src/HOL/Probability/Projective_Limit.thy
 author hoelzl Fri Feb 19 13:40:50 2016 +0100 (2016-02-19) changeset 62378 85ed00c1fe7c parent 61973 0c7e865fa7cb child 62397 5ae24f33d343 permissions -rw-r--r--
generalize more theorems to support enat and ennreal
```     1 (*  Title:      HOL/Probability/Projective_Limit.thy
```
```     2     Author:     Fabian Immler, TU München
```
```     3 *)
```
```     4
```
```     5 section \<open>Projective Limit\<close>
```
```     6
```
```     7 theory Projective_Limit
```
```     8   imports
```
```     9     Caratheodory
```
```    10     Fin_Map
```
```    11     Regularity
```
```    12     Projective_Family
```
```    13     Infinite_Product_Measure
```
```    14     "~~/src/HOL/Library/Diagonal_Subsequence"
```
```    15 begin
```
```    16
```
```    17 subsection \<open>Sequences of Finite Maps in Compact Sets\<close>
```
```    18
```
```    19 locale finmap_seqs_into_compact =
```
```    20   fixes K::"nat \<Rightarrow> (nat \<Rightarrow>\<^sub>F 'a::metric_space) set" and f::"nat \<Rightarrow> (nat \<Rightarrow>\<^sub>F 'a)" and M
```
```    21   assumes compact: "\<And>n. compact (K n)"
```
```    22   assumes f_in_K: "\<And>n. K n \<noteq> {}"
```
```    23   assumes domain_K: "\<And>n. k \<in> K n \<Longrightarrow> domain k = domain (f n)"
```
```    24   assumes proj_in_K:
```
```    25     "\<And>t n m. m \<ge> n \<Longrightarrow> t \<in> domain (f n) \<Longrightarrow> (f m)\<^sub>F t \<in> (\<lambda>k. (k)\<^sub>F t) ` K n"
```
```    26 begin
```
```    27
```
```    28 lemma proj_in_K': "(\<exists>n. \<forall>m \<ge> n. (f m)\<^sub>F t \<in> (\<lambda>k. (k)\<^sub>F t) ` K n)"
```
```    29   using proj_in_K f_in_K
```
```    30 proof cases
```
```    31   obtain k where "k \<in> K (Suc 0)" using f_in_K by auto
```
```    32   assume "\<forall>n. t \<notin> domain (f n)"
```
```    33   thus ?thesis
```
```    34     by (auto intro!: exI[where x=1] image_eqI[OF _ \<open>k \<in> K (Suc 0)\<close>]
```
```    35       simp: domain_K[OF \<open>k \<in> K (Suc 0)\<close>])
```
```    36 qed blast
```
```    37
```
```    38 lemma proj_in_KE:
```
```    39   obtains n where "\<And>m. m \<ge> n \<Longrightarrow> (f m)\<^sub>F t \<in> (\<lambda>k. (k)\<^sub>F t) ` K n"
```
```    40   using proj_in_K' by blast
```
```    41
```
```    42 lemma compact_projset:
```
```    43   shows "compact ((\<lambda>k. (k)\<^sub>F i) ` K n)"
```
```    44   using continuous_proj compact by (rule compact_continuous_image)
```
```    45
```
```    46 end
```
```    47
```
```    48 lemma compactE':
```
```    49   fixes S :: "'a :: metric_space set"
```
```    50   assumes "compact S" "\<forall>n\<ge>m. f n \<in> S"
```
```    51   obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) \<longlongrightarrow> l) sequentially"
```
```    52 proof atomize_elim
```
```    53   have "subseq (op + m)" by (simp add: subseq_def)
```
```    54   have "\<forall>n. (f o (\<lambda>i. m + i)) n \<in> S" using assms by auto
```
```    55   from seq_compactE[OF \<open>compact S\<close>[unfolded compact_eq_seq_compact_metric] this] guess l r .
```
```    56   hence "l \<in> S" "subseq ((\<lambda>i. m + i) o r) \<and> (f \<circ> ((\<lambda>i. m + i) o r)) \<longlonglongrightarrow> l"
```
```    57     using subseq_o[OF \<open>subseq (op + m)\<close> \<open>subseq r\<close>] by (auto simp: o_def)
```
```    58   thus "\<exists>l r. l \<in> S \<and> subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> l" by blast
```
```    59 qed
```
```    60
```
```    61 sublocale finmap_seqs_into_compact \<subseteq> subseqs "\<lambda>n s. (\<exists>l. (\<lambda>i. ((f o s) i)\<^sub>F n) \<longlonglongrightarrow> l)"
```
```    62 proof
```
```    63   fix n s
```
```    64   assume "subseq s"
```
```    65   from proj_in_KE[of n] guess n0 . note n0 = this
```
```    66   have "\<forall>i \<ge> n0. ((f \<circ> s) i)\<^sub>F n \<in> (\<lambda>k. (k)\<^sub>F n) ` K n0"
```
```    67   proof safe
```
```    68     fix i assume "n0 \<le> i"
```
```    69     also have "\<dots> \<le> s i" by (rule seq_suble) fact
```
```    70     finally have "n0 \<le> s i" .
```
```    71     with n0 show "((f \<circ> s) i)\<^sub>F n \<in> (\<lambda>k. (k)\<^sub>F n) ` K n0 "
```
```    72       by auto
```
```    73   qed
```
```    74   from compactE'[OF compact_projset this] guess ls rs .
```
```    75   thus "\<exists>r'. subseq r' \<and> (\<exists>l. (\<lambda>i. ((f \<circ> (s \<circ> r')) i)\<^sub>F n) \<longlonglongrightarrow> l)" by (auto simp: o_def)
```
```    76 qed
```
```    77
```
```    78 lemma (in finmap_seqs_into_compact) diagonal_tendsto: "\<exists>l. (\<lambda>i. (f (diagseq i))\<^sub>F n) \<longlonglongrightarrow> l"
```
```    79 proof -
```
```    80   obtain l where "(\<lambda>i. ((f o (diagseq o op + (Suc n))) i)\<^sub>F n) \<longlonglongrightarrow> l"
```
```    81   proof (atomize_elim, rule diagseq_holds)
```
```    82     fix r s n
```
```    83     assume "subseq r"
```
```    84     assume "\<exists>l. (\<lambda>i. ((f \<circ> s) i)\<^sub>F n) \<longlonglongrightarrow> l"
```
```    85     then obtain l where "((\<lambda>i. (f i)\<^sub>F n) o s) \<longlonglongrightarrow> l"
```
```    86       by (auto simp: o_def)
```
```    87     hence "((\<lambda>i. (f i)\<^sub>F n) o s o r) \<longlonglongrightarrow> l" using \<open>subseq r\<close>
```
```    88       by (rule LIMSEQ_subseq_LIMSEQ)
```
```    89     thus "\<exists>l. (\<lambda>i. ((f \<circ> (s \<circ> r)) i)\<^sub>F n) \<longlonglongrightarrow> l" by (auto simp add: o_def)
```
```    90   qed
```
```    91   hence "(\<lambda>i. ((f (diagseq (i + Suc n))))\<^sub>F n) \<longlonglongrightarrow> l" by (simp add: ac_simps)
```
```    92   hence "(\<lambda>i. (f (diagseq i))\<^sub>F n) \<longlonglongrightarrow> l" by (rule LIMSEQ_offset)
```
```    93   thus ?thesis ..
```
```    94 qed
```
```    95
```
```    96 subsection \<open>Daniell-Kolmogorov Theorem\<close>
```
```    97
```
```    98 text \<open>Existence of Projective Limit\<close>
```
```    99
```
```   100 locale polish_projective = projective_family I P "\<lambda>_. borel::'a::polish_space measure"
```
```   101   for I::"'i set" and P
```
```   102 begin
```
```   103
```
```   104 lemma emeasure_lim_emb:
```
```   105   assumes X: "J \<subseteq> I" "finite J" "X \<in> sets (\<Pi>\<^sub>M i\<in>J. borel)"
```
```   106   shows "lim (emb I J X) = P J X"
```
```   107 proof (rule emeasure_lim)
```
```   108   write mu_G ("\<mu>G")
```
```   109   interpret generator: algebra "space (PiM I (\<lambda>i. borel))" generator
```
```   110     by (rule algebra_generator)
```
```   111
```
```   112   fix J and B :: "nat \<Rightarrow> ('i \<Rightarrow> 'a) set"
```
```   113   assume J: "\<And>n. finite (J n)" "\<And>n. J n \<subseteq> I" "\<And>n. B n \<in> sets (\<Pi>\<^sub>M i\<in>J n. borel)" "incseq J"
```
```   114     and B: "decseq (\<lambda>n. emb I (J n) (B n))"
```
```   115     and "0 < (INF i. P (J i) (B i))" (is "0 < ?a")
```
```   116   moreover have "?a \<le> 1"
```
```   117     using J by (auto intro!: INF_lower2[of 0] prob_space_P[THEN prob_space.measure_le_1])
```
```   118   ultimately obtain r where r: "?a = ereal r" "0 < r" "r \<le> 1"
```
```   119     by (cases "?a") auto
```
```   120   def Z \<equiv> "\<lambda>n. emb I (J n) (B n)"
```
```   121   have Z_mono: "n \<le> m \<Longrightarrow> Z m \<subseteq> Z n" for n m
```
```   122     unfolding Z_def using B[THEN antimonoD, of n m] .
```
```   123   have J_mono: "\<And>n m. n \<le> m \<Longrightarrow> J n \<subseteq> J m"
```
```   124     using \<open>incseq J\<close> by (force simp: incseq_def)
```
```   125   note [simp] = \<open>\<And>n. finite (J n)\<close>
```
```   126   interpret prob_space "P (J i)" for i using J prob_space_P by simp
```
```   127
```
```   128   have P_eq[simp]:
```
```   129       "sets (P (J i)) = sets (\<Pi>\<^sub>M i\<in>J i. borel)" "space (P (J i)) = space (\<Pi>\<^sub>M i\<in>J i. borel)" for i
```
```   130     using J by (auto simp: sets_P space_P)
```
```   131
```
```   132   have "Z i \<in> generator" for i
```
```   133     unfolding Z_def by (auto intro!: generator.intros J)
```
```   134
```
```   135   have countable_UN_J: "countable (\<Union>n. J n)" by (simp add: countable_finite)
```
```   136   def Utn \<equiv> "to_nat_on (\<Union>n. J n)"
```
```   137   interpret function_to_finmap "J n" Utn "from_nat_into (\<Union>n. J n)" for n
```
```   138     by unfold_locales (auto simp: Utn_def intro: from_nat_into_to_nat_on[OF countable_UN_J])
```
```   139   have inj_on_Utn: "inj_on Utn (\<Union>n. J n)"
```
```   140     unfolding Utn_def using countable_UN_J by (rule inj_on_to_nat_on)
```
```   141   hence inj_on_Utn_J: "\<And>n. inj_on Utn (J n)" by (rule subset_inj_on) auto
```
```   142   def P' \<equiv> "\<lambda>n. mapmeasure n (P (J n)) (\<lambda>_. borel)"
```
```   143   interpret P': prob_space "P' n" for n
```
```   144     unfolding P'_def mapmeasure_def using J
```
```   145     by (auto intro!: prob_space_distr fm_measurable simp: measurable_cong_sets[OF sets_P])
```
```   146
```
```   147   let ?SUP = "\<lambda>n. SUP K : {K. K \<subseteq> fm n ` (B n) \<and> compact K}. emeasure (P' n) K"
```
```   148   { fix n
```
```   149     have "emeasure (P (J n)) (B n) = emeasure (P' n) (fm n ` (B n))"
```
```   150       using J by (auto simp: P'_def mapmeasure_PiM space_P sets_P)
```
```   151     also
```
```   152     have "\<dots> = ?SUP n"
```
```   153     proof (rule inner_regular)
```
```   154       show "sets (P' n) = sets borel" by (simp add: borel_eq_PiF_borel P'_def)
```
```   155     next
```
```   156       show "fm n ` B n \<in> sets borel"
```
```   157         unfolding borel_eq_PiF_borel by (auto simp: P'_def fm_image_measurable_finite sets_P J(3))
```
```   158     qed simp
```
```   159     finally have "emeasure (P (J n)) (B n) = ?SUP n" "?SUP n \<noteq> \<infinity>" "?SUP n \<noteq> - \<infinity>" by auto
```
```   160   } note R = this
```
```   161   have "\<forall>n. \<exists>K. emeasure (P (J n)) (B n) - emeasure (P' n) K \<le> 2 powr (-n) * ?a \<and> compact K \<and> K \<subseteq> fm n ` B n"
```
```   162   proof
```
```   163     fix n show "\<exists>K. emeasure (P (J n)) (B n) - emeasure (P' n) K \<le> ereal (2 powr - real n) * ?a \<and>
```
```   164         compact K \<and> K \<subseteq> fm n ` B n"
```
```   165       unfolding R[of n]
```
```   166     proof (rule ccontr)
```
```   167       assume H: "\<not> (\<exists>K'. ?SUP n - emeasure (P' n) K' \<le> ereal (2 powr - real n)  * ?a \<and>
```
```   168         compact K' \<and> K' \<subseteq> fm n ` B n)"
```
```   169       have "?SUP n \<le> ?SUP n - 2 powr (-n) * ?a"
```
```   170       proof (intro SUP_least)
```
```   171         fix K
```
```   172         assume "K \<in> {K. K \<subseteq> fm n ` B n \<and> compact K}"
```
```   173         with H have "\<not> ?SUP n - emeasure (P' n) K \<le> 2 powr (-n) * ?a"
```
```   174           by auto
```
```   175         hence "?SUP n - emeasure (P' n) K > 2 powr (-n) * ?a"
```
```   176           unfolding not_less[symmetric] by simp
```
```   177         hence "?SUP n - 2 powr (-n) * ?a > emeasure (P' n) K"
```
```   178           using \<open>0 < ?a\<close> by (auto simp add: ereal_less_minus_iff ac_simps)
```
```   179         thus "?SUP n - 2 powr (-n) * ?a \<ge> emeasure (P' n) K" by simp
```
```   180       qed
```
```   181       hence "?SUP n + 0 \<le> ?SUP n - (2 powr (-n) * ?a)" using \<open>0 < ?a\<close> by simp
```
```   182       hence "?SUP n + 0 \<le> ?SUP n + - (2 powr (-n) * ?a)" unfolding minus_ereal_def .
```
```   183       hence "0 \<le> - (2 powr (-n) * ?a)"
```
```   184         using \<open>?SUP n \<noteq> \<infinity>\<close> \<open>?SUP n \<noteq> - \<infinity>\<close>
```
```   185         by (subst (asm) ereal_add_le_add_iff) (auto simp:)
```
```   186       moreover have "ereal (2 powr - real n) * ?a > 0" using \<open>0 < ?a\<close>
```
```   187         by (auto simp: ereal_zero_less_0_iff)
```
```   188       ultimately show False by simp
```
```   189     qed
```
```   190   qed
```
```   191   then obtain K' where K':
```
```   192     "\<And>n. emeasure (P (J n)) (B n) - emeasure (P' n) (K' n) \<le> ereal (2 powr - real n) * ?a"
```
```   193     "\<And>n. compact (K' n)" "\<And>n. K' n \<subseteq> fm n ` B n"
```
```   194     unfolding choice_iff by blast
```
```   195   def K \<equiv> "\<lambda>n. fm n -` K' n \<inter> space (Pi\<^sub>M (J n) (\<lambda>_. borel))"
```
```   196   have K_sets: "\<And>n. K n \<in> sets (Pi\<^sub>M (J n) (\<lambda>_. borel))"
```
```   197     unfolding K_def
```
```   198     using compact_imp_closed[OF \<open>compact (K' _)\<close>]
```
```   199     by (intro measurable_sets[OF fm_measurable, of _ "Collect finite"])
```
```   200        (auto simp: borel_eq_PiF_borel[symmetric])
```
```   201   have K_B: "\<And>n. K n \<subseteq> B n"
```
```   202   proof
```
```   203     fix x n assume "x \<in> K n"
```
```   204     then have fm_in: "fm n x \<in> fm n ` B n"
```
```   205       using K' by (force simp: K_def)
```
```   206     show "x \<in> B n"
```
```   207       using \<open>x \<in> K n\<close> K_sets sets.sets_into_space J(1,2,3)[of n] inj_on_image_mem_iff[OF inj_on_fm]
```
```   208     by (metis (no_types) Int_iff K_def fm_in space_borel)
```
```   209   qed
```
```   210   def Z' \<equiv> "\<lambda>n. emb I (J n) (K n)"
```
```   211   have Z': "\<And>n. Z' n \<subseteq> Z n"
```
```   212     unfolding Z'_def Z_def
```
```   213   proof (rule prod_emb_mono, safe)
```
```   214     fix n x assume "x \<in> K n"
```
```   215     hence "fm n x \<in> K' n" "x \<in> space (Pi\<^sub>M (J n) (\<lambda>_. borel))"
```
```   216       by (simp_all add: K_def space_P)
```
```   217     note this(1)
```
```   218     also have "K' n \<subseteq> fm n ` B n" by (simp add: K')
```
```   219     finally have "fm n x \<in> fm n ` B n" .
```
```   220     thus "x \<in> B n"
```
```   221     proof safe
```
```   222       fix y assume y: "y \<in> B n"
```
```   223       hence "y \<in> space (Pi\<^sub>M (J n) (\<lambda>_. borel))" using J sets.sets_into_space[of "B n" "P (J n)"]
```
```   224         by (auto simp add: space_P sets_P)
```
```   225       assume "fm n x = fm n y"
```
```   226       note inj_onD[OF inj_on_fm[OF space_borel],
```
```   227         OF \<open>fm n x = fm n y\<close> \<open>x \<in> space _\<close> \<open>y \<in> space _\<close>]
```
```   228       with y show "x \<in> B n" by simp
```
```   229     qed
```
```   230   qed
```
```   231   have "\<And>n. Z' n \<in> generator" using J K'(2) unfolding Z'_def
```
```   232     by (auto intro!: generator.intros measurable_sets[OF fm_measurable[of _ "Collect finite"]]
```
```   233              simp: K_def borel_eq_PiF_borel[symmetric] compact_imp_closed)
```
```   234   def Y \<equiv> "\<lambda>n. \<Inter>i\<in>{1..n}. Z' i"
```
```   235   hence "\<And>n k. Y (n + k) \<subseteq> Y n" by (induct_tac k) (auto simp: Y_def)
```
```   236   hence Y_mono: "\<And>n m. n \<le> m \<Longrightarrow> Y m \<subseteq> Y n" by (auto simp: le_iff_add)
```
```   237   have Y_Z': "\<And>n. n \<ge> 1 \<Longrightarrow> Y n \<subseteq> Z' n" by (auto simp: Y_def)
```
```   238   hence Y_Z: "\<And>n. n \<ge> 1 \<Longrightarrow> Y n \<subseteq> Z n" using Z' by auto
```
```   239
```
```   240   have Y_notempty: "\<And>n. n \<ge> 1 \<Longrightarrow> (Y n) \<noteq> {}"
```
```   241   proof -
```
```   242     fix n::nat assume "n \<ge> 1" hence "Y n \<subseteq> Z n" by fact
```
```   243     have "Y n = (\<Inter>i\<in>{1..n}. emb I (J n) (emb (J n) (J i) (K i)))" using J J_mono
```
```   244       by (auto simp: Y_def Z'_def)
```
```   245     also have "\<dots> = prod_emb I (\<lambda>_. borel) (J n) (\<Inter>i\<in>{1..n}. emb (J n) (J i) (K i))"
```
```   246       using \<open>n \<ge> 1\<close>
```
```   247       by (subst prod_emb_INT) auto
```
```   248     finally
```
```   249     have Y_emb:
```
```   250       "Y n = prod_emb I (\<lambda>_. borel) (J n) (\<Inter>i\<in>{1..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))" .
```
```   251     hence "Y n \<in> generator" using J J_mono K_sets \<open>n \<ge> 1\<close>
```
```   252       by (auto simp del: prod_emb_INT intro!: generator.intros)
```
```   253     have *: "\<mu>G (Z n) = P (J n) (B n)"
```
```   254       unfolding Z_def using J by (intro mu_G_spec) auto
```
```   255     then have "\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>" by auto
```
```   256     note *
```
```   257     moreover have *: "\<mu>G (Y n) = P (J n) (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))"
```
```   258       unfolding Y_emb using J J_mono K_sets \<open>n \<ge> 1\<close> by (subst mu_G_spec) auto
```
```   259     then have "\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>" by auto
```
```   260     note *
```
```   261     moreover have "\<mu>G (Z n - Y n) =
```
```   262         P (J n) (B n - (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i)))"
```
```   263       unfolding Z_def Y_emb prod_emb_Diff[symmetric] using J J_mono K_sets \<open>n \<ge> 1\<close>
```
```   264       by (subst mu_G_spec) (auto intro!: sets.Diff)
```
```   265     ultimately
```
```   266     have "\<mu>G (Z n) - \<mu>G (Y n) = \<mu>G (Z n - Y n)"
```
```   267       using J J_mono K_sets \<open>n \<ge> 1\<close>
```
```   268       by (simp only: emeasure_eq_measure Z_def)
```
```   269         (auto dest!: bspec[where x=n] intro!: measure_Diff[symmetric] set_mp[OF K_B]
```
```   270           simp: extensional_restrict emeasure_eq_measure prod_emb_iff sets_P space_P)
```
```   271     also have subs: "Z n - Y n \<subseteq> (\<Union>i\<in>{1..n}. (Z i - Z' i))"
```
```   272       using \<open>n \<ge> 1\<close> unfolding Y_def UN_extend_simps(7) by (intro UN_mono Diff_mono Z_mono order_refl) auto
```
```   273     have "Z n - Y n \<in> generator" "(\<Union>i\<in>{1..n}. (Z i - Z' i)) \<in> generator"
```
```   274       using \<open>Z' _ \<in> generator\<close> \<open>Z _ \<in> generator\<close> \<open>Y _ \<in> generator\<close> by auto
```
```   275     hence "\<mu>G (Z n - Y n) \<le> \<mu>G (\<Union>i\<in>{1..n}. (Z i - Z' i))"
```
```   276       using subs generator.additive_increasing[OF positive_mu_G additive_mu_G]
```
```   277       unfolding increasing_def by auto
```
```   278     also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. \<mu>G (Z i - Z' i))" using \<open>Z _ \<in> generator\<close> \<open>Z' _ \<in> generator\<close>
```
```   279       by (intro generator.subadditive[OF positive_mu_G additive_mu_G]) auto
```
```   280     also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. 2 powr -real i * ?a)"
```
```   281     proof (rule setsum_mono)
```
```   282       fix i assume "i \<in> {1..n}" hence "i \<le> n" by simp
```
```   283       have "\<mu>G (Z i - Z' i) = \<mu>G (prod_emb I (\<lambda>_. borel) (J i) (B i - K i))"
```
```   284         unfolding Z'_def Z_def by simp
```
```   285       also have "\<dots> = P (J i) (B i - K i)"
```
```   286         using J K_sets by (subst mu_G_spec) auto
```
```   287       also have "\<dots> = P (J i) (B i) - P (J i) (K i)"
```
```   288         using K_sets J \<open>K _ \<subseteq> B _\<close> by (simp add: emeasure_Diff)
```
```   289       also have "\<dots> = P (J i) (B i) - P' i (K' i)"
```
```   290         unfolding K_def P'_def
```
```   291         by (auto simp: mapmeasure_PiF borel_eq_PiF_borel[symmetric]
```
```   292           compact_imp_closed[OF \<open>compact (K' _)\<close>] space_PiM PiE_def)
```
```   293       also have "\<dots> \<le> ereal (2 powr - real i) * ?a" using K'(1)[of i] .
```
```   294       finally show "\<mu>G (Z i - Z' i) \<le> (2 powr - real i) * ?a" .
```
```   295     qed
```
```   296     also have "\<dots> = ereal ((\<Sum> i\<in>{1..n}. (2 powr -real_of_ereal i)) * real_of_ereal ?a)"
```
```   297       by (simp add: setsum_left_distrib r)
```
```   298     also have "\<dots> < ereal (1 * real_of_ereal ?a)" unfolding less_ereal.simps
```
```   299     proof (rule mult_strict_right_mono)
```
```   300       have "(\<Sum>i = 1..n. 2 powr - real i) = (\<Sum>i = 1..<Suc n. (1/2) ^ i)"
```
```   301         by (rule setsum.cong) (auto simp: powr_realpow powr_divide power_divide powr_minus_divide)
```
```   302       also have "{1..<Suc n} = {..<Suc n} - {0}" by auto
```
```   303       also have "setsum (op ^ (1 / 2::real)) ({..<Suc n} - {0}) =
```
```   304         setsum (op ^ (1 / 2)) ({..<Suc n}) - 1" by (auto simp: setsum_diff1)
```
```   305       also have "\<dots> < 1" by (subst geometric_sum) auto
```
```   306       finally show "(\<Sum>i = 1..n. 2 powr - real_of_ereal i) < 1" by simp
```
```   307     qed (auto simp: r ereal_less_real_iff zero_ereal_def[symmetric])
```
```   308     also have "\<dots> = ?a" by (auto simp: r)
```
```   309     also have "\<dots> \<le> \<mu>G (Z n)"
```
```   310       using J by (auto intro: INF_lower simp: Z_def mu_G_spec)
```
```   311     finally have "\<mu>G (Z n) - \<mu>G (Y n) < \<mu>G (Z n)" .
```
```   312     hence R: "\<mu>G (Z n) < \<mu>G (Z n) + \<mu>G (Y n)"
```
```   313       using \<open>\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>\<close> by (simp add: ereal_minus_less)
```
```   314     have "0 \<le> (- \<mu>G (Z n)) + \<mu>G (Z n)" using \<open>\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>\<close> by auto
```
```   315     also have "\<dots> < (- \<mu>G (Z n)) + (\<mu>G (Z n) + \<mu>G (Y n))"
```
```   316       apply (rule ereal_less_add[OF _ R]) using \<open>\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>\<close> by auto
```
```   317     finally have "\<mu>G (Y n) > 0"
```
```   318       using \<open>\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>\<close> by (auto simp: ac_simps zero_ereal_def[symmetric])
```
```   319     thus "Y n \<noteq> {}" using positive_mu_G by (auto simp add: positive_def)
```
```   320   qed
```
```   321   hence "\<forall>n\<in>{1..}. \<exists>y. y \<in> Y n" by auto
```
```   322   then obtain y where y: "\<And>n. n \<ge> 1 \<Longrightarrow> y n \<in> Y n" unfolding bchoice_iff by force
```
```   323   {
```
```   324     fix t and n m::nat
```
```   325     assume "1 \<le> n" "n \<le> m" hence "1 \<le> m" by simp
```
```   326     from Y_mono[OF \<open>m \<ge> n\<close>] y[OF \<open>1 \<le> m\<close>] have "y m \<in> Y n" by auto
```
```   327     also have "\<dots> \<subseteq> Z' n" using Y_Z'[OF \<open>1 \<le> n\<close>] .
```
```   328     finally
```
```   329     have "fm n (restrict (y m) (J n)) \<in> K' n"
```
```   330       unfolding Z'_def K_def prod_emb_iff by (simp add: Z'_def K_def prod_emb_iff)
```
```   331     moreover have "finmap_of (J n) (restrict (y m) (J n)) = finmap_of (J n) (y m)"
```
```   332       using J by (simp add: fm_def)
```
```   333     ultimately have "fm n (y m) \<in> K' n" by simp
```
```   334   } note fm_in_K' = this
```
```   335   interpret finmap_seqs_into_compact "\<lambda>n. K' (Suc n)" "\<lambda>k. fm (Suc k) (y (Suc k))" borel
```
```   336   proof
```
```   337     fix n show "compact (K' n)" by fact
```
```   338   next
```
```   339     fix n
```
```   340     from Y_mono[of n "Suc n"] y[of "Suc n"] have "y (Suc n) \<in> Y (Suc n)" by auto
```
```   341     also have "\<dots> \<subseteq> Z' (Suc n)" using Y_Z' by auto
```
```   342     finally
```
```   343     have "fm (Suc n) (restrict (y (Suc n)) (J (Suc n))) \<in> K' (Suc n)"
```
```   344       unfolding Z'_def K_def prod_emb_iff by (simp add: Z'_def K_def prod_emb_iff)
```
```   345     thus "K' (Suc n) \<noteq> {}" by auto
```
```   346     fix k
```
```   347     assume "k \<in> K' (Suc n)"
```
```   348     with K'[of "Suc n"] sets.sets_into_space have "k \<in> fm (Suc n) ` B (Suc n)" by auto
```
```   349     then obtain b where "k = fm (Suc n) b" by auto
```
```   350     thus "domain k = domain (fm (Suc n) (y (Suc n)))"
```
```   351       by (simp_all add: fm_def)
```
```   352   next
```
```   353     fix t and n m::nat
```
```   354     assume "n \<le> m" hence "Suc n \<le> Suc m" by simp
```
```   355     assume "t \<in> domain (fm (Suc n) (y (Suc n)))"
```
```   356     then obtain j where j: "t = Utn j" "j \<in> J (Suc n)" by auto
```
```   357     hence "j \<in> J (Suc m)" using J_mono[OF \<open>Suc n \<le> Suc m\<close>] by auto
```
```   358     have img: "fm (Suc n) (y (Suc m)) \<in> K' (Suc n)" using \<open>n \<le> m\<close>
```
```   359       by (intro fm_in_K') simp_all
```
```   360     show "(fm (Suc m) (y (Suc m)))\<^sub>F t \<in> (\<lambda>k. (k)\<^sub>F t) ` K' (Suc n)"
```
```   361       apply (rule image_eqI[OF _ img])
```
```   362       using \<open>j \<in> J (Suc n)\<close> \<open>j \<in> J (Suc m)\<close>
```
```   363       unfolding j by (subst proj_fm, auto)+
```
```   364   qed
```
```   365   have "\<forall>t. \<exists>z. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) \<longlonglongrightarrow> z"
```
```   366     using diagonal_tendsto ..
```
```   367   then obtain z where z:
```
```   368     "\<And>t. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) \<longlonglongrightarrow> z t"
```
```   369     unfolding choice_iff by blast
```
```   370   {
```
```   371     fix n :: nat assume "n \<ge> 1"
```
```   372     have "\<And>i. domain (fm n (y (Suc (diagseq i)))) = domain (finmap_of (Utn ` J n) z)"
```
```   373       by simp
```
```   374     moreover
```
```   375     {
```
```   376       fix t
```
```   377       assume t: "t \<in> domain (finmap_of (Utn ` J n) z)"
```
```   378       hence "t \<in> Utn ` J n" by simp
```
```   379       then obtain j where j: "t = Utn j" "j \<in> J n" by auto
```
```   380       have "(\<lambda>i. (fm n (y (Suc (diagseq i))))\<^sub>F t) \<longlonglongrightarrow> z t"
```
```   381         apply (subst (2) tendsto_iff, subst eventually_sequentially)
```
```   382       proof safe
```
```   383         fix e :: real assume "0 < e"
```
```   384         { fix i and x :: "'i \<Rightarrow> 'a" assume i: "i \<ge> n"
```
```   385           assume "t \<in> domain (fm n x)"
```
```   386           hence "t \<in> domain (fm i x)" using J_mono[OF \<open>i \<ge> n\<close>] by auto
```
```   387           with i have "(fm i x)\<^sub>F t = (fm n x)\<^sub>F t"
```
```   388             using j by (auto simp: proj_fm dest!: inj_onD[OF inj_on_Utn])
```
```   389         } note index_shift = this
```
```   390         have I: "\<And>i. i \<ge> n \<Longrightarrow> Suc (diagseq i) \<ge> n"
```
```   391           apply (rule le_SucI)
```
```   392           apply (rule order_trans) apply simp
```
```   393           apply (rule seq_suble[OF subseq_diagseq])
```
```   394           done
```
```   395         from z
```
```   396         have "\<exists>N. \<forall>i\<ge>N. dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) (z t) < e"
```
```   397           unfolding tendsto_iff eventually_sequentially using \<open>0 < e\<close> by auto
```
```   398         then obtain N where N: "\<And>i. i \<ge> N \<Longrightarrow>
```
```   399           dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) (z t) < e" by auto
```
```   400         show "\<exists>N. \<forall>na\<ge>N. dist ((fm n (y (Suc (diagseq na))))\<^sub>F t) (z t) < e "
```
```   401         proof (rule exI[where x="max N n"], safe)
```
```   402           fix na assume "max N n \<le> na"
```
```   403           hence  "dist ((fm n (y (Suc (diagseq na))))\<^sub>F t) (z t) =
```
```   404                   dist ((fm (Suc (diagseq na)) (y (Suc (diagseq na))))\<^sub>F t) (z t)" using t
```
```   405             by (subst index_shift[OF I]) auto
```
```   406           also have "\<dots> < e" using \<open>max N n \<le> na\<close> by (intro N) simp
```
```   407           finally show "dist ((fm n (y (Suc (diagseq na))))\<^sub>F t) (z t) < e" .
```
```   408         qed
```
```   409       qed
```
```   410       hence "(\<lambda>i. (fm n (y (Suc (diagseq i))))\<^sub>F t) \<longlonglongrightarrow> (finmap_of (Utn ` J n) z)\<^sub>F t"
```
```   411         by (simp add: tendsto_intros)
```
```   412     } ultimately
```
```   413     have "(\<lambda>i. fm n (y (Suc (diagseq i)))) \<longlonglongrightarrow> finmap_of (Utn ` J n) z"
```
```   414       by (rule tendsto_finmap)
```
```   415     hence "((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) \<longlonglongrightarrow> finmap_of (Utn ` J n) z"
```
```   416       by (intro lim_subseq) (simp add: subseq_def)
```
```   417     moreover
```
```   418     have "(\<forall>i. ((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) i \<in> K' n)"
```
```   419       apply (auto simp add: o_def intro!: fm_in_K' \<open>1 \<le> n\<close> le_SucI)
```
```   420       apply (rule le_trans)
```
```   421       apply (rule le_add2)
```
```   422       using seq_suble[OF subseq_diagseq]
```
```   423       apply auto
```
```   424       done
```
```   425     moreover
```
```   426     from \<open>compact (K' n)\<close> have "closed (K' n)" by (rule compact_imp_closed)
```
```   427     ultimately
```
```   428     have "finmap_of (Utn ` J n) z \<in> K' n"
```
```   429       unfolding closed_sequential_limits by blast
```
```   430     also have "finmap_of (Utn ` J n) z  = fm n (\<lambda>i. z (Utn i))"
```
```   431       unfolding finmap_eq_iff
```
```   432     proof clarsimp
```
```   433       fix i assume i: "i \<in> J n"
```
```   434       hence "from_nat_into (\<Union>n. J n) (Utn i) = i"
```
```   435         unfolding Utn_def
```
```   436         by (subst from_nat_into_to_nat_on[OF countable_UN_J]) auto
```
```   437       with i show "z (Utn i) = (fm n (\<lambda>i. z (Utn i)))\<^sub>F (Utn i)"
```
```   438         by (simp add: finmap_eq_iff fm_def compose_def)
```
```   439     qed
```
```   440     finally have "fm n (\<lambda>i. z (Utn i)) \<in> K' n" .
```
```   441     moreover
```
```   442     let ?J = "\<Union>n. J n"
```
```   443     have "(?J \<inter> J n) = J n" by auto
```
```   444     ultimately have "restrict (\<lambda>i. z (Utn i)) (?J \<inter> J n) \<in> K n"
```
```   445       unfolding K_def by (auto simp: space_P space_PiM)
```
```   446     hence "restrict (\<lambda>i. z (Utn i)) ?J \<in> Z' n" unfolding Z'_def
```
```   447       using J by (auto simp: prod_emb_def PiE_def extensional_def)
```
```   448     also have "\<dots> \<subseteq> Z n" using Z' by simp
```
```   449     finally have "restrict (\<lambda>i. z (Utn i)) ?J \<in> Z n" .
```
```   450   } note in_Z = this
```
```   451   hence "(\<Inter>i\<in>{1..}. Z i) \<noteq> {}" by auto
```
```   452   thus "(\<Inter>i. Z i) \<noteq> {}"
```
```   453     using INT_decseq_offset[OF antimonoI[OF Z_mono]] by simp
```
```   454 qed fact+
```
```   455
```
```   456 lemma measure_lim_emb:
```
```   457   "J \<subseteq> I \<Longrightarrow> finite J \<Longrightarrow> X \<in> sets (\<Pi>\<^sub>M i\<in>J. borel) \<Longrightarrow> measure lim (emb I J X) = measure (P J) X"
```
```   458   unfolding measure_def by (subst emeasure_lim_emb) auto
```
```   459
```
```   460 end
```
```   461
```
```   462 hide_const (open) PiF
```
```   463 hide_const (open) Pi\<^sub>F
```
```   464 hide_const (open) Pi'
```
```   465 hide_const (open) Abs_finmap
```
```   466 hide_const (open) Rep_finmap
```
```   467 hide_const (open) finmap_of
```
```   468 hide_const (open) proj
```
```   469 hide_const (open) domain
```
```   470 hide_const (open) basis_finmap
```
```   471
```
```   472 sublocale polish_projective \<subseteq> P: prob_space lim
```
```   473 proof
```
```   474   have *: "emb I {} {\<lambda>x. undefined} = space (\<Pi>\<^sub>M i\<in>I. borel)"
```
```   475     by (auto simp: prod_emb_def space_PiM)
```
```   476   interpret prob_space "P {}"
```
```   477     using prob_space_P by simp
```
```   478   show "emeasure lim (space lim) = 1"
```
```   479     using emeasure_lim_emb[of "{}" "{\<lambda>x. undefined}"] emeasure_space_1
```
```   480     by (simp add: * PiM_empty space_P)
```
```   481 qed
```
```   482
```
```   483 locale polish_product_prob_space =
```
```   484   product_prob_space "\<lambda>_. borel::('a::polish_space) measure" I for I::"'i set"
```
```   485
```
```   486 sublocale polish_product_prob_space \<subseteq> P: polish_projective I "\<lambda>J. PiM J (\<lambda>_. borel::('a) measure)"
```
```   487 proof qed
```
```   488
```
```   489 lemma (in polish_product_prob_space) limP_eq_PiM: "lim = PiM I (\<lambda>_. borel)"
```
```   490   by (rule PiM_eq) (auto simp: emeasure_PiM emeasure_lim_emb)
```
```   491
```
```   492 end
```