(* Title: HOL/Probability/Independent_Family.thy
Author: Johannes Hölzl, TU München
*)
header {* Independent families of events, event sets, and random variables *}
theory Independent_Family
imports Probability_Measure
begin
definition (in prob_space)
"indep_events A I \<longleftrightarrow> (A`I \<subseteq> sets M) \<and>
(\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j)))"
definition (in prob_space)
"indep_event A B \<longleftrightarrow> indep_events (bool_case A B) UNIV"
definition (in prob_space)
"indep_sets F I \<longleftrightarrow> (\<forall>i\<in>I. F i \<subseteq> sets M) \<and>
(\<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 \<longleftrightarrow> indep_sets (bool_case A B) UNIV"
definition (in prob_space)
"indep_rv M' X I \<longleftrightarrow>
(\<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"
lemma (in prob_space) indep_sets_cong:
"I = J \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> F i = G i) \<Longrightarrow> indep_sets F I \<longleftrightarrow> 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 \<longleftrightarrow> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_events F J)"
by (auto simp: indep_events_def)
lemma (in prob_space) indep_sets_finite_index_sets:
"indep_sets F I \<longleftrightarrow> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_sets F J)"
proof (intro iffI allI impI)
assume *: "\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_sets F J"
show "indep_sets F I" unfolding indep_sets_def
proof (intro conjI ballI allI impI)
fix i assume "i \<in> I"
with *[THEN spec, of "{i}"] show "F i \<subseteq> 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 \<subseteq> I \<Longrightarrow> indep_sets F I \<Longrightarrow> 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: "\<And>i. i\<in>I \<Longrightarrow> G i \<subseteq> F i"
shows "indep_sets G I"
proof -
have "(\<forall>i\<in>I. F i \<subseteq> events) \<Longrightarrow> (\<forall>i\<in>I. G i \<subseteq> events)"
using mono by auto
moreover have "\<And>A J. J \<subseteq> I \<Longrightarrow> A \<in> (\<Pi> j\<in>J. G j) \<Longrightarrow> A \<in> (\<Pi> j\<in>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_setsI:
assumes "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events"
and "\<And>A J. J \<noteq> {} \<Longrightarrow> J \<subseteq> I \<Longrightarrow> finite J \<Longrightarrow> (\<forall>j\<in>J. A j \<in> F j) \<Longrightarrow> prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>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 \<subseteq> I" "J \<noteq> {}" "finite J" "\<forall>j\<in>J. A j \<in> F j"
shows "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
using assms unfolding indep_sets_def by auto
lemma (in prob_space) indep_setI:
assumes ev: "A \<subseteq> events" "B \<subseteq> events"
and indep: "\<And>a b. a \<in> A \<Longrightarrow> b \<in> B \<Longrightarrow> prob (a \<inter> b) = prob a * prob b"
shows "indep_set A B"
unfolding indep_set_def
proof (rule indep_setsI)
fix F J assume "J \<noteq> {}" "J \<subseteq> UNIV"
and F: "\<forall>j\<in>J. F j \<in> (case j of True \<Rightarrow> A | False \<Rightarrow> B)"
have "J \<in> Pow UNIV" by auto
with F `J \<noteq> {}` indep[of "F True" "F False"]
show "prob (\<Inter>j\<in>J. F j) = (\<Prod>j\<in>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 \<in> A" "b \<in> B"
shows "prob (a \<inter> b) = prob a * prob b"
using indep[unfolded indep_set_def, THEN indep_setsD, of UNIV "bool_case 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 \<subseteq> sets M"
and indep_setD_ev2: "B \<subseteq> sets M"
using indep unfolding indep_set_def indep_sets_def UNIV_bool by auto
lemma dynkin_systemI':
assumes 1: "\<And> A. A \<in> sets M \<Longrightarrow> A \<subseteq> space M"
assumes empty: "{} \<in> sets M"
assumes Diff: "\<And> A. A \<in> sets M \<Longrightarrow> space M - A \<in> sets M"
assumes 2: "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> sets M
\<Longrightarrow> (\<Union>i::nat. A i) \<in> sets M"
shows "dynkin_system M"
proof -
from Diff[OF empty] have "space M \<in> sets M" by auto
from 1 this Diff 2 show ?thesis
by (intro dynkin_systemI) auto
qed
lemma (in prob_space) indep_sets_dynkin:
assumes indep: "indep_sets F I"
shows "indep_sets (\<lambda>i. sets (dynkin \<lparr> space = space M, sets = F i \<rparr>)) I"
(is "indep_sets ?F I")
proof (subst indep_sets_finite_index_sets, intro allI impI ballI)
fix J assume "finite J" "J \<subseteq> I" "J \<noteq> {}"
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 \<in> S then ?F i else F i"
assume "finite J" "J \<subseteq> K"
then have "indep_sets (?G J) K"
proof induct
case (insert j J)
moreover def G \<equiv> "?G J"
ultimately have G: "indep_sets G K" "\<And>i. i \<in> K \<Longrightarrow> G i \<subseteq> events" and "j \<in> K"
by (auto simp: indep_sets_def)
let ?D = "{E\<in>events. indep_sets (G(j := {E})) K }"
{ fix X assume X: "X \<in> events"
assume indep: "\<And>J A. J \<noteq> {} \<Longrightarrow> J \<subseteq> K \<Longrightarrow> finite J \<Longrightarrow> j \<notin> J \<Longrightarrow> (\<forall>i\<in>J. A i \<in> 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 \<in> K" then show "(G(j:={X})) i \<subseteq> events"
using G X by auto
next
fix A J assume J: "J \<noteq> {}" "J \<subseteq> K" "finite J" "\<forall>i\<in>J. A i \<in> (G(j := {X})) i"
show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
proof cases
assume "j \<in> J"
with J have "A j = X" by auto
show ?thesis
proof cases
assume "J = {j}" then show ?thesis by simp
next
assume "J \<noteq> {j}"
have "prob (\<Inter>i\<in>J. A i) = prob ((\<Inter>i\<in>J-{j}. A i) \<inter> X)"
using `j \<in> J` `A j = X` by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
also have "\<dots> = prob X * (\<Prod>i\<in>J-{j}. prob (A i))"
proof (rule indep)
show "J - {j} \<noteq> {}" "J - {j} \<subseteq> K" "finite (J - {j})" "j \<notin> J - {j}"
using J `J \<noteq> {j}` `j \<in> J` by auto
show "\<forall>i\<in>J - {j}. A i \<in> G i"
using J by auto
qed
also have "\<dots> = prob (A j) * (\<Prod>i\<in>J-{j}. prob (A i))"
using `A j = X` by simp
also have "\<dots> = (\<Prod>i\<in>J. prob (A i))"
unfolding setprod.insert_remove[OF `finite J`, symmetric, of "\<lambda>i. prob (A i)"]
using `j \<in> J` by (simp add: insert_absorb)
finally show ?thesis .
qed
next
assume "j \<notin> J"
with J have "\<forall>i\<in>J. A i \<in> G i" by (auto split: split_if_asm)
with J show ?thesis
by (intro indep_setsD[OF G(1)]) auto
qed
qed }
note indep_sets_insert = this
have "dynkin_system \<lparr> space = space M, sets = ?D \<rparr>"
proof (rule dynkin_systemI', simp_all, safe)
show "indep_sets (G(j := {{}})) K"
by (rule indep_sets_insert) auto
next
fix X assume X: "X \<in> 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 \<noteq> {}" "J \<subseteq> K" "finite J" "j \<notin> J" and A: "\<forall>i\<in>J. A i \<in> G i"
then have A_sets: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> events"
using G by auto
have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) =
prob ((\<Inter>j\<in>J. A j) - (\<Inter>i\<in>insert j J. (A(j := X)) i))"
using A_sets sets_into_space X `J \<noteq> {}`
by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
also have "\<dots> = prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)"
using J `J \<noteq> {}` `j \<notin> J` A_sets X sets_into_space
by (auto intro!: finite_measure_Diff finite_INT split: split_if_asm)
finally have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) =
prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)" .
moreover {
have "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
using J A `finite J` by (intro indep_setsD[OF G(1)]) auto
then have "prob (\<Inter>j\<in>J. A j) = prob (space M) * (\<Prod>i\<in>J. prob (A i))"
using prob_space by simp }
moreover {
have "prob (\<Inter>i\<in>insert j J. (A(j := X)) i) = (\<Prod>i\<in>insert j J. prob ((A(j := X)) i))"
using J A `j \<in> K` by (intro indep_setsD[OF G']) auto
then have "prob (\<Inter>i\<in>insert j J. (A(j := X)) i) = prob X * (\<Prod>i\<in>J. prob (A i))"
using `finite J` `j \<notin> J` by (auto intro!: setprod_cong) }
ultimately have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) = (prob (space M) - prob X) * (\<Prod>i\<in>J. prob (A i))"
by (simp add: field_simps)
also have "\<dots> = prob (space M - X) * (\<Prod>i\<in>J. prob (A i))"
using X A by (simp add: finite_measure_compl)
finally show "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) = prob (space M - X) * (\<Prod>i\<in>J. prob (A i))" .
qed (insert X, auto)
next
fix F :: "nat \<Rightarrow> 'a set" assume disj: "disjoint_family F" and "range F \<subseteq> ?D"
then have F: "\<And>i. F i \<in> events" "\<And>i. indep_sets (G(j:={F i})) K" by auto
show "indep_sets (G(j := {\<Union>k. F k})) K"
proof (rule indep_sets_insert)
fix J A assume J: "j \<notin> J" "J \<noteq> {}" "J \<subseteq> K" "finite J" and A: "\<forall>i\<in>J. A i \<in> G i"
then have A_sets: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> events"
using G by auto
have "prob ((\<Inter>j\<in>J. A j) \<inter> (\<Union>k. F k)) = prob (\<Union>k. (\<Inter>i\<in>insert j J. (A(j := F k)) i))"
using `J \<noteq> {}` `j \<notin> J` `j \<in> K` by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
moreover have "(\<lambda>k. prob (\<Inter>i\<in>insert j J. (A(j := F k)) i)) sums prob (\<Union>k. (\<Inter>i\<in>insert j J. (A(j := F k)) i))"
proof (rule finite_measure_UNION)
show "disjoint_family (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i)"
using disj by (rule disjoint_family_on_bisimulation) auto
show "range (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i) \<subseteq> events"
using A_sets F `finite J` `J \<noteq> {}` `j \<notin> J` by (auto intro!: Int)
qed
moreover { fix k
from J A `j \<in> K` 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!: setprod_cong split: split_if_asm)
also have "\<dots> = prob (F k) * prob (\<Inter>i\<in>J. A i)"
using J A `j \<in> K` by (subst indep_setsD[OF G(1)]) auto
finally have "prob (\<Inter>i\<in>insert j J. (A(j := F k)) i) = prob (F k) * prob (\<Inter>i\<in>J. A i)" . }
ultimately have "(\<lambda>k. prob (F k) * prob (\<Inter>i\<in>J. A i)) sums (prob ((\<Inter>j\<in>J. A j) \<inter> (\<Union>k. F k)))"
by simp
moreover
have "(\<lambda>k. prob (F k) * prob (\<Inter>i\<in>J. A i)) sums (prob (\<Union>k. F k) * prob (\<Inter>i\<in>J. A i))"
using disj F(1) by (intro finite_measure_UNION sums_mult2) auto
then have "(\<lambda>k. prob (F k) * prob (\<Inter>i\<in>J. A i)) sums (prob (\<Union>k. F k) * (\<Prod>i\<in>J. prob (A i)))"
using J A `j \<in> K` by (subst indep_setsD[OF G(1), symmetric]) auto
ultimately
show "prob ((\<Inter>j\<in>J. A j) \<inter> (\<Union>k. F k)) = prob (\<Union>k. F k) * (\<Prod>j\<in>J. prob (A j))"
by (auto dest!: sums_unique)
qed (insert F, auto)
qed (insert sets_into_space, auto)
then have mono: "sets (dynkin \<lparr>space = space M, sets = G j\<rparr>) \<subseteq>
sets \<lparr>space = space M, sets = {E \<in> events. indep_sets (G(j := {E})) K}\<rparr>"
proof (rule dynkin_system.dynkin_subset, simp_all, safe)
fix X assume "X \<in> G j"
then show "X \<in> events" using G `j \<in> K` by auto
from `indep_sets G K`
show "indep_sets (G(j := {X})) K"
by (rule indep_sets_mono_sets) (insert `X \<in> G j`, auto)
qed
have "indep_sets (G(j:=?D)) K"
proof (rule indep_setsI)
fix i assume "i \<in> K" then show "(G(j := ?D)) i \<subseteq> events"
using G(2) by auto
next
fix A J assume J: "J\<noteq>{}" "J \<subseteq> K" "finite J" and A: "\<forall>i\<in>J. A i \<in> (G(j := ?D)) i"
show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
proof cases
assume "j \<in> 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 \<notin> J"
with J A have "\<forall>i\<in>J. A i \<in> G i" by (auto split: split_if_asm)
with J show ?thesis
by (intro indep_setsD[OF G(1)]) auto
qed
qed
then have "indep_sets (G(j:=sets (dynkin \<lparr>space = space M, sets = G j\<rparr>))) K"
by (rule indep_sets_mono_sets) (insert mono, auto)
then show ?case
by (rule indep_sets_mono_sets) (insert `j \<in> K` `j \<notin> J`, auto simp: G_def)
qed (insert `indep_sets F K`, simp) }
from this[OF `indep_sets F J` `finite J` subset_refl]
show "indep_sets (\<lambda>i. sets (dynkin \<lparr> space = space M, sets = F i \<rparr>)) J"
by (rule indep_sets_mono_sets) auto
qed
lemma (in prob_space) indep_sets_sigma:
assumes indep: "indep_sets F I"
assumes stable: "\<And>i. i \<in> I \<Longrightarrow> Int_stable \<lparr> space = space M, sets = F i \<rparr>"
shows "indep_sets (\<lambda>i. sets (sigma \<lparr> space = space M, sets = F i \<rparr>)) 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 \<in> I"
with indep have "F i \<subseteq> events" by (auto simp: indep_sets_def)
with sets_into_space show "F i \<subseteq> Pow (space M)" by auto
qed
qed
lemma (in prob_space) indep_sets_sigma_sets:
assumes "indep_sets F I"
assumes "\<And>i. i \<in> I \<Longrightarrow> Int_stable \<lparr> space = space M, sets = F i \<rparr>"
shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I"
using indep_sets_sigma[OF assms] by (simp add: sets_sigma)
lemma (in prob_space) indep_sets2_eq:
"indep_set A B \<longleftrightarrow> A \<subseteq> events \<and> B \<subseteq> events \<and> (\<forall>a\<in>A. \<forall>b\<in>B. prob (a \<inter> b) = prob a * prob b)"
unfolding indep_set_def
proof (intro iffI ballI conjI)
assume indep: "indep_sets (bool_case A B) UNIV"
{ fix a b assume "a \<in> A" "b \<in> B"
with indep_setsD[OF indep, of UNIV "bool_case a b"]
show "prob (a \<inter> b) = prob a * prob b"
unfolding UNIV_bool by (simp add: ac_simps) }
from indep show "A \<subseteq> events" "B \<subseteq> events"
unfolding indep_sets_def UNIV_bool by auto
next
assume *: "A \<subseteq> events \<and> B \<subseteq> events \<and> (\<forall>a\<in>A. \<forall>b\<in>B. prob (a \<inter> b) = prob a * prob b)"
show "indep_sets (bool_case A B) UNIV"
proof (rule indep_setsI)
fix i show "(case i of True \<Rightarrow> A | False \<Rightarrow> B) \<subseteq> events"
using * by (auto split: bool.split)
next
fix J X assume "J \<noteq> {}" "J \<subseteq> UNIV" and X: "\<forall>j\<in>J. X j \<in> (case j of True \<Rightarrow> A | False \<Rightarrow> B)"
then have "J = {True} \<or> J = {False} \<or> J = {True,False}"
by (auto simp: UNIV_bool)
then show "prob (\<Inter>j\<in>J. X j) = (\<Prod>j\<in>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 \<lparr> space = space M, sets = A \<rparr>"
assumes B: "Int_stable \<lparr> space = space M, sets = B \<rparr>"
shows "indep_set (sigma_sets (space M) A) (sigma_sets (space M) B)"
proof -
have "indep_sets (\<lambda>i. sigma_sets (space M) (case i of True \<Rightarrow> A | False \<Rightarrow> B)) UNIV"
proof (rule indep_sets_sigma_sets)
show "indep_sets (bool_case A B) UNIV"
by (rule `indep_set A B`[unfolded indep_set_def])
fix i show "Int_stable \<lparr>space = space M, sets = case i of True \<Rightarrow> A | False \<Rightarrow> B\<rparr>"
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_sets_collect_sigma:
fixes I :: "'j \<Rightarrow> 'i set" and J :: "'j set" and E :: "'i \<Rightarrow> 'a set set"
assumes indep: "indep_sets E (\<Union>j\<in>J. I j)"
assumes Int_stable: "\<And>i j. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> Int_stable \<lparr>space = space M, sets = E i\<rparr>"
assumes disjoint: "disjoint_family_on I J"
shows "indep_sets (\<lambda>j. sigma_sets (space M) (\<Union>i\<in>I j. E i)) J"
proof -
let "?E j" = "{\<Inter>k\<in>K. E' k| E' K. finite K \<and> K \<noteq> {} \<and> K \<subseteq> I j \<and> (\<forall>k\<in>K. E' k \<in> E k) }"
from indep have E: "\<And>j i. j \<in> J \<Longrightarrow> i \<in> I j \<Longrightarrow> E i \<subseteq> sets M"
unfolding indep_sets_def by auto
{ fix j
let ?S = "sigma \<lparr> space = space M, sets = (\<Union>i\<in>I j. E i) \<rparr>"
assume "j \<in> J"
from E[OF this] interpret S: sigma_algebra ?S
using sets_into_space by (intro sigma_algebra_sigma) auto
have "sigma_sets (space M) (\<Union>i\<in>I j. E i) = sigma_sets (space M) (?E j)"
proof (rule sigma_sets_eqI)
fix A assume "A \<in> (\<Union>i\<in>I j. E i)"
then guess i ..
then show "A \<in> sigma_sets (space M) (?E j)"
by (auto intro!: sigma_sets.intros exI[of _ "{i}"] exI[of _ "\<lambda>i. A"])
next
fix A assume "A \<in> ?E j"
then obtain E' K where "finite K" "K \<noteq> {}" "K \<subseteq> I j" "\<And>k. k \<in> K \<Longrightarrow> E' k \<in> E k"
and A: "A = (\<Inter>k\<in>K. E' k)"
by auto
then have "A \<in> sets ?S" unfolding A
by (safe intro!: S.finite_INT)
(auto simp: sets_sigma intro!: sigma_sets.Basic)
then show "A \<in> sigma_sets (space M) (\<Union>i\<in>I j. E i)"
by (simp add: sets_sigma)
qed }
moreover have "indep_sets (\<lambda>j. sigma_sets (space M) (?E j)) J"
proof (rule indep_sets_sigma_sets)
show "indep_sets ?E J"
proof (intro indep_setsI)
fix j assume "j \<in> J" with E show "?E j \<subseteq> events" by (force intro!: finite_INT)
next
fix K A assume K: "K \<noteq> {}" "K \<subseteq> J" "finite K"
and "\<forall>j\<in>K. A j \<in> ?E j"
then have "\<forall>j\<in>K. \<exists>E' L. A j = (\<Inter>l\<in>L. E' l) \<and> finite L \<and> L \<noteq> {} \<and> L \<subseteq> I j \<and> (\<forall>l\<in>L. E' l \<in> E l)"
by simp
from bchoice[OF this] guess E' ..
from bchoice[OF this] obtain L
where A: "\<And>j. j\<in>K \<Longrightarrow> A j = (\<Inter>l\<in>L j. E' j l)"
and L: "\<And>j. j\<in>K \<Longrightarrow> finite (L j)" "\<And>j. j\<in>K \<Longrightarrow> L j \<noteq> {}" "\<And>j. j\<in>K \<Longrightarrow> L j \<subseteq> I j"
and E': "\<And>j l. j\<in>K \<Longrightarrow> l \<in> L j \<Longrightarrow> E' j l \<in> E l"
by auto
{ fix k l j assume "k \<in> K" "j \<in> K" "l \<in> L j" "l \<in> L k"
have "k = j"
proof (rule ccontr)
assume "k \<noteq> j"
with disjoint `K \<subseteq> J` `k \<in> K` `j \<in> K` have "I k \<inter> I j = {}"
unfolding disjoint_family_on_def by auto
with L(2,3)[OF `j \<in> K`] L(2,3)[OF `k \<in> K`]
show False using `l \<in> L k` `l \<in> L j` by auto
qed }
note L_inj = this
def k \<equiv> "\<lambda>l. (SOME k. k \<in> K \<and> l \<in> L k)"
{ fix x j l assume *: "j \<in> K" "l \<in> L j"
have "k l = j" unfolding k_def
proof (rule some_equality)
fix k assume "k \<in> K \<and> l \<in> 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 (\<Inter>j\<in>K. A j) = prob (\<Inter>l\<in>(\<Union>k\<in>K. L k). ?E' l)"
by (auto simp: A intro!: arg_cong[where f=prob])
also have "\<dots> = (\<Prod>l\<in>(\<Union>k\<in>K. L k). prob (?E' l))"
using L K E' by (intro indep_setsD[OF indep]) (simp_all add: UN_mono)
also have "\<dots> = (\<Prod>j\<in>K. \<Prod>l\<in>L j. prob (E' j l))"
using K L L_inj by (subst setprod_UN_disjoint) auto
also have "\<dots> = (\<Prod>j\<in>K. prob (A j))"
using K L E' by (auto simp add: A intro!: setprod_cong indep_setsD[OF indep, symmetric]) blast
finally show "prob (\<Inter>j\<in>K. A j) = (\<Prod>j\<in>K. prob (A j))" .
qed
next
fix j assume "j \<in> J"
show "Int_stable \<lparr> space = space M, sets = ?E j \<rparr>"
proof (rule Int_stableI)
fix a assume "a \<in> ?E j" then obtain Ka Ea
where a: "a = (\<Inter>k\<in>Ka. Ea k)" "finite Ka" "Ka \<noteq> {}" "Ka \<subseteq> I j" "\<And>k. k\<in>Ka \<Longrightarrow> Ea k \<in> E k" by auto
fix b assume "b \<in> ?E j" then obtain Kb Eb
where b: "b = (\<Inter>k\<in>Kb. Eb k)" "finite Kb" "Kb \<noteq> {}" "Kb \<subseteq> I j" "\<And>k. k\<in>Kb \<Longrightarrow> Eb k \<in> E k" by auto
let ?A = "\<lambda>k. (if k \<in> Ka \<inter> Kb then Ea k \<inter> Eb k else if k \<in> Kb then Eb k else if k \<in> Ka then Ea k else {})"
have "a \<inter> b = INTER (Ka \<union> Kb) ?A"
by (simp add: a b set_eq_iff) auto
with a b `j \<in> J` Int_stableD[OF Int_stable] show "a \<inter> b \<in> ?E j"
by (intro CollectI exI[of _ "Ka \<union> Kb"] exI[of _ ?A]) auto
qed
qed
ultimately show ?thesis
by (simp cong: indep_sets_cong)
qed
definition (in prob_space) terminal_events where
"terminal_events A = (\<Inter>n. sigma_sets (space M) (UNION {n..} A))"
lemma (in prob_space) terminal_events_sets:
assumes A: "\<And>i. A i \<subseteq> sets M"
assumes "\<And>i::nat. sigma_algebra \<lparr>space = space M, sets = A i\<rparr>"
assumes X: "X \<in> terminal_events A"
shows "X \<in> sets M"
proof -
let ?A = "(\<Inter>n. sigma_sets (space M) (UNION {n..} A))"
interpret A: sigma_algebra "\<lparr>space = space M, sets = A i\<rparr>" for i by fact
from X have "\<And>n. X \<in> sigma_sets (space M) (UNION {n..} A)" by (auto simp: terminal_events_def)
from this[of 0] have "X \<in> sigma_sets (space M) (UNION UNIV A)" by simp
then show "X \<in> sets M"
by induct (insert A, auto)
qed
lemma (in prob_space) sigma_algebra_terminal_events:
assumes "\<And>i::nat. sigma_algebra \<lparr>space = space M, sets = A i\<rparr>"
shows "sigma_algebra \<lparr> space = space M, sets = terminal_events A \<rparr>"
unfolding terminal_events_def
proof (simp add: sigma_algebra_iff2, safe)
let ?A = "(\<Inter>n. sigma_sets (space M) (UNION {n..} A))"
interpret A: sigma_algebra "\<lparr>space = space M, sets = A i\<rparr>" for i by fact
{ fix X x assume "X \<in> ?A" "x \<in> X"
then have "\<And>n. X \<in> sigma_sets (space M) (UNION {n..} A)" by auto
from this[of 0] have "X \<in> sigma_sets (space M) (UNION UNIV A)" by simp
then have "X \<subseteq> space M"
by induct (insert A.sets_into_space, auto)
with `x \<in> X` show "x \<in> space M" by auto }
{ fix F :: "nat \<Rightarrow> 'a set" and n assume "range F \<subseteq> ?A"
then show "(UNION UNIV F) \<in> sigma_sets (space M) (UNION {n..} A)"
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 \<Rightarrow> 'a set set"
assumes A: "\<And>i. A i \<subseteq> sets M"
assumes "\<And>i::nat. sigma_algebra \<lparr>space = space M, sets = A i\<rparr>"
assumes indep: "indep_sets A UNIV"
and X: "X \<in> terminal_events A"
shows "prob X = 0 \<or> prob X = 1"
proof -
let ?D = "\<lparr> space = space M, sets = {D \<in> sets M. prob (X \<inter> D) = prob X * prob D} \<rparr>"
interpret A: sigma_algebra "\<lparr>space = space M, sets = A i\<rparr>" for i by fact
interpret T: sigma_algebra "\<lparr> space = space M, sets = terminal_events A \<rparr>"
by (rule sigma_algebra_terminal_events) fact
have "X \<subseteq> space M" using T.space_closed X by auto
have X_in: "X \<in> sets M"
by (rule terminal_events_sets) fact+
interpret D: dynkin_system ?D
proof (rule dynkin_systemI)
fix D assume "D \<in> sets ?D" then show "D \<subseteq> space ?D"
using sets_into_space by auto
next
show "space ?D \<in> sets ?D"
using prob_space `X \<subseteq> space M` by (simp add: Int_absorb2)
next
fix A assume A: "A \<in> sets ?D"
have "prob (X \<inter> (space M - A)) = prob (X - (X \<inter> A))"
using `X \<subseteq> space M` by (auto intro!: arg_cong[where f=prob])
also have "\<dots> = prob X - prob (X \<inter> A)"
using X_in A by (intro finite_measure_Diff) auto
also have "\<dots> = prob X * prob (space M) - prob X * prob A"
using A prob_space by auto
also have "\<dots> = prob X * prob (space M - A)"
using X_in A sets_into_space
by (subst finite_measure_Diff) (auto simp: field_simps)
finally show "space ?D - A \<in> sets ?D"
using A `X \<subseteq> space M` by auto
next
fix F :: "nat \<Rightarrow> 'a set" assume dis: "disjoint_family F" and "range F \<subseteq> sets ?D"
then have F: "range F \<subseteq> events" "\<And>i. prob (X \<inter> F i) = prob X * prob (F i)"
by auto
have "(\<lambda>i. prob (X \<inter> F i)) sums prob (\<Union>i. X \<inter> F i)"
proof (rule finite_measure_UNION)
show "range (\<lambda>i. X \<inter> F i) \<subseteq> events"
using F X_in by auto
show "disjoint_family (\<lambda>i. X \<inter> F i)"
using dis by (rule disjoint_family_on_bisimulation) auto
qed
with F have "(\<lambda>i. prob X * prob (F i)) sums prob (X \<inter> (\<Union>i. F i))"
by simp
moreover have "(\<lambda>i. prob X * prob (F i)) sums (prob X * prob (\<Union>i. F i))"
by (intro mult_right.sums finite_measure_UNION F dis)
ultimately have "prob (X \<inter> (\<Union>i. F i)) = prob X * prob (\<Union>i. F i)"
by (auto dest!: sums_unique)
with F show "(\<Union>i. F i) \<in> sets ?D"
by auto
qed
{ fix n
have "indep_sets (\<lambda>b. sigma_sets (space M) (\<Union>m\<in>bool_case {..n} {Suc n..} b. A m)) UNIV"
proof (rule indep_sets_collect_sigma)
have *: "(\<Union>b. case b of True \<Rightarrow> {..n} | False \<Rightarrow> {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 (bool_case {..n} {Suc n..})"
unfolding disjoint_family_on_def by (auto split: bool.split)
fix m
show "Int_stable \<lparr>space = space M, sets = A m\<rparr>"
unfolding Int_stable_def using A.Int by auto
qed
also have "(\<lambda>b. sigma_sets (space M) (\<Union>m\<in>bool_case {..n} {Suc n..} b. A m)) =
bool_case (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) (\<Union>m\<in>{..n}. A m)) (sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m))"
unfolding indep_set_def by simp
have "sigma_sets (space M) (\<Union>m\<in>{..n}. A m) \<subseteq> sets ?D"
proof (simp add: subset_eq, rule)
fix D assume D: "D \<in> sigma_sets (space M) (\<Union>m\<in>{..n}. A m)"
have "X \<in> sigma_sets (space M) (\<Union>m\<in>{Suc n..}. A m)"
using X unfolding terminal_events_def by simp
from indep_setD[OF indep D this] indep_setD_ev1[OF indep] D
show "D \<in> events \<and> prob (X \<inter> D) = prob X * prob D"
by (auto simp add: ac_simps)
qed }
then have "(\<Union>n. sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) \<subseteq> sets ?D" (is "?A \<subseteq> _")
by auto
have "sigma \<lparr> space = space M, sets = ?A \<rparr> =
dynkin \<lparr> space = space M, sets = ?A \<rparr>" (is "sigma ?UA = dynkin ?UA")
proof (rule sigma_eq_dynkin)
{ fix B n assume "B \<in> sigma_sets (space M) (\<Union>m\<in>{..n}. A m)"
then have "B \<subseteq> space M"
by induct (insert A sets_into_space, auto) }
then show "sets ?UA \<subseteq> Pow (space ?UA)" by auto
show "Int_stable ?UA"
proof (rule Int_stableI)
fix a assume "a \<in> ?A" then guess n .. note a = this
fix b assume "b \<in> ?A" then guess m .. note b = this
interpret Amn: sigma_algebra "sigma \<lparr>space = space M, sets = (\<Union>i\<in>{..max m n}. A i)\<rparr>"
using A sets_into_space by (intro sigma_algebra_sigma) auto
have "sigma_sets (space M) (\<Union>i\<in>{..n}. A i) \<subseteq> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
by (intro sigma_sets_subseteq UN_mono) auto
with a have "a \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" by auto
moreover
have "sigma_sets (space M) (\<Union>i\<in>{..m}. A i) \<subseteq> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
by (intro sigma_sets_subseteq UN_mono) auto
with b have "b \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)" by auto
ultimately have "a \<inter> b \<in> sigma_sets (space M) (\<Union>i\<in>{..max m n}. A i)"
using Amn.Int[of a b] by (simp add: sets_sigma)
then show "a \<inter> b \<in> (\<Union>n. sigma_sets (space M) (\<Union>i\<in>{..n}. A i))" by auto
qed
qed
moreover have "sets (dynkin ?UA) \<subseteq> sets ?D"
proof (rule D.dynkin_subset)
show "sets ?UA \<subseteq> sets ?D" using `?A \<subseteq> sets ?D` by auto
qed simp
ultimately have "sets (sigma ?UA) \<subseteq> sets ?D" by simp
moreover
have "\<And>n. sigma_sets (space M) (\<Union>i\<in>{n..}. A i) \<subseteq> sigma_sets (space M) ?A"
by (intro sigma_sets_subseteq UN_mono) (auto intro: sigma_sets.Basic)
then have "terminal_events A \<subseteq> sets (sigma ?UA)"
unfolding sets_sigma terminal_events_def by auto
moreover note `X \<in> terminal_events A`
ultimately have "X \<in> sets ?D" by auto
then show ?thesis by auto
qed
end