src/HOL/Probability/Fin_Map.thy
author haftmann
Sun Nov 18 18:07:51 2018 +0000 (8 months ago)
changeset 69313 b021008c5397
parent 69260 0a9688695a1b
child 69597 ff784d5a5bfb
permissions -rw-r--r--
removed legacy input syntax
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(*  Title:      HOL/Probability/Fin_Map.thy
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    Author:     Fabian Immler, TU M√ľnchen
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*)
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section \<open>Finite Maps\<close>
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theory Fin_Map
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  imports "HOL-Analysis.Finite_Product_Measure" "HOL-Library.Finite_Map"
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begin
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text \<open>The @{type fmap} type can be 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\<^sub>E} and their sigma-algebra
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  @{const Pi\<^sub>M}.\<close>
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type_notation fmap ("(_ \<Rightarrow>\<^sub>F /_)" [22, 21] 21)
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unbundle fmap.lifting
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subsection \<open>Domain and Application\<close>
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lift_definition domain::"('i \<Rightarrow>\<^sub>F 'a) \<Rightarrow> 'i set" is dom .
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lemma finite_domain[simp, intro]: "finite (domain P)"
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  by transfer simp
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lift_definition proj :: "('i \<Rightarrow>\<^sub>F 'a) \<Rightarrow> 'i \<Rightarrow> 'a" ("'((_)')\<^sub>F" [0] 1000) is
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  "\<lambda>f x. if x \<in> dom f then the (f x) else undefined" .
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declare [[coercion proj]]
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lemma extensional_proj[simp, intro]: "(P)\<^sub>F \<in> extensional (domain P)"
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  by transfer (auto simp: extensional_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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  apply transfer
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  apply (safe intro!: ext)
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  subgoal for P Q x
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    by (cases "x \<in> dom P"; cases "P x") (auto dest!: bspec[where x=x])
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  done
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subsection \<open>Constructor of Finite Maps\<close>
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lift_definition finmap_of::"'i set \<Rightarrow> ('i \<Rightarrow> 'a) \<Rightarrow> ('i \<Rightarrow>\<^sub>F 'a)" is
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  "\<lambda>I f x. if x \<in> I \<and> finite I then Some (f x) else None"
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  by (simp add: dom_def)
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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)\<^sub>F = restrict f inds"
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  using assms
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  by transfer force
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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 transfer (auto split: if_splits)
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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)\<^sub>F = (finmap_of K y)\<^sub>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 \<open>Product set of Finite Maps\<close>
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text \<open>This is @{term Pi} for Finite Maps, most of this is copied\<close>
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definition Pi' :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a set) \<Rightarrow> ('i \<Rightarrow>\<^sub>F 'a) set" where
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  "Pi' I A = { P. domain P = I \<and> (\<forall>i. i \<in> I \<longrightarrow> (P)\<^sub>F i \<in> A i) } "
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syntax
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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\<in>A. B" == "CONST Pi' A (\<lambda>x. B)"
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subsubsection\<open>Basic Properties of @{term Pi'}\<close>
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lemma Pi'_I[intro!]: "domain f = A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi' A B"
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  by (simp add: Pi'_def)
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lemma Pi'_I'[simp]: "domain f = A \<Longrightarrow> (\<And>x. x \<in> A \<longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi' A B"
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  by (simp add:Pi'_def)
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lemma Pi'_mem: "f\<in> Pi' A B \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<in> B x"
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  by (simp add: Pi'_def)
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lemma Pi'_iff: "f \<in> Pi' I X \<longleftrightarrow> domain f = I \<and> (\<forall>i\<in>I. f i \<in> X i)"
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  unfolding Pi'_def by auto
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lemma Pi'E [elim]:
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  "f \<in> Pi' A B \<Longrightarrow> (f x \<in> B x \<Longrightarrow> domain f = A \<Longrightarrow> Q) \<Longrightarrow> (x \<notin> A \<Longrightarrow> Q) \<Longrightarrow> Q"
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  by(auto simp: Pi'_def)
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lemma in_Pi'_cong:
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  "domain f = domain g \<Longrightarrow> (\<And> w. w \<in> A \<Longrightarrow> f w = g w) \<Longrightarrow> f \<in> Pi' A B \<longleftrightarrow> g \<in> Pi' A B"
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  by (auto simp: Pi'_def)
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lemma Pi'_eq_empty[simp]:
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  assumes "finite A" shows "(Pi' A B) = {} \<longleftrightarrow> (\<exists>x\<in>A. B x = {})"
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  using assms
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  apply (simp add: Pi'_def, auto)
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  apply (drule_tac x = "finmap_of A (\<lambda>u. SOME y. y \<in> B u)" in spec, auto)
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  apply (cut_tac P= "%y. y \<in> B i" in some_eq_ex, auto)
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  done
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lemma Pi'_mono: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x) \<Longrightarrow> Pi' A B \<subseteq> Pi' A C"
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  by (auto simp: Pi'_def)
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lemma Pi_Pi': "finite A \<Longrightarrow> (Pi\<^sub>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 \<open>Topological Space of Finite Maps\<close>
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instantiation fmap :: (type, topological_space) topological_space
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begin
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definition open_fmap :: "('a \<Rightarrow>\<^sub>F 'b) set \<Rightarrow> bool" where
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   [code del]: "open_fmap = generate_topology {Pi' a b|a 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_fmap_def)
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instance using topological_space_generate_topology
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  by intro_classes (auto simp: open_fmap_def class.topological_space_def)
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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: \<open>finite i\<close>)
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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) \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. (x)\<^sub>F i) \<longlongrightarrow> (a)\<^sub>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)\<^sub>F i \<in> S) F"
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    by eventually_elim (insert \<open>a i \<in> S\<close>, force simp: Pi'_iff split: if_split_asm)
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qed
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lemma continuous_proj:
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  shows "continuous_on s (\<lambda>x. (x)\<^sub>F i)"
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  unfolding continuous_on_def by (safe intro!: tendsto_proj tendsto_ident_at)
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instance fmap :: (type, first_countable_topology) first_countable_topology
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proof
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  fix x::"'a\<Rightarrow>\<^sub>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\<^sub>E (domain x) A)"
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  show "\<exists>A::nat \<Rightarrow> ('a\<Rightarrow>\<^sub>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[of "?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>\<^sub>F 'b) set" assume "open S" "x \<in> S"
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    thus "\<exists>a\<in>?A. a \<subseteq> S" unfolding open_fmap_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\<^sub>E (domain x) A" "Pi' (domain x) f \<subseteq> a" "g \<in> Pi\<^sub>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"])
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    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 \<open>b \<in> B\<close> 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)\<^sub>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)\<^sub>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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subsection \<open>Metric Space of Finite Maps\<close>
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(* TODO: Product of uniform spaces and compatibility with metric_spaces! *)
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instantiation fmap :: (type, metric_space) dist
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begin
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definition dist_fmap where
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  "dist P Q = Max (range (\<lambda>i. dist ((P)\<^sub>F i) ((Q)\<^sub>F i))) + (if domain P = domain Q then 0 else 1)"
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instance ..
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end
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instantiation fmap :: (type, metric_space) uniformity_dist
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begin
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definition [code del]:
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  "(uniformity :: (('a, 'b) fmap \<times> ('a \<Rightarrow>\<^sub>F 'b)) filter) =
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    (INF e\<in>{0 <..}. principal {(x, y). dist x y < e})"
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instance
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  by standard (rule uniformity_fmap_def)
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end
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declare uniformity_Abort[where 'a="('a \<Rightarrow>\<^sub>F 'b::metric_space)", code]
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instantiation fmap :: (type, metric_space) metric_space
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begin
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lemma finite_proj_image': "x \<notin> domain P \<Longrightarrow> finite ((P)\<^sub>F ` S)"
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  by (rule finite_subset[of _ "proj P ` (domain P \<inter> S \<union> {x})"]) auto
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lemma finite_proj_image: "finite ((P)\<^sub>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"])
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lemma finite_proj_diag: "finite ((\<lambda>i. d ((P)\<^sub>F i) ((Q)\<^sub>F i)) ` S)"
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proof -
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   280
  have "(\<lambda>i. d ((P)\<^sub>F i) ((Q)\<^sub>F i)) ` S = (\<lambda>(i, j). d i j) ` ((\<lambda>i. ((P)\<^sub>F i, (Q)\<^sub>F i)) ` S)" by auto
wenzelm@53015
   281
  moreover have "((\<lambda>i. ((P)\<^sub>F i, (Q)\<^sub>F i)) ` S) \<subseteq> (\<lambda>i. (P)\<^sub>F i) ` S \<times> (\<lambda>i. (Q)\<^sub>F i) ` S" by auto
immler@51104
   282
  moreover have "finite \<dots>" using finite_proj_image[of P S] finite_proj_image[of Q S]
immler@51104
   283
    by (intro finite_cartesian_product) simp_all
immler@51104
   284
  ultimately show ?thesis by (simp add: finite_subset)
immler@50088
   285
qed
immler@50088
   286
immler@51104
   287
lemma dist_le_1_imp_domain_eq:
immler@51104
   288
  shows "dist P Q < 1 \<Longrightarrow> domain P = domain Q"
lars@63885
   289
  by (simp add: dist_fmap_def finite_proj_diag split: if_split_asm)
immler@51104
   290
immler@50088
   291
lemma dist_proj:
wenzelm@53015
   292
  shows "dist ((x)\<^sub>F i) ((y)\<^sub>F i) \<le> dist x y"
immler@50088
   293
proof -
immler@51104
   294
  have "dist (x i) (y i) \<le> Max (range (\<lambda>i. dist (x i) (y i)))"
immler@51104
   295
    by (simp add: Max_ge_iff finite_proj_diag)
lars@63885
   296
  also have "\<dots> \<le> dist x y" by (simp add: dist_fmap_def)
immler@51104
   297
  finally show ?thesis .
immler@51104
   298
qed
immler@51104
   299
immler@51104
   300
lemma dist_finmap_lessI:
immler@51105
   301
  assumes "domain P = domain Q"
immler@51105
   302
  assumes "0 < e"
immler@51105
   303
  assumes "\<And>i. i \<in> domain P \<Longrightarrow> dist (P i) (Q i) < e"
immler@51104
   304
  shows "dist P Q < e"
immler@51104
   305
proof -
immler@51104
   306
  have "dist P Q = Max (range (\<lambda>i. dist (P i) (Q i)))"
lars@63885
   307
    using assms by (simp add: dist_fmap_def finite_proj_diag)
immler@51104
   308
  also have "\<dots> < e"
immler@51104
   309
  proof (subst Max_less_iff, safe)
immler@51105
   310
    fix i
wenzelm@53015
   311
    show "dist ((P)\<^sub>F i) ((Q)\<^sub>F i) < e" using assms
immler@51105
   312
      by (cases "i \<in> domain P") simp_all
immler@51104
   313
  qed (simp add: finite_proj_diag)
immler@51104
   314
  finally show ?thesis .
immler@50088
   315
qed
immler@50088
   316
immler@50088
   317
instance
immler@50088
   318
proof
wenzelm@53015
   319
  fix S::"('a \<Rightarrow>\<^sub>F 'b) set"
hoelzl@62101
   320
  have *: "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)" (is "_ = ?od")
immler@51105
   321
  proof
immler@51105
   322
    assume "open S"
immler@51105
   323
    thus ?od
lars@63885
   324
      unfolding open_fmap_def
immler@51105
   325
    proof (induct rule: generate_topology.induct)
immler@51105
   326
      case UNIV thus ?case by (auto intro: zero_less_one)
immler@51105
   327
    next
immler@51105
   328
      case (Int a b)
immler@51105
   329
      show ?case
immler@51105
   330
      proof safe
immler@51105
   331
        fix x assume x: "x \<in> a" "x \<in> b"
immler@51105
   332
        with Int x obtain e1 e2 where
immler@51105
   333
          "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
immler@51105
   334
        thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> a \<inter> b"
immler@51105
   335
          by (auto intro!: exI[where x="min e1 e2"])
immler@51105
   336
      qed
immler@51105
   337
    next
immler@51105
   338
      case (UN K)
immler@51105
   339
      show ?case
immler@51105
   340
      proof safe
wenzelm@53374
   341
        fix x X assume "x \<in> X" and X: "X \<in> K"
wenzelm@53374
   342
        with UN obtain e where "e>0" "\<And>y. dist y x < e \<longrightarrow> y \<in> X" by force
wenzelm@53374
   343
        with X show "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> \<Union>K" by auto
immler@51105
   344
      qed
immler@51105
   345
    next
immler@51105
   346
      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
immler@51105
   347
      show ?case
immler@51105
   348
      proof safe
immler@51105
   349
        fix x assume "x \<in> s"
immler@51105
   350
        hence [simp]: "finite a" and a_dom: "a = domain x" using s by (auto simp: Pi'_iff)
immler@51105
   351
        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)"
wenzelm@61808
   352
          using b \<open>x \<in> s\<close> by atomize_elim (intro bchoice, auto simp: open_dist s)
immler@51105
   353
        hence in_b: "\<And>i y. i \<in> a \<Longrightarrow> dist y (proj x i) < es i \<Longrightarrow> y \<in> b i" by auto
immler@51105
   354
        show "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> s"
immler@51105
   355
        proof (cases, rule, safe)
immler@51105
   356
          assume "a \<noteq> {}"
wenzelm@61808
   357
          show "0 < min 1 (Min (es ` a))" using es by (auto simp: \<open>a \<noteq> {}\<close>)
immler@51105
   358
          fix y assume d: "dist y x < min 1 (Min (es ` a))"
immler@51105
   359
          show "y \<in> s" unfolding s
immler@51105
   360
          proof
wenzelm@61808
   361
            show "domain y = a" using d s \<open>a \<noteq> {}\<close> by (auto simp: dist_le_1_imp_domain_eq a_dom)
wenzelm@53374
   362
            fix i assume i: "i \<in> a"
wenzelm@53015
   363
            hence "dist ((y)\<^sub>F i) ((x)\<^sub>F i) < es i" using d
lars@63885
   364
              by (auto simp: dist_fmap_def \<open>a \<noteq> {}\<close> intro!: le_less_trans[OF dist_proj])
wenzelm@53374
   365
            with i show "y i \<in> b i" by (rule in_b)
immler@51105
   366
          qed
immler@51105
   367
        next
immler@51105
   368
          assume "\<not>a \<noteq> {}"
immler@51105
   369
          thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> s"
wenzelm@61808
   370
            using s \<open>x \<in> s\<close> by (auto simp: Pi'_def dist_le_1_imp_domain_eq intro!: exI[where x=1])
immler@51105
   371
        qed
immler@51105
   372
      qed
immler@51105
   373
    qed
immler@51105
   374
  next
immler@51105
   375
    assume "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S"
immler@51105
   376
    then obtain e where e_pos: "\<And>x. x \<in> S \<Longrightarrow> e x > 0" and
immler@51105
   377
      e_in:  "\<And>x y . x \<in> S \<Longrightarrow> dist y x < e x \<Longrightarrow> y \<in> S"
immler@51105
   378
      unfolding bchoice_iff
immler@51105
   379
      by auto
immler@51105
   380
    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))}"
immler@51105
   381
    proof safe
immler@51105
   382
      fix x assume "x \<in> S"
immler@51105
   383
      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))}"
immler@51105
   384
        using e_pos by (auto intro!: exI[where x="Pi' (domain x) (\<lambda>i. ball (x i) (e x))"])
immler@51105
   385
    next
immler@51105
   386
      fix x y
immler@51105
   387
      assume "y \<in> S"
immler@51105
   388
      moreover
immler@51105
   389
      assume "x \<in> (\<Pi>' i\<in>domain y. ball (y i) (e y))"
wenzelm@61808
   390
      hence "dist x y < e y" using e_pos \<open>y \<in> S\<close>
lars@63885
   391
        by (auto simp: dist_fmap_def Pi'_iff finite_proj_diag dist_commute)
immler@51105
   392
      ultimately show "x \<in> S" by (rule e_in)
immler@51105
   393
    qed
immler@51105
   394
    also have "open \<dots>"
lars@63885
   395
      unfolding open_fmap_def
immler@51105
   396
      by (intro generate_topology.UN) (auto intro: generate_topology.Basis)
immler@51105
   397
    finally show "open S" .
immler@51105
   398
  qed
hoelzl@62101
   399
  show "open S = (\<forall>x\<in>S. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> S)"
hoelzl@62101
   400
    unfolding * eventually_uniformity_metric
lars@63885
   401
    by (simp del: split_paired_All add: dist_fmap_def dist_commute eq_commute)
immler@50088
   402
next
wenzelm@53015
   403
  fix P Q::"'a \<Rightarrow>\<^sub>F 'b"
immler@51104
   404
  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))"
haftmann@51489
   405
    by (auto intro: Max_in Max_eqI)
immler@50088
   406
  show "dist P Q = 0 \<longleftrightarrow> P = Q"
lars@63885
   407
    by (auto simp: finmap_eq_iff dist_fmap_def Max_ge_iff finite_proj_diag Max_eq_iff
hoelzl@56633
   408
        add_nonneg_eq_0_iff
immler@51104
   409
      intro!: Max_eqI image_eqI[where x=undefined])
immler@50088
   410
next
wenzelm@53015
   411
  fix P Q R::"'a \<Rightarrow>\<^sub>F 'b"
wenzelm@53015
   412
  let ?dists = "\<lambda>P Q i. dist ((P)\<^sub>F i) ((Q)\<^sub>F i)"
immler@51104
   413
  let ?dpq = "?dists P Q" and ?dpr = "?dists P R" and ?dqr = "?dists Q R"
immler@51104
   414
  let ?dom = "\<lambda>P Q. (if domain P = domain Q then 0 else 1::real)"
immler@51104
   415
  have "dist P Q = Max (range ?dpq) + ?dom P Q"
lars@63885
   416
    by (simp add: dist_fmap_def)
immler@51104
   417
  also obtain t where "t \<in> range ?dpq" "t = Max (range ?dpq)" by (simp add: finite_proj_diag)
immler@51104
   418
  then obtain i where "Max (range ?dpq) = ?dpq i" by auto
immler@51104
   419
  also have "?dpq i \<le> ?dpr i + ?dqr i" by (rule dist_triangle2)
immler@51104
   420
  also have "?dpr i \<le> Max (range ?dpr)" by (simp add: finite_proj_diag)
immler@51104
   421
  also have "?dqr i \<le> Max (range ?dqr)" by (simp add: finite_proj_diag)
immler@51104
   422
  also have "?dom P Q \<le> ?dom P R + ?dom Q R" by simp
lars@63885
   423
  finally show "dist P Q \<le> dist P R + dist Q R" by (simp add: dist_fmap_def ac_simps)
immler@50088
   424
qed
immler@50088
   425
immler@50088
   426
end
immler@50088
   427
wenzelm@61808
   428
subsection \<open>Complete Space of Finite Maps\<close>
immler@50088
   429
immler@50088
   430
lemma tendsto_finmap:
wenzelm@53015
   431
  fixes f::"nat \<Rightarrow> ('i \<Rightarrow>\<^sub>F ('a::metric_space))"
immler@50088
   432
  assumes ind_f:  "\<And>n. domain (f n) = domain g"
wenzelm@61969
   433
  assumes proj_g:  "\<And>i. i \<in> domain g \<Longrightarrow> (\<lambda>n. (f n) i) \<longlonglongrightarrow> g i"
wenzelm@61969
   434
  shows "f \<longlonglongrightarrow> g"
immler@51104
   435
  unfolding tendsto_iff
immler@51104
   436
proof safe
immler@51104
   437
  fix e::real assume "0 < e"
wenzelm@53015
   438
  let ?dists = "\<lambda>x i. dist ((f x)\<^sub>F i) ((g)\<^sub>F i)"
immler@51104
   439
  have "eventually (\<lambda>x. \<forall>i\<in>domain g. ?dists x i < e) sequentially"
immler@51104
   440
    using finite_domain[of g] proj_g
immler@51104
   441
  proof induct
immler@51104
   442
    case (insert i G)
wenzelm@61808
   443
    with \<open>0 < e\<close> have "eventually (\<lambda>x. ?dists x i < e) sequentially" by (auto simp add: tendsto_iff)
immler@51104
   444
    moreover
wenzelm@53015
   445
    from insert have "eventually (\<lambda>x. \<forall>i\<in>G. dist ((f x)\<^sub>F i) ((g)\<^sub>F i) < e) sequentially" by simp
immler@51104
   446
    ultimately show ?case by eventually_elim auto
immler@51104
   447
  qed simp
immler@51104
   448
  thus "eventually (\<lambda>x. dist (f x) g < e) sequentially"
lars@63885
   449
    by eventually_elim (auto simp add: dist_fmap_def finite_proj_diag ind_f \<open>0 < e\<close>)
immler@51104
   450
qed
immler@50088
   451
lars@63885
   452
instance fmap :: (type, complete_space) complete_space
immler@50088
   453
proof
wenzelm@53015
   454
  fix P::"nat \<Rightarrow> 'a \<Rightarrow>\<^sub>F 'b"
immler@50088
   455
  assume "Cauchy P"
immler@50088
   456
  then obtain Nd where Nd: "\<And>n. n \<ge> Nd \<Longrightarrow> dist (P n) (P Nd) < 1"
lp15@66089
   457
    by (force simp: Cauchy_altdef2)
wenzelm@63040
   458
  define d where "d = domain (P Nd)"
immler@50088
   459
  with Nd have dim: "\<And>n. n \<ge> Nd \<Longrightarrow> domain (P n) = d" using dist_le_1_imp_domain_eq by auto
immler@50088
   460
  have [simp]: "finite d" unfolding d_def by simp
wenzelm@63040
   461
  define p where "p i n = P n i" for i n
wenzelm@63040
   462
  define q where "q i = lim (p i)" for i
wenzelm@63040
   463
  define Q where "Q = finmap_of d q"
lars@63885
   464
  have q: "\<And>i. i \<in> d \<Longrightarrow> q i = Q i" by (auto simp add: Q_def Abs_fmap_inverse)
immler@50088
   465
  {
immler@50088
   466
    fix i assume "i \<in> d"
lp15@66089
   467
    have "Cauchy (p i)" unfolding Cauchy_altdef2 p_def
immler@50088
   468
    proof safe
immler@50088
   469
      fix e::real assume "0 < e"
wenzelm@61808
   470
      with \<open>Cauchy P\<close> obtain N where N: "\<And>n. n\<ge>N \<Longrightarrow> dist (P n) (P N) < min e 1"
lp15@66089
   471
        by (force simp: Cauchy_altdef2 min_def)
immler@50088
   472
      hence "\<And>n. n \<ge> N \<Longrightarrow> domain (P n) = domain (P N)" using dist_le_1_imp_domain_eq by auto
immler@50088
   473
      with dim have dim: "\<And>n. n \<ge> N \<Longrightarrow> domain (P n) = d" by (metis nat_le_linear)
immler@50088
   474
      show "\<exists>N. \<forall>n\<ge>N. dist ((P n) i) ((P N) i) < e"
immler@50088
   475
      proof (safe intro!: exI[where x="N"])
immler@50088
   476
        fix n assume "N \<le> n" have "N \<le> N" by simp
immler@50088
   477
        have "dist ((P n) i) ((P N) i) \<le> dist (P n) (P N)"
wenzelm@61808
   478
          using dim[OF \<open>N \<le> n\<close>]  dim[OF \<open>N \<le> N\<close>] \<open>i \<in> d\<close>
immler@50088
   479
          by (auto intro!: dist_proj)
wenzelm@61808
   480
        also have "\<dots> < e" using N[OF \<open>N \<le> n\<close>] by simp
immler@50088
   481
        finally show "dist ((P n) i) ((P N) i) < e" .
immler@50088
   482
      qed
immler@50088
   483
    qed
immler@50088
   484
    hence "convergent (p i)" by (metis Cauchy_convergent_iff)
wenzelm@61969
   485
    hence "p i \<longlonglongrightarrow> q i" unfolding q_def convergent_def by (metis limI)
immler@50088
   486
  } note p = this
wenzelm@61969
   487
  have "P \<longlonglongrightarrow> Q"
immler@50088
   488
  proof (rule metric_LIMSEQ_I)
immler@50088
   489
    fix e::real assume "0 < e"
immler@51104
   490
    have "\<exists>ni. \<forall>i\<in>d. \<forall>n\<ge>ni i. dist (p i n) (q i) < e"
immler@50088
   491
    proof (safe intro!: bchoice)
immler@50088
   492
      fix i assume "i \<in> d"
wenzelm@61808
   493
      from p[OF \<open>i \<in> d\<close>, THEN metric_LIMSEQ_D, OF \<open>0 < e\<close>]
immler@51104
   494
      show "\<exists>no. \<forall>n\<ge>no. dist (p i n) (q i) < e" .
immler@50088
   495
    qed then guess ni .. note ni = this
wenzelm@63040
   496
    define N where "N = max Nd (Max (ni ` d))"
immler@50088
   497
    show "\<exists>N. \<forall>n\<ge>N. dist (P n) Q < e"
immler@50088
   498
    proof (safe intro!: exI[where x="N"])
immler@50088
   499
      fix n assume "N \<le> n"
immler@51104
   500
      hence dom: "domain (P n) = d" "domain Q = d" "domain (P n) = domain Q"
lars@63885
   501
        using dim by (simp_all add: N_def Q_def dim_def Abs_fmap_inverse)
immler@51104
   502
      show "dist (P n) Q < e"
wenzelm@61808
   503
      proof (rule dist_finmap_lessI[OF dom(3) \<open>0 < e\<close>])
immler@51104
   504
        fix i
immler@51104
   505
        assume "i \<in> domain (P n)"
immler@51104
   506
        hence "ni i \<le> Max (ni ` d)" using dom by simp
immler@50088
   507
        also have "\<dots> \<le> N" by (simp add: N_def)
wenzelm@61808
   508
        finally show "dist ((P n)\<^sub>F i) ((Q)\<^sub>F i) < e" using ni \<open>i \<in> domain (P n)\<close> \<open>N \<le> n\<close> dom
immler@51104
   509
          by (auto simp: p_def q N_def less_imp_le)
immler@50088
   510
      qed
immler@50088
   511
    qed
immler@50088
   512
  qed
immler@50088
   513
  thus "convergent P" by (auto simp: convergent_def)
immler@50088
   514
qed
immler@50088
   515
wenzelm@61808
   516
subsection \<open>Second Countable Space of Finite Maps\<close>
immler@50088
   517
lars@63885
   518
instantiation fmap :: (countable, second_countable_topology) second_countable_topology
immler@50088
   519
begin
immler@50088
   520
immler@51106
   521
definition basis_proj::"'b set set"
immler@51106
   522
  where "basis_proj = (SOME B. countable B \<and> topological_basis B)"
immler@51106
   523
immler@51106
   524
lemma countable_basis_proj: "countable basis_proj" and basis_proj: "topological_basis basis_proj"
immler@51106
   525
  unfolding basis_proj_def by (intro is_basis countable_basis)+
immler@51106
   526
wenzelm@53015
   527
definition basis_finmap::"('a \<Rightarrow>\<^sub>F 'b) set set"
immler@51106
   528
  where "basis_finmap = {Pi' I S|I S. finite I \<and> (\<forall>i \<in> I. S i \<in> basis_proj)}"
immler@50245
   529
immler@50245
   530
lemma in_basis_finmapI:
immler@51106
   531
  assumes "finite I" assumes "\<And>i. i \<in> I \<Longrightarrow> S i \<in> basis_proj"
immler@50245
   532
  shows "Pi' I S \<in> basis_finmap"
immler@50245
   533
  using assms unfolding basis_finmap_def by auto
immler@50245
   534
immler@50245
   535
lemma basis_finmap_eq:
immler@51106
   536
  assumes "basis_proj \<noteq> {}"
wenzelm@53015
   537
  shows "basis_finmap = (\<lambda>f. Pi' (domain f) (\<lambda>i. from_nat_into basis_proj ((f)\<^sub>F i))) `
wenzelm@53015
   538
    (UNIV::('a \<Rightarrow>\<^sub>F nat) set)" (is "_ = ?f ` _")
immler@50245
   539
  unfolding basis_finmap_def
immler@50245
   540
proof safe
immler@50245
   541
  fix I::"'a set" and S::"'a \<Rightarrow> 'b set"
immler@51106
   542
  assume "finite I" "\<forall>i\<in>I. S i \<in> basis_proj"
immler@51106
   543
  hence "Pi' I S = ?f (finmap_of I (\<lambda>x. to_nat_on basis_proj (S x)))"
immler@51106
   544
    by (force simp: Pi'_def countable_basis_proj)
immler@50245
   545
  thus "Pi' I S \<in> range ?f" by simp
immler@51106
   546
next
wenzelm@53015
   547
  fix x and f::"'a \<Rightarrow>\<^sub>F nat"
wenzelm@56222
   548
  show "\<exists>I S. (\<Pi>' i\<in>domain f. from_nat_into basis_proj ((f)\<^sub>F i)) = Pi' I S \<and>
wenzelm@56222
   549
    finite I \<and> (\<forall>i\<in>I. S i \<in> basis_proj)"
immler@51106
   550
    using assms by (auto intro: from_nat_into)
immler@51106
   551
qed
immler@51106
   552
immler@51106
   553
lemma basis_finmap_eq_empty: "basis_proj = {} \<Longrightarrow> basis_finmap = {Pi' {} undefined}"
immler@51106
   554
  by (auto simp: Pi'_iff basis_finmap_def)
immler@50088
   555
immler@50245
   556
lemma countable_basis_finmap: "countable basis_finmap"
immler@51106
   557
  by (cases "basis_proj = {}") (auto simp: basis_finmap_eq basis_finmap_eq_empty)
immler@50088
   558
immler@50088
   559
lemma finmap_topological_basis:
immler@50245
   560
  "topological_basis basis_finmap"
immler@50088
   561
proof (subst topological_basis_iff, safe)
immler@50245
   562
  fix B' assume "B' \<in> basis_finmap"
immler@50245
   563
  thus "open B'"
immler@51106
   564
    by (auto intro!: open_Pi'I topological_basis_open[OF basis_proj]
immler@50245
   565
      simp: topological_basis_def basis_finmap_def Let_def)
immler@50088
   566
next
wenzelm@53015
   567
  fix O'::"('a \<Rightarrow>\<^sub>F 'b) set" and x
immler@51105
   568
  assume O': "open O'" "x \<in> O'"
immler@51105
   569
  then obtain a where a:
immler@51105
   570
    "x \<in> Pi' (domain x) a" "Pi' (domain x) a \<subseteq> O'" "\<And>i. i\<in>domain x \<Longrightarrow> open (a i)"
lars@63885
   571
    unfolding open_fmap_def
immler@51105
   572
  proof (atomize_elim, induct rule: generate_topology.induct)
immler@51105
   573
    case (Int a b)
immler@51105
   574
    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))"
immler@51105
   575
    from Int obtain f g where "?p a f" "?p b g" by auto
immler@51105
   576
    thus ?case by (force intro!: exI[where x="\<lambda>i. f i \<inter> g i"] simp: Pi'_def)
immler@51105
   577
  next
immler@51105
   578
    case (UN k)
immler@51105
   579
    then obtain kk a where "x \<in> kk" "kk \<in> k" "x \<in> Pi' (domain x) a" "Pi' (domain x) a \<subseteq> kk"
immler@51105
   580
      "\<And>i. i\<in>domain x \<Longrightarrow> open (a i)"
immler@51105
   581
      by force
immler@51105
   582
    thus ?case by blast
immler@51105
   583
  qed (auto simp: Pi'_def)
immler@50088
   584
  have "\<exists>B.
immler@51106
   585
    (\<forall>i\<in>domain x. x i \<in> B i \<and> B i \<subseteq> a i \<and> B i \<in> basis_proj)"
immler@50088
   586
  proof (rule bchoice, safe)
immler@50088
   587
    fix i assume "i \<in> domain x"
immler@51105
   588
    hence "open (a i)" "x i \<in> a i" using a by auto
immler@51106
   589
    from topological_basisE[OF basis_proj this] guess b' .
immler@51106
   590
    thus "\<exists>y. x i \<in> y \<and> y \<subseteq> a i \<and> y \<in> basis_proj" by auto
immler@50088
   591
  qed
immler@50088
   592
  then guess B .. note B = this
wenzelm@63040
   593
  define B' where "B' = Pi' (domain x) (\<lambda>i. (B i)::'b set)"
immler@51105
   594
  have "B' \<subseteq> Pi' (domain x) a" using B by (auto intro!: Pi'_mono simp: B'_def)
wenzelm@61808
   595
  also note \<open>\<dots> \<subseteq> O'\<close>
immler@51105
   596
  finally show "\<exists>B'\<in>basis_finmap. x \<in> B' \<and> B' \<subseteq> O'" using B
immler@51105
   597
    by (auto intro!: bexI[where x=B'] Pi'_mono in_basis_finmapI simp: B'_def)
immler@50088
   598
qed
immler@50088
   599
immler@50088
   600
lemma range_enum_basis_finmap_imp_open:
immler@50245
   601
  assumes "x \<in> basis_finmap"
immler@50088
   602
  shows "open x"
immler@50088
   603
  using finmap_topological_basis assms by (auto simp: topological_basis_def)
immler@50088
   604
hoelzl@51343
   605
instance proof qed (blast intro: finmap_topological_basis countable_basis_finmap topological_basis_imp_subbasis)
immler@50088
   606
immler@50088
   607
end
immler@50088
   608
wenzelm@61808
   609
subsection \<open>Polish Space of Finite Maps\<close>
immler@51105
   610
lars@63885
   611
instance fmap :: (countable, polish_space) polish_space proof qed
immler@51105
   612
immler@51105
   613
wenzelm@61808
   614
subsection \<open>Product Measurable Space of Finite Maps\<close>
immler@50088
   615
immler@50088
   616
definition "PiF I M \<equiv>
hoelzl@50124
   617
  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))}"
immler@50088
   618
immler@50088
   619
abbreviation
wenzelm@53015
   620
  "Pi\<^sub>F I M \<equiv> PiF I M"
immler@50088
   621
immler@50088
   622
syntax
wenzelm@53015
   623
  "_PiF" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3\<Pi>\<^sub>F _\<in>_./ _)"  10)
immler@50088
   624
translations
wenzelm@61988
   625
  "\<Pi>\<^sub>F x\<in>I. M" == "CONST PiF I (%x. M)"
immler@50088
   626
immler@50088
   627
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>
immler@50088
   628
    Pow (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j)))"
immler@50244
   629
  by (auto simp: Pi'_def) (blast dest: sets.sets_into_space)
immler@50088
   630
immler@50088
   631
lemma space_PiF: "space (PiF I M) = (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j)))"
immler@50088
   632
  unfolding PiF_def using PiF_gen_subset by (rule space_measure_of)
immler@50088
   633
immler@50088
   634
lemma sets_PiF:
immler@50088
   635
  "sets (PiF I M) = sigma_sets (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j)))
immler@50088
   636
    {(\<Pi>' j\<in>J. X j) |X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"
immler@50088
   637
  unfolding PiF_def using PiF_gen_subset by (rule sets_measure_of)
immler@50088
   638
immler@50088
   639
lemma sets_PiF_singleton:
immler@50088
   640
  "sets (PiF {I} M) = sigma_sets (\<Pi>' j\<in>I. space (M j))
immler@50088
   641
    {(\<Pi>' j\<in>I. X j) |X. X \<in> (\<Pi> j\<in>I. sets (M j))}"
immler@50088
   642
  unfolding sets_PiF by simp
immler@50088
   643
immler@50088
   644
lemma in_sets_PiFI:
immler@50088
   645
  assumes "X = (Pi' J S)" "J \<in> I" "\<And>i. i\<in>J \<Longrightarrow> S i \<in> sets (M i)"
immler@50088
   646
  shows "X \<in> sets (PiF I M)"
immler@50088
   647
  unfolding sets_PiF
immler@50088
   648
  using assms by blast
immler@50088
   649
immler@50088
   650
lemma product_in_sets_PiFI:
immler@50088
   651
  assumes "J \<in> I" "\<And>i. i\<in>J \<Longrightarrow> S i \<in> sets (M i)"
immler@50088
   652
  shows "(Pi' J S) \<in> sets (PiF I M)"
immler@50088
   653
  unfolding sets_PiF
immler@50088
   654
  using assms by blast
immler@50088
   655
immler@50088
   656
lemma singleton_space_subset_in_sets:
immler@50088
   657
  fixes J
immler@50088
   658
  assumes "J \<in> I"
immler@50088
   659
  assumes "finite J"
immler@50088
   660
  shows "space (PiF {J} M) \<in> sets (PiF I M)"
immler@50088
   661
  using assms
immler@50088
   662
  by (intro in_sets_PiFI[where J=J and S="\<lambda>i. space (M i)"])
immler@50088
   663
      (auto simp: product_def space_PiF)
immler@50088
   664
immler@50088
   665
lemma singleton_subspace_set_in_sets:
immler@50088
   666
  assumes A: "A \<in> sets (PiF {J} M)"
immler@50088
   667
  assumes "finite J"
immler@50088
   668
  assumes "J \<in> I"
immler@50088
   669
  shows "A \<in> sets (PiF I M)"
immler@50088
   670
  using A[unfolded sets_PiF]
immler@50088
   671
  apply (induct A)
immler@50088
   672
  unfolding sets_PiF[symmetric] unfolding space_PiF[symmetric]
immler@50088
   673
  using assms
immler@50088
   674
  by (auto intro: in_sets_PiFI intro!: singleton_space_subset_in_sets)
immler@50088
   675
hoelzl@50124
   676
lemma finite_measurable_singletonI:
immler@50088
   677
  assumes "finite I"
immler@50088
   678
  assumes "\<And>J. J \<in> I \<Longrightarrow> finite J"
immler@50088
   679
  assumes MN: "\<And>J. J \<in> I \<Longrightarrow> A \<in> measurable (PiF {J} M) N"
immler@50088
   680
  shows "A \<in> measurable (PiF I M) N"
immler@50088
   681
  unfolding measurable_def
immler@50088
   682
proof safe
immler@50088
   683
  fix y assume "y \<in> sets N"
immler@50088
   684
  have "A -` y \<inter> space (PiF I M) = (\<Union>J\<in>I. A -` y \<inter> space (PiF {J} M))"
immler@50088
   685
    by (auto simp: space_PiF)
immler@50088
   686
  also have "\<dots> \<in> sets (PiF I M)"
haftmann@62343
   687
  proof (rule sets.finite_UN)
immler@50088
   688
    show "finite I" by fact
immler@50088
   689
    fix J assume "J \<in> I"
immler@50088
   690
    with assms have "finite J" by simp
immler@50088
   691
    show "A -` y \<inter> space (PiF {J} M) \<in> sets (PiF I M)"
immler@50088
   692
      by (rule singleton_subspace_set_in_sets[OF measurable_sets[OF assms(3)]]) fact+
immler@50088
   693
  qed
immler@50088
   694
  finally show "A -` y \<inter> space (PiF I M) \<in> sets (PiF I M)" .
immler@50088
   695
next
immler@50088
   696
  fix x assume "x \<in> space (PiF I M)" thus "A x \<in> space N"
immler@50088
   697
    using MN[of "domain x"]
immler@50088
   698
    by (auto simp: space_PiF measurable_space Pi'_def)
immler@50088
   699
qed
immler@50088
   700
hoelzl@50124
   701
lemma countable_finite_comprehension:
immler@50088
   702
  fixes f :: "'a::countable set \<Rightarrow> _"
immler@50088
   703
  assumes "\<And>s. P s \<Longrightarrow> finite s"
immler@50088
   704
  assumes "\<And>s. P s \<Longrightarrow> f s \<in> sets M"
immler@50088
   705
  shows "\<Union>{f s|s. P s} \<in> sets M"
immler@50088
   706
proof -
immler@50088
   707
  have "\<Union>{f s|s. P s} = (\<Union>n::nat. let s = set (from_nat n) in if P s then f s else {})"
immler@50088
   708
  proof safe
wenzelm@53374
   709
    fix x X s assume *: "x \<in> f s" "P s"
wenzelm@53374
   710
    with assms obtain l where "s = set l" using finite_list by blast
wenzelm@61808
   711
    with * show "x \<in> (\<Union>n. let s = set (from_nat n) in if P s then f s else {})" using \<open>P s\<close>
immler@50088
   712
      by (auto intro!: exI[where x="to_nat l"])
immler@50088
   713
  next
immler@50088
   714
    fix x n assume "x \<in> (let s = set (from_nat n) in if P s then f s else {})"
nipkow@62390
   715
    thus "x \<in> \<Union>{f s|s. P s}" using assms by (auto simp: Let_def split: if_split_asm)
immler@50088
   716
  qed
immler@50088
   717
  hence "\<Union>{f s|s. P s} = (\<Union>n. let s = set (from_nat n) in if P s then f s else {})" by simp
immler@50088
   718
  also have "\<dots> \<in> sets M" using assms by (auto simp: Let_def)
immler@50088
   719
  finally show ?thesis .
immler@50088
   720
qed
immler@50088
   721
immler@50088
   722
lemma space_subset_in_sets:
immler@50088
   723
  fixes J::"'a::countable set set"
immler@50088
   724
  assumes "J \<subseteq> I"
immler@50088
   725
  assumes "\<And>j. j \<in> J \<Longrightarrow> finite j"
immler@50088
   726
  shows "space (PiF J M) \<in> sets (PiF I M)"
immler@50088
   727
proof -
immler@50088
   728
  have "space (PiF J M) = \<Union>{space (PiF {j} M)|j. j \<in> J}"
immler@50088
   729
    unfolding space_PiF by blast
immler@50088
   730
  also have "\<dots> \<in> sets (PiF I M)" using assms
immler@50088
   731
    by (intro countable_finite_comprehension) (auto simp: singleton_space_subset_in_sets)
immler@50088
   732
  finally show ?thesis .
immler@50088
   733
qed
immler@50088
   734
immler@50088
   735
lemma subspace_set_in_sets:
immler@50088
   736
  fixes J::"'a::countable set set"
immler@50088
   737
  assumes A: "A \<in> sets (PiF J M)"
immler@50088
   738
  assumes "J \<subseteq> I"
immler@50088
   739
  assumes "\<And>j. j \<in> J \<Longrightarrow> finite j"
immler@50088
   740
  shows "A \<in> sets (PiF I M)"
immler@50088
   741
  using A[unfolded sets_PiF]
immler@50088
   742
  apply (induct A)
immler@50088
   743
  unfolding sets_PiF[symmetric] unfolding space_PiF[symmetric]
immler@50088
   744
  using assms
immler@50088
   745
  by (auto intro: in_sets_PiFI intro!: space_subset_in_sets)
immler@50088
   746
hoelzl@50124
   747
lemma countable_measurable_PiFI:
immler@50088
   748
  fixes I::"'a::countable set set"
immler@50088
   749
  assumes MN: "\<And>J. J \<in> I \<Longrightarrow> finite J \<Longrightarrow> A \<in> measurable (PiF {J} M) N"
immler@50088
   750
  shows "A \<in> measurable (PiF I M) N"
immler@50088
   751
  unfolding measurable_def
immler@50088
   752
proof safe
immler@50088
   753
  fix y assume "y \<in> sets N"
immler@50088
   754
  have "A -` y = (\<Union>{A -` y \<inter> {x. domain x = J}|J. finite J})" by auto
wenzelm@53015
   755
  { fix x::"'a \<Rightarrow>\<^sub>F 'b"
immler@50088
   756
    from finite_list[of "domain x"] obtain xs where "set xs = domain x" by auto
immler@50245
   757
    hence "\<exists>n. domain x = set (from_nat n)"
immler@50245
   758
      by (intro exI[where x="to_nat xs"]) auto }
immler@50245
   759
  hence "A -` y \<inter> space (PiF I M) = (\<Union>n. A -` y \<inter> space (PiF ({set (from_nat n)}\<inter>I) M))"
immler@50245
   760
    by (auto simp: space_PiF Pi'_def)
immler@50088
   761
  also have "\<dots> \<in> sets (PiF I M)"
immler@50244
   762
    apply (intro sets.Int sets.countable_nat_UN subsetI, safe)
immler@50088
   763
    apply (case_tac "set (from_nat i) \<in> I")
immler@50088
   764
    apply simp_all
immler@50088
   765
    apply (rule singleton_subspace_set_in_sets[OF measurable_sets[OF MN]])
wenzelm@61808
   766
    using assms \<open>y \<in> sets N\<close>
immler@50088
   767
    apply (auto simp: space_PiF)
immler@50088
   768
    done
immler@50088
   769
  finally show "A -` y \<inter> space (PiF I M) \<in> sets (PiF I M)" .
immler@50088
   770
next
immler@50088
   771
  fix x assume "x \<in> space (PiF I M)" thus "A x \<in> space N"
immler@50088
   772
    using MN[of "domain x"] by (auto simp: space_PiF measurable_space Pi'_def)
immler@50088
   773
qed
immler@50088
   774
immler@50088
   775
lemma measurable_PiF:
immler@50088
   776
  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))"
immler@50088
   777
  assumes S: "\<And>J S. J \<in> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> S i \<in> sets (M i)) \<Longrightarrow>
immler@50088
   778
    f -` (Pi' J S) \<inter> space N \<in> sets N"
immler@50088
   779
  shows "f \<in> measurable N (PiF I M)"
immler@50088
   780
  unfolding PiF_def
immler@50088
   781
  using PiF_gen_subset
immler@50088
   782
  apply (rule measurable_measure_of)
immler@50088
   783
  using f apply force
immler@50088
   784
  apply (insert S, auto)
immler@50088
   785
  done
immler@50088
   786
hoelzl@50124
   787
lemma restrict_sets_measurable:
immler@50088
   788
  assumes A: "A \<in> sets (PiF I M)" and "J \<subseteq> I"
immler@50088
   789
  shows "A \<inter> {m. domain m \<in> J} \<in> sets (PiF J M)"
immler@50088
   790
  using A[unfolded sets_PiF]
hoelzl@50124
   791
proof (induct A)
hoelzl@50124
   792
  case (Basic a)
immler@50088
   793
  then obtain K S where S: "a = Pi' K S" "K \<in> I" "(\<forall>i\<in>K. S i \<in> sets (M i))"
immler@50088
   794
    by auto
hoelzl@50124
   795
  show ?case
immler@50088
   796
  proof cases
immler@50088
   797
    assume "K \<in> J"
immler@50088
   798
    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
immler@50088
   799
      by (auto intro!: exI[where x=K] exI[where x=S] simp: Pi'_def)
immler@50088
   800
    also have "\<dots> \<subseteq> sets (PiF J M)" unfolding sets_PiF by auto
immler@50088
   801
    finally show ?thesis .
immler@50088
   802
  next
immler@50088
   803
    assume "K \<notin> J"
immler@50088
   804
    hence "a \<inter> {m. domain m \<in> J} = {}" using S by (auto simp: Pi'_def)
immler@50088
   805
    also have "\<dots> \<in> sets (PiF J M)" by simp
immler@50088
   806
    finally show ?thesis .
immler@50088
   807
  qed
immler@50088
   808
next
hoelzl@50124
   809
  case (Union a)
haftmann@69313
   810
  have "\<Union>(a ` UNIV) \<inter> {m. domain m \<in> J} = (\<Union>i. (a i \<inter> {m. domain m \<in> J}))"
immler@50088
   811
    by simp
immler@50244
   812
  also have "\<dots> \<in> sets (PiF J M)" using Union by (intro sets.countable_nat_UN) auto
hoelzl@50124
   813
  finally show ?case .
immler@50088
   814
next
hoelzl@50124
   815
  case (Compl a)
immler@50088
   816
  have "(space (PiF I M) - a) \<inter> {m. domain m \<in> J} = (space (PiF J M) - (a \<inter> {m. domain m \<in> J}))"
wenzelm@61808
   817
    using \<open>J \<subseteq> I\<close> by (auto simp: space_PiF Pi'_def)
hoelzl@50124
   818
  also have "\<dots> \<in> sets (PiF J M)" using Compl by auto
hoelzl@50124
   819
  finally show ?case by (simp add: space_PiF)
hoelzl@50124
   820
qed simp
immler@50088
   821
immler@50088
   822
lemma measurable_finmap_of:
immler@50088
   823
  assumes f: "\<And>i. (\<exists>x \<in> space N. i \<in> J x) \<Longrightarrow> (\<lambda>x. f x i) \<in> measurable N (M i)"
immler@50088
   824
  assumes J: "\<And>x. x \<in> space N \<Longrightarrow> J x \<in> I" "\<And>x. x \<in> space N \<Longrightarrow> finite (J x)"
immler@50088
   825
  assumes JN: "\<And>S. {x. J x = S} \<inter> space N \<in> sets N"
immler@50088
   826
  shows "(\<lambda>x. finmap_of (J x) (f x)) \<in> measurable N (PiF I M)"
immler@50088
   827
proof (rule measurable_PiF)
immler@50088
   828
  fix x assume "x \<in> space N"
immler@50088
   829
  with J[of x] measurable_space[OF f]
immler@50088
   830
  show "domain (finmap_of (J x) (f x)) \<in> I \<and>
immler@50088
   831
        (\<forall>i\<in>domain (finmap_of (J x) (f x)). (finmap_of (J x) (f x)) i \<in> space (M i))"
immler@50088
   832
    by auto
immler@50088
   833
next
immler@50088
   834
  fix K S assume "K \<in> I" and *: "\<And>i. i \<in> K \<Longrightarrow> S i \<in> sets (M i)"
immler@50088
   835
  with J have eq: "(\<lambda>x. finmap_of (J x) (f x)) -` Pi' K S \<inter> space N =
immler@50088
   836
    (if \<exists>x \<in> space N. K = J x \<and> finite K then if K = {} then {x \<in> space N. J x = K}
immler@50088
   837
      else (\<Inter>i\<in>K. (\<lambda>x. f x i) -` S i \<inter> {x \<in> space N. J x = K}) else {})"
immler@50088
   838
    by (auto simp: Pi'_def)
immler@50088
   839
  have r: "{x \<in> space N. J x = K} = space N \<inter> ({x. J x = K} \<inter> space N)" by auto
immler@50088
   840
  show "(\<lambda>x. finmap_of (J x) (f x)) -` Pi' K S \<inter> space N \<in> sets N"
immler@50088
   841
    unfolding eq r
immler@50088
   842
    apply (simp del: INT_simps add: )
immler@50244
   843
    apply (intro conjI impI sets.finite_INT JN sets.Int[OF sets.top])
immler@50088
   844
    apply simp apply assumption
immler@50088
   845
    apply (subst Int_assoc[symmetric])
immler@50244
   846
    apply (rule sets.Int)
immler@50088
   847
    apply (intro measurable_sets[OF f] *) apply force apply assumption
immler@50088
   848
    apply (intro JN)
immler@50088
   849
    done
immler@50088
   850
qed
immler@50088
   851
immler@50088
   852
lemma measurable_PiM_finmap_of:
immler@50088
   853
  assumes "finite J"
wenzelm@53015
   854
  shows "finmap_of J \<in> measurable (Pi\<^sub>M J M) (PiF {J} M)"
immler@50088
   855
  apply (rule measurable_finmap_of)
immler@50088
   856
  apply (rule measurable_component_singleton)
immler@50088
   857
  apply simp
immler@50088
   858
  apply rule
wenzelm@61808
   859
  apply (rule \<open>finite J\<close>)
immler@50088
   860
  apply simp
immler@50088
   861
  done
immler@50088
   862
immler@50088
   863
lemma proj_measurable_singleton:
hoelzl@50124
   864
  assumes "A \<in> sets (M i)"
wenzelm@53015
   865
  shows "(\<lambda>x. (x)\<^sub>F i) -` A \<inter> space (PiF {I} M) \<in> sets (PiF {I} M)"
immler@50088
   866
proof cases
immler@50088
   867
  assume "i \<in> I"
wenzelm@53015
   868
  hence "(\<lambda>x. (x)\<^sub>F i) -` A \<inter> space (PiF {I} M) =
immler@50088
   869
    Pi' I (\<lambda>x. if x = i then A else space (M x))"
wenzelm@61808
   870
    using sets.sets_into_space[OF ] \<open>A \<in> sets (M i)\<close> assms
immler@50088
   871
    by (auto simp: space_PiF Pi'_def)
wenzelm@61808
   872
  thus ?thesis  using assms \<open>A \<in> sets (M i)\<close>
immler@50088
   873
    by (intro in_sets_PiFI) auto
immler@50088
   874
next
immler@50088
   875
  assume "i \<notin> I"
wenzelm@53015
   876
  hence "(\<lambda>x. (x)\<^sub>F i) -` A \<inter> space (PiF {I} M) =
immler@50088
   877
    (if undefined \<in> A then space (PiF {I} M) else {})" by (auto simp: space_PiF Pi'_def)
immler@50088
   878
  thus ?thesis by simp
immler@50088
   879
qed
immler@50088
   880
immler@50088
   881
lemma measurable_proj_singleton:
hoelzl@50124
   882
  assumes "i \<in> I"
wenzelm@53015
   883
  shows "(\<lambda>x. (x)\<^sub>F i) \<in> measurable (PiF {I} M) (M i)"
hoelzl@50124
   884
  by (unfold measurable_def, intro CollectI conjI ballI proj_measurable_singleton assms)
wenzelm@61808
   885
     (insert \<open>i \<in> I\<close>, auto simp: space_PiF)
immler@50088
   886
immler@50088
   887
lemma measurable_proj_countable:
immler@50088
   888
  fixes I::"'a::countable set set"
immler@50088
   889
  assumes "y \<in> space (M i)"
wenzelm@53015
   890
  shows "(\<lambda>x. if i \<in> domain x then (x)\<^sub>F i else y) \<in> measurable (PiF I M) (M i)"
immler@50088
   891
proof (rule countable_measurable_PiFI)
immler@50088
   892
  fix J assume "J \<in> I" "finite J"
immler@50088
   893
  show "(\<lambda>x. if i \<in> domain x then x i else y) \<in> measurable (PiF {J} M) (M i)"
immler@50088
   894
    unfolding measurable_def
immler@50088
   895
  proof safe
immler@50088
   896
    fix z assume "z \<in> sets (M i)"
immler@50088
   897
    have "(\<lambda>x. if i \<in> domain x then x i else y) -` z \<inter> space (PiF {J} M) =
wenzelm@53015
   898
      (\<lambda>x. if i \<in> J then (x)\<^sub>F i else y) -` z \<inter> space (PiF {J} M)"
immler@50088
   899
      by (auto simp: space_PiF Pi'_def)
wenzelm@61808
   900
    also have "\<dots> \<in> sets (PiF {J} M)" using \<open>z \<in> sets (M i)\<close> \<open>finite J\<close>
immler@50088
   901
      by (cases "i \<in> J") (auto intro!: measurable_sets[OF measurable_proj_singleton])
immler@50088
   902
    finally show "(\<lambda>x. if i \<in> domain x then x i else y) -` z \<inter> space (PiF {J} M) \<in>
immler@50088
   903
      sets (PiF {J} M)" .
wenzelm@61808
   904
  qed (insert \<open>y \<in> space (M i)\<close>, auto simp: space_PiF Pi'_def)
immler@50088
   905
qed
immler@50088
   906
immler@50088
   907
lemma measurable_restrict_proj:
immler@50088
   908
  assumes "J \<in> II" "finite J"
immler@50088
   909
  shows "finmap_of J \<in> measurable (PiM J M) (PiF II M)"
immler@50088
   910
  using assms
immler@50088
   911
  by (intro measurable_finmap_of measurable_component_singleton) auto
immler@50088
   912
hoelzl@50124
   913
lemma measurable_proj_PiM:
immler@50088
   914
  fixes J K ::"'a::countable set" and I::"'a set set"
immler@50088
   915
  assumes "finite J" "J \<in> I"
immler@50088
   916
  assumes "x \<in> space (PiM J M)"
hoelzl@50124
   917
  shows "proj \<in> measurable (PiF {J} M) (PiM J M)"
immler@50088
   918
proof (rule measurable_PiM_single)
wenzelm@53015
   919
  show "proj \<in> space (PiF {J} M) \<rightarrow> (\<Pi>\<^sub>E i \<in> J. space (M i))"
immler@50088
   920
    using assms by (auto simp add: space_PiM space_PiF extensional_def sets_PiF Pi'_def)
immler@50088
   921
next
immler@50088
   922
  fix A i assume A: "i \<in> J" "A \<in> sets (M i)"
wenzelm@53015
   923
  show "{\<omega> \<in> space (PiF {J} M). (\<omega>)\<^sub>F i \<in> A} \<in> sets (PiF {J} M)"
immler@50088
   924
  proof
wenzelm@53015
   925
    have "{\<omega> \<in> space (PiF {J} M). (\<omega>)\<^sub>F i \<in> A} =
wenzelm@53015
   926
      (\<lambda>\<omega>. (\<omega>)\<^sub>F i) -` A \<inter> space (PiF {J} M)" by auto
immler@50088
   927
    also have "\<dots> \<in> sets (PiF {J} M)"
immler@50088
   928
      using assms A by (auto intro: measurable_sets[OF measurable_proj_singleton] simp: space_PiM)
immler@50088
   929
    finally show ?thesis .
immler@50088
   930
  qed simp
immler@50088
   931
qed
immler@50088
   932
immler@50088
   933
lemma space_PiF_singleton_eq_product:
immler@50088
   934
  assumes "finite I"
immler@50088
   935
  shows "space (PiF {I} M) = (\<Pi>' i\<in>I. space (M i))"
immler@50088
   936
  by (auto simp: product_def space_PiF assms)
immler@50088
   937
wenzelm@61808
   938
text \<open>adapted from @{thm sets_PiM_single}\<close>
immler@50088
   939
immler@50088
   940
lemma sets_PiF_single:
immler@50088
   941
  assumes "finite I" "I \<noteq> {}"
immler@50088
   942
  shows "sets (PiF {I} M) =
immler@50088
   943
    sigma_sets (\<Pi>' i\<in>I. space (M i))
immler@50088
   944
      {{f\<in>\<Pi>' i\<in>I. space (M i). f i \<in> A} | i A. i \<in> I \<and> A \<in> sets (M i)}"
immler@50088
   945
    (is "_ = sigma_sets ?\<Omega> ?R")
immler@50088
   946
  unfolding sets_PiF_singleton
immler@50088
   947
proof (rule sigma_sets_eqI)
immler@50088
   948
  interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto
immler@50088
   949
  fix A assume "A \<in> {Pi' I X |X. X \<in> (\<Pi> j\<in>I. sets (M j))}"
immler@50088
   950
  then obtain X where X: "A = Pi' I X" "X \<in> (\<Pi> j\<in>I. sets (M j))" by auto
immler@50088
   951
  show "A \<in> sigma_sets ?\<Omega> ?R"
immler@50088
   952
  proof -
wenzelm@61808
   953
    from \<open>I \<noteq> {}\<close> X have "A = (\<Inter>j\<in>I. {f\<in>space (PiF {I} M). f j \<in> X j})"
immler@50244
   954
      using sets.sets_into_space
immler@50088
   955
      by (auto simp: space_PiF product_def) blast
immler@50088
   956
    also have "\<dots> \<in> sigma_sets ?\<Omega> ?R"
wenzelm@61808
   957
      using X \<open>I \<noteq> {}\<close> assms by (intro R.finite_INT) (auto simp: space_PiF)
immler@50088
   958
    finally show "A \<in> sigma_sets ?\<Omega> ?R" .
immler@50088
   959
  qed
immler@50088
   960
next
immler@50088
   961
  fix A assume "A \<in> ?R"
immler@50088
   962
  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)"
immler@50088
   963
    by auto
immler@50088
   964
  then have "A = (\<Pi>' j \<in> I. if j = i then B else space (M j))"
immler@50244
   965
    using sets.sets_into_space[OF A(3)]
nipkow@62390
   966
    apply (auto simp: Pi'_iff split: if_split_asm)
immler@50088
   967
    apply blast
immler@50088
   968
    done
immler@50088
   969
  also have "\<dots> \<in> sigma_sets ?\<Omega> {Pi' I X |X. X \<in> (\<Pi> j\<in>I. sets (M j))}"
immler@50088
   970
    using A
immler@50088
   971
    by (intro sigma_sets.Basic )
immler@50088
   972
       (auto intro: exI[where x="\<lambda>j. if j = i then B else space (M j)"])
immler@50088
   973
  finally show "A \<in> sigma_sets ?\<Omega> {Pi' I X |X. X \<in> (\<Pi> j\<in>I. sets (M j))}" .
immler@50088
   974
qed
immler@50088
   975
wenzelm@61808
   976
text \<open>adapted from @{thm PiE_cong}\<close>
immler@50088
   977
immler@50088
   978
lemma Pi'_cong:
immler@50088
   979
  assumes "finite I"
immler@50088
   980
  assumes "\<And>i. i \<in> I \<Longrightarrow> f i = g i"
immler@50088
   981
  shows "Pi' I f = Pi' I g"
immler@50088
   982
using assms by (auto simp: Pi'_def)
immler@50088
   983
wenzelm@61808
   984
text \<open>adapted from @{thm Pi_UN}\<close>
immler@50088
   985
immler@50088
   986
lemma Pi'_UN:
immler@50088
   987
  fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set"
immler@50088
   988
  assumes "finite I"
immler@50088
   989
  assumes mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i"
immler@50088
   990
  shows "(\<Union>n. Pi' I (A n)) = Pi' I (\<lambda>i. \<Union>n. A n i)"
immler@50088
   991
proof (intro set_eqI iffI)
immler@50088
   992
  fix f assume "f \<in> Pi' I (\<lambda>i. \<Union>n. A n i)"
wenzelm@61808
   993
  then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i" "domain f = I" by (auto simp: \<open>finite I\<close> Pi'_def)
immler@50088
   994
  from bchoice[OF this(1)] obtain n where n: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> (A (n i) i)" by auto
immler@50088
   995
  obtain k where k: "\<And>i. i \<in> I \<Longrightarrow> n i \<le> k"
wenzelm@61808
   996
    using \<open>finite I\<close> finite_nat_set_iff_bounded_le[of "n`I"] by auto
immler@50088
   997
  have "f \<in> Pi' I (\<lambda>i. A k i)"
immler@50088
   998
  proof
immler@50088
   999
    fix i assume "i \<in> I"
wenzelm@61808
  1000
    from mono[OF this, of "n i" k] k[OF this] n[OF this] \<open>domain f = I\<close> \<open>i \<in> I\<close>
wenzelm@61808
  1001
    show "f i \<in> A k i " by (auto simp: \<open>finite I\<close>)
wenzelm@61808
  1002
  qed (simp add: \<open>domain f = I\<close> \<open>finite I\<close>)
immler@50088
  1003
  then show "f \<in> (\<Union>n. Pi' I (A n))" by auto
wenzelm@61808
  1004
qed (auto simp: Pi'_def \<open>finite I\<close>)
immler@50088
  1005
wenzelm@61808
  1006
text \<open>adapted from @{thm sets_PiM_sigma}\<close>
immler@50088
  1007
immler@50088
  1008
lemma sigma_fprod_algebra_sigma_eq:
immler@51106
  1009
  fixes E :: "'i \<Rightarrow> 'a set set" and S :: "'i \<Rightarrow> nat \<Rightarrow> 'a set"
immler@50088
  1010
  assumes [simp]: "finite I" "I \<noteq> {}"
immler@50088
  1011
    and S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)"
immler@50088
  1012
    and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i"
immler@50088
  1013
  assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))"
immler@50088
  1014
    and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)"
immler@50088
  1015
  defines "P == { Pi' I F | F. \<forall>i\<in>I. F i \<in> E i }"
immler@50088
  1016
  shows "sets (PiF {I} M) = sigma_sets (space (PiF {I} M)) P"
immler@50088
  1017
proof
wenzelm@53015
  1018
  let ?P = "sigma (space (Pi\<^sub>F {I} M)) P"
wenzelm@61808
  1019
  from \<open>finite I\<close>[THEN ex_bij_betw_finite_nat] guess T ..
immler@51106
  1020
  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"
wenzelm@61808
  1021
    by (auto simp add: bij_betw_def set_eq_iff image_iff the_inv_into_f_f simp del: \<open>finite I\<close>)
wenzelm@53015
  1022
  have P_closed: "P \<subseteq> Pow (space (Pi\<^sub>F {I} M))"
immler@50088
  1023
    using E_closed by (auto simp: space_PiF P_def Pi'_iff subset_eq)
immler@50088
  1024
  then have space_P: "space ?P = (\<Pi>' i\<in>I. space (M i))"
immler@50088
  1025
    by (simp add: space_PiF)
immler@50088
  1026
  have "sets (PiF {I} M) =
immler@50088
  1027
      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)}"
immler@50088
  1028
    using sets_PiF_single[of I M] by (simp add: space_P)
immler@50088
  1029
  also have "\<dots> \<subseteq> sets (sigma (space (PiF {I} M)) P)"
immler@50244
  1030
  proof (safe intro!: sets.sigma_sets_subset)
immler@50088
  1031
    fix i A assume "i \<in> I" and A: "A \<in> sets (M i)"
wenzelm@53015
  1032
    have "(\<lambda>x. (x)\<^sub>F i) \<in> measurable ?P (sigma (space (M i)) (E i))"
immler@50088
  1033
    proof (subst measurable_iff_measure_of)
wenzelm@61808
  1034
      show "E i \<subseteq> Pow (space (M i))" using \<open>i \<in> I\<close> by fact
wenzelm@61808
  1035
      from space_P \<open>i \<in> I\<close> show "(\<lambda>x. (x)\<^sub>F i) \<in> space ?P \<rightarrow> space (M i)"
immler@50088
  1036
        by auto
wenzelm@53015
  1037
      show "\<forall>A\<in>E i. (\<lambda>x. (x)\<^sub>F i) -` A \<inter> space ?P \<in> sets ?P"
immler@50088
  1038
      proof
immler@50088
  1039
        fix A assume A: "A \<in> E i"
wenzelm@53015
  1040
        then have "(\<lambda>x. (x)\<^sub>F i) -` A \<inter> space ?P = (\<Pi>' j\<in>I. if i = j then A else space (M j))"
nipkow@62390
  1041
          using E_closed \<open>i \<in> I\<close> by (auto simp: space_P Pi_iff subset_eq split: if_split_asm)
immler@50088
  1042
        also have "\<dots> = (\<Pi>' j\<in>I. \<Union>n. if i = j then A else S j n)"
immler@50088
  1043
          by (intro Pi'_cong) (simp_all add: S_union)
immler@51106
  1044
        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))"
immler@51106
  1045
          using T
haftmann@62343
  1046
          apply (auto simp del: Union_iff)
haftmann@62343
  1047
          apply (simp_all add: Pi'_iff bchoice_iff del: Union_iff)
immler@51106
  1048
          apply (erule conjE exE)+
immler@51106
  1049
          apply (rule_tac x="map (\<lambda>n. f (the_inv_into I T n)) [0..<card I]" in exI)
immler@51106
  1050
          apply (auto simp: bij_betw_def)
immler@51106
  1051
          done
immler@50088
  1052
        also have "\<dots> \<in> sets ?P"
immler@50244
  1053
        proof (safe intro!: sets.countable_UN)
immler@51106
  1054
          fix xs show "(\<Pi>' j\<in>I. if i = j then A else S j (xs ! T j)) \<in> sets ?P"
immler@50088
  1055
            using A S_in_E
immler@50088
  1056
            by (simp add: P_closed)
immler@51106
  1057
               (auto simp: P_def subset_eq intro!: exI[of _ "\<lambda>j. if i = j then A else S j (xs ! T j)"])
immler@50088
  1058
        qed
wenzelm@53015
  1059
        finally show "(\<lambda>x. (x)\<^sub>F i) -` A \<inter> space ?P \<in> sets ?P"
immler@50088
  1060
          using P_closed by simp
immler@50088
  1061
      qed
immler@50088
  1062
    qed
wenzelm@61808
  1063
    from measurable_sets[OF this, of A] A \<open>i \<in> I\<close> E_closed
wenzelm@53015
  1064
    have "(\<lambda>x. (x)\<^sub>F i) -` A \<inter> space ?P \<in> sets ?P"
immler@50088
  1065
      by (simp add: E_generates)
wenzelm@53015
  1066
    also have "(\<lambda>x. (x)\<^sub>F i) -` A \<inter> space ?P = {f \<in> \<Pi>' i\<in>I. space (M i). f i \<in> A}"
immler@50088
  1067
      using P_closed by (auto simp: space_PiF)
immler@50088
  1068
    finally show "\<dots> \<in> sets ?P" .
immler@50088
  1069
  qed
immler@50088
  1070
  finally show "sets (PiF {I} M) \<subseteq> sigma_sets (space (PiF {I} M)) P"
immler@50088
  1071
    by (simp add: P_closed)
immler@50088
  1072
  show "sigma_sets (space (PiF {I} M)) P \<subseteq> sets (PiF {I} M)"
wenzelm@61808
  1073
    using \<open>finite I\<close> \<open>I \<noteq> {}\<close>
immler@50244
  1074
    by (auto intro!: sets.sigma_sets_subset product_in_sets_PiFI simp: E_generates P_def)
immler@50088
  1075
qed
immler@50088
  1076
immler@50088
  1077
lemma product_open_generates_sets_PiF_single:
immler@50088
  1078
  assumes "I \<noteq> {}"
immler@50088
  1079
  assumes [simp]: "finite I"
hoelzl@50881
  1080
  shows "sets (PiF {I} (\<lambda>_. borel::'b::second_countable_topology measure)) =
immler@50088
  1081
    sigma_sets (space (PiF {I} (\<lambda>_. borel))) {Pi' I F |F. (\<forall>i\<in>I. F i \<in> Collect open)}"
immler@50088
  1082
proof -
immler@51106
  1083
  from open_countable_basisE[OF open_UNIV] guess S::"'b set set" . note S = this
immler@50088
  1084
  show ?thesis
immler@50088
  1085
  proof (rule sigma_fprod_algebra_sigma_eq)
immler@50088
  1086
    show "finite I" by simp
immler@50088
  1087
    show "I \<noteq> {}" by fact
wenzelm@63040
  1088
    define S' where "S' = from_nat_into S"
immler@51106
  1089
    show "(\<Union>j. S' j) = space borel"
immler@51106
  1090
      using S
immler@51106
  1091
      apply (auto simp add: from_nat_into countable_basis_proj S'_def basis_proj_def)
immler@51106
  1092
      apply (metis (lifting, mono_tags) UNIV_I UnionE basis_proj_def countable_basis_proj countable_subset from_nat_into_surj)
immler@51106
  1093
      done
immler@51106
  1094
    show "range S' \<subseteq> Collect open"
immler@51106
  1095
      using S
immler@51106
  1096
      apply (auto simp add: from_nat_into countable_basis_proj S'_def)
immler@51106
  1097
      apply (metis UNIV_not_empty Union_empty from_nat_into set_mp topological_basis_open[OF basis_proj] basis_proj_def)
immler@51106
  1098
      done
immler@50088
  1099
    show "Collect open \<subseteq> Pow (space borel)" by simp
immler@50088
  1100
    show "sets borel = sigma_sets (space borel) (Collect open)"
immler@50088
  1101
      by (simp add: borel_def)
immler@50088
  1102
  qed
immler@50088
  1103
qed
immler@50088
  1104
wenzelm@61988
  1105
lemma finmap_UNIV[simp]: "(\<Union>J\<in>Collect finite. \<Pi>' j\<in>J. UNIV) = UNIV" by auto
immler@50088
  1106
immler@50088
  1107
lemma borel_eq_PiF_borel:
wenzelm@53015
  1108
  shows "(borel :: ('i::countable \<Rightarrow>\<^sub>F 'a::polish_space) measure) =
immler@50245
  1109
    PiF (Collect finite) (\<lambda>_. borel :: 'a measure)"
immler@50245
  1110
  unfolding borel_def PiF_def
immler@50245
  1111
proof (rule measure_eqI, clarsimp, rule sigma_sets_eqI)
wenzelm@53015
  1112
  fix a::"('i \<Rightarrow>\<^sub>F 'a) set" assume "a \<in> Collect open" hence "open a" by simp
immler@50245
  1113
  then obtain B' where B': "B'\<subseteq>basis_finmap" "a = \<Union>B'"
immler@50245
  1114
    using finmap_topological_basis by (force simp add: topological_basis_def)
immler@50245
  1115
  have "a \<in> sigma UNIV {Pi' J X |X J. finite J \<and> X \<in> J \<rightarrow> sigma_sets UNIV (Collect open)}"
wenzelm@61808
  1116
    unfolding \<open>a = \<Union>B'\<close>
immler@50245
  1117
  proof (rule sets.countable_Union)
immler@50245
  1118
    from B' countable_basis_finmap show "countable B'" by (metis countable_subset)
immler@50088
  1119
  next
immler@50245
  1120
    show "B' \<subseteq> sets (sigma UNIV
immler@50245
  1121
      {Pi' J X |X J. finite J \<and> X \<in> J \<rightarrow> sigma_sets UNIV (Collect open)})" (is "_ \<subseteq> sets ?s")
immler@50088
  1122
    proof
immler@50245
  1123
      fix x assume "x \<in> B'" with B' have "x \<in> basis_finmap" by auto
immler@50245
  1124
      then obtain J X where "x = Pi' J X" "finite J" "X \<in> J \<rightarrow> sigma_sets UNIV (Collect open)"
immler@51106
  1125
        by (auto simp: basis_finmap_def topological_basis_open[OF basis_proj])
immler@50245
  1126
      thus "x \<in> sets ?s" by auto
immler@50088
  1127
    qed
immler@50088
  1128
  qed
immler@50245
  1129
  thus "a \<in> sigma_sets UNIV {Pi' J X |X J. finite J \<and> X \<in> J \<rightarrow> sigma_sets UNIV (Collect open)}"
immler@50245
  1130
    by simp
immler@50245
  1131
next
wenzelm@53015
  1132
  fix b::"('i \<Rightarrow>\<^sub>F 'a) set"
immler@50245
  1133
  assume "b \<in> {Pi' J X |X J. finite J \<and> X \<in> J \<rightarrow> sigma_sets UNIV (Collect open)}"
wenzelm@53015
  1134
  hence b': "b \<in> sets (Pi\<^sub>F (Collect finite) (\<lambda>_. borel))" by (auto simp: sets_PiF borel_def)
immler@50245
  1135
  let ?b = "\<lambda>J. b \<inter> {x. domain x = J}"
immler@50245
  1136
  have "b = \<Union>((\<lambda>J. ?b J) ` Collect finite)" by auto
immler@50245
  1137
  also have "\<dots> \<in> sets borel"
immler@50245
  1138
  proof (rule sets.countable_Union, safe)
immler@50245
  1139
    fix J::"'i set" assume "finite J"
immler@50245
  1140
    { assume ef: "J = {}"
immler@50245
  1141
      have "?b J \<in> sets borel"
immler@50245
  1142
      proof cases
immler@50245
  1143
        assume "?b J \<noteq> {}"
immler@50245
  1144
        then obtain f where "f \<in> b" "domain f = {}" using ef by auto
wenzelm@61808
  1145
        hence "?b J = {f}" using \<open>J = {}\<close>
immler@50245
  1146
          by (auto simp: finmap_eq_iff)
immler@50245
  1147
        also have "{f} \<in> sets borel" by simp
immler@50245
  1148
        finally show ?thesis .
immler@50245
  1149
      qed simp
immler@50245
  1150
    } moreover {
immler@50245
  1151
      assume "J \<noteq> ({}::'i set)"
immler@50245
  1152
      have "(?b J) = b \<inter> {m. domain m \<in> {J}}" by auto
immler@50245
  1153
      also have "\<dots> \<in> sets (PiF {J} (\<lambda>_. borel))"
wenzelm@61808
  1154
        using b' by (rule restrict_sets_measurable) (auto simp: \<open>finite J\<close>)
immler@50245
  1155
      also have "\<dots> = sigma_sets (space (PiF {J} (\<lambda>_. borel)))
immler@50245
  1156
        {Pi' (J) F |F. (\<forall>j\<in>J. F j \<in> Collect open)}"
immler@50245
  1157
        (is "_ = sigma_sets _ ?P")
wenzelm@61808
  1158
       by (rule product_open_generates_sets_PiF_single[OF \<open>J \<noteq> {}\<close> \<open>finite J\<close>])
immler@50245
  1159
      also have "\<dots> \<subseteq> sigma_sets UNIV (Collect open)"
immler@50245
  1160
        by (intro sigma_sets_mono'') (auto intro!: open_Pi'I simp: space_PiF)
immler@50245
  1161
      finally have "(?b J) \<in> sets borel" by (simp add: borel_def)
immler@50245
  1162
    } ultimately show "(?b J) \<in> sets borel" by blast
immler@50245
  1163
  qed (simp add: countable_Collect_finite)
immler@50245
  1164
  finally show "b \<in> sigma_sets UNIV (Collect open)" by (simp add: borel_def)
immler@50088
  1165
qed (simp add: emeasure_sigma borel_def PiF_def)
immler@50088
  1166
wenzelm@61808
  1167
subsection \<open>Isomorphism between Functions and Finite Maps\<close>
immler@50088
  1168
hoelzl@50124
  1169
lemma measurable_finmap_compose:
immler@50088
  1170
  shows "(\<lambda>m. compose J m f) \<in> measurable (PiM (f ` J) (\<lambda>_. M)) (PiM J (\<lambda>_. M))"
hoelzl@50124
  1171
  unfolding compose_def by measurable
immler@50088
  1172
hoelzl@50124
  1173
lemma measurable_compose_inv:
immler@50088
  1174
  assumes inj: "\<And>j. j \<in> J \<Longrightarrow> f' (f j) = j"
immler@50088
  1175
  shows "(\<lambda>m. compose (f ` J) m f') \<in> measurable (PiM J (\<lambda>_. M)) (PiM (f ` J) (\<lambda>_. M))"
hoelzl@50124
  1176
  unfolding compose_def by (rule measurable_restrict) (auto simp: inj)
immler@50088
  1177
immler@50088
  1178
locale function_to_finmap =
immler@50088
  1179
  fixes J::"'a set" and f :: "'a \<Rightarrow> 'b::countable" and f'
immler@50088
  1180
  assumes [simp]: "finite J"
immler@50088
  1181
  assumes inv: "i \<in> J \<Longrightarrow> f' (f i) = i"
immler@50088
  1182
begin
immler@50088
  1183
wenzelm@61808
  1184
text \<open>to measure finmaps\<close>
immler@50088
  1185
immler@50088
  1186
definition "fm = (finmap_of (f ` J)) o (\<lambda>g. compose (f ` J) g f')"
immler@50088
  1187
immler@50088
  1188
lemma domain_fm[simp]: "domain (fm x) = f ` J"
immler@50088
  1189
  unfolding fm_def by simp
immler@50088
  1190
immler@50088
  1191
lemma fm_restrict[simp]: "fm (restrict y J) = fm y"
immler@50088
  1192
  unfolding fm_def by (auto simp: compose_def inv intro: restrict_ext)
immler@50088
  1193
immler@50088
  1194
lemma fm_product:
immler@50088
  1195
  assumes "\<And>i. space (M i) = UNIV"
wenzelm@53015
  1196
  shows "fm -` Pi' (f ` J) S \<inter> space (Pi\<^sub>M J M) = (\<Pi>\<^sub>E j \<in> J. S (f j))"
immler@50088
  1197
  using assms
immler@50088
  1198
  by (auto simp: inv fm_def compose_def space_PiM Pi'_def)
immler@50088
  1199
immler@50088
  1200
lemma fm_measurable:
immler@50088
  1201
  assumes "f ` J \<in> N"
wenzelm@53015
  1202
  shows "fm \<in> measurable (Pi\<^sub>M J (\<lambda>_. M)) (Pi\<^sub>F N (\<lambda>_. M))"
immler@50088
  1203
  unfolding fm_def
immler@50088
  1204
proof (rule measurable_comp, rule measurable_compose_inv)
wenzelm@53015
  1205
  show "finmap_of (f ` J) \<in> measurable (Pi\<^sub>M (f ` J) (\<lambda>_. M)) (PiF N (\<lambda>_. M)) "
immler@50088
  1206
    using assms by (intro measurable_finmap_of measurable_component_singleton) auto
immler@50088
  1207
qed (simp_all add: inv)
immler@50088
  1208
immler@50088
  1209
lemma proj_fm:
immler@50088
  1210
  assumes "x \<in> J"
immler@50088
  1211
  shows "fm m (f x) = m x"
immler@50088
  1212
  using assms by (auto simp: fm_def compose_def o_def inv)
immler@50088
  1213
immler@50088
  1214
lemma inj_on_compose_f': "inj_on (\<lambda>g. compose (f ` J) g f') (extensional J)"
immler@50088
  1215
proof (rule inj_on_inverseI)
immler@50088
  1216
  fix x::"'a \<Rightarrow> 'c" assume "x \<in> extensional J"
immler@50088
  1217
  thus "(\<lambda>x. compose J x f) (compose (f ` J) x f') = x"
immler@50088
  1218
    by (auto simp: compose_def inv extensional_def)
immler@50088
  1219
qed
immler@50088
  1220
immler@50088
  1221
lemma inj_on_fm:
immler@50088
  1222
  assumes "\<And>i. space (M i) = UNIV"
wenzelm@53015
  1223
  shows "inj_on fm (space (Pi\<^sub>M J M))"
immler@50088
  1224
  using assms
hoelzl@50123
  1225
  apply (auto simp: fm_def space_PiM PiE_def)
immler@50088
  1226
  apply (rule comp_inj_on)
immler@50088
  1227
  apply (rule inj_on_compose_f')
immler@50088
  1228
  apply (rule finmap_of_inj_on_extensional_finite)
immler@50088
  1229
  apply simp
immler@50088
  1230
  apply (auto)
immler@50088
  1231
  done
immler@50088
  1232
wenzelm@61808
  1233
text \<open>to measure functions\<close>
immler@50088
  1234
immler@50088
  1235
definition "mf = (\<lambda>g. compose J g f) o proj"
immler@50088
  1236
immler@50088
  1237
lemma mf_fm:
wenzelm@53015
  1238
  assumes "x \<in> space (Pi\<^sub>M J (\<lambda>_. M))"
immler@50088
  1239
  shows "mf (fm x) = x"
immler@50088
  1240
proof -
immler@50088
  1241
  have "mf (fm x) \<in> extensional J"
immler@50088
  1242
    by (auto simp: mf_def extensional_def compose_def)
immler@50088
  1243
  moreover
immler@50244
  1244
  have "x \<in> extensional J" using assms sets.sets_into_space
hoelzl@50123
  1245
    by (force simp: space_PiM PiE_def)
immler@50088
  1246
  moreover
immler@50088
  1247
  { fix i assume "i \<in> J"
immler@50088
  1248
    hence "mf (fm x) i = x i"
immler@50088
  1249
      by (auto simp: inv mf_def compose_def fm_def)
immler@50088
  1250
  }
immler@50088
  1251
  ultimately
immler@50088
  1252
  show ?thesis by (rule extensionalityI)
immler@50088
  1253
qed
immler@50088
  1254
immler@50088
  1255
lemma mf_measurable:
immler@50088
  1256
  assumes "space M = UNIV"
immler@50088
  1257
  shows "mf \<in> measurable (PiF {f ` J} (\<lambda>_. M)) (PiM J (\<lambda>_. M))"
immler@50088
  1258
  unfolding mf_def
immler@50088
  1259
proof (rule measurable_comp, rule measurable_proj_PiM)
wenzelm@53015
  1260
  show "(\<lambda>g. compose J g f) \<in> measurable (Pi\<^sub>M (f ` J) (\<lambda>x. M)) (Pi\<^sub>M J (\<lambda>_. M))"
hoelzl@50124
  1261
    by (rule measurable_finmap_compose)
immler@50088
  1262
qed (auto simp add: space_PiM extensional_def assms)
immler@50088
  1263
immler@50088
  1264
lemma fm_image_measurable:
immler@50088
  1265
  assumes "space M = UNIV"
wenzelm@53015
  1266
  assumes "X \<in> sets (Pi\<^sub>M J (\<lambda>_. M))"
immler@50088
  1267
  shows "fm ` X \<in> sets (PiF {f ` J} (\<lambda>_. M))"
immler@50088
  1268
proof -
immler@50088
  1269
  have "fm ` X = (mf) -` X \<inter> space (PiF {f ` J} (\<lambda>_. M))"
immler@50088
  1270
  proof safe
immler@50088
  1271
    fix x assume "x \<in> X"
immler@50244
  1272
    with mf_fm[of x] sets.sets_into_space[OF assms(2)] show "fm x \<in> mf -` X" by auto
immler@50088
  1273
    show "fm x \<in> space (PiF {f ` J} (\<lambda>_. M))" by (simp add: space_PiF assms)
immler@50088
  1274
  next
immler@50088
  1275
    fix y x
immler@50088
  1276
    assume x: "mf y \<in> X"
immler@50088
  1277
    assume y: "y \<in> space (PiF {f ` J} (\<lambda>_. M))"
immler@50088
  1278
    thus "y \<in> fm ` X"
immler@50088
  1279
      by (intro image_eqI[OF _ x], unfold finmap_eq_iff)
immler@50088
  1280
         (auto simp: space_PiF fm_def mf_def compose_def inv Pi'_def)
immler@50088
  1281
  qed
immler@50088
  1282
  also have "\<dots> \<in> sets (PiF {f ` J} (\<lambda>_. M))"
immler@50088
  1283
    using assms
immler@50088
  1284
    by (intro measurable_sets[OF mf_measurable]) auto
immler@50088
  1285
  finally show ?thesis .
immler@50088
  1286
qed
immler@50088
  1287
immler@50088
  1288
lemma fm_image_measurable_finite:
immler@50088
  1289
  assumes "space M = UNIV"
wenzelm@53015
  1290
  assumes "X \<in> sets (Pi\<^sub>M J (\<lambda>_. M::'c measure))"
immler@50088
  1291
  shows "fm ` X \<in> sets (PiF (Collect finite) (\<lambda>_. M::'c measure))"
immler@50088
  1292
  using fm_image_measurable[OF assms]
immler@50088
  1293
  by (rule subspace_set_in_sets) (auto simp: finite_subset)
immler@50088
  1294
wenzelm@61808
  1295
text \<open>measure on finmaps\<close>
immler@50088
  1296
immler@50088
  1297
definition "mapmeasure M N = distr M (PiF (Collect finite) N) (fm)"
immler@50088
  1298
immler@50088
  1299
lemma sets_mapmeasure[simp]: "sets (mapmeasure M N) = sets (PiF (Collect finite) N)"
immler@50088
  1300
  unfolding mapmeasure_def by simp
immler@50088
  1301
immler@50088
  1302
lemma space_mapmeasure[simp]: "space (mapmeasure M N) = space (PiF (Collect finite) N)"
immler@50088
  1303
  unfolding mapmeasure_def by simp
immler@50088
  1304
immler@50088
  1305
lemma mapmeasure_PiF:
wenzelm@53015
  1306
  assumes s1: "space M = space (Pi\<^sub>M J (\<lambda>_. N))"
wenzelm@53015
  1307
  assumes s2: "sets M = sets (Pi\<^sub>M J (\<lambda>_. N))"
immler@50088
  1308
  assumes "space N = UNIV"
immler@50088
  1309
  assumes "X \<in> sets (PiF (Collect finite) (\<lambda>_. N))"
immler@50088
  1310
  shows "emeasure (mapmeasure M (\<lambda>_. N)) X = emeasure M ((fm -` X \<inter> extensional J))"
immler@50088
  1311
  using assms
hoelzl@59048
  1312
  by (auto simp: measurable_cong_sets[OF s2 refl] mapmeasure_def emeasure_distr
hoelzl@50123
  1313
    fm_measurable space_PiM PiE_def)
immler@50088
  1314
immler@50088
  1315
lemma mapmeasure_PiM:
immler@50088
  1316
  fixes N::"'c measure"
wenzelm@53015
  1317
  assumes s1: "space M = space (Pi\<^sub>M J (\<lambda>_. N))"
wenzelm@53015
  1318
  assumes s2: "sets M = (Pi\<^sub>M J (\<lambda>_. N))"
immler@50088
  1319
  assumes N: "space N = UNIV"
immler@50088
  1320
  assumes X: "X \<in> sets M"
immler@50088
  1321
  shows "emeasure M X = emeasure (mapmeasure M (\<lambda>_. N)) (fm ` X)"
immler@50088
  1322
  unfolding mapmeasure_def
hoelzl@59048
  1323
proof (subst emeasure_distr, subst measurable_cong_sets[OF s2 refl], rule fm_measurable)
wenzelm@53015
  1324
  have "X \<subseteq> space (Pi\<^sub>M J (\<lambda>_. N))" using assms by (simp add: sets.sets_into_space)
wenzelm@53015
  1325
  from assms inj_on_fm[of "\<lambda>_. N"] set_mp[OF this] have "fm -` fm ` X \<inter> space (Pi\<^sub>M J (\<lambda>_. N)) = X"
immler@50088
  1326
    by (auto simp: vimage_image_eq inj_on_def)
immler@50088
  1327
  thus "emeasure M X = emeasure M (fm -` fm ` X \<inter> space M)" using s1
immler@50088
  1328
    by simp
immler@50088
  1329
  show "fm ` X \<in> sets (PiF (Collect finite) (\<lambda>_. N))"
immler@50088
  1330
    by (rule fm_image_measurable_finite[OF N X[simplified s2]])
immler@50088
  1331
qed simp
immler@50088
  1332
immler@50088
  1333
end
immler@50088
  1334
nipkow@67399
  1335
end