diff --git a/src/HOL/Wellfounded.thy b/src/HOL/Wellfounded.thy
--- a/src/HOL/Wellfounded.thy
+++ b/src/HOL/Wellfounded.thy
@@ -1,947 +1,953 @@
 (*  Title:      HOL/Wellfounded.thy
     Author:     Tobias Nipkow
     Author:     Lawrence C Paulson
     Author:     Konrad Slind
     Author:     Alexander Krauss
     Author:     Andrei Popescu, TU Muenchen
 *)
 
 section \<open>Well-founded Recursion\<close>
 
 theory Wellfounded
   imports Transitive_Closure
 begin
 
 subsection \<open>Basic Definitions\<close>
 
 definition wf :: "('a \<times> 'a) set \<Rightarrow> bool"
   where "wf r \<longleftrightarrow> (\<forall>P. (\<forall>x. (\<forall>y. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x) \<longrightarrow> (\<forall>x. P x))"
 
 definition wfP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
   where "wfP r \<longleftrightarrow> wf {(x, y). r x y}"
 
 lemma wfP_wf_eq [pred_set_conv]: "wfP (\<lambda>x y. (x, y) \<in> r) = wf r"
   by (simp add: wfP_def)
 
 lemma wfUNIVI: "(\<And>P x. (\<forall>x. (\<forall>y. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x) \<Longrightarrow> P x) \<Longrightarrow> wf r"
   unfolding wf_def by blast
 
 lemmas wfPUNIVI = wfUNIVI [to_pred]
 
 text \<open>Restriction to domain \<open>A\<close> and range \<open>B\<close>.
   If \<open>r\<close> is well-founded over their intersection, then \<open>wf r\<close>.\<close>
 lemma wfI:
   assumes "r \<subseteq> A \<times> B"
     and "\<And>x P. \<lbrakk>\<forall>x. (\<forall>y. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x;  x \<in> A; x \<in> B\<rbrakk> \<Longrightarrow> P x"
   shows "wf r"
   using assms unfolding wf_def by blast
 
 lemma wf_induct:
   assumes "wf r"
     and "\<And>x. \<forall>y. (y, x) \<in> r \<longrightarrow> P y \<Longrightarrow> P x"
   shows "P a"
   using assms unfolding wf_def by blast
 
 lemmas wfP_induct = wf_induct [to_pred]
 
 lemmas wf_induct_rule = wf_induct [rule_format, consumes 1, case_names less, induct set: wf]
 
 lemmas wfP_induct_rule = wf_induct_rule [to_pred, induct set: wfP]
 
 lemma wf_not_sym: "wf r \<Longrightarrow> (a, x) \<in> r \<Longrightarrow> (x, a) \<notin> r"
   by (induct a arbitrary: x set: wf) blast
 
 lemma wf_asym:
   assumes "wf r" "(a, x) \<in> r"
   obtains "(x, a) \<notin> r"
   by (drule wf_not_sym[OF assms])
 
 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"
   by (drule wf_not_refl[OF assms])
 
 lemma wf_wellorderI:
   assumes wf: "wf {(x::'a::ord, y). x < y}"
     and lin: "OFCLASS('a::ord, linorder_class)"
   shows "OFCLASS('a::ord, wellorder_class)"
-  using lin
-  apply (rule wellorder_class.intro)
-  apply (rule class.wellorder_axioms.intro)
-  apply (rule wf_induct_rule [OF wf])
-  apply simp
+  apply (rule wellorder_class.intro [OF lin])
+  apply (simp add: wellorder_class.intro class.wellorder_axioms.intro wf_induct_rule [OF wf])
   done
 
 lemma (in wellorder) wf: "wf {(x, y). x < y}"
   unfolding wf_def by (blast intro: less_induct)
 
 
 subsection \<open>Basic Results\<close>
 
 text \<open>Point-free characterization of well-foundedness\<close>
 
 lemma wfE_pf:
   assumes wf: "wf R"
     and a: "A \<subseteq> R `` A"
   shows "A = {}"
 proof -
   from wf have "x \<notin> A" for x
   proof induct
     fix x assume "\<And>y. (y, x) \<in> R \<Longrightarrow> y \<notin> A"
     then have "x \<notin> R `` A" by blast
     with a show "x \<notin> A" by blast
   qed
   then show ?thesis by auto
 qed
 
 lemma wfI_pf:
   assumes a: "\<And>A. A \<subseteq> R `` A \<Longrightarrow> A = {}"
   shows "wf R"
 proof (rule wfUNIVI)
   fix P :: "'a \<Rightarrow> bool" and x
   let ?A = "{x. \<not> P x}"
   assume "\<forall>x. (\<forall>y. (y, x) \<in> R \<longrightarrow> P y) \<longrightarrow> P x"
   then have "?A \<subseteq> R `` ?A" by blast
   with a show "P x" by blast
 qed
 
 
 subsubsection \<open>Minimal-element characterization of well-foundedness\<close>
 
 lemma wfE_min:
   assumes wf: "wf R" and Q: "x \<in> Q"
   obtains z where "z \<in> Q" "\<And>y. (y, z) \<in> R \<Longrightarrow> y \<notin> Q"
   using Q wfE_pf[OF wf, of Q] by blast
 
 lemma wfE_min':
   "wf R \<Longrightarrow> Q \<noteq> {} \<Longrightarrow> (\<And>z. z \<in> Q \<Longrightarrow> (\<And>y. (y, z) \<in> R \<Longrightarrow> y \<notin> Q) \<Longrightarrow> thesis) \<Longrightarrow> thesis"
   using wfE_min[of R _ Q] by blast
 
 lemma wfI_min:
   assumes a: "\<And>x Q. x \<in> Q \<Longrightarrow> \<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q"
   shows "wf R"
 proof (rule wfI_pf)
   fix A
   assume b: "A \<subseteq> R `` A"
   have False if "x \<in> A" for x
     using a[OF that] b by blast
   then show "A = {}" by blast
 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 (rule iffI)
    apply (blast intro:  elim!: wfE_min)
   by (rule wfI_min) auto
 
 lemmas wfP_eq_minimal = wf_eq_minimal [to_pred]
 
 
 subsubsection \<open>Well-foundedness of transitive closure\<close>
 
 lemma wf_trancl:
   assumes "wf r"
   shows "wf (r\<^sup>+)"
 proof -
   have "P x" if induct_step: "\<And>x. (\<And>y. (y, x) \<in> r\<^sup>+ \<Longrightarrow> P y) \<Longrightarrow> P x" for P x
   proof (rule induct_step)
     show "P y" if "(y, x) \<in> r\<^sup>+" for y
       using \<open>wf r\<close> and that
     proof (induct x arbitrary: y)
       case (less x)
       note hyp = \<open>\<And>x' y'. (x', x) \<in> r \<Longrightarrow> (y', x') \<in> r\<^sup>+ \<Longrightarrow> P y'\<close>
       from \<open>(y, x) \<in> r\<^sup>+\<close> show "P y"
       proof cases
         case base
         show "P y"
         proof (rule induct_step)
           fix y'
           assume "(y', y) \<in> r\<^sup>+"
           with \<open>(y, x) \<in> r\<close> show "P y'"
             by (rule hyp [of y y'])
         qed
       next
         case step
         then obtain x' where "(x', x) \<in> r" and "(y, x') \<in> r\<^sup>+"
           by simp
         then show "P y" by (rule hyp [of x' y])
       qed
     qed
   qed
   then show ?thesis unfolding wf_def by blast
 qed
 
 lemmas wfP_trancl = wf_trancl [to_pred]
 
 lemma wf_converse_trancl: "wf (r\<inverse>) \<Longrightarrow> wf ((r\<^sup>+)\<inverse>)"
   apply (subst trancl_converse [symmetric])
   apply (erule wf_trancl)
   done
 
 text \<open>Well-foundedness of subsets\<close>
 
 lemma wf_subset: "wf r \<Longrightarrow> p \<subseteq> r \<Longrightarrow> wf p"
   by (simp add: wf_eq_minimal) fast
 
 lemmas wfP_subset = wf_subset [to_pred]
 
 text \<open>Well-foundedness of the empty relation\<close>
 
 lemma wf_empty [iff]: "wf {}"
   by (simp add: wf_def)
 
 lemma wfP_empty [iff]: "wfP (\<lambda>x y. False)"
 proof -
   have "wfP bot"
     by (fact wf_empty[to_pred bot_empty_eq2])
   then show ?thesis
     by (simp add: bot_fun_def)
 qed
 
 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' \<inter> r)"
   by (erule wf_subset) (rule Int_lower2)
 
 text \<open>Exponentiation.\<close>
 lemma wf_exp:
   assumes "wf (R ^^ n)"
   shows "wf R"
 proof (rule wfI_pf)
   fix A assume "A \<subseteq> R `` A"
   then have "A \<subseteq> (R ^^ n) `` A"
     by (induct n) force+
   with \<open>wf (R ^^ n)\<close> show "A = {}"
     by (rule wfE_pf)
 qed
 
 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>*" (is "?lhs = ?rhs")
 proof
   assume ?lhs then show ?rhs
     by (blast elim: wf_trancl [THEN wf_irrefl]
         intro: rtrancl_into_trancl1 wf_subset rtrancl_mono [THEN subsetD])
 next
-  assume R: ?rhs 
+  assume R: ?rhs
   then have R': "Q \<noteq> {} \<Longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q)" for Q
     by (auto simp: wf_eq_minimal)
   show ?lhs
     unfolding wf_eq_minimal
   proof clarify
     fix Q :: "'a set" and q
     assume "q \<in> Q"
     then obtain a where "a \<in> Q" and a: "\<And>y. (y, a) \<in> r \<Longrightarrow> y \<notin> Q"
       using R by (auto simp: wf_eq_minimal)
     show "\<exists>z\<in>Q. \<forall>y'. (y', z) \<in> insert (y, x) r \<longrightarrow> y' \<notin> Q"
     proof (cases "a=x")
       case True
       show ?thesis
       proof (cases "y \<in> Q")
         case True
         then obtain z where "z \<in> Q" "(z, y) \<in> r\<^sup>*"
                             "\<And>z'. (z', z) \<in> r \<longrightarrow> z' \<in> Q \<longrightarrow> (z', y) \<notin> r\<^sup>*"
           using R' [of "{z \<in> Q. (z,y) \<in> r\<^sup>*}"] by auto
         with R show ?thesis
           by (rule_tac x="z" in bexI) (blast intro: rtrancl_trans)
       next
         case False
         then show ?thesis
           using a \<open>a \<in> Q\<close> by blast
       qed
     next
       case False
       with a \<open>a \<in> Q\<close> show ?thesis
         by blast
     qed
   qed
 qed
 
 
 subsubsection \<open>Well-foundedness of image\<close>
 
 lemma wf_map_prod_image_Dom_Ran:
   fixes r:: "('a \<times> 'a) set"
     and f:: "'a \<Rightarrow> 'b"
   assumes wf_r: "wf r"
     and inj: "\<And> a a'. a \<in> Domain r \<Longrightarrow> a' \<in> Range r \<Longrightarrow> f a = f a' \<Longrightarrow> a = a'"
   shows "wf (map_prod f f ` r)"
 proof (unfold wf_eq_minimal, clarify)
   fix B :: "'b set" and b::"'b"
   assume "b \<in> B"
   define A where "A = f -` B \<inter> Domain r"
   show "\<exists>z\<in>B. \<forall>y. (y, z) \<in> map_prod f f ` r \<longrightarrow> y \<notin> B"
   proof (cases "A = {}")
     case False
     then obtain a0 where "a0 \<in> A" and "\<forall>a. (a, a0) \<in> r \<longrightarrow> a \<notin> A"
       using wfE_min[OF wf_r] by auto
-    thus ?thesis 
+    thus ?thesis
       using inj unfolding A_def
       by (intro bexI[of _ "f a0"]) auto
-  qed (insert \<open>b \<in> B\<close>, unfold A_def, auto) 
+  qed (insert \<open>b \<in> B\<close>, unfold A_def, auto)
 qed
 
 lemma wf_map_prod_image: "wf r \<Longrightarrow> inj f \<Longrightarrow> wf (map_prod f f ` r)"
 by(rule wf_map_prod_image_Dom_Ran) (auto dest: inj_onD)
 
 
 subsection \<open>Well-Foundedness Results for Unions\<close>
 
 lemma wf_union_compatible:
   assumes "wf R" "wf S"
   assumes "R O S \<subseteq> R"
   shows "wf (R \<union> S)"
 proof (rule wfI_min)
   fix x :: 'a and Q
   let ?Q' = "{x \<in> Q. \<forall>y. (y, x) \<in> R \<longrightarrow> y \<notin> Q}"
   assume "x \<in> Q"
   obtain a where "a \<in> ?Q'"
     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)
   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>
 
 lemma wf_UN:
   assumes r: "\<And>i. i \<in> I \<Longrightarrow> wf (r i)"
     and disj: "\<And>i j. \<lbrakk>i \<in> I; j \<in> I; r i \<noteq> r j\<rbrakk> \<Longrightarrow> Domain (r i) \<inter> Range (r j) = {}"
   shows "wf (\<Union>i\<in>I. r i)"
   unfolding wf_eq_minimal
 proof clarify
   fix A and a :: "'b"
   assume "a \<in> A"
   show "\<exists>z\<in>A. \<forall>y. (y, z) \<in> \<Union>(r ` I) \<longrightarrow> y \<notin> A"
   proof (cases "\<exists>i\<in>I. \<exists>a\<in>A. \<exists>b\<in>A. (b, a) \<in> r i")
     case True
     then obtain i b c where ibc: "i \<in> I" "b \<in> A" "c \<in> A" "(c,b) \<in> r i"
       by blast
     have ri: "\<And>Q. Q \<noteq> {} \<Longrightarrow> \<exists>z\<in>Q. \<forall>y. (y, z) \<in> r i \<longrightarrow> y \<notin> Q"
       using r [OF \<open>i \<in> I\<close>] unfolding wf_eq_minimal by auto
     show ?thesis
-      using ri [of "{a. a \<in> A \<and> (\<exists>b\<in>A. (b, a) \<in> r i) }"] ibc disj 
+      using ri [of "{a. a \<in> A \<and> (\<exists>b\<in>A. (b, a) \<in> r i) }"] ibc disj
       by blast
   next
     case False
     with \<open>a \<in> A\<close> show ?thesis
       by blast
   qed
 qed
 
 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 (\<Squnion>(range r))"
   by (rule wf_UN[to_pred]) simp_all
 
 lemma wf_Union:
   assumes "\<forall>r\<in>R. wf r"
     and "\<forall>r\<in>R. \<forall>s\<in>R. r \<noteq> s \<longrightarrow> Domain r \<inter> Range s = {}"
   shows "wf (\<Union>R)"
   using assms wf_UN[of R "\<lambda>i. i"] by simp
 
 text \<open>
   Intuition: We find an \<open>R \<union> S\<close>-min element of a nonempty subset \<open>A\<close> by case distinction.
   \<^enum> There is a step \<open>a \<midarrow>R\<rightarrow> b\<close> with \<open>a, b \<in> A\<close>.
     Pick an \<open>R\<close>-min element \<open>z\<close> of the (nonempty) set \<open>{a\<in>A | \<exists>b\<in>A. a \<midarrow>R\<rightarrow> b}\<close>.
     By definition, there is \<open>z' \<in> A\<close> s.t. \<open>z \<midarrow>R\<rightarrow> z'\<close>. Because \<open>z\<close> is \<open>R\<close>-min in the
     subset, \<open>z'\<close> must be \<open>R\<close>-min in \<open>A\<close>. Because \<open>z'\<close> has an \<open>R\<close>-predecessor, it cannot
     have an \<open>S\<close>-successor and is thus \<open>S\<close>-min in \<open>A\<close> as well.
   \<^enum> There is no such step.
     Pick an \<open>S\<close>-min element of \<open>A\<close>. In this case it must be an \<open>R\<close>-min
     element of \<open>A\<close> as well.
 \<close>
 lemma wf_Un: "wf r \<Longrightarrow> wf s \<Longrightarrow> Domain r \<inter> Range s = {} \<Longrightarrow> wf (r \<union> s)"
   using wf_union_compatible[of s r]
   by (auto simp: Un_ac)
 
 lemma wf_union_merge: "wf (R \<union> S) = wf (R O R \<union> S O R \<union> S)"
   (is "wf ?A = wf ?B")
 proof
   assume "wf ?A"
   with wf_trancl have wfT: "wf (?A\<^sup>+)" .
   moreover have "?B \<subseteq> ?A\<^sup>+"
     by (subst trancl_unfold, subst trancl_unfold) blast
   ultimately show "wf ?B" by (rule wf_subset)
 next
   assume "wf ?B"
   show "wf ?A"
   proof (rule wfI_min)
     fix Q :: "'a set" and x
     assume "x \<in> Q"
     with \<open>wf ?B\<close> obtain z where "z \<in> Q" and "\<And>y. (y, z) \<in> ?B \<Longrightarrow> y \<notin> Q"
       by (erule wfE_min)
     then have 1: "\<And>y. (y, z) \<in> R O R \<Longrightarrow> y \<notin> Q"
       and 2: "\<And>y. (y, z) \<in> S O R \<Longrightarrow> y \<notin> Q"
       and 3: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> Q"
       by auto
     show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> ?A \<longrightarrow> y \<notin> Q"
     proof (cases "\<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q")
       case True
       with \<open>z \<in> Q\<close> 3 show ?thesis by blast
     next
       case False
       then obtain z' where "z'\<in>Q" "(z', z) \<in> R" by blast
       have "\<forall>y. (y, z') \<in> ?A \<longrightarrow> y \<notin> Q"
       proof (intro allI impI)
         fix y assume "(y, z') \<in> ?A"
         then show "y \<notin> Q"
         proof
           assume "(y, z') \<in> R"
           then have "(y, z) \<in> R O R" using \<open>(z', z) \<in> R\<close> ..
           with 1 show "y \<notin> Q" .
         next
           assume "(y, z') \<in> S"
           then have "(y, z) \<in> S O R" using  \<open>(z', z) \<in> R\<close> ..
           with 2 show "y \<notin> Q" .
         qed
       qed
       with \<open>z' \<in> Q\<close> show ?thesis ..
     qed
   qed
 qed
 
 lemma wf_comp_self: "wf R \<longleftrightarrow> wf (R O R)"  \<comment> \<open>special case\<close>
   by (rule wf_union_merge [where S = "{}", simplified])
 
 
 subsection \<open>Well-Foundedness of Composition\<close>
 
 text \<open>Bachmair and Dershowitz 1986, Lemma 2. [Provided by Tjark Weber]\<close>
 
 lemma qc_wf_relto_iff:
   assumes "R O S \<subseteq> (R \<union> S)\<^sup>* O R" \<comment> \<open>R quasi-commutes over S\<close>
   shows "wf (S\<^sup>* O R O S\<^sup>*) \<longleftrightarrow> wf R"
     (is "wf ?S \<longleftrightarrow> _")
 proof
   show "wf R" if "wf ?S"
   proof -
     have "R \<subseteq> ?S" by auto
     with wf_subset [of ?S] that show "wf R"
       by auto
   qed
 next
   show "wf ?S" if "wf R"
   proof (rule wfI_pf)
     fix A
     assume A: "A \<subseteq> ?S `` A"
     let ?X = "(R \<union> S)\<^sup>* `` A"
     have *: "R O (R \<union> S)\<^sup>* \<subseteq> (R \<union> S)\<^sup>* O R"
     proof -
       have "(x, z) \<in> (R \<union> S)\<^sup>* O R" if "(y, z) \<in> (R \<union> S)\<^sup>*" and "(x, y) \<in> R" for x y z
         using that
       proof (induct y z)
         case rtrancl_refl
         then show ?case by auto
       next
         case (rtrancl_into_rtrancl a b c)
         then have "(x, c) \<in> ((R \<union> S)\<^sup>* O (R \<union> S)\<^sup>*) O R"
           using assms by blast
         then show ?case by simp
       qed
       then show ?thesis by auto
     qed
     then have "R O S\<^sup>* \<subseteq> (R \<union> S)\<^sup>* O R"
       using rtrancl_Un_subset by blast
     then have "?S \<subseteq> (R \<union> S)\<^sup>* O (R \<union> S)\<^sup>* O R"
       by (simp add: relcomp_mono rtrancl_mono)
     also have "\<dots> = (R \<union> S)\<^sup>* O R"
       by (simp add: O_assoc[symmetric])
     finally have "?S O (R \<union> S)\<^sup>* \<subseteq> (R \<union> S)\<^sup>* O R O (R \<union> S)\<^sup>*"
       by (simp add: O_assoc[symmetric] relcomp_mono)
     also have "\<dots> \<subseteq> (R \<union> S)\<^sup>* O (R \<union> S)\<^sup>* O R"
       using * by (simp add: relcomp_mono)
     finally have "?S O (R \<union> S)\<^sup>* \<subseteq> (R \<union> S)\<^sup>* O R"
       by (simp add: O_assoc[symmetric])
     then have "(?S O (R \<union> S)\<^sup>*) `` A \<subseteq> ((R \<union> S)\<^sup>* O R) `` A"
       by (simp add: Image_mono)
     moreover have "?X \<subseteq> (?S O (R \<union> S)\<^sup>*) `` A"
       using A by (auto simp: relcomp_Image)
     ultimately have "?X \<subseteq> R `` ?X"
       by (auto simp: relcomp_Image)
     then have "?X = {}"
       using \<open>wf R\<close> by (simp add: wfE_pf)
     moreover have "A \<subseteq> ?X" by auto
     ultimately show "A = {}" by simp
   qed
 qed
 
 corollary wf_relcomp_compatible:
   assumes "wf R" and "R O S \<subseteq> S O R"
   shows "wf (S O R)"
 proof -
   have "R O S \<subseteq> (R \<union> S)\<^sup>* O R"
     using assms by blast
   then have "wf (S\<^sup>* O R O S\<^sup>*)"
     by (simp add: assms qc_wf_relto_iff)
   then show ?thesis
     by (rule Wellfounded.wf_subset) blast
 qed
 
 
 subsection \<open>Acyclic relations\<close>
 
 lemma wf_acyclic: "wf r \<Longrightarrow> acyclic r"
   by (simp add: acyclic_def) (blast elim: wf_trancl [THEN wf_irrefl])
 
 lemmas wfP_acyclicP = wf_acyclic [to_pred]
 
 
 subsubsection \<open>Wellfoundedness of finite acyclic relations\<close>
 
 lemma finite_acyclic_wf:
   assumes "finite r" "acyclic r" shows "wf r"
   using assms
 proof (induction r rule: finite_induct)
   case (insert x r)
   then show ?case
     by (cases x) simp
 qed simp
 
 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
 
 text \<open>
   Observe that the converse of an irreflexive, transitive,
   and finite relation is again well-founded. Thus, we may
   employ it for well-founded induction.
 \<close>
 lemma wf_converse:
   assumes "irrefl r" and "trans r" and "finite r"
   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)
 qed
 
 lemma wf_iff_acyclic_if_finite: "finite r \<Longrightarrow> wf r = acyclic r"
   by (blast intro: finite_acyclic_wf wf_acyclic)
 
 
 subsection \<open>\<^typ>\<open>nat\<close> is well-founded\<close>
 
 lemma less_nat_rel: "(<) = (\<lambda>m n. n = Suc m)\<^sup>+\<^sup>+"
 proof (rule ext, rule ext, rule iffI)
   fix n m :: nat
   show "(\<lambda>m n. n = Suc m)\<^sup>+\<^sup>+ m n" if "m < n"
     using that
   proof (induct n)
     case 0
     then show ?case by auto
   next
     case (Suc n)
     then show ?case
       by (auto simp add: less_Suc_eq_le le_less intro: tranclp.trancl_into_trancl)
   qed
   show "m < n" if "(\<lambda>m n. n = Suc m)\<^sup>+\<^sup>+ m n"
     using that by (induct n) (simp_all add: less_Suc_eq_le reflexive le_less)
 qed
 
 definition pred_nat :: "(nat \<times> nat) set"
   where "pred_nat = {(m, n). n = Suc m}"
 
 definition less_than :: "(nat \<times> nat) set"
   where "less_than = pred_nat\<^sup>+"
 
 lemma less_eq: "(m, n) \<in> pred_nat\<^sup>+ \<longleftrightarrow> m < n"
   unfolding less_nat_rel pred_nat_def trancl_def by simp
 
 lemma pred_nat_trancl_eq_le: "(m, n) \<in> pred_nat\<^sup>* \<longleftrightarrow> m \<le> n"
   unfolding less_eq rtrancl_eq_or_trancl by auto
 
 lemma wf_pred_nat: "wf pred_nat"
   apply (unfold wf_def pred_nat_def)
   apply clarify
   apply (induct_tac x)
    apply blast+
   done
 
 lemma wf_less_than [iff]: "wf less_than"
   by (simp add: less_than_def wf_pred_nat [THEN wf_trancl])
 
 lemma trans_less_than [iff]: "trans less_than"
   by (simp add: less_than_def)
 
 lemma less_than_iff [iff]: "((x,y) \<in> less_than) = (x<y)"
   by (simp add: less_than_def less_eq)
 
 lemma total_less_than: "total less_than"
   using total_on_def by force
 
 lemma wf_less: "wf {(x, y::nat). x < y}"
   by (rule Wellfounded.wellorder_class.wf)
 
 
 subsection \<open>Accessible Part\<close>
 
 text \<open>
   Inductive definition of the accessible part \<open>acc r\<close> of a
   relation; see also @{cite "paulin-tlca"}.
 \<close>
 
 inductive_set acc :: "('a \<times> 'a) set \<Rightarrow> 'a set" for r :: "('a \<times> 'a) set"
   where accI: "(\<And>y. (y, x) \<in> r \<Longrightarrow> y \<in> acc r) \<Longrightarrow> x \<in> acc r"
 
 abbreviation termip :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
   where "termip r \<equiv> accp (r\<inverse>\<inverse>)"
 
 abbreviation termi :: "('a \<times> 'a) set \<Rightarrow> 'a set"
   where "termi r \<equiv> acc (r\<inverse>)"
 
 lemmas accpI = accp.accI
 
 lemma accp_eq_acc [code]: "accp r = (\<lambda>x. x \<in> Wellfounded.acc {(x, y). r x y})"
   by (simp add: acc_def)
 
 
 text \<open>Induction rules\<close>
 
 theorem accp_induct:
   assumes major: "accp r a"
   assumes hyp: "\<And>x. accp r x \<Longrightarrow> \<forall>y. r y x \<longrightarrow> P y \<Longrightarrow> P x"
   shows "P a"
   apply (rule major [THEN accp.induct])
   apply (rule hyp)
    apply (rule accp.accI)
    apply auto
   done
 
 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"
   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. 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
     then have "\<And>z. R z x \<Longrightarrow> accp R z" by auto
     then have "accp R x" by (rule accp.accI)
     with na show thesis ..
   next
     case False then obtain z where "R z x" and "\<not> accp R z"
       by auto
     with a show thesis .
   qed
 qed
 
 lemma accp_downwards_aux: "r\<^sup>*\<^sup>* b a \<Longrightarrow> accp r a \<longrightarrow> accp r b"
   by (erule rtranclp_induct) (blast dest: accp_downward)+
 
 theorem accp_downwards: "accp r a \<Longrightarrow> r\<^sup>*\<^sup>* b a \<Longrightarrow> accp r b"
   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 blast+
   done
 
 theorem accp_wfPD: "wfP r \<Longrightarrow> accp r x"
   apply (erule wfP_induct_rule)
   apply (rule accp.accI)
   apply blast
   done
 
 theorem wfP_accp_iff: "wfP r = (\<forall>x. accp r x)"
   by (blast intro: accp_wfPI dest: accp_wfPD)
 
 
 text \<open>Smaller relations have bigger accessible parts:\<close>
 
 lemma accp_subset:
   assumes "R1 \<le> R2"
   shows "accp R2 \<le> accp R1"
 proof (rule predicate1I)
   fix x
   assume "accp R2 x"
   then show "accp R1 x"
   proof (induct x)
     fix x
     assume "\<And>y. R2 y x \<Longrightarrow> accp R1 y"
     with assms show "accp R1 x"
       by (blast intro: accp.accI)
   qed
 qed
 
 
 text \<open>This is a generalized induction theorem that works on
   subsets of the accessible part.\<close>
 
 lemma accp_subset_induct:
   assumes subset: "D \<le> accp R"
     and dcl: "\<And>x z. D x \<Longrightarrow> R z x \<Longrightarrow> D z"
     and "D 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>
   have "accp R x" ..
   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"
     with dcl and istep show "P x" by blast
   qed
 qed
 
 
 text \<open>Set versions of the above theorems\<close>
 
 lemmas acc_induct = accp_induct [to_set]
 lemmas acc_induct_rule = acc_induct [rule_format, induct set: acc]
 lemmas acc_downward = accp_downward [to_set]
 lemmas not_acc_down = not_accp_down [to_set]
 lemmas acc_downwards_aux = accp_downwards_aux [to_set]
 lemmas acc_downwards = accp_downwards [to_set]
 lemmas acc_wfI = accp_wfPI [to_set]
 lemmas acc_wfD = accp_wfPD [to_set]
 lemmas wf_acc_iff = wfP_accp_iff [to_set]
 lemmas acc_subset = accp_subset [to_set]
 lemmas acc_subset_induct = accp_subset_induct [to_set]
 
 
 subsection \<open>Tools for building wellfounded relations\<close>
 
 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
 
 text \<open>Measure functions into \<^typ>\<open>nat\<close>\<close>
 
 definition measure :: "('a \<Rightarrow> nat) \<Rightarrow> ('a \<times> 'a) set"
   where "measure = inv_image less_than"
 
 lemma in_measure[simp, code_unfold]: "(x, y) \<in> measure f \<longleftrightarrow> f x < f y"
   by (simp add:measure_def)
 
 lemma wf_measure [iff]: "wf (measure f)"
   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"
   using wf_measure[of f] unfolding measure_def inv_image_def less_than_def less_eq
   by (rule wf_subset) auto
 
 
 subsubsection \<open>Lexicographic combinations\<close>
 
 definition lex_prod :: "('a \<times>'a) set \<Rightarrow> ('b \<times> 'b) set \<Rightarrow> (('a \<times> 'b) \<times> ('a \<times> 'b)) set"
     (infixr "<*lex*>" 80)
-  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)"
-  unfolding 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
+    where "ra <*lex*> rb = {((a, b), (a', b')). (a, a') \<in> ra \<or> a = a' \<and> (b, b') \<in> rb}"
 
 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)
 
+lemma wf_lex_prod [intro!]:
+  assumes "wf ra" "wf rb"
+  shows "wf (ra <*lex*> rb)"
+proof (rule wfI)
+  fix z :: "'a \<times> 'b" and P
+  assume * [rule_format]: "\<forall>u. (\<forall>v. (v, u) \<in> ra <*lex*> rb \<longrightarrow> P v) \<longrightarrow> P u"
+  obtain x y where zeq: "z = (x,y)"
+    by fastforce
+  have "P(x,y)" using \<open>wf ra\<close>
+  proof (induction x arbitrary: y rule: wf_induct_rule)
+    case (less x)
+    note lessx = less
+    show ?case using \<open>wf rb\<close> less
+    proof (induction y rule: wf_induct_rule)
+      case (less y)
+      show ?case
+        by (force intro: * less.IH lessx)
+    qed
+  qed
+  then show "P z"
+    by (simp add: zeq)
+qed auto
+
 text \<open>\<open><*lex*>\<close> preserves transitivity\<close>
 lemma trans_lex_prod [simp,intro!]: "trans R1 \<Longrightarrow> trans R2 \<Longrightarrow> trans (R1 <*lex*> R2)"
   unfolding trans_def lex_prod_def by blast
 
 lemma total_on_lex_prod [simp]: "total_on A r \<Longrightarrow> total_on B s \<Longrightarrow> total_on (A \<times> B) (r <*lex*> s)"
   by (auto simp: total_on_def)
 
 lemma total_lex_prod [simp]: "total r \<Longrightarrow> total s \<Longrightarrow> total (r <*lex*> s)"
   by (auto simp: total_on_def)
 
 text \<open>lexicographic combinations with measure functions\<close>
 
 definition mlex_prod :: "('a \<Rightarrow> nat) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" (infixr "<*mlex*>" 80)
   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)" and
   mlex_less: "f x < f y \<Longrightarrow> (x, y) \<in> f <*mlex*> R" and
   mlex_leq: "f x \<le> f y \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (x, y) \<in> f <*mlex*> R" and
   mlex_iff: "(x, y) \<in> f <*mlex*> R \<longleftrightarrow> f x < f y \<or> f x = f y \<and> (x, y) \<in> R"
   by (auto simp: mlex_prod_def)
 
 text \<open>Proper subset relation on finite sets.\<close>
 definition finite_psubset :: "('a set \<times> 'a set) set"
   where "finite_psubset = {(A, B). A \<subset> B \<and> finite B}"
 
 lemma wf_finite_psubset[simp]: "wf finite_psubset"
   apply (unfold finite_psubset_def)
   apply (rule wf_measure [THEN wf_subset])
   apply (simp add: measure_def inv_image_def less_than_def less_eq)
   apply (fast elim!: psubset_card_mono)
   done
 
 lemma trans_finite_psubset: "trans finite_psubset"
   by (auto simp: finite_psubset_def less_le trans_def)
 
 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"
 
 lemma max_ext_wf:
   assumes wf: "wf r"
   shows "wf (max_ext r)"
 proof (rule acc_wfI, intro allI)
   show "M \<in> acc (max_ext r)" (is "_ \<in> ?W") for M
   proof (induct M rule: infinite_finite_induct)
     case empty
     show ?case
       by (rule accI) (auto elim: max_ext.cases)
   next
     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"
       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: "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
         assume Nless: "(N, insert a M) \<in> max_ext r"
         then have *: "\<And>x. x \<in> N \<Longrightarrow> (x, a) \<in> r \<or> (\<exists>y \<in> M. (x, y) \<in> r)"
           by (auto elim!: max_ext.cases)
 
         let ?N1 = "{n \<in> N. (n, a) \<in> r}"
         let ?N2 = "{n \<in> N. (n, a) \<notin> r}"
         have N: "?N1 \<union> ?N2 = N" by (rule set_eqI) auto
         from Nless have "finite N" by (auto elim: max_ext.cases)
         then have finites: "finite ?N1" "finite ?N2" by auto
 
         have "?N2 \<in> ?W"
         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
           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)
         qed
         with finites have "?N1 \<union> ?N2 \<in> ?W"
           by (rule add_less) simp
         then show "N \<in> ?W" by (simp only: N)
       qed
     qed
   next
     case infinite
     show ?case
       by (rule accI) (auto elim: max_ext.cases simp: infinite)
   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"
   by (force elim!: max_ext.cases)
 
 definition min_ext :: "('a \<times> 'a) set \<Rightarrow> ('a set \<times> 'a set) set"
   where "min_ext r = {(X, Y) | X Y. X \<noteq> {} \<and> (\<forall>y \<in> Y. (\<exists>x \<in> X. (x, y) \<in> r))}"
 
 lemma min_ext_wf:
   assumes "wf r"
   shows "wf (min_ext r)"
 proof (rule wfI_min)
   show "\<exists>m \<in> Q. (\<forall>n. (n, m) \<in> min_ext r \<longrightarrow> n \<notin> Q)" if nonempty: "x \<in> Q"
     for Q :: "'a set set" and x
   proof (cases "Q = {{}}")
     case True
     then show ?thesis by (simp add: min_ext_def)
   next
     case False
     with nonempty obtain e x where "x \<in> Q" "e \<in> x" by force
     then have eU: "e \<in> \<Union>Q" by auto
     with \<open>wf r\<close>
     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 (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 \<in> n" "(y, z) \<in> r"
         by (auto simp: min_ext_def)
       with z(2) show "n \<notin> Q" by auto
     qed
   qed
 qed
 
 
 subsubsection \<open>Bounded increase must terminate\<close>
 
 lemma wf_bounded_measure:
   fixes ub :: "'a \<Rightarrow> nat"
     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"
   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 (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}"
   by (rule wf_subset[OF wf_finite_psubset[unfolded finite_psubset_def]])
     (auto intro: finite_subset[OF _ assms])
 
 hide_const (open) acc accp
 
 end