src/HOL/Probability/Measurable.thy
author hoelzl
Mon, 19 May 2014 12:04:45 +0200
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introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
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(*  Title:      HOL/Probability/Measurable.thy
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    Author:     Johannes Hölzl <hoelzl@in.tum.de>
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*)
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theory Measurable
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  imports
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    Sigma_Algebra
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    "~~/src/HOL/Library/Order_Continuity"
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begin
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hide_const (open) Order_Continuity.continuous
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subsection {* Measurability prover *}
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lemma (in algebra) sets_Collect_finite_All:
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  assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S"
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  shows "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} \<in> M"
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proof -
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  have "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} = (if S = {} then \<Omega> else \<Inter>i\<in>S. {x\<in>\<Omega>. P i x})"
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    by auto
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  with assms show ?thesis by (auto intro!: sets_Collect_finite_All')
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qed
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abbreviation "pred M P \<equiv> P \<in> measurable M (count_space (UNIV::bool set))"
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lemma pred_def: "pred M P \<longleftrightarrow> {x\<in>space M. P x} \<in> sets M"
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proof
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  assume "pred M P"
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  then have "P -` {True} \<inter> space M \<in> sets M"
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    by (auto simp: measurable_count_space_eq2)
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  also have "P -` {True} \<inter> space M = {x\<in>space M. P x}" by auto
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  finally show "{x\<in>space M. P x} \<in> sets M" .
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next
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  assume P: "{x\<in>space M. P x} \<in> sets M"
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  moreover
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  { fix X
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    have "X \<in> Pow (UNIV :: bool set)" by simp
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    then have "P -` X \<inter> space M = {x\<in>space M. ((X = {True} \<longrightarrow> P x) \<and> (X = {False} \<longrightarrow> \<not> P x) \<and> X \<noteq> {})}"
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      unfolding UNIV_bool Pow_insert Pow_empty by auto
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    then have "P -` X \<inter> space M \<in> sets M"
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      by (auto intro!: sets.sets_Collect_neg sets.sets_Collect_imp sets.sets_Collect_conj sets.sets_Collect_const P) }
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  then show "pred M P"
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    by (auto simp: measurable_def)
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qed
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lemma pred_sets1: "{x\<in>space M. P x} \<in> sets M \<Longrightarrow> f \<in> measurable N M \<Longrightarrow> pred N (\<lambda>x. P (f x))"
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  by (rule measurable_compose[where f=f and N=M]) (auto simp: pred_def)
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lemma pred_sets2: "A \<in> sets N \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> pred M (\<lambda>x. f x \<in> A)"
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  by (rule measurable_compose[where f=f and N=N]) (auto simp: pred_def Int_def[symmetric])
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ML_file "measurable.ML"
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attribute_setup measurable = {*
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  Scan.lift (Scan.optional (Args.parens (Scan.optional (Args.$$$ "raw" >> K true) false --
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    Scan.optional (Args.$$$ "generic" >> K Measurable.Generic) Measurable.Concrete))
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    (false, Measurable.Concrete) >> (Thm.declaration_attribute o Measurable.add_thm))
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*} "declaration of measurability theorems"
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attribute_setup measurable_dest = {*
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  Scan.lift (Scan.succeed (Thm.declaration_attribute Measurable.add_dest))
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*} "add dest rule for measurability prover"
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attribute_setup measurable_app = {*
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  Scan.lift (Scan.succeed (Thm.declaration_attribute Measurable.add_app))
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*} "add application rule for measurability prover"
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method_setup measurable = {*
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  Scan.lift (Scan.succeed (fn ctxt => METHOD (fn facts => Measurable.measurable_tac ctxt facts)))
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*} "measurability prover"
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simproc_setup measurable ("A \<in> sets M" | "f \<in> measurable M N") = {* K Measurable.simproc *}
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declare
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  measurable_compose_rev[measurable_dest]
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  pred_sets1[measurable_dest]
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  pred_sets2[measurable_dest]
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  sets.sets_into_space[measurable_dest]
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declare
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  sets.top[measurable]
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  sets.empty_sets[measurable (raw)]
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  sets.Un[measurable (raw)]
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  sets.Diff[measurable (raw)]
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declare
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  measurable_count_space[measurable (raw)]
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  measurable_ident[measurable (raw)]
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  measurable_ident_sets[measurable (raw)]
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  measurable_const[measurable (raw)]
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  measurable_If[measurable (raw)]
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  measurable_comp[measurable (raw)]
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  measurable_sets[measurable (raw)]
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lemma predE[measurable (raw)]: 
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  "pred M P \<Longrightarrow> {x\<in>space M. P x} \<in> sets M"
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  unfolding pred_def .
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lemma pred_intros_imp'[measurable (raw)]:
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  "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<longrightarrow> P x)"
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  by (cases K) auto
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lemma pred_intros_conj1'[measurable (raw)]:
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  "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<and> P x)"
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  by (cases K) auto
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lemma pred_intros_conj2'[measurable (raw)]:
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  "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. P x \<and> K)"
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  by (cases K) auto
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lemma pred_intros_disj1'[measurable (raw)]:
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  "(\<not> K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<or> P x)"
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  by (cases K) auto
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lemma pred_intros_disj2'[measurable (raw)]:
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  "(\<not> K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. P x \<or> K)"
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  by (cases K) auto
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lemma pred_intros_logic[measurable (raw)]:
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  "pred M (\<lambda>x. x \<in> space M)"
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  "pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. \<not> P x)"
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  "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<and> P x)"
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  "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<longrightarrow> P x)"
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  "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<or> P x)"
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  "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x = P x)"
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  "pred M (\<lambda>x. f x \<in> UNIV)"
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  "pred M (\<lambda>x. f x \<in> {})"
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  "pred M (\<lambda>x. P' (f x) x) \<Longrightarrow> pred M (\<lambda>x. f x \<in> {y. P' y x})"
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  "pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> - (B x))"
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  "pred M (\<lambda>x. f x \<in> (A x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (A x) - (B x))"
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  "pred M (\<lambda>x. f x \<in> (A x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (A x) \<inter> (B x))"
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  "pred M (\<lambda>x. f x \<in> (A x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (A x) \<union> (B x))"
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  "pred M (\<lambda>x. g x (f x) \<in> (X x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (g x) -` (X x))"
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  by (auto simp: iff_conv_conj_imp pred_def)
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   135
lemma pred_intros_countable[measurable (raw)]:
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   136
  fixes P :: "'a \<Rightarrow> 'i :: countable \<Rightarrow> bool"
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   137
  shows 
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   138
    "(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<forall>i. P x i)"
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parents:
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   139
    "(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<exists>i. P x i)"
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   140
  by (auto intro!: sets.sets_Collect_countable_All sets.sets_Collect_countable_Ex simp: pred_def)
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parents:
diff changeset
   141
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   142
lemma pred_intros_countable_bounded[measurable (raw)]:
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   143
  fixes X :: "'i :: countable set"
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   144
  shows 
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   145
    "(\<And>i. i \<in> X \<Longrightarrow> pred M (\<lambda>x. x \<in> N x i)) \<Longrightarrow> pred M (\<lambda>x. x \<in> (\<Inter>i\<in>X. N x i))"
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parents:
diff changeset
   146
    "(\<And>i. i \<in> X \<Longrightarrow> pred M (\<lambda>x. x \<in> N x i)) \<Longrightarrow> pred M (\<lambda>x. x \<in> (\<Union>i\<in>X. N x i))"
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   147
    "(\<And>i. i \<in> X \<Longrightarrow> pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<forall>i\<in>X. P x i)"
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   148
    "(\<And>i. i \<in> X \<Longrightarrow> pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<exists>i\<in>X. P x i)"
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   149
  by (auto simp: Bex_def Ball_def)
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   150
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   151
lemma pred_intros_finite[measurable (raw)]:
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   152
  "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> pred M (\<lambda>x. x \<in> N x i)) \<Longrightarrow> pred M (\<lambda>x. x \<in> (\<Inter>i\<in>I. N x i))"
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   153
  "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> pred M (\<lambda>x. x \<in> N x i)) \<Longrightarrow> pred M (\<lambda>x. x \<in> (\<Union>i\<in>I. N x i))"
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   154
  "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<forall>i\<in>I. P x i)"
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   155
  "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<exists>i\<in>I. P x i)"
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parents:
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   156
  by (auto intro!: sets.sets_Collect_finite_Ex sets.sets_Collect_finite_All simp: iff_conv_conj_imp pred_def)
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diff changeset
   157
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   158
lemma countable_Un_Int[measurable (raw)]:
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   159
  "(\<And>i :: 'i :: countable. i \<in> I \<Longrightarrow> N i \<in> sets M) \<Longrightarrow> (\<Union>i\<in>I. N i) \<in> sets M"
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   160
  "I \<noteq> {} \<Longrightarrow> (\<And>i :: 'i :: countable. i \<in> I \<Longrightarrow> N i \<in> sets M) \<Longrightarrow> (\<Inter>i\<in>I. N i) \<in> sets M"
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   161
  by auto
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   162
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   163
declare
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   164
  finite_UN[measurable (raw)]
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   165
  finite_INT[measurable (raw)]
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   166
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   167
lemma sets_Int_pred[measurable (raw)]:
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   168
  assumes space: "A \<inter> B \<subseteq> space M" and [measurable]: "pred M (\<lambda>x. x \<in> A)" "pred M (\<lambda>x. x \<in> B)"
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   169
  shows "A \<inter> B \<in> sets M"
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   170
proof -
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   171
  have "{x\<in>space M. x \<in> A \<inter> B} \<in> sets M" by auto
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   172
  also have "{x\<in>space M. x \<in> A \<inter> B} = A \<inter> B"
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   173
    using space by auto
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   174
  finally show ?thesis .
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   175
qed
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   176
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   177
lemma [measurable (raw generic)]:
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   178
  assumes f: "f \<in> measurable M N" and c: "c \<in> space N \<Longrightarrow> {c} \<in> sets N"
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   179
  shows pred_eq_const1: "pred M (\<lambda>x. f x = c)"
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   180
    and pred_eq_const2: "pred M (\<lambda>x. c = f x)"
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   181
proof -
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   182
  show "pred M (\<lambda>x. f x = c)"
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   183
  proof cases
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   184
    assume "c \<in> space N"
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   185
    with measurable_sets[OF f c] show ?thesis
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   186
      by (auto simp: Int_def conj_commute pred_def)
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   187
  next
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   188
    assume "c \<notin> space N"
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   189
    with f[THEN measurable_space] have "{x \<in> space M. f x = c} = {}" by auto
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   190
    then show ?thesis by (auto simp: pred_def cong: conj_cong)
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   191
  qed
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   192
  then show "pred M (\<lambda>x. c = f x)"
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   193
    by (simp add: eq_commute)
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   194
qed
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   195
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   196
lemma pred_le_const[measurable (raw generic)]:
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   197
  assumes f: "f \<in> measurable M N" and c: "{.. c} \<in> sets N" shows "pred M (\<lambda>x. f x \<le> c)"
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   198
  using measurable_sets[OF f c]
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   199
  by (auto simp: Int_def conj_commute eq_commute pred_def)
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diff changeset
   200
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   201
lemma pred_const_le[measurable (raw generic)]:
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   202
  assumes f: "f \<in> measurable M N" and c: "{c ..} \<in> sets N" shows "pred M (\<lambda>x. c \<le> f x)"
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   203
  using measurable_sets[OF f c]
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   204
  by (auto simp: Int_def conj_commute eq_commute pred_def)
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diff changeset
   205
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   206
lemma pred_less_const[measurable (raw generic)]:
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   207
  assumes f: "f \<in> measurable M N" and c: "{..< c} \<in> sets N" shows "pred M (\<lambda>x. f x < c)"
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diff changeset
   208
  using measurable_sets[OF f c]
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diff changeset
   209
  by (auto simp: Int_def conj_commute eq_commute pred_def)
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diff changeset
   210
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diff changeset
   211
lemma pred_const_less[measurable (raw generic)]:
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   212
  assumes f: "f \<in> measurable M N" and c: "{c <..} \<in> sets N" shows "pred M (\<lambda>x. c < f x)"
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diff changeset
   213
  using measurable_sets[OF f c]
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   214
  by (auto simp: Int_def conj_commute eq_commute pred_def)
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diff changeset
   215
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   216
declare
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   217
  sets.Int[measurable (raw)]
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diff changeset
   218
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diff changeset
   219
lemma pred_in_If[measurable (raw)]:
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   220
  "(P \<Longrightarrow> pred M (\<lambda>x. x \<in> A x)) \<Longrightarrow> (\<not> P \<Longrightarrow> pred M (\<lambda>x. x \<in> B x)) \<Longrightarrow>
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   221
    pred M (\<lambda>x. x \<in> (if P then A x else B x))"
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diff changeset
   222
  by auto
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diff changeset
   223
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   224
lemma sets_range[measurable_dest]:
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   225
  "A ` I \<subseteq> sets M \<Longrightarrow> i \<in> I \<Longrightarrow> A i \<in> sets M"
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parents:
diff changeset
   226
  by auto
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diff changeset
   227
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diff changeset
   228
lemma pred_sets_range[measurable_dest]:
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   229
  "A ` I \<subseteq> sets N \<Longrightarrow> i \<in> I \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> pred M (\<lambda>x. f x \<in> A i)"
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   230
  using pred_sets2[OF sets_range] by auto
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diff changeset
   231
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   232
lemma sets_All[measurable_dest]:
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   233
  "\<forall>i. A i \<in> sets (M i) \<Longrightarrow> A i \<in> sets (M i)"
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diff changeset
   234
  by auto
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parents:
diff changeset
   235
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   236
lemma pred_sets_All[measurable_dest]:
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   237
  "\<forall>i. A i \<in> sets (N i) \<Longrightarrow> f \<in> measurable M (N i) \<Longrightarrow> pred M (\<lambda>x. f x \<in> A i)"
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parents:
diff changeset
   238
  using pred_sets2[OF sets_All, of A N f] by auto
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diff changeset
   239
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   240
lemma sets_Ball[measurable_dest]:
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   241
  "\<forall>i\<in>I. A i \<in> sets (M i) \<Longrightarrow> i\<in>I \<Longrightarrow> A i \<in> sets (M i)"
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   242
  by auto
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diff changeset
   243
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   244
lemma pred_sets_Ball[measurable_dest]:
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   245
  "\<forall>i\<in>I. A i \<in> sets (N i) \<Longrightarrow> i\<in>I \<Longrightarrow> f \<in> measurable M (N i) \<Longrightarrow> pred M (\<lambda>x. f x \<in> A i)"
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parents:
diff changeset
   246
  using pred_sets2[OF sets_Ball, of _ _ _ f] by auto
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diff changeset
   247
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   248
lemma measurable_finite[measurable (raw)]:
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   249
  fixes S :: "'a \<Rightarrow> nat set"
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   250
  assumes [measurable]: "\<And>i. {x\<in>space M. i \<in> S x} \<in> sets M"
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   251
  shows "pred M (\<lambda>x. finite (S x))"
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   252
  unfolding finite_nat_set_iff_bounded by (simp add: Ball_def)
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diff changeset
   253
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   254
lemma measurable_Least[measurable]:
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   255
  assumes [measurable]: "(\<And>i::nat. (\<lambda>x. P i x) \<in> measurable M (count_space UNIV))"q
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parents:
diff changeset
   256
  shows "(\<lambda>x. LEAST i. P i x) \<in> measurable M (count_space UNIV)"
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parents:
diff changeset
   257
  unfolding measurable_def by (safe intro!: sets_Least) simp_all
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diff changeset
   258
56993
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
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diff changeset
   259
lemma measurable_Max_nat[measurable (raw)]: 
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   260
  fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"
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diff changeset
   261
  assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   262
  shows "(\<lambda>x. Max {i. P i x}) \<in> measurable M (count_space UNIV)"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
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diff changeset
   263
  unfolding measurable_count_space_eq2_countable
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
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diff changeset
   264
proof safe
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
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diff changeset
   265
  fix n
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diff changeset
   266
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
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diff changeset
   267
  { fix x assume "\<forall>i. \<exists>n\<ge>i. P n x"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
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diff changeset
   268
    then have "infinite {i. P i x}"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   269
      unfolding infinite_nat_iff_unbounded_le by auto
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   270
    then have "Max {i. P i x} = the None"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   271
      by (rule Max.infinite) }
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   272
  note 1 = this
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   273
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   274
  { fix x i j assume "P i x" "\<forall>n\<ge>j. \<not> P n x"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   275
    then have "finite {i. P i x}"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   276
      by (auto simp: subset_eq not_le[symmetric] finite_nat_iff_bounded)
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   277
    with `P i x` have "P (Max {i. P i x}) x" "i \<le> Max {i. P i x}" "finite {i. P i x}"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   278
      using Max_in[of "{i. P i x}"] by auto }
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   279
  note 2 = this
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   280
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   281
  have "(\<lambda>x. Max {i. P i x}) -` {n} \<inter> space M = {x\<in>space M. Max {i. P i x} = n}"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   282
    by auto
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   283
  also have "\<dots> = 
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   284
    {x\<in>space M. if (\<forall>i. \<exists>n\<ge>i. P n x) then the None = n else 
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   285
      if (\<exists>i. P i x) then P n x \<and> (\<forall>i>n. \<not> P i x)
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   286
      else Max {} = n}"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   287
    by (intro arg_cong[where f=Collect] ext conj_cong)
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   288
       (auto simp add: 1 2 not_le[symmetric] intro!: Max_eqI)
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   289
  also have "\<dots> \<in> sets M"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   290
    by measurable
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   291
  finally show "(\<lambda>x. Max {i. P i x}) -` {n} \<inter> space M \<in> sets M" .
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   292
qed simp
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   293
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   294
lemma measurable_Min_nat[measurable (raw)]: 
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   295
  fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   296
  assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   297
  shows "(\<lambda>x. Min {i. P i x}) \<in> measurable M (count_space UNIV)"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   298
  unfolding measurable_count_space_eq2_countable
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   299
proof safe
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   300
  fix n
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   301
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   302
  { fix x assume "\<forall>i. \<exists>n\<ge>i. P n x"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   303
    then have "infinite {i. P i x}"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   304
      unfolding infinite_nat_iff_unbounded_le by auto
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   305
    then have "Min {i. P i x} = the None"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   306
      by (rule Min.infinite) }
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   307
  note 1 = this
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   308
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   309
  { fix x i j assume "P i x" "\<forall>n\<ge>j. \<not> P n x"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   310
    then have "finite {i. P i x}"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   311
      by (auto simp: subset_eq not_le[symmetric] finite_nat_iff_bounded)
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   312
    with `P i x` have "P (Min {i. P i x}) x" "Min {i. P i x} \<le> i" "finite {i. P i x}"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   313
      using Min_in[of "{i. P i x}"] by auto }
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   314
  note 2 = this
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   315
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   316
  have "(\<lambda>x. Min {i. P i x}) -` {n} \<inter> space M = {x\<in>space M. Min {i. P i x} = n}"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   317
    by auto
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   318
  also have "\<dots> = 
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   319
    {x\<in>space M. if (\<forall>i. \<exists>n\<ge>i. P n x) then the None = n else 
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   320
      if (\<exists>i. P i x) then P n x \<and> (\<forall>i<n. \<not> P i x)
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   321
      else Min {} = n}"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   322
    by (intro arg_cong[where f=Collect] ext conj_cong)
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   323
       (auto simp add: 1 2 not_le[symmetric] intro!: Min_eqI)
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   324
  also have "\<dots> \<in> sets M"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   325
    by measurable
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   326
  finally show "(\<lambda>x. Min {i. P i x}) -` {n} \<inter> space M \<in> sets M" .
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   327
qed simp
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   328
50387
3d8863c41fe8 Move the measurability prover to its own file
hoelzl
parents:
diff changeset
   329
lemma measurable_count_space_insert[measurable (raw)]:
3d8863c41fe8 Move the measurability prover to its own file
hoelzl
parents:
diff changeset
   330
  "s \<in> S \<Longrightarrow> A \<in> sets (count_space S) \<Longrightarrow> insert s A \<in> sets (count_space S)"
3d8863c41fe8 Move the measurability prover to its own file
hoelzl
parents:
diff changeset
   331
  by simp
3d8863c41fe8 Move the measurability prover to its own file
hoelzl
parents:
diff changeset
   332
56021
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   333
subsection {* Measurability for (co)inductive predicates *}
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   334
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   335
lemma measurable_lfp:
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   336
  assumes "Order_Continuity.continuous F"
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   337
  assumes *: "\<And>A. pred M A \<Longrightarrow> pred M (F A)"
56045
1ca060139a59 measurable_lfp/gfp: indirection not necessary
hoelzl
parents: 56021
diff changeset
   338
  shows "pred M (lfp F)"
56021
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   339
proof -
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   340
  { fix i have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>x. False) x)"
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   341
      by (induct i) (auto intro!: *) }
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   342
  then have "Measurable.pred M (\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>x. False) x)"
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   343
    by measurable
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   344
  also have "(\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>x. False) x) = (SUP i. (F ^^ i) bot)"
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   345
    by (auto simp add: bot_fun_def)
56045
1ca060139a59 measurable_lfp/gfp: indirection not necessary
hoelzl
parents: 56021
diff changeset
   346
  also have "\<dots> = lfp F"
1ca060139a59 measurable_lfp/gfp: indirection not necessary
hoelzl
parents: 56021
diff changeset
   347
    by (rule continuous_lfp[symmetric]) fact
56021
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   348
  finally show ?thesis .
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   349
qed
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   350
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   351
lemma measurable_gfp:
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   352
  assumes "Order_Continuity.down_continuous F"
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   353
  assumes *: "\<And>A. pred M A \<Longrightarrow> pred M (F A)"
56045
1ca060139a59 measurable_lfp/gfp: indirection not necessary
hoelzl
parents: 56021
diff changeset
   354
  shows "pred M (gfp F)"
56021
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   355
proof -
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   356
  { fix i have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>x. True) x)"
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   357
      by (induct i) (auto intro!: *) }
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   358
  then have "Measurable.pred M (\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>x. True) x)"
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   359
    by measurable
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   360
  also have "(\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>x. True) x) = (INF i. (F ^^ i) top)"
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   361
    by (auto simp add: top_fun_def)
56045
1ca060139a59 measurable_lfp/gfp: indirection not necessary
hoelzl
parents: 56021
diff changeset
   362
  also have "\<dots> = gfp F"
1ca060139a59 measurable_lfp/gfp: indirection not necessary
hoelzl
parents: 56021
diff changeset
   363
    by (rule down_continuous_gfp[symmetric]) fact
56021
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   364
  finally show ?thesis .
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   365
qed
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   366
50387
3d8863c41fe8 Move the measurability prover to its own file
hoelzl
parents:
diff changeset
   367
hide_const (open) pred
3d8863c41fe8 Move the measurability prover to its own file
hoelzl
parents:
diff changeset
   368
3d8863c41fe8 Move the measurability prover to its own file
hoelzl
parents:
diff changeset
   369
end