src/HOL/Analysis/Infinite_Products.thy
author paulson <lp15@cam.ac.uk>
Thu May 10 15:41:34 2018 +0100 (14 months ago)
changeset 68136 f022083489d0
parent 68127 137d5d0112bb
child 68138 c738f40e88d4
permissions -rw-r--r--
more on infinite products
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(*File:      HOL/Analysis/Infinite_Product.thy
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  Author:    Manuel Eberl & LC Paulson
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  Basic results about convergence and absolute convergence of infinite products
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  and their connection to summability.
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*)
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section \<open>Infinite Products\<close>
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theory Infinite_Products
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  imports Complex_Main
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begin
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lemma sum_le_prod:
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  fixes f :: "'a \<Rightarrow> 'b :: linordered_semidom"
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  assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<ge> 0"
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  shows   "sum f A \<le> (\<Prod>x\<in>A. 1 + f x)"
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  using assms
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proof (induction A rule: infinite_finite_induct)
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  case (insert x A)
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  from insert.hyps have "sum f A + f x * (\<Prod>x\<in>A. 1) \<le> (\<Prod>x\<in>A. 1 + f x) + f x * (\<Prod>x\<in>A. 1 + f x)"
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    by (intro add_mono insert mult_left_mono prod_mono) (auto intro: insert.prems)
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  with insert.hyps show ?case by (simp add: algebra_simps)
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qed simp_all
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lemma prod_le_exp_sum:
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  fixes f :: "'a \<Rightarrow> real"
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  assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<ge> 0"
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  shows   "prod (\<lambda>x. 1 + f x) A \<le> exp (sum f A)"
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  using assms
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proof (induction A rule: infinite_finite_induct)
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  case (insert x A)
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  have "(1 + f x) * (\<Prod>x\<in>A. 1 + f x) \<le> exp (f x) * exp (sum f A)"
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    using insert.prems by (intro mult_mono insert prod_nonneg exp_ge_add_one_self) auto
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  with insert.hyps show ?case by (simp add: algebra_simps exp_add)
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qed simp_all
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lemma lim_ln_1_plus_x_over_x_at_0: "(\<lambda>x::real. ln (1 + x) / x) \<midarrow>0\<rightarrow> 1"
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proof (rule lhopital)
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  show "(\<lambda>x::real. ln (1 + x)) \<midarrow>0\<rightarrow> 0"
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    by (rule tendsto_eq_intros refl | simp)+
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  have "eventually (\<lambda>x::real. x \<in> {-1/2<..<1/2}) (nhds 0)"
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    by (rule eventually_nhds_in_open) auto
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  hence *: "eventually (\<lambda>x::real. x \<in> {-1/2<..<1/2}) (at 0)"
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    by (rule filter_leD [rotated]) (simp_all add: at_within_def)   
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  show "eventually (\<lambda>x::real. ((\<lambda>x. ln (1 + x)) has_field_derivative inverse (1 + x)) (at x)) (at 0)"
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    using * by eventually_elim (auto intro!: derivative_eq_intros simp: field_simps)
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  show "eventually (\<lambda>x::real. ((\<lambda>x. x) has_field_derivative 1) (at x)) (at 0)"
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    using * by eventually_elim (auto intro!: derivative_eq_intros simp: field_simps)
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  show "\<forall>\<^sub>F x in at 0. x \<noteq> 0" by (auto simp: at_within_def eventually_inf_principal)
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  show "(\<lambda>x::real. inverse (1 + x) / 1) \<midarrow>0\<rightarrow> 1"
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    by (rule tendsto_eq_intros refl | simp)+
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qed auto
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definition gen_has_prod :: "[nat \<Rightarrow> 'a::{t2_space, comm_semiring_1}, nat, 'a] \<Rightarrow> bool" 
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  where "gen_has_prod f M p \<equiv> (\<lambda>n. \<Prod>i\<le>n. f (i+M)) \<longlonglongrightarrow> p \<and> p \<noteq> 0"
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text\<open>The nonzero and zero cases, as in \emph{Complex Analysis} by Joseph Bak and Donald J.Newman, page 241\<close>
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definition has_prod :: "(nat \<Rightarrow> 'a::{t2_space, comm_semiring_1}) \<Rightarrow> 'a \<Rightarrow> bool" (infixr "has'_prod" 80)
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  where "f has_prod p \<equiv> gen_has_prod f 0 p \<or> (\<exists>i q. p = 0 \<and> f i = 0 \<and> gen_has_prod f (Suc i) q)"
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definition convergent_prod :: "(nat \<Rightarrow> 'a :: {t2_space,comm_semiring_1}) \<Rightarrow> bool" where
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  "convergent_prod f \<equiv> \<exists>M p. gen_has_prod f M p"
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definition prodinf :: "(nat \<Rightarrow> 'a::{t2_space, comm_semiring_1}) \<Rightarrow> 'a"
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    (binder "\<Prod>" 10)
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  where "prodinf f = (THE p. f has_prod p)"
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lemmas prod_defs = gen_has_prod_def has_prod_def convergent_prod_def prodinf_def
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lemma has_prod_subst[trans]: "f = g \<Longrightarrow> g has_prod z \<Longrightarrow> f has_prod z"
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  by simp
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lemma has_prod_cong: "(\<And>n. f n = g n) \<Longrightarrow> f has_prod c \<longleftrightarrow> g has_prod c"
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  by presburger
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lemma gen_has_prod_nonzero [simp]: "\<not> gen_has_prod f M 0"
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  by (simp add: gen_has_prod_def)
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lemma gen_has_prod_eq_0:
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  fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
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  assumes p: "gen_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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    by (metis i that atMost_atLeast0 atMost_iff diff_add finite_atLeastAtMost prod_zero_iff)
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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 gen_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. gen_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::{idom,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 gen_has_prod_eq_0) (meson gen_has_prod_def sequentially_bot tendsto_unique)
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lemma has_prod_unique:
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  fixes f :: "nat \<Rightarrow> 'a :: {idom,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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definition abs_convergent_prod :: "(nat \<Rightarrow> _) \<Rightarrow> bool" where
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  "abs_convergent_prod f \<longleftrightarrow> convergent_prod (\<lambda>i. 1 + norm (f i - 1))"
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lemma abs_convergent_prodI:
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  assumes "convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
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  shows   "abs_convergent_prod f"
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proof -
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  from assms obtain L where L: "(\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1)) \<longlonglongrightarrow> L"
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    by (auto simp: convergent_def)
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  have "L \<ge> 1"
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  proof (rule tendsto_le)
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    show "eventually (\<lambda>n. (\<Prod>i\<le>n. 1 + norm (f i - 1)) \<ge> 1) sequentially"
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    proof (intro always_eventually allI)
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      fix n
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      have "(\<Prod>i\<le>n. 1 + norm (f i - 1)) \<ge> (\<Prod>i\<le>n. 1)"
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        by (intro prod_mono) auto
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      thus "(\<Prod>i\<le>n. 1 + norm (f i - 1)) \<ge> 1" by simp
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    qed
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  qed (use L in simp_all)
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  hence "L \<noteq> 0" by auto
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  with L show ?thesis unfolding abs_convergent_prod_def prod_defs
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    by (intro exI[of _ "0::nat"] exI[of _ L]) auto
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qed
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lemma
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  fixes f :: "nat \<Rightarrow> 'a :: {topological_semigroup_mult,t2_space,idom}"
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  assumes "convergent_prod f"
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  shows   convergent_prod_imp_convergent: "convergent (\<lambda>n. \<Prod>i\<le>n. f i)"
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    and   convergent_prod_to_zero_iff:    "(\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> 0 \<longleftrightarrow> (\<exists>i. f i = 0)"
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proof -
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  from assms obtain M L 
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    where M: "\<And>n. n \<ge> M \<Longrightarrow> f n \<noteq> 0" and "(\<lambda>n. \<Prod>i\<le>n. f (i + M)) \<longlonglongrightarrow> L" and "L \<noteq> 0"
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    by (auto simp: convergent_prod_altdef)
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  note this(2)
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  also have "(\<lambda>n. \<Prod>i\<le>n. f (i + M)) = (\<lambda>n. \<Prod>i=M..M+n. f i)"
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    by (intro ext prod.reindex_bij_witness[of _ "\<lambda>n. n - M" "\<lambda>n. n + M"]) auto
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  finally have "(\<lambda>n. (\<Prod>i<M. f i) * (\<Prod>i=M..M+n. f i)) \<longlonglongrightarrow> (\<Prod>i<M. f i) * L"
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    by (intro tendsto_mult tendsto_const)
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  also have "(\<lambda>n. (\<Prod>i<M. f i) * (\<Prod>i=M..M+n. f i)) = (\<lambda>n. (\<Prod>i\<in>{..<M}\<union>{M..M+n}. f i))"
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    by (subst prod.union_disjoint) auto
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  also have "(\<lambda>n. {..<M} \<union> {M..M+n}) = (\<lambda>n. {..n+M})" by auto
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  finally have lim: "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> prod f {..<M} * L" 
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    by (rule LIMSEQ_offset)
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  thus "convergent (\<lambda>n. \<Prod>i\<le>n. f i)"
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    by (auto simp: convergent_def)
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  show "(\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> 0 \<longleftrightarrow> (\<exists>i. f i = 0)"
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  proof
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    assume "\<exists>i. f i = 0"
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    then obtain i where "f i = 0" by auto
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    moreover with M have "i < M" by (cases "i < M") auto
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    ultimately have "(\<Prod>i<M. f i) = 0" by auto
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    with lim show "(\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> 0" by simp
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  next
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    assume "(\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> 0"
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    from tendsto_unique[OF _ this lim] and \<open>L \<noteq> 0\<close>
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    show "\<exists>i. f i = 0" by auto
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  qed
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qed
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lemma convergent_prod_iff_nz_lim:
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  fixes f :: "nat \<Rightarrow> 'a :: {topological_semigroup_mult,t2_space,idom}"
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  assumes "\<And>i. f i \<noteq> 0"
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  shows "convergent_prod f \<longleftrightarrow> (\<exists>L. (\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> L \<and> L \<noteq> 0)"
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    (is "?lhs \<longleftrightarrow> ?rhs")
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proof
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  assume ?lhs then show ?rhs
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    using assms convergentD convergent_prod_imp_convergent convergent_prod_to_zero_iff by blast
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next
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  assume ?rhs then show ?lhs
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    unfolding prod_defs
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    by (rule_tac x="0" in exI) (auto simp: )
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qed
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lemma convergent_prod_iff_convergent: 
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  fixes f :: "nat \<Rightarrow> 'a :: {topological_semigroup_mult,t2_space,idom}"
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  assumes "\<And>i. f i \<noteq> 0"
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  shows "convergent_prod f \<longleftrightarrow> convergent (\<lambda>n. \<Prod>i\<le>n. f i) \<and> lim (\<lambda>n. \<Prod>i\<le>n. f i) \<noteq> 0"
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  by (force simp add: convergent_prod_iff_nz_lim assms convergent_def limI)
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lemma abs_convergent_prod_altdef:
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  fixes f :: "nat \<Rightarrow> 'a :: {one,real_normed_vector}"
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  shows  "abs_convergent_prod f \<longleftrightarrow> convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
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proof
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  assume "abs_convergent_prod f"
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  thus "convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
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    by (auto simp: abs_convergent_prod_def intro!: convergent_prod_imp_convergent)
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qed (auto intro: abs_convergent_prodI)
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lemma weierstrass_prod_ineq:
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  fixes f :: "'a \<Rightarrow> real" 
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  assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> {0..1}"
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  shows   "1 - sum f A \<le> (\<Prod>x\<in>A. 1 - f x)"
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  using assms
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proof (induction A rule: infinite_finite_induct)
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  case (insert x A)
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  from insert.hyps and insert.prems 
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    have "1 - sum f A + f x * (\<Prod>x\<in>A. 1 - f x) \<le> (\<Prod>x\<in>A. 1 - f x) + f x * (\<Prod>x\<in>A. 1)"
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    by (intro insert.IH add_mono mult_left_mono prod_mono) auto
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  with insert.hyps show ?case by (simp add: algebra_simps)
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qed simp_all
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lemma norm_prod_minus1_le_prod_minus1:
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  fixes f :: "nat \<Rightarrow> 'a :: {real_normed_div_algebra,comm_ring_1}"  
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  shows "norm (prod (\<lambda>n. 1 + f n) A - 1) \<le> prod (\<lambda>n. 1 + norm (f n)) A - 1"
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proof (induction A rule: infinite_finite_induct)
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  case (insert x A)
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  from insert.hyps have 
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    "norm ((\<Prod>n\<in>insert x A. 1 + f n) - 1) = 
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       norm ((\<Prod>n\<in>A. 1 + f n) - 1 + f x * (\<Prod>n\<in>A. 1 + f n))"
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    by (simp add: algebra_simps)
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  also have "\<dots> \<le> norm ((\<Prod>n\<in>A. 1 + f n) - 1) + norm (f x * (\<Prod>n\<in>A. 1 + f n))"
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    by (rule norm_triangle_ineq)
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  also have "norm (f x * (\<Prod>n\<in>A. 1 + f n)) = norm (f x) * (\<Prod>x\<in>A. norm (1 + f x))"
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    by (simp add: prod_norm norm_mult)
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  also have "(\<Prod>x\<in>A. norm (1 + f x)) \<le> (\<Prod>x\<in>A. norm (1::'a) + norm (f x))"
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    by (intro prod_mono norm_triangle_ineq ballI conjI) auto
eberlm@66277
   251
  also have "norm (1::'a) = 1" by simp
eberlm@66277
   252
  also note insert.IH
eberlm@66277
   253
  also have "(\<Prod>n\<in>A. 1 + norm (f n)) - 1 + norm (f x) * (\<Prod>x\<in>A. 1 + norm (f x)) =
lp15@68064
   254
             (\<Prod>n\<in>insert x A. 1 + norm (f n)) - 1"
eberlm@66277
   255
    using insert.hyps by (simp add: algebra_simps)
eberlm@66277
   256
  finally show ?case by - (simp_all add: mult_left_mono)
eberlm@66277
   257
qed simp_all
eberlm@66277
   258
eberlm@66277
   259
lemma convergent_prod_imp_ev_nonzero:
eberlm@66277
   260
  fixes f :: "nat \<Rightarrow> 'a :: {t2_space,comm_semiring_1}"
eberlm@66277
   261
  assumes "convergent_prod f"
eberlm@66277
   262
  shows   "eventually (\<lambda>n. f n \<noteq> 0) sequentially"
eberlm@66277
   263
  using assms by (auto simp: eventually_at_top_linorder convergent_prod_altdef)
eberlm@66277
   264
eberlm@66277
   265
lemma convergent_prod_imp_LIMSEQ:
eberlm@66277
   266
  fixes f :: "nat \<Rightarrow> 'a :: {real_normed_field}"
eberlm@66277
   267
  assumes "convergent_prod f"
eberlm@66277
   268
  shows   "f \<longlonglongrightarrow> 1"
eberlm@66277
   269
proof -
eberlm@66277
   270
  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
   271
    by (auto simp: convergent_prod_altdef)
eberlm@66277
   272
  hence L': "(\<lambda>n. \<Prod>i\<le>Suc n. f (i+M)) \<longlonglongrightarrow> L" by (subst filterlim_sequentially_Suc)
eberlm@66277
   273
  have "(\<lambda>n. (\<Prod>i\<le>Suc n. f (i+M)) / (\<Prod>i\<le>n. f (i+M))) \<longlonglongrightarrow> L / L"
eberlm@66277
   274
    using L L' by (intro tendsto_divide) simp_all
eberlm@66277
   275
  also from L have "L / L = 1" by simp
eberlm@66277
   276
  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
   277
    using assms L by (auto simp: fun_eq_iff atMost_Suc)
eberlm@66277
   278
  finally show ?thesis by (rule LIMSEQ_offset)
eberlm@66277
   279
qed
eberlm@66277
   280
eberlm@66277
   281
lemma abs_convergent_prod_imp_summable:
eberlm@66277
   282
  fixes f :: "nat \<Rightarrow> 'a :: real_normed_div_algebra"
eberlm@66277
   283
  assumes "abs_convergent_prod f"
eberlm@66277
   284
  shows "summable (\<lambda>i. norm (f i - 1))"
eberlm@66277
   285
proof -
eberlm@66277
   286
  from assms have "convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))" 
eberlm@66277
   287
    unfolding abs_convergent_prod_def by (rule convergent_prod_imp_convergent)
eberlm@66277
   288
  then obtain L where L: "(\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1)) \<longlonglongrightarrow> L"
eberlm@66277
   289
    unfolding convergent_def by blast
eberlm@66277
   290
  have "convergent (\<lambda>n. \<Sum>i\<le>n. norm (f i - 1))"
eberlm@66277
   291
  proof (rule Bseq_monoseq_convergent)
eberlm@66277
   292
    have "eventually (\<lambda>n. (\<Prod>i\<le>n. 1 + norm (f i - 1)) < L + 1) sequentially"
eberlm@66277
   293
      using L(1) by (rule order_tendstoD) simp_all
eberlm@66277
   294
    hence "\<forall>\<^sub>F x in sequentially. norm (\<Sum>i\<le>x. norm (f i - 1)) \<le> L + 1"
eberlm@66277
   295
    proof eventually_elim
eberlm@66277
   296
      case (elim n)
eberlm@66277
   297
      have "norm (\<Sum>i\<le>n. norm (f i - 1)) = (\<Sum>i\<le>n. norm (f i - 1))"
eberlm@66277
   298
        unfolding real_norm_def by (intro abs_of_nonneg sum_nonneg) simp_all
eberlm@66277
   299
      also have "\<dots> \<le> (\<Prod>i\<le>n. 1 + norm (f i - 1))" by (rule sum_le_prod) auto
eberlm@66277
   300
      also have "\<dots> < L + 1" by (rule elim)
eberlm@66277
   301
      finally show ?case by simp
eberlm@66277
   302
    qed
eberlm@66277
   303
    thus "Bseq (\<lambda>n. \<Sum>i\<le>n. norm (f i - 1))" by (rule BfunI)
eberlm@66277
   304
  next
eberlm@66277
   305
    show "monoseq (\<lambda>n. \<Sum>i\<le>n. norm (f i - 1))"
eberlm@66277
   306
      by (rule mono_SucI1) auto
eberlm@66277
   307
  qed
eberlm@66277
   308
  thus "summable (\<lambda>i. norm (f i - 1))" by (simp add: summable_iff_convergent')
eberlm@66277
   309
qed
eberlm@66277
   310
eberlm@66277
   311
lemma summable_imp_abs_convergent_prod:
eberlm@66277
   312
  fixes f :: "nat \<Rightarrow> 'a :: real_normed_div_algebra"
eberlm@66277
   313
  assumes "summable (\<lambda>i. norm (f i - 1))"
eberlm@66277
   314
  shows   "abs_convergent_prod f"
eberlm@66277
   315
proof (intro abs_convergent_prodI Bseq_monoseq_convergent)
eberlm@66277
   316
  show "monoseq (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
eberlm@66277
   317
    by (intro mono_SucI1) 
eberlm@66277
   318
       (auto simp: atMost_Suc algebra_simps intro!: mult_nonneg_nonneg prod_nonneg)
eberlm@66277
   319
next
eberlm@66277
   320
  show "Bseq (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
eberlm@66277
   321
  proof (rule Bseq_eventually_mono)
eberlm@66277
   322
    show "eventually (\<lambda>n. norm (\<Prod>i\<le>n. 1 + norm (f i - 1)) \<le> 
eberlm@66277
   323
            norm (exp (\<Sum>i\<le>n. norm (f i - 1)))) sequentially"
eberlm@66277
   324
      by (intro always_eventually allI) (auto simp: abs_prod exp_sum intro!: prod_mono)
eberlm@66277
   325
  next
eberlm@66277
   326
    from assms have "(\<lambda>n. \<Sum>i\<le>n. norm (f i - 1)) \<longlonglongrightarrow> (\<Sum>i. norm (f i - 1))"
eberlm@66277
   327
      using sums_def_le by blast
eberlm@66277
   328
    hence "(\<lambda>n. exp (\<Sum>i\<le>n. norm (f i - 1))) \<longlonglongrightarrow> exp (\<Sum>i. norm (f i - 1))"
eberlm@66277
   329
      by (rule tendsto_exp)
eberlm@66277
   330
    hence "convergent (\<lambda>n. exp (\<Sum>i\<le>n. norm (f i - 1)))"
eberlm@66277
   331
      by (rule convergentI)
eberlm@66277
   332
    thus "Bseq (\<lambda>n. exp (\<Sum>i\<le>n. norm (f i - 1)))"
eberlm@66277
   333
      by (rule convergent_imp_Bseq)
eberlm@66277
   334
  qed
eberlm@66277
   335
qed
eberlm@66277
   336
eberlm@66277
   337
lemma abs_convergent_prod_conv_summable:
eberlm@66277
   338
  fixes f :: "nat \<Rightarrow> 'a :: real_normed_div_algebra"
eberlm@66277
   339
  shows "abs_convergent_prod f \<longleftrightarrow> summable (\<lambda>i. norm (f i - 1))"
eberlm@66277
   340
  by (blast intro: abs_convergent_prod_imp_summable summable_imp_abs_convergent_prod)
eberlm@66277
   341
eberlm@66277
   342
lemma abs_convergent_prod_imp_LIMSEQ:
eberlm@66277
   343
  fixes f :: "nat \<Rightarrow> 'a :: {comm_ring_1,real_normed_div_algebra}"
eberlm@66277
   344
  assumes "abs_convergent_prod f"
eberlm@66277
   345
  shows   "f \<longlonglongrightarrow> 1"
eberlm@66277
   346
proof -
eberlm@66277
   347
  from assms have "summable (\<lambda>n. norm (f n - 1))"
eberlm@66277
   348
    by (rule abs_convergent_prod_imp_summable)
eberlm@66277
   349
  from summable_LIMSEQ_zero[OF this] have "(\<lambda>n. f n - 1) \<longlonglongrightarrow> 0"
eberlm@66277
   350
    by (simp add: tendsto_norm_zero_iff)
eberlm@66277
   351
  from tendsto_add[OF this tendsto_const[of 1]] show ?thesis by simp
eberlm@66277
   352
qed
eberlm@66277
   353
eberlm@66277
   354
lemma abs_convergent_prod_imp_ev_nonzero:
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   "eventually (\<lambda>n. f n \<noteq> 0) sequentially"
eberlm@66277
   358
proof -
eberlm@66277
   359
  from assms have "f \<longlonglongrightarrow> 1" 
eberlm@66277
   360
    by (rule abs_convergent_prod_imp_LIMSEQ)
eberlm@66277
   361
  hence "eventually (\<lambda>n. dist (f n) 1 < 1) at_top"
eberlm@66277
   362
    by (auto simp: tendsto_iff)
eberlm@66277
   363
  thus ?thesis by eventually_elim auto
eberlm@66277
   364
qed
eberlm@66277
   365
eberlm@66277
   366
lemma convergent_prod_offset:
eberlm@66277
   367
  assumes "convergent_prod (\<lambda>n. f (n + m))"  
eberlm@66277
   368
  shows   "convergent_prod f"
eberlm@66277
   369
proof -
eberlm@66277
   370
  from assms obtain M L where "(\<lambda>n. \<Prod>k\<le>n. f (k + (M + m))) \<longlonglongrightarrow> L" "L \<noteq> 0"
lp15@68064
   371
    by (auto simp: prod_defs add.assoc)
lp15@68064
   372
  thus "convergent_prod f" 
lp15@68064
   373
    unfolding prod_defs by blast
eberlm@66277
   374
qed
eberlm@66277
   375
eberlm@66277
   376
lemma abs_convergent_prod_offset:
eberlm@66277
   377
  assumes "abs_convergent_prod (\<lambda>n. f (n + m))"  
eberlm@66277
   378
  shows   "abs_convergent_prod f"
eberlm@66277
   379
  using assms unfolding abs_convergent_prod_def by (rule convergent_prod_offset)
eberlm@66277
   380
eberlm@66277
   381
lemma convergent_prod_ignore_initial_segment:
eberlm@66277
   382
  fixes f :: "nat \<Rightarrow> 'a :: {real_normed_field}"
eberlm@66277
   383
  assumes "convergent_prod f"
eberlm@66277
   384
  shows   "convergent_prod (\<lambda>n. f (n + m))"
eberlm@66277
   385
proof -
eberlm@66277
   386
  from assms obtain M L 
eberlm@66277
   387
    where L: "(\<lambda>n. \<Prod>k\<le>n. f (k + M)) \<longlonglongrightarrow> L" "L \<noteq> 0" and nz: "\<And>n. n \<ge> M \<Longrightarrow> f n \<noteq> 0"
eberlm@66277
   388
    by (auto simp: convergent_prod_altdef)
eberlm@66277
   389
  define C where "C = (\<Prod>k<m. f (k + M))"
eberlm@66277
   390
  from nz have [simp]: "C \<noteq> 0" 
eberlm@66277
   391
    by (auto simp: C_def)
eberlm@66277
   392
eberlm@66277
   393
  from L(1) have "(\<lambda>n. \<Prod>k\<le>n+m. f (k + M)) \<longlonglongrightarrow> L" 
eberlm@66277
   394
    by (rule LIMSEQ_ignore_initial_segment)
eberlm@66277
   395
  also have "(\<lambda>n. \<Prod>k\<le>n+m. f (k + M)) = (\<lambda>n. C * (\<Prod>k\<le>n. f (k + M + m)))"
eberlm@66277
   396
  proof (rule ext, goal_cases)
eberlm@66277
   397
    case (1 n)
eberlm@66277
   398
    have "{..n+m} = {..<m} \<union> {m..n+m}" by auto
eberlm@66277
   399
    also have "(\<Prod>k\<in>\<dots>. f (k + M)) = C * (\<Prod>k=m..n+m. f (k + M))"
eberlm@66277
   400
      unfolding C_def by (rule prod.union_disjoint) auto
eberlm@66277
   401
    also have "(\<Prod>k=m..n+m. f (k + M)) = (\<Prod>k\<le>n. f (k + m + M))"
eberlm@66277
   402
      by (intro ext prod.reindex_bij_witness[of _ "\<lambda>k. k + m" "\<lambda>k. k - m"]) auto
eberlm@66277
   403
    finally show ?case by (simp add: add_ac)
eberlm@66277
   404
  qed
eberlm@66277
   405
  finally have "(\<lambda>n. C * (\<Prod>k\<le>n. f (k + M + m)) / C) \<longlonglongrightarrow> L / C"
eberlm@66277
   406
    by (intro tendsto_divide tendsto_const) auto
eberlm@66277
   407
  hence "(\<lambda>n. \<Prod>k\<le>n. f (k + M + m)) \<longlonglongrightarrow> L / C" by simp
eberlm@66277
   408
  moreover from \<open>L \<noteq> 0\<close> have "L / C \<noteq> 0" by simp
lp15@68064
   409
  ultimately show ?thesis 
lp15@68064
   410
    unfolding prod_defs by blast
eberlm@66277
   411
qed
eberlm@66277
   412
lp15@68136
   413
corollary convergent_prod_ignore_nonzero_segment:
lp15@68136
   414
  fixes f :: "nat \<Rightarrow> 'a :: real_normed_field"
lp15@68136
   415
  assumes f: "convergent_prod f" and nz: "\<And>i. i \<ge> M \<Longrightarrow> f i \<noteq> 0"
lp15@68136
   416
  shows "\<exists>p. gen_has_prod f M p"
lp15@68136
   417
  using convergent_prod_ignore_initial_segment [OF f]
lp15@68136
   418
  by (metis convergent_LIMSEQ_iff convergent_prod_iff_convergent le_add_same_cancel2 nz prod_defs(1) zero_order(1))
lp15@68136
   419
lp15@68136
   420
corollary abs_convergent_prod_ignore_initial_segment:
eberlm@66277
   421
  assumes "abs_convergent_prod f"
eberlm@66277
   422
  shows   "abs_convergent_prod (\<lambda>n. f (n + m))"
eberlm@66277
   423
  using assms unfolding abs_convergent_prod_def 
eberlm@66277
   424
  by (rule convergent_prod_ignore_initial_segment)
eberlm@66277
   425
eberlm@66277
   426
lemma abs_convergent_prod_imp_convergent_prod:
eberlm@66277
   427
  fixes f :: "nat \<Rightarrow> 'a :: {real_normed_div_algebra,complete_space,comm_ring_1}"
eberlm@66277
   428
  assumes "abs_convergent_prod f"
eberlm@66277
   429
  shows   "convergent_prod f"
eberlm@66277
   430
proof -
eberlm@66277
   431
  from assms have "eventually (\<lambda>n. f n \<noteq> 0) sequentially"
eberlm@66277
   432
    by (rule abs_convergent_prod_imp_ev_nonzero)
eberlm@66277
   433
  then obtain N where N: "f n \<noteq> 0" if "n \<ge> N" for n 
eberlm@66277
   434
    by (auto simp: eventually_at_top_linorder)
eberlm@66277
   435
  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
   436
eberlm@66277
   437
  have "Cauchy ?P"
eberlm@66277
   438
  proof (rule CauchyI', goal_cases)
eberlm@66277
   439
    case (1 \<epsilon>)
eberlm@66277
   440
    from assms have "abs_convergent_prod (\<lambda>n. f (n + N))"
eberlm@66277
   441
      by (rule abs_convergent_prod_ignore_initial_segment)
eberlm@66277
   442
    hence "Cauchy ?Q"
eberlm@66277
   443
      unfolding abs_convergent_prod_def
eberlm@66277
   444
      by (intro convergent_Cauchy convergent_prod_imp_convergent)
eberlm@66277
   445
    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
   446
      by blast
eberlm@66277
   447
    show ?case
eberlm@66277
   448
    proof (rule exI[of _ M], safe, goal_cases)
eberlm@66277
   449
      case (1 m n)
eberlm@66277
   450
      have "dist (?P m) (?P n) = norm (?P n - ?P m)"
eberlm@66277
   451
        by (simp add: dist_norm norm_minus_commute)
eberlm@66277
   452
      also from 1 have "{..n} = {..m} \<union> {m<..n}" by auto
eberlm@66277
   453
      hence "norm (?P n - ?P m) = norm (?P m * (\<Prod>k\<in>{m<..n}. f (k + N)) - ?P m)"
eberlm@66277
   454
        by (subst prod.union_disjoint [symmetric]) (auto simp: algebra_simps)
eberlm@66277
   455
      also have "\<dots> = norm (?P m * ((\<Prod>k\<in>{m<..n}. f (k + N)) - 1))"
eberlm@66277
   456
        by (simp add: algebra_simps)
eberlm@66277
   457
      also have "\<dots> = (\<Prod>k\<le>m. norm (f (k + N))) * norm ((\<Prod>k\<in>{m<..n}. f (k + N)) - 1)"
eberlm@66277
   458
        by (simp add: norm_mult prod_norm)
eberlm@66277
   459
      also have "\<dots> \<le> ?Q m * ((\<Prod>k\<in>{m<..n}. 1 + norm (f (k + N) - 1)) - 1)"
eberlm@66277
   460
        using norm_prod_minus1_le_prod_minus1[of "\<lambda>k. f (k + N) - 1" "{m<..n}"]
eberlm@66277
   461
              norm_triangle_ineq[of 1 "f k - 1" for k]
eberlm@66277
   462
        by (intro mult_mono prod_mono ballI conjI norm_prod_minus1_le_prod_minus1 prod_nonneg) auto
eberlm@66277
   463
      also have "\<dots> = ?Q m * (\<Prod>k\<in>{m<..n}. 1 + norm (f (k + N) - 1)) - ?Q m"
eberlm@66277
   464
        by (simp add: algebra_simps)
eberlm@66277
   465
      also have "?Q m * (\<Prod>k\<in>{m<..n}. 1 + norm (f (k + N) - 1)) = 
eberlm@66277
   466
                   (\<Prod>k\<in>{..m}\<union>{m<..n}. 1 + norm (f (k + N) - 1))"
eberlm@66277
   467
        by (rule prod.union_disjoint [symmetric]) auto
eberlm@66277
   468
      also from 1 have "{..m}\<union>{m<..n} = {..n}" by auto
eberlm@66277
   469
      also have "?Q n - ?Q m \<le> norm (?Q n - ?Q m)" by simp
eberlm@66277
   470
      also from 1 have "\<dots> < \<epsilon>" by (intro M) auto
eberlm@66277
   471
      finally show ?case .
eberlm@66277
   472
    qed
eberlm@66277
   473
  qed
eberlm@66277
   474
  hence conv: "convergent ?P" by (rule Cauchy_convergent)
eberlm@66277
   475
  then obtain L where L: "?P \<longlonglongrightarrow> L"
eberlm@66277
   476
    by (auto simp: convergent_def)
eberlm@66277
   477
eberlm@66277
   478
  have "L \<noteq> 0"
eberlm@66277
   479
  proof
eberlm@66277
   480
    assume [simp]: "L = 0"
eberlm@66277
   481
    from tendsto_norm[OF L] have limit: "(\<lambda>n. \<Prod>k\<le>n. norm (f (k + N))) \<longlonglongrightarrow> 0" 
eberlm@66277
   482
      by (simp add: prod_norm)
eberlm@66277
   483
eberlm@66277
   484
    from assms have "(\<lambda>n. f (n + N)) \<longlonglongrightarrow> 1"
eberlm@66277
   485
      by (intro abs_convergent_prod_imp_LIMSEQ abs_convergent_prod_ignore_initial_segment)
eberlm@66277
   486
    hence "eventually (\<lambda>n. norm (f (n + N) - 1) < 1) sequentially"
eberlm@66277
   487
      by (auto simp: tendsto_iff dist_norm)
eberlm@66277
   488
    then obtain M0 where M0: "norm (f (n + N) - 1) < 1" if "n \<ge> M0" for n
eberlm@66277
   489
      by (auto simp: eventually_at_top_linorder)
eberlm@66277
   490
eberlm@66277
   491
    {
eberlm@66277
   492
      fix M assume M: "M \<ge> M0"
eberlm@66277
   493
      with M0 have M: "norm (f (n + N) - 1) < 1" if "n \<ge> M" for n using that by simp
eberlm@66277
   494
eberlm@66277
   495
      have "(\<lambda>n. \<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1)) \<longlonglongrightarrow> 0"
eberlm@66277
   496
      proof (rule tendsto_sandwich)
eberlm@66277
   497
        show "eventually (\<lambda>n. (\<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1)) \<ge> 0) sequentially"
eberlm@66277
   498
          using M by (intro always_eventually prod_nonneg allI ballI) (auto intro: less_imp_le)
eberlm@66277
   499
        have "norm (1::'a) - norm (f (i + M + N) - 1) \<le> norm (f (i + M + N))" for i
eberlm@66277
   500
          using norm_triangle_ineq3[of "f (i + M + N)" 1] by simp
eberlm@66277
   501
        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
   502
          using M by (intro always_eventually allI prod_mono ballI conjI) (auto intro: less_imp_le)
eberlm@66277
   503
        
eberlm@66277
   504
        define C where "C = (\<Prod>k<M. norm (f (k + N)))"
eberlm@66277
   505
        from N have [simp]: "C \<noteq> 0" by (auto simp: C_def)
eberlm@66277
   506
        from L have "(\<lambda>n. norm (\<Prod>k\<le>n+M. f (k + N))) \<longlonglongrightarrow> 0"
eberlm@66277
   507
          by (intro LIMSEQ_ignore_initial_segment) (simp add: tendsto_norm_zero_iff)
eberlm@66277
   508
        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
   509
        proof (rule ext, goal_cases)
eberlm@66277
   510
          case (1 n)
eberlm@66277
   511
          have "{..n+M} = {..<M} \<union> {M..n+M}" by auto
eberlm@66277
   512
          also have "norm (\<Prod>k\<in>\<dots>. f (k + N)) = C * norm (\<Prod>k=M..n+M. f (k + N))"
eberlm@66277
   513
            unfolding C_def by (subst prod.union_disjoint) (auto simp: norm_mult prod_norm)
eberlm@66277
   514
          also have "(\<Prod>k=M..n+M. f (k + N)) = (\<Prod>k\<le>n. f (k + N + M))"
eberlm@66277
   515
            by (intro prod.reindex_bij_witness[of _ "\<lambda>i. i + M" "\<lambda>i. i - M"]) auto
eberlm@66277
   516
          finally show ?case by (simp add: add_ac prod_norm)
eberlm@66277
   517
        qed
eberlm@66277
   518
        finally have "(\<lambda>n. C * (\<Prod>k\<le>n. norm (f (k + M + N))) / C) \<longlonglongrightarrow> 0 / C"
eberlm@66277
   519
          by (intro tendsto_divide tendsto_const) auto
eberlm@66277
   520
        thus "(\<lambda>n. \<Prod>k\<le>n. norm (f (k + M + N))) \<longlonglongrightarrow> 0" by simp
eberlm@66277
   521
      qed simp_all
eberlm@66277
   522
eberlm@66277
   523
      have "1 - (\<Sum>i. norm (f (i + M + N) - 1)) \<le> 0"
eberlm@66277
   524
      proof (rule tendsto_le)
eberlm@66277
   525
        show "eventually (\<lambda>n. 1 - (\<Sum>k\<le>n. norm (f (k+M+N) - 1)) \<le> 
eberlm@66277
   526
                                (\<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1))) at_top"
eberlm@66277
   527
          using M by (intro always_eventually allI weierstrass_prod_ineq) (auto intro: less_imp_le)
eberlm@66277
   528
        show "(\<lambda>n. \<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1)) \<longlonglongrightarrow> 0" by fact
eberlm@66277
   529
        show "(\<lambda>n. 1 - (\<Sum>k\<le>n. norm (f (k + M + N) - 1)))
eberlm@66277
   530
                  \<longlonglongrightarrow> 1 - (\<Sum>i. norm (f (i + M + N) - 1))"
eberlm@66277
   531
          by (intro tendsto_intros summable_LIMSEQ' summable_ignore_initial_segment 
eberlm@66277
   532
                abs_convergent_prod_imp_summable assms)
eberlm@66277
   533
      qed simp_all
eberlm@66277
   534
      hence "(\<Sum>i. norm (f (i + M + N) - 1)) \<ge> 1" by simp
eberlm@66277
   535
      also have "\<dots> + (\<Sum>i<M. norm (f (i + N) - 1)) = (\<Sum>i. norm (f (i + N) - 1))"
eberlm@66277
   536
        by (intro suminf_split_initial_segment [symmetric] summable_ignore_initial_segment
eberlm@66277
   537
              abs_convergent_prod_imp_summable assms)
eberlm@66277
   538
      finally have "1 + (\<Sum>i<M. norm (f (i + N) - 1)) \<le> (\<Sum>i. norm (f (i + N) - 1))" by simp
eberlm@66277
   539
    } note * = this
eberlm@66277
   540
eberlm@66277
   541
    have "1 + (\<Sum>i. norm (f (i + N) - 1)) \<le> (\<Sum>i. norm (f (i + N) - 1))"
eberlm@66277
   542
    proof (rule tendsto_le)
eberlm@66277
   543
      show "(\<lambda>M. 1 + (\<Sum>i<M. norm (f (i + N) - 1))) \<longlonglongrightarrow> 1 + (\<Sum>i. norm (f (i + N) - 1))"
eberlm@66277
   544
        by (intro tendsto_intros summable_LIMSEQ summable_ignore_initial_segment 
eberlm@66277
   545
                abs_convergent_prod_imp_summable assms)
eberlm@66277
   546
      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
   547
        using eventually_ge_at_top[of M0] by eventually_elim (use * in auto)
eberlm@66277
   548
    qed simp_all
eberlm@66277
   549
    thus False by simp
eberlm@66277
   550
  qed
lp15@68064
   551
  with L show ?thesis by (auto simp: prod_defs)
lp15@68064
   552
qed
lp15@68064
   553
lp15@68136
   554
lemma gen_has_prod_cases:
lp15@68064
   555
  fixes f :: "nat \<Rightarrow> 'a :: {idom,topological_semigroup_mult,t2_space}"
lp15@68136
   556
  assumes "gen_has_prod f M p"
lp15@68136
   557
  obtains i where "i<M" "f i = 0" | p where "gen_has_prod f 0 p"
lp15@68136
   558
proof -
lp15@68136
   559
  have "(\<lambda>n. \<Prod>i\<le>n. f (i + M)) \<longlonglongrightarrow> p" "p \<noteq> 0"
lp15@68136
   560
    using assms unfolding gen_has_prod_def by blast+
lp15@68064
   561
  then have "(\<lambda>n. prod f {..<M} * (\<Prod>i\<le>n. f (i + M))) \<longlonglongrightarrow> prod f {..<M} * p"
lp15@68064
   562
    by (metis tendsto_mult_left)
lp15@68064
   563
  moreover have "prod f {..<M} * (\<Prod>i\<le>n. f (i + M)) = prod f {..n+M}" for n
lp15@68064
   564
  proof -
lp15@68064
   565
    have "{..n+M} = {..<M} \<union> {M..n+M}"
lp15@68064
   566
      by auto
lp15@68064
   567
    then have "prod f {..n+M} = prod f {..<M} * prod f {M..n+M}"
lp15@68064
   568
      by simp (subst prod.union_disjoint; force)
lp15@68064
   569
    also have "... = prod f {..<M} * (\<Prod>i\<le>n. f (i + M))"
lp15@68064
   570
      by (metis (mono_tags, lifting) add.left_neutral atMost_atLeast0 prod_shift_bounds_cl_nat_ivl)
lp15@68064
   571
    finally show ?thesis by metis
lp15@68064
   572
  qed
lp15@68064
   573
  ultimately have "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> prod f {..<M} * p"
lp15@68064
   574
    by (auto intro: LIMSEQ_offset [where k=M])
lp15@68136
   575
  then have "gen_has_prod f 0 (prod f {..<M} * p)" if "\<forall>i<M. f i \<noteq> 0"
lp15@68136
   576
    using \<open>p \<noteq> 0\<close> assms that by (auto simp: gen_has_prod_def)
lp15@68136
   577
  then show thesis
lp15@68136
   578
    using that by blast
lp15@68064
   579
qed
lp15@68064
   580
lp15@68136
   581
corollary convergent_prod_offset_0:
lp15@68136
   582
  fixes f :: "nat \<Rightarrow> 'a :: {idom,topological_semigroup_mult,t2_space}"
lp15@68136
   583
  assumes "convergent_prod f" "\<And>i. f i \<noteq> 0"
lp15@68136
   584
  shows "\<exists>p. gen_has_prod f 0 p"
lp15@68136
   585
  using assms convergent_prod_def gen_has_prod_cases by blast
lp15@68136
   586
lp15@68064
   587
lemma prodinf_eq_lim:
lp15@68064
   588
  fixes f :: "nat \<Rightarrow> 'a :: {idom,topological_semigroup_mult,t2_space}"
lp15@68064
   589
  assumes "convergent_prod f" "\<And>i. f i \<noteq> 0"
lp15@68064
   590
  shows "prodinf f = lim (\<lambda>n. \<Prod>i\<le>n. f i)"
lp15@68064
   591
  using assms convergent_prod_offset_0 [OF assms]
lp15@68064
   592
  by (simp add: prod_defs lim_def) (metis (no_types) assms(1) convergent_prod_to_zero_iff)
lp15@68064
   593
lp15@68064
   594
lemma has_prod_one[simp, intro]: "(\<lambda>n. 1) has_prod 1"
lp15@68064
   595
  unfolding prod_defs by auto
lp15@68064
   596
lp15@68064
   597
lemma convergent_prod_one[simp, intro]: "convergent_prod (\<lambda>n. 1)"
lp15@68064
   598
  unfolding prod_defs by auto
lp15@68064
   599
lp15@68064
   600
lemma prodinf_cong: "(\<And>n. f n = g n) \<Longrightarrow> prodinf f = prodinf g"
lp15@68064
   601
  by presburger
lp15@68064
   602
lp15@68064
   603
lemma convergent_prod_cong:
lp15@68064
   604
  fixes f g :: "nat \<Rightarrow> 'a::{field,topological_semigroup_mult,t2_space}"
lp15@68064
   605
  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
   606
  shows "convergent_prod f = convergent_prod g"
lp15@68064
   607
proof -
lp15@68064
   608
  from assms obtain N where N: "\<forall>n\<ge>N. f n = g n"
lp15@68064
   609
    by (auto simp: eventually_at_top_linorder)
lp15@68064
   610
  define C where "C = (\<Prod>k<N. f k / g k)"
lp15@68064
   611
  with g have "C \<noteq> 0"
lp15@68064
   612
    by (simp add: f)
lp15@68064
   613
  have *: "eventually (\<lambda>n. prod f {..n} = C * prod g {..n}) sequentially"
lp15@68064
   614
    using eventually_ge_at_top[of N]
lp15@68064
   615
  proof eventually_elim
lp15@68064
   616
    case (elim n)
lp15@68064
   617
    then have "{..n} = {..<N} \<union> {N..n}"
lp15@68064
   618
      by auto
lp15@68064
   619
    also have "prod f ... = prod f {..<N} * prod f {N..n}"
lp15@68064
   620
      by (intro prod.union_disjoint) auto
lp15@68064
   621
    also from N have "prod f {N..n} = prod g {N..n}"
lp15@68064
   622
      by (intro prod.cong) simp_all
lp15@68064
   623
    also have "prod f {..<N} * prod g {N..n} = C * (prod g {..<N} * prod g {N..n})"
lp15@68064
   624
      unfolding C_def by (simp add: g prod_dividef)
lp15@68064
   625
    also have "prod g {..<N} * prod g {N..n} = prod g ({..<N} \<union> {N..n})"
lp15@68064
   626
      by (intro prod.union_disjoint [symmetric]) auto
lp15@68064
   627
    also from elim have "{..<N} \<union> {N..n} = {..n}"
lp15@68064
   628
      by auto                                                                    
lp15@68064
   629
    finally show "prod f {..n} = C * prod g {..n}" .
lp15@68064
   630
  qed
lp15@68064
   631
  then have cong: "convergent (\<lambda>n. prod f {..n}) = convergent (\<lambda>n. C * prod g {..n})"
lp15@68064
   632
    by (rule convergent_cong)
lp15@68064
   633
  show ?thesis
lp15@68064
   634
  proof
lp15@68064
   635
    assume cf: "convergent_prod f"
lp15@68064
   636
    then have "\<not> (\<lambda>n. prod g {..n}) \<longlonglongrightarrow> 0"
lp15@68064
   637
      using tendsto_mult_left * convergent_prod_to_zero_iff f filterlim_cong by fastforce
lp15@68064
   638
    then show "convergent_prod g"
lp15@68064
   639
      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
   640
  next
lp15@68064
   641
    assume cg: "convergent_prod g"
lp15@68064
   642
    have "\<exists>a. C * a \<noteq> 0 \<and> (\<lambda>n. prod g {..n}) \<longlonglongrightarrow> a"
lp15@68064
   643
      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
   644
    then show "convergent_prod f"
lp15@68064
   645
      using "*" tendsto_mult_left filterlim_cong
lp15@68064
   646
      by (fastforce simp add: convergent_prod_iff_nz_lim f)
lp15@68064
   647
  qed
eberlm@66277
   648
qed
eberlm@66277
   649
lp15@68071
   650
lemma has_prod_finite:
lp15@68071
   651
  fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
lp15@68071
   652
  assumes [simp]: "finite N"
lp15@68071
   653
    and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 1"
lp15@68071
   654
  shows "f has_prod (\<Prod>n\<in>N. f n)"
lp15@68071
   655
proof -
lp15@68071
   656
  have eq: "prod f {..n + Suc (Max N)} = prod f N" for n
lp15@68071
   657
  proof (rule prod.mono_neutral_right)
lp15@68071
   658
    show "N \<subseteq> {..n + Suc (Max N)}"
lp15@68071
   659
      by (auto simp add: le_Suc_eq trans_le_add2)
lp15@68071
   660
    show "\<forall>i\<in>{..n + Suc (Max N)} - N. f i = 1"
lp15@68071
   661
      using f by blast
lp15@68071
   662
  qed auto
lp15@68071
   663
  show ?thesis
lp15@68071
   664
  proof (cases "\<forall>n\<in>N. f n \<noteq> 0")
lp15@68071
   665
    case True
lp15@68071
   666
    then have "prod f N \<noteq> 0"
lp15@68071
   667
      by simp
lp15@68071
   668
    moreover have "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> prod f N"
lp15@68071
   669
      by (rule LIMSEQ_offset[of _ "Suc (Max N)"]) (simp add: eq atLeast0LessThan del: add_Suc_right)
lp15@68071
   670
    ultimately show ?thesis
lp15@68071
   671
      by (simp add: gen_has_prod_def has_prod_def)
lp15@68071
   672
  next
lp15@68071
   673
    case False
lp15@68071
   674
    then obtain k where "k \<in> N" "f k = 0"
lp15@68071
   675
      by auto
lp15@68071
   676
    let ?Z = "{n \<in> N. f n = 0}"
lp15@68071
   677
    have maxge: "Max ?Z \<ge> n" if "f n = 0" for n
lp15@68071
   678
      using Max_ge [of ?Z] \<open>finite N\<close> \<open>f n = 0\<close>
lp15@68071
   679
      by (metis (mono_tags) Collect_mem_eq f finite_Collect_conjI mem_Collect_eq zero_neq_one)
lp15@68071
   680
    let ?q = "prod f {Suc (Max ?Z)..Max N}"
lp15@68071
   681
    have [simp]: "?q \<noteq> 0"
lp15@68071
   682
      using maxge Suc_n_not_le_n le_trans by force
lp15@68076
   683
    have eq: "(\<Prod>i\<le>n + Max N. f (Suc (i + Max ?Z))) = ?q" for n
lp15@68076
   684
    proof -
lp15@68076
   685
      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
   686
      proof (rule prod.reindex_cong [where l = "\<lambda>i. i + Suc (Max ?Z)", THEN sym])
lp15@68076
   687
        show "{Suc (Max ?Z)..n + Max N + Suc (Max ?Z)} = (\<lambda>i. i + Suc (Max ?Z)) ` {..n + Max N}"
lp15@68076
   688
          using le_Suc_ex by fastforce
lp15@68076
   689
      qed (auto simp: inj_on_def)
lp15@68076
   690
      also have "... = ?q"
lp15@68076
   691
        by (rule prod.mono_neutral_right)
lp15@68076
   692
           (use Max.coboundedI [OF \<open>finite N\<close>] f in \<open>force+\<close>)
lp15@68076
   693
      finally show ?thesis .
lp15@68076
   694
    qed
lp15@68071
   695
    have q: "gen_has_prod f (Suc (Max ?Z)) ?q"
lp15@68076
   696
    proof (simp add: gen_has_prod_def)
lp15@68076
   697
      show "(\<lambda>n. \<Prod>i\<le>n. f (Suc (i + Max ?Z))) \<longlonglongrightarrow> ?q"
lp15@68076
   698
        by (rule LIMSEQ_offset[of _ "(Max N)"]) (simp add: eq)
lp15@68076
   699
    qed
lp15@68071
   700
    show ?thesis
lp15@68071
   701
      unfolding has_prod_def
lp15@68071
   702
    proof (intro disjI2 exI conjI)      
lp15@68071
   703
      show "prod f N = 0"
lp15@68071
   704
        using \<open>f k = 0\<close> \<open>k \<in> N\<close> \<open>finite N\<close> prod_zero by blast
lp15@68071
   705
      show "f (Max ?Z) = 0"
lp15@68071
   706
        using Max_in [of ?Z] \<open>finite N\<close> \<open>f k = 0\<close> \<open>k \<in> N\<close> by auto
lp15@68071
   707
    qed (use q in auto)
lp15@68071
   708
  qed
lp15@68071
   709
qed
lp15@68071
   710
lp15@68071
   711
corollary has_prod_0:
lp15@68071
   712
  fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
lp15@68071
   713
  assumes "\<And>n. f n = 1"
lp15@68071
   714
  shows "f has_prod 1"
lp15@68071
   715
  by (simp add: assms has_prod_cong)
lp15@68071
   716
lp15@68071
   717
lemma convergent_prod_finite:
lp15@68071
   718
  fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
lp15@68071
   719
  assumes "finite N" "\<And>n. n \<notin> N \<Longrightarrow> f n = 1"
lp15@68071
   720
  shows "convergent_prod f"
lp15@68071
   721
proof -
lp15@68071
   722
  have "\<exists>n p. gen_has_prod f n p"
lp15@68071
   723
    using assms has_prod_def has_prod_finite by blast
lp15@68071
   724
  then show ?thesis
lp15@68071
   725
    by (simp add: convergent_prod_def)
lp15@68071
   726
qed
lp15@68071
   727
lp15@68127
   728
lemma has_prod_If_finite_set:
lp15@68127
   729
  fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
lp15@68127
   730
  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
   731
  using has_prod_finite[of A "(\<lambda>r. if r \<in> A then f r else 1)"]
lp15@68127
   732
  by simp
lp15@68127
   733
lp15@68127
   734
lemma has_prod_If_finite:
lp15@68127
   735
  fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
lp15@68127
   736
  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
   737
  using has_prod_If_finite_set[of "{r. P r}"] by simp
lp15@68127
   738
lp15@68127
   739
lemma convergent_prod_If_finite_set[simp, intro]:
lp15@68127
   740
  fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
lp15@68127
   741
  shows "finite A \<Longrightarrow> convergent_prod (\<lambda>r. if r \<in> A then f r else 1)"
lp15@68127
   742
  by (simp add: convergent_prod_finite)
lp15@68127
   743
lp15@68127
   744
lemma convergent_prod_If_finite[simp, intro]:
lp15@68127
   745
  fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
lp15@68127
   746
  shows "finite {r. P r} \<Longrightarrow> convergent_prod (\<lambda>r. if P r then f r else 1)"
lp15@68127
   747
  using convergent_prod_def has_prod_If_finite has_prod_def by fastforce
lp15@68127
   748
lp15@68127
   749
lemma has_prod_single:
lp15@68127
   750
  fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
lp15@68127
   751
  shows "(\<lambda>r. if r = i then f r else 1) has_prod f i"
lp15@68127
   752
  using has_prod_If_finite[of "\<lambda>r. r = i"] by simp
lp15@68127
   753
lp15@68136
   754
context
lp15@68136
   755
  fixes f :: "nat \<Rightarrow> 'a :: real_normed_field"
lp15@68136
   756
begin
lp15@68136
   757
lp15@68136
   758
lemma convergent_prod_imp_has_prod: 
lp15@68136
   759
  assumes "convergent_prod f"
lp15@68136
   760
  shows "\<exists>p. f has_prod p"
lp15@68136
   761
proof -
lp15@68136
   762
  obtain M p where p: "gen_has_prod f M p"
lp15@68136
   763
    using assms convergent_prod_def by blast
lp15@68136
   764
  then have "p \<noteq> 0"
lp15@68136
   765
    using gen_has_prod_nonzero by blast
lp15@68136
   766
  with p have fnz: "f i \<noteq> 0" if "i \<ge> M" for i
lp15@68136
   767
    using gen_has_prod_eq_0 that by blast
lp15@68136
   768
  define C where "C = (\<Prod>n<M. f n)"
lp15@68136
   769
  show ?thesis
lp15@68136
   770
  proof (cases "\<forall>n\<le>M. f n \<noteq> 0")
lp15@68136
   771
    case True
lp15@68136
   772
    then have "C \<noteq> 0"
lp15@68136
   773
      by (simp add: C_def)
lp15@68136
   774
    then show ?thesis
lp15@68136
   775
      by (meson True assms convergent_prod_offset_0 fnz has_prod_def nat_le_linear)
lp15@68136
   776
  next
lp15@68136
   777
    case False
lp15@68136
   778
    let ?N = "GREATEST n. f n = 0"
lp15@68136
   779
    have 0: "f ?N = 0"
lp15@68136
   780
      using fnz False
lp15@68136
   781
      by (metis (mono_tags, lifting) GreatestI_ex_nat nat_le_linear)
lp15@68136
   782
    have "f i \<noteq> 0" if "i > ?N" for i
lp15@68136
   783
      by (metis (mono_tags, lifting) Greatest_le_nat fnz leD linear that)
lp15@68136
   784
    then have "\<exists>p. gen_has_prod f (Suc ?N) p"
lp15@68136
   785
      using assms by (auto simp: intro!: convergent_prod_ignore_nonzero_segment)
lp15@68136
   786
    then show ?thesis
lp15@68136
   787
      unfolding has_prod_def using 0 by blast
lp15@68136
   788
  qed
lp15@68136
   789
qed
lp15@68136
   790
lp15@68136
   791
lemma convergent_prod_has_prod [intro]:
lp15@68136
   792
  shows "convergent_prod f \<Longrightarrow> f has_prod (prodinf f)"
lp15@68136
   793
  unfolding prodinf_def
lp15@68136
   794
  by (metis convergent_prod_imp_has_prod has_prod_unique theI')
lp15@68136
   795
lp15@68136
   796
lemma convergent_prod_LIMSEQ:
lp15@68136
   797
  shows "convergent_prod f \<Longrightarrow> (\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> prodinf f"
lp15@68136
   798
  by (metis convergent_LIMSEQ_iff convergent_prod_has_prod convergent_prod_imp_convergent 
lp15@68136
   799
      convergent_prod_to_zero_iff gen_has_prod_eq_0 has_prod_def prodinf_eq_lim zero_le)
lp15@68136
   800
lp15@68136
   801
lemma has_prod_iff: "f has_prod x \<longleftrightarrow> convergent_prod f \<and> prodinf f = x"
lp15@68136
   802
proof
lp15@68136
   803
  assume "f has_prod x"
lp15@68136
   804
  then show "convergent_prod f \<and> prodinf f = x"
lp15@68136
   805
    apply safe
lp15@68136
   806
    using convergent_prod_def has_prod_def apply blast
lp15@68136
   807
    using has_prod_unique by blast
lp15@68136
   808
qed auto
lp15@68136
   809
lp15@68136
   810
lemma convergent_prod_has_prod_iff: "convergent_prod f \<longleftrightarrow> f has_prod prodinf f"
lp15@68136
   811
  by (auto simp: has_prod_iff convergent_prod_has_prod)
lp15@68136
   812
lp15@68136
   813
lemma prodinf_finite:
lp15@68136
   814
  assumes N: "finite N"
lp15@68136
   815
    and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 1"
lp15@68136
   816
  shows "prodinf f = (\<Prod>n\<in>N. f n)"
lp15@68136
   817
  using has_prod_finite[OF assms, THEN has_prod_unique] by simp
lp15@68127
   818
eberlm@66277
   819
end
lp15@68136
   820
lp15@68136
   821
end