src/HOL/Analysis/Infinite_Products.thy
 author paulson Wed May 02 12:47:56 2018 +0100 (14 months ago) changeset 68064 b249fab48c76 parent 66277 512b0dc09061 child 68071 c18af2b0f83e permissions -rw-r--r--
type class generalisations; some work on infinite products
     1 (*File:      HOL/Analysis/Infinite_Product.thy

     2   Author:    Manuel Eberl & LC Paulson

     3

     4   Basic results about convergence and absolute convergence of infinite products

     5   and their connection to summability.

     6 *)

     7 section \<open>Infinite Products\<close>

     8 theory Infinite_Products

     9   imports Complex_Main

    10 begin

    11

    12 lemma sum_le_prod:

    13   fixes f :: "'a \<Rightarrow> 'b :: linordered_semidom"

    14   assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<ge> 0"

    15   shows   "sum f A \<le> (\<Prod>x\<in>A. 1 + f x)"

    16   using assms

    17 proof (induction A rule: infinite_finite_induct)

    18   case (insert x A)

    19   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)"

    20     by (intro add_mono insert mult_left_mono prod_mono) (auto intro: insert.prems)

    21   with insert.hyps show ?case by (simp add: algebra_simps)

    22 qed simp_all

    23

    24 lemma prod_le_exp_sum:

    25   fixes f :: "'a \<Rightarrow> real"

    26   assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<ge> 0"

    27   shows   "prod (\<lambda>x. 1 + f x) A \<le> exp (sum f A)"

    28   using assms

    29 proof (induction A rule: infinite_finite_induct)

    30   case (insert x A)

    31   have "(1 + f x) * (\<Prod>x\<in>A. 1 + f x) \<le> exp (f x) * exp (sum f A)"

    32     using insert.prems by (intro mult_mono insert prod_nonneg exp_ge_add_one_self) auto

    33   with insert.hyps show ?case by (simp add: algebra_simps exp_add)

    34 qed simp_all

    35

    36 lemma lim_ln_1_plus_x_over_x_at_0: "(\<lambda>x::real. ln (1 + x) / x) \<midarrow>0\<rightarrow> 1"

    37 proof (rule lhopital)

    38   show "(\<lambda>x::real. ln (1 + x)) \<midarrow>0\<rightarrow> 0"

    39     by (rule tendsto_eq_intros refl | simp)+

    40   have "eventually (\<lambda>x::real. x \<in> {-1/2<..<1/2}) (nhds 0)"

    41     by (rule eventually_nhds_in_open) auto

    42   hence *: "eventually (\<lambda>x::real. x \<in> {-1/2<..<1/2}) (at 0)"

    43     by (rule filter_leD [rotated]) (simp_all add: at_within_def)

    44   show "eventually (\<lambda>x::real. ((\<lambda>x. ln (1 + x)) has_field_derivative inverse (1 + x)) (at x)) (at 0)"

    45     using * by eventually_elim (auto intro!: derivative_eq_intros simp: field_simps)

    46   show "eventually (\<lambda>x::real. ((\<lambda>x. x) has_field_derivative 1) (at x)) (at 0)"

    47     using * by eventually_elim (auto intro!: derivative_eq_intros simp: field_simps)

    48   show "\<forall>\<^sub>F x in at 0. x \<noteq> 0" by (auto simp: at_within_def eventually_inf_principal)

    49   show "(\<lambda>x::real. inverse (1 + x) / 1) \<midarrow>0\<rightarrow> 1"

    50     by (rule tendsto_eq_intros refl | simp)+

    51 qed auto

    52

    53 definition gen_has_prod :: "[nat \<Rightarrow> 'a::{t2_space, comm_semiring_1}, nat, 'a] \<Rightarrow> bool"

    54   where "gen_has_prod f M p \<equiv> (\<lambda>n. \<Prod>i\<le>n. f (i+M)) \<longlonglongrightarrow> p \<and> p \<noteq> 0"

    55

    56 text\<open>The nonzero and zero cases, as in \emph{Complex Analysis} by Joseph Bak and Donald J.Newman, page 241\<close>

    57 definition has_prod :: "(nat \<Rightarrow> 'a::{t2_space, comm_semiring_1}) \<Rightarrow> 'a \<Rightarrow> bool" (infixr "has'_prod" 80)

    58   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)"

    59

    60 definition convergent_prod :: "(nat \<Rightarrow> 'a :: {t2_space,comm_semiring_1}) \<Rightarrow> bool" where

    61   "convergent_prod f \<equiv> \<exists>M p. gen_has_prod f M p"

    62

    63 definition prodinf :: "(nat \<Rightarrow> 'a::{t2_space, comm_semiring_1}) \<Rightarrow> 'a"

    64     (binder "\<Prod>" 10)

    65   where "prodinf f = (THE p. f has_prod p)"

    66

    67 lemmas prod_defs = gen_has_prod_def has_prod_def convergent_prod_def prodinf_def

    68

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

    70   by simp

    71

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

    73   by presburger

    74

    75 lemma convergent_prod_altdef:

    76   fixes f :: "nat \<Rightarrow> 'a :: {t2_space,comm_semiring_1}"

    77   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)"

    78 proof

    79   assume "convergent_prod f"

    80   then obtain M L where *: "(\<lambda>n. \<Prod>i\<le>n. f (i+M)) \<longlonglongrightarrow> L" "L \<noteq> 0"

    81     by (auto simp: prod_defs)

    82   have "f i \<noteq> 0" if "i \<ge> M" for i

    83   proof

    84     assume "f i = 0"

    85     have **: "eventually (\<lambda>n. (\<Prod>i\<le>n. f (i+M)) = 0) sequentially"

    86       using eventually_ge_at_top[of "i - M"]

    87     proof eventually_elim

    88       case (elim n)

    89       with \<open>f i = 0\<close> and \<open>i \<ge> M\<close> show ?case

    90         by (auto intro!: bexI[of _ "i - M"] prod_zero)

    91     qed

    92     have "(\<lambda>n. (\<Prod>i\<le>n. f (i+M))) \<longlonglongrightarrow> 0"

    93       unfolding filterlim_iff

    94       by (auto dest!: eventually_nhds_x_imp_x intro!: eventually_mono[OF **])

    95     from tendsto_unique[OF _ this *(1)] and *(2)

    96       show False by simp

    97   qed

    98   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)"

    99     by blast

   100 qed (auto simp: prod_defs)

   101

   102 definition abs_convergent_prod :: "(nat \<Rightarrow> _) \<Rightarrow> bool" where

   103   "abs_convergent_prod f \<longleftrightarrow> convergent_prod (\<lambda>i. 1 + norm (f i - 1))"

   104

   105 lemma abs_convergent_prodI:

   106   assumes "convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"

   107   shows   "abs_convergent_prod f"

   108 proof -

   109   from assms obtain L where L: "(\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1)) \<longlonglongrightarrow> L"

   110     by (auto simp: convergent_def)

   111   have "L \<ge> 1"

   112   proof (rule tendsto_le)

   113     show "eventually (\<lambda>n. (\<Prod>i\<le>n. 1 + norm (f i - 1)) \<ge> 1) sequentially"

   114     proof (intro always_eventually allI)

   115       fix n

   116       have "(\<Prod>i\<le>n. 1 + norm (f i - 1)) \<ge> (\<Prod>i\<le>n. 1)"

   117         by (intro prod_mono) auto

   118       thus "(\<Prod>i\<le>n. 1 + norm (f i - 1)) \<ge> 1" by simp

   119     qed

   120   qed (use L in simp_all)

   121   hence "L \<noteq> 0" by auto

   122   with L show ?thesis unfolding abs_convergent_prod_def prod_defs

   123     by (intro exI[of _ "0::nat"] exI[of _ L]) auto

   124 qed

   125

   126 lemma

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

   128   assumes "convergent_prod f"

   129   shows   convergent_prod_imp_convergent: "convergent (\<lambda>n. \<Prod>i\<le>n. f i)"

   130     and   convergent_prod_to_zero_iff:    "(\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> 0 \<longleftrightarrow> (\<exists>i. f i = 0)"

   131 proof -

   132   from assms obtain M L

   133     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"

   134     by (auto simp: convergent_prod_altdef)

   135   note this(2)

   136   also have "(\<lambda>n. \<Prod>i\<le>n. f (i + M)) = (\<lambda>n. \<Prod>i=M..M+n. f i)"

   137     by (intro ext prod.reindex_bij_witness[of _ "\<lambda>n. n - M" "\<lambda>n. n + M"]) auto

   138   finally have "(\<lambda>n. (\<Prod>i<M. f i) * (\<Prod>i=M..M+n. f i)) \<longlonglongrightarrow> (\<Prod>i<M. f i) * L"

   139     by (intro tendsto_mult tendsto_const)

   140   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))"

   141     by (subst prod.union_disjoint) auto

   142   also have "(\<lambda>n. {..<M} \<union> {M..M+n}) = (\<lambda>n. {..n+M})" by auto

   143   finally have lim: "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> prod f {..<M} * L"

   144     by (rule LIMSEQ_offset)

   145   thus "convergent (\<lambda>n. \<Prod>i\<le>n. f i)"

   146     by (auto simp: convergent_def)

   147

   148   show "(\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> 0 \<longleftrightarrow> (\<exists>i. f i = 0)"

   149   proof

   150     assume "\<exists>i. f i = 0"

   151     then obtain i where "f i = 0" by auto

   152     moreover with M have "i < M" by (cases "i < M") auto

   153     ultimately have "(\<Prod>i<M. f i) = 0" by auto

   154     with lim show "(\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> 0" by simp

   155   next

   156     assume "(\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> 0"

   157     from tendsto_unique[OF _ this lim] and \<open>L \<noteq> 0\<close>

   158     show "\<exists>i. f i = 0" by auto

   159   qed

   160 qed

   161

   162 lemma convergent_prod_iff_nz_lim:

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

   164   assumes "\<And>i. f i \<noteq> 0"

   165   shows "convergent_prod f \<longleftrightarrow> (\<exists>L. (\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> L \<and> L \<noteq> 0)"

   166     (is "?lhs \<longleftrightarrow> ?rhs")

   167 proof

   168   assume ?lhs then show ?rhs

   169     using assms convergentD convergent_prod_imp_convergent convergent_prod_to_zero_iff by blast

   170 next

   171   assume ?rhs then show ?lhs

   172     unfolding prod_defs

   173     by (rule_tac x="0" in exI) (auto simp: )

   174 qed

   175

   176 lemma convergent_prod_iff_convergent:

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

   178   assumes "\<And>i. f i \<noteq> 0"

   179   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"

   180   by (force simp add: convergent_prod_iff_nz_lim assms convergent_def limI)

   181

   182

   183 lemma abs_convergent_prod_altdef:

   184   fixes f :: "nat \<Rightarrow> 'a :: {one,real_normed_vector}"

   185   shows  "abs_convergent_prod f \<longleftrightarrow> convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"

   186 proof

   187   assume "abs_convergent_prod f"

   188   thus "convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"

   189     by (auto simp: abs_convergent_prod_def intro!: convergent_prod_imp_convergent)

   190 qed (auto intro: abs_convergent_prodI)

   191

   192 lemma weierstrass_prod_ineq:

   193   fixes f :: "'a \<Rightarrow> real"

   194   assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> {0..1}"

   195   shows   "1 - sum f A \<le> (\<Prod>x\<in>A. 1 - f x)"

   196   using assms

   197 proof (induction A rule: infinite_finite_induct)

   198   case (insert x A)

   199   from insert.hyps and insert.prems

   200     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)"

   201     by (intro insert.IH add_mono mult_left_mono prod_mono) auto

   202   with insert.hyps show ?case by (simp add: algebra_simps)

   203 qed simp_all

   204

   205 lemma norm_prod_minus1_le_prod_minus1:

   206   fixes f :: "nat \<Rightarrow> 'a :: {real_normed_div_algebra,comm_ring_1}"

   207   shows "norm (prod (\<lambda>n. 1 + f n) A - 1) \<le> prod (\<lambda>n. 1 + norm (f n)) A - 1"

   208 proof (induction A rule: infinite_finite_induct)

   209   case (insert x A)

   210   from insert.hyps have

   211     "norm ((\<Prod>n\<in>insert x A. 1 + f n) - 1) =

   212        norm ((\<Prod>n\<in>A. 1 + f n) - 1 + f x * (\<Prod>n\<in>A. 1 + f n))"

   213     by (simp add: algebra_simps)

   214   also have "\<dots> \<le> norm ((\<Prod>n\<in>A. 1 + f n) - 1) + norm (f x * (\<Prod>n\<in>A. 1 + f n))"

   215     by (rule norm_triangle_ineq)

   216   also have "norm (f x * (\<Prod>n\<in>A. 1 + f n)) = norm (f x) * (\<Prod>x\<in>A. norm (1 + f x))"

   217     by (simp add: prod_norm norm_mult)

   218   also have "(\<Prod>x\<in>A. norm (1 + f x)) \<le> (\<Prod>x\<in>A. norm (1::'a) + norm (f x))"

   219     by (intro prod_mono norm_triangle_ineq ballI conjI) auto

   220   also have "norm (1::'a) = 1" by simp

   221   also note insert.IH

   222   also have "(\<Prod>n\<in>A. 1 + norm (f n)) - 1 + norm (f x) * (\<Prod>x\<in>A. 1 + norm (f x)) =

   223              (\<Prod>n\<in>insert x A. 1 + norm (f n)) - 1"

   224     using insert.hyps by (simp add: algebra_simps)

   225   finally show ?case by - (simp_all add: mult_left_mono)

   226 qed simp_all

   227

   228 lemma convergent_prod_imp_ev_nonzero:

   229   fixes f :: "nat \<Rightarrow> 'a :: {t2_space,comm_semiring_1}"

   230   assumes "convergent_prod f"

   231   shows   "eventually (\<lambda>n. f n \<noteq> 0) sequentially"

   232   using assms by (auto simp: eventually_at_top_linorder convergent_prod_altdef)

   233

   234 lemma convergent_prod_imp_LIMSEQ:

   235   fixes f :: "nat \<Rightarrow> 'a :: {real_normed_field}"

   236   assumes "convergent_prod f"

   237   shows   "f \<longlonglongrightarrow> 1"

   238 proof -

   239   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"

   240     by (auto simp: convergent_prod_altdef)

   241   hence L': "(\<lambda>n. \<Prod>i\<le>Suc n. f (i+M)) \<longlonglongrightarrow> L" by (subst filterlim_sequentially_Suc)

   242   have "(\<lambda>n. (\<Prod>i\<le>Suc n. f (i+M)) / (\<Prod>i\<le>n. f (i+M))) \<longlonglongrightarrow> L / L"

   243     using L L' by (intro tendsto_divide) simp_all

   244   also from L have "L / L = 1" by simp

   245   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))"

   246     using assms L by (auto simp: fun_eq_iff atMost_Suc)

   247   finally show ?thesis by (rule LIMSEQ_offset)

   248 qed

   249

   250 lemma abs_convergent_prod_imp_summable:

   251   fixes f :: "nat \<Rightarrow> 'a :: real_normed_div_algebra"

   252   assumes "abs_convergent_prod f"

   253   shows "summable (\<lambda>i. norm (f i - 1))"

   254 proof -

   255   from assms have "convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"

   256     unfolding abs_convergent_prod_def by (rule convergent_prod_imp_convergent)

   257   then obtain L where L: "(\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1)) \<longlonglongrightarrow> L"

   258     unfolding convergent_def by blast

   259   have "convergent (\<lambda>n. \<Sum>i\<le>n. norm (f i - 1))"

   260   proof (rule Bseq_monoseq_convergent)

   261     have "eventually (\<lambda>n. (\<Prod>i\<le>n. 1 + norm (f i - 1)) < L + 1) sequentially"

   262       using L(1) by (rule order_tendstoD) simp_all

   263     hence "\<forall>\<^sub>F x in sequentially. norm (\<Sum>i\<le>x. norm (f i - 1)) \<le> L + 1"

   264     proof eventually_elim

   265       case (elim n)

   266       have "norm (\<Sum>i\<le>n. norm (f i - 1)) = (\<Sum>i\<le>n. norm (f i - 1))"

   267         unfolding real_norm_def by (intro abs_of_nonneg sum_nonneg) simp_all

   268       also have "\<dots> \<le> (\<Prod>i\<le>n. 1 + norm (f i - 1))" by (rule sum_le_prod) auto

   269       also have "\<dots> < L + 1" by (rule elim)

   270       finally show ?case by simp

   271     qed

   272     thus "Bseq (\<lambda>n. \<Sum>i\<le>n. norm (f i - 1))" by (rule BfunI)

   273   next

   274     show "monoseq (\<lambda>n. \<Sum>i\<le>n. norm (f i - 1))"

   275       by (rule mono_SucI1) auto

   276   qed

   277   thus "summable (\<lambda>i. norm (f i - 1))" by (simp add: summable_iff_convergent')

   278 qed

   279

   280 lemma summable_imp_abs_convergent_prod:

   281   fixes f :: "nat \<Rightarrow> 'a :: real_normed_div_algebra"

   282   assumes "summable (\<lambda>i. norm (f i - 1))"

   283   shows   "abs_convergent_prod f"

   284 proof (intro abs_convergent_prodI Bseq_monoseq_convergent)

   285   show "monoseq (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"

   286     by (intro mono_SucI1)

   287        (auto simp: atMost_Suc algebra_simps intro!: mult_nonneg_nonneg prod_nonneg)

   288 next

   289   show "Bseq (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"

   290   proof (rule Bseq_eventually_mono)

   291     show "eventually (\<lambda>n. norm (\<Prod>i\<le>n. 1 + norm (f i - 1)) \<le>

   292             norm (exp (\<Sum>i\<le>n. norm (f i - 1)))) sequentially"

   293       by (intro always_eventually allI) (auto simp: abs_prod exp_sum intro!: prod_mono)

   294   next

   295     from assms have "(\<lambda>n. \<Sum>i\<le>n. norm (f i - 1)) \<longlonglongrightarrow> (\<Sum>i. norm (f i - 1))"

   296       using sums_def_le by blast

   297     hence "(\<lambda>n. exp (\<Sum>i\<le>n. norm (f i - 1))) \<longlonglongrightarrow> exp (\<Sum>i. norm (f i - 1))"

   298       by (rule tendsto_exp)

   299     hence "convergent (\<lambda>n. exp (\<Sum>i\<le>n. norm (f i - 1)))"

   300       by (rule convergentI)

   301     thus "Bseq (\<lambda>n. exp (\<Sum>i\<le>n. norm (f i - 1)))"

   302       by (rule convergent_imp_Bseq)

   303   qed

   304 qed

   305

   306 lemma abs_convergent_prod_conv_summable:

   307   fixes f :: "nat \<Rightarrow> 'a :: real_normed_div_algebra"

   308   shows "abs_convergent_prod f \<longleftrightarrow> summable (\<lambda>i. norm (f i - 1))"

   309   by (blast intro: abs_convergent_prod_imp_summable summable_imp_abs_convergent_prod)

   310

   311 lemma abs_convergent_prod_imp_LIMSEQ:

   312   fixes f :: "nat \<Rightarrow> 'a :: {comm_ring_1,real_normed_div_algebra}"

   313   assumes "abs_convergent_prod f"

   314   shows   "f \<longlonglongrightarrow> 1"

   315 proof -

   316   from assms have "summable (\<lambda>n. norm (f n - 1))"

   317     by (rule abs_convergent_prod_imp_summable)

   318   from summable_LIMSEQ_zero[OF this] have "(\<lambda>n. f n - 1) \<longlonglongrightarrow> 0"

   319     by (simp add: tendsto_norm_zero_iff)

   320   from tendsto_add[OF this tendsto_const[of 1]] show ?thesis by simp

   321 qed

   322

   323 lemma abs_convergent_prod_imp_ev_nonzero:

   324   fixes f :: "nat \<Rightarrow> 'a :: {comm_ring_1,real_normed_div_algebra}"

   325   assumes "abs_convergent_prod f"

   326   shows   "eventually (\<lambda>n. f n \<noteq> 0) sequentially"

   327 proof -

   328   from assms have "f \<longlonglongrightarrow> 1"

   329     by (rule abs_convergent_prod_imp_LIMSEQ)

   330   hence "eventually (\<lambda>n. dist (f n) 1 < 1) at_top"

   331     by (auto simp: tendsto_iff)

   332   thus ?thesis by eventually_elim auto

   333 qed

   334

   335 lemma convergent_prod_offset:

   336   assumes "convergent_prod (\<lambda>n. f (n + m))"

   337   shows   "convergent_prod f"

   338 proof -

   339   from assms obtain M L where "(\<lambda>n. \<Prod>k\<le>n. f (k + (M + m))) \<longlonglongrightarrow> L" "L \<noteq> 0"

   340     by (auto simp: prod_defs add.assoc)

   341   thus "convergent_prod f"

   342     unfolding prod_defs by blast

   343 qed

   344

   345 lemma abs_convergent_prod_offset:

   346   assumes "abs_convergent_prod (\<lambda>n. f (n + m))"

   347   shows   "abs_convergent_prod f"

   348   using assms unfolding abs_convergent_prod_def by (rule convergent_prod_offset)

   349

   350 lemma convergent_prod_ignore_initial_segment:

   351   fixes f :: "nat \<Rightarrow> 'a :: {real_normed_field}"

   352   assumes "convergent_prod f"

   353   shows   "convergent_prod (\<lambda>n. f (n + m))"

   354 proof -

   355   from assms obtain M L

   356     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"

   357     by (auto simp: convergent_prod_altdef)

   358   define C where "C = (\<Prod>k<m. f (k + M))"

   359   from nz have [simp]: "C \<noteq> 0"

   360     by (auto simp: C_def)

   361

   362   from L(1) have "(\<lambda>n. \<Prod>k\<le>n+m. f (k + M)) \<longlonglongrightarrow> L"

   363     by (rule LIMSEQ_ignore_initial_segment)

   364   also have "(\<lambda>n. \<Prod>k\<le>n+m. f (k + M)) = (\<lambda>n. C * (\<Prod>k\<le>n. f (k + M + m)))"

   365   proof (rule ext, goal_cases)

   366     case (1 n)

   367     have "{..n+m} = {..<m} \<union> {m..n+m}" by auto

   368     also have "(\<Prod>k\<in>\<dots>. f (k + M)) = C * (\<Prod>k=m..n+m. f (k + M))"

   369       unfolding C_def by (rule prod.union_disjoint) auto

   370     also have "(\<Prod>k=m..n+m. f (k + M)) = (\<Prod>k\<le>n. f (k + m + M))"

   371       by (intro ext prod.reindex_bij_witness[of _ "\<lambda>k. k + m" "\<lambda>k. k - m"]) auto

   372     finally show ?case by (simp add: add_ac)

   373   qed

   374   finally have "(\<lambda>n. C * (\<Prod>k\<le>n. f (k + M + m)) / C) \<longlonglongrightarrow> L / C"

   375     by (intro tendsto_divide tendsto_const) auto

   376   hence "(\<lambda>n. \<Prod>k\<le>n. f (k + M + m)) \<longlonglongrightarrow> L / C" by simp

   377   moreover from \<open>L \<noteq> 0\<close> have "L / C \<noteq> 0" by simp

   378   ultimately show ?thesis

   379     unfolding prod_defs by blast

   380 qed

   381

   382 lemma abs_convergent_prod_ignore_initial_segment:

   383   assumes "abs_convergent_prod f"

   384   shows   "abs_convergent_prod (\<lambda>n. f (n + m))"

   385   using assms unfolding abs_convergent_prod_def

   386   by (rule convergent_prod_ignore_initial_segment)

   387

   388 lemma abs_convergent_prod_imp_convergent_prod:

   389   fixes f :: "nat \<Rightarrow> 'a :: {real_normed_div_algebra,complete_space,comm_ring_1}"

   390   assumes "abs_convergent_prod f"

   391   shows   "convergent_prod f"

   392 proof -

   393   from assms have "eventually (\<lambda>n. f n \<noteq> 0) sequentially"

   394     by (rule abs_convergent_prod_imp_ev_nonzero)

   395   then obtain N where N: "f n \<noteq> 0" if "n \<ge> N" for n

   396     by (auto simp: eventually_at_top_linorder)

   397   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)"

   398

   399   have "Cauchy ?P"

   400   proof (rule CauchyI', goal_cases)

   401     case (1 \<epsilon>)

   402     from assms have "abs_convergent_prod (\<lambda>n. f (n + N))"

   403       by (rule abs_convergent_prod_ignore_initial_segment)

   404     hence "Cauchy ?Q"

   405       unfolding abs_convergent_prod_def

   406       by (intro convergent_Cauchy convergent_prod_imp_convergent)

   407     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

   408       by blast

   409     show ?case

   410     proof (rule exI[of _ M], safe, goal_cases)

   411       case (1 m n)

   412       have "dist (?P m) (?P n) = norm (?P n - ?P m)"

   413         by (simp add: dist_norm norm_minus_commute)

   414       also from 1 have "{..n} = {..m} \<union> {m<..n}" by auto

   415       hence "norm (?P n - ?P m) = norm (?P m * (\<Prod>k\<in>{m<..n}. f (k + N)) - ?P m)"

   416         by (subst prod.union_disjoint [symmetric]) (auto simp: algebra_simps)

   417       also have "\<dots> = norm (?P m * ((\<Prod>k\<in>{m<..n}. f (k + N)) - 1))"

   418         by (simp add: algebra_simps)

   419       also have "\<dots> = (\<Prod>k\<le>m. norm (f (k + N))) * norm ((\<Prod>k\<in>{m<..n}. f (k + N)) - 1)"

   420         by (simp add: norm_mult prod_norm)

   421       also have "\<dots> \<le> ?Q m * ((\<Prod>k\<in>{m<..n}. 1 + norm (f (k + N) - 1)) - 1)"

   422         using norm_prod_minus1_le_prod_minus1[of "\<lambda>k. f (k + N) - 1" "{m<..n}"]

   423               norm_triangle_ineq[of 1 "f k - 1" for k]

   424         by (intro mult_mono prod_mono ballI conjI norm_prod_minus1_le_prod_minus1 prod_nonneg) auto

   425       also have "\<dots> = ?Q m * (\<Prod>k\<in>{m<..n}. 1 + norm (f (k + N) - 1)) - ?Q m"

   426         by (simp add: algebra_simps)

   427       also have "?Q m * (\<Prod>k\<in>{m<..n}. 1 + norm (f (k + N) - 1)) =

   428                    (\<Prod>k\<in>{..m}\<union>{m<..n}. 1 + norm (f (k + N) - 1))"

   429         by (rule prod.union_disjoint [symmetric]) auto

   430       also from 1 have "{..m}\<union>{m<..n} = {..n}" by auto

   431       also have "?Q n - ?Q m \<le> norm (?Q n - ?Q m)" by simp

   432       also from 1 have "\<dots> < \<epsilon>" by (intro M) auto

   433       finally show ?case .

   434     qed

   435   qed

   436   hence conv: "convergent ?P" by (rule Cauchy_convergent)

   437   then obtain L where L: "?P \<longlonglongrightarrow> L"

   438     by (auto simp: convergent_def)

   439

   440   have "L \<noteq> 0"

   441   proof

   442     assume [simp]: "L = 0"

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

   444       by (simp add: prod_norm)

   445

   446     from assms have "(\<lambda>n. f (n + N)) \<longlonglongrightarrow> 1"

   447       by (intro abs_convergent_prod_imp_LIMSEQ abs_convergent_prod_ignore_initial_segment)

   448     hence "eventually (\<lambda>n. norm (f (n + N) - 1) < 1) sequentially"

   449       by (auto simp: tendsto_iff dist_norm)

   450     then obtain M0 where M0: "norm (f (n + N) - 1) < 1" if "n \<ge> M0" for n

   451       by (auto simp: eventually_at_top_linorder)

   452

   453     {

   454       fix M assume M: "M \<ge> M0"

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

   456

   457       have "(\<lambda>n. \<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1)) \<longlonglongrightarrow> 0"

   458       proof (rule tendsto_sandwich)

   459         show "eventually (\<lambda>n. (\<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1)) \<ge> 0) sequentially"

   460           using M by (intro always_eventually prod_nonneg allI ballI) (auto intro: less_imp_le)

   461         have "norm (1::'a) - norm (f (i + M + N) - 1) \<le> norm (f (i + M + N))" for i

   462           using norm_triangle_ineq3[of "f (i + M + N)" 1] by simp

   463         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"

   464           using M by (intro always_eventually allI prod_mono ballI conjI) (auto intro: less_imp_le)

   465

   466         define C where "C = (\<Prod>k<M. norm (f (k + N)))"

   467         from N have [simp]: "C \<noteq> 0" by (auto simp: C_def)

   468         from L have "(\<lambda>n. norm (\<Prod>k\<le>n+M. f (k + N))) \<longlonglongrightarrow> 0"

   469           by (intro LIMSEQ_ignore_initial_segment) (simp add: tendsto_norm_zero_iff)

   470         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))))"

   471         proof (rule ext, goal_cases)

   472           case (1 n)

   473           have "{..n+M} = {..<M} \<union> {M..n+M}" by auto

   474           also have "norm (\<Prod>k\<in>\<dots>. f (k + N)) = C * norm (\<Prod>k=M..n+M. f (k + N))"

   475             unfolding C_def by (subst prod.union_disjoint) (auto simp: norm_mult prod_norm)

   476           also have "(\<Prod>k=M..n+M. f (k + N)) = (\<Prod>k\<le>n. f (k + N + M))"

   477             by (intro prod.reindex_bij_witness[of _ "\<lambda>i. i + M" "\<lambda>i. i - M"]) auto

   478           finally show ?case by (simp add: add_ac prod_norm)

   479         qed

   480         finally have "(\<lambda>n. C * (\<Prod>k\<le>n. norm (f (k + M + N))) / C) \<longlonglongrightarrow> 0 / C"

   481           by (intro tendsto_divide tendsto_const) auto

   482         thus "(\<lambda>n. \<Prod>k\<le>n. norm (f (k + M + N))) \<longlonglongrightarrow> 0" by simp

   483       qed simp_all

   484

   485       have "1 - (\<Sum>i. norm (f (i + M + N) - 1)) \<le> 0"

   486       proof (rule tendsto_le)

   487         show "eventually (\<lambda>n. 1 - (\<Sum>k\<le>n. norm (f (k+M+N) - 1)) \<le>

   488                                 (\<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1))) at_top"

   489           using M by (intro always_eventually allI weierstrass_prod_ineq) (auto intro: less_imp_le)

   490         show "(\<lambda>n. \<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1)) \<longlonglongrightarrow> 0" by fact

   491         show "(\<lambda>n. 1 - (\<Sum>k\<le>n. norm (f (k + M + N) - 1)))

   492                   \<longlonglongrightarrow> 1 - (\<Sum>i. norm (f (i + M + N) - 1))"

   493           by (intro tendsto_intros summable_LIMSEQ' summable_ignore_initial_segment

   494                 abs_convergent_prod_imp_summable assms)

   495       qed simp_all

   496       hence "(\<Sum>i. norm (f (i + M + N) - 1)) \<ge> 1" by simp

   497       also have "\<dots> + (\<Sum>i<M. norm (f (i + N) - 1)) = (\<Sum>i. norm (f (i + N) - 1))"

   498         by (intro suminf_split_initial_segment [symmetric] summable_ignore_initial_segment

   499               abs_convergent_prod_imp_summable assms)

   500       finally have "1 + (\<Sum>i<M. norm (f (i + N) - 1)) \<le> (\<Sum>i. norm (f (i + N) - 1))" by simp

   501     } note * = this

   502

   503     have "1 + (\<Sum>i. norm (f (i + N) - 1)) \<le> (\<Sum>i. norm (f (i + N) - 1))"

   504     proof (rule tendsto_le)

   505       show "(\<lambda>M. 1 + (\<Sum>i<M. norm (f (i + N) - 1))) \<longlonglongrightarrow> 1 + (\<Sum>i. norm (f (i + N) - 1))"

   506         by (intro tendsto_intros summable_LIMSEQ summable_ignore_initial_segment

   507                 abs_convergent_prod_imp_summable assms)

   508       show "eventually (\<lambda>M. 1 + (\<Sum>i<M. norm (f (i + N) - 1)) \<le> (\<Sum>i. norm (f (i + N) - 1))) at_top"

   509         using eventually_ge_at_top[of M0] by eventually_elim (use * in auto)

   510     qed simp_all

   511     thus False by simp

   512   qed

   513   with L show ?thesis by (auto simp: prod_defs)

   514 qed

   515

   516 lemma convergent_prod_offset_0:

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

   518   assumes "convergent_prod f" "\<And>i. f i \<noteq> 0"

   519   shows "\<exists>p. gen_has_prod f 0 p"

   520   using assms

   521   unfolding convergent_prod_def

   522 proof (clarsimp simp: prod_defs)

   523   fix M p

   524   assume "(\<lambda>n. \<Prod>i\<le>n. f (i + M)) \<longlonglongrightarrow> p" "p \<noteq> 0"

   525   then have "(\<lambda>n. prod f {..<M} * (\<Prod>i\<le>n. f (i + M))) \<longlonglongrightarrow> prod f {..<M} * p"

   526     by (metis tendsto_mult_left)

   527   moreover have "prod f {..<M} * (\<Prod>i\<le>n. f (i + M)) = prod f {..n+M}" for n

   528   proof -

   529     have "{..n+M} = {..<M} \<union> {M..n+M}"

   530       by auto

   531     then have "prod f {..n+M} = prod f {..<M} * prod f {M..n+M}"

   532       by simp (subst prod.union_disjoint; force)

   533     also have "... = prod f {..<M} * (\<Prod>i\<le>n. f (i + M))"

   534       by (metis (mono_tags, lifting) add.left_neutral atMost_atLeast0 prod_shift_bounds_cl_nat_ivl)

   535     finally show ?thesis by metis

   536   qed

   537   ultimately have "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> prod f {..<M} * p"

   538     by (auto intro: LIMSEQ_offset [where k=M])

   539   then show "\<exists>p. (\<lambda>n. prod f {..n}) \<longlonglongrightarrow> p \<and> p \<noteq> 0"

   540     using \<open>p \<noteq> 0\<close> assms

   541     by (rule_tac x="prod f {..<M} * p" in exI) auto

   542 qed

   543

   544 lemma prodinf_eq_lim:

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

   546   assumes "convergent_prod f" "\<And>i. f i \<noteq> 0"

   547   shows "prodinf f = lim (\<lambda>n. \<Prod>i\<le>n. f i)"

   548   using assms convergent_prod_offset_0 [OF assms]

   549   by (simp add: prod_defs lim_def) (metis (no_types) assms(1) convergent_prod_to_zero_iff)

   550

   551 lemma has_prod_one[simp, intro]: "(\<lambda>n. 1) has_prod 1"

   552   unfolding prod_defs by auto

   553

   554 lemma convergent_prod_one[simp, intro]: "convergent_prod (\<lambda>n. 1)"

   555   unfolding prod_defs by auto

   556

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

   558   by presburger

   559

   560 lemma convergent_prod_cong:

   561   fixes f g :: "nat \<Rightarrow> 'a::{field,topological_semigroup_mult,t2_space}"

   562   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"

   563   shows "convergent_prod f = convergent_prod g"

   564 proof -

   565   from assms obtain N where N: "\<forall>n\<ge>N. f n = g n"

   566     by (auto simp: eventually_at_top_linorder)

   567   define C where "C = (\<Prod>k<N. f k / g k)"

   568   with g have "C \<noteq> 0"

   569     by (simp add: f)

   570   have *: "eventually (\<lambda>n. prod f {..n} = C * prod g {..n}) sequentially"

   571     using eventually_ge_at_top[of N]

   572   proof eventually_elim

   573     case (elim n)

   574     then have "{..n} = {..<N} \<union> {N..n}"

   575       by auto

   576     also have "prod f ... = prod f {..<N} * prod f {N..n}"

   577       by (intro prod.union_disjoint) auto

   578     also from N have "prod f {N..n} = prod g {N..n}"

   579       by (intro prod.cong) simp_all

   580     also have "prod f {..<N} * prod g {N..n} = C * (prod g {..<N} * prod g {N..n})"

   581       unfolding C_def by (simp add: g prod_dividef)

   582     also have "prod g {..<N} * prod g {N..n} = prod g ({..<N} \<union> {N..n})"

   583       by (intro prod.union_disjoint [symmetric]) auto

   584     also from elim have "{..<N} \<union> {N..n} = {..n}"

   585       by auto

   586     finally show "prod f {..n} = C * prod g {..n}" .

   587   qed

   588   then have cong: "convergent (\<lambda>n. prod f {..n}) = convergent (\<lambda>n. C * prod g {..n})"

   589     by (rule convergent_cong)

   590   show ?thesis

   591   proof

   592     assume cf: "convergent_prod f"

   593     then have "\<not> (\<lambda>n. prod g {..n}) \<longlonglongrightarrow> 0"

   594       using tendsto_mult_left * convergent_prod_to_zero_iff f filterlim_cong by fastforce

   595     then show "convergent_prod g"

   596       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)

   597   next

   598     assume cg: "convergent_prod g"

   599     have "\<exists>a. C * a \<noteq> 0 \<and> (\<lambda>n. prod g {..n}) \<longlonglongrightarrow> a"

   600       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)

   601     then show "convergent_prod f"

   602       using "*" tendsto_mult_left filterlim_cong

   603       by (fastforce simp add: convergent_prod_iff_nz_lim f)

   604   qed

   605 qed

   606

   607 end