# Theory Independent_Family

```(*  Title:      HOL/Probability/Independent_Family.thy
Author:     Johannes Hölzl, TU München
Author:     Sudeep Kanav, TU München
*)

section ‹Independent families of events, event sets, and random variables›

theory Independent_Family
imports Infinite_Product_Measure
begin

definition (in prob_space)
"indep_sets F I ⟷ (∀i∈I. F i ⊆ events) ∧
(∀J⊆I. J ≠ {} ⟶ finite J ⟶ (∀A∈Pi J F. prob (⋂j∈J. A j) = (∏j∈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 (λi. {A i}) I"

lemma (in prob_space) indep_events_def:
"indep_events A I ⟷ (A`I ⊆ events) ∧
(∀J⊆I. J ≠ {} ⟶ finite J ⟶ prob (⋂j∈J. A j) = (∏j∈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 ∈ (Π j∈J. G j) ⟹ A ∈ (Π 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 ‹J ≠ {}› 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

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 (λi. 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 = "λS 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)
⟹ prob ((⋂i∈J. A i) ∩ X) = prob X * (∏i∈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 ‹j ∈ J› ‹A j = X› 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 ‹J ≠ {j}› ‹j ∈ J› 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 ‹A j = X› by simp
also have "… = (∏i∈J. prob (A i))"
unfolding prod.insert_remove[OF ‹finite J›, symmetric, of "λi. prob  (A i)"]
using ‹j ∈ J› 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 ((⋂j∈J. A j) - (⋂i∈insert j J. (A(j := X)) i))"
using A_sets sets.sets_into_space[of _ M] X ‹J ≠ {}›
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 ‹J ≠ {}› ‹j ∉ J› 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 (⋂j∈J. A j) - prob (⋂i∈insert j J. (A(j := X)) i)" .
moreover {
have "prob (⋂j∈J. A j) = (∏j∈J. prob (A j))"
using J A ‹finite J› 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 ‹j ∈ K› 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 ‹finite J› ‹j ∉ J› 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))"
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 ‹J ≠ {}› ‹j ∉ J› ‹j ∈ K› by (auto intro!: arg_cong[where f=prob] split: if_split_asm)
moreover have "(λk. 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 (λk. ⋂i∈insert j J. (A(j := F k)) i)"
using disj by (rule disjoint_family_on_bisimulation) auto
show "range (λk. ⋂i∈insert j J. (A(j := F k)) i) ⊆ events"
using A_sets F ‹finite J› ‹J ≠ {}› ‹j ∉ J› by (auto intro!: sets.Int)
qed
moreover { fix k
from J A ‹j ∈ K› have "prob (⋂i∈insert j J. (A(j := F k)) i) = prob (F k) * (∏i∈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 ‹j ∈ K› 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 "(λk. prob (F k) * prob (⋂i∈J. A i)) sums (prob ((⋂j∈J. A j) ∩ (⋃k. F k)))"
by simp
moreover
have "(λk. 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 "(λk. prob (F k) * prob (⋂i∈J. A i)) sums (prob (⋃k. F k) * (∏i∈J. prob (A i)))"
using J A ‹j ∈ K› 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 ‹indep_sets G K›
show "indep_sets (G(j := {X})) K"
by (rule indep_sets_mono_sets) (insert ‹X ∈ G j›, 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 ‹j ∈ K› ‹j ∉ 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 ?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 (λi. 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 (λi. sigma_sets (space M) (F i)) I ⟷ indep_sets F I"
proof
assume "indep_sets F I" then show "indep_sets (λi. sigma_sets (space M) (F i)) I"
by (rule indep_sets_sigma) fact
next
assume "indep_sets (λi. 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 ⟷
(∀i∈I. random_variable (M' i) (X i)) ∧
indep_sets (λi. { X i -` A ∩ space M | A. A ∈ 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 ⟷
(∀i∈I. random_variable (M' i) (X i)) ∧
indep_sets (λi. sigma_sets (space M) { X i -` A ∩ space M | A. A ∈ 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 ∧ random_variable T Y) ∧
indep_set
(sigma_sets (space M) { X -` A ∩ space M | A. A ∈ sets S})
(sigma_sets (space M) { Y -` A ∩ space M | A. A ∈ 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 (λi. 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 ‹indep_set A B›[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 (λi. {x∈space M. P i (X i x)}) I"
proof -
have "indep_sets (λi. {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 ‹i ∈ I›] 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 (λj. sigma_sets (space M) (⋃i∈I j. E i)) J"
proof -
let ?E = "λj. {⋂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 obtain i where "i ∈ I j" "A ∈ E i" ..
then show "A ∈ sigma_sets (space M) (?E j)"
by (auto intro!: sigma_sets.intros(2-) exI[of _ "{i}"] exI[of _ "λi. 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 (λj. 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] obtain E'
where "∀x∈K. ∃L. A x = ⋂ (E' x ` L) ∧ finite L ∧ L ≠ {} ∧ L ⊆ I x ∧ (∀l∈L. E' x l ∈ E l)"
..
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 ‹K ⊆ J› ‹k ∈ K› ‹j ∈ K› have "I k ∩ I j = {}"
unfolding disjoint_family_on_def by auto
with L(2,3)[OF ‹j ∈ K›] L(2,3)[OF ‹k ∈ K›]
show False using ‹l ∈ L k› ‹l ∈ L j› 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' = "λl. 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 = "λk. (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) ∩
(⋂x∈Kb - Ka. Eb x) ∩ (⋂x∈Ka - Kb. Ea x) = (⋂k∈Ka. Ea k) ∩ (⋂k∈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 ‹j ∈ J› Int_stableD[OF Int_stable] show "a ∩ b ∈ ?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 (λj. PiM (K j) M') (λj ω. restrict (λi. X i ω) (K j)) L"
unfolding indep_vars_def
proof safe
fix j assume "j ∈ L" then show "random_variable (Pi⇩M (K j) M') (λω. λi∈K j. X i ω)"
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 = "λj S. {(λω. λi∈K j. X i ω) -` A ∩ space M |A. A ∈ S}"
let ?UN = "λj. sigma_sets (space M) (⋃i∈K j. { X i -` A ∩ space M| A. A ∈ sets (M' i) })"
show "indep_sets (λi. sigma_sets (space M) (?proj i (sets (Pi⇩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⇩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 "(λω. λi∈K j. X i ω) -` prod_emb (K j) M' J (Pi⇩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 ‹finite J› ‹J ≠ {}› by (rule finite_INT) blast }
note INT = this

from ‹J ≠ {}› J K E[rule_format, THEN sets.sets_into_space] j
have "(λω. λi∈K j. X i ω) -` prod_emb (K j) M' J (Pi⇩E J E) ∩ space M
= (⋂i∈J. X i -` E i ∩ 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 ‹J ⊆ K j› E
apply auto
done
finally show ?thesis .
qed
qed
finally show "sigma_sets (space M) (?proj j (sets (Pi⇩M (K j) M'))) ⊆ ?UN j" .
next
show "indep_sets ?UN L"
proof (rule indep_sets_collect_sigma)
show "indep_sets (λi. {X i -` A ∩ space M |A. A ∈ sets (M' i)}) (⋃j∈L. K j)"
proof (rule indep_sets_mono_index)
show "indep_sets (λi. {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') (λω. restrict (λi. X i ω) A) (PiM B M') (λω. restrict (λi. X i ω) B)"
proof -
have *:
"case_bool (Pi⇩M A M') (Pi⇩M B M') = (λb. PiM (case_bool A B b) M')"
"case_bool (λω. λi∈A. X i ω) (λω. λi∈B. X i ω) = (λb ω. λi∈case_bool A B b. X i ω)"
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
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 ‹x ∈ X› show "x ∈ 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 ‹X ⊆ space M› by (simp add: Int_absorb2)
next
fix A assume A: "A ∈ ?D"
have "prob (X ∩ (space M - A)) = prob (X - (X ∩ A))"
using ‹X ⊆ space M› 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 ‹X ⊆ space M› 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 "(λi. prob (X ∩ F i)) sums prob (⋃i. X ∩ F i)"
proof (rule finite_measure_UNION)
show "range (λi. X ∩ F i) ⊆ events"
using F X_in by auto
show "disjoint_family (λi. X ∩ F i)"
using dis by (rule disjoint_family_on_bisimulation) auto
qed
with F have "(λi. prob X * prob (F i)) sums prob (X ∩ (⋃i. F i))"
by simp
moreover have "(λi. 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 (λb. 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 "(λb. sigma_sets (space M) (⋃m∈case_bool {..n} {Suc n..} b. A m)) =
case_bool (sigma_sets (space M) (⋃m∈{..n}. A m)) (sigma_sets (space M) (⋃m∈{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"
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"
qed }
then have "(⋃n. sigma_sets (space M) (⋃m∈{..n}. A m)) ⊆ ?D" (is "?A ⊆ _")
by auto

note ‹X ∈ tail_events A›
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 b assume "a ∈ ?A" "b ∈ ?A" then obtain n m
where a: "n ∈ UNIV" "a ∈ sigma_sets (space M) (⋃ (A ` {..n}))"
and b: "m ∈ UNIV" "b ∈ sigma_sets (space M) (⋃ (A ` {..m}))" by auto
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 ‹?A ⊆ ?D› 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 "λi. 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 (λi. sigma_sets (space M) {F i}) UNIV"
proof (rule indep_sets_sigma)
show "indep_sets (λi. {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 = "λn. ⋃i∈{n..}. F i"
show "(⋂n. ⋃m∈{n..}. F m) ∈ tail_events (λi. 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 (λm. {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)
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 = "λj. 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 "(λj. 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 ‹finite I›]
using prob_space ‹J ⊆ I› 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 ⟷
(∀A∈(Π i∈I. E i). prob (⋂j∈I. X j -` A j ∩ space M) = (∏j∈I. prob (X j -` A j ∩ 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 ‹i ∈ I›]
have "sigma_sets (space M) {X i -` A ∩ space M |A. A ∈ sets (M' i)}
= sigma_sets (space M) {X i -` A ∩ space M |A. A ∈ E i}"
unfolding sigma_sets_vimage_commute[OF X, OF ‹i ∈ I›, symmetric] M'[OF ‹i ∈ I›]
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 ‹i ∈ I›]
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 ‹i∈I›] have "X i ∈ measurable M (M' i)" by auto
ultimately
have "X i -` A ∩ space M ∈ sets M" by (auto intro: measurable_sets) }
with X[OF ‹i∈I›] space[OF ‹i∈I›]
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∈Π i∈I. {X i -` A ∩ space M |A. A ∈ E i}. prob (⋂(A ` I)) = (∏j∈I. prob (A j))) =
(∀A∈Π i∈I. E i. prob ((⋂j∈I. X j -` A j) ∩ space M) = (∏x∈I. prob (X x -` A x ∩ space M)))"
(is "?L = ?R")
proof safe
fix A assume ?L and A: "A ∈ (Π i∈I. E i)"
from ‹?L›[THEN bspec, of "λi. X i -` A i ∩ space M"] A ‹I ≠ {}›
show "prob ((⋂j∈I. X j -` A j) ∩ space M) = (∏x∈I. prob (X x -` A x ∩ space M))"
next
fix A assume ?R and A: "A ∈ (Π 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 ∈ (Π i∈I. E i)" by auto
from ‹?R›[THEN bspec, OF B(2)] B(1) ‹I ≠ {}›
show "prob (⋂(A ` I)) = (∏j∈I. prob (A j))"
by simp
qed
then show ?thesis using ‹I ≠ {}›
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 (λi. Y i ∘ X i) I"
unfolding indep_vars_def
proof
from rv ‹indep_vars M' X I›
show "∀i∈I. random_variable (N i) (Y i ∘ X i)"
by (auto simp: indep_vars_def)

have "indep_sets (λi. sigma_sets (space M) {X i -` A ∩ space M |A. A ∈ sets (M' i)}) I"
using ‹indep_vars M' X I› by (simp add: indep_vars_def)
then show "indep_sets (λi. 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 ‹indep_vars M' X I› have X: "X i ∈ space M → 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 ‹i ∈ I› 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 ∩ space M |A. A ∈ 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 (λi 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) (λb. 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 "(λb. 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 (λ_. borel) X (insert i I)"
shows "indep_var borel (X i) borel (λω. Min ((λi. X i ω)`I))"
proof -
have "indep_var
borel ((λf. f i) ∘ (λω. restrict (λi. X i ω) {i}))
borel ((λf. Min (f`I)) ∘ (λω. restrict (λi. X i ω) I))"
using I by (intro indep_var_compose[OF indep_var_restrict[OF indep]] borel_measurable_Min) auto
also have "((λf. f i) ∘ (λω. restrict (λi. X i ω) {i})) = X i"
by auto
also have "((λf. Min (f`I)) ∘ (λω. restrict (λi. X i ω) I)) = (λω. Min ((λi. X i ω)`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 (λ_. borel) X (insert i I)"
shows "indep_var borel (X i) borel (λω. ∑i∈I. X i ω)"
proof -
have "indep_var
borel ((λf. f i) ∘ (λω. restrict (λi. X i ω) {i}))
borel ((λf. ∑i∈I. f i) ∘ (λω. restrict (λi. X i ω) I))"
using I by (intro indep_var_compose[OF indep_var_restrict[OF indep]] ) auto
also have "((λf. f i) ∘ (λω. restrict (λi. X i ω) {i})) = X i"
by auto
also have "((λf. ∑i∈I. f i) ∘ (λω. restrict (λi. X i ω) I)) = (λω. ∑i∈I. X i ω)"
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 (λ_. borel) X (insert i I)"
shows "indep_var borel (X i) borel (λω. ∏i∈I. X i ω)"
proof -
have "indep_var
borel ((λf. f i) ∘ (λω. restrict (λi. X i ω) {i}))
borel ((λf. ∏i∈I. f i) ∘ (λω. restrict (λi. X i ω) I))"
using I by (intro indep_var_compose[OF indep_var_restrict[OF indep]] ) auto
also have "((λf. f i) ∘ (λω. restrict (λi. X i ω) {i})) = X i"
by auto
also have "((λf. ∏i∈I. f i) ∘ (λω. restrict (λi. X i ω) I)) = (λω. ∏i∈I. X i ω)"
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 (λi. 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 (λi. 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 (Π⇩M i∈I. M' i) (λx. λi∈I. X i x) = (Π⇩M i∈I. distr M (M' i) (X i))"
proof -
let ?P = "Π⇩M i∈I. M' i"
let ?X = "λx. λi∈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' = "λi. distr M (M' i) (X i)"
let ?P' = "Π⇩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')"
show "sets ?P' = sigma_sets (space ?P) (prod_algebra I M')"
by (simp add: sets_PiM space_PiM cong: prod_algebra_cong)
let ?A = "λi. Π⇩E i∈I. space (M' i)"
show "range ?A ⊆ prod_algebra I M'" "(⋃i. ?A i) = space (Pi⇩M I M')"
by (auto simp: space_PiM intro!: space_in_prod_algebra cong: prod_algebra_cong)
{ fix i show "emeasure ?D (Π⇩E i∈I. space (M' i)) ≠ ∞" by auto }
next
fix E assume E: "E ∈ prod_algebra I M'"
from prod_algebraE[OF E] obtain J Y
where J:
"E = prod_emb I M' J (Pi⇩E J Y)"
"finite J"
"J ≠ {} ∨ I = {}"
"J ⊆ I"
"⋀i. i ∈ J ⟹ Y i ∈ sets (M' i)"
by auto
from E have "E ∈ sets ?P" by (auto simp: sets_PiM)
then have "emeasure ?D E = emeasure M (?X -` E ∩ space M)"
also have "?X -` E ∩ space M = (⋂i∈J. X i -` Y i ∩ space M)"
using J ‹I ≠ {}› 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 ‹indep_vars M' X I› J ‹I ≠ {}› 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⇩E J Y)"
from Y have "(⋂j∈J. Y' j) = ?X -` ?E ∩ space M"
using J ‹I ≠ {}› 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</```