src/HOL/Library/Zorn.thy
author Andreas Lochbihler
Fri Sep 20 10:09:16 2013 +0200 (2013-09-20)
changeset 53745 788730ab7da4
parent 53374 a14d2a854c02
child 54482 a2874c8b3558
permissions -rw-r--r--
prefer Code.abort over code_abort
     1 (*  Title:      HOL/Library/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 The extension of any well-founded relation to a well-order. 
     9 *)
    10 
    11 header {* Zorn's Lemma *}
    12 
    13 theory Zorn
    14 imports Order_Union
    15 begin
    16 
    17 subsection {* Zorn's Lemma for the Subset Relation *}
    18 
    19 subsubsection {* Results that do not require an order *}
    20 
    21 text {*Let @{text P} be a binary predicate on the set @{text A}.*}
    22 locale pred_on =
    23   fixes A :: "'a set"
    24     and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<sqsubset>" 50)
    25 begin
    26 
    27 abbreviation Peq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<sqsubseteq>" 50) where
    28   "x \<sqsubseteq> y \<equiv> P\<^sup>=\<^sup>= x y"
    29 
    30 text {*A chain is a totally ordered subset of @{term A}.*}
    31 definition chain :: "'a set \<Rightarrow> bool" where
    32   "chain C \<longleftrightarrow> C \<subseteq> A \<and> (\<forall>x\<in>C. \<forall>y\<in>C. x \<sqsubseteq> y \<or> y \<sqsubseteq> x)"
    33 
    34 text {*We call a chain that is a proper superset of some set @{term X},
    35 but not necessarily a chain itself, a superchain of @{term X}.*}
    36 abbreviation superchain :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" (infix "<c" 50) where
    37   "X <c C \<equiv> chain C \<and> X \<subset> C"
    38 
    39 text {*A maximal chain is a chain that does not have a superchain.*}
    40 definition maxchain :: "'a set \<Rightarrow> bool" where
    41   "maxchain C \<longleftrightarrow> chain C \<and> \<not> (\<exists>S. C <c S)"
    42 
    43 text {*We define the successor of a set to be an arbitrary
    44 superchain, if such exists, or the set itself, otherwise.*}
    45 definition suc :: "'a set \<Rightarrow> 'a set" where
    46   "suc C = (if \<not> chain C \<or> maxchain C then C else (SOME D. C <c D))"
    47 
    48 lemma chainI [Pure.intro?]:
    49   "\<lbrakk>C \<subseteq> A; \<And>x y. \<lbrakk>x \<in> C; y \<in> C\<rbrakk> \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x\<rbrakk> \<Longrightarrow> chain C"
    50   unfolding chain_def by blast
    51 
    52 lemma chain_total:
    53   "chain C \<Longrightarrow> x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
    54   by (simp add: chain_def)
    55 
    56 lemma not_chain_suc [simp]: "\<not> chain X \<Longrightarrow> suc X = X"
    57   by (simp add: suc_def)
    58 
    59 lemma maxchain_suc [simp]: "maxchain X \<Longrightarrow> suc X = X"
    60   by (simp add: suc_def)
    61 
    62 lemma suc_subset: "X \<subseteq> suc X"
    63   by (auto simp: suc_def maxchain_def intro: someI2)
    64 
    65 lemma chain_empty [simp]: "chain {}"
    66   by (auto simp: chain_def)
    67 
    68 lemma not_maxchain_Some:
    69   "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:
    73   "chain C \<Longrightarrow> \<not> maxchain C \<Longrightarrow> suc C \<noteq> C"
    74   by (auto simp: suc_def) (metis less_irrefl not_maxchain_Some)
    75 
    76 lemma subset_suc:
    77   assumes "X \<subseteq> Y" shows "X \<subseteq> suc Y"
    78   using assms by (rule subset_trans) (rule suc_subset)
    79 
    80 text {*We build a set @{term \<C>} that is closed under applications
    81 of @{term suc} and contains the union of all its subsets.*}
    82 inductive_set suc_Union_closed ("\<C>") where
    83   suc: "X \<in> \<C> \<Longrightarrow> suc X \<in> \<C>" |
    84   Union [unfolded Pow_iff]: "X \<in> Pow \<C> \<Longrightarrow> \<Union>X \<in> \<C>"
    85 
    86 text {*Since the empty set as well as the set itself is a subset of
    87 every set, @{term \<C>} contains at least @{term "{} \<in> \<C>"} and
    88 @{term "\<Union>\<C> \<in> \<C>"}.*}
    89 lemma
    90   suc_Union_closed_empty: "{} \<in> \<C>" and
    91   suc_Union_closed_Union: "\<Union>\<C> \<in> \<C>"
    92   using Union [of "{}"] and Union [of "\<C>"] by simp+
    93 text {*Thus closure under @{term suc} will hit a maximal chain
    94 eventually, as is shown below.*}
    95 
    96 lemma suc_Union_closed_induct [consumes 1, case_names suc Union,
    97   induct pred: suc_Union_closed]:
    98   assumes "X \<in> \<C>"
    99     and "\<And>X. \<lbrakk>X \<in> \<C>; Q X\<rbrakk> \<Longrightarrow> Q (suc X)"
   100     and "\<And>X. \<lbrakk>X \<subseteq> \<C>; \<forall>x\<in>X. Q x\<rbrakk> \<Longrightarrow> Q (\<Union>X)"
   101   shows "Q X"
   102   using assms by (induct) blast+
   103 
   104 lemma suc_Union_closed_cases [consumes 1, case_names suc Union,
   105   cases pred: suc_Union_closed]:
   106   assumes "X \<in> \<C>"
   107     and "\<And>Y. \<lbrakk>X = suc Y; Y \<in> \<C>\<rbrakk> \<Longrightarrow> Q"
   108     and "\<And>Y. \<lbrakk>X = \<Union>Y; Y \<subseteq> \<C>\<rbrakk> \<Longrightarrow> Q"
   109   shows "Q"
   110   using assms by (cases) simp+
   111 
   112 text {*On chains, @{term suc} yields a chain.*}
   113 lemma chain_suc:
   114   assumes "chain X" shows "chain (suc X)"
   115   using assms
   116   by (cases "\<not> chain X \<or> maxchain X")
   117      (force simp: suc_def dest: not_maxchain_Some)+
   118 
   119 lemma chain_sucD:
   120   assumes "chain X" shows "suc X \<subseteq> A \<and> chain (suc X)"
   121 proof -
   122   from `chain X` have *: "chain (suc X)" by (rule chain_suc)
   123   then have "suc X \<subseteq> A" unfolding chain_def by blast
   124   with * show ?thesis by blast
   125 qed
   126 
   127 lemma suc_Union_closed_total':
   128   assumes "X \<in> \<C>" and "Y \<in> \<C>"
   129     and *: "\<And>Z. Z \<in> \<C> \<Longrightarrow> Z \<subseteq> Y \<Longrightarrow> Z = Y \<or> suc Z \<subseteq> Y"
   130   shows "X \<subseteq> Y \<or> suc Y \<subseteq> X"
   131   using `X \<in> \<C>`
   132 proof (induct)
   133   case (suc X)
   134   with * show ?case by (blast del: subsetI intro: subset_suc)
   135 qed blast
   136 
   137 lemma suc_Union_closed_subsetD:
   138   assumes "Y \<subseteq> X" and "X \<in> \<C>" and "Y \<in> \<C>"
   139   shows "X = Y \<or> suc Y \<subseteq> X"
   140   using assms(2-, 1)
   141 proof (induct arbitrary: Y)
   142   case (suc X)
   143   note * = `\<And>Y. \<lbrakk>Y \<in> \<C>; Y \<subseteq> X\<rbrakk> \<Longrightarrow> X = Y \<or> suc Y \<subseteq> X`
   144   with suc_Union_closed_total' [OF `Y \<in> \<C>` `X \<in> \<C>`]
   145     have "Y \<subseteq> X \<or> suc X \<subseteq> Y" by blast
   146   then show ?case
   147   proof
   148     assume "Y \<subseteq> X"
   149     with * and `Y \<in> \<C>` have "X = Y \<or> suc Y \<subseteq> X" by blast
   150     then show ?thesis
   151     proof
   152       assume "X = Y" then show ?thesis by simp
   153     next
   154       assume "suc Y \<subseteq> X"
   155       then have "suc Y \<subseteq> suc X" by (rule subset_suc)
   156       then show ?thesis by simp
   157     qed
   158   next
   159     assume "suc X \<subseteq> Y"
   160     with `Y \<subseteq> suc X` show ?thesis by blast
   161   qed
   162 next
   163   case (Union X)
   164   show ?case
   165   proof (rule ccontr)
   166     assume "\<not> ?thesis"
   167     with `Y \<subseteq> \<Union>X` obtain x y z
   168     where "\<not> suc Y \<subseteq> \<Union>X"
   169       and "x \<in> X" and "y \<in> x" and "y \<notin> Y"
   170       and "z \<in> suc Y" and "\<forall>x\<in>X. z \<notin> x" by blast
   171     with `X \<subseteq> \<C>` have "x \<in> \<C>" by blast
   172     from Union and `x \<in> X`
   173       have *: "\<And>y. \<lbrakk>y \<in> \<C>; y \<subseteq> x\<rbrakk> \<Longrightarrow> x = y \<or> suc y \<subseteq> x" by blast
   174     with suc_Union_closed_total' [OF `Y \<in> \<C>` `x \<in> \<C>`]
   175       have "Y \<subseteq> x \<or> suc x \<subseteq> Y" by blast
   176     then show False
   177     proof
   178       assume "Y \<subseteq> x"
   179       with * [OF `Y \<in> \<C>`] have "x = Y \<or> suc Y \<subseteq> x" by blast
   180       then show False
   181       proof
   182         assume "x = Y" with `y \<in> x` and `y \<notin> Y` show False by blast
   183       next
   184         assume "suc Y \<subseteq> x"
   185         with `x \<in> X` have "suc Y \<subseteq> \<Union>X" by blast
   186         with `\<not> suc Y \<subseteq> \<Union>X` show False by contradiction
   187       qed
   188     next
   189       assume "suc x \<subseteq> Y"
   190       moreover from suc_subset and `y \<in> x` have "y \<in> suc x" by blast
   191       ultimately show False using `y \<notin> Y` by blast
   192     qed
   193   qed
   194 qed
   195 
   196 text {*The elements of @{term \<C>} are totally ordered by the subset relation.*}
   197 lemma suc_Union_closed_total:
   198   assumes "X \<in> \<C>" and "Y \<in> \<C>"
   199   shows "X \<subseteq> Y \<or> Y \<subseteq> X"
   200 proof (cases "\<forall>Z\<in>\<C>. Z \<subseteq> Y \<longrightarrow> Z = Y \<or> suc Z \<subseteq> Y")
   201   case True
   202   with suc_Union_closed_total' [OF assms]
   203     have "X \<subseteq> Y \<or> suc Y \<subseteq> X" by blast
   204   then show ?thesis using suc_subset [of Y] by blast
   205 next
   206   case False
   207   then obtain Z
   208     where "Z \<in> \<C>" and "Z \<subseteq> Y" and "Z \<noteq> Y" and "\<not> suc Z \<subseteq> Y" by blast
   209   with suc_Union_closed_subsetD and `Y \<in> \<C>` show ?thesis by blast
   210 qed
   211 
   212 text {*Once we hit a fixed point w.r.t. @{term suc}, all other elements
   213 of @{term \<C>} are subsets of this fixed point.*}
   214 lemma suc_Union_closed_suc:
   215   assumes "X \<in> \<C>" and "Y \<in> \<C>" and "suc Y = Y"
   216   shows "X \<subseteq> Y"
   217 using `X \<in> \<C>`
   218 proof (induct)
   219   case (suc X)
   220   with `Y \<in> \<C>` and suc_Union_closed_subsetD
   221     have "X = Y \<or> suc X \<subseteq> Y" by blast
   222   then show ?case by (auto simp: `suc Y = Y`)
   223 qed blast
   224 
   225 lemma eq_suc_Union:
   226   assumes "X \<in> \<C>"
   227   shows "suc X = X \<longleftrightarrow> X = \<Union>\<C>"
   228 proof
   229   assume "suc X = X"
   230   with suc_Union_closed_suc [OF suc_Union_closed_Union `X \<in> \<C>`]
   231     have "\<Union>\<C> \<subseteq> X" .
   232   with `X \<in> \<C>` show "X = \<Union>\<C>" by blast
   233 next
   234   from `X \<in> \<C>` have "suc X \<in> \<C>" by (rule suc)
   235   then have "suc X \<subseteq> \<Union>\<C>" by blast
   236   moreover assume "X = \<Union>\<C>"
   237   ultimately have "suc X \<subseteq> X" by simp
   238   moreover have "X \<subseteq> suc X" by (rule suc_subset)
   239   ultimately show "suc X = X" ..
   240 qed
   241 
   242 lemma suc_in_carrier:
   243   assumes "X \<subseteq> A"
   244   shows "suc X \<subseteq> A"
   245   using assms
   246   by (cases "\<not> chain X \<or> maxchain X")
   247      (auto dest: chain_sucD)
   248 
   249 lemma suc_Union_closed_in_carrier:
   250   assumes "X \<in> \<C>"
   251   shows "X \<subseteq> A"
   252   using assms
   253   by (induct) (auto dest: suc_in_carrier)
   254 
   255 text {*All elements of @{term \<C>} are chains.*}
   256 lemma suc_Union_closed_chain:
   257   assumes "X \<in> \<C>"
   258   shows "chain X"
   259 using assms
   260 proof (induct)
   261   case (suc X) then show ?case by (simp add: suc_def) (metis not_maxchain_Some)
   262 next
   263   case (Union X)
   264   then have "\<Union>X \<subseteq> A" by (auto dest: suc_Union_closed_in_carrier)
   265   moreover have "\<forall>x\<in>\<Union>X. \<forall>y\<in>\<Union>X. x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
   266   proof (intro ballI)
   267     fix x y
   268     assume "x \<in> \<Union>X" and "y \<in> \<Union>X"
   269     then obtain u v where "x \<in> u" and "u \<in> X" and "y \<in> v" and "v \<in> X" by blast
   270     with Union have "u \<in> \<C>" and "v \<in> \<C>" and "chain u" and "chain v" by blast+
   271     with suc_Union_closed_total have "u \<subseteq> v \<or> v \<subseteq> u" by blast
   272     then show "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
   273     proof
   274       assume "u \<subseteq> v"
   275       from `chain v` show ?thesis
   276       proof (rule chain_total)
   277         show "y \<in> v" by fact
   278         show "x \<in> v" using `u \<subseteq> v` and `x \<in> u` by blast
   279       qed
   280     next
   281       assume "v \<subseteq> u"
   282       from `chain u` show ?thesis
   283       proof (rule chain_total)
   284         show "x \<in> u" by fact
   285         show "y \<in> u" using `v \<subseteq> u` and `y \<in> v` by blast
   286       qed
   287     qed
   288   qed
   289   ultimately show ?case unfolding chain_def ..
   290 qed
   291 
   292 subsubsection {* Hausdorff's Maximum Principle *}
   293 
   294 text {*There exists a maximal totally ordered subset of @{term A}. (Note that we do not
   295 require @{term A} to be partially ordered.)*}
   296 
   297 theorem Hausdorff: "\<exists>C. maxchain C"
   298 proof -
   299   let ?M = "\<Union>\<C>"
   300   have "maxchain ?M"
   301   proof (rule ccontr)
   302     assume "\<not> maxchain ?M"
   303     then have "suc ?M \<noteq> ?M"
   304       using suc_not_equals and
   305       suc_Union_closed_chain [OF suc_Union_closed_Union] by simp
   306     moreover have "suc ?M = ?M"
   307       using eq_suc_Union [OF suc_Union_closed_Union] by simp
   308     ultimately show False by contradiction
   309   qed
   310   then show ?thesis by blast
   311 qed
   312 
   313 text {*Make notation @{term \<C>} available again.*}
   314 no_notation suc_Union_closed ("\<C>")
   315 
   316 lemma chain_extend:
   317   "chain C \<Longrightarrow> z \<in> A \<Longrightarrow> \<forall>x\<in>C. x \<sqsubseteq> z \<Longrightarrow> chain ({z} \<union> C)"
   318   unfolding chain_def by blast
   319 
   320 lemma maxchain_imp_chain:
   321   "maxchain C \<Longrightarrow> chain C"
   322   by (simp add: maxchain_def)
   323 
   324 end
   325 
   326 text {*Hide constant @{const pred_on.suc_Union_closed}, which was just needed
   327 for the proof of Hausforff's maximum principle.*}
   328 hide_const pred_on.suc_Union_closed
   329 
   330 lemma chain_mono:
   331   assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; P x y\<rbrakk> \<Longrightarrow> Q x y"
   332     and "pred_on.chain A P C"
   333   shows "pred_on.chain A Q C"
   334   using assms unfolding pred_on.chain_def by blast
   335 
   336 subsubsection {* Results for the proper subset relation *}
   337 
   338 interpretation subset: pred_on "A" "op \<subset>" for A .
   339 
   340 lemma subset_maxchain_max:
   341   assumes "subset.maxchain A C" and "X \<in> A" and "\<Union>C \<subseteq> X"
   342   shows "\<Union>C = X"
   343 proof (rule ccontr)
   344   let ?C = "{X} \<union> C"
   345   from `subset.maxchain A C` have "subset.chain A C"
   346     and *: "\<And>S. subset.chain A S \<Longrightarrow> \<not> C \<subset> S"
   347     by (auto simp: subset.maxchain_def)
   348   moreover have "\<forall>x\<in>C. x \<subseteq> X" using `\<Union>C \<subseteq> X` by auto
   349   ultimately have "subset.chain A ?C"
   350     using subset.chain_extend [of A C X] and `X \<in> A` by auto
   351   moreover assume **: "\<Union>C \<noteq> X"
   352   moreover from ** have "C \<subset> ?C" using `\<Union>C \<subseteq> X` by auto
   353   ultimately show False using * by blast
   354 qed
   355 
   356 subsubsection {* Zorn's lemma *}
   357 
   358 text {*If every chain has an upper bound, then there is a maximal set.*}
   359 lemma subset_Zorn:
   360   assumes "\<And>C. subset.chain A C \<Longrightarrow> \<exists>U\<in>A. \<forall>X\<in>C. X \<subseteq> U"
   361   shows "\<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
   362 proof -
   363   from subset.Hausdorff [of A] obtain M where "subset.maxchain A M" ..
   364   then have "subset.chain A M" by (rule subset.maxchain_imp_chain)
   365   with assms obtain Y where "Y \<in> A" and "\<forall>X\<in>M. X \<subseteq> Y" by blast
   366   moreover have "\<forall>X\<in>A. Y \<subseteq> X \<longrightarrow> Y = X"
   367   proof (intro ballI impI)
   368     fix X
   369     assume "X \<in> A" and "Y \<subseteq> X"
   370     show "Y = X"
   371     proof (rule ccontr)
   372       assume "Y \<noteq> X"
   373       with `Y \<subseteq> X` have "\<not> X \<subseteq> Y" by blast
   374       from subset.chain_extend [OF `subset.chain A M` `X \<in> A`] and `\<forall>X\<in>M. X \<subseteq> Y`
   375         have "subset.chain A ({X} \<union> M)" using `Y \<subseteq> X` by auto
   376       moreover have "M \<subset> {X} \<union> M" using `\<forall>X\<in>M. X \<subseteq> Y` and `\<not> X \<subseteq> Y` by auto
   377       ultimately show False
   378         using `subset.maxchain A M` by (auto simp: subset.maxchain_def)
   379     qed
   380   qed
   381   ultimately show ?thesis by blast
   382 qed
   383 
   384 text{*Alternative version of Zorn's lemma for the subset relation.*}
   385 lemma subset_Zorn':
   386   assumes "\<And>C. subset.chain A C \<Longrightarrow> \<Union>C \<in> A"
   387   shows "\<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
   388 proof -
   389   from subset.Hausdorff [of A] obtain M where "subset.maxchain A M" ..
   390   then have "subset.chain A M" by (rule subset.maxchain_imp_chain)
   391   with assms have "\<Union>M \<in> A" .
   392   moreover have "\<forall>Z\<in>A. \<Union>M \<subseteq> Z \<longrightarrow> \<Union>M = Z"
   393   proof (intro ballI impI)
   394     fix Z
   395     assume "Z \<in> A" and "\<Union>M \<subseteq> Z"
   396     with subset_maxchain_max [OF `subset.maxchain A M`]
   397       show "\<Union>M = Z" .
   398   qed
   399   ultimately show ?thesis by blast
   400 qed
   401 
   402 
   403 subsection {* Zorn's Lemma for Partial Orders *}
   404 
   405 text {*Relate old to new definitions.*}
   406 
   407 (* Define globally? In Set.thy? *)
   408 definition chain_subset :: "'a set set \<Rightarrow> bool" ("chain\<^sub>\<subseteq>") where
   409   "chain\<^sub>\<subseteq> C \<longleftrightarrow> (\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A)"
   410 
   411 definition chains :: "'a set set \<Rightarrow> 'a set set set" where
   412   "chains A = {C. C \<subseteq> A \<and> chain\<^sub>\<subseteq> C}"
   413 
   414 (* Define globally? In Relation.thy? *)
   415 definition Chains :: "('a \<times> 'a) set \<Rightarrow> 'a set set" where
   416   "Chains r = {C. \<forall>a\<in>C. \<forall>b\<in>C. (a, b) \<in> r \<or> (b, a) \<in> r}"
   417 
   418 lemma chains_extend:
   419   "[| c \<in> chains S; z \<in> S; \<forall>x \<in> c. x \<subseteq> (z:: 'a set) |] ==> {z} Un c \<in> chains S"
   420   by (unfold chains_def chain_subset_def) blast
   421 
   422 lemma mono_Chains: "r \<subseteq> s \<Longrightarrow> Chains r \<subseteq> Chains s"
   423   unfolding Chains_def by blast
   424 
   425 lemma chain_subset_alt_def: "chain\<^sub>\<subseteq> C = subset.chain UNIV C"
   426   by (auto simp add: chain_subset_def subset.chain_def)
   427 
   428 lemma chains_alt_def: "chains A = {C. subset.chain A C}"
   429   by (simp add: chains_def chain_subset_alt_def subset.chain_def)
   430 
   431 lemma Chains_subset:
   432   "Chains r \<subseteq> {C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C}"
   433   by (force simp add: Chains_def pred_on.chain_def)
   434 
   435 lemma Chains_subset':
   436   assumes "refl r"
   437   shows "{C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C} \<subseteq> Chains r"
   438   using assms
   439   by (auto simp add: Chains_def pred_on.chain_def refl_on_def)
   440 
   441 lemma Chains_alt_def:
   442   assumes "refl r"
   443   shows "Chains r = {C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C}"
   444   using assms
   445   by (metis Chains_subset Chains_subset' subset_antisym)
   446 
   447 lemma Zorn_Lemma:
   448   "\<forall>C\<in>chains A. \<Union>C \<in> A \<Longrightarrow> \<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
   449   using subset_Zorn' [of A] by (force simp: chains_alt_def)
   450 
   451 lemma Zorn_Lemma2:
   452   "\<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"
   453   using subset_Zorn [of A] by (auto simp: chains_alt_def)
   454 
   455 text{*Various other lemmas*}
   456 
   457 lemma chainsD: "[| c \<in> chains S; x \<in> c; y \<in> c |] ==> x \<subseteq> y | y \<subseteq> x"
   458 by (unfold chains_def chain_subset_def) blast
   459 
   460 lemma chainsD2: "!!(c :: 'a set set). c \<in> chains S ==> c \<subseteq> S"
   461 by (unfold chains_def) blast
   462 
   463 lemma Zorns_po_lemma:
   464   assumes po: "Partial_order r"
   465     and u: "\<forall>C\<in>Chains r. \<exists>u\<in>Field r. \<forall>a\<in>C. (a, u) \<in> r"
   466   shows "\<exists>m\<in>Field r. \<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m"
   467 proof -
   468   have "Preorder r" using po by (simp add: partial_order_on_def)
   469 --{* Mirror r in the set of subsets below (wrt r) elements of A*}
   470   let ?B = "%x. r\<inverse> `` {x}" let ?S = "?B ` Field r"
   471   {
   472     fix C assume 1: "C \<subseteq> ?S" and 2: "\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A"
   473     let ?A = "{x\<in>Field r. \<exists>M\<in>C. M = ?B x}"
   474     have "C = ?B ` ?A" using 1 by (auto simp: image_def)
   475     have "?A \<in> Chains r"
   476     proof (simp add: Chains_def, intro allI impI, elim conjE)
   477       fix a b
   478       assume "a \<in> Field r" and "?B a \<in> C" and "b \<in> Field r" and "?B b \<in> C"
   479       hence "?B a \<subseteq> ?B b \<or> ?B b \<subseteq> ?B a" using 2 by auto
   480       thus "(a, b) \<in> r \<or> (b, a) \<in> r"
   481         using `Preorder r` and `a \<in> Field r` and `b \<in> Field r`
   482         by (simp add:subset_Image1_Image1_iff)
   483     qed
   484     then obtain u where uA: "u \<in> Field r" "\<forall>a\<in>?A. (a, u) \<in> r" using u by auto
   485     have "\<forall>A\<in>C. A \<subseteq> r\<inverse> `` {u}" (is "?P u")
   486     proof auto
   487       fix a B assume aB: "B \<in> C" "a \<in> B"
   488       with 1 obtain x where "x \<in> Field r" and "B = r\<inverse> `` {x}" by auto
   489       thus "(a, u) \<in> r" using uA and aB and `Preorder r`
   490         by (auto simp add: preorder_on_def refl_on_def) (metis transD)
   491     qed
   492     then have "\<exists>u\<in>Field r. ?P u" using `u \<in> Field r` by blast
   493   }
   494   then have "\<forall>C\<in>chains ?S. \<exists>U\<in>?S. \<forall>A\<in>C. A \<subseteq> U"
   495     by (auto simp: chains_def chain_subset_def)
   496   from Zorn_Lemma2 [OF this]
   497   obtain m B where "m \<in> Field r" and "B = r\<inverse> `` {m}"
   498     and "\<forall>x\<in>Field r. B \<subseteq> r\<inverse> `` {x} \<longrightarrow> r\<inverse> `` {x} = B"
   499     by auto
   500   hence "\<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m"
   501     using po and `Preorder r` and `m \<in> Field r`
   502     by (auto simp: subset_Image1_Image1_iff Partial_order_eq_Image1_Image1_iff)
   503   thus ?thesis using `m \<in> Field r` by blast
   504 qed
   505 
   506 
   507 subsection {* The Well Ordering Theorem *}
   508 
   509 (* The initial segment of a relation appears generally useful.
   510    Move to Relation.thy?
   511    Definition correct/most general?
   512    Naming?
   513 *)
   514 definition init_seg_of :: "(('a \<times> 'a) set \<times> ('a \<times> 'a) set) set" where
   515   "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)}"
   516 
   517 abbreviation
   518   initialSegmentOf :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool" (infix "initial'_segment'_of" 55)
   519 where
   520   "r initial_segment_of s \<equiv> (r, s) \<in> init_seg_of"
   521 
   522 lemma refl_on_init_seg_of [simp]: "r initial_segment_of r"
   523   by (simp add: init_seg_of_def)
   524 
   525 lemma trans_init_seg_of:
   526   "r initial_segment_of s \<Longrightarrow> s initial_segment_of t \<Longrightarrow> r initial_segment_of t"
   527   by (simp (no_asm_use) add: init_seg_of_def)
   528      (metis UnCI Un_absorb2 subset_trans)
   529 
   530 lemma antisym_init_seg_of:
   531   "r initial_segment_of s \<Longrightarrow> s initial_segment_of r \<Longrightarrow> r = s"
   532   unfolding init_seg_of_def by safe
   533 
   534 lemma Chains_init_seg_of_Union:
   535   "R \<in> Chains init_seg_of \<Longrightarrow> r\<in>R \<Longrightarrow> r initial_segment_of \<Union>R"
   536   by (auto simp: init_seg_of_def Ball_def Chains_def) blast
   537 
   538 lemma chain_subset_trans_Union:
   539   "chain\<^sub>\<subseteq> R \<Longrightarrow> \<forall>r\<in>R. trans r \<Longrightarrow> trans (\<Union>R)"
   540 apply (auto simp add: chain_subset_def)
   541 apply (simp (no_asm_use) add: trans_def)
   542 apply (metis subsetD)
   543 done
   544 
   545 lemma chain_subset_antisym_Union:
   546   "chain\<^sub>\<subseteq> R \<Longrightarrow> \<forall>r\<in>R. antisym r \<Longrightarrow> antisym (\<Union>R)"
   547 apply (auto simp add: chain_subset_def antisym_def)
   548 apply (metis subsetD)
   549 done
   550 
   551 lemma chain_subset_Total_Union:
   552   assumes "chain\<^sub>\<subseteq> R" and "\<forall>r\<in>R. Total r"
   553   shows "Total (\<Union>R)"
   554 proof (simp add: total_on_def Ball_def, auto del: disjCI)
   555   fix r s a b assume A: "r \<in> R" "s \<in> R" "a \<in> Field r" "b \<in> Field s" "a \<noteq> b"
   556   from `chain\<^sub>\<subseteq> R` and `r \<in> R` and `s \<in> R` have "r \<subseteq> s \<or> s \<subseteq> r"
   557     by (auto simp add: chain_subset_def)
   558   thus "(\<exists>r\<in>R. (a, b) \<in> r) \<or> (\<exists>r\<in>R. (b, a) \<in> r)"
   559   proof
   560     assume "r \<subseteq> s" hence "(a, b) \<in> s \<or> (b, a) \<in> s" using assms(2) A
   561       by (simp add: total_on_def) (metis mono_Field subsetD)
   562     thus ?thesis using `s \<in> R` by blast
   563   next
   564     assume "s \<subseteq> r" hence "(a, b) \<in> r \<or> (b, a) \<in> r" using assms(2) A
   565       by (simp add: total_on_def) (metis mono_Field subsetD)
   566     thus ?thesis using `r \<in> R` by blast
   567   qed
   568 qed
   569 
   570 lemma wf_Union_wf_init_segs:
   571   assumes "R \<in> Chains init_seg_of" and "\<forall>r\<in>R. wf r"
   572   shows "wf (\<Union>R)"
   573 proof(simp add: wf_iff_no_infinite_down_chain, rule ccontr, auto)
   574   fix f assume 1: "\<forall>i. \<exists>r\<in>R. (f (Suc i), f i) \<in> r"
   575   then obtain r where "r \<in> R" and "(f (Suc 0), f 0) \<in> r" by auto
   576   { fix i have "(f (Suc i), f i) \<in> r"
   577     proof (induct i)
   578       case 0 show ?case by fact
   579     next
   580       case (Suc i)
   581       then obtain s where s: "s \<in> R" "(f (Suc (Suc i)), f(Suc i)) \<in> s"
   582         using 1 by auto
   583       then have "s initial_segment_of r \<or> r initial_segment_of s"
   584         using assms(1) `r \<in> R` by (simp add: Chains_def)
   585       with Suc s show ?case by (simp add: init_seg_of_def) blast
   586     qed
   587   }
   588   thus False using assms(2) and `r \<in> R`
   589     by (simp add: wf_iff_no_infinite_down_chain) blast
   590 qed
   591 
   592 lemma initial_segment_of_Diff:
   593   "p initial_segment_of q \<Longrightarrow> p - s initial_segment_of q - s"
   594   unfolding init_seg_of_def by blast
   595 
   596 lemma Chains_inits_DiffI:
   597   "R \<in> Chains init_seg_of \<Longrightarrow> {r - s |r. r \<in> R} \<in> Chains init_seg_of"
   598   unfolding Chains_def by (blast intro: initial_segment_of_Diff)
   599 
   600 theorem well_ordering: "\<exists>r::'a rel. Well_order r \<and> Field r = UNIV"
   601 proof -
   602 -- {*The initial segment relation on well-orders: *}
   603   let ?WO = "{r::'a rel. Well_order r}"
   604   def I \<equiv> "init_seg_of \<inter> ?WO \<times> ?WO"
   605   have I_init: "I \<subseteq> init_seg_of" by (auto simp: I_def)
   606   hence subch: "\<And>R. R \<in> Chains I \<Longrightarrow> chain\<^sub>\<subseteq> R"
   607     by (auto simp: init_seg_of_def chain_subset_def Chains_def)
   608   have Chains_wo: "\<And>R r. R \<in> Chains I \<Longrightarrow> r \<in> R \<Longrightarrow> Well_order r"
   609     by (simp add: Chains_def I_def) blast
   610   have FI: "Field I = ?WO" by (auto simp add: I_def init_seg_of_def Field_def)
   611   hence 0: "Partial_order I"
   612     by (auto simp: partial_order_on_def preorder_on_def antisym_def antisym_init_seg_of refl_on_def
   613       trans_def I_def elim!: trans_init_seg_of)
   614 -- {*I-chains have upper bounds in ?WO wrt I: their Union*}
   615   { fix R assume "R \<in> Chains I"
   616     hence Ris: "R \<in> Chains init_seg_of" using mono_Chains [OF I_init] by blast
   617     have subch: "chain\<^sub>\<subseteq> R" using `R : Chains I` I_init
   618       by (auto simp: init_seg_of_def chain_subset_def Chains_def)
   619     have "\<forall>r\<in>R. Refl r" and "\<forall>r\<in>R. trans r" and "\<forall>r\<in>R. antisym r"
   620       and "\<forall>r\<in>R. Total r" and "\<forall>r\<in>R. wf (r - Id)"
   621       using Chains_wo [OF `R \<in> Chains I`] by (simp_all add: order_on_defs)
   622     have "Refl (\<Union>R)" using `\<forall>r\<in>R. Refl r` by (auto simp: refl_on_def)
   623     moreover have "trans (\<Union>R)"
   624       by (rule chain_subset_trans_Union [OF subch `\<forall>r\<in>R. trans r`])
   625     moreover have "antisym (\<Union>R)"
   626       by (rule chain_subset_antisym_Union [OF subch `\<forall>r\<in>R. antisym r`])
   627     moreover have "Total (\<Union>R)"
   628       by (rule chain_subset_Total_Union [OF subch `\<forall>r\<in>R. Total r`])
   629     moreover have "wf ((\<Union>R) - Id)"
   630     proof -
   631       have "(\<Union>R) - Id = \<Union>{r - Id | r. r \<in> R}" by blast
   632       with `\<forall>r\<in>R. wf (r - Id)` and wf_Union_wf_init_segs [OF Chains_inits_DiffI [OF Ris]]
   633       show ?thesis by (simp (no_asm_simp)) blast
   634     qed
   635     ultimately have "Well_order (\<Union>R)" by(simp add:order_on_defs)
   636     moreover have "\<forall>r \<in> R. r initial_segment_of \<Union>R" using Ris
   637       by(simp add: Chains_init_seg_of_Union)
   638     ultimately have "\<Union>R \<in> ?WO \<and> (\<forall>r\<in>R. (r, \<Union>R) \<in> I)"
   639       using mono_Chains [OF I_init] and `R \<in> Chains I`
   640       by (simp (no_asm) add: I_def del: Field_Union) (metis Chains_wo)
   641   }
   642   hence 1: "\<forall>R \<in> Chains I. \<exists>u\<in>Field I. \<forall>r\<in>R. (r, u) \<in> I" by (subst FI) blast
   643 --{*Zorn's Lemma yields a maximal well-order m:*}
   644   then obtain m::"'a rel" where "Well_order m" and
   645     max: "\<forall>r. Well_order r \<and> (m, r) \<in> I \<longrightarrow> r = m"
   646     using Zorns_po_lemma[OF 0 1] by (auto simp:FI)
   647 --{*Now show by contradiction that m covers the whole type:*}
   648   { fix x::'a assume "x \<notin> Field m"
   649 --{*We assume that x is not covered and extend m at the top with x*}
   650     have "m \<noteq> {}"
   651     proof
   652       assume "m = {}"
   653       moreover have "Well_order {(x, x)}"
   654         by (simp add: order_on_defs refl_on_def trans_def antisym_def total_on_def Field_def)
   655       ultimately show False using max
   656         by (auto simp: I_def init_seg_of_def simp del: Field_insert)
   657     qed
   658     hence "Field m \<noteq> {}" by(auto simp:Field_def)
   659     moreover have "wf (m - Id)" using `Well_order m`
   660       by (simp add: well_order_on_def)
   661 --{*The extension of m by x:*}
   662     let ?s = "{(a, x) | a. a \<in> Field m}"
   663     let ?m = "insert (x, x) m \<union> ?s"
   664     have Fm: "Field ?m = insert x (Field m)"
   665       by (auto simp: Field_def)
   666     have "Refl m" and "trans m" and "antisym m" and "Total m" and "wf (m - Id)"
   667       using `Well_order m` by (simp_all add: order_on_defs)
   668 --{*We show that the extension is a well-order*}
   669     have "Refl ?m" using `Refl m` Fm by (auto simp: refl_on_def)
   670     moreover have "trans ?m" using `trans m` and `x \<notin> Field m`
   671       unfolding trans_def Field_def by blast
   672     moreover have "antisym ?m" using `antisym m` and `x \<notin> Field m`
   673       unfolding antisym_def Field_def by blast
   674     moreover have "Total ?m" using `Total m` and Fm by (auto simp: total_on_def)
   675     moreover have "wf (?m - Id)"
   676     proof -
   677       have "wf ?s" using `x \<notin> Field m`
   678         by (auto simp add: wf_eq_minimal Field_def) metis
   679       thus ?thesis using `wf (m - Id)` and `x \<notin> Field m`
   680         wf_subset [OF `wf ?s` Diff_subset]
   681         by (fastforce intro!: wf_Un simp add: Un_Diff Field_def)
   682     qed
   683     ultimately have "Well_order ?m" by (simp add: order_on_defs)
   684 --{*We show that the extension is above m*}
   685     moreover have "(m, ?m) \<in> I" using `Well_order ?m` and `Well_order m` and `x \<notin> Field m`
   686       by (fastforce simp: I_def init_seg_of_def Field_def)
   687     ultimately
   688 --{*This contradicts maximality of m:*}
   689     have False using max and `x \<notin> Field m` unfolding Field_def by blast
   690   }
   691   hence "Field m = UNIV" by auto
   692   with `Well_order m` show ?thesis by blast
   693 qed
   694 
   695 corollary well_order_on: "\<exists>r::'a rel. well_order_on A r"
   696 proof -
   697   obtain r::"'a rel" where wo: "Well_order r" and univ: "Field r = UNIV"
   698     using well_ordering [where 'a = "'a"] by blast
   699   let ?r = "{(x, y). x \<in> A \<and> y \<in> A \<and> (x, y) \<in> r}"
   700   have 1: "Field ?r = A" using wo univ
   701     by (fastforce simp: Field_def order_on_defs refl_on_def)
   702   have "Refl r" and "trans r" and "antisym r" and "Total r" and "wf (r - Id)"
   703     using `Well_order r` by (simp_all add: order_on_defs)
   704   have "Refl ?r" using `Refl r` by (auto simp: refl_on_def 1 univ)
   705   moreover have "trans ?r" using `trans r`
   706     unfolding trans_def by blast
   707   moreover have "antisym ?r" using `antisym r`
   708     unfolding antisym_def by blast
   709   moreover have "Total ?r" using `Total r` by (simp add:total_on_def 1 univ)
   710   moreover have "wf (?r - Id)" by (rule wf_subset [OF `wf (r - Id)`]) blast
   711   ultimately have "Well_order ?r" by (simp add: order_on_defs)
   712   with 1 show ?thesis by metis
   713 qed
   714 
   715 subsection {* Extending Well-founded Relations to Well-Orders *}
   716 
   717 text {*A \emph{downset} (also lower set, decreasing set, initial segment, or
   718 downward closed set) is closed w.r.t.\ smaller elements.*}
   719 definition downset_on where
   720   "downset_on A r = (\<forall>x y. (x, y) \<in> r \<and> y \<in> A \<longrightarrow> x \<in> A)"
   721 
   722 (*
   723 text {*Connection to order filters of the @{theory Cardinals} theory.*}
   724 lemma (in wo_rel) ofilter_downset_on_conv:
   725   "ofilter A \<longleftrightarrow> downset_on A r \<and> A \<subseteq> Field r"
   726   by (auto simp: downset_on_def ofilter_def under_def)
   727 *)
   728 
   729 lemma downset_onI:
   730   "(\<And>x y. (x, y) \<in> r \<Longrightarrow> y \<in> A \<Longrightarrow> x \<in> A) \<Longrightarrow> downset_on A r"
   731   by (auto simp: downset_on_def)
   732 
   733 lemma downset_onD:
   734   "downset_on A r \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> y \<in> A \<Longrightarrow> x \<in> A"
   735   by (auto simp: downset_on_def)
   736 
   737 text {*Extensions of relations w.r.t.\ a given set.*}
   738 definition extension_on where
   739   "extension_on A r s = (\<forall>x\<in>A. \<forall>y\<in>A. (x, y) \<in> s \<longrightarrow> (x, y) \<in> r)"
   740 
   741 lemma extension_onI:
   742   "(\<And>x y. \<lbrakk>x \<in> A; y \<in> A; (x, y) \<in> s\<rbrakk> \<Longrightarrow> (x, y) \<in> r) \<Longrightarrow> extension_on A r s"
   743   by (auto simp: extension_on_def)
   744 
   745 lemma extension_onD:
   746   "extension_on A r s \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> (x, y) \<in> s \<Longrightarrow> (x, y) \<in> r"
   747   by (auto simp: extension_on_def)
   748 
   749 lemma downset_on_Union:
   750   assumes "\<And>r. r \<in> R \<Longrightarrow> downset_on (Field r) p"
   751   shows "downset_on (Field (\<Union>R)) p"
   752   using assms by (auto intro: downset_onI dest: downset_onD)
   753 
   754 lemma chain_subset_extension_on_Union:
   755   assumes "chain\<^sub>\<subseteq> R" and "\<And>r. r \<in> R \<Longrightarrow> extension_on (Field r) r p"
   756   shows "extension_on (Field (\<Union>R)) (\<Union>R) p"
   757   using assms
   758   by (simp add: chain_subset_def extension_on_def) (metis mono_Field set_mp)
   759 
   760 lemma downset_on_empty [simp]: "downset_on {} p"
   761   by (auto simp: downset_on_def)
   762 
   763 lemma extension_on_empty [simp]: "extension_on {} p q"
   764   by (auto simp: extension_on_def)
   765 
   766 text {*Every well-founded relation can be extended to a well-order.*}
   767 theorem well_order_extension:
   768   assumes "wf p"
   769   shows "\<exists>w. p \<subseteq> w \<and> Well_order w"
   770 proof -
   771   let ?K = "{r. Well_order r \<and> downset_on (Field r) p \<and> extension_on (Field r) r p}"
   772   def I \<equiv> "init_seg_of \<inter> ?K \<times> ?K"
   773   have I_init: "I \<subseteq> init_seg_of" by (simp add: I_def)
   774   then have subch: "\<And>R. R \<in> Chains I \<Longrightarrow> chain\<^sub>\<subseteq> R"
   775     by (auto simp: init_seg_of_def chain_subset_def Chains_def)
   776   have Chains_wo: "\<And>R r. R \<in> Chains I \<Longrightarrow> r \<in> R \<Longrightarrow>
   777       Well_order r \<and> downset_on (Field r) p \<and> extension_on (Field r) r p"
   778     by (simp add: Chains_def I_def) blast
   779   have FI: "Field I = ?K" by (auto simp: I_def init_seg_of_def Field_def)
   780   then have 0: "Partial_order I"
   781     by (auto simp: partial_order_on_def preorder_on_def antisym_def antisym_init_seg_of refl_on_def
   782       trans_def I_def elim: trans_init_seg_of)
   783   { fix R assume "R \<in> Chains I"
   784     then have Ris: "R \<in> Chains init_seg_of" using mono_Chains [OF I_init] by blast
   785     have subch: "chain\<^sub>\<subseteq> R" using `R \<in> Chains I` I_init
   786       by (auto simp: init_seg_of_def chain_subset_def Chains_def)
   787     have "\<forall>r\<in>R. Refl r" and "\<forall>r\<in>R. trans r" and "\<forall>r\<in>R. antisym r" and
   788       "\<forall>r\<in>R. Total r" and "\<forall>r\<in>R. wf (r - Id)" and
   789       "\<And>r. r \<in> R \<Longrightarrow> downset_on (Field r) p" and
   790       "\<And>r. r \<in> R \<Longrightarrow> extension_on (Field r) r p"
   791       using Chains_wo [OF `R \<in> Chains I`] by (simp_all add: order_on_defs)
   792     have "Refl (\<Union>R)" using `\<forall>r\<in>R. Refl r` by (auto simp: refl_on_def)
   793     moreover have "trans (\<Union>R)"
   794       by (rule chain_subset_trans_Union [OF subch `\<forall>r\<in>R. trans r`])
   795     moreover have "antisym (\<Union>R)"
   796       by (rule chain_subset_antisym_Union [OF subch `\<forall>r\<in>R. antisym r`])
   797     moreover have "Total (\<Union>R)"
   798       by (rule chain_subset_Total_Union [OF subch `\<forall>r\<in>R. Total r`])
   799     moreover have "wf ((\<Union>R) - Id)"
   800     proof -
   801       have "(\<Union>R) - Id = \<Union>{r - Id | r. r \<in> R}" by blast
   802       with `\<forall>r\<in>R. wf (r - Id)` wf_Union_wf_init_segs [OF Chains_inits_DiffI [OF Ris]]
   803       show ?thesis by (simp (no_asm_simp)) blast
   804     qed
   805     ultimately have "Well_order (\<Union>R)" by (simp add: order_on_defs)
   806     moreover have "\<forall>r\<in>R. r initial_segment_of \<Union>R" using Ris
   807       by (simp add: Chains_init_seg_of_Union)
   808     moreover have "downset_on (Field (\<Union>R)) p"
   809       by (rule downset_on_Union [OF `\<And>r. r \<in> R \<Longrightarrow> downset_on (Field r) p`])
   810     moreover have "extension_on (Field (\<Union>R)) (\<Union>R) p"
   811       by (rule chain_subset_extension_on_Union [OF subch `\<And>r. r \<in> R \<Longrightarrow> extension_on (Field r) r p`])
   812     ultimately have "\<Union>R \<in> ?K \<and> (\<forall>r\<in>R. (r,\<Union>R) \<in> I)"
   813       using mono_Chains [OF I_init] and `R \<in> Chains I`
   814       by (simp (no_asm) add: I_def del: Field_Union) (metis Chains_wo)
   815   }
   816   then have 1: "\<forall>R\<in>Chains I. \<exists>u\<in>Field I. \<forall>r\<in>R. (r, u) \<in> I" by (subst FI) blast
   817   txt {*Zorn's Lemma yields a maximal well-order m.*}
   818   from Zorns_po_lemma [OF 0 1] obtain m :: "('a \<times> 'a) set"
   819     where "Well_order m" and "downset_on (Field m) p" and "extension_on (Field m) m p" and
   820     max: "\<forall>r. Well_order r \<and> downset_on (Field r) p \<and> extension_on (Field r) r p \<and>
   821       (m, r) \<in> I \<longrightarrow> r = m"
   822     by (auto simp: FI)
   823   have "Field p \<subseteq> Field m"
   824   proof (rule ccontr)
   825     let ?Q = "Field p - Field m"
   826     assume "\<not> (Field p \<subseteq> Field m)"
   827     with assms [unfolded wf_eq_minimal, THEN spec, of ?Q]
   828       obtain x where "x \<in> Field p" and "x \<notin> Field m" and
   829       min: "\<forall>y. (y, x) \<in> p \<longrightarrow> y \<notin> ?Q" by blast
   830     txt {*Add @{term x} as topmost element to @{term m}.*}
   831     let ?s = "{(y, x) | y. y \<in> Field m}"
   832     let ?m = "insert (x, x) m \<union> ?s"
   833     have Fm: "Field ?m = insert x (Field m)" by (auto simp: Field_def)
   834     have "Refl m" and "trans m" and "antisym m" and "Total m" and "wf (m - Id)"
   835       using `Well_order m` by (simp_all add: order_on_defs)
   836     txt {*We show that the extension is a well-order.*}
   837     have "Refl ?m" using `Refl m` Fm by (auto simp: refl_on_def)
   838     moreover have "trans ?m" using `trans m` `x \<notin> Field m`
   839       unfolding trans_def Field_def Domain_unfold Domain_converse [symmetric] by blast
   840     moreover have "antisym ?m" using `antisym m` `x \<notin> Field m`
   841       unfolding antisym_def Field_def Domain_unfold Domain_converse [symmetric] by blast
   842     moreover have "Total ?m" using `Total m` Fm by (auto simp: Relation.total_on_def)
   843     moreover have "wf (?m - Id)"
   844     proof -
   845       have "wf ?s" using `x \<notin> Field m`
   846         by (simp add: wf_eq_minimal Field_def Domain_unfold Domain_converse [symmetric]) metis
   847       thus ?thesis using `wf (m - Id)` `x \<notin> Field m`
   848         wf_subset [OF `wf ?s` Diff_subset]
   849         by (fastforce intro!: wf_Un simp add: Un_Diff Field_def)
   850     qed
   851     ultimately have "Well_order ?m" by (simp add: order_on_defs)
   852     moreover have "extension_on (Field ?m) ?m p"
   853       using `extension_on (Field m) m p` `downset_on (Field m) p`
   854       by (subst Fm) (auto simp: extension_on_def dest: downset_onD)
   855     moreover have "downset_on (Field ?m) p"
   856       using `downset_on (Field m) p` and min
   857       by (subst Fm, simp add: downset_on_def Field_def) (metis Domain_iff)
   858     moreover have "(m, ?m) \<in> I"
   859       using `Well_order m` and `Well_order ?m` and
   860       `downset_on (Field m) p` and `downset_on (Field ?m) p` and
   861       `extension_on (Field m) m p` and `extension_on (Field ?m) ?m p` and
   862       `Refl m` and `x \<notin> Field m`
   863       by (auto simp: I_def init_seg_of_def refl_on_def)
   864     ultimately
   865     --{*This contradicts maximality of m:*}
   866     show False using max and `x \<notin> Field m` unfolding Field_def by blast
   867   qed
   868   have "p \<subseteq> m"
   869     using `Field p \<subseteq> Field m` and `extension_on (Field m) m p`
   870     by (force simp: Field_def extension_on_def)
   871   with `Well_order m` show ?thesis by blast
   872 qed
   873 
   874 text {*Every well-founded relation can be extended to a total well-order.*}
   875 corollary total_well_order_extension:
   876   assumes "wf p"
   877   shows "\<exists>w. p \<subseteq> w \<and> Well_order w \<and> Field w = UNIV"
   878 proof -
   879   from well_order_extension [OF assms] obtain w
   880     where "p \<subseteq> w" and wo: "Well_order w" by blast
   881   let ?A = "UNIV - Field w"
   882   from well_order_on [of ?A] obtain w' where wo': "well_order_on ?A w'" ..
   883   have [simp]: "Field w' = ?A" using rel.well_order_on_Well_order [OF wo'] by simp
   884   have *: "Field w \<inter> Field w' = {}" by simp
   885   let ?w = "w \<union>o w'"
   886   have "p \<subseteq> ?w" using `p \<subseteq> w` by (auto simp: Osum_def)
   887   moreover have "Well_order ?w" using Osum_Well_order [OF * wo] and wo' by simp
   888   moreover have "Field ?w = UNIV" by (simp add: Field_Osum)
   889   ultimately show ?thesis by blast
   890 qed
   891 
   892 corollary well_order_on_extension:
   893   assumes "wf p" and "Field p \<subseteq> A"
   894   shows "\<exists>w. p \<subseteq> w \<and> well_order_on A w"
   895 proof -
   896   from total_well_order_extension [OF `wf p`] obtain r
   897     where "p \<subseteq> r" and wo: "Well_order r" and univ: "Field r = UNIV" by blast
   898   let ?r = "{(x, y). x \<in> A \<and> y \<in> A \<and> (x, y) \<in> r}"
   899   from `p \<subseteq> r` have "p \<subseteq> ?r" using `Field p \<subseteq> A` by (auto simp: Field_def)
   900   have 1: "Field ?r = A" using wo univ
   901     by (fastforce simp: Field_def order_on_defs refl_on_def)
   902   have "Refl r" "trans r" "antisym r" "Total r" "wf (r - Id)"
   903     using `Well_order r` by (simp_all add: order_on_defs)
   904   have "refl_on A ?r" using `Refl r` by (auto simp: refl_on_def univ)
   905   moreover have "trans ?r" using `trans r`
   906     unfolding trans_def by blast
   907   moreover have "antisym ?r" using `antisym r`
   908     unfolding antisym_def by blast
   909   moreover have "total_on A ?r" using `Total r` by (simp add: total_on_def univ)
   910   moreover have "wf (?r - Id)" by (rule wf_subset [OF `wf(r - Id)`]) blast
   911   ultimately have "well_order_on A ?r" by (simp add: order_on_defs)
   912   with `p \<subseteq> ?r` show ?thesis by blast
   913 qed
   914 
   915 end
   916