author  immler 
Thu, 15 Nov 2012 17:36:08 +0100  
changeset 50091  b3b5dc2350b7 
parent 50088  32d1795cc77a 
child 50094  84ddcf5364b4 
permissions  rwrr 
50091  1 
(* 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" [1000] 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 

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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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abbreviation 

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finmapset :: "['a set, 'b set] => ('a \<Rightarrow>\<^isub>F 'b) set" 

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(infixr "~>" 60) where 

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"A ~> B \<equiv> Pi' A (%_. B)" 

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notation (xsymbols) 

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finmapset (infixr "\<leadsto>" 60) 

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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 finmapsetI: "domain f = A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<in> B) \<Longrightarrow> f \<in> A \<leadsto> 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 funcset_mem: "[f \<in> A \<leadsto> B; x \<in> A] ==> f x \<in> B" 

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by (simp add: Pi'_def) 

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lemma funcset_image: "f \<in> A \<leadsto> B ==> f ` A \<subseteq> B" 

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by auto 

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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 

362 
moreover 

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

364 
hence "y i = undefined" using `i \<notin> domain x` by simp 

365 
ultimately 

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

367 
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 *} 

373 

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lemma tendsto_dist_zero: 

375 
assumes "(\<lambda>i. dist (f i) g) > 0" 

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shows "f > g" 

377 
using assms by (auto simp: tendsto_iff dist_real_def) 

378 

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lemma tendsto_dist_zero': 

380 
assumes "(\<lambda>i. dist (f i) g) > x" 

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assumes "0 = x" 

382 
shows "f > g" 

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using assms tendsto_dist_zero by simp 

384 

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lemma tendsto_finmap: 

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

387 
assumes ind_f: "\<And>n. domain (f n) = domain g" 

388 
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') 

391 
unfolding dist_finmap_def assms 

392 
apply (rule tendsto_intros proj_g  simp)+ 

393 
done 

394 

395 
instance finmap :: (type, complete_space) complete_space 

396 
proof 

397 
fix P::"nat \<Rightarrow> 'a \<Rightarrow>\<^isub>F 'b" 

398 
assume "Cauchy P" 

399 
then obtain Nd where Nd: "\<And>n. n \<ge> Nd \<Longrightarrow> dist (P n) (P Nd) < 1" 

400 
by (force simp: cauchy) 

401 
def d \<equiv> "domain (P Nd)" 

402 
with Nd have dim: "\<And>n. n \<ge> Nd \<Longrightarrow> domain (P n) = d" using dist_le_1_imp_domain_eq by auto 

403 
have [simp]: "finite d" unfolding d_def by simp 

404 
def p \<equiv> "\<lambda>i n. (P n) i" 

405 
def q \<equiv> "\<lambda>i. lim (p i)" 

406 
def Q \<equiv> "finmap_of d q" 

407 
have q: "\<And>i. i \<in> d \<Longrightarrow> q i = Q i" by (auto simp add: Q_def Abs_finmap_inverse) 

408 
{ 

409 
fix i assume "i \<in> d" 

410 
have "Cauchy (p i)" unfolding cauchy p_def 

411 
proof safe 

412 
fix e::real assume "0 < e" 

413 
with `Cauchy P` obtain N where N: "\<And>n. n\<ge>N \<Longrightarrow> dist (P n) (P N) < min e 1" 

414 
by (force simp: cauchy min_def) 

415 
hence "\<And>n. n \<ge> N \<Longrightarrow> domain (P n) = domain (P N)" using dist_le_1_imp_domain_eq by auto 

416 
with dim have dim: "\<And>n. n \<ge> N \<Longrightarrow> domain (P n) = d" by (metis nat_le_linear) 

417 
show "\<exists>N. \<forall>n\<ge>N. dist ((P n) i) ((P N) i) < e" 

418 
proof (safe intro!: exI[where x="N"]) 

419 
fix n assume "N \<le> n" have "N \<le> N" by simp 

420 
have "dist ((P n) i) ((P N) i) \<le> dist (P n) (P N)" 

421 
using dim[OF `N \<le> n`] dim[OF `N \<le> N`] `i \<in> d` 

422 
by (auto intro!: dist_proj) 

423 
also have "\<dots> < e" using N[OF `N \<le> n`] by simp 

424 
finally show "dist ((P n) i) ((P N) i) < e" . 

425 
qed 

426 
qed 

427 
hence "convergent (p i)" by (metis Cauchy_convergent_iff) 

428 
hence "p i > q i" unfolding q_def convergent_def by (metis limI) 

429 
} note p = this 

430 
have "P > Q" 

431 
proof (rule metric_LIMSEQ_I) 

432 
fix e::real assume "0 < e" 

433 
def e' \<equiv> "min 1 (e / (card d + 1))" 

434 
hence "0 < e'" using `0 < e` by (auto simp: e'_def intro: divide_pos_pos) 

435 
have "\<exists>ni. \<forall>i\<in>d. \<forall>n\<ge>ni i. dist (p i n) (q i) < e'" 

436 
proof (safe intro!: bchoice) 

437 
fix i assume "i \<in> d" 

438 
from p[OF `i \<in> d`, THEN metric_LIMSEQ_D, OF `0 < e'`] 

439 
show "\<exists>no. \<forall>n\<ge>no. dist (p i n) (q i) < e'" . 

440 
qed then guess ni .. note ni = this 

441 
def N \<equiv> "max Nd (Max (ni ` d))" 

442 
show "\<exists>N. \<forall>n\<ge>N. dist (P n) Q < e" 

443 
proof (safe intro!: exI[where x="N"]) 

444 
fix n assume "N \<le> n" 

445 
hence "domain (P n) = d" "domain Q = d" "domain (P n) = domain Q" 

446 
using dim by (simp_all add: N_def Q_def dim_def Abs_finmap_inverse) 

447 
hence "dist (P n) Q = (\<Sum>i\<in>d. dist ((P n) i) (Q i))" by (simp add: dist_finmap_def) 

448 
also have "\<dots> \<le> (\<Sum>i\<in>d. e')" 

449 
proof (intro setsum_mono less_imp_le) 

450 
fix i assume "i \<in> d" 

451 
hence "ni i \<le> Max (ni ` d)" by simp 

452 
also have "\<dots> \<le> N" by (simp add: N_def) 

453 
also have "\<dots> \<le> n" using `N \<le> n` . 

454 
finally 

455 
show "dist ((P n) i) (Q i) < e'" 

456 
using ni `i \<in> d` by (auto simp: p_def q N_def) 

457 
qed 

458 
also have "\<dots> = card d * e'" by (simp add: real_eq_of_nat) 

459 
also have "\<dots> < e" using `0 < e` by (simp add: e'_def field_simps min_def) 

460 
finally show "dist (P n) Q < e" . 

461 
qed 

462 
qed 

463 
thus "convergent P" by (auto simp: convergent_def) 

464 
qed 

465 

466 
subsection {* Polish Space of Finite Maps *} 

467 

468 
instantiation finmap :: (countable, polish_space) polish_space 

469 
begin 

470 

471 
definition enum_basis_finmap :: "nat \<Rightarrow> ('a \<Rightarrow>\<^isub>F 'b) set" where 

472 
"enum_basis_finmap n = 

473 
(let m = from_nat n::('a \<Rightarrow>\<^isub>F nat) in Pi' (domain m) (enum_basis o (m)\<^isub>F))" 

474 

475 
lemma range_enum_basis_eq: 

476 
"range enum_basis_finmap = {Pi' I SI S. finite I \<and> (\<forall>i \<in> I. S i \<in> range enum_basis)}" 

477 
proof (auto simp: enum_basis_finmap_def[abs_def]) 

478 
fix S::"('a \<Rightarrow> 'b set)" and I 

479 
assume "\<forall>i\<in>I. S i \<in> range enum_basis" 

480 
hence "\<forall>i\<in>I. \<exists>n. S i = enum_basis n" by auto 

481 
then obtain n where n: "\<forall>i\<in>I. S i = enum_basis (n i)" 

482 
unfolding bchoice_iff by blast 

483 
assume [simp]: "finite I" 

484 
have "\<exists>fm. domain fm = I \<and> (\<forall>i\<in>I. n i = (fm i))" 

485 
by (rule finmap_choice) auto 

486 
then obtain m where "Pi' I S = Pi' (domain m) (enum_basis o m)" 

487 
using n by (auto simp: Pi'_def) 

488 
hence "Pi' I S = (let m = from_nat (to_nat m) in Pi' (domain m) (enum_basis \<circ> m))" 

489 
by simp 

490 
thus "Pi' I S \<in> range (\<lambda>n. let m = from_nat n in Pi' (domain m) (enum_basis \<circ> m))" 

491 
by blast 

492 
qed (metis finite_domain o_apply rangeI) 

493 

494 
lemma in_enum_basis_finmapI: 

495 
assumes "finite I" assumes "\<And>i. i \<in> I \<Longrightarrow> S i \<in> range enum_basis" 

496 
shows "Pi' I S \<in> range enum_basis_finmap" 

497 
using assms unfolding range_enum_basis_eq by auto 

498 

499 
lemma finmap_topological_basis: 

500 
"topological_basis (range (enum_basis_finmap))" 

501 
proof (subst topological_basis_iff, safe) 

502 
fix n::nat 

503 
show "open (enum_basis_finmap n::('a \<Rightarrow>\<^isub>F 'b) set)" using enumerable_basis 

504 
by (auto intro!: open_Pi'I simp: topological_basis_def enum_basis_finmap_def Let_def) 

505 
next 

506 
fix O'::"('a \<Rightarrow>\<^isub>F 'b) set" and x 

507 
assume "open O'" "x \<in> O'" 

508 
then obtain e where e: "e > 0" "\<And>y. dist y x < e \<Longrightarrow> y \<in> O'" unfolding open_dist by blast 

509 
def e' \<equiv> "e / (card (domain x) + 1)" 

510 

511 
have "\<exists>B. 

512 
(\<forall>i\<in>domain x. x i \<in> enum_basis (B i) \<and> enum_basis (B i) \<subseteq> ball (x i) e')" 

513 
proof (rule bchoice, safe) 

514 
fix i assume "i \<in> domain x" 

515 
have "open (ball (x i) e')" "x i \<in> ball (x i) e'" using e 

516 
by (auto simp add: e'_def intro!: divide_pos_pos) 

517 
from enumerable_basisE[OF this] guess b' . 

518 
thus "\<exists>y. x i \<in> enum_basis y \<and> 

519 
enum_basis y \<subseteq> ball (x i) e'" by auto 

520 
qed 

521 
then guess B .. note B = this 

522 
def B' \<equiv> "Pi' (domain x) (\<lambda>i. enum_basis (B i)::'b set)" 

523 
hence "B' \<in> range enum_basis_finmap" unfolding B'_def 

524 
by (intro in_enum_basis_finmapI) auto 

525 
moreover have "x \<in> B'" unfolding B'_def using B by auto 

526 
moreover have "B' \<subseteq> O'" 

527 
proof 

528 
fix y assume "y \<in> B'" with B have "domain y = domain x" unfolding B'_def 

529 
by (simp add: Pi'_def) 

530 
show "y \<in> O'" 

531 
proof (rule e) 

532 
have "dist y x = (\<Sum>i \<in> domain x. dist (y i) (x i))" 

533 
using `domain y = domain x` by (simp add: dist_finmap_def) 

534 
also have "\<dots> \<le> (\<Sum>i \<in> domain x. e')" 

535 
proof (rule setsum_mono) 

536 
fix i assume "i \<in> domain x" 

537 
with `y \<in> B'` B have "y i \<in> enum_basis (B i)" 

538 
by (simp add: Pi'_def B'_def) 

539 
hence "y i \<in> ball (x i) e'" using B `domain y = domain x` `i \<in> domain x` 

540 
by force 

541 
thus "dist (y i) (x i) \<le> e'" by (simp add: dist_commute) 

542 
qed 

543 
also have "\<dots> = card (domain x) * e'" by (simp add: real_eq_of_nat) 

544 
also have "\<dots> < e" using e by (simp add: e'_def field_simps) 

545 
finally show "dist y x < e" . 

546 
qed 

547 
qed 

548 
ultimately 

549 
show "\<exists>B'\<in>range enum_basis_finmap. x \<in> B' \<and> B' \<subseteq> O'" by blast 

550 
qed 

551 

552 
lemma range_enum_basis_finmap_imp_open: 

553 
assumes "x \<in> range enum_basis_finmap" 

554 
shows "open x" 

555 
using finmap_topological_basis assms by (auto simp: topological_basis_def) 

556 

557 
lemma 

558 
open_imp_ex_UNION_of_enum: 

559 
fixes X::"('a \<Rightarrow>\<^isub>F 'b) set" 

560 
assumes "open X" assumes "X \<noteq> {}" 

561 
shows "\<exists>A::nat\<Rightarrow>'a set. \<exists>B::nat\<Rightarrow>('a \<Rightarrow> 'b set) . X = UNION UNIV (\<lambda>i. Pi' (A i) (B i)) \<and> 

562 
(\<forall>n. \<forall>i\<in>A n. (B n) i \<in> range enum_basis) \<and> (\<forall>n. finite (A n))" 

563 
proof  

564 
from `open X` obtain B' where B': "B'\<subseteq>range enum_basis_finmap" "\<Union>B' = X" 

565 
using finmap_topological_basis by (force simp add: topological_basis_def) 

566 
then obtain B where B: "B' = enum_basis_finmap ` B" by (auto simp: subset_image_iff) 

567 
show ?thesis 

568 
proof cases 

569 
assume "B = {}" with B have "B' = {}" by simp hence False using B' assms by simp 

570 
thus ?thesis by simp 

571 
next 

572 
assume "B \<noteq> {}" then obtain b where b: "b \<in> B" by auto 

573 
def NA \<equiv> "\<lambda>n::nat. if n \<in> B 

574 
then domain ((from_nat::_\<Rightarrow>'a \<Rightarrow>\<^isub>F nat) n) 

575 
else domain ((from_nat::_\<Rightarrow>'a\<Rightarrow>\<^isub>F nat) b)" 

576 
def NB \<equiv> "\<lambda>n::nat. if n \<in> B 

577 
then (\<lambda>i. (enum_basis::nat\<Rightarrow>'b set) (((from_nat::_\<Rightarrow>'a \<Rightarrow>\<^isub>F nat) n) i)) 

578 
else (\<lambda>i. (enum_basis::nat\<Rightarrow>'b set) (((from_nat::_\<Rightarrow>'a \<Rightarrow>\<^isub>F nat) b) i))" 

579 
have "X = UNION UNIV (\<lambda>i. Pi' (NA i) (NB i))" unfolding B'(2)[symmetric] using b 

580 
unfolding B 

581 
by safe 

582 
(auto simp add: NA_def NB_def enum_basis_finmap_def Let_def o_def split: split_if_asm) 

583 
moreover 

584 
have "(\<forall>n. \<forall>i\<in>NA n. (NB n) i \<in> range enum_basis)" 

585 
using enumerable_basis by (auto simp: topological_basis_def NA_def NB_def) 

586 
moreover have "(\<forall>n. finite (NA n))" by (simp add: NA_def) 

587 
ultimately show ?thesis by auto 

588 
qed 

589 
qed 

590 

591 
lemma 

592 
open_imp_ex_UNION: 

593 
fixes X::"('a \<Rightarrow>\<^isub>F 'b) set" 

594 
assumes "open X" assumes "X \<noteq> {}" 

595 
shows "\<exists>A::nat\<Rightarrow>'a set. \<exists>B::nat\<Rightarrow>('a \<Rightarrow> 'b set) . X = UNION UNIV (\<lambda>i. Pi' (A i) (B i)) \<and> 

596 
(\<forall>n. \<forall>i\<in>A n. open ((B n) i)) \<and> (\<forall>n. finite (A n))" 

597 
using open_imp_ex_UNION_of_enum[OF assms] 

598 
apply auto 

599 
apply (rule_tac x = A in exI) 

600 
apply (rule_tac x = B in exI) 

601 
apply (auto simp: open_enum_basis) 

602 
done 

603 

604 
lemma 

605 
open_basisE: 

606 
assumes "open X" assumes "X \<noteq> {}" 

607 
obtains A::"nat\<Rightarrow>'a set" and B::"nat\<Rightarrow>('a \<Rightarrow> 'b set)" where 

608 
"X = UNION UNIV (\<lambda>i. Pi' (A i) (B i))" "\<And>n i. i\<in>A n \<Longrightarrow> open ((B n) i)" "\<And>n. finite (A n)" 

609 
using open_imp_ex_UNION[OF assms] by auto 

610 

611 
lemma 

612 
open_basis_of_enumE: 

613 
assumes "open X" assumes "X \<noteq> {}" 

614 
obtains A::"nat\<Rightarrow>'a set" and B::"nat\<Rightarrow>('a \<Rightarrow> 'b set)" where 

615 
"X = UNION UNIV (\<lambda>i. Pi' (A i) (B i))" "\<And>n i. i\<in>A n \<Longrightarrow> (B n) i \<in> range enum_basis" 

616 
"\<And>n. finite (A n)" 

617 
using open_imp_ex_UNION_of_enum[OF assms] by auto 

618 

619 
instance proof qed (blast intro: finmap_topological_basis) 

620 

621 
end 

622 

623 
subsection {* Product Measurable Space of Finite Maps *} 

624 

625 
definition "PiF I M \<equiv> 

626 
sigma 

627 
(\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j))) 

628 
{(\<Pi>' j\<in>J. X j) X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}" 

629 

630 
abbreviation 

631 
"Pi\<^isub>F I M \<equiv> PiF I M" 

632 

633 
syntax 

634 
"_PiF" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3PIF _:_./ _)" 10) 

635 

636 
syntax (xsymbols) 

637 
"_PiF" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3\<Pi>\<^isub>F _\<in>_./ _)" 10) 

638 

639 
syntax (HTML output) 

640 
"_PiF" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3\<Pi>\<^isub>F _\<in>_./ _)" 10) 

641 

642 
translations 

643 
"PIF x:I. M" == "CONST PiF I (%x. M)" 

644 

645 
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> 

646 
Pow (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j)))" 

647 
by (auto simp: Pi'_def) (blast dest: sets_into_space) 

648 

649 
lemma space_PiF: "space (PiF I M) = (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j)))" 

650 
unfolding PiF_def using PiF_gen_subset by (rule space_measure_of) 

651 

652 
lemma sets_PiF: 

653 
"sets (PiF I M) = sigma_sets (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j))) 

654 
{(\<Pi>' j\<in>J. X j) X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}" 

655 
unfolding PiF_def using PiF_gen_subset by (rule sets_measure_of) 

656 

657 
lemma sets_PiF_singleton: 

658 
"sets (PiF {I} M) = sigma_sets (\<Pi>' j\<in>I. space (M j)) 

659 
{(\<Pi>' j\<in>I. X j) X. X \<in> (\<Pi> j\<in>I. sets (M j))}" 

660 
unfolding sets_PiF by simp 

661 

662 
lemma in_sets_PiFI: 

663 
assumes "X = (Pi' J S)" "J \<in> I" "\<And>i. i\<in>J \<Longrightarrow> S i \<in> sets (M i)" 

664 
shows "X \<in> sets (PiF I M)" 

665 
unfolding sets_PiF 

666 
using assms by blast 

667 

668 
lemma product_in_sets_PiFI: 

669 
assumes "J \<in> I" "\<And>i. i\<in>J \<Longrightarrow> S i \<in> sets (M i)" 

670 
shows "(Pi' J S) \<in> sets (PiF I M)" 

671 
unfolding sets_PiF 

672 
using assms by blast 

673 

674 
lemma singleton_space_subset_in_sets: 

675 
fixes J 

676 
assumes "J \<in> I" 

677 
assumes "finite J" 

678 
shows "space (PiF {J} M) \<in> sets (PiF I M)" 

679 
using assms 

680 
by (intro in_sets_PiFI[where J=J and S="\<lambda>i. space (M i)"]) 

681 
(auto simp: product_def space_PiF) 

682 

683 
lemma singleton_subspace_set_in_sets: 

684 
assumes A: "A \<in> sets (PiF {J} M)" 

685 
assumes "finite J" 

686 
assumes "J \<in> I" 

687 
shows "A \<in> sets (PiF I M)" 

688 
using A[unfolded sets_PiF] 

689 
apply (induct A) 

690 
unfolding sets_PiF[symmetric] unfolding space_PiF[symmetric] 

691 
using assms 

692 
by (auto intro: in_sets_PiFI intro!: singleton_space_subset_in_sets) 

693 

694 
lemma 

695 
finite_measurable_singletonI: 

696 
assumes "finite I" 

697 
assumes "\<And>J. J \<in> I \<Longrightarrow> finite J" 

698 
assumes MN: "\<And>J. J \<in> I \<Longrightarrow> A \<in> measurable (PiF {J} M) N" 

699 
shows "A \<in> measurable (PiF I M) N" 

700 
unfolding measurable_def 

701 
proof safe 

702 
fix y assume "y \<in> sets N" 

703 
have "A ` y \<inter> space (PiF I M) = (\<Union>J\<in>I. A ` y \<inter> space (PiF {J} M))" 

704 
by (auto simp: space_PiF) 

705 
also have "\<dots> \<in> sets (PiF I M)" 

706 
proof 

707 
show "finite I" by fact 

708 
fix J assume "J \<in> I" 

709 
with assms have "finite J" by simp 

710 
show "A ` y \<inter> space (PiF {J} M) \<in> sets (PiF I M)" 

711 
by (rule singleton_subspace_set_in_sets[OF measurable_sets[OF assms(3)]]) fact+ 

712 
qed 

713 
finally show "A ` y \<inter> space (PiF I M) \<in> sets (PiF I M)" . 

714 
next 

715 
fix x assume "x \<in> space (PiF I M)" thus "A x \<in> space N" 

716 
using MN[of "domain x"] 

717 
by (auto simp: space_PiF measurable_space Pi'_def) 

718 
qed 

719 

720 
lemma 

721 
countable_finite_comprehension: 

722 
fixes f :: "'a::countable set \<Rightarrow> _" 

723 
assumes "\<And>s. P s \<Longrightarrow> finite s" 

724 
assumes "\<And>s. P s \<Longrightarrow> f s \<in> sets M" 

725 
shows "\<Union>{f ss. P s} \<in> sets M" 

726 
proof  

727 
have "\<Union>{f ss. P s} = (\<Union>n::nat. let s = set (from_nat n) in if P s then f s else {})" 

728 
proof safe 

729 
fix x X s assume "x \<in> f s" "P s" 

730 
moreover with assms obtain l where "s = set l" using finite_list by blast 

731 
ultimately show "x \<in> (\<Union>n. let s = set (from_nat n) in if P s then f s else {})" using `P s` 

732 
by (auto intro!: exI[where x="to_nat l"]) 

733 
next 

734 
fix x n assume "x \<in> (let s = set (from_nat n) in if P s then f s else {})" 

735 
thus "x \<in> \<Union>{f ss. P s}" using assms by (auto simp: Let_def split: split_if_asm) 

736 
qed 

737 
hence "\<Union>{f ss. P s} = (\<Union>n. let s = set (from_nat n) in if P s then f s else {})" by simp 

738 
also have "\<dots> \<in> sets M" using assms by (auto simp: Let_def) 

739 
finally show ?thesis . 

740 
qed 

741 

742 
lemma space_subset_in_sets: 

743 
fixes J::"'a::countable set set" 

744 
assumes "J \<subseteq> I" 

745 
assumes "\<And>j. j \<in> J \<Longrightarrow> finite j" 

746 
shows "space (PiF J M) \<in> sets (PiF I M)" 

747 
proof  

748 
have "space (PiF J M) = \<Union>{space (PiF {j} M)j. j \<in> J}" 

749 
unfolding space_PiF by blast 

750 
also have "\<dots> \<in> sets (PiF I M)" using assms 

751 
by (intro countable_finite_comprehension) (auto simp: singleton_space_subset_in_sets) 

752 
finally show ?thesis . 

753 
qed 

754 

755 
lemma subspace_set_in_sets: 

756 
fixes J::"'a::countable set set" 

757 
assumes A: "A \<in> sets (PiF J M)" 

758 
assumes "J \<subseteq> I" 

759 
assumes "\<And>j. j \<in> J \<Longrightarrow> finite j" 

760 
shows "A \<in> sets (PiF I M)" 

761 
using A[unfolded sets_PiF] 

762 
apply (induct A) 

763 
unfolding sets_PiF[symmetric] unfolding space_PiF[symmetric] 

764 
using assms 

765 
by (auto intro: in_sets_PiFI intro!: space_subset_in_sets) 

766 

767 
lemma 

768 
countable_measurable_PiFI: 

769 
fixes I::"'a::countable set set" 

770 
assumes MN: "\<And>J. J \<in> I \<Longrightarrow> finite J \<Longrightarrow> A \<in> measurable (PiF {J} M) N" 

771 
shows "A \<in> measurable (PiF I M) N" 

772 
unfolding measurable_def 

773 
proof safe 

774 
fix y assume "y \<in> sets N" 

775 
have "A ` y = (\<Union>{A ` y \<inter> {x. domain x = J}J. finite J})" by auto 

776 
hence "A ` y \<inter> space (PiF I M) = (\<Union>n. A ` y \<inter> space (PiF ({set (from_nat n)}\<inter>I) M))" 

777 
apply (auto simp: space_PiF Pi'_def) 

778 
proof  

779 
case goal1 

780 
from finite_list[of "domain x"] obtain xs where "set xs = domain x" by auto 

781 
thus ?case 

782 
apply (intro exI[where x="to_nat xs"]) 

783 
apply auto 

784 
done 

785 
qed 

786 
also have "\<dots> \<in> sets (PiF I M)" 

787 
apply (intro Int countable_nat_UN subsetI, safe) 

788 
apply (case_tac "set (from_nat i) \<in> I") 

789 
apply simp_all 

790 
apply (rule singleton_subspace_set_in_sets[OF measurable_sets[OF MN]]) 

791 
using assms `y \<in> sets N` 

792 
apply (auto simp: space_PiF) 

793 
done 

794 
finally show "A ` y \<inter> space (PiF I M) \<in> sets (PiF I M)" . 

795 
next 

796 
fix x assume "x \<in> space (PiF I M)" thus "A x \<in> space N" 

797 
using MN[of "domain x"] by (auto simp: space_PiF measurable_space Pi'_def) 

798 
qed 

799 

800 
lemma measurable_PiF: 

801 
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))" 

802 
assumes S: "\<And>J S. J \<in> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> S i \<in> sets (M i)) \<Longrightarrow> 

803 
f ` (Pi' J S) \<inter> space N \<in> sets N" 

804 
shows "f \<in> measurable N (PiF I M)" 

805 
unfolding PiF_def 

806 
using PiF_gen_subset 

807 
apply (rule measurable_measure_of) 

808 
using f apply force 

809 
apply (insert S, auto) 

810 
done 

811 

812 
lemma 

813 
restrict_sets_measurable: 

814 
assumes A: "A \<in> sets (PiF I M)" and "J \<subseteq> I" 

815 
shows "A \<inter> {m. domain m \<in> J} \<in> sets (PiF J M)" 

816 
using A[unfolded sets_PiF] 

817 
apply (induct A) 

818 
unfolding sets_PiF[symmetric] unfolding space_PiF[symmetric] 

819 
proof  

820 
fix a assume "a \<in> {Pi' J X X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}" 

821 
then obtain K S where S: "a = Pi' K S" "K \<in> I" "(\<forall>i\<in>K. S i \<in> sets (M i))" 

822 
by auto 

823 
show "a \<inter> {m. domain m \<in> J} \<in> sets (PiF J M)" 

824 
proof cases 

825 
assume "K \<in> J" 

826 
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 

827 
by (auto intro!: exI[where x=K] exI[where x=S] simp: Pi'_def) 

828 
also have "\<dots> \<subseteq> sets (PiF J M)" unfolding sets_PiF by auto 

829 
finally show ?thesis . 

830 
next 

831 
assume "K \<notin> J" 

832 
hence "a \<inter> {m. domain m \<in> J} = {}" using S by (auto simp: Pi'_def) 

833 
also have "\<dots> \<in> sets (PiF J M)" by simp 

834 
finally show ?thesis . 

835 
qed 

836 
next 

837 
show "{} \<inter> {m. domain m \<in> J} \<in> sets (PiF J M)" by simp 

838 
next 

839 
fix a :: "nat \<Rightarrow> _" 

840 
assume a: "(\<And>i. a i \<inter> {m. domain m \<in> J} \<in> sets (PiF J M))" 

841 
have "UNION UNIV a \<inter> {m. domain m \<in> J} = (\<Union>i. (a i \<inter> {m. domain m \<in> J}))" 

842 
by simp 

843 
also have "\<dots> \<in> sets (PiF J M)" using a by (intro countable_nat_UN) auto 

844 
finally show "UNION UNIV a \<inter> {m. domain m \<in> J} \<in> sets (PiF J M)" . 

845 
next 

846 
fix a assume a: "a \<inter> {m. domain m \<in> J} \<in> sets (PiF J M)" 

847 
have "(space (PiF I M)  a) \<inter> {m. domain m \<in> J} = (space (PiF J M)  (a \<inter> {m. domain m \<in> J}))" 

848 
using `J \<subseteq> I` by (auto simp: space_PiF Pi'_def) 

849 
also have "\<dots> \<in> sets (PiF J M)" using a by auto 

850 
finally show "(space (PiF I M)  a) \<inter> {m. domain m \<in> J} \<in> sets (PiF J M)" . 

851 
qed 

852 

853 
lemma measurable_finmap_of: 

854 
assumes f: "\<And>i. (\<exists>x \<in> space N. i \<in> J x) \<Longrightarrow> (\<lambda>x. f x i) \<in> measurable N (M i)" 

855 
assumes J: "\<And>x. x \<in> space N \<Longrightarrow> J x \<in> I" "\<And>x. x \<in> space N \<Longrightarrow> finite (J x)" 

856 
assumes JN: "\<And>S. {x. J x = S} \<inter> space N \<in> sets N" 

857 
shows "(\<lambda>x. finmap_of (J x) (f x)) \<in> measurable N (PiF I M)" 

858 
proof (rule measurable_PiF) 

859 
fix x assume "x \<in> space N" 

860 
with J[of x] measurable_space[OF f] 

861 
show "domain (finmap_of (J x) (f x)) \<in> I \<and> 

862 
(\<forall>i\<in>domain (finmap_of (J x) (f x)). (finmap_of (J x) (f x)) i \<in> space (M i))" 

863 
by auto 

864 
next 

865 
fix K S assume "K \<in> I" and *: "\<And>i. i \<in> K \<Longrightarrow> S i \<in> sets (M i)" 

866 
with J have eq: "(\<lambda>x. finmap_of (J x) (f x)) ` Pi' K S \<inter> space N = 

867 
(if \<exists>x \<in> space N. K = J x \<and> finite K then if K = {} then {x \<in> space N. J x = K} 

868 
else (\<Inter>i\<in>K. (\<lambda>x. f x i) ` S i \<inter> {x \<in> space N. J x = K}) else {})" 

869 
by (auto simp: Pi'_def) 

870 
have r: "{x \<in> space N. J x = K} = space N \<inter> ({x. J x = K} \<inter> space N)" by auto 

871 
show "(\<lambda>x. finmap_of (J x) (f x)) ` Pi' K S \<inter> space N \<in> sets N" 

872 
unfolding eq r 

873 
apply (simp del: INT_simps add: ) 

874 
apply (intro conjI impI finite_INT JN Int[OF top]) 

875 
apply simp apply assumption 

876 
apply (subst Int_assoc[symmetric]) 

877 
apply (rule Int) 

878 
apply (intro measurable_sets[OF f] *) apply force apply assumption 

879 
apply (intro JN) 

880 
done 

881 
qed 

882 

883 
lemma measurable_PiM_finmap_of: 

884 
assumes "finite J" 

885 
shows "finmap_of J \<in> measurable (Pi\<^isub>M J M) (PiF {J} M)" 

886 
apply (rule measurable_finmap_of) 

887 
apply (rule measurable_component_singleton) 

888 
apply simp 

889 
apply rule 

890 
apply (rule `finite J`) 

891 
apply simp 

892 
done 

893 

894 
lemma proj_measurable_singleton: 

895 
assumes "A \<in> sets (M i)" "finite I" 

896 
shows "(\<lambda>x. (x)\<^isub>F i) ` A \<inter> space (PiF {I} M) \<in> sets (PiF {I} M)" 

897 
proof cases 

898 
assume "i \<in> I" 

899 
hence "(\<lambda>x. (x)\<^isub>F i) ` A \<inter> space (PiF {I} M) = 

900 
Pi' I (\<lambda>x. if x = i then A else space (M x))" 

901 
using sets_into_space[OF ] `A \<in> sets (M i)` assms 

902 
by (auto simp: space_PiF Pi'_def) 

903 
thus ?thesis using assms `A \<in> sets (M i)` 

904 
by (intro in_sets_PiFI) auto 

905 
next 

906 
assume "i \<notin> I" 

907 
hence "(\<lambda>x. (x)\<^isub>F i) ` A \<inter> space (PiF {I} M) = 

908 
(if undefined \<in> A then space (PiF {I} M) else {})" by (auto simp: space_PiF Pi'_def) 

909 
thus ?thesis by simp 

910 
qed 

911 

912 
lemma measurable_proj_singleton: 

913 
fixes I 

914 
assumes "finite I" "i \<in> I" 

915 
shows "(\<lambda>x. (x)\<^isub>F i) \<in> measurable (PiF {I} M) (M i)" 

916 
proof (unfold measurable_def, intro CollectI conjI ballI proj_measurable_singleton assms) 

917 
qed (insert `i \<in> I`, auto simp: space_PiF) 

918 

919 
lemma measurable_proj_countable: 

920 
fixes I::"'a::countable set set" 

921 
assumes "y \<in> space (M i)" 

922 
shows "(\<lambda>x. if i \<in> domain x then (x)\<^isub>F i else y) \<in> measurable (PiF I M) (M i)" 

923 
proof (rule countable_measurable_PiFI) 

924 
fix J assume "J \<in> I" "finite J" 

925 
show "(\<lambda>x. if i \<in> domain x then x i else y) \<in> measurable (PiF {J} M) (M i)" 

926 
unfolding measurable_def 

927 
proof safe 

928 
fix z assume "z \<in> sets (M i)" 

929 
have "(\<lambda>x. if i \<in> domain x then x i else y) ` z \<inter> space (PiF {J} M) = 

930 
(\<lambda>x. if i \<in> J then (x)\<^isub>F i else y) ` z \<inter> space (PiF {J} M)" 

931 
by (auto simp: space_PiF Pi'_def) 

932 
also have "\<dots> \<in> sets (PiF {J} M)" using `z \<in> sets (M i)` `finite J` 

933 
by (cases "i \<in> J") (auto intro!: measurable_sets[OF measurable_proj_singleton]) 

934 
finally show "(\<lambda>x. if i \<in> domain x then x i else y) ` z \<inter> space (PiF {J} M) \<in> 

935 
sets (PiF {J} M)" . 

936 
qed (insert `y \<in> space (M i)`, auto simp: space_PiF Pi'_def) 

937 
qed 

938 

939 
lemma measurable_restrict_proj: 

940 
assumes "J \<in> II" "finite J" 

941 
shows "finmap_of J \<in> measurable (PiM J M) (PiF II M)" 

942 
using assms 

943 
by (intro measurable_finmap_of measurable_component_singleton) auto 

944 

945 
lemma 

946 
measurable_proj_PiM: 

947 
fixes J K ::"'a::countable set" and I::"'a set set" 

948 
assumes "finite J" "J \<in> I" 

949 
assumes "x \<in> space (PiM J M)" 

950 
shows "proj \<in> 

951 
measurable (PiF {J} M) (PiM J M)" 

952 
proof (rule measurable_PiM_single) 

953 
show "proj \<in> space (PiF {J} M) \<rightarrow> (\<Pi>\<^isub>E i \<in> J. space (M i))" 

954 
using assms by (auto simp add: space_PiM space_PiF extensional_def sets_PiF Pi'_def) 

955 
next 

956 
fix A i assume A: "i \<in> J" "A \<in> sets (M i)" 

957 
show "{\<omega> \<in> space (PiF {J} M). (\<omega>)\<^isub>F i \<in> A} \<in> sets (PiF {J} M)" 

958 
proof 

959 
have "{\<omega> \<in> space (PiF {J} M). (\<omega>)\<^isub>F i \<in> A} = 

960 
(\<lambda>\<omega>. (\<omega>)\<^isub>F i) ` A \<inter> space (PiF {J} M)" by auto 

961 
also have "\<dots> \<in> sets (PiF {J} M)" 

962 
using assms A by (auto intro: measurable_sets[OF measurable_proj_singleton] simp: space_PiM) 

963 
finally show ?thesis . 

964 
qed simp 

965 
qed 

966 

967 
lemma sets_subspaceI: 

968 
assumes "A \<inter> space M \<in> sets M" 

969 
assumes "B \<in> sets M" 

970 
shows "A \<inter> B \<in> sets M" using assms 

971 
proof  

972 
have "A \<inter> B = (A \<inter> space M) \<inter> B" 

973 
using assms sets_into_space by auto 

974 
thus ?thesis using assms by auto 

975 
qed 

976 

977 
lemma space_PiF_singleton_eq_product: 

978 
assumes "finite I" 

979 
shows "space (PiF {I} M) = (\<Pi>' i\<in>I. space (M i))" 

980 
by (auto simp: product_def space_PiF assms) 

981 

982 
text {* adapted from @{thm sets_PiM_single} *} 

983 

984 
lemma sets_PiF_single: 

985 
assumes "finite I" "I \<noteq> {}" 

986 
shows "sets (PiF {I} M) = 

987 
sigma_sets (\<Pi>' i\<in>I. space (M i)) 

988 
{{f\<in>\<Pi>' i\<in>I. space (M i). f i \<in> A}  i A. i \<in> I \<and> A \<in> sets (M i)}" 

989 
(is "_ = sigma_sets ?\<Omega> ?R") 

990 
unfolding sets_PiF_singleton 

991 
proof (rule sigma_sets_eqI) 

992 
interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto 

993 
fix A assume "A \<in> {Pi' I X X. X \<in> (\<Pi> j\<in>I. sets (M j))}" 

994 
then obtain X where X: "A = Pi' I X" "X \<in> (\<Pi> j\<in>I. sets (M j))" by auto 

995 
show "A \<in> sigma_sets ?\<Omega> ?R" 

996 
proof  

997 
from `I \<noteq> {}` X have "A = (\<Inter>j\<in>I. {f\<in>space (PiF {I} M). f j \<in> X j})" 

998 
using sets_into_space 

999 
by (auto simp: space_PiF product_def) blast 

1000 
also have "\<dots> \<in> sigma_sets ?\<Omega> ?R" 

1001 
using X `I \<noteq> {}` assms by (intro R.finite_INT) (auto simp: space_PiF) 

1002 
finally show "A \<in> sigma_sets ?\<Omega> ?R" . 

1003 
qed 

1004 
next 

1005 
fix A assume "A \<in> ?R" 

1006 
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)" 

1007 
by auto 

1008 
then have "A = (\<Pi>' j \<in> I. if j = i then B else space (M j))" 

1009 
using sets_into_space[OF A(3)] 

1010 
apply (auto simp: Pi'_iff split: split_if_asm) 

1011 
apply blast 

1012 
done 

1013 
also have "\<dots> \<in> sigma_sets ?\<Omega> {Pi' I X X. X \<in> (\<Pi> j\<in>I. sets (M j))}" 

1014 
using A 

1015 
by (intro sigma_sets.Basic ) 

1016 
(auto intro: exI[where x="\<lambda>j. if j = i then B else space (M j)"]) 

1017 
finally show "A \<in> sigma_sets ?\<Omega> {Pi' I X X. X \<in> (\<Pi> j\<in>I. sets (M j))}" . 

1018 
qed 

1019 

1020 
text {* adapted from @{thm PiE_cong} *} 

1021 

1022 
lemma Pi'_cong: 

1023 
assumes "finite I" 

1024 
assumes "\<And>i. i \<in> I \<Longrightarrow> f i = g i" 

1025 
shows "Pi' I f = Pi' I g" 

1026 
using assms by (auto simp: Pi'_def) 

1027 

1028 
text {* adapted from @{thm Pi_UN} *} 

1029 

1030 
lemma Pi'_UN: 

1031 
fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set" 

1032 
assumes "finite I" 

1033 
assumes mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i" 

1034 
shows "(\<Union>n. Pi' I (A n)) = Pi' I (\<lambda>i. \<Union>n. A n i)" 

1035 
proof (intro set_eqI iffI) 

1036 
fix f assume "f \<in> Pi' I (\<lambda>i. \<Union>n. A n i)" 

1037 
then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i" "domain f = I" by (auto simp: `finite I` Pi'_def) 

1038 
from bchoice[OF this(1)] obtain n where n: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> (A (n i) i)" by auto 

1039 
obtain k where k: "\<And>i. i \<in> I \<Longrightarrow> n i \<le> k" 

1040 
using `finite I` finite_nat_set_iff_bounded_le[of "n`I"] by auto 

1041 
have "f \<in> Pi' I (\<lambda>i. A k i)" 

1042 
proof 

1043 
fix i assume "i \<in> I" 

1044 
from mono[OF this, of "n i" k] k[OF this] n[OF this] `domain f = I` `i \<in> I` 

1045 
show "f i \<in> A k i " by (auto simp: `finite I`) 

1046 
qed (simp add: `domain f = I` `finite I`) 

1047 
then show "f \<in> (\<Union>n. Pi' I (A n))" by auto 

1048 
qed (auto simp: Pi'_def `finite I`) 

1049 

1050 
text {* adapted from @{thm sigma_prod_algebra_sigma_eq} *} 

1051 

1052 
lemma sigma_fprod_algebra_sigma_eq: 

1053 
fixes E :: "'i \<Rightarrow> 'a set set" 

1054 
assumes [simp]: "finite I" "I \<noteq> {}" 

1055 
assumes S_mono: "\<And>i. i \<in> I \<Longrightarrow> incseq (S i)" 

1056 
and S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)" 

1057 
and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i" 

1058 
assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))" 

1059 
and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)" 

1060 
defines "P == { Pi' I F  F. \<forall>i\<in>I. F i \<in> E i }" 

1061 
shows "sets (PiF {I} M) = sigma_sets (space (PiF {I} M)) P" 

1062 
proof 

1063 
let ?P = "sigma (space (Pi\<^isub>F {I} M)) P" 

1064 
have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>F {I} M))" 

1065 
using E_closed by (auto simp: space_PiF P_def Pi'_iff subset_eq) 

1066 
then have space_P: "space ?P = (\<Pi>' i\<in>I. space (M i))" 

1067 
by (simp add: space_PiF) 

1068 
have "sets (PiF {I} M) = 

1069 
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)}" 

1070 
using sets_PiF_single[of I M] by (simp add: space_P) 

1071 
also have "\<dots> \<subseteq> sets (sigma (space (PiF {I} M)) P)" 

1072 
proof (safe intro!: sigma_sets_subset) 

1073 
fix i A assume "i \<in> I" and A: "A \<in> sets (M i)" 

1074 
have "(\<lambda>x. (x)\<^isub>F i) \<in> measurable ?P (sigma (space (M i)) (E i))" 

1075 
proof (subst measurable_iff_measure_of) 

1076 
show "E i \<subseteq> Pow (space (M i))" using `i \<in> I` by fact 

1077 
from space_P `i \<in> I` show "(\<lambda>x. (x)\<^isub>F i) \<in> space ?P \<rightarrow> space (M i)" 

1078 
by auto 

1079 
show "\<forall>A\<in>E i. (\<lambda>x. (x)\<^isub>F i) ` A \<inter> space ?P \<in> sets ?P" 

1080 
proof 

1081 
fix A assume A: "A \<in> E i" 

1082 
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))" 

1083 
using E_closed `i \<in> I` by (auto simp: space_P Pi_iff subset_eq split: split_if_asm) 

1084 
also have "\<dots> = (\<Pi>' j\<in>I. \<Union>n. if i = j then A else S j n)" 

1085 
by (intro Pi'_cong) (simp_all add: S_union) 

1086 
also have "\<dots> = (\<Union>n. \<Pi>' j\<in>I. if i = j then A else S j n)" 

1087 
using S_mono 

1088 
by (subst Pi'_UN[symmetric, OF `finite I`]) (auto simp: incseq_def) 

1089 
also have "\<dots> \<in> sets ?P" 

1090 
proof (safe intro!: countable_UN) 

1091 
fix n show "(\<Pi>' j\<in>I. if i = j then A else S j n) \<in> sets ?P" 

1092 
using A S_in_E 

1093 
by (simp add: P_closed) 

1094 
(auto simp: P_def subset_eq intro!: exI[of _ "\<lambda>j. if i = j then A else S j n"]) 

1095 
qed 

1096 
finally show "(\<lambda>x. (x)\<^isub>F i) ` A \<inter> space ?P \<in> sets ?P" 

1097 
using P_closed by simp 

1098 
qed 

1099 
qed 

1100 
from measurable_sets[OF this, of A] A `i \<in> I` E_closed 

1101 
have "(\<lambda>x. (x)\<^isub>F i) ` A \<inter> space ?P \<in> sets ?P" 

1102 
by (simp add: E_generates) 

1103 
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}" 

1104 
using P_closed by (auto simp: space_PiF) 

1105 
finally show "\<dots> \<in> sets ?P" . 

1106 
qed 

1107 
finally show "sets (PiF {I} M) \<subseteq> sigma_sets (space (PiF {I} M)) P" 

1108 
by (simp add: P_closed) 

1109 
show "sigma_sets (space (PiF {I} M)) P \<subseteq> sets (PiF {I} M)" 

1110 
using `finite I` `I \<noteq> {}` 

1111 
by (auto intro!: sigma_sets_subset product_in_sets_PiFI simp: E_generates P_def) 

1112 
qed 

1113 

1114 
lemma enumerable_sigma_fprod_algebra_sigma_eq: 

1115 
assumes "I \<noteq> {}" 

1116 
assumes [simp]: "finite I" 

1117 
shows "sets (PiF {I} (\<lambda>_. borel)) = sigma_sets (space (PiF {I} (\<lambda>_. borel))) 

1118 
{Pi' I F F. (\<forall>i\<in>I. F i \<in> range enum_basis)}" 

1119 
proof  

1120 
from open_incseqE[OF open_UNIV] guess S::"nat \<Rightarrow> 'b set" . note S = this 

1121 
show ?thesis 

1122 
proof (rule sigma_fprod_algebra_sigma_eq) 

1123 
show "finite I" by simp 

1124 
show "I \<noteq> {}" by fact 

1125 
show "incseq S" "(\<Union>j. S j) = space borel" "range S \<subseteq> range enum_basis" 

1126 
using S by simp_all 

1127 
show "range enum_basis \<subseteq> Pow (space borel)" by simp 

1128 
show "sets borel = sigma_sets (space borel) (range enum_basis)" 

1129 
by (simp add: borel_eq_enum_basis) 

1130 
qed 

1131 
qed 

1132 

1133 
text {* adapted from @{thm enumerable_sigma_fprod_algebra_sigma_eq} *} 

1134 

1135 
lemma enumerable_sigma_prod_algebra_sigma_eq: 

1136 
assumes "I \<noteq> {}" 

1137 
assumes [simp]: "finite I" 

1138 
shows "sets (PiM I (\<lambda>_. borel)) = sigma_sets (space (PiM I (\<lambda>_. borel))) 

1139 
{Pi\<^isub>E I F F. \<forall>i\<in>I. F i \<in> range enum_basis}" 

1140 
proof  

1141 
from open_incseqE[OF open_UNIV] guess S::"nat \<Rightarrow> 'b set" . note S = this 

1142 
show ?thesis 

1143 
proof (rule sigma_prod_algebra_sigma_eq) 

1144 
show "finite I" by simp note[[show_types]] 

1145 
fix i show "(\<Union>j. S j) = space borel" "range S \<subseteq> range enum_basis" 

1146 
using S by simp_all 

1147 
show "range enum_basis \<subseteq> Pow (space borel)" by simp 

1148 
show "sets borel = sigma_sets (space borel) (range enum_basis)" 

1149 
by (simp add: borel_eq_enum_basis) 

1150 
qed 

1151 
qed 

1152 

1153 
lemma product_open_generates_sets_PiF_single: 

1154 
assumes "I \<noteq> {}" 

1155 
assumes [simp]: "finite I" 

1156 
shows "sets (PiF {I} (\<lambda>_. borel::'b::enumerable_basis measure)) = 

1157 
sigma_sets (space (PiF {I} (\<lambda>_. borel))) {Pi' I F F. (\<forall>i\<in>I. F i \<in> Collect open)}" 

1158 
proof  

1159 
from open_incseqE[OF open_UNIV] guess S::"nat \<Rightarrow> 'b set" . note S = this 

1160 
show ?thesis 

1161 
proof (rule sigma_fprod_algebra_sigma_eq) 

1162 
show "finite I" by simp 

1163 
show "I \<noteq> {}" by fact 

1164 
show "incseq S" "(\<Union>j. S j) = space borel" "range S \<subseteq> Collect open" 

1165 
using S by (auto simp: open_enum_basis) 

1166 
show "Collect open \<subseteq> Pow (space borel)" by simp 

1167 
show "sets borel = sigma_sets (space borel) (Collect open)" 

1168 
by (simp add: borel_def) 

1169 
qed 

1170 
qed 

1171 

1172 
lemma product_open_generates_sets_PiM: 

1173 
assumes "I \<noteq> {}" 

1174 
assumes [simp]: "finite I" 

1175 
shows "sets (PiM I (\<lambda>_. borel::'b::enumerable_basis measure)) = 

1176 
sigma_sets (space (PiM I (\<lambda>_. borel))) {Pi\<^isub>E I F F. \<forall>i\<in>I. F i \<in> Collect open}" 

1177 
proof  

1178 
from open_incseqE[OF open_UNIV] guess S::"nat \<Rightarrow> 'b set" . note S = this 

1179 
show ?thesis 

1180 
proof (rule sigma_prod_algebra_sigma_eq) 

1181 
show "finite I" by simp note[[show_types]] 

1182 
fix i show "(\<Union>j. S j) = space borel" "range S \<subseteq> Collect open" 

1183 
using S by (auto simp: open_enum_basis) 

1184 
show "Collect open \<subseteq> Pow (space borel)" by simp 

1185 
show "sets borel = sigma_sets (space borel) (Collect open)" 

1186 
by (simp add: borel_def) 

1187 
qed 

1188 
qed 

1189 

1190 
lemma finmap_UNIV[simp]: "(\<Union>J\<in>Collect finite. J \<leadsto> UNIV) = UNIV" by auto 

1191 

1192 
lemma borel_eq_PiF_borel: 

1193 
shows "(borel :: ('i::countable \<Rightarrow>\<^isub>F 'a::polish_space) measure) = 

1194 
PiF (Collect finite) (\<lambda>_. borel :: 'a measure)" 

1195 
proof (rule measure_eqI) 

1196 
have C: "Collect finite \<noteq> {}" by auto 

1197 
show "sets (borel::('i \<Rightarrow>\<^isub>F 'a) measure) = sets (PiF (Collect finite) (\<lambda>_. borel))" 

1198 
proof 

1199 
show "sets (borel::('i \<Rightarrow>\<^isub>F 'a) measure) \<subseteq> sets (PiF (Collect finite) (\<lambda>_. borel))" 

1200 
apply (simp add: borel_def sets_PiF) 

1201 
proof (rule sigma_sets_mono, safe, cases) 

1202 
fix X::"('i \<Rightarrow>\<^isub>F 'a) set" assume "open X" "X \<noteq> {}" 

1203 
from open_basisE[OF this] guess NA NB . note N = this 

1204 
hence "X = (\<Union>i. Pi' (NA i) (NB i))" by simp 

1205 
also have "\<dots> \<in> 

1206 
sigma_sets UNIV {Pi' J S S J. finite J \<and> S \<in> J \<rightarrow> sigma_sets UNIV (Collect open)}" 

1207 
using N by (intro Union sigma_sets.Basic) blast 

1208 
finally show "X \<in> sigma_sets UNIV 

1209 
{Pi' J X X J. finite J \<and> X \<in> J \<rightarrow> sigma_sets UNIV (Collect open)}" . 

1210 
qed (auto simp: Empty) 

1211 
next 

1212 
show "sets (PiF (Collect finite) (\<lambda>_. borel)) \<subseteq> sets (borel::('i \<Rightarrow>\<^isub>F 'a) measure)" 

1213 
proof 

1214 
fix x assume x: "x \<in> sets (PiF (Collect finite::'i set set) (\<lambda>_. borel::'a measure))" 

1215 
hence x_sp: "x \<subseteq> space (PiF (Collect finite) (\<lambda>_. borel))" by (rule sets_into_space) 

1216 
let ?x = "\<lambda>J. x \<inter> {x. domain x = J}" 

1217 
have "x = \<Union>{?x J J. finite J}" by auto 

1218 
also have "\<dots> \<in> sets borel" 

1219 
proof (rule countable_finite_comprehension, assumption) 

1220 
fix J::"'i set" assume "finite J" 

1221 
{ assume ef: "J = {}" 

1222 
{ assume e: "?x J = {}" 

1223 
hence "?x J \<in> sets borel" by simp 

1224 
} moreover { 

1225 
assume "?x J \<noteq> {}" 

1226 
then obtain f where "f \<in> x" "domain f = {}" using ef by auto 

1227 
hence "?x J = {f}" using `J = {}` 

1228 
by (auto simp: finmap_eq_iff) 

1229 
also have "{f} \<in> sets borel" by simp 

1230 
finally have "?x J \<in> sets borel" . 

1231 
} ultimately have "?x J \<in> sets borel" by blast 

1232 
} moreover { 

1233 
assume "J \<noteq> ({}::'i set)" 

1234 
from open_incseqE[OF open_UNIV] guess S::"nat \<Rightarrow> 'a set" . note S = this 

1235 
have "(?x J) = x \<inter> {m. domain m \<in> {J}}" by auto 

1236 
also have "\<dots> \<in> sets (PiF {J} (\<lambda>_. borel))" 

1237 
using x by (rule restrict_sets_measurable) (auto simp: `finite J`) 

1238 
also have "\<dots> = sigma_sets (space (PiF {J} (\<lambda>_. borel))) 

1239 
{Pi' (J) F F. (\<forall>j\<in>J. F j \<in> range enum_basis)}" 

1240 
(is "_ = sigma_sets _ ?P") 

1241 
by (rule enumerable_sigma_fprod_algebra_sigma_eq[OF `J \<noteq> {}` `finite J`]) 

1242 
also have "\<dots> \<subseteq> sets borel" 

1243 
proof 

1244 
fix x 

1245 
assume "x \<in> sigma_sets (space (PiF {J} (\<lambda>_. borel))) ?P" 

1246 
thus "x \<in> sets borel" 

1247 
proof (rule sigma_sets.induct, safe) 

1248 
fix F::"'i \<Rightarrow> 'a set" 

1249 
assume "\<forall>j\<in>J. F j \<in> range enum_basis" 

1250 
hence "Pi' J F \<in> range enum_basis_finmap" 

1251 
unfolding range_enum_basis_eq 

1252 
by (auto simp: `finite J` intro!: exI[where x=J] exI[where x=F]) 

1253 
hence "open (Pi' (J) F)" by (rule range_enum_basis_finmap_imp_open) 

1254 
thus "Pi' (J) F \<in> sets borel" by simp 

1255 
next 

1256 
fix a::"('i \<Rightarrow>\<^isub>F 'a) set" 

1257 
have "space (PiF {J::'i set} (\<lambda>_. borel::'a measure)) = 

1258 
Pi' (J) (\<lambda>_. UNIV)" 

1259 
by (auto simp: space_PiF product_def) 

1260 
moreover have "open (Pi' (J::'i set) (\<lambda>_. UNIV::'a set))" 

1261 
by (intro open_Pi'I) auto 

1262 
ultimately 

1263 
have "space (PiF {J::'i set} (\<lambda>_. borel::'a measure)) \<in> sets borel" 

1264 
by simp 

1265 
moreover 

1266 
assume "a \<in> sets borel" 

1267 
ultimately show "space (PiF {J} (\<lambda>_. borel))  a \<in> sets borel" .. 

1268 
qed auto 

1269 
qed 

1270 
finally have "(?x J) \<in> sets borel" . 

1271 
} ultimately show "(?x J) \<in> sets borel" by blast 

1272 
qed 

1273 
finally show "x \<in> sets (borel)" . 

1274 
qed 

1275 
qed 

1276 
qed (simp add: emeasure_sigma borel_def PiF_def) 

1277 

1278 
subsection {* Isomorphism between Functions and Finite Maps *} 

1279 

1280 
lemma 

1281 
measurable_compose: 

1282 
fixes f::"'a \<Rightarrow> 'b" 

1283 
assumes inj: "\<And>j. j \<in> J \<Longrightarrow> f' (f j) = j" 

1284 
assumes "finite J" 

1285 
shows "(\<lambda>m. compose J m f) \<in> measurable (PiM (f ` J) (\<lambda>_. M)) (PiM J (\<lambda>_. M))" 

1286 
proof (rule measurable_PiM) 

1287 
show "(\<lambda>m. compose J m f) 

1288 
\<in> space (Pi\<^isub>M (f ` J) (\<lambda>_. M)) \<rightarrow> 

1289 
(J \<rightarrow> space M) \<inter> extensional J" 

1290 
proof safe 

1291 
fix x and i 

1292 
assume x: "x \<in> space (PiM (f ` J) (\<lambda>_. M))" "i \<in> J" 

1293 
with inj show "compose J x f i \<in> space M" 

1294 
by (auto simp: space_PiM compose_def) 

1295 
next 

1296 
fix x assume "x \<in> space (PiM (f ` J) (\<lambda>_. M))" 

1297 
show "(compose J x f) \<in> extensional J" by (rule compose_extensional) 

1298 
qed 

1299 
next 

1300 
fix S X 

1301 
have inv: "\<And>j. j \<in> f ` J \<Longrightarrow> f (f' j) = j" using assms by auto 

1302 
assume S: "S \<noteq> {} \<or> J = {}" "finite S" "S \<subseteq> J" and P: "\<And>i. i \<in> S \<Longrightarrow> X i \<in> sets M" 

1303 
have "(\<lambda>m. compose J m f) ` prod_emb J (\<lambda>_. M) S (Pi\<^isub>E S X) \<inter> 

1304 
space (Pi\<^isub>M (f ` J) (\<lambda>_. M)) = prod_emb (f ` J) (\<lambda>_. M) (f ` S) (Pi\<^isub>E (f ` S) (\<lambda>b. X (f' b)))" 

1305 
using assms inv S sets_into_space[OF P] 

1306 
by (force simp: prod_emb_iff compose_def space_PiM extensional_def Pi_def intro: imageI) 

1307 
also have "\<dots> \<in> sets (Pi\<^isub>M (f ` J) (\<lambda>_. M))" 

1308 
proof 

1309 
from S show "f ` S \<subseteq> f ` J" by auto 

1310 
show "(\<Pi>\<^isub>E b\<in>f ` S. X (f' b)) \<in> sets (Pi\<^isub>M (f ` S) (\<lambda>_. M))" 

1311 
proof (rule sets_PiM_I_finite) 

1312 
show "finite (f ` S)" using S by simp 

1313 
fix i assume "i \<in> f ` S" hence "f' i \<in> S" using S assms by auto 

1314 
thus "X (f' i) \<in> sets M" by (rule P) 

1315 
qed 

1316 
qed 

1317 
finally show "(\<lambda>m. compose J m f) ` prod_emb J (\<lambda>_. M) S (Pi\<^isub>E S X) \<inter> 

1318 
space (Pi\<^isub>M (f ` J) (\<lambda>_. M)) \<in> sets (Pi\<^isub>M (f ` J) (\<lambda>_. M))" . 

1319 
qed 

1320 

1321 
lemma 

1322 
measurable_compose_inv: 

1323 
fixes f::"'a \<Rightarrow> 'b" 

1324 
assumes inj: "\<And>j. j \<in> J \<Longrightarrow> f' (f j) = j" 

1325 
assumes "finite J" 

1326 
shows "(\<lambda>m. compose (f ` J) m f') \<in> measurable (PiM J (\<lambda>_. M)) (PiM (f ` J) (\<lambda>_. M))" 

1327 
proof  

1328 
have "(\<lambda>m. compose (f ` J) m f') \<in> measurable (Pi\<^isub>M (f' ` f ` J) (\<lambda>_. M)) (Pi\<^isub>M (f ` J) (\<lambda>_. M))" 

1329 
using assms by (auto intro: measurable_compose) 

1330 
moreover 

1331 
from inj have "f' ` f ` J = J" by (metis (hide_lams, mono_tags) image_iff set_eqI) 

1332 
ultimately show ?thesis by simp 

1333 
qed 

1334 

1335 
locale function_to_finmap = 

1336 
fixes J::"'a set" and f :: "'a \<Rightarrow> 'b::countable" and f' 

1337 
assumes [simp]: "finite J" 

1338 
assumes inv: "i \<in> J \<Longrightarrow> f' (f i) = i" 

1339 
begin 

1340 

1341 
text {* to measure finmaps *} 

1342 

1343 
definition "fm = (finmap_of (f ` J)) o (\<lambda>g. compose (f ` J) g f')" 

1344 

1345 
lemma domain_fm[simp]: "domain (fm x) = f ` J" 

1346 
unfolding fm_def by simp 

1347 

1348 
lemma fm_restrict[simp]: "fm (restrict y J) = fm y" 

1349 
unfolding fm_def by (auto simp: compose_def inv intro: restrict_ext) 

1350 

1351 
lemma fm_product: 

1352 
assumes "\<And>i. space (M i) = UNIV" 

1353 
shows "fm ` Pi' (f ` J) S \<inter> space (Pi\<^isub>M J M) = (\<Pi>\<^isub>E j \<in> J. S (f j))" 

1354 
using assms 

1355 
by (auto simp: inv fm_def compose_def space_PiM Pi'_def) 

1356 

1357 
lemma fm_measurable: 

1358 
assumes "f ` J \<in> N" 

1359 
shows "fm \<in> measurable (Pi\<^isub>M J (\<lambda>_. M)) (Pi\<^isub>F N (\<lambda>_. M))" 

1360 
unfolding fm_def 

1361 
proof (rule measurable_comp, rule measurable_compose_inv) 

1362 
show "finmap_of (f ` J) \<in> measurable (Pi\<^isub>M (f ` J) (\<lambda>_. M)) (PiF N (\<lambda>_. M)) " 

1363 
using assms by (intro measurable_finmap_of measurable_component_singleton) auto 

1364 
qed (simp_all add: inv) 

1365 

1366 
lemma proj_fm: 

1367 
assumes "x \<in> J" 

1368 
shows "fm m (f x) = m x" 

1369 
using assms by (auto simp: fm_def compose_def o_def inv) 

1370 

1371 
lemma inj_on_compose_f': "inj_on (\<lambda>g. compose (f ` J) g f') (extensional J)" 

1372 
proof (rule inj_on_inverseI) 

1373 
fix x::"'a \<Rightarrow> 'c" assume "x \<in> extensional J" 

1374 
thus "(\<lambda>x. compose J x f) (compose (f ` J) x f') = x" 

1375 
by (auto simp: compose_def inv extensional_def) 

1376 
qed 

1377 

1378 
lemma inj_on_fm: 

1379 
assumes "\<And>i. space (M i) = UNIV" 

1380 
shows "inj_on fm (space (Pi\<^isub>M J M))" 

1381 
using assms 

1382 
apply (auto simp: fm_def space_PiM) 

1383 
apply (rule comp_inj_on) 

1384 
apply (rule inj_on_compose_f') 

1385 
apply (rule finmap_of_inj_on_extensional_finite) 

1386 
apply simp 

1387 
apply (auto) 

1388 
done 

1389 

1390 
text {* to measure functions *} 

1391 

1392 
definition "mf = (\<lambda>g. compose J g f) o proj" 

1393 

1394 
lemma 

1395 
assumes "x \<in> space (Pi\<^isub>M J (\<lambda>_. M))" "finite J" 

1396 
shows "proj (finmap_of J x) = x" 

1397 
using assms by (auto simp: space_PiM extensional_def) 

1398 

1399 
lemma 

1400 
assumes "x \<in> space (Pi\<^isub>F {J} (\<lambda>_. M))" 

1401 
shows "finmap_of J (proj x) = x" 

1402 
using assms by (auto simp: space_PiF Pi'_def finmap_eq_iff) 

1403 

1404 
lemma mf_fm: 

1405 
assumes "x \<in> space (Pi\<^isub>M J (\<lambda>_. M))" 

1406 
shows "mf (fm x) = x" 

1407 
proof  

1408 
have "mf (fm x) \<in> extensional J" 

1409 
by (auto simp: mf_def extensional_def compose_def) 

1410 
moreover 

1411 
have "x \<in> extensional J" using assms sets_into_space 

1412 
by (force simp: space_PiM) 

1413 
moreover 

1414 
{ fix i assume "i \<in> J" 

1415 
hence "mf (fm x) i = x i" 

1416 
by (auto simp: inv mf_def compose_def fm_def) 

1417 
} 

1418 
ultimately 

1419 
show ?thesis by (rule extensionalityI) 

1420 
qed 

1421 

1422 
lemma mf_measurable: 

1423 
assumes "space M = UNIV" 

1424 
shows "mf \<in> measurable (PiF {f ` J} (\<lambda>_. M)) (PiM J (\<lambda>_. M))" 

1425 
unfolding mf_def 

1426 
proof (rule measurable_comp, rule measurable_proj_PiM) 

1427 
show "(\<lambda>g. compose J g f) \<in> 

1428 
measurable (Pi\<^isub>M (f ` J) (\<lambda>x. M)) (Pi\<^isub>M J (\<lambda>_. M))" 

1429 
by (rule measurable_compose, rule inv) auto 

1430 
qed (auto simp add: space_PiM extensional_def assms) 

1431 

1432 
lemma fm_image_measurable: 

1433 
assumes "space M = UNIV" 

1434 
assumes "X \<in> sets (Pi\<^isub>M J (\<lambda>_. M))" 

1435 
shows "fm ` X \<in> sets (PiF {f ` J} (\<lambda>_. M))" 

1436 
proof  

1437 
have "fm ` X = (mf) ` X \<inter> space (PiF {f ` J} (\<lambda>_. M))" 

1438 
proof safe 

1439 
fix x assume "x \<in> X" 

1440 
with mf_fm[of x] sets_into_space[OF assms(2)] show "fm x \<in> mf ` X" by auto 

1441 
show "fm x \<in> space (PiF {f ` J} (\<lambda>_. M))" by (simp add: space_PiF assms) 

1442 
next 

1443 
fix y x 

1444 
assume x: "mf y \<in> X" 

1445 
assume y: "y \<in> space (PiF {f ` J} (\<lambda>_. M))" 

1446 
thus "y \<in> fm ` X" 

1447 
by (intro image_eqI[OF _ x], unfold finmap_eq_iff) 

1448 
(auto simp: space_PiF fm_def mf_def compose_def inv Pi'_def) 

1449 
qed 

1450 
also have "\<dots> \<in> sets (PiF {f ` J} (\<lambda>_. M))" 

1451 
using assms 

1452 
by (intro measurable_sets[OF mf_measurable]) auto 

1453 
finally show ?thesis . 

1454 
qed 

1455 

1456 
lemma fm_image_measurable_finite: 

1457 
assumes "space M = UNIV" 

1458 
assumes "X \<in> sets (Pi\<^isub>M J (\<lambda>_. M::'c measure))" 

1459 
shows "fm ` X \<in> sets (PiF (Collect finite) (\<lambda>_. M::'c measure))" 

1460 
using fm_image_measurable[OF assms] 

1461 
by (rule subspace_set_in_sets) (auto simp: finite_subset) 

1462 

1463 
text {* measure on finmaps *} 

1464 

1465 
definition "mapmeasure M N = distr M (PiF (Collect finite) N) (fm)" 

1466 

1467 
lemma sets_mapmeasure[simp]: "sets (mapmeasure M N) = sets (PiF (Collect finite) N)" 

1468 
unfolding mapmeasure_def by simp 

1469 

1470 
lemma space_mapmeasure[simp]: "space (mapmeasure M N) = space (PiF (Collect finite) N)" 

1471 
unfolding mapmeasure_def by simp 

1472 

1473 
lemma mapmeasure_PiF: 

1474 
assumes s1: "space M = space (Pi\<^isub>M J (\<lambda>_. N))" 

1475 
assumes s2: "sets M = (Pi\<^isub>M J (\<lambda>_. N))" 

1476 
assumes "space N = UNIV" 

1477 
assumes "X \<in> sets (PiF (Collect finite) (\<lambda>_. N))" 

1478 
shows "emeasure (mapmeasure M (\<lambda>_. N)) X = emeasure M ((fm ` X \<inter> extensional J))" 

1479 
using assms 

1480 
by (auto simp: measurable_eqI[OF s1 refl s2 refl] mapmeasure_def emeasure_distr 

1481 
fm_measurable space_PiM) 

1482 

1483 
lemma mapmeasure_PiM: 

1484 
fixes N::"'c measure" 

1485 
assumes s1: "space M = space (Pi\<^isub>M J (\<lambda>_. N))" 

1486 
assumes s2: "sets M = (Pi\<^isub>M J (\<lambda>_. N))" 

1487 
assumes N: "space N = UNIV" 

1488 
assumes X: "X \<in> sets M" 

32d1795cc77a
added projective limit;
&# 