src/HOL/Series.thy
author hoelzl
Mon Feb 08 19:53:49 2016 +0100 (2016-02-08)
changeset 62368 106569399cd6
parent 62217 527488dc8b90
child 62376 85f38d5f8807
permissions -rw-r--r--
add type class for topological monoids
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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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corollary sums_0:
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   "(\<And>n. f n = 0) \<Longrightarrow> (f sums 0)"
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    by (metis (no_types) finite.emptyI setsum.empty sums_finite)
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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 topological monoids\<close>
hoelzl@62368
   275
hoelzl@62368
   276
lemma Zero_notin_Suc: "0 \<notin> Suc ` A"
hoelzl@62368
   277
  by auto
hoelzl@62368
   278
hoelzl@62368
   279
context
hoelzl@62368
   280
  fixes f g :: "nat \<Rightarrow> 'a :: {t2_space, topological_comm_monoid_add}"
hoelzl@62368
   281
begin
hoelzl@62368
   282
hoelzl@62368
   283
lemma sums_Suc:
hoelzl@62368
   284
  assumes "(\<lambda>n. f (Suc n)) sums l" shows "f sums (l + f 0)"
hoelzl@62368
   285
proof  -
hoelzl@62368
   286
  have "(\<lambda>n. (\<Sum>i<n. f (Suc i)) + f 0) \<longlonglongrightarrow> l + f 0"
hoelzl@62368
   287
    using assms by (auto intro!: tendsto_add simp: sums_def)
hoelzl@62368
   288
  moreover have "(\<Sum>i<n. f (Suc i)) + f 0 = (\<Sum>i<Suc n. f i)" for n
hoelzl@62368
   289
    unfolding lessThan_Suc_eq_insert_0 by (simp add: Zero_notin_Suc ac_simps setsum.reindex)
hoelzl@62368
   290
  ultimately show ?thesis
hoelzl@62368
   291
    by (auto simp add: sums_def simp del: setsum_lessThan_Suc intro: LIMSEQ_Suc_iff[THEN iffD1])
hoelzl@62368
   292
qed
hoelzl@62368
   293
hoelzl@62368
   294
lemma sums_add: "f sums a \<Longrightarrow> g sums b \<Longrightarrow> (\<lambda>n. f n + g n) sums (a + b)"
hoelzl@62368
   295
  unfolding sums_def by (simp add: setsum.distrib tendsto_add)
hoelzl@62368
   296
hoelzl@62368
   297
lemma summable_add: "summable f \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. f n + g n)"
hoelzl@62368
   298
  unfolding summable_def by (auto intro: sums_add)
hoelzl@62368
   299
hoelzl@62368
   300
lemma suminf_add: "summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f + suminf g = (\<Sum>n. f n + g n)"
hoelzl@62368
   301
  by (intro sums_unique sums_add summable_sums)
hoelzl@62368
   302
hoelzl@62368
   303
end
hoelzl@62368
   304
hoelzl@62368
   305
context
hoelzl@62368
   306
  fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::{t2_space, topological_comm_monoid_add}" and I :: "'i set"
hoelzl@62368
   307
begin
hoelzl@62368
   308
hoelzl@62368
   309
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@62368
   310
  by (induct I rule: infinite_finite_induct) (auto intro!: sums_add)
hoelzl@62368
   311
hoelzl@62368
   312
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@62368
   313
  using sums_unique[OF sums_setsum, OF summable_sums] by simp
hoelzl@62368
   314
hoelzl@62368
   315
lemma summable_setsum: "(\<And>i. i \<in> I \<Longrightarrow> summable (f i)) \<Longrightarrow> summable (\<lambda>n. \<Sum>i\<in>I. f i n)"
hoelzl@62368
   316
  using sums_summable[OF sums_setsum[OF summable_sums]] .
hoelzl@62368
   317
hoelzl@62368
   318
end
eberlm@61531
   319
wenzelm@60758
   320
subsection \<open>Infinite summability on real normed vector spaces\<close>
hoelzl@56193
   321
hoelzl@62368
   322
context
hoelzl@56193
   323
  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@62368
   324
begin
hoelzl@62368
   325
hoelzl@62368
   326
lemma sums_Suc_iff: "(\<lambda>n. f (Suc n)) sums s \<longleftrightarrow> f sums (s + f 0)"
hoelzl@56193
   327
proof -
wenzelm@61969
   328
  have "f sums (s + f 0) \<longleftrightarrow> (\<lambda>i. \<Sum>j<Suc i. f j) \<longlonglongrightarrow> s + f 0"
hoelzl@56193
   329
    by (subst LIMSEQ_Suc_iff) (simp add: sums_def)
wenzelm@61969
   330
  also have "\<dots> \<longleftrightarrow> (\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) \<longlonglongrightarrow> s + f 0"
haftmann@57418
   331
    by (simp add: ac_simps setsum.reindex image_iff lessThan_Suc_eq_insert_0)
hoelzl@56193
   332
  also have "\<dots> \<longleftrightarrow> (\<lambda>n. f (Suc n)) sums s"
hoelzl@56193
   333
  proof
wenzelm@61969
   334
    assume "(\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) \<longlonglongrightarrow> s + f 0"
hoelzl@56193
   335
    with tendsto_add[OF this tendsto_const, of "- f 0"]
hoelzl@56193
   336
    show "(\<lambda>i. f (Suc i)) sums s"
hoelzl@56193
   337
      by (simp add: sums_def)
hoelzl@58729
   338
  qed (auto intro: tendsto_add simp: sums_def)
hoelzl@56193
   339
  finally show ?thesis ..
hoelzl@50999
   340
qed
hoelzl@50999
   341
hoelzl@62368
   342
lemma summable_Suc_iff: "summable (\<lambda>n. f (Suc n)) = summable f"
eberlm@61531
   343
proof
eberlm@61531
   344
  assume "summable f"
eberlm@61531
   345
  hence "f sums suminf f" by (rule summable_sums)
eberlm@61531
   346
  hence "(\<lambda>n. f (Suc n)) sums (suminf f - f 0)" by (simp add: sums_Suc_iff)
eberlm@61531
   347
  thus "summable (\<lambda>n. f (Suc n))" unfolding summable_def by blast
eberlm@61531
   348
qed (auto simp: sums_Suc_iff summable_def)
eberlm@61531
   349
hoelzl@62368
   350
end
hoelzl@62368
   351
hoelzl@56193
   352
context
hoelzl@56193
   353
  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@56193
   354
begin
hoelzl@56193
   355
hoelzl@56193
   356
lemma sums_diff: "f sums a \<Longrightarrow> g sums b \<Longrightarrow> (\<lambda>n. f n - g n) sums (a - b)"
hoelzl@56193
   357
  unfolding sums_def by (simp add: setsum_subtractf tendsto_diff)
hoelzl@56193
   358
hoelzl@56193
   359
lemma summable_diff: "summable f \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. f n - g n)"
hoelzl@56193
   360
  unfolding summable_def by (auto intro: sums_diff)
hoelzl@56193
   361
hoelzl@56193
   362
lemma suminf_diff: "summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f - suminf g = (\<Sum>n. f n - g n)"
hoelzl@56193
   363
  by (intro sums_unique sums_diff summable_sums)
hoelzl@56193
   364
hoelzl@56193
   365
lemma sums_minus: "f sums a \<Longrightarrow> (\<lambda>n. - f n) sums (- a)"
hoelzl@56193
   366
  unfolding sums_def by (simp add: setsum_negf tendsto_minus)
hoelzl@56193
   367
hoelzl@56193
   368
lemma summable_minus: "summable f \<Longrightarrow> summable (\<lambda>n. - f n)"
hoelzl@56193
   369
  unfolding summable_def by (auto intro: sums_minus)
huffman@20692
   370
hoelzl@56193
   371
lemma suminf_minus: "summable f \<Longrightarrow> (\<Sum>n. - f n) = - (\<Sum>n. f n)"
hoelzl@56193
   372
  by (intro sums_unique [symmetric] sums_minus summable_sums)
hoelzl@56193
   373
hoelzl@56193
   374
lemma sums_iff_shift: "(\<lambda>i. f (i + n)) sums s \<longleftrightarrow> f sums (s + (\<Sum>i<n. f i))"
hoelzl@56193
   375
proof (induct n arbitrary: s)
hoelzl@56193
   376
  case (Suc n)
hoelzl@56193
   377
  moreover have "(\<lambda>i. f (Suc i + n)) sums s \<longleftrightarrow> (\<lambda>i. f (i + n)) sums (s + f n)"
hoelzl@56193
   378
    by (subst sums_Suc_iff) simp
hoelzl@56193
   379
  ultimately show ?case
hoelzl@56193
   380
    by (simp add: ac_simps)
hoelzl@56193
   381
qed simp
huffman@20692
   382
hoelzl@56193
   383
lemma summable_iff_shift: "summable (\<lambda>n. f (n + k)) \<longleftrightarrow> summable f"
hoelzl@56193
   384
  by (metis diff_add_cancel summable_def sums_iff_shift[abs_def])
hoelzl@56193
   385
hoelzl@56193
   386
lemma sums_split_initial_segment: "f sums s \<Longrightarrow> (\<lambda>i. f (i + n)) sums (s - (\<Sum>i<n. f i))"
hoelzl@56193
   387
  by (simp add: sums_iff_shift)
hoelzl@56193
   388
hoelzl@56193
   389
lemma summable_ignore_initial_segment: "summable f \<Longrightarrow> summable (\<lambda>n. f(n + k))"
hoelzl@56193
   390
  by (simp add: summable_iff_shift)
hoelzl@56193
   391
hoelzl@56193
   392
lemma suminf_minus_initial_segment: "summable f \<Longrightarrow> (\<Sum>n. f (n + k)) = (\<Sum>n. f n) - (\<Sum>i<k. f i)"
hoelzl@56193
   393
  by (rule sums_unique[symmetric]) (auto simp: sums_iff_shift)
hoelzl@56193
   394
hoelzl@56193
   395
lemma suminf_split_initial_segment: "summable f \<Longrightarrow> suminf f = (\<Sum>n. f(n + k)) + (\<Sum>i<k. f i)"
hoelzl@56193
   396
  by (auto simp add: suminf_minus_initial_segment)
huffman@20692
   397
eberlm@61531
   398
lemma suminf_split_head: "summable f \<Longrightarrow> (\<Sum>n. f (Suc n)) = suminf f - f 0"
eberlm@61531
   399
  using suminf_split_initial_segment[of 1] by simp
eberlm@61531
   400
lp15@61609
   401
lemma suminf_exist_split:
hoelzl@56193
   402
  fixes r :: real assumes "0 < r" and "summable f"
hoelzl@56193
   403
  shows "\<exists>N. \<forall>n\<ge>N. norm (\<Sum>i. f (i + n)) < r"
hoelzl@56193
   404
proof -
wenzelm@60758
   405
  from LIMSEQ_D[OF summable_LIMSEQ[OF \<open>summable f\<close>] \<open>0 < r\<close>]
hoelzl@56193
   406
  obtain N :: nat where "\<forall> n \<ge> N. norm (setsum f {..<n} - suminf f) < r" by auto
hoelzl@56193
   407
  thus ?thesis
wenzelm@60758
   408
    by (auto simp: norm_minus_commute suminf_minus_initial_segment[OF \<open>summable f\<close>])
hoelzl@56193
   409
qed
hoelzl@56193
   410
wenzelm@61969
   411
lemma summable_LIMSEQ_zero: "summable f \<Longrightarrow> f \<longlonglongrightarrow> 0"
hoelzl@56193
   412
  apply (drule summable_iff_convergent [THEN iffD1])
hoelzl@56193
   413
  apply (drule convergent_Cauchy)
hoelzl@56193
   414
  apply (simp only: Cauchy_iff LIMSEQ_iff, safe)
hoelzl@56193
   415
  apply (drule_tac x="r" in spec, safe)
hoelzl@56193
   416
  apply (rule_tac x="M" in exI, safe)
hoelzl@56193
   417
  apply (drule_tac x="Suc n" in spec, simp)
hoelzl@56193
   418
  apply (drule_tac x="n" in spec, simp)
hoelzl@56193
   419
  done
hoelzl@56193
   420
hoelzl@62368
   421
lemma summable_imp_convergent: "summable f \<Longrightarrow> convergent f"
eberlm@61531
   422
  by (force dest!: summable_LIMSEQ_zero simp: convergent_def)
eberlm@61531
   423
hoelzl@62368
   424
lemma summable_imp_Bseq: "summable f \<Longrightarrow> Bseq f"
eberlm@61531
   425
  by (simp add: convergent_imp_Bseq summable_imp_convergent)
eberlm@61531
   426
hoelzl@56193
   427
end
hoelzl@56193
   428
lp15@59613
   429
lemma summable_minus_iff:
lp15@59613
   430
  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
lp15@59613
   431
  shows "summable (\<lambda>n. - f n) \<longleftrightarrow> summable f"
wenzelm@61799
   432
  by (auto dest: summable_minus) \<comment>\<open>used two ways, hence must be outside the context above\<close>
lp15@59613
   433
hoelzl@56193
   434
lemma (in bounded_linear) sums: "(\<lambda>n. X n) sums a \<Longrightarrow> (\<lambda>n. f (X n)) sums (f a)"
hoelzl@56193
   435
  unfolding sums_def by (drule tendsto, simp only: setsum)
hoelzl@56193
   436
hoelzl@56193
   437
lemma (in bounded_linear) summable: "summable (\<lambda>n. X n) \<Longrightarrow> summable (\<lambda>n. f (X n))"
hoelzl@56193
   438
  unfolding summable_def by (auto intro: sums)
hoelzl@56193
   439
hoelzl@56193
   440
lemma (in bounded_linear) suminf: "summable (\<lambda>n. X n) \<Longrightarrow> f (\<Sum>n. X n) = (\<Sum>n. f (X n))"
hoelzl@56193
   441
  by (intro sums_unique sums summable_sums)
hoelzl@56193
   442
hoelzl@56193
   443
lemmas sums_of_real = bounded_linear.sums [OF bounded_linear_of_real]
hoelzl@56193
   444
lemmas summable_of_real = bounded_linear.summable [OF bounded_linear_of_real]
hoelzl@56193
   445
lemmas suminf_of_real = bounded_linear.suminf [OF bounded_linear_of_real]
hoelzl@56193
   446
hoelzl@57275
   447
lemmas sums_scaleR_left = bounded_linear.sums[OF bounded_linear_scaleR_left]
hoelzl@57275
   448
lemmas summable_scaleR_left = bounded_linear.summable[OF bounded_linear_scaleR_left]
hoelzl@57275
   449
lemmas suminf_scaleR_left = bounded_linear.suminf[OF bounded_linear_scaleR_left]
hoelzl@57275
   450
hoelzl@57275
   451
lemmas sums_scaleR_right = bounded_linear.sums[OF bounded_linear_scaleR_right]
hoelzl@57275
   452
lemmas summable_scaleR_right = bounded_linear.summable[OF bounded_linear_scaleR_right]
hoelzl@57275
   453
lemmas suminf_scaleR_right = bounded_linear.suminf[OF bounded_linear_scaleR_right]
hoelzl@57275
   454
eberlm@61531
   455
lemma summable_const_iff: "summable (\<lambda>_. c) \<longleftrightarrow> (c :: 'a :: real_normed_vector) = 0"
eberlm@61531
   456
proof -
eberlm@61531
   457
  {
eberlm@61531
   458
    assume "c \<noteq> 0"
eberlm@61531
   459
    hence "filterlim (\<lambda>n. of_nat n * norm c) at_top sequentially"
eberlm@61531
   460
      by (subst mult.commute)
eberlm@61531
   461
         (auto intro!: filterlim_tendsto_pos_mult_at_top filterlim_real_sequentially)
eberlm@61531
   462
    hence "\<not>convergent (\<lambda>n. norm (\<Sum>k<n. c))"
lp15@61609
   463
      by (intro filterlim_at_infinity_imp_not_convergent filterlim_at_top_imp_at_infinity)
eberlm@61531
   464
         (simp_all add: setsum_constant_scaleR)
eberlm@61531
   465
    hence "\<not>summable (\<lambda>_. c)" unfolding summable_iff_convergent using convergent_norm by blast
eberlm@61531
   466
  }
eberlm@61531
   467
  thus ?thesis by auto
eberlm@61531
   468
qed
eberlm@61531
   469
eberlm@61531
   470
wenzelm@60758
   471
subsection \<open>Infinite summability on real normed algebras\<close>
hoelzl@56213
   472
hoelzl@56193
   473
context
hoelzl@56193
   474
  fixes f :: "nat \<Rightarrow> 'a::real_normed_algebra"
hoelzl@56193
   475
begin
hoelzl@56193
   476
hoelzl@56193
   477
lemma sums_mult: "f sums a \<Longrightarrow> (\<lambda>n. c * f n) sums (c * a)"
hoelzl@56193
   478
  by (rule bounded_linear.sums [OF bounded_linear_mult_right])
hoelzl@56193
   479
hoelzl@56193
   480
lemma summable_mult: "summable f \<Longrightarrow> summable (\<lambda>n. c * f n)"
hoelzl@56193
   481
  by (rule bounded_linear.summable [OF bounded_linear_mult_right])
hoelzl@56193
   482
hoelzl@56193
   483
lemma suminf_mult: "summable f \<Longrightarrow> suminf (\<lambda>n. c * f n) = c * suminf f"
hoelzl@56193
   484
  by (rule bounded_linear.suminf [OF bounded_linear_mult_right, symmetric])
hoelzl@56193
   485
hoelzl@56193
   486
lemma sums_mult2: "f sums a \<Longrightarrow> (\<lambda>n. f n * c) sums (a * c)"
hoelzl@56193
   487
  by (rule bounded_linear.sums [OF bounded_linear_mult_left])
hoelzl@56193
   488
hoelzl@56193
   489
lemma summable_mult2: "summable f \<Longrightarrow> summable (\<lambda>n. f n * c)"
hoelzl@56193
   490
  by (rule bounded_linear.summable [OF bounded_linear_mult_left])
hoelzl@56193
   491
hoelzl@56193
   492
lemma suminf_mult2: "summable f \<Longrightarrow> suminf f * c = (\<Sum>n. f n * c)"
hoelzl@56193
   493
  by (rule bounded_linear.suminf [OF bounded_linear_mult_left])
hoelzl@56193
   494
hoelzl@56193
   495
end
hoelzl@56193
   496
eberlm@61531
   497
lemma sums_mult_iff:
eberlm@61531
   498
  assumes "c \<noteq> 0"
eberlm@61531
   499
  shows   "(\<lambda>n. c * f n :: 'a :: {real_normed_algebra,field}) sums (c * d) \<longleftrightarrow> f sums d"
eberlm@61531
   500
  using sums_mult[of f d c] sums_mult[of "\<lambda>n. c * f n" "c * d" "inverse c"]
eberlm@61531
   501
  by (force simp: field_simps assms)
eberlm@61531
   502
eberlm@61531
   503
lemma sums_mult2_iff:
eberlm@61531
   504
  assumes "c \<noteq> (0 :: 'a :: {real_normed_algebra, field})"
eberlm@61531
   505
  shows   "(\<lambda>n. f n * c) sums (d * c) \<longleftrightarrow> f sums d"
eberlm@61531
   506
  using sums_mult_iff[OF assms, of f d] by (simp add: mult.commute)
eberlm@61531
   507
eberlm@61531
   508
lemma sums_of_real_iff:
eberlm@61531
   509
  "(\<lambda>n. of_real (f n) :: 'a :: real_normed_div_algebra) sums of_real c \<longleftrightarrow> f sums c"
eberlm@61531
   510
  by (simp add: sums_def of_real_setsum[symmetric] tendsto_of_real_iff del: of_real_setsum)
eberlm@61531
   511
eberlm@61531
   512
wenzelm@60758
   513
subsection \<open>Infinite summability on real normed fields\<close>
hoelzl@56213
   514
hoelzl@56193
   515
context
hoelzl@56193
   516
  fixes c :: "'a::real_normed_field"
hoelzl@56193
   517
begin
hoelzl@56193
   518
hoelzl@56193
   519
lemma sums_divide: "f sums a \<Longrightarrow> (\<lambda>n. f n / c) sums (a / c)"
hoelzl@56193
   520
  by (rule bounded_linear.sums [OF bounded_linear_divide])
hoelzl@56193
   521
hoelzl@56193
   522
lemma summable_divide: "summable f \<Longrightarrow> summable (\<lambda>n. f n / c)"
hoelzl@56193
   523
  by (rule bounded_linear.summable [OF bounded_linear_divide])
hoelzl@56193
   524
hoelzl@56193
   525
lemma suminf_divide: "summable f \<Longrightarrow> suminf (\<lambda>n. f n / c) = suminf f / c"
hoelzl@56193
   526
  by (rule bounded_linear.suminf [OF bounded_linear_divide, symmetric])
paulson@14416
   527
wenzelm@60758
   528
text\<open>Sum of a geometric progression.\<close>
paulson@14416
   529
hoelzl@56193
   530
lemma geometric_sums: "norm c < 1 \<Longrightarrow> (\<lambda>n. c^n) sums (1 / (1 - c))"
huffman@20692
   531
proof -
hoelzl@56193
   532
  assume less_1: "norm c < 1"
hoelzl@56193
   533
  hence neq_1: "c \<noteq> 1" by auto
hoelzl@56193
   534
  hence neq_0: "c - 1 \<noteq> 0" by simp
wenzelm@61969
   535
  from less_1 have lim_0: "(\<lambda>n. c^n) \<longlonglongrightarrow> 0"
huffman@20692
   536
    by (rule LIMSEQ_power_zero)
wenzelm@61969
   537
  hence "(\<lambda>n. c ^ n / (c - 1) - 1 / (c - 1)) \<longlonglongrightarrow> 0 / (c - 1) - 1 / (c - 1)"
huffman@44568
   538
    using neq_0 by (intro tendsto_intros)
wenzelm@61969
   539
  hence "(\<lambda>n. (c ^ n - 1) / (c - 1)) \<longlonglongrightarrow> 1 / (1 - c)"
huffman@20692
   540
    by (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib)
hoelzl@56193
   541
  thus "(\<lambda>n. c ^ n) sums (1 / (1 - c))"
huffman@20692
   542
    by (simp add: sums_def geometric_sum neq_1)
huffman@20692
   543
qed
huffman@20692
   544
hoelzl@56193
   545
lemma summable_geometric: "norm c < 1 \<Longrightarrow> summable (\<lambda>n. c^n)"
hoelzl@56193
   546
  by (rule geometric_sums [THEN sums_summable])
paulson@14416
   547
hoelzl@56193
   548
lemma suminf_geometric: "norm c < 1 \<Longrightarrow> suminf (\<lambda>n. c^n) = 1 / (1 - c)"
hoelzl@56193
   549
  by (rule sums_unique[symmetric]) (rule geometric_sums)
hoelzl@56193
   550
eberlm@61531
   551
lemma summable_geometric_iff: "summable (\<lambda>n. c ^ n) \<longleftrightarrow> norm c < 1"
eberlm@61531
   552
proof
eberlm@61531
   553
  assume "summable (\<lambda>n. c ^ n :: 'a :: real_normed_field)"
wenzelm@61969
   554
  hence "(\<lambda>n. norm c ^ n) \<longlonglongrightarrow> 0"
eberlm@61531
   555
    by (simp add: norm_power [symmetric] tendsto_norm_zero_iff summable_LIMSEQ_zero)
eberlm@61531
   556
  from order_tendstoD(2)[OF this zero_less_one] obtain n where "norm c ^ n < 1"
eberlm@61531
   557
    by (auto simp: eventually_at_top_linorder)
eberlm@61531
   558
  thus "norm c < 1" using one_le_power[of "norm c" n] by (cases "norm c \<ge> 1") (linarith, simp)
eberlm@61531
   559
qed (rule summable_geometric)
lp15@61609
   560
hoelzl@56193
   561
end
paulson@33271
   562
paulson@33271
   563
lemma power_half_series: "(\<lambda>n. (1/2::real)^Suc n) sums 1"
paulson@33271
   564
proof -
paulson@33271
   565
  have 2: "(\<lambda>n. (1/2::real)^n) sums 2" using geometric_sums [of "1/2::real"]
paulson@33271
   566
    by auto
paulson@33271
   567
  have "(\<lambda>n. (1/2::real)^Suc n) = (\<lambda>n. (1 / 2) ^ n / 2)"
lp15@59741
   568
    by (simp add: mult.commute)
huffman@44282
   569
  thus ?thesis using sums_divide [OF 2, of 2]
paulson@33271
   570
    by simp
paulson@33271
   571
qed
paulson@33271
   572
eberlm@61531
   573
eberlm@61531
   574
subsection \<open>Telescoping\<close>
eberlm@61531
   575
eberlm@61531
   576
lemma telescope_sums:
wenzelm@61969
   577
  assumes "f \<longlonglongrightarrow> (c :: 'a :: real_normed_vector)"
eberlm@61531
   578
  shows   "(\<lambda>n. f (Suc n) - f n) sums (c - f 0)"
eberlm@61531
   579
  unfolding sums_def
eberlm@61531
   580
proof (subst LIMSEQ_Suc_iff [symmetric])
eberlm@61531
   581
  have "(\<lambda>n. \<Sum>k<Suc n. f (Suc k) - f k) = (\<lambda>n. f (Suc n) - f 0)"
eberlm@61531
   582
    by (simp add: lessThan_Suc_atMost atLeast0AtMost [symmetric] setsum_Suc_diff)
wenzelm@61969
   583
  also have "\<dots> \<longlonglongrightarrow> c - f 0" by (intro tendsto_diff LIMSEQ_Suc[OF assms] tendsto_const)
wenzelm@61969
   584
  finally show "(\<lambda>n. \<Sum>n<Suc n. f (Suc n) - f n) \<longlonglongrightarrow> c - f 0" .
eberlm@61531
   585
qed
eberlm@61531
   586
eberlm@61531
   587
lemma telescope_sums':
wenzelm@61969
   588
  assumes "f \<longlonglongrightarrow> (c :: 'a :: real_normed_vector)"
eberlm@61531
   589
  shows   "(\<lambda>n. f n - f (Suc n)) sums (f 0 - c)"
eberlm@61531
   590
  using sums_minus[OF telescope_sums[OF assms]] by (simp add: algebra_simps)
eberlm@61531
   591
eberlm@61531
   592
lemma telescope_summable:
wenzelm@61969
   593
  assumes "f \<longlonglongrightarrow> (c :: 'a :: real_normed_vector)"
eberlm@61531
   594
  shows   "summable (\<lambda>n. f (Suc n) - f n)"
eberlm@61531
   595
  using telescope_sums[OF assms] by (simp add: sums_iff)
eberlm@61531
   596
eberlm@61531
   597
lemma telescope_summable':
wenzelm@61969
   598
  assumes "f \<longlonglongrightarrow> (c :: 'a :: real_normed_vector)"
eberlm@61531
   599
  shows   "summable (\<lambda>n. f n - f (Suc n))"
eberlm@61531
   600
  using summable_minus[OF telescope_summable[OF assms]] by (simp add: algebra_simps)
eberlm@61531
   601
eberlm@61531
   602
wenzelm@60758
   603
subsection \<open>Infinite summability on Banach spaces\<close>
hoelzl@56213
   604
wenzelm@60758
   605
text\<open>Cauchy-type criterion for convergence of series (c.f. Harrison)\<close>
paulson@15085
   606
hoelzl@56193
   607
lemma summable_Cauchy:
hoelzl@56193
   608
  fixes f :: "nat \<Rightarrow> 'a::banach"
hoelzl@56193
   609
  shows "summable f \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n. norm (setsum f {m..<n}) < e)"
hoelzl@56193
   610
  apply (simp only: summable_iff_convergent Cauchy_convergent_iff [symmetric] Cauchy_iff, safe)
hoelzl@56193
   611
  apply (drule spec, drule (1) mp)
hoelzl@56193
   612
  apply (erule exE, rule_tac x="M" in exI, clarify)
hoelzl@56193
   613
  apply (rule_tac x="m" and y="n" in linorder_le_cases)
hoelzl@56193
   614
  apply (frule (1) order_trans)
hoelzl@56193
   615
  apply (drule_tac x="n" in spec, drule (1) mp)
hoelzl@56193
   616
  apply (drule_tac x="m" in spec, drule (1) mp)
hoelzl@56193
   617
  apply (simp_all add: setsum_diff [symmetric])
hoelzl@56193
   618
  apply (drule spec, drule (1) mp)
hoelzl@56193
   619
  apply (erule exE, rule_tac x="N" in exI, clarify)
hoelzl@56193
   620
  apply (rule_tac x="m" and y="n" in linorder_le_cases)
hoelzl@56193
   621
  apply (subst norm_minus_commute)
hoelzl@56193
   622
  apply (simp_all add: setsum_diff [symmetric])
hoelzl@56193
   623
  done
paulson@14416
   624
hoelzl@56193
   625
context
hoelzl@56193
   626
  fixes f :: "nat \<Rightarrow> 'a::banach"
eberlm@61531
   627
begin
hoelzl@56193
   628
wenzelm@60758
   629
text\<open>Absolute convergence imples normal convergence\<close>
huffman@20689
   630
hoelzl@56194
   631
lemma summable_norm_cancel: "summable (\<lambda>n. norm (f n)) \<Longrightarrow> summable f"
hoelzl@56193
   632
  apply (simp only: summable_Cauchy, safe)
hoelzl@56193
   633
  apply (drule_tac x="e" in spec, safe)
hoelzl@56193
   634
  apply (rule_tac x="N" in exI, safe)
hoelzl@56193
   635
  apply (drule_tac x="m" in spec, safe)
hoelzl@56193
   636
  apply (rule order_le_less_trans [OF norm_setsum])
hoelzl@56193
   637
  apply (rule order_le_less_trans [OF abs_ge_self])
hoelzl@56193
   638
  apply simp
hoelzl@50999
   639
  done
paulson@32707
   640
hoelzl@56193
   641
lemma summable_norm: "summable (\<lambda>n. norm (f n)) \<Longrightarrow> norm (suminf f) \<le> (\<Sum>n. norm (f n))"
hoelzl@56193
   642
  by (auto intro: LIMSEQ_le tendsto_norm summable_norm_cancel summable_LIMSEQ norm_setsum)
hoelzl@56193
   643
wenzelm@60758
   644
text \<open>Comparison tests\<close>
paulson@14416
   645
hoelzl@56194
   646
lemma summable_comparison_test: "\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n \<Longrightarrow> summable g \<Longrightarrow> summable f"
hoelzl@56193
   647
  apply (simp add: summable_Cauchy, safe)
hoelzl@56193
   648
  apply (drule_tac x="e" in spec, safe)
hoelzl@56193
   649
  apply (rule_tac x = "N + Na" in exI, safe)
hoelzl@56193
   650
  apply (rotate_tac 2)
hoelzl@56193
   651
  apply (drule_tac x = m in spec)
hoelzl@56193
   652
  apply (auto, rotate_tac 2, drule_tac x = n in spec)
hoelzl@56193
   653
  apply (rule_tac y = "\<Sum>k=m..<n. norm (f k)" in order_le_less_trans)
hoelzl@56193
   654
  apply (rule norm_setsum)
hoelzl@56193
   655
  apply (rule_tac y = "setsum g {m..<n}" in order_le_less_trans)
hoelzl@56193
   656
  apply (auto intro: setsum_mono simp add: abs_less_iff)
hoelzl@56193
   657
  done
hoelzl@56193
   658
eberlm@61531
   659
lemma summable_comparison_test_ev:
eberlm@61531
   660
  shows "eventually (\<lambda>n. norm (f n) \<le> g n) sequentially \<Longrightarrow> summable g \<Longrightarrow> summable f"
eberlm@61531
   661
  by (rule summable_comparison_test) (auto simp: eventually_at_top_linorder)
eberlm@61531
   662
lp15@56217
   663
(*A better argument order*)
lp15@56217
   664
lemma summable_comparison_test': "summable g \<Longrightarrow> (\<And>n. n \<ge> N \<Longrightarrow> norm(f n) \<le> g n) \<Longrightarrow> summable f"
hoelzl@56369
   665
  by (rule summable_comparison_test) auto
lp15@56217
   666
wenzelm@60758
   667
subsection \<open>The Ratio Test\<close>
paulson@15085
   668
lp15@61609
   669
lemma summable_ratio_test:
hoelzl@56193
   670
  assumes "c < 1" "\<And>n. n \<ge> N \<Longrightarrow> norm (f (Suc n)) \<le> c * norm (f n)"
hoelzl@56193
   671
  shows "summable f"
hoelzl@56193
   672
proof cases
hoelzl@56193
   673
  assume "0 < c"
hoelzl@56193
   674
  show "summable f"
hoelzl@56193
   675
  proof (rule summable_comparison_test)
hoelzl@56193
   676
    show "\<exists>N'. \<forall>n\<ge>N'. norm (f n) \<le> (norm (f N) / (c ^ N)) * c ^ n"
hoelzl@56193
   677
    proof (intro exI allI impI)
hoelzl@56193
   678
      fix n assume "N \<le> n" then show "norm (f n) \<le> (norm (f N) / (c ^ N)) * c ^ n"
hoelzl@56193
   679
      proof (induct rule: inc_induct)
hoelzl@56193
   680
        case (step m)
hoelzl@56193
   681
        moreover have "norm (f (Suc m)) / c ^ Suc m * c ^ n \<le> norm (f m) / c ^ m * c ^ n"
wenzelm@60758
   682
          using \<open>0 < c\<close> \<open>c < 1\<close> assms(2)[OF \<open>N \<le> m\<close>] by (simp add: field_simps)
hoelzl@56193
   683
        ultimately show ?case by simp
wenzelm@60758
   684
      qed (insert \<open>0 < c\<close>, simp)
hoelzl@56193
   685
    qed
hoelzl@56193
   686
    show "summable (\<lambda>n. norm (f N) / c ^ N * c ^ n)"
wenzelm@60758
   687
      using \<open>0 < c\<close> \<open>c < 1\<close> by (intro summable_mult summable_geometric) simp
hoelzl@56193
   688
  qed
hoelzl@56193
   689
next
hoelzl@56193
   690
  assume c: "\<not> 0 < c"
hoelzl@56193
   691
  { fix n assume "n \<ge> N"
hoelzl@56193
   692
    then have "norm (f (Suc n)) \<le> c * norm (f n)"
hoelzl@56193
   693
      by fact
hoelzl@56193
   694
    also have "\<dots> \<le> 0"
hoelzl@56193
   695
      using c by (simp add: not_less mult_nonpos_nonneg)
hoelzl@56193
   696
    finally have "f (Suc n) = 0"
hoelzl@56193
   697
      by auto }
hoelzl@56193
   698
  then show "summable f"
hoelzl@56194
   699
    by (intro sums_summable[OF sums_finite, of "{.. Suc N}"]) (auto simp: not_le Suc_less_eq2)
lp15@56178
   700
qed
lp15@56178
   701
hoelzl@56193
   702
end
paulson@14416
   703
wenzelm@60758
   704
text\<open>Relations among convergence and absolute convergence for power series.\<close>
hoelzl@56369
   705
paulson@62087
   706
lemma Abel_lemma:
hoelzl@56369
   707
  fixes a :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@56369
   708
  assumes r: "0 \<le> r" and r0: "r < r0" and M: "\<And>n. norm (a n) * r0^n \<le> M"
hoelzl@56369
   709
    shows "summable (\<lambda>n. norm (a n) * r^n)"
hoelzl@56369
   710
proof (rule summable_comparison_test')
hoelzl@56369
   711
  show "summable (\<lambda>n. M * (r / r0) ^ n)"
lp15@61609
   712
    using assms
hoelzl@56369
   713
    by (auto simp add: summable_mult summable_geometric)
hoelzl@56369
   714
next
hoelzl@56369
   715
  fix n
hoelzl@56369
   716
  show "norm (norm (a n) * r ^ n) \<le> M * (r / r0) ^ n"
hoelzl@56369
   717
    using r r0 M [of n]
haftmann@60867
   718
    apply (auto simp add: abs_mult field_simps)
hoelzl@56369
   719
    apply (cases "r=0", simp)
hoelzl@56369
   720
    apply (cases n, auto)
hoelzl@56369
   721
    done
hoelzl@56369
   722
qed
hoelzl@56369
   723
hoelzl@56369
   724
wenzelm@60758
   725
text\<open>Summability of geometric series for real algebras\<close>
huffman@23084
   726
huffman@23084
   727
lemma complete_algebra_summable_geometric:
haftmann@31017
   728
  fixes x :: "'a::{real_normed_algebra_1,banach}"
huffman@23084
   729
  shows "norm x < 1 \<Longrightarrow> summable (\<lambda>n. x ^ n)"
huffman@23084
   730
proof (rule summable_comparison_test)
huffman@23084
   731
  show "\<exists>N. \<forall>n\<ge>N. norm (x ^ n) \<le> norm x ^ n"
huffman@23084
   732
    by (simp add: norm_power_ineq)
huffman@23084
   733
  show "norm x < 1 \<Longrightarrow> summable (\<lambda>n. norm x ^ n)"
huffman@23084
   734
    by (simp add: summable_geometric)
huffman@23084
   735
qed
huffman@23084
   736
wenzelm@60758
   737
subsection \<open>Cauchy Product Formula\<close>
huffman@23111
   738
wenzelm@60758
   739
text \<open>
wenzelm@54703
   740
  Proof based on Analysis WebNotes: Chapter 07, Class 41
wenzelm@54703
   741
  @{url "http://www.math.unl.edu/~webnotes/classes/class41/prp77.htm"}
wenzelm@60758
   742
\<close>
huffman@23111
   743
huffman@23111
   744
lemma Cauchy_product_sums:
huffman@23111
   745
  fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}"
huffman@23111
   746
  assumes a: "summable (\<lambda>k. norm (a k))"
huffman@23111
   747
  assumes b: "summable (\<lambda>k. norm (b k))"
hoelzl@56213
   748
  shows "(\<lambda>k. \<Sum>i\<le>k. a i * b (k - i)) sums ((\<Sum>k. a k) * (\<Sum>k. b k))"
huffman@23111
   749
proof -
hoelzl@56193
   750
  let ?S1 = "\<lambda>n::nat. {..<n} \<times> {..<n}"
huffman@23111
   751
  let ?S2 = "\<lambda>n::nat. {(i,j). i + j < n}"
huffman@23111
   752
  have S1_mono: "\<And>m n. m \<le> n \<Longrightarrow> ?S1 m \<subseteq> ?S1 n" by auto
huffman@23111
   753
  have S2_le_S1: "\<And>n. ?S2 n \<subseteq> ?S1 n" by auto
huffman@23111
   754
  have S1_le_S2: "\<And>n. ?S1 (n div 2) \<subseteq> ?S2 n" by auto
huffman@23111
   755
  have finite_S1: "\<And>n. finite (?S1 n)" by simp
huffman@23111
   756
  with S2_le_S1 have finite_S2: "\<And>n. finite (?S2 n)" by (rule finite_subset)
huffman@23111
   757
huffman@23111
   758
  let ?g = "\<lambda>(i,j). a i * b j"
huffman@23111
   759
  let ?f = "\<lambda>(i,j). norm (a i) * norm (b j)"
nipkow@56536
   760
  have f_nonneg: "\<And>x. 0 \<le> ?f x" by (auto)
huffman@23111
   761
  hence norm_setsum_f: "\<And>A. norm (setsum ?f A) = setsum ?f A"
huffman@23111
   762
    unfolding real_norm_def
huffman@23111
   763
    by (simp only: abs_of_nonneg setsum_nonneg [rule_format])
huffman@23111
   764
wenzelm@61969
   765
  have "(\<lambda>n. (\<Sum>k<n. a k) * (\<Sum>k<n. b k)) \<longlonglongrightarrow> (\<Sum>k. a k) * (\<Sum>k. b k)"
hoelzl@56193
   766
    by (intro tendsto_mult summable_LIMSEQ summable_norm_cancel [OF a] summable_norm_cancel [OF b])
wenzelm@61969
   767
  hence 1: "(\<lambda>n. setsum ?g (?S1 n)) \<longlonglongrightarrow> (\<Sum>k. a k) * (\<Sum>k. b k)"
haftmann@57418
   768
    by (simp only: setsum_product setsum.Sigma [rule_format] finite_lessThan)
huffman@23111
   769
wenzelm@61969
   770
  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
   771
    using a b by (intro tendsto_mult summable_LIMSEQ)
wenzelm@61969
   772
  hence "(\<lambda>n. setsum ?f (?S1 n)) \<longlonglongrightarrow> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
haftmann@57418
   773
    by (simp only: setsum_product setsum.Sigma [rule_format] finite_lessThan)
huffman@23111
   774
  hence "convergent (\<lambda>n. setsum ?f (?S1 n))"
huffman@23111
   775
    by (rule convergentI)
huffman@23111
   776
  hence Cauchy: "Cauchy (\<lambda>n. setsum ?f (?S1 n))"
huffman@23111
   777
    by (rule convergent_Cauchy)
huffman@36657
   778
  have "Zfun (\<lambda>n. setsum ?f (?S1 n - ?S2 n)) sequentially"
huffman@36657
   779
  proof (rule ZfunI, simp only: eventually_sequentially norm_setsum_f)
huffman@23111
   780
    fix r :: real
huffman@23111
   781
    assume r: "0 < r"
huffman@23111
   782
    from CauchyD [OF Cauchy r] obtain N
huffman@23111
   783
    where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (setsum ?f (?S1 m) - setsum ?f (?S1 n)) < r" ..
huffman@23111
   784
    hence "\<And>m n. \<lbrakk>N \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> norm (setsum ?f (?S1 m - ?S1 n)) < r"
huffman@23111
   785
      by (simp only: setsum_diff finite_S1 S1_mono)
huffman@23111
   786
    hence N: "\<And>m n. \<lbrakk>N \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> setsum ?f (?S1 m - ?S1 n) < r"
huffman@23111
   787
      by (simp only: norm_setsum_f)
huffman@23111
   788
    show "\<exists>N. \<forall>n\<ge>N. setsum ?f (?S1 n - ?S2 n) < r"
huffman@23111
   789
    proof (intro exI allI impI)
huffman@23111
   790
      fix n assume "2 * N \<le> n"
huffman@23111
   791
      hence n: "N \<le> n div 2" by simp
huffman@23111
   792
      have "setsum ?f (?S1 n - ?S2 n) \<le> setsum ?f (?S1 n - ?S1 (n div 2))"
huffman@23111
   793
        by (intro setsum_mono2 finite_Diff finite_S1 f_nonneg
huffman@23111
   794
                  Diff_mono subset_refl S1_le_S2)
huffman@23111
   795
      also have "\<dots> < r"
huffman@23111
   796
        using n div_le_dividend by (rule N)
huffman@23111
   797
      finally show "setsum ?f (?S1 n - ?S2 n) < r" .
huffman@23111
   798
    qed
huffman@23111
   799
  qed
huffman@36657
   800
  hence "Zfun (\<lambda>n. setsum ?g (?S1 n - ?S2 n)) sequentially"
huffman@36657
   801
    apply (rule Zfun_le [rule_format])
huffman@23111
   802
    apply (simp only: norm_setsum_f)
huffman@23111
   803
    apply (rule order_trans [OF norm_setsum setsum_mono])
huffman@23111
   804
    apply (auto simp add: norm_mult_ineq)
huffman@23111
   805
    done
wenzelm@61969
   806
  hence 2: "(\<lambda>n. setsum ?g (?S1 n) - setsum ?g (?S2 n)) \<longlonglongrightarrow> 0"
huffman@36660
   807
    unfolding tendsto_Zfun_iff diff_0_right
huffman@36657
   808
    by (simp only: setsum_diff finite_S1 S2_le_S1)
huffman@23111
   809
wenzelm@61969
   810
  with 1 have "(\<lambda>n. setsum ?g (?S2 n)) \<longlonglongrightarrow> (\<Sum>k. a k) * (\<Sum>k. b k)"
lp15@60141
   811
    by (rule Lim_transform2)
huffman@23111
   812
  thus ?thesis by (simp only: sums_def setsum_triangle_reindex)
huffman@23111
   813
qed
huffman@23111
   814
huffman@23111
   815
lemma Cauchy_product:
huffman@23111
   816
  fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}"
huffman@23111
   817
  assumes a: "summable (\<lambda>k. norm (a k))"
huffman@23111
   818
  assumes b: "summable (\<lambda>k. norm (b k))"
hoelzl@56213
   819
  shows "(\<Sum>k. a k) * (\<Sum>k. b k) = (\<Sum>k. \<Sum>i\<le>k. a i * b (k - i))"
hoelzl@56213
   820
  using a b
hoelzl@56213
   821
  by (rule Cauchy_product_sums [THEN sums_unique])
hoelzl@56213
   822
eberlm@62049
   823
lemma summable_Cauchy_product:
paulson@62087
   824
  assumes "summable (\<lambda>k. norm (a k :: 'a :: {real_normed_algebra,banach}))"
eberlm@62049
   825
          "summable (\<lambda>k. norm (b k))"
eberlm@62049
   826
  shows   "summable (\<lambda>k. \<Sum>i\<le>k. a i * b (k - i))"
paulson@62087
   827
  using Cauchy_product_sums[OF assms] by (simp add: sums_iff)
eberlm@62049
   828
wenzelm@60758
   829
subsection \<open>Series on @{typ real}s\<close>
hoelzl@56213
   830
hoelzl@56213
   831
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
   832
  by (rule summable_comparison_test) auto
hoelzl@56213
   833
hoelzl@56213
   834
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
   835
  by (rule summable_comparison_test) auto
hoelzl@56213
   836
hoelzl@56213
   837
lemma summable_rabs_cancel: "summable (\<lambda>n. \<bar>f n :: real\<bar>) \<Longrightarrow> summable f"
hoelzl@56213
   838
  by (rule summable_norm_cancel) simp
hoelzl@56213
   839
hoelzl@56213
   840
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
   841
  by (fold real_norm_def) (rule summable_norm)
huffman@23111
   842
eberlm@61531
   843
lemma summable_zero_power [simp]: "summable (\<lambda>n. 0 ^ n :: 'a :: {comm_ring_1,topological_space})"
eberlm@61531
   844
proof -
eberlm@61531
   845
  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
   846
  moreover have "summable \<dots>" by simp
eberlm@61531
   847
  ultimately show ?thesis by simp
eberlm@61531
   848
qed
eberlm@61531
   849
eberlm@61531
   850
lemma summable_zero_power' [simp]: "summable (\<lambda>n. f n * 0 ^ n :: 'a :: {ring_1,topological_space})"
eberlm@61531
   851
proof -
lp15@61609
   852
  have "(\<lambda>n. f n * 0 ^ n :: 'a) = (\<lambda>n. if n = 0 then f 0 * 0^0 else 0)"
eberlm@61531
   853
    by (intro ext) (simp add: zero_power)
eberlm@61531
   854
  moreover have "summable \<dots>" by simp
eberlm@61531
   855
  ultimately show ?thesis by simp
eberlm@61531
   856
qed
eberlm@61531
   857
hoelzl@59000
   858
lemma summable_power_series:
hoelzl@59000
   859
  fixes z :: real
hoelzl@59000
   860
  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
   861
  shows "summable (\<lambda>i. f i * z^i)"
hoelzl@59000
   862
proof (rule summable_comparison_test[OF _ summable_geometric])
hoelzl@59000
   863
  show "norm z < 1" using z by (auto simp: less_imp_le)
hoelzl@59000
   864
  show "\<And>n. \<exists>N. \<forall>na\<ge>N. norm (f na * z ^ na) \<le> z ^ na"
hoelzl@59000
   865
    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
   866
qed
hoelzl@59000
   867
eberlm@61531
   868
lemma summable_0_powser:
eberlm@61531
   869
  "summable (\<lambda>n. f n * 0 ^ n :: 'a :: real_normed_div_algebra)"
eberlm@61531
   870
proof -
eberlm@61531
   871
  have A: "(\<lambda>n. f n * 0 ^ n) = (\<lambda>n. if n = 0 then f n else 0)"
eberlm@61531
   872
    by (intro ext) auto
eberlm@61531
   873
  thus ?thesis by (subst A) simp_all
eberlm@61531
   874
qed
eberlm@61531
   875
eberlm@61531
   876
lemma summable_powser_split_head:
eberlm@61531
   877
  "summable (\<lambda>n. f (Suc n) * z ^ n :: 'a :: real_normed_div_algebra) = summable (\<lambda>n. f n * z ^ n)"
eberlm@61531
   878
proof -
eberlm@61531
   879
  have "summable (\<lambda>n. f (Suc n) * z ^ n) \<longleftrightarrow> summable (\<lambda>n. f (Suc n) * z ^ Suc n)"
eberlm@61531
   880
  proof
eberlm@61531
   881
    assume "summable (\<lambda>n. f (Suc n) * z ^ n)"
lp15@61609
   882
    from summable_mult2[OF this, of z] show "summable (\<lambda>n. f (Suc n) * z ^ Suc n)"
eberlm@61531
   883
      by (simp add: power_commutes algebra_simps)
eberlm@61531
   884
  next
eberlm@61531
   885
    assume "summable (\<lambda>n. f (Suc n) * z ^ Suc n)"
eberlm@61531
   886
    from summable_mult2[OF this, of "inverse z"] show "summable (\<lambda>n. f (Suc n) * z ^ n)"
eberlm@61531
   887
      by (cases "z \<noteq> 0", subst (asm) power_Suc2) (simp_all add: algebra_simps)
eberlm@61531
   888
  qed
eberlm@61531
   889
  also have "\<dots> \<longleftrightarrow> summable (\<lambda>n. f n * z ^ n)" by (rule summable_Suc_iff)
eberlm@61531
   890
  finally show ?thesis .
eberlm@61531
   891
qed
eberlm@61531
   892
eberlm@61531
   893
lemma powser_split_head:
eberlm@61531
   894
  assumes "summable (\<lambda>n. f n * z ^ n :: 'a :: {real_normed_div_algebra,banach})"
eberlm@61531
   895
  shows   "suminf (\<lambda>n. f n * z ^ n) = f 0 + suminf (\<lambda>n. f (Suc n) * z ^ n) * z"
eberlm@61531
   896
          "suminf (\<lambda>n. f (Suc n) * z ^ n) * z = suminf (\<lambda>n. f n * z ^ n) - f 0"
eberlm@61531
   897
          "summable (\<lambda>n. f (Suc n) * z ^ n)"
eberlm@61531
   898
proof -
eberlm@61531
   899
  from assms show "summable (\<lambda>n. f (Suc n) * z ^ n)" by (subst summable_powser_split_head)
eberlm@61531
   900
lp15@61609
   901
  from suminf_mult2[OF this, of z]
eberlm@61531
   902
    have "(\<Sum>n. f (Suc n) * z ^ n) * z = (\<Sum>n. f (Suc n) * z ^ Suc n)"
eberlm@61531
   903
    by (simp add: power_commutes algebra_simps)
eberlm@61531
   904
  also from assms have "\<dots> = suminf (\<lambda>n. f n * z ^ n) - f 0"
eberlm@61531
   905
    by (subst suminf_split_head) simp_all
eberlm@61531
   906
  finally show "suminf (\<lambda>n. f n * z ^ n) = f 0 + suminf (\<lambda>n. f (Suc n) * z ^ n) * z" by simp
eberlm@61531
   907
  thus "suminf (\<lambda>n. f (Suc n) * z ^ n) * z = suminf (\<lambda>n. f n * z ^ n) - f 0" by simp
eberlm@61531
   908
qed
eberlm@61531
   909
eberlm@61531
   910
lemma summable_partial_sum_bound:
eberlm@61531
   911
  fixes f :: "nat \<Rightarrow> 'a :: banach"
eberlm@61531
   912
  assumes summable: "summable f" and e: "e > (0::real)"
eberlm@61531
   913
  obtains N where "\<And>m n. m \<ge> N \<Longrightarrow> norm (\<Sum>k=m..n. f k) < e"
eberlm@61531
   914
proof -
lp15@61609
   915
  from summable have "Cauchy (\<lambda>n. \<Sum>k<n. f k)"
eberlm@61531
   916
    by (simp add: Cauchy_convergent_iff summable_iff_convergent)
lp15@61609
   917
  from CauchyD[OF this e] obtain N
eberlm@61531
   918
    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
   919
  {
eberlm@61531
   920
    fix m n :: nat assume m: "m \<ge> N"
eberlm@61531
   921
    have "norm (\<Sum>k=m..n. f k) < e"
eberlm@61531
   922
    proof (cases "n \<ge> m")
eberlm@61531
   923
      assume n: "n \<ge> m"
eberlm@61531
   924
      with m have "norm ((\<Sum>k<Suc n. f k) - (\<Sum>k<m. f k)) < e" by (intro N) simp_all
eberlm@61531
   925
      also from n have "(\<Sum>k<Suc n. f k) - (\<Sum>k<m. f k) = (\<Sum>k=m..n. f k)"
eberlm@61531
   926
        by (subst setsum_diff [symmetric]) (simp_all add: setsum_last_plus)
eberlm@61531
   927
      finally show ?thesis .
eberlm@61531
   928
    qed (insert e, simp_all)
eberlm@61531
   929
  }
eberlm@61531
   930
  thus ?thesis by (rule that)
eberlm@61531
   931
qed
eberlm@61531
   932
lp15@61609
   933
lemma powser_sums_if:
eberlm@61531
   934
  "(\<lambda>n. (if n = m then (1 :: 'a :: {ring_1,topological_space}) else 0) * z^n) sums z^m"
eberlm@61531
   935
proof -
lp15@61609
   936
  have "(\<lambda>n. (if n = m then 1 else 0) * z^n) = (\<lambda>n. if n = m then z^n else 0)"
eberlm@61531
   937
    by (intro ext) auto
eberlm@61531
   938
  thus ?thesis by (simp add: sums_single)
eberlm@61531
   939
qed
eberlm@61531
   940
Andreas@59025
   941
lemma
Andreas@59025
   942
   fixes f :: "nat \<Rightarrow> real"
Andreas@59025
   943
   assumes "summable f"
Andreas@59025
   944
   and "inj g"
hoelzl@62368
   945
   and pos: "\<And>x. 0 \<le> f x"
Andreas@59025
   946
   shows summable_reindex: "summable (f o g)"
Andreas@59025
   947
   and suminf_reindex_mono: "suminf (f o g) \<le> suminf f"
Andreas@59025
   948
   and suminf_reindex: "(\<And>x. x \<notin> range g \<Longrightarrow> f x = 0) \<Longrightarrow> suminf (f \<circ> g) = suminf f"
Andreas@59025
   949
proof -
Andreas@59025
   950
  from \<open>inj g\<close> have [simp]: "\<And>A. inj_on g A" by(rule subset_inj_on) simp
Andreas@59025
   951
Andreas@59025
   952
  have smaller: "\<forall>n. (\<Sum>i<n. (f \<circ> g) i) \<le> suminf f"
Andreas@59025
   953
  proof
Andreas@59025
   954
    fix n
lp15@61609
   955
    have "\<forall> n' \<in> (g ` {..<n}). n' < Suc (Max (g ` {..<n}))"
Andreas@59025
   956
      by(metis Max_ge finite_imageI finite_lessThan not_le not_less_eq)
Andreas@59025
   957
    then obtain m where n: "\<And>n'. n' < n \<Longrightarrow> g n' < m" by blast
Andreas@59025
   958
Andreas@59025
   959
    have "(\<Sum>i<n. f (g i)) = setsum f (g ` {..<n})"
Andreas@59025
   960
      by (simp add: setsum.reindex)
Andreas@59025
   961
    also have "\<dots> \<le> (\<Sum>i<m. f i)"
Andreas@59025
   962
      by (rule setsum_mono3) (auto simp add: pos n[rule_format])
Andreas@59025
   963
    also have "\<dots> \<le> suminf f"
lp15@61609
   964
      using \<open>summable f\<close>
Andreas@59025
   965
      by (rule setsum_le_suminf) (simp add: pos)
Andreas@59025
   966
    finally show "(\<Sum>i<n. (f \<circ>  g) i) \<le> suminf f" by simp
Andreas@59025
   967
  qed
Andreas@59025
   968
Andreas@59025
   969
  have "incseq (\<lambda>n. \<Sum>i<n. (f \<circ> g) i)"
Andreas@59025
   970
    by (rule incseq_SucI) (auto simp add: pos)
wenzelm@61969
   971
  then obtain  L where L: "(\<lambda> n. \<Sum>i<n. (f \<circ> g) i) \<longlonglongrightarrow> L"
Andreas@59025
   972
    using smaller by(rule incseq_convergent)
Andreas@59025
   973
  hence "(f \<circ> g) sums L" by (simp add: sums_def)
Andreas@59025
   974
  thus "summable (f o g)" by (auto simp add: sums_iff)
Andreas@59025
   975
wenzelm@61969
   976
  hence "(\<lambda>n. \<Sum>i<n. (f \<circ> g) i) \<longlonglongrightarrow> suminf (f \<circ> g)"
Andreas@59025
   977
    by(rule summable_LIMSEQ)
Andreas@59025
   978
  thus le: "suminf (f \<circ> g) \<le> suminf f"
Andreas@59025
   979
    by(rule LIMSEQ_le_const2)(blast intro: smaller[rule_format])
Andreas@59025
   980
Andreas@59025
   981
  assume f: "\<And>x. x \<notin> range g \<Longrightarrow> f x = 0"
Andreas@59025
   982
Andreas@59025
   983
  from \<open>summable f\<close> have "suminf f \<le> suminf (f \<circ> g)"
Andreas@59025
   984
  proof(rule suminf_le_const)
Andreas@59025
   985
    fix n
Andreas@59025
   986
    have "\<forall> n' \<in> (g -` {..<n}). n' < Suc (Max (g -` {..<n}))"
Andreas@59025
   987
      by(auto intro: Max_ge simp add: finite_vimageI less_Suc_eq_le)
Andreas@59025
   988
    then obtain m where n: "\<And>n'. g n' < n \<Longrightarrow> n' < m" by blast
Andreas@59025
   989
Andreas@59025
   990
    have "(\<Sum>i<n. f i) = (\<Sum>i\<in>{..<n} \<inter> range g. f i)"
Andreas@59025
   991
      using f by(auto intro: setsum.mono_neutral_cong_right)
Andreas@59025
   992
    also have "\<dots> = (\<Sum>i\<in>g -` {..<n}. (f \<circ> g) i)"
Andreas@59025
   993
      by(rule setsum.reindex_cong[where l=g])(auto)
Andreas@59025
   994
    also have "\<dots> \<le> (\<Sum>i<m. (f \<circ> g) i)"
Andreas@59025
   995
      by(rule setsum_mono3)(auto simp add: pos n)
Andreas@59025
   996
    also have "\<dots> \<le> suminf (f \<circ> g)"
Andreas@59025
   997
      using \<open>summable (f o g)\<close>
Andreas@59025
   998
      by(rule setsum_le_suminf)(simp add: pos)
Andreas@59025
   999
    finally show "setsum f {..<n} \<le> suminf (f \<circ> g)" .
Andreas@59025
  1000
  qed
Andreas@59025
  1001
  with le show "suminf (f \<circ> g) = suminf f" by(rule antisym)
Andreas@59025
  1002
qed
Andreas@59025
  1003
eberlm@61531
  1004
lemma sums_mono_reindex:
eberlm@61531
  1005
  assumes subseq: "subseq g" and zero: "\<And>n. n \<notin> range g \<Longrightarrow> f n = 0"
eberlm@61531
  1006
  shows   "(\<lambda>n. f (g n)) sums c \<longleftrightarrow> f sums c"
eberlm@61531
  1007
unfolding sums_def
eberlm@61531
  1008
proof
wenzelm@61969
  1009
  assume lim: "(\<lambda>n. \<Sum>k<n. f k) \<longlonglongrightarrow> c"
eberlm@61531
  1010
  have "(\<lambda>n. \<Sum>k<n. f (g k)) = (\<lambda>n. \<Sum>k<g n. f k)"
eberlm@61531
  1011
  proof
eberlm@61531
  1012
    fix n :: nat
eberlm@61531
  1013
    from subseq have "(\<Sum>k<n. f (g k)) = (\<Sum>k\<in>g`{..<n}. f k)"
eberlm@61531
  1014
      by (subst setsum.reindex) (auto intro: subseq_imp_inj_on)
eberlm@61531
  1015
    also from subseq have "\<dots> = (\<Sum>k<g n. f k)"
eberlm@61531
  1016
      by (intro setsum.mono_neutral_left ballI zero)
eberlm@61531
  1017
         (auto dest: subseq_strict_mono simp: strict_mono_less strict_mono_less_eq)
eberlm@61531
  1018
    finally show "(\<Sum>k<n. f (g k)) = (\<Sum>k<g n. f k)" .
eberlm@61531
  1019
  qed
wenzelm@61969
  1020
  also from LIMSEQ_subseq_LIMSEQ[OF lim subseq] have "\<dots> \<longlonglongrightarrow> c" unfolding o_def .
wenzelm@61969
  1021
  finally show "(\<lambda>n. \<Sum>k<n. f (g k)) \<longlonglongrightarrow> c" .
eberlm@61531
  1022
next
wenzelm@61969
  1023
  assume lim: "(\<lambda>n. \<Sum>k<n. f (g k)) \<longlonglongrightarrow> c"
eberlm@61531
  1024
  def g_inv \<equiv> "\<lambda>n. LEAST m. g m \<ge> n"
eberlm@61531
  1025
  from filterlim_subseq[OF subseq] have g_inv_ex: "\<exists>m. g m \<ge> n" for n
eberlm@61531
  1026
    by (auto simp: filterlim_at_top eventually_at_top_linorder)
eberlm@61531
  1027
  hence g_inv: "g (g_inv n) \<ge> n" for n unfolding g_inv_def by (rule LeastI_ex)
lp15@61609
  1028
  have g_inv_least: "m \<ge> g_inv n" if "g m \<ge> n" for m n using that
eberlm@61531
  1029
    unfolding g_inv_def by (rule Least_le)
lp15@61609
  1030
  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
  1031
  have "(\<lambda>n. \<Sum>k<n. f k) = (\<lambda>n. \<Sum>k<g_inv n. f (g k))"
eberlm@61531
  1032
  proof
eberlm@61531
  1033
    fix n :: nat
eberlm@61531
  1034
    {
eberlm@61531
  1035
      fix k assume k: "k \<in> {..<n} - g`{..<g_inv n}"
eberlm@61531
  1036
      have "k \<notin> range g"
eberlm@61531
  1037
      proof (rule notI, elim imageE)
eberlm@61531
  1038
        fix l assume l: "k = g l"
eberlm@61531
  1039
        have "g l < g (g_inv n)" by (rule less_le_trans[OF _ g_inv]) (insert k l, simp_all)
eberlm@61531
  1040
        with subseq have "l < g_inv n" by (simp add: subseq_strict_mono strict_mono_less)
eberlm@61531
  1041
        with k l show False by simp
eberlm@61531
  1042
      qed
eberlm@61531
  1043
      hence "f k = 0" by (rule zero)
eberlm@61531
  1044
    }
eberlm@61531
  1045
    with g_inv_least' g_inv have "(\<Sum>k<n. f k) = (\<Sum>k\<in>g`{..<g_inv n}. f k)"
eberlm@61531
  1046
      by (intro setsum.mono_neutral_right) auto
lp15@61609
  1047
    also from subseq have "\<dots> = (\<Sum>k<g_inv n. f (g k))" using subseq_imp_inj_on
eberlm@61531
  1048
      by (subst setsum.reindex) simp_all
eberlm@61531
  1049
    finally show "(\<Sum>k<n. f k) = (\<Sum>k<g_inv n. f (g k))" .
eberlm@61531
  1050
  qed
eberlm@61531
  1051
  also {
eberlm@61531
  1052
    fix K n :: nat assume "g K \<le> n"
eberlm@61531
  1053
    also have "n \<le> g (g_inv n)" by (rule g_inv)
eberlm@61531
  1054
    finally have "K \<le> g_inv n" using subseq by (simp add: strict_mono_less_eq subseq_strict_mono)
eberlm@61531
  1055
  }
lp15@61609
  1056
  hence "filterlim g_inv at_top sequentially"
eberlm@61531
  1057
    by (auto simp: filterlim_at_top eventually_at_top_linorder)
wenzelm@61969
  1058
  from lim and this have "(\<lambda>n. \<Sum>k<g_inv n. f (g k)) \<longlonglongrightarrow> c" by (rule filterlim_compose)
wenzelm@61969
  1059
  finally show "(\<lambda>n. \<Sum>k<n. f k) \<longlonglongrightarrow> c" .
eberlm@61531
  1060
qed
eberlm@61531
  1061
eberlm@61531
  1062
lemma summable_mono_reindex:
eberlm@61531
  1063
  assumes subseq: "subseq g" and zero: "\<And>n. n \<notin> range g \<Longrightarrow> f n = 0"
eberlm@61531
  1064
  shows   "summable (\<lambda>n. f (g n)) \<longleftrightarrow> summable f"
eberlm@61531
  1065
  using sums_mono_reindex[of g f, OF assms] by (simp add: summable_def)
eberlm@61531
  1066
lp15@61609
  1067
lemma suminf_mono_reindex:
eberlm@61531
  1068
  assumes "subseq g" "\<And>n. n \<notin> range g \<Longrightarrow> f n = (0 :: 'a :: {t2_space,comm_monoid_add})"
eberlm@61531
  1069
  shows   "suminf (\<lambda>n. f (g n)) = suminf f"
eberlm@61531
  1070
proof (cases "summable f")
eberlm@61531
  1071
  case False
eberlm@61531
  1072
  hence "\<not>(\<exists>c. f sums c)" unfolding summable_def by blast
eberlm@61531
  1073
  hence "suminf f = The (\<lambda>_. False)" by (simp add: suminf_def)
eberlm@61531
  1074
  moreover from False have "\<not>summable (\<lambda>n. f (g n))"
eberlm@61531
  1075
    using summable_mono_reindex[of g f, OF assms] by simp
eberlm@61531
  1076
  hence "\<not>(\<exists>c. (\<lambda>n. f (g n)) sums c)" unfolding summable_def by blast
eberlm@61531
  1077
  hence "suminf (\<lambda>n. f (g n)) = The (\<lambda>_. False)" by (simp add: suminf_def)
eberlm@61531
  1078
  ultimately show ?thesis by simp
lp15@61609
  1079
qed (insert sums_mono_reindex[of g f, OF assms] summable_mono_reindex[of g f, OF assms],
eberlm@61531
  1080
     simp_all add: sums_iff)
eberlm@61531
  1081
paulson@14416
  1082
end