src/HOL/Zorn.thy

-(*  Title:      HOL/Zorn.thy
-    Author:     Jacques D. Fleuriot
-    Author:     Tobias Nipkow, TUM
-    Author:     Christian Sternagel, JAIST
+(*  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>
 
 theory Zorn
-imports Order_Relation Hilbert_Choice
+  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)
+    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 @{term A}.\<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 @{term X},
-but not necessarily a chain itself, a superchain of @{term X}.\<close>
-abbreviation superchain :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" (infix "<c" 50) where
-  "X <c C \<equiv> chain C \<and> X \<subset> C"
+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> \<not> (\<exists>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?]:
-  "\<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"
+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"
+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)
 

   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)"
+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"
+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"
+  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 \<C>} that is closed under applications
-of @{term suc} 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 \<C>} contains at least @{term "{} \<in> \<C>"} and
-@{term "\<Union>\<C> \<in> \<C>"}.\<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+
+text \<open>
+  We build a set @{term \<C>} that is closed under applications
+  of @{term suc} 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 \<C>} contains at least @{term "{} \<in> \<C>"} and
+  @{term "\<Union>\<C> \<in> \<C>"}.
+\<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 suc} 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]:
+  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. \<lbrakk>X \<in> \<C>; Q X\<rbrakk> \<Longrightarrow> Q (suc X)"
-    and "\<And>X. \<lbrakk>X \<subseteq> \<C>; \<forall>x\<in>X. Q x\<rbrakk> \<Longrightarrow> Q (\<Union>X)"
+    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]:
+  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. \<lbrakk>X = suc Y; Y \<in> \<C>\<rbrakk> \<Longrightarrow> Q"
-    and "\<And>Y. \<lbrakk>X = \<Union>Y; Y \<subseteq> \<C>\<rbrakk> \<Longrightarrow> Q"
+    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+
+  using assms by cases simp_all
 
 text \<open>On chains, @{term suc} yields a chain.\<close>
 lemma chain_suc:
-  assumes "chain X" shows "chain (suc X)"
+  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)+
+  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)"
+  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
+  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)
+proof induct
   case (suc X)
   with * show ?case by (blast del: subsetI intro: subset_suc)
-qed blast
+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-, 1)
+  using assms(2,3,1)
 proof (induct arbitrary: Y)
   case (suc X)
-  note * = \<open>\<And>Y. \<lbrakk>Y \<in> \<C>; Y \<subseteq> X\<rbrakk> \<Longrightarrow> X = Y \<or> suc Y \<subseteq> X\<close>
+  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
+  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
+      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

   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
+      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. \<lbrakk>y \<in> \<C>; y \<subseteq> x\<rbrakk> \<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
+    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
+        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

   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
-  then show ?thesis using suc_subset [of Y] by blast
+  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
+  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 suc}, all other elements
-of @{term \<C>} are subsets of this fixed point.\<close>
+  of @{term \<C>} 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)
+  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>)
-qed blast
+  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 "suc X = X"
-  with suc_Union_closed_suc [OF suc_Union_closed_Union \<open>X \<in> \<C>\<close>]
-    have "\<Union>\<C> \<subseteq> X" .
-  with \<open>X \<in> \<C>\<close> show "X = \<Union>\<C>" by blast
+  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 "X = \<Union>\<C>"
+  moreover assume ?rhs
   ultimately have "suc X \<subseteq> X" by simp
   moreover have "X \<subseteq> suc X" by (rule suc_subset)
-  ultimately show "suc X = X" ..
+  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)
+  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)
+  by induct (auto dest: suc_in_carrier)
 
 text \<open>All elements of @{term \<C>} 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)
+  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)
+  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 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)

   ultimately show ?case unfolding chain_def ..
 qed
 
 subsubsection \<open>Hausdorff's Maximum Principle\<close>
 
-text \<open>There exists a maximal totally ordered subset of @{term A}. (Note that we do not
-require @{term A} to be partially ordered.)\<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> maxchain ?M"
+    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
+      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 \<C>} 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)"
+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"
+lemma maxchain_imp_chain: "maxchain C \<Longrightarrow> chain C"
   by (simp add: maxchain_def)
 
 end
 
 text \<open>Hide constant @{const pred_on.suc_Union_closed}, which was just needed
-for the proof of Hausforff's maximum principle.\<close>
+  for the proof of Hausforff's maximum principle.\<close>
 hide_const pred_on.suc_Union_closed
 
 lemma chain_mono:
-  assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; P x y\<rbrakk> \<Longrightarrow> Q x y"
+  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" "op \<subset>" for A .
 
 lemma subset_maxchain_max:
-  assumes "subset.maxchain A C" and "X \<in> A" and "\<Union>C \<subseteq> X"
+  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"

   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
 
+
 subsubsection \<open>Zorn's lemma\<close>
 
 text \<open>If every chain has an upper bound, then there is a maximal set.\<close>
 lemma 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
+  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 "Y \<noteq> X"
+      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
+      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>
+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)
+  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"

 
 subsection \<open>Zorn's Lemma for Partial Orders\<close>
 
 text \<open>Relate old to new definitions.\<close>
 
-(* Define globally? In Set.thy? *)
-definition chain_subset :: "'a set set \<Rightarrow> bool" ("chain\<^sub>\<subseteq>") 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}"
-
-(* Define globally? In Relation.thy? *)
-definition Chains :: "('a \<times> 'a) set \<Rightarrow> 'a set set" 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; z \<in> S; \<forall>x \<in> c. x \<subseteq> (z:: 'a set) |] ==> {z} Un c \<in> chains S"
+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
 

   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}"
+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"

 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 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"
+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"
+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)
 
-text\<open>Various other lemmas\<close>
-
-lemma chainsD: "[| c \<in> chains S; x \<in> c; y \<in> c |] ==> x \<subseteq> y | y \<subseteq> x"
+text \<open>Various other lemmas\<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 :: 'a set set). c \<in> chains S ==> c \<subseteq> S"
+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: "\<forall>C\<in>Chains r. \<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)
-\<comment>\<open>Mirror r in the set of subsets below (wrt r) elements of A\<close>
-  let ?B = "%x. r\<inverse> `` {x}" let ?S = "?B ` Field r"
-  {
-    fix C assume 1: "C \<subseteq> ?S" and 2: "\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A"
+  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}"
-    have "C = ?B ` ?A" using 1 by (auto simp: image_def)
+    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"
-      hence "?B a \<subseteq> ?B b \<or> ?B b \<subseteq> ?B a" using 2 by auto
-      thus "(a, b) \<in> r \<or> (b, a) \<in> r"
+      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" using u by auto
-    have "\<forall>A\<in>C. A \<subseteq> r\<inverse> `` {u}" (is "?P u")
+    with u obtain u where uA: "u \<in> Field r" "\<forall>a\<in>?A. (a, u) \<in> r" by auto
+    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
-      thus "(a, u) \<in> r" using uA and aB and \<open>Preorder r\<close>
+      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 have "\<exists>u\<in>Field r. ?P u" using \<open>u \<in> Field r\<close> by blast
-  }
+    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"
+  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
-  hence "\<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m"
+  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)
-  thus ?thesis using \<open>m \<in> Field r\<close> by blast
+  then show ?thesis
+    using \<open>m \<in> Field r\<close> by blast
 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
-  initialSegmentOf :: "('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"
+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"
+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"
+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
+  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
+  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)
+  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
+  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
+  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
+  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"
+  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)
-  thus "(\<exists>r\<in>R. (a, b) \<in> r) \<or> (\<exists>r\<in>R. (b, a) \<in> r)"
+  then show "(\<exists>r\<in>R. (a, b) \<in> r) \<or> (\<exists>r\<in>R. (b, a) \<in> r)"
   proof
-    assume "r \<subseteq> s" hence "(a, b) \<in> s \<or> (b, a) \<in> s" using assms(2) A mono_Field[of r s]
+    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)
-    thus ?thesis using \<open>s \<in> R\<close> by blast
+    then show ?thesis
+      using \<open>s \<in> R\<close> by blast
   next
-    assume "s \<subseteq> r" hence "(a, b) \<in> r \<or> (b, a) \<in> r" using assms(2) A mono_Field[of s r]
+    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)
-    thus ?thesis using \<open>r \<in> R\<close> by blast
+    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"
+  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"
+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
-  { fix i have "(f (Suc i), f i) \<in> r"
-    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
-  }
-  thus False using assms(2) and \<open>r \<in> R\<close>
+  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"
+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"
+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"
-  have I_init: "I \<subseteq> init_seg_of" by (auto simp: I_def)
-  hence subch: "\<And>R. R \<in> Chains I \<Longrightarrow> chain\<^sub>\<subseteq> R"
+  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)
-  hence 0: "Partial_order I"
+  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>I-chains have upper bounds in ?WO wrt I: their Union\<close>
-  { fix R assume "R \<in> Chains I"
-    hence Ris: "R \<in> Chains init_seg_of" using mono_Chains [OF I_init] by blast
-    have subch: "chain\<^sub>\<subseteq> R" using \<open>R : Chains I\<close> I_init
-      by (auto simp: init_seg_of_def chain_subset_def Chains_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 : 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
+    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)"

     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 have "\<Union>R \<in> ?WO \<and> (\<forall>r\<in>R. (r, \<Union>R) \<in> I)"
+    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
-  }
-  hence 1: "\<forall>R \<in> Chains I. \<exists>u\<in>Field I. \<forall>r\<in>R. (r, u) \<in> I" by (subst FI) blast
-\<comment>\<open>Zorn's Lemma yields a maximal well-order m:\<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"
+  qed
+  then have 1: "\<forall>R \<in> Chains I. \<exists>u\<in>Field I. \<forall>r\<in>R. (r, u) \<in> I"
+    by (subst FI) blast
+\<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 m covers the whole type:\<close>
-  { fix x::'a assume "x \<notin> Field m"
-\<comment>\<open>We assume that x is not covered and extend m at the top with x\<close>
+\<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
-    hence "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 m by x:\<close>
+    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
+    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 "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)
-      thus ?thesis using \<open>wf (m - Id)\<close> and \<open>x \<notin> Field m\<close>
-        wf_subset [OF \<open>wf ?s\<close> Diff_subset]
+      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 m\<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>
+    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 m:\<close>
-    have False using max and \<open>x \<notin> Field m\<close> unfolding Field_def by blast
-  }
-  hence "Field m = UNIV" by auto
+\<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"
+  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)
-  have "Refl r" and "trans r" and "antisym r" and "Total r" and "wf (r - Id)"
-    using \<open>Well_order r\<close> by (simp_all add: order_on_defs)
-  have "Refl ?r" using \<open>Refl r\<close> by (auto simp: refl_on_def 1 univ)
-  moreover have "trans ?r" using \<open>trans r\<close>
+  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 have "antisym ?r" using \<open>antisym r\<close>
+  moreover from \<open>antisym r\<close> have "antisym ?r"
     unfolding antisym_def by blast
-  moreover have "Total ?r" using \<open>Total r\<close> 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)
+  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"
-  assumes P: "\<And>x f. (\<And>y. (y, x) \<in> R \<Longrightarrow> P f y (f y)) \<Longrightarrow> \<exists>r. P f x r"
+  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 
+  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)
-    fix x assume "\<And>y. (y, x) \<in> R \<Longrightarrow> P f y (f y)"
+    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)
       show "P f x (Eps (P f x))"

   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"
-  assumes P: "\<And>x f. (\<And>y. y < x \<Longrightarrow> P f y (f y)) \<Longrightarrow> \<exists>r. P f 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