author  hoelzl 
Fri, 22 Mar 2013 10:41:43 +0100  
changeset 51473  1210309fddab 
parent 51343  b61b32f62c78 
child 51489  f738e6dbd844 
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> (\<forall>k\<in>i. m k= n k)" 
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using assms by (auto simp: finmap_eq_iff) 

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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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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) 

132 

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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 {* Topological Space of Finite Maps *} 
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instantiation finmap :: (type, topological_space) topological_space 

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begin 

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

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"open_finmap = generate_topology {Pi' a ba b. \<forall>i\<in>a. open (b i)}" 

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lemma open_Pi'I: "(\<And>i. i \<in> I \<Longrightarrow> open (A i)) \<Longrightarrow> open (Pi' I A)" 

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by (auto intro: generate_topology.Basis simp: open_finmap_def) 

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instance using topological_space_generate_topology 

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by intro_classes (auto simp: open_finmap_def class.topological_space_def) 

167 

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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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using open_restricted_space[of "\<lambda>x. \<not> P x"] 

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unfolding closed_def by (rule back_subst) auto 

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lemma tendsto_proj: "((\<lambda>x. x) > a) F \<Longrightarrow> ((\<lambda>x. (x)\<^isub>F i) > (a)\<^isub>F i) F" 

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unfolding tendsto_def 

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proof safe 

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fix S::"'b set" 

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

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assume "open S" hence "open ?S" by (auto intro!: open_Pi'I) 

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moreover assume "\<forall>S. open S \<longrightarrow> a \<in> S \<longrightarrow> eventually (\<lambda>x. x \<in> S) F" "a i \<in> S" 

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ultimately have "eventually (\<lambda>x. x \<in> ?S) F" by auto 

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thus "eventually (\<lambda>x. (x)\<^isub>F i \<in> S) F" 

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by eventually_elim (insert `a i \<in> S`, force simp: Pi'_iff split: split_if_asm) 

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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_def 

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by (safe intro!: tendsto_proj tendsto_ident_at_within) 

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instance finmap :: (type, first_countable_topology) first_countable_topology 

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proof 

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

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have "\<forall>i. \<exists>A. countable A \<and> (\<forall>a\<in>A. x i \<in> a) \<and> (\<forall>a\<in>A. open a) \<and> 

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(\<forall>S. open S \<and> x i \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)) \<and> (\<forall>a b. a \<in> A \<longrightarrow> b \<in> A \<longrightarrow> a \<inter> b \<in> A)" (is "\<forall>i. ?th i") 

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proof 

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fix i from first_countable_basis_Int_stableE[of "x i"] guess A . 

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thus "?th i" by (intro exI[where x=A]) simp 

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qed 

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then guess A unfolding choice_iff .. note A = this 

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hence open_sub: "\<And>i S. i\<in>domain x \<Longrightarrow> open (S i) \<Longrightarrow> x i\<in>(S i) \<Longrightarrow> (\<exists>a\<in>A i. a\<subseteq>(S i))" by auto 

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have A_notempty: "\<And>i. i \<in> domain x \<Longrightarrow> A i \<noteq> {}" using open_sub[of _ "\<lambda>_. UNIV"] by auto 

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let ?A = "(\<lambda>f. Pi' (domain x) f) ` (Pi\<^isub>E (domain x) A)" 

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show "\<exists>A::nat \<Rightarrow> ('a\<Rightarrow>\<^isub>F'b) set. (\<forall>i. x \<in> (A i) \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))" 
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proof (rule first_countableI[where A="?A"], safe) 

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show "countable ?A" using A by (simp add: countable_PiE) 
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next 

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

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thus "\<exists>a\<in>?A. a \<subseteq> S" unfolding open_finmap_def 

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proof (induct rule: generate_topology.induct) 

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case UNIV thus ?case by (auto simp add: ex_in_conv PiE_eq_empty_iff A_notempty) 

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next 

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case (Int a b) 

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then obtain f g where 

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"f \<in> Pi\<^isub>E (domain x) A" "Pi' (domain x) f \<subseteq> a" "g \<in> Pi\<^isub>E (domain x) A" "Pi' (domain x) g \<subseteq> b" 

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by auto 

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thus ?case using A 

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by (auto simp: Pi'_iff PiE_iff extensional_def Int_stable_def 

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intro!: bexI[where x="\<lambda>i. f i \<inter> g i"]) 

241 
next 

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case (UN B) 

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then obtain b where "x \<in> b" "b \<in> B" by auto 

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hence "\<exists>a\<in>?A. a \<subseteq> b" using UN by simp 

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thus ?case using `b \<in> B` by blast 

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next 

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case (Basis s) 

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then obtain a b where xs: "x\<in> Pi' a b" "s = Pi' a b" "\<And>i. i\<in>a \<Longrightarrow> open (b i)" by auto 

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have "\<forall>i. \<exists>a. (i \<in> domain x \<and> open (b i) \<and> (x)\<^isub>F i \<in> b i) \<longrightarrow> (a\<in>A i \<and> a \<subseteq> b i)" 

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using open_sub[of _ b] by auto 

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then obtain b' 

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where "\<And>i. i \<in> domain x \<Longrightarrow> open (b i) \<Longrightarrow> (x)\<^isub>F i \<in> b i \<Longrightarrow> (b' i \<in>A i \<and> b' i \<subseteq> b i)" 

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unfolding choice_iff by auto 

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with xs have "\<And>i. i \<in> a \<Longrightarrow> (b' i \<in>A i \<and> b' i \<subseteq> b i)" "Pi' a b' \<subseteq> Pi' a b" 

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by (auto simp: Pi'_iff intro!: Pi'_mono) 

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thus ?case using xs 

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by (intro bexI[where x="Pi' a b'"]) 

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(auto simp: Pi'_iff intro!: image_eqI[where x="restrict b' (domain x)"]) 

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qed 

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qed (insert A,auto simp: PiE_iff intro!: open_Pi'I) 

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qed 

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50088  263 
subsection {* Metric Space of Finite Maps *} 
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instantiation finmap :: (type, metric_space) metric_space 

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begin 

267 

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definition dist_finmap where 

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"dist P Q = Max (range (\<lambda>i. dist ((P)\<^isub>F i) ((Q)\<^isub>F i))) + (if domain P = domain Q then 0 else 1)" 
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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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51104  277 
lemma finite_proj_image': "x \<notin> domain P \<Longrightarrow> finite ((P)\<^isub>F ` S)" 
278 
by (rule finite_subset[of _ "proj P ` (domain P \<inter> S \<union> {x})"]) auto 

279 

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lemma finite_proj_image: "finite ((P)\<^isub>F ` S)" 

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by (cases "\<exists>x. x \<notin> domain P") (auto intro: finite_proj_image' finite_subset[where B="domain P"]) 

282 

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lemma finite_proj_diag: "finite ((\<lambda>i. d ((P)\<^isub>F i) ((Q)\<^isub>F i)) ` S)" 

50088  284 
proof  
51104  285 
have "(\<lambda>i. d ((P)\<^isub>F i) ((Q)\<^isub>F i)) ` S = (\<lambda>(i, j). d i j) ` ((\<lambda>i. ((P)\<^isub>F i, (Q)\<^isub>F i)) ` S)" by auto 
286 
moreover have "((\<lambda>i. ((P)\<^isub>F i, (Q)\<^isub>F i)) ` S) \<subseteq> (\<lambda>i. (P)\<^isub>F i) ` S \<times> (\<lambda>i. (Q)\<^isub>F i) ` S" by auto 

287 
moreover have "finite \<dots>" using finite_proj_image[of P S] finite_proj_image[of Q S] 

288 
by (intro finite_cartesian_product) simp_all 

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ultimately show ?thesis by (simp add: finite_subset) 

50088  290 
qed 
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51104  292 
lemma dist_le_1_imp_domain_eq: 
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shows "dist P Q < 1 \<Longrightarrow> domain P = domain Q" 

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by (simp add: dist_finmap_def finite_proj_diag split: split_if_asm) 

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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) \<le> Max (range (\<lambda>i. dist (x i) (y i)))" 
300 
by (simp add: Max_ge_iff finite_proj_diag) 

301 
also have "\<dots> \<le> dist x y" by (simp add: dist_finmap_def) 

302 
finally show ?thesis . 

303 
qed 

304 

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lemma dist_finmap_lessI: 

51105  306 
assumes "domain P = domain Q" 
307 
assumes "0 < e" 

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assumes "\<And>i. i \<in> domain P \<Longrightarrow> dist (P i) (Q i) < e" 

51104  309 
shows "dist P Q < e" 
310 
proof  

311 
have "dist P Q = Max (range (\<lambda>i. dist (P i) (Q i)))" 

312 
using assms by (simp add: dist_finmap_def finite_proj_diag) 

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also have "\<dots> < e" 

314 
proof (subst Max_less_iff, safe) 

51105  315 
fix i 
316 
show "dist ((P)\<^isub>F i) ((Q)\<^isub>F i) < e" using assms 

317 
by (cases "i \<in> domain P") simp_all 

51104  318 
qed (simp add: finite_proj_diag) 
319 
finally show ?thesis . 

50088  320 
qed 
321 

322 
instance 

323 
proof 

324 
fix S::"('a \<Rightarrow>\<^isub>F 'b) set" 

51105  325 
show "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)" (is "_ = ?od") 
326 
proof 

327 
assume "open S" 

328 
thus ?od 

329 
unfolding open_finmap_def 

330 
proof (induct rule: generate_topology.induct) 

331 
case UNIV thus ?case by (auto intro: zero_less_one) 

332 
next 

333 
case (Int a b) 

334 
show ?case 

335 
proof safe 

336 
fix x assume x: "x \<in> a" "x \<in> b" 

337 
with Int x obtain e1 e2 where 

338 
"e1>0" "\<forall>y. dist y x < e1 \<longrightarrow> y \<in> a" "e2>0" "\<forall>y. dist y x < e2 \<longrightarrow> y \<in> b" by force 

339 
thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> a \<inter> b" 

340 
by (auto intro!: exI[where x="min e1 e2"]) 

341 
qed 

342 
next 

343 
case (UN K) 

344 
show ?case 

345 
proof safe 

346 
fix x X assume "x \<in> X" "X \<in> K" 

347 
moreover with UN obtain e where "e>0" "\<And>y. dist y x < e \<longrightarrow> y \<in> X" by force 

348 
ultimately show "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> \<Union>K" by auto 

349 
qed 

350 
next 

351 
case (Basis s) then obtain a b where s: "s = Pi' a b" and b: "\<And>i. i\<in>a \<Longrightarrow> open (b i)" by auto 

352 
show ?case 

353 
proof safe 

354 
fix x assume "x \<in> s" 

355 
hence [simp]: "finite a" and a_dom: "a = domain x" using s by (auto simp: Pi'_iff) 

356 
obtain es where es: "\<forall>i \<in> a. es i > 0 \<and> (\<forall>y. dist y (proj x i) < es i \<longrightarrow> y \<in> b i)" 

357 
using b `x \<in> s` by atomize_elim (intro bchoice, auto simp: open_dist s) 

358 
hence in_b: "\<And>i y. i \<in> a \<Longrightarrow> dist y (proj x i) < es i \<Longrightarrow> y \<in> b i" by auto 

359 
show "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> s" 

360 
proof (cases, rule, safe) 

361 
assume "a \<noteq> {}" 

362 
show "0 < min 1 (Min (es ` a))" using es by (auto simp: `a \<noteq> {}`) 

363 
fix y assume d: "dist y x < min 1 (Min (es ` a))" 

364 
show "y \<in> s" unfolding s 

365 
proof 

366 
show "domain y = a" using d s `a \<noteq> {}` by (auto simp: dist_le_1_imp_domain_eq a_dom) 

367 
fix i assume "i \<in> a" 

368 
moreover 

369 
hence "dist ((y)\<^isub>F i) ((x)\<^isub>F i) < es i" using d 

370 
by (auto simp: dist_finmap_def `a \<noteq> {}` intro!: le_less_trans[OF dist_proj]) 

371 
ultimately 

372 
show "y i \<in> b i" by (rule in_b) 

373 
qed 

374 
next 

375 
assume "\<not>a \<noteq> {}" 

376 
thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> s" 

377 
using s `x \<in> s` by (auto simp: Pi'_def dist_le_1_imp_domain_eq intro!: exI[where x=1]) 

378 
qed 

379 
qed 

380 
qed 

381 
next 

382 
assume "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" 

383 
then obtain e where e_pos: "\<And>x. x \<in> S \<Longrightarrow> e x > 0" and 

384 
e_in: "\<And>x y . x \<in> S \<Longrightarrow> dist y x < e x \<Longrightarrow> y \<in> S" 

385 
unfolding bchoice_iff 

386 
by auto 

387 
have S_eq: "S = \<Union>{Pi' a b a b. \<exists>x\<in>S. domain x = a \<and> b = (\<lambda>i. ball (x i) (e x))}" 

388 
proof safe 

389 
fix x assume "x \<in> S" 

390 
thus "x \<in> \<Union>{Pi' a b a b. \<exists>x\<in>S. domain x = a \<and> b = (\<lambda>i. ball (x i) (e x))}" 

391 
using e_pos by (auto intro!: exI[where x="Pi' (domain x) (\<lambda>i. ball (x i) (e x))"]) 

392 
next 

393 
fix x y 

394 
assume "y \<in> S" 

395 
moreover 

396 
assume "x \<in> (\<Pi>' i\<in>domain y. ball (y i) (e y))" 

397 
hence "dist x y < e y" using e_pos `y \<in> S` 

398 
by (auto simp: dist_finmap_def Pi'_iff finite_proj_diag dist_commute) 

399 
ultimately show "x \<in> S" by (rule e_in) 

400 
qed 

401 
also have "open \<dots>" 

402 
unfolding open_finmap_def 

403 
by (intro generate_topology.UN) (auto intro: generate_topology.Basis) 

404 
finally show "open S" . 

405 
qed 

50088  406 
next 
407 
fix P Q::"'a \<Rightarrow>\<^isub>F 'b" 

51104  408 
have Max_eq_iff: "\<And>A m. finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> (Max A = m) = (m \<in> A \<and> (\<forall>a\<in>A. a \<le> m))" 
409 
by (metis Max.in_idem Max_in max_def min_max.sup.commute order_refl) 

50088  410 
show "dist P Q = 0 \<longleftrightarrow> P = Q" 
51104  411 
by (auto simp: finmap_eq_iff dist_finmap_def Max_ge_iff finite_proj_diag Max_eq_iff 
412 
intro!: Max_eqI image_eqI[where x=undefined]) 

50088  413 
next 
414 
fix P Q R::"'a \<Rightarrow>\<^isub>F 'b" 

51104  415 
let ?dists = "\<lambda>P Q i. dist ((P)\<^isub>F i) ((Q)\<^isub>F i)" 
416 
let ?dpq = "?dists P Q" and ?dpr = "?dists P R" and ?dqr = "?dists Q R" 

417 
let ?dom = "\<lambda>P Q. (if domain P = domain Q then 0 else 1::real)" 

418 
have "dist P Q = Max (range ?dpq) + ?dom P Q" 

419 
by (simp add: dist_finmap_def) 

420 
also obtain t where "t \<in> range ?dpq" "t = Max (range ?dpq)" by (simp add: finite_proj_diag) 

421 
then obtain i where "Max (range ?dpq) = ?dpq i" by auto 

422 
also have "?dpq i \<le> ?dpr i + ?dqr i" by (rule dist_triangle2) 

423 
also have "?dpr i \<le> Max (range ?dpr)" by (simp add: finite_proj_diag) 

424 
also have "?dqr i \<le> Max (range ?dqr)" by (simp add: finite_proj_diag) 

425 
also have "?dom P Q \<le> ?dom P R + ?dom Q R" by simp 

426 
finally show "dist P Q \<le> dist P R + dist Q R" by (simp add: dist_finmap_def ac_simps) 

50088  427 
qed 
428 

429 
end 

430 

431 
subsection {* Complete Space of Finite Maps *} 

432 

433 
lemma tendsto_finmap: 

434 
fixes f::"nat \<Rightarrow> ('i \<Rightarrow>\<^isub>F ('a::metric_space))" 

435 
assumes ind_f: "\<And>n. domain (f n) = domain g" 

436 
assumes proj_g: "\<And>i. i \<in> domain g \<Longrightarrow> (\<lambda>n. (f n) i) > g i" 

437 
shows "f > g" 

51104  438 
unfolding tendsto_iff 
439 
proof safe 

440 
fix e::real assume "0 < e" 

441 
let ?dists = "\<lambda>x i. dist ((f x)\<^isub>F i) ((g)\<^isub>F i)" 

442 
have "eventually (\<lambda>x. \<forall>i\<in>domain g. ?dists x i < e) sequentially" 

443 
using finite_domain[of g] proj_g 

444 
proof induct 

445 
case (insert i G) 

446 
with `0 < e` have "eventually (\<lambda>x. ?dists x i < e) sequentially" by (auto simp add: tendsto_iff) 

447 
moreover 

448 
from insert have "eventually (\<lambda>x. \<forall>i\<in>G. dist ((f x)\<^isub>F i) ((g)\<^isub>F i) < e) sequentially" by simp 

449 
ultimately show ?case by eventually_elim auto 

450 
qed simp 

451 
thus "eventually (\<lambda>x. dist (f x) g < e) sequentially" 

452 
by eventually_elim (auto simp add: dist_finmap_def finite_proj_diag ind_f `0 < e`) 

453 
qed 

50088  454 

455 
instance finmap :: (type, complete_space) complete_space 

456 
proof 

457 
fix P::"nat \<Rightarrow> 'a \<Rightarrow>\<^isub>F 'b" 

458 
assume "Cauchy P" 

459 
then obtain Nd where Nd: "\<And>n. n \<ge> Nd \<Longrightarrow> dist (P n) (P Nd) < 1" 

460 
by (force simp: cauchy) 

461 
def d \<equiv> "domain (P Nd)" 

462 
with Nd have dim: "\<And>n. n \<ge> Nd \<Longrightarrow> domain (P n) = d" using dist_le_1_imp_domain_eq by auto 

463 
have [simp]: "finite d" unfolding d_def by simp 

464 
def p \<equiv> "\<lambda>i n. (P n) i" 

465 
def q \<equiv> "\<lambda>i. lim (p i)" 

466 
def Q \<equiv> "finmap_of d q" 

467 
have q: "\<And>i. i \<in> d \<Longrightarrow> q i = Q i" by (auto simp add: Q_def Abs_finmap_inverse) 

468 
{ 

469 
fix i assume "i \<in> d" 

470 
have "Cauchy (p i)" unfolding cauchy p_def 

471 
proof safe 

472 
fix e::real assume "0 < e" 

473 
with `Cauchy P` obtain N where N: "\<And>n. n\<ge>N \<Longrightarrow> dist (P n) (P N) < min e 1" 

474 
by (force simp: cauchy min_def) 

475 
hence "\<And>n. n \<ge> N \<Longrightarrow> domain (P n) = domain (P N)" using dist_le_1_imp_domain_eq by auto 

476 
with dim have dim: "\<And>n. n \<ge> N \<Longrightarrow> domain (P n) = d" by (metis nat_le_linear) 

477 
show "\<exists>N. \<forall>n\<ge>N. dist ((P n) i) ((P N) i) < e" 

478 
proof (safe intro!: exI[where x="N"]) 

479 
fix n assume "N \<le> n" have "N \<le> N" by simp 

480 
have "dist ((P n) i) ((P N) i) \<le> dist (P n) (P N)" 

481 
using dim[OF `N \<le> n`] dim[OF `N \<le> N`] `i \<in> d` 

482 
by (auto intro!: dist_proj) 

483 
also have "\<dots> < e" using N[OF `N \<le> n`] by simp 

484 
finally show "dist ((P n) i) ((P N) i) < e" . 

485 
qed 

486 
qed 

487 
hence "convergent (p i)" by (metis Cauchy_convergent_iff) 

488 
hence "p i > q i" unfolding q_def convergent_def by (metis limI) 

489 
} note p = this 

490 
have "P > Q" 

491 
proof (rule metric_LIMSEQ_I) 

492 
fix e::real assume "0 < e" 

51104  493 
have "\<exists>ni. \<forall>i\<in>d. \<forall>n\<ge>ni i. dist (p i n) (q i) < e" 
50088  494 
proof (safe intro!: bchoice) 
495 
fix i assume "i \<in> d" 

51104  496 
from p[OF `i \<in> d`, THEN metric_LIMSEQ_D, OF `0 < e`] 
497 
show "\<exists>no. \<forall>n\<ge>no. dist (p i n) (q i) < e" . 

50088  498 
qed then guess ni .. note ni = this 
499 
def N \<equiv> "max Nd (Max (ni ` d))" 

500 
show "\<exists>N. \<forall>n\<ge>N. dist (P n) Q < e" 

501 
proof (safe intro!: exI[where x="N"]) 

502 
fix n assume "N \<le> n" 

51104  503 
hence dom: "domain (P n) = d" "domain Q = d" "domain (P n) = domain Q" 
50088  504 
using dim by (simp_all add: N_def Q_def dim_def Abs_finmap_inverse) 
51104  505 
show "dist (P n) Q < e" 
506 
proof (rule dist_finmap_lessI[OF dom(3) `0 < e`]) 

507 
fix i 

508 
assume "i \<in> domain (P n)" 

509 
hence "ni i \<le> Max (ni ` d)" using dom by simp 

50088  510 
also have "\<dots> \<le> N" by (simp add: N_def) 
51104  511 
finally show "dist ((P n)\<^isub>F i) ((Q)\<^isub>F i) < e" using ni `i \<in> domain (P n)` `N \<le> n` dom 
512 
by (auto simp: p_def q N_def less_imp_le) 

50088  513 
qed 
514 
qed 

515 
qed 

516 
thus "convergent P" by (auto simp: convergent_def) 

517 
qed 

518 

51105  519 
subsection {* Second Countable Space of Finite Maps *} 
50088  520 

51105  521 
instantiation finmap :: (countable, second_countable_topology) second_countable_topology 
50088  522 
begin 
523 

51106  524 
definition basis_proj::"'b set set" 
525 
where "basis_proj = (SOME B. countable B \<and> topological_basis B)" 

526 

527 
lemma countable_basis_proj: "countable basis_proj" and basis_proj: "topological_basis basis_proj" 

528 
unfolding basis_proj_def by (intro is_basis countable_basis)+ 

529 

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530 
definition basis_finmap::"('a \<Rightarrow>\<^isub>F 'b) set set" 
51106  531 
where "basis_finmap = {Pi' I SI S. finite I \<and> (\<forall>i \<in> I. S i \<in> basis_proj)}" 
50245
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532 

dea9363887a6
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533 
lemma in_basis_finmapI: 
51106  534 
assumes "finite I" assumes "\<And>i. i \<in> I \<Longrightarrow> S i \<in> basis_proj" 
50245
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535 
shows "Pi' I S \<in> basis_finmap" 
dea9363887a6
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immler
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536 
using assms unfolding basis_finmap_def by auto 
dea9363887a6
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537 

dea9363887a6
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538 
lemma basis_finmap_eq: 
51106  539 
assumes "basis_proj \<noteq> {}" 
540 
shows "basis_finmap = (\<lambda>f. Pi' (domain f) (\<lambda>i. from_nat_into basis_proj ((f)\<^isub>F i))) ` 

50245
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immler
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changeset

541 
(UNIV::('a \<Rightarrow>\<^isub>F nat) set)" (is "_ = ?f ` _") 
dea9363887a6
based countable topological basis on Countable_Set
immler
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542 
unfolding basis_finmap_def 
dea9363887a6
based countable topological basis on Countable_Set
immler
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diff
changeset

543 
proof safe 
dea9363887a6
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immler
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changeset

544 
fix I::"'a set" and S::"'a \<Rightarrow> 'b set" 
51106  545 
assume "finite I" "\<forall>i\<in>I. S i \<in> basis_proj" 
546 
hence "Pi' I S = ?f (finmap_of I (\<lambda>x. to_nat_on basis_proj (S x)))" 

547 
by (force simp: Pi'_def countable_basis_proj) 

50245
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diff
changeset

548 
thus "Pi' I S \<in> range ?f" by simp 
51106  549 
next 
550 
fix x and f::"'a \<Rightarrow>\<^isub>F nat" 

551 
show "\<exists>I S. (\<Pi>' i\<in>domain f. from_nat_into local.basis_proj ((f)\<^isub>F i)) = Pi' I S \<and> 

552 
finite I \<and> (\<forall>i\<in>I. S i \<in> local.basis_proj)" 

553 
using assms by (auto intro: from_nat_into) 

554 
qed 

555 

556 
lemma basis_finmap_eq_empty: "basis_proj = {} \<Longrightarrow> basis_finmap = {Pi' {} undefined}" 

557 
by (auto simp: Pi'_iff basis_finmap_def) 

50088  558 

50245
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559 
lemma countable_basis_finmap: "countable basis_finmap" 
51106  560 
by (cases "basis_proj = {}") (auto simp: basis_finmap_eq basis_finmap_eq_empty) 
50088  561 

562 
lemma finmap_topological_basis: 

50245
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diff
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563 
"topological_basis basis_finmap" 
50088  564 
proof (subst topological_basis_iff, safe) 
50245
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immler
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diff
changeset

565 
fix B' assume "B' \<in> basis_finmap" 
dea9363887a6
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diff
changeset

566 
thus "open B'" 
51106  567 
by (auto intro!: open_Pi'I topological_basis_open[OF basis_proj] 
50245
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immler
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changeset

568 
simp: topological_basis_def basis_finmap_def Let_def) 
50088  569 
next 
570 
fix O'::"('a \<Rightarrow>\<^isub>F 'b) set" and x 

51105  571 
assume O': "open O'" "x \<in> O'" 
572 
then obtain a where a: 

573 
"x \<in> Pi' (domain x) a" "Pi' (domain x) a \<subseteq> O'" "\<And>i. i\<in>domain x \<Longrightarrow> open (a i)" 

574 
unfolding open_finmap_def 

575 
proof (atomize_elim, induct rule: generate_topology.induct) 

576 
case (Int a b) 

577 
let ?p="\<lambda>a f. x \<in> Pi' (domain x) f \<and> Pi' (domain x) f \<subseteq> a \<and> (\<forall>i. i \<in> domain x \<longrightarrow> open (f i))" 

578 
from Int obtain f g where "?p a f" "?p b g" by auto 

579 
thus ?case by (force intro!: exI[where x="\<lambda>i. f i \<inter> g i"] simp: Pi'_def) 

580 
next 

581 
case (UN k) 

582 
then obtain kk a where "x \<in> kk" "kk \<in> k" "x \<in> Pi' (domain x) a" "Pi' (domain x) a \<subseteq> kk" 

583 
"\<And>i. i\<in>domain x \<Longrightarrow> open (a i)" 

584 
by force 

585 
thus ?case by blast 

586 
qed (auto simp: Pi'_def) 

50088  587 
have "\<exists>B. 
51106  588 
(\<forall>i\<in>domain x. x i \<in> B i \<and> B i \<subseteq> a i \<and> B i \<in> basis_proj)" 
50088  589 
proof (rule bchoice, safe) 
590 
fix i assume "i \<in> domain x" 

51105  591 
hence "open (a i)" "x i \<in> a i" using a by auto 
51106  592 
from topological_basisE[OF basis_proj this] guess b' . 
593 
thus "\<exists>y. x i \<in> y \<and> y \<subseteq> a i \<and> y \<in> basis_proj" by auto 

50088  594 
qed 
595 
then guess B .. note B = this 

50245
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immler
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50244
diff
changeset

596 
def B' \<equiv> "Pi' (domain x) (\<lambda>i. (B i)::'b set)" 
51105  597 
have "B' \<subseteq> Pi' (domain x) a" using B by (auto intro!: Pi'_mono simp: B'_def) 
598 
also note `\<dots> \<subseteq> O'` 

599 
finally show "\<exists>B'\<in>basis_finmap. x \<in> B' \<and> B' \<subseteq> O'" using B 

600 
by (auto intro!: bexI[where x=B'] Pi'_mono in_basis_finmapI simp: B'_def) 

50088  601 
qed 
602 

603 
lemma range_enum_basis_finmap_imp_open: 

50245
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50244
diff
changeset

604 
assumes "x \<in> basis_finmap" 
50088  605 
shows "open x" 
606 
using finmap_topological_basis assms by (auto simp: topological_basis_def) 

607 

51343
b61b32f62c78
use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents:
51106
diff
changeset

608 
instance proof qed (blast intro: finmap_topological_basis countable_basis_finmap topological_basis_imp_subbasis) 
50088  609 

610 
end 

611 

51105  612 
subsection {* Polish Space of Finite Maps *} 
613 

614 
instance finmap :: (countable, polish_space) polish_space proof qed 

615 

616 

50088  617 
subsection {* Product Measurable Space of Finite Maps *} 
618 

619 
definition "PiF I M \<equiv> 

50124  620 
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  621 

622 
abbreviation 

623 
"Pi\<^isub>F I M \<equiv> PiF I M" 

624 

625 
syntax 

626 
"_PiF" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3PIF _:_./ _)" 10) 

627 

628 
syntax (xsymbols) 

629 
"_PiF" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3\<Pi>\<^isub>F _\<in>_./ _)" 10) 

630 

631 
syntax (HTML output) 

632 
"_PiF" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3\<Pi>\<^isub>F _\<in>_./ _)" 10) 

633 

634 
translations 

635 
"PIF x:I. M" == "CONST PiF I (%x. M)" 

636 

637 
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> 

638 
Pow (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j)))" 

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

639 
by (auto simp: Pi'_def) (blast dest: sets.sets_into_space) 
50088  640 

641 
lemma space_PiF: "space (PiF I M) = (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j)))" 

642 
unfolding PiF_def using PiF_gen_subset by (rule space_measure_of) 

643 

644 
lemma sets_PiF: 

645 
"sets (PiF I M) = sigma_sets (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j))) 

646 
{(\<Pi>' j\<in>J. X j) X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}" 

647 
unfolding PiF_def using PiF_gen_subset by (rule sets_measure_of) 

648 

649 
lemma sets_PiF_singleton: 

650 
"sets (PiF {I} M) = sigma_sets (\<Pi>' j\<in>I. space (M j)) 

651 
{(\<Pi>' j\<in>I. X j) X. X \<in> (\<Pi> j\<in>I. sets (M j))}" 

652 
unfolding sets_PiF by simp 

653 

654 
lemma in_sets_PiFI: 

655 
assumes "X = (Pi' J S)" "J \<in> I" "\<And>i. i\<in>J \<Longrightarrow> S i \<in> sets (M i)" 

656 
shows "X \<in> sets (PiF I M)" 

657 
unfolding sets_PiF 

658 
using assms by blast 

659 

660 
lemma product_in_sets_PiFI: 

661 
assumes "J \<in> I" "\<And>i. i\<in>J \<Longrightarrow> S i \<in> sets (M i)" 

662 
shows "(Pi' J S) \<in> sets (PiF I M)" 

663 
unfolding sets_PiF 

664 
using assms by blast 

665 

666 
lemma singleton_space_subset_in_sets: 

667 
fixes J 

668 
assumes "J \<in> I" 

669 
assumes "finite J" 

670 
shows "space (PiF {J} M) \<in> sets (PiF I M)" 

671 
using assms 

672 
by (intro in_sets_PiFI[where J=J and S="\<lambda>i. space (M i)"]) 

673 
(auto simp: product_def space_PiF) 

674 

675 
lemma singleton_subspace_set_in_sets: 

676 
assumes A: "A \<in> sets (PiF {J} M)" 

677 
assumes "finite J" 

678 
assumes "J \<in> I" 

679 
shows "A \<in> sets (PiF I M)" 

680 
using A[unfolded sets_PiF] 

681 
apply (induct A) 

682 
unfolding sets_PiF[symmetric] unfolding space_PiF[symmetric] 

683 
using assms 

684 
by (auto intro: in_sets_PiFI intro!: singleton_space_subset_in_sets) 

685 

50124  686 
lemma finite_measurable_singletonI: 
50088  687 
assumes "finite I" 
688 
assumes "\<And>J. J \<in> I \<Longrightarrow> finite J" 

689 
assumes MN: "\<And>J. J \<in> I \<Longrightarrow> A \<in> measurable (PiF {J} M) N" 

690 
shows "A \<in> measurable (PiF I M) N" 

691 
unfolding measurable_def 

692 
proof safe 

693 
fix y assume "y \<in> sets N" 

694 
have "A ` y \<inter> space (PiF I M) = (\<Union>J\<in>I. A ` y \<inter> space (PiF {J} M))" 

695 
by (auto simp: space_PiF) 

696 
also have "\<dots> \<in> sets (PiF I M)" 

697 
proof 

698 
show "finite I" by fact 

699 
fix J assume "J \<in> I" 

700 
with assms have "finite J" by simp 

701 
show "A ` y \<inter> space (PiF {J} M) \<in> sets (PiF I M)" 

702 
by (rule singleton_subspace_set_in_sets[OF measurable_sets[OF assms(3)]]) fact+ 

703 
qed 

704 
finally show "A ` y \<inter> space (PiF I M) \<in> sets (PiF I M)" . 

705 
next 

706 
fix x assume "x \<in> space (PiF I M)" thus "A x \<in> space N" 

707 
using MN[of "domain x"] 

708 
by (auto simp: space_PiF measurable_space Pi'_def) 

709 
qed 

710 

50124  711 
lemma countable_finite_comprehension: 
50088  712 
fixes f :: "'a::countable set \<Rightarrow> _" 
713 
assumes "\<And>s. P s \<Longrightarrow> finite s" 

714 
assumes "\<And>s. P s \<Longrightarrow> f s \<in> sets M" 

715 
shows "\<Union>{f ss. P s} \<in> sets M" 

716 
proof  

717 
have "\<Union>{f ss. P s} = (\<Union>n::nat. let s = set (from_nat n) in if P s then f s else {})" 

718 
proof safe 

719 
fix x X s assume "x \<in> f s" "P s" 

720 
moreover with assms obtain l where "s = set l" using finite_list by blast 

721 
ultimately show "x \<in> (\<Union>n. let s = set (from_nat n) in if P s then f s else {})" using `P s` 

722 
by (auto intro!: exI[where x="to_nat l"]) 

723 
next 

724 
fix x n assume "x \<in> (let s = set (from_nat n) in if P s then f s else {})" 

725 
thus "x \<in> \<Union>{f ss. P s}" using assms by (auto simp: Let_def split: split_if_asm) 

726 
qed 

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

728 
also have "\<dots> \<in> sets M" using assms by (auto simp: Let_def) 

729 
finally show ?thesis . 

730 
qed 

731 

732 
lemma space_subset_in_sets: 

733 
fixes J::"'a::countable set set" 

734 
assumes "J \<subseteq> I" 

735 
assumes "\<And>j. j \<in> J \<Longrightarrow> finite j" 

736 
shows "space (PiF J M) \<in> sets (PiF I M)" 

737 
proof  

738 
have "space (PiF J M) = \<Union>{space (PiF {j} M)j. j \<in> J}" 

739 
unfolding space_PiF by blast 

740 
also have "\<dots> \<in> sets (PiF I M)" using assms 

741 
by (intro countable_finite_comprehension) (auto simp: singleton_space_subset_in_sets) 

742 
finally show ?thesis . 

743 
qed 

744 

745 
lemma subspace_set_in_sets: 

746 
fixes J::"'a::countable set set" 

747 
assumes A: "A \<in> sets (PiF J M)" 

748 
assumes "J \<subseteq> I" 

749 
assumes "\<And>j. j \<in> J \<Longrightarrow> finite j" 

750 
shows "A \<in> sets (PiF I M)" 

751 
using A[unfolded sets_PiF] 

752 
apply (induct A) 

753 
unfolding sets_PiF[symmetric] unfolding space_PiF[symmetric] 

754 
using assms 

755 
by (auto intro: in_sets_PiFI intro!: space_subset_in_sets) 

756 

50124  757 
lemma countable_measurable_PiFI: 
50088  758 
fixes I::"'a::countable set set" 
759 
assumes MN: "\<And>J. J \<in> I \<Longrightarrow> finite J \<Longrightarrow> A \<in> measurable (PiF {J} M) N" 

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

761 
unfolding measurable_def 

762 
proof safe 

763 
fix y assume "y \<in> sets N" 

764 
have "A ` y = (\<Union>{A ` y \<inter> {x. domain x = J}J. finite J})" by auto 

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

765 
{ fix x::"'a \<Rightarrow>\<^isub>F 'b" 
50088  766 
from finite_list[of "domain x"] obtain xs where "set xs = domain x" by auto 
50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

767 
hence "\<exists>n. domain x = set (from_nat n)" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

768 
by (intro exI[where x="to_nat xs"]) auto } 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

769 
hence "A ` y \<inter> space (PiF I M) = (\<Union>n. A ` y \<inter> space (PiF ({set (from_nat n)}\<inter>I) M))" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

770 
by (auto simp: space_PiF Pi'_def) 
50088  771 
also have "\<dots> \<in> sets (PiF I M)" 
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50124
diff
changeset

772 
apply (intro sets.Int sets.countable_nat_UN subsetI, safe) 
50088  773 
apply (case_tac "set (from_nat i) \<in> I") 
774 
apply simp_all 

775 
apply (rule singleton_subspace_set_in_sets[OF measurable_sets[OF MN]]) 

776 
using assms `y \<in> sets N` 

777 
apply (auto simp: space_PiF) 

778 
done 

779 
finally show "A ` y \<inter> space (PiF I M) \<in> sets (PiF I M)" . 

780 
next 

781 
fix x assume "x \<in> space (PiF I M)" thus "A x \<in> space N" 

782 
using MN[of "domain x"] by (auto simp: space_PiF measurable_space Pi'_def) 

783 
qed 

784 

785 
lemma measurable_PiF: 

786 
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))" 

787 
assumes S: "\<And>J S. J \<in> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> S i \<in> sets (M i)) \<Longrightarrow> 

788 
f ` (Pi' J S) \<inter> space N \<in> sets N" 

789 
shows "f \<in> measurable N (PiF I M)" 

790 
unfolding PiF_def 

791 
using PiF_gen_subset 

792 
apply (rule measurable_measure_of) 

793 
using f apply force 

794 
apply (insert S, auto) 

795 
done 

796 

50124  797 
lemma restrict_sets_measurable: 
50088  798 
assumes A: "A \<in> sets (PiF I M)" and "J \<subseteq> I" 
799 
shows "A \<inter> {m. domain m \<in> J} \<in> sets (PiF J M)" 

800 
using A[unfolded sets_PiF] 

50124  801 
proof (induct A) 
802 
case (Basic a) 

50088  803 
then obtain K S where S: "a = Pi' K S" "K \<in> I" "(\<forall>i\<in>K. S i \<in> sets (M i))" 
804 
by auto 

50124  805 
show ?case 
50088  806 
proof cases 
807 
assume "K \<in> J" 

808 
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 

809 
by (auto intro!: exI[where x=K] exI[where x=S] simp: Pi'_def) 

810 
also have "\<dots> \<subseteq> sets (PiF J M)" unfolding sets_PiF by auto 

811 
finally show ?thesis . 

812 
next 

813 
assume "K \<notin> J" 

814 
hence "a \<inter> {m. domain m \<in> J} = {}" using S by (auto simp: Pi'_def) 

815 
also have "\<dots> \<in> sets (PiF J M)" by simp 

816 
finally show ?thesis . 

817 
qed 

818 
next 

50124  819 
case (Union a) 
50088  820 
have "UNION UNIV a \<inter> {m. domain m \<in> J} = (\<Union>i. (a i \<inter> {m. domain m \<in> J}))" 
821 
by simp 

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

822 
also have "\<dots> \<in> sets (PiF J M)" using Union by (intro sets.countable_nat_UN) auto 
50124  823 
finally show ?case . 
50088  824 
next 
50124  825 
case (Compl a) 
50088  826 
have "(space (PiF I M)  a) \<inter> {m. domain m \<in> J} = (space (PiF J M)  (a \<inter> {m. domain m \<in> J}))" 
827 
using `J \<subseteq> I` by (auto simp: space_PiF Pi'_def) 

50124  828 
also have "\<dots> \<in> sets (PiF J M)" using Compl by auto 
829 
finally show ?case by (simp add: space_PiF) 

830 
qed simp 

50088  831 

832 
lemma measurable_finmap_of: 

833 
assumes f: "\<And>i. (\<exists>x \<in> space N. i \<in> J x) \<Longrightarrow> (\<lambda>x. f x i) \<in> measurable N (M i)" 

834 
assumes J: "\<And>x. x \<in> space N \<Longrightarrow> J x \<in> I" "\<And>x. x \<in> space N \<Longrightarrow> finite (J x)" 

835 
assumes JN: "\<And>S. {x. J x = S} \<inter> space N \<in> sets N" 

836 
shows "(\<lambda>x. finmap_of (J x) (f x)) \<in> measurable N (PiF I M)" 

837 
proof (rule measurable_PiF) 

838 
fix x assume "x \<in> space N" 

839 
with J[of x] measurable_space[OF f] 

840 
show "domain (finmap_of (J x) (f x)) \<in> I \<and> 

841 
(\<forall>i\<in>domain (finmap_of (J x) (f x)). (finmap_of (J x) (f x)) i \<in> space (M i))" 

842 
by auto 

843 
next 

844 
fix K S assume "K \<in> I" and *: "\<And>i. i \<in> K \<Longrightarrow> S i \<in> sets (M i)" 

845 
with J have eq: "(\<lambda>x. finmap_of (J x) (f x)) ` Pi' K S \<inter> space N = 

846 
(if \<exists>x \<in> space N. K = J x \<and> finite K then if K = {} then {x \<in> space N. J x = K} 

847 
else (\<Inter>i\<in>K. (\<lambda>x. f x i) ` S i \<inter> {x \<in> space N. J x = K}) else {})" 

848 
by (auto simp: Pi'_def) 

849 
have r: "{x \<in> space N. J x = K} = space N \<inter> ({x. J x = K} \<inter> space N)" by auto 

850 
show "(\<lambda>x. finmap_of (J x) (f x)) ` Pi' K S \<inter> space N \<in> sets N" 

851 
unfolding eq r 

852 
apply (simp del: INT_simps add: ) 

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

853 
apply (intro conjI impI sets.finite_INT JN sets.Int[OF sets.top]) 
50088  854 
apply simp apply assumption 
855 
apply (subst Int_assoc[symmetric]) 

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

856 
apply (rule sets.Int) 
50088  857 
apply (intro measurable_sets[OF f] *) apply force apply assumption 
858 
apply (intro JN) 

859 
done 

860 
qed 

861 

862 
lemma measurable_PiM_finmap_of: 

863 
assumes "finite J" 

864 
shows "finmap_of J \<in> measurable (Pi\<^isub>M J M) (PiF {J} M)" 

865 
apply (rule measurable_finmap_of) 

866 
apply (rule measurable_component_singleton) 

867 
apply simp 

868 
apply rule 

869 
apply (rule `finite J`) 

870 
apply simp 

871 
done 

872 

873 
lemma proj_measurable_singleton: 

50124  874 
assumes "A \<in> sets (M i)" 
50088  875 
shows "(\<lambda>x. (x)\<^isub>F i) ` A \<inter> space (PiF {I} M) \<in> sets (PiF {I} M)" 
876 
proof cases 

877 
assume "i \<in> I" 

878 
hence "(\<lambda>x. (x)\<^isub>F i) ` A \<inter> space (PiF {I} M) = 

879 
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

880 
using sets.sets_into_space[OF ] `A \<in> sets (M i)` assms 
50088  881 
by (auto simp: space_PiF Pi'_def) 
882 
thus ?thesis using assms `A \<in> sets (M i)` 

883 
by (intro in_sets_PiFI) auto 

884 
next 

885 
assume "i \<notin> I" 

886 
hence "(\<lambda>x. (x)\<^isub>F i) ` A \<inter> space (PiF {I} M) = 

887 
(if undefined \<in> A then space (PiF {I} M) else {})" by (auto simp: space_PiF Pi'_def) 

888 
thus ?thesis by simp 

889 
qed 

890 

891 
lemma measurable_proj_singleton: 

50124  892 
assumes "i \<in> I" 
50088  893 
shows "(\<lambda>x. (x)\<^isub>F i) \<in> measurable (PiF {I} M) (M i)" 
50124  894 
by (unfold measurable_def, intro CollectI conjI ballI proj_measurable_singleton assms) 
895 
(insert `i \<in> I`, auto simp: space_PiF) 

50088  896 

897 
lemma measurable_proj_countable: 

898 
fixes I::"'a::countable set set" 

899 
assumes "y \<in> space (M i)" 

900 
shows "(\<lambda>x. if i \<in> domain x then (x)\<^isub>F i else y) \<in> measurable (PiF I M) (M i)" 

901 
proof (rule countable_measurable_PiFI) 

902 
fix J assume "J \<in> I" "finite J" 

903 
show "(\<lambda>x. if i \<in> domain x then x i else y) \<in> measurable (PiF {J} M) (M i)" 

904 
unfolding measurable_def 

905 
proof safe 

906 
fix z assume "z \<in> sets (M i)" 

907 
have "(\<lambda>x. if i \<in> domain x then x i else y) ` z \<inter> space (PiF {J} M) = 

908 
(\<lambda>x. if i \<in> J then (x)\<^isub>F i else y) ` z \<inter> space (PiF {J} M)" 

909 
by (auto simp: space_PiF Pi'_def) 

910 
also have "\<dots> \<in> sets (PiF {J} M)" using `z \<in> sets (M i)` `finite J` 

911 
by (cases "i \<in> J") (auto intro!: measurable_sets[OF measurable_proj_singleton]) 

912 
finally show "(\<lambda>x. if i \<in> domain x then x i else y) ` z \<inter> space (PiF {J} M) \<in> 

913 
sets (PiF {J} M)" . 

914 
qed (insert `y \<in> space (M i)`, auto simp: space_PiF Pi'_def) 

915 
qed 

916 

917 
lemma measurable_restrict_proj: 

918 
assumes "J \<in> II" "finite J" 

919 
shows "finmap_of J \<in> measurable (PiM J M) (PiF II M)" 

920 
using assms 

921 
by (intro measurable_finmap_of measurable_component_singleton) auto 

922 

50124  923 
lemma measurable_proj_PiM: 
50088  924 
fixes J K ::"'a::countable set" and I::"'a set set" 
925 
assumes "finite J" "J \<in> I" 

926 
assumes "x \<in> space (PiM J M)" 

50124  927 
shows "proj \<in> measurable (PiF {J} M) (PiM J M)" 
50088  928 
proof (rule measurable_PiM_single) 
929 
show "proj \<in> space (PiF {J} M) \<rightarrow> (\<Pi>\<^isub>E i \<in> J. space (M i))" 

930 
using assms by (auto simp add: space_PiM space_PiF extensional_def sets_PiF Pi'_def) 

931 
next 

932 
fix A i assume A: "i \<in> J" "A \<in> sets (M i)" 

933 
show "{\<omega> \<in> space (PiF {J} M). (\<omega>)\<^isub>F i \<in> A} \<in> sets (PiF {J} M)" 

934 
proof 

935 
have "{\<omega> \<in> space (PiF {J} M). (\<omega>)\<^isub>F i \<in> A} = 

936 
(\<lambda>\<omega>. (\<omega>)\<^isub>F i) ` A \<inter> space (PiF {J} M)" by auto 

937 
also have "\<dots> \<in> sets (PiF {J} M)" 

938 
using assms A by (auto intro: measurable_sets[OF measurable_proj_singleton] simp: space_PiM) 

939 
finally show ?thesis . 

940 
qed simp 

941 
qed 

942 

943 
lemma space_PiF_singleton_eq_product: 

944 
assumes "finite I" 

945 
shows "space (PiF {I} M) = (\<Pi>' i\<in>I. space (M i))" 

946 
by (auto simp: product_def space_PiF assms) 

947 

948 
text {* adapted from @{thm sets_PiM_single} *} 

949 

950 
lemma sets_PiF_single: 

951 
assumes "finite I" "I \<noteq> {}" 

952 
shows "sets (PiF {I} M) = 

953 
sigma_sets (\<Pi>' i\<in>I. space (M i)) 

954 
{{f\<in>\<Pi>' i\<in>I. space (M i). f i \<in> A}  i A. i \<in> I \<and> A \<in> sets (M i)}" 

955 
(is "_ = sigma_sets ?\<Omega> ?R") 

956 
unfolding sets_PiF_singleton 

957 
proof (rule sigma_sets_eqI) 

958 
interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto 

959 
fix A assume "A \<in> {Pi' I X X. X \<in> (\<Pi> j\<in>I. sets (M j))}" 

960 
then obtain X where X: "A = Pi' I X" "X \<in> (\<Pi> j\<in>I. sets (M j))" by auto 

961 
show "A \<in> sigma_sets ?\<Omega> ?R" 

962 
proof  

963 
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

964 
using sets.sets_into_space 
50088  965 
by (auto simp: space_PiF product_def) blast 
966 
also have "\<dots> \<in> sigma_sets ?\<Omega> ?R" 

967 
using X `I \<noteq> {}` assms by (intro R.finite_INT) (auto simp: space_PiF) 

968 
finally show "A \<in> sigma_sets ?\<Omega> ?R" . 

969 
qed 

970 
next 

971 
fix A assume "A \<in> ?R" 

972 
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)" 

973 
by auto 

974 
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

975 
using sets.sets_into_space[OF A(3)] 
50088  976 
apply (auto simp: Pi'_iff split: split_if_asm) 
977 
apply blast 

978 
done 

979 
also have "\<dots> \<in> sigma_sets ?\<Omega> {Pi' I X X. X \<in> (\<Pi> j\<in>I. sets (M j))}" 

980 
using A 

981 
by (intro sigma_sets.Basic ) 

982 
(auto intro: exI[where x="\<lambda>j. if j = i then B else space (M j)"]) 

983 
finally show "A \<in> sigma_sets ?\<Omega> {Pi' I X X. X \<in> (\<Pi> j\<in>I. sets (M j))}" . 

984 
qed 

985 

986 
text {* adapted from @{thm PiE_cong} *} 

987 

988 
lemma Pi'_cong: 

989 
assumes "finite I" 

990 
assumes "\<And>i. i \<in> I \<Longrightarrow> f i = g i" 

991 
shows "Pi' I f = Pi' I g" 

992 
using assms by (auto simp: Pi'_def) 

993 

994 
text {* adapted from @{thm Pi_UN} *} 

995 

996 
lemma Pi'_UN: 

997 
fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set" 

998 
assumes "finite I" 

999 
assumes mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i" 

1000 
shows "(\<Union>n. Pi' I (A n)) = Pi' I (\<lambda>i. \<Union>n. A n i)" 

1001 
proof (intro set_eqI iffI) 

1002 
fix f assume "f \<in> Pi' I (\<lambda>i. \<Union>n. A n i)" 

1003 
then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i" "domain f = I" by (auto simp: `finite I` Pi'_def) 

1004 
from bchoice[OF this(1)] obtain n where n: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> (A (n i) i)" by auto 

1005 
obtain k where k: "\<And>i. i \<in> I \<Longrightarrow> n i \<le> k" 

1006 
using `finite I` finite_nat_set_iff_bounded_le[of "n`I"] by auto 

1007 
have "f \<in> Pi' I (\<lambda>i. A k i)" 

1008 
proof 

1009 
fix i assume "i \<in> I" 

1010 
from mono[OF this, of "n i" k] k[OF this] n[OF this] `domain f = I` `i \<in> I` 

1011 
show "f i \<in> A k i " by (auto simp: `finite I`) 

1012 
qed (simp add: `domain f = I` `finite I`) 

1013 
then show "f \<in> (\<Union>n. Pi' I (A n))" by auto 

1014 
qed (auto simp: Pi'_def `finite I`) 

1015 

1016 
text {* adapted from @{thm sigma_prod_algebra_sigma_eq} *} 

1017 

1018 
lemma sigma_fprod_algebra_sigma_eq: 

51106  1019 
fixes E :: "'i \<Rightarrow> 'a set set" and S :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" 
50088  1020 
assumes [simp]: "finite I" "I \<noteq> {}" 
1021 
and S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)" 

1022 
and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i" 

1023 
assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))" 

1024 
and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)" 

1025 
defines "P == { Pi' I F  F. \<forall>i\<in>I. F i \<in> E i }" 

1026 
shows "sets (PiF {I} M) = sigma_sets (space (PiF {I} M)) P" 

1027 
proof 

1028 
let ?P = "sigma (space (Pi\<^isub>F {I} M)) P" 

51106  1029 
from `finite I`[THEN ex_bij_betw_finite_nat] guess T .. 
1030 
then have T: "\<And>i. i \<in> I \<Longrightarrow> T i < card I" "\<And>i. i\<in>I \<Longrightarrow> the_inv_into I T (T i) = i" 

1031 
by (auto simp add: bij_betw_def set_eq_iff image_iff the_inv_into_f_f simp del: `finite I`) 

50088  1032 
have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>F {I} M))" 
1033 
using E_closed by (auto simp: space_PiF P_def Pi'_iff subset_eq) 

1034 
then have space_P: "space ?P = (\<Pi>' i\<in>I. space (M i))" 

1035 
by (simp add: space_PiF) 

1036 
have "sets (PiF {I} M) = 

1037 
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)}" 

1038 
using sets_PiF_single[of I M] by (simp add: space_P) 

1039 
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

1040 
proof (safe intro!: sets.sigma_sets_subset) 
50088  1041 
fix i A assume "i \<in> I" and A: "A \<in> sets (M i)" 
1042 
have "(\<lambda>x. (x)\<^isub>F i) \<in> measurable ?P (sigma (space (M i)) (E i))" 

1043 
proof (subst measurable_iff_measure_of) 

1044 
show "E i \<subseteq> Pow (space (M i))" using `i \<in> I` by fact 

1045 
from space_P `i \<in> I` show "(\<lambda>x. (x)\<^isub>F i) \<in> space ?P \<rightarrow> space (M i)" 

1046 
by auto 

1047 
show "\<forall>A\<in>E i. (\<lambda>x. (x)\<^isub>F i) ` A \<inter> space ?P \<in> sets ?P" 

1048 
proof 

1049 
fix A assume A: "A \<in> E i" 

1050 
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))" 

1051 
using E_closed `i \<in> I` by (auto simp: space_P Pi_iff subset_eq split: split_if_asm) 

1052 
also have "\<dots> = (\<Pi>' j\<in>I. \<Union>n. if i = j then A else S j n)" 

1053 
by (intro Pi'_cong) (simp_all add: S_union) 

51106  1054 
also have "\<dots> = (\<Union>xs\<in>{xs. length xs = card I}. \<Pi>' j\<in>I. if i = j then A else S j (xs ! T j))" 
1055 
using T 

1056 
apply auto 

1057 
apply (simp_all add: Pi'_iff bchoice_iff) 

1058 
apply (erule conjE exE)+ 

1059 
apply (rule_tac x="map (\<lambda>n. f (the_inv_into I T n)) [0..<card I]" in exI) 

1060 
apply (auto simp: bij_betw_def) 

1061 
done 

50088  1062 
also have "\<dots> \<in> sets ?P" 
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50124
diff
changeset

1063 
proof (safe intro!: sets.countable_UN) 
51106  1064 
fix xs show "(\<Pi>' j\<in>I. if i = j then A else S j (xs ! T j)) \<in> sets ?P" 
50088  1065 
using A S_in_E 
1066 
by (simp add: P_closed) 

51106  1067 
(auto simp: P_def subset_eq intro!: exI[of _ "\<lambda>j. if i = j then A else S j (xs ! T j)"]) 
50088  1068 
qed 
1069 
finally show "(\<lambda>x. (x)\<^isub>F i) ` A \<inter> space ?P \<in> sets ?P" 

1070 
using P_closed by simp 

1071 
qed 

1072 
qed 

1073 
from measurable_sets[OF this, of A] A `i \<in> I` E_closed 

1074 
have "(\<lambda>x. (x)\<^isub>F i) ` A \<inter> space ?P \<in> sets ?P" 

1075 
by (simp add: E_generates) 

1076 
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}" 

1077 
using P_closed by (auto simp: space_PiF) 

1078 
finally show "\<dots> \<in> sets ?P" . 

1079 
qed 

1080 
finally show "sets (PiF {I} M) \<subseteq> sigma_sets (space (PiF {I} M)) P" 

1081 
by (simp add: P_closed) 

1082 
show "sigma_sets (space (PiF {I} M)) P \<subseteq> sets (PiF {I} M)" 

1083 
using `finite I` `I \<noteq> {}` 

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

1084 
by (auto intro!: sets.sigma_sets_subset product_in_sets_PiFI simp: E_generates P_def) 
50088  1085 
qed 
1086 

1087 
lemma product_open_generates_sets_PiF_single: 

1088 
assumes "I \<noteq> {}" 

1089 
assumes [simp]: "finite I" 

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

1090 
shows "sets (PiF {I} (\<lambda>_. borel::'b::second_countable_topology measure)) = 
50088  1091 
sigma_sets (space (PiF {I} (\<lambda>_. borel))) {Pi' I F F. (\<forall>i\<in>I. F i \<in> Collect open)}" 
1092 
proof  

51106  1093 
from open_countable_basisE[OF open_UNIV] guess S::"'b set set" . note S = this 
50088  1094 
show ?thesis 
1095 
proof (rule sigma_fprod_algebra_sigma_eq) 

1096 
show "finite I" by simp 

1097 
show "I \<noteq> {}" by fact 

51106  1098 
def S'\<equiv>"from_nat_into S" 
1099 
show "(\<Union>j. S' j) = space borel" 

1100 
using S 

1101 
apply (auto simp add: from_nat_into countable_basis_proj S'_def basis_proj_def) 

1102 
apply (metis (lifting, mono_tags) UNIV_I UnionE basis_proj_def countable_basis_proj countable_subset from_nat_into_surj) 

1103 
done 

1104 
show "range S' \<subseteq> Collect open" 

1105 
using S 

1106 
apply (auto simp add: from_nat_into countable_basis_proj S'_def) 

1107 
apply (metis UNIV_not_empty Union_empty from_nat_into set_mp topological_basis_open[OF basis_proj] basis_proj_def) 

1108 
done 

50088  1109 
show "Collect open \<subseteq> Pow (space borel)" by simp 
1110 
show "sets borel = sigma_sets (space borel) (Collect open)" 

1111 
by (simp add: borel_def) 

1112 
qed 

1113 
qed 

1114 

50124  1115 
lemma finmap_UNIV[simp]: "(\<Union>J\<in>Collect finite. PI' j : J. UNIV) = UNIV" by auto 
50088  1116 

1117 
lemma borel_eq_PiF_borel: 

1118 
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

1119 
PiF (Collect finite) (\<lambda>_. borel :: 'a measure)" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1120 
unfolding borel_def PiF_def 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1121 
proof (rule measure_eqI, clarsimp, rule sigma_sets_eqI) 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1122 
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

1123 
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

1124 
using finmap_topological_basis by (force simp add: topological_basis_def) 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1125 
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

1126 
unfolding `a = \<Union>B'` 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1127 
proof (rule sets.countable_Union) 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1128 
from B' countable_basis_finmap show "countable B'" by (metis countable_subset) 
50088  1129 
next 
50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1130 
show "B' \<subseteq> sets (sigma UNIV 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1131 
{Pi' J X X J. finite J \<and> X \<in> J \<rightarrow> sigma_sets UNIV (Collect open)})" (is "_ \<subseteq> sets ?s") 
50088  1132 
proof 
50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1133 
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

1134 
then obtain J X where "x = Pi' J X" "finite J" "X \<in> J \<rightarrow> sigma_sets UNIV (Collect open)" 
51106  1135 
by (auto simp: basis_finmap_def topological_basis_open[OF basis_proj]) 
50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1136 
thus "x \<in> sets ?s" by auto 
50088  1137 
qed 
1138 
qed 

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

1139 
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

1140 
by simp 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1141 
next 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1142 
fix b::"('i \<Rightarrow>\<^isub>F 'a) set" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1143 
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

1144 
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

1145 
let ?b = "\<lambda>J. b \<inter> {x. domain x = J}" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1146 
have "b = \<Union>((\<lambda>J. ?b J) ` Collect finite)" by auto 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1147 
also have "\<dots> \<in> sets borel" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1148 
proof (rule sets.countable_Union, safe) 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1149 
fix J::"'i set" assume "finite J" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1150 
{ assume ef: "J = {}" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1151 
have "?b J \<in> sets borel" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1152 
proof cases 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1153 
assume "?b J \<noteq> {}" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1154 
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

1155 
hence "?b J = {f}" using `J = {}` 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1156 
by (auto simp: finmap_eq_iff) 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1157 
also have "{f} \<in> sets borel" by simp 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1158 
finally show ?thesis . 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1159 
qed simp 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1160 
} moreover { 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1161 
assume "J \<noteq> ({}::'i set)" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1162 
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

1163 
also have "\<dots> \<in> sets (PiF {J} (\<lambda>_. borel))" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1164 
using b' by (rule restrict_sets_measurable) (auto simp: `finite J`) 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1165 
also have "\<dots> = sigma_sets (space (PiF {J} (\<lambda>_. borel))) 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1166 
{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

1167 
(is "_ = sigma_sets _ ?P") 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1168 
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

1169 
also have "\<dots> \<subseteq> sigma_sets UNIV (Collect open)" 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1170 
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

1171 
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

1172 
} ultimately show "(?b J) \<in> sets borel" by blast 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1173 
qed (simp add: countable_Collect_finite) 
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset

1174 
finally show "b \<in> sigma_sets UNIV (Collect open)" by (simp add: borel_def) 
50088  1175 
qed (simp add: emeasure_sigma borel_def PiF_def) 
1176 

1177 
subsection {* Isomorphism between Functions and Finite Maps *} 

1178 

50124  1179 
lemma measurable_finmap_compose: 
50088  1180 
shows "(\<lambda>m. compose J m f) \<in> measurable (PiM (f ` J) (\<lambda>_. M)) (PiM J (\<lambda>_. M))" 
50124  1181 
unfolding compose_def by measurable 
50088  1182 

50124  1183 
lemma measurable_compose_inv: 
50088  1184 
assumes inj: "\<And>j. j \<in> J \<Longrightarrow> f' (f j) = j" 
1185 
shows "(\<lambda>m. compose (f ` J) m f') \<in> measurable (PiM J (\<lambda>_. M)) (PiM (f ` J) (\<lambda>_. M))" 

50124  1186 
unfolding compose_def by (rule measurable_restrict) (auto simp: inj) 
50088  1187 

1188 
locale function_to_finmap = 

1189 
fixes J::"'a set" and f :: "'a \<Rightarrow> 'b::countable" and f' 

1190 
assumes [simp]: "finite J" 

1191 
assumes inv: "i \<in> J \<Longrightarrow> f' (f i) = i" 

1192 
begin 

1193 

1194 
text {* to measure finmaps *} 

1195 

1196 
definition "fm = (finmap_of (f ` J)) o (\<lambda>g. compose (f ` J) g f')" 

1197 

1198 
lemma domain_fm[simp]: "domain (fm x) = f ` J" 

1199 
unfolding fm_def by simp 

1200 

1201 
lemma fm_restrict[simp]: "fm (restrict y J) = fm y" 

1202 
unfolding fm_def by (auto simp: compose_def inv intro: restrict_ext) 

1203 

1204 
lemma fm_product: 

1205 
assumes "\<And>i. space (M i) = UNIV" 

1206 
shows "fm ` Pi' (f ` J) S \<inter> space (Pi\<^isub>M J M) = (\<Pi>\<^isub>E j \<in> J. S (f j))" 

1207 
using assms 

1208 
by (auto simp: inv fm_def compose_def space_PiM Pi'_def) 

1209 

1210 
lemma fm_measurable: 

1211 
assumes "f ` J \<in> N" 

1212 
shows "fm \<in> measurable (Pi\<^isub>M J (\<lambda>_. M)) (Pi\<^isub>F N (\<lambda>_. M))" 

1213 
unfolding fm_def 

1214 
proof (rule measurable_comp, rule measurable_compose_inv) 

1215 
show "finmap_of (f ` J) \<in> measurable (Pi\<^isub>M (f ` J) (\<lambda>_. M)) (PiF N (\<lambda>_. M)) " 

1216 
using assms by (intro measurable_finmap_of measurable_component_singleton) auto 

1217 
qed (simp_all add: inv) 

1218 

1219 
lemma proj_fm: 

1220 
assumes "x \<in> J" 

1221 
shows "fm m (f x) = m x" 

1222 
using assms by (auto simp: fm_def compose_def o_def inv) 

1223 

1224 
lemma inj_on_compose_f': "inj_on (\<lambda>g. compose (f ` J) g f') (extensional J)" 

1225 
proof (rule inj_on_inverseI) 

1226 
fix x::"'a \<Rightarrow> 'c" assume "x \<in> extensional J" 

1227 
thus "(\<lambda>x. compose J x f) (compose (f ` J) x f') = x" 

1228 
by (auto simp: compose_def inv extensional_def) 

1229 
qed 

1230 

1231 
lemma inj_on_fm: 

1232 
assumes "\<And>i. space (M i) = UNIV" 

1233 
shows "inj_on fm (space (Pi\<^isub>M J M))" 

1234 
using assms 

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

1235 
apply (auto simp: fm_def space_PiM PiE_def) 
50088  1236 
apply (rule comp_inj_on) 
1237 
apply (rule inj_on_compose_f') 

1238 
apply (rule finmap_of_inj_on_extensional_finite) 

1239 
apply simp 

1240 
apply (auto) 

1241 
done 

1242 

1243 
text {* to measure functions *} 

1244 

1245 
definition "mf = (\<lambda>g. compose J g f) o proj" 

1246 

1247 
lemma mf_fm: 

1248 
assumes "x \<in> space (Pi\<^isub>M J (\<lambda>_. M))" 

1249 
shows "mf (fm x) = x" 

1250 
proof  

1251 
have "mf (fm x) \<in> extensional J" 

1252 
by (auto simp: mf_def extensional_def compose_def) 

1253 
moreover 

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

1254 
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

1255 
by (force simp: space_PiM PiE_def) 
50088  1256 
moreover 
1257 
{ fix i assume "i \<in> J" 

1258 
hence "mf (fm x) i = x i" 

1259 
by (auto simp: inv mf_def compose_def fm_def) 

1260 
} 

1261 
ultimately 

1262 
show ?thesis by (rule extensionalityI) 

1263 
qed 

1264 

1265 
lemma mf_measurable: 

1266 
assumes "space M = UNIV" 

1267 
shows "mf \<in> measurable (PiF {f ` J} (\<lambda>_. M)) (PiM J (\<lambda>_. M))" 

1268 
unfolding mf_def 

1269 
proof (rule measurable_comp, rule measurable_proj_PiM) 

50124  1270 
show "(\<lambda>g. compose J g f) \<in> measurable (Pi\<^isub>M (f ` J) (\<lambda>x. M)) (Pi\<^isub>M J (\<lambda>_. M))" 
1271 
by (rule measurable_finmap_compose) 

50088  1272 
qed (auto simp add: space_PiM extensional_def assms) 
1273 

1274 
lemma fm_image_measurable: 

1275 
assumes "space M = UNIV" 

1276 
assumes "X \<in> sets (Pi\<^isub>M J (\<lambda>_. M))" 

1277 
shows "fm ` X \<in> sets (PiF {f ` J} (\<lambda>_. M))" 

1278 
proof  

1279 
have "fm ` X = (mf) ` X \<inter> space (PiF {f ` J} (\<lambda>_. M))" 

1280 
proof safe 

1281 
fix x assume "x \<in> X" 

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

1282 
with mf_fm[of x] sets.sets_into_space[OF assms(2)] show "fm x \<in> mf ` X" by auto 
50088  1283 
show "fm x \<in> space (PiF {f ` J} (\<lambda>_. M))" by (simp add: space_PiF assms) 
1284 
next 

1285 
fix y x 

1286 
assume x: "mf y \<in> X" 

1287 
assume y: "y \<in> space (PiF {f ` J} (\<lambda>_. M))" 

1288 
thus "y \<in> fm ` X" 

1289 
by (intro image_eqI[OF _ x], unfold finmap_eq_iff) 

1290 
(auto simp: space_PiF fm_def mf_def compose_def inv Pi'_def) 

1291 
qed 

1292 
also have "\<dots> \<in> sets (PiF {f ` J} (\<lambda>_. M))" 

1293 
using assms 

1294 
by (intro measurable_sets[OF mf_measurable]) auto 

1295 
finally show ?thesis . 

1296 
qed 

1297 

1298 
lemma fm_image_measurable_finite: 

1299 
assumes "space M = UNIV" 

1300 
assumes "X \<in> sets (Pi\<^isub>M J (\<lambda>_. M::'c measure))" 

1301 
shows "fm ` X \<in> sets (PiF (Collect finite) (\<lambda>_. M::'c measure))" 

1302 
using fm_image_measurable[OF assms] 

1303 
by (rule subspace_set_in_sets) (auto simp: finite_subset) 

1304 

1305 
text {* measure on finmaps *} 

1306 

1307 
definition "mapmeasure M N = distr M (PiF (Collect finite) N) (fm)" 

1308 

1309 
lemma sets_mapmeasure[simp]: "sets (mapmeasure M N) = sets (PiF (Collect finite) N)" 

1310 
unfolding mapmeasure_def by simp 

1311 

1312 
lemma space_mapmeasure[simp]: "space (mapmeasure M N) = space (PiF (Collect finite) N)" 

1313 
unfolding mapmeasure_def by simp 

1314 

1315 
lemma mapmeasure_PiF: 

1316 
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

1317 
assumes s2: "sets M = sets (Pi\<^isub>M J (\<lambda>_. N))" 
50088  1318 
assumes "space N = UNIV" 
1319 
assumes "X \<in> sets (PiF (Collect finite) (\<lambda>_. N))" 

1320 
shows "emeasure (mapmeasure M (\<lambda>_. N)) X = emeasure M ((fm ` X \<inter> extensional J))" 

1321 
using assms 

1322 
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

1323 
fm_measurable space_PiM PiE_def) 
50088  1324 

1325 
lemma mapmeasure_PiM: 

1326 
fixes N::"'c measure" 

1327 
assumes s1: "space M = space (Pi\<^isub>M J (\<lambda>_. N))" 

1328 
assumes s2: "sets M = (Pi\<^isub>M J (\<lambda>_. N))" 

1329 
assumes N: "space N = UNIV" 

1330 
assumes X: "X \<in> sets M" 

1331 
shows "emeasure M X = emeasure (mapmeasure M (\<lambda>_. N)) (fm ` X)" 

1332 
unfolding mapmeasure_def 

1333 
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

1334 
have "X \<subseteq> space (Pi\<^isub>M J (\<lambda>_. N))" using assms by (simp add: sets.sets_into_space) 
50088  1335 
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" 
1336 
by (auto simp: vimage_image_eq inj_on_def) 

1337 
thus "emeasure M X = emeasure M (fm ` fm ` X \<inter> space M)" using s1 

1338 
by simp 

1339 
show "fm ` X \<in> sets (PiF (Collect finite) (\<lambda>_. N))" 

1340 
by (rule fm_image_measurable_finite[OF N X[simplified s2]]) 

1341 
qed simp 

1342 

1343 
end 

1344 

1345 
end 