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Quellcode-Bibliothek
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Datei:
Independent_Family.thy
Sprache: Isabelle
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(* Title: HOL/Probability/Independent_Family.thy
Author: Johannes Hölzl, TU München
Author: Sudeep Kanav, TU München
*)
section \<open>Independent families of events, event sets, and random variables\<close>
theory Independent_Family
imports Infinite_Product_Measure
begin
definition (in prob_space)
"indep_sets F I \ (\i\I. F i \ events) \
(\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> (\<forall>A\<in>Pi J F. prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))))"
definition (in prob_space)
"indep_set A B \ indep_sets (case_bool A B) UNIV"
definition (in prob_space)
indep_events_def_alt: "indep_events A I \ indep_sets (\i. {A i}) I"
lemma (in prob_space) indep_events_def:
"indep_events A I \ (A`I \ events) \
(\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j)))"
unfolding indep_events_def_alt indep_sets_def
apply (simp add: Ball_def Pi_iff image_subset_iff_funcset)
apply (intro conj_cong refl arg_cong[where f=All] ext imp_cong)
apply auto
done
lemma (in prob_space) indep_eventsI:
"(\i. i \ I \ F i \ sets M) \ (\J. J \ I \ finite J \ J \ {} \ prob (\i\J. F i) = (\i\J. prob (F i))) \ indep_events F I"
by (auto simp: indep_events_def)
definition (in prob_space)
"indep_event A B \ indep_events (case_bool A B) UNIV"
lemma (in prob_space) indep_sets_cong:
"I = J \ (\i. i \ I \ F i = G i) \ indep_sets F I \ indep_sets G J"
by (simp add: indep_sets_def, intro conj_cong all_cong imp_cong ball_cong) blast+
lemma (in prob_space) indep_events_finite_index_events:
"indep_events F I \ (\J\I. J \ {} \ finite J \ indep_events F J)"
by (auto simp: indep_events_def)
lemma (in prob_space) indep_sets_finite_index_sets:
"indep_sets F I \ (\J\I. J \ {} \ finite J \ indep_sets F J)"
proof (intro iffI allI impI)
assume *: "\J\I. J \ {} \ finite J \ indep_sets F J"
show "indep_sets F I" unfolding indep_sets_def
proof (intro conjI ballI allI impI)
fix i assume "i \ I"
with *[THEN spec, of "{i}"] show "F i \ events"
by (auto simp: indep_sets_def)
qed (insert *, auto simp: indep_sets_def)
qed (auto simp: indep_sets_def)
lemma (in prob_space) indep_sets_mono_index:
"J \ I \ indep_sets F I \ indep_sets F J"
unfolding indep_sets_def by auto
lemma (in prob_space) indep_sets_mono_sets:
assumes indep: "indep_sets F I"
assumes mono: "\i. i\I \ G i \ F i"
shows "indep_sets G I"
proof -
have "(\i\I. F i \ events) \ (\i\I. G i \ events)"
using mono by auto
moreover have "\A J. J \ I \ A \ (\ j\J. G j) \ A \ (\ j\J. F j)"
using mono by (auto simp: Pi_iff)
ultimately show ?thesis
using indep by (auto simp: indep_sets_def)
qed
lemma (in prob_space) indep_sets_mono:
assumes indep: "indep_sets F I"
assumes mono: "J \ I" "\i. i\J \ G i \ F i"
shows "indep_sets G J"
apply (rule indep_sets_mono_sets)
apply (rule indep_sets_mono_index)
apply (fact +)
done
lemma (in prob_space) indep_setsI:
assumes "\i. i \ I \ F i \ events"
and "\A J. J \ {} \ J \ I \ finite J \ (\j\J. A j \ F j) \ prob (\j\J. A j) = (\j\J. prob (A j))"
shows "indep_sets F I"
using assms unfolding indep_sets_def by (auto simp: Pi_iff)
lemma (in prob_space) indep_setsD:
assumes "indep_sets F I" and "J \ I" "J \ {}" "finite J" "\j\J. A j \ F j"
shows "prob (\j\J. A j) = (\j\J. prob (A j))"
using assms unfolding indep_sets_def by auto
lemma (in prob_space) indep_setI:
assumes ev: "A \ events" "B \ events"
and indep: "\a b. a \ A \ b \ B \ prob (a \ b) = prob a * prob b"
shows "indep_set A B"
unfolding indep_set_def
proof (rule indep_setsI)
fix F J assume "J \ {}" "J \ UNIV"
and F: "\j\J. F j \ (case j of True \ A | False \ B)"
have "J \ Pow UNIV" by auto
with F \<open>J \<noteq> {}\<close> indep[of "F True" "F False"]
show "prob (\j\J. F j) = (\j\J. prob (F j))"
unfolding UNIV_bool Pow_insert by (auto simp: ac_simps)
qed (auto split: bool.split simp: ev)
lemma (in prob_space) indep_setD:
assumes indep: "indep_set A B" and ev: "a \ A" "b \ B"
shows "prob (a \ b) = prob a * prob b"
using indep[unfolded indep_set_def, THEN indep_setsD, of UNIV "case_bool a b"] ev
by (simp add: ac_simps UNIV_bool)
lemma (in prob_space)
assumes indep: "indep_set A B"
shows indep_setD_ev1: "A \ events"
and indep_setD_ev2: "B \ events"
using indep unfolding indep_set_def indep_sets_def UNIV_bool by auto
lemma (in prob_space) indep_sets_Dynkin:
assumes indep: "indep_sets F I"
shows "indep_sets (\i. Dynkin (space M) (F i)) I"
(is "indep_sets ?F I")
proof (subst indep_sets_finite_index_sets, intro allI impI ballI)
fix J assume "finite J" "J \ I" "J \ {}"
with indep have "indep_sets F J"
by (subst (asm) indep_sets_finite_index_sets) auto
{ fix J K assume "indep_sets F K"
let ?G = "\S i. if i \ S then ?F i else F i"
assume "finite J" "J \ K"
then have "indep_sets (?G J) K"
proof induct
case (insert j J)
moreover define G where "G = ?G J"
ultimately have G: "indep_sets G K" "\i. i \ K \ G i \ events" and "j \ K"
by (auto simp: indep_sets_def)
let ?D = "{E\events. indep_sets (G(j := {E})) K }"
{ fix X assume X: "X \ events"
assume indep: "\J A. J \ {} \ J \ K \ finite J \ j \ J \ (\i\J. A i \ G i)
\<Longrightarrow> prob ((\<Inter>i\<in>J. A i) \<inter> X) = prob X * (\<Prod>i\<in>J. prob (A i))"
have "indep_sets (G(j := {X})) K"
proof (rule indep_setsI)
fix i assume "i \ K" then show "(G(j:={X})) i \ events"
using G X by auto
next
fix A J assume J: "J \ {}" "J \ K" "finite J" "\i\J. A i \ (G(j := {X})) i"
show "prob (\j\J. A j) = (\j\J. prob (A j))"
proof cases
assume "j \ J"
with J have "A j = X" by auto
show ?thesis
proof cases
assume "J = {j}" then show ?thesis by simp
next
assume "J \ {j}"
have "prob (\i\J. A i) = prob ((\i\J-{j}. A i) \ X)"
using \<open>j \<in> J\<close> \<open>A j = X\<close> by (auto intro!: arg_cong[where f=prob] split: if_split_asm)
also have "\ = prob X * (\i\J-{j}. prob (A i))"
proof (rule indep)
show "J - {j} \ {}" "J - {j} \ K" "finite (J - {j})" "j \ J - {j}"
using J \<open>J \<noteq> {j}\<close> \<open>j \<in> J\<close> by auto
show "\i\J - {j}. A i \ G i"
using J by auto
qed
also have "\ = prob (A j) * (\i\J-{j}. prob (A i))"
using \<open>A j = X\<close> by simp
also have "\ = (\i\J. prob (A i))"
unfolding prod.insert_remove[OF \<open>finite J\<close>, symmetric, of "\<lambda>i. prob (A i)"]
using \<open>j \<in> J\<close> by (simp add: insert_absorb)
finally show ?thesis .
qed
next
assume "j \ J"
with J have "\i\J. A i \ G i" by (auto split: if_split_asm)
with J show ?thesis
by (intro indep_setsD[OF G(1)]) auto
qed
qed }
note indep_sets_insert = this
have "Dynkin_system (space M) ?D"
proof (rule Dynkin_systemI', simp_all cong del: indep_sets_cong, safe)
show "indep_sets (G(j := {{}})) K"
by (rule indep_sets_insert) auto
next
fix X assume X: "X \ events" and G': "indep_sets (G(j := {X})) K"
show "indep_sets (G(j := {space M - X})) K"
proof (rule indep_sets_insert)
fix J A assume J: "J \ {}" "J \ K" "finite J" "j \ J" and A: "\i\J. A i \ G i"
then have A_sets: "\i. i\J \ A i \ events"
using G by auto
have "prob ((\j\J. A j) \ (space M - X)) =
prob ((\<Inter>j\<in>J. A j) - (\<Inter>i\<in>insert j J. (A(j := X)) i))"
using A_sets sets.sets_into_space[of _ M] X \<open>J \<noteq> {}\<close>
by (auto intro!: arg_cong[where f=prob] split: if_split_asm)
also have "\ = prob (\j\J. A j) - prob (\i\insert j J. (A(j := X)) i)"
using J \<open>J \<noteq> {}\<close> \<open>j \<notin> J\<close> A_sets X sets.sets_into_space
by (auto intro!: finite_measure_Diff sets.finite_INT split: if_split_asm)
finally have "prob ((\j\J. A j) \ (space M - X)) =
prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)" .
moreover {
have "prob (\j\J. A j) = (\j\J. prob (A j))"
using J A \<open>finite J\<close> by (intro indep_setsD[OF G(1)]) auto
then have "prob (\j\J. A j) = prob (space M) * (\i\J. prob (A i))"
using prob_space by simp }
moreover {
have "prob (\i\insert j J. (A(j := X)) i) = (\i\insert j J. prob ((A(j := X)) i))"
using J A \<open>j \<in> K\<close> by (intro indep_setsD[OF G']) auto
then have "prob (\i\insert j J. (A(j := X)) i) = prob X * (\i\J. prob (A i))"
using \<open>finite J\<close> \<open>j \<notin> J\<close> by (auto intro!: prod.cong) }
ultimately have "prob ((\j\J. A j) \ (space M - X)) = (prob (space M) - prob X) * (\i\J. prob (A i))"
by (simp add: field_simps)
also have "\ = prob (space M - X) * (\i\J. prob (A i))"
using X A by (simp add: finite_measure_compl)
finally show "prob ((\j\J. A j) \ (space M - X)) = prob (space M - X) * (\i\J. prob (A i))" .
qed (insert X, auto)
next
fix F :: "nat \ 'a set" assume disj: "disjoint_family F" and "range F \ ?D"
then have F: "\i. F i \ events" "\i. indep_sets (G(j:={F i})) K" by auto
show "indep_sets (G(j := {\k. F k})) K"
proof (rule indep_sets_insert)
fix J A assume J: "j \ J" "J \ {}" "J \ K" "finite J" and A: "\i\J. A i \ G i"
then have A_sets: "\i. i\J \ A i \ events"
using G by auto
have "prob ((\j\J. A j) \ (\k. F k)) = prob (\k. (\i\insert j J. (A(j := F k)) i))"
using \<open>J \<noteq> {}\<close> \<open>j \<notin> J\<close> \<open>j \<in> K\<close> by (auto intro!: arg_cong[where f=prob] split: if_split_asm)
moreover have "(\k. prob (\i\insert j J. (A(j := F k)) i)) sums prob (\k. (\i\insert j J. (A(j := F k)) i))"
proof (rule finite_measure_UNION)
show "disjoint_family (\k. \i\insert j J. (A(j := F k)) i)"
using disj by (rule disjoint_family_on_bisimulation) auto
show "range (\k. \i\insert j J. (A(j := F k)) i) \ events"
using A_sets F \<open>finite J\<close> \<open>J \<noteq> {}\<close> \<open>j \<notin> J\<close> by (auto intro!: sets.Int)
qed
moreover { fix k
from J A \<open>j \<in> K\<close> have "prob (\<Inter>i\<in>insert j J. (A(j := F k)) i) = prob (F k) * (\<Prod>i\<in>J. prob (A i))"
by (subst indep_setsD[OF F(2)]) (auto intro!: prod.cong split: if_split_asm)
also have "\ = prob (F k) * prob (\i\J. A i)"
using J A \<open>j \<in> K\<close> by (subst indep_setsD[OF G(1)]) auto
finally have "prob (\i\insert j J. (A(j := F k)) i) = prob (F k) * prob (\i\J. A i)" . }
ultimately have "(\k. prob (F k) * prob (\i\J. A i)) sums (prob ((\j\J. A j) \ (\k. F k)))"
by simp
moreover
have "(\k. prob (F k) * prob (\i\J. A i)) sums (prob (\k. F k) * prob (\i\J. A i))"
using disj F(1) by (intro finite_measure_UNION sums_mult2) auto
then have "(\k. prob (F k) * prob (\i\J. A i)) sums (prob (\k. F k) * (\i\J. prob (A i)))"
using J A \<open>j \<in> K\<close> by (subst indep_setsD[OF G(1), symmetric]) auto
ultimately
show "prob ((\j\J. A j) \ (\k. F k)) = prob (\k. F k) * (\j\J. prob (A j))"
by (auto dest!: sums_unique)
qed (insert F, auto)
qed (insert sets.sets_into_space, auto)
then have mono: "Dynkin (space M) (G j) \ {E \ events. indep_sets (G(j := {E})) K}"
proof (rule Dynkin_system.Dynkin_subset, safe)
fix X assume "X \ G j"
then show "X \ events" using G \j \ K\ by auto
from \<open>indep_sets G K\<close>
show "indep_sets (G(j := {X})) K"
by (rule indep_sets_mono_sets) (insert \<open>X \<in> G j\<close>, auto)
qed
have "indep_sets (G(j:=?D)) K"
proof (rule indep_setsI)
fix i assume "i \ K" then show "(G(j := ?D)) i \ events"
using G(2) by auto
next
fix A J assume J: "J\{}" "J \ K" "finite J" and A: "\i\J. A i \ (G(j := ?D)) i"
show "prob (\j\J. A j) = (\j\J. prob (A j))"
proof cases
assume "j \ J"
with A have indep: "indep_sets (G(j := {A j})) K" by auto
from J A show ?thesis
by (intro indep_setsD[OF indep]) auto
next
assume "j \ J"
with J A have "\i\J. A i \ G i" by (auto split: if_split_asm)
with J show ?thesis
by (intro indep_setsD[OF G(1)]) auto
qed
qed
then have "indep_sets (G(j := Dynkin (space M) (G j))) K"
by (rule indep_sets_mono_sets) (insert mono, auto)
then show ?case
by (rule indep_sets_mono_sets) (insert \<open>j \<in> K\<close> \<open>j \<notin> J\<close>, auto simp: G_def)
qed (insert \<open>indep_sets F K\<close>, simp) }
from this[OF \<open>indep_sets F J\<close> \<open>finite J\<close> subset_refl]
show "indep_sets ?F J"
by (rule indep_sets_mono_sets) auto
qed
lemma (in prob_space) indep_sets_sigma:
assumes indep: "indep_sets F I"
assumes stable: "\i. i \ I \ Int_stable (F i)"
shows "indep_sets (\i. sigma_sets (space M) (F i)) I"
proof -
from indep_sets_Dynkin[OF indep]
show ?thesis
proof (rule indep_sets_mono_sets, subst sigma_eq_Dynkin, simp_all add: stable)
fix i assume "i \ I"
with indep have "F i \ events" by (auto simp: indep_sets_def)
with sets.sets_into_space show "F i \ Pow (space M)" by auto
qed
qed
lemma (in prob_space) indep_sets_sigma_sets_iff:
assumes "\i. i \ I \ Int_stable (F i)"
shows "indep_sets (\i. sigma_sets (space M) (F i)) I \ indep_sets F I"
proof
assume "indep_sets F I" then show "indep_sets (\i. sigma_sets (space M) (F i)) I"
by (rule indep_sets_sigma) fact
next
assume "indep_sets (\i. sigma_sets (space M) (F i)) I" then show "indep_sets F I"
by (rule indep_sets_mono_sets) (intro subsetI sigma_sets.Basic)
qed
definition (in prob_space)
indep_vars_def2: "indep_vars M' X I \
(\<forall>i\<in>I. random_variable (M' i) (X i)) \<and>
indep_sets (\<lambda>i. { X i -` A \<inter> space M | A. A \<in> sets (M' i)}) I"
definition (in prob_space)
"indep_var Ma A Mb B \ indep_vars (case_bool Ma Mb) (case_bool A B) UNIV"
lemma (in prob_space) indep_vars_def:
"indep_vars M' X I \
(\<forall>i\<in>I. random_variable (M' i) (X i)) \<and>
indep_sets (\<lambda>i. sigma_sets (space M) { X i -` A \<inter> space M | A. A \<in> sets (M' i)}) I"
unfolding indep_vars_def2
apply (rule conj_cong[OF refl])
apply (rule indep_sets_sigma_sets_iff[symmetric])
apply (auto simp: Int_stable_def)
apply (rule_tac x="A \ Aa" in exI)
apply auto
done
lemma (in prob_space) indep_var_eq:
"indep_var S X T Y \
(random_variable S X \<and> random_variable T Y) \<and>
indep_set
(sigma_sets (space M) { X -` A \<inter> space M | A. A \<in> sets S})
(sigma_sets (space M) { Y -` A \<inter> space M | A. A \<in> sets T})"
unfolding indep_var_def indep_vars_def indep_set_def UNIV_bool
by (intro arg_cong2[where f="(\)"] arg_cong2[where f=indep_sets] ext)
(auto split: bool.split)
lemma (in prob_space) indep_sets2_eq:
"indep_set A B \ A \ events \ B \ events \ (\a\A. \b\B. prob (a \ b) = prob a * prob b)"
unfolding indep_set_def
proof (intro iffI ballI conjI)
assume indep: "indep_sets (case_bool A B) UNIV"
{ fix a b assume "a \ A" "b \ B"
with indep_setsD[OF indep, of UNIV "case_bool a b"]
show "prob (a \ b) = prob a * prob b"
unfolding UNIV_bool by (simp add: ac_simps) }
from indep show "A \ events" "B \ events"
unfolding indep_sets_def UNIV_bool by auto
next
assume *: "A \ events \ B \ events \ (\a\A. \b\B. prob (a \ b) = prob a * prob b)"
show "indep_sets (case_bool A B) UNIV"
proof (rule indep_setsI)
fix i show "(case i of True \ A | False \ B) \ events"
using * by (auto split: bool.split)
next
fix J X assume "J \ {}" "J \ UNIV" and X: "\j\J. X j \ (case j of True \ A | False \ B)"
then have "J = {True} \ J = {False} \ J = {True,False}"
by (auto simp: UNIV_bool)
then show "prob (\j\J. X j) = (\j\J. prob (X j))"
using X * by auto
qed
qed
lemma (in prob_space) indep_set_sigma_sets:
assumes "indep_set A B"
assumes A: "Int_stable A" and B: "Int_stable B"
shows "indep_set (sigma_sets (space M) A) (sigma_sets (space M) B)"
proof -
have "indep_sets (\i. sigma_sets (space M) (case i of True \ A | False \ B)) UNIV"
proof (rule indep_sets_sigma)
show "indep_sets (case_bool A B) UNIV"
by (rule \<open>indep_set A B\<close>[unfolded indep_set_def])
fix i show "Int_stable (case i of True \ A | False \ B)"
using A B by (cases i) auto
qed
then show ?thesis
unfolding indep_set_def
by (rule indep_sets_mono_sets) (auto split: bool.split)
qed
lemma (in prob_space) indep_eventsI_indep_vars:
assumes indep: "indep_vars N X I"
assumes P: "\i. i \ I \ {x\space (N i). P i x} \ sets (N i)"
shows "indep_events (\i. {x\space M. P i (X i x)}) I"
proof -
have "indep_sets (\i. {X i -` A \ space M |A. A \ sets (N i)}) I"
using indep unfolding indep_vars_def2 by auto
then show ?thesis
unfolding indep_events_def_alt
proof (rule indep_sets_mono_sets)
fix i assume "i \ I"
then have "{{x \ space M. P i (X i x)}} = {X i -` {x\space (N i). P i x} \ space M}"
using indep by (auto simp: indep_vars_def dest: measurable_space)
also have "\ \ {X i -` A \ space M |A. A \ sets (N i)}"
using P[OF \<open>i \<in> I\<close>] by blast
finally show "{{x \ space M. P i (X i x)}} \ {X i -` A \ space M |A. A \ sets (N i)}" .
qed
qed
lemma (in prob_space) indep_sets_collect_sigma:
fixes I :: "'j \ 'i set" and J :: "'j set" and E :: "'i \ 'a set set"
assumes indep: "indep_sets E (\j\J. I j)"
assumes Int_stable: "\i j. j \ J \ i \ I j \ Int_stable (E i)"
assumes disjoint: "disjoint_family_on I J"
shows "indep_sets (\j. sigma_sets (space M) (\i\I j. E i)) J"
proof -
let ?E = "\j. {\k\K. E' k| E' K. finite K \ K \ {} \ K \ I j \ (\k\K. E' k \ E k) }"
from indep have E: "\j i. j \ J \ i \ I j \ E i \ events"
unfolding indep_sets_def by auto
{ fix j
let ?S = "sigma_sets (space M) (\i\I j. E i)"
assume "j \ J"
from E[OF this] interpret S: sigma_algebra "space M" ?S
using sets.sets_into_space[of _ M] by (intro sigma_algebra_sigma_sets) auto
have "sigma_sets (space M) (\i\I j. E i) = sigma_sets (space M) (?E j)"
proof (rule sigma_sets_eqI)
fix A assume "A \ (\i\I j. E i)"
then guess i ..
then show "A \ sigma_sets (space M) (?E j)"
by (auto intro!: sigma_sets.intros(2-) exI[of _ "{i}"] exI[of _ "\i. A"])
next
fix A assume "A \ ?E j"
then obtain E' K where "finite K" "K \ {}" "K \ I j" "\k. k \ K \ E' k \ E k"
and A: "A = (\k\K. E' k)"
by auto
then have "A \ ?S" unfolding A
by (safe intro!: S.finite_INT) auto
then show "A \ sigma_sets (space M) (\i\I j. E i)"
by simp
qed }
moreover have "indep_sets (\j. sigma_sets (space M) (?E j)) J"
proof (rule indep_sets_sigma)
show "indep_sets ?E J"
proof (intro indep_setsI)
fix j assume "j \ J" with E show "?E j \ events" by (force intro!: sets.finite_INT)
next
fix K A assume K: "K \ {}" "K \ J" "finite K"
and "\j\K. A j \ ?E j"
then have "\j\K. \E' L. A j = (\l\L. E' l) \ finite L \ L \ {} \ L \ I j \ (\l\L. E' l \ E l)"
by simp
from bchoice[OF this] guess E' ..
from bchoice[OF this] obtain L
where A: "\j. j\K \ A j = (\l\L j. E' j l)"
and L: "\j. j\K \ finite (L j)" "\j. j\K \ L j \ {}" "\j. j\K \ L j \ I j"
and E': "\j l. j\K \ l \ L j \ E' j l \ E l"
by auto
{ fix k l j assume "k \ K" "j \ K" "l \ L j" "l \ L k"
have "k = j"
proof (rule ccontr)
assume "k \ j"
with disjoint \<open>K \<subseteq> J\<close> \<open>k \<in> K\<close> \<open>j \<in> K\<close> have "I k \<inter> I j = {}"
unfolding disjoint_family_on_def by auto
with L(2,3)[OF \<open>j \<in> K\<close>] L(2,3)[OF \<open>k \<in> K\<close>]
show False using \<open>l \<in> L k\<close> \<open>l \<in> L j\<close> by auto
qed }
note L_inj = this
define k where "k l = (SOME k. k \ K \ l \ L k)" for l
{ fix x j l assume *: "j \ K" "l \ L j"
have "k l = j" unfolding k_def
proof (rule some_equality)
fix k assume "k \ K \ l \ L k"
with * L_inj show "k = j" by auto
qed (insert *, simp) }
note k_simp[simp] = this
let ?E' = "\l. E' (k l) l"
have "prob (\j\K. A j) = prob (\l\(\k\K. L k). ?E' l)"
by (auto simp: A intro!: arg_cong[where f=prob])
also have "\ = (\l\(\k\K. L k). prob (?E' l))"
using L K E' by (intro indep_setsD[OF indep]) (simp_all add: UN_mono)
also have "\ = (\j\K. \l\L j. prob (E' j l))"
using K L L_inj by (subst prod.UNION_disjoint) auto
also have "\ = (\j\K. prob (A j))"
using K L E' by (auto simp add: A intro!: prod.cong indep_setsD[OF indep, symmetric]) blast
finally show "prob (\j\K. A j) = (\j\K. prob (A j))" .
qed
next
fix j assume "j \ J"
show "Int_stable (?E j)"
proof (rule Int_stableI)
fix a assume "a \ ?E j" then obtain Ka Ea
where a: "a = (\k\Ka. Ea k)" "finite Ka" "Ka \ {}" "Ka \ I j" "\k. k\Ka \ Ea k \ E k" by auto
fix b assume "b \ ?E j" then obtain Kb Eb
where b: "b = (\k\Kb. Eb k)" "finite Kb" "Kb \ {}" "Kb \ I j" "\k. k\Kb \ Eb k \ E k" by auto
let ?f = "\k. (if k \ Ka \ Kb then Ea k \ Eb k else if k \ Kb then Eb k else if k \ Ka then Ea k else {})"
have "Ka \ Kb = (Ka \ Kb) \ (Kb - Ka) \ (Ka - Kb)"
by blast
moreover have "(\x\Ka \ Kb. Ea x \ Eb x) \
(\<Inter>x\<in>Kb - Ka. Eb x) \<inter> (\<Inter>x\<in>Ka - Kb. Ea x) = (\<Inter>k\<in>Ka. Ea k) \<inter> (\<Inter>k\<in>Kb. Eb k)"
by auto
ultimately have "(\k\Ka \ Kb. ?f k) = (\k\Ka. Ea k) \ (\k\Kb. Eb k)" (is "?lhs = ?rhs")
by (simp only: image_Un Inter_Un_distrib) simp
then have "a \ b = (\k\Ka \ Kb. ?f k)"
by (simp only: a(1) b(1))
with a b \<open>j \<in> J\<close> Int_stableD[OF Int_stable] show "a \<inter> b \<in> ?E j"
by (intro CollectI exI[of _ "Ka \ Kb"] exI[of _ ?f]) auto
qed
qed
ultimately show ?thesis
by (simp cong: indep_sets_cong)
qed
lemma (in prob_space) indep_vars_restrict:
assumes ind: "indep_vars M' X I" and K: "\j. j \ L \ K j \ I" and J: "disjoint_family_on K L"
shows "indep_vars (\j. PiM (K j) M') (\j \. restrict (\i. X i \) (K j)) L"
unfolding indep_vars_def
proof safe
fix j assume "j \ L" then show "random_variable (Pi\<^sub>M (K j) M') (\\. \i\K j. X i \)"
using K ind by (auto simp: indep_vars_def intro!: measurable_restrict)
next
have X: "\i. i \ I \ X i \ measurable M (M' i)"
using ind by (auto simp: indep_vars_def)
let ?proj = "\j S. {(\\. \i\K j. X i \) -` A \ space M |A. A \ S}"
let ?UN = "\j. sigma_sets (space M) (\i\K j. { X i -` A \ space M| A. A \ sets (M' i) })"
show "indep_sets (\i. sigma_sets (space M) (?proj i (sets (Pi\<^sub>M (K i) M')))) L"
proof (rule indep_sets_mono_sets)
fix j assume j: "j \ L"
have "sigma_sets (space M) (?proj j (sets (Pi\<^sub>M (K j) M'))) =
sigma_sets (space M) (sigma_sets (space M) (?proj j (prod_algebra (K j) M')))"
using j K X[THEN measurable_space] unfolding sets_PiM
by (subst sigma_sets_vimage_commute) (auto simp add: Pi_iff)
also have "\ = sigma_sets (space M) (?proj j (prod_algebra (K j) M'))"
by (rule sigma_sets_sigma_sets_eq) auto
also have "\ \ ?UN j"
proof (rule sigma_sets_mono, safe del: disjE elim!: prod_algebraE)
fix J E assume J: "finite J" "J \ {} \ K j = {}" "J \ K j" and E: "\i. i \ J \ E i \ sets (M' i)"
show "(\\. \i\K j. X i \) -` prod_emb (K j) M' J (Pi\<^sub>E J E) \ space M \ ?UN j"
proof cases
assume "K j = {}" with J show ?thesis
by (auto simp add: sigma_sets_empty_eq prod_emb_def)
next
assume "K j \ {}" with J have "J \ {}"
by auto
{ interpret sigma_algebra "space M" "?UN j"
by (rule sigma_algebra_sigma_sets) auto
have "\A. (\i. i \ J \ A i \ ?UN j) \ \(A ` J) \ ?UN j"
using \<open>finite J\<close> \<open>J \<noteq> {}\<close> by (rule finite_INT) blast }
note INT = this
from \<open>J \<noteq> {}\<close> J K E[rule_format, THEN sets.sets_into_space] j
have "(\\. \i\K j. X i \) -` prod_emb (K j) M' J (Pi\<^sub>E J E) \ space M
= (\<Inter>i\<in>J. X i -` E i \<inter> space M)"
apply (subst prod_emb_PiE[OF _ ])
apply auto []
apply auto []
apply (auto simp add: Pi_iff intro!: X[THEN measurable_space])
apply (erule_tac x=i in ballE)
apply auto
done
also have "\ \ ?UN j"
apply (rule INT)
apply (rule sigma_sets.Basic)
using \<open>J \<subseteq> K j\<close> E
apply auto
done
finally show ?thesis .
qed
qed
finally show "sigma_sets (space M) (?proj j (sets (Pi\<^sub>M (K j) M'))) \ ?UN j" .
next
show "indep_sets ?UN L"
proof (rule indep_sets_collect_sigma)
show "indep_sets (\i. {X i -` A \ space M |A. A \ sets (M' i)}) (\j\L. K j)"
proof (rule indep_sets_mono_index)
show "indep_sets (\i. {X i -` A \ space M |A. A \ sets (M' i)}) I"
using ind unfolding indep_vars_def2 by auto
show "(\l\L. K l) \ I"
using K by auto
qed
next
fix l i assume "l \ L" "i \ K l"
show "Int_stable {X i -` A \ space M |A. A \ sets (M' i)}"
apply (auto simp: Int_stable_def)
apply (rule_tac x="A \ Aa" in exI)
apply auto
done
qed fact
qed
qed
lemma (in prob_space) indep_var_restrict:
assumes ind: "indep_vars M' X I" and AB: "A \ B = {}" "A \ I" "B \ I"
shows "indep_var (PiM A M') (\\. restrict (\i. X i \) A) (PiM B M') (\\. restrict (\i. X i \) B)"
proof -
have *:
"case_bool (Pi\<^sub>M A M') (Pi\<^sub>M B M') = (\b. PiM (case_bool A B b) M')"
"case_bool (\\. \i\A. X i \) (\\. \i\B. X i \) = (\b \. \i\case_bool A B b. X i \)"
by (simp_all add: fun_eq_iff split: bool.split)
show ?thesis
unfolding indep_var_def * using AB
by (intro indep_vars_restrict[OF ind]) (auto simp: disjoint_family_on_def split: bool.split)
qed
lemma (in prob_space) indep_vars_subset:
assumes "indep_vars M' X I" "J \ I"
shows "indep_vars M' X J"
using assms unfolding indep_vars_def indep_sets_def
by auto
lemma (in prob_space) indep_vars_cong:
"I = J \ (\i. i \ I \ X i = Y i) \ (\i. i \ I \ M' i = N' i) \ indep_vars M' X I \ indep_vars N' Y J"
unfolding indep_vars_def2 by (intro conj_cong indep_sets_cong) auto
definition (in prob_space) tail_events where
"tail_events A = (\n. sigma_sets (space M) (\ (A ` {n..})))"
lemma (in prob_space) tail_events_sets:
assumes A: "\i::nat. A i \ events"
shows "tail_events A \ events"
proof
fix X assume X: "X \ tail_events A"
let ?A = "(\n. sigma_sets (space M) (\ (A ` {n..})))"
from X have "\n::nat. X \ sigma_sets (space M) (\ (A ` {n..}))" by (auto simp: tail_events_def)
from this[of 0] have "X \ sigma_sets (space M) (\(A ` UNIV))" by simp
then show "X \ events"
by induct (insert A, auto)
qed
lemma (in prob_space) sigma_algebra_tail_events:
assumes "\i::nat. sigma_algebra (space M) (A i)"
shows "sigma_algebra (space M) (tail_events A)"
unfolding tail_events_def
proof (simp add: sigma_algebra_iff2, safe)
let ?A = "(\n. sigma_sets (space M) (\ (A ` {n..})))"
interpret A: sigma_algebra "space M" "A i" for i by fact
{ fix X x assume "X \ ?A" "x \ X"
then have "\n. X \ sigma_sets (space M) (\ (A ` {n..}))" by auto
from this[of 0] have "X \ sigma_sets (space M) (\(A ` UNIV))" by simp
then have "X \ space M"
by induct (insert A.sets_into_space, auto)
with \<open>x \<in> X\<close> show "x \<in> space M" by auto }
{ fix F :: "nat \ 'a set" and n assume "range F \ ?A"
then show "(\(F ` UNIV)) \ sigma_sets (space M) (\ (A ` {n..}))"
by (intro sigma_sets.Union) auto }
qed (auto intro!: sigma_sets.Compl sigma_sets.Empty)
lemma (in prob_space) kolmogorov_0_1_law:
fixes A :: "nat \ 'a set set"
assumes "\i::nat. sigma_algebra (space M) (A i)"
assumes indep: "indep_sets A UNIV"
and X: "X \ tail_events A"
shows "prob X = 0 \ prob X = 1"
proof -
have A: "\i. A i \ events"
using indep unfolding indep_sets_def by simp
let ?D = "{D \ events. prob (X \ D) = prob X * prob D}"
interpret A: sigma_algebra "space M" "A i" for i by fact
interpret T: sigma_algebra "space M" "tail_events A"
by (rule sigma_algebra_tail_events) fact
have "X \ space M" using T.space_closed X by auto
have X_in: "X \ events"
using tail_events_sets A X by auto
interpret D: Dynkin_system "space M" ?D
proof (rule Dynkin_systemI)
fix D assume "D \ ?D" then show "D \ space M"
using sets.sets_into_space by auto
next
show "space M \ ?D"
using prob_space \<open>X \<subseteq> space M\<close> by (simp add: Int_absorb2)
next
fix A assume A: "A \ ?D"
have "prob (X \ (space M - A)) = prob (X - (X \ A))"
using \<open>X \<subseteq> space M\<close> by (auto intro!: arg_cong[where f=prob])
also have "\ = prob X - prob (X \ A)"
using X_in A by (intro finite_measure_Diff) auto
also have "\ = prob X * prob (space M) - prob X * prob A"
using A prob_space by auto
also have "\ = prob X * prob (space M - A)"
using X_in A sets.sets_into_space
by (subst finite_measure_Diff) (auto simp: field_simps)
finally show "space M - A \ ?D"
using A \<open>X \<subseteq> space M\<close> by auto
next
fix F :: "nat \ 'a set" assume dis: "disjoint_family F" and "range F \ ?D"
then have F: "range F \ events" "\i. prob (X \ F i) = prob X * prob (F i)"
by auto
have "(\i. prob (X \ F i)) sums prob (\i. X \ F i)"
proof (rule finite_measure_UNION)
show "range (\i. X \ F i) \ events"
using F X_in by auto
show "disjoint_family (\i. X \ F i)"
using dis by (rule disjoint_family_on_bisimulation) auto
qed
with F have "(\i. prob X * prob (F i)) sums prob (X \ (\i. F i))"
by simp
moreover have "(\i. prob X * prob (F i)) sums (prob X * prob (\i. F i))"
by (intro sums_mult finite_measure_UNION F dis)
ultimately have "prob (X \ (\i. F i)) = prob X * prob (\i. F i)"
by (auto dest!: sums_unique)
with F show "(\i. F i) \ ?D"
by auto
qed
{ fix n
have "indep_sets (\b. sigma_sets (space M) (\m\case_bool {..n} {Suc n..} b. A m)) UNIV"
proof (rule indep_sets_collect_sigma)
have *: "(\b. case b of True \ {..n} | False \ {Suc n..}) = UNIV" (is "?U = _")
by (simp split: bool.split add: set_eq_iff) (metis not_less_eq_eq)
with indep show "indep_sets A ?U" by simp
show "disjoint_family (case_bool {..n} {Suc n..})"
unfolding disjoint_family_on_def by (auto split: bool.split)
fix m
show "Int_stable (A m)"
unfolding Int_stable_def using A.Int by auto
qed
also have "(\b. sigma_sets (space M) (\m\case_bool {..n} {Suc n..} b. A m)) =
case_bool (sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) (sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m))"
by (auto intro!: ext split: bool.split)
finally have indep: "indep_set (sigma_sets (space M) (\m\{..n}. A m)) (sigma_sets (space M) (\m\{Suc n..}. A m))"
unfolding indep_set_def by simp
have "sigma_sets (space M) (\m\{..n}. A m) \ ?D"
proof (simp add: subset_eq, rule)
fix D assume D: "D \ sigma_sets (space M) (\m\{..n}. A m)"
have "X \ sigma_sets (space M) (\m\{Suc n..}. A m)"
using X unfolding tail_events_def by simp
from indep_setD[OF indep D this] indep_setD_ev1[OF indep] D
show "D \ events \ prob (X \ D) = prob X * prob D"
by (auto simp add: ac_simps)
qed }
then have "(\n. sigma_sets (space M) (\m\{..n}. A m)) \ ?D" (is "?A \ _")
by auto
note \<open>X \<in> tail_events A\<close>
also {
have "\n. sigma_sets (space M) (\i\{n..}. A i) \ sigma_sets (space M) ?A"
by (intro sigma_sets_subseteq UN_mono) auto
then have "tail_events A \ sigma_sets (space M) ?A"
unfolding tail_events_def by auto }
also have "sigma_sets (space M) ?A = Dynkin (space M) ?A"
proof (rule sigma_eq_Dynkin)
{ fix B n assume "B \ sigma_sets (space M) (\m\{..n}. A m)"
then have "B \ space M"
by induct (insert A sets.sets_into_space[of _ M], auto) }
then show "?A \ Pow (space M)" by auto
show "Int_stable ?A"
proof (rule Int_stableI)
fix a assume "a \ ?A" then guess n .. note a = this
fix b assume "b \ ?A" then guess m .. note b = this
interpret Amn: sigma_algebra "space M" "sigma_sets (space M) (\i\{..max m n}. A i)"
using A sets.sets_into_space[of _ M] by (intro sigma_algebra_sigma_sets) auto
have "sigma_sets (space M) (\i\{..n}. A i) \ sigma_sets (space M) (\i\{..max m n}. A i)"
by (intro sigma_sets_subseteq UN_mono) auto
with a have "a \ sigma_sets (space M) (\i\{..max m n}. A i)" by auto
moreover
have "sigma_sets (space M) (\i\{..m}. A i) \ sigma_sets (space M) (\i\{..max m n}. A i)"
by (intro sigma_sets_subseteq UN_mono) auto
with b have "b \ sigma_sets (space M) (\i\{..max m n}. A i)" by auto
ultimately have "a \ b \ sigma_sets (space M) (\i\{..max m n}. A i)"
using Amn.Int[of a b] by simp
then show "a \ b \ (\n. sigma_sets (space M) (\i\{..n}. A i))" by auto
qed
qed
also have "Dynkin (space M) ?A \ ?D"
using \<open>?A \<subseteq> ?D\<close> by (auto intro!: D.Dynkin_subset)
finally show ?thesis by auto
qed
lemma (in prob_space) borel_0_1_law:
fixes F :: "nat \ 'a set"
assumes F2: "indep_events F UNIV"
shows "prob (\n. \m\{n..}. F m) = 0 \ prob (\n. \m\{n..}. F m) = 1"
proof (rule kolmogorov_0_1_law[of "\i. sigma_sets (space M) { F i }"])
have F1: "range F \ events"
using F2 by (simp add: indep_events_def subset_eq)
{ fix i show "sigma_algebra (space M) (sigma_sets (space M) {F i})"
using sigma_algebra_sigma_sets[of "{F i}" "space M"] F1 sets.sets_into_space
by auto }
show "indep_sets (\i. sigma_sets (space M) {F i}) UNIV"
proof (rule indep_sets_sigma)
show "indep_sets (\i. {F i}) UNIV"
unfolding indep_events_def_alt[symmetric] by fact
fix i show "Int_stable {F i}"
unfolding Int_stable_def by simp
qed
let ?Q = "\n. \i\{n..}. F i"
show "(\n. \m\{n..}. F m) \ tail_events (\i. sigma_sets (space M) {F i})"
unfolding tail_events_def
proof
fix j
interpret S: sigma_algebra "space M" "sigma_sets (space M) (\i\{j..}. sigma_sets (space M) {F i})"
using order_trans[OF F1 sets.space_closed]
by (intro sigma_algebra_sigma_sets) (simp add: sigma_sets_singleton subset_eq)
have "(\n. ?Q n) = (\n\{j..}. ?Q n)"
by (intro decseq_SucI INT_decseq_offset UN_mono) auto
also have "\ \ sigma_sets (space M) (\i\{j..}. sigma_sets (space M) {F i})"
using order_trans[OF F1 sets.space_closed]
by (safe intro!: S.countable_INT S.countable_UN)
(auto simp: sigma_sets_singleton intro!: sigma_sets.Basic bexI)
finally show "(\n. ?Q n) \ sigma_sets (space M) (\i\{j..}. sigma_sets (space M) {F i})"
by simp
qed
qed
lemma (in prob_space) borel_0_1_law_AE:
fixes P :: "nat \ 'a \ bool"
assumes "indep_events (\m. {x\space M. P m x}) UNIV" (is "indep_events ?P _")
shows "(AE x in M. infinite {m. P m x}) \ (AE x in M. finite {m. P m x})"
proof -
have [measurable]: "\m. {x\space M. P m x} \ sets M"
using assms by (auto simp: indep_events_def)
have *: "(\n. \m\{n..}. {x \ space M. P m x}) \ events"
by simp
from assms have "prob (\n. \m\{n..}. ?P m) = 0 \ prob (\n. \m\{n..}. ?P m) = 1"
by (rule borel_0_1_law)
also have "prob (\n. \m\{n..}. ?P m) = 1 \ (AE x in M. infinite {m. P m x})"
using * by (simp add: prob_eq_1)
(simp add: Bex_def infinite_nat_iff_unbounded_le)
also have "prob (\n. \m\{n..}. ?P m) = 0 \ (AE x in M. finite {m. P m x})"
using * by (simp add: prob_eq_0)
(auto simp add: Ball_def finite_nat_iff_bounded not_less [symmetric])
finally show ?thesis
by blast
qed
lemma (in prob_space) indep_sets_finite:
assumes I: "I \ {}" "finite I"
and F: "\i. i \ I \ F i \ events" "\i. i \ I \ space M \ F i"
shows "indep_sets F I \ (\A\Pi I F. prob (\j\I. A j) = (\j\I. prob (A j)))"
proof
assume *: "indep_sets F I"
from I show "\A\Pi I F. prob (\j\I. A j) = (\j\I. prob (A j))"
by (intro indep_setsD[OF *] ballI) auto
next
assume indep: "\A\Pi I F. prob (\j\I. A j) = (\j\I. prob (A j))"
show "indep_sets F I"
proof (rule indep_setsI[OF F(1)])
fix A J assume J: "J \ {}" "J \ I" "finite J"
assume A: "\j\J. A j \ F j"
let ?A = "\j. if j \ J then A j else space M"
have "prob (\j\I. ?A j) = prob (\j\J. A j)"
using subset_trans[OF F(1) sets.space_closed] J A
by (auto intro!: arg_cong[where f=prob] split: if_split_asm) blast
also
from A F have "(\j. if j \ J then A j else space M) \ Pi I F" (is "?A \ _")
by (auto split: if_split_asm)
with indep have "prob (\j\I. ?A j) = (\j\I. prob (?A j))"
by auto
also have "\ = (\j\J. prob (A j))"
unfolding if_distrib prod.If_cases[OF \<open>finite I\<close>]
using prob_space \<open>J \<subseteq> I\<close> by (simp add: Int_absorb1 prod.neutral_const)
finally show "prob (\j\J. A j) = (\j\J. prob (A j))" ..
qed
qed
lemma (in prob_space) indep_vars_finite:
fixes I :: "'i set"
assumes I: "I \ {}" "finite I"
and M': "\i. i \ I \ sets (M' i) = sigma_sets (space (M' i)) (E i)"
and rv: "\i. i \ I \ random_variable (M' i) (X i)"
and Int_stable: "\i. i \ I \ Int_stable (E i)"
and space: "\i. i \ I \ space (M' i) \ E i" and closed: "\i. i \ I \ E i \ Pow (space (M' i))"
shows "indep_vars M' X I \
(\<forall>A\<in>(\<Pi> i\<in>I. E i). prob (\<Inter>j\<in>I. X j -` A j \<inter> space M) = (\<Prod>j\<in>I. prob (X j -` A j \<inter> space M)))"
proof -
from rv have X: "\i. i \ I \ X i \ space M \ space (M' i)"
unfolding measurable_def by simp
{ fix i assume "i\I"
from closed[OF \<open>i \<in> I\<close>]
have "sigma_sets (space M) {X i -` A \ space M |A. A \ sets (M' i)}
= sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> E i}"
unfolding sigma_sets_vimage_commute[OF X, OF \<open>i \<in> I\<close>, symmetric] M'[OF \<open>i \<in> I\<close>]
by (subst sigma_sets_sigma_sets_eq) auto }
note sigma_sets_X = this
{ fix i assume "i\I"
have "Int_stable {X i -` A \ space M |A. A \ E i}"
proof (rule Int_stableI)
fix a assume "a \ {X i -` A \ space M |A. A \ E i}"
then obtain A where "a = X i -` A \ space M" "A \ E i" by auto
moreover
fix b assume "b \ {X i -` A \ space M |A. A \ E i}"
then obtain B where "b = X i -` B \ space M" "B \ E i" by auto
moreover
have "(X i -` A \ space M) \ (X i -` B \ space M) = X i -` (A \ B) \ space M" by auto
moreover note Int_stable[OF \<open>i \<in> I\<close>]
ultimately
show "a \ b \ {X i -` A \ space M |A. A \ E i}"
by (auto simp del: vimage_Int intro!: exI[of _ "A \ B"] dest: Int_stableD)
qed }
note indep_sets_X = indep_sets_sigma_sets_iff[OF this]
{ fix i assume "i \ I"
{ fix A assume "A \ E i"
with M'[OF \i \ I\] have "A \ sets (M' i)" by auto
moreover
from rv[OF \<open>i\<in>I\<close>] have "X i \<in> measurable M (M' i)" by auto
ultimately
have "X i -` A \ space M \ sets M" by (auto intro: measurable_sets) }
with X[OF \<open>i\<in>I\<close>] space[OF \<open>i\<in>I\<close>]
have "{X i -` A \ space M |A. A \ E i} \ events"
"space M \ {X i -` A \ space M |A. A \ E i}"
by (auto intro!: exI[of _ "space (M' i)"]) }
note indep_sets_finite_X = indep_sets_finite[OF I this]
have "(\A\\ i\I. {X i -` A \ space M |A. A \ E i}. prob (\(A ` I)) = (\j\I. prob (A j))) =
(\<forall>A\<in>\<Pi> i\<in>I. E i. prob ((\<Inter>j\<in>I. X j -` A j) \<inter> space M) = (\<Prod>x\<in>I. prob (X x -` A x \<inter> space M)))"
(is "?L = ?R")
proof safe
fix A assume ?L and A: "A \ (\ i\I. E i)"
from \<open>?L\<close>[THEN bspec, of "\<lambda>i. X i -` A i \<inter> space M"] A \<open>I \<noteq> {}\<close>
show "prob ((\j\I. X j -` A j) \ space M) = (\x\I. prob (X x -` A x \ space M))"
by (auto simp add: Pi_iff)
next
fix A assume ?R and A: "A \ (\ i\I. {X i -` A \ space M |A. A \ E i})"
from A have "\i\I. \B. A i = X i -` B \ space M \ B \ E i" by auto
from bchoice[OF this] obtain B where B: "\i\I. A i = X i -` B i \ space M"
"B \ (\ i\I. E i)" by auto
from \<open>?R\<close>[THEN bspec, OF B(2)] B(1) \<open>I \<noteq> {}\<close>
show "prob (\(A ` I)) = (\j\I. prob (A j))"
by simp
qed
then show ?thesis using \<open>I \<noteq> {}\<close>
by (simp add: rv indep_vars_def indep_sets_X sigma_sets_X indep_sets_finite_X cong: indep_sets_cong)
qed
lemma (in prob_space) indep_vars_compose:
assumes "indep_vars M' X I"
assumes rv: "\i. i \ I \ Y i \ measurable (M' i) (N i)"
shows "indep_vars N (\i. Y i \ X i) I"
unfolding indep_vars_def
proof
from rv \<open>indep_vars M' X I\<close>
show "\i\I. random_variable (N i) (Y i \ X i)"
by (auto simp: indep_vars_def)
have "indep_sets (\i. sigma_sets (space M) {X i -` A \ space M |A. A \ sets (M' i)}) I"
using \<open>indep_vars M' X I\<close> by (simp add: indep_vars_def)
then show "indep_sets (\i. sigma_sets (space M) {(Y i \ X i) -` A \ space M |A. A \ sets (N i)}) I"
proof (rule indep_sets_mono_sets)
fix i assume "i \ I"
with \<open>indep_vars M' X I\<close> have X: "X i \<in> space M \<rightarrow> space (M' i)"
unfolding indep_vars_def measurable_def by auto
{ fix A assume "A \ sets (N i)"
then have "\B. (Y i \ X i) -` A \ space M = X i -` B \ space M \ B \ sets (M' i)"
by (intro exI[of _ "Y i -` A \ space (M' i)"])
(auto simp: vimage_comp intro!: measurable_sets rv \<open>i \<in> I\<close> funcset_mem[OF X]) }
then show "sigma_sets (space M) {(Y i \ X i) -` A \ space M |A. A \ sets (N i)} \
sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"
by (intro sigma_sets_subseteq) (auto simp: vimage_comp)
qed
qed
lemma (in prob_space) indep_vars_compose2:
assumes "indep_vars M' X I"
assumes rv: "\i. i \ I \ Y i \ measurable (M' i) (N i)"
shows "indep_vars N (\i x. Y i (X i x)) I"
using indep_vars_compose [OF assms] by (simp add: comp_def)
lemma (in prob_space) indep_var_compose:
assumes "indep_var M1 X1 M2 X2" "Y1 \ measurable M1 N1" "Y2 \ measurable M2 N2"
shows "indep_var N1 (Y1 \ X1) N2 (Y2 \ X2)"
proof -
have "indep_vars (case_bool N1 N2) (\b. case_bool Y1 Y2 b \ case_bool X1 X2 b) UNIV"
using assms
by (intro indep_vars_compose[where M'="case_bool M1 M2"])
(auto simp: indep_var_def split: bool.split)
also have "(\b. case_bool Y1 Y2 b \ case_bool X1 X2 b) = case_bool (Y1 \ X1) (Y2 \ X2)"
by (simp add: fun_eq_iff split: bool.split)
finally show ?thesis
unfolding indep_var_def .
qed
lemma (in prob_space) indep_vars_Min:
fixes X :: "'i \ 'a \ real"
assumes I: "finite I" "i \ I" and indep: "indep_vars (\_. borel) X (insert i I)"
shows "indep_var borel (X i) borel (\\. Min ((\i. X i \)`I))"
proof -
have "indep_var
borel ((\<lambda>f. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) {i}))
borel ((\<lambda>f. Min (f`I)) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) I))"
using I by (intro indep_var_compose[OF indep_var_restrict[OF indep]] borel_measurable_Min) auto
also have "((\f. f i) \ (\\. restrict (\i. X i \) {i})) = X i"
by auto
also have "((\f. Min (f`I)) \ (\\. restrict (\i. X i \) I)) = (\\. Min ((\i. X i \)`I))"
by (auto cong: rev_conj_cong)
finally show ?thesis
unfolding indep_var_def .
qed
lemma (in prob_space) indep_vars_sum:
fixes X :: "'i \ 'a \ real"
assumes I: "finite I" "i \ I" and indep: "indep_vars (\_. borel) X (insert i I)"
shows "indep_var borel (X i) borel (\\. \i\I. X i \)"
proof -
have "indep_var
borel ((\<lambda>f. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) {i}))
borel ((\<lambda>f. \<Sum>i\<in>I. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) I))"
using I by (intro indep_var_compose[OF indep_var_restrict[OF indep]] ) auto
also have "((\f. f i) \ (\\. restrict (\i. X i \) {i})) = X i"
by auto
also have "((\f. \i\I. f i) \ (\\. restrict (\i. X i \) I)) = (\\. \i\I. X i \)"
by (auto cong: rev_conj_cong)
finally show ?thesis .
qed
lemma (in prob_space) indep_vars_prod:
fixes X :: "'i \ 'a \ real"
assumes I: "finite I" "i \ I" and indep: "indep_vars (\_. borel) X (insert i I)"
shows "indep_var borel (X i) borel (\\. \i\I. X i \)"
proof -
have "indep_var
borel ((\<lambda>f. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) {i}))
borel ((\<lambda>f. \<Prod>i\<in>I. f i) \<circ> (\<lambda>\<omega>. restrict (\<lambda>i. X i \<omega>) I))"
using I by (intro indep_var_compose[OF indep_var_restrict[OF indep]] ) auto
also have "((\f. f i) \ (\\. restrict (\i. X i \) {i})) = X i"
by auto
also have "((\f. \i\I. f i) \ (\\. restrict (\i. X i \) I)) = (\\. \i\I. X i \)"
by (auto cong: rev_conj_cong)
finally show ?thesis .
qed
lemma (in prob_space) indep_varsD_finite:
assumes X: "indep_vars M' X I"
assumes I: "I \ {}" "finite I" "\i. i \ I \ A i \ sets (M' i)"
shows "prob (\i\I. X i -` A i \ space M) = (\i\I. prob (X i -` A i \ space M))"
proof (rule indep_setsD)
show "indep_sets (\i. sigma_sets (space M) {X i -` A \ space M |A. A \ sets (M' i)}) I"
using X by (auto simp: indep_vars_def)
show "I \ I" "I \ {}" "finite I" using I by auto
show "\i\I. X i -` A i \ space M \ sigma_sets (space M) {X i -` A \ space M |A. A \ sets (M' i)}"
using I by auto
qed
lemma (in prob_space) indep_varsD:
assumes X: "indep_vars M' X I"
assumes I: "J \ {}" "finite J" "J \ I" "\i. i \ J \ A i \ sets (M' i)"
shows "prob (\i\J. X i -` A i \ space M) = (\i\J. prob (X i -` A i \ space M))"
proof (rule indep_setsD)
show "indep_sets (\i. sigma_sets (space M) {X i -` A \ space M |A. A \ sets (M' i)}) I"
using X by (auto simp: indep_vars_def)
show "\i\J. X i -` A i \ space M \ sigma_sets (space M) {X i -` A \ space M |A. A \ sets (M' i)}"
using I by auto
qed fact+
lemma (in prob_space) indep_vars_iff_distr_eq_PiM:
fixes I :: "'i set" and X :: "'i \ 'a \ 'b"
assumes "I \ {}"
assumes rv: "\i. random_variable (M' i) (X i)"
shows "indep_vars M' X I \
distr M (\<Pi>\<^sub>M i\<in>I. M' i) (\<lambda>x. \<lambda>i\<in>I. X i x) = (\<Pi>\<^sub>M i\<in>I. distr M (M' i) (X i))"
proof -
let ?P = "\\<^sub>M i\I. M' i"
let ?X = "\x. \i\I. X i x"
let ?D = "distr M ?P ?X"
have X: "random_variable ?P ?X" by (intro measurable_restrict rv)
interpret D: prob_space ?D by (intro prob_space_distr X)
let ?D' = "\i. distr M (M' i) (X i)"
let ?P' = "\\<^sub>M i\I. distr M (M' i) (X i)"
interpret D': prob_space "?D' i" for i by (intro prob_space_distr rv)
interpret P: product_prob_space ?D' I ..
show ?thesis
proof
assume "indep_vars M' X I"
show "?D = ?P'"
proof (rule measure_eqI_generator_eq)
show "Int_stable (prod_algebra I M')"
by (rule Int_stable_prod_algebra)
show "prod_algebra I M' \ Pow (space ?P)"
using prod_algebra_sets_into_space by (simp add: space_PiM)
show "sets ?D = sigma_sets (space ?P) (prod_algebra I M')"
by (simp add: sets_PiM space_PiM)
show "sets ?P' = sigma_sets (space ?P) (prod_algebra I M')"
by (simp add: sets_PiM space_PiM cong: prod_algebra_cong)
let ?A = "\i. \\<^sub>E i\I. space (M' i)"
show "range ?A \ prod_algebra I M'" "(\i. ?A i) = space (Pi\<^sub>M I M')"
by (auto simp: space_PiM intro!: space_in_prod_algebra cong: prod_algebra_cong)
{ fix i show "emeasure ?D (\\<^sub>E i\I. space (M' i)) \ \" by auto }
next
fix E assume E: "E \ prod_algebra I M'"
from prod_algebraE[OF E] guess J Y . note J = this
from E have "E \ sets ?P" by (auto simp: sets_PiM)
then have "emeasure ?D E = emeasure M (?X -` E \ space M)"
by (simp add: emeasure_distr X)
also have "?X -` E \ space M = (\i\J. X i -` Y i \ space M)"
using J \<open>I \<noteq> {}\<close> measurable_space[OF rv] by (auto simp: prod_emb_def PiE_iff split: if_split_asm)
also have "emeasure M (\i\J. X i -` Y i \ space M) = (\ i\J. emeasure M (X i -` Y i \ space M))"
using \<open>indep_vars M' X I\<close> J \<open>I \<noteq> {}\<close> using indep_varsD[of M' X I J]
by (auto simp: emeasure_eq_measure prod_ennreal measure_nonneg prod_nonneg)
also have "\ = (\ i\J. emeasure (?D' i) (Y i))"
using rv J by (simp add: emeasure_distr)
also have "\ = emeasure ?P' E"
using P.emeasure_PiM_emb[of J Y] J by (simp add: prod_emb_def)
finally show "emeasure ?D E = emeasure ?P' E" .
qed
next
assume "?D = ?P'"
show "indep_vars M' X I" unfolding indep_vars_def
proof (intro conjI indep_setsI ballI rv)
fix i show "sigma_sets (space M) {X i -` A \ space M |A. A \ sets (M' i)} \ events"
by (auto intro!: sets.sigma_sets_subset measurable_sets rv)
next
fix J Y' assume J: "J \ {}" "J \ I" "finite J"
assume Y': "\j\J. Y' j \ sigma_sets (space M) {X j -` A \ space M |A. A \ sets (M' j)}"
have "\j\J. \Y. Y' j = X j -` Y \ space M \ Y \ sets (M' j)"
proof
fix j assume "j \ J"
from Y'[rule_format, OF this] rv[of j]
show "\Y. Y' j = X j -` Y \ space M \ Y \ sets (M' j)"
by (subst (asm) sigma_sets_vimage_commute[symmetric, of _ _ "space (M' j)"])
(auto dest: measurable_space simp: sets.sigma_sets_eq)
qed
from bchoice[OF this] obtain Y where
Y: "\j. j \ J \ Y' j = X j -` Y j \ space M" "\j. j \ J \ Y j \ sets (M' j)" by auto
let ?E = "prod_emb I M' J (Pi\<^sub>E J Y)"
from Y have "(\j\J. Y' j) = ?X -` ?E \ space M"
using J \<open>I \<noteq> {}\<close> measurable_space[OF rv] by (auto simp: prod_emb_def PiE_iff split: if_split_asm)
then have "emeasure M (\j\J. Y' j) = emeasure M (?X -` ?E \ space M)"
by simp
also have "\ = emeasure ?D ?E"
using Y J by (intro emeasure_distr[symmetric] X sets_PiM_I) auto
also have "\ = emeasure ?P' ?E"
using \<open>?D = ?P'\<close> by simp
also have "\ = (\ i\J. emeasure (?D' i) (Y i))"
using P.emeasure_PiM_emb[of J Y] J Y by (simp add: prod_emb_def)
also have "\ = (\ i\J. emeasure M (Y' i))"
using rv J Y by (simp add: emeasure_distr)
finally have "emeasure M (\j\J. Y' j) = (\ i\J. emeasure M (Y' i))" .
then show "prob (\j\J. Y' j) = (\ i\J. prob (Y' i))"
by (auto simp: emeasure_eq_measure prod_ennreal measure_nonneg prod_nonneg)
qed
qed
qed
lemma (in prob_space) indep_varD:
assumes indep: "indep_var Ma A Mb B"
assumes sets: "Xa \ sets Ma" "Xb \ sets Mb"
shows "prob ((\x. (A x, B x)) -` (Xa \ Xb) \ space M) =
prob (A -` Xa \<inter> space M) * prob (B -` Xb \<inter> space M)"
proof -
have "prob ((\x. (A x, B x)) -` (Xa \ Xb) \ space M) =
prob (\<Inter>i\<in>UNIV. (case_bool A B i -` case_bool Xa Xb i \<inter> space M))"
by (auto intro!: arg_cong[where f=prob] simp: UNIV_bool)
also have "\ = (\i\UNIV. prob (case_bool A B i -` case_bool Xa Xb i \ space M))"
using indep unfolding indep_var_def
by (rule indep_varsD) (auto split: bool.split intro: sets)
also have "\ = prob (A -` Xa \ space M) * prob (B -` Xb \ space M)"
unfolding UNIV_bool by simp
finally show ?thesis .
qed
lemma (in prob_space) prob_indep_random_variable:
assumes ind[simp]: "indep_var N X N Y"
assumes [simp]: "A \ sets N" "B \ sets N"
shows "\(x in M. X x \ A \ Y x \ B) = \(x in M. X x \ A) * \(x in M. Y x \ B)"
proof-
have " \(x in M. (X x)\A \ (Y x)\ B ) = prob ((\x. (X x, Y x)) -` (A \ B) \ space M)"
by (auto intro!: arg_cong[where f= prob])
also have "...= prob (X -` A \ space M) * prob (Y -` B \ space M)"
by (auto intro!: indep_varD[where Ma=N and Mb=N])
also have "... = \(x in M. X x \ A) * \(x in M. Y x \ B)"
by (auto intro!: arg_cong2[where f= "(*)"] arg_cong[where f= prob])
finally show ?thesis .
qed
lemma (in prob_space)
assumes "indep_var S X T Y"
shows indep_var_rv1: "random_variable S X"
and indep_var_rv2: "random_variable T Y"
proof -
have "\i\UNIV. random_variable (case_bool S T i) (case_bool X Y i)"
using assms unfolding indep_var_def indep_vars_def by auto
then show "random_variable S X" "random_variable T Y"
unfolding UNIV_bool by auto
qed
lemma (in prob_space) indep_var_distribution_eq:
"indep_var S X T Y \ random_variable S X \ random_variable T Y \
distr M S X \<Otimes>\<^sub>M distr M T Y = distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))" (is "_ \<longleftrightarrow> _ \<and> _ \<and> ?S \<Otimes>\<^sub>M ?T = ?J")
proof safe
assume "indep_var S X T Y"
then show rvs: "random_variable S X" "random_variable T Y"
by (blast dest: indep_var_rv1 indep_var_rv2)+
then have XY: "random_variable (S \\<^sub>M T) (\x. (X x, Y x))"
by (rule measurable_Pair)
interpret X: prob_space ?S by (rule prob_space_distr) fact
interpret Y: prob_space ?T by (rule prob_space_distr) fact
interpret XY: pair_prob_space ?S ?T ..
show "?S \\<^sub>M ?T = ?J"
proof (rule pair_measure_eqI)
show "sigma_finite_measure ?S" ..
show "sigma_finite_measure ?T" ..
fix A B assume A: "A \ sets ?S" and B: "B \ sets ?T"
have "emeasure ?J (A \ B) = emeasure M ((\x. (X x, Y x)) -` (A \ B) \ space M)"
using A B by (intro emeasure_distr[OF XY]) auto
also have "\ = emeasure M (X -` A \ space M) * emeasure M (Y -` B \ space M)"
using indep_varD[OF \<open>indep_var S X T Y\<close>, of A B] A B
by (simp add: emeasure_eq_measure measure_nonneg ennreal_mult)
also have "\ = emeasure ?S A * emeasure ?T B"
using rvs A B by (simp add: emeasure_distr)
finally show "emeasure ?S A * emeasure ?T B = emeasure ?J (A \ B)" by simp
qed simp
next
assume rvs: "random_variable S X" "random_variable T Y"
then have XY: "random_variable (S \\<^sub>M T) (\x. (X x, Y x))"
by (rule measurable_Pair)
let ?S = "distr M S X" and ?T = "distr M T Y"
interpret X: prob_space ?S by (rule prob_space_distr) fact
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