src/HOL/Series.thy
author Andreas Lochbihler
Fri, 21 Nov 2014 13:18:56 +0100
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add lemma following a proof suggestion by Joachim Breitner
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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 {* Infinite Series *}
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theory Series
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imports Limits
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begin
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subsection {* Definition of infinite summability *}
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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) ----> 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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subsection {* Infinite summability on topological monoids *}
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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_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 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 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) ----> 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 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 {* Infinite summability on ordered, topological monoids *}
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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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   152
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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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   155
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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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   158
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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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   163
    setsum_mono2[of "{..<i}" "{..<n}" f]
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   164
  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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   174
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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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   179
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) ----> 0"
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   182
    using summable_LIMSEQ[of f] by simp
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   183
  then have "\<And>i. (\<Sum>n\<in>{i}. f n) \<le> 0"
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   184
  proof (rule LIMSEQ_le_const)
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   185
    fix i show "\<exists>N. \<forall>n\<ge>N. (\<Sum>n\<in>{i}. f n) \<le> setsum f {..<n}"
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   186
      using pos by (intro exI[of _ "Suc i"] allI impI setsum_mono2) auto
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   187
  qed
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   188
  with pos show "\<forall>n. f n = 0"
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   189
    by (auto intro!: antisym)
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   190
qed (metis suminf_zero fun_eq_iff)
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   191
56213
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   192
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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   193
  using setsum_le_suminf[of 0] suminf_eq_zero_iff by (simp add: less_le)
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end
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   197
lemma summableI_nonneg_bounded:
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   198
  fixes f:: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology, conditionally_complete_linorder}"
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   199
  assumes pos[simp]: "\<And>n. 0 \<le> f n" and le: "\<And>n. (\<Sum>i<n. f i) \<le> x"
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   200
  shows "summable f"
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   201
  unfolding summable_def sums_def[abs_def]
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   202
proof (intro exI order_tendstoI)
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   203
  have [simp, intro]: "bdd_above (range (\<lambda>n. \<Sum>i<n. f i))"
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   204
    using le by (auto simp: bdd_above_def)
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   205
  { fix a assume "a < (SUP n. \<Sum>i<n. f i)"
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   206
    then obtain n where "a < (\<Sum>i<n. f i)"
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   207
      by (auto simp add: less_cSUP_iff)
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   208
    then have "\<And>m. n \<le> m \<Longrightarrow> a < (\<Sum>i<m. f i)"
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   209
      by (rule less_le_trans) (auto intro!: setsum_mono2)
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   210
    then show "eventually (\<lambda>n. a < (\<Sum>i<n. f i)) sequentially"
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   211
      by (auto simp: eventually_sequentially) }
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   212
  { fix a assume "(SUP n. \<Sum>i<n. f i) < a"
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   213
    moreover have "\<And>n. (\<Sum>i<n. f i) \<le> (SUP n. \<Sum>i<n. f i)"
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   214
      by (auto intro: cSUP_upper)
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   215
    ultimately show "eventually (\<lambda>n. (\<Sum>i<n. f i) < a) sequentially"
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   216
      by (auto intro: le_less_trans simp: eventually_sequentially) }
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   217
qed
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   218
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subsection {* Infinite summability on real normed vector spaces *}
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lemma sums_Suc_iff:
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  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
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   223
  shows "(\<lambda>n. f (Suc n)) sums s \<longleftrightarrow> f sums (s + f 0)"
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   224
proof -
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   225
  have "f sums (s + f 0) \<longleftrightarrow> (\<lambda>i. \<Sum>j<Suc i. f j) ----> s + f 0"
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   226
    by (subst LIMSEQ_Suc_iff) (simp add: sums_def)
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   227
  also have "\<dots> \<longleftrightarrow> (\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) ----> s + f 0"
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   228
    by (simp add: ac_simps setsum.reindex image_iff lessThan_Suc_eq_insert_0)
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   229
  also have "\<dots> \<longleftrightarrow> (\<lambda>n. f (Suc n)) sums s"
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   230
  proof
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   231
    assume "(\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) ----> s + f 0"
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   232
    with tendsto_add[OF this tendsto_const, of "- f 0"]
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   233
    show "(\<lambda>i. f (Suc i)) sums s"
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   234
      by (simp add: sums_def)
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   235
  qed (auto intro: tendsto_add simp: sums_def)
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   236
  finally show ?thesis ..
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   237
qed
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   238
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   239
context
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  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
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   241
begin
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   242
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   243
lemma sums_add: "f sums a \<Longrightarrow> g sums b \<Longrightarrow> (\<lambda>n. f n + g n) sums (a + b)"
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   244
  unfolding sums_def by (simp add: setsum.distrib tendsto_add)
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   245
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   246
lemma summable_add: "summable f \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. f n + g n)"
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   247
  unfolding summable_def by (auto intro: sums_add)
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   248
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   249
lemma suminf_add: "summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f + suminf g = (\<Sum>n. f n + g n)"
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diff changeset
   250
  by (intro sums_unique sums_add summable_sums)
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   251
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   252
lemma sums_diff: "f sums a \<Longrightarrow> g sums b \<Longrightarrow> (\<lambda>n. f n - g n) sums (a - b)"
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diff changeset
   253
  unfolding sums_def by (simp add: setsum_subtractf tendsto_diff)
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   254
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   255
lemma summable_diff: "summable f \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. f n - g n)"
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diff changeset
   256
  unfolding summable_def by (auto intro: sums_diff)
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diff changeset
   257
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   258
lemma suminf_diff: "summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f - suminf g = (\<Sum>n. f n - g n)"
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diff changeset
   259
  by (intro sums_unique sums_diff summable_sums)
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diff changeset
   260
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diff changeset
   261
lemma sums_minus: "f sums a \<Longrightarrow> (\<lambda>n. - f n) sums (- a)"
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diff changeset
   262
  unfolding sums_def by (simp add: setsum_negf tendsto_minus)
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diff changeset
   263
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diff changeset
   264
lemma summable_minus: "summable f \<Longrightarrow> summable (\<lambda>n. - f n)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
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diff changeset
   265
  unfolding summable_def by (auto intro: sums_minus)
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   266
56193
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diff changeset
   267
lemma suminf_minus: "summable f \<Longrightarrow> (\<Sum>n. - f n) = - (\<Sum>n. f n)"
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  by (intro sums_unique [symmetric] sums_minus summable_sums)
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lemma sums_Suc: "(\<lambda> n. f (Suc n)) sums l \<Longrightarrow> f sums (l + f 0)"
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  by (simp add: sums_Suc_iff)
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lemma sums_iff_shift: "(\<lambda>i. f (i + n)) sums s \<longleftrightarrow> f sums (s + (\<Sum>i<n. f i))"
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proof (induct n arbitrary: s)
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  case (Suc n)
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  moreover have "(\<lambda>i. f (Suc i + n)) sums s \<longleftrightarrow> (\<lambda>i. f (i + n)) sums (s + f n)"
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    by (subst sums_Suc_iff) simp
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  ultimately show ?case
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    by (simp add: ac_simps)
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qed simp
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lemma summable_iff_shift: "summable (\<lambda>n. f (n + k)) \<longleftrightarrow> summable f"
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  by (metis diff_add_cancel summable_def sums_iff_shift[abs_def])
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lemma sums_split_initial_segment: "f sums s \<Longrightarrow> (\<lambda>i. f (i + n)) sums (s - (\<Sum>i<n. f i))"
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  by (simp add: sums_iff_shift)
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lemma summable_ignore_initial_segment: "summable f \<Longrightarrow> summable (\<lambda>n. f(n + k))"
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  by (simp add: summable_iff_shift)
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lemma suminf_minus_initial_segment: "summable f \<Longrightarrow> (\<Sum>n. f (n + k)) = (\<Sum>n. f n) - (\<Sum>i<k. f i)"
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  by (rule sums_unique[symmetric]) (auto simp: sums_iff_shift)
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lemma suminf_split_initial_segment: "summable f \<Longrightarrow> suminf f = (\<Sum>n. f(n + k)) + (\<Sum>i<k. f i)"
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  by (auto simp add: suminf_minus_initial_segment)
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lemma suminf_exist_split: 
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  fixes r :: real assumes "0 < r" and "summable f"
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  shows "\<exists>N. \<forall>n\<ge>N. norm (\<Sum>i. f (i + n)) < r"
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proof -
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  from LIMSEQ_D[OF summable_LIMSEQ[OF `summable f`] `0 < r`]
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  obtain N :: nat where "\<forall> n \<ge> N. norm (setsum f {..<n} - suminf f) < r" by auto
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  thus ?thesis
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    by (auto simp: norm_minus_commute suminf_minus_initial_segment[OF `summable f`])
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qed
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lemma summable_LIMSEQ_zero: "summable f \<Longrightarrow> f ----> 0"
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  apply (drule summable_iff_convergent [THEN iffD1])
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  apply (drule convergent_Cauchy)
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  apply (simp only: Cauchy_iff LIMSEQ_iff, safe)
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  apply (drule_tac x="r" in spec, safe)
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  apply (rule_tac x="M" in exI, safe)
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  apply (drule_tac x="Suc n" in spec, simp)
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  apply (drule_tac x="n" in spec, simp)
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  done
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end
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57025
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context
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  fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::real_normed_vector" and I :: "'i set"
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begin
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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)"
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  by (induct I rule: infinite_finite_induct) (auto intro!: sums_add)
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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)"
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  using sums_unique[OF sums_setsum, OF summable_sums] by simp
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lemma summable_setsum: "(\<And>i. i \<in> I \<Longrightarrow> summable (f i)) \<Longrightarrow> summable (\<lambda>n. \<Sum>i\<in>I. f i n)"
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  using sums_summable[OF sums_setsum[OF summable_sums]] .
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end
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lemma (in bounded_linear) sums: "(\<lambda>n. X n) sums a \<Longrightarrow> (\<lambda>n. f (X n)) sums (f a)"
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  unfolding sums_def by (drule tendsto, simp only: setsum)
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lemma (in bounded_linear) summable: "summable (\<lambda>n. X n) \<Longrightarrow> summable (\<lambda>n. f (X n))"
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  unfolding summable_def by (auto intro: sums)
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lemma (in bounded_linear) suminf: "summable (\<lambda>n. X n) \<Longrightarrow> f (\<Sum>n. X n) = (\<Sum>n. f (X n))"
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  by (intro sums_unique sums summable_sums)
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lemmas sums_of_real = bounded_linear.sums [OF bounded_linear_of_real]
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lemmas summable_of_real = bounded_linear.summable [OF bounded_linear_of_real]
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lemmas suminf_of_real = bounded_linear.suminf [OF bounded_linear_of_real]
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57275
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lemmas sums_scaleR_left = bounded_linear.sums[OF bounded_linear_scaleR_left]
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lemmas summable_scaleR_left = bounded_linear.summable[OF bounded_linear_scaleR_left]
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lemmas suminf_scaleR_left = bounded_linear.suminf[OF bounded_linear_scaleR_left]
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lemmas sums_scaleR_right = bounded_linear.sums[OF bounded_linear_scaleR_right]
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lemmas summable_scaleR_right = bounded_linear.summable[OF bounded_linear_scaleR_right]
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   353
lemmas suminf_scaleR_right = bounded_linear.suminf[OF bounded_linear_scaleR_right]
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56213
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subsection {* Infinite summability on real normed algebras *}
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context
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  fixes f :: "nat \<Rightarrow> 'a::real_normed_algebra"
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begin
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lemma sums_mult: "f sums a \<Longrightarrow> (\<lambda>n. c * f n) sums (c * a)"
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  by (rule bounded_linear.sums [OF bounded_linear_mult_right])
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lemma summable_mult: "summable f \<Longrightarrow> summable (\<lambda>n. c * f n)"
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  by (rule bounded_linear.summable [OF bounded_linear_mult_right])
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lemma suminf_mult: "summable f \<Longrightarrow> suminf (\<lambda>n. c * f n) = c * suminf f"
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  by (rule bounded_linear.suminf [OF bounded_linear_mult_right, symmetric])
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   369
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lemma sums_mult2: "f sums a \<Longrightarrow> (\<lambda>n. f n * c) sums (a * c)"
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   371
  by (rule bounded_linear.sums [OF bounded_linear_mult_left])
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lemma summable_mult2: "summable f \<Longrightarrow> summable (\<lambda>n. f n * c)"
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  by (rule bounded_linear.summable [OF bounded_linear_mult_left])
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lemma suminf_mult2: "summable f \<Longrightarrow> suminf f * c = (\<Sum>n. f n * c)"
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   377
  by (rule bounded_linear.suminf [OF bounded_linear_mult_left])
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end
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subsection {* Infinite summability on real normed fields *}
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context
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  fixes c :: "'a::real_normed_field"
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   385
begin
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   386
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lemma sums_divide: "f sums a \<Longrightarrow> (\<lambda>n. f n / c) sums (a / c)"
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   388
  by (rule bounded_linear.sums [OF bounded_linear_divide])
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diff changeset
   389
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   390
lemma summable_divide: "summable f \<Longrightarrow> summable (\<lambda>n. f n / c)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   391
  by (rule bounded_linear.summable [OF bounded_linear_divide])
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   392
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   393
lemma suminf_divide: "summable f \<Longrightarrow> suminf (\<lambda>n. f n / c) = suminf f / c"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   394
  by (rule bounded_linear.suminf [OF bounded_linear_divide, symmetric])
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   395
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   396
text{*Sum of a geometric progression.*}
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   397
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   398
lemma geometric_sums: "norm c < 1 \<Longrightarrow> (\<lambda>n. c^n) sums (1 / (1 - c))"
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   399
proof -
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   400
  assume less_1: "norm c < 1"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   401
  hence neq_1: "c \<noteq> 1" by auto
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   402
  hence neq_0: "c - 1 \<noteq> 0" by simp
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   403
  from less_1 have lim_0: "(\<lambda>n. c^n) ----> 0"
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   404
    by (rule LIMSEQ_power_zero)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   405
  hence "(\<lambda>n. c ^ n / (c - 1) - 1 / (c - 1)) ----> 0 / (c - 1) - 1 / (c - 1)"
44568
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44289
diff changeset
   406
    using neq_0 by (intro tendsto_intros)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   407
  hence "(\<lambda>n. (c ^ n - 1) / (c - 1)) ----> 1 / (1 - c)"
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   408
    by (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   409
  thus "(\<lambda>n. c ^ n) sums (1 / (1 - c))"
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   410
    by (simp add: sums_def geometric_sum neq_1)
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   411
qed
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   412
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   413
lemma summable_geometric: "norm c < 1 \<Longrightarrow> summable (\<lambda>n. c^n)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   414
  by (rule geometric_sums [THEN sums_summable])
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   415
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   416
lemma suminf_geometric: "norm c < 1 \<Longrightarrow> suminf (\<lambda>n. c^n) = 1 / (1 - c)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   417
  by (rule sums_unique[symmetric]) (rule geometric_sums)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   418
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   419
end
33271
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 32877
diff changeset
   420
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 32877
diff changeset
   421
lemma power_half_series: "(\<lambda>n. (1/2::real)^Suc n) sums 1"
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 32877
diff changeset
   422
proof -
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 32877
diff changeset
   423
  have 2: "(\<lambda>n. (1/2::real)^n) sums 2" using geometric_sums [of "1/2::real"]
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 32877
diff changeset
   424
    by auto
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 32877
diff changeset
   425
  have "(\<lambda>n. (1/2::real)^Suc n) = (\<lambda>n. (1 / 2) ^ n / 2)"
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 32877
diff changeset
   426
    by simp
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 41970
diff changeset
   427
  thus ?thesis using sums_divide [OF 2, of 2]
33271
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 32877
diff changeset
   428
    by simp
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 32877
diff changeset
   429
qed
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 32877
diff changeset
   430
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   431
subsection {* Infinite summability on Banach spaces *}
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   432
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   433
text{*Cauchy-type criterion for convergence of series (c.f. Harrison)*}
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   434
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   435
lemma summable_Cauchy:
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   436
  fixes f :: "nat \<Rightarrow> 'a::banach"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   437
  shows "summable f \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n. norm (setsum f {m..<n}) < e)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   438
  apply (simp only: summable_iff_convergent Cauchy_convergent_iff [symmetric] Cauchy_iff, safe)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   439
  apply (drule spec, drule (1) mp)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   440
  apply (erule exE, rule_tac x="M" in exI, clarify)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   441
  apply (rule_tac x="m" and y="n" in linorder_le_cases)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   442
  apply (frule (1) order_trans)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   443
  apply (drule_tac x="n" in spec, drule (1) mp)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   444
  apply (drule_tac x="m" in spec, drule (1) mp)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   445
  apply (simp_all add: setsum_diff [symmetric])
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   446
  apply (drule spec, drule (1) mp)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   447
  apply (erule exE, rule_tac x="N" in exI, clarify)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   448
  apply (rule_tac x="m" and y="n" in linorder_le_cases)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   449
  apply (subst norm_minus_commute)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   450
  apply (simp_all add: setsum_diff [symmetric])
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   451
  done
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   452
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   453
context
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   454
  fixes f :: "nat \<Rightarrow> 'a::banach"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   455
begin  
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   456
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   457
text{*Absolute convergence imples normal convergence*}
20689
4950e45442b8 add proof of summable_LIMSEQ_zero
huffman
parents: 20688
diff changeset
   458
56194
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56193
diff changeset
   459
lemma summable_norm_cancel: "summable (\<lambda>n. norm (f n)) \<Longrightarrow> summable f"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   460
  apply (simp only: summable_Cauchy, safe)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   461
  apply (drule_tac x="e" in spec, safe)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   462
  apply (rule_tac x="N" in exI, safe)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   463
  apply (drule_tac x="m" in spec, safe)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   464
  apply (rule order_le_less_trans [OF norm_setsum])
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   465
  apply (rule order_le_less_trans [OF abs_ge_self])
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   466
  apply simp
50999
3de230ed0547 introduce order topology
hoelzl
parents: 50331
diff changeset
   467
  done
32707
836ec9d0a0c8 New lemmas involving the real numbers, especially limits and series
paulson
parents: 31336
diff changeset
   468
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   469
lemma summable_norm: "summable (\<lambda>n. norm (f n)) \<Longrightarrow> norm (suminf f) \<le> (\<Sum>n. norm (f n))"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   470
  by (auto intro: LIMSEQ_le tendsto_norm summable_norm_cancel summable_LIMSEQ norm_setsum)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   471
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   472
text {* Comparison tests *}
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   473
56194
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56193
diff changeset
   474
lemma summable_comparison_test: "\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n \<Longrightarrow> summable g \<Longrightarrow> summable f"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   475
  apply (simp add: summable_Cauchy, safe)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   476
  apply (drule_tac x="e" in spec, safe)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   477
  apply (rule_tac x = "N + Na" in exI, safe)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   478
  apply (rotate_tac 2)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   479
  apply (drule_tac x = m in spec)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   480
  apply (auto, rotate_tac 2, drule_tac x = n in spec)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   481
  apply (rule_tac y = "\<Sum>k=m..<n. norm (f k)" in order_le_less_trans)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   482
  apply (rule norm_setsum)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   483
  apply (rule_tac y = "setsum g {m..<n}" in order_le_less_trans)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   484
  apply (auto intro: setsum_mono simp add: abs_less_iff)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   485
  done
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   486
56217
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 56213
diff changeset
   487
(*A better argument order*)
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 56213
diff changeset
   488
lemma summable_comparison_test': "summable g \<Longrightarrow> (\<And>n. n \<ge> N \<Longrightarrow> norm(f n) \<le> g n) \<Longrightarrow> summable f"
56369
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   489
  by (rule summable_comparison_test) auto
56217
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 56213
diff changeset
   490
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   491
subsection {* The Ratio Test*}
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   492
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   493
lemma summable_ratio_test: 
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   494
  assumes "c < 1" "\<And>n. n \<ge> N \<Longrightarrow> norm (f (Suc n)) \<le> c * norm (f n)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   495
  shows "summable f"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   496
proof cases
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   497
  assume "0 < c"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   498
  show "summable f"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   499
  proof (rule summable_comparison_test)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   500
    show "\<exists>N'. \<forall>n\<ge>N'. norm (f n) \<le> (norm (f N) / (c ^ N)) * c ^ n"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   501
    proof (intro exI allI impI)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   502
      fix n assume "N \<le> n" then show "norm (f n) \<le> (norm (f N) / (c ^ N)) * c ^ n"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   503
      proof (induct rule: inc_induct)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   504
        case (step m)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   505
        moreover have "norm (f (Suc m)) / c ^ Suc m * c ^ n \<le> norm (f m) / c ^ m * c ^ n"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   506
          using `0 < c` `c < 1` assms(2)[OF `N \<le> m`] by (simp add: field_simps)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   507
        ultimately show ?case by simp
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   508
      qed (insert `0 < c`, simp)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   509
    qed
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   510
    show "summable (\<lambda>n. norm (f N) / c ^ N * c ^ n)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   511
      using `0 < c` `c < 1` by (intro summable_mult summable_geometric) simp
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   512
  qed
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   513
next
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   514
  assume c: "\<not> 0 < c"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   515
  { fix n assume "n \<ge> N"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   516
    then have "norm (f (Suc n)) \<le> c * norm (f n)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   517
      by fact
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   518
    also have "\<dots> \<le> 0"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   519
      using c by (simp add: not_less mult_nonpos_nonneg)
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   520
    finally have "f (Suc n) = 0"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   521
      by auto }
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   522
  then show "summable f"
56194
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56193
diff changeset
   523
    by (intro sums_summable[OF sums_finite, of "{.. Suc N}"]) (auto simp: not_le Suc_less_eq2)
56178
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 54703
diff changeset
   524
qed
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 54703
diff changeset
   525
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   526
end
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   527
56369
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   528
text{*Relations among convergence and absolute convergence for power series.*}
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   529
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   530
lemma abel_lemma:
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   531
  fixes a :: "nat \<Rightarrow> 'a::real_normed_vector"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   532
  assumes r: "0 \<le> r" and r0: "r < r0" and M: "\<And>n. norm (a n) * r0^n \<le> M"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   533
    shows "summable (\<lambda>n. norm (a n) * r^n)"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   534
proof (rule summable_comparison_test')
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   535
  show "summable (\<lambda>n. M * (r / r0) ^ n)"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   536
    using assms 
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   537
    by (auto simp add: summable_mult summable_geometric)
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   538
next
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   539
  fix n
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   540
  show "norm (norm (a n) * r ^ n) \<le> M * (r / r0) ^ n"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   541
    using r r0 M [of n]
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   542
    apply (auto simp add: abs_mult field_simps power_divide)
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   543
    apply (cases "r=0", simp)
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   544
    apply (cases n, auto)
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   545
    done
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   546
qed
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   547
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56217
diff changeset
   548
23084
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   549
text{*Summability of geometric series for real algebras*}
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   550
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   551
lemma complete_algebra_summable_geometric:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30649
diff changeset
   552
  fixes x :: "'a::{real_normed_algebra_1,banach}"
23084
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   553
  shows "norm x < 1 \<Longrightarrow> summable (\<lambda>n. x ^ n)"
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   554
proof (rule summable_comparison_test)
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   555
  show "\<exists>N. \<forall>n\<ge>N. norm (x ^ n) \<le> norm x ^ n"
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   556
    by (simp add: norm_power_ineq)
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   557
  show "norm x < 1 \<Longrightarrow> summable (\<lambda>n. norm x ^ n)"
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   558
    by (simp add: summable_geometric)
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   559
qed
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   560
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   561
subsection {* Cauchy Product Formula *}
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   562
54703
499f92dc6e45 more antiquotations;
wenzelm
parents: 54230
diff changeset
   563
text {*
499f92dc6e45 more antiquotations;
wenzelm
parents: 54230
diff changeset
   564
  Proof based on Analysis WebNotes: Chapter 07, Class 41
499f92dc6e45 more antiquotations;
wenzelm
parents: 54230
diff changeset
   565
  @{url "http://www.math.unl.edu/~webnotes/classes/class41/prp77.htm"}
499f92dc6e45 more antiquotations;
wenzelm
parents: 54230
diff changeset
   566
*}
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   567
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   568
lemma setsum_triangle_reindex:
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   569
  fixes n :: nat
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   570
  shows "(\<Sum>(i,j)\<in>{(i,j). i+j < n}. f i j) = (\<Sum>k<n. \<Sum>i\<le>k. f i (k - i))"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57275
diff changeset
   571
  apply (simp add: setsum.Sigma)
57129
7edb7550663e introduce more powerful reindexing rules for big operators
hoelzl
parents: 57025
diff changeset
   572
  apply (rule setsum.reindex_bij_witness[where j="\<lambda>(i, j). (i+j, i)" and i="\<lambda>(k, i). (i, k - i)"])
7edb7550663e introduce more powerful reindexing rules for big operators
hoelzl
parents: 57025
diff changeset
   573
  apply auto
7edb7550663e introduce more powerful reindexing rules for big operators
hoelzl
parents: 57025
diff changeset
   574
  done
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   575
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   576
lemma Cauchy_product_sums:
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   577
  fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   578
  assumes a: "summable (\<lambda>k. norm (a k))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   579
  assumes b: "summable (\<lambda>k. norm (b k))"
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   580
  shows "(\<lambda>k. \<Sum>i\<le>k. a i * b (k - i)) sums ((\<Sum>k. a k) * (\<Sum>k. b k))"
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   581
proof -
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   582
  let ?S1 = "\<lambda>n::nat. {..<n} \<times> {..<n}"
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   583
  let ?S2 = "\<lambda>n::nat. {(i,j). i + j < n}"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   584
  have S1_mono: "\<And>m n. m \<le> n \<Longrightarrow> ?S1 m \<subseteq> ?S1 n" by auto
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   585
  have S2_le_S1: "\<And>n. ?S2 n \<subseteq> ?S1 n" by auto
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   586
  have S1_le_S2: "\<And>n. ?S1 (n div 2) \<subseteq> ?S2 n" by auto
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   587
  have finite_S1: "\<And>n. finite (?S1 n)" by simp
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   588
  with S2_le_S1 have finite_S2: "\<And>n. finite (?S2 n)" by (rule finite_subset)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   589
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   590
  let ?g = "\<lambda>(i,j). a i * b j"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   591
  let ?f = "\<lambda>(i,j). norm (a i) * norm (b j)"
56536
aefb4a8da31f made mult_nonneg_nonneg a simp rule
nipkow
parents: 56369
diff changeset
   592
  have f_nonneg: "\<And>x. 0 \<le> ?f x" by (auto)
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   593
  hence norm_setsum_f: "\<And>A. norm (setsum ?f A) = setsum ?f A"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   594
    unfolding real_norm_def
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   595
    by (simp only: abs_of_nonneg setsum_nonneg [rule_format])
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   596
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   597
  have "(\<lambda>n. (\<Sum>k<n. a k) * (\<Sum>k<n. b k)) ----> (\<Sum>k. a k) * (\<Sum>k. b k)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   598
    by (intro tendsto_mult summable_LIMSEQ summable_norm_cancel [OF a] summable_norm_cancel [OF b])
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   599
  hence 1: "(\<lambda>n. setsum ?g (?S1 n)) ----> (\<Sum>k. a k) * (\<Sum>k. b k)"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57275
diff changeset
   600
    by (simp only: setsum_product setsum.Sigma [rule_format] finite_lessThan)
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   601
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   602
  have "(\<lambda>n. (\<Sum>k<n. norm (a k)) * (\<Sum>k<n. norm (b k))) ----> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56178
diff changeset
   603
    using a b by (intro tendsto_mult summable_LIMSEQ)
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   604
  hence "(\<lambda>n. setsum ?f (?S1 n)) ----> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57275
diff changeset
   605
    by (simp only: setsum_product setsum.Sigma [rule_format] finite_lessThan)
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   606
  hence "convergent (\<lambda>n. setsum ?f (?S1 n))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   607
    by (rule convergentI)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   608
  hence Cauchy: "Cauchy (\<lambda>n. setsum ?f (?S1 n))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   609
    by (rule convergent_Cauchy)
36657
f376af79f6b7 remove unneeded constant Zseq
huffman
parents: 36409
diff changeset
   610
  have "Zfun (\<lambda>n. setsum ?f (?S1 n - ?S2 n)) sequentially"
f376af79f6b7 remove unneeded constant Zseq
huffman
parents: 36409
diff changeset
   611
  proof (rule ZfunI, simp only: eventually_sequentially norm_setsum_f)
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   612
    fix r :: real
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   613
    assume r: "0 < r"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   614
    from CauchyD [OF Cauchy r] obtain N
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   615
    where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (setsum ?f (?S1 m) - setsum ?f (?S1 n)) < r" ..
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   616
    hence "\<And>m n. \<lbrakk>N \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> norm (setsum ?f (?S1 m - ?S1 n)) < r"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   617
      by (simp only: setsum_diff finite_S1 S1_mono)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   618
    hence N: "\<And>m n. \<lbrakk>N \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> setsum ?f (?S1 m - ?S1 n) < r"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   619
      by (simp only: norm_setsum_f)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   620
    show "\<exists>N. \<forall>n\<ge>N. setsum ?f (?S1 n - ?S2 n) < r"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   621
    proof (intro exI allI impI)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   622
      fix n assume "2 * N \<le> n"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   623
      hence n: "N \<le> n div 2" by simp
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   624
      have "setsum ?f (?S1 n - ?S2 n) \<le> setsum ?f (?S1 n - ?S1 (n div 2))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   625
        by (intro setsum_mono2 finite_Diff finite_S1 f_nonneg
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   626
                  Diff_mono subset_refl S1_le_S2)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   627
      also have "\<dots> < r"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   628
        using n div_le_dividend by (rule N)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   629
      finally show "setsum ?f (?S1 n - ?S2 n) < r" .
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   630
    qed
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   631
  qed
36657
f376af79f6b7 remove unneeded constant Zseq
huffman
parents: 36409
diff changeset
   632
  hence "Zfun (\<lambda>n. setsum ?g (?S1 n - ?S2 n)) sequentially"
f376af79f6b7 remove unneeded constant Zseq
huffman
parents: 36409
diff changeset
   633
    apply (rule Zfun_le [rule_format])
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   634
    apply (simp only: norm_setsum_f)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   635
    apply (rule order_trans [OF norm_setsum setsum_mono])
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   636
    apply (auto simp add: norm_mult_ineq)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   637
    done
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   638
  hence 2: "(\<lambda>n. setsum ?g (?S1 n) - setsum ?g (?S2 n)) ----> 0"
36660
1cc4ab4b7ff7 make (X ----> L) an abbreviation for (X ---> L) sequentially
huffman
parents: 36657
diff changeset
   639
    unfolding tendsto_Zfun_iff diff_0_right
36657
f376af79f6b7 remove unneeded constant Zseq
huffman
parents: 36409
diff changeset
   640
    by (simp only: setsum_diff finite_S1 S2_le_S1)
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   641
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   642
  with 1 have "(\<lambda>n. setsum ?g (?S2 n)) ----> (\<Sum>k. a k) * (\<Sum>k. b k)"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   643
    by (rule LIMSEQ_diff_approach_zero2)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   644
  thus ?thesis by (simp only: sums_def setsum_triangle_reindex)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   645
qed
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   646
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   647
lemma Cauchy_product:
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   648
  fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   649
  assumes a: "summable (\<lambda>k. norm (a k))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   650
  assumes b: "summable (\<lambda>k. norm (b k))"
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   651
  shows "(\<Sum>k. a k) * (\<Sum>k. b k) = (\<Sum>k. \<Sum>i\<le>k. a i * b (k - i))"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   652
  using a b
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   653
  by (rule Cauchy_product_sums [THEN sums_unique])
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   654
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   655
subsection {* Series on @{typ real}s *}
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   656
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   657
lemma summable_norm_comparison_test: "\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. norm (f n))"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   658
  by (rule summable_comparison_test) auto
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   659
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   660
lemma summable_rabs_comparison_test: "\<lbrakk>\<exists>N. \<forall>n\<ge>N. \<bar>f n\<bar> \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable (\<lambda>n. \<bar>f n :: real\<bar>)"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   661
  by (rule summable_comparison_test) auto
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   662
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   663
lemma summable_rabs_cancel: "summable (\<lambda>n. \<bar>f n :: real\<bar>) \<Longrightarrow> summable f"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   664
  by (rule summable_norm_cancel) simp
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   665
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   666
lemma summable_rabs: "summable (\<lambda>n. \<bar>f n :: real\<bar>) \<Longrightarrow> \<bar>suminf f\<bar> \<le> (\<Sum>n. \<bar>f n\<bar>)"
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56194
diff changeset
   667
  by (fold real_norm_def) (rule summable_norm)
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   668
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   669
lemma summable_power_series:
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   670
  fixes z :: real
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   671
  assumes le_1: "\<And>i. f i \<le> 1" and nonneg: "\<And>i. 0 \<le> f i" and z: "0 \<le> z" "z < 1"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   672
  shows "summable (\<lambda>i. f i * z^i)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   673
proof (rule summable_comparison_test[OF _ summable_geometric])
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   674
  show "norm z < 1" using z by (auto simp: less_imp_le)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   675
  show "\<And>n. \<exists>N. \<forall>na\<ge>N. norm (f na * z ^ na) \<le> z ^ na"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   676
    using z by (auto intro!: exI[of _ 0] mult_left_le_one_le simp: abs_mult nonneg power_abs less_imp_le le_1)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   677
qed
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   678
59025
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   679
lemma
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   680
   fixes f :: "nat \<Rightarrow> real"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   681
   assumes "summable f"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   682
   and "inj g"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   683
   and pos: "!!x. 0 \<le> f x"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   684
   shows summable_reindex: "summable (f o g)"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   685
   and suminf_reindex_mono: "suminf (f o g) \<le> suminf f"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   686
   and suminf_reindex: "(\<And>x. x \<notin> range g \<Longrightarrow> f x = 0) \<Longrightarrow> suminf (f \<circ> g) = suminf f"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   687
proof -
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   688
  from \<open>inj g\<close> have [simp]: "\<And>A. inj_on g A" by(rule subset_inj_on) simp
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   689
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   690
  have smaller: "\<forall>n. (\<Sum>i<n. (f \<circ> g) i) \<le> suminf f"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   691
  proof
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   692
    fix n
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   693
    have "\<forall> n' \<in> (g ` {..<n}). n' < Suc (Max (g ` {..<n}))" 
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   694
      by(metis Max_ge finite_imageI finite_lessThan not_le not_less_eq)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   695
    then obtain m where n: "\<And>n'. n' < n \<Longrightarrow> g n' < m" by blast
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   696
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   697
    have "(\<Sum>i<n. f (g i)) = setsum f (g ` {..<n})"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   698
      by (simp add: setsum.reindex)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   699
    also have "\<dots> \<le> (\<Sum>i<m. f i)"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   700
      by (rule setsum_mono3) (auto simp add: pos n[rule_format])
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   701
    also have "\<dots> \<le> suminf f"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   702
      using `summable f` 
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   703
      by (rule setsum_le_suminf) (simp add: pos)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   704
    finally show "(\<Sum>i<n. (f \<circ>  g) i) \<le> suminf f" by simp
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   705
  qed
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   706
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   707
  have "incseq (\<lambda>n. \<Sum>i<n. (f \<circ> g) i)"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   708
    by (rule incseq_SucI) (auto simp add: pos)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   709
  then obtain  L where L: "(\<lambda> n. \<Sum>i<n. (f \<circ> g) i) ----> L"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   710
    using smaller by(rule incseq_convergent)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   711
  hence "(f \<circ> g) sums L" by (simp add: sums_def)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   712
  thus "summable (f o g)" by (auto simp add: sums_iff)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   713
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   714
  hence "(\<lambda>n. \<Sum>i<n. (f \<circ> g) i) ----> suminf (f \<circ> g)"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   715
    by(rule summable_LIMSEQ)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   716
  thus le: "suminf (f \<circ> g) \<le> suminf f"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   717
    by(rule LIMSEQ_le_const2)(blast intro: smaller[rule_format])
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   718
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   719
  assume f: "\<And>x. x \<notin> range g \<Longrightarrow> f x = 0"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   720
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   721
  from \<open>summable f\<close> have "suminf f \<le> suminf (f \<circ> g)"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   722
  proof(rule suminf_le_const)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   723
    fix n
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   724
    have "\<forall> n' \<in> (g -` {..<n}). n' < Suc (Max (g -` {..<n}))"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   725
      by(auto intro: Max_ge simp add: finite_vimageI less_Suc_eq_le)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   726
    then obtain m where n: "\<And>n'. g n' < n \<Longrightarrow> n' < m" by blast
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   727
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   728
    have "(\<Sum>i<n. f i) = (\<Sum>i\<in>{..<n} \<inter> range g. f i)"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   729
      using f by(auto intro: setsum.mono_neutral_cong_right)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   730
    also have "\<dots> = (\<Sum>i\<in>g -` {..<n}. (f \<circ> g) i)"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   731
      by(rule setsum.reindex_cong[where l=g])(auto)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   732
    also have "\<dots> \<le> (\<Sum>i<m. (f \<circ> g) i)"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   733
      by(rule setsum_mono3)(auto simp add: pos n)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   734
    also have "\<dots> \<le> suminf (f \<circ> g)"
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   735
      using \<open>summable (f o g)\<close>
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   736
      by(rule setsum_le_suminf)(simp add: pos)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   737
    finally show "setsum f {..<n} \<le> suminf (f \<circ> g)" .
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   738
  qed
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   739
  with le show "suminf (f \<circ> g) = suminf f" by(rule antisym)
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   740
qed
d885cff91200 add lemma following a proof suggestion by Joachim Breitner
Andreas Lochbihler
parents: 59000
diff changeset
   741
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   742
end