src/HOL/Analysis/Infinite_Products.thy
 author paulson Mon Jun 11 14:34:17 2018 +0100 (12 months ago) changeset 68424 02e5a44ffe7d parent 68361 20375f232f3b child 68426 e0b5f2d14bf9 permissions -rw-r--r--
the last of the infinite product proofs
     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 Topology_Euclidean_Space

    10 begin

    11

    12 subsection\<open>Preliminaries\<close>

    13

    14 lemma sum_le_prod:

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

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

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

    18   using assms

    19 proof (induction A rule: infinite_finite_induct)

    20   case (insert x A)

    21   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)"

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

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

    24 qed simp_all

    25

    26 lemma prod_le_exp_sum:

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

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

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

    30   using assms

    31 proof (induction A rule: infinite_finite_induct)

    32   case (insert x A)

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

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

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

    36 qed simp_all

    37

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

    39 proof (rule lhopital)

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

    41     by (rule tendsto_eq_intros refl | simp)+

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

    43     by (rule eventually_nhds_in_open) auto

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

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

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

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

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

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

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

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

    52     by (rule tendsto_eq_intros refl | simp)+

    53 qed auto

    54

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

    56

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

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

    59

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

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

    62   where "f has_prod p \<equiv> raw_has_prod f 0 p \<or> (\<exists>i q. p = 0 \<and> f i = 0 \<and> raw_has_prod f (Suc i) q)"

    63

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

    65   "convergent_prod f \<equiv> \<exists>M p. raw_has_prod f M p"

    66

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

    68     (binder "\<Prod>" 10)

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

    70

    71 lemmas prod_defs = raw_has_prod_def has_prod_def convergent_prod_def prodinf_def

    72

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

    74   by simp

    75

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

    77   by presburger

    78

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

    80   by (simp add: raw_has_prod_def)

    81

    82 lemma raw_has_prod_eq_0:

    83   fixes f :: "nat \<Rightarrow> 'a::{semidom,t2_space}"

    84   assumes p: "raw_has_prod f m p" and i: "f i = 0" "i \<ge> m"

    85   shows "p = 0"

    86 proof -

    87   have eq0: "(\<Prod>k\<le>n. f (k+m)) = 0" if "i - m \<le> n" for n

    88   proof -

    89     have "\<exists>k\<le>n. f (k + m) = 0"

    90       using i that by auto

    91     then show ?thesis

    92       by auto

    93   qed

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

    95     by (rule LIMSEQ_offset [where k = "i-m"]) (simp add: eq0)

    96     with p show ?thesis

    97       unfolding raw_has_prod_def

    98     using LIMSEQ_unique by blast

    99 qed

   100

   101 lemma has_prod_0_iff: "f has_prod 0 \<longleftrightarrow> (\<exists>i. f i = 0 \<and> (\<exists>p. raw_has_prod f (Suc i) p))"

   102   by (simp add: has_prod_def)

   103

   104 lemma has_prod_unique2:

   105   fixes f :: "nat \<Rightarrow> 'a::{semidom,t2_space}"

   106   assumes "f has_prod a" "f has_prod b" shows "a = b"

   107   using assms

   108   by (auto simp: has_prod_def raw_has_prod_eq_0) (meson raw_has_prod_def sequentially_bot tendsto_unique)

   109

   110 lemma has_prod_unique:

   111   fixes f :: "nat \<Rightarrow> 'a :: {semidom,t2_space}"

   112   shows "f has_prod s \<Longrightarrow> s = prodinf f"

   113   by (simp add: has_prod_unique2 prodinf_def the_equality)

   114

   115 lemma convergent_prod_altdef:

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

   117   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)"

   118 proof

   119   assume "convergent_prod f"

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

   121     by (auto simp: prod_defs)

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

   123   proof

   124     assume "f i = 0"

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

   126       using eventually_ge_at_top[of "i - M"]

   127     proof eventually_elim

   128       case (elim n)

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

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

   131     qed

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

   133       unfolding filterlim_iff

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

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

   136       show False by simp

   137   qed

   138   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)"

   139     by blast

   140 qed (auto simp: prod_defs)

   141

   142

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

   144

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

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

   147

   148 lemma abs_convergent_prodI:

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

   150   shows   "abs_convergent_prod f"

   151 proof -

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

   153     by (auto simp: convergent_def)

   154   have "L \<ge> 1"

   155   proof (rule tendsto_le)

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

   157     proof (intro always_eventually allI)

   158       fix n

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

   160         by (intro prod_mono) auto

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

   162     qed

   163   qed (use L in simp_all)

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

   165   with L show ?thesis unfolding abs_convergent_prod_def prod_defs

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

   167 qed

   168

   169 lemma

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

   171   assumes "convergent_prod f"

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

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

   174 proof -

   175   from assms obtain M L

   176     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"

   177     by (auto simp: convergent_prod_altdef)

   178   note this(2)

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

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

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

   182     by (intro tendsto_mult tendsto_const)

   183   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))"

   184     by (subst prod.union_disjoint) auto

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

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

   187     by (rule LIMSEQ_offset)

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

   189     by (auto simp: convergent_def)

   190

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

   192   proof

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

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

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

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

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

   198   next

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

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

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

   202   qed

   203 qed

   204

   205 lemma convergent_prod_iff_nz_lim:

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

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

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

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

   210 proof

   211   assume ?lhs then show ?rhs

   212     using assms convergentD convergent_prod_imp_convergent convergent_prod_to_zero_iff by blast

   213 next

   214   assume ?rhs then show ?lhs

   215     unfolding prod_defs

   216     by (rule_tac x=0 in exI) auto

   217 qed

   218

   219 lemma convergent_prod_iff_convergent:

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

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

   222   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"

   223   by (force simp: convergent_prod_iff_nz_lim assms convergent_def limI)

   224

   225

   226 lemma abs_convergent_prod_altdef:

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

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

   229 proof

   230   assume "abs_convergent_prod f"

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

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

   233 qed (auto intro: abs_convergent_prodI)

   234

   235 lemma weierstrass_prod_ineq:

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

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

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

   239   using assms

   240 proof (induction A rule: infinite_finite_induct)

   241   case (insert x A)

   242   from insert.hyps and insert.prems

   243     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)"

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

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

   246 qed simp_all

   247

   248 lemma norm_prod_minus1_le_prod_minus1:

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

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

   251 proof (induction A rule: infinite_finite_induct)

   252   case (insert x A)

   253   from insert.hyps have

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

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

   256     by (simp add: algebra_simps)

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

   258     by (rule norm_triangle_ineq)

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

   260     by (simp add: prod_norm norm_mult)

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

   262     by (intro prod_mono norm_triangle_ineq ballI conjI) auto

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

   264   also note insert.IH

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

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

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

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

   269 qed simp_all

   270

   271 lemma convergent_prod_imp_ev_nonzero:

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

   273   assumes "convergent_prod f"

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

   275   using assms by (auto simp: eventually_at_top_linorder convergent_prod_altdef)

   276

   277 lemma convergent_prod_imp_LIMSEQ:

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

   279   assumes "convergent_prod f"

   280   shows   "f \<longlonglongrightarrow> 1"

   281 proof -

   282   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"

   283     by (auto simp: convergent_prod_altdef)

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

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

   286     using L L' by (intro tendsto_divide) simp_all

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

   288   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))"

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

   290   finally show ?thesis by (rule LIMSEQ_offset)

   291 qed

   292

   293 lemma abs_convergent_prod_imp_summable:

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

   295   assumes "abs_convergent_prod f"

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

   297 proof -

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

   299     unfolding abs_convergent_prod_def by (rule convergent_prod_imp_convergent)

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

   301     unfolding convergent_def by blast

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

   303   proof (rule Bseq_monoseq_convergent)

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

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

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

   307     proof eventually_elim

   308       case (elim n)

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

   310         unfolding real_norm_def by (intro abs_of_nonneg sum_nonneg) simp_all

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

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

   313       finally show ?case by simp

   314     qed

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

   316   next

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

   318       by (rule mono_SucI1) auto

   319   qed

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

   321 qed

   322

   323 lemma summable_imp_abs_convergent_prod:

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

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

   326   shows   "abs_convergent_prod f"

   327 proof (intro abs_convergent_prodI Bseq_monoseq_convergent)

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

   329     by (intro mono_SucI1)

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

   331 next

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

   333   proof (rule Bseq_eventually_mono)

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

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

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

   337   next

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

   339       using sums_def_le by blast

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

   341       by (rule tendsto_exp)

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

   343       by (rule convergentI)

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

   345       by (rule convergent_imp_Bseq)

   346   qed

   347 qed

   348

   349 lemma abs_convergent_prod_conv_summable:

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

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

   352   by (blast intro: abs_convergent_prod_imp_summable summable_imp_abs_convergent_prod)

   353

   354 lemma abs_convergent_prod_imp_LIMSEQ:

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

   356   assumes "abs_convergent_prod f"

   357   shows   "f \<longlonglongrightarrow> 1"

   358 proof -

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

   360     by (rule abs_convergent_prod_imp_summable)

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

   362     by (simp add: tendsto_norm_zero_iff)

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

   364 qed

   365

   366 lemma abs_convergent_prod_imp_ev_nonzero:

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

   368   assumes "abs_convergent_prod f"

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

   370 proof -

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

   372     by (rule abs_convergent_prod_imp_LIMSEQ)

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

   374     by (auto simp: tendsto_iff)

   375   thus ?thesis by eventually_elim auto

   376 qed

   377

   378 lemma convergent_prod_offset:

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

   380   shows   "convergent_prod f"

   381 proof -

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

   383     by (auto simp: prod_defs add.assoc)

   384   thus "convergent_prod f"

   385     unfolding prod_defs by blast

   386 qed

   387

   388 lemma abs_convergent_prod_offset:

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

   390   shows   "abs_convergent_prod f"

   391   using assms unfolding abs_convergent_prod_def by (rule convergent_prod_offset)

   392

   393 subsection\<open>Ignoring initial segments\<close>

   394

   395 lemma raw_has_prod_ignore_initial_segment:

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

   397   assumes "raw_has_prod f M p" "N \<ge> M"

   398   obtains q where  "raw_has_prod f N q"

   399 proof -

   400   have p: "(\<lambda>n. \<Prod>k\<le>n. f (k + M)) \<longlonglongrightarrow> p" and "p \<noteq> 0"

   401     using assms by (auto simp: raw_has_prod_def)

   402   then have nz: "\<And>n. n \<ge> M \<Longrightarrow> f n \<noteq> 0"

   403     using assms by (auto simp: raw_has_prod_eq_0)

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

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

   406     by (auto simp: C_def)

   407

   408   from p have "(\<lambda>i. \<Prod>k\<le>i + (N-M). f (k + M)) \<longlonglongrightarrow> p"

   409     by (rule LIMSEQ_ignore_initial_segment)

   410   also have "(\<lambda>i. \<Prod>k\<le>i + (N-M). f (k + M)) = (\<lambda>n. C * (\<Prod>k\<le>n. f (k + N)))"

   411   proof (rule ext, goal_cases)

   412     case (1 n)

   413     have "{..n+(N-M)} = {..<(N-M)} \<union> {(N-M)..n+(N-M)}" by auto

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

   415       unfolding C_def by (rule prod.union_disjoint) auto

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

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

   418     finally show ?case

   419       using \<open>N \<ge> M\<close> by (simp add: add_ac)

   420   qed

   421   finally have "(\<lambda>n. C * (\<Prod>k\<le>n. f (k + N)) / C) \<longlonglongrightarrow> p / C"

   422     by (intro tendsto_divide tendsto_const) auto

   423   hence "(\<lambda>n. \<Prod>k\<le>n. f (k + N)) \<longlonglongrightarrow> p / C" by simp

   424   moreover from \<open>p \<noteq> 0\<close> have "p / C \<noteq> 0" by simp

   425   ultimately show ?thesis

   426     using raw_has_prod_def that by blast

   427 qed

   428

   429 corollary convergent_prod_ignore_initial_segment:

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

   431   assumes "convergent_prod f"

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

   433   using assms

   434   unfolding convergent_prod_def

   435   apply clarify

   436   apply (erule_tac N="M+m" in raw_has_prod_ignore_initial_segment)

   437   apply (auto simp add: raw_has_prod_def add_ac)

   438   done

   439

   440 corollary convergent_prod_ignore_nonzero_segment:

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

   442   assumes f: "convergent_prod f" and nz: "\<And>i. i \<ge> M \<Longrightarrow> f i \<noteq> 0"

   443   shows "\<exists>p. raw_has_prod f M p"

   444   using convergent_prod_ignore_initial_segment [OF f]

   445   by (metis convergent_LIMSEQ_iff convergent_prod_iff_convergent le_add_same_cancel2 nz prod_defs(1) zero_order(1))

   446

   447 corollary abs_convergent_prod_ignore_initial_segment:

   448   assumes "abs_convergent_prod f"

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

   450   using assms unfolding abs_convergent_prod_def

   451   by (rule convergent_prod_ignore_initial_segment)

   452

   453 lemma abs_convergent_prod_imp_convergent_prod:

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

   455   assumes "abs_convergent_prod f"

   456   shows   "convergent_prod f"

   457 proof -

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

   459     by (rule abs_convergent_prod_imp_ev_nonzero)

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

   461     by (auto simp: eventually_at_top_linorder)

   462   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)"

   463

   464   have "Cauchy ?P"

   465   proof (rule CauchyI', goal_cases)

   466     case (1 \<epsilon>)

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

   468       by (rule abs_convergent_prod_ignore_initial_segment)

   469     hence "Cauchy ?Q"

   470       unfolding abs_convergent_prod_def

   471       by (intro convergent_Cauchy convergent_prod_imp_convergent)

   472     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

   473       by blast

   474     show ?case

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

   476       case (1 m n)

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

   478         by (simp add: dist_norm norm_minus_commute)

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

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

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

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

   483         by (simp add: algebra_simps)

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

   485         by (simp add: norm_mult prod_norm)

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

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

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

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

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

   491         by (simp add: algebra_simps)

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

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

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

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

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

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

   498       finally show ?case .

   499     qed

   500   qed

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

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

   503     by (auto simp: convergent_def)

   504

   505   have "L \<noteq> 0"

   506   proof

   507     assume [simp]: "L = 0"

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

   509       by (simp add: prod_norm)

   510

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

   512       by (intro abs_convergent_prod_imp_LIMSEQ abs_convergent_prod_ignore_initial_segment)

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

   514       by (auto simp: tendsto_iff dist_norm)

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

   516       by (auto simp: eventually_at_top_linorder)

   517

   518     {

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

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

   521

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

   523       proof (rule tendsto_sandwich)

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

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

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

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

   528         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"

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

   530

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

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

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

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

   535         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))))"

   536         proof (rule ext, goal_cases)

   537           case (1 n)

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

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

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

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

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

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

   544         qed

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

   546           by (intro tendsto_divide tendsto_const) auto

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

   548       qed simp_all

   549

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

   551       proof (rule tendsto_le)

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

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

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

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

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

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

   558           by (intro tendsto_intros summable_LIMSEQ' summable_ignore_initial_segment

   559                 abs_convergent_prod_imp_summable assms)

   560       qed simp_all

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

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

   563         by (intro suminf_split_initial_segment [symmetric] summable_ignore_initial_segment

   564               abs_convergent_prod_imp_summable assms)

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

   566     } note * = this

   567

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

   569     proof (rule tendsto_le)

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

   571         by (intro tendsto_intros summable_LIMSEQ summable_ignore_initial_segment

   572                 abs_convergent_prod_imp_summable assms)

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

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

   575     qed simp_all

   576     thus False by simp

   577   qed

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

   579 qed

   580

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

   582

   583 lemma raw_has_prod_cases:

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

   585   assumes "raw_has_prod f M p"

   586   obtains i where "i<M" "f i = 0" | p where "raw_has_prod f 0 p"

   587 proof -

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

   589     using assms unfolding raw_has_prod_def by blast+

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

   591     by (metis tendsto_mult_left)

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

   593   proof -

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

   595       by auto

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

   597       by simp (subst prod.union_disjoint; force)

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

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

   600     finally show ?thesis by metis

   601   qed

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

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

   604   then have "raw_has_prod f 0 (prod f {..<M} * p)" if "\<forall>i<M. f i \<noteq> 0"

   605     using \<open>p \<noteq> 0\<close> assms that by (auto simp: raw_has_prod_def)

   606   then show thesis

   607     using that by blast

   608 qed

   609

   610 corollary convergent_prod_offset_0:

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

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

   613   shows "\<exists>p. raw_has_prod f 0 p"

   614   using assms convergent_prod_def raw_has_prod_cases by blast

   615

   616 lemma prodinf_eq_lim:

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

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

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

   620   using assms convergent_prod_offset_0 [OF assms]

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

   622

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

   624   unfolding prod_defs by auto

   625

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

   627   unfolding prod_defs by auto

   628

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

   630   by presburger

   631

   632 lemma convergent_prod_cong:

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

   634   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"

   635   shows "convergent_prod f = convergent_prod g"

   636 proof -

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

   638     by (auto simp: eventually_at_top_linorder)

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

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

   641     by (simp add: f)

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

   643     using eventually_ge_at_top[of N]

   644   proof eventually_elim

   645     case (elim n)

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

   647       by auto

   648     also have "prod f \<dots> = prod f {..<N} * prod f {N..n}"

   649       by (intro prod.union_disjoint) auto

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

   651       by (intro prod.cong) simp_all

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

   653       unfolding C_def by (simp add: g prod_dividef)

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

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

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

   657       by auto

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

   659   qed

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

   661     by (rule convergent_cong)

   662   show ?thesis

   663   proof

   664     assume cf: "convergent_prod f"

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

   666       using tendsto_mult_left * convergent_prod_to_zero_iff f filterlim_cong by fastforce

   667     then show "convergent_prod g"

   668       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)

   669   next

   670     assume cg: "convergent_prod g"

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

   672       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)

   673     then show "convergent_prod f"

   674       using "*" tendsto_mult_left filterlim_cong

   675       by (fastforce simp add: convergent_prod_iff_nz_lim f)

   676   qed

   677 qed

   678

   679 lemma has_prod_finite:

   680   fixes f :: "nat \<Rightarrow> 'a::{semidom,t2_space}"

   681   assumes [simp]: "finite N"

   682     and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 1"

   683   shows "f has_prod (\<Prod>n\<in>N. f n)"

   684 proof -

   685   have eq: "prod f {..n + Suc (Max N)} = prod f N" for n

   686   proof (rule prod.mono_neutral_right)

   687     show "N \<subseteq> {..n + Suc (Max N)}"

   688       by (auto simp: le_Suc_eq trans_le_add2)

   689     show "\<forall>i\<in>{..n + Suc (Max N)} - N. f i = 1"

   690       using f by blast

   691   qed auto

   692   show ?thesis

   693   proof (cases "\<forall>n\<in>N. f n \<noteq> 0")

   694     case True

   695     then have "prod f N \<noteq> 0"

   696       by simp

   697     moreover have "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> prod f N"

   698       by (rule LIMSEQ_offset[of _ "Suc (Max N)"]) (simp add: eq atLeast0LessThan del: add_Suc_right)

   699     ultimately show ?thesis

   700       by (simp add: raw_has_prod_def has_prod_def)

   701   next

   702     case False

   703     then obtain k where "k \<in> N" "f k = 0"

   704       by auto

   705     let ?Z = "{n \<in> N. f n = 0}"

   706     have maxge: "Max ?Z \<ge> n" if "f n = 0" for n

   707       using Max_ge [of ?Z] \<open>finite N\<close> \<open>f n = 0\<close>

   708       by (metis (mono_tags) Collect_mem_eq f finite_Collect_conjI mem_Collect_eq zero_neq_one)

   709     let ?q = "prod f {Suc (Max ?Z)..Max N}"

   710     have [simp]: "?q \<noteq> 0"

   711       using maxge Suc_n_not_le_n le_trans by force

   712     have eq: "(\<Prod>i\<le>n + Max N. f (Suc (i + Max ?Z))) = ?q" for n

   713     proof -

   714       have "(\<Prod>i\<le>n + Max N. f (Suc (i + Max ?Z))) = prod f {Suc (Max ?Z)..n + Max N + Suc (Max ?Z)}"

   715       proof (rule prod.reindex_cong [where l = "\<lambda>i. i + Suc (Max ?Z)", THEN sym])

   716         show "{Suc (Max ?Z)..n + Max N + Suc (Max ?Z)} = (\<lambda>i. i + Suc (Max ?Z))  {..n + Max N}"

   717           using le_Suc_ex by fastforce

   718       qed (auto simp: inj_on_def)

   719       also have "\<dots> = ?q"

   720         by (rule prod.mono_neutral_right)

   721            (use Max.coboundedI [OF \<open>finite N\<close>] f in \<open>force+\<close>)

   722       finally show ?thesis .

   723     qed

   724     have q: "raw_has_prod f (Suc (Max ?Z)) ?q"

   725     proof (simp add: raw_has_prod_def)

   726       show "(\<lambda>n. \<Prod>i\<le>n. f (Suc (i + Max ?Z))) \<longlonglongrightarrow> ?q"

   727         by (rule LIMSEQ_offset[of _ "(Max N)"]) (simp add: eq)

   728     qed

   729     show ?thesis

   730       unfolding has_prod_def

   731     proof (intro disjI2 exI conjI)

   732       show "prod f N = 0"

   733         using \<open>f k = 0\<close> \<open>k \<in> N\<close> \<open>finite N\<close> prod_zero by blast

   734       show "f (Max ?Z) = 0"

   735         using Max_in [of ?Z] \<open>finite N\<close> \<open>f k = 0\<close> \<open>k \<in> N\<close> by auto

   736     qed (use q in auto)

   737   qed

   738 qed

   739

   740 corollary has_prod_0:

   741   fixes f :: "nat \<Rightarrow> 'a::{semidom,t2_space}"

   742   assumes "\<And>n. f n = 1"

   743   shows "f has_prod 1"

   744   by (simp add: assms has_prod_cong)

   745

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

   747   using has_prod_unique by force

   748

   749 lemma convergent_prod_finite:

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

   751   assumes "finite N" "\<And>n. n \<notin> N \<Longrightarrow> f n = 1"

   752   shows "convergent_prod f"

   753 proof -

   754   have "\<exists>n p. raw_has_prod f n p"

   755     using assms has_prod_def has_prod_finite by blast

   756   then show ?thesis

   757     by (simp add: convergent_prod_def)

   758 qed

   759

   760 lemma has_prod_If_finite_set:

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

   762   shows "finite A \<Longrightarrow> (\<lambda>r. if r \<in> A then f r else 1) has_prod (\<Prod>r\<in>A. f r)"

   763   using has_prod_finite[of A "(\<lambda>r. if r \<in> A then f r else 1)"]

   764   by simp

   765

   766 lemma has_prod_If_finite:

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

   768   shows "finite {r. P r} \<Longrightarrow> (\<lambda>r. if P r then f r else 1) has_prod (\<Prod>r | P r. f r)"

   769   using has_prod_If_finite_set[of "{r. P r}"] by simp

   770

   771 lemma convergent_prod_If_finite_set[simp, intro]:

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

   773   shows "finite A \<Longrightarrow> convergent_prod (\<lambda>r. if r \<in> A then f r else 1)"

   774   by (simp add: convergent_prod_finite)

   775

   776 lemma convergent_prod_If_finite[simp, intro]:

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

   778   shows "finite {r. P r} \<Longrightarrow> convergent_prod (\<lambda>r. if P r then f r else 1)"

   779   using convergent_prod_def has_prod_If_finite has_prod_def by fastforce

   780

   781 lemma has_prod_single:

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

   783   shows "(\<lambda>r. if r = i then f r else 1) has_prod f i"

   784   using has_prod_If_finite[of "\<lambda>r. r = i"] by simp

   785

   786 context

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

   788 begin

   789

   790 lemma convergent_prod_imp_has_prod:

   791   assumes "convergent_prod f"

   792   shows "\<exists>p. f has_prod p"

   793 proof -

   794   obtain M p where p: "raw_has_prod f M p"

   795     using assms convergent_prod_def by blast

   796   then have "p \<noteq> 0"

   797     using raw_has_prod_nonzero by blast

   798   with p have fnz: "f i \<noteq> 0" if "i \<ge> M" for i

   799     using raw_has_prod_eq_0 that by blast

   800   define C where "C = (\<Prod>n<M. f n)"

   801   show ?thesis

   802   proof (cases "\<forall>n\<le>M. f n \<noteq> 0")

   803     case True

   804     then have "C \<noteq> 0"

   805       by (simp add: C_def)

   806     then show ?thesis

   807       by (meson True assms convergent_prod_offset_0 fnz has_prod_def nat_le_linear)

   808   next

   809     case False

   810     let ?N = "GREATEST n. f n = 0"

   811     have 0: "f ?N = 0"

   812       using fnz False

   813       by (metis (mono_tags, lifting) GreatestI_ex_nat nat_le_linear)

   814     have "f i \<noteq> 0" if "i > ?N" for i

   815       by (metis (mono_tags, lifting) Greatest_le_nat fnz leD linear that)

   816     then have "\<exists>p. raw_has_prod f (Suc ?N) p"

   817       using assms by (auto simp: intro!: convergent_prod_ignore_nonzero_segment)

   818     then show ?thesis

   819       unfolding has_prod_def using 0 by blast

   820   qed

   821 qed

   822

   823 lemma convergent_prod_has_prod [intro]:

   824   shows "convergent_prod f \<Longrightarrow> f has_prod (prodinf f)"

   825   unfolding prodinf_def

   826   by (metis convergent_prod_imp_has_prod has_prod_unique theI')

   827

   828 lemma convergent_prod_LIMSEQ:

   829   shows "convergent_prod f \<Longrightarrow> (\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> prodinf f"

   830   by (metis convergent_LIMSEQ_iff convergent_prod_has_prod convergent_prod_imp_convergent

   831       convergent_prod_to_zero_iff raw_has_prod_eq_0 has_prod_def prodinf_eq_lim zero_le)

   832

   833 lemma has_prod_iff: "f has_prod x \<longleftrightarrow> convergent_prod f \<and> prodinf f = x"

   834 proof

   835   assume "f has_prod x"

   836   then show "convergent_prod f \<and> prodinf f = x"

   837     apply safe

   838     using convergent_prod_def has_prod_def apply blast

   839     using has_prod_unique by blast

   840 qed auto

   841

   842 lemma convergent_prod_has_prod_iff: "convergent_prod f \<longleftrightarrow> f has_prod prodinf f"

   843   by (auto simp: has_prod_iff convergent_prod_has_prod)

   844

   845 lemma prodinf_finite:

   846   assumes N: "finite N"

   847     and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 1"

   848   shows "prodinf f = (\<Prod>n\<in>N. f n)"

   849   using has_prod_finite[OF assms, THEN has_prod_unique] by simp

   850

   851 end

   852

   853 subsection \<open>Infinite products on ordered, topological monoids\<close>

   854

   855 lemma LIMSEQ_prod_0:

   856   fixes f :: "nat \<Rightarrow> 'a::{semidom,topological_space}"

   857   assumes "f i = 0"

   858   shows "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> 0"

   859 proof (subst tendsto_cong)

   860   show "\<forall>\<^sub>F n in sequentially. prod f {..n} = 0"

   861   proof

   862     show "prod f {..n} = 0" if "n \<ge> i" for n

   863       using that assms by auto

   864   qed

   865 qed auto

   866

   867 lemma LIMSEQ_prod_nonneg:

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

   869   assumes 0: "\<And>n. 0 \<le> f n" and a: "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> a"

   870   shows "a \<ge> 0"

   871   by (simp add: "0" prod_nonneg LIMSEQ_le_const [OF a])

   872

   873

   874 context

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

   876 begin

   877

   878 lemma has_prod_le:

   879   assumes f: "f has_prod a" and g: "g has_prod b" and le: "\<And>n. 0 \<le> f n \<and> f n \<le> g n"

   880   shows "a \<le> b"

   881 proof (cases "a=0 \<or> b=0")

   882   case True

   883   then show ?thesis

   884   proof

   885     assume [simp]: "a=0"

   886     have "b \<ge> 0"

   887     proof (rule LIMSEQ_prod_nonneg)

   888       show "(\<lambda>n. prod g {..n}) \<longlonglongrightarrow> b"

   889         using g by (auto simp: has_prod_def raw_has_prod_def LIMSEQ_prod_0)

   890     qed (use le order_trans in auto)

   891     then show ?thesis

   892       by auto

   893   next

   894     assume [simp]: "b=0"

   895     then obtain i where "g i = 0"

   896       using g by (auto simp: prod_defs)

   897     then have "f i = 0"

   898       using antisym le by force

   899     then have "a=0"

   900       using f by (auto simp: prod_defs LIMSEQ_prod_0 LIMSEQ_unique)

   901     then show ?thesis

   902       by auto

   903   qed

   904 next

   905   case False

   906   then show ?thesis

   907     using assms

   908     unfolding has_prod_def raw_has_prod_def

   909     by (force simp: LIMSEQ_prod_0 intro!: LIMSEQ_le prod_mono)

   910 qed

   911

   912 lemma prodinf_le:

   913   assumes f: "f has_prod a" and g: "g has_prod b" and le: "\<And>n. 0 \<le> f n \<and> f n \<le> g n"

   914   shows "prodinf f \<le> prodinf g"

   915   using has_prod_le [OF assms] has_prod_unique f g  by blast

   916

   917 end

   918

   919

   920 lemma prod_le_prodinf:

   921   fixes f :: "nat \<Rightarrow> 'a::{linordered_idom,linorder_topology}"

   922   assumes "f has_prod a" "\<And>i. 0 \<le> f i" "\<And>i. i\<ge>n \<Longrightarrow> 1 \<le> f i"

   923   shows "prod f {..<n} \<le> prodinf f"

   924   by(rule has_prod_le[OF has_prod_If_finite_set]) (use assms has_prod_unique in auto)

   925

   926 lemma prodinf_nonneg:

   927   fixes f :: "nat \<Rightarrow> 'a::{linordered_idom,linorder_topology}"

   928   assumes "f has_prod a" "\<And>i. 1 \<le> f i"

   929   shows "1 \<le> prodinf f"

   930   using prod_le_prodinf[of f a 0] assms

   931   by (metis order_trans prod_ge_1 zero_le_one)

   932

   933 lemma prodinf_le_const:

   934   fixes f :: "nat \<Rightarrow> real"

   935   assumes "convergent_prod f" "\<And>n. prod f {..<n} \<le> x"

   936   shows "prodinf f \<le> x"

   937   by (metis lessThan_Suc_atMost assms convergent_prod_LIMSEQ LIMSEQ_le_const2)

   938

   939 lemma prodinf_eq_one_iff:

   940   fixes f :: "nat \<Rightarrow> real"

   941   assumes f: "convergent_prod f" and ge1: "\<And>n. 1 \<le> f n"

   942   shows "prodinf f = 1 \<longleftrightarrow> (\<forall>n. f n = 1)"

   943 proof

   944   assume "prodinf f = 1"

   945   then have "(\<lambda>n. \<Prod>i<n. f i) \<longlonglongrightarrow> 1"

   946     using convergent_prod_LIMSEQ[of f] assms by (simp add: LIMSEQ_lessThan_iff_atMost)

   947   then have "\<And>i. (\<Prod>n\<in>{i}. f n) \<le> 1"

   948   proof (rule LIMSEQ_le_const)

   949     have "1 \<le> prod f n" for n

   950       by (simp add: ge1 prod_ge_1)

   951     have "prod f {..<n} = 1" for n

   952       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)

   953     then have "(\<Prod>n\<in>{i}. f n) \<le> prod f {..<n}" if "n \<ge> Suc i" for i n

   954       by (metis mult.left_neutral order_refl prod.cong prod.neutral_const prod_lessThan_Suc)

   955     then show "\<exists>N. \<forall>n\<ge>N. (\<Prod>n\<in>{i}. f n) \<le> prod f {..<n}" for i

   956       by blast

   957   qed

   958   with ge1 show "\<forall>n. f n = 1"

   959     by (auto intro!: antisym)

   960 qed (metis prodinf_zero fun_eq_iff)

   961

   962 lemma prodinf_pos_iff:

   963   fixes f :: "nat \<Rightarrow> real"

   964   assumes "convergent_prod f" "\<And>n. 1 \<le> f n"

   965   shows "1 < prodinf f \<longleftrightarrow> (\<exists>i. 1 < f i)"

   966   using prod_le_prodinf[of f 1] prodinf_eq_one_iff

   967   by (metis convergent_prod_has_prod assms less_le prodinf_nonneg)

   968

   969 lemma less_1_prodinf2:

   970   fixes f :: "nat \<Rightarrow> real"

   971   assumes "convergent_prod f" "\<And>n. 1 \<le> f n" "1 < f i"

   972   shows "1 < prodinf f"

   973 proof -

   974   have "1 < (\<Prod>n<Suc i. f n)"

   975     using assms  by (intro less_1_prod2[where i=i]) auto

   976   also have "\<dots> \<le> prodinf f"

   977     by (intro prod_le_prodinf) (use assms order_trans zero_le_one in \<open>blast+\<close>)

   978   finally show ?thesis .

   979 qed

   980

   981 lemma less_1_prodinf:

   982   fixes f :: "nat \<Rightarrow> real"

   983   shows "\<lbrakk>convergent_prod f; \<And>n. 1 < f n\<rbrakk> \<Longrightarrow> 1 < prodinf f"

   984   by (intro less_1_prodinf2[where i=1]) (auto intro: less_imp_le)

   985

   986 lemma prodinf_nonzero:

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

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

   989   shows "prodinf f \<noteq> 0"

   990   by (metis assms convergent_prod_offset_0 has_prod_unique raw_has_prod_def has_prod_def)

   991

   992 lemma less_0_prodinf:

   993   fixes f :: "nat \<Rightarrow> real"

   994   assumes f: "convergent_prod f" and 0: "\<And>i. f i > 0"

   995   shows "0 < prodinf f"

   996 proof -

   997   have "prodinf f \<noteq> 0"

   998     by (metis assms less_irrefl prodinf_nonzero)

   999   moreover have "0 < (\<Prod>n<i. f n)" for i

  1000     by (simp add: 0 prod_pos)

  1001   then have "prodinf f \<ge> 0"

  1002     using convergent_prod_LIMSEQ [OF f] LIMSEQ_prod_nonneg 0 less_le by blast

  1003   ultimately show ?thesis

  1004     by auto

  1005 qed

  1006

  1007 lemma prod_less_prodinf2:

  1008   fixes f :: "nat \<Rightarrow> real"

  1009   assumes f: "convergent_prod f" and 1: "\<And>m. m\<ge>n \<Longrightarrow> 1 \<le> f m" and 0: "\<And>m. 0 < f m" and i: "n \<le> i" "1 < f i"

  1010   shows "prod f {..<n} < prodinf f"

  1011 proof -

  1012   have "prod f {..<n} \<le> prod f {..<i}"

  1013     by (rule prod_mono2) (use assms less_le in auto)

  1014   then have "prod f {..<n} < f i * prod f {..<i}"

  1015     using mult_less_le_imp_less[of 1 "f i" "prod f {..<n}" "prod f {..<i}"] assms

  1016     by (simp add: prod_pos)

  1017   moreover have "prod f {..<Suc i} \<le> prodinf f"

  1018     using prod_le_prodinf[of f _ "Suc i"]

  1019     by (meson "0" "1" Suc_leD convergent_prod_has_prod f \<open>n \<le> i\<close> le_trans less_eq_real_def)

  1020   ultimately show ?thesis

  1021     by (metis le_less_trans mult.commute not_le prod_lessThan_Suc)

  1022 qed

  1023

  1024 lemma prod_less_prodinf:

  1025   fixes f :: "nat \<Rightarrow> real"

  1026   assumes f: "convergent_prod f" and 1: "\<And>m. m\<ge>n \<Longrightarrow> 1 < f m" and 0: "\<And>m. 0 < f m"

  1027   shows "prod f {..<n} < prodinf f"

  1028   by (meson "0" "1" f le_less prod_less_prodinf2)

  1029

  1030 lemma raw_has_prodI_bounded:

  1031   fixes f :: "nat \<Rightarrow> real"

  1032   assumes pos: "\<And>n. 1 \<le> f n"

  1033     and le: "\<And>n. (\<Prod>i<n. f i) \<le> x"

  1034   shows "\<exists>p. raw_has_prod f 0 p"

  1035   unfolding raw_has_prod_def add_0_right

  1036 proof (rule exI LIMSEQ_incseq_SUP conjI)+

  1037   show "bdd_above (range (\<lambda>n. prod f {..n}))"

  1038     by (metis bdd_aboveI2 le lessThan_Suc_atMost)

  1039   then have "(SUP i. prod f {..i}) > 0"

  1040     by (metis UNIV_I cSUP_upper less_le_trans pos prod_pos zero_less_one)

  1041   then show "(SUP i. prod f {..i}) \<noteq> 0"

  1042     by auto

  1043   show "incseq (\<lambda>n. prod f {..n})"

  1044     using pos order_trans [OF zero_le_one] by (auto simp: mono_def intro!: prod_mono2)

  1045 qed

  1046

  1047 lemma convergent_prodI_nonneg_bounded:

  1048   fixes f :: "nat \<Rightarrow> real"

  1049   assumes "\<And>n. 1 \<le> f n" "\<And>n. (\<Prod>i<n. f i) \<le> x"

  1050   shows "convergent_prod f"

  1051   using convergent_prod_def raw_has_prodI_bounded [OF assms] by blast

  1052

  1053

  1054 subsection \<open>Infinite products on topological spaces\<close>

  1055

  1056 context

  1057   fixes f g :: "nat \<Rightarrow> 'a::{t2_space,topological_semigroup_mult,idom}"

  1058 begin

  1059

  1060 lemma raw_has_prod_mult: "\<lbrakk>raw_has_prod f M a; raw_has_prod g M b\<rbrakk> \<Longrightarrow> raw_has_prod (\<lambda>n. f n * g n) M (a * b)"

  1061   by (force simp add: prod.distrib tendsto_mult raw_has_prod_def)

  1062

  1063 lemma has_prod_mult_nz: "\<lbrakk>f has_prod a; g has_prod b; a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>n. f n * g n) has_prod (a * b)"

  1064   by (simp add: raw_has_prod_mult has_prod_def)

  1065

  1066 end

  1067

  1068

  1069 context

  1070   fixes f g :: "nat \<Rightarrow> 'a::real_normed_field"

  1071 begin

  1072

  1073 lemma has_prod_mult:

  1074   assumes f: "f has_prod a" and g: "g has_prod b"

  1075   shows "(\<lambda>n. f n * g n) has_prod (a * b)"

  1076   using f [unfolded has_prod_def]

  1077 proof (elim disjE exE conjE)

  1078   assume f0: "raw_has_prod f 0 a"

  1079   show ?thesis

  1080     using g [unfolded has_prod_def]

  1081   proof (elim disjE exE conjE)

  1082     assume g0: "raw_has_prod g 0 b"

  1083     with f0 show ?thesis

  1084       by (force simp add: has_prod_def prod.distrib tendsto_mult raw_has_prod_def)

  1085   next

  1086     fix j q

  1087     assume "b = 0" and "g j = 0" and q: "raw_has_prod g (Suc j) q"

  1088     obtain p where p: "raw_has_prod f (Suc j) p"

  1089       using f0 raw_has_prod_ignore_initial_segment by blast

  1090     then have "Ex (raw_has_prod (\<lambda>n. f n * g n) (Suc j))"

  1091       using q raw_has_prod_mult by blast

  1092     then show ?thesis

  1093       using \<open>b = 0\<close> \<open>g j = 0\<close> has_prod_0_iff by fastforce

  1094   qed

  1095 next

  1096   fix i p

  1097   assume "a = 0" and "f i = 0" and p: "raw_has_prod f (Suc i) p"

  1098   show ?thesis

  1099     using g [unfolded has_prod_def]

  1100   proof (elim disjE exE conjE)

  1101     assume g0: "raw_has_prod g 0 b"

  1102     obtain q where q: "raw_has_prod g (Suc i) q"

  1103       using g0 raw_has_prod_ignore_initial_segment by blast

  1104     then have "Ex (raw_has_prod (\<lambda>n. f n * g n) (Suc i))"

  1105       using raw_has_prod_mult p by blast

  1106     then show ?thesis

  1107       using \<open>a = 0\<close> \<open>f i = 0\<close> has_prod_0_iff by fastforce

  1108   next

  1109     fix j q

  1110     assume "b = 0" and "g j = 0" and q: "raw_has_prod g (Suc j) q"

  1111     obtain p' where p': "raw_has_prod f (Suc (max i j)) p'"

  1112       by (metis raw_has_prod_ignore_initial_segment max_Suc_Suc max_def p)

  1113     moreover

  1114     obtain q' where q': "raw_has_prod g (Suc (max i j)) q'"

  1115       by (metis raw_has_prod_ignore_initial_segment max.cobounded2 max_Suc_Suc q)

  1116     ultimately show ?thesis

  1117       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)

  1118   qed

  1119 qed

  1120

  1121 lemma convergent_prod_mult:

  1122   assumes f: "convergent_prod f" and g: "convergent_prod g"

  1123   shows "convergent_prod (\<lambda>n. f n * g n)"

  1124   unfolding convergent_prod_def

  1125 proof -

  1126   obtain M p N q where p: "raw_has_prod f M p" and q: "raw_has_prod g N q"

  1127     using convergent_prod_def f g by blast+

  1128   then obtain p' q' where p': "raw_has_prod f (max M N) p'" and q': "raw_has_prod g (max M N) q'"

  1129     by (meson raw_has_prod_ignore_initial_segment max.cobounded1 max.cobounded2)

  1130   then show "\<exists>M p. raw_has_prod (\<lambda>n. f n * g n) M p"

  1131     using raw_has_prod_mult by blast

  1132 qed

  1133

  1134 lemma prodinf_mult: "convergent_prod f \<Longrightarrow> convergent_prod g \<Longrightarrow> prodinf f * prodinf g = (\<Prod>n. f n * g n)"

  1135   by (intro has_prod_unique has_prod_mult convergent_prod_has_prod)

  1136

  1137 end

  1138

  1139 context

  1140   fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::real_normed_field"

  1141     and I :: "'i set"

  1142 begin

  1143

  1144 lemma has_prod_prod: "(\<And>i. i \<in> I \<Longrightarrow> (f i) has_prod (x i)) \<Longrightarrow> (\<lambda>n. \<Prod>i\<in>I. f i n) has_prod (\<Prod>i\<in>I. x i)"

  1145   by (induct I rule: infinite_finite_induct) (auto intro!: has_prod_mult)

  1146

  1147 lemma prodinf_prod: "(\<And>i. i \<in> I \<Longrightarrow> convergent_prod (f i)) \<Longrightarrow> (\<Prod>n. \<Prod>i\<in>I. f i n) = (\<Prod>i\<in>I. \<Prod>n. f i n)"

  1148   using has_prod_unique[OF has_prod_prod, OF convergent_prod_has_prod] by simp

  1149

  1150 lemma convergent_prod_prod: "(\<And>i. i \<in> I \<Longrightarrow> convergent_prod (f i)) \<Longrightarrow> convergent_prod (\<lambda>n. \<Prod>i\<in>I. f i n)"

  1151   using convergent_prod_has_prod_iff has_prod_prod prodinf_prod by force

  1152

  1153 end

  1154

  1155 subsection \<open>Infinite summability on real normed fields\<close>

  1156

  1157 context

  1158   fixes f :: "nat \<Rightarrow> 'a::real_normed_field"

  1159 begin

  1160

  1161 lemma raw_has_prod_Suc_iff: "raw_has_prod f M (a * f M) \<longleftrightarrow> raw_has_prod (\<lambda>n. f (Suc n)) M a \<and> f M \<noteq> 0"

  1162 proof -

  1163   have "raw_has_prod f M (a * f M) \<longleftrightarrow> (\<lambda>i. \<Prod>j\<le>Suc i. f (j+M)) \<longlonglongrightarrow> a * f M \<and> a * f M \<noteq> 0"

  1164     by (subst LIMSEQ_Suc_iff) (simp add: raw_has_prod_def)

  1165   also have "\<dots> \<longleftrightarrow> (\<lambda>i. (\<Prod>j\<le>i. f (Suc j + M)) * f M) \<longlonglongrightarrow> a * f M \<and> a * f M \<noteq> 0"

  1166     by (simp add: ac_simps atMost_Suc_eq_insert_0 image_Suc_atMost prod_atLeast1_atMost_eq lessThan_Suc_atMost)

  1167   also have "\<dots> \<longleftrightarrow> raw_has_prod (\<lambda>n. f (Suc n)) M a \<and> f M \<noteq> 0"

  1168   proof safe

  1169     assume tends: "(\<lambda>i. (\<Prod>j\<le>i. f (Suc j + M)) * f M) \<longlonglongrightarrow> a * f M" and 0: "a * f M \<noteq> 0"

  1170     with tendsto_divide[OF tends tendsto_const, of "f M"]

  1171     show "raw_has_prod (\<lambda>n. f (Suc n)) M a"

  1172       by (simp add: raw_has_prod_def)

  1173   qed (auto intro: tendsto_mult_right simp:  raw_has_prod_def)

  1174   finally show ?thesis .

  1175 qed

  1176

  1177 lemma has_prod_Suc_iff:

  1178   assumes "f 0 \<noteq> 0" shows "(\<lambda>n. f (Suc n)) has_prod a \<longleftrightarrow> f has_prod (a * f 0)"

  1179 proof (cases "a = 0")

  1180   case True

  1181   then show ?thesis

  1182   proof (simp add: has_prod_def, safe)

  1183     fix i x

  1184     assume "f (Suc i) = 0" and "raw_has_prod (\<lambda>n. f (Suc n)) (Suc i) x"

  1185     then obtain y where "raw_has_prod f (Suc (Suc i)) y"

  1186       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)

  1187     then show "\<exists>i. f i = 0 \<and> Ex (raw_has_prod f (Suc i))"

  1188       using \<open>f (Suc i) = 0\<close> by blast

  1189   next

  1190     fix i x

  1191     assume "f i = 0" and x: "raw_has_prod f (Suc i) x"

  1192     then obtain j where j: "i = Suc j"

  1193       by (metis assms not0_implies_Suc)

  1194     moreover have "\<exists> y. raw_has_prod (\<lambda>n. f (Suc n)) i y"

  1195       using x by (auto simp: raw_has_prod_def)

  1196     then show "\<exists>i. f (Suc i) = 0 \<and> Ex (raw_has_prod (\<lambda>n. f (Suc n)) (Suc i))"

  1197       using \<open>f i = 0\<close> j by blast

  1198   qed

  1199 next

  1200   case False

  1201   then show ?thesis

  1202     by (auto simp: has_prod_def raw_has_prod_Suc_iff assms)

  1203 qed

  1204

  1205 lemma convergent_prod_Suc_iff:

  1206   assumes "\<And>k. f k \<noteq> 0" shows "convergent_prod (\<lambda>n. f (Suc n)) = convergent_prod f"

  1207 proof

  1208   assume "convergent_prod f"

  1209   then have "f has_prod prodinf f"

  1210     by (rule convergent_prod_has_prod)

  1211   moreover have "prodinf f \<noteq> 0"

  1212     by (simp add: \<open>convergent_prod f\<close> assms prodinf_nonzero)

  1213   ultimately have "(\<lambda>n. f (Suc n)) has_prod (prodinf f * inverse (f 0))"

  1214     by (simp add: has_prod_Suc_iff inverse_eq_divide assms)

  1215   then show "convergent_prod (\<lambda>n. f (Suc n))"

  1216     using has_prod_iff by blast

  1217 next

  1218   assume "convergent_prod (\<lambda>n. f (Suc n))"

  1219   then show "convergent_prod f"

  1220     using assms convergent_prod_def raw_has_prod_Suc_iff by blast

  1221 qed

  1222

  1223 lemma raw_has_prod_inverse:

  1224   assumes "raw_has_prod f M a" shows "raw_has_prod (\<lambda>n. inverse (f n)) M (inverse a)"

  1225   using assms unfolding raw_has_prod_def by (auto dest: tendsto_inverse simp: prod_inversef [symmetric])

  1226

  1227 lemma has_prod_inverse:

  1228   assumes "f has_prod a" shows "(\<lambda>n. inverse (f n)) has_prod (inverse a)"

  1229 using assms raw_has_prod_inverse unfolding has_prod_def by auto

  1230

  1231 lemma convergent_prod_inverse:

  1232   assumes "convergent_prod f"

  1233   shows "convergent_prod (\<lambda>n. inverse (f n))"

  1234   using assms unfolding convergent_prod_def  by (blast intro: raw_has_prod_inverse elim: )

  1235

  1236 end

  1237

  1238 context

  1239   fixes f :: "nat \<Rightarrow> 'a::real_normed_field"

  1240 begin

  1241

  1242 lemma raw_has_prod_Suc_iff': "raw_has_prod f M a \<longleftrightarrow> raw_has_prod (\<lambda>n. f (Suc n)) M (a / f M) \<and> f M \<noteq> 0"

  1243   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)

  1244

  1245 lemma has_prod_divide: "f has_prod a \<Longrightarrow> g has_prod b \<Longrightarrow> (\<lambda>n. f n / g n) has_prod (a / b)"

  1246   unfolding divide_inverse by (intro has_prod_inverse has_prod_mult)

  1247

  1248 lemma convergent_prod_divide:

  1249   assumes f: "convergent_prod f" and g: "convergent_prod g"

  1250   shows "convergent_prod (\<lambda>n. f n / g n)"

  1251   using f g has_prod_divide has_prod_iff by blast

  1252

  1253 lemma prodinf_divide: "convergent_prod f \<Longrightarrow> convergent_prod g \<Longrightarrow> prodinf f / prodinf g = (\<Prod>n. f n / g n)"

  1254   by (intro has_prod_unique has_prod_divide convergent_prod_has_prod)

  1255

  1256 lemma prodinf_inverse: "convergent_prod f \<Longrightarrow> (\<Prod>n. inverse (f n)) = inverse (\<Prod>n. f n)"

  1257   by (intro has_prod_unique [symmetric] has_prod_inverse convergent_prod_has_prod)

  1258

  1259 lemma has_prod_iff_shift:

  1260   assumes "\<And>i. i < n \<Longrightarrow> f i \<noteq> 0"

  1261   shows "(\<lambda>i. f (i + n)) has_prod a \<longleftrightarrow> f has_prod (a * (\<Prod>i<n. f i))"

  1262   using assms

  1263 proof (induct n arbitrary: a)

  1264   case 0

  1265   then show ?case by simp

  1266 next

  1267   case (Suc n)

  1268   then have "(\<lambda>i. f (Suc i + n)) has_prod a \<longleftrightarrow> (\<lambda>i. f (i + n)) has_prod (a * f n)"

  1269     by (subst has_prod_Suc_iff) auto

  1270   with Suc show ?case

  1271     by (simp add: ac_simps)

  1272 qed

  1273

  1274 corollary has_prod_iff_shift':

  1275   assumes "\<And>i. i < n \<Longrightarrow> f i \<noteq> 0"

  1276   shows "(\<lambda>i. f (i + n)) has_prod (a / (\<Prod>i<n. f i)) \<longleftrightarrow> f has_prod a"

  1277   by (simp add: assms has_prod_iff_shift)

  1278

  1279 lemma has_prod_one_iff_shift:

  1280   assumes "\<And>i. i < n \<Longrightarrow> f i = 1"

  1281   shows "(\<lambda>i. f (i+n)) has_prod a \<longleftrightarrow> (\<lambda>i. f i) has_prod a"

  1282   by (simp add: assms has_prod_iff_shift)

  1283

  1284 lemma convergent_prod_iff_shift:

  1285   shows "convergent_prod (\<lambda>i. f (i + n)) \<longleftrightarrow> convergent_prod f"

  1286   apply safe

  1287   using convergent_prod_offset apply blast

  1288   using convergent_prod_ignore_initial_segment convergent_prod_def by blast

  1289

  1290 lemma has_prod_split_initial_segment:

  1291   assumes "f has_prod a" "\<And>i. i < n \<Longrightarrow> f i \<noteq> 0"

  1292   shows "(\<lambda>i. f (i + n)) has_prod (a / (\<Prod>i<n. f i))"

  1293   using assms has_prod_iff_shift' by blast

  1294

  1295 lemma prodinf_divide_initial_segment:

  1296   assumes "convergent_prod f" "\<And>i. i < n \<Longrightarrow> f i \<noteq> 0"

  1297   shows "(\<Prod>i. f (i + n)) = (\<Prod>i. f i) / (\<Prod>i<n. f i)"

  1298   by (rule has_prod_unique[symmetric]) (auto simp: assms has_prod_iff_shift)

  1299

  1300 lemma prodinf_split_initial_segment:

  1301   assumes "convergent_prod f" "\<And>i. i < n \<Longrightarrow> f i \<noteq> 0"

  1302   shows "prodinf f = (\<Prod>i. f (i + n)) * (\<Prod>i<n. f i)"

  1303   by (auto simp add: assms prodinf_divide_initial_segment)

  1304

  1305 lemma prodinf_split_head:

  1306   assumes "convergent_prod f" "f 0 \<noteq> 0"

  1307   shows "(\<Prod>n. f (Suc n)) = prodinf f / f 0"

  1308   using prodinf_split_initial_segment[of 1] assms by simp

  1309

  1310 end

  1311

  1312 context

  1313   fixes f :: "nat \<Rightarrow> 'a::real_normed_field"

  1314 begin

  1315

  1316 lemma convergent_prod_inverse_iff: "convergent_prod (\<lambda>n. inverse (f n)) \<longleftrightarrow> convergent_prod f"

  1317   by (auto dest: convergent_prod_inverse)

  1318

  1319 lemma convergent_prod_const_iff:

  1320   fixes c :: "'a :: {real_normed_field}"

  1321   shows "convergent_prod (\<lambda>_. c) \<longleftrightarrow> c = 1"

  1322 proof

  1323   assume "convergent_prod (\<lambda>_. c)"

  1324   then show "c = 1"

  1325     using convergent_prod_imp_LIMSEQ LIMSEQ_unique by blast

  1326 next

  1327   assume "c = 1"

  1328   then show "convergent_prod (\<lambda>_. c)"

  1329     by auto

  1330 qed

  1331

  1332 lemma has_prod_power: "f has_prod a \<Longrightarrow> (\<lambda>i. f i ^ n) has_prod (a ^ n)"

  1333   by (induction n) (auto simp: has_prod_mult)

  1334

  1335 lemma convergent_prod_power: "convergent_prod f \<Longrightarrow> convergent_prod (\<lambda>i. f i ^ n)"

  1336   by (induction n) (auto simp: convergent_prod_mult)

  1337

  1338 lemma prodinf_power: "convergent_prod f \<Longrightarrow> prodinf (\<lambda>i. f i ^ n) = prodinf f ^ n"

  1339   by (metis has_prod_unique convergent_prod_imp_has_prod has_prod_power)

  1340

  1341 end

  1342

  1343

  1344 subsection\<open>Exponentials and logarithms\<close>

  1345

  1346 context

  1347   fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"

  1348 begin

  1349

  1350 lemma sums_imp_has_prod_exp:

  1351   assumes "f sums s"

  1352   shows "raw_has_prod (\<lambda>i. exp (f i)) 0 (exp s)"

  1353   using assms continuous_on_exp [of UNIV "\<lambda>x::'a. x"]

  1354   using continuous_on_tendsto_compose [of UNIV exp "(\<lambda>n. sum f {..n})" s]

  1355   by (simp add: prod_defs sums_def_le exp_sum)

  1356

  1357 lemma convergent_prod_exp:

  1358   assumes "summable f"

  1359   shows "convergent_prod (\<lambda>i. exp (f i))"

  1360   using sums_imp_has_prod_exp assms unfolding summable_def convergent_prod_def  by blast

  1361

  1362 lemma prodinf_exp:

  1363   assumes "summable f"

  1364   shows "prodinf (\<lambda>i. exp (f i)) = exp (suminf f)"

  1365 proof -

  1366   have "f sums suminf f"

  1367     using assms by blast

  1368   then have "(\<lambda>i. exp (f i)) has_prod exp (suminf f)"

  1369     by (simp add: has_prod_def sums_imp_has_prod_exp)

  1370   then show ?thesis

  1371     by (rule has_prod_unique [symmetric])

  1372 qed

  1373

  1374 end

  1375

  1376 lemma has_prod_imp_sums_ln_real:

  1377   fixes f :: "nat \<Rightarrow> real"

  1378   assumes "raw_has_prod f 0 p" and 0: "\<And>x. f x > 0"

  1379   shows "(\<lambda>i. ln (f i)) sums (ln p)"

  1380 proof -

  1381   have "p > 0"

  1382     using assms unfolding prod_defs by (metis LIMSEQ_prod_nonneg less_eq_real_def)

  1383   then show ?thesis

  1384   using assms continuous_on_ln [of "{0<..}" "\<lambda>x. x"]

  1385   using continuous_on_tendsto_compose [of "{0<..}" ln "(\<lambda>n. prod f {..n})" p]

  1386   by (auto simp: prod_defs sums_def_le ln_prod order_tendstoD)

  1387 qed

  1388

  1389 lemma summable_ln_real:

  1390   fixes f :: "nat \<Rightarrow> real"

  1391   assumes f: "convergent_prod f" and 0: "\<And>x. f x > 0"

  1392   shows "summable (\<lambda>i. ln (f i))"

  1393 proof -

  1394   obtain M p where "raw_has_prod f M p"

  1395     using f convergent_prod_def by blast

  1396   then consider i where "i<M" "f i = 0" | p where "raw_has_prod f 0 p"

  1397     using raw_has_prod_cases by blast

  1398   then show ?thesis

  1399   proof cases

  1400     case 1

  1401     with 0 show ?thesis

  1402       by (metis less_irrefl)

  1403   next

  1404     case 2

  1405     then show ?thesis

  1406       using "0" has_prod_imp_sums_ln_real summable_def by blast

  1407   qed

  1408 qed

  1409

  1410 lemma suminf_ln_real:

  1411   fixes f :: "nat \<Rightarrow> real"

  1412   assumes f: "convergent_prod f" and 0: "\<And>x. f x > 0"

  1413   shows "suminf (\<lambda>i. ln (f i)) = ln (prodinf f)"

  1414 proof -

  1415   have "f has_prod prodinf f"

  1416     by (simp add: f has_prod_iff)

  1417   then have "raw_has_prod f 0 (prodinf f)"

  1418     by (metis "0" has_prod_def less_irrefl)

  1419   then have "(\<lambda>i. ln (f i)) sums ln (prodinf f)"

  1420     using "0" has_prod_imp_sums_ln_real by blast

  1421   then show ?thesis

  1422     by (rule sums_unique [symmetric])

  1423 qed

  1424

  1425 lemma prodinf_exp_real:

  1426   fixes f :: "nat \<Rightarrow> real"

  1427   assumes f: "convergent_prod f" and 0: "\<And>x. f x > 0"

  1428   shows "prodinf f = exp (suminf (\<lambda>i. ln (f i)))"

  1429   by (simp add: "0" f less_0_prodinf suminf_ln_real)

  1430

  1431

  1432 subsection\<open>Embeddings from the reals into some complete real normed field\<close>

  1433

  1434 lemma tendsto_of_real:

  1435   assumes "(\<lambda>n. of_real (f n) :: 'a::{complete_space,real_normed_field}) \<longlonglongrightarrow> q"

  1436   shows "q = of_real (lim f)"

  1437 proof -

  1438   have "convergent (\<lambda>n. of_real (f n) :: 'a)"

  1439     using assms convergent_def by blast

  1440   then have "convergent f"

  1441     unfolding convergent_def

  1442     by (simp add: convergent_eq_Cauchy Cauchy_def)

  1443   then show ?thesis

  1444     by (metis LIMSEQ_unique assms convergentD sequentially_bot tendsto_Lim tendsto_of_real)

  1445 qed

  1446

  1447 lemma tendsto_of_real':

  1448   assumes "(\<lambda>n. of_real (f n) :: 'a::{complete_space,real_normed_field}) \<longlonglongrightarrow> q"

  1449   obtains r where "q = of_real r"

  1450   using tendsto_of_real assms by blast

  1451

  1452 lemma has_prod_of_real_iff:

  1453   "(\<lambda>n. of_real (f n) :: 'a::{complete_space,real_normed_field}) has_prod of_real c \<longleftrightarrow> f has_prod c"

  1454   (is "?lhs = ?rhs")

  1455 proof

  1456   assume ?lhs

  1457   then show ?rhs

  1458     apply (auto simp: prod_defs LIMSEQ_prod_0 tendsto_of_real_iff simp flip: of_real_prod)

  1459     using tendsto_of_real'

  1460     by (metis of_real_0 tendsto_of_real_iff)

  1461 next

  1462   assume ?rhs

  1463   with tendsto_of_real_iff show ?lhs

  1464     by (fastforce simp: prod_defs simp flip: of_real_prod)

  1465 qed

  1466

  1467 end
`