src/HOL/Series.thy
author paulson <lp15@cam.ac.uk>
Tue Nov 10 14:18:41 2015 +0000 (2015-11-10)
changeset 61609 77b453bd616f
parent 61531 ab2e862263e7
child 61799 4cf66f21b764
permissions -rw-r--r--
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson@10751
     1
(*  Title       : Series.thy
paulson@10751
     2
    Author      : Jacques D. Fleuriot
paulson@10751
     3
    Copyright   : 1998  University of Cambridge
paulson@14416
     4
paulson@14416
     5
Converted to Isar and polished by lcp
nipkow@15539
     6
Converted to setsum and polished yet more by TNN
avigad@16819
     7
Additional contributions by Jeremy Avigad
hoelzl@41970
     8
*)
paulson@10751
     9
wenzelm@60758
    10
section \<open>Infinite Series\<close>
paulson@10751
    11
nipkow@15131
    12
theory Series
hoelzl@59712
    13
imports Limits Inequalities
lp15@61609
    14
begin
nipkow@15561
    15
wenzelm@60758
    16
subsection \<open>Definition of infinite summability\<close>
hoelzl@56213
    17
hoelzl@56193
    18
definition
hoelzl@56193
    19
  sums :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a \<Rightarrow> bool"
hoelzl@56193
    20
  (infixr "sums" 80)
hoelzl@56193
    21
where
hoelzl@56193
    22
  "f sums s \<longleftrightarrow> (\<lambda>n. \<Sum>i<n. f i) ----> s"
paulson@14416
    23
hoelzl@56193
    24
definition summable :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> bool" where
hoelzl@56193
    25
   "summable f \<longleftrightarrow> (\<exists>s. f sums s)"
hoelzl@56193
    26
hoelzl@56193
    27
definition
hoelzl@56193
    28
  suminf :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a"
hoelzl@56193
    29
  (binder "\<Sum>" 10)
hoelzl@56193
    30
where
hoelzl@56193
    31
  "suminf f = (THE s. f sums s)"
hoelzl@56193
    32
wenzelm@60758
    33
subsection \<open>Infinite summability on topological monoids\<close>
hoelzl@56213
    34
hoelzl@56193
    35
lemma sums_subst[trans]: "f = g \<Longrightarrow> g sums z \<Longrightarrow> f sums z"
hoelzl@56193
    36
  by simp
hoelzl@56193
    37
eberlm@61531
    38
lemma sums_cong: "(\<And>n. f n = g n) \<Longrightarrow> f sums c \<longleftrightarrow> g sums c"
eberlm@61531
    39
  by (drule ext) simp
eberlm@61531
    40
hoelzl@56193
    41
lemma sums_summable: "f sums l \<Longrightarrow> summable f"
hoelzl@41970
    42
  by (simp add: sums_def summable_def, blast)
paulson@14416
    43
hoelzl@56193
    44
lemma summable_iff_convergent: "summable f \<longleftrightarrow> convergent (\<lambda>n. \<Sum>i<n. f i)"
hoelzl@56193
    45
  by (simp add: summable_def sums_def convergent_def)
paulson@14416
    46
eberlm@61531
    47
lemma summable_iff_convergent':
eberlm@61531
    48
  "summable f \<longleftrightarrow> convergent (\<lambda>n. setsum f {..n})"
lp15@61609
    49
  by (simp_all only: summable_iff_convergent convergent_def
eberlm@61531
    50
        lessThan_Suc_atMost [symmetric] LIMSEQ_Suc_iff[of "\<lambda>n. setsum f {..<n}"])
eberlm@61531
    51
hoelzl@56193
    52
lemma suminf_eq_lim: "suminf f = lim (\<lambda>n. \<Sum>i<n. f i)"
hoelzl@41970
    53
  by (simp add: suminf_def sums_def lim_def)
paulson@32707
    54
hoelzl@56213
    55
lemma sums_zero[simp, intro]: "(\<lambda>n. 0) sums 0"
hoelzl@58729
    56
  unfolding sums_def by simp
hoelzl@56213
    57
hoelzl@56213
    58
lemma summable_zero[simp, intro]: "summable (\<lambda>n. 0)"
hoelzl@56213
    59
  by (rule sums_zero [THEN sums_summable])
hoelzl@56213
    60
hoelzl@56213
    61
lemma sums_group: "f sums s \<Longrightarrow> 0 < k \<Longrightarrow> (\<lambda>n. setsum f {n * k ..< n * k + k}) sums s"
hoelzl@56213
    62
  apply (simp only: sums_def setsum_nat_group tendsto_def eventually_sequentially)
hoelzl@56213
    63
  apply safe
hoelzl@56213
    64
  apply (erule_tac x=S in allE)
hoelzl@56213
    65
  apply safe
hoelzl@56213
    66
  apply (rule_tac x="N" in exI, safe)
hoelzl@56213
    67
  apply (drule_tac x="n*k" in spec)
hoelzl@56213
    68
  apply (erule mp)
hoelzl@56213
    69
  apply (erule order_trans)
hoelzl@56213
    70
  apply simp
hoelzl@56213
    71
  done
hoelzl@56213
    72
eberlm@61531
    73
lemma suminf_cong: "(\<And>n. f n = g n) \<Longrightarrow> suminf f = suminf g"
eberlm@61531
    74
  by (rule arg_cong[of f g], rule ext) simp
eberlm@61531
    75
eberlm@61531
    76
lemma summable_cong:
eberlm@61531
    77
  assumes "eventually (\<lambda>x. f x = (g x :: 'a :: real_normed_vector)) sequentially"
eberlm@61531
    78
  shows   "summable f = summable g"
eberlm@61531
    79
proof -
eberlm@61531
    80
  from assms obtain N where N: "\<forall>n\<ge>N. f n = g n" by (auto simp: eventually_at_top_linorder)
eberlm@61531
    81
  def C \<equiv> "(\<Sum>k<N. f k - g k)"
lp15@61609
    82
  from eventually_ge_at_top[of N]
eberlm@61531
    83
    have "eventually (\<lambda>n. setsum f {..<n} = C + setsum g {..<n}) sequentially"
eberlm@61531
    84
  proof eventually_elim
eberlm@61531
    85
    fix n assume n: "n \<ge> N"
eberlm@61531
    86
    from n have "{..<n} = {..<N} \<union> {N..<n}" by auto
eberlm@61531
    87
    also have "setsum f ... = setsum f {..<N} + setsum f {N..<n}"
eberlm@61531
    88
      by (intro setsum.union_disjoint) auto
eberlm@61531
    89
    also from N have "setsum f {N..<n} = setsum g {N..<n}" by (intro setsum.cong) simp_all
lp15@61609
    90
    also have "setsum f {..<N} + setsum g {N..<n} = C + (setsum g {..<N} + setsum g {N..<n})"
eberlm@61531
    91
      unfolding C_def by (simp add: algebra_simps setsum_subtractf)
eberlm@61531
    92
    also have "setsum g {..<N} + setsum g {N..<n} = setsum g ({..<N} \<union> {N..<n})"
eberlm@61531
    93
      by (intro setsum.union_disjoint [symmetric]) auto
eberlm@61531
    94
    also from n have "{..<N} \<union> {N..<n} = {..<n}" by auto
eberlm@61531
    95
    finally show "setsum f {..<n} = C + setsum g {..<n}" .
eberlm@61531
    96
  qed
eberlm@61531
    97
  from convergent_cong[OF this] show ?thesis
eberlm@61531
    98
    by (simp add: summable_iff_convergent convergent_add_const_iff)
eberlm@61531
    99
qed
eberlm@61531
   100
hoelzl@47761
   101
lemma sums_finite:
hoelzl@56193
   102
  assumes [simp]: "finite N" and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 0"
hoelzl@47761
   103
  shows "f sums (\<Sum>n\<in>N. f n)"
hoelzl@47761
   104
proof -
hoelzl@47761
   105
  { fix n
hoelzl@47761
   106
    have "setsum f {..<n + Suc (Max N)} = setsum f N"
hoelzl@47761
   107
    proof cases
hoelzl@47761
   108
      assume "N = {}"
hoelzl@47761
   109
      with f have "f = (\<lambda>x. 0)" by auto
hoelzl@47761
   110
      then show ?thesis by simp
hoelzl@47761
   111
    next
hoelzl@47761
   112
      assume [simp]: "N \<noteq> {}"
hoelzl@47761
   113
      show ?thesis
haftmann@57418
   114
      proof (safe intro!: setsum.mono_neutral_right f)
hoelzl@47761
   115
        fix i assume "i \<in> N"
hoelzl@47761
   116
        then have "i \<le> Max N" by simp
hoelzl@47761
   117
        then show "i < n + Suc (Max N)" by simp
hoelzl@47761
   118
      qed
hoelzl@47761
   119
    qed }
hoelzl@47761
   120
  note eq = this
hoelzl@47761
   121
  show ?thesis unfolding sums_def
hoelzl@47761
   122
    by (rule LIMSEQ_offset[of _ "Suc (Max N)"])
hoelzl@58729
   123
       (simp add: eq atLeast0LessThan del: add_Suc_right)
hoelzl@47761
   124
qed
hoelzl@47761
   125
hoelzl@56213
   126
lemma summable_finite: "finite N \<Longrightarrow> (\<And>n. n \<notin> N \<Longrightarrow> f n = 0) \<Longrightarrow> summable f"
hoelzl@56213
   127
  by (rule sums_summable) (rule sums_finite)
hoelzl@56213
   128
hoelzl@56193
   129
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)"
hoelzl@47761
   130
  using sums_finite[of A "(\<lambda>r. if r \<in> A then f r else 0)"] by simp
hoelzl@47761
   131
hoelzl@56213
   132
lemma summable_If_finite_set[simp, intro]: "finite A \<Longrightarrow> summable (\<lambda>r. if r \<in> A then f r else 0)"
hoelzl@56213
   133
  by (rule sums_summable) (rule sums_If_finite_set)
hoelzl@56213
   134
hoelzl@56193
   135
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)"
hoelzl@56193
   136
  using sums_If_finite_set[of "{r. P r}"] by simp
avigad@16819
   137
hoelzl@56213
   138
lemma summable_If_finite[simp, intro]: "finite {r. P r} \<Longrightarrow> summable (\<lambda>r. if P r then f r else 0)"
hoelzl@56213
   139
  by (rule sums_summable) (rule sums_If_finite)
hoelzl@56213
   140
hoelzl@56193
   141
lemma sums_single: "(\<lambda>r. if r = i then f r else 0) sums f i"
hoelzl@56193
   142
  using sums_If_finite[of "\<lambda>r. r = i"] by simp
hoelzl@29803
   143
hoelzl@56213
   144
lemma summable_single[simp, intro]: "summable (\<lambda>r. if r = i then f r else 0)"
hoelzl@56213
   145
  by (rule sums_summable) (rule sums_single)
hoelzl@56193
   146
hoelzl@56193
   147
context
hoelzl@56193
   148
  fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}"
hoelzl@56193
   149
begin
hoelzl@56193
   150
hoelzl@56193
   151
lemma summable_sums[intro]: "summable f \<Longrightarrow> f sums (suminf f)"
hoelzl@56193
   152
  by (simp add: summable_def sums_def suminf_def)
hoelzl@56193
   153
     (metis convergent_LIMSEQ_iff convergent_def lim_def)
hoelzl@56193
   154
hoelzl@56193
   155
lemma summable_LIMSEQ: "summable f \<Longrightarrow> (\<lambda>n. \<Sum>i<n. f i) ----> suminf f"
hoelzl@56193
   156
  by (rule summable_sums [unfolded sums_def])
hoelzl@56193
   157
hoelzl@56193
   158
lemma sums_unique: "f sums s \<Longrightarrow> s = suminf f"
hoelzl@56193
   159
  by (metis limI suminf_eq_lim sums_def)
hoelzl@56193
   160
hoelzl@56193
   161
lemma sums_iff: "f sums x \<longleftrightarrow> summable f \<and> (suminf f = x)"
hoelzl@56193
   162
  by (metis summable_sums sums_summable sums_unique)
hoelzl@56193
   163
lp15@61609
   164
lemma summable_sums_iff:
eberlm@61531
   165
  "summable (f :: nat \<Rightarrow> 'a :: {comm_monoid_add,t2_space}) \<longleftrightarrow> f sums suminf f"
eberlm@61531
   166
  by (auto simp: sums_iff summable_sums)
eberlm@61531
   167
lp15@59613
   168
lemma sums_unique2:
lp15@59613
   169
  fixes a b :: "'a::{comm_monoid_add,t2_space}"
lp15@59613
   170
  shows "f sums a \<Longrightarrow> f sums b \<Longrightarrow> a = b"
lp15@59613
   171
by (simp add: sums_iff)
lp15@59613
   172
hoelzl@56193
   173
lemma suminf_finite:
hoelzl@56193
   174
  assumes N: "finite N" and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 0"
hoelzl@56193
   175
  shows "suminf f = (\<Sum>n\<in>N. f n)"
hoelzl@56193
   176
  using sums_finite[OF assms, THEN sums_unique] by simp
hoelzl@56193
   177
hoelzl@56193
   178
end
avigad@16819
   179
hoelzl@41970
   180
lemma suminf_zero[simp]: "suminf (\<lambda>n. 0::'a::{t2_space, comm_monoid_add}) = 0"
hoelzl@56193
   181
  by (rule sums_zero [THEN sums_unique, symmetric])
avigad@16819
   182
hoelzl@56213
   183
wenzelm@60758
   184
subsection \<open>Infinite summability on ordered, topological monoids\<close>
hoelzl@56213
   185
hoelzl@56213
   186
lemma sums_le:
hoelzl@56213
   187
  fixes f g :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology}"
hoelzl@56213
   188
  shows "\<forall>n. f n \<le> g n \<Longrightarrow> f sums s \<Longrightarrow> g sums t \<Longrightarrow> s \<le> t"
hoelzl@56213
   189
  by (rule LIMSEQ_le) (auto intro: setsum_mono simp: sums_def)
hoelzl@56213
   190
hoelzl@56193
   191
context
hoelzl@56193
   192
  fixes f :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology}"
hoelzl@56193
   193
begin
paulson@14416
   194
hoelzl@56213
   195
lemma suminf_le: "\<lbrakk>\<forall>n. f n \<le> g n; summable f; summable g\<rbrakk> \<Longrightarrow> suminf f \<le> suminf g"
hoelzl@56213
   196
  by (auto dest: sums_summable intro: sums_le)
hoelzl@56213
   197
hoelzl@56213
   198
lemma setsum_le_suminf: "summable f \<Longrightarrow> \<forall>m\<ge>n. 0 \<le> f m \<Longrightarrow> setsum f {..<n} \<le> suminf f"
hoelzl@56213
   199
  by (rule sums_le[OF _ sums_If_finite_set summable_sums]) auto
hoelzl@56213
   200
hoelzl@56213
   201
lemma suminf_nonneg: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> 0 \<le> suminf f"
hoelzl@56213
   202
  using setsum_le_suminf[of 0] by simp
hoelzl@56213
   203
hoelzl@56213
   204
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"
hoelzl@56213
   205
  using
hoelzl@56213
   206
    setsum_le_suminf[of "Suc i"]
hoelzl@56213
   207
    add_strict_increasing[of "f i" "setsum f {..<n}" "setsum f {..<i}"]
hoelzl@56213
   208
    setsum_mono2[of "{..<i}" "{..<n}" f]
hoelzl@56213
   209
  by (auto simp: less_imp_le ac_simps)
hoelzl@56213
   210
hoelzl@56213
   211
lemma setsum_less_suminf: "summable f \<Longrightarrow> \<forall>m\<ge>n. 0 < f m \<Longrightarrow> setsum f {..<n} < suminf f"
hoelzl@56213
   212
  using setsum_less_suminf2[of n n] by (simp add: less_imp_le)
hoelzl@56213
   213
hoelzl@56213
   214
lemma suminf_pos2: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> 0 < f i \<Longrightarrow> 0 < suminf f"
hoelzl@56213
   215
  using setsum_less_suminf2[of 0 i] by simp
hoelzl@56213
   216
hoelzl@56213
   217
lemma suminf_pos: "summable f \<Longrightarrow> \<forall>n. 0 < f n \<Longrightarrow> 0 < suminf f"
hoelzl@56213
   218
  using suminf_pos2[of 0] by (simp add: less_imp_le)
hoelzl@56213
   219
hoelzl@56213
   220
lemma suminf_le_const: "summable f \<Longrightarrow> (\<And>n. setsum f {..<n} \<le> x) \<Longrightarrow> suminf f \<le> x"
hoelzl@56213
   221
  by (metis LIMSEQ_le_const2 summable_LIMSEQ)
paulson@14416
   222
hoelzl@56193
   223
lemma suminf_eq_zero_iff: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> suminf f = 0 \<longleftrightarrow> (\<forall>n. f n = 0)"
hoelzl@50999
   224
proof
hoelzl@50999
   225
  assume "summable f" "suminf f = 0" and pos: "\<forall>n. 0 \<le> f n"
hoelzl@50999
   226
  then have f: "(\<lambda>n. \<Sum>i<n. f i) ----> 0"
hoelzl@56213
   227
    using summable_LIMSEQ[of f] by simp
hoelzl@56213
   228
  then have "\<And>i. (\<Sum>n\<in>{i}. f n) \<le> 0"
hoelzl@56213
   229
  proof (rule LIMSEQ_le_const)
hoelzl@50999
   230
    fix i show "\<exists>N. \<forall>n\<ge>N. (\<Sum>n\<in>{i}. f n) \<le> setsum f {..<n}"
hoelzl@50999
   231
      using pos by (intro exI[of _ "Suc i"] allI impI setsum_mono2) auto
hoelzl@50999
   232
  qed
hoelzl@50999
   233
  with pos show "\<forall>n. f n = 0"
hoelzl@50999
   234
    by (auto intro!: antisym)
hoelzl@56193
   235
qed (metis suminf_zero fun_eq_iff)
hoelzl@56193
   236
hoelzl@56213
   237
lemma suminf_pos_iff: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> 0 < suminf f \<longleftrightarrow> (\<exists>i. 0 < f i)"
hoelzl@56213
   238
  using setsum_le_suminf[of 0] suminf_eq_zero_iff by (simp add: less_le)
hoelzl@56193
   239
hoelzl@56193
   240
end
hoelzl@56193
   241
hoelzl@56213
   242
lemma summableI_nonneg_bounded:
hoelzl@56213
   243
  fixes f:: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology, conditionally_complete_linorder}"
hoelzl@56213
   244
  assumes pos[simp]: "\<And>n. 0 \<le> f n" and le: "\<And>n. (\<Sum>i<n. f i) \<le> x"
hoelzl@56213
   245
  shows "summable f"
hoelzl@56213
   246
  unfolding summable_def sums_def[abs_def]
hoelzl@56213
   247
proof (intro exI order_tendstoI)
hoelzl@56213
   248
  have [simp, intro]: "bdd_above (range (\<lambda>n. \<Sum>i<n. f i))"
hoelzl@56213
   249
    using le by (auto simp: bdd_above_def)
hoelzl@56213
   250
  { fix a assume "a < (SUP n. \<Sum>i<n. f i)"
hoelzl@56213
   251
    then obtain n where "a < (\<Sum>i<n. f i)"
hoelzl@56213
   252
      by (auto simp add: less_cSUP_iff)
hoelzl@56213
   253
    then have "\<And>m. n \<le> m \<Longrightarrow> a < (\<Sum>i<m. f i)"
hoelzl@56213
   254
      by (rule less_le_trans) (auto intro!: setsum_mono2)
hoelzl@56213
   255
    then show "eventually (\<lambda>n. a < (\<Sum>i<n. f i)) sequentially"
hoelzl@56213
   256
      by (auto simp: eventually_sequentially) }
hoelzl@56213
   257
  { fix a assume "(SUP n. \<Sum>i<n. f i) < a"
hoelzl@56213
   258
    moreover have "\<And>n. (\<Sum>i<n. f i) \<le> (SUP n. \<Sum>i<n. f i)"
hoelzl@56213
   259
      by (auto intro: cSUP_upper)
hoelzl@56213
   260
    ultimately show "eventually (\<lambda>n. (\<Sum>i<n. f i) < a) sequentially"
hoelzl@56213
   261
      by (auto intro: le_less_trans simp: eventually_sequentially) }
hoelzl@56213
   262
qed
hoelzl@56213
   263
eberlm@61531
   264
wenzelm@60758
   265
subsection \<open>Infinite summability on real normed vector spaces\<close>
hoelzl@56193
   266
hoelzl@56193
   267
lemma sums_Suc_iff:
hoelzl@56193
   268
  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@56193
   269
  shows "(\<lambda>n. f (Suc n)) sums s \<longleftrightarrow> f sums (s + f 0)"
hoelzl@56193
   270
proof -
hoelzl@56193
   271
  have "f sums (s + f 0) \<longleftrightarrow> (\<lambda>i. \<Sum>j<Suc i. f j) ----> s + f 0"
hoelzl@56193
   272
    by (subst LIMSEQ_Suc_iff) (simp add: sums_def)
hoelzl@56193
   273
  also have "\<dots> \<longleftrightarrow> (\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) ----> s + f 0"
haftmann@57418
   274
    by (simp add: ac_simps setsum.reindex image_iff lessThan_Suc_eq_insert_0)
hoelzl@56193
   275
  also have "\<dots> \<longleftrightarrow> (\<lambda>n. f (Suc n)) sums s"
hoelzl@56193
   276
  proof
hoelzl@56193
   277
    assume "(\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) ----> s + f 0"
hoelzl@56193
   278
    with tendsto_add[OF this tendsto_const, of "- f 0"]
hoelzl@56193
   279
    show "(\<lambda>i. f (Suc i)) sums s"
hoelzl@56193
   280
      by (simp add: sums_def)
hoelzl@58729
   281
  qed (auto intro: tendsto_add simp: sums_def)
hoelzl@56193
   282
  finally show ?thesis ..
hoelzl@50999
   283
qed
hoelzl@50999
   284
eberlm@61531
   285
lemma summable_Suc_iff: "summable (\<lambda>n. f (Suc n) :: 'a :: real_normed_vector) = summable f"
eberlm@61531
   286
proof
eberlm@61531
   287
  assume "summable f"
eberlm@61531
   288
  hence "f sums suminf f" by (rule summable_sums)
eberlm@61531
   289
  hence "(\<lambda>n. f (Suc n)) sums (suminf f - f 0)" by (simp add: sums_Suc_iff)
eberlm@61531
   290
  thus "summable (\<lambda>n. f (Suc n))" unfolding summable_def by blast
eberlm@61531
   291
qed (auto simp: sums_Suc_iff summable_def)
eberlm@61531
   292
hoelzl@56193
   293
context
hoelzl@56193
   294
  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@56193
   295
begin
hoelzl@56193
   296
hoelzl@56193
   297
lemma sums_add: "f sums a \<Longrightarrow> g sums b \<Longrightarrow> (\<lambda>n. f n + g n) sums (a + b)"
haftmann@57418
   298
  unfolding sums_def by (simp add: setsum.distrib tendsto_add)
hoelzl@56193
   299
hoelzl@56193
   300
lemma summable_add: "summable f \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. f n + g n)"
hoelzl@56193
   301
  unfolding summable_def by (auto intro: sums_add)
hoelzl@56193
   302
hoelzl@56193
   303
lemma suminf_add: "summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f + suminf g = (\<Sum>n. f n + g n)"
hoelzl@56193
   304
  by (intro sums_unique sums_add summable_sums)
hoelzl@56193
   305
hoelzl@56193
   306
lemma sums_diff: "f sums a \<Longrightarrow> g sums b \<Longrightarrow> (\<lambda>n. f n - g n) sums (a - b)"
hoelzl@56193
   307
  unfolding sums_def by (simp add: setsum_subtractf tendsto_diff)
hoelzl@56193
   308
hoelzl@56193
   309
lemma summable_diff: "summable f \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. f n - g n)"
hoelzl@56193
   310
  unfolding summable_def by (auto intro: sums_diff)
hoelzl@56193
   311
hoelzl@56193
   312
lemma suminf_diff: "summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f - suminf g = (\<Sum>n. f n - g n)"
hoelzl@56193
   313
  by (intro sums_unique sums_diff summable_sums)
hoelzl@56193
   314
hoelzl@56193
   315
lemma sums_minus: "f sums a \<Longrightarrow> (\<lambda>n. - f n) sums (- a)"
hoelzl@56193
   316
  unfolding sums_def by (simp add: setsum_negf tendsto_minus)
hoelzl@56193
   317
hoelzl@56193
   318
lemma summable_minus: "summable f \<Longrightarrow> summable (\<lambda>n. - f n)"
hoelzl@56193
   319
  unfolding summable_def by (auto intro: sums_minus)
huffman@20692
   320
hoelzl@56193
   321
lemma suminf_minus: "summable f \<Longrightarrow> (\<Sum>n. - f n) = - (\<Sum>n. f n)"
hoelzl@56193
   322
  by (intro sums_unique [symmetric] sums_minus summable_sums)
hoelzl@56193
   323
hoelzl@56193
   324
lemma sums_Suc: "(\<lambda> n. f (Suc n)) sums l \<Longrightarrow> f sums (l + f 0)"
hoelzl@56193
   325
  by (simp add: sums_Suc_iff)
hoelzl@56193
   326
hoelzl@56193
   327
lemma sums_iff_shift: "(\<lambda>i. f (i + n)) sums s \<longleftrightarrow> f sums (s + (\<Sum>i<n. f i))"
hoelzl@56193
   328
proof (induct n arbitrary: s)
hoelzl@56193
   329
  case (Suc n)
hoelzl@56193
   330
  moreover have "(\<lambda>i. f (Suc i + n)) sums s \<longleftrightarrow> (\<lambda>i. f (i + n)) sums (s + f n)"
hoelzl@56193
   331
    by (subst sums_Suc_iff) simp
hoelzl@56193
   332
  ultimately show ?case
hoelzl@56193
   333
    by (simp add: ac_simps)
hoelzl@56193
   334
qed simp
huffman@20692
   335
hoelzl@56193
   336
lemma summable_iff_shift: "summable (\<lambda>n. f (n + k)) \<longleftrightarrow> summable f"
hoelzl@56193
   337
  by (metis diff_add_cancel summable_def sums_iff_shift[abs_def])
hoelzl@56193
   338
hoelzl@56193
   339
lemma sums_split_initial_segment: "f sums s \<Longrightarrow> (\<lambda>i. f (i + n)) sums (s - (\<Sum>i<n. f i))"
hoelzl@56193
   340
  by (simp add: sums_iff_shift)
hoelzl@56193
   341
hoelzl@56193
   342
lemma summable_ignore_initial_segment: "summable f \<Longrightarrow> summable (\<lambda>n. f(n + k))"
hoelzl@56193
   343
  by (simp add: summable_iff_shift)
hoelzl@56193
   344
hoelzl@56193
   345
lemma suminf_minus_initial_segment: "summable f \<Longrightarrow> (\<Sum>n. f (n + k)) = (\<Sum>n. f n) - (\<Sum>i<k. f i)"
hoelzl@56193
   346
  by (rule sums_unique[symmetric]) (auto simp: sums_iff_shift)
hoelzl@56193
   347
hoelzl@56193
   348
lemma suminf_split_initial_segment: "summable f \<Longrightarrow> suminf f = (\<Sum>n. f(n + k)) + (\<Sum>i<k. f i)"
hoelzl@56193
   349
  by (auto simp add: suminf_minus_initial_segment)
huffman@20692
   350
eberlm@61531
   351
lemma suminf_split_head: "summable f \<Longrightarrow> (\<Sum>n. f (Suc n)) = suminf f - f 0"
eberlm@61531
   352
  using suminf_split_initial_segment[of 1] by simp
eberlm@61531
   353
lp15@61609
   354
lemma suminf_exist_split:
hoelzl@56193
   355
  fixes r :: real assumes "0 < r" and "summable f"
hoelzl@56193
   356
  shows "\<exists>N. \<forall>n\<ge>N. norm (\<Sum>i. f (i + n)) < r"
hoelzl@56193
   357
proof -
wenzelm@60758
   358
  from LIMSEQ_D[OF summable_LIMSEQ[OF \<open>summable f\<close>] \<open>0 < r\<close>]
hoelzl@56193
   359
  obtain N :: nat where "\<forall> n \<ge> N. norm (setsum f {..<n} - suminf f) < r" by auto
hoelzl@56193
   360
  thus ?thesis
wenzelm@60758
   361
    by (auto simp: norm_minus_commute suminf_minus_initial_segment[OF \<open>summable f\<close>])
hoelzl@56193
   362
qed
hoelzl@56193
   363
hoelzl@56193
   364
lemma summable_LIMSEQ_zero: "summable f \<Longrightarrow> f ----> 0"
hoelzl@56193
   365
  apply (drule summable_iff_convergent [THEN iffD1])
hoelzl@56193
   366
  apply (drule convergent_Cauchy)
hoelzl@56193
   367
  apply (simp only: Cauchy_iff LIMSEQ_iff, safe)
hoelzl@56193
   368
  apply (drule_tac x="r" in spec, safe)
hoelzl@56193
   369
  apply (rule_tac x="M" in exI, safe)
hoelzl@56193
   370
  apply (drule_tac x="Suc n" in spec, simp)
hoelzl@56193
   371
  apply (drule_tac x="n" in spec, simp)
hoelzl@56193
   372
  done
hoelzl@56193
   373
eberlm@61531
   374
lemma summable_imp_convergent:
eberlm@61531
   375
  "summable (f :: nat \<Rightarrow> 'a) \<Longrightarrow> convergent f"
eberlm@61531
   376
  by (force dest!: summable_LIMSEQ_zero simp: convergent_def)
eberlm@61531
   377
eberlm@61531
   378
lemma summable_imp_Bseq:
eberlm@61531
   379
  "summable f \<Longrightarrow> Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector)"
eberlm@61531
   380
  by (simp add: convergent_imp_Bseq summable_imp_convergent)
eberlm@61531
   381
hoelzl@56193
   382
end
hoelzl@56193
   383
lp15@59613
   384
lemma summable_minus_iff:
lp15@59613
   385
  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
lp15@59613
   386
  shows "summable (\<lambda>n. - f n) \<longleftrightarrow> summable f"
wenzelm@60758
   387
  by (auto dest: summable_minus) --\<open>used two ways, hence must be outside the context above\<close>
lp15@59613
   388
lp15@59613
   389
hoelzl@57025
   390
context
hoelzl@57025
   391
  fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::real_normed_vector" and I :: "'i set"
hoelzl@57025
   392
begin
hoelzl@57025
   393
hoelzl@57025
   394
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)"
hoelzl@57025
   395
  by (induct I rule: infinite_finite_induct) (auto intro!: sums_add)
hoelzl@57025
   396
hoelzl@57025
   397
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)"
hoelzl@57025
   398
  using sums_unique[OF sums_setsum, OF summable_sums] by simp
hoelzl@57025
   399
hoelzl@57025
   400
lemma summable_setsum: "(\<And>i. i \<in> I \<Longrightarrow> summable (f i)) \<Longrightarrow> summable (\<lambda>n. \<Sum>i\<in>I. f i n)"
hoelzl@57025
   401
  using sums_summable[OF sums_setsum[OF summable_sums]] .
hoelzl@57025
   402
hoelzl@57025
   403
end
hoelzl@57025
   404
hoelzl@56193
   405
lemma (in bounded_linear) sums: "(\<lambda>n. X n) sums a \<Longrightarrow> (\<lambda>n. f (X n)) sums (f a)"
hoelzl@56193
   406
  unfolding sums_def by (drule tendsto, simp only: setsum)
hoelzl@56193
   407
hoelzl@56193
   408
lemma (in bounded_linear) summable: "summable (\<lambda>n. X n) \<Longrightarrow> summable (\<lambda>n. f (X n))"
hoelzl@56193
   409
  unfolding summable_def by (auto intro: sums)
hoelzl@56193
   410
hoelzl@56193
   411
lemma (in bounded_linear) suminf: "summable (\<lambda>n. X n) \<Longrightarrow> f (\<Sum>n. X n) = (\<Sum>n. f (X n))"
hoelzl@56193
   412
  by (intro sums_unique sums summable_sums)
hoelzl@56193
   413
hoelzl@56193
   414
lemmas sums_of_real = bounded_linear.sums [OF bounded_linear_of_real]
hoelzl@56193
   415
lemmas summable_of_real = bounded_linear.summable [OF bounded_linear_of_real]
hoelzl@56193
   416
lemmas suminf_of_real = bounded_linear.suminf [OF bounded_linear_of_real]
hoelzl@56193
   417
hoelzl@57275
   418
lemmas sums_scaleR_left = bounded_linear.sums[OF bounded_linear_scaleR_left]
hoelzl@57275
   419
lemmas summable_scaleR_left = bounded_linear.summable[OF bounded_linear_scaleR_left]
hoelzl@57275
   420
lemmas suminf_scaleR_left = bounded_linear.suminf[OF bounded_linear_scaleR_left]
hoelzl@57275
   421
hoelzl@57275
   422
lemmas sums_scaleR_right = bounded_linear.sums[OF bounded_linear_scaleR_right]
hoelzl@57275
   423
lemmas summable_scaleR_right = bounded_linear.summable[OF bounded_linear_scaleR_right]
hoelzl@57275
   424
lemmas suminf_scaleR_right = bounded_linear.suminf[OF bounded_linear_scaleR_right]
hoelzl@57275
   425
eberlm@61531
   426
lemma summable_const_iff: "summable (\<lambda>_. c) \<longleftrightarrow> (c :: 'a :: real_normed_vector) = 0"
eberlm@61531
   427
proof -
eberlm@61531
   428
  {
eberlm@61531
   429
    assume "c \<noteq> 0"
eberlm@61531
   430
    hence "filterlim (\<lambda>n. of_nat n * norm c) at_top sequentially"
eberlm@61531
   431
      by (subst mult.commute)
eberlm@61531
   432
         (auto intro!: filterlim_tendsto_pos_mult_at_top filterlim_real_sequentially)
eberlm@61531
   433
    hence "\<not>convergent (\<lambda>n. norm (\<Sum>k<n. c))"
lp15@61609
   434
      by (intro filterlim_at_infinity_imp_not_convergent filterlim_at_top_imp_at_infinity)
eberlm@61531
   435
         (simp_all add: setsum_constant_scaleR)
eberlm@61531
   436
    hence "\<not>summable (\<lambda>_. c)" unfolding summable_iff_convergent using convergent_norm by blast
eberlm@61531
   437
  }
eberlm@61531
   438
  thus ?thesis by auto
eberlm@61531
   439
qed
eberlm@61531
   440
eberlm@61531
   441
wenzelm@60758
   442
subsection \<open>Infinite summability on real normed algebras\<close>
hoelzl@56213
   443
hoelzl@56193
   444
context
hoelzl@56193
   445
  fixes f :: "nat \<Rightarrow> 'a::real_normed_algebra"
hoelzl@56193
   446
begin
hoelzl@56193
   447
hoelzl@56193
   448
lemma sums_mult: "f sums a \<Longrightarrow> (\<lambda>n. c * f n) sums (c * a)"
hoelzl@56193
   449
  by (rule bounded_linear.sums [OF bounded_linear_mult_right])
hoelzl@56193
   450
hoelzl@56193
   451
lemma summable_mult: "summable f \<Longrightarrow> summable (\<lambda>n. c * f n)"
hoelzl@56193
   452
  by (rule bounded_linear.summable [OF bounded_linear_mult_right])
hoelzl@56193
   453
hoelzl@56193
   454
lemma suminf_mult: "summable f \<Longrightarrow> suminf (\<lambda>n. c * f n) = c * suminf f"
hoelzl@56193
   455
  by (rule bounded_linear.suminf [OF bounded_linear_mult_right, symmetric])
hoelzl@56193
   456
hoelzl@56193
   457
lemma sums_mult2: "f sums a \<Longrightarrow> (\<lambda>n. f n * c) sums (a * c)"
hoelzl@56193
   458
  by (rule bounded_linear.sums [OF bounded_linear_mult_left])
hoelzl@56193
   459
hoelzl@56193
   460
lemma summable_mult2: "summable f \<Longrightarrow> summable (\<lambda>n. f n * c)"
hoelzl@56193
   461
  by (rule bounded_linear.summable [OF bounded_linear_mult_left])
hoelzl@56193
   462
hoelzl@56193
   463
lemma suminf_mult2: "summable f \<Longrightarrow> suminf f * c = (\<Sum>n. f n * c)"
hoelzl@56193
   464
  by (rule bounded_linear.suminf [OF bounded_linear_mult_left])
hoelzl@56193
   465
hoelzl@56193
   466
end
hoelzl@56193
   467
eberlm@61531
   468
lemma sums_mult_iff:
eberlm@61531
   469
  assumes "c \<noteq> 0"
eberlm@61531
   470
  shows   "(\<lambda>n. c * f n :: 'a :: {real_normed_algebra,field}) sums (c * d) \<longleftrightarrow> f sums d"
eberlm@61531
   471
  using sums_mult[of f d c] sums_mult[of "\<lambda>n. c * f n" "c * d" "inverse c"]
eberlm@61531
   472
  by (force simp: field_simps assms)
eberlm@61531
   473
eberlm@61531
   474
lemma sums_mult2_iff:
eberlm@61531
   475
  assumes "c \<noteq> (0 :: 'a :: {real_normed_algebra, field})"
eberlm@61531
   476
  shows   "(\<lambda>n. f n * c) sums (d * c) \<longleftrightarrow> f sums d"
eberlm@61531
   477
  using sums_mult_iff[OF assms, of f d] by (simp add: mult.commute)
eberlm@61531
   478
eberlm@61531
   479
lemma sums_of_real_iff:
eberlm@61531
   480
  "(\<lambda>n. of_real (f n) :: 'a :: real_normed_div_algebra) sums of_real c \<longleftrightarrow> f sums c"
eberlm@61531
   481
  by (simp add: sums_def of_real_setsum[symmetric] tendsto_of_real_iff del: of_real_setsum)
eberlm@61531
   482
eberlm@61531
   483
wenzelm@60758
   484
subsection \<open>Infinite summability on real normed fields\<close>
hoelzl@56213
   485
hoelzl@56193
   486
context
hoelzl@56193
   487
  fixes c :: "'a::real_normed_field"
hoelzl@56193
   488
begin
hoelzl@56193
   489
hoelzl@56193
   490
lemma sums_divide: "f sums a \<Longrightarrow> (\<lambda>n. f n / c) sums (a / c)"
hoelzl@56193
   491
  by (rule bounded_linear.sums [OF bounded_linear_divide])
hoelzl@56193
   492
hoelzl@56193
   493
lemma summable_divide: "summable f \<Longrightarrow> summable (\<lambda>n. f n / c)"
hoelzl@56193
   494
  by (rule bounded_linear.summable [OF bounded_linear_divide])
hoelzl@56193
   495
hoelzl@56193
   496
lemma suminf_divide: "summable f \<Longrightarrow> suminf (\<lambda>n. f n / c) = suminf f / c"
hoelzl@56193
   497
  by (rule bounded_linear.suminf [OF bounded_linear_divide, symmetric])
paulson@14416
   498
wenzelm@60758
   499
text\<open>Sum of a geometric progression.\<close>
paulson@14416
   500
hoelzl@56193
   501
lemma geometric_sums: "norm c < 1 \<Longrightarrow> (\<lambda>n. c^n) sums (1 / (1 - c))"
huffman@20692
   502
proof -
hoelzl@56193
   503
  assume less_1: "norm c < 1"
hoelzl@56193
   504
  hence neq_1: "c \<noteq> 1" by auto
hoelzl@56193
   505
  hence neq_0: "c - 1 \<noteq> 0" by simp
hoelzl@56193
   506
  from less_1 have lim_0: "(\<lambda>n. c^n) ----> 0"
huffman@20692
   507
    by (rule LIMSEQ_power_zero)
hoelzl@56193
   508
  hence "(\<lambda>n. c ^ n / (c - 1) - 1 / (c - 1)) ----> 0 / (c - 1) - 1 / (c - 1)"
huffman@44568
   509
    using neq_0 by (intro tendsto_intros)
hoelzl@56193
   510
  hence "(\<lambda>n. (c ^ n - 1) / (c - 1)) ----> 1 / (1 - c)"
huffman@20692
   511
    by (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib)
hoelzl@56193
   512
  thus "(\<lambda>n. c ^ n) sums (1 / (1 - c))"
huffman@20692
   513
    by (simp add: sums_def geometric_sum neq_1)
huffman@20692
   514
qed
huffman@20692
   515
hoelzl@56193
   516
lemma summable_geometric: "norm c < 1 \<Longrightarrow> summable (\<lambda>n. c^n)"
hoelzl@56193
   517
  by (rule geometric_sums [THEN sums_summable])
paulson@14416
   518
hoelzl@56193
   519
lemma suminf_geometric: "norm c < 1 \<Longrightarrow> suminf (\<lambda>n. c^n) = 1 / (1 - c)"
hoelzl@56193
   520
  by (rule sums_unique[symmetric]) (rule geometric_sums)
hoelzl@56193
   521
eberlm@61531
   522
lemma summable_geometric_iff: "summable (\<lambda>n. c ^ n) \<longleftrightarrow> norm c < 1"
eberlm@61531
   523
proof
eberlm@61531
   524
  assume "summable (\<lambda>n. c ^ n :: 'a :: real_normed_field)"
eberlm@61531
   525
  hence "(\<lambda>n. norm c ^ n) ----> 0"
eberlm@61531
   526
    by (simp add: norm_power [symmetric] tendsto_norm_zero_iff summable_LIMSEQ_zero)
eberlm@61531
   527
  from order_tendstoD(2)[OF this zero_less_one] obtain n where "norm c ^ n < 1"
eberlm@61531
   528
    by (auto simp: eventually_at_top_linorder)
eberlm@61531
   529
  thus "norm c < 1" using one_le_power[of "norm c" n] by (cases "norm c \<ge> 1") (linarith, simp)
eberlm@61531
   530
qed (rule summable_geometric)
lp15@61609
   531
hoelzl@56193
   532
end
paulson@33271
   533
paulson@33271
   534
lemma power_half_series: "(\<lambda>n. (1/2::real)^Suc n) sums 1"
paulson@33271
   535
proof -
paulson@33271
   536
  have 2: "(\<lambda>n. (1/2::real)^n) sums 2" using geometric_sums [of "1/2::real"]
paulson@33271
   537
    by auto
paulson@33271
   538
  have "(\<lambda>n. (1/2::real)^Suc n) = (\<lambda>n. (1 / 2) ^ n / 2)"
lp15@59741
   539
    by (simp add: mult.commute)
huffman@44282
   540
  thus ?thesis using sums_divide [OF 2, of 2]
paulson@33271
   541
    by simp
paulson@33271
   542
qed
paulson@33271
   543
eberlm@61531
   544
eberlm@61531
   545
subsection \<open>Telescoping\<close>
eberlm@61531
   546
eberlm@61531
   547
lemma telescope_sums:
eberlm@61531
   548
  assumes "f ----> (c :: 'a :: real_normed_vector)"
eberlm@61531
   549
  shows   "(\<lambda>n. f (Suc n) - f n) sums (c - f 0)"
eberlm@61531
   550
  unfolding sums_def
eberlm@61531
   551
proof (subst LIMSEQ_Suc_iff [symmetric])
eberlm@61531
   552
  have "(\<lambda>n. \<Sum>k<Suc n. f (Suc k) - f k) = (\<lambda>n. f (Suc n) - f 0)"
eberlm@61531
   553
    by (simp add: lessThan_Suc_atMost atLeast0AtMost [symmetric] setsum_Suc_diff)
eberlm@61531
   554
  also have "\<dots> ----> c - f 0" by (intro tendsto_diff LIMSEQ_Suc[OF assms] tendsto_const)
eberlm@61531
   555
  finally show "(\<lambda>n. \<Sum>n<Suc n. f (Suc n) - f n) ----> c - f 0" .
eberlm@61531
   556
qed
eberlm@61531
   557
eberlm@61531
   558
lemma telescope_sums':
eberlm@61531
   559
  assumes "f ----> (c :: 'a :: real_normed_vector)"
eberlm@61531
   560
  shows   "(\<lambda>n. f n - f (Suc n)) sums (f 0 - c)"
eberlm@61531
   561
  using sums_minus[OF telescope_sums[OF assms]] by (simp add: algebra_simps)
eberlm@61531
   562
eberlm@61531
   563
lemma telescope_summable:
eberlm@61531
   564
  assumes "f ----> (c :: 'a :: real_normed_vector)"
eberlm@61531
   565
  shows   "summable (\<lambda>n. f (Suc n) - f n)"
eberlm@61531
   566
  using telescope_sums[OF assms] by (simp add: sums_iff)
eberlm@61531
   567
eberlm@61531
   568
lemma telescope_summable':
eberlm@61531
   569
  assumes "f ----> (c :: 'a :: real_normed_vector)"
eberlm@61531
   570
  shows   "summable (\<lambda>n. f n - f (Suc n))"
eberlm@61531
   571
  using summable_minus[OF telescope_summable[OF assms]] by (simp add: algebra_simps)
eberlm@61531
   572
eberlm@61531
   573
wenzelm@60758
   574
subsection \<open>Infinite summability on Banach spaces\<close>
hoelzl@56213
   575
wenzelm@60758
   576
text\<open>Cauchy-type criterion for convergence of series (c.f. Harrison)\<close>
paulson@15085
   577
hoelzl@56193
   578
lemma summable_Cauchy:
hoelzl@56193
   579
  fixes f :: "nat \<Rightarrow> 'a::banach"
hoelzl@56193
   580
  shows "summable f \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n. norm (setsum f {m..<n}) < e)"
hoelzl@56193
   581
  apply (simp only: summable_iff_convergent Cauchy_convergent_iff [symmetric] Cauchy_iff, safe)
hoelzl@56193
   582
  apply (drule spec, drule (1) mp)
hoelzl@56193
   583
  apply (erule exE, rule_tac x="M" in exI, clarify)
hoelzl@56193
   584
  apply (rule_tac x="m" and y="n" in linorder_le_cases)
hoelzl@56193
   585
  apply (frule (1) order_trans)
hoelzl@56193
   586
  apply (drule_tac x="n" in spec, drule (1) mp)
hoelzl@56193
   587
  apply (drule_tac x="m" in spec, drule (1) mp)
hoelzl@56193
   588
  apply (simp_all add: setsum_diff [symmetric])
hoelzl@56193
   589
  apply (drule spec, drule (1) mp)
hoelzl@56193
   590
  apply (erule exE, rule_tac x="N" in exI, clarify)
hoelzl@56193
   591
  apply (rule_tac x="m" and y="n" in linorder_le_cases)
hoelzl@56193
   592
  apply (subst norm_minus_commute)
hoelzl@56193
   593
  apply (simp_all add: setsum_diff [symmetric])
hoelzl@56193
   594
  done
paulson@14416
   595
hoelzl@56193
   596
context
hoelzl@56193
   597
  fixes f :: "nat \<Rightarrow> 'a::banach"
eberlm@61531
   598
begin
hoelzl@56193
   599
wenzelm@60758
   600
text\<open>Absolute convergence imples normal convergence\<close>
huffman@20689
   601
hoelzl@56194
   602
lemma summable_norm_cancel: "summable (\<lambda>n. norm (f n)) \<Longrightarrow> summable f"
hoelzl@56193
   603
  apply (simp only: summable_Cauchy, safe)
hoelzl@56193
   604
  apply (drule_tac x="e" in spec, safe)
hoelzl@56193
   605
  apply (rule_tac x="N" in exI, safe)
hoelzl@56193
   606
  apply (drule_tac x="m" in spec, safe)
hoelzl@56193
   607
  apply (rule order_le_less_trans [OF norm_setsum])
hoelzl@56193
   608
  apply (rule order_le_less_trans [OF abs_ge_self])
hoelzl@56193
   609
  apply simp
hoelzl@50999
   610
  done
paulson@32707
   611
hoelzl@56193
   612
lemma summable_norm: "summable (\<lambda>n. norm (f n)) \<Longrightarrow> norm (suminf f) \<le> (\<Sum>n. norm (f n))"
hoelzl@56193
   613
  by (auto intro: LIMSEQ_le tendsto_norm summable_norm_cancel summable_LIMSEQ norm_setsum)
hoelzl@56193
   614
wenzelm@60758
   615
text \<open>Comparison tests\<close>
paulson@14416
   616
hoelzl@56194
   617
lemma summable_comparison_test: "\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n \<Longrightarrow> summable g \<Longrightarrow> summable f"
hoelzl@56193
   618
  apply (simp add: summable_Cauchy, safe)
hoelzl@56193
   619
  apply (drule_tac x="e" in spec, safe)
hoelzl@56193
   620
  apply (rule_tac x = "N + Na" in exI, safe)
hoelzl@56193
   621
  apply (rotate_tac 2)
hoelzl@56193
   622
  apply (drule_tac x = m in spec)
hoelzl@56193
   623
  apply (auto, rotate_tac 2, drule_tac x = n in spec)
hoelzl@56193
   624
  apply (rule_tac y = "\<Sum>k=m..<n. norm (f k)" in order_le_less_trans)
hoelzl@56193
   625
  apply (rule norm_setsum)
hoelzl@56193
   626
  apply (rule_tac y = "setsum g {m..<n}" in order_le_less_trans)
hoelzl@56193
   627
  apply (auto intro: setsum_mono simp add: abs_less_iff)
hoelzl@56193
   628
  done
hoelzl@56193
   629
eberlm@61531
   630
lemma summable_comparison_test_ev:
eberlm@61531
   631
  shows "eventually (\<lambda>n. norm (f n) \<le> g n) sequentially \<Longrightarrow> summable g \<Longrightarrow> summable f"
eberlm@61531
   632
  by (rule summable_comparison_test) (auto simp: eventually_at_top_linorder)
eberlm@61531
   633
lp15@56217
   634
(*A better argument order*)
lp15@56217
   635
lemma summable_comparison_test': "summable g \<Longrightarrow> (\<And>n. n \<ge> N \<Longrightarrow> norm(f n) \<le> g n) \<Longrightarrow> summable f"
hoelzl@56369
   636
  by (rule summable_comparison_test) auto
lp15@56217
   637
wenzelm@60758
   638
subsection \<open>The Ratio Test\<close>
paulson@15085
   639
lp15@61609
   640
lemma summable_ratio_test:
hoelzl@56193
   641
  assumes "c < 1" "\<And>n. n \<ge> N \<Longrightarrow> norm (f (Suc n)) \<le> c * norm (f n)"
hoelzl@56193
   642
  shows "summable f"
hoelzl@56193
   643
proof cases
hoelzl@56193
   644
  assume "0 < c"
hoelzl@56193
   645
  show "summable f"
hoelzl@56193
   646
  proof (rule summable_comparison_test)
hoelzl@56193
   647
    show "\<exists>N'. \<forall>n\<ge>N'. norm (f n) \<le> (norm (f N) / (c ^ N)) * c ^ n"
hoelzl@56193
   648
    proof (intro exI allI impI)
hoelzl@56193
   649
      fix n assume "N \<le> n" then show "norm (f n) \<le> (norm (f N) / (c ^ N)) * c ^ n"
hoelzl@56193
   650
      proof (induct rule: inc_induct)
hoelzl@56193
   651
        case (step m)
hoelzl@56193
   652
        moreover have "norm (f (Suc m)) / c ^ Suc m * c ^ n \<le> norm (f m) / c ^ m * c ^ n"
wenzelm@60758
   653
          using \<open>0 < c\<close> \<open>c < 1\<close> assms(2)[OF \<open>N \<le> m\<close>] by (simp add: field_simps)
hoelzl@56193
   654
        ultimately show ?case by simp
wenzelm@60758
   655
      qed (insert \<open>0 < c\<close>, simp)
hoelzl@56193
   656
    qed
hoelzl@56193
   657
    show "summable (\<lambda>n. norm (f N) / c ^ N * c ^ n)"
wenzelm@60758
   658
      using \<open>0 < c\<close> \<open>c < 1\<close> by (intro summable_mult summable_geometric) simp
hoelzl@56193
   659
  qed
hoelzl@56193
   660
next
hoelzl@56193
   661
  assume c: "\<not> 0 < c"
hoelzl@56193
   662
  { fix n assume "n \<ge> N"
hoelzl@56193
   663
    then have "norm (f (Suc n)) \<le> c * norm (f n)"
hoelzl@56193
   664
      by fact
hoelzl@56193
   665
    also have "\<dots> \<le> 0"
hoelzl@56193
   666
      using c by (simp add: not_less mult_nonpos_nonneg)
hoelzl@56193
   667
    finally have "f (Suc n) = 0"
hoelzl@56193
   668
      by auto }
hoelzl@56193
   669
  then show "summable f"
hoelzl@56194
   670
    by (intro sums_summable[OF sums_finite, of "{.. Suc N}"]) (auto simp: not_le Suc_less_eq2)
lp15@56178
   671
qed
lp15@56178
   672
hoelzl@56193
   673
end
paulson@14416
   674
wenzelm@60758
   675
text\<open>Relations among convergence and absolute convergence for power series.\<close>
hoelzl@56369
   676
hoelzl@56369
   677
lemma abel_lemma:
hoelzl@56369
   678
  fixes a :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@56369
   679
  assumes r: "0 \<le> r" and r0: "r < r0" and M: "\<And>n. norm (a n) * r0^n \<le> M"
hoelzl@56369
   680
    shows "summable (\<lambda>n. norm (a n) * r^n)"
hoelzl@56369
   681
proof (rule summable_comparison_test')
hoelzl@56369
   682
  show "summable (\<lambda>n. M * (r / r0) ^ n)"
lp15@61609
   683
    using assms
hoelzl@56369
   684
    by (auto simp add: summable_mult summable_geometric)
hoelzl@56369
   685
next
hoelzl@56369
   686
  fix n
hoelzl@56369
   687
  show "norm (norm (a n) * r ^ n) \<le> M * (r / r0) ^ n"
hoelzl@56369
   688
    using r r0 M [of n]
haftmann@60867
   689
    apply (auto simp add: abs_mult field_simps)
hoelzl@56369
   690
    apply (cases "r=0", simp)
hoelzl@56369
   691
    apply (cases n, auto)
hoelzl@56369
   692
    done
hoelzl@56369
   693
qed
hoelzl@56369
   694
hoelzl@56369
   695
wenzelm@60758
   696
text\<open>Summability of geometric series for real algebras\<close>
huffman@23084
   697
huffman@23084
   698
lemma complete_algebra_summable_geometric:
haftmann@31017
   699
  fixes x :: "'a::{real_normed_algebra_1,banach}"
huffman@23084
   700
  shows "norm x < 1 \<Longrightarrow> summable (\<lambda>n. x ^ n)"
huffman@23084
   701
proof (rule summable_comparison_test)
huffman@23084
   702
  show "\<exists>N. \<forall>n\<ge>N. norm (x ^ n) \<le> norm x ^ n"
huffman@23084
   703
    by (simp add: norm_power_ineq)
huffman@23084
   704
  show "norm x < 1 \<Longrightarrow> summable (\<lambda>n. norm x ^ n)"
huffman@23084
   705
    by (simp add: summable_geometric)
huffman@23084
   706
qed
huffman@23084
   707
wenzelm@60758
   708
subsection \<open>Cauchy Product Formula\<close>
huffman@23111
   709
wenzelm@60758
   710
text \<open>
wenzelm@54703
   711
  Proof based on Analysis WebNotes: Chapter 07, Class 41
wenzelm@54703
   712
  @{url "http://www.math.unl.edu/~webnotes/classes/class41/prp77.htm"}
wenzelm@60758
   713
\<close>
huffman@23111
   714
huffman@23111
   715
lemma Cauchy_product_sums:
huffman@23111
   716
  fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}"
huffman@23111
   717
  assumes a: "summable (\<lambda>k. norm (a k))"
huffman@23111
   718
  assumes b: "summable (\<lambda>k. norm (b k))"
hoelzl@56213
   719
  shows "(\<lambda>k. \<Sum>i\<le>k. a i * b (k - i)) sums ((\<Sum>k. a k) * (\<Sum>k. b k))"
huffman@23111
   720
proof -
hoelzl@56193
   721
  let ?S1 = "\<lambda>n::nat. {..<n} \<times> {..<n}"
huffman@23111
   722
  let ?S2 = "\<lambda>n::nat. {(i,j). i + j < n}"
huffman@23111
   723
  have S1_mono: "\<And>m n. m \<le> n \<Longrightarrow> ?S1 m \<subseteq> ?S1 n" by auto
huffman@23111
   724
  have S2_le_S1: "\<And>n. ?S2 n \<subseteq> ?S1 n" by auto
huffman@23111
   725
  have S1_le_S2: "\<And>n. ?S1 (n div 2) \<subseteq> ?S2 n" by auto
huffman@23111
   726
  have finite_S1: "\<And>n. finite (?S1 n)" by simp
huffman@23111
   727
  with S2_le_S1 have finite_S2: "\<And>n. finite (?S2 n)" by (rule finite_subset)
huffman@23111
   728
huffman@23111
   729
  let ?g = "\<lambda>(i,j). a i * b j"
huffman@23111
   730
  let ?f = "\<lambda>(i,j). norm (a i) * norm (b j)"
nipkow@56536
   731
  have f_nonneg: "\<And>x. 0 \<le> ?f x" by (auto)
huffman@23111
   732
  hence norm_setsum_f: "\<And>A. norm (setsum ?f A) = setsum ?f A"
huffman@23111
   733
    unfolding real_norm_def
huffman@23111
   734
    by (simp only: abs_of_nonneg setsum_nonneg [rule_format])
huffman@23111
   735
hoelzl@56193
   736
  have "(\<lambda>n. (\<Sum>k<n. a k) * (\<Sum>k<n. b k)) ----> (\<Sum>k. a k) * (\<Sum>k. b k)"
hoelzl@56193
   737
    by (intro tendsto_mult summable_LIMSEQ summable_norm_cancel [OF a] summable_norm_cancel [OF b])
huffman@23111
   738
  hence 1: "(\<lambda>n. setsum ?g (?S1 n)) ----> (\<Sum>k. a k) * (\<Sum>k. b k)"
haftmann@57418
   739
    by (simp only: setsum_product setsum.Sigma [rule_format] finite_lessThan)
huffman@23111
   740
hoelzl@56193
   741
  have "(\<lambda>n. (\<Sum>k<n. norm (a k)) * (\<Sum>k<n. norm (b k))) ----> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
hoelzl@56193
   742
    using a b by (intro tendsto_mult summable_LIMSEQ)
huffman@23111
   743
  hence "(\<lambda>n. setsum ?f (?S1 n)) ----> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
haftmann@57418
   744
    by (simp only: setsum_product setsum.Sigma [rule_format] finite_lessThan)
huffman@23111
   745
  hence "convergent (\<lambda>n. setsum ?f (?S1 n))"
huffman@23111
   746
    by (rule convergentI)
huffman@23111
   747
  hence Cauchy: "Cauchy (\<lambda>n. setsum ?f (?S1 n))"
huffman@23111
   748
    by (rule convergent_Cauchy)
huffman@36657
   749
  have "Zfun (\<lambda>n. setsum ?f (?S1 n - ?S2 n)) sequentially"
huffman@36657
   750
  proof (rule ZfunI, simp only: eventually_sequentially norm_setsum_f)
huffman@23111
   751
    fix r :: real
huffman@23111
   752
    assume r: "0 < r"
huffman@23111
   753
    from CauchyD [OF Cauchy r] obtain N
huffman@23111
   754
    where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (setsum ?f (?S1 m) - setsum ?f (?S1 n)) < r" ..
huffman@23111
   755
    hence "\<And>m n. \<lbrakk>N \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> norm (setsum ?f (?S1 m - ?S1 n)) < r"
huffman@23111
   756
      by (simp only: setsum_diff finite_S1 S1_mono)
huffman@23111
   757
    hence N: "\<And>m n. \<lbrakk>N \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> setsum ?f (?S1 m - ?S1 n) < r"
huffman@23111
   758
      by (simp only: norm_setsum_f)
huffman@23111
   759
    show "\<exists>N. \<forall>n\<ge>N. setsum ?f (?S1 n - ?S2 n) < r"
huffman@23111
   760
    proof (intro exI allI impI)
huffman@23111
   761
      fix n assume "2 * N \<le> n"
huffman@23111
   762
      hence n: "N \<le> n div 2" by simp
huffman@23111
   763
      have "setsum ?f (?S1 n - ?S2 n) \<le> setsum ?f (?S1 n - ?S1 (n div 2))"
huffman@23111
   764
        by (intro setsum_mono2 finite_Diff finite_S1 f_nonneg
huffman@23111
   765
                  Diff_mono subset_refl S1_le_S2)
huffman@23111
   766
      also have "\<dots> < r"
huffman@23111
   767
        using n div_le_dividend by (rule N)
huffman@23111
   768
      finally show "setsum ?f (?S1 n - ?S2 n) < r" .
huffman@23111
   769
    qed
huffman@23111
   770
  qed
huffman@36657
   771
  hence "Zfun (\<lambda>n. setsum ?g (?S1 n - ?S2 n)) sequentially"
huffman@36657
   772
    apply (rule Zfun_le [rule_format])
huffman@23111
   773
    apply (simp only: norm_setsum_f)
huffman@23111
   774
    apply (rule order_trans [OF norm_setsum setsum_mono])
huffman@23111
   775
    apply (auto simp add: norm_mult_ineq)
huffman@23111
   776
    done
huffman@23111
   777
  hence 2: "(\<lambda>n. setsum ?g (?S1 n) - setsum ?g (?S2 n)) ----> 0"
huffman@36660
   778
    unfolding tendsto_Zfun_iff diff_0_right
huffman@36657
   779
    by (simp only: setsum_diff finite_S1 S2_le_S1)
huffman@23111
   780
huffman@23111
   781
  with 1 have "(\<lambda>n. setsum ?g (?S2 n)) ----> (\<Sum>k. a k) * (\<Sum>k. b k)"
lp15@60141
   782
    by (rule Lim_transform2)
huffman@23111
   783
  thus ?thesis by (simp only: sums_def setsum_triangle_reindex)
huffman@23111
   784
qed
huffman@23111
   785
huffman@23111
   786
lemma Cauchy_product:
huffman@23111
   787
  fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}"
huffman@23111
   788
  assumes a: "summable (\<lambda>k. norm (a k))"
huffman@23111
   789
  assumes b: "summable (\<lambda>k. norm (b k))"
hoelzl@56213
   790
  shows "(\<Sum>k. a k) * (\<Sum>k. b k) = (\<Sum>k. \<Sum>i\<le>k. a i * b (k - i))"
hoelzl@56213
   791
  using a b
hoelzl@56213
   792
  by (rule Cauchy_product_sums [THEN sums_unique])
hoelzl@56213
   793
wenzelm@60758
   794
subsection \<open>Series on @{typ real}s\<close>
hoelzl@56213
   795
hoelzl@56213
   796
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))"
hoelzl@56213
   797
  by (rule summable_comparison_test) auto
hoelzl@56213
   798
hoelzl@56213
   799
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>)"
hoelzl@56213
   800
  by (rule summable_comparison_test) auto
hoelzl@56213
   801
hoelzl@56213
   802
lemma summable_rabs_cancel: "summable (\<lambda>n. \<bar>f n :: real\<bar>) \<Longrightarrow> summable f"
hoelzl@56213
   803
  by (rule summable_norm_cancel) simp
hoelzl@56213
   804
hoelzl@56213
   805
lemma summable_rabs: "summable (\<lambda>n. \<bar>f n :: real\<bar>) \<Longrightarrow> \<bar>suminf f\<bar> \<le> (\<Sum>n. \<bar>f n\<bar>)"
hoelzl@56213
   806
  by (fold real_norm_def) (rule summable_norm)
huffman@23111
   807
eberlm@61531
   808
lemma summable_zero_power [simp]: "summable (\<lambda>n. 0 ^ n :: 'a :: {comm_ring_1,topological_space})"
eberlm@61531
   809
proof -
eberlm@61531
   810
  have "(\<lambda>n. 0 ^ n :: 'a) = (\<lambda>n. if n = 0 then 0^0 else 0)" by (intro ext) (simp add: zero_power)
eberlm@61531
   811
  moreover have "summable \<dots>" by simp
eberlm@61531
   812
  ultimately show ?thesis by simp
eberlm@61531
   813
qed
eberlm@61531
   814
eberlm@61531
   815
lemma summable_zero_power' [simp]: "summable (\<lambda>n. f n * 0 ^ n :: 'a :: {ring_1,topological_space})"
eberlm@61531
   816
proof -
lp15@61609
   817
  have "(\<lambda>n. f n * 0 ^ n :: 'a) = (\<lambda>n. if n = 0 then f 0 * 0^0 else 0)"
eberlm@61531
   818
    by (intro ext) (simp add: zero_power)
eberlm@61531
   819
  moreover have "summable \<dots>" by simp
eberlm@61531
   820
  ultimately show ?thesis by simp
eberlm@61531
   821
qed
eberlm@61531
   822
hoelzl@59000
   823
lemma summable_power_series:
hoelzl@59000
   824
  fixes z :: real
hoelzl@59000
   825
  assumes le_1: "\<And>i. f i \<le> 1" and nonneg: "\<And>i. 0 \<le> f i" and z: "0 \<le> z" "z < 1"
hoelzl@59000
   826
  shows "summable (\<lambda>i. f i * z^i)"
hoelzl@59000
   827
proof (rule summable_comparison_test[OF _ summable_geometric])
hoelzl@59000
   828
  show "norm z < 1" using z by (auto simp: less_imp_le)
hoelzl@59000
   829
  show "\<And>n. \<exists>N. \<forall>na\<ge>N. norm (f na * z ^ na) \<le> z ^ na"
hoelzl@59000
   830
    using z by (auto intro!: exI[of _ 0] mult_left_le_one_le simp: abs_mult nonneg power_abs less_imp_le le_1)
hoelzl@59000
   831
qed
hoelzl@59000
   832
eberlm@61531
   833
lemma summable_0_powser:
eberlm@61531
   834
  "summable (\<lambda>n. f n * 0 ^ n :: 'a :: real_normed_div_algebra)"
eberlm@61531
   835
proof -
eberlm@61531
   836
  have A: "(\<lambda>n. f n * 0 ^ n) = (\<lambda>n. if n = 0 then f n else 0)"
eberlm@61531
   837
    by (intro ext) auto
eberlm@61531
   838
  thus ?thesis by (subst A) simp_all
eberlm@61531
   839
qed
eberlm@61531
   840
eberlm@61531
   841
lemma summable_powser_split_head:
eberlm@61531
   842
  "summable (\<lambda>n. f (Suc n) * z ^ n :: 'a :: real_normed_div_algebra) = summable (\<lambda>n. f n * z ^ n)"
eberlm@61531
   843
proof -
eberlm@61531
   844
  have "summable (\<lambda>n. f (Suc n) * z ^ n) \<longleftrightarrow> summable (\<lambda>n. f (Suc n) * z ^ Suc n)"
eberlm@61531
   845
  proof
eberlm@61531
   846
    assume "summable (\<lambda>n. f (Suc n) * z ^ n)"
lp15@61609
   847
    from summable_mult2[OF this, of z] show "summable (\<lambda>n. f (Suc n) * z ^ Suc n)"
eberlm@61531
   848
      by (simp add: power_commutes algebra_simps)
eberlm@61531
   849
  next
eberlm@61531
   850
    assume "summable (\<lambda>n. f (Suc n) * z ^ Suc n)"
eberlm@61531
   851
    from summable_mult2[OF this, of "inverse z"] show "summable (\<lambda>n. f (Suc n) * z ^ n)"
eberlm@61531
   852
      by (cases "z \<noteq> 0", subst (asm) power_Suc2) (simp_all add: algebra_simps)
eberlm@61531
   853
  qed
eberlm@61531
   854
  also have "\<dots> \<longleftrightarrow> summable (\<lambda>n. f n * z ^ n)" by (rule summable_Suc_iff)
eberlm@61531
   855
  finally show ?thesis .
eberlm@61531
   856
qed
eberlm@61531
   857
eberlm@61531
   858
lemma powser_split_head:
eberlm@61531
   859
  assumes "summable (\<lambda>n. f n * z ^ n :: 'a :: {real_normed_div_algebra,banach})"
eberlm@61531
   860
  shows   "suminf (\<lambda>n. f n * z ^ n) = f 0 + suminf (\<lambda>n. f (Suc n) * z ^ n) * z"
eberlm@61531
   861
          "suminf (\<lambda>n. f (Suc n) * z ^ n) * z = suminf (\<lambda>n. f n * z ^ n) - f 0"
eberlm@61531
   862
          "summable (\<lambda>n. f (Suc n) * z ^ n)"
eberlm@61531
   863
proof -
eberlm@61531
   864
  from assms show "summable (\<lambda>n. f (Suc n) * z ^ n)" by (subst summable_powser_split_head)
eberlm@61531
   865
lp15@61609
   866
  from suminf_mult2[OF this, of z]
eberlm@61531
   867
    have "(\<Sum>n. f (Suc n) * z ^ n) * z = (\<Sum>n. f (Suc n) * z ^ Suc n)"
eberlm@61531
   868
    by (simp add: power_commutes algebra_simps)
eberlm@61531
   869
  also from assms have "\<dots> = suminf (\<lambda>n. f n * z ^ n) - f 0"
eberlm@61531
   870
    by (subst suminf_split_head) simp_all
eberlm@61531
   871
  finally show "suminf (\<lambda>n. f n * z ^ n) = f 0 + suminf (\<lambda>n. f (Suc n) * z ^ n) * z" by simp
eberlm@61531
   872
  thus "suminf (\<lambda>n. f (Suc n) * z ^ n) * z = suminf (\<lambda>n. f n * z ^ n) - f 0" by simp
eberlm@61531
   873
qed
eberlm@61531
   874
eberlm@61531
   875
lemma summable_partial_sum_bound:
eberlm@61531
   876
  fixes f :: "nat \<Rightarrow> 'a :: banach"
eberlm@61531
   877
  assumes summable: "summable f" and e: "e > (0::real)"
eberlm@61531
   878
  obtains N where "\<And>m n. m \<ge> N \<Longrightarrow> norm (\<Sum>k=m..n. f k) < e"
eberlm@61531
   879
proof -
lp15@61609
   880
  from summable have "Cauchy (\<lambda>n. \<Sum>k<n. f k)"
eberlm@61531
   881
    by (simp add: Cauchy_convergent_iff summable_iff_convergent)
lp15@61609
   882
  from CauchyD[OF this e] obtain N
eberlm@61531
   883
    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
eberlm@61531
   884
  {
eberlm@61531
   885
    fix m n :: nat assume m: "m \<ge> N"
eberlm@61531
   886
    have "norm (\<Sum>k=m..n. f k) < e"
eberlm@61531
   887
    proof (cases "n \<ge> m")
eberlm@61531
   888
      assume n: "n \<ge> m"
eberlm@61531
   889
      with m have "norm ((\<Sum>k<Suc n. f k) - (\<Sum>k<m. f k)) < e" by (intro N) simp_all
eberlm@61531
   890
      also from n have "(\<Sum>k<Suc n. f k) - (\<Sum>k<m. f k) = (\<Sum>k=m..n. f k)"
eberlm@61531
   891
        by (subst setsum_diff [symmetric]) (simp_all add: setsum_last_plus)
eberlm@61531
   892
      finally show ?thesis .
eberlm@61531
   893
    qed (insert e, simp_all)
eberlm@61531
   894
  }
eberlm@61531
   895
  thus ?thesis by (rule that)
eberlm@61531
   896
qed
eberlm@61531
   897
lp15@61609
   898
lemma powser_sums_if:
eberlm@61531
   899
  "(\<lambda>n. (if n = m then (1 :: 'a :: {ring_1,topological_space}) else 0) * z^n) sums z^m"
eberlm@61531
   900
proof -
lp15@61609
   901
  have "(\<lambda>n. (if n = m then 1 else 0) * z^n) = (\<lambda>n. if n = m then z^n else 0)"
eberlm@61531
   902
    by (intro ext) auto
eberlm@61531
   903
  thus ?thesis by (simp add: sums_single)
eberlm@61531
   904
qed
eberlm@61531
   905
Andreas@59025
   906
lemma
Andreas@59025
   907
   fixes f :: "nat \<Rightarrow> real"
Andreas@59025
   908
   assumes "summable f"
Andreas@59025
   909
   and "inj g"
Andreas@59025
   910
   and pos: "!!x. 0 \<le> f x"
Andreas@59025
   911
   shows summable_reindex: "summable (f o g)"
Andreas@59025
   912
   and suminf_reindex_mono: "suminf (f o g) \<le> suminf f"
Andreas@59025
   913
   and suminf_reindex: "(\<And>x. x \<notin> range g \<Longrightarrow> f x = 0) \<Longrightarrow> suminf (f \<circ> g) = suminf f"
Andreas@59025
   914
proof -
Andreas@59025
   915
  from \<open>inj g\<close> have [simp]: "\<And>A. inj_on g A" by(rule subset_inj_on) simp
Andreas@59025
   916
Andreas@59025
   917
  have smaller: "\<forall>n. (\<Sum>i<n. (f \<circ> g) i) \<le> suminf f"
Andreas@59025
   918
  proof
Andreas@59025
   919
    fix n
lp15@61609
   920
    have "\<forall> n' \<in> (g ` {..<n}). n' < Suc (Max (g ` {..<n}))"
Andreas@59025
   921
      by(metis Max_ge finite_imageI finite_lessThan not_le not_less_eq)
Andreas@59025
   922
    then obtain m where n: "\<And>n'. n' < n \<Longrightarrow> g n' < m" by blast
Andreas@59025
   923
Andreas@59025
   924
    have "(\<Sum>i<n. f (g i)) = setsum f (g ` {..<n})"
Andreas@59025
   925
      by (simp add: setsum.reindex)
Andreas@59025
   926
    also have "\<dots> \<le> (\<Sum>i<m. f i)"
Andreas@59025
   927
      by (rule setsum_mono3) (auto simp add: pos n[rule_format])
Andreas@59025
   928
    also have "\<dots> \<le> suminf f"
lp15@61609
   929
      using \<open>summable f\<close>
Andreas@59025
   930
      by (rule setsum_le_suminf) (simp add: pos)
Andreas@59025
   931
    finally show "(\<Sum>i<n. (f \<circ>  g) i) \<le> suminf f" by simp
Andreas@59025
   932
  qed
Andreas@59025
   933
Andreas@59025
   934
  have "incseq (\<lambda>n. \<Sum>i<n. (f \<circ> g) i)"
Andreas@59025
   935
    by (rule incseq_SucI) (auto simp add: pos)
Andreas@59025
   936
  then obtain  L where L: "(\<lambda> n. \<Sum>i<n. (f \<circ> g) i) ----> L"
Andreas@59025
   937
    using smaller by(rule incseq_convergent)
Andreas@59025
   938
  hence "(f \<circ> g) sums L" by (simp add: sums_def)
Andreas@59025
   939
  thus "summable (f o g)" by (auto simp add: sums_iff)
Andreas@59025
   940
Andreas@59025
   941
  hence "(\<lambda>n. \<Sum>i<n. (f \<circ> g) i) ----> suminf (f \<circ> g)"
Andreas@59025
   942
    by(rule summable_LIMSEQ)
Andreas@59025
   943
  thus le: "suminf (f \<circ> g) \<le> suminf f"
Andreas@59025
   944
    by(rule LIMSEQ_le_const2)(blast intro: smaller[rule_format])
Andreas@59025
   945
Andreas@59025
   946
  assume f: "\<And>x. x \<notin> range g \<Longrightarrow> f x = 0"
Andreas@59025
   947
Andreas@59025
   948
  from \<open>summable f\<close> have "suminf f \<le> suminf (f \<circ> g)"
Andreas@59025
   949
  proof(rule suminf_le_const)
Andreas@59025
   950
    fix n
Andreas@59025
   951
    have "\<forall> n' \<in> (g -` {..<n}). n' < Suc (Max (g -` {..<n}))"
Andreas@59025
   952
      by(auto intro: Max_ge simp add: finite_vimageI less_Suc_eq_le)
Andreas@59025
   953
    then obtain m where n: "\<And>n'. g n' < n \<Longrightarrow> n' < m" by blast
Andreas@59025
   954
Andreas@59025
   955
    have "(\<Sum>i<n. f i) = (\<Sum>i\<in>{..<n} \<inter> range g. f i)"
Andreas@59025
   956
      using f by(auto intro: setsum.mono_neutral_cong_right)
Andreas@59025
   957
    also have "\<dots> = (\<Sum>i\<in>g -` {..<n}. (f \<circ> g) i)"
Andreas@59025
   958
      by(rule setsum.reindex_cong[where l=g])(auto)
Andreas@59025
   959
    also have "\<dots> \<le> (\<Sum>i<m. (f \<circ> g) i)"
Andreas@59025
   960
      by(rule setsum_mono3)(auto simp add: pos n)
Andreas@59025
   961
    also have "\<dots> \<le> suminf (f \<circ> g)"
Andreas@59025
   962
      using \<open>summable (f o g)\<close>
Andreas@59025
   963
      by(rule setsum_le_suminf)(simp add: pos)
Andreas@59025
   964
    finally show "setsum f {..<n} \<le> suminf (f \<circ> g)" .
Andreas@59025
   965
  qed
Andreas@59025
   966
  with le show "suminf (f \<circ> g) = suminf f" by(rule antisym)
Andreas@59025
   967
qed
Andreas@59025
   968
eberlm@61531
   969
lemma sums_mono_reindex:
eberlm@61531
   970
  assumes subseq: "subseq g" and zero: "\<And>n. n \<notin> range g \<Longrightarrow> f n = 0"
eberlm@61531
   971
  shows   "(\<lambda>n. f (g n)) sums c \<longleftrightarrow> f sums c"
eberlm@61531
   972
unfolding sums_def
eberlm@61531
   973
proof
eberlm@61531
   974
  assume lim: "(\<lambda>n. \<Sum>k<n. f k) ----> c"
eberlm@61531
   975
  have "(\<lambda>n. \<Sum>k<n. f (g k)) = (\<lambda>n. \<Sum>k<g n. f k)"
eberlm@61531
   976
  proof
eberlm@61531
   977
    fix n :: nat
eberlm@61531
   978
    from subseq have "(\<Sum>k<n. f (g k)) = (\<Sum>k\<in>g`{..<n}. f k)"
eberlm@61531
   979
      by (subst setsum.reindex) (auto intro: subseq_imp_inj_on)
eberlm@61531
   980
    also from subseq have "\<dots> = (\<Sum>k<g n. f k)"
eberlm@61531
   981
      by (intro setsum.mono_neutral_left ballI zero)
eberlm@61531
   982
         (auto dest: subseq_strict_mono simp: strict_mono_less strict_mono_less_eq)
eberlm@61531
   983
    finally show "(\<Sum>k<n. f (g k)) = (\<Sum>k<g n. f k)" .
eberlm@61531
   984
  qed
eberlm@61531
   985
  also from LIMSEQ_subseq_LIMSEQ[OF lim subseq] have "\<dots> ----> c" unfolding o_def .
eberlm@61531
   986
  finally show "(\<lambda>n. \<Sum>k<n. f (g k)) ----> c" .
eberlm@61531
   987
next
eberlm@61531
   988
  assume lim: "(\<lambda>n. \<Sum>k<n. f (g k)) ----> c"
eberlm@61531
   989
  def g_inv \<equiv> "\<lambda>n. LEAST m. g m \<ge> n"
eberlm@61531
   990
  from filterlim_subseq[OF subseq] have g_inv_ex: "\<exists>m. g m \<ge> n" for n
eberlm@61531
   991
    by (auto simp: filterlim_at_top eventually_at_top_linorder)
eberlm@61531
   992
  hence g_inv: "g (g_inv n) \<ge> n" for n unfolding g_inv_def by (rule LeastI_ex)
lp15@61609
   993
  have g_inv_least: "m \<ge> g_inv n" if "g m \<ge> n" for m n using that
eberlm@61531
   994
    unfolding g_inv_def by (rule Least_le)
lp15@61609
   995
  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
eberlm@61531
   996
  have "(\<lambda>n. \<Sum>k<n. f k) = (\<lambda>n. \<Sum>k<g_inv n. f (g k))"
eberlm@61531
   997
  proof
eberlm@61531
   998
    fix n :: nat
eberlm@61531
   999
    {
eberlm@61531
  1000
      fix k assume k: "k \<in> {..<n} - g`{..<g_inv n}"
eberlm@61531
  1001
      have "k \<notin> range g"
eberlm@61531
  1002
      proof (rule notI, elim imageE)
eberlm@61531
  1003
        fix l assume l: "k = g l"
eberlm@61531
  1004
        have "g l < g (g_inv n)" by (rule less_le_trans[OF _ g_inv]) (insert k l, simp_all)
eberlm@61531
  1005
        with subseq have "l < g_inv n" by (simp add: subseq_strict_mono strict_mono_less)
eberlm@61531
  1006
        with k l show False by simp
eberlm@61531
  1007
      qed
eberlm@61531
  1008
      hence "f k = 0" by (rule zero)
eberlm@61531
  1009
    }
eberlm@61531
  1010
    with g_inv_least' g_inv have "(\<Sum>k<n. f k) = (\<Sum>k\<in>g`{..<g_inv n}. f k)"
eberlm@61531
  1011
      by (intro setsum.mono_neutral_right) auto
lp15@61609
  1012
    also from subseq have "\<dots> = (\<Sum>k<g_inv n. f (g k))" using subseq_imp_inj_on
eberlm@61531
  1013
      by (subst setsum.reindex) simp_all
eberlm@61531
  1014
    finally show "(\<Sum>k<n. f k) = (\<Sum>k<g_inv n. f (g k))" .
eberlm@61531
  1015
  qed
eberlm@61531
  1016
  also {
eberlm@61531
  1017
    fix K n :: nat assume "g K \<le> n"
eberlm@61531
  1018
    also have "n \<le> g (g_inv n)" by (rule g_inv)
eberlm@61531
  1019
    finally have "K \<le> g_inv n" using subseq by (simp add: strict_mono_less_eq subseq_strict_mono)
eberlm@61531
  1020
  }
lp15@61609
  1021
  hence "filterlim g_inv at_top sequentially"
eberlm@61531
  1022
    by (auto simp: filterlim_at_top eventually_at_top_linorder)
eberlm@61531
  1023
  from lim and this have "(\<lambda>n. \<Sum>k<g_inv n. f (g k)) ----> c" by (rule filterlim_compose)
eberlm@61531
  1024
  finally show "(\<lambda>n. \<Sum>k<n. f k) ----> c" .
eberlm@61531
  1025
qed
eberlm@61531
  1026
eberlm@61531
  1027
lemma summable_mono_reindex:
eberlm@61531
  1028
  assumes subseq: "subseq g" and zero: "\<And>n. n \<notin> range g \<Longrightarrow> f n = 0"
eberlm@61531
  1029
  shows   "summable (\<lambda>n. f (g n)) \<longleftrightarrow> summable f"
eberlm@61531
  1030
  using sums_mono_reindex[of g f, OF assms] by (simp add: summable_def)
eberlm@61531
  1031
lp15@61609
  1032
lemma suminf_mono_reindex:
eberlm@61531
  1033
  assumes "subseq g" "\<And>n. n \<notin> range g \<Longrightarrow> f n = (0 :: 'a :: {t2_space,comm_monoid_add})"
eberlm@61531
  1034
  shows   "suminf (\<lambda>n. f (g n)) = suminf f"
eberlm@61531
  1035
proof (cases "summable f")
eberlm@61531
  1036
  case False
eberlm@61531
  1037
  hence "\<not>(\<exists>c. f sums c)" unfolding summable_def by blast
eberlm@61531
  1038
  hence "suminf f = The (\<lambda>_. False)" by (simp add: suminf_def)
eberlm@61531
  1039
  moreover from False have "\<not>summable (\<lambda>n. f (g n))"
eberlm@61531
  1040
    using summable_mono_reindex[of g f, OF assms] by simp
eberlm@61531
  1041
  hence "\<not>(\<exists>c. (\<lambda>n. f (g n)) sums c)" unfolding summable_def by blast
eberlm@61531
  1042
  hence "suminf (\<lambda>n. f (g n)) = The (\<lambda>_. False)" by (simp add: suminf_def)
eberlm@61531
  1043
  ultimately show ?thesis by simp
lp15@61609
  1044
qed (insert sums_mono_reindex[of g f, OF assms] summable_mono_reindex[of g f, OF assms],
eberlm@61531
  1045
     simp_all add: sums_iff)
eberlm@61531
  1046
paulson@14416
  1047
end