1(* Title: HOL/Zorn.thy 2 Author: Jacques D. Fleuriot 3 Author: Tobias Nipkow, TUM 4 Author: Christian Sternagel, JAIST 5 6Zorn's Lemma (ported from Larry Paulson's Zorn.thy in ZF). 7The well-ordering theorem. 8*) 9 10section \<open>Zorn's Lemma\<close> 11 12theory Zorn 13 imports Order_Relation Hilbert_Choice 14begin 15 16subsection \<open>Zorn's Lemma for the Subset Relation\<close> 17 18subsubsection \<open>Results that do not require an order\<close> 19 20text \<open>Let \<open>P\<close> be a binary predicate on the set \<open>A\<close>.\<close> 21locale pred_on = 22 fixes A :: "'a set" 23 and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<sqsubset>" 50) 24begin 25 26abbreviation Peq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<sqsubseteq>" 50) 27 where "x \<sqsubseteq> y \<equiv> P\<^sup>=\<^sup>= x y" 28 29text \<open>A chain is a totally ordered subset of \<open>A\<close>.\<close> 30definition chain :: "'a set \<Rightarrow> bool" 31 where "chain C \<longleftrightarrow> C \<subseteq> A \<and> (\<forall>x\<in>C. \<forall>y\<in>C. x \<sqsubseteq> y \<or> y \<sqsubseteq> x)" 32 33text \<open> 34 We call a chain that is a proper superset of some set \<open>X\<close>, 35 but not necessarily a chain itself, a superchain of \<open>X\<close>. 36\<close> 37abbreviation superchain :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" (infix "<c" 50) 38 where "X <c C \<equiv> chain C \<and> X \<subset> C" 39 40text \<open>A maximal chain is a chain that does not have a superchain.\<close> 41definition maxchain :: "'a set \<Rightarrow> bool" 42 where "maxchain C \<longleftrightarrow> chain C \<and> (\<nexists>S. C <c S)" 43 44text \<open> 45 We define the successor of a set to be an arbitrary 46 superchain, if such exists, or the set itself, otherwise. 47\<close> 48definition suc :: "'a set \<Rightarrow> 'a set" 49 where "suc C = (if \<not> chain C \<or> maxchain C then C else (SOME D. C <c D))" 50 51lemma chainI [Pure.intro?]: "C \<subseteq> A \<Longrightarrow> (\<And>x y. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x) \<Longrightarrow> chain C" 52 unfolding chain_def by blast 53 54lemma chain_total: "chain C \<Longrightarrow> x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x" 55 by (simp add: chain_def) 56 57lemma not_chain_suc [simp]: "\<not> chain X \<Longrightarrow> suc X = X" 58 by (simp add: suc_def) 59 60lemma maxchain_suc [simp]: "maxchain X \<Longrightarrow> suc X = X" 61 by (simp add: suc_def) 62 63lemma suc_subset: "X \<subseteq> suc X" 64 by (auto simp: suc_def maxchain_def intro: someI2) 65 66lemma chain_empty [simp]: "chain {}" 67 by (auto simp: chain_def) 68 69lemma not_maxchain_Some: "chain C \<Longrightarrow> \<not> maxchain C \<Longrightarrow> C <c (SOME D. C <c D)" 70 by (rule someI_ex) (auto simp: maxchain_def) 71 72lemma suc_not_equals: "chain C \<Longrightarrow> \<not> maxchain C \<Longrightarrow> suc C \<noteq> C" 73 using not_maxchain_Some by (auto simp: suc_def) 74 75lemma subset_suc: 76 assumes "X \<subseteq> Y" 77 shows "X \<subseteq> suc Y" 78 using assms by (rule subset_trans) (rule suc_subset) 79 80text \<open> 81 We build a set \<^term>\<open>\<C>\<close> that is closed under applications 82 of \<^term>\<open>suc\<close> and contains the union of all its subsets. 83\<close> 84inductive_set suc_Union_closed ("\<C>") 85 where 86 suc: "X \<in> \<C> \<Longrightarrow> suc X \<in> \<C>" 87 | Union [unfolded Pow_iff]: "X \<in> Pow \<C> \<Longrightarrow> \<Union>X \<in> \<C>" 88 89text \<open> 90 Since the empty set as well as the set itself is a subset of 91 every set, \<^term>\<open>\<C>\<close> contains at least \<^term>\<open>{} \<in> \<C>\<close> and 92 \<^term>\<open>\<Union>\<C> \<in> \<C>\<close>. 93\<close> 94lemma suc_Union_closed_empty: "{} \<in> \<C>" 95 and suc_Union_closed_Union: "\<Union>\<C> \<in> \<C>" 96 using Union [of "{}"] and Union [of "\<C>"] by simp_all 97 98text \<open>Thus closure under \<^term>\<open>suc\<close> will hit a maximal chain 99 eventually, as is shown below.\<close> 100 101lemma suc_Union_closed_induct [consumes 1, case_names suc Union, induct pred: suc_Union_closed]: 102 assumes "X \<in> \<C>" 103 and "\<And>X. X \<in> \<C> \<Longrightarrow> Q X \<Longrightarrow> Q (suc X)" 104 and "\<And>X. X \<subseteq> \<C> \<Longrightarrow> \<forall>x\<in>X. Q x \<Longrightarrow> Q (\<Union>X)" 105 shows "Q X" 106 using assms by induct blast+ 107 108lemma suc_Union_closed_cases [consumes 1, case_names suc Union, cases pred: suc_Union_closed]: 109 assumes "X \<in> \<C>" 110 and "\<And>Y. X = suc Y \<Longrightarrow> Y \<in> \<C> \<Longrightarrow> Q" 111 and "\<And>Y. X = \<Union>Y \<Longrightarrow> Y \<subseteq> \<C> \<Longrightarrow> Q" 112 shows "Q" 113 using assms by cases simp_all 114 115text \<open>On chains, \<^term>\<open>suc\<close> yields a chain.\<close> 116lemma chain_suc: 117 assumes "chain X" 118 shows "chain (suc X)" 119 using assms 120 by (cases "\<not> chain X \<or> maxchain X") (force simp: suc_def dest: not_maxchain_Some)+ 121 122lemma chain_sucD: 123 assumes "chain X" 124 shows "suc X \<subseteq> A \<and> chain (suc X)" 125proof - 126 from \<open>chain X\<close> have *: "chain (suc X)" 127 by (rule chain_suc) 128 then have "suc X \<subseteq> A" 129 unfolding chain_def by blast 130 with * show ?thesis by blast 131qed 132 133lemma suc_Union_closed_total': 134 assumes "X \<in> \<C>" and "Y \<in> \<C>" 135 and *: "\<And>Z. Z \<in> \<C> \<Longrightarrow> Z \<subseteq> Y \<Longrightarrow> Z = Y \<or> suc Z \<subseteq> Y" 136 shows "X \<subseteq> Y \<or> suc Y \<subseteq> X" 137 using \<open>X \<in> \<C>\<close> 138proof induct 139 case (suc X) 140 with * show ?case by (blast del: subsetI intro: subset_suc) 141next 142 case Union 143 then show ?case by blast 144qed 145 146lemma suc_Union_closed_subsetD: 147 assumes "Y \<subseteq> X" and "X \<in> \<C>" and "Y \<in> \<C>" 148 shows "X = Y \<or> suc Y \<subseteq> X" 149 using assms(2,3,1) 150proof (induct arbitrary: Y) 151 case (suc X) 152 note * = \<open>\<And>Y. Y \<in> \<C> \<Longrightarrow> Y \<subseteq> X \<Longrightarrow> X = Y \<or> suc Y \<subseteq> X\<close> 153 with suc_Union_closed_total' [OF \<open>Y \<in> \<C>\<close> \<open>X \<in> \<C>\<close>] 154 have "Y \<subseteq> X \<or> suc X \<subseteq> Y" by blast 155 then show ?case 156 proof 157 assume "Y \<subseteq> X" 158 with * and \<open>Y \<in> \<C>\<close> have "X = Y \<or> suc Y \<subseteq> X" by blast 159 then show ?thesis 160 proof 161 assume "X = Y" 162 then show ?thesis by simp 163 next 164 assume "suc Y \<subseteq> X" 165 then have "suc Y \<subseteq> suc X" by (rule subset_suc) 166 then show ?thesis by simp 167 qed 168 next 169 assume "suc X \<subseteq> Y" 170 with \<open>Y \<subseteq> suc X\<close> show ?thesis by blast 171 qed 172next 173 case (Union X) 174 show ?case 175 proof (rule ccontr) 176 assume "\<not> ?thesis" 177 with \<open>Y \<subseteq> \<Union>X\<close> obtain x y z 178 where "\<not> suc Y \<subseteq> \<Union>X" 179 and "x \<in> X" and "y \<in> x" and "y \<notin> Y" 180 and "z \<in> suc Y" and "\<forall>x\<in>X. z \<notin> x" by blast 181 with \<open>X \<subseteq> \<C>\<close> have "x \<in> \<C>" by blast 182 from Union and \<open>x \<in> X\<close> have *: "\<And>y. y \<in> \<C> \<Longrightarrow> y \<subseteq> x \<Longrightarrow> x = y \<or> suc y \<subseteq> x" 183 by blast 184 with suc_Union_closed_total' [OF \<open>Y \<in> \<C>\<close> \<open>x \<in> \<C>\<close>] have "Y \<subseteq> x \<or> suc x \<subseteq> Y" 185 by blast 186 then show False 187 proof 188 assume "Y \<subseteq> x" 189 with * [OF \<open>Y \<in> \<C>\<close>] have "x = Y \<or> suc Y \<subseteq> x" by blast 190 then show False 191 proof 192 assume "x = Y" 193 with \<open>y \<in> x\<close> and \<open>y \<notin> Y\<close> show False by blast 194 next 195 assume "suc Y \<subseteq> x" 196 with \<open>x \<in> X\<close> have "suc Y \<subseteq> \<Union>X" by blast 197 with \<open>\<not> suc Y \<subseteq> \<Union>X\<close> show False by contradiction 198 qed 199 next 200 assume "suc x \<subseteq> Y" 201 moreover from suc_subset and \<open>y \<in> x\<close> have "y \<in> suc x" by blast 202 ultimately show False using \<open>y \<notin> Y\<close> by blast 203 qed 204 qed 205qed 206 207text \<open>The elements of \<^term>\<open>\<C>\<close> are totally ordered by the subset relation.\<close> 208lemma suc_Union_closed_total: 209 assumes "X \<in> \<C>" and "Y \<in> \<C>" 210 shows "X \<subseteq> Y \<or> Y \<subseteq> X" 211proof (cases "\<forall>Z\<in>\<C>. Z \<subseteq> Y \<longrightarrow> Z = Y \<or> suc Z \<subseteq> Y") 212 case True 213 with suc_Union_closed_total' [OF assms] 214 have "X \<subseteq> Y \<or> suc Y \<subseteq> X" by blast 215 with suc_subset [of Y] show ?thesis by blast 216next 217 case False 218 then obtain Z where "Z \<in> \<C>" and "Z \<subseteq> Y" and "Z \<noteq> Y" and "\<not> suc Z \<subseteq> Y" 219 by blast 220 with suc_Union_closed_subsetD and \<open>Y \<in> \<C>\<close> show ?thesis 221 by blast 222qed 223 224text \<open>Once we hit a fixed point w.r.t. \<^term>\<open>suc\<close>, all other elements 225 of \<^term>\<open>\<C>\<close> are subsets of this fixed point.\<close> 226lemma suc_Union_closed_suc: 227 assumes "X \<in> \<C>" and "Y \<in> \<C>" and "suc Y = Y" 228 shows "X \<subseteq> Y" 229 using \<open>X \<in> \<C>\<close> 230proof induct 231 case (suc X) 232 with \<open>Y \<in> \<C>\<close> and suc_Union_closed_subsetD have "X = Y \<or> suc X \<subseteq> Y" 233 by blast 234 then show ?case 235 by (auto simp: \<open>suc Y = Y\<close>) 236next 237 case Union 238 then show ?case by blast 239qed 240 241lemma eq_suc_Union: 242 assumes "X \<in> \<C>" 243 shows "suc X = X \<longleftrightarrow> X = \<Union>\<C>" 244 (is "?lhs \<longleftrightarrow> ?rhs") 245proof 246 assume ?lhs 247 then have "\<Union>\<C> \<subseteq> X" 248 by (rule suc_Union_closed_suc [OF suc_Union_closed_Union \<open>X \<in> \<C>\<close>]) 249 with \<open>X \<in> \<C>\<close> show ?rhs 250 by blast 251next 252 from \<open>X \<in> \<C>\<close> have "suc X \<in> \<C>" by (rule suc) 253 then have "suc X \<subseteq> \<Union>\<C>" by blast 254 moreover assume ?rhs 255 ultimately have "suc X \<subseteq> X" by simp 256 moreover have "X \<subseteq> suc X" by (rule suc_subset) 257 ultimately show ?lhs .. 258qed 259 260lemma suc_in_carrier: 261 assumes "X \<subseteq> A" 262 shows "suc X \<subseteq> A" 263 using assms 264 by (cases "\<not> chain X \<or> maxchain X") (auto dest: chain_sucD) 265 266lemma suc_Union_closed_in_carrier: 267 assumes "X \<in> \<C>" 268 shows "X \<subseteq> A" 269 using assms 270 by induct (auto dest: suc_in_carrier) 271 272text \<open>All elements of \<^term>\<open>\<C>\<close> are chains.\<close> 273lemma suc_Union_closed_chain: 274 assumes "X \<in> \<C>" 275 shows "chain X" 276 using assms 277proof induct 278 case (suc X) 279 then show ?case 280 using not_maxchain_Some by (simp add: suc_def) 281next 282 case (Union X) 283 then have "\<Union>X \<subseteq> A" 284 by (auto dest: suc_Union_closed_in_carrier) 285 moreover have "\<forall>x\<in>\<Union>X. \<forall>y\<in>\<Union>X. x \<sqsubseteq> y \<or> y \<sqsubseteq> x" 286 proof (intro ballI) 287 fix x y 288 assume "x \<in> \<Union>X" and "y \<in> \<Union>X" 289 then obtain u v where "x \<in> u" and "u \<in> X" and "y \<in> v" and "v \<in> X" 290 by blast 291 with Union have "u \<in> \<C>" and "v \<in> \<C>" and "chain u" and "chain v" 292 by blast+ 293 with suc_Union_closed_total have "u \<subseteq> v \<or> v \<subseteq> u" 294 by blast 295 then show "x \<sqsubseteq> y \<or> y \<sqsubseteq> x" 296 proof 297 assume "u \<subseteq> v" 298 from \<open>chain v\<close> show ?thesis 299 proof (rule chain_total) 300 show "y \<in> v" by fact 301 show "x \<in> v" using \<open>u \<subseteq> v\<close> and \<open>x \<in> u\<close> by blast 302 qed 303 next 304 assume "v \<subseteq> u" 305 from \<open>chain u\<close> show ?thesis 306 proof (rule chain_total) 307 show "x \<in> u" by fact 308 show "y \<in> u" using \<open>v \<subseteq> u\<close> and \<open>y \<in> v\<close> by blast 309 qed 310 qed 311 qed 312 ultimately show ?case unfolding chain_def .. 313qed 314 315subsubsection \<open>Hausdorff's Maximum Principle\<close> 316 317text \<open>There exists a maximal totally ordered subset of \<open>A\<close>. (Note that we do not 318 require \<open>A\<close> to be partially ordered.)\<close> 319 320theorem Hausdorff: "\<exists>C. maxchain C" 321proof - 322 let ?M = "\<Union>\<C>" 323 have "maxchain ?M" 324 proof (rule ccontr) 325 assume "\<not> ?thesis" 326 then have "suc ?M \<noteq> ?M" 327 using suc_not_equals and suc_Union_closed_chain [OF suc_Union_closed_Union] by simp 328 moreover have "suc ?M = ?M" 329 using eq_suc_Union [OF suc_Union_closed_Union] by simp 330 ultimately show False by contradiction 331 qed 332 then show ?thesis by blast 333qed 334 335text \<open>Make notation \<^term>\<open>\<C>\<close> available again.\<close> 336no_notation suc_Union_closed ("\<C>") 337 338lemma chain_extend: "chain C \<Longrightarrow> z \<in> A \<Longrightarrow> \<forall>x\<in>C. x \<sqsubseteq> z \<Longrightarrow> chain ({z} \<union> C)" 339 unfolding chain_def by blast 340 341lemma maxchain_imp_chain: "maxchain C \<Longrightarrow> chain C" 342 by (simp add: maxchain_def) 343 344end 345 346text \<open>Hide constant \<^const>\<open>pred_on.suc_Union_closed\<close>, which was just needed 347 for the proof of Hausforff's maximum principle.\<close> 348hide_const pred_on.suc_Union_closed 349 350lemma chain_mono: 351 assumes "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> P x y \<Longrightarrow> Q x y" 352 and "pred_on.chain A P C" 353 shows "pred_on.chain A Q C" 354 using assms unfolding pred_on.chain_def by blast 355 356 357subsubsection \<open>Results for the proper subset relation\<close> 358 359interpretation subset: pred_on "A" "(\<subset>)" for A . 360 361lemma subset_maxchain_max: 362 assumes "subset.maxchain A C" 363 and "X \<in> A" 364 and "\<Union>C \<subseteq> X" 365 shows "\<Union>C = X" 366proof (rule ccontr) 367 let ?C = "{X} \<union> C" 368 from \<open>subset.maxchain A C\<close> have "subset.chain A C" 369 and *: "\<And>S. subset.chain A S \<Longrightarrow> \<not> C \<subset> S" 370 by (auto simp: subset.maxchain_def) 371 moreover have "\<forall>x\<in>C. x \<subseteq> X" using \<open>\<Union>C \<subseteq> X\<close> by auto 372 ultimately have "subset.chain A ?C" 373 using subset.chain_extend [of A C X] and \<open>X \<in> A\<close> by auto 374 moreover assume **: "\<Union>C \<noteq> X" 375 moreover from ** have "C \<subset> ?C" using \<open>\<Union>C \<subseteq> X\<close> by auto 376 ultimately show False using * by blast 377qed 378 379lemma subset_chain_def: "\<And>\<A>. subset.chain \<A> \<C> = (\<C> \<subseteq> \<A> \<and> (\<forall>X\<in>\<C>. \<forall>Y\<in>\<C>. X \<subseteq> Y \<or> Y \<subseteq> X))" 380 by (auto simp: subset.chain_def) 381 382lemma subset_chain_insert: 383 "subset.chain \<A> (insert B \<B>) \<longleftrightarrow> B \<in> \<A> \<and> (\<forall>X\<in>\<B>. X \<subseteq> B \<or> B \<subseteq> X) \<and> subset.chain \<A> \<B>" 384 by (fastforce simp add: subset_chain_def) 385 386subsubsection \<open>Zorn's lemma\<close> 387 388text \<open>If every chain has an upper bound, then there is a maximal set.\<close> 389theorem subset_Zorn: 390 assumes "\<And>C. subset.chain A C \<Longrightarrow> \<exists>U\<in>A. \<forall>X\<in>C. X \<subseteq> U" 391 shows "\<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M" 392proof - 393 from subset.Hausdorff [of A] obtain M where "subset.maxchain A M" .. 394 then have "subset.chain A M" 395 by (rule subset.maxchain_imp_chain) 396 with assms obtain Y where "Y \<in> A" and "\<forall>X\<in>M. X \<subseteq> Y" 397 by blast 398 moreover have "\<forall>X\<in>A. Y \<subseteq> X \<longrightarrow> Y = X" 399 proof (intro ballI impI) 400 fix X 401 assume "X \<in> A" and "Y \<subseteq> X" 402 show "Y = X" 403 proof (rule ccontr) 404 assume "\<not> ?thesis" 405 with \<open>Y \<subseteq> X\<close> have "\<not> X \<subseteq> Y" by blast 406 from subset.chain_extend [OF \<open>subset.chain A M\<close> \<open>X \<in> A\<close>] and \<open>\<forall>X\<in>M. X \<subseteq> Y\<close> 407 have "subset.chain A ({X} \<union> M)" 408 using \<open>Y \<subseteq> X\<close> by auto 409 moreover have "M \<subset> {X} \<union> M" 410 using \<open>\<forall>X\<in>M. X \<subseteq> Y\<close> and \<open>\<not> X \<subseteq> Y\<close> by auto 411 ultimately show False 412 using \<open>subset.maxchain A M\<close> by (auto simp: subset.maxchain_def) 413 qed 414 qed 415 ultimately show ?thesis by blast 416qed 417 418text \<open>Alternative version of Zorn's lemma for the subset relation.\<close> 419lemma subset_Zorn': 420 assumes "\<And>C. subset.chain A C \<Longrightarrow> \<Union>C \<in> A" 421 shows "\<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M" 422proof - 423 from subset.Hausdorff [of A] obtain M where "subset.maxchain A M" .. 424 then have "subset.chain A M" 425 by (rule subset.maxchain_imp_chain) 426 with assms have "\<Union>M \<in> A" . 427 moreover have "\<forall>Z\<in>A. \<Union>M \<subseteq> Z \<longrightarrow> \<Union>M = Z" 428 proof (intro ballI impI) 429 fix Z 430 assume "Z \<in> A" and "\<Union>M \<subseteq> Z" 431 with subset_maxchain_max [OF \<open>subset.maxchain A M\<close>] 432 show "\<Union>M = Z" . 433 qed 434 ultimately show ?thesis by blast 435qed 436 437 438subsection \<open>Zorn's Lemma for Partial Orders\<close> 439 440text \<open>Relate old to new definitions.\<close> 441 442definition chain_subset :: "'a set set \<Rightarrow> bool" ("chain\<^sub>\<subseteq>") (* Define globally? In Set.thy? *) 443 where "chain\<^sub>\<subseteq> C \<longleftrightarrow> (\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A)" 444 445definition chains :: "'a set set \<Rightarrow> 'a set set set" 446 where "chains A = {C. C \<subseteq> A \<and> chain\<^sub>\<subseteq> C}" 447 448definition Chains :: "('a \<times> 'a) set \<Rightarrow> 'a set set" (* Define globally? In Relation.thy? *) 449 where "Chains r = {C. \<forall>a\<in>C. \<forall>b\<in>C. (a, b) \<in> r \<or> (b, a) \<in> r}" 450 451lemma chains_extend: "c \<in> chains S \<Longrightarrow> z \<in> S \<Longrightarrow> \<forall>x \<in> c. x \<subseteq> z \<Longrightarrow> {z} \<union> c \<in> chains S" 452 for z :: "'a set" 453 unfolding chains_def chain_subset_def by blast 454 455lemma mono_Chains: "r \<subseteq> s \<Longrightarrow> Chains r \<subseteq> Chains s" 456 unfolding Chains_def by blast 457 458lemma chain_subset_alt_def: "chain\<^sub>\<subseteq> C = subset.chain UNIV C" 459 unfolding chain_subset_def subset.chain_def by fast 460 461lemma chains_alt_def: "chains A = {C. subset.chain A C}" 462 by (simp add: chains_def chain_subset_alt_def subset.chain_def) 463 464lemma Chains_subset: "Chains r \<subseteq> {C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C}" 465 by (force simp add: Chains_def pred_on.chain_def) 466 467lemma Chains_subset': 468 assumes "refl r" 469 shows "{C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C} \<subseteq> Chains r" 470 using assms 471 by (auto simp add: Chains_def pred_on.chain_def refl_on_def) 472 473lemma Chains_alt_def: 474 assumes "refl r" 475 shows "Chains r = {C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C}" 476 using assms Chains_subset Chains_subset' by blast 477 478lemma Chains_relation_of: 479 assumes "C \<in> Chains (relation_of P A)" shows "C \<subseteq> A" 480 using assms unfolding Chains_def relation_of_def by auto 481 482lemma pairwise_chain_Union: 483 assumes P: "\<And>S. S \<in> \<C> \<Longrightarrow> pairwise R S" and "chain\<^sub>\<subseteq> \<C>" 484 shows "pairwise R (\<Union>\<C>)" 485 using \<open>chain\<^sub>\<subseteq> \<C>\<close> unfolding pairwise_def chain_subset_def 486 by (blast intro: P [unfolded pairwise_def, rule_format]) 487 488lemma Zorn_Lemma: "\<forall>C\<in>chains A. \<Union>C \<in> A \<Longrightarrow> \<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M" 489 using subset_Zorn' [of A] by (force simp: chains_alt_def) 490 491lemma Zorn_Lemma2: "\<forall>C\<in>chains A. \<exists>U\<in>A. \<forall>X\<in>C. X \<subseteq> U \<Longrightarrow> \<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M" 492 using subset_Zorn [of A] by (auto simp: chains_alt_def) 493 494subsection \<open>Other variants of Zorn's Lemma\<close> 495 496lemma chainsD: "c \<in> chains S \<Longrightarrow> x \<in> c \<Longrightarrow> y \<in> c \<Longrightarrow> x \<subseteq> y \<or> y \<subseteq> x" 497 unfolding chains_def chain_subset_def by blast 498 499lemma chainsD2: "c \<in> chains S \<Longrightarrow> c \<subseteq> S" 500 unfolding chains_def by blast 501 502lemma Zorns_po_lemma: 503 assumes po: "Partial_order r" 504 and u: "\<And>C. C \<in> Chains r \<Longrightarrow> \<exists>u\<in>Field r. \<forall>a\<in>C. (a, u) \<in> r" 505 shows "\<exists>m\<in>Field r. \<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m" 506proof - 507 have "Preorder r" 508 using po by (simp add: partial_order_on_def) 509 txt \<open>Mirror \<open>r\<close> in the set of subsets below (wrt \<open>r\<close>) elements of \<open>A\<close>.\<close> 510 let ?B = "\<lambda>x. r\<inverse> `` {x}" 511 let ?S = "?B ` Field r" 512 have "\<exists>u\<in>Field r. \<forall>A\<in>C. A \<subseteq> r\<inverse> `` {u}" (is "\<exists>u\<in>Field r. ?P u") 513 if 1: "C \<subseteq> ?S" and 2: "\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A" for C 514 proof - 515 let ?A = "{x\<in>Field r. \<exists>M\<in>C. M = ?B x}" 516 from 1 have "C = ?B ` ?A" by (auto simp: image_def) 517 have "?A \<in> Chains r" 518 proof (simp add: Chains_def, intro allI impI, elim conjE) 519 fix a b 520 assume "a \<in> Field r" and "?B a \<in> C" and "b \<in> Field r" and "?B b \<in> C" 521 with 2 have "?B a \<subseteq> ?B b \<or> ?B b \<subseteq> ?B a" by auto 522 then show "(a, b) \<in> r \<or> (b, a) \<in> r" 523 using \<open>Preorder r\<close> and \<open>a \<in> Field r\<close> and \<open>b \<in> Field r\<close> 524 by (simp add:subset_Image1_Image1_iff) 525 qed 526 then obtain u where uA: "u \<in> Field r" "\<forall>a\<in>?A. (a, u) \<in> r" 527 by (auto simp: dest: u) 528 have "?P u" 529 proof auto 530 fix a B assume aB: "B \<in> C" "a \<in> B" 531 with 1 obtain x where "x \<in> Field r" and "B = r\<inverse> `` {x}" by auto 532 then show "(a, u) \<in> r" 533 using uA and aB and \<open>Preorder r\<close> 534 unfolding preorder_on_def refl_on_def by simp (fast dest: transD) 535 qed 536 then show ?thesis 537 using \<open>u \<in> Field r\<close> by blast 538 qed 539 then have "\<forall>C\<in>chains ?S. \<exists>U\<in>?S. \<forall>A\<in>C. A \<subseteq> U" 540 by (auto simp: chains_def chain_subset_def) 541 from Zorn_Lemma2 [OF this] obtain m B 542 where "m \<in> Field r" 543 and "B = r\<inverse> `` {m}" 544 and "\<forall>x\<in>Field r. B \<subseteq> r\<inverse> `` {x} \<longrightarrow> r\<inverse> `` {x} = B" 545 by auto 546 then have "\<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m" 547 using po and \<open>Preorder r\<close> and \<open>m \<in> Field r\<close> 548 by (auto simp: subset_Image1_Image1_iff Partial_order_eq_Image1_Image1_iff) 549 then show ?thesis 550 using \<open>m \<in> Field r\<close> by blast 551qed 552 553lemma predicate_Zorn: 554 assumes po: "partial_order_on A (relation_of P A)" 555 and ch: "\<And>C. C \<in> Chains (relation_of P A) \<Longrightarrow> \<exists>u \<in> A. \<forall>a \<in> C. P a u" 556 shows "\<exists>m \<in> A. \<forall>a \<in> A. P m a \<longrightarrow> a = m" 557proof - 558 have "a \<in> A" if "C \<in> Chains (relation_of P A)" and "a \<in> C" for C a 559 using that unfolding Chains_def relation_of_def by auto 560 moreover have "(a, u) \<in> relation_of P A" if "a \<in> A" and "u \<in> A" and "P a u" for a u 561 unfolding relation_of_def using that by auto 562 ultimately have "\<exists>m\<in>A. \<forall>a\<in>A. (m, a) \<in> relation_of P A \<longrightarrow> a = m" 563 using Zorns_po_lemma[OF Partial_order_relation_ofI[OF po], rule_format] ch 564 unfolding Field_relation_of[OF partial_order_onD(1)[OF po]] by blast 565 then show ?thesis 566 by (auto simp: relation_of_def) 567qed 568 569lemma Union_in_chain: "\<lbrakk>finite \<B>; \<B> \<noteq> {}; subset.chain \<A> \<B>\<rbrakk> \<Longrightarrow> \<Union>\<B> \<in> \<B>" 570proof (induction \<B> rule: finite_induct) 571 case (insert B \<B>) 572 show ?case 573 proof (cases "\<B> = {}") 574 case False 575 then show ?thesis 576 using insert sup.absorb2 by (auto simp: subset_chain_insert dest!: bspec [where x="\<Union>\<B>"]) 577 qed auto 578qed simp 579 580lemma Inter_in_chain: "\<lbrakk>finite \<B>; \<B> \<noteq> {}; subset.chain \<A> \<B>\<rbrakk> \<Longrightarrow> \<Inter>\<B> \<in> \<B>" 581proof (induction \<B> rule: finite_induct) 582 case (insert B \<B>) 583 show ?case 584 proof (cases "\<B> = {}") 585 case False 586 then show ?thesis 587 using insert inf.absorb2 by (auto simp: subset_chain_insert dest!: bspec [where x="\<Inter>\<B>"]) 588 qed auto 589qed simp 590 591lemma finite_subset_Union_chain: 592 assumes "finite A" "A \<subseteq> \<Union>\<B>" "\<B> \<noteq> {}" and sub: "subset.chain \<A> \<B>" 593 obtains B where "B \<in> \<B>" "A \<subseteq> B" 594proof - 595 obtain \<F> where \<F>: "finite \<F>" "\<F> \<subseteq> \<B>" "A \<subseteq> \<Union>\<F>" 596 using assms by (auto intro: finite_subset_Union) 597 show thesis 598 proof (cases "\<F> = {}") 599 case True 600 then show ?thesis 601 using \<open>A \<subseteq> \<Union>\<F>\<close> \<open>\<B> \<noteq> {}\<close> that by fastforce 602 next 603 case False 604 show ?thesis 605 proof 606 show "\<Union>\<F> \<in> \<B>" 607 using sub \<open>\<F> \<subseteq> \<B>\<close> \<open>finite \<F>\<close> 608 by (simp add: Union_in_chain False subset.chain_def subset_iff) 609 show "A \<subseteq> \<Union>\<F>" 610 using \<open>A \<subseteq> \<Union>\<F>\<close> by blast 611 qed 612 qed 613qed 614 615lemma subset_Zorn_nonempty: 616 assumes "\<A> \<noteq> {}" and ch: "\<And>\<C>. \<lbrakk>\<C>\<noteq>{}; subset.chain \<A> \<C>\<rbrakk> \<Longrightarrow> \<Union>\<C> \<in> \<A>" 617 shows "\<exists>M\<in>\<A>. \<forall>X\<in>\<A>. M \<subseteq> X \<longrightarrow> X = M" 618proof (rule subset_Zorn) 619 show "\<exists>U\<in>\<A>. \<forall>X\<in>\<C>. X \<subseteq> U" if "subset.chain \<A> \<C>" for \<C> 620 proof (cases "\<C> = {}") 621 case True 622 then show ?thesis 623 using \<open>\<A> \<noteq> {}\<close> by blast 624 next 625 case False 626 show ?thesis 627 by (blast intro!: ch False that Union_upper) 628 qed 629qed 630 631subsection \<open>The Well Ordering Theorem\<close> 632 633(* The initial segment of a relation appears generally useful. 634 Move to Relation.thy? 635 Definition correct/most general? 636 Naming? 637*) 638definition init_seg_of :: "(('a \<times> 'a) set \<times> ('a \<times> 'a) set) set" 639 where "init_seg_of = {(r, s). r \<subseteq> s \<and> (\<forall>a b c. (a, b) \<in> s \<and> (b, c) \<in> r \<longrightarrow> (a, b) \<in> r)}" 640 641abbreviation initial_segment_of_syntax :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool" 642 (infix "initial'_segment'_of" 55) 643 where "r initial_segment_of s \<equiv> (r, s) \<in> init_seg_of" 644 645lemma refl_on_init_seg_of [simp]: "r initial_segment_of r" 646 by (simp add: init_seg_of_def) 647 648lemma trans_init_seg_of: 649 "r initial_segment_of s \<Longrightarrow> s initial_segment_of t \<Longrightarrow> r initial_segment_of t" 650 by (simp (no_asm_use) add: init_seg_of_def) blast 651 652lemma antisym_init_seg_of: "r initial_segment_of s \<Longrightarrow> s initial_segment_of r \<Longrightarrow> r = s" 653 unfolding init_seg_of_def by safe 654 655lemma Chains_init_seg_of_Union: "R \<in> Chains init_seg_of \<Longrightarrow> r\<in>R \<Longrightarrow> r initial_segment_of \<Union>R" 656 by (auto simp: init_seg_of_def Ball_def Chains_def) blast 657 658lemma chain_subset_trans_Union: 659 assumes "chain\<^sub>\<subseteq> R" "\<forall>r\<in>R. trans r" 660 shows "trans (\<Union>R)" 661proof (intro transI, elim UnionE) 662 fix S1 S2 :: "'a rel" and x y z :: 'a 663 assume "S1 \<in> R" "S2 \<in> R" 664 with assms(1) have "S1 \<subseteq> S2 \<or> S2 \<subseteq> S1" 665 unfolding chain_subset_def by blast 666 moreover assume "(x, y) \<in> S1" "(y, z) \<in> S2" 667 ultimately have "((x, y) \<in> S1 \<and> (y, z) \<in> S1) \<or> ((x, y) \<in> S2 \<and> (y, z) \<in> S2)" 668 by blast 669 with \<open>S1 \<in> R\<close> \<open>S2 \<in> R\<close> assms(2) show "(x, z) \<in> \<Union>R" 670 by (auto elim: transE) 671qed 672 673lemma chain_subset_antisym_Union: 674 assumes "chain\<^sub>\<subseteq> R" "\<forall>r\<in>R. antisym r" 675 shows "antisym (\<Union>R)" 676proof (intro antisymI, elim UnionE) 677 fix S1 S2 :: "'a rel" and x y :: 'a 678 assume "S1 \<in> R" "S2 \<in> R" 679 with assms(1) have "S1 \<subseteq> S2 \<or> S2 \<subseteq> S1" 680 unfolding chain_subset_def by blast 681 moreover assume "(x, y) \<in> S1" "(y, x) \<in> S2" 682 ultimately have "((x, y) \<in> S1 \<and> (y, x) \<in> S1) \<or> ((x, y) \<in> S2 \<and> (y, x) \<in> S2)" 683 by blast 684 with \<open>S1 \<in> R\<close> \<open>S2 \<in> R\<close> assms(2) show "x = y" 685 unfolding antisym_def by auto 686qed 687 688lemma chain_subset_Total_Union: 689 assumes "chain\<^sub>\<subseteq> R" and "\<forall>r\<in>R. Total r" 690 shows "Total (\<Union>R)" 691proof (simp add: total_on_def Ball_def, auto del: disjCI) 692 fix r s a b 693 assume A: "r \<in> R" "s \<in> R" "a \<in> Field r" "b \<in> Field s" "a \<noteq> b" 694 from \<open>chain\<^sub>\<subseteq> R\<close> and \<open>r \<in> R\<close> and \<open>s \<in> R\<close> have "r \<subseteq> s \<or> s \<subseteq> r" 695 by (auto simp add: chain_subset_def) 696 then show "(\<exists>r\<in>R. (a, b) \<in> r) \<or> (\<exists>r\<in>R. (b, a) \<in> r)" 697 proof 698 assume "r \<subseteq> s" 699 then have "(a, b) \<in> s \<or> (b, a) \<in> s" 700 using assms(2) A mono_Field[of r s] 701 by (auto simp add: total_on_def) 702 then show ?thesis 703 using \<open>s \<in> R\<close> by blast 704 next 705 assume "s \<subseteq> r" 706 then have "(a, b) \<in> r \<or> (b, a) \<in> r" 707 using assms(2) A mono_Field[of s r] 708 by (fastforce simp add: total_on_def) 709 then show ?thesis 710 using \<open>r \<in> R\<close> by blast 711 qed 712qed 713 714lemma wf_Union_wf_init_segs: 715 assumes "R \<in> Chains init_seg_of" 716 and "\<forall>r\<in>R. wf r" 717 shows "wf (\<Union>R)" 718proof (simp add: wf_iff_no_infinite_down_chain, rule ccontr, auto) 719 fix f 720 assume 1: "\<forall>i. \<exists>r\<in>R. (f (Suc i), f i) \<in> r" 721 then obtain r where "r \<in> R" and "(f (Suc 0), f 0) \<in> r" by auto 722 have "(f (Suc i), f i) \<in> r" for i 723 proof (induct i) 724 case 0 725 show ?case by fact 726 next 727 case (Suc i) 728 then obtain s where s: "s \<in> R" "(f (Suc (Suc i)), f(Suc i)) \<in> s" 729 using 1 by auto 730 then have "s initial_segment_of r \<or> r initial_segment_of s" 731 using assms(1) \<open>r \<in> R\<close> by (simp add: Chains_def) 732 with Suc s show ?case by (simp add: init_seg_of_def) blast 733 qed 734 then show False 735 using assms(2) and \<open>r \<in> R\<close> 736 by (simp add: wf_iff_no_infinite_down_chain) blast 737qed 738 739lemma initial_segment_of_Diff: "p initial_segment_of q \<Longrightarrow> p - s initial_segment_of q - s" 740 unfolding init_seg_of_def by blast 741 742lemma Chains_inits_DiffI: "R \<in> Chains init_seg_of \<Longrightarrow> {r - s |r. r \<in> R} \<in> Chains init_seg_of" 743 unfolding Chains_def by (blast intro: initial_segment_of_Diff) 744 745theorem well_ordering: "\<exists>r::'a rel. Well_order r \<and> Field r = UNIV" 746proof - 747\<comment> \<open>The initial segment relation on well-orders:\<close> 748 let ?WO = "{r::'a rel. Well_order r}" 749 define I where "I = init_seg_of \<inter> ?WO \<times> ?WO" 750 then have I_init: "I \<subseteq> init_seg_of" by simp 751 then have subch: "\<And>R. R \<in> Chains I \<Longrightarrow> chain\<^sub>\<subseteq> R" 752 unfolding init_seg_of_def chain_subset_def Chains_def by blast 753 have Chains_wo: "\<And>R r. R \<in> Chains I \<Longrightarrow> r \<in> R \<Longrightarrow> Well_order r" 754 by (simp add: Chains_def I_def) blast 755 have FI: "Field I = ?WO" 756 by (auto simp add: I_def init_seg_of_def Field_def) 757 then have 0: "Partial_order I" 758 by (auto simp: partial_order_on_def preorder_on_def antisym_def antisym_init_seg_of refl_on_def 759 trans_def I_def elim!: trans_init_seg_of) 760\<comment> \<open>\<open>I\<close>-chains have upper bounds in \<open>?WO\<close> wrt \<open>I\<close>: their Union\<close> 761 have "\<Union>R \<in> ?WO \<and> (\<forall>r\<in>R. (r, \<Union>R) \<in> I)" if "R \<in> Chains I" for R 762 proof - 763 from that have Ris: "R \<in> Chains init_seg_of" 764 using mono_Chains [OF I_init] by blast 765 have subch: "chain\<^sub>\<subseteq> R" 766 using \<open>R \<in> Chains I\<close> I_init by (auto simp: init_seg_of_def chain_subset_def Chains_def) 767 have "\<forall>r\<in>R. Refl r" and "\<forall>r\<in>R. trans r" and "\<forall>r\<in>R. antisym r" 768 and "\<forall>r\<in>R. Total r" and "\<forall>r\<in>R. wf (r - Id)" 769 using Chains_wo [OF \<open>R \<in> Chains I\<close>] by (simp_all add: order_on_defs) 770 have "Refl (\<Union>R)" 771 using \<open>\<forall>r\<in>R. Refl r\<close> unfolding refl_on_def by fastforce 772 moreover have "trans (\<Union>R)" 773 by (rule chain_subset_trans_Union [OF subch \<open>\<forall>r\<in>R. trans r\<close>]) 774 moreover have "antisym (\<Union>R)" 775 by (rule chain_subset_antisym_Union [OF subch \<open>\<forall>r\<in>R. antisym r\<close>]) 776 moreover have "Total (\<Union>R)" 777 by (rule chain_subset_Total_Union [OF subch \<open>\<forall>r\<in>R. Total r\<close>]) 778 moreover have "wf ((\<Union>R) - Id)" 779 proof - 780 have "(\<Union>R) - Id = \<Union>{r - Id | r. r \<in> R}" by blast 781 with \<open>\<forall>r\<in>R. wf (r - Id)\<close> and wf_Union_wf_init_segs [OF Chains_inits_DiffI [OF Ris]] 782 show ?thesis by fastforce 783 qed 784 ultimately have "Well_order (\<Union>R)" 785 by (simp add:order_on_defs) 786 moreover have "\<forall>r \<in> R. r initial_segment_of \<Union>R" 787 using Ris by (simp add: Chains_init_seg_of_Union) 788 ultimately show ?thesis 789 using mono_Chains [OF I_init] Chains_wo[of R] and \<open>R \<in> Chains I\<close> 790 unfolding I_def by blast 791 qed 792 then have 1: "\<exists>u\<in>Field I. \<forall>r\<in>R. (r, u) \<in> I" if "R \<in> Chains I" for R 793 using that by (subst FI) blast 794\<comment> \<open>Zorn's Lemma yields a maximal well-order \<open>m\<close>:\<close> 795 then obtain m :: "'a rel" 796 where "Well_order m" 797 and max: "\<forall>r. Well_order r \<and> (m, r) \<in> I \<longrightarrow> r = m" 798 using Zorns_po_lemma[OF 0 1] unfolding FI by fastforce 799\<comment> \<open>Now show by contradiction that \<open>m\<close> covers the whole type:\<close> 800 have False if "x \<notin> Field m" for x :: 'a 801 proof - 802\<comment> \<open>Assuming that \<open>x\<close> is not covered and extend \<open>m\<close> at the top with \<open>x\<close>\<close> 803 have "m \<noteq> {}" 804 proof 805 assume "m = {}" 806 moreover have "Well_order {(x, x)}" 807 by (simp add: order_on_defs refl_on_def trans_def antisym_def total_on_def Field_def) 808 ultimately show False using max 809 by (auto simp: I_def init_seg_of_def simp del: Field_insert) 810 qed 811 then have "Field m \<noteq> {}" by (auto simp: Field_def) 812 moreover have "wf (m - Id)" 813 using \<open>Well_order m\<close> by (simp add: well_order_on_def) 814\<comment> \<open>The extension of \<open>m\<close> by \<open>x\<close>:\<close> 815 let ?s = "{(a, x) | a. a \<in> Field m}" 816 let ?m = "insert (x, x) m \<union> ?s" 817 have Fm: "Field ?m = insert x (Field m)" 818 by (auto simp: Field_def) 819 have "Refl m" and "trans m" and "antisym m" and "Total m" and "wf (m - Id)" 820 using \<open>Well_order m\<close> by (simp_all add: order_on_defs) 821\<comment> \<open>We show that the extension is a well-order\<close> 822 have "Refl ?m" 823 using \<open>Refl m\<close> Fm unfolding refl_on_def by blast 824 moreover have "trans ?m" using \<open>trans m\<close> and \<open>x \<notin> Field m\<close> 825 unfolding trans_def Field_def by blast 826 moreover have "antisym ?m" 827 using \<open>antisym m\<close> and \<open>x \<notin> Field m\<close> unfolding antisym_def Field_def by blast 828 moreover have "Total ?m" 829 using \<open>Total m\<close> and Fm by (auto simp: total_on_def) 830 moreover have "wf (?m - Id)" 831 proof - 832 have "wf ?s" 833 using \<open>x \<notin> Field m\<close> by (auto simp: wf_eq_minimal Field_def Bex_def) 834 then show ?thesis 835 using \<open>wf (m - Id)\<close> and \<open>x \<notin> Field m\<close> wf_subset [OF \<open>wf ?s\<close> Diff_subset] 836 by (auto simp: Un_Diff Field_def intro: wf_Un) 837 qed 838 ultimately have "Well_order ?m" 839 by (simp add: order_on_defs) 840\<comment> \<open>We show that the extension is above \<open>m\<close>\<close> 841 moreover have "(m, ?m) \<in> I" 842 using \<open>Well_order ?m\<close> and \<open>Well_order m\<close> and \<open>x \<notin> Field m\<close> 843 by (fastforce simp: I_def init_seg_of_def Field_def) 844 ultimately 845\<comment> \<open>This contradicts maximality of \<open>m\<close>:\<close> 846 show False 847 using max and \<open>x \<notin> Field m\<close> unfolding Field_def by blast 848 qed 849 then have "Field m = UNIV" by auto 850 with \<open>Well_order m\<close> show ?thesis by blast 851qed 852 853corollary well_order_on: "\<exists>r::'a rel. well_order_on A r" 854proof - 855 obtain r :: "'a rel" where wo: "Well_order r" and univ: "Field r = UNIV" 856 using well_ordering [where 'a = "'a"] by blast 857 let ?r = "{(x, y). x \<in> A \<and> y \<in> A \<and> (x, y) \<in> r}" 858 have 1: "Field ?r = A" 859 using wo univ by (fastforce simp: Field_def order_on_defs refl_on_def) 860 from \<open>Well_order r\<close> have "Refl r" "trans r" "antisym r" "Total r" "wf (r - Id)" 861 by (simp_all add: order_on_defs) 862 from \<open>Refl r\<close> have "Refl ?r" 863 by (auto simp: refl_on_def 1 univ) 864 moreover from \<open>trans r\<close> have "trans ?r" 865 unfolding trans_def by blast 866 moreover from \<open>antisym r\<close> have "antisym ?r" 867 unfolding antisym_def by blast 868 moreover from \<open>Total r\<close> have "Total ?r" 869 by (simp add:total_on_def 1 univ) 870 moreover have "wf (?r - Id)" 871 by (rule wf_subset [OF \<open>wf (r - Id)\<close>]) blast 872 ultimately have "Well_order ?r" 873 by (simp add: order_on_defs) 874 with 1 show ?thesis by auto 875qed 876 877(* Move this to Hilbert Choice and wfrec to Wellfounded*) 878 879lemma wfrec_def_adm: "f \<equiv> wfrec R F \<Longrightarrow> wf R \<Longrightarrow> adm_wf R F \<Longrightarrow> f = F f" 880 using wfrec_fixpoint by simp 881 882lemma dependent_wf_choice: 883 fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" 884 assumes "wf R" 885 and adm: "\<And>f g x r. (\<And>z. (z, x) \<in> R \<Longrightarrow> f z = g z) \<Longrightarrow> P f x r = P g x r" 886 and P: "\<And>x f. (\<And>y. (y, x) \<in> R \<Longrightarrow> P f y (f y)) \<Longrightarrow> \<exists>r. P f x r" 887 shows "\<exists>f. \<forall>x. P f x (f x)" 888proof (intro exI allI) 889 fix x 890 define f where "f \<equiv> wfrec R (\<lambda>f x. SOME r. P f x r)" 891 from \<open>wf R\<close> show "P f x (f x)" 892 proof (induct x) 893 case (less x) 894 show "P f x (f x)" 895 proof (subst (2) wfrec_def_adm[OF f_def \<open>wf R\<close>]) 896 show "adm_wf R (\<lambda>f x. SOME r. P f x r)" 897 by (auto simp add: adm_wf_def intro!: arg_cong[where f=Eps] ext adm) 898 show "P f x (Eps (P f x))" 899 using P by (rule someI_ex) fact 900 qed 901 qed 902qed 903 904lemma (in wellorder) dependent_wellorder_choice: 905 assumes "\<And>r f g x. (\<And>y. y < x \<Longrightarrow> f y = g y) \<Longrightarrow> P f x r = P g x r" 906 and P: "\<And>x f. (\<And>y. y < x \<Longrightarrow> P f y (f y)) \<Longrightarrow> \<exists>r. P f x r" 907 shows "\<exists>f. \<forall>x. P f x (f x)" 908 using wf by (rule dependent_wf_choice) (auto intro!: assms) 909 910end 911