src/HOL/Analysis/Infinite_Products.thy
author eberlm <eberlm@in.tum.de>
Sat Jul 15 14:33:56 2017 +0100 (2017-07-15)
changeset 66277 512b0dc09061
child 68064 b249fab48c76
permissions -rw-r--r--
HOL-Analysis: Infinite products
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(*
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  File:      HOL/Analysis/Infinite_Product.thy
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  Author:    Manuel Eberl
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  Basic results about convergence and absolute convergence of infinite products
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  and their connection to summability.
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*)
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section \<open>Infinite Products\<close>
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theory Infinite_Products
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  imports Complex_Main
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begin
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lemma sum_le_prod:
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  fixes f :: "'a \<Rightarrow> 'b :: linordered_semidom"
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  assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<ge> 0"
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  shows   "sum f A \<le> (\<Prod>x\<in>A. 1 + f x)"
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  using assms
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proof (induction A rule: infinite_finite_induct)
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  case (insert x A)
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  from insert.hyps have "sum f A + f x * (\<Prod>x\<in>A. 1) \<le> (\<Prod>x\<in>A. 1 + f x) + f x * (\<Prod>x\<in>A. 1 + f x)"
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    by (intro add_mono insert mult_left_mono prod_mono) (auto intro: insert.prems)
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  with insert.hyps show ?case by (simp add: algebra_simps)
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qed simp_all
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lemma prod_le_exp_sum:
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  fixes f :: "'a \<Rightarrow> real"
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  assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<ge> 0"
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  shows   "prod (\<lambda>x. 1 + f x) A \<le> exp (sum f A)"
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  using assms
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proof (induction A rule: infinite_finite_induct)
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  case (insert x A)
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  have "(1 + f x) * (\<Prod>x\<in>A. 1 + f x) \<le> exp (f x) * exp (sum f A)"
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    using insert.prems by (intro mult_mono insert prod_nonneg exp_ge_add_one_self) auto
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  with insert.hyps show ?case by (simp add: algebra_simps exp_add)
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qed simp_all
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lemma lim_ln_1_plus_x_over_x_at_0: "(\<lambda>x::real. ln (1 + x) / x) \<midarrow>0\<rightarrow> 1"
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proof (rule lhopital)
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  show "(\<lambda>x::real. ln (1 + x)) \<midarrow>0\<rightarrow> 0"
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    by (rule tendsto_eq_intros refl | simp)+
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  have "eventually (\<lambda>x::real. x \<in> {-1/2<..<1/2}) (nhds 0)"
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    by (rule eventually_nhds_in_open) auto
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  hence *: "eventually (\<lambda>x::real. x \<in> {-1/2<..<1/2}) (at 0)"
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    by (rule filter_leD [rotated]) (simp_all add: at_within_def)   
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  show "eventually (\<lambda>x::real. ((\<lambda>x. ln (1 + x)) has_field_derivative inverse (1 + x)) (at x)) (at 0)"
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    using * by eventually_elim (auto intro!: derivative_eq_intros simp: field_simps)
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  show "eventually (\<lambda>x::real. ((\<lambda>x. x) has_field_derivative 1) (at x)) (at 0)"
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    using * by eventually_elim (auto intro!: derivative_eq_intros simp: field_simps)
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  show "\<forall>\<^sub>F x in at 0. x \<noteq> 0" by (auto simp: at_within_def eventually_inf_principal)
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  show "(\<lambda>x::real. inverse (1 + x) / 1) \<midarrow>0\<rightarrow> 1"
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    by (rule tendsto_eq_intros refl | simp)+
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qed auto
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definition convergent_prod :: "(nat \<Rightarrow> 'a :: {t2_space,comm_semiring_1}) \<Rightarrow> bool" where
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  "convergent_prod f \<longleftrightarrow> (\<exists>M L. (\<lambda>n. \<Prod>i\<le>n. f (i+M)) \<longlonglongrightarrow> L \<and> L \<noteq> 0)"
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lemma convergent_prod_altdef:
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  fixes f :: "nat \<Rightarrow> 'a :: {t2_space,comm_semiring_1}"
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  shows "convergent_prod f \<longleftrightarrow> (\<exists>M L. (\<forall>n\<ge>M. f n \<noteq> 0) \<and> (\<lambda>n. \<Prod>i\<le>n. f (i+M)) \<longlonglongrightarrow> L \<and> L \<noteq> 0)"
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proof
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  assume "convergent_prod f"
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  then obtain M L where *: "(\<lambda>n. \<Prod>i\<le>n. f (i+M)) \<longlonglongrightarrow> L" "L \<noteq> 0"
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    by (auto simp: convergent_prod_def)
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  have "f i \<noteq> 0" if "i \<ge> M" for i
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  proof
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    assume "f i = 0"
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    have **: "eventually (\<lambda>n. (\<Prod>i\<le>n. f (i+M)) = 0) sequentially"
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      using eventually_ge_at_top[of "i - M"]
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    proof eventually_elim
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      case (elim n)
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      with \<open>f i = 0\<close> and \<open>i \<ge> M\<close> show ?case
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        by (auto intro!: bexI[of _ "i - M"] prod_zero)
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    qed
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    have "(\<lambda>n. (\<Prod>i\<le>n. f (i+M))) \<longlonglongrightarrow> 0"
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      unfolding filterlim_iff
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      by (auto dest!: eventually_nhds_x_imp_x intro!: eventually_mono[OF **])
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    from tendsto_unique[OF _ this *(1)] and *(2)
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      show False by simp
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  qed
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  with * show "(\<exists>M L. (\<forall>n\<ge>M. f n \<noteq> 0) \<and> (\<lambda>n. \<Prod>i\<le>n. f (i+M)) \<longlonglongrightarrow> L \<and> L \<noteq> 0)" 
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    by blast
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qed (auto simp: convergent_prod_def)
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definition abs_convergent_prod :: "(nat \<Rightarrow> _) \<Rightarrow> bool" where
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  "abs_convergent_prod f \<longleftrightarrow> convergent_prod (\<lambda>i. 1 + norm (f i - 1))"
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lemma abs_convergent_prodI:
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  assumes "convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
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  shows   "abs_convergent_prod f"
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proof -
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  from assms obtain L where L: "(\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1)) \<longlonglongrightarrow> L"
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    by (auto simp: convergent_def)
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  have "L \<ge> 1"
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  proof (rule tendsto_le)
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    show "eventually (\<lambda>n. (\<Prod>i\<le>n. 1 + norm (f i - 1)) \<ge> 1) sequentially"
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    proof (intro always_eventually allI)
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      fix n
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      have "(\<Prod>i\<le>n. 1 + norm (f i - 1)) \<ge> (\<Prod>i\<le>n. 1)"
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        by (intro prod_mono) auto
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      thus "(\<Prod>i\<le>n. 1 + norm (f i - 1)) \<ge> 1" by simp
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    qed
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  qed (use L in simp_all)
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  hence "L \<noteq> 0" by auto
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  with L show ?thesis unfolding abs_convergent_prod_def convergent_prod_def
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    by (intro exI[of _ "0::nat"] exI[of _ L]) auto
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qed
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lemma
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  fixes f :: "nat \<Rightarrow> 'a :: {real_normed_div_algebra,idom}"
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  assumes "convergent_prod f"
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  shows   convergent_prod_imp_convergent: "convergent (\<lambda>n. \<Prod>i\<le>n. f i)"
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    and   convergent_prod_to_zero_iff:    "(\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> 0 \<longleftrightarrow> (\<exists>i. f i = 0)"
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proof -
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  from assms obtain M L 
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    where M: "\<And>n. n \<ge> M \<Longrightarrow> f n \<noteq> 0" and "(\<lambda>n. \<Prod>i\<le>n. f (i + M)) \<longlonglongrightarrow> L" and "L \<noteq> 0"
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    by (auto simp: convergent_prod_altdef)
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  note this(2)
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  also have "(\<lambda>n. \<Prod>i\<le>n. f (i + M)) = (\<lambda>n. \<Prod>i=M..M+n. f i)"
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    by (intro ext prod.reindex_bij_witness[of _ "\<lambda>n. n - M" "\<lambda>n. n + M"]) auto
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  finally have "(\<lambda>n. (\<Prod>i<M. f i) * (\<Prod>i=M..M+n. f i)) \<longlonglongrightarrow> (\<Prod>i<M. f i) * L"
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    by (intro tendsto_mult tendsto_const)
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  also have "(\<lambda>n. (\<Prod>i<M. f i) * (\<Prod>i=M..M+n. f i)) = (\<lambda>n. (\<Prod>i\<in>{..<M}\<union>{M..M+n}. f i))"
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    by (subst prod.union_disjoint) auto
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  also have "(\<lambda>n. {..<M} \<union> {M..M+n}) = (\<lambda>n. {..n+M})" by auto
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  finally have lim: "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> prod f {..<M} * L" 
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    by (rule LIMSEQ_offset)
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  thus "convergent (\<lambda>n. \<Prod>i\<le>n. f i)"
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    by (auto simp: convergent_def)
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  show "(\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> 0 \<longleftrightarrow> (\<exists>i. f i = 0)"
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  proof
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    assume "\<exists>i. f i = 0"
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    then obtain i where "f i = 0" by auto
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    moreover with M have "i < M" by (cases "i < M") auto
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    ultimately have "(\<Prod>i<M. f i) = 0" by auto
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    with lim show "(\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> 0" by simp
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  next
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    assume "(\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> 0"
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    from tendsto_unique[OF _ this lim] and \<open>L \<noteq> 0\<close>
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    show "\<exists>i. f i = 0" by auto
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  qed
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qed
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lemma abs_convergent_prod_altdef:
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  "abs_convergent_prod f \<longleftrightarrow> convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
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proof
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  assume "abs_convergent_prod f"
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  thus "convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
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    by (auto simp: abs_convergent_prod_def intro!: convergent_prod_imp_convergent)
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qed (auto intro: abs_convergent_prodI)
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lemma weierstrass_prod_ineq:
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  fixes f :: "'a \<Rightarrow> real" 
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  assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> {0..1}"
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  shows   "1 - sum f A \<le> (\<Prod>x\<in>A. 1 - f x)"
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  using assms
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proof (induction A rule: infinite_finite_induct)
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  case (insert x A)
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  from insert.hyps and insert.prems 
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    have "1 - sum f A + f x * (\<Prod>x\<in>A. 1 - f x) \<le> (\<Prod>x\<in>A. 1 - f x) + f x * (\<Prod>x\<in>A. 1)"
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    by (intro insert.IH add_mono mult_left_mono prod_mono) auto
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  with insert.hyps show ?case by (simp add: algebra_simps)
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qed simp_all
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lemma norm_prod_minus1_le_prod_minus1:
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  fixes f :: "nat \<Rightarrow> 'a :: {real_normed_div_algebra,comm_ring_1}"  
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  shows "norm (prod (\<lambda>n. 1 + f n) A - 1) \<le> prod (\<lambda>n. 1 + norm (f n)) A - 1"
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proof (induction A rule: infinite_finite_induct)
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  case (insert x A)
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  from insert.hyps have 
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    "norm ((\<Prod>n\<in>insert x A. 1 + f n) - 1) = 
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       norm ((\<Prod>n\<in>A. 1 + f n) - 1 + f x * (\<Prod>n\<in>A. 1 + f n))"
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    by (simp add: algebra_simps)
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  also have "\<dots> \<le> norm ((\<Prod>n\<in>A. 1 + f n) - 1) + norm (f x * (\<Prod>n\<in>A. 1 + f n))"
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    by (rule norm_triangle_ineq)
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  also have "norm (f x * (\<Prod>n\<in>A. 1 + f n)) = norm (f x) * (\<Prod>x\<in>A. norm (1 + f x))"
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    by (simp add: prod_norm norm_mult)
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  also have "(\<Prod>x\<in>A. norm (1 + f x)) \<le> (\<Prod>x\<in>A. norm (1::'a) + norm (f x))"
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    by (intro prod_mono norm_triangle_ineq ballI conjI) auto
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  also have "norm (1::'a) = 1" by simp
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  also note insert.IH
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  also have "(\<Prod>n\<in>A. 1 + norm (f n)) - 1 + norm (f x) * (\<Prod>x\<in>A. 1 + norm (f x)) =
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               (\<Prod>n\<in>insert x A. 1 + norm (f n)) - 1"
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    using insert.hyps by (simp add: algebra_simps)
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  finally show ?case by - (simp_all add: mult_left_mono)
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qed simp_all
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lemma convergent_prod_imp_ev_nonzero:
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  fixes f :: "nat \<Rightarrow> 'a :: {t2_space,comm_semiring_1}"
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  assumes "convergent_prod f"
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  shows   "eventually (\<lambda>n. f n \<noteq> 0) sequentially"
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  using assms by (auto simp: eventually_at_top_linorder convergent_prod_altdef)
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lemma convergent_prod_imp_LIMSEQ:
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  fixes f :: "nat \<Rightarrow> 'a :: {real_normed_field}"
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  assumes "convergent_prod f"
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  shows   "f \<longlonglongrightarrow> 1"
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proof -
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  from assms obtain M L where L: "(\<lambda>n. \<Prod>i\<le>n. f (i+M)) \<longlonglongrightarrow> L" "\<And>n. n \<ge> M \<Longrightarrow> f n \<noteq> 0" "L \<noteq> 0"
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    by (auto simp: convergent_prod_altdef)
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  hence L': "(\<lambda>n. \<Prod>i\<le>Suc n. f (i+M)) \<longlonglongrightarrow> L" by (subst filterlim_sequentially_Suc)
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  have "(\<lambda>n. (\<Prod>i\<le>Suc n. f (i+M)) / (\<Prod>i\<le>n. f (i+M))) \<longlonglongrightarrow> L / L"
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    using L L' by (intro tendsto_divide) simp_all
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  also from L have "L / L = 1" by simp
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  also have "(\<lambda>n. (\<Prod>i\<le>Suc n. f (i+M)) / (\<Prod>i\<le>n. f (i+M))) = (\<lambda>n. f (n + Suc M))"
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    using assms L by (auto simp: fun_eq_iff atMost_Suc)
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  finally show ?thesis by (rule LIMSEQ_offset)
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qed
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lemma abs_convergent_prod_imp_summable:
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  fixes f :: "nat \<Rightarrow> 'a :: real_normed_div_algebra"
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  assumes "abs_convergent_prod f"
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  shows "summable (\<lambda>i. norm (f i - 1))"
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proof -
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  from assms have "convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))" 
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    unfolding abs_convergent_prod_def by (rule convergent_prod_imp_convergent)
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  then obtain L where L: "(\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1)) \<longlonglongrightarrow> L"
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    unfolding convergent_def by blast
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  have "convergent (\<lambda>n. \<Sum>i\<le>n. norm (f i - 1))"
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  proof (rule Bseq_monoseq_convergent)
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    have "eventually (\<lambda>n. (\<Prod>i\<le>n. 1 + norm (f i - 1)) < L + 1) sequentially"
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      using L(1) by (rule order_tendstoD) simp_all
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    hence "\<forall>\<^sub>F x in sequentially. norm (\<Sum>i\<le>x. norm (f i - 1)) \<le> L + 1"
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    proof eventually_elim
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      case (elim n)
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      have "norm (\<Sum>i\<le>n. norm (f i - 1)) = (\<Sum>i\<le>n. norm (f i - 1))"
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        unfolding real_norm_def by (intro abs_of_nonneg sum_nonneg) simp_all
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      also have "\<dots> \<le> (\<Prod>i\<le>n. 1 + norm (f i - 1))" by (rule sum_le_prod) auto
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      also have "\<dots> < L + 1" by (rule elim)
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      finally show ?case by simp
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    qed
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    thus "Bseq (\<lambda>n. \<Sum>i\<le>n. norm (f i - 1))" by (rule BfunI)
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  next
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    show "monoseq (\<lambda>n. \<Sum>i\<le>n. norm (f i - 1))"
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      by (rule mono_SucI1) auto
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  qed
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  thus "summable (\<lambda>i. norm (f i - 1))" by (simp add: summable_iff_convergent')
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qed
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lemma summable_imp_abs_convergent_prod:
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  fixes f :: "nat \<Rightarrow> 'a :: real_normed_div_algebra"
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  assumes "summable (\<lambda>i. norm (f i - 1))"
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  shows   "abs_convergent_prod f"
eberlm@66277
   244
proof (intro abs_convergent_prodI Bseq_monoseq_convergent)
eberlm@66277
   245
  show "monoseq (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
eberlm@66277
   246
    by (intro mono_SucI1) 
eberlm@66277
   247
       (auto simp: atMost_Suc algebra_simps intro!: mult_nonneg_nonneg prod_nonneg)
eberlm@66277
   248
next
eberlm@66277
   249
  show "Bseq (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
eberlm@66277
   250
  proof (rule Bseq_eventually_mono)
eberlm@66277
   251
    show "eventually (\<lambda>n. norm (\<Prod>i\<le>n. 1 + norm (f i - 1)) \<le> 
eberlm@66277
   252
            norm (exp (\<Sum>i\<le>n. norm (f i - 1)))) sequentially"
eberlm@66277
   253
      by (intro always_eventually allI) (auto simp: abs_prod exp_sum intro!: prod_mono)
eberlm@66277
   254
  next
eberlm@66277
   255
    from assms have "(\<lambda>n. \<Sum>i\<le>n. norm (f i - 1)) \<longlonglongrightarrow> (\<Sum>i. norm (f i - 1))"
eberlm@66277
   256
      using sums_def_le by blast
eberlm@66277
   257
    hence "(\<lambda>n. exp (\<Sum>i\<le>n. norm (f i - 1))) \<longlonglongrightarrow> exp (\<Sum>i. norm (f i - 1))"
eberlm@66277
   258
      by (rule tendsto_exp)
eberlm@66277
   259
    hence "convergent (\<lambda>n. exp (\<Sum>i\<le>n. norm (f i - 1)))"
eberlm@66277
   260
      by (rule convergentI)
eberlm@66277
   261
    thus "Bseq (\<lambda>n. exp (\<Sum>i\<le>n. norm (f i - 1)))"
eberlm@66277
   262
      by (rule convergent_imp_Bseq)
eberlm@66277
   263
  qed
eberlm@66277
   264
qed
eberlm@66277
   265
eberlm@66277
   266
lemma abs_convergent_prod_conv_summable:
eberlm@66277
   267
  fixes f :: "nat \<Rightarrow> 'a :: real_normed_div_algebra"
eberlm@66277
   268
  shows "abs_convergent_prod f \<longleftrightarrow> summable (\<lambda>i. norm (f i - 1))"
eberlm@66277
   269
  by (blast intro: abs_convergent_prod_imp_summable summable_imp_abs_convergent_prod)
eberlm@66277
   270
eberlm@66277
   271
lemma abs_convergent_prod_imp_LIMSEQ:
eberlm@66277
   272
  fixes f :: "nat \<Rightarrow> 'a :: {comm_ring_1,real_normed_div_algebra}"
eberlm@66277
   273
  assumes "abs_convergent_prod f"
eberlm@66277
   274
  shows   "f \<longlonglongrightarrow> 1"
eberlm@66277
   275
proof -
eberlm@66277
   276
  from assms have "summable (\<lambda>n. norm (f n - 1))"
eberlm@66277
   277
    by (rule abs_convergent_prod_imp_summable)
eberlm@66277
   278
  from summable_LIMSEQ_zero[OF this] have "(\<lambda>n. f n - 1) \<longlonglongrightarrow> 0"
eberlm@66277
   279
    by (simp add: tendsto_norm_zero_iff)
eberlm@66277
   280
  from tendsto_add[OF this tendsto_const[of 1]] show ?thesis by simp
eberlm@66277
   281
qed
eberlm@66277
   282
eberlm@66277
   283
lemma abs_convergent_prod_imp_ev_nonzero:
eberlm@66277
   284
  fixes f :: "nat \<Rightarrow> 'a :: {comm_ring_1,real_normed_div_algebra}"
eberlm@66277
   285
  assumes "abs_convergent_prod f"
eberlm@66277
   286
  shows   "eventually (\<lambda>n. f n \<noteq> 0) sequentially"
eberlm@66277
   287
proof -
eberlm@66277
   288
  from assms have "f \<longlonglongrightarrow> 1" 
eberlm@66277
   289
    by (rule abs_convergent_prod_imp_LIMSEQ)
eberlm@66277
   290
  hence "eventually (\<lambda>n. dist (f n) 1 < 1) at_top"
eberlm@66277
   291
    by (auto simp: tendsto_iff)
eberlm@66277
   292
  thus ?thesis by eventually_elim auto
eberlm@66277
   293
qed
eberlm@66277
   294
eberlm@66277
   295
lemma convergent_prod_offset:
eberlm@66277
   296
  assumes "convergent_prod (\<lambda>n. f (n + m))"  
eberlm@66277
   297
  shows   "convergent_prod f"
eberlm@66277
   298
proof -
eberlm@66277
   299
  from assms obtain M L where "(\<lambda>n. \<Prod>k\<le>n. f (k + (M + m))) \<longlonglongrightarrow> L" "L \<noteq> 0"
eberlm@66277
   300
    by (auto simp: convergent_prod_def add.assoc)
eberlm@66277
   301
  thus "convergent_prod f" unfolding convergent_prod_def by blast
eberlm@66277
   302
qed
eberlm@66277
   303
eberlm@66277
   304
lemma abs_convergent_prod_offset:
eberlm@66277
   305
  assumes "abs_convergent_prod (\<lambda>n. f (n + m))"  
eberlm@66277
   306
  shows   "abs_convergent_prod f"
eberlm@66277
   307
  using assms unfolding abs_convergent_prod_def by (rule convergent_prod_offset)
eberlm@66277
   308
eberlm@66277
   309
lemma convergent_prod_ignore_initial_segment:
eberlm@66277
   310
  fixes f :: "nat \<Rightarrow> 'a :: {real_normed_field}"
eberlm@66277
   311
  assumes "convergent_prod f"
eberlm@66277
   312
  shows   "convergent_prod (\<lambda>n. f (n + m))"
eberlm@66277
   313
proof -
eberlm@66277
   314
  from assms obtain M L 
eberlm@66277
   315
    where L: "(\<lambda>n. \<Prod>k\<le>n. f (k + M)) \<longlonglongrightarrow> L" "L \<noteq> 0" and nz: "\<And>n. n \<ge> M \<Longrightarrow> f n \<noteq> 0"
eberlm@66277
   316
    by (auto simp: convergent_prod_altdef)
eberlm@66277
   317
  define C where "C = (\<Prod>k<m. f (k + M))"
eberlm@66277
   318
  from nz have [simp]: "C \<noteq> 0" 
eberlm@66277
   319
    by (auto simp: C_def)
eberlm@66277
   320
eberlm@66277
   321
  from L(1) have "(\<lambda>n. \<Prod>k\<le>n+m. f (k + M)) \<longlonglongrightarrow> L" 
eberlm@66277
   322
    by (rule LIMSEQ_ignore_initial_segment)
eberlm@66277
   323
  also have "(\<lambda>n. \<Prod>k\<le>n+m. f (k + M)) = (\<lambda>n. C * (\<Prod>k\<le>n. f (k + M + m)))"
eberlm@66277
   324
  proof (rule ext, goal_cases)
eberlm@66277
   325
    case (1 n)
eberlm@66277
   326
    have "{..n+m} = {..<m} \<union> {m..n+m}" by auto
eberlm@66277
   327
    also have "(\<Prod>k\<in>\<dots>. f (k + M)) = C * (\<Prod>k=m..n+m. f (k + M))"
eberlm@66277
   328
      unfolding C_def by (rule prod.union_disjoint) auto
eberlm@66277
   329
    also have "(\<Prod>k=m..n+m. f (k + M)) = (\<Prod>k\<le>n. f (k + m + M))"
eberlm@66277
   330
      by (intro ext prod.reindex_bij_witness[of _ "\<lambda>k. k + m" "\<lambda>k. k - m"]) auto
eberlm@66277
   331
    finally show ?case by (simp add: add_ac)
eberlm@66277
   332
  qed
eberlm@66277
   333
  finally have "(\<lambda>n. C * (\<Prod>k\<le>n. f (k + M + m)) / C) \<longlonglongrightarrow> L / C"
eberlm@66277
   334
    by (intro tendsto_divide tendsto_const) auto
eberlm@66277
   335
  hence "(\<lambda>n. \<Prod>k\<le>n. f (k + M + m)) \<longlonglongrightarrow> L / C" by simp
eberlm@66277
   336
  moreover from \<open>L \<noteq> 0\<close> have "L / C \<noteq> 0" by simp
eberlm@66277
   337
  ultimately show ?thesis unfolding convergent_prod_def by blast
eberlm@66277
   338
qed
eberlm@66277
   339
eberlm@66277
   340
lemma abs_convergent_prod_ignore_initial_segment:
eberlm@66277
   341
  assumes "abs_convergent_prod f"
eberlm@66277
   342
  shows   "abs_convergent_prod (\<lambda>n. f (n + m))"
eberlm@66277
   343
  using assms unfolding abs_convergent_prod_def 
eberlm@66277
   344
  by (rule convergent_prod_ignore_initial_segment)
eberlm@66277
   345
eberlm@66277
   346
lemma summable_LIMSEQ': 
eberlm@66277
   347
  assumes "summable (f::nat\<Rightarrow>'a::{t2_space,comm_monoid_add})"
eberlm@66277
   348
  shows   "(\<lambda>n. \<Sum>i\<le>n. f i) \<longlonglongrightarrow> suminf f"
eberlm@66277
   349
  using assms sums_def_le by blast
eberlm@66277
   350
eberlm@66277
   351
lemma abs_convergent_prod_imp_convergent_prod:
eberlm@66277
   352
  fixes f :: "nat \<Rightarrow> 'a :: {real_normed_div_algebra,complete_space,comm_ring_1}"
eberlm@66277
   353
  assumes "abs_convergent_prod f"
eberlm@66277
   354
  shows   "convergent_prod f"
eberlm@66277
   355
proof -
eberlm@66277
   356
  from assms have "eventually (\<lambda>n. f n \<noteq> 0) sequentially"
eberlm@66277
   357
    by (rule abs_convergent_prod_imp_ev_nonzero)
eberlm@66277
   358
  then obtain N where N: "f n \<noteq> 0" if "n \<ge> N" for n 
eberlm@66277
   359
    by (auto simp: eventually_at_top_linorder)
eberlm@66277
   360
  let ?P = "\<lambda>n. \<Prod>i\<le>n. f (i + N)" and ?Q = "\<lambda>n. \<Prod>i\<le>n. 1 + norm (f (i + N) - 1)"
eberlm@66277
   361
eberlm@66277
   362
  have "Cauchy ?P"
eberlm@66277
   363
  proof (rule CauchyI', goal_cases)
eberlm@66277
   364
    case (1 \<epsilon>)
eberlm@66277
   365
    from assms have "abs_convergent_prod (\<lambda>n. f (n + N))"
eberlm@66277
   366
      by (rule abs_convergent_prod_ignore_initial_segment)
eberlm@66277
   367
    hence "Cauchy ?Q"
eberlm@66277
   368
      unfolding abs_convergent_prod_def
eberlm@66277
   369
      by (intro convergent_Cauchy convergent_prod_imp_convergent)
eberlm@66277
   370
    from CauchyD[OF this 1] obtain M where M: "norm (?Q m - ?Q n) < \<epsilon>" if "m \<ge> M" "n \<ge> M" for m n
eberlm@66277
   371
      by blast
eberlm@66277
   372
    show ?case
eberlm@66277
   373
    proof (rule exI[of _ M], safe, goal_cases)
eberlm@66277
   374
      case (1 m n)
eberlm@66277
   375
      have "dist (?P m) (?P n) = norm (?P n - ?P m)"
eberlm@66277
   376
        by (simp add: dist_norm norm_minus_commute)
eberlm@66277
   377
      also from 1 have "{..n} = {..m} \<union> {m<..n}" by auto
eberlm@66277
   378
      hence "norm (?P n - ?P m) = norm (?P m * (\<Prod>k\<in>{m<..n}. f (k + N)) - ?P m)"
eberlm@66277
   379
        by (subst prod.union_disjoint [symmetric]) (auto simp: algebra_simps)
eberlm@66277
   380
      also have "\<dots> = norm (?P m * ((\<Prod>k\<in>{m<..n}. f (k + N)) - 1))"
eberlm@66277
   381
        by (simp add: algebra_simps)
eberlm@66277
   382
      also have "\<dots> = (\<Prod>k\<le>m. norm (f (k + N))) * norm ((\<Prod>k\<in>{m<..n}. f (k + N)) - 1)"
eberlm@66277
   383
        by (simp add: norm_mult prod_norm)
eberlm@66277
   384
      also have "\<dots> \<le> ?Q m * ((\<Prod>k\<in>{m<..n}. 1 + norm (f (k + N) - 1)) - 1)"
eberlm@66277
   385
        using norm_prod_minus1_le_prod_minus1[of "\<lambda>k. f (k + N) - 1" "{m<..n}"]
eberlm@66277
   386
              norm_triangle_ineq[of 1 "f k - 1" for k]
eberlm@66277
   387
        by (intro mult_mono prod_mono ballI conjI norm_prod_minus1_le_prod_minus1 prod_nonneg) auto
eberlm@66277
   388
      also have "\<dots> = ?Q m * (\<Prod>k\<in>{m<..n}. 1 + norm (f (k + N) - 1)) - ?Q m"
eberlm@66277
   389
        by (simp add: algebra_simps)
eberlm@66277
   390
      also have "?Q m * (\<Prod>k\<in>{m<..n}. 1 + norm (f (k + N) - 1)) = 
eberlm@66277
   391
                   (\<Prod>k\<in>{..m}\<union>{m<..n}. 1 + norm (f (k + N) - 1))"
eberlm@66277
   392
        by (rule prod.union_disjoint [symmetric]) auto
eberlm@66277
   393
      also from 1 have "{..m}\<union>{m<..n} = {..n}" by auto
eberlm@66277
   394
      also have "?Q n - ?Q m \<le> norm (?Q n - ?Q m)" by simp
eberlm@66277
   395
      also from 1 have "\<dots> < \<epsilon>" by (intro M) auto
eberlm@66277
   396
      finally show ?case .
eberlm@66277
   397
    qed
eberlm@66277
   398
  qed
eberlm@66277
   399
  hence conv: "convergent ?P" by (rule Cauchy_convergent)
eberlm@66277
   400
  then obtain L where L: "?P \<longlonglongrightarrow> L"
eberlm@66277
   401
    by (auto simp: convergent_def)
eberlm@66277
   402
eberlm@66277
   403
  have "L \<noteq> 0"
eberlm@66277
   404
  proof
eberlm@66277
   405
    assume [simp]: "L = 0"
eberlm@66277
   406
    from tendsto_norm[OF L] have limit: "(\<lambda>n. \<Prod>k\<le>n. norm (f (k + N))) \<longlonglongrightarrow> 0" 
eberlm@66277
   407
      by (simp add: prod_norm)
eberlm@66277
   408
eberlm@66277
   409
    from assms have "(\<lambda>n. f (n + N)) \<longlonglongrightarrow> 1"
eberlm@66277
   410
      by (intro abs_convergent_prod_imp_LIMSEQ abs_convergent_prod_ignore_initial_segment)
eberlm@66277
   411
    hence "eventually (\<lambda>n. norm (f (n + N) - 1) < 1) sequentially"
eberlm@66277
   412
      by (auto simp: tendsto_iff dist_norm)
eberlm@66277
   413
    then obtain M0 where M0: "norm (f (n + N) - 1) < 1" if "n \<ge> M0" for n
eberlm@66277
   414
      by (auto simp: eventually_at_top_linorder)
eberlm@66277
   415
eberlm@66277
   416
    {
eberlm@66277
   417
      fix M assume M: "M \<ge> M0"
eberlm@66277
   418
      with M0 have M: "norm (f (n + N) - 1) < 1" if "n \<ge> M" for n using that by simp
eberlm@66277
   419
eberlm@66277
   420
      have "(\<lambda>n. \<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1)) \<longlonglongrightarrow> 0"
eberlm@66277
   421
      proof (rule tendsto_sandwich)
eberlm@66277
   422
        show "eventually (\<lambda>n. (\<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1)) \<ge> 0) sequentially"
eberlm@66277
   423
          using M by (intro always_eventually prod_nonneg allI ballI) (auto intro: less_imp_le)
eberlm@66277
   424
        have "norm (1::'a) - norm (f (i + M + N) - 1) \<le> norm (f (i + M + N))" for i
eberlm@66277
   425
          using norm_triangle_ineq3[of "f (i + M + N)" 1] by simp
eberlm@66277
   426
        thus "eventually (\<lambda>n. (\<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1)) \<le> (\<Prod>k\<le>n. norm (f (k+M+N)))) at_top"
eberlm@66277
   427
          using M by (intro always_eventually allI prod_mono ballI conjI) (auto intro: less_imp_le)
eberlm@66277
   428
        
eberlm@66277
   429
        define C where "C = (\<Prod>k<M. norm (f (k + N)))"
eberlm@66277
   430
        from N have [simp]: "C \<noteq> 0" by (auto simp: C_def)
eberlm@66277
   431
        from L have "(\<lambda>n. norm (\<Prod>k\<le>n+M. f (k + N))) \<longlonglongrightarrow> 0"
eberlm@66277
   432
          by (intro LIMSEQ_ignore_initial_segment) (simp add: tendsto_norm_zero_iff)
eberlm@66277
   433
        also have "(\<lambda>n. norm (\<Prod>k\<le>n+M. f (k + N))) = (\<lambda>n. C * (\<Prod>k\<le>n. norm (f (k + M + N))))"
eberlm@66277
   434
        proof (rule ext, goal_cases)
eberlm@66277
   435
          case (1 n)
eberlm@66277
   436
          have "{..n+M} = {..<M} \<union> {M..n+M}" by auto
eberlm@66277
   437
          also have "norm (\<Prod>k\<in>\<dots>. f (k + N)) = C * norm (\<Prod>k=M..n+M. f (k + N))"
eberlm@66277
   438
            unfolding C_def by (subst prod.union_disjoint) (auto simp: norm_mult prod_norm)
eberlm@66277
   439
          also have "(\<Prod>k=M..n+M. f (k + N)) = (\<Prod>k\<le>n. f (k + N + M))"
eberlm@66277
   440
            by (intro prod.reindex_bij_witness[of _ "\<lambda>i. i + M" "\<lambda>i. i - M"]) auto
eberlm@66277
   441
          finally show ?case by (simp add: add_ac prod_norm)
eberlm@66277
   442
        qed
eberlm@66277
   443
        finally have "(\<lambda>n. C * (\<Prod>k\<le>n. norm (f (k + M + N))) / C) \<longlonglongrightarrow> 0 / C"
eberlm@66277
   444
          by (intro tendsto_divide tendsto_const) auto
eberlm@66277
   445
        thus "(\<lambda>n. \<Prod>k\<le>n. norm (f (k + M + N))) \<longlonglongrightarrow> 0" by simp
eberlm@66277
   446
      qed simp_all
eberlm@66277
   447
eberlm@66277
   448
      have "1 - (\<Sum>i. norm (f (i + M + N) - 1)) \<le> 0"
eberlm@66277
   449
      proof (rule tendsto_le)
eberlm@66277
   450
        show "eventually (\<lambda>n. 1 - (\<Sum>k\<le>n. norm (f (k+M+N) - 1)) \<le> 
eberlm@66277
   451
                                (\<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1))) at_top"
eberlm@66277
   452
          using M by (intro always_eventually allI weierstrass_prod_ineq) (auto intro: less_imp_le)
eberlm@66277
   453
        show "(\<lambda>n. \<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1)) \<longlonglongrightarrow> 0" by fact
eberlm@66277
   454
        show "(\<lambda>n. 1 - (\<Sum>k\<le>n. norm (f (k + M + N) - 1)))
eberlm@66277
   455
                  \<longlonglongrightarrow> 1 - (\<Sum>i. norm (f (i + M + N) - 1))"
eberlm@66277
   456
          by (intro tendsto_intros summable_LIMSEQ' summable_ignore_initial_segment 
eberlm@66277
   457
                abs_convergent_prod_imp_summable assms)
eberlm@66277
   458
      qed simp_all
eberlm@66277
   459
      hence "(\<Sum>i. norm (f (i + M + N) - 1)) \<ge> 1" by simp
eberlm@66277
   460
      also have "\<dots> + (\<Sum>i<M. norm (f (i + N) - 1)) = (\<Sum>i. norm (f (i + N) - 1))"
eberlm@66277
   461
        by (intro suminf_split_initial_segment [symmetric] summable_ignore_initial_segment
eberlm@66277
   462
              abs_convergent_prod_imp_summable assms)
eberlm@66277
   463
      finally have "1 + (\<Sum>i<M. norm (f (i + N) - 1)) \<le> (\<Sum>i. norm (f (i + N) - 1))" by simp
eberlm@66277
   464
    } note * = this
eberlm@66277
   465
eberlm@66277
   466
    have "1 + (\<Sum>i. norm (f (i + N) - 1)) \<le> (\<Sum>i. norm (f (i + N) - 1))"
eberlm@66277
   467
    proof (rule tendsto_le)
eberlm@66277
   468
      show "(\<lambda>M. 1 + (\<Sum>i<M. norm (f (i + N) - 1))) \<longlonglongrightarrow> 1 + (\<Sum>i. norm (f (i + N) - 1))"
eberlm@66277
   469
        by (intro tendsto_intros summable_LIMSEQ summable_ignore_initial_segment 
eberlm@66277
   470
                abs_convergent_prod_imp_summable assms)
eberlm@66277
   471
      show "eventually (\<lambda>M. 1 + (\<Sum>i<M. norm (f (i + N) - 1)) \<le> (\<Sum>i. norm (f (i + N) - 1))) at_top"
eberlm@66277
   472
        using eventually_ge_at_top[of M0] by eventually_elim (use * in auto)
eberlm@66277
   473
    qed simp_all
eberlm@66277
   474
    thus False by simp
eberlm@66277
   475
  qed
eberlm@66277
   476
  with L show ?thesis by (auto simp: convergent_prod_def)
eberlm@66277
   477
qed
eberlm@66277
   478
eberlm@66277
   479
end