diff --git a/src/HOL/Relation.thy b/src/HOL/Relation.thy
--- a/src/HOL/Relation.thy
+++ b/src/HOL/Relation.thy
@@ -1,1446 +1,1446 @@
 (*  Title:      HOL/Relation.thy
     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     Author:     Stefan Berghofer, TU Muenchen
     Author:     Martin Desharnais, MPI-INF Saarbruecken
 *)
 
 section \<open>Relations -- as sets of pairs, and binary predicates\<close>
 
 theory Relation
   imports Finite_Set
 begin
 
 text \<open>A preliminary: classical rules for reasoning on predicates\<close>
 
 declare predicate1I [Pure.intro!, intro!]
 declare predicate1D [Pure.dest, dest]
 declare predicate2I [Pure.intro!, intro!]
 declare predicate2D [Pure.dest, dest]
 declare bot1E [elim!]
 declare bot2E [elim!]
 declare top1I [intro!]
 declare top2I [intro!]
 declare inf1I [intro!]
 declare inf2I [intro!]
 declare inf1E [elim!]
 declare inf2E [elim!]
 declare sup1I1 [intro?]
 declare sup2I1 [intro?]
 declare sup1I2 [intro?]
 declare sup2I2 [intro?]
 declare sup1E [elim!]
 declare sup2E [elim!]
 declare sup1CI [intro!]
 declare sup2CI [intro!]
 declare Inf1_I [intro!]
 declare INF1_I [intro!]
 declare Inf2_I [intro!]
 declare INF2_I [intro!]
 declare Inf1_D [elim]
 declare INF1_D [elim]
 declare Inf2_D [elim]
 declare INF2_D [elim]
 declare Inf1_E [elim]
 declare INF1_E [elim]
 declare Inf2_E [elim]
 declare INF2_E [elim]
 declare Sup1_I [intro]
 declare SUP1_I [intro]
 declare Sup2_I [intro]
 declare SUP2_I [intro]
 declare Sup1_E [elim!]
 declare SUP1_E [elim!]
 declare Sup2_E [elim!]
 declare SUP2_E [elim!]
 
 
 subsection \<open>Fundamental\<close>
 
 subsubsection \<open>Relations as sets of pairs\<close>
 
 type_synonym 'a rel = "('a \<times> 'a) set"
 
 lemma subrelI: "(\<And>x y. (x, y) \<in> r \<Longrightarrow> (x, y) \<in> s) \<Longrightarrow> r \<subseteq> s"
   \<comment> \<open>Version of @{thm [source] subsetI} for binary relations\<close>
   by auto
 
 lemma lfp_induct2:
   "(a, b) \<in> lfp f \<Longrightarrow> mono f \<Longrightarrow>
     (\<And>a b. (a, b) \<in> f (lfp f \<inter> {(x, y). P x y}) \<Longrightarrow> P a b) \<Longrightarrow> P a b"
   \<comment> \<open>Version of @{thm [source] lfp_induct} for binary relations\<close>
   using lfp_induct_set [of "(a, b)" f "case_prod P"] by auto
 
 
 subsubsection \<open>Conversions between set and predicate relations\<close>
 
 lemma pred_equals_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) = (\<lambda>x. x \<in> S) \<longleftrightarrow> R = S"
   by (simp add: set_eq_iff fun_eq_iff)
 
 lemma pred_equals_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) = (\<lambda>x y. (x, y) \<in> S) \<longleftrightarrow> R = S"
   by (simp add: set_eq_iff fun_eq_iff)
 
 lemma pred_subset_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) \<le> (\<lambda>x. x \<in> S) \<longleftrightarrow> R \<subseteq> S"
   by (simp add: subset_iff le_fun_def)
 
 lemma pred_subset_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<le> (\<lambda>x y. (x, y) \<in> S) \<longleftrightarrow> R \<subseteq> S"
   by (simp add: subset_iff le_fun_def)
 
 lemma bot_empty_eq [pred_set_conv]: "\<bottom> = (\<lambda>x. x \<in> {})"
   by (auto simp add: fun_eq_iff)
 
 lemma bot_empty_eq2 [pred_set_conv]: "\<bottom> = (\<lambda>x y. (x, y) \<in> {})"
   by (auto simp add: fun_eq_iff)
 
 lemma top_empty_eq: "\<top> = (\<lambda>x. x \<in> UNIV)"
   by (auto simp add: fun_eq_iff)
 
 lemma top_empty_eq2: "\<top> = (\<lambda>x y. (x, y) \<in> UNIV)"
   by (auto simp add: fun_eq_iff)
 
 lemma inf_Int_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) \<sqinter> (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)"
   by (simp add: inf_fun_def)
 
 lemma inf_Int_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<sqinter> (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<inter> S)"
   by (simp add: inf_fun_def)
 
 lemma sup_Un_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) \<squnion> (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<union> S)"
   by (simp add: sup_fun_def)
 
 lemma sup_Un_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<squnion> (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<union> S)"
   by (simp add: sup_fun_def)
 
 lemma INF_INT_eq [pred_set_conv]: "(\<Sqinter>i\<in>S. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (\<Inter>i\<in>S. r i))"
   by (simp add: fun_eq_iff)
 
 lemma INF_INT_eq2 [pred_set_conv]: "(\<Sqinter>i\<in>S. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (\<Inter>i\<in>S. r i))"
   by (simp add: fun_eq_iff)
 
 lemma SUP_UN_eq [pred_set_conv]: "(\<Squnion>i\<in>S. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (\<Union>i\<in>S. r i))"
   by (simp add: fun_eq_iff)
 
 lemma SUP_UN_eq2 [pred_set_conv]: "(\<Squnion>i\<in>S. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (\<Union>i\<in>S. r i))"
   by (simp add: fun_eq_iff)
 
 lemma Inf_INT_eq [pred_set_conv]: "\<Sqinter>S = (\<lambda>x. x \<in> (\<Inter>(Collect ` S)))"
   by (simp add: fun_eq_iff)
 
 lemma INF_Int_eq [pred_set_conv]: "(\<Sqinter>i\<in>S. (\<lambda>x. x \<in> i)) = (\<lambda>x. x \<in> \<Inter>S)"
   by (simp add: fun_eq_iff)
 
 lemma Inf_INT_eq2 [pred_set_conv]: "\<Sqinter>S = (\<lambda>x y. (x, y) \<in> (\<Inter>(Collect ` case_prod ` S)))"
   by (simp add: fun_eq_iff)
 
 lemma INF_Int_eq2 [pred_set_conv]: "(\<Sqinter>i\<in>S. (\<lambda>x y. (x, y) \<in> i)) = (\<lambda>x y. (x, y) \<in> \<Inter>S)"
   by (simp add: fun_eq_iff)
 
 lemma Sup_SUP_eq [pred_set_conv]: "\<Squnion>S = (\<lambda>x. x \<in> \<Union>(Collect ` S))"
   by (simp add: fun_eq_iff)
 
 lemma SUP_Sup_eq [pred_set_conv]: "(\<Squnion>i\<in>S. (\<lambda>x. x \<in> i)) = (\<lambda>x. x \<in> \<Union>S)"
   by (simp add: fun_eq_iff)
 
 lemma Sup_SUP_eq2 [pred_set_conv]: "\<Squnion>S = (\<lambda>x y. (x, y) \<in> (\<Union>(Collect ` case_prod ` S)))"
   by (simp add: fun_eq_iff)
 
 lemma SUP_Sup_eq2 [pred_set_conv]: "(\<Squnion>i\<in>S. (\<lambda>x y. (x, y) \<in> i)) = (\<lambda>x y. (x, y) \<in> \<Union>S)"
   by (simp add: fun_eq_iff)
 
 
 subsection \<open>Properties of relations\<close>
 
 subsubsection \<open>Reflexivity\<close>
 
 definition refl_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool"
   where "refl_on A r \<longleftrightarrow> r \<subseteq> A \<times> A \<and> (\<forall>x\<in>A. (x, x) \<in> r)"
 
 abbreviation refl :: "'a rel \<Rightarrow> bool" \<comment> \<open>reflexivity over a type\<close>
   where "refl \<equiv> refl_on UNIV"
 
 definition reflp_on :: "'a set \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
   where "reflp_on A R \<longleftrightarrow> (\<forall>x\<in>A. R x x)"
 
 abbreviation reflp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
   where "reflp \<equiv> reflp_on UNIV"
 
 lemma reflp_def[no_atp]: "reflp R \<longleftrightarrow> (\<forall>x. R x x)"
   by (simp add: reflp_on_def)
 
 text \<open>@{thm [source] reflp_def} is for backward compatibility.\<close>
 
 lemma reflp_refl_eq [pred_set_conv]: "reflp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> refl r"
   by (simp add: refl_on_def reflp_def)
 
 lemma refl_onI [intro?]: "r \<subseteq> A \<times> A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> (x, x) \<in> r) \<Longrightarrow> refl_on A r"
   unfolding refl_on_def by (iprover intro!: ballI)
 
 lemma reflp_onI:
   "(\<And>x. x \<in> A \<Longrightarrow> R x x) \<Longrightarrow> reflp_on A R"
   by (simp add: reflp_on_def)
 
 lemma reflpI[intro?]: "(\<And>x. R x x) \<Longrightarrow> reflp R"
   by (rule reflp_onI)
 
 lemma refl_onD: "refl_on A r \<Longrightarrow> a \<in> A \<Longrightarrow> (a, a) \<in> r"
   unfolding refl_on_def by blast
 
 lemma refl_onD1: "refl_on A r \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> x \<in> A"
   unfolding refl_on_def by blast
 
 lemma refl_onD2: "refl_on A r \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> y \<in> A"
   unfolding refl_on_def by blast
 
 lemma reflp_onD:
   "reflp_on A R \<Longrightarrow> x \<in> A \<Longrightarrow> R x x"
   by (simp add: reflp_on_def)
 
 lemma reflpD[dest?]: "reflp R \<Longrightarrow> R x x"
   by (simp add: reflp_onD)
 
 lemma reflpE:
   assumes "reflp r"
   obtains "r x x"
   using assms by (auto dest: refl_onD simp add: reflp_def)
 
 lemma reflp_on_subset: "reflp_on A R \<Longrightarrow> B \<subseteq> A \<Longrightarrow> reflp_on B R"
   by (auto intro: reflp_onI dest: reflp_onD)
 
 lemma refl_on_Int: "refl_on A r \<Longrightarrow> refl_on B s \<Longrightarrow> refl_on (A \<inter> B) (r \<inter> s)"
   unfolding refl_on_def by blast
 
 lemma reflp_on_inf: "reflp_on A R \<Longrightarrow> reflp_on B S \<Longrightarrow> reflp_on (A \<inter> B) (R \<sqinter> S)"
   by (auto intro: reflp_onI dest: reflp_onD)
 
 lemma reflp_inf: "reflp r \<Longrightarrow> reflp s \<Longrightarrow> reflp (r \<sqinter> s)"
   by (rule reflp_on_inf[of UNIV _ UNIV, unfolded Int_absorb])
 
 lemma refl_on_Un: "refl_on A r \<Longrightarrow> refl_on B s \<Longrightarrow> refl_on (A \<union> B) (r \<union> s)"
   unfolding refl_on_def by blast
 
 lemma reflp_on_sup: "reflp_on A R \<Longrightarrow> reflp_on B S \<Longrightarrow> reflp_on (A \<union> B) (R \<squnion> S)"
   by (auto intro: reflp_onI dest: reflp_onD)
 
 lemma reflp_sup: "reflp r \<Longrightarrow> reflp s \<Longrightarrow> reflp (r \<squnion> s)"
   by (rule reflp_on_sup[of UNIV _ UNIV, unfolded Un_absorb])
 
 lemma refl_on_INTER: "\<forall>x\<in>S. refl_on (A x) (r x) \<Longrightarrow> refl_on (\<Inter>(A ` S)) (\<Inter>(r ` S))"
   unfolding refl_on_def by fast
 
 lemma reflp_on_Inf: "\<forall>x\<in>S. reflp_on (A x) (R x) \<Longrightarrow> reflp_on (\<Inter>(A ` S)) (\<Sqinter>(R ` S))"
   by (auto intro: reflp_onI dest: reflp_onD)
 
 lemma refl_on_UNION: "\<forall>x\<in>S. refl_on (A x) (r x) \<Longrightarrow> refl_on (\<Union>(A ` S)) (\<Union>(r ` S))"
   unfolding refl_on_def by blast
 
 lemma reflp_on_Sup: "\<forall>x\<in>S. reflp_on (A x) (R x) \<Longrightarrow> reflp_on (\<Union>(A ` S)) (\<Squnion>(R ` S))"
   by (auto intro: reflp_onI dest: reflp_onD)
 
 lemma refl_on_empty [simp]: "refl_on {} {}"
   by (simp add: refl_on_def)
 
 lemma reflp_on_empty [simp]: "reflp_on {} R"
   by (auto intro: reflp_onI)
 
 lemma refl_on_singleton [simp]: "refl_on {x} {(x, x)}"
 by (blast intro: refl_onI)
 
 lemma refl_on_def' [nitpick_unfold, code]:
   "refl_on A r \<longleftrightarrow> (\<forall>(x, y) \<in> r. x \<in> A \<and> y \<in> A) \<and> (\<forall>x \<in> A. (x, x) \<in> r)"
   by (auto intro: refl_onI dest: refl_onD refl_onD1 refl_onD2)
 
 lemma reflp_on_equality [simp]: "reflp_on A (=)"
   by (simp add: reflp_on_def)
 
 lemma reflp_on_mono:
   "reflp_on A R \<Longrightarrow> (\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> R x y \<Longrightarrow> Q x y) \<Longrightarrow> reflp_on A Q"
   by (auto intro: reflp_onI dest: reflp_onD)
 
 lemma reflp_mono: "reflp R \<Longrightarrow> (\<And>x y. R x y \<Longrightarrow> Q x y) \<Longrightarrow> reflp Q"
   by (rule reflp_on_mono[of UNIV R Q]) simp_all
 
 lemma (in preorder) reflp_on_le[simp]: "reflp_on A (\<le>)"
   by (simp add: reflp_onI)
 
 lemma (in preorder) reflp_on_ge[simp]: "reflp_on A (\<ge>)"
   by (simp add: reflp_onI)
 
 
 subsubsection \<open>Irreflexivity\<close>
 
 definition irrefl_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool" where
   "irrefl_on A r \<longleftrightarrow> (\<forall>a \<in> A. (a, a) \<notin> r)"
 
 abbreviation irrefl :: "'a rel \<Rightarrow> bool" where
   "irrefl \<equiv> irrefl_on UNIV"
 
 definition irreflp_on :: "'a set \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
   "irreflp_on A R \<longleftrightarrow> (\<forall>a \<in> A. \<not> R a a)"
 
 abbreviation irreflp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
   "irreflp \<equiv> irreflp_on UNIV"
 
 lemma irrefl_def[no_atp]: "irrefl r \<longleftrightarrow> (\<forall>a. (a, a) \<notin> r)"
   by (simp add: irrefl_on_def)
 
 lemma irreflp_def[no_atp]: "irreflp R \<longleftrightarrow> (\<forall>a. \<not> R a a)"
   by (simp add: irreflp_on_def)
 
 text \<open>@{thm [source] irrefl_def} and @{thm [source] irreflp_def} are for backward compatibility.\<close>
 
 lemma irreflp_on_irrefl_on_eq [pred_set_conv]: "irreflp_on A (\<lambda>a b. (a, b) \<in> r) \<longleftrightarrow> irrefl_on A r"
   by (simp add: irrefl_on_def irreflp_on_def)
 
 lemmas irreflp_irrefl_eq = irreflp_on_irrefl_on_eq[of UNIV]
 
 lemma irrefl_onI: "(\<And>a. a \<in> A \<Longrightarrow> (a, a) \<notin> r) \<Longrightarrow> irrefl_on A r"
   by (simp add: irrefl_on_def)
 
 lemma irreflI[intro?]: "(\<And>a. (a, a) \<notin> r) \<Longrightarrow> irrefl r"
   by (rule irrefl_onI[of UNIV, simplified])
 
 lemma irreflp_onI: "(\<And>a. a \<in> A \<Longrightarrow> \<not> R a a) \<Longrightarrow> irreflp_on A R"
-  by (simp add: irreflp_on_def)
+  by (rule irrefl_onI[to_pred])
 
 lemma irreflpI[intro?]: "(\<And>a. \<not> R a a) \<Longrightarrow> irreflp R"
-  by (rule irreflp_onI[of UNIV, simplified])
+  by (rule irreflI[to_pred])
 
 lemma irrefl_onD: "irrefl_on A r \<Longrightarrow> a \<in> A \<Longrightarrow> (a, a) \<notin> r"
   by (simp add: irrefl_on_def)
 
 lemma irreflD: "irrefl r \<Longrightarrow> (x, x) \<notin> r"
   by (rule irrefl_onD[of UNIV, simplified])
 
 lemma irreflp_onD: "irreflp_on A R \<Longrightarrow> a \<in> A \<Longrightarrow> \<not> R a a"
-  by (simp add: irreflp_on_def)
+  by (rule irrefl_onD[to_pred])
 
 lemma irreflpD: "irreflp R \<Longrightarrow> \<not> R x x"
-  by (rule irreflp_onD[of UNIV, simplified])
+  by (rule irreflD[to_pred])
 
 lemma irrefl_on_distinct [code]: "irrefl_on A r \<longleftrightarrow> (\<forall>(a, b) \<in> r. a \<in> A \<longrightarrow> b \<in> A \<longrightarrow> a \<noteq> b)"
   by (auto simp add: irrefl_on_def)
 
 lemmas irrefl_distinct = irrefl_on_distinct \<comment> \<open>For backward compatibility\<close>
 
 lemma irrefl_on_subset: "irrefl_on A r \<Longrightarrow> B \<subseteq> A \<Longrightarrow> irrefl_on B r"
   by (auto simp: irrefl_on_def)
 
 lemma irreflp_on_subset: "irreflp_on A R \<Longrightarrow> B \<subseteq> A \<Longrightarrow> irreflp_on B R"
   by (auto simp: irreflp_on_def)
 
 lemma (in preorder) irreflp_on_less[simp]: "irreflp_on A (<)"
   by (simp add: irreflp_onI)
 
 lemma (in preorder) irreflp_on_greater[simp]: "irreflp_on A (>)"
   by (simp add: irreflp_onI)
 
 subsubsection \<open>Asymmetry\<close>
 
 inductive asym :: "'a rel \<Rightarrow> bool"
   where asymI: "(\<And>a b. (a, b) \<in> R \<Longrightarrow> (b, a) \<notin> R) \<Longrightarrow> asym R"
 
 inductive asymp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
   where asympI: "(\<And>a b. R a b \<Longrightarrow> \<not> R b a) \<Longrightarrow> asymp R"
 
 lemma asymp_asym_eq [pred_set_conv]: "asymp (\<lambda>a b. (a, b) \<in> R) \<longleftrightarrow> asym R"
   by (auto intro!: asymI asympI elim: asym.cases asymp.cases simp add: irreflp_irrefl_eq)
 
 lemma asymD: "\<lbrakk>asym R; (x,y) \<in> R\<rbrakk> \<Longrightarrow> (y,x) \<notin> R"
   by (simp add: asym.simps)
 
 lemma asympD: "asymp R \<Longrightarrow> R x y \<Longrightarrow> \<not> R y x"
   by (rule asymD[to_pred])
 
 lemma asym_iff: "asym R \<longleftrightarrow> (\<forall>x y. (x,y) \<in> R \<longrightarrow> (y,x) \<notin> R)"
   by (blast intro: asymI dest: asymD)
 
 lemma (in preorder) asymp_less[simp]: "asymp (<)"
   by (auto intro: asympI dual_order.asym)
 
 lemma (in preorder) asymp_greater[simp]: "asymp (>)"
   by (auto intro: asympI dual_order.asym)
 
 
 subsubsection \<open>Symmetry\<close>
 
 definition sym :: "'a rel \<Rightarrow> bool"
   where "sym r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r)"
 
 definition symp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
   where "symp r \<longleftrightarrow> (\<forall>x y. r x y \<longrightarrow> r y x)"
 
 lemma symp_sym_eq [pred_set_conv]: "symp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> sym r"
   by (simp add: sym_def symp_def)
 
 lemma symI [intro?]: "(\<And>a b. (a, b) \<in> r \<Longrightarrow> (b, a) \<in> r) \<Longrightarrow> sym r"
   by (unfold sym_def) iprover
 
 lemma sympI [intro?]: "(\<And>a b. r a b \<Longrightarrow> r b a) \<Longrightarrow> symp r"
   by (fact symI [to_pred])
 
 lemma symE:
   assumes "sym r" and "(b, a) \<in> r"
   obtains "(a, b) \<in> r"
   using assms by (simp add: sym_def)
 
 lemma sympE:
   assumes "symp r" and "r b a"
   obtains "r a b"
   using assms by (rule symE [to_pred])
 
 lemma symD [dest?]:
   assumes "sym r" and "(b, a) \<in> r"
   shows "(a, b) \<in> r"
   using assms by (rule symE)
 
 lemma sympD [dest?]:
   assumes "symp r" and "r b a"
   shows "r a b"
   using assms by (rule symD [to_pred])
 
 lemma sym_Int: "sym r \<Longrightarrow> sym s \<Longrightarrow> sym (r \<inter> s)"
   by (fast intro: symI elim: symE)
 
 lemma symp_inf: "symp r \<Longrightarrow> symp s \<Longrightarrow> symp (r \<sqinter> s)"
   by (fact sym_Int [to_pred])
 
 lemma sym_Un: "sym r \<Longrightarrow> sym s \<Longrightarrow> sym (r \<union> s)"
   by (fast intro: symI elim: symE)
 
 lemma symp_sup: "symp r \<Longrightarrow> symp s \<Longrightarrow> symp (r \<squnion> s)"
   by (fact sym_Un [to_pred])
 
 lemma sym_INTER: "\<forall>x\<in>S. sym (r x) \<Longrightarrow> sym (\<Inter>(r ` S))"
   by (fast intro: symI elim: symE)
 
 lemma symp_INF: "\<forall>x\<in>S. symp (r x) \<Longrightarrow> symp (\<Sqinter>(r ` S))"
   by (fact sym_INTER [to_pred])
 
 lemma sym_UNION: "\<forall>x\<in>S. sym (r x) \<Longrightarrow> sym (\<Union>(r ` S))"
   by (fast intro: symI elim: symE)
 
 lemma symp_SUP: "\<forall>x\<in>S. symp (r x) \<Longrightarrow> symp (\<Squnion>(r ` S))"
   by (fact sym_UNION [to_pred])
 
 
 subsubsection \<open>Antisymmetry\<close>
 
 definition antisym :: "'a rel \<Rightarrow> bool"
   where "antisym r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r \<longrightarrow> x = y)"
 
 definition antisymp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
   where "antisymp r \<longleftrightarrow> (\<forall>x y. r x y \<longrightarrow> r y x \<longrightarrow> x = y)"
 
 lemma antisymp_antisym_eq [pred_set_conv]: "antisymp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> antisym r"
   by (simp add: antisym_def antisymp_def)
 
 lemma antisymI [intro?]:
   "(\<And>x y. (x, y) \<in> r \<Longrightarrow> (y, x) \<in> r \<Longrightarrow> x = y) \<Longrightarrow> antisym r"
   unfolding antisym_def by iprover
 
 lemma antisympI [intro?]:
   "(\<And>x y. r x y \<Longrightarrow> r y x \<Longrightarrow> x = y) \<Longrightarrow> antisymp r"
   by (fact antisymI [to_pred])
     
 lemma antisymD [dest?]:
   "antisym r \<Longrightarrow> (a, b) \<in> r \<Longrightarrow> (b, a) \<in> r \<Longrightarrow> a = b"
   unfolding antisym_def by iprover
 
 lemma antisympD [dest?]:
   "antisymp r \<Longrightarrow> r a b \<Longrightarrow> r b a \<Longrightarrow> a = b"
   by (fact antisymD [to_pred])
 
 lemma antisym_subset:
   "r \<subseteq> s \<Longrightarrow> antisym s \<Longrightarrow> antisym r"
   unfolding antisym_def by blast
 
 lemma antisymp_less_eq:
   "r \<le> s \<Longrightarrow> antisymp s \<Longrightarrow> antisymp r"
   by (fact antisym_subset [to_pred])
     
 lemma antisym_empty [simp]:
   "antisym {}"
   unfolding antisym_def by blast
 
 lemma antisym_bot [simp]:
   "antisymp \<bottom>"
   by (fact antisym_empty [to_pred])
     
 lemma antisymp_equality [simp]:
   "antisymp HOL.eq"
   by (auto intro: antisympI)
 
 lemma antisym_singleton [simp]:
   "antisym {x}"
   by (blast intro: antisymI)
 
 lemma antisym_if_asym: "asym r \<Longrightarrow> antisym r"
   by (auto intro: antisymI elim: asym.cases)
 
 lemma antisymp_if_asymp: "asymp R \<Longrightarrow> antisymp R"
   by (rule antisym_if_asym[to_pred])
 
 lemma (in preorder) antisymp_less[simp]: "antisymp (<)"
   by (rule antisymp_if_asymp[OF asymp_less])
 
 lemma (in preorder) antisymp_greater[simp]: "antisymp (>)"
   by (rule antisymp_if_asymp[OF asymp_greater])
 
 lemma (in order) antisymp_le[simp]: "antisymp (\<le>)"
   by (simp add: antisympI)
 
 lemma (in order) antisymp_ge[simp]: "antisymp (\<ge>)"
   by (simp add: antisympI)
 
 
 subsubsection \<open>Transitivity\<close>
 
 definition trans :: "'a rel \<Rightarrow> bool"
   where "trans r \<longleftrightarrow> (\<forall>x y z. (x, y) \<in> r \<longrightarrow> (y, z) \<in> r \<longrightarrow> (x, z) \<in> r)"
 
 definition transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
   where "transp r \<longleftrightarrow> (\<forall>x y z. r x y \<longrightarrow> r y z \<longrightarrow> r x z)"
 
 lemma transp_trans_eq [pred_set_conv]: "transp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> trans r"
   by (simp add: trans_def transp_def)
 
 lemma transI [intro?]: "(\<And>x y z. (x, y) \<in> r \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> (x, z) \<in> r) \<Longrightarrow> trans r"
   by (unfold trans_def) iprover
 
 lemma transpI [intro?]: "(\<And>x y z. r x y \<Longrightarrow> r y z \<Longrightarrow> r x z) \<Longrightarrow> transp r"
   by (fact transI [to_pred])
 
 lemma transE:
   assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r"
   obtains "(x, z) \<in> r"
   using assms by (unfold trans_def) iprover
 
 lemma transpE:
   assumes "transp r" and "r x y" and "r y z"
   obtains "r x z"
   using assms by (rule transE [to_pred])
 
 lemma transD [dest?]:
   assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r"
   shows "(x, z) \<in> r"
   using assms by (rule transE)
 
 lemma transpD [dest?]:
   assumes "transp r" and "r x y" and "r y z"
   shows "r x z"
   using assms by (rule transD [to_pred])
 
 lemma trans_Int: "trans r \<Longrightarrow> trans s \<Longrightarrow> trans (r \<inter> s)"
   by (fast intro: transI elim: transE)
 
 lemma transp_inf: "transp r \<Longrightarrow> transp s \<Longrightarrow> transp (r \<sqinter> s)"
   by (fact trans_Int [to_pred])
 
 lemma trans_INTER: "\<forall>x\<in>S. trans (r x) \<Longrightarrow> trans (\<Inter>(r ` S))"
   by (fast intro: transI elim: transD)
 
 lemma transp_INF: "\<forall>x\<in>S. transp (r x) \<Longrightarrow> transp (\<Sqinter>(r ` S))"
   by (fact trans_INTER [to_pred])
     
 lemma trans_join [code]: "trans r \<longleftrightarrow> (\<forall>(x, y1) \<in> r. \<forall>(y2, z) \<in> r. y1 = y2 \<longrightarrow> (x, z) \<in> r)"
   by (auto simp add: trans_def)
 
 lemma transp_trans: "transp r \<longleftrightarrow> trans {(x, y). r x y}"
   by (simp add: trans_def transp_def)
 
 lemma transp_equality [simp]: "transp (=)"
   by (auto intro: transpI)
 
 lemma trans_empty [simp]: "trans {}"
   by (blast intro: transI)
 
 lemma transp_empty [simp]: "transp (\<lambda>x y. False)"
   using trans_empty[to_pred] by (simp add: bot_fun_def)
 
 lemma trans_singleton [simp]: "trans {(a, a)}"
   by (blast intro: transI)
 
 lemma transp_singleton [simp]: "transp (\<lambda>x y. x = a \<and> y = a)"
   by (simp add: transp_def)
 
 lemma asym_if_irrefl_and_trans: "irrefl R \<Longrightarrow> trans R \<Longrightarrow> asym R"
   by (auto intro: asymI dest: transD irreflD)
 
 lemma asymp_if_irreflp_and_transp: "irreflp R \<Longrightarrow> transp R \<Longrightarrow> asymp R"
   by (rule asym_if_irrefl_and_trans[to_pred])
 
 context preorder
 begin
 
 lemma transp_le[simp]: "transp (\<le>)"
 by(auto simp add: transp_def intro: order_trans)
 
 lemma transp_less[simp]: "transp (<)"
 by(auto simp add: transp_def intro: less_trans)
 
 lemma transp_ge[simp]: "transp (\<ge>)"
 by(auto simp add: transp_def intro: order_trans)
 
 lemma transp_gr[simp]: "transp (>)"
 by(auto simp add: transp_def intro: less_trans)
 
 end
 
 subsubsection \<open>Totality\<close>
 
 definition total_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool" where
   "total_on A r \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. x \<noteq> y \<longrightarrow> (x, y) \<in> r \<or> (y, x) \<in> r)"
 
 abbreviation total :: "'a rel \<Rightarrow> bool" where
   "total \<equiv> total_on UNIV"
 
 definition totalp_on :: "'a set \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
   "totalp_on A R \<longleftrightarrow> (\<forall>x \<in> A. \<forall>y \<in> A. x \<noteq> y \<longrightarrow> R x y \<or> R y x)"
 
 abbreviation totalp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
   "totalp \<equiv> totalp_on UNIV"
 
 lemma totalp_on_refl_on_eq[pred_set_conv]: "totalp_on A (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> total_on A r"
   by (simp add: totalp_on_def total_on_def)
 
 lemma total_onI [intro?]:
   "(\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> (x, y) \<in> r \<or> (y, x) \<in> r) \<Longrightarrow> total_on A r"
   unfolding total_on_def by blast
 
 lemma totalI: "(\<And>x y. x \<noteq> y \<Longrightarrow> (x, y) \<in> r \<or> (y, x) \<in> r) \<Longrightarrow> total r"
   by (rule total_onI)
 
 lemma totalp_onI: "(\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> R x y \<or> R y x) \<Longrightarrow> totalp_on A R"
-  by (simp add: totalp_on_def)
+  by (rule total_onI[to_pred])
 
 lemma totalpI: "(\<And>x y. x \<noteq> y \<Longrightarrow> R x y \<or> R y x) \<Longrightarrow> totalp R"
-  by (rule totalp_onI)
+  by (rule totalI[to_pred])
 
 lemma totalp_onD:
   "totalp_on A R \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> R x y \<or> R y x"
   by (simp add: totalp_on_def)
 
 lemma totalpD: "totalp R \<Longrightarrow> x \<noteq> y \<Longrightarrow> R x y \<or> R y x"
   by (simp add: totalp_onD)
 
 lemma total_on_subset: "total_on A r \<Longrightarrow> B \<subseteq> A \<Longrightarrow> total_on B r"
   by (auto simp: total_on_def)
 
 lemma totalp_on_subset: "totalp_on A R \<Longrightarrow> B \<subseteq> A \<Longrightarrow> totalp_on B R"
   by (auto intro: totalp_onI dest: totalp_onD)
 
 lemma total_on_empty [simp]: "total_on {} r"
   by (simp add: total_on_def)
 
 lemma totalp_on_empty [simp]: "totalp_on {} R"
   by (simp add: totalp_on_def)
 
 lemma total_on_singleton [simp]: "total_on {x} r"
   by (simp add: total_on_def)
 
 lemma totalp_on_singleton [simp]: "totalp_on {x} R"
   by (simp add: totalp_on_def)
 
 lemma (in linorder) totalp_on_less[simp]: "totalp_on A (<)"
   by (auto intro: totalp_onI)
 
 lemma (in linorder) totalp_on_greater[simp]: "totalp_on A (>)"
   by (auto intro: totalp_onI)
 
 lemma (in linorder) totalp_on_le[simp]: "totalp_on A (\<le>)"
   by (rule totalp_onI, rule linear)
 
 lemma (in linorder) totalp_on_ge[simp]: "totalp_on A (\<ge>)"
   by (rule totalp_onI, rule linear)
 
 
 subsubsection \<open>Single valued relations\<close>
 
 definition single_valued :: "('a \<times> 'b) set \<Rightarrow> bool"
   where "single_valued r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (\<forall>z. (x, z) \<in> r \<longrightarrow> y = z))"
 
 definition single_valuedp :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
   where "single_valuedp r \<longleftrightarrow> (\<forall>x y. r x y \<longrightarrow> (\<forall>z. r x z \<longrightarrow> y = z))"
 
 lemma single_valuedp_single_valued_eq [pred_set_conv]:
   "single_valuedp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> single_valued r"
   by (simp add: single_valued_def single_valuedp_def)
 
 lemma single_valuedp_iff_Uniq:
   "single_valuedp r \<longleftrightarrow> (\<forall>x. \<exists>\<^sub>\<le>\<^sub>1y. r x y)"
   unfolding Uniq_def single_valuedp_def by auto
 
 lemma single_valuedI:
   "(\<And>x y. (x, y) \<in> r \<Longrightarrow> (\<And>z. (x, z) \<in> r \<Longrightarrow> y = z)) \<Longrightarrow> single_valued r"
   unfolding single_valued_def by blast
 
 lemma single_valuedpI:
   "(\<And>x y. r x y \<Longrightarrow> (\<And>z. r x z \<Longrightarrow> y = z)) \<Longrightarrow> single_valuedp r"
   by (fact single_valuedI [to_pred])
 
 lemma single_valuedD:
   "single_valued r \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> (x, z) \<in> r \<Longrightarrow> y = z"
   by (simp add: single_valued_def)
 
 lemma single_valuedpD:
   "single_valuedp r \<Longrightarrow> r x y \<Longrightarrow> r x z \<Longrightarrow> y = z"
   by (fact single_valuedD [to_pred])
 
 lemma single_valued_empty [simp]:
   "single_valued {}"
   by (simp add: single_valued_def)
 
 lemma single_valuedp_bot [simp]:
   "single_valuedp \<bottom>"
   by (fact single_valued_empty [to_pred])
 
 lemma single_valued_subset:
   "r \<subseteq> s \<Longrightarrow> single_valued s \<Longrightarrow> single_valued r"
   unfolding single_valued_def by blast
 
 lemma single_valuedp_less_eq:
   "r \<le> s \<Longrightarrow> single_valuedp s \<Longrightarrow> single_valuedp r"
   by (fact single_valued_subset [to_pred])
 
 
 subsection \<open>Relation operations\<close>
 
 subsubsection \<open>The identity relation\<close>
 
 definition Id :: "'a rel"
   where "Id = {p. \<exists>x. p = (x, x)}"
 
 lemma IdI [intro]: "(a, a) \<in> Id"
   by (simp add: Id_def)
 
 lemma IdE [elim!]: "p \<in> Id \<Longrightarrow> (\<And>x. p = (x, x) \<Longrightarrow> P) \<Longrightarrow> P"
   unfolding Id_def by (iprover elim: CollectE)
 
 lemma pair_in_Id_conv [iff]: "(a, b) \<in> Id \<longleftrightarrow> a = b"
   unfolding Id_def by blast
 
 lemma refl_Id: "refl Id"
   by (simp add: refl_on_def)
 
 lemma antisym_Id: "antisym Id"
   \<comment> \<open>A strange result, since \<open>Id\<close> is also symmetric.\<close>
   by (simp add: antisym_def)
 
 lemma sym_Id: "sym Id"
   by (simp add: sym_def)
 
 lemma trans_Id: "trans Id"
   by (simp add: trans_def)
 
 lemma single_valued_Id [simp]: "single_valued Id"
   by (unfold single_valued_def) blast
 
 lemma irrefl_diff_Id [simp]: "irrefl (r - Id)"
   by (simp add: irrefl_def)
 
 lemma trans_diff_Id: "trans r \<Longrightarrow> antisym r \<Longrightarrow> trans (r - Id)"
   unfolding antisym_def trans_def by blast
 
 lemma total_on_diff_Id [simp]: "total_on A (r - Id) = total_on A r"
   by (simp add: total_on_def)
 
 lemma Id_fstsnd_eq: "Id = {x. fst x = snd x}"
   by force
 
 
 subsubsection \<open>Diagonal: identity over a set\<close>
 
 definition Id_on :: "'a set \<Rightarrow> 'a rel"
   where "Id_on A = (\<Union>x\<in>A. {(x, x)})"
 
 lemma Id_on_empty [simp]: "Id_on {} = {}"
   by (simp add: Id_on_def)
 
 lemma Id_on_eqI: "a = b \<Longrightarrow> a \<in> A \<Longrightarrow> (a, b) \<in> Id_on A"
   by (simp add: Id_on_def)
 
 lemma Id_onI [intro!]: "a \<in> A \<Longrightarrow> (a, a) \<in> Id_on A"
   by (rule Id_on_eqI) (rule refl)
 
 lemma Id_onE [elim!]: "c \<in> Id_on A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> c = (x, x) \<Longrightarrow> P) \<Longrightarrow> P"
   \<comment> \<open>The general elimination rule.\<close>
   unfolding Id_on_def by (iprover elim!: UN_E singletonE)
 
 lemma Id_on_iff: "(x, y) \<in> Id_on A \<longleftrightarrow> x = y \<and> x \<in> A"
   by blast
 
 lemma Id_on_def' [nitpick_unfold]: "Id_on {x. A x} = Collect (\<lambda>(x, y). x = y \<and> A x)"
   by auto
 
 lemma Id_on_subset_Times: "Id_on A \<subseteq> A \<times> A"
   by blast
 
 lemma refl_on_Id_on: "refl_on A (Id_on A)"
   by (rule refl_onI [OF Id_on_subset_Times Id_onI])
 
 lemma antisym_Id_on [simp]: "antisym (Id_on A)"
   unfolding antisym_def by blast
 
 lemma sym_Id_on [simp]: "sym (Id_on A)"
   by (rule symI) clarify
 
 lemma trans_Id_on [simp]: "trans (Id_on A)"
   by (fast intro: transI elim: transD)
 
 lemma single_valued_Id_on [simp]: "single_valued (Id_on A)"
   unfolding single_valued_def by blast
 
 
 subsubsection \<open>Composition\<close>
 
 inductive_set relcomp  :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'c) set \<Rightarrow> ('a \<times> 'c) set"  (infixr "O" 75)
   for r :: "('a \<times> 'b) set" and s :: "('b \<times> 'c) set"
   where relcompI [intro]: "(a, b) \<in> r \<Longrightarrow> (b, c) \<in> s \<Longrightarrow> (a, c) \<in> r O s"
 
 notation relcompp (infixr "OO" 75)
 
 lemmas relcomppI = relcompp.intros
 
 text \<open>
   For historic reasons, the elimination rules are not wholly corresponding.
   Feel free to consolidate this.
 \<close>
 
 inductive_cases relcompEpair: "(a, c) \<in> r O s"
 inductive_cases relcomppE [elim!]: "(r OO s) a c"
 
 lemma relcompE [elim!]: "xz \<in> r O s \<Longrightarrow>
   (\<And>x y z. xz = (x, z) \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> (y, z) \<in> s  \<Longrightarrow> P) \<Longrightarrow> P"
   apply (cases xz)
   apply simp
   apply (erule relcompEpair)
   apply iprover
   done
 
 lemma R_O_Id [simp]: "R O Id = R"
   by fast
 
 lemma Id_O_R [simp]: "Id O R = R"
   by fast
 
 lemma relcomp_empty1 [simp]: "{} O R = {}"
   by blast
 
 lemma relcompp_bot1 [simp]: "\<bottom> OO R = \<bottom>"
   by (fact relcomp_empty1 [to_pred])
 
 lemma relcomp_empty2 [simp]: "R O {} = {}"
   by blast
 
 lemma relcompp_bot2 [simp]: "R OO \<bottom> = \<bottom>"
   by (fact relcomp_empty2 [to_pred])
 
 lemma O_assoc: "(R O S) O T = R O (S O T)"
   by blast
 
 lemma relcompp_assoc: "(r OO s) OO t = r OO (s OO t)"
   by (fact O_assoc [to_pred])
 
 lemma trans_O_subset: "trans r \<Longrightarrow> r O r \<subseteq> r"
   by (unfold trans_def) blast
 
 lemma transp_relcompp_less_eq: "transp r \<Longrightarrow> r OO r \<le> r "
   by (fact trans_O_subset [to_pred])
 
 lemma relcomp_mono: "r' \<subseteq> r \<Longrightarrow> s' \<subseteq> s \<Longrightarrow> r' O s' \<subseteq> r O s"
   by blast
 
 lemma relcompp_mono: "r' \<le> r \<Longrightarrow> s' \<le> s \<Longrightarrow> r' OO s' \<le> r OO s "
   by (fact relcomp_mono [to_pred])
 
 lemma relcomp_subset_Sigma: "r \<subseteq> A \<times> B \<Longrightarrow> s \<subseteq> B \<times> C \<Longrightarrow> r O s \<subseteq> A \<times> C"
   by blast
 
 lemma relcomp_distrib [simp]: "R O (S \<union> T) = (R O S) \<union> (R O T)"
   by auto
 
 lemma relcompp_distrib [simp]: "R OO (S \<squnion> T) = R OO S \<squnion> R OO T"
   by (fact relcomp_distrib [to_pred])
 
 lemma relcomp_distrib2 [simp]: "(S \<union> T) O R = (S O R) \<union> (T O R)"
   by auto
 
 lemma relcompp_distrib2 [simp]: "(S \<squnion> T) OO R = S OO R \<squnion> T OO R"
   by (fact relcomp_distrib2 [to_pred])
 
 lemma relcomp_UNION_distrib: "s O \<Union>(r ` I) = (\<Union>i\<in>I. s O r i) "
   by auto
 
 lemma relcompp_SUP_distrib: "s OO \<Squnion>(r ` I) = (\<Squnion>i\<in>I. s OO r i)"
   by (fact relcomp_UNION_distrib [to_pred])
     
 lemma relcomp_UNION_distrib2: "\<Union>(r ` I) O s = (\<Union>i\<in>I. r i O s) "
   by auto
 
 lemma relcompp_SUP_distrib2: "\<Squnion>(r ` I) OO s = (\<Squnion>i\<in>I. r i OO s)"
   by (fact relcomp_UNION_distrib2 [to_pred])
     
 lemma single_valued_relcomp: "single_valued r \<Longrightarrow> single_valued s \<Longrightarrow> single_valued (r O s)"
   unfolding single_valued_def by blast
 
 lemma relcomp_unfold: "r O s = {(x, z). \<exists>y. (x, y) \<in> r \<and> (y, z) \<in> s}"
   by (auto simp add: set_eq_iff)
 
 lemma relcompp_apply: "(R OO S) a c \<longleftrightarrow> (\<exists>b. R a b \<and> S b c)"
   unfolding relcomp_unfold [to_pred] ..
 
 lemma eq_OO: "(=) OO R = R"
   by blast
 
 lemma OO_eq: "R OO (=) = R"
   by blast
 
 
 subsubsection \<open>Converse\<close>
 
 inductive_set converse :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'a) set"  ("(_\<inverse>)" [1000] 999)
   for r :: "('a \<times> 'b) set"
   where "(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r\<inverse>"
 
 notation conversep  ("(_\<inverse>\<inverse>)" [1000] 1000)
 
 notation (ASCII)
   converse  ("(_^-1)" [1000] 999) and
   conversep ("(_^--1)" [1000] 1000)
 
 lemma converseI [sym]: "(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r\<inverse>"
   by (fact converse.intros)
 
 lemma conversepI (* CANDIDATE [sym] *): "r a b \<Longrightarrow> r\<inverse>\<inverse> b a"
   by (fact conversep.intros)
 
 lemma converseD [sym]: "(a, b) \<in> r\<inverse> \<Longrightarrow> (b, a) \<in> r"
   by (erule converse.cases) iprover
 
 lemma conversepD (* CANDIDATE [sym] *): "r\<inverse>\<inverse> b a \<Longrightarrow> r a b"
   by (fact converseD [to_pred])
 
 lemma converseE [elim!]: "yx \<in> r\<inverse> \<Longrightarrow> (\<And>x y. yx = (y, x) \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> P) \<Longrightarrow> P"
   \<comment> \<open>More general than \<open>converseD\<close>, as it ``splits'' the member of the relation.\<close>
   apply (cases yx)
   apply simp
   apply (erule converse.cases)
   apply iprover
   done
 
 lemmas conversepE [elim!] = conversep.cases
 
 lemma converse_iff [iff]: "(a, b) \<in> r\<inverse> \<longleftrightarrow> (b, a) \<in> r"
   by (auto intro: converseI)
 
 lemma conversep_iff [iff]: "r\<inverse>\<inverse> a b = r b a"
   by (fact converse_iff [to_pred])
 
 lemma converse_converse [simp]: "(r\<inverse>)\<inverse> = r"
   by (simp add: set_eq_iff)
 
 lemma conversep_conversep [simp]: "(r\<inverse>\<inverse>)\<inverse>\<inverse> = r"
   by (fact converse_converse [to_pred])
 
 lemma converse_empty[simp]: "{}\<inverse> = {}"
   by auto
 
 lemma converse_UNIV[simp]: "UNIV\<inverse> = UNIV"
   by auto
 
 lemma converse_relcomp: "(r O s)\<inverse> = s\<inverse> O r\<inverse>"
   by blast
 
 lemma converse_relcompp: "(r OO s)\<inverse>\<inverse> = s\<inverse>\<inverse> OO r\<inverse>\<inverse>"
   by (iprover intro: order_antisym conversepI relcomppI elim: relcomppE dest: conversepD)
 
 lemma converse_Int: "(r \<inter> s)\<inverse> = r\<inverse> \<inter> s\<inverse>"
   by blast
 
 lemma converse_meet: "(r \<sqinter> s)\<inverse>\<inverse> = r\<inverse>\<inverse> \<sqinter> s\<inverse>\<inverse>"
   by (simp add: inf_fun_def) (iprover intro: conversepI ext dest: conversepD)
 
 lemma converse_Un: "(r \<union> s)\<inverse> = r\<inverse> \<union> s\<inverse>"
   by blast
 
 lemma converse_join: "(r \<squnion> s)\<inverse>\<inverse> = r\<inverse>\<inverse> \<squnion> s\<inverse>\<inverse>"
   by (simp add: sup_fun_def) (iprover intro: conversepI ext dest: conversepD)
 
 lemma converse_INTER: "(\<Inter>(r ` S))\<inverse> = (\<Inter>x\<in>S. (r x)\<inverse>)"
   by fast
 
 lemma converse_UNION: "(\<Union>(r ` S))\<inverse> = (\<Union>x\<in>S. (r x)\<inverse>)"
   by blast
 
 lemma converse_mono[simp]: "r\<inverse> \<subseteq> s \<inverse> \<longleftrightarrow> r \<subseteq> s"
   by auto
 
 lemma conversep_mono[simp]: "r\<inverse>\<inverse> \<le> s \<inverse>\<inverse> \<longleftrightarrow> r \<le> s"
   by (fact converse_mono[to_pred])
 
 lemma converse_inject[simp]: "r\<inverse> = s \<inverse> \<longleftrightarrow> r = s"
   by auto
 
 lemma conversep_inject[simp]: "r\<inverse>\<inverse> = s \<inverse>\<inverse> \<longleftrightarrow> r = s"
   by (fact converse_inject[to_pred])
 
 lemma converse_subset_swap: "r \<subseteq> s \<inverse> \<longleftrightarrow> r \<inverse> \<subseteq> s"
   by auto
 
 lemma conversep_le_swap: "r \<le> s \<inverse>\<inverse> \<longleftrightarrow> r \<inverse>\<inverse> \<le> s"
   by (fact converse_subset_swap[to_pred])
 
 lemma converse_Id [simp]: "Id\<inverse> = Id"
   by blast
 
 lemma converse_Id_on [simp]: "(Id_on A)\<inverse> = Id_on A"
   by blast
 
 lemma refl_on_converse [simp]: "refl_on A (r\<inverse>) = refl_on A r"
   by (auto simp: refl_on_def)
 
 lemma reflp_on_conversp [simp]: "reflp_on A R\<inverse>\<inverse> \<longleftrightarrow> reflp_on A R"
   by (auto simp: reflp_on_def)
 
 lemma irrefl_on_converse [simp]: "irrefl_on A (r\<inverse>) = irrefl_on A r"
   by (simp add: irrefl_on_def)
 
 lemma irreflp_on_converse [simp]: "irreflp_on A (r\<inverse>\<inverse>) = irreflp_on A r"
   by (rule irrefl_on_converse[to_pred])
 
 lemma sym_converse [simp]: "sym (converse r) = sym r"
   unfolding sym_def by blast
 
 lemma antisym_converse [simp]: "antisym (converse r) = antisym r"
   unfolding antisym_def by blast
 
 lemma trans_converse [simp]: "trans (converse r) = trans r"
   unfolding trans_def by blast
 
 lemma sym_conv_converse_eq: "sym r \<longleftrightarrow> r\<inverse> = r"
   unfolding sym_def by fast
 
 lemma sym_Un_converse: "sym (r \<union> r\<inverse>)"
   unfolding sym_def by blast
 
 lemma sym_Int_converse: "sym (r \<inter> r\<inverse>)"
   unfolding sym_def by blast
 
 lemma total_on_converse [simp]: "total_on A (r\<inverse>) = total_on A r"
   by (auto simp: total_on_def)
 
 lemma totalp_on_converse [simp]: "totalp_on A R\<inverse>\<inverse> = totalp_on A R"
   by (rule total_on_converse[to_pred])
 
 lemma finite_converse [iff]: "finite (r\<inverse>) = finite r"
 unfolding converse_def conversep_iff using [[simproc add: finite_Collect]]
 by (auto elim: finite_imageD simp: inj_on_def)
 
 lemma card_inverse[simp]: "card (R\<inverse>) = card R"
 proof -
   have *: "\<And>R. prod.swap ` R = R\<inverse>" by auto
   {
     assume "\<not>finite R"
     hence ?thesis
       by auto
   } moreover {
     assume "finite R"
     with card_image_le[of R prod.swap] card_image_le[of "R\<inverse>" prod.swap]
     have ?thesis by (auto simp: *)
   } ultimately show ?thesis by blast
 qed  
 
 lemma conversep_noteq [simp]: "(\<noteq>)\<inverse>\<inverse> = (\<noteq>)"
   by (auto simp add: fun_eq_iff)
 
 lemma conversep_eq [simp]: "(=)\<inverse>\<inverse> = (=)"
   by (auto simp add: fun_eq_iff)
 
 lemma converse_unfold [code]: "r\<inverse> = {(y, x). (x, y) \<in> r}"
   by (simp add: set_eq_iff)
 
 
 subsubsection \<open>Domain, range and field\<close>
 
 inductive_set Domain :: "('a \<times> 'b) set \<Rightarrow> 'a set" for r :: "('a \<times> 'b) set"
   where DomainI [intro]: "(a, b) \<in> r \<Longrightarrow> a \<in> Domain r"
 
 lemmas DomainPI = Domainp.DomainI
 
 inductive_cases DomainE [elim!]: "a \<in> Domain r"
 inductive_cases DomainpE [elim!]: "Domainp r a"
 
 inductive_set Range :: "('a \<times> 'b) set \<Rightarrow> 'b set" for r :: "('a \<times> 'b) set"
   where RangeI [intro]: "(a, b) \<in> r \<Longrightarrow> b \<in> Range r"
 
 lemmas RangePI = Rangep.RangeI
 
 inductive_cases RangeE [elim!]: "b \<in> Range r"
 inductive_cases RangepE [elim!]: "Rangep r b"
 
 definition Field :: "'a rel \<Rightarrow> 'a set"
   where "Field r = Domain r \<union> Range r"
 
 lemma FieldI1: "(i, j) \<in> R \<Longrightarrow> i \<in> Field R"
   unfolding Field_def by blast
 
 lemma FieldI2: "(i, j) \<in> R \<Longrightarrow> j \<in> Field R"
   unfolding Field_def by auto
 
 lemma Domain_fst [code]: "Domain r = fst ` r"
   by force
 
 lemma Range_snd [code]: "Range r = snd ` r"
   by force
 
 lemma fst_eq_Domain: "fst ` R = Domain R"
   by force
 
 lemma snd_eq_Range: "snd ` R = Range R"
   by force
 
 lemma range_fst [simp]: "range fst = UNIV"
   by (auto simp: fst_eq_Domain)
 
 lemma range_snd [simp]: "range snd = UNIV"
   by (auto simp: snd_eq_Range)
 
 lemma Domain_empty [simp]: "Domain {} = {}"
   by auto
 
 lemma Range_empty [simp]: "Range {} = {}"
   by auto
 
 lemma Field_empty [simp]: "Field {} = {}"
   by (simp add: Field_def)
 
 lemma Domain_empty_iff: "Domain r = {} \<longleftrightarrow> r = {}"
   by auto
 
 lemma Range_empty_iff: "Range r = {} \<longleftrightarrow> r = {}"
   by auto
 
 lemma Domain_insert [simp]: "Domain (insert (a, b) r) = insert a (Domain r)"
   by blast
 
 lemma Range_insert [simp]: "Range (insert (a, b) r) = insert b (Range r)"
   by blast
 
 lemma Field_insert [simp]: "Field (insert (a, b) r) = {a, b} \<union> Field r"
   by (auto simp add: Field_def)
 
 lemma Domain_iff: "a \<in> Domain r \<longleftrightarrow> (\<exists>y. (a, y) \<in> r)"
   by blast
 
 lemma Range_iff: "a \<in> Range r \<longleftrightarrow> (\<exists>y. (y, a) \<in> r)"
   by blast
 
 lemma Domain_Id [simp]: "Domain Id = UNIV"
   by blast
 
 lemma Range_Id [simp]: "Range Id = UNIV"
   by blast
 
 lemma Domain_Id_on [simp]: "Domain (Id_on A) = A"
   by blast
 
 lemma Range_Id_on [simp]: "Range (Id_on A) = A"
   by blast
 
 lemma Domain_Un_eq: "Domain (A \<union> B) = Domain A \<union> Domain B"
   by blast
 
 lemma Range_Un_eq: "Range (A \<union> B) = Range A \<union> Range B"
   by blast
 
 lemma Field_Un [simp]: "Field (r \<union> s) = Field r \<union> Field s"
   by (auto simp: Field_def)
 
 lemma Domain_Int_subset: "Domain (A \<inter> B) \<subseteq> Domain A \<inter> Domain B"
   by blast
 
 lemma Range_Int_subset: "Range (A \<inter> B) \<subseteq> Range A \<inter> Range B"
   by blast
 
 lemma Domain_Diff_subset: "Domain A - Domain B \<subseteq> Domain (A - B)"
   by blast
 
 lemma Range_Diff_subset: "Range A - Range B \<subseteq> Range (A - B)"
   by blast
 
 lemma Domain_Union: "Domain (\<Union>S) = (\<Union>A\<in>S. Domain A)"
   by blast
 
 lemma Range_Union: "Range (\<Union>S) = (\<Union>A\<in>S. Range A)"
   by blast
 
 lemma Field_Union [simp]: "Field (\<Union>R) = \<Union>(Field ` R)"
   by (auto simp: Field_def)
 
 lemma Domain_converse [simp]: "Domain (r\<inverse>) = Range r"
   by auto
 
 lemma Range_converse [simp]: "Range (r\<inverse>) = Domain r"
   by blast
 
 lemma Field_converse [simp]: "Field (r\<inverse>) = Field r"
   by (auto simp: Field_def)
 
 lemma Domain_Collect_case_prod [simp]: "Domain {(x, y). P x y} = {x. \<exists>y. P x y}"
   by auto
 
 lemma Range_Collect_case_prod [simp]: "Range {(x, y). P x y} = {y. \<exists>x. P x y}"
   by auto
 
 lemma finite_Domain: "finite r \<Longrightarrow> finite (Domain r)"
   by (induct set: finite) auto
 
 lemma finite_Range: "finite r \<Longrightarrow> finite (Range r)"
   by (induct set: finite) auto
 
 lemma finite_Field: "finite r \<Longrightarrow> finite (Field r)"
   by (simp add: Field_def finite_Domain finite_Range)
 
 lemma Domain_mono: "r \<subseteq> s \<Longrightarrow> Domain r \<subseteq> Domain s"
   by blast
 
 lemma Range_mono: "r \<subseteq> s \<Longrightarrow> Range r \<subseteq> Range s"
   by blast
 
 lemma mono_Field: "r \<subseteq> s \<Longrightarrow> Field r \<subseteq> Field s"
   by (auto simp: Field_def Domain_def Range_def)
 
 lemma Domain_unfold: "Domain r = {x. \<exists>y. (x, y) \<in> r}"
   by blast
 
 lemma Field_square [simp]: "Field (x \<times> x) = x"
   unfolding Field_def by blast
 
 
 subsubsection \<open>Image of a set under a relation\<close>
 
 definition Image :: "('a \<times> 'b) set \<Rightarrow> 'a set \<Rightarrow> 'b set"  (infixr "``" 90)
   where "r `` s = {y. \<exists>x\<in>s. (x, y) \<in> r}"
 
 lemma Image_iff: "b \<in> r``A \<longleftrightarrow> (\<exists>x\<in>A. (x, b) \<in> r)"
   by (simp add: Image_def)
 
 lemma Image_singleton: "r``{a} = {b. (a, b) \<in> r}"
   by (simp add: Image_def)
 
 lemma Image_singleton_iff [iff]: "b \<in> r``{a} \<longleftrightarrow> (a, b) \<in> r"
   by (rule Image_iff [THEN trans]) simp
 
 lemma ImageI [intro]: "(a, b) \<in> r \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> r``A"
   unfolding Image_def by blast
 
 lemma ImageE [elim!]: "b \<in> r `` A \<Longrightarrow> (\<And>x. (x, b) \<in> r \<Longrightarrow> x \<in> A \<Longrightarrow> P) \<Longrightarrow> P"
   unfolding Image_def by (iprover elim!: CollectE bexE)
 
 lemma rev_ImageI: "a \<in> A \<Longrightarrow> (a, b) \<in> r \<Longrightarrow> b \<in> r `` A"
   \<comment> \<open>This version's more effective when we already have the required \<open>a\<close>\<close>
   by blast
 
 lemma Image_empty1 [simp]: "{} `` X = {}"
 by auto
 
 lemma Image_empty2 [simp]: "R``{} = {}"
 by blast
 
 lemma Image_Id [simp]: "Id `` A = A"
   by blast
 
 lemma Image_Id_on [simp]: "Id_on A `` B = A \<inter> B"
   by blast
 
 lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B"
   by blast
 
 lemma Image_Int_eq: "single_valued (converse R) \<Longrightarrow> R `` (A \<inter> B) = R `` A \<inter> R `` B"
   by (auto simp: single_valued_def)
 
 lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B"
   by blast
 
 lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A"
   by blast
 
 lemma Image_subset: "r \<subseteq> A \<times> B \<Longrightarrow> r``C \<subseteq> B"
   by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)
 
 lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})"
   \<comment> \<open>NOT suitable for rewriting\<close>
   by blast
 
 lemma Image_mono: "r' \<subseteq> r \<Longrightarrow> A' \<subseteq> A \<Longrightarrow> (r' `` A') \<subseteq> (r `` A)"
   by blast
 
 lemma Image_UN: "r `` (\<Union>(B ` A)) = (\<Union>x\<in>A. r `` (B x))"
   by blast
 
 lemma UN_Image: "(\<Union>i\<in>I. X i) `` S = (\<Union>i\<in>I. X i `` S)"
   by auto
 
 lemma Image_INT_subset: "(r `` (\<Inter>(B ` A))) \<subseteq> (\<Inter>x\<in>A. r `` (B x))"
   by blast
 
 text \<open>Converse inclusion requires some assumptions\<close>
 lemma Image_INT_eq:
   assumes "single_valued (r\<inverse>)"
     and "A \<noteq> {}"
   shows "r `` (\<Inter>(B ` A)) = (\<Inter>x\<in>A. r `` B x)"
 proof(rule equalityI, rule Image_INT_subset)
   show "(\<Inter>x\<in>A. r `` B x) \<subseteq> r `` \<Inter> (B ` A)"
   proof
     fix x
     assume "x \<in> (\<Inter>x\<in>A. r `` B x)"
     then show "x \<in> r `` \<Inter> (B ` A)"
       using assms unfolding single_valued_def by simp blast
   qed
 qed
 
 lemma Image_subset_eq: "r``A \<subseteq> B \<longleftrightarrow> A \<subseteq> - ((r\<inverse>) `` (- B))"
   by blast
 
 lemma Image_Collect_case_prod [simp]: "{(x, y). P x y} `` A = {y. \<exists>x\<in>A. P x y}"
   by auto
 
 lemma Sigma_Image: "(SIGMA x:A. B x) `` X = (\<Union>x\<in>X \<inter> A. B x)"
   by auto
 
 lemma relcomp_Image: "(X O Y) `` Z = Y `` (X `` Z)"
   by auto
 
 lemma finite_Image[simp]: assumes "finite R" shows "finite (R `` A)"
 by(rule finite_subset[OF _ finite_Range[OF assms]]) auto
 
 
 subsubsection \<open>Inverse image\<close>
 
 definition inv_image :: "'b rel \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a rel"
   where "inv_image r f = {(x, y). (f x, f y) \<in> r}"
 
 definition inv_imagep :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
   where "inv_imagep r f = (\<lambda>x y. r (f x) (f y))"
 
 lemma [pred_set_conv]: "inv_imagep (\<lambda>x y. (x, y) \<in> r) f = (\<lambda>x y. (x, y) \<in> inv_image r f)"
   by (simp add: inv_image_def inv_imagep_def)
 
 lemma sym_inv_image: "sym r \<Longrightarrow> sym (inv_image r f)"
   unfolding sym_def inv_image_def by blast
 
 lemma trans_inv_image: "trans r \<Longrightarrow> trans (inv_image r f)"
   unfolding trans_def inv_image_def
   by (simp (no_asm)) blast
 
 lemma total_inv_image: "\<lbrakk>inj f; total r\<rbrakk> \<Longrightarrow> total (inv_image r f)"
   unfolding inv_image_def total_on_def by (auto simp: inj_eq)
 
 lemma asym_inv_image: "asym R \<Longrightarrow> asym (inv_image R f)"
   by (simp add: inv_image_def asym_iff)
 
 lemma in_inv_image[simp]: "(x, y) \<in> inv_image r f \<longleftrightarrow> (f x, f y) \<in> r"
   by (auto simp: inv_image_def)
 
 lemma converse_inv_image[simp]: "(inv_image R f)\<inverse> = inv_image (R\<inverse>) f"
   unfolding inv_image_def converse_unfold by auto
 
 lemma in_inv_imagep [simp]: "inv_imagep r f x y = r (f x) (f y)"
   by (simp add: inv_imagep_def)
 
 
 subsubsection \<open>Powerset\<close>
 
 definition Powp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
   where "Powp A = (\<lambda>B. \<forall>x \<in> B. A x)"
 
 lemma Powp_Pow_eq [pred_set_conv]: "Powp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> Pow A)"
   by (auto simp add: Powp_def fun_eq_iff)
 
 lemmas Powp_mono [mono] = Pow_mono [to_pred]
 
 
 subsubsection \<open>Expressing relation operations via \<^const>\<open>Finite_Set.fold\<close>\<close>
 
 lemma Id_on_fold:
   assumes "finite A"
   shows "Id_on A = Finite_Set.fold (\<lambda>x. Set.insert (Pair x x)) {} A"
 proof -
   interpret comp_fun_commute "\<lambda>x. Set.insert (Pair x x)"
     by standard auto
   from assms show ?thesis
     unfolding Id_on_def by (induct A) simp_all
 qed
 
 lemma comp_fun_commute_Image_fold:
   "comp_fun_commute (\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A)"
 proof -
   interpret comp_fun_idem Set.insert
     by (fact comp_fun_idem_insert)
   show ?thesis
     by standard (auto simp: fun_eq_iff comp_fun_commute split: prod.split)
 qed
 
 lemma Image_fold:
   assumes "finite R"
   shows "R `` S = Finite_Set.fold (\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A) {} R"
 proof -
   interpret comp_fun_commute "(\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A)"
     by (rule comp_fun_commute_Image_fold)
   have *: "\<And>x F. Set.insert x F `` S = (if fst x \<in> S then Set.insert (snd x) (F `` S) else (F `` S))"
     by (force intro: rev_ImageI)
   show ?thesis
     using assms by (induct R) (auto simp: *)
 qed
 
 lemma insert_relcomp_union_fold:
   assumes "finite S"
   shows "{x} O S \<union> X = Finite_Set.fold (\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A') X S"
 proof -
   interpret comp_fun_commute "\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A'"
   proof -
     interpret comp_fun_idem Set.insert
       by (fact comp_fun_idem_insert)
     show "comp_fun_commute (\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A')"
       by standard (auto simp add: fun_eq_iff split: prod.split)
   qed
   have *: "{x} O S = {(x', z). x' = fst x \<and> (snd x, z) \<in> S}"
     by (auto simp: relcomp_unfold intro!: exI)
   show ?thesis
     unfolding * using \<open>finite S\<close> by (induct S) (auto split: prod.split)
 qed
 
 lemma insert_relcomp_fold:
   assumes "finite S"
   shows "Set.insert x R O S =
     Finite_Set.fold (\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A') (R O S) S"
 proof -
   have "Set.insert x R O S = ({x} O S) \<union> (R O S)"
     by auto
   then show ?thesis
     by (auto simp: insert_relcomp_union_fold [OF assms])
 qed
 
 lemma comp_fun_commute_relcomp_fold:
   assumes "finite S"
   shows "comp_fun_commute (\<lambda>(x,y) A.
     Finite_Set.fold (\<lambda>(w,z) A'. if y = w then Set.insert (x,z) A' else A') A S)"
 proof -
   have *: "\<And>a b A.
     Finite_Set.fold (\<lambda>(w, z) A'. if b = w then Set.insert (a, z) A' else A') A S = {(a,b)} O S \<union> A"
     by (auto simp: insert_relcomp_union_fold[OF assms] cong: if_cong)
   show ?thesis
     by standard (auto simp: *)
 qed
 
 lemma relcomp_fold:
   assumes "finite R" "finite S"
   shows "R O S = Finite_Set.fold
     (\<lambda>(x,y) A. Finite_Set.fold (\<lambda>(w,z) A'. if y = w then Set.insert (x,z) A' else A') A S) {} R"
 proof -
   interpret commute_relcomp_fold: comp_fun_commute
     "(\<lambda>(x, y) A. Finite_Set.fold (\<lambda>(w, z) A'. if y = w then insert (x, z) A' else A') A S)"
     by (fact comp_fun_commute_relcomp_fold[OF \<open>finite S\<close>])
   from assms show ?thesis
     by (induct R) (auto simp: comp_fun_commute_relcomp_fold insert_relcomp_fold cong: if_cong)
 qed
 
 end
diff --git a/src/HOL/Wellfounded.thy b/src/HOL/Wellfounded.thy
--- a/src/HOL/Wellfounded.thy
+++ b/src/HOL/Wellfounded.thy
@@ -1,1020 +1,1020 @@
 (*  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_imp_asym: "wf r \<Longrightarrow> asym r"
   by (auto intro: asymI elim: wf_asym)
 
 lemma wfP_imp_asymp: "wfP r \<Longrightarrow> asymp r"
   by (rule wf_imp_asym[to_pred])
 
 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_imp_irrefl:
   assumes "wf r" shows "irrefl r" 
   using wf_irrefl [OF assms] by (auto simp add: irrefl_def)
 
 lemma wfP_imp_irreflp: "wfP r \<Longrightarrow> irreflp r"
-  by (rule wf_imp_irrefl[to_pred, folded top_set_def])
+  by (rule wf_imp_irrefl[to_pred])
 
 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)"
   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)
 
 lemma (in wellorder) wfP_less[simp]: "wfP (<)"
   by (simp add: wf wfP_def)
 
 
 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
   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
         then have "\<forall>y'. (y', z) \<in> insert (y, x) r \<longrightarrow> y' \<notin> Q"
           using R by(blast intro: rtrancl_trans)+
         then show ?thesis
           by (rule bexI) fact
       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
       using inj unfolding A_def
       by (intro bexI[of _ "f a0"]) auto
   qed (use \<open>b \<in> B\<close> in  \<open>unfold A_def, auto\<close>)
 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
       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"
   unfolding wf_def
 proof clarify
   fix P x
   assume "\<forall>x'. (\<forall>y. (y, x') \<in> pred_nat \<longrightarrow> P y) \<longrightarrow> P x'"
   then show "P x"
     unfolding pred_nat_def by (induction x) blast+
 qed
 
 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 irrefl_less_than: "irrefl less_than"
   using irrefl_def by blast
 
 lemma asym_less_than: "asym less_than"
   by (simp add: asym.simps irrefl_less_than)
 
 lemma total_less_than: "total less_than" and total_on_less_than [simp]: "total_on A 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"
 proof (rule wfPUNIVI)
   fix P x
   assume "\<forall>x. accp r x" "\<forall>x. (\<forall>y. r y x \<longrightarrow> P y) \<longrightarrow> P x"
   then show "P x"
     using accp_induct[where P = P] by blast
 qed
 
 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!]: 
   fixes f :: "'a \<Rightarrow> 'b"
   assumes "wf r"
   shows "wf (inv_image r f)"
 proof -
   have "\<And>x P. x \<in> P \<Longrightarrow> \<exists>z\<in>P. \<forall>y. (f y, f z) \<in> r \<longrightarrow> y \<notin> P"
   proof -
     fix P and x::'a
     assume "x \<in> P"
     then obtain w where w: "w \<in> {w. \<exists>x::'a. x \<in> P \<and> f x = w}"
       by auto
     have *: "\<And>Q u. u \<in> Q \<Longrightarrow> \<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q"
       using assms by (auto simp add: wf_eq_minimal)
     show "\<exists>z\<in>P. \<forall>y. (f y, f z) \<in> r \<longrightarrow> y \<notin> P"
       using * [OF w] by auto
   qed
   then show ?thesis
     by (clarsimp simp: inv_image_def wf_eq_minimal)
 qed
 
 
 subsubsection \<open>Conversion to a known well-founded relation\<close>
 
 lemma wf_if_convertible_to_wf:
   fixes r :: "'a rel" and s :: "'b rel" and f :: "'a \<Rightarrow> 'b"
   assumes "wf s" and convertible: "\<And>x y. (x, y) \<in> r \<Longrightarrow> (f x, f y) \<in> s"
   shows "wf r"
 proof (rule wfI_min[of r])
   fix x :: 'a and Q :: "'a set"
   assume "x \<in> Q"
   then obtain y where "y \<in> Q" and "\<And>z. (f z, f y) \<in> s \<Longrightarrow> z \<notin> Q"
     by (auto elim: wfE_min[OF wf_inv_image[of s f, OF \<open>wf s\<close>], unfolded in_inv_image])
   thus "\<exists>z \<in> Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q"
     by (auto intro: convertible)
 qed
 
 lemma wfP_if_convertible_to_wfP: "wfP S \<Longrightarrow> (\<And>x y. R x y \<Longrightarrow> S (f x) (f y)) \<Longrightarrow> wfP R"
   using wf_if_convertible_to_wf[to_pred, of S R f] by simp
 
 text \<open>Converting to @{typ nat} is a very common special case that might be found more easily by
   Sledgehammer.\<close>
 
 lemma wfP_if_convertible_to_nat:
   fixes f :: "_ \<Rightarrow> nat"
   shows "(\<And>x y. R x y \<Longrightarrow> f x < f y) \<Longrightarrow> wfP R"
   by (rule wfP_if_convertible_to_wfP[of "(<) :: nat \<Rightarrow> nat \<Rightarrow> bool", simplified])
 
 
 subsubsection \<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 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 asym_lex_prod: "\<lbrakk>asym R; asym S\<rbrakk> \<Longrightarrow> asym (R <*lex*> S)"
   by (auto simp add: asym_iff lex_prod_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"
             using max_extI[OF _ _ \<open>M \<noteq> {}\<close>, where ?X = ?N2] by 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