(*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 Complex_Main
begin
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
definition gen_has_prod :: "[nat \<Rightarrow> 'a::{t2_space, comm_semiring_1}, nat, 'a] \<Rightarrow> bool"
where "gen_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>
definition has_prod :: "(nat \<Rightarrow> 'a::{t2_space, comm_semiring_1}) \<Rightarrow> 'a \<Rightarrow> bool" (infixr "has'_prod" 80)
where "f has_prod p \<equiv> gen_has_prod f 0 p \<or> (\<exists>i q. p = 0 \<and> f i = 0 \<and> gen_has_prod f (Suc i) q)"
definition convergent_prod :: "(nat \<Rightarrow> 'a :: {t2_space,comm_semiring_1}) \<Rightarrow> bool" where
"convergent_prod f \<equiv> \<exists>M p. gen_has_prod f M p"
definition prodinf :: "(nat \<Rightarrow> 'a::{t2_space, comm_semiring_1}) \<Rightarrow> 'a"
(binder "\<Prod>" 10)
where "prodinf f = (THE p. f has_prod p)"
lemmas prod_defs = gen_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 gen_has_prod_nonzero [simp]: "\<not> gen_has_prod f M 0"
by (simp add: gen_has_prod_def)
lemma has_prod_0_iff: "f has_prod 0 \<longleftrightarrow> (\<exists>i. f i = 0 \<and> (\<exists>p. gen_has_prod f (Suc i) p))"
by (simp add: has_prod_def)
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)
definition abs_convergent_prod :: "(nat \<Rightarrow> _) \<Rightarrow> bool" where
"abs_convergent_prod f \<longleftrightarrow> convergent_prod (\<lambda>i. 1 + norm (f i - 1))"
lemma abs_convergent_prodI:
assumes "convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
shows "abs_convergent_prod f"
proof -
from assms obtain L where L: "(\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1)) \<longlonglongrightarrow> L"
by (auto simp: convergent_def)
have "L \<ge> 1"
proof (rule tendsto_le)
show "eventually (\<lambda>n. (\<Prod>i\<le>n. 1 + norm (f i - 1)) \<ge> 1) sequentially"
proof (intro always_eventually allI)
fix n
have "(\<Prod>i\<le>n. 1 + norm (f i - 1)) \<ge> (\<Prod>i\<le>n. 1)"
by (intro prod_mono) auto
thus "(\<Prod>i\<le>n. 1 + norm (f i - 1)) \<ge> 1" by simp
qed
qed (use L in simp_all)
hence "L \<noteq> 0" by auto
with L show ?thesis unfolding abs_convergent_prod_def prod_defs
by (intro exI[of _ "0::nat"] exI[of _ L]) auto
qed
lemma
fixes f :: "nat \<Rightarrow> 'a :: {topological_semigroup_mult,t2_space,idom}"
assumes "convergent_prod f"
shows convergent_prod_imp_convergent: "convergent (\<lambda>n. \<Prod>i\<le>n. f i)"
and convergent_prod_to_zero_iff: "(\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> 0 \<longleftrightarrow> (\<exists>i. f i = 0)"
proof -
from assms obtain M L
where M: "\<And>n. n \<ge> M \<Longrightarrow> f n \<noteq> 0" and "(\<lambda>n. \<Prod>i\<le>n. f (i + M)) \<longlonglongrightarrow> L" and "L \<noteq> 0"
by (auto simp: convergent_prod_altdef)
note this(2)
also have "(\<lambda>n. \<Prod>i\<le>n. f (i + M)) = (\<lambda>n. \<Prod>i=M..M+n. f i)"
by (intro ext prod.reindex_bij_witness[of _ "\<lambda>n. n - M" "\<lambda>n. n + M"]) auto
finally have "(\<lambda>n. (\<Prod>i<M. f i) * (\<Prod>i=M..M+n. f i)) \<longlonglongrightarrow> (\<Prod>i<M. f i) * L"
by (intro tendsto_mult tendsto_const)
also have "(\<lambda>n. (\<Prod>i<M. f i) * (\<Prod>i=M..M+n. f i)) = (\<lambda>n. (\<Prod>i\<in>{..<M}\<union>{M..M+n}. f i))"
by (subst prod.union_disjoint) auto
also have "(\<lambda>n. {..<M} \<union> {M..M+n}) = (\<lambda>n. {..n+M})" by auto
finally have lim: "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> prod f {..<M} * L"
by (rule LIMSEQ_offset)
thus "convergent (\<lambda>n. \<Prod>i\<le>n. f i)"
by (auto simp: convergent_def)
show "(\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> 0 \<longleftrightarrow> (\<exists>i. f i = 0)"
proof
assume "\<exists>i. f i = 0"
then obtain i where "f i = 0" by auto
moreover with M have "i < M" by (cases "i < M") auto
ultimately have "(\<Prod>i<M. f i) = 0" by auto
with lim show "(\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> 0" by simp
next
assume "(\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> 0"
from tendsto_unique[OF _ this lim] and \<open>L \<noteq> 0\<close>
show "\<exists>i. f i = 0" by auto
qed
qed
lemma convergent_prod_iff_nz_lim:
fixes f :: "nat \<Rightarrow> 'a :: {topological_semigroup_mult,t2_space,idom}"
assumes "\<And>i. f i \<noteq> 0"
shows "convergent_prod f \<longleftrightarrow> (\<exists>L. (\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> L \<and> L \<noteq> 0)"
(is "?lhs \<longleftrightarrow> ?rhs")
proof
assume ?lhs then show ?rhs
using assms convergentD convergent_prod_imp_convergent convergent_prod_to_zero_iff by blast
next
assume ?rhs then show ?lhs
unfolding prod_defs
by (rule_tac x="0" in exI) (auto simp: )
qed
lemma 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 add: convergent_prod_iff_nz_lim assms convergent_def limI)
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
lemma 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
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 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))"
proof -
from assms obtain M L
where L: "(\<lambda>n. \<Prod>k\<le>n. f (k + M)) \<longlonglongrightarrow> L" "L \<noteq> 0" and nz: "\<And>n. n \<ge> M \<Longrightarrow> f n \<noteq> 0"
by (auto simp: convergent_prod_altdef)
define C where "C = (\<Prod>k<m. f (k + M))"
from nz have [simp]: "C \<noteq> 0"
by (auto simp: C_def)
from L(1) have "(\<lambda>n. \<Prod>k\<le>n+m. f (k + M)) \<longlonglongrightarrow> L"
by (rule LIMSEQ_ignore_initial_segment)
also have "(\<lambda>n. \<Prod>k\<le>n+m. f (k + M)) = (\<lambda>n. C * (\<Prod>k\<le>n. f (k + M + m)))"
proof (rule ext, goal_cases)
case (1 n)
have "{..n+m} = {..<m} \<union> {m..n+m}" by auto
also have "(\<Prod>k\<in>\<dots>. f (k + M)) = C * (\<Prod>k=m..n+m. f (k + M))"
unfolding C_def by (rule prod.union_disjoint) auto
also have "(\<Prod>k=m..n+m. f (k + M)) = (\<Prod>k\<le>n. f (k + m + M))"
by (intro ext prod.reindex_bij_witness[of _ "\<lambda>k. k + m" "\<lambda>k. k - m"]) auto
finally show ?case by (simp add: add_ac)
qed
finally have "(\<lambda>n. C * (\<Prod>k\<le>n. f (k + M + m)) / C) \<longlonglongrightarrow> L / C"
by (intro tendsto_divide tendsto_const) auto
hence "(\<lambda>n. \<Prod>k\<le>n. f (k + M + m)) \<longlonglongrightarrow> L / C" by simp
moreover from \<open>L \<noteq> 0\<close> have "L / C \<noteq> 0" by simp
ultimately show ?thesis
unfolding prod_defs by blast
qed
lemma 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)
lemma 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 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. gen_has_prod f 0 p"
using assms
unfolding convergent_prod_def
proof (clarsimp simp: prod_defs)
fix M p
assume "(\<lambda>n. \<Prod>i\<le>n. f (i + M)) \<longlonglongrightarrow> p" "p \<noteq> 0"
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 "... = 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 show "\<exists>p. (\<lambda>n. prod f {..n}) \<longlonglongrightarrow> p \<and> p \<noteq> 0"
using \<open>p \<noteq> 0\<close> assms
by (rule_tac x="prod f {..<M} * p" in exI) auto
qed
lemma prodinf_eq_lim:
fixes f :: "nat \<Rightarrow> 'a :: {idom,topological_semigroup_mult,t2_space}"
assumes "convergent_prod f" "\<And>i. f i \<noteq> 0"
shows "prodinf f = lim (\<lambda>n. \<Prod>i\<le>n. f i)"
using assms convergent_prod_offset_0 [OF assms]
by (simp add: prod_defs lim_def) (metis (no_types) assms(1) convergent_prod_to_zero_iff)
lemma has_prod_one[simp, intro]: "(\<lambda>n. 1) has_prod 1"
unfolding prod_defs by auto
lemma convergent_prod_one[simp, intro]: "convergent_prod (\<lambda>n. 1)"
unfolding prod_defs by auto
lemma prodinf_cong: "(\<And>n. f n = g n) \<Longrightarrow> prodinf f = prodinf g"
by presburger
lemma convergent_prod_cong:
fixes f g :: "nat \<Rightarrow> 'a::{field,topological_semigroup_mult,t2_space}"
assumes ev: "eventually (\<lambda>x. f x = g x) sequentially" and f: "\<And>i. f i \<noteq> 0" and g: "\<And>i. g i \<noteq> 0"
shows "convergent_prod f = convergent_prod g"
proof -
from assms obtain N where N: "\<forall>n\<ge>N. f n = g n"
by (auto simp: eventually_at_top_linorder)
define C where "C = (\<Prod>k<N. f k / g k)"
with g have "C \<noteq> 0"
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 ... = prod f {..<N} * prod f {N..n}"
by (intro prod.union_disjoint) auto
also from N have "prod f {N..n} = prod g {N..n}"
by (intro prod.cong) simp_all
also have "prod f {..<N} * prod g {N..n} = C * (prod g {..<N} * prod g {N..n})"
unfolding C_def by (simp add: g prod_dividef)
also have "prod g {..<N} * prod g {N..n} = prod g ({..<N} \<union> {N..n})"
by (intro prod.union_disjoint [symmetric]) auto
also from elim have "{..<N} \<union> {N..n} = {..n}"
by auto
finally show "prod f {..n} = C * prod g {..n}" .
qed
then have cong: "convergent (\<lambda>n. prod f {..n}) = convergent (\<lambda>n. C * prod g {..n})"
by (rule convergent_cong)
show ?thesis
proof
assume cf: "convergent_prod f"
then have "\<not> (\<lambda>n. prod g {..n}) \<longlonglongrightarrow> 0"
using tendsto_mult_left * convergent_prod_to_zero_iff f filterlim_cong by fastforce
then show "convergent_prod g"
by (metis convergent_mult_const_iff \<open>C \<noteq> 0\<close> cong cf convergent_LIMSEQ_iff convergent_prod_iff_convergent convergent_prod_imp_convergent g)
next
assume cg: "convergent_prod g"
have "\<exists>a. C * a \<noteq> 0 \<and> (\<lambda>n. prod g {..n}) \<longlonglongrightarrow> a"
by (metis (no_types) \<open>C \<noteq> 0\<close> cg convergent_prod_iff_nz_lim divide_eq_0_iff g nonzero_mult_div_cancel_right)
then show "convergent_prod f"
using "*" tendsto_mult_left filterlim_cong
by (fastforce simp add: convergent_prod_iff_nz_lim f)
qed
qed
lemma has_prod_finite:
fixes f :: "nat \<Rightarrow> 'a::{idom,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 add: 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: gen_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 eq1: "(\<Prod>i\<le>n + Max N. f (Suc (i + Max ?Z))) = prod f {Suc (Max ?Z)..n + (Max N) + Suc (Max ?Z)}" for n
apply (rule sym)
apply (rule prod.reindex_cong [where l = "\<lambda>i. i + Suc (Max ?Z)"])
apply (auto simp: )
apply (simp add: inj_on_def)
apply (auto simp: image_iff)
using le_Suc_ex by fastforce
have eq: "prod f {Suc (Max ?Z)..n + (Max N) + Suc (Max ?Z)} = ?q" for n
apply (rule prod.mono_neutral_right)
apply (auto simp: )
using Max.coboundedI assms(1) f by blast
have q: "gen_has_prod f (Suc (Max ?Z)) ?q"
apply (auto simp: gen_has_prod_def)
apply (rule LIMSEQ_offset[of _ "(Max N)"])
apply (simp add: eq1 eq atLeast0LessThan del: add_Suc_right)
done
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 has_prod_0:
fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
assumes "\<And>n. f n = 1"
shows "f has_prod 1"
by (simp add: assms has_prod_cong)
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. gen_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
end