src/HOL/Analysis/Infinite_Products.thy
author paulson <lp15@cam.ac.uk>
Wed Jul 11 23:24:25 2018 +0100 (12 months ago)
changeset 68616 cedf3480fdad
parent 68586 006da53a8ac1
child 68651 16d98ef49a2c
permissions -rw-r--r--
de-applying (mostly Quotient)
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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 Topology_Euclidean_Space Complex_Transcendental
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begin
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subsection\<open>Preliminaries\<close>
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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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subsection\<open>Definitions and basic properties\<close>
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definition raw_has_prod :: "[nat \<Rightarrow> 'a::{t2_space, comm_semiring_1}, nat, 'a] \<Rightarrow> bool" 
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  where "raw_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> raw_has_prod f 0 p \<or> (\<exists>i q. p = 0 \<and> f i = 0 \<and> raw_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. raw_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 = raw_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 raw_has_prod_nonzero [simp]: "\<not> raw_has_prod f M 0"
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  by (simp add: raw_has_prod_def)
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lemma raw_has_prod_eq_0:
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  fixes f :: "nat \<Rightarrow> 'a::{semidom,t2_space}"
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  assumes p: "raw_has_prod f m p" and i: "f i = 0" "i \<ge> m"
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  shows "p = 0"
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proof -
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  have eq0: "(\<Prod>k\<le>n. f (k+m)) = 0" if "i - m \<le> n" for n
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  proof -
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    have "\<exists>k\<le>n. f (k + m) = 0"
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      using i that by auto
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    then show ?thesis
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      by auto
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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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    by (rule LIMSEQ_offset [where k = "i-m"]) (simp add: eq0)
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    with p show ?thesis
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      unfolding raw_has_prod_def
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    using LIMSEQ_unique by blast
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qed
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lemma raw_has_prod_Suc: 
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  "raw_has_prod f (Suc M) a \<longleftrightarrow> raw_has_prod (\<lambda>n. f (Suc n)) M a"
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  unfolding raw_has_prod_def by auto
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lemma has_prod_0_iff: "f has_prod 0 \<longleftrightarrow> (\<exists>i. f i = 0 \<and> (\<exists>p. raw_has_prod f (Suc i) p))"
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  by (simp add: has_prod_def)
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lemma has_prod_unique2: 
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  fixes f :: "nat \<Rightarrow> 'a::{semidom,t2_space}"
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  assumes "f has_prod a" "f has_prod b" shows "a = b"
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  using assms
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  by (auto simp: has_prod_def raw_has_prod_eq_0) (meson raw_has_prod_def sequentially_bot tendsto_unique)
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lemma has_prod_unique:
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  fixes f :: "nat \<Rightarrow> 'a :: {semidom,t2_space}"
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  shows "f has_prod s \<Longrightarrow> s = prodinf f"
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  by (simp add: has_prod_unique2 prodinf_def the_equality)
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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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subsection\<open>Absolutely convergent products\<close>
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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
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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: convergent_prod_iff_nz_lim assms convergent_def limI)
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lemma bounded_imp_convergent_prod:
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  fixes a :: "nat \<Rightarrow> real"
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  assumes 1: "\<And>n. a n \<ge> 1" and bounded: "\<And>n. (\<Prod>i\<le>n. a i) \<le> B"
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  shows "convergent_prod a"
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proof -
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  have "bdd_above (range(\<lambda>n. \<Prod>i\<le>n. a i))"
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    by (meson bdd_aboveI2 bounded)
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  moreover have "incseq (\<lambda>n. \<Prod>i\<le>n. a i)"
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    unfolding mono_def by (metis 1 prod_mono2 atMost_subset_iff dual_order.trans finite_atMost zero_le_one)
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  ultimately obtain p where p: "(\<lambda>n. \<Prod>i\<le>n. a i) \<longlonglongrightarrow> p"
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    using LIMSEQ_incseq_SUP by blast
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  then have "p \<noteq> 0"
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    by (metis "1" not_one_le_zero prod_ge_1 LIMSEQ_le_const)
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  with 1 p show ?thesis
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    by (metis convergent_prod_iff_nz_lim not_one_le_zero)
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qed
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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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   261
proof (induction A rule: infinite_finite_induct)
eberlm@66277
   262
  case (insert x A)
eberlm@66277
   263
  from insert.hyps and insert.prems 
eberlm@66277
   264
    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)"
eberlm@66277
   265
    by (intro insert.IH add_mono mult_left_mono prod_mono) auto
eberlm@66277
   266
  with insert.hyps show ?case by (simp add: algebra_simps)
eberlm@66277
   267
qed simp_all
eberlm@66277
   268
eberlm@66277
   269
lemma norm_prod_minus1_le_prod_minus1:
eberlm@66277
   270
  fixes f :: "nat \<Rightarrow> 'a :: {real_normed_div_algebra,comm_ring_1}"  
eberlm@66277
   271
  shows "norm (prod (\<lambda>n. 1 + f n) A - 1) \<le> prod (\<lambda>n. 1 + norm (f n)) A - 1"
eberlm@66277
   272
proof (induction A rule: infinite_finite_induct)
eberlm@66277
   273
  case (insert x A)
eberlm@66277
   274
  from insert.hyps have 
eberlm@66277
   275
    "norm ((\<Prod>n\<in>insert x A. 1 + f n) - 1) = 
eberlm@66277
   276
       norm ((\<Prod>n\<in>A. 1 + f n) - 1 + f x * (\<Prod>n\<in>A. 1 + f n))"
eberlm@66277
   277
    by (simp add: algebra_simps)
eberlm@66277
   278
  also have "\<dots> \<le> norm ((\<Prod>n\<in>A. 1 + f n) - 1) + norm (f x * (\<Prod>n\<in>A. 1 + f n))"
eberlm@66277
   279
    by (rule norm_triangle_ineq)
eberlm@66277
   280
  also have "norm (f x * (\<Prod>n\<in>A. 1 + f n)) = norm (f x) * (\<Prod>x\<in>A. norm (1 + f x))"
eberlm@66277
   281
    by (simp add: prod_norm norm_mult)
eberlm@66277
   282
  also have "(\<Prod>x\<in>A. norm (1 + f x)) \<le> (\<Prod>x\<in>A. norm (1::'a) + norm (f x))"
eberlm@66277
   283
    by (intro prod_mono norm_triangle_ineq ballI conjI) auto
eberlm@66277
   284
  also have "norm (1::'a) = 1" by simp
eberlm@66277
   285
  also note insert.IH
eberlm@66277
   286
  also have "(\<Prod>n\<in>A. 1 + norm (f n)) - 1 + norm (f x) * (\<Prod>x\<in>A. 1 + norm (f x)) =
lp15@68064
   287
             (\<Prod>n\<in>insert x A. 1 + norm (f n)) - 1"
eberlm@66277
   288
    using insert.hyps by (simp add: algebra_simps)
eberlm@66277
   289
  finally show ?case by - (simp_all add: mult_left_mono)
eberlm@66277
   290
qed simp_all
eberlm@66277
   291
eberlm@66277
   292
lemma convergent_prod_imp_ev_nonzero:
eberlm@66277
   293
  fixes f :: "nat \<Rightarrow> 'a :: {t2_space,comm_semiring_1}"
eberlm@66277
   294
  assumes "convergent_prod f"
eberlm@66277
   295
  shows   "eventually (\<lambda>n. f n \<noteq> 0) sequentially"
eberlm@66277
   296
  using assms by (auto simp: eventually_at_top_linorder convergent_prod_altdef)
eberlm@66277
   297
eberlm@66277
   298
lemma convergent_prod_imp_LIMSEQ:
eberlm@66277
   299
  fixes f :: "nat \<Rightarrow> 'a :: {real_normed_field}"
eberlm@66277
   300
  assumes "convergent_prod f"
eberlm@66277
   301
  shows   "f \<longlonglongrightarrow> 1"
eberlm@66277
   302
proof -
eberlm@66277
   303
  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"
eberlm@66277
   304
    by (auto simp: convergent_prod_altdef)
eberlm@66277
   305
  hence L': "(\<lambda>n. \<Prod>i\<le>Suc n. f (i+M)) \<longlonglongrightarrow> L" by (subst filterlim_sequentially_Suc)
eberlm@66277
   306
  have "(\<lambda>n. (\<Prod>i\<le>Suc n. f (i+M)) / (\<Prod>i\<le>n. f (i+M))) \<longlonglongrightarrow> L / L"
eberlm@66277
   307
    using L L' by (intro tendsto_divide) simp_all
eberlm@66277
   308
  also from L have "L / L = 1" by simp
eberlm@66277
   309
  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
   310
    using assms L by (auto simp: fun_eq_iff atMost_Suc)
eberlm@66277
   311
  finally show ?thesis by (rule LIMSEQ_offset)
eberlm@66277
   312
qed
eberlm@66277
   313
eberlm@66277
   314
lemma abs_convergent_prod_imp_summable:
eberlm@66277
   315
  fixes f :: "nat \<Rightarrow> 'a :: real_normed_div_algebra"
eberlm@66277
   316
  assumes "abs_convergent_prod f"
eberlm@66277
   317
  shows "summable (\<lambda>i. norm (f i - 1))"
eberlm@66277
   318
proof -
eberlm@66277
   319
  from assms have "convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))" 
eberlm@66277
   320
    unfolding abs_convergent_prod_def by (rule convergent_prod_imp_convergent)
eberlm@66277
   321
  then obtain L where L: "(\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1)) \<longlonglongrightarrow> L"
eberlm@66277
   322
    unfolding convergent_def by blast
eberlm@66277
   323
  have "convergent (\<lambda>n. \<Sum>i\<le>n. norm (f i - 1))"
eberlm@66277
   324
  proof (rule Bseq_monoseq_convergent)
eberlm@66277
   325
    have "eventually (\<lambda>n. (\<Prod>i\<le>n. 1 + norm (f i - 1)) < L + 1) sequentially"
eberlm@66277
   326
      using L(1) by (rule order_tendstoD) simp_all
eberlm@66277
   327
    hence "\<forall>\<^sub>F x in sequentially. norm (\<Sum>i\<le>x. norm (f i - 1)) \<le> L + 1"
eberlm@66277
   328
    proof eventually_elim
eberlm@66277
   329
      case (elim n)
eberlm@66277
   330
      have "norm (\<Sum>i\<le>n. norm (f i - 1)) = (\<Sum>i\<le>n. norm (f i - 1))"
eberlm@66277
   331
        unfolding real_norm_def by (intro abs_of_nonneg sum_nonneg) simp_all
eberlm@66277
   332
      also have "\<dots> \<le> (\<Prod>i\<le>n. 1 + norm (f i - 1))" by (rule sum_le_prod) auto
eberlm@66277
   333
      also have "\<dots> < L + 1" by (rule elim)
eberlm@66277
   334
      finally show ?case by simp
eberlm@66277
   335
    qed
eberlm@66277
   336
    thus "Bseq (\<lambda>n. \<Sum>i\<le>n. norm (f i - 1))" by (rule BfunI)
eberlm@66277
   337
  next
eberlm@66277
   338
    show "monoseq (\<lambda>n. \<Sum>i\<le>n. norm (f i - 1))"
eberlm@66277
   339
      by (rule mono_SucI1) auto
eberlm@66277
   340
  qed
eberlm@66277
   341
  thus "summable (\<lambda>i. norm (f i - 1))" by (simp add: summable_iff_convergent')
eberlm@66277
   342
qed
eberlm@66277
   343
eberlm@66277
   344
lemma summable_imp_abs_convergent_prod:
eberlm@66277
   345
  fixes f :: "nat \<Rightarrow> 'a :: real_normed_div_algebra"
eberlm@66277
   346
  assumes "summable (\<lambda>i. norm (f i - 1))"
eberlm@66277
   347
  shows   "abs_convergent_prod f"
eberlm@66277
   348
proof (intro abs_convergent_prodI Bseq_monoseq_convergent)
eberlm@66277
   349
  show "monoseq (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
eberlm@66277
   350
    by (intro mono_SucI1) 
eberlm@66277
   351
       (auto simp: atMost_Suc algebra_simps intro!: mult_nonneg_nonneg prod_nonneg)
eberlm@66277
   352
next
eberlm@66277
   353
  show "Bseq (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
eberlm@66277
   354
  proof (rule Bseq_eventually_mono)
eberlm@66277
   355
    show "eventually (\<lambda>n. norm (\<Prod>i\<le>n. 1 + norm (f i - 1)) \<le> 
eberlm@66277
   356
            norm (exp (\<Sum>i\<le>n. norm (f i - 1)))) sequentially"
eberlm@66277
   357
      by (intro always_eventually allI) (auto simp: abs_prod exp_sum intro!: prod_mono)
eberlm@66277
   358
  next
eberlm@66277
   359
    from assms have "(\<lambda>n. \<Sum>i\<le>n. norm (f i - 1)) \<longlonglongrightarrow> (\<Sum>i. norm (f i - 1))"
eberlm@66277
   360
      using sums_def_le by blast
eberlm@66277
   361
    hence "(\<lambda>n. exp (\<Sum>i\<le>n. norm (f i - 1))) \<longlonglongrightarrow> exp (\<Sum>i. norm (f i - 1))"
eberlm@66277
   362
      by (rule tendsto_exp)
eberlm@66277
   363
    hence "convergent (\<lambda>n. exp (\<Sum>i\<le>n. norm (f i - 1)))"
eberlm@66277
   364
      by (rule convergentI)
eberlm@66277
   365
    thus "Bseq (\<lambda>n. exp (\<Sum>i\<le>n. norm (f i - 1)))"
eberlm@66277
   366
      by (rule convergent_imp_Bseq)
eberlm@66277
   367
  qed
eberlm@66277
   368
qed
eberlm@66277
   369
eberlm@66277
   370
lemma abs_convergent_prod_conv_summable:
eberlm@66277
   371
  fixes f :: "nat \<Rightarrow> 'a :: real_normed_div_algebra"
eberlm@66277
   372
  shows "abs_convergent_prod f \<longleftrightarrow> summable (\<lambda>i. norm (f i - 1))"
eberlm@66277
   373
  by (blast intro: abs_convergent_prod_imp_summable summable_imp_abs_convergent_prod)
eberlm@66277
   374
eberlm@66277
   375
lemma abs_convergent_prod_imp_LIMSEQ:
eberlm@66277
   376
  fixes f :: "nat \<Rightarrow> 'a :: {comm_ring_1,real_normed_div_algebra}"
eberlm@66277
   377
  assumes "abs_convergent_prod f"
eberlm@66277
   378
  shows   "f \<longlonglongrightarrow> 1"
eberlm@66277
   379
proof -
eberlm@66277
   380
  from assms have "summable (\<lambda>n. norm (f n - 1))"
eberlm@66277
   381
    by (rule abs_convergent_prod_imp_summable)
eberlm@66277
   382
  from summable_LIMSEQ_zero[OF this] have "(\<lambda>n. f n - 1) \<longlonglongrightarrow> 0"
eberlm@66277
   383
    by (simp add: tendsto_norm_zero_iff)
eberlm@66277
   384
  from tendsto_add[OF this tendsto_const[of 1]] show ?thesis by simp
eberlm@66277
   385
qed
eberlm@66277
   386
eberlm@66277
   387
lemma abs_convergent_prod_imp_ev_nonzero:
eberlm@66277
   388
  fixes f :: "nat \<Rightarrow> 'a :: {comm_ring_1,real_normed_div_algebra}"
eberlm@66277
   389
  assumes "abs_convergent_prod f"
eberlm@66277
   390
  shows   "eventually (\<lambda>n. f n \<noteq> 0) sequentially"
eberlm@66277
   391
proof -
eberlm@66277
   392
  from assms have "f \<longlonglongrightarrow> 1" 
eberlm@66277
   393
    by (rule abs_convergent_prod_imp_LIMSEQ)
eberlm@66277
   394
  hence "eventually (\<lambda>n. dist (f n) 1 < 1) at_top"
eberlm@66277
   395
    by (auto simp: tendsto_iff)
eberlm@66277
   396
  thus ?thesis by eventually_elim auto
eberlm@66277
   397
qed
eberlm@66277
   398
eberlm@66277
   399
lemma convergent_prod_offset:
eberlm@66277
   400
  assumes "convergent_prod (\<lambda>n. f (n + m))"  
eberlm@66277
   401
  shows   "convergent_prod f"
eberlm@66277
   402
proof -
eberlm@66277
   403
  from assms obtain M L where "(\<lambda>n. \<Prod>k\<le>n. f (k + (M + m))) \<longlonglongrightarrow> L" "L \<noteq> 0"
lp15@68064
   404
    by (auto simp: prod_defs add.assoc)
lp15@68064
   405
  thus "convergent_prod f" 
lp15@68064
   406
    unfolding prod_defs by blast
eberlm@66277
   407
qed
eberlm@66277
   408
eberlm@66277
   409
lemma abs_convergent_prod_offset:
eberlm@66277
   410
  assumes "abs_convergent_prod (\<lambda>n. f (n + m))"  
eberlm@66277
   411
  shows   "abs_convergent_prod f"
eberlm@66277
   412
  using assms unfolding abs_convergent_prod_def by (rule convergent_prod_offset)
eberlm@66277
   413
lp15@68424
   414
subsection\<open>Ignoring initial segments\<close>
lp15@68424
   415
lp15@68361
   416
lemma raw_has_prod_ignore_initial_segment:
lp15@68361
   417
  fixes f :: "nat \<Rightarrow> 'a :: real_normed_field"
lp15@68361
   418
  assumes "raw_has_prod f M p" "N \<ge> M"
lp15@68361
   419
  obtains q where  "raw_has_prod f N q"
eberlm@66277
   420
proof -
lp15@68361
   421
  have p: "(\<lambda>n. \<Prod>k\<le>n. f (k + M)) \<longlonglongrightarrow> p" and "p \<noteq> 0" 
lp15@68361
   422
    using assms by (auto simp: raw_has_prod_def)
lp15@68361
   423
  then have nz: "\<And>n. n \<ge> M \<Longrightarrow> f n \<noteq> 0"
lp15@68361
   424
    using assms by (auto simp: raw_has_prod_eq_0)
lp15@68361
   425
  define C where "C = (\<Prod>k<N-M. f (k + M))"
eberlm@66277
   426
  from nz have [simp]: "C \<noteq> 0" 
eberlm@66277
   427
    by (auto simp: C_def)
eberlm@66277
   428
lp15@68361
   429
  from p have "(\<lambda>i. \<Prod>k\<le>i + (N-M). f (k + M)) \<longlonglongrightarrow> p" 
eberlm@66277
   430
    by (rule LIMSEQ_ignore_initial_segment)
lp15@68361
   431
  also have "(\<lambda>i. \<Prod>k\<le>i + (N-M). f (k + M)) = (\<lambda>n. C * (\<Prod>k\<le>n. f (k + N)))"
eberlm@66277
   432
  proof (rule ext, goal_cases)
eberlm@66277
   433
    case (1 n)
lp15@68361
   434
    have "{..n+(N-M)} = {..<(N-M)} \<union> {(N-M)..n+(N-M)}" by auto
lp15@68361
   435
    also have "(\<Prod>k\<in>\<dots>. f (k + M)) = C * (\<Prod>k=(N-M)..n+(N-M). f (k + M))"
eberlm@66277
   436
      unfolding C_def by (rule prod.union_disjoint) auto
lp15@68361
   437
    also have "(\<Prod>k=(N-M)..n+(N-M). f (k + M)) = (\<Prod>k\<le>n. f (k + (N-M) + M))"
lp15@68361
   438
      by (intro ext prod.reindex_bij_witness[of _ "\<lambda>k. k + (N-M)" "\<lambda>k. k - (N-M)"]) auto
lp15@68361
   439
    finally show ?case
lp15@68361
   440
      using \<open>N \<ge> M\<close> by (simp add: add_ac)
eberlm@66277
   441
  qed
lp15@68361
   442
  finally have "(\<lambda>n. C * (\<Prod>k\<le>n. f (k + N)) / C) \<longlonglongrightarrow> p / C"
eberlm@66277
   443
    by (intro tendsto_divide tendsto_const) auto
lp15@68361
   444
  hence "(\<lambda>n. \<Prod>k\<le>n. f (k + N)) \<longlonglongrightarrow> p / C" by simp
lp15@68361
   445
  moreover from \<open>p \<noteq> 0\<close> have "p / C \<noteq> 0" by simp
lp15@68361
   446
  ultimately show ?thesis
lp15@68361
   447
    using raw_has_prod_def that by blast 
eberlm@66277
   448
qed
eberlm@66277
   449
lp15@68361
   450
corollary convergent_prod_ignore_initial_segment:
lp15@68361
   451
  fixes f :: "nat \<Rightarrow> 'a :: real_normed_field"
lp15@68361
   452
  assumes "convergent_prod f"
lp15@68361
   453
  shows   "convergent_prod (\<lambda>n. f (n + m))"
lp15@68361
   454
  using assms
lp15@68361
   455
  unfolding convergent_prod_def 
lp15@68361
   456
  apply clarify
lp15@68361
   457
  apply (erule_tac N="M+m" in raw_has_prod_ignore_initial_segment)
lp15@68361
   458
  apply (auto simp add: raw_has_prod_def add_ac)
lp15@68361
   459
  done
lp15@68361
   460
lp15@68136
   461
corollary convergent_prod_ignore_nonzero_segment:
lp15@68136
   462
  fixes f :: "nat \<Rightarrow> 'a :: real_normed_field"
lp15@68136
   463
  assumes f: "convergent_prod f" and nz: "\<And>i. i \<ge> M \<Longrightarrow> f i \<noteq> 0"
lp15@68361
   464
  shows "\<exists>p. raw_has_prod f M p"
lp15@68136
   465
  using convergent_prod_ignore_initial_segment [OF f]
lp15@68136
   466
  by (metis convergent_LIMSEQ_iff convergent_prod_iff_convergent le_add_same_cancel2 nz prod_defs(1) zero_order(1))
lp15@68136
   467
lp15@68136
   468
corollary abs_convergent_prod_ignore_initial_segment:
eberlm@66277
   469
  assumes "abs_convergent_prod f"
eberlm@66277
   470
  shows   "abs_convergent_prod (\<lambda>n. f (n + m))"
eberlm@66277
   471
  using assms unfolding abs_convergent_prod_def 
eberlm@66277
   472
  by (rule convergent_prod_ignore_initial_segment)
eberlm@66277
   473
eberlm@66277
   474
lemma abs_convergent_prod_imp_convergent_prod:
eberlm@66277
   475
  fixes f :: "nat \<Rightarrow> 'a :: {real_normed_div_algebra,complete_space,comm_ring_1}"
eberlm@66277
   476
  assumes "abs_convergent_prod f"
eberlm@66277
   477
  shows   "convergent_prod f"
eberlm@66277
   478
proof -
eberlm@66277
   479
  from assms have "eventually (\<lambda>n. f n \<noteq> 0) sequentially"
eberlm@66277
   480
    by (rule abs_convergent_prod_imp_ev_nonzero)
eberlm@66277
   481
  then obtain N where N: "f n \<noteq> 0" if "n \<ge> N" for n 
eberlm@66277
   482
    by (auto simp: eventually_at_top_linorder)
eberlm@66277
   483
  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
   484
eberlm@66277
   485
  have "Cauchy ?P"
eberlm@66277
   486
  proof (rule CauchyI', goal_cases)
eberlm@66277
   487
    case (1 \<epsilon>)
eberlm@66277
   488
    from assms have "abs_convergent_prod (\<lambda>n. f (n + N))"
eberlm@66277
   489
      by (rule abs_convergent_prod_ignore_initial_segment)
eberlm@66277
   490
    hence "Cauchy ?Q"
eberlm@66277
   491
      unfolding abs_convergent_prod_def
eberlm@66277
   492
      by (intro convergent_Cauchy convergent_prod_imp_convergent)
eberlm@66277
   493
    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
   494
      by blast
eberlm@66277
   495
    show ?case
eberlm@66277
   496
    proof (rule exI[of _ M], safe, goal_cases)
eberlm@66277
   497
      case (1 m n)
eberlm@66277
   498
      have "dist (?P m) (?P n) = norm (?P n - ?P m)"
eberlm@66277
   499
        by (simp add: dist_norm norm_minus_commute)
eberlm@66277
   500
      also from 1 have "{..n} = {..m} \<union> {m<..n}" by auto
eberlm@66277
   501
      hence "norm (?P n - ?P m) = norm (?P m * (\<Prod>k\<in>{m<..n}. f (k + N)) - ?P m)"
eberlm@66277
   502
        by (subst prod.union_disjoint [symmetric]) (auto simp: algebra_simps)
eberlm@66277
   503
      also have "\<dots> = norm (?P m * ((\<Prod>k\<in>{m<..n}. f (k + N)) - 1))"
eberlm@66277
   504
        by (simp add: algebra_simps)
eberlm@66277
   505
      also have "\<dots> = (\<Prod>k\<le>m. norm (f (k + N))) * norm ((\<Prod>k\<in>{m<..n}. f (k + N)) - 1)"
eberlm@66277
   506
        by (simp add: norm_mult prod_norm)
eberlm@66277
   507
      also have "\<dots> \<le> ?Q m * ((\<Prod>k\<in>{m<..n}. 1 + norm (f (k + N) - 1)) - 1)"
eberlm@66277
   508
        using norm_prod_minus1_le_prod_minus1[of "\<lambda>k. f (k + N) - 1" "{m<..n}"]
eberlm@66277
   509
              norm_triangle_ineq[of 1 "f k - 1" for k]
eberlm@66277
   510
        by (intro mult_mono prod_mono ballI conjI norm_prod_minus1_le_prod_minus1 prod_nonneg) auto
eberlm@66277
   511
      also have "\<dots> = ?Q m * (\<Prod>k\<in>{m<..n}. 1 + norm (f (k + N) - 1)) - ?Q m"
eberlm@66277
   512
        by (simp add: algebra_simps)
eberlm@66277
   513
      also have "?Q m * (\<Prod>k\<in>{m<..n}. 1 + norm (f (k + N) - 1)) = 
eberlm@66277
   514
                   (\<Prod>k\<in>{..m}\<union>{m<..n}. 1 + norm (f (k + N) - 1))"
eberlm@66277
   515
        by (rule prod.union_disjoint [symmetric]) auto
eberlm@66277
   516
      also from 1 have "{..m}\<union>{m<..n} = {..n}" by auto
eberlm@66277
   517
      also have "?Q n - ?Q m \<le> norm (?Q n - ?Q m)" by simp
eberlm@66277
   518
      also from 1 have "\<dots> < \<epsilon>" by (intro M) auto
eberlm@66277
   519
      finally show ?case .
eberlm@66277
   520
    qed
eberlm@66277
   521
  qed
eberlm@66277
   522
  hence conv: "convergent ?P" by (rule Cauchy_convergent)
eberlm@66277
   523
  then obtain L where L: "?P \<longlonglongrightarrow> L"
eberlm@66277
   524
    by (auto simp: convergent_def)
eberlm@66277
   525
eberlm@66277
   526
  have "L \<noteq> 0"
eberlm@66277
   527
  proof
eberlm@66277
   528
    assume [simp]: "L = 0"
eberlm@66277
   529
    from tendsto_norm[OF L] have limit: "(\<lambda>n. \<Prod>k\<le>n. norm (f (k + N))) \<longlonglongrightarrow> 0" 
eberlm@66277
   530
      by (simp add: prod_norm)
eberlm@66277
   531
eberlm@66277
   532
    from assms have "(\<lambda>n. f (n + N)) \<longlonglongrightarrow> 1"
eberlm@66277
   533
      by (intro abs_convergent_prod_imp_LIMSEQ abs_convergent_prod_ignore_initial_segment)
eberlm@66277
   534
    hence "eventually (\<lambda>n. norm (f (n + N) - 1) < 1) sequentially"
eberlm@66277
   535
      by (auto simp: tendsto_iff dist_norm)
eberlm@66277
   536
    then obtain M0 where M0: "norm (f (n + N) - 1) < 1" if "n \<ge> M0" for n
eberlm@66277
   537
      by (auto simp: eventually_at_top_linorder)
eberlm@66277
   538
eberlm@66277
   539
    {
eberlm@66277
   540
      fix M assume M: "M \<ge> M0"
eberlm@66277
   541
      with M0 have M: "norm (f (n + N) - 1) < 1" if "n \<ge> M" for n using that by simp
eberlm@66277
   542
eberlm@66277
   543
      have "(\<lambda>n. \<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1)) \<longlonglongrightarrow> 0"
eberlm@66277
   544
      proof (rule tendsto_sandwich)
eberlm@66277
   545
        show "eventually (\<lambda>n. (\<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1)) \<ge> 0) sequentially"
eberlm@66277
   546
          using M by (intro always_eventually prod_nonneg allI ballI) (auto intro: less_imp_le)
eberlm@66277
   547
        have "norm (1::'a) - norm (f (i + M + N) - 1) \<le> norm (f (i + M + N))" for i
eberlm@66277
   548
          using norm_triangle_ineq3[of "f (i + M + N)" 1] by simp
eberlm@66277
   549
        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
   550
          using M by (intro always_eventually allI prod_mono ballI conjI) (auto intro: less_imp_le)
eberlm@66277
   551
        
eberlm@66277
   552
        define C where "C = (\<Prod>k<M. norm (f (k + N)))"
eberlm@66277
   553
        from N have [simp]: "C \<noteq> 0" by (auto simp: C_def)
eberlm@66277
   554
        from L have "(\<lambda>n. norm (\<Prod>k\<le>n+M. f (k + N))) \<longlonglongrightarrow> 0"
eberlm@66277
   555
          by (intro LIMSEQ_ignore_initial_segment) (simp add: tendsto_norm_zero_iff)
eberlm@66277
   556
        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
   557
        proof (rule ext, goal_cases)
eberlm@66277
   558
          case (1 n)
eberlm@66277
   559
          have "{..n+M} = {..<M} \<union> {M..n+M}" by auto
eberlm@66277
   560
          also have "norm (\<Prod>k\<in>\<dots>. f (k + N)) = C * norm (\<Prod>k=M..n+M. f (k + N))"
eberlm@66277
   561
            unfolding C_def by (subst prod.union_disjoint) (auto simp: norm_mult prod_norm)
eberlm@66277
   562
          also have "(\<Prod>k=M..n+M. f (k + N)) = (\<Prod>k\<le>n. f (k + N + M))"
eberlm@66277
   563
            by (intro prod.reindex_bij_witness[of _ "\<lambda>i. i + M" "\<lambda>i. i - M"]) auto
eberlm@66277
   564
          finally show ?case by (simp add: add_ac prod_norm)
eberlm@66277
   565
        qed
eberlm@66277
   566
        finally have "(\<lambda>n. C * (\<Prod>k\<le>n. norm (f (k + M + N))) / C) \<longlonglongrightarrow> 0 / C"
eberlm@66277
   567
          by (intro tendsto_divide tendsto_const) auto
eberlm@66277
   568
        thus "(\<lambda>n. \<Prod>k\<le>n. norm (f (k + M + N))) \<longlonglongrightarrow> 0" by simp
eberlm@66277
   569
      qed simp_all
eberlm@66277
   570
eberlm@66277
   571
      have "1 - (\<Sum>i. norm (f (i + M + N) - 1)) \<le> 0"
eberlm@66277
   572
      proof (rule tendsto_le)
eberlm@66277
   573
        show "eventually (\<lambda>n. 1 - (\<Sum>k\<le>n. norm (f (k+M+N) - 1)) \<le> 
eberlm@66277
   574
                                (\<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1))) at_top"
eberlm@66277
   575
          using M by (intro always_eventually allI weierstrass_prod_ineq) (auto intro: less_imp_le)
eberlm@66277
   576
        show "(\<lambda>n. \<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1)) \<longlonglongrightarrow> 0" by fact
eberlm@66277
   577
        show "(\<lambda>n. 1 - (\<Sum>k\<le>n. norm (f (k + M + N) - 1)))
eberlm@66277
   578
                  \<longlonglongrightarrow> 1 - (\<Sum>i. norm (f (i + M + N) - 1))"
eberlm@66277
   579
          by (intro tendsto_intros summable_LIMSEQ' summable_ignore_initial_segment 
eberlm@66277
   580
                abs_convergent_prod_imp_summable assms)
eberlm@66277
   581
      qed simp_all
eberlm@66277
   582
      hence "(\<Sum>i. norm (f (i + M + N) - 1)) \<ge> 1" by simp
eberlm@66277
   583
      also have "\<dots> + (\<Sum>i<M. norm (f (i + N) - 1)) = (\<Sum>i. norm (f (i + N) - 1))"
eberlm@66277
   584
        by (intro suminf_split_initial_segment [symmetric] summable_ignore_initial_segment
eberlm@66277
   585
              abs_convergent_prod_imp_summable assms)
eberlm@66277
   586
      finally have "1 + (\<Sum>i<M. norm (f (i + N) - 1)) \<le> (\<Sum>i. norm (f (i + N) - 1))" by simp
eberlm@66277
   587
    } note * = this
eberlm@66277
   588
eberlm@66277
   589
    have "1 + (\<Sum>i. norm (f (i + N) - 1)) \<le> (\<Sum>i. norm (f (i + N) - 1))"
eberlm@66277
   590
    proof (rule tendsto_le)
eberlm@66277
   591
      show "(\<lambda>M. 1 + (\<Sum>i<M. norm (f (i + N) - 1))) \<longlonglongrightarrow> 1 + (\<Sum>i. norm (f (i + N) - 1))"
eberlm@66277
   592
        by (intro tendsto_intros summable_LIMSEQ summable_ignore_initial_segment 
eberlm@66277
   593
                abs_convergent_prod_imp_summable assms)
eberlm@66277
   594
      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
   595
        using eventually_ge_at_top[of M0] by eventually_elim (use * in auto)
eberlm@66277
   596
    qed simp_all
eberlm@66277
   597
    thus False by simp
eberlm@66277
   598
  qed
lp15@68064
   599
  with L show ?thesis by (auto simp: prod_defs)
lp15@68064
   600
qed
lp15@68064
   601
lp15@68424
   602
subsection\<open>More elementary properties\<close>
lp15@68424
   603
lp15@68361
   604
lemma raw_has_prod_cases:
lp15@68064
   605
  fixes f :: "nat \<Rightarrow> 'a :: {idom,topological_semigroup_mult,t2_space}"
lp15@68361
   606
  assumes "raw_has_prod f M p"
lp15@68361
   607
  obtains i where "i<M" "f i = 0" | p where "raw_has_prod f 0 p"
lp15@68136
   608
proof -
lp15@68136
   609
  have "(\<lambda>n. \<Prod>i\<le>n. f (i + M)) \<longlonglongrightarrow> p" "p \<noteq> 0"
lp15@68361
   610
    using assms unfolding raw_has_prod_def by blast+
lp15@68064
   611
  then have "(\<lambda>n. prod f {..<M} * (\<Prod>i\<le>n. f (i + M))) \<longlonglongrightarrow> prod f {..<M} * p"
lp15@68064
   612
    by (metis tendsto_mult_left)
lp15@68064
   613
  moreover have "prod f {..<M} * (\<Prod>i\<le>n. f (i + M)) = prod f {..n+M}" for n
lp15@68064
   614
  proof -
lp15@68064
   615
    have "{..n+M} = {..<M} \<union> {M..n+M}"
lp15@68064
   616
      by auto
lp15@68064
   617
    then have "prod f {..n+M} = prod f {..<M} * prod f {M..n+M}"
lp15@68064
   618
      by simp (subst prod.union_disjoint; force)
lp15@68138
   619
    also have "\<dots> = prod f {..<M} * (\<Prod>i\<le>n. f (i + M))"
lp15@68064
   620
      by (metis (mono_tags, lifting) add.left_neutral atMost_atLeast0 prod_shift_bounds_cl_nat_ivl)
lp15@68064
   621
    finally show ?thesis by metis
lp15@68064
   622
  qed
lp15@68064
   623
  ultimately have "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> prod f {..<M} * p"
lp15@68064
   624
    by (auto intro: LIMSEQ_offset [where k=M])
lp15@68361
   625
  then have "raw_has_prod f 0 (prod f {..<M} * p)" if "\<forall>i<M. f i \<noteq> 0"
lp15@68361
   626
    using \<open>p \<noteq> 0\<close> assms that by (auto simp: raw_has_prod_def)
lp15@68136
   627
  then show thesis
lp15@68136
   628
    using that by blast
lp15@68064
   629
qed
lp15@68064
   630
lp15@68136
   631
corollary convergent_prod_offset_0:
lp15@68136
   632
  fixes f :: "nat \<Rightarrow> 'a :: {idom,topological_semigroup_mult,t2_space}"
lp15@68136
   633
  assumes "convergent_prod f" "\<And>i. f i \<noteq> 0"
lp15@68361
   634
  shows "\<exists>p. raw_has_prod f 0 p"
lp15@68361
   635
  using assms convergent_prod_def raw_has_prod_cases by blast
lp15@68136
   636
lp15@68064
   637
lemma prodinf_eq_lim:
lp15@68064
   638
  fixes f :: "nat \<Rightarrow> 'a :: {idom,topological_semigroup_mult,t2_space}"
lp15@68064
   639
  assumes "convergent_prod f" "\<And>i. f i \<noteq> 0"
lp15@68064
   640
  shows "prodinf f = lim (\<lambda>n. \<Prod>i\<le>n. f i)"
lp15@68064
   641
  using assms convergent_prod_offset_0 [OF assms]
lp15@68064
   642
  by (simp add: prod_defs lim_def) (metis (no_types) assms(1) convergent_prod_to_zero_iff)
lp15@68064
   643
lp15@68064
   644
lemma has_prod_one[simp, intro]: "(\<lambda>n. 1) has_prod 1"
lp15@68064
   645
  unfolding prod_defs by auto
lp15@68064
   646
lp15@68064
   647
lemma convergent_prod_one[simp, intro]: "convergent_prod (\<lambda>n. 1)"
lp15@68064
   648
  unfolding prod_defs by auto
lp15@68064
   649
lp15@68064
   650
lemma prodinf_cong: "(\<And>n. f n = g n) \<Longrightarrow> prodinf f = prodinf g"
lp15@68064
   651
  by presburger
lp15@68064
   652
lp15@68064
   653
lemma convergent_prod_cong:
lp15@68064
   654
  fixes f g :: "nat \<Rightarrow> 'a::{field,topological_semigroup_mult,t2_space}"
lp15@68064
   655
  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
   656
  shows "convergent_prod f = convergent_prod g"
lp15@68064
   657
proof -
lp15@68064
   658
  from assms obtain N where N: "\<forall>n\<ge>N. f n = g n"
lp15@68064
   659
    by (auto simp: eventually_at_top_linorder)
lp15@68064
   660
  define C where "C = (\<Prod>k<N. f k / g k)"
lp15@68064
   661
  with g have "C \<noteq> 0"
lp15@68064
   662
    by (simp add: f)
lp15@68064
   663
  have *: "eventually (\<lambda>n. prod f {..n} = C * prod g {..n}) sequentially"
lp15@68064
   664
    using eventually_ge_at_top[of N]
lp15@68064
   665
  proof eventually_elim
lp15@68064
   666
    case (elim n)
lp15@68064
   667
    then have "{..n} = {..<N} \<union> {N..n}"
lp15@68064
   668
      by auto
lp15@68138
   669
    also have "prod f \<dots> = prod f {..<N} * prod f {N..n}"
lp15@68064
   670
      by (intro prod.union_disjoint) auto
lp15@68064
   671
    also from N have "prod f {N..n} = prod g {N..n}"
lp15@68064
   672
      by (intro prod.cong) simp_all
lp15@68064
   673
    also have "prod f {..<N} * prod g {N..n} = C * (prod g {..<N} * prod g {N..n})"
lp15@68064
   674
      unfolding C_def by (simp add: g prod_dividef)
lp15@68064
   675
    also have "prod g {..<N} * prod g {N..n} = prod g ({..<N} \<union> {N..n})"
lp15@68064
   676
      by (intro prod.union_disjoint [symmetric]) auto
lp15@68064
   677
    also from elim have "{..<N} \<union> {N..n} = {..n}"
lp15@68064
   678
      by auto                                                                    
lp15@68064
   679
    finally show "prod f {..n} = C * prod g {..n}" .
lp15@68064
   680
  qed
lp15@68064
   681
  then have cong: "convergent (\<lambda>n. prod f {..n}) = convergent (\<lambda>n. C * prod g {..n})"
lp15@68064
   682
    by (rule convergent_cong)
lp15@68064
   683
  show ?thesis
lp15@68064
   684
  proof
lp15@68064
   685
    assume cf: "convergent_prod f"
lp15@68064
   686
    then have "\<not> (\<lambda>n. prod g {..n}) \<longlonglongrightarrow> 0"
lp15@68064
   687
      using tendsto_mult_left * convergent_prod_to_zero_iff f filterlim_cong by fastforce
lp15@68064
   688
    then show "convergent_prod g"
lp15@68064
   689
      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
   690
  next
lp15@68064
   691
    assume cg: "convergent_prod g"
lp15@68064
   692
    have "\<exists>a. C * a \<noteq> 0 \<and> (\<lambda>n. prod g {..n}) \<longlonglongrightarrow> a"
lp15@68064
   693
      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
   694
    then show "convergent_prod f"
lp15@68064
   695
      using "*" tendsto_mult_left filterlim_cong
lp15@68064
   696
      by (fastforce simp add: convergent_prod_iff_nz_lim f)
lp15@68064
   697
  qed
eberlm@66277
   698
qed
eberlm@66277
   699
lp15@68071
   700
lemma has_prod_finite:
lp15@68361
   701
  fixes f :: "nat \<Rightarrow> 'a::{semidom,t2_space}"
lp15@68071
   702
  assumes [simp]: "finite N"
lp15@68071
   703
    and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 1"
lp15@68071
   704
  shows "f has_prod (\<Prod>n\<in>N. f n)"
lp15@68071
   705
proof -
lp15@68071
   706
  have eq: "prod f {..n + Suc (Max N)} = prod f N" for n
lp15@68071
   707
  proof (rule prod.mono_neutral_right)
lp15@68071
   708
    show "N \<subseteq> {..n + Suc (Max N)}"
lp15@68138
   709
      by (auto simp: le_Suc_eq trans_le_add2)
lp15@68071
   710
    show "\<forall>i\<in>{..n + Suc (Max N)} - N. f i = 1"
lp15@68071
   711
      using f by blast
lp15@68071
   712
  qed auto
lp15@68071
   713
  show ?thesis
lp15@68071
   714
  proof (cases "\<forall>n\<in>N. f n \<noteq> 0")
lp15@68071
   715
    case True
lp15@68071
   716
    then have "prod f N \<noteq> 0"
lp15@68071
   717
      by simp
lp15@68071
   718
    moreover have "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> prod f N"
lp15@68071
   719
      by (rule LIMSEQ_offset[of _ "Suc (Max N)"]) (simp add: eq atLeast0LessThan del: add_Suc_right)
lp15@68071
   720
    ultimately show ?thesis
lp15@68361
   721
      by (simp add: raw_has_prod_def has_prod_def)
lp15@68071
   722
  next
lp15@68071
   723
    case False
lp15@68071
   724
    then obtain k where "k \<in> N" "f k = 0"
lp15@68071
   725
      by auto
lp15@68071
   726
    let ?Z = "{n \<in> N. f n = 0}"
lp15@68071
   727
    have maxge: "Max ?Z \<ge> n" if "f n = 0" for n
lp15@68071
   728
      using Max_ge [of ?Z] \<open>finite N\<close> \<open>f n = 0\<close>
lp15@68071
   729
      by (metis (mono_tags) Collect_mem_eq f finite_Collect_conjI mem_Collect_eq zero_neq_one)
lp15@68071
   730
    let ?q = "prod f {Suc (Max ?Z)..Max N}"
lp15@68071
   731
    have [simp]: "?q \<noteq> 0"
lp15@68071
   732
      using maxge Suc_n_not_le_n le_trans by force
lp15@68076
   733
    have eq: "(\<Prod>i\<le>n + Max N. f (Suc (i + Max ?Z))) = ?q" for n
lp15@68076
   734
    proof -
lp15@68076
   735
      have "(\<Prod>i\<le>n + Max N. f (Suc (i + Max ?Z))) = prod f {Suc (Max ?Z)..n + Max N + Suc (Max ?Z)}" 
lp15@68076
   736
      proof (rule prod.reindex_cong [where l = "\<lambda>i. i + Suc (Max ?Z)", THEN sym])
lp15@68076
   737
        show "{Suc (Max ?Z)..n + Max N + Suc (Max ?Z)} = (\<lambda>i. i + Suc (Max ?Z)) ` {..n + Max N}"
lp15@68076
   738
          using le_Suc_ex by fastforce
lp15@68076
   739
      qed (auto simp: inj_on_def)
lp15@68138
   740
      also have "\<dots> = ?q"
lp15@68076
   741
        by (rule prod.mono_neutral_right)
lp15@68076
   742
           (use Max.coboundedI [OF \<open>finite N\<close>] f in \<open>force+\<close>)
lp15@68076
   743
      finally show ?thesis .
lp15@68076
   744
    qed
lp15@68361
   745
    have q: "raw_has_prod f (Suc (Max ?Z)) ?q"
lp15@68361
   746
    proof (simp add: raw_has_prod_def)
lp15@68076
   747
      show "(\<lambda>n. \<Prod>i\<le>n. f (Suc (i + Max ?Z))) \<longlonglongrightarrow> ?q"
lp15@68076
   748
        by (rule LIMSEQ_offset[of _ "(Max N)"]) (simp add: eq)
lp15@68076
   749
    qed
lp15@68071
   750
    show ?thesis
lp15@68071
   751
      unfolding has_prod_def
lp15@68071
   752
    proof (intro disjI2 exI conjI)      
lp15@68071
   753
      show "prod f N = 0"
lp15@68071
   754
        using \<open>f k = 0\<close> \<open>k \<in> N\<close> \<open>finite N\<close> prod_zero by blast
lp15@68071
   755
      show "f (Max ?Z) = 0"
lp15@68071
   756
        using Max_in [of ?Z] \<open>finite N\<close> \<open>f k = 0\<close> \<open>k \<in> N\<close> by auto
lp15@68071
   757
    qed (use q in auto)
lp15@68071
   758
  qed
lp15@68071
   759
qed
lp15@68071
   760
lp15@68071
   761
corollary has_prod_0:
lp15@68361
   762
  fixes f :: "nat \<Rightarrow> 'a::{semidom,t2_space}"
lp15@68071
   763
  assumes "\<And>n. f n = 1"
lp15@68071
   764
  shows "f has_prod 1"
lp15@68071
   765
  by (simp add: assms has_prod_cong)
lp15@68071
   766
lp15@68361
   767
lemma prodinf_zero[simp]: "prodinf (\<lambda>n. 1::'a::real_normed_field) = 1"
lp15@68361
   768
  using has_prod_unique by force
lp15@68361
   769
lp15@68071
   770
lemma convergent_prod_finite:
lp15@68071
   771
  fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
lp15@68071
   772
  assumes "finite N" "\<And>n. n \<notin> N \<Longrightarrow> f n = 1"
lp15@68071
   773
  shows "convergent_prod f"
lp15@68071
   774
proof -
lp15@68361
   775
  have "\<exists>n p. raw_has_prod f n p"
lp15@68071
   776
    using assms has_prod_def has_prod_finite by blast
lp15@68071
   777
  then show ?thesis
lp15@68071
   778
    by (simp add: convergent_prod_def)
lp15@68071
   779
qed
lp15@68071
   780
lp15@68127
   781
lemma has_prod_If_finite_set:
lp15@68127
   782
  fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
lp15@68127
   783
  shows "finite A \<Longrightarrow> (\<lambda>r. if r \<in> A then f r else 1) has_prod (\<Prod>r\<in>A. f r)"
lp15@68127
   784
  using has_prod_finite[of A "(\<lambda>r. if r \<in> A then f r else 1)"]
lp15@68127
   785
  by simp
lp15@68127
   786
lp15@68127
   787
lemma has_prod_If_finite:
lp15@68127
   788
  fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
lp15@68127
   789
  shows "finite {r. P r} \<Longrightarrow> (\<lambda>r. if P r then f r else 1) has_prod (\<Prod>r | P r. f r)"
lp15@68127
   790
  using has_prod_If_finite_set[of "{r. P r}"] by simp
lp15@68127
   791
lp15@68127
   792
lemma convergent_prod_If_finite_set[simp, intro]:
lp15@68127
   793
  fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
lp15@68127
   794
  shows "finite A \<Longrightarrow> convergent_prod (\<lambda>r. if r \<in> A then f r else 1)"
lp15@68127
   795
  by (simp add: convergent_prod_finite)
lp15@68127
   796
lp15@68127
   797
lemma convergent_prod_If_finite[simp, intro]:
lp15@68127
   798
  fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
lp15@68127
   799
  shows "finite {r. P r} \<Longrightarrow> convergent_prod (\<lambda>r. if P r then f r else 1)"
lp15@68127
   800
  using convergent_prod_def has_prod_If_finite has_prod_def by fastforce
lp15@68127
   801
lp15@68127
   802
lemma has_prod_single:
lp15@68127
   803
  fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
lp15@68127
   804
  shows "(\<lambda>r. if r = i then f r else 1) has_prod f i"
lp15@68127
   805
  using has_prod_If_finite[of "\<lambda>r. r = i"] by simp
lp15@68127
   806
lp15@68136
   807
context
lp15@68136
   808
  fixes f :: "nat \<Rightarrow> 'a :: real_normed_field"
lp15@68136
   809
begin
lp15@68136
   810
lp15@68136
   811
lemma convergent_prod_imp_has_prod: 
lp15@68136
   812
  assumes "convergent_prod f"
lp15@68136
   813
  shows "\<exists>p. f has_prod p"
lp15@68136
   814
proof -
lp15@68361
   815
  obtain M p where p: "raw_has_prod f M p"
lp15@68136
   816
    using assms convergent_prod_def by blast
lp15@68136
   817
  then have "p \<noteq> 0"
lp15@68361
   818
    using raw_has_prod_nonzero by blast
lp15@68136
   819
  with p have fnz: "f i \<noteq> 0" if "i \<ge> M" for i
lp15@68361
   820
    using raw_has_prod_eq_0 that by blast
lp15@68136
   821
  define C where "C = (\<Prod>n<M. f n)"
lp15@68136
   822
  show ?thesis
lp15@68136
   823
  proof (cases "\<forall>n\<le>M. f n \<noteq> 0")
lp15@68136
   824
    case True
lp15@68136
   825
    then have "C \<noteq> 0"
lp15@68136
   826
      by (simp add: C_def)
lp15@68136
   827
    then show ?thesis
lp15@68136
   828
      by (meson True assms convergent_prod_offset_0 fnz has_prod_def nat_le_linear)
lp15@68136
   829
  next
lp15@68136
   830
    case False
lp15@68136
   831
    let ?N = "GREATEST n. f n = 0"
lp15@68136
   832
    have 0: "f ?N = 0"
lp15@68136
   833
      using fnz False
lp15@68136
   834
      by (metis (mono_tags, lifting) GreatestI_ex_nat nat_le_linear)
lp15@68136
   835
    have "f i \<noteq> 0" if "i > ?N" for i
lp15@68136
   836
      by (metis (mono_tags, lifting) Greatest_le_nat fnz leD linear that)
lp15@68361
   837
    then have "\<exists>p. raw_has_prod f (Suc ?N) p"
lp15@68136
   838
      using assms by (auto simp: intro!: convergent_prod_ignore_nonzero_segment)
lp15@68136
   839
    then show ?thesis
lp15@68136
   840
      unfolding has_prod_def using 0 by blast
lp15@68136
   841
  qed
lp15@68136
   842
qed
lp15@68136
   843
lp15@68136
   844
lemma convergent_prod_has_prod [intro]:
lp15@68136
   845
  shows "convergent_prod f \<Longrightarrow> f has_prod (prodinf f)"
lp15@68136
   846
  unfolding prodinf_def
lp15@68136
   847
  by (metis convergent_prod_imp_has_prod has_prod_unique theI')
lp15@68136
   848
lp15@68136
   849
lemma convergent_prod_LIMSEQ:
lp15@68136
   850
  shows "convergent_prod f \<Longrightarrow> (\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> prodinf f"
lp15@68136
   851
  by (metis convergent_LIMSEQ_iff convergent_prod_has_prod convergent_prod_imp_convergent 
lp15@68361
   852
      convergent_prod_to_zero_iff raw_has_prod_eq_0 has_prod_def prodinf_eq_lim zero_le)
lp15@68136
   853
lp15@68136
   854
lemma has_prod_iff: "f has_prod x \<longleftrightarrow> convergent_prod f \<and> prodinf f = x"
lp15@68136
   855
proof
lp15@68136
   856
  assume "f has_prod x"
lp15@68136
   857
  then show "convergent_prod f \<and> prodinf f = x"
lp15@68136
   858
    apply safe
lp15@68136
   859
    using convergent_prod_def has_prod_def apply blast
lp15@68136
   860
    using has_prod_unique by blast
lp15@68136
   861
qed auto
lp15@68136
   862
lp15@68136
   863
lemma convergent_prod_has_prod_iff: "convergent_prod f \<longleftrightarrow> f has_prod prodinf f"
lp15@68136
   864
  by (auto simp: has_prod_iff convergent_prod_has_prod)
lp15@68136
   865
lp15@68136
   866
lemma prodinf_finite:
lp15@68136
   867
  assumes N: "finite N"
lp15@68136
   868
    and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 1"
lp15@68136
   869
  shows "prodinf f = (\<Prod>n\<in>N. f n)"
lp15@68136
   870
  using has_prod_finite[OF assms, THEN has_prod_unique] by simp
lp15@68127
   871
eberlm@66277
   872
end
lp15@68136
   873
lp15@68361
   874
subsection \<open>Infinite products on ordered, topological monoids\<close>
lp15@68361
   875
lp15@68361
   876
lemma LIMSEQ_prod_0: 
lp15@68361
   877
  fixes f :: "nat \<Rightarrow> 'a::{semidom,topological_space}"
lp15@68361
   878
  assumes "f i = 0"
lp15@68361
   879
  shows "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> 0"
lp15@68361
   880
proof (subst tendsto_cong)
lp15@68361
   881
  show "\<forall>\<^sub>F n in sequentially. prod f {..n} = 0"
lp15@68361
   882
  proof
lp15@68361
   883
    show "prod f {..n} = 0" if "n \<ge> i" for n
lp15@68361
   884
      using that assms by auto
lp15@68361
   885
  qed
lp15@68361
   886
qed auto
lp15@68361
   887
lp15@68361
   888
lemma LIMSEQ_prod_nonneg: 
lp15@68361
   889
  fixes f :: "nat \<Rightarrow> 'a::{linordered_semidom,linorder_topology}"
lp15@68361
   890
  assumes 0: "\<And>n. 0 \<le> f n" and a: "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> a"
lp15@68361
   891
  shows "a \<ge> 0"
lp15@68361
   892
  by (simp add: "0" prod_nonneg LIMSEQ_le_const [OF a])
lp15@68361
   893
lp15@68361
   894
lp15@68361
   895
context
lp15@68361
   896
  fixes f :: "nat \<Rightarrow> 'a::{linordered_semidom,linorder_topology}"
lp15@68361
   897
begin
lp15@68361
   898
lp15@68361
   899
lemma has_prod_le:
lp15@68361
   900
  assumes f: "f has_prod a" and g: "g has_prod b" and le: "\<And>n. 0 \<le> f n \<and> f n \<le> g n"
lp15@68361
   901
  shows "a \<le> b"
lp15@68361
   902
proof (cases "a=0 \<or> b=0")
lp15@68361
   903
  case True
lp15@68361
   904
  then show ?thesis
lp15@68361
   905
  proof
lp15@68361
   906
    assume [simp]: "a=0"
lp15@68361
   907
    have "b \<ge> 0"
lp15@68361
   908
    proof (rule LIMSEQ_prod_nonneg)
lp15@68361
   909
      show "(\<lambda>n. prod g {..n}) \<longlonglongrightarrow> b"
lp15@68361
   910
        using g by (auto simp: has_prod_def raw_has_prod_def LIMSEQ_prod_0)
lp15@68361
   911
    qed (use le order_trans in auto)
lp15@68361
   912
    then show ?thesis
lp15@68361
   913
      by auto
lp15@68361
   914
  next
lp15@68361
   915
    assume [simp]: "b=0"
lp15@68361
   916
    then obtain i where "g i = 0"    
lp15@68361
   917
      using g by (auto simp: prod_defs)
lp15@68361
   918
    then have "f i = 0"
lp15@68361
   919
      using antisym le by force
lp15@68361
   920
    then have "a=0"
lp15@68361
   921
      using f by (auto simp: prod_defs LIMSEQ_prod_0 LIMSEQ_unique)
lp15@68361
   922
    then show ?thesis
lp15@68361
   923
      by auto
lp15@68361
   924
  qed
lp15@68361
   925
next
lp15@68361
   926
  case False
lp15@68361
   927
  then show ?thesis
lp15@68361
   928
    using assms
lp15@68361
   929
    unfolding has_prod_def raw_has_prod_def
lp15@68361
   930
    by (force simp: LIMSEQ_prod_0 intro!: LIMSEQ_le prod_mono)
lp15@68361
   931
qed
lp15@68361
   932
lp15@68361
   933
lemma prodinf_le: 
lp15@68361
   934
  assumes f: "f has_prod a" and g: "g has_prod b" and le: "\<And>n. 0 \<le> f n \<and> f n \<le> g n"
lp15@68361
   935
  shows "prodinf f \<le> prodinf g"
lp15@68361
   936
  using has_prod_le [OF assms] has_prod_unique f g  by blast
lp15@68361
   937
lp15@68136
   938
end
lp15@68361
   939
lp15@68361
   940
lp15@68361
   941
lemma prod_le_prodinf: 
lp15@68361
   942
  fixes f :: "nat \<Rightarrow> 'a::{linordered_idom,linorder_topology}"
lp15@68361
   943
  assumes "f has_prod a" "\<And>i. 0 \<le> f i" "\<And>i. i\<ge>n \<Longrightarrow> 1 \<le> f i"
lp15@68361
   944
  shows "prod f {..<n} \<le> prodinf f"
lp15@68361
   945
  by(rule has_prod_le[OF has_prod_If_finite_set]) (use assms has_prod_unique in auto)
lp15@68361
   946
lp15@68361
   947
lemma prodinf_nonneg:
lp15@68361
   948
  fixes f :: "nat \<Rightarrow> 'a::{linordered_idom,linorder_topology}"
lp15@68361
   949
  assumes "f has_prod a" "\<And>i. 1 \<le> f i" 
lp15@68361
   950
  shows "1 \<le> prodinf f"
lp15@68361
   951
  using prod_le_prodinf[of f a 0] assms
lp15@68361
   952
  by (metis order_trans prod_ge_1 zero_le_one)
lp15@68361
   953
lp15@68361
   954
lemma prodinf_le_const:
lp15@68361
   955
  fixes f :: "nat \<Rightarrow> real"
lp15@68361
   956
  assumes "convergent_prod f" "\<And>n. prod f {..<n} \<le> x" 
lp15@68361
   957
  shows "prodinf f \<le> x"
lp15@68361
   958
  by (metis lessThan_Suc_atMost assms convergent_prod_LIMSEQ LIMSEQ_le_const2)
lp15@68361
   959
lp15@68361
   960
lemma prodinf_eq_one_iff: 
lp15@68361
   961
  fixes f :: "nat \<Rightarrow> real"
lp15@68361
   962
  assumes f: "convergent_prod f" and ge1: "\<And>n. 1 \<le> f n"
lp15@68361
   963
  shows "prodinf f = 1 \<longleftrightarrow> (\<forall>n. f n = 1)"
lp15@68361
   964
proof
lp15@68361
   965
  assume "prodinf f = 1" 
lp15@68361
   966
  then have "(\<lambda>n. \<Prod>i<n. f i) \<longlonglongrightarrow> 1"
lp15@68361
   967
    using convergent_prod_LIMSEQ[of f] assms by (simp add: LIMSEQ_lessThan_iff_atMost)
lp15@68361
   968
  then have "\<And>i. (\<Prod>n\<in>{i}. f n) \<le> 1"
lp15@68361
   969
  proof (rule LIMSEQ_le_const)
lp15@68361
   970
    have "1 \<le> prod f n" for n
lp15@68361
   971
      by (simp add: ge1 prod_ge_1)
lp15@68361
   972
    have "prod f {..<n} = 1" for n
lp15@68361
   973
      by (metis \<open>\<And>n. 1 \<le> prod f n\<close> \<open>prodinf f = 1\<close> antisym f convergent_prod_has_prod ge1 order_trans prod_le_prodinf zero_le_one)
lp15@68361
   974
    then have "(\<Prod>n\<in>{i}. f n) \<le> prod f {..<n}" if "n \<ge> Suc i" for i n
lp15@68361
   975
      by (metis mult.left_neutral order_refl prod.cong prod.neutral_const prod_lessThan_Suc)
lp15@68361
   976
    then show "\<exists>N. \<forall>n\<ge>N. (\<Prod>n\<in>{i}. f n) \<le> prod f {..<n}" for i
lp15@68361
   977
      by blast      
lp15@68361
   978
  qed
lp15@68361
   979
  with ge1 show "\<forall>n. f n = 1"
lp15@68361
   980
    by (auto intro!: antisym)
lp15@68361
   981
qed (metis prodinf_zero fun_eq_iff)
lp15@68361
   982
lp15@68361
   983
lemma prodinf_pos_iff:
lp15@68361
   984
  fixes f :: "nat \<Rightarrow> real"
lp15@68361
   985
  assumes "convergent_prod f" "\<And>n. 1 \<le> f n"
lp15@68361
   986
  shows "1 < prodinf f \<longleftrightarrow> (\<exists>i. 1 < f i)"
lp15@68361
   987
  using prod_le_prodinf[of f 1] prodinf_eq_one_iff
lp15@68361
   988
  by (metis convergent_prod_has_prod assms less_le prodinf_nonneg)
lp15@68361
   989
lp15@68361
   990
lemma less_1_prodinf2:
lp15@68361
   991
  fixes f :: "nat \<Rightarrow> real"
lp15@68361
   992
  assumes "convergent_prod f" "\<And>n. 1 \<le> f n" "1 < f i"
lp15@68361
   993
  shows "1 < prodinf f"
lp15@68361
   994
proof -
lp15@68361
   995
  have "1 < (\<Prod>n<Suc i. f n)"
lp15@68361
   996
    using assms  by (intro less_1_prod2[where i=i]) auto
lp15@68361
   997
  also have "\<dots> \<le> prodinf f"
lp15@68361
   998
    by (intro prod_le_prodinf) (use assms order_trans zero_le_one in \<open>blast+\<close>)
lp15@68361
   999
  finally show ?thesis .
lp15@68361
  1000
qed
lp15@68361
  1001
lp15@68361
  1002
lemma less_1_prodinf:
lp15@68361
  1003
  fixes f :: "nat \<Rightarrow> real"
lp15@68361
  1004
  shows "\<lbrakk>convergent_prod f; \<And>n. 1 < f n\<rbrakk> \<Longrightarrow> 1 < prodinf f"
lp15@68361
  1005
  by (intro less_1_prodinf2[where i=1]) (auto intro: less_imp_le)
lp15@68361
  1006
lp15@68361
  1007
lemma prodinf_nonzero:
lp15@68361
  1008
  fixes f :: "nat \<Rightarrow> 'a :: {idom,topological_semigroup_mult,t2_space}"
lp15@68361
  1009
  assumes "convergent_prod f" "\<And>i. f i \<noteq> 0"
lp15@68361
  1010
  shows "prodinf f \<noteq> 0"
lp15@68361
  1011
  by (metis assms convergent_prod_offset_0 has_prod_unique raw_has_prod_def has_prod_def)
lp15@68361
  1012
lp15@68361
  1013
lemma less_0_prodinf:
lp15@68361
  1014
  fixes f :: "nat \<Rightarrow> real"
lp15@68361
  1015
  assumes f: "convergent_prod f" and 0: "\<And>i. f i > 0"
lp15@68361
  1016
  shows "0 < prodinf f"
lp15@68361
  1017
proof -
lp15@68361
  1018
  have "prodinf f \<noteq> 0"
lp15@68361
  1019
    by (metis assms less_irrefl prodinf_nonzero)
lp15@68361
  1020
  moreover have "0 < (\<Prod>n<i. f n)" for i
lp15@68361
  1021
    by (simp add: 0 prod_pos)
lp15@68361
  1022
  then have "prodinf f \<ge> 0"
lp15@68361
  1023
    using convergent_prod_LIMSEQ [OF f] LIMSEQ_prod_nonneg 0 less_le by blast
lp15@68361
  1024
  ultimately show ?thesis
lp15@68361
  1025
    by auto
lp15@68361
  1026
qed
lp15@68361
  1027
lp15@68361
  1028
lemma prod_less_prodinf2:
lp15@68361
  1029
  fixes f :: "nat \<Rightarrow> real"
lp15@68361
  1030
  assumes f: "convergent_prod f" and 1: "\<And>m. m\<ge>n \<Longrightarrow> 1 \<le> f m" and 0: "\<And>m. 0 < f m" and i: "n \<le> i" "1 < f i"
lp15@68361
  1031
  shows "prod f {..<n} < prodinf f"
lp15@68361
  1032
proof -
lp15@68361
  1033
  have "prod f {..<n} \<le> prod f {..<i}"
lp15@68361
  1034
    by (rule prod_mono2) (use assms less_le in auto)
lp15@68361
  1035
  then have "prod f {..<n} < f i * prod f {..<i}"
lp15@68361
  1036
    using mult_less_le_imp_less[of 1 "f i" "prod f {..<n}" "prod f {..<i}"] assms
lp15@68361
  1037
    by (simp add: prod_pos)
lp15@68361
  1038
  moreover have "prod f {..<Suc i} \<le> prodinf f"
lp15@68361
  1039
    using prod_le_prodinf[of f _ "Suc i"]
lp15@68361
  1040
    by (meson "0" "1" Suc_leD convergent_prod_has_prod f \<open>n \<le> i\<close> le_trans less_eq_real_def)
lp15@68361
  1041
  ultimately show ?thesis
lp15@68361
  1042
    by (metis le_less_trans mult.commute not_le prod_lessThan_Suc)
lp15@68361
  1043
qed
lp15@68361
  1044
lp15@68361
  1045
lemma prod_less_prodinf:
lp15@68361
  1046
  fixes f :: "nat \<Rightarrow> real"
lp15@68361
  1047
  assumes f: "convergent_prod f" and 1: "\<And>m. m\<ge>n \<Longrightarrow> 1 < f m" and 0: "\<And>m. 0 < f m" 
lp15@68361
  1048
  shows "prod f {..<n} < prodinf f"
lp15@68361
  1049
  by (meson "0" "1" f le_less prod_less_prodinf2)
lp15@68361
  1050
lp15@68361
  1051
lemma raw_has_prodI_bounded:
lp15@68361
  1052
  fixes f :: "nat \<Rightarrow> real"
lp15@68361
  1053
  assumes pos: "\<And>n. 1 \<le> f n"
lp15@68361
  1054
    and le: "\<And>n. (\<Prod>i<n. f i) \<le> x"
lp15@68361
  1055
  shows "\<exists>p. raw_has_prod f 0 p"
lp15@68361
  1056
  unfolding raw_has_prod_def add_0_right
lp15@68361
  1057
proof (rule exI LIMSEQ_incseq_SUP conjI)+
lp15@68361
  1058
  show "bdd_above (range (\<lambda>n. prod f {..n}))"
lp15@68361
  1059
    by (metis bdd_aboveI2 le lessThan_Suc_atMost)
lp15@68361
  1060
  then have "(SUP i. prod f {..i}) > 0"
lp15@68361
  1061
    by (metis UNIV_I cSUP_upper less_le_trans pos prod_pos zero_less_one)
lp15@68361
  1062
  then show "(SUP i. prod f {..i}) \<noteq> 0"
lp15@68361
  1063
    by auto
lp15@68361
  1064
  show "incseq (\<lambda>n. prod f {..n})"
lp15@68361
  1065
    using pos order_trans [OF zero_le_one] by (auto simp: mono_def intro!: prod_mono2)
lp15@68361
  1066
qed
lp15@68361
  1067
lp15@68361
  1068
lemma convergent_prodI_nonneg_bounded:
lp15@68361
  1069
  fixes f :: "nat \<Rightarrow> real"
lp15@68361
  1070
  assumes "\<And>n. 1 \<le> f n" "\<And>n. (\<Prod>i<n. f i) \<le> x"
lp15@68361
  1071
  shows "convergent_prod f"
lp15@68361
  1072
  using convergent_prod_def raw_has_prodI_bounded [OF assms] by blast
lp15@68361
  1073
lp15@68361
  1074
lp15@68424
  1075
subsection \<open>Infinite products on topological spaces\<close>
lp15@68361
  1076
lp15@68361
  1077
context
lp15@68361
  1078
  fixes f g :: "nat \<Rightarrow> 'a::{t2_space,topological_semigroup_mult,idom}"
lp15@68361
  1079
begin
lp15@68361
  1080
lp15@68361
  1081
lemma raw_has_prod_mult: "\<lbrakk>raw_has_prod f M a; raw_has_prod g M b\<rbrakk> \<Longrightarrow> raw_has_prod (\<lambda>n. f n * g n) M (a * b)"
lp15@68361
  1082
  by (force simp add: prod.distrib tendsto_mult raw_has_prod_def)
lp15@68361
  1083
lp15@68361
  1084
lemma has_prod_mult_nz: "\<lbrakk>f has_prod a; g has_prod b; a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>n. f n * g n) has_prod (a * b)"
lp15@68361
  1085
  by (simp add: raw_has_prod_mult has_prod_def)
lp15@68361
  1086
lp15@68361
  1087
end
lp15@68361
  1088
lp15@68361
  1089
lp15@68361
  1090
context
lp15@68361
  1091
  fixes f g :: "nat \<Rightarrow> 'a::real_normed_field"
lp15@68361
  1092
begin
lp15@68361
  1093
lp15@68361
  1094
lemma has_prod_mult:
lp15@68361
  1095
  assumes f: "f has_prod a" and g: "g has_prod b"
lp15@68361
  1096
  shows "(\<lambda>n. f n * g n) has_prod (a * b)"
lp15@68361
  1097
  using f [unfolded has_prod_def]
lp15@68361
  1098
proof (elim disjE exE conjE)
lp15@68361
  1099
  assume f0: "raw_has_prod f 0 a"
lp15@68361
  1100
  show ?thesis
lp15@68361
  1101
    using g [unfolded has_prod_def]
lp15@68361
  1102
  proof (elim disjE exE conjE)
lp15@68361
  1103
    assume g0: "raw_has_prod g 0 b"
lp15@68361
  1104
    with f0 show ?thesis
lp15@68361
  1105
      by (force simp add: has_prod_def prod.distrib tendsto_mult raw_has_prod_def)
lp15@68361
  1106
  next
lp15@68361
  1107
    fix j q
lp15@68361
  1108
    assume "b = 0" and "g j = 0" and q: "raw_has_prod g (Suc j) q"
lp15@68361
  1109
    obtain p where p: "raw_has_prod f (Suc j) p"
lp15@68361
  1110
      using f0 raw_has_prod_ignore_initial_segment by blast
lp15@68361
  1111
    then have "Ex (raw_has_prod (\<lambda>n. f n * g n) (Suc j))"
lp15@68361
  1112
      using q raw_has_prod_mult by blast
lp15@68361
  1113
    then show ?thesis
lp15@68361
  1114
      using \<open>b = 0\<close> \<open>g j = 0\<close> has_prod_0_iff by fastforce
lp15@68361
  1115
  qed
lp15@68361
  1116
next
lp15@68361
  1117
  fix i p
lp15@68361
  1118
  assume "a = 0" and "f i = 0" and p: "raw_has_prod f (Suc i) p"
lp15@68361
  1119
  show ?thesis
lp15@68361
  1120
    using g [unfolded has_prod_def]
lp15@68361
  1121
  proof (elim disjE exE conjE)
lp15@68361
  1122
    assume g0: "raw_has_prod g 0 b"
lp15@68361
  1123
    obtain q where q: "raw_has_prod g (Suc i) q"
lp15@68361
  1124
      using g0 raw_has_prod_ignore_initial_segment by blast
lp15@68361
  1125
    then have "Ex (raw_has_prod (\<lambda>n. f n * g n) (Suc i))"
lp15@68361
  1126
      using raw_has_prod_mult p by blast
lp15@68361
  1127
    then show ?thesis
lp15@68361
  1128
      using \<open>a = 0\<close> \<open>f i = 0\<close> has_prod_0_iff by fastforce
lp15@68361
  1129
  next
lp15@68361
  1130
    fix j q
lp15@68361
  1131
    assume "b = 0" and "g j = 0" and q: "raw_has_prod g (Suc j) q"
lp15@68361
  1132
    obtain p' where p': "raw_has_prod f (Suc (max i j)) p'"
lp15@68361
  1133
      by (metis raw_has_prod_ignore_initial_segment max_Suc_Suc max_def p)
lp15@68361
  1134
    moreover
lp15@68361
  1135
    obtain q' where q': "raw_has_prod g (Suc (max i j)) q'"
lp15@68361
  1136
      by (metis raw_has_prod_ignore_initial_segment max.cobounded2 max_Suc_Suc q)
lp15@68361
  1137
    ultimately show ?thesis
lp15@68361
  1138
      using \<open>b = 0\<close> by (simp add: has_prod_def) (metis \<open>f i = 0\<close> \<open>g j = 0\<close> raw_has_prod_mult max_def)
lp15@68361
  1139
  qed
lp15@68361
  1140
qed
lp15@68361
  1141
lp15@68361
  1142
lemma convergent_prod_mult:
lp15@68361
  1143
  assumes f: "convergent_prod f" and g: "convergent_prod g"
lp15@68361
  1144
  shows "convergent_prod (\<lambda>n. f n * g n)"
lp15@68361
  1145
  unfolding convergent_prod_def
lp15@68361
  1146
proof -
lp15@68361
  1147
  obtain M p N q where p: "raw_has_prod f M p" and q: "raw_has_prod g N q"
lp15@68361
  1148
    using convergent_prod_def f g by blast+
lp15@68361
  1149
  then obtain p' q' where p': "raw_has_prod f (max M N) p'" and q': "raw_has_prod g (max M N) q'"
lp15@68361
  1150
    by (meson raw_has_prod_ignore_initial_segment max.cobounded1 max.cobounded2)
lp15@68361
  1151
  then show "\<exists>M p. raw_has_prod (\<lambda>n. f n * g n) M p"
lp15@68361
  1152
    using raw_has_prod_mult by blast
lp15@68361
  1153
qed
lp15@68361
  1154
lp15@68361
  1155
lemma prodinf_mult: "convergent_prod f \<Longrightarrow> convergent_prod g \<Longrightarrow> prodinf f * prodinf g = (\<Prod>n. f n * g n)"
lp15@68361
  1156
  by (intro has_prod_unique has_prod_mult convergent_prod_has_prod)
lp15@68361
  1157
lp15@68361
  1158
end
lp15@68361
  1159
lp15@68361
  1160
context
lp15@68361
  1161
  fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::real_normed_field"
lp15@68361
  1162
    and I :: "'i set"
lp15@68361
  1163
begin
lp15@68361
  1164
lp15@68361
  1165
lemma has_prod_prod: "(\<And>i. i \<in> I \<Longrightarrow> (f i) has_prod (x i)) \<Longrightarrow> (\<lambda>n. \<Prod>i\<in>I. f i n) has_prod (\<Prod>i\<in>I. x i)"
lp15@68361
  1166
  by (induct I rule: infinite_finite_induct) (auto intro!: has_prod_mult)
lp15@68361
  1167
lp15@68361
  1168
lemma prodinf_prod: "(\<And>i. i \<in> I \<Longrightarrow> convergent_prod (f i)) \<Longrightarrow> (\<Prod>n. \<Prod>i\<in>I. f i n) = (\<Prod>i\<in>I. \<Prod>n. f i n)"
lp15@68361
  1169
  using has_prod_unique[OF has_prod_prod, OF convergent_prod_has_prod] by simp
lp15@68361
  1170
lp15@68361
  1171
lemma convergent_prod_prod: "(\<And>i. i \<in> I \<Longrightarrow> convergent_prod (f i)) \<Longrightarrow> convergent_prod (\<lambda>n. \<Prod>i\<in>I. f i n)"
lp15@68361
  1172
  using convergent_prod_has_prod_iff has_prod_prod prodinf_prod by force
lp15@68361
  1173
lp15@68361
  1174
end
lp15@68361
  1175
lp15@68424
  1176
subsection \<open>Infinite summability on real normed fields\<close>
lp15@68361
  1177
lp15@68361
  1178
context
lp15@68361
  1179
  fixes f :: "nat \<Rightarrow> 'a::real_normed_field"
lp15@68361
  1180
begin
lp15@68361
  1181
lp15@68361
  1182
lemma raw_has_prod_Suc_iff: "raw_has_prod f M (a * f M) \<longleftrightarrow> raw_has_prod (\<lambda>n. f (Suc n)) M a \<and> f M \<noteq> 0"
lp15@68361
  1183
proof -
lp15@68361
  1184
  have "raw_has_prod f M (a * f M) \<longleftrightarrow> (\<lambda>i. \<Prod>j\<le>Suc i. f (j+M)) \<longlonglongrightarrow> a * f M \<and> a * f M \<noteq> 0"
lp15@68361
  1185
    by (subst LIMSEQ_Suc_iff) (simp add: raw_has_prod_def)
lp15@68361
  1186
  also have "\<dots> \<longleftrightarrow> (\<lambda>i. (\<Prod>j\<le>i. f (Suc j + M)) * f M) \<longlonglongrightarrow> a * f M \<and> a * f M \<noteq> 0"
lp15@68361
  1187
    by (simp add: ac_simps atMost_Suc_eq_insert_0 image_Suc_atMost prod_atLeast1_atMost_eq lessThan_Suc_atMost)
lp15@68361
  1188
  also have "\<dots> \<longleftrightarrow> raw_has_prod (\<lambda>n. f (Suc n)) M a \<and> f M \<noteq> 0"
lp15@68361
  1189
  proof safe
lp15@68361
  1190
    assume tends: "(\<lambda>i. (\<Prod>j\<le>i. f (Suc j + M)) * f M) \<longlonglongrightarrow> a * f M" and 0: "a * f M \<noteq> 0"
lp15@68361
  1191
    with tendsto_divide[OF tends tendsto_const, of "f M"]    
lp15@68361
  1192
    show "raw_has_prod (\<lambda>n. f (Suc n)) M a"
lp15@68361
  1193
      by (simp add: raw_has_prod_def)
lp15@68361
  1194
  qed (auto intro: tendsto_mult_right simp:  raw_has_prod_def)
lp15@68361
  1195
  finally show ?thesis .
lp15@68361
  1196
qed
lp15@68361
  1197
lp15@68361
  1198
lemma has_prod_Suc_iff:
lp15@68361
  1199
  assumes "f 0 \<noteq> 0" shows "(\<lambda>n. f (Suc n)) has_prod a \<longleftrightarrow> f has_prod (a * f 0)"
lp15@68361
  1200
proof (cases "a = 0")
lp15@68361
  1201
  case True
lp15@68361
  1202
  then show ?thesis
lp15@68361
  1203
  proof (simp add: has_prod_def, safe)
lp15@68361
  1204
    fix i x
lp15@68361
  1205
    assume "f (Suc i) = 0" and "raw_has_prod (\<lambda>n. f (Suc n)) (Suc i) x"
lp15@68361
  1206
    then obtain y where "raw_has_prod f (Suc (Suc i)) y"
lp15@68361
  1207
      by (metis (no_types) raw_has_prod_eq_0 Suc_n_not_le_n raw_has_prod_Suc_iff raw_has_prod_ignore_initial_segment raw_has_prod_nonzero linear)
lp15@68361
  1208
    then show "\<exists>i. f i = 0 \<and> Ex (raw_has_prod f (Suc i))"
lp15@68361
  1209
      using \<open>f (Suc i) = 0\<close> by blast
lp15@68361
  1210
  next
lp15@68361
  1211
    fix i x
lp15@68361
  1212
    assume "f i = 0" and x: "raw_has_prod f (Suc i) x"
lp15@68361
  1213
    then obtain j where j: "i = Suc j"
lp15@68361
  1214
      by (metis assms not0_implies_Suc)
lp15@68361
  1215
    moreover have "\<exists> y. raw_has_prod (\<lambda>n. f (Suc n)) i y"
lp15@68361
  1216
      using x by (auto simp: raw_has_prod_def)
lp15@68361
  1217
    then show "\<exists>i. f (Suc i) = 0 \<and> Ex (raw_has_prod (\<lambda>n. f (Suc n)) (Suc i))"
lp15@68361
  1218
      using \<open>f i = 0\<close> j by blast
lp15@68361
  1219
  qed
lp15@68361
  1220
next
lp15@68361
  1221
  case False
lp15@68361
  1222
  then show ?thesis
lp15@68361
  1223
    by (auto simp: has_prod_def raw_has_prod_Suc_iff assms)
lp15@68361
  1224
qed
lp15@68361
  1225
lp15@68361
  1226
lemma convergent_prod_Suc_iff:
lp15@68452
  1227
  shows "convergent_prod (\<lambda>n. f (Suc n)) = convergent_prod f"
lp15@68361
  1228
proof
lp15@68361
  1229
  assume "convergent_prod f"
lp15@68452
  1230
  then obtain M L where M_nz:"\<forall>n\<ge>M. f n \<noteq> 0" and 
lp15@68452
  1231
        M_L:"(\<lambda>n. \<Prod>i\<le>n. f (i + M)) \<longlonglongrightarrow> L" and "L \<noteq> 0" 
lp15@68452
  1232
    unfolding convergent_prod_altdef by auto
lp15@68452
  1233
  have "(\<lambda>n. \<Prod>i\<le>n. f (Suc (i + M))) \<longlonglongrightarrow> L / f M"
lp15@68452
  1234
  proof -
lp15@68452
  1235
    have "(\<lambda>n. \<Prod>i\<in>{0..Suc n}. f (i + M)) \<longlonglongrightarrow> L"
lp15@68452
  1236
      using M_L 
lp15@68452
  1237
      apply (subst (asm) LIMSEQ_Suc_iff[symmetric]) 
lp15@68452
  1238
      using atLeast0AtMost by auto
lp15@68452
  1239
    then have "(\<lambda>n. f M * (\<Prod>i\<in>{0..n}. f (Suc (i + M)))) \<longlonglongrightarrow> L"
lp15@68452
  1240
      apply (subst (asm) prod.atLeast0_atMost_Suc_shift)
lp15@68452
  1241
      by simp
lp15@68452
  1242
    then have "(\<lambda>n. (\<Prod>i\<in>{0..n}. f (Suc (i + M)))) \<longlonglongrightarrow> L/f M"
lp15@68452
  1243
      apply (drule_tac tendsto_divide)
lp15@68452
  1244
      using M_nz[rule_format,of M,simplified] by auto
lp15@68452
  1245
    then show ?thesis unfolding atLeast0AtMost .
lp15@68452
  1246
  qed
lp15@68452
  1247
  then show "convergent_prod (\<lambda>n. f (Suc n))" unfolding convergent_prod_altdef
lp15@68452
  1248
    apply (rule_tac exI[where x=M])
lp15@68452
  1249
    apply (rule_tac exI[where x="L/f M"])
lp15@68452
  1250
    using M_nz \<open>L\<noteq>0\<close> by auto
lp15@68361
  1251
next
lp15@68361
  1252
  assume "convergent_prod (\<lambda>n. f (Suc n))"
lp15@68452
  1253
  then obtain M where "\<exists>L. (\<forall>n\<ge>M. f (Suc n) \<noteq> 0) \<and> (\<lambda>n. \<Prod>i\<le>n. f (Suc (i + M))) \<longlonglongrightarrow> L \<and> L \<noteq> 0"
lp15@68452
  1254
    unfolding convergent_prod_altdef by auto
lp15@68452
  1255
  then show "convergent_prod f" unfolding convergent_prod_altdef
lp15@68452
  1256
    apply (rule_tac exI[where x="Suc M"])
lp15@68452
  1257
    using Suc_le_D by auto
lp15@68361
  1258
qed
lp15@68361
  1259
lp15@68361
  1260
lemma raw_has_prod_inverse: 
lp15@68361
  1261
  assumes "raw_has_prod f M a" shows "raw_has_prod (\<lambda>n. inverse (f n)) M (inverse a)"
lp15@68361
  1262
  using assms unfolding raw_has_prod_def by (auto dest: tendsto_inverse simp: prod_inversef [symmetric])
lp15@68361
  1263
lp15@68361
  1264
lemma has_prod_inverse: 
lp15@68361
  1265
  assumes "f has_prod a" shows "(\<lambda>n. inverse (f n)) has_prod (inverse a)"
lp15@68361
  1266
using assms raw_has_prod_inverse unfolding has_prod_def by auto 
lp15@68361
  1267
lp15@68361
  1268
lemma convergent_prod_inverse:
lp15@68361
  1269
  assumes "convergent_prod f" 
lp15@68361
  1270
  shows "convergent_prod (\<lambda>n. inverse (f n))"
lp15@68361
  1271
  using assms unfolding convergent_prod_def  by (blast intro: raw_has_prod_inverse elim: )
lp15@68361
  1272
lp15@68361
  1273
end
lp15@68361
  1274
lp15@68424
  1275
context 
lp15@68361
  1276
  fixes f :: "nat \<Rightarrow> 'a::real_normed_field"
lp15@68361
  1277
begin
lp15@68361
  1278
lp15@68361
  1279
lemma raw_has_prod_Suc_iff': "raw_has_prod f M a \<longleftrightarrow> raw_has_prod (\<lambda>n. f (Suc n)) M (a / f M) \<and> f M \<noteq> 0"
lp15@68361
  1280
  by (metis raw_has_prod_eq_0 add.commute add.left_neutral raw_has_prod_Suc_iff raw_has_prod_nonzero le_add1 nonzero_mult_div_cancel_right times_divide_eq_left)
lp15@68361
  1281
lp15@68361
  1282
lemma has_prod_divide: "f has_prod a \<Longrightarrow> g has_prod b \<Longrightarrow> (\<lambda>n. f n / g n) has_prod (a / b)"
lp15@68361
  1283
  unfolding divide_inverse by (intro has_prod_inverse has_prod_mult)
lp15@68361
  1284
lp15@68361
  1285
lemma convergent_prod_divide:
lp15@68361
  1286
  assumes f: "convergent_prod f" and g: "convergent_prod g"
lp15@68361
  1287
  shows "convergent_prod (\<lambda>n. f n / g n)"
lp15@68361
  1288
  using f g has_prod_divide has_prod_iff by blast
lp15@68361
  1289
lp15@68361
  1290
lemma prodinf_divide: "convergent_prod f \<Longrightarrow> convergent_prod g \<Longrightarrow> prodinf f / prodinf g = (\<Prod>n. f n / g n)"
lp15@68361
  1291
  by (intro has_prod_unique has_prod_divide convergent_prod_has_prod)
lp15@68361
  1292
lp15@68361
  1293
lemma prodinf_inverse: "convergent_prod f \<Longrightarrow> (\<Prod>n. inverse (f n)) = inverse (\<Prod>n. f n)"
lp15@68361
  1294
  by (intro has_prod_unique [symmetric] has_prod_inverse convergent_prod_has_prod)
lp15@68361
  1295
lp15@68452
  1296
lemma has_prod_Suc_imp: 
lp15@68452
  1297
  assumes "(\<lambda>n. f (Suc n)) has_prod a"
lp15@68452
  1298
  shows "f has_prod (a * f 0)"
lp15@68452
  1299
proof -
lp15@68452
  1300
  have "f has_prod (a * f 0)" when "raw_has_prod (\<lambda>n. f (Suc n)) 0 a" 
lp15@68452
  1301
    apply (cases "f 0=0")
lp15@68452
  1302
    using that unfolding has_prod_def raw_has_prod_Suc 
lp15@68452
  1303
    by (auto simp add: raw_has_prod_Suc_iff)
lp15@68452
  1304
  moreover have "f has_prod (a * f 0)" when 
lp15@68452
  1305
    "(\<exists>i q. a = 0 \<and> f (Suc i) = 0 \<and> raw_has_prod (\<lambda>n. f (Suc n)) (Suc i) q)" 
lp15@68452
  1306
  proof -
lp15@68452
  1307
    from that 
lp15@68452
  1308
    obtain i q where "a = 0" "f (Suc i) = 0" "raw_has_prod (\<lambda>n. f (Suc n)) (Suc i) q"
lp15@68452
  1309
      by auto
lp15@68452
  1310
    then show ?thesis unfolding has_prod_def 
lp15@68452
  1311
      by (auto intro!:exI[where x="Suc i"] simp:raw_has_prod_Suc)
lp15@68452
  1312
  qed
lp15@68452
  1313
  ultimately show "f has_prod (a * f 0)" using assms unfolding has_prod_def by auto
lp15@68452
  1314
qed
lp15@68452
  1315
lp15@68361
  1316
lemma has_prod_iff_shift: 
lp15@68361
  1317
  assumes "\<And>i. i < n \<Longrightarrow> f i \<noteq> 0"
lp15@68361
  1318
  shows "(\<lambda>i. f (i + n)) has_prod a \<longleftrightarrow> f has_prod (a * (\<Prod>i<n. f i))"
lp15@68361
  1319
  using assms
lp15@68361
  1320
proof (induct n arbitrary: a)
lp15@68361
  1321
  case 0
lp15@68361
  1322
  then show ?case by simp
lp15@68361
  1323
next
lp15@68361
  1324
  case (Suc n)
lp15@68361
  1325
  then have "(\<lambda>i. f (Suc i + n)) has_prod a \<longleftrightarrow> (\<lambda>i. f (i + n)) has_prod (a * f n)"
lp15@68361
  1326
    by (subst has_prod_Suc_iff) auto
lp15@68361
  1327
  with Suc show ?case
lp15@68361
  1328
    by (simp add: ac_simps)
lp15@68361
  1329
qed
lp15@68361
  1330
lp15@68361
  1331
corollary has_prod_iff_shift':
lp15@68361
  1332
  assumes "\<And>i. i < n \<Longrightarrow> f i \<noteq> 0"
lp15@68361
  1333
  shows "(\<lambda>i. f (i + n)) has_prod (a / (\<Prod>i<n. f i)) \<longleftrightarrow> f has_prod a"
lp15@68361
  1334
  by (simp add: assms has_prod_iff_shift)
lp15@68361
  1335
lp15@68361
  1336
lemma has_prod_one_iff_shift:
lp15@68361
  1337
  assumes "\<And>i. i < n \<Longrightarrow> f i = 1"
lp15@68361
  1338
  shows "(\<lambda>i. f (i+n)) has_prod a \<longleftrightarrow> (\<lambda>i. f i) has_prod a"
lp15@68361
  1339
  by (simp add: assms has_prod_iff_shift)
lp15@68361
  1340
lp15@68361
  1341
lemma convergent_prod_iff_shift:
lp15@68361
  1342
  shows "convergent_prod (\<lambda>i. f (i + n)) \<longleftrightarrow> convergent_prod f"
lp15@68361
  1343
  apply safe
lp15@68361
  1344
  using convergent_prod_offset apply blast
lp15@68361
  1345
  using convergent_prod_ignore_initial_segment convergent_prod_def by blast
lp15@68361
  1346
lp15@68361
  1347
lemma has_prod_split_initial_segment:
lp15@68361
  1348
  assumes "f has_prod a" "\<And>i. i < n \<Longrightarrow> f i \<noteq> 0"
lp15@68361
  1349
  shows "(\<lambda>i. f (i + n)) has_prod (a / (\<Prod>i<n. f i))"
lp15@68361
  1350
  using assms has_prod_iff_shift' by blast
lp15@68361
  1351
lp15@68361
  1352
lemma prodinf_divide_initial_segment:
lp15@68361
  1353
  assumes "convergent_prod f" "\<And>i. i < n \<Longrightarrow> f i \<noteq> 0"
lp15@68361
  1354
  shows "(\<Prod>i. f (i + n)) = (\<Prod>i. f i) / (\<Prod>i<n. f i)"
lp15@68361
  1355
  by (rule has_prod_unique[symmetric]) (auto simp: assms has_prod_iff_shift)
lp15@68361
  1356
lp15@68361
  1357
lemma prodinf_split_initial_segment:
lp15@68361
  1358
  assumes "convergent_prod f" "\<And>i. i < n \<Longrightarrow> f i \<noteq> 0"
lp15@68361
  1359
  shows "prodinf f = (\<Prod>i. f (i + n)) * (\<Prod>i<n. f i)"
lp15@68361
  1360
  by (auto simp add: assms prodinf_divide_initial_segment)
lp15@68361
  1361
lp15@68361
  1362
lemma prodinf_split_head:
lp15@68361
  1363
  assumes "convergent_prod f" "f 0 \<noteq> 0"
lp15@68361
  1364
  shows "(\<Prod>n. f (Suc n)) = prodinf f / f 0"
lp15@68361
  1365
  using prodinf_split_initial_segment[of 1] assms by simp
lp15@68361
  1366
lp15@68361
  1367
end
lp15@68361
  1368
lp15@68424
  1369
context 
lp15@68361
  1370
  fixes f :: "nat \<Rightarrow> 'a::real_normed_field"
lp15@68361
  1371
begin
lp15@68361
  1372
lp15@68361
  1373
lemma convergent_prod_inverse_iff: "convergent_prod (\<lambda>n. inverse (f n)) \<longleftrightarrow> convergent_prod f"
lp15@68361
  1374
  by (auto dest: convergent_prod_inverse)
lp15@68361
  1375
lp15@68361
  1376
lemma convergent_prod_const_iff:
lp15@68361
  1377
  fixes c :: "'a :: {real_normed_field}"
lp15@68361
  1378
  shows "convergent_prod (\<lambda>_. c) \<longleftrightarrow> c = 1"
lp15@68361
  1379
proof
lp15@68361
  1380
  assume "convergent_prod (\<lambda>_. c)"
lp15@68361
  1381
  then show "c = 1"
lp15@68361
  1382
    using convergent_prod_imp_LIMSEQ LIMSEQ_unique by blast 
lp15@68361
  1383
next
lp15@68361
  1384
  assume "c = 1"
lp15@68361
  1385
  then show "convergent_prod (\<lambda>_. c)"
lp15@68361
  1386
    by auto
lp15@68361
  1387
qed
lp15@68361
  1388
lp15@68361
  1389
lemma has_prod_power: "f has_prod a \<Longrightarrow> (\<lambda>i. f i ^ n) has_prod (a ^ n)"
lp15@68361
  1390
  by (induction n) (auto simp: has_prod_mult)
lp15@68361
  1391
lp15@68361
  1392
lemma convergent_prod_power: "convergent_prod f \<Longrightarrow> convergent_prod (\<lambda>i. f i ^ n)"
lp15@68361
  1393
  by (induction n) (auto simp: convergent_prod_mult)
lp15@68361
  1394
lp15@68361
  1395
lemma prodinf_power: "convergent_prod f \<Longrightarrow> prodinf (\<lambda>i. f i ^ n) = prodinf f ^ n"
lp15@68361
  1396
  by (metis has_prod_unique convergent_prod_imp_has_prod has_prod_power)
lp15@68361
  1397
lp15@68361
  1398
end
lp15@68361
  1399
lp15@68424
  1400
lp15@68424
  1401
subsection\<open>Exponentials and logarithms\<close>
lp15@68424
  1402
lp15@68424
  1403
context 
lp15@68424
  1404
  fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
lp15@68424
  1405
begin
lp15@68424
  1406
lp15@68424
  1407
lemma sums_imp_has_prod_exp: 
lp15@68424
  1408
  assumes "f sums s"
lp15@68424
  1409
  shows "raw_has_prod (\<lambda>i. exp (f i)) 0 (exp s)"
lp15@68424
  1410
  using assms continuous_on_exp [of UNIV "\<lambda>x::'a. x"]
lp15@68424
  1411
  using continuous_on_tendsto_compose [of UNIV exp "(\<lambda>n. sum f {..n})" s]
lp15@68424
  1412
  by (simp add: prod_defs sums_def_le exp_sum)
lp15@68424
  1413
lp15@68424
  1414
lemma convergent_prod_exp: 
lp15@68424
  1415
  assumes "summable f"
lp15@68424
  1416
  shows "convergent_prod (\<lambda>i. exp (f i))"
lp15@68424
  1417
  using sums_imp_has_prod_exp assms unfolding summable_def convergent_prod_def  by blast
lp15@68424
  1418
lp15@68424
  1419
lemma prodinf_exp: 
lp15@68424
  1420
  assumes "summable f"
lp15@68424
  1421
  shows "prodinf (\<lambda>i. exp (f i)) = exp (suminf f)"
lp15@68424
  1422
proof -
lp15@68424
  1423
  have "f sums suminf f"
lp15@68424
  1424
    using assms by blast
lp15@68424
  1425
  then have "(\<lambda>i. exp (f i)) has_prod exp (suminf f)"
lp15@68424
  1426
    by (simp add: has_prod_def sums_imp_has_prod_exp)
lp15@68424
  1427
  then show ?thesis
lp15@68424
  1428
    by (rule has_prod_unique [symmetric])
lp15@68424
  1429
qed
lp15@68424
  1430
lp15@68361
  1431
end
lp15@68424
  1432
lp15@68585
  1433
lemma convergent_prod_iff_summable_real:
lp15@68585
  1434
  fixes a :: "nat \<Rightarrow> real"
lp15@68585
  1435
  assumes "\<And>n. a n > 0"
lp15@68585
  1436
  shows "convergent_prod (\<lambda>k. 1 + a k) \<longleftrightarrow> summable a" (is "?lhs = ?rhs")
lp15@68585
  1437
proof
lp15@68585
  1438
  assume ?lhs
lp15@68585
  1439
  then obtain p where "raw_has_prod (\<lambda>k. 1 + a k) 0 p"
lp15@68585
  1440
    by (metis assms add_less_same_cancel2 convergent_prod_offset_0 not_one_less_zero)
lp15@68585
  1441
  then have to_p: "(\<lambda>n. \<Prod>k\<le>n. 1 + a k) \<longlonglongrightarrow> p"
lp15@68585
  1442
    by (auto simp: raw_has_prod_def)
lp15@68585
  1443
  moreover have le: "(\<Sum>k\<le>n. a k) \<le> (\<Prod>k\<le>n. 1 + a k)" for n
lp15@68585
  1444
    by (rule sum_le_prod) (use assms less_le in force)
lp15@68585
  1445
  have "(\<Prod>k\<le>n. 1 + a k) \<le> p" for n
lp15@68585
  1446
  proof (rule incseq_le [OF _ to_p])
lp15@68585
  1447
    show "incseq (\<lambda>n. \<Prod>k\<le>n. 1 + a k)"
lp15@68585
  1448
      using assms by (auto simp: mono_def order.strict_implies_order intro!: prod_mono2)
lp15@68585
  1449
  qed
lp15@68585
  1450
  with le have "(\<Sum>k\<le>n. a k) \<le> p" for n
lp15@68585
  1451
    by (metis order_trans)
lp15@68585
  1452
  with assms bounded_imp_summable show ?rhs
lp15@68585
  1453
    by (metis not_less order.asym)
lp15@68585
  1454
next
lp15@68585
  1455
  assume R: ?rhs
lp15@68585
  1456
  have "(\<Prod>k\<le>n. 1 + a k) \<le> exp (suminf a)" for n
lp15@68585
  1457
  proof -
lp15@68585
  1458
    have "(\<Prod>k\<le>n. 1 + a k) \<le> exp (\<Sum>k\<le>n. a k)" for n
lp15@68585
  1459
      by (rule prod_le_exp_sum) (use assms less_le in force)
lp15@68585
  1460
    moreover have "exp (\<Sum>k\<le>n. a k) \<le> exp (suminf a)" for n
lp15@68585
  1461
      unfolding exp_le_cancel_iff
lp15@68585
  1462
      by (meson sum_le_suminf R assms finite_atMost less_eq_real_def)
lp15@68585
  1463
    ultimately show ?thesis
lp15@68585
  1464
      by (meson order_trans)
lp15@68585
  1465
  qed
lp15@68585
  1466
  then obtain L where L: "(\<lambda>n. \<Prod>k\<le>n. 1 + a k) \<longlonglongrightarrow> L"
lp15@68585
  1467
    by (metis assms bounded_imp_convergent_prod convergent_prod_iff_nz_lim le_add_same_cancel1 le_add_same_cancel2 less_le not_le zero_le_one)
lp15@68585
  1468
  moreover have "L \<noteq> 0"
lp15@68585
  1469
  proof
lp15@68585
  1470
    assume "L = 0"
lp15@68585
  1471
    with L have "(\<lambda>n. \<Prod>k\<le>n. 1 + a k) \<longlonglongrightarrow> 0"
lp15@68585
  1472
      by simp
lp15@68585
  1473
    moreover have "(\<Prod>k\<le>n. 1 + a k) > 1" for n
lp15@68585
  1474
      by (simp add: assms less_1_prod)
lp15@68585
  1475
    ultimately show False
lp15@68585
  1476
      by (meson Lim_bounded2 not_one_le_zero less_imp_le)
lp15@68585
  1477
  qed
lp15@68585
  1478
  ultimately show ?lhs
lp15@68585
  1479
    using assms convergent_prod_iff_nz_lim
lp15@68585
  1480
    by (metis add_less_same_cancel1 less_le not_le zero_less_one)
lp15@68585
  1481
qed
lp15@68585
  1482
lp15@68452
  1483
lemma exp_suminf_prodinf_real:
lp15@68452
  1484
  fixes f :: "nat \<Rightarrow> real"
lp15@68452
  1485
  assumes ge0:"\<And>n. f n \<ge> 0" and ac: "abs_convergent_prod (\<lambda>n. exp (f n))"
lp15@68452
  1486
  shows "prodinf (\<lambda>i. exp (f i)) = exp (suminf f)"
lp15@68452
  1487
proof -
lp15@68517
  1488
  have "summable f"
lp15@68452
  1489
    using ac unfolding abs_convergent_prod_conv_summable
lp15@68452
  1490
  proof (elim summable_comparison_test')
lp15@68452
  1491
    fix n
lp15@68517
  1492
    have "\<bar>f n\<bar> = f n"
lp15@68517
  1493
      by (simp add: ge0)
lp15@68517
  1494
    also have "\<dots> \<le> exp (f n) - 1"
lp15@68517
  1495
      by (metis diff_diff_add exp_ge_add_one_self ge_iff_diff_ge_0)
lp15@68517
  1496
    finally show "norm (f n) \<le> norm (exp (f n) - 1)"
lp15@68517
  1497
      by simp
lp15@68452
  1498
  qed
lp15@68452
  1499
  then show ?thesis
lp15@68452
  1500
    by (simp add: prodinf_exp)
lp15@68452
  1501
qed
lp15@68452
  1502
lp15@68424
  1503
lemma has_prod_imp_sums_ln_real: 
lp15@68424
  1504
  fixes f :: "nat \<Rightarrow> real"
lp15@68424
  1505
  assumes "raw_has_prod f 0 p" and 0: "\<And>x. f x > 0"
lp15@68424
  1506
  shows "(\<lambda>i. ln (f i)) sums (ln p)"
lp15@68424
  1507
proof -
lp15@68424
  1508
  have "p > 0"
lp15@68424
  1509
    using assms unfolding prod_defs by (metis LIMSEQ_prod_nonneg less_eq_real_def)
lp15@68424
  1510
  then show ?thesis
lp15@68424
  1511
  using assms continuous_on_ln [of "{0<..}" "\<lambda>x. x"]
lp15@68424
  1512
  using continuous_on_tendsto_compose [of "{0<..}" ln "(\<lambda>n. prod f {..n})" p]
lp15@68424
  1513
  by (auto simp: prod_defs sums_def_le ln_prod order_tendstoD)
lp15@68424
  1514
qed
lp15@68424
  1515
lp15@68424
  1516
lemma summable_ln_real: 
lp15@68424
  1517
  fixes f :: "nat \<Rightarrow> real"
lp15@68424
  1518
  assumes f: "convergent_prod f" and 0: "\<And>x. f x > 0"
lp15@68424
  1519
  shows "summable (\<lambda>i. ln (f i))"
lp15@68424
  1520
proof -
lp15@68424
  1521
  obtain M p where "raw_has_prod f M p"
lp15@68424
  1522
    using f convergent_prod_def by blast
lp15@68424
  1523
  then consider i where "i<M" "f i = 0" | p where "raw_has_prod f 0 p"
lp15@68424
  1524
    using raw_has_prod_cases by blast
lp15@68424
  1525
  then show ?thesis
lp15@68424
  1526
  proof cases
lp15@68424
  1527
    case 1
lp15@68424
  1528
    with 0 show ?thesis
lp15@68424
  1529
      by (metis less_irrefl)
lp15@68424
  1530
  next
lp15@68424
  1531
    case 2
lp15@68424
  1532
    then show ?thesis
lp15@68424
  1533
      using "0" has_prod_imp_sums_ln_real summable_def by blast
lp15@68424
  1534
  qed
lp15@68424
  1535
qed
lp15@68424
  1536
lp15@68424
  1537
lemma suminf_ln_real: 
lp15@68424
  1538
  fixes f :: "nat \<Rightarrow> real"
lp15@68424
  1539
  assumes f: "convergent_prod f" and 0: "\<And>x. f x > 0"
lp15@68424
  1540
  shows "suminf (\<lambda>i. ln (f i)) = ln (prodinf f)"
lp15@68424
  1541
proof -
lp15@68424
  1542
  have "f has_prod prodinf f"
lp15@68424
  1543
    by (simp add: f has_prod_iff)
lp15@68424
  1544
  then have "raw_has_prod f 0 (prodinf f)"
lp15@68424
  1545
    by (metis "0" has_prod_def less_irrefl)
lp15@68424
  1546
  then have "(\<lambda>i. ln (f i)) sums ln (prodinf f)"
lp15@68424
  1547
    using "0" has_prod_imp_sums_ln_real by blast
lp15@68424
  1548
  then show ?thesis
lp15@68424
  1549
    by (rule sums_unique [symmetric])
lp15@68424
  1550
qed
lp15@68424
  1551
lp15@68424
  1552
lemma prodinf_exp_real: 
lp15@68424
  1553
  fixes f :: "nat \<Rightarrow> real"
lp15@68424
  1554
  assumes f: "convergent_prod f" and 0: "\<And>x. f x > 0"
lp15@68424
  1555
  shows "prodinf f = exp (suminf (\<lambda>i. ln (f i)))"
lp15@68424
  1556
  by (simp add: "0" f less_0_prodinf suminf_ln_real)
lp15@68424
  1557
lp15@68424
  1558
lp15@68585
  1559
lemma Ln_prodinf_complex:
lp15@68585
  1560
  fixes z :: "nat \<Rightarrow> complex"
lp15@68585
  1561
  assumes z: "\<And>j. z j \<noteq> 0" and \<xi>: "\<xi> \<noteq> 0"
lp15@68585
  1562
  shows "((\<lambda>n. \<Prod>j\<le>n. z j) \<longlonglongrightarrow> \<xi>) \<longleftrightarrow> (\<exists>k. (\<lambda>n. (\<Sum>j\<le>n. Ln (z j))) \<longlonglongrightarrow> Ln \<xi> + of_int k * (of_real(2*pi) * \<i>))" (is "?lhs = ?rhs")
lp15@68585
  1563
proof
lp15@68585
  1564
  assume L: ?lhs
lp15@68585
  1565
  have pnz: "(\<Prod>j\<le>n. z j) \<noteq> 0" for n
lp15@68585
  1566
    using z by auto
lp15@68585
  1567
  define \<Theta> where "\<Theta> \<equiv> Arg \<xi> + 2*pi"
lp15@68585
  1568
  then have "\<Theta> > pi"
lp15@68585
  1569
    using Arg_def mpi_less_Im_Ln by fastforce
lp15@68585
  1570
  have \<xi>_eq: "\<xi> = cmod \<xi> * exp (\<i> * \<Theta>)"
lp15@68585
  1571
    using Arg_def Arg_eq \<xi> unfolding \<Theta>_def by (simp add: algebra_simps exp_add)
lp15@68585
  1572
  define \<theta> where "\<theta> \<equiv> \<lambda>n. THE t. is_Arg (\<Prod>j\<le>n. z j) t \<and> t \<in> {\<Theta>-pi<..\<Theta>+pi}"
lp15@68585
  1573
  have uniq: "\<exists>!s. is_Arg (\<Prod>j\<le>n. z j) s \<and> s \<in> {\<Theta>-pi<..\<Theta>+pi}" for n
lp15@68585
  1574
    using Argument_exists_unique [OF pnz] by metis
lp15@68585
  1575
  have \<theta>: "is_Arg (\<Prod>j\<le>n. z j) (\<theta> n)" and \<theta>_interval: "\<theta> n \<in> {\<Theta>-pi<..\<Theta>+pi}" for n
lp15@68585
  1576
    unfolding \<theta>_def
lp15@68585
  1577
    using theI' [OF uniq] by metis+
lp15@68585
  1578
  have \<theta>_pos: "\<And>j. \<theta> j > 0"
lp15@68585
  1579
    using \<theta>_interval \<open>\<Theta> > pi\<close> by simp (meson diff_gt_0_iff_gt less_trans)
lp15@68585
  1580
  have "(\<Prod>j\<le>n. z j) = cmod (\<Prod>j\<le>n. z j) * exp (\<i> * \<theta> n)" for n
lp15@68585
  1581
    using \<theta> by (auto simp: is_Arg_def)
lp15@68585
  1582
  then have eq: "(\<lambda>n. \<Prod>j\<le>n. z j) = (\<lambda>n. cmod (\<Prod>j\<le>n. z j) * exp (\<i> * \<theta> n))"
lp15@68585
  1583
    by simp
lp15@68585
  1584
  then have "(\<lambda>n. (cmod (\<Prod>j\<le>n. z j)) * exp (\<i> * (\<theta> n))) \<longlonglongrightarrow> \<xi>"
lp15@68585
  1585
    using L by force
lp15@68585
  1586
  then obtain k where k: "(\<lambda>j. \<theta> j - of_int (k j) * (2 * pi)) \<longlonglongrightarrow> \<Theta>"
lp15@68585
  1587
    using L by (subst (asm) \<xi>_eq) (auto simp add: eq z \<xi> polar_convergence)
lp15@68585
  1588
  moreover have "\<forall>\<^sub>F n in sequentially. k n = 0"
lp15@68585
  1589
  proof -
lp15@68585
  1590
    have *: "kj = 0" if "dist (vj - real_of_int kj * 2) V < 1" "vj \<in> {V - 1<..V + 1}" for kj vj V
lp15@68585
  1591
      using that  by (auto simp: dist_norm)
lp15@68585
  1592
    have "\<forall>\<^sub>F j in sequentially. dist (\<theta> j - of_int (k j) * (2 * pi)) \<Theta> < pi"
lp15@68585
  1593
      using tendstoD [OF k] pi_gt_zero by blast
lp15@68585
  1594
    then show ?thesis
lp15@68585
  1595
    proof (rule eventually_mono)
lp15@68585
  1596
      fix j
lp15@68585
  1597
      assume d: "dist (\<theta> j - real_of_int (k j) * (2 * pi)) \<Theta> < pi"
lp15@68585
  1598
      show "k j = 0"
lp15@68585
  1599
        by (rule * [of "\<theta> j/pi" _ "\<Theta>/pi"])
lp15@68585
  1600
           (use \<theta>_interval [of j] d in \<open>simp_all add: divide_simps dist_norm\<close>)
lp15@68585
  1601
    qed
lp15@68585
  1602
  qed
lp15@68585
  1603
  ultimately have \<theta>to\<Theta>: "\<theta> \<longlonglongrightarrow> \<Theta>"
lp15@68585
  1604
    apply (simp only: tendsto_def)
lp15@68585
  1605
    apply (erule all_forward imp_forward asm_rl)+
lp15@68585
  1606
    apply (drule (1) eventually_conj)
lp15@68585
  1607
    apply (auto elim: eventually_mono)
lp15@68585
  1608
    done
lp15@68585
  1609
  then have to0: "(\<lambda>n. \<bar>\<theta> (Suc n) - \<theta> n\<bar>) \<longlonglongrightarrow> 0"
lp15@68585
  1610
    by (metis (full_types) diff_self filterlim_sequentially_Suc tendsto_diff tendsto_rabs_zero)
lp15@68585
  1611
  have "\<exists>k. Im (\<Sum>j\<le>n. Ln (z j)) - of_int k * (2*pi) = \<theta> n" for n
lp15@68585
  1612
  proof (rule is_Arg_exp_diff_2pi)
lp15@68585
  1613
    show "is_Arg (exp (\<Sum>j\<le>n. Ln (z j))) (\<theta> n)"
lp15@68585
  1614
      using pnz \<theta> by (simp add: is_Arg_def exp_sum prod_norm)
lp15@68585
  1615
  qed
lp15@68585
  1616
  then have "\<exists>k. (\<Sum>j\<le>n. Im (Ln (z j))) = \<theta> n + of_int k * (2*pi)" for n
lp15@68585
  1617
    by (simp add: algebra_simps)
lp15@68585
  1618
  then obtain k where k: "\<And>n. (\<Sum>j\<le>n. Im (Ln (z j))) = \<theta> n + of_int (k n) * (2*pi)"
lp15@68585
  1619
    by metis
lp15@68585
  1620
  obtain K where "\<forall>\<^sub>F n in sequentially. k n = K"
lp15@68585
  1621
  proof -
lp15@68585
  1622
    have k_le: "(2*pi) * \<bar>k (Suc n) - k n\<bar> \<le> \<bar>\<theta> (Suc n) - \<theta> n\<bar> + \<bar>Im (Ln (z (Suc n)))\<bar>" for n
lp15@68585
  1623
    proof -
lp15@68585
  1624
      have "(\<Sum>j\<le>Suc n. Im (Ln (z j))) - (\<Sum>j\<le>n. Im (Ln (z j))) = Im (Ln (z (Suc n)))"
lp15@68585
  1625
        by simp
lp15@68585
  1626
      then show ?thesis
lp15@68585
  1627
        using k [of "Suc n"] k [of n] by (auto simp: abs_if algebra_simps)
lp15@68585
  1628
    qed
lp15@68585
  1629
    have "z \<longlonglongrightarrow> 1"
lp15@68585
  1630
      using L \<xi> convergent_prod_iff_nz_lim z by (blast intro: convergent_prod_imp_LIMSEQ)
lp15@68585
  1631
    with z have "(\<lambda>n. Ln (z n)) \<longlonglongrightarrow> Ln 1"
lp15@68585
  1632
      using isCont_tendsto_compose [OF continuous_at_Ln] nonpos_Reals_one_I by blast
lp15@68585
  1633
    then have "(\<lambda>n. Ln (z n)) \<longlonglongrightarrow> 0"
lp15@68585
  1634
      by simp
lp15@68585
  1635
    then have "(\<lambda>n. \<bar>Im (Ln (z (Suc n)))\<bar>) \<longlonglongrightarrow> 0"
lp15@68585
  1636
      by (metis LIMSEQ_unique \<open>z \<longlonglongrightarrow> 1\<close> continuous_at_Ln filterlim_sequentially_Suc isCont_tendsto_compose nonpos_Reals_one_I tendsto_Im tendsto_rabs_zero_iff zero_complex.simps(2))
lp15@68585
  1637
    then have "\<forall>\<^sub>F n in sequentially. \<bar>Im (Ln (z (Suc n)))\<bar> < 1"
lp15@68585
  1638
      by (simp add: order_tendsto_iff)
lp15@68585
  1639
    moreover have "\<forall>\<^sub>F n in sequentially. \<bar>\<theta> (Suc n) - \<theta> n\<bar> < 1"
lp15@68585
  1640
      using to0 by (simp add: order_tendsto_iff)
lp15@68585
  1641
    ultimately have "\<forall>\<^sub>F n in sequentially. (2*pi) * \<bar>k (Suc n) - k n\<bar> < 1 + 1" 
lp15@68585
  1642
    proof (rule eventually_elim2) 
lp15@68585
  1643
      fix n 
lp15@68585
  1644
      assume "\<bar>Im (Ln (z (Suc n)))\<bar> < 1" and "\<bar>\<theta> (Suc n) - \<theta> n\<bar> < 1"
lp15@68585
  1645
      with k_le [of n] show "2 * pi * real_of_int \<bar>k (Suc n) - k n\<bar> < 1 + 1"
lp15@68585
  1646
        by linarith
lp15@68585
  1647
    qed
lp15@68585
  1648
    then have "\<forall>\<^sub>F n in sequentially. real_of_int\<bar>k (Suc n) - k n\<bar> < 1" 
lp15@68585
  1649
    proof (rule eventually_mono)
lp15@68585
  1650
      fix n :: "nat"
lp15@68585
  1651
      assume "2 * pi * \<bar>k (Suc n) - k n\<bar> < 1 + 1"
lp15@68585
  1652
      then have "\<bar>k (Suc n) - k n\<bar> < 2 / (2*pi)"
lp15@68585
  1653
        by (simp add: field_simps)
lp15@68585
  1654
      also have "... < 1"
lp15@68585
  1655
        using pi_ge_two by auto
lp15@68585
  1656
      finally show "real_of_int \<bar>k (Suc n) - k n\<bar> < 1" .
lp15@68585
  1657
    qed
lp15@68585
  1658
  then obtain N where N: "\<And>n. n\<ge>N \<Longrightarrow> \<bar>k (Suc n) - k n\<bar> = 0"
lp15@68585
  1659
    using eventually_sequentially less_irrefl of_int_abs by fastforce
lp15@68585
  1660
  have "k (N+i) = k N" for i
lp15@68585
  1661
  proof (induction i)
lp15@68585
  1662
    case (Suc i)
lp15@68585
  1663
    with N [of "N+i"] show ?case
lp15@68585
  1664
      by auto
lp15@68585
  1665
  qed simp
lp15@68585
  1666
  then have "\<And>n. n\<ge>N \<Longrightarrow> k n = k N"
lp15@68585
  1667
    using le_Suc_ex by auto
lp15@68585
  1668
  then show ?thesis
lp15@68585
  1669
    by (force simp add: eventually_sequentially intro: that)
lp15@68585
  1670
  qed
lp15@68585
  1671
  with \<theta>to\<Theta> have "(\<lambda>n. (\<Sum>j\<le>n. Im (Ln (z j)))) \<longlonglongrightarrow> \<Theta> + of_int K * (2*pi)"
lp15@68585
  1672
    by (simp add: k tendsto_add tendsto_mult Lim_eventually)
lp15@68585
  1673
  moreover have "(\<lambda>n. (\<Sum>k\<le>n. Re (Ln (z k)))) \<longlonglongrightarrow> Re (Ln \<xi>)"
lp15@68585
  1674
    using assms continuous_imp_tendsto [OF isCont_ln tendsto_norm [OF L]]
lp15@68585
  1675
    by (simp add: o_def flip: prod_norm ln_prod)
lp15@68585
  1676
  ultimately show ?rhs
lp15@68585
  1677
    by (rule_tac x="K+1" in exI) (auto simp: tendsto_complex_iff \<Theta>_def Arg_def assms algebra_simps)
lp15@68585
  1678
next
lp15@68585
  1679
  assume ?rhs
lp15@68585
  1680
  then obtain r where r: "(\<lambda>n. (\<Sum>k\<le>n. Ln (z k))) \<longlonglongrightarrow> Ln \<xi> + of_int r * (of_real(2*pi) * \<i>)" ..
lp15@68585
  1681
  have "(\<lambda>n. exp (\<Sum>k\<le>n. Ln (z k))) \<longlonglongrightarrow> \<xi>"
lp15@68585
  1682
    using assms continuous_imp_tendsto [OF isCont_exp r] exp_integer_2pi [of r]
lp15@68585
  1683
    by (simp add: o_def exp_add algebra_simps)
lp15@68585
  1684
  moreover have "exp (\<Sum>k\<le>n. Ln (z k)) = (\<Prod>k\<le>n. z k)" for n
lp15@68585
  1685
    by (simp add: exp_sum add_eq_0_iff assms)
lp15@68585
  1686
  ultimately show ?lhs
lp15@68585
  1687
    by auto
lp15@68585
  1688
qed
lp15@68585
  1689
lp15@68585
  1690
text\<open>Prop 17.2 of Bak and Newman, Complex Analysis, p.242\<close>
lp15@68585
  1691
proposition convergent_prod_iff_summable_complex:
lp15@68585
  1692
  fixes z :: "nat \<Rightarrow> complex"
lp15@68585
  1693
  assumes "\<And>k. z k \<noteq> 0"
lp15@68585
  1694
  shows "convergent_prod (\<lambda>k. z k) \<longleftrightarrow> summable (\<lambda>k. Ln (z k))" (is "?lhs = ?rhs")
lp15@68585
  1695
proof
lp15@68585
  1696
  assume ?lhs
lp15@68585
  1697
  then obtain p where p: "(\<lambda>n. \<Prod>k\<le>n. z k) \<longlonglongrightarrow> p" and "p \<noteq> 0"
lp15@68585
  1698
    using convergent_prod_LIMSEQ prodinf_nonzero add_eq_0_iff assms by fastforce
lp15@68585
  1699
  then show ?rhs
lp15@68585
  1700
    using Ln_prodinf_complex assms
lp15@68585
  1701
    by (auto simp: prodinf_nonzero summable_def sums_def_le)
lp15@68585
  1702
next
lp15@68585
  1703
  assume R: ?rhs
lp15@68585
  1704
  have "(\<Prod>k\<le>n. z k) = exp (\<Sum>k\<le>n. Ln (z k))" for n
lp15@68585
  1705
    by (simp add: exp_sum add_eq_0_iff assms)
lp15@68585
  1706
  then have "(\<lambda>n. \<Prod>k\<le>n. z k) \<longlonglongrightarrow> exp (suminf (\<lambda>k. Ln (z k)))"
lp15@68585
  1707
    using continuous_imp_tendsto [OF isCont_exp summable_LIMSEQ' [OF R]] by (simp add: o_def)
lp15@68585
  1708
  then show ?lhs
lp15@68585
  1709
    by (subst convergent_prod_iff_convergent) (auto simp: convergent_def tendsto_Lim assms add_eq_0_iff)
lp15@68585
  1710
qed
lp15@68585
  1711
lp15@68586
  1712
text\<open>Prop 17.3 of Bak and Newman, Complex Analysis\<close>
lp15@68586
  1713
proposition summable_imp_convergent_prod_complex:
lp15@68586
  1714
  fixes z :: "nat \<Rightarrow> complex"
lp15@68586
  1715
  assumes z: "summable (\<lambda>k. norm (z k))" and non0: "\<And>k. z k \<noteq> -1"
lp15@68586
  1716
  shows "convergent_prod (\<lambda>k. 1 + z k)" 
lp15@68586
  1717
proof -
lp15@68586
  1718
  note if_cong [cong] power_Suc [simp del]
lp15@68586
  1719
  obtain N where N: "\<And>k. k\<ge>N \<Longrightarrow> norm (z k) < 1/2"
lp15@68586
  1720
    using summable_LIMSEQ_zero [OF z]
lp15@68586
  1721
    by (metis diff_zero dist_norm half_gt_zero_iff less_numeral_extra(1) lim_sequentially tendsto_norm_zero_iff)
lp15@68586
  1722
  have "norm (Ln (1 + z k)) \<le> 2 * norm (z k)" if "k \<ge> N" for k
lp15@68586
  1723
  proof (cases "z k = 0")
lp15@68586
  1724
    case False
lp15@68586
  1725
    let ?f = "\<lambda>i. cmod ((- 1) ^ i * z k ^ i / of_nat (Suc i))"
lp15@68586
  1726
    have normf: "norm (?f n) \<le> (1 / 2) ^ n" for n
lp15@68586
  1727
    proof -
lp15@68586
  1728
      have "norm (?f n) = cmod (z k) ^ n / cmod (1 + of_nat n)"
lp15@68586
  1729
        by (auto simp: norm_divide norm_mult norm_power)
lp15@68586
  1730
      also have "\<dots> \<le> cmod (z k) ^ n"
lp15@68586
  1731
        by (auto simp: divide_simps mult_le_cancel_left1 in_Reals_norm)
lp15@68586
  1732
      also have "\<dots> \<le> (1 / 2) ^ n"
lp15@68586
  1733
        using N [OF that] by (simp add: power_mono)
lp15@68586
  1734
      finally show "norm (?f n) \<le> (1 / 2) ^ n" .
lp15@68586
  1735
    qed
lp15@68586
  1736
    have summablef: "summable ?f"
lp15@68586
  1737
      by (intro normf summable_comparison_test' [OF summable_geometric [of "1/2"]]) auto
lp15@68586
  1738
    have "(\<lambda>n. (- 1) ^ Suc n / of_nat n * z k ^ n) sums Ln (1 + z k)"
lp15@68586
  1739
      using Ln_series [of "z k"] N that by fastforce
lp15@68586
  1740
    then have *: "(\<lambda>i. z k * (((- 1) ^ i * z k ^ i) / (Suc i))) sums Ln (1 + z k)"
lp15@68586
  1741
      using sums_split_initial_segment [where n= 1]  by (force simp: power_Suc mult_ac)
lp15@68586
  1742
    then have "norm (Ln (1 + z k)) = norm (suminf (\<lambda>i. z k * (((- 1) ^ i * z k ^ i) / (Suc i))))"
lp15@68586
  1743
      using sums_unique by force
lp15@68586
  1744
    also have "\<dots> = norm (z k * suminf (\<lambda>i. ((- 1) ^ i * z k ^ i) / (Suc i)))"
lp15@68586
  1745
      apply (subst suminf_mult)
lp15@68586
  1746
      using * False
lp15@68586
  1747
      by (auto simp: sums_summable intro: summable_mult_D [of "z k"])
lp15@68586
  1748
    also have "\<dots> = norm (z k) * norm (suminf (\<lambda>i. ((- 1) ^ i * z k ^ i) / (Suc i)))"
lp15@68586
  1749
      by (simp add: norm_mult)
lp15@68586
  1750
    also have "\<dots> \<le> norm (z k) * suminf (\<lambda>i. norm (((- 1) ^ i * z k ^ i) / (Suc i)))"
lp15@68586
  1751
      by (intro mult_left_mono summable_norm summablef) auto
lp15@68586
  1752
    also have "\<dots> \<le> norm (z k) * suminf (\<lambda>i. (1/2) ^ i)"
lp15@68586
  1753
      by (intro mult_left_mono suminf_le) (use summable_geometric [of "1/2"] summablef normf in auto)
lp15@68586
  1754
    also have "\<dots> \<le> norm (z k) * 2"
lp15@68586
  1755
      using suminf_geometric [of "1/2::real"] by simp
lp15@68586
  1756
    finally show ?thesis
lp15@68586
  1757
      by (simp add: mult_ac)
lp15@68586
  1758
  qed simp
lp15@68586
  1759
  then have "summable (\<lambda>k. Ln (1 + z k))"
lp15@68586
  1760
    by (metis summable_comparison_test summable_mult z)
lp15@68586
  1761
  with non0 show ?thesis
lp15@68586
  1762
    by (simp add: add_eq_0_iff convergent_prod_iff_summable_complex)
lp15@68586
  1763
qed
lp15@68586
  1764
lp15@68616
  1765
lemma summable_Ln_complex:
lp15@68616
  1766
  fixes z :: "nat \<Rightarrow> complex"
lp15@68616
  1767
  assumes "convergent_prod z" "\<And>k. z k \<noteq> 0"
lp15@68616
  1768
  shows "summable (\<lambda>k. Ln (z k))"
lp15@68616
  1769
  using convergent_prod_def assms convergent_prod_iff_summable_complex by blast
lp15@68616
  1770
lp15@68586
  1771
lp15@68424
  1772
subsection\<open>Embeddings from the reals into some complete real normed field\<close>
lp15@68424
  1773
lp15@68426
  1774
lemma tendsto_eq_of_real_lim:
lp15@68424
  1775
  assumes "(\<lambda>n. of_real (f n) :: 'a::{complete_space,real_normed_field}) \<longlonglongrightarrow> q"
lp15@68424
  1776
  shows "q = of_real (lim f)"
lp15@68424
  1777
proof -
lp15@68424
  1778
  have "convergent (\<lambda>n. of_real (f n) :: 'a)"
lp15@68424
  1779
    using assms convergent_def by blast 
lp15@68424
  1780
  then have "convergent f"
lp15@68424
  1781
    unfolding convergent_def
lp15@68424
  1782
    by (simp add: convergent_eq_Cauchy Cauchy_def)
lp15@68424
  1783
  then show ?thesis
lp15@68424
  1784
    by (metis LIMSEQ_unique assms convergentD sequentially_bot tendsto_Lim tendsto_of_real)
lp15@68424
  1785
qed
lp15@68424
  1786
lp15@68426
  1787
lemma tendsto_eq_of_real:
lp15@68424
  1788
  assumes "(\<lambda>n. of_real (f n) :: 'a::{complete_space,real_normed_field}) \<longlonglongrightarrow> q"
lp15@68424
  1789
  obtains r where "q = of_real r"
lp15@68426
  1790
  using tendsto_eq_of_real_lim assms by blast
lp15@68424
  1791
lp15@68424
  1792
lemma has_prod_of_real_iff:
lp15@68424
  1793
  "(\<lambda>n. of_real (f n) :: 'a::{complete_space,real_normed_field}) has_prod of_real c \<longleftrightarrow> f has_prod c"
lp15@68424
  1794
  (is "?lhs = ?rhs")
lp15@68424
  1795
proof
lp15@68424
  1796
  assume ?lhs
lp15@68424
  1797
  then show ?rhs
lp15@68424
  1798
    apply (auto simp: prod_defs LIMSEQ_prod_0 tendsto_of_real_iff simp flip: of_real_prod)
lp15@68426
  1799
    using tendsto_eq_of_real
lp15@68424
  1800
    by (metis of_real_0 tendsto_of_real_iff)
lp15@68424
  1801
next
lp15@68424
  1802
  assume ?rhs
lp15@68424
  1803
  with tendsto_of_real_iff show ?lhs
lp15@68424
  1804
    by (fastforce simp: prod_defs simp flip: of_real_prod)
lp15@68424
  1805
qed
lp15@68424
  1806
lp15@68424
  1807
end