src/HOL/Series.thy
author hoelzl
Fri, 30 May 2014 14:55:10 +0200
changeset 57129 7edb7550663e
parent 57025 e7fd64f82876
child 57275 0ddb5b755cdc
permissions -rw-r--r--
introduce more powerful reindexing rules for big operators
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
10751
a81ea5d3dd41 separation of HOL-Hyperreal from HOL-Real
paulson
parents:
diff changeset
     1
(*  Title       : Series.thy
a81ea5d3dd41 separation of HOL-Hyperreal from HOL-Real
paulson
parents:
diff changeset
     2
    Author      : Jacques D. Fleuriot
a81ea5d3dd41 separation of HOL-Hyperreal from HOL-Real
paulson
parents:
diff changeset
     3
    Copyright   : 1998  University of Cambridge
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
     4
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
     5
Converted to Isar and polished by lcp
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15537
diff changeset
     6
Converted to setsum and polished yet more by TNN
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16733
diff changeset
     7
Additional contributions by Jeremy Avigad
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 36660
diff changeset
     8
*)
10751
a81ea5d3dd41 separation of HOL-Hyperreal from HOL-Real
paulson
parents:
diff changeset
     9
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    10
header {* Infinite Series *}
10751
a81ea5d3dd41 separation of HOL-Hyperreal from HOL-Real
paulson
parents:
diff changeset
    11
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15085
diff changeset
    12
theory Series
51528
66c3a7589de7 Series.thy is based on Limits.thy and not Deriv.thy
hoelzl
parents: 51526
diff changeset
    13
imports Limits
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15085
diff changeset
    14
begin
15561
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15546
diff changeset
    15
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    16
subsection {* Definition of infinite summability *}
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    17
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
    18
definition
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
    19
  sums :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a \<Rightarrow> bool"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
    20
  (infixr "sums" 80)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
    21
where
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
    22
  "f sums s \<longleftrightarrow> (\<lambda>n. \<Sum>i<n. f i) ----> s"
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
    23
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
    24
definition summable :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> bool" where
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
    25
   "summable f \<longleftrightarrow> (\<exists>s. f sums s)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
    26
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
    27
definition
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
    28
  suminf :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
    29
  (binder "\<Sum>" 10)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
    30
where
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
    31
  "suminf f = (THE s. f sums s)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
    32
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    33
subsection {* Infinite summability on topological monoids *}
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    34
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
    35
lemma sums_subst[trans]: "f = g \<Longrightarrow> g sums z \<Longrightarrow> f sums z"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
    36
  by simp
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
    37
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
    38
lemma sums_summable: "f sums l \<Longrightarrow> summable f"
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 36660
diff changeset
    39
  by (simp add: sums_def summable_def, blast)
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
    40
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
    41
lemma summable_iff_convergent: "summable f \<longleftrightarrow> convergent (\<lambda>n. \<Sum>i<n. f i)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
    42
  by (simp add: summable_def sums_def convergent_def)
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
    43
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
    44
lemma suminf_eq_lim: "suminf f = lim (\<lambda>n. \<Sum>i<n. f i)"
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 36660
diff changeset
    45
  by (simp add: suminf_def sums_def lim_def)
32707
836ec9d0a0c8 New lemmas involving the real numbers, especially limits and series
paulson
parents: 31336
diff changeset
    46
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    47
lemma sums_zero[simp, intro]: "(\<lambda>n. 0) sums 0"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    48
  unfolding sums_def by (simp add: tendsto_const)
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    49
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    50
lemma summable_zero[simp, intro]: "summable (\<lambda>n. 0)"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    51
  by (rule sums_zero [THEN sums_summable])
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    52
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    53
lemma sums_group: "f sums s \<Longrightarrow> 0 < k \<Longrightarrow> (\<lambda>n. setsum f {n * k ..< n * k + k}) sums s"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    54
  apply (simp only: sums_def setsum_nat_group tendsto_def eventually_sequentially)
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    55
  apply safe
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    56
  apply (erule_tac x=S in allE)
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    57
  apply safe
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    58
  apply (rule_tac x="N" in exI, safe)
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    59
  apply (drule_tac x="n*k" in spec)
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    60
  apply (erule mp)
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    61
  apply (erule order_trans)
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    62
  apply simp
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    63
  done
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    64
47761
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47108
diff changeset
    65
lemma sums_finite:
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
    66
  assumes [simp]: "finite N" and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 0"
47761
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47108
diff changeset
    67
  shows "f sums (\<Sum>n\<in>N. f n)"
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47108
diff changeset
    68
proof -
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47108
diff changeset
    69
  { fix n
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47108
diff changeset
    70
    have "setsum f {..<n + Suc (Max N)} = setsum f N"
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47108
diff changeset
    71
    proof cases
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47108
diff changeset
    72
      assume "N = {}"
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47108
diff changeset
    73
      with f have "f = (\<lambda>x. 0)" by auto
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47108
diff changeset
    74
      then show ?thesis by simp
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47108
diff changeset
    75
    next
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47108
diff changeset
    76
      assume [simp]: "N \<noteq> {}"
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47108
diff changeset
    77
      show ?thesis
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47108
diff changeset
    78
      proof (safe intro!: setsum_mono_zero_right f)
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47108
diff changeset
    79
        fix i assume "i \<in> N"
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47108
diff changeset
    80
        then have "i \<le> Max N" by simp
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47108
diff changeset
    81
        then show "i < n + Suc (Max N)" by simp
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47108
diff changeset
    82
      qed
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47108
diff changeset
    83
    qed }
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47108
diff changeset
    84
  note eq = this
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47108
diff changeset
    85
  show ?thesis unfolding sums_def
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47108
diff changeset
    86
    by (rule LIMSEQ_offset[of _ "Suc (Max N)"])
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47108
diff changeset
    87
       (simp add: eq atLeast0LessThan tendsto_const del: add_Suc_right)
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47108
diff changeset
    88
qed
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47108
diff changeset
    89
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    90
lemma summable_finite: "finite N \<Longrightarrow> (\<And>n. n \<notin> N \<Longrightarrow> f n = 0) \<Longrightarrow> summable f"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    91
  by (rule sums_summable) (rule sums_finite)
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    92
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
    93
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)"
47761
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47108
diff changeset
    94
  using sums_finite[of A "(\<lambda>r. if r \<in> A then f r else 0)"] by simp
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47108
diff changeset
    95
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    96
lemma summable_If_finite_set[simp, intro]: "finite A \<Longrightarrow> summable (\<lambda>r. if r \<in> A then f r else 0)"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    97
  by (rule sums_summable) (rule sums_If_finite_set)
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
    98
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
    99
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)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   100
  using sums_If_finite_set[of "{r. P r}"] by simp
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16733
diff changeset
   101
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   102
lemma summable_If_finite[simp, intro]: "finite {r. P r} \<Longrightarrow> summable (\<lambda>r. if P r then f r else 0)"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   103
  by (rule sums_summable) (rule sums_If_finite)
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   104
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   105
lemma sums_single: "(\<lambda>r. if r = i then f r else 0) sums f i"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   106
  using sums_If_finite[of "\<lambda>r. r = i"] by simp
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29197
diff changeset
   107
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   108
lemma summable_single[simp, intro]: "summable (\<lambda>r. if r = i then f r else 0)"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   109
  by (rule sums_summable) (rule sums_single)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   110
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   111
context
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   112
  fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   113
begin
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   114
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   115
lemma summable_sums[intro]: "summable f \<Longrightarrow> f sums (suminf f)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   116
  by (simp add: summable_def sums_def suminf_def)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   117
     (metis convergent_LIMSEQ_iff convergent_def lim_def)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   118
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   119
lemma summable_LIMSEQ: "summable f \<Longrightarrow> (\<lambda>n. \<Sum>i<n. f i) ----> suminf f"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   120
  by (rule summable_sums [unfolded sums_def])
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   121
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   122
lemma sums_unique: "f sums s \<Longrightarrow> s = suminf f"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   123
  by (metis limI suminf_eq_lim sums_def)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   124
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   125
lemma sums_iff: "f sums x \<longleftrightarrow> summable f \<and> (suminf f = x)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   126
  by (metis summable_sums sums_summable sums_unique)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   127
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   128
lemma suminf_finite:
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   129
  assumes N: "finite N" and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 0"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   130
  shows "suminf f = (\<Sum>n\<in>N. f n)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   131
  using sums_finite[OF assms, THEN sums_unique] by simp
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   132
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   133
end
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16733
diff changeset
   134
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 36660
diff changeset
   135
lemma suminf_zero[simp]: "suminf (\<lambda>n. 0::'a::{t2_space, comm_monoid_add}) = 0"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   136
  by (rule sums_zero [THEN sums_unique, symmetric])
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16733
diff changeset
   137
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   138
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   139
subsection {* Infinite summability on ordered, topological monoids *}
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   140
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   141
lemma sums_le:
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   142
  fixes f g :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology}"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   143
  shows "\<forall>n. f n \<le> g n \<Longrightarrow> f sums s \<Longrightarrow> g sums t \<Longrightarrow> s \<le> t"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   144
  by (rule LIMSEQ_le) (auto intro: setsum_mono simp: sums_def)
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   145
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   146
context
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   147
  fixes f :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology}"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   148
begin
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   149
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   150
lemma suminf_le: "\<lbrakk>\<forall>n. f n \<le> g n; summable f; summable g\<rbrakk> \<Longrightarrow> suminf f \<le> suminf g"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   151
  by (auto dest: sums_summable intro: sums_le)
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   152
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   153
lemma setsum_le_suminf: "summable f \<Longrightarrow> \<forall>m\<ge>n. 0 \<le> f m \<Longrightarrow> setsum f {..<n} \<le> suminf f"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   154
  by (rule sums_le[OF _ sums_If_finite_set summable_sums]) auto
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   155
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   156
lemma suminf_nonneg: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> 0 \<le> suminf f"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   157
  using setsum_le_suminf[of 0] by simp
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   158
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   159
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"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   160
  using
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   161
    setsum_le_suminf[of "Suc i"]
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   162
    add_strict_increasing[of "f i" "setsum f {..<n}" "setsum f {..<i}"]
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   163
    setsum_mono2[of "{..<i}" "{..<n}" f]
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   164
  by (auto simp: less_imp_le ac_simps)
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   165
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   166
lemma setsum_less_suminf: "summable f \<Longrightarrow> \<forall>m\<ge>n. 0 < f m \<Longrightarrow> setsum f {..<n} < suminf f"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   167
  using setsum_less_suminf2[of n n] by (simp add: less_imp_le)
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   168
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   169
lemma suminf_pos2: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> 0 < f i \<Longrightarrow> 0 < suminf f"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   170
  using setsum_less_suminf2[of 0 i] by simp
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   171
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   172
lemma suminf_pos: "summable f \<Longrightarrow> \<forall>n. 0 < f n \<Longrightarrow> 0 < suminf f"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   173
  using suminf_pos2[of 0] by (simp add: less_imp_le)
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   174
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   175
lemma suminf_le_const: "summable f \<Longrightarrow> (\<And>n. setsum f {..<n} \<le> x) \<Longrightarrow> suminf f \<le> x"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   176
  by (metis LIMSEQ_le_const2 summable_LIMSEQ)
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   177
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   178
lemma suminf_eq_zero_iff: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> suminf f = 0 \<longleftrightarrow> (\<forall>n. f n = 0)"
50999
3de230ed0547 introduce order topology
hoelzl
parents: 50331
diff changeset
   179
proof
3de230ed0547 introduce order topology
hoelzl
parents: 50331
diff changeset
   180
  assume "summable f" "suminf f = 0" and pos: "\<forall>n. 0 \<le> f n"
3de230ed0547 introduce order topology
hoelzl
parents: 50331
diff changeset
   181
  then have f: "(\<lambda>n. \<Sum>i<n. f i) ----> 0"
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   182
    using summable_LIMSEQ[of f] by simp
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   183
  then have "\<And>i. (\<Sum>n\<in>{i}. f n) \<le> 0"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   184
  proof (rule LIMSEQ_le_const)
50999
3de230ed0547 introduce order topology
hoelzl
parents: 50331
diff changeset
   185
    fix i show "\<exists>N. \<forall>n\<ge>N. (\<Sum>n\<in>{i}. f n) \<le> setsum f {..<n}"
3de230ed0547 introduce order topology
hoelzl
parents: 50331
diff changeset
   186
      using pos by (intro exI[of _ "Suc i"] allI impI setsum_mono2) auto
3de230ed0547 introduce order topology
hoelzl
parents: 50331
diff changeset
   187
  qed
3de230ed0547 introduce order topology
hoelzl
parents: 50331
diff changeset
   188
  with pos show "\<forall>n. f n = 0"
3de230ed0547 introduce order topology
hoelzl
parents: 50331
diff changeset
   189
    by (auto intro!: antisym)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   190
qed (metis suminf_zero fun_eq_iff)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   191
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   192
lemma suminf_pos_iff: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> 0 < suminf f \<longleftrightarrow> (\<exists>i. 0 < f i)"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   193
  using setsum_le_suminf[of 0] suminf_eq_zero_iff by (simp add: less_le)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   194
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   195
end
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   196
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   197
lemma summableI_nonneg_bounded:
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   198
  fixes f:: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology, conditionally_complete_linorder}"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   199
  assumes pos[simp]: "\<And>n. 0 \<le> f n" and le: "\<And>n. (\<Sum>i<n. f i) \<le> x"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   200
  shows "summable f"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   201
  unfolding summable_def sums_def[abs_def]
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   202
proof (intro exI order_tendstoI)
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   203
  have [simp, intro]: "bdd_above (range (\<lambda>n. \<Sum>i<n. f i))"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   204
    using le by (auto simp: bdd_above_def)
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   205
  { fix a assume "a < (SUP n. \<Sum>i<n. f i)"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   206
    then obtain n where "a < (\<Sum>i<n. f i)"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   207
      by (auto simp add: less_cSUP_iff)
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   208
    then have "\<And>m. n \<le> m \<Longrightarrow> a < (\<Sum>i<m. f i)"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   209
      by (rule less_le_trans) (auto intro!: setsum_mono2)
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   210
    then show "eventually (\<lambda>n. a < (\<Sum>i<n. f i)) sequentially"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   211
      by (auto simp: eventually_sequentially) }
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   212
  { fix a assume "(SUP n. \<Sum>i<n. f i) < a"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   213
    moreover have "\<And>n. (\<Sum>i<n. f i) \<le> (SUP n. \<Sum>i<n. f i)"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   214
      by (auto intro: cSUP_upper)
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   215
    ultimately show "eventually (\<lambda>n. (\<Sum>i<n. f i) < a) sequentially"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   216
      by (auto intro: le_less_trans simp: eventually_sequentially) }
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   217
qed
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   218
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   219
subsection {* Infinite summability on real normed vector spaces *}
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   220
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   221
lemma sums_Suc_iff:
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   222
  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   223
  shows "(\<lambda>n. f (Suc n)) sums s \<longleftrightarrow> f sums (s + f 0)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   224
proof -
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   225
  have "f sums (s + f 0) \<longleftrightarrow> (\<lambda>i. \<Sum>j<Suc i. f j) ----> s + f 0"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   226
    by (subst LIMSEQ_Suc_iff) (simp add: sums_def)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   227
  also have "\<dots> \<longleftrightarrow> (\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) ----> s + f 0"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   228
    by (simp add: ac_simps setsum_reindex image_iff lessThan_Suc_eq_insert_0)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   229
  also have "\<dots> \<longleftrightarrow> (\<lambda>n. f (Suc n)) sums s"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   230
  proof
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   231
    assume "(\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) ----> s + f 0"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   232
    with tendsto_add[OF this tendsto_const, of "- f 0"]
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   233
    show "(\<lambda>i. f (Suc i)) sums s"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   234
      by (simp add: sums_def)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   235
  qed (auto intro: tendsto_add tendsto_const simp: sums_def)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   236
  finally show ?thesis ..
50999
3de230ed0547 introduce order topology
hoelzl
parents: 50331
diff changeset
   237
qed
3de230ed0547 introduce order topology
hoelzl
parents: 50331
diff changeset
   238
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   239
context
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   240
  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   241
begin
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   242
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   243
lemma sums_add: "f sums a \<Longrightarrow> g sums b \<Longrightarrow> (\<lambda>n. f n + g n) sums (a + b)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   244
  unfolding sums_def by (simp add: setsum_addf tendsto_add)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   245
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   246
lemma summable_add: "summable f \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. f n + g n)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   247
  unfolding summable_def by (auto intro: sums_add)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   248
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   249
lemma suminf_add: "summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f + suminf g = (\<Sum>n. f n + g n)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   250
  by (intro sums_unique sums_add summable_sums)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   251
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   252
lemma sums_diff: "f sums a \<Longrightarrow> g sums b \<Longrightarrow> (\<lambda>n. f n - g n) sums (a - b)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   253
  unfolding sums_def by (simp add: setsum_subtractf tendsto_diff)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   254
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   255
lemma summable_diff: "summable f \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. f n - g n)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   256
  unfolding summable_def by (auto intro: sums_diff)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   257
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   258
lemma suminf_diff: "summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f - suminf g = (\<Sum>n. f n - g n)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   259
  by (intro sums_unique sums_diff summable_sums)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   260
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   261
lemma sums_minus: "f sums a \<Longrightarrow> (\<lambda>n. - f n) sums (- a)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   262
  unfolding sums_def by (simp add: setsum_negf tendsto_minus)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   263
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   264
lemma summable_minus: "summable f \<Longrightarrow> summable (\<lambda>n. - f n)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   265
  unfolding summable_def by (auto intro: sums_minus)
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   266
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   267
lemma suminf_minus: "summable f \<Longrightarrow> (\<Sum>n. - f n) = - (\<Sum>n. f n)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   268
  by (intro sums_unique [symmetric] sums_minus summable_sums)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   269
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   270
lemma sums_Suc: "(\<lambda> n. f (Suc n)) sums l \<Longrightarrow> f sums (l + f 0)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   271
  by (simp add: sums_Suc_iff)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   272
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   273
lemma sums_iff_shift: "(\<lambda>i. f (i + n)) sums s \<longleftrightarrow> f sums (s + (\<Sum>i<n. f i))"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   274
proof (induct n arbitrary: s)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   275
  case (Suc n)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   276
  moreover have "(\<lambda>i. f (Suc i + n)) sums s \<longleftrightarrow> (\<lambda>i. f (i + n)) sums (s + f n)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   277
    by (subst sums_Suc_iff) simp
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   278
  ultimately show ?case
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   279
    by (simp add: ac_simps)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   280
qed simp
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   281
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   282
lemma summable_iff_shift: "summable (\<lambda>n. f (n + k)) \<longleftrightarrow> summable f"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   283
  by (metis diff_add_cancel summable_def sums_iff_shift[abs_def])
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   284
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   285
lemma sums_split_initial_segment: "f sums s \<Longrightarrow> (\<lambda>i. f (i + n)) sums (s - (\<Sum>i<n. f i))"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   286
  by (simp add: sums_iff_shift)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   287
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   288
lemma summable_ignore_initial_segment: "summable f \<Longrightarrow> summable (\<lambda>n. f(n + k))"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   289
  by (simp add: summable_iff_shift)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   290
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   291
lemma suminf_minus_initial_segment: "summable f \<Longrightarrow> (\<Sum>n. f (n + k)) = (\<Sum>n. f n) - (\<Sum>i<k. f i)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   292
  by (rule sums_unique[symmetric]) (auto simp: sums_iff_shift)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   293
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   294
lemma suminf_split_initial_segment: "summable f \<Longrightarrow> suminf f = (\<Sum>n. f(n + k)) + (\<Sum>i<k. f i)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   295
  by (auto simp add: suminf_minus_initial_segment)
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   296
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   297
lemma suminf_exist_split: 
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   298
  fixes r :: real assumes "0 < r" and "summable f"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   299
  shows "\<exists>N. \<forall>n\<ge>N. norm (\<Sum>i. f (i + n)) < r"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   300
proof -
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   301
  from LIMSEQ_D[OF summable_LIMSEQ[OF `summable f`] `0 < r`]
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   302
  obtain N :: nat where "\<forall> n \<ge> N. norm (setsum f {..<n} - suminf f) < r" by auto
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   303
  thus ?thesis
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   304
    by (auto simp: norm_minus_commute suminf_minus_initial_segment[OF `summable f`])
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   305
qed
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   306
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   307
lemma summable_LIMSEQ_zero: "summable f \<Longrightarrow> f ----> 0"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   308
  apply (drule summable_iff_convergent [THEN iffD1])
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   309
  apply (drule convergent_Cauchy)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   310
  apply (simp only: Cauchy_iff LIMSEQ_iff, safe)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   311
  apply (drule_tac x="r" in spec, safe)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   312
  apply (rule_tac x="M" in exI, safe)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   313
  apply (drule_tac x="Suc n" in spec, simp)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   314
  apply (drule_tac x="n" in spec, simp)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   315
  done
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   316
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   317
end
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   318
57025
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   319
context
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   320
  fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::real_normed_vector" and I :: "'i set"
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   321
begin
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   322
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   323
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)"
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   324
  by (induct I rule: infinite_finite_induct) (auto intro!: sums_add)
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   325
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   326
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)"
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   327
  using sums_unique[OF sums_setsum, OF summable_sums] by simp
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   328
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   329
lemma summable_setsum: "(\<And>i. i \<in> I \<Longrightarrow> summable (f i)) \<Longrightarrow> summable (\<lambda>n. \<Sum>i\<in>I. f i n)"
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   330
  using sums_summable[OF sums_setsum[OF summable_sums]] .
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   331
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   332
end
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   333
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   334
lemma (in bounded_linear) sums: "(\<lambda>n. X n) sums a \<Longrightarrow> (\<lambda>n. f (X n)) sums (f a)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   335
  unfolding sums_def by (drule tendsto, simp only: setsum)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   336
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   337
lemma (in bounded_linear) summable: "summable (\<lambda>n. X n) \<Longrightarrow> summable (\<lambda>n. f (X n))"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   338
  unfolding summable_def by (auto intro: sums)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   339
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   340
lemma (in bounded_linear) suminf: "summable (\<lambda>n. X n) \<Longrightarrow> f (\<Sum>n. X n) = (\<Sum>n. f (X n))"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   341
  by (intro sums_unique sums summable_sums)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   342
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   343
lemmas sums_of_real = bounded_linear.sums [OF bounded_linear_of_real]
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   344
lemmas summable_of_real = bounded_linear.summable [OF bounded_linear_of_real]
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   345
lemmas suminf_of_real = bounded_linear.suminf [OF bounded_linear_of_real]
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   346
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   347
subsection {* Infinite summability on real normed algebras *}
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   348
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   349
context
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   350
  fixes f :: "nat \<Rightarrow> 'a::real_normed_algebra"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   351
begin
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   352
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   353
lemma sums_mult: "f sums a \<Longrightarrow> (\<lambda>n. c * f n) sums (c * a)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   354
  by (rule bounded_linear.sums [OF bounded_linear_mult_right])
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   355
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   356
lemma summable_mult: "summable f \<Longrightarrow> summable (\<lambda>n. c * f n)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   357
  by (rule bounded_linear.summable [OF bounded_linear_mult_right])
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   358
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   359
lemma suminf_mult: "summable f \<Longrightarrow> suminf (\<lambda>n. c * f n) = c * suminf f"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   360
  by (rule bounded_linear.suminf [OF bounded_linear_mult_right, symmetric])
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   361
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   362
lemma sums_mult2: "f sums a \<Longrightarrow> (\<lambda>n. f n * c) sums (a * c)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   363
  by (rule bounded_linear.sums [OF bounded_linear_mult_left])
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   364
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   365
lemma summable_mult2: "summable f \<Longrightarrow> summable (\<lambda>n. f n * c)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   366
  by (rule bounded_linear.summable [OF bounded_linear_mult_left])
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   367
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   368
lemma suminf_mult2: "summable f \<Longrightarrow> suminf f * c = (\<Sum>n. f n * c)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   369
  by (rule bounded_linear.suminf [OF bounded_linear_mult_left])
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   370
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   371
end
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   372
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   373
subsection {* Infinite summability on real normed fields *}
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   374
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   375
context
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   376
  fixes c :: "'a::real_normed_field"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   377
begin
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   378
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   379
lemma sums_divide: "f sums a \<Longrightarrow> (\<lambda>n. f n / c) sums (a / c)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   380
  by (rule bounded_linear.sums [OF bounded_linear_divide])
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   381
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   382
lemma summable_divide: "summable f \<Longrightarrow> summable (\<lambda>n. f n / c)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   383
  by (rule bounded_linear.summable [OF bounded_linear_divide])
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   384
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   385
lemma suminf_divide: "summable f \<Longrightarrow> suminf (\<lambda>n. f n / c) = suminf f / c"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   386
  by (rule bounded_linear.suminf [OF bounded_linear_divide, symmetric])
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   387
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   388
text{*Sum of a geometric progression.*}
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   389
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   390
lemma geometric_sums: "norm c < 1 \<Longrightarrow> (\<lambda>n. c^n) sums (1 / (1 - c))"
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   391
proof -
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   392
  assume less_1: "norm c < 1"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   393
  hence neq_1: "c \<noteq> 1" by auto
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   394
  hence neq_0: "c - 1 \<noteq> 0" by simp
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   395
  from less_1 have lim_0: "(\<lambda>n. c^n) ----> 0"
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   396
    by (rule LIMSEQ_power_zero)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   397
  hence "(\<lambda>n. c ^ n / (c - 1) - 1 / (c - 1)) ----> 0 / (c - 1) - 1 / (c - 1)"
44568
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44289
diff changeset
   398
    using neq_0 by (intro tendsto_intros)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   399
  hence "(\<lambda>n. (c ^ n - 1) / (c - 1)) ----> 1 / (1 - c)"
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   400
    by (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   401
  thus "(\<lambda>n. c ^ n) sums (1 / (1 - c))"
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   402
    by (simp add: sums_def geometric_sum neq_1)
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   403
qed
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   404
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   405
lemma summable_geometric: "norm c < 1 \<Longrightarrow> summable (\<lambda>n. c^n)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   406
  by (rule geometric_sums [THEN sums_summable])
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   407
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   408
lemma suminf_geometric: "norm c < 1 \<Longrightarrow> suminf (\<lambda>n. c^n) = 1 / (1 - c)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   409
  by (rule sums_unique[symmetric]) (rule geometric_sums)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   410
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   411
end
33271
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 32877
diff changeset
   412
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 32877
diff changeset
   413
lemma power_half_series: "(\<lambda>n. (1/2::real)^Suc n) sums 1"
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 32877
diff changeset
   414
proof -
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 32877
diff changeset
   415
  have 2: "(\<lambda>n. (1/2::real)^n) sums 2" using geometric_sums [of "1/2::real"]
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 32877
diff changeset
   416
    by auto
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 32877
diff changeset
   417
  have "(\<lambda>n. (1/2::real)^Suc n) = (\<lambda>n. (1 / 2) ^ n / 2)"
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 32877
diff changeset
   418
    by simp
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 41970
diff changeset
   419
  thus ?thesis using sums_divide [OF 2, of 2]
33271
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 32877
diff changeset
   420
    by simp
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 32877
diff changeset
   421
qed
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 32877
diff changeset
   422
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   423
subsection {* Infinite summability on Banach spaces *}
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   424
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   425
text{*Cauchy-type criterion for convergence of series (c.f. Harrison)*}
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   426
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   427
lemma summable_Cauchy:
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   428
  fixes f :: "nat \<Rightarrow> 'a::banach"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   429
  shows "summable f \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n. norm (setsum f {m..<n}) < e)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   430
  apply (simp only: summable_iff_convergent Cauchy_convergent_iff [symmetric] Cauchy_iff, safe)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   431
  apply (drule spec, drule (1) mp)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   432
  apply (erule exE, rule_tac x="M" in exI, clarify)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   433
  apply (rule_tac x="m" and y="n" in linorder_le_cases)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   434
  apply (frule (1) order_trans)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   435
  apply (drule_tac x="n" in spec, drule (1) mp)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   436
  apply (drule_tac x="m" in spec, drule (1) mp)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   437
  apply (simp_all add: setsum_diff [symmetric])
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   438
  apply (drule spec, drule (1) mp)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   439
  apply (erule exE, rule_tac x="N" in exI, clarify)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   440
  apply (rule_tac x="m" and y="n" in linorder_le_cases)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   441
  apply (subst norm_minus_commute)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   442
  apply (simp_all add: setsum_diff [symmetric])
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   443
  done
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   444
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   445
context
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   446
  fixes f :: "nat \<Rightarrow> 'a::banach"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   447
begin  
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   448
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   449
text{*Absolute convergence imples normal convergence*}
20689
4950e45442b8 add proof of summable_LIMSEQ_zero
huffman
parents: 20688
diff changeset
   450
56194
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56193
diff changeset
   451
lemma summable_norm_cancel: "summable (\<lambda>n. norm (f n)) \<Longrightarrow> summable f"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   452
  apply (simp only: summable_Cauchy, safe)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   453
  apply (drule_tac x="e" in spec, safe)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   454
  apply (rule_tac x="N" in exI, safe)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   455
  apply (drule_tac x="m" in spec, safe)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   456
  apply (rule order_le_less_trans [OF norm_setsum])
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   457
  apply (rule order_le_less_trans [OF abs_ge_self])
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   458
  apply simp
50999
3de230ed0547 introduce order topology
hoelzl
parents: 50331
diff changeset
   459
  done
32707
836ec9d0a0c8 New lemmas involving the real numbers, especially limits and series
paulson
parents: 31336
diff changeset
   460
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   461
lemma summable_norm: "summable (\<lambda>n. norm (f n)) \<Longrightarrow> norm (suminf f) \<le> (\<Sum>n. norm (f n))"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   462
  by (auto intro: LIMSEQ_le tendsto_norm summable_norm_cancel summable_LIMSEQ norm_setsum)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   463
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   464
text {* Comparison tests *}
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   465
56194
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56193
diff changeset
   466
lemma summable_comparison_test: "\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n \<Longrightarrow> summable g \<Longrightarrow> summable f"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   467
  apply (simp add: summable_Cauchy, safe)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   468
  apply (drule_tac x="e" in spec, safe)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   469
  apply (rule_tac x = "N + Na" in exI, safe)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   470
  apply (rotate_tac 2)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   471
  apply (drule_tac x = m in spec)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   472
  apply (auto, rotate_tac 2, drule_tac x = n in spec)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   473
  apply (rule_tac y = "\<Sum>k=m..<n. norm (f k)" in order_le_less_trans)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   474
  apply (rule norm_setsum)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   475
  apply (rule_tac y = "setsum g {m..<n}" in order_le_less_trans)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   476
  apply (auto intro: setsum_mono simp add: abs_less_iff)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   477
  done
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   478
56217
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 56213
diff changeset
   479
(*A better argument order*)
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 56213
diff changeset
   480
lemma summable_comparison_test': "summable g \<Longrightarrow> (\<And>n. n \<ge> N \<Longrightarrow> norm(f n) \<le> g n) \<Longrightarrow> summable f"
56369
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   481
  by (rule summable_comparison_test) auto
56217
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 56213
diff changeset
   482
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   483
subsection {* The Ratio Test*}
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   484
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   485
lemma summable_ratio_test: 
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   486
  assumes "c < 1" "\<And>n. n \<ge> N \<Longrightarrow> norm (f (Suc n)) \<le> c * norm (f n)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   487
  shows "summable f"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   488
proof cases
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   489
  assume "0 < c"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   490
  show "summable f"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   491
  proof (rule summable_comparison_test)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   492
    show "\<exists>N'. \<forall>n\<ge>N'. norm (f n) \<le> (norm (f N) / (c ^ N)) * c ^ n"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   493
    proof (intro exI allI impI)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   494
      fix n assume "N \<le> n" then show "norm (f n) \<le> (norm (f N) / (c ^ N)) * c ^ n"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   495
      proof (induct rule: inc_induct)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   496
        case (step m)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   497
        moreover have "norm (f (Suc m)) / c ^ Suc m * c ^ n \<le> norm (f m) / c ^ m * c ^ n"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   498
          using `0 < c` `c < 1` assms(2)[OF `N \<le> m`] by (simp add: field_simps)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   499
        ultimately show ?case by simp
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   500
      qed (insert `0 < c`, simp)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   501
    qed
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   502
    show "summable (\<lambda>n. norm (f N) / c ^ N * c ^ n)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   503
      using `0 < c` `c < 1` by (intro summable_mult summable_geometric) simp
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   504
  qed
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   505
next
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   506
  assume c: "\<not> 0 < c"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   507
  { fix n assume "n \<ge> N"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   508
    then have "norm (f (Suc n)) \<le> c * norm (f n)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   509
      by fact
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   510
    also have "\<dots> \<le> 0"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   511
      using c by (simp add: not_less mult_nonpos_nonneg)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   512
    finally have "f (Suc n) = 0"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   513
      by auto }
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   514
  then show "summable f"
56194
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56193
diff changeset
   515
    by (intro sums_summable[OF sums_finite, of "{.. Suc N}"]) (auto simp: not_le Suc_less_eq2)
56178
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 54703
diff changeset
   516
qed
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 54703
diff changeset
   517
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   518
end
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   519
56369
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   520
text{*Relations among convergence and absolute convergence for power series.*}
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   521
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   522
lemma abel_lemma:
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   523
  fixes a :: "nat \<Rightarrow> 'a::real_normed_vector"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   524
  assumes r: "0 \<le> r" and r0: "r < r0" and M: "\<And>n. norm (a n) * r0^n \<le> M"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   525
    shows "summable (\<lambda>n. norm (a n) * r^n)"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   526
proof (rule summable_comparison_test')
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   527
  show "summable (\<lambda>n. M * (r / r0) ^ n)"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   528
    using assms 
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   529
    by (auto simp add: summable_mult summable_geometric)
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   530
next
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   531
  fix n
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   532
  show "norm (norm (a n) * r ^ n) \<le> M * (r / r0) ^ n"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   533
    using r r0 M [of n]
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   534
    apply (auto simp add: abs_mult field_simps power_divide)
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   535
    apply (cases "r=0", simp)
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   536
    apply (cases n, auto)
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   537
    done
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   538
qed
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   539
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   540
23084
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   541
text{*Summability of geometric series for real algebras*}
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   542
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   543
lemma complete_algebra_summable_geometric:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30649
diff changeset
   544
  fixes x :: "'a::{real_normed_algebra_1,banach}"
23084
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   545
  shows "norm x < 1 \<Longrightarrow> summable (\<lambda>n. x ^ n)"
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   546
proof (rule summable_comparison_test)
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   547
  show "\<exists>N. \<forall>n\<ge>N. norm (x ^ n) \<le> norm x ^ n"
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   548
    by (simp add: norm_power_ineq)
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   549
  show "norm x < 1 \<Longrightarrow> summable (\<lambda>n. norm x ^ n)"
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   550
    by (simp add: summable_geometric)
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   551
qed
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   552
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   553
subsection {* Cauchy Product Formula *}
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   554
54703
499f92dc6e45 more antiquotations;
wenzelm
parents: 54230
diff changeset
   555
text {*
499f92dc6e45 more antiquotations;
wenzelm
parents: 54230
diff changeset
   556
  Proof based on Analysis WebNotes: Chapter 07, Class 41
499f92dc6e45 more antiquotations;
wenzelm
parents: 54230
diff changeset
   557
  @{url "http://www.math.unl.edu/~webnotes/classes/class41/prp77.htm"}
499f92dc6e45 more antiquotations;
wenzelm
parents: 54230
diff changeset
   558
*}
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   559
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   560
lemma setsum_triangle_reindex:
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   561
  fixes n :: nat
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   562
  shows "(\<Sum>(i,j)\<in>{(i,j). i+j < n}. f i j) = (\<Sum>k<n. \<Sum>i\<le>k. f i (k - i))"
57129
7edb7550663e introduce more powerful reindexing rules for big operators
hoelzl
parents: 57025
diff changeset
   563
  apply (simp add: setsum_Sigma)
7edb7550663e introduce more powerful reindexing rules for big operators
hoelzl
parents: 57025
diff changeset
   564
  apply (rule setsum.reindex_bij_witness[where j="\<lambda>(i, j). (i+j, i)" and i="\<lambda>(k, i). (i, k - i)"])
7edb7550663e introduce more powerful reindexing rules for big operators
hoelzl
parents: 57025
diff changeset
   565
  apply auto
7edb7550663e introduce more powerful reindexing rules for big operators
hoelzl
parents: 57025
diff changeset
   566
  done
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   567
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   568
lemma Cauchy_product_sums:
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   569
  fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   570
  assumes a: "summable (\<lambda>k. norm (a k))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   571
  assumes b: "summable (\<lambda>k. norm (b k))"
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   572
  shows "(\<lambda>k. \<Sum>i\<le>k. a i * b (k - i)) sums ((\<Sum>k. a k) * (\<Sum>k. b k))"
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   573
proof -
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   574
  let ?S1 = "\<lambda>n::nat. {..<n} \<times> {..<n}"
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   575
  let ?S2 = "\<lambda>n::nat. {(i,j). i + j < n}"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   576
  have S1_mono: "\<And>m n. m \<le> n \<Longrightarrow> ?S1 m \<subseteq> ?S1 n" by auto
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   577
  have S2_le_S1: "\<And>n. ?S2 n \<subseteq> ?S1 n" by auto
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   578
  have S1_le_S2: "\<And>n. ?S1 (n div 2) \<subseteq> ?S2 n" by auto
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   579
  have finite_S1: "\<And>n. finite (?S1 n)" by simp
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   580
  with S2_le_S1 have finite_S2: "\<And>n. finite (?S2 n)" by (rule finite_subset)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   581
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   582
  let ?g = "\<lambda>(i,j). a i * b j"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   583
  let ?f = "\<lambda>(i,j). norm (a i) * norm (b j)"
56536
aefb4a8da31f made mult_nonneg_nonneg a simp rule
nipkow
parents: 56369
diff changeset
   584
  have f_nonneg: "\<And>x. 0 \<le> ?f x" by (auto)
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   585
  hence norm_setsum_f: "\<And>A. norm (setsum ?f A) = setsum ?f A"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   586
    unfolding real_norm_def
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   587
    by (simp only: abs_of_nonneg setsum_nonneg [rule_format])
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   588
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   589
  have "(\<lambda>n. (\<Sum>k<n. a k) * (\<Sum>k<n. b k)) ----> (\<Sum>k. a k) * (\<Sum>k. b k)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   590
    by (intro tendsto_mult summable_LIMSEQ summable_norm_cancel [OF a] summable_norm_cancel [OF b])
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   591
  hence 1: "(\<lambda>n. setsum ?g (?S1 n)) ----> (\<Sum>k. a k) * (\<Sum>k. b k)"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   592
    by (simp only: setsum_product setsum_Sigma [rule_format] finite_lessThan)
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   593
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   594
  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))"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   595
    using a b by (intro tendsto_mult summable_LIMSEQ)
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   596
  hence "(\<lambda>n. setsum ?f (?S1 n)) ----> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   597
    by (simp only: setsum_product setsum_Sigma [rule_format] finite_lessThan)
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   598
  hence "convergent (\<lambda>n. setsum ?f (?S1 n))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   599
    by (rule convergentI)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   600
  hence Cauchy: "Cauchy (\<lambda>n. setsum ?f (?S1 n))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   601
    by (rule convergent_Cauchy)
36657
f376af79f6b7 remove unneeded constant Zseq
huffman
parents: 36409
diff changeset
   602
  have "Zfun (\<lambda>n. setsum ?f (?S1 n - ?S2 n)) sequentially"
f376af79f6b7 remove unneeded constant Zseq
huffman
parents: 36409
diff changeset
   603
  proof (rule ZfunI, simp only: eventually_sequentially norm_setsum_f)
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   604
    fix r :: real
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   605
    assume r: "0 < r"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   606
    from CauchyD [OF Cauchy r] obtain N
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   607
    where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (setsum ?f (?S1 m) - setsum ?f (?S1 n)) < r" ..
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   608
    hence "\<And>m n. \<lbrakk>N \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> norm (setsum ?f (?S1 m - ?S1 n)) < r"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   609
      by (simp only: setsum_diff finite_S1 S1_mono)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   610
    hence N: "\<And>m n. \<lbrakk>N \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> setsum ?f (?S1 m - ?S1 n) < r"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   611
      by (simp only: norm_setsum_f)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   612
    show "\<exists>N. \<forall>n\<ge>N. setsum ?f (?S1 n - ?S2 n) < r"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   613
    proof (intro exI allI impI)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   614
      fix n assume "2 * N \<le> n"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   615
      hence n: "N \<le> n div 2" by simp
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   616
      have "setsum ?f (?S1 n - ?S2 n) \<le> setsum ?f (?S1 n - ?S1 (n div 2))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   617
        by (intro setsum_mono2 finite_Diff finite_S1 f_nonneg
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   618
                  Diff_mono subset_refl S1_le_S2)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   619
      also have "\<dots> < r"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   620
        using n div_le_dividend by (rule N)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   621
      finally show "setsum ?f (?S1 n - ?S2 n) < r" .
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   622
    qed
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   623
  qed
36657
f376af79f6b7 remove unneeded constant Zseq
huffman
parents: 36409
diff changeset
   624
  hence "Zfun (\<lambda>n. setsum ?g (?S1 n - ?S2 n)) sequentially"
f376af79f6b7 remove unneeded constant Zseq
huffman
parents: 36409
diff changeset
   625
    apply (rule Zfun_le [rule_format])
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   626
    apply (simp only: norm_setsum_f)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   627
    apply (rule order_trans [OF norm_setsum setsum_mono])
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   628
    apply (auto simp add: norm_mult_ineq)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   629
    done
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   630
  hence 2: "(\<lambda>n. setsum ?g (?S1 n) - setsum ?g (?S2 n)) ----> 0"
36660
1cc4ab4b7ff7 make (X ----> L) an abbreviation for (X ---> L) sequentially
huffman
parents: 36657
diff changeset
   631
    unfolding tendsto_Zfun_iff diff_0_right
36657
f376af79f6b7 remove unneeded constant Zseq
huffman
parents: 36409
diff changeset
   632
    by (simp only: setsum_diff finite_S1 S2_le_S1)
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   633
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   634
  with 1 have "(\<lambda>n. setsum ?g (?S2 n)) ----> (\<Sum>k. a k) * (\<Sum>k. b k)"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   635
    by (rule LIMSEQ_diff_approach_zero2)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   636
  thus ?thesis by (simp only: sums_def setsum_triangle_reindex)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   637
qed
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   638
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   639
lemma Cauchy_product:
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   640
  fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   641
  assumes a: "summable (\<lambda>k. norm (a k))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   642
  assumes b: "summable (\<lambda>k. norm (b k))"
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   643
  shows "(\<Sum>k. a k) * (\<Sum>k. b k) = (\<Sum>k. \<Sum>i\<le>k. a i * b (k - i))"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   644
  using a b
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   645
  by (rule Cauchy_product_sums [THEN sums_unique])
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   646
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   647
subsection {* Series on @{typ real}s *}
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   648
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   649
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))"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   650
  by (rule summable_comparison_test) auto
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   651
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   652
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>)"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   653
  by (rule summable_comparison_test) auto
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   654
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   655
lemma summable_rabs_cancel: "summable (\<lambda>n. \<bar>f n :: real\<bar>) \<Longrightarrow> summable f"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   656
  by (rule summable_norm_cancel) simp
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   657
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   658
lemma summable_rabs: "summable (\<lambda>n. \<bar>f n :: real\<bar>) \<Longrightarrow> \<bar>suminf f\<bar> \<le> (\<Sum>n. \<bar>f n\<bar>)"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   659
  by (fold real_norm_def) (rule summable_norm)
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   660
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   661
end