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Datei:
Measurable.thy
Sprache: Isabelle
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(* Title: HOL/Analysis/Measurable.thy
Author: Johannes Hölzl <[email protected]>
*)
section \<open>Measurability Prover\<close>
theory Measurable
imports
Sigma_Algebra
"HOL-Library.Order_Continuity"
begin
lemma (in algebra) sets_Collect_finite_All:
assumes "\i. i \ S \ {x\\. P i x} \ M" "finite S"
shows "{x\\. \i\S. P i x} \ M"
proof -
have "{x\\. \i\S. P i x} = (if S = {} then \ else \i\S. {x\\. P i x})"
by auto
with assms show ?thesis by (auto intro!: sets_Collect_finite_All')
qed
abbreviation "pred M P \ P \ measurable M (count_space (UNIV::bool set))"
lemma pred_def: "pred M P \ {x\space M. P x} \ sets M"
proof
assume "pred M P"
then have "P -` {True} \ space M \ sets M"
by (auto simp: measurable_count_space_eq2)
also have "P -` {True} \ space M = {x\space M. P x}" by auto
finally show "{x\space M. P x} \ sets M" .
next
assume P: "{x\space M. P x} \ sets M"
moreover
{ fix X
have "X \ Pow (UNIV :: bool set)" by simp
then have "P -` X \ space M = {x\space M. ((X = {True} \ P x) \ (X = {False} \ \ P x) \ X \ {})}"
unfolding UNIV_bool Pow_insert Pow_empty by auto
then have "P -` X \ space M \ 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\space M. P x} \ sets M \ f \ measurable N M \ pred N (\x. P (f x))"
by (rule measurable_compose[where f=f and N=M]) (auto simp: pred_def)
lemma pred_sets2: "A \ sets N \ f \ measurable M N \ pred M (\x. f x \ A)"
by (rule measurable_compose[where f=f and N=N]) (auto simp: pred_def Int_def[symmetric])
ML_file \<open>measurable.ML\<close>
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\<^marker>\<open>tag important\<close> measurable = \<open> Scan.lift (Scan.succeed (METHOD o Measurable.measurable_tac)) \<close>
"measurability prover"
simproc_setup\<^marker>\<open>tag important\<close> measurable ("A \<in> sets M" | "f \<in> measurable M N") = \<open>K Measurable.simproc\<close>
setup \<open>
Global_Theory.add_thms_dynamic (\<^binding>\<open>measurable\<close>, 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 \ {x\space M. P x} \ sets M"
unfolding pred_def .
lemma pred_intros_imp'[measurable (raw)]:
"(K \ pred M (\x. P x)) \ pred M (\x. K \ P x)"
by (cases K) auto
lemma pred_intros_conj1'[measurable (raw)]:
"(K \ pred M (\x. P x)) \ pred M (\x. K \ P x)"
by (cases K) auto
lemma pred_intros_conj2'[measurable (raw)]:
"(K \ pred M (\x. P x)) \ pred M (\x. P x \ K)"
by (cases K) auto
lemma pred_intros_disj1'[measurable (raw)]:
"(\ K \ pred M (\x. P x)) \ pred M (\x. K \ P x)"
by (cases K) auto
lemma pred_intros_disj2'[measurable (raw)]:
"(\ K \ pred M (\x. P x)) \ pred M (\x. P x \ K)"
by (cases K) auto
lemma pred_intros_logic[measurable (raw)]:
"pred M (\x. x \ space M)"
"pred M (\x. P x) \ pred M (\x. \ P x)"
"pred M (\x. Q x) \ pred M (\x. P x) \ pred M (\x. Q x \ P x)"
"pred M (\x. Q x) \ pred M (\x. P x) \ pred M (\x. Q x \ P x)"
"pred M (\x. Q x) \ pred M (\x. P x) \ pred M (\x. Q x \ P x)"
"pred M (\x. Q x) \ pred M (\x. P x) \ pred M (\x. Q x = P x)"
"pred M (\x. f x \ UNIV)"
"pred M (\x. f x \ {})"
"pred M (\x. P' (f x) x) \ pred M (\x. f x \ {y. P' y x})"
"pred M (\x. f x \ (B x)) \ pred M (\x. f x \ - (B x))"
"pred M (\x. f x \ (A x)) \ pred M (\x. f x \ (B x)) \ pred M (\x. f x \ (A x) - (B x))"
"pred M (\x. f x \ (A x)) \ pred M (\x. f x \ (B x)) \ pred M (\x. f x \ (A x) \ (B x))"
"pred M (\x. f x \ (A x)) \ pred M (\x. f x \ (B x)) \ pred M (\x. f x \ (A x) \ (B x))"
"pred M (\x. g x (f x) \ (X x)) \ pred M (\x. f x \ (g x) -` (X x))"
by (auto simp: iff_conv_conj_imp pred_def)
lemma pred_intros_countable[measurable (raw)]:
fixes P :: "'a \ 'i :: countable \ bool"
shows
"(\i. pred M (\x. P x i)) \ pred M (\x. \i. P x i)"
"(\i. pred M (\x. P x i)) \ pred M (\x. \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
"(\i. i \ X \ pred M (\x. x \ N x i)) \ pred M (\x. x \ (\i\X. N x i))"
"(\i. i \ X \ pred M (\x. x \ N x i)) \ pred M (\x. x \ (\i\X. N x i))"
"(\i. i \ X \ pred M (\x. P x i)) \ pred M (\x. \i\X. P x i)"
"(\i. i \ X \ pred M (\x. P x i)) \ pred M (\x. \i\X. P x i)"
by simp_all (auto simp: Bex_def Ball_def)
lemma pred_intros_finite[measurable (raw)]:
"finite I \ (\i. i \ I \ pred M (\x. x \ N x i)) \ pred M (\x. x \ (\i\I. N x i))"
"finite I \ (\i. i \ I \ pred M (\x. x \ N x i)) \ pred M (\x. x \ (\i\I. N x i))"
"finite I \ (\i. i \ I \ pred M (\x. P x i)) \ pred M (\x. \i\I. P x i)"
"finite I \ (\i. i \ I \ pred M (\x. P x i)) \ pred M (\x. \i\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)]:
"(\i :: 'i :: countable. i \ I \ N i \ sets M) \ (\i\I. N i) \ sets M"
"I \ {} \ (\i :: 'i :: countable. i \ I \ N i \ sets M) \ (\i\I. N i) \ sets M"
by auto
declare
finite_UN[measurable (raw)]
finite_INT[measurable (raw)]
lemma sets_Int_pred[measurable (raw)]:
assumes space: "A \ B \ space M" and [measurable]: "pred M (\x. x \ A)" "pred M (\x. x \ B)"
shows "A \ B \ sets M"
proof -
have "{x\space M. x \ A \ B} \ sets M" by auto
also have "{x\space M. x \ A \ B} = A \ B"
using space by auto
finally show ?thesis .
qed
lemma [measurable (raw generic)]:
assumes f: "f \ measurable M N" and c: "c \ space N \ {c} \ sets N"
shows pred_eq_const1: "pred M (\x. f x = c)"
and pred_eq_const2: "pred M (\x. c = f x)"
proof -
show "pred M (\x. f x = c)"
proof cases
assume "c \ space N"
with measurable_sets[OF f c] show ?thesis
by (auto simp: Int_def conj_commute pred_def)
next
assume "c \ space N"
with f[THEN measurable_space] have "{x \ space M. f x = c} = {}" by auto
then show ?thesis by (auto simp: pred_def cong: conj_cong)
qed
then show "pred M (\x. c = f x)"
by (simp add: eq_commute)
qed
lemma pred_count_space_const1[measurable (raw)]:
"f \ measurable M (count_space UNIV) \ Measurable.pred M (\x. f x = c)"
by (intro pred_eq_const1[where N="count_space UNIV"]) (auto )
lemma pred_count_space_const2[measurable (raw)]:
"f \ measurable M (count_space UNIV) \ Measurable.pred M (\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 \ measurable M N" and c: "{.. c} \ sets N" shows "pred M (\x. f x \ 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 \ measurable M N" and c: "{c ..} \ sets N" shows "pred M (\x. c \ 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 \ measurable M N" and c: "{..< c} \ sets N" shows "pred M (\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 \ measurable M N" and c: "{c <..} \ sets N" shows "pred M (\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 \ pred M (\x. x \ A x)) \ (\ P \ pred M (\x. x \ B x)) \
pred M (\<lambda>x. x \<in> (if P then A x else B x))"
by auto
lemma sets_range[measurable_dest]:
"A ` I \ sets M \ i \ I \ A i \ sets M"
by auto
lemma pred_sets_range[measurable_dest]:
"A ` I \ sets N \ i \ I \ f \ measurable M N \ pred M (\x. f x \ A i)"
using pred_sets2[OF sets_range] by auto
lemma sets_All[measurable_dest]:
"\i. A i \ sets (M i) \ A i \ sets (M i)"
by auto
lemma pred_sets_All[measurable_dest]:
"\i. A i \ sets (N i) \ f \ measurable M (N i) \ pred M (\x. f x \ A i)"
using pred_sets2[OF sets_All, of A N f] by auto
lemma sets_Ball[measurable_dest]:
"\i\I. A i \ sets (M i) \ i\I \ A i \ sets (M i)"
by auto
lemma pred_sets_Ball[measurable_dest]:
"\i\I. A i \ sets (N i) \ i\I \ f \ measurable M (N i) \ pred M (\x. f x \ A i)"
using pred_sets2[OF sets_Ball, of _ _ _ f] by auto
lemma measurable_finite[measurable (raw)]:
fixes S :: "'a \ nat set"
assumes [measurable]: "\i. {x\space M. i \ S x} \ sets M"
shows "pred M (\x. finite (S x))"
unfolding finite_nat_set_iff_bounded by (simp add: Ball_def)
lemma measurable_Least[measurable]:
assumes [measurable]: "(\i::nat. (\x. P i x) \ measurable M (count_space UNIV))"
shows "(\x. LEAST i. P i x) \ measurable M (count_space UNIV)"
unfolding measurable_def by (safe intro!: sets_Least) simp_all
lemma measurable_Max_nat[measurable (raw)]:
fixes P :: "nat \ 'a \ bool"
assumes [measurable]: "\i. Measurable.pred M (P i)"
shows "(\x. Max {i. P i x}) \ measurable M (count_space UNIV)"
unfolding measurable_count_space_eq2_countable
proof safe
fix n
{ fix x assume "\i. \n\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" "\n\j. \ 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 "(\x. Max {i. P i x}) -` {n} \ space M = {x\space M. Max {i. P i x} = n}"
by auto
also have "\ =
{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 "\ \ sets M"
by measurable
finally show "(\x. Max {i. P i x}) -` {n} \ space M \ sets M" .
qed simp
lemma measurable_Min_nat[measurable (raw)]:
fixes P :: "nat \ 'a \ bool"
assumes [measurable]: "\i. Measurable.pred M (P i)"
shows "(\x. Min {i. P i x}) \ measurable M (count_space UNIV)"
unfolding measurable_count_space_eq2_countable
proof safe
fix n
{ fix x assume "\i. \n\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" "\n\j. \ 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 "(\x. Min {i. P i x}) -` {n} \ space M = {x\space M. Min {i. P i x} = n}"
by auto
also have "\ =
{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 "\ \ sets M"
by measurable
finally show "(\x. Min {i. P i x}) -` {n} \ space M \ sets M" .
qed simp
lemma measurable_count_space_insert[measurable (raw)]:
"s \ S \ A \ sets (count_space S) \ insert s A \ sets (count_space S)"
by simp
lemma sets_UNIV [measurable (raw)]: "A \ sets (count_space UNIV)"
by simp
lemma measurable_card[measurable]:
fixes S :: "'a \ nat set"
assumes [measurable]: "\i. {x\space M. i \ S x} \ sets M"
shows "(\x. card (S x)) \ measurable M (count_space UNIV)"
unfolding measurable_count_space_eq2_countable
proof safe
fix n show "(\x. card (S x)) -` {n} \ space M \ sets M"
proof (cases n)
case 0
then have "(\x. card (S x)) -` {n} \ space M = {x\space M. infinite (S x) \ (\i. i \ S x)}"
by auto
also have "\ \ sets M"
by measurable
finally show ?thesis .
next
case (Suc i)
then have "(\x. card (S x)) -` {n} \ 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 "\ \ 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
"(\i. i \ X \ Measurable.pred M (\x. P x i)) \ Measurable.pred M (\x. \i\X. P x i)"
"(\i. i \ X \ Measurable.pred M (\x. P x i)) \ Measurable.pred M (\x. \i\X. P x i)"
unfolding pred_def
by (auto intro!: sets.sets_Collect_countable_All' sets.sets_Collect_countable_Ex' assms)
subsection\<^marker>\<open>tag unimportant\<close> \<open>Measurability for (co)inductive predicates\<close>
lemma measurable_bot[measurable]: "bot \ measurable M (count_space UNIV)"
by (simp add: bot_fun_def)
lemma measurable_top[measurable]: "top \ measurable M (count_space UNIV)"
by (simp add: top_fun_def)
lemma measurable_SUP[measurable]:
fixes F :: "'i \ 'a \ 'b::{complete_lattice, countable}"
assumes [simp]: "countable I"
assumes [measurable]: "\i. i \ I \ F i \ measurable M (count_space UNIV)"
shows "(\x. SUP i\I. F i x) \ measurable M (count_space UNIV)"
unfolding measurable_count_space_eq2_countable
proof (safe intro!: UNIV_I)
fix a
have "(\x. SUP i\I. F i x) -` {a} \ 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 "\ \ sets M"
by measurable
finally show "(\x. SUP i\I. F i x) -` {a} \ space M \ sets M" .
qed
lemma measurable_INF[measurable]:
fixes F :: "'i \ 'a \ 'b::{complete_lattice, countable}"
assumes [simp]: "countable I"
assumes [measurable]: "\i. i \ I \ F i \ measurable M (count_space UNIV)"
shows "(\x. INF i\I. F i x) \ measurable M (count_space UNIV)"
unfolding measurable_count_space_eq2_countable
proof (safe intro!: UNIV_I)
fix a
have "(\x. INF i\I. F i x) -` {a} \ 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 "\ \ sets M"
by measurable
finally show "(\x. INF i\I. F i x) -` {a} \ space M \ sets M" .
qed
lemma measurable_lfp_coinduct[consumes 1, case_names continuity step]:
fixes F :: "('a \ 'b) \ ('a \ 'b::{complete_lattice, countable})"
assumes "P M"
assumes F: "sup_continuous F"
assumes *: "\M A. P M \ (\N. P N \ A \ measurable N (count_space UNIV)) \ F A \ measurable M (count_space UNIV)"
shows "lfp F \ 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 "(\x. SUP i. (F ^^ i) bot x) \ measurable M (count_space UNIV)"
by measurable
also have "(\x. SUP i. (F ^^ i) bot x) = lfp F"
by (subst sup_continuous_lfp) (auto intro: F simp: image_comp)
finally show ?thesis .
qed
lemma measurable_lfp:
fixes F :: "('a \ 'b) \ ('a \ 'b::{complete_lattice, countable})"
assumes F: "sup_continuous F"
assumes *: "\A. A \ measurable M (count_space UNIV) \ F A \ measurable M (count_space UNIV)"
shows "lfp F \ 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 \ 'b) \ ('a \ 'b::{complete_lattice, countable})"
assumes "P M"
assumes F: "inf_continuous F"
assumes *: "\M A. P M \ (\N. P N \ A \ measurable N (count_space UNIV)) \ F A \ measurable M (count_space UNIV)"
shows "gfp F \ 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 "(\x. INF i. (F ^^ i) top x) \ measurable M (count_space UNIV)"
by measurable
also have "(\x. INF i. (F ^^ i) top x) = gfp F"
by (subst inf_continuous_gfp) (auto intro: F simp: image_comp)
finally show ?thesis .
qed
lemma measurable_gfp:
fixes F :: "('a \ 'b) \ ('a \ 'b::{complete_lattice, countable})"
assumes F: "inf_continuous F"
assumes *: "\A. A \ measurable M (count_space UNIV) \ F A \ measurable M (count_space UNIV)"
shows "gfp F \ 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 \ 'c \ 'b) \ ('a \ 'c \ 'b::{complete_lattice, countable})"
assumes "P M s"
assumes F: "sup_continuous F"
assumes *: "\M A s. P M s \ (\N t. P N t \ A t \ measurable N (count_space UNIV)) \ F A s \ measurable M (count_space UNIV)"
shows "lfp F s \ 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 "(\x. SUP i. (F ^^ i) bot s x) \ measurable M (count_space UNIV)"
by measurable
also have "(\x. SUP i. (F ^^ i) bot s x) = lfp F s"
by (subst sup_continuous_lfp) (auto simp: F simp: image_comp)
finally show ?thesis .
qed
lemma measurable_gfp2_coinduct[consumes 1, case_names continuity step]:
fixes F :: "('a \ 'c \ 'b) \ ('a \ 'c \ 'b::{complete_lattice, countable})"
assumes "P M s"
assumes F: "inf_continuous F"
assumes *: "\M A s. P M s \ (\N t. P N t \ A t \ measurable N (count_space UNIV)) \ F A s \ measurable M (count_space UNIV)"
shows "gfp F s \ 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 "(\x. INF i. (F ^^ i) top s x) \ measurable M (count_space UNIV)"
by measurable
also have "(\x. INF i. (F ^^ i) top s x) = gfp F s"
by (subst inf_continuous_gfp) (auto simp: F simp: image_comp)
finally show ?thesis .
qed
lemma measurable_enat_coinduct:
fixes f :: "'a \ enat"
assumes "R f"
assumes *: "\f. R f \ \g h i P. R g \ f = (\x. if P x then h x else eSuc (g (i x))) \
Measurable.pred M P \<and>
i \<in> measurable M M \<and>
h \<in> measurable M (count_space UNIV)"
shows "f \ measurable M (count_space UNIV)"
proof (simp add: measurable_count_space_eq2_countable, rule )
fix a :: enat
have "f -` {a} \ space M = {x\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 = (\x. if P x then h x else eSuc (g (i x)))" and
[measurable]: "Measurable.pred M P" "i \ measurable M M" "h \ measurable M (count_space UNIV)"
by auto
have "Measurable.pred M (\x. P x \ h x = 0)"
by measurable
also have "(\x. P x \ h x = 0) = (\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 = (\x. if P x then h x else eSuc (g (i x)))" and "R g" and
M[measurable]: "Measurable.pred M P" "i \ measurable M M" "h \ measurable M (count_space UNIV)"
by auto
have "(\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 \"
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} \ space M \ sets M"
by (simp add: pred_def Int_def conj_commute) }
note fin = this
show "f -` {a} \ space M \ sets M"
proof (cases a)
case infinity
then have "f -` {a} \ space M = space M - (\n. f -` {enat n} \ space M)"
by auto
also have "\ \ 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 \ 'b \ bool"
assumes [measurable]: "\i. Measurable.pred M (P i)"
assumes I[simp]: "countable I" "\i x. x \ space M \ P i x \ i \ I"
assumes unique: "\x i j. x \ space M \ P i x \ P j x \ i = j"
shows "(\x. THE i. P i x) \ measurable M (count_space UNIV)"
unfolding measurable_def
proof safe
fix X
define f where "f x = (THE i. P i x)" for x
define undef where "undef = (THE i::'a. False)"
{ fix i x assume "x \ 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 \ space M" "\i\I. \ P i x"
then have "\i. \ 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 \ space M = (\i\I \ X. {x\space M. P i x}) \
(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 "\ \ sets M"
by (auto intro!: sets.Diff sets.countable_UN')
finally show "f -` X \ space M \ sets M" .
qed simp
lemma measurable_Ex1[measurable (raw)]:
assumes [simp]: "countable I" and [measurable]: "\i. i \ I \ Measurable.pred M (P i)"
shows "Measurable.pred M (\x. \!i\I. P i x)"
unfolding bex1_def by measurable
lemma measurable_Sup_nat[measurable (raw)]:
fixes F :: "'a \ nat set"
assumes [measurable]: "\i. Measurable.pred M (\x. i \ F x)"
shows "(\x. Sup (F x)) \ M \\<^sub>M count_space UNIV"
proof (clarsimp simp add: measurable_count_space_eq2_countable)
fix a
have F_empty_iff: "F x = {} \ (\i. i \ F x)" for x
by auto
have "Measurable.pred M (\x. if finite (F x) then if F x = {} then a = 0
else a \<in> F x \<and> (\<forall>j. j \<in> F x \<longrightarrow> j \<le> a) else a = the None)"
unfolding finite_nat_set_iff_bounded Ball_def F_empty_iff by measurable
moreover have "(\x. Sup (F x)) -` {a} \ space M =
{x\<in>space M. if finite (F x) then if F x = {} then a = 0
else a \<in> F x \<and> (\<forall>j. j \<in> F x \<longrightarrow> j \<le> a) else a = the None}"
by (intro set_eqI)
(auto simp: Sup_nat_def Max.infinite intro!: Max_in Max_eqI)
ultimately show "(\x. Sup (F x)) -` {a} \ space M \ sets M"
by auto
qed
lemma measurable_if_split[measurable (raw)]:
"(c \ Measurable.pred M f) \ (\ c \ Measurable.pred M g) \
Measurable.pred M (if c then f else g)"
by simp
lemma pred_restrict_space:
assumes "S \ sets M"
shows "Measurable.pred (restrict_space M S) P \ Measurable.pred M (\x. x \ S \ P x)"
unfolding pred_def sets_Collect_restrict_space_iff[OF assms] ..
lemma measurable_predpow[measurable]:
assumes "Measurable.pred M T"
assumes "\Q. Measurable.pred M Q \ Measurable.pred M (R Q)"
shows "Measurable.pred M ((R ^^ n) T)"
by (induct n) (auto intro: assms)
lemma measurable_compose_countable_restrict:
assumes P: "countable {i. P i}"
and f: "f \ M \\<^sub>M count_space UNIV"
and Q: "\i. P i \ pred M (Q i)"
shows "pred M (\x. P (f x) \ Q (f x) x)"
proof -
have P_f: "{x \ space M. P (f x)} \ sets M"
unfolding pred_def[symmetric] by (rule measurable_compose[OF f]) simp
have "pred (restrict_space M {x\space M. P (f x)}) (\x. Q (f x) x)"
proof (rule measurable_compose_countable'[where g=f, OF _ _ P])
show "f \ restrict_space M {x\space M. P (f x)} \\<^sub>M count_space {i. P i}"
by (rule measurable_count_space_extend[OF subset_UNIV])
(auto simp: space_restrict_space intro!: measurable_restrict_space1 f)
qed (auto intro!: measurable_restrict_space1 Q)
then show ?thesis
unfolding pred_restrict_space[OF P_f] by (simp cong: measurable_cong)
qed
lemma measurable_limsup [measurable (raw)]:
assumes [measurable]: "\n. A n \ sets M"
shows "limsup A \ sets M"
by (subst limsup_INF_SUP, auto)
lemma measurable_liminf [measurable (raw)]:
assumes [measurable]: "\n. A n \ sets M"
shows "liminf A \ sets M"
by (subst liminf_SUP_INF, auto)
lemma measurable_case_enat[measurable (raw)]:
assumes f: "f \ M \\<^sub>M count_space UNIV" and g: "\i. g i \ M \\<^sub>M N" and h: "h \ M \\<^sub>M N"
shows "(\x. case f x of enat i \ g i x | \ \ h x) \ M \\<^sub>M N"
apply (rule measurable_compose_countable[OF _ f])
subgoal for i
by (cases i) (auto intro: g h)
done
hide_const (open) pred
end
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