src/HOL/Probability/Projective_Family.thy
 author wenzelm Thu, 18 Apr 2013 17:07:01 +0200 changeset 51717 9e7d1c139569 parent 50252 4aa34bd43228 child 53015 a1119cf551e8 permissions -rw-r--r--
simplifier uses proper Proof.context instead of historic type simpset;
```
(*  Title:      HOL/Probability/Projective_Family.thy
Author:     Fabian Immler, TU München
Author:     Johannes Hölzl, TU München
*)

theory Projective_Family
imports Finite_Product_Measure Probability_Measure
begin

lemma (in product_prob_space) distr_restrict:
assumes "J \<noteq> {}" "J \<subseteq> K" "finite K"
shows "(\<Pi>\<^isub>M i\<in>J. M i) = distr (\<Pi>\<^isub>M i\<in>K. M i) (\<Pi>\<^isub>M i\<in>J. M i) (\<lambda>f. restrict f J)" (is "?P = ?D")
proof (rule measure_eqI_generator_eq)
have "finite J" using `J \<subseteq> K` `finite K` by (auto simp add: finite_subset)
interpret J: finite_product_prob_space M J proof qed fact
interpret K: finite_product_prob_space M K proof qed fact

let ?J = "{Pi\<^isub>E J E | E. \<forall>i\<in>J. E i \<in> sets (M i)}"
let ?F = "\<lambda>i. \<Pi>\<^isub>E k\<in>J. space (M k)"
let ?\<Omega> = "(\<Pi>\<^isub>E k\<in>J. space (M k))"
show "Int_stable ?J"
by (rule Int_stable_PiE)
show "range ?F \<subseteq> ?J" "(\<Union>i. ?F i) = ?\<Omega>"
using `finite J` by (auto intro!: prod_algebraI_finite)
{ fix i show "emeasure ?P (?F i) \<noteq> \<infinity>" by simp }
show "?J \<subseteq> Pow ?\<Omega>" by (auto simp: Pi_iff dest: sets.sets_into_space)
show "sets (\<Pi>\<^isub>M i\<in>J. M i) = sigma_sets ?\<Omega> ?J" "sets ?D = sigma_sets ?\<Omega> ?J"
using `finite J` by (simp_all add: sets_PiM prod_algebra_eq_finite Pi_iff)

fix X assume "X \<in> ?J"
then obtain E where [simp]: "X = Pi\<^isub>E J E" and E: "\<forall>i\<in>J. E i \<in> sets (M i)" by auto
with `finite J` have X: "X \<in> sets (Pi\<^isub>M J M)"
by simp

have "emeasure ?P X = (\<Prod> i\<in>J. emeasure (M i) (E i))"
using E by (simp add: J.measure_times)
also have "\<dots> = (\<Prod> i\<in>J. emeasure (M i) (if i \<in> J then E i else space (M i)))"
by simp
also have "\<dots> = (\<Prod> i\<in>K. emeasure (M i) (if i \<in> J then E i else space (M i)))"
using `finite K` `J \<subseteq> K`
by (intro setprod_mono_one_left) (auto simp: M.emeasure_space_1)
also have "\<dots> = emeasure (Pi\<^isub>M K M) (\<Pi>\<^isub>E i\<in>K. if i \<in> J then E i else space (M i))"
using E by (simp add: K.measure_times)
also have "(\<Pi>\<^isub>E i\<in>K. if i \<in> J then E i else space (M i)) = (\<lambda>f. restrict f J) -` Pi\<^isub>E J E \<inter> (\<Pi>\<^isub>E i\<in>K. space (M i))"
using `J \<subseteq> K` sets.sets_into_space E by (force simp: Pi_iff PiE_def split: split_if_asm)
finally show "emeasure (Pi\<^isub>M J M) X = emeasure ?D X"
using X `J \<subseteq> K` apply (subst emeasure_distr)
by (auto intro!: measurable_restrict_subset simp: space_PiM)
qed

lemma (in product_prob_space) emeasure_prod_emb[simp]:
assumes L: "J \<noteq> {}" "J \<subseteq> L" "finite L" and X: "X \<in> sets (Pi\<^isub>M J M)"
shows "emeasure (Pi\<^isub>M L M) (prod_emb L M J X) = emeasure (Pi\<^isub>M J M) X"
by (subst distr_restrict[OF L])
(simp add: prod_emb_def space_PiM emeasure_distr measurable_restrict_subset L X)

definition
limP :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i set \<Rightarrow> ('i \<Rightarrow> 'a) measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where
"limP I M P = extend_measure (\<Pi>\<^isub>E i\<in>I. space (M i))
{(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}
(\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j))
(\<lambda>(J, X). emeasure (P J) (Pi\<^isub>E J X))"

abbreviation "lim\<^isub>P \<equiv> limP"

lemma space_limP[simp]: "space (limP I M P) = space (PiM I M)"
by (auto simp add: limP_def space_PiM prod_emb_def intro!: space_extend_measure)

lemma sets_limP[simp]: "sets (limP I M P) = sets (PiM I M)"
by (auto simp add: limP_def sets_PiM prod_algebra_def prod_emb_def intro!: sets_extend_measure)

lemma measurable_limP1[simp]: "measurable (limP I M P) M' = measurable (\<Pi>\<^isub>M i\<in>I. M i) M'"
unfolding measurable_def by auto

lemma measurable_limP2[simp]: "measurable M' (limP I M P) = measurable M' (\<Pi>\<^isub>M i\<in>I. M i)"
unfolding measurable_def by auto

locale projective_family =
fixes I::"'i set" and P::"'i set \<Rightarrow> ('i \<Rightarrow> 'a) measure" and M::"('i \<Rightarrow> 'a measure)"
assumes projective: "\<And>J H X. J \<noteq> {} \<Longrightarrow> J \<subseteq> H \<Longrightarrow> H \<subseteq> I \<Longrightarrow> finite H \<Longrightarrow> X \<in> sets (PiM J M) \<Longrightarrow>
(P H) (prod_emb H M J X) = (P J) X"
assumes proj_prob_space: "\<And>J. finite J \<Longrightarrow> prob_space (P J)"
assumes proj_space: "\<And>J. finite J \<Longrightarrow> space (P J) = space (PiM J M)"
assumes proj_sets: "\<And>J. finite J \<Longrightarrow> sets (P J) = sets (PiM J M)"
begin

lemma emeasure_limP:
assumes "finite J"
assumes "J \<subseteq> I"
assumes A: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> sets (M i)"
shows "emeasure (limP J M P) (Pi\<^isub>E J A) = emeasure (P J) (Pi\<^isub>E J A)"
proof -
have "Pi\<^isub>E J (restrict A J) \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))"
using sets.sets_into_space[OF A] by (auto simp: PiE_iff) blast
hence "emeasure (limP J M P) (Pi\<^isub>E J A) =
emeasure (limP J M P) (prod_emb J M J (Pi\<^isub>E J A))"
using assms(1-3) sets.sets_into_space by (auto simp add: prod_emb_id PiE_def Pi_def)
also have "\<dots> = emeasure (P J) (Pi\<^isub>E J A)"
proof (rule emeasure_extend_measure_Pair[OF limP_def])
show "positive (sets (limP J M P)) (P J)" unfolding positive_def by auto
by (auto simp: suminf_emeasure proj_sets[OF `finite J`])
show "(J \<noteq> {} \<or> J = {}) \<and> finite J \<and> J \<subseteq> J \<and> A \<in> (\<Pi> j\<in>J. sets (M j))"
using assms by auto
fix K and X::"'i \<Rightarrow> 'a set"
show "prod_emb J M K (Pi\<^isub>E K X) \<in> Pow (\<Pi>\<^isub>E i\<in>J. space (M i))"
by (auto simp: prod_emb_def)
assume JX: "(K \<noteq> {} \<or> J = {}) \<and> finite K \<and> K \<subseteq> J \<and> X \<in> (\<Pi> j\<in>K. sets (M j))"
thus "emeasure (P J) (prod_emb J M K (Pi\<^isub>E K X)) = emeasure (P K) (Pi\<^isub>E K X)"
using assms
apply (cases "J = {}")
apply (fastforce simp add: intro!: projective sets_PiM_I_finite)
done
qed
finally show ?thesis .
qed

lemma limP_finite:
assumes "finite J"
assumes "J \<subseteq> I"
shows "limP J M P = P J" (is "?P = _")
proof (rule measure_eqI_generator_eq)
let ?J = "{Pi\<^isub>E J E | E. \<forall>i\<in>J. E i \<in> sets (M i)}"
let ?\<Omega> = "(\<Pi>\<^isub>E k\<in>J. space (M k))"
interpret prob_space "P J" using proj_prob_space `finite J` by simp
show "emeasure ?P (\<Pi>\<^isub>E k\<in>J. space (M k)) \<noteq> \<infinity>" using assms `finite J` by (auto simp: emeasure_limP)
show "sets (limP J M P) = sigma_sets ?\<Omega> ?J" "sets (P J) = sigma_sets ?\<Omega> ?J"
using `finite J` proj_sets by (simp_all add: sets_PiM prod_algebra_eq_finite Pi_iff)
fix X assume "X \<in> ?J"
then obtain E where X: "X = Pi\<^isub>E J E" and E: "\<forall>i\<in>J. E i \<in> sets (M i)" by auto
with `finite J` have "X \<in> sets (limP J M P)" by simp
have emb_self: "prod_emb J M J (Pi\<^isub>E J E) = Pi\<^isub>E J E"
using E sets.sets_into_space
by (auto intro!: prod_emb_PiE_same_index)
show "emeasure (limP J M P) X = emeasure (P J) X"
unfolding X using E
by (intro emeasure_limP assms) simp
qed (auto simp: Pi_iff dest: sets.sets_into_space intro: Int_stable_PiE)

lemma emeasure_fun_emb[simp]:
assumes L: "J \<noteq> {}" "J \<subseteq> L" "finite L" "L \<subseteq> I" and X: "X \<in> sets (PiM J M)"
shows "emeasure (limP L M P) (prod_emb L M J X) = emeasure (limP J M P) X"
using assms
by (subst limP_finite) (auto simp: limP_finite finite_subset projective)

abbreviation
"emb L K X \<equiv> prod_emb L M K X"

lemma prod_emb_injective:
assumes "J \<subseteq> L" and sets: "X \<in> sets (Pi\<^isub>M J M)" "Y \<in> sets (Pi\<^isub>M J M)"
assumes "emb L J X = emb L J Y"
shows "X = Y"
proof (rule injective_vimage_restrict)
show "X \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))" "Y \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))"
using sets[THEN sets.sets_into_space] by (auto simp: space_PiM)
have "\<forall>i\<in>L. \<exists>x. x \<in> space (M i)"
proof
fix i assume "i \<in> L"
interpret prob_space "P {i}" using proj_prob_space by simp
from not_empty show "\<exists>x. x \<in> space (M i)" by (auto simp add: proj_space space_PiM)
qed
from bchoice[OF this]
show "(\<Pi>\<^isub>E i\<in>L. space (M i)) \<noteq> {}" by (auto simp: PiE_def)
show "(\<lambda>x. restrict x J) -` X \<inter> (\<Pi>\<^isub>E i\<in>L. space (M i)) = (\<lambda>x. restrict x J) -` Y \<inter> (\<Pi>\<^isub>E i\<in>L. space (M i))"
using `prod_emb L M J X = prod_emb L M J Y` by (simp add: prod_emb_def)
qed fact

definition generator :: "('i \<Rightarrow> 'a) set set" where
"generator = (\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` sets (Pi\<^isub>M J M))"

lemma generatorI':
"J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> emb I J X \<in> generator"
unfolding generator_def by auto

lemma algebra_generator:
assumes "I \<noteq> {}" shows "algebra (\<Pi>\<^isub>E i\<in>I. space (M i)) generator" (is "algebra ?\<Omega> ?G")
unfolding algebra_def algebra_axioms_def ring_of_sets_iff
proof (intro conjI ballI)
let ?G = generator
show "?G \<subseteq> Pow ?\<Omega>"
by (auto simp: generator_def prod_emb_def)
from `I \<noteq> {}` obtain i where "i \<in> I" by auto
then show "{} \<in> ?G"
by (auto intro!: exI[of _ "{i}"] image_eqI[where x="\<lambda>i. {}"]
simp: sigma_sets.Empty generator_def prod_emb_def)
from `i \<in> I` show "?\<Omega> \<in> ?G"
by (auto intro!: exI[of _ "{i}"] image_eqI[where x="Pi\<^isub>E {i} (\<lambda>i. space (M i))"]
simp: generator_def prod_emb_def)
fix A assume "A \<in> ?G"
then obtain JA XA where XA: "JA \<noteq> {}" "finite JA" "JA \<subseteq> I" "XA \<in> sets (Pi\<^isub>M JA M)" and A: "A = emb I JA XA"
by (auto simp: generator_def)
fix B assume "B \<in> ?G"
then obtain JB XB where XB: "JB \<noteq> {}" "finite JB" "JB \<subseteq> I" "XB \<in> sets (Pi\<^isub>M JB M)" and B: "B = emb I JB XB"
by (auto simp: generator_def)
let ?RA = "emb (JA \<union> JB) JA XA"
let ?RB = "emb (JA \<union> JB) JB XB"
have *: "A - B = emb I (JA \<union> JB) (?RA - ?RB)" "A \<union> B = emb I (JA \<union> JB) (?RA \<union> ?RB)"
using XA A XB B by auto
show "A - B \<in> ?G" "A \<union> B \<in> ?G"
unfolding * using XA XB by (safe intro!: generatorI') auto
qed

lemma sets_PiM_generator:
"sets (PiM I M) = sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) generator"
proof cases
assume "I = {}" then show ?thesis
unfolding generator_def
by (auto simp: sets_PiM_empty sigma_sets_empty_eq cong: conj_cong)
next
assume "I \<noteq> {}"
show ?thesis
proof
show "sets (Pi\<^isub>M I M) \<subseteq> sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) generator"
unfolding sets_PiM
proof (safe intro!: sigma_sets_subseteq)
fix A assume "A \<in> prod_algebra I M" with `I \<noteq> {}` show "A \<in> generator"
by (auto intro!: generatorI' sets_PiM_I_finite elim!: prod_algebraE)
qed
qed (auto simp: generator_def space_PiM[symmetric] intro!: sets.sigma_sets_subset)
qed

lemma generatorI:
"J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> A = emb I J X \<Longrightarrow> A \<in> generator"
unfolding generator_def by auto

definition mu_G ("\<mu>G") where
"\<mu>G A =
(THE x. \<forall>J. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> J \<subseteq> I \<longrightarrow> (\<forall>X\<in>sets (Pi\<^isub>M J M). A = emb I J X \<longrightarrow> x = emeasure (limP J M P) X))"

lemma mu_G_spec:
assumes J: "J \<noteq> {}" "finite J" "J \<subseteq> I" "A = emb I J X" "X \<in> sets (Pi\<^isub>M J M)"
shows "\<mu>G A = emeasure (limP J M P) X"
unfolding mu_G_def
proof (intro the_equality allI impI ballI)
fix K Y assume K: "K \<noteq> {}" "finite K" "K \<subseteq> I" "A = emb I K Y" "Y \<in> sets (Pi\<^isub>M K M)"
have "emeasure (limP K M P) Y = emeasure (limP (K \<union> J) M P) (emb (K \<union> J) K Y)"
using K J by simp
also have "emb (K \<union> J) K Y = emb (K \<union> J) J X"
using K J by (simp add: prod_emb_injective[of "K \<union> J" I])
also have "emeasure (limP (K \<union> J) M P) (emb (K \<union> J) J X) = emeasure (limP J M P) X"
using K J by simp
finally show "emeasure (limP J M P) X = emeasure (limP K M P) Y" ..
qed (insert J, force)

lemma mu_G_eq:
"J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> \<mu>G (emb I J X) = emeasure (limP J M P) X"
by (intro mu_G_spec) auto

lemma generator_Ex:
assumes *: "A \<in> generator"
shows "\<exists>J X. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> X \<in> sets (Pi\<^isub>M J M) \<and> A = emb I J X \<and> \<mu>G A = emeasure (limP J M P) X"
proof -
from * obtain J X where J: "J \<noteq> {}" "finite J" "J \<subseteq> I" "A = emb I J X" "X \<in> sets (Pi\<^isub>M J M)"
unfolding generator_def by auto
with mu_G_spec[OF this] show ?thesis by auto
qed

lemma generatorE:
assumes A: "A \<in> generator"
obtains J X where "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)" "emb I J X = A" "\<mu>G A = emeasure (limP J M P) X"
using generator_Ex[OF A] by atomize_elim auto

lemma merge_sets:
"J \<inter> K = {} \<Longrightarrow> A \<in> sets (Pi\<^isub>M (J \<union> K) M) \<Longrightarrow> x \<in> space (Pi\<^isub>M J M) \<Longrightarrow> (\<lambda>y. merge J K (x,y)) -` A \<inter> space (Pi\<^isub>M K M) \<in> sets (Pi\<^isub>M K M)"
by simp

lemma merge_emb:
assumes "K \<subseteq> I" "J \<subseteq> I" and y: "y \<in> space (Pi\<^isub>M J M)"
shows "((\<lambda>x. merge J (I - J) (y, x)) -` emb I K X \<inter> space (Pi\<^isub>M I M)) =
emb I (K - J) ((\<lambda>x. merge J (K - J) (y, x)) -` emb (J \<union> K) K X \<inter> space (Pi\<^isub>M (K - J) M))"
proof -
have [simp]: "\<And>x J K L. merge J K (y, restrict x L) = merge J (K \<inter> L) (y, x)"
by (auto simp: restrict_def merge_def)
have [simp]: "\<And>x J K L. restrict (merge J K (y, x)) L = merge (J \<inter> L) (K \<inter> L) (y, x)"
by (auto simp: restrict_def merge_def)
have [simp]: "(I - J) \<inter> K = K - J" using `K \<subseteq> I` `J \<subseteq> I` by auto
have [simp]: "(K - J) \<inter> (K \<union> J) = K - J" by auto
have [simp]: "(K - J) \<inter> K = K - J" by auto
from y `K \<subseteq> I` `J \<subseteq> I` show ?thesis
by (simp split: split_merge add: prod_emb_def Pi_iff PiE_def extensional_merge_sub set_eq_iff space_PiM)
auto
qed

lemma positive_mu_G:
assumes "I \<noteq> {}"
shows "positive generator \<mu>G"
proof -
interpret G!: algebra "\<Pi>\<^isub>E i\<in>I. space (M i)" generator by (rule algebra_generator) fact
show ?thesis
proof (intro positive_def[THEN iffD2] conjI ballI)
from generatorE[OF G.empty_sets] guess J X . note this[simp]
have "X = {}"
by (rule prod_emb_injective[of J I]) simp_all
then show "\<mu>G {} = 0" by simp
next
fix A assume "A \<in> generator"
from generatorE[OF this] guess J X . note this[simp]
show "0 \<le> \<mu>G A" by (simp add: emeasure_nonneg)
qed
qed

assumes "I \<noteq> {}"
proof -
interpret G!: algebra "\<Pi>\<^isub>E i\<in>I. space (M i)" generator by (rule algebra_generator) fact
show ?thesis
proof (intro additive_def[THEN iffD2] ballI impI)
fix A assume "A \<in> generator" with generatorE guess J X . note J = this
fix B assume "B \<in> generator" with generatorE guess K Y . note K = this
assume "A \<inter> B = {}"
have JK: "J \<union> K \<noteq> {}" "J \<union> K \<subseteq> I" "finite (J \<union> K)"
using J K by auto
have JK_disj: "emb (J \<union> K) J X \<inter> emb (J \<union> K) K Y = {}"
apply (rule prod_emb_injective[of "J \<union> K" I])
apply (insert `A \<inter> B = {}` JK J K)
done
have AB: "A = emb I (J \<union> K) (emb (J \<union> K) J X)" "B = emb I (J \<union> K) (emb (J \<union> K) K Y)"
using J K by simp_all
then have "\<mu>G (A \<union> B) = \<mu>G (emb I (J \<union> K) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y))"
by simp
also have "\<dots> = emeasure (limP (J \<union> K) M P) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y)"
using JK J(1, 4) K(1, 4) by (simp add: mu_G_eq sets.Un del: prod_emb_Un)
also have "\<dots> = \<mu>G A + \<mu>G B"
using J K JK_disj by (simp add: plus_emeasure[symmetric])
finally show "\<mu>G (A \<union> B) = \<mu>G A + \<mu>G B" .
qed
qed

end

sublocale product_prob_space \<subseteq> projective_family I "\<lambda>J. PiM J M" M
proof
fix J::"'i set" assume "finite J"
interpret f: finite_product_prob_space M J proof qed fact
show "emeasure (Pi\<^isub>M J M) (space (Pi\<^isub>M J M)) \<noteq> \<infinity>" by simp
show "\<exists>A. range A \<subseteq> sets (Pi\<^isub>M J M) \<and>
(\<Union>i. A i) = space (Pi\<^isub>M J M) \<and>
(\<forall>i. emeasure (Pi\<^isub>M J M) (A i) \<noteq> \<infinity>)" using sigma_finite[OF `finite J`]