moved 'Zorn' into 'Main', since it's a BNF dependency
authorblanchet
Thu Jan 16 16:33:19 2014 +0100 (2014-01-16)
changeset 550182a526bd279ed
parent 55017 2df6ad1dbd66
child 55019 0d5e831175de
moved 'Zorn' into 'Main', since it's a BNF dependency
src/HOL/Cardinals/Wellorder_Embedding_FP.thy
src/HOL/Cardinals/Wellorder_Extension.thy
src/HOL/Hahn_Banach/Vector_Space.thy
src/HOL/Hahn_Banach/Zorn_Lemma.thy
src/HOL/Library/Library.thy
src/HOL/Library/Zorn.thy
src/HOL/Main.thy
src/HOL/NSA/Filter.thy
src/HOL/ROOT
src/HOL/Zorn.thy
     1.1 --- a/src/HOL/Cardinals/Wellorder_Embedding_FP.thy	Thu Jan 16 16:20:17 2014 +0100
     1.2 +++ b/src/HOL/Cardinals/Wellorder_Embedding_FP.thy	Thu Jan 16 16:33:19 2014 +0100
     1.3 @@ -8,7 +8,7 @@
     1.4  header {* Well-Order Embeddings (FP) *}
     1.5  
     1.6  theory Wellorder_Embedding_FP
     1.7 -imports "~~/src/HOL/Library/Zorn" Fun_More_FP Wellorder_Relation_FP
     1.8 +imports Zorn Fun_More_FP Wellorder_Relation_FP
     1.9  begin
    1.10  
    1.11  
     2.1 --- a/src/HOL/Cardinals/Wellorder_Extension.thy	Thu Jan 16 16:20:17 2014 +0100
     2.2 +++ b/src/HOL/Cardinals/Wellorder_Extension.thy	Thu Jan 16 16:33:19 2014 +0100
     2.3 @@ -5,7 +5,7 @@
     2.4  header {* Extending Well-founded Relations to Wellorders *}
     2.5  
     2.6  theory Wellorder_Extension
     2.7 -imports "~~/src/HOL/Library/Zorn" Order_Union
     2.8 +imports Zorn Order_Union
     2.9  begin
    2.10  
    2.11  subsection {* Extending Well-founded Relations to Wellorders *}
     3.1 --- a/src/HOL/Hahn_Banach/Vector_Space.thy	Thu Jan 16 16:20:17 2014 +0100
     3.2 +++ b/src/HOL/Hahn_Banach/Vector_Space.thy	Thu Jan 16 16:33:19 2014 +0100
     3.3 @@ -5,7 +5,7 @@
     3.4  header {* Vector spaces *}
     3.5  
     3.6  theory Vector_Space
     3.7 -imports Complex_Main Bounds "~~/src/HOL/Library/Zorn"
     3.8 +imports Complex_Main Bounds
     3.9  begin
    3.10  
    3.11  subsection {* Signature *}
     4.1 --- a/src/HOL/Hahn_Banach/Zorn_Lemma.thy	Thu Jan 16 16:20:17 2014 +0100
     4.2 +++ b/src/HOL/Hahn_Banach/Zorn_Lemma.thy	Thu Jan 16 16:33:19 2014 +0100
     4.3 @@ -5,7 +5,7 @@
     4.4  header {* Zorn's Lemma *}
     4.5  
     4.6  theory Zorn_Lemma
     4.7 -imports "~~/src/HOL/Library/Zorn"
     4.8 +imports Main
     4.9  begin
    4.10  
    4.11  text {*
     5.1 --- a/src/HOL/Library/Library.thy	Thu Jan 16 16:20:17 2014 +0100
     5.2 +++ b/src/HOL/Library/Library.thy	Thu Jan 16 16:33:19 2014 +0100
     5.3 @@ -64,7 +64,6 @@
     5.4    Sum_of_Squares
     5.5    Transitive_Closure_Table
     5.6    While_Combinator
     5.7 -  Zorn
     5.8  begin
     5.9  end
    5.10  (*>*)
     6.1 --- a/src/HOL/Library/Zorn.thy	Thu Jan 16 16:20:17 2014 +0100
     6.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
     6.3 @@ -1,712 +0,0 @@
     6.4 -(*  Title:      HOL/Library/Zorn.thy
     6.5 -    Author:     Jacques D. Fleuriot
     6.6 -    Author:     Tobias Nipkow, TUM
     6.7 -    Author:     Christian Sternagel, JAIST
     6.8 -
     6.9 -Zorn's Lemma (ported from Larry Paulson's Zorn.thy in ZF).
    6.10 -The well-ordering theorem.
    6.11 -*)
    6.12 -
    6.13 -header {* Zorn's Lemma *}
    6.14 -
    6.15 -theory Zorn
    6.16 -imports Main
    6.17 -begin
    6.18 -
    6.19 -subsection {* Zorn's Lemma for the Subset Relation *}
    6.20 -
    6.21 -subsubsection {* Results that do not require an order *}
    6.22 -
    6.23 -text {*Let @{text P} be a binary predicate on the set @{text A}.*}
    6.24 -locale pred_on =
    6.25 -  fixes A :: "'a set"
    6.26 -    and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<sqsubset>" 50)
    6.27 -begin
    6.28 -
    6.29 -abbreviation Peq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<sqsubseteq>" 50) where
    6.30 -  "x \<sqsubseteq> y \<equiv> P\<^sup>=\<^sup>= x y"
    6.31 -
    6.32 -text {*A chain is a totally ordered subset of @{term A}.*}
    6.33 -definition chain :: "'a set \<Rightarrow> bool" where
    6.34 -  "chain C \<longleftrightarrow> C \<subseteq> A \<and> (\<forall>x\<in>C. \<forall>y\<in>C. x \<sqsubseteq> y \<or> y \<sqsubseteq> x)"
    6.35 -
    6.36 -text {*We call a chain that is a proper superset of some set @{term X},
    6.37 -but not necessarily a chain itself, a superchain of @{term X}.*}
    6.38 -abbreviation superchain :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" (infix "<c" 50) where
    6.39 -  "X <c C \<equiv> chain C \<and> X \<subset> C"
    6.40 -
    6.41 -text {*A maximal chain is a chain that does not have a superchain.*}
    6.42 -definition maxchain :: "'a set \<Rightarrow> bool" where
    6.43 -  "maxchain C \<longleftrightarrow> chain C \<and> \<not> (\<exists>S. C <c S)"
    6.44 -
    6.45 -text {*We define the successor of a set to be an arbitrary
    6.46 -superchain, if such exists, or the set itself, otherwise.*}
    6.47 -definition suc :: "'a set \<Rightarrow> 'a set" where
    6.48 -  "suc C = (if \<not> chain C \<or> maxchain C then C else (SOME D. C <c D))"
    6.49 -
    6.50 -lemma chainI [Pure.intro?]:
    6.51 -  "\<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"
    6.52 -  unfolding chain_def by blast
    6.53 -
    6.54 -lemma chain_total:
    6.55 -  "chain C \<Longrightarrow> x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
    6.56 -  by (simp add: chain_def)
    6.57 -
    6.58 -lemma not_chain_suc [simp]: "\<not> chain X \<Longrightarrow> suc X = X"
    6.59 -  by (simp add: suc_def)
    6.60 -
    6.61 -lemma maxchain_suc [simp]: "maxchain X \<Longrightarrow> suc X = X"
    6.62 -  by (simp add: suc_def)
    6.63 -
    6.64 -lemma suc_subset: "X \<subseteq> suc X"
    6.65 -  by (auto simp: suc_def maxchain_def intro: someI2)
    6.66 -
    6.67 -lemma chain_empty [simp]: "chain {}"
    6.68 -  by (auto simp: chain_def)
    6.69 -
    6.70 -lemma not_maxchain_Some:
    6.71 -  "chain C \<Longrightarrow> \<not> maxchain C \<Longrightarrow> C <c (SOME D. C <c D)"
    6.72 -  by (rule someI_ex) (auto simp: maxchain_def)
    6.73 -
    6.74 -lemma suc_not_equals:
    6.75 -  "chain C \<Longrightarrow> \<not> maxchain C \<Longrightarrow> suc C \<noteq> C"
    6.76 -  by (auto simp: suc_def) (metis (no_types) less_irrefl not_maxchain_Some)
    6.77 -
    6.78 -lemma subset_suc:
    6.79 -  assumes "X \<subseteq> Y" shows "X \<subseteq> suc Y"
    6.80 -  using assms by (rule subset_trans) (rule suc_subset)
    6.81 -
    6.82 -text {*We build a set @{term \<C>} that is closed under applications
    6.83 -of @{term suc} and contains the union of all its subsets.*}
    6.84 -inductive_set suc_Union_closed ("\<C>") where
    6.85 -  suc: "X \<in> \<C> \<Longrightarrow> suc X \<in> \<C>" |
    6.86 -  Union [unfolded Pow_iff]: "X \<in> Pow \<C> \<Longrightarrow> \<Union>X \<in> \<C>"
    6.87 -
    6.88 -text {*Since the empty set as well as the set itself is a subset of
    6.89 -every set, @{term \<C>} contains at least @{term "{} \<in> \<C>"} and
    6.90 -@{term "\<Union>\<C> \<in> \<C>"}.*}
    6.91 -lemma
    6.92 -  suc_Union_closed_empty: "{} \<in> \<C>" and
    6.93 -  suc_Union_closed_Union: "\<Union>\<C> \<in> \<C>"
    6.94 -  using Union [of "{}"] and Union [of "\<C>"] by simp+
    6.95 -text {*Thus closure under @{term suc} will hit a maximal chain
    6.96 -eventually, as is shown below.*}
    6.97 -
    6.98 -lemma suc_Union_closed_induct [consumes 1, case_names suc Union,
    6.99 -  induct pred: suc_Union_closed]:
   6.100 -  assumes "X \<in> \<C>"
   6.101 -    and "\<And>X. \<lbrakk>X \<in> \<C>; Q X\<rbrakk> \<Longrightarrow> Q (suc X)"
   6.102 -    and "\<And>X. \<lbrakk>X \<subseteq> \<C>; \<forall>x\<in>X. Q x\<rbrakk> \<Longrightarrow> Q (\<Union>X)"
   6.103 -  shows "Q X"
   6.104 -  using assms by (induct) blast+
   6.105 -
   6.106 -lemma suc_Union_closed_cases [consumes 1, case_names suc Union,
   6.107 -  cases pred: suc_Union_closed]:
   6.108 -  assumes "X \<in> \<C>"
   6.109 -    and "\<And>Y. \<lbrakk>X = suc Y; Y \<in> \<C>\<rbrakk> \<Longrightarrow> Q"
   6.110 -    and "\<And>Y. \<lbrakk>X = \<Union>Y; Y \<subseteq> \<C>\<rbrakk> \<Longrightarrow> Q"
   6.111 -  shows "Q"
   6.112 -  using assms by (cases) simp+
   6.113 -
   6.114 -text {*On chains, @{term suc} yields a chain.*}
   6.115 -lemma chain_suc:
   6.116 -  assumes "chain X" shows "chain (suc X)"
   6.117 -  using assms
   6.118 -  by (cases "\<not> chain X \<or> maxchain X")
   6.119 -     (force simp: suc_def dest: not_maxchain_Some)+
   6.120 -
   6.121 -lemma chain_sucD:
   6.122 -  assumes "chain X" shows "suc X \<subseteq> A \<and> chain (suc X)"
   6.123 -proof -
   6.124 -  from `chain X` have *: "chain (suc X)" by (rule chain_suc)
   6.125 -  then have "suc X \<subseteq> A" unfolding chain_def by blast
   6.126 -  with * show ?thesis by blast
   6.127 -qed
   6.128 -
   6.129 -lemma suc_Union_closed_total':
   6.130 -  assumes "X \<in> \<C>" and "Y \<in> \<C>"
   6.131 -    and *: "\<And>Z. Z \<in> \<C> \<Longrightarrow> Z \<subseteq> Y \<Longrightarrow> Z = Y \<or> suc Z \<subseteq> Y"
   6.132 -  shows "X \<subseteq> Y \<or> suc Y \<subseteq> X"
   6.133 -  using `X \<in> \<C>`
   6.134 -proof (induct)
   6.135 -  case (suc X)
   6.136 -  with * show ?case by (blast del: subsetI intro: subset_suc)
   6.137 -qed blast
   6.138 -
   6.139 -lemma suc_Union_closed_subsetD:
   6.140 -  assumes "Y \<subseteq> X" and "X \<in> \<C>" and "Y \<in> \<C>"
   6.141 -  shows "X = Y \<or> suc Y \<subseteq> X"
   6.142 -  using assms(2-, 1)
   6.143 -proof (induct arbitrary: Y)
   6.144 -  case (suc X)
   6.145 -  note * = `\<And>Y. \<lbrakk>Y \<in> \<C>; Y \<subseteq> X\<rbrakk> \<Longrightarrow> X = Y \<or> suc Y \<subseteq> X`
   6.146 -  with suc_Union_closed_total' [OF `Y \<in> \<C>` `X \<in> \<C>`]
   6.147 -    have "Y \<subseteq> X \<or> suc X \<subseteq> Y" by blast
   6.148 -  then show ?case
   6.149 -  proof
   6.150 -    assume "Y \<subseteq> X"
   6.151 -    with * and `Y \<in> \<C>` have "X = Y \<or> suc Y \<subseteq> X" by blast
   6.152 -    then show ?thesis
   6.153 -    proof
   6.154 -      assume "X = Y" then show ?thesis by simp
   6.155 -    next
   6.156 -      assume "suc Y \<subseteq> X"
   6.157 -      then have "suc Y \<subseteq> suc X" by (rule subset_suc)
   6.158 -      then show ?thesis by simp
   6.159 -    qed
   6.160 -  next
   6.161 -    assume "suc X \<subseteq> Y"
   6.162 -    with `Y \<subseteq> suc X` show ?thesis by blast
   6.163 -  qed
   6.164 -next
   6.165 -  case (Union X)
   6.166 -  show ?case
   6.167 -  proof (rule ccontr)
   6.168 -    assume "\<not> ?thesis"
   6.169 -    with `Y \<subseteq> \<Union>X` obtain x y z
   6.170 -    where "\<not> suc Y \<subseteq> \<Union>X"
   6.171 -      and "x \<in> X" and "y \<in> x" and "y \<notin> Y"
   6.172 -      and "z \<in> suc Y" and "\<forall>x\<in>X. z \<notin> x" by blast
   6.173 -    with `X \<subseteq> \<C>` have "x \<in> \<C>" by blast
   6.174 -    from Union and `x \<in> X`
   6.175 -      have *: "\<And>y. \<lbrakk>y \<in> \<C>; y \<subseteq> x\<rbrakk> \<Longrightarrow> x = y \<or> suc y \<subseteq> x" by blast
   6.176 -    with suc_Union_closed_total' [OF `Y \<in> \<C>` `x \<in> \<C>`]
   6.177 -      have "Y \<subseteq> x \<or> suc x \<subseteq> Y" by blast
   6.178 -    then show False
   6.179 -    proof
   6.180 -      assume "Y \<subseteq> x"
   6.181 -      with * [OF `Y \<in> \<C>`] have "x = Y \<or> suc Y \<subseteq> x" by blast
   6.182 -      then show False
   6.183 -      proof
   6.184 -        assume "x = Y" with `y \<in> x` and `y \<notin> Y` show False by blast
   6.185 -      next
   6.186 -        assume "suc Y \<subseteq> x"
   6.187 -        with `x \<in> X` have "suc Y \<subseteq> \<Union>X" by blast
   6.188 -        with `\<not> suc Y \<subseteq> \<Union>X` show False by contradiction
   6.189 -      qed
   6.190 -    next
   6.191 -      assume "suc x \<subseteq> Y"
   6.192 -      moreover from suc_subset and `y \<in> x` have "y \<in> suc x" by blast
   6.193 -      ultimately show False using `y \<notin> Y` by blast
   6.194 -    qed
   6.195 -  qed
   6.196 -qed
   6.197 -
   6.198 -text {*The elements of @{term \<C>} are totally ordered by the subset relation.*}
   6.199 -lemma suc_Union_closed_total:
   6.200 -  assumes "X \<in> \<C>" and "Y \<in> \<C>"
   6.201 -  shows "X \<subseteq> Y \<or> Y \<subseteq> X"
   6.202 -proof (cases "\<forall>Z\<in>\<C>. Z \<subseteq> Y \<longrightarrow> Z = Y \<or> suc Z \<subseteq> Y")
   6.203 -  case True
   6.204 -  with suc_Union_closed_total' [OF assms]
   6.205 -    have "X \<subseteq> Y \<or> suc Y \<subseteq> X" by blast
   6.206 -  then show ?thesis using suc_subset [of Y] by blast
   6.207 -next
   6.208 -  case False
   6.209 -  then obtain Z
   6.210 -    where "Z \<in> \<C>" and "Z \<subseteq> Y" and "Z \<noteq> Y" and "\<not> suc Z \<subseteq> Y" by blast
   6.211 -  with suc_Union_closed_subsetD and `Y \<in> \<C>` show ?thesis by blast
   6.212 -qed
   6.213 -
   6.214 -text {*Once we hit a fixed point w.r.t. @{term suc}, all other elements
   6.215 -of @{term \<C>} are subsets of this fixed point.*}
   6.216 -lemma suc_Union_closed_suc:
   6.217 -  assumes "X \<in> \<C>" and "Y \<in> \<C>" and "suc Y = Y"
   6.218 -  shows "X \<subseteq> Y"
   6.219 -using `X \<in> \<C>`
   6.220 -proof (induct)
   6.221 -  case (suc X)
   6.222 -  with `Y \<in> \<C>` and suc_Union_closed_subsetD
   6.223 -    have "X = Y \<or> suc X \<subseteq> Y" by blast
   6.224 -  then show ?case by (auto simp: `suc Y = Y`)
   6.225 -qed blast
   6.226 -
   6.227 -lemma eq_suc_Union:
   6.228 -  assumes "X \<in> \<C>"
   6.229 -  shows "suc X = X \<longleftrightarrow> X = \<Union>\<C>"
   6.230 -proof
   6.231 -  assume "suc X = X"
   6.232 -  with suc_Union_closed_suc [OF suc_Union_closed_Union `X \<in> \<C>`]
   6.233 -    have "\<Union>\<C> \<subseteq> X" .
   6.234 -  with `X \<in> \<C>` show "X = \<Union>\<C>" by blast
   6.235 -next
   6.236 -  from `X \<in> \<C>` have "suc X \<in> \<C>" by (rule suc)
   6.237 -  then have "suc X \<subseteq> \<Union>\<C>" by blast
   6.238 -  moreover assume "X = \<Union>\<C>"
   6.239 -  ultimately have "suc X \<subseteq> X" by simp
   6.240 -  moreover have "X \<subseteq> suc X" by (rule suc_subset)
   6.241 -  ultimately show "suc X = X" ..
   6.242 -qed
   6.243 -
   6.244 -lemma suc_in_carrier:
   6.245 -  assumes "X \<subseteq> A"
   6.246 -  shows "suc X \<subseteq> A"
   6.247 -  using assms
   6.248 -  by (cases "\<not> chain X \<or> maxchain X")
   6.249 -     (auto dest: chain_sucD)
   6.250 -
   6.251 -lemma suc_Union_closed_in_carrier:
   6.252 -  assumes "X \<in> \<C>"
   6.253 -  shows "X \<subseteq> A"
   6.254 -  using assms
   6.255 -  by (induct) (auto dest: suc_in_carrier)
   6.256 -
   6.257 -text {*All elements of @{term \<C>} are chains.*}
   6.258 -lemma suc_Union_closed_chain:
   6.259 -  assumes "X \<in> \<C>"
   6.260 -  shows "chain X"
   6.261 -using assms
   6.262 -proof (induct)
   6.263 -  case (suc X) then show ?case by (simp add: suc_def) (metis (no_types) not_maxchain_Some)
   6.264 -next
   6.265 -  case (Union X)
   6.266 -  then have "\<Union>X \<subseteq> A" by (auto dest: suc_Union_closed_in_carrier)
   6.267 -  moreover have "\<forall>x\<in>\<Union>X. \<forall>y\<in>\<Union>X. x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
   6.268 -  proof (intro ballI)
   6.269 -    fix x y
   6.270 -    assume "x \<in> \<Union>X" and "y \<in> \<Union>X"
   6.271 -    then obtain u v where "x \<in> u" and "u \<in> X" and "y \<in> v" and "v \<in> X" by blast
   6.272 -    with Union have "u \<in> \<C>" and "v \<in> \<C>" and "chain u" and "chain v" by blast+
   6.273 -    with suc_Union_closed_total have "u \<subseteq> v \<or> v \<subseteq> u" by blast
   6.274 -    then show "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
   6.275 -    proof
   6.276 -      assume "u \<subseteq> v"
   6.277 -      from `chain v` show ?thesis
   6.278 -      proof (rule chain_total)
   6.279 -        show "y \<in> v" by fact
   6.280 -        show "x \<in> v" using `u \<subseteq> v` and `x \<in> u` by blast
   6.281 -      qed
   6.282 -    next
   6.283 -      assume "v \<subseteq> u"
   6.284 -      from `chain u` show ?thesis
   6.285 -      proof (rule chain_total)
   6.286 -        show "x \<in> u" by fact
   6.287 -        show "y \<in> u" using `v \<subseteq> u` and `y \<in> v` by blast
   6.288 -      qed
   6.289 -    qed
   6.290 -  qed
   6.291 -  ultimately show ?case unfolding chain_def ..
   6.292 -qed
   6.293 -
   6.294 -subsubsection {* Hausdorff's Maximum Principle *}
   6.295 -
   6.296 -text {*There exists a maximal totally ordered subset of @{term A}. (Note that we do not
   6.297 -require @{term A} to be partially ordered.)*}
   6.298 -
   6.299 -theorem Hausdorff: "\<exists>C. maxchain C"
   6.300 -proof -
   6.301 -  let ?M = "\<Union>\<C>"
   6.302 -  have "maxchain ?M"
   6.303 -  proof (rule ccontr)
   6.304 -    assume "\<not> maxchain ?M"
   6.305 -    then have "suc ?M \<noteq> ?M"
   6.306 -      using suc_not_equals and
   6.307 -      suc_Union_closed_chain [OF suc_Union_closed_Union] by simp
   6.308 -    moreover have "suc ?M = ?M"
   6.309 -      using eq_suc_Union [OF suc_Union_closed_Union] by simp
   6.310 -    ultimately show False by contradiction
   6.311 -  qed
   6.312 -  then show ?thesis by blast
   6.313 -qed
   6.314 -
   6.315 -text {*Make notation @{term \<C>} available again.*}
   6.316 -no_notation suc_Union_closed ("\<C>")
   6.317 -
   6.318 -lemma chain_extend:
   6.319 -  "chain C \<Longrightarrow> z \<in> A \<Longrightarrow> \<forall>x\<in>C. x \<sqsubseteq> z \<Longrightarrow> chain ({z} \<union> C)"
   6.320 -  unfolding chain_def by blast
   6.321 -
   6.322 -lemma maxchain_imp_chain:
   6.323 -  "maxchain C \<Longrightarrow> chain C"
   6.324 -  by (simp add: maxchain_def)
   6.325 -
   6.326 -end
   6.327 -
   6.328 -text {*Hide constant @{const pred_on.suc_Union_closed}, which was just needed
   6.329 -for the proof of Hausforff's maximum principle.*}
   6.330 -hide_const pred_on.suc_Union_closed
   6.331 -
   6.332 -lemma chain_mono:
   6.333 -  assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; P x y\<rbrakk> \<Longrightarrow> Q x y"
   6.334 -    and "pred_on.chain A P C"
   6.335 -  shows "pred_on.chain A Q C"
   6.336 -  using assms unfolding pred_on.chain_def by blast
   6.337 -
   6.338 -subsubsection {* Results for the proper subset relation *}
   6.339 -
   6.340 -interpretation subset: pred_on "A" "op \<subset>" for A .
   6.341 -
   6.342 -lemma subset_maxchain_max:
   6.343 -  assumes "subset.maxchain A C" and "X \<in> A" and "\<Union>C \<subseteq> X"
   6.344 -  shows "\<Union>C = X"
   6.345 -proof (rule ccontr)
   6.346 -  let ?C = "{X} \<union> C"
   6.347 -  from `subset.maxchain A C` have "subset.chain A C"
   6.348 -    and *: "\<And>S. subset.chain A S \<Longrightarrow> \<not> C \<subset> S"
   6.349 -    by (auto simp: subset.maxchain_def)
   6.350 -  moreover have "\<forall>x\<in>C. x \<subseteq> X" using `\<Union>C \<subseteq> X` by auto
   6.351 -  ultimately have "subset.chain A ?C"
   6.352 -    using subset.chain_extend [of A C X] and `X \<in> A` by auto
   6.353 -  moreover assume **: "\<Union>C \<noteq> X"
   6.354 -  moreover from ** have "C \<subset> ?C" using `\<Union>C \<subseteq> X` by auto
   6.355 -  ultimately show False using * by blast
   6.356 -qed
   6.357 -
   6.358 -subsubsection {* Zorn's lemma *}
   6.359 -
   6.360 -text {*If every chain has an upper bound, then there is a maximal set.*}
   6.361 -lemma subset_Zorn:
   6.362 -  assumes "\<And>C. subset.chain A C \<Longrightarrow> \<exists>U\<in>A. \<forall>X\<in>C. X \<subseteq> U"
   6.363 -  shows "\<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
   6.364 -proof -
   6.365 -  from subset.Hausdorff [of A] obtain M where "subset.maxchain A M" ..
   6.366 -  then have "subset.chain A M" by (rule subset.maxchain_imp_chain)
   6.367 -  with assms obtain Y where "Y \<in> A" and "\<forall>X\<in>M. X \<subseteq> Y" by blast
   6.368 -  moreover have "\<forall>X\<in>A. Y \<subseteq> X \<longrightarrow> Y = X"
   6.369 -  proof (intro ballI impI)
   6.370 -    fix X
   6.371 -    assume "X \<in> A" and "Y \<subseteq> X"
   6.372 -    show "Y = X"
   6.373 -    proof (rule ccontr)
   6.374 -      assume "Y \<noteq> X"
   6.375 -      with `Y \<subseteq> X` have "\<not> X \<subseteq> Y" by blast
   6.376 -      from subset.chain_extend [OF `subset.chain A M` `X \<in> A`] and `\<forall>X\<in>M. X \<subseteq> Y`
   6.377 -        have "subset.chain A ({X} \<union> M)" using `Y \<subseteq> X` by auto
   6.378 -      moreover have "M \<subset> {X} \<union> M" using `\<forall>X\<in>M. X \<subseteq> Y` and `\<not> X \<subseteq> Y` by auto
   6.379 -      ultimately show False
   6.380 -        using `subset.maxchain A M` by (auto simp: subset.maxchain_def)
   6.381 -    qed
   6.382 -  qed
   6.383 -  ultimately show ?thesis by metis
   6.384 -qed
   6.385 -
   6.386 -text{*Alternative version of Zorn's lemma for the subset relation.*}
   6.387 -lemma subset_Zorn':
   6.388 -  assumes "\<And>C. subset.chain A C \<Longrightarrow> \<Union>C \<in> A"
   6.389 -  shows "\<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
   6.390 -proof -
   6.391 -  from subset.Hausdorff [of A] obtain M where "subset.maxchain A M" ..
   6.392 -  then have "subset.chain A M" by (rule subset.maxchain_imp_chain)
   6.393 -  with assms have "\<Union>M \<in> A" .
   6.394 -  moreover have "\<forall>Z\<in>A. \<Union>M \<subseteq> Z \<longrightarrow> \<Union>M = Z"
   6.395 -  proof (intro ballI impI)
   6.396 -    fix Z
   6.397 -    assume "Z \<in> A" and "\<Union>M \<subseteq> Z"
   6.398 -    with subset_maxchain_max [OF `subset.maxchain A M`]
   6.399 -      show "\<Union>M = Z" .
   6.400 -  qed
   6.401 -  ultimately show ?thesis by blast
   6.402 -qed
   6.403 -
   6.404 -
   6.405 -subsection {* Zorn's Lemma for Partial Orders *}
   6.406 -
   6.407 -text {*Relate old to new definitions.*}
   6.408 -
   6.409 -(* Define globally? In Set.thy? *)
   6.410 -definition chain_subset :: "'a set set \<Rightarrow> bool" ("chain\<^sub>\<subseteq>") where
   6.411 -  "chain\<^sub>\<subseteq> C \<longleftrightarrow> (\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A)"
   6.412 -
   6.413 -definition chains :: "'a set set \<Rightarrow> 'a set set set" where
   6.414 -  "chains A = {C. C \<subseteq> A \<and> chain\<^sub>\<subseteq> C}"
   6.415 -
   6.416 -(* Define globally? In Relation.thy? *)
   6.417 -definition Chains :: "('a \<times> 'a) set \<Rightarrow> 'a set set" where
   6.418 -  "Chains r = {C. \<forall>a\<in>C. \<forall>b\<in>C. (a, b) \<in> r \<or> (b, a) \<in> r}"
   6.419 -
   6.420 -lemma chains_extend:
   6.421 -  "[| c \<in> chains S; z \<in> S; \<forall>x \<in> c. x \<subseteq> (z:: 'a set) |] ==> {z} Un c \<in> chains S"
   6.422 -  by (unfold chains_def chain_subset_def) blast
   6.423 -
   6.424 -lemma mono_Chains: "r \<subseteq> s \<Longrightarrow> Chains r \<subseteq> Chains s"
   6.425 -  unfolding Chains_def by blast
   6.426 -
   6.427 -lemma chain_subset_alt_def: "chain\<^sub>\<subseteq> C = subset.chain UNIV C"
   6.428 -  unfolding chain_subset_def subset.chain_def by fast
   6.429 -
   6.430 -lemma chains_alt_def: "chains A = {C. subset.chain A C}"
   6.431 -  by (simp add: chains_def chain_subset_alt_def subset.chain_def)
   6.432 -
   6.433 -lemma Chains_subset:
   6.434 -  "Chains r \<subseteq> {C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C}"
   6.435 -  by (force simp add: Chains_def pred_on.chain_def)
   6.436 -
   6.437 -lemma Chains_subset':
   6.438 -  assumes "refl r"
   6.439 -  shows "{C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C} \<subseteq> Chains r"
   6.440 -  using assms
   6.441 -  by (auto simp add: Chains_def pred_on.chain_def refl_on_def)
   6.442 -
   6.443 -lemma Chains_alt_def:
   6.444 -  assumes "refl r"
   6.445 -  shows "Chains r = {C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C}"
   6.446 -  using assms
   6.447 -  by (metis Chains_subset Chains_subset' subset_antisym)
   6.448 -
   6.449 -lemma Zorn_Lemma:
   6.450 -  "\<forall>C\<in>chains A. \<Union>C \<in> A \<Longrightarrow> \<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
   6.451 -  using subset_Zorn' [of A] by (force simp: chains_alt_def)
   6.452 -
   6.453 -lemma Zorn_Lemma2:
   6.454 -  "\<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"
   6.455 -  using subset_Zorn [of A] by (auto simp: chains_alt_def)
   6.456 -
   6.457 -text{*Various other lemmas*}
   6.458 -
   6.459 -lemma chainsD: "[| c \<in> chains S; x \<in> c; y \<in> c |] ==> x \<subseteq> y | y \<subseteq> x"
   6.460 -by (unfold chains_def chain_subset_def) blast
   6.461 -
   6.462 -lemma chainsD2: "!!(c :: 'a set set). c \<in> chains S ==> c \<subseteq> S"
   6.463 -by (unfold chains_def) blast
   6.464 -
   6.465 -lemma Zorns_po_lemma:
   6.466 -  assumes po: "Partial_order r"
   6.467 -    and u: "\<forall>C\<in>Chains r. \<exists>u\<in>Field r. \<forall>a\<in>C. (a, u) \<in> r"
   6.468 -  shows "\<exists>m\<in>Field r. \<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m"
   6.469 -proof -
   6.470 -  have "Preorder r" using po by (simp add: partial_order_on_def)
   6.471 ---{* Mirror r in the set of subsets below (wrt r) elements of A*}
   6.472 -  let ?B = "%x. r\<inverse> `` {x}" let ?S = "?B ` Field r"
   6.473 -  {
   6.474 -    fix C assume 1: "C \<subseteq> ?S" and 2: "\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A"
   6.475 -    let ?A = "{x\<in>Field r. \<exists>M\<in>C. M = ?B x}"
   6.476 -    have "C = ?B ` ?A" using 1 by (auto simp: image_def)
   6.477 -    have "?A \<in> Chains r"
   6.478 -    proof (simp add: Chains_def, intro allI impI, elim conjE)
   6.479 -      fix a b
   6.480 -      assume "a \<in> Field r" and "?B a \<in> C" and "b \<in> Field r" and "?B b \<in> C"
   6.481 -      hence "?B a \<subseteq> ?B b \<or> ?B b \<subseteq> ?B a" using 2 by auto
   6.482 -      thus "(a, b) \<in> r \<or> (b, a) \<in> r"
   6.483 -        using `Preorder r` and `a \<in> Field r` and `b \<in> Field r`
   6.484 -        by (simp add:subset_Image1_Image1_iff)
   6.485 -    qed
   6.486 -    then obtain u where uA: "u \<in> Field r" "\<forall>a\<in>?A. (a, u) \<in> r" using u by auto
   6.487 -    have "\<forall>A\<in>C. A \<subseteq> r\<inverse> `` {u}" (is "?P u")
   6.488 -    proof auto
   6.489 -      fix a B assume aB: "B \<in> C" "a \<in> B"
   6.490 -      with 1 obtain x where "x \<in> Field r" and "B = r\<inverse> `` {x}" by auto
   6.491 -      thus "(a, u) \<in> r" using uA and aB and `Preorder r`
   6.492 -        unfolding preorder_on_def refl_on_def by simp (fast dest: transD)
   6.493 -    qed
   6.494 -    then have "\<exists>u\<in>Field r. ?P u" using `u \<in> Field r` by blast
   6.495 -  }
   6.496 -  then have "\<forall>C\<in>chains ?S. \<exists>U\<in>?S. \<forall>A\<in>C. A \<subseteq> U"
   6.497 -    by (auto simp: chains_def chain_subset_def)
   6.498 -  from Zorn_Lemma2 [OF this]
   6.499 -  obtain m B where "m \<in> Field r" and "B = r\<inverse> `` {m}"
   6.500 -    and "\<forall>x\<in>Field r. B \<subseteq> r\<inverse> `` {x} \<longrightarrow> r\<inverse> `` {x} = B"
   6.501 -    by auto
   6.502 -  hence "\<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m"
   6.503 -    using po and `Preorder r` and `m \<in> Field r`
   6.504 -    by (auto simp: subset_Image1_Image1_iff Partial_order_eq_Image1_Image1_iff)
   6.505 -  thus ?thesis using `m \<in> Field r` by blast
   6.506 -qed
   6.507 -
   6.508 -
   6.509 -subsection {* The Well Ordering Theorem *}
   6.510 -
   6.511 -(* The initial segment of a relation appears generally useful.
   6.512 -   Move to Relation.thy?
   6.513 -   Definition correct/most general?
   6.514 -   Naming?
   6.515 -*)
   6.516 -definition init_seg_of :: "(('a \<times> 'a) set \<times> ('a \<times> 'a) set) set" where
   6.517 -  "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)}"
   6.518 -
   6.519 -abbreviation
   6.520 -  initialSegmentOf :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool" (infix "initial'_segment'_of" 55)
   6.521 -where
   6.522 -  "r initial_segment_of s \<equiv> (r, s) \<in> init_seg_of"
   6.523 -
   6.524 -lemma refl_on_init_seg_of [simp]: "r initial_segment_of r"
   6.525 -  by (simp add: init_seg_of_def)
   6.526 -
   6.527 -lemma trans_init_seg_of:
   6.528 -  "r initial_segment_of s \<Longrightarrow> s initial_segment_of t \<Longrightarrow> r initial_segment_of t"
   6.529 -  by (simp (no_asm_use) add: init_seg_of_def) blast
   6.530 -
   6.531 -lemma antisym_init_seg_of:
   6.532 -  "r initial_segment_of s \<Longrightarrow> s initial_segment_of r \<Longrightarrow> r = s"
   6.533 -  unfolding init_seg_of_def by safe
   6.534 -
   6.535 -lemma Chains_init_seg_of_Union:
   6.536 -  "R \<in> Chains init_seg_of \<Longrightarrow> r\<in>R \<Longrightarrow> r initial_segment_of \<Union>R"
   6.537 -  by (auto simp: init_seg_of_def Ball_def Chains_def) blast
   6.538 -
   6.539 -lemma chain_subset_trans_Union:
   6.540 -  "chain\<^sub>\<subseteq> R \<Longrightarrow> \<forall>r\<in>R. trans r \<Longrightarrow> trans (\<Union>R)"
   6.541 -apply (auto simp add: chain_subset_def)
   6.542 -apply (simp (no_asm_use) add: trans_def)
   6.543 -by (metis subsetD)
   6.544 -
   6.545 -lemma chain_subset_antisym_Union:
   6.546 -  "chain\<^sub>\<subseteq> R \<Longrightarrow> \<forall>r\<in>R. antisym r \<Longrightarrow> antisym (\<Union>R)"
   6.547 -unfolding chain_subset_def antisym_def
   6.548 -apply simp
   6.549 -by (metis (no_types) subsetD)
   6.550 -
   6.551 -lemma chain_subset_Total_Union:
   6.552 -  assumes "chain\<^sub>\<subseteq> R" and "\<forall>r\<in>R. Total r"
   6.553 -  shows "Total (\<Union>R)"
   6.554 -proof (simp add: total_on_def Ball_def, auto del: disjCI)
   6.555 -  fix r s a b assume A: "r \<in> R" "s \<in> R" "a \<in> Field r" "b \<in> Field s" "a \<noteq> b"
   6.556 -  from `chain\<^sub>\<subseteq> R` and `r \<in> R` and `s \<in> R` have "r \<subseteq> s \<or> s \<subseteq> r"
   6.557 -    by (auto simp add: chain_subset_def)
   6.558 -  thus "(\<exists>r\<in>R. (a, b) \<in> r) \<or> (\<exists>r\<in>R. (b, a) \<in> r)"
   6.559 -  proof
   6.560 -    assume "r \<subseteq> s" hence "(a, b) \<in> s \<or> (b, a) \<in> s" using assms(2) A
   6.561 -      by (simp add: total_on_def) (metis (no_types) mono_Field subsetD)
   6.562 -    thus ?thesis using `s \<in> R` by blast
   6.563 -  next
   6.564 -    assume "s \<subseteq> r" hence "(a, b) \<in> r \<or> (b, a) \<in> r" using assms(2) A
   6.565 -      by (simp add: total_on_def) (metis (no_types) mono_Field subsetD)
   6.566 -    thus ?thesis using `r \<in> R` by blast
   6.567 -  qed
   6.568 -qed
   6.569 -
   6.570 -lemma wf_Union_wf_init_segs:
   6.571 -  assumes "R \<in> Chains init_seg_of" and "\<forall>r\<in>R. wf r"
   6.572 -  shows "wf (\<Union>R)"
   6.573 -proof(simp add: wf_iff_no_infinite_down_chain, rule ccontr, auto)
   6.574 -  fix f assume 1: "\<forall>i. \<exists>r\<in>R. (f (Suc i), f i) \<in> r"
   6.575 -  then obtain r where "r \<in> R" and "(f (Suc 0), f 0) \<in> r" by auto
   6.576 -  { fix i have "(f (Suc i), f i) \<in> r"
   6.577 -    proof (induct i)
   6.578 -      case 0 show ?case by fact
   6.579 -    next
   6.580 -      case (Suc i)
   6.581 -      then obtain s where s: "s \<in> R" "(f (Suc (Suc i)), f(Suc i)) \<in> s"
   6.582 -        using 1 by auto
   6.583 -      then have "s initial_segment_of r \<or> r initial_segment_of s"
   6.584 -        using assms(1) `r \<in> R` by (simp add: Chains_def)
   6.585 -      with Suc s show ?case by (simp add: init_seg_of_def) blast
   6.586 -    qed
   6.587 -  }
   6.588 -  thus False using assms(2) and `r \<in> R`
   6.589 -    by (simp add: wf_iff_no_infinite_down_chain) blast
   6.590 -qed
   6.591 -
   6.592 -lemma initial_segment_of_Diff:
   6.593 -  "p initial_segment_of q \<Longrightarrow> p - s initial_segment_of q - s"
   6.594 -  unfolding init_seg_of_def by blast
   6.595 -
   6.596 -lemma Chains_inits_DiffI:
   6.597 -  "R \<in> Chains init_seg_of \<Longrightarrow> {r - s |r. r \<in> R} \<in> Chains init_seg_of"
   6.598 -  unfolding Chains_def by (blast intro: initial_segment_of_Diff)
   6.599 -
   6.600 -theorem well_ordering: "\<exists>r::'a rel. Well_order r \<and> Field r = UNIV"
   6.601 -proof -
   6.602 --- {*The initial segment relation on well-orders: *}
   6.603 -  let ?WO = "{r::'a rel. Well_order r}"
   6.604 -  def I \<equiv> "init_seg_of \<inter> ?WO \<times> ?WO"
   6.605 -  have I_init: "I \<subseteq> init_seg_of" by (auto simp: I_def)
   6.606 -  hence subch: "\<And>R. R \<in> Chains I \<Longrightarrow> chain\<^sub>\<subseteq> R"
   6.607 -    unfolding init_seg_of_def chain_subset_def Chains_def by blast
   6.608 -  have Chains_wo: "\<And>R r. R \<in> Chains I \<Longrightarrow> r \<in> R \<Longrightarrow> Well_order r"
   6.609 -    by (simp add: Chains_def I_def) blast
   6.610 -  have FI: "Field I = ?WO" by (auto simp add: I_def init_seg_of_def Field_def)
   6.611 -  hence 0: "Partial_order I"
   6.612 -    by (auto simp: partial_order_on_def preorder_on_def antisym_def antisym_init_seg_of refl_on_def
   6.613 -      trans_def I_def elim!: trans_init_seg_of)
   6.614 --- {*I-chains have upper bounds in ?WO wrt I: their Union*}
   6.615 -  { fix R assume "R \<in> Chains I"
   6.616 -    hence Ris: "R \<in> Chains init_seg_of" using mono_Chains [OF I_init] by blast
   6.617 -    have subch: "chain\<^sub>\<subseteq> R" using `R : Chains I` I_init
   6.618 -      by (auto simp: init_seg_of_def chain_subset_def Chains_def)
   6.619 -    have "\<forall>r\<in>R. Refl r" and "\<forall>r\<in>R. trans r" and "\<forall>r\<in>R. antisym r"
   6.620 -      and "\<forall>r\<in>R. Total r" and "\<forall>r\<in>R. wf (r - Id)"
   6.621 -      using Chains_wo [OF `R \<in> Chains I`] by (simp_all add: order_on_defs)
   6.622 -    have "Refl (\<Union>R)" using `\<forall>r\<in>R. Refl r` unfolding refl_on_def by fastforce
   6.623 -    moreover have "trans (\<Union>R)"
   6.624 -      by (rule chain_subset_trans_Union [OF subch `\<forall>r\<in>R. trans r`])
   6.625 -    moreover have "antisym (\<Union>R)"
   6.626 -      by (rule chain_subset_antisym_Union [OF subch `\<forall>r\<in>R. antisym r`])
   6.627 -    moreover have "Total (\<Union>R)"
   6.628 -      by (rule chain_subset_Total_Union [OF subch `\<forall>r\<in>R. Total r`])
   6.629 -    moreover have "wf ((\<Union>R) - Id)"
   6.630 -    proof -
   6.631 -      have "(\<Union>R) - Id = \<Union>{r - Id | r. r \<in> R}" by blast
   6.632 -      with `\<forall>r\<in>R. wf (r - Id)` and wf_Union_wf_init_segs [OF Chains_inits_DiffI [OF Ris]]
   6.633 -      show ?thesis by fastforce
   6.634 -    qed
   6.635 -    ultimately have "Well_order (\<Union>R)" by(simp add:order_on_defs)
   6.636 -    moreover have "\<forall>r \<in> R. r initial_segment_of \<Union>R" using Ris
   6.637 -      by(simp add: Chains_init_seg_of_Union)
   6.638 -    ultimately have "\<Union>R \<in> ?WO \<and> (\<forall>r\<in>R. (r, \<Union>R) \<in> I)"
   6.639 -      using mono_Chains [OF I_init] and `R \<in> Chains I`
   6.640 -      by (simp (no_asm) add: I_def del: Field_Union) (metis Chains_wo)
   6.641 -  }
   6.642 -  hence 1: "\<forall>R \<in> Chains I. \<exists>u\<in>Field I. \<forall>r\<in>R. (r, u) \<in> I" by (subst FI) blast
   6.643 ---{*Zorn's Lemma yields a maximal well-order m:*}
   6.644 -  then obtain m::"'a rel" where "Well_order m" and
   6.645 -    max: "\<forall>r. Well_order r \<and> (m, r) \<in> I \<longrightarrow> r = m"
   6.646 -    using Zorns_po_lemma[OF 0 1] unfolding FI by fastforce
   6.647 ---{*Now show by contradiction that m covers the whole type:*}
   6.648 -  { fix x::'a assume "x \<notin> Field m"
   6.649 ---{*We assume that x is not covered and extend m at the top with x*}
   6.650 -    have "m \<noteq> {}"
   6.651 -    proof
   6.652 -      assume "m = {}"
   6.653 -      moreover have "Well_order {(x, x)}"
   6.654 -        by (simp add: order_on_defs refl_on_def trans_def antisym_def total_on_def Field_def)
   6.655 -      ultimately show False using max
   6.656 -        by (auto simp: I_def init_seg_of_def simp del: Field_insert)
   6.657 -    qed
   6.658 -    hence "Field m \<noteq> {}" by(auto simp:Field_def)
   6.659 -    moreover have "wf (m - Id)" using `Well_order m`
   6.660 -      by (simp add: well_order_on_def)
   6.661 ---{*The extension of m by x:*}
   6.662 -    let ?s = "{(a, x) | a. a \<in> Field m}"
   6.663 -    let ?m = "insert (x, x) m \<union> ?s"
   6.664 -    have Fm: "Field ?m = insert x (Field m)"
   6.665 -      by (auto simp: Field_def)
   6.666 -    have "Refl m" and "trans m" and "antisym m" and "Total m" and "wf (m - Id)"
   6.667 -      using `Well_order m` by (simp_all add: order_on_defs)
   6.668 ---{*We show that the extension is a well-order*}
   6.669 -    have "Refl ?m" using `Refl m` Fm unfolding refl_on_def by blast
   6.670 -    moreover have "trans ?m" using `trans m` and `x \<notin> Field m`
   6.671 -      unfolding trans_def Field_def by blast
   6.672 -    moreover have "antisym ?m" using `antisym m` and `x \<notin> Field m`
   6.673 -      unfolding antisym_def Field_def by blast
   6.674 -    moreover have "Total ?m" using `Total m` and Fm by (auto simp: total_on_def)
   6.675 -    moreover have "wf (?m - Id)"
   6.676 -    proof -
   6.677 -      have "wf ?s" using `x \<notin> Field m`
   6.678 -        by (auto simp add: wf_eq_minimal Field_def) metis
   6.679 -      thus ?thesis using `wf (m - Id)` and `x \<notin> Field m`
   6.680 -        wf_subset [OF `wf ?s` Diff_subset]
   6.681 -        unfolding Un_Diff Field_def by (auto intro: wf_Un)
   6.682 -    qed
   6.683 -    ultimately have "Well_order ?m" by (simp add: order_on_defs)
   6.684 ---{*We show that the extension is above m*}
   6.685 -    moreover have "(m, ?m) \<in> I" using `Well_order ?m` and `Well_order m` and `x \<notin> Field m`
   6.686 -      by (fastforce simp: I_def init_seg_of_def Field_def)
   6.687 -    ultimately
   6.688 ---{*This contradicts maximality of m:*}
   6.689 -    have False using max and `x \<notin> Field m` unfolding Field_def by blast
   6.690 -  }
   6.691 -  hence "Field m = UNIV" by auto
   6.692 -  with `Well_order m` show ?thesis by blast
   6.693 -qed
   6.694 -
   6.695 -corollary well_order_on: "\<exists>r::'a rel. well_order_on A r"
   6.696 -proof -
   6.697 -  obtain r::"'a rel" where wo: "Well_order r" and univ: "Field r = UNIV"
   6.698 -    using well_ordering [where 'a = "'a"] by blast
   6.699 -  let ?r = "{(x, y). x \<in> A \<and> y \<in> A \<and> (x, y) \<in> r}"
   6.700 -  have 1: "Field ?r = A" using wo univ
   6.701 -    by (fastforce simp: Field_def order_on_defs refl_on_def)
   6.702 -  have "Refl r" and "trans r" and "antisym r" and "Total r" and "wf (r - Id)"
   6.703 -    using `Well_order r` by (simp_all add: order_on_defs)
   6.704 -  have "Refl ?r" using `Refl r` by (auto simp: refl_on_def 1 univ)
   6.705 -  moreover have "trans ?r" using `trans r`
   6.706 -    unfolding trans_def by blast
   6.707 -  moreover have "antisym ?r" using `antisym r`
   6.708 -    unfolding antisym_def by blast
   6.709 -  moreover have "Total ?r" using `Total r` by (simp add:total_on_def 1 univ)
   6.710 -  moreover have "wf (?r - Id)" by (rule wf_subset [OF `wf (r - Id)`]) blast
   6.711 -  ultimately have "Well_order ?r" by (simp add: order_on_defs)
   6.712 -  with 1 show ?thesis by auto
   6.713 -qed
   6.714 -
   6.715 -end
     7.1 --- a/src/HOL/Main.thy	Thu Jan 16 16:20:17 2014 +0100
     7.2 +++ b/src/HOL/Main.thy	Thu Jan 16 16:33:19 2014 +0100
     7.3 @@ -1,7 +1,7 @@
     7.4  header {* Main HOL *}
     7.5  
     7.6  theory Main
     7.7 -imports Predicate_Compile Nitpick Extraction Lifting_Sum List_Prefix Coinduction Order_Relation
     7.8 +imports Predicate_Compile Nitpick Extraction Lifting_Sum List_Prefix Coinduction Zorn
     7.9  begin
    7.10  
    7.11  text {*
     8.1 --- a/src/HOL/NSA/Filter.thy	Thu Jan 16 16:20:17 2014 +0100
     8.2 +++ b/src/HOL/NSA/Filter.thy	Thu Jan 16 16:33:19 2014 +0100
     8.3 @@ -7,7 +7,7 @@
     8.4  header {* Filters and Ultrafilters *}
     8.5  
     8.6  theory Filter
     8.7 -imports "~~/src/HOL/Library/Zorn" "~~/src/HOL/Library/Infinite_Set"
     8.8 +imports "~~/src/HOL/Library/Infinite_Set"
     8.9  begin
    8.10  
    8.11  subsection {* Definitions and basic properties *}
     9.1 --- a/src/HOL/ROOT	Thu Jan 16 16:20:17 2014 +0100
     9.2 +++ b/src/HOL/ROOT	Thu Jan 16 16:33:19 2014 +0100
     9.3 @@ -61,7 +61,7 @@
     9.4  
     9.5      This is the proof of the Hahn-Banach theorem for real vectorspaces,
     9.6      following H. Heuser, Funktionalanalysis, p. 228 -232. The Hahn-Banach
     9.7 -    theorem is one of the fundamental theorems of functioal analysis. It is a
     9.8 +    theorem is one of the fundamental theorems of functional analysis. It is a
     9.9      conclusion of Zorn's lemma.
    9.10  
    9.11      Two different formaulations of the theorem are presented, one for general
    10.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
    10.2 +++ b/src/HOL/Zorn.thy	Thu Jan 16 16:33:19 2014 +0100
    10.3 @@ -0,0 +1,712 @@
    10.4 +(*  Title:      HOL/Zorn.thy
    10.5 +    Author:     Jacques D. Fleuriot
    10.6 +    Author:     Tobias Nipkow, TUM
    10.7 +    Author:     Christian Sternagel, JAIST
    10.8 +
    10.9 +Zorn's Lemma (ported from Larry Paulson's Zorn.thy in ZF).
   10.10 +The well-ordering theorem.
   10.11 +*)
   10.12 +
   10.13 +header {* Zorn's Lemma *}
   10.14 +
   10.15 +theory Zorn
   10.16 +imports Order_Relation Hilbert_Choice
   10.17 +begin
   10.18 +
   10.19 +subsection {* Zorn's Lemma for the Subset Relation *}
   10.20 +
   10.21 +subsubsection {* Results that do not require an order *}
   10.22 +
   10.23 +text {*Let @{text P} be a binary predicate on the set @{text A}.*}
   10.24 +locale pred_on =
   10.25 +  fixes A :: "'a set"
   10.26 +    and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<sqsubset>" 50)
   10.27 +begin
   10.28 +
   10.29 +abbreviation Peq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<sqsubseteq>" 50) where
   10.30 +  "x \<sqsubseteq> y \<equiv> P\<^sup>=\<^sup>= x y"
   10.31 +
   10.32 +text {*A chain is a totally ordered subset of @{term A}.*}
   10.33 +definition chain :: "'a set \<Rightarrow> bool" where
   10.34 +  "chain C \<longleftrightarrow> C \<subseteq> A \<and> (\<forall>x\<in>C. \<forall>y\<in>C. x \<sqsubseteq> y \<or> y \<sqsubseteq> x)"
   10.35 +
   10.36 +text {*We call a chain that is a proper superset of some set @{term X},
   10.37 +but not necessarily a chain itself, a superchain of @{term X}.*}
   10.38 +abbreviation superchain :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" (infix "<c" 50) where
   10.39 +  "X <c C \<equiv> chain C \<and> X \<subset> C"
   10.40 +
   10.41 +text {*A maximal chain is a chain that does not have a superchain.*}
   10.42 +definition maxchain :: "'a set \<Rightarrow> bool" where
   10.43 +  "maxchain C \<longleftrightarrow> chain C \<and> \<not> (\<exists>S. C <c S)"
   10.44 +
   10.45 +text {*We define the successor of a set to be an arbitrary
   10.46 +superchain, if such exists, or the set itself, otherwise.*}
   10.47 +definition suc :: "'a set \<Rightarrow> 'a set" where
   10.48 +  "suc C = (if \<not> chain C \<or> maxchain C then C else (SOME D. C <c D))"
   10.49 +
   10.50 +lemma chainI [Pure.intro?]:
   10.51 +  "\<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"
   10.52 +  unfolding chain_def by blast
   10.53 +
   10.54 +lemma chain_total:
   10.55 +  "chain C \<Longrightarrow> x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
   10.56 +  by (simp add: chain_def)
   10.57 +
   10.58 +lemma not_chain_suc [simp]: "\<not> chain X \<Longrightarrow> suc X = X"
   10.59 +  by (simp add: suc_def)
   10.60 +
   10.61 +lemma maxchain_suc [simp]: "maxchain X \<Longrightarrow> suc X = X"
   10.62 +  by (simp add: suc_def)
   10.63 +
   10.64 +lemma suc_subset: "X \<subseteq> suc X"
   10.65 +  by (auto simp: suc_def maxchain_def intro: someI2)
   10.66 +
   10.67 +lemma chain_empty [simp]: "chain {}"
   10.68 +  by (auto simp: chain_def)
   10.69 +
   10.70 +lemma not_maxchain_Some:
   10.71 +  "chain C \<Longrightarrow> \<not> maxchain C \<Longrightarrow> C <c (SOME D. C <c D)"
   10.72 +  by (rule someI_ex) (auto simp: maxchain_def)
   10.73 +
   10.74 +lemma suc_not_equals:
   10.75 +  "chain C \<Longrightarrow> \<not> maxchain C \<Longrightarrow> suc C \<noteq> C"
   10.76 +  by (auto simp: suc_def) (metis (no_types) less_irrefl not_maxchain_Some)
   10.77 +
   10.78 +lemma subset_suc:
   10.79 +  assumes "X \<subseteq> Y" shows "X \<subseteq> suc Y"
   10.80 +  using assms by (rule subset_trans) (rule suc_subset)
   10.81 +
   10.82 +text {*We build a set @{term \<C>} that is closed under applications
   10.83 +of @{term suc} and contains the union of all its subsets.*}
   10.84 +inductive_set suc_Union_closed ("\<C>") where
   10.85 +  suc: "X \<in> \<C> \<Longrightarrow> suc X \<in> \<C>" |
   10.86 +  Union [unfolded Pow_iff]: "X \<in> Pow \<C> \<Longrightarrow> \<Union>X \<in> \<C>"
   10.87 +
   10.88 +text {*Since the empty set as well as the set itself is a subset of
   10.89 +every set, @{term \<C>} contains at least @{term "{} \<in> \<C>"} and
   10.90 +@{term "\<Union>\<C> \<in> \<C>"}.*}
   10.91 +lemma
   10.92 +  suc_Union_closed_empty: "{} \<in> \<C>" and
   10.93 +  suc_Union_closed_Union: "\<Union>\<C> \<in> \<C>"
   10.94 +  using Union [of "{}"] and Union [of "\<C>"] by simp+
   10.95 +text {*Thus closure under @{term suc} will hit a maximal chain
   10.96 +eventually, as is shown below.*}
   10.97 +
   10.98 +lemma suc_Union_closed_induct [consumes 1, case_names suc Union,
   10.99 +  induct pred: suc_Union_closed]:
  10.100 +  assumes "X \<in> \<C>"
  10.101 +    and "\<And>X. \<lbrakk>X \<in> \<C>; Q X\<rbrakk> \<Longrightarrow> Q (suc X)"
  10.102 +    and "\<And>X. \<lbrakk>X \<subseteq> \<C>; \<forall>x\<in>X. Q x\<rbrakk> \<Longrightarrow> Q (\<Union>X)"
  10.103 +  shows "Q X"
  10.104 +  using assms by (induct) blast+
  10.105 +
  10.106 +lemma suc_Union_closed_cases [consumes 1, case_names suc Union,
  10.107 +  cases pred: suc_Union_closed]:
  10.108 +  assumes "X \<in> \<C>"
  10.109 +    and "\<And>Y. \<lbrakk>X = suc Y; Y \<in> \<C>\<rbrakk> \<Longrightarrow> Q"
  10.110 +    and "\<And>Y. \<lbrakk>X = \<Union>Y; Y \<subseteq> \<C>\<rbrakk> \<Longrightarrow> Q"
  10.111 +  shows "Q"
  10.112 +  using assms by (cases) simp+
  10.113 +
  10.114 +text {*On chains, @{term suc} yields a chain.*}
  10.115 +lemma chain_suc:
  10.116 +  assumes "chain X" shows "chain (suc X)"
  10.117 +  using assms
  10.118 +  by (cases "\<not> chain X \<or> maxchain X")
  10.119 +     (force simp: suc_def dest: not_maxchain_Some)+
  10.120 +
  10.121 +lemma chain_sucD:
  10.122 +  assumes "chain X" shows "suc X \<subseteq> A \<and> chain (suc X)"
  10.123 +proof -
  10.124 +  from `chain X` have *: "chain (suc X)" by (rule chain_suc)
  10.125 +  then have "suc X \<subseteq> A" unfolding chain_def by blast
  10.126 +  with * show ?thesis by blast
  10.127 +qed
  10.128 +
  10.129 +lemma suc_Union_closed_total':
  10.130 +  assumes "X \<in> \<C>" and "Y \<in> \<C>"
  10.131 +    and *: "\<And>Z. Z \<in> \<C> \<Longrightarrow> Z \<subseteq> Y \<Longrightarrow> Z = Y \<or> suc Z \<subseteq> Y"
  10.132 +  shows "X \<subseteq> Y \<or> suc Y \<subseteq> X"
  10.133 +  using `X \<in> \<C>`
  10.134 +proof (induct)
  10.135 +  case (suc X)
  10.136 +  with * show ?case by (blast del: subsetI intro: subset_suc)
  10.137 +qed blast
  10.138 +
  10.139 +lemma suc_Union_closed_subsetD:
  10.140 +  assumes "Y \<subseteq> X" and "X \<in> \<C>" and "Y \<in> \<C>"
  10.141 +  shows "X = Y \<or> suc Y \<subseteq> X"
  10.142 +  using assms(2-, 1)
  10.143 +proof (induct arbitrary: Y)
  10.144 +  case (suc X)
  10.145 +  note * = `\<And>Y. \<lbrakk>Y \<in> \<C>; Y \<subseteq> X\<rbrakk> \<Longrightarrow> X = Y \<or> suc Y \<subseteq> X`
  10.146 +  with suc_Union_closed_total' [OF `Y \<in> \<C>` `X \<in> \<C>`]
  10.147 +    have "Y \<subseteq> X \<or> suc X \<subseteq> Y" by blast
  10.148 +  then show ?case
  10.149 +  proof
  10.150 +    assume "Y \<subseteq> X"
  10.151 +    with * and `Y \<in> \<C>` have "X = Y \<or> suc Y \<subseteq> X" by blast
  10.152 +    then show ?thesis
  10.153 +    proof
  10.154 +      assume "X = Y" then show ?thesis by simp
  10.155 +    next
  10.156 +      assume "suc Y \<subseteq> X"
  10.157 +      then have "suc Y \<subseteq> suc X" by (rule subset_suc)
  10.158 +      then show ?thesis by simp
  10.159 +    qed
  10.160 +  next
  10.161 +    assume "suc X \<subseteq> Y"
  10.162 +    with `Y \<subseteq> suc X` show ?thesis by blast
  10.163 +  qed
  10.164 +next
  10.165 +  case (Union X)
  10.166 +  show ?case
  10.167 +  proof (rule ccontr)
  10.168 +    assume "\<not> ?thesis"
  10.169 +    with `Y \<subseteq> \<Union>X` obtain x y z
  10.170 +    where "\<not> suc Y \<subseteq> \<Union>X"
  10.171 +      and "x \<in> X" and "y \<in> x" and "y \<notin> Y"
  10.172 +      and "z \<in> suc Y" and "\<forall>x\<in>X. z \<notin> x" by blast
  10.173 +    with `X \<subseteq> \<C>` have "x \<in> \<C>" by blast
  10.174 +    from Union and `x \<in> X`
  10.175 +      have *: "\<And>y. \<lbrakk>y \<in> \<C>; y \<subseteq> x\<rbrakk> \<Longrightarrow> x = y \<or> suc y \<subseteq> x" by blast
  10.176 +    with suc_Union_closed_total' [OF `Y \<in> \<C>` `x \<in> \<C>`]
  10.177 +      have "Y \<subseteq> x \<or> suc x \<subseteq> Y" by blast
  10.178 +    then show False
  10.179 +    proof
  10.180 +      assume "Y \<subseteq> x"
  10.181 +      with * [OF `Y \<in> \<C>`] have "x = Y \<or> suc Y \<subseteq> x" by blast
  10.182 +      then show False
  10.183 +      proof
  10.184 +        assume "x = Y" with `y \<in> x` and `y \<notin> Y` show False by blast
  10.185 +      next
  10.186 +        assume "suc Y \<subseteq> x"
  10.187 +        with `x \<in> X` have "suc Y \<subseteq> \<Union>X" by blast
  10.188 +        with `\<not> suc Y \<subseteq> \<Union>X` show False by contradiction
  10.189 +      qed
  10.190 +    next
  10.191 +      assume "suc x \<subseteq> Y"
  10.192 +      moreover from suc_subset and `y \<in> x` have "y \<in> suc x" by blast
  10.193 +      ultimately show False using `y \<notin> Y` by blast
  10.194 +    qed
  10.195 +  qed
  10.196 +qed
  10.197 +
  10.198 +text {*The elements of @{term \<C>} are totally ordered by the subset relation.*}
  10.199 +lemma suc_Union_closed_total:
  10.200 +  assumes "X \<in> \<C>" and "Y \<in> \<C>"
  10.201 +  shows "X \<subseteq> Y \<or> Y \<subseteq> X"
  10.202 +proof (cases "\<forall>Z\<in>\<C>. Z \<subseteq> Y \<longrightarrow> Z = Y \<or> suc Z \<subseteq> Y")
  10.203 +  case True
  10.204 +  with suc_Union_closed_total' [OF assms]
  10.205 +    have "X \<subseteq> Y \<or> suc Y \<subseteq> X" by blast
  10.206 +  then show ?thesis using suc_subset [of Y] by blast
  10.207 +next
  10.208 +  case False
  10.209 +  then obtain Z
  10.210 +    where "Z \<in> \<C>" and "Z \<subseteq> Y" and "Z \<noteq> Y" and "\<not> suc Z \<subseteq> Y" by blast
  10.211 +  with suc_Union_closed_subsetD and `Y \<in> \<C>` show ?thesis by blast
  10.212 +qed
  10.213 +
  10.214 +text {*Once we hit a fixed point w.r.t. @{term suc}, all other elements
  10.215 +of @{term \<C>} are subsets of this fixed point.*}
  10.216 +lemma suc_Union_closed_suc:
  10.217 +  assumes "X \<in> \<C>" and "Y \<in> \<C>" and "suc Y = Y"
  10.218 +  shows "X \<subseteq> Y"
  10.219 +using `X \<in> \<C>`
  10.220 +proof (induct)
  10.221 +  case (suc X)
  10.222 +  with `Y \<in> \<C>` and suc_Union_closed_subsetD
  10.223 +    have "X = Y \<or> suc X \<subseteq> Y" by blast
  10.224 +  then show ?case by (auto simp: `suc Y = Y`)
  10.225 +qed blast
  10.226 +
  10.227 +lemma eq_suc_Union:
  10.228 +  assumes "X \<in> \<C>"
  10.229 +  shows "suc X = X \<longleftrightarrow> X = \<Union>\<C>"
  10.230 +proof
  10.231 +  assume "suc X = X"
  10.232 +  with suc_Union_closed_suc [OF suc_Union_closed_Union `X \<in> \<C>`]
  10.233 +    have "\<Union>\<C> \<subseteq> X" .
  10.234 +  with `X \<in> \<C>` show "X = \<Union>\<C>" by blast
  10.235 +next
  10.236 +  from `X \<in> \<C>` have "suc X \<in> \<C>" by (rule suc)
  10.237 +  then have "suc X \<subseteq> \<Union>\<C>" by blast
  10.238 +  moreover assume "X = \<Union>\<C>"
  10.239 +  ultimately have "suc X \<subseteq> X" by simp
  10.240 +  moreover have "X \<subseteq> suc X" by (rule suc_subset)
  10.241 +  ultimately show "suc X = X" ..
  10.242 +qed
  10.243 +
  10.244 +lemma suc_in_carrier:
  10.245 +  assumes "X \<subseteq> A"
  10.246 +  shows "suc X \<subseteq> A"
  10.247 +  using assms
  10.248 +  by (cases "\<not> chain X \<or> maxchain X")
  10.249 +     (auto dest: chain_sucD)
  10.250 +
  10.251 +lemma suc_Union_closed_in_carrier:
  10.252 +  assumes "X \<in> \<C>"
  10.253 +  shows "X \<subseteq> A"
  10.254 +  using assms
  10.255 +  by (induct) (auto dest: suc_in_carrier)
  10.256 +
  10.257 +text {*All elements of @{term \<C>} are chains.*}
  10.258 +lemma suc_Union_closed_chain:
  10.259 +  assumes "X \<in> \<C>"
  10.260 +  shows "chain X"
  10.261 +using assms
  10.262 +proof (induct)
  10.263 +  case (suc X) then show ?case by (simp add: suc_def) (metis (no_types) not_maxchain_Some)
  10.264 +next
  10.265 +  case (Union X)
  10.266 +  then have "\<Union>X \<subseteq> A" by (auto dest: suc_Union_closed_in_carrier)
  10.267 +  moreover have "\<forall>x\<in>\<Union>X. \<forall>y\<in>\<Union>X. x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
  10.268 +  proof (intro ballI)
  10.269 +    fix x y
  10.270 +    assume "x \<in> \<Union>X" and "y \<in> \<Union>X"
  10.271 +    then obtain u v where "x \<in> u" and "u \<in> X" and "y \<in> v" and "v \<in> X" by blast
  10.272 +    with Union have "u \<in> \<C>" and "v \<in> \<C>" and "chain u" and "chain v" by blast+
  10.273 +    with suc_Union_closed_total have "u \<subseteq> v \<or> v \<subseteq> u" by blast
  10.274 +    then show "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
  10.275 +    proof
  10.276 +      assume "u \<subseteq> v"
  10.277 +      from `chain v` show ?thesis
  10.278 +      proof (rule chain_total)
  10.279 +        show "y \<in> v" by fact
  10.280 +        show "x \<in> v" using `u \<subseteq> v` and `x \<in> u` by blast
  10.281 +      qed
  10.282 +    next
  10.283 +      assume "v \<subseteq> u"
  10.284 +      from `chain u` show ?thesis
  10.285 +      proof (rule chain_total)
  10.286 +        show "x \<in> u" by fact
  10.287 +        show "y \<in> u" using `v \<subseteq> u` and `y \<in> v` by blast
  10.288 +      qed
  10.289 +    qed
  10.290 +  qed
  10.291 +  ultimately show ?case unfolding chain_def ..
  10.292 +qed
  10.293 +
  10.294 +subsubsection {* Hausdorff's Maximum Principle *}
  10.295 +
  10.296 +text {*There exists a maximal totally ordered subset of @{term A}. (Note that we do not
  10.297 +require @{term A} to be partially ordered.)*}
  10.298 +
  10.299 +theorem Hausdorff: "\<exists>C. maxchain C"
  10.300 +proof -
  10.301 +  let ?M = "\<Union>\<C>"
  10.302 +  have "maxchain ?M"
  10.303 +  proof (rule ccontr)
  10.304 +    assume "\<not> maxchain ?M"
  10.305 +    then have "suc ?M \<noteq> ?M"
  10.306 +      using suc_not_equals and
  10.307 +      suc_Union_closed_chain [OF suc_Union_closed_Union] by simp
  10.308 +    moreover have "suc ?M = ?M"
  10.309 +      using eq_suc_Union [OF suc_Union_closed_Union] by simp
  10.310 +    ultimately show False by contradiction
  10.311 +  qed
  10.312 +  then show ?thesis by blast
  10.313 +qed
  10.314 +
  10.315 +text {*Make notation @{term \<C>} available again.*}
  10.316 +no_notation suc_Union_closed ("\<C>")
  10.317 +
  10.318 +lemma chain_extend:
  10.319 +  "chain C \<Longrightarrow> z \<in> A \<Longrightarrow> \<forall>x\<in>C. x \<sqsubseteq> z \<Longrightarrow> chain ({z} \<union> C)"
  10.320 +  unfolding chain_def by blast
  10.321 +
  10.322 +lemma maxchain_imp_chain:
  10.323 +  "maxchain C \<Longrightarrow> chain C"
  10.324 +  by (simp add: maxchain_def)
  10.325 +
  10.326 +end
  10.327 +
  10.328 +text {*Hide constant @{const pred_on.suc_Union_closed}, which was just needed
  10.329 +for the proof of Hausforff's maximum principle.*}
  10.330 +hide_const pred_on.suc_Union_closed
  10.331 +
  10.332 +lemma chain_mono:
  10.333 +  assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; P x y\<rbrakk> \<Longrightarrow> Q x y"
  10.334 +    and "pred_on.chain A P C"
  10.335 +  shows "pred_on.chain A Q C"
  10.336 +  using assms unfolding pred_on.chain_def by blast
  10.337 +
  10.338 +subsubsection {* Results for the proper subset relation *}
  10.339 +
  10.340 +interpretation subset: pred_on "A" "op \<subset>" for A .
  10.341 +
  10.342 +lemma subset_maxchain_max:
  10.343 +  assumes "subset.maxchain A C" and "X \<in> A" and "\<Union>C \<subseteq> X"
  10.344 +  shows "\<Union>C = X"
  10.345 +proof (rule ccontr)
  10.346 +  let ?C = "{X} \<union> C"
  10.347 +  from `subset.maxchain A C` have "subset.chain A C"
  10.348 +    and *: "\<And>S. subset.chain A S \<Longrightarrow> \<not> C \<subset> S"
  10.349 +    by (auto simp: subset.maxchain_def)
  10.350 +  moreover have "\<forall>x\<in>C. x \<subseteq> X" using `\<Union>C \<subseteq> X` by auto
  10.351 +  ultimately have "subset.chain A ?C"
  10.352 +    using subset.chain_extend [of A C X] and `X \<in> A` by auto
  10.353 +  moreover assume **: "\<Union>C \<noteq> X"
  10.354 +  moreover from ** have "C \<subset> ?C" using `\<Union>C \<subseteq> X` by auto
  10.355 +  ultimately show False using * by blast
  10.356 +qed
  10.357 +
  10.358 +subsubsection {* Zorn's lemma *}
  10.359 +
  10.360 +text {*If every chain has an upper bound, then there is a maximal set.*}
  10.361 +lemma subset_Zorn:
  10.362 +  assumes "\<And>C. subset.chain A C \<Longrightarrow> \<exists>U\<in>A. \<forall>X\<in>C. X \<subseteq> U"
  10.363 +  shows "\<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
  10.364 +proof -
  10.365 +  from subset.Hausdorff [of A] obtain M where "subset.maxchain A M" ..
  10.366 +  then have "subset.chain A M" by (rule subset.maxchain_imp_chain)
  10.367 +  with assms obtain Y where "Y \<in> A" and "\<forall>X\<in>M. X \<subseteq> Y" by blast
  10.368 +  moreover have "\<forall>X\<in>A. Y \<subseteq> X \<longrightarrow> Y = X"
  10.369 +  proof (intro ballI impI)
  10.370 +    fix X
  10.371 +    assume "X \<in> A" and "Y \<subseteq> X"
  10.372 +    show "Y = X"
  10.373 +    proof (rule ccontr)
  10.374 +      assume "Y \<noteq> X"
  10.375 +      with `Y \<subseteq> X` have "\<not> X \<subseteq> Y" by blast
  10.376 +      from subset.chain_extend [OF `subset.chain A M` `X \<in> A`] and `\<forall>X\<in>M. X \<subseteq> Y`
  10.377 +        have "subset.chain A ({X} \<union> M)" using `Y \<subseteq> X` by auto
  10.378 +      moreover have "M \<subset> {X} \<union> M" using `\<forall>X\<in>M. X \<subseteq> Y` and `\<not> X \<subseteq> Y` by auto
  10.379 +      ultimately show False
  10.380 +        using `subset.maxchain A M` by (auto simp: subset.maxchain_def)
  10.381 +    qed
  10.382 +  qed
  10.383 +  ultimately show ?thesis by metis
  10.384 +qed
  10.385 +
  10.386 +text{*Alternative version of Zorn's lemma for the subset relation.*}
  10.387 +lemma subset_Zorn':
  10.388 +  assumes "\<And>C. subset.chain A C \<Longrightarrow> \<Union>C \<in> A"
  10.389 +  shows "\<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
  10.390 +proof -
  10.391 +  from subset.Hausdorff [of A] obtain M where "subset.maxchain A M" ..
  10.392 +  then have "subset.chain A M" by (rule subset.maxchain_imp_chain)
  10.393 +  with assms have "\<Union>M \<in> A" .
  10.394 +  moreover have "\<forall>Z\<in>A. \<Union>M \<subseteq> Z \<longrightarrow> \<Union>M = Z"
  10.395 +  proof (intro ballI impI)
  10.396 +    fix Z
  10.397 +    assume "Z \<in> A" and "\<Union>M \<subseteq> Z"
  10.398 +    with subset_maxchain_max [OF `subset.maxchain A M`]
  10.399 +      show "\<Union>M = Z" .
  10.400 +  qed
  10.401 +  ultimately show ?thesis by blast
  10.402 +qed
  10.403 +
  10.404 +
  10.405 +subsection {* Zorn's Lemma for Partial Orders *}
  10.406 +
  10.407 +text {*Relate old to new definitions.*}
  10.408 +
  10.409 +(* Define globally? In Set.thy? *)
  10.410 +definition chain_subset :: "'a set set \<Rightarrow> bool" ("chain\<^sub>\<subseteq>") where
  10.411 +  "chain\<^sub>\<subseteq> C \<longleftrightarrow> (\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A)"
  10.412 +
  10.413 +definition chains :: "'a set set \<Rightarrow> 'a set set set" where
  10.414 +  "chains A = {C. C \<subseteq> A \<and> chain\<^sub>\<subseteq> C}"
  10.415 +
  10.416 +(* Define globally? In Relation.thy? *)
  10.417 +definition Chains :: "('a \<times> 'a) set \<Rightarrow> 'a set set" where
  10.418 +  "Chains r = {C. \<forall>a\<in>C. \<forall>b\<in>C. (a, b) \<in> r \<or> (b, a) \<in> r}"
  10.419 +
  10.420 +lemma chains_extend:
  10.421 +  "[| c \<in> chains S; z \<in> S; \<forall>x \<in> c. x \<subseteq> (z:: 'a set) |] ==> {z} Un c \<in> chains S"
  10.422 +  by (unfold chains_def chain_subset_def) blast
  10.423 +
  10.424 +lemma mono_Chains: "r \<subseteq> s \<Longrightarrow> Chains r \<subseteq> Chains s"
  10.425 +  unfolding Chains_def by blast
  10.426 +
  10.427 +lemma chain_subset_alt_def: "chain\<^sub>\<subseteq> C = subset.chain UNIV C"
  10.428 +  unfolding chain_subset_def subset.chain_def by fast
  10.429 +
  10.430 +lemma chains_alt_def: "chains A = {C. subset.chain A C}"
  10.431 +  by (simp add: chains_def chain_subset_alt_def subset.chain_def)
  10.432 +
  10.433 +lemma Chains_subset:
  10.434 +  "Chains r \<subseteq> {C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C}"
  10.435 +  by (force simp add: Chains_def pred_on.chain_def)
  10.436 +
  10.437 +lemma Chains_subset':
  10.438 +  assumes "refl r"
  10.439 +  shows "{C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C} \<subseteq> Chains r"
  10.440 +  using assms
  10.441 +  by (auto simp add: Chains_def pred_on.chain_def refl_on_def)
  10.442 +
  10.443 +lemma Chains_alt_def:
  10.444 +  assumes "refl r"
  10.445 +  shows "Chains r = {C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C}"
  10.446 +  using assms
  10.447 +  by (metis Chains_subset Chains_subset' subset_antisym)
  10.448 +
  10.449 +lemma Zorn_Lemma:
  10.450 +  "\<forall>C\<in>chains A. \<Union>C \<in> A \<Longrightarrow> \<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
  10.451 +  using subset_Zorn' [of A] by (force simp: chains_alt_def)
  10.452 +
  10.453 +lemma Zorn_Lemma2:
  10.454 +  "\<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"
  10.455 +  using subset_Zorn [of A] by (auto simp: chains_alt_def)
  10.456 +
  10.457 +text{*Various other lemmas*}
  10.458 +
  10.459 +lemma chainsD: "[| c \<in> chains S; x \<in> c; y \<in> c |] ==> x \<subseteq> y | y \<subseteq> x"
  10.460 +by (unfold chains_def chain_subset_def) blast
  10.461 +
  10.462 +lemma chainsD2: "!!(c :: 'a set set). c \<in> chains S ==> c \<subseteq> S"
  10.463 +by (unfold chains_def) blast
  10.464 +
  10.465 +lemma Zorns_po_lemma:
  10.466 +  assumes po: "Partial_order r"
  10.467 +    and u: "\<forall>C\<in>Chains r. \<exists>u\<in>Field r. \<forall>a\<in>C. (a, u) \<in> r"
  10.468 +  shows "\<exists>m\<in>Field r. \<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m"
  10.469 +proof -
  10.470 +  have "Preorder r" using po by (simp add: partial_order_on_def)
  10.471 +--{* Mirror r in the set of subsets below (wrt r) elements of A*}
  10.472 +  let ?B = "%x. r\<inverse> `` {x}" let ?S = "?B ` Field r"
  10.473 +  {
  10.474 +    fix C assume 1: "C \<subseteq> ?S" and 2: "\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A"
  10.475 +    let ?A = "{x\<in>Field r. \<exists>M\<in>C. M = ?B x}"
  10.476 +    have "C = ?B ` ?A" using 1 by (auto simp: image_def)
  10.477 +    have "?A \<in> Chains r"
  10.478 +    proof (simp add: Chains_def, intro allI impI, elim conjE)
  10.479 +      fix a b
  10.480 +      assume "a \<in> Field r" and "?B a \<in> C" and "b \<in> Field r" and "?B b \<in> C"
  10.481 +      hence "?B a \<subseteq> ?B b \<or> ?B b \<subseteq> ?B a" using 2 by auto
  10.482 +      thus "(a, b) \<in> r \<or> (b, a) \<in> r"
  10.483 +        using `Preorder r` and `a \<in> Field r` and `b \<in> Field r`
  10.484 +        by (simp add:subset_Image1_Image1_iff)
  10.485 +    qed
  10.486 +    then obtain u where uA: "u \<in> Field r" "\<forall>a\<in>?A. (a, u) \<in> r" using u by auto
  10.487 +    have "\<forall>A\<in>C. A \<subseteq> r\<inverse> `` {u}" (is "?P u")
  10.488 +    proof auto
  10.489 +      fix a B assume aB: "B \<in> C" "a \<in> B"
  10.490 +      with 1 obtain x where "x \<in> Field r" and "B = r\<inverse> `` {x}" by auto
  10.491 +      thus "(a, u) \<in> r" using uA and aB and `Preorder r`
  10.492 +        unfolding preorder_on_def refl_on_def by simp (fast dest: transD)
  10.493 +    qed
  10.494 +    then have "\<exists>u\<in>Field r. ?P u" using `u \<in> Field r` by blast
  10.495 +  }
  10.496 +  then have "\<forall>C\<in>chains ?S. \<exists>U\<in>?S. \<forall>A\<in>C. A \<subseteq> U"
  10.497 +    by (auto simp: chains_def chain_subset_def)
  10.498 +  from Zorn_Lemma2 [OF this]
  10.499 +  obtain m B where "m \<in> Field r" and "B = r\<inverse> `` {m}"
  10.500 +    and "\<forall>x\<in>Field r. B \<subseteq> r\<inverse> `` {x} \<longrightarrow> r\<inverse> `` {x} = B"
  10.501 +    by auto
  10.502 +  hence "\<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m"
  10.503 +    using po and `Preorder r` and `m \<in> Field r`
  10.504 +    by (auto simp: subset_Image1_Image1_iff Partial_order_eq_Image1_Image1_iff)
  10.505 +  thus ?thesis using `m \<in> Field r` by blast
  10.506 +qed
  10.507 +
  10.508 +
  10.509 +subsection {* The Well Ordering Theorem *}
  10.510 +
  10.511 +(* The initial segment of a relation appears generally useful.
  10.512 +   Move to Relation.thy?
  10.513 +   Definition correct/most general?
  10.514 +   Naming?
  10.515 +*)
  10.516 +definition init_seg_of :: "(('a \<times> 'a) set \<times> ('a \<times> 'a) set) set" where
  10.517 +  "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)}"
  10.518 +
  10.519 +abbreviation
  10.520 +  initialSegmentOf :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool" (infix "initial'_segment'_of" 55)
  10.521 +where
  10.522 +  "r initial_segment_of s \<equiv> (r, s) \<in> init_seg_of"
  10.523 +
  10.524 +lemma refl_on_init_seg_of [simp]: "r initial_segment_of r"
  10.525 +  by (simp add: init_seg_of_def)
  10.526 +
  10.527 +lemma trans_init_seg_of:
  10.528 +  "r initial_segment_of s \<Longrightarrow> s initial_segment_of t \<Longrightarrow> r initial_segment_of t"
  10.529 +  by (simp (no_asm_use) add: init_seg_of_def) blast
  10.530 +
  10.531 +lemma antisym_init_seg_of:
  10.532 +  "r initial_segment_of s \<Longrightarrow> s initial_segment_of r \<Longrightarrow> r = s"
  10.533 +  unfolding init_seg_of_def by safe
  10.534 +
  10.535 +lemma Chains_init_seg_of_Union:
  10.536 +  "R \<in> Chains init_seg_of \<Longrightarrow> r\<in>R \<Longrightarrow> r initial_segment_of \<Union>R"
  10.537 +  by (auto simp: init_seg_of_def Ball_def Chains_def) blast
  10.538 +
  10.539 +lemma chain_subset_trans_Union:
  10.540 +  "chain\<^sub>\<subseteq> R \<Longrightarrow> \<forall>r\<in>R. trans r \<Longrightarrow> trans (\<Union>R)"
  10.541 +apply (auto simp add: chain_subset_def)
  10.542 +apply (simp (no_asm_use) add: trans_def)
  10.543 +by (metis subsetD)
  10.544 +
  10.545 +lemma chain_subset_antisym_Union:
  10.546 +  "chain\<^sub>\<subseteq> R \<Longrightarrow> \<forall>r\<in>R. antisym r \<Longrightarrow> antisym (\<Union>R)"
  10.547 +unfolding chain_subset_def antisym_def
  10.548 +apply simp
  10.549 +by (metis (no_types) subsetD)
  10.550 +
  10.551 +lemma chain_subset_Total_Union:
  10.552 +  assumes "chain\<^sub>\<subseteq> R" and "\<forall>r\<in>R. Total r"
  10.553 +  shows "Total (\<Union>R)"
  10.554 +proof (simp add: total_on_def Ball_def, auto del: disjCI)
  10.555 +  fix r s a b assume A: "r \<in> R" "s \<in> R" "a \<in> Field r" "b \<in> Field s" "a \<noteq> b"
  10.556 +  from `chain\<^sub>\<subseteq> R` and `r \<in> R` and `s \<in> R` have "r \<subseteq> s \<or> s \<subseteq> r"
  10.557 +    by (auto simp add: chain_subset_def)
  10.558 +  thus "(\<exists>r\<in>R. (a, b) \<in> r) \<or> (\<exists>r\<in>R. (b, a) \<in> r)"
  10.559 +  proof
  10.560 +    assume "r \<subseteq> s" hence "(a, b) \<in> s \<or> (b, a) \<in> s" using assms(2) A
  10.561 +      by (simp add: total_on_def) (metis (no_types) mono_Field subsetD)
  10.562 +    thus ?thesis using `s \<in> R` by blast
  10.563 +  next
  10.564 +    assume "s \<subseteq> r" hence "(a, b) \<in> r \<or> (b, a) \<in> r" using assms(2) A
  10.565 +      by (simp add: total_on_def) (metis (no_types) mono_Field subsetD)
  10.566 +    thus ?thesis using `r \<in> R` by blast
  10.567 +  qed
  10.568 +qed
  10.569 +
  10.570 +lemma wf_Union_wf_init_segs:
  10.571 +  assumes "R \<in> Chains init_seg_of" and "\<forall>r\<in>R. wf r"
  10.572 +  shows "wf (\<Union>R)"
  10.573 +proof(simp add: wf_iff_no_infinite_down_chain, rule ccontr, auto)
  10.574 +  fix f assume 1: "\<forall>i. \<exists>r\<in>R. (f (Suc i), f i) \<in> r"
  10.575 +  then obtain r where "r \<in> R" and "(f (Suc 0), f 0) \<in> r" by auto
  10.576 +  { fix i have "(f (Suc i), f i) \<in> r"
  10.577 +    proof (induct i)
  10.578 +      case 0 show ?case by fact
  10.579 +    next
  10.580 +      case (Suc i)
  10.581 +      then obtain s where s: "s \<in> R" "(f (Suc (Suc i)), f(Suc i)) \<in> s"
  10.582 +        using 1 by auto
  10.583 +      then have "s initial_segment_of r \<or> r initial_segment_of s"
  10.584 +        using assms(1) `r \<in> R` by (simp add: Chains_def)
  10.585 +      with Suc s show ?case by (simp add: init_seg_of_def) blast
  10.586 +    qed
  10.587 +  }
  10.588 +  thus False using assms(2) and `r \<in> R`
  10.589 +    by (simp add: wf_iff_no_infinite_down_chain) blast
  10.590 +qed
  10.591 +
  10.592 +lemma initial_segment_of_Diff:
  10.593 +  "p initial_segment_of q \<Longrightarrow> p - s initial_segment_of q - s"
  10.594 +  unfolding init_seg_of_def by blast
  10.595 +
  10.596 +lemma Chains_inits_DiffI:
  10.597 +  "R \<in> Chains init_seg_of \<Longrightarrow> {r - s |r. r \<in> R} \<in> Chains init_seg_of"
  10.598 +  unfolding Chains_def by (blast intro: initial_segment_of_Diff)
  10.599 +
  10.600 +theorem well_ordering: "\<exists>r::'a rel. Well_order r \<and> Field r = UNIV"
  10.601 +proof -
  10.602 +-- {*The initial segment relation on well-orders: *}
  10.603 +  let ?WO = "{r::'a rel. Well_order r}"
  10.604 +  def I \<equiv> "init_seg_of \<inter> ?WO \<times> ?WO"
  10.605 +  have I_init: "I \<subseteq> init_seg_of" by (auto simp: I_def)
  10.606 +  hence subch: "\<And>R. R \<in> Chains I \<Longrightarrow> chain\<^sub>\<subseteq> R"
  10.607 +    unfolding init_seg_of_def chain_subset_def Chains_def by blast
  10.608 +  have Chains_wo: "\<And>R r. R \<in> Chains I \<Longrightarrow> r \<in> R \<Longrightarrow> Well_order r"
  10.609 +    by (simp add: Chains_def I_def) blast
  10.610 +  have FI: "Field I = ?WO" by (auto simp add: I_def init_seg_of_def Field_def)
  10.611 +  hence 0: "Partial_order I"
  10.612 +    by (auto simp: partial_order_on_def preorder_on_def antisym_def antisym_init_seg_of refl_on_def
  10.613 +      trans_def I_def elim!: trans_init_seg_of)
  10.614 +-- {*I-chains have upper bounds in ?WO wrt I: their Union*}
  10.615 +  { fix R assume "R \<in> Chains I"
  10.616 +    hence Ris: "R \<in> Chains init_seg_of" using mono_Chains [OF I_init] by blast
  10.617 +    have subch: "chain\<^sub>\<subseteq> R" using `R : Chains I` I_init
  10.618 +      by (auto simp: init_seg_of_def chain_subset_def Chains_def)
  10.619 +    have "\<forall>r\<in>R. Refl r" and "\<forall>r\<in>R. trans r" and "\<forall>r\<in>R. antisym r"
  10.620 +      and "\<forall>r\<in>R. Total r" and "\<forall>r\<in>R. wf (r - Id)"
  10.621 +      using Chains_wo [OF `R \<in> Chains I`] by (simp_all add: order_on_defs)
  10.622 +    have "Refl (\<Union>R)" using `\<forall>r\<in>R. Refl r` unfolding refl_on_def by fastforce
  10.623 +    moreover have "trans (\<Union>R)"
  10.624 +      by (rule chain_subset_trans_Union [OF subch `\<forall>r\<in>R. trans r`])
  10.625 +    moreover have "antisym (\<Union>R)"
  10.626 +      by (rule chain_subset_antisym_Union [OF subch `\<forall>r\<in>R. antisym r`])
  10.627 +    moreover have "Total (\<Union>R)"
  10.628 +      by (rule chain_subset_Total_Union [OF subch `\<forall>r\<in>R. Total r`])
  10.629 +    moreover have "wf ((\<Union>R) - Id)"
  10.630 +    proof -
  10.631 +      have "(\<Union>R) - Id = \<Union>{r - Id | r. r \<in> R}" by blast
  10.632 +      with `\<forall>r\<in>R. wf (r - Id)` and wf_Union_wf_init_segs [OF Chains_inits_DiffI [OF Ris]]
  10.633 +      show ?thesis by fastforce
  10.634 +    qed
  10.635 +    ultimately have "Well_order (\<Union>R)" by(simp add:order_on_defs)
  10.636 +    moreover have "\<forall>r \<in> R. r initial_segment_of \<Union>R" using Ris
  10.637 +      by(simp add: Chains_init_seg_of_Union)
  10.638 +    ultimately have "\<Union>R \<in> ?WO \<and> (\<forall>r\<in>R. (r, \<Union>R) \<in> I)"
  10.639 +      using mono_Chains [OF I_init] and `R \<in> Chains I`
  10.640 +      by (simp (no_asm) add: I_def del: Field_Union) (metis Chains_wo)
  10.641 +  }
  10.642 +  hence 1: "\<forall>R \<in> Chains I. \<exists>u\<in>Field I. \<forall>r\<in>R. (r, u) \<in> I" by (subst FI) blast
  10.643 +--{*Zorn's Lemma yields a maximal well-order m:*}
  10.644 +  then obtain m::"'a rel" where "Well_order m" and
  10.645 +    max: "\<forall>r. Well_order r \<and> (m, r) \<in> I \<longrightarrow> r = m"
  10.646 +    using Zorns_po_lemma[OF 0 1] unfolding FI by fastforce
  10.647 +--{*Now show by contradiction that m covers the whole type:*}
  10.648 +  { fix x::'a assume "x \<notin> Field m"
  10.649 +--{*We assume that x is not covered and extend m at the top with x*}
  10.650 +    have "m \<noteq> {}"
  10.651 +    proof
  10.652 +      assume "m = {}"
  10.653 +      moreover have "Well_order {(x, x)}"
  10.654 +        by (simp add: order_on_defs refl_on_def trans_def antisym_def total_on_def Field_def)
  10.655 +      ultimately show False using max
  10.656 +        by (auto simp: I_def init_seg_of_def simp del: Field_insert)
  10.657 +    qed
  10.658 +    hence "Field m \<noteq> {}" by(auto simp:Field_def)
  10.659 +    moreover have "wf (m - Id)" using `Well_order m`
  10.660 +      by (simp add: well_order_on_def)
  10.661 +--{*The extension of m by x:*}
  10.662 +    let ?s = "{(a, x) | a. a \<in> Field m}"
  10.663 +    let ?m = "insert (x, x) m \<union> ?s"
  10.664 +    have Fm: "Field ?m = insert x (Field m)"
  10.665 +      by (auto simp: Field_def)
  10.666 +    have "Refl m" and "trans m" and "antisym m" and "Total m" and "wf (m - Id)"
  10.667 +      using `Well_order m` by (simp_all add: order_on_defs)
  10.668 +--{*We show that the extension is a well-order*}
  10.669 +    have "Refl ?m" using `Refl m` Fm unfolding refl_on_def by blast
  10.670 +    moreover have "trans ?m" using `trans m` and `x \<notin> Field m`
  10.671 +      unfolding trans_def Field_def by blast
  10.672 +    moreover have "antisym ?m" using `antisym m` and `x \<notin> Field m`
  10.673 +      unfolding antisym_def Field_def by blast
  10.674 +    moreover have "Total ?m" using `Total m` and Fm by (auto simp: total_on_def)
  10.675 +    moreover have "wf (?m - Id)"
  10.676 +    proof -
  10.677 +      have "wf ?s" using `x \<notin> Field m`
  10.678 +        by (auto simp add: wf_eq_minimal Field_def) metis
  10.679 +      thus ?thesis using `wf (m - Id)` and `x \<notin> Field m`
  10.680 +        wf_subset [OF `wf ?s` Diff_subset]
  10.681 +        unfolding Un_Diff Field_def by (auto intro: wf_Un)
  10.682 +    qed
  10.683 +    ultimately have "Well_order ?m" by (simp add: order_on_defs)
  10.684 +--{*We show that the extension is above m*}
  10.685 +    moreover have "(m, ?m) \<in> I" using `Well_order ?m` and `Well_order m` and `x \<notin> Field m`
  10.686 +      by (fastforce simp: I_def init_seg_of_def Field_def)
  10.687 +    ultimately
  10.688 +--{*This contradicts maximality of m:*}
  10.689 +    have False using max and `x \<notin> Field m` unfolding Field_def by blast
  10.690 +  }
  10.691 +  hence "Field m = UNIV" by auto
  10.692 +  with `Well_order m` show ?thesis by blast
  10.693 +qed
  10.694 +
  10.695 +corollary well_order_on: "\<exists>r::'a rel. well_order_on A r"
  10.696 +proof -
  10.697 +  obtain r::"'a rel" where wo: "Well_order r" and univ: "Field r = UNIV"
  10.698 +    using well_ordering [where 'a = "'a"] by blast
  10.699 +  let ?r = "{(x, y). x \<in> A \<and> y \<in> A \<and> (x, y) \<in> r}"
  10.700 +  have 1: "Field ?r = A" using wo univ
  10.701 +    by (fastforce simp: Field_def order_on_defs refl_on_def)
  10.702 +  have "Refl r" and "trans r" and "antisym r" and "Total r" and "wf (r - Id)"
  10.703 +    using `Well_order r` by (simp_all add: order_on_defs)
  10.704 +  have "Refl ?r" using `Refl r` by (auto simp: refl_on_def 1 univ)
  10.705 +  moreover have "trans ?r" using `trans r`
  10.706 +    unfolding trans_def by blast
  10.707 +  moreover have "antisym ?r" using `antisym r`
  10.708 +    unfolding antisym_def by blast
  10.709 +  moreover have "Total ?r" using `Total r` by (simp add:total_on_def 1 univ)
  10.710 +  moreover have "wf (?r - Id)" by (rule wf_subset [OF `wf (r - Id)`]) blast
  10.711 +  ultimately have "Well_order ?r" by (simp add: order_on_defs)
  10.712 +  with 1 show ?thesis by auto
  10.713 +qed
  10.714 +
  10.715 +end