# HG changeset patch
# User wenzelm
# Date 1469998578 -7200
# Node ID c0cbfd2b5a45f68a1183f6ce3bdcd63e89c897a7
# Parent  aee0d92995b69f307d1b7b6087b3ac639f1f6955
misc tuning and modernization;

diff -r aee0d92995b6 -r c0cbfd2b5a45 src/HOL/Order_Relation.thy
--- a/src/HOL/Order_Relation.thy	Sun Jul 31 19:09:21 2016 +0200
+++ b/src/HOL/Order_Relation.thy	Sun Jul 31 22:56:18 2016 +0200
@@ -9,7 +9,7 @@
 imports Wfrec
 begin
 
-subsection\<open>Orders on a set\<close>
+subsection \<open>Orders on a set\<close>
 
 definition "preorder_on A r \<equiv> refl_on A r \<and> trans r"
 
@@ -27,51 +27,48 @@
 
 
 lemma preorder_on_empty[simp]: "preorder_on {} {}"
-by(simp add:preorder_on_def trans_def)
+  by (simp add: preorder_on_def trans_def)
 
 lemma partial_order_on_empty[simp]: "partial_order_on {} {}"
-by(simp add:partial_order_on_def)
+  by (simp add: partial_order_on_def)
 
 lemma lnear_order_on_empty[simp]: "linear_order_on {} {}"
-by(simp add:linear_order_on_def)
+  by (simp add: linear_order_on_def)
 
 lemma well_order_on_empty[simp]: "well_order_on {} {}"
-by(simp add:well_order_on_def)
+  by (simp add: well_order_on_def)
 
 
-lemma preorder_on_converse[simp]: "preorder_on A (r^-1) = preorder_on A r"
-by (simp add:preorder_on_def)
+lemma preorder_on_converse[simp]: "preorder_on A (r\<inverse>) = preorder_on A r"
+  by (simp add: preorder_on_def)
 
-lemma partial_order_on_converse[simp]:
-  "partial_order_on A (r^-1) = partial_order_on A r"
-by (simp add: partial_order_on_def)
+lemma partial_order_on_converse[simp]: "partial_order_on A (r\<inverse>) = partial_order_on A r"
+  by (simp add: partial_order_on_def)
 
-lemma linear_order_on_converse[simp]:
-  "linear_order_on A (r^-1) = linear_order_on A r"
-by (simp add: linear_order_on_def)
+lemma linear_order_on_converse[simp]: "linear_order_on A (r\<inverse>) = linear_order_on A r"
+  by (simp add: linear_order_on_def)
 
 
-lemma strict_linear_order_on_diff_Id:
-  "linear_order_on A r \<Longrightarrow> strict_linear_order_on A (r-Id)"
-by(simp add: order_on_defs trans_diff_Id)
+lemma strict_linear_order_on_diff_Id: "linear_order_on A r \<Longrightarrow> strict_linear_order_on A (r - Id)"
+  by (simp add: order_on_defs trans_diff_Id)
 
 lemma linear_order_on_singleton [simp]: "linear_order_on {x} {(x, x)}"
-unfolding order_on_defs by simp
+  by (simp add: order_on_defs)
 
 lemma linear_order_on_acyclic:
   assumes "linear_order_on A r"
   shows "acyclic (r - Id)"
-using strict_linear_order_on_diff_Id[OF assms] 
-by(auto simp add: acyclic_irrefl strict_linear_order_on_def)
+  using strict_linear_order_on_diff_Id[OF assms]
+  by (auto simp add: acyclic_irrefl strict_linear_order_on_def)
 
 lemma linear_order_on_well_order_on:
   assumes "finite r"
   shows "linear_order_on A r \<longleftrightarrow> well_order_on A r"
-unfolding well_order_on_def
-using assms finite_acyclic_wf[OF _ linear_order_on_acyclic, of r] by blast
+  unfolding well_order_on_def
+  using assms finite_acyclic_wf[OF _ linear_order_on_acyclic, of r] by blast
 
 
-subsection\<open>Orders on the field\<close>
+subsection \<open>Orders on the field\<close>
 
 abbreviation "Refl r \<equiv> refl_on (Field r) r"
 
@@ -87,50 +84,57 @@
 
 
 lemma subset_Image_Image_iff:
-  "\<lbrakk> Preorder r; A \<subseteq> Field r; B \<subseteq> Field r\<rbrakk> \<Longrightarrow>
-   r `` A \<subseteq> r `` B \<longleftrightarrow> (\<forall>a\<in>A.\<exists>b\<in>B. (b,a):r)"
-unfolding preorder_on_def refl_on_def Image_def
-apply (simp add: subset_eq)
-unfolding trans_def by fast
+  "Preorder r \<Longrightarrow> A \<subseteq> Field r \<Longrightarrow> B \<subseteq> Field r \<Longrightarrow>
+    r `` A \<subseteq> r `` B \<longleftrightarrow> (\<forall>a\<in>A.\<exists>b\<in>B. (b, a) \<in> r)"
+  apply (simp add: preorder_on_def refl_on_def Image_def subset_eq)
+  apply (simp only: trans_def)
+  apply fast
+  done
 
 lemma subset_Image1_Image1_iff:
-  "\<lbrakk> Preorder r; a : Field r; b : Field r\<rbrakk> \<Longrightarrow> r `` {a} \<subseteq> r `` {b} \<longleftrightarrow> (b,a):r"
-by(simp add:subset_Image_Image_iff)
+  "Preorder r \<Longrightarrow> a \<in> Field r \<Longrightarrow> b \<in> Field r \<Longrightarrow> r `` {a} \<subseteq> r `` {b} \<longleftrightarrow> (b, a) \<in> r"
+  by (simp add: subset_Image_Image_iff)
 
 lemma Refl_antisym_eq_Image1_Image1_iff:
-  assumes r: "Refl r" and as: "antisym r" and abf: "a \<in> Field r" "b \<in> Field r"
+  assumes "Refl r"
+    and as: "antisym r"
+    and abf: "a \<in> Field r" "b \<in> Field r"
   shows "r `` {a} = r `` {b} \<longleftrightarrow> a = b"
+    (is "?lhs \<longleftrightarrow> ?rhs")
 proof
-  assume "r `` {a} = r `` {b}"
-  hence e: "\<And>x. (a, x) \<in> r \<longleftrightarrow> (b, x) \<in> r" by (simp add: set_eq_iff)
-  have "(a, a) \<in> r" "(b, b) \<in> r" using r abf by (simp_all add: refl_on_def)
-  hence "(a, b) \<in> r" "(b, a) \<in> r" using e[of a] e[of b] by simp_all
-  thus "a = b" using as[unfolded antisym_def] by blast
-qed fast
+  assume ?lhs
+  then have *: "\<And>x. (a, x) \<in> r \<longleftrightarrow> (b, x) \<in> r"
+    by (simp add: set_eq_iff)
+  have "(a, a) \<in> r" "(b, b) \<in> r" using \<open>Refl r\<close> abf by (simp_all add: refl_on_def)
+  then have "(a, b) \<in> r" "(b, a) \<in> r" using *[of a] *[of b] by simp_all
+  then show ?rhs
+    using \<open>antisym r\<close>[unfolded antisym_def] by blast
+next
+  assume ?rhs
+  then show ?lhs by fast
+qed
 
 lemma Partial_order_eq_Image1_Image1_iff:
-  "\<lbrakk>Partial_order r; a:Field r; b:Field r\<rbrakk> \<Longrightarrow> r `` {a} = r `` {b} \<longleftrightarrow> a=b"
-by(auto simp:order_on_defs Refl_antisym_eq_Image1_Image1_iff)
+  "Partial_order r \<Longrightarrow> a \<in> Field r \<Longrightarrow> b \<in> Field r \<Longrightarrow> r `` {a} = r `` {b} \<longleftrightarrow> a = b"
+  by (auto simp: order_on_defs Refl_antisym_eq_Image1_Image1_iff)
 
 lemma Total_Id_Field:
-assumes TOT: "Total r" and NID: "\<not> (r <= Id)"
-shows "Field r = Field(r - Id)"
-using mono_Field[of "r - Id" r] Diff_subset[of r Id]
-proof(auto)
-  have "r \<noteq> {}" using NID by fast
-  then obtain b and c where "b \<noteq> c \<and> (b,c) \<in> r" using NID by auto
-  hence 1: "b \<noteq> c \<and> {b,c} \<le> Field r" by (auto simp: Field_def)
-
+  assumes "Total r"
+    and not_Id: "\<not> r \<subseteq> Id"
+  shows "Field r = Field (r - Id)"
+  using mono_Field[of "r - Id" r] Diff_subset[of r Id]
+proof auto
   fix a assume *: "a \<in> Field r"
-  obtain d where 2: "d \<in> Field r" and 3: "d \<noteq> a"
-  using * 1 by auto
-  hence "(a,d) \<in> r \<or> (d,a) \<in> r" using * TOT
-  by (simp add: total_on_def)
-  thus "a \<in> Field(r - Id)" using 3 unfolding Field_def by blast
+  from not_Id have "r \<noteq> {}" by fast
+  with not_Id obtain b and c where "b \<noteq> c \<and> (b,c) \<in> r" by auto
+  then have "b \<noteq> c \<and> {b, c} \<subseteq> Field r" by (auto simp: Field_def)
+  with * obtain d where "d \<in> Field r" "d \<noteq> a" by auto
+  with * \<open>Total r\<close> have "(a, d) \<in> r \<or> (d, a) \<in> r" by (simp add: total_on_def)
+  with \<open>d \<noteq> a\<close> show "a \<in> Field (r - Id)" unfolding Field_def by blast
 qed
 
 
-subsection\<open>Orders on a type\<close>
+subsection \<open>Orders on a type\<close>
 
 abbreviation "strict_linear_order \<equiv> strict_linear_order_on UNIV"
 
@@ -141,297 +145,303 @@
 
 subsection \<open>Order-like relations\<close>
 
-text\<open>In this subsection, we develop basic concepts and results pertaining
-to order-like relations, i.e., to reflexive and/or transitive and/or symmetric and/or
-total relations. We also further define upper and lower bounds operators.\<close>
+text \<open>
+  In this subsection, we develop basic concepts and results pertaining
+  to order-like relations, i.e., to reflexive and/or transitive and/or symmetric and/or
+  total relations. We also further define upper and lower bounds operators.
+\<close>
 
 
 subsubsection \<open>Auxiliaries\<close>
 
-lemma refl_on_domain:
-"\<lbrakk>refl_on A r; (a,b) : r\<rbrakk> \<Longrightarrow> a \<in> A \<and> b \<in> A"
-by(auto simp add: refl_on_def)
+lemma refl_on_domain: "refl_on A r \<Longrightarrow> (a, b) \<in> r \<Longrightarrow> a \<in> A \<and> b \<in> A"
+  by (auto simp add: refl_on_def)
 
-corollary well_order_on_domain:
-"\<lbrakk>well_order_on A r; (a,b) \<in> r\<rbrakk> \<Longrightarrow> a \<in> A \<and> b \<in> A"
-by (auto simp add: refl_on_domain order_on_defs)
+corollary well_order_on_domain: "well_order_on A r \<Longrightarrow> (a, b) \<in> r \<Longrightarrow> a \<in> A \<and> b \<in> A"
+  by (auto simp add: refl_on_domain order_on_defs)
 
-lemma well_order_on_Field:
-"well_order_on A r \<Longrightarrow> A = Field r"
-by(auto simp add: refl_on_def Field_def order_on_defs)
+lemma well_order_on_Field: "well_order_on A r \<Longrightarrow> A = Field r"
+  by (auto simp add: refl_on_def Field_def order_on_defs)
 
-lemma well_order_on_Well_order:
-"well_order_on A r \<Longrightarrow> A = Field r \<and> Well_order r"
-using well_order_on_Field by auto
+lemma well_order_on_Well_order: "well_order_on A r \<Longrightarrow> A = Field r \<and> Well_order r"
+  using well_order_on_Field [of A] by auto
 
 lemma Total_subset_Id:
-assumes TOT: "Total r" and SUB: "r \<le> Id"
-shows "r = {} \<or> (\<exists>a. r = {(a,a)})"
-proof-
-  {assume "r \<noteq> {}"
-   then obtain a b where 1: "(a,b) \<in> r" by fast
-   hence "a = b" using SUB by blast
-   hence 2: "(a,a) \<in> r" using 1 by simp
-   {fix c d assume "(c,d) \<in> r"
-    hence "{a,c,d} \<le> Field r" using 1 unfolding Field_def by blast
-    hence "((a,c) \<in> r \<or> (c,a) \<in> r \<or> a = c) \<and>
-           ((a,d) \<in> r \<or> (d,a) \<in> r \<or> a = d)"
-    using TOT unfolding total_on_def by blast
-    hence "a = c \<and> a = d" using SUB by blast
-   }
-   hence "r \<le> {(a,a)}" by auto
-   with 2 have "\<exists>a. r = {(a,a)}" by blast
-  }
-  thus ?thesis by blast
+  assumes "Total r"
+    and "r \<subseteq> Id"
+  shows "r = {} \<or> (\<exists>a. r = {(a, a)})"
+proof -
+  have "\<exists>a. r = {(a, a)}" if "r \<noteq> {}"
+  proof -
+    from that obtain a b where ab: "(a, b) \<in> r" by fast
+    with \<open>r \<subseteq> Id\<close> have "a = b" by blast
+    with ab have aa: "(a, a) \<in> r" by simp
+    have "a = c \<and> a = d" if "(c, d) \<in> r" for c d
+    proof -
+      from that have "{a, c, d} \<subseteq> Field r"
+        using ab unfolding Field_def by blast
+      then have "((a, c) \<in> r \<or> (c, a) \<in> r \<or> a = c) \<and> ((a, d) \<in> r \<or> (d, a) \<in> r \<or> a = d)"
+        using \<open>Total r\<close> unfolding total_on_def by blast
+      with \<open>r \<subseteq> Id\<close> show ?thesis by blast
+    qed
+    then have "r \<subseteq> {(a, a)}" by auto
+    with aa show ?thesis by blast
+  qed
+  then show ?thesis by blast
 qed
 
 lemma Linear_order_in_diff_Id:
-assumes LI: "Linear_order r" and
-        IN1: "a \<in> Field r" and IN2: "b \<in> Field r"
-shows "((a,b) \<in> r) = ((b,a) \<notin> r - Id)"
-using assms unfolding order_on_defs total_on_def antisym_def Id_def refl_on_def by force
+  assumes "Linear_order r"
+    and "a \<in> Field r"
+    and "b \<in> Field r"
+  shows "(a, b) \<in> r \<longleftrightarrow> (b, a) \<notin> r - Id"
+  using assms unfolding order_on_defs total_on_def antisym_def Id_def refl_on_def by force
 
 
 subsubsection \<open>The upper and lower bounds operators\<close>
 
-text\<open>Here we define upper (``above") and lower (``below") bounds operators.
-We think of \<open>r\<close> as a {\em non-strict} relation.  The suffix ``S"
-at the names of some operators indicates that the bounds are strict -- e.g.,
-\<open>underS a\<close> is the set of all strict lower bounds of \<open>a\<close> (w.r.t. \<open>r\<close>).
-Capitalization of the first letter in the name reminds that the operator acts on sets, rather
-than on individual elements.\<close>
+text \<open>
+  Here we define upper (``above") and lower (``below") bounds operators. We
+  think of \<open>r\<close> as a \<^emph>\<open>non-strict\<close> relation. The suffix \<open>S\<close> at the names of
+  some operators indicates that the bounds are strict -- e.g., \<open>underS a\<close> is
+  the set of all strict lower bounds of \<open>a\<close> (w.r.t. \<open>r\<close>). Capitalization of
+  the first letter in the name reminds that the operator acts on sets, rather
+  than on individual elements.
+\<close>
 
-definition under::"'a rel \<Rightarrow> 'a \<Rightarrow> 'a set"
-where "under r a \<equiv> {b. (b,a) \<in> r}"
+definition under :: "'a rel \<Rightarrow> 'a \<Rightarrow> 'a set"
+  where "under r a \<equiv> {b. (b, a) \<in> r}"
 
-definition underS::"'a rel \<Rightarrow> 'a \<Rightarrow> 'a set"
-where "underS r a \<equiv> {b. b \<noteq> a \<and> (b,a) \<in> r}"
+definition underS :: "'a rel \<Rightarrow> 'a \<Rightarrow> 'a set"
+  where "underS r a \<equiv> {b. b \<noteq> a \<and> (b, a) \<in> r}"
 
-definition Under::"'a rel \<Rightarrow> 'a set \<Rightarrow> 'a set"
-where "Under r A \<equiv> {b \<in> Field r. \<forall>a \<in> A. (b,a) \<in> r}"
+definition Under :: "'a rel \<Rightarrow> 'a set \<Rightarrow> 'a set"
+  where "Under r A \<equiv> {b \<in> Field r. \<forall>a \<in> A. (b, a) \<in> r}"
 
-definition UnderS::"'a rel \<Rightarrow> 'a set \<Rightarrow> 'a set"
-where "UnderS r A \<equiv> {b \<in> Field r. \<forall>a \<in> A. b \<noteq> a \<and> (b,a) \<in> r}"
+definition UnderS :: "'a rel \<Rightarrow> 'a set \<Rightarrow> 'a set"
+  where "UnderS r A \<equiv> {b \<in> Field r. \<forall>a \<in> A. b \<noteq> a \<and> (b, a) \<in> r}"
 
-definition above::"'a rel \<Rightarrow> 'a \<Rightarrow> 'a set"
-where "above r a \<equiv> {b. (a,b) \<in> r}"
+definition above :: "'a rel \<Rightarrow> 'a \<Rightarrow> 'a set"
+  where "above r a \<equiv> {b. (a, b) \<in> r}"
 
-definition aboveS::"'a rel \<Rightarrow> 'a \<Rightarrow> 'a set"
-where "aboveS r a \<equiv> {b. b \<noteq> a \<and> (a,b) \<in> r}"
+definition aboveS :: "'a rel \<Rightarrow> 'a \<Rightarrow> 'a set"
+  where "aboveS r a \<equiv> {b. b \<noteq> a \<and> (a, b) \<in> r}"
 
-definition Above::"'a rel \<Rightarrow> 'a set \<Rightarrow> 'a set"
-where "Above r A \<equiv> {b \<in> Field r. \<forall>a \<in> A. (a,b) \<in> r}"
+definition Above :: "'a rel \<Rightarrow> 'a set \<Rightarrow> 'a set"
+  where "Above r A \<equiv> {b \<in> Field r. \<forall>a \<in> A. (a, b) \<in> r}"
 
-definition AboveS::"'a rel \<Rightarrow> 'a set \<Rightarrow> 'a set"
-where "AboveS r A \<equiv> {b \<in> Field r. \<forall>a \<in> A. b \<noteq> a \<and> (a,b) \<in> r}"
+definition AboveS :: "'a rel \<Rightarrow> 'a set \<Rightarrow> 'a set"
+  where "AboveS r A \<equiv> {b \<in> Field r. \<forall>a \<in> A. b \<noteq> a \<and> (a, b) \<in> r}"
 
 definition ofilter :: "'a rel \<Rightarrow> 'a set \<Rightarrow> bool"
-where "ofilter r A \<equiv> (A \<le> Field r) \<and> (\<forall>a \<in> A. under r a \<le> A)"
+  where "ofilter r A \<equiv> A \<subseteq> Field r \<and> (\<forall>a \<in> A. under r a \<subseteq> A)"
 
-text\<open>Note:  In the definitions of \<open>Above[S]\<close> and \<open>Under[S]\<close>,
-  we bounded comprehension by \<open>Field r\<close> in order to properly cover
-  the case of \<open>A\<close> being empty.\<close>
+text \<open>
+  Note: In the definitions of \<open>Above[S]\<close> and \<open>Under[S]\<close>, we bounded
+  comprehension by \<open>Field r\<close> in order to properly cover the case of \<open>A\<close> being
+  empty.
+\<close>
 
-lemma underS_subset_under: "underS r a \<le> under r a"
-by(auto simp add: underS_def under_def)
+lemma underS_subset_under: "underS r a \<subseteq> under r a"
+  by (auto simp add: underS_def under_def)
 
 lemma underS_notIn: "a \<notin> underS r a"
-by(simp add: underS_def)
+  by (simp add: underS_def)
 
-lemma Refl_under_in: "\<lbrakk>Refl r; a \<in> Field r\<rbrakk> \<Longrightarrow> a \<in> under r a"
-by(simp add: refl_on_def under_def)
+lemma Refl_under_in: "Refl r \<Longrightarrow> a \<in> Field r \<Longrightarrow> a \<in> under r a"
+  by (simp add: refl_on_def under_def)
 
-lemma AboveS_disjoint: "A Int (AboveS r A) = {}"
-by(auto simp add: AboveS_def)
+lemma AboveS_disjoint: "A \<inter> (AboveS r A) = {}"
+  by (auto simp add: AboveS_def)
 
 lemma in_AboveS_underS: "a \<in> Field r \<Longrightarrow> a \<in> AboveS r (underS r a)"
-by(auto simp add: AboveS_def underS_def)
+  by (auto simp add: AboveS_def underS_def)
 
-lemma Refl_under_underS:
-  assumes "Refl r" "a \<in> Field r"
-  shows "under r a = underS r a \<union> {a}"
-unfolding under_def underS_def
-using assms refl_on_def[of _ r] by fastforce
+lemma Refl_under_underS: "Refl r \<Longrightarrow> a \<in> Field r \<Longrightarrow> under r a = underS r a \<union> {a}"
+  unfolding under_def underS_def
+  using refl_on_def[of _ r] by fastforce
 
 lemma underS_empty: "a \<notin> Field r \<Longrightarrow> underS r a = {}"
-by (auto simp: Field_def underS_def)
+  by (auto simp: Field_def underS_def)
 
-lemma under_Field: "under r a \<le> Field r"
-by(unfold under_def Field_def, auto)
+lemma under_Field: "under r a \<subseteq> Field r"
+  by (auto simp: under_def Field_def)
 
-lemma underS_Field: "underS r a \<le> Field r"
-by(unfold underS_def Field_def, auto)
+lemma underS_Field: "underS r a \<subseteq> Field r"
+  by (auto simp: underS_def Field_def)
 
-lemma underS_Field2:
-"a \<in> Field r \<Longrightarrow> underS r a < Field r"
-using underS_notIn underS_Field by fast
+lemma underS_Field2: "a \<in> Field r \<Longrightarrow> underS r a \<subset> Field r"
+  using underS_notIn underS_Field by fast
 
-lemma underS_Field3:
-"Field r \<noteq> {} \<Longrightarrow> underS r a < Field r"
-by(cases "a \<in> Field r", simp add: underS_Field2, auto simp add: underS_empty)
+lemma underS_Field3: "Field r \<noteq> {} \<Longrightarrow> underS r a \<subset> Field r"
+  by (cases "a \<in> Field r") (auto simp: underS_Field2 underS_empty)
 
-lemma AboveS_Field: "AboveS r A \<le> Field r"
-by(unfold AboveS_def Field_def, auto)
+lemma AboveS_Field: "AboveS r A \<subseteq> Field r"
+  by (auto simp: AboveS_def Field_def)
 
 lemma under_incr:
-  assumes TRANS: "trans r" and REL: "(a,b) \<in> r"
-  shows "under r a \<le> under r b"
-proof(unfold under_def, auto)
-  fix x assume "(x,a) \<in> r"
-  with REL TRANS trans_def[of r]
-  show "(x,b) \<in> r" by blast
+  assumes "trans r"
+    and "(a, b) \<in> r"
+  shows "under r a \<subseteq> under r b"
+  unfolding under_def
+proof auto
+  fix x assume "(x, a) \<in> r"
+  with assms trans_def[of r] show "(x, b) \<in> r" by blast
 qed
 
 lemma underS_incr:
-assumes TRANS: "trans r" and ANTISYM: "antisym r" and
-        REL: "(a,b) \<in> r"
-shows "underS r a \<le> underS r b"
-proof(unfold underS_def, auto)
-  assume *: "b \<noteq> a" and **: "(b,a) \<in> r"
-  with ANTISYM antisym_def[of r] REL
-  show False by blast
+  assumes "trans r"
+    and "antisym r"
+    and ab: "(a, b) \<in> r"
+  shows "underS r a \<subseteq> underS r b"
+  unfolding underS_def
+proof auto
+  assume *: "b \<noteq> a" and **: "(b, a) \<in> r"
+  with \<open>antisym r\<close> antisym_def[of r] ab show False
+    by blast
 next
-  fix x assume "x \<noteq> a" "(x,a) \<in> r"
-  with REL TRANS trans_def[of r]
-  show "(x,b) \<in> r" by blast
+  fix x assume "x \<noteq> a" "(x, a) \<in> r"
+  with ab \<open>trans r\<close> trans_def[of r] show "(x, b) \<in> r"
+    by blast
 qed
 
 lemma underS_incl_iff:
-assumes LO: "Linear_order r" and
-        INa: "a \<in> Field r" and INb: "b \<in> Field r"
-shows "(underS r a \<le> underS r b) = ((a,b) \<in> r)"
+  assumes LO: "Linear_order r"
+    and INa: "a \<in> Field r"
+    and INb: "b \<in> Field r"
+  shows "underS r a \<subseteq> underS r b \<longleftrightarrow> (a, b) \<in> r"
+    (is "?lhs \<longleftrightarrow> ?rhs")
 proof
-  assume "(a,b) \<in> r"
-  thus "underS r a \<le> underS r b" using LO
-  by (simp add: order_on_defs underS_incr)
+  assume ?rhs
+  with \<open>Linear_order r\<close> show ?lhs
+    by (simp add: order_on_defs underS_incr)
 next
-  assume *: "underS r a \<le> underS r b"
-  {assume "a = b"
-   hence "(a,b) \<in> r" using assms
-   by (simp add: order_on_defs refl_on_def)
-  }
-  moreover
-  {assume "a \<noteq> b \<and> (b,a) \<in> r"
-   hence "b \<in> underS r a" unfolding underS_def by blast
-   hence "b \<in> underS r b" using * by blast
-   hence False by (simp add: underS_notIn)
-  }
-  ultimately
-  show "(a,b) \<in> r" using assms
-  order_on_defs[of "Field r" r] total_on_def[of "Field r" r] by blast
+  assume *: ?lhs
+  have "(a, b) \<in> r" if "a = b"
+    using assms that by (simp add: order_on_defs refl_on_def)
+  moreover have False if "a \<noteq> b" "(b, a) \<in> r"
+  proof -
+    from that have "b \<in> underS r a" unfolding underS_def by blast
+    with * have "b \<in> underS r b" by blast
+    then show ?thesis by (simp add: underS_notIn)
+  qed
+  ultimately show "(a,b) \<in> r"
+    using assms order_on_defs[of "Field r" r] total_on_def[of "Field r" r] by blast
 qed
 
 lemma finite_Linear_order_induct[consumes 3, case_names step]:
   assumes "Linear_order r"
-  and "x \<in> Field r"
-  and "finite r"
-  and step: "\<And>x. \<lbrakk>x \<in> Field r; \<And>y. y \<in> aboveS r x \<Longrightarrow> P y\<rbrakk> \<Longrightarrow> P x"
+    and "x \<in> Field r"
+    and "finite r"
+    and step: "\<And>x. x \<in> Field r \<Longrightarrow> (\<And>y. y \<in> aboveS r x \<Longrightarrow> P y) \<Longrightarrow> P x"
   shows "P x"
-using assms(2)
-proof(induct rule: wf_induct[of "r\<inverse> - Id"])
+  using assms(2)
+proof (induct rule: wf_induct[of "r\<inverse> - Id"])
+  case 1
   from assms(1,3) show "wf (r\<inverse> - Id)"
     using linear_order_on_well_order_on linear_order_on_converse
     unfolding well_order_on_def by blast
 next
-  case (2 x) then show ?case
-    by - (rule step; auto simp: aboveS_def intro: FieldI2)
+  case prems: (2 x)
+  show ?case
+    by (rule step) (use prems in \<open>auto simp: aboveS_def intro: FieldI2\<close>)
 qed
 
 
 subsection \<open>Variations on Well-Founded Relations\<close>
 
 text \<open>
-This subsection contains some variations of the results from @{theory Wellfounded}:
-\begin{itemize}
-\item means for slightly more direct definitions by well-founded recursion;
-\item variations of well-founded induction;
-\item means for proving a linear order to be a well-order.
-\end{itemize}
+  This subsection contains some variations of the results from @{theory Wellfounded}:
+    \<^item> means for slightly more direct definitions by well-founded recursion;
+    \<^item> variations of well-founded induction;
+    \<^item> means for proving a linear order to be a well-order.
 \<close>
 
 
 subsubsection \<open>Characterizations of well-foundedness\<close>
 
-text \<open>A transitive relation is well-founded iff it is ``locally'' well-founded,
-i.e., iff its restriction to the lower bounds of of any element is well-founded.\<close>
+text \<open>
+  A transitive relation is well-founded iff it is ``locally'' well-founded,
+  i.e., iff its restriction to the lower bounds of of any element is
+  well-founded.
+\<close>
 
 lemma trans_wf_iff:
-assumes "trans r"
-shows "wf r = (\<forall>a. wf(r Int (r^-1``{a} \<times> r^-1``{a})))"
-proof-
-  obtain R where R_def: "R = (\<lambda> a. r Int (r^-1``{a} \<times> r^-1``{a}))" by blast
-  {assume *: "wf r"
-   {fix a
-    have "wf(R a)"
-    using * R_def wf_subset[of r "R a"] by auto
-   }
-  }
-  (*  *)
+  assumes "trans r"
+  shows "wf r \<longleftrightarrow> (\<forall>a. wf (r \<inter> (r\<inverse>``{a} \<times> r\<inverse>``{a})))"
+proof -
+  define R where "R a = r \<inter> (r\<inverse>``{a} \<times> r\<inverse>``{a})" for a
+  have "wf (R a)" if "wf r" for a
+    using that R_def wf_subset[of r "R a"] by auto
   moreover
-  {assume *: "\<forall>a. wf(R a)"
-   have "wf r"
-   proof(unfold wf_def, clarify)
-     fix phi a
-     assume **: "\<forall>a. (\<forall>b. (b,a) \<in> r \<longrightarrow> phi b) \<longrightarrow> phi a"
-     obtain chi where chi_def: "chi = (\<lambda>b. (b,a) \<in> r \<longrightarrow> phi b)" by blast
-     with * have "wf (R a)" by auto
-     hence "(\<forall>b. (\<forall>c. (c,b) \<in> R a \<longrightarrow> chi c) \<longrightarrow> chi b) \<longrightarrow> (\<forall>b. chi b)"
-     unfolding wf_def by blast
-     moreover
-     have "\<forall>b. (\<forall>c. (c,b) \<in> R a \<longrightarrow> chi c) \<longrightarrow> chi b"
-     proof(auto simp add: chi_def R_def)
-       fix b
-       assume 1: "(b,a) \<in> r" and 2: "\<forall>c. (c, b) \<in> r \<and> (c, a) \<in> r \<longrightarrow> phi c"
-       hence "\<forall>c. (c, b) \<in> r \<longrightarrow> phi c"
-       using assms trans_def[of r] by blast
-       thus "phi b" using ** by blast
-     qed
-     ultimately have  "\<forall>b. chi b" by (rule mp)
-     with ** chi_def show "phi a" by blast
-   qed
-  }
-  ultimately show ?thesis using R_def by blast
+  have "wf r" if *: "\<forall>a. wf(R a)"
+    unfolding wf_def
+  proof clarify
+    fix phi a
+    assume **: "\<forall>a. (\<forall>b. (b, a) \<in> r \<longrightarrow> phi b) \<longrightarrow> phi a"
+    define chi where "chi b \<longleftrightarrow> (b, a) \<in> r \<longrightarrow> phi b" for b
+    with * have "wf (R a)" by auto
+    then have "(\<forall>b. (\<forall>c. (c, b) \<in> R a \<longrightarrow> chi c) \<longrightarrow> chi b) \<longrightarrow> (\<forall>b. chi b)"
+      unfolding wf_def by blast
+    also have "\<forall>b. (\<forall>c. (c, b) \<in> R a \<longrightarrow> chi c) \<longrightarrow> chi b"
+    proof (auto simp add: chi_def R_def)
+      fix b
+      assume "(b, a) \<in> r" and "\<forall>c. (c, b) \<in> r \<and> (c, a) \<in> r \<longrightarrow> phi c"
+      then have "\<forall>c. (c, b) \<in> r \<longrightarrow> phi c"
+        using assms trans_def[of r] by blast
+      with ** show "phi b" by blast
+    qed
+    finally have  "\<forall>b. chi b" .
+    with ** chi_def show "phi a" by blast
+  qed
+  ultimately show ?thesis unfolding R_def by blast
 qed
 
 text \<open>The next lemma is a variation of \<open>wf_eq_minimal\<close> from Wellfounded,
-allowing one to assume the set included in the field.\<close>
+  allowing one to assume the set included in the field.\<close>
 
-lemma wf_eq_minimal2:
-"wf r = (\<forall>A. A <= Field r \<and> A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. \<not> (a',a) \<in> r))"
+lemma wf_eq_minimal2: "wf r \<longleftrightarrow> (\<forall>A. A \<subseteq> Field r \<and> A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a', a) \<notin> r))"
 proof-
-  let ?phi = "\<lambda> A. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. \<not> (a',a) \<in> r)"
-  have "wf r = (\<forall>A. ?phi A)"
-  by (auto simp: ex_in_conv [THEN sym], erule wfE_min, assumption, blast)
-     (rule wfI_min, fast)
-  (*  *)
-  also have "(\<forall>A. ?phi A) = (\<forall>B \<le> Field r. ?phi B)"
+  let ?phi = "\<lambda>A. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a',a) \<notin> r)"
+  have "wf r \<longleftrightarrow> (\<forall>A. ?phi A)"
+    apply (auto simp: ex_in_conv [THEN sym])
+     apply (erule wfE_min)
+      apply assumption
+     apply blast
+    apply (rule wfI_min)
+    apply fast
+    done
+  also have "(\<forall>A. ?phi A) \<longleftrightarrow> (\<forall>B \<subseteq> Field r. ?phi B)"
   proof
     assume "\<forall>A. ?phi A"
-    thus "\<forall>B \<le> Field r. ?phi B" by simp
+    then show "\<forall>B \<subseteq> Field r. ?phi B" by simp
   next
-    assume *: "\<forall>B \<le> Field r. ?phi B"
+    assume *: "\<forall>B \<subseteq> Field r. ?phi B"
     show "\<forall>A. ?phi A"
-    proof(clarify)
-      fix A::"'a set" assume **: "A \<noteq> {}"
-      obtain B where B_def: "B = A Int (Field r)" by blast
-      show "\<exists>a \<in> A. \<forall>a' \<in> A. (a',a) \<notin> r"
-      proof(cases "B = {}")
-        assume Case1: "B = {}"
-        obtain a where 1: "a \<in> A \<and> a \<notin> Field r"
-        using ** Case1 unfolding B_def by blast
-        hence "\<forall>a' \<in> A. (a',a) \<notin> r" using 1 unfolding Field_def by blast
-        thus ?thesis using 1 by blast
+    proof clarify
+      fix A :: "'a set"
+      assume **: "A \<noteq> {}"
+      define B where "B = A \<inter> Field r"
+      show "\<exists>a \<in> A. \<forall>a' \<in> A. (a', a) \<notin> r"
+      proof (cases "B = {}")
+        case True
+        with ** obtain a where a: "a \<in> A" "a \<notin> Field r"
+          unfolding B_def by blast
+        with a have "\<forall>a' \<in> A. (a',a) \<notin> r"
+          unfolding Field_def by blast
+        with a show ?thesis by blast
       next
-        assume Case2: "B \<noteq> {}" have 1: "B \<le> Field r" unfolding B_def by blast
-        obtain a where 2: "a \<in> B \<and> (\<forall>a' \<in> B. (a',a) \<notin> r)"
-        using Case2 1 * by blast
-        have "\<forall>a' \<in> A. (a',a) \<notin> r"
-        proof(clarify)
-          fix a' assume "a' \<in> A" and **: "(a',a) \<in> r"
-          hence "a' \<in> B" unfolding B_def Field_def by blast
-          thus False using 2 ** by blast
+        case False
+        have "B \<subseteq> Field r" unfolding B_def by blast
+        with False * obtain a where a: "a \<in> B" "\<forall>a' \<in> B. (a', a) \<notin> r"
+          by blast
+        have "(a', a) \<notin> r" if "a' \<in> A" for a'
+        proof
+          assume a'a: "(a', a) \<in> r"
+          with that have "a' \<in> B" unfolding B_def Field_def by blast
+          with a a'a show False by blast
         qed
-        thus ?thesis using 2 unfolding B_def by blast
+        with a show ?thesis unfolding B_def by blast
       qed
     qed
   qed
@@ -441,58 +451,67 @@
 
 subsubsection \<open>Characterizations of well-foundedness\<close>
 
-text \<open>The next lemma and its corollary enable one to prove that
-a linear order is a well-order in a way which is more standard than
-via well-foundedness of the strict version of the relation.\<close>
+text \<open>
+  The next lemma and its corollary enable one to prove that a linear order is
+  a well-order in a way which is more standard than via well-foundedness of
+  the strict version of the relation.
+\<close>
 
 lemma Linear_order_wf_diff_Id:
-assumes LI: "Linear_order r"
-shows "wf(r - Id) = (\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r))"
-proof(cases "r \<le> Id")
-  assume Case1: "r \<le> Id"
-  hence temp: "r - Id = {}" by blast
-  hence "wf(r - Id)" by (simp add: temp)
-  moreover
-  {fix A assume *: "A \<le> Field r" and **: "A \<noteq> {}"
-   obtain a where 1: "r = {} \<or> r = {(a,a)}" using LI
-   unfolding order_on_defs using Case1 Total_subset_Id by auto
-   hence "A = {a} \<and> r = {(a,a)}" using * ** unfolding Field_def by blast
-   hence "\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r" using 1 by blast
-  }
+  assumes "Linear_order r"
+  shows "wf (r - Id) \<longleftrightarrow> (\<forall>A \<subseteq> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a, a') \<in> r))"
+proof (cases "r \<subseteq> Id")
+  case True
+  then have *: "r - Id = {}" by blast
+  have "wf (r - Id)" by (simp add: *)
+  moreover have "\<exists>a \<in> A. \<forall>a' \<in> A. (a, a') \<in> r"
+    if *: "A \<subseteq> Field r" and **: "A \<noteq> {}" for A
+  proof -
+    from \<open>Linear_order r\<close> True
+    obtain a where a: "r = {} \<or> r = {(a, a)}"
+      unfolding order_on_defs using Total_subset_Id [of r] by blast
+    with * ** have "A = {a} \<and> r = {(a, a)}"
+      unfolding Field_def by blast
+    with a show ?thesis by blast
+  qed
   ultimately show ?thesis by blast
 next
-  assume Case2: "\<not> r \<le> Id"
-  hence 1: "Field r = Field(r - Id)" using Total_Id_Field LI
-  unfolding order_on_defs by blast
+  case False
+  with \<open>Linear_order r\<close> have Field: "Field r = Field (r - Id)"
+    unfolding order_on_defs using Total_Id_Field [of r] by blast
   show ?thesis
   proof
-    assume *: "wf(r - Id)"
-    show "\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r)"
-    proof(clarify)
-      fix A assume **: "A \<le> Field r" and ***: "A \<noteq> {}"
-      hence "\<exists>a \<in> A. \<forall>a' \<in> A. (a',a) \<notin> r - Id"
-      using 1 * unfolding wf_eq_minimal2 by simp
-      moreover have "\<forall>a \<in> A. \<forall>a' \<in> A. ((a,a') \<in> r) = ((a',a) \<notin> r - Id)"
-      using Linear_order_in_diff_Id[of r] ** LI by blast
-      ultimately show "\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r" by blast
+    assume *: "wf (r - Id)"
+    show "\<forall>A \<subseteq> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a, a') \<in> r)"
+    proof clarify
+      fix A
+      assume **: "A \<subseteq> Field r" and ***: "A \<noteq> {}"
+      then have "\<exists>a \<in> A. \<forall>a' \<in> A. (a',a) \<notin> r - Id"
+        using Field * unfolding wf_eq_minimal2 by simp
+      moreover have "\<forall>a \<in> A. \<forall>a' \<in> A. (a, a') \<in> r \<longleftrightarrow> (a', a) \<notin> r - Id"
+        using Linear_order_in_diff_Id [OF \<open>Linear_order r\<close>] ** by blast
+      ultimately show "\<exists>a \<in> A. \<forall>a' \<in> A. (a, a') \<in> r" by blast
     qed
   next
-    assume *: "\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r)"
-    show "wf(r - Id)"
-    proof(unfold wf_eq_minimal2, clarify)
-      fix A assume **: "A \<le> Field(r - Id)" and ***: "A \<noteq> {}"
-      hence "\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r"
-      using 1 * by simp
-      moreover have "\<forall>a \<in> A. \<forall>a' \<in> A. ((a,a') \<in> r) = ((a',a) \<notin> r - Id)"
-      using Linear_order_in_diff_Id[of r] ** LI mono_Field[of "r - Id" r] by blast
-      ultimately show "\<exists>a \<in> A. \<forall>a' \<in> A. (a',a) \<notin> r - Id" by blast
+    assume *: "\<forall>A \<subseteq> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a, a') \<in> r)"
+    show "wf (r - Id)"
+      unfolding wf_eq_minimal2
+    proof clarify
+      fix A
+      assume **: "A \<subseteq> Field(r - Id)" and ***: "A \<noteq> {}"
+      then have "\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r"
+        using Field * by simp
+      moreover have "\<forall>a \<in> A. \<forall>a' \<in> A. (a, a') \<in> r \<longleftrightarrow> (a', a) \<notin> r - Id"
+        using Linear_order_in_diff_Id [OF \<open>Linear_order r\<close>] ** mono_Field[of "r - Id" r] by blast
+      ultimately show "\<exists>a \<in> A. \<forall>a' \<in> A. (a',a) \<notin> r - Id"
+        by blast
     qed
   qed
 qed
 
 corollary Linear_order_Well_order_iff:
-assumes "Linear_order r"
-shows "Well_order r = (\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r))"
-using assms unfolding well_order_on_def using Linear_order_wf_diff_Id[of r] by blast
+  "Linear_order r \<Longrightarrow>
+    Well_order r \<longleftrightarrow> (\<forall>A \<subseteq> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a, a') \<in> r))"
+  unfolding well_order_on_def using Linear_order_wf_diff_Id[of r] by blast
 
 end
diff -r aee0d92995b6 -r c0cbfd2b5a45 src/HOL/Wellfounded.thy
--- a/src/HOL/Wellfounded.thy	Sun Jul 31 19:09:21 2016 +0200
+++ b/src/HOL/Wellfounded.thy	Sun Jul 31 22:56:18 2016 +0200
@@ -9,7 +9,7 @@
 section \<open>Well-founded Recursion\<close>
 
 theory Wellfounded
-imports Transitive_Closure
+  imports Transitive_Closure
 begin
 
 subsection \<open>Basic Definitions\<close>
@@ -59,12 +59,14 @@
 lemma wf_not_refl [simp]: "wf r \<Longrightarrow> (a, a) \<notin> r"
   by (blast elim: wf_asym)
 
-lemma wf_irrefl: assumes "wf r" obtains "(a, a) \<notin> r"
+lemma wf_irrefl:
+  assumes "wf r"
+  obtains "(a, a) \<notin> r"
   by (drule wf_not_refl[OF assms])
 
 lemma wf_wellorderI:
   assumes wf: "wf {(x::'a::ord, y). x < y}"
-  assumes lin: "OFCLASS('a::ord, linorder_class)"
+    and lin: "OFCLASS('a::ord, linorder_class)"
   shows "OFCLASS('a::ord, wellorder_class)"
   using lin
   apply (rule wellorder_class.intro)
@@ -83,7 +85,7 @@
 
 lemma wfE_pf:
   assumes wf: "wf R"
-  assumes a: "A \<subseteq> R `` A"
+    and a: "A \<subseteq> R `` A"
   shows "A = {}"
 proof -
   from wf have "x \<notin> A" for x
@@ -130,10 +132,13 @@
 qed
 
 lemma wf_eq_minimal: "wf r \<longleftrightarrow> (\<forall>Q x. x \<in> Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q))"
-apply auto
-apply (erule wfE_min, assumption, blast)
-apply (rule wfI_min, auto)
-done
+  apply auto
+   apply (erule wfE_min)
+    apply assumption
+   apply blast
+  apply (rule wfI_min)
+  apply auto
+  done
 
 lemmas wfP_eq_minimal = wf_eq_minimal [to_pred]
 
@@ -200,18 +205,13 @@
   then show ?thesis by (simp add: bot_fun_def)
 qed
 
-lemma wf_Int1: "wf r \<Longrightarrow> wf (r Int r')"
-  apply (erule wf_subset)
-  apply (rule Int_lower1)
-  done
+lemma wf_Int1: "wf r \<Longrightarrow> wf (r \<inter> r')"
+  by (erule wf_subset) (rule Int_lower1)
 
-lemma wf_Int2: "wf r \<Longrightarrow> wf (r' Int r)"
-  apply (erule wf_subset)
-  apply (rule Int_lower2)
-  done
+lemma wf_Int2: "wf r \<Longrightarrow> wf (r' \<inter> r)"
+  by (erule wf_subset) (rule Int_lower2)
 
-text \<open>Exponentiation\<close>
-
+text \<open>Exponentiation.\<close>
 lemma wf_exp:
   assumes "wf (R ^^ n)"
   shows "wf R"
@@ -222,38 +222,43 @@
   show "A = {}" by (rule wfE_pf)
 qed
 
-text \<open>Well-foundedness of insert\<close>
-
+text \<open>Well-foundedness of \<open>insert\<close>.\<close>
 lemma wf_insert [iff]: "wf (insert (y, x) r) \<longleftrightarrow> wf r \<and> (x, y) \<notin> r\<^sup>*"
-apply (rule iffI)
- apply (blast elim: wf_trancl [THEN wf_irrefl]
-              intro: rtrancl_into_trancl1 wf_subset
-                     rtrancl_mono [THEN [2] rev_subsetD])
-apply (simp add: wf_eq_minimal, safe)
-apply (rule allE, assumption, erule impE, blast)
-apply (erule bexE)
-apply (rename_tac "a", case_tac "a = x")
- prefer 2
-apply blast
-apply (case_tac "y \<in> Q")
- prefer 2 apply blast
-apply (rule_tac x = "{z. z \<in> Q \<and> (z,y) \<in> r\<^sup>*}" in allE)
- apply assumption
-apply (erule_tac V = "\<forall>Q. (\<exists>x. x \<in> Q) \<longrightarrow> P Q" for P in thin_rl)
+  apply (rule iffI)
+   apply (blast elim: wf_trancl [THEN wf_irrefl]
+      intro: rtrancl_into_trancl1 wf_subset rtrancl_mono [THEN [2] rev_subsetD])
+  apply (simp add: wf_eq_minimal)
+  apply safe
+  apply (rule allE)
+   apply assumption
+  apply (erule impE)
+   apply blast
+  apply (erule bexE)
+  apply (rename_tac a, case_tac "a = x")
+   prefer 2
+   apply blast
+  apply (case_tac "y \<in> Q")
+   prefer 2
+   apply blast
+  apply (rule_tac x = "{z. z \<in> Q \<and> (z,y) \<in> r\<^sup>*}" in allE)
+   apply assumption
+  apply (erule_tac V = "\<forall>Q. (\<exists>x. x \<in> Q) \<longrightarrow> P Q" for P in thin_rl)
   (*essential for speed*)
-(*blast with new substOccur fails*)
-apply (fast intro: converse_rtrancl_into_rtrancl)
-done
+  (*blast with new substOccur fails*)
+  apply (fast intro: converse_rtrancl_into_rtrancl)
+  done
 
 
 subsubsection \<open>Well-foundedness of image\<close>
 
 lemma wf_map_prod_image: "wf r \<Longrightarrow> inj f \<Longrightarrow> wf (map_prod f f ` r)"
-apply (simp only: wf_eq_minimal, clarify)
-apply (case_tac "\<exists>p. f p \<in> Q")
-apply (erule_tac x = "{p. f p \<in> Q}" in allE)
-apply (fast dest: inj_onD, blast)
-done
+  apply (simp only: wf_eq_minimal)
+  apply clarify
+  apply (case_tac "\<exists>p. f p \<in> Q")
+   apply (erule_tac x = "{p. f p \<in> Q}" in allE)
+   apply (fast dest: inj_onD)
+apply blast
+  done
 
 
 subsection \<open>Well-Foundedness Results for Unions\<close>
@@ -270,24 +275,21 @@
     by (rule wfE_min [OF \<open>wf R\<close> \<open>x \<in> Q\<close>]) blast
   with \<open>wf S\<close> obtain z where "z \<in> ?Q'" and zmin: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> ?Q'"
     by (erule wfE_min)
-  {
-    fix y assume "(y, z) \<in> S"
-    then have "y \<notin> ?Q'" by (rule zmin)
-    have "y \<notin> Q"
-    proof
-      assume "y \<in> Q"
-      with \<open>y \<notin> ?Q'\<close> obtain w where "(w, y) \<in> R" and "w \<in> Q" by auto
-      from \<open>(w, y) \<in> R\<close> \<open>(y, z) \<in> S\<close> have "(w, z) \<in> R O S" by (rule relcompI)
-      with \<open>R O S \<subseteq> R\<close> have "(w, z) \<in> R" ..
-      with \<open>z \<in> ?Q'\<close> have "w \<notin> Q" by blast
-      with \<open>w \<in> Q\<close> show False by contradiction
-    qed
-  }
+  have "y \<notin> Q" if "(y, z) \<in> S" for y
+  proof
+    from that have "y \<notin> ?Q'" by (rule zmin)
+    assume "y \<in> Q"
+    with \<open>y \<notin> ?Q'\<close> obtain w where "(w, y) \<in> R" and "w \<in> Q" by auto
+    from \<open>(w, y) \<in> R\<close> \<open>(y, z) \<in> S\<close> have "(w, z) \<in> R O S" by (rule relcompI)
+    with \<open>R O S \<subseteq> R\<close> have "(w, z) \<in> R" ..
+    with \<open>z \<in> ?Q'\<close> have "w \<notin> Q" by blast
+    with \<open>w \<in> Q\<close> show False by contradiction
+  qed
   with \<open>z \<in> ?Q'\<close> show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<union> S \<longrightarrow> y \<notin> Q" by blast
 qed
 
 
-text \<open>Well-foundedness of indexed union with disjoint domains and ranges\<close>
+text \<open>Well-foundedness of indexed union with disjoint domains and ranges.\<close>
 
 lemma wf_UN:
   assumes "\<forall>i\<in>I. wf (r i)"
@@ -306,10 +308,9 @@
   done
 
 lemma wfP_SUP:
-  "\<forall>i. wfP (r i) \<Longrightarrow> \<forall>i j. r i \<noteq> r j \<longrightarrow> inf (DomainP (r i)) (RangeP (r j)) = bot \<Longrightarrow> wfP (SUPREMUM UNIV r)"
-  apply (rule wf_UN[to_pred])
-  apply simp_all
-  done
+  "\<forall>i. wfP (r i) \<Longrightarrow> \<forall>i j. r i \<noteq> r j \<longrightarrow> inf (DomainP (r i)) (RangeP (r j)) = bot \<Longrightarrow>
+    wfP (SUPREMUM UNIV r)"
+  by (rule wf_UN[to_pred]) simp_all
 
 lemma wf_Union:
   assumes "\<forall>r\<in>R. wf r"
@@ -458,9 +459,7 @@
 subsection \<open>Acyclic relations\<close>
 
 lemma wf_acyclic: "wf r \<Longrightarrow> acyclic r"
-apply (simp add: acyclic_def)
-apply (blast elim: wf_trancl [THEN wf_irrefl])
-done
+  by (simp add: acyclic_def) (blast elim: wf_trancl [THEN wf_irrefl])
 
 lemmas wfP_acyclicP = wf_acyclic [to_pred]
 
@@ -468,15 +467,15 @@
 subsubsection \<open>Wellfoundedness of finite acyclic relations\<close>
 
 lemma finite_acyclic_wf [rule_format]: "finite r \<Longrightarrow> acyclic r \<longrightarrow> wf r"
-apply (erule finite_induct, blast)
-apply (simp only: split_tupled_all)
-apply simp
-done
+  apply (erule finite_induct)
+   apply blast
+  apply (simp add: split_tupled_all)
+  done
 
 lemma finite_acyclic_wf_converse: "finite r \<Longrightarrow> acyclic r \<Longrightarrow> wf (r\<inverse>)"
-apply (erule finite_converse [THEN iffD2, THEN finite_acyclic_wf])
-apply (erule acyclic_converse [THEN iffD2])
-done
+  apply (erule finite_converse [THEN iffD2, THEN finite_acyclic_wf])
+  apply (erule acyclic_converse [THEN iffD2])
+  done
 
 text \<open>
   Observe that the converse of an irreflexive, transitive,
@@ -488,12 +487,14 @@
   shows "wf (r\<inverse>)"
 proof -
   have "acyclic r"
-    using \<open>irrefl r\<close> and \<open>trans r\<close> by (simp add: irrefl_def acyclic_irrefl)
-  with \<open>finite r\<close> show ?thesis by (rule finite_acyclic_wf_converse)
+    using \<open>irrefl r\<close> and \<open>trans r\<close>
+    by (simp add: irrefl_def acyclic_irrefl)
+  with \<open>finite r\<close> show ?thesis
+    by (rule finite_acyclic_wf_converse)
 qed
 
 lemma wf_iff_acyclic_if_finite: "finite r \<Longrightarrow> wf r = acyclic r"
-by (blast intro: finite_acyclic_wf wf_acyclic)
+  by (blast intro: finite_acyclic_wf wf_acyclic)
 
 
 subsection \<open>@{typ nat} is well-founded\<close>
@@ -528,8 +529,10 @@
   unfolding less_eq rtrancl_eq_or_trancl by auto
 
 lemma wf_pred_nat: "wf pred_nat"
-  apply (unfold wf_def pred_nat_def, clarify)
-  apply (induct_tac x, blast+)
+  apply (unfold wf_def pred_nat_def)
+  apply clarify
+  apply (induct_tac x)
+   apply blast+
   done
 
 lemma wf_less_than [iff]: "wf less_than"
@@ -583,15 +586,13 @@
 lemmas accp_induct_rule = accp_induct [rule_format, induct set: accp]
 
 theorem accp_downward: "accp r b \<Longrightarrow> r a b \<Longrightarrow> accp r a"
-  apply (erule accp.cases)
-  apply fast
-  done
+  by (cases rule: accp.cases)
 
 lemma not_accp_down:
   assumes na: "\<not> accp R x"
   obtains z where "R z x" and "\<not> accp R z"
 proof -
-  assume a: "\<And>z. \<lbrakk>R z x; \<not> accp R z\<rbrakk> \<Longrightarrow> thesis"
+  assume a: "\<And>z. R z x \<Longrightarrow> \<not> accp R z \<Longrightarrow> thesis"
   show thesis
   proof (cases "\<forall>z. R z x \<longrightarrow> accp R z")
     case True
@@ -612,12 +613,11 @@
   done
 
 theorem accp_downwards: "accp r a \<Longrightarrow> r\<^sup>*\<^sup>* b a \<Longrightarrow> accp r b"
-  apply (blast dest: accp_downwards_aux)
-  done
+  by (blast dest: accp_downwards_aux)
 
 theorem accp_wfPI: "\<forall>x. accp r x \<Longrightarrow> wfP r"
   apply (rule wfPUNIVI)
-  apply (rule_tac P=P in accp_induct)
+  apply (rule_tac P = P in accp_induct)
    apply blast
   apply blast
   done
@@ -629,22 +629,22 @@
   done
 
 theorem wfP_accp_iff: "wfP r = (\<forall>x. accp r x)"
-  apply (blast intro: accp_wfPI dest: accp_wfPD)
-  done
+  by (blast intro: accp_wfPI dest: accp_wfPD)
 
 
 text \<open>Smaller relations have bigger accessible parts:\<close>
 
 lemma accp_subset:
-  assumes sub: "R1 \<le> R2"
+  assumes "R1 \<le> R2"
   shows "accp R2 \<le> accp R1"
 proof (rule predicate1I)
-  fix x assume "accp R2 x"
+  fix x
+  assume "accp R2 x"
   then show "accp R1 x"
   proof (induct x)
     fix x
-    assume ih: "\<And>y. R2 y x \<Longrightarrow> accp R1 y"
-    with sub show "accp R1 x"
+    assume "\<And>y. R2 y x \<Longrightarrow> accp R1 y"
+    with assms show "accp R1 x"
       by (blast intro: accp.accI)
   qed
 qed
@@ -655,9 +655,9 @@
 
 lemma accp_subset_induct:
   assumes subset: "D \<le> accp R"
-    and dcl: "\<And>x z. \<lbrakk>D x; R z x\<rbrakk> \<Longrightarrow> D z"
+    and dcl: "\<And>x z. D x \<Longrightarrow> R z x \<Longrightarrow> D z"
     and "D x"
-    and istep: "\<And>x. \<lbrakk>D x; (\<And>z. R z x \<Longrightarrow> P z)\<rbrakk> \<Longrightarrow> P x"
+    and istep: "\<And>x. D x \<Longrightarrow> (\<And>z. R z x \<Longrightarrow> P z) \<Longrightarrow> P x"
   shows "P x"
 proof -
   from subset and \<open>D x\<close>
@@ -665,8 +665,7 @@
   then show "P x" using \<open>D x\<close>
   proof (induct x)
     fix x
-    assume "D x"
-      and "\<And>y. R y x \<Longrightarrow> D y \<Longrightarrow> P y"
+    assume "D x" and "\<And>y. R y x \<Longrightarrow> D y \<Longrightarrow> P y"
     with dcl and istep show "P x" by blast
   qed
 qed
@@ -691,15 +690,17 @@
 
 text \<open>Inverse Image\<close>
 
-lemma wf_inv_image [simp,intro!]: "wf r \<Longrightarrow> wf (inv_image r f)" for f :: "'a \<Rightarrow> 'b"
-apply (simp add: inv_image_def wf_eq_minimal)
-apply clarify
-apply (subgoal_tac "\<exists>w::'b. w \<in> {w. \<exists>x::'a. x \<in> Q \<and> f x = w}")
-prefer 2 apply (blast del: allE)
-apply (erule allE)
-apply (erule (1) notE impE)
-apply blast
-done
+lemma wf_inv_image [simp,intro!]: "wf r \<Longrightarrow> wf (inv_image r f)"
+ for f :: "'a \<Rightarrow> 'b"
+  apply (simp add: inv_image_def wf_eq_minimal)
+  apply clarify
+  apply (subgoal_tac "\<exists>w::'b. w \<in> {w. \<exists>x::'a. x \<in> Q \<and> f x = w}")
+   prefer 2
+   apply (blast del: allE)
+  apply (erule allE)
+  apply (erule (1) notE impE)
+  apply blast
+  done
 
 text \<open>Measure functions into @{typ nat}\<close>
 
@@ -710,17 +711,15 @@
   by (simp add:measure_def)
 
 lemma wf_measure [iff]: "wf (measure f)"
-apply (unfold measure_def)
-apply (rule wf_less_than [THEN wf_inv_image])
-done
+  unfolding measure_def by (rule wf_less_than [THEN wf_inv_image])
 
 lemma wf_if_measure: "(\<And>x. P x \<Longrightarrow> f(g x) < f x) \<Longrightarrow> wf {(y,x). P x \<and> y = g x}"
   for f :: "'a \<Rightarrow> nat"
-apply(insert wf_measure[of f])
-apply(simp only: measure_def inv_image_def less_than_def less_eq)
-apply(erule wf_subset)
-apply auto
-done
+  apply (insert wf_measure[of f])
+  apply (simp only: measure_def inv_image_def less_than_def less_eq)
+  apply (erule wf_subset)
+  apply auto
+  done
 
 
 subsubsection \<open>Lexicographic combinations\<close>
@@ -730,13 +729,18 @@
   where "ra <*lex*> rb = {((a, b), (a', b')). (a, a') \<in> ra \<or> a = a' \<and> (b, b') \<in> rb}"
 
 lemma wf_lex_prod [intro!]: "wf ra \<Longrightarrow> wf rb \<Longrightarrow> wf (ra <*lex*> rb)"
-apply (unfold wf_def lex_prod_def)
-apply (rule allI, rule impI)
-apply (simp only: split_paired_All)
-apply (drule spec, erule mp)
-apply (rule allI, rule impI)
-apply (drule spec, erule mp, blast)
-done
+  apply (unfold wf_def lex_prod_def)
+  apply (rule allI)
+  apply (rule impI)
+  apply (simp only: split_paired_All)
+  apply (drule spec)
+  apply (erule mp)
+  apply (rule allI)
+  apply (rule impI)
+  apply (drule spec)
+  apply (erule mp)
+  apply blast
+  done
 
 lemma in_lex_prod[simp]: "((a, b), (a', b')) \<in> r <*lex*> s \<longleftrightarrow> (a, a') \<in> r \<or> a = a' \<and> (b, b') \<in> s"
   by (auto simp:lex_prod_def)
@@ -752,19 +756,17 @@
   where "f <*mlex*> R = inv_image (less_than <*lex*> R) (\<lambda>x. (f x, x))"
 
 lemma wf_mlex: "wf R \<Longrightarrow> wf (f <*mlex*> R)"
-  unfolding mlex_prod_def
-  by auto
+  by (auto simp: mlex_prod_def)
 
 lemma mlex_less: "f x < f y \<Longrightarrow> (x, y) \<in> f <*mlex*> R"
-  unfolding mlex_prod_def by simp
+  by (simp add: mlex_prod_def)
 
 lemma mlex_leq: "f x \<le> f y \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (x, y) \<in> f <*mlex*> R"
-  unfolding mlex_prod_def by auto
+  by (auto simp: mlex_prod_def)
 
-text \<open>proper subset relation on finite sets\<close>
-
+text \<open>Proper subset relation on finite sets.\<close>
 definition finite_psubset :: "('a set \<times> 'a set) set"
-  where "finite_psubset = {(A,B). A < B \<and> finite B}"
+  where "finite_psubset = {(A, B). A \<subset> B \<and> finite B}"
 
 lemma wf_finite_psubset[simp]: "wf finite_psubset"
   apply (unfold finite_psubset_def)
@@ -776,15 +778,15 @@
 lemma trans_finite_psubset: "trans finite_psubset"
   by (auto simp add: finite_psubset_def less_le trans_def)
 
-lemma in_finite_psubset[simp]: "(A, B) \<in> finite_psubset \<longleftrightarrow> A < B \<and> finite B"
+lemma in_finite_psubset[simp]: "(A, B) \<in> finite_psubset \<longleftrightarrow> A \<subset> B \<and> finite B"
   unfolding finite_psubset_def by auto
 
 text \<open>max- and min-extension of order to finite sets\<close>
 
 inductive_set max_ext :: "('a \<times> 'a) set \<Rightarrow> ('a set \<times> 'a set) set"
   for R :: "('a \<times> 'a) set"
-where
-  max_extI[intro]: "finite X \<Longrightarrow> finite Y \<Longrightarrow> Y \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> \<exists>y\<in>Y. (x, y) \<in> R) \<Longrightarrow> (X, Y) \<in> max_ext R"
+  where max_extI[intro]:
+    "finite X \<Longrightarrow> finite Y \<Longrightarrow> Y \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> \<exists>y\<in>Y. (x, y) \<in> R) \<Longrightarrow> (X, Y) \<in> max_ext R"
 
 lemma max_ext_wf:
   assumes wf: "wf r"
@@ -792,23 +794,24 @@
 proof (rule acc_wfI, intro allI)
   fix M
   show "M \<in> acc (max_ext r)" (is "_ \<in> ?W")
-  proof cases
-    assume "finite M"
+  proof (cases "finite M")
+    case True
     then show ?thesis
     proof (induct M)
-      show "{} \<in> ?W"
+      case empty
+      show ?case
         by (rule accI) (auto elim: max_ext.cases)
     next
-      fix M a assume "M \<in> ?W" "finite M"
-      with wf show "insert a M \<in> ?W"
+      case (insert a M)
+      from wf \<open>M \<in> ?W\<close> \<open>finite M\<close> show "insert a M \<in> ?W"
       proof (induct arbitrary: M)
         fix M a
-        assume "M \<in> ?W"  and  [intro]: "finite M"
+        assume "M \<in> ?W"
+        assume [intro]: "finite M"
         assume hyp: "\<And>b M. (b, a) \<in> r \<Longrightarrow> M \<in> ?W \<Longrightarrow> finite M \<Longrightarrow> insert b M \<in> ?W"
-        have add_less: "\<lbrakk>M \<in> ?W ; (\<And>y. y \<in> N \<Longrightarrow> (y, a) \<in> r)\<rbrakk> \<Longrightarrow> N \<union> M \<in> ?W"
+        have add_less: "M \<in> ?W \<Longrightarrow> (\<And>y. y \<in> N \<Longrightarrow> (y, a) \<in> r) \<Longrightarrow> N \<union> M \<in> ?W"
           if "finite N" "finite M" for N M :: "'a set"
           using that by (induct N arbitrary: M) (auto simp: hyp)
-
         show "insert a M \<in> ?W"
         proof (rule accI)
           fix N
@@ -823,14 +826,13 @@
           then have finites: "finite ?N1" "finite ?N2" by auto
 
           have "?N2 \<in> ?W"
-          proof cases
-            assume [simp]: "M = {}"
+          proof (cases "M = {}")
+            case [simp]: True
             have Mw: "{} \<in> ?W" by (rule accI) (auto elim: max_ext.cases)
-
             from * have "?N2 = {}" by auto
             with Mw show "?N2 \<in> ?W" by (simp only:)
           next
-            assume "M \<noteq> {}"
+            case False
             from * finites have N2: "(?N2, M) \<in> max_ext r"
               by (rule_tac max_extI[OF _ _ \<open>M \<noteq> {}\<close>]) auto
             with \<open>M \<in> ?W\<close> show "?N2 \<in> ?W" by (rule acc_downward)
@@ -842,15 +844,13 @@
       qed
     qed
   next
-    assume [simp]: "\<not> finite M"
+    case [simp]: False
     show ?thesis
       by (rule accI) (auto elim: max_ext.cases)
   qed
 qed
 
-lemma max_ext_additive:
-  "(A, B) \<in> max_ext R \<Longrightarrow> (C, D) \<in> max_ext R \<Longrightarrow>
-    (A \<union> C, B \<union> D) \<in> max_ext R"
+lemma max_ext_additive: "(A, B) \<in> max_ext R \<Longrightarrow> (C, D) \<in> max_ext R \<Longrightarrow> (A \<union> C, B \<union> D) \<in> max_ext R"
   by (force elim!: max_ext.cases)
 
 
@@ -874,13 +874,13 @@
     obtain z where z: "z \<in> \<Union>Q" "\<And>y. (y, z) \<in> r \<Longrightarrow> y \<notin> \<Union>Q"
       by (erule wfE_min)
     from z obtain m where "m \<in> Q" "z \<in> m" by auto
-    from \<open>m \<in> Q\<close>
-    show ?thesis
-    proof (rule, intro bexI allI impI)
+    from \<open>m \<in> Q\<close> show ?thesis
+    proof (intro rev_bexI allI impI)
       fix n
       assume smaller: "(n, m) \<in> min_ext r"
-      with \<open>z \<in> m\<close> obtain y where y: "y \<in> n" "(y, z) \<in> r" by (auto simp: min_ext_def)
-      then show "n \<notin> Q" using z(2) by auto
+      with \<open>z \<in> m\<close> obtain y where "y \<in> n" "(y, z) \<in> r"
+        by (auto simp: min_ext_def)
+      with z(2) show "n \<notin> Q" by auto
     qed
   qed
 qed
@@ -893,32 +893,33 @@
     and f :: "'a \<Rightarrow> nat"
   assumes "\<And>a b. (b, a) \<in> r \<Longrightarrow> ub b \<le> ub a \<and> ub a \<ge> f b \<and> f b > f a"
   shows "wf r"
-  apply (rule wf_subset[OF wf_measure[of "\<lambda>a. ub a - f a"]])
-  apply (auto dest: assms)
-  done
+  by (rule wf_subset[OF wf_measure[of "\<lambda>a. ub a - f a"]]) (auto dest: assms)
 
 lemma wf_bounded_set:
   fixes ub :: "'a \<Rightarrow> 'b set"
     and f :: "'a \<Rightarrow> 'b set"
   assumes "\<And>a b. (b,a) \<in> r \<Longrightarrow> finite (ub a) \<and> ub b \<subseteq> ub a \<and> ub a \<supseteq> f b \<and> f b \<supset> f a"
   shows "wf r"
-  apply(rule wf_bounded_measure[of r "\<lambda>a. card(ub a)" "\<lambda>a. card(f a)"])
-  apply(drule assms)
+  apply (rule wf_bounded_measure[of r "\<lambda>a. card (ub a)" "\<lambda>a. card (f a)"])
+  apply (drule assms)
   apply (blast intro: card_mono finite_subset psubset_card_mono dest: psubset_eq[THEN iffD2])
   done
 
 lemma finite_subset_wf:
   assumes "finite A"
-  shows   "wf {(X,Y). X \<subset> Y \<and> Y \<subseteq> A}"
+  shows "wf {(X,Y). X \<subset> Y \<and> Y \<subseteq> A}"
 proof (intro finite_acyclic_wf)
-  have "{(X,Y). X \<subset> Y \<and> Y \<subseteq> A} \<subseteq> Pow A \<times> Pow A" by blast
+  have "{(X,Y). X \<subset> Y \<and> Y \<subseteq> A} \<subseteq> Pow A \<times> Pow A"
+    by blast
   then show "finite {(X,Y). X \<subset> Y \<and> Y \<subseteq> A}"
     by (rule finite_subset) (auto simp: assms finite_cartesian_product)
 next
   have "{(X, Y). X \<subset> Y \<and> Y \<subseteq> A}\<^sup>+ = {(X, Y). X \<subset> Y \<and> Y \<subseteq> A}"
     by (intro trancl_id transI) blast
-  also have " \<forall>x. (x, x) \<notin> \<dots>" by blast
-  finally show "acyclic {(X,Y). X \<subset> Y \<and> Y \<subseteq> A}" by (rule acyclicI)
+  also have " \<forall>x. (x, x) \<notin> \<dots>"
+    by blast
+  finally show "acyclic {(X,Y). X \<subset> Y \<and> Y \<subseteq> A}"
+    by (rule acyclicI)
 qed
 
 hide_const (open) acc accp
diff -r aee0d92995b6 -r c0cbfd2b5a45 src/HOL/Wfrec.thy
--- a/src/HOL/Wfrec.thy	Sun Jul 31 19:09:21 2016 +0200
+++ b/src/HOL/Wfrec.thy	Sun Jul 31 22:56:18 2016 +0200
@@ -7,20 +7,20 @@
 section \<open>Well-Founded Recursion Combinator\<close>
 
 theory Wfrec
-imports Wellfounded
+  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)"
+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 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 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)"
+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)
@@ -28,13 +28,17 @@
 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 wfrec combinator; for details see:
-John Harrison, "Inductive definitions: automation and application"\<close>
+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"
+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
@@ -46,44 +50,46 @@
 qed
 
 lemma adm_lemma: "adm_wf R (\<lambda>f x. F (cut f R x) x)"
-  by (auto simp add: adm_wf_def
-           intro!: arg_cong[where f="\<lambda>x. F x y" for y] cuts_eq[THEN iffD2])
+  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], assumption)
-apply (rule wfrec_rel.wfrecI)
-apply (erule adm_lemma [THEN wfrec_unique, THEN theI'])
-done
+  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 ==\<close>
+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)
+  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"
+  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
+    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)
-    hence "F (cut (wfrec R F) R x) x = F (wfrec R F) x"
-      using ADM adm_wf_def[of R F] by auto }
+  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
 
+
 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 wfrec} declarations where the first n parameters
+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 wfrec} 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"
@@ -92,12 +98,13 @@
 lemma wf_same_fst:
   assumes prem: "\<And>x. P x \<Longrightarrow> wf (R x)"
   shows "wf (same_fst P R)"
-apply (simp cong del: imp_cong add: wf_def same_fst_def)
-apply (intro strip)
-apply (rename_tac a b)
-apply (case_tac "wf (R a)")
- apply (erule_tac a = b in wf_induct, blast)
-apply (blast intro: prem)
-done
+  apply (simp cong del: imp_cong add: wf_def same_fst_def)
+  apply (intro strip)
+  apply (rename_tac a b)
+  apply (case_tac "wf (R a)")
+   apply (erule_tac a = b in wf_induct)
+   apply blast
+  apply (blast intro: prem)
+  done
 
 end
diff -r aee0d92995b6 -r c0cbfd2b5a45 src/HOL/Zorn.thy
--- a/src/HOL/Zorn.thy	Sun Jul 31 19:09:21 2016 +0200
+++ b/src/HOL/Zorn.thy	Sun Jul 31 22:56:18 2016 +0200
@@ -1,7 +1,7 @@
-(*  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.
@@ -10,7 +10,7 @@
 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>
@@ -20,36 +20,38 @@
 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"
+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>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"
+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)"
+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))"
+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"
+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"
@@ -64,62 +66,67 @@
 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>Thus closure under @{term suc} will hit a maximal chain
-eventually, as is shown below.\<close>
+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>"
 
-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)"
-  shows "Q X"
-  using assms by (induct) blast+
+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_cases [consumes 1, case_names suc Union,
-  cases pred: suc_Union_closed]:
+lemma suc_Union_closed_induct [consumes 1, case_names suc Union, induct 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>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+
+  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
 
@@ -128,27 +135,31 @@
     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)
@@ -164,21 +175,22 @@
   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
@@ -199,75 +211,87 @@
 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"
@@ -290,18 +314,17 @@
 
 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
@@ -310,34 +333,35 @@
 qed
 
 text \<open>Make notation @{term \<C>} available again.\<close>
-no_notation suc_Union_closed ("\<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)"
+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"
@@ -352,6 +376,7 @@
   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>
@@ -360,19 +385,23 @@
   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
@@ -380,13 +409,14 @@
   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)
@@ -403,19 +433,17 @@
 
 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 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 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}"
+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; z \<in> S; \<forall>x \<in> c. x \<subseteq> (z:: 'a set) |] ==> {z} Un c \<in> chains S"
+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"
@@ -427,8 +455,7 @@
 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':
@@ -442,20 +469,18 @@
   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>
+text \<open>Various other lemmas\<close>
 
-lemma chainsD: "[| c \<in> chains S; x \<in> c; y \<in> c |] ==> x \<subseteq> y | y \<subseteq> x"
+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:
@@ -463,42 +488,49 @@
     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
 
 
@@ -509,13 +541,12 @@
    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)}"
+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"
+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)
@@ -524,85 +555,97 @@
   "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"
@@ -610,24 +653,28 @@
 \<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)"
@@ -640,21 +687,25 @@
       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 = {}"
@@ -663,10 +714,10 @@
       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)"
@@ -674,49 +725,58 @@
     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
 
@@ -727,15 +787,16 @@
 
 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)"
@@ -748,7 +809,7 @@
 
 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)