src/HOL/Analysis/Infinite_Products.thy
 author paulson Fri, 25 Dec 2020 11:44:18 +0000 changeset 73004 cf14976d4fdb parent 71827 5e315defb038 child 73005 83b114a6545f permissions -rw-r--r--
infinite products iff simprule

(*File:      HOL/Analysis/Infinite_Product.thy
Author:    Manuel Eberl & LC Paulson

Basic results about convergence and absolute convergence of infinite products
and their connection to summability.
*)
section \<open>Infinite Products\<close>
theory Infinite_Products
imports Topology_Euclidean_Space Complex_Transcendental
begin

subsection\<^marker>\<open>tag unimportant\<close> \<open>Preliminaries\<close>

lemma sum_le_prod:
fixes f :: "'a \<Rightarrow> 'b :: linordered_semidom"
assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<ge> 0"
shows   "sum f A \<le> (\<Prod>x\<in>A. 1 + f x)"
using assms
proof (induction A rule: infinite_finite_induct)
case (insert x A)
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)"
by (intro add_mono insert mult_left_mono prod_mono) (auto intro: insert.prems)
with insert.hyps show ?case by (simp add: algebra_simps)
qed simp_all

lemma prod_le_exp_sum:
fixes f :: "'a \<Rightarrow> real"
assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<ge> 0"
shows   "prod (\<lambda>x. 1 + f x) A \<le> exp (sum f A)"
using assms
proof (induction A rule: infinite_finite_induct)
case (insert x A)
have "(1 + f x) * (\<Prod>x\<in>A. 1 + f x) \<le> exp (f x) * exp (sum f A)"
using insert.prems by (intro mult_mono insert prod_nonneg exp_ge_add_one_self) auto
qed simp_all

lemma lim_ln_1_plus_x_over_x_at_0: "(\<lambda>x::real. ln (1 + x) / x) \<midarrow>0\<rightarrow> 1"
proof (rule lhopital)
show "(\<lambda>x::real. ln (1 + x)) \<midarrow>0\<rightarrow> 0"
by (rule tendsto_eq_intros refl | simp)+
have "eventually (\<lambda>x::real. x \<in> {-1/2<..<1/2}) (nhds 0)"
by (rule eventually_nhds_in_open) auto
hence *: "eventually (\<lambda>x::real. x \<in> {-1/2<..<1/2}) (at 0)"
by (rule filter_leD [rotated]) (simp_all add: at_within_def)
show "eventually (\<lambda>x::real. ((\<lambda>x. ln (1 + x)) has_field_derivative inverse (1 + x)) (at x)) (at 0)"
using * by eventually_elim (auto intro!: derivative_eq_intros simp: field_simps)
show "eventually (\<lambda>x::real. ((\<lambda>x. x) has_field_derivative 1) (at x)) (at 0)"
using * by eventually_elim (auto intro!: derivative_eq_intros simp: field_simps)
show "\<forall>\<^sub>F x in at 0. x \<noteq> 0" by (auto simp: at_within_def eventually_inf_principal)
show "(\<lambda>x::real. inverse (1 + x) / 1) \<midarrow>0\<rightarrow> 1"
by (rule tendsto_eq_intros refl | simp)+
qed auto

subsection\<open>Definitions and basic properties\<close>

definition\<^marker>\<open>tag important\<close> raw_has_prod :: "[nat \<Rightarrow> 'a::{t2_space, comm_semiring_1}, nat, 'a] \<Rightarrow> bool"
where "raw_has_prod f M p \<equiv> (\<lambda>n. \<Prod>i\<le>n. f (i+M)) \<longlonglongrightarrow> p \<and> p \<noteq> 0"

text\<open>The nonzero and zero cases, as in \emph{Complex Analysis} by Joseph Bak and Donald J.Newman, page 241\<close>
text\<^marker>\<open>tag important\<close> \<open>%whitespace\<close>
definition\<^marker>\<open>tag important\<close>
has_prod :: "(nat \<Rightarrow> 'a::{t2_space, comm_semiring_1}) \<Rightarrow> 'a \<Rightarrow> bool" (infixr "has'_prod" 80)
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)"

definition\<^marker>\<open>tag important\<close> convergent_prod :: "(nat \<Rightarrow> 'a :: {t2_space,comm_semiring_1}) \<Rightarrow> bool" where
"convergent_prod f \<equiv> \<exists>M p. raw_has_prod f M p"

definition\<^marker>\<open>tag important\<close> prodinf :: "(nat \<Rightarrow> 'a::{t2_space, comm_semiring_1}) \<Rightarrow> 'a"
(binder "\<Prod>" 10)
where "prodinf f = (THE p. f has_prod p)"

lemmas prod_defs = raw_has_prod_def has_prod_def convergent_prod_def prodinf_def

lemma has_prod_subst[trans]: "f = g \<Longrightarrow> g has_prod z \<Longrightarrow> f has_prod z"
by simp

lemma has_prod_cong: "(\<And>n. f n = g n) \<Longrightarrow> f has_prod c \<longleftrightarrow> g has_prod c"
by presburger

lemma raw_has_prod_nonzero [simp]: "\<not> raw_has_prod f M 0"

lemma raw_has_prod_eq_0:
fixes f :: "nat \<Rightarrow> 'a::{semidom,t2_space}"
assumes p: "raw_has_prod f m p" and i: "f i = 0" "i \<ge> m"
shows "p = 0"
proof -
have eq0: "(\<Prod>k\<le>n. f (k+m)) = 0" if "i - m \<le> n" for n
proof -
have "\<exists>k\<le>n. f (k + m) = 0"
using i that by auto
then show ?thesis
by auto
qed
have "(\<lambda>n. \<Prod>i\<le>n. f (i + m)) \<longlonglongrightarrow> 0"
by (rule LIMSEQ_offset [where k = "i-m"]) (simp add: eq0)
with p show ?thesis
unfolding raw_has_prod_def
using LIMSEQ_unique by blast
qed

lemma raw_has_prod_Suc:
"raw_has_prod f (Suc M) a \<longleftrightarrow> raw_has_prod (\<lambda>n. f (Suc n)) M a"
unfolding raw_has_prod_def by auto

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))"

lemma has_prod_unique2:
fixes f :: "nat \<Rightarrow> 'a::{semidom,t2_space}"
assumes "f has_prod a" "f has_prod b" shows "a = b"
using assms
by (auto simp: has_prod_def raw_has_prod_eq_0) (meson raw_has_prod_def sequentially_bot tendsto_unique)

lemma has_prod_unique:
fixes f :: "nat \<Rightarrow> 'a :: {semidom,t2_space}"
shows "f has_prod s \<Longrightarrow> s = prodinf f"
by (simp add: has_prod_unique2 prodinf_def the_equality)

lemma convergent_prod_altdef:
fixes f :: "nat \<Rightarrow> 'a :: {t2_space,comm_semiring_1}"
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)"
proof
assume "convergent_prod f"
then obtain M L where *: "(\<lambda>n. \<Prod>i\<le>n. f (i+M)) \<longlonglongrightarrow> L" "L \<noteq> 0"
by (auto simp: prod_defs)
have "f i \<noteq> 0" if "i \<ge> M" for i
proof
assume "f i = 0"
have **: "eventually (\<lambda>n. (\<Prod>i\<le>n. f (i+M)) = 0) sequentially"
using eventually_ge_at_top[of "i - M"]
proof eventually_elim
case (elim n)
with \<open>f i = 0\<close> and \<open>i \<ge> M\<close> show ?case
by (auto intro!: bexI[of _ "i - M"] prod_zero)
qed
have "(\<lambda>n. (\<Prod>i\<le>n. f (i+M))) \<longlonglongrightarrow> 0"
unfolding filterlim_iff
by (auto dest!: eventually_nhds_x_imp_x intro!: eventually_mono[OF **])
from tendsto_unique[OF _ this *(1)] and *(2)
show False by simp
qed
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)"
by blast
qed (auto simp: prod_defs)

subsection\<open>Absolutely convergent products\<close>

definition\<^marker>\<open>tag important\<close> abs_convergent_prod :: "(nat \<Rightarrow> _) \<Rightarrow> bool" where
"abs_convergent_prod f \<longleftrightarrow> convergent_prod (\<lambda>i. 1 + norm (f i - 1))"

lemma abs_convergent_prodI:
assumes "convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
shows   "abs_convergent_prod f"
proof -
from assms obtain L where L: "(\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1)) \<longlonglongrightarrow> L"
by (auto simp: convergent_def)
have "L \<ge> 1"
proof (rule tendsto_le)
show "eventually (\<lambda>n. (\<Prod>i\<le>n. 1 + norm (f i - 1)) \<ge> 1) sequentially"
proof (intro always_eventually allI)
fix n
have "(\<Prod>i\<le>n. 1 + norm (f i - 1)) \<ge> (\<Prod>i\<le>n. 1)"
by (intro prod_mono) auto
thus "(\<Prod>i\<le>n. 1 + norm (f i - 1)) \<ge> 1" by simp
qed
qed (use L in simp_all)
hence "L \<noteq> 0" by auto
with L show ?thesis unfolding abs_convergent_prod_def prod_defs
by (intro exI[of _ "0::nat"] exI[of _ L]) auto
qed

lemma
fixes f :: "nat \<Rightarrow> 'a :: {topological_semigroup_mult,t2_space,idom}"
assumes "convergent_prod f"
shows   convergent_prod_imp_convergent: "convergent (\<lambda>n. \<Prod>i\<le>n. f i)"
and   convergent_prod_to_zero_iff:    "(\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> 0 \<longleftrightarrow> (\<exists>i. f i = 0)"
proof -
from assms obtain M L
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"
by (auto simp: convergent_prod_altdef)
note this(2)
also have "(\<lambda>n. \<Prod>i\<le>n. f (i + M)) = (\<lambda>n. \<Prod>i=M..M+n. f i)"
by (intro ext prod.reindex_bij_witness[of _ "\<lambda>n. n - M" "\<lambda>n. n + M"]) auto
finally have "(\<lambda>n. (\<Prod>i<M. f i) * (\<Prod>i=M..M+n. f i)) \<longlonglongrightarrow> (\<Prod>i<M. f i) * L"
by (intro tendsto_mult tendsto_const)
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))"
by (subst prod.union_disjoint) auto
also have "(\<lambda>n. {..<M} \<union> {M..M+n}) = (\<lambda>n. {..n+M})" by auto
finally have lim: "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> prod f {..<M} * L"
by (rule LIMSEQ_offset)
thus "convergent (\<lambda>n. \<Prod>i\<le>n. f i)"
by (auto simp: convergent_def)

show "(\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> 0 \<longleftrightarrow> (\<exists>i. f i = 0)"
proof
assume "\<exists>i. f i = 0"
then obtain i where "f i = 0" by auto
moreover with M have "i < M" by (cases "i < M") auto
ultimately have "(\<Prod>i<M. f i) = 0" by auto
with lim show "(\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> 0" by simp
next
assume "(\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> 0"
from tendsto_unique[OF _ this lim] and \<open>L \<noteq> 0\<close>
show "\<exists>i. f i = 0" by auto
qed
qed

lemma convergent_prod_iff_nz_lim:
fixes f :: "nat \<Rightarrow> 'a :: {topological_semigroup_mult,t2_space,idom}"
assumes "\<And>i. f i \<noteq> 0"
shows "convergent_prod f \<longleftrightarrow> (\<exists>L. (\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> L \<and> L \<noteq> 0)"
(is "?lhs \<longleftrightarrow> ?rhs")
proof
assume ?lhs then show ?rhs
using assms convergentD convergent_prod_imp_convergent convergent_prod_to_zero_iff by blast
next
assume ?rhs then show ?lhs
unfolding prod_defs
by (rule_tac x=0 in exI) auto
qed

lemma\<^marker>\<open>tag important\<close> convergent_prod_iff_convergent:
fixes f :: "nat \<Rightarrow> 'a :: {topological_semigroup_mult,t2_space,idom}"
assumes "\<And>i. f i \<noteq> 0"
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"
by (force simp: convergent_prod_iff_nz_lim assms convergent_def limI)

lemma bounded_imp_convergent_prod:
fixes a :: "nat \<Rightarrow> real"
assumes 1: "\<And>n. a n \<ge> 1" and bounded: "\<And>n. (\<Prod>i\<le>n. a i) \<le> B"
shows "convergent_prod a"
proof -
have "bdd_above (range(\<lambda>n. \<Prod>i\<le>n. a i))"
by (meson bdd_aboveI2 bounded)
moreover have "incseq (\<lambda>n. \<Prod>i\<le>n. a i)"
unfolding mono_def by (metis 1 prod_mono2 atMost_subset_iff dual_order.trans finite_atMost zero_le_one)
ultimately obtain p where p: "(\<lambda>n. \<Prod>i\<le>n. a i) \<longlonglongrightarrow> p"
using LIMSEQ_incseq_SUP by blast
then have "p \<noteq> 0"
by (metis "1" not_one_le_zero prod_ge_1 LIMSEQ_le_const)
with 1 p show ?thesis
by (metis convergent_prod_iff_nz_lim not_one_le_zero)
qed

lemma abs_convergent_prod_altdef:
fixes f :: "nat \<Rightarrow> 'a :: {one,real_normed_vector}"
shows  "abs_convergent_prod f \<longleftrightarrow> convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
proof
assume "abs_convergent_prod f"
thus "convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
by (auto simp: abs_convergent_prod_def intro!: convergent_prod_imp_convergent)
qed (auto intro: abs_convergent_prodI)

lemma Weierstrass_prod_ineq:
fixes f :: "'a \<Rightarrow> real"
assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> {0..1}"
shows   "1 - sum f A \<le> (\<Prod>x\<in>A. 1 - f x)"
using assms
proof (induction A rule: infinite_finite_induct)
case (insert x A)
from insert.hyps and insert.prems
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)"
by (intro insert.IH add_mono mult_left_mono prod_mono) auto
with insert.hyps show ?case by (simp add: algebra_simps)
qed simp_all

lemma norm_prod_minus1_le_prod_minus1:
fixes f :: "nat \<Rightarrow> 'a :: {real_normed_div_algebra,comm_ring_1}"
shows "norm (prod (\<lambda>n. 1 + f n) A - 1) \<le> prod (\<lambda>n. 1 + norm (f n)) A - 1"
proof (induction A rule: infinite_finite_induct)
case (insert x A)
from insert.hyps have
"norm ((\<Prod>n\<in>insert x A. 1 + f n) - 1) =
norm ((\<Prod>n\<in>A. 1 + f n) - 1 + f x * (\<Prod>n\<in>A. 1 + f n))"
also have "\<dots> \<le> norm ((\<Prod>n\<in>A. 1 + f n) - 1) + norm (f x * (\<Prod>n\<in>A. 1 + f n))"
by (rule norm_triangle_ineq)
also have "norm (f x * (\<Prod>n\<in>A. 1 + f n)) = norm (f x) * (\<Prod>x\<in>A. norm (1 + f x))"
also have "(\<Prod>x\<in>A. norm (1 + f x)) \<le> (\<Prod>x\<in>A. norm (1::'a) + norm (f x))"
by (intro prod_mono norm_triangle_ineq ballI conjI) auto
also have "norm (1::'a) = 1" by simp
also note insert.IH
also have "(\<Prod>n\<in>A. 1 + norm (f n)) - 1 + norm (f x) * (\<Prod>x\<in>A. 1 + norm (f x)) =
(\<Prod>n\<in>insert x A. 1 + norm (f n)) - 1"
using insert.hyps by (simp add: algebra_simps)
finally show ?case by - (simp_all add: mult_left_mono)
qed simp_all

lemma convergent_prod_imp_ev_nonzero:
fixes f :: "nat \<Rightarrow> 'a :: {t2_space,comm_semiring_1}"
assumes "convergent_prod f"
shows   "eventually (\<lambda>n. f n \<noteq> 0) sequentially"
using assms by (auto simp: eventually_at_top_linorder convergent_prod_altdef)

lemma convergent_prod_imp_LIMSEQ:
fixes f :: "nat \<Rightarrow> 'a :: {real_normed_field}"
assumes "convergent_prod f"
shows   "f \<longlonglongrightarrow> 1"
proof -
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"
by (auto simp: convergent_prod_altdef)
hence L': "(\<lambda>n. \<Prod>i\<le>Suc n. f (i+M)) \<longlonglongrightarrow> L" by (subst filterlim_sequentially_Suc)
have "(\<lambda>n. (\<Prod>i\<le>Suc n. f (i+M)) / (\<Prod>i\<le>n. f (i+M))) \<longlonglongrightarrow> L / L"
using L L' by (intro tendsto_divide) simp_all
also from L have "L / L = 1" by simp
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))"
using assms L by (auto simp: fun_eq_iff atMost_Suc)
finally show ?thesis by (rule LIMSEQ_offset)
qed

lemma abs_convergent_prod_imp_summable:
fixes f :: "nat \<Rightarrow> 'a :: real_normed_div_algebra"
assumes "abs_convergent_prod f"
shows "summable (\<lambda>i. norm (f i - 1))"
proof -
from assms have "convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
unfolding abs_convergent_prod_def by (rule convergent_prod_imp_convergent)
then obtain L where L: "(\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1)) \<longlonglongrightarrow> L"
unfolding convergent_def by blast
have "convergent (\<lambda>n. \<Sum>i\<le>n. norm (f i - 1))"
proof (rule Bseq_monoseq_convergent)
have "eventually (\<lambda>n. (\<Prod>i\<le>n. 1 + norm (f i - 1)) < L + 1) sequentially"
using L(1) by (rule order_tendstoD) simp_all
hence "\<forall>\<^sub>F x in sequentially. norm (\<Sum>i\<le>x. norm (f i - 1)) \<le> L + 1"
proof eventually_elim
case (elim n)
have "norm (\<Sum>i\<le>n. norm (f i - 1)) = (\<Sum>i\<le>n. norm (f i - 1))"
unfolding real_norm_def by (intro abs_of_nonneg sum_nonneg) simp_all
also have "\<dots> \<le> (\<Prod>i\<le>n. 1 + norm (f i - 1))" by (rule sum_le_prod) auto
also have "\<dots> < L + 1" by (rule elim)
finally show ?case by simp
qed
thus "Bseq (\<lambda>n. \<Sum>i\<le>n. norm (f i - 1))" by (rule BfunI)
next
show "monoseq (\<lambda>n. \<Sum>i\<le>n. norm (f i - 1))"
by (rule mono_SucI1) auto
qed
thus "summable (\<lambda>i. norm (f i - 1))" by (simp add: summable_iff_convergent')
qed

lemma summable_imp_abs_convergent_prod:
fixes f :: "nat \<Rightarrow> 'a :: real_normed_div_algebra"
assumes "summable (\<lambda>i. norm (f i - 1))"
shows   "abs_convergent_prod f"
proof (intro abs_convergent_prodI Bseq_monoseq_convergent)
show "monoseq (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
by (intro mono_SucI1)
(auto simp: atMost_Suc algebra_simps intro!: mult_nonneg_nonneg prod_nonneg)
next
show "Bseq (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
proof (rule Bseq_eventually_mono)
show "eventually (\<lambda>n. norm (\<Prod>i\<le>n. 1 + norm (f i - 1)) \<le>
norm (exp (\<Sum>i\<le>n. norm (f i - 1)))) sequentially"
by (intro always_eventually allI) (auto simp: abs_prod exp_sum intro!: prod_mono)
next
from assms have "(\<lambda>n. \<Sum>i\<le>n. norm (f i - 1)) \<longlonglongrightarrow> (\<Sum>i. norm (f i - 1))"
using sums_def_le by blast
hence "(\<lambda>n. exp (\<Sum>i\<le>n. norm (f i - 1))) \<longlonglongrightarrow> exp (\<Sum>i. norm (f i - 1))"
by (rule tendsto_exp)
hence "convergent (\<lambda>n. exp (\<Sum>i\<le>n. norm (f i - 1)))"
by (rule convergentI)
thus "Bseq (\<lambda>n. exp (\<Sum>i\<le>n. norm (f i - 1)))"
by (rule convergent_imp_Bseq)
qed
qed

theorem abs_convergent_prod_conv_summable:
fixes f :: "nat \<Rightarrow> 'a :: real_normed_div_algebra"
shows "abs_convergent_prod f \<longleftrightarrow> summable (\<lambda>i. norm (f i - 1))"
by (blast intro: abs_convergent_prod_imp_summable summable_imp_abs_convergent_prod)

lemma abs_convergent_prod_imp_LIMSEQ:
fixes f :: "nat \<Rightarrow> 'a :: {comm_ring_1,real_normed_div_algebra}"
assumes "abs_convergent_prod f"
shows   "f \<longlonglongrightarrow> 1"
proof -
from assms have "summable (\<lambda>n. norm (f n - 1))"
by (rule abs_convergent_prod_imp_summable)
from summable_LIMSEQ_zero[OF this] have "(\<lambda>n. f n - 1) \<longlonglongrightarrow> 0"
from tendsto_add[OF this tendsto_const[of 1]] show ?thesis by simp
qed

lemma abs_convergent_prod_imp_ev_nonzero:
fixes f :: "nat \<Rightarrow> 'a :: {comm_ring_1,real_normed_div_algebra}"
assumes "abs_convergent_prod f"
shows   "eventually (\<lambda>n. f n \<noteq> 0) sequentially"
proof -
from assms have "f \<longlonglongrightarrow> 1"
by (rule abs_convergent_prod_imp_LIMSEQ)
hence "eventually (\<lambda>n. dist (f n) 1 < 1) at_top"
by (auto simp: tendsto_iff)
thus ?thesis by eventually_elim auto
qed

subsection\<^marker>\<open>tag unimportant\<close> \<open>Ignoring initial segments\<close>

lemma convergent_prod_offset:
assumes "convergent_prod (\<lambda>n. f (n + m))"
shows   "convergent_prod f"
proof -
from assms obtain M L where "(\<lambda>n. \<Prod>k\<le>n. f (k + (M + m))) \<longlonglongrightarrow> L" "L \<noteq> 0"
thus "convergent_prod f"
unfolding prod_defs by blast
qed

lemma abs_convergent_prod_offset:
assumes "abs_convergent_prod (\<lambda>n. f (n + m))"
shows   "abs_convergent_prod f"
using assms unfolding abs_convergent_prod_def by (rule convergent_prod_offset)

lemma raw_has_prod_ignore_initial_segment:
fixes f :: "nat \<Rightarrow> 'a :: real_normed_field"
assumes "raw_has_prod f M p" "N \<ge> M"
obtains q where  "raw_has_prod f N q"
proof -
have p: "(\<lambda>n. \<Prod>k\<le>n. f (k + M)) \<longlonglongrightarrow> p" and "p \<noteq> 0"
using assms by (auto simp: raw_has_prod_def)
then have nz: "\<And>n. n \<ge> M \<Longrightarrow> f n \<noteq> 0"
using assms by (auto simp: raw_has_prod_eq_0)
define C where "C = (\<Prod>k<N-M. f (k + M))"
from nz have [simp]: "C \<noteq> 0"
by (auto simp: C_def)

from p have "(\<lambda>i. \<Prod>k\<le>i + (N-M). f (k + M)) \<longlonglongrightarrow> p"
by (rule LIMSEQ_ignore_initial_segment)
also have "(\<lambda>i. \<Prod>k\<le>i + (N-M). f (k + M)) = (\<lambda>n. C * (\<Prod>k\<le>n. f (k + N)))"
proof (rule ext, goal_cases)
case (1 n)
have "{..n+(N-M)} = {..<(N-M)} \<union> {(N-M)..n+(N-M)}" by auto
also have "(\<Prod>k\<in>\<dots>. f (k + M)) = C * (\<Prod>k=(N-M)..n+(N-M). f (k + M))"
unfolding C_def by (rule prod.union_disjoint) auto
also have "(\<Prod>k=(N-M)..n+(N-M). f (k + M)) = (\<Prod>k\<le>n. f (k + (N-M) + M))"
by (intro ext prod.reindex_bij_witness[of _ "\<lambda>k. k + (N-M)" "\<lambda>k. k - (N-M)"]) auto
finally show ?case
qed
finally have "(\<lambda>n. C * (\<Prod>k\<le>n. f (k + N)) / C) \<longlonglongrightarrow> p / C"
by (intro tendsto_divide tendsto_const) auto
hence "(\<lambda>n. \<Prod>k\<le>n. f (k + N)) \<longlonglongrightarrow> p / C" by simp
moreover from \<open>p \<noteq> 0\<close> have "p / C \<noteq> 0" by simp
ultimately show ?thesis
using raw_has_prod_def that by blast
qed

corollary\<^marker>\<open>tag unimportant\<close> convergent_prod_ignore_initial_segment:
fixes f :: "nat \<Rightarrow> 'a :: real_normed_field"
assumes "convergent_prod f"
shows   "convergent_prod (\<lambda>n. f (n + m))"
using assms
unfolding convergent_prod_def
apply clarify
apply (erule_tac N="M+m" in raw_has_prod_ignore_initial_segment)
done

corollary\<^marker>\<open>tag unimportant\<close> convergent_prod_ignore_nonzero_segment:
fixes f :: "nat \<Rightarrow> 'a :: real_normed_field"
assumes f: "convergent_prod f" and nz: "\<And>i. i \<ge> M \<Longrightarrow> f i \<noteq> 0"
shows "\<exists>p. raw_has_prod f M p"
using convergent_prod_ignore_initial_segment [OF f]
by (metis convergent_LIMSEQ_iff convergent_prod_iff_convergent le_add_same_cancel2 nz prod_defs(1) zero_order(1))

corollary\<^marker>\<open>tag unimportant\<close> abs_convergent_prod_ignore_initial_segment:
assumes "abs_convergent_prod f"
shows   "abs_convergent_prod (\<lambda>n. f (n + m))"
using assms unfolding abs_convergent_prod_def
by (rule convergent_prod_ignore_initial_segment)

subsection\<open>More elementary properties\<close>

theorem abs_convergent_prod_imp_convergent_prod:
fixes f :: "nat \<Rightarrow> 'a :: {real_normed_div_algebra,complete_space,comm_ring_1}"
assumes "abs_convergent_prod f"
shows   "convergent_prod f"
proof -
from assms have "eventually (\<lambda>n. f n \<noteq> 0) sequentially"
by (rule abs_convergent_prod_imp_ev_nonzero)
then obtain N where N: "f n \<noteq> 0" if "n \<ge> N" for n
by (auto simp: eventually_at_top_linorder)
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)"

have "Cauchy ?P"
proof (rule CauchyI', goal_cases)
case (1 \<epsilon>)
from assms have "abs_convergent_prod (\<lambda>n. f (n + N))"
by (rule abs_convergent_prod_ignore_initial_segment)
hence "Cauchy ?Q"
unfolding abs_convergent_prod_def
by (intro convergent_Cauchy convergent_prod_imp_convergent)
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
by blast
show ?case
proof (rule exI[of _ M], safe, goal_cases)
case (1 m n)
have "dist (?P m) (?P n) = norm (?P n - ?P m)"
also from 1 have "{..n} = {..m} \<union> {m<..n}" by auto
hence "norm (?P n - ?P m) = norm (?P m * (\<Prod>k\<in>{m<..n}. f (k + N)) - ?P m)"
by (subst prod.union_disjoint [symmetric]) (auto simp: algebra_simps)
also have "\<dots> = norm (?P m * ((\<Prod>k\<in>{m<..n}. f (k + N)) - 1))"
also have "\<dots> = (\<Prod>k\<le>m. norm (f (k + N))) * norm ((\<Prod>k\<in>{m<..n}. f (k + N)) - 1)"
also have "\<dots> \<le> ?Q m * ((\<Prod>k\<in>{m<..n}. 1 + norm (f (k + N) - 1)) - 1)"
using norm_prod_minus1_le_prod_minus1[of "\<lambda>k. f (k + N) - 1" "{m<..n}"]
norm_triangle_ineq[of 1 "f k - 1" for k]
by (intro mult_mono prod_mono ballI conjI norm_prod_minus1_le_prod_minus1 prod_nonneg) auto
also have "\<dots> = ?Q m * (\<Prod>k\<in>{m<..n}. 1 + norm (f (k + N) - 1)) - ?Q m"
also have "?Q m * (\<Prod>k\<in>{m<..n}. 1 + norm (f (k + N) - 1)) =
(\<Prod>k\<in>{..m}\<union>{m<..n}. 1 + norm (f (k + N) - 1))"
by (rule prod.union_disjoint [symmetric]) auto
also from 1 have "{..m}\<union>{m<..n} = {..n}" by auto
also have "?Q n - ?Q m \<le> norm (?Q n - ?Q m)" by simp
also from 1 have "\<dots> < \<epsilon>" by (intro M) auto
finally show ?case .
qed
qed
hence conv: "convergent ?P" by (rule Cauchy_convergent)
then obtain L where L: "?P \<longlonglongrightarrow> L"
by (auto simp: convergent_def)

have "L \<noteq> 0"
proof
assume [simp]: "L = 0"
from tendsto_norm[OF L] have limit: "(\<lambda>n. \<Prod>k\<le>n. norm (f (k + N))) \<longlonglongrightarrow> 0"

from assms have "(\<lambda>n. f (n + N)) \<longlonglongrightarrow> 1"
by (intro abs_convergent_prod_imp_LIMSEQ abs_convergent_prod_ignore_initial_segment)
hence "eventually (\<lambda>n. norm (f (n + N) - 1) < 1) sequentially"
by (auto simp: tendsto_iff dist_norm)
then obtain M0 where M0: "norm (f (n + N) - 1) < 1" if "n \<ge> M0" for n
by (auto simp: eventually_at_top_linorder)

{
fix M assume M: "M \<ge> M0"
with M0 have M: "norm (f (n + N) - 1) < 1" if "n \<ge> M" for n using that by simp

have "(\<lambda>n. \<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1)) \<longlonglongrightarrow> 0"
proof (rule tendsto_sandwich)
show "eventually (\<lambda>n. (\<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1)) \<ge> 0) sequentially"
using M by (intro always_eventually prod_nonneg allI ballI) (auto intro: less_imp_le)
have "norm (1::'a) - norm (f (i + M + N) - 1) \<le> norm (f (i + M + N))" for i
using norm_triangle_ineq3[of "f (i + M + N)" 1] by simp
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"
using M by (intro always_eventually allI prod_mono ballI conjI) (auto intro: less_imp_le)

define C where "C = (\<Prod>k<M. norm (f (k + N)))"
from N have [simp]: "C \<noteq> 0" by (auto simp: C_def)
from L have "(\<lambda>n. norm (\<Prod>k\<le>n+M. f (k + N))) \<longlonglongrightarrow> 0"
by (intro LIMSEQ_ignore_initial_segment) (simp add: tendsto_norm_zero_iff)
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))))"
proof (rule ext, goal_cases)
case (1 n)
have "{..n+M} = {..<M} \<union> {M..n+M}" by auto
also have "norm (\<Prod>k\<in>\<dots>. f (k + N)) = C * norm (\<Prod>k=M..n+M. f (k + N))"
unfolding C_def by (subst prod.union_disjoint) (auto simp: norm_mult prod_norm)
also have "(\<Prod>k=M..n+M. f (k + N)) = (\<Prod>k\<le>n. f (k + N + M))"
by (intro prod.reindex_bij_witness[of _ "\<lambda>i. i + M" "\<lambda>i. i - M"]) auto
qed
finally have "(\<lambda>n. C * (\<Prod>k\<le>n. norm (f (k + M + N))) / C) \<longlonglongrightarrow> 0 / C"
by (intro tendsto_divide tendsto_const) auto
thus "(\<lambda>n. \<Prod>k\<le>n. norm (f (k + M + N))) \<longlonglongrightarrow> 0" by simp
qed simp_all

have "1 - (\<Sum>i. norm (f (i + M + N) - 1)) \<le> 0"
proof (rule tendsto_le)
show "eventually (\<lambda>n. 1 - (\<Sum>k\<le>n. norm (f (k+M+N) - 1)) \<le>
(\<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1))) at_top"
using M by (intro always_eventually allI Weierstrass_prod_ineq) (auto intro: less_imp_le)
show "(\<lambda>n. \<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1)) \<longlonglongrightarrow> 0" by fact
show "(\<lambda>n. 1 - (\<Sum>k\<le>n. norm (f (k + M + N) - 1)))
\<longlonglongrightarrow> 1 - (\<Sum>i. norm (f (i + M + N) - 1))"
by (intro tendsto_intros summable_LIMSEQ' summable_ignore_initial_segment
abs_convergent_prod_imp_summable assms)
qed simp_all
hence "(\<Sum>i. norm (f (i + M + N) - 1)) \<ge> 1" by simp
also have "\<dots> + (\<Sum>i<M. norm (f (i + N) - 1)) = (\<Sum>i. norm (f (i + N) - 1))"
by (intro suminf_split_initial_segment [symmetric] summable_ignore_initial_segment
abs_convergent_prod_imp_summable assms)
finally have "1 + (\<Sum>i<M. norm (f (i + N) - 1)) \<le> (\<Sum>i. norm (f (i + N) - 1))" by simp
} note * = this

have "1 + (\<Sum>i. norm (f (i + N) - 1)) \<le> (\<Sum>i. norm (f (i + N) - 1))"
proof (rule tendsto_le)
show "(\<lambda>M. 1 + (\<Sum>i<M. norm (f (i + N) - 1))) \<longlonglongrightarrow> 1 + (\<Sum>i. norm (f (i + N) - 1))"
by (intro tendsto_intros summable_LIMSEQ summable_ignore_initial_segment
abs_convergent_prod_imp_summable assms)
show "eventually (\<lambda>M. 1 + (\<Sum>i<M. norm (f (i + N) - 1)) \<le> (\<Sum>i. norm (f (i + N) - 1))) at_top"
using eventually_ge_at_top[of M0] by eventually_elim (use * in auto)
qed simp_all
thus False by simp
qed
with L show ?thesis by (auto simp: prod_defs)
qed

lemma raw_has_prod_cases:
fixes f :: "nat \<Rightarrow> 'a :: {idom,topological_semigroup_mult,t2_space}"
assumes "raw_has_prod f M p"
obtains i where "i<M" "f i = 0" | p where "raw_has_prod f 0 p"
proof -
have "(\<lambda>n. \<Prod>i\<le>n. f (i + M)) \<longlonglongrightarrow> p" "p \<noteq> 0"
using assms unfolding raw_has_prod_def by blast+
then have "(\<lambda>n. prod f {..<M} * (\<Prod>i\<le>n. f (i + M))) \<longlonglongrightarrow> prod f {..<M} * p"
by (metis tendsto_mult_left)
moreover have "prod f {..<M} * (\<Prod>i\<le>n. f (i + M)) = prod f {..n+M}" for n
proof -
have "{..n+M} = {..<M} \<union> {M..n+M}"
by auto
then have "prod f {..n+M} = prod f {..<M} * prod f {M..n+M}"
by simp (subst prod.union_disjoint; force)
also have "\<dots> = prod f {..<M} * (\<Prod>i\<le>n. f (i + M))"
by (metis (mono_tags, lifting) add.left_neutral atMost_atLeast0 prod.shift_bounds_cl_nat_ivl)
finally show ?thesis by metis
qed
ultimately have "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> prod f {..<M} * p"
by (auto intro: LIMSEQ_offset [where k=M])
then have "raw_has_prod f 0 (prod f {..<M} * p)" if "\<forall>i<M. f i \<noteq> 0"
using \<open>p \<noteq> 0\<close> assms that by (auto simp: raw_has_prod_def)
then show thesis
using that by blast
qed

corollary convergent_prod_offset_0:
fixes f :: "nat \<Rightarrow> 'a :: {idom,topological_semigroup_mult,t2_space}"
assumes "convergent_prod f" "\<And>i. f i \<noteq> 0"
shows "\<exists>p. raw_has_prod f 0 p"
using assms convergent_prod_def raw_has_prod_cases by blast

lemma prodinf_eq_lim:
fixes f :: "nat \<Rightarrow> 'a :: {idom,topological_semigroup_mult,t2_space}"
assumes "convergent_prod f" "\<And>i. f i \<noteq> 0"
shows "prodinf f = lim (\<lambda>n. \<Prod>i\<le>n. f i)"
using assms convergent_prod_offset_0 [OF assms]
by (simp add: prod_defs lim_def) (metis (no_types) assms(1) convergent_prod_to_zero_iff)

lemma has_prod_one[simp, intro]: "(\<lambda>n. 1) has_prod 1"
unfolding prod_defs by auto

lemma convergent_prod_one[simp, intro]: "convergent_prod (\<lambda>n. 1)"
unfolding prod_defs by auto

lemma prodinf_cong: "(\<And>n. f n = g n) \<Longrightarrow> prodinf f = prodinf g"
by presburger

lemma convergent_prod_cong:
fixes f g :: "nat \<Rightarrow> 'a::{field,topological_semigroup_mult,t2_space}"
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"
shows "convergent_prod f = convergent_prod g"
proof -
from assms obtain N where N: "\<forall>n\<ge>N. f n = g n"
by (auto simp: eventually_at_top_linorder)
define C where "C = (\<Prod>k<N. f k / g k)"
with g have "C \<noteq> 0"
have *: "eventually (\<lambda>n. prod f {..n} = C * prod g {..n}) sequentially"
using eventually_ge_at_top[of N]
proof eventually_elim
case (elim n)
then have "{..n} = {..<N} \<union> {N..n}"
by auto
also have "prod f \<dots> = prod f {..<N} * prod f {N..n}"
by (intro prod.union_disjoint) auto
also from N have "prod f {N..n} = prod g {N..n}"
by (intro prod.cong) simp_all
also have "prod f {..<N} * prod g {N..n} = C * (prod g {..<N} * prod g {N..n})"
unfolding C_def by (simp add: g prod_dividef)
also have "prod g {..<N} * prod g {N..n} = prod g ({..<N} \<union> {N..n})"
by (intro prod.union_disjoint [symmetric]) auto
also from elim have "{..<N} \<union> {N..n} = {..n}"
by auto
finally show "prod f {..n} = C * prod g {..n}" .
qed
then have cong: "convergent (\<lambda>n. prod f {..n}) = convergent (\<lambda>n. C * prod g {..n})"
by (rule convergent_cong)
show ?thesis
proof
assume cf: "convergent_prod f"
then have "\<not> (\<lambda>n. prod g {..n}) \<longlonglongrightarrow> 0"
using tendsto_mult_left * convergent_prod_to_zero_iff f filterlim_cong by fastforce
then show "convergent_prod g"
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)
next
assume cg: "convergent_prod g"
have "\<exists>a. C * a \<noteq> 0 \<and> (\<lambda>n. prod g {..n}) \<longlonglongrightarrow> a"
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)
then show "convergent_prod f"
using "*" tendsto_mult_left filterlim_cong
by (fastforce simp add: convergent_prod_iff_nz_lim f)
qed
qed

lemma has_prod_finite:
fixes f :: "nat \<Rightarrow> 'a::{semidom,t2_space}"
assumes [simp]: "finite N"
and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 1"
shows "f has_prod (\<Prod>n\<in>N. f n)"
proof -
have eq: "prod f {..n + Suc (Max N)} = prod f N" for n
proof (rule prod.mono_neutral_right)
show "N \<subseteq> {..n + Suc (Max N)}"
show "\<forall>i\<in>{..n + Suc (Max N)} - N. f i = 1"
using f by blast
qed auto
show ?thesis
proof (cases "\<forall>n\<in>N. f n \<noteq> 0")
case True
then have "prod f N \<noteq> 0"
by simp
moreover have "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> prod f N"
by (rule LIMSEQ_offset[of _ "Suc (Max N)"]) (simp add: eq atLeast0LessThan del: add_Suc_right)
ultimately show ?thesis
next
case False
then obtain k where "k \<in> N" "f k = 0"
by auto
let ?Z = "{n \<in> N. f n = 0}"
have maxge: "Max ?Z \<ge> n" if "f n = 0" for n
using Max_ge [of ?Z] \<open>finite N\<close> \<open>f n = 0\<close>
by (metis (mono_tags) Collect_mem_eq f finite_Collect_conjI mem_Collect_eq zero_neq_one)
let ?q = "prod f {Suc (Max ?Z)..Max N}"
have [simp]: "?q \<noteq> 0"
using maxge Suc_n_not_le_n le_trans by force
have eq: "(\<Prod>i\<le>n + Max N. f (Suc (i + Max ?Z))) = ?q" for n
proof -
have "(\<Prod>i\<le>n + Max N. f (Suc (i + Max ?Z))) = prod f {Suc (Max ?Z)..n + Max N + Suc (Max ?Z)}"
proof (rule prod.reindex_cong [where l = "\<lambda>i. i + Suc (Max ?Z)", THEN sym])
show "{Suc (Max ?Z)..n + Max N + Suc (Max ?Z)} = (\<lambda>i. i + Suc (Max ?Z)) ` {..n + Max N}"
using le_Suc_ex by fastforce
qed (auto simp: inj_on_def)
also have "\<dots> = ?q"
by (rule prod.mono_neutral_right)
(use Max.coboundedI [OF \<open>finite N\<close>] f in \<open>force+\<close>)
finally show ?thesis .
qed
have q: "raw_has_prod f (Suc (Max ?Z)) ?q"
show "(\<lambda>n. \<Prod>i\<le>n. f (Suc (i + Max ?Z))) \<longlonglongrightarrow> ?q"
by (rule LIMSEQ_offset[of _ "(Max N)"]) (simp add: eq)
qed
show ?thesis
unfolding has_prod_def
proof (intro disjI2 exI conjI)
show "prod f N = 0"
using \<open>f k = 0\<close> \<open>k \<in> N\<close> \<open>finite N\<close> prod_zero by blast
show "f (Max ?Z) = 0"
using Max_in [of ?Z] \<open>finite N\<close> \<open>f k = 0\<close> \<open>k \<in> N\<close> by auto
qed (use q in auto)
qed
qed

corollary\<^marker>\<open>tag unimportant\<close> has_prod_0:
fixes f :: "nat \<Rightarrow> 'a::{semidom,t2_space}"
assumes "\<And>n. f n = 1"
shows "f has_prod 1"

lemma prodinf_zero[simp]: "prodinf (\<lambda>n. 1::'a::real_normed_field) = 1"
using has_prod_unique by force

lemma convergent_prod_finite:
fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
assumes "finite N" "\<And>n. n \<notin> N \<Longrightarrow> f n = 1"
shows "convergent_prod f"
proof -
have "\<exists>n p. raw_has_prod f n p"
using assms has_prod_def has_prod_finite by blast
then show ?thesis
qed

lemma has_prod_If_finite_set:
fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
shows "finite A \<Longrightarrow> (\<lambda>r. if r \<in> A then f r else 1) has_prod (\<Prod>r\<in>A. f r)"
using has_prod_finite[of A "(\<lambda>r. if r \<in> A then f r else 1)"]
by simp

lemma has_prod_If_finite:
fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
shows "finite {r. P r} \<Longrightarrow> (\<lambda>r. if P r then f r else 1) has_prod (\<Prod>r | P r. f r)"
using has_prod_If_finite_set[of "{r. P r}"] by simp

lemma convergent_prod_If_finite_set[simp, intro]:
fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
shows "finite A \<Longrightarrow> convergent_prod (\<lambda>r. if r \<in> A then f r else 1)"

lemma convergent_prod_If_finite[simp, intro]:
fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
shows "finite {r. P r} \<Longrightarrow> convergent_prod (\<lambda>r. if P r then f r else 1)"
using convergent_prod_def has_prod_If_finite has_prod_def by fastforce

lemma has_prod_single:
fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
shows "(\<lambda>r. if r = i then f r else 1) has_prod f i"
using has_prod_If_finite[of "\<lambda>r. r = i"] by simp

context
fixes f :: "nat \<Rightarrow> 'a :: real_normed_field"
begin

lemma convergent_prod_imp_has_prod:
assumes "convergent_prod f"
shows "\<exists>p. f has_prod p"
proof -
obtain M p where p: "raw_has_prod f M p"
using assms convergent_prod_def by blast
then have "p \<noteq> 0"
using raw_has_prod_nonzero by blast
with p have fnz: "f i \<noteq> 0" if "i \<ge> M" for i
using raw_has_prod_eq_0 that by blast
define C where "C = (\<Prod>n<M. f n)"
show ?thesis
proof (cases "\<forall>n\<le>M. f n \<noteq> 0")
case True
then have "C \<noteq> 0"
then show ?thesis
by (meson True assms convergent_prod_offset_0 fnz has_prod_def nat_le_linear)
next
case False
let ?N = "GREATEST n. f n = 0"
have 0: "f ?N = 0"
using fnz False
by (metis (mono_tags, lifting) GreatestI_ex_nat nat_le_linear)
have "f i \<noteq> 0" if "i > ?N" for i
by (metis (mono_tags, lifting) Greatest_le_nat fnz leD linear that)
then have "\<exists>p. raw_has_prod f (Suc ?N) p"
using assms by (auto simp: intro!: convergent_prod_ignore_nonzero_segment)
then show ?thesis
unfolding has_prod_def using 0 by blast
qed
qed

lemma convergent_prod_has_prod [intro]:
shows "convergent_prod f \<Longrightarrow> f has_prod (prodinf f)"
unfolding prodinf_def
by (metis convergent_prod_imp_has_prod has_prod_unique theI')

lemma convergent_prod_LIMSEQ:
shows "convergent_prod f \<Longrightarrow> (\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> prodinf f"
by (metis convergent_LIMSEQ_iff convergent_prod_has_prod convergent_prod_imp_convergent
convergent_prod_to_zero_iff raw_has_prod_eq_0 has_prod_def prodinf_eq_lim zero_le)

theorem has_prod_iff: "f has_prod x \<longleftrightarrow> convergent_prod f \<and> prodinf f = x"
proof
assume "f has_prod x"
then show "convergent_prod f \<and> prodinf f = x"
apply safe
using convergent_prod_def has_prod_def apply blast
using has_prod_unique by blast
qed auto

lemma convergent_prod_has_prod_iff: "convergent_prod f \<longleftrightarrow> f has_prod prodinf f"
by (auto simp: has_prod_iff convergent_prod_has_prod)

lemma prodinf_finite:
assumes N: "finite N"
and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 1"
shows "prodinf f = (\<Prod>n\<in>N. f n)"
using has_prod_finite[OF assms, THEN has_prod_unique] by simp

end

subsection\<^marker>\<open>tag unimportant\<close> \<open>Infinite products on ordered topological monoids\<close>

lemma LIMSEQ_prod_0:
fixes f :: "nat \<Rightarrow> 'a::{semidom,topological_space}"
assumes "f i = 0"
shows "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> 0"
proof (subst tendsto_cong)
show "\<forall>\<^sub>F n in sequentially. prod f {..n} = 0"
proof
show "prod f {..n} = 0" if "n \<ge> i" for n
using that assms by auto
qed
qed auto

lemma LIMSEQ_prod_nonneg:
fixes f :: "nat \<Rightarrow> 'a::{linordered_semidom,linorder_topology}"
assumes 0: "\<And>n. 0 \<le> f n" and a: "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> a"
shows "a \<ge> 0"
by (simp add: "0" prod_nonneg LIMSEQ_le_const [OF a])

context
fixes f :: "nat \<Rightarrow> 'a::{linordered_semidom,linorder_topology}"
begin

lemma has_prod_le:
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"
shows "a \<le> b"
proof (cases "a=0 \<or> b=0")
case True
then show ?thesis
proof
assume [simp]: "a=0"
have "b \<ge> 0"
proof (rule LIMSEQ_prod_nonneg)
show "(\<lambda>n. prod g {..n}) \<longlonglongrightarrow> b"
using g by (auto simp: has_prod_def raw_has_prod_def LIMSEQ_prod_0)
qed (use le order_trans in auto)
then show ?thesis
by auto
next
assume [simp]: "b=0"
then obtain i where "g i = 0"
using g by (auto simp: prod_defs)
then have "f i = 0"
using antisym le by force
then have "a=0"
using f by (auto simp: prod_defs LIMSEQ_prod_0 LIMSEQ_unique)
then show ?thesis
by auto
qed
next
case False
then show ?thesis
using assms
unfolding has_prod_def raw_has_prod_def
by (force simp: LIMSEQ_prod_0 intro!: LIMSEQ_le prod_mono)
qed

lemma prodinf_le:
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"
shows "prodinf f \<le> prodinf g"
using has_prod_le [OF assms] has_prod_unique f g  by blast

end

lemma prod_le_prodinf:
fixes f :: "nat \<Rightarrow> 'a::{linordered_idom,linorder_topology}"
assumes "f has_prod a" "\<And>i. 0 \<le> f i" "\<And>i. i\<ge>n \<Longrightarrow> 1 \<le> f i"
shows "prod f {..<n} \<le> prodinf f"
by(rule has_prod_le[OF has_prod_If_finite_set]) (use assms has_prod_unique in auto)

lemma prodinf_nonneg:
fixes f :: "nat \<Rightarrow> 'a::{linordered_idom,linorder_topology}"
assumes "f has_prod a" "\<And>i. 1 \<le> f i"
shows "1 \<le> prodinf f"
using prod_le_prodinf[of f a 0] assms
by (metis order_trans prod_ge_1 zero_le_one)

lemma prodinf_le_const:
fixes f :: "nat \<Rightarrow> real"
assumes "convergent_prod f" "\<And>n. prod f {..<n} \<le> x"
shows "prodinf f \<le> x"
by (metis lessThan_Suc_atMost assms convergent_prod_LIMSEQ LIMSEQ_le_const2)

lemma prodinf_eq_one_iff:
fixes f :: "nat \<Rightarrow> real"
assumes f: "convergent_prod f" and ge1: "\<And>n. 1 \<le> f n"
shows "prodinf f = 1 \<longleftrightarrow> (\<forall>n. f n = 1)"
proof
assume "prodinf f = 1"
then have "(\<lambda>n. \<Prod>i<n. f i) \<longlonglongrightarrow> 1"
using convergent_prod_LIMSEQ[of f] assms by (simp add: LIMSEQ_lessThan_iff_atMost)
then have "\<And>i. (\<Prod>n\<in>{i}. f n) \<le> 1"
proof (rule LIMSEQ_le_const)
have "1 \<le> prod f n" for n
have "prod f {..<n} = 1" for n
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)
then have "(\<Prod>n\<in>{i}. f n) \<le> prod f {..<n}" if "n \<ge> Suc i" for i n
by (metis mult.left_neutral order_refl prod.cong prod.neutral_const prod.lessThan_Suc)
then show "\<exists>N. \<forall>n\<ge>N. (\<Prod>n\<in>{i}. f n) \<le> prod f {..<n}" for i
by blast
qed
with ge1 show "\<forall>n. f n = 1"
by (auto intro!: antisym)
qed (metis prodinf_zero fun_eq_iff)

lemma prodinf_pos_iff:
fixes f :: "nat \<Rightarrow> real"
assumes "convergent_prod f" "\<And>n. 1 \<le> f n"
shows "1 < prodinf f \<longleftrightarrow> (\<exists>i. 1 < f i)"
using prod_le_prodinf[of f 1] prodinf_eq_one_iff
by (metis convergent_prod_has_prod assms less_le prodinf_nonneg)

lemma less_1_prodinf2:
fixes f :: "nat \<Rightarrow> real"
assumes "convergent_prod f" "\<And>n. 1 \<le> f n" "1 < f i"
shows "1 < prodinf f"
proof -
have "1 < (\<Prod>n<Suc i. f n)"
using assms  by (intro less_1_prod2[where i=i]) auto
also have "\<dots> \<le> prodinf f"
by (intro prod_le_prodinf) (use assms order_trans zero_le_one in \<open>blast+\<close>)
finally show ?thesis .
qed

lemma less_1_prodinf:
fixes f :: "nat \<Rightarrow> real"
shows "\<lbrakk>convergent_prod f; \<And>n. 1 < f n\<rbrakk> \<Longrightarrow> 1 < prodinf f"
by (intro less_1_prodinf2[where i=1]) (auto intro: less_imp_le)

lemma prodinf_nonzero:
fixes f :: "nat \<Rightarrow> 'a :: {idom,topological_semigroup_mult,t2_space}"
assumes "convergent_prod f" "\<And>i. f i \<noteq> 0"
shows "prodinf f \<noteq> 0"
by (metis assms convergent_prod_offset_0 has_prod_unique raw_has_prod_def has_prod_def)

lemma less_0_prodinf:
fixes f :: "nat \<Rightarrow> real"
assumes f: "convergent_prod f" and 0: "\<And>i. f i > 0"
shows "0 < prodinf f"
proof -
have "prodinf f \<noteq> 0"
by (metis assms less_irrefl prodinf_nonzero)
moreover have "0 < (\<Prod>n<i. f n)" for i
then have "prodinf f \<ge> 0"
using convergent_prod_LIMSEQ [OF f] LIMSEQ_prod_nonneg 0 less_le by blast
ultimately show ?thesis
by auto
qed

lemma prod_less_prodinf2:
fixes f :: "nat \<Rightarrow> real"
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"
shows "prod f {..<n} < prodinf f"
proof -
have "prod f {..<n} \<le> prod f {..<i}"
by (rule prod_mono2) (use assms less_le in auto)
then have "prod f {..<n} < f i * prod f {..<i}"
using mult_less_le_imp_less[of 1 "f i" "prod f {..<n}" "prod f {..<i}"] assms
moreover have "prod f {..<Suc i} \<le> prodinf f"
using prod_le_prodinf[of f _ "Suc i"]
by (meson "0" "1" Suc_leD convergent_prod_has_prod f \<open>n \<le> i\<close> le_trans less_eq_real_def)
ultimately show ?thesis
by (metis le_less_trans mult.commute not_le prod.lessThan_Suc)
qed

lemma prod_less_prodinf:
fixes f :: "nat \<Rightarrow> real"
assumes f: "convergent_prod f" and 1: "\<And>m. m\<ge>n \<Longrightarrow> 1 < f m" and 0: "\<And>m. 0 < f m"
shows "prod f {..<n} < prodinf f"
by (meson "0" "1" f le_less prod_less_prodinf2)

lemma raw_has_prodI_bounded:
fixes f :: "nat \<Rightarrow> real"
assumes pos: "\<And>n. 1 \<le> f n"
and le: "\<And>n. (\<Prod>i<n. f i) \<le> x"
shows "\<exists>p. raw_has_prod f 0 p"
proof (rule exI LIMSEQ_incseq_SUP conjI)+
show "bdd_above (range (\<lambda>n. prod f {..n}))"
by (metis bdd_aboveI2 le lessThan_Suc_atMost)
then have "(SUP i. prod f {..i}) > 0"
by (metis UNIV_I cSUP_upper less_le_trans pos prod_pos zero_less_one)
then show "(SUP i. prod f {..i}) \<noteq> 0"
by auto
show "incseq (\<lambda>n. prod f {..n})"
using pos order_trans [OF zero_le_one] by (auto simp: mono_def intro!: prod_mono2)
qed

lemma convergent_prodI_nonneg_bounded:
fixes f :: "nat \<Rightarrow> real"
assumes "\<And>n. 1 \<le> f n" "\<And>n. (\<Prod>i<n. f i) \<le> x"
shows "convergent_prod f"
using convergent_prod_def raw_has_prodI_bounded [OF assms] by blast

subsection\<^marker>\<open>tag unimportant\<close> \<open>Infinite products on topological spaces\<close>

context
fixes f g :: "nat \<Rightarrow> 'a::{t2_space,topological_semigroup_mult,idom}"
begin

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)"
by (force simp add: prod.distrib tendsto_mult raw_has_prod_def)

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)"

end

context
fixes f g :: "nat \<Rightarrow> 'a::real_normed_field"
begin

lemma has_prod_mult:
assumes f: "f has_prod a" and g: "g has_prod b"
shows "(\<lambda>n. f n * g n) has_prod (a * b)"
using f [unfolded has_prod_def]
proof (elim disjE exE conjE)
assume f0: "raw_has_prod f 0 a"
show ?thesis
using g [unfolded has_prod_def]
proof (elim disjE exE conjE)
assume g0: "raw_has_prod g 0 b"
with f0 show ?thesis
by (force simp add: has_prod_def prod.distrib tendsto_mult raw_has_prod_def)
next
fix j q
assume "b = 0" and "g j = 0" and q: "raw_has_prod g (Suc j) q"
obtain p where p: "raw_has_prod f (Suc j) p"
using f0 raw_has_prod_ignore_initial_segment by blast
then have "Ex (raw_has_prod (\<lambda>n. f n * g n) (Suc j))"
using q raw_has_prod_mult by blast
then show ?thesis
using \<open>b = 0\<close> \<open>g j = 0\<close> has_prod_0_iff by fastforce
qed
next
fix i p
assume "a = 0" and "f i = 0" and p: "raw_has_prod f (Suc i) p"
show ?thesis
using g [unfolded has_prod_def]
proof (elim disjE exE conjE)
assume g0: "raw_has_prod g 0 b"
obtain q where q: "raw_has_prod g (Suc i) q"
using g0 raw_has_prod_ignore_initial_segment by blast
then have "Ex (raw_has_prod (\<lambda>n. f n * g n) (Suc i))"
using raw_has_prod_mult p by blast
then show ?thesis
using \<open>a = 0\<close> \<open>f i = 0\<close> has_prod_0_iff by fastforce
next
fix j q
assume "b = 0" and "g j = 0" and q: "raw_has_prod g (Suc j) q"
obtain p' where p': "raw_has_prod f (Suc (max i j)) p'"
by (metis raw_has_prod_ignore_initial_segment max_Suc_Suc max_def p)
moreover
obtain q' where q': "raw_has_prod g (Suc (max i j)) q'"
by (metis raw_has_prod_ignore_initial_segment max.cobounded2 max_Suc_Suc q)
ultimately show ?thesis
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)
qed
qed

lemma convergent_prod_mult:
assumes f: "convergent_prod f" and g: "convergent_prod g"
shows "convergent_prod (\<lambda>n. f n * g n)"
unfolding convergent_prod_def
proof -
obtain M p N q where p: "raw_has_prod f M p" and q: "raw_has_prod g N q"
using convergent_prod_def f g by blast+
then obtain p' q' where p': "raw_has_prod f (max M N) p'" and q': "raw_has_prod g (max M N) q'"
by (meson raw_has_prod_ignore_initial_segment max.cobounded1 max.cobounded2)
then show "\<exists>M p. raw_has_prod (\<lambda>n. f n * g n) M p"
using raw_has_prod_mult by blast
qed

lemma prodinf_mult: "convergent_prod f \<Longrightarrow> convergent_prod g \<Longrightarrow> prodinf f * prodinf g = (\<Prod>n. f n * g n)"
by (intro has_prod_unique has_prod_mult convergent_prod_has_prod)

end

context
fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::real_normed_field"
and I :: "'i set"
begin

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)"
by (induct I rule: infinite_finite_induct) (auto intro!: has_prod_mult)

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)"
using has_prod_unique[OF has_prod_prod, OF convergent_prod_has_prod] by simp

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)"
using convergent_prod_has_prod_iff has_prod_prod prodinf_prod by force

end

subsection\<^marker>\<open>tag unimportant\<close> \<open>Infinite summability on real normed fields\<close>

context
fixes f :: "nat \<Rightarrow> 'a::real_normed_field"
begin

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"
proof -
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"
by (subst filterlim_sequentially_Suc) (simp add: raw_has_prod_def)
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"
by (simp add: ac_simps atMost_Suc_eq_insert_0 image_Suc_atMost prod.atLeast1_atMost_eq lessThan_Suc_atMost
del: prod.cl_ivl_Suc)
also have "\<dots> \<longleftrightarrow> raw_has_prod (\<lambda>n. f (Suc n)) M a \<and> f M \<noteq> 0"
proof safe
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"
with tendsto_divide[OF tends tendsto_const, of "f M"]
show "raw_has_prod (\<lambda>n. f (Suc n)) M a"
qed (auto intro: tendsto_mult_right simp:  raw_has_prod_def)
finally show ?thesis .
qed

lemma has_prod_Suc_iff:
assumes "f 0 \<noteq> 0" shows "(\<lambda>n. f (Suc n)) has_prod a \<longleftrightarrow> f has_prod (a * f 0)"
proof (cases "a = 0")
case True
then show ?thesis
fix i x
assume "f (Suc i) = 0" and "raw_has_prod (\<lambda>n. f (Suc n)) (Suc i) x"
then obtain y where "raw_has_prod f (Suc (Suc i)) y"
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)
then show "\<exists>i. f i = 0 \<and> Ex (raw_has_prod f (Suc i))"
using \<open>f (Suc i) = 0\<close> by blast
next
fix i x
assume "f i = 0" and x: "raw_has_prod f (Suc i) x"
then obtain j where j: "i = Suc j"
by (metis assms not0_implies_Suc)
moreover have "\<exists> y. raw_has_prod (\<lambda>n. f (Suc n)) i y"
using x by (auto simp: raw_has_prod_def)
then show "\<exists>i. f (Suc i) = 0 \<and> Ex (raw_has_prod (\<lambda>n. f (Suc n)) (Suc i))"
using \<open>f i = 0\<close> j by blast
qed
next
case False
then show ?thesis
by (auto simp: has_prod_def raw_has_prod_Suc_iff assms)
qed

lemma convergent_prod_Suc_iff:
shows "convergent_prod (\<lambda>n. f (Suc n)) = convergent_prod f"
proof
assume "convergent_prod f"
then obtain M L where M_nz:"\<forall>n\<ge>M. f n \<noteq> 0" and
M_L:"(\<lambda>n. \<Prod>i\<le>n. f (i + M)) \<longlonglongrightarrow> L" and "L \<noteq> 0"
unfolding convergent_prod_altdef by auto
have "(\<lambda>n. \<Prod>i\<le>n. f (Suc (i + M))) \<longlonglongrightarrow> L / f M"
proof -
have "(\<lambda>n. \<Prod>i\<in>{0..Suc n}. f (i + M)) \<longlonglongrightarrow> L"
using M_L
apply (subst (asm) filterlim_sequentially_Suc[symmetric])
using atLeast0AtMost by auto
then have "(\<lambda>n. f M * (\<Prod>i\<in>{0..n}. f (Suc (i + M)))) \<longlonglongrightarrow> L"
apply (subst (asm) prod.atLeast0_atMost_Suc_shift)
by simp
then have "(\<lambda>n. (\<Prod>i\<in>{0..n}. f (Suc (i + M)))) \<longlonglongrightarrow> L/f M"
apply (drule_tac tendsto_divide)
using M_nz[rule_format,of M,simplified] by auto
then show ?thesis unfolding atLeast0AtMost .
qed
then show "convergent_prod (\<lambda>n. f (Suc n))" unfolding convergent_prod_altdef
apply (rule_tac exI[where x=M])
apply (rule_tac exI[where x="L/f M"])
using M_nz \<open>L\<noteq>0\<close> by auto
next
assume "convergent_prod (\<lambda>n. f (Suc n))"
then obtain M where "\<exists>L. (\<forall>n\<ge>M. f (Suc n) \<noteq> 0) \<and> (\<lambda>n. \<Prod>i\<le>n. f (Suc (i + M))) \<longlonglongrightarrow> L \<and> L \<noteq> 0"
unfolding convergent_prod_altdef by auto
then show "convergent_prod f" unfolding convergent_prod_altdef
apply (rule_tac exI[where x="Suc M"])
using Suc_le_D by auto
qed

lemma raw_has_prod_inverse:
assumes "raw_has_prod f M a" shows "raw_has_prod (\<lambda>n. inverse (f n)) M (inverse a)"
using assms unfolding raw_has_prod_def by (auto dest: tendsto_inverse simp: prod_inversef [symmetric])

lemma has_prod_inverse:
assumes "f has_prod a" shows "(\<lambda>n. inverse (f n)) has_prod (inverse a)"
using assms raw_has_prod_inverse unfolding has_prod_def by auto

lemma convergent_prod_inverse:
assumes "convergent_prod f"
shows "convergent_prod (\<lambda>n. inverse (f n))"
using assms unfolding convergent_prod_def  by (blast intro: raw_has_prod_inverse elim: )

end

context
fixes f :: "nat \<Rightarrow> 'a::real_normed_field"
begin

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"

lemma has_prod_divide: "f has_prod a \<Longrightarrow> g has_prod b \<Longrightarrow> (\<lambda>n. f n / g n) has_prod (a / b)"
unfolding divide_inverse by (intro has_prod_inverse has_prod_mult)

lemma convergent_prod_divide:
assumes f: "convergent_prod f" and g: "convergent_prod g"
shows "convergent_prod (\<lambda>n. f n / g n)"
using f g has_prod_divide has_prod_iff by blast

lemma prodinf_divide: "convergent_prod f \<Longrightarrow> convergent_prod g \<Longrightarrow> prodinf f / prodinf g = (\<Prod>n. f n / g n)"
by (intro has_prod_unique has_prod_divide convergent_prod_has_prod)

lemma prodinf_inverse: "convergent_prod f \<Longrightarrow> (\<Prod>n. inverse (f n)) = inverse (\<Prod>n. f n)"
by (intro has_prod_unique [symmetric] has_prod_inverse convergent_prod_has_prod)

lemma has_prod_Suc_imp:
assumes "(\<lambda>n. f (Suc n)) has_prod a"
shows "f has_prod (a * f 0)"
proof -
have "f has_prod (a * f 0)" when "raw_has_prod (\<lambda>n. f (Suc n)) 0 a"
apply (cases "f 0=0")
using that unfolding has_prod_def raw_has_prod_Suc
moreover have "f has_prod (a * f 0)" when
"(\<exists>i q. a = 0 \<and> f (Suc i) = 0 \<and> raw_has_prod (\<lambda>n. f (Suc n)) (Suc i) q)"
proof -
from that
obtain i q where "a = 0" "f (Suc i) = 0" "raw_has_prod (\<lambda>n. f (Suc n)) (Suc i) q"
by auto
then show ?thesis unfolding has_prod_def
by (auto intro!:exI[where x="Suc i"] simp:raw_has_prod_Suc)
qed
ultimately show "f has_prod (a * f 0)" using assms unfolding has_prod_def by auto
qed

lemma has_prod_iff_shift:
assumes "\<And>i. i < n \<Longrightarrow> f i \<noteq> 0"
shows "(\<lambda>i. f (i + n)) has_prod a \<longleftrightarrow> f has_prod (a * (\<Prod>i<n. f i))"
using assms
proof (induct n arbitrary: a)
case 0
then show ?case by simp
next
case (Suc n)
then have "(\<lambda>i. f (Suc i + n)) has_prod a \<longleftrightarrow> (\<lambda>i. f (i + n)) has_prod (a * f n)"
by (subst has_prod_Suc_iff) auto
with Suc show ?case
qed

corollary\<^marker>\<open>tag unimportant\<close> has_prod_iff_shift':
assumes "\<And>i. i < n \<Longrightarrow> f i \<noteq> 0"
shows "(\<lambda>i. f (i + n)) has_prod (a / (\<Prod>i<n. f i)) \<longleftrightarrow> f has_prod a"

lemma has_prod_one_iff_shift:
assumes "\<And>i. i < n \<Longrightarrow> f i = 1"
shows "(\<lambda>i. f (i+n)) has_prod a \<longleftrightarrow> (\<lambda>i. f i) has_prod a"

lemma convergent_prod_iff_shift [simp]:
shows "convergent_prod (\<lambda>i. f (i + n)) \<longleftrightarrow> convergent_prod f"
apply safe
using convergent_prod_offset apply blast
using convergent_prod_ignore_initial_segment convergent_prod_def by blast

lemma has_prod_split_initial_segment:
assumes "f has_prod a" "\<And>i. i < n \<Longrightarrow> f i \<noteq> 0"
shows "(\<lambda>i. f (i + n)) has_prod (a / (\<Prod>i<n. f i))"
using assms has_prod_iff_shift' by blast

lemma prodinf_divide_initial_segment:
assumes "convergent_prod f" "\<And>i. i < n \<Longrightarrow> f i \<noteq> 0"
shows "(\<Prod>i. f (i + n)) = (\<Prod>i. f i) / (\<Prod>i<n. f i)"
by (rule has_prod_unique[symmetric]) (auto simp: assms has_prod_iff_shift)

lemma prodinf_split_initial_segment:
assumes "convergent_prod f" "\<And>i. i < n \<Longrightarrow> f i \<noteq> 0"
shows "prodinf f = (\<Prod>i. f (i + n)) * (\<Prod>i<n. f i)"
by (auto simp add: assms prodinf_divide_initial_segment)

assumes "convergent_prod f" "f 0 \<noteq> 0"
shows "(\<Prod>n. f (Suc n)) = prodinf f / f 0"
using prodinf_split_initial_segment[of 1] assms by simp

end

context
fixes f :: "nat \<Rightarrow> 'a::real_normed_field"
begin

lemma convergent_prod_inverse_iff: "convergent_prod (\<lambda>n. inverse (f n)) \<longleftrightarrow> convergent_prod f"
by (auto dest: convergent_prod_inverse)

lemma convergent_prod_const_iff:
fixes c :: "'a :: {real_normed_field}"
shows "convergent_prod (\<lambda>_. c) \<longleftrightarrow> c = 1"
proof
assume "convergent_prod (\<lambda>_. c)"
then show "c = 1"
using convergent_prod_imp_LIMSEQ LIMSEQ_unique by blast
next
assume "c = 1"
then show "convergent_prod (\<lambda>_. c)"
by auto
qed

lemma has_prod_power: "f has_prod a \<Longrightarrow> (\<lambda>i. f i ^ n) has_prod (a ^ n)"
by (induction n) (auto simp: has_prod_mult)

lemma convergent_prod_power: "convergent_prod f \<Longrightarrow> convergent_prod (\<lambda>i. f i ^ n)"
by (induction n) (auto simp: convergent_prod_mult)

lemma prodinf_power: "convergent_prod f \<Longrightarrow> prodinf (\<lambda>i. f i ^ n) = prodinf f ^ n"
by (metis has_prod_unique convergent_prod_imp_has_prod has_prod_power)

end

subsection\<open>Exponentials and logarithms\<close>

context
fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
begin

lemma sums_imp_has_prod_exp:
assumes "f sums s"
shows "raw_has_prod (\<lambda>i. exp (f i)) 0 (exp s)"
using assms continuous_on_exp [of UNIV "\<lambda>x::'a. x"]
using continuous_on_tendsto_compose [of UNIV exp "(\<lambda>n. sum f {..n})" s]
by (simp add: prod_defs sums_def_le exp_sum)

lemma convergent_prod_exp:
assumes "summable f"
shows "convergent_prod (\<lambda>i. exp (f i))"
using sums_imp_has_prod_exp assms unfolding summable_def convergent_prod_def  by blast

lemma prodinf_exp:
assumes "summable f"
shows "prodinf (\<lambda>i. exp (f i)) = exp (suminf f)"
proof -
have "f sums suminf f"
using assms by blast
then have "(\<lambda>i. exp (f i)) has_prod exp (suminf f)"
then show ?thesis
by (rule has_prod_unique [symmetric])
qed

end

theorem convergent_prod_iff_summable_real:
fixes a :: "nat \<Rightarrow> real"
assumes "\<And>n. a n > 0"
shows "convergent_prod (\<lambda>k. 1 + a k) \<longleftrightarrow> summable a" (is "?lhs = ?rhs")
proof
assume ?lhs
then obtain p where "raw_has_prod (\<lambda>k. 1 + a k) 0 p"
by (metis assms add_less_same_cancel2 convergent_prod_offset_0 not_one_less_zero)
then have to_p: "(\<lambda>n. \<Prod>k\<le>n. 1 + a k) \<longlonglongrightarrow> p"
by (auto simp: raw_has_prod_def)
moreover have le: "(\<Sum>k\<le>n. a k) \<le> (\<Prod>k\<le>n. 1 + a k)" for n
by (rule sum_le_prod) (use assms less_le in force)
have "(\<Prod>k\<le>n. 1 + a k) \<le> p" for n
proof (rule incseq_le [OF _ to_p])
show "incseq (\<lambda>n. \<Prod>k\<le>n. 1 + a k)"
using assms by (auto simp: mono_def order.strict_implies_order intro!: prod_mono2)
qed
with le have "(\<Sum>k\<le>n. a k) \<le> p" for n
by (metis order_trans)
with assms bounded_imp_summable show ?rhs
by (metis not_less order.asym)
next
assume R: ?rhs
have "(\<Prod>k\<le>n. 1 + a k) \<le> exp (suminf a)" for n
proof -
have "(\<Prod>k\<le>n. 1 + a k) \<le> exp (\<Sum>k\<le>n. a k)" for n
by (rule prod_le_exp_sum) (use assms less_le in force)
moreover have "exp (\<Sum>k\<le>n. a k) \<le> exp (suminf a)" for n
unfolding exp_le_cancel_iff
by (meson sum_le_suminf R assms finite_atMost less_eq_real_def)
ultimately show ?thesis
by (meson order_trans)
qed
then obtain L where L: "(\<lambda>n. \<Prod>k\<le>n. 1 + a k) \<longlonglongrightarrow> L"
moreover have "L \<noteq> 0"
proof
assume "L = 0"
with L have "(\<lambda>n. \<Prod>k\<le>n. 1 + a k) \<longlonglongrightarrow> 0"
by simp
moreover have "(\<Prod>k\<le>n. 1 + a k) > 1" for n
ultimately show False
by (meson Lim_bounded2 not_one_le_zero less_imp_le)
qed
ultimately show ?lhs
using assms convergent_prod_iff_nz_lim
by (metis add_less_same_cancel1 less_le not_le zero_less_one)
qed

lemma exp_suminf_prodinf_real:
fixes f :: "nat \<Rightarrow> real"
assumes ge0:"\<And>n. f n \<ge> 0" and ac: "abs_convergent_prod (\<lambda>n. exp (f n))"
shows "prodinf (\<lambda>i. exp (f i)) = exp (suminf f)"
proof -
have "summable f"
using ac unfolding abs_convergent_prod_conv_summable
proof (elim summable_comparison_test')
fix n
have "\<bar>f n\<bar> = f n"
also have "\<dots> \<le> exp (f n) - 1"
finally show "norm (f n) \<le> norm (exp (f n) - 1)"
by simp
qed
then show ?thesis
qed

lemma has_prod_imp_sums_ln_real:
fixes f :: "nat \<Rightarrow> real"
assumes "raw_has_prod f 0 p" and 0: "\<And>x. f x > 0"
shows "(\<lambda>i. ln (f i)) sums (ln p)"
proof -
have "p > 0"
using assms unfolding prod_defs by (metis LIMSEQ_prod_nonneg less_eq_real_def)
then show ?thesis
using assms continuous_on_ln [of "{0<..}" "\<lambda>x. x"]
using continuous_on_tendsto_compose [of "{0<..}" ln "(\<lambda>n. prod f {..n})" p]
by (auto simp: prod_defs sums_def_le ln_prod order_tendstoD)
qed

lemma summable_ln_real:
fixes f :: "nat \<Rightarrow> real"
assumes f: "convergent_prod f" and 0: "\<And>x. f x > 0"
shows "summable (\<lambda>i. ln (f i))"
proof -
obtain M p where "raw_has_prod f M p"
using f convergent_prod_def by blast
then consider i where "i<M" "f i = 0" | p where "raw_has_prod f 0 p"
using raw_has_prod_cases by blast
then show ?thesis
proof cases
case 1
with 0 show ?thesis
by (metis less_irrefl)
next
case 2
then show ?thesis
using "0" has_prod_imp_sums_ln_real summable_def by blast
qed
qed

lemma suminf_ln_real:
fixes f :: "nat \<Rightarrow> real"
assumes f: "convergent_prod f" and 0: "\<And>x. f x > 0"
shows "suminf (\<lambda>i. ln (f i)) = ln (prodinf f)"
proof -
have "f has_prod prodinf f"
then have "raw_has_prod f 0 (prodinf f)"
by (metis "0" has_prod_def less_irrefl)
then have "(\<lambda>i. ln (f i)) sums ln (prodinf f)"
using "0" has_prod_imp_sums_ln_real by blast
then show ?thesis
by (rule sums_unique [symmetric])
qed

lemma prodinf_exp_real:
fixes f :: "nat \<Rightarrow> real"
assumes f: "convergent_prod f" and 0: "\<And>x. f x > 0"
shows "prodinf f = exp (suminf (\<lambda>i. ln (f i)))"
by (simp add: "0" f less_0_prodinf suminf_ln_real)

theorem Ln_prodinf_complex:
fixes z :: "nat \<Rightarrow> complex"
assumes z: "\<And>j. z j \<noteq> 0" and \<xi>: "\<xi> \<noteq> 0"
shows "((\<lambda>n. \<Prod>j\<le>n. z j) \<longlonglongrightarrow> \<xi>) \<longleftrightarrow> (\<exists>k. (\<lambda>n. (\<Sum>j\<le>n. Ln (z j))) \<longlonglongrightarrow> Ln \<xi> + of_int k * (of_real(2*pi) * \<i>))" (is "?lhs = ?rhs")
proof
assume L: ?lhs
have pnz: "(\<Prod>j\<le>n. z j) \<noteq> 0" for n
using z by auto
define \<Theta> where "\<Theta> \<equiv> Arg \<xi> + 2*pi"
then have "\<Theta> > pi"
using Arg_def mpi_less_Im_Ln by fastforce
have \<xi>_eq: "\<xi> = cmod \<xi> * exp (\<i> * \<Theta>)"
define \<theta> where "\<theta> \<equiv> \<lambda>n. THE t. is_Arg (\<Prod>j\<le>n. z j) t \<and> t \<in> {\<Theta>-pi<..\<Theta>+pi}"
have uniq: "\<exists>!s. is_Arg (\<Prod>j\<le>n. z j) s \<and> s \<in> {\<Theta>-pi<..\<Theta>+pi}" for n
using Argument_exists_unique [OF pnz] by metis
have \<theta>: "is_Arg (\<Prod>j\<le>n. z j) (\<theta> n)" and \<theta>_interval: "\<theta> n \<in> {\<Theta>-pi<..\<Theta>+pi}" for n
unfolding \<theta>_def
using theI' [OF uniq] by metis+
have \<theta>_pos: "\<And>j. \<theta> j > 0"
using \<theta>_interval \<open>\<Theta> > pi\<close> by simp (meson diff_gt_0_iff_gt less_trans)
have "(\<Prod>j\<le>n. z j) = cmod (\<Prod>j\<le>n. z j) * exp (\<i> * \<theta> n)" for n
using \<theta> by (auto simp: is_Arg_def)
then have eq: "(\<lambda>n. \<Prod>j\<le>n. z j) = (\<lambda>n. cmod (\<Prod>j\<le>n. z j) * exp (\<i> * \<theta> n))"
by simp
then have "(\<lambda>n. (cmod (\<Prod>j\<le>n. z j)) * exp (\<i> * (\<theta> n))) \<longlonglongrightarrow> \<xi>"
using L by force
then obtain k where k: "(\<lambda>j. \<theta> j - of_int (k j) * (2 * pi)) \<longlonglongrightarrow> \<Theta>"
using L by (subst (asm) \<xi>_eq) (auto simp add: eq z \<xi> polar_convergence)
moreover have "\<forall>\<^sub>F n in sequentially. k n = 0"
proof -
have *: "kj = 0" if "dist (vj - real_of_int kj * 2) V < 1" "vj \<in> {V - 1<..V + 1}" for kj vj V
using that  by (auto simp: dist_norm)
have "\<forall>\<^sub>F j in sequentially. dist (\<theta> j - of_int (k j) * (2 * pi)) \<Theta> < pi"
using tendstoD [OF k] pi_gt_zero by blast
then show ?thesis
proof (rule eventually_mono)
fix j
assume d: "dist (\<theta> j - real_of_int (k j) * (2 * pi)) \<Theta> < pi"
show "k j = 0"
by (rule * [of "\<theta> j/pi" _ "\<Theta>/pi"])
(use \<theta>_interval [of j] d in \<open>simp_all add: divide_simps dist_norm\<close>)
qed
qed
ultimately have \<theta>to\<Theta>: "\<theta> \<longlonglongrightarrow> \<Theta>"
apply (simp only: tendsto_def)
apply (erule all_forward imp_forward asm_rl)+
apply (drule (1) eventually_conj)
apply (auto elim: eventually_mono)
done
then have to0: "(\<lambda>n. \<bar>\<theta> (Suc n) - \<theta> n\<bar>) \<longlonglongrightarrow> 0"
by (metis (full_types) diff_self filterlim_sequentially_Suc tendsto_diff tendsto_rabs_zero)
have "\<exists>k. Im (\<Sum>j\<le>n. Ln (z j)) - of_int k * (2*pi) = \<theta> n" for n
proof (rule is_Arg_exp_diff_2pi)
show "is_Arg (exp (\<Sum>j\<le>n. Ln (z j))) (\<theta> n)"
using pnz \<theta> by (simp add: is_Arg_def exp_sum prod_norm)
qed
then have "\<exists>k. (\<Sum>j\<le>n. Im (Ln (z j))) = \<theta> n + of_int k * (2*pi)" for n
then obtain k where k: "\<And>n. (\<Sum>j\<le>n. Im (Ln (z j))) = \<theta> n + of_int (k n) * (2*pi)"
by metis
obtain K where "\<forall>\<^sub>F n in sequentially. k n = K"
proof -
have k_le: "(2*pi) * \<bar>k (Suc n) - k n\<bar> \<le> \<bar>\<theta> (Suc n) - \<theta> n\<bar> + \<bar>Im (Ln (z (Suc n)))\<bar>" for n
proof -
have "(\<Sum>j\<le>Suc n. Im (Ln (z j))) - (\<Sum>j\<le>n. Im (Ln (z j))) = Im (Ln (z (Suc n)))"
by simp
then show ?thesis
using k [of "Suc n"] k [of n] by (auto simp: abs_if algebra_simps)
qed
have "z \<longlonglongrightarrow> 1"
using L \<xi> convergent_prod_iff_nz_lim z by (blast intro: convergent_prod_imp_LIMSEQ)
with z have "(\<lambda>n. Ln (z n)) \<longlonglongrightarrow> Ln 1"
using isCont_tendsto_compose [OF continuous_at_Ln] nonpos_Reals_one_I by blast
then have "(\<lambda>n. Ln (z n)) \<longlonglongrightarrow> 0"
by simp
then have "(\<lambda>n. \<bar>Im (Ln (z (Suc n)))\<bar>) \<longlonglongrightarrow> 0"
by (metis LIMSEQ_unique \<open>z \<longlonglongrightarrow> 1\<close> continuous_at_Ln filterlim_sequentially_Suc isCont_tendsto_compose nonpos_Reals_one_I tendsto_Im tendsto_rabs_zero_iff zero_complex.simps(2))
then have "\<forall>\<^sub>F n in sequentially. \<bar>Im (Ln (z (Suc n)))\<bar> < 1"
moreover have "\<forall>\<^sub>F n in sequentially. \<bar>\<theta> (Suc n) - \<theta> n\<bar> < 1"
using to0 by (simp add: order_tendsto_iff)
ultimately have "\<forall>\<^sub>F n in sequentially. (2*pi) * \<bar>k (Suc n) - k n\<bar> < 1 + 1"
proof (rule eventually_elim2)
fix n
assume "\<bar>Im (Ln (z (Suc n)))\<bar> < 1" and "\<bar>\<theta> (Suc n) - \<theta> n\<bar> < 1"
with k_le [of n] show "2 * pi * real_of_int \<bar>k (Suc n) - k n\<bar> < 1 + 1"
by linarith
qed
then have "\<forall>\<^sub>F n in sequentially. real_of_int\<bar>k (Suc n) - k n\<bar> < 1"
proof (rule eventually_mono)
fix n :: "nat"
assume "2 * pi * \<bar>k (Suc n) - k n\<bar> < 1 + 1"
then have "\<bar>k (Suc n) - k n\<bar> < 2 / (2*pi)"
also have "... < 1"
using pi_ge_two by auto
finally show "real_of_int \<bar>k (Suc n) - k n\<bar> < 1" .
qed
then obtain N where N: "\<And>n. n\<ge>N \<Longrightarrow> \<bar>k (Suc n) - k n\<bar> = 0"
using eventually_sequentially less_irrefl of_int_abs by fastforce
have "k (N+i) = k N" for i
proof (induction i)
case (Suc i)
with N [of "N+i"] show ?case
by auto
qed simp
then have "\<And>n. n\<ge>N \<Longrightarrow> k n = k N"
using le_Suc_ex by auto
then show ?thesis
by (force simp add: eventually_sequentially intro: that)
qed
with \<theta>to\<Theta> have "(\<lambda>n. (\<Sum>j\<le>n. Im (Ln (z j)))) \<longlonglongrightarrow> \<Theta> + of_int K * (2*pi)"
moreover have "(\<lambda>n. (\<Sum>k\<le>n. Re (Ln (z k)))) \<longlonglongrightarrow> Re (Ln \<xi>)"
using assms continuous_imp_tendsto [OF isCont_ln tendsto_norm [OF L]]
by (simp add: o_def flip: prod_norm ln_prod)
ultimately show ?rhs
by (rule_tac x="K+1" in exI) (auto simp: tendsto_complex_iff \<Theta>_def Arg_def assms algebra_simps)
next
assume ?rhs
then obtain r where r: "(\<lambda>n. (\<Sum>k\<le>n. Ln (z k))) \<longlonglongrightarrow> Ln \<xi> + of_int r * (of_real(2*pi) * \<i>)" ..
have "(\<lambda>n. exp (\<Sum>k\<le>n. Ln (z k))) \<longlonglongrightarrow> \<xi>"
using assms continuous_imp_tendsto [OF isCont_exp r] exp_integer_2pi [of r]
moreover have "exp (\<Sum>k\<le>n. Ln (z k)) = (\<Prod>k\<le>n. z k)" for n
ultimately show ?lhs
by auto
qed

text\<open>Prop 17.2 of Bak and Newman, Complex Analysis, p.242\<close>
proposition convergent_prod_iff_summable_complex:
fixes z :: "nat \<Rightarrow> complex"
assumes "\<And>k. z k \<noteq> 0"
shows "convergent_prod (\<lambda>k. z k) \<longleftrightarrow> summable (\<lambda>k. Ln (z k))" (is "?lhs = ?rhs")
proof
assume ?lhs
then obtain p where p: "(\<lambda>n. \<Prod>k\<le>n. z k) \<longlonglongrightarrow> p" and "p \<noteq> 0"
using convergent_prod_LIMSEQ prodinf_nonzero add_eq_0_iff assms by fastforce
then show ?rhs
using Ln_prodinf_complex assms
by (auto simp: prodinf_nonzero summable_def sums_def_le)
next
assume R: ?rhs
have "(\<Prod>k\<le>n. z k) = exp (\<Sum>k\<le>n. Ln (z k))" for n
then have "(\<lambda>n. \<Prod>k\<le>n. z k) \<longlonglongrightarrow> exp (suminf (\<lambda>k. Ln (z k)))"
using continuous_imp_tendsto [OF isCont_exp summable_LIMSEQ' [OF R]] by (simp add: o_def)
then show ?lhs
by (subst convergent_prod_iff_convergent) (auto simp: convergent_def tendsto_Lim assms add_eq_0_iff)
qed

text\<open>Prop 17.3 of Bak and Newman, Complex Analysis\<close>
proposition summable_imp_convergent_prod_complex:
fixes z :: "nat \<Rightarrow> complex"
assumes z: "summable (\<lambda>k. norm (z k))" and non0: "\<And>k. z k \<noteq> -1"
shows "convergent_prod (\<lambda>k. 1 + z k)"
proof -
note if_cong [cong] power_Suc [simp del]
obtain N where N: "\<And>k. k\<ge>N \<Longrightarrow> norm (z k) < 1/2"
using summable_LIMSEQ_zero [OF z]
by (metis diff_zero dist_norm half_gt_zero_iff less_numeral_extra(1) lim_sequentially tendsto_norm_zero_iff)
have "norm (Ln (1 + z k)) \<le> 2 * norm (z k)" if "k \<ge> N" for k
proof (cases "z k = 0")
case False
let ?f = "\<lambda>i. cmod ((- 1) ^ i * z k ^ i / of_nat (Suc i))"
have normf: "norm (?f n) \<le> (1 / 2) ^ n" for n
proof -
have "norm (?f n) = cmod (z k) ^ n / cmod (1 + of_nat n)"
by (auto simp: norm_divide norm_mult norm_power)
also have "\<dots> \<le> cmod (z k) ^ n"
by (auto simp: field_split_simps mult_le_cancel_left1 in_Reals_norm)
also have "\<dots> \<le> (1 / 2) ^ n"
using N [OF that] by (simp add: power_mono)
finally show "norm (?f n) \<le> (1 / 2) ^ n" .
qed
have summablef: "summable ?f"
by (intro normf summable_comparison_test' [OF summable_geometric [of "1/2"]]) auto
have "(\<lambda>n. (- 1) ^ Suc n / of_nat n * z k ^ n) sums Ln (1 + z k)"
using Ln_series [of "z k"] N that by fastforce
then have *: "(\<lambda>i. z k * (((- 1) ^ i * z k ^ i) / (Suc i))) sums Ln (1 + z k)"
using sums_split_initial_segment [where n= 1]  by (force simp: power_Suc mult_ac)
then have "norm (Ln (1 + z k)) = norm (suminf (\<lambda>i. z k * (((- 1) ^ i * z k ^ i) / (Suc i))))"
using sums_unique by force
also have "\<dots> = norm (z k * suminf (\<lambda>i. ((- 1) ^ i * z k ^ i) / (Suc i)))"
apply (subst suminf_mult)
using * False
by (auto simp: sums_summable intro: summable_mult_D [of "z k"])
also have "\<dots> = norm (z k) * norm (suminf (\<lambda>i. ((- 1) ^ i * z k ^ i) / (Suc i)))"
also have "\<dots> \<le> norm (z k) * suminf (\<lambda>i. norm (((- 1) ^ i * z k ^ i) / (Suc i)))"
by (intro mult_left_mono summable_norm summablef) auto
also have "\<dots> \<le> norm (z k) * suminf (\<lambda>i. (1/2) ^ i)"
by (intro mult_left_mono suminf_le) (use summable_geometric [of "1/2"] summablef normf in auto)
also have "\<dots> \<le> norm (z k) * 2"
using suminf_geometric [of "1/2::real"] by simp
finally show ?thesis
qed simp
then have "summable (\<lambda>k. Ln (1 + z k))"
by (metis summable_comparison_test summable_mult z)
with non0 show ?thesis
qed

lemma summable_Ln_complex:
fixes z :: "nat \<Rightarrow> complex"
assumes "convergent_prod z" "\<And>k. z k \<noteq> 0"
shows "summable (\<lambda>k. Ln (z k))"
using convergent_prod_def assms convergent_prod_iff_summable_complex by blast

subsection\<^marker>\<open>tag unimportant\<close> \<open>Embeddings from the reals into some complete real normed field\<close>

lemma tendsto_eq_of_real_lim:
assumes "(\<lambda>n. of_real (f n) :: 'a::{complete_space,real_normed_field}) \<longlonglongrightarrow> q"
shows "q = of_real (lim f)"
proof -
have "convergent (\<lambda>n. of_real (f n) :: 'a)"
using assms convergent_def by blast
then have "convergent f"
unfolding convergent_def
then show ?thesis
by (metis LIMSEQ_unique assms convergentD sequentially_bot tendsto_Lim tendsto_of_real)
qed

lemma tendsto_eq_of_real:
assumes "(\<lambda>n. of_real (f n) :: 'a::{complete_space,real_normed_field}) \<longlonglongrightarrow> q"
obtains r where "q = of_real r"
using tendsto_eq_of_real_lim assms by blast

lemma has_prod_of_real_iff:
"(\<lambda>n. of_real (f n) :: 'a::{complete_space,real_normed_field}) has_prod of_real c \<longleftrightarrow> f has_prod c"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
apply (auto simp: prod_defs LIMSEQ_prod_0 tendsto_of_real_iff simp flip: of_real_prod)
using tendsto_eq_of_real
by (metis of_real_0 tendsto_of_real_iff)
next
assume ?rhs
with tendsto_of_real_iff show ?lhs
by (fastforce simp: prod_defs simp flip: of_real_prod)
qed

end