on A\<close>
   then have \<open>(\<lambda>x. inverse c * (c * f x)) summable_on A\<close>
     by (rule summable_on_cmult_right)
   then show \<open>f summable_on A\<close>
     by (metis (no_types, lifting) assms summable_on_cong left_inverse mult.assoc mult.left_neutral)
 qed
 
 lemma infsum_cmult_left':
   fixes f :: "'a \<Rightarrow> 'b :: {t2_space, topological_semigroup_mult, division_ring}"
   shows "infsum (\<lambda>x. f x * c) A = infsum f A * c"
   by (metis (full_types) infsum_cmult_left infsum_not_exists mult_eq_0_iff summable_on_cmult_left')
 
 lemma infsum_cmult_right':
   fixes f :: "'a \<Rightarrow> 'b :: {t2_space,topological_semigroup_mult,division_ring}"
   shows "infsum (\<lambda>x. c * f x) A = c * infsum f A"
   by (metis (full_types) infsum_cmult_right infsum_not_exists mult_eq_0_iff summable_on_cmult_right')
 
 
 lemma has_sum_constant[simp]:
   assumes \<open>finite F\<close>
   shows \<open>has_sum (\<lambda>_. c) F (of_nat (card F) * c)\<close>
   by (metis assms has_sum_finite sum_constant)
 
 lemma infsum_constant[simp]:
   assumes \<open>finite F\<close>
   shows \<open>infsum (\<lambda>_. c) F = of_nat (card F) * c\<close>
   by (simp add: assms)
 
 lemma infsum_diverge_constant:
   \<comment> \<open>This probably does not really need all of \<^class>\<open>archimedean_field\<close> but Isabelle/HOL
        has no type class such as, e.g., "archimedean ring".\<close>
   fixes c :: \<open>'a::{archimedean_field, comm_monoid_add, linorder_topology, topological_semigroup_mult}\<close>
   assumes \<open>infinite A\<close> and \<open>c \<noteq> 0\<close>
   shows \<open>\<not> (\<lambda>_. c) summable_on A\<close>
 proof (rule notI)
   assume \<open>(\<lambda>_. c) summable_on A\<close>
   then have \<open>(\<lambda>_. inverse c * c) summable_on A\<close>
     by (rule summable_on_cmult_right)
   then have [simp]: \<open>(\<lambda>_. 1::'a) summable_on A\<close>
     using assms by auto
   have \<open>infsum (\<lambda>_. 1) A \<ge> d\<close> for d :: 'a
   proof -
     obtain n :: nat where \<open>of_nat n \<ge> d\<close>
       by (meson real_arch_simple)
     from assms
     obtain F where \<open>F \<subseteq> A\<close> and \<open>finite F\<close> and \<open>card F = n\<close>
       by (meson infinite_arbitrarily_large)
     note \<open>d \<le> of_nat n\<close>
     also have \<open>of_nat n = infsum (\<lambda>_. 1::'a) F\<close>
       by (simp add: \<open>card F = n\<close> \<open>finite F\<close>)
     also have \<open>\<dots> \<le> infsum (\<lambda>_. 1::'a) A\<close>
       apply (rule infsum_mono_neutral)
       using \<open>finite F\<close> \<open>F \<subseteq> A\<close> by auto
     finally show ?thesis .
   qed                                                        
   then show False
     by (meson linordered_field_no_ub not_less)
 qed
 
 lemma has_sum_constant_archimedean[simp]:
   \<comment> \<open>This probably does not really need all of \<^class>\<open>archimedean_field\<close> but Isabelle/HOL
        has no type class such as, e.g., "archimedean ring".\<close>
   fixes c :: \<open>'a::{archimedean_field, comm_monoid_add, linorder_topology, topological_semigroup_mult}\<close>
   shows \<open>infsum (\<lambda>_. c) A = of_nat (card A) * c\<close>
   by (metis infsum_0 infsum_constant infsum_diverge_constant infsum_not_exists sum.infinite sum_constant)
 
 lemma has_sum_uminus:
   fixes f :: \<open>'a \<Rightarrow> 'b::topological_ab_group_add\<close>
   shows \<open>has_sum (\<lambda>x. - f x) A a \<longleftrightarrow> has_sum f A (- a)\<close>
   by (auto simp add: sum_negf[abs_def] tendsto_minus_cancel_left has_sum_def)
 
 lemma summable_on_uminus:
   fixes f :: \<open>'a \<Rightarrow> 'b::topological_ab_group_add\<close>
   shows\<open>(\<lambda>x. - f x) summable_on A \<longleftrightarrow> f summable_on A\<close>
   by (metis summable_on_def has_sum_uminus verit_minus_simplify(4))
 
 lemma infsum_uminus:
   fixes f :: \<open>'a \<Rightarrow> 'b::{topological_ab_group_add, t2_space}\<close>
   shows \<open>infsum (\<lambda>x. - f x) A = - infsum f A\<close>
   by (metis (full_types) add.inverse_inverse add.inverse_neutral infsumI infsum_def has_sum_infsum has_sum_uminus)
 
 lemma has_sum_le_finite_sums:
   fixes a :: \<open>'a::{comm_monoid_add,topological_space,linorder_topology}\<close>
   assumes \<open>has_sum f A a\<close>
   assumes \<open>\<And>F. finite F \<Longrightarrow> F \<subseteq> A \<Longrightarrow> sum f F \<le> b\<close>
   shows \<open>a \<le> b\<close>
   by (metis assms eventually_finite_subsets_at_top_weakI finite_subsets_at_top_neq_bot has_sum_def tendsto_upperbound)
 
 lemma infsum_le_finite_sums:
   fixes b :: \<open>'a::{comm_monoid_add,topological_space,linorder_topology}\<close>
   assumes \<open>f summable_on A\<close>
   assumes \<open>\<And>F. finite F \<Longrightarrow> F \<subseteq> A \<Longrightarrow> sum f F \<le> b\<close>
   shows \<open>infsum f A \<le> b\<close>
   by (meson assms(1) assms(2) has_sum_infsum has_sum_le_finite_sums)
 
 
 lemma summable_on_scaleR_left [intro]:
   fixes c :: \<open>'a :: real_normed_vector\<close>
   assumes "c \<noteq> 0 \<Longrightarrow> f summable_on A"
   shows   "(\<lambda>x. f x *\<^sub>R c) summable_on A"
 proof (cases \<open>c = 0\<close>)
   case False
   then have "(\<lambda>y. y *\<^sub>R c) \<circ> f summable_on A"
     using assms by (auto simp add: scaleR_left.additive_axioms summable_on_comm_additive)
   then show ?thesis
     by (metis (mono_tags, lifting) comp_apply summable_on_cong)
 qed auto
 
 
 lemma summable_on_scaleR_right [intro]:
   fixes f :: \<open>'a \<Rightarrow> 'b :: real_normed_vector\<close>
   assumes "c \<noteq> 0 \<Longrightarrow> f summable_on A"
   shows   "(\<lambda>x. c *\<^sub>R f x) summable_on A"
 proof (cases \<open>c = 0\<close>)
   case False
   then have "(*\<^sub>R) c \<circ> f summable_on A"
   using assms by (auto simp add: scaleR_right.additive_axioms summable_on_comm_additive)
   then show ?thesis
     by (metis (mono_tags, lifting) comp_apply summable_on_cong)
 qed auto
 
 lemma infsum_scaleR_left:
   fixes c :: \<open>'a :: real_normed_vector\<close>
   assumes "c \<noteq> 0 \<Longrightarrow> f summable_on A"
   shows   "infsum (\<lambda>x. f x *\<^sub>R c) A = infsum f A *\<^sub>R c"
 proof (cases \<open>c = 0\<close>)
   case False
   then have "infsum ((\<lambda>y. y *\<^sub>R c) \<circ> f) A = infsum f A *\<^sub>R c"
   using assms by (auto simp add: scaleR_left.additive_axioms infsum_comm_additive)
   then show ?thesis
     by (metis (mono_tags, lifting) comp_apply infsum_cong)
 qed auto
 
 lemma infsum_scaleR_right:
   fixes f :: \<open>'a \<Rightarrow> 'b :: real_normed_vector\<close>
   shows   "infsum (\<lambda>x. c *\<^sub>R f x) A = c *\<^sub>R infsum f A"
 proof -
   consider (summable) \<open>f summable_on A\<close> | (c0) \<open>c = 0\<close> | (not_summable) \<open>\<not> f summable_on A\<close> \<open>c \<noteq> 0\<close>
     by auto
   then show ?thesis
   proof cases
     case summable
     then have "infsum ((*\<^sub>R) c \<circ> f) A = c *\<^sub>R infsum f A"
       by (auto simp add: scaleR_right.additive_axioms infsum_comm_additive)
     then show ?thesis
       by (metis (mono_tags, lifting) comp_apply infsum_cong)
   next
     case c0
     then show ?thesis by auto
   next
     case not_summable
     have \<open>\<not> (\<lambda>x. c *\<^sub>R f x) summable_on A\<close>
     proof (rule notI)
       assume \<open>(\<lambda>x. c *\<^sub>R f x) summable_on A\<close>
       then have \<open>(\<lambda>x. inverse c *\<^sub>R c *\<^sub>R f x) summable_on A\<close>
         using summable_on_scaleR_right by blast
       with not_summable show False
         by simp
     qed
     then show ?thesis
       by (simp add: infsum_not_exists not_summable(1)) 
   qed
 qed
 
 
 lemma infsum_Un_Int:
   fixes f :: "'a \<Rightarrow> 'b::{topological_ab_group_add, t2_space}"
   assumes "f summable_on A - B" "f summable_on B - A" \<open>f summable_on A \<inter> B\<close>
   shows   "infsum f (A \<union> B) = infsum f A + infsum f B - infsum f (A \<inter> B)"
 
 proof -
   have \<open>f summable_on A\<close>
     using assms by (metis Diff_Diff_Int Diff_disjoint Un_Diff_Int summable_on_Un_disjoint)
   have \<open>infsum f (A \<union> B) = infsum f A + infsum f (B - A)\<close>
     by (metis Diff_disjoint Un_Diff_cancel \<open>f summable_on A\<close> assms(2) infsum_Un_disjoint)
   moreover have \<open>infsum f (B - A \<union> A \<inter> B) = infsum f (B - A) + infsum f (A \<inter> B)\<close>
     using assms by (metis Int_Diff_disjoint inf_commute infsum_Un_disjoint)
   ultimately show ?thesis
     by (metis Un_Diff_Int add_diff_cancel_right' add_diff_eq inf_commute)
 qed
 
 lemma inj_combinator':
   assumes "x \<notin> F"
   shows \<open>inj_on (\<lambda>(g, y). g(x := y)) (Pi\<^sub>E F B \<times> B x)\<close>
 proof -
   have "inj_on ((\<lambda>(y, g). g(x := y)) \<circ> prod.swap) (Pi\<^sub>E F B \<times> B x)"
     using inj_combinator[of x F B] assms by (intro comp_inj_on) (auto simp: product_swap)
   thus ?thesis
     by (simp add: o_def)
 qed
 
 lemma infsum_prod_PiE:
   \<comment> \<open>See also \<open>infsum_prod_PiE_abs\<close> below with incomparable premises.\<close>
   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c :: {comm_monoid_mult, topological_semigroup_mult, division_ring, banach}"
   assumes finite: "finite A"
   assumes "\<And>x. x \<in> A \<Longrightarrow> f x summable_on B x"
   assumes "(\<lambda>g. \<Prod>x\<in>A. f x (g x)) summable_on (PiE A B)"
   shows   "infsum (\<lambda>g. \<Prod>x\<in>A. f x (g x)) (PiE A B) = (\<Prod>x\<in>A. infsum (f x) (B x))"
 proof (use finite assms(2-) in induction)
   case empty
   then show ?case 
     by auto
 next
   case (insert x F)
   have pi: \<open>Pi\<^sub>E (insert x F) B = (\<lambda>(g,y). g(x:=y)) ` (Pi\<^sub>E F B \<times> B x)\<close>
     unfolding PiE_insert_eq 
     by (subst swap_product [symmetric]) (simp add: image_image case_prod_unfold)
   have prod: \<open>(\<Prod>x'\<in>F. f x' ((p(x:=y)) x')) = (\<Prod>x'\<in>F. f x' (p x'))\<close> for p y
     by (rule prod.cong) (use insert.hyps in auto)
   have inj: \<open>inj_on (\<lambda>(g, y). g(x := y)) (Pi\<^sub>E F B \<times> B x)\<close>
     using \<open>x \<notin> F\<close> by (rule inj_combinator')
 
   have summable1: \<open>(\<lambda>g. \<Prod>x\<in>insert x F. f x (g x)) summable_on Pi\<^sub>E (insert x F) B\<close>
     using insert.prems(2) .
   also have \<open>Pi\<^sub>E (insert x F) B = (\<lambda>(g,y). g(x:=y)) ` (Pi\<^sub>E F B \<times> B x)\<close>
     by (simp only: pi)
   also have "(\<lambda>g. \<Prod>x\<in>insert x F. f x (g x)) summable_on \<dots> \<longleftrightarrow>
                ((\<lambda>g. \<Prod>x\<in>insert x F. f x (g x)) \<circ> (\<lambda>(g,y). g(x:=y))) summable_on (Pi\<^sub>E F B \<times> B x)"
     using inj by (rule summable_on_reindex)
   also have "(\<Prod>z\<in>F. f z ((g(x := y)) z)) = (\<Prod>z\<in>F. f z (g z))" for g y
     using insert.hyps by (intro prod.cong) auto
   hence "((\<lambda>g. \<Prod>x\<in>insert x F. f x (g x)) \<circ> (\<lambda>(g,y). g(x:=y))) =
              (\<lambda>(p, y). f x y * (\<Prod>x'\<in>F. f x' (p x')))"
     using insert.hyps by (auto simp: fun_eq_iff cong: prod.cong_simp)
   finally have summable2: \<open>(\<lambda>(p, y). f x y * (\<Prod>x'\<in>F. f x' (p x'))) summable_on Pi\<^sub>E F B \<times> B x\<close> .
 
   then have \<open>(\<lambda>p. \<Sum>\<^sub>\<infinity>y\<in>B x. f x y * (\<Prod>x'\<in>F. f x' (p x'))) summable_on Pi\<^sub>E F B\<close>
     by (rule summable_on_Sigma_banach)
   then have \<open>(\<lambda>p. (\<Sum>\<^sub>\<infinity>y\<in>B x. f x y) * (\<Prod>x'\<in>F. f x' (p x'))) summable_on Pi\<^sub>E F B\<close>
     by (metis (mono_tags, lifting) infsum_cmult_left' infsum_cong summable_on_cong)
   then have summable3: \<open>(\<lambda>p. (\<Prod>x'\<in>F. f x' (p x'))) summable_on Pi\<^sub>E F B\<close> if \<open>(\<Sum>\<^sub>\<infinity>y\<in>B x. f x y) \<noteq> 0\<close>
     using summable_on_cmult_right' that by blast
 
   have \<open>(\<Sum>\<^sub>\<infinity>g\<in>Pi\<^sub>E (insert x F) B. \<Prod>x\<in>insert x F. f x (g x))
      = (\<Sum>\<^sub>\<infinity>(p,y)\<in>Pi\<^sub>E F B \<times> B x. \<Prod>x'\<in>insert x F. f x' ((p(x:=y)) x'))\<close>
     by (smt (verit, ccfv_SIG) comp_apply infsum_cong infsum_reindex inj pi prod.cong split_def)
   also have \<open>\<dots> = (\<Sum>\<^sub>\<infinity>(p, y)\<in>Pi\<^sub>E F B \<times> B x. f x y * (\<Prod>x'\<in>F. f x' ((p(x:=y)) x')))\<close>
     using insert.hyps by auto
   also have \<open>\<dots> = (\<Sum>\<^sub>\<infinity>(p, y)\<in>Pi\<^sub>E F B \<times> B x. f x y * (\<Prod>x'\<in>F. f x' (p x')))\<close>
     using prod by presburger
   also have \<open>\<dots> = (\<Sum>\<^sub>\<infinity>p\<in>Pi\<^sub>E F B. \<Sum>\<^sub>\<infinity>y\<in>B x. f x y * (\<Prod>x'\<in>F. f x' (p x')))\<close>
     using infsum_Sigma'_banach summable2 by force
   also have \<open>\<dots> = (\<Sum>\<^sub>\<infinity>y\<in>B x. f x y) * (\<Sum>\<^sub>\<infinity>p\<in>Pi\<^sub>E F B. \<Prod>x'\<in>F. f x' (p x'))\<close>
     by (smt (verit) infsum_cmult_left' infsum_cmult_right' infsum_cong)
   also have \<open>\<dots> = (\<Prod>x\<in>insert x F. infsum (f x) (B x))\<close>
     using insert summable3 by auto
   finally show ?case
     by simp
 qed
 
 lemma infsum_prod_PiE_abs:
   \<comment> \<open>See also @{thm [source] infsum_prod_PiE} above with incomparable premises.\<close>
   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c :: {banach, real_normed_div_algebra, comm_semiring_1}"
   assumes finite: "finite A"
   assumes "\<And>x. x \<in> A \<Longrightarrow> f x abs_summable_on B x"
   shows   "infsum (\<lambda>g. \<Prod>x\<in>A. f x (g x)) (PiE A B) = (\<Prod>x\<in>A. infsum (f x) (B x))"
 proof (use finite assms(2) in induction)
   case empty
   then show ?case 
     by auto
 next
   case (insert x A)
   
   have pi: \<open>Pi\<^sub>E (insert x F) B = (\<lambda>(g,y). g(x:=y)) ` (Pi\<^sub>E F B \<times> B x)\<close> for x F and B :: "'a \<Rightarrow> 'b set"
     unfolding PiE_insert_eq 
     by (subst swap_product [symmetric]) (simp add: image_image case_prod_unfold)
   have prod: \<open>(\<Prod>x'\<in>A. f x' ((p(x:=y)) x')) = (\<Prod>x'\<in>A. f x' (p x'))\<close> for p y
     by (rule prod.cong) (use insert.hyps in auto)
   have inj: \<open>inj_on (\<lambda>(g, y). g(x := y)) (Pi\<^sub>E A B \<times> B x)\<close>
     using \<open>x \<notin> A\<close> by (rule inj_combinator')
 
   define s where \<open>s x = infsum (\<lambda>y. norm (f x y)) (B x)\<close> for x
 
   have *: \<open>(\<Sum>p\<in>P. norm (\<Prod>x\<in>F. f x (p x))) \<le> prod s F\<close> 
     if P: \<open>P \<subseteq> Pi\<^sub>E F B\<close> and [simp]: \<open>finite P\<close> \<open>finite F\<close> 
       and sum: \<open>\<And>x. x \<in> F \<Longrightarrow> f x abs_summable_on B x\<close> for P F
   proof -
     define B' where \<open>B' x = {p x| p. p\<in>P}\<close> for x
     have fin_B'[simp]: \<open>finite (B' x)\<close> for x
       using that by (auto simp: B'_def)
     have [simp]: \<open>finite (Pi\<^sub>E F B')\<close>
       by (simp add: finite_PiE)
     have [simp]: \<open>P \<subseteq> Pi\<^sub>E F B'\<close>
       using that by (auto simp: B'_def)
     have B'B: \<open>B' x \<subseteq> B x\<close> if \<open>x \<in> F\<close> for x
       unfolding B'_def using P that 
       by auto
     have s_bound: \<open>(\<Sum>y\<in>B' x. norm (f x y)) \<le> s x\<close> if \<open>x \<in> F\<close> for x
       by (metis B'B fin_B' finite_sum_le_has_sum has_sum_infsum norm_ge_zero s_def sum that)
     have \<open>(\<Sum>p\<in>P. norm (\<Prod>x\<in>F. f x (p x))) \<le> (\<Sum>p\<in>Pi\<^sub>E F B'. norm (\<Prod>x\<in>F. f x (p x)))\<close>
       by (simp add: sum_mono2)
     also have \<open>\<dots> = (\<Sum>p\<in>Pi\<^sub>E F B'. \<Prod>x\<in>F. norm (f x (p x)))\<close>
       by (simp add: prod_norm)
     also have \<open>\<dots> = (\<Prod>x\<in>F. \<Sum>y\<in>B' x. norm (f x y))\<close>
     proof (use \<open>finite F\<close> in induction)
       case empty
       then show ?case by simp
     next
       case (insert x F)
       have aux: \<open>a = b \<Longrightarrow> c * a = c * b\<close> for a b c :: real
         by auto
       have inj: \<open>inj_on (\<lambda>(g, y). g(x := y)) (Pi\<^sub>E F B' \<times> B' x)\<close>
         by (simp add: inj_combinator' insert.hyps)
       then have \<open>(\<Sum>p\<in>Pi\<^sub>E (insert x F) B'. \<Prod>x\<in>insert x F. norm (f x (p x)))
          =  (\<Sum>(p,y)\<in>Pi\<^sub>E F B' \<times> B' x. \<Prod>x'\<in>insert x F. norm (f x' ((p(x := y)) x')))\<close>
         by (simp add: pi sum.reindex case_prod_unfold)
       also have \<open>\<dots> = (\<Sum>(p,y)\<in>Pi\<^sub>E F B' \<times> B' x. norm (f x y) * (\<Prod>x'\<in>F. norm (f x' ((p(x := y)) x'))))\<close>
         using insert.hyps by auto
       also have \<open>\<dots> = (\<Sum>(p, y)\<in>Pi\<^sub>E F B' \<times> B' x. norm (f x y) * (\<Prod>x'\<in>F. norm (f x' (p x'))))\<close>
         by (smt (verit) fun_upd_apply insert.hyps(2) prod.cong split_def sum.cong)
       also have \<open>\<dots> = (\<Sum>y\<in>B' x. norm (f x y)) * (\<Sum>p\<in>Pi\<^sub>E F B'. \<Prod>x'\<in>F. norm (f x' (p x')))\<close>
         by (simp add: sum_product sum.swap [of _ "Pi\<^sub>E F B'"] sum.cartesian_product)
       also have \<open>\<dots> = (\<Sum>y\<in>B' x. norm (f x y)) * (\<Prod>x\<in>F. \<Sum>y\<in>B' x. norm (f x y))\<close>
         by (simp add: insert.IH)
       also have \<open>\<dots> = (\<Prod>x\<in>insert x F. \<Sum>y\<in>B' x. norm (f x y))\<close>
         using insert.hyps(1) insert.hyps(2) by force
       finally show ?case .
     qed
     also have \<open>\<dots> = (\<Prod>x\<in>F. \<Sum>\<^sub>\<infinity>y\<in>B' x. norm (f x y))\<close>
       by auto
     also have \<open>\<dots> \<le> (\<Prod>x\<in>F. s x)\<close>
       using s_bound by (simp add: prod_mono sum_nonneg)
     finally show ?thesis .
   qed
   have "bdd_above
      (sum (\<lambda>g. norm (\<Prod>x\<in>insert x A. f x (g x))) ` {F. F \<subseteq> Pi\<^sub>E (insert x A) B \<and> finite F})"
     apply (rule bdd_aboveI)
     using * insert.hyps insert.prems by blast
   then have \<open>(\<lambda>g. \<Prod>x\<in>insert x A. f x (g x)) abs_summable_on Pi\<^sub>E (insert x A) B\<close>
     using nonneg_bdd_above_summable_on
     by (metis (mono_tags, lifting) Collect_cong norm_ge_zero)
   also have \<open>Pi\<^sub>E (insert x A) B = (\<lambda>(g,y). g(x:=y)) ` (Pi\<^sub>E A B \<times> B x)\<close>
     by (simp only: pi)
   also have "(\<lambda>g. \<Prod>x\<in>insert x A. f x (g x)) abs_summable_on \<dots> \<longleftrightarrow>
                ((\<lambda>g. \<Prod>x\<in>insert x A. f x (g x)) \<circ> (\<lambda>(g,y). g(x:=y))) abs_summable_on (Pi\<^sub>E A B \<times> B x)"
     using inj by (subst summable_on_reindex) (auto simp: o_def)
   also have "(\<Prod>z\<in>A. f z ((g(x := y)) z)) = (\<Prod>z\<in>A. f z (g z))" for g y
     using insert.hyps by (intro prod.cong) auto
   hence "((\<lambda>g. \<Prod>x\<in>insert x A. f x (g x)) \<circ> (\<lambda>(g,y). g(x:=y))) =
              (\<lambda>(p, y). f x y * (\<Prod>x'\<in>A. f x' (p x')))"
     using insert.hyps by (auto simp: fun_eq_iff cong: prod.cong_simp)
   finally have summable2: \<open>(\<lambda>(p, y). f x y * (\<Prod>x'\<in>A. f x' (p x'))) abs_summable_on Pi\<^sub>E A B \<times> B x\<close> .
 
   have \<open>(\<Sum>\<^sub>\<infinity>g\<in>Pi\<^sub>E (insert x A) B. \<Prod>x\<in>insert x A. f x (g x))
      = (\<Sum>\<^sub>\<infinity>(p,y)\<in>Pi\<^sub>E A B \<times> B x. \<Prod>x'\<in>insert x A. f x' ((p(x:=y)) x'))\<close>
     using inj by (simp add: pi infsum_reindex o_def case_prod_unfold)
   also have \<open>\<dots> = (\<Sum>\<^sub>\<infinity>(p,y) \<in> Pi\<^sub>E A B \<times> B x. f x y * (\<Prod>x'\<in>A. f x' ((p(x:=y)) x')))\<close>
     using insert.hyps by auto
   also have \<open>\<dots> = (\<Sum>\<^sub>\<infinity>(p,y) \<in> Pi\<^sub>E A B \<times> B x. f x y * (\<Prod>x'\<in>A. f x' (p x')))\<close>
     using prod by presburger
   also have \<open>\<dots> = (\<Sum>\<^sub>\<infinity>p\<in>Pi\<^sub>E A B. \<Sum>\<^sub>\<infinity>y\<in>B x. f x y * (\<Prod>x'\<in>A. f x' (p x')))\<close>
     using abs_summable_summable infsum_Sigma'_banach summable2 by fastforce
   also have \<open>\<dots> = (\<Sum>\<^sub>\<infinity>y\<in>B x. f x y) * (\<Sum>\<^sub>\<infinity>p\<in>Pi\<^sub>E A B. \<Prod>x'\<in>A. f x' (p x'))\<close>
     by (smt (verit, best) infsum_cmult_left' infsum_cmult_right' infsum_cong)
   also have \<open>\<dots> = (\<Prod>x\<in>insert x A. infsum (f x) (B x))\<close>
     by (simp add: insert)
   finally show ?case
     by simp
 qed
 
 
 
 subsection \<open>Absolute convergence\<close>
 
 lemma abs_summable_countable:
   assumes \<open>f abs_summable_on A\<close>
   shows \<open>countable {x\<in>A. f x \<noteq> 0}\<close>
 proof -
   have fin: \<open>finite {x\<in>A. norm (f x) \<ge> t}\<close> if \<open>t > 0\<close> for t
   proof (rule ccontr)
     assume *: \<open>infinite {x \<in> A. t \<le> norm (f x)}\<close>
     have \<open>infsum (\<lambda>x. norm (f x)) A \<ge> b\<close> for b
     proof -
       obtain b' where b': \<open>of_nat b' \<ge> b / t\<close>
         by (meson real_arch_simple)
       from *
       obtain F where cardF: \<open>card F \<ge> b'\<close> and \<open>finite F\<close> and F: \<open>F \<subseteq> {x \<in> A. t \<le> norm (f x)}\<close>
         by (meson finite_if_finite_subsets_card_bdd nle_le)
       have \<open>b \<le> of_nat b' * t\<close>
         using b' \<open>t > 0\<close> by (simp add: field_simps split: if_splits)
       also have \<open>\<dots> \<le> of_nat (card F) * t\<close>
         by (simp add: cardF that)
       also have \<open>\<dots> = sum (\<lambda>x. t) F\<close>
         by simp
       also have \<open>\<dots> \<le> sum (\<lambda>x. norm (f x)) F\<close>
         by (metis (mono_tags, lifting) F in_mono mem_Collect_eq sum_mono)
       also have \<open>\<dots> = infsum (\<lambda>x. norm (f x)) F\<close>
         using \<open>finite F\<close> by (rule infsum_finite[symmetric])
       also have \<open>\<dots> \<le> infsum (\<lambda>x. norm (f x)) A\<close>
         by (rule infsum_mono_neutral) (use \<open>finite F\<close> assms F in auto)
       finally show ?thesis .
     qed
     then show False
       by (meson gt_ex linorder_not_less)
   qed
   have \<open>countable (\<Union>i\<in>{1..}. {x\<in>A. norm (f x) \<ge> 1/of_nat i})\<close>
     by (rule countable_UN) (use fin in \<open>auto intro!: countable_finite\<close>)
   also have \<open>\<dots> = {x\<in>A. f x \<noteq> 0}\<close>
   proof safe
     fix x assume x: "x \<in> A" "f x \<noteq> 0"
     define i where "i = max 1 (nat (ceiling (1 / norm (f x))))"
     have "i \<ge> 1"
       by (simp add: i_def)
     moreover have "real i \<ge> 1 / norm (f x)"
       unfolding i_def by linarith
     hence "1 / real i \<le> norm (f x)" using \<open>f x \<noteq> 0\<close>
       by (auto simp: divide_simps mult_ac)
     ultimately show "x \<in> (\<Union>i\<in>{1..}. {x \<in> A. 1 / real i \<le> norm (f x)})"
       using \<open>x \<in> A\<close> by auto
   qed auto
   finally show ?thesis .
 qed
 
 (* Logically belongs in the section about reals, but needed as a dependency here *)
 lemma summable_on_iff_abs_summable_on_real:
   fixes f :: \<open>'a \<Rightarrow> real\<close>
   shows \<open>f summable_on A \<longleftrightarrow> f abs_summable_on A\<close>
 proof (rule iffI)
   assume \<open>f summable_on A\<close>
   define n A\<^sub>p A\<^sub>n
     where \<open>n x = norm (f x)\<close> and \<open>A\<^sub>p = {x\<in>A. f x \<ge> 0}\<close> and \<open>A\<^sub>n = {x\<in>A. f x < 0}\<close> for x
   have A: \<open>A\<^sub>p \<union> A\<^sub>n = A\<close> \<open>A\<^sub>p \<inter> A\<^sub>n = {}\<close>
     by (auto simp: A\<^sub>p_def A\<^sub>n_def)
   from \<open>f summable_on A\<close> have \<open>f summable_on A\<^sub>p\<close> \<open>f summable_on A\<^sub>n\<close>
     using A\<^sub>p_def A\<^sub>n_def summable_on_subset_banach by fastforce+
   then have \<open>n summable_on A\<^sub>p\<close>
     by (smt (verit) A\<^sub>p_def n_def mem_Collect_eq real_norm_def summable_on_cong)
   moreover have \<open>n summable_on A\<^sub>n\<close>
     by (smt (verit, best) \<open>f summable_on A\<^sub>n\<close>  summable_on_uminus A\<^sub>n_def n_def summable_on_cong mem_Collect_eq real_norm_def)
   ultimately show \<open>n summable_on A\<close>
     using A summable_on_Un_disjoint by blast
 next
   show \<open>f abs_summable_on A \<Longrightarrow> f summable_on A\<close>
     using abs_summable_summable by blast
 qed
 
 lemma abs_summable_on_Sigma_iff:
   shows   "f abs_summable_on Sigma A B \<longleftrightarrow>
              (\<forall>x\<in>A. (\<lambda>y. f (x, y)) abs_summable_on B x) \<and>
              ((\<lambda>x. infsum (\<lambda>y. norm (f (x, y))) (B x)) abs_summable_on A)"
 proof (intro iffI conjI ballI)
   assume asm: \<open>f abs_summable_on Sigma A B\<close>
   then have \<open>(\<lambda>x. infsum (\<lambda>y. norm (f (x,y))) (B x)) summable_on A\<close>
     by (simp add: cond_case_prod_eta summable_on_Sigma_banach)
   then show \<open>(\<lambda>x. \<Sum>\<^sub>\<infinity>y\<in>B x. norm (f (x, y))) abs_summable_on A\<close>
     using summable_on_iff_abs_summable_on_real by force
 
   show \<open>(\<lambda>y. f (x, y)) abs_summable_on B x\<close> if \<open>x \<in> A\<close> for x
   proof -
     from asm have \<open>f abs_summable_on Pair x ` B x\<close>
       by (simp add: image_subset_iff summable_on_subset_banach that)
     then show ?thesis
       by (metis (mono_tags, lifting) o_def inj_on_def summable_on_reindex prod.inject summable_on_cong)
   qed
 next
   assume asm: \<open>(\<forall>x\<in>A. (\<lambda>xa. f (x, xa)) abs_summable_on B x) \<and>
     (\<lambda>x. \<Sum>\<^sub>\<infinity>y\<in>B x. norm (f (x, y))) abs_summable_on A\<close>
   have \<open>(\<Sum>xy\<in>F. norm (f xy)) \<le> (\<Sum>\<^sub>\<infinity>x\<in>A. \<Sum>\<^sub>\<infinity>y\<in>B x. norm (f (x, y)))\<close>
     if \<open>F \<subseteq> Sigma A B\<close> and [simp]: \<open>finite F\<close> for F
   proof -
     have [simp]: \<open>(SIGMA x:fst ` F. {y. (x, y) \<in> F}) = F\<close>
       by (auto intro!: set_eqI simp add: Domain.DomainI fst_eq_Domain)
     have [simp]: \<open>finite {y. (x, y) \<in> F}\<close> for x
       by (metis \<open>finite F\<close> Range.intros finite_Range finite_subset mem_Collect_eq subsetI)
     have \<open>(\<Sum>xy\<in>F. norm (f xy)) = (\<Sum>x\<in>fst ` F. \<Sum>y\<in>{y. (x,y)\<in>F}. norm (f (x,y)))\<close>
       by (simp add: sum.Sigma)
     also have \<open>\<dots> = (\<Sum>\<^sub>\<infinity>x\<in>fst ` F. \<Sum>\<^sub>\<infinity>y\<in>{y. (x,y)\<in>F}. norm (f (x,y)))\<close>
       by auto
     also have \<open>\<dots> \<le> (\<Sum>\<^sub>\<infinity>x\<in>fst ` F. \<Sum>\<^sub>\<infinity>y\<in>B x. norm (f (x,y)))\<close>
       using asm that(1) by (intro infsum_mono infsum_mono_neutral) auto
     also have \<open>\<dots> \<le> (\<Sum>\<^sub>\<infinity>x\<in>A. \<Sum>\<^sub>\<infinity>y\<in>B x. norm (f (x,y)))\<close>
       by (rule infsum_mono_neutral) (use asm that(1) in \<open>auto simp add: infsum_nonneg\<close>)
     finally show ?thesis .
   qed
   then show \<open>f abs_summable_on Sigma A B\<close>
     by (intro nonneg_bdd_above_summable_on) (auto simp: bdd_above_def)
 qed
 
 lemma abs_summable_on_comparison_test:
   assumes "g abs_summable_on A"
   assumes "\<And>x. x \<in> A \<Longrightarrow> norm (f x) \<le> norm (g x)"
   shows   "f abs_summable_on A"
 proof (rule nonneg_bdd_above_summable_on)
   show "bdd_above (sum (\<lambda>x. norm (f x)) ` {F. F \<subseteq> A \<and> finite F})"
   proof (rule bdd_aboveI2)
     fix F assume F: "F \<in> {F. F \<subseteq> A \<and> finite F}"
     have \<open>sum (\<lambda>x. norm (f x)) F \<le> sum (\<lambda>x. norm (g x)) F\<close>
       using assms F by (intro sum_mono) auto
     also have \<open>\<dots> = infsum (\<lambda>x. norm (g x)) F\<close>
       using F by simp
     also have \<open>\<dots> \<le> infsum (\<lambda>x. norm (g x)) A\<close>
     proof (rule infsum_mono_neutral)
       show "g abs_summable_on F"
         by (rule summable_on_subset_banach[OF assms(1)]) (use F in auto)
     qed (use F assms in auto)
     finally show "(\<Sum>x\<in>F. norm (f x)) \<le> (\<Sum>\<^sub>\<infinity>x\<in>A. norm (g x))" .
   qed
 qed auto
 
 lemma abs_summable_iff_bdd_above:
   fixes f :: \<open>'a \<Rightarrow> 'b::real_normed_vector\<close>
   shows \<open>f abs_summable_on A \<longleftrightarrow> bdd_above (sum (\<lambda>x. norm (f x)) ` {F. F\<subseteq>A \<and> finite F})\<close>
 proof (rule iffI)
   assume \<open>f abs_summable_on A\<close>
   show \<open>bdd_above (sum (\<lambda>x. norm (f x)) ` {F. F \<subseteq> A \<and> finite F})\<close>
   proof (rule bdd_aboveI2)
     fix F assume F: "F \<in> {F. F \<subseteq> A \<and> finite F}"
     show "(\<Sum>x\<in>F. norm (f x)) \<le> (\<Sum>\<^sub>\<infinity>x\<in>A. norm (f x))"
       by (rule finite_sum_le_infsum) (use \<open>f abs_summable_on A\<close> F in auto)
   qed
 next
   assume \<open>bdd_above (sum (\<lambda>x. norm (f x)) ` {F. F\<subseteq>A \<and> finite F})\<close>
   then show \<open>f abs_summable_on A\<close>
     by (simp add: nonneg_bdd_above_summable_on)
 qed
 
 lemma abs_summable_product:
   fixes x :: "'a \<Rightarrow> 'b::{real_normed_div_algebra,banach,second_countable_topology}"
   assumes x2_sum: "(\<lambda>i. (x i) * (x i)) abs_summable_on A"
     and y2_sum: "(\<lambda>i. (y i) * (y i)) abs_summable_on A"
   shows "(\<lambda>i. x i * y i) abs_summable_on A"
 proof (rule nonneg_bdd_above_summable_on)
   show "bdd_above (sum (\<lambda>xa. norm (x xa * y xa)) ` {F. F \<subseteq> A \<and> finite F})"
   proof (rule bdd_aboveI2)
     fix F assume F: \<open>F \<in> {F. F \<subseteq> A \<and> finite F}\<close>
     then have r1: "finite F" and b4: "F \<subseteq> A"
       by auto
   
     have a1: "(\<Sum>\<^sub>\<infinity>i\<in>F. norm (x i * x i)) \<le> (\<Sum>\<^sub>\<infinity>i\<in>A. norm (x i * x i))"
       by (metis (no_types, lifting) b4 infsum_mono2 norm_ge_zero summable_on_subset_banach x2_sum)
 
     have "norm (x i * y i) \<le> norm (x i * x i) + norm (y i * y i)" for i
       unfolding norm_mult by (smt mult_left_mono mult_nonneg_nonneg mult_right_mono norm_ge_zero)
     hence "(\<Sum>i\<in>F. norm (x i * y i)) \<le> (\<Sum>i\<in>F. norm (x i * x i) + norm (y i * y i))"
       by (simp add: sum_mono)
     also have "\<dots> = (\<Sum>i\<in>F. norm (x i * x i)) + (\<Sum>i\<in>F. norm (y i * y i))"
       by (simp add: sum.distrib)
     also have "\<dots> = (\<Sum>\<^sub>\<infinity>i\<in>F. norm (x i * x i)) + (\<Sum>\<^sub>\<infinity>i\<in>F. norm (y i * y i))"
       by (simp add: \<open>finite F\<close>)
     also have "\<dots> \<le> (\<Sum>\<^sub>\<infinity>i\<in>A. norm (x i * x i)) + (\<Sum>\<^sub>\<infinity>i\<in>A. norm (y i * y i))"
       using F assms
       by (intro add_mono infsum_mono2) auto
     finally show \<open>(\<Sum>xa\<in>F. norm (x xa * y xa)) \<le> (\<Sum>\<^sub>\<infinity>i\<in>A. norm (x i * x i)) + (\<Sum>\<^sub>\<infinity>i\<in>A. norm (y i * y i))\<close>
       by simp
   qed
 qed auto
 
 subsection \<open>Extended reals and nats\<close>
 
 lemma summable_on_ennreal[simp]: \<open>(f::_ \<Rightarrow> ennreal) summable_on S\<close>
   by (rule nonneg_summable_on_complete) simp
 
 lemma summable_on_enat[simp]: \<open>(f::_ \<Rightarrow> enat) summable_on S\<close>
   by (rule nonneg_summable_on_complete) simp
 
 lemma has_sum_superconst_infinite_ennreal:
   fixes f :: \<open>'a \<Rightarrow> ennreal\<close>
   assumes geqb: \<open>\<And>x. x \<in> S \<Longrightarrow> f x \<ge> b\<close>
   assumes b: \<open>b > 0\<close>
   assumes \<open>infinite S\<close>
   shows "has_sum f S \<infinity>"
 proof -
   have \<open>(sum f \<longlongrightarrow> \<infinity>) (finite_subsets_at_top S)\<close>
   proof (rule order_tendstoI[rotated], simp)
     fix y :: ennreal assume \<open>y < \<infinity>\<close>
     then have \<open>y / b < \<infinity>\<close>
       by (metis b ennreal_divide_eq_top_iff gr_implies_not_zero infinity_ennreal_def top.not_eq_extremum)
     then obtain F where \<open>finite F\<close> and \<open>F \<subseteq> S\<close> and cardF: \<open>card F > y / b\<close>
       using \<open>infinite S\<close>
       by (metis ennreal_Ex_less_of_nat infinite_arbitrarily_large infinity_ennreal_def)
     moreover have \<open>sum f Y > y\<close> if \<open>finite Y\<close> and \<open>F \<subseteq> Y\<close> and \<open>Y \<subseteq> S\<close> for Y
     proof -
       have \<open>y < b * card F\<close>
         by (metis \<open>y < \<infinity>\<close> b cardF divide_less_ennreal ennreal_mult_eq_top_iff gr_implies_not_zero infinity_ennreal_def mult.commute top.not_eq_extremum)
       also have \<open>\<dots> \<le> b * card Y\<close>
         by (meson b card_mono less_imp_le mult_left_mono of_nat_le_iff that(1) that(2))
       also have \<open>\<dots> = sum (\<lambda>_. b) Y\<close>
         by (simp add: mult.commute)
       also have \<open>\<dots> \<le> sum f Y\<close>
         using geqb by (meson subset_eq sum_mono that(3))
       finally show ?thesis .
     qed
     ultimately show \<open>\<forall>\<^sub>F x in finite_subsets_at_top S. y < sum f x\<close>
       unfolding eventually_finite_subsets_at_top 
       by auto
   qed
   then show ?thesis
     by (simp add: has_sum_def)
 qed
 
 lemma infsum_superconst_infinite_ennreal:
   fixes f :: \<open>'a \<Rightarrow> ennreal\<close>
   assumes \<open>\<And>x. x \<in> S \<Longrightarrow> f x \<ge> b\<close>
   assumes \<open>b > 0\<close>
   assumes \<open>infinite S\<close>
   shows "infsum f S = \<infinity>"
   using assms infsumI has_sum_superconst_infinite_ennreal by blast
 
 lemma infsum_superconst_infinite_ereal:
   fixes f :: \<open>'a \<Rightarrow> ereal\<close>
   assumes geqb: \<open>\<And>x. x \<in> S \<Longrightarrow> f x \<ge> b\<close>
   assumes b: \<open>b > 0\<close>
   assumes \<open>infinite S\<close>
   shows "infsum f S = \<infinity>"
 proof -
   obtain b' where b': \<open>e2ennreal b' = b\<close> and \<open>b' > 0\<close>
     using b by blast
   have "0 < e2ennreal b"
     using b' b
     by (metis dual_order.refl enn2ereal_e2ennreal gr_zeroI order_less_le zero_ennreal.abs_eq)
   hence *: \<open>infsum (e2ennreal \<circ> f) S = \<infinity>\<close>
     using assms b'
     by (intro infsum_superconst_infinite_ennreal[where b=b']) (auto intro!: e2ennreal_mono)
   have \<open>infsum f S = infsum (enn2ereal \<circ> (e2ennreal \<circ> f)) S\<close>
     using geqb b by (intro infsum_cong) (fastforce simp: enn2ereal_e2ennreal)
   also have \<open>\<dots> = enn2ereal \<infinity>\<close>
     using * by (simp add: infsum_comm_additive_general continuous_at_enn2ereal)
   also have \<open>\<dots> = \<infinity>\<close>
     by simp
   finally show ?thesis .
 qed
 
 lemma has_sum_superconst_infinite_ereal:
   fixes f :: \<open>'a \<Rightarrow> ereal\<close>
   assumes \<open>\<And>x. x \<in> S \<Longrightarrow> f x \<ge> b\<close>
   assumes \<open>b > 0\<close>
   assumes \<open>infinite S\<close>
   shows "has_sum f S \<infinity>"
   by (metis Infty_neq_0(1) assms infsum_def has_sum_infsum infsum_superconst_infinite_ereal)
 
 lemma infsum_superconst_infinite_enat:
   fixes f :: \<open>'a \<Rightarrow> enat\<close>
   assumes geqb: \<open>\<And>x. x \<in> S \<Longrightarrow> f x \<ge> b\<close>
   assumes b: \<open>b > 0\<close>
   assumes \<open>infinite S\<close>
   shows "infsum f S = \<infinity>"
 proof -
   have \<open>ennreal_of_enat (infsum f S) = infsum (ennreal_of_enat \<circ> f) S\<close>
     by (simp flip: infsum_comm_additive_general)
   also have \<open>\<dots> = \<infinity>\<close>
     by (metis assms(3) b comp_def ennreal_of_enat_0 ennreal_of_enat_le_iff geqb infsum_superconst_infinite_ennreal leD leI)
   also have \<open>\<dots> = ennreal_of_enat \<infinity>\<close>
     by simp
   finally show ?thesis
     by (rule ennreal_of_enat_inj[THEN iffD1])
 qed
 
 lemma has_sum_superconst_infinite_enat:
   fixes f :: \<open>'a \<Rightarrow> enat\<close>
   assumes \<open>\<And>x. x \<in> S \<Longrightarrow> f x \<ge> b\<close>
   assumes \<open>b > 0\<close>
   assumes \<open>infinite S\<close>
   shows "has_sum f S \<infinity>"
   by (metis assms i0_lb has_sum_infsum infsum_superconst_infinite_enat nonneg_summable_on_complete)
 
 text \<open>This lemma helps to relate a real-valued infsum to a supremum over extended nonnegative reals.\<close>
 
 lemma infsum_nonneg_is_SUPREMUM_ennreal:
   fixes f :: "'a \<Rightarrow> real"
   assumes summable: "f summable_on A"
     and fnn: "\<And>x. x\<in>A \<Longrightarrow> f x \<ge> 0"
   shows "ennreal (infsum f A) = (SUP F\<in>{F. finite F \<and> F \<subseteq> A}. (ennreal (sum f F)))"
 proof -
   have \<section>: "\<And>F. \<lbrakk>finite F; F \<subseteq> A\<rbrakk> \<Longrightarrow> sum (ennreal \<circ> f) F = ennreal (sum f F)"
     by (metis (mono_tags, lifting) comp_def fnn subsetD sum.cong sum_ennreal)
   then have \<open>ennreal (infsum f A) = infsum (ennreal \<circ> f) A\<close>
     by (simp add: infsum_comm_additive_general local.summable)
   also have \<open>\<dots> = (SUP F\<in>{F. finite F \<and> F \<subseteq> A}. (ennreal (sum f F)))\<close>
     by (metis (mono_tags, lifting) \<section> image_cong mem_Collect_eq nonneg_infsum_complete zero_le)
   finally show ?thesis .
 qed
 
 text \<open>This lemma helps to related a real-valued infsum to a supremum over extended reals.\<close>
 
 lemma infsum_nonneg_is_SUPREMUM_ereal:
   fixes f :: "'a \<Rightarrow> real"
   assumes summable: "f summable_on A"
     and fnn: "\<And>x. x\<in>A \<Longrightarrow> f x \<ge> 0"
   shows "ereal (infsum f A) = (SUP F\<in>{F. finite F \<and> F \<subseteq> A}. (ereal (sum f F)))"
 proof -
   have "\<And>F. \<lbrakk>finite F; F \<subseteq> A\<rbrakk> \<Longrightarrow> sum (ereal \<circ> f) F = ereal (sum f F)"
     by auto
   then have \<open>ereal (infsum f A) = infsum (ereal \<circ> f) A\<close>
     by (simp add: infsum_comm_additive_general local.summable)
   also have \<open>\<dots> = (SUP F\<in>{F. finite F \<and> F \<subseteq> A}. (ereal (sum f F)))\<close>
     by (subst nonneg_infsum_complete) (simp_all add: assms)
   finally show ?thesis .
 qed
 
 
 subsection \<open>Real numbers\<close>
 
 text \<open>Most lemmas in the general property section already apply to real numbers.
       A few ones that are specific to reals are given here.\<close>
 
 lemma infsum_nonneg_is_SUPREMUM_real:
   fixes f :: "'a \<Rightarrow> real"
   assumes summable: "f summable_on A"
     and fnn: "\<And>x. x\<in>A \<Longrightarrow> f x \<ge> 0"
   shows "infsum f A = (SUP F\<in>{F. finite F \<and> F \<subseteq> A}. (sum f F))"
 proof -
   have "ereal (infsum f A) = (SUP F\<in>{F. finite F \<and> F \<subseteq> A}. (ereal (sum f F)))"
     using assms by (rule infsum_nonneg_is_SUPREMUM_ereal)
   also have "\<dots> = ereal (SUP F\<in>{F. finite F \<and> F \<subseteq> A}. (sum f F))"
   proof (subst ereal_SUP)
     show "\<bar>SUP a\<in>{F. finite F \<and> F \<subseteq> A}. ereal (sum f a)\<bar> \<noteq> \<infinity>"
       using calculation by fastforce      
     show "(SUP F\<in>{F. finite F \<and> F \<subseteq> A}. ereal (sum f F)) = (SUP a\<in>{F. finite F \<and> F \<subseteq> A}. ereal (sum f a))"
       by simp      
   qed
   finally show ?thesis by simp
 qed
 
 
 lemma has_sum_nonneg_SUPREMUM_real:
   fixes f :: "'a \<Rightarrow> real"
   assumes "f summable_on A" and "\<And>x. x\<in>A \<Longrightarrow> f x \<ge> 0"
   shows "has_sum f A (SUP F\<in>{F. finite F \<and> F \<subseteq> A}. (sum f F))"
   by (metis (mono_tags, lifting) assms has_sum_infsum infsum_nonneg_is_SUPREMUM_real)
 
 lemma summable_countable_real:
   fixes f :: \<open>'a \<Rightarrow> real\<close>
   assumes \<open>f summable_on A\<close>
   shows \<open>countable {x\<in>A. f x \<noteq> 0}\<close>
   using abs_summable_countable assms summable_on_iff_abs_summable_on_real by blast
 
 subsection \<open>Complex numbers\<close>
 
 lemma has_sum_cnj_iff[simp]: 
   fixes f :: \<open>'a \<Rightarrow> complex\<close>
   shows \<open>has_sum (\<lambda>x. cnj (f x)) M (cnj a) \<longleftrightarrow> has_sum f M a\<close>
   by (simp add: has_sum_def lim_cnj del: cnj_sum add: cnj_sum[symmetric, abs_def, of f])
 
 lemma summable_on_cnj_iff[simp]:
   "(\<lambda>i. cnj (f i)) summable_on A \<longleftrightarrow> f summable_on A"
   by (metis complex_cnj_cnj summable_on_def has_sum_cnj_iff)
 
 lemma infsum_cnj[simp]: \<open>infsum (\<lambda>x. cnj (f x)) M = cnj (infsum f M)\<close>
   by (metis complex_cnj_zero infsumI has_sum_cnj_iff infsum_def summable_on_cnj_iff has_sum_infsum)
 
 lemma has_sum_Re:
   assumes "has_sum f M a"
   shows "has_sum (\<lambda>x. Re (f x)) M (Re a)"
   using has_sum_comm_additive[where f=Re]
   using  assms tendsto_Re by (fastforce simp add: o_def additive_def)
 
 lemma infsum_Re:
   assumes "f summable_on M"
   shows "infsum (\<lambda>x. Re (f x)) M = Re (infsum f M)"
   by (simp add: assms has_sum_Re infsumI)
 
 lemma summable_on_Re: 
   assumes "f summable_on M"
   shows "(\<lambda>x. Re (f x)) summable_on M"
   by (metis assms has_sum_Re summable_on_def)
 
 lemma has_sum_Im:
   assumes "has_sum f M a"
   shows "has_sum (\<lambda>x. Im (f x)) M (Im a)"
   using has_sum_comm_additive[where f=Im]
   using  assms tendsto_Im by (fastforce simp add: o_def additive_def)
 
 lemma infsum_Im: 
   assumes "f summable_on M"
   shows "infsum (\<lambda>x. Im (f x)) M = Im (infsum f M)"
   by (simp add: assms has_sum_Im infsumI)
 
 lemma summable_on_Im: 
   assumes "f summable_on M"
   shows "(\<lambda>x. Im (f x)) summable_on M"
   by (metis assms has_sum_Im summable_on_def)
 
 lemma nonneg_infsum_le_0D_complex:
   fixes f :: "'a \<Rightarrow> complex"
   assumes "infsum f A \<le> 0"
     and abs_sum: "f summable_on A"
     and nneg: "\<And>x. x \<in> A \<Longrightarrow> f x \<ge> 0"
     and "x \<in> A"
   shows "f x = 0"
 proof -
   have \<open>Im (f x) = 0\<close>
     using assms(4) less_eq_complex_def nneg by auto
   moreover have \<open>Re (f x) = 0\<close>
     using assms by (auto simp add: summable_on_Re infsum_Re less_eq_complex_def intro: nonneg_infsum_le_0D[where A=A])
   ultimately show ?thesis
     by (simp add: complex_eqI)
 qed
 
 lemma nonneg_has_sum_le_0D_complex:
   fixes f :: "'a \<Rightarrow> complex"
   assumes "has_sum f A a" and \<open>a \<le> 0\<close>
     and "\<And>x. x \<in> A \<Longrightarrow> f x \<ge> 0" and "x \<in> A"
   shows "f x = 0"
   by (metis assms infsumI nonneg_infsum_le_0D_complex summable_on_def)
 
 text \<open>The lemma @{thm [source] infsum_mono_neutral} above applies to various linear ordered monoids such as the reals but not to the complex numbers.
       Thus we have a separate corollary for those:\<close>
 
 lemma infsum_mono_neutral_complex:
   fixes f :: "'a \<Rightarrow> complex"
   assumes [simp]: "f summable_on A"
     and [simp]: "g summable_on B"
   assumes \<open>\<And>x. x \<in> A\<inter>B \<Longrightarrow> f x \<le> g x\<close>
   assumes \<open>\<And>x. x \<in> A-B \<Longrightarrow> f x \<le> 0\<close>
   assumes \<open>\<And>x. x \<in> B-A \<Longrightarrow> g x \<ge> 0\<close>
   shows \<open>infsum f A \<le> infsum g B\<close>
 proof -
   have \<open>infsum (\<lambda>x. Re (f x)) A \<le> infsum (\<lambda>x. Re (g x)) B\<close>
     by (smt (verit) assms infsum_cong infsum_mono_neutral less_eq_complex_def summable_on_Re zero_complex.simps(1))
   then have Re: \<open>Re (infsum f A) \<le> Re (infsum g B)\<close>
     by (metis assms(1-2) infsum_Re)
   have \<open>infsum (\<lambda>x. Im (f x)) A = infsum (\<lambda>x. Im (g x)) B\<close>
     by (smt (verit, best) assms(3-5) infsum_cong_neutral less_eq_complex_def zero_complex.simps(2))
   then have Im: \<open>Im (infsum f A) = Im (infsum g B)\<close>
     by (metis assms(1-2) infsum_Im)
   from Re Im show ?thesis
     by (auto simp: less_eq_complex_def)
 qed
 
 lemma infsum_mono_complex:
   \<comment> \<open>For \<^typ>\<open>real\<close>, @{thm [source] infsum_mono} can be used. 
       But \<^typ>\<open>complex\<close> does not have the right typeclass.\<close>
   fixes f g :: "'a \<Rightarrow> complex"
   assumes f_sum: "f summable_on A" and g_sum: "g summable_on A"
   assumes leq: "\<And>x. x \<in> A \<Longrightarrow> f x \<le> g x"
   shows   "infsum f A \<le> infsum g A"
   by (metis DiffE IntD1 f_sum g_sum infsum_mono_neutral_complex leq)
 
 
 lemma infsum_nonneg_complex:
   fixes f :: "'a \<Rightarrow> complex"
   assumes "f summable_on M"
     and "\<And>x. x \<in> M \<Longrightarrow> 0 \<le> f x"
   shows "infsum f M \<ge> 0" (is "?lhs \<ge> _")
   by (metis assms infsum_0_simp summable_on_0_simp infsum_mono_complex)
 
 lemma infsum_cmod:
   assumes "f summable_on M"
     and fnn: "\<And>x. x \<in> M \<Longrightarrow> 0 \<le> f x"
   shows "infsum (\<lambda>x. cmod (f x)) M = cmod (infsum f M)"
 proof -
   have \<open>complex_of_real (infsum (\<lambda>x. cmod (f x)) M) = infsum (\<lambda>x. complex_of_real (cmod (f x))) M\<close>
   proof (rule infsum_comm_additive[symmetric, unfolded o_def])
     have "(\<lambda>z. Re (f z)) summable_on M"
       using assms summable_on_Re by blast
     also have "?this \<longleftrightarrow> f abs_summable_on M"
       using fnn by (intro summable_on_cong) (auto simp: less_eq_complex_def cmod_def)
     finally show \<dots> .
   qed (auto simp: additive_def)
   also have \<open>\<dots> = infsum f M\<close>
     using fnn cmod_eq_Re complex_is_Real_iff less_eq_complex_def by (force cong: infsum_cong)
   finally show ?thesis
     by (metis abs_of_nonneg infsum_def le_less_trans norm_ge_zero norm_infsum_bound norm_of_real not_le order_refl)
 qed
 
 
 lemma summable_on_iff_abs_summable_on_complex:
   fixes f :: \<open>'a \<Rightarrow> complex\<close>
   shows \<open>f summable_on A \<longleftrightarrow> f abs_summable_on A\<close>
 proof (rule iffI)
   assume \<open>f summable_on A\<close>
   define i r ni nr n where \<open>i x = Im (f x)\<close> and \<open>r x = Re (f x)\<close>
     and \<open>ni x = norm (i x)\<close> and \<open>nr x = norm (r x)\<close> and \<open>n x = norm (f x)\<close> for x
   from \<open>f summable_on A\<close> have \<open>i summable_on A\<close>
     by (simp add: i_def[abs_def] summable_on_Im)
   then have [simp]: \<open>ni summable_on A\<close>
     using ni_def[abs_def] summable_on_iff_abs_summable_on_real by force
 
   from \<open>f summable_on A\<close> have \<open>r summable_on A\<close>
     by (simp add: r_def[abs_def] summable_on_Re)
   then have [simp]: \<open>nr summable_on A\<close>
     by (metis nr_def summable_on_cong summable_on_iff_abs_summable_on_real)
 
   have n_sum: \<open>n x \<le> nr x + ni x\<close> for x
     by (simp add: n_def nr_def ni_def r_def i_def cmod_le)
 
   have *: \<open>(\<lambda>x. nr x + ni x) summable_on A\<close>
     by (simp add: summable_on_add)
   have "bdd_above (sum n ` {F. F \<subseteq> A \<and> finite F})"
     apply (rule bdd_aboveI[where M=\<open>infsum (\<lambda>x. nr x + ni x) A\<close>])
     using * n_sum by (auto simp flip: infsum_finite simp: ni_def nr_def intro!: infsum_mono_neutral)
   then show \<open>n summable_on A\<close>
     by (simp add: n_def nonneg_bdd_above_summable_on)
 next
   show \<open>f abs_summable_on A \<Longrightarrow> f summable_on A\<close>
     using abs_summable_summable by blast
 qed
 
 lemma summable_countable_complex:
   fixes f :: \<open>'a \<Rightarrow> complex\<close>
   assumes \<open>f summable_on A\<close>
   shows \<open>countable {x\<in>A. f x \<noteq> 0}\<close>
   using abs_summable_countable assms summable_on_iff_abs_summable_on_complex by blast
 
 end