src/HOL/Series.thy
author paulson
Thu, 07 Jan 2016 17:40:55 +0000
changeset 62087 44841d07ef1d
parent 62049 b0f941e207cf
child 62217 527488dc8b90
permissions -rw-r--r--
revisions to limits and derivatives, plus new lemmas
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(*  Title       : Series.thy
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    Author      : Jacques D. Fleuriot
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    Copyright   : 1998  University of Cambridge
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1f256287d4f0 converted Hyperreal/Series to Isar script
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Converted to Isar and polished by lcp
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
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Converted to setsum and polished yet more by TNN
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Additional contributions by Jeremy Avigad
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*)
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section \<open>Infinite Series\<close>
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theory Series
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imports Limits Inequalities
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begin
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subsection \<open>Definition of infinite summability\<close>
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definition
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  sums :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a \<Rightarrow> bool"
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  (infixr "sums" 80)
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where
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  "f sums s \<longleftrightarrow> (\<lambda>n. \<Sum>i<n. f i) \<longlonglongrightarrow> s"
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definition summable :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> bool" where
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   "summable f \<longleftrightarrow> (\<exists>s. f sums s)"
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definition
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  suminf :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a"
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  (binder "\<Sum>" 10)
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where
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  "suminf f = (THE s. f sums s)"
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lemma sums_def': "f sums s \<longleftrightarrow> (\<lambda>n. \<Sum>i = 0..n. f i) \<longlonglongrightarrow> s"
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  apply (simp add: sums_def)
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  apply (subst LIMSEQ_Suc_iff [symmetric])
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  apply (simp only: lessThan_Suc_atMost atLeast0AtMost)
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  done
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subsection \<open>Infinite summability on topological monoids\<close>
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lemma sums_subst[trans]: "f = g \<Longrightarrow> g sums z \<Longrightarrow> f sums z"
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  by simp
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lemma sums_cong: "(\<And>n. f n = g n) \<Longrightarrow> f sums c \<longleftrightarrow> g sums c"
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  by (drule ext) simp
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lemma sums_summable: "f sums l \<Longrightarrow> summable f"
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    48
  by (simp add: sums_def summable_def, blast)
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lemma summable_iff_convergent: "summable f \<longleftrightarrow> convergent (\<lambda>n. \<Sum>i<n. f i)"
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  by (simp add: summable_def sums_def convergent_def)
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lemma summable_iff_convergent':
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  "summable f \<longleftrightarrow> convergent (\<lambda>n. setsum f {..n})"
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paulson <lp15@cam.ac.uk>
parents: 61531
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    55
  by (simp_all only: summable_iff_convergent convergent_def
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    56
        lessThan_Suc_atMost [symmetric] LIMSEQ_Suc_iff[of "\<lambda>n. setsum f {..<n}"])
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    57
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lemma suminf_eq_lim: "suminf f = lim (\<lambda>n. \<Sum>i<n. f i)"
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  by (simp add: suminf_def sums_def lim_def)
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    60
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lemma sums_zero[simp, intro]: "(\<lambda>n. 0) sums 0"
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  unfolding sums_def by simp
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lemma summable_zero[simp, intro]: "summable (\<lambda>n. 0)"
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  by (rule sums_zero [THEN sums_summable])
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lemma sums_group: "f sums s \<Longrightarrow> 0 < k \<Longrightarrow> (\<lambda>n. setsum f {n * k ..< n * k + k}) sums s"
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  apply (simp only: sums_def setsum_nat_group tendsto_def eventually_sequentially)
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    69
  apply safe
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  apply (erule_tac x=S in allE)
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  apply safe
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  apply (rule_tac x="N" in exI, safe)
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    73
  apply (drule_tac x="n*k" in spec)
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    74
  apply (erule mp)
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  apply (erule order_trans)
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  apply simp
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  done
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lemma suminf_cong: "(\<And>n. f n = g n) \<Longrightarrow> suminf f = suminf g"
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    80
  by (rule arg_cong[of f g], rule ext) simp
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lemma summable_cong:
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  assumes "eventually (\<lambda>x. f x = (g x :: 'a :: real_normed_vector)) sequentially"
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    84
  shows   "summable f = summable g"
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    85
proof -
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    86
  from assms obtain N where N: "\<forall>n\<ge>N. f n = g n" by (auto simp: eventually_at_top_linorder)
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eberlm
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    87
  def C \<equiv> "(\<Sum>k<N. f k - g k)"
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77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61531
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    88
  from eventually_ge_at_top[of N]
61531
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eberlm
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    89
    have "eventually (\<lambda>n. setsum f {..<n} = C + setsum g {..<n}) sequentially"
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eberlm
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    90
  proof eventually_elim
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    91
    fix n assume n: "n \<ge> N"
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    92
    from n have "{..<n} = {..<N} \<union> {N..<n}" by auto
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    93
    also have "setsum f ... = setsum f {..<N} + setsum f {N..<n}"
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    94
      by (intro setsum.union_disjoint) auto
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    95
    also from N have "setsum f {N..<n} = setsum g {N..<n}" by (intro setsum.cong) simp_all
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61531
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    96
    also have "setsum f {..<N} + setsum g {N..<n} = C + (setsum g {..<N} + setsum g {N..<n})"
61531
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eberlm
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    97
      unfolding C_def by (simp add: algebra_simps setsum_subtractf)
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eberlm
parents: 60867
diff changeset
    98
    also have "setsum g {..<N} + setsum g {N..<n} = setsum g ({..<N} \<union> {N..<n})"
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eberlm
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    99
      by (intro setsum.union_disjoint [symmetric]) auto
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eberlm
parents: 60867
diff changeset
   100
    also from n have "{..<N} \<union> {N..<n} = {..<n}" by auto
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eberlm
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   101
    finally show "setsum f {..<n} = C + setsum g {..<n}" .
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eberlm
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   102
  qed
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eberlm
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   103
  from convergent_cong[OF this] show ?thesis
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eberlm
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   104
    by (simp add: summable_iff_convergent convergent_add_const_iff)
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eberlm
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   105
qed
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eberlm
parents: 60867
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   106
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   107
lemma sums_finite:
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   108
  assumes [simp]: "finite N" and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 0"
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   109
  shows "f sums (\<Sum>n\<in>N. f n)"
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hoelzl
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   110
proof -
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
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   111
  { fix n
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hoelzl
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   112
    have "setsum f {..<n + Suc (Max N)} = setsum f N"
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hoelzl
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   113
    proof cases
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hoelzl
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   114
      assume "N = {}"
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   115
      with f have "f = (\<lambda>x. 0)" by auto
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hoelzl
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   116
      then show ?thesis by simp
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   117
    next
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hoelzl
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   118
      assume [simp]: "N \<noteq> {}"
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hoelzl
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   119
      show ?thesis
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haftmann
parents: 57275
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   120
      proof (safe intro!: setsum.mono_neutral_right f)
47761
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hoelzl
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   121
        fix i assume "i \<in> N"
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hoelzl
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   122
        then have "i \<le> Max N" by simp
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hoelzl
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   123
        then show "i < n + Suc (Max N)" by simp
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hoelzl
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   124
      qed
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   125
    qed }
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   126
  note eq = this
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   127
  show ?thesis unfolding sums_def
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   128
    by (rule LIMSEQ_offset[of _ "Suc (Max N)"])
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hoelzl
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diff changeset
   129
       (simp add: eq atLeast0LessThan del: add_Suc_right)
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   130
qed
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hoelzl
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   131
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lemma summable_finite: "finite N \<Longrightarrow> (\<And>n. n \<notin> N \<Longrightarrow> f n = 0) \<Longrightarrow> summable f"
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   133
  by (rule sums_summable) (rule sums_finite)
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   134
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   135
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
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hoelzl
parents: 47108
diff changeset
   136
  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
   137
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   138
lemma summable_If_finite_set[simp, intro]: "finite A \<Longrightarrow> summable (\<lambda>r. if r \<in> A then f r else 0)"
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   139
  by (rule sums_summable) (rule sums_If_finite_set)
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diff changeset
   140
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   141
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)"
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   142
  using sums_If_finite_set[of "{r. P r}"] by simp
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00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16733
diff changeset
   143
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   144
lemma summable_If_finite[simp, intro]: "finite {r. P r} \<Longrightarrow> summable (\<lambda>r. if P r then f r else 0)"
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   145
  by (rule sums_summable) (rule sums_If_finite)
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   146
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   147
lemma sums_single: "(\<lambda>r. if r = i then f r else 0) sums f i"
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diff changeset
   148
  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
   149
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   150
lemma summable_single[simp, intro]: "summable (\<lambda>r. if r = i then f r else 0)"
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   151
  by (rule sums_summable) (rule sums_single)
56193
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   152
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   153
context
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   154
  fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}"
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   155
begin
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diff changeset
   156
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
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   157
lemma summable_sums[intro]: "summable f \<Longrightarrow> f sums (suminf f)"
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diff changeset
   158
  by (simp add: summable_def sums_def suminf_def)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
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diff changeset
   159
     (metis convergent_LIMSEQ_iff convergent_def lim_def)
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   160
61969
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wenzelm
parents: 61799
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   161
lemma summable_LIMSEQ: "summable f \<Longrightarrow> (\<lambda>n. \<Sum>i<n. f i) \<longlonglongrightarrow> suminf f"
56193
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parents: 56178
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   162
  by (rule summable_sums [unfolded sums_def])
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diff changeset
   163
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   164
lemma sums_unique: "f sums s \<Longrightarrow> s = suminf f"
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   165
  by (metis limI suminf_eq_lim sums_def)
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diff changeset
   166
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
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   167
lemma sums_iff: "f sums x \<longleftrightarrow> summable f \<and> (suminf f = x)"
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diff changeset
   168
  by (metis summable_sums sums_summable sums_unique)
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   169
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61531
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   170
lemma summable_sums_iff:
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
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   171
  "summable (f :: nat \<Rightarrow> 'a :: {comm_monoid_add,t2_space}) \<longleftrightarrow> f sums suminf f"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   172
  by (auto simp: sums_iff summable_sums)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   173
59613
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59025
diff changeset
   174
lemma sums_unique2:
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59025
diff changeset
   175
  fixes a b :: "'a::{comm_monoid_add,t2_space}"
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59025
diff changeset
   176
  shows "f sums a \<Longrightarrow> f sums b \<Longrightarrow> a = b"
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59025
diff changeset
   177
by (simp add: sums_iff)
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59025
diff changeset
   178
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   179
lemma suminf_finite:
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   180
  assumes N: "finite N" and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 0"
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   181
  shows "suminf f = (\<Sum>n\<in>N. f n)"
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   182
  using sums_finite[OF assms, THEN sums_unique] by simp
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diff changeset
   183
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   184
end
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00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
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   185
41970
47d6e13d1710 generalize infinite sums
hoelzl
parents: 36660
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   186
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..<_}
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   187
  by (rule sums_zero [THEN sums_unique, symmetric])
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parents: 16733
diff changeset
   188
56213
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   189
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60141
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   190
subsection \<open>Infinite summability on ordered, topological monoids\<close>
56213
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   191
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   192
lemma sums_le:
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   193
  fixes f g :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology}"
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diff changeset
   194
  shows "\<forall>n. f n \<le> g n \<Longrightarrow> f sums s \<Longrightarrow> g sums t \<Longrightarrow> s \<le> t"
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diff changeset
   195
  by (rule LIMSEQ_le) (auto intro: setsum_mono simp: sums_def)
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diff changeset
   196
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   197
context
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   198
  fixes f :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology}"
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diff changeset
   199
begin
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paulson
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diff changeset
   200
56213
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   201
lemma suminf_le: "\<lbrakk>\<forall>n. f n \<le> g n; summable f; summable g\<rbrakk> \<Longrightarrow> suminf f \<le> suminf g"
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diff changeset
   202
  by (auto dest: sums_summable intro: sums_le)
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diff changeset
   203
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   204
lemma setsum_le_suminf: "summable f \<Longrightarrow> \<forall>m\<ge>n. 0 \<le> f m \<Longrightarrow> setsum f {..<n} \<le> suminf f"
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diff changeset
   205
  by (rule sums_le[OF _ sums_If_finite_set summable_sums]) auto
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diff changeset
   206
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   207
lemma suminf_nonneg: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> 0 \<le> suminf f"
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diff changeset
   208
  using setsum_le_suminf[of 0] by simp
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diff changeset
   209
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   210
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"
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parents: 56194
diff changeset
   211
  using
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diff changeset
   212
    setsum_le_suminf[of "Suc i"]
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diff changeset
   213
    add_strict_increasing[of "f i" "setsum f {..<n}" "setsum f {..<i}"]
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diff changeset
   214
    setsum_mono2[of "{..<i}" "{..<n}" f]
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parents: 56194
diff changeset
   215
  by (auto simp: less_imp_le ac_simps)
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parents: 56194
diff changeset
   216
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diff changeset
   217
lemma setsum_less_suminf: "summable f \<Longrightarrow> \<forall>m\<ge>n. 0 < f m \<Longrightarrow> setsum f {..<n} < suminf f"
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parents: 56194
diff changeset
   218
  using setsum_less_suminf2[of n n] by (simp add: less_imp_le)
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parents: 56194
diff changeset
   219
e5720d3c18f0 further renaming in Series
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diff changeset
   220
lemma suminf_pos2: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> 0 < f i \<Longrightarrow> 0 < suminf f"
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hoelzl
parents: 56194
diff changeset
   221
  using setsum_less_suminf2[of 0 i] by simp
e5720d3c18f0 further renaming in Series
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diff changeset
   222
e5720d3c18f0 further renaming in Series
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diff changeset
   223
lemma suminf_pos: "summable f \<Longrightarrow> \<forall>n. 0 < f n \<Longrightarrow> 0 < suminf f"
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parents: 56194
diff changeset
   224
  using suminf_pos2[of 0] by (simp add: less_imp_le)
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diff changeset
   225
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diff changeset
   226
lemma suminf_le_const: "summable f \<Longrightarrow> (\<And>n. setsum f {..<n} \<le> x) \<Longrightarrow> suminf f \<le> x"
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diff changeset
   227
  by (metis LIMSEQ_le_const2 summable_LIMSEQ)
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   228
56193
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   229
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
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diff changeset
   230
proof
3de230ed0547 introduce order topology
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diff changeset
   231
  assume "summable f" "suminf f = 0" and pos: "\<forall>n. 0 \<le> f n"
61969
e01015e49041 more symbols;
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parents: 61799
diff changeset
   232
  then have f: "(\<lambda>n. \<Sum>i<n. f i) \<longlonglongrightarrow> 0"
56213
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diff changeset
   233
    using summable_LIMSEQ[of f] by simp
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hoelzl
parents: 56194
diff changeset
   234
  then have "\<And>i. (\<Sum>n\<in>{i}. f n) \<le> 0"
e5720d3c18f0 further renaming in Series
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parents: 56194
diff changeset
   235
  proof (rule LIMSEQ_le_const)
50999
3de230ed0547 introduce order topology
hoelzl
parents: 50331
diff changeset
   236
    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
   237
      using pos by (intro exI[of _ "Suc i"] allI impI setsum_mono2) auto
3de230ed0547 introduce order topology
hoelzl
parents: 50331
diff changeset
   238
  qed
3de230ed0547 introduce order topology
hoelzl
parents: 50331
diff changeset
   239
  with pos show "\<forall>n. f n = 0"
3de230ed0547 introduce order topology
hoelzl
parents: 50331
diff changeset
   240
    by (auto intro!: antisym)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   241
qed (metis suminf_zero fun_eq_iff)
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hoelzl
parents: 56178
diff changeset
   242
56213
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diff changeset
   243
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
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parents: 56194
diff changeset
   244
  using setsum_le_suminf[of 0] suminf_eq_zero_iff by (simp add: less_le)
56193
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hoelzl
parents: 56178
diff changeset
   245
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   246
end
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
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parents: 56178
diff changeset
   247
56213
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diff changeset
   248
lemma summableI_nonneg_bounded:
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diff changeset
   249
  fixes f:: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology, conditionally_complete_linorder}"
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hoelzl
parents: 56194
diff changeset
   250
  assumes pos[simp]: "\<And>n. 0 \<le> f n" and le: "\<And>n. (\<Sum>i<n. f i) \<le> x"
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parents: 56194
diff changeset
   251
  shows "summable f"
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diff changeset
   252
  unfolding summable_def sums_def[abs_def]
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parents: 56194
diff changeset
   253
proof (intro exI order_tendstoI)
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hoelzl
parents: 56194
diff changeset
   254
  have [simp, intro]: "bdd_above (range (\<lambda>n. \<Sum>i<n. f i))"
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hoelzl
parents: 56194
diff changeset
   255
    using le by (auto simp: bdd_above_def)
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parents: 56194
diff changeset
   256
  { fix a assume "a < (SUP n. \<Sum>i<n. f i)"
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parents: 56194
diff changeset
   257
    then obtain n where "a < (\<Sum>i<n. f i)"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   258
      by (auto simp add: less_cSUP_iff)
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hoelzl
parents: 56194
diff changeset
   259
    then have "\<And>m. n \<le> m \<Longrightarrow> a < (\<Sum>i<m. f i)"
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hoelzl
parents: 56194
diff changeset
   260
      by (rule less_le_trans) (auto intro!: setsum_mono2)
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   261
    then show "eventually (\<lambda>n. a < (\<Sum>i<n. f i)) sequentially"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   262
      by (auto simp: eventually_sequentially) }
e5720d3c18f0 further renaming in Series
hoelzl
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diff changeset
   263
  { fix a assume "(SUP n. \<Sum>i<n. f i) < a"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   264
    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
   265
      by (auto intro: cSUP_upper)
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   266
    ultimately show "eventually (\<lambda>n. (\<Sum>i<n. f i) < a) sequentially"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   267
      by (auto intro: le_less_trans simp: eventually_sequentially) }
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hoelzl
parents: 56194
diff changeset
   268
qed
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   269
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   270
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60141
diff changeset
   271
subsection \<open>Infinite summability on real normed vector spaces\<close>
56193
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hoelzl
parents: 56178
diff changeset
   272
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
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   273
lemma sums_Suc_iff:
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hoelzl
parents: 56178
diff changeset
   274
  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
   275
  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
   276
proof -
61969
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
   277
  have "f sums (s + f 0) \<longleftrightarrow> (\<lambda>i. \<Sum>j<Suc i. f j) \<longlonglongrightarrow> s + f 0"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   278
    by (subst LIMSEQ_Suc_iff) (simp add: sums_def)
61969
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
   279
  also have "\<dots> \<longleftrightarrow> (\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) \<longlonglongrightarrow> s + f 0"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57275
diff changeset
   280
    by (simp add: ac_simps setsum.reindex image_iff lessThan_Suc_eq_insert_0)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   281
  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
   282
  proof
61969
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
   283
    assume "(\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) \<longlonglongrightarrow> s + f 0"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   284
    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
   285
    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
   286
      by (simp add: sums_def)
58729
e8ecc79aee43 add tendsto_const and tendsto_ident_at as simp and intro rules
hoelzl
parents: 57418
diff changeset
   287
  qed (auto intro: tendsto_add simp: sums_def)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   288
  finally show ?thesis ..
50999
3de230ed0547 introduce order topology
hoelzl
parents: 50331
diff changeset
   289
qed
3de230ed0547 introduce order topology
hoelzl
parents: 50331
diff changeset
   290
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   291
lemma summable_Suc_iff: "summable (\<lambda>n. f (Suc n) :: 'a :: real_normed_vector) = summable f"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   292
proof
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   293
  assume "summable f"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   294
  hence "f sums suminf f" by (rule summable_sums)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   295
  hence "(\<lambda>n. f (Suc n)) sums (suminf f - f 0)" by (simp add: sums_Suc_iff)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   296
  thus "summable (\<lambda>n. f (Suc n))" unfolding summable_def by blast
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   297
qed (auto simp: sums_Suc_iff summable_def)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   298
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   299
context
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   300
  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
   301
begin
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   302
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   303
lemma sums_add: "f sums a \<Longrightarrow> g sums b \<Longrightarrow> (\<lambda>n. f n + g n) sums (a + b)"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57275
diff changeset
   304
  unfolding sums_def by (simp add: setsum.distrib tendsto_add)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   305
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   306
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
   307
  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
   308
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   309
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
   310
  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
   311
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   312
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
   313
  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
   314
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   315
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
   316
  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
   317
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   318
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
   319
  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
   320
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   321
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
   322
  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
   323
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   324
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
   325
  unfolding summable_def by (auto intro: sums_minus)
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   326
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   327
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
   328
  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
   329
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   330
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
   331
  by (simp add: sums_Suc_iff)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   332
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   333
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
   334
proof (induct n arbitrary: s)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   335
  case (Suc n)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   336
  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
   337
    by (subst sums_Suc_iff) simp
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   338
  ultimately show ?case
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   339
    by (simp add: ac_simps)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   340
qed simp
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   341
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   342
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
   343
  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
   344
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   345
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
   346
  by (simp add: sums_iff_shift)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   347
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   348
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
   349
  by (simp add: summable_iff_shift)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   350
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   351
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
   352
  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
   353
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   354
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
   355
  by (auto simp add: suminf_minus_initial_segment)
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   356
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   357
lemma suminf_split_head: "summable f \<Longrightarrow> (\<Sum>n. f (Suc n)) = suminf f - f 0"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   358
  using suminf_split_initial_segment[of 1] by simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   359
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   360
lemma suminf_exist_split:
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   361
  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
   362
  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
   363
proof -
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60141
diff changeset
   364
  from LIMSEQ_D[OF summable_LIMSEQ[OF \<open>summable f\<close>] \<open>0 < r\<close>]
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   365
  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
   366
  thus ?thesis
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60141
diff changeset
   367
    by (auto simp: norm_minus_commute suminf_minus_initial_segment[OF \<open>summable f\<close>])
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   368
qed
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   369
61969
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
   370
lemma summable_LIMSEQ_zero: "summable f \<Longrightarrow> f \<longlonglongrightarrow> 0"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   371
  apply (drule summable_iff_convergent [THEN iffD1])
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   372
  apply (drule convergent_Cauchy)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   373
  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
   374
  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
   375
  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
   376
  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
   377
  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
   378
  done
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   379
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   380
lemma summable_imp_convergent:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   381
  "summable (f :: nat \<Rightarrow> 'a) \<Longrightarrow> convergent f"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   382
  by (force dest!: summable_LIMSEQ_zero simp: convergent_def)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   383
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   384
lemma summable_imp_Bseq:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   385
  "summable f \<Longrightarrow> Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   386
  by (simp add: convergent_imp_Bseq summable_imp_convergent)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   387
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   388
end
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   389
59613
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59025
diff changeset
   390
lemma summable_minus_iff:
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59025
diff changeset
   391
  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59025
diff changeset
   392
  shows "summable (\<lambda>n. - f n) \<longleftrightarrow> summable f"
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61609
diff changeset
   393
  by (auto dest: summable_minus) \<comment>\<open>used two ways, hence must be outside the context above\<close>
59613
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59025
diff changeset
   394
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59025
diff changeset
   395
57025
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   396
context
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   397
  fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::real_normed_vector" and I :: "'i set"
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   398
begin
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   399
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   400
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
   401
  by (induct I rule: infinite_finite_induct) (auto intro!: sums_add)
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   402
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   403
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
   404
  using sums_unique[OF sums_setsum, OF summable_sums] by simp
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   405
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   406
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
   407
  using sums_summable[OF sums_setsum[OF summable_sums]] .
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   408
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   409
end
e7fd64f82876 add various lemmas
hoelzl
parents: 56536
diff changeset
   410
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   411
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
   412
  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
   413
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   414
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
   415
  unfolding summable_def by (auto intro: sums)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   416
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   417
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
   418
  by (intro sums_unique sums summable_sums)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   419
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   420
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
   421
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
   422
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
   423
57275
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57129
diff changeset
   424
lemmas sums_scaleR_left = bounded_linear.sums[OF bounded_linear_scaleR_left]
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57129
diff changeset
   425
lemmas summable_scaleR_left = bounded_linear.summable[OF bounded_linear_scaleR_left]
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57129
diff changeset
   426
lemmas suminf_scaleR_left = bounded_linear.suminf[OF bounded_linear_scaleR_left]
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57129
diff changeset
   427
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57129
diff changeset
   428
lemmas sums_scaleR_right = bounded_linear.sums[OF bounded_linear_scaleR_right]
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57129
diff changeset
   429
lemmas summable_scaleR_right = bounded_linear.summable[OF bounded_linear_scaleR_right]
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57129
diff changeset
   430
lemmas suminf_scaleR_right = bounded_linear.suminf[OF bounded_linear_scaleR_right]
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57129
diff changeset
   431
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   432
lemma summable_const_iff: "summable (\<lambda>_. c) \<longleftrightarrow> (c :: 'a :: real_normed_vector) = 0"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   433
proof -
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   434
  {
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   435
    assume "c \<noteq> 0"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   436
    hence "filterlim (\<lambda>n. of_nat n * norm c) at_top sequentially"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   437
      by (subst mult.commute)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   438
         (auto intro!: filterlim_tendsto_pos_mult_at_top filterlim_real_sequentially)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   439
    hence "\<not>convergent (\<lambda>n. norm (\<Sum>k<n. c))"
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   440
      by (intro filterlim_at_infinity_imp_not_convergent filterlim_at_top_imp_at_infinity)
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   441
         (simp_all add: setsum_constant_scaleR)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   442
    hence "\<not>summable (\<lambda>_. c)" unfolding summable_iff_convergent using convergent_norm by blast
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   443
  }
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   444
  thus ?thesis by auto
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   445
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   446
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   447
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60141
diff changeset
   448
subsection \<open>Infinite summability on real normed algebras\<close>
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   449
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   450
context
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   451
  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
   452
begin
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   453
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   454
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
   455
  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
   456
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   457
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
   458
  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
   459
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   460
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
   461
  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
   462
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   463
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
   464
  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
   465
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   466
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
   467
  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
   468
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   469
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
   470
  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
   471
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   472
end
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   473
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   474
lemma sums_mult_iff:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   475
  assumes "c \<noteq> 0"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   476
  shows   "(\<lambda>n. c * f n :: 'a :: {real_normed_algebra,field}) sums (c * d) \<longleftrightarrow> f sums d"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   477
  using sums_mult[of f d c] sums_mult[of "\<lambda>n. c * f n" "c * d" "inverse c"]
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   478
  by (force simp: field_simps assms)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   479
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   480
lemma sums_mult2_iff:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   481
  assumes "c \<noteq> (0 :: 'a :: {real_normed_algebra, field})"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   482
  shows   "(\<lambda>n. f n * c) sums (d * c) \<longleftrightarrow> f sums d"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   483
  using sums_mult_iff[OF assms, of f d] by (simp add: mult.commute)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   484
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   485
lemma sums_of_real_iff:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   486
  "(\<lambda>n. of_real (f n) :: 'a :: real_normed_div_algebra) sums of_real c \<longleftrightarrow> f sums c"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   487
  by (simp add: sums_def of_real_setsum[symmetric] tendsto_of_real_iff del: of_real_setsum)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   488
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   489
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60141
diff changeset
   490
subsection \<open>Infinite summability on real normed fields\<close>
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   491
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   492
context
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   493
  fixes c :: "'a::real_normed_field"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   494
begin
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   495
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   496
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
   497
  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
   498
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   499
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
   500
  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
   501
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   502
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
   503
  by (rule bounded_linear.suminf [OF bounded_linear_divide, symmetric])
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   504
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60141
diff changeset
   505
text\<open>Sum of a geometric progression.\<close>
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   506
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   507
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
   508
proof -
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   509
  assume less_1: "norm c < 1"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   510
  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
   511
  hence neq_0: "c - 1 \<noteq> 0" by simp
61969
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
   512
  from less_1 have lim_0: "(\<lambda>n. c^n) \<longlonglongrightarrow> 0"
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   513
    by (rule LIMSEQ_power_zero)
61969
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
   514
  hence "(\<lambda>n. c ^ n / (c - 1) - 1 / (c - 1)) \<longlonglongrightarrow> 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
   515
    using neq_0 by (intro tendsto_intros)
61969
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
   516
  hence "(\<lambda>n. (c ^ n - 1) / (c - 1)) \<longlonglongrightarrow> 1 / (1 - c)"
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   517
    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
   518
  thus "(\<lambda>n. c ^ n) sums (1 / (1 - c))"
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   519
    by (simp add: sums_def geometric_sum neq_1)
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   520
qed
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   521
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   522
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
   523
  by (rule geometric_sums [THEN sums_summable])
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   524
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   525
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
   526
  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
   527
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   528
lemma summable_geometric_iff: "summable (\<lambda>n. c ^ n) \<longleftrightarrow> norm c < 1"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   529
proof
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   530
  assume "summable (\<lambda>n. c ^ n :: 'a :: real_normed_field)"
61969
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
   531
  hence "(\<lambda>n. norm c ^ n) \<longlonglongrightarrow> 0"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   532
    by (simp add: norm_power [symmetric] tendsto_norm_zero_iff summable_LIMSEQ_zero)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   533
  from order_tendstoD(2)[OF this zero_less_one] obtain n where "norm c ^ n < 1"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   534
    by (auto simp: eventually_at_top_linorder)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   535
  thus "norm c < 1" using one_le_power[of "norm c" n] by (cases "norm c \<ge> 1") (linarith, simp)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   536
qed (rule summable_geometric)
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   537
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   538
end
33271
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 32877
diff changeset
   539
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 32877
diff changeset
   540
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
   541
proof -
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 32877
diff changeset
   542
  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
   543
    by auto
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 32877
diff changeset
   544
  have "(\<lambda>n. (1/2::real)^Suc n) = (\<lambda>n. (1 / 2) ^ n / 2)"
59741
5b762cd73a8e Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents: 59712
diff changeset
   545
    by (simp add: mult.commute)
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 41970
diff changeset
   546
  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
   547
    by simp
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 32877
diff changeset
   548
qed
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 32877
diff changeset
   549
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   550
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   551
subsection \<open>Telescoping\<close>
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   552
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   553
lemma telescope_sums:
61969
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
   554
  assumes "f \<longlonglongrightarrow> (c :: 'a :: real_normed_vector)"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   555
  shows   "(\<lambda>n. f (Suc n) - f n) sums (c - f 0)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   556
  unfolding sums_def
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   557
proof (subst LIMSEQ_Suc_iff [symmetric])
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   558
  have "(\<lambda>n. \<Sum>k<Suc n. f (Suc k) - f k) = (\<lambda>n. f (Suc n) - f 0)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   559
    by (simp add: lessThan_Suc_atMost atLeast0AtMost [symmetric] setsum_Suc_diff)
61969
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
   560
  also have "\<dots> \<longlonglongrightarrow> c - f 0" by (intro tendsto_diff LIMSEQ_Suc[OF assms] tendsto_const)
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
   561
  finally show "(\<lambda>n. \<Sum>n<Suc n. f (Suc n) - f n) \<longlonglongrightarrow> c - f 0" .
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   562
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   563
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   564
lemma telescope_sums':
61969
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
   565
  assumes "f \<longlonglongrightarrow> (c :: 'a :: real_normed_vector)"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   566
  shows   "(\<lambda>n. f n - f (Suc n)) sums (f 0 - c)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   567
  using sums_minus[OF telescope_sums[OF assms]] by (simp add: algebra_simps)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   568
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   569
lemma telescope_summable:
61969
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
   570
  assumes "f \<longlonglongrightarrow> (c :: 'a :: real_normed_vector)"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   571
  shows   "summable (\<lambda>n. f (Suc n) - f n)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   572
  using telescope_sums[OF assms] by (simp add: sums_iff)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   573
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   574
lemma telescope_summable':
61969
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
   575
  assumes "f \<longlonglongrightarrow> (c :: 'a :: real_normed_vector)"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   576
  shows   "summable (\<lambda>n. f n - f (Suc n))"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   577
  using summable_minus[OF telescope_summable[OF assms]] by (simp add: algebra_simps)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   578
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   579
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60141
diff changeset
   580
subsection \<open>Infinite summability on Banach spaces\<close>
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   581
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60141
diff changeset
   582
text\<open>Cauchy-type criterion for convergence of series (c.f. Harrison)\<close>
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   583
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   584
lemma summable_Cauchy:
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   585
  fixes f :: "nat \<Rightarrow> 'a::banach"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   586
  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
   587
  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
   588
  apply (drule spec, drule (1) mp)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   589
  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
   590
  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
   591
  apply (frule (1) order_trans)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   592
  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
   593
  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
   594
  apply (simp_all add: setsum_diff [symmetric])
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   595
  apply (drule spec, drule (1) mp)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   596
  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
   597
  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
   598
  apply (subst norm_minus_commute)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   599
  apply (simp_all add: setsum_diff [symmetric])
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   600
  done
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   601
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   602
context
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   603
  fixes f :: "nat \<Rightarrow> 'a::banach"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   604
begin
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   605
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60141
diff changeset
   606
text\<open>Absolute convergence imples normal convergence\<close>
20689
4950e45442b8 add proof of summable_LIMSEQ_zero
huffman
parents: 20688
diff changeset
   607
56194
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56193
diff changeset
   608
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
   609
  apply (simp only: summable_Cauchy, safe)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   610
  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
   611
  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
   612
  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
   613
  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
   614
  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
   615
  apply simp
50999
3de230ed0547 introduce order topology
hoelzl
parents: 50331
diff changeset
   616
  done
32707
836ec9d0a0c8 New lemmas involving the real numbers, especially limits and series
paulson
parents: 31336
diff changeset
   617
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   618
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
   619
  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
   620
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60141
diff changeset
   621
text \<open>Comparison tests\<close>
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   622
56194
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56193
diff changeset
   623
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
   624
  apply (simp add: summable_Cauchy, safe)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   625
  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
   626
  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
   627
  apply (rotate_tac 2)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   628
  apply (drule_tac x = m in spec)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   629
  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
   630
  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
   631
  apply (rule norm_setsum)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   632
  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
   633
  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
   634
  done
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   635
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   636
lemma summable_comparison_test_ev:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   637
  shows "eventually (\<lambda>n. norm (f n) \<le> g n) sequentially \<Longrightarrow> summable g \<Longrightarrow> summable f"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   638
  by (rule summable_comparison_test) (auto simp: eventually_at_top_linorder)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   639
56217
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 56213
diff changeset
   640
(*A better argument order*)
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 56213
diff changeset
   641
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
   642
  by (rule summable_comparison_test) auto
56217
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 56213
diff changeset
   643
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60141
diff changeset
   644
subsection \<open>The Ratio Test\<close>
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   645
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   646
lemma summable_ratio_test:
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   647
  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
   648
  shows "summable f"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   649
proof cases
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   650
  assume "0 < c"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   651
  show "summable f"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   652
  proof (rule summable_comparison_test)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   653
    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
   654
    proof (intro exI allI impI)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   655
      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
   656
      proof (induct rule: inc_induct)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   657
        case (step m)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   658
        moreover have "norm (f (Suc m)) / c ^ Suc m * c ^ n \<le> norm (f m) / c ^ m * c ^ n"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60141
diff changeset
   659
          using \<open>0 < c\<close> \<open>c < 1\<close> assms(2)[OF \<open>N \<le> m\<close>] by (simp add: field_simps)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   660
        ultimately show ?case by simp
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60141
diff changeset
   661
      qed (insert \<open>0 < c\<close>, simp)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   662
    qed
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   663
    show "summable (\<lambda>n. norm (f N) / c ^ N * c ^ n)"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60141
diff changeset
   664
      using \<open>0 < c\<close> \<open>c < 1\<close> by (intro summable_mult summable_geometric) simp
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   665
  qed
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   666
next
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   667
  assume c: "\<not> 0 < c"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   668
  { fix n assume "n \<ge> N"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   669
    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
   670
      by fact
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   671
    also have "\<dots> \<le> 0"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   672
      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
   673
    finally have "f (Suc n) = 0"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   674
      by auto }
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   675
  then show "summable f"
56194
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56193
diff changeset
   676
    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
   677
qed
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 54703
diff changeset
   678
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   679
end
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   680
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60141
diff changeset
   681
text\<open>Relations among convergence and absolute convergence for power series.\<close>
56369
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   682
62087
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 62049
diff changeset
   683
lemma Abel_lemma:
56369
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   684
  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
   685
  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
   686
    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
   687
proof (rule summable_comparison_test')
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   688
  show "summable (\<lambda>n. M * (r / r0) ^ n)"
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   689
    using assms
56369
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   690
    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
   691
next
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   692
  fix n
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   693
  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
   694
    using r r0 M [of n]
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   695
    apply (auto simp add: abs_mult field_simps)
56369
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   696
    apply (cases "r=0", simp)
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   697
    apply (cases n, auto)
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   698
    done
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   699
qed
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   700
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   701
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60141
diff changeset
   702
text\<open>Summability of geometric series for real algebras\<close>
23084
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   703
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   704
lemma complete_algebra_summable_geometric:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30649
diff changeset
   705
  fixes x :: "'a::{real_normed_algebra_1,banach}"
23084
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   706
  shows "norm x < 1 \<Longrightarrow> summable (\<lambda>n. x ^ n)"
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   707
proof (rule summable_comparison_test)
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   708
  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
   709
    by (simp add: norm_power_ineq)
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   710
  show "norm x < 1 \<Longrightarrow> summable (\<lambda>n. norm x ^ n)"
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   711
    by (simp add: summable_geometric)
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   712
qed
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   713
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60141
diff changeset
   714
subsection \<open>Cauchy Product Formula\<close>
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   715
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60141
diff changeset
   716
text \<open>
54703
499f92dc6e45 more antiquotations;
wenzelm
parents: 54230
diff changeset
   717
  Proof based on Analysis WebNotes: Chapter 07, Class 41
499f92dc6e45 more antiquotations;
wenzelm
parents: 54230
diff changeset
   718
  @{url "http://www.math.unl.edu/~webnotes/classes/class41/prp77.htm"}
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60141
diff changeset
   719
\<close>
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   720
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   721
lemma Cauchy_product_sums:
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   722
  fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   723
  assumes a: "summable (\<lambda>k. norm (a k))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   724
  assumes b: "summable (\<lambda>k. norm (b k))"
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   725
  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
   726
proof -
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   727
  let ?S1 = "\<lambda>n::nat. {..<n} \<times> {..<n}"
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   728
  let ?S2 = "\<lambda>n::nat. {(i,j). i + j < n}"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   729
  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
   730
  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
   731
  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
   732
  have finite_S1: "\<And>n. finite (?S1 n)" by simp
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   733
  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
   734
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   735
  let ?g = "\<lambda>(i,j). a i * b j"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   736
  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
   737
  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
   738
  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
   739
    unfolding real_norm_def
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   740
    by (simp only: abs_of_nonneg setsum_nonneg [rule_format])
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   741
61969
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
   742
  have "(\<lambda>n. (\<Sum>k<n. a k) * (\<Sum>k<n. b k)) \<longlonglongrightarrow> (\<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
   743
    by (intro tendsto_mult summable_LIMSEQ summable_norm_cancel [OF a] summable_norm_cancel [OF b])
61969
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
   744
  hence 1: "(\<lambda>n. setsum ?g (?S1 n)) \<longlonglongrightarrow> (\<Sum>k. a k) * (\<Sum>k. b k)"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57275
diff changeset
   745
    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
   746
61969
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
   747
  have "(\<lambda>n. (\<Sum>k<n. norm (a k)) * (\<Sum>k<n. norm (b k))) \<longlonglongrightarrow> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   748
    using a b by (intro tendsto_mult summable_LIMSEQ)
61969
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
   749
  hence "(\<lambda>n. setsum ?f (?S1 n)) \<longlonglongrightarrow> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57275
diff changeset
   750
    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
   751
  hence "convergent (\<lambda>n. setsum ?f (?S1 n))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   752
    by (rule convergentI)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   753
  hence Cauchy: "Cauchy (\<lambda>n. setsum ?f (?S1 n))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   754
    by (rule convergent_Cauchy)
36657
f376af79f6b7 remove unneeded constant Zseq
huffman
parents: 36409
diff changeset
   755
  have "Zfun (\<lambda>n. setsum ?f (?S1 n - ?S2 n)) sequentially"
f376af79f6b7 remove unneeded constant Zseq
huffman
parents: 36409
diff changeset
   756
  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
   757
    fix r :: real
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   758
    assume r: "0 < r"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   759
    from CauchyD [OF Cauchy r] obtain N
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   760
    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
   761
    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
   762
      by (simp only: setsum_diff finite_S1 S1_mono)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   763
    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
   764
      by (simp only: norm_setsum_f)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   765
    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
   766
    proof (intro exI allI impI)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   767
      fix n assume "2 * N \<le> n"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   768
      hence n: "N \<le> n div 2" by simp
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   769
      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
   770
        by (intro setsum_mono2 finite_Diff finite_S1 f_nonneg
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   771
                  Diff_mono subset_refl S1_le_S2)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   772
      also have "\<dots> < r"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   773
        using n div_le_dividend by (rule N)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   774
      finally show "setsum ?f (?S1 n - ?S2 n) < r" .
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   775
    qed
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   776
  qed
36657
f376af79f6b7 remove unneeded constant Zseq
huffman
parents: 36409
diff changeset
   777
  hence "Zfun (\<lambda>n. setsum ?g (?S1 n - ?S2 n)) sequentially"
f376af79f6b7 remove unneeded constant Zseq
huffman
parents: 36409
diff changeset
   778
    apply (rule Zfun_le [rule_format])
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   779
    apply (simp only: norm_setsum_f)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   780
    apply (rule order_trans [OF norm_setsum setsum_mono])
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   781
    apply (auto simp add: norm_mult_ineq)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   782
    done
61969
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
   783
  hence 2: "(\<lambda>n. setsum ?g (?S1 n) - setsum ?g (?S2 n)) \<longlonglongrightarrow> 0"
36660
1cc4ab4b7ff7 make (X ----> L) an abbreviation for (X ---> L) sequentially
huffman
parents: 36657
diff changeset
   784
    unfolding tendsto_Zfun_iff diff_0_right
36657
f376af79f6b7 remove unneeded constant Zseq
huffman
parents: 36409
diff changeset
   785
    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
   786
61969
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
   787
  with 1 have "(\<lambda>n. setsum ?g (?S2 n)) \<longlonglongrightarrow> (\<Sum>k. a k) * (\<Sum>k. b k)"
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 59741
diff changeset
   788
    by (rule Lim_transform2)
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   789
  thus ?thesis by (simp only: sums_def setsum_triangle_reindex)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   790
qed
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   791
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   792
lemma Cauchy_product:
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   793
  fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   794
  assumes a: "summable (\<lambda>k. norm (a k))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   795
  assumes b: "summable (\<lambda>k. norm (b k))"
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   796
  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
   797
  using a b
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   798
  by (rule Cauchy_product_sums [THEN sums_unique])
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   799
62049
b0f941e207cf Added lots of material on infinite sums, convergence radii, harmonic numbers, Gamma function
eberlm
parents: 61969
diff changeset
   800
lemma summable_Cauchy_product:
62087
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 62049
diff changeset
   801
  assumes "summable (\<lambda>k. norm (a k :: 'a :: {real_normed_algebra,banach}))"
62049
b0f941e207cf Added lots of material on infinite sums, convergence radii, harmonic numbers, Gamma function
eberlm
parents: 61969
diff changeset
   802
          "summable (\<lambda>k. norm (b k))"
b0f941e207cf Added lots of material on infinite sums, convergence radii, harmonic numbers, Gamma function
eberlm
parents: 61969
diff changeset
   803
  shows   "summable (\<lambda>k. \<Sum>i\<le>k. a i * b (k - i))"
62087
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 62049
diff changeset
   804
  using Cauchy_product_sums[OF assms] by (simp add: sums_iff)
62049
b0f941e207cf Added lots of material on infinite sums, convergence radii, harmonic numbers, Gamma function
eberlm
parents: 61969
diff changeset
   805
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60141
diff changeset
   806
subsection \<open>Series on @{typ real}s\<close>
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   807
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   808
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
   809
  by (rule summable_comparison_test) auto
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   810
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   811
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
   812
  by (rule summable_comparison_test) auto
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   813
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   814
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
   815
  by (rule summable_norm_cancel) simp
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   816
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   817
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
   818
  by (fold real_norm_def) (rule summable_norm)
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   819
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   820
lemma summable_zero_power [simp]: "summable (\<lambda>n. 0 ^ n :: 'a :: {comm_ring_1,topological_space})"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   821
proof -
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   822
  have "(\<lambda>n. 0 ^ n :: 'a) = (\<lambda>n. if n = 0 then 0^0 else 0)" by (intro ext) (simp add: zero_power)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   823
  moreover have "summable \<dots>" by simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   824
  ultimately show ?thesis by simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   825
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   826
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   827
lemma summable_zero_power' [simp]: "summable (\<lambda>n. f n * 0 ^ n :: 'a :: {ring_1,topological_space})"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   828
proof -
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   829
  have "(\<lambda>n. f n * 0 ^ n :: 'a) = (\<lambda>n. if n = 0 then f 0 * 0^0 else 0)"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   830
    by (intro ext) (simp add: zero_power)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   831
  moreover have "summable \<dots>" by simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   832
  ultimately show ?thesis by simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   833
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   834
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   835
lemma summable_power_series:
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   836
  fixes z :: real
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   837
  assumes le_1: "\<And>i. f i \<le> 1" and nonneg: "\<And>i. 0 \<le> f i" and z: "0 \<le> z" "z < 1"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   838
  shows "summable (\<lambda>i. f i * z^i)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   839
proof (rule summable_comparison_test[OF _ summable_geometric])
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   840
  show "norm z < 1" using z by (auto simp: less_imp_le)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   841
  show "\<And>n. \<exists>N. \<forall>na\<ge>N. norm (f na * z ^ na) \<le> z ^ na"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   842
    using z by (auto intro!: exI[of _ 0] mult_left_le_one_le simp: abs_mult nonneg power_abs less_imp_le le_1)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   843
qed
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   844
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   845
lemma summable_0_powser:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   846
  "summable (\<lambda>n. f n * 0 ^ n :: 'a :: real_normed_div_algebra)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   847
proof -
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   848
  have A: "(\<lambda>n. f n * 0 ^ n) = (\<lambda>n. if n = 0 then f n else 0)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   849
    by (intro ext) auto
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   850
  thus ?thesis by (subst A) simp_all
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   851
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   852
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   853
lemma summable_powser_split_head:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   854
  "summable (\<lambda>n. f (Suc n) * z ^ n :: 'a :: real_normed_div_algebra) = summable (\<lambda>n. f n * z ^ n)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   855
proof -
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   856
  have "summable (\<lambda>n. f (Suc n) * z ^ n) \<longleftrightarrow> summable (\<lambda>n. f (Suc n) * z ^ Suc n)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   857
  proof
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   858
    assume "summable (\<lambda>n. f (Suc n) * z ^ n)"
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   859
    from summable_mult2[OF this, of z] show "summable (\<lambda>n. f (Suc n) * z ^ Suc n)"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   860
      by (simp add: power_commutes algebra_simps)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   861
  next
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   862
    assume "summable (\<lambda>n. f (Suc n) * z ^ Suc n)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   863
    from summable_mult2[OF this, of "inverse z"] show "summable (\<lambda>n. f (Suc n) * z ^ n)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   864
      by (cases "z \<noteq> 0", subst (asm) power_Suc2) (simp_all add: algebra_simps)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   865
  qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   866
  also have "\<dots> \<longleftrightarrow> summable (\<lambda>n. f n * z ^ n)" by (rule summable_Suc_iff)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   867
  finally show ?thesis .
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   868
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   869
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   870
lemma powser_split_head:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   871
  assumes "summable (\<lambda>n. f n * z ^ n :: 'a :: {real_normed_div_algebra,banach})"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   872
  shows   "suminf (\<lambda>n. f n * z ^ n) = f 0 + suminf (\<lambda>n. f (Suc n) * z ^ n) * z"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   873
          "suminf (\<lambda>n. f (Suc n) * z ^ n) * z = suminf (\<lambda>n. f n * z ^ n) - f 0"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   874
          "summable (\<lambda>n. f (Suc n) * z ^ n)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   875
proof -
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   876
  from assms show "summable (\<lambda>n. f (Suc n) * z ^ n)" by (subst summable_powser_split_head)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   877
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   878
  from suminf_mult2[OF this, of z]
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   879
    have "(\<Sum>n. f (Suc n) * z ^ n) * z = (\<Sum>n. f (Suc n) * z ^ Suc n)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   880
    by (simp add: power_commutes algebra_simps)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   881
  also from assms have "\<dots> = suminf (\<lambda>n. f n * z ^ n) - f 0"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   882
    by (subst suminf_split_head) simp_all
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   883
  finally show "suminf (\<lambda>n. f n * z ^ n) = f 0 + suminf (\<lambda>n. f (Suc n) * z ^ n) * z" by simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   884
  thus "suminf (\<lambda>n. f (Suc n) * z ^ n) * z = suminf (\<lambda>n. f n * z ^ n) - f 0" by simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   885
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   886
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   887
lemma summable_partial_sum_bound:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   888
  fixes f :: "nat \<Rightarrow> 'a :: banach"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   889
  assumes summable: "summable f" and e: "e > (0::real)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   890
  obtains N where "\<And>m n. m \<ge> N \<Longrightarrow> norm (\<Sum>k=m..n. f k) < e"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   891
proof -
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   892
  from summable have "Cauchy (\<lambda>n. \<Sum>k<n. f k)"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   893
    by (simp add: Cauchy_convergent_iff summable_iff_convergent)
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   894
  from CauchyD[OF this e] obtain N
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   895
    where N: "\<And>m n. m \<ge> N \<Longrightarrow> n \<ge> N \<Longrightarrow> norm ((\<Sum>k<m. f k) - (\<Sum>k<n. f k)) < e" by blast
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   896
  {
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   897
    fix m n :: nat assume m: "m \<ge> N"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   898
    have "norm (\<Sum>k=m..n. f k) < e"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   899
    proof (cases "n \<ge> m")
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   900
      assume n: "n \<ge> m"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   901
      with m have "norm ((\<Sum>k<Suc n. f k) - (\<Sum>k<m. f k)) < e" by (intro N) simp_all
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   902
      also from n have "(\<Sum>k<Suc n. f k) - (\<Sum>k<m. f k) = (\<Sum>k=m..n. f k)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   903
        by (subst setsum_diff [symmetric]) (simp_all add: setsum_last_plus)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   904
      finally show ?thesis .
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   905
    qed (insert e, simp_all)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   906
  }
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   907
  thus ?thesis by (rule that)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   908
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   909
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   910
lemma powser_sums_if:
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   911
  "(\<lambda>n. (if n = m then (1 :: 'a :: {ring_1,topological_space}) else 0) * z^n) sums z^m"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   912
proof -
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   913
  have "(\<lambda>n. (if n = m then 1 else 0) * z^n) = (\<lambda>n. if n = m then z^n else 0)"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   914
    by (intro ext) auto
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   915
  thus ?thesis by (simp add: sums_single)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   916
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   917
59025
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   918
lemma
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   919
   fixes f :: "nat \<Rightarrow> real"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   920
   assumes "summable f"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   921
   and "inj g"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   922
   and pos: "!!x. 0 \<le> f x"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   923
   shows summable_reindex: "summable (f o g)"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   924
   and suminf_reindex_mono: "suminf (f o g) \<le> suminf f"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   925
   and suminf_reindex: "(\<And>x. x \<notin> range g \<Longrightarrow> f x = 0) \<Longrightarrow> suminf (f \<circ> g) = suminf f"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   926
proof -
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   927
  from \<open>inj g\<close> have [simp]: "\<And>A. inj_on g A" by(rule subset_inj_on) simp
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   928
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   929
  have smaller: "\<forall>n. (\<Sum>i<n. (f \<circ> g) i) \<le> suminf f"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   930
  proof
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   931
    fix n
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   932
    have "\<forall> n' \<in> (g ` {..<n}). n' < Suc (Max (g ` {..<n}))"
59025
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   933
      by(metis Max_ge finite_imageI finite_lessThan not_le not_less_eq)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   934
    then obtain m where n: "\<And>n'. n' < n \<Longrightarrow> g n' < m" by blast
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   935
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   936
    have "(\<Sum>i<n. f (g i)) = setsum f (g ` {..<n})"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   937
      by (simp add: setsum.reindex)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   938
    also have "\<dots> \<le> (\<Sum>i<m. f i)"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   939
      by (rule setsum_mono3) (auto simp add: pos n[rule_format])
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   940
    also have "\<dots> \<le> suminf f"
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
   941
      using \<open>summable f\<close>
59025
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   942
      by (rule setsum_le_suminf) (simp add: pos)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   943
    finally show "(\<Sum>i<n. (f \<circ>  g) i) \<le> suminf f" by simp
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   944
  qed
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   945
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   946
  have "incseq (\<lambda>n. \<Sum>i<n. (f \<circ> g) i)"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   947
    by (rule incseq_SucI) (auto simp add: pos)
61969
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
   948
  then obtain  L where L: "(\<lambda> n. \<Sum>i<n. (f \<circ> g) i) \<longlonglongrightarrow> L"
59025
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   949
    using smaller by(rule incseq_convergent)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   950
  hence "(f \<circ> g) sums L" by (simp add: sums_def)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   951
  thus "summable (f o g)" by (auto simp add: sums_iff)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   952
61969
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
   953
  hence "(\<lambda>n. \<Sum>i<n. (f \<circ> g) i) \<longlonglongrightarrow> suminf (f \<circ> g)"
59025
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   954
    by(rule summable_LIMSEQ)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   955
  thus le: "suminf (f \<circ> g) \<le> suminf f"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   956
    by(rule LIMSEQ_le_const2)(blast intro: smaller[rule_format])
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   957
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   958
  assume f: "\<And>x. x \<notin> range g \<Longrightarrow> f x = 0"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   959
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   960
  from \<open>summable f\<close> have "suminf f \<le> suminf (f \<circ> g)"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   961
  proof(rule suminf_le_const)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   962
    fix n
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   963
    have "\<forall> n' \<in> (g -` {..<n}). n' < Suc (Max (g -` {..<n}))"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   964
      by(auto intro: Max_ge simp add: finite_vimageI less_Suc_eq_le)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   965
    then obtain m where n: "\<And>n'. g n' < n \<Longrightarrow> n' < m" by blast
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   966
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   967
    have "(\<Sum>i<n. f i) = (\<Sum>i\<in>{..<n} \<inter> range g. f i)"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   968
      using f by(auto intro: setsum.mono_neutral_cong_right)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   969
    also have "\<dots> = (\<Sum>i\<in>g -` {..<n}. (f \<circ> g) i)"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   970
      by(rule setsum.reindex_cong[where l=g])(auto)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   971
    also have "\<dots> \<le> (\<Sum>i<m. (f \<circ> g) i)"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   972
      by(rule setsum_mono3)(auto simp add: pos n)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   973
    also have "\<dots> \<le> suminf (f \<circ> g)"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   974
      using \<open>summable (f o g)\<close>
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   975
      by(rule setsum_le_suminf)(simp add: pos)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   976
    finally show "setsum f {..<n} \<le> suminf (f \<circ> g)" .
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   977
  qed
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   978
  with le show "suminf (f \<circ> g) = suminf f" by(rule antisym)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   979
qed
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   980
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   981
lemma sums_mono_reindex:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   982
  assumes subseq: "subseq g" and zero: "\<And>n. n \<notin> range g \<Longrightarrow> f n = 0"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   983
  shows   "(\<lambda>n. f (g n)) sums c \<longleftrightarrow> f sums c"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   984
unfolding sums_def
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   985
proof
61969
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
   986
  assume lim: "(\<lambda>n. \<Sum>k<n. f k) \<longlonglongrightarrow> c"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   987
  have "(\<lambda>n. \<Sum>k<n. f (g k)) = (\<lambda>n. \<Sum>k<g n. f k)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   988
  proof
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   989
    fix n :: nat
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   990
    from subseq have "(\<Sum>k<n. f (g k)) = (\<Sum>k\<in>g`{..<n}. f k)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   991
      by (subst setsum.reindex) (auto intro: subseq_imp_inj_on)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   992
    also from subseq have "\<dots> = (\<Sum>k<g n. f k)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   993
      by (intro setsum.mono_neutral_left ballI zero)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   994
         (auto dest: subseq_strict_mono simp: strict_mono_less strict_mono_less_eq)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   995
    finally show "(\<Sum>k<n. f (g k)) = (\<Sum>k<g n. f k)" .
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   996
  qed
61969
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
   997
  also from LIMSEQ_subseq_LIMSEQ[OF lim subseq] have "\<dots> \<longlonglongrightarrow> c" unfolding o_def .
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
   998
  finally show "(\<lambda>n. \<Sum>k<n. f (g k)) \<longlonglongrightarrow> c" .
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
   999
next
61969
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
  1000
  assume lim: "(\<lambda>n. \<Sum>k<n. f (g k)) \<longlonglongrightarrow> c"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1001
  def g_inv \<equiv> "\<lambda>n. LEAST m. g m \<ge> n"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1002
  from filterlim_subseq[OF subseq] have g_inv_ex: "\<exists>m. g m \<ge> n" for n
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1003
    by (auto simp: filterlim_at_top eventually_at_top_linorder)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1004
  hence g_inv: "g (g_inv n) \<ge> n" for n unfolding g_inv_def by (rule LeastI_ex)
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
  1005
  have g_inv_least: "m \<ge> g_inv n" if "g m \<ge> n" for m n using that
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1006
    unfolding g_inv_def by (rule Least_le)
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
  1007
  have g_inv_least': "g m < n" if "m < g_inv n" for m n using that g_inv_least[of n m] by linarith
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1008
  have "(\<lambda>n. \<Sum>k<n. f k) = (\<lambda>n. \<Sum>k<g_inv n. f (g k))"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1009
  proof
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1010
    fix n :: nat
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1011
    {
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1012
      fix k assume k: "k \<in> {..<n} - g`{..<g_inv n}"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1013
      have "k \<notin> range g"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1014
      proof (rule notI, elim imageE)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1015
        fix l assume l: "k = g l"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1016
        have "g l < g (g_inv n)" by (rule less_le_trans[OF _ g_inv]) (insert k l, simp_all)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1017
        with subseq have "l < g_inv n" by (simp add: subseq_strict_mono strict_mono_less)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1018
        with k l show False by simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1019
      qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1020
      hence "f k = 0" by (rule zero)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1021
    }
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1022
    with g_inv_least' g_inv have "(\<Sum>k<n. f k) = (\<Sum>k\<in>g`{..<g_inv n}. f k)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1023
      by (intro setsum.mono_neutral_right) auto
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
  1024
    also from subseq have "\<dots> = (\<Sum>k<g_inv n. f (g k))" using subseq_imp_inj_on
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1025
      by (subst setsum.reindex) simp_all
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1026
    finally show "(\<Sum>k<n. f k) = (\<Sum>k<g_inv n. f (g k))" .
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1027
  qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1028
  also {
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1029
    fix K n :: nat assume "g K \<le> n"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1030
    also have "n \<le> g (g_inv n)" by (rule g_inv)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1031
    finally have "K \<le> g_inv n" using subseq by (simp add: strict_mono_less_eq subseq_strict_mono)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1032
  }
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
  1033
  hence "filterlim g_inv at_top sequentially"
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1034
    by (auto simp: filterlim_at_top eventually_at_top_linorder)
61969
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
  1035
  from lim and this have "(\<lambda>n. \<Sum>k<g_inv n. f (g k)) \<longlonglongrightarrow> c" by (rule filterlim_compose)
e01015e49041 more symbols;
wenzelm
parents: 61799
diff changeset
  1036
  finally show "(\<lambda>n. \<Sum>k<n. f k) \<longlonglongrightarrow> c" .
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1037
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1038
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1039
lemma summable_mono_reindex:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1040
  assumes subseq: "subseq g" and zero: "\<And>n. n \<notin> range g \<Longrightarrow> f n = 0"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1041
  shows   "summable (\<lambda>n. f (g n)) \<longleftrightarrow> summable f"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1042
  using sums_mono_reindex[of g f, OF assms] by (simp add: summable_def)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1043
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
  1044
lemma suminf_mono_reindex:
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1045
  assumes "subseq g" "\<And>n. n \<notin> range g \<Longrightarrow> f n = (0 :: 'a :: {t2_space,comm_monoid_add})"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1046
  shows   "suminf (\<lambda>n. f (g n)) = suminf f"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1047
proof (cases "summable f")
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1048
  case False
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1049
  hence "\<not>(\<exists>c. f sums c)" unfolding summable_def by blast
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1050
  hence "suminf f = The (\<lambda>_. False)" by (simp add: suminf_def)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1051
  moreover from False have "\<not>summable (\<lambda>n. f (g n))"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1052
    using summable_mono_reindex[of g f, OF assms] by simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1053
  hence "\<not>(\<exists>c. (\<lambda>n. f (g n)) sums c)" unfolding summable_def by blast
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1054
  hence "suminf (\<lambda>n. f (g n)) = The (\<lambda>_. False)" by (simp add: suminf_def)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1055
  ultimately show ?thesis by simp
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61531
diff changeset
  1056
qed (insert sums_mono_reindex[of g f, OF assms] summable_mono_reindex[of g f, OF assms],
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1057
     simp_all add: sums_iff)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 60867
diff changeset
  1058
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
  1059
end