src/HOL/Analysis/Infinite_Products.thy
author paulson <lp15@cam.ac.uk>
Fri Jun 15 12:18:06 2018 +0100 (13 months ago)
changeset 68452 c027dfbfad30
parent 68426 e0b5f2d14bf9
child 68517 6b5f15387353
permissions -rw-r--r--
more on infinite products. Also subgroup_imp_subset -> subgroup.subset
     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 raw_has_prod_Suc: 
   102   "raw_has_prod f (Suc M) a \<longleftrightarrow> raw_has_prod (\<lambda>n. f (Suc n)) M a"
   103   unfolding raw_has_prod_def by auto
   104 
   105 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))"
   106   by (simp add: has_prod_def)
   107       
   108 lemma has_prod_unique2: 
   109   fixes f :: "nat \<Rightarrow> 'a::{semidom,t2_space}"
   110   assumes "f has_prod a" "f has_prod b" shows "a = b"
   111   using assms
   112   by (auto simp: has_prod_def raw_has_prod_eq_0) (meson raw_has_prod_def sequentially_bot tendsto_unique)
   113 
   114 lemma has_prod_unique:
   115   fixes f :: "nat \<Rightarrow> 'a :: {semidom,t2_space}"
   116   shows "f has_prod s \<Longrightarrow> s = prodinf f"
   117   by (simp add: has_prod_unique2 prodinf_def the_equality)
   118 
   119 lemma convergent_prod_altdef:
   120   fixes f :: "nat \<Rightarrow> 'a :: {t2_space,comm_semiring_1}"
   121   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)"
   122 proof
   123   assume "convergent_prod f"
   124   then obtain M L where *: "(\<lambda>n. \<Prod>i\<le>n. f (i+M)) \<longlonglongrightarrow> L" "L \<noteq> 0"
   125     by (auto simp: prod_defs)
   126   have "f i \<noteq> 0" if "i \<ge> M" for i
   127   proof
   128     assume "f i = 0"
   129     have **: "eventually (\<lambda>n. (\<Prod>i\<le>n. f (i+M)) = 0) sequentially"
   130       using eventually_ge_at_top[of "i - M"]
   131     proof eventually_elim
   132       case (elim n)
   133       with \<open>f i = 0\<close> and \<open>i \<ge> M\<close> show ?case
   134         by (auto intro!: bexI[of _ "i - M"] prod_zero)
   135     qed
   136     have "(\<lambda>n. (\<Prod>i\<le>n. f (i+M))) \<longlonglongrightarrow> 0"
   137       unfolding filterlim_iff
   138       by (auto dest!: eventually_nhds_x_imp_x intro!: eventually_mono[OF **])
   139     from tendsto_unique[OF _ this *(1)] and *(2)
   140       show False by simp
   141   qed
   142   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)" 
   143     by blast
   144 qed (auto simp: prod_defs)
   145 
   146 
   147 subsection\<open>Absolutely convergent products\<close>
   148 
   149 definition abs_convergent_prod :: "(nat \<Rightarrow> _) \<Rightarrow> bool" where
   150   "abs_convergent_prod f \<longleftrightarrow> convergent_prod (\<lambda>i. 1 + norm (f i - 1))"
   151 
   152 lemma abs_convergent_prodI:
   153   assumes "convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
   154   shows   "abs_convergent_prod f"
   155 proof -
   156   from assms obtain L where L: "(\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1)) \<longlonglongrightarrow> L"
   157     by (auto simp: convergent_def)
   158   have "L \<ge> 1"
   159   proof (rule tendsto_le)
   160     show "eventually (\<lambda>n. (\<Prod>i\<le>n. 1 + norm (f i - 1)) \<ge> 1) sequentially"
   161     proof (intro always_eventually allI)
   162       fix n
   163       have "(\<Prod>i\<le>n. 1 + norm (f i - 1)) \<ge> (\<Prod>i\<le>n. 1)"
   164         by (intro prod_mono) auto
   165       thus "(\<Prod>i\<le>n. 1 + norm (f i - 1)) \<ge> 1" by simp
   166     qed
   167   qed (use L in simp_all)
   168   hence "L \<noteq> 0" by auto
   169   with L show ?thesis unfolding abs_convergent_prod_def prod_defs
   170     by (intro exI[of _ "0::nat"] exI[of _ L]) auto
   171 qed
   172 
   173 lemma
   174   fixes f :: "nat \<Rightarrow> 'a :: {topological_semigroup_mult,t2_space,idom}"
   175   assumes "convergent_prod f"
   176   shows   convergent_prod_imp_convergent: "convergent (\<lambda>n. \<Prod>i\<le>n. f i)"
   177     and   convergent_prod_to_zero_iff:    "(\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> 0 \<longleftrightarrow> (\<exists>i. f i = 0)"
   178 proof -
   179   from assms obtain M L 
   180     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"
   181     by (auto simp: convergent_prod_altdef)
   182   note this(2)
   183   also have "(\<lambda>n. \<Prod>i\<le>n. f (i + M)) = (\<lambda>n. \<Prod>i=M..M+n. f i)"
   184     by (intro ext prod.reindex_bij_witness[of _ "\<lambda>n. n - M" "\<lambda>n. n + M"]) auto
   185   finally have "(\<lambda>n. (\<Prod>i<M. f i) * (\<Prod>i=M..M+n. f i)) \<longlonglongrightarrow> (\<Prod>i<M. f i) * L"
   186     by (intro tendsto_mult tendsto_const)
   187   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))"
   188     by (subst prod.union_disjoint) auto
   189   also have "(\<lambda>n. {..<M} \<union> {M..M+n}) = (\<lambda>n. {..n+M})" by auto
   190   finally have lim: "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> prod f {..<M} * L" 
   191     by (rule LIMSEQ_offset)
   192   thus "convergent (\<lambda>n. \<Prod>i\<le>n. f i)"
   193     by (auto simp: convergent_def)
   194 
   195   show "(\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> 0 \<longleftrightarrow> (\<exists>i. f i = 0)"
   196   proof
   197     assume "\<exists>i. f i = 0"
   198     then obtain i where "f i = 0" by auto
   199     moreover with M have "i < M" by (cases "i < M") auto
   200     ultimately have "(\<Prod>i<M. f i) = 0" by auto
   201     with lim show "(\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> 0" by simp
   202   next
   203     assume "(\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> 0"
   204     from tendsto_unique[OF _ this lim] and \<open>L \<noteq> 0\<close>
   205     show "\<exists>i. f i = 0" by auto
   206   qed
   207 qed
   208 
   209 lemma convergent_prod_iff_nz_lim:
   210   fixes f :: "nat \<Rightarrow> 'a :: {topological_semigroup_mult,t2_space,idom}"
   211   assumes "\<And>i. f i \<noteq> 0"
   212   shows "convergent_prod f \<longleftrightarrow> (\<exists>L. (\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> L \<and> L \<noteq> 0)"
   213     (is "?lhs \<longleftrightarrow> ?rhs")
   214 proof
   215   assume ?lhs then show ?rhs
   216     using assms convergentD convergent_prod_imp_convergent convergent_prod_to_zero_iff by blast
   217 next
   218   assume ?rhs then show ?lhs
   219     unfolding prod_defs
   220     by (rule_tac x=0 in exI) auto
   221 qed
   222 
   223 lemma convergent_prod_iff_convergent: 
   224   fixes f :: "nat \<Rightarrow> 'a :: {topological_semigroup_mult,t2_space,idom}"
   225   assumes "\<And>i. f i \<noteq> 0"
   226   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"
   227   by (force simp: convergent_prod_iff_nz_lim assms convergent_def limI)
   228 
   229 
   230 lemma abs_convergent_prod_altdef:
   231   fixes f :: "nat \<Rightarrow> 'a :: {one,real_normed_vector}"
   232   shows  "abs_convergent_prod f \<longleftrightarrow> convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
   233 proof
   234   assume "abs_convergent_prod f"
   235   thus "convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
   236     by (auto simp: abs_convergent_prod_def intro!: convergent_prod_imp_convergent)
   237 qed (auto intro: abs_convergent_prodI)
   238 
   239 lemma weierstrass_prod_ineq:
   240   fixes f :: "'a \<Rightarrow> real" 
   241   assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> {0..1}"
   242   shows   "1 - sum f A \<le> (\<Prod>x\<in>A. 1 - f x)"
   243   using assms
   244 proof (induction A rule: infinite_finite_induct)
   245   case (insert x A)
   246   from insert.hyps and insert.prems 
   247     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)"
   248     by (intro insert.IH add_mono mult_left_mono prod_mono) auto
   249   with insert.hyps show ?case by (simp add: algebra_simps)
   250 qed simp_all
   251 
   252 lemma norm_prod_minus1_le_prod_minus1:
   253   fixes f :: "nat \<Rightarrow> 'a :: {real_normed_div_algebra,comm_ring_1}"  
   254   shows "norm (prod (\<lambda>n. 1 + f n) A - 1) \<le> prod (\<lambda>n. 1 + norm (f n)) A - 1"
   255 proof (induction A rule: infinite_finite_induct)
   256   case (insert x A)
   257   from insert.hyps have 
   258     "norm ((\<Prod>n\<in>insert x A. 1 + f n) - 1) = 
   259        norm ((\<Prod>n\<in>A. 1 + f n) - 1 + f x * (\<Prod>n\<in>A. 1 + f n))"
   260     by (simp add: algebra_simps)
   261   also have "\<dots> \<le> norm ((\<Prod>n\<in>A. 1 + f n) - 1) + norm (f x * (\<Prod>n\<in>A. 1 + f n))"
   262     by (rule norm_triangle_ineq)
   263   also have "norm (f x * (\<Prod>n\<in>A. 1 + f n)) = norm (f x) * (\<Prod>x\<in>A. norm (1 + f x))"
   264     by (simp add: prod_norm norm_mult)
   265   also have "(\<Prod>x\<in>A. norm (1 + f x)) \<le> (\<Prod>x\<in>A. norm (1::'a) + norm (f x))"
   266     by (intro prod_mono norm_triangle_ineq ballI conjI) auto
   267   also have "norm (1::'a) = 1" by simp
   268   also note insert.IH
   269   also have "(\<Prod>n\<in>A. 1 + norm (f n)) - 1 + norm (f x) * (\<Prod>x\<in>A. 1 + norm (f x)) =
   270              (\<Prod>n\<in>insert x A. 1 + norm (f n)) - 1"
   271     using insert.hyps by (simp add: algebra_simps)
   272   finally show ?case by - (simp_all add: mult_left_mono)
   273 qed simp_all
   274 
   275 lemma convergent_prod_imp_ev_nonzero:
   276   fixes f :: "nat \<Rightarrow> 'a :: {t2_space,comm_semiring_1}"
   277   assumes "convergent_prod f"
   278   shows   "eventually (\<lambda>n. f n \<noteq> 0) sequentially"
   279   using assms by (auto simp: eventually_at_top_linorder convergent_prod_altdef)
   280 
   281 lemma convergent_prod_imp_LIMSEQ:
   282   fixes f :: "nat \<Rightarrow> 'a :: {real_normed_field}"
   283   assumes "convergent_prod f"
   284   shows   "f \<longlonglongrightarrow> 1"
   285 proof -
   286   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"
   287     by (auto simp: convergent_prod_altdef)
   288   hence L': "(\<lambda>n. \<Prod>i\<le>Suc n. f (i+M)) \<longlonglongrightarrow> L" by (subst filterlim_sequentially_Suc)
   289   have "(\<lambda>n. (\<Prod>i\<le>Suc n. f (i+M)) / (\<Prod>i\<le>n. f (i+M))) \<longlonglongrightarrow> L / L"
   290     using L L' by (intro tendsto_divide) simp_all
   291   also from L have "L / L = 1" by simp
   292   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))"
   293     using assms L by (auto simp: fun_eq_iff atMost_Suc)
   294   finally show ?thesis by (rule LIMSEQ_offset)
   295 qed
   296 
   297 lemma abs_convergent_prod_imp_summable:
   298   fixes f :: "nat \<Rightarrow> 'a :: real_normed_div_algebra"
   299   assumes "abs_convergent_prod f"
   300   shows "summable (\<lambda>i. norm (f i - 1))"
   301 proof -
   302   from assms have "convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))" 
   303     unfolding abs_convergent_prod_def by (rule convergent_prod_imp_convergent)
   304   then obtain L where L: "(\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1)) \<longlonglongrightarrow> L"
   305     unfolding convergent_def by blast
   306   have "convergent (\<lambda>n. \<Sum>i\<le>n. norm (f i - 1))"
   307   proof (rule Bseq_monoseq_convergent)
   308     have "eventually (\<lambda>n. (\<Prod>i\<le>n. 1 + norm (f i - 1)) < L + 1) sequentially"
   309       using L(1) by (rule order_tendstoD) simp_all
   310     hence "\<forall>\<^sub>F x in sequentially. norm (\<Sum>i\<le>x. norm (f i - 1)) \<le> L + 1"
   311     proof eventually_elim
   312       case (elim n)
   313       have "norm (\<Sum>i\<le>n. norm (f i - 1)) = (\<Sum>i\<le>n. norm (f i - 1))"
   314         unfolding real_norm_def by (intro abs_of_nonneg sum_nonneg) simp_all
   315       also have "\<dots> \<le> (\<Prod>i\<le>n. 1 + norm (f i - 1))" by (rule sum_le_prod) auto
   316       also have "\<dots> < L + 1" by (rule elim)
   317       finally show ?case by simp
   318     qed
   319     thus "Bseq (\<lambda>n. \<Sum>i\<le>n. norm (f i - 1))" by (rule BfunI)
   320   next
   321     show "monoseq (\<lambda>n. \<Sum>i\<le>n. norm (f i - 1))"
   322       by (rule mono_SucI1) auto
   323   qed
   324   thus "summable (\<lambda>i. norm (f i - 1))" by (simp add: summable_iff_convergent')
   325 qed
   326 
   327 lemma summable_imp_abs_convergent_prod:
   328   fixes f :: "nat \<Rightarrow> 'a :: real_normed_div_algebra"
   329   assumes "summable (\<lambda>i. norm (f i - 1))"
   330   shows   "abs_convergent_prod f"
   331 proof (intro abs_convergent_prodI Bseq_monoseq_convergent)
   332   show "monoseq (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
   333     by (intro mono_SucI1) 
   334        (auto simp: atMost_Suc algebra_simps intro!: mult_nonneg_nonneg prod_nonneg)
   335 next
   336   show "Bseq (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
   337   proof (rule Bseq_eventually_mono)
   338     show "eventually (\<lambda>n. norm (\<Prod>i\<le>n. 1 + norm (f i - 1)) \<le> 
   339             norm (exp (\<Sum>i\<le>n. norm (f i - 1)))) sequentially"
   340       by (intro always_eventually allI) (auto simp: abs_prod exp_sum intro!: prod_mono)
   341   next
   342     from assms have "(\<lambda>n. \<Sum>i\<le>n. norm (f i - 1)) \<longlonglongrightarrow> (\<Sum>i. norm (f i - 1))"
   343       using sums_def_le by blast
   344     hence "(\<lambda>n. exp (\<Sum>i\<le>n. norm (f i - 1))) \<longlonglongrightarrow> exp (\<Sum>i. norm (f i - 1))"
   345       by (rule tendsto_exp)
   346     hence "convergent (\<lambda>n. exp (\<Sum>i\<le>n. norm (f i - 1)))"
   347       by (rule convergentI)
   348     thus "Bseq (\<lambda>n. exp (\<Sum>i\<le>n. norm (f i - 1)))"
   349       by (rule convergent_imp_Bseq)
   350   qed
   351 qed
   352 
   353 lemma abs_convergent_prod_conv_summable:
   354   fixes f :: "nat \<Rightarrow> 'a :: real_normed_div_algebra"
   355   shows "abs_convergent_prod f \<longleftrightarrow> summable (\<lambda>i. norm (f i - 1))"
   356   by (blast intro: abs_convergent_prod_imp_summable summable_imp_abs_convergent_prod)
   357 
   358 lemma abs_convergent_prod_imp_LIMSEQ:
   359   fixes f :: "nat \<Rightarrow> 'a :: {comm_ring_1,real_normed_div_algebra}"
   360   assumes "abs_convergent_prod f"
   361   shows   "f \<longlonglongrightarrow> 1"
   362 proof -
   363   from assms have "summable (\<lambda>n. norm (f n - 1))"
   364     by (rule abs_convergent_prod_imp_summable)
   365   from summable_LIMSEQ_zero[OF this] have "(\<lambda>n. f n - 1) \<longlonglongrightarrow> 0"
   366     by (simp add: tendsto_norm_zero_iff)
   367   from tendsto_add[OF this tendsto_const[of 1]] show ?thesis by simp
   368 qed
   369 
   370 lemma abs_convergent_prod_imp_ev_nonzero:
   371   fixes f :: "nat \<Rightarrow> 'a :: {comm_ring_1,real_normed_div_algebra}"
   372   assumes "abs_convergent_prod f"
   373   shows   "eventually (\<lambda>n. f n \<noteq> 0) sequentially"
   374 proof -
   375   from assms have "f \<longlonglongrightarrow> 1" 
   376     by (rule abs_convergent_prod_imp_LIMSEQ)
   377   hence "eventually (\<lambda>n. dist (f n) 1 < 1) at_top"
   378     by (auto simp: tendsto_iff)
   379   thus ?thesis by eventually_elim auto
   380 qed
   381 
   382 lemma convergent_prod_offset:
   383   assumes "convergent_prod (\<lambda>n. f (n + m))"  
   384   shows   "convergent_prod f"
   385 proof -
   386   from assms obtain M L where "(\<lambda>n. \<Prod>k\<le>n. f (k + (M + m))) \<longlonglongrightarrow> L" "L \<noteq> 0"
   387     by (auto simp: prod_defs add.assoc)
   388   thus "convergent_prod f" 
   389     unfolding prod_defs by blast
   390 qed
   391 
   392 lemma abs_convergent_prod_offset:
   393   assumes "abs_convergent_prod (\<lambda>n. f (n + m))"  
   394   shows   "abs_convergent_prod f"
   395   using assms unfolding abs_convergent_prod_def by (rule convergent_prod_offset)
   396 
   397 subsection\<open>Ignoring initial segments\<close>
   398 
   399 lemma raw_has_prod_ignore_initial_segment:
   400   fixes f :: "nat \<Rightarrow> 'a :: real_normed_field"
   401   assumes "raw_has_prod f M p" "N \<ge> M"
   402   obtains q where  "raw_has_prod f N q"
   403 proof -
   404   have p: "(\<lambda>n. \<Prod>k\<le>n. f (k + M)) \<longlonglongrightarrow> p" and "p \<noteq> 0" 
   405     using assms by (auto simp: raw_has_prod_def)
   406   then have nz: "\<And>n. n \<ge> M \<Longrightarrow> f n \<noteq> 0"
   407     using assms by (auto simp: raw_has_prod_eq_0)
   408   define C where "C = (\<Prod>k<N-M. f (k + M))"
   409   from nz have [simp]: "C \<noteq> 0" 
   410     by (auto simp: C_def)
   411 
   412   from p have "(\<lambda>i. \<Prod>k\<le>i + (N-M). f (k + M)) \<longlonglongrightarrow> p" 
   413     by (rule LIMSEQ_ignore_initial_segment)
   414   also have "(\<lambda>i. \<Prod>k\<le>i + (N-M). f (k + M)) = (\<lambda>n. C * (\<Prod>k\<le>n. f (k + N)))"
   415   proof (rule ext, goal_cases)
   416     case (1 n)
   417     have "{..n+(N-M)} = {..<(N-M)} \<union> {(N-M)..n+(N-M)}" by auto
   418     also have "(\<Prod>k\<in>\<dots>. f (k + M)) = C * (\<Prod>k=(N-M)..n+(N-M). f (k + M))"
   419       unfolding C_def by (rule prod.union_disjoint) auto
   420     also have "(\<Prod>k=(N-M)..n+(N-M). f (k + M)) = (\<Prod>k\<le>n. f (k + (N-M) + M))"
   421       by (intro ext prod.reindex_bij_witness[of _ "\<lambda>k. k + (N-M)" "\<lambda>k. k - (N-M)"]) auto
   422     finally show ?case
   423       using \<open>N \<ge> M\<close> by (simp add: add_ac)
   424   qed
   425   finally have "(\<lambda>n. C * (\<Prod>k\<le>n. f (k + N)) / C) \<longlonglongrightarrow> p / C"
   426     by (intro tendsto_divide tendsto_const) auto
   427   hence "(\<lambda>n. \<Prod>k\<le>n. f (k + N)) \<longlonglongrightarrow> p / C" by simp
   428   moreover from \<open>p \<noteq> 0\<close> have "p / C \<noteq> 0" by simp
   429   ultimately show ?thesis
   430     using raw_has_prod_def that by blast 
   431 qed
   432 
   433 corollary convergent_prod_ignore_initial_segment:
   434   fixes f :: "nat \<Rightarrow> 'a :: real_normed_field"
   435   assumes "convergent_prod f"
   436   shows   "convergent_prod (\<lambda>n. f (n + m))"
   437   using assms
   438   unfolding convergent_prod_def 
   439   apply clarify
   440   apply (erule_tac N="M+m" in raw_has_prod_ignore_initial_segment)
   441   apply (auto simp add: raw_has_prod_def add_ac)
   442   done
   443 
   444 corollary convergent_prod_ignore_nonzero_segment:
   445   fixes f :: "nat \<Rightarrow> 'a :: real_normed_field"
   446   assumes f: "convergent_prod f" and nz: "\<And>i. i \<ge> M \<Longrightarrow> f i \<noteq> 0"
   447   shows "\<exists>p. raw_has_prod f M p"
   448   using convergent_prod_ignore_initial_segment [OF f]
   449   by (metis convergent_LIMSEQ_iff convergent_prod_iff_convergent le_add_same_cancel2 nz prod_defs(1) zero_order(1))
   450 
   451 corollary abs_convergent_prod_ignore_initial_segment:
   452   assumes "abs_convergent_prod f"
   453   shows   "abs_convergent_prod (\<lambda>n. f (n + m))"
   454   using assms unfolding abs_convergent_prod_def 
   455   by (rule convergent_prod_ignore_initial_segment)
   456 
   457 lemma abs_convergent_prod_imp_convergent_prod:
   458   fixes f :: "nat \<Rightarrow> 'a :: {real_normed_div_algebra,complete_space,comm_ring_1}"
   459   assumes "abs_convergent_prod f"
   460   shows   "convergent_prod f"
   461 proof -
   462   from assms have "eventually (\<lambda>n. f n \<noteq> 0) sequentially"
   463     by (rule abs_convergent_prod_imp_ev_nonzero)
   464   then obtain N where N: "f n \<noteq> 0" if "n \<ge> N" for n 
   465     by (auto simp: eventually_at_top_linorder)
   466   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)"
   467 
   468   have "Cauchy ?P"
   469   proof (rule CauchyI', goal_cases)
   470     case (1 \<epsilon>)
   471     from assms have "abs_convergent_prod (\<lambda>n. f (n + N))"
   472       by (rule abs_convergent_prod_ignore_initial_segment)
   473     hence "Cauchy ?Q"
   474       unfolding abs_convergent_prod_def
   475       by (intro convergent_Cauchy convergent_prod_imp_convergent)
   476     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
   477       by blast
   478     show ?case
   479     proof (rule exI[of _ M], safe, goal_cases)
   480       case (1 m n)
   481       have "dist (?P m) (?P n) = norm (?P n - ?P m)"
   482         by (simp add: dist_norm norm_minus_commute)
   483       also from 1 have "{..n} = {..m} \<union> {m<..n}" by auto
   484       hence "norm (?P n - ?P m) = norm (?P m * (\<Prod>k\<in>{m<..n}. f (k + N)) - ?P m)"
   485         by (subst prod.union_disjoint [symmetric]) (auto simp: algebra_simps)
   486       also have "\<dots> = norm (?P m * ((\<Prod>k\<in>{m<..n}. f (k + N)) - 1))"
   487         by (simp add: algebra_simps)
   488       also have "\<dots> = (\<Prod>k\<le>m. norm (f (k + N))) * norm ((\<Prod>k\<in>{m<..n}. f (k + N)) - 1)"
   489         by (simp add: norm_mult prod_norm)
   490       also have "\<dots> \<le> ?Q m * ((\<Prod>k\<in>{m<..n}. 1 + norm (f (k + N) - 1)) - 1)"
   491         using norm_prod_minus1_le_prod_minus1[of "\<lambda>k. f (k + N) - 1" "{m<..n}"]
   492               norm_triangle_ineq[of 1 "f k - 1" for k]
   493         by (intro mult_mono prod_mono ballI conjI norm_prod_minus1_le_prod_minus1 prod_nonneg) auto
   494       also have "\<dots> = ?Q m * (\<Prod>k\<in>{m<..n}. 1 + norm (f (k + N) - 1)) - ?Q m"
   495         by (simp add: algebra_simps)
   496       also have "?Q m * (\<Prod>k\<in>{m<..n}. 1 + norm (f (k + N) - 1)) = 
   497                    (\<Prod>k\<in>{..m}\<union>{m<..n}. 1 + norm (f (k + N) - 1))"
   498         by (rule prod.union_disjoint [symmetric]) auto
   499       also from 1 have "{..m}\<union>{m<..n} = {..n}" by auto
   500       also have "?Q n - ?Q m \<le> norm (?Q n - ?Q m)" by simp
   501       also from 1 have "\<dots> < \<epsilon>" by (intro M) auto
   502       finally show ?case .
   503     qed
   504   qed
   505   hence conv: "convergent ?P" by (rule Cauchy_convergent)
   506   then obtain L where L: "?P \<longlonglongrightarrow> L"
   507     by (auto simp: convergent_def)
   508 
   509   have "L \<noteq> 0"
   510   proof
   511     assume [simp]: "L = 0"
   512     from tendsto_norm[OF L] have limit: "(\<lambda>n. \<Prod>k\<le>n. norm (f (k + N))) \<longlonglongrightarrow> 0" 
   513       by (simp add: prod_norm)
   514 
   515     from assms have "(\<lambda>n. f (n + N)) \<longlonglongrightarrow> 1"
   516       by (intro abs_convergent_prod_imp_LIMSEQ abs_convergent_prod_ignore_initial_segment)
   517     hence "eventually (\<lambda>n. norm (f (n + N) - 1) < 1) sequentially"
   518       by (auto simp: tendsto_iff dist_norm)
   519     then obtain M0 where M0: "norm (f (n + N) - 1) < 1" if "n \<ge> M0" for n
   520       by (auto simp: eventually_at_top_linorder)
   521 
   522     {
   523       fix M assume M: "M \<ge> M0"
   524       with M0 have M: "norm (f (n + N) - 1) < 1" if "n \<ge> M" for n using that by simp
   525 
   526       have "(\<lambda>n. \<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1)) \<longlonglongrightarrow> 0"
   527       proof (rule tendsto_sandwich)
   528         show "eventually (\<lambda>n. (\<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1)) \<ge> 0) sequentially"
   529           using M by (intro always_eventually prod_nonneg allI ballI) (auto intro: less_imp_le)
   530         have "norm (1::'a) - norm (f (i + M + N) - 1) \<le> norm (f (i + M + N))" for i
   531           using norm_triangle_ineq3[of "f (i + M + N)" 1] by simp
   532         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"
   533           using M by (intro always_eventually allI prod_mono ballI conjI) (auto intro: less_imp_le)
   534         
   535         define C where "C = (\<Prod>k<M. norm (f (k + N)))"
   536         from N have [simp]: "C \<noteq> 0" by (auto simp: C_def)
   537         from L have "(\<lambda>n. norm (\<Prod>k\<le>n+M. f (k + N))) \<longlonglongrightarrow> 0"
   538           by (intro LIMSEQ_ignore_initial_segment) (simp add: tendsto_norm_zero_iff)
   539         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))))"
   540         proof (rule ext, goal_cases)
   541           case (1 n)
   542           have "{..n+M} = {..<M} \<union> {M..n+M}" by auto
   543           also have "norm (\<Prod>k\<in>\<dots>. f (k + N)) = C * norm (\<Prod>k=M..n+M. f (k + N))"
   544             unfolding C_def by (subst prod.union_disjoint) (auto simp: norm_mult prod_norm)
   545           also have "(\<Prod>k=M..n+M. f (k + N)) = (\<Prod>k\<le>n. f (k + N + M))"
   546             by (intro prod.reindex_bij_witness[of _ "\<lambda>i. i + M" "\<lambda>i. i - M"]) auto
   547           finally show ?case by (simp add: add_ac prod_norm)
   548         qed
   549         finally have "(\<lambda>n. C * (\<Prod>k\<le>n. norm (f (k + M + N))) / C) \<longlonglongrightarrow> 0 / C"
   550           by (intro tendsto_divide tendsto_const) auto
   551         thus "(\<lambda>n. \<Prod>k\<le>n. norm (f (k + M + N))) \<longlonglongrightarrow> 0" by simp
   552       qed simp_all
   553 
   554       have "1 - (\<Sum>i. norm (f (i + M + N) - 1)) \<le> 0"
   555       proof (rule tendsto_le)
   556         show "eventually (\<lambda>n. 1 - (\<Sum>k\<le>n. norm (f (k+M+N) - 1)) \<le> 
   557                                 (\<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1))) at_top"
   558           using M by (intro always_eventually allI weierstrass_prod_ineq) (auto intro: less_imp_le)
   559         show "(\<lambda>n. \<Prod>k\<le>n. 1 - norm (f (k+M+N) - 1)) \<longlonglongrightarrow> 0" by fact
   560         show "(\<lambda>n. 1 - (\<Sum>k\<le>n. norm (f (k + M + N) - 1)))
   561                   \<longlonglongrightarrow> 1 - (\<Sum>i. norm (f (i + M + N) - 1))"
   562           by (intro tendsto_intros summable_LIMSEQ' summable_ignore_initial_segment 
   563                 abs_convergent_prod_imp_summable assms)
   564       qed simp_all
   565       hence "(\<Sum>i. norm (f (i + M + N) - 1)) \<ge> 1" by simp
   566       also have "\<dots> + (\<Sum>i<M. norm (f (i + N) - 1)) = (\<Sum>i. norm (f (i + N) - 1))"
   567         by (intro suminf_split_initial_segment [symmetric] summable_ignore_initial_segment
   568               abs_convergent_prod_imp_summable assms)
   569       finally have "1 + (\<Sum>i<M. norm (f (i + N) - 1)) \<le> (\<Sum>i. norm (f (i + N) - 1))" by simp
   570     } note * = this
   571 
   572     have "1 + (\<Sum>i. norm (f (i + N) - 1)) \<le> (\<Sum>i. norm (f (i + N) - 1))"
   573     proof (rule tendsto_le)
   574       show "(\<lambda>M. 1 + (\<Sum>i<M. norm (f (i + N) - 1))) \<longlonglongrightarrow> 1 + (\<Sum>i. norm (f (i + N) - 1))"
   575         by (intro tendsto_intros summable_LIMSEQ summable_ignore_initial_segment 
   576                 abs_convergent_prod_imp_summable assms)
   577       show "eventually (\<lambda>M. 1 + (\<Sum>i<M. norm (f (i + N) - 1)) \<le> (\<Sum>i. norm (f (i + N) - 1))) at_top"
   578         using eventually_ge_at_top[of M0] by eventually_elim (use * in auto)
   579     qed simp_all
   580     thus False by simp
   581   qed
   582   with L show ?thesis by (auto simp: prod_defs)
   583 qed
   584 
   585 subsection\<open>More elementary properties\<close>
   586 
   587 lemma raw_has_prod_cases:
   588   fixes f :: "nat \<Rightarrow> 'a :: {idom,topological_semigroup_mult,t2_space}"
   589   assumes "raw_has_prod f M p"
   590   obtains i where "i<M" "f i = 0" | p where "raw_has_prod f 0 p"
   591 proof -
   592   have "(\<lambda>n. \<Prod>i\<le>n. f (i + M)) \<longlonglongrightarrow> p" "p \<noteq> 0"
   593     using assms unfolding raw_has_prod_def by blast+
   594   then have "(\<lambda>n. prod f {..<M} * (\<Prod>i\<le>n. f (i + M))) \<longlonglongrightarrow> prod f {..<M} * p"
   595     by (metis tendsto_mult_left)
   596   moreover have "prod f {..<M} * (\<Prod>i\<le>n. f (i + M)) = prod f {..n+M}" for n
   597   proof -
   598     have "{..n+M} = {..<M} \<union> {M..n+M}"
   599       by auto
   600     then have "prod f {..n+M} = prod f {..<M} * prod f {M..n+M}"
   601       by simp (subst prod.union_disjoint; force)
   602     also have "\<dots> = prod f {..<M} * (\<Prod>i\<le>n. f (i + M))"
   603       by (metis (mono_tags, lifting) add.left_neutral atMost_atLeast0 prod_shift_bounds_cl_nat_ivl)
   604     finally show ?thesis by metis
   605   qed
   606   ultimately have "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> prod f {..<M} * p"
   607     by (auto intro: LIMSEQ_offset [where k=M])
   608   then have "raw_has_prod f 0 (prod f {..<M} * p)" if "\<forall>i<M. f i \<noteq> 0"
   609     using \<open>p \<noteq> 0\<close> assms that by (auto simp: raw_has_prod_def)
   610   then show thesis
   611     using that by blast
   612 qed
   613 
   614 corollary convergent_prod_offset_0:
   615   fixes f :: "nat \<Rightarrow> 'a :: {idom,topological_semigroup_mult,t2_space}"
   616   assumes "convergent_prod f" "\<And>i. f i \<noteq> 0"
   617   shows "\<exists>p. raw_has_prod f 0 p"
   618   using assms convergent_prod_def raw_has_prod_cases by blast
   619 
   620 lemma prodinf_eq_lim:
   621   fixes f :: "nat \<Rightarrow> 'a :: {idom,topological_semigroup_mult,t2_space}"
   622   assumes "convergent_prod f" "\<And>i. f i \<noteq> 0"
   623   shows "prodinf f = lim (\<lambda>n. \<Prod>i\<le>n. f i)"
   624   using assms convergent_prod_offset_0 [OF assms]
   625   by (simp add: prod_defs lim_def) (metis (no_types) assms(1) convergent_prod_to_zero_iff)
   626 
   627 lemma has_prod_one[simp, intro]: "(\<lambda>n. 1) has_prod 1"
   628   unfolding prod_defs by auto
   629 
   630 lemma convergent_prod_one[simp, intro]: "convergent_prod (\<lambda>n. 1)"
   631   unfolding prod_defs by auto
   632 
   633 lemma prodinf_cong: "(\<And>n. f n = g n) \<Longrightarrow> prodinf f = prodinf g"
   634   by presburger
   635 
   636 lemma convergent_prod_cong:
   637   fixes f g :: "nat \<Rightarrow> 'a::{field,topological_semigroup_mult,t2_space}"
   638   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"
   639   shows "convergent_prod f = convergent_prod g"
   640 proof -
   641   from assms obtain N where N: "\<forall>n\<ge>N. f n = g n"
   642     by (auto simp: eventually_at_top_linorder)
   643   define C where "C = (\<Prod>k<N. f k / g k)"
   644   with g have "C \<noteq> 0"
   645     by (simp add: f)
   646   have *: "eventually (\<lambda>n. prod f {..n} = C * prod g {..n}) sequentially"
   647     using eventually_ge_at_top[of N]
   648   proof eventually_elim
   649     case (elim n)
   650     then have "{..n} = {..<N} \<union> {N..n}"
   651       by auto
   652     also have "prod f \<dots> = prod f {..<N} * prod f {N..n}"
   653       by (intro prod.union_disjoint) auto
   654     also from N have "prod f {N..n} = prod g {N..n}"
   655       by (intro prod.cong) simp_all
   656     also have "prod f {..<N} * prod g {N..n} = C * (prod g {..<N} * prod g {N..n})"
   657       unfolding C_def by (simp add: g prod_dividef)
   658     also have "prod g {..<N} * prod g {N..n} = prod g ({..<N} \<union> {N..n})"
   659       by (intro prod.union_disjoint [symmetric]) auto
   660     also from elim have "{..<N} \<union> {N..n} = {..n}"
   661       by auto                                                                    
   662     finally show "prod f {..n} = C * prod g {..n}" .
   663   qed
   664   then have cong: "convergent (\<lambda>n. prod f {..n}) = convergent (\<lambda>n. C * prod g {..n})"
   665     by (rule convergent_cong)
   666   show ?thesis
   667   proof
   668     assume cf: "convergent_prod f"
   669     then have "\<not> (\<lambda>n. prod g {..n}) \<longlonglongrightarrow> 0"
   670       using tendsto_mult_left * convergent_prod_to_zero_iff f filterlim_cong by fastforce
   671     then show "convergent_prod g"
   672       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)
   673   next
   674     assume cg: "convergent_prod g"
   675     have "\<exists>a. C * a \<noteq> 0 \<and> (\<lambda>n. prod g {..n}) \<longlonglongrightarrow> a"
   676       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)
   677     then show "convergent_prod f"
   678       using "*" tendsto_mult_left filterlim_cong
   679       by (fastforce simp add: convergent_prod_iff_nz_lim f)
   680   qed
   681 qed
   682 
   683 lemma has_prod_finite:
   684   fixes f :: "nat \<Rightarrow> 'a::{semidom,t2_space}"
   685   assumes [simp]: "finite N"
   686     and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 1"
   687   shows "f has_prod (\<Prod>n\<in>N. f n)"
   688 proof -
   689   have eq: "prod f {..n + Suc (Max N)} = prod f N" for n
   690   proof (rule prod.mono_neutral_right)
   691     show "N \<subseteq> {..n + Suc (Max N)}"
   692       by (auto simp: le_Suc_eq trans_le_add2)
   693     show "\<forall>i\<in>{..n + Suc (Max N)} - N. f i = 1"
   694       using f by blast
   695   qed auto
   696   show ?thesis
   697   proof (cases "\<forall>n\<in>N. f n \<noteq> 0")
   698     case True
   699     then have "prod f N \<noteq> 0"
   700       by simp
   701     moreover have "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> prod f N"
   702       by (rule LIMSEQ_offset[of _ "Suc (Max N)"]) (simp add: eq atLeast0LessThan del: add_Suc_right)
   703     ultimately show ?thesis
   704       by (simp add: raw_has_prod_def has_prod_def)
   705   next
   706     case False
   707     then obtain k where "k \<in> N" "f k = 0"
   708       by auto
   709     let ?Z = "{n \<in> N. f n = 0}"
   710     have maxge: "Max ?Z \<ge> n" if "f n = 0" for n
   711       using Max_ge [of ?Z] \<open>finite N\<close> \<open>f n = 0\<close>
   712       by (metis (mono_tags) Collect_mem_eq f finite_Collect_conjI mem_Collect_eq zero_neq_one)
   713     let ?q = "prod f {Suc (Max ?Z)..Max N}"
   714     have [simp]: "?q \<noteq> 0"
   715       using maxge Suc_n_not_le_n le_trans by force
   716     have eq: "(\<Prod>i\<le>n + Max N. f (Suc (i + Max ?Z))) = ?q" for n
   717     proof -
   718       have "(\<Prod>i\<le>n + Max N. f (Suc (i + Max ?Z))) = prod f {Suc (Max ?Z)..n + Max N + Suc (Max ?Z)}" 
   719       proof (rule prod.reindex_cong [where l = "\<lambda>i. i + Suc (Max ?Z)", THEN sym])
   720         show "{Suc (Max ?Z)..n + Max N + Suc (Max ?Z)} = (\<lambda>i. i + Suc (Max ?Z)) ` {..n + Max N}"
   721           using le_Suc_ex by fastforce
   722       qed (auto simp: inj_on_def)
   723       also have "\<dots> = ?q"
   724         by (rule prod.mono_neutral_right)
   725            (use Max.coboundedI [OF \<open>finite N\<close>] f in \<open>force+\<close>)
   726       finally show ?thesis .
   727     qed
   728     have q: "raw_has_prod f (Suc (Max ?Z)) ?q"
   729     proof (simp add: raw_has_prod_def)
   730       show "(\<lambda>n. \<Prod>i\<le>n. f (Suc (i + Max ?Z))) \<longlonglongrightarrow> ?q"
   731         by (rule LIMSEQ_offset[of _ "(Max N)"]) (simp add: eq)
   732     qed
   733     show ?thesis
   734       unfolding has_prod_def
   735     proof (intro disjI2 exI conjI)      
   736       show "prod f N = 0"
   737         using \<open>f k = 0\<close> \<open>k \<in> N\<close> \<open>finite N\<close> prod_zero by blast
   738       show "f (Max ?Z) = 0"
   739         using Max_in [of ?Z] \<open>finite N\<close> \<open>f k = 0\<close> \<open>k \<in> N\<close> by auto
   740     qed (use q in auto)
   741   qed
   742 qed
   743 
   744 corollary has_prod_0:
   745   fixes f :: "nat \<Rightarrow> 'a::{semidom,t2_space}"
   746   assumes "\<And>n. f n = 1"
   747   shows "f has_prod 1"
   748   by (simp add: assms has_prod_cong)
   749 
   750 lemma prodinf_zero[simp]: "prodinf (\<lambda>n. 1::'a::real_normed_field) = 1"
   751   using has_prod_unique by force
   752 
   753 lemma convergent_prod_finite:
   754   fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
   755   assumes "finite N" "\<And>n. n \<notin> N \<Longrightarrow> f n = 1"
   756   shows "convergent_prod f"
   757 proof -
   758   have "\<exists>n p. raw_has_prod f n p"
   759     using assms has_prod_def has_prod_finite by blast
   760   then show ?thesis
   761     by (simp add: convergent_prod_def)
   762 qed
   763 
   764 lemma has_prod_If_finite_set:
   765   fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
   766   shows "finite A \<Longrightarrow> (\<lambda>r. if r \<in> A then f r else 1) has_prod (\<Prod>r\<in>A. f r)"
   767   using has_prod_finite[of A "(\<lambda>r. if r \<in> A then f r else 1)"]
   768   by simp
   769 
   770 lemma has_prod_If_finite:
   771   fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
   772   shows "finite {r. P r} \<Longrightarrow> (\<lambda>r. if P r then f r else 1) has_prod (\<Prod>r | P r. f r)"
   773   using has_prod_If_finite_set[of "{r. P r}"] by simp
   774 
   775 lemma convergent_prod_If_finite_set[simp, intro]:
   776   fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
   777   shows "finite A \<Longrightarrow> convergent_prod (\<lambda>r. if r \<in> A then f r else 1)"
   778   by (simp add: convergent_prod_finite)
   779 
   780 lemma convergent_prod_If_finite[simp, intro]:
   781   fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
   782   shows "finite {r. P r} \<Longrightarrow> convergent_prod (\<lambda>r. if P r then f r else 1)"
   783   using convergent_prod_def has_prod_If_finite has_prod_def by fastforce
   784 
   785 lemma has_prod_single:
   786   fixes f :: "nat \<Rightarrow> 'a::{idom,t2_space}"
   787   shows "(\<lambda>r. if r = i then f r else 1) has_prod f i"
   788   using has_prod_If_finite[of "\<lambda>r. r = i"] by simp
   789 
   790 context
   791   fixes f :: "nat \<Rightarrow> 'a :: real_normed_field"
   792 begin
   793 
   794 lemma convergent_prod_imp_has_prod: 
   795   assumes "convergent_prod f"
   796   shows "\<exists>p. f has_prod p"
   797 proof -
   798   obtain M p where p: "raw_has_prod f M p"
   799     using assms convergent_prod_def by blast
   800   then have "p \<noteq> 0"
   801     using raw_has_prod_nonzero by blast
   802   with p have fnz: "f i \<noteq> 0" if "i \<ge> M" for i
   803     using raw_has_prod_eq_0 that by blast
   804   define C where "C = (\<Prod>n<M. f n)"
   805   show ?thesis
   806   proof (cases "\<forall>n\<le>M. f n \<noteq> 0")
   807     case True
   808     then have "C \<noteq> 0"
   809       by (simp add: C_def)
   810     then show ?thesis
   811       by (meson True assms convergent_prod_offset_0 fnz has_prod_def nat_le_linear)
   812   next
   813     case False
   814     let ?N = "GREATEST n. f n = 0"
   815     have 0: "f ?N = 0"
   816       using fnz False
   817       by (metis (mono_tags, lifting) GreatestI_ex_nat nat_le_linear)
   818     have "f i \<noteq> 0" if "i > ?N" for i
   819       by (metis (mono_tags, lifting) Greatest_le_nat fnz leD linear that)
   820     then have "\<exists>p. raw_has_prod f (Suc ?N) p"
   821       using assms by (auto simp: intro!: convergent_prod_ignore_nonzero_segment)
   822     then show ?thesis
   823       unfolding has_prod_def using 0 by blast
   824   qed
   825 qed
   826 
   827 lemma convergent_prod_has_prod [intro]:
   828   shows "convergent_prod f \<Longrightarrow> f has_prod (prodinf f)"
   829   unfolding prodinf_def
   830   by (metis convergent_prod_imp_has_prod has_prod_unique theI')
   831 
   832 lemma convergent_prod_LIMSEQ:
   833   shows "convergent_prod f \<Longrightarrow> (\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> prodinf f"
   834   by (metis convergent_LIMSEQ_iff convergent_prod_has_prod convergent_prod_imp_convergent 
   835       convergent_prod_to_zero_iff raw_has_prod_eq_0 has_prod_def prodinf_eq_lim zero_le)
   836 
   837 lemma has_prod_iff: "f has_prod x \<longleftrightarrow> convergent_prod f \<and> prodinf f = x"
   838 proof
   839   assume "f has_prod x"
   840   then show "convergent_prod f \<and> prodinf f = x"
   841     apply safe
   842     using convergent_prod_def has_prod_def apply blast
   843     using has_prod_unique by blast
   844 qed auto
   845 
   846 lemma convergent_prod_has_prod_iff: "convergent_prod f \<longleftrightarrow> f has_prod prodinf f"
   847   by (auto simp: has_prod_iff convergent_prod_has_prod)
   848 
   849 lemma prodinf_finite:
   850   assumes N: "finite N"
   851     and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 1"
   852   shows "prodinf f = (\<Prod>n\<in>N. f n)"
   853   using has_prod_finite[OF assms, THEN has_prod_unique] by simp
   854 
   855 end
   856 
   857 subsection \<open>Infinite products on ordered, topological monoids\<close>
   858 
   859 lemma LIMSEQ_prod_0: 
   860   fixes f :: "nat \<Rightarrow> 'a::{semidom,topological_space}"
   861   assumes "f i = 0"
   862   shows "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> 0"
   863 proof (subst tendsto_cong)
   864   show "\<forall>\<^sub>F n in sequentially. prod f {..n} = 0"
   865   proof
   866     show "prod f {..n} = 0" if "n \<ge> i" for n
   867       using that assms by auto
   868   qed
   869 qed auto
   870 
   871 lemma LIMSEQ_prod_nonneg: 
   872   fixes f :: "nat \<Rightarrow> 'a::{linordered_semidom,linorder_topology}"
   873   assumes 0: "\<And>n. 0 \<le> f n" and a: "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> a"
   874   shows "a \<ge> 0"
   875   by (simp add: "0" prod_nonneg LIMSEQ_le_const [OF a])
   876 
   877 
   878 context
   879   fixes f :: "nat \<Rightarrow> 'a::{linordered_semidom,linorder_topology}"
   880 begin
   881 
   882 lemma has_prod_le:
   883   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"
   884   shows "a \<le> b"
   885 proof (cases "a=0 \<or> b=0")
   886   case True
   887   then show ?thesis
   888   proof
   889     assume [simp]: "a=0"
   890     have "b \<ge> 0"
   891     proof (rule LIMSEQ_prod_nonneg)
   892       show "(\<lambda>n. prod g {..n}) \<longlonglongrightarrow> b"
   893         using g by (auto simp: has_prod_def raw_has_prod_def LIMSEQ_prod_0)
   894     qed (use le order_trans in auto)
   895     then show ?thesis
   896       by auto
   897   next
   898     assume [simp]: "b=0"
   899     then obtain i where "g i = 0"    
   900       using g by (auto simp: prod_defs)
   901     then have "f i = 0"
   902       using antisym le by force
   903     then have "a=0"
   904       using f by (auto simp: prod_defs LIMSEQ_prod_0 LIMSEQ_unique)
   905     then show ?thesis
   906       by auto
   907   qed
   908 next
   909   case False
   910   then show ?thesis
   911     using assms
   912     unfolding has_prod_def raw_has_prod_def
   913     by (force simp: LIMSEQ_prod_0 intro!: LIMSEQ_le prod_mono)
   914 qed
   915 
   916 lemma prodinf_le: 
   917   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"
   918   shows "prodinf f \<le> prodinf g"
   919   using has_prod_le [OF assms] has_prod_unique f g  by blast
   920 
   921 end
   922 
   923 
   924 lemma prod_le_prodinf: 
   925   fixes f :: "nat \<Rightarrow> 'a::{linordered_idom,linorder_topology}"
   926   assumes "f has_prod a" "\<And>i. 0 \<le> f i" "\<And>i. i\<ge>n \<Longrightarrow> 1 \<le> f i"
   927   shows "prod f {..<n} \<le> prodinf f"
   928   by(rule has_prod_le[OF has_prod_If_finite_set]) (use assms has_prod_unique in auto)
   929 
   930 lemma prodinf_nonneg:
   931   fixes f :: "nat \<Rightarrow> 'a::{linordered_idom,linorder_topology}"
   932   assumes "f has_prod a" "\<And>i. 1 \<le> f i" 
   933   shows "1 \<le> prodinf f"
   934   using prod_le_prodinf[of f a 0] assms
   935   by (metis order_trans prod_ge_1 zero_le_one)
   936 
   937 lemma prodinf_le_const:
   938   fixes f :: "nat \<Rightarrow> real"
   939   assumes "convergent_prod f" "\<And>n. prod f {..<n} \<le> x" 
   940   shows "prodinf f \<le> x"
   941   by (metis lessThan_Suc_atMost assms convergent_prod_LIMSEQ LIMSEQ_le_const2)
   942 
   943 lemma prodinf_eq_one_iff: 
   944   fixes f :: "nat \<Rightarrow> real"
   945   assumes f: "convergent_prod f" and ge1: "\<And>n. 1 \<le> f n"
   946   shows "prodinf f = 1 \<longleftrightarrow> (\<forall>n. f n = 1)"
   947 proof
   948   assume "prodinf f = 1" 
   949   then have "(\<lambda>n. \<Prod>i<n. f i) \<longlonglongrightarrow> 1"
   950     using convergent_prod_LIMSEQ[of f] assms by (simp add: LIMSEQ_lessThan_iff_atMost)
   951   then have "\<And>i. (\<Prod>n\<in>{i}. f n) \<le> 1"
   952   proof (rule LIMSEQ_le_const)
   953     have "1 \<le> prod f n" for n
   954       by (simp add: ge1 prod_ge_1)
   955     have "prod f {..<n} = 1" for n
   956       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)
   957     then have "(\<Prod>n\<in>{i}. f n) \<le> prod f {..<n}" if "n \<ge> Suc i" for i n
   958       by (metis mult.left_neutral order_refl prod.cong prod.neutral_const prod_lessThan_Suc)
   959     then show "\<exists>N. \<forall>n\<ge>N. (\<Prod>n\<in>{i}. f n) \<le> prod f {..<n}" for i
   960       by blast      
   961   qed
   962   with ge1 show "\<forall>n. f n = 1"
   963     by (auto intro!: antisym)
   964 qed (metis prodinf_zero fun_eq_iff)
   965 
   966 lemma prodinf_pos_iff:
   967   fixes f :: "nat \<Rightarrow> real"
   968   assumes "convergent_prod f" "\<And>n. 1 \<le> f n"
   969   shows "1 < prodinf f \<longleftrightarrow> (\<exists>i. 1 < f i)"
   970   using prod_le_prodinf[of f 1] prodinf_eq_one_iff
   971   by (metis convergent_prod_has_prod assms less_le prodinf_nonneg)
   972 
   973 lemma less_1_prodinf2:
   974   fixes f :: "nat \<Rightarrow> real"
   975   assumes "convergent_prod f" "\<And>n. 1 \<le> f n" "1 < f i"
   976   shows "1 < prodinf f"
   977 proof -
   978   have "1 < (\<Prod>n<Suc i. f n)"
   979     using assms  by (intro less_1_prod2[where i=i]) auto
   980   also have "\<dots> \<le> prodinf f"
   981     by (intro prod_le_prodinf) (use assms order_trans zero_le_one in \<open>blast+\<close>)
   982   finally show ?thesis .
   983 qed
   984 
   985 lemma less_1_prodinf:
   986   fixes f :: "nat \<Rightarrow> real"
   987   shows "\<lbrakk>convergent_prod f; \<And>n. 1 < f n\<rbrakk> \<Longrightarrow> 1 < prodinf f"
   988   by (intro less_1_prodinf2[where i=1]) (auto intro: less_imp_le)
   989 
   990 lemma prodinf_nonzero:
   991   fixes f :: "nat \<Rightarrow> 'a :: {idom,topological_semigroup_mult,t2_space}"
   992   assumes "convergent_prod f" "\<And>i. f i \<noteq> 0"
   993   shows "prodinf f \<noteq> 0"
   994   by (metis assms convergent_prod_offset_0 has_prod_unique raw_has_prod_def has_prod_def)
   995 
   996 lemma less_0_prodinf:
   997   fixes f :: "nat \<Rightarrow> real"
   998   assumes f: "convergent_prod f" and 0: "\<And>i. f i > 0"
   999   shows "0 < prodinf f"
  1000 proof -
  1001   have "prodinf f \<noteq> 0"
  1002     by (metis assms less_irrefl prodinf_nonzero)
  1003   moreover have "0 < (\<Prod>n<i. f n)" for i
  1004     by (simp add: 0 prod_pos)
  1005   then have "prodinf f \<ge> 0"
  1006     using convergent_prod_LIMSEQ [OF f] LIMSEQ_prod_nonneg 0 less_le by blast
  1007   ultimately show ?thesis
  1008     by auto
  1009 qed
  1010 
  1011 lemma prod_less_prodinf2:
  1012   fixes f :: "nat \<Rightarrow> real"
  1013   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"
  1014   shows "prod f {..<n} < prodinf f"
  1015 proof -
  1016   have "prod f {..<n} \<le> prod f {..<i}"
  1017     by (rule prod_mono2) (use assms less_le in auto)
  1018   then have "prod f {..<n} < f i * prod f {..<i}"
  1019     using mult_less_le_imp_less[of 1 "f i" "prod f {..<n}" "prod f {..<i}"] assms
  1020     by (simp add: prod_pos)
  1021   moreover have "prod f {..<Suc i} \<le> prodinf f"
  1022     using prod_le_prodinf[of f _ "Suc i"]
  1023     by (meson "0" "1" Suc_leD convergent_prod_has_prod f \<open>n \<le> i\<close> le_trans less_eq_real_def)
  1024   ultimately show ?thesis
  1025     by (metis le_less_trans mult.commute not_le prod_lessThan_Suc)
  1026 qed
  1027 
  1028 lemma prod_less_prodinf:
  1029   fixes f :: "nat \<Rightarrow> real"
  1030   assumes f: "convergent_prod f" and 1: "\<And>m. m\<ge>n \<Longrightarrow> 1 < f m" and 0: "\<And>m. 0 < f m" 
  1031   shows "prod f {..<n} < prodinf f"
  1032   by (meson "0" "1" f le_less prod_less_prodinf2)
  1033 
  1034 lemma raw_has_prodI_bounded:
  1035   fixes f :: "nat \<Rightarrow> real"
  1036   assumes pos: "\<And>n. 1 \<le> f n"
  1037     and le: "\<And>n. (\<Prod>i<n. f i) \<le> x"
  1038   shows "\<exists>p. raw_has_prod f 0 p"
  1039   unfolding raw_has_prod_def add_0_right
  1040 proof (rule exI LIMSEQ_incseq_SUP conjI)+
  1041   show "bdd_above (range (\<lambda>n. prod f {..n}))"
  1042     by (metis bdd_aboveI2 le lessThan_Suc_atMost)
  1043   then have "(SUP i. prod f {..i}) > 0"
  1044     by (metis UNIV_I cSUP_upper less_le_trans pos prod_pos zero_less_one)
  1045   then show "(SUP i. prod f {..i}) \<noteq> 0"
  1046     by auto
  1047   show "incseq (\<lambda>n. prod f {..n})"
  1048     using pos order_trans [OF zero_le_one] by (auto simp: mono_def intro!: prod_mono2)
  1049 qed
  1050 
  1051 lemma convergent_prodI_nonneg_bounded:
  1052   fixes f :: "nat \<Rightarrow> real"
  1053   assumes "\<And>n. 1 \<le> f n" "\<And>n. (\<Prod>i<n. f i) \<le> x"
  1054   shows "convergent_prod f"
  1055   using convergent_prod_def raw_has_prodI_bounded [OF assms] by blast
  1056 
  1057 
  1058 subsection \<open>Infinite products on topological spaces\<close>
  1059 
  1060 context
  1061   fixes f g :: "nat \<Rightarrow> 'a::{t2_space,topological_semigroup_mult,idom}"
  1062 begin
  1063 
  1064 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)"
  1065   by (force simp add: prod.distrib tendsto_mult raw_has_prod_def)
  1066 
  1067 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)"
  1068   by (simp add: raw_has_prod_mult has_prod_def)
  1069 
  1070 end
  1071 
  1072 
  1073 context
  1074   fixes f g :: "nat \<Rightarrow> 'a::real_normed_field"
  1075 begin
  1076 
  1077 lemma has_prod_mult:
  1078   assumes f: "f has_prod a" and g: "g has_prod b"
  1079   shows "(\<lambda>n. f n * g n) has_prod (a * b)"
  1080   using f [unfolded has_prod_def]
  1081 proof (elim disjE exE conjE)
  1082   assume f0: "raw_has_prod f 0 a"
  1083   show ?thesis
  1084     using g [unfolded has_prod_def]
  1085   proof (elim disjE exE conjE)
  1086     assume g0: "raw_has_prod g 0 b"
  1087     with f0 show ?thesis
  1088       by (force simp add: has_prod_def prod.distrib tendsto_mult raw_has_prod_def)
  1089   next
  1090     fix j q
  1091     assume "b = 0" and "g j = 0" and q: "raw_has_prod g (Suc j) q"
  1092     obtain p where p: "raw_has_prod f (Suc j) p"
  1093       using f0 raw_has_prod_ignore_initial_segment by blast
  1094     then have "Ex (raw_has_prod (\<lambda>n. f n * g n) (Suc j))"
  1095       using q raw_has_prod_mult by blast
  1096     then show ?thesis
  1097       using \<open>b = 0\<close> \<open>g j = 0\<close> has_prod_0_iff by fastforce
  1098   qed
  1099 next
  1100   fix i p
  1101   assume "a = 0" and "f i = 0" and p: "raw_has_prod f (Suc i) p"
  1102   show ?thesis
  1103     using g [unfolded has_prod_def]
  1104   proof (elim disjE exE conjE)
  1105     assume g0: "raw_has_prod g 0 b"
  1106     obtain q where q: "raw_has_prod g (Suc i) q"
  1107       using g0 raw_has_prod_ignore_initial_segment by blast
  1108     then have "Ex (raw_has_prod (\<lambda>n. f n * g n) (Suc i))"
  1109       using raw_has_prod_mult p by blast
  1110     then show ?thesis
  1111       using \<open>a = 0\<close> \<open>f i = 0\<close> has_prod_0_iff by fastforce
  1112   next
  1113     fix j q
  1114     assume "b = 0" and "g j = 0" and q: "raw_has_prod g (Suc j) q"
  1115     obtain p' where p': "raw_has_prod f (Suc (max i j)) p'"
  1116       by (metis raw_has_prod_ignore_initial_segment max_Suc_Suc max_def p)
  1117     moreover
  1118     obtain q' where q': "raw_has_prod g (Suc (max i j)) q'"
  1119       by (metis raw_has_prod_ignore_initial_segment max.cobounded2 max_Suc_Suc q)
  1120     ultimately show ?thesis
  1121       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)
  1122   qed
  1123 qed
  1124 
  1125 lemma convergent_prod_mult:
  1126   assumes f: "convergent_prod f" and g: "convergent_prod g"
  1127   shows "convergent_prod (\<lambda>n. f n * g n)"
  1128   unfolding convergent_prod_def
  1129 proof -
  1130   obtain M p N q where p: "raw_has_prod f M p" and q: "raw_has_prod g N q"
  1131     using convergent_prod_def f g by blast+
  1132   then obtain p' q' where p': "raw_has_prod f (max M N) p'" and q': "raw_has_prod g (max M N) q'"
  1133     by (meson raw_has_prod_ignore_initial_segment max.cobounded1 max.cobounded2)
  1134   then show "\<exists>M p. raw_has_prod (\<lambda>n. f n * g n) M p"
  1135     using raw_has_prod_mult by blast
  1136 qed
  1137 
  1138 lemma prodinf_mult: "convergent_prod f \<Longrightarrow> convergent_prod g \<Longrightarrow> prodinf f * prodinf g = (\<Prod>n. f n * g n)"
  1139   by (intro has_prod_unique has_prod_mult convergent_prod_has_prod)
  1140 
  1141 end
  1142 
  1143 context
  1144   fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::real_normed_field"
  1145     and I :: "'i set"
  1146 begin
  1147 
  1148 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)"
  1149   by (induct I rule: infinite_finite_induct) (auto intro!: has_prod_mult)
  1150 
  1151 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)"
  1152   using has_prod_unique[OF has_prod_prod, OF convergent_prod_has_prod] by simp
  1153 
  1154 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)"
  1155   using convergent_prod_has_prod_iff has_prod_prod prodinf_prod by force
  1156 
  1157 end
  1158 
  1159 subsection \<open>Infinite summability on real normed fields\<close>
  1160 
  1161 context
  1162   fixes f :: "nat \<Rightarrow> 'a::real_normed_field"
  1163 begin
  1164 
  1165 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"
  1166 proof -
  1167   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"
  1168     by (subst LIMSEQ_Suc_iff) (simp add: raw_has_prod_def)
  1169   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"
  1170     by (simp add: ac_simps atMost_Suc_eq_insert_0 image_Suc_atMost prod_atLeast1_atMost_eq lessThan_Suc_atMost)
  1171   also have "\<dots> \<longleftrightarrow> raw_has_prod (\<lambda>n. f (Suc n)) M a \<and> f M \<noteq> 0"
  1172   proof safe
  1173     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"
  1174     with tendsto_divide[OF tends tendsto_const, of "f M"]    
  1175     show "raw_has_prod (\<lambda>n. f (Suc n)) M a"
  1176       by (simp add: raw_has_prod_def)
  1177   qed (auto intro: tendsto_mult_right simp:  raw_has_prod_def)
  1178   finally show ?thesis .
  1179 qed
  1180 
  1181 lemma has_prod_Suc_iff:
  1182   assumes "f 0 \<noteq> 0" shows "(\<lambda>n. f (Suc n)) has_prod a \<longleftrightarrow> f has_prod (a * f 0)"
  1183 proof (cases "a = 0")
  1184   case True
  1185   then show ?thesis
  1186   proof (simp add: has_prod_def, safe)
  1187     fix i x
  1188     assume "f (Suc i) = 0" and "raw_has_prod (\<lambda>n. f (Suc n)) (Suc i) x"
  1189     then obtain y where "raw_has_prod f (Suc (Suc i)) y"
  1190       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)
  1191     then show "\<exists>i. f i = 0 \<and> Ex (raw_has_prod f (Suc i))"
  1192       using \<open>f (Suc i) = 0\<close> by blast
  1193   next
  1194     fix i x
  1195     assume "f i = 0" and x: "raw_has_prod f (Suc i) x"
  1196     then obtain j where j: "i = Suc j"
  1197       by (metis assms not0_implies_Suc)
  1198     moreover have "\<exists> y. raw_has_prod (\<lambda>n. f (Suc n)) i y"
  1199       using x by (auto simp: raw_has_prod_def)
  1200     then show "\<exists>i. f (Suc i) = 0 \<and> Ex (raw_has_prod (\<lambda>n. f (Suc n)) (Suc i))"
  1201       using \<open>f i = 0\<close> j by blast
  1202   qed
  1203 next
  1204   case False
  1205   then show ?thesis
  1206     by (auto simp: has_prod_def raw_has_prod_Suc_iff assms)
  1207 qed
  1208 
  1209 lemma convergent_prod_Suc_iff:
  1210   shows "convergent_prod (\<lambda>n. f (Suc n)) = convergent_prod f"
  1211 proof
  1212   assume "convergent_prod f"
  1213   then obtain M L where M_nz:"\<forall>n\<ge>M. f n \<noteq> 0" and 
  1214         M_L:"(\<lambda>n. \<Prod>i\<le>n. f (i + M)) \<longlonglongrightarrow> L" and "L \<noteq> 0" 
  1215     unfolding convergent_prod_altdef by auto
  1216   have "(\<lambda>n. \<Prod>i\<le>n. f (Suc (i + M))) \<longlonglongrightarrow> L / f M"
  1217   proof -
  1218     have "(\<lambda>n. \<Prod>i\<in>{0..Suc n}. f (i + M)) \<longlonglongrightarrow> L"
  1219       using M_L 
  1220       apply (subst (asm) LIMSEQ_Suc_iff[symmetric]) 
  1221       using atLeast0AtMost by auto
  1222     then have "(\<lambda>n. f M * (\<Prod>i\<in>{0..n}. f (Suc (i + M)))) \<longlonglongrightarrow> L"
  1223       apply (subst (asm) prod.atLeast0_atMost_Suc_shift)
  1224       by simp
  1225     then have "(\<lambda>n. (\<Prod>i\<in>{0..n}. f (Suc (i + M)))) \<longlonglongrightarrow> L/f M"
  1226       apply (drule_tac tendsto_divide)
  1227       using M_nz[rule_format,of M,simplified] by auto
  1228     then show ?thesis unfolding atLeast0AtMost .
  1229   qed
  1230   then show "convergent_prod (\<lambda>n. f (Suc n))" unfolding convergent_prod_altdef
  1231     apply (rule_tac exI[where x=M])
  1232     apply (rule_tac exI[where x="L/f M"])
  1233     using M_nz \<open>L\<noteq>0\<close> by auto
  1234 next
  1235   assume "convergent_prod (\<lambda>n. f (Suc n))"
  1236   then obtain M where "\<exists>L. (\<forall>n\<ge>M. f (Suc n) \<noteq> 0) \<and> (\<lambda>n. \<Prod>i\<le>n. f (Suc (i + M))) \<longlonglongrightarrow> L \<and> L \<noteq> 0"
  1237     unfolding convergent_prod_altdef by auto
  1238   then show "convergent_prod f" unfolding convergent_prod_altdef
  1239     apply (rule_tac exI[where x="Suc M"])
  1240     using Suc_le_D by auto
  1241 qed
  1242 
  1243 lemma raw_has_prod_inverse: 
  1244   assumes "raw_has_prod f M a" shows "raw_has_prod (\<lambda>n. inverse (f n)) M (inverse a)"
  1245   using assms unfolding raw_has_prod_def by (auto dest: tendsto_inverse simp: prod_inversef [symmetric])
  1246 
  1247 lemma has_prod_inverse: 
  1248   assumes "f has_prod a" shows "(\<lambda>n. inverse (f n)) has_prod (inverse a)"
  1249 using assms raw_has_prod_inverse unfolding has_prod_def by auto 
  1250 
  1251 lemma convergent_prod_inverse:
  1252   assumes "convergent_prod f" 
  1253   shows "convergent_prod (\<lambda>n. inverse (f n))"
  1254   using assms unfolding convergent_prod_def  by (blast intro: raw_has_prod_inverse elim: )
  1255 
  1256 end
  1257 
  1258 context 
  1259   fixes f :: "nat \<Rightarrow> 'a::real_normed_field"
  1260 begin
  1261 
  1262 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"
  1263   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)
  1264 
  1265 lemma has_prod_divide: "f has_prod a \<Longrightarrow> g has_prod b \<Longrightarrow> (\<lambda>n. f n / g n) has_prod (a / b)"
  1266   unfolding divide_inverse by (intro has_prod_inverse has_prod_mult)
  1267 
  1268 lemma convergent_prod_divide:
  1269   assumes f: "convergent_prod f" and g: "convergent_prod g"
  1270   shows "convergent_prod (\<lambda>n. f n / g n)"
  1271   using f g has_prod_divide has_prod_iff by blast
  1272 
  1273 lemma prodinf_divide: "convergent_prod f \<Longrightarrow> convergent_prod g \<Longrightarrow> prodinf f / prodinf g = (\<Prod>n. f n / g n)"
  1274   by (intro has_prod_unique has_prod_divide convergent_prod_has_prod)
  1275 
  1276 lemma prodinf_inverse: "convergent_prod f \<Longrightarrow> (\<Prod>n. inverse (f n)) = inverse (\<Prod>n. f n)"
  1277   by (intro has_prod_unique [symmetric] has_prod_inverse convergent_prod_has_prod)
  1278 
  1279 lemma has_prod_Suc_imp: 
  1280   assumes "(\<lambda>n. f (Suc n)) has_prod a"
  1281   shows "f has_prod (a * f 0)"
  1282 proof -
  1283   have "f has_prod (a * f 0)" when "raw_has_prod (\<lambda>n. f (Suc n)) 0 a" 
  1284     apply (cases "f 0=0")
  1285     using that unfolding has_prod_def raw_has_prod_Suc 
  1286     by (auto simp add: raw_has_prod_Suc_iff)
  1287   moreover have "f has_prod (a * f 0)" when 
  1288     "(\<exists>i q. a = 0 \<and> f (Suc i) = 0 \<and> raw_has_prod (\<lambda>n. f (Suc n)) (Suc i) q)" 
  1289   proof -
  1290     from that 
  1291     obtain i q where "a = 0" "f (Suc i) = 0" "raw_has_prod (\<lambda>n. f (Suc n)) (Suc i) q"
  1292       by auto
  1293     then show ?thesis unfolding has_prod_def 
  1294       by (auto intro!:exI[where x="Suc i"] simp:raw_has_prod_Suc)
  1295   qed
  1296   ultimately show "f has_prod (a * f 0)" using assms unfolding has_prod_def by auto
  1297 qed
  1298 
  1299 lemma has_prod_iff_shift: 
  1300   assumes "\<And>i. i < n \<Longrightarrow> f i \<noteq> 0"
  1301   shows "(\<lambda>i. f (i + n)) has_prod a \<longleftrightarrow> f has_prod (a * (\<Prod>i<n. f i))"
  1302   using assms
  1303 proof (induct n arbitrary: a)
  1304   case 0
  1305   then show ?case by simp
  1306 next
  1307   case (Suc n)
  1308   then have "(\<lambda>i. f (Suc i + n)) has_prod a \<longleftrightarrow> (\<lambda>i. f (i + n)) has_prod (a * f n)"
  1309     by (subst has_prod_Suc_iff) auto
  1310   with Suc show ?case
  1311     by (simp add: ac_simps)
  1312 qed
  1313 
  1314 corollary has_prod_iff_shift':
  1315   assumes "\<And>i. i < n \<Longrightarrow> f i \<noteq> 0"
  1316   shows "(\<lambda>i. f (i + n)) has_prod (a / (\<Prod>i<n. f i)) \<longleftrightarrow> f has_prod a"
  1317   by (simp add: assms has_prod_iff_shift)
  1318 
  1319 lemma has_prod_one_iff_shift:
  1320   assumes "\<And>i. i < n \<Longrightarrow> f i = 1"
  1321   shows "(\<lambda>i. f (i+n)) has_prod a \<longleftrightarrow> (\<lambda>i. f i) has_prod a"
  1322   by (simp add: assms has_prod_iff_shift)
  1323 
  1324 lemma convergent_prod_iff_shift:
  1325   shows "convergent_prod (\<lambda>i. f (i + n)) \<longleftrightarrow> convergent_prod f"
  1326   apply safe
  1327   using convergent_prod_offset apply blast
  1328   using convergent_prod_ignore_initial_segment convergent_prod_def by blast
  1329 
  1330 lemma has_prod_split_initial_segment:
  1331   assumes "f has_prod a" "\<And>i. i < n \<Longrightarrow> f i \<noteq> 0"
  1332   shows "(\<lambda>i. f (i + n)) has_prod (a / (\<Prod>i<n. f i))"
  1333   using assms has_prod_iff_shift' by blast
  1334 
  1335 lemma prodinf_divide_initial_segment:
  1336   assumes "convergent_prod f" "\<And>i. i < n \<Longrightarrow> f i \<noteq> 0"
  1337   shows "(\<Prod>i. f (i + n)) = (\<Prod>i. f i) / (\<Prod>i<n. f i)"
  1338   by (rule has_prod_unique[symmetric]) (auto simp: assms has_prod_iff_shift)
  1339 
  1340 lemma prodinf_split_initial_segment:
  1341   assumes "convergent_prod f" "\<And>i. i < n \<Longrightarrow> f i \<noteq> 0"
  1342   shows "prodinf f = (\<Prod>i. f (i + n)) * (\<Prod>i<n. f i)"
  1343   by (auto simp add: assms prodinf_divide_initial_segment)
  1344 
  1345 lemma prodinf_split_head:
  1346   assumes "convergent_prod f" "f 0 \<noteq> 0"
  1347   shows "(\<Prod>n. f (Suc n)) = prodinf f / f 0"
  1348   using prodinf_split_initial_segment[of 1] assms by simp
  1349 
  1350 end
  1351 
  1352 context 
  1353   fixes f :: "nat \<Rightarrow> 'a::real_normed_field"
  1354 begin
  1355 
  1356 lemma convergent_prod_inverse_iff: "convergent_prod (\<lambda>n. inverse (f n)) \<longleftrightarrow> convergent_prod f"
  1357   by (auto dest: convergent_prod_inverse)
  1358 
  1359 lemma convergent_prod_const_iff:
  1360   fixes c :: "'a :: {real_normed_field}"
  1361   shows "convergent_prod (\<lambda>_. c) \<longleftrightarrow> c = 1"
  1362 proof
  1363   assume "convergent_prod (\<lambda>_. c)"
  1364   then show "c = 1"
  1365     using convergent_prod_imp_LIMSEQ LIMSEQ_unique by blast 
  1366 next
  1367   assume "c = 1"
  1368   then show "convergent_prod (\<lambda>_. c)"
  1369     by auto
  1370 qed
  1371 
  1372 lemma has_prod_power: "f has_prod a \<Longrightarrow> (\<lambda>i. f i ^ n) has_prod (a ^ n)"
  1373   by (induction n) (auto simp: has_prod_mult)
  1374 
  1375 lemma convergent_prod_power: "convergent_prod f \<Longrightarrow> convergent_prod (\<lambda>i. f i ^ n)"
  1376   by (induction n) (auto simp: convergent_prod_mult)
  1377 
  1378 lemma prodinf_power: "convergent_prod f \<Longrightarrow> prodinf (\<lambda>i. f i ^ n) = prodinf f ^ n"
  1379   by (metis has_prod_unique convergent_prod_imp_has_prod has_prod_power)
  1380 
  1381 end
  1382 
  1383 
  1384 subsection\<open>Exponentials and logarithms\<close>
  1385 
  1386 context 
  1387   fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
  1388 begin
  1389 
  1390 lemma sums_imp_has_prod_exp: 
  1391   assumes "f sums s"
  1392   shows "raw_has_prod (\<lambda>i. exp (f i)) 0 (exp s)"
  1393   using assms continuous_on_exp [of UNIV "\<lambda>x::'a. x"]
  1394   using continuous_on_tendsto_compose [of UNIV exp "(\<lambda>n. sum f {..n})" s]
  1395   by (simp add: prod_defs sums_def_le exp_sum)
  1396 
  1397 lemma convergent_prod_exp: 
  1398   assumes "summable f"
  1399   shows "convergent_prod (\<lambda>i. exp (f i))"
  1400   using sums_imp_has_prod_exp assms unfolding summable_def convergent_prod_def  by blast
  1401 
  1402 lemma prodinf_exp: 
  1403   assumes "summable f"
  1404   shows "prodinf (\<lambda>i. exp (f i)) = exp (suminf f)"
  1405 proof -
  1406   have "f sums suminf f"
  1407     using assms by blast
  1408   then have "(\<lambda>i. exp (f i)) has_prod exp (suminf f)"
  1409     by (simp add: has_prod_def sums_imp_has_prod_exp)
  1410   then show ?thesis
  1411     by (rule has_prod_unique [symmetric])
  1412 qed
  1413 
  1414 end
  1415 
  1416 lemma exp_suminf_prodinf_real:
  1417   fixes f :: "nat \<Rightarrow> real"
  1418   assumes ge0:"\<And>n. f n \<ge> 0" and ac: "abs_convergent_prod (\<lambda>n. exp (f n))"
  1419   shows "prodinf (\<lambda>i. exp (f i)) = exp (suminf f)"
  1420 proof -
  1421   have "summable f" 
  1422     using ac unfolding abs_convergent_prod_conv_summable
  1423   proof (elim summable_comparison_test')
  1424     fix n
  1425     show "norm (f n) \<le> norm (exp (f n) - 1)" 
  1426       using ge0[of n] 
  1427       by (metis abs_of_nonneg add.commute diff_add_cancel diff_ge_0_iff_ge exp_ge_add_one_self 
  1428           exp_le_cancel_iff one_le_exp_iff real_norm_def)
  1429   qed
  1430   then show ?thesis
  1431     by (simp add: prodinf_exp)
  1432 qed
  1433 
  1434 lemma has_prod_imp_sums_ln_real: 
  1435   fixes f :: "nat \<Rightarrow> real"
  1436   assumes "raw_has_prod f 0 p" and 0: "\<And>x. f x > 0"
  1437   shows "(\<lambda>i. ln (f i)) sums (ln p)"
  1438 proof -
  1439   have "p > 0"
  1440     using assms unfolding prod_defs by (metis LIMSEQ_prod_nonneg less_eq_real_def)
  1441   then show ?thesis
  1442   using assms continuous_on_ln [of "{0<..}" "\<lambda>x. x"]
  1443   using continuous_on_tendsto_compose [of "{0<..}" ln "(\<lambda>n. prod f {..n})" p]
  1444   by (auto simp: prod_defs sums_def_le ln_prod order_tendstoD)
  1445 qed
  1446 
  1447 lemma summable_ln_real: 
  1448   fixes f :: "nat \<Rightarrow> real"
  1449   assumes f: "convergent_prod f" and 0: "\<And>x. f x > 0"
  1450   shows "summable (\<lambda>i. ln (f i))"
  1451 proof -
  1452   obtain M p where "raw_has_prod f M p"
  1453     using f convergent_prod_def by blast
  1454   then consider i where "i<M" "f i = 0" | p where "raw_has_prod f 0 p"
  1455     using raw_has_prod_cases by blast
  1456   then show ?thesis
  1457   proof cases
  1458     case 1
  1459     with 0 show ?thesis
  1460       by (metis less_irrefl)
  1461   next
  1462     case 2
  1463     then show ?thesis
  1464       using "0" has_prod_imp_sums_ln_real summable_def by blast
  1465   qed
  1466 qed
  1467 
  1468 lemma suminf_ln_real: 
  1469   fixes f :: "nat \<Rightarrow> real"
  1470   assumes f: "convergent_prod f" and 0: "\<And>x. f x > 0"
  1471   shows "suminf (\<lambda>i. ln (f i)) = ln (prodinf f)"
  1472 proof -
  1473   have "f has_prod prodinf f"
  1474     by (simp add: f has_prod_iff)
  1475   then have "raw_has_prod f 0 (prodinf f)"
  1476     by (metis "0" has_prod_def less_irrefl)
  1477   then have "(\<lambda>i. ln (f i)) sums ln (prodinf f)"
  1478     using "0" has_prod_imp_sums_ln_real by blast
  1479   then show ?thesis
  1480     by (rule sums_unique [symmetric])
  1481 qed
  1482 
  1483 lemma prodinf_exp_real: 
  1484   fixes f :: "nat \<Rightarrow> real"
  1485   assumes f: "convergent_prod f" and 0: "\<And>x. f x > 0"
  1486   shows "prodinf f = exp (suminf (\<lambda>i. ln (f i)))"
  1487   by (simp add: "0" f less_0_prodinf suminf_ln_real)
  1488 
  1489 
  1490 subsection\<open>Embeddings from the reals into some complete real normed field\<close>
  1491 
  1492 lemma tendsto_eq_of_real_lim:
  1493   assumes "(\<lambda>n. of_real (f n) :: 'a::{complete_space,real_normed_field}) \<longlonglongrightarrow> q"
  1494   shows "q = of_real (lim f)"
  1495 proof -
  1496   have "convergent (\<lambda>n. of_real (f n) :: 'a)"
  1497     using assms convergent_def by blast 
  1498   then have "convergent f"
  1499     unfolding convergent_def
  1500     by (simp add: convergent_eq_Cauchy Cauchy_def)
  1501   then show ?thesis
  1502     by (metis LIMSEQ_unique assms convergentD sequentially_bot tendsto_Lim tendsto_of_real)
  1503 qed
  1504 
  1505 lemma tendsto_eq_of_real:
  1506   assumes "(\<lambda>n. of_real (f n) :: 'a::{complete_space,real_normed_field}) \<longlonglongrightarrow> q"
  1507   obtains r where "q = of_real r"
  1508   using tendsto_eq_of_real_lim assms by blast
  1509 
  1510 lemma has_prod_of_real_iff:
  1511   "(\<lambda>n. of_real (f n) :: 'a::{complete_space,real_normed_field}) has_prod of_real c \<longleftrightarrow> f has_prod c"
  1512   (is "?lhs = ?rhs")
  1513 proof
  1514   assume ?lhs
  1515   then show ?rhs
  1516     apply (auto simp: prod_defs LIMSEQ_prod_0 tendsto_of_real_iff simp flip: of_real_prod)
  1517     using tendsto_eq_of_real
  1518     by (metis of_real_0 tendsto_of_real_iff)
  1519 next
  1520   assume ?rhs
  1521   with tendsto_of_real_iff show ?lhs
  1522     by (fastforce simp: prod_defs simp flip: of_real_prod)
  1523 qed
  1524 
  1525 end