src/HOL/Analysis/Infinite_Products.thy
author paulson <lp15@cam.ac.uk>
Wed May 02 12:47:56 2018 +0100 (14 months ago)
changeset 68064 b249fab48c76
parent 66277 512b0dc09061
child 68071 c18af2b0f83e
permissions -rw-r--r--
type class generalisations; some work on infinite products
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(*File:      HOL/Analysis/Infinite_Product.thy
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  Author:    Manuel Eberl & LC Paulson
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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 gen_has_prod :: "[nat \<Rightarrow> 'a::{t2_space, comm_semiring_1}, nat, 'a] \<Rightarrow> bool" 
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  where "gen_has_prod f M p \<equiv> (\<lambda>n. \<Prod>i\<le>n. f (i+M)) \<longlonglongrightarrow> p \<and> p \<noteq> 0"
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text\<open>The nonzero and zero cases, as in \emph{Complex Analysis} by Joseph Bak and Donald J.Newman, page 241\<close>
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definition has_prod :: "(nat \<Rightarrow> 'a::{t2_space, comm_semiring_1}) \<Rightarrow> 'a \<Rightarrow> bool" (infixr "has'_prod" 80)
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  where "f has_prod p \<equiv> gen_has_prod f 0 p \<or> (\<exists>i q. p = 0 \<and> f i = 0 \<and> gen_has_prod f (Suc i) q)"
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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 \<equiv> \<exists>M p. gen_has_prod f M p"
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definition prodinf :: "(nat \<Rightarrow> 'a::{t2_space, comm_semiring_1}) \<Rightarrow> 'a"
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    (binder "\<Prod>" 10)
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  where "prodinf f = (THE p. f has_prod p)"
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lemmas prod_defs = gen_has_prod_def has_prod_def convergent_prod_def prodinf_def
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lemma has_prod_subst[trans]: "f = g \<Longrightarrow> g has_prod z \<Longrightarrow> f has_prod z"
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  by simp
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lemma has_prod_cong: "(\<And>n. f n = g n) \<Longrightarrow> f has_prod c \<longleftrightarrow> g has_prod c"
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  by presburger
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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: prod_defs)
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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: prod_defs)
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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 prod_defs
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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 :: {topological_semigroup_mult,t2_space,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 convergent_prod_iff_nz_lim:
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  fixes f :: "nat \<Rightarrow> 'a :: {topological_semigroup_mult,t2_space,idom}"
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  assumes "\<And>i. f i \<noteq> 0"
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  shows "convergent_prod f \<longleftrightarrow> (\<exists>L. (\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> L \<and> L \<noteq> 0)"
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    (is "?lhs \<longleftrightarrow> ?rhs")
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proof
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  assume ?lhs then show ?rhs
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    using assms convergentD convergent_prod_imp_convergent convergent_prod_to_zero_iff by blast
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next
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  assume ?rhs then show ?lhs
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    unfolding prod_defs
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    by (rule_tac x="0" in exI) (auto simp: )
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qed
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lemma convergent_prod_iff_convergent: 
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  fixes f :: "nat \<Rightarrow> 'a :: {topological_semigroup_mult,t2_space,idom}"
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  assumes "\<And>i. f i \<noteq> 0"
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  shows "convergent_prod f \<longleftrightarrow> convergent (\<lambda>n. \<Prod>i\<le>n. f i) \<and> lim (\<lambda>n. \<Prod>i\<le>n. f i) \<noteq> 0"
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  by (force simp add: convergent_prod_iff_nz_lim assms convergent_def limI)
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lemma abs_convergent_prod_altdef:
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  fixes f :: "nat \<Rightarrow> 'a :: {one,real_normed_vector}"
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  shows  "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
eberlm@66277
   245
  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))"
eberlm@66277
   246
    using assms L by (auto simp: fun_eq_iff atMost_Suc)
eberlm@66277
   247
  finally show ?thesis by (rule LIMSEQ_offset)
eberlm@66277
   248
qed
eberlm@66277
   249
eberlm@66277
   250
lemma abs_convergent_prod_imp_summable:
eberlm@66277
   251
  fixes f :: "nat \<Rightarrow> 'a :: real_normed_div_algebra"
eberlm@66277
   252
  assumes "abs_convergent_prod f"
eberlm@66277
   253
  shows "summable (\<lambda>i. norm (f i - 1))"
eberlm@66277
   254
proof -
eberlm@66277
   255
  from assms have "convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))" 
eberlm@66277
   256
    unfolding abs_convergent_prod_def by (rule convergent_prod_imp_convergent)
eberlm@66277
   257
  then obtain L where L: "(\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1)) \<longlonglongrightarrow> L"
eberlm@66277
   258
    unfolding convergent_def by blast
eberlm@66277
   259
  have "convergent (\<lambda>n. \<Sum>i\<le>n. norm (f i - 1))"
eberlm@66277
   260
  proof (rule Bseq_monoseq_convergent)
eberlm@66277
   261
    have "eventually (\<lambda>n. (\<Prod>i\<le>n. 1 + norm (f i - 1)) < L + 1) sequentially"
eberlm@66277
   262
      using L(1) by (rule order_tendstoD) simp_all
eberlm@66277
   263
    hence "\<forall>\<^sub>F x in sequentially. norm (\<Sum>i\<le>x. norm (f i - 1)) \<le> L + 1"
eberlm@66277
   264
    proof eventually_elim
eberlm@66277
   265
      case (elim n)
eberlm@66277
   266
      have "norm (\<Sum>i\<le>n. norm (f i - 1)) = (\<Sum>i\<le>n. norm (f i - 1))"
eberlm@66277
   267
        unfolding real_norm_def by (intro abs_of_nonneg sum_nonneg) simp_all
eberlm@66277
   268
      also have "\<dots> \<le> (\<Prod>i\<le>n. 1 + norm (f i - 1))" by (rule sum_le_prod) auto
eberlm@66277
   269
      also have "\<dots> < L + 1" by (rule elim)
eberlm@66277
   270
      finally show ?case by simp
eberlm@66277
   271
    qed
eberlm@66277
   272
    thus "Bseq (\<lambda>n. \<Sum>i\<le>n. norm (f i - 1))" by (rule BfunI)
eberlm@66277
   273
  next
eberlm@66277
   274
    show "monoseq (\<lambda>n. \<Sum>i\<le>n. norm (f i - 1))"
eberlm@66277
   275
      by (rule mono_SucI1) auto
eberlm@66277
   276
  qed
eberlm@66277
   277
  thus "summable (\<lambda>i. norm (f i - 1))" by (simp add: summable_iff_convergent')
eberlm@66277
   278
qed
eberlm@66277
   279
eberlm@66277
   280
lemma summable_imp_abs_convergent_prod:
eberlm@66277
   281
  fixes f :: "nat \<Rightarrow> 'a :: real_normed_div_algebra"
eberlm@66277
   282
  assumes "summable (\<lambda>i. norm (f i - 1))"
eberlm@66277
   283
  shows   "abs_convergent_prod f"
eberlm@66277
   284
proof (intro abs_convergent_prodI Bseq_monoseq_convergent)
eberlm@66277
   285
  show "monoseq (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
eberlm@66277
   286
    by (intro mono_SucI1) 
eberlm@66277
   287
       (auto simp: atMost_Suc algebra_simps intro!: mult_nonneg_nonneg prod_nonneg)
eberlm@66277
   288
next
eberlm@66277
   289
  show "Bseq (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
eberlm@66277
   290
  proof (rule Bseq_eventually_mono)
eberlm@66277
   291
    show "eventually (\<lambda>n. norm (\<Prod>i\<le>n. 1 + norm (f i - 1)) \<le> 
eberlm@66277
   292
            norm (exp (\<Sum>i\<le>n. norm (f i - 1)))) sequentially"
eberlm@66277
   293
      by (intro always_eventually allI) (auto simp: abs_prod exp_sum intro!: prod_mono)
eberlm@66277
   294
  next
eberlm@66277
   295
    from assms have "(\<lambda>n. \<Sum>i\<le>n. norm (f i - 1)) \<longlonglongrightarrow> (\<Sum>i. norm (f i - 1))"
eberlm@66277
   296
      using sums_def_le by blast
eberlm@66277
   297
    hence "(\<lambda>n. exp (\<Sum>i\<le>n. norm (f i - 1))) \<longlonglongrightarrow> exp (\<Sum>i. norm (f i - 1))"
eberlm@66277
   298
      by (rule tendsto_exp)
eberlm@66277
   299
    hence "convergent (\<lambda>n. exp (\<Sum>i\<le>n. norm (f i - 1)))"
eberlm@66277
   300
      by (rule convergentI)
eberlm@66277
   301
    thus "Bseq (\<lambda>n. exp (\<Sum>i\<le>n. norm (f i - 1)))"
eberlm@66277
   302
      by (rule convergent_imp_Bseq)
eberlm@66277
   303
  qed
eberlm@66277
   304
qed
eberlm@66277
   305
eberlm@66277
   306
lemma abs_convergent_prod_conv_summable:
eberlm@66277
   307
  fixes f :: "nat \<Rightarrow> 'a :: real_normed_div_algebra"
eberlm@66277
   308
  shows "abs_convergent_prod f \<longleftrightarrow> summable (\<lambda>i. norm (f i - 1))"
eberlm@66277
   309
  by (blast intro: abs_convergent_prod_imp_summable summable_imp_abs_convergent_prod)
eberlm@66277
   310
eberlm@66277
   311
lemma abs_convergent_prod_imp_LIMSEQ:
eberlm@66277
   312
  fixes f :: "nat \<Rightarrow> 'a :: {comm_ring_1,real_normed_div_algebra}"
eberlm@66277
   313
  assumes "abs_convergent_prod f"
eberlm@66277
   314
  shows   "f \<longlonglongrightarrow> 1"
eberlm@66277
   315
proof -
eberlm@66277
   316
  from assms have "summable (\<lambda>n. norm (f n - 1))"
eberlm@66277
   317
    by (rule abs_convergent_prod_imp_summable)
eberlm@66277
   318
  from summable_LIMSEQ_zero[OF this] have "(\<lambda>n. f n - 1) \<longlonglongrightarrow> 0"
eberlm@66277
   319
    by (simp add: tendsto_norm_zero_iff)
eberlm@66277
   320
  from tendsto_add[OF this tendsto_const[of 1]] show ?thesis by simp
eberlm@66277
   321
qed
eberlm@66277
   322
eberlm@66277
   323
lemma abs_convergent_prod_imp_ev_nonzero:
eberlm@66277
   324
  fixes f :: "nat \<Rightarrow> 'a :: {comm_ring_1,real_normed_div_algebra}"
eberlm@66277
   325
  assumes "abs_convergent_prod f"
eberlm@66277
   326
  shows   "eventually (\<lambda>n. f n \<noteq> 0) sequentially"
eberlm@66277
   327
proof -
eberlm@66277
   328
  from assms have "f \<longlonglongrightarrow> 1" 
eberlm@66277
   329
    by (rule abs_convergent_prod_imp_LIMSEQ)
eberlm@66277
   330
  hence "eventually (\<lambda>n. dist (f n) 1 < 1) at_top"
eberlm@66277
   331
    by (auto simp: tendsto_iff)
eberlm@66277
   332
  thus ?thesis by eventually_elim auto
eberlm@66277
   333
qed
eberlm@66277
   334
eberlm@66277
   335
lemma convergent_prod_offset:
eberlm@66277
   336
  assumes "convergent_prod (\<lambda>n. f (n + m))"  
eberlm@66277
   337
  shows   "convergent_prod f"
eberlm@66277
   338
proof -
eberlm@66277
   339
  from assms obtain M L where "(\<lambda>n. \<Prod>k\<le>n. f (k + (M + m))) \<longlonglongrightarrow> L" "L \<noteq> 0"
lp15@68064
   340
    by (auto simp: prod_defs add.assoc)
lp15@68064
   341
  thus "convergent_prod f" 
lp15@68064
   342
    unfolding prod_defs by blast
eberlm@66277
   343
qed
eberlm@66277
   344
eberlm@66277
   345
lemma abs_convergent_prod_offset:
eberlm@66277
   346
  assumes "abs_convergent_prod (\<lambda>n. f (n + m))"  
eberlm@66277
   347
  shows   "abs_convergent_prod f"
eberlm@66277
   348
  using assms unfolding abs_convergent_prod_def by (rule convergent_prod_offset)
eberlm@66277
   349
eberlm@66277
   350
lemma convergent_prod_ignore_initial_segment:
eberlm@66277
   351
  fixes f :: "nat \<Rightarrow> 'a :: {real_normed_field}"
eberlm@66277
   352
  assumes "convergent_prod f"
eberlm@66277
   353
  shows   "convergent_prod (\<lambda>n. f (n + m))"
eberlm@66277
   354
proof -
eberlm@66277
   355
  from assms obtain M L 
eberlm@66277
   356
    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
   357
    by (auto simp: convergent_prod_altdef)
eberlm@66277
   358
  define C where "C = (\<Prod>k<m. f (k + M))"
eberlm@66277
   359
  from nz have [simp]: "C \<noteq> 0" 
eberlm@66277
   360
    by (auto simp: C_def)
eberlm@66277
   361
eberlm@66277
   362
  from L(1) have "(\<lambda>n. \<Prod>k\<le>n+m. f (k + M)) \<longlonglongrightarrow> L" 
eberlm@66277
   363
    by (rule LIMSEQ_ignore_initial_segment)
eberlm@66277
   364
  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
   365
  proof (rule ext, goal_cases)
eberlm@66277
   366
    case (1 n)
eberlm@66277
   367
    have "{..n+m} = {..<m} \<union> {m..n+m}" by auto
eberlm@66277
   368
    also have "(\<Prod>k\<in>\<dots>. f (k + M)) = C * (\<Prod>k=m..n+m. f (k + M))"
eberlm@66277
   369
      unfolding C_def by (rule prod.union_disjoint) auto
eberlm@66277
   370
    also have "(\<Prod>k=m..n+m. f (k + M)) = (\<Prod>k\<le>n. f (k + m + M))"
eberlm@66277
   371
      by (intro ext prod.reindex_bij_witness[of _ "\<lambda>k. k + m" "\<lambda>k. k - m"]) auto
eberlm@66277
   372
    finally show ?case by (simp add: add_ac)
eberlm@66277
   373
  qed
eberlm@66277
   374
  finally have "(\<lambda>n. C * (\<Prod>k\<le>n. f (k + M + m)) / C) \<longlonglongrightarrow> L / C"
eberlm@66277
   375
    by (intro tendsto_divide tendsto_const) auto
eberlm@66277
   376
  hence "(\<lambda>n. \<Prod>k\<le>n. f (k + M + m)) \<longlonglongrightarrow> L / C" by simp
eberlm@66277
   377
  moreover from \<open>L \<noteq> 0\<close> have "L / C \<noteq> 0" by simp
lp15@68064
   378
  ultimately show ?thesis 
lp15@68064
   379
    unfolding prod_defs by blast
eberlm@66277
   380
qed
eberlm@66277
   381
eberlm@66277
   382
lemma abs_convergent_prod_ignore_initial_segment:
eberlm@66277
   383
  assumes "abs_convergent_prod f"
eberlm@66277
   384
  shows   "abs_convergent_prod (\<lambda>n. f (n + m))"
eberlm@66277
   385
  using assms unfolding abs_convergent_prod_def 
eberlm@66277
   386
  by (rule convergent_prod_ignore_initial_segment)
eberlm@66277
   387
eberlm@66277
   388
lemma abs_convergent_prod_imp_convergent_prod:
eberlm@66277
   389
  fixes f :: "nat \<Rightarrow> 'a :: {real_normed_div_algebra,complete_space,comm_ring_1}"
eberlm@66277
   390
  assumes "abs_convergent_prod f"
eberlm@66277
   391
  shows   "convergent_prod f"
eberlm@66277
   392
proof -
eberlm@66277
   393
  from assms have "eventually (\<lambda>n. f n \<noteq> 0) sequentially"
eberlm@66277
   394
    by (rule abs_convergent_prod_imp_ev_nonzero)
eberlm@66277
   395
  then obtain N where N: "f n \<noteq> 0" if "n \<ge> N" for n 
eberlm@66277
   396
    by (auto simp: eventually_at_top_linorder)
eberlm@66277
   397
  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
   398
eberlm@66277
   399
  have "Cauchy ?P"
eberlm@66277
   400
  proof (rule CauchyI', goal_cases)
eberlm@66277
   401
    case (1 \<epsilon>)
eberlm@66277
   402
    from assms have "abs_convergent_prod (\<lambda>n. f (n + N))"
eberlm@66277
   403
      by (rule abs_convergent_prod_ignore_initial_segment)
eberlm@66277
   404
    hence "Cauchy ?Q"
eberlm@66277
   405
      unfolding abs_convergent_prod_def
eberlm@66277
   406
      by (intro convergent_Cauchy convergent_prod_imp_convergent)
eberlm@66277
   407
    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
   408
      by blast
eberlm@66277
   409
    show ?case
eberlm@66277
   410
    proof (rule exI[of _ M], safe, goal_cases)
eberlm@66277
   411
      case (1 m n)
eberlm@66277
   412
      have "dist (?P m) (?P n) = norm (?P n - ?P m)"
eberlm@66277
   413
        by (simp add: dist_norm norm_minus_commute)
eberlm@66277
   414
      also from 1 have "{..n} = {..m} \<union> {m<..n}" by auto
eberlm@66277
   415
      hence "norm (?P n - ?P m) = norm (?P m * (\<Prod>k\<in>{m<..n}. f (k + N)) - ?P m)"
eberlm@66277
   416
        by (subst prod.union_disjoint [symmetric]) (auto simp: algebra_simps)
eberlm@66277
   417
      also have "\<dots> = norm (?P m * ((\<Prod>k\<in>{m<..n}. f (k + N)) - 1))"
eberlm@66277
   418
        by (simp add: algebra_simps)
eberlm@66277
   419
      also have "\<dots> = (\<Prod>k\<le>m. norm (f (k + N))) * norm ((\<Prod>k\<in>{m<..n}. f (k + N)) - 1)"
eberlm@66277
   420
        by (simp add: norm_mult prod_norm)
eberlm@66277
   421
      also have "\<dots> \<le> ?Q m * ((\<Prod>k\<in>{m<..n}. 1 + norm (f (k + N) - 1)) - 1)"
eberlm@66277
   422
        using norm_prod_minus1_le_prod_minus1[of "\<lambda>k. f (k + N) - 1" "{m<..n}"]
eberlm@66277
   423
              norm_triangle_ineq[of 1 "f k - 1" for k]
eberlm@66277
   424
        by (intro mult_mono prod_mono ballI conjI norm_prod_minus1_le_prod_minus1 prod_nonneg) auto
eberlm@66277
   425
      also have "\<dots> = ?Q m * (\<Prod>k\<in>{m<..n}. 1 + norm (f (k + N) - 1)) - ?Q m"
eberlm@66277
   426
        by (simp add: algebra_simps)
eberlm@66277
   427
      also have "?Q m * (\<Prod>k\<in>{m<..n}. 1 + norm (f (k + N) - 1)) = 
eberlm@66277
   428
                   (\<Prod>k\<in>{..m}\<union>{m<..n}. 1 + norm (f (k + N) - 1))"
eberlm@66277
   429
        by (rule prod.union_disjoint [symmetric]) auto
eberlm@66277
   430
      also from 1 have "{..m}\<union>{m<..n} = {..n}" by auto
eberlm@66277
   431
      also have "?Q n - ?Q m \<le> norm (?Q n - ?Q m)" by simp
eberlm@66277
   432
      also from 1 have "\<dots> < \<epsilon>" by (intro M) auto
eberlm@66277
   433
      finally show ?case .
eberlm@66277
   434
    qed
eberlm@66277
   435
  qed
eberlm@66277
   436
  hence conv: "convergent ?P" by (rule Cauchy_convergent)
eberlm@66277
   437
  then obtain L where L: "?P \<longlonglongrightarrow> L"
eberlm@66277
   438
    by (auto simp: convergent_def)
eberlm@66277
   439
eberlm@66277
   440
  have "L \<noteq> 0"
eberlm@66277
   441
  proof
eberlm@66277
   442
    assume [simp]: "L = 0"
eberlm@66277
   443
    from tendsto_norm[OF L] have limit: "(\<lambda>n. \<Prod>k\<le>n. norm (f (k + N))) \<longlonglongrightarrow> 0" 
eberlm@66277
   444
      by (simp add: prod_norm)
eberlm@66277
   445
eberlm@66277
   446
    from assms have "(\<lambda>n. f (n + N)) \<longlonglongrightarrow> 1"
eberlm@66277
   447
      by (intro abs_convergent_prod_imp_LIMSEQ abs_convergent_prod_ignore_initial_segment)
eberlm@66277
   448
    hence "eventually (\<lambda>n. norm (f (n + N) - 1) < 1) sequentially"
eberlm@66277
   449
      by (auto simp: tendsto_iff dist_norm)
eberlm@66277
   450
    then obtain M0 where M0: "norm (f (n + N) - 1) < 1" if "n \<ge> M0" for n
eberlm@66277
   451
      by (auto simp: eventually_at_top_linorder)
eberlm@66277
   452
eberlm@66277
   453
    {
eberlm@66277
   454
      fix M assume M: "M \<ge> M0"
eberlm@66277
   455
      with M0 have M: "norm (f (n + N) - 1) < 1" if "n \<ge> M" for n using that by simp
eberlm@66277
   456
eberlm@66277
   457
      have "(\<lambda>n. \<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1)) \<longlonglongrightarrow> 0"
eberlm@66277
   458
      proof (rule tendsto_sandwich)
eberlm@66277
   459
        show "eventually (\<lambda>n. (\<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1)) \<ge> 0) sequentially"
eberlm@66277
   460
          using M by (intro always_eventually prod_nonneg allI ballI) (auto intro: less_imp_le)
eberlm@66277
   461
        have "norm (1::'a) - norm (f (i + M + N) - 1) \<le> norm (f (i + M + N))" for i
eberlm@66277
   462
          using norm_triangle_ineq3[of "f (i + M + N)" 1] by simp
eberlm@66277
   463
        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
   464
          using M by (intro always_eventually allI prod_mono ballI conjI) (auto intro: less_imp_le)
eberlm@66277
   465
        
eberlm@66277
   466
        define C where "C = (\<Prod>k<M. norm (f (k + N)))"
eberlm@66277
   467
        from N have [simp]: "C \<noteq> 0" by (auto simp: C_def)
eberlm@66277
   468
        from L have "(\<lambda>n. norm (\<Prod>k\<le>n+M. f (k + N))) \<longlonglongrightarrow> 0"
eberlm@66277
   469
          by (intro LIMSEQ_ignore_initial_segment) (simp add: tendsto_norm_zero_iff)
eberlm@66277
   470
        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
   471
        proof (rule ext, goal_cases)
eberlm@66277
   472
          case (1 n)
eberlm@66277
   473
          have "{..n+M} = {..<M} \<union> {M..n+M}" by auto
eberlm@66277
   474
          also have "norm (\<Prod>k\<in>\<dots>. f (k + N)) = C * norm (\<Prod>k=M..n+M. f (k + N))"
eberlm@66277
   475
            unfolding C_def by (subst prod.union_disjoint) (auto simp: norm_mult prod_norm)
eberlm@66277
   476
          also have "(\<Prod>k=M..n+M. f (k + N)) = (\<Prod>k\<le>n. f (k + N + M))"
eberlm@66277
   477
            by (intro prod.reindex_bij_witness[of _ "\<lambda>i. i + M" "\<lambda>i. i - M"]) auto
eberlm@66277
   478
          finally show ?case by (simp add: add_ac prod_norm)
eberlm@66277
   479
        qed
eberlm@66277
   480
        finally have "(\<lambda>n. C * (\<Prod>k\<le>n. norm (f (k + M + N))) / C) \<longlonglongrightarrow> 0 / C"
eberlm@66277
   481
          by (intro tendsto_divide tendsto_const) auto
eberlm@66277
   482
        thus "(\<lambda>n. \<Prod>k\<le>n. norm (f (k + M + N))) \<longlonglongrightarrow> 0" by simp
eberlm@66277
   483
      qed simp_all
eberlm@66277
   484
eberlm@66277
   485
      have "1 - (\<Sum>i. norm (f (i + M + N) - 1)) \<le> 0"
eberlm@66277
   486
      proof (rule tendsto_le)
eberlm@66277
   487
        show "eventually (\<lambda>n. 1 - (\<Sum>k\<le>n. norm (f (k+M+N) - 1)) \<le> 
eberlm@66277
   488
                                (\<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1))) at_top"
eberlm@66277
   489
          using M by (intro always_eventually allI weierstrass_prod_ineq) (auto intro: less_imp_le)
eberlm@66277
   490
        show "(\<lambda>n. \<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1)) \<longlonglongrightarrow> 0" by fact
eberlm@66277
   491
        show "(\<lambda>n. 1 - (\<Sum>k\<le>n. norm (f (k + M + N) - 1)))
eberlm@66277
   492
                  \<longlonglongrightarrow> 1 - (\<Sum>i. norm (f (i + M + N) - 1))"
eberlm@66277
   493
          by (intro tendsto_intros summable_LIMSEQ' summable_ignore_initial_segment 
eberlm@66277
   494
                abs_convergent_prod_imp_summable assms)
eberlm@66277
   495
      qed simp_all
eberlm@66277
   496
      hence "(\<Sum>i. norm (f (i + M + N) - 1)) \<ge> 1" by simp
eberlm@66277
   497
      also have "\<dots> + (\<Sum>i<M. norm (f (i + N) - 1)) = (\<Sum>i. norm (f (i + N) - 1))"
eberlm@66277
   498
        by (intro suminf_split_initial_segment [symmetric] summable_ignore_initial_segment
eberlm@66277
   499
              abs_convergent_prod_imp_summable assms)
eberlm@66277
   500
      finally have "1 + (\<Sum>i<M. norm (f (i + N) - 1)) \<le> (\<Sum>i. norm (f (i + N) - 1))" by simp
eberlm@66277
   501
    } note * = this
eberlm@66277
   502
eberlm@66277
   503
    have "1 + (\<Sum>i. norm (f (i + N) - 1)) \<le> (\<Sum>i. norm (f (i + N) - 1))"
eberlm@66277
   504
    proof (rule tendsto_le)
eberlm@66277
   505
      show "(\<lambda>M. 1 + (\<Sum>i<M. norm (f (i + N) - 1))) \<longlonglongrightarrow> 1 + (\<Sum>i. norm (f (i + N) - 1))"
eberlm@66277
   506
        by (intro tendsto_intros summable_LIMSEQ summable_ignore_initial_segment 
eberlm@66277
   507
                abs_convergent_prod_imp_summable assms)
eberlm@66277
   508
      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
   509
        using eventually_ge_at_top[of M0] by eventually_elim (use * in auto)
eberlm@66277
   510
    qed simp_all
eberlm@66277
   511
    thus False by simp
eberlm@66277
   512
  qed
lp15@68064
   513
  with L show ?thesis by (auto simp: prod_defs)
lp15@68064
   514
qed
lp15@68064
   515
lp15@68064
   516
lemma convergent_prod_offset_0:
lp15@68064
   517
  fixes f :: "nat \<Rightarrow> 'a :: {idom,topological_semigroup_mult,t2_space}"
lp15@68064
   518
  assumes "convergent_prod f" "\<And>i. f i \<noteq> 0"
lp15@68064
   519
  shows "\<exists>p. gen_has_prod f 0 p"
lp15@68064
   520
  using assms
lp15@68064
   521
  unfolding convergent_prod_def
lp15@68064
   522
proof (clarsimp simp: prod_defs)
lp15@68064
   523
  fix M p
lp15@68064
   524
  assume "(\<lambda>n. \<Prod>i\<le>n. f (i + M)) \<longlonglongrightarrow> p" "p \<noteq> 0"
lp15@68064
   525
  then have "(\<lambda>n. prod f {..<M} * (\<Prod>i\<le>n. f (i + M))) \<longlonglongrightarrow> prod f {..<M} * p"
lp15@68064
   526
    by (metis tendsto_mult_left)
lp15@68064
   527
  moreover have "prod f {..<M} * (\<Prod>i\<le>n. f (i + M)) = prod f {..n+M}" for n
lp15@68064
   528
  proof -
lp15@68064
   529
    have "{..n+M} = {..<M} \<union> {M..n+M}"
lp15@68064
   530
      by auto
lp15@68064
   531
    then have "prod f {..n+M} = prod f {..<M} * prod f {M..n+M}"
lp15@68064
   532
      by simp (subst prod.union_disjoint; force)
lp15@68064
   533
    also have "... = prod f {..<M} * (\<Prod>i\<le>n. f (i + M))"
lp15@68064
   534
      by (metis (mono_tags, lifting) add.left_neutral atMost_atLeast0 prod_shift_bounds_cl_nat_ivl)
lp15@68064
   535
    finally show ?thesis by metis
lp15@68064
   536
  qed
lp15@68064
   537
  ultimately have "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> prod f {..<M} * p"
lp15@68064
   538
    by (auto intro: LIMSEQ_offset [where k=M])
lp15@68064
   539
  then show "\<exists>p. (\<lambda>n. prod f {..n}) \<longlonglongrightarrow> p \<and> p \<noteq> 0"
lp15@68064
   540
    using \<open>p \<noteq> 0\<close> assms
lp15@68064
   541
    by (rule_tac x="prod f {..<M} * p" in exI) auto
lp15@68064
   542
qed
lp15@68064
   543
lp15@68064
   544
lemma prodinf_eq_lim:
lp15@68064
   545
  fixes f :: "nat \<Rightarrow> 'a :: {idom,topological_semigroup_mult,t2_space}"
lp15@68064
   546
  assumes "convergent_prod f" "\<And>i. f i \<noteq> 0"
lp15@68064
   547
  shows "prodinf f = lim (\<lambda>n. \<Prod>i\<le>n. f i)"
lp15@68064
   548
  using assms convergent_prod_offset_0 [OF assms]
lp15@68064
   549
  by (simp add: prod_defs lim_def) (metis (no_types) assms(1) convergent_prod_to_zero_iff)
lp15@68064
   550
lp15@68064
   551
lemma has_prod_one[simp, intro]: "(\<lambda>n. 1) has_prod 1"
lp15@68064
   552
  unfolding prod_defs by auto
lp15@68064
   553
lp15@68064
   554
lemma convergent_prod_one[simp, intro]: "convergent_prod (\<lambda>n. 1)"
lp15@68064
   555
  unfolding prod_defs by auto
lp15@68064
   556
lp15@68064
   557
lemma prodinf_cong: "(\<And>n. f n = g n) \<Longrightarrow> prodinf f = prodinf g"
lp15@68064
   558
  by presburger
lp15@68064
   559
lp15@68064
   560
lemma convergent_prod_cong:
lp15@68064
   561
  fixes f g :: "nat \<Rightarrow> 'a::{field,topological_semigroup_mult,t2_space}"
lp15@68064
   562
  assumes ev: "eventually (\<lambda>x. f x = g x) sequentially" and f: "\<And>i. f i \<noteq> 0" and g: "\<And>i. g i \<noteq> 0"
lp15@68064
   563
  shows "convergent_prod f = convergent_prod g"
lp15@68064
   564
proof -
lp15@68064
   565
  from assms obtain N where N: "\<forall>n\<ge>N. f n = g n"
lp15@68064
   566
    by (auto simp: eventually_at_top_linorder)
lp15@68064
   567
  define C where "C = (\<Prod>k<N. f k / g k)"
lp15@68064
   568
  with g have "C \<noteq> 0"
lp15@68064
   569
    by (simp add: f)
lp15@68064
   570
  have *: "eventually (\<lambda>n. prod f {..n} = C * prod g {..n}) sequentially"
lp15@68064
   571
    using eventually_ge_at_top[of N]
lp15@68064
   572
  proof eventually_elim
lp15@68064
   573
    case (elim n)
lp15@68064
   574
    then have "{..n} = {..<N} \<union> {N..n}"
lp15@68064
   575
      by auto
lp15@68064
   576
    also have "prod f ... = prod f {..<N} * prod f {N..n}"
lp15@68064
   577
      by (intro prod.union_disjoint) auto
lp15@68064
   578
    also from N have "prod f {N..n} = prod g {N..n}"
lp15@68064
   579
      by (intro prod.cong) simp_all
lp15@68064
   580
    also have "prod f {..<N} * prod g {N..n} = C * (prod g {..<N} * prod g {N..n})"
lp15@68064
   581
      unfolding C_def by (simp add: g prod_dividef)
lp15@68064
   582
    also have "prod g {..<N} * prod g {N..n} = prod g ({..<N} \<union> {N..n})"
lp15@68064
   583
      by (intro prod.union_disjoint [symmetric]) auto
lp15@68064
   584
    also from elim have "{..<N} \<union> {N..n} = {..n}"
lp15@68064
   585
      by auto                                                                    
lp15@68064
   586
    finally show "prod f {..n} = C * prod g {..n}" .
lp15@68064
   587
  qed
lp15@68064
   588
  then have cong: "convergent (\<lambda>n. prod f {..n}) = convergent (\<lambda>n. C * prod g {..n})"
lp15@68064
   589
    by (rule convergent_cong)
lp15@68064
   590
  show ?thesis
lp15@68064
   591
  proof
lp15@68064
   592
    assume cf: "convergent_prod f"
lp15@68064
   593
    then have "\<not> (\<lambda>n. prod g {..n}) \<longlonglongrightarrow> 0"
lp15@68064
   594
      using tendsto_mult_left * convergent_prod_to_zero_iff f filterlim_cong by fastforce
lp15@68064
   595
    then show "convergent_prod g"
lp15@68064
   596
      by (metis convergent_mult_const_iff \<open>C \<noteq> 0\<close> cong cf convergent_LIMSEQ_iff convergent_prod_iff_convergent convergent_prod_imp_convergent g)
lp15@68064
   597
  next
lp15@68064
   598
    assume cg: "convergent_prod g"
lp15@68064
   599
    have "\<exists>a. C * a \<noteq> 0 \<and> (\<lambda>n. prod g {..n}) \<longlonglongrightarrow> a"
lp15@68064
   600
      by (metis (no_types) \<open>C \<noteq> 0\<close> cg convergent_prod_iff_nz_lim divide_eq_0_iff g nonzero_mult_div_cancel_right)
lp15@68064
   601
    then show "convergent_prod f"
lp15@68064
   602
      using "*" tendsto_mult_left filterlim_cong
lp15@68064
   603
      by (fastforce simp add: convergent_prod_iff_nz_lim f)
lp15@68064
   604
  qed
eberlm@66277
   605
qed
eberlm@66277
   606
eberlm@66277
   607
end