src/HOL/Probability/Measurable.thy
author wenzelm
Thu, 18 Apr 2013 17:07:01 +0200
changeset 51717 9e7d1c139569
parent 50530 6266e44b3396
child 53043 8cbfbeb566a4
permissions -rw-r--r--
simplifier uses proper Proof.context instead of historic type simpset;
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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 Sigma_Algebra
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begin
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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 = {* Measurable.attr *} "declaration of measurability theorems"
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attribute_setup measurable_dest = {* Measurable.dest_attr *} "add dest rule for measurability prover"
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attribute_setup measurable_app = {* Measurable.app_attr *} "add application rule for measurability prover"
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method_setup measurable = {* Measurable.method *} "measurability prover"
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simproc_setup measurable ("A \<in> sets M" | "f \<in> measurable M N") = {* K Measurable.simproc *}
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declare
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  measurable_compose_rev[measurable_dest]
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  pred_sets1[measurable_dest]
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  pred_sets2[measurable_dest]
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  sets.sets_into_space[measurable_dest]
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declare
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  sets.top[measurable]
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  sets.empty_sets[measurable (raw)]
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  sets.Un[measurable (raw)]
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  sets.Diff[measurable (raw)]
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declare
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  measurable_count_space[measurable (raw)]
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  measurable_ident[measurable (raw)]
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  measurable_ident_sets[measurable (raw)]
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  measurable_const[measurable (raw)]
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  measurable_If[measurable (raw)]
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  measurable_comp[measurable (raw)]
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  measurable_sets[measurable (raw)]
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lemma predE[measurable (raw)]: 
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  "pred M P \<Longrightarrow> {x\<in>space M. P x} \<in> sets M"
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  unfolding pred_def .
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lemma pred_intros_imp'[measurable (raw)]:
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  "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<longrightarrow> P x)"
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  by (cases K) auto
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lemma pred_intros_conj1'[measurable (raw)]:
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  "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<and> P x)"
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  by (cases K) auto
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lemma pred_intros_conj2'[measurable (raw)]:
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  "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. P x \<and> K)"
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  by (cases K) auto
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lemma pred_intros_disj1'[measurable (raw)]:
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  "(\<not> K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<or> P x)"
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  by (cases K) auto
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lemma pred_intros_disj2'[measurable (raw)]:
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  "(\<not> K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. P x \<or> K)"
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  by (cases K) auto
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lemma pred_intros_logic[measurable (raw)]:
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  "pred M (\<lambda>x. x \<in> space M)"
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  "pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. \<not> P x)"
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  "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<and> P x)"
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  "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<longrightarrow> P x)"
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  "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<or> P x)"
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  "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x = P x)"
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  "pred M (\<lambda>x. f x \<in> UNIV)"
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  "pred M (\<lambda>x. f x \<in> {})"
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  "pred M (\<lambda>x. P' (f x) x) \<Longrightarrow> pred M (\<lambda>x. f x \<in> {y. P' y x})"
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  "pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> - (B x))"
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  "pred M (\<lambda>x. f x \<in> (A x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (A x) - (B x))"
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  "pred M (\<lambda>x. f x \<in> (A x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (A x) \<inter> (B x))"
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  "pred M (\<lambda>x. f x \<in> (A x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (A x) \<union> (B x))"
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  "pred M (\<lambda>x. g x (f x) \<in> (X x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (g x) -` (X x))"
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  by (auto simp: iff_conv_conj_imp pred_def)
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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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   136
  "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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   137
  "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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   138
  by (auto intro!: sets.sets_Collect_finite_Ex sets.sets_Collect_finite_All simp: iff_conv_conj_imp pred_def)
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   139
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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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   143
  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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   148
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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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   158
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   159
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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   164
  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_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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   181
  by (auto simp: Int_def conj_commute eq_commute pred_def)
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   182
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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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   185
  using measurable_sets[OF f c]
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   186
  by (auto simp: Int_def conj_commute eq_commute pred_def)
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   187
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   188
lemma pred_less_const[measurable (raw generic)]:
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   189
  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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   190
  using measurable_sets[OF f c]
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   191
  by (auto simp: Int_def conj_commute eq_commute pred_def)
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   192
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   193
lemma pred_const_less[measurable (raw generic)]:
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   194
  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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   195
  using measurable_sets[OF f c]
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   196
  by (auto simp: Int_def conj_commute eq_commute pred_def)
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   197
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declare
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  sets.Int[measurable (raw)]
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   200
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   201
lemma pred_in_If[measurable (raw)]:
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   202
  "(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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   203
    pred M (\<lambda>x. x \<in> (if P then A x else B x))"
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   204
  by auto
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   205
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   206
lemma sets_range[measurable_dest]:
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   207
  "A ` I \<subseteq> sets M \<Longrightarrow> i \<in> I \<Longrightarrow> A i \<in> sets M"
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   208
  by auto
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   209
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   210
lemma pred_sets_range[measurable_dest]:
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   211
  "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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   212
  using pred_sets2[OF sets_range] by auto
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   213
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   214
lemma sets_All[measurable_dest]:
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   215
  "\<forall>i. A i \<in> sets (M i) \<Longrightarrow> A i \<in> sets (M i)"
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   216
  by auto
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   217
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   218
lemma pred_sets_All[measurable_dest]:
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   219
  "\<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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   220
  using pred_sets2[OF sets_All, of A N f] by auto
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   221
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   222
lemma sets_Ball[measurable_dest]:
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   223
  "\<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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   224
  by auto
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   225
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   226
lemma pred_sets_Ball[measurable_dest]:
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   227
  "\<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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   228
  using pred_sets2[OF sets_Ball, of _ _ _ f] by auto
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   229
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   230
lemma measurable_finite[measurable (raw)]:
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   231
  fixes S :: "'a \<Rightarrow> nat set"
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   232
  assumes [measurable]: "\<And>i. {x\<in>space M. i \<in> S x} \<in> sets M"
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   233
  shows "pred M (\<lambda>x. finite (S x))"
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   234
  unfolding finite_nat_set_iff_bounded by (simp add: Ball_def)
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   235
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   236
lemma measurable_Least[measurable]:
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   237
  assumes [measurable]: "(\<And>i::nat. (\<lambda>x. P i x) \<in> measurable M (count_space UNIV))"q
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   238
  shows "(\<lambda>x. LEAST i. P i x) \<in> measurable M (count_space UNIV)"
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parents:
diff changeset
   239
  unfolding measurable_def by (safe intro!: sets_Least) simp_all
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diff changeset
   240
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   241
lemma measurable_count_space_insert[measurable (raw)]:
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   242
  "s \<in> S \<Longrightarrow> A \<in> sets (count_space S) \<Longrightarrow> insert s A \<in> sets (count_space S)"
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   243
  by simp
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   244
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   245
hide_const (open) pred
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   246
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   247
end