src/HOL/Probability/Measurable.thy
author hoelzl
Thu Nov 13 17:19:52 2014 +0100 (2014-11-13)
changeset 59000 6eb0725503fc
parent 58965 a62cdcc5344b
child 59047 8d7cec9b861d
permissions -rw-r--r--
import general theorems from AFP/Markov_Models
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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.$$$ "del" >> K false) true --
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    Scan.optional (Args.parens (Scan.optional (Args.$$$ "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 uncurry Measurable.add_del_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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setup {*
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  Global_Theory.add_thms_dynamic (@{binding measurable}, Measurable.get_all o Context.proof_of)
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*}
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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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lemma pred_intros_countable[measurable (raw)]:
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  fixes P :: "'a \<Rightarrow> 'i :: countable \<Rightarrow> bool"
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  shows 
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    "(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<forall>i. P x i)"
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    "(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<exists>i. P x i)"
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  by (auto intro!: sets.sets_Collect_countable_All sets.sets_Collect_countable_Ex simp: pred_def)
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lemma pred_intros_countable_bounded[measurable (raw)]:
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  fixes X :: "'i :: countable set"
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  shows 
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    "(\<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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    "(\<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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    "(\<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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    "(\<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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  by (auto simp: Bex_def Ball_def)
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lemma pred_intros_finite[measurable (raw)]:
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  "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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  "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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  "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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  "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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  by (auto intro!: sets.sets_Collect_finite_Ex sets.sets_Collect_finite_All simp: iff_conv_conj_imp pred_def)
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lemma countable_Un_Int[measurable (raw)]:
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  "(\<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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  "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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  by auto
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declare
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  finite_UN[measurable (raw)]
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  finite_INT[measurable (raw)]
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lemma sets_Int_pred[measurable (raw)]:
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  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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  shows "A \<inter> B \<in> sets M"
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proof -
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  have "{x\<in>space M. x \<in> A \<inter> B} \<in> sets M" by auto
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  also have "{x\<in>space M. x \<in> A \<inter> B} = A \<inter> B"
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    using space by auto
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  finally show ?thesis .
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qed
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lemma [measurable (raw generic)]:
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  assumes f: "f \<in> measurable M N" and c: "c \<in> space N \<Longrightarrow> {c} \<in> sets N"
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  shows pred_eq_const1: "pred M (\<lambda>x. f x = c)"
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    and pred_eq_const2: "pred M (\<lambda>x. c = f x)"
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proof -
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  show "pred M (\<lambda>x. f x = c)"
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  proof cases
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    assume "c \<in> space N"
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    with measurable_sets[OF f c] show ?thesis
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      by (auto simp: Int_def conj_commute pred_def)
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  next
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    assume "c \<notin> space N"
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    with f[THEN measurable_space] have "{x \<in> space M. f x = c} = {}" by auto
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    then show ?thesis by (auto simp: pred_def cong: conj_cong)
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  qed
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  then show "pred M (\<lambda>x. c = f x)"
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    by (simp add: eq_commute)
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qed
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lemma pred_count_space_const1[measurable (raw)]:
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  "f \<in> measurable M (count_space UNIV) \<Longrightarrow> Measurable.pred M (\<lambda>x. f x = c)"
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  by (intro pred_eq_const1[where N="count_space UNIV"]) (auto )
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lemma pred_count_space_const2[measurable (raw)]:
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  "f \<in> measurable M (count_space UNIV) \<Longrightarrow> Measurable.pred M (\<lambda>x. c = f x)"
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  by (intro pred_eq_const2[where N="count_space UNIV"]) (auto )
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lemma pred_le_const[measurable (raw generic)]:
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  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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  using measurable_sets[OF f c]
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  by (auto simp: Int_def conj_commute eq_commute pred_def)
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lemma pred_const_le[measurable (raw generic)]:
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  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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  using measurable_sets[OF f c]
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  by (auto simp: Int_def conj_commute eq_commute pred_def)
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lemma pred_less_const[measurable (raw generic)]:
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  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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  using measurable_sets[OF f c]
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  by (auto simp: Int_def conj_commute eq_commute pred_def)
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lemma pred_const_less[measurable (raw generic)]:
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  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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  using measurable_sets[OF f c]
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  by (auto simp: Int_def conj_commute eq_commute pred_def)
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declare
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  sets.Int[measurable (raw)]
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lemma pred_in_If[measurable (raw)]:
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  "(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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    pred M (\<lambda>x. x \<in> (if P then A x else B x))"
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  by auto
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lemma sets_range[measurable_dest]:
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  "A ` I \<subseteq> sets M \<Longrightarrow> i \<in> I \<Longrightarrow> A i \<in> sets M"
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  by auto
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lemma pred_sets_range[measurable_dest]:
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  "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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  using pred_sets2[OF sets_range] by auto
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lemma sets_All[measurable_dest]:
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  "\<forall>i. A i \<in> sets (M i) \<Longrightarrow> A i \<in> sets (M i)"
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  by auto
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lemma pred_sets_All[measurable_dest]:
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  "\<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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  using pred_sets2[OF sets_All, of A N f] by auto
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lemma sets_Ball[measurable_dest]:
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  "\<forall>i\<in>I. A i \<in> sets (M i) \<Longrightarrow> i\<in>I \<Longrightarrow> A i \<in> sets (M i)"
hoelzl@50387
   255
  by auto
hoelzl@50387
   256
hoelzl@50387
   257
lemma pred_sets_Ball[measurable_dest]:
hoelzl@50387
   258
  "\<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)"
hoelzl@50387
   259
  using pred_sets2[OF sets_Ball, of _ _ _ f] by auto
hoelzl@50387
   260
hoelzl@50387
   261
lemma measurable_finite[measurable (raw)]:
hoelzl@50387
   262
  fixes S :: "'a \<Rightarrow> nat set"
hoelzl@50387
   263
  assumes [measurable]: "\<And>i. {x\<in>space M. i \<in> S x} \<in> sets M"
hoelzl@50387
   264
  shows "pred M (\<lambda>x. finite (S x))"
hoelzl@50387
   265
  unfolding finite_nat_set_iff_bounded by (simp add: Ball_def)
hoelzl@50387
   266
hoelzl@50387
   267
lemma measurable_Least[measurable]:
hoelzl@50387
   268
  assumes [measurable]: "(\<And>i::nat. (\<lambda>x. P i x) \<in> measurable M (count_space UNIV))"q
hoelzl@50387
   269
  shows "(\<lambda>x. LEAST i. P i x) \<in> measurable M (count_space UNIV)"
hoelzl@50387
   270
  unfolding measurable_def by (safe intro!: sets_Least) simp_all
hoelzl@50387
   271
hoelzl@56993
   272
lemma measurable_Max_nat[measurable (raw)]: 
hoelzl@56993
   273
  fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"
hoelzl@56993
   274
  assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
hoelzl@56993
   275
  shows "(\<lambda>x. Max {i. P i x}) \<in> measurable M (count_space UNIV)"
hoelzl@56993
   276
  unfolding measurable_count_space_eq2_countable
hoelzl@56993
   277
proof safe
hoelzl@56993
   278
  fix n
hoelzl@56993
   279
hoelzl@56993
   280
  { fix x assume "\<forall>i. \<exists>n\<ge>i. P n x"
hoelzl@56993
   281
    then have "infinite {i. P i x}"
hoelzl@56993
   282
      unfolding infinite_nat_iff_unbounded_le by auto
hoelzl@56993
   283
    then have "Max {i. P i x} = the None"
hoelzl@56993
   284
      by (rule Max.infinite) }
hoelzl@56993
   285
  note 1 = this
hoelzl@56993
   286
hoelzl@56993
   287
  { fix x i j assume "P i x" "\<forall>n\<ge>j. \<not> P n x"
hoelzl@56993
   288
    then have "finite {i. P i x}"
hoelzl@56993
   289
      by (auto simp: subset_eq not_le[symmetric] finite_nat_iff_bounded)
hoelzl@56993
   290
    with `P i x` have "P (Max {i. P i x}) x" "i \<le> Max {i. P i x}" "finite {i. P i x}"
hoelzl@56993
   291
      using Max_in[of "{i. P i x}"] by auto }
hoelzl@56993
   292
  note 2 = this
hoelzl@56993
   293
hoelzl@56993
   294
  have "(\<lambda>x. Max {i. P i x}) -` {n} \<inter> space M = {x\<in>space M. Max {i. P i x} = n}"
hoelzl@56993
   295
    by auto
hoelzl@56993
   296
  also have "\<dots> = 
hoelzl@56993
   297
    {x\<in>space M. if (\<forall>i. \<exists>n\<ge>i. P n x) then the None = n else 
hoelzl@56993
   298
      if (\<exists>i. P i x) then P n x \<and> (\<forall>i>n. \<not> P i x)
hoelzl@56993
   299
      else Max {} = n}"
hoelzl@56993
   300
    by (intro arg_cong[where f=Collect] ext conj_cong)
hoelzl@56993
   301
       (auto simp add: 1 2 not_le[symmetric] intro!: Max_eqI)
hoelzl@56993
   302
  also have "\<dots> \<in> sets M"
hoelzl@56993
   303
    by measurable
hoelzl@56993
   304
  finally show "(\<lambda>x. Max {i. P i x}) -` {n} \<inter> space M \<in> sets M" .
hoelzl@56993
   305
qed simp
hoelzl@56993
   306
hoelzl@56993
   307
lemma measurable_Min_nat[measurable (raw)]: 
hoelzl@56993
   308
  fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"
hoelzl@56993
   309
  assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
hoelzl@56993
   310
  shows "(\<lambda>x. Min {i. P i x}) \<in> measurable M (count_space UNIV)"
hoelzl@56993
   311
  unfolding measurable_count_space_eq2_countable
hoelzl@56993
   312
proof safe
hoelzl@56993
   313
  fix n
hoelzl@56993
   314
hoelzl@56993
   315
  { fix x assume "\<forall>i. \<exists>n\<ge>i. P n x"
hoelzl@56993
   316
    then have "infinite {i. P i x}"
hoelzl@56993
   317
      unfolding infinite_nat_iff_unbounded_le by auto
hoelzl@56993
   318
    then have "Min {i. P i x} = the None"
hoelzl@56993
   319
      by (rule Min.infinite) }
hoelzl@56993
   320
  note 1 = this
hoelzl@56993
   321
hoelzl@56993
   322
  { fix x i j assume "P i x" "\<forall>n\<ge>j. \<not> P n x"
hoelzl@56993
   323
    then have "finite {i. P i x}"
hoelzl@56993
   324
      by (auto simp: subset_eq not_le[symmetric] finite_nat_iff_bounded)
hoelzl@56993
   325
    with `P i x` have "P (Min {i. P i x}) x" "Min {i. P i x} \<le> i" "finite {i. P i x}"
hoelzl@56993
   326
      using Min_in[of "{i. P i x}"] by auto }
hoelzl@56993
   327
  note 2 = this
hoelzl@56993
   328
hoelzl@56993
   329
  have "(\<lambda>x. Min {i. P i x}) -` {n} \<inter> space M = {x\<in>space M. Min {i. P i x} = n}"
hoelzl@56993
   330
    by auto
hoelzl@56993
   331
  also have "\<dots> = 
hoelzl@56993
   332
    {x\<in>space M. if (\<forall>i. \<exists>n\<ge>i. P n x) then the None = n else 
hoelzl@56993
   333
      if (\<exists>i. P i x) then P n x \<and> (\<forall>i<n. \<not> P i x)
hoelzl@56993
   334
      else Min {} = n}"
hoelzl@56993
   335
    by (intro arg_cong[where f=Collect] ext conj_cong)
hoelzl@56993
   336
       (auto simp add: 1 2 not_le[symmetric] intro!: Min_eqI)
hoelzl@56993
   337
  also have "\<dots> \<in> sets M"
hoelzl@56993
   338
    by measurable
hoelzl@56993
   339
  finally show "(\<lambda>x. Min {i. P i x}) -` {n} \<inter> space M \<in> sets M" .
hoelzl@56993
   340
qed simp
hoelzl@56993
   341
hoelzl@50387
   342
lemma measurable_count_space_insert[measurable (raw)]:
hoelzl@50387
   343
  "s \<in> S \<Longrightarrow> A \<in> sets (count_space S) \<Longrightarrow> insert s A \<in> sets (count_space S)"
hoelzl@50387
   344
  by simp
hoelzl@50387
   345
hoelzl@59000
   346
lemma sets_UNIV [measurable (raw)]: "A \<in> sets (count_space UNIV)"
hoelzl@59000
   347
  by simp
hoelzl@59000
   348
hoelzl@57025
   349
lemma measurable_card[measurable]:
hoelzl@57025
   350
  fixes S :: "'a \<Rightarrow> nat set"
hoelzl@57025
   351
  assumes [measurable]: "\<And>i. {x\<in>space M. i \<in> S x} \<in> sets M"
hoelzl@57025
   352
  shows "(\<lambda>x. card (S x)) \<in> measurable M (count_space UNIV)"
hoelzl@57025
   353
  unfolding measurable_count_space_eq2_countable
hoelzl@57025
   354
proof safe
hoelzl@57025
   355
  fix n show "(\<lambda>x. card (S x)) -` {n} \<inter> space M \<in> sets M"
hoelzl@57025
   356
  proof (cases n)
hoelzl@57025
   357
    case 0
hoelzl@57025
   358
    then have "(\<lambda>x. card (S x)) -` {n} \<inter> space M = {x\<in>space M. infinite (S x) \<or> (\<forall>i. i \<notin> S x)}"
hoelzl@57025
   359
      by auto
hoelzl@57025
   360
    also have "\<dots> \<in> sets M"
hoelzl@57025
   361
      by measurable
hoelzl@57025
   362
    finally show ?thesis .
hoelzl@57025
   363
  next
hoelzl@57025
   364
    case (Suc i)
hoelzl@57025
   365
    then have "(\<lambda>x. card (S x)) -` {n} \<inter> space M =
hoelzl@57025
   366
      (\<Union>F\<in>{A\<in>{A. finite A}. card A = n}. {x\<in>space M. (\<forall>i. i \<in> S x \<longleftrightarrow> i \<in> F)})"
hoelzl@57025
   367
      unfolding set_eq_iff[symmetric] Collect_bex_eq[symmetric] by (auto intro: card_ge_0_finite)
hoelzl@57025
   368
    also have "\<dots> \<in> sets M"
hoelzl@57025
   369
      by (intro sets.countable_UN' countable_Collect countable_Collect_finite) auto
hoelzl@57025
   370
    finally show ?thesis .
hoelzl@57025
   371
  qed
hoelzl@57025
   372
qed rule
hoelzl@57025
   373
hoelzl@56021
   374
subsection {* Measurability for (co)inductive predicates *}
hoelzl@56021
   375
hoelzl@56021
   376
lemma measurable_lfp:
hoelzl@56021
   377
  assumes "Order_Continuity.continuous F"
hoelzl@56021
   378
  assumes *: "\<And>A. pred M A \<Longrightarrow> pred M (F A)"
hoelzl@56045
   379
  shows "pred M (lfp F)"
hoelzl@56021
   380
proof -
hoelzl@56021
   381
  { fix i have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>x. False) x)"
hoelzl@56021
   382
      by (induct i) (auto intro!: *) }
hoelzl@56021
   383
  then have "Measurable.pred M (\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>x. False) x)"
hoelzl@56021
   384
    by measurable
hoelzl@56021
   385
  also have "(\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>x. False) x) = (SUP i. (F ^^ i) bot)"
hoelzl@56021
   386
    by (auto simp add: bot_fun_def)
hoelzl@56045
   387
  also have "\<dots> = lfp F"
hoelzl@56045
   388
    by (rule continuous_lfp[symmetric]) fact
hoelzl@56021
   389
  finally show ?thesis .
hoelzl@56021
   390
qed
hoelzl@56021
   391
hoelzl@56021
   392
lemma measurable_gfp:
hoelzl@56021
   393
  assumes "Order_Continuity.down_continuous F"
hoelzl@56021
   394
  assumes *: "\<And>A. pred M A \<Longrightarrow> pred M (F A)"
hoelzl@56045
   395
  shows "pred M (gfp F)"
hoelzl@56021
   396
proof -
hoelzl@56021
   397
  { fix i have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>x. True) x)"
hoelzl@56021
   398
      by (induct i) (auto intro!: *) }
hoelzl@56021
   399
  then have "Measurable.pred M (\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>x. True) x)"
hoelzl@56021
   400
    by measurable
hoelzl@56021
   401
  also have "(\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>x. True) x) = (INF i. (F ^^ i) top)"
hoelzl@56021
   402
    by (auto simp add: top_fun_def)
hoelzl@56045
   403
  also have "\<dots> = gfp F"
hoelzl@56045
   404
    by (rule down_continuous_gfp[symmetric]) fact
hoelzl@56021
   405
  finally show ?thesis .
hoelzl@56021
   406
qed
hoelzl@56021
   407
hoelzl@59000
   408
lemma measurable_lfp_coinduct[consumes 1, case_names continuity step]:
hoelzl@59000
   409
  assumes "P M"
hoelzl@59000
   410
  assumes "Order_Continuity.continuous F"
hoelzl@59000
   411
  assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> Measurable.pred N A) \<Longrightarrow> Measurable.pred M (F A)"
hoelzl@59000
   412
  shows "Measurable.pred M (lfp F)"
hoelzl@59000
   413
proof -
hoelzl@59000
   414
  { fix i from `P M` have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>x. False) x)"
hoelzl@59000
   415
      by (induct i arbitrary: M) (auto intro!: *) }
hoelzl@59000
   416
  then have "Measurable.pred M (\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>x. False) x)"
hoelzl@59000
   417
    by measurable
hoelzl@59000
   418
  also have "(\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>x. False) x) = (SUP i. (F ^^ i) bot)"
hoelzl@59000
   419
    by (auto simp add: bot_fun_def)
hoelzl@59000
   420
  also have "\<dots> = lfp F"
hoelzl@59000
   421
    by (rule continuous_lfp[symmetric]) fact
hoelzl@59000
   422
  finally show ?thesis .
hoelzl@59000
   423
qed
hoelzl@59000
   424
hoelzl@59000
   425
lemma measurable_gfp_coinduct[consumes 1, case_names continuity step]:
hoelzl@59000
   426
  assumes "P M"
hoelzl@59000
   427
  assumes "Order_Continuity.down_continuous F"
hoelzl@59000
   428
  assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> Measurable.pred N A) \<Longrightarrow> Measurable.pred M (F A)"
hoelzl@59000
   429
  shows "Measurable.pred M (gfp F)"
hoelzl@59000
   430
proof -
hoelzl@59000
   431
  { fix i from `P M` have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>x. True) x)"
hoelzl@59000
   432
      by (induct i arbitrary: M) (auto intro!: *) }
hoelzl@59000
   433
  then have "Measurable.pred M (\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>x. True) x)"
hoelzl@59000
   434
    by measurable
hoelzl@59000
   435
  also have "(\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>x. True) x) = (INF i. (F ^^ i) top)"
hoelzl@59000
   436
    by (auto simp add: top_fun_def)
hoelzl@59000
   437
  also have "\<dots> = gfp F"
hoelzl@59000
   438
    by (rule down_continuous_gfp[symmetric]) fact
hoelzl@59000
   439
  finally show ?thesis .
hoelzl@59000
   440
qed
hoelzl@59000
   441
hoelzl@59000
   442
lemma measurable_lfp2_coinduct[consumes 1, case_names continuity step]:
hoelzl@59000
   443
  assumes "P M s"
hoelzl@59000
   444
  assumes "Order_Continuity.continuous F"
hoelzl@59000
   445
  assumes *: "\<And>M A s. P M s \<Longrightarrow> (\<And>N t. P N t \<Longrightarrow> Measurable.pred N (A t)) \<Longrightarrow> Measurable.pred M (F A s)"
hoelzl@59000
   446
  shows "Measurable.pred M (lfp F s)"
hoelzl@59000
   447
proof -
hoelzl@59000
   448
  { fix i from `P M s` have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>t x. False) s x)"
hoelzl@59000
   449
      by (induct i arbitrary: M s) (auto intro!: *) }
hoelzl@59000
   450
  then have "Measurable.pred M (\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>t x. False) s x)"
hoelzl@59000
   451
    by measurable
hoelzl@59000
   452
  also have "(\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>t x. False) s x) = (SUP i. (F ^^ i) bot) s"
hoelzl@59000
   453
    by (auto simp add: bot_fun_def)
hoelzl@59000
   454
  also have "(SUP i. (F ^^ i) bot) = lfp F"
hoelzl@59000
   455
    by (rule continuous_lfp[symmetric]) fact
hoelzl@59000
   456
  finally show ?thesis .
hoelzl@59000
   457
qed
hoelzl@59000
   458
hoelzl@59000
   459
lemma measurable_gfp2_coinduct[consumes 1, case_names continuity step]:
hoelzl@59000
   460
  assumes "P M s"
hoelzl@59000
   461
  assumes "Order_Continuity.down_continuous F"
hoelzl@59000
   462
  assumes *: "\<And>M A s. P M s \<Longrightarrow> (\<And>N t. P N t \<Longrightarrow> Measurable.pred N (A t)) \<Longrightarrow> Measurable.pred M (F A s)"
hoelzl@59000
   463
  shows "Measurable.pred M (gfp F s)"
hoelzl@59000
   464
proof -
hoelzl@59000
   465
  { fix i from `P M s` have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>t x. True) s x)"
hoelzl@59000
   466
      by (induct i arbitrary: M s) (auto intro!: *) }
hoelzl@59000
   467
  then have "Measurable.pred M (\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>t x. True) s x)"
hoelzl@59000
   468
    by measurable
hoelzl@59000
   469
  also have "(\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>t x. True) s x) = (INF i. (F ^^ i) top) s"
hoelzl@59000
   470
    by (auto simp add: top_fun_def)
hoelzl@59000
   471
  also have "(INF i. (F ^^ i) top) = gfp F"
hoelzl@59000
   472
    by (rule down_continuous_gfp[symmetric]) fact
hoelzl@59000
   473
  finally show ?thesis .
hoelzl@59000
   474
qed
hoelzl@59000
   475
hoelzl@59000
   476
lemma measurable_enat_coinduct:
hoelzl@59000
   477
  fixes f :: "'a \<Rightarrow> enat"
hoelzl@59000
   478
  assumes "R f"
hoelzl@59000
   479
  assumes *: "\<And>f. R f \<Longrightarrow> \<exists>g h i P. R g \<and> f = (\<lambda>x. if P x then h x else eSuc (g (i x))) \<and> 
hoelzl@59000
   480
    Measurable.pred M P \<and>
hoelzl@59000
   481
    i \<in> measurable M M \<and>
hoelzl@59000
   482
    h \<in> measurable M (count_space UNIV)"
hoelzl@59000
   483
  shows "f \<in> measurable M (count_space UNIV)"
hoelzl@59000
   484
proof (simp add: measurable_count_space_eq2_countable, rule )
hoelzl@59000
   485
  fix a :: enat
hoelzl@59000
   486
  have "f -` {a} \<inter> space M = {x\<in>space M. f x = a}"
hoelzl@59000
   487
    by auto
hoelzl@59000
   488
  { fix i :: nat
hoelzl@59000
   489
    from `R f` have "Measurable.pred M (\<lambda>x. f x = enat i)"
hoelzl@59000
   490
    proof (induction i arbitrary: f)
hoelzl@59000
   491
      case 0
hoelzl@59000
   492
      from *[OF this] obtain g h i P
hoelzl@59000
   493
        where f: "f = (\<lambda>x. if P x then h x else eSuc (g (i x)))" and
hoelzl@59000
   494
          [measurable]: "Measurable.pred M P" "i \<in> measurable M M" "h \<in> measurable M (count_space UNIV)"
hoelzl@59000
   495
        by auto
hoelzl@59000
   496
      have "Measurable.pred M (\<lambda>x. P x \<and> h x = 0)"
hoelzl@59000
   497
        by measurable
hoelzl@59000
   498
      also have "(\<lambda>x. P x \<and> h x = 0) = (\<lambda>x. f x = enat 0)"
hoelzl@59000
   499
        by (auto simp: f zero_enat_def[symmetric])
hoelzl@59000
   500
      finally show ?case .
hoelzl@59000
   501
    next
hoelzl@59000
   502
      case (Suc n)
hoelzl@59000
   503
      from *[OF Suc.prems] obtain g h i P
hoelzl@59000
   504
        where f: "f = (\<lambda>x. if P x then h x else eSuc (g (i x)))" and "R g" and
hoelzl@59000
   505
          M[measurable]: "Measurable.pred M P" "i \<in> measurable M M" "h \<in> measurable M (count_space UNIV)"
hoelzl@59000
   506
        by auto
hoelzl@59000
   507
      have "(\<lambda>x. f x = enat (Suc n)) =
hoelzl@59000
   508
        (\<lambda>x. (P x \<longrightarrow> h x = enat (Suc n)) \<and> (\<not> P x \<longrightarrow> g (i x) = enat n))"
hoelzl@59000
   509
        by (auto simp: f zero_enat_def[symmetric] eSuc_enat[symmetric])
hoelzl@59000
   510
      also have "Measurable.pred M \<dots>"
hoelzl@59000
   511
        by (intro pred_intros_logic measurable_compose[OF M(2)] Suc `R g`) measurable
hoelzl@59000
   512
      finally show ?case .
hoelzl@59000
   513
    qed
hoelzl@59000
   514
    then have "f -` {enat i} \<inter> space M \<in> sets M"
hoelzl@59000
   515
      by (simp add: pred_def Int_def conj_commute) }
hoelzl@59000
   516
  note fin = this
hoelzl@59000
   517
  show "f -` {a} \<inter> space M \<in> sets M"
hoelzl@59000
   518
  proof (cases a)
hoelzl@59000
   519
    case infinity
hoelzl@59000
   520
    then have "f -` {a} \<inter> space M = space M - (\<Union>n. f -` {enat n} \<inter> space M)"
hoelzl@59000
   521
      by auto
hoelzl@59000
   522
    also have "\<dots> \<in> sets M"
hoelzl@59000
   523
      by (intro sets.Diff sets.top sets.Un sets.countable_UN) (auto intro!: fin)
hoelzl@59000
   524
    finally show ?thesis .
hoelzl@59000
   525
  qed (simp add: fin)
hoelzl@59000
   526
qed
hoelzl@59000
   527
hoelzl@59000
   528
lemma measurable_pred_countable[measurable (raw)]:
hoelzl@59000
   529
  assumes "countable X"
hoelzl@59000
   530
  shows 
hoelzl@59000
   531
    "(\<And>i. i \<in> X \<Longrightarrow> Measurable.pred M (\<lambda>x. P x i)) \<Longrightarrow> Measurable.pred M (\<lambda>x. \<forall>i\<in>X. P x i)"
hoelzl@59000
   532
    "(\<And>i. i \<in> X \<Longrightarrow> Measurable.pred M (\<lambda>x. P x i)) \<Longrightarrow> Measurable.pred M (\<lambda>x. \<exists>i\<in>X. P x i)"
hoelzl@59000
   533
  unfolding pred_def
hoelzl@59000
   534
  by (auto intro!: sets.sets_Collect_countable_All' sets.sets_Collect_countable_Ex' assms)
hoelzl@59000
   535
hoelzl@59000
   536
lemma measurable_THE:
hoelzl@59000
   537
  fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
hoelzl@59000
   538
  assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
hoelzl@59000
   539
  assumes I[simp]: "countable I" "\<And>i x. x \<in> space M \<Longrightarrow> P i x \<Longrightarrow> i \<in> I"
hoelzl@59000
   540
  assumes unique: "\<And>x i j. x \<in> space M \<Longrightarrow> P i x \<Longrightarrow> P j x \<Longrightarrow> i = j"
hoelzl@59000
   541
  shows "(\<lambda>x. THE i. P i x) \<in> measurable M (count_space UNIV)"
hoelzl@59000
   542
  unfolding measurable_def
hoelzl@59000
   543
proof safe
hoelzl@59000
   544
  fix X
hoelzl@59000
   545
  def f \<equiv> "\<lambda>x. THE i. P i x" def undef \<equiv> "THE i::'a. False"
hoelzl@59000
   546
  { fix i x assume "x \<in> space M" "P i x" then have "f x = i"
hoelzl@59000
   547
      unfolding f_def using unique by auto }
hoelzl@59000
   548
  note f_eq = this
hoelzl@59000
   549
  { fix x assume "x \<in> space M" "\<forall>i\<in>I. \<not> P i x"
hoelzl@59000
   550
    then have "\<And>i. \<not> P i x"
hoelzl@59000
   551
      using I(2)[of x] by auto
hoelzl@59000
   552
    then have "f x = undef"
hoelzl@59000
   553
      by (auto simp: undef_def f_def) }
hoelzl@59000
   554
  then have "f -` X \<inter> space M = (\<Union>i\<in>I \<inter> X. {x\<in>space M. P i x}) \<union>
hoelzl@59000
   555
     (if undef \<in> X then space M - (\<Union>i\<in>I. {x\<in>space M. P i x}) else {})"
hoelzl@59000
   556
    by (auto dest: f_eq)
hoelzl@59000
   557
  also have "\<dots> \<in> sets M"
hoelzl@59000
   558
    by (auto intro!: sets.Diff sets.countable_UN')
hoelzl@59000
   559
  finally show "f -` X \<inter> space M \<in> sets M" .
hoelzl@59000
   560
qed simp
hoelzl@59000
   561
hoelzl@59000
   562
lemma measurable_bot[measurable]: "Measurable.pred M bot"
hoelzl@59000
   563
  by (simp add: bot_fun_def)
hoelzl@59000
   564
hoelzl@59000
   565
lemma measurable_top[measurable]: "Measurable.pred M top"
hoelzl@59000
   566
  by (simp add: top_fun_def)
hoelzl@59000
   567
hoelzl@59000
   568
lemma measurable_Ex1[measurable (raw)]:
hoelzl@59000
   569
  assumes [simp]: "countable I" and [measurable]: "\<And>i. i \<in> I \<Longrightarrow> Measurable.pred M (P i)"
hoelzl@59000
   570
  shows "Measurable.pred M (\<lambda>x. \<exists>!i\<in>I. P i x)"
hoelzl@59000
   571
  unfolding bex1_def by measurable
hoelzl@59000
   572
hoelzl@59000
   573
lemma measurable_split_if[measurable (raw)]:
hoelzl@59000
   574
  "(c \<Longrightarrow> Measurable.pred M f) \<Longrightarrow> (\<not> c \<Longrightarrow> Measurable.pred M g) \<Longrightarrow>
hoelzl@59000
   575
   Measurable.pred M (if c then f else g)"
hoelzl@59000
   576
  by simp
hoelzl@59000
   577
hoelzl@59000
   578
lemma pred_restrict_space:
hoelzl@59000
   579
  assumes "S \<in> sets M"
hoelzl@59000
   580
  shows "Measurable.pred (restrict_space M S) P \<longleftrightarrow> Measurable.pred M (\<lambda>x. x \<in> S \<and> P x)"
hoelzl@59000
   581
  unfolding pred_def sets_Collect_restrict_space_iff[OF assms] ..
hoelzl@59000
   582
hoelzl@59000
   583
lemma measurable_predpow[measurable]:
hoelzl@59000
   584
  assumes "Measurable.pred M T"
hoelzl@59000
   585
  assumes "\<And>Q. Measurable.pred M Q \<Longrightarrow> Measurable.pred M (R Q)"
hoelzl@59000
   586
  shows "Measurable.pred M ((R ^^ n) T)"
hoelzl@59000
   587
  by (induct n) (auto intro: assms)
hoelzl@59000
   588
hoelzl@50387
   589
hide_const (open) pred
hoelzl@50387
   590
hoelzl@50387
   591
end