src/HOL/Analysis/Infinite_Products.thy

some material on power series and infinite products

(*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
  with insert.hyps show ?case by (simp add: algebra_simps exp_add)
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 \<open>has'_prod\<close> 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 \<open>\<Prod>\<close> 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"
  by (simp add: raw_has_prod_def)

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))"
  by (simp add: has_prod_def)
      
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 has_prod_eq_0_iff:
  fixes f :: "nat \<Rightarrow> 'a :: {semidom, comm_semiring_1, t2_space}"
  assumes "f has_prod P"
  shows   "P = 0 \<longleftrightarrow> 0 \<in> range f"
proof
  assume "0 \<in> range f"
  then obtain N where N: "f N = 0"
    by auto
  have "eventually (\<lambda>n. n > N) at_top"
    by (rule eventually_gt_at_top)
  hence "eventually (\<lambda>n. (\<Prod>k<n. f k) = 0) at_top"
    by eventually_elim (use N in auto)
  hence "(\<lambda>n. \<Prod>k<n. f k) \<longlonglongrightarrow> 0"
    by (simp add: tendsto_eventually)
  moreover have "(\<lambda>n. \<Prod>k<n. f k) \<longlonglongrightarrow> P"
    using assms by (metis N calculation prod_defs(2) raw_has_prod_eq_0 zero_le)
  ultimately show "P = 0"
    using tendsto_unique by force
qed (use assms in \<open>auto simp: has_prod_def\<close>)

lemma has_prod_0D:
  fixes f :: "nat \<Rightarrow> 'a :: {semidom, comm_semiring_1, t2_space}"
  shows "f has_prod 0 \<Longrightarrow> 0 \<in> range f"
  using has_prod_eq_0_iff[of f 0] by auto

lemma has_prod_zeroI:
  fixes f :: "nat \<Rightarrow> 'a :: {semidom, comm_semiring_1, t2_space}"
  assumes "f has_prod P" "f n = 0"
  shows   "P = 0"
  using assms by (auto simp: has_prod_eq_0_iff)  

lemma raw_has_prod_in_Reals:
  assumes "raw_has_prod (complex_of_real \<circ> z) M p"
  shows "p \<in> \<real>"
  using assms by (auto simp: raw_has_prod_def real_lim_sequentially)

lemma raw_has_prod_of_real_iff: "raw_has_prod (complex_of_real \<circ> z) M (of_real p) \<longleftrightarrow> raw_has_prod z M p"
  by (auto simp: raw_has_prod_def tendsto_of_real_iff simp flip: of_real_prod)

lemma convergent_prod_of_real_iff: "convergent_prod (complex_of_real \<circ> z) \<longleftrightarrow> convergent_prod z"
  by (smt (verit, best) Reals_cases convergent_prod_def raw_has_prod_in_Reals raw_has_prod_of_real_iff)

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)

lemma raw_has_prod_norm:
  fixes a :: "'a ::real_normed_field"
  assumes "raw_has_prod f M a"
  shows "raw_has_prod (\<lambda>n. norm (f n)) M (norm a)"
  using assms by (auto simp: raw_has_prod_def prod_norm tendsto_norm)

lemma has_prod_norm:
  fixes a :: "'a ::real_normed_field"
  assumes f: "f has_prod a" 
  shows "(\<lambda>n. norm (f n)) has_prod (norm a)"
  using f [unfolded has_prod_def]
proof (elim disjE exE conjE)
  assume f0: "raw_has_prod f 0 a"
  then show "(\<lambda>n. norm (f n)) has_prod norm a"
    using has_prod_def raw_has_prod_norm by blast
next
  fix i p
  assume "a = 0" and "f i = 0" and p: "raw_has_prod f (Suc i) p"
  then have "Ex (raw_has_prod (\<lambda>n. norm (f n)) (Suc i))"
    using raw_has_prod_norm by blast
  then show ?thesis
    by (metis \<open>a = 0\<close> \<open>f i = 0\<close> has_prod_0_iff norm_zero)
qed

lemma raw_has_prod_imp_nonzero:
  assumes "raw_has_prod f N P" "n \<ge> N"
  shows   "f n \<noteq> 0"
proof
  assume "f n = 0"
  from assms(1) have lim: "(\<lambda>m. (\<Prod>k\<le>m. f (k + N))) \<longlonglongrightarrow> P" and "P \<noteq> 0"
    unfolding raw_has_prod_def by blast+
  have "eventually (\<lambda>m. m \<ge> n - N) at_top"
    by (rule eventually_ge_at_top)
  hence "eventually (\<lambda>m. (\<Prod>k\<le>m. f (k + N)) = 0) at_top"
  proof eventually_elim
    case (elim m)
    have "f ((n - N) + N) = 0" "n - N \<in> {..m}" "finite {..m}"
      using \<open>n \<ge> N\<close> \<open>f n = 0\<close> elim by auto
    thus "(\<Prod>k\<le>m. f (k + N)) = 0"
      using prod_zero[of "{..m}" "\<lambda>k. f (k + N)"] by blast
  qed
  with lim have "P = 0"
    by (simp add: LIMSEQ_const_iff tendsto_cong)
  thus False
    using \<open>P \<noteq> 0\<close> by contradiction
qed

lemma has_prod_imp_tendsto:
  fixes f :: "nat \<Rightarrow> 'a :: {semidom, t2_space}"
  assumes "f has_prod P"
  shows   "(\<lambda>n. \<Prod>k\<le>n. f k) \<longlonglongrightarrow> P"
proof (cases "P = 0")
  case False
  with assms show ?thesis
    by (auto simp: has_prod_def raw_has_prod_def)
next
  case True
  with assms obtain N P' where "f N = 0" "raw_has_prod f (Suc N) P'"
    by (auto simp: has_prod_def)
  thus ?thesis
    using LIMSEQ_prod_0 True \<open>f N = 0\<close> by blast
qed

lemma has_prod_imp_tendsto':
  fixes f :: "nat \<Rightarrow> 'a :: {semidom, t2_space}"
  assumes "f has_prod P"
  shows   "(\<lambda>n. \<Prod>k<n. f k) \<longlonglongrightarrow> P"
  using has_prod_imp_tendsto[OF assms] LIMSEQ_lessThan_iff_atMost by blast

lemma has_prod_nonneg:
  assumes "f has_prod P" "\<And>n. f n \<ge> (0::real)"
  shows   "P \<ge> 0"
proof (rule tendsto_le)
  show "((\<lambda>n. \<Prod>i\<le>n. f i)) \<longlonglongrightarrow> P"
    using assms(1) by (rule has_prod_imp_tendsto)
  show "(\<lambda>n. 0::real) \<longlonglongrightarrow> 0"
    by auto
qed (use assms in \<open>auto intro!: always_eventually prod_nonneg\<close>)

lemma has_prod_pos:
  assumes "f has_prod P" "\<And>n. f n > (0::real)"
  shows   "P > 0"
proof -
  have "P \<ge> 0"
    by (rule has_prod_nonneg[OF assms(1)]) (auto intro!: less_imp_le assms(2))
  moreover have "f n \<noteq> 0" for n
    using assms(2)[of n] by auto
  hence "P \<noteq> 0"
    using has_prod_0_iff[of f] assms by auto
  ultimately show ?thesis
    by linarith
qed


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 [simp]: "(\<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))"
    by (simp add: algebra_simps)
  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))"
    by (simp add: prod_norm norm_mult)
  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"
    by (simp add: tendsto_norm_zero_iff)
  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"
    by (auto simp: prod_defs add.assoc)
  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
      using \<open>N \<ge> M\<close> by (simp add: add_ac)
  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)
  apply (auto simp add: raw_has_prod_def add_ac)
  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)"
        by (simp add: dist_norm norm_minus_commute)
      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))"
        by (simp add: algebra_simps)
      also have "\<dots> = (\<Prod>k\<le>m. norm (f (k + N))) * norm ((\<Prod>k\<in>{m<..n}. f (k + N)) - 1)"
        by (simp add: norm_mult prod_norm)
      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"
        by (simp add: algebra_simps)
      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" 
      by (simp add: prod_norm)

    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
          finally show ?case by (simp add: add_ac prod_norm)
        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 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<n. f i)"
  by (metis assms prodinf_eq_lim LIMSEQ_lessThan_iff_atMost convergent_prod_iff_nz_lim limI)

lemma prodinf_eq_prod_lim:
  fixes a:: "'a :: {topological_semigroup_mult,t2_space,idom}"
  assumes "(\<lambda>n. \<Prod>k\<le>n. f k) \<longlonglongrightarrow> a" "a \<noteq> 0"
  shows"(\<Prod>k. f k) = a"
  by (metis LIMSEQ_prod_0 LIMSEQ_unique assms convergent_prod_iff_nz_lim limI prodinf_eq_lim)

lemma prodinf_eq_prod_lim':
  fixes a:: "'a :: {topological_semigroup_mult,t2_space,idom}"
  assumes "(\<lambda>n. \<Prod>k<n. f k) \<longlonglongrightarrow> a" "a \<noteq> 0"
  shows"(\<Prod>k. f k) = a"
  using LIMSEQ_lessThan_iff_atMost assms prodinf_eq_prod_lim by blast

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"
    by (simp add: f)
  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"
    with f have "\<not> (\<lambda>n. prod f {..n}) \<longlonglongrightarrow> 0"
      by simp
    then have "\<not> (\<lambda>n. prod g {..n}) \<longlonglongrightarrow> 0"
      using * \<open>C \<noteq> 0\<close> 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 **: "eventually (\<lambda>n. prod g {..n} = prod f {..n} / C) sequentially"
      using * by eventually_elim (use \<open>C \<noteq> 0\<close> in auto)
    from cg and g have "\<not> (\<lambda>n. prod g {..n}) \<longlonglongrightarrow> 0"
      by simp
    then have "\<not> (\<lambda>n. prod f {..n}) \<longlonglongrightarrow> 0"
      using ** \<open>C \<noteq> 0\<close> filterlim_cong by fastforce
    then show "convergent_prod f"
      by (metis \<open>C \<noteq> 0\<close> cg convergent_LIMSEQ_iff
          convergent_mult_const_iff convergent_prod_iff_convergent
          convergent_prod_imp_convergent f local.cong)
  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)}"
      by (auto simp: le_Suc_eq trans_le_add2)
    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
      by (simp add: raw_has_prod_def has_prod_def)
  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"
    proof (simp add: raw_has_prod_def)
      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"
  by (simp add: assms has_prod_cong)

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
    by (simp add: convergent_prod_def)
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)"
  by (simp add: convergent_prod_finite)

lemma convergent_prod_If_finite[simp, intro]:
  fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
  assumes "finite {r. P r}"
  shows   "convergent_prod (\<lambda>r. if P r then f r else 1)"
proof -
  have "(\<lambda>r. if P r then f r else 1) has_prod (\<Prod>r | P r. f r)"
    by (rule has_prod_If_finite) fact
  thus ?thesis
    by (meson convergent_prod_def has_prod_def)
qed

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

text \<open>The ge1 assumption can probably be weakened, at the expense of extra work\<close>
lemma uniform_limit_prodinf:
  fixes f:: "nat \<Rightarrow> real \<Rightarrow> real"
  assumes "uniformly_convergent_on X (\<lambda>n x. \<Prod>k<n. f k x)" 
    and ge1: "\<And>x k . x \<in> X \<Longrightarrow> f k x \<ge> 1"
  shows "uniform_limit X (\<lambda>n x. \<Prod>k<n. f k x) (\<lambda>x. \<Prod>k. f k x) sequentially"
proof -
  have ul: "uniform_limit X (\<lambda>n x. \<Prod>k<n. f k x) (\<lambda>x. lim (\<lambda>n. \<Prod>k<n. f k x)) sequentially"
    using assms uniformly_convergent_uniform_limit_iff by blast
  moreover have "(\<Prod>k. f k x) = lim (\<lambda>n. \<Prod>k<n. f k x)" if "x \<in> X" for x
  proof (intro prodinf_eq_lim')
    have tends: "(\<lambda>n. \<Prod>k<n. f k x) \<longlonglongrightarrow> lim (\<lambda>n. \<Prod>k<n. f k x)"
      using tendsto_uniform_limitI [OF ul] that by metis
    moreover have "(\<Prod>k<n. f k x) \<ge> 1" for n
      using ge1 by (simp add: prod_ge_1 that)
    ultimately have "lim (\<lambda>n. \<Prod>k<n. f k x) \<ge> 1"
      by (meson LIMSEQ_le_const)
    then have "raw_has_prod (\<lambda>k. f k x) 0 (lim (\<lambda>n. \<Prod>k<n. f k x))"
      using LIMSEQ_lessThan_iff_atMost tends by (auto simp: raw_has_prod_def)
    then show "convergent_prod (\<lambda>k. f k x)"
      unfolding convergent_prod_def by blast
    show "\<And>k. f k x \<noteq> 0"
      by (smt (verit) ge1 that)
  qed
  ultimately show ?thesis
    by (metis (mono_tags, lifting) uniform_limit_cong')
qed

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"
      by (simp add: C_def)
    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

lemma convergent_prod_tendsto_imp_has_prod:
  assumes "convergent_prod f" "(\<lambda>n. (\<Prod>i\<le>n. f i)) \<longlonglongrightarrow> P"
  shows   "f has_prod P"
  using assms by (metis convergent_prod_imp_has_prod has_prod_imp_tendsto limI)
    
end

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

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

lemma has_prod_nonzero:
  assumes "f has_prod a" "a \<noteq> 0"
  shows "f k \<noteq> 0"
  using assms by (auto simp: has_prod_def raw_has_prod_def LIMSEQ_prod_0 LIMSEQ_unique)

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. n \<ge> N \<Longrightarrow> prod f {..<n} \<le> x" 
  shows "prodinf f \<le> x"
  by (metis lessThan_Suc_atMost assms convergent_prod_LIMSEQ LIMSEQ_le_const2 atMost_iff lessThan_iff less_le)

lemma prodinf_eq_one_iff [simp]: 
  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
      by (simp add: ge1 prod_ge_1)
    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
    by (simp add: 0 prod_pos)
  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
    by (simp add: prod_pos)
  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"
  unfolding raw_has_prod_def add_0_right
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)"
  by (simp add: raw_has_prod_mult has_prod_def)

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"
      by (simp add: raw_has_prod_def)
  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
  proof (simp add: has_prod_def, safe)
    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 [simp]:
  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"
  by (metis raw_has_prod_eq_0 add.commute add.left_neutral raw_has_prod_Suc_iff raw_has_prod_nonzero le_add1 nonzero_mult_div_cancel_right times_divide_eq_left)

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 
    by (auto simp add: raw_has_prod_Suc_iff)
  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
    by (simp add: ac_simps)
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"
  by (simp add: assms has_prod_iff_shift)

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"
  by (simp add: assms has_prod_iff_shift)

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)

lemma prodinf_split_head:
  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

lemma has_prod_ignore_initial_segment':
  assumes "convergent_prod f"
  shows   "f has_prod ((\<Prod>k<n. f k) * (\<Prod>k. f (k + n)))"
proof (cases "\<exists>k<n. f k = 0")
  case True
  hence [simp]: "(\<Prod>k<n. f k) = 0"
    by (meson finite_lessThan lessThan_iff prod_zero)
  thus ?thesis using True assms
    by (metis convergent_prod_has_prod_iff has_prod_zeroI mult_not_zero)
next
  case False
  hence "(\<lambda>i. f (i + n)) has_prod (prodinf f / prod f {..<n})"
    using assms by (intro has_prod_split_initial_segment) (auto simp: convergent_prod_has_prod_iff)
  hence "prodinf f = prod f {..<n} * (\<Prod>k. f (k + n))"
    using False by (simp add: has_prod_iff divide_simps mult_ac)
  thus ?thesis
    using assms by (simp add: convergent_prod_has_prod_iff)
qed

end

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

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

lemma convergent_prod_const_iff [simp]:
  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

lemma prod_ge_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> f i \<le> 1"
  shows "prod f {..<n} \<ge> prodinf f"
proof (rule has_prod_le; (intro conjI)?)
  show "f has_prod prodinf f"
    using assms(1) has_prod_unique by blast
  show "(\<lambda>r. if r \<in> {..<n} then f r else 1) has_prod prod f {..<n}"
    by (rule has_prod_If_finite_set) auto
next
  fix i 
  show "f i \<ge> 0"
    by (rule assms)
  show "f i \<le> (if i \<in> {..<n} then f i else 1)"
    using assms(3)[of i] by auto
qed

lemma has_prod_less:
  fixes F G :: real
  assumes less: "f m < g m"
  assumes f: "f has_prod F" and g: "g has_prod G"
  assumes pos: "\<And>n. 0 < f n" and le: "\<And>n. f n \<le> g n"
  shows   "F < G"
proof -
  define F' G' where "F' = (\<Prod>n<Suc m. f n)" and "G' = (\<Prod>n<Suc m. g n)"
  have [simp]: "f n \<noteq> 0" "g n \<noteq> 0" for n
    using pos[of n] le[of n] by auto
  have [simp]: "F' \<noteq> 0" "G' \<noteq> 0"
    by (auto simp: F'_def G'_def)
  have f': "(\<lambda>n. f (n + Suc m)) has_prod (F / F')"
    unfolding F'_def using f
    by (intro has_prod_split_initial_segment) auto
  have g': "(\<lambda>n. g (n + Suc m)) has_prod (G / G')"
    unfolding G'_def using g
    by (intro has_prod_split_initial_segment) auto
  have "F' * (F / F') < G' * (F / F')"
  proof (rule mult_strict_right_mono)
    show "F' < G'"
      unfolding F'_def G'_def
      by (rule prod_mono_strict[of m])
         (auto intro: le less_imp_le[OF pos] less_le_trans[OF pos le] less)
    show "F / F' > 0"
      using f' by (rule has_prod_pos) (use pos in auto)
  qed
  also have "\<dots> \<le> G' * (G / G')"
  proof (rule mult_left_mono)
    show "F / F' \<le> G / G'"
      using f' g' by (rule has_prod_le) (auto intro: less_imp_le[OF pos] le)
    show "G' \<ge> 0"
      unfolding G'_def by (intro prod_nonneg order.trans[OF less_imp_le[OF pos] le])
  qed
  finally show ?thesis
    by simp
qed


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)"
    by (simp add: has_prod_def sums_imp_has_prod_exp)
  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"
    by (metis assms bounded_imp_convergent_prod convergent_prod_iff_nz_lim le_add_same_cancel1 le_add_same_cancel2 less_le not_le zero_le_one)
  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
      by (simp add: assms less_1_prod)
    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"
      by (simp add: ge0)
    also have "\<dots> \<le> exp (f n) - 1"
      by (metis diff_diff_add exp_ge_add_one_self ge_iff_diff_ge_0)
    finally show "norm (f n) \<le> norm (exp (f n) - 1)"
      by simp
  qed
  then show ?thesis
    by (simp add: prodinf_exp)
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)
  moreover have "\<And>x. f x \<noteq> 0"
    by (smt (verit, best) "0")
  ultimately 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 has_prod_imp_sums_ln_real':
  fixes P :: real
  assumes "f has_prod P" "\<And>n. f n > 0"
  shows   "(\<lambda>n. ln (f n)) sums (ln P)"
proof -
  have nz: "f n \<noteq> 0" for n
    using assms(2)[of n] by simp
  have "P \<noteq> 0"
    using has_prod_eq_0_iff[OF assms(1)] by (auto simp: nz)

  have "(\<lambda>n. \<Prod>k<n. f k) \<longlonglongrightarrow> P"
    using has_prod_imp_tendsto'[OF assms(1)] by simp
  hence "(\<lambda>n. ln (\<Prod>k<n. f k)) \<longlonglongrightarrow> ln P"
    by (intro tendsto_intros \<open>P \<noteq> 0\<close>)
  also have "(\<lambda>n. ln (\<Prod>k<n. f k)) = (\<lambda>n. \<Sum>k<n. ln (f k))"
    by (subst ln_prod) (auto simp: nz)
  finally show ?thesis
    by (simp add: sums_def)
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"
    by (simp add: f has_prod_iff)
  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>)"
    using Arg_def Arg_eq \<xi> unfolding \<Theta>_def by (simp add: algebra_simps exp_add)
  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
    by (simp add: algebra_simps)
  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"
      by (simp add: order_tendsto_iff)
    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)"
        by (simp add: field_simps)
      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)"
    by (simp add: k tendsto_add tendsto_mult tendsto_eventually)
  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]
    by (simp add: o_def exp_add algebra_simps)
  moreover have "exp (\<Sum>k\<le>n. Ln (z k)) = (\<Prod>k\<le>n. z k)" for n
    by (simp add: exp_sum add_eq_0_iff assms)
  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
    by (simp add: exp_sum add_eq_0_iff assms)
  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 -
  obtain N where "\<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)
  then have "summable (\<lambda>k. Ln (1 + z k))"
    by (metis norm_Ln_le summable_comparison_test summable_mult z)
  with non0 show ?thesis
    by (simp add: add_eq_0_iff convergent_prod_iff_summable_complex)
qed

corollary summable_imp_convergent_prod_real:
  fixes z :: "nat \<Rightarrow> real"
  assumes z: "summable (\<lambda>k. \<bar>z k\<bar>)" and non0: "\<And>k. z k \<noteq> -1"
  shows "convergent_prod (\<lambda>k. 1 + z k)" 
proof -
  have "\<And>k. (complex_of_real \<circ> z) k \<noteq> - 1"
    by (metis non0 o_apply of_real_1 of_real_eq_iff of_real_minus)
  with z 
  have "convergent_prod (\<lambda>k. 1 + (complex_of_real \<circ> z) k)"
    by (auto intro: summable_imp_convergent_prod_complex)
  then show ?thesis 
    using convergent_prod_of_real_iff [of "\<lambda>k. 1 + z k"] by (simp add: o_def)
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
    by (simp add: convergent_eq_Cauchy Cauchy_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 [simp]:
  "(\<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

subsection \<open>Convergence criteria: especially uniform convergence of infinite products\<close>

text \<open>
  Cauchy's criterion for the convergence of infinite products, adapted to proving
  uniform convergence: let $f_k(x)$ be a sequence of functions such that

    \<^enum> $f_k(x)$ has uniformly bounded partial products, i.e.\ there exists a constant \<open>C\<close>
      such that $\prod_{k=0}^m f_k(x) \leq C$ for all $m$ and $x\in A$.

    \<^enum> For any $\varepsilon > 0$ there exists a number $M \in \mathbb{N}$ such that, for any
      $m, n \geq M$ and all $x\in A$ we have $|(\prod_{k=m}^n f_k(x)) - 1| < \varepsilon$

  Then $\prod_{k=0}^n f_k(x)$ converges to $\prod_{k=0}^\infty f_k(x)$ uniformly for all
  $x \in A$.
\<close>
lemma uniformly_convergent_prod_Cauchy:
  fixes f :: "nat \<Rightarrow> 'a :: topological_space \<Rightarrow> 'b :: {real_normed_div_algebra, comm_ring_1, banach}"
  assumes C: "\<And>x m. x \<in> A \<Longrightarrow> norm (\<Prod>k<m. f k x) \<le> C"
  assumes "\<And>e. e > 0 \<Longrightarrow> \<exists>M. \<forall>x\<in>A. \<forall>m\<ge>M. \<forall>n\<ge>m. dist (\<Prod>k=m..n. f k x) 1 < e"
  shows   "uniformly_convergent_on A (\<lambda>N x. \<Prod>n<N. f n x)"
proof (rule Cauchy_uniformly_convergent, rule uniformly_Cauchy_onI')
  fix \<epsilon> :: real assume \<epsilon>: "\<epsilon> > 0"
  define C' where "C' = max C 1"
  have C': "C' > 0"
    by (auto simp: C'_def)
  define \<delta> where "\<delta> = Min {2 / 3 * \<epsilon> / C', 1 / 2}"
  from \<epsilon> have "\<delta> > 0"
    using \<open>C' > 0\<close> by (auto simp: \<delta>_def)
  obtain M where M: "\<And>x m n. x \<in> A \<Longrightarrow> m \<ge> M \<Longrightarrow> n \<ge> m \<Longrightarrow> dist (\<Prod>k=m..n. f k x) 1 < \<delta>"
    using \<open>\<delta> > 0\<close> assms by fast

  show "\<exists>M. \<forall>x\<in>A. \<forall>m\<ge>M. \<forall>n>m. dist (\<Prod>k<m. f k x) (\<Prod>k<n. f k x) < \<epsilon>"
  proof (rule exI, intro ballI allI impI)
    fix x m n
    assume x: "x \<in> A" and mn: "M + 1 \<le> m" "m < n"
    show "dist (\<Prod>k<m. f k x) (\<Prod>k<n. f k x) < \<epsilon>"
    proof (cases "\<exists>k<m. f k x = 0")
      case True
      hence "(\<Prod>k<m. f k x) = 0" and "(\<Prod>k<n. f k x) = 0"
        using mn x by (auto intro!: prod_zero)
      thus ?thesis
        using \<epsilon> by simp
    next
      case False
      have *: "{..<n} = {..<m} \<union> {m..n-1}"
        using mn by auto
      have "dist (\<Prod>k<m. f k x) (\<Prod>k<n. f k x) = norm ((\<Prod>k<m. f k x) * ((\<Prod>k=m..n-1. f k x) - 1))"
        unfolding * by (subst prod.union_disjoint)
                       (use mn in \<open>auto simp: dist_norm algebra_simps norm_minus_commute\<close>)
      also have "\<dots> = (\<Prod>k<m. norm (f k x)) * dist (\<Prod>k=m..n-1. f k x) 1"
        by (simp add: norm_mult dist_norm prod_norm)
      also have "\<dots> < (\<Prod>k<m. norm (f k x)) * (2 / 3 * \<epsilon> / C')"
      proof (rule mult_strict_left_mono)
        show "dist (\<Prod>k = m..n - 1. f k x) 1 < 2 / 3 * \<epsilon> / C'"
          using M[of x m "n-1"] x mn unfolding \<delta>_def by fastforce
      qed (use False in \<open>auto intro!: prod_pos\<close>)
      also have "(\<Prod>k<m. norm (f k x)) = (\<Prod>k<M. norm (f k x)) * norm (\<Prod>k=M..<m. (f k x))"
      proof -
        have *: "{..<m} = {..<M} \<union> {M..<m}"
          using mn by auto
        show ?thesis
          unfolding * using mn by (subst prod.union_disjoint) (auto simp: prod_norm)
      qed
      also have "norm (\<Prod>k=M..<m. (f k x)) \<le> 3 / 2"
      proof -
        have "dist (\<Prod>k=M..m-1. f k x) 1 < \<delta>"
          using M[of x M "m-1"] x mn \<open>\<delta> > 0\<close> by auto
        also have "\<dots> \<le> 1 / 2"
          by (simp add: \<delta>_def)
        also have "{M..m-1} = {M..<m}"
          using mn by auto
        finally have "norm (\<Prod>k=M..<m. f k x) \<le> norm (1 :: 'b) + 1 / 2"
          by norm
        thus ?thesis
          by simp
      qed
      hence "(\<Prod>k<M. norm (f k x)) * norm (\<Prod>k = M..<m. f k x) * (2 / 3 * \<epsilon> / C') \<le>
             (\<Prod>k<M. norm (f k x)) * (3 / 2) * (2 / 3 * \<epsilon> / C')"
        using \<epsilon> C' by (intro mult_left_mono mult_right_mono prod_nonneg) auto
      also have "\<dots> \<le> C' * (3 / 2) * (2 / 3 * \<epsilon> / C')"
      proof (intro mult_right_mono)
        have "(\<Prod>k<M. norm (f k x)) \<le> C"
          using C[of x M] x by (simp add: prod_norm)
        also have "\<dots> \<le> C'"
          by (simp add: C'_def)
        finally show "(\<Prod>k<M. norm (f k x)) \<le> C'" .
      qed (use \<epsilon> C' in auto)
      finally show "dist (\<Prod>k<m. f k x) (\<Prod>k<n. f k x) < \<epsilon>"
        using \<open>C' > 0\<close> by (simp add: field_simps)
    qed
  qed
qed

text \<open>
  By instantiating the set $A$ in this result with a singleton set, we obtain the ``normal''
  Cauchy criterion for infinite products:
\<close>
lemma convergent_prod_Cauchy_sufficient:
  fixes f :: "nat \<Rightarrow> 'b :: {real_normed_div_algebra, comm_ring_1, banach}"
  assumes "\<And>e. e > 0 \<Longrightarrow> \<exists>M. \<forall>m n. M \<le> m \<longrightarrow> m \<le> n \<longrightarrow> dist (\<Prod>k=m..n. f k) 1 < e"
  shows   "convergent_prod f"
proof -
  obtain M where M: "\<And>m n. m \<ge> M \<Longrightarrow> n \<ge> m \<Longrightarrow> dist (prod f {m..n}) 1 < 1 / 2"
    using assms(1)[of "1 / 2"] by auto
  have nz: "f m \<noteq> 0" if "m \<ge> M" for m
    using M[of m m] that by auto

  have M': "dist (prod (\<lambda>k. f (k + M)) {m..<n}) 1 < 1 / 2" for m n
  proof (cases "m < n")
    case True
    have "dist (prod f {m+M..n-1+M}) 1 < 1 / 2"
      by (rule M) (use True in auto)
    also have "prod f {m+M..n-1+M} = prod (\<lambda>k. f (k + M)) {m..<n}"
      by (rule prod.reindex_bij_witness[of _ "\<lambda>k. k + M" "\<lambda>k. k - M"]) (use True in auto)
    finally show ?thesis .
  qed auto 

  have "uniformly_convergent_on {0::'b} (\<lambda>N x. \<Prod>n<N. f (n + M))"
  proof (rule uniformly_convergent_prod_Cauchy)
    fix m :: nat
    have "norm (\<Prod>k=0..<m. f (k + M)) < norm (1 :: 'b) + 1 / 2"
      using M'[of 0 m] by norm
    thus "norm (\<Prod>k<m. f (k + M)) \<le> 3 / 2"
      by (simp add: atLeast0LessThan)
  next
    fix e :: real assume e: "e > 0"
    obtain M' where M': "\<And>m n. M' \<le> m \<longrightarrow> m \<le> n \<longrightarrow> dist (\<Prod>k=m..n. f k) 1 < e"
      using assms e by blast
    show "\<exists>M'. \<forall>x\<in>{0}. \<forall>m\<ge>M'. \<forall>n\<ge>m. dist (\<Prod>k=m..n. f (k + M)) 1 < e"
    proof (rule exI[of _ M'], intro ballI impI allI)
      fix m n :: nat assume "M' \<le> m" "m \<le> n"
      thus "dist (\<Prod>k=m..n. f (k + M)) 1 < e"
        using M' by (metis add.commute add_left_mono prod.shift_bounds_cl_nat_ivl trans_le_add1)
    qed
  qed
  hence "convergent (\<lambda>N. \<Prod>n<N. f (n + M))"
    by (rule uniformly_convergent_imp_convergent[of _ _ 0]) auto
  then obtain L where L: "(\<lambda>N. \<Prod>n<N. f (n + M)) \<longlonglongrightarrow> L"
    unfolding convergent_def by blast

  show ?thesis
    unfolding convergent_prod_altdef
  proof (rule exI[of _ M], rule exI[of _ L], intro conjI)
    show "\<forall>n\<ge>M. f n \<noteq> 0"
      using nz by auto
  next
    show "(\<lambda>n. \<Prod>i\<le>n. f (i + M)) \<longlonglongrightarrow> L"
      using LIMSEQ_Suc[OF L] by (subst (asm) lessThan_Suc_atMost)
  next
    have "norm L \<ge> 1 / 2"
    proof (rule tendsto_lowerbound)
      show "(\<lambda>n. norm (\<Prod>i<n. f (i + M))) \<longlonglongrightarrow> norm L"
        by (intro tendsto_intros L)
      show "\<forall>\<^sub>F n in sequentially. 1 / 2 \<le> norm (\<Prod>i<n. f (i + M))"
      proof (intro always_eventually allI)
        fix m :: nat
        have "norm (\<Prod>k=0..<m. f (k + M)) \<ge> norm (1 :: 'b) - 1 / 2"
          using M'[of 0 m] by norm
        thus "norm (\<Prod>k<m. f (k + M)) \<ge> 1 / 2"
          by (simp add: atLeast0LessThan)
      qed
    qed auto
    thus "L \<noteq> 0"
      by auto
  qed
qed


text \<open>
  We now prove that the Cauchy criterion for pointwise convergence is both necessary
  and sufficient.
\<close>
lemma convergent_prod_Cauchy_necessary:
  fixes f :: "nat \<Rightarrow> 'b :: {real_normed_field, banach}"
  assumes "convergent_prod f" "e > 0"
  shows   "\<exists>M. \<forall>m n. M \<le> m \<longrightarrow> m \<le> n \<longrightarrow> dist (\<Prod>k=m..n. f k) 1 < e"
proof -
  have *: "\<exists>M. \<forall>m n. M \<le> m \<longrightarrow> m \<le> n \<longrightarrow> dist (\<Prod>k=m..n. f k) 1 < e"
    if f: "convergent_prod f" "0 \<notin> range f" and e: "e > 0"
    for f :: "nat \<Rightarrow> 'b" and e :: real
  proof -
    have *: "(\<lambda>n. norm (\<Prod>k<n. f k)) \<longlonglongrightarrow> norm (\<Prod>k. f k)"
      using has_prod_imp_tendsto' f(1) by (intro tendsto_norm) blast
    from f(1,2) have [simp]: "(\<Prod>k. f k) \<noteq> 0"
      using prodinf_nonzero by fastforce
    obtain M' where M': "norm (\<Prod>k<m. f k) > norm (\<Prod>k. f k) / 2" if "m \<ge> M'" for m
      using order_tendstoD(1)[OF *, of "norm (\<Prod>k. f k) / 2"]
      by (auto simp: eventually_at_top_linorder)
    define M where "M = Min (insert (norm (\<Prod>k. f k) / 2) ((\<lambda>m. norm (\<Prod>k<m. f k)) ` {..<M'}))"
    have "M > 0"
      unfolding M_def using f(2) by (subst Min_gr_iff) auto
    have norm_ge: "norm (\<Prod>k<m. f k) \<ge> M" for m
    proof (cases "m \<ge> M'")
      case True
      have "M \<le> norm (\<Prod>k. f k) / 2"
        unfolding M_def by (intro Min.coboundedI) auto
      also from True have "norm (\<Prod>k<m. f k) > norm (\<Prod>k. f k) / 2"
        by (intro M')
      finally show ?thesis by linarith
    next
      case False
      thus ?thesis
        unfolding M_def
        by (intro Min.coboundedI) auto
    qed

    have "convergent (\<lambda>n. \<Prod>k<n. f k)"
      using f(1) convergent_def has_prod_imp_tendsto' by blast
    hence "Cauchy (\<lambda>n. \<Prod>k<n. f k)"
      by (rule convergent_Cauchy)
    moreover have "e * M > 0"
      using e \<open>M > 0\<close> by auto
    ultimately obtain N where N: "dist (\<Prod>k<m. f k) (\<Prod>k<n. f k) < e * M" if "m \<ge> N" "n \<ge> N" for m n
      unfolding Cauchy_def by fast

    show "\<exists>M. \<forall>m n. M \<le> m \<longrightarrow> m \<le> n \<longrightarrow> dist (prod f {m..n}) 1 < e"
    proof (rule exI[of _ N], intro allI impI, goal_cases)
      case (1 m n)
      have "dist (\<Prod>k<m. f k) (\<Prod>k<Suc n. f k) < e * M"
        by (rule N) (use 1 in auto)
      also have "dist (\<Prod>k<m. f k) (\<Prod>k<Suc n. f k) = norm ((\<Prod>k<Suc n. f k) - (\<Prod>k<m. f k))"
        by (simp add: dist_norm norm_minus_commute)
      also have "(\<Prod>k<Suc n. f k) = (\<Prod>k\<in>{..<m}\<union>{m..n}. f k)"
        using 1 by (intro prod.cong) auto
      also have "\<dots> = (\<Prod>k\<in>{..<m}. f k) * (\<Prod>k\<in>{m..n}. f k)"
        by (subst prod.union_disjoint) auto
      also have "\<dots> - (\<Prod>k<m. f k) = (\<Prod>k<m. f k) * ((\<Prod>k\<in>{m..n}. f k) - 1)"
        by (simp add: algebra_simps)
      finally have "norm (prod f {m..n} - 1) < e * M / norm (prod f {..<m})"
        using f(2) by (auto simp add: norm_mult divide_simps mult_ac)
      also have "\<dots> \<le> e * M / M"
        using e \<open>M > 0\<close> f(2) by (intro divide_left_mono norm_ge mult_pos_pos) auto
      also have "\<dots> = e"
        using \<open>M > 0\<close> by simp
      finally show ?case
        by (simp add: dist_norm)
    qed
  qed

  obtain M where M: "f m \<noteq> 0" if "m \<ge> M" for m
    using convergent_prod_imp_ev_nonzero[OF assms(1)]
    by (auto simp: eventually_at_top_linorder)

  have "\<exists>M'. \<forall>m n. M' \<le> m \<longrightarrow> m \<le> n \<longrightarrow> dist (\<Prod>k=m..n. f (k + M)) 1 < e"
    by (rule *) (use assms M in auto)
  then obtain M' where M': "dist (\<Prod>k=m..n. f (k + M)) 1 < e" if "M' \<le> m" "m \<le> n" for m n
    by blast

  show "\<exists>M. \<forall>m n. M \<le> m \<longrightarrow> m \<le> n \<longrightarrow> dist (prod f {m..n}) 1 < e"
  proof (rule exI[of _ "M + M'"], safe, goal_cases)
    case (1 m n)
    have "dist (\<Prod>k=m-M..n-M. f (k + M)) 1 < e"
      by (rule M') (use 1 in auto)
    also have "(\<Prod>k=m-M..n-M. f (k + M)) = (\<Prod>k=m..n. f k)"
      using 1 by (intro prod.reindex_bij_witness[of _ "\<lambda>k. k - M" "\<lambda>k. k + M"]) auto
    finally show ?case .
  qed
qed

lemma convergent_prod_Cauchy_iff:
  fixes f :: "nat \<Rightarrow> 'b :: {real_normed_field, banach}"
  shows "convergent_prod f \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m n. M \<le> m \<longrightarrow> m \<le> n \<longrightarrow> dist (\<Prod>k=m..n. f k) 1 < e)"
  using convergent_prod_Cauchy_necessary[of f] convergent_prod_Cauchy_sufficient[of f]
  by blast

lemma uniformly_convergent_on_prod:
  fixes f :: "nat \<Rightarrow> 'a :: topological_space \<Rightarrow> 'b :: {real_normed_div_algebra, comm_ring_1, banach}"
  assumes cont: "\<And>n. continuous_on A (f n)"
  assumes A: "compact A"
  assumes conv_sum: "uniformly_convergent_on A (\<lambda>N x. \<Sum>n<N. norm (f n x))"
  shows   "uniformly_convergent_on A (\<lambda>N x. \<Prod>n<N. 1 + f n x)"
proof -
  have lim: "uniform_limit A (\<lambda>n x. \<Sum>k<n. norm (f k x)) (\<lambda>x. \<Sum>k. norm (f k x)) sequentially"
    by (rule uniform_limit_suminf) fact
  have cont': "\<forall>\<^sub>F n in sequentially. continuous_on A (\<lambda>x. \<Sum>k<n. norm (f k x))"
    using cont by (auto intro!: continuous_intros always_eventually cont)
  have "continuous_on A (\<lambda>x. \<Sum>k. norm (f k x))"
    by (rule uniform_limit_theorem[OF cont' lim]) auto
  hence "compact ((\<lambda>x. \<Sum>k. norm (f k x)) ` A)"
    by (intro compact_continuous_image A)
  hence "bounded ((\<lambda>x. \<Sum>k. norm (f k x)) ` A)"
    by (rule compact_imp_bounded)
  then obtain C where C: "norm (\<Sum>k. norm (f k x)) \<le> C" if "x \<in> A" for x
    unfolding bounded_iff by blast
  show ?thesis
  proof (rule uniformly_convergent_prod_Cauchy)
    fix x :: 'a and m :: nat
    assume x: "x \<in> A"
    have "norm (\<Prod>k<m. 1 + f k x) = (\<Prod>k<m. norm (1 + f k x))"
      by (simp add: prod_norm)
    also have "\<dots> \<le> (\<Prod>k<m. norm (1 :: 'b) + norm (f k x))"
      by (intro prod_mono) norm
    also have "\<dots> = (\<Prod>k<m. 1 + norm (f k x))"
      by simp
    also have "\<dots> \<le> exp (\<Sum>k<m. norm (f k x))"
      by (rule prod_le_exp_sum) auto
    also have "(\<Sum>k<m. norm (f k x)) \<le> (\<Sum>k. norm (f k x))"
    proof (rule sum_le_suminf)
      have "(\<lambda>n. \<Sum>k<n. norm (f k x)) \<longlonglongrightarrow> (\<Sum>k. norm (f k x))"
        by (rule tendsto_uniform_limitI[OF lim]) fact
      thus "summable (\<lambda>k. norm (f k x))"
        using sums_def sums_iff by blast
    qed auto
    also have "exp (\<Sum>k. norm (f k x)) \<le> exp (norm (\<Sum>k. norm (f k x)))"
      by simp
    also have "norm (\<Sum>k. norm (f k x)) \<le> C"
      by (rule C) fact
    finally show "norm (\<Prod>k<m. 1 + f k x) \<le> exp C"
      by - simp_all
  next
    fix \<epsilon> :: real assume \<epsilon>: "\<epsilon> > 0"
    have "uniformly_Cauchy_on A (\<lambda>N x. \<Sum>n<N. norm (f n x))"
      by (rule uniformly_convergent_Cauchy) fact
    moreover have "ln (1 + \<epsilon>) > 0"
      using \<epsilon> by simp
    ultimately obtain M where M: "\<And>m n x. x \<in> A \<Longrightarrow> M \<le> m \<Longrightarrow> M \<le> n \<Longrightarrow>
        dist (\<Sum>k<m. norm (f k x)) (\<Sum>k<n. norm (f k x)) < ln (1 + \<epsilon>)"
      using \<epsilon> unfolding uniformly_Cauchy_on_def by metis
  
    show "\<exists>M. \<forall>x\<in>A. \<forall>m\<ge>M. \<forall>n\<ge>m. dist (\<Prod>k = m..n. 1 + f k x) 1 < \<epsilon>"
    proof (rule exI, intro ballI allI impI)
      fix x m n
      assume x: "x \<in> A" and mn: "M \<le> m" "m \<le> n"
      have "dist (\<Sum>k<m. norm (f k x)) (\<Sum>k<Suc n. norm (f k x)) < ln (1 + \<epsilon>)"
        by (rule M) (use x mn in auto)
      also have "dist (\<Sum>k<m. norm (f k x)) (\<Sum>k<Suc n. norm (f k x)) =
                 \<bar>\<Sum>k\<in>{..<Suc n}-{..<m}. norm (f k x)\<bar>"
        using mn by (subst sum_diff) (auto simp: dist_norm)
      also have "{..<Suc n}-{..<m} = {m..n}"
        using mn by auto
      also have "\<bar>\<Sum>k=m..n. norm (f k x)\<bar> = (\<Sum>k=m..n. norm (f k x))"
        by (intro abs_of_nonneg sum_nonneg) auto
      finally have *: "(\<Sum>k=m..n. norm (f k x)) < ln (1 + \<epsilon>)" .
  
      have "dist (\<Prod>k=m..n. 1 + f k x) 1 = norm ((\<Prod>k=m..n. 1 + f k x) - 1)"
        by (simp add: dist_norm)
      also have "norm ((\<Prod>k=m..n. 1 + f k x) - 1) \<le> (\<Prod>n=m..n. 1 + norm (f n x)) - 1"
        by (rule norm_prod_minus1_le_prod_minus1)
      also have "(\<Prod>n=m..n. 1 + norm (f n x)) \<le> exp (\<Sum>k=m..n. norm (f k x))"
        by (rule prod_le_exp_sum) auto
      also note *
      finally show "dist (\<Prod>k = m..n. 1 + f k x) 1 < \<epsilon>"
        using \<epsilon> by - simp_all
    qed
  qed
qed

lemma uniformly_convergent_on_prod':
  fixes f :: "nat \<Rightarrow> 'a :: topological_space \<Rightarrow> 'b :: {real_normed_div_algebra, comm_ring_1, banach}"
  assumes cont: "\<And>n. continuous_on A (f n)"
  assumes A: "compact A"
  assumes conv_sum: "uniformly_convergent_on A (\<lambda>N x. \<Sum>n<N. norm (f n x - 1))"
  shows "uniformly_convergent_on A (\<lambda>N x. \<Prod>n<N. f n x)"
proof -
  have "uniformly_convergent_on A (\<lambda>N x. \<Prod>n<N. 1 + (f n x - 1))"
    by (rule uniformly_convergent_on_prod) (use assms in \<open>auto intro!: continuous_intros\<close>)
  thus ?thesis
    by simp
qed

end