1(* Title: HOL/Probability/Fin_Map.thy 2 Author: Fabian Immler, TU M��nchen 3*) 4 5section \<open>Finite Maps\<close> 6 7theory Fin_Map 8 imports "HOL-Analysis.Finite_Product_Measure" "HOL-Library.Finite_Map" 9begin 10 11text \<open>The \<^type>\<open>fmap\<close> type can be instantiated to \<^class>\<open>polish_space\<close>, needed for the proof of 12 projective limit. \<^const>\<open>extensional\<close> functions are used for the representation in order to 13 stay close to the developments of (finite) products \<^const>\<open>Pi\<^sub>E\<close> and their sigma-algebra 14 \<^const>\<open>Pi\<^sub>M\<close>.\<close> 15 16type_notation fmap ("(_ \<Rightarrow>\<^sub>F /_)" [22, 21] 21) 17 18unbundle fmap.lifting 19 20 21subsection \<open>Domain and Application\<close> 22 23lift_definition domain::"('i \<Rightarrow>\<^sub>F 'a) \<Rightarrow> 'i set" is dom . 24 25lemma finite_domain[simp, intro]: "finite (domain P)" 26 by transfer simp 27 28lift_definition proj :: "('i \<Rightarrow>\<^sub>F 'a) \<Rightarrow> 'i \<Rightarrow> 'a" ("'((_)')\<^sub>F" [0] 1000) is 29 "\<lambda>f x. if x \<in> dom f then the (f x) else undefined" . 30 31declare [[coercion proj]] 32 33lemma extensional_proj[simp, intro]: "(P)\<^sub>F \<in> extensional (domain P)" 34 by transfer (auto simp: extensional_def) 35 36lemma proj_undefined[simp, intro]: "i \<notin> domain P \<Longrightarrow> P i = undefined" 37 using extensional_proj[of P] unfolding extensional_def by auto 38 39lemma finmap_eq_iff: "P = Q \<longleftrightarrow> (domain P = domain Q \<and> (\<forall>i\<in>domain P. P i = Q i))" 40 apply transfer 41 apply (safe intro!: ext) 42 subgoal for P Q x 43 by (cases "x \<in> dom P"; cases "P x") (auto dest!: bspec[where x=x]) 44 done 45 46 47subsection \<open>Constructor of Finite Maps\<close> 48 49lift_definition finmap_of::"'i set \<Rightarrow> ('i \<Rightarrow> 'a) \<Rightarrow> ('i \<Rightarrow>\<^sub>F 'a)" is 50 "\<lambda>I f x. if x \<in> I \<and> finite I then Some (f x) else None" 51 by (simp add: dom_def) 52 53lemma proj_finmap_of[simp]: 54 assumes "finite inds" 55 shows "(finmap_of inds f)\<^sub>F = restrict f inds" 56 using assms 57 by transfer force 58 59lemma domain_finmap_of[simp]: 60 assumes "finite inds" 61 shows "domain (finmap_of inds f) = inds" 62 using assms 63 by transfer (auto split: if_splits) 64 65lemma finmap_of_eq_iff[simp]: 66 assumes "finite i" "finite j" 67 shows "finmap_of i m = finmap_of j n \<longleftrightarrow> i = j \<and> (\<forall>k\<in>i. m k= n k)" 68 using assms by (auto simp: finmap_eq_iff) 69 70lemma finmap_of_inj_on_extensional_finite: 71 assumes "finite K" 72 assumes "S \<subseteq> extensional K" 73 shows "inj_on (finmap_of K) S" 74proof (rule inj_onI) 75 fix x y::"'a \<Rightarrow> 'b" 76 assume "finmap_of K x = finmap_of K y" 77 hence "(finmap_of K x)\<^sub>F = (finmap_of K y)\<^sub>F" by simp 78 moreover 79 assume "x \<in> S" "y \<in> S" hence "x \<in> extensional K" "y \<in> extensional K" using assms by auto 80 ultimately 81 show "x = y" using assms by (simp add: extensional_restrict) 82qed 83 84subsection \<open>Product set of Finite Maps\<close> 85 86text \<open>This is \<^term>\<open>Pi\<close> for Finite Maps, most of this is copied\<close> 87 88definition Pi' :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a set) \<Rightarrow> ('i \<Rightarrow>\<^sub>F 'a) set" where 89 "Pi' I A = { P. domain P = I \<and> (\<forall>i. i \<in> I \<longrightarrow> (P)\<^sub>F i \<in> A i) } " 90 91syntax 92 "_Pi'" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3\<Pi>'' _\<in>_./ _)" 10) 93translations 94 "\<Pi>' x\<in>A. B" == "CONST Pi' A (\<lambda>x. B)" 95 96subsubsection\<open>Basic Properties of \<^term>\<open>Pi'\<close>\<close> 97 98lemma Pi'_I[intro!]: "domain f = A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi' A B" 99 by (simp add: Pi'_def) 100 101lemma Pi'_I'[simp]: "domain f = A \<Longrightarrow> (\<And>x. x \<in> A \<longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi' A B" 102 by (simp add:Pi'_def) 103 104lemma Pi'_mem: "f\<in> Pi' A B \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<in> B x" 105 by (simp add: Pi'_def) 106 107lemma Pi'_iff: "f \<in> Pi' I X \<longleftrightarrow> domain f = I \<and> (\<forall>i\<in>I. f i \<in> X i)" 108 unfolding Pi'_def by auto 109 110lemma Pi'E [elim]: 111 "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" 112 by(auto simp: Pi'_def) 113 114lemma in_Pi'_cong: 115 "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" 116 by (auto simp: Pi'_def) 117 118lemma Pi'_eq_empty[simp]: 119 assumes "finite A" shows "(Pi' A B) = {} \<longleftrightarrow> (\<exists>x\<in>A. B x = {})" 120 using assms 121 apply (simp add: Pi'_def, auto) 122 apply (drule_tac x = "finmap_of A (\<lambda>u. SOME y. y \<in> B u)" in spec, auto) 123 apply (cut_tac P= "%y. y \<in> B i" in some_eq_ex, auto) 124 done 125 126lemma Pi'_mono: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x) \<Longrightarrow> Pi' A B \<subseteq> Pi' A C" 127 by (auto simp: Pi'_def) 128 129lemma Pi_Pi': "finite A \<Longrightarrow> (Pi\<^sub>E A B) = proj ` Pi' A B" 130 apply (auto simp: Pi'_def Pi_def extensional_def) 131 apply (rule_tac x = "finmap_of A (restrict x A)" in image_eqI) 132 apply auto 133 done 134 135subsection \<open>Topological Space of Finite Maps\<close> 136 137instantiation fmap :: (type, topological_space) topological_space 138begin 139 140definition open_fmap :: "('a \<Rightarrow>\<^sub>F 'b) set \<Rightarrow> bool" where 141 [code del]: "open_fmap = generate_topology {Pi' a b|a b. \<forall>i\<in>a. open (b i)}" 142 143lemma open_Pi'I: "(\<And>i. i \<in> I \<Longrightarrow> open (A i)) \<Longrightarrow> open (Pi' I A)" 144 by (auto intro: generate_topology.Basis simp: open_fmap_def) 145 146instance using topological_space_generate_topology 147 by intro_classes (auto simp: open_fmap_def class.topological_space_def) 148 149end 150 151lemma open_restricted_space: 152 shows "open {m. P (domain m)}" 153proof - 154 have "{m. P (domain m)} = (\<Union>i \<in> Collect P. {m. domain m = i})" by auto 155 also have "open \<dots>" 156 proof (rule, safe, cases) 157 fix i::"'a set" 158 assume "finite i" 159 hence "{m. domain m = i} = Pi' i (\<lambda>_. UNIV)" by (auto simp: Pi'_def) 160 also have "open \<dots>" by (auto intro: open_Pi'I simp: \<open>finite i\<close>) 161 finally show "open {m. domain m = i}" . 162 next 163 fix i::"'a set" 164 assume "\<not> finite i" hence "{m. domain m = i} = {}" by auto 165 also have "open \<dots>" by simp 166 finally show "open {m. domain m = i}" . 167 qed 168 finally show ?thesis . 169qed 170 171lemma closed_restricted_space: 172 shows "closed {m. P (domain m)}" 173 using open_restricted_space[of "\<lambda>x. \<not> P x"] 174 unfolding closed_def by (rule back_subst) auto 175 176lemma tendsto_proj: "((\<lambda>x. x) \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. (x)\<^sub>F i) \<longlongrightarrow> (a)\<^sub>F i) F" 177 unfolding tendsto_def 178proof safe 179 fix S::"'b set" 180 let ?S = "Pi' (domain a) (\<lambda>x. if x = i then S else UNIV)" 181 assume "open S" hence "open ?S" by (auto intro!: open_Pi'I) 182 moreover assume "\<forall>S. open S \<longrightarrow> a \<in> S \<longrightarrow> eventually (\<lambda>x. x \<in> S) F" "a i \<in> S" 183 ultimately have "eventually (\<lambda>x. x \<in> ?S) F" by auto 184 thus "eventually (\<lambda>x. (x)\<^sub>F i \<in> S) F" 185 by eventually_elim (insert \<open>a i \<in> S\<close>, force simp: Pi'_iff split: if_split_asm) 186qed 187 188lemma continuous_proj: 189 shows "continuous_on s (\<lambda>x. (x)\<^sub>F i)" 190 unfolding continuous_on_def by (safe intro!: tendsto_proj tendsto_ident_at) 191 192instance fmap :: (type, first_countable_topology) first_countable_topology 193proof 194 fix x::"'a\<Rightarrow>\<^sub>F'b" 195 have "\<forall>i. \<exists>A. countable A \<and> (\<forall>a\<in>A. x i \<in> a) \<and> (\<forall>a\<in>A. open a) \<and> 196 (\<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") 197 proof 198 fix i from first_countable_basis_Int_stableE[of "x i"] guess A . 199 thus "?th i" by (intro exI[where x=A]) simp 200 qed 201 then guess A unfolding choice_iff .. note A = this 202 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 203 have A_notempty: "\<And>i. i \<in> domain x \<Longrightarrow> A i \<noteq> {}" using open_sub[of _ "\<lambda>_. UNIV"] by auto 204 let ?A = "(\<lambda>f. Pi' (domain x) f) ` (Pi\<^sub>E (domain x) A)" 205 show "\<exists>A::nat \<Rightarrow> ('a\<Rightarrow>\<^sub>F'b) set. (\<forall>i. x \<in> (A i) \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))" 206 proof (rule first_countableI[of "?A"], safe) 207 show "countable ?A" using A by (simp add: countable_PiE) 208 next 209 fix S::"('a \<Rightarrow>\<^sub>F 'b) set" assume "open S" "x \<in> S" 210 thus "\<exists>a\<in>?A. a \<subseteq> S" unfolding open_fmap_def 211 proof (induct rule: generate_topology.induct) 212 case UNIV thus ?case by (auto simp add: ex_in_conv PiE_eq_empty_iff A_notempty) 213 next 214 case (Int a b) 215 then obtain f g where 216 "f \<in> Pi\<^sub>E (domain x) A" "Pi' (domain x) f \<subseteq> a" "g \<in> Pi\<^sub>E (domain x) A" "Pi' (domain x) g \<subseteq> b" 217 by auto 218 thus ?case using A 219 by (auto simp: Pi'_iff PiE_iff extensional_def Int_stable_def 220 intro!: bexI[where x="\<lambda>i. f i \<inter> g i"]) 221 next 222 case (UN B) 223 then obtain b where "x \<in> b" "b \<in> B" by auto 224 hence "\<exists>a\<in>?A. a \<subseteq> b" using UN by simp 225 thus ?case using \<open>b \<in> B\<close> by blast 226 next 227 case (Basis s) 228 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 229 have "\<forall>i. \<exists>a. (i \<in> domain x \<and> open (b i) \<and> (x)\<^sub>F i \<in> b i) \<longrightarrow> (a\<in>A i \<and> a \<subseteq> b i)" 230 using open_sub[of _ b] by auto 231 then obtain b' 232 where "\<And>i. i \<in> domain x \<Longrightarrow> open (b i) \<Longrightarrow> (x)\<^sub>F i \<in> b i \<Longrightarrow> (b' i \<in>A i \<and> b' i \<subseteq> b i)" 233 unfolding choice_iff by auto 234 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" 235 by (auto simp: Pi'_iff intro!: Pi'_mono) 236 thus ?case using xs 237 by (intro bexI[where x="Pi' a b'"]) 238 (auto simp: Pi'_iff intro!: image_eqI[where x="restrict b' (domain x)"]) 239 qed 240 qed (insert A,auto simp: PiE_iff intro!: open_Pi'I) 241qed 242 243subsection \<open>Metric Space of Finite Maps\<close> 244 245(* TODO: Product of uniform spaces and compatibility with metric_spaces! *) 246 247instantiation fmap :: (type, metric_space) dist 248begin 249 250definition dist_fmap where 251 "dist P Q = Max (range (\<lambda>i. dist ((P)\<^sub>F i) ((Q)\<^sub>F i))) + (if domain P = domain Q then 0 else 1)" 252 253instance .. 254end 255 256instantiation fmap :: (type, metric_space) uniformity_dist 257begin 258 259definition [code del]: 260 "(uniformity :: (('a, 'b) fmap \<times> ('a \<Rightarrow>\<^sub>F 'b)) filter) = 261 (INF e\<in>{0 <..}. principal {(x, y). dist x y < e})" 262 263instance 264 by standard (rule uniformity_fmap_def) 265end 266 267declare uniformity_Abort[where 'a="('a \<Rightarrow>\<^sub>F 'b::metric_space)", code] 268 269instantiation fmap :: (type, metric_space) metric_space 270begin 271 272lemma finite_proj_image': "x \<notin> domain P \<Longrightarrow> finite ((P)\<^sub>F ` S)" 273 by (rule finite_subset[of _ "proj P ` (domain P \<inter> S \<union> {x})"]) auto 274 275lemma finite_proj_image: "finite ((P)\<^sub>F ` S)" 276 by (cases "\<exists>x. x \<notin> domain P") (auto intro: finite_proj_image' finite_subset[where B="domain P"]) 277 278lemma finite_proj_diag: "finite ((\<lambda>i. d ((P)\<^sub>F i) ((Q)\<^sub>F i)) ` S)" 279proof - 280 have "(\<lambda>i. d ((P)\<^sub>F i) ((Q)\<^sub>F i)) ` S = (\<lambda>(i, j). d i j) ` ((\<lambda>i. ((P)\<^sub>F i, (Q)\<^sub>F i)) ` S)" by auto 281 moreover have "((\<lambda>i. ((P)\<^sub>F i, (Q)\<^sub>F i)) ` S) \<subseteq> (\<lambda>i. (P)\<^sub>F i) ` S \<times> (\<lambda>i. (Q)\<^sub>F i) ` S" by auto 282 moreover have "finite \<dots>" using finite_proj_image[of P S] finite_proj_image[of Q S] 283 by (intro finite_cartesian_product) simp_all 284 ultimately show ?thesis by (simp add: finite_subset) 285qed 286 287lemma dist_le_1_imp_domain_eq: 288 shows "dist P Q < 1 \<Longrightarrow> domain P = domain Q" 289 by (simp add: dist_fmap_def finite_proj_diag split: if_split_asm) 290 291lemma dist_proj: 292 shows "dist ((x)\<^sub>F i) ((y)\<^sub>F i) \<le> dist x y" 293proof - 294 have "dist (x i) (y i) \<le> Max (range (\<lambda>i. dist (x i) (y i)))" 295 by (simp add: Max_ge_iff finite_proj_diag) 296 also have "\<dots> \<le> dist x y" by (simp add: dist_fmap_def) 297 finally show ?thesis . 298qed 299 300lemma dist_finmap_lessI: 301 assumes "domain P = domain Q" 302 assumes "0 < e" 303 assumes "\<And>i. i \<in> domain P \<Longrightarrow> dist (P i) (Q i) < e" 304 shows "dist P Q < e" 305proof - 306 have "dist P Q = Max (range (\<lambda>i. dist (P i) (Q i)))" 307 using assms by (simp add: dist_fmap_def finite_proj_diag) 308 also have "\<dots> < e" 309 proof (subst Max_less_iff, safe) 310 fix i 311 show "dist ((P)\<^sub>F i) ((Q)\<^sub>F i) < e" using assms 312 by (cases "i \<in> domain P") simp_all 313 qed (simp add: finite_proj_diag) 314 finally show ?thesis . 315qed 316 317instance 318proof 319 fix S::"('a \<Rightarrow>\<^sub>F 'b) set" 320 have *: "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)" (is "_ = ?od") 321 proof 322 assume "open S" 323 thus ?od 324 unfolding open_fmap_def 325 proof (induct rule: generate_topology.induct) 326 case UNIV thus ?case by (auto intro: zero_less_one) 327 next 328 case (Int a b) 329 show ?case 330 proof safe 331 fix x assume x: "x \<in> a" "x \<in> b" 332 with Int x obtain e1 e2 where 333 "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 334 thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> a \<inter> b" 335 by (auto intro!: exI[where x="min e1 e2"]) 336 qed 337 next 338 case (UN K) 339 show ?case 340 proof safe 341 fix x X assume "x \<in> X" and X: "X \<in> K" 342 with UN obtain e where "e>0" "\<And>y. dist y x < e \<longrightarrow> y \<in> X" by force 343 with X show "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> \<Union>K" by auto 344 qed 345 next 346 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 347 show ?case 348 proof safe 349 fix x assume "x \<in> s" 350 hence [simp]: "finite a" and a_dom: "a = domain x" using s by (auto simp: Pi'_iff) 351 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)" 352 using b \<open>x \<in> s\<close> by atomize_elim (intro bchoice, auto simp: open_dist s) 353 hence in_b: "\<And>i y. i \<in> a \<Longrightarrow> dist y (proj x i) < es i \<Longrightarrow> y \<in> b i" by auto 354 show "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> s" 355 proof (cases, rule, safe) 356 assume "a \<noteq> {}" 357 show "0 < min 1 (Min (es ` a))" using es by (auto simp: \<open>a \<noteq> {}\<close>) 358 fix y assume d: "dist y x < min 1 (Min (es ` a))" 359 show "y \<in> s" unfolding s 360 proof 361 show "domain y = a" using d s \<open>a \<noteq> {}\<close> by (auto simp: dist_le_1_imp_domain_eq a_dom) 362 fix i assume i: "i \<in> a" 363 hence "dist ((y)\<^sub>F i) ((x)\<^sub>F i) < es i" using d 364 by (auto simp: dist_fmap_def \<open>a \<noteq> {}\<close> intro!: le_less_trans[OF dist_proj]) 365 with i show "y i \<in> b i" by (rule in_b) 366 qed 367 next 368 assume "\<not>a \<noteq> {}" 369 thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> s" 370 using s \<open>x \<in> s\<close> by (auto simp: Pi'_def dist_le_1_imp_domain_eq intro!: exI[where x=1]) 371 qed 372 qed 373 qed 374 next 375 assume "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" 376 then obtain e where e_pos: "\<And>x. x \<in> S \<Longrightarrow> e x > 0" and 377 e_in: "\<And>x y . x \<in> S \<Longrightarrow> dist y x < e x \<Longrightarrow> y \<in> S" 378 unfolding bchoice_iff 379 by auto 380 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))}" 381 proof safe 382 fix x assume "x \<in> S" 383 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))}" 384 using e_pos by (auto intro!: exI[where x="Pi' (domain x) (\<lambda>i. ball (x i) (e x))"]) 385 next 386 fix x y 387 assume "y \<in> S" 388 moreover 389 assume "x \<in> (\<Pi>' i\<in>domain y. ball (y i) (e y))" 390 hence "dist x y < e y" using e_pos \<open>y \<in> S\<close> 391 by (auto simp: dist_fmap_def Pi'_iff finite_proj_diag dist_commute) 392 ultimately show "x \<in> S" by (rule e_in) 393 qed 394 also have "open \<dots>" 395 unfolding open_fmap_def 396 by (intro generate_topology.UN) (auto intro: generate_topology.Basis) 397 finally show "open S" . 398 qed 399 show "open S = (\<forall>x\<in>S. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> S)" 400 unfolding * eventually_uniformity_metric 401 by (simp del: split_paired_All add: dist_fmap_def dist_commute eq_commute) 402next 403 fix P Q::"'a \<Rightarrow>\<^sub>F 'b" 404 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))" 405 by (auto intro: Max_in Max_eqI) 406 show "dist P Q = 0 \<longleftrightarrow> P = Q" 407 by (auto simp: finmap_eq_iff dist_fmap_def Max_ge_iff finite_proj_diag Max_eq_iff 408 add_nonneg_eq_0_iff 409 intro!: Max_eqI image_eqI[where x=undefined]) 410next 411 fix P Q R::"'a \<Rightarrow>\<^sub>F 'b" 412 let ?dists = "\<lambda>P Q i. dist ((P)\<^sub>F i) ((Q)\<^sub>F i)" 413 let ?dpq = "?dists P Q" and ?dpr = "?dists P R" and ?dqr = "?dists Q R" 414 let ?dom = "\<lambda>P Q. (if domain P = domain Q then 0 else 1::real)" 415 have "dist P Q = Max (range ?dpq) + ?dom P Q" 416 by (simp add: dist_fmap_def) 417 also obtain t where "t \<in> range ?dpq" "t = Max (range ?dpq)" by (simp add: finite_proj_diag) 418 then obtain i where "Max (range ?dpq) = ?dpq i" by auto 419 also have "?dpq i \<le> ?dpr i + ?dqr i" by (rule dist_triangle2) 420 also have "?dpr i \<le> Max (range ?dpr)" by (simp add: finite_proj_diag) 421 also have "?dqr i \<le> Max (range ?dqr)" by (simp add: finite_proj_diag) 422 also have "?dom P Q \<le> ?dom P R + ?dom Q R" by simp 423 finally show "dist P Q \<le> dist P R + dist Q R" by (simp add: dist_fmap_def ac_simps) 424qed 425 426end 427 428subsection \<open>Complete Space of Finite Maps\<close> 429 430lemma tendsto_finmap: 431 fixes f::"nat \<Rightarrow> ('i \<Rightarrow>\<^sub>F ('a::metric_space))" 432 assumes ind_f: "\<And>n. domain (f n) = domain g" 433 assumes proj_g: "\<And>i. i \<in> domain g \<Longrightarrow> (\<lambda>n. (f n) i) \<longlonglongrightarrow> g i" 434 shows "f \<longlonglongrightarrow> g" 435 unfolding tendsto_iff 436proof safe 437 fix e::real assume "0 < e" 438 let ?dists = "\<lambda>x i. dist ((f x)\<^sub>F i) ((g)\<^sub>F i)" 439 have "eventually (\<lambda>x. \<forall>i\<in>domain g. ?dists x i < e) sequentially" 440 using finite_domain[of g] proj_g 441 proof induct 442 case (insert i G) 443 with \<open>0 < e\<close> have "eventually (\<lambda>x. ?dists x i < e) sequentially" by (auto simp add: tendsto_iff) 444 moreover 445 from insert have "eventually (\<lambda>x. \<forall>i\<in>G. dist ((f x)\<^sub>F i) ((g)\<^sub>F i) < e) sequentially" by simp 446 ultimately show ?case by eventually_elim auto 447 qed simp 448 thus "eventually (\<lambda>x. dist (f x) g < e) sequentially" 449 by eventually_elim (auto simp add: dist_fmap_def finite_proj_diag ind_f \<open>0 < e\<close>) 450qed 451 452instance fmap :: (type, complete_space) complete_space 453proof 454 fix P::"nat \<Rightarrow> 'a \<Rightarrow>\<^sub>F 'b" 455 assume "Cauchy P" 456 then obtain Nd where Nd: "\<And>n. n \<ge> Nd \<Longrightarrow> dist (P n) (P Nd) < 1" 457 by (force simp: Cauchy_altdef2) 458 define d where "d = domain (P Nd)" 459 with Nd have dim: "\<And>n. n \<ge> Nd \<Longrightarrow> domain (P n) = d" using dist_le_1_imp_domain_eq by auto 460 have [simp]: "finite d" unfolding d_def by simp 461 define p where "p i n = P n i" for i n 462 define q where "q i = lim (p i)" for i 463 define Q where "Q = finmap_of d q" 464 have q: "\<And>i. i \<in> d \<Longrightarrow> q i = Q i" by (auto simp add: Q_def Abs_fmap_inverse) 465 { 466 fix i assume "i \<in> d" 467 have "Cauchy (p i)" unfolding Cauchy_altdef2 p_def 468 proof safe 469 fix e::real assume "0 < e" 470 with \<open>Cauchy P\<close> obtain N where N: "\<And>n. n\<ge>N \<Longrightarrow> dist (P n) (P N) < min e 1" 471 by (force simp: Cauchy_altdef2 min_def) 472 hence "\<And>n. n \<ge> N \<Longrightarrow> domain (P n) = domain (P N)" using dist_le_1_imp_domain_eq by auto 473 with dim have dim: "\<And>n. n \<ge> N \<Longrightarrow> domain (P n) = d" by (metis nat_le_linear) 474 show "\<exists>N. \<forall>n\<ge>N. dist ((P n) i) ((P N) i) < e" 475 proof (safe intro!: exI[where x="N"]) 476 fix n assume "N \<le> n" have "N \<le> N" by simp 477 have "dist ((P n) i) ((P N) i) \<le> dist (P n) (P N)" 478 using dim[OF \<open>N \<le> n\<close>] dim[OF \<open>N \<le> N\<close>] \<open>i \<in> d\<close> 479 by (auto intro!: dist_proj) 480 also have "\<dots> < e" using N[OF \<open>N \<le> n\<close>] by simp 481 finally show "dist ((P n) i) ((P N) i) < e" . 482 qed 483 qed 484 hence "convergent (p i)" by (metis Cauchy_convergent_iff) 485 hence "p i \<longlonglongrightarrow> q i" unfolding q_def convergent_def by (metis limI) 486 } note p = this 487 have "P \<longlonglongrightarrow> Q" 488 proof (rule metric_LIMSEQ_I) 489 fix e::real assume "0 < e" 490 have "\<exists>ni. \<forall>i\<in>d. \<forall>n\<ge>ni i. dist (p i n) (q i) < e" 491 proof (safe intro!: bchoice) 492 fix i assume "i \<in> d" 493 from p[OF \<open>i \<in> d\<close>, THEN metric_LIMSEQ_D, OF \<open>0 < e\<close>] 494 show "\<exists>no. \<forall>n\<ge>no. dist (p i n) (q i) < e" . 495 qed then guess ni .. note ni = this 496 define N where "N = max Nd (Max (ni ` d))" 497 show "\<exists>N. \<forall>n\<ge>N. dist (P n) Q < e" 498 proof (safe intro!: exI[where x="N"]) 499 fix n assume "N \<le> n" 500 hence dom: "domain (P n) = d" "domain Q = d" "domain (P n) = domain Q" 501 using dim by (simp_all add: N_def Q_def dim_def Abs_fmap_inverse) 502 show "dist (P n) Q < e" 503 proof (rule dist_finmap_lessI[OF dom(3) \<open>0 < e\<close>]) 504 fix i 505 assume "i \<in> domain (P n)" 506 hence "ni i \<le> Max (ni ` d)" using dom by simp 507 also have "\<dots> \<le> N" by (simp add: N_def) 508 finally show "dist ((P n)\<^sub>F i) ((Q)\<^sub>F i) < e" using ni \<open>i \<in> domain (P n)\<close> \<open>N \<le> n\<close> dom 509 by (auto simp: p_def q N_def less_imp_le) 510 qed 511 qed 512 qed 513 thus "convergent P" by (auto simp: convergent_def) 514qed 515 516subsection \<open>Second Countable Space of Finite Maps\<close> 517 518instantiation fmap :: (countable, second_countable_topology) second_countable_topology 519begin 520 521definition basis_proj::"'b set set" 522 where "basis_proj = (SOME B. countable B \<and> topological_basis B)" 523 524lemma countable_basis_proj: "countable basis_proj" and basis_proj: "topological_basis basis_proj" 525 unfolding basis_proj_def by (intro is_basis countable_basis)+ 526 527definition basis_finmap::"('a \<Rightarrow>\<^sub>F 'b) set set" 528 where "basis_finmap = {Pi' I S|I S. finite I \<and> (\<forall>i \<in> I. S i \<in> basis_proj)}" 529 530lemma in_basis_finmapI: 531 assumes "finite I" assumes "\<And>i. i \<in> I \<Longrightarrow> S i \<in> basis_proj" 532 shows "Pi' I S \<in> basis_finmap" 533 using assms unfolding basis_finmap_def by auto 534 535lemma basis_finmap_eq: 536 assumes "basis_proj \<noteq> {}" 537 shows "basis_finmap = (\<lambda>f. Pi' (domain f) (\<lambda>i. from_nat_into basis_proj ((f)\<^sub>F i))) ` 538 (UNIV::('a \<Rightarrow>\<^sub>F nat) set)" (is "_ = ?f ` _") 539 unfolding basis_finmap_def 540proof safe 541 fix I::"'a set" and S::"'a \<Rightarrow> 'b set" 542 assume "finite I" "\<forall>i\<in>I. S i \<in> basis_proj" 543 hence "Pi' I S = ?f (finmap_of I (\<lambda>x. to_nat_on basis_proj (S x)))" 544 by (force simp: Pi'_def countable_basis_proj) 545 thus "Pi' I S \<in> range ?f" by simp 546next 547 fix x and f::"'a \<Rightarrow>\<^sub>F nat" 548 show "\<exists>I S. (\<Pi>' i\<in>domain f. from_nat_into basis_proj ((f)\<^sub>F i)) = Pi' I S \<and> 549 finite I \<and> (\<forall>i\<in>I. S i \<in> basis_proj)" 550 using assms by (auto intro: from_nat_into) 551qed 552 553lemma basis_finmap_eq_empty: "basis_proj = {} \<Longrightarrow> basis_finmap = {Pi' {} undefined}" 554 by (auto simp: Pi'_iff basis_finmap_def) 555 556lemma countable_basis_finmap: "countable basis_finmap" 557 by (cases "basis_proj = {}") (auto simp: basis_finmap_eq basis_finmap_eq_empty) 558 559lemma finmap_topological_basis: 560 "topological_basis basis_finmap" 561proof (subst topological_basis_iff, safe) 562 fix B' assume "B' \<in> basis_finmap" 563 thus "open B'" 564 by (auto intro!: open_Pi'I topological_basis_open[OF basis_proj] 565 simp: topological_basis_def basis_finmap_def Let_def) 566next 567 fix O'::"('a \<Rightarrow>\<^sub>F 'b) set" and x 568 assume O': "open O'" "x \<in> O'" 569 then obtain a where a: 570 "x \<in> Pi' (domain x) a" "Pi' (domain x) a \<subseteq> O'" "\<And>i. i\<in>domain x \<Longrightarrow> open (a i)" 571 unfolding open_fmap_def 572 proof (atomize_elim, induct rule: generate_topology.induct) 573 case (Int a b) 574 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))" 575 from Int obtain f g where "?p a f" "?p b g" by auto 576 thus ?case by (force intro!: exI[where x="\<lambda>i. f i \<inter> g i"] simp: Pi'_def) 577 next 578 case (UN k) 579 then obtain kk a where "x \<in> kk" "kk \<in> k" "x \<in> Pi' (domain x) a" "Pi' (domain x) a \<subseteq> kk" 580 "\<And>i. i\<in>domain x \<Longrightarrow> open (a i)" 581 by force 582 thus ?case by blast 583 qed (auto simp: Pi'_def) 584 have "\<exists>B. 585 (\<forall>i\<in>domain x. x i \<in> B i \<and> B i \<subseteq> a i \<and> B i \<in> basis_proj)" 586 proof (rule bchoice, safe) 587 fix i assume "i \<in> domain x" 588 hence "open (a i)" "x i \<in> a i" using a by auto 589 from topological_basisE[OF basis_proj this] guess b' . 590 thus "\<exists>y. x i \<in> y \<and> y \<subseteq> a i \<and> y \<in> basis_proj" by auto 591 qed 592 then guess B .. note B = this 593 define B' where "B' = Pi' (domain x) (\<lambda>i. (B i)::'b set)" 594 have "B' \<subseteq> Pi' (domain x) a" using B by (auto intro!: Pi'_mono simp: B'_def) 595 also note \<open>\<dots> \<subseteq> O'\<close> 596 finally show "\<exists>B'\<in>basis_finmap. x \<in> B' \<and> B' \<subseteq> O'" using B 597 by (auto intro!: bexI[where x=B'] Pi'_mono in_basis_finmapI simp: B'_def) 598qed 599 600lemma range_enum_basis_finmap_imp_open: 601 assumes "x \<in> basis_finmap" 602 shows "open x" 603 using finmap_topological_basis assms by (auto simp: topological_basis_def) 604 605instance proof qed (blast intro: finmap_topological_basis countable_basis_finmap topological_basis_imp_subbasis) 606 607end 608 609subsection \<open>Polish Space of Finite Maps\<close> 610 611instance fmap :: (countable, polish_space) polish_space proof qed 612 613 614subsection \<open>Product Measurable Space of Finite Maps\<close> 615 616definition "PiF I M \<equiv> 617 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))}" 618 619abbreviation 620 "Pi\<^sub>F I M \<equiv> PiF I M" 621 622syntax 623 "_PiF" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3\<Pi>\<^sub>F _\<in>_./ _)" 10) 624translations 625 "\<Pi>\<^sub>F x\<in>I. M" == "CONST PiF I (%x. M)" 626 627lemma 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> 628 Pow (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j)))" 629 by (auto simp: Pi'_def) (blast dest: sets.sets_into_space) 630 631lemma space_PiF: "space (PiF I M) = (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j)))" 632 unfolding PiF_def using PiF_gen_subset by (rule space_measure_of) 633 634lemma sets_PiF: 635 "sets (PiF I M) = sigma_sets (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j))) 636 {(\<Pi>' j\<in>J. X j) |X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}" 637 unfolding PiF_def using PiF_gen_subset by (rule sets_measure_of) 638 639lemma sets_PiF_singleton: 640 "sets (PiF {I} M) = sigma_sets (\<Pi>' j\<in>I. space (M j)) 641 {(\<Pi>' j\<in>I. X j) |X. X \<in> (\<Pi> j\<in>I. sets (M j))}" 642 unfolding sets_PiF by simp 643 644lemma in_sets_PiFI: 645 assumes "X = (Pi' J S)" "J \<in> I" "\<And>i. i\<in>J \<Longrightarrow> S i \<in> sets (M i)" 646 shows "X \<in> sets (PiF I M)" 647 unfolding sets_PiF 648 using assms by blast 649 650lemma product_in_sets_PiFI: 651 assumes "J \<in> I" "\<And>i. i\<in>J \<Longrightarrow> S i \<in> sets (M i)" 652 shows "(Pi' J S) \<in> sets (PiF I M)" 653 unfolding sets_PiF 654 using assms by blast 655 656lemma singleton_space_subset_in_sets: 657 fixes J 658 assumes "J \<in> I" 659 assumes "finite J" 660 shows "space (PiF {J} M) \<in> sets (PiF I M)" 661 using assms 662 by (intro in_sets_PiFI[where J=J and S="\<lambda>i. space (M i)"]) 663 (auto simp: product_def space_PiF) 664 665lemma singleton_subspace_set_in_sets: 666 assumes A: "A \<in> sets (PiF {J} M)" 667 assumes "finite J" 668 assumes "J \<in> I" 669 shows "A \<in> sets (PiF I M)" 670 using A[unfolded sets_PiF] 671 apply (induct A) 672 unfolding sets_PiF[symmetric] unfolding space_PiF[symmetric] 673 using assms 674 by (auto intro: in_sets_PiFI intro!: singleton_space_subset_in_sets) 675 676lemma finite_measurable_singletonI: 677 assumes "finite I" 678 assumes "\<And>J. J \<in> I \<Longrightarrow> finite J" 679 assumes MN: "\<And>J. J \<in> I \<Longrightarrow> A \<in> measurable (PiF {J} M) N" 680 shows "A \<in> measurable (PiF I M) N" 681 unfolding measurable_def 682proof safe 683 fix y assume "y \<in> sets N" 684 have "A -` y \<inter> space (PiF I M) = (\<Union>J\<in>I. A -` y \<inter> space (PiF {J} M))" 685 by (auto simp: space_PiF) 686 also have "\<dots> \<in> sets (PiF I M)" 687 proof (rule sets.finite_UN) 688 show "finite I" by fact 689 fix J assume "J \<in> I" 690 with assms have "finite J" by simp 691 show "A -` y \<inter> space (PiF {J} M) \<in> sets (PiF I M)" 692 by (rule singleton_subspace_set_in_sets[OF measurable_sets[OF assms(3)]]) fact+ 693 qed 694 finally show "A -` y \<inter> space (PiF I M) \<in> sets (PiF I M)" . 695next 696 fix x assume "x \<in> space (PiF I M)" thus "A x \<in> space N" 697 using MN[of "domain x"] 698 by (auto simp: space_PiF measurable_space Pi'_def) 699qed 700 701lemma countable_finite_comprehension: 702 fixes f :: "'a::countable set \<Rightarrow> _" 703 assumes "\<And>s. P s \<Longrightarrow> finite s" 704 assumes "\<And>s. P s \<Longrightarrow> f s \<in> sets M" 705 shows "\<Union>{f s|s. P s} \<in> sets M" 706proof - 707 have "\<Union>{f s|s. P s} = (\<Union>n::nat. let s = set (from_nat n) in if P s then f s else {})" 708 proof safe 709 fix x X s assume *: "x \<in> f s" "P s" 710 with assms obtain l where "s = set l" using finite_list by blast 711 with * show "x \<in> (\<Union>n. let s = set (from_nat n) in if P s then f s else {})" using \<open>P s\<close> 712 by (auto intro!: exI[where x="to_nat l"]) 713 next 714 fix x n assume "x \<in> (let s = set (from_nat n) in if P s then f s else {})" 715 thus "x \<in> \<Union>{f s|s. P s}" using assms by (auto simp: Let_def split: if_split_asm) 716 qed 717 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 718 also have "\<dots> \<in> sets M" using assms by (auto simp: Let_def) 719 finally show ?thesis . 720qed 721 722lemma space_subset_in_sets: 723 fixes J::"'a::countable set set" 724 assumes "J \<subseteq> I" 725 assumes "\<And>j. j \<in> J \<Longrightarrow> finite j" 726 shows "space (PiF J M) \<in> sets (PiF I M)" 727proof - 728 have "space (PiF J M) = \<Union>{space (PiF {j} M)|j. j \<in> J}" 729 unfolding space_PiF by blast 730 also have "\<dots> \<in> sets (PiF I M)" using assms 731 by (intro countable_finite_comprehension) (auto simp: singleton_space_subset_in_sets) 732 finally show ?thesis . 733qed 734 735lemma subspace_set_in_sets: 736 fixes J::"'a::countable set set" 737 assumes A: "A \<in> sets (PiF J M)" 738 assumes "J \<subseteq> I" 739 assumes "\<And>j. j \<in> J \<Longrightarrow> finite j" 740 shows "A \<in> sets (PiF I M)" 741 using A[unfolded sets_PiF] 742 apply (induct A) 743 unfolding sets_PiF[symmetric] unfolding space_PiF[symmetric] 744 using assms 745 by (auto intro: in_sets_PiFI intro!: space_subset_in_sets) 746 747lemma countable_measurable_PiFI: 748 fixes I::"'a::countable set set" 749 assumes MN: "\<And>J. J \<in> I \<Longrightarrow> finite J \<Longrightarrow> A \<in> measurable (PiF {J} M) N" 750 shows "A \<in> measurable (PiF I M) N" 751 unfolding measurable_def 752proof safe 753 fix y assume "y \<in> sets N" 754 have "A -` y = (\<Union>{A -` y \<inter> {x. domain x = J}|J. finite J})" by auto 755 { fix x::"'a \<Rightarrow>\<^sub>F 'b" 756 from finite_list[of "domain x"] obtain xs where "set xs = domain x" by auto 757 hence "\<exists>n. domain x = set (from_nat n)" 758 by (intro exI[where x="to_nat xs"]) auto } 759 hence "A -` y \<inter> space (PiF I M) = (\<Union>n. A -` y \<inter> space (PiF ({set (from_nat n)}\<inter>I) M))" 760 by (auto simp: space_PiF Pi'_def) 761 also have "\<dots> \<in> sets (PiF I M)" 762 apply (intro sets.Int sets.countable_nat_UN subsetI, safe) 763 apply (case_tac "set (from_nat i) \<in> I") 764 apply simp_all 765 apply (rule singleton_subspace_set_in_sets[OF measurable_sets[OF MN]]) 766 using assms \<open>y \<in> sets N\<close> 767 apply (auto simp: space_PiF) 768 done 769 finally show "A -` y \<inter> space (PiF I M) \<in> sets (PiF I M)" . 770next 771 fix x assume "x \<in> space (PiF I M)" thus "A x \<in> space N" 772 using MN[of "domain x"] by (auto simp: space_PiF measurable_space Pi'_def) 773qed 774 775lemma measurable_PiF: 776 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))" 777 assumes S: "\<And>J S. J \<in> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> S i \<in> sets (M i)) \<Longrightarrow> 778 f -` (Pi' J S) \<inter> space N \<in> sets N" 779 shows "f \<in> measurable N (PiF I M)" 780 unfolding PiF_def 781 using PiF_gen_subset 782 apply (rule measurable_measure_of) 783 using f apply force 784 apply (insert S, auto) 785 done 786 787lemma restrict_sets_measurable: 788 assumes A: "A \<in> sets (PiF I M)" and "J \<subseteq> I" 789 shows "A \<inter> {m. domain m \<in> J} \<in> sets (PiF J M)" 790 using A[unfolded sets_PiF] 791proof (induct A) 792 case (Basic a) 793 then obtain K S where S: "a = Pi' K S" "K \<in> I" "(\<forall>i\<in>K. S i \<in> sets (M i))" 794 by auto 795 show ?case 796 proof cases 797 assume "K \<in> J" 798 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 799 by (auto intro!: exI[where x=K] exI[where x=S] simp: Pi'_def) 800 also have "\<dots> \<subseteq> sets (PiF J M)" unfolding sets_PiF by auto 801 finally show ?thesis . 802 next 803 assume "K \<notin> J" 804 hence "a \<inter> {m. domain m \<in> J} = {}" using S by (auto simp: Pi'_def) 805 also have "\<dots> \<in> sets (PiF J M)" by simp 806 finally show ?thesis . 807 qed 808next 809 case (Union a) 810 have "\<Union>(a ` UNIV) \<inter> {m. domain m \<in> J} = (\<Union>i. (a i \<inter> {m. domain m \<in> J}))" 811 by simp 812 also have "\<dots> \<in> sets (PiF J M)" using Union by (intro sets.countable_nat_UN) auto 813 finally show ?case . 814next 815 case (Compl a) 816 have "(space (PiF I M) - a) \<inter> {m. domain m \<in> J} = (space (PiF J M) - (a \<inter> {m. domain m \<in> J}))" 817 using \<open>J \<subseteq> I\<close> by (auto simp: space_PiF Pi'_def) 818 also have "\<dots> \<in> sets (PiF J M)" using Compl by auto 819 finally show ?case by (simp add: space_PiF) 820qed simp 821 822lemma measurable_finmap_of: 823 assumes f: "\<And>i. (\<exists>x \<in> space N. i \<in> J x) \<Longrightarrow> (\<lambda>x. f x i) \<in> measurable N (M i)" 824 assumes J: "\<And>x. x \<in> space N \<Longrightarrow> J x \<in> I" "\<And>x. x \<in> space N \<Longrightarrow> finite (J x)" 825 assumes JN: "\<And>S. {x. J x = S} \<inter> space N \<in> sets N" 826 shows "(\<lambda>x. finmap_of (J x) (f x)) \<in> measurable N (PiF I M)" 827proof (rule measurable_PiF) 828 fix x assume "x \<in> space N" 829 with J[of x] measurable_space[OF f] 830 show "domain (finmap_of (J x) (f x)) \<in> I \<and> 831 (\<forall>i\<in>domain (finmap_of (J x) (f x)). (finmap_of (J x) (f x)) i \<in> space (M i))" 832 by auto 833next 834 fix K S assume "K \<in> I" and *: "\<And>i. i \<in> K \<Longrightarrow> S i \<in> sets (M i)" 835 with J have eq: "(\<lambda>x. finmap_of (J x) (f x)) -` Pi' K S \<inter> space N = 836 (if \<exists>x \<in> space N. K = J x \<and> finite K then if K = {} then {x \<in> space N. J x = K} 837 else (\<Inter>i\<in>K. (\<lambda>x. f x i) -` S i \<inter> {x \<in> space N. J x = K}) else {})" 838 by (auto simp: Pi'_def) 839 have r: "{x \<in> space N. J x = K} = space N \<inter> ({x. J x = K} \<inter> space N)" by auto 840 show "(\<lambda>x. finmap_of (J x) (f x)) -` Pi' K S \<inter> space N \<in> sets N" 841 unfolding eq r 842 apply (simp del: INT_simps add: ) 843 apply (intro conjI impI sets.finite_INT JN sets.Int[OF sets.top]) 844 apply simp apply assumption 845 apply (subst Int_assoc[symmetric]) 846 apply (rule sets.Int) 847 apply (intro measurable_sets[OF f] *) apply force apply assumption 848 apply (intro JN) 849 done 850qed 851 852lemma measurable_PiM_finmap_of: 853 assumes "finite J" 854 shows "finmap_of J \<in> measurable (Pi\<^sub>M J M) (PiF {J} M)" 855 apply (rule measurable_finmap_of) 856 apply (rule measurable_component_singleton) 857 apply simp 858 apply rule 859 apply (rule \<open>finite J\<close>) 860 apply simp 861 done 862 863lemma proj_measurable_singleton: 864 assumes "A \<in> sets (M i)" 865 shows "(\<lambda>x. (x)\<^sub>F i) -` A \<inter> space (PiF {I} M) \<in> sets (PiF {I} M)" 866proof cases 867 assume "i \<in> I" 868 hence "(\<lambda>x. (x)\<^sub>F i) -` A \<inter> space (PiF {I} M) = 869 Pi' I (\<lambda>x. if x = i then A else space (M x))" 870 using sets.sets_into_space[OF ] \<open>A \<in> sets (M i)\<close> assms 871 by (auto simp: space_PiF Pi'_def) 872 thus ?thesis using assms \<open>A \<in> sets (M i)\<close> 873 by (intro in_sets_PiFI) auto 874next 875 assume "i \<notin> I" 876 hence "(\<lambda>x. (x)\<^sub>F i) -` A \<inter> space (PiF {I} M) = 877 (if undefined \<in> A then space (PiF {I} M) else {})" by (auto simp: space_PiF Pi'_def) 878 thus ?thesis by simp 879qed 880 881lemma measurable_proj_singleton: 882 assumes "i \<in> I" 883 shows "(\<lambda>x. (x)\<^sub>F i) \<in> measurable (PiF {I} M) (M i)" 884 by (unfold measurable_def, intro CollectI conjI ballI proj_measurable_singleton assms) 885 (insert \<open>i \<in> I\<close>, auto simp: space_PiF) 886 887lemma measurable_proj_countable: 888 fixes I::"'a::countable set set" 889 assumes "y \<in> space (M i)" 890 shows "(\<lambda>x. if i \<in> domain x then (x)\<^sub>F i else y) \<in> measurable (PiF I M) (M i)" 891proof (rule countable_measurable_PiFI) 892 fix J assume "J \<in> I" "finite J" 893 show "(\<lambda>x. if i \<in> domain x then x i else y) \<in> measurable (PiF {J} M) (M i)" 894 unfolding measurable_def 895 proof safe 896 fix z assume "z \<in> sets (M i)" 897 have "(\<lambda>x. if i \<in> domain x then x i else y) -` z \<inter> space (PiF {J} M) = 898 (\<lambda>x. if i \<in> J then (x)\<^sub>F i else y) -` z \<inter> space (PiF {J} M)" 899 by (auto simp: space_PiF Pi'_def) 900 also have "\<dots> \<in> sets (PiF {J} M)" using \<open>z \<in> sets (M i)\<close> \<open>finite J\<close> 901 by (cases "i \<in> J") (auto intro!: measurable_sets[OF measurable_proj_singleton]) 902 finally show "(\<lambda>x. if i \<in> domain x then x i else y) -` z \<inter> space (PiF {J} M) \<in> 903 sets (PiF {J} M)" . 904 qed (insert \<open>y \<in> space (M i)\<close>, auto simp: space_PiF Pi'_def) 905qed 906 907lemma measurable_restrict_proj: 908 assumes "J \<in> II" "finite J" 909 shows "finmap_of J \<in> measurable (PiM J M) (PiF II M)" 910 using assms 911 by (intro measurable_finmap_of measurable_component_singleton) auto 912 913lemma measurable_proj_PiM: 914 fixes J K ::"'a::countable set" and I::"'a set set" 915 assumes "finite J" "J \<in> I" 916 assumes "x \<in> space (PiM J M)" 917 shows "proj \<in> measurable (PiF {J} M) (PiM J M)" 918proof (rule measurable_PiM_single) 919 show "proj \<in> space (PiF {J} M) \<rightarrow> (\<Pi>\<^sub>E i \<in> J. space (M i))" 920 using assms by (auto simp add: space_PiM space_PiF extensional_def sets_PiF Pi'_def) 921next 922 fix A i assume A: "i \<in> J" "A \<in> sets (M i)" 923 show "{\<omega> \<in> space (PiF {J} M). (\<omega>)\<^sub>F i \<in> A} \<in> sets (PiF {J} M)" 924 proof 925 have "{\<omega> \<in> space (PiF {J} M). (\<omega>)\<^sub>F i \<in> A} = 926 (\<lambda>\<omega>. (\<omega>)\<^sub>F i) -` A \<inter> space (PiF {J} M)" by auto 927 also have "\<dots> \<in> sets (PiF {J} M)" 928 using assms A by (auto intro: measurable_sets[OF measurable_proj_singleton] simp: space_PiM) 929 finally show ?thesis . 930 qed simp 931qed 932 933lemma space_PiF_singleton_eq_product: 934 assumes "finite I" 935 shows "space (PiF {I} M) = (\<Pi>' i\<in>I. space (M i))" 936 by (auto simp: product_def space_PiF assms) 937 938text \<open>adapted from @{thm sets_PiM_single}\<close> 939 940lemma sets_PiF_single: 941 assumes "finite I" "I \<noteq> {}" 942 shows "sets (PiF {I} M) = 943 sigma_sets (\<Pi>' i\<in>I. space (M i)) 944 {{f\<in>\<Pi>' i\<in>I. space (M i). f i \<in> A} | i A. i \<in> I \<and> A \<in> sets (M i)}" 945 (is "_ = sigma_sets ?\<Omega> ?R") 946 unfolding sets_PiF_singleton 947proof (rule sigma_sets_eqI) 948 interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto 949 fix A assume "A \<in> {Pi' I X |X. X \<in> (\<Pi> j\<in>I. sets (M j))}" 950 then obtain X where X: "A = Pi' I X" "X \<in> (\<Pi> j\<in>I. sets (M j))" by auto 951 show "A \<in> sigma_sets ?\<Omega> ?R" 952 proof - 953 from \<open>I \<noteq> {}\<close> X have "A = (\<Inter>j\<in>I. {f\<in>space (PiF {I} M). f j \<in> X j})" 954 using sets.sets_into_space 955 by (auto simp: space_PiF product_def) blast 956 also have "\<dots> \<in> sigma_sets ?\<Omega> ?R" 957 using X \<open>I \<noteq> {}\<close> assms by (intro R.finite_INT) (auto simp: space_PiF) 958 finally show "A \<in> sigma_sets ?\<Omega> ?R" . 959 qed 960next 961 fix A assume "A \<in> ?R" 962 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)" 963 by auto 964 then have "A = (\<Pi>' j \<in> I. if j = i then B else space (M j))" 965 using sets.sets_into_space[OF A(3)] 966 apply (auto simp: Pi'_iff split: if_split_asm) 967 apply blast 968 done 969 also have "\<dots> \<in> sigma_sets ?\<Omega> {Pi' I X |X. X \<in> (\<Pi> j\<in>I. sets (M j))}" 970 using A 971 by (intro sigma_sets.Basic ) 972 (auto intro: exI[where x="\<lambda>j. if j = i then B else space (M j)"]) 973 finally show "A \<in> sigma_sets ?\<Omega> {Pi' I X |X. X \<in> (\<Pi> j\<in>I. sets (M j))}" . 974qed 975 976text \<open>adapted from @{thm PiE_cong}\<close> 977 978lemma Pi'_cong: 979 assumes "finite I" 980 assumes "\<And>i. i \<in> I \<Longrightarrow> f i = g i" 981 shows "Pi' I f = Pi' I g" 982using assms by (auto simp: Pi'_def) 983 984text \<open>adapted from @{thm Pi_UN}\<close> 985 986lemma Pi'_UN: 987 fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set" 988 assumes "finite I" 989 assumes mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i" 990 shows "(\<Union>n. Pi' I (A n)) = Pi' I (\<lambda>i. \<Union>n. A n i)" 991proof (intro set_eqI iffI) 992 fix f assume "f \<in> Pi' I (\<lambda>i. \<Union>n. A n i)" 993 then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i" "domain f = I" by (auto simp: \<open>finite I\<close> Pi'_def) 994 from bchoice[OF this(1)] obtain n where n: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> (A (n i) i)" by auto 995 obtain k where k: "\<And>i. i \<in> I \<Longrightarrow> n i \<le> k" 996 using \<open>finite I\<close> finite_nat_set_iff_bounded_le[of "n`I"] by auto 997 have "f \<in> Pi' I (\<lambda>i. A k i)" 998 proof 999 fix i assume "i \<in> I" 1000 from mono[OF this, of "n i" k] k[OF this] n[OF this] \<open>domain f = I\<close> \<open>i \<in> I\<close> 1001 show "f i \<in> A k i " by (auto simp: \<open>finite I\<close>) 1002 qed (simp add: \<open>domain f = I\<close> \<open>finite I\<close>) 1003 then show "f \<in> (\<Union>n. Pi' I (A n))" by auto 1004qed (auto simp: Pi'_def \<open>finite I\<close>) 1005 1006text \<open>adapted from @{thm sets_PiM_sigma}\<close> 1007 1008lemma sigma_fprod_algebra_sigma_eq: 1009 fixes E :: "'i \<Rightarrow> 'a set set" and S :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" 1010 assumes [simp]: "finite I" "I \<noteq> {}" 1011 and S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)" 1012 and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i" 1013 assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))" 1014 and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)" 1015 defines "P == { Pi' I F | F. \<forall>i\<in>I. F i \<in> E i }" 1016 shows "sets (PiF {I} M) = sigma_sets (space (PiF {I} M)) P" 1017proof 1018 let ?P = "sigma (space (Pi\<^sub>F {I} M)) P" 1019 from \<open>finite I\<close>[THEN ex_bij_betw_finite_nat] guess T .. 1020 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" 1021 by (auto simp add: bij_betw_def set_eq_iff image_iff the_inv_into_f_f simp del: \<open>finite I\<close>) 1022 have P_closed: "P \<subseteq> Pow (space (Pi\<^sub>F {I} M))" 1023 using E_closed by (auto simp: space_PiF P_def Pi'_iff subset_eq) 1024 then have space_P: "space ?P = (\<Pi>' i\<in>I. space (M i))" 1025 by (simp add: space_PiF) 1026 have "sets (PiF {I} M) = 1027 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)}" 1028 using sets_PiF_single[of I M] by (simp add: space_P) 1029 also have "\<dots> \<subseteq> sets (sigma (space (PiF {I} M)) P)" 1030 proof (safe intro!: sets.sigma_sets_subset) 1031 fix i A assume "i \<in> I" and A: "A \<in> sets (M i)" 1032 have "(\<lambda>x. (x)\<^sub>F i) \<in> measurable ?P (sigma (space (M i)) (E i))" 1033 proof (subst measurable_iff_measure_of) 1034 show "E i \<subseteq> Pow (space (M i))" using \<open>i \<in> I\<close> by fact 1035 from space_P \<open>i \<in> I\<close> show "(\<lambda>x. (x)\<^sub>F i) \<in> space ?P \<rightarrow> space (M i)" 1036 by auto 1037 show "\<forall>A\<in>E i. (\<lambda>x. (x)\<^sub>F i) -` A \<inter> space ?P \<in> sets ?P" 1038 proof 1039 fix A assume A: "A \<in> E i" 1040 from T have *: "\<exists>xs. length xs = card I 1041 \<and> (\<forall>j\<in>I. (g)\<^sub>F j \<in> (if i = j then A else S j (xs ! T j)))" 1042 if "domain g = I" and "\<forall>j\<in>I. (g)\<^sub>F j \<in> (if i = j then A else S j (f j))" for g f 1043 using that by (auto intro!: exI [of _ "map (\<lambda>n. f (the_inv_into I T n)) [0..<card I]"]) 1044 from A have "(\<lambda>x. (x)\<^sub>F i) -` A \<inter> space ?P = (\<Pi>' j\<in>I. if i = j then A else space (M j))" 1045 using E_closed \<open>i \<in> I\<close> by (auto simp: space_P Pi_iff subset_eq split: if_split_asm) 1046 also have "\<dots> = (\<Pi>' j\<in>I. \<Union>n. if i = j then A else S j n)" 1047 by (intro Pi'_cong) (simp_all add: S_union) 1048 also have "\<dots> = {g. domain g = I \<and> (\<exists>f. \<forall>j\<in>I. (g)\<^sub>F j \<in> (if i = j then A else S j (f j)))}" 1049 by (rule set_eqI) (simp del: if_image_distrib add: Pi'_iff bchoice_iff) 1050 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))" 1051 using * by (auto simp add: Pi'_iff split del: if_split) 1052 also have "\<dots> \<in> sets ?P" 1053 proof (safe intro!: sets.countable_UN) 1054 fix xs show "(\<Pi>' j\<in>I. if i = j then A else S j (xs ! T j)) \<in> sets ?P" 1055 using A S_in_E 1056 by (simp add: P_closed) 1057 (auto simp: P_def subset_eq intro!: exI[of _ "\<lambda>j. if i = j then A else S j (xs ! T j)"]) 1058 qed 1059 finally show "(\<lambda>x. (x)\<^sub>F i) -` A \<inter> space ?P \<in> sets ?P" 1060 using P_closed by simp 1061 qed 1062 qed 1063 from measurable_sets[OF this, of A] A \<open>i \<in> I\<close> E_closed 1064 have "(\<lambda>x. (x)\<^sub>F i) -` A \<inter> space ?P \<in> sets ?P" 1065 by (simp add: E_generates) 1066 also have "(\<lambda>x. (x)\<^sub>F i) -` A \<inter> space ?P = {f \<in> \<Pi>' i\<in>I. space (M i). f i \<in> A}" 1067 using P_closed by (auto simp: space_PiF) 1068 finally show "\<dots> \<in> sets ?P" . 1069 qed 1070 finally show "sets (PiF {I} M) \<subseteq> sigma_sets (space (PiF {I} M)) P" 1071 by (simp add: P_closed) 1072 show "sigma_sets (space (PiF {I} M)) P \<subseteq> sets (PiF {I} M)" 1073 using \<open>finite I\<close> \<open>I \<noteq> {}\<close> 1074 by (auto intro!: sets.sigma_sets_subset product_in_sets_PiFI simp: E_generates P_def) 1075qed 1076 1077lemma product_open_generates_sets_PiF_single: 1078 assumes "I \<noteq> {}" 1079 assumes [simp]: "finite I" 1080 shows "sets (PiF {I} (\<lambda>_. borel::'b::second_countable_topology measure)) = 1081 sigma_sets (space (PiF {I} (\<lambda>_. borel))) {Pi' I F |F. (\<forall>i\<in>I. F i \<in> Collect open)}" 1082proof - 1083 from open_countable_basisE[OF open_UNIV] guess S::"'b set set" . note S = this 1084 show ?thesis 1085 proof (rule sigma_fprod_algebra_sigma_eq) 1086 show "finite I" by simp 1087 show "I \<noteq> {}" by fact 1088 define S' where "S' = from_nat_into S" 1089 show "(\<Union>j. S' j) = space borel" 1090 using S 1091 apply (auto simp add: from_nat_into countable_basis_proj S'_def basis_proj_def) 1092 apply (metis (lifting, mono_tags) UNIV_I UnionE basis_proj_def countable_basis_proj countable_subset from_nat_into_surj) 1093 done 1094 show "range S' \<subseteq> Collect open" 1095 using S 1096 apply (auto simp add: from_nat_into countable_basis_proj S'_def) 1097 apply (metis UNIV_not_empty Union_empty from_nat_into subsetD topological_basis_open[OF basis_proj] basis_proj_def) 1098 done 1099 show "Collect open \<subseteq> Pow (space borel)" by simp 1100 show "sets borel = sigma_sets (space borel) (Collect open)" 1101 by (simp add: borel_def) 1102 qed 1103qed 1104 1105lemma finmap_UNIV[simp]: "(\<Union>J\<in>Collect finite. \<Pi>' j\<in>J. UNIV) = UNIV" by auto 1106 1107lemma borel_eq_PiF_borel: 1108 shows "(borel :: ('i::countable \<Rightarrow>\<^sub>F 'a::polish_space) measure) = 1109 PiF (Collect finite) (\<lambda>_. borel :: 'a measure)" 1110 unfolding borel_def PiF_def 1111proof (rule measure_eqI, clarsimp, rule sigma_sets_eqI) 1112 fix a::"('i \<Rightarrow>\<^sub>F 'a) set" assume "a \<in> Collect open" hence "open a" by simp 1113 then obtain B' where B': "B'\<subseteq>basis_finmap" "a = \<Union>B'" 1114 using finmap_topological_basis by (force simp add: topological_basis_def) 1115 have "a \<in> sigma UNIV {Pi' J X |X J. finite J \<and> X \<in> J \<rightarrow> sigma_sets UNIV (Collect open)}" 1116 unfolding \<open>a = \<Union>B'\<close> 1117 proof (rule sets.countable_Union) 1118 from B' countable_basis_finmap show "countable B'" by (metis countable_subset) 1119 next 1120 show "B' \<subseteq> sets (sigma UNIV 1121 {Pi' J X |X J. finite J \<and> X \<in> J \<rightarrow> sigma_sets UNIV (Collect open)})" (is "_ \<subseteq> sets ?s") 1122 proof 1123 fix x assume "x \<in> B'" with B' have "x \<in> basis_finmap" by auto 1124 then obtain J X where "x = Pi' J X" "finite J" "X \<in> J \<rightarrow> sigma_sets UNIV (Collect open)" 1125 by (auto simp: basis_finmap_def topological_basis_open[OF basis_proj]) 1126 thus "x \<in> sets ?s" by auto 1127 qed 1128 qed 1129 thus "a \<in> sigma_sets UNIV {Pi' J X |X J. finite J \<and> X \<in> J \<rightarrow> sigma_sets UNIV (Collect open)}" 1130 by simp 1131next 1132 fix b::"('i \<Rightarrow>\<^sub>F 'a) set" 1133 assume "b \<in> {Pi' J X |X J. finite J \<and> X \<in> J \<rightarrow> sigma_sets UNIV (Collect open)}" 1134 hence b': "b \<in> sets (Pi\<^sub>F (Collect finite) (\<lambda>_. borel))" by (auto simp: sets_PiF borel_def) 1135 let ?b = "\<lambda>J. b \<inter> {x. domain x = J}" 1136 have "b = \<Union>((\<lambda>J. ?b J) ` Collect finite)" by auto 1137 also have "\<dots> \<in> sets borel" 1138 proof (rule sets.countable_Union, safe) 1139 fix J::"'i set" assume "finite J" 1140 { assume ef: "J = {}" 1141 have "?b J \<in> sets borel" 1142 proof cases 1143 assume "?b J \<noteq> {}" 1144 then obtain f where "f \<in> b" "domain f = {}" using ef by auto 1145 hence "?b J = {f}" using \<open>J = {}\<close> 1146 by (auto simp: finmap_eq_iff) 1147 also have "{f} \<in> sets borel" by simp 1148 finally show ?thesis . 1149 qed simp 1150 } moreover { 1151 assume "J \<noteq> ({}::'i set)" 1152 have "(?b J) = b \<inter> {m. domain m \<in> {J}}" by auto 1153 also have "\<dots> \<in> sets (PiF {J} (\<lambda>_. borel))" 1154 using b' by (rule restrict_sets_measurable) (auto simp: \<open>finite J\<close>) 1155 also have "\<dots> = sigma_sets (space (PiF {J} (\<lambda>_. borel))) 1156 {Pi' (J) F |F. (\<forall>j\<in>J. F j \<in> Collect open)}" 1157 (is "_ = sigma_sets _ ?P") 1158 by (rule product_open_generates_sets_PiF_single[OF \<open>J \<noteq> {}\<close> \<open>finite J\<close>]) 1159 also have "\<dots> \<subseteq> sigma_sets UNIV (Collect open)" 1160 by (intro sigma_sets_mono'') (auto intro!: open_Pi'I simp: space_PiF) 1161 finally have "(?b J) \<in> sets borel" by (simp add: borel_def) 1162 } ultimately show "(?b J) \<in> sets borel" by blast 1163 qed (simp add: countable_Collect_finite) 1164 finally show "b \<in> sigma_sets UNIV (Collect open)" by (simp add: borel_def) 1165qed (simp add: emeasure_sigma borel_def PiF_def) 1166 1167subsection \<open>Isomorphism between Functions and Finite Maps\<close> 1168 1169lemma measurable_finmap_compose: 1170 shows "(\<lambda>m. compose J m f) \<in> measurable (PiM (f ` J) (\<lambda>_. M)) (PiM J (\<lambda>_. M))" 1171 unfolding compose_def by measurable 1172 1173lemma measurable_compose_inv: 1174 assumes inj: "\<And>j. j \<in> J \<Longrightarrow> f' (f j) = j" 1175 shows "(\<lambda>m. compose (f ` J) m f') \<in> measurable (PiM J (\<lambda>_. M)) (PiM (f ` J) (\<lambda>_. M))" 1176 unfolding compose_def by (rule measurable_restrict) (auto simp: inj) 1177 1178locale function_to_finmap = 1179 fixes J::"'a set" and f :: "'a \<Rightarrow> 'b::countable" and f' 1180 assumes [simp]: "finite J" 1181 assumes inv: "i \<in> J \<Longrightarrow> f' (f i) = i" 1182begin 1183 1184text \<open>to measure finmaps\<close> 1185 1186definition "fm = (finmap_of (f ` J)) o (\<lambda>g. compose (f ` J) g f')" 1187 1188lemma domain_fm[simp]: "domain (fm x) = f ` J" 1189 unfolding fm_def by simp 1190 1191lemma fm_restrict[simp]: "fm (restrict y J) = fm y" 1192 unfolding fm_def by (auto simp: compose_def inv intro: restrict_ext) 1193 1194lemma fm_product: 1195 assumes "\<And>i. space (M i) = UNIV" 1196 shows "fm -` Pi' (f ` J) S \<inter> space (Pi\<^sub>M J M) = (\<Pi>\<^sub>E j \<in> J. S (f j))" 1197 using assms 1198 by (auto simp: inv fm_def compose_def space_PiM Pi'_def) 1199 1200lemma fm_measurable: 1201 assumes "f ` J \<in> N" 1202 shows "fm \<in> measurable (Pi\<^sub>M J (\<lambda>_. M)) (Pi\<^sub>F N (\<lambda>_. M))" 1203 unfolding fm_def 1204proof (rule measurable_comp, rule measurable_compose_inv) 1205 show "finmap_of (f ` J) \<in> measurable (Pi\<^sub>M (f ` J) (\<lambda>_. M)) (PiF N (\<lambda>_. M)) " 1206 using assms by (intro measurable_finmap_of measurable_component_singleton) auto 1207qed (simp_all add: inv) 1208 1209lemma proj_fm: 1210 assumes "x \<in> J" 1211 shows "fm m (f x) = m x" 1212 using assms by (auto simp: fm_def compose_def o_def inv) 1213 1214lemma inj_on_compose_f': "inj_on (\<lambda>g. compose (f ` J) g f') (extensional J)" 1215proof (rule inj_on_inverseI) 1216 fix x::"'a \<Rightarrow> 'c" assume "x \<in> extensional J" 1217 thus "(\<lambda>x. compose J x f) (compose (f ` J) x f') = x" 1218 by (auto simp: compose_def inv extensional_def) 1219qed 1220 1221lemma inj_on_fm: 1222 assumes "\<And>i. space (M i) = UNIV" 1223 shows "inj_on fm (space (Pi\<^sub>M J M))" 1224 using assms 1225 apply (auto simp: fm_def space_PiM PiE_def) 1226 apply (rule comp_inj_on) 1227 apply (rule inj_on_compose_f') 1228 apply (rule finmap_of_inj_on_extensional_finite) 1229 apply simp 1230 apply (auto) 1231 done 1232 1233text \<open>to measure functions\<close> 1234 1235definition "mf = (\<lambda>g. compose J g f) o proj" 1236 1237lemma mf_fm: 1238 assumes "x \<in> space (Pi\<^sub>M J (\<lambda>_. M))" 1239 shows "mf (fm x) = x" 1240proof - 1241 have "mf (fm x) \<in> extensional J" 1242 by (auto simp: mf_def extensional_def compose_def) 1243 moreover 1244 have "x \<in> extensional J" using assms sets.sets_into_space 1245 by (force simp: space_PiM PiE_def) 1246 moreover 1247 { fix i assume "i \<in> J" 1248 hence "mf (fm x) i = x i" 1249 by (auto simp: inv mf_def compose_def fm_def) 1250 } 1251 ultimately 1252 show ?thesis by (rule extensionalityI) 1253qed 1254 1255lemma mf_measurable: 1256 assumes "space M = UNIV" 1257 shows "mf \<in> measurable (PiF {f ` J} (\<lambda>_. M)) (PiM J (\<lambda>_. M))" 1258 unfolding mf_def 1259proof (rule measurable_comp, rule measurable_proj_PiM) 1260 show "(\<lambda>g. compose J g f) \<in> measurable (Pi\<^sub>M (f ` J) (\<lambda>x. M)) (Pi\<^sub>M J (\<lambda>_. M))" 1261 by (rule measurable_finmap_compose) 1262qed (auto simp add: space_PiM extensional_def assms) 1263 1264lemma fm_image_measurable: 1265 assumes "space M = UNIV" 1266 assumes "X \<in> sets (Pi\<^sub>M J (\<lambda>_. M))" 1267 shows "fm ` X \<in> sets (PiF {f ` J} (\<lambda>_. M))" 1268proof - 1269 have "fm ` X = (mf) -` X \<inter> space (PiF {f ` J} (\<lambda>_. M))" 1270 proof safe 1271 fix x assume "x \<in> X" 1272 with mf_fm[of x] sets.sets_into_space[OF assms(2)] show "fm x \<in> mf -` X" by auto 1273 show "fm x \<in> space (PiF {f ` J} (\<lambda>_. M))" by (simp add: space_PiF assms) 1274 next 1275 fix y x 1276 assume x: "mf y \<in> X" 1277 assume y: "y \<in> space (PiF {f ` J} (\<lambda>_. M))" 1278 thus "y \<in> fm ` X" 1279 by (intro image_eqI[OF _ x], unfold finmap_eq_iff) 1280 (auto simp: space_PiF fm_def mf_def compose_def inv Pi'_def) 1281 qed 1282 also have "\<dots> \<in> sets (PiF {f ` J} (\<lambda>_. M))" 1283 using assms 1284 by (intro measurable_sets[OF mf_measurable]) auto 1285 finally show ?thesis . 1286qed 1287 1288lemma fm_image_measurable_finite: 1289 assumes "space M = UNIV" 1290 assumes "X \<in> sets (Pi\<^sub>M J (\<lambda>_. M::'c measure))" 1291 shows "fm ` X \<in> sets (PiF (Collect finite) (\<lambda>_. M::'c measure))" 1292 using fm_image_measurable[OF assms] 1293 by (rule subspace_set_in_sets) (auto simp: finite_subset) 1294 1295text \<open>measure on finmaps\<close> 1296 1297definition "mapmeasure M N = distr M (PiF (Collect finite) N) (fm)" 1298 1299lemma sets_mapmeasure[simp]: "sets (mapmeasure M N) = sets (PiF (Collect finite) N)" 1300 unfolding mapmeasure_def by simp 1301 1302lemma space_mapmeasure[simp]: "space (mapmeasure M N) = space (PiF (Collect finite) N)" 1303 unfolding mapmeasure_def by simp 1304 1305lemma mapmeasure_PiF: 1306 assumes s1: "space M = space (Pi\<^sub>M J (\<lambda>_. N))" 1307 assumes s2: "sets M = sets (Pi\<^sub>M J (\<lambda>_. N))" 1308 assumes "space N = UNIV" 1309 assumes "X \<in> sets (PiF (Collect finite) (\<lambda>_. N))" 1310 shows "emeasure (mapmeasure M (\<lambda>_. N)) X = emeasure M ((fm -` X \<inter> extensional J))" 1311 using assms 1312 by (auto simp: measurable_cong_sets[OF s2 refl] mapmeasure_def emeasure_distr 1313 fm_measurable space_PiM PiE_def) 1314 1315lemma mapmeasure_PiM: 1316 fixes N::"'c measure" 1317 assumes s1: "space M = space (Pi\<^sub>M J (\<lambda>_. N))" 1318 assumes s2: "sets M = (Pi\<^sub>M J (\<lambda>_. N))" 1319 assumes N: "space N = UNIV" 1320 assumes X: "X \<in> sets M" 1321 shows "emeasure M X = emeasure (mapmeasure M (\<lambda>_. N)) (fm ` X)" 1322 unfolding mapmeasure_def 1323proof (subst emeasure_distr, subst measurable_cong_sets[OF s2 refl], rule fm_measurable) 1324 have "X \<subseteq> space (Pi\<^sub>M J (\<lambda>_. N))" using assms by (simp add: sets.sets_into_space) 1325 from assms inj_on_fm[of "\<lambda>_. N"] subsetD[OF this] have "fm -` fm ` X \<inter> space (Pi\<^sub>M J (\<lambda>_. N)) = X" 1326 by (auto simp: vimage_image_eq inj_on_def) 1327 thus "emeasure M X = emeasure M (fm -` fm ` X \<inter> space M)" using s1 1328 by simp 1329 show "fm ` X \<in> sets (PiF (Collect finite) (\<lambda>_. N))" 1330 by (rule fm_image_measurable_finite[OF N X[simplified s2]]) 1331qed simp 1332 1333end 1334 1335end 1336