author  hoelzl 
Mon, 14 Jan 2013 17:29:04 +0100  
changeset 50881  ae630bab13da 
parent 50251  227477f17c26 
child 51104  59b574c6f803 
permissions  rwrr 
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(* Title: HOL/Probability/Fin_Map.thy 
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Author: Fabian Immler, TU München 
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*) 

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header {* Finite Maps *} 
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theory Fin_Map 
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imports Finite_Product_Measure 

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begin 

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text {* Auxiliary type that is instantiated to @{class polish_space}, needed for the proof of 

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projective limit. @{const extensional} functions are used for the representation in order to 

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stay close to the developments of (finite) products @{const Pi\<^isub>E} and their sigmaalgebra 

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@{const Pi\<^isub>M}. *} 

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typedef ('i, 'a) finmap ("(_ \<Rightarrow>\<^isub>F /_)" [22, 21] 21) = 

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"{(I::'i set, f::'i \<Rightarrow> 'a). finite I \<and> f \<in> extensional I}" by auto 

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subsection {* Domain and Application *} 

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definition domain where "domain P = fst (Rep_finmap P)" 

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lemma finite_domain[simp, intro]: "finite (domain P)" 

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by (cases P) (auto simp: domain_def Abs_finmap_inverse) 

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definition proj ("'((_)')\<^isub>F" [0] 1000) where "proj P i = snd (Rep_finmap P) i" 
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declare [[coercion proj]] 

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lemma extensional_proj[simp, intro]: "(P)\<^isub>F \<in> extensional (domain P)" 

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by (cases P) (auto simp: domain_def Abs_finmap_inverse proj_def[abs_def]) 

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lemma proj_undefined[simp, intro]: "i \<notin> domain P \<Longrightarrow> P i = undefined" 

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using extensional_proj[of P] unfolding extensional_def by auto 

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lemma finmap_eq_iff: "P = Q \<longleftrightarrow> (domain P = domain Q \<and> (\<forall>i\<in>domain P. P i = Q i))" 

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by (cases P, cases Q) 

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(auto simp add: Abs_finmap_inject extensional_def domain_def proj_def Abs_finmap_inverse 

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intro: extensionalityI) 

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subsection {* Countable Finite Maps *} 

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instance finmap :: (countable, countable) countable 

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proof 

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obtain mapper where mapper: "\<And>fm :: 'a \<Rightarrow>\<^isub>F 'b. set (mapper fm) = domain fm" 

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by (metis finite_list[OF finite_domain]) 

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have "inj (\<lambda>fm. map (\<lambda>i. (i, (fm)\<^isub>F i)) (mapper fm))" (is "inj ?F") 

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proof (rule inj_onI) 

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fix f1 f2 assume "?F f1 = ?F f2" 

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then have "map fst (?F f1) = map fst (?F f2)" by simp 

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then have "mapper f1 = mapper f2" by (simp add: comp_def) 

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then have "domain f1 = domain f2" by (simp add: mapper[symmetric]) 

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with `?F f1 = ?F f2` show "f1 = f2" 

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unfolding `mapper f1 = mapper f2` map_eq_conv mapper 

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by (simp add: finmap_eq_iff) 

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qed 

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then show "\<exists>to_nat :: 'a \<Rightarrow>\<^isub>F 'b \<Rightarrow> nat. inj to_nat" 

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by (intro exI[of _ "to_nat \<circ> ?F"] inj_comp) auto 

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qed 

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subsection {* Constructor of Finite Maps *} 

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definition "finmap_of inds f = Abs_finmap (inds, restrict f inds)" 

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lemma proj_finmap_of[simp]: 

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assumes "finite inds" 

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shows "(finmap_of inds f)\<^isub>F = restrict f inds" 

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using assms 

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by (auto simp: Abs_finmap_inverse finmap_of_def proj_def) 

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lemma domain_finmap_of[simp]: 

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assumes "finite inds" 

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shows "domain (finmap_of inds f) = inds" 

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using assms 

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by (auto simp: Abs_finmap_inverse finmap_of_def domain_def) 

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lemma finmap_of_eq_iff[simp]: 

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assumes "finite i" "finite j" 

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shows "finmap_of i m = finmap_of j n \<longleftrightarrow> i = j \<and> restrict m i = restrict n i" 

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using assms 

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apply (auto simp: finmap_eq_iff restrict_def) by metis 

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lemma finmap_of_inj_on_extensional_finite: 
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assumes "finite K" 
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assumes "S \<subseteq> extensional K" 

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shows "inj_on (finmap_of K) S" 

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proof (rule inj_onI) 

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fix x y::"'a \<Rightarrow> 'b" 

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assume "finmap_of K x = finmap_of K y" 

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hence "(finmap_of K x)\<^isub>F = (finmap_of K y)\<^isub>F" by simp 

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moreover 

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assume "x \<in> S" "y \<in> S" hence "x \<in> extensional K" "y \<in> extensional K" using assms by auto 

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ultimately 

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show "x = y" using assms by (simp add: extensional_restrict) 

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qed 

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lemma finmap_choice: 

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assumes *: "\<And>i. i \<in> I \<Longrightarrow> \<exists>x. P i x" and I: "finite I" 

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shows "\<exists>fm. domain fm = I \<and> (\<forall>i\<in>I. P i (fm i))" 

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proof  

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have "\<exists>f. \<forall>i\<in>I. P i (f i)" 

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unfolding bchoice_iff[symmetric] using * by auto 

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then guess f .. 

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with I show ?thesis 

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by (intro exI[of _ "finmap_of I f"]) auto 

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qed 

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subsection {* Product set of Finite Maps *} 

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text {* This is @{term Pi} for Finite Maps, most of this is copied *} 

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definition Pi' :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a set) \<Rightarrow> ('i \<Rightarrow>\<^isub>F 'a) set" where 

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"Pi' I A = { P. domain P = I \<and> (\<forall>i. i \<in> I \<longrightarrow> (P)\<^isub>F i \<in> A i) } " 

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syntax 

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"_Pi'" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3PI' _:_./ _)" 10) 

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syntax (xsymbols) 

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"_Pi'" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3\<Pi>' _\<in>_./ _)" 10) 

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syntax (HTML output) 

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"_Pi'" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3\<Pi>' _\<in>_./ _)" 10) 

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translations 

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"PI' x:A. B" == "CONST Pi' A (%x. B)" 

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subsubsection{*Basic Properties of @{term Pi'}*} 

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lemma Pi'_I[intro!]: "domain f = A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi' A B" 

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by (simp add: Pi'_def) 

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lemma Pi'_I'[simp]: "domain f = A \<Longrightarrow> (\<And>x. x \<in> A \<longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi' A B" 

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by (simp add:Pi'_def) 

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lemma Pi'_mem: "f\<in> Pi' A B \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<in> B x" 

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by (simp add: Pi'_def) 

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lemma Pi'_iff: "f \<in> Pi' I X \<longleftrightarrow> domain f = I \<and> (\<forall>i\<in>I. f i \<in> X i)" 

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unfolding Pi'_def by auto 

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lemma Pi'E [elim]: 

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"f \<in> Pi' A B \<Longrightarrow> (f x \<in> B x \<Longrightarrow> domain f = A \<Longrightarrow> Q) \<Longrightarrow> (x \<notin> A \<Longrightarrow> Q) \<Longrightarrow> Q" 

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by(auto simp: Pi'_def) 

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lemma in_Pi'_cong: 

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"domain f = domain g \<Longrightarrow> (\<And> w. w \<in> A \<Longrightarrow> f w = g w) \<Longrightarrow> f \<in> Pi' A B \<longleftrightarrow> g \<in> Pi' A B" 

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by (auto simp: Pi'_def) 

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lemma Pi'_eq_empty[simp]: 

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assumes "finite A" shows "(Pi' A B) = {} \<longleftrightarrow> (\<exists>x\<in>A. B x = {})" 

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using assms 

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apply (simp add: Pi'_def, auto) 

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apply (drule_tac x = "finmap_of A (\<lambda>u. SOME y. y \<in> B u)" in spec, auto) 

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apply (cut_tac P= "%y. y \<in> B i" in some_eq_ex, auto) 

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done 

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lemma Pi'_mono: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x) \<Longrightarrow> Pi' A B \<subseteq> Pi' A C" 

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by (auto simp: Pi'_def) 

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lemma Pi_Pi': "finite A \<Longrightarrow> (Pi\<^isub>E A B) = proj ` Pi' A B" 

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apply (auto simp: Pi'_def Pi_def extensional_def) 

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apply (rule_tac x = "finmap_of A (restrict x A)" in image_eqI) 

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apply auto 

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done 

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subsection {* Metric Space of Finite Maps *} 

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instantiation finmap :: (type, metric_space) metric_space 

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begin 

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definition dist_finmap where 

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"dist P Q = (\<Sum>i\<in>domain P \<union> domain Q. dist ((P)\<^isub>F i) ((Q)\<^isub>F i)) + 

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card ((domain P  domain Q) \<union> (domain Q  domain P))" 

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lemma dist_finmap_extend: 

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assumes "finite X" 

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shows "dist P Q = (\<Sum>i\<in>domain P \<union> domain Q \<union> X. dist ((P)\<^isub>F i) ((Q)\<^isub>F i)) + 

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card ((domain P  domain Q) \<union> (domain Q  domain P))" 

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unfolding dist_finmap_def add_right_cancel 

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using assms extensional_arb[of "(P)\<^isub>F"] extensional_arb[of "(Q)\<^isub>F" "domain Q"] 

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by (intro setsum_mono_zero_cong_left) auto 

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definition open_finmap :: "('a \<Rightarrow>\<^isub>F 'b) set \<Rightarrow> bool" where 

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"open_finmap S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)" 

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lemma add_eq_zero_iff[simp]: 

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fixes a b::real 

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assumes "a \<ge> 0" "b \<ge> 0" 

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shows "a + b = 0 \<longleftrightarrow> a = 0 \<and> b = 0" 

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using assms by auto 

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lemma dist_le_1_imp_domain_eq: 

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assumes "dist P Q < 1" 

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shows "domain P = domain Q" 

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proof  

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have "0 \<le> (\<Sum>i\<in>domain P \<union> domain Q. dist (P i) (Q i))" 

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by (simp add: setsum_nonneg) 

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with assms have "card (domain P  domain Q \<union> (domain Q  domain P)) = 0" 

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unfolding dist_finmap_def by arith 

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thus "domain P = domain Q" by auto 

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qed 

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lemma dist_proj: 

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shows "dist ((x)\<^isub>F i) ((y)\<^isub>F i) \<le> dist x y" 

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proof  

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have "dist (x i) (y i) = (\<Sum>i\<in>{i}. dist (x i) (y i))" by simp 

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also have "\<dots> \<le> (\<Sum>i\<in>domain x \<union> domain y \<union> {i}. dist (x i) (y i))" 

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by (intro setsum_mono2) auto 

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also have "\<dots> \<le> dist x y" by (simp add: dist_finmap_extend[of "{i}"]) 

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finally show ?thesis by simp 

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qed 

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lemma open_Pi'I: 

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assumes open_component: "\<And>i. i \<in> I \<Longrightarrow> open (A i)" 

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shows "open (Pi' I A)" 

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proof (subst open_finmap_def, safe) 

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fix x assume x: "x \<in> Pi' I A" 

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hence dim_x: "domain x = I" by (simp add: Pi'_def) 

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hence [simp]: "finite I" unfolding dim_x[symmetric] by simp 

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have "\<exists>ei. \<forall>i\<in>I. 0 < ei i \<and> (\<forall>y. dist y (x i) < ei i \<longrightarrow> y \<in> A i)" 

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proof (safe intro!: bchoice) 

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fix i assume i: "i \<in> I" 

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moreover with open_component have "open (A i)" by simp 

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moreover have "x i \<in> A i" using x i 

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by (auto simp: proj_def) 

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ultimately show "\<exists>e>0. \<forall>y. dist y (x i) < e \<longrightarrow> y \<in> A i" 

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using x by (auto simp: open_dist Ball_def) 

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qed 

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then guess ei .. note ei = this 

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def es \<equiv> "ei ` I" 

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def e \<equiv> "if es = {} then 0.5 else min 0.5 (Min es)" 

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from ei have "e > 0" using x 

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by (auto simp add: e_def es_def Pi'_def Ball_def) 

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moreover have "\<forall>y. dist y x < e \<longrightarrow> y \<in> Pi' I A" 

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proof (intro allI impI) 

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fix y 

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assume "dist y x < e" 

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also have "\<dots> < 1" by (auto simp: e_def) 

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finally have "domain y = domain x" by (rule dist_le_1_imp_domain_eq) 

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with dim_x have dims: "domain y = domain x" "domain x = I" by auto 

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show "y \<in> Pi' I A" 

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proof 

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show "domain y = I" using dims by simp 

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next 

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fix i 

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assume "i \<in> I" 

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have "dist (y i) (x i) \<le> dist y x" using dims `i \<in> I` 

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by (auto intro: dist_proj) 

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also have "\<dots> < e" using `dist y x < e` dims 

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by (simp add: dist_finmap_def) 

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also have "e \<le> Min (ei ` I)" using dims `i \<in> I` 

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by (auto simp: e_def es_def) 

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also have "\<dots> \<le> ei i" using `i \<in> I` by (simp add: e_def) 

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finally have "dist (y i) (x i) < ei i" . 

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with ei `i \<in> I` show "y i \<in> A i" by simp 

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qed 

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qed 

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ultimately 

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show "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> Pi' I A" by blast 

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qed 

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instance 

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proof 

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fix S::"('a \<Rightarrow>\<^isub>F 'b) set" 

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show "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)" 

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unfolding open_finmap_def .. 

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next 

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fix P Q::"'a \<Rightarrow>\<^isub>F 'b" 

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show "dist P Q = 0 \<longleftrightarrow> P = Q" 

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by (auto simp: finmap_eq_iff dist_finmap_def setsum_nonneg setsum_nonneg_eq_0_iff) 

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next 

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fix P Q R::"'a \<Rightarrow>\<^isub>F 'b" 

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let ?symdiff = "\<lambda>a b. domain a  domain b \<union> (domain b  domain a)" 

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def E \<equiv> "domain P \<union> domain Q \<union> domain R" 

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hence "finite E" by (simp add: E_def) 

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have "card (?symdiff P Q) \<le> card (?symdiff P R \<union> ?symdiff Q R)" 

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by (auto intro: card_mono) 

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also have "\<dots> \<le> card (?symdiff P R) + card (?symdiff Q R)" 

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by (subst card_Un_Int) auto 

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finally have "dist P Q \<le> (\<Sum>i\<in>E. dist (P i) (R i) + dist (Q i) (R i)) + 

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real (card (?symdiff P R) + card (?symdiff Q R))" 

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unfolding dist_finmap_extend[OF `finite E`] 

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by (intro add_mono) (auto simp: E_def intro: setsum_mono dist_triangle_le) 

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also have "\<dots> \<le> dist P R + dist Q R" 

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unfolding dist_finmap_extend[OF `finite E`] by (simp add: ac_simps E_def setsum_addf[symmetric]) 

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finally show "dist P Q \<le> dist P R + dist Q R" by simp 

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qed 

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end 

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lemma open_restricted_space: 

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shows "open {m. P (domain m)}" 

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proof  

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have "{m. P (domain m)} = (\<Union>i \<in> Collect P. {m. domain m = i})" by auto 

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also have "open \<dots>" 

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proof (rule, safe, cases) 

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fix i::"'a set" 

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assume "finite i" 

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hence "{m. domain m = i} = Pi' i (\<lambda>_. UNIV)" by (auto simp: Pi'_def) 

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also have "open \<dots>" by (auto intro: open_Pi'I simp: `finite i`) 

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finally show "open {m. domain m = i}" . 

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next 

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fix i::"'a set" 

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assume "\<not> finite i" hence "{m. domain m = i} = {}" by auto 

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also have "open \<dots>" by simp 

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finally show "open {m. domain m = i}" . 

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qed 

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finally show ?thesis . 

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qed 

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lemma closed_restricted_space: 

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shows "closed {m. P (domain m)}" 

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proof  

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have "{m. P (domain m)} =  (\<Union>i \<in>  Collect P. {m. domain m = i})" by auto 

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also have "closed \<dots>" 

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proof (rule, rule, rule, cases) 

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fix i::"'a set" 

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assume "finite i" 

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hence "{m. domain m = i} = Pi' i (\<lambda>_. UNIV)" by (auto simp: Pi'_def) 

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also have "open \<dots>" by (auto intro: open_Pi'I simp: `finite i`) 

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finally show "open {m. domain m = i}" . 

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next 

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fix i::"'a set" 

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assume "\<not> finite i" hence "{m. domain m = i} = {}" by auto 

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also have "open \<dots>" by simp 

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finally show "open {m. domain m = i}" . 

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qed 

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finally show ?thesis . 

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qed 

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lemma continuous_proj: 

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shows "continuous_on s (\<lambda>x. (x)\<^isub>F i)" 

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unfolding continuous_on_topological 

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proof safe 

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fix x B assume "x \<in> s" "open B" "x i \<in> B" 

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let ?A = "Pi' (domain x) (\<lambda>j. if i = j then B else UNIV)" 

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have "open ?A" using `open B` by (auto intro: open_Pi'I) 

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moreover have "x \<in> ?A" using `x i \<in> B` by auto 

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moreover have "(\<forall>y\<in>s. y \<in> ?A \<longrightarrow> y i \<in> B)" 

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proof (cases, safe) 

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fix y assume "y \<in> s" 

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assume "i \<notin> domain x" hence "undefined \<in> B" using `x i \<in> B` 

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by simp 

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moreover 

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assume "y \<in> ?A" hence "domain y = domain x" by (simp add: Pi'_def) 

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hence "y i = undefined" using `i \<notin> domain x` by simp 

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ultimately 

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show "y i \<in> B" by simp 

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qed force 

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ultimately 

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show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> y i \<in> B)" by blast 

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qed 

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subsection {* Complete Space of Finite Maps *} 

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lemma tendsto_dist_zero: 

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assumes "(\<lambda>i. dist (f i) g) > 0" 

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shows "f > g" 

359 
using assms by (auto simp: tendsto_iff dist_real_def) 

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lemma tendsto_dist_zero': 

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assumes "(\<lambda>i. dist (f i) g) > x" 

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assumes "0 = x" 

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shows "f > g" 

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using assms tendsto_dist_zero by simp 

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lemma tendsto_finmap: 

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fixes f::"nat \<Rightarrow> ('i \<Rightarrow>\<^isub>F ('a::metric_space))" 

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assumes ind_f: "\<And>n. domain (f n) = domain g" 

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assumes proj_g: "\<And>i. i \<in> domain g \<Longrightarrow> (\<lambda>n. (f n) i) > g i" 

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shows "f > g" 

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apply (rule tendsto_dist_zero') 

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unfolding dist_finmap_def assms 

374 
apply (rule tendsto_intros proj_g  simp)+ 

375 
done 

376 

377 
instance finmap :: (type, complete_space) complete_space 

378 
proof 

379 
fix P::"nat \<Rightarrow> 'a \<Rightarrow>\<^isub>F 'b" 

380 
assume "Cauchy P" 

381 
then obtain Nd where Nd: "\<And>n. n \<ge> Nd \<Longrightarrow> dist (P n) (P Nd) < 1" 

382 
by (force simp: cauchy) 

383 
def d \<equiv> "domain (P Nd)" 

384 
with Nd have dim: "\<And>n. n \<ge> Nd \<Longrightarrow> domain (P n) = d" using dist_le_1_imp_domain_eq by auto 

385 
have [simp]: "finite d" unfolding d_def by simp 

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def p \<equiv> "\<lambda>i n. (P n) i" 

387 
def q \<equiv> "\<lambda>i. lim (p i)" 

388 
def Q \<equiv> "finmap_of d q" 

389 
have q: "\<And>i. i \<in> d \<Longrightarrow> q i = Q i" by (auto simp add: Q_def Abs_finmap_inverse) 

390 
{ 

391 
fix i assume "i \<in> d" 

392 
have "Cauchy (p i)" unfolding cauchy p_def 

393 
proof safe 

394 
fix e::real assume "0 < e" 

395 
with `Cauchy P` obtain N where N: "\<And>n. n\<ge>N \<Longrightarrow> dist (P n) (P N) < min e 1" 

396 
by (force simp: cauchy min_def) 

397 
hence "\<And>n. n \<ge> N \<Longrightarrow> domain (P n) = domain (P N)" using dist_le_1_imp_domain_eq by auto 

398 
with dim have dim: "\<And>n. n \<ge> N \<Longrightarrow> domain (P n) = d" by (metis nat_le_linear) 

399 
show "\<exists>N. \<forall>n\<ge>N. dist ((P n) i) ((P N) i) < e" 

400 
proof (safe intro!: exI[where x="N"]) 

401 
fix n assume "N \<le> n" have "N \<le> N" by simp 

402 
have "dist ((P n) i) ((P N) i) \<le> dist (P n) (P N)" 

403 
using dim[OF `N \<le> n`] dim[OF `N \<le> N`] `i \<in> d` 

404 
by (auto intro!: dist_proj) 

405 
also have "\<dots> < e" using N[OF `N \<le> n`] by simp 

406 
finally show "dist ((P n) i) ((P N) i) < e" . 

407 
qed 

408 
qed 

409 
hence "convergent (p i)" by (metis Cauchy_convergent_iff) 

410 
hence "p i > q i" unfolding q_def convergent_def by (metis limI) 

411 
} note p = this 

412 
have "P > Q" 

413 
proof (rule metric_LIMSEQ_I) 

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fix e::real assume "0 < e" 

415 
def e' \<equiv> "min 1 (e / (card d + 1))" 

416 
hence "0 < e'" using `0 < e` by (auto simp: e'_def intro: divide_pos_pos) 

417 
have "\<exists>ni. \<forall>i\<in>d. \<forall>n\<ge>ni i. dist (p i n) (q i) < e'" 

418 
proof (safe intro!: bchoice) 

419 
fix i assume "i \<in> d" 

420 
from p[OF `i \<in> d`, THEN metric_LIMSEQ_D, OF `0 < e'`] 

421 
show "\<exists>no. \<forall>n\<ge>no. dist (p i n) (q i) < e'" . 

422 
qed then guess ni .. note ni = this 

423 
def N \<equiv> "max Nd (Max (ni ` d))" 

424 
show "\<exists>N. \<forall>n\<ge>N. dist (P n) Q < e" 

425 
proof (safe intro!: exI[where x="N"]) 

426 
fix n assume "N \<le> n" 

427 
hence "domain (P n) = d" "domain Q = d" "domain (P n) = domain Q" 

428 
using dim by (simp_all add: N_def Q_def dim_def Abs_finmap_inverse) 

429 
hence "dist (P n) Q = (\<Sum>i\<in>d. dist ((P n) i) (Q i))" by (simp add: dist_finmap_def) 

430 
also have "\<dots> \<le> (\<Sum>i\<in>d. e')" 

431 
proof (intro setsum_mono less_imp_le) 

432 
fix i assume "i \<in> d" 

433 
hence "ni i \<le> Max (ni ` d)" by simp 

434 
also have "\<dots> \<le> N" by (simp add: N_def) 

435 
also have "\<dots> \<le> n" using `N \<le> n` . 

436 
finally 

437 
show "dist ((P n) i) (Q i) < e'" 

438 
using ni `i \<in> d` by (auto simp: p_def q N_def) 

439 
qed 

440 
also have "\<dots> = card d * e'" by (simp add: real_eq_of_nat) 

441 
also have "\<dots> < e" using `0 < e` by (simp add: e'_def field_simps min_def) 

442 
finally show "dist (P n) Q < e" . 

443 
qed 

444 
qed 

445 
thus "convergent P" by (auto simp: convergent_def) 

446 
qed 

447 

448 
subsection {* Polish Space of Finite Maps *} 

449 

450 
instantiation finmap :: (countable, polish_space) polish_space 

451 
begin 

452 

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453 
definition basis_finmap::"('a \<Rightarrow>\<^isub>F 'b) set set" 
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454 
where "basis_finmap = {Pi' I SI S. finite I \<and> (\<forall>i \<in> I. S i \<in> union_closed_basis)}" 
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455 

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456 
lemma in_basis_finmapI: 
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457 
assumes "finite I" assumes "\<And>i. i \<in> I \<Longrightarrow> S i \<in> union_closed_basis" 
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458 
shows "Pi' I S \<in> basis_finmap" 
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459 
using assms unfolding basis_finmap_def by auto 
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460 

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461 
lemma in_basis_finmapE: 
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462 
assumes "x \<in> basis_finmap" 
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463 
obtains I S where "x = Pi' I S" "finite I" "\<And>i. i \<in> I \<Longrightarrow> S i \<in> union_closed_basis" 
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464 
using assms unfolding basis_finmap_def by auto 
50088  465 

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466 
lemma basis_finmap_eq: 
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467 
"basis_finmap = (\<lambda>f. Pi' (domain f) (\<lambda>i. from_nat_into union_closed_basis ((f)\<^isub>F i))) ` 
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468 
(UNIV::('a \<Rightarrow>\<^isub>F nat) set)" (is "_ = ?f ` _") 
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469 
unfolding basis_finmap_def 
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470 
proof safe 
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471 
fix I::"'a set" and S::"'a \<Rightarrow> 'b set" 
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472 
assume "finite I" "\<forall>i\<in>I. S i \<in> union_closed_basis" 
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473 
hence "Pi' I S = ?f (finmap_of I (\<lambda>x. to_nat_on union_closed_basis (S x)))" 
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474 
by (force simp: Pi'_def countable_union_closed_basis) 
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475 
thus "Pi' I S \<in> range ?f" by simp 
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476 
qed (metis (mono_tags) empty_basisI equals0D finite_domain from_nat_into) 
50088  477 

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478 
lemma countable_basis_finmap: "countable basis_finmap" 
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479 
unfolding basis_finmap_eq by simp 
50088  480 

481 
lemma finmap_topological_basis: 

50245
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482 
"topological_basis basis_finmap" 
50088  483 
proof (subst topological_basis_iff, safe) 
50245
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484 
fix B' assume "B' \<in> basis_finmap" 
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485 
thus "open B'" 
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486 
by (auto intro!: open_Pi'I topological_basis_open[OF basis_union_closed_basis] 
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487 
simp: topological_basis_def basis_finmap_def Let_def) 
50088  488 
next 
489 
fix O'::"('a \<Rightarrow>\<^isub>F 'b) set" and x 

490 
assume "open O'" "x \<in> O'" 

491 
then obtain e where e: "e > 0" "\<And>y. dist y x < e \<Longrightarrow> y \<in> O'" unfolding open_dist by blast 

492 
def e' \<equiv> "e / (card (domain x) + 1)" 

493 

494 
have "\<exists>B. 

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495 
(\<forall>i\<in>domain x. x i \<in> B i \<and> B i \<subseteq> ball (x i) e' \<and> B i \<in> union_closed_basis)" 
50088  496 
proof (rule bchoice, safe) 
497 
fix i assume "i \<in> domain x" 

498 
have "open (ball (x i) e')" "x i \<in> ball (x i) e'" using e 

499 
by (auto simp add: e'_def intro!: divide_pos_pos) 

50245
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500 
from topological_basisE[OF basis_union_closed_basis this] guess b' . 
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501 
thus "\<exists>y. x i \<in> y \<and> y \<subseteq> ball (x i) e' \<and> y \<in> union_closed_basis" by auto 
50088  502 
qed 
503 
then guess B .. note B = this 

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504 
def B' \<equiv> "Pi' (domain x) (\<lambda>i. (B i)::'b set)" 
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505 
hence "B' \<in> basis_finmap" unfolding B'_def using B 
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506 
by (intro in_basis_finmapI) auto 
50088  507 
moreover have "x \<in> B'" unfolding B'_def using B by auto 
508 
moreover have "B' \<subseteq> O'" 

509 
proof 

510 
fix y assume "y \<in> B'" with B have "domain y = domain x" unfolding B'_def 

511 
by (simp add: Pi'_def) 

512 
show "y \<in> O'" 

513 
proof (rule e) 

514 
have "dist y x = (\<Sum>i \<in> domain x. dist (y i) (x i))" 

515 
using `domain y = domain x` by (simp add: dist_finmap_def) 

516 
also have "\<dots> \<le> (\<Sum>i \<in> domain x. e')" 

517 
proof (rule setsum_mono) 

518 
fix i assume "i \<in> domain x" 

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519 
with `y \<in> B'` B have "y i \<in> B i" 
50088  520 
by (simp add: Pi'_def B'_def) 
521 
hence "y i \<in> ball (x i) e'" using B `domain y = domain x` `i \<in> domain x` 

522 
by force 

523 
thus "dist (y i) (x i) \<le> e'" by (simp add: dist_commute) 

524 
qed 

525 
also have "\<dots> = card (domain x) * e'" by (simp add: real_eq_of_nat) 

526 
also have "\<dots> < e" using e by (simp add: e'_def field_simps) 

527 
finally show "dist y x < e" . 

528 
qed 

529 
qed 

530 
ultimately 

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531 
show "\<exists>B'\<in>basis_finmap. x \<in> B' \<and> B' \<subseteq> O'" by blast 
50088  532 
qed 
533 

534 
lemma range_enum_basis_finmap_imp_open: 

50245
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535 
assumes "x \<in> basis_finmap" 
50088  536 
shows "open x" 
537 
using finmap_topological_basis assms by (auto simp: topological_basis_def) 

538 

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539 
instance proof qed (blast intro: finmap_topological_basis countable_basis_finmap) 
50088  540 

541 
end 

542 

543 
subsection {* Product Measurable Space of Finite Maps *} 

544 

545 
definition "PiF I M \<equiv> 

50124  546 
sigma (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j))) {(\<Pi>' j\<in>J. X j) X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}" 
50088  547 

548 
abbreviation 

549 
"Pi\<^isub>F I M \<equiv> PiF I M" 

550 

551 
syntax 

552 
"_PiF" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3PIF _:_./ _)" 10) 

553 

554 
syntax (xsymbols) 

555 
"_PiF" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3\<Pi>\<^isub>F _\<in>_./ _)" 10) 

556 

557 
syntax (HTML output) 

558 
"_PiF" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3\<Pi>\<^isub>F _\<in>_./ _)" 10) 

559 

560 
translations 

561 
"PIF x:I. M" == "CONST PiF I (%x. M)" 

562 

563 
lemma PiF_gen_subset: "{(\<Pi>' j\<in>J. X j) X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))} \<subseteq> 

564 
Pow (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j)))" 

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565 
by (auto simp: Pi'_def) (blast dest: sets.sets_into_space) 
50088  566 

567 
lemma space_PiF: "space (PiF I M) = (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j)))" 

568 
unfolding PiF_def using PiF_gen_subset by (rule space_measure_of) 

569 

570 
lemma sets_PiF: 

571 
"sets (PiF I M) = sigma_sets (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j))) 

572 
{(\<Pi>' j\<in>J. X j) X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}" 

573 
unfolding PiF_def using PiF_gen_subset by (rule sets_measure_of) 

574 

575 
lemma sets_PiF_singleton: 

576 
"sets (PiF {I} M) = sigma_sets (\<Pi>' j\<in>I. space (M j)) 

577 
{(\<Pi>' j\<in>I. X j) X. X \<in> (\<Pi> j\<in>I. sets (M j))}" 

578 
unfolding sets_PiF by simp 

579 

580 
lemma in_sets_PiFI: 

581 
assumes "X = (Pi' J S)" "J \<in> I" "\<And>i. i\<in>J \<Longrightarrow> S i \<in> sets (M i)" 

582 
shows "X \<in> sets (PiF I M)" 

583 
unfolding sets_PiF 

584 
using assms by blast 

585 

586 
lemma product_in_sets_PiFI: 

587 
assumes "J \<in> I" "\<And>i. i\<in>J \<Longrightarrow> S i \<in> sets (M i)" 

588 
shows "(Pi' J S) \<in> sets (PiF I M)" 

589 
unfolding sets_PiF 

590 
using assms by blast 

591 

592 
lemma singleton_space_subset_in_sets: 

593 
fixes J 

594 
assumes "J \<in> I" 

595 
assumes "finite J" 

596 
shows "space (PiF {J} M) \<in> sets (PiF I M)" 

597 
using assms 

598 
by (intro in_sets_PiFI[where J=J and S="\<lambda>i. space (M i)"]) 

599 
(auto simp: product_def space_PiF) 

600 

601 
lemma singleton_subspace_set_in_sets: 

602 
assumes A: "A \<in> sets (PiF {J} M)" 

603 
assumes "finite J" 

604 
assumes "J \<in> I" 

605 
shows "A \<in> sets (PiF I M)" 

606 
using A[unfolded sets_PiF] 

607 
apply (induct A) 

608 
unfolding sets_PiF[symmetric] unfolding space_PiF[symmetric] 

609 
using assms 

610 
by (auto intro: in_sets_PiFI intro!: singleton_space_subset_in_sets) 

611 

50124  612 
lemma finite_measurable_singletonI: 
50088  613 
assumes "finite I" 
614 
assumes "\<And>J. J \<in> I \<Longrightarrow> finite J" 

615 
assumes MN: "\<And>J. J \<in> I \<Longrightarrow> A \<in> measurable (PiF {J} M) N" 

616 
shows "A \<in> measurable (PiF I M) N" 

617 
unfolding measurable_def 

618 
proof safe 

619 
fix y assume "y \<in> sets N" 

620 
have "A ` y \<inter> space (PiF I M) = (\<Union>J\<in>I. A ` y \<inter> space (PiF {J} M))" 

621 
by (auto simp: space_PiF) 

622 
also have "\<dots> \<in> sets (PiF I M)" 

623 
proof 

624 
show "finite I" by fact 

625 
fix J assume "J \<in> I" 

626 
with assms have "finite J" by simp 

627 
show "A ` y \<inter> space (PiF {J} M) \<in> sets (PiF I M)" 

628 
by (rule singleton_subspace_set_in_sets[OF measurable_sets[OF assms(3)]]) fact+ 

629 
qed 

630 
finally show "A ` y \<inter> space (PiF I M) \<in> sets (PiF I M)" . 

631 
next 

632 
fix x assume "x \<in> space (PiF I M)" thus "A x \<in> space N" 

633 
using MN[of "domain x"] 

634 
by (auto simp: space_PiF measurable_space Pi'_def) 

635 
qed 

636 

50124  637 
lemma countable_finite_comprehension: 
50088  638 
fixes f :: "'a::countable set \<Rightarrow> _" 
639 
assumes "\<And>s. P s \<Longrightarrow> finite s" 

640 
assumes "\<And>s. P s \<Longrightarrow> f s \<in> sets M" 

641 
shows "\<Union>{f ss. P s} \<in> sets M" 

642 
proof  

643 
have "\<Union>{f ss. P s} = (\<Union>n::nat. let s = set (from_nat n) in if P s then f s else {})" 

644 
proof safe 

645 
fix x X s assume "x \<in> f s" "P s" 

646 
moreover with assms obtain l where "s = set l" using finite_list by blast 

647 
ultimately show "x \<in> (\<Union>n. let s = set (from_nat n) in if P s then f s else {})" using `P s` 

648 
by (auto intro!: exI[where x="to_nat l"]) 

649 
next 

650 
fix x n assume "x \<in> (let s = set (from_nat n) in if P s then f s else {})" 

651 
thus "x \<in> \<Union>{f ss. P s}" using assms by (auto simp: Let_def split: split_if_asm) 

652 
qed 

653 
hence "\<Union>{f ss. P s} = (\<Union>n. let s = set (from_nat n) in if P s then f s else {})" by simp 

654 
also have "\<dots> \<in> sets M" using assms by (auto simp: Let_def) 

655 
finally show ?thesis . 

656 
qed 

657 

658 
lemma space_subset_in_sets: 

659 
fixes J::"'a::countable set set" 

660 
assumes "J \<subseteq> I" 

661 
assumes "\<And>j. j \<in> J \<Longrightarrow> finite j" 

662 
shows "space (PiF J M) \<in> sets (PiF I M)" 

663 
proof  

664 
have "space (PiF J M) = \<Union>{space (PiF {j} M)j. j \<in> J}" 

665 
unfolding space_PiF by blast 

666 
also have "\<dots> \<in> sets (PiF I M)" using assms 

667 
by (intro countable_finite_comprehension) (auto simp: singleton_space_subset_in_sets) 

668 
finally show ?thesis . 

669 
qed 

670 

671 
lemma subspace_set_in_sets: 

672 
fixes J::"'a::countable set set" 

673 
assumes A: "A \<in> sets (PiF J M)" 

674 
assumes "J \<subseteq> I" 

675 
assumes "\<And>j. j \<in> J \<Longrightarrow> finite j" 

676 
shows "A \<in> sets (PiF I M)" 

677 
using A[unfolded sets_PiF] 

678 
apply (induct A) 

679 
unfolding sets_PiF[symmetric] unfolding space_PiF[symmetric] 

680 
using assms 

681 
by (auto intro: in_sets_PiFI intro!: space_subset_in_sets) 

682 

50124  683 
lemma countable_measurable_PiFI: 
50088  684 
fixes I::"'a::countable set set" 
685 
assumes MN: "\<And>J. J \<in> I \<Longrightarrow> finite J \<Longrightarrow> A \<in> measurable (PiF {J} M) N" 

686 
shows "A \<in> measurable (PiF I M) N" 

687 
unfolding measurable_def 

688 
proof safe 

689 
fix y assume "y \<in> sets N" 

690 
have "A ` y = (\<Union>{A ` y \<inter> {x. domain x = J}J. finite J})" by auto 

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691 
{ fix x::"'a \<Rightarrow>\<^isub>F 'b" 
50088  692 
from finite_list[of "domain x"] obtain xs where "set xs = domain x" by auto 
50245
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693 
hence "\<exists>n. domain x = set (from_nat n)" 
dea9363887a6
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694 
by (intro exI[where x="to_nat xs"]) auto } 
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695 
hence "A ` y \<inter> space (PiF I M) = (\<Union>n. A ` y \<inter> space (PiF ({set (from_nat n)}\<inter>I) M))" 
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696 
by (auto simp: space_PiF Pi'_def) 
50088  697 
also have "\<dots> \<in> sets (PiF I M)" 
50244
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698 
apply (intro sets.Int sets.countable_nat_UN subsetI, safe) 
50088  699 
apply (case_tac "set (from_nat i) \<in> I") 
700 
apply simp_all 

701 
apply (rule singleton_subspace_set_in_sets[OF measurable_sets[OF MN]]) 

702 
using assms `y \<in> sets N` 

703 
apply (auto simp: space_PiF) 

704 
done 

705 
finally show "A ` y \<inter> space (PiF I M) \<in> sets (PiF I M)" . 

706 
next 

707 
fix x assume "x \<in> space (PiF I M)" thus "A x \<in> space N" 

708 
using MN[of "domain x"] by (auto simp: space_PiF measurable_space Pi'_def) 

709 
qed 

710 

711 
lemma measurable_PiF: 

712 
assumes f: "\<And>x. x \<in> space N \<Longrightarrow> domain (f x) \<in> I \<and> (\<forall>i\<in>domain (f x). (f x) i \<in> space (M i))" 

713 
assumes S: "\<And>J S. J \<in> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> S i \<in> sets (M i)) \<Longrightarrow> 

714 
f ` (Pi' J S) \<inter> space N \<in> sets N" 

715 
shows "f \<in> measurable N (PiF I M)" 

716 
unfolding PiF_def 

717 
using PiF_gen_subset 

718 
apply (rule measurable_measure_of) 

719 
using f apply force 

720 
apply (insert S, auto) 

721 
done 

722 

50124  723 
lemma restrict_sets_measurable: 
50088  724 
assumes A: "A \<in> sets (PiF I M)" and "J \<subseteq> I" 
725 
shows "A \<inter> {m. domain m \<in> J} \<in> sets (PiF J M)" 

726 
using A[unfolded sets_PiF] 

50124  727 
proof (induct A) 
728 
case (Basic a) 

50088  729 
then obtain K S where S: "a = Pi' K S" "K \<in> I" "(\<forall>i\<in>K. S i \<in> sets (M i))" 
730 
by auto 

50124  731 
show ?case 
50088  732 
proof cases 
733 
assume "K \<in> J" 

734 
hence "a \<inter> {m. domain m \<in> J} \<in> {Pi' K X X K. K \<in> J \<and> X \<in> (\<Pi> j\<in>K. sets (M j))}" using S 

735 
by (auto intro!: exI[where x=K] exI[where x=S] simp: Pi'_def) 

736 
also have "\<dots> \<subseteq> sets (PiF J M)" unfolding sets_PiF by auto 

737 
finally show ?thesis . 

738 
next 

739 
assume "K \<notin> J" 

740 
hence "a \<inter> {m. domain m \<in> J} = {}" using S by (auto simp: Pi'_def) 

741 
also have "\<dots> \<in> sets (PiF J M)" by simp 

742 
finally show ?thesis . 

743 
qed 

744 
next 

50124  745 
case (Union a) 
50088  746 
have "UNION UNIV a \<inter> {m. domain m \<in> J} = (\<Union>i. (a i \<inter> {m. domain m \<in> J}))" 
747 
by simp 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50124
diff
changeset

748 
also have "\<dots> \<in> sets (PiF J M)" using Union by (intro sets.countable_nat_UN) auto 
50124  749 
finally show ?case . 
50088  750 
next 
50124  751 
case (Compl a) 
50088  752 
have "(space (PiF I M)  a) \<inter> {m. domain m \<in> J} = (space (PiF J M)  (a \<inter> {m. domain m \<in> J}))" 
753 
using `J \<subseteq> I` by (auto simp: space_PiF Pi'_def) 

50124  754 
also have "\<dots> \<in> sets (PiF J M)" using Compl by auto 
755 
finally show ?case by (simp add: space_PiF) 

756 
qed simp 

50088  757 

758 
lemma measurable_finmap_of: 

759 
assumes f: "\<And>i. (\<exists>x \<in> space N. i \<in> J x) \<Longrightarrow> (\<lambda>x. f x i) \<in> measurable N (M i)" 

760 
assumes J: "\<And>x. x \<in> space N \<Longrightarrow> J x \<in> I" "\<And>x. x \<in> space N \<Longrightarrow> finite (J x)" 

761 
assumes JN: "\<And>S. {x. J x = S} \<inter> space N \<in> sets N" 

762 
shows "(\<lambda>x. finmap_of (J x) (f x)) \<in> measurable N (PiF I M)" 

763 
proof (rule measurable_PiF) 

764 
fix x assume "x \<in> space N" 

765 
with J[of x] measurable_space[OF f] 

766 
show "domain (finmap_of (J x) (f x)) \<in> I \<and> 

767 
(\<forall>i\<in>domain (finmap_of (J x) (f x)). (finmap_of (J x) (f x)) i \<in> space (M i))" 

768 
by auto 

769 
next 

770 
fix K S assume "K \<in> I" and *: "\<And>i. i \<in> K \<Longrightarrow> S i \<in> sets (M i)" 

771 
with J have eq: "(\<lambda>x. finmap_of (J x) (f x)) ` Pi' K S \<inter> space N = 

772 
(if \<exists>x \<in> space N. K = J x \<and> finite K then if K = {} then {x \<in> space N. J x = K} 

773 
else (\<Inter>i\<in>K. (\<lambda>x. f x i) ` S i \<inter> {x \<in> space N. J x = K}) else {})" 

774 
by (auto simp: Pi'_def) 

775 
have r: "{x \<in> space N. J x = K} = space N \<inter> ({x. J x = K} \<inter> space N)" by auto 

776 
show "(\<lambda>x. finmap_of (J x) (f x)) ` Pi' K S \<inter> space N \<in> sets N" 

777 
unfolding eq r 

778 
apply (simp del: INT_simps add: ) 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50124
diff
changeset

779 
apply (intro conjI impI sets.finite_INT JN sets.Int[OF sets.top]) 
50088  780 
apply simp apply assumption 
781 
apply (subst Int_assoc[symmetric]) 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50124
diff
changeset

782 
apply (rule sets.Int) 
50088  783 
apply (intro measurable_sets[OF f] *) apply force apply assumption 
784 
apply (intro JN) 

785 
done 

786 
qed 

787 

788 
lemma measurable_PiM_finmap_of: 

789 
assumes "finite J" 

790 
shows "finmap_of J \<in> measurable (Pi\<^isub>M J M) (PiF {J} M)" 

791 
apply (rule measurable_finmap_of) 

792 
apply (rule measurable_component_singleton) 

793 
apply simp 

794 
apply rule 

795 
apply (rule `finite J`) 

796 
apply simp 

797 
done 

798 

799 
lemma proj_measurable_singleton: 

50124  800 
assumes "A \<in> sets (M i)" 
50088  801 
shows "(\<lambda>x. (x)\<^isub>F i) ` A \<inter> space (PiF {I} M) \<in> sets (PiF {I} M)" 
802 
proof cases 

803 
assume "i \<in> I" 

804 
hence "(\<lambda>x. (x)\<^isub>F i) ` A \<inter> space (PiF {I} M) = 

805 
Pi' I (\<lambda>x. if x = i then A else space (M x))" 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50124
diff
changeset

806 
using sets.sets_into_space[OF ] `A \<in> sets (M i)` assms 
50088  807 
by (auto simp: space_PiF Pi'_def) 
808 
thus ?thesis using assms `A \<in> sets (M i)` 

809 
by (intro in_sets_PiFI) auto 

810 
next 

811 
assume "i \<notin> I" 

812 
hence "(\<lambda>x. (x)\<^isub>F i) ` A \<inter> space (PiF {I} M) = 

813 
(if undefined \<in> A then space (PiF {I} M) else {})" by (auto simp: space_PiF Pi'_def) 

814 
thus ?thesis by simp 

815 
qed 

816 

817 
lemma measurable_proj_singleton: 

50124  818 
assumes "i \<in> I" 
50088  819 
shows "(\<lambda>x. (x)\<^isub>F i) \<in> measurable (PiF {I} M) (M i)" 
50124  820 
by (unfold measurable_def, intro CollectI conjI ballI proj_measurable_singleton assms) 
821 
(insert `i \<in> I`, auto simp: space_PiF) 

50088  822 

823 
lemma measurable_proj_countable: 

824 
fixes I::"'a::countable set set" 

825 
assumes "y \<in> space (M i)" 

826 
shows "(\<lambda>x. if i \<in> domain x then (x)\<^isub>F i else y) \<in> measurable (PiF I M) (M i)" 

827 
proof (rule countable_measurable_PiFI) 

828 
fix J assume "J \<in> I" "finite J" 

829 
show "(\<lambda>x. if i \<in> domain x then x i else y) \<in> measurable (PiF {J} M) (M i)" 

830 
unfolding measurable_def 

831 
proof safe 

832 
fix z assume "z \<in> sets (M i)" 

833 
have "(\<lambda>x. if i \<in> domain x then x i else y) ` z \<inter> space (PiF {J} M) = 

834 
(\<lambda>x. if i \<in> J then (x)\<^isub>F i else y) ` z \<inter> space (PiF {J} M)" 

835 
by (auto simp: space_PiF Pi'_def) 

836 
also have "\<dots> \<in> sets (PiF {J} M)" using `z \<in> sets (M i)` `finite J` 

837 
by (cases "i \<in> J") (auto intro!: measurable_sets[OF measurable_proj_singleton]) 

838 
finally show "(\<lambda>x. if i \<in> domain x then x i else y) ` z \<inter> space (PiF {J} M) \<in> 

839 
sets (PiF {J} M)" . 

840 
qed (insert `y \<in> space (M i)`, auto simp: space_PiF Pi'_def) 

841 
qed 

842 

843 
lemma measurable_restrict_proj: 

844 
assumes "J \<in> II" "finite J" 

845 
shows "finmap_of J \<in> measurable (PiM J M) (PiF II M)" 

846 
using assms 

847 
by (intro measurable_finmap_of measurable_component_singleton) auto 

848 

50124  849 
lemma measurable_proj_PiM: 
50088  850 
fixes J K ::"'a::countable set" and I::"'a set set" 
851 
assumes "finite J" "J \<in> I" 

852 
assumes "x \<in> space (PiM J M)" 

50124  853 
shows "proj \<in> measurable (PiF {J} M) (PiM J M)" 
50088  854 
proof (rule measurable_PiM_single) 
855 
show "proj \<in> space (PiF {J} M) \<rightarrow> (\<Pi>\<^isub>E i \<in> J. space (M i))" 

856 
using assms by (auto simp add: space_PiM space_PiF extensional_def sets_PiF Pi'_def) 

857 
next 

858 
fix A i assume A: "i \<in> J" "A \<in> sets (M i)" 

859 
show "{\<omega> \<in> space (PiF {J} M). (\<omega>)\<^isub>F i \<in> A} \<in> sets (PiF {J} M)" 

860 
proof 

861 
have "{\<omega> \<in> space (PiF {J} M). (\<omega>)\<^isub>F i \<in> A} = 

862 
(\<lambda>\<omega>. (\<omega>)\<^isub>F i) ` A \<inter> space (PiF {J} M)" by auto 

863 
also have "\<dots> \<in> sets (PiF {J} M)" 

864 
using assms A by (auto intro: measurable_sets[OF measurable_proj_singleton] simp: space_PiM) 

865 
finally show ?thesis . 

866 
qed simp 

867 
qed 

868 

869 
lemma space_PiF_singleton_eq_product: 

870 
assumes "finite I" 

871 
shows "space (PiF {I} M) = (\<Pi>' i\<in>I. space (M i))" 

872 
by (auto simp: product_def space_PiF assms) 

873 

874 
text {* adapted from @{thm sets_PiM_single} *} 

875 

876 
lemma sets_PiF_single: 

877 
assumes "finite I" "I \<noteq> {}" 

878 
shows "sets (PiF {I} M) = 

879 
sigma_sets (\<Pi>' i\<in>I. space (M i)) 

880 
{{f\<in>\<Pi>' i\<in>I. space (M i). f i \<in> A}  i A. i \<in> I \<and> A \<in> sets (M i)}" 

881 
(is "_ = sigma_sets ?\<Omega> ?R") 

882 
unfolding sets_PiF_singleton 

883 
proof (rule sigma_sets_eqI) 

884 
interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto 

885 
fix A assume "A \<in> {Pi' I X X. X \<in> (\<Pi> j\<in>I. sets (M j))}" 

886 
then obtain X where X: "A = Pi' I X" "X \<in> (\<Pi> j\<in>I. sets (M j))" by auto 

887 
show "A \<in> sigma_sets ?\<Omega> ?R" 

888 
proof  

889 
from `I \<noteq> {}` X have "A = (\<Inter>j\<in>I. {f\<in>space (PiF {I} M). f j \<in> X j})" 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50124
diff
changeset

890 
using sets.sets_into_space 
50088  891 
by (auto simp: space_PiF product_def) blast 
892 
also have "\<dots> \<in> sigma_sets ?\<Omega> ?R" 

893 
using X `I \<noteq> {}` assms by (intro R.finite_INT) (auto simp: space_PiF) 

894 
finally show "A \<in> sigma_sets ?\<Omega> ?R" . 

895 
qed 

896 
next 

897 
fix A assume "A \<in> ?R" 

898 
then obtain i B where A: "A = {f\<in>\<Pi>' i\<in>I. space (M i). f i \<in> B}" "i \<in> I" "B \<in> sets (M i)" 

899 
by auto 

900 
then have "A = (\<Pi>' j \<in> I. if j = i then B else space (M j))" 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50124
diff
changeset

901 
using sets.sets_into_space[OF A(3)] 
50088  902 
apply (auto simp: Pi'_iff split: split_if_asm) 
903 
apply blast 

904 
done 

905 
also have "\<dots> \<in> sigma_sets ?\<Omega> {Pi' I X X. X \<in> (\<Pi> j\<in>I. sets (M j))}" 

906 
using A 

907 
by (intro sigma_sets.Basic ) 

908 
(auto intro: exI[where x="\<lambda>j. if j = i then B else space (M j)"]) 

909 
finally show "A \<in> sigma_sets ?\<Omega> {Pi' I X X. X \<in> (\<Pi> j\<in>I. sets (M j))}" . 

910 
qed 

911 

912 
text {* adapted from @{thm PiE_cong} *} 

913 

914 
lemma Pi'_cong: 

915 
assumes "finite I" 

916 
assumes "\<And>i. i \<in> I \<Longrightarrow> f i = g i" 

917 
shows "Pi' I f = Pi' I g" 

918 
using assms by (auto simp: Pi'_def) 

919 

920 
text {* adapted from @{thm Pi_UN} *} 

921 

922 
lemma Pi'_UN: 

923 
fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set" 

924 
assumes "finite I" 

925 
assumes mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i" 

926 
shows "(\<Union>n. Pi' I (A n)) = Pi' I (\<lambda>i. \<Union>n. A n i)" 

927 
proof (intro set_eqI iffI) 

928 
fix f assume "f \<in> Pi' I (\<lambda>i. \<Union>n. A n i)" 

929 
then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i" "domain f = I" by (auto simp: `finite I` Pi'_def) 

930 
from bchoice[OF this(1)] obtain n where n: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> (A (n i) i)" by auto 

931 
obtain k where k: "\<And>i. i \<in> I \<Longrightarrow> n i \<le> k" 

932 
using `finite I` finite_nat_set_iff_bounded_le[of "n`I"] by auto 

933 
have "f \<in> Pi' I (\<lambda>i. A k i)" 

934 
proof 

935 
fix i assume "i \<in> I" 

936 
from mono[OF this, of "n i" k] k[OF this] n[OF this] `domain f = I` `i \<in> I` 

937 
show "f i \<in> A k i " by (auto simp: `finite I`) 

938 
qed (simp add: `domain f = I` `finite I`) 

939 
then show "f \<in> (\<Union>n. Pi' I (A n))" by auto 

940 
qed (auto simp: Pi'_def `finite I`) 

941 

942 
text {* adapted from @{thm sigma_prod_algebra_sigma_eq} *} 

943 

944 
lemma sigma_fprod_algebra_sigma_eq: 

945 
fixes E :: "'i \<Rightarrow> 'a set set" 

946 
assumes [simp]: "finite I" "I \<noteq> {}" 

947 
assumes S_mono: "\<And>i. i \<in> I \<Longrightarrow> incseq (S i)" 

948 
and S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)" 

949 
and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i" 

950 
assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))" 

951 
and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)" 

952 
defines "P == { Pi' I F  F. \<forall>i\<in>I. F i \<in> E i }" 

953 
shows "sets (PiF {I} M) = sigma_sets (space (PiF {I} M)) P" 

954 
proof 

955 
let ?P = "sigma (space (Pi\<^isub>F {I} M)) P" 

956 
have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>F {I} M))" 

957 
using E_closed by (auto simp: space_PiF P_def Pi'_iff subset_eq) 

958 
then have space_P: "space ?P = (\<Pi>' i\<in>I. space (M i))" 

959 
by (simp add: space_PiF) 

960 
have "sets (PiF {I} M) = 

961 
sigma_sets (space ?P) {{f \<in> \<Pi>' i\<in>I. space (M i). f i \<in> A} i A. i \<in> I \<and> A \<in> sets (M i)}" 

962 
using sets_PiF_single[of I M] by (simp add: space_P) 

963 
also have "\<dots> \<subseteq> sets (sigma (space (PiF {I} M)) P)" 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50124
diff
changeset

964 
proof (safe intro!: sets.sigma_sets_subset) 
50088  965 
fix i A assume "i \<in> I" and A: "A \<in> sets (M i)" 
966 
have "(\<lambda>x. (x)\<^isub>F i) \<in> measurable ?P (sigma (space (M i)) (E i))" 

967 
proof (subst measurable_iff_measure_of) 

968 
show "E i \<subseteq> Pow (space (M i))" using `i \<in> I` by fact 

969 
from space_P `i \<in> I` show "(\<lambda>x. (x)\<^isub>F i) \<in> space ?P \<rightarrow> space (M i)" 

970 
by auto 

971 
show "\<forall>A\<in>E i. (\<lambda>x. (x)\<^isub>F i) ` A \<inter> space ?P \<in> sets ?P" 

972 
proof 

973 
fix A assume A: "A \<in> E i" 

974 
then have "(\<lambda>x. (x)\<^isub>F i) ` A \<inter> space ?P = (\<Pi>' j\<in>I. if i = j then A else space (M j))" 

975 
using E_closed `i \<in> I` by (auto simp: space_P Pi_iff subset_eq split: split_if_asm) 

976 
also have "\<dots> = (\<Pi>' j\<in>I. \<Union>n. if i = j then A else S j n)" 

977 
by (intro Pi'_cong) (simp_all add: S_union) 

978 
also have "\<dots> = (\<Union>n. \<Pi>' j\<in>I. if i = j then A else S j n)" 

979 
using S_mono 

980 
by (subst Pi'_UN[symmetric, OF `finite I`]) (auto simp: incseq_def) 

981 
also have "\<dots> \<in> sets ?P" 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50124
diff
changeset

982 
proof (safe intro!: sets.countable_UN) 
50088  983 
fix n show "(\<Pi>' j\<in>I. if i = j then A else S j n) \<in> sets ?P" 
984 
using A S_in_E 

985 
by (simp add: P_closed) 

986 
(auto simp: P_def subset_eq intro!: exI[of _ "\<lambda>j. if i = j then A else S j n"]) 

987 
qed 

988 
finally show "(\<lambda>x. (x)\<^isub>F i) ` A \<inter> space ?P \<in> sets ?P" 

989 
using P_closed by simp 

990 
qed 

991 
qed 

992 
from measurable_sets[OF this, of A] A `i \<in> I` E_closed 

993 
have "(\<lambda>x. (x)\<^isub>F i) ` A \<inter> space ?P \<in> sets ?P" 

994 
by (simp add: E_generates) 

995 
also have "(\<lambda>x. (x)\<^isub>F i) ` A \<inter> space ?P = {f \<in> \<Pi>' i\<in>I. space (M i). f i \<in> A}" 

996 
using P_closed by (auto simp: space_PiF) 

997 
finally show "\<dots> \<in> sets ?P" . 

998 
qed 

999 
finally show "sets (PiF {I} M) \<subseteq> sigma_sets (space (PiF {I} M)) P" 

1000 
by (simp add: P_closed) 

1001 
show "sigma_sets (space (PiF {I} M)) P \<subseteq> sets (PiF {I} M)" 

1002 
using `finite I` `I \<noteq> {}` 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50124
diff
changeset

1003 
by (auto intro!: sets.sigma_sets_subset product_in_sets_PiFI simp: E_generates P_def) 
50088  1004 
qed 
1005 

50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1006 
lemma sets_PiF_eq_sigma_union_closed_basis_single: 
50088  1007 
assumes "I \<noteq> {}" 
1008 
assumes [simp]: "finite I" 

1009 
shows "sets (PiF {I} (\<lambda>_. borel)) = sigma_sets (space (PiF {I} (\<lambda>_. borel))) 

50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1010 
{Pi' I F F. (\<forall>i\<in>I. F i \<in> union_closed_basis)}" 
50088  1011 
proof  
1012 
from open_incseqE[OF open_UNIV] guess S::"nat \<Rightarrow> 'b set" . note S = this 

1013 
show ?thesis 

1014 
proof (rule sigma_fprod_algebra_sigma_eq) 

1015 
show "finite I" by simp 

1016 
show "I \<noteq> {}" by fact 

50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1017 
show "incseq S" "(\<Union>j. S j) = space borel" "range S \<subseteq> union_closed_basis" 
50088  1018 
using S by simp_all 
50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1019 
show "union_closed_basis \<subseteq> Pow (space borel)" by simp 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1020 
show "sets borel = sigma_sets (space borel) union_closed_basis" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1021 
by (simp add: borel_eq_union_closed_basis) 
50088  1022 
qed 
1023 
qed 

1024 

50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1025 
text {* adapted from @{thm sets_PiF_eq_sigma_union_closed_basis_single} *} 
50088  1026 

50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1027 
lemma sets_PiM_eq_sigma_union_closed_basis: 
50088  1028 
assumes "I \<noteq> {}" 
1029 
assumes [simp]: "finite I" 

1030 
shows "sets (PiM I (\<lambda>_. borel)) = sigma_sets (space (PiM I (\<lambda>_. borel))) 

50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1031 
{Pi\<^isub>E I F F. \<forall>i\<in>I. F i \<in> union_closed_basis}" 
50088  1032 
proof  
1033 
from open_incseqE[OF open_UNIV] guess S::"nat \<Rightarrow> 'b set" . note S = this 

1034 
show ?thesis 

1035 
proof (rule sigma_prod_algebra_sigma_eq) 

1036 
show "finite I" by simp note[[show_types]] 

50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1037 
fix i show "(\<Union>j. S j) = space borel" "range S \<subseteq> union_closed_basis" 
50088  1038 
using S by simp_all 
50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1039 
show "union_closed_basis \<subseteq> Pow (space borel)" by simp 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1040 
show "sets borel = sigma_sets (space borel) union_closed_basis" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1041 
by (simp add: borel_eq_union_closed_basis) 
50088  1042 
qed 
1043 
qed 

1044 

1045 
lemma product_open_generates_sets_PiF_single: 

1046 
assumes "I \<noteq> {}" 

1047 
assumes [simp]: "finite I" 

50881
ae630bab13da
renamed countable_basis_space to second_countable_topology
hoelzl
parents:
50251
diff
changeset

1048 
shows "sets (PiF {I} (\<lambda>_. borel::'b::second_countable_topology measure)) = 
50088  1049 
sigma_sets (space (PiF {I} (\<lambda>_. borel))) {Pi' I F F. (\<forall>i\<in>I. F i \<in> Collect open)}" 
1050 
proof  

1051 
from open_incseqE[OF open_UNIV] guess S::"nat \<Rightarrow> 'b set" . note S = this 

1052 
show ?thesis 

1053 
proof (rule sigma_fprod_algebra_sigma_eq) 

1054 
show "finite I" by simp 

1055 
show "I \<noteq> {}" by fact 

1056 
show "incseq S" "(\<Union>j. S j) = space borel" "range S \<subseteq> Collect open" 

50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1057 
using S by (auto simp: open_union_closed_basis) 
50088  1058 
show "Collect open \<subseteq> Pow (space borel)" by simp 
1059 
show "sets borel = sigma_sets (space borel) (Collect open)" 

1060 
by (simp add: borel_def) 

1061 
qed 

1062 
qed 

1063 

1064 
lemma product_open_generates_sets_PiM: 

1065 
assumes "I \<noteq> {}" 

1066 
assumes [simp]: "finite I" 

50881
ae630bab13da
renamed countable_basis_space to second_countable_topology
hoelzl
parents:
50251
diff
changeset

1067 
shows "sets (PiM I (\<lambda>_. borel::'b::second_countable_topology measure)) = 
50088  1068 
sigma_sets (space (PiM I (\<lambda>_. borel))) {Pi\<^isub>E I F F. \<forall>i\<in>I. F i \<in> Collect open}" 
1069 
proof  

1070 
from open_incseqE[OF open_UNIV] guess S::"nat \<Rightarrow> 'b set" . note S = this 

1071 
show ?thesis 

1072 
proof (rule sigma_prod_algebra_sigma_eq) 

1073 
show "finite I" by simp note[[show_types]] 

1074 
fix i show "(\<Union>j. S j) = space borel" "range S \<subseteq> Collect open" 

50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1075 
using S by (auto simp: open_union_closed_basis) 
50088  1076 
show "Collect open \<subseteq> Pow (space borel)" by simp 
1077 
show "sets borel = sigma_sets (space borel) (Collect open)" 

1078 
by (simp add: borel_def) 

1079 
qed 

1080 
qed 

1081 

50124  1082 
lemma finmap_UNIV[simp]: "(\<Union>J\<in>Collect finite. PI' j : J. UNIV) = UNIV" by auto 
50088  1083 

1084 
lemma borel_eq_PiF_borel: 

1085 
shows "(borel :: ('i::countable \<Rightarrow>\<^isub>F 'a::polish_space) measure) = 

50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1086 
PiF (Collect finite) (\<lambda>_. borel :: 'a measure)" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1087 
unfolding borel_def PiF_def 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1088 
proof (rule measure_eqI, clarsimp, rule sigma_sets_eqI) 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1089 
fix a::"('i \<Rightarrow>\<^isub>F 'a) set" assume "a \<in> Collect open" hence "open a" by simp 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1090 
then obtain B' where B': "B'\<subseteq>basis_finmap" "a = \<Union>B'" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1091 
using finmap_topological_basis by (force simp add: topological_basis_def) 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1092 
have "a \<in> sigma UNIV {Pi' J X X J. finite J \<and> X \<in> J \<rightarrow> sigma_sets UNIV (Collect open)}" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1093 
unfolding `a = \<Union>B'` 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1094 
proof (rule sets.countable_Union) 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1095 
from B' countable_basis_finmap show "countable B'" by (metis countable_subset) 
50088  1096 
next 
50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1097 
show "B' \<subseteq> sets (sigma UNIV 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1098 
{Pi' J X X J. finite J \<and> X \<in> J \<rightarrow> sigma_sets UNIV (Collect open)})" (is "_ \<subseteq> sets ?s") 
50088  1099 
proof 
50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1100 
fix x assume "x \<in> B'" with B' have "x \<in> basis_finmap" by auto 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1101 
then obtain J X where "x = Pi' J X" "finite J" "X \<in> J \<rightarrow> sigma_sets UNIV (Collect open)" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1102 
by (auto simp: basis_finmap_def open_union_closed_basis) 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1103 
thus "x \<in> sets ?s" by auto 
50088  1104 
qed 
1105 
qed 

50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1106 
thus "a \<in> sigma_sets UNIV {Pi' J X X J. finite J \<and> X \<in> J \<rightarrow> sigma_sets UNIV (Collect open)}" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1107 
by simp 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1108 
next 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1109 
fix b::"('i \<Rightarrow>\<^isub>F 'a) set" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1110 
assume "b \<in> {Pi' J X X J. finite J \<and> X \<in> J \<rightarrow> sigma_sets UNIV (Collect open)}" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1111 
hence b': "b \<in> sets (Pi\<^isub>F (Collect finite) (\<lambda>_. borel))" by (auto simp: sets_PiF borel_def) 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1112 
let ?b = "\<lambda>J. b \<inter> {x. domain x = J}" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1113 
have "b = \<Union>((\<lambda>J. ?b J) ` Collect finite)" by auto 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1114 
also have "\<dots> \<in> sets borel" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1115 
proof (rule sets.countable_Union, safe) 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1116 
fix J::"'i set" assume "finite J" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1117 
{ assume ef: "J = {}" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1118 
have "?b J \<in> sets borel" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1119 
proof cases 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1120 
assume "?b J \<noteq> {}" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1121 
then obtain f where "f \<in> b" "domain f = {}" using ef by auto 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1122 
hence "?b J = {f}" using `J = {}` 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1123 
by (auto simp: finmap_eq_iff) 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1124 
also have "{f} \<in> sets borel" by simp 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1125 
finally show ?thesis . 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1126 
qed simp 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1127 
} moreover { 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1128 
assume "J \<noteq> ({}::'i set)" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1129 
have "(?b J) = b \<inter> {m. domain m \<in> {J}}" by auto 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1130 
also have "\<dots> \<in> sets (PiF {J} (\<lambda>_. borel))" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1131 
using b' by (rule restrict_sets_measurable) (auto simp: `finite J`) 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1132 
also have "\<dots> = sigma_sets (space (PiF {J} (\<lambda>_. borel))) 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1133 
{Pi' (J) F F. (\<forall>j\<in>J. F j \<in> Collect open)}" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1134 
(is "_ = sigma_sets _ ?P") 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1135 
by (rule product_open_generates_sets_PiF_single[OF `J \<noteq> {}` `finite J`]) 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1136 
also have "\<dots> \<subseteq> sigma_sets UNIV (Collect open)" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1137 
by (intro sigma_sets_mono'') (auto intro!: open_Pi'I simp: space_PiF) 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1138 
finally have "(?b J) \<in> sets borel" by (simp add: borel_def) 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1139 
} ultimately show "(?b J) \<in> sets borel" by blast 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1140 
qed (simp add: countable_Collect_finite) 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1141 
finally show "b \<in> sigma_sets UNIV (Collect open)" by (simp add: borel_def) 
50088  1142 
qed (simp add: emeasure_sigma borel_def PiF_def) 
1143 

1144 
subsection {* Isomorphism between Functions and Finite Maps *} 

1145 

50124  1146 
lemma measurable_finmap_compose: 
50088  1147 
shows "(\<lambda>m. compose J m f) \<in> measurable (PiM (f ` J) (\<lambda>_. M)) (PiM J (\<lambda>_. M))" 
50124  1148 
unfolding compose_def by measurable 
50088  1149 

50124  1150 
lemma measurable_compose_inv: 
50088  1151 
assumes inj: "\<And>j. j \<in> J \<Longrightarrow> f' (f j) = j" 
1152 
shows "(\<lambda>m. compose (f ` J) m f') \<in> measurable (PiM J (\<lambda>_. M)) (PiM (f ` J) (\<lambda>_. M))" 

50124  1153 
unfolding compose_def by (rule measurable_restrict) (auto simp: inj) 
50088  1154 

1155 
locale function_to_finmap = 

1156 
fixes J::"'a set" and f :: "'a \<Rightarrow> 'b::countable" and f' 

1157 
assumes [simp]: "finite J" 

1158 
assumes inv: "i \<in> J \<Longrightarrow> f' (f i) = i" 

1159 
begin 

1160 

1161 
text {* to measure finmaps *} 

1162 

1163 
definition "fm = (finmap_of (f ` J)) o (\<lambda>g. compose (f ` J) g f')" 

1164 

1165 
lemma domain_fm[simp]: "domain (fm x) = f ` J" 

1166 
unfolding fm_def by simp 

1167 

1168 
lemma fm_restrict[simp]: "fm (restrict y J) = fm y" 

1169 
unfolding fm_def by (auto simp: compose_def inv intro: restrict_ext) 

1170 

1171 
lemma fm_product: 

1172 
assumes "\<And>i. space (M i) = UNIV" 

1173 
shows "fm ` Pi' (f ` J) S \<inter> space (Pi\<^isub>M J M) = (\<Pi>\<^isub>E j \<in> J. S (f j))" 

1174 
using assms 

1175 
by (auto simp: inv fm_def compose_def space_PiM Pi'_def) 

1176 

1177 
lemma fm_measurable: 

1178 
assumes "f ` J \<in> N" 

1179 
shows "fm \<in> measurable (Pi\<^isub>M J (\<lambda>_. M)) (Pi\<^isub>F N (\<lambda>_. M))" 

1180 
unfolding fm_def 

1181 
proof (rule measurable_comp, rule measurable_compose_inv) 

1182 
show "finmap_of (f ` J) \<in> measurable (Pi\<^isub>M (f ` J) (\<lambda>_. M)) (PiF N (\<lambda>_. M)) " 

1183 
using assms by (intro measurable_finmap_of measurable_component_singleton) auto 

1184 
qed (simp_all add: inv) 

1185 

1186 
lemma proj_fm: 

1187 
assumes "x \<in> J" 

1188 
shows "fm m (f x) = m x" 

1189 
using assms by (auto simp: fm_def compose_def o_def inv) 

1190 

1191 
lemma inj_on_compose_f': "inj_on (\<lambda>g. compose (f ` J) g f') (extensional J)" 

1192 
proof (rule inj_on_inverseI) 

1193 
fix x::"'a \<Rightarrow> 'c" assume "x \<in> extensional J" 

1194 
thus "(\<lambda>x. compose J x f) (compose (f ` J) x f') = x" 

1195 
by (auto simp: compose_def inv extensional_def) 

1196 
qed 

1197 

1198 
lemma inj_on_fm: 

1199 
assumes "\<And>i. space (M i) = UNIV" 

1200 
shows "inj_on fm (space (Pi\<^isub>M J M))" 

1201 
using assms 

50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50100
diff
changeset

1202 
apply (auto simp: fm_def space_PiM PiE_def) 
50088  1203 
apply (rule comp_inj_on) 
1204 
apply (rule inj_on_compose_f') 

1205 
apply (rule finmap_of_inj_on_extensional_finite) 

1206 
apply simp 

1207 
apply (auto) 

1208 
done 

1209 

1210 
text {* to measure functions *} 

1211 

1212 
definition "mf = (\<lambda>g. compose J g f) o proj" 

1213 

1214 
lemma mf_fm: 

1215 
assumes "x \<in> space (Pi\<^isub>M J (\<lambda>_. M))" 

1216 
shows "mf (fm x) = x" 

1217 
proof  

1218 
have "mf (fm x) \<in> extensional J" 

1219 
by (auto simp: mf_def extensional_def compose_def) 

1220 
moreover 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50124
diff
changeset

1221 
have "x \<in> extensional J" using assms sets.sets_into_space 
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50100
diff
changeset

1222 
by (force simp: space_PiM PiE_def) 
50088  1223 
moreover 
1224 
{ fix i assume "i \<in> J" 

1225 
hence "mf (fm x) i = x i" 

1226 
by (auto simp: inv mf_def compose_def fm_def) 

1227 
} 

1228 
ultimately 

1229 
show ?thesis by (rule extensionalityI) 

1230 
qed 

1231 

1232 
lemma mf_measurable: 

1233 
assumes "space M = UNIV" 

1234 
shows "mf \<in> measurable (PiF {f ` J} (\<lambda>_. M)) (PiM J (\<lambda>_. M))" 

1235 
unfolding mf_def 

1236 
proof (rule measurable_comp, rule measurable_proj_PiM) 

50124  1237 
show "(\<lambda>g. compose J g f) \<in> measurable (Pi\<^isub>M (f ` J) (\<lambda>x. M)) (Pi\<^isub>M J (\<lambda>_. M))" 
1238 
by (rule measurable_finmap_compose) 

50088  1239 
qed (auto simp add: space_PiM extensional_def assms) 
1240 

1241 
lemma fm_image_measurable: 

1242 
assumes "space M = UNIV" 

1243 
assumes "X \<in> sets (Pi\<^isub>M J (\<lambda>_. M))" 

1244 
shows "fm ` X \<in> sets (PiF {f ` J} (\<lambda>_. M))" 

1245 
proof  

1246 
have "fm ` X = (mf) ` X \<inter> space (PiF {f ` J} (\<lambda>_. M))" 

1247 
proof safe 

1248 
fix x assume "x \<in> X" 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50124
diff
changeset

1249 
with mf_fm[of x] sets.sets_into_space[OF assms(2)] show "fm x \<in> mf ` X" by auto 
50088  1250 
show "fm x \<in> space (PiF {f ` J} (\<lambda>_. M))" by (simp add: space_PiF assms) 
1251 
next 

1252 
fix y x 

1253 
assume x: "mf y \<in> X" 

1254 
assume y: "y \<in> space (PiF {f ` J} (\<lambda>_. M))" 

1255 
thus "y \<in> fm ` X" 

1256 
by (intro image_eqI[OF _ x], unfold finmap_eq_iff) 

1257 
(auto simp: space_PiF fm_def mf_def compose_def inv Pi'_def) 

1258 
qed 

1259 
also have "\<dots> \<in> sets (PiF {f ` J} (\<lambda>_. M))" 

1260 
using assms 

1261 
by (intro measurable_sets[OF mf_measurable]) auto 

1262 
finally show ?thesis . 

1263 
qed 

1264 

1265 
lemma fm_image_measurable_finite: 

1266 
assumes "space M = UNIV" 

1267 
assumes "X \<in> sets (Pi\<^isub>M J (\<lambda>_. M::'c measure))" 

1268 
shows "fm ` X \<in> sets (PiF (Collect finite) (\<lambda>_. M::'c measure))" 

1269 
using fm_image_measurable[OF assms] 

1270 
by (rule subspace_set_in_sets) (auto simp: finite_subset) 

1271 

1272 
text {* measure on finmaps *} 

1273 

1274 
definition "mapmeasure M N = distr M (PiF (Collect finite) N) (fm)" 

1275 

1276 
lemma sets_mapmeasure[simp]: "sets (mapmeasure M N) = sets (PiF (Collect finite) N)" 

1277 
unfolding mapmeasure_def by simp 

1278 

1279 
lemma space_mapmeasure[simp]: "space (mapmeasure M N) = space (PiF (Collect finite) N)" 

1280 
unfolding mapmeasure_def by simp 

1281 

1282 
lemma mapmeasure_PiF: 

1283 
assumes s1: "space M = space (Pi\<^isub>M J (\<lambda>_. N))" 

50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50100
diff
changeset

1284 
assumes s2: "sets M = sets (Pi\<^isub>M J (\<lambda>_. N))" 
50088  1285 
assumes "space N = UNIV" 
1286 
assumes "X \<in> sets (PiF (Collect finite) (\<lambda>_. N))" 

1287 
shows "emeasure (mapmeasure M (\<lambda>_. N)) X = emeasure M ((fm ` X \<inter> extensional J))" 

1288 
using assms 

1289 
by (auto simp: measurable_eqI[OF s1 refl s2 refl] mapmeasure_def emeasure_distr 

50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50100
diff
changeset

1290 
fm_measurable space_PiM PiE_def) 
50088  1291 

1292 
lemma mapmeasure_PiM: 

1293 
fixes N::"'c measure" 

1294 
assumes s1: "space M = space (Pi\<^isub>M J (\<lambda>_. N))" 

1295 
assumes s2: "sets M = (Pi\<^isub>M J (\<lambda>_. N))" 

1296 
assumes N: "space N = UNIV" 

1297 
assumes X: "X \<in> sets M" 

1298 
shows "emeasure M X = emeasure (mapmeasure M (\<lambda>_. N)) (fm ` X)" 

1299 
unfolding mapmeasure_def 

1300 
proof (subst emeasure_distr, subst measurable_eqI[OF s1 refl s2 refl], rule fm_measurable) 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50124
diff
changeset

1301 
have "X \<subseteq> space (Pi\<^isub>M J (\<lambda>_. N))" using assms by (simp add: sets.sets_into_space) 
50088  1302 
from assms inj_on_fm[of "\<lambda>_. N"] set_mp[OF this] have "fm ` fm ` X \<inter> space (Pi\<^isub>M J (\<lambda>_. N)) = X" 
1303 
by (auto simp: vimage_image_eq inj_on_def) 

1304 
thus "emeasure M X = emeasure M (fm ` fm ` X \<inter> space M)" using s1 

1305 
by simp 

1306 
show "fm ` X \<in> sets (PiF (Collect finite) (\<lambda>_. N))" 

1307 
by (rule fm_image_measurable_finite[OF N X[simplified s2]]) 

1308 
qed simp 

1309 

1310 
end 

1311 

1312 
end 