src/HOL/Analysis/Infinite_Products.thy
author paulson <lp15@cam.ac.uk>
Mon Jun 11 16:23:21 2018 +0100 (13 months ago)
changeset 68426 e0b5f2d14bf9
parent 68424 02e5a44ffe7d
child 68452 c027dfbfad30
permissions -rw-r--r--
fixed a name clash
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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
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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 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 abs_convergent_prod_altdef:
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  fixes f :: "nat \<Rightarrow> 'a :: {one,real_normed_vector}"
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  shows  "abs_convergent_prod f \<longleftrightarrow> convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
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proof
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  assume "abs_convergent_prod f"
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  thus "convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
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    by (auto simp: abs_convergent_prod_def intro!: convergent_prod_imp_convergent)
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qed (auto intro: abs_convergent_prodI)
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lemma weierstrass_prod_ineq:
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  fixes f :: "'a \<Rightarrow> real" 
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  assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> {0..1}"
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  shows   "1 - sum f A \<le> (\<Prod>x\<in>A. 1 - f x)"
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  using assms
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proof (induction A rule: infinite_finite_induct)
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  case (insert x A)
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  from insert.hyps and insert.prems 
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    have "1 - sum f A + f x * (\<Prod>x\<in>A. 1 - f x) \<le> (\<Prod>x\<in>A. 1 - f x) + f x * (\<Prod>x\<in>A. 1)"
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    by (intro insert.IH add_mono mult_left_mono prod_mono) auto
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  with insert.hyps show ?case by (simp add: algebra_simps)
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qed simp_all
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lemma norm_prod_minus1_le_prod_minus1:
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  fixes f :: "nat \<Rightarrow> 'a :: {real_normed_div_algebra,comm_ring_1}"  
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  shows "norm (prod (\<lambda>n. 1 + f n) A - 1) \<le> prod (\<lambda>n. 1 + norm (f n)) A - 1"
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proof (induction A rule: infinite_finite_induct)
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  case (insert x A)
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  from insert.hyps have 
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    "norm ((\<Prod>n\<in>insert x A. 1 + f n) - 1) = 
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       norm ((\<Prod>n\<in>A. 1 + f n) - 1 + f x * (\<Prod>n\<in>A. 1 + f n))"
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    by (simp add: algebra_simps)
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  also have "\<dots> \<le> norm ((\<Prod>n\<in>A. 1 + f n) - 1) + norm (f x * (\<Prod>n\<in>A. 1 + f n))"
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    by (rule norm_triangle_ineq)
eberlm@66277
   259
  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
   260
    by (simp add: prod_norm norm_mult)
eberlm@66277
   261
  also have "(\<Prod>x\<in>A. norm (1 + f x)) \<le> (\<Prod>x\<in>A. norm (1::'a) + norm (f x))"
eberlm@66277
   262
    by (intro prod_mono norm_triangle_ineq ballI conjI) auto
eberlm@66277
   263
  also have "norm (1::'a) = 1" by simp
eberlm@66277
   264
  also note insert.IH
eberlm@66277
   265
  also have "(\<Prod>n\<in>A. 1 + norm (f n)) - 1 + norm (f x) * (\<Prod>x\<in>A. 1 + norm (f x)) =
lp15@68064
   266
             (\<Prod>n\<in>insert x A. 1 + norm (f n)) - 1"
eberlm@66277
   267
    using insert.hyps by (simp add: algebra_simps)
eberlm@66277
   268
  finally show ?case by - (simp_all add: mult_left_mono)
eberlm@66277
   269
qed simp_all
eberlm@66277
   270
eberlm@66277
   271
lemma convergent_prod_imp_ev_nonzero:
eberlm@66277
   272
  fixes f :: "nat \<Rightarrow> 'a :: {t2_space,comm_semiring_1}"
eberlm@66277
   273
  assumes "convergent_prod f"
eberlm@66277
   274
  shows   "eventually (\<lambda>n. f n \<noteq> 0) sequentially"
eberlm@66277
   275
  using assms by (auto simp: eventually_at_top_linorder convergent_prod_altdef)
eberlm@66277
   276
eberlm@66277
   277
lemma convergent_prod_imp_LIMSEQ:
eberlm@66277
   278
  fixes f :: "nat \<Rightarrow> 'a :: {real_normed_field}"
eberlm@66277
   279
  assumes "convergent_prod f"
eberlm@66277
   280
  shows   "f \<longlonglongrightarrow> 1"
eberlm@66277
   281
proof -
eberlm@66277
   282
  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
   283
    by (auto simp: convergent_prod_altdef)
eberlm@66277
   284
  hence L': "(\<lambda>n. \<Prod>i\<le>Suc n. f (i+M)) \<longlonglongrightarrow> L" by (subst filterlim_sequentially_Suc)
eberlm@66277
   285
  have "(\<lambda>n. (\<Prod>i\<le>Suc n. f (i+M)) / (\<Prod>i\<le>n. f (i+M))) \<longlonglongrightarrow> L / L"
eberlm@66277
   286
    using L L' by (intro tendsto_divide) simp_all
eberlm@66277
   287
  also from L have "L / L = 1" by simp
eberlm@66277
   288
  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
   289
    using assms L by (auto simp: fun_eq_iff atMost_Suc)
eberlm@66277
   290
  finally show ?thesis by (rule LIMSEQ_offset)
eberlm@66277
   291
qed
eberlm@66277
   292
eberlm@66277
   293
lemma abs_convergent_prod_imp_summable:
eberlm@66277
   294
  fixes f :: "nat \<Rightarrow> 'a :: real_normed_div_algebra"
eberlm@66277
   295
  assumes "abs_convergent_prod f"
eberlm@66277
   296
  shows "summable (\<lambda>i. norm (f i - 1))"
eberlm@66277
   297
proof -
eberlm@66277
   298
  from assms have "convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))" 
eberlm@66277
   299
    unfolding abs_convergent_prod_def by (rule convergent_prod_imp_convergent)
eberlm@66277
   300
  then obtain L where L: "(\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1)) \<longlonglongrightarrow> L"
eberlm@66277
   301
    unfolding convergent_def by blast
eberlm@66277
   302
  have "convergent (\<lambda>n. \<Sum>i\<le>n. norm (f i - 1))"
eberlm@66277
   303
  proof (rule Bseq_monoseq_convergent)
eberlm@66277
   304
    have "eventually (\<lambda>n. (\<Prod>i\<le>n. 1 + norm (f i - 1)) < L + 1) sequentially"
eberlm@66277
   305
      using L(1) by (rule order_tendstoD) simp_all
eberlm@66277
   306
    hence "\<forall>\<^sub>F x in sequentially. norm (\<Sum>i\<le>x. norm (f i - 1)) \<le> L + 1"
eberlm@66277
   307
    proof eventually_elim
eberlm@66277
   308
      case (elim n)
eberlm@66277
   309
      have "norm (\<Sum>i\<le>n. norm (f i - 1)) = (\<Sum>i\<le>n. norm (f i - 1))"
eberlm@66277
   310
        unfolding real_norm_def by (intro abs_of_nonneg sum_nonneg) simp_all
eberlm@66277
   311
      also have "\<dots> \<le> (\<Prod>i\<le>n. 1 + norm (f i - 1))" by (rule sum_le_prod) auto
eberlm@66277
   312
      also have "\<dots> < L + 1" by (rule elim)
eberlm@66277
   313
      finally show ?case by simp
eberlm@66277
   314
    qed
eberlm@66277
   315
    thus "Bseq (\<lambda>n. \<Sum>i\<le>n. norm (f i - 1))" by (rule BfunI)
eberlm@66277
   316
  next
eberlm@66277
   317
    show "monoseq (\<lambda>n. \<Sum>i\<le>n. norm (f i - 1))"
eberlm@66277
   318
      by (rule mono_SucI1) auto
eberlm@66277
   319
  qed
eberlm@66277
   320
  thus "summable (\<lambda>i. norm (f i - 1))" by (simp add: summable_iff_convergent')
eberlm@66277
   321
qed
eberlm@66277
   322
eberlm@66277
   323
lemma summable_imp_abs_convergent_prod:
eberlm@66277
   324
  fixes f :: "nat \<Rightarrow> 'a :: real_normed_div_algebra"
eberlm@66277
   325
  assumes "summable (\<lambda>i. norm (f i - 1))"
eberlm@66277
   326
  shows   "abs_convergent_prod f"
eberlm@66277
   327
proof (intro abs_convergent_prodI Bseq_monoseq_convergent)
eberlm@66277
   328
  show "monoseq (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
eberlm@66277
   329
    by (intro mono_SucI1) 
eberlm@66277
   330
       (auto simp: atMost_Suc algebra_simps intro!: mult_nonneg_nonneg prod_nonneg)
eberlm@66277
   331
next
eberlm@66277
   332
  show "Bseq (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
eberlm@66277
   333
  proof (rule Bseq_eventually_mono)
eberlm@66277
   334
    show "eventually (\<lambda>n. norm (\<Prod>i\<le>n. 1 + norm (f i - 1)) \<le> 
eberlm@66277
   335
            norm (exp (\<Sum>i\<le>n. norm (f i - 1)))) sequentially"
eberlm@66277
   336
      by (intro always_eventually allI) (auto simp: abs_prod exp_sum intro!: prod_mono)
eberlm@66277
   337
  next
eberlm@66277
   338
    from assms have "(\<lambda>n. \<Sum>i\<le>n. norm (f i - 1)) \<longlonglongrightarrow> (\<Sum>i. norm (f i - 1))"
eberlm@66277
   339
      using sums_def_le by blast
eberlm@66277
   340
    hence "(\<lambda>n. exp (\<Sum>i\<le>n. norm (f i - 1))) \<longlonglongrightarrow> exp (\<Sum>i. norm (f i - 1))"
eberlm@66277
   341
      by (rule tendsto_exp)
eberlm@66277
   342
    hence "convergent (\<lambda>n. exp (\<Sum>i\<le>n. norm (f i - 1)))"
eberlm@66277
   343
      by (rule convergentI)
eberlm@66277
   344
    thus "Bseq (\<lambda>n. exp (\<Sum>i\<le>n. norm (f i - 1)))"
eberlm@66277
   345
      by (rule convergent_imp_Bseq)
eberlm@66277
   346
  qed
eberlm@66277
   347
qed
eberlm@66277
   348
eberlm@66277
   349
lemma abs_convergent_prod_conv_summable:
eberlm@66277
   350
  fixes f :: "nat \<Rightarrow> 'a :: real_normed_div_algebra"
eberlm@66277
   351
  shows "abs_convergent_prod f \<longleftrightarrow> summable (\<lambda>i. norm (f i - 1))"
eberlm@66277
   352
  by (blast intro: abs_convergent_prod_imp_summable summable_imp_abs_convergent_prod)
eberlm@66277
   353
eberlm@66277
   354
lemma abs_convergent_prod_imp_LIMSEQ:
eberlm@66277
   355
  fixes f :: "nat \<Rightarrow> 'a :: {comm_ring_1,real_normed_div_algebra}"
eberlm@66277
   356
  assumes "abs_convergent_prod f"
eberlm@66277
   357
  shows   "f \<longlonglongrightarrow> 1"
eberlm@66277
   358
proof -
eberlm@66277
   359
  from assms have "summable (\<lambda>n. norm (f n - 1))"
eberlm@66277
   360
    by (rule abs_convergent_prod_imp_summable)
eberlm@66277
   361
  from summable_LIMSEQ_zero[OF this] have "(\<lambda>n. f n - 1) \<longlonglongrightarrow> 0"
eberlm@66277
   362
    by (simp add: tendsto_norm_zero_iff)
eberlm@66277
   363
  from tendsto_add[OF this tendsto_const[of 1]] show ?thesis by simp
eberlm@66277
   364
qed
eberlm@66277
   365
eberlm@66277
   366
lemma abs_convergent_prod_imp_ev_nonzero:
eberlm@66277
   367
  fixes f :: "nat \<Rightarrow> 'a :: {comm_ring_1,real_normed_div_algebra}"
eberlm@66277
   368
  assumes "abs_convergent_prod f"
eberlm@66277
   369
  shows   "eventually (\<lambda>n. f n \<noteq> 0) sequentially"
eberlm@66277
   370
proof -
eberlm@66277
   371
  from assms have "f \<longlonglongrightarrow> 1" 
eberlm@66277
   372
    by (rule abs_convergent_prod_imp_LIMSEQ)
eberlm@66277
   373
  hence "eventually (\<lambda>n. dist (f n) 1 < 1) at_top"
eberlm@66277
   374
    by (auto simp: tendsto_iff)
eberlm@66277
   375
  thus ?thesis by eventually_elim auto
eberlm@66277
   376
qed
eberlm@66277
   377
eberlm@66277
   378
lemma convergent_prod_offset:
eberlm@66277
   379
  assumes "convergent_prod (\<lambda>n. f (n + m))"  
eberlm@66277
   380
  shows   "convergent_prod f"
eberlm@66277
   381
proof -
eberlm@66277
   382
  from assms obtain M L where "(\<lambda>n. \<Prod>k\<le>n. f (k + (M + m))) \<longlonglongrightarrow> L" "L \<noteq> 0"
lp15@68064
   383
    by (auto simp: prod_defs add.assoc)
lp15@68064
   384
  thus "convergent_prod f" 
lp15@68064
   385
    unfolding prod_defs by blast
eberlm@66277
   386
qed
eberlm@66277
   387
eberlm@66277
   388
lemma abs_convergent_prod_offset:
eberlm@66277
   389
  assumes "abs_convergent_prod (\<lambda>n. f (n + m))"  
eberlm@66277
   390
  shows   "abs_convergent_prod f"
eberlm@66277
   391
  using assms unfolding abs_convergent_prod_def by (rule convergent_prod_offset)
eberlm@66277
   392
lp15@68424
   393
subsection\<open>Ignoring initial segments\<close>
lp15@68424
   394
lp15@68361
   395
lemma raw_has_prod_ignore_initial_segment:
lp15@68361
   396
  fixes f :: "nat \<Rightarrow> 'a :: real_normed_field"
lp15@68361
   397
  assumes "raw_has_prod f M p" "N \<ge> M"
lp15@68361
   398
  obtains q where  "raw_has_prod f N q"
eberlm@66277
   399
proof -
lp15@68361
   400
  have p: "(\<lambda>n. \<Prod>k\<le>n. f (k + M)) \<longlonglongrightarrow> p" and "p \<noteq> 0" 
lp15@68361
   401
    using assms by (auto simp: raw_has_prod_def)
lp15@68361
   402
  then have nz: "\<And>n. n \<ge> M \<Longrightarrow> f n \<noteq> 0"
lp15@68361
   403
    using assms by (auto simp: raw_has_prod_eq_0)
lp15@68361
   404
  define C where "C = (\<Prod>k<N-M. f (k + M))"
eberlm@66277
   405
  from nz have [simp]: "C \<noteq> 0" 
eberlm@66277
   406
    by (auto simp: C_def)
eberlm@66277
   407
lp15@68361
   408
  from p have "(\<lambda>i. \<Prod>k\<le>i + (N-M). f (k + M)) \<longlonglongrightarrow> p" 
eberlm@66277
   409
    by (rule LIMSEQ_ignore_initial_segment)
lp15@68361
   410
  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
   411
  proof (rule ext, goal_cases)
eberlm@66277
   412
    case (1 n)
lp15@68361
   413
    have "{..n+(N-M)} = {..<(N-M)} \<union> {(N-M)..n+(N-M)}" by auto
lp15@68361
   414
    also have "(\<Prod>k\<in>\<dots>. f (k + M)) = C * (\<Prod>k=(N-M)..n+(N-M). f (k + M))"
eberlm@66277
   415
      unfolding C_def by (rule prod.union_disjoint) auto
lp15@68361
   416
    also have "(\<Prod>k=(N-M)..n+(N-M). f (k + M)) = (\<Prod>k\<le>n. f (k + (N-M) + M))"
lp15@68361
   417
      by (intro ext prod.reindex_bij_witness[of _ "\<lambda>k. k + (N-M)" "\<lambda>k. k - (N-M)"]) auto
lp15@68361
   418
    finally show ?case
lp15@68361
   419
      using \<open>N \<ge> M\<close> by (simp add: add_ac)
eberlm@66277
   420
  qed
lp15@68361
   421
  finally have "(\<lambda>n. C * (\<Prod>k\<le>n. f (k + N)) / C) \<longlonglongrightarrow> p / C"
eberlm@66277
   422
    by (intro tendsto_divide tendsto_const) auto
lp15@68361
   423
  hence "(\<lambda>n. \<Prod>k\<le>n. f (k + N)) \<longlonglongrightarrow> p / C" by simp
lp15@68361
   424
  moreover from \<open>p \<noteq> 0\<close> have "p / C \<noteq> 0" by simp
lp15@68361
   425
  ultimately show ?thesis
lp15@68361
   426
    using raw_has_prod_def that by blast 
eberlm@66277
   427
qed
eberlm@66277
   428
lp15@68361
   429
corollary convergent_prod_ignore_initial_segment:
lp15@68361
   430
  fixes f :: "nat \<Rightarrow> 'a :: real_normed_field"
lp15@68361
   431
  assumes "convergent_prod f"
lp15@68361
   432
  shows   "convergent_prod (\<lambda>n. f (n + m))"
lp15@68361
   433
  using assms
lp15@68361
   434
  unfolding convergent_prod_def 
lp15@68361
   435
  apply clarify
lp15@68361
   436
  apply (erule_tac N="M+m" in raw_has_prod_ignore_initial_segment)
lp15@68361
   437
  apply (auto simp add: raw_has_prod_def add_ac)
lp15@68361
   438
  done
lp15@68361
   439
lp15@68136
   440
corollary convergent_prod_ignore_nonzero_segment:
lp15@68136
   441
  fixes f :: "nat \<Rightarrow> 'a :: real_normed_field"
lp15@68136
   442
  assumes f: "convergent_prod f" and nz: "\<And>i. i \<ge> M \<Longrightarrow> f i \<noteq> 0"
lp15@68361
   443
  shows "\<exists>p. raw_has_prod f M p"
lp15@68136
   444
  using convergent_prod_ignore_initial_segment [OF f]
lp15@68136
   445
  by (metis convergent_LIMSEQ_iff convergent_prod_iff_convergent le_add_same_cancel2 nz prod_defs(1) zero_order(1))
lp15@68136
   446
lp15@68136
   447
corollary abs_convergent_prod_ignore_initial_segment:
eberlm@66277
   448
  assumes "abs_convergent_prod f"
eberlm@66277
   449
  shows   "abs_convergent_prod (\<lambda>n. f (n + m))"
eberlm@66277
   450
  using assms unfolding abs_convergent_prod_def 
eberlm@66277
   451
  by (rule convergent_prod_ignore_initial_segment)
eberlm@66277
   452
eberlm@66277
   453
lemma abs_convergent_prod_imp_convergent_prod:
eberlm@66277
   454
  fixes f :: "nat \<Rightarrow> 'a :: {real_normed_div_algebra,complete_space,comm_ring_1}"
eberlm@66277
   455
  assumes "abs_convergent_prod f"
eberlm@66277
   456
  shows   "convergent_prod f"
eberlm@66277
   457
proof -
eberlm@66277
   458
  from assms have "eventually (\<lambda>n. f n \<noteq> 0) sequentially"
eberlm@66277
   459
    by (rule abs_convergent_prod_imp_ev_nonzero)
eberlm@66277
   460
  then obtain N where N: "f n \<noteq> 0" if "n \<ge> N" for n 
eberlm@66277
   461
    by (auto simp: eventually_at_top_linorder)
eberlm@66277
   462
  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
   463
eberlm@66277
   464
  have "Cauchy ?P"
eberlm@66277
   465
  proof (rule CauchyI', goal_cases)
eberlm@66277
   466
    case (1 \<epsilon>)
eberlm@66277
   467
    from assms have "abs_convergent_prod (\<lambda>n. f (n + N))"
eberlm@66277
   468
      by (rule abs_convergent_prod_ignore_initial_segment)
eberlm@66277
   469
    hence "Cauchy ?Q"
eberlm@66277
   470
      unfolding abs_convergent_prod_def
eberlm@66277
   471
      by (intro convergent_Cauchy convergent_prod_imp_convergent)
eberlm@66277
   472
    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
   473
      by blast
eberlm@66277
   474
    show ?case
eberlm@66277
   475
    proof (rule exI[of _ M], safe, goal_cases)
eberlm@66277
   476
      case (1 m n)
eberlm@66277
   477
      have "dist (?P m) (?P n) = norm (?P n - ?P m)"
eberlm@66277
   478
        by (simp add: dist_norm norm_minus_commute)
eberlm@66277
   479
      also from 1 have "{..n} = {..m} \<union> {m<..n}" by auto
eberlm@66277
   480
      hence "norm (?P n - ?P m) = norm (?P m * (\<Prod>k\<in>{m<..n}. f (k + N)) - ?P m)"
eberlm@66277
   481
        by (subst prod.union_disjoint [symmetric]) (auto simp: algebra_simps)
eberlm@66277
   482
      also have "\<dots> = norm (?P m * ((\<Prod>k\<in>{m<..n}. f (k + N)) - 1))"
eberlm@66277
   483
        by (simp add: algebra_simps)
eberlm@66277
   484
      also have "\<dots> = (\<Prod>k\<le>m. norm (f (k + N))) * norm ((\<Prod>k\<in>{m<..n}. f (k + N)) - 1)"
eberlm@66277
   485
        by (simp add: norm_mult prod_norm)
eberlm@66277
   486
      also have "\<dots> \<le> ?Q m * ((\<Prod>k\<in>{m<..n}. 1 + norm (f (k + N) - 1)) - 1)"
eberlm@66277
   487
        using norm_prod_minus1_le_prod_minus1[of "\<lambda>k. f (k + N) - 1" "{m<..n}"]
eberlm@66277
   488
              norm_triangle_ineq[of 1 "f k - 1" for k]
eberlm@66277
   489
        by (intro mult_mono prod_mono ballI conjI norm_prod_minus1_le_prod_minus1 prod_nonneg) auto
eberlm@66277
   490
      also have "\<dots> = ?Q m * (\<Prod>k\<in>{m<..n}. 1 + norm (f (k + N) - 1)) - ?Q m"
eberlm@66277
   491
        by (simp add: algebra_simps)
eberlm@66277
   492
      also have "?Q m * (\<Prod>k\<in>{m<..n}. 1 + norm (f (k + N) - 1)) = 
eberlm@66277
   493
                   (\<Prod>k\<in>{..m}\<union>{m<..n}. 1 + norm (f (k + N) - 1))"
eberlm@66277
   494
        by (rule prod.union_disjoint [symmetric]) auto
eberlm@66277
   495
      also from 1 have "{..m}\<union>{m<..n} = {..n}" by auto
eberlm@66277
   496
      also have "?Q n - ?Q m \<le> norm (?Q n - ?Q m)" by simp
eberlm@66277
   497
      also from 1 have "\<dots> < \<epsilon>" by (intro M) auto
eberlm@66277
   498
      finally show ?case .
eberlm@66277
   499
    qed
eberlm@66277
   500
  qed
eberlm@66277
   501
  hence conv: "convergent ?P" by (rule Cauchy_convergent)
eberlm@66277
   502
  then obtain L where L: "?P \<longlonglongrightarrow> L"
eberlm@66277
   503
    by (auto simp: convergent_def)
eberlm@66277
   504
eberlm@66277
   505
  have "L \<noteq> 0"
eberlm@66277
   506
  proof
eberlm@66277
   507
    assume [simp]: "L = 0"
eberlm@66277
   508
    from tendsto_norm[OF L] have limit: "(\<lambda>n. \<Prod>k\<le>n. norm (f (k + N))) \<longlonglongrightarrow> 0" 
eberlm@66277
   509
      by (simp add: prod_norm)
eberlm@66277
   510
eberlm@66277
   511
    from assms have "(\<lambda>n. f (n + N)) \<longlonglongrightarrow> 1"
eberlm@66277
   512
      by (intro abs_convergent_prod_imp_LIMSEQ abs_convergent_prod_ignore_initial_segment)
eberlm@66277
   513
    hence "eventually (\<lambda>n. norm (f (n + N) - 1) < 1) sequentially"
eberlm@66277
   514
      by (auto simp: tendsto_iff dist_norm)
eberlm@66277
   515
    then obtain M0 where M0: "norm (f (n + N) - 1) < 1" if "n \<ge> M0" for n
eberlm@66277
   516
      by (auto simp: eventually_at_top_linorder)
eberlm@66277
   517
eberlm@66277
   518
    {
eberlm@66277
   519
      fix M assume M: "M \<ge> M0"
eberlm@66277
   520
      with M0 have M: "norm (f (n + N) - 1) < 1" if "n \<ge> M" for n using that by simp
eberlm@66277
   521
eberlm@66277
   522
      have "(\<lambda>n. \<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1)) \<longlonglongrightarrow> 0"
eberlm@66277
   523
      proof (rule tendsto_sandwich)
eberlm@66277
   524
        show "eventually (\<lambda>n. (\<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1)) \<ge> 0) sequentially"
eberlm@66277
   525
          using M by (intro always_eventually prod_nonneg allI ballI) (auto intro: less_imp_le)
eberlm@66277
   526
        have "norm (1::'a) - norm (f (i + M + N) - 1) \<le> norm (f (i + M + N))" for i
eberlm@66277
   527
          using norm_triangle_ineq3[of "f (i + M + N)" 1] by simp
eberlm@66277
   528
        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
   529
          using M by (intro always_eventually allI prod_mono ballI conjI) (auto intro: less_imp_le)
eberlm@66277
   530
        
eberlm@66277
   531
        define C where "C = (\<Prod>k<M. norm (f (k + N)))"
eberlm@66277
   532
        from N have [simp]: "C \<noteq> 0" by (auto simp: C_def)
eberlm@66277
   533
        from L have "(\<lambda>n. norm (\<Prod>k\<le>n+M. f (k + N))) \<longlonglongrightarrow> 0"
eberlm@66277
   534
          by (intro LIMSEQ_ignore_initial_segment) (simp add: tendsto_norm_zero_iff)
eberlm@66277
   535
        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
   536
        proof (rule ext, goal_cases)
eberlm@66277
   537
          case (1 n)
eberlm@66277
   538
          have "{..n+M} = {..<M} \<union> {M..n+M}" by auto
eberlm@66277
   539
          also have "norm (\<Prod>k\<in>\<dots>. f (k + N)) = C * norm (\<Prod>k=M..n+M. f (k + N))"
eberlm@66277
   540
            unfolding C_def by (subst prod.union_disjoint) (auto simp: norm_mult prod_norm)
eberlm@66277
   541
          also have "(\<Prod>k=M..n+M. f (k + N)) = (\<Prod>k\<le>n. f (k + N + M))"
eberlm@66277
   542
            by (intro prod.reindex_bij_witness[of _ "\<lambda>i. i + M" "\<lambda>i. i - M"]) auto
eberlm@66277
   543
          finally show ?case by (simp add: add_ac prod_norm)
eberlm@66277
   544
        qed
eberlm@66277
   545
        finally have "(\<lambda>n. C * (\<Prod>k\<le>n. norm (f (k + M + N))) / C) \<longlonglongrightarrow> 0 / C"
eberlm@66277
   546
          by (intro tendsto_divide tendsto_const) auto
eberlm@66277
   547
        thus "(\<lambda>n. \<Prod>k\<le>n. norm (f (k + M + N))) \<longlonglongrightarrow> 0" by simp
eberlm@66277
   548
      qed simp_all
eberlm@66277
   549
eberlm@66277
   550
      have "1 - (\<Sum>i. norm (f (i + M + N) - 1)) \<le> 0"
eberlm@66277
   551
      proof (rule tendsto_le)
eberlm@66277
   552
        show "eventually (\<lambda>n. 1 - (\<Sum>k\<le>n. norm (f (k+M+N) - 1)) \<le> 
eberlm@66277
   553
                                (\<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1))) at_top"
eberlm@66277
   554
          using M by (intro always_eventually allI weierstrass_prod_ineq) (auto intro: less_imp_le)
eberlm@66277
   555
        show "(\<lambda>n. \<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1)) \<longlonglongrightarrow> 0" by fact
eberlm@66277
   556
        show "(\<lambda>n. 1 - (\<Sum>k\<le>n. norm (f (k + M + N) - 1)))
eberlm@66277
   557
                  \<longlonglongrightarrow> 1 - (\<Sum>i. norm (f (i + M + N) - 1))"
eberlm@66277
   558
          by (intro tendsto_intros summable_LIMSEQ' summable_ignore_initial_segment 
eberlm@66277
   559
                abs_convergent_prod_imp_summable assms)
eberlm@66277
   560
      qed simp_all
eberlm@66277
   561
      hence "(\<Sum>i. norm (f (i + M + N) - 1)) \<ge> 1" by simp
eberlm@66277
   562
      also have "\<dots> + (\<Sum>i<M. norm (f (i + N) - 1)) = (\<Sum>i. norm (f (i + N) - 1))"
eberlm@66277
   563
        by (intro suminf_split_initial_segment [symmetric] summable_ignore_initial_segment
eberlm@66277
   564
              abs_convergent_prod_imp_summable assms)
eberlm@66277
   565
      finally have "1 + (\<Sum>i<M. norm (f (i + N) - 1)) \<le> (\<Sum>i. norm (f (i + N) - 1))" by simp
eberlm@66277
   566
    } note * = this
eberlm@66277
   567
eberlm@66277
   568
    have "1 + (\<Sum>i. norm (f (i + N) - 1)) \<le> (\<Sum>i. norm (f (i + N) - 1))"
eberlm@66277
   569
    proof (rule tendsto_le)
eberlm@66277
   570
      show "(\<lambda>M. 1 + (\<Sum>i<M. norm (f (i + N) - 1))) \<longlonglongrightarrow> 1 + (\<Sum>i. norm (f (i + N) - 1))"
eberlm@66277
   571
        by (intro tendsto_intros summable_LIMSEQ summable_ignore_initial_segment 
eberlm@66277
   572
                abs_convergent_prod_imp_summable assms)
eberlm@66277
   573
      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
   574
        using eventually_ge_at_top[of M0] by eventually_elim (use * in auto)
eberlm@66277
   575
    qed simp_all
eberlm@66277
   576
    thus False by simp
eberlm@66277
   577
  qed
lp15@68064
   578
  with L show ?thesis by (auto simp: prod_defs)
lp15@68064
   579
qed
lp15@68064
   580
lp15@68424
   581
subsection\<open>More elementary properties\<close>
lp15@68424
   582
lp15@68361
   583
lemma raw_has_prod_cases:
lp15@68064
   584
  fixes f :: "nat \<Rightarrow> 'a :: {idom,topological_semigroup_mult,t2_space}"
lp15@68361
   585
  assumes "raw_has_prod f M p"
lp15@68361
   586
  obtains i where "i<M" "f i = 0" | p where "raw_has_prod f 0 p"
lp15@68136
   587
proof -
lp15@68136
   588
  have "(\<lambda>n. \<Prod>i\<le>n. f (i + M)) \<longlonglongrightarrow> p" "p \<noteq> 0"
lp15@68361
   589
    using assms unfolding raw_has_prod_def by blast+
lp15@68064
   590
  then have "(\<lambda>n. prod f {..<M} * (\<Prod>i\<le>n. f (i + M))) \<longlonglongrightarrow> prod f {..<M} * p"
lp15@68064
   591
    by (metis tendsto_mult_left)
lp15@68064
   592
  moreover have "prod f {..<M} * (\<Prod>i\<le>n. f (i + M)) = prod f {..n+M}" for n
lp15@68064
   593
  proof -
lp15@68064
   594
    have "{..n+M} = {..<M} \<union> {M..n+M}"
lp15@68064
   595
      by auto
lp15@68064
   596
    then have "prod f {..n+M} = prod f {..<M} * prod f {M..n+M}"
lp15@68064
   597
      by simp (subst prod.union_disjoint; force)
lp15@68138
   598
    also have "\<dots> = prod f {..<M} * (\<Prod>i\<le>n. f (i + M))"
lp15@68064
   599
      by (metis (mono_tags, lifting) add.left_neutral atMost_atLeast0 prod_shift_bounds_cl_nat_ivl)
lp15@68064
   600
    finally show ?thesis by metis
lp15@68064
   601
  qed
lp15@68064
   602
  ultimately have "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> prod f {..<M} * p"
lp15@68064
   603
    by (auto intro: LIMSEQ_offset [where k=M])
lp15@68361
   604
  then have "raw_has_prod f 0 (prod f {..<M} * p)" if "\<forall>i<M. f i \<noteq> 0"
lp15@68361
   605
    using \<open>p \<noteq> 0\<close> assms that by (auto simp: raw_has_prod_def)
lp15@68136
   606
  then show thesis
lp15@68136
   607
    using that by blast
lp15@68064
   608
qed
lp15@68064
   609
lp15@68136
   610
corollary convergent_prod_offset_0:
lp15@68136
   611
  fixes f :: "nat \<Rightarrow> 'a :: {idom,topological_semigroup_mult,t2_space}"
lp15@68136
   612
  assumes "convergent_prod f" "\<And>i. f i \<noteq> 0"
lp15@68361
   613
  shows "\<exists>p. raw_has_prod f 0 p"
lp15@68361
   614
  using assms convergent_prod_def raw_has_prod_cases by blast
lp15@68136
   615
lp15@68064
   616
lemma prodinf_eq_lim:
lp15@68064
   617
  fixes f :: "nat \<Rightarrow> 'a :: {idom,topological_semigroup_mult,t2_space}"
lp15@68064
   618
  assumes "convergent_prod f" "\<And>i. f i \<noteq> 0"
lp15@68064
   619
  shows "prodinf f = lim (\<lambda>n. \<Prod>i\<le>n. f i)"
lp15@68064
   620
  using assms convergent_prod_offset_0 [OF assms]
lp15@68064
   621
  by (simp add: prod_defs lim_def) (metis (no_types) assms(1) convergent_prod_to_zero_iff)
lp15@68064
   622
lp15@68064
   623
lemma has_prod_one[simp, intro]: "(\<lambda>n. 1) has_prod 1"
lp15@68064
   624
  unfolding prod_defs by auto
lp15@68064
   625
lp15@68064
   626
lemma convergent_prod_one[simp, intro]: "convergent_prod (\<lambda>n. 1)"
lp15@68064
   627
  unfolding prod_defs by auto
lp15@68064
   628
lp15@68064
   629
lemma prodinf_cong: "(\<And>n. f n = g n) \<Longrightarrow> prodinf f = prodinf g"
lp15@68064
   630
  by presburger
lp15@68064
   631
lp15@68064
   632
lemma convergent_prod_cong:
lp15@68064
   633
  fixes f g :: "nat \<Rightarrow> 'a::{field,topological_semigroup_mult,t2_space}"
lp15@68064
   634
  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
   635
  shows "convergent_prod f = convergent_prod g"
lp15@68064
   636
proof -
lp15@68064
   637
  from assms obtain N where N: "\<forall>n\<ge>N. f n = g n"
lp15@68064
   638
    by (auto simp: eventually_at_top_linorder)
lp15@68064
   639
  define C where "C = (\<Prod>k<N. f k / g k)"
lp15@68064
   640
  with g have "C \<noteq> 0"
lp15@68064
   641
    by (simp add: f)
lp15@68064
   642
  have *: "eventually (\<lambda>n. prod f {..n} = C * prod g {..n}) sequentially"
lp15@68064
   643
    using eventually_ge_at_top[of N]
lp15@68064
   644
  proof eventually_elim
lp15@68064
   645
    case (elim n)
lp15@68064
   646
    then have "{..n} = {..<N} \<union> {N..n}"
lp15@68064
   647
      by auto
lp15@68138
   648
    also have "prod f \<dots> = prod f {..<N} * prod f {N..n}"
lp15@68064
   649
      by (intro prod.union_disjoint) auto
lp15@68064
   650
    also from N have "prod f {N..n} = prod g {N..n}"
lp15@68064
   651
      by (intro prod.cong) simp_all
lp15@68064
   652
    also have "prod f {..<N} * prod g {N..n} = C * (prod g {..<N} * prod g {N..n})"
lp15@68064
   653
      unfolding C_def by (simp add: g prod_dividef)
lp15@68064
   654
    also have "prod g {..<N} * prod g {N..n} = prod g ({..<N} \<union> {N..n})"
lp15@68064
   655
      by (intro prod.union_disjoint [symmetric]) auto
lp15@68064
   656
    also from elim have "{..<N} \<union> {N..n} = {..n}"
lp15@68064
   657
      by auto                                                                    
lp15@68064
   658
    finally show "prod f {..n} = C * prod g {..n}" .
lp15@68064
   659
  qed
lp15@68064
   660
  then have cong: "convergent (\<lambda>n. prod f {..n}) = convergent (\<lambda>n. C * prod g {..n})"
lp15@68064
   661
    by (rule convergent_cong)
lp15@68064
   662
  show ?thesis
lp15@68064
   663
  proof
lp15@68064
   664
    assume cf: "convergent_prod f"
lp15@68064
   665
    then have "\<not> (\<lambda>n. prod g {..n}) \<longlonglongrightarrow> 0"
lp15@68064
   666
      using tendsto_mult_left * convergent_prod_to_zero_iff f filterlim_cong by fastforce
lp15@68064
   667
    then show "convergent_prod g"
lp15@68064
   668
      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
   669
  next
lp15@68064
   670
    assume cg: "convergent_prod g"
lp15@68064
   671
    have "\<exists>a. C * a \<noteq> 0 \<and> (\<lambda>n. prod g {..n}) \<longlonglongrightarrow> a"
lp15@68064
   672
      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
   673
    then show "convergent_prod f"
lp15@68064
   674
      using "*" tendsto_mult_left filterlim_cong
lp15@68064
   675
      by (fastforce simp add: convergent_prod_iff_nz_lim f)
lp15@68064
   676
  qed
eberlm@66277
   677
qed
eberlm@66277
   678
lp15@68071
   679
lemma has_prod_finite:
lp15@68361
   680
  fixes f :: "nat \<Rightarrow> 'a::{semidom,t2_space}"
lp15@68071
   681
  assumes [simp]: "finite N"
lp15@68071
   682
    and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 1"
lp15@68071
   683
  shows "f has_prod (\<Prod>n\<in>N. f n)"
lp15@68071
   684
proof -
lp15@68071
   685
  have eq: "prod f {..n + Suc (Max N)} = prod f N" for n
lp15@68071
   686
  proof (rule prod.mono_neutral_right)
lp15@68071
   687
    show "N \<subseteq> {..n + Suc (Max N)}"
lp15@68138
   688
      by (auto simp: le_Suc_eq trans_le_add2)
lp15@68071
   689
    show "\<forall>i\<in>{..n + Suc (Max N)} - N. f i = 1"
lp15@68071
   690
      using f by blast
lp15@68071
   691
  qed auto
lp15@68071
   692
  show ?thesis
lp15@68071
   693
  proof (cases "\<forall>n\<in>N. f n \<noteq> 0")
lp15@68071
   694
    case True
lp15@68071
   695
    then have "prod f N \<noteq> 0"
lp15@68071
   696
      by simp
lp15@68071
   697
    moreover have "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> prod f N"
lp15@68071
   698
      by (rule LIMSEQ_offset[of _ "Suc (Max N)"]) (simp add: eq atLeast0LessThan del: add_Suc_right)
lp15@68071
   699
    ultimately show ?thesis
lp15@68361
   700
      by (simp add: raw_has_prod_def has_prod_def)
lp15@68071
   701
  next
lp15@68071
   702
    case False
lp15@68071
   703
    then obtain k where "k \<in> N" "f k = 0"
lp15@68071
   704
      by auto
lp15@68071
   705
    let ?Z = "{n \<in> N. f n = 0}"
lp15@68071
   706
    have maxge: "Max ?Z \<ge> n" if "f n = 0" for n
lp15@68071
   707
      using Max_ge [of ?Z] \<open>finite N\<close> \<open>f n = 0\<close>
lp15@68071
   708
      by (metis (mono_tags) Collect_mem_eq f finite_Collect_conjI mem_Collect_eq zero_neq_one)
lp15@68071
   709
    let ?q = "prod f {Suc (Max ?Z)..Max N}"
lp15@68071
   710
    have [simp]: "?q \<noteq> 0"
lp15@68071
   711
      using maxge Suc_n_not_le_n le_trans by force
lp15@68076
   712
    have eq: "(\<Prod>i\<le>n + Max N. f (Suc (i + Max ?Z))) = ?q" for n
lp15@68076
   713
    proof -
lp15@68076
   714
      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
   715
      proof (rule prod.reindex_cong [where l = "\<lambda>i. i + Suc (Max ?Z)", THEN sym])
lp15@68076
   716
        show "{Suc (Max ?Z)..n + Max N + Suc (Max ?Z)} = (\<lambda>i. i + Suc (Max ?Z)) ` {..n + Max N}"
lp15@68076
   717
          using le_Suc_ex by fastforce
lp15@68076
   718
      qed (auto simp: inj_on_def)
lp15@68138
   719
      also have "\<dots> = ?q"
lp15@68076
   720
        by (rule prod.mono_neutral_right)
lp15@68076
   721
           (use Max.coboundedI [OF \<open>finite N\<close>] f in \<open>force+\<close>)
lp15@68076
   722
      finally show ?thesis .
lp15@68076
   723
    qed
lp15@68361
   724
    have q: "raw_has_prod f (Suc (Max ?Z)) ?q"
lp15@68361
   725
    proof (simp add: raw_has_prod_def)
lp15@68076
   726
      show "(\<lambda>n. \<Prod>i\<le>n. f (Suc (i + Max ?Z))) \<longlonglongrightarrow> ?q"
lp15@68076
   727
        by (rule LIMSEQ_offset[of _ "(Max N)"]) (simp add: eq)
lp15@68076
   728
    qed
lp15@68071
   729
    show ?thesis
lp15@68071
   730
      unfolding has_prod_def
lp15@68071
   731
    proof (intro disjI2 exI conjI)      
lp15@68071
   732
      show "prod f N = 0"
lp15@68071
   733
        using \<open>f k = 0\<close> \<open>k \<in> N\<close> \<open>finite N\<close> prod_zero by blast
lp15@68071
   734
      show "f (Max ?Z) = 0"
lp15@68071
   735
        using Max_in [of ?Z] \<open>finite N\<close> \<open>f k = 0\<close> \<open>k \<in> N\<close> by auto
lp15@68071
   736
    qed (use q in auto)
lp15@68071
   737
  qed
lp15@68071
   738
qed
lp15@68071
   739
lp15@68071
   740
corollary has_prod_0:
lp15@68361
   741
  fixes f :: "nat \<Rightarrow> 'a::{semidom,t2_space}"
lp15@68071
   742
  assumes "\<And>n. f n = 1"
lp15@68071
   743
  shows "f has_prod 1"
lp15@68071
   744
  by (simp add: assms has_prod_cong)
lp15@68071
   745
lp15@68361
   746
lemma prodinf_zero[simp]: "prodinf (\<lambda>n. 1::'a::real_normed_field) = 1"
lp15@68361
   747
  using has_prod_unique by force
lp15@68361
   748
lp15@68071
   749
lemma convergent_prod_finite:
lp15@68071
   750
  fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
lp15@68071
   751
  assumes "finite N" "\<And>n. n \<notin> N \<Longrightarrow> f n = 1"
lp15@68071
   752
  shows "convergent_prod f"
lp15@68071
   753
proof -
lp15@68361
   754
  have "\<exists>n p. raw_has_prod f n p"
lp15@68071
   755
    using assms has_prod_def has_prod_finite by blast
lp15@68071
   756
  then show ?thesis
lp15@68071
   757
    by (simp add: convergent_prod_def)
lp15@68071
   758
qed
lp15@68071
   759
lp15@68127
   760
lemma has_prod_If_finite_set:
lp15@68127
   761
  fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
lp15@68127
   762
  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
   763
  using has_prod_finite[of A "(\<lambda>r. if r \<in> A then f r else 1)"]
lp15@68127
   764
  by simp
lp15@68127
   765
lp15@68127
   766
lemma has_prod_If_finite:
lp15@68127
   767
  fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
lp15@68127
   768
  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
   769
  using has_prod_If_finite_set[of "{r. P r}"] by simp
lp15@68127
   770
lp15@68127
   771
lemma convergent_prod_If_finite_set[simp, intro]:
lp15@68127
   772
  fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
lp15@68127
   773
  shows "finite A \<Longrightarrow> convergent_prod (\<lambda>r. if r \<in> A then f r else 1)"
lp15@68127
   774
  by (simp add: convergent_prod_finite)
lp15@68127
   775
lp15@68127
   776
lemma convergent_prod_If_finite[simp, intro]:
lp15@68127
   777
  fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
lp15@68127
   778
  shows "finite {r. P r} \<Longrightarrow> convergent_prod (\<lambda>r. if P r then f r else 1)"
lp15@68127
   779
  using convergent_prod_def has_prod_If_finite has_prod_def by fastforce
lp15@68127
   780
lp15@68127
   781
lemma has_prod_single:
lp15@68127
   782
  fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
lp15@68127
   783
  shows "(\<lambda>r. if r = i then f r else 1) has_prod f i"
lp15@68127
   784
  using has_prod_If_finite[of "\<lambda>r. r = i"] by simp
lp15@68127
   785
lp15@68136
   786
context
lp15@68136
   787
  fixes f :: "nat \<Rightarrow> 'a :: real_normed_field"
lp15@68136
   788
begin
lp15@68136
   789
lp15@68136
   790
lemma convergent_prod_imp_has_prod: 
lp15@68136
   791
  assumes "convergent_prod f"
lp15@68136
   792
  shows "\<exists>p. f has_prod p"
lp15@68136
   793
proof -
lp15@68361
   794
  obtain M p where p: "raw_has_prod f M p"
lp15@68136
   795
    using assms convergent_prod_def by blast
lp15@68136
   796
  then have "p \<noteq> 0"
lp15@68361
   797
    using raw_has_prod_nonzero by blast
lp15@68136
   798
  with p have fnz: "f i \<noteq> 0" if "i \<ge> M" for i
lp15@68361
   799
    using raw_has_prod_eq_0 that by blast
lp15@68136
   800
  define C where "C = (\<Prod>n<M. f n)"
lp15@68136
   801
  show ?thesis
lp15@68136
   802
  proof (cases "\<forall>n\<le>M. f n \<noteq> 0")
lp15@68136
   803
    case True
lp15@68136
   804
    then have "C \<noteq> 0"
lp15@68136
   805
      by (simp add: C_def)
lp15@68136
   806
    then show ?thesis
lp15@68136
   807
      by (meson True assms convergent_prod_offset_0 fnz has_prod_def nat_le_linear)
lp15@68136
   808
  next
lp15@68136
   809
    case False
lp15@68136
   810
    let ?N = "GREATEST n. f n = 0"
lp15@68136
   811
    have 0: "f ?N = 0"
lp15@68136
   812
      using fnz False
lp15@68136
   813
      by (metis (mono_tags, lifting) GreatestI_ex_nat nat_le_linear)
lp15@68136
   814
    have "f i \<noteq> 0" if "i > ?N" for i
lp15@68136
   815
      by (metis (mono_tags, lifting) Greatest_le_nat fnz leD linear that)
lp15@68361
   816
    then have "\<exists>p. raw_has_prod f (Suc ?N) p"
lp15@68136
   817
      using assms by (auto simp: intro!: convergent_prod_ignore_nonzero_segment)
lp15@68136
   818
    then show ?thesis
lp15@68136
   819
      unfolding has_prod_def using 0 by blast
lp15@68136
   820
  qed
lp15@68136
   821
qed
lp15@68136
   822
lp15@68136
   823
lemma convergent_prod_has_prod [intro]:
lp15@68136
   824
  shows "convergent_prod f \<Longrightarrow> f has_prod (prodinf f)"
lp15@68136
   825
  unfolding prodinf_def
lp15@68136
   826
  by (metis convergent_prod_imp_has_prod has_prod_unique theI')
lp15@68136
   827
lp15@68136
   828
lemma convergent_prod_LIMSEQ:
lp15@68136
   829
  shows "convergent_prod f \<Longrightarrow> (\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> prodinf f"
lp15@68136
   830
  by (metis convergent_LIMSEQ_iff convergent_prod_has_prod convergent_prod_imp_convergent 
lp15@68361
   831
      convergent_prod_to_zero_iff raw_has_prod_eq_0 has_prod_def prodinf_eq_lim zero_le)
lp15@68136
   832
lp15@68136
   833
lemma has_prod_iff: "f has_prod x \<longleftrightarrow> convergent_prod f \<and> prodinf f = x"
lp15@68136
   834
proof
lp15@68136
   835
  assume "f has_prod x"
lp15@68136
   836
  then show "convergent_prod f \<and> prodinf f = x"
lp15@68136
   837
    apply safe
lp15@68136
   838
    using convergent_prod_def has_prod_def apply blast
lp15@68136
   839
    using has_prod_unique by blast
lp15@68136
   840
qed auto
lp15@68136
   841
lp15@68136
   842
lemma convergent_prod_has_prod_iff: "convergent_prod f \<longleftrightarrow> f has_prod prodinf f"
lp15@68136
   843
  by (auto simp: has_prod_iff convergent_prod_has_prod)
lp15@68136
   844
lp15@68136
   845
lemma prodinf_finite:
lp15@68136
   846
  assumes N: "finite N"
lp15@68136
   847
    and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 1"
lp15@68136
   848
  shows "prodinf f = (\<Prod>n\<in>N. f n)"
lp15@68136
   849
  using has_prod_finite[OF assms, THEN has_prod_unique] by simp
lp15@68127
   850
eberlm@66277
   851
end
lp15@68136
   852
lp15@68361
   853
subsection \<open>Infinite products on ordered, topological monoids\<close>
lp15@68361
   854
lp15@68361
   855
lemma LIMSEQ_prod_0: 
lp15@68361
   856
  fixes f :: "nat \<Rightarrow> 'a::{semidom,topological_space}"
lp15@68361
   857
  assumes "f i = 0"
lp15@68361
   858
  shows "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> 0"
lp15@68361
   859
proof (subst tendsto_cong)
lp15@68361
   860
  show "\<forall>\<^sub>F n in sequentially. prod f {..n} = 0"
lp15@68361
   861
  proof
lp15@68361
   862
    show "prod f {..n} = 0" if "n \<ge> i" for n
lp15@68361
   863
      using that assms by auto
lp15@68361
   864
  qed
lp15@68361
   865
qed auto
lp15@68361
   866
lp15@68361
   867
lemma LIMSEQ_prod_nonneg: 
lp15@68361
   868
  fixes f :: "nat \<Rightarrow> 'a::{linordered_semidom,linorder_topology}"
lp15@68361
   869
  assumes 0: "\<And>n. 0 \<le> f n" and a: "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> a"
lp15@68361
   870
  shows "a \<ge> 0"
lp15@68361
   871
  by (simp add: "0" prod_nonneg LIMSEQ_le_const [OF a])
lp15@68361
   872
lp15@68361
   873
lp15@68361
   874
context
lp15@68361
   875
  fixes f :: "nat \<Rightarrow> 'a::{linordered_semidom,linorder_topology}"
lp15@68361
   876
begin
lp15@68361
   877
lp15@68361
   878
lemma has_prod_le:
lp15@68361
   879
  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
   880
  shows "a \<le> b"
lp15@68361
   881
proof (cases "a=0 \<or> b=0")
lp15@68361
   882
  case True
lp15@68361
   883
  then show ?thesis
lp15@68361
   884
  proof
lp15@68361
   885
    assume [simp]: "a=0"
lp15@68361
   886
    have "b \<ge> 0"
lp15@68361
   887
    proof (rule LIMSEQ_prod_nonneg)
lp15@68361
   888
      show "(\<lambda>n. prod g {..n}) \<longlonglongrightarrow> b"
lp15@68361
   889
        using g by (auto simp: has_prod_def raw_has_prod_def LIMSEQ_prod_0)
lp15@68361
   890
    qed (use le order_trans in auto)
lp15@68361
   891
    then show ?thesis
lp15@68361
   892
      by auto
lp15@68361
   893
  next
lp15@68361
   894
    assume [simp]: "b=0"
lp15@68361
   895
    then obtain i where "g i = 0"    
lp15@68361
   896
      using g by (auto simp: prod_defs)
lp15@68361
   897
    then have "f i = 0"
lp15@68361
   898
      using antisym le by force
lp15@68361
   899
    then have "a=0"
lp15@68361
   900
      using f by (auto simp: prod_defs LIMSEQ_prod_0 LIMSEQ_unique)
lp15@68361
   901
    then show ?thesis
lp15@68361
   902
      by auto
lp15@68361
   903
  qed
lp15@68361
   904
next
lp15@68361
   905
  case False
lp15@68361
   906
  then show ?thesis
lp15@68361
   907
    using assms
lp15@68361
   908
    unfolding has_prod_def raw_has_prod_def
lp15@68361
   909
    by (force simp: LIMSEQ_prod_0 intro!: LIMSEQ_le prod_mono)
lp15@68361
   910
qed
lp15@68361
   911
lp15@68361
   912
lemma prodinf_le: 
lp15@68361
   913
  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
   914
  shows "prodinf f \<le> prodinf g"
lp15@68361
   915
  using has_prod_le [OF assms] has_prod_unique f g  by blast
lp15@68361
   916
lp15@68136
   917
end
lp15@68361
   918
lp15@68361
   919
lp15@68361
   920
lemma prod_le_prodinf: 
lp15@68361
   921
  fixes f :: "nat \<Rightarrow> 'a::{linordered_idom,linorder_topology}"
lp15@68361
   922
  assumes "f has_prod a" "\<And>i. 0 \<le> f i" "\<And>i. i\<ge>n \<Longrightarrow> 1 \<le> f i"
lp15@68361
   923
  shows "prod f {..<n} \<le> prodinf f"
lp15@68361
   924
  by(rule has_prod_le[OF has_prod_If_finite_set]) (use assms has_prod_unique in auto)
lp15@68361
   925
lp15@68361
   926
lemma prodinf_nonneg:
lp15@68361
   927
  fixes f :: "nat \<Rightarrow> 'a::{linordered_idom,linorder_topology}"
lp15@68361
   928
  assumes "f has_prod a" "\<And>i. 1 \<le> f i" 
lp15@68361
   929
  shows "1 \<le> prodinf f"
lp15@68361
   930
  using prod_le_prodinf[of f a 0] assms
lp15@68361
   931
  by (metis order_trans prod_ge_1 zero_le_one)
lp15@68361
   932
lp15@68361
   933
lemma prodinf_le_const:
lp15@68361
   934
  fixes f :: "nat \<Rightarrow> real"
lp15@68361
   935
  assumes "convergent_prod f" "\<And>n. prod f {..<n} \<le> x" 
lp15@68361
   936
  shows "prodinf f \<le> x"
lp15@68361
   937
  by (metis lessThan_Suc_atMost assms convergent_prod_LIMSEQ LIMSEQ_le_const2)
lp15@68361
   938
lp15@68361
   939
lemma prodinf_eq_one_iff: 
lp15@68361
   940
  fixes f :: "nat \<Rightarrow> real"
lp15@68361
   941
  assumes f: "convergent_prod f" and ge1: "\<And>n. 1 \<le> f n"
lp15@68361
   942
  shows "prodinf f = 1 \<longleftrightarrow> (\<forall>n. f n = 1)"
lp15@68361
   943
proof
lp15@68361
   944
  assume "prodinf f = 1" 
lp15@68361
   945
  then have "(\<lambda>n. \<Prod>i<n. f i) \<longlonglongrightarrow> 1"
lp15@68361
   946
    using convergent_prod_LIMSEQ[of f] assms by (simp add: LIMSEQ_lessThan_iff_atMost)
lp15@68361
   947
  then have "\<And>i. (\<Prod>n\<in>{i}. f n) \<le> 1"
lp15@68361
   948
  proof (rule LIMSEQ_le_const)
lp15@68361
   949
    have "1 \<le> prod f n" for n
lp15@68361
   950
      by (simp add: ge1 prod_ge_1)
lp15@68361
   951
    have "prod f {..<n} = 1" for n
lp15@68361
   952
      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
   953
    then have "(\<Prod>n\<in>{i}. f n) \<le> prod f {..<n}" if "n \<ge> Suc i" for i n
lp15@68361
   954
      by (metis mult.left_neutral order_refl prod.cong prod.neutral_const prod_lessThan_Suc)
lp15@68361
   955
    then show "\<exists>N. \<forall>n\<ge>N. (\<Prod>n\<in>{i}. f n) \<le> prod f {..<n}" for i
lp15@68361
   956
      by blast      
lp15@68361
   957
  qed
lp15@68361
   958
  with ge1 show "\<forall>n. f n = 1"
lp15@68361
   959
    by (auto intro!: antisym)
lp15@68361
   960
qed (metis prodinf_zero fun_eq_iff)
lp15@68361
   961
lp15@68361
   962
lemma prodinf_pos_iff:
lp15@68361
   963
  fixes f :: "nat \<Rightarrow> real"
lp15@68361
   964
  assumes "convergent_prod f" "\<And>n. 1 \<le> f n"
lp15@68361
   965
  shows "1 < prodinf f \<longleftrightarrow> (\<exists>i. 1 < f i)"
lp15@68361
   966
  using prod_le_prodinf[of f 1] prodinf_eq_one_iff
lp15@68361
   967
  by (metis convergent_prod_has_prod assms less_le prodinf_nonneg)
lp15@68361
   968
lp15@68361
   969
lemma less_1_prodinf2:
lp15@68361
   970
  fixes f :: "nat \<Rightarrow> real"
lp15@68361
   971
  assumes "convergent_prod f" "\<And>n. 1 \<le> f n" "1 < f i"
lp15@68361
   972
  shows "1 < prodinf f"
lp15@68361
   973
proof -
lp15@68361
   974
  have "1 < (\<Prod>n<Suc i. f n)"
lp15@68361
   975
    using assms  by (intro less_1_prod2[where i=i]) auto
lp15@68361
   976
  also have "\<dots> \<le> prodinf f"
lp15@68361
   977
    by (intro prod_le_prodinf) (use assms order_trans zero_le_one in \<open>blast+\<close>)
lp15@68361
   978
  finally show ?thesis .
lp15@68361
   979
qed
lp15@68361
   980
lp15@68361
   981
lemma less_1_prodinf:
lp15@68361
   982
  fixes f :: "nat \<Rightarrow> real"
lp15@68361
   983
  shows "\<lbrakk>convergent_prod f; \<And>n. 1 < f n\<rbrakk> \<Longrightarrow> 1 < prodinf f"
lp15@68361
   984
  by (intro less_1_prodinf2[where i=1]) (auto intro: less_imp_le)
lp15@68361
   985
lp15@68361
   986
lemma prodinf_nonzero:
lp15@68361
   987
  fixes f :: "nat \<Rightarrow> 'a :: {idom,topological_semigroup_mult,t2_space}"
lp15@68361
   988
  assumes "convergent_prod f" "\<And>i. f i \<noteq> 0"
lp15@68361
   989
  shows "prodinf f \<noteq> 0"
lp15@68361
   990
  by (metis assms convergent_prod_offset_0 has_prod_unique raw_has_prod_def has_prod_def)
lp15@68361
   991
lp15@68361
   992
lemma less_0_prodinf:
lp15@68361
   993
  fixes f :: "nat \<Rightarrow> real"
lp15@68361
   994
  assumes f: "convergent_prod f" and 0: "\<And>i. f i > 0"
lp15@68361
   995
  shows "0 < prodinf f"
lp15@68361
   996
proof -
lp15@68361
   997
  have "prodinf f \<noteq> 0"
lp15@68361
   998
    by (metis assms less_irrefl prodinf_nonzero)
lp15@68361
   999
  moreover have "0 < (\<Prod>n<i. f n)" for i
lp15@68361
  1000
    by (simp add: 0 prod_pos)
lp15@68361
  1001
  then have "prodinf f \<ge> 0"
lp15@68361
  1002
    using convergent_prod_LIMSEQ [OF f] LIMSEQ_prod_nonneg 0 less_le by blast
lp15@68361
  1003
  ultimately show ?thesis
lp15@68361
  1004
    by auto
lp15@68361
  1005
qed
lp15@68361
  1006
lp15@68361
  1007
lemma prod_less_prodinf2:
lp15@68361
  1008
  fixes f :: "nat \<Rightarrow> real"
lp15@68361
  1009
  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
  1010
  shows "prod f {..<n} < prodinf f"
lp15@68361
  1011
proof -
lp15@68361
  1012
  have "prod f {..<n} \<le> prod f {..<i}"
lp15@68361
  1013
    by (rule prod_mono2) (use assms less_le in auto)
lp15@68361
  1014
  then have "prod f {..<n} < f i * prod f {..<i}"
lp15@68361
  1015
    using mult_less_le_imp_less[of 1 "f i" "prod f {..<n}" "prod f {..<i}"] assms
lp15@68361
  1016
    by (simp add: prod_pos)
lp15@68361
  1017
  moreover have "prod f {..<Suc i} \<le> prodinf f"
lp15@68361
  1018
    using prod_le_prodinf[of f _ "Suc i"]
lp15@68361
  1019
    by (meson "0" "1" Suc_leD convergent_prod_has_prod f \<open>n \<le> i\<close> le_trans less_eq_real_def)
lp15@68361
  1020
  ultimately show ?thesis
lp15@68361
  1021
    by (metis le_less_trans mult.commute not_le prod_lessThan_Suc)
lp15@68361
  1022
qed
lp15@68361
  1023
lp15@68361
  1024
lemma prod_less_prodinf:
lp15@68361
  1025
  fixes f :: "nat \<Rightarrow> real"
lp15@68361
  1026
  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
  1027
  shows "prod f {..<n} < prodinf f"
lp15@68361
  1028
  by (meson "0" "1" f le_less prod_less_prodinf2)
lp15@68361
  1029
lp15@68361
  1030
lemma raw_has_prodI_bounded:
lp15@68361
  1031
  fixes f :: "nat \<Rightarrow> real"
lp15@68361
  1032
  assumes pos: "\<And>n. 1 \<le> f n"
lp15@68361
  1033
    and le: "\<And>n. (\<Prod>i<n. f i) \<le> x"
lp15@68361
  1034
  shows "\<exists>p. raw_has_prod f 0 p"
lp15@68361
  1035
  unfolding raw_has_prod_def add_0_right
lp15@68361
  1036
proof (rule exI LIMSEQ_incseq_SUP conjI)+
lp15@68361
  1037
  show "bdd_above (range (\<lambda>n. prod f {..n}))"
lp15@68361
  1038
    by (metis bdd_aboveI2 le lessThan_Suc_atMost)
lp15@68361
  1039
  then have "(SUP i. prod f {..i}) > 0"
lp15@68361
  1040
    by (metis UNIV_I cSUP_upper less_le_trans pos prod_pos zero_less_one)
lp15@68361
  1041
  then show "(SUP i. prod f {..i}) \<noteq> 0"
lp15@68361
  1042
    by auto
lp15@68361
  1043
  show "incseq (\<lambda>n. prod f {..n})"
lp15@68361
  1044
    using pos order_trans [OF zero_le_one] by (auto simp: mono_def intro!: prod_mono2)
lp15@68361
  1045
qed
lp15@68361
  1046
lp15@68361
  1047
lemma convergent_prodI_nonneg_bounded:
lp15@68361
  1048
  fixes f :: "nat \<Rightarrow> real"
lp15@68361
  1049
  assumes "\<And>n. 1 \<le> f n" "\<And>n. (\<Prod>i<n. f i) \<le> x"
lp15@68361
  1050
  shows "convergent_prod f"
lp15@68361
  1051
  using convergent_prod_def raw_has_prodI_bounded [OF assms] by blast
lp15@68361
  1052
lp15@68361
  1053
lp15@68424
  1054
subsection \<open>Infinite products on topological spaces\<close>
lp15@68361
  1055
lp15@68361
  1056
context
lp15@68361
  1057
  fixes f g :: "nat \<Rightarrow> 'a::{t2_space,topological_semigroup_mult,idom}"
lp15@68361
  1058
begin
lp15@68361
  1059
lp15@68361
  1060
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
  1061
  by (force simp add: prod.distrib tendsto_mult raw_has_prod_def)
lp15@68361
  1062
lp15@68361
  1063
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
  1064
  by (simp add: raw_has_prod_mult has_prod_def)
lp15@68361
  1065
lp15@68361
  1066
end
lp15@68361
  1067
lp15@68361
  1068
lp15@68361
  1069
context
lp15@68361
  1070
  fixes f g :: "nat \<Rightarrow> 'a::real_normed_field"
lp15@68361
  1071
begin
lp15@68361
  1072
lp15@68361
  1073
lemma has_prod_mult:
lp15@68361
  1074
  assumes f: "f has_prod a" and g: "g has_prod b"
lp15@68361
  1075
  shows "(\<lambda>n. f n * g n) has_prod (a * b)"
lp15@68361
  1076
  using f [unfolded has_prod_def]
lp15@68361
  1077
proof (elim disjE exE conjE)
lp15@68361
  1078
  assume f0: "raw_has_prod f 0 a"
lp15@68361
  1079
  show ?thesis
lp15@68361
  1080
    using g [unfolded has_prod_def]
lp15@68361
  1081
  proof (elim disjE exE conjE)
lp15@68361
  1082
    assume g0: "raw_has_prod g 0 b"
lp15@68361
  1083
    with f0 show ?thesis
lp15@68361
  1084
      by (force simp add: has_prod_def prod.distrib tendsto_mult raw_has_prod_def)
lp15@68361
  1085
  next
lp15@68361
  1086
    fix j q
lp15@68361
  1087
    assume "b = 0" and "g j = 0" and q: "raw_has_prod g (Suc j) q"
lp15@68361
  1088
    obtain p where p: "raw_has_prod f (Suc j) p"
lp15@68361
  1089
      using f0 raw_has_prod_ignore_initial_segment by blast
lp15@68361
  1090
    then have "Ex (raw_has_prod (\<lambda>n. f n * g n) (Suc j))"
lp15@68361
  1091
      using q raw_has_prod_mult by blast
lp15@68361
  1092
    then show ?thesis
lp15@68361
  1093
      using \<open>b = 0\<close> \<open>g j = 0\<close> has_prod_0_iff by fastforce
lp15@68361
  1094
  qed
lp15@68361
  1095
next
lp15@68361
  1096
  fix i p
lp15@68361
  1097
  assume "a = 0" and "f i = 0" and p: "raw_has_prod f (Suc i) p"
lp15@68361
  1098
  show ?thesis
lp15@68361
  1099
    using g [unfolded has_prod_def]
lp15@68361
  1100
  proof (elim disjE exE conjE)
lp15@68361
  1101
    assume g0: "raw_has_prod g 0 b"
lp15@68361
  1102
    obtain q where q: "raw_has_prod g (Suc i) q"
lp15@68361
  1103
      using g0 raw_has_prod_ignore_initial_segment by blast
lp15@68361
  1104
    then have "Ex (raw_has_prod (\<lambda>n. f n * g n) (Suc i))"
lp15@68361
  1105
      using raw_has_prod_mult p by blast
lp15@68361
  1106
    then show ?thesis
lp15@68361
  1107
      using \<open>a = 0\<close> \<open>f i = 0\<close> has_prod_0_iff by fastforce
lp15@68361
  1108
  next
lp15@68361
  1109
    fix j q
lp15@68361
  1110
    assume "b = 0" and "g j = 0" and q: "raw_has_prod g (Suc j) q"
lp15@68361
  1111
    obtain p' where p': "raw_has_prod f (Suc (max i j)) p'"
lp15@68361
  1112
      by (metis raw_has_prod_ignore_initial_segment max_Suc_Suc max_def p)
lp15@68361
  1113
    moreover
lp15@68361
  1114
    obtain q' where q': "raw_has_prod g (Suc (max i j)) q'"
lp15@68361
  1115
      by (metis raw_has_prod_ignore_initial_segment max.cobounded2 max_Suc_Suc q)
lp15@68361
  1116
    ultimately show ?thesis
lp15@68361
  1117
      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
  1118
  qed
lp15@68361
  1119
qed
lp15@68361
  1120
lp15@68361
  1121
lemma convergent_prod_mult:
lp15@68361
  1122
  assumes f: "convergent_prod f" and g: "convergent_prod g"
lp15@68361
  1123
  shows "convergent_prod (\<lambda>n. f n * g n)"
lp15@68361
  1124
  unfolding convergent_prod_def
lp15@68361
  1125
proof -
lp15@68361
  1126
  obtain M p N q where p: "raw_has_prod f M p" and q: "raw_has_prod g N q"
lp15@68361
  1127
    using convergent_prod_def f g by blast+
lp15@68361
  1128
  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
  1129
    by (meson raw_has_prod_ignore_initial_segment max.cobounded1 max.cobounded2)
lp15@68361
  1130
  then show "\<exists>M p. raw_has_prod (\<lambda>n. f n * g n) M p"
lp15@68361
  1131
    using raw_has_prod_mult by blast
lp15@68361
  1132
qed
lp15@68361
  1133
lp15@68361
  1134
lemma prodinf_mult: "convergent_prod f \<Longrightarrow> convergent_prod g \<Longrightarrow> prodinf f * prodinf g = (\<Prod>n. f n * g n)"
lp15@68361
  1135
  by (intro has_prod_unique has_prod_mult convergent_prod_has_prod)
lp15@68361
  1136
lp15@68361
  1137
end
lp15@68361
  1138
lp15@68361
  1139
context
lp15@68361
  1140
  fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::real_normed_field"
lp15@68361
  1141
    and I :: "'i set"
lp15@68361
  1142
begin
lp15@68361
  1143
lp15@68361
  1144
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
  1145
  by (induct I rule: infinite_finite_induct) (auto intro!: has_prod_mult)
lp15@68361
  1146
lp15@68361
  1147
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
  1148
  using has_prod_unique[OF has_prod_prod, OF convergent_prod_has_prod] by simp
lp15@68361
  1149
lp15@68361
  1150
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
  1151
  using convergent_prod_has_prod_iff has_prod_prod prodinf_prod by force
lp15@68361
  1152
lp15@68361
  1153
end
lp15@68361
  1154
lp15@68424
  1155
subsection \<open>Infinite summability on real normed fields\<close>
lp15@68361
  1156
lp15@68361
  1157
context
lp15@68361
  1158
  fixes f :: "nat \<Rightarrow> 'a::real_normed_field"
lp15@68361
  1159
begin
lp15@68361
  1160
lp15@68361
  1161
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
  1162
proof -
lp15@68361
  1163
  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
  1164
    by (subst LIMSEQ_Suc_iff) (simp add: raw_has_prod_def)
lp15@68361
  1165
  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
  1166
    by (simp add: ac_simps atMost_Suc_eq_insert_0 image_Suc_atMost prod_atLeast1_atMost_eq lessThan_Suc_atMost)
lp15@68361
  1167
  also have "\<dots> \<longleftrightarrow> raw_has_prod (\<lambda>n. f (Suc n)) M a \<and> f M \<noteq> 0"
lp15@68361
  1168
  proof safe
lp15@68361
  1169
    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
  1170
    with tendsto_divide[OF tends tendsto_const, of "f M"]    
lp15@68361
  1171
    show "raw_has_prod (\<lambda>n. f (Suc n)) M a"
lp15@68361
  1172
      by (simp add: raw_has_prod_def)
lp15@68361
  1173
  qed (auto intro: tendsto_mult_right simp:  raw_has_prod_def)
lp15@68361
  1174
  finally show ?thesis .
lp15@68361
  1175
qed
lp15@68361
  1176
lp15@68361
  1177
lemma has_prod_Suc_iff:
lp15@68361
  1178
  assumes "f 0 \<noteq> 0" shows "(\<lambda>n. f (Suc n)) has_prod a \<longleftrightarrow> f has_prod (a * f 0)"
lp15@68361
  1179
proof (cases "a = 0")
lp15@68361
  1180
  case True
lp15@68361
  1181
  then show ?thesis
lp15@68361
  1182
  proof (simp add: has_prod_def, safe)
lp15@68361
  1183
    fix i x
lp15@68361
  1184
    assume "f (Suc i) = 0" and "raw_has_prod (\<lambda>n. f (Suc n)) (Suc i) x"
lp15@68361
  1185
    then obtain y where "raw_has_prod f (Suc (Suc i)) y"
lp15@68361
  1186
      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
  1187
    then show "\<exists>i. f i = 0 \<and> Ex (raw_has_prod f (Suc i))"
lp15@68361
  1188
      using \<open>f (Suc i) = 0\<close> by blast
lp15@68361
  1189
  next
lp15@68361
  1190
    fix i x
lp15@68361
  1191
    assume "f i = 0" and x: "raw_has_prod f (Suc i) x"
lp15@68361
  1192
    then obtain j where j: "i = Suc j"
lp15@68361
  1193
      by (metis assms not0_implies_Suc)
lp15@68361
  1194
    moreover have "\<exists> y. raw_has_prod (\<lambda>n. f (Suc n)) i y"
lp15@68361
  1195
      using x by (auto simp: raw_has_prod_def)
lp15@68361
  1196
    then show "\<exists>i. f (Suc i) = 0 \<and> Ex (raw_has_prod (\<lambda>n. f (Suc n)) (Suc i))"
lp15@68361
  1197
      using \<open>f i = 0\<close> j by blast
lp15@68361
  1198
  qed
lp15@68361
  1199
next
lp15@68361
  1200
  case False
lp15@68361
  1201
  then show ?thesis
lp15@68361
  1202
    by (auto simp: has_prod_def raw_has_prod_Suc_iff assms)
lp15@68361
  1203
qed
lp15@68361
  1204
lp15@68361
  1205
lemma convergent_prod_Suc_iff:
lp15@68361
  1206
  assumes "\<And>k. f k \<noteq> 0" shows "convergent_prod (\<lambda>n. f (Suc n)) = convergent_prod f"
lp15@68361
  1207
proof
lp15@68361
  1208
  assume "convergent_prod f"
lp15@68361
  1209
  then have "f has_prod prodinf f"
lp15@68361
  1210
    by (rule convergent_prod_has_prod)
lp15@68361
  1211
  moreover have "prodinf f \<noteq> 0"
lp15@68361
  1212
    by (simp add: \<open>convergent_prod f\<close> assms prodinf_nonzero)
lp15@68361
  1213
  ultimately have "(\<lambda>n. f (Suc n)) has_prod (prodinf f * inverse (f 0))"
lp15@68361
  1214
    by (simp add: has_prod_Suc_iff inverse_eq_divide assms)
lp15@68361
  1215
  then show "convergent_prod (\<lambda>n. f (Suc n))"
lp15@68361
  1216
    using has_prod_iff by blast
lp15@68361
  1217
next
lp15@68361
  1218
  assume "convergent_prod (\<lambda>n. f (Suc n))"
lp15@68361
  1219
  then show "convergent_prod f"
lp15@68361
  1220
    using assms convergent_prod_def raw_has_prod_Suc_iff by blast
lp15@68361
  1221
qed
lp15@68361
  1222
lp15@68361
  1223
lemma raw_has_prod_inverse: 
lp15@68361
  1224
  assumes "raw_has_prod f M a" shows "raw_has_prod (\<lambda>n. inverse (f n)) M (inverse a)"
lp15@68361
  1225
  using assms unfolding raw_has_prod_def by (auto dest: tendsto_inverse simp: prod_inversef [symmetric])
lp15@68361
  1226
lp15@68361
  1227
lemma has_prod_inverse: 
lp15@68361
  1228
  assumes "f has_prod a" shows "(\<lambda>n. inverse (f n)) has_prod (inverse a)"
lp15@68361
  1229
using assms raw_has_prod_inverse unfolding has_prod_def by auto 
lp15@68361
  1230
lp15@68361
  1231
lemma convergent_prod_inverse:
lp15@68361
  1232
  assumes "convergent_prod f" 
lp15@68361
  1233
  shows "convergent_prod (\<lambda>n. inverse (f n))"
lp15@68361
  1234
  using assms unfolding convergent_prod_def  by (blast intro: raw_has_prod_inverse elim: )
lp15@68361
  1235
lp15@68361
  1236
end
lp15@68361
  1237
lp15@68424
  1238
context 
lp15@68361
  1239
  fixes f :: "nat \<Rightarrow> 'a::real_normed_field"
lp15@68361
  1240
begin
lp15@68361
  1241
lp15@68361
  1242
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
  1243
  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
  1244
lp15@68361
  1245
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
  1246
  unfolding divide_inverse by (intro has_prod_inverse has_prod_mult)
lp15@68361
  1247
lp15@68361
  1248
lemma convergent_prod_divide:
lp15@68361
  1249
  assumes f: "convergent_prod f" and g: "convergent_prod g"
lp15@68361
  1250
  shows "convergent_prod (\<lambda>n. f n / g n)"
lp15@68361
  1251
  using f g has_prod_divide has_prod_iff by blast
lp15@68361
  1252
lp15@68361
  1253
lemma prodinf_divide: "convergent_prod f \<Longrightarrow> convergent_prod g \<Longrightarrow> prodinf f / prodinf g = (\<Prod>n. f n / g n)"
lp15@68361
  1254
  by (intro has_prod_unique has_prod_divide convergent_prod_has_prod)
lp15@68361
  1255
lp15@68361
  1256
lemma prodinf_inverse: "convergent_prod f \<Longrightarrow> (\<Prod>n. inverse (f n)) = inverse (\<Prod>n. f n)"
lp15@68361
  1257
  by (intro has_prod_unique [symmetric] has_prod_inverse convergent_prod_has_prod)
lp15@68361
  1258
lp15@68361
  1259
lemma has_prod_iff_shift: 
lp15@68361
  1260
  assumes "\<And>i. i < n \<Longrightarrow> f i \<noteq> 0"
lp15@68361
  1261
  shows "(\<lambda>i. f (i + n)) has_prod a \<longleftrightarrow> f has_prod (a * (\<Prod>i<n. f i))"
lp15@68361
  1262
  using assms
lp15@68361
  1263
proof (induct n arbitrary: a)
lp15@68361
  1264
  case 0
lp15@68361
  1265
  then show ?case by simp
lp15@68361
  1266
next
lp15@68361
  1267
  case (Suc n)
lp15@68361
  1268
  then have "(\<lambda>i. f (Suc i + n)) has_prod a \<longleftrightarrow> (\<lambda>i. f (i + n)) has_prod (a * f n)"
lp15@68361
  1269
    by (subst has_prod_Suc_iff) auto
lp15@68361
  1270
  with Suc show ?case
lp15@68361
  1271
    by (simp add: ac_simps)
lp15@68361
  1272
qed
lp15@68361
  1273
lp15@68361
  1274
corollary has_prod_iff_shift':
lp15@68361
  1275
  assumes "\<And>i. i < n \<Longrightarrow> f i \<noteq> 0"
lp15@68361
  1276
  shows "(\<lambda>i. f (i + n)) has_prod (a / (\<Prod>i<n. f i)) \<longleftrightarrow> f has_prod a"
lp15@68361
  1277
  by (simp add: assms has_prod_iff_shift)
lp15@68361
  1278
lp15@68361
  1279
lemma has_prod_one_iff_shift:
lp15@68361
  1280
  assumes "\<And>i. i < n \<Longrightarrow> f i = 1"
lp15@68361
  1281
  shows "(\<lambda>i. f (i+n)) has_prod a \<longleftrightarrow> (\<lambda>i. f i) has_prod a"
lp15@68361
  1282
  by (simp add: assms has_prod_iff_shift)
lp15@68361
  1283
lp15@68361
  1284
lemma convergent_prod_iff_shift:
lp15@68361
  1285
  shows "convergent_prod (\<lambda>i. f (i + n)) \<longleftrightarrow> convergent_prod f"
lp15@68361
  1286
  apply safe
lp15@68361
  1287
  using convergent_prod_offset apply blast
lp15@68361
  1288
  using convergent_prod_ignore_initial_segment convergent_prod_def by blast
lp15@68361
  1289
lp15@68361
  1290
lemma has_prod_split_initial_segment:
lp15@68361
  1291
  assumes "f has_prod a" "\<And>i. i < n \<Longrightarrow> f i \<noteq> 0"
lp15@68361
  1292
  shows "(\<lambda>i. f (i + n)) has_prod (a / (\<Prod>i<n. f i))"
lp15@68361
  1293
  using assms has_prod_iff_shift' by blast
lp15@68361
  1294
lp15@68361
  1295
lemma prodinf_divide_initial_segment:
lp15@68361
  1296
  assumes "convergent_prod f" "\<And>i. i < n \<Longrightarrow> f i \<noteq> 0"
lp15@68361
  1297
  shows "(\<Prod>i. f (i + n)) = (\<Prod>i. f i) / (\<Prod>i<n. f i)"
lp15@68361
  1298
  by (rule has_prod_unique[symmetric]) (auto simp: assms has_prod_iff_shift)
lp15@68361
  1299
lp15@68361
  1300
lemma prodinf_split_initial_segment:
lp15@68361
  1301
  assumes "convergent_prod f" "\<And>i. i < n \<Longrightarrow> f i \<noteq> 0"
lp15@68361
  1302
  shows "prodinf f = (\<Prod>i. f (i + n)) * (\<Prod>i<n. f i)"
lp15@68361
  1303
  by (auto simp add: assms prodinf_divide_initial_segment)
lp15@68361
  1304
lp15@68361
  1305
lemma prodinf_split_head:
lp15@68361
  1306
  assumes "convergent_prod f" "f 0 \<noteq> 0"
lp15@68361
  1307
  shows "(\<Prod>n. f (Suc n)) = prodinf f / f 0"
lp15@68361
  1308
  using prodinf_split_initial_segment[of 1] assms by simp
lp15@68361
  1309
lp15@68361
  1310
end
lp15@68361
  1311
lp15@68424
  1312
context 
lp15@68361
  1313
  fixes f :: "nat \<Rightarrow> 'a::real_normed_field"
lp15@68361
  1314
begin
lp15@68361
  1315
lp15@68361
  1316
lemma convergent_prod_inverse_iff: "convergent_prod (\<lambda>n. inverse (f n)) \<longleftrightarrow> convergent_prod f"
lp15@68361
  1317
  by (auto dest: convergent_prod_inverse)
lp15@68361
  1318
lp15@68361
  1319
lemma convergent_prod_const_iff:
lp15@68361
  1320
  fixes c :: "'a :: {real_normed_field}"
lp15@68361
  1321
  shows "convergent_prod (\<lambda>_. c) \<longleftrightarrow> c = 1"
lp15@68361
  1322
proof
lp15@68361
  1323
  assume "convergent_prod (\<lambda>_. c)"
lp15@68361
  1324
  then show "c = 1"
lp15@68361
  1325
    using convergent_prod_imp_LIMSEQ LIMSEQ_unique by blast 
lp15@68361
  1326
next
lp15@68361
  1327
  assume "c = 1"
lp15@68361
  1328
  then show "convergent_prod (\<lambda>_. c)"
lp15@68361
  1329
    by auto
lp15@68361
  1330
qed
lp15@68361
  1331
lp15@68361
  1332
lemma has_prod_power: "f has_prod a \<Longrightarrow> (\<lambda>i. f i ^ n) has_prod (a ^ n)"
lp15@68361
  1333
  by (induction n) (auto simp: has_prod_mult)
lp15@68361
  1334
lp15@68361
  1335
lemma convergent_prod_power: "convergent_prod f \<Longrightarrow> convergent_prod (\<lambda>i. f i ^ n)"
lp15@68361
  1336
  by (induction n) (auto simp: convergent_prod_mult)
lp15@68361
  1337
lp15@68361
  1338
lemma prodinf_power: "convergent_prod f \<Longrightarrow> prodinf (\<lambda>i. f i ^ n) = prodinf f ^ n"
lp15@68361
  1339
  by (metis has_prod_unique convergent_prod_imp_has_prod has_prod_power)
lp15@68361
  1340
lp15@68361
  1341
end
lp15@68361
  1342
lp15@68424
  1343
lp15@68424
  1344
subsection\<open>Exponentials and logarithms\<close>
lp15@68424
  1345
lp15@68424
  1346
context 
lp15@68424
  1347
  fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
lp15@68424
  1348
begin
lp15@68424
  1349
lp15@68424
  1350
lemma sums_imp_has_prod_exp: 
lp15@68424
  1351
  assumes "f sums s"
lp15@68424
  1352
  shows "raw_has_prod (\<lambda>i. exp (f i)) 0 (exp s)"
lp15@68424
  1353
  using assms continuous_on_exp [of UNIV "\<lambda>x::'a. x"]
lp15@68424
  1354
  using continuous_on_tendsto_compose [of UNIV exp "(\<lambda>n. sum f {..n})" s]
lp15@68424
  1355
  by (simp add: prod_defs sums_def_le exp_sum)
lp15@68424
  1356
lp15@68424
  1357
lemma convergent_prod_exp: 
lp15@68424
  1358
  assumes "summable f"
lp15@68424
  1359
  shows "convergent_prod (\<lambda>i. exp (f i))"
lp15@68424
  1360
  using sums_imp_has_prod_exp assms unfolding summable_def convergent_prod_def  by blast
lp15@68424
  1361
lp15@68424
  1362
lemma prodinf_exp: 
lp15@68424
  1363
  assumes "summable f"
lp15@68424
  1364
  shows "prodinf (\<lambda>i. exp (f i)) = exp (suminf f)"
lp15@68424
  1365
proof -
lp15@68424
  1366
  have "f sums suminf f"
lp15@68424
  1367
    using assms by blast
lp15@68424
  1368
  then have "(\<lambda>i. exp (f i)) has_prod exp (suminf f)"
lp15@68424
  1369
    by (simp add: has_prod_def sums_imp_has_prod_exp)
lp15@68424
  1370
  then show ?thesis
lp15@68424
  1371
    by (rule has_prod_unique [symmetric])
lp15@68424
  1372
qed
lp15@68424
  1373
lp15@68361
  1374
end
lp15@68424
  1375
lp15@68424
  1376
lemma has_prod_imp_sums_ln_real: 
lp15@68424
  1377
  fixes f :: "nat \<Rightarrow> real"
lp15@68424
  1378
  assumes "raw_has_prod f 0 p" and 0: "\<And>x. f x > 0"
lp15@68424
  1379
  shows "(\<lambda>i. ln (f i)) sums (ln p)"
lp15@68424
  1380
proof -
lp15@68424
  1381
  have "p > 0"
lp15@68424
  1382
    using assms unfolding prod_defs by (metis LIMSEQ_prod_nonneg less_eq_real_def)
lp15@68424
  1383
  then show ?thesis
lp15@68424
  1384
  using assms continuous_on_ln [of "{0<..}" "\<lambda>x. x"]
lp15@68424
  1385
  using continuous_on_tendsto_compose [of "{0<..}" ln "(\<lambda>n. prod f {..n})" p]
lp15@68424
  1386
  by (auto simp: prod_defs sums_def_le ln_prod order_tendstoD)
lp15@68424
  1387
qed
lp15@68424
  1388
lp15@68424
  1389
lemma summable_ln_real: 
lp15@68424
  1390
  fixes f :: "nat \<Rightarrow> real"
lp15@68424
  1391
  assumes f: "convergent_prod f" and 0: "\<And>x. f x > 0"
lp15@68424
  1392
  shows "summable (\<lambda>i. ln (f i))"
lp15@68424
  1393
proof -
lp15@68424
  1394
  obtain M p where "raw_has_prod f M p"
lp15@68424
  1395
    using f convergent_prod_def by blast
lp15@68424
  1396
  then consider i where "i<M" "f i = 0" | p where "raw_has_prod f 0 p"
lp15@68424
  1397
    using raw_has_prod_cases by blast
lp15@68424
  1398
  then show ?thesis
lp15@68424
  1399
  proof cases
lp15@68424
  1400
    case 1
lp15@68424
  1401
    with 0 show ?thesis
lp15@68424
  1402
      by (metis less_irrefl)
lp15@68424
  1403
  next
lp15@68424
  1404
    case 2
lp15@68424
  1405
    then show ?thesis
lp15@68424
  1406
      using "0" has_prod_imp_sums_ln_real summable_def by blast
lp15@68424
  1407
  qed
lp15@68424
  1408
qed
lp15@68424
  1409
lp15@68424
  1410
lemma suminf_ln_real: 
lp15@68424
  1411
  fixes f :: "nat \<Rightarrow> real"
lp15@68424
  1412
  assumes f: "convergent_prod f" and 0: "\<And>x. f x > 0"
lp15@68424
  1413
  shows "suminf (\<lambda>i. ln (f i)) = ln (prodinf f)"
lp15@68424
  1414
proof -
lp15@68424
  1415
  have "f has_prod prodinf f"
lp15@68424
  1416
    by (simp add: f has_prod_iff)
lp15@68424
  1417
  then have "raw_has_prod f 0 (prodinf f)"
lp15@68424
  1418
    by (metis "0" has_prod_def less_irrefl)
lp15@68424
  1419
  then have "(\<lambda>i. ln (f i)) sums ln (prodinf f)"
lp15@68424
  1420
    using "0" has_prod_imp_sums_ln_real by blast
lp15@68424
  1421
  then show ?thesis
lp15@68424
  1422
    by (rule sums_unique [symmetric])
lp15@68424
  1423
qed
lp15@68424
  1424
lp15@68424
  1425
lemma prodinf_exp_real: 
lp15@68424
  1426
  fixes f :: "nat \<Rightarrow> real"
lp15@68424
  1427
  assumes f: "convergent_prod f" and 0: "\<And>x. f x > 0"
lp15@68424
  1428
  shows "prodinf f = exp (suminf (\<lambda>i. ln (f i)))"
lp15@68424
  1429
  by (simp add: "0" f less_0_prodinf suminf_ln_real)
lp15@68424
  1430
lp15@68424
  1431
lp15@68424
  1432
subsection\<open>Embeddings from the reals into some complete real normed field\<close>
lp15@68424
  1433
lp15@68426
  1434
lemma tendsto_eq_of_real_lim:
lp15@68424
  1435
  assumes "(\<lambda>n. of_real (f n) :: 'a::{complete_space,real_normed_field}) \<longlonglongrightarrow> q"
lp15@68424
  1436
  shows "q = of_real (lim f)"
lp15@68424
  1437
proof -
lp15@68424
  1438
  have "convergent (\<lambda>n. of_real (f n) :: 'a)"
lp15@68424
  1439
    using assms convergent_def by blast 
lp15@68424
  1440
  then have "convergent f"
lp15@68424
  1441
    unfolding convergent_def
lp15@68424
  1442
    by (simp add: convergent_eq_Cauchy Cauchy_def)
lp15@68424
  1443
  then show ?thesis
lp15@68424
  1444
    by (metis LIMSEQ_unique assms convergentD sequentially_bot tendsto_Lim tendsto_of_real)
lp15@68424
  1445
qed
lp15@68424
  1446
lp15@68426
  1447
lemma tendsto_eq_of_real:
lp15@68424
  1448
  assumes "(\<lambda>n. of_real (f n) :: 'a::{complete_space,real_normed_field}) \<longlonglongrightarrow> q"
lp15@68424
  1449
  obtains r where "q = of_real r"
lp15@68426
  1450
  using tendsto_eq_of_real_lim assms by blast
lp15@68424
  1451
lp15@68424
  1452
lemma has_prod_of_real_iff:
lp15@68424
  1453
  "(\<lambda>n. of_real (f n) :: 'a::{complete_space,real_normed_field}) has_prod of_real c \<longleftrightarrow> f has_prod c"
lp15@68424
  1454
  (is "?lhs = ?rhs")
lp15@68424
  1455
proof
lp15@68424
  1456
  assume ?lhs
lp15@68424
  1457
  then show ?rhs
lp15@68424
  1458
    apply (auto simp: prod_defs LIMSEQ_prod_0 tendsto_of_real_iff simp flip: of_real_prod)
lp15@68426
  1459
    using tendsto_eq_of_real
lp15@68424
  1460
    by (metis of_real_0 tendsto_of_real_iff)
lp15@68424
  1461
next
lp15@68424
  1462
  assume ?rhs
lp15@68424
  1463
  with tendsto_of_real_iff show ?lhs
lp15@68424
  1464
    by (fastforce simp: prod_defs simp flip: of_real_prod)
lp15@68424
  1465
qed
lp15@68424
  1466
lp15@68424
  1467
end