src/HOL/Series.thy
author paulson
Thu Jan 07 17:40:55 2016 +0000 (2016-01-07)
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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Converted to Isar and polished by lcp
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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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  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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  by (simp_all only: summable_iff_convergent convergent_def
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        lessThan_Suc_atMost [symmetric] LIMSEQ_Suc_iff[of "\<lambda>n. setsum f {..<n}"])
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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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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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  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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  apply (drule_tac x="n*k" in spec)
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  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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  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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  shows   "summable f = summable g"
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proof -
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  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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  def C \<equiv> "(\<Sum>k<N. f k - g k)"
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  from eventually_ge_at_top[of N]
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    have "eventually (\<lambda>n. setsum f {..<n} = C + setsum g {..<n}) sequentially"
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  proof eventually_elim
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    fix n assume n: "n \<ge> N"
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    from n have "{..<n} = {..<N} \<union> {N..<n}" by auto
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    also have "setsum f ... = setsum f {..<N} + setsum f {N..<n}"
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      by (intro setsum.union_disjoint) auto
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    also from N have "setsum f {N..<n} = setsum g {N..<n}" by (intro setsum.cong) simp_all
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    also have "setsum f {..<N} + setsum g {N..<n} = C + (setsum g {..<N} + setsum g {N..<n})"
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      unfolding C_def by (simp add: algebra_simps setsum_subtractf)
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    also have "setsum g {..<N} + setsum g {N..<n} = setsum g ({..<N} \<union> {N..<n})"
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      by (intro setsum.union_disjoint [symmetric]) auto
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    also from n have "{..<N} \<union> {N..<n} = {..<n}" by auto
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    finally show "setsum f {..<n} = C + setsum g {..<n}" .
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  qed
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  from convergent_cong[OF this] show ?thesis
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    by (simp add: summable_iff_convergent convergent_add_const_iff)
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qed
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lemma sums_finite:
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  assumes [simp]: "finite N" and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 0"
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  shows "f sums (\<Sum>n\<in>N. f n)"
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proof -
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  { fix n
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    have "setsum f {..<n + Suc (Max N)} = setsum f N"
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    proof cases
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      assume "N = {}"
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      with f have "f = (\<lambda>x. 0)" by auto
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      then show ?thesis by simp
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    next
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      assume [simp]: "N \<noteq> {}"
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      show ?thesis
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      proof (safe intro!: setsum.mono_neutral_right f)
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        fix i assume "i \<in> N"
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        then have "i \<le> Max N" by simp
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        then show "i < n + Suc (Max N)" by simp
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      qed
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    qed }
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  note eq = this
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  show ?thesis unfolding sums_def
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    by (rule LIMSEQ_offset[of _ "Suc (Max N)"])
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       (simp add: eq atLeast0LessThan del: add_Suc_right)
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qed
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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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  by (rule sums_summable) (rule sums_finite)
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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)"
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  using sums_finite[of A "(\<lambda>r. if r \<in> A then f r else 0)"] by simp
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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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  by (rule sums_summable) (rule sums_If_finite_set)
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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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  using sums_If_finite_set[of "{r. P r}"] by simp
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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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  by (rule sums_summable) (rule sums_If_finite)
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lemma sums_single: "(\<lambda>r. if r = i then f r else 0) sums f i"
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  using sums_If_finite[of "\<lambda>r. r = i"] by simp
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lemma summable_single[simp, intro]: "summable (\<lambda>r. if r = i then f r else 0)"
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  by (rule sums_summable) (rule sums_single)
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context
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  fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}"
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begin
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lemma summable_sums[intro]: "summable f \<Longrightarrow> f sums (suminf f)"
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  by (simp add: summable_def sums_def suminf_def)
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     (metis convergent_LIMSEQ_iff convergent_def lim_def)
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lemma summable_LIMSEQ: "summable f \<Longrightarrow> (\<lambda>n. \<Sum>i<n. f i) \<longlonglongrightarrow> suminf f"
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  by (rule summable_sums [unfolded sums_def])
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lemma sums_unique: "f sums s \<Longrightarrow> s = suminf f"
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  by (metis limI suminf_eq_lim sums_def)
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lemma sums_iff: "f sums x \<longleftrightarrow> summable f \<and> (suminf f = x)"
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  by (metis summable_sums sums_summable sums_unique)
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lemma summable_sums_iff:
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  "summable (f :: nat \<Rightarrow> 'a :: {comm_monoid_add,t2_space}) \<longleftrightarrow> f sums suminf f"
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  by (auto simp: sums_iff summable_sums)
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lemma sums_unique2:
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  fixes a b :: "'a::{comm_monoid_add,t2_space}"
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  shows "f sums a \<Longrightarrow> f sums b \<Longrightarrow> a = b"
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by (simp add: sums_iff)
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lemma suminf_finite:
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  assumes N: "finite N" and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 0"
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  shows "suminf f = (\<Sum>n\<in>N. f n)"
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  using sums_finite[OF assms, THEN sums_unique] by simp
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end
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lemma suminf_zero[simp]: "suminf (\<lambda>n. 0::'a::{t2_space, comm_monoid_add}) = 0"
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  by (rule sums_zero [THEN sums_unique, symmetric])
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subsection \<open>Infinite summability on ordered, topological monoids\<close>
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lemma sums_le:
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  fixes f g :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology}"
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  shows "\<forall>n. f n \<le> g n \<Longrightarrow> f sums s \<Longrightarrow> g sums t \<Longrightarrow> s \<le> t"
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  by (rule LIMSEQ_le) (auto intro: setsum_mono simp: sums_def)
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context
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  fixes f :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology}"
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begin
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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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  by (auto dest: sums_summable intro: sums_le)
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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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  by (rule sums_le[OF _ sums_If_finite_set summable_sums]) auto
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lemma suminf_nonneg: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> 0 \<le> suminf f"
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  using setsum_le_suminf[of 0] by simp
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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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  using
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    setsum_le_suminf[of "Suc i"]
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    add_strict_increasing[of "f i" "setsum f {..<n}" "setsum f {..<i}"]
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    setsum_mono2[of "{..<i}" "{..<n}" f]
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  by (auto simp: less_imp_le ac_simps)
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lemma setsum_less_suminf: "summable f \<Longrightarrow> \<forall>m\<ge>n. 0 < f m \<Longrightarrow> setsum f {..<n} < suminf f"
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  using setsum_less_suminf2[of n n] by (simp add: less_imp_le)
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lemma suminf_pos2: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> 0 < f i \<Longrightarrow> 0 < suminf f"
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  using setsum_less_suminf2[of 0 i] by simp
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lemma suminf_pos: "summable f \<Longrightarrow> \<forall>n. 0 < f n \<Longrightarrow> 0 < suminf f"
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  using suminf_pos2[of 0] by (simp add: less_imp_le)
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lemma suminf_le_const: "summable f \<Longrightarrow> (\<And>n. setsum f {..<n} \<le> x) \<Longrightarrow> suminf f \<le> x"
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  by (metis LIMSEQ_le_const2 summable_LIMSEQ)
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lemma suminf_eq_zero_iff: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> suminf f = 0 \<longleftrightarrow> (\<forall>n. f n = 0)"
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proof
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  assume "summable f" "suminf f = 0" and pos: "\<forall>n. 0 \<le> f n"
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  then have f: "(\<lambda>n. \<Sum>i<n. f i) \<longlonglongrightarrow> 0"
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    using summable_LIMSEQ[of f] by simp
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  then have "\<And>i. (\<Sum>n\<in>{i}. f n) \<le> 0"
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  proof (rule LIMSEQ_le_const)
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    fix i show "\<exists>N. \<forall>n\<ge>N. (\<Sum>n\<in>{i}. f n) \<le> setsum f {..<n}"
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      using pos by (intro exI[of _ "Suc i"] allI impI setsum_mono2) auto
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  qed
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  with pos show "\<forall>n. f n = 0"
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    by (auto intro!: antisym)
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qed (metis suminf_zero fun_eq_iff)
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lemma suminf_pos_iff: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> 0 < suminf f \<longleftrightarrow> (\<exists>i. 0 < f i)"
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  using setsum_le_suminf[of 0] suminf_eq_zero_iff by (simp add: less_le)
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end
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lemma summableI_nonneg_bounded:
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  fixes f:: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology, conditionally_complete_linorder}"
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  assumes pos[simp]: "\<And>n. 0 \<le> f n" and le: "\<And>n. (\<Sum>i<n. f i) \<le> x"
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  shows "summable f"
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  unfolding summable_def sums_def[abs_def]
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proof (intro exI order_tendstoI)
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  have [simp, intro]: "bdd_above (range (\<lambda>n. \<Sum>i<n. f i))"
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    using le by (auto simp: bdd_above_def)
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  { fix a assume "a < (SUP n. \<Sum>i<n. f i)"
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    then obtain n where "a < (\<Sum>i<n. f i)"
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      by (auto simp add: less_cSUP_iff)
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    then have "\<And>m. n \<le> m \<Longrightarrow> a < (\<Sum>i<m. f i)"
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      by (rule less_le_trans) (auto intro!: setsum_mono2)
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    then show "eventually (\<lambda>n. a < (\<Sum>i<n. f i)) sequentially"
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      by (auto simp: eventually_sequentially) }
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  { fix a assume "(SUP n. \<Sum>i<n. f i) < a"
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    moreover have "\<And>n. (\<Sum>i<n. f i) \<le> (SUP n. \<Sum>i<n. f i)"
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      by (auto intro: cSUP_upper)
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    ultimately show "eventually (\<lambda>n. (\<Sum>i<n. f i) < a) sequentially"
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      by (auto intro: le_less_trans simp: eventually_sequentially) }
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qed
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subsection \<open>Infinite summability on real normed vector spaces\<close>
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lemma sums_Suc_iff:
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   274
  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@56193
   275
  shows "(\<lambda>n. f (Suc n)) sums s \<longleftrightarrow> f sums (s + f 0)"
hoelzl@56193
   276
proof -
wenzelm@61969
   277
  have "f sums (s + f 0) \<longleftrightarrow> (\<lambda>i. \<Sum>j<Suc i. f j) \<longlonglongrightarrow> s + f 0"
hoelzl@56193
   278
    by (subst LIMSEQ_Suc_iff) (simp add: sums_def)
wenzelm@61969
   279
  also have "\<dots> \<longleftrightarrow> (\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) \<longlonglongrightarrow> s + f 0"
haftmann@57418
   280
    by (simp add: ac_simps setsum.reindex image_iff lessThan_Suc_eq_insert_0)
hoelzl@56193
   281
  also have "\<dots> \<longleftrightarrow> (\<lambda>n. f (Suc n)) sums s"
hoelzl@56193
   282
  proof
wenzelm@61969
   283
    assume "(\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) \<longlonglongrightarrow> s + f 0"
hoelzl@56193
   284
    with tendsto_add[OF this tendsto_const, of "- f 0"]
hoelzl@56193
   285
    show "(\<lambda>i. f (Suc i)) sums s"
hoelzl@56193
   286
      by (simp add: sums_def)
hoelzl@58729
   287
  qed (auto intro: tendsto_add simp: sums_def)
hoelzl@56193
   288
  finally show ?thesis ..
hoelzl@50999
   289
qed
hoelzl@50999
   290
eberlm@61531
   291
lemma summable_Suc_iff: "summable (\<lambda>n. f (Suc n) :: 'a :: real_normed_vector) = summable f"
eberlm@61531
   292
proof
eberlm@61531
   293
  assume "summable f"
eberlm@61531
   294
  hence "f sums suminf f" by (rule summable_sums)
eberlm@61531
   295
  hence "(\<lambda>n. f (Suc n)) sums (suminf f - f 0)" by (simp add: sums_Suc_iff)
eberlm@61531
   296
  thus "summable (\<lambda>n. f (Suc n))" unfolding summable_def by blast
eberlm@61531
   297
qed (auto simp: sums_Suc_iff summable_def)
eberlm@61531
   298
hoelzl@56193
   299
context
hoelzl@56193
   300
  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@56193
   301
begin
hoelzl@56193
   302
hoelzl@56193
   303
lemma sums_add: "f sums a \<Longrightarrow> g sums b \<Longrightarrow> (\<lambda>n. f n + g n) sums (a + b)"
haftmann@57418
   304
  unfolding sums_def by (simp add: setsum.distrib tendsto_add)
hoelzl@56193
   305
hoelzl@56193
   306
lemma summable_add: "summable f \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. f n + g n)"
hoelzl@56193
   307
  unfolding summable_def by (auto intro: sums_add)
hoelzl@56193
   308
hoelzl@56193
   309
lemma suminf_add: "summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f + suminf g = (\<Sum>n. f n + g n)"
hoelzl@56193
   310
  by (intro sums_unique sums_add summable_sums)
hoelzl@56193
   311
hoelzl@56193
   312
lemma sums_diff: "f sums a \<Longrightarrow> g sums b \<Longrightarrow> (\<lambda>n. f n - g n) sums (a - b)"
hoelzl@56193
   313
  unfolding sums_def by (simp add: setsum_subtractf tendsto_diff)
hoelzl@56193
   314
hoelzl@56193
   315
lemma summable_diff: "summable f \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. f n - g n)"
hoelzl@56193
   316
  unfolding summable_def by (auto intro: sums_diff)
hoelzl@56193
   317
hoelzl@56193
   318
lemma suminf_diff: "summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f - suminf g = (\<Sum>n. f n - g n)"
hoelzl@56193
   319
  by (intro sums_unique sums_diff summable_sums)
hoelzl@56193
   320
hoelzl@56193
   321
lemma sums_minus: "f sums a \<Longrightarrow> (\<lambda>n. - f n) sums (- a)"
hoelzl@56193
   322
  unfolding sums_def by (simp add: setsum_negf tendsto_minus)
hoelzl@56193
   323
hoelzl@56193
   324
lemma summable_minus: "summable f \<Longrightarrow> summable (\<lambda>n. - f n)"
hoelzl@56193
   325
  unfolding summable_def by (auto intro: sums_minus)
huffman@20692
   326
hoelzl@56193
   327
lemma suminf_minus: "summable f \<Longrightarrow> (\<Sum>n. - f n) = - (\<Sum>n. f n)"
hoelzl@56193
   328
  by (intro sums_unique [symmetric] sums_minus summable_sums)
hoelzl@56193
   329
hoelzl@56193
   330
lemma sums_Suc: "(\<lambda> n. f (Suc n)) sums l \<Longrightarrow> f sums (l + f 0)"
hoelzl@56193
   331
  by (simp add: sums_Suc_iff)
hoelzl@56193
   332
hoelzl@56193
   333
lemma sums_iff_shift: "(\<lambda>i. f (i + n)) sums s \<longleftrightarrow> f sums (s + (\<Sum>i<n. f i))"
hoelzl@56193
   334
proof (induct n arbitrary: s)
hoelzl@56193
   335
  case (Suc n)
hoelzl@56193
   336
  moreover have "(\<lambda>i. f (Suc i + n)) sums s \<longleftrightarrow> (\<lambda>i. f (i + n)) sums (s + f n)"
hoelzl@56193
   337
    by (subst sums_Suc_iff) simp
hoelzl@56193
   338
  ultimately show ?case
hoelzl@56193
   339
    by (simp add: ac_simps)
hoelzl@56193
   340
qed simp
huffman@20692
   341
hoelzl@56193
   342
lemma summable_iff_shift: "summable (\<lambda>n. f (n + k)) \<longleftrightarrow> summable f"
hoelzl@56193
   343
  by (metis diff_add_cancel summable_def sums_iff_shift[abs_def])
hoelzl@56193
   344
hoelzl@56193
   345
lemma sums_split_initial_segment: "f sums s \<Longrightarrow> (\<lambda>i. f (i + n)) sums (s - (\<Sum>i<n. f i))"
hoelzl@56193
   346
  by (simp add: sums_iff_shift)
hoelzl@56193
   347
hoelzl@56193
   348
lemma summable_ignore_initial_segment: "summable f \<Longrightarrow> summable (\<lambda>n. f(n + k))"
hoelzl@56193
   349
  by (simp add: summable_iff_shift)
hoelzl@56193
   350
hoelzl@56193
   351
lemma suminf_minus_initial_segment: "summable f \<Longrightarrow> (\<Sum>n. f (n + k)) = (\<Sum>n. f n) - (\<Sum>i<k. f i)"
hoelzl@56193
   352
  by (rule sums_unique[symmetric]) (auto simp: sums_iff_shift)
hoelzl@56193
   353
hoelzl@56193
   354
lemma suminf_split_initial_segment: "summable f \<Longrightarrow> suminf f = (\<Sum>n. f(n + k)) + (\<Sum>i<k. f i)"
hoelzl@56193
   355
  by (auto simp add: suminf_minus_initial_segment)
huffman@20692
   356
eberlm@61531
   357
lemma suminf_split_head: "summable f \<Longrightarrow> (\<Sum>n. f (Suc n)) = suminf f - f 0"
eberlm@61531
   358
  using suminf_split_initial_segment[of 1] by simp
eberlm@61531
   359
lp15@61609
   360
lemma suminf_exist_split:
hoelzl@56193
   361
  fixes r :: real assumes "0 < r" and "summable f"
hoelzl@56193
   362
  shows "\<exists>N. \<forall>n\<ge>N. norm (\<Sum>i. f (i + n)) < r"
hoelzl@56193
   363
proof -
wenzelm@60758
   364
  from LIMSEQ_D[OF summable_LIMSEQ[OF \<open>summable f\<close>] \<open>0 < r\<close>]
hoelzl@56193
   365
  obtain N :: nat where "\<forall> n \<ge> N. norm (setsum f {..<n} - suminf f) < r" by auto
hoelzl@56193
   366
  thus ?thesis
wenzelm@60758
   367
    by (auto simp: norm_minus_commute suminf_minus_initial_segment[OF \<open>summable f\<close>])
hoelzl@56193
   368
qed
hoelzl@56193
   369
wenzelm@61969
   370
lemma summable_LIMSEQ_zero: "summable f \<Longrightarrow> f \<longlonglongrightarrow> 0"
hoelzl@56193
   371
  apply (drule summable_iff_convergent [THEN iffD1])
hoelzl@56193
   372
  apply (drule convergent_Cauchy)
hoelzl@56193
   373
  apply (simp only: Cauchy_iff LIMSEQ_iff, safe)
hoelzl@56193
   374
  apply (drule_tac x="r" in spec, safe)
hoelzl@56193
   375
  apply (rule_tac x="M" in exI, safe)
hoelzl@56193
   376
  apply (drule_tac x="Suc n" in spec, simp)
hoelzl@56193
   377
  apply (drule_tac x="n" in spec, simp)
hoelzl@56193
   378
  done
hoelzl@56193
   379
eberlm@61531
   380
lemma summable_imp_convergent:
eberlm@61531
   381
  "summable (f :: nat \<Rightarrow> 'a) \<Longrightarrow> convergent f"
eberlm@61531
   382
  by (force dest!: summable_LIMSEQ_zero simp: convergent_def)
eberlm@61531
   383
eberlm@61531
   384
lemma summable_imp_Bseq:
eberlm@61531
   385
  "summable f \<Longrightarrow> Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector)"
eberlm@61531
   386
  by (simp add: convergent_imp_Bseq summable_imp_convergent)
eberlm@61531
   387
hoelzl@56193
   388
end
hoelzl@56193
   389
lp15@59613
   390
lemma summable_minus_iff:
lp15@59613
   391
  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
lp15@59613
   392
  shows "summable (\<lambda>n. - f n) \<longleftrightarrow> summable f"
wenzelm@61799
   393
  by (auto dest: summable_minus) \<comment>\<open>used two ways, hence must be outside the context above\<close>
lp15@59613
   394
lp15@59613
   395
hoelzl@57025
   396
context
hoelzl@57025
   397
  fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::real_normed_vector" and I :: "'i set"
hoelzl@57025
   398
begin
hoelzl@57025
   399
hoelzl@57025
   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)"
hoelzl@57025
   401
  by (induct I rule: infinite_finite_induct) (auto intro!: sums_add)
hoelzl@57025
   402
hoelzl@57025
   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)"
hoelzl@57025
   404
  using sums_unique[OF sums_setsum, OF summable_sums] by simp
hoelzl@57025
   405
hoelzl@57025
   406
lemma summable_setsum: "(\<And>i. i \<in> I \<Longrightarrow> summable (f i)) \<Longrightarrow> summable (\<lambda>n. \<Sum>i\<in>I. f i n)"
hoelzl@57025
   407
  using sums_summable[OF sums_setsum[OF summable_sums]] .
hoelzl@57025
   408
hoelzl@57025
   409
end
hoelzl@57025
   410
hoelzl@56193
   411
lemma (in bounded_linear) sums: "(\<lambda>n. X n) sums a \<Longrightarrow> (\<lambda>n. f (X n)) sums (f a)"
hoelzl@56193
   412
  unfolding sums_def by (drule tendsto, simp only: setsum)
hoelzl@56193
   413
hoelzl@56193
   414
lemma (in bounded_linear) summable: "summable (\<lambda>n. X n) \<Longrightarrow> summable (\<lambda>n. f (X n))"
hoelzl@56193
   415
  unfolding summable_def by (auto intro: sums)
hoelzl@56193
   416
hoelzl@56193
   417
lemma (in bounded_linear) suminf: "summable (\<lambda>n. X n) \<Longrightarrow> f (\<Sum>n. X n) = (\<Sum>n. f (X n))"
hoelzl@56193
   418
  by (intro sums_unique sums summable_sums)
hoelzl@56193
   419
hoelzl@56193
   420
lemmas sums_of_real = bounded_linear.sums [OF bounded_linear_of_real]
hoelzl@56193
   421
lemmas summable_of_real = bounded_linear.summable [OF bounded_linear_of_real]
hoelzl@56193
   422
lemmas suminf_of_real = bounded_linear.suminf [OF bounded_linear_of_real]
hoelzl@56193
   423
hoelzl@57275
   424
lemmas sums_scaleR_left = bounded_linear.sums[OF bounded_linear_scaleR_left]
hoelzl@57275
   425
lemmas summable_scaleR_left = bounded_linear.summable[OF bounded_linear_scaleR_left]
hoelzl@57275
   426
lemmas suminf_scaleR_left = bounded_linear.suminf[OF bounded_linear_scaleR_left]
hoelzl@57275
   427
hoelzl@57275
   428
lemmas sums_scaleR_right = bounded_linear.sums[OF bounded_linear_scaleR_right]
hoelzl@57275
   429
lemmas summable_scaleR_right = bounded_linear.summable[OF bounded_linear_scaleR_right]
hoelzl@57275
   430
lemmas suminf_scaleR_right = bounded_linear.suminf[OF bounded_linear_scaleR_right]
hoelzl@57275
   431
eberlm@61531
   432
lemma summable_const_iff: "summable (\<lambda>_. c) \<longleftrightarrow> (c :: 'a :: real_normed_vector) = 0"
eberlm@61531
   433
proof -
eberlm@61531
   434
  {
eberlm@61531
   435
    assume "c \<noteq> 0"
eberlm@61531
   436
    hence "filterlim (\<lambda>n. of_nat n * norm c) at_top sequentially"
eberlm@61531
   437
      by (subst mult.commute)
eberlm@61531
   438
         (auto intro!: filterlim_tendsto_pos_mult_at_top filterlim_real_sequentially)
eberlm@61531
   439
    hence "\<not>convergent (\<lambda>n. norm (\<Sum>k<n. c))"
lp15@61609
   440
      by (intro filterlim_at_infinity_imp_not_convergent filterlim_at_top_imp_at_infinity)
eberlm@61531
   441
         (simp_all add: setsum_constant_scaleR)
eberlm@61531
   442
    hence "\<not>summable (\<lambda>_. c)" unfolding summable_iff_convergent using convergent_norm by blast
eberlm@61531
   443
  }
eberlm@61531
   444
  thus ?thesis by auto
eberlm@61531
   445
qed
eberlm@61531
   446
eberlm@61531
   447
wenzelm@60758
   448
subsection \<open>Infinite summability on real normed algebras\<close>
hoelzl@56213
   449
hoelzl@56193
   450
context
hoelzl@56193
   451
  fixes f :: "nat \<Rightarrow> 'a::real_normed_algebra"
hoelzl@56193
   452
begin
hoelzl@56193
   453
hoelzl@56193
   454
lemma sums_mult: "f sums a \<Longrightarrow> (\<lambda>n. c * f n) sums (c * a)"
hoelzl@56193
   455
  by (rule bounded_linear.sums [OF bounded_linear_mult_right])
hoelzl@56193
   456
hoelzl@56193
   457
lemma summable_mult: "summable f \<Longrightarrow> summable (\<lambda>n. c * f n)"
hoelzl@56193
   458
  by (rule bounded_linear.summable [OF bounded_linear_mult_right])
hoelzl@56193
   459
hoelzl@56193
   460
lemma suminf_mult: "summable f \<Longrightarrow> suminf (\<lambda>n. c * f n) = c * suminf f"
hoelzl@56193
   461
  by (rule bounded_linear.suminf [OF bounded_linear_mult_right, symmetric])
hoelzl@56193
   462
hoelzl@56193
   463
lemma sums_mult2: "f sums a \<Longrightarrow> (\<lambda>n. f n * c) sums (a * c)"
hoelzl@56193
   464
  by (rule bounded_linear.sums [OF bounded_linear_mult_left])
hoelzl@56193
   465
hoelzl@56193
   466
lemma summable_mult2: "summable f \<Longrightarrow> summable (\<lambda>n. f n * c)"
hoelzl@56193
   467
  by (rule bounded_linear.summable [OF bounded_linear_mult_left])
hoelzl@56193
   468
hoelzl@56193
   469
lemma suminf_mult2: "summable f \<Longrightarrow> suminf f * c = (\<Sum>n. f n * c)"
hoelzl@56193
   470
  by (rule bounded_linear.suminf [OF bounded_linear_mult_left])
hoelzl@56193
   471
hoelzl@56193
   472
end
hoelzl@56193
   473
eberlm@61531
   474
lemma sums_mult_iff:
eberlm@61531
   475
  assumes "c \<noteq> 0"
eberlm@61531
   476
  shows   "(\<lambda>n. c * f n :: 'a :: {real_normed_algebra,field}) sums (c * d) \<longleftrightarrow> f sums d"
eberlm@61531
   477
  using sums_mult[of f d c] sums_mult[of "\<lambda>n. c * f n" "c * d" "inverse c"]
eberlm@61531
   478
  by (force simp: field_simps assms)
eberlm@61531
   479
eberlm@61531
   480
lemma sums_mult2_iff:
eberlm@61531
   481
  assumes "c \<noteq> (0 :: 'a :: {real_normed_algebra, field})"
eberlm@61531
   482
  shows   "(\<lambda>n. f n * c) sums (d * c) \<longleftrightarrow> f sums d"
eberlm@61531
   483
  using sums_mult_iff[OF assms, of f d] by (simp add: mult.commute)
eberlm@61531
   484
eberlm@61531
   485
lemma sums_of_real_iff:
eberlm@61531
   486
  "(\<lambda>n. of_real (f n) :: 'a :: real_normed_div_algebra) sums of_real c \<longleftrightarrow> f sums c"
eberlm@61531
   487
  by (simp add: sums_def of_real_setsum[symmetric] tendsto_of_real_iff del: of_real_setsum)
eberlm@61531
   488
eberlm@61531
   489
wenzelm@60758
   490
subsection \<open>Infinite summability on real normed fields\<close>
hoelzl@56213
   491
hoelzl@56193
   492
context
hoelzl@56193
   493
  fixes c :: "'a::real_normed_field"
hoelzl@56193
   494
begin
hoelzl@56193
   495
hoelzl@56193
   496
lemma sums_divide: "f sums a \<Longrightarrow> (\<lambda>n. f n / c) sums (a / c)"
hoelzl@56193
   497
  by (rule bounded_linear.sums [OF bounded_linear_divide])
hoelzl@56193
   498
hoelzl@56193
   499
lemma summable_divide: "summable f \<Longrightarrow> summable (\<lambda>n. f n / c)"
hoelzl@56193
   500
  by (rule bounded_linear.summable [OF bounded_linear_divide])
hoelzl@56193
   501
hoelzl@56193
   502
lemma suminf_divide: "summable f \<Longrightarrow> suminf (\<lambda>n. f n / c) = suminf f / c"
hoelzl@56193
   503
  by (rule bounded_linear.suminf [OF bounded_linear_divide, symmetric])
paulson@14416
   504
wenzelm@60758
   505
text\<open>Sum of a geometric progression.\<close>
paulson@14416
   506
hoelzl@56193
   507
lemma geometric_sums: "norm c < 1 \<Longrightarrow> (\<lambda>n. c^n) sums (1 / (1 - c))"
huffman@20692
   508
proof -
hoelzl@56193
   509
  assume less_1: "norm c < 1"
hoelzl@56193
   510
  hence neq_1: "c \<noteq> 1" by auto
hoelzl@56193
   511
  hence neq_0: "c - 1 \<noteq> 0" by simp
wenzelm@61969
   512
  from less_1 have lim_0: "(\<lambda>n. c^n) \<longlonglongrightarrow> 0"
huffman@20692
   513
    by (rule LIMSEQ_power_zero)
wenzelm@61969
   514
  hence "(\<lambda>n. c ^ n / (c - 1) - 1 / (c - 1)) \<longlonglongrightarrow> 0 / (c - 1) - 1 / (c - 1)"
huffman@44568
   515
    using neq_0 by (intro tendsto_intros)
wenzelm@61969
   516
  hence "(\<lambda>n. (c ^ n - 1) / (c - 1)) \<longlonglongrightarrow> 1 / (1 - c)"
huffman@20692
   517
    by (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib)
hoelzl@56193
   518
  thus "(\<lambda>n. c ^ n) sums (1 / (1 - c))"
huffman@20692
   519
    by (simp add: sums_def geometric_sum neq_1)
huffman@20692
   520
qed
huffman@20692
   521
hoelzl@56193
   522
lemma summable_geometric: "norm c < 1 \<Longrightarrow> summable (\<lambda>n. c^n)"
hoelzl@56193
   523
  by (rule geometric_sums [THEN sums_summable])
paulson@14416
   524
hoelzl@56193
   525
lemma suminf_geometric: "norm c < 1 \<Longrightarrow> suminf (\<lambda>n. c^n) = 1 / (1 - c)"
hoelzl@56193
   526
  by (rule sums_unique[symmetric]) (rule geometric_sums)
hoelzl@56193
   527
eberlm@61531
   528
lemma summable_geometric_iff: "summable (\<lambda>n. c ^ n) \<longleftrightarrow> norm c < 1"
eberlm@61531
   529
proof
eberlm@61531
   530
  assume "summable (\<lambda>n. c ^ n :: 'a :: real_normed_field)"
wenzelm@61969
   531
  hence "(\<lambda>n. norm c ^ n) \<longlonglongrightarrow> 0"
eberlm@61531
   532
    by (simp add: norm_power [symmetric] tendsto_norm_zero_iff summable_LIMSEQ_zero)
eberlm@61531
   533
  from order_tendstoD(2)[OF this zero_less_one] obtain n where "norm c ^ n < 1"
eberlm@61531
   534
    by (auto simp: eventually_at_top_linorder)
eberlm@61531
   535
  thus "norm c < 1" using one_le_power[of "norm c" n] by (cases "norm c \<ge> 1") (linarith, simp)
eberlm@61531
   536
qed (rule summable_geometric)
lp15@61609
   537
hoelzl@56193
   538
end
paulson@33271
   539
paulson@33271
   540
lemma power_half_series: "(\<lambda>n. (1/2::real)^Suc n) sums 1"
paulson@33271
   541
proof -
paulson@33271
   542
  have 2: "(\<lambda>n. (1/2::real)^n) sums 2" using geometric_sums [of "1/2::real"]
paulson@33271
   543
    by auto
paulson@33271
   544
  have "(\<lambda>n. (1/2::real)^Suc n) = (\<lambda>n. (1 / 2) ^ n / 2)"
lp15@59741
   545
    by (simp add: mult.commute)
huffman@44282
   546
  thus ?thesis using sums_divide [OF 2, of 2]
paulson@33271
   547
    by simp
paulson@33271
   548
qed
paulson@33271
   549
eberlm@61531
   550
eberlm@61531
   551
subsection \<open>Telescoping\<close>
eberlm@61531
   552
eberlm@61531
   553
lemma telescope_sums:
wenzelm@61969
   554
  assumes "f \<longlonglongrightarrow> (c :: 'a :: real_normed_vector)"
eberlm@61531
   555
  shows   "(\<lambda>n. f (Suc n) - f n) sums (c - f 0)"
eberlm@61531
   556
  unfolding sums_def
eberlm@61531
   557
proof (subst LIMSEQ_Suc_iff [symmetric])
eberlm@61531
   558
  have "(\<lambda>n. \<Sum>k<Suc n. f (Suc k) - f k) = (\<lambda>n. f (Suc n) - f 0)"
eberlm@61531
   559
    by (simp add: lessThan_Suc_atMost atLeast0AtMost [symmetric] setsum_Suc_diff)
wenzelm@61969
   560
  also have "\<dots> \<longlonglongrightarrow> c - f 0" by (intro tendsto_diff LIMSEQ_Suc[OF assms] tendsto_const)
wenzelm@61969
   561
  finally show "(\<lambda>n. \<Sum>n<Suc n. f (Suc n) - f n) \<longlonglongrightarrow> c - f 0" .
eberlm@61531
   562
qed
eberlm@61531
   563
eberlm@61531
   564
lemma telescope_sums':
wenzelm@61969
   565
  assumes "f \<longlonglongrightarrow> (c :: 'a :: real_normed_vector)"
eberlm@61531
   566
  shows   "(\<lambda>n. f n - f (Suc n)) sums (f 0 - c)"
eberlm@61531
   567
  using sums_minus[OF telescope_sums[OF assms]] by (simp add: algebra_simps)
eberlm@61531
   568
eberlm@61531
   569
lemma telescope_summable:
wenzelm@61969
   570
  assumes "f \<longlonglongrightarrow> (c :: 'a :: real_normed_vector)"
eberlm@61531
   571
  shows   "summable (\<lambda>n. f (Suc n) - f n)"
eberlm@61531
   572
  using telescope_sums[OF assms] by (simp add: sums_iff)
eberlm@61531
   573
eberlm@61531
   574
lemma telescope_summable':
wenzelm@61969
   575
  assumes "f \<longlonglongrightarrow> (c :: 'a :: real_normed_vector)"
eberlm@61531
   576
  shows   "summable (\<lambda>n. f n - f (Suc n))"
eberlm@61531
   577
  using summable_minus[OF telescope_summable[OF assms]] by (simp add: algebra_simps)
eberlm@61531
   578
eberlm@61531
   579
wenzelm@60758
   580
subsection \<open>Infinite summability on Banach spaces\<close>
hoelzl@56213
   581
wenzelm@60758
   582
text\<open>Cauchy-type criterion for convergence of series (c.f. Harrison)\<close>
paulson@15085
   583
hoelzl@56193
   584
lemma summable_Cauchy:
hoelzl@56193
   585
  fixes f :: "nat \<Rightarrow> 'a::banach"
hoelzl@56193
   586
  shows "summable f \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n. norm (setsum f {m..<n}) < e)"
hoelzl@56193
   587
  apply (simp only: summable_iff_convergent Cauchy_convergent_iff [symmetric] Cauchy_iff, safe)
hoelzl@56193
   588
  apply (drule spec, drule (1) mp)
hoelzl@56193
   589
  apply (erule exE, rule_tac x="M" in exI, clarify)
hoelzl@56193
   590
  apply (rule_tac x="m" and y="n" in linorder_le_cases)
hoelzl@56193
   591
  apply (frule (1) order_trans)
hoelzl@56193
   592
  apply (drule_tac x="n" in spec, drule (1) mp)
hoelzl@56193
   593
  apply (drule_tac x="m" in spec, drule (1) mp)
hoelzl@56193
   594
  apply (simp_all add: setsum_diff [symmetric])
hoelzl@56193
   595
  apply (drule spec, drule (1) mp)
hoelzl@56193
   596
  apply (erule exE, rule_tac x="N" in exI, clarify)
hoelzl@56193
   597
  apply (rule_tac x="m" and y="n" in linorder_le_cases)
hoelzl@56193
   598
  apply (subst norm_minus_commute)
hoelzl@56193
   599
  apply (simp_all add: setsum_diff [symmetric])
hoelzl@56193
   600
  done
paulson@14416
   601
hoelzl@56193
   602
context
hoelzl@56193
   603
  fixes f :: "nat \<Rightarrow> 'a::banach"
eberlm@61531
   604
begin
hoelzl@56193
   605
wenzelm@60758
   606
text\<open>Absolute convergence imples normal convergence\<close>
huffman@20689
   607
hoelzl@56194
   608
lemma summable_norm_cancel: "summable (\<lambda>n. norm (f n)) \<Longrightarrow> summable f"
hoelzl@56193
   609
  apply (simp only: summable_Cauchy, safe)
hoelzl@56193
   610
  apply (drule_tac x="e" in spec, safe)
hoelzl@56193
   611
  apply (rule_tac x="N" in exI, safe)
hoelzl@56193
   612
  apply (drule_tac x="m" in spec, safe)
hoelzl@56193
   613
  apply (rule order_le_less_trans [OF norm_setsum])
hoelzl@56193
   614
  apply (rule order_le_less_trans [OF abs_ge_self])
hoelzl@56193
   615
  apply simp
hoelzl@50999
   616
  done
paulson@32707
   617
hoelzl@56193
   618
lemma summable_norm: "summable (\<lambda>n. norm (f n)) \<Longrightarrow> norm (suminf f) \<le> (\<Sum>n. norm (f n))"
hoelzl@56193
   619
  by (auto intro: LIMSEQ_le tendsto_norm summable_norm_cancel summable_LIMSEQ norm_setsum)
hoelzl@56193
   620
wenzelm@60758
   621
text \<open>Comparison tests\<close>
paulson@14416
   622
hoelzl@56194
   623
lemma summable_comparison_test: "\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n \<Longrightarrow> summable g \<Longrightarrow> summable f"
hoelzl@56193
   624
  apply (simp add: summable_Cauchy, safe)
hoelzl@56193
   625
  apply (drule_tac x="e" in spec, safe)
hoelzl@56193
   626
  apply (rule_tac x = "N + Na" in exI, safe)
hoelzl@56193
   627
  apply (rotate_tac 2)
hoelzl@56193
   628
  apply (drule_tac x = m in spec)
hoelzl@56193
   629
  apply (auto, rotate_tac 2, drule_tac x = n in spec)
hoelzl@56193
   630
  apply (rule_tac y = "\<Sum>k=m..<n. norm (f k)" in order_le_less_trans)
hoelzl@56193
   631
  apply (rule norm_setsum)
hoelzl@56193
   632
  apply (rule_tac y = "setsum g {m..<n}" in order_le_less_trans)
hoelzl@56193
   633
  apply (auto intro: setsum_mono simp add: abs_less_iff)
hoelzl@56193
   634
  done
hoelzl@56193
   635
eberlm@61531
   636
lemma summable_comparison_test_ev:
eberlm@61531
   637
  shows "eventually (\<lambda>n. norm (f n) \<le> g n) sequentially \<Longrightarrow> summable g \<Longrightarrow> summable f"
eberlm@61531
   638
  by (rule summable_comparison_test) (auto simp: eventually_at_top_linorder)
eberlm@61531
   639
lp15@56217
   640
(*A better argument order*)
lp15@56217
   641
lemma summable_comparison_test': "summable g \<Longrightarrow> (\<And>n. n \<ge> N \<Longrightarrow> norm(f n) \<le> g n) \<Longrightarrow> summable f"
hoelzl@56369
   642
  by (rule summable_comparison_test) auto
lp15@56217
   643
wenzelm@60758
   644
subsection \<open>The Ratio Test\<close>
paulson@15085
   645
lp15@61609
   646
lemma summable_ratio_test:
hoelzl@56193
   647
  assumes "c < 1" "\<And>n. n \<ge> N \<Longrightarrow> norm (f (Suc n)) \<le> c * norm (f n)"
hoelzl@56193
   648
  shows "summable f"
hoelzl@56193
   649
proof cases
hoelzl@56193
   650
  assume "0 < c"
hoelzl@56193
   651
  show "summable f"
hoelzl@56193
   652
  proof (rule summable_comparison_test)
hoelzl@56193
   653
    show "\<exists>N'. \<forall>n\<ge>N'. norm (f n) \<le> (norm (f N) / (c ^ N)) * c ^ n"
hoelzl@56193
   654
    proof (intro exI allI impI)
hoelzl@56193
   655
      fix n assume "N \<le> n" then show "norm (f n) \<le> (norm (f N) / (c ^ N)) * c ^ n"
hoelzl@56193
   656
      proof (induct rule: inc_induct)
hoelzl@56193
   657
        case (step m)
hoelzl@56193
   658
        moreover have "norm (f (Suc m)) / c ^ Suc m * c ^ n \<le> norm (f m) / c ^ m * c ^ n"
wenzelm@60758
   659
          using \<open>0 < c\<close> \<open>c < 1\<close> assms(2)[OF \<open>N \<le> m\<close>] by (simp add: field_simps)
hoelzl@56193
   660
        ultimately show ?case by simp
wenzelm@60758
   661
      qed (insert \<open>0 < c\<close>, simp)
hoelzl@56193
   662
    qed
hoelzl@56193
   663
    show "summable (\<lambda>n. norm (f N) / c ^ N * c ^ n)"
wenzelm@60758
   664
      using \<open>0 < c\<close> \<open>c < 1\<close> by (intro summable_mult summable_geometric) simp
hoelzl@56193
   665
  qed
hoelzl@56193
   666
next
hoelzl@56193
   667
  assume c: "\<not> 0 < c"
hoelzl@56193
   668
  { fix n assume "n \<ge> N"
hoelzl@56193
   669
    then have "norm (f (Suc n)) \<le> c * norm (f n)"
hoelzl@56193
   670
      by fact
hoelzl@56193
   671
    also have "\<dots> \<le> 0"
hoelzl@56193
   672
      using c by (simp add: not_less mult_nonpos_nonneg)
hoelzl@56193
   673
    finally have "f (Suc n) = 0"
hoelzl@56193
   674
      by auto }
hoelzl@56193
   675
  then show "summable f"
hoelzl@56194
   676
    by (intro sums_summable[OF sums_finite, of "{.. Suc N}"]) (auto simp: not_le Suc_less_eq2)
lp15@56178
   677
qed
lp15@56178
   678
hoelzl@56193
   679
end
paulson@14416
   680
wenzelm@60758
   681
text\<open>Relations among convergence and absolute convergence for power series.\<close>
hoelzl@56369
   682
paulson@62087
   683
lemma Abel_lemma:
hoelzl@56369
   684
  fixes a :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@56369
   685
  assumes r: "0 \<le> r" and r0: "r < r0" and M: "\<And>n. norm (a n) * r0^n \<le> M"
hoelzl@56369
   686
    shows "summable (\<lambda>n. norm (a n) * r^n)"
hoelzl@56369
   687
proof (rule summable_comparison_test')
hoelzl@56369
   688
  show "summable (\<lambda>n. M * (r / r0) ^ n)"
lp15@61609
   689
    using assms
hoelzl@56369
   690
    by (auto simp add: summable_mult summable_geometric)
hoelzl@56369
   691
next
hoelzl@56369
   692
  fix n
hoelzl@56369
   693
  show "norm (norm (a n) * r ^ n) \<le> M * (r / r0) ^ n"
hoelzl@56369
   694
    using r r0 M [of n]
haftmann@60867
   695
    apply (auto simp add: abs_mult field_simps)
hoelzl@56369
   696
    apply (cases "r=0", simp)
hoelzl@56369
   697
    apply (cases n, auto)
hoelzl@56369
   698
    done
hoelzl@56369
   699
qed
hoelzl@56369
   700
hoelzl@56369
   701
wenzelm@60758
   702
text\<open>Summability of geometric series for real algebras\<close>
huffman@23084
   703
huffman@23084
   704
lemma complete_algebra_summable_geometric:
haftmann@31017
   705
  fixes x :: "'a::{real_normed_algebra_1,banach}"
huffman@23084
   706
  shows "norm x < 1 \<Longrightarrow> summable (\<lambda>n. x ^ n)"
huffman@23084
   707
proof (rule summable_comparison_test)
huffman@23084
   708
  show "\<exists>N. \<forall>n\<ge>N. norm (x ^ n) \<le> norm x ^ n"
huffman@23084
   709
    by (simp add: norm_power_ineq)
huffman@23084
   710
  show "norm x < 1 \<Longrightarrow> summable (\<lambda>n. norm x ^ n)"
huffman@23084
   711
    by (simp add: summable_geometric)
huffman@23084
   712
qed
huffman@23084
   713
wenzelm@60758
   714
subsection \<open>Cauchy Product Formula\<close>
huffman@23111
   715
wenzelm@60758
   716
text \<open>
wenzelm@54703
   717
  Proof based on Analysis WebNotes: Chapter 07, Class 41
wenzelm@54703
   718
  @{url "http://www.math.unl.edu/~webnotes/classes/class41/prp77.htm"}
wenzelm@60758
   719
\<close>
huffman@23111
   720
huffman@23111
   721
lemma Cauchy_product_sums:
huffman@23111
   722
  fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}"
huffman@23111
   723
  assumes a: "summable (\<lambda>k. norm (a k))"
huffman@23111
   724
  assumes b: "summable (\<lambda>k. norm (b k))"
hoelzl@56213
   725
  shows "(\<lambda>k. \<Sum>i\<le>k. a i * b (k - i)) sums ((\<Sum>k. a k) * (\<Sum>k. b k))"
huffman@23111
   726
proof -
hoelzl@56193
   727
  let ?S1 = "\<lambda>n::nat. {..<n} \<times> {..<n}"
huffman@23111
   728
  let ?S2 = "\<lambda>n::nat. {(i,j). i + j < n}"
huffman@23111
   729
  have S1_mono: "\<And>m n. m \<le> n \<Longrightarrow> ?S1 m \<subseteq> ?S1 n" by auto
huffman@23111
   730
  have S2_le_S1: "\<And>n. ?S2 n \<subseteq> ?S1 n" by auto
huffman@23111
   731
  have S1_le_S2: "\<And>n. ?S1 (n div 2) \<subseteq> ?S2 n" by auto
huffman@23111
   732
  have finite_S1: "\<And>n. finite (?S1 n)" by simp
huffman@23111
   733
  with S2_le_S1 have finite_S2: "\<And>n. finite (?S2 n)" by (rule finite_subset)
huffman@23111
   734
huffman@23111
   735
  let ?g = "\<lambda>(i,j). a i * b j"
huffman@23111
   736
  let ?f = "\<lambda>(i,j). norm (a i) * norm (b j)"
nipkow@56536
   737
  have f_nonneg: "\<And>x. 0 \<le> ?f x" by (auto)
huffman@23111
   738
  hence norm_setsum_f: "\<And>A. norm (setsum ?f A) = setsum ?f A"
huffman@23111
   739
    unfolding real_norm_def
huffman@23111
   740
    by (simp only: abs_of_nonneg setsum_nonneg [rule_format])
huffman@23111
   741
wenzelm@61969
   742
  have "(\<lambda>n. (\<Sum>k<n. a k) * (\<Sum>k<n. b k)) \<longlonglongrightarrow> (\<Sum>k. a k) * (\<Sum>k. b k)"
hoelzl@56193
   743
    by (intro tendsto_mult summable_LIMSEQ summable_norm_cancel [OF a] summable_norm_cancel [OF b])
wenzelm@61969
   744
  hence 1: "(\<lambda>n. setsum ?g (?S1 n)) \<longlonglongrightarrow> (\<Sum>k. a k) * (\<Sum>k. b k)"
haftmann@57418
   745
    by (simp only: setsum_product setsum.Sigma [rule_format] finite_lessThan)
huffman@23111
   746
wenzelm@61969
   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))"
hoelzl@56193
   748
    using a b by (intro tendsto_mult summable_LIMSEQ)
wenzelm@61969
   749
  hence "(\<lambda>n. setsum ?f (?S1 n)) \<longlonglongrightarrow> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
haftmann@57418
   750
    by (simp only: setsum_product setsum.Sigma [rule_format] finite_lessThan)
huffman@23111
   751
  hence "convergent (\<lambda>n. setsum ?f (?S1 n))"
huffman@23111
   752
    by (rule convergentI)
huffman@23111
   753
  hence Cauchy: "Cauchy (\<lambda>n. setsum ?f (?S1 n))"
huffman@23111
   754
    by (rule convergent_Cauchy)
huffman@36657
   755
  have "Zfun (\<lambda>n. setsum ?f (?S1 n - ?S2 n)) sequentially"
huffman@36657
   756
  proof (rule ZfunI, simp only: eventually_sequentially norm_setsum_f)
huffman@23111
   757
    fix r :: real
huffman@23111
   758
    assume r: "0 < r"
huffman@23111
   759
    from CauchyD [OF Cauchy r] obtain N
huffman@23111
   760
    where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (setsum ?f (?S1 m) - setsum ?f (?S1 n)) < r" ..
huffman@23111
   761
    hence "\<And>m n. \<lbrakk>N \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> norm (setsum ?f (?S1 m - ?S1 n)) < r"
huffman@23111
   762
      by (simp only: setsum_diff finite_S1 S1_mono)
huffman@23111
   763
    hence N: "\<And>m n. \<lbrakk>N \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> setsum ?f (?S1 m - ?S1 n) < r"
huffman@23111
   764
      by (simp only: norm_setsum_f)
huffman@23111
   765
    show "\<exists>N. \<forall>n\<ge>N. setsum ?f (?S1 n - ?S2 n) < r"
huffman@23111
   766
    proof (intro exI allI impI)
huffman@23111
   767
      fix n assume "2 * N \<le> n"
huffman@23111
   768
      hence n: "N \<le> n div 2" by simp
huffman@23111
   769
      have "setsum ?f (?S1 n - ?S2 n) \<le> setsum ?f (?S1 n - ?S1 (n div 2))"
huffman@23111
   770
        by (intro setsum_mono2 finite_Diff finite_S1 f_nonneg
huffman@23111
   771
                  Diff_mono subset_refl S1_le_S2)
huffman@23111
   772
      also have "\<dots> < r"
huffman@23111
   773
        using n div_le_dividend by (rule N)
huffman@23111
   774
      finally show "setsum ?f (?S1 n - ?S2 n) < r" .
huffman@23111
   775
    qed
huffman@23111
   776
  qed
huffman@36657
   777
  hence "Zfun (\<lambda>n. setsum ?g (?S1 n - ?S2 n)) sequentially"
huffman@36657
   778
    apply (rule Zfun_le [rule_format])
huffman@23111
   779
    apply (simp only: norm_setsum_f)
huffman@23111
   780
    apply (rule order_trans [OF norm_setsum setsum_mono])
huffman@23111
   781
    apply (auto simp add: norm_mult_ineq)
huffman@23111
   782
    done
wenzelm@61969
   783
  hence 2: "(\<lambda>n. setsum ?g (?S1 n) - setsum ?g (?S2 n)) \<longlonglongrightarrow> 0"
huffman@36660
   784
    unfolding tendsto_Zfun_iff diff_0_right
huffman@36657
   785
    by (simp only: setsum_diff finite_S1 S2_le_S1)
huffman@23111
   786
wenzelm@61969
   787
  with 1 have "(\<lambda>n. setsum ?g (?S2 n)) \<longlonglongrightarrow> (\<Sum>k. a k) * (\<Sum>k. b k)"
lp15@60141
   788
    by (rule Lim_transform2)
huffman@23111
   789
  thus ?thesis by (simp only: sums_def setsum_triangle_reindex)
huffman@23111
   790
qed
huffman@23111
   791
huffman@23111
   792
lemma Cauchy_product:
huffman@23111
   793
  fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}"
huffman@23111
   794
  assumes a: "summable (\<lambda>k. norm (a k))"
huffman@23111
   795
  assumes b: "summable (\<lambda>k. norm (b k))"
hoelzl@56213
   796
  shows "(\<Sum>k. a k) * (\<Sum>k. b k) = (\<Sum>k. \<Sum>i\<le>k. a i * b (k - i))"
hoelzl@56213
   797
  using a b
hoelzl@56213
   798
  by (rule Cauchy_product_sums [THEN sums_unique])
hoelzl@56213
   799
eberlm@62049
   800
lemma summable_Cauchy_product:
paulson@62087
   801
  assumes "summable (\<lambda>k. norm (a k :: 'a :: {real_normed_algebra,banach}))"
eberlm@62049
   802
          "summable (\<lambda>k. norm (b k))"
eberlm@62049
   803
  shows   "summable (\<lambda>k. \<Sum>i\<le>k. a i * b (k - i))"
paulson@62087
   804
  using Cauchy_product_sums[OF assms] by (simp add: sums_iff)
eberlm@62049
   805
wenzelm@60758
   806
subsection \<open>Series on @{typ real}s\<close>
hoelzl@56213
   807
hoelzl@56213
   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))"
hoelzl@56213
   809
  by (rule summable_comparison_test) auto
hoelzl@56213
   810
hoelzl@56213
   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>)"
hoelzl@56213
   812
  by (rule summable_comparison_test) auto
hoelzl@56213
   813
hoelzl@56213
   814
lemma summable_rabs_cancel: "summable (\<lambda>n. \<bar>f n :: real\<bar>) \<Longrightarrow> summable f"
hoelzl@56213
   815
  by (rule summable_norm_cancel) simp
hoelzl@56213
   816
hoelzl@56213
   817
lemma summable_rabs: "summable (\<lambda>n. \<bar>f n :: real\<bar>) \<Longrightarrow> \<bar>suminf f\<bar> \<le> (\<Sum>n. \<bar>f n\<bar>)"
hoelzl@56213
   818
  by (fold real_norm_def) (rule summable_norm)
huffman@23111
   819
eberlm@61531
   820
lemma summable_zero_power [simp]: "summable (\<lambda>n. 0 ^ n :: 'a :: {comm_ring_1,topological_space})"
eberlm@61531
   821
proof -
eberlm@61531
   822
  have "(\<lambda>n. 0 ^ n :: 'a) = (\<lambda>n. if n = 0 then 0^0 else 0)" by (intro ext) (simp add: zero_power)
eberlm@61531
   823
  moreover have "summable \<dots>" by simp
eberlm@61531
   824
  ultimately show ?thesis by simp
eberlm@61531
   825
qed
eberlm@61531
   826
eberlm@61531
   827
lemma summable_zero_power' [simp]: "summable (\<lambda>n. f n * 0 ^ n :: 'a :: {ring_1,topological_space})"
eberlm@61531
   828
proof -
lp15@61609
   829
  have "(\<lambda>n. f n * 0 ^ n :: 'a) = (\<lambda>n. if n = 0 then f 0 * 0^0 else 0)"
eberlm@61531
   830
    by (intro ext) (simp add: zero_power)
eberlm@61531
   831
  moreover have "summable \<dots>" by simp
eberlm@61531
   832
  ultimately show ?thesis by simp
eberlm@61531
   833
qed
eberlm@61531
   834
hoelzl@59000
   835
lemma summable_power_series:
hoelzl@59000
   836
  fixes z :: real
hoelzl@59000
   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"
hoelzl@59000
   838
  shows "summable (\<lambda>i. f i * z^i)"
hoelzl@59000
   839
proof (rule summable_comparison_test[OF _ summable_geometric])
hoelzl@59000
   840
  show "norm z < 1" using z by (auto simp: less_imp_le)
hoelzl@59000
   841
  show "\<And>n. \<exists>N. \<forall>na\<ge>N. norm (f na * z ^ na) \<le> z ^ na"
hoelzl@59000
   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)
hoelzl@59000
   843
qed
hoelzl@59000
   844
eberlm@61531
   845
lemma summable_0_powser:
eberlm@61531
   846
  "summable (\<lambda>n. f n * 0 ^ n :: 'a :: real_normed_div_algebra)"
eberlm@61531
   847
proof -
eberlm@61531
   848
  have A: "(\<lambda>n. f n * 0 ^ n) = (\<lambda>n. if n = 0 then f n else 0)"
eberlm@61531
   849
    by (intro ext) auto
eberlm@61531
   850
  thus ?thesis by (subst A) simp_all
eberlm@61531
   851
qed
eberlm@61531
   852
eberlm@61531
   853
lemma summable_powser_split_head:
eberlm@61531
   854
  "summable (\<lambda>n. f (Suc n) * z ^ n :: 'a :: real_normed_div_algebra) = summable (\<lambda>n. f n * z ^ n)"
eberlm@61531
   855
proof -
eberlm@61531
   856
  have "summable (\<lambda>n. f (Suc n) * z ^ n) \<longleftrightarrow> summable (\<lambda>n. f (Suc n) * z ^ Suc n)"
eberlm@61531
   857
  proof
eberlm@61531
   858
    assume "summable (\<lambda>n. f (Suc n) * z ^ n)"
lp15@61609
   859
    from summable_mult2[OF this, of z] show "summable (\<lambda>n. f (Suc n) * z ^ Suc n)"
eberlm@61531
   860
      by (simp add: power_commutes algebra_simps)
eberlm@61531
   861
  next
eberlm@61531
   862
    assume "summable (\<lambda>n. f (Suc n) * z ^ Suc n)"
eberlm@61531
   863
    from summable_mult2[OF this, of "inverse z"] show "summable (\<lambda>n. f (Suc n) * z ^ n)"
eberlm@61531
   864
      by (cases "z \<noteq> 0", subst (asm) power_Suc2) (simp_all add: algebra_simps)
eberlm@61531
   865
  qed
eberlm@61531
   866
  also have "\<dots> \<longleftrightarrow> summable (\<lambda>n. f n * z ^ n)" by (rule summable_Suc_iff)
eberlm@61531
   867
  finally show ?thesis .
eberlm@61531
   868
qed
eberlm@61531
   869
eberlm@61531
   870
lemma powser_split_head:
eberlm@61531
   871
  assumes "summable (\<lambda>n. f n * z ^ n :: 'a :: {real_normed_div_algebra,banach})"
eberlm@61531
   872
  shows   "suminf (\<lambda>n. f n * z ^ n) = f 0 + suminf (\<lambda>n. f (Suc n) * z ^ n) * z"
eberlm@61531
   873
          "suminf (\<lambda>n. f (Suc n) * z ^ n) * z = suminf (\<lambda>n. f n * z ^ n) - f 0"
eberlm@61531
   874
          "summable (\<lambda>n. f (Suc n) * z ^ n)"
eberlm@61531
   875
proof -
eberlm@61531
   876
  from assms show "summable (\<lambda>n. f (Suc n) * z ^ n)" by (subst summable_powser_split_head)
eberlm@61531
   877
lp15@61609
   878
  from suminf_mult2[OF this, of z]
eberlm@61531
   879
    have "(\<Sum>n. f (Suc n) * z ^ n) * z = (\<Sum>n. f (Suc n) * z ^ Suc n)"
eberlm@61531
   880
    by (simp add: power_commutes algebra_simps)
eberlm@61531
   881
  also from assms have "\<dots> = suminf (\<lambda>n. f n * z ^ n) - f 0"
eberlm@61531
   882
    by (subst suminf_split_head) simp_all
eberlm@61531
   883
  finally show "suminf (\<lambda>n. f n * z ^ n) = f 0 + suminf (\<lambda>n. f (Suc n) * z ^ n) * z" by simp
eberlm@61531
   884
  thus "suminf (\<lambda>n. f (Suc n) * z ^ n) * z = suminf (\<lambda>n. f n * z ^ n) - f 0" by simp
eberlm@61531
   885
qed
eberlm@61531
   886
eberlm@61531
   887
lemma summable_partial_sum_bound:
eberlm@61531
   888
  fixes f :: "nat \<Rightarrow> 'a :: banach"
eberlm@61531
   889
  assumes summable: "summable f" and e: "e > (0::real)"
eberlm@61531
   890
  obtains N where "\<And>m n. m \<ge> N \<Longrightarrow> norm (\<Sum>k=m..n. f k) < e"
eberlm@61531
   891
proof -
lp15@61609
   892
  from summable have "Cauchy (\<lambda>n. \<Sum>k<n. f k)"
eberlm@61531
   893
    by (simp add: Cauchy_convergent_iff summable_iff_convergent)
lp15@61609
   894
  from CauchyD[OF this e] obtain N
eberlm@61531
   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
eberlm@61531
   896
  {
eberlm@61531
   897
    fix m n :: nat assume m: "m \<ge> N"
eberlm@61531
   898
    have "norm (\<Sum>k=m..n. f k) < e"
eberlm@61531
   899
    proof (cases "n \<ge> m")
eberlm@61531
   900
      assume n: "n \<ge> m"
eberlm@61531
   901
      with m have "norm ((\<Sum>k<Suc n. f k) - (\<Sum>k<m. f k)) < e" by (intro N) simp_all
eberlm@61531
   902
      also from n have "(\<Sum>k<Suc n. f k) - (\<Sum>k<m. f k) = (\<Sum>k=m..n. f k)"
eberlm@61531
   903
        by (subst setsum_diff [symmetric]) (simp_all add: setsum_last_plus)
eberlm@61531
   904
      finally show ?thesis .
eberlm@61531
   905
    qed (insert e, simp_all)
eberlm@61531
   906
  }
eberlm@61531
   907
  thus ?thesis by (rule that)
eberlm@61531
   908
qed
eberlm@61531
   909
lp15@61609
   910
lemma powser_sums_if:
eberlm@61531
   911
  "(\<lambda>n. (if n = m then (1 :: 'a :: {ring_1,topological_space}) else 0) * z^n) sums z^m"
eberlm@61531
   912
proof -
lp15@61609
   913
  have "(\<lambda>n. (if n = m then 1 else 0) * z^n) = (\<lambda>n. if n = m then z^n else 0)"
eberlm@61531
   914
    by (intro ext) auto
eberlm@61531
   915
  thus ?thesis by (simp add: sums_single)
eberlm@61531
   916
qed
eberlm@61531
   917
Andreas@59025
   918
lemma
Andreas@59025
   919
   fixes f :: "nat \<Rightarrow> real"
Andreas@59025
   920
   assumes "summable f"
Andreas@59025
   921
   and "inj g"
Andreas@59025
   922
   and pos: "!!x. 0 \<le> f x"
Andreas@59025
   923
   shows summable_reindex: "summable (f o g)"
Andreas@59025
   924
   and suminf_reindex_mono: "suminf (f o g) \<le> suminf f"
Andreas@59025
   925
   and suminf_reindex: "(\<And>x. x \<notin> range g \<Longrightarrow> f x = 0) \<Longrightarrow> suminf (f \<circ> g) = suminf f"
Andreas@59025
   926
proof -
Andreas@59025
   927
  from \<open>inj g\<close> have [simp]: "\<And>A. inj_on g A" by(rule subset_inj_on) simp
Andreas@59025
   928
Andreas@59025
   929
  have smaller: "\<forall>n. (\<Sum>i<n. (f \<circ> g) i) \<le> suminf f"
Andreas@59025
   930
  proof
Andreas@59025
   931
    fix n
lp15@61609
   932
    have "\<forall> n' \<in> (g ` {..<n}). n' < Suc (Max (g ` {..<n}))"
Andreas@59025
   933
      by(metis Max_ge finite_imageI finite_lessThan not_le not_less_eq)
Andreas@59025
   934
    then obtain m where n: "\<And>n'. n' < n \<Longrightarrow> g n' < m" by blast
Andreas@59025
   935
Andreas@59025
   936
    have "(\<Sum>i<n. f (g i)) = setsum f (g ` {..<n})"
Andreas@59025
   937
      by (simp add: setsum.reindex)
Andreas@59025
   938
    also have "\<dots> \<le> (\<Sum>i<m. f i)"
Andreas@59025
   939
      by (rule setsum_mono3) (auto simp add: pos n[rule_format])
Andreas@59025
   940
    also have "\<dots> \<le> suminf f"
lp15@61609
   941
      using \<open>summable f\<close>
Andreas@59025
   942
      by (rule setsum_le_suminf) (simp add: pos)
Andreas@59025
   943
    finally show "(\<Sum>i<n. (f \<circ>  g) i) \<le> suminf f" by simp
Andreas@59025
   944
  qed
Andreas@59025
   945
Andreas@59025
   946
  have "incseq (\<lambda>n. \<Sum>i<n. (f \<circ> g) i)"
Andreas@59025
   947
    by (rule incseq_SucI) (auto simp add: pos)
wenzelm@61969
   948
  then obtain  L where L: "(\<lambda> n. \<Sum>i<n. (f \<circ> g) i) \<longlonglongrightarrow> L"
Andreas@59025
   949
    using smaller by(rule incseq_convergent)
Andreas@59025
   950
  hence "(f \<circ> g) sums L" by (simp add: sums_def)
Andreas@59025
   951
  thus "summable (f o g)" by (auto simp add: sums_iff)
Andreas@59025
   952
wenzelm@61969
   953
  hence "(\<lambda>n. \<Sum>i<n. (f \<circ> g) i) \<longlonglongrightarrow> suminf (f \<circ> g)"
Andreas@59025
   954
    by(rule summable_LIMSEQ)
Andreas@59025
   955
  thus le: "suminf (f \<circ> g) \<le> suminf f"
Andreas@59025
   956
    by(rule LIMSEQ_le_const2)(blast intro: smaller[rule_format])
Andreas@59025
   957
Andreas@59025
   958
  assume f: "\<And>x. x \<notin> range g \<Longrightarrow> f x = 0"
Andreas@59025
   959
Andreas@59025
   960
  from \<open>summable f\<close> have "suminf f \<le> suminf (f \<circ> g)"
Andreas@59025
   961
  proof(rule suminf_le_const)
Andreas@59025
   962
    fix n
Andreas@59025
   963
    have "\<forall> n' \<in> (g -` {..<n}). n' < Suc (Max (g -` {..<n}))"
Andreas@59025
   964
      by(auto intro: Max_ge simp add: finite_vimageI less_Suc_eq_le)
Andreas@59025
   965
    then obtain m where n: "\<And>n'. g n' < n \<Longrightarrow> n' < m" by blast
Andreas@59025
   966
Andreas@59025
   967
    have "(\<Sum>i<n. f i) = (\<Sum>i\<in>{..<n} \<inter> range g. f i)"
Andreas@59025
   968
      using f by(auto intro: setsum.mono_neutral_cong_right)
Andreas@59025
   969
    also have "\<dots> = (\<Sum>i\<in>g -` {..<n}. (f \<circ> g) i)"
Andreas@59025
   970
      by(rule setsum.reindex_cong[where l=g])(auto)
Andreas@59025
   971
    also have "\<dots> \<le> (\<Sum>i<m. (f \<circ> g) i)"
Andreas@59025
   972
      by(rule setsum_mono3)(auto simp add: pos n)
Andreas@59025
   973
    also have "\<dots> \<le> suminf (f \<circ> g)"
Andreas@59025
   974
      using \<open>summable (f o g)\<close>
Andreas@59025
   975
      by(rule setsum_le_suminf)(simp add: pos)
Andreas@59025
   976
    finally show "setsum f {..<n} \<le> suminf (f \<circ> g)" .
Andreas@59025
   977
  qed
Andreas@59025
   978
  with le show "suminf (f \<circ> g) = suminf f" by(rule antisym)
Andreas@59025
   979
qed
Andreas@59025
   980
eberlm@61531
   981
lemma sums_mono_reindex:
eberlm@61531
   982
  assumes subseq: "subseq g" and zero: "\<And>n. n \<notin> range g \<Longrightarrow> f n = 0"
eberlm@61531
   983
  shows   "(\<lambda>n. f (g n)) sums c \<longleftrightarrow> f sums c"
eberlm@61531
   984
unfolding sums_def
eberlm@61531
   985
proof
wenzelm@61969
   986
  assume lim: "(\<lambda>n. \<Sum>k<n. f k) \<longlonglongrightarrow> c"
eberlm@61531
   987
  have "(\<lambda>n. \<Sum>k<n. f (g k)) = (\<lambda>n. \<Sum>k<g n. f k)"
eberlm@61531
   988
  proof
eberlm@61531
   989
    fix n :: nat
eberlm@61531
   990
    from subseq have "(\<Sum>k<n. f (g k)) = (\<Sum>k\<in>g`{..<n}. f k)"
eberlm@61531
   991
      by (subst setsum.reindex) (auto intro: subseq_imp_inj_on)
eberlm@61531
   992
    also from subseq have "\<dots> = (\<Sum>k<g n. f k)"
eberlm@61531
   993
      by (intro setsum.mono_neutral_left ballI zero)
eberlm@61531
   994
         (auto dest: subseq_strict_mono simp: strict_mono_less strict_mono_less_eq)
eberlm@61531
   995
    finally show "(\<Sum>k<n. f (g k)) = (\<Sum>k<g n. f k)" .
eberlm@61531
   996
  qed
wenzelm@61969
   997
  also from LIMSEQ_subseq_LIMSEQ[OF lim subseq] have "\<dots> \<longlonglongrightarrow> c" unfolding o_def .
wenzelm@61969
   998
  finally show "(\<lambda>n. \<Sum>k<n. f (g k)) \<longlonglongrightarrow> c" .
eberlm@61531
   999
next
wenzelm@61969
  1000
  assume lim: "(\<lambda>n. \<Sum>k<n. f (g k)) \<longlonglongrightarrow> c"
eberlm@61531
  1001
  def g_inv \<equiv> "\<lambda>n. LEAST m. g m \<ge> n"
eberlm@61531
  1002
  from filterlim_subseq[OF subseq] have g_inv_ex: "\<exists>m. g m \<ge> n" for n
eberlm@61531
  1003
    by (auto simp: filterlim_at_top eventually_at_top_linorder)
eberlm@61531
  1004
  hence g_inv: "g (g_inv n) \<ge> n" for n unfolding g_inv_def by (rule LeastI_ex)
lp15@61609
  1005
  have g_inv_least: "m \<ge> g_inv n" if "g m \<ge> n" for m n using that
eberlm@61531
  1006
    unfolding g_inv_def by (rule Least_le)
lp15@61609
  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
eberlm@61531
  1008
  have "(\<lambda>n. \<Sum>k<n. f k) = (\<lambda>n. \<Sum>k<g_inv n. f (g k))"
eberlm@61531
  1009
  proof
eberlm@61531
  1010
    fix n :: nat
eberlm@61531
  1011
    {
eberlm@61531
  1012
      fix k assume k: "k \<in> {..<n} - g`{..<g_inv n}"
eberlm@61531
  1013
      have "k \<notin> range g"
eberlm@61531
  1014
      proof (rule notI, elim imageE)
eberlm@61531
  1015
        fix l assume l: "k = g l"
eberlm@61531
  1016
        have "g l < g (g_inv n)" by (rule less_le_trans[OF _ g_inv]) (insert k l, simp_all)
eberlm@61531
  1017
        with subseq have "l < g_inv n" by (simp add: subseq_strict_mono strict_mono_less)
eberlm@61531
  1018
        with k l show False by simp
eberlm@61531
  1019
      qed
eberlm@61531
  1020
      hence "f k = 0" by (rule zero)
eberlm@61531
  1021
    }
eberlm@61531
  1022
    with g_inv_least' g_inv have "(\<Sum>k<n. f k) = (\<Sum>k\<in>g`{..<g_inv n}. f k)"
eberlm@61531
  1023
      by (intro setsum.mono_neutral_right) auto
lp15@61609
  1024
    also from subseq have "\<dots> = (\<Sum>k<g_inv n. f (g k))" using subseq_imp_inj_on
eberlm@61531
  1025
      by (subst setsum.reindex) simp_all
eberlm@61531
  1026
    finally show "(\<Sum>k<n. f k) = (\<Sum>k<g_inv n. f (g k))" .
eberlm@61531
  1027
  qed
eberlm@61531
  1028
  also {
eberlm@61531
  1029
    fix K n :: nat assume "g K \<le> n"
eberlm@61531
  1030
    also have "n \<le> g (g_inv n)" by (rule g_inv)
eberlm@61531
  1031
    finally have "K \<le> g_inv n" using subseq by (simp add: strict_mono_less_eq subseq_strict_mono)
eberlm@61531
  1032
  }
lp15@61609
  1033
  hence "filterlim g_inv at_top sequentially"
eberlm@61531
  1034
    by (auto simp: filterlim_at_top eventually_at_top_linorder)
wenzelm@61969
  1035
  from lim and this have "(\<lambda>n. \<Sum>k<g_inv n. f (g k)) \<longlonglongrightarrow> c" by (rule filterlim_compose)
wenzelm@61969
  1036
  finally show "(\<lambda>n. \<Sum>k<n. f k) \<longlonglongrightarrow> c" .
eberlm@61531
  1037
qed
eberlm@61531
  1038
eberlm@61531
  1039
lemma summable_mono_reindex:
eberlm@61531
  1040
  assumes subseq: "subseq g" and zero: "\<And>n. n \<notin> range g \<Longrightarrow> f n = 0"
eberlm@61531
  1041
  shows   "summable (\<lambda>n. f (g n)) \<longleftrightarrow> summable f"
eberlm@61531
  1042
  using sums_mono_reindex[of g f, OF assms] by (simp add: summable_def)
eberlm@61531
  1043
lp15@61609
  1044
lemma suminf_mono_reindex:
eberlm@61531
  1045
  assumes "subseq g" "\<And>n. n \<notin> range g \<Longrightarrow> f n = (0 :: 'a :: {t2_space,comm_monoid_add})"
eberlm@61531
  1046
  shows   "suminf (\<lambda>n. f (g n)) = suminf f"
eberlm@61531
  1047
proof (cases "summable f")
eberlm@61531
  1048
  case False
eberlm@61531
  1049
  hence "\<not>(\<exists>c. f sums c)" unfolding summable_def by blast
eberlm@61531
  1050
  hence "suminf f = The (\<lambda>_. False)" by (simp add: suminf_def)
eberlm@61531
  1051
  moreover from False have "\<not>summable (\<lambda>n. f (g n))"
eberlm@61531
  1052
    using summable_mono_reindex[of g f, OF assms] by simp
eberlm@61531
  1053
  hence "\<not>(\<exists>c. (\<lambda>n. f (g n)) sums c)" unfolding summable_def by blast
eberlm@61531
  1054
  hence "suminf (\<lambda>n. f (g n)) = The (\<lambda>_. False)" by (simp add: suminf_def)
eberlm@61531
  1055
  ultimately show ?thesis by simp
lp15@61609
  1056
qed (insert sums_mono_reindex[of g f, OF assms] summable_mono_reindex[of g f, OF assms],
eberlm@61531
  1057
     simp_all add: sums_iff)
eberlm@61531
  1058
paulson@14416
  1059
end