(* Title: HOL/Probability/Fin_Map.thy Author: Fabian Immler, TU München
*)
section \<open>Finite Maps\<close>
theory Fin_Map imports"HOL-Analysis.Finite_Product_Measure""HOL-Library.Finite_Map" begin
text\<open>The \<^type>\<open>fmap\<close> type can be instantiated to \<^class>\<open>polish_space\<close>, needed for the proof of
projective limit. \<^const>\<open>extensional\<close> functions are used for the representation in order to
stay close to the developments of (finite) products \<^const>\<open>Pi\<^sub>E\<close> and their sigma-algebra \<^const>\<open>Pi\<^sub>M\<close>.\<close>
lift_definition domain::"('i \\<^sub>F 'a) \ 'i set" is dom .
lemma finite_domain[simp, intro]: "finite (domain P)" by transfer simp
lift_definition proj :: "('i \\<^sub>F 'a) \ 'i \ 'a"
(\<open>(\<open>indent=1 notation=\<open>mixfix proj\<close>\<close>'(_')\<^sub>F)\<close> [0] 1000) is "\f x. if x \ dom f then the (f x) else undefined" .
declare [[coercion proj]]
lemma extensional_proj[simp, intro]: "(P)\<^sub>F \ extensional (domain P)" by transfer (auto simp: extensional_def)
lemma proj_undefined[simp, intro]: "i \ domain P \ P i = undefined" using extensional_proj[of P] unfolding extensional_def by auto
lemma finmap_eq_iff: "P = Q \ (domain P = domain Q \ (\i\domain P. P i = Q i))" apply transfer apply (safe intro!: ext)
subgoal for P Q x by (cases "x \ dom P"; cases "P x") (auto dest!: bspec[where x=x]) done
subsection \<open>Constructor of Finite Maps\<close>
lift_definition finmap_of::"'i set \ ('i \ 'a) \ ('i \\<^sub>F 'a)" is "\I f x. if x \ I \ finite I then Some (f x) else None" by (simp add: dom_def)
lemma proj_finmap_of[simp]: assumes"finite inds" shows"(finmap_of inds f)\<^sub>F = restrict f inds" using assms by transfer force
lemma domain_finmap_of[simp]: assumes"finite inds" shows"domain (finmap_of inds f) = inds" using assms by transfer (auto split: if_splits)
lemma finmap_of_eq_iff[simp]: assumes"finite i""finite j" shows"finmap_of i m = finmap_of j n \ i = j \ (\k\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 \ extensional K" shows"inj_on (finmap_of K) S" proof (rule inj_onI) fix x y::"'a \ 'b" assume"finmap_of K x = finmap_of K y" hence"(finmap_of K x)\<^sub>F = (finmap_of K y)\<^sub>F" by simp moreover assume"x \ S" "y \ S" hence "x \ extensional K" "y \ extensional K" using assms by auto ultimately show"x = y"using assms by (simp add: extensional_restrict) qed
subsection \<open>Product set of Finite Maps\<close>
text\<open>This is \<^term>\<open>Pi\<close> for Finite Maps, most of this is copied\<close>
definition Pi' :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a set) \<Rightarrow> ('i \<Rightarrow>\<^sub>F 'a) set" where "Pi' I A = { P. domain P = I \ (\i. i \ I \ (P)\<^sub>F i \ A i) } "
subsubsection\<open>Basic Properties of \<^term>\<open>Pi'\<close>\<close>
lemma Pi'_I[intro!]: "domain f = A \ (\x. x \ A \ f x \ B x) \ f \ Pi' A B" by (simp add: Pi'_def)
lemma Pi'_I'[simp]: "domain f = A \ (\x. x \ A \ f x \ B x) \ f \ Pi' A B" by (simp add:Pi'_def)
lemma Pi'_mem: "f\ Pi' A B \ x \ A \ f x \ B x" by (simp add: Pi'_def)
lemma Pi'_iff: "f \ Pi' I X \ domain f = I \ (\i\I. f i \ X i)" unfolding Pi'_def by auto
lemma Pi'E [elim]: "f \ Pi' A B \ (f x \ B x \ domain f = A \ Q) \ (x \ A \ Q) \ Q" by(auto simp: Pi'_def)
lemma in_Pi'_cong: "domain f = domain g \ (\ w. w \ A \ f w = g w) \ f \ Pi' A B \ g \ Pi' A B" by (auto simp: Pi'_def)
lemma Pi'_eq_empty[simp]: assumes"finite A"shows"(Pi' A B) = {} \ (\x\A. B x = {})" using assms apply (simp add: Pi'_def, auto) apply (drule_tac x = "finmap_of A (\u. SOME y. y \ B u)" in spec, auto) apply (cut_tac P= "%y. y \ B i" in some_eq_ex, auto) done
lemma Pi'_mono: "(\x. x \ A \ B x \ C x) \ Pi' A B \ Pi' A C" by (auto simp: Pi'_def)
lemma Pi_Pi': "finite A \ (Pi\<^sub>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 \<open>Topological Space of Finite Maps\<close>
instantiation fmap :: (type, topological_space) topological_space begin
definition open_fmap :: "('a \\<^sub>F 'b) set \ bool" where
[code del]: "open_fmap = generate_topology {Pi' a b|a b. \i\a. open (b i)}"
lemma open_Pi'I: "(\i. i \ I \ open (A i)) \ open (Pi' I A)" by (auto intro: generate_topology.Basis simp: open_fmap_def)
instanceusing topological_space_generate_topology by intro_classes (auto simp: open_fmap_def class.topological_space_def)
end
lemma open_restricted_space: shows"open {m. P (domain m)}" proof - have"{m. P (domain m)} = (\i \ Collect P. {m. domain m = i})" by auto alsohave"open \" proof (rule, safe, cases) fix i::"'a set" assume"finite i" hence"{m. domain m = i} = Pi' i (\_. UNIV)" by (auto simp: Pi'_def) alsohave"open \" by (auto intro: open_Pi'I simp: \finite i\) finallyshow"open {m. domain m = i}" . next fix i::"'a set" assume"\ finite i" hence "{m. domain m = i} = {}" by auto alsohave"open \" by simp finallyshow"open {m. domain m = i}" . qed finallyshow ?thesis . qed
lemma closed_restricted_space: shows"closed {m. P (domain m)}" using open_restricted_space[of "\x. \ P x"] unfolding closed_def by (rule back_subst) auto
lemma tendsto_proj: "((\x. x) \ a) F \ ((\x. (x)\<^sub>F i) \ (a)\<^sub>F i) F" unfolding tendsto_def proof safe fix S::"'b set" let ?S = "Pi' (domain a) (\x. if x = i then S else UNIV)" assume"open S"hence"open ?S"by (auto intro!: open_Pi'I) moreoverassume"\S. open S \ a \ S \ eventually (\x. x \ S) F" "a i \ S" ultimatelyhave"eventually (\x. x \ ?S) F" by auto thus"eventually (\x. (x)\<^sub>F i \ S) F" by eventually_elim (insert \<open>a i \<in> S\<close>, force simp: Pi'_iff split: if_split_asm) qed
lemma continuous_proj: shows"continuous_on s (\x. (x)\<^sub>F i)" unfolding continuous_on_def by (safe intro!: tendsto_proj tendsto_ident_at)
instance fmap :: (type, first_countable_topology) first_countable_topology proof fix x::"'a\\<^sub>F'b" have"\i. \A. countable A \ (\a\A. x i \ a) \ (\a\A. open a) \
(\<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"] obtain A where "countable A" "\C. C \ A \ (x)\<^sub>F i \ C" "\C. C \ A \ open C" "\S. open S \ (x)\<^sub>F i \ S \ \A\A. A \ S" "\C D. C \ A \ D \ A \ C \ D \ A" by auto thus"?th i"by (intro exI[where x=A]) simp qed thenobtain A where A: "\i. countable (A i) \ Ball (A i) ((\) ((x)\<^sub>F i)) \ Ball (A i) open \
(\<forall>S. open S \<and> (x)\<^sub>F i \<in> S \<longrightarrow> (\<exists>a\<in>A i. a \<subseteq> S)) \<and> (\<forall>a b. a \<in> A i \<longrightarrow> b \<in> A i \<longrightarrow> a \<inter> b \<in> A i)" by (auto simp: choice_iff) hence open_sub: "\i S. i\domain x \ open (S i) \ x i\(S i) \ (\a\A i. a\(S i))" by auto have A_notempty: "\i. i \ domain x \ A i \ {}" using open_sub[of _ "\_. UNIV"] by auto let ?A = "(\f. Pi' (domain x) f) ` (Pi\<^sub>E (domain x) A)" show"\A::nat \ ('a\\<^sub>F'b) set. (\i. x \ (A i) \ open (A i)) \ (\S. open S \ x \ S \ (\i. A i \ S))" proof (rule first_countableI[of "?A"], safe) show"countable ?A"using A by (simp add: countable_PiE) next fix S::"('a \\<^sub>F 'b) set" assume "open S" "x \ S" thus"\a\?A. a \ S" unfolding open_fmap_def proof (induct rule: generate_topology.induct) case UNIV thus ?caseby (auto simp add: ex_in_conv PiE_eq_empty_iff A_notempty) next case (Int a b) thenobtain f g where "f \ Pi\<^sub>E (domain x) A" "Pi' (domain x) f \ a" "g \ Pi\<^sub>E (domain x) A" "Pi' (domain x) g \ b" by auto thus ?caseusing A by (auto simp: Pi'_iff PiE_iff extensional_def Int_stable_def
intro!: bexI[where x="\i. f i \ g i"]) next case (UN B) thenobtain b where"x \ b" "b \ B" by auto hence"\a\?A. a \ b" using UN by simp thus ?caseusing\<open>b \<in> B\<close> by (metis Sup_upper2) next case (Basis s) thenobtain a b where xs: "x\ Pi' a b" "s = Pi' a b" "\i. i\a \ open (b i)" by auto have"\i. \a. (i \ domain x \ open (b i) \ (x)\<^sub>F i \ b i) \ (a\A i \ a \ b i)" using open_sub[of _ b] by auto thenobtain b' where"\i. i \ domain x \ open (b i) \ (x)\<^sub>F i \ b i \ (b' i \A i \ b' i \ b i)" unfolding choice_iff by auto with xs have"\i. i \ a \ (b' i \A i \ b' i \ b i)" "Pi' a b' \ Pi' a b" by (auto simp: Pi'_iff intro!: Pi'_mono) thus ?caseusing 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 \<open>Metric Space of Finite Maps\<close>
(* TODO: Product of uniform spaces and compatibility with metric_spaces! *)
instantiation fmap :: (type, metric_space) dist begin
definition dist_fmap where "dist P Q = Max (range (\i. dist ((P)\<^sub>F i) ((Q)\<^sub>F i))) + (if domain P = domain Q then 0 else 1)"
instance .. end
instantiation fmap :: (type, metric_space) uniformity_dist begin
definition [code del]: "(uniformity :: (('a, 'b) fmap \ ('a \\<^sub>F 'b)) filter) =
(INF e\<in>{0 <..}. principal {(x, y). dist x y < e})"
instance by standard (rule uniformity_fmap_def) end
instantiation fmap :: (type, metric_space) metric_space begin
lemma finite_proj_image': "x \ domain P \ finite ((P)\<^sub>F ` S)" by (rule finite_subset[of _ "proj P ` (domain P \ S \ {x})"]) auto
lemma finite_proj_image: "finite ((P)\<^sub>F ` S)" by (cases "\x. x \ domain P") (auto intro: finite_proj_image' finite_subset[where B="domain P"])
lemma finite_proj_diag: "finite ((\i. d ((P)\<^sub>F i) ((Q)\<^sub>F i)) ` S)" proof - have"(\i. d ((P)\<^sub>F i) ((Q)\<^sub>F i)) ` S = (\(i, j). d i j) ` ((\i. ((P)\<^sub>F i, (Q)\<^sub>F i)) ` S)" by auto moreoverhave"((\i. ((P)\<^sub>F i, (Q)\<^sub>F i)) ` S) \ (\i. (P)\<^sub>F i) ` S \ (\i. (Q)\<^sub>F i) ` S" by auto moreoverhave"finite \" using finite_proj_image[of P S] finite_proj_image[of Q S] by (intro finite_cartesian_product) simp_all ultimatelyshow ?thesis by (simp add: finite_subset) qed
lemma dist_le_1_imp_domain_eq: shows"dist P Q < 1 \ domain P = domain Q" by (simp add: dist_fmap_def finite_proj_diag split: if_split_asm)
lemma dist_proj: shows"dist ((x)\<^sub>F i) ((y)\<^sub>F i) \ dist x y" proof - have"dist (x i) (y i) \ Max (range (\i. dist (x i) (y i)))" by (simp add: Max_ge_iff finite_proj_diag) alsohave"\ \ dist x y" by (simp add: dist_fmap_def) finallyshow ?thesis . qed
lemma dist_finmap_lessI: assumes"domain P = domain Q" assumes"0 < e" assumes"\i. i \ domain P \ dist (P i) (Q i) < e" shows"dist P Q < e" proof - have"dist P Q = Max (range (\i. dist (P i) (Q i)))" using assms by (simp add: dist_fmap_def finite_proj_diag) alsohave"\ < e" proof (subst Max_less_iff, safe) fix i show"dist ((P)\<^sub>F i) ((Q)\<^sub>F i) < e" using assms by (cases "i \ domain P") simp_all qed (simp add: finite_proj_diag) finallyshow ?thesis . qed
instance proof fix S::"('a \\<^sub>F 'b) set" have *: "open S = (\x\S. \e>0. \y. dist y x < e \ y \ S)" (is "_ = ?od") proof assume"open S" thus ?od unfolding open_fmap_def proof (induct rule: generate_topology.induct) case UNIV thus ?caseby (auto intro: zero_less_one) next case (Int a b) show ?case proof safe fix x assume x: "x \ a" "x \ b" with Int x obtain e1 e2 where "e1>0""\y. dist y x < e1 \ y \ a" "e2>0" "\y. dist y x < e2 \ y \ b" by force thus"\e>0. \y. dist y x < e \ y \ a \ b" by (auto intro!: exI[where x="min e1 e2"]) qed next case (UN K) show ?case proof safe fix x X assume"x \ X" and X: "X \ K" with UN obtain e where"e>0""\y. dist y x < e \ y \ X" by force with X show"\e>0. \y. dist y x < e \ y \ \K" by auto qed next case (Basis s) thenobtain a b where s: "s = Pi' a b"and b: "\i. i\a \ open (b i)" by auto show ?case proof safe fix x assume"x \ s" hence [simp]: "finite a"and a_dom: "a = domain x"using s by (auto simp: Pi'_iff) obtain es where es: "\i \ a. es i > 0 \ (\y. dist y (proj x i) < es i \ y \ b i)" using b \<open>x \<in> s\<close> by atomize_elim (intro bchoice, auto simp: open_dist s) hence in_b: "\i y. i \ a \ dist y (proj x i) < es i \ y \ b i" by auto show"\e>0. \y. dist y x < e \ y \ s" proof (cases, rule, safe) assume"a \ {}" show"0 < min 1 (Min (es ` a))"using es by (auto simp: \<open>a \<noteq> {}\<close>) fix y assume d: "dist y x < min 1 (Min (es ` a))" show"y \ s" unfolding s proof show"domain y = a"using d s \<open>a \<noteq> {}\<close> by (auto simp: dist_le_1_imp_domain_eq a_dom) fix i assume i: "i \ a" hence"dist ((y)\<^sub>F i) ((x)\<^sub>F i) < es i" using d by (auto simp: dist_fmap_def \<open>a \<noteq> {}\<close> intro!: le_less_trans[OF dist_proj]) with i show"y i \ b i" by (rule in_b) qed next assume"\a \ {}" thus"\e>0. \y. dist y x < e \ y \ s" using s \<open>x \<in> s\<close> by (auto simp: Pi'_def dist_le_1_imp_domain_eq intro!: exI[where x=1]) qed qed qed next assume"\x\S. \e>0. \y. dist y x < e \ y \ S" thenobtain e where e_pos: "\x. x \ S \ e x > 0" and
e_in: "\x y . x \ S \ dist y x < e x \ y \ S" unfolding bchoice_iff by auto have S_eq: "S = \{Pi' a b| a b. \x\S. domain x = a \ b = (\i. ball (x i) (e x))}" proof safe fix x assume"x \ S" thus"x \ \{Pi' a b| a b. \x\S. domain x = a \ b = (\i. ball (x i) (e x))}" using e_pos by (auto intro!: exI[where x="Pi' (domain x) (\i. ball (x i) (e x))"]) next fix x y assume"y \ S" moreover assume"x \ (\' i\domain y. ball (y i) (e y))" hence"dist x y < e y"using e_pos \<open>y \<in> S\<close> by (auto simp: dist_fmap_def Pi'_iff finite_proj_diag dist_commute) ultimatelyshow"x \ S" by (rule e_in) qed alsohave"open \" unfolding open_fmap_def by (intro generate_topology.UN) (auto intro: generate_topology.Basis) finallyshow"open S" . qed show"open S = (\x\S. \\<^sub>F (x', y) in uniformity. x' = x \ y \ S)" unfolding * eventually_uniformity_metric by (simp del: split_paired_All add: dist_fmap_def dist_commute eq_commute) next fix P Q::"'a \\<^sub>F 'b" have Max_eq_iff: "\A m. finite A \ A \ {} \ (Max A = m) = (m \ A \ (\a\A. a \ m))" by (auto intro: Max_in Max_eqI) show"dist P Q = 0 \ P = Q" by (auto simp: finmap_eq_iff dist_fmap_def Max_ge_iff finite_proj_diag Max_eq_iff
add_nonneg_eq_0_iff
intro!: Max_eqI image_eqI[where x=undefined]) next fix P Q R::"'a \\<^sub>F 'b" let ?dists = "\P Q i. dist ((P)\<^sub>F i) ((Q)\<^sub>F i)" let ?dpq = "?dists P Q"and ?dpr = "?dists P R"and ?dqr = "?dists Q R" let ?dom = "\P Q. (if domain P = domain Q then 0 else 1::real)" have"dist P Q = Max (range ?dpq) + ?dom P Q" by (simp add: dist_fmap_def) alsoobtain t where"t \ range ?dpq" "t = Max (range ?dpq)" by (simp add: finite_proj_diag) thenobtain i where"Max (range ?dpq) = ?dpq i"by auto alsohave"?dpq i \ ?dpr i + ?dqr i" by (rule dist_triangle2) alsohave"?dpr i \ Max (range ?dpr)" by (simp add: finite_proj_diag) alsohave"?dqr i \ Max (range ?dqr)" by (simp add: finite_proj_diag) alsohave"?dom P Q \ ?dom P R + ?dom Q R" by simp finallyshow"dist P Q \ dist P R + dist Q R" by (simp add: dist_fmap_def ac_simps) qed
end
subsection \<open>Complete Space of Finite Maps\<close>
lemma tendsto_finmap: fixes f::"nat \ ('i \\<^sub>F ('a::metric_space))" assumes ind_f: "\n. domain (f n) = domain g" assumes proj_g: "\i. i \ domain g \ (\n. (f n) i) \ g i" shows"f \ g" unfolding tendsto_iff proof safe fix e::real assume"0 < e" let ?dists = "\x i. dist ((f x)\<^sub>F i) ((g)\<^sub>F i)" have"eventually (\x. \i\domain g. ?dists x i < e) sequentially" using finite_domain[of g] proj_g proof induct case (insert i G) with\<open>0 < e\<close> have "eventually (\<lambda>x. ?dists x i < e) sequentially" by (auto simp add: tendsto_iff) moreover from insert have"eventually (\x. \i\G. dist ((f x)\<^sub>F i) ((g)\<^sub>F i) < e) sequentially" by simp ultimatelyshow ?caseby eventually_elim auto qed simp thus"eventually (\x. dist (f x) g < e) sequentially" by eventually_elim (auto simp add: dist_fmap_def finite_proj_diag ind_f \<open>0 < e\<close>) qed
instance fmap :: (type, complete_space) complete_space proof fix P::"nat \ 'a \\<^sub>F 'b" assume"Cauchy P" thenobtain Nd where Nd: "\n. n \ Nd \ dist (P n) (P Nd) < 1" by (force simp: Cauchy_altdef2)
define d where"d = domain (P Nd)" with Nd have dim: "\n. n \ Nd \ domain (P n) = d" using dist_le_1_imp_domain_eq by auto have [simp]: "finite d"unfolding d_def by simp
define p where"p i n = P n i"for i n
define q where"q i = lim (p i)"for i
define Q where"Q = finmap_of d q" have q: "\i. i \ d \ q i = Q i" by (auto simp add: Q_def Abs_fmap_inverse)
{ fix i assume"i \ d" have"Cauchy (p i)"unfolding Cauchy_altdef2 p_def proof safe fix e::real assume"0 < e" with\<open>Cauchy P\<close> obtain N where N: "\<And>n. n\<ge>N \<Longrightarrow> dist (P n) (P N) < min e 1" by (force simp: Cauchy_altdef2 min_def) hence"\n. n \ N \ domain (P n) = domain (P N)" using dist_le_1_imp_domain_eq by auto with dim have dim: "\n. n \ N \ domain (P n) = d" by (metis nat_le_linear) show"\N. \n\N. dist ((P n) i) ((P N) i) < e" proof (safe intro!: exI[where x="N"]) fix n assume"N \ n" have "N \ N" by simp have"dist ((P n) i) ((P N) i) \ dist (P n) (P N)" using dim[OF \<open>N \<le> n\<close>] dim[OF \<open>N \<le> N\<close>] \<open>i \<in> d\<close> by (auto intro!: dist_proj) alsohave"\ < e" using N[OF \N \ n\] by simp finallyshow"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"\ni. \i\d. \n\ni i. dist (p i n) (q i) < e" proof (safe intro!: bchoice) fix i assume"i \ d" from p[OF \<open>i \<in> d\<close>, THEN metric_LIMSEQ_D, OF \<open>0 < e\<close>] show"\no. \n\no. dist (p i n) (q i) < e" . qed thenobtain ni where ni: "\i\d. \n\ni i. dist (p i n) (q i) < e" ..
define N where"N = max Nd (Max (ni ` d))" show"\N. \n\N. dist (P n) Q < e" proof (safe intro!: exI[where x="N"]) fix n assume"N \ 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_fmap_inverse) show"dist (P n) Q < e" proof (rule dist_finmap_lessI[OF dom(3) \<open>0 < e\<close>]) fix i assume"i \ domain (P n)" hence"ni i \ Max (ni ` d)" using dom by simp alsohave"\ \ N" by (simp add: N_def) finallyshow"dist ((P n)\<^sub>F i) ((Q)\<^sub>F i) < e" using ni \i \ domain (P n)\ \N \ 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 \<open>Second Countable Space of Finite Maps\<close>
instantiation fmap :: (countable, second_countable_topology) second_countable_topology begin
definition basis_proj::"'b set set" where"basis_proj = (SOME B. countable B \ topological_basis B)"
definition basis_finmap::"('a \\<^sub>F 'b) set set" where"basis_finmap = {Pi' I S|I S. finite I \ (\i \ I. S i \ basis_proj)}"
lemma in_basis_finmapI: assumes"finite I"assumes"\i. i \ I \ S i \ basis_proj" shows"Pi' I S \ basis_finmap" using assms unfolding basis_finmap_def by auto
lemma basis_finmap_eq: assumes"basis_proj \ {}" shows"basis_finmap = (\f. Pi' (domain f) (\i. from_nat_into basis_proj ((f)\<^sub>F i))) `
(UNIV::('a \\<^sub>F nat) set)" (is "_ = ?f ` _") unfolding basis_finmap_def proof safe fix I::"'a set"and S::"'a \ 'b set" assume"finite I""\i\I. S i \ basis_proj" hence"Pi' I S = ?f (finmap_of I (\x. to_nat_on basis_proj (S x)))" by (force simp: Pi'_def countable_basis_proj) thus"Pi' I S \ range ?f" by simp next fix x and f::"'a \\<^sub>F nat" show"\I S. (\' i\domain f. from_nat_into basis_proj ((f)\<^sub>F i)) = Pi' I S \
finite I \<and> (\<forall>i\<in>I. S i \<in> basis_proj)" using assms by (auto intro: from_nat_into) qed
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>\<^sub>F 'b) set" and x assume O': "open O'" "x \<in> O'" thenobtain a where a: "x \ Pi' (domain x) a" "Pi' (domain x) a \ O'" "\i. i\domain x \ open (a i)" unfolding open_fmap_def proof (atomize_elim, induct rule: generate_topology.induct) case (Int a b) let ?p="\a f. x \ Pi' (domain x) f \ Pi' (domain x) f \ a \ (\i. i \ domain x \ open (f i))" from Int obtain f g where"?p a f""?p b g"by auto thus ?caseby (force intro!: exI[where x="\i. f i \ g i"] simp: Pi'_def) next case (UN k) thenobtain kk a where"x \ kk" "kk \ k" "x \ Pi' (domain x) a" "Pi' (domain x) a \ kk" "\i. i\domain x \ open (a i)" by force thus ?caseby blast qed (auto simp: Pi'_def) have"\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 \ domain x" hence"open (a i)""x i \ a i" using a by auto from topological_basisE[OF basis_proj this] obtain b' where"b' \ basis_proj" "(x)\<^sub>F i \ b'" "b' \ a i" by blast thus"\y. x i \ y \ y \ a i \ y \ basis_proj" by auto qed thenobtain B where B: "\i\domain x. (x)\<^sub>F i \ B i \ B i \ a i \ B i \ basis_proj" by auto
define B' where "B' = Pi' (domain x) (\i. (B i)::'b set)" have"B' \ Pi' (domain x) a" using B by (auto intro!: Pi'_mono simp: B'_def) alsonote\<open>\<dots> \<subseteq> O'\<close> finallyshow"\B'\basis_finmap. x \ B' \ B' \ 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 \ basis_finmap" shows"open x" using finmap_topological_basis assms by (auto simp: topological_basis_def)
subsection \<open>Product Measurable Space of Finite Maps\<close>
definition"PiF I M \
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))}"
lemma PiF_gen_subset: "{(\' j\J. X j) |X J. J \ I \ X \ (\ j\J. sets (M j))} \
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) = (\J \ I. (\' j\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 (\J \ I. (\' j\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 (\' j\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 \ I" "\i. i\J \ S i \ sets (M i)" shows"X \ sets (PiF I M)" unfolding sets_PiF using assms by blast
lemma product_in_sets_PiFI: assumes"J \ I" "\i. i\J \ S i \ sets (M i)" shows"(Pi' J S) \ sets (PiF I M)" unfolding sets_PiF using assms by blast
lemma singleton_space_subset_in_sets: fixes J assumes"J \ I" assumes"finite J" shows"space (PiF {J} M) \ sets (PiF I M)" using assms by (intro in_sets_PiFI[where J=J and S="\i. space (M i)"])
(auto simp: product_def space_PiF)
lemma singleton_subspace_set_in_sets: assumes A: "A \ sets (PiF {J} M)" assumes"finite J" assumes"J \ I" shows"A \ 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"\J. J \ I \ finite J" assumes MN: "\J. J \ I \ A \ measurable (PiF {J} M) N" shows"A \ measurable (PiF I M) N" unfolding measurable_def proof safe fix y assume"y \ sets N" have"A -` y \ space (PiF I M) = (\J\I. A -` y \ space (PiF {J} M))" by (auto simp: space_PiF) alsohave"\ \ sets (PiF I M)" proof (rule sets.finite_UN) show"finite I"by fact fix J assume"J \ I" with assms have"finite J"by simp show"A -` y \ space (PiF {J} M) \ sets (PiF I M)" by (rule singleton_subspace_set_in_sets[OF measurable_sets[OF assms(3)]]) fact+ qed finallyshow"A -` y \ space (PiF I M) \ sets (PiF I M)" . next fix x assume"x \ space (PiF I M)" thus "A x \ 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 \ _" assumes"\s. P s \ finite s" assumes"\s. P s \ f s \ sets M" shows"\{f s|s. P s} \ sets M" proof - have"\{f s|s. P s} = (\n::nat. let s = set (from_nat n) in if P s then f s else {})" proof safe fix x X s assume *: "x \ f s" "P s" with assms obtain l where"s = set l"using finite_list by blast with * show"x \ (\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 \ (let s = set (from_nat n) in if P s then f s else {})" thus"x \ \{f s|s. P s}" using assms by (auto simp: Let_def split: if_split_asm) qed hence"\{f s|s. P s} = (\n. let s = set (from_nat n) in if P s then f s else {})" by simp alsohave"\ \ sets M" using assms by (auto simp: Let_def) finallyshow ?thesis . qed
lemma space_subset_in_sets: fixes J::"'a::countable set set" assumes"J \ I" assumes"\j. j \ J \ finite j" shows"space (PiF J M) \ sets (PiF I M)" proof - have"space (PiF J M) = \{space (PiF {j} M)|j. j \ J}" unfolding space_PiF by blast alsohave"\ \ sets (PiF I M)" using assms by (intro countable_finite_comprehension) (auto simp: singleton_space_subset_in_sets) finallyshow ?thesis . qed
lemma subspace_set_in_sets: fixes J::"'a::countable set set" assumes A: "A \ sets (PiF J M)" assumes"J \ I" assumes"\j. j \ J \ finite j" shows"A \ 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: "\J. J \ I \ finite J \ A \ measurable (PiF {J} M) N" shows"A \ measurable (PiF I M) N" unfolding measurable_def proof safe fix y assume"y \ sets N" have"A -` y = (\{A -` y \ {x. domain x = J}|J. finite J})" by auto
{ fix x::"'a \\<^sub>F 'b" from finite_list[of "domain x"] obtain xs where"set xs = domain x"by auto hence"\n. domain x = set (from_nat n)" by (intro exI[where x="to_nat xs"]) auto } hence"A -` y \ space (PiF I M) = (\n. A -` y \ space (PiF ({set (from_nat n)}\I) M))" by (auto simp: space_PiF Pi'_def) alsohave"\ \ sets (PiF I M)" apply (intro sets.Int sets.countable_nat_UN subsetI, safe) apply (case_tac "set (from_nat i) \ I") apply simp_all apply (rule singleton_subspace_set_in_sets[OF measurable_sets[OF MN]]) using assms \<open>y \<in> sets N\<close> apply (auto simp: space_PiF) done finallyshow"A -` y \ space (PiF I M) \ sets (PiF I M)" . next fix x assume"x \ space (PiF I M)" thus "A x \ space N" using MN[of "domain x"] by (auto simp: space_PiF measurable_space Pi'_def) qed
lemma measurable_PiF: assumes f: "\x. x \ space N \ domain (f x) \ I \ (\i\domain (f x). (f x) i \ space (M i))" assumes S: "\J S. J \ I \ (\i. i \ J \ S i \ sets (M i)) \
f -` (Pi' J S) \ space N \ sets N" shows"f \ 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 \ sets (PiF I M)" and "J \ I" shows"A \ {m. domain m \ J} \ sets (PiF J M)" using A[unfolded sets_PiF] proof (induct A) case (Basic a) thenobtain K S where S: "a = Pi' K S""K \ I" "(\i\K. S i \ sets (M i))" by auto show ?case proof cases assume"K \ J" hence"a \ {m. domain m \ J} \ {Pi' K X |X K. K \ J \ X \ (\ j\K. sets (M j))}" using S by (auto intro!: exI[where x=K] exI[where x=S] simp: Pi'_def) alsohave"\ \ sets (PiF J M)" unfolding sets_PiF by auto finallyshow ?thesis . next assume"K \ J" hence"a \ {m. domain m \ J} = {}" using S by (auto simp: Pi'_def) alsohave"\ \ sets (PiF J M)" by simp finallyshow ?thesis . qed next case (Union a) have"\(a ` UNIV) \ {m. domain m \ J} = (\i. (a i \ {m. domain m \ J}))" by simp alsohave"\ \ sets (PiF J M)" using Union by (intro sets.countable_nat_UN) auto finallyshow ?case . next case (Compl a) have"(space (PiF I M) - a) \ {m. domain m \ J} = (space (PiF J M) - (a \ {m. domain m \ J}))" using\<open>J \<subseteq> I\<close> by (auto simp: space_PiF Pi'_def) alsohave"\ \ sets (PiF J M)" using Compl by auto finallyshow ?caseby (simp add: space_PiF) qed simp
lemma measurable_finmap_of: assumes f: "\i. (\x \ space N. i \ J x) \ (\x. f x i) \ measurable N (M i)" assumes J: "\x. x \ space N \ J x \ I" "\x. x \ space N \ finite (J x)" assumes JN: "\S. {x. J x = S} \ space N \ sets N" shows"(\x. finmap_of (J x) (f x)) \ measurable N (PiF I M)" proof (rule measurable_PiF) fix x assume"x \ space N" with J[of x] measurable_space[OF f] show"domain (finmap_of (J x) (f x)) \ I \
(\<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 \ I" and *: "\i. i \ K \ S i \ sets (M i)" with J have eq: "(\x. finmap_of (J x) (f x)) -` Pi' K S \ 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 \ space N. J x = K} = space N \ ({x. J x = K} \ space N)" by auto show"(\x. finmap_of (J x) (f x)) -` Pi' K S \ space N \ 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 proj_measurable_singleton: assumes"A \ sets (M i)" shows"(\x. (x)\<^sub>F i) -` A \ space (PiF {I} M) \ sets (PiF {I} M)" proof cases assume"i \ I" hence"(\x. (x)\<^sub>F i) -` A \ space (PiF {I} M) =
Pi' I (\x. if x = i then A else space (M x))" using sets.sets_into_space[OF ] \<open>A \<in> sets (M i)\<close> assms by (auto simp: space_PiF Pi'_def) thus ?thesis using assms \<open>A \<in> sets (M i)\<close> by (intro in_sets_PiFI) auto next assume"i \ I" hence"(\x. (x)\<^sub>F i) -` A \ 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_countable: fixes I::"'a::countable set set" assumes"y \ space (M i)" shows"(\x. if i \ domain x then (x)\<^sub>F i else y) \ measurable (PiF I M) (M i)" proof (rule countable_measurable_PiFI) fix J assume"J \ I" "finite J" show"(\x. if i \ domain x then x i else y) \ measurable (PiF {J} M) (M i)" unfolding measurable_def proof safe fix z assume"z \ sets (M i)" have"(\x. if i \ domain x then x i else y) -` z \ space (PiF {J} M) =
(\<lambda>x. if i \<in> J then (x)\<^sub>F i else y) -` z \<inter> space (PiF {J} M)" by (auto simp: space_PiF Pi'_def) alsohave"\ \ sets (PiF {J} M)" using \z \ sets (M i)\ \finite J\ by (cases "i \ J") (auto intro!: measurable_sets[OF measurable_proj_singleton]) finallyshow"(\x. if i \ domain x then x i else y) -` z \ space (PiF {J} M) \
sets (PiF {J} M)" . qed (insert \<open>y \<in> space (M i)\<close>, auto simp: space_PiF Pi'_def) qed
lemma measurable_restrict_proj: assumes"J \ II" "finite J" shows"finmap_of J \ 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 \ I" assumes"x \ space (PiM J M)" shows"proj \ measurable (PiF {J} M) (PiM J M)" proof (rule measurable_PiM_single) show"proj \ space (PiF {J} M) \ (\\<^sub>E i \ 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 \ J" "A \ sets (M i)" show"{\ \ space (PiF {J} M). (\)\<^sub>F i \ A} \ sets (PiF {J} M)" proof have"{\ \ space (PiF {J} M). (\)\<^sub>F i \ A} =
(\<lambda>\<omega>. (\<omega>)\<^sub>F i) -` A \<inter> space (PiF {J} M)" by auto alsohave"\ \ sets (PiF {J} M)" using assms A by (auto intro: measurable_sets[OF measurable_proj_singleton] simp: space_PiM) finallyshow ?thesis . qed simp qed
lemma space_PiF_singleton_eq_product: assumes"finite I" shows"space (PiF {I} M) = (\' i\I. space (M i))" by (auto simp: product_def space_PiF assms)
text\<open>adapted from @{thm sets_PiM_single}\<close>
lemma sets_PiF_single: assumes"finite I""I \ {}" 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 ?\ ?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 \ {Pi' I X |X. X \ (\ j\I. sets (M j))}" thenobtain X where X: "A = Pi' I X""X \ (\ j\I. sets (M j))" by auto show"A \ sigma_sets ?\ ?R" proof - from\<open>I \<noteq> {}\<close> 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 alsohave"\ \ sigma_sets ?\ ?R" using X \<open>I \<noteq> {}\<close> assms by (intro R.finite_INT) (auto simp: space_PiF) finallyshow"A \ sigma_sets ?\ ?R" . qed next fix A assume"A \ ?R" thenobtain i B where A: "A = {f\\' i\I. space (M i). f i \ B}" "i \ I" "B \ sets (M i)" by auto thenhave"A = (\' j \ I. if j = i then B else space (M j))" using sets.sets_into_space[OF A(3)] apply (auto simp: Pi'_iff split: if_split_asm) apply blast done alsohave"\ \ sigma_sets ?\ {Pi' I X |X. X \ (\ j\I. sets (M j))}" using A by (intro sigma_sets.Basic )
(auto intro: exI[where x="\j. if j = i then B else space (M j)"]) finallyshow"A \ sigma_sets ?\ {Pi' I X |X. X \ (\ j\I. sets (M j))}" . qed
text\<open>adapted from @{thm PiE_cong}\<close>
lemma Pi'_cong: assumes"finite I" assumes"\i. i \ I \ f i = g i" shows"Pi' I f = Pi' I g" using assms by (auto simp: Pi'_def)
text\<open>adapted from @{thm Pi_UN}\<close>
lemma Pi'_UN: fixes A :: "nat \ 'i \ 'a set" assumes"finite I" assumes mono: "\i n m. i \ I \ n \ m \ A n i \ A m i" shows"(\n. Pi' I (A n)) = Pi' I (\i. \n. A n i)" proof (intro set_eqI iffI) fix f assume"f \ Pi' I (\i. \n. A n i)" thenhave"\i\I. \n. f i \ A n i" "domain f = I" by (auto simp: \finite I\ Pi'_def) from bchoice[OF this(1)] obtain n where n: "\i. i \ I \ f i \ (A (n i) i)" by auto obtain k where k: "\i. i \ I \ n i \ k" using\<open>finite I\<close> finite_nat_set_iff_bounded_le[of "n`I"] by auto have"f \ Pi' I (\i. A k i)" proof fix i assume"i \ I" from mono[OF this, of "n i" k] k[OF this] n[OF this] \<open>domain f = I\<close> \<open>i \<in> I\<close> show"f i \ A k i " by (auto simp: \finite I\) qed (simp add: \<open>domain f = I\<close> \<open>finite I\<close>) thenshow"f \ (\n. Pi' I (A n))" by auto qed (auto simp: Pi'_def \finite I\)
text\<open>adapted from @{thm sets_PiM_sigma}\<close>
lemma sigma_fprod_algebra_sigma_eq: fixes E :: "'i \ 'a set set" and S :: "'i \ nat \ 'a set" assumes [simp]: "finite I""I \ {}" and S_union: "\i. i \ I \ (\j. S i j) = space (M i)" and S_in_E: "\i. i \ I \ range (S i) \ E i" assumes E_closed: "\i. i \ I \ E i \ Pow (space (M i))" and E_generates: "\i. i \ I \ sets (M i) = sigma_sets (space (M i)) (E i)" defines"P == { Pi' I F | F. \i\I. F i \ E i }" shows"sets (PiF {I} M) = sigma_sets (space (PiF {I} M)) P" proof let ?P = "sigma (space (Pi\<^sub>F {I} M)) P" from\<open>finite I\<close>[THEN ex_bij_betw_finite_nat] obtain T where "bij_betw T I {0..<card I}" .. thenhave T: "\i. i \ I \ T i < card I" "\i. i\I \ 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: \<open>finite I\<close>) have P_closed: "P \ Pow (space (Pi\<^sub>F {I} M))" using E_closed by (auto simp: space_PiF P_def Pi'_iff subset_eq) thenhave space_P: "space ?P = (\' i\I. space (M i))" by (simp add: space_PiF) 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) alsohave"\ \ sets (sigma (space (PiF {I} M)) P)" proof (safe intro!: sets.sigma_sets_subset) fix i A assume"i \ I" and A: "A \ sets (M i)" have"(\x. (x)\<^sub>F i) \ measurable ?P (sigma (space (M i)) (E i))" proof (subst measurable_iff_measure_of) show"E i \ Pow (space (M i))" using \i \ I\ by fact from space_P \<open>i \<in> I\<close> show "(\<lambda>x. (x)\<^sub>F i) \<in> space ?P \<rightarrow> space (M i)" by auto show"\A\E i. (\x. (x)\<^sub>F i) -` A \ space ?P \ sets ?P" proof fix A assume A: "A \ E i" from T have *: "\xs. length xs = card I \<and> (\<forall>j\<in>I. (g)\<^sub>F j \<in> (if i = j then A else S j (xs ! T j)))" if"domain g = I"and"\j\I. (g)\<^sub>F j \ (if i = j then A else S j (f j))" for g f using that by (auto intro!: exI [of _ "map (\n. f (the_inv_into I T n)) [0.. from A have"(\x. (x)\<^sub>F i) -` A \ space ?P = (\' j\I. if i = j then A else space (M j))" using E_closed \<open>i \<in> I\<close> by (auto simp: space_P Pi_iff subset_eq split: if_split_asm) alsohave"\ = (\' j\I. \n. if i = j then A else S j n)" by (intro Pi'_cong) (simp_all add: S_union) alsohave"\ = {g. domain g = I \ (\f. \j\I. (g)\<^sub>F j \ (if i = j then A else S j (f j)))}" by (rule set_eqI) (simp del: if_image_distrib add: Pi'_iff bchoice_iff) alsohave"\ = (\xs\{xs. length xs = card I}. \' j\I. if i = j then A else S j (xs ! T j))" using * by (auto simp add: Pi'_iff split del: if_split) alsohave"\ \ sets ?P" proof (safe intro!: sets.countable_UN) fix xs show"(\' j\I. if i = j then A else S j (xs ! T j)) \ sets ?P" using A S_in_E by (simp add: P_closed)
(auto simp: P_def subset_eq intro!: exI[of _ "\j. if i = j then A else S j (xs ! T j)"]) qed finallyshow"(\x. (x)\<^sub>F i) -` A \ space ?P \ sets ?P" using P_closed by simp qed qed from measurable_sets[OF this, of A] A \<open>i \<in> I\<close> E_closed have"(\x. (x)\<^sub>F i) -` A \ space ?P \ sets ?P" by (simp add: E_generates) alsohave"(\x. (x)\<^sub>F i) -` A \ space ?P = {f \ \' i\I. space (M i). f i \ A}" using P_closed by (auto simp: space_PiF) finallyshow"\ \ sets ?P" . qed finallyshow"sets (PiF {I} M) \ sigma_sets (space (PiF {I} M)) P" by (simp add: P_closed) show"sigma_sets (space (PiF {I} M)) P \ sets (PiF {I} M)" using\<open>finite I\<close> \<open>I \<noteq> {}\<close> 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 \ {}" assumes [simp]: "finite I" shows"sets (PiF {I} (\_. 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] obtain S::"'b set set" where S: "S \ (SOME B. countable B \ topological_basis B)" "UNIV = \ S" by auto show ?thesis proof (rule sigma_fprod_algebra_sigma_eq) show"finite I"by simp show"I \ {}" by fact
define S' where "S' = from_nat_into S" show"(\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' \ 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 subsetD topological_basis_open[OF basis_proj] basis_proj_def) done show"Collect open \ Pow (space borel)" by simp show"sets borel = sigma_sets (space borel) (Collect open)" by (simp add: borel_def) qed qed
lemma finmap_UNIV[simp]: "(\J\Collect finite. \' j\J. UNIV) = UNIV" by auto
lemma borel_eq_PiF_borel: shows"(borel :: ('i::countable \\<^sub>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 \\<^sub>F 'a) set" assume "a \ Collect open" hence "open a" by simp thenobtain B' where B': "B'\basis_finmap" "a = \B'" using finmap_topological_basis by (force simp add: topological_basis_def) have"a \ sigma UNIV {Pi' J X |X J. finite J \ X \ J \ sigma_sets UNIV (Collect open)}" unfolding\<open>a = \<Union>B'\<close> proof (rule sets.countable_Union) from B' countable_basis_finmap show "countable B'" by (metis countable_subset) next show"B' \ sets (sigma UNIV
{Pi' J X |X J. finite J \ X \ J \ sigma_sets UNIV (Collect open)})" (is "_ \ sets ?s") proof fix x assume"x \ B'" with B' have "x \ basis_finmap" by auto thenobtain J X where"x = Pi' J X""finite J""X \ J \ sigma_sets UNIV (Collect open)" by (auto simp: basis_finmap_def topological_basis_open[OF basis_proj]) thus"x \ sets ?s" by auto qed qed thus"a \ sigma_sets UNIV {Pi' J X |X J. finite J \ X \ J \ sigma_sets UNIV (Collect open)}" by simp next fix b::"('i \\<^sub>F 'a) set" assume"b \ {Pi' J X |X J. finite J \ X \ J \ sigma_sets UNIV (Collect open)}" hence b': "b \ sets (Pi\<^sub>F (Collect finite) (\_. borel))" by (auto simp: sets_PiF borel_def) let ?b = "\J. b \ {x. domain x = J}" have"b = \((\J. ?b J) ` Collect finite)" by auto alsohave"\ \ sets borel" proof (rule sets.countable_Union, safe) fix J::"'i set"assume"finite J"
{ assume ef: "J = {}" have"?b J \ sets borel" proof cases assume"?b J \ {}" thenobtain f where"f \ b" "domain f = {}" using ef by auto hence"?b J = {f}"using\<open>J = {}\<close> by (auto simp: finmap_eq_iff) alsohave"{f} \ sets borel" by simp finallyshow ?thesis . qed simp
} moreover { assume"J \ ({}::'i set)" have"(?b J) = b \ {m. domain m \ {J}}" by auto alsohave"\ \ sets (PiF {J} (\_. borel))" using b' by (rule restrict_sets_measurable) (auto simp: \finite J\) alsohave"\ = sigma_sets (space (PiF {J} (\_. borel)))
{Pi' (J) F |F. (\j\J. F j \ Collect open)}"
(is"_ = sigma_sets _ ?P") by (rule product_open_generates_sets_PiF_single[OF \<open>J \<noteq> {}\<close> \<open>finite J\<close>]) alsohave"\ \ sigma_sets UNIV (Collect open)" by (intro sigma_sets_mono'') (auto intro!: open_Pi'I simp: space_PiF) finallyhave"(?b J) \ sets borel" by (simp add: borel_def)
} ultimatelyshow"(?b J) \ sets borel" by blast qed (simp add: countable_Collect_finite) finallyshow"b \ sigma_sets UNIV (Collect open)" by (simp add: borel_def) qed (simp add: emeasure_sigma borel_def PiF_def)
subsection \<open>Isomorphism between Functions and Finite Maps\<close>
locale function_to_finmap = fixes J::"'a set"and f :: "'a \ 'b::countable" and f' assumes [simp]: "finite J" assumes inv: "i \ J \ f' (f i) = i" begin
text\<open>to measure finmaps\<close>
definition"fm = (finmap_of (f ` J)) o (\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"\i. space (M i) = UNIV" shows"fm -` Pi' (f ` J) S \ space (Pi\<^sub>M J M) = (\\<^sub>E j \ J. S (f j))" using assms by (auto simp: inv fm_def compose_def space_PiM Pi'_def)
lemma fm_measurable: assumes"f ` J \ N" shows"fm \ measurable (Pi\<^sub>M J (\_. M)) (Pi\<^sub>F N (\_. M))" unfolding fm_def proof (rule measurable_comp, rule measurable_compose_inv) show"finmap_of (f ` J) \ measurable (Pi\<^sub>M (f ` J) (\_. M)) (PiF N (\_. M)) " using assms by (intro measurable_finmap_of measurable_component_singleton) auto qed (simp_all add: inv)
lemma proj_fm: assumes"x \ 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 (\g. compose (f ` J) g f') (extensional J)" proof (rule inj_on_inverseI) fix x::"'a \ 'c" assume "x \ extensional J" thus"(\x. compose J x f) (compose (f ` J) x f') = x" by (auto simp: compose_def inv extensional_def) qed
lemma fm_image_measurable: assumes"space M = UNIV" assumes"X \ sets (Pi\<^sub>M J (\_. M))" shows"fm ` X \ sets (PiF {f ` J} (\_. M))" proof - have"fm ` X = (mf) -` X \ space (PiF {f ` J} (\_. M))" proof safe fix x assume"x \ X" with mf_fm[of x] sets.sets_into_space[OF assms(2)] show"fm x \ mf -` X" by auto show"fm x \ space (PiF {f ` J} (\_. M))" by (simp add: space_PiF assms) next fix y x assume x: "mf y \ X" assume y: "y \ space (PiF {f ` J} (\_. M))" thus"y \ 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 alsohave"\ \ sets (PiF {f ` J} (\_. M))" using assms by (intro measurable_sets[OF mf_measurable]) auto finallyshow ?thesis . qed
lemma fm_image_measurable_finite: assumes"space M = UNIV" assumes"X \ sets (Pi\<^sub>M J (\_. M::'c measure))" shows"fm ` X \ sets (PiF (Collect finite) (\_. M::'c measure))" using fm_image_measurable[OF assms] by (rule subspace_set_in_sets) (auto simp: finite_subset)
text\<open>measure on finmaps\<close>
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\<^sub>M J (\_. N))" assumes s2: "sets M = sets (Pi\<^sub>M J (\_. N))" assumes"space N = UNIV" assumes"X \ sets (PiF (Collect finite) (\_. N))" shows"emeasure (mapmeasure M (\_. N)) X = emeasure M ((fm -` X \ extensional J))" using assms by (auto simp: measurable_cong_sets[OF 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\<^sub>M J (\_. N))" assumes s2: "sets M = (Pi\<^sub>M J (\_. N))" assumes N: "space N = UNIV" assumes X: "X \ sets M" shows"emeasure M X = emeasure (mapmeasure M (\_. N)) (fm ` X)" unfolding mapmeasure_def proof (subst emeasure_distr, subst measurable_cong_sets[OF s2 refl], rule fm_measurable) have"X \ space (Pi\<^sub>M J (\_. N))" using assms by (simp add: sets.sets_into_space) from assms inj_on_fm[of "\_. N"] subsetD[OF this] have "fm -` fm ` X \ space (Pi\<^sub>M J (\_. N)) = X" by (auto simp: vimage_image_eq inj_on_def) thus"emeasure M X = emeasure M (fm -` fm ` X \ space M)" using s1 by simp show"fm ` X \ sets (PiF (Collect finite) (\_. N))" by (rule fm_image_measurable_finite[OF N X[simplified s2]]) qed simp
end
end
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