src/HOL/Probability/Measurable.thy
author hoelzl
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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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   137
  "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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   138
  by (auto simp: iff_conv_conj_imp pred_def)
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diff changeset
   139
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   140
lemma pred_intros_countable[measurable (raw)]:
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   141
  fixes P :: "'a \<Rightarrow> 'i :: countable \<Rightarrow> bool"
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   142
  shows 
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   143
    "(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<forall>i. P x i)"
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   144
    "(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<exists>i. P x i)"
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diff changeset
   145
  by (auto intro!: sets.sets_Collect_countable_All sets.sets_Collect_countable_Ex simp: pred_def)
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diff changeset
   146
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   147
lemma pred_intros_countable_bounded[measurable (raw)]:
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   148
  fixes X :: "'i :: countable set"
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   149
  shows 
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   150
    "(\<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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diff changeset
   151
    "(\<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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   152
    "(\<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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diff changeset
   153
    "(\<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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   154
  by (auto simp: Bex_def Ball_def)
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   155
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   156
lemma pred_intros_finite[measurable (raw)]:
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   157
  "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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   158
  "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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   159
  "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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   160
  "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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   161
  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
   162
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   163
lemma countable_Un_Int[measurable (raw)]:
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   164
  "(\<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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   165
  "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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   166
  by auto
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   167
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   168
declare
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   169
  finite_UN[measurable (raw)]
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   170
  finite_INT[measurable (raw)]
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   171
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   172
lemma sets_Int_pred[measurable (raw)]:
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   173
  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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   174
  shows "A \<inter> B \<in> sets M"
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   175
proof -
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   176
  have "{x\<in>space M. x \<in> A \<inter> B} \<in> sets M" by auto
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   177
  also have "{x\<in>space M. x \<in> A \<inter> B} = A \<inter> B"
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   178
    using space by auto
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   179
  finally show ?thesis .
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   180
qed
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   181
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   182
lemma [measurable (raw generic)]:
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   183
  assumes f: "f \<in> measurable M N" and c: "c \<in> space N \<Longrightarrow> {c} \<in> sets N"
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   184
  shows pred_eq_const1: "pred M (\<lambda>x. f x = c)"
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   185
    and pred_eq_const2: "pred M (\<lambda>x. c = f x)"
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   186
proof -
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   187
  show "pred M (\<lambda>x. f x = c)"
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   188
  proof cases
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   189
    assume "c \<in> space N"
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   190
    with measurable_sets[OF f c] show ?thesis
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   191
      by (auto simp: Int_def conj_commute pred_def)
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   192
  next
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   193
    assume "c \<notin> space N"
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   194
    with f[THEN measurable_space] have "{x \<in> space M. f x = c} = {}" by auto
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   195
    then show ?thesis by (auto simp: pred_def cong: conj_cong)
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   196
  qed
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   197
  then show "pred M (\<lambda>x. c = f x)"
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   198
    by (simp add: eq_commute)
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   199
qed
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   200
59000
6eb0725503fc import general theorems from AFP/Markov_Models
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   201
lemma pred_count_space_const1[measurable (raw)]:
6eb0725503fc import general theorems from AFP/Markov_Models
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   202
  "f \<in> measurable M (count_space UNIV) \<Longrightarrow> Measurable.pred M (\<lambda>x. f x = c)"
6eb0725503fc import general theorems from AFP/Markov_Models
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parents: 58965
diff changeset
   203
  by (intro pred_eq_const1[where N="count_space UNIV"]) (auto )
6eb0725503fc import general theorems from AFP/Markov_Models
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diff changeset
   204
6eb0725503fc import general theorems from AFP/Markov_Models
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diff changeset
   205
lemma pred_count_space_const2[measurable (raw)]:
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diff changeset
   206
  "f \<in> measurable M (count_space UNIV) \<Longrightarrow> Measurable.pred M (\<lambda>x. c = f x)"
6eb0725503fc import general theorems from AFP/Markov_Models
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parents: 58965
diff changeset
   207
  by (intro pred_eq_const2[where N="count_space UNIV"]) (auto )
6eb0725503fc import general theorems from AFP/Markov_Models
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diff changeset
   208
50387
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   209
lemma pred_le_const[measurable (raw generic)]:
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   210
  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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diff changeset
   211
  using measurable_sets[OF f c]
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   212
  by (auto simp: Int_def conj_commute eq_commute pred_def)
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diff changeset
   213
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   214
lemma pred_const_le[measurable (raw generic)]:
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   215
  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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   216
  using measurable_sets[OF f c]
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diff changeset
   217
  by (auto simp: Int_def conj_commute eq_commute pred_def)
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diff changeset
   218
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   219
lemma pred_less_const[measurable (raw generic)]:
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   220
  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
   221
  using measurable_sets[OF f c]
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diff changeset
   222
  by (auto simp: Int_def conj_commute eq_commute pred_def)
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parents:
diff changeset
   223
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   224
lemma pred_const_less[measurable (raw generic)]:
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   225
  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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parents:
diff changeset
   226
  using measurable_sets[OF f c]
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diff changeset
   227
  by (auto simp: Int_def conj_commute eq_commute pred_def)
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   228
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   229
declare
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   230
  sets.Int[measurable (raw)]
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diff changeset
   231
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   232
lemma pred_in_If[measurable (raw)]:
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   233
  "(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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   234
    pred M (\<lambda>x. x \<in> (if P then A x else B x))"
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parents:
diff changeset
   235
  by auto
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diff changeset
   236
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   237
lemma sets_range[measurable_dest]:
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   238
  "A ` I \<subseteq> sets M \<Longrightarrow> i \<in> I \<Longrightarrow> A i \<in> sets M"
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parents:
diff changeset
   239
  by auto
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parents:
diff changeset
   240
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   241
lemma pred_sets_range[measurable_dest]:
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   242
  "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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parents:
diff changeset
   243
  using pred_sets2[OF sets_range] by auto
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diff changeset
   244
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   245
lemma sets_All[measurable_dest]:
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   246
  "\<forall>i. A i \<in> sets (M i) \<Longrightarrow> A i \<in> sets (M i)"
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parents:
diff changeset
   247
  by auto
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diff changeset
   248
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   249
lemma pred_sets_All[measurable_dest]:
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   250
  "\<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
   251
  using pred_sets2[OF sets_All, of A N f] by auto
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diff changeset
   252
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   253
lemma sets_Ball[measurable_dest]:
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   254
  "\<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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parents:
diff changeset
   255
  by auto
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parents:
diff changeset
   256
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   257
lemma pred_sets_Ball[measurable_dest]:
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   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)"
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parents:
diff changeset
   259
  using pred_sets2[OF sets_Ball, of _ _ _ f] by auto
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diff changeset
   260
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diff changeset
   261
lemma measurable_finite[measurable (raw)]:
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   262
  fixes S :: "'a \<Rightarrow> nat set"
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diff changeset
   263
  assumes [measurable]: "\<And>i. {x\<in>space M. i \<in> S x} \<in> sets M"
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parents:
diff changeset
   264
  shows "pred M (\<lambda>x. finite (S x))"
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parents:
diff changeset
   265
  unfolding finite_nat_set_iff_bounded by (simp add: Ball_def)
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parents:
diff changeset
   266
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   267
lemma measurable_Least[measurable]:
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diff changeset
   268
  assumes [measurable]: "(\<And>i::nat. (\<lambda>x. P i x) \<in> measurable M (count_space UNIV))"q
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parents:
diff changeset
   269
  shows "(\<lambda>x. LEAST i. P i x) \<in> measurable M (count_space UNIV)"
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diff changeset
   270
  unfolding measurable_def by (safe intro!: sets_Least) simp_all
3d8863c41fe8 Move the measurability prover to its own file
hoelzl
parents:
diff changeset
   271
56993
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   272
lemma measurable_Max_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
   273
  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
   274
  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
   275
  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
parents: 56045
diff changeset
   276
  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
   277
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
   278
  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
   279
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   280
  { 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
   281
    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
   282
      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
   283
    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
   284
      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
   285
  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
   286
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   287
  { 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
   288
    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
   289
      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
   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}"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   291
      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
   292
  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
   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
  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
   295
    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
   296
  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
   297
    {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
   298
      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
   299
      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
   300
    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
   301
       (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
   302
  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
   303
    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
   304
  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
   305
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
   306
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   307
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
   308
  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
   309
  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
   310
  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
   311
  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
   312
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
   313
  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
   314
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   315
  { 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
   316
    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
   317
      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
   318
    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
   319
      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
   320
  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
   321
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   322
  { 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
   323
    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
   324
      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
   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}"
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   326
      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
   327
  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
   328
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56045
diff changeset
   329
  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
   330
    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
   331
  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
   332
    {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
   333
      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
   334
      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
   335
    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
   336
       (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
   337
  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
   338
    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
   339
  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
   340
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
   341
50387
3d8863c41fe8 Move the measurability prover to its own file
hoelzl
parents:
diff changeset
   342
lemma measurable_count_space_insert[measurable (raw)]:
3d8863c41fe8 Move the measurability prover to its own file
hoelzl
parents:
diff changeset
   343
  "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
   344
  by simp
3d8863c41fe8 Move the measurability prover to its own file
hoelzl
parents:
diff changeset
   345
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   346
lemma sets_UNIV [measurable (raw)]: "A \<in> sets (count_space UNIV)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   347
  by simp
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   348
57025
e7fd64f82876 add various lemmas
hoelzl
parents: 56993
diff changeset
   349
lemma measurable_card[measurable]:
e7fd64f82876 add various lemmas
hoelzl
parents: 56993
diff changeset
   350
  fixes S :: "'a \<Rightarrow> nat set"
e7fd64f82876 add various lemmas
hoelzl
parents: 56993
diff changeset
   351
  assumes [measurable]: "\<And>i. {x\<in>space M. i \<in> S x} \<in> sets M"
e7fd64f82876 add various lemmas
hoelzl
parents: 56993
diff changeset
   352
  shows "(\<lambda>x. card (S x)) \<in> measurable M (count_space UNIV)"
e7fd64f82876 add various lemmas
hoelzl
parents: 56993
diff changeset
   353
  unfolding measurable_count_space_eq2_countable
e7fd64f82876 add various lemmas
hoelzl
parents: 56993
diff changeset
   354
proof safe
e7fd64f82876 add various lemmas
hoelzl
parents: 56993
diff changeset
   355
  fix n show "(\<lambda>x. card (S x)) -` {n} \<inter> space M \<in> sets M"
e7fd64f82876 add various lemmas
hoelzl
parents: 56993
diff changeset
   356
  proof (cases n)
e7fd64f82876 add various lemmas
hoelzl
parents: 56993
diff changeset
   357
    case 0
e7fd64f82876 add various lemmas
hoelzl
parents: 56993
diff changeset
   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)}"
e7fd64f82876 add various lemmas
hoelzl
parents: 56993
diff changeset
   359
      by auto
e7fd64f82876 add various lemmas
hoelzl
parents: 56993
diff changeset
   360
    also have "\<dots> \<in> sets M"
e7fd64f82876 add various lemmas
hoelzl
parents: 56993
diff changeset
   361
      by measurable
e7fd64f82876 add various lemmas
hoelzl
parents: 56993
diff changeset
   362
    finally show ?thesis .
e7fd64f82876 add various lemmas
hoelzl
parents: 56993
diff changeset
   363
  next
e7fd64f82876 add various lemmas
hoelzl
parents: 56993
diff changeset
   364
    case (Suc i)
e7fd64f82876 add various lemmas
hoelzl
parents: 56993
diff changeset
   365
    then have "(\<lambda>x. card (S x)) -` {n} \<inter> space M =
e7fd64f82876 add various lemmas
hoelzl
parents: 56993
diff changeset
   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)})"
e7fd64f82876 add various lemmas
hoelzl
parents: 56993
diff changeset
   367
      unfolding set_eq_iff[symmetric] Collect_bex_eq[symmetric] by (auto intro: card_ge_0_finite)
e7fd64f82876 add various lemmas
hoelzl
parents: 56993
diff changeset
   368
    also have "\<dots> \<in> sets M"
e7fd64f82876 add various lemmas
hoelzl
parents: 56993
diff changeset
   369
      by (intro sets.countable_UN' countable_Collect countable_Collect_finite) auto
e7fd64f82876 add various lemmas
hoelzl
parents: 56993
diff changeset
   370
    finally show ?thesis .
e7fd64f82876 add various lemmas
hoelzl
parents: 56993
diff changeset
   371
  qed
e7fd64f82876 add various lemmas
hoelzl
parents: 56993
diff changeset
   372
qed rule
e7fd64f82876 add various lemmas
hoelzl
parents: 56993
diff changeset
   373
56021
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   374
subsection {* Measurability for (co)inductive predicates *}
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   375
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   376
lemma measurable_lfp:
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   377
  assumes "Order_Continuity.continuous F"
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   378
  assumes *: "\<And>A. pred M A \<Longrightarrow> pred M (F A)"
56045
1ca060139a59 measurable_lfp/gfp: indirection not necessary
hoelzl
parents: 56021
diff changeset
   379
  shows "pred M (lfp F)"
56021
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   380
proof -
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   381
  { 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
   382
      by (induct i) (auto intro!: *) }
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   383
  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
   384
    by measurable
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   385
  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
   386
    by (auto simp add: bot_fun_def)
56045
1ca060139a59 measurable_lfp/gfp: indirection not necessary
hoelzl
parents: 56021
diff changeset
   387
  also have "\<dots> = lfp F"
1ca060139a59 measurable_lfp/gfp: indirection not necessary
hoelzl
parents: 56021
diff changeset
   388
    by (rule continuous_lfp[symmetric]) fact
56021
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   389
  finally show ?thesis .
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   390
qed
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   391
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   392
lemma measurable_gfp:
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   393
  assumes "Order_Continuity.down_continuous F"
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   394
  assumes *: "\<And>A. pred M A \<Longrightarrow> pred M (F A)"
56045
1ca060139a59 measurable_lfp/gfp: indirection not necessary
hoelzl
parents: 56021
diff changeset
   395
  shows "pred M (gfp F)"
56021
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   396
proof -
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   397
  { 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
   398
      by (induct i) (auto intro!: *) }
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   399
  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
   400
    by measurable
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   401
  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
   402
    by (auto simp add: top_fun_def)
56045
1ca060139a59 measurable_lfp/gfp: indirection not necessary
hoelzl
parents: 56021
diff changeset
   403
  also have "\<dots> = gfp F"
1ca060139a59 measurable_lfp/gfp: indirection not necessary
hoelzl
parents: 56021
diff changeset
   404
    by (rule down_continuous_gfp[symmetric]) fact
56021
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   405
  finally show ?thesis .
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   406
qed
e0c9d76c2a6d add measurability rule for (co)inductive predicates
hoelzl
parents: 53043
diff changeset
   407
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   408
lemma measurable_lfp_coinduct[consumes 1, case_names continuity step]:
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   409
  assumes "P M"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   410
  assumes "Order_Continuity.continuous F"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   411
  assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> Measurable.pred N A) \<Longrightarrow> Measurable.pred M (F A)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   412
  shows "Measurable.pred M (lfp F)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   413
proof -
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   414
  { fix i from `P M` have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>x. False) x)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   415
      by (induct i arbitrary: M) (auto intro!: *) }
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   416
  then have "Measurable.pred M (\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>x. False) x)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   417
    by measurable
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   418
  also have "(\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>x. False) x) = (SUP i. (F ^^ i) bot)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   419
    by (auto simp add: bot_fun_def)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   420
  also have "\<dots> = lfp F"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   421
    by (rule continuous_lfp[symmetric]) fact
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   422
  finally show ?thesis .
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   423
qed
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   424
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   425
lemma measurable_gfp_coinduct[consumes 1, case_names continuity step]:
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   426
  assumes "P M"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   427
  assumes "Order_Continuity.down_continuous F"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   428
  assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> Measurable.pred N A) \<Longrightarrow> Measurable.pred M (F A)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   429
  shows "Measurable.pred M (gfp F)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   430
proof -
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   431
  { fix i from `P M` have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>x. True) x)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   432
      by (induct i arbitrary: M) (auto intro!: *) }
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   433
  then have "Measurable.pred M (\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>x. True) x)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   434
    by measurable
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   435
  also have "(\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>x. True) x) = (INF i. (F ^^ i) top)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   436
    by (auto simp add: top_fun_def)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   437
  also have "\<dots> = gfp F"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   438
    by (rule down_continuous_gfp[symmetric]) fact
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   439
  finally show ?thesis .
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   440
qed
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   441
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   442
lemma measurable_lfp2_coinduct[consumes 1, case_names continuity step]:
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   443
  assumes "P M s"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   444
  assumes "Order_Continuity.continuous F"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   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)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   446
  shows "Measurable.pred M (lfp F s)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   447
proof -
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   448
  { fix i from `P M s` have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>t x. False) s x)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   449
      by (induct i arbitrary: M s) (auto intro!: *) }
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   450
  then have "Measurable.pred M (\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>t x. False) s x)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   451
    by measurable
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   452
  also have "(\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>t x. False) s x) = (SUP i. (F ^^ i) bot) s"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   453
    by (auto simp add: bot_fun_def)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   454
  also have "(SUP i. (F ^^ i) bot) = lfp F"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   455
    by (rule continuous_lfp[symmetric]) fact
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   456
  finally show ?thesis .
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   457
qed
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   458
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   459
lemma measurable_gfp2_coinduct[consumes 1, case_names continuity step]:
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   460
  assumes "P M s"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   461
  assumes "Order_Continuity.down_continuous F"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   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)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   463
  shows "Measurable.pred M (gfp F s)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   464
proof -
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   465
  { fix i from `P M s` have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>t x. True) s x)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   466
      by (induct i arbitrary: M s) (auto intro!: *) }
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   467
  then have "Measurable.pred M (\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>t x. True) s x)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   468
    by measurable
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   469
  also have "(\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>t x. True) s x) = (INF i. (F ^^ i) top) s"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   470
    by (auto simp add: top_fun_def)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   471
  also have "(INF i. (F ^^ i) top) = gfp F"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   472
    by (rule down_continuous_gfp[symmetric]) fact
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   473
  finally show ?thesis .
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   474
qed
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   475
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   476
lemma measurable_enat_coinduct:
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   477
  fixes f :: "'a \<Rightarrow> enat"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   478
  assumes "R f"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   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> 
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   480
    Measurable.pred M P \<and>
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   481
    i \<in> measurable M M \<and>
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   482
    h \<in> measurable M (count_space UNIV)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   483
  shows "f \<in> measurable M (count_space UNIV)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   484
proof (simp add: measurable_count_space_eq2_countable, rule )
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   485
  fix a :: enat
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   486
  have "f -` {a} \<inter> space M = {x\<in>space M. f x = a}"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   487
    by auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   488
  { fix i :: nat
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   489
    from `R f` have "Measurable.pred M (\<lambda>x. f x = enat i)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   490
    proof (induction i arbitrary: f)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   491
      case 0
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   492
      from *[OF this] obtain g h i P
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   493
        where f: "f = (\<lambda>x. if P x then h x else eSuc (g (i x)))" and
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   494
          [measurable]: "Measurable.pred M P" "i \<in> measurable M M" "h \<in> measurable M (count_space UNIV)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   495
        by auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   496
      have "Measurable.pred M (\<lambda>x. P x \<and> h x = 0)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   497
        by measurable
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   498
      also have "(\<lambda>x. P x \<and> h x = 0) = (\<lambda>x. f x = enat 0)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   499
        by (auto simp: f zero_enat_def[symmetric])
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   500
      finally show ?case .
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   501
    next
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   502
      case (Suc n)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   503
      from *[OF Suc.prems] obtain g h i P
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   504
        where f: "f = (\<lambda>x. if P x then h x else eSuc (g (i x)))" and "R g" and
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   505
          M[measurable]: "Measurable.pred M P" "i \<in> measurable M M" "h \<in> measurable M (count_space UNIV)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   506
        by auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   507
      have "(\<lambda>x. f x = enat (Suc n)) =
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   508
        (\<lambda>x. (P x \<longrightarrow> h x = enat (Suc n)) \<and> (\<not> P x \<longrightarrow> g (i x) = enat n))"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   509
        by (auto simp: f zero_enat_def[symmetric] eSuc_enat[symmetric])
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   510
      also have "Measurable.pred M \<dots>"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   511
        by (intro pred_intros_logic measurable_compose[OF M(2)] Suc `R g`) measurable
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   512
      finally show ?case .
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   513
    qed
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   514
    then have "f -` {enat i} \<inter> space M \<in> sets M"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   515
      by (simp add: pred_def Int_def conj_commute) }
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   516
  note fin = this
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   517
  show "f -` {a} \<inter> space M \<in> sets M"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   518
  proof (cases a)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   519
    case infinity
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   520
    then have "f -` {a} \<inter> space M = space M - (\<Union>n. f -` {enat n} \<inter> space M)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   521
      by auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   522
    also have "\<dots> \<in> sets M"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   523
      by (intro sets.Diff sets.top sets.Un sets.countable_UN) (auto intro!: fin)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   524
    finally show ?thesis .
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   525
  qed (simp add: fin)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   526
qed
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   527
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   528
lemma measurable_pred_countable[measurable (raw)]:
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   529
  assumes "countable X"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   530
  shows 
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   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)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   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)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   533
  unfolding pred_def
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   534
  by (auto intro!: sets.sets_Collect_countable_All' sets.sets_Collect_countable_Ex' assms)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   535
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   536
lemma measurable_THE:
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   537
  fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   538
  assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   539
  assumes I[simp]: "countable I" "\<And>i x. x \<in> space M \<Longrightarrow> P i x \<Longrightarrow> i \<in> I"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   540
  assumes unique: "\<And>x i j. x \<in> space M \<Longrightarrow> P i x \<Longrightarrow> P j x \<Longrightarrow> i = j"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   541
  shows "(\<lambda>x. THE i. P i x) \<in> measurable M (count_space UNIV)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   542
  unfolding measurable_def
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   543
proof safe
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   544
  fix X
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   545
  def f \<equiv> "\<lambda>x. THE i. P i x" def undef \<equiv> "THE i::'a. False"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   546
  { fix i x assume "x \<in> space M" "P i x" then have "f x = i"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   547
      unfolding f_def using unique by auto }
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   548
  note f_eq = this
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   549
  { fix x assume "x \<in> space M" "\<forall>i\<in>I. \<not> P i x"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   550
    then have "\<And>i. \<not> P i x"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   551
      using I(2)[of x] by auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   552
    then have "f x = undef"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   553
      by (auto simp: undef_def f_def) }
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   554
  then have "f -` X \<inter> space M = (\<Union>i\<in>I \<inter> X. {x\<in>space M. P i x}) \<union>
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   555
     (if undef \<in> X then space M - (\<Union>i\<in>I. {x\<in>space M. P i x}) else {})"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   556
    by (auto dest: f_eq)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   557
  also have "\<dots> \<in> sets M"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   558
    by (auto intro!: sets.Diff sets.countable_UN')
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   559
  finally show "f -` X \<inter> space M \<in> sets M" .
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   560
qed simp
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   561
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   562
lemma measurable_bot[measurable]: "Measurable.pred M bot"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   563
  by (simp add: bot_fun_def)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   564
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   565
lemma measurable_top[measurable]: "Measurable.pred M top"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   566
  by (simp add: top_fun_def)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   567
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   568
lemma measurable_Ex1[measurable (raw)]:
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   569
  assumes [simp]: "countable I" and [measurable]: "\<And>i. i \<in> I \<Longrightarrow> Measurable.pred M (P i)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   570
  shows "Measurable.pred M (\<lambda>x. \<exists>!i\<in>I. P i x)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   571
  unfolding bex1_def by measurable
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   572
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   573
lemma measurable_split_if[measurable (raw)]:
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   574
  "(c \<Longrightarrow> Measurable.pred M f) \<Longrightarrow> (\<not> c \<Longrightarrow> Measurable.pred M g) \<Longrightarrow>
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   575
   Measurable.pred M (if c then f else g)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   576
  by simp
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   577
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   578
lemma pred_restrict_space:
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   579
  assumes "S \<in> sets M"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   580
  shows "Measurable.pred (restrict_space M S) P \<longleftrightarrow> Measurable.pred M (\<lambda>x. x \<in> S \<and> P x)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   581
  unfolding pred_def sets_Collect_restrict_space_iff[OF assms] ..
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   582
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   583
lemma measurable_predpow[measurable]:
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   584
  assumes "Measurable.pred M T"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   585
  assumes "\<And>Q. Measurable.pred M Q \<Longrightarrow> Measurable.pred M (R Q)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   586
  shows "Measurable.pred M ((R ^^ n) T)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   587
  by (induct n) (auto intro: assms)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58965
diff changeset
   588
50387
3d8863c41fe8 Move the measurability prover to its own file
hoelzl
parents:
diff changeset
   589
hide_const (open) pred
3d8863c41fe8 Move the measurability prover to its own file
hoelzl
parents:
diff changeset
   590
3d8863c41fe8 Move the measurability prover to its own file
hoelzl
parents:
diff changeset
   591
end