(* Title: HOL/Probability/Measurable.thy
Author: Johannes Hölzl <hoelzl@in.tum.de>
*)
theory Measurable
imports
Sigma_Algebra
"~~/src/HOL/Library/Order_Continuity"
begin
subsection \<open>Measurability prover\<close>
lemma (in algebra) sets_Collect_finite_All:
assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S"
shows "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} \<in> M"
proof -
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})"
by auto
with assms show ?thesis by (auto intro!: sets_Collect_finite_All')
qed
abbreviation "pred M P \<equiv> P \<in> measurable M (count_space (UNIV::bool set))"
lemma pred_def: "pred M P \<longleftrightarrow> {x\<in>space M. P x} \<in> sets M"
proof
assume "pred M P"
then have "P -` {True} \<inter> space M \<in> sets M"
by (auto simp: measurable_count_space_eq2)
also have "P -` {True} \<inter> space M = {x\<in>space M. P x}" by auto
finally show "{x\<in>space M. P x} \<in> sets M" .
next
assume P: "{x\<in>space M. P x} \<in> sets M"
moreover
{ fix X
have "X \<in> Pow (UNIV :: bool set)" by simp
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> {})}"
unfolding UNIV_bool Pow_insert Pow_empty by auto
then have "P -` X \<inter> space M \<in> sets M"
by (auto intro!: sets.sets_Collect_neg sets.sets_Collect_imp sets.sets_Collect_conj sets.sets_Collect_const P) }
then show "pred M P"
by (auto simp: measurable_def)
qed
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))"
by (rule measurable_compose[where f=f and N=M]) (auto simp: pred_def)
lemma pred_sets2: "A \<in> sets N \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> pred M (\<lambda>x. f x \<in> A)"
by (rule measurable_compose[where f=f and N=N]) (auto simp: pred_def Int_def[symmetric])
ML_file "measurable.ML"
attribute_setup measurable = \<open>
Scan.lift (
(Args.add >> K true || Args.del >> K false || Scan.succeed true) --
Scan.optional (Args.parens (
Scan.optional (Args.$$$ "raw" >> K true) false --
Scan.optional (Args.$$$ "generic" >> K Measurable.Generic) Measurable.Concrete))
(false, Measurable.Concrete) >>
Measurable.measurable_thm_attr)
\<close> "declaration of measurability theorems"
attribute_setup measurable_dest = Measurable.dest_thm_attr
"add dest rule to measurability prover"
attribute_setup measurable_cong = Measurable.cong_thm_attr
"add congurence rules to measurability prover"
method_setup measurable = \<open> Scan.lift (Scan.succeed (METHOD o Measurable.measurable_tac)) \<close>
"measurability prover"
simproc_setup measurable ("A \<in> sets M" | "f \<in> measurable M N") = \<open>K Measurable.simproc\<close>
setup \<open>
Global_Theory.add_thms_dynamic (@{binding measurable}, Measurable.get_all)
\<close>
declare
pred_sets1[measurable_dest]
pred_sets2[measurable_dest]
sets.sets_into_space[measurable_dest]
declare
sets.top[measurable]
sets.empty_sets[measurable (raw)]
sets.Un[measurable (raw)]
sets.Diff[measurable (raw)]
declare
measurable_count_space[measurable (raw)]
measurable_ident[measurable (raw)]
measurable_id[measurable (raw)]
measurable_const[measurable (raw)]
measurable_If[measurable (raw)]
measurable_comp[measurable (raw)]
measurable_sets[measurable (raw)]
declare measurable_cong_sets[measurable_cong]
declare sets_restrict_space_cong[measurable_cong]
declare sets_restrict_UNIV[measurable_cong]
lemma predE[measurable (raw)]:
"pred M P \<Longrightarrow> {x\<in>space M. P x} \<in> sets M"
unfolding pred_def .
lemma pred_intros_imp'[measurable (raw)]:
"(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<longrightarrow> P x)"
by (cases K) auto
lemma pred_intros_conj1'[measurable (raw)]:
"(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<and> P x)"
by (cases K) auto
lemma pred_intros_conj2'[measurable (raw)]:
"(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. P x \<and> K)"
by (cases K) auto
lemma pred_intros_disj1'[measurable (raw)]:
"(\<not> K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<or> P x)"
by (cases K) auto
lemma pred_intros_disj2'[measurable (raw)]:
"(\<not> K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. P x \<or> K)"
by (cases K) auto
lemma pred_intros_logic[measurable (raw)]:
"pred M (\<lambda>x. x \<in> space M)"
"pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. \<not> P x)"
"pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<and> P x)"
"pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<longrightarrow> P x)"
"pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<or> P x)"
"pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x = P x)"
"pred M (\<lambda>x. f x \<in> UNIV)"
"pred M (\<lambda>x. f x \<in> {})"
"pred M (\<lambda>x. P' (f x) x) \<Longrightarrow> pred M (\<lambda>x. f x \<in> {y. P' y x})"
"pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> - (B x))"
"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))"
"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))"
"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))"
"pred M (\<lambda>x. g x (f x) \<in> (X x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (g x) -` (X x))"
by (auto simp: iff_conv_conj_imp pred_def)
lemma pred_intros_countable[measurable (raw)]:
fixes P :: "'a \<Rightarrow> 'i :: countable \<Rightarrow> bool"
shows
"(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<forall>i. P x i)"
"(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<exists>i. P x i)"
by (auto intro!: sets.sets_Collect_countable_All sets.sets_Collect_countable_Ex simp: pred_def)
lemma pred_intros_countable_bounded[measurable (raw)]:
fixes X :: "'i :: countable set"
shows
"(\<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))"
"(\<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))"
"(\<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)"
"(\<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)"
by simp_all (auto simp: Bex_def Ball_def)
lemma pred_intros_finite[measurable (raw)]:
"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))"
"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))"
"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)"
"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)"
by (auto intro!: sets.sets_Collect_finite_Ex sets.sets_Collect_finite_All simp: iff_conv_conj_imp pred_def)
lemma countable_Un_Int[measurable (raw)]:
"(\<And>i :: 'i :: countable. i \<in> I \<Longrightarrow> N i \<in> sets M) \<Longrightarrow> (\<Union>i\<in>I. N i) \<in> sets M"
"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"
by auto
declare
finite_UN[measurable (raw)]
finite_INT[measurable (raw)]
lemma sets_Int_pred[measurable (raw)]:
assumes space: "A \<inter> B \<subseteq> space M" and [measurable]: "pred M (\<lambda>x. x \<in> A)" "pred M (\<lambda>x. x \<in> B)"
shows "A \<inter> B \<in> sets M"
proof -
have "{x\<in>space M. x \<in> A \<inter> B} \<in> sets M" by auto
also have "{x\<in>space M. x \<in> A \<inter> B} = A \<inter> B"
using space by auto
finally show ?thesis .
qed
lemma [measurable (raw generic)]:
assumes f: "f \<in> measurable M N" and c: "c \<in> space N \<Longrightarrow> {c} \<in> sets N"
shows pred_eq_const1: "pred M (\<lambda>x. f x = c)"
and pred_eq_const2: "pred M (\<lambda>x. c = f x)"
proof -
show "pred M (\<lambda>x. f x = c)"
proof cases
assume "c \<in> space N"
with measurable_sets[OF f c] show ?thesis
by (auto simp: Int_def conj_commute pred_def)
next
assume "c \<notin> space N"
with f[THEN measurable_space] have "{x \<in> space M. f x = c} = {}" by auto
then show ?thesis by (auto simp: pred_def cong: conj_cong)
qed
then show "pred M (\<lambda>x. c = f x)"
by (simp add: eq_commute)
qed
lemma pred_count_space_const1[measurable (raw)]:
"f \<in> measurable M (count_space UNIV) \<Longrightarrow> Measurable.pred M (\<lambda>x. f x = c)"
by (intro pred_eq_const1[where N="count_space UNIV"]) (auto )
lemma pred_count_space_const2[measurable (raw)]:
"f \<in> measurable M (count_space UNIV) \<Longrightarrow> Measurable.pred M (\<lambda>x. c = f x)"
by (intro pred_eq_const2[where N="count_space UNIV"]) (auto )
lemma pred_le_const[measurable (raw generic)]:
assumes f: "f \<in> measurable M N" and c: "{.. c} \<in> sets N" shows "pred M (\<lambda>x. f x \<le> c)"
using measurable_sets[OF f c]
by (auto simp: Int_def conj_commute eq_commute pred_def)
lemma pred_const_le[measurable (raw generic)]:
assumes f: "f \<in> measurable M N" and c: "{c ..} \<in> sets N" shows "pred M (\<lambda>x. c \<le> f x)"
using measurable_sets[OF f c]
by (auto simp: Int_def conj_commute eq_commute pred_def)
lemma pred_less_const[measurable (raw generic)]:
assumes f: "f \<in> measurable M N" and c: "{..< c} \<in> sets N" shows "pred M (\<lambda>x. f x < c)"
using measurable_sets[OF f c]
by (auto simp: Int_def conj_commute eq_commute pred_def)
lemma pred_const_less[measurable (raw generic)]:
assumes f: "f \<in> measurable M N" and c: "{c <..} \<in> sets N" shows "pred M (\<lambda>x. c < f x)"
using measurable_sets[OF f c]
by (auto simp: Int_def conj_commute eq_commute pred_def)
declare
sets.Int[measurable (raw)]
lemma pred_in_If[measurable (raw)]:
"(P \<Longrightarrow> pred M (\<lambda>x. x \<in> A x)) \<Longrightarrow> (\<not> P \<Longrightarrow> pred M (\<lambda>x. x \<in> B x)) \<Longrightarrow>
pred M (\<lambda>x. x \<in> (if P then A x else B x))"
by auto
lemma sets_range[measurable_dest]:
"A ` I \<subseteq> sets M \<Longrightarrow> i \<in> I \<Longrightarrow> A i \<in> sets M"
by auto
lemma pred_sets_range[measurable_dest]:
"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)"
using pred_sets2[OF sets_range] by auto
lemma sets_All[measurable_dest]:
"\<forall>i. A i \<in> sets (M i) \<Longrightarrow> A i \<in> sets (M i)"
by auto
lemma pred_sets_All[measurable_dest]:
"\<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)"
using pred_sets2[OF sets_All, of A N f] by auto
lemma sets_Ball[measurable_dest]:
"\<forall>i\<in>I. A i \<in> sets (M i) \<Longrightarrow> i\<in>I \<Longrightarrow> A i \<in> sets (M i)"
by auto
lemma pred_sets_Ball[measurable_dest]:
"\<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)"
using pred_sets2[OF sets_Ball, of _ _ _ f] by auto
lemma measurable_finite[measurable (raw)]:
fixes S :: "'a \<Rightarrow> nat set"
assumes [measurable]: "\<And>i. {x\<in>space M. i \<in> S x} \<in> sets M"
shows "pred M (\<lambda>x. finite (S x))"
unfolding finite_nat_set_iff_bounded by (simp add: Ball_def)
lemma measurable_Least[measurable]:
assumes [measurable]: "(\<And>i::nat. (\<lambda>x. P i x) \<in> measurable M (count_space UNIV))"q
shows "(\<lambda>x. LEAST i. P i x) \<in> measurable M (count_space UNIV)"
unfolding measurable_def by (safe intro!: sets_Least) simp_all
lemma measurable_Max_nat[measurable (raw)]:
fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"
assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
shows "(\<lambda>x. Max {i. P i x}) \<in> measurable M (count_space UNIV)"
unfolding measurable_count_space_eq2_countable
proof safe
fix n
{ fix x assume "\<forall>i. \<exists>n\<ge>i. P n x"
then have "infinite {i. P i x}"
unfolding infinite_nat_iff_unbounded_le by auto
then have "Max {i. P i x} = the None"
by (rule Max.infinite) }
note 1 = this
{ fix x i j assume "P i x" "\<forall>n\<ge>j. \<not> P n x"
then have "finite {i. P i x}"
by (auto simp: subset_eq not_le[symmetric] finite_nat_iff_bounded)
with \<open>P i x\<close> have "P (Max {i. P i x}) x" "i \<le> Max {i. P i x}" "finite {i. P i x}"
using Max_in[of "{i. P i x}"] by auto }
note 2 = this
have "(\<lambda>x. Max {i. P i x}) -` {n} \<inter> space M = {x\<in>space M. Max {i. P i x} = n}"
by auto
also have "\<dots> =
{x\<in>space M. if (\<forall>i. \<exists>n\<ge>i. P n x) then the None = n else
if (\<exists>i. P i x) then P n x \<and> (\<forall>i>n. \<not> P i x)
else Max {} = n}"
by (intro arg_cong[where f=Collect] ext conj_cong)
(auto simp add: 1 2 not_le[symmetric] intro!: Max_eqI)
also have "\<dots> \<in> sets M"
by measurable
finally show "(\<lambda>x. Max {i. P i x}) -` {n} \<inter> space M \<in> sets M" .
qed simp
lemma measurable_Min_nat[measurable (raw)]:
fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"
assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
shows "(\<lambda>x. Min {i. P i x}) \<in> measurable M (count_space UNIV)"
unfolding measurable_count_space_eq2_countable
proof safe
fix n
{ fix x assume "\<forall>i. \<exists>n\<ge>i. P n x"
then have "infinite {i. P i x}"
unfolding infinite_nat_iff_unbounded_le by auto
then have "Min {i. P i x} = the None"
by (rule Min.infinite) }
note 1 = this
{ fix x i j assume "P i x" "\<forall>n\<ge>j. \<not> P n x"
then have "finite {i. P i x}"
by (auto simp: subset_eq not_le[symmetric] finite_nat_iff_bounded)
with \<open>P i x\<close> have "P (Min {i. P i x}) x" "Min {i. P i x} \<le> i" "finite {i. P i x}"
using Min_in[of "{i. P i x}"] by auto }
note 2 = this
have "(\<lambda>x. Min {i. P i x}) -` {n} \<inter> space M = {x\<in>space M. Min {i. P i x} = n}"
by auto
also have "\<dots> =
{x\<in>space M. if (\<forall>i. \<exists>n\<ge>i. P n x) then the None = n else
if (\<exists>i. P i x) then P n x \<and> (\<forall>i<n. \<not> P i x)
else Min {} = n}"
by (intro arg_cong[where f=Collect] ext conj_cong)
(auto simp add: 1 2 not_le[symmetric] intro!: Min_eqI)
also have "\<dots> \<in> sets M"
by measurable
finally show "(\<lambda>x. Min {i. P i x}) -` {n} \<inter> space M \<in> sets M" .
qed simp
lemma measurable_count_space_insert[measurable (raw)]:
"s \<in> S \<Longrightarrow> A \<in> sets (count_space S) \<Longrightarrow> insert s A \<in> sets (count_space S)"
by simp
lemma sets_UNIV [measurable (raw)]: "A \<in> sets (count_space UNIV)"
by simp
lemma measurable_card[measurable]:
fixes S :: "'a \<Rightarrow> nat set"
assumes [measurable]: "\<And>i. {x\<in>space M. i \<in> S x} \<in> sets M"
shows "(\<lambda>x. card (S x)) \<in> measurable M (count_space UNIV)"
unfolding measurable_count_space_eq2_countable
proof safe
fix n show "(\<lambda>x. card (S x)) -` {n} \<inter> space M \<in> sets M"
proof (cases n)
case 0
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)}"
by auto
also have "\<dots> \<in> sets M"
by measurable
finally show ?thesis .
next
case (Suc i)
then have "(\<lambda>x. card (S x)) -` {n} \<inter> space M =
(\<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)})"
unfolding set_eq_iff[symmetric] Collect_bex_eq[symmetric] by (auto intro: card_ge_0_finite)
also have "\<dots> \<in> sets M"
by (intro sets.countable_UN' countable_Collect countable_Collect_finite) auto
finally show ?thesis .
qed
qed rule
lemma measurable_pred_countable[measurable (raw)]:
assumes "countable X"
shows
"(\<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)"
"(\<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)"
unfolding pred_def
by (auto intro!: sets.sets_Collect_countable_All' sets.sets_Collect_countable_Ex' assms)
subsection \<open>Measurability for (co)inductive predicates\<close>
lemma measurable_bot[measurable]: "bot \<in> measurable M (count_space UNIV)"
by (simp add: bot_fun_def)
lemma measurable_top[measurable]: "top \<in> measurable M (count_space UNIV)"
by (simp add: top_fun_def)
lemma measurable_SUP[measurable]:
fixes F :: "'i \<Rightarrow> 'a \<Rightarrow> 'b::{complete_lattice, countable}"
assumes [simp]: "countable I"
assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> measurable M (count_space UNIV)"
shows "(\<lambda>x. SUP i:I. F i x) \<in> measurable M (count_space UNIV)"
unfolding measurable_count_space_eq2_countable
proof (safe intro!: UNIV_I)
fix a
have "(\<lambda>x. SUP i:I. F i x) -` {a} \<inter> space M =
{x\<in>space M. (\<forall>i\<in>I. F i x \<le> a) \<and> (\<forall>b. (\<forall>i\<in>I. F i x \<le> b) \<longrightarrow> a \<le> b)}"
unfolding SUP_le_iff[symmetric] by auto
also have "\<dots> \<in> sets M"
by measurable
finally show "(\<lambda>x. SUP i:I. F i x) -` {a} \<inter> space M \<in> sets M" .
qed
lemma measurable_INF[measurable]:
fixes F :: "'i \<Rightarrow> 'a \<Rightarrow> 'b::{complete_lattice, countable}"
assumes [simp]: "countable I"
assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> measurable M (count_space UNIV)"
shows "(\<lambda>x. INF i:I. F i x) \<in> measurable M (count_space UNIV)"
unfolding measurable_count_space_eq2_countable
proof (safe intro!: UNIV_I)
fix a
have "(\<lambda>x. INF i:I. F i x) -` {a} \<inter> space M =
{x\<in>space M. (\<forall>i\<in>I. a \<le> F i x) \<and> (\<forall>b. (\<forall>i\<in>I. b \<le> F i x) \<longrightarrow> b \<le> a)}"
unfolding le_INF_iff[symmetric] by auto
also have "\<dots> \<in> sets M"
by measurable
finally show "(\<lambda>x. INF i:I. F i x) -` {a} \<inter> space M \<in> sets M" .
qed
lemma measurable_lfp_coinduct[consumes 1, case_names continuity step]:
fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_lattice, countable})"
assumes "P M"
assumes F: "sup_continuous F"
assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> A \<in> measurable N (count_space UNIV)) \<Longrightarrow> F A \<in> measurable M (count_space UNIV)"
shows "lfp F \<in> measurable M (count_space UNIV)"
proof -
{ fix i from \<open>P M\<close> have "((F ^^ i) bot) \<in> measurable M (count_space UNIV)"
by (induct i arbitrary: M) (auto intro!: *) }
then have "(\<lambda>x. SUP i. (F ^^ i) bot x) \<in> measurable M (count_space UNIV)"
by measurable
also have "(\<lambda>x. SUP i. (F ^^ i) bot x) = lfp F"
by (subst sup_continuous_lfp) (auto intro: F)
finally show ?thesis .
qed
lemma measurable_lfp:
fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_lattice, countable})"
assumes F: "sup_continuous F"
assumes *: "\<And>A. A \<in> measurable M (count_space UNIV) \<Longrightarrow> F A \<in> measurable M (count_space UNIV)"
shows "lfp F \<in> measurable M (count_space UNIV)"
by (coinduction rule: measurable_lfp_coinduct[OF _ F]) (blast intro: *)
lemma measurable_gfp_coinduct[consumes 1, case_names continuity step]:
fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_lattice, countable})"
assumes "P M"
assumes F: "inf_continuous F"
assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> A \<in> measurable N (count_space UNIV)) \<Longrightarrow> F A \<in> measurable M (count_space UNIV)"
shows "gfp F \<in> measurable M (count_space UNIV)"
proof -
{ fix i from \<open>P M\<close> have "((F ^^ i) top) \<in> measurable M (count_space UNIV)"
by (induct i arbitrary: M) (auto intro!: *) }
then have "(\<lambda>x. INF i. (F ^^ i) top x) \<in> measurable M (count_space UNIV)"
by measurable
also have "(\<lambda>x. INF i. (F ^^ i) top x) = gfp F"
by (subst inf_continuous_gfp) (auto intro: F)
finally show ?thesis .
qed
lemma measurable_gfp:
fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_lattice, countable})"
assumes F: "inf_continuous F"
assumes *: "\<And>A. A \<in> measurable M (count_space UNIV) \<Longrightarrow> F A \<in> measurable M (count_space UNIV)"
shows "gfp F \<in> measurable M (count_space UNIV)"
by (coinduction rule: measurable_gfp_coinduct[OF _ F]) (blast intro: *)
lemma measurable_lfp2_coinduct[consumes 1, case_names continuity step]:
fixes F :: "('a \<Rightarrow> 'c \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'c \<Rightarrow> 'b::{complete_lattice, countable})"
assumes "P M s"
assumes F: "sup_continuous F"
assumes *: "\<And>M A s. P M s \<Longrightarrow> (\<And>N t. P N t \<Longrightarrow> A t \<in> measurable N (count_space UNIV)) \<Longrightarrow> F A s \<in> measurable M (count_space UNIV)"
shows "lfp F s \<in> measurable M (count_space UNIV)"
proof -
{ fix i from \<open>P M s\<close> have "(\<lambda>x. (F ^^ i) bot s x) \<in> measurable M (count_space UNIV)"
by (induct i arbitrary: M s) (auto intro!: *) }
then have "(\<lambda>x. SUP i. (F ^^ i) bot s x) \<in> measurable M (count_space UNIV)"
by measurable
also have "(\<lambda>x. SUP i. (F ^^ i) bot s x) = lfp F s"
by (subst sup_continuous_lfp) (auto simp: F)
finally show ?thesis .
qed
lemma measurable_gfp2_coinduct[consumes 1, case_names continuity step]:
fixes F :: "('a \<Rightarrow> 'c \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'c \<Rightarrow> 'b::{complete_lattice, countable})"
assumes "P M s"
assumes F: "inf_continuous F"
assumes *: "\<And>M A s. P M s \<Longrightarrow> (\<And>N t. P N t \<Longrightarrow> A t \<in> measurable N (count_space UNIV)) \<Longrightarrow> F A s \<in> measurable M (count_space UNIV)"
shows "gfp F s \<in> measurable M (count_space UNIV)"
proof -
{ fix i from \<open>P M s\<close> have "(\<lambda>x. (F ^^ i) top s x) \<in> measurable M (count_space UNIV)"
by (induct i arbitrary: M s) (auto intro!: *) }
then have "(\<lambda>x. INF i. (F ^^ i) top s x) \<in> measurable M (count_space UNIV)"
by measurable
also have "(\<lambda>x. INF i. (F ^^ i) top s x) = gfp F s"
by (subst inf_continuous_gfp) (auto simp: F)
finally show ?thesis .
qed
lemma measurable_enat_coinduct:
fixes f :: "'a \<Rightarrow> enat"
assumes "R f"
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>
Measurable.pred M P \<and>
i \<in> measurable M M \<and>
h \<in> measurable M (count_space UNIV)"
shows "f \<in> measurable M (count_space UNIV)"
proof (simp add: measurable_count_space_eq2_countable, rule )
fix a :: enat
have "f -` {a} \<inter> space M = {x\<in>space M. f x = a}"
by auto
{ fix i :: nat
from \<open>R f\<close> have "Measurable.pred M (\<lambda>x. f x = enat i)"
proof (induction i arbitrary: f)
case 0
from *[OF this] obtain g h i P
where f: "f = (\<lambda>x. if P x then h x else eSuc (g (i x)))" and
[measurable]: "Measurable.pred M P" "i \<in> measurable M M" "h \<in> measurable M (count_space UNIV)"
by auto
have "Measurable.pred M (\<lambda>x. P x \<and> h x = 0)"
by measurable
also have "(\<lambda>x. P x \<and> h x = 0) = (\<lambda>x. f x = enat 0)"
by (auto simp: f zero_enat_def[symmetric])
finally show ?case .
next
case (Suc n)
from *[OF Suc.prems] obtain g h i P
where f: "f = (\<lambda>x. if P x then h x else eSuc (g (i x)))" and "R g" and
M[measurable]: "Measurable.pred M P" "i \<in> measurable M M" "h \<in> measurable M (count_space UNIV)"
by auto
have "(\<lambda>x. f x = enat (Suc n)) =
(\<lambda>x. (P x \<longrightarrow> h x = enat (Suc n)) \<and> (\<not> P x \<longrightarrow> g (i x) = enat n))"
by (auto simp: f zero_enat_def[symmetric] eSuc_enat[symmetric])
also have "Measurable.pred M \<dots>"
by (intro pred_intros_logic measurable_compose[OF M(2)] Suc \<open>R g\<close>) measurable
finally show ?case .
qed
then have "f -` {enat i} \<inter> space M \<in> sets M"
by (simp add: pred_def Int_def conj_commute) }
note fin = this
show "f -` {a} \<inter> space M \<in> sets M"
proof (cases a)
case infinity
then have "f -` {a} \<inter> space M = space M - (\<Union>n. f -` {enat n} \<inter> space M)"
by auto
also have "\<dots> \<in> sets M"
by (intro sets.Diff sets.top sets.Un sets.countable_UN) (auto intro!: fin)
finally show ?thesis .
qed (simp add: fin)
qed
lemma measurable_THE:
fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
assumes I[simp]: "countable I" "\<And>i x. x \<in> space M \<Longrightarrow> P i x \<Longrightarrow> i \<in> I"
assumes unique: "\<And>x i j. x \<in> space M \<Longrightarrow> P i x \<Longrightarrow> P j x \<Longrightarrow> i = j"
shows "(\<lambda>x. THE i. P i x) \<in> measurable M (count_space UNIV)"
unfolding measurable_def
proof safe
fix X
def f \<equiv> "\<lambda>x. THE i. P i x" def undef \<equiv> "THE i::'a. False"
{ fix i x assume "x \<in> space M" "P i x" then have "f x = i"
unfolding f_def using unique by auto }
note f_eq = this
{ fix x assume "x \<in> space M" "\<forall>i\<in>I. \<not> P i x"
then have "\<And>i. \<not> P i x"
using I(2)[of x] by auto
then have "f x = undef"
by (auto simp: undef_def f_def) }
then have "f -` X \<inter> space M = (\<Union>i\<in>I \<inter> X. {x\<in>space M. P i x}) \<union>
(if undef \<in> X then space M - (\<Union>i\<in>I. {x\<in>space M. P i x}) else {})"
by (auto dest: f_eq)
also have "\<dots> \<in> sets M"
by (auto intro!: sets.Diff sets.countable_UN')
finally show "f -` X \<inter> space M \<in> sets M" .
qed simp
lemma measurable_Ex1[measurable (raw)]:
assumes [simp]: "countable I" and [measurable]: "\<And>i. i \<in> I \<Longrightarrow> Measurable.pred M (P i)"
shows "Measurable.pred M (\<lambda>x. \<exists>!i\<in>I. P i x)"
unfolding bex1_def by measurable
lemma measurable_split_if[measurable (raw)]:
"(c \<Longrightarrow> Measurable.pred M f) \<Longrightarrow> (\<not> c \<Longrightarrow> Measurable.pred M g) \<Longrightarrow>
Measurable.pred M (if c then f else g)"
by simp
lemma pred_restrict_space:
assumes "S \<in> sets M"
shows "Measurable.pred (restrict_space M S) P \<longleftrightarrow> Measurable.pred M (\<lambda>x. x \<in> S \<and> P x)"
unfolding pred_def sets_Collect_restrict_space_iff[OF assms] ..
lemma measurable_predpow[measurable]:
assumes "Measurable.pred M T"
assumes "\<And>Q. Measurable.pred M Q \<Longrightarrow> Measurable.pred M (R Q)"
shows "Measurable.pred M ((R ^^ n) T)"
by (induct n) (auto intro: assms)
hide_const (open) pred
end