diff --git a/src/HOL/Wfrec.thy b/src/HOL/Wfrec.thy
--- a/src/HOL/Wfrec.thy
+++ b/src/HOL/Wfrec.thy
@@ -1,112 +1,115 @@
 (*  Title:      HOL/Wfrec.thy
     Author:     Tobias Nipkow
     Author:     Lawrence C Paulson
     Author:     Konrad Slind
 *)
 
 section \<open>Well-Founded Recursion Combinator\<close>
 
 theory Wfrec
   imports Wellfounded
 begin
 
 inductive wfrec_rel :: "('a \<times> 'a) set \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" for R F
   where wfrecI: "(\<And>z. (z, x) \<in> R \<Longrightarrow> wfrec_rel R F z (g z)) \<Longrightarrow> wfrec_rel R F x (F g x)"
 
 definition cut :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b"
   where "cut f R x = (\<lambda>y. if (y, x) \<in> R then f y else undefined)"
 
 definition adm_wf :: "('a \<times> 'a) set \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)) \<Rightarrow> bool"
   where "adm_wf R F \<longleftrightarrow> (\<forall>f g x. (\<forall>z. (z, x) \<in> R \<longrightarrow> f z = g z) \<longrightarrow> F f x = F g x)"
 
 definition wfrec :: "('a \<times> 'a) set \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)) \<Rightarrow> ('a \<Rightarrow> 'b)"
   where "wfrec R F = (\<lambda>x. THE y. wfrec_rel R (\<lambda>f x. F (cut f R x) x) x y)"
 
 lemma cuts_eq: "(cut f R x = cut g R x) \<longleftrightarrow> (\<forall>y. (y, x) \<in> R \<longrightarrow> f y = g y)"
   by (simp add: fun_eq_iff cut_def)
 
 lemma cut_apply: "(x, a) \<in> R \<Longrightarrow> cut f R a x = f x"
   by (simp add: cut_def)
 
 text \<open>
   Inductive characterization of \<open>wfrec\<close> combinator; for details see:
   John Harrison, "Inductive definitions: automation and application".
 \<close>
 
 lemma theI_unique: "\<exists>!x. P x \<Longrightarrow> P x \<longleftrightarrow> x = The P"
   by (auto intro: the_equality[symmetric] theI)
 
 lemma wfrec_unique:
   assumes "adm_wf R F" "wf R"
   shows "\<exists>!y. wfrec_rel R F x y"
   using \<open>wf R\<close>
 proof induct
   define f where "f y = (THE z. wfrec_rel R F y z)" for y
   case (less x)
   then have "\<And>y z. (y, x) \<in> R \<Longrightarrow> wfrec_rel R F y z \<longleftrightarrow> z = f y"
     unfolding f_def by (rule theI_unique)
   with \<open>adm_wf R F\<close> show ?case
     by (subst wfrec_rel.simps) (auto simp: adm_wf_def)
 qed
 
 lemma adm_lemma: "adm_wf R (\<lambda>f x. F (cut f R x) x)"
   by (auto simp: adm_wf_def intro!: arg_cong[where f="\<lambda>x. F x y" for y] cuts_eq[THEN iffD2])
 
 lemma wfrec: "wf R \<Longrightarrow> wfrec R F a = F (cut (wfrec R F) R a) a"
   apply (simp add: wfrec_def)
   apply (rule adm_lemma [THEN wfrec_unique, THEN the1_equality])
    apply assumption
   apply (rule wfrec_rel.wfrecI)
   apply (erule adm_lemma [THEN wfrec_unique, THEN theI'])
   done
 
 
 text \<open>This form avoids giant explosions in proofs.  NOTE USE OF \<open>\<equiv>\<close>.\<close>
 lemma def_wfrec: "f \<equiv> wfrec R F \<Longrightarrow> wf R \<Longrightarrow> f a = F (cut f R a) a"
   by (auto intro: wfrec)
 
 
 subsubsection \<open>Well-founded recursion via genuine fixpoints\<close>
 
 lemma wfrec_fixpoint:
   assumes wf: "wf R"
     and adm: "adm_wf R F"
   shows "wfrec R F = F (wfrec R F)"
 proof (rule ext)
   fix x
   have "wfrec R F x = F (cut (wfrec R F) R x) x"
     using wfrec[of R F] wf by simp
   also
   have "\<And>y. (y, x) \<in> R \<Longrightarrow> cut (wfrec R F) R x y = wfrec R F y"
     by (auto simp add: cut_apply)
   then have "F (cut (wfrec R F) R x) x = F (wfrec R F) x"
     using adm adm_wf_def[of R F] by auto
   finally show "wfrec R F x = F (wfrec R F) x" .
 qed
 
+lemma wfrec_def_adm: "f \<equiv> wfrec R F \<Longrightarrow> wf R \<Longrightarrow> adm_wf R F \<Longrightarrow> f = F f"
+  using wfrec_fixpoint by simp
+
 
 subsection \<open>Wellfoundedness of \<open>same_fst\<close>\<close>
 
 definition same_fst :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> ('b \<times> 'b) set) \<Rightarrow> (('a \<times> 'b) \<times> ('a \<times> 'b)) set"
   where "same_fst P R = {((x', y'), (x, y)) . x' = x \<and> P x \<and> (y',y) \<in> R x}"
    \<comment> \<open>For \<^const>\<open>wfrec\<close> declarations where the first n parameters
        stay unchanged in the recursive call.\<close>
 
 lemma same_fstI [intro!]: "P x \<Longrightarrow> (y', y) \<in> R x \<Longrightarrow> ((x, y'), (x, y)) \<in> same_fst P R"
   by (simp add: same_fst_def)
 
 lemma wf_same_fst:
   assumes "\<And>x. P x \<Longrightarrow> wf (R x)"
   shows "wf (same_fst P R)"
 proof (clarsimp simp add: wf_def same_fst_def)
   fix Q a b
   assume *: "\<forall>a b. (\<forall>x. P a \<and> (x,b) \<in> R a \<longrightarrow> Q (a,x)) \<longrightarrow> Q (a,b)"
   show "Q(a,b)"
   proof (cases "wf (R a)")
     case True
     then show ?thesis
       by (induction b rule: wf_induct_rule) (use * in blast)
   qed (use * assms in blast)
 qed
 
 end
diff --git a/src/HOL/Zorn.thy b/src/HOL/Zorn.thy
--- a/src/HOL/Zorn.thy
+++ b/src/HOL/Zorn.thy
@@ -1,910 +1,887 @@
 (*  Title:       HOL/Zorn.thy
     Author:      Jacques D. Fleuriot
     Author:      Tobias Nipkow, TUM
     Author:      Christian Sternagel, JAIST
 
 Zorn's Lemma (ported from Larry Paulson's Zorn.thy in ZF).
-The well-ordering theorem.
 *)
 
-section \<open>Zorn's Lemma\<close>
+section \<open>Zorn's Lemma and the Well-ordering Theorem\<close>
 
 theory Zorn
   imports Order_Relation Hilbert_Choice
 begin
 
 subsection \<open>Zorn's Lemma for the Subset Relation\<close>
 
 subsubsection \<open>Results that do not require an order\<close>
 
 text \<open>Let \<open>P\<close> be a binary predicate on the set \<open>A\<close>.\<close>
 locale pred_on =
   fixes A :: "'a set"
     and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"  (infix "\<sqsubset>" 50)
 begin
 
 abbreviation Peq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"  (infix "\<sqsubseteq>" 50)
   where "x \<sqsubseteq> y \<equiv> P\<^sup>=\<^sup>= x y"
 
 text \<open>A chain is a totally ordered subset of \<open>A\<close>.\<close>
 definition chain :: "'a set \<Rightarrow> bool"
   where "chain C \<longleftrightarrow> C \<subseteq> A \<and> (\<forall>x\<in>C. \<forall>y\<in>C. x \<sqsubseteq> y \<or> y \<sqsubseteq> x)"
 
 text \<open>
   We call a chain that is a proper superset of some set \<open>X\<close>,
   but not necessarily a chain itself, a superchain of \<open>X\<close>.
 \<close>
 abbreviation superchain :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"  (infix "<c" 50)
   where "X <c C \<equiv> chain C \<and> X \<subset> C"
 
 text \<open>A maximal chain is a chain that does not have a superchain.\<close>
 definition maxchain :: "'a set \<Rightarrow> bool"
   where "maxchain C \<longleftrightarrow> chain C \<and> (\<nexists>S. C <c S)"
 
 text \<open>
   We define the successor of a set to be an arbitrary
   superchain, if such exists, or the set itself, otherwise.
 \<close>
 definition suc :: "'a set \<Rightarrow> 'a set"
   where "suc C = (if \<not> chain C \<or> maxchain C then C else (SOME D. C <c D))"
 
 lemma chainI [Pure.intro?]: "C \<subseteq> A \<Longrightarrow> (\<And>x y. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x) \<Longrightarrow> chain C"
   unfolding chain_def by blast
 
 lemma chain_total: "chain C \<Longrightarrow> x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
   by (simp add: chain_def)
 
 lemma not_chain_suc [simp]: "\<not> chain X \<Longrightarrow> suc X = X"
   by (simp add: suc_def)
 
 lemma maxchain_suc [simp]: "maxchain X \<Longrightarrow> suc X = X"
   by (simp add: suc_def)
 
 lemma suc_subset: "X \<subseteq> suc X"
   by (auto simp: suc_def maxchain_def intro: someI2)
 
 lemma chain_empty [simp]: "chain {}"
   by (auto simp: chain_def)
 
 lemma not_maxchain_Some: "chain C \<Longrightarrow> \<not> maxchain C \<Longrightarrow> C <c (SOME D. C <c D)"
   by (rule someI_ex) (auto simp: maxchain_def)
 
 lemma suc_not_equals: "chain C \<Longrightarrow> \<not> maxchain C \<Longrightarrow> suc C \<noteq> C"
   using not_maxchain_Some by (auto simp: suc_def)
 
 lemma subset_suc:
   assumes "X \<subseteq> Y"
   shows "X \<subseteq> suc Y"
   using assms by (rule subset_trans) (rule suc_subset)
 
 text \<open>
   We build a set \<^term>\<open>\<C>\<close> that is closed under applications
   of \<^term>\<open>suc\<close> and contains the union of all its subsets.
 \<close>
 inductive_set suc_Union_closed ("\<C>")
   where
     suc: "X \<in> \<C> \<Longrightarrow> suc X \<in> \<C>"
   | Union [unfolded Pow_iff]: "X \<in> Pow \<C> \<Longrightarrow> \<Union>X \<in> \<C>"
 
 text \<open>
   Since the empty set as well as the set itself is a subset of
   every set, \<^term>\<open>\<C>\<close> contains at least \<^term>\<open>{} \<in> \<C>\<close> and
   \<^term>\<open>\<Union>\<C> \<in> \<C>\<close>.
 \<close>
 lemma suc_Union_closed_empty: "{} \<in> \<C>"
   and suc_Union_closed_Union: "\<Union>\<C> \<in> \<C>"
   using Union [of "{}"] and Union [of "\<C>"] by simp_all
 
 text \<open>Thus closure under \<^term>\<open>suc\<close> will hit a maximal chain
   eventually, as is shown below.\<close>
 
 lemma suc_Union_closed_induct [consumes 1, case_names suc Union, induct pred: suc_Union_closed]:
   assumes "X \<in> \<C>"
     and "\<And>X. X \<in> \<C> \<Longrightarrow> Q X \<Longrightarrow> Q (suc X)"
     and "\<And>X. X \<subseteq> \<C> \<Longrightarrow> \<forall>x\<in>X. Q x \<Longrightarrow> Q (\<Union>X)"
   shows "Q X"
   using assms by induct blast+
 
 lemma suc_Union_closed_cases [consumes 1, case_names suc Union, cases pred: suc_Union_closed]:
   assumes "X \<in> \<C>"
     and "\<And>Y. X = suc Y \<Longrightarrow> Y \<in> \<C> \<Longrightarrow> Q"
     and "\<And>Y. X = \<Union>Y \<Longrightarrow> Y \<subseteq> \<C> \<Longrightarrow> Q"
   shows "Q"
   using assms by cases simp_all
 
 text \<open>On chains, \<^term>\<open>suc\<close> yields a chain.\<close>
 lemma chain_suc:
   assumes "chain X"
   shows "chain (suc X)"
   using assms
   by (cases "\<not> chain X \<or> maxchain X") (force simp: suc_def dest: not_maxchain_Some)+
 
 lemma chain_sucD:
   assumes "chain X"
   shows "suc X \<subseteq> A \<and> chain (suc X)"
 proof -
   from \<open>chain X\<close> have *: "chain (suc X)"
     by (rule chain_suc)
   then have "suc X \<subseteq> A"
     unfolding chain_def by blast
   with * show ?thesis by blast
 qed
 
 lemma suc_Union_closed_total':
   assumes "X \<in> \<C>" and "Y \<in> \<C>"
     and *: "\<And>Z. Z \<in> \<C> \<Longrightarrow> Z \<subseteq> Y \<Longrightarrow> Z = Y \<or> suc Z \<subseteq> Y"
   shows "X \<subseteq> Y \<or> suc Y \<subseteq> X"
   using \<open>X \<in> \<C>\<close>
 proof induct
   case (suc X)
   with * show ?case by (blast del: subsetI intro: subset_suc)
 next
   case Union
   then show ?case by blast
 qed
 
 lemma suc_Union_closed_subsetD:
   assumes "Y \<subseteq> X" and "X \<in> \<C>" and "Y \<in> \<C>"
   shows "X = Y \<or> suc Y \<subseteq> X"
   using assms(2,3,1)
 proof (induct arbitrary: Y)
   case (suc X)
   note * = \<open>\<And>Y. Y \<in> \<C> \<Longrightarrow> Y \<subseteq> X \<Longrightarrow> X = Y \<or> suc Y \<subseteq> X\<close>
   with suc_Union_closed_total' [OF \<open>Y \<in> \<C>\<close> \<open>X \<in> \<C>\<close>]
   have "Y \<subseteq> X \<or> suc X \<subseteq> Y" by blast
   then show ?case
   proof
     assume "Y \<subseteq> X"
-    with * and \<open>Y \<in> \<C>\<close> have "X = Y \<or> suc Y \<subseteq> X" by blast
-    then show ?thesis
-    proof
-      assume "X = Y"
-      then show ?thesis by simp
-    next
-      assume "suc Y \<subseteq> X"
-      then have "suc Y \<subseteq> suc X" by (rule subset_suc)
-      then show ?thesis by simp
-    qed
+    with * and \<open>Y \<in> \<C>\<close> subset_suc show ?thesis
+      by fastforce
   next
     assume "suc X \<subseteq> Y"
     with \<open>Y \<subseteq> suc X\<close> show ?thesis by blast
   qed
 next
   case (Union X)
   show ?case
   proof (rule ccontr)
     assume "\<not> ?thesis"
     with \<open>Y \<subseteq> \<Union>X\<close> obtain x y z
       where "\<not> suc Y \<subseteq> \<Union>X"
         and "x \<in> X" and "y \<in> x" and "y \<notin> Y"
         and "z \<in> suc Y" and "\<forall>x\<in>X. z \<notin> x" by blast
     with \<open>X \<subseteq> \<C>\<close> have "x \<in> \<C>" by blast
     from Union and \<open>x \<in> X\<close> have *: "\<And>y. y \<in> \<C> \<Longrightarrow> y \<subseteq> x \<Longrightarrow> x = y \<or> suc y \<subseteq> x"
       by blast
     with suc_Union_closed_total' [OF \<open>Y \<in> \<C>\<close> \<open>x \<in> \<C>\<close>] have "Y \<subseteq> x \<or> suc x \<subseteq> Y"
       by blast
     then show False
     proof
       assume "Y \<subseteq> x"
-      with * [OF \<open>Y \<in> \<C>\<close>] have "x = Y \<or> suc Y \<subseteq> x" by blast
-      then show False
-      proof
-        assume "x = Y"
-        with \<open>y \<in> x\<close> and \<open>y \<notin> Y\<close> show False by blast
-      next
-        assume "suc Y \<subseteq> x"
-        with \<open>x \<in> X\<close> have "suc Y \<subseteq> \<Union>X" by blast
-        with \<open>\<not> suc Y \<subseteq> \<Union>X\<close> show False by contradiction
-      qed
+      with * [OF \<open>Y \<in> \<C>\<close>] \<open>y \<in> x\<close> \<open>y \<notin> Y\<close> \<open>x \<in> X\<close> \<open>\<not> suc Y \<subseteq> \<Union>X\<close> show False
+        by blast
     next
       assume "suc x \<subseteq> Y"
-      moreover from suc_subset and \<open>y \<in> x\<close> have "y \<in> suc x" by blast
-      ultimately show False using \<open>y \<notin> Y\<close> by blast
+      with \<open>y \<notin> Y\<close> suc_subset \<open>y \<in> x\<close> show False by blast
     qed
   qed
 qed
 
 text \<open>The elements of \<^term>\<open>\<C>\<close> are totally ordered by the subset relation.\<close>
 lemma suc_Union_closed_total:
   assumes "X \<in> \<C>" and "Y \<in> \<C>"
   shows "X \<subseteq> Y \<or> Y \<subseteq> X"
 proof (cases "\<forall>Z\<in>\<C>. Z \<subseteq> Y \<longrightarrow> Z = Y \<or> suc Z \<subseteq> Y")
   case True
   with suc_Union_closed_total' [OF assms]
   have "X \<subseteq> Y \<or> suc Y \<subseteq> X" by blast
   with suc_subset [of Y] show ?thesis by blast
 next
   case False
   then obtain Z where "Z \<in> \<C>" and "Z \<subseteq> Y" and "Z \<noteq> Y" and "\<not> suc Z \<subseteq> Y"
     by blast
   with suc_Union_closed_subsetD and \<open>Y \<in> \<C>\<close> show ?thesis
     by blast
 qed
 
 text \<open>Once we hit a fixed point w.r.t. \<^term>\<open>suc\<close>, all other elements
   of \<^term>\<open>\<C>\<close> are subsets of this fixed point.\<close>
 lemma suc_Union_closed_suc:
   assumes "X \<in> \<C>" and "Y \<in> \<C>" and "suc Y = Y"
   shows "X \<subseteq> Y"
   using \<open>X \<in> \<C>\<close>
 proof induct
   case (suc X)
   with \<open>Y \<in> \<C>\<close> and suc_Union_closed_subsetD have "X = Y \<or> suc X \<subseteq> Y"
     by blast
   then show ?case
     by (auto simp: \<open>suc Y = Y\<close>)
 next
   case Union
   then show ?case by blast
 qed
 
 lemma eq_suc_Union:
   assumes "X \<in> \<C>"
   shows "suc X = X \<longleftrightarrow> X = \<Union>\<C>"
     (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   assume ?lhs
   then have "\<Union>\<C> \<subseteq> X"
     by (rule suc_Union_closed_suc [OF suc_Union_closed_Union \<open>X \<in> \<C>\<close>])
   with \<open>X \<in> \<C>\<close> show ?rhs
     by blast
 next
   from \<open>X \<in> \<C>\<close> have "suc X \<in> \<C>" by (rule suc)
   then have "suc X \<subseteq> \<Union>\<C>" by blast
   moreover assume ?rhs
   ultimately have "suc X \<subseteq> X" by simp
   moreover have "X \<subseteq> suc X" by (rule suc_subset)
   ultimately show ?lhs ..
 qed
 
 lemma suc_in_carrier:
   assumes "X \<subseteq> A"
   shows "suc X \<subseteq> A"
   using assms
   by (cases "\<not> chain X \<or> maxchain X") (auto dest: chain_sucD)
 
 lemma suc_Union_closed_in_carrier:
   assumes "X \<in> \<C>"
   shows "X \<subseteq> A"
   using assms
   by induct (auto dest: suc_in_carrier)
 
 text \<open>All elements of \<^term>\<open>\<C>\<close> are chains.\<close>
 lemma suc_Union_closed_chain:
   assumes "X \<in> \<C>"
   shows "chain X"
   using assms
 proof induct
   case (suc X)
   then show ?case
     using not_maxchain_Some by (simp add: suc_def)
 next
   case (Union X)
   then have "\<Union>X \<subseteq> A"
     by (auto dest: suc_Union_closed_in_carrier)
   moreover have "\<forall>x\<in>\<Union>X. \<forall>y\<in>\<Union>X. x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
   proof (intro ballI)
     fix x y
     assume "x \<in> \<Union>X" and "y \<in> \<Union>X"
     then obtain u v where "x \<in> u" and "u \<in> X" and "y \<in> v" and "v \<in> X"
       by blast
     with Union have "u \<in> \<C>" and "v \<in> \<C>" and "chain u" and "chain v"
       by blast+
     with suc_Union_closed_total have "u \<subseteq> v \<or> v \<subseteq> u"
       by blast
     then show "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
     proof
       assume "u \<subseteq> v"
       from \<open>chain v\<close> show ?thesis
       proof (rule chain_total)
         show "y \<in> v" by fact
         show "x \<in> v" using \<open>u \<subseteq> v\<close> and \<open>x \<in> u\<close> by blast
       qed
     next
       assume "v \<subseteq> u"
       from \<open>chain u\<close> show ?thesis
       proof (rule chain_total)
         show "x \<in> u" by fact
         show "y \<in> u" using \<open>v \<subseteq> u\<close> and \<open>y \<in> v\<close> by blast
       qed
     qed
   qed
   ultimately show ?case unfolding chain_def ..
 qed
 
 subsubsection \<open>Hausdorff's Maximum Principle\<close>
 
 text \<open>There exists a maximal totally ordered subset of \<open>A\<close>. (Note that we do not
   require \<open>A\<close> to be partially ordered.)\<close>
 
 theorem Hausdorff: "\<exists>C. maxchain C"
 proof -
   let ?M = "\<Union>\<C>"
   have "maxchain ?M"
   proof (rule ccontr)
     assume "\<not> ?thesis"
     then have "suc ?M \<noteq> ?M"
       using suc_not_equals and suc_Union_closed_chain [OF suc_Union_closed_Union] by simp
     moreover have "suc ?M = ?M"
       using eq_suc_Union [OF suc_Union_closed_Union] by simp
     ultimately show False by contradiction
   qed
   then show ?thesis by blast
 qed
 
 text \<open>Make notation \<^term>\<open>\<C>\<close> available again.\<close>
 no_notation suc_Union_closed  ("\<C>")
 
 lemma chain_extend: "chain C \<Longrightarrow> z \<in> A \<Longrightarrow> \<forall>x\<in>C. x \<sqsubseteq> z \<Longrightarrow> chain ({z} \<union> C)"
   unfolding chain_def by blast
 
 lemma maxchain_imp_chain: "maxchain C \<Longrightarrow> chain C"
   by (simp add: maxchain_def)
 
 end
 
 text \<open>Hide constant \<^const>\<open>pred_on.suc_Union_closed\<close>, which was just needed
   for the proof of Hausforff's maximum principle.\<close>
 hide_const pred_on.suc_Union_closed
 
 lemma chain_mono:
   assumes "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> P x y \<Longrightarrow> Q x y"
     and "pred_on.chain A P C"
   shows "pred_on.chain A Q C"
   using assms unfolding pred_on.chain_def by blast
 
 
 subsubsection \<open>Results for the proper subset relation\<close>
 
 interpretation subset: pred_on "A" "(\<subset>)" for A .
 
 lemma subset_maxchain_max:
   assumes "subset.maxchain A C"
     and "X \<in> A"
     and "\<Union>C \<subseteq> X"
   shows "\<Union>C = X"
 proof (rule ccontr)
   let ?C = "{X} \<union> C"
   from \<open>subset.maxchain A C\<close> have "subset.chain A C"
     and *: "\<And>S. subset.chain A S \<Longrightarrow> \<not> C \<subset> S"
     by (auto simp: subset.maxchain_def)
   moreover have "\<forall>x\<in>C. x \<subseteq> X" using \<open>\<Union>C \<subseteq> X\<close> by auto
   ultimately have "subset.chain A ?C"
     using subset.chain_extend [of A C X] and \<open>X \<in> A\<close> by auto
   moreover assume **: "\<Union>C \<noteq> X"
   moreover from ** have "C \<subset> ?C" using \<open>\<Union>C \<subseteq> X\<close> by auto
   ultimately show False using * by blast
 qed
 
 lemma subset_chain_def: "\<And>\<A>. subset.chain \<A> \<C> = (\<C> \<subseteq> \<A> \<and> (\<forall>X\<in>\<C>. \<forall>Y\<in>\<C>. X \<subseteq> Y \<or> Y \<subseteq> X))"
   by (auto simp: subset.chain_def)
 
 lemma subset_chain_insert:
   "subset.chain \<A> (insert B \<B>) \<longleftrightarrow> B \<in> \<A> \<and> (\<forall>X\<in>\<B>. X \<subseteq> B \<or> B \<subseteq> X) \<and> subset.chain \<A> \<B>"
   by (fastforce simp add: subset_chain_def)
 
 subsubsection \<open>Zorn's lemma\<close>
 
 text \<open>If every chain has an upper bound, then there is a maximal set.\<close>
 theorem subset_Zorn:
   assumes "\<And>C. subset.chain A C \<Longrightarrow> \<exists>U\<in>A. \<forall>X\<in>C. X \<subseteq> U"
   shows "\<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
 proof -
   from subset.Hausdorff [of A] obtain M where "subset.maxchain A M" ..
   then have "subset.chain A M"
     by (rule subset.maxchain_imp_chain)
   with assms obtain Y where "Y \<in> A" and "\<forall>X\<in>M. X \<subseteq> Y"
     by blast
   moreover have "\<forall>X\<in>A. Y \<subseteq> X \<longrightarrow> Y = X"
   proof (intro ballI impI)
     fix X
     assume "X \<in> A" and "Y \<subseteq> X"
     show "Y = X"
     proof (rule ccontr)
       assume "\<not> ?thesis"
       with \<open>Y \<subseteq> X\<close> have "\<not> X \<subseteq> Y" by blast
       from subset.chain_extend [OF \<open>subset.chain A M\<close> \<open>X \<in> A\<close>] and \<open>\<forall>X\<in>M. X \<subseteq> Y\<close>
       have "subset.chain A ({X} \<union> M)"
         using \<open>Y \<subseteq> X\<close> by auto
       moreover have "M \<subset> {X} \<union> M"
         using \<open>\<forall>X\<in>M. X \<subseteq> Y\<close> and \<open>\<not> X \<subseteq> Y\<close> by auto
       ultimately show False
         using \<open>subset.maxchain A M\<close> by (auto simp: subset.maxchain_def)
     qed
   qed
   ultimately show ?thesis by blast
 qed
 
 text \<open>Alternative version of Zorn's lemma for the subset relation.\<close>
 lemma subset_Zorn':
   assumes "\<And>C. subset.chain A C \<Longrightarrow> \<Union>C \<in> A"
   shows "\<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
 proof -
   from subset.Hausdorff [of A] obtain M where "subset.maxchain A M" ..
   then have "subset.chain A M"
     by (rule subset.maxchain_imp_chain)
   with assms have "\<Union>M \<in> A" .
   moreover have "\<forall>Z\<in>A. \<Union>M \<subseteq> Z \<longrightarrow> \<Union>M = Z"
   proof (intro ballI impI)
     fix Z
     assume "Z \<in> A" and "\<Union>M \<subseteq> Z"
     with subset_maxchain_max [OF \<open>subset.maxchain A M\<close>]
       show "\<Union>M = Z" .
   qed
   ultimately show ?thesis by blast
 qed
 
 
 subsection \<open>Zorn's Lemma for Partial Orders\<close>
 
 text \<open>Relate old to new definitions.\<close>
 
 definition chain_subset :: "'a set set \<Rightarrow> bool"  ("chain\<^sub>\<subseteq>")  (* Define globally? In Set.thy? *)
   where "chain\<^sub>\<subseteq> C \<longleftrightarrow> (\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A)"
 
 definition chains :: "'a set set \<Rightarrow> 'a set set set"
   where "chains A = {C. C \<subseteq> A \<and> chain\<^sub>\<subseteq> C}"
 
 definition Chains :: "('a \<times> 'a) set \<Rightarrow> 'a set set"  (* Define globally? In Relation.thy? *)
   where "Chains r = {C. \<forall>a\<in>C. \<forall>b\<in>C. (a, b) \<in> r \<or> (b, a) \<in> r}"
 
 lemma chains_extend: "c \<in> chains S \<Longrightarrow> z \<in> S \<Longrightarrow> \<forall>x \<in> c. x \<subseteq> z \<Longrightarrow> {z} \<union> c \<in> chains S"
   for z :: "'a set"
   unfolding chains_def chain_subset_def by blast
 
 lemma mono_Chains: "r \<subseteq> s \<Longrightarrow> Chains r \<subseteq> Chains s"
   unfolding Chains_def by blast
 
 lemma chain_subset_alt_def: "chain\<^sub>\<subseteq> C = subset.chain UNIV C"
   unfolding chain_subset_def subset.chain_def by fast
 
 lemma chains_alt_def: "chains A = {C. subset.chain A C}"
   by (simp add: chains_def chain_subset_alt_def subset.chain_def)
 
 lemma Chains_subset: "Chains r \<subseteq> {C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C}"
   by (force simp add: Chains_def pred_on.chain_def)
 
 lemma Chains_subset':
   assumes "refl r"
   shows "{C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C} \<subseteq> Chains r"
   using assms
   by (auto simp add: Chains_def pred_on.chain_def refl_on_def)
 
 lemma Chains_alt_def:
   assumes "refl r"
   shows "Chains r = {C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C}"
   using assms Chains_subset Chains_subset' by blast
 
 lemma Chains_relation_of:
   assumes "C \<in> Chains (relation_of P A)" shows "C \<subseteq> A"
   using assms unfolding Chains_def relation_of_def by auto
 
 lemma pairwise_chain_Union:
   assumes P: "\<And>S. S \<in> \<C> \<Longrightarrow> pairwise R S" and "chain\<^sub>\<subseteq> \<C>"
   shows "pairwise R (\<Union>\<C>)"
   using \<open>chain\<^sub>\<subseteq> \<C>\<close> unfolding pairwise_def chain_subset_def
   by (blast intro: P [unfolded pairwise_def, rule_format])
 
 lemma Zorn_Lemma: "\<forall>C\<in>chains A. \<Union>C \<in> A \<Longrightarrow> \<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
   using subset_Zorn' [of A] by (force simp: chains_alt_def)
 
 lemma Zorn_Lemma2: "\<forall>C\<in>chains A. \<exists>U\<in>A. \<forall>X\<in>C. X \<subseteq> U \<Longrightarrow> \<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
   using subset_Zorn [of A] by (auto simp: chains_alt_def)
 
 subsection \<open>Other variants of Zorn's Lemma\<close>
 
 lemma chainsD: "c \<in> chains S \<Longrightarrow> x \<in> c \<Longrightarrow> y \<in> c \<Longrightarrow> x \<subseteq> y \<or> y \<subseteq> x"
   unfolding chains_def chain_subset_def by blast
 
 lemma chainsD2: "c \<in> chains S \<Longrightarrow> c \<subseteq> S"
   unfolding chains_def by blast
 
 lemma Zorns_po_lemma:
   assumes po: "Partial_order r"
     and u: "\<And>C. C \<in> Chains r \<Longrightarrow> \<exists>u\<in>Field r. \<forall>a\<in>C. (a, u) \<in> r"
   shows "\<exists>m\<in>Field r. \<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m"
 proof -
   have "Preorder r"
     using po by (simp add: partial_order_on_def)
   txt \<open>Mirror \<open>r\<close> in the set of subsets below (wrt \<open>r\<close>) elements of \<open>A\<close>.\<close>
   let ?B = "\<lambda>x. r\<inverse> `` {x}"
   let ?S = "?B ` Field r"
   have "\<exists>u\<in>Field r. \<forall>A\<in>C. A \<subseteq> r\<inverse> `` {u}"  (is "\<exists>u\<in>Field r. ?P u")
     if 1: "C \<subseteq> ?S" and 2: "\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A" for C
   proof -
     let ?A = "{x\<in>Field r. \<exists>M\<in>C. M = ?B x}"
     from 1 have "C = ?B ` ?A" by (auto simp: image_def)
     have "?A \<in> Chains r"
     proof (simp add: Chains_def, intro allI impI, elim conjE)
       fix a b
       assume "a \<in> Field r" and "?B a \<in> C" and "b \<in> Field r" and "?B b \<in> C"
       with 2 have "?B a \<subseteq> ?B b \<or> ?B b \<subseteq> ?B a" by auto
       then show "(a, b) \<in> r \<or> (b, a) \<in> r"
         using \<open>Preorder r\<close> and \<open>a \<in> Field r\<close> and \<open>b \<in> Field r\<close>
         by (simp add:subset_Image1_Image1_iff)
     qed
     then obtain u where uA: "u \<in> Field r" "\<forall>a\<in>?A. (a, u) \<in> r"
       by (auto simp: dest: u)
     have "?P u"
     proof auto
       fix a B assume aB: "B \<in> C" "a \<in> B"
       with 1 obtain x where "x \<in> Field r" and "B = r\<inverse> `` {x}" by auto
       then show "(a, u) \<in> r"
         using uA and aB and \<open>Preorder r\<close>
         unfolding preorder_on_def refl_on_def by simp (fast dest: transD)
     qed
     then show ?thesis
       using \<open>u \<in> Field r\<close> by blast
   qed
   then have "\<forall>C\<in>chains ?S. \<exists>U\<in>?S. \<forall>A\<in>C. A \<subseteq> U"
     by (auto simp: chains_def chain_subset_def)
   from Zorn_Lemma2 [OF this] obtain m B
     where "m \<in> Field r"
       and "B = r\<inverse> `` {m}"
       and "\<forall>x\<in>Field r. B \<subseteq> r\<inverse> `` {x} \<longrightarrow> r\<inverse> `` {x} = B"
     by auto
   then have "\<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m"
     using po and \<open>Preorder r\<close> and \<open>m \<in> Field r\<close>
     by (auto simp: subset_Image1_Image1_iff Partial_order_eq_Image1_Image1_iff)
   then show ?thesis
     using \<open>m \<in> Field r\<close> by blast
 qed
 
 lemma predicate_Zorn:
   assumes po: "partial_order_on A (relation_of P A)"
     and ch: "\<And>C. C \<in> Chains (relation_of P A) \<Longrightarrow> \<exists>u \<in> A. \<forall>a \<in> C. P a u"
   shows "\<exists>m \<in> A. \<forall>a \<in> A. P m a \<longrightarrow> a = m"
 proof -
   have "a \<in> A" if "C \<in> Chains (relation_of P A)" and "a \<in> C" for C a
     using that unfolding Chains_def relation_of_def by auto
   moreover have "(a, u) \<in> relation_of P A" if "a \<in> A" and "u \<in> A" and "P a u" for a u
     unfolding relation_of_def using that by auto
   ultimately have "\<exists>m\<in>A. \<forall>a\<in>A. (m, a) \<in> relation_of P A \<longrightarrow> a = m"
     using Zorns_po_lemma[OF Partial_order_relation_ofI[OF po], rule_format] ch
     unfolding Field_relation_of[OF partial_order_onD(1)[OF po]] by blast
   then show ?thesis
     by (auto simp: relation_of_def)
 qed
 
 lemma Union_in_chain: "\<lbrakk>finite \<B>; \<B> \<noteq> {}; subset.chain \<A> \<B>\<rbrakk> \<Longrightarrow> \<Union>\<B> \<in> \<B>"
 proof (induction \<B> rule: finite_induct)
   case (insert B \<B>)
   show ?case
   proof (cases "\<B> = {}")
     case False
     then show ?thesis
       using insert sup.absorb2 by (auto simp: subset_chain_insert dest!: bspec [where x="\<Union>\<B>"])
   qed auto
 qed simp
 
 lemma Inter_in_chain: "\<lbrakk>finite \<B>; \<B> \<noteq> {}; subset.chain \<A> \<B>\<rbrakk> \<Longrightarrow> \<Inter>\<B> \<in> \<B>"
 proof (induction \<B> rule: finite_induct)
   case (insert B \<B>)
   show ?case
   proof (cases "\<B> = {}")
     case False
     then show ?thesis
       using insert inf.absorb2 by (auto simp: subset_chain_insert dest!: bspec [where x="\<Inter>\<B>"])
   qed auto
 qed simp
 
 lemma finite_subset_Union_chain:
   assumes "finite A" "A \<subseteq> \<Union>\<B>" "\<B> \<noteq> {}" and sub: "subset.chain \<A> \<B>"
   obtains B where "B \<in> \<B>" "A \<subseteq> B"
 proof -
   obtain \<F> where \<F>: "finite \<F>" "\<F> \<subseteq> \<B>" "A \<subseteq> \<Union>\<F>"
     using assms by (auto intro: finite_subset_Union)
   show thesis
   proof (cases "\<F> = {}")
     case True
     then show ?thesis
       using \<open>A \<subseteq> \<Union>\<F>\<close> \<open>\<B> \<noteq> {}\<close> that by fastforce
   next
     case False
     show ?thesis
     proof
       show "\<Union>\<F> \<in> \<B>"
         using sub \<open>\<F> \<subseteq> \<B>\<close> \<open>finite \<F>\<close>
         by (simp add: Union_in_chain False subset.chain_def subset_iff)
       show "A \<subseteq> \<Union>\<F>"
         using \<open>A \<subseteq> \<Union>\<F>\<close> by blast
     qed
   qed
 qed
 
 lemma subset_Zorn_nonempty:
   assumes "\<A> \<noteq> {}" and ch: "\<And>\<C>. \<lbrakk>\<C>\<noteq>{}; subset.chain \<A> \<C>\<rbrakk> \<Longrightarrow> \<Union>\<C> \<in> \<A>"
   shows "\<exists>M\<in>\<A>. \<forall>X\<in>\<A>. M \<subseteq> X \<longrightarrow> X = M"
 proof (rule subset_Zorn)
   show "\<exists>U\<in>\<A>. \<forall>X\<in>\<C>. X \<subseteq> U" if "subset.chain \<A> \<C>" for \<C>
   proof (cases "\<C> = {}")
     case True
     then show ?thesis
       using \<open>\<A> \<noteq> {}\<close> by blast
   next
     case False
     show ?thesis
       by (blast intro!: ch False that Union_upper)
   qed
 qed
 
 subsection \<open>The Well Ordering Theorem\<close>
 
 (* The initial segment of a relation appears generally useful.
    Move to Relation.thy?
    Definition correct/most general?
    Naming?
 *)
 definition init_seg_of :: "(('a \<times> 'a) set \<times> ('a \<times> 'a) set) set"
   where "init_seg_of = {(r, s). r \<subseteq> s \<and> (\<forall>a b c. (a, b) \<in> s \<and> (b, c) \<in> r \<longrightarrow> (a, b) \<in> r)}"
 
 abbreviation initial_segment_of_syntax :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool"
     (infix "initial'_segment'_of" 55)
   where "r initial_segment_of s \<equiv> (r, s) \<in> init_seg_of"
 
 lemma refl_on_init_seg_of [simp]: "r initial_segment_of r"
   by (simp add: init_seg_of_def)
 
 lemma trans_init_seg_of:
   "r initial_segment_of s \<Longrightarrow> s initial_segment_of t \<Longrightarrow> r initial_segment_of t"
   by (simp (no_asm_use) add: init_seg_of_def) blast
 
 lemma antisym_init_seg_of: "r initial_segment_of s \<Longrightarrow> s initial_segment_of r \<Longrightarrow> r = s"
   unfolding init_seg_of_def by safe
 
 lemma Chains_init_seg_of_Union: "R \<in> Chains init_seg_of \<Longrightarrow> r\<in>R \<Longrightarrow> r initial_segment_of \<Union>R"
   by (auto simp: init_seg_of_def Ball_def Chains_def) blast
 
 lemma chain_subset_trans_Union:
   assumes "chain\<^sub>\<subseteq> R" "\<forall>r\<in>R. trans r"
   shows "trans (\<Union>R)"
 proof (intro transI, elim UnionE)
   fix S1 S2 :: "'a rel" and x y z :: 'a
   assume "S1 \<in> R" "S2 \<in> R"
   with assms(1) have "S1 \<subseteq> S2 \<or> S2 \<subseteq> S1"
     unfolding chain_subset_def by blast
   moreover assume "(x, y) \<in> S1" "(y, z) \<in> S2"
   ultimately have "((x, y) \<in> S1 \<and> (y, z) \<in> S1) \<or> ((x, y) \<in> S2 \<and> (y, z) \<in> S2)"
     by blast
   with \<open>S1 \<in> R\<close> \<open>S2 \<in> R\<close> assms(2) show "(x, z) \<in> \<Union>R"
     by (auto elim: transE)
 qed
 
 lemma chain_subset_antisym_Union:
   assumes "chain\<^sub>\<subseteq> R" "\<forall>r\<in>R. antisym r"
   shows "antisym (\<Union>R)"
 proof (intro antisymI, elim UnionE)
   fix S1 S2 :: "'a rel" and x y :: 'a
   assume "S1 \<in> R" "S2 \<in> R"
   with assms(1) have "S1 \<subseteq> S2 \<or> S2 \<subseteq> S1"
     unfolding chain_subset_def by blast
   moreover assume "(x, y) \<in> S1" "(y, x) \<in> S2"
   ultimately have "((x, y) \<in> S1 \<and> (y, x) \<in> S1) \<or> ((x, y) \<in> S2 \<and> (y, x) \<in> S2)"
     by blast
   with \<open>S1 \<in> R\<close> \<open>S2 \<in> R\<close> assms(2) show "x = y"
     unfolding antisym_def by auto
 qed
 
 lemma chain_subset_Total_Union:
   assumes "chain\<^sub>\<subseteq> R" and "\<forall>r\<in>R. Total r"
   shows "Total (\<Union>R)"
 proof (simp add: total_on_def Ball_def, auto del: disjCI)
   fix r s a b
   assume A: "r \<in> R" "s \<in> R" "a \<in> Field r" "b \<in> Field s" "a \<noteq> b"
   from \<open>chain\<^sub>\<subseteq> R\<close> and \<open>r \<in> R\<close> and \<open>s \<in> R\<close> have "r \<subseteq> s \<or> s \<subseteq> r"
     by (auto simp add: chain_subset_def)
   then show "(\<exists>r\<in>R. (a, b) \<in> r) \<or> (\<exists>r\<in>R. (b, a) \<in> r)"
   proof
     assume "r \<subseteq> s"
     then have "(a, b) \<in> s \<or> (b, a) \<in> s"
       using assms(2) A mono_Field[of r s]
       by (auto simp add: total_on_def)
     then show ?thesis
       using \<open>s \<in> R\<close> by blast
   next
     assume "s \<subseteq> r"
     then have "(a, b) \<in> r \<or> (b, a) \<in> r"
       using assms(2) A mono_Field[of s r]
       by (fastforce simp add: total_on_def)
     then show ?thesis
       using \<open>r \<in> R\<close> by blast
   qed
 qed
 
 lemma wf_Union_wf_init_segs:
   assumes "R \<in> Chains init_seg_of"
     and "\<forall>r\<in>R. wf r"
   shows "wf (\<Union>R)"
 proof (simp add: wf_iff_no_infinite_down_chain, rule ccontr, auto)
   fix f
   assume 1: "\<forall>i. \<exists>r\<in>R. (f (Suc i), f i) \<in> r"
   then obtain r where "r \<in> R" and "(f (Suc 0), f 0) \<in> r" by auto
   have "(f (Suc i), f i) \<in> r" for i
   proof (induct i)
     case 0
     show ?case by fact
   next
     case (Suc i)
     then obtain s where s: "s \<in> R" "(f (Suc (Suc i)), f(Suc i)) \<in> s"
       using 1 by auto
     then have "s initial_segment_of r \<or> r initial_segment_of s"
       using assms(1) \<open>r \<in> R\<close> by (simp add: Chains_def)
     with Suc s show ?case by (simp add: init_seg_of_def) blast
   qed
   then show False
     using assms(2) and \<open>r \<in> R\<close>
     by (simp add: wf_iff_no_infinite_down_chain) blast
 qed
 
 lemma initial_segment_of_Diff: "p initial_segment_of q \<Longrightarrow> p - s initial_segment_of q - s"
   unfolding init_seg_of_def by blast
 
 lemma Chains_inits_DiffI: "R \<in> Chains init_seg_of \<Longrightarrow> {r - s |r. r \<in> R} \<in> Chains init_seg_of"
   unfolding Chains_def by (blast intro: initial_segment_of_Diff)
 
 theorem well_ordering: "\<exists>r::'a rel. Well_order r \<and> Field r = UNIV"
 proof -
 \<comment> \<open>The initial segment relation on well-orders:\<close>
   let ?WO = "{r::'a rel. Well_order r}"
   define I where "I = init_seg_of \<inter> ?WO \<times> ?WO"
   then have I_init: "I \<subseteq> init_seg_of" by simp
   then have subch: "\<And>R. R \<in> Chains I \<Longrightarrow> chain\<^sub>\<subseteq> R"
     unfolding init_seg_of_def chain_subset_def Chains_def by blast
   have Chains_wo: "\<And>R r. R \<in> Chains I \<Longrightarrow> r \<in> R \<Longrightarrow> Well_order r"
     by (simp add: Chains_def I_def) blast
   have FI: "Field I = ?WO"
     by (auto simp add: I_def init_seg_of_def Field_def)
   then have 0: "Partial_order I"
     by (auto simp: partial_order_on_def preorder_on_def antisym_def antisym_init_seg_of refl_on_def
         trans_def I_def elim!: trans_init_seg_of)
 \<comment> \<open>\<open>I\<close>-chains have upper bounds in \<open>?WO\<close> wrt \<open>I\<close>: their Union\<close>
   have "\<Union>R \<in> ?WO \<and> (\<forall>r\<in>R. (r, \<Union>R) \<in> I)" if "R \<in> Chains I" for R
   proof -
     from that have Ris: "R \<in> Chains init_seg_of"
       using mono_Chains [OF I_init] by blast
     have subch: "chain\<^sub>\<subseteq> R"
       using \<open>R \<in> Chains I\<close> I_init by (auto simp: init_seg_of_def chain_subset_def Chains_def)
     have "\<forall>r\<in>R. Refl r" and "\<forall>r\<in>R. trans r" and "\<forall>r\<in>R. antisym r"
       and "\<forall>r\<in>R. Total r" and "\<forall>r\<in>R. wf (r - Id)"
       using Chains_wo [OF \<open>R \<in> Chains I\<close>] by (simp_all add: order_on_defs)
     have "Refl (\<Union>R)"
       using \<open>\<forall>r\<in>R. Refl r\<close> unfolding refl_on_def by fastforce
     moreover have "trans (\<Union>R)"
       by (rule chain_subset_trans_Union [OF subch \<open>\<forall>r\<in>R. trans r\<close>])
     moreover have "antisym (\<Union>R)"
       by (rule chain_subset_antisym_Union [OF subch \<open>\<forall>r\<in>R. antisym r\<close>])
     moreover have "Total (\<Union>R)"
       by (rule chain_subset_Total_Union [OF subch \<open>\<forall>r\<in>R. Total r\<close>])
     moreover have "wf ((\<Union>R) - Id)"
     proof -
       have "(\<Union>R) - Id = \<Union>{r - Id | r. r \<in> R}" by blast
       with \<open>\<forall>r\<in>R. wf (r - Id)\<close> and wf_Union_wf_init_segs [OF Chains_inits_DiffI [OF Ris]]
       show ?thesis by fastforce
     qed
     ultimately have "Well_order (\<Union>R)"
       by (simp add:order_on_defs)
     moreover have "\<forall>r \<in> R. r initial_segment_of \<Union>R"
       using Ris by (simp add: Chains_init_seg_of_Union)
     ultimately show ?thesis
       using mono_Chains [OF I_init] Chains_wo[of R] and \<open>R \<in> Chains I\<close>
       unfolding I_def by blast
   qed
   then have 1: "\<exists>u\<in>Field I. \<forall>r\<in>R. (r, u) \<in> I" if "R \<in> Chains I" for R
     using that by (subst FI) blast
 \<comment> \<open>Zorn's Lemma yields a maximal well-order \<open>m\<close>:\<close>
   then obtain m :: "'a rel"
     where "Well_order m"
       and max: "\<forall>r. Well_order r \<and> (m, r) \<in> I \<longrightarrow> r = m"
     using Zorns_po_lemma[OF 0 1] unfolding FI by fastforce
 \<comment> \<open>Now show by contradiction that \<open>m\<close> covers the whole type:\<close>
   have False if "x \<notin> Field m" for x :: 'a
   proof -
 \<comment> \<open>Assuming that \<open>x\<close> is not covered and extend \<open>m\<close> at the top with \<open>x\<close>\<close>
     have "m \<noteq> {}"
     proof
       assume "m = {}"
       moreover have "Well_order {(x, x)}"
         by (simp add: order_on_defs refl_on_def trans_def antisym_def total_on_def Field_def)
       ultimately show False using max
         by (auto simp: I_def init_seg_of_def simp del: Field_insert)
     qed
     then have "Field m \<noteq> {}" by (auto simp: Field_def)
     moreover have "wf (m - Id)"
       using \<open>Well_order m\<close> by (simp add: well_order_on_def)
 \<comment> \<open>The extension of \<open>m\<close> by \<open>x\<close>:\<close>
     let ?s = "{(a, x) | a. a \<in> Field m}"
     let ?m = "insert (x, x) m \<union> ?s"
     have Fm: "Field ?m = insert x (Field m)"
       by (auto simp: Field_def)
     have "Refl m" and "trans m" and "antisym m" and "Total m" and "wf (m - Id)"
       using \<open>Well_order m\<close> by (simp_all add: order_on_defs)
 \<comment> \<open>We show that the extension is a well-order\<close>
     have "Refl ?m"
       using \<open>Refl m\<close> Fm unfolding refl_on_def by blast
     moreover have "trans ?m" using \<open>trans m\<close> and \<open>x \<notin> Field m\<close>
       unfolding trans_def Field_def by blast
     moreover have "antisym ?m"
       using \<open>antisym m\<close> and \<open>x \<notin> Field m\<close> unfolding antisym_def Field_def by blast
     moreover have "Total ?m"
       using \<open>Total m\<close> and Fm by (auto simp: total_on_def)
     moreover have "wf (?m - Id)"
     proof -
       have "wf ?s"
         using \<open>x \<notin> Field m\<close> by (auto simp: wf_eq_minimal Field_def Bex_def)
       then show ?thesis
         using \<open>wf (m - Id)\<close> and \<open>x \<notin> Field m\<close> wf_subset [OF \<open>wf ?s\<close> Diff_subset]
         by (auto simp: Un_Diff Field_def intro: wf_Un)
     qed
     ultimately have "Well_order ?m"
       by (simp add: order_on_defs)
 \<comment> \<open>We show that the extension is above \<open>m\<close>\<close>
     moreover have "(m, ?m) \<in> I"
       using \<open>Well_order ?m\<close> and \<open>Well_order m\<close> and \<open>x \<notin> Field m\<close>
       by (fastforce simp: I_def init_seg_of_def Field_def)
     ultimately
 \<comment> \<open>This contradicts maximality of \<open>m\<close>:\<close>
     show False
       using max and \<open>x \<notin> Field m\<close> unfolding Field_def by blast
   qed
   then have "Field m = UNIV" by auto
   with \<open>Well_order m\<close> show ?thesis by blast
 qed
 
 corollary well_order_on: "\<exists>r::'a rel. well_order_on A r"
 proof -
   obtain r :: "'a rel" where wo: "Well_order r" and univ: "Field r = UNIV"
     using well_ordering [where 'a = "'a"] by blast
   let ?r = "{(x, y). x \<in> A \<and> y \<in> A \<and> (x, y) \<in> r}"
   have 1: "Field ?r = A"
     using wo univ by (fastforce simp: Field_def order_on_defs refl_on_def)
   from \<open>Well_order r\<close> have "Refl r" "trans r" "antisym r" "Total r" "wf (r - Id)"
     by (simp_all add: order_on_defs)
   from \<open>Refl r\<close> have "Refl ?r"
     by (auto simp: refl_on_def 1 univ)
   moreover from \<open>trans r\<close> have "trans ?r"
     unfolding trans_def by blast
   moreover from \<open>antisym r\<close> have "antisym ?r"
     unfolding antisym_def by blast
   moreover from \<open>Total r\<close> have "Total ?r"
     by (simp add:total_on_def 1 univ)
   moreover have "wf (?r - Id)"
     by (rule wf_subset [OF \<open>wf (r - Id)\<close>]) blast
   ultimately have "Well_order ?r"
     by (simp add: order_on_defs)
   with 1 show ?thesis by auto
 qed
 
-(* Move this to Hilbert Choice and wfrec to Wellfounded*)
-
-lemma wfrec_def_adm: "f \<equiv> wfrec R F \<Longrightarrow> wf R \<Longrightarrow> adm_wf R F \<Longrightarrow> f = F f"
-  using wfrec_fixpoint by simp
-
 lemma dependent_wf_choice:
   fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
   assumes "wf R"
     and adm: "\<And>f g x r. (\<And>z. (z, x) \<in> R \<Longrightarrow> f z = g z) \<Longrightarrow> P f x r = P g x r"
     and P: "\<And>x f. (\<And>y. (y, x) \<in> R \<Longrightarrow> P f y (f y)) \<Longrightarrow> \<exists>r. P f x r"
   shows "\<exists>f. \<forall>x. P f x (f x)"
 proof (intro exI allI)
   fix x
   define f where "f \<equiv> wfrec R (\<lambda>f x. SOME r. P f x r)"
   from \<open>wf R\<close> show "P f x (f x)"
   proof (induct x)
     case (less x)
     show "P f x (f x)"
     proof (subst (2) wfrec_def_adm[OF f_def \<open>wf R\<close>])
       show "adm_wf R (\<lambda>f x. SOME r. P f x r)"
-        by (auto simp add: adm_wf_def intro!: arg_cong[where f=Eps] ext adm)
+        by (auto simp: adm_wf_def intro!: arg_cong[where f=Eps] adm)
       show "P f x (Eps (P f x))"
         using P by (rule someI_ex) fact
     qed
   qed
 qed
 
 lemma (in wellorder) dependent_wellorder_choice:
   assumes "\<And>r f g x. (\<And>y. y < x \<Longrightarrow> f y = g y) \<Longrightarrow> P f x r = P g x r"
     and P: "\<And>x f. (\<And>y. y < x \<Longrightarrow> P f y (f y)) \<Longrightarrow> \<exists>r. P f x r"
   shows "\<exists>f. \<forall>x. P f x (f x)"
   using wf by (rule dependent_wf_choice) (auto intro!: assms)
 
 end