diff --git a/src/HOL/Series.thy b/src/HOL/Series.thy
--- a/src/HOL/Series.thy
+++ b/src/HOL/Series.thy
@@ -1,1333 +1,1329 @@
 (*  Title       : Series.thy
     Author      : Jacques D. Fleuriot
     Copyright   : 1998  University of Cambridge
 
 Converted to Isar and polished by lcp
 Converted to sum and polished yet more by TNN
 Additional contributions by Jeremy Avigad
 *)
 
 section \<open>Infinite Series\<close>
 
 theory Series
 imports Limits Inequalities
 begin
 
 subsection \<open>Definition of infinite summability\<close>
 
 definition sums :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a \<Rightarrow> bool"
     (infixr "sums" 80)
   where "f sums s \<longleftrightarrow> (\<lambda>n. \<Sum>i<n. f i) \<longlonglongrightarrow> s"
 
 definition summable :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> bool"
   where "summable f \<longleftrightarrow> (\<exists>s. f sums s)"
 
 definition suminf :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a"
     (binder "\<Sum>" 10)
   where "suminf f = (THE s. f sums s)"
 
 text\<open>Variants of the definition\<close>
 lemma sums_def': "f sums s \<longleftrightarrow> (\<lambda>n. \<Sum>i = 0..n. f i) \<longlonglongrightarrow> s"
   unfolding sums_def
   apply (subst filterlim_sequentially_Suc [symmetric])
   apply (simp only: lessThan_Suc_atMost atLeast0AtMost)
   done
 
 lemma sums_def_le: "f sums s \<longleftrightarrow> (\<lambda>n. \<Sum>i\<le>n. f i) \<longlonglongrightarrow> s"
   by (simp add: sums_def' atMost_atLeast0)
 
 lemma bounded_imp_summable:
   fixes a :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder,linorder_topology,linordered_comm_semiring_strict}"
   assumes 0: "\<And>n. a n \<ge> 0" and bounded: "\<And>n. (\<Sum>k\<le>n. a k) \<le> B"
   shows "summable a" 
 proof -
   have "bdd_above (range(\<lambda>n. \<Sum>k\<le>n. a k))"
     by (meson bdd_aboveI2 bounded)
   moreover have "incseq (\<lambda>n. \<Sum>k\<le>n. a k)"
     by (simp add: mono_def "0" sum_mono2)
   ultimately obtain s where "(\<lambda>n. \<Sum>k\<le>n. a k) \<longlonglongrightarrow> s"
     using LIMSEQ_incseq_SUP by blast
   then show ?thesis
     by (auto simp: sums_def_le summable_def)
 qed
 
 
 subsection \<open>Infinite summability on topological monoids\<close>
 
 lemma sums_subst[trans]: "f = g \<Longrightarrow> g sums z \<Longrightarrow> f sums z"
   by simp
 
 lemma sums_cong: "(\<And>n. f n = g n) \<Longrightarrow> f sums c \<longleftrightarrow> g sums c"
-  by (drule ext) simp
+  by presburger
 
 lemma sums_summable: "f sums l \<Longrightarrow> summable f"
   by (simp add: sums_def summable_def, blast)
 
 lemma summable_iff_convergent: "summable f \<longleftrightarrow> convergent (\<lambda>n. \<Sum>i<n. f i)"
   by (simp add: summable_def sums_def convergent_def)
 
 lemma summable_iff_convergent': "summable f \<longleftrightarrow> convergent (\<lambda>n. sum f {..n})"
-  by (simp_all only: summable_iff_convergent convergent_def
-        lessThan_Suc_atMost [symmetric] filterlim_sequentially_Suc[of "\<lambda>n. sum f {..<n}"])
+  by (simp add: convergent_def summable_def sums_def_le)
 
 lemma suminf_eq_lim: "suminf f = lim (\<lambda>n. \<Sum>i<n. f i)"
   by (simp add: suminf_def sums_def lim_def)
 
 lemma sums_zero[simp, intro]: "(\<lambda>n. 0) sums 0"
   unfolding sums_def by simp
 
 lemma summable_zero[simp, intro]: "summable (\<lambda>n. 0)"
   by (rule sums_zero [THEN sums_summable])
 
 lemma sums_group: "f sums s \<Longrightarrow> 0 < k \<Longrightarrow> (\<lambda>n. sum f {n * k ..< n * k + k}) sums s"
   apply (simp only: sums_def sum.nat_group tendsto_def eventually_sequentially)
   apply (erule all_forward imp_forward exE| assumption)+
-  apply (rule_tac x="N" in exI)
   by (metis le_square mult.commute mult.left_neutral mult_le_cancel2 mult_le_mono)
 
 lemma suminf_cong: "(\<And>n. f n = g n) \<Longrightarrow> suminf f = suminf g"
-  by (rule arg_cong[of f g], rule ext) simp
+  by presburger
 
 lemma summable_cong:
   fixes f g :: "nat \<Rightarrow> 'a::real_normed_vector"
   assumes "eventually (\<lambda>x. f x = g x) sequentially"
   shows "summable f = summable g"
 proof -
   from assms obtain N where N: "\<forall>n\<ge>N. f n = g n"
     by (auto simp: eventually_at_top_linorder)
   define C where "C = (\<Sum>k<N. f k - g k)"
   from eventually_ge_at_top[of N]
   have "eventually (\<lambda>n. sum f {..<n} = C + sum g {..<n}) sequentially"
   proof eventually_elim
     case (elim n)
     then have "{..<n} = {..<N} \<union> {N..<n}"
       by auto
     also have "sum f ... = sum f {..<N} + sum f {N..<n}"
       by (intro sum.union_disjoint) auto
     also from N have "sum f {N..<n} = sum g {N..<n}"
       by (intro sum.cong) simp_all
     also have "sum f {..<N} + sum g {N..<n} = C + (sum g {..<N} + sum g {N..<n})"
       unfolding C_def by (simp add: algebra_simps sum_subtractf)
     also have "sum g {..<N} + sum g {N..<n} = sum g ({..<N} \<union> {N..<n})"
       by (intro sum.union_disjoint [symmetric]) auto
     also from elim have "{..<N} \<union> {N..<n} = {..<n}"
       by auto
     finally show "sum f {..<n} = C + sum g {..<n}" .
   qed
   from convergent_cong[OF this] show ?thesis
     by (simp add: summable_iff_convergent convergent_add_const_iff)
 qed
 
 lemma sums_finite:
   assumes [simp]: "finite N"
     and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 0"
   shows "f sums (\<Sum>n\<in>N. f n)"
 proof -
   have eq: "sum f {..<n + Suc (Max N)} = sum f N" for n
     by (rule sum.mono_neutral_right) (auto simp: add_increasing less_Suc_eq_le f)
   show ?thesis
     unfolding sums_def
     by (rule LIMSEQ_offset[of _ "Suc (Max N)"])
       (simp add: eq atLeast0LessThan del: add_Suc_right)
 qed
 
 corollary sums_0: "(\<And>n. f n = 0) \<Longrightarrow> (f sums 0)"
     by (metis (no_types) finite.emptyI sum.empty sums_finite)
 
 lemma summable_finite: "finite N \<Longrightarrow> (\<And>n. n \<notin> N \<Longrightarrow> f n = 0) \<Longrightarrow> summable f"
   by (rule sums_summable) (rule sums_finite)
 
 lemma sums_If_finite_set: "finite A \<Longrightarrow> (\<lambda>r. if r \<in> A then f r else 0) sums (\<Sum>r\<in>A. f r)"
   using sums_finite[of A "(\<lambda>r. if r \<in> A then f r else 0)"] by simp
 
 lemma summable_If_finite_set[simp, intro]: "finite A \<Longrightarrow> summable (\<lambda>r. if r \<in> A then f r else 0)"
   by (rule sums_summable) (rule sums_If_finite_set)
 
 lemma sums_If_finite: "finite {r. P r} \<Longrightarrow> (\<lambda>r. if P r then f r else 0) sums (\<Sum>r | P r. f r)"
   using sums_If_finite_set[of "{r. P r}"] by simp
 
 lemma summable_If_finite[simp, intro]: "finite {r. P r} \<Longrightarrow> summable (\<lambda>r. if P r then f r else 0)"
   by (rule sums_summable) (rule sums_If_finite)
 
 lemma sums_single: "(\<lambda>r. if r = i then f r else 0) sums f i"
   using sums_If_finite[of "\<lambda>r. r = i"] by simp
 
 lemma summable_single[simp, intro]: "summable (\<lambda>r. if r = i then f r else 0)"
   by (rule sums_summable) (rule sums_single)
 
 context
   fixes f :: "nat \<Rightarrow> 'a::{t2_space,comm_monoid_add}"
 begin
 
 lemma summable_sums[intro]: "summable f \<Longrightarrow> f sums (suminf f)"
   by (simp add: summable_def sums_def suminf_def)
      (metis convergent_LIMSEQ_iff convergent_def lim_def)
 
 lemma summable_LIMSEQ: "summable f \<Longrightarrow> (\<lambda>n. \<Sum>i<n. f i) \<longlonglongrightarrow> suminf f"
   by (rule summable_sums [unfolded sums_def])
 
 lemma summable_LIMSEQ': "summable f \<Longrightarrow> (\<lambda>n. \<Sum>i\<le>n. f i) \<longlonglongrightarrow> suminf f"
   using sums_def_le by blast
 
 lemma sums_unique: "f sums s \<Longrightarrow> s = suminf f"
   by (metis limI suminf_eq_lim sums_def)
 
 lemma sums_iff: "f sums x \<longleftrightarrow> summable f \<and> suminf f = x"
   by (metis summable_sums sums_summable sums_unique)
 
 lemma summable_sums_iff: "summable f \<longleftrightarrow> f sums suminf f"
   by (auto simp: sums_iff summable_sums)
 
 lemma sums_unique2: "f sums a \<Longrightarrow> f sums b \<Longrightarrow> a = b"
   for a b :: 'a
   by (simp add: sums_iff)
 
 lemma sums_Uniq: "\<exists>\<^sub>\<le>\<^sub>1a. f sums a"
   for a b :: 'a
   by (simp add: sums_unique2 Uniq_def)
 
 lemma suminf_finite:
   assumes N: "finite N"
     and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 0"
   shows "suminf f = (\<Sum>n\<in>N. f n)"
   using sums_finite[OF assms, THEN sums_unique] by simp
 
 end
 
 lemma suminf_zero[simp]: "suminf (\<lambda>n. 0::'a::{t2_space, comm_monoid_add}) = 0"
   by (rule sums_zero [THEN sums_unique, symmetric])
 
 
 subsection \<open>Infinite summability on ordered, topological monoids\<close>
 
 lemma sums_le: "(\<And>n. f n \<le> g n) \<Longrightarrow> f sums s \<Longrightarrow> g sums t \<Longrightarrow> s \<le> t"
   for f g :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add,linorder_topology}"
   by (rule LIMSEQ_le) (auto intro: sum_mono simp: sums_def)
 
 context
   fixes f :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add,linorder_topology}"
 begin
 
 lemma suminf_le: "(\<And>n. f n \<le> g n) \<Longrightarrow> summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f \<le> suminf g"
   using sums_le by blast
 
 lemma sum_le_suminf:
   shows "summable f \<Longrightarrow> finite I \<Longrightarrow> (\<And>n. n \<in>- I \<Longrightarrow> 0 \<le> f n) \<Longrightarrow> sum f I \<le> suminf f"
   by (rule sums_le[OF _ sums_If_finite_set summable_sums]) auto
 
 lemma suminf_nonneg: "summable f \<Longrightarrow> (\<And>n. 0 \<le> f n) \<Longrightarrow> 0 \<le> suminf f"
   using sum_le_suminf by force
 
 lemma suminf_le_const: "summable f \<Longrightarrow> (\<And>n. sum f {..<n} \<le> x) \<Longrightarrow> suminf f \<le> x"
   by (metis LIMSEQ_le_const2 summable_LIMSEQ)
 
 lemma suminf_eq_zero_iff: 
   assumes "summable f" and pos: "\<And>n. 0 \<le> f n"
   shows "suminf f = 0 \<longleftrightarrow> (\<forall>n. f n = 0)"
 proof
-  assume "suminf f = 0" 
+  assume L: "suminf f = 0" 
   then have f: "(\<lambda>n. \<Sum>i<n. f i) \<longlonglongrightarrow> 0"
     using summable_LIMSEQ[of f] assms by simp
   then have "\<And>i. (\<Sum>n\<in>{i}. f n) \<le> 0"
-  proof (rule LIMSEQ_le_const)
-    show "\<exists>N. \<forall>n\<ge>N. (\<Sum>n\<in>{i}. f n) \<le> sum f {..<n}" for i
-      using pos by (intro exI[of _ "Suc i"] allI impI sum_mono2) auto
-  qed
+    by (metis L \<open>summable f\<close> order_refl pos sum.infinite sum_le_suminf)
   with pos show "\<forall>n. f n = 0"
-    by (auto intro!: antisym)
+    by (simp add: order.antisym)
 qed (metis suminf_zero fun_eq_iff)
 
 lemma suminf_pos_iff: "summable f \<Longrightarrow> (\<And>n. 0 \<le> f n) \<Longrightarrow> 0 < suminf f \<longleftrightarrow> (\<exists>i. 0 < f i)"
   using sum_le_suminf[of "{}"] suminf_eq_zero_iff by (simp add: less_le)
 
 lemma suminf_pos2:
   assumes "summable f" "\<And>n. 0 \<le> f n" "0 < f i"
   shows "0 < suminf f"
 proof -
   have "0 < (\<Sum>n<Suc i. f n)"
     using assms by (intro sum_pos2[where i=i]) auto
   also have "\<dots> \<le> suminf f"
     using assms by (intro sum_le_suminf) auto
   finally show ?thesis .
 qed
 
 lemma suminf_pos: "summable f \<Longrightarrow> (\<And>n. 0 < f n) \<Longrightarrow> 0 < suminf f"
   by (intro suminf_pos2[where i=0]) (auto intro: less_imp_le)
 
 end
 
 context
   fixes f :: "nat \<Rightarrow> 'a::{ordered_cancel_comm_monoid_add,linorder_topology}"
 begin
 
 lemma sum_less_suminf2:
   "summable f \<Longrightarrow> (\<And>m. m\<ge>n \<Longrightarrow> 0 \<le> f m) \<Longrightarrow> n \<le> i \<Longrightarrow> 0 < f i \<Longrightarrow> sum f {..<n} < suminf f"
   using sum_le_suminf[of f "{..< Suc i}"]
     and add_strict_increasing[of "f i" "sum f {..<n}" "sum f {..<i}"]
     and sum_mono2[of "{..<i}" "{..<n}" f]
   by (auto simp: less_imp_le ac_simps)
 
 lemma sum_less_suminf: "summable f \<Longrightarrow> (\<And>m. m\<ge>n \<Longrightarrow> 0 < f m) \<Longrightarrow> sum f {..<n} < suminf f"
   using sum_less_suminf2[of n n] by (simp add: less_imp_le)
 
 end
 
 lemma summableI_nonneg_bounded:
   fixes f :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add,linorder_topology,conditionally_complete_linorder}"
   assumes pos[simp]: "\<And>n. 0 \<le> f n"
     and le: "\<And>n. (\<Sum>i<n. f i) \<le> x"
   shows "summable f"
   unfolding summable_def sums_def [abs_def]
 proof (rule exI LIMSEQ_incseq_SUP)+
   show "bdd_above (range (\<lambda>n. sum f {..<n}))"
     using le by (auto simp: bdd_above_def)
   show "incseq (\<lambda>n. sum f {..<n})"
     by (auto simp: mono_def intro!: sum_mono2)
 qed
 
 lemma summableI[intro, simp]: "summable f"
   for f :: "nat \<Rightarrow> 'a::{canonically_ordered_monoid_add,linorder_topology,complete_linorder}"
   by (intro summableI_nonneg_bounded[where x=top] zero_le top_greatest)
 
 lemma suminf_eq_SUP_real:
   assumes X: "summable X" "\<And>i. 0 \<le> X i" shows "suminf X = (SUP i. \<Sum>n<i. X n::real)"
   by (intro LIMSEQ_unique[OF summable_LIMSEQ] X LIMSEQ_incseq_SUP)
      (auto intro!: bdd_aboveI2[where M="\<Sum>i. X i"] sum_le_suminf X monoI sum_mono2)
 
 
 subsection \<open>Infinite summability on topological monoids\<close>
 
 context
   fixes f g :: "nat \<Rightarrow> 'a::{t2_space,topological_comm_monoid_add}"
 begin
 
 lemma sums_Suc:
   assumes "(\<lambda>n. f (Suc n)) sums l"
   shows "f sums (l + f 0)"
 proof  -
   have "(\<lambda>n. (\<Sum>i<n. f (Suc i)) + f 0) \<longlonglongrightarrow> l + f 0"
     using assms by (auto intro!: tendsto_add simp: sums_def)
   moreover have "(\<Sum>i<n. f (Suc i)) + f 0 = (\<Sum>i<Suc n. f i)" for n
     unfolding lessThan_Suc_eq_insert_0
     by (simp add: ac_simps sum.atLeast1_atMost_eq image_Suc_lessThan)
   ultimately show ?thesis
     by (auto simp: sums_def simp del: sum.lessThan_Suc intro: filterlim_sequentially_Suc[THEN iffD1])
 qed
 
 lemma sums_add: "f sums a \<Longrightarrow> g sums b \<Longrightarrow> (\<lambda>n. f n + g n) sums (a + b)"
   unfolding sums_def by (simp add: sum.distrib tendsto_add)
 
 lemma summable_add: "summable f \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. f n + g n)"
   unfolding summable_def by (auto intro: sums_add)
 
 lemma suminf_add: "summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f + suminf g = (\<Sum>n. f n + g n)"
   by (intro sums_unique sums_add summable_sums)
 
 end
 
 context
   fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::{t2_space,topological_comm_monoid_add}"
     and I :: "'i set"
 begin
 
 lemma sums_sum: "(\<And>i. i \<in> I \<Longrightarrow> (f i) sums (x i)) \<Longrightarrow> (\<lambda>n. \<Sum>i\<in>I. f i n) sums (\<Sum>i\<in>I. x i)"
   by (induct I rule: infinite_finite_induct) (auto intro!: sums_add)
 
 lemma suminf_sum: "(\<And>i. i \<in> I \<Longrightarrow> summable (f i)) \<Longrightarrow> (\<Sum>n. \<Sum>i\<in>I. f i n) = (\<Sum>i\<in>I. \<Sum>n. f i n)"
   using sums_unique[OF sums_sum, OF summable_sums] by simp
 
 lemma summable_sum: "(\<And>i. i \<in> I \<Longrightarrow> summable (f i)) \<Longrightarrow> summable (\<lambda>n. \<Sum>i\<in>I. f i n)"
   using sums_summable[OF sums_sum[OF summable_sums]] .
 
 end
 
 lemma sums_If_finite_set':
   fixes f g :: "nat \<Rightarrow> 'a::{t2_space,topological_ab_group_add}"
   assumes "g sums S" and "finite A" and "S' = S + (\<Sum>n\<in>A. f n - g n)"
   shows   "(\<lambda>n. if n \<in> A then f n else g n) sums S'"
 proof -
   have "(\<lambda>n. g n + (if n \<in> A then f n - g n else 0)) sums (S + (\<Sum>n\<in>A. f n - g n))"
     by (intro sums_add assms sums_If_finite_set)
   also have "(\<lambda>n. g n + (if n \<in> A then f n - g n else 0)) = (\<lambda>n. if n \<in> A then f n else g n)"
     by (simp add: fun_eq_iff)
   finally show ?thesis using assms by simp
 qed
 
 subsection \<open>Infinite summability on real normed vector spaces\<close>
 
 context
   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
 begin
 
 lemma sums_Suc_iff: "(\<lambda>n. f (Suc n)) sums s \<longleftrightarrow> f sums (s + f 0)"
 proof -
   have "f sums (s + f 0) \<longleftrightarrow> (\<lambda>i. \<Sum>j<Suc i. f j) \<longlonglongrightarrow> s + f 0"
     by (subst filterlim_sequentially_Suc) (simp add: sums_def)
   also have "\<dots> \<longleftrightarrow> (\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) \<longlonglongrightarrow> s + f 0"
     by (simp add: ac_simps lessThan_Suc_eq_insert_0 image_Suc_lessThan sum.atLeast1_atMost_eq)
   also have "\<dots> \<longleftrightarrow> (\<lambda>n. f (Suc n)) sums s"
   proof
     assume "(\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) \<longlonglongrightarrow> s + f 0"
     with tendsto_add[OF this tendsto_const, of "- f 0"] show "(\<lambda>i. f (Suc i)) sums s"
       by (simp add: sums_def)
   qed (auto intro: tendsto_add simp: sums_def)
   finally show ?thesis ..
 qed
 
 lemma summable_Suc_iff: "summable (\<lambda>n. f (Suc n)) = summable f"
 proof
   assume "summable f"
   then have "f sums suminf f"
     by (rule summable_sums)
   then have "(\<lambda>n. f (Suc n)) sums (suminf f - f 0)"
     by (simp add: sums_Suc_iff)
   then show "summable (\<lambda>n. f (Suc n))"
     unfolding summable_def by blast
 qed (auto simp: sums_Suc_iff summable_def)
 
 lemma sums_Suc_imp: "f 0 = 0 \<Longrightarrow> (\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambda>n. f n) sums s"
   using sums_Suc_iff by simp
 
 end
 
 context (* Separate contexts are necessary to allow general use of the results above, here. *)
   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
 begin
 
 lemma sums_diff: "f sums a \<Longrightarrow> g sums b \<Longrightarrow> (\<lambda>n. f n - g n) sums (a - b)"
   unfolding sums_def by (simp add: sum_subtractf tendsto_diff)
 
 lemma summable_diff: "summable f \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. f n - g n)"
   unfolding summable_def by (auto intro: sums_diff)
 
 lemma suminf_diff: "summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f - suminf g = (\<Sum>n. f n - g n)"
   by (intro sums_unique sums_diff summable_sums)
 
 lemma sums_minus: "f sums a \<Longrightarrow> (\<lambda>n. - f n) sums (- a)"
   unfolding sums_def by (simp add: sum_negf tendsto_minus)
 
 lemma summable_minus: "summable f \<Longrightarrow> summable (\<lambda>n. - f n)"
   unfolding summable_def by (auto intro: sums_minus)
 
 lemma suminf_minus: "summable f \<Longrightarrow> (\<Sum>n. - f n) = - (\<Sum>n. f n)"
   by (intro sums_unique [symmetric] sums_minus summable_sums)
 
 lemma sums_iff_shift: "(\<lambda>i. f (i + n)) sums s \<longleftrightarrow> f sums (s + (\<Sum>i<n. f i))"
 proof (induct n arbitrary: s)
   case 0
   then show ?case by simp
 next
   case (Suc n)
   then have "(\<lambda>i. f (Suc i + n)) sums s \<longleftrightarrow> (\<lambda>i. f (i + n)) sums (s + f n)"
     by (subst sums_Suc_iff) simp
   with Suc show ?case
     by (simp add: ac_simps)
 qed
 
 corollary sums_iff_shift': "(\<lambda>i. f (i + n)) sums (s - (\<Sum>i<n. f i)) \<longleftrightarrow> f sums s"
   by (simp add: sums_iff_shift)
 
 lemma sums_zero_iff_shift:
   assumes "\<And>i. i < n \<Longrightarrow> f i = 0"
   shows "(\<lambda>i. f (i+n)) sums s \<longleftrightarrow> (\<lambda>i. f i) sums s"
   by (simp add: assms sums_iff_shift)
 
 lemma summable_iff_shift [simp]: "summable (\<lambda>n. f (n + k)) \<longleftrightarrow> summable f"
   by (metis diff_add_cancel summable_def sums_iff_shift [abs_def])
 
 lemma sums_split_initial_segment: "f sums s \<Longrightarrow> (\<lambda>i. f (i + n)) sums (s - (\<Sum>i<n. f i))"
   by (simp add: sums_iff_shift)
 
 lemma summable_ignore_initial_segment: "summable f \<Longrightarrow> summable (\<lambda>n. f(n + k))"
   by (simp add: summable_iff_shift)
 
 lemma suminf_minus_initial_segment: "summable f \<Longrightarrow> (\<Sum>n. f (n + k)) = (\<Sum>n. f n) - (\<Sum>i<k. f i)"
   by (rule sums_unique[symmetric]) (auto simp: sums_iff_shift)
 
 lemma suminf_split_initial_segment: "summable f \<Longrightarrow> suminf f = (\<Sum>n. f(n + k)) + (\<Sum>i<k. f i)"
   by (auto simp add: suminf_minus_initial_segment)
 
 lemma suminf_split_head: "summable f \<Longrightarrow> (\<Sum>n. f (Suc n)) = suminf f - f 0"
   using suminf_split_initial_segment[of 1] by simp
 
 lemma suminf_exist_split:
   fixes r :: real
   assumes "0 < r" and "summable f"
   shows "\<exists>N. \<forall>n\<ge>N. norm (\<Sum>i. f (i + n)) < r"
 proof -
   from LIMSEQ_D[OF summable_LIMSEQ[OF \<open>summable f\<close>] \<open>0 < r\<close>]
   obtain N :: nat where "\<forall> n \<ge> N. norm (sum f {..<n} - suminf f) < r"
     by auto
   then show ?thesis
     by (auto simp: norm_minus_commute suminf_minus_initial_segment[OF \<open>summable f\<close>])
 qed
 
 lemma summable_LIMSEQ_zero: 
   assumes "summable f" shows "f \<longlonglongrightarrow> 0"
 proof -
   have "Cauchy (\<lambda>n. sum f {..<n})"
     using LIMSEQ_imp_Cauchy assms summable_LIMSEQ by blast
   then show ?thesis
     unfolding  Cauchy_iff LIMSEQ_iff
     by (metis add.commute add_diff_cancel_right' diff_zero le_SucI sum.lessThan_Suc)
 qed
 
 lemma summable_imp_convergent: "summable f \<Longrightarrow> convergent f"
   by (force dest!: summable_LIMSEQ_zero simp: convergent_def)
 
 lemma summable_imp_Bseq: "summable f \<Longrightarrow> Bseq f"
   by (simp add: convergent_imp_Bseq summable_imp_convergent)
 
 end
 
 lemma summable_minus_iff: "summable (\<lambda>n. - f n) \<longleftrightarrow> summable f"
   for f :: "nat \<Rightarrow> 'a::real_normed_vector"
   by (auto dest: summable_minus)  (* used two ways, hence must be outside the context above *)
 
 lemma (in bounded_linear) sums: "(\<lambda>n. X n) sums a \<Longrightarrow> (\<lambda>n. f (X n)) sums (f a)"
   unfolding sums_def by (drule tendsto) (simp only: sum)
 
 lemma (in bounded_linear) summable: "summable (\<lambda>n. X n) \<Longrightarrow> summable (\<lambda>n. f (X n))"
   unfolding summable_def by (auto intro: sums)
 
 lemma (in bounded_linear) suminf: "summable (\<lambda>n. X n) \<Longrightarrow> f (\<Sum>n. X n) = (\<Sum>n. f (X n))"
   by (intro sums_unique sums summable_sums)
 
 lemmas sums_of_real = bounded_linear.sums [OF bounded_linear_of_real]
 lemmas summable_of_real = bounded_linear.summable [OF bounded_linear_of_real]
 lemmas suminf_of_real = bounded_linear.suminf [OF bounded_linear_of_real]
 
 lemmas sums_scaleR_left = bounded_linear.sums[OF bounded_linear_scaleR_left]
 lemmas summable_scaleR_left = bounded_linear.summable[OF bounded_linear_scaleR_left]
 lemmas suminf_scaleR_left = bounded_linear.suminf[OF bounded_linear_scaleR_left]
 
 lemmas sums_scaleR_right = bounded_linear.sums[OF bounded_linear_scaleR_right]
 lemmas summable_scaleR_right = bounded_linear.summable[OF bounded_linear_scaleR_right]
 lemmas suminf_scaleR_right = bounded_linear.suminf[OF bounded_linear_scaleR_right]
 
 lemma summable_const_iff: "summable (\<lambda>_. c) \<longleftrightarrow> c = 0"
   for c :: "'a::real_normed_vector"
 proof -
   have "\<not> summable (\<lambda>_. c)" if "c \<noteq> 0"
   proof -
     from that have "filterlim (\<lambda>n. of_nat n * norm c) at_top sequentially"
       by (subst mult.commute)
         (auto intro!: filterlim_tendsto_pos_mult_at_top filterlim_real_sequentially)
     then have "\<not> convergent (\<lambda>n. norm (\<Sum>k<n. c))"
       by (intro filterlim_at_infinity_imp_not_convergent filterlim_at_top_imp_at_infinity)
         (simp_all add: sum_constant_scaleR)
     then show ?thesis
       unfolding summable_iff_convergent using convergent_norm by blast
   qed
   then show ?thesis by auto
 qed
 
 
 subsection \<open>Infinite summability on real normed algebras\<close>
 
 context
   fixes f :: "nat \<Rightarrow> 'a::real_normed_algebra"
 begin
 
 lemma sums_mult: "f sums a \<Longrightarrow> (\<lambda>n. c * f n) sums (c * a)"
   by (rule bounded_linear.sums [OF bounded_linear_mult_right])
 
 lemma summable_mult: "summable f \<Longrightarrow> summable (\<lambda>n. c * f n)"
   by (rule bounded_linear.summable [OF bounded_linear_mult_right])
 
 lemma suminf_mult: "summable f \<Longrightarrow> suminf (\<lambda>n. c * f n) = c * suminf f"
   by (rule bounded_linear.suminf [OF bounded_linear_mult_right, symmetric])
 
 lemma sums_mult2: "f sums a \<Longrightarrow> (\<lambda>n. f n * c) sums (a * c)"
   by (rule bounded_linear.sums [OF bounded_linear_mult_left])
 
 lemma summable_mult2: "summable f \<Longrightarrow> summable (\<lambda>n. f n * c)"
   by (rule bounded_linear.summable [OF bounded_linear_mult_left])
 
 lemma suminf_mult2: "summable f \<Longrightarrow> suminf f * c = (\<Sum>n. f n * c)"
   by (rule bounded_linear.suminf [OF bounded_linear_mult_left])
 
 end
 
 lemma sums_mult_iff:
   fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra,field}"
   assumes "c \<noteq> 0"
   shows "(\<lambda>n. c * f n) sums (c * d) \<longleftrightarrow> f sums d"
   using sums_mult[of f d c] sums_mult[of "\<lambda>n. c * f n" "c * d" "inverse c"]
   by (force simp: field_simps assms)
 
 lemma sums_mult2_iff:
   fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra,field}"
   assumes "c \<noteq> 0"
   shows   "(\<lambda>n. f n * c) sums (d * c) \<longleftrightarrow> f sums d"
   using sums_mult_iff[OF assms, of f d] by (simp add: mult.commute)
 
 lemma sums_of_real_iff:
   "(\<lambda>n. of_real (f n) :: 'a::real_normed_div_algebra) sums of_real c \<longleftrightarrow> f sums c"
   by (simp add: sums_def of_real_sum[symmetric] tendsto_of_real_iff del: of_real_sum)
 
 
 subsection \<open>Infinite summability on real normed fields\<close>
 
 context
   fixes c :: "'a::real_normed_field"
 begin
 
 lemma sums_divide: "f sums a \<Longrightarrow> (\<lambda>n. f n / c) sums (a / c)"
   by (rule bounded_linear.sums [OF bounded_linear_divide])
 
 lemma summable_divide: "summable f \<Longrightarrow> summable (\<lambda>n. f n / c)"
   by (rule bounded_linear.summable [OF bounded_linear_divide])
 
 lemma suminf_divide: "summable f \<Longrightarrow> suminf (\<lambda>n. f n / c) = suminf f / c"
   by (rule bounded_linear.suminf [OF bounded_linear_divide, symmetric])
 
 lemma summable_inverse_divide: "summable (inverse \<circ> f) \<Longrightarrow> summable (\<lambda>n. c / f n)"
   by (auto dest: summable_mult [of _ c] simp: field_simps)
 
 lemma sums_mult_D: "(\<lambda>n. c * f n) sums a \<Longrightarrow> c \<noteq> 0 \<Longrightarrow> f sums (a/c)"
   using sums_mult_iff by fastforce
 
 lemma summable_mult_D: "summable (\<lambda>n. c * f n) \<Longrightarrow> c \<noteq> 0 \<Longrightarrow> summable f"
   by (auto dest: summable_divide)
 
 
 text \<open>Sum of a geometric progression.\<close>
 
 lemma geometric_sums:
   assumes "norm c < 1"
   shows "(\<lambda>n. c^n) sums (1 / (1 - c))"
 proof -
   have neq_0: "c - 1 \<noteq> 0"
     using assms by auto
   then have "(\<lambda>n. c ^ n / (c - 1) - 1 / (c - 1)) \<longlonglongrightarrow> 0 / (c - 1) - 1 / (c - 1)"
     by (intro tendsto_intros assms)
   then have "(\<lambda>n. (c ^ n - 1) / (c - 1)) \<longlonglongrightarrow> 1 / (1 - c)"
     by (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib)
   with neq_0 show "(\<lambda>n. c ^ n) sums (1 / (1 - c))"
     by (simp add: sums_def geometric_sum)
 qed
 
 lemma summable_geometric: "norm c < 1 \<Longrightarrow> summable (\<lambda>n. c^n)"
   by (rule geometric_sums [THEN sums_summable])
 
 lemma suminf_geometric: "norm c < 1 \<Longrightarrow> suminf (\<lambda>n. c^n) = 1 / (1 - c)"
   by (rule sums_unique[symmetric]) (rule geometric_sums)
 
 lemma summable_geometric_iff [simp]: "summable (\<lambda>n. c ^ n) \<longleftrightarrow> norm c < 1"
 proof
   assume "summable (\<lambda>n. c ^ n :: 'a :: real_normed_field)"
   then have "(\<lambda>n. norm c ^ n) \<longlonglongrightarrow> 0"
     by (simp add: norm_power [symmetric] tendsto_norm_zero_iff summable_LIMSEQ_zero)
   from order_tendstoD(2)[OF this zero_less_one] obtain n where "norm c ^ n < 1"
     by (auto simp: eventually_at_top_linorder)
   then show "norm c < 1" using one_le_power[of "norm c" n]
     by (cases "norm c \<ge> 1") (linarith, simp)
 qed (rule summable_geometric)
 
 end
 
 text \<open>Biconditional versions for constant factors\<close>
 context
   fixes c :: "'a::real_normed_field"
 begin
 
 lemma summable_cmult_iff [simp]: "summable (\<lambda>n. c * f n) \<longleftrightarrow> c=0 \<or> summable f"
 proof -
   have "\<lbrakk>summable (\<lambda>n. c * f n); c \<noteq> 0\<rbrakk> \<Longrightarrow> summable f"
     using summable_mult_D by blast
   then show ?thesis
     by (auto simp: summable_mult)
 qed
 
 lemma summable_divide_iff [simp]: "summable (\<lambda>n. f n / c) \<longleftrightarrow> c=0 \<or> summable f"
 proof -
   have "\<lbrakk>summable (\<lambda>n. f n / c); c \<noteq> 0\<rbrakk> \<Longrightarrow> summable f"
     by (auto dest: summable_divide [where c = "1/c"])
   then show ?thesis
     by (auto simp: summable_divide)
 qed
 
 end
 
 lemma power_half_series: "(\<lambda>n. (1/2::real)^Suc n) sums 1"
 proof -
   have 2: "(\<lambda>n. (1/2::real)^n) sums 2"
     using geometric_sums [of "1/2::real"] by auto
   have "(\<lambda>n. (1/2::real)^Suc n) = (\<lambda>n. (1 / 2) ^ n / 2)"
     by (simp add: mult.commute)
   then show ?thesis
     using sums_divide [OF 2, of 2] by simp
 qed
 
 
 subsection \<open>Telescoping\<close>
 
 lemma telescope_sums:
   fixes c :: "'a::real_normed_vector"
   assumes "f \<longlonglongrightarrow> c"
   shows "(\<lambda>n. f (Suc n) - f n) sums (c - f 0)"
   unfolding sums_def
 proof (subst filterlim_sequentially_Suc [symmetric])
   have "(\<lambda>n. \<Sum>k<Suc n. f (Suc k) - f k) = (\<lambda>n. f (Suc n) - f 0)"
     by (simp add: lessThan_Suc_atMost atLeast0AtMost [symmetric] sum_Suc_diff)
   also have "\<dots> \<longlonglongrightarrow> c - f 0"
     by (intro tendsto_diff LIMSEQ_Suc[OF assms] tendsto_const)
   finally show "(\<lambda>n. \<Sum>n<Suc n. f (Suc n) - f n) \<longlonglongrightarrow> c - f 0" .
 qed
 
 lemma telescope_sums':
   fixes c :: "'a::real_normed_vector"
   assumes "f \<longlonglongrightarrow> c"
   shows "(\<lambda>n. f n - f (Suc n)) sums (f 0 - c)"
   using sums_minus[OF telescope_sums[OF assms]] by (simp add: algebra_simps)
 
 lemma telescope_summable:
   fixes c :: "'a::real_normed_vector"
   assumes "f \<longlonglongrightarrow> c"
   shows "summable (\<lambda>n. f (Suc n) - f n)"
   using telescope_sums[OF assms] by (simp add: sums_iff)
 
 lemma telescope_summable':
   fixes c :: "'a::real_normed_vector"
   assumes "f \<longlonglongrightarrow> c"
   shows "summable (\<lambda>n. f n - f (Suc n))"
   using summable_minus[OF telescope_summable[OF assms]] by (simp add: algebra_simps)
 
 
 subsection \<open>Infinite summability on Banach spaces\<close>
 
 text \<open>Cauchy-type criterion for convergence of series (c.f. Harrison).\<close>
 
 lemma summable_Cauchy: "summable f \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n. norm (sum f {m..<n}) < e)" (is "_ = ?rhs")
   for f :: "nat \<Rightarrow> 'a::banach"
 proof
   assume f: "summable f"
   show ?rhs
   proof clarify
     fix e :: real
     assume "0 < e"
     then obtain M where M: "\<And>m n. \<lbrakk>m\<ge>M; n\<ge>M\<rbrakk> \<Longrightarrow> norm (sum f {..<m} - sum f {..<n}) < e"
       using f by (force simp add: summable_iff_convergent Cauchy_convergent_iff [symmetric] Cauchy_iff)
     have "norm (sum f {m..<n}) < e" if "m \<ge> M" for m n
     proof (cases m n rule: linorder_class.le_cases)
       assume "m \<le> n"
       then show ?thesis
         by (metis (mono_tags, hide_lams) M atLeast0LessThan order_trans sum_diff_nat_ivl that zero_le)
     next
       assume "n \<le> m"
       then show ?thesis
         by (simp add: \<open>0 < e\<close>)
     qed
     then show "\<exists>N. \<forall>m\<ge>N. \<forall>n. norm (sum f {m..<n}) < e"
       by blast
   qed
 next
   assume r: ?rhs
   then show "summable f"
     unfolding summable_iff_convergent Cauchy_convergent_iff [symmetric] Cauchy_iff
   proof clarify
     fix e :: real
     assume "0 < e"
     with r obtain N where N: "\<And>m n. m \<ge> N \<Longrightarrow> norm (sum f {m..<n}) < e"
       by blast
     have "norm (sum f {..<m} - sum f {..<n}) < e" if "m\<ge>N" "n\<ge>N" for m n
     proof (cases m n rule: linorder_class.le_cases)
       assume "m \<le> n"
       then show ?thesis
-        by (metis Groups_Big.sum_diff N finite_lessThan lessThan_minus_lessThan lessThan_subset_iff norm_minus_commute \<open>m\<ge>N\<close>)
+        by (metis N finite_lessThan lessThan_minus_lessThan lessThan_subset_iff norm_minus_commute sum_diff \<open>m\<ge>N\<close>)
     next
       assume "n \<le> m"
       then show ?thesis
-        by (metis Groups_Big.sum_diff N finite_lessThan lessThan_minus_lessThan lessThan_subset_iff \<open>n\<ge>N\<close>)
+        by (metis N finite_lessThan lessThan_minus_lessThan lessThan_subset_iff sum_diff \<open>n\<ge>N\<close>)
     qed
     then show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (sum f {..<m} - sum f {..<n}) < e"
       by blast
   qed
 qed
 
 lemma summable_Cauchy':
   fixes f :: "nat \<Rightarrow> 'a :: banach"
-  assumes "eventually (\<lambda>m. \<forall>n\<ge>m. norm (sum f {m..<n}) \<le> g m) sequentially"
-  assumes "filterlim g (nhds 0) sequentially"
+  assumes ev: "eventually (\<lambda>m. \<forall>n\<ge>m. norm (sum f {m..<n}) \<le> g m) sequentially"
+  assumes g0: "g \<longlonglongrightarrow> 0"
   shows "summable f"
 proof (subst summable_Cauchy, intro allI impI, goal_cases)
   case (1 e)
-  from order_tendstoD(2)[OF assms(2) this] and assms(1)
-  have "eventually (\<lambda>m. \<forall>n. norm (sum f {m..<n}) < e) at_top"
+  then have "\<forall>\<^sub>F x in sequentially. g x < e"
+    using g0 order_tendstoD(2) by blast
+  with ev have "eventually (\<lambda>m. \<forall>n. norm (sum f {m..<n}) < e) at_top"
   proof eventually_elim
     case (elim m)
     show ?case
     proof
       fix n
       from elim show "norm (sum f {m..<n}) < e"
         by (cases "n \<ge> m") auto
     qed
   qed
   thus ?case by (auto simp: eventually_at_top_linorder)
 qed
 
 context
   fixes f :: "nat \<Rightarrow> 'a::banach"
 begin
 
 text \<open>Absolute convergence imples normal convergence.\<close>
 
 lemma summable_norm_cancel: "summable (\<lambda>n. norm (f n)) \<Longrightarrow> summable f"
   unfolding summable_Cauchy
   apply (erule all_forward imp_forward ex_forward | assumption)+
   apply (fastforce simp add: order_le_less_trans [OF norm_sum] order_le_less_trans [OF abs_ge_self])
   done
 
 lemma summable_norm: "summable (\<lambda>n. norm (f n)) \<Longrightarrow> norm (suminf f) \<le> (\<Sum>n. norm (f n))"
   by (auto intro: LIMSEQ_le tendsto_norm summable_norm_cancel summable_LIMSEQ norm_sum)
 
 text \<open>Comparison tests.\<close>
 
 lemma summable_comparison_test: 
   assumes fg: "\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n" and g: "summable g"
   shows "summable f"
 proof -
   obtain N where N: "\<And>n. n\<ge>N \<Longrightarrow> norm (f n) \<le> g n" 
     using assms by blast
   show ?thesis
   proof (clarsimp simp add: summable_Cauchy)
     fix e :: real
     assume "0 < e"
     then obtain Ng where Ng: "\<And>m n. m \<ge> Ng \<Longrightarrow> norm (sum g {m..<n}) < e" 
       using g by (fastforce simp: summable_Cauchy)
     with N have "norm (sum f {m..<n}) < e" if "m\<ge>max N Ng" for m n
     proof -
       have "norm (sum f {m..<n}) \<le> sum g {m..<n}"
         using N that by (force intro: sum_norm_le)
       also have "... \<le> norm (sum g {m..<n})"
         by simp
       also have "... < e"
         using Ng that by auto
       finally show ?thesis .
     qed
     then show "\<exists>N. \<forall>m\<ge>N. \<forall>n. norm (sum f {m..<n}) < e" 
       by blast
   qed
 qed
 
 lemma summable_comparison_test_ev:
   "eventually (\<lambda>n. norm (f n) \<le> g n) sequentially \<Longrightarrow> summable g \<Longrightarrow> summable f"
   by (rule summable_comparison_test) (auto simp: eventually_at_top_linorder)
 
 text \<open>A better argument order.\<close>
 lemma summable_comparison_test': "summable g \<Longrightarrow> (\<And>n. n \<ge> N \<Longrightarrow> norm (f n) \<le> g n) \<Longrightarrow> summable f"
   by (rule summable_comparison_test) auto
 
 
 subsection \<open>The Ratio Test\<close>
 
 lemma summable_ratio_test:
   assumes "c < 1" "\<And>n. n \<ge> N \<Longrightarrow> norm (f (Suc n)) \<le> c * norm (f n)"
   shows "summable f"
 proof (cases "0 < c")
   case True
   show "summable f"
   proof (rule summable_comparison_test)
     show "\<exists>N'. \<forall>n\<ge>N'. norm (f n) \<le> (norm (f N) / (c ^ N)) * c ^ n"
     proof (intro exI allI impI)
       fix n
       assume "N \<le> n"
       then show "norm (f n) \<le> (norm (f N) / (c ^ N)) * c ^ n"
       proof (induct rule: inc_induct)
         case base
         with True show ?case by simp
       next
         case (step m)
         have "norm (f (Suc m)) / c ^ Suc m * c ^ n \<le> norm (f m) / c ^ m * c ^ n"
           using \<open>0 < c\<close> \<open>c < 1\<close> assms(2)[OF \<open>N \<le> m\<close>] by (simp add: field_simps)
         with step show ?case by simp
       qed
     qed
     show "summable (\<lambda>n. norm (f N) / c ^ N * c ^ n)"
       using \<open>0 < c\<close> \<open>c < 1\<close> by (intro summable_mult summable_geometric) simp
   qed
 next
   case False
   have "f (Suc n) = 0" if "n \<ge> N" for n
   proof -
     from that have "norm (f (Suc n)) \<le> c * norm (f n)"
       by (rule assms(2))
     also have "\<dots> \<le> 0"
       using False by (simp add: not_less mult_nonpos_nonneg)
     finally show ?thesis
       by auto
   qed
   then show "summable f"
     by (intro sums_summable[OF sums_finite, of "{.. Suc N}"]) (auto simp: not_le Suc_less_eq2)
 qed
 
 end
 
 
 text \<open>Relations among convergence and absolute convergence for power series.\<close>
 
 lemma Abel_lemma:
   fixes a :: "nat \<Rightarrow> 'a::real_normed_vector"
   assumes r: "0 \<le> r"
     and r0: "r < r0"
     and M: "\<And>n. norm (a n) * r0^n \<le> M"
   shows "summable (\<lambda>n. norm (a n) * r^n)"
 proof (rule summable_comparison_test')
   show "summable (\<lambda>n. M * (r / r0) ^ n)"
     using assms by (auto simp add: summable_mult summable_geometric)
   show "norm (norm (a n) * r ^ n) \<le> M * (r / r0) ^ n" for n
     using r r0 M [of n] dual_order.order_iff_strict
     by (fastforce simp add: abs_mult field_simps)
 qed
 
 
 text \<open>Summability of geometric series for real algebras.\<close>
 
 lemma complete_algebra_summable_geometric:
   fixes x :: "'a::{real_normed_algebra_1,banach}"
   assumes "norm x < 1"
   shows "summable (\<lambda>n. x ^ n)"
 proof (rule summable_comparison_test)
   show "\<exists>N. \<forall>n\<ge>N. norm (x ^ n) \<le> norm x ^ n"
     by (simp add: norm_power_ineq)
   from assms show "summable (\<lambda>n. norm x ^ n)"
     by (simp add: summable_geometric)
 qed
 
 
 subsection \<open>Cauchy Product Formula\<close>
 
 text \<open>
   Proof based on Analysis WebNotes: Chapter 07, Class 41
   \<^url>\<open>http://www.math.unl.edu/~webnotes/classes/class41/prp77.htm\<close>
 \<close>
 
 lemma Cauchy_product_sums:
   fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}"
   assumes a: "summable (\<lambda>k. norm (a k))"
     and b: "summable (\<lambda>k. norm (b k))"
   shows "(\<lambda>k. \<Sum>i\<le>k. a i * b (k - i)) sums ((\<Sum>k. a k) * (\<Sum>k. b k))"
 proof -
   let ?S1 = "\<lambda>n::nat. {..<n} \<times> {..<n}"
   let ?S2 = "\<lambda>n::nat. {(i,j). i + j < n}"
   have S1_mono: "\<And>m n. m \<le> n \<Longrightarrow> ?S1 m \<subseteq> ?S1 n" by auto
   have S2_le_S1: "\<And>n. ?S2 n \<subseteq> ?S1 n" by auto
   have S1_le_S2: "\<And>n. ?S1 (n div 2) \<subseteq> ?S2 n" by auto
   have finite_S1: "\<And>n. finite (?S1 n)" by simp
   with S2_le_S1 have finite_S2: "\<And>n. finite (?S2 n)" by (rule finite_subset)
 
   let ?g = "\<lambda>(i,j). a i * b j"
   let ?f = "\<lambda>(i,j). norm (a i) * norm (b j)"
   have f_nonneg: "\<And>x. 0 \<le> ?f x" by auto
   then have norm_sum_f: "\<And>A. norm (sum ?f A) = sum ?f A"
     unfolding real_norm_def
     by (simp only: abs_of_nonneg sum_nonneg [rule_format])
 
   have "(\<lambda>n. (\<Sum>k<n. a k) * (\<Sum>k<n. b k)) \<longlonglongrightarrow> (\<Sum>k. a k) * (\<Sum>k. b k)"
     by (intro tendsto_mult summable_LIMSEQ summable_norm_cancel [OF a] summable_norm_cancel [OF b])
   then have 1: "(\<lambda>n. sum ?g (?S1 n)) \<longlonglongrightarrow> (\<Sum>k. a k) * (\<Sum>k. b k)"
     by (simp only: sum_product sum.Sigma [rule_format] finite_lessThan)
 
   have "(\<lambda>n. (\<Sum>k<n. norm (a k)) * (\<Sum>k<n. norm (b k))) \<longlonglongrightarrow> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
     using a b by (intro tendsto_mult summable_LIMSEQ)
   then have "(\<lambda>n. sum ?f (?S1 n)) \<longlonglongrightarrow> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
     by (simp only: sum_product sum.Sigma [rule_format] finite_lessThan)
   then have "convergent (\<lambda>n. sum ?f (?S1 n))"
     by (rule convergentI)
   then have Cauchy: "Cauchy (\<lambda>n. sum ?f (?S1 n))"
     by (rule convergent_Cauchy)
   have "Zfun (\<lambda>n. sum ?f (?S1 n - ?S2 n)) sequentially"
   proof (rule ZfunI, simp only: eventually_sequentially norm_sum_f)
     fix r :: real
     assume r: "0 < r"
     from CauchyD [OF Cauchy r] obtain N
       where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (sum ?f (?S1 m) - sum ?f (?S1 n)) < r" ..
     then have "\<And>m n. N \<le> n \<Longrightarrow> n \<le> m \<Longrightarrow> norm (sum ?f (?S1 m - ?S1 n)) < r"
       by (simp only: sum_diff finite_S1 S1_mono)
     then have N: "\<And>m n. N \<le> n \<Longrightarrow> n \<le> m \<Longrightarrow> sum ?f (?S1 m - ?S1 n) < r"
       by (simp only: norm_sum_f)
     show "\<exists>N. \<forall>n\<ge>N. sum ?f (?S1 n - ?S2 n) < r"
     proof (intro exI allI impI)
       fix n
       assume "2 * N \<le> n"
       then have n: "N \<le> n div 2" by simp
       have "sum ?f (?S1 n - ?S2 n) \<le> sum ?f (?S1 n - ?S1 (n div 2))"
         by (intro sum_mono2 finite_Diff finite_S1 f_nonneg Diff_mono subset_refl S1_le_S2)
       also have "\<dots> < r"
         using n div_le_dividend by (rule N)
       finally show "sum ?f (?S1 n - ?S2 n) < r" .
     qed
   qed
   then have "Zfun (\<lambda>n. sum ?g (?S1 n - ?S2 n)) sequentially"
     apply (rule Zfun_le [rule_format])
     apply (simp only: norm_sum_f)
     apply (rule order_trans [OF norm_sum sum_mono])
     apply (auto simp add: norm_mult_ineq)
     done
   then have 2: "(\<lambda>n. sum ?g (?S1 n) - sum ?g (?S2 n)) \<longlonglongrightarrow> 0"
     unfolding tendsto_Zfun_iff diff_0_right
     by (simp only: sum_diff finite_S1 S2_le_S1)
   with 1 have "(\<lambda>n. sum ?g (?S2 n)) \<longlonglongrightarrow> (\<Sum>k. a k) * (\<Sum>k. b k)"
     by (rule Lim_transform2)
   then show ?thesis
     by (simp only: sums_def sum.triangle_reindex)
 qed
 
 lemma Cauchy_product:
   fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}"
   assumes "summable (\<lambda>k. norm (a k))"
     and "summable (\<lambda>k. norm (b k))"
   shows "(\<Sum>k. a k) * (\<Sum>k. b k) = (\<Sum>k. \<Sum>i\<le>k. a i * b (k - i))"
   using assms by (rule Cauchy_product_sums [THEN sums_unique])
 
 lemma summable_Cauchy_product:
   fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}"
   assumes "summable (\<lambda>k. norm (a k))"
     and "summable (\<lambda>k. norm (b k))"
   shows "summable (\<lambda>k. \<Sum>i\<le>k. a i * b (k - i))"
   using Cauchy_product_sums[OF assms] by (simp add: sums_iff)
 
 
 subsection \<open>Series on \<^typ>\<open>real\<close>s\<close>
 
 lemma summable_norm_comparison_test:
   "\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. norm (f n))"
   by (rule summable_comparison_test) auto
 
 lemma summable_rabs_comparison_test: "\<exists>N. \<forall>n\<ge>N. \<bar>f n\<bar> \<le> g n \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. \<bar>f n\<bar>)"
   for f :: "nat \<Rightarrow> real"
   by (rule summable_comparison_test) auto
 
 lemma summable_rabs_cancel: "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> summable f"
   for f :: "nat \<Rightarrow> real"
   by (rule summable_norm_cancel) simp
 
 lemma summable_rabs: "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> \<bar>suminf f\<bar> \<le> (\<Sum>n. \<bar>f n\<bar>)"
   for f :: "nat \<Rightarrow> real"
   by (fold real_norm_def) (rule summable_norm)
 
 lemma summable_zero_power [simp]: "summable (\<lambda>n. 0 ^ n :: 'a::{comm_ring_1,topological_space})"
 proof -
   have "(\<lambda>n. 0 ^ n :: 'a) = (\<lambda>n. if n = 0 then 0^0 else 0)"
     by (intro ext) (simp add: zero_power)
   moreover have "summable \<dots>" by simp
   ultimately show ?thesis by simp
 qed
 
 lemma summable_zero_power' [simp]: "summable (\<lambda>n. f n * 0 ^ n :: 'a::{ring_1,topological_space})"
 proof -
   have "(\<lambda>n. f n * 0 ^ n :: 'a) = (\<lambda>n. if n = 0 then f 0 * 0^0 else 0)"
     by (intro ext) (simp add: zero_power)
   moreover have "summable \<dots>" by simp
   ultimately show ?thesis by simp
 qed
 
 lemma summable_power_series:
   fixes z :: real
   assumes le_1: "\<And>i. f i \<le> 1"
     and nonneg: "\<And>i. 0 \<le> f i"
     and z: "0 \<le> z" "z < 1"
   shows "summable (\<lambda>i. f i * z^i)"
 proof (rule summable_comparison_test[OF _ summable_geometric])
   show "norm z < 1"
     using z by (auto simp: less_imp_le)
   show "\<And>n. \<exists>N. \<forall>na\<ge>N. norm (f na * z ^ na) \<le> z ^ na"
     using z
     by (auto intro!: exI[of _ 0] mult_left_le_one_le simp: abs_mult nonneg power_abs less_imp_le le_1)
 qed
 
 lemma summable_0_powser: "summable (\<lambda>n. f n * 0 ^ n :: 'a::real_normed_div_algebra)"
 proof -
   have A: "(\<lambda>n. f n * 0 ^ n) = (\<lambda>n. if n = 0 then f n else 0)"
     by (intro ext) auto
   then show ?thesis
     by (subst A) simp_all
 qed
 
 lemma summable_powser_split_head:
   "summable (\<lambda>n. f (Suc n) * z ^ n :: 'a::real_normed_div_algebra) = summable (\<lambda>n. f n * z ^ n)"
 proof -
   have "summable (\<lambda>n. f (Suc n) * z ^ n) \<longleftrightarrow> summable (\<lambda>n. f (Suc n) * z ^ Suc n)"
     (is "?lhs \<longleftrightarrow> ?rhs")
   proof
     show ?rhs if ?lhs
       using summable_mult2[OF that, of z]
       by (simp add: power_commutes algebra_simps)
     show ?lhs if ?rhs
       using summable_mult2[OF that, of "inverse z"]
       by (cases "z \<noteq> 0", subst (asm) power_Suc2) (simp_all add: algebra_simps)
   qed
   also have "\<dots> \<longleftrightarrow> summable (\<lambda>n. f n * z ^ n)" by (rule summable_Suc_iff)
   finally show ?thesis .
 qed
 
 lemma summable_powser_ignore_initial_segment:
   fixes f :: "nat \<Rightarrow> 'a :: real_normed_div_algebra"
   shows "summable (\<lambda>n. f (n + m) * z ^ n) \<longleftrightarrow> summable (\<lambda>n. f n * z ^ n)"
 proof (induction m)
   case (Suc m)
   have "summable (\<lambda>n. f (n + Suc m) * z ^ n) = summable (\<lambda>n. f (Suc n + m) * z ^ n)"
     by simp
   also have "\<dots> = summable (\<lambda>n. f (n + m) * z ^ n)"
     by (rule summable_powser_split_head)
   also have "\<dots> = summable (\<lambda>n. f n * z ^ n)"
     by (rule Suc.IH)
   finally show ?case .
 qed simp_all
 
 lemma powser_split_head:
   fixes f :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}"
   assumes "summable (\<lambda>n. f n * z ^ n)"
   shows "suminf (\<lambda>n. f n * z ^ n) = f 0 + suminf (\<lambda>n. f (Suc n) * z ^ n) * z"
     and "suminf (\<lambda>n. f (Suc n) * z ^ n) * z = suminf (\<lambda>n. f n * z ^ n) - f 0"
     and "summable (\<lambda>n. f (Suc n) * z ^ n)"
 proof -
   from assms show "summable (\<lambda>n. f (Suc n) * z ^ n)"
     by (subst summable_powser_split_head)
   from suminf_mult2[OF this, of z]
     have "(\<Sum>n. f (Suc n) * z ^ n) * z = (\<Sum>n. f (Suc n) * z ^ Suc n)"
     by (simp add: power_commutes algebra_simps)
   also from assms have "\<dots> = suminf (\<lambda>n. f n * z ^ n) - f 0"
     by (subst suminf_split_head) simp_all
   finally show "suminf (\<lambda>n. f n * z ^ n) = f 0 + suminf (\<lambda>n. f (Suc n) * z ^ n) * z"
     by simp
   then show "suminf (\<lambda>n. f (Suc n) * z ^ n) * z = suminf (\<lambda>n. f n * z ^ n) - f 0"
     by simp
 qed
 
 lemma summable_partial_sum_bound:
   fixes f :: "nat \<Rightarrow> 'a :: banach"
     and e :: real
   assumes summable: "summable f"
     and e: "e > 0"
   obtains N where "\<And>m n. m \<ge> N \<Longrightarrow> norm (\<Sum>k=m..n. f k) < e"
 proof -
   from summable have "Cauchy (\<lambda>n. \<Sum>k<n. f k)"
     by (simp add: Cauchy_convergent_iff summable_iff_convergent)
   from CauchyD [OF this e] obtain N
     where N: "\<And>m n. m \<ge> N \<Longrightarrow> n \<ge> N \<Longrightarrow> norm ((\<Sum>k<m. f k) - (\<Sum>k<n. f k)) < e"
     by blast
   have "norm (\<Sum>k=m..n. f k) < e" if m: "m \<ge> N" for m n
   proof (cases "n \<ge> m")
     case True
     with m have "norm ((\<Sum>k<Suc n. f k) - (\<Sum>k<m. f k)) < e"
       by (intro N) simp_all
     also from True have "(\<Sum>k<Suc n. f k) - (\<Sum>k<m. f k) = (\<Sum>k=m..n. f k)"
       by (subst sum_diff [symmetric]) (simp_all add: sum.last_plus)
     finally show ?thesis .
   next
     case False
     with e show ?thesis by simp_all
   qed
   then show ?thesis by (rule that)
 qed
 
 lemma powser_sums_if:
   "(\<lambda>n. (if n = m then (1 :: 'a::{ring_1,topological_space}) else 0) * z^n) sums z^m"
 proof -
   have "(\<lambda>n. (if n = m then 1 else 0) * z^n) = (\<lambda>n. if n = m then z^n else 0)"
     by (intro ext) auto
   then show ?thesis
     by (simp add: sums_single)
 qed
 
 lemma
   fixes f :: "nat \<Rightarrow> real"
   assumes "summable f"
     and "inj g"
     and pos: "\<And>x. 0 \<le> f x"
   shows summable_reindex: "summable (f \<circ> g)"
     and suminf_reindex_mono: "suminf (f \<circ> g) \<le> suminf f"
     and suminf_reindex: "(\<And>x. x \<notin> range g \<Longrightarrow> f x = 0) \<Longrightarrow> suminf (f \<circ> g) = suminf f"
 proof -
   from \<open>inj g\<close> have [simp]: "\<And>A. inj_on g A"
     by (rule subset_inj_on) simp
 
   have smaller: "\<forall>n. (\<Sum>i<n. (f \<circ> g) i) \<le> suminf f"
   proof
     fix n
     have "\<forall> n' \<in> (g ` {..<n}). n' < Suc (Max (g ` {..<n}))"
       by (metis Max_ge finite_imageI finite_lessThan not_le not_less_eq)
     then obtain m where n: "\<And>n'. n' < n \<Longrightarrow> g n' < m"
       by blast
 
     have "(\<Sum>i<n. f (g i)) = sum f (g ` {..<n})"
       by (simp add: sum.reindex)
     also have "\<dots> \<le> (\<Sum>i<m. f i)"
       by (rule sum_mono2) (auto simp add: pos n[rule_format])
     also have "\<dots> \<le> suminf f"
       using \<open>summable f\<close>
       by (rule sum_le_suminf) (simp_all add: pos)
     finally show "(\<Sum>i<n. (f \<circ>  g) i) \<le> suminf f"
       by simp
   qed
 
   have "incseq (\<lambda>n. \<Sum>i<n. (f \<circ> g) i)"
     by (rule incseq_SucI) (auto simp add: pos)
   then obtain  L where L: "(\<lambda> n. \<Sum>i<n. (f \<circ> g) i) \<longlonglongrightarrow> L"
     using smaller by(rule incseq_convergent)
   then have "(f \<circ> g) sums L"
     by (simp add: sums_def)
   then show "summable (f \<circ> g)"
     by (auto simp add: sums_iff)
 
   then have "(\<lambda>n. \<Sum>i<n. (f \<circ> g) i) \<longlonglongrightarrow> suminf (f \<circ> g)"
     by (rule summable_LIMSEQ)
   then show le: "suminf (f \<circ> g) \<le> suminf f"
     by(rule LIMSEQ_le_const2)(blast intro: smaller[rule_format])
 
   assume f: "\<And>x. x \<notin> range g \<Longrightarrow> f x = 0"
 
   from \<open>summable f\<close> have "suminf f \<le> suminf (f \<circ> g)"
   proof (rule suminf_le_const)
     fix n
     have "\<forall> n' \<in> (g -` {..<n}). n' < Suc (Max (g -` {..<n}))"
       by(auto intro: Max_ge simp add: finite_vimageI less_Suc_eq_le)
     then obtain m where n: "\<And>n'. g n' < n \<Longrightarrow> n' < m"
       by blast
     have "(\<Sum>i<n. f i) = (\<Sum>i\<in>{..<n} \<inter> range g. f i)"
       using f by(auto intro: sum.mono_neutral_cong_right)
     also have "\<dots> = (\<Sum>i\<in>g -` {..<n}. (f \<circ> g) i)"
       by (rule sum.reindex_cong[where l=g])(auto)
     also have "\<dots> \<le> (\<Sum>i<m. (f \<circ> g) i)"
       by (rule sum_mono2)(auto simp add: pos n)
     also have "\<dots> \<le> suminf (f \<circ> g)"
       using \<open>summable (f \<circ> g)\<close> by (rule sum_le_suminf) (simp_all add: pos)
     finally show "sum f {..<n} \<le> suminf (f \<circ> g)" .
   qed
   with le show "suminf (f \<circ> g) = suminf f"
     by (rule antisym)
 qed
 
 lemma sums_mono_reindex:
   assumes subseq: "strict_mono g"
     and zero: "\<And>n. n \<notin> range g \<Longrightarrow> f n = 0"
   shows "(\<lambda>n. f (g n)) sums c \<longleftrightarrow> f sums c"
   unfolding sums_def
 proof
   assume lim: "(\<lambda>n. \<Sum>k<n. f k) \<longlonglongrightarrow> c"
   have "(\<lambda>n. \<Sum>k<n. f (g k)) = (\<lambda>n. \<Sum>k<g n. f k)"
   proof
     fix n :: nat
     from subseq have "(\<Sum>k<n. f (g k)) = (\<Sum>k\<in>g`{..<n}. f k)"
       by (subst sum.reindex) (auto intro: strict_mono_imp_inj_on)
     also from subseq have "\<dots> = (\<Sum>k<g n. f k)"
       by (intro sum.mono_neutral_left ballI zero)
         (auto simp: strict_mono_less strict_mono_less_eq)
     finally show "(\<Sum>k<n. f (g k)) = (\<Sum>k<g n. f k)" .
   qed
   also from LIMSEQ_subseq_LIMSEQ[OF lim subseq] have "\<dots> \<longlonglongrightarrow> c"
     by (simp only: o_def)
   finally show "(\<lambda>n. \<Sum>k<n. f (g k)) \<longlonglongrightarrow> c" .
 next
   assume lim: "(\<lambda>n. \<Sum>k<n. f (g k)) \<longlonglongrightarrow> c"
   define g_inv where "g_inv n = (LEAST m. g m \<ge> n)" for n
   from filterlim_subseq[OF subseq] have g_inv_ex: "\<exists>m. g m \<ge> n" for n
     by (auto simp: filterlim_at_top eventually_at_top_linorder)
   then have g_inv: "g (g_inv n) \<ge> n" for n
     unfolding g_inv_def by (rule LeastI_ex)
   have g_inv_least: "m \<ge> g_inv n" if "g m \<ge> n" for m n
     using that unfolding g_inv_def by (rule Least_le)
   have g_inv_least': "g m < n" if "m < g_inv n" for m n
     using that g_inv_least[of n m] by linarith
   have "(\<lambda>n. \<Sum>k<n. f k) = (\<lambda>n. \<Sum>k<g_inv n. f (g k))"
   proof
     fix n :: nat
     {
       fix k
       assume k: "k \<in> {..<n} - g`{..<g_inv n}"
       have "k \<notin> range g"
       proof (rule notI, elim imageE)
         fix l
         assume l: "k = g l"
         have "g l < g (g_inv n)"
           by (rule less_le_trans[OF _ g_inv]) (use k l in simp_all)
         with subseq have "l < g_inv n"
           by (simp add: strict_mono_less)
         with k l show False
           by simp
       qed
       then have "f k = 0"
         by (rule zero)
     }
     with g_inv_least' g_inv have "(\<Sum>k<n. f k) = (\<Sum>k\<in>g`{..<g_inv n}. f k)"
       by (intro sum.mono_neutral_right) auto
     also from subseq have "\<dots> = (\<Sum>k<g_inv n. f (g k))"
       using strict_mono_imp_inj_on by (subst sum.reindex) simp_all
     finally show "(\<Sum>k<n. f k) = (\<Sum>k<g_inv n. f (g k))" .
   qed
   also {
     fix K n :: nat
     assume "g K \<le> n"
     also have "n \<le> g (g_inv n)"
       by (rule g_inv)
     finally have "K \<le> g_inv n"
       using subseq by (simp add: strict_mono_less_eq)
   }
   then have "filterlim g_inv at_top sequentially"
     by (auto simp: filterlim_at_top eventually_at_top_linorder)
   with lim have "(\<lambda>n. \<Sum>k<g_inv n. f (g k)) \<longlonglongrightarrow> c"
     by (rule filterlim_compose)
   finally show "(\<lambda>n. \<Sum>k<n. f k) \<longlonglongrightarrow> c" .
 qed
 
 lemma summable_mono_reindex:
   assumes subseq: "strict_mono g"
     and zero: "\<And>n. n \<notin> range g \<Longrightarrow> f n = 0"
   shows "summable (\<lambda>n. f (g n)) \<longleftrightarrow> summable f"
   using sums_mono_reindex[of g f, OF assms] by (simp add: summable_def)
 
 lemma suminf_mono_reindex:
   fixes f :: "nat \<Rightarrow> 'a::{t2_space,comm_monoid_add}"
   assumes "strict_mono g" "\<And>n. n \<notin> range g \<Longrightarrow> f n = 0"
   shows   "suminf (\<lambda>n. f (g n)) = suminf f"
 proof (cases "summable f")
   case True
   with sums_mono_reindex [of g f, OF assms]
     and summable_mono_reindex [of g f, OF assms]
   show ?thesis
     by (simp add: sums_iff)
 next
   case False
   then have "\<not>(\<exists>c. f sums c)"
     unfolding summable_def by blast
   then have "suminf f = The (\<lambda>_. False)"
     by (simp add: suminf_def)
   moreover from False have "\<not> summable (\<lambda>n. f (g n))"
     using summable_mono_reindex[of g f, OF assms] by simp
   then have "\<not>(\<exists>c. (\<lambda>n. f (g n)) sums c)"
     unfolding summable_def by blast
   then have "suminf (\<lambda>n. f (g n)) = The (\<lambda>_. False)"
     by (simp add: suminf_def)
   ultimately show ?thesis by simp
 qed
 
 lemma summable_bounded_partials:
   fixes f :: "nat \<Rightarrow> 'a :: {real_normed_vector,complete_space}"
   assumes bound: "eventually (\<lambda>x0. \<forall>a\<ge>x0. \<forall>b>a. norm (sum f {a<..b}) \<le> g a) sequentially"
   assumes g: "g \<longlonglongrightarrow> 0"
   shows   "summable f" unfolding summable_iff_convergent'
 proof (intro Cauchy_convergent CauchyI', goal_cases)
   case (1 \<epsilon>)
   with g have "eventually (\<lambda>x. \<bar>g x\<bar> < \<epsilon>) sequentially"
     by (auto simp: tendsto_iff)
   from eventually_conj[OF this bound] obtain x0 where x0:
     "\<And>x. x \<ge> x0 \<Longrightarrow> \<bar>g x\<bar> < \<epsilon>" "\<And>a b. x0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> norm (sum f {a<..b}) \<le> g a" 
     unfolding eventually_at_top_linorder by auto
   show ?case
   proof (intro exI[of _ x0] allI impI)
     fix m n assume mn: "x0 \<le> m" "m < n"
     have "dist (sum f {..m}) (sum f {..n}) = norm (sum f {..n} - sum f {..m})"
       by (simp add: dist_norm norm_minus_commute)
     also have "sum f {..n} - sum f {..m} = sum f ({..n} - {..m})"
       using mn by (intro Groups_Big.sum_diff [symmetric]) auto
     also have "{..n} - {..m} = {m<..n}" using mn by auto
     also have "norm (sum f {m<..n}) \<le> g m" using mn by (intro x0) auto
     also have "\<dots> \<le> \<bar>g m\<bar>" by simp
     also have "\<dots> < \<epsilon>" using mn by (intro x0) auto
     finally show "dist (sum f {..m}) (sum f {..n}) < \<epsilon>" .
   qed
 qed
 
 end