src/HOL/Zorn.thy
author wenzelm
Tue Oct 10 19:23:03 2017 +0200 (23 months ago)
changeset 66831 29ea2b900a05
parent 63572 c0cbfd2b5a45
child 67399 eab6ce8368fa
permissions -rw-r--r--
tuned: each session has at most one defining entry;
     1 (*  Title:       HOL/Zorn.thy
     2     Author:      Jacques D. Fleuriot
     3     Author:      Tobias Nipkow, TUM
     4     Author:      Christian Sternagel, JAIST
     5 
     6 Zorn's Lemma (ported from Larry Paulson's Zorn.thy in ZF).
     7 The well-ordering theorem.
     8 *)
     9 
    10 section \<open>Zorn's Lemma\<close>
    11 
    12 theory Zorn
    13   imports Order_Relation Hilbert_Choice
    14 begin
    15 
    16 subsection \<open>Zorn's Lemma for the Subset Relation\<close>
    17 
    18 subsubsection \<open>Results that do not require an order\<close>
    19 
    20 text \<open>Let \<open>P\<close> be a binary predicate on the set \<open>A\<close>.\<close>
    21 locale pred_on =
    22   fixes A :: "'a set"
    23     and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"  (infix "\<sqsubset>" 50)
    24 begin
    25 
    26 abbreviation Peq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"  (infix "\<sqsubseteq>" 50)
    27   where "x \<sqsubseteq> y \<equiv> P\<^sup>=\<^sup>= x y"
    28 
    29 text \<open>A chain is a totally ordered subset of \<open>A\<close>.\<close>
    30 definition 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 
    33 text \<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>
    37 abbreviation superchain :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"  (infix "<c" 50)
    38   where "X <c C \<equiv> chain C \<and> X \<subset> C"
    39 
    40 text \<open>A maximal chain is a chain that does not have a superchain.\<close>
    41 definition maxchain :: "'a set \<Rightarrow> bool"
    42   where "maxchain C \<longleftrightarrow> chain C \<and> (\<nexists>S. C <c S)"
    43 
    44 text \<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>
    48 definition 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 
    51 lemma 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 
    54 lemma 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 
    57 lemma not_chain_suc [simp]: "\<not> chain X \<Longrightarrow> suc X = X"
    58   by (simp add: suc_def)
    59 
    60 lemma maxchain_suc [simp]: "maxchain X \<Longrightarrow> suc X = X"
    61   by (simp add: suc_def)
    62 
    63 lemma suc_subset: "X \<subseteq> suc X"
    64   by (auto simp: suc_def maxchain_def intro: someI2)
    65 
    66 lemma chain_empty [simp]: "chain {}"
    67   by (auto simp: chain_def)
    68 
    69 lemma 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 
    72 lemma 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 
    75 lemma subset_suc:
    76   assumes "X \<subseteq> Y"
    77   shows "X \<subseteq> suc Y"
    78   using assms by (rule subset_trans) (rule suc_subset)
    79 
    80 text \<open>
    81   We build a set @{term \<C>} that is closed under applications
    82   of @{term suc} and contains the union of all its subsets.
    83 \<close>
    84 inductive_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 
    89 text \<open>
    90   Since the empty set as well as the set itself is a subset of
    91   every set, @{term \<C>} contains at least @{term "{} \<in> \<C>"} and
    92   @{term "\<Union>\<C> \<in> \<C>"}.
    93 \<close>
    94 lemma 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 
    98 text \<open>Thus closure under @{term suc} will hit a maximal chain
    99   eventually, as is shown below.\<close>
   100 
   101 lemma 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 
   108 lemma 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 
   115 text \<open>On chains, @{term suc} yields a chain.\<close>
   116 lemma 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 
   122 lemma chain_sucD:
   123   assumes "chain X"
   124   shows "suc X \<subseteq> A \<and> chain (suc X)"
   125 proof -
   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
   131 qed
   132 
   133 lemma 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>
   138 proof induct
   139   case (suc X)
   140   with * show ?case by (blast del: subsetI intro: subset_suc)
   141 next
   142   case Union
   143   then show ?case by blast
   144 qed
   145 
   146 lemma 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)
   150 proof (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
   172 next
   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
   205 qed
   206 
   207 text \<open>The elements of @{term \<C>} are totally ordered by the subset relation.\<close>
   208 lemma suc_Union_closed_total:
   209   assumes "X \<in> \<C>" and "Y \<in> \<C>"
   210   shows "X \<subseteq> Y \<or> Y \<subseteq> X"
   211 proof (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
   216 next
   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
   222 qed
   223 
   224 text \<open>Once we hit a fixed point w.r.t. @{term suc}, all other elements
   225   of @{term \<C>} are subsets of this fixed point.\<close>
   226 lemma 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>
   230 proof 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>)
   236 next
   237   case Union
   238   then show ?case by blast
   239 qed
   240 
   241 lemma eq_suc_Union:
   242   assumes "X \<in> \<C>"
   243   shows "suc X = X \<longleftrightarrow> X = \<Union>\<C>"
   244     (is "?lhs \<longleftrightarrow> ?rhs")
   245 proof
   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
   251 next
   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 ..
   258 qed
   259 
   260 lemma 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 
   266 lemma 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 
   272 text \<open>All elements of @{term \<C>} are chains.\<close>
   273 lemma suc_Union_closed_chain:
   274   assumes "X \<in> \<C>"
   275   shows "chain X"
   276   using assms
   277 proof induct
   278   case (suc X)
   279   then show ?case
   280     using not_maxchain_Some by (simp add: suc_def)
   281 next
   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 ..
   313 qed
   314 
   315 subsubsection \<open>Hausdorff's Maximum Principle\<close>
   316 
   317 text \<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 
   320 theorem Hausdorff: "\<exists>C. maxchain C"
   321 proof -
   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
   333 qed
   334 
   335 text \<open>Make notation @{term \<C>} available again.\<close>
   336 no_notation suc_Union_closed  ("\<C>")
   337 
   338 lemma 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 
   341 lemma maxchain_imp_chain: "maxchain C \<Longrightarrow> chain C"
   342   by (simp add: maxchain_def)
   343 
   344 end
   345 
   346 text \<open>Hide constant @{const pred_on.suc_Union_closed}, which was just needed
   347   for the proof of Hausforff's maximum principle.\<close>
   348 hide_const pred_on.suc_Union_closed
   349 
   350 lemma 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 
   357 subsubsection \<open>Results for the proper subset relation\<close>
   358 
   359 interpretation subset: pred_on "A" "op \<subset>" for A .
   360 
   361 lemma 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"
   366 proof (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
   377 qed
   378 
   379 
   380 subsubsection \<open>Zorn's lemma\<close>
   381 
   382 text \<open>If every chain has an upper bound, then there is a maximal set.\<close>
   383 lemma subset_Zorn:
   384   assumes "\<And>C. subset.chain A C \<Longrightarrow> \<exists>U\<in>A. \<forall>X\<in>C. X \<subseteq> U"
   385   shows "\<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
   386 proof -
   387   from subset.Hausdorff [of A] obtain M where "subset.maxchain A M" ..
   388   then have "subset.chain A M"
   389     by (rule subset.maxchain_imp_chain)
   390   with assms obtain Y where "Y \<in> A" and "\<forall>X\<in>M. X \<subseteq> Y"
   391     by blast
   392   moreover have "\<forall>X\<in>A. Y \<subseteq> X \<longrightarrow> Y = X"
   393   proof (intro ballI impI)
   394     fix X
   395     assume "X \<in> A" and "Y \<subseteq> X"
   396     show "Y = X"
   397     proof (rule ccontr)
   398       assume "\<not> ?thesis"
   399       with \<open>Y \<subseteq> X\<close> have "\<not> X \<subseteq> Y" by blast
   400       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>
   401       have "subset.chain A ({X} \<union> M)"
   402         using \<open>Y \<subseteq> X\<close> by auto
   403       moreover have "M \<subset> {X} \<union> M"
   404         using \<open>\<forall>X\<in>M. X \<subseteq> Y\<close> and \<open>\<not> X \<subseteq> Y\<close> by auto
   405       ultimately show False
   406         using \<open>subset.maxchain A M\<close> by (auto simp: subset.maxchain_def)
   407     qed
   408   qed
   409   ultimately show ?thesis by blast
   410 qed
   411 
   412 text \<open>Alternative version of Zorn's lemma for the subset relation.\<close>
   413 lemma subset_Zorn':
   414   assumes "\<And>C. subset.chain A C \<Longrightarrow> \<Union>C \<in> A"
   415   shows "\<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
   416 proof -
   417   from subset.Hausdorff [of A] obtain M where "subset.maxchain A M" ..
   418   then have "subset.chain A M"
   419     by (rule subset.maxchain_imp_chain)
   420   with assms have "\<Union>M \<in> A" .
   421   moreover have "\<forall>Z\<in>A. \<Union>M \<subseteq> Z \<longrightarrow> \<Union>M = Z"
   422   proof (intro ballI impI)
   423     fix Z
   424     assume "Z \<in> A" and "\<Union>M \<subseteq> Z"
   425     with subset_maxchain_max [OF \<open>subset.maxchain A M\<close>]
   426       show "\<Union>M = Z" .
   427   qed
   428   ultimately show ?thesis by blast
   429 qed
   430 
   431 
   432 subsection \<open>Zorn's Lemma for Partial Orders\<close>
   433 
   434 text \<open>Relate old to new definitions.\<close>
   435 
   436 definition chain_subset :: "'a set set \<Rightarrow> bool"  ("chain\<^sub>\<subseteq>")  (* Define globally? In Set.thy? *)
   437   where "chain\<^sub>\<subseteq> C \<longleftrightarrow> (\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A)"
   438 
   439 definition chains :: "'a set set \<Rightarrow> 'a set set set"
   440   where "chains A = {C. C \<subseteq> A \<and> chain\<^sub>\<subseteq> C}"
   441 
   442 definition Chains :: "('a \<times> 'a) set \<Rightarrow> 'a set set"  (* Define globally? In Relation.thy? *)
   443   where "Chains r = {C. \<forall>a\<in>C. \<forall>b\<in>C. (a, b) \<in> r \<or> (b, a) \<in> r}"
   444 
   445 lemma 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"
   446   for z :: "'a set"
   447   unfolding chains_def chain_subset_def by blast
   448 
   449 lemma mono_Chains: "r \<subseteq> s \<Longrightarrow> Chains r \<subseteq> Chains s"
   450   unfolding Chains_def by blast
   451 
   452 lemma chain_subset_alt_def: "chain\<^sub>\<subseteq> C = subset.chain UNIV C"
   453   unfolding chain_subset_def subset.chain_def by fast
   454 
   455 lemma chains_alt_def: "chains A = {C. subset.chain A C}"
   456   by (simp add: chains_def chain_subset_alt_def subset.chain_def)
   457 
   458 lemma Chains_subset: "Chains r \<subseteq> {C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C}"
   459   by (force simp add: Chains_def pred_on.chain_def)
   460 
   461 lemma Chains_subset':
   462   assumes "refl r"
   463   shows "{C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C} \<subseteq> Chains r"
   464   using assms
   465   by (auto simp add: Chains_def pred_on.chain_def refl_on_def)
   466 
   467 lemma Chains_alt_def:
   468   assumes "refl r"
   469   shows "Chains r = {C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C}"
   470   using assms Chains_subset Chains_subset' by blast
   471 
   472 lemma 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"
   473   using subset_Zorn' [of A] by (force simp: chains_alt_def)
   474 
   475 lemma 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"
   476   using subset_Zorn [of A] by (auto simp: chains_alt_def)
   477 
   478 text \<open>Various other lemmas\<close>
   479 
   480 lemma chainsD: "c \<in> chains S \<Longrightarrow> x \<in> c \<Longrightarrow> y \<in> c \<Longrightarrow> x \<subseteq> y \<or> y \<subseteq> x"
   481   unfolding chains_def chain_subset_def by blast
   482 
   483 lemma chainsD2: "c \<in> chains S \<Longrightarrow> c \<subseteq> S"
   484   unfolding chains_def by blast
   485 
   486 lemma Zorns_po_lemma:
   487   assumes po: "Partial_order r"
   488     and u: "\<forall>C\<in>Chains r. \<exists>u\<in>Field r. \<forall>a\<in>C. (a, u) \<in> r"
   489   shows "\<exists>m\<in>Field r. \<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m"
   490 proof -
   491   have "Preorder r"
   492     using po by (simp add: partial_order_on_def)
   493   txt \<open>Mirror \<open>r\<close> in the set of subsets below (wrt \<open>r\<close>) elements of \<open>A\<close>.\<close>
   494   let ?B = "\<lambda>x. r\<inverse> `` {x}"
   495   let ?S = "?B ` Field r"
   496   have "\<exists>u\<in>Field r. \<forall>A\<in>C. A \<subseteq> r\<inverse> `` {u}"  (is "\<exists>u\<in>Field r. ?P u")
   497     if 1: "C \<subseteq> ?S" and 2: "\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A" for C
   498   proof -
   499     let ?A = "{x\<in>Field r. \<exists>M\<in>C. M = ?B x}"
   500     from 1 have "C = ?B ` ?A" by (auto simp: image_def)
   501     have "?A \<in> Chains r"
   502     proof (simp add: Chains_def, intro allI impI, elim conjE)
   503       fix a b
   504       assume "a \<in> Field r" and "?B a \<in> C" and "b \<in> Field r" and "?B b \<in> C"
   505       with 2 have "?B a \<subseteq> ?B b \<or> ?B b \<subseteq> ?B a" by auto
   506       then show "(a, b) \<in> r \<or> (b, a) \<in> r"
   507         using \<open>Preorder r\<close> and \<open>a \<in> Field r\<close> and \<open>b \<in> Field r\<close>
   508         by (simp add:subset_Image1_Image1_iff)
   509     qed
   510     with u obtain u where uA: "u \<in> Field r" "\<forall>a\<in>?A. (a, u) \<in> r" by auto
   511     have "?P u"
   512     proof auto
   513       fix a B assume aB: "B \<in> C" "a \<in> B"
   514       with 1 obtain x where "x \<in> Field r" and "B = r\<inverse> `` {x}" by auto
   515       then show "(a, u) \<in> r"
   516         using uA and aB and \<open>Preorder r\<close>
   517         unfolding preorder_on_def refl_on_def by simp (fast dest: transD)
   518     qed
   519     then show ?thesis
   520       using \<open>u \<in> Field r\<close> by blast
   521   qed
   522   then have "\<forall>C\<in>chains ?S. \<exists>U\<in>?S. \<forall>A\<in>C. A \<subseteq> U"
   523     by (auto simp: chains_def chain_subset_def)
   524   from Zorn_Lemma2 [OF this] obtain m B
   525     where "m \<in> Field r"
   526       and "B = r\<inverse> `` {m}"
   527       and "\<forall>x\<in>Field r. B \<subseteq> r\<inverse> `` {x} \<longrightarrow> r\<inverse> `` {x} = B"
   528     by auto
   529   then have "\<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m"
   530     using po and \<open>Preorder r\<close> and \<open>m \<in> Field r\<close>
   531     by (auto simp: subset_Image1_Image1_iff Partial_order_eq_Image1_Image1_iff)
   532   then show ?thesis
   533     using \<open>m \<in> Field r\<close> by blast
   534 qed
   535 
   536 
   537 subsection \<open>The Well Ordering Theorem\<close>
   538 
   539 (* The initial segment of a relation appears generally useful.
   540    Move to Relation.thy?
   541    Definition correct/most general?
   542    Naming?
   543 *)
   544 definition init_seg_of :: "(('a \<times> 'a) set \<times> ('a \<times> 'a) set) set"
   545   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)}"
   546 
   547 abbreviation initial_segment_of_syntax :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool"
   548     (infix "initial'_segment'_of" 55)
   549   where "r initial_segment_of s \<equiv> (r, s) \<in> init_seg_of"
   550 
   551 lemma refl_on_init_seg_of [simp]: "r initial_segment_of r"
   552   by (simp add: init_seg_of_def)
   553 
   554 lemma trans_init_seg_of:
   555   "r initial_segment_of s \<Longrightarrow> s initial_segment_of t \<Longrightarrow> r initial_segment_of t"
   556   by (simp (no_asm_use) add: init_seg_of_def) blast
   557 
   558 lemma antisym_init_seg_of: "r initial_segment_of s \<Longrightarrow> s initial_segment_of r \<Longrightarrow> r = s"
   559   unfolding init_seg_of_def by safe
   560 
   561 lemma Chains_init_seg_of_Union: "R \<in> Chains init_seg_of \<Longrightarrow> r\<in>R \<Longrightarrow> r initial_segment_of \<Union>R"
   562   by (auto simp: init_seg_of_def Ball_def Chains_def) blast
   563 
   564 lemma chain_subset_trans_Union:
   565   assumes "chain\<^sub>\<subseteq> R" "\<forall>r\<in>R. trans r"
   566   shows "trans (\<Union>R)"
   567 proof (intro transI, elim UnionE)
   568   fix S1 S2 :: "'a rel" and x y z :: 'a
   569   assume "S1 \<in> R" "S2 \<in> R"
   570   with assms(1) have "S1 \<subseteq> S2 \<or> S2 \<subseteq> S1"
   571     unfolding chain_subset_def by blast
   572   moreover assume "(x, y) \<in> S1" "(y, z) \<in> S2"
   573   ultimately have "((x, y) \<in> S1 \<and> (y, z) \<in> S1) \<or> ((x, y) \<in> S2 \<and> (y, z) \<in> S2)"
   574     by blast
   575   with \<open>S1 \<in> R\<close> \<open>S2 \<in> R\<close> assms(2) show "(x, z) \<in> \<Union>R"
   576     by (auto elim: transE)
   577 qed
   578 
   579 lemma chain_subset_antisym_Union:
   580   assumes "chain\<^sub>\<subseteq> R" "\<forall>r\<in>R. antisym r"
   581   shows "antisym (\<Union>R)"
   582 proof (intro antisymI, elim UnionE)
   583   fix S1 S2 :: "'a rel" and x y :: 'a
   584   assume "S1 \<in> R" "S2 \<in> R"
   585   with assms(1) have "S1 \<subseteq> S2 \<or> S2 \<subseteq> S1"
   586     unfolding chain_subset_def by blast
   587   moreover assume "(x, y) \<in> S1" "(y, x) \<in> S2"
   588   ultimately have "((x, y) \<in> S1 \<and> (y, x) \<in> S1) \<or> ((x, y) \<in> S2 \<and> (y, x) \<in> S2)"
   589     by blast
   590   with \<open>S1 \<in> R\<close> \<open>S2 \<in> R\<close> assms(2) show "x = y"
   591     unfolding antisym_def by auto
   592 qed
   593 
   594 lemma chain_subset_Total_Union:
   595   assumes "chain\<^sub>\<subseteq> R" and "\<forall>r\<in>R. Total r"
   596   shows "Total (\<Union>R)"
   597 proof (simp add: total_on_def Ball_def, auto del: disjCI)
   598   fix r s a b
   599   assume A: "r \<in> R" "s \<in> R" "a \<in> Field r" "b \<in> Field s" "a \<noteq> b"
   600   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"
   601     by (auto simp add: chain_subset_def)
   602   then show "(\<exists>r\<in>R. (a, b) \<in> r) \<or> (\<exists>r\<in>R. (b, a) \<in> r)"
   603   proof
   604     assume "r \<subseteq> s"
   605     then have "(a, b) \<in> s \<or> (b, a) \<in> s"
   606       using assms(2) A mono_Field[of r s]
   607       by (auto simp add: total_on_def)
   608     then show ?thesis
   609       using \<open>s \<in> R\<close> by blast
   610   next
   611     assume "s \<subseteq> r"
   612     then have "(a, b) \<in> r \<or> (b, a) \<in> r"
   613       using assms(2) A mono_Field[of s r]
   614       by (fastforce simp add: total_on_def)
   615     then show ?thesis
   616       using \<open>r \<in> R\<close> by blast
   617   qed
   618 qed
   619 
   620 lemma wf_Union_wf_init_segs:
   621   assumes "R \<in> Chains init_seg_of"
   622     and "\<forall>r\<in>R. wf r"
   623   shows "wf (\<Union>R)"
   624 proof (simp add: wf_iff_no_infinite_down_chain, rule ccontr, auto)
   625   fix f
   626   assume 1: "\<forall>i. \<exists>r\<in>R. (f (Suc i), f i) \<in> r"
   627   then obtain r where "r \<in> R" and "(f (Suc 0), f 0) \<in> r" by auto
   628   have "(f (Suc i), f i) \<in> r" for i
   629   proof (induct i)
   630     case 0
   631     show ?case by fact
   632   next
   633     case (Suc i)
   634     then obtain s where s: "s \<in> R" "(f (Suc (Suc i)), f(Suc i)) \<in> s"
   635       using 1 by auto
   636     then have "s initial_segment_of r \<or> r initial_segment_of s"
   637       using assms(1) \<open>r \<in> R\<close> by (simp add: Chains_def)
   638     with Suc s show ?case by (simp add: init_seg_of_def) blast
   639   qed
   640   then show False
   641     using assms(2) and \<open>r \<in> R\<close>
   642     by (simp add: wf_iff_no_infinite_down_chain) blast
   643 qed
   644 
   645 lemma initial_segment_of_Diff: "p initial_segment_of q \<Longrightarrow> p - s initial_segment_of q - s"
   646   unfolding init_seg_of_def by blast
   647 
   648 lemma Chains_inits_DiffI: "R \<in> Chains init_seg_of \<Longrightarrow> {r - s |r. r \<in> R} \<in> Chains init_seg_of"
   649   unfolding Chains_def by (blast intro: initial_segment_of_Diff)
   650 
   651 theorem well_ordering: "\<exists>r::'a rel. Well_order r \<and> Field r = UNIV"
   652 proof -
   653 \<comment> \<open>The initial segment relation on well-orders:\<close>
   654   let ?WO = "{r::'a rel. Well_order r}"
   655   define I where "I = init_seg_of \<inter> ?WO \<times> ?WO"
   656   then have I_init: "I \<subseteq> init_seg_of" by simp
   657   then have subch: "\<And>R. R \<in> Chains I \<Longrightarrow> chain\<^sub>\<subseteq> R"
   658     unfolding init_seg_of_def chain_subset_def Chains_def by blast
   659   have Chains_wo: "\<And>R r. R \<in> Chains I \<Longrightarrow> r \<in> R \<Longrightarrow> Well_order r"
   660     by (simp add: Chains_def I_def) blast
   661   have FI: "Field I = ?WO"
   662     by (auto simp add: I_def init_seg_of_def Field_def)
   663   then have 0: "Partial_order I"
   664     by (auto simp: partial_order_on_def preorder_on_def antisym_def antisym_init_seg_of refl_on_def
   665         trans_def I_def elim!: trans_init_seg_of)
   666 \<comment> \<open>\<open>I\<close>-chains have upper bounds in \<open>?WO\<close> wrt \<open>I\<close>: their Union\<close>
   667   have "\<Union>R \<in> ?WO \<and> (\<forall>r\<in>R. (r, \<Union>R) \<in> I)" if "R \<in> Chains I" for R
   668   proof -
   669     from that have Ris: "R \<in> Chains init_seg_of"
   670       using mono_Chains [OF I_init] by blast
   671     have subch: "chain\<^sub>\<subseteq> R"
   672       using \<open>R : Chains I\<close> I_init by (auto simp: init_seg_of_def chain_subset_def Chains_def)
   673     have "\<forall>r\<in>R. Refl r" and "\<forall>r\<in>R. trans r" and "\<forall>r\<in>R. antisym r"
   674       and "\<forall>r\<in>R. Total r" and "\<forall>r\<in>R. wf (r - Id)"
   675       using Chains_wo [OF \<open>R \<in> Chains I\<close>] by (simp_all add: order_on_defs)
   676     have "Refl (\<Union>R)"
   677       using \<open>\<forall>r\<in>R. Refl r\<close> unfolding refl_on_def by fastforce
   678     moreover have "trans (\<Union>R)"
   679       by (rule chain_subset_trans_Union [OF subch \<open>\<forall>r\<in>R. trans r\<close>])
   680     moreover have "antisym (\<Union>R)"
   681       by (rule chain_subset_antisym_Union [OF subch \<open>\<forall>r\<in>R. antisym r\<close>])
   682     moreover have "Total (\<Union>R)"
   683       by (rule chain_subset_Total_Union [OF subch \<open>\<forall>r\<in>R. Total r\<close>])
   684     moreover have "wf ((\<Union>R) - Id)"
   685     proof -
   686       have "(\<Union>R) - Id = \<Union>{r - Id | r. r \<in> R}" by blast
   687       with \<open>\<forall>r\<in>R. wf (r - Id)\<close> and wf_Union_wf_init_segs [OF Chains_inits_DiffI [OF Ris]]
   688       show ?thesis by fastforce
   689     qed
   690     ultimately have "Well_order (\<Union>R)"
   691       by (simp add:order_on_defs)
   692     moreover have "\<forall>r \<in> R. r initial_segment_of \<Union>R"
   693       using Ris by (simp add: Chains_init_seg_of_Union)
   694     ultimately show ?thesis
   695       using mono_Chains [OF I_init] Chains_wo[of R] and \<open>R \<in> Chains I\<close>
   696       unfolding I_def by blast
   697   qed
   698   then have 1: "\<forall>R \<in> Chains I. \<exists>u\<in>Field I. \<forall>r\<in>R. (r, u) \<in> I"
   699     by (subst FI) blast
   700 \<comment>\<open>Zorn's Lemma yields a maximal well-order \<open>m\<close>:\<close>
   701   then obtain m :: "'a rel"
   702     where "Well_order m"
   703       and max: "\<forall>r. Well_order r \<and> (m, r) \<in> I \<longrightarrow> r = m"
   704     using Zorns_po_lemma[OF 0 1] unfolding FI by fastforce
   705 \<comment>\<open>Now show by contradiction that \<open>m\<close> covers the whole type:\<close>
   706   have False if "x \<notin> Field m" for x :: 'a
   707   proof -
   708 \<comment>\<open>Assuming that \<open>x\<close> is not covered and extend \<open>m\<close> at the top with \<open>x\<close>\<close>
   709     have "m \<noteq> {}"
   710     proof
   711       assume "m = {}"
   712       moreover have "Well_order {(x, x)}"
   713         by (simp add: order_on_defs refl_on_def trans_def antisym_def total_on_def Field_def)
   714       ultimately show False using max
   715         by (auto simp: I_def init_seg_of_def simp del: Field_insert)
   716     qed
   717     then have "Field m \<noteq> {}" by (auto simp: Field_def)
   718     moreover have "wf (m - Id)"
   719       using \<open>Well_order m\<close> by (simp add: well_order_on_def)
   720 \<comment>\<open>The extension of \<open>m\<close> by \<open>x\<close>:\<close>
   721     let ?s = "{(a, x) | a. a \<in> Field m}"
   722     let ?m = "insert (x, x) m \<union> ?s"
   723     have Fm: "Field ?m = insert x (Field m)"
   724       by (auto simp: Field_def)
   725     have "Refl m" and "trans m" and "antisym m" and "Total m" and "wf (m - Id)"
   726       using \<open>Well_order m\<close> by (simp_all add: order_on_defs)
   727 \<comment>\<open>We show that the extension is a well-order\<close>
   728     have "Refl ?m"
   729       using \<open>Refl m\<close> Fm unfolding refl_on_def by blast
   730     moreover have "trans ?m" using \<open>trans m\<close> and \<open>x \<notin> Field m\<close>
   731       unfolding trans_def Field_def by blast
   732     moreover have "antisym ?m"
   733       using \<open>antisym m\<close> and \<open>x \<notin> Field m\<close> unfolding antisym_def Field_def by blast
   734     moreover have "Total ?m"
   735       using \<open>Total m\<close> and Fm by (auto simp: total_on_def)
   736     moreover have "wf (?m - Id)"
   737     proof -
   738       have "wf ?s"
   739         using \<open>x \<notin> Field m\<close> by (auto simp: wf_eq_minimal Field_def Bex_def)
   740       then show ?thesis
   741         using \<open>wf (m - Id)\<close> and \<open>x \<notin> Field m\<close> wf_subset [OF \<open>wf ?s\<close> Diff_subset]
   742         by (auto simp: Un_Diff Field_def intro: wf_Un)
   743     qed
   744     ultimately have "Well_order ?m"
   745       by (simp add: order_on_defs)
   746 \<comment>\<open>We show that the extension is above \<open>m\<close>\<close>
   747     moreover have "(m, ?m) \<in> I"
   748       using \<open>Well_order ?m\<close> and \<open>Well_order m\<close> and \<open>x \<notin> Field m\<close>
   749       by (fastforce simp: I_def init_seg_of_def Field_def)
   750     ultimately
   751 \<comment>\<open>This contradicts maximality of \<open>m\<close>:\<close>
   752     show False
   753       using max and \<open>x \<notin> Field m\<close> unfolding Field_def by blast
   754   qed
   755   then have "Field m = UNIV" by auto
   756   with \<open>Well_order m\<close> show ?thesis by blast
   757 qed
   758 
   759 corollary well_order_on: "\<exists>r::'a rel. well_order_on A r"
   760 proof -
   761   obtain r :: "'a rel" where wo: "Well_order r" and univ: "Field r = UNIV"
   762     using well_ordering [where 'a = "'a"] by blast
   763   let ?r = "{(x, y). x \<in> A \<and> y \<in> A \<and> (x, y) \<in> r}"
   764   have 1: "Field ?r = A"
   765     using wo univ by (fastforce simp: Field_def order_on_defs refl_on_def)
   766   from \<open>Well_order r\<close> have "Refl r" "trans r" "antisym r" "Total r" "wf (r - Id)"
   767     by (simp_all add: order_on_defs)
   768   from \<open>Refl r\<close> have "Refl ?r"
   769     by (auto simp: refl_on_def 1 univ)
   770   moreover from \<open>trans r\<close> have "trans ?r"
   771     unfolding trans_def by blast
   772   moreover from \<open>antisym r\<close> have "antisym ?r"
   773     unfolding antisym_def by blast
   774   moreover from \<open>Total r\<close> have "Total ?r"
   775     by (simp add:total_on_def 1 univ)
   776   moreover have "wf (?r - Id)"
   777     by (rule wf_subset [OF \<open>wf (r - Id)\<close>]) blast
   778   ultimately have "Well_order ?r"
   779     by (simp add: order_on_defs)
   780   with 1 show ?thesis by auto
   781 qed
   782 
   783 (* Move this to Hilbert Choice and wfrec to Wellfounded*)
   784 
   785 lemma wfrec_def_adm: "f \<equiv> wfrec R F \<Longrightarrow> wf R \<Longrightarrow> adm_wf R F \<Longrightarrow> f = F f"
   786   using wfrec_fixpoint by simp
   787 
   788 lemma dependent_wf_choice:
   789   fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
   790   assumes "wf R"
   791     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"
   792     and P: "\<And>x f. (\<And>y. (y, x) \<in> R \<Longrightarrow> P f y (f y)) \<Longrightarrow> \<exists>r. P f x r"
   793   shows "\<exists>f. \<forall>x. P f x (f x)"
   794 proof (intro exI allI)
   795   fix x
   796   define f where "f \<equiv> wfrec R (\<lambda>f x. SOME r. P f x r)"
   797   from \<open>wf R\<close> show "P f x (f x)"
   798   proof (induct x)
   799     case (less x)
   800     show "P f x (f x)"
   801     proof (subst (2) wfrec_def_adm[OF f_def \<open>wf R\<close>])
   802       show "adm_wf R (\<lambda>f x. SOME r. P f x r)"
   803         by (auto simp add: adm_wf_def intro!: arg_cong[where f=Eps] ext adm)
   804       show "P f x (Eps (P f x))"
   805         using P by (rule someI_ex) fact
   806     qed
   807   qed
   808 qed
   809 
   810 lemma (in wellorder) dependent_wellorder_choice:
   811   assumes "\<And>r f g x. (\<And>y. y < x \<Longrightarrow> f y = g y) \<Longrightarrow> P f x r = P g x r"
   812     and P: "\<And>x f. (\<And>y. y < x \<Longrightarrow> P f y (f y)) \<Longrightarrow> \<exists>r. P f x r"
   813   shows "\<exists>f. \<forall>x. P f x (f x)"
   814   using wf by (rule dependent_wf_choice) (auto intro!: assms)
   815 
   816 end