src/HOL/Series.thy
 author paulson Mon May 23 15:33:24 2016 +0100 (2016-05-23) changeset 63114 27afe7af7379 parent 63040 eb4ddd18d635 child 63145 703edebd1d92 permissions -rw-r--r--
Lots of new material for multivariate analysis
```     1 (*  Title       : Series.thy
```
```     2     Author      : Jacques D. Fleuriot
```
```     3     Copyright   : 1998  University of Cambridge
```
```     4
```
```     5 Converted to Isar and polished by lcp
```
```     6 Converted to setsum and polished yet more by TNN
```
```     7 Additional contributions by Jeremy Avigad
```
```     8 *)
```
```     9
```
```    10 section \<open>Infinite Series\<close>
```
```    11
```
```    12 theory Series
```
```    13 imports Limits Inequalities
```
```    14 begin
```
```    15
```
```    16 subsection \<open>Definition of infinite summability\<close>
```
```    17
```
```    18 definition
```
```    19   sums :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a \<Rightarrow> bool"
```
```    20   (infixr "sums" 80)
```
```    21 where
```
```    22   "f sums s \<longleftrightarrow> (\<lambda>n. \<Sum>i<n. f i) \<longlonglongrightarrow> s"
```
```    23
```
```    24 definition summable :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> bool" where
```
```    25    "summable f \<longleftrightarrow> (\<exists>s. f sums s)"
```
```    26
```
```    27 definition
```
```    28   suminf :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a"
```
```    29   (binder "\<Sum>" 10)
```
```    30 where
```
```    31   "suminf f = (THE s. f sums s)"
```
```    32
```
```    33 lemma sums_def': "f sums s \<longleftrightarrow> (\<lambda>n. \<Sum>i = 0..n. f i) \<longlonglongrightarrow> s"
```
```    34   apply (simp add: sums_def)
```
```    35   apply (subst LIMSEQ_Suc_iff [symmetric])
```
```    36   apply (simp only: lessThan_Suc_atMost atLeast0AtMost)
```
```    37   done
```
```    38
```
```    39 subsection \<open>Infinite summability on topological monoids\<close>
```
```    40
```
```    41 lemma sums_subst[trans]: "f = g \<Longrightarrow> g sums z \<Longrightarrow> f sums z"
```
```    42   by simp
```
```    43
```
```    44 lemma sums_cong: "(\<And>n. f n = g n) \<Longrightarrow> f sums c \<longleftrightarrow> g sums c"
```
```    45   by (drule ext) simp
```
```    46
```
```    47 lemma sums_summable: "f sums l \<Longrightarrow> summable f"
```
```    48   by (simp add: sums_def summable_def, blast)
```
```    49
```
```    50 lemma summable_iff_convergent: "summable f \<longleftrightarrow> convergent (\<lambda>n. \<Sum>i<n. f i)"
```
```    51   by (simp add: summable_def sums_def convergent_def)
```
```    52
```
```    53 lemma summable_iff_convergent':
```
```    54   "summable f \<longleftrightarrow> convergent (\<lambda>n. setsum f {..n})"
```
```    55   by (simp_all only: summable_iff_convergent convergent_def
```
```    56         lessThan_Suc_atMost [symmetric] LIMSEQ_Suc_iff[of "\<lambda>n. setsum f {..<n}"])
```
```    57
```
```    58 lemma suminf_eq_lim: "suminf f = lim (\<lambda>n. \<Sum>i<n. f i)"
```
```    59   by (simp add: suminf_def sums_def lim_def)
```
```    60
```
```    61 lemma sums_zero[simp, intro]: "(\<lambda>n. 0) sums 0"
```
```    62   unfolding sums_def by simp
```
```    63
```
```    64 lemma summable_zero[simp, intro]: "summable (\<lambda>n. 0)"
```
```    65   by (rule sums_zero [THEN sums_summable])
```
```    66
```
```    67 lemma sums_group: "f sums s \<Longrightarrow> 0 < k \<Longrightarrow> (\<lambda>n. setsum f {n * k ..< n * k + k}) sums s"
```
```    68   apply (simp only: sums_def setsum_nat_group tendsto_def eventually_sequentially)
```
```    69   apply safe
```
```    70   apply (erule_tac x=S in allE)
```
```    71   apply safe
```
```    72   apply (rule_tac x="N" in exI, safe)
```
```    73   apply (drule_tac x="n*k" in spec)
```
```    74   apply (erule mp)
```
```    75   apply (erule order_trans)
```
```    76   apply simp
```
```    77   done
```
```    78
```
```    79 lemma suminf_cong: "(\<And>n. f n = g n) \<Longrightarrow> suminf f = suminf g"
```
```    80   by (rule arg_cong[of f g], rule ext) simp
```
```    81
```
```    82 lemma summable_cong:
```
```    83   assumes "eventually (\<lambda>x. f x = (g x :: 'a :: real_normed_vector)) sequentially"
```
```    84   shows   "summable f = summable g"
```
```    85 proof -
```
```    86   from assms obtain N where N: "\<forall>n\<ge>N. f n = g n" by (auto simp: eventually_at_top_linorder)
```
```    87   define C where "C = (\<Sum>k<N. f k - g k)"
```
```    88   from eventually_ge_at_top[of N]
```
```    89     have "eventually (\<lambda>n. setsum f {..<n} = C + setsum g {..<n}) sequentially"
```
```    90   proof eventually_elim
```
```    91     fix n assume n: "n \<ge> N"
```
```    92     from n have "{..<n} = {..<N} \<union> {N..<n}" by auto
```
```    93     also have "setsum f ... = setsum f {..<N} + setsum f {N..<n}"
```
```    94       by (intro setsum.union_disjoint) auto
```
```    95     also from N have "setsum f {N..<n} = setsum g {N..<n}" by (intro setsum.cong) simp_all
```
```    96     also have "setsum f {..<N} + setsum g {N..<n} = C + (setsum g {..<N} + setsum g {N..<n})"
```
```    97       unfolding C_def by (simp add: algebra_simps setsum_subtractf)
```
```    98     also have "setsum g {..<N} + setsum g {N..<n} = setsum g ({..<N} \<union> {N..<n})"
```
```    99       by (intro setsum.union_disjoint [symmetric]) auto
```
```   100     also from n have "{..<N} \<union> {N..<n} = {..<n}" by auto
```
```   101     finally show "setsum f {..<n} = C + setsum g {..<n}" .
```
```   102   qed
```
```   103   from convergent_cong[OF this] show ?thesis
```
```   104     by (simp add: summable_iff_convergent convergent_add_const_iff)
```
```   105 qed
```
```   106
```
```   107 lemma sums_finite:
```
```   108   assumes [simp]: "finite N" and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 0"
```
```   109   shows "f sums (\<Sum>n\<in>N. f n)"
```
```   110 proof -
```
```   111   { fix n
```
```   112     have "setsum f {..<n + Suc (Max N)} = setsum f N"
```
```   113     proof cases
```
```   114       assume "N = {}"
```
```   115       with f have "f = (\<lambda>x. 0)" by auto
```
```   116       then show ?thesis by simp
```
```   117     next
```
```   118       assume [simp]: "N \<noteq> {}"
```
```   119       show ?thesis
```
```   120       proof (safe intro!: setsum.mono_neutral_right f)
```
```   121         fix i assume "i \<in> N"
```
```   122         then have "i \<le> Max N" by simp
```
```   123         then show "i < n + Suc (Max N)" by simp
```
```   124       qed
```
```   125     qed }
```
```   126   note eq = this
```
```   127   show ?thesis unfolding sums_def
```
```   128     by (rule LIMSEQ_offset[of _ "Suc (Max N)"])
```
```   129        (simp add: eq atLeast0LessThan del: add_Suc_right)
```
```   130 qed
```
```   131
```
```   132 corollary sums_0:
```
```   133    "(\<And>n. f n = 0) \<Longrightarrow> (f sums 0)"
```
```   134     by (metis (no_types) finite.emptyI setsum.empty sums_finite)
```
```   135
```
```   136 lemma summable_finite: "finite N \<Longrightarrow> (\<And>n. n \<notin> N \<Longrightarrow> f n = 0) \<Longrightarrow> summable f"
```
```   137   by (rule sums_summable) (rule sums_finite)
```
```   138
```
```   139 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)"
```
```   140   using sums_finite[of A "(\<lambda>r. if r \<in> A then f r else 0)"] by simp
```
```   141
```
```   142 lemma summable_If_finite_set[simp, intro]: "finite A \<Longrightarrow> summable (\<lambda>r. if r \<in> A then f r else 0)"
```
```   143   by (rule sums_summable) (rule sums_If_finite_set)
```
```   144
```
```   145 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)"
```
```   146   using sums_If_finite_set[of "{r. P r}"] by simp
```
```   147
```
```   148 lemma summable_If_finite[simp, intro]: "finite {r. P r} \<Longrightarrow> summable (\<lambda>r. if P r then f r else 0)"
```
```   149   by (rule sums_summable) (rule sums_If_finite)
```
```   150
```
```   151 lemma sums_single: "(\<lambda>r. if r = i then f r else 0) sums f i"
```
```   152   using sums_If_finite[of "\<lambda>r. r = i"] by simp
```
```   153
```
```   154 lemma summable_single[simp, intro]: "summable (\<lambda>r. if r = i then f r else 0)"
```
```   155   by (rule sums_summable) (rule sums_single)
```
```   156
```
```   157 context
```
```   158   fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}"
```
```   159 begin
```
```   160
```
```   161 lemma summable_sums[intro]: "summable f \<Longrightarrow> f sums (suminf f)"
```
```   162   by (simp add: summable_def sums_def suminf_def)
```
```   163      (metis convergent_LIMSEQ_iff convergent_def lim_def)
```
```   164
```
```   165 lemma summable_LIMSEQ: "summable f \<Longrightarrow> (\<lambda>n. \<Sum>i<n. f i) \<longlonglongrightarrow> suminf f"
```
```   166   by (rule summable_sums [unfolded sums_def])
```
```   167
```
```   168 lemma sums_unique: "f sums s \<Longrightarrow> s = suminf f"
```
```   169   by (metis limI suminf_eq_lim sums_def)
```
```   170
```
```   171 lemma sums_iff: "f sums x \<longleftrightarrow> summable f \<and> (suminf f = x)"
```
```   172   by (metis summable_sums sums_summable sums_unique)
```
```   173
```
```   174 lemma summable_sums_iff:
```
```   175   "summable (f :: nat \<Rightarrow> 'a :: {comm_monoid_add,t2_space}) \<longleftrightarrow> f sums suminf f"
```
```   176   by (auto simp: sums_iff summable_sums)
```
```   177
```
```   178 lemma sums_unique2:
```
```   179   fixes a b :: "'a::{comm_monoid_add,t2_space}"
```
```   180   shows "f sums a \<Longrightarrow> f sums b \<Longrightarrow> a = b"
```
```   181 by (simp add: sums_iff)
```
```   182
```
```   183 lemma suminf_finite:
```
```   184   assumes N: "finite N" and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 0"
```
```   185   shows "suminf f = (\<Sum>n\<in>N. f n)"
```
```   186   using sums_finite[OF assms, THEN sums_unique] by simp
```
```   187
```
```   188 end
```
```   189
```
```   190 lemma suminf_zero[simp]: "suminf (\<lambda>n. 0::'a::{t2_space, comm_monoid_add}) = 0"
```
```   191   by (rule sums_zero [THEN sums_unique, symmetric])
```
```   192
```
```   193
```
```   194 subsection \<open>Infinite summability on ordered, topological monoids\<close>
```
```   195
```
```   196 lemma sums_le:
```
```   197   fixes f g :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology}"
```
```   198   shows "\<forall>n. f n \<le> g n \<Longrightarrow> f sums s \<Longrightarrow> g sums t \<Longrightarrow> s \<le> t"
```
```   199   by (rule LIMSEQ_le) (auto intro: setsum_mono simp: sums_def)
```
```   200
```
```   201 context
```
```   202   fixes f :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology}"
```
```   203 begin
```
```   204
```
```   205 lemma suminf_le: "\<lbrakk>\<forall>n. f n \<le> g n; summable f; summable g\<rbrakk> \<Longrightarrow> suminf f \<le> suminf g"
```
```   206   by (auto dest: sums_summable intro: sums_le)
```
```   207
```
```   208 lemma setsum_le_suminf: "summable f \<Longrightarrow> \<forall>m\<ge>n. 0 \<le> f m \<Longrightarrow> setsum f {..<n} \<le> suminf f"
```
```   209   by (rule sums_le[OF _ sums_If_finite_set summable_sums]) auto
```
```   210
```
```   211 lemma suminf_nonneg: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> 0 \<le> suminf f"
```
```   212   using setsum_le_suminf[of 0] by simp
```
```   213
```
```   214 lemma suminf_le_const: "summable f \<Longrightarrow> (\<And>n. setsum f {..<n} \<le> x) \<Longrightarrow> suminf f \<le> x"
```
```   215   by (metis LIMSEQ_le_const2 summable_LIMSEQ)
```
```   216
```
```   217 lemma suminf_eq_zero_iff: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> suminf f = 0 \<longleftrightarrow> (\<forall>n. f n = 0)"
```
```   218 proof
```
```   219   assume "summable f" "suminf f = 0" and pos: "\<forall>n. 0 \<le> f n"
```
```   220   then have f: "(\<lambda>n. \<Sum>i<n. f i) \<longlonglongrightarrow> 0"
```
```   221     using summable_LIMSEQ[of f] by simp
```
```   222   then have "\<And>i. (\<Sum>n\<in>{i}. f n) \<le> 0"
```
```   223   proof (rule LIMSEQ_le_const)
```
```   224     fix i show "\<exists>N. \<forall>n\<ge>N. (\<Sum>n\<in>{i}. f n) \<le> setsum f {..<n}"
```
```   225       using pos by (intro exI[of _ "Suc i"] allI impI setsum_mono2) auto
```
```   226   qed
```
```   227   with pos show "\<forall>n. f n = 0"
```
```   228     by (auto intro!: antisym)
```
```   229 qed (metis suminf_zero fun_eq_iff)
```
```   230
```
```   231 lemma suminf_pos_iff:
```
```   232   "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> 0 < suminf f \<longleftrightarrow> (\<exists>i. 0 < f i)"
```
```   233   using setsum_le_suminf[of 0] suminf_eq_zero_iff by (simp add: less_le)
```
```   234
```
```   235 lemma suminf_pos2:
```
```   236   assumes "summable f" "\<forall>n. 0 \<le> f n" "0 < f i"
```
```   237   shows "0 < suminf f"
```
```   238 proof -
```
```   239   have "0 < (\<Sum>n<Suc i. f n)"
```
```   240     using assms by (intro setsum_pos2[where i=i]) auto
```
```   241   also have "\<dots> \<le> suminf f"
```
```   242     using assms by (intro setsum_le_suminf) auto
```
```   243   finally show ?thesis .
```
```   244 qed
```
```   245
```
```   246 lemma suminf_pos: "summable f \<Longrightarrow> \<forall>n. 0 < f n \<Longrightarrow> 0 < suminf f"
```
```   247   by (intro suminf_pos2[where i=0]) (auto intro: less_imp_le)
```
```   248
```
```   249 end
```
```   250
```
```   251 context
```
```   252   fixes f :: "nat \<Rightarrow> 'a::{ordered_cancel_comm_monoid_add, linorder_topology}"
```
```   253 begin
```
```   254
```
```   255 lemma setsum_less_suminf2: "summable f \<Longrightarrow> \<forall>m\<ge>n. 0 \<le> f m \<Longrightarrow> n \<le> i \<Longrightarrow> 0 < f i \<Longrightarrow> setsum f {..<n} < suminf f"
```
```   256   using
```
```   257     setsum_le_suminf[of f "Suc i"]
```
```   258     add_strict_increasing[of "f i" "setsum f {..<n}" "setsum f {..<i}"]
```
```   259     setsum_mono2[of "{..<i}" "{..<n}" f]
```
```   260   by (auto simp: less_imp_le ac_simps)
```
```   261
```
```   262 lemma setsum_less_suminf: "summable f \<Longrightarrow> \<forall>m\<ge>n. 0 < f m \<Longrightarrow> setsum f {..<n} < suminf f"
```
```   263   using setsum_less_suminf2[of n n] by (simp add: less_imp_le)
```
```   264
```
```   265 end
```
```   266
```
```   267 lemma summableI_nonneg_bounded:
```
```   268   fixes f:: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology, conditionally_complete_linorder}"
```
```   269   assumes pos[simp]: "\<And>n. 0 \<le> f n" and le: "\<And>n. (\<Sum>i<n. f i) \<le> x"
```
```   270   shows "summable f"
```
```   271   unfolding summable_def sums_def[abs_def]
```
```   272 proof (rule exI LIMSEQ_incseq_SUP)+
```
```   273   show "bdd_above (range (\<lambda>n. setsum f {..<n}))"
```
```   274     using le by (auto simp: bdd_above_def)
```
```   275   show "incseq (\<lambda>n. setsum f {..<n})"
```
```   276     by (auto simp: mono_def intro!: setsum_mono2)
```
```   277 qed
```
```   278
```
```   279 lemma summableI[intro, simp]:
```
```   280   fixes f:: "nat \<Rightarrow> 'a::{canonically_ordered_monoid_add, linorder_topology, complete_linorder}"
```
```   281   shows "summable f"
```
```   282   by (intro summableI_nonneg_bounded[where x=top] zero_le top_greatest)
```
```   283
```
```   284 subsection \<open>Infinite summability on topological monoids\<close>
```
```   285
```
```   286 lemma Zero_notin_Suc: "0 \<notin> Suc ` A"
```
```   287   by auto
```
```   288
```
```   289 context
```
```   290   fixes f g :: "nat \<Rightarrow> 'a :: {t2_space, topological_comm_monoid_add}"
```
```   291 begin
```
```   292
```
```   293 lemma sums_Suc:
```
```   294   assumes "(\<lambda>n. f (Suc n)) sums l" shows "f sums (l + f 0)"
```
```   295 proof  -
```
```   296   have "(\<lambda>n. (\<Sum>i<n. f (Suc i)) + f 0) \<longlonglongrightarrow> l + f 0"
```
```   297     using assms by (auto intro!: tendsto_add simp: sums_def)
```
```   298   moreover have "(\<Sum>i<n. f (Suc i)) + f 0 = (\<Sum>i<Suc n. f i)" for n
```
```   299     unfolding lessThan_Suc_eq_insert_0 by (simp add: Zero_notin_Suc ac_simps setsum.reindex)
```
```   300   ultimately show ?thesis
```
```   301     by (auto simp add: sums_def simp del: setsum_lessThan_Suc intro: LIMSEQ_Suc_iff[THEN iffD1])
```
```   302 qed
```
```   303
```
```   304 lemma sums_add: "f sums a \<Longrightarrow> g sums b \<Longrightarrow> (\<lambda>n. f n + g n) sums (a + b)"
```
```   305   unfolding sums_def by (simp add: setsum.distrib tendsto_add)
```
```   306
```
```   307 lemma summable_add: "summable f \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. f n + g n)"
```
```   308   unfolding summable_def by (auto intro: sums_add)
```
```   309
```
```   310 lemma suminf_add: "summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f + suminf g = (\<Sum>n. f n + g n)"
```
```   311   by (intro sums_unique sums_add summable_sums)
```
```   312
```
```   313 end
```
```   314
```
```   315 context
```
```   316   fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::{t2_space, topological_comm_monoid_add}" and I :: "'i set"
```
```   317 begin
```
```   318
```
```   319 lemma sums_setsum: "(\<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)"
```
```   320   by (induct I rule: infinite_finite_induct) (auto intro!: sums_add)
```
```   321
```
```   322 lemma suminf_setsum: "(\<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)"
```
```   323   using sums_unique[OF sums_setsum, OF summable_sums] by simp
```
```   324
```
```   325 lemma summable_setsum: "(\<And>i. i \<in> I \<Longrightarrow> summable (f i)) \<Longrightarrow> summable (\<lambda>n. \<Sum>i\<in>I. f i n)"
```
```   326   using sums_summable[OF sums_setsum[OF summable_sums]] .
```
```   327
```
```   328 end
```
```   329
```
```   330 subsection \<open>Infinite summability on real normed vector spaces\<close>
```
```   331
```
```   332 context
```
```   333   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```   334 begin
```
```   335
```
```   336 lemma sums_Suc_iff: "(\<lambda>n. f (Suc n)) sums s \<longleftrightarrow> f sums (s + f 0)"
```
```   337 proof -
```
```   338   have "f sums (s + f 0) \<longleftrightarrow> (\<lambda>i. \<Sum>j<Suc i. f j) \<longlonglongrightarrow> s + f 0"
```
```   339     by (subst LIMSEQ_Suc_iff) (simp add: sums_def)
```
```   340   also have "\<dots> \<longleftrightarrow> (\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) \<longlonglongrightarrow> s + f 0"
```
```   341     by (simp add: ac_simps setsum.reindex image_iff lessThan_Suc_eq_insert_0)
```
```   342   also have "\<dots> \<longleftrightarrow> (\<lambda>n. f (Suc n)) sums s"
```
```   343   proof
```
```   344     assume "(\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) \<longlonglongrightarrow> s + f 0"
```
```   345     with tendsto_add[OF this tendsto_const, of "- f 0"]
```
```   346     show "(\<lambda>i. f (Suc i)) sums s"
```
```   347       by (simp add: sums_def)
```
```   348   qed (auto intro: tendsto_add simp: sums_def)
```
```   349   finally show ?thesis ..
```
```   350 qed
```
```   351
```
```   352 lemma summable_Suc_iff: "summable (\<lambda>n. f (Suc n)) = summable f"
```
```   353 proof
```
```   354   assume "summable f"
```
```   355   hence "f sums suminf f" by (rule summable_sums)
```
```   356   hence "(\<lambda>n. f (Suc n)) sums (suminf f - f 0)" by (simp add: sums_Suc_iff)
```
```   357   thus "summable (\<lambda>n. f (Suc n))" unfolding summable_def by blast
```
```   358 qed (auto simp: sums_Suc_iff summable_def)
```
```   359
```
```   360 lemma sums_Suc_imp: "f 0 = 0 \<Longrightarrow> (\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambda>n. f n) sums s"
```
```   361   using sums_Suc_iff by simp
```
```   362
```
```   363 end
```
```   364
```
```   365 context --\<open>Separate contexts are necessary to allow general use of the results above, here.\<close>
```
```   366   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```   367 begin
```
```   368
```
```   369 lemma sums_diff: "f sums a \<Longrightarrow> g sums b \<Longrightarrow> (\<lambda>n. f n - g n) sums (a - b)"
```
```   370   unfolding sums_def by (simp add: setsum_subtractf tendsto_diff)
```
```   371
```
```   372 lemma summable_diff: "summable f \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. f n - g n)"
```
```   373   unfolding summable_def by (auto intro: sums_diff)
```
```   374
```
```   375 lemma suminf_diff: "summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f - suminf g = (\<Sum>n. f n - g n)"
```
```   376   by (intro sums_unique sums_diff summable_sums)
```
```   377
```
```   378 lemma sums_minus: "f sums a \<Longrightarrow> (\<lambda>n. - f n) sums (- a)"
```
```   379   unfolding sums_def by (simp add: setsum_negf tendsto_minus)
```
```   380
```
```   381 lemma summable_minus: "summable f \<Longrightarrow> summable (\<lambda>n. - f n)"
```
```   382   unfolding summable_def by (auto intro: sums_minus)
```
```   383
```
```   384 lemma suminf_minus: "summable f \<Longrightarrow> (\<Sum>n. - f n) = - (\<Sum>n. f n)"
```
```   385   by (intro sums_unique [symmetric] sums_minus summable_sums)
```
```   386
```
```   387 lemma sums_iff_shift: "(\<lambda>i. f (i + n)) sums s \<longleftrightarrow> f sums (s + (\<Sum>i<n. f i))"
```
```   388 proof (induct n arbitrary: s)
```
```   389   case (Suc n)
```
```   390   moreover have "(\<lambda>i. f (Suc i + n)) sums s \<longleftrightarrow> (\<lambda>i. f (i + n)) sums (s + f n)"
```
```   391     by (subst sums_Suc_iff) simp
```
```   392   ultimately show ?case
```
```   393     by (simp add: ac_simps)
```
```   394 qed simp
```
```   395
```
```   396 corollary sums_iff_shift': "(\<lambda>i. f (i + n)) sums (s - (\<Sum>i<n. f i)) \<longleftrightarrow> f sums s"
```
```   397   by (simp add: sums_iff_shift)
```
```   398
```
```   399 lemma sums_zero_iff_shift:
```
```   400   assumes "\<And>i. i < n \<Longrightarrow> f i = 0"
```
```   401   shows "(\<lambda>i. f (i+n)) sums s \<longleftrightarrow> (\<lambda>i. f i) sums s"
```
```   402 by (simp add: assms sums_iff_shift)
```
```   403
```
```   404 lemma summable_iff_shift: "summable (\<lambda>n. f (n + k)) \<longleftrightarrow> summable f"
```
```   405   by (metis diff_add_cancel summable_def sums_iff_shift[abs_def])
```
```   406
```
```   407 lemma sums_split_initial_segment: "f sums s \<Longrightarrow> (\<lambda>i. f (i + n)) sums (s - (\<Sum>i<n. f i))"
```
```   408   by (simp add: sums_iff_shift)
```
```   409
```
```   410 lemma summable_ignore_initial_segment: "summable f \<Longrightarrow> summable (\<lambda>n. f(n + k))"
```
```   411   by (simp add: summable_iff_shift)
```
```   412
```
```   413 lemma suminf_minus_initial_segment: "summable f \<Longrightarrow> (\<Sum>n. f (n + k)) = (\<Sum>n. f n) - (\<Sum>i<k. f i)"
```
```   414   by (rule sums_unique[symmetric]) (auto simp: sums_iff_shift)
```
```   415
```
```   416 lemma suminf_split_initial_segment: "summable f \<Longrightarrow> suminf f = (\<Sum>n. f(n + k)) + (\<Sum>i<k. f i)"
```
```   417   by (auto simp add: suminf_minus_initial_segment)
```
```   418
```
```   419 lemma suminf_split_head: "summable f \<Longrightarrow> (\<Sum>n. f (Suc n)) = suminf f - f 0"
```
```   420   using suminf_split_initial_segment[of 1] by simp
```
```   421
```
```   422 lemma suminf_exist_split:
```
```   423   fixes r :: real assumes "0 < r" and "summable f"
```
```   424   shows "\<exists>N. \<forall>n\<ge>N. norm (\<Sum>i. f (i + n)) < r"
```
```   425 proof -
```
```   426   from LIMSEQ_D[OF summable_LIMSEQ[OF \<open>summable f\<close>] \<open>0 < r\<close>]
```
```   427   obtain N :: nat where "\<forall> n \<ge> N. norm (setsum f {..<n} - suminf f) < r" by auto
```
```   428   thus ?thesis
```
```   429     by (auto simp: norm_minus_commute suminf_minus_initial_segment[OF \<open>summable f\<close>])
```
```   430 qed
```
```   431
```
```   432 lemma summable_LIMSEQ_zero: "summable f \<Longrightarrow> f \<longlonglongrightarrow> 0"
```
```   433   apply (drule summable_iff_convergent [THEN iffD1])
```
```   434   apply (drule convergent_Cauchy)
```
```   435   apply (simp only: Cauchy_iff LIMSEQ_iff, safe)
```
```   436   apply (drule_tac x="r" in spec, safe)
```
```   437   apply (rule_tac x="M" in exI, safe)
```
```   438   apply (drule_tac x="Suc n" in spec, simp)
```
```   439   apply (drule_tac x="n" in spec, simp)
```
```   440   done
```
```   441
```
```   442 lemma summable_imp_convergent: "summable f \<Longrightarrow> convergent f"
```
```   443   by (force dest!: summable_LIMSEQ_zero simp: convergent_def)
```
```   444
```
```   445 lemma summable_imp_Bseq: "summable f \<Longrightarrow> Bseq f"
```
```   446   by (simp add: convergent_imp_Bseq summable_imp_convergent)
```
```   447
```
```   448 end
```
```   449
```
```   450 lemma summable_minus_iff:
```
```   451   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```   452   shows "summable (\<lambda>n. - f n) \<longleftrightarrow> summable f"
```
```   453   by (auto dest: summable_minus) \<comment>\<open>used two ways, hence must be outside the context above\<close>
```
```   454
```
```   455 lemma (in bounded_linear) sums: "(\<lambda>n. X n) sums a \<Longrightarrow> (\<lambda>n. f (X n)) sums (f a)"
```
```   456   unfolding sums_def by (drule tendsto, simp only: setsum)
```
```   457
```
```   458 lemma (in bounded_linear) summable: "summable (\<lambda>n. X n) \<Longrightarrow> summable (\<lambda>n. f (X n))"
```
```   459   unfolding summable_def by (auto intro: sums)
```
```   460
```
```   461 lemma (in bounded_linear) suminf: "summable (\<lambda>n. X n) \<Longrightarrow> f (\<Sum>n. X n) = (\<Sum>n. f (X n))"
```
```   462   by (intro sums_unique sums summable_sums)
```
```   463
```
```   464 lemmas sums_of_real = bounded_linear.sums [OF bounded_linear_of_real]
```
```   465 lemmas summable_of_real = bounded_linear.summable [OF bounded_linear_of_real]
```
```   466 lemmas suminf_of_real = bounded_linear.suminf [OF bounded_linear_of_real]
```
```   467
```
```   468 lemmas sums_scaleR_left = bounded_linear.sums[OF bounded_linear_scaleR_left]
```
```   469 lemmas summable_scaleR_left = bounded_linear.summable[OF bounded_linear_scaleR_left]
```
```   470 lemmas suminf_scaleR_left = bounded_linear.suminf[OF bounded_linear_scaleR_left]
```
```   471
```
```   472 lemmas sums_scaleR_right = bounded_linear.sums[OF bounded_linear_scaleR_right]
```
```   473 lemmas summable_scaleR_right = bounded_linear.summable[OF bounded_linear_scaleR_right]
```
```   474 lemmas suminf_scaleR_right = bounded_linear.suminf[OF bounded_linear_scaleR_right]
```
```   475
```
```   476 lemma summable_const_iff: "summable (\<lambda>_. c) \<longleftrightarrow> (c :: 'a :: real_normed_vector) = 0"
```
```   477 proof -
```
```   478   {
```
```   479     assume "c \<noteq> 0"
```
```   480     hence "filterlim (\<lambda>n. of_nat n * norm c) at_top sequentially"
```
```   481       by (subst mult.commute)
```
```   482          (auto intro!: filterlim_tendsto_pos_mult_at_top filterlim_real_sequentially)
```
```   483     hence "\<not>convergent (\<lambda>n. norm (\<Sum>k<n. c))"
```
```   484       by (intro filterlim_at_infinity_imp_not_convergent filterlim_at_top_imp_at_infinity)
```
```   485          (simp_all add: setsum_constant_scaleR)
```
```   486     hence "\<not>summable (\<lambda>_. c)" unfolding summable_iff_convergent using convergent_norm by blast
```
```   487   }
```
```   488   thus ?thesis by auto
```
```   489 qed
```
```   490
```
```   491
```
```   492 subsection \<open>Infinite summability on real normed algebras\<close>
```
```   493
```
```   494 context
```
```   495   fixes f :: "nat \<Rightarrow> 'a::real_normed_algebra"
```
```   496 begin
```
```   497
```
```   498 lemma sums_mult: "f sums a \<Longrightarrow> (\<lambda>n. c * f n) sums (c * a)"
```
```   499   by (rule bounded_linear.sums [OF bounded_linear_mult_right])
```
```   500
```
```   501 lemma summable_mult: "summable f \<Longrightarrow> summable (\<lambda>n. c * f n)"
```
```   502   by (rule bounded_linear.summable [OF bounded_linear_mult_right])
```
```   503
```
```   504 lemma suminf_mult: "summable f \<Longrightarrow> suminf (\<lambda>n. c * f n) = c * suminf f"
```
```   505   by (rule bounded_linear.suminf [OF bounded_linear_mult_right, symmetric])
```
```   506
```
```   507 lemma sums_mult2: "f sums a \<Longrightarrow> (\<lambda>n. f n * c) sums (a * c)"
```
```   508   by (rule bounded_linear.sums [OF bounded_linear_mult_left])
```
```   509
```
```   510 lemma summable_mult2: "summable f \<Longrightarrow> summable (\<lambda>n. f n * c)"
```
```   511   by (rule bounded_linear.summable [OF bounded_linear_mult_left])
```
```   512
```
```   513 lemma suminf_mult2: "summable f \<Longrightarrow> suminf f * c = (\<Sum>n. f n * c)"
```
```   514   by (rule bounded_linear.suminf [OF bounded_linear_mult_left])
```
```   515
```
```   516 end
```
```   517
```
```   518 lemma sums_mult_iff:
```
```   519   assumes "c \<noteq> 0"
```
```   520   shows   "(\<lambda>n. c * f n :: 'a :: {real_normed_algebra,field}) sums (c * d) \<longleftrightarrow> f sums d"
```
```   521   using sums_mult[of f d c] sums_mult[of "\<lambda>n. c * f n" "c * d" "inverse c"]
```
```   522   by (force simp: field_simps assms)
```
```   523
```
```   524 lemma sums_mult2_iff:
```
```   525   assumes "c \<noteq> (0 :: 'a :: {real_normed_algebra, field})"
```
```   526   shows   "(\<lambda>n. f n * c) sums (d * c) \<longleftrightarrow> f sums d"
```
```   527   using sums_mult_iff[OF assms, of f d] by (simp add: mult.commute)
```
```   528
```
```   529 lemma sums_of_real_iff:
```
```   530   "(\<lambda>n. of_real (f n) :: 'a :: real_normed_div_algebra) sums of_real c \<longleftrightarrow> f sums c"
```
```   531   by (simp add: sums_def of_real_setsum[symmetric] tendsto_of_real_iff del: of_real_setsum)
```
```   532
```
```   533
```
```   534 subsection \<open>Infinite summability on real normed fields\<close>
```
```   535
```
```   536 context
```
```   537   fixes c :: "'a::real_normed_field"
```
```   538 begin
```
```   539
```
```   540 lemma sums_divide: "f sums a \<Longrightarrow> (\<lambda>n. f n / c) sums (a / c)"
```
```   541   by (rule bounded_linear.sums [OF bounded_linear_divide])
```
```   542
```
```   543 lemma summable_divide: "summable f \<Longrightarrow> summable (\<lambda>n. f n / c)"
```
```   544   by (rule bounded_linear.summable [OF bounded_linear_divide])
```
```   545
```
```   546 lemma suminf_divide: "summable f \<Longrightarrow> suminf (\<lambda>n. f n / c) = suminf f / c"
```
```   547   by (rule bounded_linear.suminf [OF bounded_linear_divide, symmetric])
```
```   548
```
```   549 lemma sums_mult_D: "\<lbrakk>(\<lambda>n. c * f n) sums a; c \<noteq> 0\<rbrakk> \<Longrightarrow> f sums (a/c)"
```
```   550   using sums_mult_iff by fastforce
```
```   551
```
```   552 lemma summable_mult_D: "\<lbrakk>summable (\<lambda>n. c * f n); c \<noteq> 0\<rbrakk> \<Longrightarrow> summable f"
```
```   553   by (auto dest: summable_divide)
```
```   554
```
```   555 text\<open>Sum of a geometric progression.\<close>
```
```   556
```
```   557 lemma geometric_sums: "norm c < 1 \<Longrightarrow> (\<lambda>n. c^n) sums (1 / (1 - c))"
```
```   558 proof -
```
```   559   assume less_1: "norm c < 1"
```
```   560   hence neq_1: "c \<noteq> 1" by auto
```
```   561   hence neq_0: "c - 1 \<noteq> 0" by simp
```
```   562   from less_1 have lim_0: "(\<lambda>n. c^n) \<longlonglongrightarrow> 0"
```
```   563     by (rule LIMSEQ_power_zero)
```
```   564   hence "(\<lambda>n. c ^ n / (c - 1) - 1 / (c - 1)) \<longlonglongrightarrow> 0 / (c - 1) - 1 / (c - 1)"
```
```   565     using neq_0 by (intro tendsto_intros)
```
```   566   hence "(\<lambda>n. (c ^ n - 1) / (c - 1)) \<longlonglongrightarrow> 1 / (1 - c)"
```
```   567     by (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib)
```
```   568   thus "(\<lambda>n. c ^ n) sums (1 / (1 - c))"
```
```   569     by (simp add: sums_def geometric_sum neq_1)
```
```   570 qed
```
```   571
```
```   572 lemma summable_geometric: "norm c < 1 \<Longrightarrow> summable (\<lambda>n. c^n)"
```
```   573   by (rule geometric_sums [THEN sums_summable])
```
```   574
```
```   575 lemma suminf_geometric: "norm c < 1 \<Longrightarrow> suminf (\<lambda>n. c^n) = 1 / (1 - c)"
```
```   576   by (rule sums_unique[symmetric]) (rule geometric_sums)
```
```   577
```
```   578 lemma summable_geometric_iff: "summable (\<lambda>n. c ^ n) \<longleftrightarrow> norm c < 1"
```
```   579 proof
```
```   580   assume "summable (\<lambda>n. c ^ n :: 'a :: real_normed_field)"
```
```   581   hence "(\<lambda>n. norm c ^ n) \<longlonglongrightarrow> 0"
```
```   582     by (simp add: norm_power [symmetric] tendsto_norm_zero_iff summable_LIMSEQ_zero)
```
```   583   from order_tendstoD(2)[OF this zero_less_one] obtain n where "norm c ^ n < 1"
```
```   584     by (auto simp: eventually_at_top_linorder)
```
```   585   thus "norm c < 1" using one_le_power[of "norm c" n] by (cases "norm c \<ge> 1") (linarith, simp)
```
```   586 qed (rule summable_geometric)
```
```   587
```
```   588 end
```
```   589
```
```   590 lemma power_half_series: "(\<lambda>n. (1/2::real)^Suc n) sums 1"
```
```   591 proof -
```
```   592   have 2: "(\<lambda>n. (1/2::real)^n) sums 2" using geometric_sums [of "1/2::real"]
```
```   593     by auto
```
```   594   have "(\<lambda>n. (1/2::real)^Suc n) = (\<lambda>n. (1 / 2) ^ n / 2)"
```
```   595     by (simp add: mult.commute)
```
```   596   thus ?thesis using sums_divide [OF 2, of 2]
```
```   597     by simp
```
```   598 qed
```
```   599
```
```   600
```
```   601 subsection \<open>Telescoping\<close>
```
```   602
```
```   603 lemma telescope_sums:
```
```   604   assumes "f \<longlonglongrightarrow> (c :: 'a :: real_normed_vector)"
```
```   605   shows   "(\<lambda>n. f (Suc n) - f n) sums (c - f 0)"
```
```   606   unfolding sums_def
```
```   607 proof (subst LIMSEQ_Suc_iff [symmetric])
```
```   608   have "(\<lambda>n. \<Sum>k<Suc n. f (Suc k) - f k) = (\<lambda>n. f (Suc n) - f 0)"
```
```   609     by (simp add: lessThan_Suc_atMost atLeast0AtMost [symmetric] setsum_Suc_diff)
```
```   610   also have "\<dots> \<longlonglongrightarrow> c - f 0" by (intro tendsto_diff LIMSEQ_Suc[OF assms] tendsto_const)
```
```   611   finally show "(\<lambda>n. \<Sum>n<Suc n. f (Suc n) - f n) \<longlonglongrightarrow> c - f 0" .
```
```   612 qed
```
```   613
```
```   614 lemma telescope_sums':
```
```   615   assumes "f \<longlonglongrightarrow> (c :: 'a :: real_normed_vector)"
```
```   616   shows   "(\<lambda>n. f n - f (Suc n)) sums (f 0 - c)"
```
```   617   using sums_minus[OF telescope_sums[OF assms]] by (simp add: algebra_simps)
```
```   618
```
```   619 lemma telescope_summable:
```
```   620   assumes "f \<longlonglongrightarrow> (c :: 'a :: real_normed_vector)"
```
```   621   shows   "summable (\<lambda>n. f (Suc n) - f n)"
```
```   622   using telescope_sums[OF assms] by (simp add: sums_iff)
```
```   623
```
```   624 lemma telescope_summable':
```
```   625   assumes "f \<longlonglongrightarrow> (c :: 'a :: real_normed_vector)"
```
```   626   shows   "summable (\<lambda>n. f n - f (Suc n))"
```
```   627   using summable_minus[OF telescope_summable[OF assms]] by (simp add: algebra_simps)
```
```   628
```
```   629
```
```   630 subsection \<open>Infinite summability on Banach spaces\<close>
```
```   631
```
```   632 text\<open>Cauchy-type criterion for convergence of series (c.f. Harrison)\<close>
```
```   633
```
```   634 lemma summable_Cauchy:
```
```   635   fixes f :: "nat \<Rightarrow> 'a::banach"
```
```   636   shows "summable f \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n. norm (setsum f {m..<n}) < e)"
```
```   637   apply (simp only: summable_iff_convergent Cauchy_convergent_iff [symmetric] Cauchy_iff, safe)
```
```   638   apply (drule spec, drule (1) mp)
```
```   639   apply (erule exE, rule_tac x="M" in exI, clarify)
```
```   640   apply (rule_tac x="m" and y="n" in linorder_le_cases)
```
```   641   apply (frule (1) order_trans)
```
```   642   apply (drule_tac x="n" in spec, drule (1) mp)
```
```   643   apply (drule_tac x="m" in spec, drule (1) mp)
```
```   644   apply (simp_all add: setsum_diff [symmetric])
```
```   645   apply (drule spec, drule (1) mp)
```
```   646   apply (erule exE, rule_tac x="N" in exI, clarify)
```
```   647   apply (rule_tac x="m" and y="n" in linorder_le_cases)
```
```   648   apply (subst norm_minus_commute)
```
```   649   apply (simp_all add: setsum_diff [symmetric])
```
```   650   done
```
```   651
```
```   652 context
```
```   653   fixes f :: "nat \<Rightarrow> 'a::banach"
```
```   654 begin
```
```   655
```
```   656 text\<open>Absolute convergence imples normal convergence\<close>
```
```   657
```
```   658 lemma summable_norm_cancel: "summable (\<lambda>n. norm (f n)) \<Longrightarrow> summable f"
```
```   659   apply (simp only: summable_Cauchy, safe)
```
```   660   apply (drule_tac x="e" in spec, safe)
```
```   661   apply (rule_tac x="N" in exI, safe)
```
```   662   apply (drule_tac x="m" in spec, safe)
```
```   663   apply (rule order_le_less_trans [OF norm_setsum])
```
```   664   apply (rule order_le_less_trans [OF abs_ge_self])
```
```   665   apply simp
```
```   666   done
```
```   667
```
```   668 lemma summable_norm: "summable (\<lambda>n. norm (f n)) \<Longrightarrow> norm (suminf f) \<le> (\<Sum>n. norm (f n))"
```
```   669   by (auto intro: LIMSEQ_le tendsto_norm summable_norm_cancel summable_LIMSEQ norm_setsum)
```
```   670
```
```   671 text \<open>Comparison tests\<close>
```
```   672
```
```   673 lemma summable_comparison_test: "\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n \<Longrightarrow> summable g \<Longrightarrow> summable f"
```
```   674   apply (simp add: summable_Cauchy, safe)
```
```   675   apply (drule_tac x="e" in spec, safe)
```
```   676   apply (rule_tac x = "N + Na" in exI, safe)
```
```   677   apply (rotate_tac 2)
```
```   678   apply (drule_tac x = m in spec)
```
```   679   apply (auto, rotate_tac 2, drule_tac x = n in spec)
```
```   680   apply (rule_tac y = "\<Sum>k=m..<n. norm (f k)" in order_le_less_trans)
```
```   681   apply (rule norm_setsum)
```
```   682   apply (rule_tac y = "setsum g {m..<n}" in order_le_less_trans)
```
```   683   apply (auto intro: setsum_mono simp add: abs_less_iff)
```
```   684   done
```
```   685
```
```   686 lemma summable_comparison_test_ev:
```
```   687   shows "eventually (\<lambda>n. norm (f n) \<le> g n) sequentially \<Longrightarrow> summable g \<Longrightarrow> summable f"
```
```   688   by (rule summable_comparison_test) (auto simp: eventually_at_top_linorder)
```
```   689
```
```   690 (*A better argument order*)
```
```   691 lemma summable_comparison_test': "summable g \<Longrightarrow> (\<And>n. n \<ge> N \<Longrightarrow> norm(f n) \<le> g n) \<Longrightarrow> summable f"
```
```   692   by (rule summable_comparison_test) auto
```
```   693
```
```   694 subsection \<open>The Ratio Test\<close>
```
```   695
```
```   696 lemma summable_ratio_test:
```
```   697   assumes "c < 1" "\<And>n. n \<ge> N \<Longrightarrow> norm (f (Suc n)) \<le> c * norm (f n)"
```
```   698   shows "summable f"
```
```   699 proof cases
```
```   700   assume "0 < c"
```
```   701   show "summable f"
```
```   702   proof (rule summable_comparison_test)
```
```   703     show "\<exists>N'. \<forall>n\<ge>N'. norm (f n) \<le> (norm (f N) / (c ^ N)) * c ^ n"
```
```   704     proof (intro exI allI impI)
```
```   705       fix n assume "N \<le> n" then show "norm (f n) \<le> (norm (f N) / (c ^ N)) * c ^ n"
```
```   706       proof (induct rule: inc_induct)
```
```   707         case (step m)
```
```   708         moreover have "norm (f (Suc m)) / c ^ Suc m * c ^ n \<le> norm (f m) / c ^ m * c ^ n"
```
```   709           using \<open>0 < c\<close> \<open>c < 1\<close> assms(2)[OF \<open>N \<le> m\<close>] by (simp add: field_simps)
```
```   710         ultimately show ?case by simp
```
```   711       qed (insert \<open>0 < c\<close>, simp)
```
```   712     qed
```
```   713     show "summable (\<lambda>n. norm (f N) / c ^ N * c ^ n)"
```
```   714       using \<open>0 < c\<close> \<open>c < 1\<close> by (intro summable_mult summable_geometric) simp
```
```   715   qed
```
```   716 next
```
```   717   assume c: "\<not> 0 < c"
```
```   718   { fix n assume "n \<ge> N"
```
```   719     then have "norm (f (Suc n)) \<le> c * norm (f n)"
```
```   720       by fact
```
```   721     also have "\<dots> \<le> 0"
```
```   722       using c by (simp add: not_less mult_nonpos_nonneg)
```
```   723     finally have "f (Suc n) = 0"
```
```   724       by auto }
```
```   725   then show "summable f"
```
```   726     by (intro sums_summable[OF sums_finite, of "{.. Suc N}"]) (auto simp: not_le Suc_less_eq2)
```
```   727 qed
```
```   728
```
```   729 end
```
```   730
```
```   731 text\<open>Relations among convergence and absolute convergence for power series.\<close>
```
```   732
```
```   733 lemma Abel_lemma:
```
```   734   fixes a :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```   735   assumes r: "0 \<le> r" and r0: "r < r0" and M: "\<And>n. norm (a n) * r0^n \<le> M"
```
```   736     shows "summable (\<lambda>n. norm (a n) * r^n)"
```
```   737 proof (rule summable_comparison_test')
```
```   738   show "summable (\<lambda>n. M * (r / r0) ^ n)"
```
```   739     using assms
```
```   740     by (auto simp add: summable_mult summable_geometric)
```
```   741 next
```
```   742   fix n
```
```   743   show "norm (norm (a n) * r ^ n) \<le> M * (r / r0) ^ n"
```
```   744     using r r0 M [of n]
```
```   745     apply (auto simp add: abs_mult field_simps)
```
```   746     apply (cases "r=0", simp)
```
```   747     apply (cases n, auto)
```
```   748     done
```
```   749 qed
```
```   750
```
```   751
```
```   752 text\<open>Summability of geometric series for real algebras\<close>
```
```   753
```
```   754 lemma complete_algebra_summable_geometric:
```
```   755   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```   756   shows "norm x < 1 \<Longrightarrow> summable (\<lambda>n. x ^ n)"
```
```   757 proof (rule summable_comparison_test)
```
```   758   show "\<exists>N. \<forall>n\<ge>N. norm (x ^ n) \<le> norm x ^ n"
```
```   759     by (simp add: norm_power_ineq)
```
```   760   show "norm x < 1 \<Longrightarrow> summable (\<lambda>n. norm x ^ n)"
```
```   761     by (simp add: summable_geometric)
```
```   762 qed
```
```   763
```
```   764 subsection \<open>Cauchy Product Formula\<close>
```
```   765
```
```   766 text \<open>
```
```   767   Proof based on Analysis WebNotes: Chapter 07, Class 41
```
```   768   @{url "http://www.math.unl.edu/~webnotes/classes/class41/prp77.htm"}
```
```   769 \<close>
```
```   770
```
```   771 lemma Cauchy_product_sums:
```
```   772   fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}"
```
```   773   assumes a: "summable (\<lambda>k. norm (a k))"
```
```   774   assumes b: "summable (\<lambda>k. norm (b k))"
```
```   775   shows "(\<lambda>k. \<Sum>i\<le>k. a i * b (k - i)) sums ((\<Sum>k. a k) * (\<Sum>k. b k))"
```
```   776 proof -
```
```   777   let ?S1 = "\<lambda>n::nat. {..<n} \<times> {..<n}"
```
```   778   let ?S2 = "\<lambda>n::nat. {(i,j). i + j < n}"
```
```   779   have S1_mono: "\<And>m n. m \<le> n \<Longrightarrow> ?S1 m \<subseteq> ?S1 n" by auto
```
```   780   have S2_le_S1: "\<And>n. ?S2 n \<subseteq> ?S1 n" by auto
```
```   781   have S1_le_S2: "\<And>n. ?S1 (n div 2) \<subseteq> ?S2 n" by auto
```
```   782   have finite_S1: "\<And>n. finite (?S1 n)" by simp
```
```   783   with S2_le_S1 have finite_S2: "\<And>n. finite (?S2 n)" by (rule finite_subset)
```
```   784
```
```   785   let ?g = "\<lambda>(i,j). a i * b j"
```
```   786   let ?f = "\<lambda>(i,j). norm (a i) * norm (b j)"
```
```   787   have f_nonneg: "\<And>x. 0 \<le> ?f x" by (auto)
```
```   788   hence norm_setsum_f: "\<And>A. norm (setsum ?f A) = setsum ?f A"
```
```   789     unfolding real_norm_def
```
```   790     by (simp only: abs_of_nonneg setsum_nonneg [rule_format])
```
```   791
```
```   792   have "(\<lambda>n. (\<Sum>k<n. a k) * (\<Sum>k<n. b k)) \<longlonglongrightarrow> (\<Sum>k. a k) * (\<Sum>k. b k)"
```
```   793     by (intro tendsto_mult summable_LIMSEQ summable_norm_cancel [OF a] summable_norm_cancel [OF b])
```
```   794   hence 1: "(\<lambda>n. setsum ?g (?S1 n)) \<longlonglongrightarrow> (\<Sum>k. a k) * (\<Sum>k. b k)"
```
```   795     by (simp only: setsum_product setsum.Sigma [rule_format] finite_lessThan)
```
```   796
```
```   797   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))"
```
```   798     using a b by (intro tendsto_mult summable_LIMSEQ)
```
```   799   hence "(\<lambda>n. setsum ?f (?S1 n)) \<longlonglongrightarrow> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
```
```   800     by (simp only: setsum_product setsum.Sigma [rule_format] finite_lessThan)
```
```   801   hence "convergent (\<lambda>n. setsum ?f (?S1 n))"
```
```   802     by (rule convergentI)
```
```   803   hence Cauchy: "Cauchy (\<lambda>n. setsum ?f (?S1 n))"
```
```   804     by (rule convergent_Cauchy)
```
```   805   have "Zfun (\<lambda>n. setsum ?f (?S1 n - ?S2 n)) sequentially"
```
```   806   proof (rule ZfunI, simp only: eventually_sequentially norm_setsum_f)
```
```   807     fix r :: real
```
```   808     assume r: "0 < r"
```
```   809     from CauchyD [OF Cauchy r] obtain N
```
```   810     where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (setsum ?f (?S1 m) - setsum ?f (?S1 n)) < r" ..
```
```   811     hence "\<And>m n. \<lbrakk>N \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> norm (setsum ?f (?S1 m - ?S1 n)) < r"
```
```   812       by (simp only: setsum_diff finite_S1 S1_mono)
```
```   813     hence N: "\<And>m n. \<lbrakk>N \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> setsum ?f (?S1 m - ?S1 n) < r"
```
```   814       by (simp only: norm_setsum_f)
```
```   815     show "\<exists>N. \<forall>n\<ge>N. setsum ?f (?S1 n - ?S2 n) < r"
```
```   816     proof (intro exI allI impI)
```
```   817       fix n assume "2 * N \<le> n"
```
```   818       hence n: "N \<le> n div 2" by simp
```
```   819       have "setsum ?f (?S1 n - ?S2 n) \<le> setsum ?f (?S1 n - ?S1 (n div 2))"
```
```   820         by (intro setsum_mono2 finite_Diff finite_S1 f_nonneg
```
```   821                   Diff_mono subset_refl S1_le_S2)
```
```   822       also have "\<dots> < r"
```
```   823         using n div_le_dividend by (rule N)
```
```   824       finally show "setsum ?f (?S1 n - ?S2 n) < r" .
```
```   825     qed
```
```   826   qed
```
```   827   hence "Zfun (\<lambda>n. setsum ?g (?S1 n - ?S2 n)) sequentially"
```
```   828     apply (rule Zfun_le [rule_format])
```
```   829     apply (simp only: norm_setsum_f)
```
```   830     apply (rule order_trans [OF norm_setsum setsum_mono])
```
```   831     apply (auto simp add: norm_mult_ineq)
```
```   832     done
```
```   833   hence 2: "(\<lambda>n. setsum ?g (?S1 n) - setsum ?g (?S2 n)) \<longlonglongrightarrow> 0"
```
```   834     unfolding tendsto_Zfun_iff diff_0_right
```
```   835     by (simp only: setsum_diff finite_S1 S2_le_S1)
```
```   836
```
```   837   with 1 have "(\<lambda>n. setsum ?g (?S2 n)) \<longlonglongrightarrow> (\<Sum>k. a k) * (\<Sum>k. b k)"
```
```   838     by (rule Lim_transform2)
```
```   839   thus ?thesis by (simp only: sums_def setsum_triangle_reindex)
```
```   840 qed
```
```   841
```
```   842 lemma Cauchy_product:
```
```   843   fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}"
```
```   844   assumes a: "summable (\<lambda>k. norm (a k))"
```
```   845   assumes b: "summable (\<lambda>k. norm (b k))"
```
```   846   shows "(\<Sum>k. a k) * (\<Sum>k. b k) = (\<Sum>k. \<Sum>i\<le>k. a i * b (k - i))"
```
```   847   using a b
```
```   848   by (rule Cauchy_product_sums [THEN sums_unique])
```
```   849
```
```   850 lemma summable_Cauchy_product:
```
```   851   assumes "summable (\<lambda>k. norm (a k :: 'a :: {real_normed_algebra,banach}))"
```
```   852           "summable (\<lambda>k. norm (b k))"
```
```   853   shows   "summable (\<lambda>k. \<Sum>i\<le>k. a i * b (k - i))"
```
```   854   using Cauchy_product_sums[OF assms] by (simp add: sums_iff)
```
```   855
```
```   856 subsection \<open>Series on @{typ real}s\<close>
```
```   857
```
```   858 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))"
```
```   859   by (rule summable_comparison_test) auto
```
```   860
```
```   861 lemma summable_rabs_comparison_test: "\<lbrakk>\<exists>N. \<forall>n\<ge>N. \<bar>f n\<bar> \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable (\<lambda>n. \<bar>f n :: real\<bar>)"
```
```   862   by (rule summable_comparison_test) auto
```
```   863
```
```   864 lemma summable_rabs_cancel: "summable (\<lambda>n. \<bar>f n :: real\<bar>) \<Longrightarrow> summable f"
```
```   865   by (rule summable_norm_cancel) simp
```
```   866
```
```   867 lemma summable_rabs: "summable (\<lambda>n. \<bar>f n :: real\<bar>) \<Longrightarrow> \<bar>suminf f\<bar> \<le> (\<Sum>n. \<bar>f n\<bar>)"
```
```   868   by (fold real_norm_def) (rule summable_norm)
```
```   869
```
```   870 lemma summable_zero_power [simp]: "summable (\<lambda>n. 0 ^ n :: 'a :: {comm_ring_1,topological_space})"
```
```   871 proof -
```
```   872   have "(\<lambda>n. 0 ^ n :: 'a) = (\<lambda>n. if n = 0 then 0^0 else 0)" by (intro ext) (simp add: zero_power)
```
```   873   moreover have "summable \<dots>" by simp
```
```   874   ultimately show ?thesis by simp
```
```   875 qed
```
```   876
```
```   877 lemma summable_zero_power' [simp]: "summable (\<lambda>n. f n * 0 ^ n :: 'a :: {ring_1,topological_space})"
```
```   878 proof -
```
```   879   have "(\<lambda>n. f n * 0 ^ n :: 'a) = (\<lambda>n. if n = 0 then f 0 * 0^0 else 0)"
```
```   880     by (intro ext) (simp add: zero_power)
```
```   881   moreover have "summable \<dots>" by simp
```
```   882   ultimately show ?thesis by simp
```
```   883 qed
```
```   884
```
```   885 lemma summable_power_series:
```
```   886   fixes z :: real
```
```   887   assumes le_1: "\<And>i. f i \<le> 1" and nonneg: "\<And>i. 0 \<le> f i" and z: "0 \<le> z" "z < 1"
```
```   888   shows "summable (\<lambda>i. f i * z^i)"
```
```   889 proof (rule summable_comparison_test[OF _ summable_geometric])
```
```   890   show "norm z < 1" using z by (auto simp: less_imp_le)
```
```   891   show "\<And>n. \<exists>N. \<forall>na\<ge>N. norm (f na * z ^ na) \<le> z ^ na"
```
```   892     using z by (auto intro!: exI[of _ 0] mult_left_le_one_le simp: abs_mult nonneg power_abs less_imp_le le_1)
```
```   893 qed
```
```   894
```
```   895 lemma summable_0_powser:
```
```   896   "summable (\<lambda>n. f n * 0 ^ n :: 'a :: real_normed_div_algebra)"
```
```   897 proof -
```
```   898   have A: "(\<lambda>n. f n * 0 ^ n) = (\<lambda>n. if n = 0 then f n else 0)"
```
```   899     by (intro ext) auto
```
```   900   thus ?thesis by (subst A) simp_all
```
```   901 qed
```
```   902
```
```   903 lemma summable_powser_split_head:
```
```   904   "summable (\<lambda>n. f (Suc n) * z ^ n :: 'a :: real_normed_div_algebra) = summable (\<lambda>n. f n * z ^ n)"
```
```   905 proof -
```
```   906   have "summable (\<lambda>n. f (Suc n) * z ^ n) \<longleftrightarrow> summable (\<lambda>n. f (Suc n) * z ^ Suc n)"
```
```   907   proof
```
```   908     assume "summable (\<lambda>n. f (Suc n) * z ^ n)"
```
```   909     from summable_mult2[OF this, of z] show "summable (\<lambda>n. f (Suc n) * z ^ Suc n)"
```
```   910       by (simp add: power_commutes algebra_simps)
```
```   911   next
```
```   912     assume "summable (\<lambda>n. f (Suc n) * z ^ Suc n)"
```
```   913     from summable_mult2[OF this, of "inverse z"] show "summable (\<lambda>n. f (Suc n) * z ^ n)"
```
```   914       by (cases "z \<noteq> 0", subst (asm) power_Suc2) (simp_all add: algebra_simps)
```
```   915   qed
```
```   916   also have "\<dots> \<longleftrightarrow> summable (\<lambda>n. f n * z ^ n)" by (rule summable_Suc_iff)
```
```   917   finally show ?thesis .
```
```   918 qed
```
```   919
```
```   920 lemma powser_split_head:
```
```   921   assumes "summable (\<lambda>n. f n * z ^ n :: 'a :: {real_normed_div_algebra,banach})"
```
```   922   shows   "suminf (\<lambda>n. f n * z ^ n) = f 0 + suminf (\<lambda>n. f (Suc n) * z ^ n) * z"
```
```   923           "suminf (\<lambda>n. f (Suc n) * z ^ n) * z = suminf (\<lambda>n. f n * z ^ n) - f 0"
```
```   924           "summable (\<lambda>n. f (Suc n) * z ^ n)"
```
```   925 proof -
```
```   926   from assms show "summable (\<lambda>n. f (Suc n) * z ^ n)" by (subst summable_powser_split_head)
```
```   927
```
```   928   from suminf_mult2[OF this, of z]
```
```   929     have "(\<Sum>n. f (Suc n) * z ^ n) * z = (\<Sum>n. f (Suc n) * z ^ Suc n)"
```
```   930     by (simp add: power_commutes algebra_simps)
```
```   931   also from assms have "\<dots> = suminf (\<lambda>n. f n * z ^ n) - f 0"
```
```   932     by (subst suminf_split_head) simp_all
```
```   933   finally show "suminf (\<lambda>n. f n * z ^ n) = f 0 + suminf (\<lambda>n. f (Suc n) * z ^ n) * z" by simp
```
```   934   thus "suminf (\<lambda>n. f (Suc n) * z ^ n) * z = suminf (\<lambda>n. f n * z ^ n) - f 0" by simp
```
```   935 qed
```
```   936
```
```   937 lemma summable_partial_sum_bound:
```
```   938   fixes f :: "nat \<Rightarrow> 'a :: banach"
```
```   939   assumes summable: "summable f" and e: "e > (0::real)"
```
```   940   obtains N where "\<And>m n. m \<ge> N \<Longrightarrow> norm (\<Sum>k=m..n. f k) < e"
```
```   941 proof -
```
```   942   from summable have "Cauchy (\<lambda>n. \<Sum>k<n. f k)"
```
```   943     by (simp add: Cauchy_convergent_iff summable_iff_convergent)
```
```   944   from CauchyD[OF this e] obtain N
```
```   945     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
```
```   946   {
```
```   947     fix m n :: nat assume m: "m \<ge> N"
```
```   948     have "norm (\<Sum>k=m..n. f k) < e"
```
```   949     proof (cases "n \<ge> m")
```
```   950       assume n: "n \<ge> m"
```
```   951       with m have "norm ((\<Sum>k<Suc n. f k) - (\<Sum>k<m. f k)) < e" by (intro N) simp_all
```
```   952       also from n have "(\<Sum>k<Suc n. f k) - (\<Sum>k<m. f k) = (\<Sum>k=m..n. f k)"
```
```   953         by (subst setsum_diff [symmetric]) (simp_all add: setsum_last_plus)
```
```   954       finally show ?thesis .
```
```   955     qed (insert e, simp_all)
```
```   956   }
```
```   957   thus ?thesis by (rule that)
```
```   958 qed
```
```   959
```
```   960 lemma powser_sums_if:
```
```   961   "(\<lambda>n. (if n = m then (1 :: 'a :: {ring_1,topological_space}) else 0) * z^n) sums z^m"
```
```   962 proof -
```
```   963   have "(\<lambda>n. (if n = m then 1 else 0) * z^n) = (\<lambda>n. if n = m then z^n else 0)"
```
```   964     by (intro ext) auto
```
```   965   thus ?thesis by (simp add: sums_single)
```
```   966 qed
```
```   967
```
```   968 lemma
```
```   969    fixes f :: "nat \<Rightarrow> real"
```
```   970    assumes "summable f"
```
```   971    and "inj g"
```
```   972    and pos: "\<And>x. 0 \<le> f x"
```
```   973    shows summable_reindex: "summable (f o g)"
```
```   974    and suminf_reindex_mono: "suminf (f o g) \<le> suminf f"
```
```   975    and suminf_reindex: "(\<And>x. x \<notin> range g \<Longrightarrow> f x = 0) \<Longrightarrow> suminf (f \<circ> g) = suminf f"
```
```   976 proof -
```
```   977   from \<open>inj g\<close> have [simp]: "\<And>A. inj_on g A" by(rule subset_inj_on) simp
```
```   978
```
```   979   have smaller: "\<forall>n. (\<Sum>i<n. (f \<circ> g) i) \<le> suminf f"
```
```   980   proof
```
```   981     fix n
```
```   982     have "\<forall> n' \<in> (g ` {..<n}). n' < Suc (Max (g ` {..<n}))"
```
```   983       by(metis Max_ge finite_imageI finite_lessThan not_le not_less_eq)
```
```   984     then obtain m where n: "\<And>n'. n' < n \<Longrightarrow> g n' < m" by blast
```
```   985
```
```   986     have "(\<Sum>i<n. f (g i)) = setsum f (g ` {..<n})"
```
```   987       by (simp add: setsum.reindex)
```
```   988     also have "\<dots> \<le> (\<Sum>i<m. f i)"
```
```   989       by (rule setsum_mono3) (auto simp add: pos n[rule_format])
```
```   990     also have "\<dots> \<le> suminf f"
```
```   991       using \<open>summable f\<close>
```
```   992       by (rule setsum_le_suminf) (simp add: pos)
```
```   993     finally show "(\<Sum>i<n. (f \<circ>  g) i) \<le> suminf f" by simp
```
```   994   qed
```
```   995
```
```   996   have "incseq (\<lambda>n. \<Sum>i<n. (f \<circ> g) i)"
```
```   997     by (rule incseq_SucI) (auto simp add: pos)
```
```   998   then obtain  L where L: "(\<lambda> n. \<Sum>i<n. (f \<circ> g) i) \<longlonglongrightarrow> L"
```
```   999     using smaller by(rule incseq_convergent)
```
```  1000   hence "(f \<circ> g) sums L" by (simp add: sums_def)
```
```  1001   thus "summable (f o g)" by (auto simp add: sums_iff)
```
```  1002
```
```  1003   hence "(\<lambda>n. \<Sum>i<n. (f \<circ> g) i) \<longlonglongrightarrow> suminf (f \<circ> g)"
```
```  1004     by(rule summable_LIMSEQ)
```
```  1005   thus le: "suminf (f \<circ> g) \<le> suminf f"
```
```  1006     by(rule LIMSEQ_le_const2)(blast intro: smaller[rule_format])
```
```  1007
```
```  1008   assume f: "\<And>x. x \<notin> range g \<Longrightarrow> f x = 0"
```
```  1009
```
```  1010   from \<open>summable f\<close> have "suminf f \<le> suminf (f \<circ> g)"
```
```  1011   proof(rule suminf_le_const)
```
```  1012     fix n
```
```  1013     have "\<forall> n' \<in> (g -` {..<n}). n' < Suc (Max (g -` {..<n}))"
```
```  1014       by(auto intro: Max_ge simp add: finite_vimageI less_Suc_eq_le)
```
```  1015     then obtain m where n: "\<And>n'. g n' < n \<Longrightarrow> n' < m" by blast
```
```  1016
```
```  1017     have "(\<Sum>i<n. f i) = (\<Sum>i\<in>{..<n} \<inter> range g. f i)"
```
```  1018       using f by(auto intro: setsum.mono_neutral_cong_right)
```
```  1019     also have "\<dots> = (\<Sum>i\<in>g -` {..<n}. (f \<circ> g) i)"
```
```  1020       by(rule setsum.reindex_cong[where l=g])(auto)
```
```  1021     also have "\<dots> \<le> (\<Sum>i<m. (f \<circ> g) i)"
```
```  1022       by(rule setsum_mono3)(auto simp add: pos n)
```
```  1023     also have "\<dots> \<le> suminf (f \<circ> g)"
```
```  1024       using \<open>summable (f o g)\<close>
```
```  1025       by(rule setsum_le_suminf)(simp add: pos)
```
```  1026     finally show "setsum f {..<n} \<le> suminf (f \<circ> g)" .
```
```  1027   qed
```
```  1028   with le show "suminf (f \<circ> g) = suminf f" by(rule antisym)
```
```  1029 qed
```
```  1030
```
```  1031 lemma sums_mono_reindex:
```
```  1032   assumes subseq: "subseq g" and zero: "\<And>n. n \<notin> range g \<Longrightarrow> f n = 0"
```
```  1033   shows   "(\<lambda>n. f (g n)) sums c \<longleftrightarrow> f sums c"
```
```  1034 unfolding sums_def
```
```  1035 proof
```
```  1036   assume lim: "(\<lambda>n. \<Sum>k<n. f k) \<longlonglongrightarrow> c"
```
```  1037   have "(\<lambda>n. \<Sum>k<n. f (g k)) = (\<lambda>n. \<Sum>k<g n. f k)"
```
```  1038   proof
```
```  1039     fix n :: nat
```
```  1040     from subseq have "(\<Sum>k<n. f (g k)) = (\<Sum>k\<in>g`{..<n}. f k)"
```
```  1041       by (subst setsum.reindex) (auto intro: subseq_imp_inj_on)
```
```  1042     also from subseq have "\<dots> = (\<Sum>k<g n. f k)"
```
```  1043       by (intro setsum.mono_neutral_left ballI zero)
```
```  1044          (auto dest: subseq_strict_mono simp: strict_mono_less strict_mono_less_eq)
```
```  1045     finally show "(\<Sum>k<n. f (g k)) = (\<Sum>k<g n. f k)" .
```
```  1046   qed
```
```  1047   also from LIMSEQ_subseq_LIMSEQ[OF lim subseq] have "\<dots> \<longlonglongrightarrow> c" unfolding o_def .
```
```  1048   finally show "(\<lambda>n. \<Sum>k<n. f (g k)) \<longlonglongrightarrow> c" .
```
```  1049 next
```
```  1050   assume lim: "(\<lambda>n. \<Sum>k<n. f (g k)) \<longlonglongrightarrow> c"
```
```  1051   define g_inv where "g_inv n = (LEAST m. g m \<ge> n)" for n
```
```  1052   from filterlim_subseq[OF subseq] have g_inv_ex: "\<exists>m. g m \<ge> n" for n
```
```  1053     by (auto simp: filterlim_at_top eventually_at_top_linorder)
```
```  1054   hence g_inv: "g (g_inv n) \<ge> n" for n unfolding g_inv_def by (rule LeastI_ex)
```
```  1055   have g_inv_least: "m \<ge> g_inv n" if "g m \<ge> n" for m n using that
```
```  1056     unfolding g_inv_def by (rule Least_le)
```
```  1057   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
```
```  1058   have "(\<lambda>n. \<Sum>k<n. f k) = (\<lambda>n. \<Sum>k<g_inv n. f (g k))"
```
```  1059   proof
```
```  1060     fix n :: nat
```
```  1061     {
```
```  1062       fix k assume k: "k \<in> {..<n} - g`{..<g_inv n}"
```
```  1063       have "k \<notin> range g"
```
```  1064       proof (rule notI, elim imageE)
```
```  1065         fix l assume l: "k = g l"
```
```  1066         have "g l < g (g_inv n)" by (rule less_le_trans[OF _ g_inv]) (insert k l, simp_all)
```
```  1067         with subseq have "l < g_inv n" by (simp add: subseq_strict_mono strict_mono_less)
```
```  1068         with k l show False by simp
```
```  1069       qed
```
```  1070       hence "f k = 0" by (rule zero)
```
```  1071     }
```
```  1072     with g_inv_least' g_inv have "(\<Sum>k<n. f k) = (\<Sum>k\<in>g`{..<g_inv n}. f k)"
```
```  1073       by (intro setsum.mono_neutral_right) auto
```
```  1074     also from subseq have "\<dots> = (\<Sum>k<g_inv n. f (g k))" using subseq_imp_inj_on
```
```  1075       by (subst setsum.reindex) simp_all
```
```  1076     finally show "(\<Sum>k<n. f k) = (\<Sum>k<g_inv n. f (g k))" .
```
```  1077   qed
```
```  1078   also {
```
```  1079     fix K n :: nat assume "g K \<le> n"
```
```  1080     also have "n \<le> g (g_inv n)" by (rule g_inv)
```
```  1081     finally have "K \<le> g_inv n" using subseq by (simp add: strict_mono_less_eq subseq_strict_mono)
```
```  1082   }
```
```  1083   hence "filterlim g_inv at_top sequentially"
```
```  1084     by (auto simp: filterlim_at_top eventually_at_top_linorder)
```
```  1085   from lim and this have "(\<lambda>n. \<Sum>k<g_inv n. f (g k)) \<longlonglongrightarrow> c" by (rule filterlim_compose)
```
```  1086   finally show "(\<lambda>n. \<Sum>k<n. f k) \<longlonglongrightarrow> c" .
```
```  1087 qed
```
```  1088
```
```  1089 lemma summable_mono_reindex:
```
```  1090   assumes subseq: "subseq g" and zero: "\<And>n. n \<notin> range g \<Longrightarrow> f n = 0"
```
```  1091   shows   "summable (\<lambda>n. f (g n)) \<longleftrightarrow> summable f"
```
```  1092   using sums_mono_reindex[of g f, OF assms] by (simp add: summable_def)
```
```  1093
```
```  1094 lemma suminf_mono_reindex:
```
```  1095   assumes "subseq g" "\<And>n. n \<notin> range g \<Longrightarrow> f n = (0 :: 'a :: {t2_space,comm_monoid_add})"
```
```  1096   shows   "suminf (\<lambda>n. f (g n)) = suminf f"
```
```  1097 proof (cases "summable f")
```
```  1098   case False
```
```  1099   hence "\<not>(\<exists>c. f sums c)" unfolding summable_def by blast
```
```  1100   hence "suminf f = The (\<lambda>_. False)" by (simp add: suminf_def)
```
```  1101   moreover from False have "\<not>summable (\<lambda>n. f (g n))"
```
```  1102     using summable_mono_reindex[of g f, OF assms] by simp
```
```  1103   hence "\<not>(\<exists>c. (\<lambda>n. f (g n)) sums c)" unfolding summable_def by blast
```
```  1104   hence "suminf (\<lambda>n. f (g n)) = The (\<lambda>_. False)" by (simp add: suminf_def)
```
```  1105   ultimately show ?thesis by simp
```
```  1106 qed (insert sums_mono_reindex[of g f, OF assms] summable_mono_reindex[of g f, OF assms],
```
```  1107      simp_all add: sums_iff)
```
```  1108
```
```  1109 end
```