src/HOL/Probability/Fin_Map.thy
 author hoelzl Tue, 05 Mar 2013 15:43:14 +0100 changeset 51343 b61b32f62c78 parent 51106 5746e671ea70 child 51473 1210309fddab permissions -rw-r--r--
use generate_topology for second countable topologies, does not require intersection stable basis
```
(*  Title:      HOL/Probability/Fin_Map.thy
Author:     Fabian Immler, TU München
*)

theory Fin_Map
imports Finite_Product_Measure
begin

text {* Auxiliary type that is instantiated to @{class polish_space}, needed for the proof of
projective limit. @{const extensional} functions are used for the representation in order to
stay close to the developments of (finite) products @{const Pi\<^isub>E} and their sigma-algebra
@{const Pi\<^isub>M}. *}

typedef ('i, 'a) finmap ("(_ \<Rightarrow>\<^isub>F /_)" [22, 21] 21) =
"{(I::'i set, f::'i \<Rightarrow> 'a). finite I \<and> f \<in> extensional I}" by auto

subsection {* Domain and Application *}

definition domain where "domain P = fst (Rep_finmap P)"

lemma finite_domain[simp, intro]: "finite (domain P)"
by (cases P) (auto simp: domain_def Abs_finmap_inverse)

definition proj ("'((_)')\<^isub>F" [0] 1000) where "proj P i = snd (Rep_finmap P) i"

declare [[coercion proj]]

lemma extensional_proj[simp, intro]: "(P)\<^isub>F \<in> extensional (domain P)"
by (cases P) (auto simp: domain_def Abs_finmap_inverse proj_def[abs_def])

lemma proj_undefined[simp, intro]: "i \<notin> domain P \<Longrightarrow> P i = undefined"
using extensional_proj[of P] unfolding extensional_def by auto

lemma finmap_eq_iff: "P = Q \<longleftrightarrow> (domain P = domain Q \<and> (\<forall>i\<in>domain P. P i = Q i))"
by (cases P, cases Q)
(auto simp add: Abs_finmap_inject extensional_def domain_def proj_def Abs_finmap_inverse
intro: extensionalityI)

subsection {* Countable Finite Maps *}

instance finmap :: (countable, countable) countable
proof
obtain mapper where mapper: "\<And>fm :: 'a \<Rightarrow>\<^isub>F 'b. set (mapper fm) = domain fm"
by (metis finite_list[OF finite_domain])
have "inj (\<lambda>fm. map (\<lambda>i. (i, (fm)\<^isub>F i)) (mapper fm))" (is "inj ?F")
proof (rule inj_onI)
fix f1 f2 assume "?F f1 = ?F f2"
then have "map fst (?F f1) = map fst (?F f2)" by simp
then have "mapper f1 = mapper f2" by (simp add: comp_def)
then have "domain f1 = domain f2" by (simp add: mapper[symmetric])
with `?F f1 = ?F f2` show "f1 = f2"
unfolding `mapper f1 = mapper f2` map_eq_conv mapper
qed
then show "\<exists>to_nat :: 'a \<Rightarrow>\<^isub>F 'b \<Rightarrow> nat. inj to_nat"
by (intro exI[of _ "to_nat \<circ> ?F"] inj_comp) auto
qed

subsection {* Constructor of Finite Maps *}

definition "finmap_of inds f = Abs_finmap (inds, restrict f inds)"

lemma proj_finmap_of[simp]:
assumes "finite inds"
shows "(finmap_of inds f)\<^isub>F = restrict f inds"
using assms
by (auto simp: Abs_finmap_inverse finmap_of_def proj_def)

lemma domain_finmap_of[simp]:
assumes "finite inds"
shows "domain (finmap_of inds f) = inds"
using assms
by (auto simp: Abs_finmap_inverse finmap_of_def domain_def)

lemma finmap_of_eq_iff[simp]:
assumes "finite i" "finite j"
shows "finmap_of i m = finmap_of j n \<longleftrightarrow> i = j \<and> (\<forall>k\<in>i. m k= n k)"
using assms by (auto simp: finmap_eq_iff)

lemma finmap_of_inj_on_extensional_finite:
assumes "finite K"
assumes "S \<subseteq> extensional K"
shows "inj_on (finmap_of K) S"
proof (rule inj_onI)
fix x y::"'a \<Rightarrow> 'b"
assume "finmap_of K x = finmap_of K y"
hence "(finmap_of K x)\<^isub>F = (finmap_of K y)\<^isub>F" by simp
moreover
assume "x \<in> S" "y \<in> S" hence "x \<in> extensional K" "y \<in> extensional K" using assms by auto
ultimately
show "x = y" using assms by (simp add: extensional_restrict)
qed

subsection {* Product set of Finite Maps *}

text {* This is @{term Pi} for Finite Maps, most of this is copied *}

definition Pi' :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a set) \<Rightarrow> ('i \<Rightarrow>\<^isub>F 'a) set" where
"Pi' I A = { P. domain P = I \<and> (\<forall>i. i \<in> I \<longrightarrow> (P)\<^isub>F i \<in> A i) } "

syntax
"_Pi'"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PI' _:_./ _)" 10)

syntax (xsymbols)
"_Pi'" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>' _\<in>_./ _)"   10)

syntax (HTML output)
"_Pi'" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>' _\<in>_./ _)"   10)

translations
"PI' x:A. B" == "CONST Pi' A (%x. B)"

subsubsection{*Basic Properties of @{term Pi'}*}

lemma Pi'_I[intro!]: "domain f = A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi' A B"

lemma Pi'_I'[simp]: "domain f = A \<Longrightarrow> (\<And>x. x \<in> A \<longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi' A B"

lemma Pi'_mem: "f\<in> Pi' A B \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<in> B x"

lemma Pi'_iff: "f \<in> Pi' I X \<longleftrightarrow> domain f = I \<and> (\<forall>i\<in>I. f i \<in> X i)"
unfolding Pi'_def by auto

lemma Pi'E [elim]:
"f \<in> Pi' A B \<Longrightarrow> (f x \<in> B x \<Longrightarrow> domain f = A \<Longrightarrow> Q) \<Longrightarrow> (x \<notin> A \<Longrightarrow> Q) \<Longrightarrow> Q"
by(auto simp: Pi'_def)

lemma in_Pi'_cong:
"domain f = domain g \<Longrightarrow> (\<And> w. w \<in> A \<Longrightarrow> f w = g w) \<Longrightarrow> f \<in> Pi' A B \<longleftrightarrow> g \<in> Pi' A B"
by (auto simp: Pi'_def)

lemma Pi'_eq_empty[simp]:
assumes "finite A" shows "(Pi' A B) = {} \<longleftrightarrow> (\<exists>x\<in>A. B x = {})"
using assms
apply (drule_tac x = "finmap_of A (\<lambda>u. SOME y. y \<in> B u)" in spec, auto)
apply (cut_tac P= "%y. y \<in> B i" in some_eq_ex, auto)
done

lemma Pi'_mono: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x) \<Longrightarrow> Pi' A B \<subseteq> Pi' A C"
by (auto simp: Pi'_def)

lemma Pi_Pi': "finite A \<Longrightarrow> (Pi\<^isub>E A B) = proj ` Pi' A B"
apply (auto simp: Pi'_def Pi_def extensional_def)
apply (rule_tac x = "finmap_of A (restrict x A)" in image_eqI)
apply auto
done

subsection {* Topological Space of Finite Maps *}

instantiation finmap :: (type, topological_space) topological_space
begin

definition open_finmap :: "('a \<Rightarrow>\<^isub>F 'b) set \<Rightarrow> bool" where
"open_finmap = generate_topology {Pi' a b|a b. \<forall>i\<in>a. open (b i)}"

lemma open_Pi'I: "(\<And>i. i \<in> I \<Longrightarrow> open (A i)) \<Longrightarrow> open (Pi' I A)"
by (auto intro: generate_topology.Basis simp: open_finmap_def)

instance using topological_space_generate_topology
by intro_classes (auto simp: open_finmap_def class.topological_space_def)

end

lemma open_restricted_space:
shows "open {m. P (domain m)}"
proof -
have "{m. P (domain m)} = (\<Union>i \<in> Collect P. {m. domain m = i})" by auto
also have "open \<dots>"
proof (rule, safe, cases)
fix i::"'a set"
assume "finite i"
hence "{m. domain m = i} = Pi' i (\<lambda>_. UNIV)" by (auto simp: Pi'_def)
also have "open \<dots>" by (auto intro: open_Pi'I simp: `finite i`)
finally show "open {m. domain m = i}" .
next
fix i::"'a set"
assume "\<not> finite i" hence "{m. domain m = i} = {}" by auto
also have "open \<dots>" by simp
finally show "open {m. domain m = i}" .
qed
finally show ?thesis .
qed

lemma closed_restricted_space:
shows "closed {m. P (domain m)}"
using open_restricted_space[of "\<lambda>x. \<not> P x"]
unfolding closed_def by (rule back_subst) auto

lemma tendsto_proj: "((\<lambda>x. x) ---> a) F \<Longrightarrow> ((\<lambda>x. (x)\<^isub>F i) ---> (a)\<^isub>F i) F"
unfolding tendsto_def
proof safe
fix S::"'b set"
let ?S = "Pi' (domain a) (\<lambda>x. if x = i then S else UNIV)"
assume "open S" hence "open ?S" by (auto intro!: open_Pi'I)
moreover assume "\<forall>S. open S \<longrightarrow> a \<in> S \<longrightarrow> eventually (\<lambda>x. x \<in> S) F" "a i \<in> S"
ultimately have "eventually (\<lambda>x. x \<in> ?S) F" by auto
thus "eventually (\<lambda>x. (x)\<^isub>F i \<in> S) F"
by eventually_elim (insert `a i \<in> S`, force simp: Pi'_iff split: split_if_asm)
qed

lemma continuous_proj:
shows "continuous_on s (\<lambda>x. (x)\<^isub>F i)"
unfolding continuous_on_def
by (safe intro!: tendsto_proj tendsto_ident_at_within)

instance finmap :: (type, first_countable_topology) first_countable_topology
proof
fix x::"'a\<Rightarrow>\<^isub>F'b"
have "\<forall>i. \<exists>A. countable A \<and> (\<forall>a\<in>A. x i \<in> a) \<and> (\<forall>a\<in>A. open a) \<and>
(\<forall>S. open S \<and> x i \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)) \<and> (\<forall>a b. a \<in> A \<longrightarrow> b \<in> A \<longrightarrow> a \<inter> b \<in> A)" (is "\<forall>i. ?th i")
proof
fix i from first_countable_basis_Int_stableE[of "x i"] guess A .
thus "?th i" by (intro exI[where x=A]) simp
qed
then guess A unfolding choice_iff .. note A = this
hence open_sub: "\<And>i S. i\<in>domain x \<Longrightarrow> open (S i) \<Longrightarrow> x i\<in>(S i) \<Longrightarrow> (\<exists>a\<in>A i. a\<subseteq>(S i))" by auto
have A_notempty: "\<And>i. i \<in> domain x \<Longrightarrow> A i \<noteq> {}" using open_sub[of _ "\<lambda>_. UNIV"] by auto
let ?A = "(\<lambda>f. Pi' (domain x) f) ` (Pi\<^isub>E (domain x) A)"
show "\<exists>A. countable A \<and> (\<forall>a\<in>A. x \<in> a \<and> open a) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S))"
proof (rule exI[where x="?A"], safe)
show "countable ?A" using A by (simp add: countable_PiE)
next
fix S::"('a \<Rightarrow>\<^isub>F 'b) set" assume "open S" "x \<in> S"
thus "\<exists>a\<in>?A. a \<subseteq> S" unfolding open_finmap_def
proof (induct rule: generate_topology.induct)
case UNIV thus ?case by (auto simp add: ex_in_conv PiE_eq_empty_iff A_notempty)
next
case (Int a b)
then obtain f g where
"f \<in> Pi\<^isub>E (domain x) A" "Pi' (domain x) f \<subseteq> a" "g \<in> Pi\<^isub>E (domain x) A" "Pi' (domain x) g \<subseteq> b"
by auto
thus ?case using A
by (auto simp: Pi'_iff PiE_iff extensional_def Int_stable_def
intro!: bexI[where x="\<lambda>i. f i \<inter> g i"])
next
case (UN B)
then obtain b where "x \<in> b" "b \<in> B" by auto
hence "\<exists>a\<in>?A. a \<subseteq> b" using UN by simp
thus ?case using `b \<in> B` by blast
next
case (Basis s)
then obtain a b where xs: "x\<in> Pi' a b" "s = Pi' a b" "\<And>i. i\<in>a \<Longrightarrow> open (b i)" by auto
have "\<forall>i. \<exists>a. (i \<in> domain x \<and> open (b i) \<and> (x)\<^isub>F i \<in> b i) \<longrightarrow> (a\<in>A i \<and> a \<subseteq> b i)"
using open_sub[of _ b] by auto
then obtain b'
where "\<And>i. i \<in> domain x \<Longrightarrow> open (b i) \<Longrightarrow> (x)\<^isub>F i \<in> b i \<Longrightarrow> (b' i \<in>A i \<and> b' i \<subseteq> b i)"
unfolding choice_iff by auto
with xs have "\<And>i. i \<in> a \<Longrightarrow> (b' i \<in>A i \<and> b' i \<subseteq> b i)" "Pi' a b' \<subseteq> Pi' a b"
by (auto simp: Pi'_iff intro!: Pi'_mono)
thus ?case using xs
by (intro bexI[where x="Pi' a b'"])
(auto simp: Pi'_iff intro!: image_eqI[where x="restrict b' (domain x)"])
qed
qed (insert A,auto simp: PiE_iff intro!: open_Pi'I)
qed

subsection {* Metric Space of Finite Maps *}

instantiation finmap :: (type, metric_space) metric_space
begin

definition dist_finmap where
"dist P Q = Max (range (\<lambda>i. dist ((P)\<^isub>F i) ((Q)\<^isub>F i))) + (if domain P = domain Q then 0 else 1)"

fixes a b::real
assumes "a \<ge> 0" "b \<ge> 0"
shows "a + b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
using assms by auto

lemma finite_proj_image': "x \<notin> domain P \<Longrightarrow> finite ((P)\<^isub>F ` S)"
by (rule finite_subset[of _ "proj P ` (domain P \<inter> S \<union> {x})"]) auto

lemma finite_proj_image: "finite ((P)\<^isub>F ` S)"
by (cases "\<exists>x. x \<notin> domain P") (auto intro: finite_proj_image' finite_subset[where B="domain P"])

lemma finite_proj_diag: "finite ((\<lambda>i. d ((P)\<^isub>F i) ((Q)\<^isub>F i)) ` S)"
proof -
have "(\<lambda>i. d ((P)\<^isub>F i) ((Q)\<^isub>F i)) ` S = (\<lambda>(i, j). d i j) ` ((\<lambda>i. ((P)\<^isub>F i, (Q)\<^isub>F i)) ` S)" by auto
moreover have "((\<lambda>i. ((P)\<^isub>F i, (Q)\<^isub>F i)) ` S) \<subseteq> (\<lambda>i. (P)\<^isub>F i) ` S \<times> (\<lambda>i. (Q)\<^isub>F i) ` S" by auto
moreover have "finite \<dots>" using finite_proj_image[of P S] finite_proj_image[of Q S]
by (intro finite_cartesian_product) simp_all
ultimately show ?thesis by (simp add: finite_subset)
qed

lemma dist_le_1_imp_domain_eq:
shows "dist P Q < 1 \<Longrightarrow> domain P = domain Q"
by (simp add: dist_finmap_def finite_proj_diag split: split_if_asm)

lemma dist_proj:
shows "dist ((x)\<^isub>F i) ((y)\<^isub>F i) \<le> dist x y"
proof -
have "dist (x i) (y i) \<le> Max (range (\<lambda>i. dist (x i) (y i)))"
also have "\<dots> \<le> dist x y" by (simp add: dist_finmap_def)
finally show ?thesis .
qed

lemma dist_finmap_lessI:
assumes "domain P = domain Q"
assumes "0 < e"
assumes "\<And>i. i \<in> domain P \<Longrightarrow> dist (P i) (Q i) < e"
shows "dist P Q < e"
proof -
have "dist P Q = Max (range (\<lambda>i. dist (P i) (Q i)))"
using assms by (simp add: dist_finmap_def finite_proj_diag)
also have "\<dots> < e"
proof (subst Max_less_iff, safe)
fix i
show "dist ((P)\<^isub>F i) ((Q)\<^isub>F i) < e" using assms
by (cases "i \<in> domain P") simp_all
finally show ?thesis .
qed

instance
proof
fix S::"('a \<Rightarrow>\<^isub>F 'b) set"
show "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)" (is "_ = ?od")
proof
assume "open S"
thus ?od
unfolding open_finmap_def
proof (induct rule: generate_topology.induct)
case UNIV thus ?case by (auto intro: zero_less_one)
next
case (Int a b)
show ?case
proof safe
fix x assume x: "x \<in> a" "x \<in> b"
with Int x obtain e1 e2 where
"e1>0" "\<forall>y. dist y x < e1 \<longrightarrow> y \<in> a" "e2>0" "\<forall>y. dist y x < e2 \<longrightarrow> y \<in> b" by force
thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> a \<inter> b"
by (auto intro!: exI[where x="min e1 e2"])
qed
next
case (UN K)
show ?case
proof safe
fix x X assume "x \<in> X" "X \<in> K"
moreover with UN obtain e where "e>0" "\<And>y. dist y x < e \<longrightarrow> y \<in> X" by force
ultimately show "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> \<Union>K" by auto
qed
next
case (Basis s) then obtain a b where s: "s = Pi' a b" and b: "\<And>i. i\<in>a \<Longrightarrow> open (b i)" by auto
show ?case
proof safe
fix x assume "x \<in> s"
hence [simp]: "finite a" and a_dom: "a = domain x" using s by (auto simp: Pi'_iff)
obtain es where es: "\<forall>i \<in> a. es i > 0 \<and> (\<forall>y. dist y (proj x i) < es i \<longrightarrow> y \<in> b i)"
using b `x \<in> s` by atomize_elim (intro bchoice, auto simp: open_dist s)
hence in_b: "\<And>i y. i \<in> a \<Longrightarrow> dist y (proj x i) < es i \<Longrightarrow> y \<in> b i" by auto
show "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> s"
proof (cases, rule, safe)
assume "a \<noteq> {}"
show "0 < min 1 (Min (es ` a))" using es by (auto simp: `a \<noteq> {}`)
fix y assume d: "dist y x < min 1 (Min (es ` a))"
show "y \<in> s" unfolding s
proof
show "domain y = a" using d s `a \<noteq> {}` by (auto simp: dist_le_1_imp_domain_eq a_dom)
fix i assume "i \<in> a"
moreover
hence "dist ((y)\<^isub>F i) ((x)\<^isub>F i) < es i" using d
by (auto simp: dist_finmap_def `a \<noteq> {}` intro!: le_less_trans[OF dist_proj])
ultimately
show "y i \<in> b i" by (rule in_b)
qed
next
assume "\<not>a \<noteq> {}"
thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> s"
using s `x \<in> s` by (auto simp: Pi'_def dist_le_1_imp_domain_eq intro!: exI[where x=1])
qed
qed
qed
next
assume "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S"
then obtain e where e_pos: "\<And>x. x \<in> S \<Longrightarrow> e x > 0" and
e_in:  "\<And>x y . x \<in> S \<Longrightarrow> dist y x < e x \<Longrightarrow> y \<in> S"
unfolding bchoice_iff
by auto
have S_eq: "S = \<Union>{Pi' a b| a b. \<exists>x\<in>S. domain x = a \<and> b = (\<lambda>i. ball (x i) (e x))}"
proof safe
fix x assume "x \<in> S"
thus "x \<in> \<Union>{Pi' a b| a b. \<exists>x\<in>S. domain x = a \<and> b = (\<lambda>i. ball (x i) (e x))}"
using e_pos by (auto intro!: exI[where x="Pi' (domain x) (\<lambda>i. ball (x i) (e x))"])
next
fix x y
assume "y \<in> S"
moreover
assume "x \<in> (\<Pi>' i\<in>domain y. ball (y i) (e y))"
hence "dist x y < e y" using e_pos `y \<in> S`
by (auto simp: dist_finmap_def Pi'_iff finite_proj_diag dist_commute)
ultimately show "x \<in> S" by (rule e_in)
qed
also have "open \<dots>"
unfolding open_finmap_def
by (intro generate_topology.UN) (auto intro: generate_topology.Basis)
finally show "open S" .
qed
next
fix P Q::"'a \<Rightarrow>\<^isub>F 'b"
have Max_eq_iff: "\<And>A m. finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> (Max A = m) = (m \<in> A \<and> (\<forall>a\<in>A. a \<le> m))"
by (metis Max.in_idem Max_in max_def min_max.sup.commute order_refl)
show "dist P Q = 0 \<longleftrightarrow> P = Q"
by (auto simp: finmap_eq_iff dist_finmap_def Max_ge_iff finite_proj_diag Max_eq_iff
intro!: Max_eqI image_eqI[where x=undefined])
next
fix P Q R::"'a \<Rightarrow>\<^isub>F 'b"
let ?dists = "\<lambda>P Q i. dist ((P)\<^isub>F i) ((Q)\<^isub>F i)"
let ?dpq = "?dists P Q" and ?dpr = "?dists P R" and ?dqr = "?dists Q R"
let ?dom = "\<lambda>P Q. (if domain P = domain Q then 0 else 1::real)"
have "dist P Q = Max (range ?dpq) + ?dom P Q"
also obtain t where "t \<in> range ?dpq" "t = Max (range ?dpq)" by (simp add: finite_proj_diag)
then obtain i where "Max (range ?dpq) = ?dpq i" by auto
also have "?dpq i \<le> ?dpr i + ?dqr i" by (rule dist_triangle2)
also have "?dpr i \<le> Max (range ?dpr)" by (simp add: finite_proj_diag)
also have "?dqr i \<le> Max (range ?dqr)" by (simp add: finite_proj_diag)
also have "?dom P Q \<le> ?dom P R + ?dom Q R" by simp
finally show "dist P Q \<le> dist P R + dist Q R" by (simp add: dist_finmap_def ac_simps)
qed

end

subsection {* Complete Space of Finite Maps *}

lemma tendsto_finmap:
fixes f::"nat \<Rightarrow> ('i \<Rightarrow>\<^isub>F ('a::metric_space))"
assumes ind_f:  "\<And>n. domain (f n) = domain g"
assumes proj_g:  "\<And>i. i \<in> domain g \<Longrightarrow> (\<lambda>n. (f n) i) ----> g i"
shows "f ----> g"
unfolding tendsto_iff
proof safe
fix e::real assume "0 < e"
let ?dists = "\<lambda>x i. dist ((f x)\<^isub>F i) ((g)\<^isub>F i)"
have "eventually (\<lambda>x. \<forall>i\<in>domain g. ?dists x i < e) sequentially"
using finite_domain[of g] proj_g
proof induct
case (insert i G)
with `0 < e` have "eventually (\<lambda>x. ?dists x i < e) sequentially" by (auto simp add: tendsto_iff)
moreover
from insert have "eventually (\<lambda>x. \<forall>i\<in>G. dist ((f x)\<^isub>F i) ((g)\<^isub>F i) < e) sequentially" by simp
ultimately show ?case by eventually_elim auto
qed simp
thus "eventually (\<lambda>x. dist (f x) g < e) sequentially"
by eventually_elim (auto simp add: dist_finmap_def finite_proj_diag ind_f `0 < e`)
qed

instance finmap :: (type, complete_space) complete_space
proof
fix P::"nat \<Rightarrow> 'a \<Rightarrow>\<^isub>F 'b"
assume "Cauchy P"
then obtain Nd where Nd: "\<And>n. n \<ge> Nd \<Longrightarrow> dist (P n) (P Nd) < 1"
by (force simp: cauchy)
def d \<equiv> "domain (P Nd)"
with Nd have dim: "\<And>n. n \<ge> Nd \<Longrightarrow> domain (P n) = d" using dist_le_1_imp_domain_eq by auto
have [simp]: "finite d" unfolding d_def by simp
def p \<equiv> "\<lambda>i n. (P n) i"
def q \<equiv> "\<lambda>i. lim (p i)"
def Q \<equiv> "finmap_of d q"
have q: "\<And>i. i \<in> d \<Longrightarrow> q i = Q i" by (auto simp add: Q_def Abs_finmap_inverse)
{
fix i assume "i \<in> d"
have "Cauchy (p i)" unfolding cauchy p_def
proof safe
fix e::real assume "0 < e"
with `Cauchy P` obtain N where N: "\<And>n. n\<ge>N \<Longrightarrow> dist (P n) (P N) < min e 1"
by (force simp: cauchy min_def)
hence "\<And>n. n \<ge> N \<Longrightarrow> domain (P n) = domain (P N)" using dist_le_1_imp_domain_eq by auto
with dim have dim: "\<And>n. n \<ge> N \<Longrightarrow> domain (P n) = d" by (metis nat_le_linear)
show "\<exists>N. \<forall>n\<ge>N. dist ((P n) i) ((P N) i) < e"
proof (safe intro!: exI[where x="N"])
fix n assume "N \<le> n" have "N \<le> N" by simp
have "dist ((P n) i) ((P N) i) \<le> dist (P n) (P N)"
using dim[OF `N \<le> n`]  dim[OF `N \<le> N`] `i \<in> d`
by (auto intro!: dist_proj)
also have "\<dots> < e" using N[OF `N \<le> n`] by simp
finally show "dist ((P n) i) ((P N) i) < e" .
qed
qed
hence "convergent (p i)" by (metis Cauchy_convergent_iff)
hence "p i ----> q i" unfolding q_def convergent_def by (metis limI)
} note p = this
have "P ----> Q"
proof (rule metric_LIMSEQ_I)
fix e::real assume "0 < e"
have "\<exists>ni. \<forall>i\<in>d. \<forall>n\<ge>ni i. dist (p i n) (q i) < e"
proof (safe intro!: bchoice)
fix i assume "i \<in> d"
from p[OF `i \<in> d`, THEN metric_LIMSEQ_D, OF `0 < e`]
show "\<exists>no. \<forall>n\<ge>no. dist (p i n) (q i) < e" .
qed then guess ni .. note ni = this
def N \<equiv> "max Nd (Max (ni ` d))"
show "\<exists>N. \<forall>n\<ge>N. dist (P n) Q < e"
proof (safe intro!: exI[where x="N"])
fix n assume "N \<le> n"
hence dom: "domain (P n) = d" "domain Q = d" "domain (P n) = domain Q"
using dim by (simp_all add: N_def Q_def dim_def Abs_finmap_inverse)
show "dist (P n) Q < e"
proof (rule dist_finmap_lessI[OF dom(3) `0 < e`])
fix i
assume "i \<in> domain (P n)"
hence "ni i \<le> Max (ni ` d)" using dom by simp
also have "\<dots> \<le> N" by (simp add: N_def)
finally show "dist ((P n)\<^isub>F i) ((Q)\<^isub>F i) < e" using ni `i \<in> domain (P n)` `N \<le> n` dom
by (auto simp: p_def q N_def less_imp_le)
qed
qed
qed
thus "convergent P" by (auto simp: convergent_def)
qed

subsection {* Second Countable Space of Finite Maps *}

instantiation finmap :: (countable, second_countable_topology) second_countable_topology
begin

definition basis_proj::"'b set set"
where "basis_proj = (SOME B. countable B \<and> topological_basis B)"

lemma countable_basis_proj: "countable basis_proj" and basis_proj: "topological_basis basis_proj"
unfolding basis_proj_def by (intro is_basis countable_basis)+

definition basis_finmap::"('a \<Rightarrow>\<^isub>F 'b) set set"
where "basis_finmap = {Pi' I S|I S. finite I \<and> (\<forall>i \<in> I. S i \<in> basis_proj)}"

lemma in_basis_finmapI:
assumes "finite I" assumes "\<And>i. i \<in> I \<Longrightarrow> S i \<in> basis_proj"
shows "Pi' I S \<in> basis_finmap"
using assms unfolding basis_finmap_def by auto

lemma basis_finmap_eq:
assumes "basis_proj \<noteq> {}"
shows "basis_finmap = (\<lambda>f. Pi' (domain f) (\<lambda>i. from_nat_into basis_proj ((f)\<^isub>F i))) `
(UNIV::('a \<Rightarrow>\<^isub>F nat) set)" (is "_ = ?f ` _")
unfolding basis_finmap_def
proof safe
fix I::"'a set" and S::"'a \<Rightarrow> 'b set"
assume "finite I" "\<forall>i\<in>I. S i \<in> basis_proj"
hence "Pi' I S = ?f (finmap_of I (\<lambda>x. to_nat_on basis_proj (S x)))"
by (force simp: Pi'_def countable_basis_proj)
thus "Pi' I S \<in> range ?f" by simp
next
fix x and f::"'a \<Rightarrow>\<^isub>F nat"
show "\<exists>I S. (\<Pi>' i\<in>domain f. from_nat_into local.basis_proj ((f)\<^isub>F i)) = Pi' I S \<and>
finite I \<and> (\<forall>i\<in>I. S i \<in> local.basis_proj)"
using assms by (auto intro: from_nat_into)
qed

lemma basis_finmap_eq_empty: "basis_proj = {} \<Longrightarrow> basis_finmap = {Pi' {} undefined}"
by (auto simp: Pi'_iff basis_finmap_def)

lemma countable_basis_finmap: "countable basis_finmap"
by (cases "basis_proj = {}") (auto simp: basis_finmap_eq basis_finmap_eq_empty)

lemma finmap_topological_basis:
"topological_basis basis_finmap"
proof (subst topological_basis_iff, safe)
fix B' assume "B' \<in> basis_finmap"
thus "open B'"
by (auto intro!: open_Pi'I topological_basis_open[OF basis_proj]
simp: topological_basis_def basis_finmap_def Let_def)
next
fix O'::"('a \<Rightarrow>\<^isub>F 'b) set" and x
assume O': "open O'" "x \<in> O'"
then obtain a where a:
"x \<in> Pi' (domain x) a" "Pi' (domain x) a \<subseteq> O'" "\<And>i. i\<in>domain x \<Longrightarrow> open (a i)"
unfolding open_finmap_def
proof (atomize_elim, induct rule: generate_topology.induct)
case (Int a b)
let ?p="\<lambda>a f. x \<in> Pi' (domain x) f \<and> Pi' (domain x) f \<subseteq> a \<and> (\<forall>i. i \<in> domain x \<longrightarrow> open (f i))"
from Int obtain f g where "?p a f" "?p b g" by auto
thus ?case by (force intro!: exI[where x="\<lambda>i. f i \<inter> g i"] simp: Pi'_def)
next
case (UN k)
then obtain kk a where "x \<in> kk" "kk \<in> k" "x \<in> Pi' (domain x) a" "Pi' (domain x) a \<subseteq> kk"
"\<And>i. i\<in>domain x \<Longrightarrow> open (a i)"
by force
thus ?case by blast
qed (auto simp: Pi'_def)
have "\<exists>B.
(\<forall>i\<in>domain x. x i \<in> B i \<and> B i \<subseteq> a i \<and> B i \<in> basis_proj)"
proof (rule bchoice, safe)
fix i assume "i \<in> domain x"
hence "open (a i)" "x i \<in> a i" using a by auto
from topological_basisE[OF basis_proj this] guess b' .
thus "\<exists>y. x i \<in> y \<and> y \<subseteq> a i \<and> y \<in> basis_proj" by auto
qed
then guess B .. note B = this
def B' \<equiv> "Pi' (domain x) (\<lambda>i. (B i)::'b set)"
have "B' \<subseteq> Pi' (domain x) a" using B by (auto intro!: Pi'_mono simp: B'_def)
also note `\<dots> \<subseteq> O'`
finally show "\<exists>B'\<in>basis_finmap. x \<in> B' \<and> B' \<subseteq> O'" using B
by (auto intro!: bexI[where x=B'] Pi'_mono in_basis_finmapI simp: B'_def)
qed

lemma range_enum_basis_finmap_imp_open:
assumes "x \<in> basis_finmap"
shows "open x"
using finmap_topological_basis assms by (auto simp: topological_basis_def)

instance proof qed (blast intro: finmap_topological_basis countable_basis_finmap topological_basis_imp_subbasis)

end

subsection {* Polish Space of Finite Maps *}

instance finmap :: (countable, polish_space) polish_space proof qed

subsection {* Product Measurable Space of Finite Maps *}

definition "PiF I M \<equiv>
sigma (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j))) {(\<Pi>' j\<in>J. X j) |X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"

abbreviation
"Pi\<^isub>F I M \<equiv> PiF I M"

syntax
"_PiF" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3PIF _:_./ _)" 10)

syntax (xsymbols)
"_PiF" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3\<Pi>\<^isub>F _\<in>_./ _)"  10)

syntax (HTML output)
"_PiF" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure"  ("(3\<Pi>\<^isub>F _\<in>_./ _)"  10)

translations
"PIF x:I. M" == "CONST PiF I (%x. M)"

lemma PiF_gen_subset: "{(\<Pi>' j\<in>J. X j) |X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))} \<subseteq>
Pow (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j)))"
by (auto simp: Pi'_def) (blast dest: sets.sets_into_space)

lemma space_PiF: "space (PiF I M) = (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j)))"
unfolding PiF_def using PiF_gen_subset by (rule space_measure_of)

lemma sets_PiF:
"sets (PiF I M) = sigma_sets (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j)))
{(\<Pi>' j\<in>J. X j) |X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"
unfolding PiF_def using PiF_gen_subset by (rule sets_measure_of)

lemma sets_PiF_singleton:
"sets (PiF {I} M) = sigma_sets (\<Pi>' j\<in>I. space (M j))
{(\<Pi>' j\<in>I. X j) |X. X \<in> (\<Pi> j\<in>I. sets (M j))}"
unfolding sets_PiF by simp

lemma in_sets_PiFI:
assumes "X = (Pi' J S)" "J \<in> I" "\<And>i. i\<in>J \<Longrightarrow> S i \<in> sets (M i)"
shows "X \<in> sets (PiF I M)"
unfolding sets_PiF
using assms by blast

lemma product_in_sets_PiFI:
assumes "J \<in> I" "\<And>i. i\<in>J \<Longrightarrow> S i \<in> sets (M i)"
shows "(Pi' J S) \<in> sets (PiF I M)"
unfolding sets_PiF
using assms by blast

lemma singleton_space_subset_in_sets:
fixes J
assumes "J \<in> I"
assumes "finite J"
shows "space (PiF {J} M) \<in> sets (PiF I M)"
using assms
by (intro in_sets_PiFI[where J=J and S="\<lambda>i. space (M i)"])
(auto simp: product_def space_PiF)

lemma singleton_subspace_set_in_sets:
assumes A: "A \<in> sets (PiF {J} M)"
assumes "finite J"
assumes "J \<in> I"
shows "A \<in> sets (PiF I M)"
using A[unfolded sets_PiF]
apply (induct A)
unfolding sets_PiF[symmetric] unfolding space_PiF[symmetric]
using assms
by (auto intro: in_sets_PiFI intro!: singleton_space_subset_in_sets)

lemma finite_measurable_singletonI:
assumes "finite I"
assumes "\<And>J. J \<in> I \<Longrightarrow> finite J"
assumes MN: "\<And>J. J \<in> I \<Longrightarrow> A \<in> measurable (PiF {J} M) N"
shows "A \<in> measurable (PiF I M) N"
unfolding measurable_def
proof safe
fix y assume "y \<in> sets N"
have "A -` y \<inter> space (PiF I M) = (\<Union>J\<in>I. A -` y \<inter> space (PiF {J} M))"
by (auto simp: space_PiF)
also have "\<dots> \<in> sets (PiF I M)"
proof
show "finite I" by fact
fix J assume "J \<in> I"
with assms have "finite J" by simp
show "A -` y \<inter> space (PiF {J} M) \<in> sets (PiF I M)"
by (rule singleton_subspace_set_in_sets[OF measurable_sets[OF assms(3)]]) fact+
qed
finally show "A -` y \<inter> space (PiF I M) \<in> sets (PiF I M)" .
next
fix x assume "x \<in> space (PiF I M)" thus "A x \<in> space N"
using MN[of "domain x"]
by (auto simp: space_PiF measurable_space Pi'_def)
qed

lemma countable_finite_comprehension:
fixes f :: "'a::countable set \<Rightarrow> _"
assumes "\<And>s. P s \<Longrightarrow> finite s"
assumes "\<And>s. P s \<Longrightarrow> f s \<in> sets M"
shows "\<Union>{f s|s. P s} \<in> sets M"
proof -
have "\<Union>{f s|s. P s} = (\<Union>n::nat. let s = set (from_nat n) in if P s then f s else {})"
proof safe
fix x X s assume "x \<in> f s" "P s"
moreover with assms obtain l where "s = set l" using finite_list by blast
ultimately show "x \<in> (\<Union>n. let s = set (from_nat n) in if P s then f s else {})" using `P s`
by (auto intro!: exI[where x="to_nat l"])
next
fix x n assume "x \<in> (let s = set (from_nat n) in if P s then f s else {})"
thus "x \<in> \<Union>{f s|s. P s}" using assms by (auto simp: Let_def split: split_if_asm)
qed
hence "\<Union>{f s|s. P s} = (\<Union>n. let s = set (from_nat n) in if P s then f s else {})" by simp
also have "\<dots> \<in> sets M" using assms by (auto simp: Let_def)
finally show ?thesis .
qed

lemma space_subset_in_sets:
fixes J::"'a::countable set set"
assumes "J \<subseteq> I"
assumes "\<And>j. j \<in> J \<Longrightarrow> finite j"
shows "space (PiF J M) \<in> sets (PiF I M)"
proof -
have "space (PiF J M) = \<Union>{space (PiF {j} M)|j. j \<in> J}"
unfolding space_PiF by blast
also have "\<dots> \<in> sets (PiF I M)" using assms
by (intro countable_finite_comprehension) (auto simp: singleton_space_subset_in_sets)
finally show ?thesis .
qed

lemma subspace_set_in_sets:
fixes J::"'a::countable set set"
assumes A: "A \<in> sets (PiF J M)"
assumes "J \<subseteq> I"
assumes "\<And>j. j \<in> J \<Longrightarrow> finite j"
shows "A \<in> sets (PiF I M)"
using A[unfolded sets_PiF]
apply (induct A)
unfolding sets_PiF[symmetric] unfolding space_PiF[symmetric]
using assms
by (auto intro: in_sets_PiFI intro!: space_subset_in_sets)

lemma countable_measurable_PiFI:
fixes I::"'a::countable set set"
assumes MN: "\<And>J. J \<in> I \<Longrightarrow> finite J \<Longrightarrow> A \<in> measurable (PiF {J} M) N"
shows "A \<in> measurable (PiF I M) N"
unfolding measurable_def
proof safe
fix y assume "y \<in> sets N"
have "A -` y = (\<Union>{A -` y \<inter> {x. domain x = J}|J. finite J})" by auto
{ fix x::"'a \<Rightarrow>\<^isub>F 'b"
from finite_list[of "domain x"] obtain xs where "set xs = domain x" by auto
hence "\<exists>n. domain x = set (from_nat n)"
by (intro exI[where x="to_nat xs"]) auto }
hence "A -` y \<inter> space (PiF I M) = (\<Union>n. A -` y \<inter> space (PiF ({set (from_nat n)}\<inter>I) M))"
by (auto simp: space_PiF Pi'_def)
also have "\<dots> \<in> sets (PiF I M)"
apply (intro sets.Int sets.countable_nat_UN subsetI, safe)
apply (case_tac "set (from_nat i) \<in> I")
apply simp_all
apply (rule singleton_subspace_set_in_sets[OF measurable_sets[OF MN]])
using assms `y \<in> sets N`
apply (auto simp: space_PiF)
done
finally show "A -` y \<inter> space (PiF I M) \<in> sets (PiF I M)" .
next
fix x assume "x \<in> space (PiF I M)" thus "A x \<in> space N"
using MN[of "domain x"] by (auto simp: space_PiF measurable_space Pi'_def)
qed

lemma measurable_PiF:
assumes f: "\<And>x. x \<in> space N \<Longrightarrow> domain (f x) \<in> I \<and> (\<forall>i\<in>domain (f x). (f x) i \<in> space (M i))"
assumes S: "\<And>J S. J \<in> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> S i \<in> sets (M i)) \<Longrightarrow>
f -` (Pi' J S) \<inter> space N \<in> sets N"
shows "f \<in> measurable N (PiF I M)"
unfolding PiF_def
using PiF_gen_subset
apply (rule measurable_measure_of)
using f apply force
apply (insert S, auto)
done

lemma restrict_sets_measurable:
assumes A: "A \<in> sets (PiF I M)" and "J \<subseteq> I"
shows "A \<inter> {m. domain m \<in> J} \<in> sets (PiF J M)"
using A[unfolded sets_PiF]
proof (induct A)
case (Basic a)
then obtain K S where S: "a = Pi' K S" "K \<in> I" "(\<forall>i\<in>K. S i \<in> sets (M i))"
by auto
show ?case
proof cases
assume "K \<in> J"
hence "a \<inter> {m. domain m \<in> J} \<in> {Pi' K X |X K. K \<in> J \<and> X \<in> (\<Pi> j\<in>K. sets (M j))}" using S
by (auto intro!: exI[where x=K] exI[where x=S] simp: Pi'_def)
also have "\<dots> \<subseteq> sets (PiF J M)" unfolding sets_PiF by auto
finally show ?thesis .
next
assume "K \<notin> J"
hence "a \<inter> {m. domain m \<in> J} = {}" using S by (auto simp: Pi'_def)
also have "\<dots> \<in> sets (PiF J M)" by simp
finally show ?thesis .
qed
next
case (Union a)
have "UNION UNIV a \<inter> {m. domain m \<in> J} = (\<Union>i. (a i \<inter> {m. domain m \<in> J}))"
by simp
also have "\<dots> \<in> sets (PiF J M)" using Union by (intro sets.countable_nat_UN) auto
finally show ?case .
next
case (Compl a)
have "(space (PiF I M) - a) \<inter> {m. domain m \<in> J} = (space (PiF J M) - (a \<inter> {m. domain m \<in> J}))"
using `J \<subseteq> I` by (auto simp: space_PiF Pi'_def)
also have "\<dots> \<in> sets (PiF J M)" using Compl by auto
finally show ?case by (simp add: space_PiF)
qed simp

lemma measurable_finmap_of:
assumes f: "\<And>i. (\<exists>x \<in> space N. i \<in> J x) \<Longrightarrow> (\<lambda>x. f x i) \<in> measurable N (M i)"
assumes J: "\<And>x. x \<in> space N \<Longrightarrow> J x \<in> I" "\<And>x. x \<in> space N \<Longrightarrow> finite (J x)"
assumes JN: "\<And>S. {x. J x = S} \<inter> space N \<in> sets N"
shows "(\<lambda>x. finmap_of (J x) (f x)) \<in> measurable N (PiF I M)"
proof (rule measurable_PiF)
fix x assume "x \<in> space N"
with J[of x] measurable_space[OF f]
show "domain (finmap_of (J x) (f x)) \<in> I \<and>
(\<forall>i\<in>domain (finmap_of (J x) (f x)). (finmap_of (J x) (f x)) i \<in> space (M i))"
by auto
next
fix K S assume "K \<in> I" and *: "\<And>i. i \<in> K \<Longrightarrow> S i \<in> sets (M i)"
with J have eq: "(\<lambda>x. finmap_of (J x) (f x)) -` Pi' K S \<inter> space N =
(if \<exists>x \<in> space N. K = J x \<and> finite K then if K = {} then {x \<in> space N. J x = K}
else (\<Inter>i\<in>K. (\<lambda>x. f x i) -` S i \<inter> {x \<in> space N. J x = K}) else {})"
by (auto simp: Pi'_def)
have r: "{x \<in> space N. J x = K} = space N \<inter> ({x. J x = K} \<inter> space N)" by auto
show "(\<lambda>x. finmap_of (J x) (f x)) -` Pi' K S \<inter> space N \<in> sets N"
unfolding eq r
apply (simp del: INT_simps add: )
apply (intro conjI impI sets.finite_INT JN sets.Int[OF sets.top])
apply simp apply assumption
apply (subst Int_assoc[symmetric])
apply (rule sets.Int)
apply (intro measurable_sets[OF f] *) apply force apply assumption
apply (intro JN)
done
qed

lemma measurable_PiM_finmap_of:
assumes "finite J"
shows "finmap_of J \<in> measurable (Pi\<^isub>M J M) (PiF {J} M)"
apply (rule measurable_finmap_of)
apply (rule measurable_component_singleton)
apply simp
apply rule
apply (rule `finite J`)
apply simp
done

lemma proj_measurable_singleton:
assumes "A \<in> sets (M i)"
shows "(\<lambda>x. (x)\<^isub>F i) -` A \<inter> space (PiF {I} M) \<in> sets (PiF {I} M)"
proof cases
assume "i \<in> I"
hence "(\<lambda>x. (x)\<^isub>F i) -` A \<inter> space (PiF {I} M) =
Pi' I (\<lambda>x. if x = i then A else space (M x))"
using sets.sets_into_space[OF ] `A \<in> sets (M i)` assms
by (auto simp: space_PiF Pi'_def)
thus ?thesis  using assms `A \<in> sets (M i)`
by (intro in_sets_PiFI) auto
next
assume "i \<notin> I"
hence "(\<lambda>x. (x)\<^isub>F i) -` A \<inter> space (PiF {I} M) =
(if undefined \<in> A then space (PiF {I} M) else {})" by (auto simp: space_PiF Pi'_def)
thus ?thesis by simp
qed

lemma measurable_proj_singleton:
assumes "i \<in> I"
shows "(\<lambda>x. (x)\<^isub>F i) \<in> measurable (PiF {I} M) (M i)"
by (unfold measurable_def, intro CollectI conjI ballI proj_measurable_singleton assms)
(insert `i \<in> I`, auto simp: space_PiF)

lemma measurable_proj_countable:
fixes I::"'a::countable set set"
assumes "y \<in> space (M i)"
shows "(\<lambda>x. if i \<in> domain x then (x)\<^isub>F i else y) \<in> measurable (PiF I M) (M i)"
proof (rule countable_measurable_PiFI)
fix J assume "J \<in> I" "finite J"
show "(\<lambda>x. if i \<in> domain x then x i else y) \<in> measurable (PiF {J} M) (M i)"
unfolding measurable_def
proof safe
fix z assume "z \<in> sets (M i)"
have "(\<lambda>x. if i \<in> domain x then x i else y) -` z \<inter> space (PiF {J} M) =
(\<lambda>x. if i \<in> J then (x)\<^isub>F i else y) -` z \<inter> space (PiF {J} M)"
by (auto simp: space_PiF Pi'_def)
also have "\<dots> \<in> sets (PiF {J} M)" using `z \<in> sets (M i)` `finite J`
by (cases "i \<in> J") (auto intro!: measurable_sets[OF measurable_proj_singleton])
finally show "(\<lambda>x. if i \<in> domain x then x i else y) -` z \<inter> space (PiF {J} M) \<in>
sets (PiF {J} M)" .
qed (insert `y \<in> space (M i)`, auto simp: space_PiF Pi'_def)
qed

lemma measurable_restrict_proj:
assumes "J \<in> II" "finite J"
shows "finmap_of J \<in> measurable (PiM J M) (PiF II M)"
using assms
by (intro measurable_finmap_of measurable_component_singleton) auto

lemma measurable_proj_PiM:
fixes J K ::"'a::countable set" and I::"'a set set"
assumes "finite J" "J \<in> I"
assumes "x \<in> space (PiM J M)"
shows "proj \<in> measurable (PiF {J} M) (PiM J M)"
proof (rule measurable_PiM_single)
show "proj \<in> space (PiF {J} M) \<rightarrow> (\<Pi>\<^isub>E i \<in> J. space (M i))"
using assms by (auto simp add: space_PiM space_PiF extensional_def sets_PiF Pi'_def)
next
fix A i assume A: "i \<in> J" "A \<in> sets (M i)"
show "{\<omega> \<in> space (PiF {J} M). (\<omega>)\<^isub>F i \<in> A} \<in> sets (PiF {J} M)"
proof
have "{\<omega> \<in> space (PiF {J} M). (\<omega>)\<^isub>F i \<in> A} =
(\<lambda>\<omega>. (\<omega>)\<^isub>F i) -` A \<inter> space (PiF {J} M)" by auto
also have "\<dots> \<in> sets (PiF {J} M)"
using assms A by (auto intro: measurable_sets[OF measurable_proj_singleton] simp: space_PiM)
finally show ?thesis .
qed simp
qed

lemma space_PiF_singleton_eq_product:
assumes "finite I"
shows "space (PiF {I} M) = (\<Pi>' i\<in>I. space (M i))"
by (auto simp: product_def space_PiF assms)

text {* adapted from @{thm sets_PiM_single} *}

lemma sets_PiF_single:
assumes "finite I" "I \<noteq> {}"
shows "sets (PiF {I} M) =
sigma_sets (\<Pi>' i\<in>I. space (M i))
{{f\<in>\<Pi>' i\<in>I. space (M i). f i \<in> A} | i A. i \<in> I \<and> A \<in> sets (M i)}"
(is "_ = sigma_sets ?\<Omega> ?R")
unfolding sets_PiF_singleton
proof (rule sigma_sets_eqI)
interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto
fix A assume "A \<in> {Pi' I X |X. X \<in> (\<Pi> j\<in>I. sets (M j))}"
then obtain X where X: "A = Pi' I X" "X \<in> (\<Pi> j\<in>I. sets (M j))" by auto
show "A \<in> sigma_sets ?\<Omega> ?R"
proof -
from `I \<noteq> {}` X have "A = (\<Inter>j\<in>I. {f\<in>space (PiF {I} M). f j \<in> X j})"
using sets.sets_into_space
by (auto simp: space_PiF product_def) blast
also have "\<dots> \<in> sigma_sets ?\<Omega> ?R"
using X `I \<noteq> {}` assms by (intro R.finite_INT) (auto simp: space_PiF)
finally show "A \<in> sigma_sets ?\<Omega> ?R" .
qed
next
fix A assume "A \<in> ?R"
then obtain i B where A: "A = {f\<in>\<Pi>' i\<in>I. space (M i). f i \<in> B}" "i \<in> I" "B \<in> sets (M i)"
by auto
then have "A = (\<Pi>' j \<in> I. if j = i then B else space (M j))"
using sets.sets_into_space[OF A(3)]
apply (auto simp: Pi'_iff split: split_if_asm)
apply blast
done
also have "\<dots> \<in> sigma_sets ?\<Omega> {Pi' I X |X. X \<in> (\<Pi> j\<in>I. sets (M j))}"
using A
by (intro sigma_sets.Basic )
(auto intro: exI[where x="\<lambda>j. if j = i then B else space (M j)"])
finally show "A \<in> sigma_sets ?\<Omega> {Pi' I X |X. X \<in> (\<Pi> j\<in>I. sets (M j))}" .
qed

text {* adapted from @{thm PiE_cong} *}

lemma Pi'_cong:
assumes "finite I"
assumes "\<And>i. i \<in> I \<Longrightarrow> f i = g i"
shows "Pi' I f = Pi' I g"
using assms by (auto simp: Pi'_def)

text {* adapted from @{thm Pi_UN} *}

lemma Pi'_UN:
fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set"
assumes "finite I"
assumes mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i"
shows "(\<Union>n. Pi' I (A n)) = Pi' I (\<lambda>i. \<Union>n. A n i)"
proof (intro set_eqI iffI)
fix f assume "f \<in> Pi' I (\<lambda>i. \<Union>n. A n i)"
then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i" "domain f = I" by (auto simp: `finite I` Pi'_def)
from bchoice[OF this(1)] obtain n where n: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> (A (n i) i)" by auto
obtain k where k: "\<And>i. i \<in> I \<Longrightarrow> n i \<le> k"
using `finite I` finite_nat_set_iff_bounded_le[of "n`I"] by auto
have "f \<in> Pi' I (\<lambda>i. A k i)"
proof
fix i assume "i \<in> I"
from mono[OF this, of "n i" k] k[OF this] n[OF this] `domain f = I` `i \<in> I`
show "f i \<in> A k i " by (auto simp: `finite I`)
qed (simp add: `domain f = I` `finite I`)
then show "f \<in> (\<Union>n. Pi' I (A n))" by auto
qed (auto simp: Pi'_def `finite I`)

text {* adapted from @{thm sigma_prod_algebra_sigma_eq} *}

lemma sigma_fprod_algebra_sigma_eq:
fixes E :: "'i \<Rightarrow> 'a set set" and S :: "'i \<Rightarrow> nat \<Rightarrow> 'a set"
assumes [simp]: "finite I" "I \<noteq> {}"
and S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)"
and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i"
assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))"
and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)"
defines "P == { Pi' I F | F. \<forall>i\<in>I. F i \<in> E i }"
shows "sets (PiF {I} M) = sigma_sets (space (PiF {I} M)) P"
proof
let ?P = "sigma (space (Pi\<^isub>F {I} M)) P"
from `finite I`[THEN ex_bij_betw_finite_nat] guess T ..
then have T: "\<And>i. i \<in> I \<Longrightarrow> T i < card I" "\<And>i. i\<in>I \<Longrightarrow> the_inv_into I T (T i) = i"
by (auto simp add: bij_betw_def set_eq_iff image_iff the_inv_into_f_f simp del: `finite I`)
have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>F {I} M))"
using E_closed by (auto simp: space_PiF P_def Pi'_iff subset_eq)
then have space_P: "space ?P = (\<Pi>' i\<in>I. space (M i))"
have "sets (PiF {I} M) =
sigma_sets (space ?P) {{f \<in> \<Pi>' i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}"
using sets_PiF_single[of I M] by (simp add: space_P)
also have "\<dots> \<subseteq> sets (sigma (space (PiF {I} M)) P)"
proof (safe intro!: sets.sigma_sets_subset)
fix i A assume "i \<in> I" and A: "A \<in> sets (M i)"
have "(\<lambda>x. (x)\<^isub>F i) \<in> measurable ?P (sigma (space (M i)) (E i))"
proof (subst measurable_iff_measure_of)
show "E i \<subseteq> Pow (space (M i))" using `i \<in> I` by fact
from space_P `i \<in> I` show "(\<lambda>x. (x)\<^isub>F i) \<in> space ?P \<rightarrow> space (M i)"
by auto
show "\<forall>A\<in>E i. (\<lambda>x. (x)\<^isub>F i) -` A \<inter> space ?P \<in> sets ?P"
proof
fix A assume A: "A \<in> E i"
then have "(\<lambda>x. (x)\<^isub>F i) -` A \<inter> space ?P = (\<Pi>' j\<in>I. if i = j then A else space (M j))"
using E_closed `i \<in> I` by (auto simp: space_P Pi_iff subset_eq split: split_if_asm)
also have "\<dots> = (\<Pi>' j\<in>I. \<Union>n. if i = j then A else S j n)"
by (intro Pi'_cong) (simp_all add: S_union)
also have "\<dots> = (\<Union>xs\<in>{xs. length xs = card I}. \<Pi>' j\<in>I. if i = j then A else S j (xs ! T j))"
using T
apply auto
apply (erule conjE exE)+
apply (rule_tac x="map (\<lambda>n. f (the_inv_into I T n)) [0..<card I]" in exI)
apply (auto simp: bij_betw_def)
done
also have "\<dots> \<in> sets ?P"
proof (safe intro!: sets.countable_UN)
fix xs show "(\<Pi>' j\<in>I. if i = j then A else S j (xs ! T j)) \<in> sets ?P"
using A S_in_E
(auto simp: P_def subset_eq intro!: exI[of _ "\<lambda>j. if i = j then A else S j (xs ! T j)"])
qed
finally show "(\<lambda>x. (x)\<^isub>F i) -` A \<inter> space ?P \<in> sets ?P"
using P_closed by simp
qed
qed
from measurable_sets[OF this, of A] A `i \<in> I` E_closed
have "(\<lambda>x. (x)\<^isub>F i) -` A \<inter> space ?P \<in> sets ?P"
also have "(\<lambda>x. (x)\<^isub>F i) -` A \<inter> space ?P = {f \<in> \<Pi>' i\<in>I. space (M i). f i \<in> A}"
using P_closed by (auto simp: space_PiF)
finally show "\<dots> \<in> sets ?P" .
qed
finally show "sets (PiF {I} M) \<subseteq> sigma_sets (space (PiF {I} M)) P"
show "sigma_sets (space (PiF {I} M)) P \<subseteq> sets (PiF {I} M)"
using `finite I` `I \<noteq> {}`
by (auto intro!: sets.sigma_sets_subset product_in_sets_PiFI simp: E_generates P_def)
qed

lemma product_open_generates_sets_PiF_single:
assumes "I \<noteq> {}"
assumes [simp]: "finite I"
shows "sets (PiF {I} (\<lambda>_. borel::'b::second_countable_topology measure)) =
sigma_sets (space (PiF {I} (\<lambda>_. borel))) {Pi' I F |F. (\<forall>i\<in>I. F i \<in> Collect open)}"
proof -
from open_countable_basisE[OF open_UNIV] guess S::"'b set set" . note S = this
show ?thesis
proof (rule sigma_fprod_algebra_sigma_eq)
show "finite I" by simp
show "I \<noteq> {}" by fact
def S'\<equiv>"from_nat_into S"
show "(\<Union>j. S' j) = space borel"
using S
apply (auto simp add: from_nat_into countable_basis_proj S'_def basis_proj_def)
apply (metis (lifting, mono_tags) UNIV_I UnionE basis_proj_def countable_basis_proj countable_subset from_nat_into_surj)
done
show "range S' \<subseteq> Collect open"
using S
apply (auto simp add: from_nat_into countable_basis_proj S'_def)
apply (metis UNIV_not_empty Union_empty from_nat_into set_mp topological_basis_open[OF basis_proj] basis_proj_def)
done
show "Collect open \<subseteq> Pow (space borel)" by simp
show "sets borel = sigma_sets (space borel) (Collect open)"
qed
qed

lemma finmap_UNIV[simp]: "(\<Union>J\<in>Collect finite. PI' j : J. UNIV) = UNIV" by auto

lemma borel_eq_PiF_borel:
shows "(borel :: ('i::countable \<Rightarrow>\<^isub>F 'a::polish_space) measure) =
PiF (Collect finite) (\<lambda>_. borel :: 'a measure)"
unfolding borel_def PiF_def
proof (rule measure_eqI, clarsimp, rule sigma_sets_eqI)
fix a::"('i \<Rightarrow>\<^isub>F 'a) set" assume "a \<in> Collect open" hence "open a" by simp
then obtain B' where B': "B'\<subseteq>basis_finmap" "a = \<Union>B'"
using finmap_topological_basis by (force simp add: topological_basis_def)
have "a \<in> sigma UNIV {Pi' J X |X J. finite J \<and> X \<in> J \<rightarrow> sigma_sets UNIV (Collect open)}"
unfolding `a = \<Union>B'`
proof (rule sets.countable_Union)
from B' countable_basis_finmap show "countable B'" by (metis countable_subset)
next
show "B' \<subseteq> sets (sigma UNIV
{Pi' J X |X J. finite J \<and> X \<in> J \<rightarrow> sigma_sets UNIV (Collect open)})" (is "_ \<subseteq> sets ?s")
proof
fix x assume "x \<in> B'" with B' have "x \<in> basis_finmap" by auto
then obtain J X where "x = Pi' J X" "finite J" "X \<in> J \<rightarrow> sigma_sets UNIV (Collect open)"
by (auto simp: basis_finmap_def topological_basis_open[OF basis_proj])
thus "x \<in> sets ?s" by auto
qed
qed
thus "a \<in> sigma_sets UNIV {Pi' J X |X J. finite J \<and> X \<in> J \<rightarrow> sigma_sets UNIV (Collect open)}"
by simp
next
fix b::"('i \<Rightarrow>\<^isub>F 'a) set"
assume "b \<in> {Pi' J X |X J. finite J \<and> X \<in> J \<rightarrow> sigma_sets UNIV (Collect open)}"
hence b': "b \<in> sets (Pi\<^isub>F (Collect finite) (\<lambda>_. borel))" by (auto simp: sets_PiF borel_def)
let ?b = "\<lambda>J. b \<inter> {x. domain x = J}"
have "b = \<Union>((\<lambda>J. ?b J) ` Collect finite)" by auto
also have "\<dots> \<in> sets borel"
proof (rule sets.countable_Union, safe)
fix J::"'i set" assume "finite J"
{ assume ef: "J = {}"
have "?b J \<in> sets borel"
proof cases
assume "?b J \<noteq> {}"
then obtain f where "f \<in> b" "domain f = {}" using ef by auto
hence "?b J = {f}" using `J = {}`
by (auto simp: finmap_eq_iff)
also have "{f} \<in> sets borel" by simp
finally show ?thesis .
qed simp
} moreover {
assume "J \<noteq> ({}::'i set)"
have "(?b J) = b \<inter> {m. domain m \<in> {J}}" by auto
also have "\<dots> \<in> sets (PiF {J} (\<lambda>_. borel))"
using b' by (rule restrict_sets_measurable) (auto simp: `finite J`)
also have "\<dots> = sigma_sets (space (PiF {J} (\<lambda>_. borel)))
{Pi' (J) F |F. (\<forall>j\<in>J. F j \<in> Collect open)}"
(is "_ = sigma_sets _ ?P")
by (rule product_open_generates_sets_PiF_single[OF `J \<noteq> {}` `finite J`])
also have "\<dots> \<subseteq> sigma_sets UNIV (Collect open)"
by (intro sigma_sets_mono'') (auto intro!: open_Pi'I simp: space_PiF)
finally have "(?b J) \<in> sets borel" by (simp add: borel_def)
} ultimately show "(?b J) \<in> sets borel" by blast
finally show "b \<in> sigma_sets UNIV (Collect open)" by (simp add: borel_def)
qed (simp add: emeasure_sigma borel_def PiF_def)

subsection {* Isomorphism between Functions and Finite Maps *}

lemma measurable_finmap_compose:
shows "(\<lambda>m. compose J m f) \<in> measurable (PiM (f ` J) (\<lambda>_. M)) (PiM J (\<lambda>_. M))"
unfolding compose_def by measurable

lemma measurable_compose_inv:
assumes inj: "\<And>j. j \<in> J \<Longrightarrow> f' (f j) = j"
shows "(\<lambda>m. compose (f ` J) m f') \<in> measurable (PiM J (\<lambda>_. M)) (PiM (f ` J) (\<lambda>_. M))"
unfolding compose_def by (rule measurable_restrict) (auto simp: inj)

locale function_to_finmap =
fixes J::"'a set" and f :: "'a \<Rightarrow> 'b::countable" and f'
assumes [simp]: "finite J"
assumes inv: "i \<in> J \<Longrightarrow> f' (f i) = i"
begin

text {* to measure finmaps *}

definition "fm = (finmap_of (f ` J)) o (\<lambda>g. compose (f ` J) g f')"

lemma domain_fm[simp]: "domain (fm x) = f ` J"
unfolding fm_def by simp

lemma fm_restrict[simp]: "fm (restrict y J) = fm y"
unfolding fm_def by (auto simp: compose_def inv intro: restrict_ext)

lemma fm_product:
assumes "\<And>i. space (M i) = UNIV"
shows "fm -` Pi' (f ` J) S \<inter> space (Pi\<^isub>M J M) = (\<Pi>\<^isub>E j \<in> J. S (f j))"
using assms
by (auto simp: inv fm_def compose_def space_PiM Pi'_def)

lemma fm_measurable:
assumes "f ` J \<in> N"
shows "fm \<in> measurable (Pi\<^isub>M J (\<lambda>_. M)) (Pi\<^isub>F N (\<lambda>_. M))"
unfolding fm_def
proof (rule measurable_comp, rule measurable_compose_inv)
show "finmap_of (f ` J) \<in> measurable (Pi\<^isub>M (f ` J) (\<lambda>_. M)) (PiF N (\<lambda>_. M)) "
using assms by (intro measurable_finmap_of measurable_component_singleton) auto

lemma proj_fm:
assumes "x \<in> J"
shows "fm m (f x) = m x"
using assms by (auto simp: fm_def compose_def o_def inv)

lemma inj_on_compose_f': "inj_on (\<lambda>g. compose (f ` J) g f') (extensional J)"
proof (rule inj_on_inverseI)
fix x::"'a \<Rightarrow> 'c" assume "x \<in> extensional J"
thus "(\<lambda>x. compose J x f) (compose (f ` J) x f') = x"
by (auto simp: compose_def inv extensional_def)
qed

lemma inj_on_fm:
assumes "\<And>i. space (M i) = UNIV"
shows "inj_on fm (space (Pi\<^isub>M J M))"
using assms
apply (auto simp: fm_def space_PiM PiE_def)
apply (rule comp_inj_on)
apply (rule inj_on_compose_f')
apply (rule finmap_of_inj_on_extensional_finite)
apply simp
apply (auto)
done

text {* to measure functions *}

definition "mf = (\<lambda>g. compose J g f) o proj"

lemma mf_fm:
assumes "x \<in> space (Pi\<^isub>M J (\<lambda>_. M))"
shows "mf (fm x) = x"
proof -
have "mf (fm x) \<in> extensional J"
by (auto simp: mf_def extensional_def compose_def)
moreover
have "x \<in> extensional J" using assms sets.sets_into_space
by (force simp: space_PiM PiE_def)
moreover
{ fix i assume "i \<in> J"
hence "mf (fm x) i = x i"
by (auto simp: inv mf_def compose_def fm_def)
}
ultimately
show ?thesis by (rule extensionalityI)
qed

lemma mf_measurable:
assumes "space M = UNIV"
shows "mf \<in> measurable (PiF {f ` J} (\<lambda>_. M)) (PiM J (\<lambda>_. M))"
unfolding mf_def
proof (rule measurable_comp, rule measurable_proj_PiM)
show "(\<lambda>g. compose J g f) \<in> measurable (Pi\<^isub>M (f ` J) (\<lambda>x. M)) (Pi\<^isub>M J (\<lambda>_. M))"
by (rule measurable_finmap_compose)
qed (auto simp add: space_PiM extensional_def assms)

lemma fm_image_measurable:
assumes "space M = UNIV"
assumes "X \<in> sets (Pi\<^isub>M J (\<lambda>_. M))"
shows "fm ` X \<in> sets (PiF {f ` J} (\<lambda>_. M))"
proof -
have "fm ` X = (mf) -` X \<inter> space (PiF {f ` J} (\<lambda>_. M))"
proof safe
fix x assume "x \<in> X"
with mf_fm[of x] sets.sets_into_space[OF assms(2)] show "fm x \<in> mf -` X" by auto
show "fm x \<in> space (PiF {f ` J} (\<lambda>_. M))" by (simp add: space_PiF assms)
next
fix y x
assume x: "mf y \<in> X"
assume y: "y \<in> space (PiF {f ` J} (\<lambda>_. M))"
thus "y \<in> fm ` X"
by (intro image_eqI[OF _ x], unfold finmap_eq_iff)
(auto simp: space_PiF fm_def mf_def compose_def inv Pi'_def)
qed
also have "\<dots> \<in> sets (PiF {f ` J} (\<lambda>_. M))"
using assms
by (intro measurable_sets[OF mf_measurable]) auto
finally show ?thesis .
qed

lemma fm_image_measurable_finite:
assumes "space M = UNIV"
assumes "X \<in> sets (Pi\<^isub>M J (\<lambda>_. M::'c measure))"
shows "fm ` X \<in> sets (PiF (Collect finite) (\<lambda>_. M::'c measure))"
using fm_image_measurable[OF assms]
by (rule subspace_set_in_sets) (auto simp: finite_subset)

text {* measure on finmaps *}

definition "mapmeasure M N = distr M (PiF (Collect finite) N) (fm)"

lemma sets_mapmeasure[simp]: "sets (mapmeasure M N) = sets (PiF (Collect finite) N)"
unfolding mapmeasure_def by simp

lemma space_mapmeasure[simp]: "space (mapmeasure M N) = space (PiF (Collect finite) N)"
unfolding mapmeasure_def by simp

lemma mapmeasure_PiF:
assumes s1: "space M = space (Pi\<^isub>M J (\<lambda>_. N))"
assumes s2: "sets M = sets (Pi\<^isub>M J (\<lambda>_. N))"
assumes "space N = UNIV"
assumes "X \<in> sets (PiF (Collect finite) (\<lambda>_. N))"
shows "emeasure (mapmeasure M (\<lambda>_. N)) X = emeasure M ((fm -` X \<inter> extensional J))"
using assms
by (auto simp: measurable_eqI[OF s1 refl s2 refl] mapmeasure_def emeasure_distr
fm_measurable space_PiM PiE_def)

lemma mapmeasure_PiM:
fixes N::"'c measure"
assumes s1: "space M = space (Pi\<^isub>M J (\<lambda>_. N))"
assumes s2: "sets M = (Pi\<^isub>M J (\<lambda>_. N))"
assumes N: "space N = UNIV"
assumes X: "X \<in> sets M"
shows "emeasure M X = emeasure (mapmeasure M (\<lambda>_. N)) (fm ` X)"
unfolding mapmeasure_def
proof (subst emeasure_distr, subst measurable_eqI[OF s1 refl s2 refl], rule fm_measurable)
have "X \<subseteq> space (Pi\<^isub>M J (\<lambda>_. N))" using assms by (simp add: sets.sets_into_space)
from assms inj_on_fm[of "\<lambda>_. N"] set_mp[OF this] have "fm -` fm ` X \<inter> space (Pi\<^isub>M J (\<lambda>_. N)) = X"
by (auto simp: vimage_image_eq inj_on_def)
thus "emeasure M X = emeasure M (fm -` fm ` X \<inter> space M)" using s1
by simp
show "fm ` X \<in> sets (PiF (Collect finite) (\<lambda>_. N))"
by (rule fm_image_measurable_finite[OF N X[simplified s2]])
qed simp

end

end
```