src/HOL/Analysis/Infinite_Products.thy
 author paulson Thu Jun 28 14:13:57 2018 +0100 (12 months ago) changeset 68527 2f4e2aab190a parent 68517 6b5f15387353 child 68585 1657b9a5dd5d permissions -rw-r--r--
Generalising and renaming some basic results
 lp15@68064  1 (*File: HOL/Analysis/Infinite_Product.thy  lp15@68064  2  Author: Manuel Eberl & LC Paulson  eberlm@66277  3 eberlm@66277  4  Basic results about convergence and absolute convergence of infinite products  eberlm@66277  5  and their connection to summability.  eberlm@66277  6 *)  eberlm@66277  7 section \Infinite Products\  eberlm@66277  8 theory Infinite_Products  lp15@68424  9  imports Topology_Euclidean_Space  eberlm@66277  10 begin  lp15@68424  11 lp15@68424  12 subsection\Preliminaries\  lp15@68424  13 eberlm@66277  14 lemma sum_le_prod:  eberlm@66277  15  fixes f :: "'a \ 'b :: linordered_semidom"  eberlm@66277  16  assumes "\x. x \ A \ f x \ 0"  eberlm@66277  17  shows "sum f A \ (\x\A. 1 + f x)"  eberlm@66277  18  using assms  eberlm@66277  19 proof (induction A rule: infinite_finite_induct)  eberlm@66277  20  case (insert x A)  eberlm@66277  21  from insert.hyps have "sum f A + f x * (\x\A. 1) \ (\x\A. 1 + f x) + f x * (\x\A. 1 + f x)"  eberlm@66277  22  by (intro add_mono insert mult_left_mono prod_mono) (auto intro: insert.prems)  eberlm@66277  23  with insert.hyps show ?case by (simp add: algebra_simps)  eberlm@66277  24 qed simp_all  eberlm@66277  25 eberlm@66277  26 lemma prod_le_exp_sum:  eberlm@66277  27  fixes f :: "'a \ real"  eberlm@66277  28  assumes "\x. x \ A \ f x \ 0"  eberlm@66277  29  shows "prod (\x. 1 + f x) A \ exp (sum f A)"  eberlm@66277  30  using assms  eberlm@66277  31 proof (induction A rule: infinite_finite_induct)  eberlm@66277  32  case (insert x A)  eberlm@66277  33  have "(1 + f x) * (\x\A. 1 + f x) \ exp (f x) * exp (sum f A)"  eberlm@66277  34  using insert.prems by (intro mult_mono insert prod_nonneg exp_ge_add_one_self) auto  eberlm@66277  35  with insert.hyps show ?case by (simp add: algebra_simps exp_add)  eberlm@66277  36 qed simp_all  eberlm@66277  37 eberlm@66277  38 lemma lim_ln_1_plus_x_over_x_at_0: "(\x::real. ln (1 + x) / x) \0\ 1"  eberlm@66277  39 proof (rule lhopital)  eberlm@66277  40  show "(\x::real. ln (1 + x)) \0\ 0"  eberlm@66277  41  by (rule tendsto_eq_intros refl | simp)+  eberlm@66277  42  have "eventually (\x::real. x \ {-1/2<..<1/2}) (nhds 0)"  eberlm@66277  43  by (rule eventually_nhds_in_open) auto  eberlm@66277  44  hence *: "eventually (\x::real. x \ {-1/2<..<1/2}) (at 0)"  eberlm@66277  45  by (rule filter_leD [rotated]) (simp_all add: at_within_def)  eberlm@66277  46  show "eventually (\x::real. ((\x. ln (1 + x)) has_field_derivative inverse (1 + x)) (at x)) (at 0)"  eberlm@66277  47  using * by eventually_elim (auto intro!: derivative_eq_intros simp: field_simps)  eberlm@66277  48  show "eventually (\x::real. ((\x. x) has_field_derivative 1) (at x)) (at 0)"  eberlm@66277  49  using * by eventually_elim (auto intro!: derivative_eq_intros simp: field_simps)  eberlm@66277  50  show "\\<^sub>F x in at 0. x \ 0" by (auto simp: at_within_def eventually_inf_principal)  eberlm@66277  51  show "(\x::real. inverse (1 + x) / 1) \0\ 1"  eberlm@66277  52  by (rule tendsto_eq_intros refl | simp)+  eberlm@66277  53 qed auto  eberlm@66277  54 lp15@68424  55 subsection\Definitions and basic properties\  lp15@68424  56 lp15@68361  57 definition raw_has_prod :: "[nat \ 'a::{t2_space, comm_semiring_1}, nat, 'a] \ bool"  lp15@68361  58  where "raw_has_prod f M p \ (\n. \i\n. f (i+M)) \ p \ p \ 0"  lp15@68064  59 lp15@68064  60 text\The nonzero and zero cases, as in \emph{Complex Analysis} by Joseph Bak and Donald J.Newman, page 241\  lp15@68064  61 definition has_prod :: "(nat \ 'a::{t2_space, comm_semiring_1}) \ 'a \ bool" (infixr "has'_prod" 80)  lp15@68361  62  where "f has_prod p \ raw_has_prod f 0 p \ (\i q. p = 0 \ f i = 0 \ raw_has_prod f (Suc i) q)"  lp15@68064  63 eberlm@66277  64 definition convergent_prod :: "(nat \ 'a :: {t2_space,comm_semiring_1}) \ bool" where  lp15@68361  65  "convergent_prod f \ \M p. raw_has_prod f M p"  lp15@68064  66 lp15@68064  67 definition prodinf :: "(nat \ 'a::{t2_space, comm_semiring_1}) \ 'a"  lp15@68064  68  (binder "\" 10)  lp15@68064  69  where "prodinf f = (THE p. f has_prod p)"  lp15@68064  70 lp15@68361  71 lemmas prod_defs = raw_has_prod_def has_prod_def convergent_prod_def prodinf_def  lp15@68064  72 lp15@68064  73 lemma has_prod_subst[trans]: "f = g \ g has_prod z \ f has_prod z"  lp15@68064  74  by simp  lp15@68064  75 lp15@68064  76 lemma has_prod_cong: "(\n. f n = g n) \ f has_prod c \ g has_prod c"  lp15@68064  77  by presburger  eberlm@66277  78 lp15@68361  79 lemma raw_has_prod_nonzero [simp]: "\ raw_has_prod f M 0"  lp15@68361  80  by (simp add: raw_has_prod_def)  lp15@68071  81 lp15@68361  82 lemma raw_has_prod_eq_0:  lp15@68361  83  fixes f :: "nat \ 'a::{semidom,t2_space}"  lp15@68361  84  assumes p: "raw_has_prod f m p" and i: "f i = 0" "i \ m"  lp15@68136  85  shows "p = 0"  lp15@68136  86 proof -  lp15@68136  87  have eq0: "(\k\n. f (k+m)) = 0" if "i - m \ n" for n  lp15@68361  88  proof -  lp15@68361  89  have "\k\n. f (k + m) = 0"  lp15@68361  90  using i that by auto  lp15@68361  91  then show ?thesis  lp15@68361  92  by auto  lp15@68361  93  qed  lp15@68136  94  have "(\n. \i\n. f (i + m)) \ 0"  lp15@68136  95  by (rule LIMSEQ_offset [where k = "i-m"]) (simp add: eq0)  lp15@68136  96  with p show ?thesis  lp15@68361  97  unfolding raw_has_prod_def  lp15@68136  98  using LIMSEQ_unique by blast  lp15@68136  99 qed  lp15@68136  100 lp15@68452  101 lemma raw_has_prod_Suc:  lp15@68452  102  "raw_has_prod f (Suc M) a \ raw_has_prod (\n. f (Suc n)) M a"  lp15@68452  103  unfolding raw_has_prod_def by auto  lp15@68452  104 lp15@68361  105 lemma has_prod_0_iff: "f has_prod 0 \ (\i. f i = 0 \ (\p. raw_has_prod f (Suc i) p))"  lp15@68071  106  by (simp add: has_prod_def)  lp15@68136  107   lp15@68136  108 lemma has_prod_unique2:  lp15@68361  109  fixes f :: "nat \ 'a::{semidom,t2_space}"  lp15@68136  110  assumes "f has_prod a" "f has_prod b" shows "a = b"  lp15@68136  111  using assms  lp15@68361  112  by (auto simp: has_prod_def raw_has_prod_eq_0) (meson raw_has_prod_def sequentially_bot tendsto_unique)  lp15@68136  113 lp15@68136  114 lemma has_prod_unique:  lp15@68361  115  fixes f :: "nat \ 'a :: {semidom,t2_space}"  lp15@68136  116  shows "f has_prod s \ s = prodinf f"  lp15@68136  117  by (simp add: has_prod_unique2 prodinf_def the_equality)  lp15@68071  118 eberlm@66277  119 lemma convergent_prod_altdef:  eberlm@66277  120  fixes f :: "nat \ 'a :: {t2_space,comm_semiring_1}"  eberlm@66277  121  shows "convergent_prod f \ (\M L. (\n\M. f n \ 0) \ (\n. \i\n. f (i+M)) \ L \ L \ 0)"  eberlm@66277  122 proof  eberlm@66277  123  assume "convergent_prod f"  eberlm@66277  124  then obtain M L where *: "(\n. \i\n. f (i+M)) \ L" "L \ 0"  lp15@68064  125  by (auto simp: prod_defs)  eberlm@66277  126  have "f i \ 0" if "i \ M" for i  eberlm@66277  127  proof  eberlm@66277  128  assume "f i = 0"  eberlm@66277  129  have **: "eventually (\n. (\i\n. f (i+M)) = 0) sequentially"  eberlm@66277  130  using eventually_ge_at_top[of "i - M"]  eberlm@66277  131  proof eventually_elim  eberlm@66277  132  case (elim n)  eberlm@66277  133  with \f i = 0\ and \i \ M\ show ?case  eberlm@66277  134  by (auto intro!: bexI[of _ "i - M"] prod_zero)  eberlm@66277  135  qed  eberlm@66277  136  have "(\n. (\i\n. f (i+M))) \ 0"  eberlm@66277  137  unfolding filterlim_iff  eberlm@66277  138  by (auto dest!: eventually_nhds_x_imp_x intro!: eventually_mono[OF **])  eberlm@66277  139  from tendsto_unique[OF _ this *(1)] and *(2)  eberlm@66277  140  show False by simp  eberlm@66277  141  qed  eberlm@66277  142  with * show "(\M L. (\n\M. f n \ 0) \ (\n. \i\n. f (i+M)) \ L \ L \ 0)"  eberlm@66277  143  by blast  lp15@68064  144 qed (auto simp: prod_defs)  eberlm@66277  145 lp15@68424  146 lp15@68424  147 subsection\Absolutely convergent products\  lp15@68424  148 eberlm@66277  149 definition abs_convergent_prod :: "(nat \ _) \ bool" where  eberlm@66277  150  "abs_convergent_prod f \ convergent_prod (\i. 1 + norm (f i - 1))"  eberlm@66277  151 eberlm@66277  152 lemma abs_convergent_prodI:  eberlm@66277  153  assumes "convergent (\n. \i\n. 1 + norm (f i - 1))"  eberlm@66277  154  shows "abs_convergent_prod f"  eberlm@66277  155 proof -  eberlm@66277  156  from assms obtain L where L: "(\n. \i\n. 1 + norm (f i - 1)) \ L"  eberlm@66277  157  by (auto simp: convergent_def)  eberlm@66277  158  have "L \ 1"  eberlm@66277  159  proof (rule tendsto_le)  eberlm@66277  160  show "eventually (\n. (\i\n. 1 + norm (f i - 1)) \ 1) sequentially"  eberlm@66277  161  proof (intro always_eventually allI)  eberlm@66277  162  fix n  eberlm@66277  163  have "(\i\n. 1 + norm (f i - 1)) \ (\i\n. 1)"  eberlm@66277  164  by (intro prod_mono) auto  eberlm@66277  165  thus "(\i\n. 1 + norm (f i - 1)) \ 1" by simp  eberlm@66277  166  qed  eberlm@66277  167  qed (use L in simp_all)  eberlm@66277  168  hence "L \ 0" by auto  lp15@68064  169  with L show ?thesis unfolding abs_convergent_prod_def prod_defs  eberlm@66277  170  by (intro exI[of _ "0::nat"] exI[of _ L]) auto  eberlm@66277  171 qed  eberlm@66277  172 eberlm@66277  173 lemma  lp15@68064  174  fixes f :: "nat \ 'a :: {topological_semigroup_mult,t2_space,idom}"  eberlm@66277  175  assumes "convergent_prod f"  eberlm@66277  176  shows convergent_prod_imp_convergent: "convergent (\n. \i\n. f i)"  eberlm@66277  177  and convergent_prod_to_zero_iff: "(\n. \i\n. f i) \ 0 \ (\i. f i = 0)"  eberlm@66277  178 proof -  eberlm@66277  179  from assms obtain M L  eberlm@66277  180  where M: "\n. n \ M \ f n \ 0" and "(\n. \i\n. f (i + M)) \ L" and "L \ 0"  eberlm@66277  181  by (auto simp: convergent_prod_altdef)  eberlm@66277  182  note this(2)  eberlm@66277  183  also have "(\n. \i\n. f (i + M)) = (\n. \i=M..M+n. f i)"  eberlm@66277  184  by (intro ext prod.reindex_bij_witness[of _ "\n. n - M" "\n. n + M"]) auto  eberlm@66277  185  finally have "(\n. (\ii=M..M+n. f i)) \ (\in. (\ii=M..M+n. f i)) = (\n. (\i\{..{M..M+n}. f i))"  eberlm@66277  188  by (subst prod.union_disjoint) auto  eberlm@66277  189  also have "(\n. {.. {M..M+n}) = (\n. {..n+M})" by auto  eberlm@66277  190  finally have lim: "(\n. prod f {..n}) \ prod f {..n. \i\n. f i)"  eberlm@66277  193  by (auto simp: convergent_def)  eberlm@66277  194 eberlm@66277  195  show "(\n. \i\n. f i) \ 0 \ (\i. f i = 0)"  eberlm@66277  196  proof  eberlm@66277  197  assume "\i. f i = 0"  eberlm@66277  198  then obtain i where "f i = 0" by auto  eberlm@66277  199  moreover with M have "i < M" by (cases "i < M") auto  eberlm@66277  200  ultimately have "(\in. \i\n. f i) \ 0" by simp  eberlm@66277  202  next  eberlm@66277  203  assume "(\n. \i\n. f i) \ 0"  eberlm@66277  204  from tendsto_unique[OF _ this lim] and \L \ 0\  eberlm@66277  205  show "\i. f i = 0" by auto  eberlm@66277  206  qed  eberlm@66277  207 qed  eberlm@66277  208 lp15@68064  209 lemma convergent_prod_iff_nz_lim:  lp15@68064  210  fixes f :: "nat \ 'a :: {topological_semigroup_mult,t2_space,idom}"  lp15@68064  211  assumes "\i. f i \ 0"  lp15@68064  212  shows "convergent_prod f \ (\L. (\n. \i\n. f i) \ L \ L \ 0)"  lp15@68064  213  (is "?lhs \ ?rhs")  lp15@68064  214 proof  lp15@68064  215  assume ?lhs then show ?rhs  lp15@68064  216  using assms convergentD convergent_prod_imp_convergent convergent_prod_to_zero_iff by blast  lp15@68064  217 next  lp15@68064  218  assume ?rhs then show ?lhs  lp15@68064  219  unfolding prod_defs  lp15@68138  220  by (rule_tac x=0 in exI) auto  lp15@68064  221 qed  lp15@68064  222 lp15@68064  223 lemma convergent_prod_iff_convergent:  lp15@68064  224  fixes f :: "nat \ 'a :: {topological_semigroup_mult,t2_space,idom}"  lp15@68064  225  assumes "\i. f i \ 0"  lp15@68064  226  shows "convergent_prod f \ convergent (\n. \i\n. f i) \ lim (\n. \i\n. f i) \ 0"  lp15@68138  227  by (force simp: convergent_prod_iff_nz_lim assms convergent_def limI)  lp15@68064  228 lp15@68527  229 lemma bounded_imp_convergent_prod:  lp15@68527  230  fixes a :: "nat \ real"  lp15@68527  231  assumes 1: "\n. a n \ 1" and bounded: "\n. (\i\n. a i) \ B"  lp15@68527  232  shows "convergent_prod a"  lp15@68527  233 proof -  lp15@68527  234  have "bdd_above (range(\n. \i\n. a i))"  lp15@68527  235  by (meson bdd_aboveI2 bounded)  lp15@68527  236  moreover have "incseq (\n. \i\n. a i)"  lp15@68527  237  unfolding mono_def by (metis 1 prod_mono2 atMost_subset_iff dual_order.trans finite_atMost zero_le_one)  lp15@68527  238  ultimately obtain p where p: "(\n. \i\n. a i) \ p"  lp15@68527  239  using LIMSEQ_incseq_SUP by blast  lp15@68527  240  then have "p \ 0"  lp15@68527  241  by (metis "1" not_one_le_zero prod_ge_1 LIMSEQ_le_const)  lp15@68527  242  with 1 p show ?thesis  lp15@68527  243  by (metis convergent_prod_iff_nz_lim not_one_le_zero)  lp15@68527  244 qed  lp15@68527  245 lp15@68064  246 eberlm@66277  247 lemma abs_convergent_prod_altdef:  lp15@68064  248  fixes f :: "nat \ 'a :: {one,real_normed_vector}"  lp15@68064  249  shows "abs_convergent_prod f \ convergent (\n. \i\n. 1 + norm (f i - 1))"  eberlm@66277  250 proof  eberlm@66277  251  assume "abs_convergent_prod f"  eberlm@66277  252  thus "convergent (\n. \i\n. 1 + norm (f i - 1))"  eberlm@66277  253  by (auto simp: abs_convergent_prod_def intro!: convergent_prod_imp_convergent)  eberlm@66277  254 qed (auto intro: abs_convergent_prodI)  eberlm@66277  255 eberlm@66277  256 lemma weierstrass_prod_ineq:  eberlm@66277  257  fixes f :: "'a \ real"  eberlm@66277  258  assumes "\x. x \ A \ f x \ {0..1}"  eberlm@66277  259  shows "1 - sum f A \ (\x\A. 1 - f x)"  eberlm@66277  260  using assms  eberlm@66277  261 proof (induction A rule: infinite_finite_induct)  eberlm@66277  262  case (insert x A)  eberlm@66277  263  from insert.hyps and insert.prems  eberlm@66277  264  have "1 - sum f A + f x * (\x\A. 1 - f x) \ (\x\A. 1 - f x) + f x * (\x\A. 1)"  eberlm@66277  265  by (intro insert.IH add_mono mult_left_mono prod_mono) auto  eberlm@66277  266  with insert.hyps show ?case by (simp add: algebra_simps)  eberlm@66277  267 qed simp_all  eberlm@66277  268 eberlm@66277  269 lemma norm_prod_minus1_le_prod_minus1:  eberlm@66277  270  fixes f :: "nat \ 'a :: {real_normed_div_algebra,comm_ring_1}"  eberlm@66277  271  shows "norm (prod (\n. 1 + f n) A - 1) \ prod (\n. 1 + norm (f n)) A - 1"  eberlm@66277  272 proof (induction A rule: infinite_finite_induct)  eberlm@66277  273  case (insert x A)  eberlm@66277  274  from insert.hyps have  eberlm@66277  275  "norm ((\n\insert x A. 1 + f n) - 1) =  eberlm@66277  276  norm ((\n\A. 1 + f n) - 1 + f x * (\n\A. 1 + f n))"  eberlm@66277  277  by (simp add: algebra_simps)  eberlm@66277  278  also have "\ \ norm ((\n\A. 1 + f n) - 1) + norm (f x * (\n\A. 1 + f n))"  eberlm@66277  279  by (rule norm_triangle_ineq)  eberlm@66277  280  also have "norm (f x * (\n\A. 1 + f n)) = norm (f x) * (\x\A. norm (1 + f x))"  eberlm@66277  281  by (simp add: prod_norm norm_mult)  eberlm@66277  282  also have "(\x\A. norm (1 + f x)) \ (\x\A. norm (1::'a) + norm (f x))"  eberlm@66277  283  by (intro prod_mono norm_triangle_ineq ballI conjI) auto  eberlm@66277  284  also have "norm (1::'a) = 1" by simp  eberlm@66277  285  also note insert.IH  eberlm@66277  286  also have "(\n\A. 1 + norm (f n)) - 1 + norm (f x) * (\x\A. 1 + norm (f x)) =  lp15@68064  287  (\n\insert x A. 1 + norm (f n)) - 1"  eberlm@66277  288  using insert.hyps by (simp add: algebra_simps)  eberlm@66277  289  finally show ?case by - (simp_all add: mult_left_mono)  eberlm@66277  290 qed simp_all  eberlm@66277  291 eberlm@66277  292 lemma convergent_prod_imp_ev_nonzero:  eberlm@66277  293  fixes f :: "nat \ 'a :: {t2_space,comm_semiring_1}"  eberlm@66277  294  assumes "convergent_prod f"  eberlm@66277  295  shows "eventually (\n. f n \ 0) sequentially"  eberlm@66277  296  using assms by (auto simp: eventually_at_top_linorder convergent_prod_altdef)  eberlm@66277  297 eberlm@66277  298 lemma convergent_prod_imp_LIMSEQ:  eberlm@66277  299  fixes f :: "nat \ 'a :: {real_normed_field}"  eberlm@66277  300  assumes "convergent_prod f"  eberlm@66277  301  shows "f \ 1"  eberlm@66277  302 proof -  eberlm@66277  303  from assms obtain M L where L: "(\n. \i\n. f (i+M)) \ L" "\n. n \ M \ f n \ 0" "L \ 0"  eberlm@66277  304  by (auto simp: convergent_prod_altdef)  eberlm@66277  305  hence L': "(\n. \i\Suc n. f (i+M)) \ L" by (subst filterlim_sequentially_Suc)  eberlm@66277  306  have "(\n. (\i\Suc n. f (i+M)) / (\i\n. f (i+M))) \ L / L"  eberlm@66277  307  using L L' by (intro tendsto_divide) simp_all  eberlm@66277  308  also from L have "L / L = 1" by simp  eberlm@66277  309  also have "(\n. (\i\Suc n. f (i+M)) / (\i\n. f (i+M))) = (\n. f (n + Suc M))"  eberlm@66277  310  using assms L by (auto simp: fun_eq_iff atMost_Suc)  eberlm@66277  311  finally show ?thesis by (rule LIMSEQ_offset)  eberlm@66277  312 qed  eberlm@66277  313 eberlm@66277  314 lemma abs_convergent_prod_imp_summable:  eberlm@66277  315  fixes f :: "nat \ 'a :: real_normed_div_algebra"  eberlm@66277  316  assumes "abs_convergent_prod f"  eberlm@66277  317  shows "summable (\i. norm (f i - 1))"  eberlm@66277  318 proof -  eberlm@66277  319  from assms have "convergent (\n. \i\n. 1 + norm (f i - 1))"  eberlm@66277  320  unfolding abs_convergent_prod_def by (rule convergent_prod_imp_convergent)  eberlm@66277  321  then obtain L where L: "(\n. \i\n. 1 + norm (f i - 1)) \ L"  eberlm@66277  322  unfolding convergent_def by blast  eberlm@66277  323  have "convergent (\n. \i\n. norm (f i - 1))"  eberlm@66277  324  proof (rule Bseq_monoseq_convergent)  eberlm@66277  325  have "eventually (\n. (\i\n. 1 + norm (f i - 1)) < L + 1) sequentially"  eberlm@66277  326  using L(1) by (rule order_tendstoD) simp_all  eberlm@66277  327  hence "\\<^sub>F x in sequentially. norm (\i\x. norm (f i - 1)) \ L + 1"  eberlm@66277  328  proof eventually_elim  eberlm@66277  329  case (elim n)  eberlm@66277  330  have "norm (\i\n. norm (f i - 1)) = (\i\n. norm (f i - 1))"  eberlm@66277  331  unfolding real_norm_def by (intro abs_of_nonneg sum_nonneg) simp_all  eberlm@66277  332  also have "\ \ (\i\n. 1 + norm (f i - 1))" by (rule sum_le_prod) auto  eberlm@66277  333  also have "\ < L + 1" by (rule elim)  eberlm@66277  334  finally show ?case by simp  eberlm@66277  335  qed  eberlm@66277  336  thus "Bseq (\n. \i\n. norm (f i - 1))" by (rule BfunI)  eberlm@66277  337  next  eberlm@66277  338  show "monoseq (\n. \i\n. norm (f i - 1))"  eberlm@66277  339  by (rule mono_SucI1) auto  eberlm@66277  340  qed  eberlm@66277  341  thus "summable (\i. norm (f i - 1))" by (simp add: summable_iff_convergent')  eberlm@66277  342 qed  eberlm@66277  343 eberlm@66277  344 lemma summable_imp_abs_convergent_prod:  eberlm@66277  345  fixes f :: "nat \ 'a :: real_normed_div_algebra"  eberlm@66277  346  assumes "summable (\i. norm (f i - 1))"  eberlm@66277  347  shows "abs_convergent_prod f"  eberlm@66277  348 proof (intro abs_convergent_prodI Bseq_monoseq_convergent)  eberlm@66277  349  show "monoseq (\n. \i\n. 1 + norm (f i - 1))"  eberlm@66277  350  by (intro mono_SucI1)  eberlm@66277  351  (auto simp: atMost_Suc algebra_simps intro!: mult_nonneg_nonneg prod_nonneg)  eberlm@66277  352 next  eberlm@66277  353  show "Bseq (\n. \i\n. 1 + norm (f i - 1))"  eberlm@66277  354  proof (rule Bseq_eventually_mono)  eberlm@66277  355  show "eventually (\n. norm (\i\n. 1 + norm (f i - 1)) \  eberlm@66277  356  norm (exp (\i\n. norm (f i - 1)))) sequentially"  eberlm@66277  357  by (intro always_eventually allI) (auto simp: abs_prod exp_sum intro!: prod_mono)  eberlm@66277  358  next  eberlm@66277  359  from assms have "(\n. \i\n. norm (f i - 1)) \ (\i. norm (f i - 1))"  eberlm@66277  360  using sums_def_le by blast  eberlm@66277  361  hence "(\n. exp (\i\n. norm (f i - 1))) \ exp (\i. norm (f i - 1))"  eberlm@66277  362  by (rule tendsto_exp)  eberlm@66277  363  hence "convergent (\n. exp (\i\n. norm (f i - 1)))"  eberlm@66277  364  by (rule convergentI)  eberlm@66277  365  thus "Bseq (\n. exp (\i\n. norm (f i - 1)))"  eberlm@66277  366  by (rule convergent_imp_Bseq)  eberlm@66277  367  qed  eberlm@66277  368 qed  eberlm@66277  369 eberlm@66277  370 lemma abs_convergent_prod_conv_summable:  eberlm@66277  371  fixes f :: "nat \ 'a :: real_normed_div_algebra"  eberlm@66277  372  shows "abs_convergent_prod f \ summable (\i. norm (f i - 1))"  eberlm@66277  373  by (blast intro: abs_convergent_prod_imp_summable summable_imp_abs_convergent_prod)  eberlm@66277  374 eberlm@66277  375 lemma abs_convergent_prod_imp_LIMSEQ:  eberlm@66277  376  fixes f :: "nat \ 'a :: {comm_ring_1,real_normed_div_algebra}"  eberlm@66277  377  assumes "abs_convergent_prod f"  eberlm@66277  378  shows "f \ 1"  eberlm@66277  379 proof -  eberlm@66277  380  from assms have "summable (\n. norm (f n - 1))"  eberlm@66277  381  by (rule abs_convergent_prod_imp_summable)  eberlm@66277  382  from summable_LIMSEQ_zero[OF this] have "(\n. f n - 1) \ 0"  eberlm@66277  383  by (simp add: tendsto_norm_zero_iff)  eberlm@66277  384  from tendsto_add[OF this tendsto_const[of 1]] show ?thesis by simp  eberlm@66277  385 qed  eberlm@66277  386 eberlm@66277  387 lemma abs_convergent_prod_imp_ev_nonzero:  eberlm@66277  388  fixes f :: "nat \ 'a :: {comm_ring_1,real_normed_div_algebra}"  eberlm@66277  389  assumes "abs_convergent_prod f"  eberlm@66277  390  shows "eventually (\n. f n \ 0) sequentially"  eberlm@66277  391 proof -  eberlm@66277  392  from assms have "f \ 1"  eberlm@66277  393  by (rule abs_convergent_prod_imp_LIMSEQ)  eberlm@66277  394  hence "eventually (\n. dist (f n) 1 < 1) at_top"  eberlm@66277  395  by (auto simp: tendsto_iff)  eberlm@66277  396  thus ?thesis by eventually_elim auto  eberlm@66277  397 qed  eberlm@66277  398 eberlm@66277  399 lemma convergent_prod_offset:  eberlm@66277  400  assumes "convergent_prod (\n. f (n + m))"  eberlm@66277  401  shows "convergent_prod f"  eberlm@66277  402 proof -  eberlm@66277  403  from assms obtain M L where "(\n. \k\n. f (k + (M + m))) \ L" "L \ 0"  lp15@68064  404  by (auto simp: prod_defs add.assoc)  lp15@68064  405  thus "convergent_prod f"  lp15@68064  406  unfolding prod_defs by blast  eberlm@66277  407 qed  eberlm@66277  408 eberlm@66277  409 lemma abs_convergent_prod_offset:  eberlm@66277  410  assumes "abs_convergent_prod (\n. f (n + m))"  eberlm@66277  411  shows "abs_convergent_prod f"  eberlm@66277  412  using assms unfolding abs_convergent_prod_def by (rule convergent_prod_offset)  eberlm@66277  413 lp15@68424  414 subsection\Ignoring initial segments\  lp15@68424  415 lp15@68361  416 lemma raw_has_prod_ignore_initial_segment:  lp15@68361  417  fixes f :: "nat \ 'a :: real_normed_field"  lp15@68361  418  assumes "raw_has_prod f M p" "N \ M"  lp15@68361  419  obtains q where "raw_has_prod f N q"  eberlm@66277  420 proof -  lp15@68361  421  have p: "(\n. \k\n. f (k + M)) \ p" and "p \ 0"  lp15@68361  422  using assms by (auto simp: raw_has_prod_def)  lp15@68361  423  then have nz: "\n. n \ M \ f n \ 0"  lp15@68361  424  using assms by (auto simp: raw_has_prod_eq_0)  lp15@68361  425  define C where "C = (\k 0"  eberlm@66277  427  by (auto simp: C_def)  eberlm@66277  428 lp15@68361  429  from p have "(\i. \k\i + (N-M). f (k + M)) \ p"  eberlm@66277  430  by (rule LIMSEQ_ignore_initial_segment)  lp15@68361  431  also have "(\i. \k\i + (N-M). f (k + M)) = (\n. C * (\k\n. f (k + N)))"  eberlm@66277  432  proof (rule ext, goal_cases)  eberlm@66277  433  case (1 n)  lp15@68361  434  have "{..n+(N-M)} = {..<(N-M)} \ {(N-M)..n+(N-M)}" by auto  lp15@68361  435  also have "(\k\\. f (k + M)) = C * (\k=(N-M)..n+(N-M). f (k + M))"  eberlm@66277  436  unfolding C_def by (rule prod.union_disjoint) auto  lp15@68361  437  also have "(\k=(N-M)..n+(N-M). f (k + M)) = (\k\n. f (k + (N-M) + M))"  lp15@68361  438  by (intro ext prod.reindex_bij_witness[of _ "\k. k + (N-M)" "\k. k - (N-M)"]) auto  lp15@68361  439  finally show ?case  lp15@68361  440  using \N \ M\ by (simp add: add_ac)  eberlm@66277  441  qed  lp15@68361  442  finally have "(\n. C * (\k\n. f (k + N)) / C) \ p / C"  eberlm@66277  443  by (intro tendsto_divide tendsto_const) auto  lp15@68361  444  hence "(\n. \k\n. f (k + N)) \ p / C" by simp  lp15@68361  445  moreover from \p \ 0\ have "p / C \ 0" by simp  lp15@68361  446  ultimately show ?thesis  lp15@68361  447  using raw_has_prod_def that by blast  eberlm@66277  448 qed  eberlm@66277  449 lp15@68361  450 corollary convergent_prod_ignore_initial_segment:  lp15@68361  451  fixes f :: "nat \ 'a :: real_normed_field"  lp15@68361  452  assumes "convergent_prod f"  lp15@68361  453  shows "convergent_prod (\n. f (n + m))"  lp15@68361  454  using assms  lp15@68361  455  unfolding convergent_prod_def  lp15@68361  456  apply clarify  lp15@68361  457  apply (erule_tac N="M+m" in raw_has_prod_ignore_initial_segment)  lp15@68361  458  apply (auto simp add: raw_has_prod_def add_ac)  lp15@68361  459  done  lp15@68361  460 lp15@68136  461 corollary convergent_prod_ignore_nonzero_segment:  lp15@68136  462  fixes f :: "nat \ 'a :: real_normed_field"  lp15@68136  463  assumes f: "convergent_prod f" and nz: "\i. i \ M \ f i \ 0"  lp15@68361  464  shows "\p. raw_has_prod f M p"  lp15@68136  465  using convergent_prod_ignore_initial_segment [OF f]  lp15@68136  466  by (metis convergent_LIMSEQ_iff convergent_prod_iff_convergent le_add_same_cancel2 nz prod_defs(1) zero_order(1))  lp15@68136  467 lp15@68136  468 corollary abs_convergent_prod_ignore_initial_segment:  eberlm@66277  469  assumes "abs_convergent_prod f"  eberlm@66277  470  shows "abs_convergent_prod (\n. f (n + m))"  eberlm@66277  471  using assms unfolding abs_convergent_prod_def  eberlm@66277  472  by (rule convergent_prod_ignore_initial_segment)  eberlm@66277  473 eberlm@66277  474 lemma abs_convergent_prod_imp_convergent_prod:  eberlm@66277  475  fixes f :: "nat \ 'a :: {real_normed_div_algebra,complete_space,comm_ring_1}"  eberlm@66277  476  assumes "abs_convergent_prod f"  eberlm@66277  477  shows "convergent_prod f"  eberlm@66277  478 proof -  eberlm@66277  479  from assms have "eventually (\n. f n \ 0) sequentially"  eberlm@66277  480  by (rule abs_convergent_prod_imp_ev_nonzero)  eberlm@66277  481  then obtain N where N: "f n \ 0" if "n \ N" for n  eberlm@66277  482  by (auto simp: eventually_at_top_linorder)  eberlm@66277  483  let ?P = "\n. \i\n. f (i + N)" and ?Q = "\n. \i\n. 1 + norm (f (i + N) - 1)"  eberlm@66277  484 eberlm@66277  485  have "Cauchy ?P"  eberlm@66277  486  proof (rule CauchyI', goal_cases)  eberlm@66277  487  case (1 \)  eberlm@66277  488  from assms have "abs_convergent_prod (\n. f (n + N))"  eberlm@66277  489  by (rule abs_convergent_prod_ignore_initial_segment)  eberlm@66277  490  hence "Cauchy ?Q"  eberlm@66277  491  unfolding abs_convergent_prod_def  eberlm@66277  492  by (intro convergent_Cauchy convergent_prod_imp_convergent)  eberlm@66277  493  from CauchyD[OF this 1] obtain M where M: "norm (?Q m - ?Q n) < \" if "m \ M" "n \ M" for m n  eberlm@66277  494  by blast  eberlm@66277  495  show ?case  eberlm@66277  496  proof (rule exI[of _ M], safe, goal_cases)  eberlm@66277  497  case (1 m n)  eberlm@66277  498  have "dist (?P m) (?P n) = norm (?P n - ?P m)"  eberlm@66277  499  by (simp add: dist_norm norm_minus_commute)  eberlm@66277  500  also from 1 have "{..n} = {..m} \ {m<..n}" by auto  eberlm@66277  501  hence "norm (?P n - ?P m) = norm (?P m * (\k\{m<..n}. f (k + N)) - ?P m)"  eberlm@66277  502  by (subst prod.union_disjoint [symmetric]) (auto simp: algebra_simps)  eberlm@66277  503  also have "\ = norm (?P m * ((\k\{m<..n}. f (k + N)) - 1))"  eberlm@66277  504  by (simp add: algebra_simps)  eberlm@66277  505  also have "\ = (\k\m. norm (f (k + N))) * norm ((\k\{m<..n}. f (k + N)) - 1)"  eberlm@66277  506  by (simp add: norm_mult prod_norm)  eberlm@66277  507  also have "\ \ ?Q m * ((\k\{m<..n}. 1 + norm (f (k + N) - 1)) - 1)"  eberlm@66277  508  using norm_prod_minus1_le_prod_minus1[of "\k. f (k + N) - 1" "{m<..n}"]  eberlm@66277  509  norm_triangle_ineq[of 1 "f k - 1" for k]  eberlm@66277  510  by (intro mult_mono prod_mono ballI conjI norm_prod_minus1_le_prod_minus1 prod_nonneg) auto  eberlm@66277  511  also have "\ = ?Q m * (\k\{m<..n}. 1 + norm (f (k + N) - 1)) - ?Q m"  eberlm@66277  512  by (simp add: algebra_simps)  eberlm@66277  513  also have "?Q m * (\k\{m<..n}. 1 + norm (f (k + N) - 1)) =  eberlm@66277  514  (\k\{..m}\{m<..n}. 1 + norm (f (k + N) - 1))"  eberlm@66277  515  by (rule prod.union_disjoint [symmetric]) auto  eberlm@66277  516  also from 1 have "{..m}\{m<..n} = {..n}" by auto  eberlm@66277  517  also have "?Q n - ?Q m \ norm (?Q n - ?Q m)" by simp  eberlm@66277  518  also from 1 have "\ < \" by (intro M) auto  eberlm@66277  519  finally show ?case .  eberlm@66277  520  qed  eberlm@66277  521  qed  eberlm@66277  522  hence conv: "convergent ?P" by (rule Cauchy_convergent)  eberlm@66277  523  then obtain L where L: "?P \ L"  eberlm@66277  524  by (auto simp: convergent_def)  eberlm@66277  525 eberlm@66277  526  have "L \ 0"  eberlm@66277  527  proof  eberlm@66277  528  assume [simp]: "L = 0"  eberlm@66277  529  from tendsto_norm[OF L] have limit: "(\n. \k\n. norm (f (k + N))) \ 0"  eberlm@66277  530  by (simp add: prod_norm)  eberlm@66277  531 eberlm@66277  532  from assms have "(\n. f (n + N)) \ 1"  eberlm@66277  533  by (intro abs_convergent_prod_imp_LIMSEQ abs_convergent_prod_ignore_initial_segment)  eberlm@66277  534  hence "eventually (\n. norm (f (n + N) - 1) < 1) sequentially"  eberlm@66277  535  by (auto simp: tendsto_iff dist_norm)  eberlm@66277  536  then obtain M0 where M0: "norm (f (n + N) - 1) < 1" if "n \ M0" for n  eberlm@66277  537  by (auto simp: eventually_at_top_linorder)  eberlm@66277  538 eberlm@66277  539  {  eberlm@66277  540  fix M assume M: "M \ M0"  eberlm@66277  541  with M0 have M: "norm (f (n + N) - 1) < 1" if "n \ M" for n using that by simp  eberlm@66277  542 eberlm@66277  543  have "(\n. \k\n. 1 - norm (f (k+M+N) - 1)) \ 0"  eberlm@66277  544  proof (rule tendsto_sandwich)  eberlm@66277  545  show "eventually (\n. (\k\n. 1 - norm (f (k+M+N) - 1)) \ 0) sequentially"  eberlm@66277  546  using M by (intro always_eventually prod_nonneg allI ballI) (auto intro: less_imp_le)  eberlm@66277  547  have "norm (1::'a) - norm (f (i + M + N) - 1) \ norm (f (i + M + N))" for i  eberlm@66277  548  using norm_triangle_ineq3[of "f (i + M + N)" 1] by simp  eberlm@66277  549  thus "eventually (\n. (\k\n. 1 - norm (f (k+M+N) - 1)) \ (\k\n. norm (f (k+M+N)))) at_top"  eberlm@66277  550  using M by (intro always_eventually allI prod_mono ballI conjI) (auto intro: less_imp_le)  eberlm@66277  551   eberlm@66277  552  define C where "C = (\k 0" by (auto simp: C_def)  eberlm@66277  554  from L have "(\n. norm (\k\n+M. f (k + N))) \ 0"  eberlm@66277  555  by (intro LIMSEQ_ignore_initial_segment) (simp add: tendsto_norm_zero_iff)  eberlm@66277  556  also have "(\n. norm (\k\n+M. f (k + N))) = (\n. C * (\k\n. norm (f (k + M + N))))"  eberlm@66277  557  proof (rule ext, goal_cases)  eberlm@66277  558  case (1 n)  eberlm@66277  559  have "{..n+M} = {.. {M..n+M}" by auto  eberlm@66277  560  also have "norm (\k\\. f (k + N)) = C * norm (\k=M..n+M. f (k + N))"  eberlm@66277  561  unfolding C_def by (subst prod.union_disjoint) (auto simp: norm_mult prod_norm)  eberlm@66277  562  also have "(\k=M..n+M. f (k + N)) = (\k\n. f (k + N + M))"  eberlm@66277  563  by (intro prod.reindex_bij_witness[of _ "\i. i + M" "\i. i - M"]) auto  eberlm@66277  564  finally show ?case by (simp add: add_ac prod_norm)  eberlm@66277  565  qed  eberlm@66277  566  finally have "(\n. C * (\k\n. norm (f (k + M + N))) / C) \ 0 / C"  eberlm@66277  567  by (intro tendsto_divide tendsto_const) auto  eberlm@66277  568  thus "(\n. \k\n. norm (f (k + M + N))) \ 0" by simp  eberlm@66277  569  qed simp_all  eberlm@66277  570 eberlm@66277  571  have "1 - (\i. norm (f (i + M + N) - 1)) \ 0"  eberlm@66277  572  proof (rule tendsto_le)  eberlm@66277  573  show "eventually (\n. 1 - (\k\n. norm (f (k+M+N) - 1)) \  eberlm@66277  574  (\k\n. 1 - norm (f (k+M+N) - 1))) at_top"  eberlm@66277  575  using M by (intro always_eventually allI weierstrass_prod_ineq) (auto intro: less_imp_le)  eberlm@66277  576  show "(\n. \k\n. 1 - norm (f (k+M+N) - 1)) \ 0" by fact  eberlm@66277  577  show "(\n. 1 - (\k\n. norm (f (k + M + N) - 1)))  eberlm@66277  578  \ 1 - (\i. norm (f (i + M + N) - 1))"  eberlm@66277  579  by (intro tendsto_intros summable_LIMSEQ' summable_ignore_initial_segment  eberlm@66277  580  abs_convergent_prod_imp_summable assms)  eberlm@66277  581  qed simp_all  eberlm@66277  582  hence "(\i. norm (f (i + M + N) - 1)) \ 1" by simp  eberlm@66277  583  also have "\ + (\ii. norm (f (i + N) - 1))"  eberlm@66277  584  by (intro suminf_split_initial_segment [symmetric] summable_ignore_initial_segment  eberlm@66277  585  abs_convergent_prod_imp_summable assms)  eberlm@66277  586  finally have "1 + (\i (\i. norm (f (i + N) - 1))" by simp  eberlm@66277  587  } note * = this  eberlm@66277  588 eberlm@66277  589  have "1 + (\i. norm (f (i + N) - 1)) \ (\i. norm (f (i + N) - 1))"  eberlm@66277  590  proof (rule tendsto_le)  eberlm@66277  591  show "(\M. 1 + (\i 1 + (\i. norm (f (i + N) - 1))"  eberlm@66277  592  by (intro tendsto_intros summable_LIMSEQ summable_ignore_initial_segment  eberlm@66277  593  abs_convergent_prod_imp_summable assms)  eberlm@66277  594  show "eventually (\M. 1 + (\i (\i. norm (f (i + N) - 1))) at_top"  eberlm@66277  595  using eventually_ge_at_top[of M0] by eventually_elim (use * in auto)  eberlm@66277  596  qed simp_all  eberlm@66277  597  thus False by simp  eberlm@66277  598  qed  lp15@68064  599  with L show ?thesis by (auto simp: prod_defs)  lp15@68064  600 qed  lp15@68064  601 lp15@68424  602 subsection\More elementary properties\  lp15@68424  603 lp15@68361  604 lemma raw_has_prod_cases:  lp15@68064  605  fixes f :: "nat \ 'a :: {idom,topological_semigroup_mult,t2_space}"  lp15@68361  606  assumes "raw_has_prod f M p"  lp15@68361  607  obtains i where "in. \i\n. f (i + M)) \ p" "p \ 0"  lp15@68361  610  using assms unfolding raw_has_prod_def by blast+  lp15@68064  611  then have "(\n. prod f {..i\n. f (i + M))) \ prod f {..i\n. f (i + M)) = prod f {..n+M}" for n  lp15@68064  614  proof -  lp15@68064  615  have "{..n+M} = {.. {M..n+M}"  lp15@68064  616  by auto  lp15@68064  617  then have "prod f {..n+M} = prod f {.. = prod f {..i\n. f (i + M))"  lp15@68064  620  by (metis (mono_tags, lifting) add.left_neutral atMost_atLeast0 prod_shift_bounds_cl_nat_ivl)  lp15@68064  621  finally show ?thesis by metis  lp15@68064  622  qed  lp15@68064  623  ultimately have "(\n. prod f {..n}) \ prod f {..i 0"  lp15@68361  626  using \p \ 0\ assms that by (auto simp: raw_has_prod_def)  lp15@68136  627  then show thesis  lp15@68136  628  using that by blast  lp15@68064  629 qed  lp15@68064  630 lp15@68136  631 corollary convergent_prod_offset_0:  lp15@68136  632  fixes f :: "nat \ 'a :: {idom,topological_semigroup_mult,t2_space}"  lp15@68136  633  assumes "convergent_prod f" "\i. f i \ 0"  lp15@68361  634  shows "\p. raw_has_prod f 0 p"  lp15@68361  635  using assms convergent_prod_def raw_has_prod_cases by blast  lp15@68136  636 lp15@68064  637 lemma prodinf_eq_lim:  lp15@68064  638  fixes f :: "nat \ 'a :: {idom,topological_semigroup_mult,t2_space}"  lp15@68064  639  assumes "convergent_prod f" "\i. f i \ 0"  lp15@68064  640  shows "prodinf f = lim (\n. \i\n. f i)"  lp15@68064  641  using assms convergent_prod_offset_0 [OF assms]  lp15@68064  642  by (simp add: prod_defs lim_def) (metis (no_types) assms(1) convergent_prod_to_zero_iff)  lp15@68064  643 lp15@68064  644 lemma has_prod_one[simp, intro]: "(\n. 1) has_prod 1"  lp15@68064  645  unfolding prod_defs by auto  lp15@68064  646 lp15@68064  647 lemma convergent_prod_one[simp, intro]: "convergent_prod (\n. 1)"  lp15@68064  648  unfolding prod_defs by auto  lp15@68064  649 lp15@68064  650 lemma prodinf_cong: "(\n. f n = g n) \ prodinf f = prodinf g"  lp15@68064  651  by presburger  lp15@68064  652 lp15@68064  653 lemma convergent_prod_cong:  lp15@68064  654  fixes f g :: "nat \ 'a::{field,topological_semigroup_mult,t2_space}"  lp15@68064  655  assumes ev: "eventually (\x. f x = g x) sequentially" and f: "\i. f i \ 0" and g: "\i. g i \ 0"  lp15@68064  656  shows "convergent_prod f = convergent_prod g"  lp15@68064  657 proof -  lp15@68064  658  from assms obtain N where N: "\n\N. f n = g n"  lp15@68064  659  by (auto simp: eventually_at_top_linorder)  lp15@68064  660  define C where "C = (\k 0"  lp15@68064  662  by (simp add: f)  lp15@68064  663  have *: "eventually (\n. prod f {..n} = C * prod g {..n}) sequentially"  lp15@68064  664  using eventually_ge_at_top[of N]  lp15@68064  665  proof eventually_elim  lp15@68064  666  case (elim n)  lp15@68064  667  then have "{..n} = {.. {N..n}"  lp15@68064  668  by auto  lp15@68138  669  also have "prod f \ = prod f {.. {N..n})"  lp15@68064  676  by (intro prod.union_disjoint [symmetric]) auto  lp15@68064  677  also from elim have "{.. {N..n} = {..n}"  lp15@68064  678  by auto  lp15@68064  679  finally show "prod f {..n} = C * prod g {..n}" .  lp15@68064  680  qed  lp15@68064  681  then have cong: "convergent (\n. prod f {..n}) = convergent (\n. C * prod g {..n})"  lp15@68064  682  by (rule convergent_cong)  lp15@68064  683  show ?thesis  lp15@68064  684  proof  lp15@68064  685  assume cf: "convergent_prod f"  lp15@68064  686  then have "\ (\n. prod g {..n}) \ 0"  lp15@68064  687  using tendsto_mult_left * convergent_prod_to_zero_iff f filterlim_cong by fastforce  lp15@68064  688  then show "convergent_prod g"  lp15@68064  689  by (metis convergent_mult_const_iff \C \ 0\ cong cf convergent_LIMSEQ_iff convergent_prod_iff_convergent convergent_prod_imp_convergent g)  lp15@68064  690  next  lp15@68064  691  assume cg: "convergent_prod g"  lp15@68064  692  have "\a. C * a \ 0 \ (\n. prod g {..n}) \ a"  lp15@68064  693  by (metis (no_types) \C \ 0\ cg convergent_prod_iff_nz_lim divide_eq_0_iff g nonzero_mult_div_cancel_right)  lp15@68064  694  then show "convergent_prod f"  lp15@68064  695  using "*" tendsto_mult_left filterlim_cong  lp15@68064  696  by (fastforce simp add: convergent_prod_iff_nz_lim f)  lp15@68064  697  qed  eberlm@66277  698 qed  eberlm@66277  699 lp15@68071  700 lemma has_prod_finite:  lp15@68361  701  fixes f :: "nat \ 'a::{semidom,t2_space}"  lp15@68071  702  assumes [simp]: "finite N"  lp15@68071  703  and f: "\n. n \ N \ f n = 1"  lp15@68071  704  shows "f has_prod (\n\N. f n)"  lp15@68071  705 proof -  lp15@68071  706  have eq: "prod f {..n + Suc (Max N)} = prod f N" for n  lp15@68071  707  proof (rule prod.mono_neutral_right)  lp15@68071  708  show "N \ {..n + Suc (Max N)}"  lp15@68138  709  by (auto simp: le_Suc_eq trans_le_add2)  lp15@68071  710  show "\i\{..n + Suc (Max N)} - N. f i = 1"  lp15@68071  711  using f by blast  lp15@68071  712  qed auto  lp15@68071  713  show ?thesis  lp15@68071  714  proof (cases "\n\N. f n \ 0")  lp15@68071  715  case True  lp15@68071  716  then have "prod f N \ 0"  lp15@68071  717  by simp  lp15@68071  718  moreover have "(\n. prod f {..n}) \ prod f N"  lp15@68071  719  by (rule LIMSEQ_offset[of _ "Suc (Max N)"]) (simp add: eq atLeast0LessThan del: add_Suc_right)  lp15@68071  720  ultimately show ?thesis  lp15@68361  721  by (simp add: raw_has_prod_def has_prod_def)  lp15@68071  722  next  lp15@68071  723  case False  lp15@68071  724  then obtain k where "k \ N" "f k = 0"  lp15@68071  725  by auto  lp15@68071  726  let ?Z = "{n \ N. f n = 0}"  lp15@68071  727  have maxge: "Max ?Z \ n" if "f n = 0" for n  lp15@68071  728  using Max_ge [of ?Z] \finite N\ \f n = 0\  lp15@68071  729  by (metis (mono_tags) Collect_mem_eq f finite_Collect_conjI mem_Collect_eq zero_neq_one)  lp15@68071  730  let ?q = "prod f {Suc (Max ?Z)..Max N}"  lp15@68071  731  have [simp]: "?q \ 0"  lp15@68071  732  using maxge Suc_n_not_le_n le_trans by force  lp15@68076  733  have eq: "(\i\n + Max N. f (Suc (i + Max ?Z))) = ?q" for n  lp15@68076  734  proof -  lp15@68076  735  have "(\i\n + Max N. f (Suc (i + Max ?Z))) = prod f {Suc (Max ?Z)..n + Max N + Suc (Max ?Z)}"  lp15@68076  736  proof (rule prod.reindex_cong [where l = "\i. i + Suc (Max ?Z)", THEN sym])  lp15@68076  737  show "{Suc (Max ?Z)..n + Max N + Suc (Max ?Z)} = (\i. i + Suc (Max ?Z))  {..n + Max N}"  lp15@68076  738  using le_Suc_ex by fastforce  lp15@68076  739  qed (auto simp: inj_on_def)  lp15@68138  740  also have "\ = ?q"  lp15@68076  741  by (rule prod.mono_neutral_right)  lp15@68076  742  (use Max.coboundedI [OF \finite N\] f in \force+\)  lp15@68076  743  finally show ?thesis .  lp15@68076  744  qed  lp15@68361  745  have q: "raw_has_prod f (Suc (Max ?Z)) ?q"  lp15@68361  746  proof (simp add: raw_has_prod_def)  lp15@68076  747  show "(\n. \i\n. f (Suc (i + Max ?Z))) \ ?q"  lp15@68076  748  by (rule LIMSEQ_offset[of _ "(Max N)"]) (simp add: eq)  lp15@68076  749  qed  lp15@68071  750  show ?thesis  lp15@68071  751  unfolding has_prod_def  lp15@68071  752  proof (intro disjI2 exI conjI)  lp15@68071  753  show "prod f N = 0"  lp15@68071  754  using \f k = 0\ \k \ N\ \finite N\ prod_zero by blast  lp15@68071  755  show "f (Max ?Z) = 0"  lp15@68071  756  using Max_in [of ?Z] \finite N\ \f k = 0\ \k \ N\ by auto  lp15@68071  757  qed (use q in auto)  lp15@68071  758  qed  lp15@68071  759 qed  lp15@68071  760 lp15@68071  761 corollary has_prod_0:  lp15@68361  762  fixes f :: "nat \ 'a::{semidom,t2_space}"  lp15@68071  763  assumes "\n. f n = 1"  lp15@68071  764  shows "f has_prod 1"  lp15@68071  765  by (simp add: assms has_prod_cong)  lp15@68071  766 lp15@68361  767 lemma prodinf_zero[simp]: "prodinf (\n. 1::'a::real_normed_field) = 1"  lp15@68361  768  using has_prod_unique by force  lp15@68361  769 lp15@68071  770 lemma convergent_prod_finite:  lp15@68071  771  fixes f :: "nat \ 'a::{idom,t2_space}"  lp15@68071  772  assumes "finite N" "\n. n \ N \ f n = 1"  lp15@68071  773  shows "convergent_prod f"  lp15@68071  774 proof -  lp15@68361  775  have "\n p. raw_has_prod f n p"  lp15@68071  776  using assms has_prod_def has_prod_finite by blast  lp15@68071  777  then show ?thesis  lp15@68071  778  by (simp add: convergent_prod_def)  lp15@68071  779 qed  lp15@68071  780 lp15@68127  781 lemma has_prod_If_finite_set:  lp15@68127  782  fixes f :: "nat \ 'a::{idom,t2_space}"  lp15@68127  783  shows "finite A \ (\r. if r \ A then f r else 1) has_prod (\r\A. f r)"  lp15@68127  784  using has_prod_finite[of A "(\r. if r \ A then f r else 1)"]  lp15@68127  785  by simp  lp15@68127  786 lp15@68127  787 lemma has_prod_If_finite:  lp15@68127  788  fixes f :: "nat \ 'a::{idom,t2_space}"  lp15@68127  789  shows "finite {r. P r} \ (\r. if P r then f r else 1) has_prod (\r | P r. f r)"  lp15@68127  790  using has_prod_If_finite_set[of "{r. P r}"] by simp  lp15@68127  791 lp15@68127  792 lemma convergent_prod_If_finite_set[simp, intro]:  lp15@68127  793  fixes f :: "nat \ 'a::{idom,t2_space}"  lp15@68127  794  shows "finite A \ convergent_prod (\r. if r \ A then f r else 1)"  lp15@68127  795  by (simp add: convergent_prod_finite)  lp15@68127  796 lp15@68127  797 lemma convergent_prod_If_finite[simp, intro]:  lp15@68127  798  fixes f :: "nat \ 'a::{idom,t2_space}"  lp15@68127  799  shows "finite {r. P r} \ convergent_prod (\r. if P r then f r else 1)"  lp15@68127  800  using convergent_prod_def has_prod_If_finite has_prod_def by fastforce  lp15@68127  801 lp15@68127  802 lemma has_prod_single:  lp15@68127  803  fixes f :: "nat \ 'a::{idom,t2_space}"  lp15@68127  804  shows "(\r. if r = i then f r else 1) has_prod f i"  lp15@68127  805  using has_prod_If_finite[of "\r. r = i"] by simp  lp15@68127  806 lp15@68136  807 context  lp15@68136  808  fixes f :: "nat \ 'a :: real_normed_field"  lp15@68136  809 begin  lp15@68136  810 lp15@68136  811 lemma convergent_prod_imp_has_prod:  lp15@68136  812  assumes "convergent_prod f"  lp15@68136  813  shows "\p. f has_prod p"  lp15@68136  814 proof -  lp15@68361  815  obtain M p where p: "raw_has_prod f M p"  lp15@68136  816  using assms convergent_prod_def by blast  lp15@68136  817  then have "p \ 0"  lp15@68361  818  using raw_has_prod_nonzero by blast  lp15@68136  819  with p have fnz: "f i \ 0" if "i \ M" for i  lp15@68361  820  using raw_has_prod_eq_0 that by blast  lp15@68136  821  define C where "C = (\nn\M. f n \ 0")  lp15@68136  824  case True  lp15@68136  825  then have "C \ 0"  lp15@68136  826  by (simp add: C_def)  lp15@68136  827  then show ?thesis  lp15@68136  828  by (meson True assms convergent_prod_offset_0 fnz has_prod_def nat_le_linear)  lp15@68136  829  next  lp15@68136  830  case False  lp15@68136  831  let ?N = "GREATEST n. f n = 0"  lp15@68136  832  have 0: "f ?N = 0"  lp15@68136  833  using fnz False  lp15@68136  834  by (metis (mono_tags, lifting) GreatestI_ex_nat nat_le_linear)  lp15@68136  835  have "f i \ 0" if "i > ?N" for i  lp15@68136  836  by (metis (mono_tags, lifting) Greatest_le_nat fnz leD linear that)  lp15@68361  837  then have "\p. raw_has_prod f (Suc ?N) p"  lp15@68136  838  using assms by (auto simp: intro!: convergent_prod_ignore_nonzero_segment)  lp15@68136  839  then show ?thesis  lp15@68136  840  unfolding has_prod_def using 0 by blast  lp15@68136  841  qed  lp15@68136  842 qed  lp15@68136  843 lp15@68136  844 lemma convergent_prod_has_prod [intro]:  lp15@68136  845  shows "convergent_prod f \ f has_prod (prodinf f)"  lp15@68136  846  unfolding prodinf_def  lp15@68136  847  by (metis convergent_prod_imp_has_prod has_prod_unique theI')  lp15@68136  848 lp15@68136  849 lemma convergent_prod_LIMSEQ:  lp15@68136  850  shows "convergent_prod f \ (\n. \i\n. f i) \ prodinf f"  lp15@68136  851  by (metis convergent_LIMSEQ_iff convergent_prod_has_prod convergent_prod_imp_convergent  lp15@68361  852  convergent_prod_to_zero_iff raw_has_prod_eq_0 has_prod_def prodinf_eq_lim zero_le)  lp15@68136  853 lp15@68136  854 lemma has_prod_iff: "f has_prod x \ convergent_prod f \ prodinf f = x"  lp15@68136  855 proof  lp15@68136  856  assume "f has_prod x"  lp15@68136  857  then show "convergent_prod f \ prodinf f = x"  lp15@68136  858  apply safe  lp15@68136  859  using convergent_prod_def has_prod_def apply blast  lp15@68136  860  using has_prod_unique by blast  lp15@68136  861 qed auto  lp15@68136  862 lp15@68136  863 lemma convergent_prod_has_prod_iff: "convergent_prod f \ f has_prod prodinf f"  lp15@68136  864  by (auto simp: has_prod_iff convergent_prod_has_prod)  lp15@68136  865 lp15@68136  866 lemma prodinf_finite:  lp15@68136  867  assumes N: "finite N"  lp15@68136  868  and f: "\n. n \ N \ f n = 1"  lp15@68136  869  shows "prodinf f = (\n\N. f n)"  lp15@68136  870  using has_prod_finite[OF assms, THEN has_prod_unique] by simp  lp15@68127  871 eberlm@66277  872 end  lp15@68136  873 lp15@68361  874 subsection \Infinite products on ordered, topological monoids\  lp15@68361  875 lp15@68361  876 lemma LIMSEQ_prod_0:  lp15@68361  877  fixes f :: "nat \ 'a::{semidom,topological_space}"  lp15@68361  878  assumes "f i = 0"  lp15@68361  879  shows "(\n. prod f {..n}) \ 0"  lp15@68361  880 proof (subst tendsto_cong)  lp15@68361  881  show "\\<^sub>F n in sequentially. prod f {..n} = 0"  lp15@68361  882  proof  lp15@68361  883  show "prod f {..n} = 0" if "n \ i" for n  lp15@68361  884  using that assms by auto  lp15@68361  885  qed  lp15@68361  886 qed auto  lp15@68361  887 lp15@68361  888 lemma LIMSEQ_prod_nonneg:  lp15@68361  889  fixes f :: "nat \ 'a::{linordered_semidom,linorder_topology}"  lp15@68361  890  assumes 0: "\n. 0 \ f n" and a: "(\n. prod f {..n}) \ a"  lp15@68361  891  shows "a \ 0"  lp15@68361  892  by (simp add: "0" prod_nonneg LIMSEQ_le_const [OF a])  lp15@68361  893 lp15@68361  894 lp15@68361  895 context  lp15@68361  896  fixes f :: "nat \ 'a::{linordered_semidom,linorder_topology}"  lp15@68361  897 begin  lp15@68361  898 lp15@68361  899 lemma has_prod_le:  lp15@68361  900  assumes f: "f has_prod a" and g: "g has_prod b" and le: "\n. 0 \ f n \ f n \ g n"  lp15@68361  901  shows "a \ b"  lp15@68361  902 proof (cases "a=0 \ b=0")  lp15@68361  903  case True  lp15@68361  904  then show ?thesis  lp15@68361  905  proof  lp15@68361  906  assume [simp]: "a=0"  lp15@68361  907  have "b \ 0"  lp15@68361  908  proof (rule LIMSEQ_prod_nonneg)  lp15@68361  909  show "(\n. prod g {..n}) \ b"  lp15@68361  910  using g by (auto simp: has_prod_def raw_has_prod_def LIMSEQ_prod_0)  lp15@68361  911  qed (use le order_trans in auto)  lp15@68361  912  then show ?thesis  lp15@68361  913  by auto  lp15@68361  914  next  lp15@68361  915  assume [simp]: "b=0"  lp15@68361  916  then obtain i where "g i = 0"  lp15@68361  917  using g by (auto simp: prod_defs)  lp15@68361  918  then have "f i = 0"  lp15@68361  919  using antisym le by force  lp15@68361  920  then have "a=0"  lp15@68361  921  using f by (auto simp: prod_defs LIMSEQ_prod_0 LIMSEQ_unique)  lp15@68361  922  then show ?thesis  lp15@68361  923  by auto  lp15@68361  924  qed  lp15@68361  925 next  lp15@68361  926  case False  lp15@68361  927  then show ?thesis  lp15@68361  928  using assms  lp15@68361  929  unfolding has_prod_def raw_has_prod_def  lp15@68361  930  by (force simp: LIMSEQ_prod_0 intro!: LIMSEQ_le prod_mono)  lp15@68361  931 qed  lp15@68361  932 lp15@68361  933 lemma prodinf_le:  lp15@68361  934  assumes f: "f has_prod a" and g: "g has_prod b" and le: "\n. 0 \ f n \ f n \ g n"  lp15@68361  935  shows "prodinf f \ prodinf g"  lp15@68361  936  using has_prod_le [OF assms] has_prod_unique f g by blast  lp15@68361  937 lp15@68136  938 end  lp15@68361  939 lp15@68361  940 lp15@68361  941 lemma prod_le_prodinf:  lp15@68361  942  fixes f :: "nat \ 'a::{linordered_idom,linorder_topology}"  lp15@68361  943  assumes "f has_prod a" "\i. 0 \ f i" "\i. i\n \ 1 \ f i"  lp15@68361  944  shows "prod f {.. prodinf f"  lp15@68361  945  by(rule has_prod_le[OF has_prod_If_finite_set]) (use assms has_prod_unique in auto)  lp15@68361  946 lp15@68361  947 lemma prodinf_nonneg:  lp15@68361  948  fixes f :: "nat \ 'a::{linordered_idom,linorder_topology}"  lp15@68361  949  assumes "f has_prod a" "\i. 1 \ f i"  lp15@68361  950  shows "1 \ prodinf f"  lp15@68361  951  using prod_le_prodinf[of f a 0] assms  lp15@68361  952  by (metis order_trans prod_ge_1 zero_le_one)  lp15@68361  953 lp15@68361  954 lemma prodinf_le_const:  lp15@68361  955  fixes f :: "nat \ real"  lp15@68361  956  assumes "convergent_prod f" "\n. prod f {.. x"  lp15@68361  957  shows "prodinf f \ x"  lp15@68361  958  by (metis lessThan_Suc_atMost assms convergent_prod_LIMSEQ LIMSEQ_le_const2)  lp15@68361  959 lp15@68361  960 lemma prodinf_eq_one_iff:  lp15@68361  961  fixes f :: "nat \ real"  lp15@68361  962  assumes f: "convergent_prod f" and ge1: "\n. 1 \ f n"  lp15@68361  963  shows "prodinf f = 1 \ (\n. f n = 1)"  lp15@68361  964 proof  lp15@68361  965  assume "prodinf f = 1"  lp15@68361  966  then have "(\n. \i 1"  lp15@68361  967  using convergent_prod_LIMSEQ[of f] assms by (simp add: LIMSEQ_lessThan_iff_atMost)  lp15@68361  968  then have "\i. (\n\{i}. f n) \ 1"  lp15@68361  969  proof (rule LIMSEQ_le_const)  lp15@68361  970  have "1 \ prod f n" for n  lp15@68361  971  by (simp add: ge1 prod_ge_1)  lp15@68361  972  have "prod f {..\n. 1 \ prod f n\ \prodinf f = 1\ antisym f convergent_prod_has_prod ge1 order_trans prod_le_prodinf zero_le_one)  lp15@68361  974  then have "(\n\{i}. f n) \ prod f {.. Suc i" for i n  lp15@68361  975  by (metis mult.left_neutral order_refl prod.cong prod.neutral_const prod_lessThan_Suc)  lp15@68361  976  then show "\N. \n\N. (\n\{i}. f n) \ prod f {..n. f n = 1"  lp15@68361  980  by (auto intro!: antisym)  lp15@68361  981 qed (metis prodinf_zero fun_eq_iff)  lp15@68361  982 lp15@68361  983 lemma prodinf_pos_iff:  lp15@68361  984  fixes f :: "nat \ real"  lp15@68361  985  assumes "convergent_prod f" "\n. 1 \ f n"  lp15@68361  986  shows "1 < prodinf f \ (\i. 1 < f i)"  lp15@68361  987  using prod_le_prodinf[of f 1] prodinf_eq_one_iff  lp15@68361  988  by (metis convergent_prod_has_prod assms less_le prodinf_nonneg)  lp15@68361  989 lp15@68361  990 lemma less_1_prodinf2:  lp15@68361  991  fixes f :: "nat \ real"  lp15@68361  992  assumes "convergent_prod f" "\n. 1 \ f n" "1 < f i"  lp15@68361  993  shows "1 < prodinf f"  lp15@68361  994 proof -  lp15@68361  995  have "1 < (\n \ prodinf f"  lp15@68361  998  by (intro prod_le_prodinf) (use assms order_trans zero_le_one in \blast+\)  lp15@68361  999  finally show ?thesis .  lp15@68361  1000 qed  lp15@68361  1001 lp15@68361  1002 lemma less_1_prodinf:  lp15@68361  1003  fixes f :: "nat \ real"  lp15@68361  1004  shows "\convergent_prod f; \n. 1 < f n\ \ 1 < prodinf f"  lp15@68361  1005  by (intro less_1_prodinf2[where i=1]) (auto intro: less_imp_le)  lp15@68361  1006 lp15@68361  1007 lemma prodinf_nonzero:  lp15@68361  1008  fixes f :: "nat \ 'a :: {idom,topological_semigroup_mult,t2_space}"  lp15@68361  1009  assumes "convergent_prod f" "\i. f i \ 0"  lp15@68361  1010  shows "prodinf f \ 0"  lp15@68361  1011  by (metis assms convergent_prod_offset_0 has_prod_unique raw_has_prod_def has_prod_def)  lp15@68361  1012 lp15@68361  1013 lemma less_0_prodinf:  lp15@68361  1014  fixes f :: "nat \ real"  lp15@68361  1015  assumes f: "convergent_prod f" and 0: "\i. f i > 0"  lp15@68361  1016  shows "0 < prodinf f"  lp15@68361  1017 proof -  lp15@68361  1018  have "prodinf f \ 0"  lp15@68361  1019  by (metis assms less_irrefl prodinf_nonzero)  lp15@68361  1020  moreover have "0 < (\n 0"  lp15@68361  1023  using convergent_prod_LIMSEQ [OF f] LIMSEQ_prod_nonneg 0 less_le by blast  lp15@68361  1024  ultimately show ?thesis  lp15@68361  1025  by auto  lp15@68361  1026 qed  lp15@68361  1027 lp15@68361  1028 lemma prod_less_prodinf2:  lp15@68361  1029  fixes f :: "nat \ real"  lp15@68361  1030  assumes f: "convergent_prod f" and 1: "\m. m\n \ 1 \ f m" and 0: "\m. 0 < f m" and i: "n \ i" "1 < f i"  lp15@68361  1031  shows "prod f {.. prod f {.. prodinf f"  lp15@68361  1039  using prod_le_prodinf[of f _ "Suc i"]  lp15@68361  1040  by (meson "0" "1" Suc_leD convergent_prod_has_prod f \n \ i\ le_trans less_eq_real_def)  lp15@68361  1041  ultimately show ?thesis  lp15@68361  1042  by (metis le_less_trans mult.commute not_le prod_lessThan_Suc)  lp15@68361  1043 qed  lp15@68361  1044 lp15@68361  1045 lemma prod_less_prodinf:  lp15@68361  1046  fixes f :: "nat \ real"  lp15@68361  1047  assumes f: "convergent_prod f" and 1: "\m. m\n \ 1 < f m" and 0: "\m. 0 < f m"  lp15@68361  1048  shows "prod f {.. real"  lp15@68361  1053  assumes pos: "\n. 1 \ f n"  lp15@68361  1054  and le: "\n. (\i x"  lp15@68361  1055  shows "\p. raw_has_prod f 0 p"  lp15@68361  1056  unfolding raw_has_prod_def add_0_right  lp15@68361  1057 proof (rule exI LIMSEQ_incseq_SUP conjI)+  lp15@68361  1058  show "bdd_above (range (\n. prod f {..n}))"  lp15@68361  1059  by (metis bdd_aboveI2 le lessThan_Suc_atMost)  lp15@68361  1060  then have "(SUP i. prod f {..i}) > 0"  lp15@68361  1061  by (metis UNIV_I cSUP_upper less_le_trans pos prod_pos zero_less_one)  lp15@68361  1062  then show "(SUP i. prod f {..i}) \ 0"  lp15@68361  1063  by auto  lp15@68361  1064  show "incseq (\n. prod f {..n})"  lp15@68361  1065  using pos order_trans [OF zero_le_one] by (auto simp: mono_def intro!: prod_mono2)  lp15@68361  1066 qed  lp15@68361  1067 lp15@68361  1068 lemma convergent_prodI_nonneg_bounded:  lp15@68361  1069  fixes f :: "nat \ real"  lp15@68361  1070  assumes "\n. 1 \ f n" "\n. (\i x"  lp15@68361  1071  shows "convergent_prod f"  lp15@68361  1072  using convergent_prod_def raw_has_prodI_bounded [OF assms] by blast  lp15@68361  1073 lp15@68361  1074 lp15@68424  1075 subsection \Infinite products on topological spaces\  lp15@68361  1076 lp15@68361  1077 context  lp15@68361  1078  fixes f g :: "nat \ 'a::{t2_space,topological_semigroup_mult,idom}"  lp15@68361  1079 begin  lp15@68361  1080 lp15@68361  1081 lemma raw_has_prod_mult: "\raw_has_prod f M a; raw_has_prod g M b\ \ raw_has_prod (\n. f n * g n) M (a * b)"  lp15@68361  1082  by (force simp add: prod.distrib tendsto_mult raw_has_prod_def)  lp15@68361  1083 lp15@68361  1084 lemma has_prod_mult_nz: "\f has_prod a; g has_prod b; a \ 0; b \ 0\ \ (\n. f n * g n) has_prod (a * b)"  lp15@68361  1085  by (simp add: raw_has_prod_mult has_prod_def)  lp15@68361  1086 lp15@68361  1087 end  lp15@68361  1088 lp15@68361  1089 lp15@68361  1090 context  lp15@68361  1091  fixes f g :: "nat \ 'a::real_normed_field"  lp15@68361  1092 begin  lp15@68361  1093 lp15@68361  1094 lemma has_prod_mult:  lp15@68361  1095  assumes f: "f has_prod a" and g: "g has_prod b"  lp15@68361  1096  shows "(\n. f n * g n) has_prod (a * b)"  lp15@68361  1097  using f [unfolded has_prod_def]  lp15@68361  1098 proof (elim disjE exE conjE)  lp15@68361  1099  assume f0: "raw_has_prod f 0 a"  lp15@68361  1100  show ?thesis  lp15@68361  1101  using g [unfolded has_prod_def]  lp15@68361  1102  proof (elim disjE exE conjE)  lp15@68361  1103  assume g0: "raw_has_prod g 0 b"  lp15@68361  1104  with f0 show ?thesis  lp15@68361  1105  by (force simp add: has_prod_def prod.distrib tendsto_mult raw_has_prod_def)  lp15@68361  1106  next  lp15@68361  1107  fix j q  lp15@68361  1108  assume "b = 0" and "g j = 0" and q: "raw_has_prod g (Suc j) q"  lp15@68361  1109  obtain p where p: "raw_has_prod f (Suc j) p"  lp15@68361  1110  using f0 raw_has_prod_ignore_initial_segment by blast  lp15@68361  1111  then have "Ex (raw_has_prod (\n. f n * g n) (Suc j))"  lp15@68361  1112  using q raw_has_prod_mult by blast  lp15@68361  1113  then show ?thesis  lp15@68361  1114  using \b = 0\ \g j = 0\ has_prod_0_iff by fastforce  lp15@68361  1115  qed  lp15@68361  1116 next  lp15@68361  1117  fix i p  lp15@68361  1118  assume "a = 0" and "f i = 0" and p: "raw_has_prod f (Suc i) p"  lp15@68361  1119  show ?thesis  lp15@68361  1120  using g [unfolded has_prod_def]  lp15@68361  1121  proof (elim disjE exE conjE)  lp15@68361  1122  assume g0: "raw_has_prod g 0 b"  lp15@68361  1123  obtain q where q: "raw_has_prod g (Suc i) q"  lp15@68361  1124  using g0 raw_has_prod_ignore_initial_segment by blast  lp15@68361  1125  then have "Ex (raw_has_prod (\n. f n * g n) (Suc i))"  lp15@68361  1126  using raw_has_prod_mult p by blast  lp15@68361  1127  then show ?thesis  lp15@68361  1128  using \a = 0\ \f i = 0\ has_prod_0_iff by fastforce  lp15@68361  1129  next  lp15@68361  1130  fix j q  lp15@68361  1131  assume "b = 0" and "g j = 0" and q: "raw_has_prod g (Suc j) q"  lp15@68361  1132  obtain p' where p': "raw_has_prod f (Suc (max i j)) p'"  lp15@68361  1133  by (metis raw_has_prod_ignore_initial_segment max_Suc_Suc max_def p)  lp15@68361  1134  moreover  lp15@68361  1135  obtain q' where q': "raw_has_prod g (Suc (max i j)) q'"  lp15@68361  1136  by (metis raw_has_prod_ignore_initial_segment max.cobounded2 max_Suc_Suc q)  lp15@68361  1137  ultimately show ?thesis  lp15@68361  1138  using \b = 0\ by (simp add: has_prod_def) (metis \f i = 0\ \g j = 0\ raw_has_prod_mult max_def)  lp15@68361  1139  qed  lp15@68361  1140 qed  lp15@68361  1141 lp15@68361  1142 lemma convergent_prod_mult:  lp15@68361  1143  assumes f: "convergent_prod f" and g: "convergent_prod g"  lp15@68361  1144  shows "convergent_prod (\n. f n * g n)"  lp15@68361  1145  unfolding convergent_prod_def  lp15@68361  1146 proof -  lp15@68361  1147  obtain M p N q where p: "raw_has_prod f M p" and q: "raw_has_prod g N q"  lp15@68361  1148  using convergent_prod_def f g by blast+  lp15@68361  1149  then obtain p' q' where p': "raw_has_prod f (max M N) p'" and q': "raw_has_prod g (max M N) q'"  lp15@68361  1150  by (meson raw_has_prod_ignore_initial_segment max.cobounded1 max.cobounded2)  lp15@68361  1151  then show "\M p. raw_has_prod (\n. f n * g n) M p"  lp15@68361  1152  using raw_has_prod_mult by blast  lp15@68361  1153 qed  lp15@68361  1154 lp15@68361  1155 lemma prodinf_mult: "convergent_prod f \ convergent_prod g \ prodinf f * prodinf g = (\n. f n * g n)"  lp15@68361  1156  by (intro has_prod_unique has_prod_mult convergent_prod_has_prod)  lp15@68361  1157 lp15@68361  1158 end  lp15@68361  1159 lp15@68361  1160 context  lp15@68361  1161  fixes f :: "'i \ nat \ 'a::real_normed_field"  lp15@68361  1162  and I :: "'i set"  lp15@68361  1163 begin  lp15@68361  1164 lp15@68361  1165 lemma has_prod_prod: "(\i. i \ I \ (f i) has_prod (x i)) \ (\n. \i\I. f i n) has_prod (\i\I. x i)"  lp15@68361  1166  by (induct I rule: infinite_finite_induct) (auto intro!: has_prod_mult)  lp15@68361  1167 lp15@68361  1168 lemma prodinf_prod: "(\i. i \ I \ convergent_prod (f i)) \ (\n. \i\I. f i n) = (\i\I. \n. f i n)"  lp15@68361  1169  using has_prod_unique[OF has_prod_prod, OF convergent_prod_has_prod] by simp  lp15@68361  1170 lp15@68361  1171 lemma convergent_prod_prod: "(\i. i \ I \ convergent_prod (f i)) \ convergent_prod (\n. \i\I. f i n)"  lp15@68361  1172  using convergent_prod_has_prod_iff has_prod_prod prodinf_prod by force  lp15@68361  1173 lp15@68361  1174 end  lp15@68361  1175 lp15@68424  1176 subsection \Infinite summability on real normed fields\  lp15@68361  1177 lp15@68361  1178 context  lp15@68361  1179  fixes f :: "nat \ 'a::real_normed_field"  lp15@68361  1180 begin  lp15@68361  1181 lp15@68361  1182 lemma raw_has_prod_Suc_iff: "raw_has_prod f M (a * f M) \ raw_has_prod (\n. f (Suc n)) M a \ f M \ 0"  lp15@68361  1183 proof -  lp15@68361  1184  have "raw_has_prod f M (a * f M) \ (\i. \j\Suc i. f (j+M)) \ a * f M \ a * f M \ 0"  lp15@68361  1185  by (subst LIMSEQ_Suc_iff) (simp add: raw_has_prod_def)  lp15@68361  1186  also have "\ \ (\i. (\j\i. f (Suc j + M)) * f M) \ a * f M \ a * f M \ 0"  lp15@68361  1187  by (simp add: ac_simps atMost_Suc_eq_insert_0 image_Suc_atMost prod_atLeast1_atMost_eq lessThan_Suc_atMost)  lp15@68361  1188  also have "\ \ raw_has_prod (\n. f (Suc n)) M a \ f M \ 0"  lp15@68361  1189  proof safe  lp15@68361  1190  assume tends: "(\i. (\j\i. f (Suc j + M)) * f M) \ a * f M" and 0: "a * f M \ 0"  lp15@68361  1191  with tendsto_divide[OF tends tendsto_const, of "f M"]  lp15@68361  1192  show "raw_has_prod (\n. f (Suc n)) M a"  lp15@68361  1193  by (simp add: raw_has_prod_def)  lp15@68361  1194  qed (auto intro: tendsto_mult_right simp: raw_has_prod_def)  lp15@68361  1195  finally show ?thesis .  lp15@68361  1196 qed  lp15@68361  1197 lp15@68361  1198 lemma has_prod_Suc_iff:  lp15@68361  1199  assumes "f 0 \ 0" shows "(\n. f (Suc n)) has_prod a \ f has_prod (a * f 0)"  lp15@68361  1200 proof (cases "a = 0")  lp15@68361  1201  case True  lp15@68361  1202  then show ?thesis  lp15@68361  1203  proof (simp add: has_prod_def, safe)  lp15@68361  1204  fix i x  lp15@68361  1205  assume "f (Suc i) = 0" and "raw_has_prod (\n. f (Suc n)) (Suc i) x"  lp15@68361  1206  then obtain y where "raw_has_prod f (Suc (Suc i)) y"  lp15@68361  1207  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)  lp15@68361  1208  then show "\i. f i = 0 \ Ex (raw_has_prod f (Suc i))"  lp15@68361  1209  using \f (Suc i) = 0\ by blast  lp15@68361  1210  next  lp15@68361  1211  fix i x  lp15@68361  1212  assume "f i = 0" and x: "raw_has_prod f (Suc i) x"  lp15@68361  1213  then obtain j where j: "i = Suc j"  lp15@68361  1214  by (metis assms not0_implies_Suc)  lp15@68361  1215  moreover have "\ y. raw_has_prod (\n. f (Suc n)) i y"  lp15@68361  1216  using x by (auto simp: raw_has_prod_def)  lp15@68361  1217  then show "\i. f (Suc i) = 0 \ Ex (raw_has_prod (\n. f (Suc n)) (Suc i))"  lp15@68361  1218  using \f i = 0\ j by blast  lp15@68361  1219  qed  lp15@68361  1220 next  lp15@68361  1221  case False  lp15@68361  1222  then show ?thesis  lp15@68361  1223  by (auto simp: has_prod_def raw_has_prod_Suc_iff assms)  lp15@68361  1224 qed  lp15@68361  1225 lp15@68361  1226 lemma convergent_prod_Suc_iff:  lp15@68452  1227  shows "convergent_prod (\n. f (Suc n)) = convergent_prod f"  lp15@68361  1228 proof  lp15@68361  1229  assume "convergent_prod f"  lp15@68452  1230  then obtain M L where M_nz:"\n\M. f n \ 0" and  lp15@68452  1231  M_L:"(\n. \i\n. f (i + M)) \ L" and "L \ 0"  lp15@68452  1232  unfolding convergent_prod_altdef by auto  lp15@68452  1233  have "(\n. \i\n. f (Suc (i + M))) \ L / f M"  lp15@68452  1234  proof -  lp15@68452  1235  have "(\n. \i\{0..Suc n}. f (i + M)) \ L"  lp15@68452  1236  using M_L  lp15@68452  1237  apply (subst (asm) LIMSEQ_Suc_iff[symmetric])  lp15@68452  1238  using atLeast0AtMost by auto  lp15@68452  1239  then have "(\n. f M * (\i\{0..n}. f (Suc (i + M)))) \ L"  lp15@68452  1240  apply (subst (asm) prod.atLeast0_atMost_Suc_shift)  lp15@68452  1241  by simp  lp15@68452  1242  then have "(\n. (\i\{0..n}. f (Suc (i + M)))) \ L/f M"  lp15@68452  1243  apply (drule_tac tendsto_divide)  lp15@68452  1244  using M_nz[rule_format,of M,simplified] by auto  lp15@68452  1245  then show ?thesis unfolding atLeast0AtMost .  lp15@68452  1246  qed  lp15@68452  1247  then show "convergent_prod (\n. f (Suc n))" unfolding convergent_prod_altdef  lp15@68452  1248  apply (rule_tac exI[where x=M])  lp15@68452  1249  apply (rule_tac exI[where x="L/f M"])  lp15@68452  1250  using M_nz \L\0\ by auto  lp15@68361  1251 next  lp15@68361  1252  assume "convergent_prod (\n. f (Suc n))"  lp15@68452  1253  then obtain M where "\L. (\n\M. f (Suc n) \ 0) \ (\n. \i\n. f (Suc (i + M))) \ L \ L \ 0"  lp15@68452  1254  unfolding convergent_prod_altdef by auto  lp15@68452  1255  then show "convergent_prod f" unfolding convergent_prod_altdef  lp15@68452  1256  apply (rule_tac exI[where x="Suc M"])  lp15@68452  1257  using Suc_le_D by auto  lp15@68361  1258 qed  lp15@68361  1259 lp15@68361  1260 lemma raw_has_prod_inverse:  lp15@68361  1261  assumes "raw_has_prod f M a" shows "raw_has_prod (\n. inverse (f n)) M (inverse a)"  lp15@68361  1262  using assms unfolding raw_has_prod_def by (auto dest: tendsto_inverse simp: prod_inversef [symmetric])  lp15@68361  1263 lp15@68361  1264 lemma has_prod_inverse:  lp15@68361  1265  assumes "f has_prod a" shows "(\n. inverse (f n)) has_prod (inverse a)"  lp15@68361  1266 using assms raw_has_prod_inverse unfolding has_prod_def by auto  lp15@68361  1267 lp15@68361  1268 lemma convergent_prod_inverse:  lp15@68361  1269  assumes "convergent_prod f"  lp15@68361  1270  shows "convergent_prod (\n. inverse (f n))"  lp15@68361  1271  using assms unfolding convergent_prod_def by (blast intro: raw_has_prod_inverse elim: )  lp15@68361  1272 lp15@68361  1273 end  lp15@68361  1274 lp15@68424  1275 context  lp15@68361  1276  fixes f :: "nat \ 'a::real_normed_field"  lp15@68361  1277 begin  lp15@68361  1278 lp15@68361  1279 lemma raw_has_prod_Suc_iff': "raw_has_prod f M a \ raw_has_prod (\n. f (Suc n)) M (a / f M) \ f M \ 0"  lp15@68361  1280  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)  lp15@68361  1281 lp15@68361  1282 lemma has_prod_divide: "f has_prod a \ g has_prod b \ (\n. f n / g n) has_prod (a / b)"  lp15@68361  1283  unfolding divide_inverse by (intro has_prod_inverse has_prod_mult)  lp15@68361  1284 lp15@68361  1285 lemma convergent_prod_divide:  lp15@68361  1286  assumes f: "convergent_prod f" and g: "convergent_prod g"  lp15@68361  1287  shows "convergent_prod (\n. f n / g n)"  lp15@68361  1288  using f g has_prod_divide has_prod_iff by blast  lp15@68361  1289 lp15@68361  1290 lemma prodinf_divide: "convergent_prod f \ convergent_prod g \ prodinf f / prodinf g = (\n. f n / g n)"  lp15@68361  1291  by (intro has_prod_unique has_prod_divide convergent_prod_has_prod)  lp15@68361  1292 lp15@68361  1293 lemma prodinf_inverse: "convergent_prod f \ (\n. inverse (f n)) = inverse (\n. f n)"  lp15@68361  1294  by (intro has_prod_unique [symmetric] has_prod_inverse convergent_prod_has_prod)  lp15@68361  1295 lp15@68452  1296 lemma has_prod_Suc_imp:  lp15@68452  1297  assumes "(\n. f (Suc n)) has_prod a"  lp15@68452  1298  shows "f has_prod (a * f 0)"  lp15@68452  1299 proof -  lp15@68452  1300  have "f has_prod (a * f 0)" when "raw_has_prod (\n. f (Suc n)) 0 a"  lp15@68452  1301  apply (cases "f 0=0")  lp15@68452  1302  using that unfolding has_prod_def raw_has_prod_Suc  lp15@68452  1303  by (auto simp add: raw_has_prod_Suc_iff)  lp15@68452  1304  moreover have "f has_prod (a * f 0)" when  lp15@68452  1305  "(\i q. a = 0 \ f (Suc i) = 0 \ raw_has_prod (\n. f (Suc n)) (Suc i) q)"  lp15@68452  1306  proof -  lp15@68452  1307  from that  lp15@68452  1308  obtain i q where "a = 0" "f (Suc i) = 0" "raw_has_prod (\n. f (Suc n)) (Suc i) q"  lp15@68452  1309  by auto  lp15@68452  1310  then show ?thesis unfolding has_prod_def  lp15@68452  1311  by (auto intro!:exI[where x="Suc i"] simp:raw_has_prod_Suc)  lp15@68452  1312  qed  lp15@68452  1313  ultimately show "f has_prod (a * f 0)" using assms unfolding has_prod_def by auto  lp15@68452  1314 qed  lp15@68452  1315 lp15@68361  1316 lemma has_prod_iff_shift:  lp15@68361  1317  assumes "\i. i < n \ f i \ 0"  lp15@68361  1318  shows "(\i. f (i + n)) has_prod a \ f has_prod (a * (\ii. f (Suc i + n)) has_prod a \ (\i. f (i + n)) has_prod (a * f n)"  lp15@68361  1326  by (subst has_prod_Suc_iff) auto  lp15@68361  1327  with Suc show ?case  lp15@68361  1328  by (simp add: ac_simps)  lp15@68361  1329 qed  lp15@68361  1330 lp15@68361  1331 corollary has_prod_iff_shift':  lp15@68361  1332  assumes "\i. i < n \ f i \ 0"  lp15@68361  1333  shows "(\i. f (i + n)) has_prod (a / (\i f has_prod a"  lp15@68361  1334  by (simp add: assms has_prod_iff_shift)  lp15@68361  1335 lp15@68361  1336 lemma has_prod_one_iff_shift:  lp15@68361  1337  assumes "\i. i < n \ f i = 1"  lp15@68361  1338  shows "(\i. f (i+n)) has_prod a \ (\i. f i) has_prod a"  lp15@68361  1339  by (simp add: assms has_prod_iff_shift)  lp15@68361  1340 lp15@68361  1341 lemma convergent_prod_iff_shift:  lp15@68361  1342  shows "convergent_prod (\i. f (i + n)) \ convergent_prod f"  lp15@68361  1343  apply safe  lp15@68361  1344  using convergent_prod_offset apply blast  lp15@68361  1345  using convergent_prod_ignore_initial_segment convergent_prod_def by blast  lp15@68361  1346 lp15@68361  1347 lemma has_prod_split_initial_segment:  lp15@68361  1348  assumes "f has_prod a" "\i. i < n \ f i \ 0"  lp15@68361  1349  shows "(\i. f (i + n)) has_prod (a / (\ii. i < n \ f i \ 0"  lp15@68361  1354  shows "(\i. f (i + n)) = (\i. f i) / (\ii. i < n \ f i \ 0"  lp15@68361  1359  shows "prodinf f = (\i. f (i + n)) * (\i 0"  lp15@68361  1364  shows "(\n. f (Suc n)) = prodinf f / f 0"  lp15@68361  1365  using prodinf_split_initial_segment[of 1] assms by simp  lp15@68361  1366 lp15@68361  1367 end  lp15@68361  1368 lp15@68424  1369 context  lp15@68361  1370  fixes f :: "nat \ 'a::real_normed_field"  lp15@68361  1371 begin  lp15@68361  1372 lp15@68361  1373 lemma convergent_prod_inverse_iff: "convergent_prod (\n. inverse (f n)) \ convergent_prod f"  lp15@68361  1374  by (auto dest: convergent_prod_inverse)  lp15@68361  1375 lp15@68361  1376 lemma convergent_prod_const_iff:  lp15@68361  1377  fixes c :: "'a :: {real_normed_field}"  lp15@68361  1378  shows "convergent_prod (\_. c) \ c = 1"  lp15@68361  1379 proof  lp15@68361  1380  assume "convergent_prod (\_. c)"  lp15@68361  1381  then show "c = 1"  lp15@68361  1382  using convergent_prod_imp_LIMSEQ LIMSEQ_unique by blast  lp15@68361  1383 next  lp15@68361  1384  assume "c = 1"  lp15@68361  1385  then show "convergent_prod (\_. c)"  lp15@68361  1386  by auto  lp15@68361  1387 qed  lp15@68361  1388 lp15@68361  1389 lemma has_prod_power: "f has_prod a \ (\i. f i ^ n) has_prod (a ^ n)"  lp15@68361  1390  by (induction n) (auto simp: has_prod_mult)  lp15@68361  1391 lp15@68361  1392 lemma convergent_prod_power: "convergent_prod f \ convergent_prod (\i. f i ^ n)"  lp15@68361  1393  by (induction n) (auto simp: convergent_prod_mult)  lp15@68361  1394 lp15@68361  1395 lemma prodinf_power: "convergent_prod f \ prodinf (\i. f i ^ n) = prodinf f ^ n"  lp15@68361  1396  by (metis has_prod_unique convergent_prod_imp_has_prod has_prod_power)  lp15@68361  1397 lp15@68361  1398 end  lp15@68361  1399 lp15@68424  1400 lp15@68424  1401 subsection\Exponentials and logarithms\  lp15@68424  1402 lp15@68424  1403 context  lp15@68424  1404  fixes f :: "nat \ 'a::{real_normed_field,banach}"  lp15@68424  1405 begin  lp15@68424  1406 lp15@68424  1407 lemma sums_imp_has_prod_exp:  lp15@68424  1408  assumes "f sums s"  lp15@68424  1409  shows "raw_has_prod (\i. exp (f i)) 0 (exp s)"  lp15@68424  1410  using assms continuous_on_exp [of UNIV "\x::'a. x"]  lp15@68424  1411  using continuous_on_tendsto_compose [of UNIV exp "(\n. sum f {..n})" s]  lp15@68424  1412  by (simp add: prod_defs sums_def_le exp_sum)  lp15@68424  1413 lp15@68424  1414 lemma convergent_prod_exp:  lp15@68424  1415  assumes "summable f"  lp15@68424  1416  shows "convergent_prod (\i. exp (f i))"  lp15@68424  1417  using sums_imp_has_prod_exp assms unfolding summable_def convergent_prod_def by blast  lp15@68424  1418 lp15@68424  1419 lemma prodinf_exp:  lp15@68424  1420  assumes "summable f"  lp15@68424  1421  shows "prodinf (\i. exp (f i)) = exp (suminf f)"  lp15@68424  1422 proof -  lp15@68424  1423  have "f sums suminf f"  lp15@68424  1424  using assms by blast  lp15@68424  1425  then have "(\i. exp (f i)) has_prod exp (suminf f)"  lp15@68424  1426  by (simp add: has_prod_def sums_imp_has_prod_exp)  lp15@68424  1427  then show ?thesis  lp15@68424  1428  by (rule has_prod_unique [symmetric])  lp15@68424  1429 qed  lp15@68424  1430 lp15@68361  1431 end  lp15@68424  1432 lp15@68452  1433 lemma exp_suminf_prodinf_real:  lp15@68452  1434  fixes f :: "nat \ real"  lp15@68452  1435  assumes ge0:"\n. f n \ 0" and ac: "abs_convergent_prod (\n. exp (f n))"  lp15@68452  1436  shows "prodinf (\i. exp (f i)) = exp (suminf f)"  lp15@68452  1437 proof -  lp15@68517  1438  have "summable f"  lp15@68452  1439  using ac unfolding abs_convergent_prod_conv_summable  lp15@68452  1440  proof (elim summable_comparison_test')  lp15@68452  1441  fix n  lp15@68517  1442  have "\f n\ = f n"  lp15@68517  1443  by (simp add: ge0)  lp15@68517  1444  also have "\ \ exp (f n) - 1"  lp15@68517  1445  by (metis diff_diff_add exp_ge_add_one_self ge_iff_diff_ge_0)  lp15@68517  1446  finally show "norm (f n) \ norm (exp (f n) - 1)"  lp15@68517  1447  by simp  lp15@68452  1448  qed  lp15@68452  1449  then show ?thesis  lp15@68452  1450  by (simp add: prodinf_exp)  lp15@68452  1451 qed  lp15@68452  1452 lp15@68424  1453 lemma has_prod_imp_sums_ln_real:  lp15@68424  1454  fixes f :: "nat \ real"  lp15@68424  1455  assumes "raw_has_prod f 0 p" and 0: "\x. f x > 0"  lp15@68424  1456  shows "(\i. ln (f i)) sums (ln p)"  lp15@68424  1457 proof -  lp15@68424  1458  have "p > 0"  lp15@68424  1459  using assms unfolding prod_defs by (metis LIMSEQ_prod_nonneg less_eq_real_def)  lp15@68424  1460  then show ?thesis  lp15@68424  1461  using assms continuous_on_ln [of "{0<..}" "\x. x"]  lp15@68424  1462  using continuous_on_tendsto_compose [of "{0<..}" ln "(\n. prod f {..n})" p]  lp15@68424  1463  by (auto simp: prod_defs sums_def_le ln_prod order_tendstoD)  lp15@68424  1464 qed  lp15@68424  1465 lp15@68424  1466 lemma summable_ln_real:  lp15@68424  1467  fixes f :: "nat \ real"  lp15@68424  1468  assumes f: "convergent_prod f" and 0: "\x. f x > 0"  lp15@68424  1469  shows "summable (\i. ln (f i))"  lp15@68424  1470 proof -  lp15@68424  1471  obtain M p where "raw_has_prod f M p"  lp15@68424  1472  using f convergent_prod_def by blast  lp15@68424  1473  then consider i where "i real"  lp15@68424  1489  assumes f: "convergent_prod f" and 0: "\x. f x > 0"  lp15@68424  1490  shows "suminf (\i. ln (f i)) = ln (prodinf f)"  lp15@68424  1491 proof -  lp15@68424  1492  have "f has_prod prodinf f"  lp15@68424  1493  by (simp add: f has_prod_iff)  lp15@68424  1494  then have "raw_has_prod f 0 (prodinf f)"  lp15@68424  1495  by (metis "0" has_prod_def less_irrefl)  lp15@68424  1496  then have "(\i. ln (f i)) sums ln (prodinf f)"  lp15@68424  1497  using "0" has_prod_imp_sums_ln_real by blast  lp15@68424  1498  then show ?thesis  lp15@68424  1499  by (rule sums_unique [symmetric])  lp15@68424  1500 qed  lp15@68424  1501 lp15@68424  1502 lemma prodinf_exp_real:  lp15@68424  1503  fixes f :: "nat \ real"  lp15@68424  1504  assumes f: "convergent_prod f" and 0: "\x. f x > 0"  lp15@68424  1505  shows "prodinf f = exp (suminf (\i. ln (f i)))"  lp15@68424  1506  by (simp add: "0" f less_0_prodinf suminf_ln_real)  lp15@68424  1507 lp15@68424  1508 lp15@68424  1509 subsection\Embeddings from the reals into some complete real normed field\  lp15@68424  1510 lp15@68426  1511 lemma tendsto_eq_of_real_lim:  lp15@68424  1512  assumes "(\n. of_real (f n) :: 'a::{complete_space,real_normed_field}) \ q"  lp15@68424  1513  shows "q = of_real (lim f)"  lp15@68424  1514 proof -  lp15@68424  1515  have "convergent (\n. of_real (f n) :: 'a)"  lp15@68424  1516  using assms convergent_def by blast  lp15@68424  1517  then have "convergent f"  lp15@68424  1518  unfolding convergent_def  lp15@68424  1519  by (simp add: convergent_eq_Cauchy Cauchy_def)  lp15@68424  1520  then show ?thesis  lp15@68424  1521  by (metis LIMSEQ_unique assms convergentD sequentially_bot tendsto_Lim tendsto_of_real)  lp15@68424  1522 qed  lp15@68424  1523 lp15@68426  1524 lemma tendsto_eq_of_real:  lp15@68424  1525  assumes "(\n. of_real (f n) :: 'a::{complete_space,real_normed_field}) \ q"  lp15@68424  1526  obtains r where "q = of_real r"  lp15@68426  1527  using tendsto_eq_of_real_lim assms by blast  lp15@68424  1528 lp15@68424  1529 lemma has_prod_of_real_iff:  lp15@68424  1530  "(\n. of_real (f n) :: 'a::{complete_space,real_normed_field}) has_prod of_real c \ f has_prod c"  lp15@68424  1531  (is "?lhs = ?rhs")  lp15@68424  1532 proof  lp15@68424  1533  assume ?lhs  lp15@68424  1534  then show ?rhs  lp15@68424  1535  apply (auto simp: prod_defs LIMSEQ_prod_0 tendsto_of_real_iff simp flip: of_real_prod)  lp15@68426  1536  using tendsto_eq_of_real  lp15@68424  1537  by (metis of_real_0 tendsto_of_real_iff)  lp15@68424  1538 next  lp15@68424  1539  assume ?rhs  lp15@68424  1540  with tendsto_of_real_iff show ?lhs  lp15@68424  1541  by (fastforce simp: prod_defs simp flip: of_real_prod)  lp15@68424  1542 qed  lp15@68424  1543 lp15@68424  1544 end `