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Datei:
Arcwise_Connected.thy
Sprache: Isabelle
(*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 \ 'b :: linordered_semidom"
assumes "\x. x \ A \ f x \ 0"
shows "sum f A \ (\x\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 * (\x\A. 1) \ (\x\A. 1 + f x) + f x * (\x\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 \ real"
assumes "\x. x \ A \ f x \ 0"
shows "prod (\x. 1 + f x) A \ exp (sum f A)"
using assms
proof (induction A rule: infinite_finite_induct)
case (insert x A)
have "(1 + f x) * (\x\A. 1 + f x) \ 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: "(\x::real. ln (1 + x) / x) \0\ 1"
proof (rule lhopital)
show "(\x::real. ln (1 + x)) \0\ 0"
by (rule tendsto_eq_intros refl | simp)+
have "eventually (\x::real. x \ {-1/2<..<1/2}) (nhds 0)"
by (rule eventually_nhds_in_open) auto
hence *: "eventually (\x::real. x \ {-1/2<..<1/2}) (at 0)"
by (rule filter_leD [rotated]) (simp_all add: at_within_def)
show "eventually (\x::real. ((\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 (\x::real. ((\x. x) has_field_derivative 1) (at x)) (at 0)"
using * by eventually_elim (auto intro!: derivative_eq_intros simp: field_simps)
show "\\<^sub>F x in at 0. x \ 0" by (auto simp: at_within_def eventually_inf_principal)
show "(\x::real. inverse (1 + x) / 1) \0\ 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 \ (\n. \i\n. f (i+M)) \ p \ p \ 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 \ 'a::{t2_space, comm_semiring_1}) \ 'a \ bool" (infixr "has'_prod" 80)
where "f has_prod p \ raw_has_prod f 0 p \ (\i q. p = 0 \ f i = 0 \ 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 \ \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 "\" 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 \ g has_prod z \ f has_prod z"
by simp
lemma has_prod_cong: "(\n. f n = g n) \ f has_prod c \ g has_prod c"
by presburger
lemma raw_has_prod_nonzero [simp]: "\ raw_has_prod f M 0"
by (simp add: raw_has_prod_def)
lemma raw_has_prod_eq_0:
fixes f :: "nat \ 'a::{semidom,t2_space}"
assumes p: "raw_has_prod f m p" and i: "f i = 0" "i \ m"
shows "p = 0"
proof -
have eq0: "(\k\n. f (k+m)) = 0" if "i - m \ n" for n
proof -
have "\k\n. f (k + m) = 0"
using i that by auto
then show ?thesis
by auto
qed
have "(\n. \i\n. f (i + m)) \ 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 \ raw_has_prod (\n. f (Suc n)) M a"
unfolding raw_has_prod_def by auto
lemma has_prod_0_iff: "f has_prod 0 \ (\i. f i = 0 \ (\p. raw_has_prod f (Suc i) p))"
by (simp add: has_prod_def)
lemma has_prod_unique2:
fixes f :: "nat \ '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 \ 'a :: {semidom,t2_space}"
shows "f has_prod s \ s = prodinf f"
by (simp add: has_prod_unique2 prodinf_def the_equality)
lemma convergent_prod_altdef:
fixes f :: "nat \ 'a :: {t2_space,comm_semiring_1}"
shows "convergent_prod f \ (\M L. (\n\M. f n \ 0) \ (\n. \i\n. f (i+M)) \ L \ L \ 0)"
proof
assume "convergent_prod f"
then obtain M L where *: "(\n. \i\n. f (i+M)) \ L" "L \ 0"
by (auto simp: prod_defs)
have "f i \ 0" if "i \ M" for i
proof
assume "f i = 0"
have **: "eventually (\n. (\i\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 "(\n. (\i\n. f (i+M))) \ 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 "(\M L. (\n\M. f n \ 0) \ (\n. \i\n. f (i+M)) \ L \ L \ 0)"
by blast
qed (auto simp: prod_defs)
subsection\<open>Absolutely convergent products\<close>
definition\<^marker>\<open>tag important\<close> abs_convergent_prod :: "(nat \<Rightarrow> _) \<Rightarrow> bool" where
"abs_convergent_prod f \ convergent_prod (\i. 1 + norm (f i - 1))"
lemma abs_convergent_prodI:
assumes "convergent (\n. \i\n. 1 + norm (f i - 1))"
shows "abs_convergent_prod f"
proof -
from assms obtain L where L: "(\n. \i\n. 1 + norm (f i - 1)) \ L"
by (auto simp: convergent_def)
have "L \ 1"
proof (rule tendsto_le)
show "eventually (\n. (\i\n. 1 + norm (f i - 1)) \ 1) sequentially"
proof (intro always_eventually allI)
fix n
have "(\i\n. 1 + norm (f i - 1)) \ (\i\n. 1)"
by (intro prod_mono) auto
thus "(\i\n. 1 + norm (f i - 1)) \ 1" by simp
qed
qed (use L in simp_all)
hence "L \ 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 \ 'a :: {topological_semigroup_mult,t2_space,idom}"
assumes "convergent_prod f"
shows convergent_prod_imp_convergent: "convergent (\n. \i\n. f i)"
and convergent_prod_to_zero_iff [simp]: "(\n. \i\n. f i) \ 0 \ (\i. f i = 0)"
proof -
from assms obtain M L
where M: "\n. n \ M \ f n \ 0" and "(\n. \i\n. f (i + M)) \ L" and "L \ 0"
by (auto simp: convergent_prod_altdef)
note this(2)
also have "(\n. \i\n. f (i + M)) = (\n. \i=M..M+n. f i)"
by (intro ext prod.reindex_bij_witness[of _ "\n. n - M" "\n. n + M"]) auto
finally have "(\n. (\ii=M..M+n. f i)) \ (\i
by (intro tendsto_mult tendsto_const)
also have "(\n. (\ii=M..M+n. f i)) = (\n. (\i\{..{M..M+n}. f i))"
by (subst prod.union_disjoint) auto
also have "(\n. {.. {M..M+n}) = (\n. {..n+M})" by auto
finally have lim: "(\n. prod f {..n}) \ prod f {..
by (rule LIMSEQ_offset)
thus "convergent (\n. \i\n. f i)"
by (auto simp: convergent_def)
show "(\n. \i\n. f i) \ 0 \ (\i. f i = 0)"
proof
assume "\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 "(\i
with lim show "(\n. \i\n. f i) \ 0" by simp
next
assume "(\n. \i\n. f i) \ 0"
from tendsto_unique[OF _ this lim] and \<open>L \<noteq> 0\<close>
show "\i. f i = 0" by auto
qed
qed
lemma convergent_prod_iff_nz_lim:
fixes f :: "nat \ 'a :: {topological_semigroup_mult,t2_space,idom}"
assumes "\i. f i \ 0"
shows "convergent_prod f \ (\L. (\n. \i\n. f i) \ L \ L \ 0)"
(is "?lhs \ ?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 \ 'a :: {topological_semigroup_mult,t2_space,idom}"
assumes "\i. f i \ 0"
shows "convergent_prod f \ convergent (\n. \i\n. f i) \ lim (\n. \i\n. f i) \ 0"
by (force simp: convergent_prod_iff_nz_lim assms convergent_def limI)
lemma bounded_imp_convergent_prod:
fixes a :: "nat \ real"
assumes 1: "\n. a n \ 1" and bounded: "\n. (\i\n. a i) \ B"
shows "convergent_prod a"
proof -
have "bdd_above (range(\n. \i\n. a i))"
by (meson bdd_aboveI2 bounded)
moreover have "incseq (\n. \i\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: "(\n. \i\n. a i) \ p"
using LIMSEQ_incseq_SUP by blast
then have "p \ 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 \ 'a :: {one,real_normed_vector}"
shows "abs_convergent_prod f \ convergent (\n. \i\n. 1 + norm (f i - 1))"
proof
assume "abs_convergent_prod f"
thus "convergent (\n. \i\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 \ real"
assumes "\x. x \ A \ f x \ {0..1}"
shows "1 - sum f A \ (\x\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 * (\x\A. 1 - f x) \ (\x\A. 1 - f x) + f x * (\x\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 \ 'a :: {real_normed_div_algebra,comm_ring_1}"
shows "norm (prod (\n. 1 + f n) A - 1) \ prod (\n. 1 + norm (f n)) A - 1"
proof (induction A rule: infinite_finite_induct)
case (insert x A)
from insert.hyps have
"norm ((\n\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 "\ \ norm ((\n\A. 1 + f n) - 1) + norm (f x * (\n\A. 1 + f n))"
by (rule norm_triangle_ineq)
also have "norm (f x * (\n\A. 1 + f n)) = norm (f x) * (\x\A. norm (1 + f x))"
by (simp add: prod_norm norm_mult)
also have "(\x\A. norm (1 + f x)) \ (\x\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 "(\n\A. 1 + norm (f n)) - 1 + norm (f x) * (\x\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 \ 'a :: {t2_space,comm_semiring_1}"
assumes "convergent_prod f"
shows "eventually (\n. f n \ 0) sequentially"
using assms by (auto simp: eventually_at_top_linorder convergent_prod_altdef)
lemma convergent_prod_imp_LIMSEQ:
fixes f :: "nat \ 'a :: {real_normed_field}"
assumes "convergent_prod f"
shows "f \ 1"
proof -
from assms obtain M L where L: "(\n. \i\n. f (i+M)) \ L" "\n. n \ M \ f n \ 0" "L \ 0"
by (auto simp: convergent_prod_altdef)
hence L': "(\n. \i\Suc n. f (i+M)) \ L" by (subst filterlim_sequentially_Suc)
have "(\n. (\i\Suc n. f (i+M)) / (\i\n. f (i+M))) \ L / L"
using L L' by (intro tendsto_divide) simp_all
also from L have "L / L = 1" by simp
also have "(\n. (\i\Suc n. f (i+M)) / (\i\n. f (i+M))) = (\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 \ 'a :: real_normed_div_algebra"
assumes "abs_convergent_prod f"
shows "summable (\i. norm (f i - 1))"
proof -
from assms have "convergent (\n. \i\n. 1 + norm (f i - 1))"
unfolding abs_convergent_prod_def by (rule convergent_prod_imp_convergent)
then obtain L where L: "(\n. \i\n. 1 + norm (f i - 1)) \ L"
unfolding convergent_def by blast
have "convergent (\n. \i\n. norm (f i - 1))"
proof (rule Bseq_monoseq_convergent)
have "eventually (\n. (\i\n. 1 + norm (f i - 1)) < L + 1) sequentially"
using L(1) by (rule order_tendstoD) simp_all
hence "\\<^sub>F x in sequentially. norm (\i\x. norm (f i - 1)) \ L + 1"
proof eventually_elim
case (elim n)
have "norm (\i\n. norm (f i - 1)) = (\i\n. norm (f i - 1))"
unfolding real_norm_def by (intro abs_of_nonneg sum_nonneg) simp_all
also have "\ \ (\i\n. 1 + norm (f i - 1))" by (rule sum_le_prod) auto
also have "\ < L + 1" by (rule elim)
finally show ?case by simp
qed
thus "Bseq (\n. \i\n. norm (f i - 1))" by (rule BfunI)
next
show "monoseq (\n. \i\n. norm (f i - 1))"
by (rule mono_SucI1) auto
qed
thus "summable (\i. norm (f i - 1))" by (simp add: summable_iff_convergent')
qed
lemma summable_imp_abs_convergent_prod:
fixes f :: "nat \ 'a :: real_normed_div_algebra"
assumes "summable (\i. norm (f i - 1))"
shows "abs_convergent_prod f"
proof (intro abs_convergent_prodI Bseq_monoseq_convergent)
show "monoseq (\n. \i\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 (\n. \i\n. 1 + norm (f i - 1))"
proof (rule Bseq_eventually_mono)
show "eventually (\n. norm (\i\n. 1 + norm (f i - 1)) \
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 "(\n. \i\n. norm (f i - 1)) \ (\i. norm (f i - 1))"
using sums_def_le by blast
hence "(\n. exp (\i\n. norm (f i - 1))) \ exp (\i. norm (f i - 1))"
by (rule tendsto_exp)
hence "convergent (\n. exp (\i\n. norm (f i - 1)))"
by (rule convergentI)
thus "Bseq (\n. exp (\i\n. norm (f i - 1)))"
by (rule convergent_imp_Bseq)
qed
qed
theorem abs_convergent_prod_conv_summable:
fixes f :: "nat \ 'a :: real_normed_div_algebra"
shows "abs_convergent_prod f \ summable (\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 \ 'a :: {comm_ring_1,real_normed_div_algebra}"
assumes "abs_convergent_prod f"
shows "f \ 1"
proof -
from assms have "summable (\n. norm (f n - 1))"
by (rule abs_convergent_prod_imp_summable)
from summable_LIMSEQ_zero[OF this] have "(\n. f n - 1) \ 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 \ 'a :: {comm_ring_1,real_normed_div_algebra}"
assumes "abs_convergent_prod f"
shows "eventually (\n. f n \ 0) sequentially"
proof -
from assms have "f \ 1"
by (rule abs_convergent_prod_imp_LIMSEQ)
hence "eventually (\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 (\n. f (n + m))"
shows "convergent_prod f"
proof -
from assms obtain M L where "(\n. \k\n. f (k + (M + m))) \ L" "L \ 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 (\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 \ 'a :: real_normed_field"
assumes "raw_has_prod f M p" "N \ M"
obtains q where "raw_has_prod f N q"
proof -
have p: "(\n. \k\n. f (k + M)) \ p" and "p \ 0"
using assms by (auto simp: raw_has_prod_def)
then have nz: "\n. n \ M \ f n \ 0"
using assms by (auto simp: raw_has_prod_eq_0)
define C where "C = (\k
from nz have [simp]: "C \ 0"
by (auto simp: C_def)
from p have "(\i. \k\i + (N-M). f (k + M)) \ p"
by (rule LIMSEQ_ignore_initial_segment)
also have "(\i. \k\i + (N-M). f (k + M)) = (\n. C * (\k\n. f (k + N)))"
proof (rule ext, goal_cases)
case (1 n)
have "{..n+(N-M)} = {..<(N-M)} \ {(N-M)..n+(N-M)}" by auto
also have "(\k\\. f (k + M)) = C * (\k=(N-M)..n+(N-M). f (k + M))"
unfolding C_def by (rule prod.union_disjoint) auto
also have "(\k=(N-M)..n+(N-M). f (k + M)) = (\k\n. f (k + (N-M) + M))"
by (intro ext prod.reindex_bij_witness[of _ "\k. k + (N-M)" "\k. k - (N-M)"]) auto
finally show ?case
using \<open>N \<ge> M\<close> by (simp add: add_ac)
qed
finally have "(\n. C * (\k\n. f (k + N)) / C) \ p / C"
by (intro tendsto_divide tendsto_const) auto
hence "(\n. \k\n. f (k + N)) \ 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 \ 'a :: real_normed_field"
assumes "convergent_prod f"
shows "convergent_prod (\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 \ 'a :: real_normed_field"
assumes f: "convergent_prod f" and nz: "\i. i \ M \ f i \ 0"
shows "\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 (\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 \ 'a :: {real_normed_div_algebra,complete_space,comm_ring_1}"
assumes "abs_convergent_prod f"
shows "convergent_prod f"
proof -
from assms have "eventually (\n. f n \ 0) sequentially"
by (rule abs_convergent_prod_imp_ev_nonzero)
then obtain N where N: "f n \ 0" if "n \ N" for n
by (auto simp: eventually_at_top_linorder)
let ?P = "\n. \i\n. f (i + N)" and ?Q = "\n. \i\n. 1 + norm (f (i + N) - 1)"
have "Cauchy ?P"
proof (rule CauchyI', goal_cases)
case (1 \<epsilon>)
from assms have "abs_convergent_prod (\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) < \" if "m \ M" "n \ 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} \ {m<..n}" by auto
hence "norm (?P n - ?P m) = norm (?P m * (\k\{m<..n}. f (k + N)) - ?P m)"
by (subst prod.union_disjoint [symmetric]) (auto simp: algebra_simps)
also have "\ = norm (?P m * ((\k\{m<..n}. f (k + N)) - 1))"
by (simp add: algebra_simps)
also have "\ = (\k\m. norm (f (k + N))) * norm ((\k\{m<..n}. f (k + N)) - 1)"
by (simp add: norm_mult prod_norm)
also have "\ \ ?Q m * ((\k\{m<..n}. 1 + norm (f (k + N) - 1)) - 1)"
using norm_prod_minus1_le_prod_minus1[of "\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 "\ = ?Q m * (\k\{m<..n}. 1 + norm (f (k + N) - 1)) - ?Q m"
by (simp add: algebra_simps)
also have "?Q m * (\k\{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}\{m<..n} = {..n}" by auto
also have "?Q n - ?Q m \ norm (?Q n - ?Q m)" by simp
also from 1 have "\ < \" by (intro M) auto
finally show ?case .
qed
qed
hence conv: "convergent ?P" by (rule Cauchy_convergent)
then obtain L where L: "?P \ L"
by (auto simp: convergent_def)
have "L \ 0"
proof
assume [simp]: "L = 0"
from tendsto_norm[OF L] have limit: "(\n. \k\n. norm (f (k + N))) \ 0"
by (simp add: prod_norm)
from assms have "(\n. f (n + N)) \ 1"
by (intro abs_convergent_prod_imp_LIMSEQ abs_convergent_prod_ignore_initial_segment)
hence "eventually (\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 \ M0" for n
by (auto simp: eventually_at_top_linorder)
{
fix M assume M: "M \ M0"
with M0 have M: "norm (f (n + N) - 1) < 1" if "n \ M" for n using that by simp
have "(\n. \k\n. 1 - norm (f (k+M+N) - 1)) \ 0"
proof (rule tendsto_sandwich)
show "eventually (\n. (\k\n. 1 - norm (f (k+M+N) - 1)) \ 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) \ norm (f (i + M + N))" for i
using norm_triangle_ineq3[of "f (i + M + N)" 1] by simp
thus "eventually (\n. (\k\n. 1 - norm (f (k+M+N) - 1)) \ (\k\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 = (\k
from N have [simp]: "C \ 0" by (auto simp: C_def)
from L have "(\n. norm (\k\n+M. f (k + N))) \ 0"
by (intro LIMSEQ_ignore_initial_segment) (simp add: tendsto_norm_zero_iff)
also have "(\n. norm (\k\n+M. f (k + N))) = (\n. C * (\k\n. norm (f (k + M + N))))"
proof (rule ext, goal_cases)
case (1 n)
have "{..n+M} = {.. {M..n+M}" by auto
also have "norm (\k\\. f (k + N)) = C * norm (\k=M..n+M. f (k + N))"
unfolding C_def by (subst prod.union_disjoint) (auto simp: norm_mult prod_norm)
also have "(\k=M..n+M. f (k + N)) = (\k\n. f (k + N + M))"
by (intro prod.reindex_bij_witness[of _ "\i. i + M" "\i. i - M"]) auto
finally show ?case by (simp add: add_ac prod_norm)
qed
finally have "(\n. C * (\k\n. norm (f (k + M + N))) / C) \ 0 / C"
by (intro tendsto_divide tendsto_const) auto
thus "(\n. \k\n. norm (f (k + M + N))) \ 0" by simp
qed simp_all
have "1 - (\i. norm (f (i + M + N) - 1)) \ 0"
proof (rule tendsto_le)
show "eventually (\n. 1 - (\k\n. norm (f (k+M+N) - 1)) \
(\<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 "(\n. \k\n. 1 - norm (f (k+M+N) - 1)) \ 0" by fact
show "(\n. 1 - (\k\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 "(\i. norm (f (i + M + N) - 1)) \ 1" by simp
also have "\ + (\ii. 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 + (\i (\i. norm (f (i + N) - 1))" by simp
} note * = this
have "1 + (\i. norm (f (i + N) - 1)) \ (\i. norm (f (i + N) - 1))"
proof (rule tendsto_le)
show "(\M. 1 + (\i 1 + (\i. norm (f (i + N) - 1))"
by (intro tendsto_intros summable_LIMSEQ summable_ignore_initial_segment
abs_convergent_prod_imp_summable assms)
show "eventually (\M. 1 + (\i (\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 \ 'a :: {idom,topological_semigroup_mult,t2_space}"
assumes "raw_has_prod f M p"
obtains i where "i "f i = 0" | p where "raw_has_prod f 0 p"
proof -
have "(\n. \i\n. f (i + M)) \ p" "p \ 0"
using assms unfolding raw_has_prod_def by blast+
then have "(\n. prod f {..i\n. f (i + M))) \ prod f {..
by (metis tendsto_mult_left)
moreover have "prod f {..i\n. f (i + M)) = prod f {..n+M}" for n
proof -
have "{..n+M} = {.. {M..n+M}"
by auto
then have "prod f {..n+M} = prod f {..
by simp (subst prod.union_disjoint; force)
also have "\ = prod f {..i\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 "(\n. prod f {..n}) \ prod f {..
by (auto intro: LIMSEQ_offset [where k=M])
then have "raw_has_prod f 0 (prod f {.. if "\i 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 \ 'a :: {idom,topological_semigroup_mult,t2_space}"
assumes "convergent_prod f" "\i. f i \ 0"
shows "\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 \ 'a :: {idom,topological_semigroup_mult,t2_space}"
assumes "convergent_prod f" "\i. f i \ 0"
shows "prodinf f = lim (\n. \i\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]: "(\n. 1) has_prod 1"
unfolding prod_defs by auto
lemma convergent_prod_one[simp, intro]: "convergent_prod (\n. 1)"
unfolding prod_defs by auto
lemma prodinf_cong: "(\n. f n = g n) \ prodinf f = prodinf g"
by presburger
lemma convergent_prod_cong:
fixes f g :: "nat \ 'a::{field,topological_semigroup_mult,t2_space}"
assumes ev: "eventually (\x. f x = g x) sequentially" and f: "\i. f i \ 0" and g: "\i. g i \ 0"
shows "convergent_prod f = convergent_prod g"
proof -
from assms obtain N where N: "\n\N. f n = g n"
by (auto simp: eventually_at_top_linorder)
define C where "C = (\k
with g have "C \ 0"
by (simp add: f)
have *: "eventually (\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..n}"
by auto
also have "prod f \ = prod f {..
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 {..
unfolding C_def by (simp add: g prod_dividef)
also have "prod g {.. {N..n})"
by (intro prod.union_disjoint [symmetric]) auto
also from elim have "{.. {N..n} = {..n}"
by auto
finally show "prod f {..n} = C * prod g {..n}" .
qed
then have cong: "convergent (\n. prod f {..n}) = convergent (\n. C * prod g {..n})"
by (rule convergent_cong)
show ?thesis
proof
assume cf: "convergent_prod f"
then have "\ (\n. prod g {..n}) \ 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 "\a. C * a \ 0 \ (\n. prod g {..n}) \ 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 \ 'a::{semidom,t2_space}"
assumes [simp]: "finite N"
and f: "\n. n \ N \ f n = 1"
shows "f has_prod (\n\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 \ {..n + Suc (Max N)}"
by (auto simp: le_Suc_eq trans_le_add2)
show "\i\{..n + Suc (Max N)} - N. f i = 1"
using f by blast
qed auto
show ?thesis
proof (cases "\n\N. f n \ 0")
case True
then have "prod f N \ 0"
by simp
moreover have "(\n. prod f {..n}) \ 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 \ N" "f k = 0"
by auto
let ?Z = "{n \ N. f n = 0}"
have maxge: "Max ?Z \ 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 \ 0"
using maxge Suc_n_not_le_n le_trans by force
have eq: "(\i\n + Max N. f (Suc (i + Max ?Z))) = ?q" for n
proof -
have "(\i\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 = "\i. i + Suc (Max ?Z)", THEN sym])
show "{Suc (Max ?Z)..n + Max N + Suc (Max ?Z)} = (\i. i + Suc (Max ?Z)) ` {..n + Max N}"
using le_Suc_ex by fastforce
qed (auto simp: inj_on_def)
also have "\ = ?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 "(\n. \i\n. f (Suc (i + Max ?Z))) \ ?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 \ 'a::{semidom,t2_space}"
assumes "\n. f n = 1"
shows "f has_prod 1"
by (simp add: assms has_prod_cong)
lemma prodinf_zero[simp]: "prodinf (\n. 1::'a::real_normed_field) = 1"
using has_prod_unique by force
lemma convergent_prod_finite:
fixes f :: "nat \ 'a::{idom,t2_space}"
assumes "finite N" "\n. n \ N \ f n = 1"
shows "convergent_prod f"
proof -
have "\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 \ 'a::{idom,t2_space}"
shows "finite A \ (\r. if r \ A then f r else 1) has_prod (\r\A. f r)"
using has_prod_finite[of A "(\r. if r \ A then f r else 1)"]
by simp
lemma has_prod_If_finite:
fixes f :: "nat \ 'a::{idom,t2_space}"
shows "finite {r. P r} \ (\r. if P r then f r else 1) has_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 \ 'a::{idom,t2_space}"
shows "finite A \ convergent_prod (\r. if r \ A then f r else 1)"
by (simp add: convergent_prod_finite)
lemma convergent_prod_If_finite[simp, intro]:
fixes f :: "nat \ 'a::{idom,t2_space}"
shows "finite {r. P r} \ convergent_prod (\r. if P r then f r else 1)"
using convergent_prod_def has_prod_If_finite has_prod_def by fastforce
lemma has_prod_single:
fixes f :: "nat \ 'a::{idom,t2_space}"
shows "(\r. if r = i then f r else 1) has_prod f i"
using has_prod_If_finite[of "\r. r = i"] by simp
context
fixes f :: "nat \ 'a :: real_normed_field"
begin
lemma convergent_prod_imp_has_prod:
assumes "convergent_prod f"
shows "\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 \ 0"
using raw_has_prod_nonzero by blast
with p have fnz: "f i \ 0" if "i \ M" for i
using raw_has_prod_eq_0 that by blast
define C where "C = (\n
show ?thesis
proof (cases "\n\M. f n \ 0")
case True
then have "C \ 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 \ 0" if "i > ?N" for i
by (metis (mono_tags, lifting) Greatest_le_nat fnz leD linear that)
then have "\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 \ 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 \ (\n. \i\n. f i) \ 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 \ convergent_prod f \ prodinf f = x"
proof
assume "f has_prod x"
then show "convergent_prod f \ 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 \ f has_prod prodinf f"
by (auto simp: has_prod_iff convergent_prod_has_prod)
lemma prodinf_finite:
assumes N: "finite N"
and f: "\n. n \ N \ f n = 1"
shows "prodinf f = (\n\N. f n)"
using has_prod_finite[OF assms, THEN has_prod_unique] by simp
end
subsection\<^marker>\<open>tag unimportant\<close> \<open>Infinite products on ordered topological monoids\<close>
lemma LIMSEQ_prod_0:
fixes f :: "nat \ 'a::{semidom,topological_space}"
assumes "f i = 0"
shows "(\n. prod f {..n}) \ 0"
proof (subst tendsto_cong)
show "\\<^sub>F n in sequentially. prod f {..n} = 0"
proof
show "prod f {..n} = 0" if "n \ i" for n
using that assms by auto
qed
qed auto
lemma LIMSEQ_prod_nonneg:
fixes f :: "nat \ 'a::{linordered_semidom,linorder_topology}"
assumes 0: "\n. 0 \ f n" and a: "(\n. prod f {..n}) \ a"
shows "a \ 0"
by (simp add: "0" prod_nonneg LIMSEQ_le_const [OF a])
context
fixes f :: "nat \ 'a::{linordered_semidom,linorder_topology}"
begin
lemma has_prod_le:
assumes f: "f has_prod a" and g: "g has_prod b" and le: "\n. 0 \ f n \ f n \ g n"
shows "a \ b"
proof (cases "a=0 \ b=0")
case True
then show ?thesis
proof
assume [simp]: "a=0"
have "b \ 0"
proof (rule LIMSEQ_prod_nonneg)
show "(\n. prod g {..n}) \ 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: "\n. 0 \ f n \ f n \ g n"
shows "prodinf f \ prodinf g"
using has_prod_le [OF assms] has_prod_unique f g by blast
end
lemma prod_le_prodinf:
fixes f :: "nat \ 'a::{linordered_idom,linorder_topology}"
assumes "f has_prod a" "\i. 0 \ f i" "\i. i\n \ 1 \ f i"
shows "prod f {.. 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 \ 'a::{linordered_idom,linorder_topology}"
assumes "f has_prod a" "\i. 1 \ f i"
shows "1 \ 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 \ real"
assumes "convergent_prod f" "\n. prod f {.. x"
shows "prodinf f \ x"
by (metis lessThan_Suc_atMost assms convergent_prod_LIMSEQ LIMSEQ_le_const2)
lemma prodinf_eq_one_iff [simp]:
fixes f :: "nat \ real"
assumes f: "convergent_prod f" and ge1: "\n. 1 \ f n"
shows "prodinf f = 1 \ (\n. f n = 1)"
proof
assume "prodinf f = 1"
then have "(\n. \i 1"
using convergent_prod_LIMSEQ[of f] assms by (simp add: LIMSEQ_lessThan_iff_atMost)
then have "\i. (\n\{i}. f n) \ 1"
proof (rule LIMSEQ_le_const)
have "1 \ prod f n" for n
by (simp add: ge1 prod_ge_1)
have "prod f {.. 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 "(\n\{i}. f n) \ prod f {.. Suc i" for i n
by (metis mult.left_neutral order_refl prod.cong prod.neutral_const prod.lessThan_Suc)
then show "\N. \n\N. (\n\{i}. f n) \ prod f {..
by blast
qed
with ge1 show "\n. f n = 1"
by (auto intro!: antisym)
qed (metis prodinf_zero fun_eq_iff)
lemma prodinf_pos_iff:
fixes f :: "nat \ real"
assumes "convergent_prod f" "\n. 1 \ f n"
shows "1 < prodinf f \ (\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 \ real"
assumes "convergent_prod f" "\n. 1 \ f n" "1 < f i"
shows "1 < prodinf f"
proof -
have "1 < (\n
using assms by (intro less_1_prod2[where i=i]) auto
also have "\ \ 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 \ real"
shows "\convergent_prod f; \n. 1 < f n\ \ 1 < prodinf f"
by (intro less_1_prodinf2[where i=1]) (auto intro: less_imp_le)
lemma prodinf_nonzero:
fixes f :: "nat \ 'a :: {idom,topological_semigroup_mult,t2_space}"
assumes "convergent_prod f" "\i. f i \ 0"
shows "prodinf f \ 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 \ real"
assumes f: "convergent_prod f" and 0: "\i. f i > 0"
shows "0 < prodinf f"
proof -
have "prodinf f \ 0"
by (metis assms less_irrefl prodinf_nonzero)
moreover have "0 < (\n
by (simp add: 0 prod_pos)
then have "prodinf f \ 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 \ real"
assumes f: "convergent_prod f" and 1: "\m. m\n \ 1 \ f m" and 0: "\m. 0 < f m" and i: "n \ i" "1 < f i"
shows "prod f {..
proof -
have "prod f {.. prod f {..
by (rule prod_mono2) (use assms less_le in auto)
then have "prod f {..
using mult_less_le_imp_less[of 1 "f i" "prod f {.. "prod f {..] assms
by (simp add: prod_pos)
moreover have "prod f {.. 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 \ real"
assumes f: "convergent_prod f" and 1: "\m. m\n \ 1 < f m" and 0: "\m. 0 < f m"
shows "prod f {..
by (meson "0" "1" f le_less prod_less_prodinf2)
lemma raw_has_prodI_bounded:
fixes f :: "nat \ real"
assumes pos: "\n. 1 \ f n"
and le: "\n. (\i x"
shows "\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 (\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}) \ 0"
by auto
show "incseq (\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 \ real"
assumes "\n. 1 \ f n" "\n. (\i 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 \ 'a::{t2_space,topological_semigroup_mult,idom}"
begin
lemma raw_has_prod_mult: "\raw_has_prod f M a; raw_has_prod g M b\ \ raw_has_prod (\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: "\f has_prod a; g has_prod b; a \ 0; b \ 0\ \ (\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 \ 'a::real_normed_field"
begin
lemma has_prod_mult:
assumes f: "f has_prod a" and g: "g has_prod b"
shows "(\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 (\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 (\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 (\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 "\M p. raw_has_prod (\n. f n * g n) M p"
using raw_has_prod_mult by blast
qed
lemma prodinf_mult: "convergent_prod f \ convergent_prod g \ prodinf f * prodinf g = (\n. f n * g n)"
by (intro has_prod_unique has_prod_mult convergent_prod_has_prod)
end
context
fixes f :: "'i \ nat \ 'a::real_normed_field"
and I :: "'i set"
begin
lemma has_prod_prod: "(\i. i \ I \ (f i) has_prod (x i)) \ (\n. \i\I. f i n) has_prod (\i\I. x i)"
by (induct I rule: infinite_finite_induct) (auto intro!: has_prod_mult)
lemma prodinf_prod: "(\i. i \ I \ convergent_prod (f i)) \ (\n. \i\I. f i n) = (\i\I. \n. f i n)"
using has_prod_unique[OF has_prod_prod, OF convergent_prod_has_prod] by simp
lemma convergent_prod_prod: "(\i. i \ I \ convergent_prod (f i)) \ convergent_prod (\n. \i\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 \ 'a::real_normed_field"
begin
lemma raw_has_prod_Suc_iff: "raw_has_prod f M (a * f M) \ raw_has_prod (\n. f (Suc n)) M a \ f M \ 0"
proof -
have "raw_has_prod f M (a * f M) \ (\i. \j\Suc i. f (j+M)) \ a * f M \ a * f M \ 0"
by (subst filterlim_sequentially_Suc) (simp add: raw_has_prod_def)
also have "\ \ (\i. (\j\i. f (Suc j + M)) * f M) \ a * f M \ a * f M \ 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 "\ \ raw_has_prod (\n. f (Suc n)) M a \ f M \ 0"
proof safe
assume tends: "(\i. (\j\i. f (Suc j + M)) * f M) \ a * f M" and 0: "a * f M \ 0"
with tendsto_divide[OF tends tendsto_const, of "f M"]
show "raw_has_prod (\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 \ 0" shows "(\n. f (Suc n)) has_prod a \ 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 (\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 "\i. f i = 0 \ 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 "\ y. raw_has_prod (\n. f (Suc n)) i y"
using x by (auto simp: raw_has_prod_def)
then show "\i. f (Suc i) = 0 \ Ex (raw_has_prod (\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 (\n. f (Suc n)) = convergent_prod f"
proof
assume "convergent_prod f"
then obtain M L where M_nz:"\n\M. f n \ 0" and
M_L:"(\n. \i\n. f (i + M)) \ L" and "L \ 0"
unfolding convergent_prod_altdef by auto
have "(\n. \i\n. f (Suc (i + M))) \ L / f M"
proof -
have "(\n. \i\{0..Suc n}. f (i + M)) \ L"
using M_L
apply (subst (asm) filterlim_sequentially_Suc[symmetric])
using atLeast0AtMost by auto
then have "(\n. f M * (\i\{0..n}. f (Suc (i + M)))) \ L"
apply (subst (asm) prod.atLeast0_atMost_Suc_shift)
by simp
then have "(\n. (\i\{0..n}. f (Suc (i + M)))) \ 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 (\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 (\n. f (Suc n))"
then obtain M where "\L. (\n\M. f (Suc n) \ 0) \ (\n. \i\n. f (Suc (i + M))) \ L \ L \ 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 (\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 "(\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 (\n. inverse (f n))"
using assms unfolding convergent_prod_def by (blast intro: raw_has_prod_inverse elim: )
end
context
fixes f :: "nat \ 'a::real_normed_field"
begin
lemma raw_has_prod_Suc_iff': "raw_has_prod f M a \ raw_has_prod (\n. f (Suc n)) M (a / f M) \ f M \ 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 \ g has_prod b \ (\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 (\n. f n / g n)"
using f g has_prod_divide has_prod_iff by blast
lemma prodinf_divide: "convergent_prod f \ convergent_prod g \ prodinf f / prodinf g = (\n. f n / g n)"
by (intro has_prod_unique has_prod_divide convergent_prod_has_prod)
lemma prodinf_inverse: "convergent_prod f \ (\n. inverse (f n)) = inverse (\n. f n)"
by (intro has_prod_unique [symmetric] has_prod_inverse convergent_prod_has_prod)
lemma has_prod_Suc_imp:
assumes "(\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 (\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
"(\i q. a = 0 \ f (Suc i) = 0 \ raw_has_prod (\n. f (Suc n)) (Suc i) q)"
proof -
from that
obtain i q where "a = 0" "f (Suc i) = 0" "raw_has_prod (\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 "\i. i < n \ f i \ 0"
shows "(\i. f (i + n)) has_prod a \ f has_prod (a * (\i
using assms
proof (induct n arbitrary: a)
case 0
then show ?case by simp
next
case (Suc n)
then have "(\i. f (Suc i + n)) has_prod a \ (\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 "\i. i < n \ f i \ 0"
shows "(\i. f (i + n)) has_prod (a / (\i f has_prod a"
by (simp add: assms has_prod_iff_shift)
lemma has_prod_one_iff_shift:
assumes "\i. i < n \ f i = 1"
shows "(\i. f (i+n)) has_prod a \ (\i. f i) has_prod a"
by (simp add: assms has_prod_iff_shift)
lemma convergent_prod_iff_shift [simp]:
shows "convergent_prod (\i. f (i + n)) \ 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" "\i. i < n \ f i \ 0"
shows "(\i. f (i + n)) has_prod (a / (\i
using assms has_prod_iff_shift' by blast
lemma prodinf_divide_initial_segment:
assumes "convergent_prod f" "\i. i < n \ f i \ 0"
shows "(\i. f (i + n)) = (\i. f i) / (\i
by (rule has_prod_unique[symmetric]) (auto simp: assms has_prod_iff_shift)
lemma prodinf_split_initial_segment:
assumes "convergent_prod f" "\i. i < n \ f i \ 0"
shows "prodinf f = (\i. f (i + n)) * (\i
by (auto simp add: assms prodinf_divide_initial_segment)
lemma prodinf_split_head:
assumes "convergent_prod f" "f 0 \ 0"
shows "(\n. f (Suc n)) = prodinf f / f 0"
using prodinf_split_initial_segment[of 1] assms by simp
end
context
fixes f :: "nat \ 'a::real_normed_field"
begin
lemma convergent_prod_inverse_iff [simp]: "convergent_prod (\n. inverse (f n)) \ convergent_prod f"
by (auto dest: convergent_prod_inverse)
lemma convergent_prod_const_iff [simp]:
fixes c :: "'a :: {real_normed_field}"
shows "convergent_prod (\_. c) \ c = 1"
proof
assume "convergent_prod (\_. c)"
then show "c = 1"
using convergent_prod_imp_LIMSEQ LIMSEQ_unique by blast
next
assume "c = 1"
then show "convergent_prod (\_. c)"
by auto
qed
lemma has_prod_power: "f has_prod a \ (\i. f i ^ n) has_prod (a ^ n)"
by (induction n) (auto simp: has_prod_mult)
lemma convergent_prod_power: "convergent_prod f \ convergent_prod (\i. f i ^ n)"
by (induction n) (auto simp: convergent_prod_mult)
lemma prodinf_power: "convergent_prod f \ prodinf (\i. f i ^ n) = prodinf f ^ n"
by (metis has_prod_unique convergent_prod_imp_has_prod has_prod_power)
end
subsection\<open>Exponentials and logarithms\<close>
context
fixes f :: "nat \ 'a::{real_normed_field,banach}"
begin
lemma sums_imp_has_prod_exp:
assumes "f sums s"
shows "raw_has_prod (\i. exp (f i)) 0 (exp s)"
using assms continuous_on_exp [of UNIV "\x::'a. x"]
using continuous_on_tendsto_compose [of UNIV exp "(\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 (\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 (\i. exp (f i)) = exp (suminf f)"
proof -
have "f sums suminf f"
using assms by blast
then have "(\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 \ real"
assumes "\n. a n > 0"
shows "convergent_prod (\k. 1 + a k) \ summable a" (is "?lhs = ?rhs")
proof
assume ?lhs
then obtain p where "raw_has_prod (\k. 1 + a k) 0 p"
--> --------------------
--> maximum size reached
--> --------------------
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