diff --git a/src/HOL/Equiv_Relations.thy b/src/HOL/Equiv_Relations.thy
--- a/src/HOL/Equiv_Relations.thy
+++ b/src/HOL/Equiv_Relations.thy
@@ -1,639 +1,642 @@
 (*  Title:      HOL/Equiv_Relations.thy
     Author:     Lawrence C Paulson, 1996 Cambridge University Computer Laboratory
 *)
 
 section \<open>Equivalence Relations in Higher-Order Set Theory\<close>
 
 theory Equiv_Relations
   imports Groups_Big
 begin
 
 subsection \<open>Equivalence relations -- set version\<close>
 
 definition equiv :: "'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool"
   where "equiv A r \<longleftrightarrow> refl_on A r \<and> sym r \<and> trans r"
 
 lemma equivI: "refl_on A r \<Longrightarrow> sym r \<Longrightarrow> trans r \<Longrightarrow> equiv A r"
   by (simp add: equiv_def)
 
 lemma equivE:
   assumes "equiv A r"
   obtains "refl_on A r" and "sym r" and "trans r"
   using assms by (simp add: equiv_def)
 
 text \<open>
   Suppes, Theorem 70: \<open>r\<close> is an equiv relation iff \<open>r\<inverse> O r = r\<close>.
 
   First half: \<open>equiv A r \<Longrightarrow> r\<inverse> O r = r\<close>.
 \<close>
 
 lemma sym_trans_comp_subset: "sym r \<Longrightarrow> trans r \<Longrightarrow> r\<inverse> O r \<subseteq> r"
   unfolding trans_def sym_def converse_unfold by blast
 
 lemma refl_on_comp_subset: "refl_on A r \<Longrightarrow> r \<subseteq> r\<inverse> O r"
   unfolding refl_on_def by blast
 
 lemma equiv_comp_eq: "equiv A r \<Longrightarrow> r\<inverse> O r = r"
   unfolding equiv_def
   by (iprover intro: sym_trans_comp_subset refl_on_comp_subset equalityI)
 
 text \<open>Second half.\<close>
 
 lemma comp_equivI:
   assumes "r\<inverse> O r = r" "Domain r = A"
   shows "equiv A r"
 proof -
   have *: "\<And>x y. (x, y) \<in> r \<Longrightarrow> (y, x) \<in> r"
     using assms by blast
   show ?thesis
     unfolding equiv_def refl_on_def sym_def trans_def
     using assms by (auto intro: *)
 qed
 
 
 subsection \<open>Equivalence classes\<close>
 
 lemma equiv_class_subset: "equiv A r \<Longrightarrow> (a, b) \<in> r \<Longrightarrow> r``{a} \<subseteq> r``{b}"
   \<comment> \<open>lemma for the next result\<close>
   unfolding equiv_def trans_def sym_def by blast
 
 theorem equiv_class_eq: "equiv A r \<Longrightarrow> (a, b) \<in> r \<Longrightarrow> r``{a} = r``{b}"
   by (intro equalityI equiv_class_subset; force simp add: equiv_def sym_def)
 
 lemma equiv_class_self: "equiv A r \<Longrightarrow> a \<in> A \<Longrightarrow> a \<in> r``{a}"
   unfolding equiv_def refl_on_def by blast
 
 lemma subset_equiv_class: "equiv A r \<Longrightarrow> r``{b} \<subseteq> r``{a} \<Longrightarrow> b \<in> A \<Longrightarrow> (a, b) \<in> r"
   \<comment> \<open>lemma for the next result\<close>
   unfolding equiv_def refl_on_def by blast
 
 lemma eq_equiv_class: "r``{a} = r``{b} \<Longrightarrow> equiv A r \<Longrightarrow> b \<in> A \<Longrightarrow> (a, b) \<in> r"
   by (iprover intro: equalityD2 subset_equiv_class)
 
 lemma equiv_class_nondisjoint: "equiv A r \<Longrightarrow> x \<in> (r``{a} \<inter> r``{b}) \<Longrightarrow> (a, b) \<in> r"
   unfolding equiv_def trans_def sym_def by blast
 
 lemma equiv_type: "equiv A r \<Longrightarrow> r \<subseteq> A \<times> A"
   unfolding equiv_def refl_on_def by blast
 
 lemma equiv_class_eq_iff: "equiv A r \<Longrightarrow> (x, y) \<in> r \<longleftrightarrow> r``{x} = r``{y} \<and> x \<in> A \<and> y \<in> A"
   by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
 
 lemma eq_equiv_class_iff: "equiv A r \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> r``{x} = r``{y} \<longleftrightarrow> (x, y) \<in> r"
   by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
 
+lemma disjnt_equiv_class: "equiv A r \<Longrightarrow> disjnt (r``{a}) (r``{b}) \<longleftrightarrow> (a, b) \<notin> r"
+  by (auto dest: equiv_class_self simp: equiv_class_eq_iff disjnt_def)
+
 
 subsection \<open>Quotients\<close>
 
 definition quotient :: "'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> 'a set set"  (infixl "'/'/" 90)
   where "A//r = (\<Union>x \<in> A. {r``{x}})"  \<comment> \<open>set of equiv classes\<close>
 
 lemma quotientI: "x \<in> A \<Longrightarrow> r``{x} \<in> A//r"
   unfolding quotient_def by blast
 
 lemma quotientE: "X \<in> A//r \<Longrightarrow> (\<And>x. X = r``{x} \<Longrightarrow> x \<in> A \<Longrightarrow> P) \<Longrightarrow> P"
   unfolding quotient_def by blast
 
 lemma Union_quotient: "equiv A r \<Longrightarrow> \<Union>(A//r) = A"
   unfolding equiv_def refl_on_def quotient_def by blast
 
 lemma quotient_disj: "equiv A r \<Longrightarrow> X \<in> A//r \<Longrightarrow> Y \<in> A//r \<Longrightarrow> X = Y \<or> X \<inter> Y = {}"
   unfolding quotient_def equiv_def trans_def sym_def by blast
 
 lemma quotient_eqI:
   assumes "equiv A r" "X \<in> A//r" "Y \<in> A//r" and xy: "x \<in> X" "y \<in> Y" "(x, y) \<in> r"
   shows "X = Y"
 proof -
   obtain a b where "a \<in> A" and a: "X = r `` {a}" and "b \<in> A" and b: "Y = r `` {b}"
     using assms by (auto elim!: quotientE)
   then have "(a,b) \<in> r"
       using xy \<open>equiv A r\<close> unfolding equiv_def sym_def trans_def by blast
   then show ?thesis
     unfolding a b by (rule equiv_class_eq [OF \<open>equiv A r\<close>])
 qed
 
 lemma quotient_eq_iff:
   assumes "equiv A r" "X \<in> A//r" "Y \<in> A//r" and xy: "x \<in> X" "y \<in> Y" 
   shows "X = Y \<longleftrightarrow> (x, y) \<in> r"
 proof
   assume L: "X = Y" 
   with assms show "(x, y) \<in> r" 
     unfolding equiv_def sym_def trans_def by (blast elim!: quotientE)
 next
   assume \<section>: "(x, y) \<in> r" show "X = Y"
     by (rule quotient_eqI) (use \<section> assms in \<open>blast+\<close>)
 qed
 
 lemma eq_equiv_class_iff2: "equiv A r \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> {x}//r = {y}//r \<longleftrightarrow> (x, y) \<in> r"
   by (simp add: quotient_def eq_equiv_class_iff)
 
 lemma quotient_empty [simp]: "{}//r = {}"
   by (simp add: quotient_def)
 
 lemma quotient_is_empty [iff]: "A//r = {} \<longleftrightarrow> A = {}"
   by (simp add: quotient_def)
 
 lemma quotient_is_empty2 [iff]: "{} = A//r \<longleftrightarrow> A = {}"
   by (simp add: quotient_def)
 
 lemma singleton_quotient: "{x}//r = {r `` {x}}"
   by (simp add: quotient_def)
 
 lemma quotient_diff1: "inj_on (\<lambda>a. {a}//r) A \<Longrightarrow> a \<in> A \<Longrightarrow> (A - {a})//r = A//r - {a}//r"
   unfolding quotient_def inj_on_def by blast
 
 
 subsection \<open>Refinement of one equivalence relation WRT another\<close>
 
 lemma refines_equiv_class_eq: "R \<subseteq> S \<Longrightarrow> equiv A R \<Longrightarrow> equiv A S \<Longrightarrow> R``(S``{a}) = S``{a}"
   by (auto simp: equiv_class_eq_iff)
 
 lemma refines_equiv_class_eq2: "R \<subseteq> S \<Longrightarrow> equiv A R \<Longrightarrow> equiv A S \<Longrightarrow> S``(R``{a}) = S``{a}"
   by (auto simp: equiv_class_eq_iff)
 
 lemma refines_equiv_image_eq: "R \<subseteq> S \<Longrightarrow> equiv A R \<Longrightarrow> equiv A S \<Longrightarrow> (\<lambda>X. S``X) ` (A//R) = A//S"
    by (auto simp: quotient_def image_UN refines_equiv_class_eq2)
 
 lemma finite_refines_finite:
   "finite (A//R) \<Longrightarrow> R \<subseteq> S \<Longrightarrow> equiv A R \<Longrightarrow> equiv A S \<Longrightarrow> finite (A//S)"
   by (erule finite_surj [where f = "\<lambda>X. S``X"]) (simp add: refines_equiv_image_eq)
 
 lemma finite_refines_card_le:
   "finite (A//R) \<Longrightarrow> R \<subseteq> S \<Longrightarrow> equiv A R \<Longrightarrow> equiv A S \<Longrightarrow> card (A//S) \<le> card (A//R)"
   by (subst refines_equiv_image_eq [of R S A, symmetric])
     (auto simp: card_image_le [where f = "\<lambda>X. S``X"])
 
 
 subsection \<open>Defining unary operations upon equivalence classes\<close>
 
 text \<open>A congruence-preserving function.\<close>
 
 definition congruent :: "('a \<times> 'a) set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
   where "congruent r f \<longleftrightarrow> (\<forall>(y, z) \<in> r. f y = f z)"
 
 lemma congruentI: "(\<And>y z. (y, z) \<in> r \<Longrightarrow> f y = f z) \<Longrightarrow> congruent r f"
   by (auto simp add: congruent_def)
 
 lemma congruentD: "congruent r f \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> f y = f z"
   by (auto simp add: congruent_def)
 
 abbreviation RESPECTS :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool"  (infixr "respects" 80)
   where "f respects r \<equiv> congruent r f"
 
 
 lemma UN_constant_eq: "a \<in> A \<Longrightarrow> \<forall>y \<in> A. f y = c \<Longrightarrow> (\<Union>y \<in> A. f y) = c"
   \<comment> \<open>lemma required to prove \<open>UN_equiv_class\<close>\<close>
   by auto
 
 lemma UN_equiv_class:
   assumes "equiv A r" "f respects r" "a \<in> A"
   shows "(\<Union>x \<in> r``{a}. f x) = f a"
   \<comment> \<open>Conversion rule\<close>
 proof -
   have \<section>: "\<forall>x\<in>r `` {a}. f x = f a"
     using assms unfolding equiv_def congruent_def sym_def by blast
   show ?thesis
     by (iprover intro: assms UN_constant_eq [OF equiv_class_self \<section>])
 qed
 
 lemma UN_equiv_class_type:
   assumes r: "equiv A r" "f respects r" and X: "X \<in> A//r" and AB: "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B"
   shows "(\<Union>x \<in> X. f x) \<in> B"
   using assms unfolding quotient_def
   by (auto simp: UN_equiv_class [OF r])
 
 text \<open>
   Sufficient conditions for injectiveness.  Could weaken premises!
   major premise could be an inclusion; \<open>bcong\<close> could be
   \<open>\<And>y. y \<in> A \<Longrightarrow> f y \<in> B\<close>.
 \<close>
 
 lemma UN_equiv_class_inject:
   assumes "equiv A r" "f respects r"
     and eq: "(\<Union>x \<in> X. f x) = (\<Union>y \<in> Y. f y)" 
     and X: "X \<in> A//r" and Y: "Y \<in> A//r" 
     and fr: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> f x = f y \<Longrightarrow> (x, y) \<in> r"
   shows "X = Y"
 proof -
   obtain a b where "a \<in> A" and a: "X = r `` {a}" and "b \<in> A" and b: "Y = r `` {b}"
     using assms by (auto elim!: quotientE)
   then have "\<Union> (f ` r `` {a}) = f a" "\<Union> (f ` r `` {b}) = f b"
     by (iprover intro: UN_equiv_class [OF \<open>equiv A r\<close>] assms)+
   then have "f a = f b"
     using eq unfolding a b by (iprover intro: trans sym)
   then have "(a,b) \<in> r"
     using fr \<open>a \<in> A\<close> \<open>b \<in> A\<close> by blast
   then show ?thesis
     unfolding a b by (rule equiv_class_eq [OF \<open>equiv A r\<close>])
 qed
 
 
 subsection \<open>Defining binary operations upon equivalence classes\<close>
 
 text \<open>A congruence-preserving function of two arguments.\<close>
 
 definition congruent2 :: "('a \<times> 'a) set \<Rightarrow> ('b \<times> 'b) set \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> bool"
   where "congruent2 r1 r2 f \<longleftrightarrow> (\<forall>(y1, z1) \<in> r1. \<forall>(y2, z2) \<in> r2. f y1 y2 = f z1 z2)"
 
 lemma congruent2I':
   assumes "\<And>y1 z1 y2 z2. (y1, z1) \<in> r1 \<Longrightarrow> (y2, z2) \<in> r2 \<Longrightarrow> f y1 y2 = f z1 z2"
   shows "congruent2 r1 r2 f"
   using assms by (auto simp add: congruent2_def)
 
 lemma congruent2D: "congruent2 r1 r2 f \<Longrightarrow> (y1, z1) \<in> r1 \<Longrightarrow> (y2, z2) \<in> r2 \<Longrightarrow> f y1 y2 = f z1 z2"
   by (auto simp add: congruent2_def)
 
 text \<open>Abbreviation for the common case where the relations are identical.\<close>
 abbreviation RESPECTS2:: "('a \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool"  (infixr "respects2" 80)
   where "f respects2 r \<equiv> congruent2 r r f"
 
 
 lemma congruent2_implies_congruent:
   "equiv A r1 \<Longrightarrow> congruent2 r1 r2 f \<Longrightarrow> a \<in> A \<Longrightarrow> congruent r2 (f a)"
   unfolding congruent_def congruent2_def equiv_def refl_on_def by blast
 
 lemma congruent2_implies_congruent_UN:
   assumes "equiv A1 r1" "equiv A2 r2" "congruent2 r1 r2 f" "a \<in> A2" 
   shows "congruent r1 (\<lambda>x1. \<Union>x2 \<in> r2``{a}. f x1 x2)"
   unfolding congruent_def
 proof clarify
   fix c d
   assume cd: "(c,d) \<in> r1"
   then have "c \<in> A1" "d \<in> A1"
     using \<open>equiv A1 r1\<close> by (auto elim!: equiv_type [THEN subsetD, THEN SigmaE2])
   with assms show "\<Union> (f c ` r2 `` {a}) = \<Union> (f d ` r2 `` {a})"
   proof (simp add: UN_equiv_class congruent2_implies_congruent)
     show "f c a = f d a"
       using assms cd unfolding congruent2_def equiv_def refl_on_def by blast
   qed
 qed
 
 lemma UN_equiv_class2:
   "equiv A1 r1 \<Longrightarrow> equiv A2 r2 \<Longrightarrow> congruent2 r1 r2 f \<Longrightarrow> a1 \<in> A1 \<Longrightarrow> a2 \<in> A2 \<Longrightarrow>
     (\<Union>x1 \<in> r1``{a1}. \<Union>x2 \<in> r2``{a2}. f x1 x2) = f a1 a2"
   by (simp add: UN_equiv_class congruent2_implies_congruent congruent2_implies_congruent_UN)
 
 lemma UN_equiv_class_type2:
   "equiv A1 r1 \<Longrightarrow> equiv A2 r2 \<Longrightarrow> congruent2 r1 r2 f
     \<Longrightarrow> X1 \<in> A1//r1 \<Longrightarrow> X2 \<in> A2//r2
     \<Longrightarrow> (\<And>x1 x2. x1 \<in> A1 \<Longrightarrow> x2 \<in> A2 \<Longrightarrow> f x1 x2 \<in> B)
     \<Longrightarrow> (\<Union>x1 \<in> X1. \<Union>x2 \<in> X2. f x1 x2) \<in> B"
   unfolding quotient_def
   by (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN
                    congruent2_implies_congruent quotientI)
 
 
 lemma UN_UN_split_split_eq:
   "(\<Union>(x1, x2) \<in> X. \<Union>(y1, y2) \<in> Y. A x1 x2 y1 y2) =
     (\<Union>x \<in> X. \<Union>y \<in> Y. (\<lambda>(x1, x2). (\<lambda>(y1, y2). A x1 x2 y1 y2) y) x)"
   \<comment> \<open>Allows a natural expression of binary operators,\<close>
   \<comment> \<open>without explicit calls to \<open>split\<close>\<close>
   by auto
 
 lemma congruent2I:
   "equiv A1 r1 \<Longrightarrow> equiv A2 r2
     \<Longrightarrow> (\<And>y z w. w \<in> A2 \<Longrightarrow> (y,z) \<in> r1 \<Longrightarrow> f y w = f z w)
     \<Longrightarrow> (\<And>y z w. w \<in> A1 \<Longrightarrow> (y,z) \<in> r2 \<Longrightarrow> f w y = f w z)
     \<Longrightarrow> congruent2 r1 r2 f"
   \<comment> \<open>Suggested by John Harrison -- the two subproofs may be\<close>
   \<comment> \<open>\<^emph>\<open>much\<close> simpler than the direct proof.\<close>
   unfolding congruent2_def equiv_def refl_on_def
   by (blast intro: trans)
 
 lemma congruent2_commuteI:
   assumes equivA: "equiv A r"
     and commute: "\<And>y z. y \<in> A \<Longrightarrow> z \<in> A \<Longrightarrow> f y z = f z y"
     and congt: "\<And>y z w. w \<in> A \<Longrightarrow> (y,z) \<in> r \<Longrightarrow> f w y = f w z"
   shows "f respects2 r"
 proof (rule congruent2I [OF equivA equivA])
   note eqv = equivA [THEN equiv_type, THEN subsetD, THEN SigmaE2]
   show "\<And>y z w. \<lbrakk>w \<in> A; (y, z) \<in> r\<rbrakk> \<Longrightarrow> f y w = f z w"
     by (iprover intro: commute [THEN trans] sym congt elim: eqv)
   show "\<And>y z w. \<lbrakk>w \<in> A; (y, z) \<in> r\<rbrakk> \<Longrightarrow> f w y = f w z"
     by (iprover intro: congt elim: eqv)
 qed
 
 
 subsection \<open>Quotients and finiteness\<close>
 
 text \<open>Suggested by Florian Kammüller\<close>
 
 lemma finite_quotient:
   assumes "finite A" "r \<subseteq> A \<times> A"
   shows "finite (A//r)"
     \<comment> \<open>recall @{thm equiv_type}\<close>
 proof -
   have "A//r \<subseteq> Pow A"
     using assms unfolding quotient_def by blast
   moreover have "finite (Pow A)"
     using assms by simp
   ultimately show ?thesis
     by (iprover intro: finite_subset)
 qed
 
 lemma finite_equiv_class: "finite A \<Longrightarrow> r \<subseteq> A \<times> A \<Longrightarrow> X \<in> A//r \<Longrightarrow> finite X"
   unfolding quotient_def
   by (erule rev_finite_subset) blast
 
 lemma equiv_imp_dvd_card:
   assumes "finite A" "equiv A r" "\<And>X. X \<in> A//r \<Longrightarrow> k dvd card X"
   shows "k dvd card A"
 proof (rule Union_quotient [THEN subst])
   show "k dvd card (\<Union> (A // r))"
     apply (rule dvd_partition)
     using assms
     by (auto simp: Union_quotient dest: quotient_disj)
 qed (use assms in blast)
 
 lemma card_quotient_disjoint:
   assumes "finite A" "inj_on (\<lambda>x. {x} // r) A"
   shows "card (A//r) = card A"
 proof -
   have "\<forall>i\<in>A. \<forall>j\<in>A. i \<noteq> j \<longrightarrow> r `` {j} \<noteq> r `` {i}"
     using assms by (fastforce simp add: quotient_def inj_on_def)
   with assms show ?thesis
     by (simp add: quotient_def card_UN_disjoint)
 qed
 
 text \<open>By Jakub Kądziołka:\<close>
 
 lemma sum_fun_comp:
   assumes "finite S" "finite R" "g ` S \<subseteq> R"
   shows "(\<Sum>x \<in> S. f (g x)) = (\<Sum>y \<in> R. of_nat (card {x \<in> S. g x = y}) * f y)"
 proof -
   let ?r = "relation_of (\<lambda>p q. g p = g q) S"
   have eqv: "equiv S ?r"
     unfolding relation_of_def by (auto intro: comp_equivI)
   have finite: "C \<in> S//?r \<Longrightarrow> finite C" for C
     by (fact finite_equiv_class[OF `finite S` equiv_type[OF `equiv S ?r`]])
   have disjoint: "A \<in> S//?r \<Longrightarrow> B \<in> S//?r \<Longrightarrow> A \<noteq> B \<Longrightarrow> A \<inter> B = {}" for A B
     using eqv quotient_disj by blast
 
   let ?cls = "\<lambda>y. {x \<in> S. y = g x}"
   have quot_as_img: "S//?r = ?cls ` g ` S"
     by (auto simp add: relation_of_def quotient_def)
   have cls_inj: "inj_on ?cls (g ` S)"
     by (auto intro: inj_onI)
 
   have rest_0: "(\<Sum>y \<in> R - g ` S. of_nat (card (?cls y)) * f y) = 0"
   proof -
     have "of_nat (card (?cls y)) * f y = 0" if asm: "y \<in> R - g ` S" for y
     proof -
       from asm have *: "?cls y = {}" by auto
       show ?thesis unfolding * by simp
     qed
     thus ?thesis by simp
   qed
 
   have "(\<Sum>x \<in> S. f (g x)) = (\<Sum>C \<in> S//?r. \<Sum>x \<in> C. f (g x))"
     using eqv finite disjoint
     by (simp flip: sum.Union_disjoint[simplified] add: Union_quotient)
   also have "... = (\<Sum>y \<in> g ` S. \<Sum>x \<in> ?cls y. f (g x))"
     unfolding quot_as_img by (simp add: sum.reindex[OF cls_inj])
   also have "... = (\<Sum>y \<in> g ` S. \<Sum>x \<in> ?cls y. f y)"
     by auto
   also have "... = (\<Sum>y \<in> g ` S. of_nat (card (?cls y)) * f y)"
     by (simp flip: sum_constant)
   also have "... = (\<Sum>y \<in> R. of_nat (card (?cls y)) * f y)"
     using rest_0 by (simp add: sum.subset_diff[OF \<open>g ` S \<subseteq> R\<close> \<open>finite R\<close>])
   finally show ?thesis
     by (simp add: eq_commute)
 qed
 
 subsection \<open>Projection\<close>
 
 definition proj :: "('b \<times> 'a) set \<Rightarrow> 'b \<Rightarrow> 'a set"
   where "proj r x = r `` {x}"
 
 lemma proj_preserves: "x \<in> A \<Longrightarrow> proj r x \<in> A//r"
   unfolding proj_def by (rule quotientI)
 
 lemma proj_in_iff:
   assumes "equiv A r"
   shows "proj r x \<in> A//r \<longleftrightarrow> x \<in> A"
     (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   assume ?rhs
   then show ?lhs by (simp add: proj_preserves)
 next
   assume ?lhs
   then show ?rhs
     unfolding proj_def quotient_def
   proof clarsimp
     fix y
     assume y: "y \<in> A" and "r `` {x} = r `` {y}"
     moreover have "y \<in> r `` {y}"
       using assms y unfolding equiv_def refl_on_def by blast
     ultimately have "(x, y) \<in> r" by blast
     then show "x \<in> A"
       using assms unfolding equiv_def refl_on_def by blast
   qed
 qed
 
 lemma proj_iff: "equiv A r \<Longrightarrow> {x, y} \<subseteq> A \<Longrightarrow> proj r x = proj r y \<longleftrightarrow> (x, y) \<in> r"
   by (simp add: proj_def eq_equiv_class_iff)
 
 (*
 lemma in_proj: "\<lbrakk>equiv A r; x \<in> A\<rbrakk> \<Longrightarrow> x \<in> proj r x"
 unfolding proj_def equiv_def refl_on_def by blast
 *)
 
 lemma proj_image: "proj r ` A = A//r"
   unfolding proj_def[abs_def] quotient_def by blast
 
 lemma in_quotient_imp_non_empty: "equiv A r \<Longrightarrow> X \<in> A//r \<Longrightarrow> X \<noteq> {}"
   unfolding quotient_def using equiv_class_self by fast
 
 lemma in_quotient_imp_in_rel: "equiv A r \<Longrightarrow> X \<in> A//r \<Longrightarrow> {x, y} \<subseteq> X \<Longrightarrow> (x, y) \<in> r"
   using quotient_eq_iff[THEN iffD1] by fastforce
 
 lemma in_quotient_imp_closed: "equiv A r \<Longrightarrow> X \<in> A//r \<Longrightarrow> x \<in> X \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> y \<in> X"
   unfolding quotient_def equiv_def trans_def by blast
 
 lemma in_quotient_imp_subset: "equiv A r \<Longrightarrow> X \<in> A//r \<Longrightarrow> X \<subseteq> A"
   using in_quotient_imp_in_rel equiv_type by fastforce
 
 
 subsection \<open>Equivalence relations -- predicate version\<close>
 
 text \<open>Partial equivalences.\<close>
 
 definition part_equivp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
   where "part_equivp R \<longleftrightarrow> (\<exists>x. R x x) \<and> (\<forall>x y. R x y \<longleftrightarrow> R x x \<and> R y y \<and> R x = R y)"
     \<comment> \<open>John-Harrison-style characterization\<close>
 
 lemma part_equivpI: "\<exists>x. R x x \<Longrightarrow> symp R \<Longrightarrow> transp R \<Longrightarrow> part_equivp R"
   by (auto simp add: part_equivp_def) (auto elim: sympE transpE)
 
 lemma part_equivpE:
   assumes "part_equivp R"
   obtains x where "R x x" and "symp R" and "transp R"
 proof -
   from assms have 1: "\<exists>x. R x x"
     and 2: "\<And>x y. R x y \<longleftrightarrow> R x x \<and> R y y \<and> R x = R y"
     unfolding part_equivp_def by blast+
   from 1 obtain x where "R x x" ..
   moreover have "symp R"
   proof (rule sympI)
     fix x y
     assume "R x y"
     with 2 [of x y] show "R y x" by auto
   qed
   moreover have "transp R"
   proof (rule transpI)
     fix x y z
     assume "R x y" and "R y z"
     with 2 [of x y] 2 [of y z] show "R x z" by auto
   qed
   ultimately show thesis by (rule that)
 qed
 
 lemma part_equivp_refl_symp_transp: "part_equivp R \<longleftrightarrow> (\<exists>x. R x x) \<and> symp R \<and> transp R"
   by (auto intro: part_equivpI elim: part_equivpE)
 
 lemma part_equivp_symp: "part_equivp R \<Longrightarrow> R x y \<Longrightarrow> R y x"
   by (erule part_equivpE, erule sympE)
 
 lemma part_equivp_transp: "part_equivp R \<Longrightarrow> R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
   by (erule part_equivpE, erule transpE)
 
 lemma part_equivp_typedef: "part_equivp R \<Longrightarrow> \<exists>d. d \<in> {c. \<exists>x. R x x \<and> c = Collect (R x)}"
   by (auto elim: part_equivpE)
 
 
 text \<open>Total equivalences.\<close>
 
 definition equivp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
   where "equivp R \<longleftrightarrow> (\<forall>x y. R x y = (R x = R y))" \<comment> \<open>John-Harrison-style characterization\<close>
 
 lemma equivpI: "reflp R \<Longrightarrow> symp R \<Longrightarrow> transp R \<Longrightarrow> equivp R"
   by (auto elim: reflpE sympE transpE simp add: equivp_def)
 
 lemma equivpE:
   assumes "equivp R"
   obtains "reflp R" and "symp R" and "transp R"
   using assms by (auto intro!: that reflpI sympI transpI simp add: equivp_def)
 
 lemma equivp_implies_part_equivp: "equivp R \<Longrightarrow> part_equivp R"
   by (auto intro: part_equivpI elim: equivpE reflpE)
 
 lemma equivp_equiv: "equiv UNIV A \<longleftrightarrow> equivp (\<lambda>x y. (x, y) \<in> A)"
   by (auto intro!: equivI equivpI [to_set] elim!: equivE equivpE [to_set])
 
 lemma equivp_reflp_symp_transp: "equivp R \<longleftrightarrow> reflp R \<and> symp R \<and> transp R"
   by (auto intro: equivpI elim: equivpE)
 
 lemma identity_equivp: "equivp (=)"
   by (auto intro: equivpI reflpI sympI transpI)
 
 lemma equivp_reflp: "equivp R \<Longrightarrow> R x x"
   by (erule equivpE, erule reflpE)
 
 lemma equivp_symp: "equivp R \<Longrightarrow> R x y \<Longrightarrow> R y x"
   by (erule equivpE, erule sympE)
 
 lemma equivp_transp: "equivp R \<Longrightarrow> R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
   by (erule equivpE, erule transpE)
 
 lemma equivp_rtranclp: "symp r \<Longrightarrow> equivp r\<^sup>*\<^sup>*"
   by(intro equivpI reflpI sympI transpI)(auto dest: sympD[OF symp_rtranclp])
 
 lemmas equivp_rtranclp_symclp [simp] = equivp_rtranclp[OF symp_symclp]
 
 lemma equivp_vimage2p: "equivp R \<Longrightarrow> equivp (vimage2p f f R)"
   by(auto simp add: equivp_def vimage2p_def dest: fun_cong)
 
 lemma equivp_imp_transp: "equivp R \<Longrightarrow> transp R"
   by(simp add: equivp_reflp_symp_transp)
 
 
 subsection \<open>Equivalence closure\<close>
 
 definition equivclp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" where
   "equivclp r = (symclp r)\<^sup>*\<^sup>*"
 
 lemma transp_equivclp [simp]: "transp (equivclp r)"
   by(simp add: equivclp_def)
 
 lemma reflp_equivclp [simp]: "reflp (equivclp r)"
   by(simp add: equivclp_def)
 
 lemma symp_equivclp [simp]: "symp (equivclp r)"
   by(simp add: equivclp_def)
 
 lemma equivp_evquivclp [simp]: "equivp (equivclp r)"
   by(simp add: equivpI)
 
 lemma tranclp_equivclp [simp]: "(equivclp r)\<^sup>+\<^sup>+ = equivclp r"
   by(simp add: equivclp_def)
 
 lemma rtranclp_equivclp [simp]: "(equivclp r)\<^sup>*\<^sup>* = equivclp r"
   by(simp add: equivclp_def)
 
 lemma symclp_equivclp [simp]: "symclp (equivclp r) = equivclp r"
   by(simp add: equivclp_def symp_symclp_eq)
 
 lemma equivclp_symclp [simp]: "equivclp (symclp r) = equivclp r"
   by(simp add: equivclp_def)
 
 lemma equivclp_conversep [simp]: "equivclp (conversep r) = equivclp r"
   by(simp add: equivclp_def)
 
 lemma equivclp_sym [sym]: "equivclp r x y \<Longrightarrow> equivclp r y x"
   by(rule sympD[OF symp_equivclp])
 
 lemma equivclp_OO_equivclp_le_equivclp: "equivclp r OO equivclp r \<le> equivclp r"
   by(rule transp_relcompp_less_eq transp_equivclp)+
 
 lemma rtranlcp_le_equivclp: "r\<^sup>*\<^sup>* \<le> equivclp r"
   unfolding equivclp_def by(rule rtranclp_mono)(simp add: symclp_pointfree)
 
 lemma rtranclp_conversep_le_equivclp: "r\<inverse>\<inverse>\<^sup>*\<^sup>* \<le> equivclp r"
   unfolding equivclp_def by(rule rtranclp_mono)(simp add: symclp_pointfree)
 
 lemma symclp_rtranclp_le_equivclp: "symclp r\<^sup>*\<^sup>* \<le> equivclp r"
   unfolding symclp_pointfree
   by(rule le_supI)(simp_all add: rtranclp_conversep[symmetric] rtranlcp_le_equivclp rtranclp_conversep_le_equivclp)
 
 lemma r_OO_conversep_into_equivclp:
   "r\<^sup>*\<^sup>* OO r\<inverse>\<inverse>\<^sup>*\<^sup>* \<le> equivclp r"
   by(blast intro: order_trans[OF _ equivclp_OO_equivclp_le_equivclp] relcompp_mono rtranlcp_le_equivclp rtranclp_conversep_le_equivclp del: predicate2I)
 
 lemma equivclp_induct [consumes 1, case_names base step, induct pred: equivclp]:
   assumes a: "equivclp r a b"
     and cases: "P a" "\<And>y z. equivclp r a y \<Longrightarrow> r y z \<or> r z y \<Longrightarrow> P y \<Longrightarrow> P z"
   shows "P b"
   using a unfolding equivclp_def
   by(induction rule: rtranclp_induct; fold equivclp_def; blast intro: cases elim: symclpE)
 
 lemma converse_equivclp_induct [consumes 1, case_names base step]:
   assumes major: "equivclp r a b"
     and cases: "P b" "\<And>y z. r y z \<or> r z y \<Longrightarrow> equivclp r z b \<Longrightarrow> P z \<Longrightarrow> P y"
   shows "P a"
   using major unfolding equivclp_def
   by(induction rule: converse_rtranclp_induct; fold equivclp_def; blast intro: cases elim: symclpE)
 
 lemma equivclp_refl [simp]: "equivclp r x x"
   by(rule reflpD[OF reflp_equivclp])
 
 lemma r_into_equivclp [intro]: "r x y \<Longrightarrow> equivclp r x y"
   unfolding equivclp_def by(blast intro: symclpI)
 
 lemma converse_r_into_equivclp [intro]: "r y x \<Longrightarrow> equivclp r x y"
   unfolding equivclp_def by(blast intro: symclpI)
 
 lemma rtranclp_into_equivclp: "r\<^sup>*\<^sup>* x y \<Longrightarrow> equivclp r x y"
   using rtranlcp_le_equivclp[of r] by blast
 
 lemma converse_rtranclp_into_equivclp: "r\<^sup>*\<^sup>* y x \<Longrightarrow> equivclp r x y"
   by(blast intro: equivclp_sym rtranclp_into_equivclp)
 
 lemma equivclp_into_equivclp: "\<lbrakk> equivclp r a b; r b c \<or> r c b \<rbrakk> \<Longrightarrow> equivclp r a c"
   unfolding equivclp_def by(erule rtranclp.rtrancl_into_rtrancl)(auto intro: symclpI)
 
 lemma equivclp_trans [trans]: "\<lbrakk> equivclp r a b; equivclp r b c \<rbrakk> \<Longrightarrow> equivclp r a c"
   using equivclp_OO_equivclp_le_equivclp[of r] by blast
 
 hide_const (open) proj
 
 end
diff --git a/src/HOL/Library/Disjoint_Sets.thy b/src/HOL/Library/Disjoint_Sets.thy
--- a/src/HOL/Library/Disjoint_Sets.thy
+++ b/src/HOL/Library/Disjoint_Sets.thy
@@ -1,499 +1,598 @@
 (*  Author:     Johannes Hölzl, TU München
 *)
 
 section \<open>Partitions and Disjoint Sets\<close>
 
 theory Disjoint_Sets
-  imports Main
+  imports FuncSet
 begin
 
-lemma range_subsetD: "range f \<subseteq> B \<Longrightarrow> f i \<in> B"
-  by blast
-
-lemma Int_Diff_disjoint: "A \<inter> B \<inter> (A - B) = {}"
-  by blast
-
-lemma Int_Diff_Un: "A \<inter> B \<union> (A - B) = A"
-  by blast
-
-lemma mono_Un: "mono A \<Longrightarrow> (\<Union>i\<le>n. A i) = A n"
+lemma mono_imp_UN_eq_last: "mono A \<Longrightarrow> (\<Union>i\<le>n. A i) = A n"
   unfolding mono_def by auto
 
-lemma disjnt_equiv_class: "equiv A r \<Longrightarrow> disjnt (r``{a}) (r``{b}) \<longleftrightarrow> (a, b) \<notin> r"
-  by (auto dest: equiv_class_self simp: equiv_class_eq_iff disjnt_def)
-
 subsection \<open>Set of Disjoint Sets\<close>
 
 abbreviation disjoint :: "'a set set \<Rightarrow> bool" where "disjoint \<equiv> pairwise disjnt"
 
 lemma disjoint_def: "disjoint A \<longleftrightarrow> (\<forall>a\<in>A. \<forall>b\<in>A. a \<noteq> b \<longrightarrow> a \<inter> b = {})"
   unfolding pairwise_def disjnt_def by auto
 
 lemma disjointI:
   "(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<inter> b = {}) \<Longrightarrow> disjoint A"
   unfolding disjoint_def by auto
 
 lemma disjointD:
   "disjoint A \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<inter> b = {}"
   unfolding disjoint_def by auto
 
 lemma disjoint_image: "inj_on f (\<Union>A) \<Longrightarrow> disjoint A \<Longrightarrow> disjoint ((`) f ` A)"
   unfolding inj_on_def disjoint_def by blast
 
 lemma assumes "disjoint (A \<union> B)"
       shows   disjoint_unionD1: "disjoint A" and disjoint_unionD2: "disjoint B"
   using assms by (simp_all add: disjoint_def)
   
 lemma disjoint_INT:
   assumes *: "\<And>i. i \<in> I \<Longrightarrow> disjoint (F i)"
   shows "disjoint {\<Inter>i\<in>I. X i | X. \<forall>i\<in>I. X i \<in> F i}"
 proof (safe intro!: disjointI del: equalityI)
   fix A B :: "'a \<Rightarrow> 'b set" assume "(\<Inter>i\<in>I. A i) \<noteq> (\<Inter>i\<in>I. B i)"
   then obtain i where "A i \<noteq> B i" "i \<in> I"
     by auto
   moreover assume "\<forall>i\<in>I. A i \<in> F i" "\<forall>i\<in>I. B i \<in> F i"
   ultimately show "(\<Inter>i\<in>I. A i) \<inter> (\<Inter>i\<in>I. B i) = {}"
     using *[OF \<open>i\<in>I\<close>, THEN disjointD, of "A i" "B i"]
     by (auto simp flip: INT_Int_distrib)
 qed
 
 lemma diff_Union_pairwise_disjoint:
   assumes "pairwise disjnt \<A>" "\<B> \<subseteq> \<A>"
   shows "\<Union>\<A> - \<Union>\<B> = \<Union>(\<A> - \<B>)"
 proof -
   have "False"
     if x: "x \<in> A" "x \<in> B" and AB: "A \<in> \<A>" "A \<notin> \<B>" "B \<in> \<B>" for x A B
   proof -
     have "A \<inter> B = {}"
       using assms disjointD AB by blast
   with x show ?thesis
     by blast
   qed
   then show ?thesis by auto
 qed
 
 lemma Int_Union_pairwise_disjoint:
   assumes "pairwise disjnt (\<A> \<union> \<B>)"
   shows "\<Union>\<A> \<inter> \<Union>\<B> = \<Union>(\<A> \<inter> \<B>)"
 proof -
   have "False"
     if x: "x \<in> A" "x \<in> B" and AB: "A \<in> \<A>" "A \<notin> \<B>" "B \<in> \<B>" for x A B
   proof -
     have "A \<inter> B = {}"
       using assms disjointD AB by blast
   with x show ?thesis
     by blast
   qed
   then show ?thesis by auto
 qed
 
 lemma psubset_Union_pairwise_disjoint:
   assumes \<B>: "pairwise disjnt \<B>" and "\<A> \<subset> \<B> - {{}}"
   shows "\<Union>\<A> \<subset> \<Union>\<B>"
   unfolding psubset_eq
 proof
   show "\<Union>\<A> \<subseteq> \<Union>\<B>"
     using assms by blast
   have "\<A> \<subseteq> \<B>" "\<Union>(\<B> - \<A> \<inter> (\<B> - {{}})) \<noteq> {}"
     using assms by blast+
   then show "\<Union>\<A> \<noteq> \<Union>\<B>"
     using diff_Union_pairwise_disjoint [OF \<B>] by blast
 qed
 
 subsubsection "Family of Disjoint Sets"
 
 definition disjoint_family_on :: "('i \<Rightarrow> 'a set) \<Rightarrow> 'i set \<Rightarrow> bool" where
   "disjoint_family_on A S \<longleftrightarrow> (\<forall>m\<in>S. \<forall>n\<in>S. m \<noteq> n \<longrightarrow> A m \<inter> A n = {})"
 
 abbreviation "disjoint_family A \<equiv> disjoint_family_on A UNIV"
 
 lemma disjoint_family_elem_disjnt:
   assumes "infinite A" "finite C"
       and df: "disjoint_family_on B A"
   obtains x where "x \<in> A" "disjnt C (B x)"
 proof -
   have "False" if *: "\<forall>x \<in> A. \<exists>y. y \<in> C \<and> y \<in> B x"
   proof -
     obtain g where g: "\<forall>x \<in> A. g x \<in> C \<and> g x \<in> B x"
       using * by metis
     with df have "inj_on g A"
       by (fastforce simp add: inj_on_def disjoint_family_on_def)
     then have "infinite (g ` A)"
       using \<open>infinite A\<close> finite_image_iff by blast
     then show False
       by (meson \<open>finite C\<close> finite_subset g image_subset_iff)
   qed
   then show ?thesis
     by (force simp: disjnt_iff intro: that)
 qed
 
 lemma disjoint_family_onD:
   "disjoint_family_on A I \<Longrightarrow> i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> i \<noteq> j \<Longrightarrow> A i \<inter> A j = {}"
   by (auto simp: disjoint_family_on_def)
 
 lemma disjoint_family_subset: "disjoint_family A \<Longrightarrow> (\<And>x. B x \<subseteq> A x) \<Longrightarrow> disjoint_family B"
   by (force simp add: disjoint_family_on_def)
 
 lemma disjoint_family_on_bisimulation:
   assumes "disjoint_family_on f S"
   and "\<And>n m. n \<in> S \<Longrightarrow> m \<in> S \<Longrightarrow> n \<noteq> m \<Longrightarrow> f n \<inter> f m = {} \<Longrightarrow> g n \<inter> g m = {}"
   shows "disjoint_family_on g S"
   using assms unfolding disjoint_family_on_def by auto
 
 lemma disjoint_family_on_mono:
   "A \<subseteq> B \<Longrightarrow> disjoint_family_on f B \<Longrightarrow> disjoint_family_on f A"
   unfolding disjoint_family_on_def by auto
 
 lemma disjoint_family_Suc:
   "(\<And>n. A n \<subseteq> A (Suc n)) \<Longrightarrow> disjoint_family (\<lambda>i. A (Suc i) - A i)"
   using lift_Suc_mono_le[of A]
   by (auto simp add: disjoint_family_on_def)
      (metis insert_absorb insert_subset le_SucE le_antisym not_le_imp_less less_imp_le)
 
 lemma disjoint_family_on_disjoint_image:
   "disjoint_family_on A I \<Longrightarrow> disjoint (A ` I)"
   unfolding disjoint_family_on_def disjoint_def by force
  
 lemma disjoint_family_on_vimageI: "disjoint_family_on F I \<Longrightarrow> disjoint_family_on (\<lambda>i. f -` F i) I"
   by (auto simp: disjoint_family_on_def)
 
 lemma disjoint_image_disjoint_family_on:
   assumes d: "disjoint (A ` I)" and i: "inj_on A I"
   shows "disjoint_family_on A I"
   unfolding disjoint_family_on_def
 proof (intro ballI impI)
   fix n m assume nm: "m \<in> I" "n \<in> I" and "n \<noteq> m"
   with i[THEN inj_onD, of n m] show "A n \<inter> A m = {}"
     by (intro disjointD[OF d]) auto
 qed
 
 lemma disjoint_family_on_iff_disjoint_image:
   assumes "\<And>i. i \<in> I \<Longrightarrow> A i \<noteq> {}"
   shows "disjoint_family_on A I \<longleftrightarrow> disjoint (A ` I) \<and> inj_on A I"
 proof
   assume "disjoint_family_on A I"
   then show "disjoint (A ` I) \<and> inj_on A I"
     by (metis (mono_tags, lifting) assms disjoint_family_onD disjoint_family_on_disjoint_image inf.idem inj_onI)
 qed (use disjoint_image_disjoint_family_on in metis)
 
 lemma card_UN_disjoint':
   assumes "disjoint_family_on A I" "\<And>i. i \<in> I \<Longrightarrow> finite (A i)" "finite I"
   shows "card (\<Union>i\<in>I. A i) = (\<Sum>i\<in>I. card (A i))"
   using assms   by (simp add: card_UN_disjoint disjoint_family_on_def)
 
 lemma disjoint_UN:
   assumes F: "\<And>i. i \<in> I \<Longrightarrow> disjoint (F i)" and *: "disjoint_family_on (\<lambda>i. \<Union>(F i)) I"
   shows "disjoint (\<Union>i\<in>I. F i)"
 proof (safe intro!: disjointI del: equalityI)
   fix A B i j assume "A \<noteq> B" "A \<in> F i" "i \<in> I" "B \<in> F j" "j \<in> I"
   show "A \<inter> B = {}"
   proof cases
     assume "i = j" with F[of i] \<open>i \<in> I\<close> \<open>A \<in> F i\<close> \<open>B \<in> F j\<close> \<open>A \<noteq> B\<close> show "A \<inter> B = {}"
       by (auto dest: disjointD)
   next
     assume "i \<noteq> j"
     with * \<open>i\<in>I\<close> \<open>j\<in>I\<close> have "(\<Union>(F i)) \<inter> (\<Union>(F j)) = {}"
       by (rule disjoint_family_onD)
     with \<open>A\<in>F i\<close> \<open>i\<in>I\<close> \<open>B\<in>F j\<close> \<open>j\<in>I\<close>
     show "A \<inter> B = {}"
       by auto
   qed
 qed
 
 lemma distinct_list_bind: 
   assumes "distinct xs" "\<And>x. x \<in> set xs \<Longrightarrow> distinct (f x)" 
           "disjoint_family_on (set \<circ> f) (set xs)"
   shows   "distinct (List.bind xs f)"
   using assms
   by (induction xs)
      (auto simp: disjoint_family_on_def distinct_map inj_on_def set_list_bind)
 
 lemma bij_betw_UNION_disjoint:
   assumes disj: "disjoint_family_on A' I"
   assumes bij: "\<And>i. i \<in> I \<Longrightarrow> bij_betw f (A i) (A' i)"
   shows   "bij_betw f (\<Union>i\<in>I. A i) (\<Union>i\<in>I. A' i)"
 unfolding bij_betw_def
 proof
   from bij show eq: "f ` \<Union>(A ` I) = \<Union>(A' ` I)"
     by (auto simp: bij_betw_def image_UN)
   show "inj_on f (\<Union>(A ` I))"
   proof (rule inj_onI, clarify)
     fix i j x y assume A: "i \<in> I" "j \<in> I" "x \<in> A i" "y \<in> A j" and B: "f x = f y"
     from A bij[of i] bij[of j] have "f x \<in> A' i" "f y \<in> A' j"
       by (auto simp: bij_betw_def)
     with B have "A' i \<inter> A' j \<noteq> {}" by auto
     with disj A have "i = j" unfolding disjoint_family_on_def by blast
     with A B bij[of i] show "x = y" by (auto simp: bij_betw_def dest: inj_onD)
   qed
 qed
 
 lemma disjoint_union: "disjoint C \<Longrightarrow> disjoint B \<Longrightarrow> \<Union>C \<inter> \<Union>B = {} \<Longrightarrow> disjoint (C \<union> B)"
   using disjoint_UN[of "{C, B}" "\<lambda>x. x"] by (auto simp add: disjoint_family_on_def)
 
 text \<open>
   Sum/product of the union of a finite disjoint family
 \<close>
 context comm_monoid_set
 begin
 
 lemma UNION_disjoint_family:
   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
     and "disjoint_family_on A I"
   shows "F g (\<Union>(A ` I)) = F (\<lambda>x. F g (A x)) I"
   using assms unfolding disjoint_family_on_def  by (rule UNION_disjoint)
 
 lemma Union_disjoint_sets:
   assumes "\<forall>A\<in>C. finite A" and "disjoint C"
   shows "F g (\<Union>C) = (F \<circ> F) g C"
   using assms unfolding disjoint_def  by (rule Union_disjoint)
 
 end
 
 text \<open>
   The union of an infinite disjoint family of non-empty sets is infinite.
 \<close>
 lemma infinite_disjoint_family_imp_infinite_UNION:
   assumes "\<not>finite A" "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> {}" "disjoint_family_on f A"
   shows   "\<not>finite (\<Union>(f ` A))"
 proof -
   define g where "g x = (SOME y. y \<in> f x)" for x
   have g: "g x \<in> f x" if "x \<in> A" for x
     unfolding g_def by (rule someI_ex, insert assms(2) that) blast
   have inj_on_g: "inj_on g A"
   proof (rule inj_onI, rule ccontr)
     fix x y assume A: "x \<in> A" "y \<in> A" "g x = g y" "x \<noteq> y"
     with g[of x] g[of y] have "g x \<in> f x" "g x \<in> f y" by auto
     with A \<open>x \<noteq> y\<close> assms show False
       by (auto simp: disjoint_family_on_def inj_on_def)
   qed
   from g have "g ` A \<subseteq> \<Union>(f ` A)" by blast
   moreover from inj_on_g \<open>\<not>finite A\<close> have "\<not>finite (g ` A)"
     using finite_imageD by blast
   ultimately show ?thesis using finite_subset by blast
 qed  
   
 
 subsection \<open>Construct Disjoint Sequences\<close>
 
 definition disjointed :: "(nat \<Rightarrow> 'a set) \<Rightarrow> nat \<Rightarrow> 'a set" where
   "disjointed A n = A n - (\<Union>i\<in>{0..<n}. A i)"
 
 lemma finite_UN_disjointed_eq: "(\<Union>i\<in>{0..<n}. disjointed A i) = (\<Union>i\<in>{0..<n}. A i)"
 proof (induct n)
   case 0 show ?case by simp
 next
   case (Suc n)
   thus ?case by (simp add: atLeastLessThanSuc disjointed_def)
 qed
 
 lemma UN_disjointed_eq: "(\<Union>i. disjointed A i) = (\<Union>i. A i)"
   by (rule UN_finite2_eq [where k=0])
      (simp add: finite_UN_disjointed_eq)
 
 lemma less_disjoint_disjointed: "m < n \<Longrightarrow> disjointed A m \<inter> disjointed A n = {}"
   by (auto simp add: disjointed_def)
 
 lemma disjoint_family_disjointed: "disjoint_family (disjointed A)"
   by (simp add: disjoint_family_on_def)
      (metis neq_iff Int_commute less_disjoint_disjointed)
 
 lemma disjointed_subset: "disjointed A n \<subseteq> A n"
   by (auto simp add: disjointed_def)
 
 lemma disjointed_0[simp]: "disjointed A 0 = A 0"
   by (simp add: disjointed_def)
 
 lemma disjointed_mono: "mono A \<Longrightarrow> disjointed A (Suc n) = A (Suc n) - A n"
-  using mono_Un[of A] by (simp add: disjointed_def atLeastLessThanSuc_atLeastAtMost atLeast0AtMost)
+  using mono_imp_UN_eq_last[of A] by (simp add: disjointed_def atLeastLessThanSuc_atLeastAtMost atLeast0AtMost)
 
 subsection \<open>Partitions\<close>
 
 text \<open>
   Partitions \<^term>\<open>P\<close> of a set \<^term>\<open>A\<close>. We explicitly disallow empty sets.
 \<close>
 
 definition partition_on :: "'a set \<Rightarrow> 'a set set \<Rightarrow> bool"
 where
   "partition_on A P \<longleftrightarrow> \<Union>P = A \<and> disjoint P \<and> {} \<notin> P"
 
 lemma partition_onI:
   "\<Union>P = A \<Longrightarrow> (\<And>p q. p \<in> P \<Longrightarrow> q \<in> P \<Longrightarrow> p \<noteq> q \<Longrightarrow> disjnt p q) \<Longrightarrow> {} \<notin> P \<Longrightarrow> partition_on A P"
   by (auto simp: partition_on_def pairwise_def)
 
 lemma partition_onD1: "partition_on A P \<Longrightarrow> A = \<Union>P"
   by (auto simp: partition_on_def)
 
 lemma partition_onD2: "partition_on A P \<Longrightarrow> disjoint P"
   by (auto simp: partition_on_def)
 
 lemma partition_onD3: "partition_on A P \<Longrightarrow> {} \<notin> P"
   by (auto simp: partition_on_def)
 
 subsection \<open>Constructions of partitions\<close>
 
 lemma partition_on_empty: "partition_on {} P \<longleftrightarrow> P = {}"
   unfolding partition_on_def by fastforce
 
 lemma partition_on_space: "A \<noteq> {} \<Longrightarrow> partition_on A {A}"
   by (auto simp: partition_on_def disjoint_def)
 
 lemma partition_on_singletons: "partition_on A ((\<lambda>x. {x}) ` A)"
   by (auto simp: partition_on_def disjoint_def)
 
 lemma partition_on_transform:
   assumes P: "partition_on A P"
   assumes F_UN: "\<Union>(F ` P) = F (\<Union>P)" and F_disjnt: "\<And>p q. p \<in> P \<Longrightarrow> q \<in> P \<Longrightarrow> disjnt p q \<Longrightarrow> disjnt (F p) (F q)"
   shows "partition_on (F A) (F ` P - {{}})"
 proof -
   have "\<Union>(F ` P - {{}}) = F A"
     unfolding P[THEN partition_onD1] F_UN[symmetric] by auto
   with P show ?thesis
     by (auto simp add: partition_on_def pairwise_def intro!: F_disjnt)
 qed
 
 lemma partition_on_restrict: "partition_on A P \<Longrightarrow> partition_on (B \<inter> A) ((\<inter>) B ` P - {{}})"
   by (intro partition_on_transform) (auto simp: disjnt_def)
 
 lemma partition_on_vimage: "partition_on A P \<Longrightarrow> partition_on (f -` A) ((-`) f ` P - {{}})"
   by (intro partition_on_transform) (auto simp: disjnt_def)
 
 lemma partition_on_inj_image:
   assumes P: "partition_on A P" and f: "inj_on f A"
   shows "partition_on (f ` A) ((`) f ` P - {{}})"
 proof (rule partition_on_transform[OF P])
   show "p \<in> P \<Longrightarrow> q \<in> P \<Longrightarrow> disjnt p q \<Longrightarrow> disjnt (f ` p) (f ` q)" for p q
     using f[THEN inj_onD] P[THEN partition_onD1] by (auto simp: disjnt_def)
 qed auto
 
 lemma partition_on_insert:
   assumes "disjnt p (\<Union>P)"
   shows "partition_on A (insert p P) \<longleftrightarrow> partition_on (A-p) P \<and> p \<subseteq> A \<and> p \<noteq> {}"
   using assms
   by (auto simp: partition_on_def disjnt_iff pairwise_insert)
 
 subsection \<open>Finiteness of partitions\<close>
 
 lemma finitely_many_partition_on:
   assumes "finite A"
   shows "finite {P. partition_on A P}"
 proof (rule finite_subset)
   show "{P. partition_on A P} \<subseteq> Pow (Pow A)"
     unfolding partition_on_def by auto
   show "finite (Pow (Pow A))"
     using assms by simp
 qed
 
 lemma finite_elements: "finite A \<Longrightarrow> partition_on A P \<Longrightarrow> finite P"
   using partition_onD1[of A P] by (simp add: finite_UnionD)
 
 lemma product_partition:
   assumes "partition_on A P" and "\<And>p. p \<in> P \<Longrightarrow> finite p" 
   shows "card A = (\<Sum>p\<in>P. card p)"
   using assms unfolding partition_on_def  by (meson card_Union_disjoint)
 
 subsection \<open>Equivalence of partitions and equivalence classes\<close>
 
 lemma partition_on_quotient:
   assumes r: "equiv A r"
   shows "partition_on A (A // r)"
 proof (rule partition_onI)
   from r have "refl_on A r"
     by (auto elim: equivE)
   then show "\<Union>(A // r) = A" "{} \<notin> A // r"
     by (auto simp: refl_on_def quotient_def)
 
   fix p q assume "p \<in> A // r" "q \<in> A // r" "p \<noteq> q"
   then obtain x y where "x \<in> A" "y \<in> A" "p = r `` {x}" "q = r `` {y}"
     by (auto simp: quotient_def)
   with r equiv_class_eq_iff[OF r, of x y] \<open>p \<noteq> q\<close> show "disjnt p q"
     by (auto simp: disjnt_equiv_class)
 qed
 
 lemma equiv_partition_on:
   assumes P: "partition_on A P"
   shows "equiv A {(x, y). \<exists>p \<in> P. x \<in> p \<and> y \<in> p}"
 proof (rule equivI)
   have "A = \<Union>P" "disjoint P" "{} \<notin> P"
     using P by (auto simp: partition_on_def)
   then show "refl_on A {(x, y). \<exists>p\<in>P. x \<in> p \<and> y \<in> p}"
     unfolding refl_on_def by auto
   show "trans {(x, y). \<exists>p\<in>P. x \<in> p \<and> y \<in> p}"
     using \<open>disjoint P\<close> by (auto simp: trans_def disjoint_def)
 qed (auto simp: sym_def)
 
 lemma partition_on_eq_quotient:
   assumes P: "partition_on A P"
   shows "A // {(x, y). \<exists>p \<in> P. x \<in> p \<and> y \<in> p} = P"
   unfolding quotient_def
 proof safe
   fix x assume "x \<in> A"
   then obtain p where "p \<in> P" "x \<in> p" "\<And>q. q \<in> P \<Longrightarrow> x \<in> q \<Longrightarrow> p = q"
     using P by (auto simp: partition_on_def disjoint_def)
   then have "{y. \<exists>p\<in>P. x \<in> p \<and> y \<in> p} = p"
     by (safe intro!: bexI[of _ p]) simp
   then show "{(x, y). \<exists>p\<in>P. x \<in> p \<and> y \<in> p} `` {x} \<in> P"
     by (simp add: \<open>p \<in> P\<close>)
 next
   fix p assume "p \<in> P"
   then have "p \<noteq> {}"
     using P by (auto simp: partition_on_def)
   then obtain x where "x \<in> p"
     by auto
   then have "x \<in> A" "\<And>q. q \<in> P \<Longrightarrow> x \<in> q \<Longrightarrow> p = q"
     using P \<open>p \<in> P\<close> by (auto simp: partition_on_def disjoint_def)
   with \<open>p\<in>P\<close> \<open>x \<in> p\<close> have "{y. \<exists>p\<in>P. x \<in> p \<and> y \<in> p} = p"
     by (safe intro!: bexI[of _ p]) simp
   then show "p \<in> (\<Union>x\<in>A. {{(x, y). \<exists>p\<in>P. x \<in> p \<and> y \<in> p} `` {x}})"
     by (auto intro: \<open>x \<in> A\<close>)
 qed
 
 lemma partition_on_alt: "partition_on A P \<longleftrightarrow> (\<exists>r. equiv A r \<and> P = A // r)"
   by (auto simp: partition_on_eq_quotient intro!: partition_on_quotient intro: equiv_partition_on)
 
 subsection \<open>Refinement of partitions\<close>
 
 definition refines :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set \<Rightarrow> bool"
   where "refines A P Q \<equiv>
         partition_on A P \<and> partition_on A Q \<and> (\<forall>X\<in>P. \<exists>Y\<in>Q. X \<subseteq> Y)" 
 
 lemma refines_refl: "partition_on A P \<Longrightarrow> refines A P P"
   using refines_def by blast
 
 lemma refines_asym1:
   assumes "refines A P Q" "refines A Q P"
   shows "P \<subseteq> Q"
 proof 
   fix X
   assume "X \<in> P"
   then obtain Y X' where "Y \<in> Q" "X \<subseteq> Y" "X' \<in> P" "Y \<subseteq> X'"
     by (meson assms refines_def)
   then have "X' = X"
     using assms(2) unfolding partition_on_def refines_def
     by (metis \<open>X \<in> P\<close> \<open>X \<subseteq> Y\<close> disjnt_self_iff_empty disjnt_subset1 pairwiseD)
   then show "X \<in> Q"
     using \<open>X \<subseteq> Y\<close> \<open>Y \<in> Q\<close> \<open>Y \<subseteq> X'\<close> by force
 qed
 
 lemma refines_asym: "\<lbrakk>refines A P Q; refines A Q P\<rbrakk> \<Longrightarrow> P=Q"
   by (meson antisym_conv refines_asym1)
 
 lemma refines_trans: "\<lbrakk>refines A P Q; refines A Q R\<rbrakk> \<Longrightarrow> refines A P R"
   by (meson order.trans refines_def)
 
 lemma refines_obtains_subset:
   assumes "refines A P Q" "q \<in> Q"
   shows "partition_on q {p \<in> P. p \<subseteq> q}"
 proof -
   have "p \<subseteq> q \<or> disjnt p q" if "p \<in> P" for p
     using that assms unfolding refines_def partition_on_def disjoint_def
     by (metis disjnt_def disjnt_subset1)
   with assms have "q \<subseteq> Union {p \<in> P. p \<subseteq> q}"
     using assms 
    by (clarsimp simp: refines_def disjnt_iff partition_on_def) (metis Union_iff)
   with assms have "q = Union {p \<in> P. p \<subseteq> q}"
     by auto
   then show ?thesis
     using assms by (auto simp: refines_def disjoint_def partition_on_def)
 qed 
 
+subsection \<open>The coarsest common refinement of a set of partitions\<close>
+
+definition common_refinement :: "'a set set set \<Rightarrow> 'a set set"
+  where "common_refinement \<P> \<equiv> (\<Union>f \<in> (\<Pi>\<^sub>E P\<in>\<P>. P). {\<Inter> (f ` \<P>)}) - {{}}" 
+
+text \<open>With non-extensional function space\<close>
+lemma common_refinement: "common_refinement \<P> = (\<Union>f \<in> (\<Pi> P\<in>\<P>. P). {\<Inter> (f ` \<P>)}) - {{}}" 
+  (is "?lhs = ?rhs")
+proof
+  show "?rhs \<subseteq> ?lhs"
+    apply (clarsimp simp add: common_refinement_def PiE_def Ball_def)
+    by (metis restrict_Pi_cancel image_restrict_eq restrict_extensional)
+qed (auto simp add: common_refinement_def PiE_def)
+
+lemma common_refinement_exists: "\<lbrakk>X \<in> common_refinement \<P>; P \<in> \<P>\<rbrakk> \<Longrightarrow> \<exists>R\<in>P. X \<subseteq> R"
+  by (auto simp add: common_refinement)
+
+lemma Union_common_refinement: "\<Union> (common_refinement \<P>) = (\<Inter> P\<in>\<P>. \<Union>P)"
+proof
+  show "(\<Inter> P\<in>\<P>. \<Union>P) \<subseteq> \<Union> (common_refinement \<P>)"
+  proof (clarsimp simp: common_refinement)
+    fix x
+    assume "\<forall>P\<in>\<P>. \<exists>X\<in>P. x \<in> X"
+    then obtain F where F: "\<And>P. P\<in>\<P> \<Longrightarrow> F P \<in> P \<and> x \<in> F P"
+      by metis
+    then have "x \<in> \<Inter> (F ` \<P>)"
+      by force
+    with F show "\<exists>X\<in>(\<Union>x\<in>\<Pi> P\<in>\<P>. P. {\<Inter> (x ` \<P>)}) - {{}}. x \<in> X"
+      by (auto simp add: Pi_iff Bex_def)
+  qed
+qed (auto simp: common_refinement_def)
+
+lemma partition_on_common_refinement:
+  assumes A: "\<And>P. P \<in> \<P> \<Longrightarrow> partition_on A P" and "\<P> \<noteq> {}"
+  shows "partition_on A (common_refinement \<P>)"
+proof (rule partition_onI)
+  show "\<Union> (common_refinement \<P>) = A"
+    using assms by (simp add: partition_on_def Union_common_refinement)
+  fix P Q
+  assume "P \<in> common_refinement \<P>" and "Q \<in> common_refinement \<P>" and "P \<noteq> Q"
+  then obtain f g where f: "f \<in> (\<Pi>\<^sub>E P\<in>\<P>. P)" and P: "P = \<Inter> (f ` \<P>)" and "P \<noteq> {}"
+                  and   g: "g \<in> (\<Pi>\<^sub>E P\<in>\<P>. P)" and Q: "Q = \<Inter> (g ` \<P>)" and "Q \<noteq> {}"
+    by (auto simp add: common_refinement_def)
+  have "f=g" if "x \<in> P" "x \<in> Q" for x
+  proof (rule extensionalityI [of _ \<P>])
+    fix R
+    assume "R \<in> \<P>"
+    with that P Q f g A [unfolded partition_on_def, OF \<open>R \<in> \<P>\<close>]
+    show "f R = g R"
+      by (metis INT_E Int_iff PiE_iff disjointD emptyE)
+  qed (use PiE_iff f g in auto)
+  then show "disjnt P Q"
+    by (metis P Q \<open>P \<noteq> Q\<close> disjnt_iff) 
+qed (simp add: common_refinement_def)
+
+lemma refines_common_refinement:
+  assumes "\<And>P. P \<in> \<P> \<Longrightarrow> partition_on A P" "P \<in> \<P>"
+  shows "refines A (common_refinement \<P>) P"
+  unfolding refines_def
+proof (intro conjI strip)
+  fix X
+  assume "X \<in> common_refinement \<P>"
+  with assms show "\<exists>Y\<in>P. X \<subseteq> Y"
+    by (auto simp: common_refinement_def)
+qed (use assms partition_on_common_refinement in auto)
+
+text \<open>The common refinement is itself refined by any other\<close>
+lemma common_refinement_coarsest:
+  assumes "\<And>P. P \<in> \<P> \<Longrightarrow> partition_on A P" "partition_on A R" "\<And>P. P \<in> \<P> \<Longrightarrow> refines A R P" "\<P> \<noteq> {}"
+  shows "refines A R (common_refinement \<P>)"
+  unfolding refines_def
+proof (intro conjI ballI partition_on_common_refinement)
+  fix X
+  assume "X \<in> R"
+  have "\<exists>p \<in> P. X \<subseteq> p" if "P \<in> \<P>" for P
+    by (meson \<open>X \<in> R\<close> assms(3) refines_def that)
+  then obtain F where f: "\<And>P. P \<in> \<P> \<Longrightarrow> F P \<in> P \<and> X \<subseteq> F P"
+    by metis
+  with \<open>partition_on A R\<close> \<open>X \<in> R\<close> \<open>\<P> \<noteq> {}\<close>
+  have "\<Inter> (F ` \<P>) \<in> common_refinement \<P>"
+    apply (simp add: partition_on_def common_refinement Pi_iff Bex_def)
+    by (metis (no_types, lifting) cINF_greatest subset_empty)
+  with f show "\<exists>Y\<in>common_refinement \<P>. X \<subseteq> Y"
+    by (metis \<open>\<P> \<noteq> {}\<close> cINF_greatest)
+qed (use assms in auto)
+
+lemma finite_common_refinement:
+  assumes "finite \<P>" "\<And>P. P \<in> \<P> \<Longrightarrow> finite P"
+  shows "finite (common_refinement \<P>)"
+proof -
+  have "finite (\<Pi>\<^sub>E P\<in>\<P>. P)"
+    by (simp add: assms finite_PiE)
+  then show ?thesis
+    by (auto simp: common_refinement_def)
+qed
+
+lemma card_common_refinement:
+  assumes "finite \<P>" "\<And>P. P \<in> \<P> \<Longrightarrow> finite P"
+  shows "card (common_refinement \<P>) \<le> (\<Prod>P \<in> \<P>. card P)"
+proof -
+  have "card (common_refinement \<P>) \<le> card (\<Union>f \<in> (\<Pi>\<^sub>E P\<in>\<P>. P). {\<Inter> (f ` \<P>)})"
+    unfolding common_refinement_def by (meson card_Diff1_le)
+  also have "\<dots> \<le> (\<Sum>f\<in>(\<Pi>\<^sub>E P\<in>\<P>. P). card{\<Inter> (f ` \<P>)})"
+    by (metis assms finite_PiE card_UN_le)
+  also have "\<dots> = card(\<Pi>\<^sub>E P\<in>\<P>. P)"
+    by simp
+  also have "\<dots> = (\<Prod>P \<in> \<P>. card P)"
+    by (simp add: assms(1) card_PiE dual_order.eq_iff)
+  finally show ?thesis .
+qed
+
 end
diff --git a/src/HOL/Set.thy b/src/HOL/Set.thy
--- a/src/HOL/Set.thy
+++ b/src/HOL/Set.thy
@@ -1,2041 +1,2050 @@
 (*  Title:      HOL/Set.thy
     Author:     Tobias Nipkow
     Author:     Lawrence C Paulson
     Author:     Markus Wenzel
 *)
 
 section \<open>Set theory for higher-order logic\<close>
 
 theory Set
   imports Lattices Boolean_Algebras
 begin
 
 subsection \<open>Sets as predicates\<close>
 
 typedecl 'a set
 
 axiomatization Collect :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set" \<comment> \<open>comprehension\<close>
   and member :: "'a \<Rightarrow> 'a set \<Rightarrow> bool" \<comment> \<open>membership\<close>
   where mem_Collect_eq [iff, code_unfold]: "member a (Collect P) = P a"
     and Collect_mem_eq [simp]: "Collect (\<lambda>x. member x A) = A"
 
 notation
   member  ("'(\<in>')") and
   member  ("(_/ \<in> _)" [51, 51] 50)
 
 abbreviation not_member
   where "not_member x A \<equiv> \<not> (x \<in> A)" \<comment> \<open>non-membership\<close>
 notation
   not_member  ("'(\<notin>')") and
   not_member  ("(_/ \<notin> _)" [51, 51] 50)
 
 notation (ASCII)
   member  ("'(:')") and
   member  ("(_/ : _)" [51, 51] 50) and
   not_member  ("'(~:')") and
   not_member  ("(_/ ~: _)" [51, 51] 50)
 
 
 text \<open>Set comprehensions\<close>
 
 syntax
   "_Coll" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a set"    ("(1{_./ _})")
 translations
   "{x. P}" \<rightleftharpoons> "CONST Collect (\<lambda>x. P)"
 
 syntax (ASCII)
   "_Collect" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> 'a set"  ("(1{(_/: _)./ _})")
 syntax
   "_Collect" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> 'a set"  ("(1{(_/ \<in> _)./ _})")
 translations
   "{p:A. P}" \<rightharpoonup> "CONST Collect (\<lambda>p. p \<in> A \<and> P)"
 
 lemma CollectI: "P a \<Longrightarrow> a \<in> {x. P x}"
   by simp
 
 lemma CollectD: "a \<in> {x. P x} \<Longrightarrow> P a"
   by simp
 
 lemma Collect_cong: "(\<And>x. P x = Q x) \<Longrightarrow> {x. P x} = {x. Q x}"
   by simp
 
 text \<open>
   Simproc for pulling \<open>x = t\<close> in \<open>{x. \<dots> \<and> x = t \<and> \<dots>}\<close>
   to the front (and similarly for \<open>t = x\<close>):
 \<close>
 
 simproc_setup defined_Collect ("{x. P x \<and> Q x}") = \<open>
   fn _ => Quantifier1.rearrange_Collect
     (fn ctxt =>
       resolve_tac ctxt @{thms Collect_cong} 1 THEN
       resolve_tac ctxt @{thms iffI} 1 THEN
       ALLGOALS
         (EVERY' [REPEAT_DETERM o eresolve_tac ctxt @{thms conjE},
           DEPTH_SOLVE_1 o (assume_tac ctxt ORELSE' resolve_tac ctxt @{thms conjI})]))
 \<close>
 
 lemmas CollectE = CollectD [elim_format]
 
 lemma set_eqI:
   assumes "\<And>x. x \<in> A \<longleftrightarrow> x \<in> B"
   shows "A = B"
 proof -
   from assms have "{x. x \<in> A} = {x. x \<in> B}"
     by simp
   then show ?thesis by simp
 qed
 
 lemma set_eq_iff: "A = B \<longleftrightarrow> (\<forall>x. x \<in> A \<longleftrightarrow> x \<in> B)"
   by (auto intro:set_eqI)
 
 lemma Collect_eqI:
   assumes "\<And>x. P x = Q x"
   shows "Collect P = Collect Q"
   using assms by (auto intro: set_eqI)
 
 text \<open>Lifting of predicate class instances\<close>
 
 instantiation set :: (type) boolean_algebra
 begin
 
 definition less_eq_set
   where "A \<le> B \<longleftrightarrow> (\<lambda>x. member x A) \<le> (\<lambda>x. member x B)"
 
 definition less_set
   where "A < B \<longleftrightarrow> (\<lambda>x. member x A) < (\<lambda>x. member x B)"
 
 definition inf_set
   where "A \<sqinter> B = Collect ((\<lambda>x. member x A) \<sqinter> (\<lambda>x. member x B))"
 
 definition sup_set
   where "A \<squnion> B = Collect ((\<lambda>x. member x A) \<squnion> (\<lambda>x. member x B))"
 
 definition bot_set
   where "\<bottom> = Collect \<bottom>"
 
 definition top_set
   where "\<top> = Collect \<top>"
 
 definition uminus_set
   where "- A = Collect (- (\<lambda>x. member x A))"
 
 definition minus_set
   where "A - B = Collect ((\<lambda>x. member x A) - (\<lambda>x. member x B))"
 
 instance
   by standard
     (simp_all add: less_eq_set_def less_set_def inf_set_def sup_set_def
       bot_set_def top_set_def uminus_set_def minus_set_def
       less_le_not_le sup_inf_distrib1 diff_eq set_eqI fun_eq_iff
       del: inf_apply sup_apply bot_apply top_apply minus_apply uminus_apply)
 
 end
 
 text \<open>Set enumerations\<close>
 
 abbreviation empty :: "'a set" ("{}")
   where "{} \<equiv> bot"
 
 definition insert :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set"
   where insert_compr: "insert a B = {x. x = a \<or> x \<in> B}"
 
 syntax
   "_Finset" :: "args \<Rightarrow> 'a set"    ("{(_)}")
 translations
   "{x, xs}" \<rightleftharpoons> "CONST insert x {xs}"
   "{x}" \<rightleftharpoons> "CONST insert x {}"
 
 
 subsection \<open>Subsets and bounded quantifiers\<close>
 
 abbreviation subset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
   where "subset \<equiv> less"
 
 abbreviation subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
   where "subset_eq \<equiv> less_eq"
 
 notation
   subset  ("'(\<subset>')") and
   subset  ("(_/ \<subset> _)" [51, 51] 50) and
   subset_eq  ("'(\<subseteq>')") and
   subset_eq  ("(_/ \<subseteq> _)" [51, 51] 50)
 
 abbreviation (input)
   supset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
   "supset \<equiv> greater"
 
 abbreviation (input)
   supset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
   "supset_eq \<equiv> greater_eq"
 
 notation
   supset  ("'(\<supset>')") and
   supset  ("(_/ \<supset> _)" [51, 51] 50) and
   supset_eq  ("'(\<supseteq>')") and
   supset_eq  ("(_/ \<supseteq> _)" [51, 51] 50)
 
 notation (ASCII output)
   subset  ("'(<')") and
   subset  ("(_/ < _)" [51, 51] 50) and
   subset_eq  ("'(<=')") and
   subset_eq  ("(_/ <= _)" [51, 51] 50)
 
 definition Ball :: "'a set \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
   where "Ball A P \<longleftrightarrow> (\<forall>x. x \<in> A \<longrightarrow> P x)"   \<comment> \<open>bounded universal quantifiers\<close>
 
 definition Bex :: "'a set \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
   where "Bex A P \<longleftrightarrow> (\<exists>x. x \<in> A \<and> P x)"   \<comment> \<open>bounded existential quantifiers\<close>
 
 syntax (ASCII)
   "_Ball"       :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool"      ("(3ALL (_/:_)./ _)" [0, 0, 10] 10)
   "_Bex"        :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool"      ("(3EX (_/:_)./ _)" [0, 0, 10] 10)
   "_Bex1"       :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool"      ("(3EX! (_/:_)./ _)" [0, 0, 10] 10)
   "_Bleast"     :: "id \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> 'a"           ("(3LEAST (_/:_)./ _)" [0, 0, 10] 10)
 
 syntax (input)
   "_Ball"       :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool"      ("(3! (_/:_)./ _)" [0, 0, 10] 10)
   "_Bex"        :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool"      ("(3? (_/:_)./ _)" [0, 0, 10] 10)
   "_Bex1"       :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool"      ("(3?! (_/:_)./ _)" [0, 0, 10] 10)
 
 syntax
   "_Ball"       :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool"      ("(3\<forall>(_/\<in>_)./ _)" [0, 0, 10] 10)
   "_Bex"        :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool"      ("(3\<exists>(_/\<in>_)./ _)" [0, 0, 10] 10)
   "_Bex1"       :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool"      ("(3\<exists>!(_/\<in>_)./ _)" [0, 0, 10] 10)
   "_Bleast"     :: "id \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> 'a"           ("(3LEAST(_/\<in>_)./ _)" [0, 0, 10] 10)
 
 translations
   "\<forall>x\<in>A. P" \<rightleftharpoons> "CONST Ball A (\<lambda>x. P)"
   "\<exists>x\<in>A. P" \<rightleftharpoons> "CONST Bex A (\<lambda>x. P)"
   "\<exists>!x\<in>A. P" \<rightharpoonup> "\<exists>!x. x \<in> A \<and> P"
   "LEAST x:A. P" \<rightharpoonup> "LEAST x. x \<in> A \<and> P"
 
 syntax (ASCII output)
   "_setlessAll" :: "[idt, 'a, bool] \<Rightarrow> bool"  ("(3ALL _<_./ _)"  [0, 0, 10] 10)
   "_setlessEx"  :: "[idt, 'a, bool] \<Rightarrow> bool"  ("(3EX _<_./ _)"  [0, 0, 10] 10)
   "_setleAll"   :: "[idt, 'a, bool] \<Rightarrow> bool"  ("(3ALL _<=_./ _)" [0, 0, 10] 10)
   "_setleEx"    :: "[idt, 'a, bool] \<Rightarrow> bool"  ("(3EX _<=_./ _)" [0, 0, 10] 10)
   "_setleEx1"   :: "[idt, 'a, bool] \<Rightarrow> bool"  ("(3EX! _<=_./ _)" [0, 0, 10] 10)
 
 syntax
   "_setlessAll" :: "[idt, 'a, bool] \<Rightarrow> bool"   ("(3\<forall>_\<subset>_./ _)"  [0, 0, 10] 10)
   "_setlessEx"  :: "[idt, 'a, bool] \<Rightarrow> bool"   ("(3\<exists>_\<subset>_./ _)"  [0, 0, 10] 10)
   "_setleAll"   :: "[idt, 'a, bool] \<Rightarrow> bool"   ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10)
   "_setleEx"    :: "[idt, 'a, bool] \<Rightarrow> bool"   ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10)
   "_setleEx1"   :: "[idt, 'a, bool] \<Rightarrow> bool"   ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10)
 
 translations
  "\<forall>A\<subset>B. P" \<rightharpoonup> "\<forall>A. A \<subset> B \<longrightarrow> P"
  "\<exists>A\<subset>B. P" \<rightharpoonup> "\<exists>A. A \<subset> B \<and> P"
  "\<forall>A\<subseteq>B. P" \<rightharpoonup> "\<forall>A. A \<subseteq> B \<longrightarrow> P"
  "\<exists>A\<subseteq>B. P" \<rightharpoonup> "\<exists>A. A \<subseteq> B \<and> P"
  "\<exists>!A\<subseteq>B. P" \<rightharpoonup> "\<exists>!A. A \<subseteq> B \<and> P"
 
 print_translation \<open>
   let
     val All_binder = Mixfix.binder_name \<^const_syntax>\<open>All\<close>;
     val Ex_binder = Mixfix.binder_name \<^const_syntax>\<open>Ex\<close>;
     val impl = \<^const_syntax>\<open>HOL.implies\<close>;
     val conj = \<^const_syntax>\<open>HOL.conj\<close>;
     val sbset = \<^const_syntax>\<open>subset\<close>;
     val sbset_eq = \<^const_syntax>\<open>subset_eq\<close>;
 
     val trans =
      [((All_binder, impl, sbset), \<^syntax_const>\<open>_setlessAll\<close>),
       ((All_binder, impl, sbset_eq), \<^syntax_const>\<open>_setleAll\<close>),
       ((Ex_binder, conj, sbset), \<^syntax_const>\<open>_setlessEx\<close>),
       ((Ex_binder, conj, sbset_eq), \<^syntax_const>\<open>_setleEx\<close>)];
 
     fun mk v (v', T) c n P =
       if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v | _ => false) n)
       then Syntax.const c $ Syntax_Trans.mark_bound_body (v', T) $ n $ P
       else raise Match;
 
     fun tr' q = (q, fn _ =>
       (fn [Const (\<^syntax_const>\<open>_bound\<close>, _) $ Free (v, Type (\<^type_name>\<open>set\<close>, _)),
           Const (c, _) $
             (Const (d, _) $ (Const (\<^syntax_const>\<open>_bound\<close>, _) $ Free (v', T)) $ n) $ P] =>
           (case AList.lookup (=) trans (q, c, d) of
             NONE => raise Match
           | SOME l => mk v (v', T) l n P)
         | _ => raise Match));
   in
     [tr' All_binder, tr' Ex_binder]
   end
 \<close>
 
 
 text \<open>
   \<^medskip>
   Translate between \<open>{e | x1\<dots>xn. P}\<close> and \<open>{u. \<exists>x1\<dots>xn. u = e \<and> P}\<close>;
   \<open>{y. \<exists>x1\<dots>xn. y = e \<and> P}\<close> is only translated if \<open>[0..n] \<subseteq> bvs e\<close>.
 \<close>
 
 syntax
   "_Setcompr" :: "'a \<Rightarrow> idts \<Rightarrow> bool \<Rightarrow> 'a set"    ("(1{_ |/_./ _})")
 
 parse_translation \<open>
   let
     val ex_tr = snd (Syntax_Trans.mk_binder_tr ("EX ", \<^const_syntax>\<open>Ex\<close>));
 
     fun nvars (Const (\<^syntax_const>\<open>_idts\<close>, _) $ _ $ idts) = nvars idts + 1
       | nvars _ = 1;
 
     fun setcompr_tr ctxt [e, idts, b] =
       let
         val eq = Syntax.const \<^const_syntax>\<open>HOL.eq\<close> $ Bound (nvars idts) $ e;
         val P = Syntax.const \<^const_syntax>\<open>HOL.conj\<close> $ eq $ b;
         val exP = ex_tr ctxt [idts, P];
       in Syntax.const \<^const_syntax>\<open>Collect\<close> $ absdummy dummyT exP end;
 
   in [(\<^syntax_const>\<open>_Setcompr\<close>, setcompr_tr)] end
 \<close>
 
 print_translation \<open>
  [Syntax_Trans.preserve_binder_abs2_tr' \<^const_syntax>\<open>Ball\<close> \<^syntax_const>\<open>_Ball\<close>,
   Syntax_Trans.preserve_binder_abs2_tr' \<^const_syntax>\<open>Bex\<close> \<^syntax_const>\<open>_Bex\<close>]
 \<close> \<comment> \<open>to avoid eta-contraction of body\<close>
 
 print_translation \<open>
 let
   val ex_tr' = snd (Syntax_Trans.mk_binder_tr' (\<^const_syntax>\<open>Ex\<close>, "DUMMY"));
 
   fun setcompr_tr' ctxt [Abs (abs as (_, _, P))] =
     let
       fun check (Const (\<^const_syntax>\<open>Ex\<close>, _) $ Abs (_, _, P), n) = check (P, n + 1)
         | check (Const (\<^const_syntax>\<open>HOL.conj\<close>, _) $
               (Const (\<^const_syntax>\<open>HOL.eq\<close>, _) $ Bound m $ e) $ P, n) =
             n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso
             subset (=) (0 upto (n - 1), add_loose_bnos (e, 0, []))
         | check _ = false;
 
         fun tr' (_ $ abs) =
           let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' ctxt [abs]
           in Syntax.const \<^syntax_const>\<open>_Setcompr\<close> $ e $ idts $ Q end;
     in
       if check (P, 0) then tr' P
       else
         let
           val (x as _ $ Free(xN, _), t) = Syntax_Trans.atomic_abs_tr' abs;
           val M = Syntax.const \<^syntax_const>\<open>_Coll\<close> $ x $ t;
         in
           case t of
             Const (\<^const_syntax>\<open>HOL.conj\<close>, _) $
               (Const (\<^const_syntax>\<open>Set.member\<close>, _) $
                 (Const (\<^syntax_const>\<open>_bound\<close>, _) $ Free (yN, _)) $ A) $ P =>
             if xN = yN then Syntax.const \<^syntax_const>\<open>_Collect\<close> $ x $ A $ P else M
           | _ => M
         end
     end;
   in [(\<^const_syntax>\<open>Collect\<close>, setcompr_tr')] end
 \<close>
 
 simproc_setup defined_Bex ("\<exists>x\<in>A. P x \<and> Q x") = \<open>
   fn _ => Quantifier1.rearrange_Bex
     (fn ctxt => unfold_tac ctxt @{thms Bex_def})
 \<close>
 
 simproc_setup defined_All ("\<forall>x\<in>A. P x \<longrightarrow> Q x") = \<open>
   fn _ => Quantifier1.rearrange_Ball
     (fn ctxt => unfold_tac ctxt @{thms Ball_def})
 \<close>
 
 lemma ballI [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> P x) \<Longrightarrow> \<forall>x\<in>A. P x"
   by (simp add: Ball_def)
 
 lemmas strip = impI allI ballI
 
 lemma bspec [dest?]: "\<forall>x\<in>A. P x \<Longrightarrow> x \<in> A \<Longrightarrow> P x"
   by (simp add: Ball_def)
 
 text \<open>Gives better instantiation for bound:\<close>
 setup \<open>
   map_theory_claset (fn ctxt =>
     ctxt addbefore ("bspec", fn ctxt' => dresolve_tac ctxt' @{thms bspec} THEN' assume_tac ctxt'))
 \<close>
 
 ML \<open>
 structure Simpdata =
 struct
   open Simpdata;
   val mksimps_pairs = [(\<^const_name>\<open>Ball\<close>, @{thms bspec})] @ mksimps_pairs;
 end;
 
 open Simpdata;
 \<close>
 
 declaration \<open>fn _ => Simplifier.map_ss (Simplifier.set_mksimps (mksimps mksimps_pairs))\<close>
 
 lemma ballE [elim]: "\<forall>x\<in>A. P x \<Longrightarrow> (P x \<Longrightarrow> Q) \<Longrightarrow> (x \<notin> A \<Longrightarrow> Q) \<Longrightarrow> Q"
   unfolding Ball_def by blast
 
 lemma bexI [intro]: "P x \<Longrightarrow> x \<in> A \<Longrightarrow> \<exists>x\<in>A. P x"
   \<comment> \<open>Normally the best argument order: \<open>P x\<close> constrains the choice of \<open>x \<in> A\<close>.\<close>
   unfolding Bex_def by blast
 
 lemma rev_bexI [intro?]: "x \<in> A \<Longrightarrow> P x \<Longrightarrow> \<exists>x\<in>A. P x"
   \<comment> \<open>The best argument order when there is only one \<open>x \<in> A\<close>.\<close>
   unfolding Bex_def by blast
 
 lemma bexCI: "(\<forall>x\<in>A. \<not> P x \<Longrightarrow> P a) \<Longrightarrow> a \<in> A \<Longrightarrow> \<exists>x\<in>A. P x"
   unfolding Bex_def by blast
 
 lemma bexE [elim!]: "\<exists>x\<in>A. P x \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> P x \<Longrightarrow> Q) \<Longrightarrow> Q"
   unfolding Bex_def by blast
 
 lemma ball_triv [simp]: "(\<forall>x\<in>A. P) \<longleftrightarrow> ((\<exists>x. x \<in> A) \<longrightarrow> P)"
   \<comment> \<open>trivial rewrite rule.\<close>
   by (simp add: Ball_def)
 
 lemma bex_triv [simp]: "(\<exists>x\<in>A. P) \<longleftrightarrow> ((\<exists>x. x \<in> A) \<and> P)"
   \<comment> \<open>Dual form for existentials.\<close>
   by (simp add: Bex_def)
 
 lemma bex_triv_one_point1 [simp]: "(\<exists>x\<in>A. x = a) \<longleftrightarrow> a \<in> A"
   by blast
 
 lemma bex_triv_one_point2 [simp]: "(\<exists>x\<in>A. a = x) \<longleftrightarrow> a \<in> A"
   by blast
 
 lemma bex_one_point1 [simp]: "(\<exists>x\<in>A. x = a \<and> P x) \<longleftrightarrow> a \<in> A \<and> P a"
   by blast
 
 lemma bex_one_point2 [simp]: "(\<exists>x\<in>A. a = x \<and> P x) \<longleftrightarrow> a \<in> A \<and> P a"
   by blast
 
 lemma ball_one_point1 [simp]: "(\<forall>x\<in>A. x = a \<longrightarrow> P x) \<longleftrightarrow> (a \<in> A \<longrightarrow> P a)"
   by blast
 
 lemma ball_one_point2 [simp]: "(\<forall>x\<in>A. a = x \<longrightarrow> P x) \<longleftrightarrow> (a \<in> A \<longrightarrow> P a)"
   by blast
 
 lemma ball_conj_distrib: "(\<forall>x\<in>A. P x \<and> Q x) \<longleftrightarrow> (\<forall>x\<in>A. P x) \<and> (\<forall>x\<in>A. Q x)"
   by blast
 
 lemma bex_disj_distrib: "(\<exists>x\<in>A. P x \<or> Q x) \<longleftrightarrow> (\<exists>x\<in>A. P x) \<or> (\<exists>x\<in>A. Q x)"
   by blast
 
 
 text \<open>Congruence rules\<close>
 
 lemma ball_cong:
   "\<lbrakk> A = B;  \<And>x. x \<in> B \<Longrightarrow> P x \<longleftrightarrow> Q x \<rbrakk> \<Longrightarrow>
     (\<forall>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>B. Q x)"
 by (simp add: Ball_def)
 
 lemma ball_cong_simp [cong]:
   "\<lbrakk> A = B;  \<And>x. x \<in> B =simp=> P x \<longleftrightarrow> Q x \<rbrakk> \<Longrightarrow>
     (\<forall>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>B. Q x)"
 by (simp add: simp_implies_def Ball_def)
 
 lemma bex_cong:
   "\<lbrakk> A = B;  \<And>x. x \<in> B \<Longrightarrow> P x \<longleftrightarrow> Q x \<rbrakk> \<Longrightarrow>
     (\<exists>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>B. Q x)"
 by (simp add: Bex_def cong: conj_cong)
 
 lemma bex_cong_simp [cong]:
   "\<lbrakk> A = B;  \<And>x. x \<in> B =simp=> P x \<longleftrightarrow> Q x \<rbrakk> \<Longrightarrow>
     (\<exists>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>B. Q x)"
 by (simp add: simp_implies_def Bex_def cong: conj_cong)
 
 lemma bex1_def: "(\<exists>!x\<in>X. P x) \<longleftrightarrow> (\<exists>x\<in>X. P x) \<and> (\<forall>x\<in>X. \<forall>y\<in>X. P x \<longrightarrow> P y \<longrightarrow> x = y)"
   by auto
 
 
 subsection \<open>Basic operations\<close>
 
 subsubsection \<open>Subsets\<close>
 
 lemma subsetI [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> x \<in> B) \<Longrightarrow> A \<subseteq> B"
   by (simp add: less_eq_set_def le_fun_def)
 
 text \<open>
   \<^medskip>
   Map the type \<open>'a set \<Rightarrow> anything\<close> to just \<open>'a\<close>; for overloading constants
   whose first argument has type \<open>'a set\<close>.
 \<close>
 
 lemma subsetD [elim, intro?]: "A \<subseteq> B \<Longrightarrow> c \<in> A \<Longrightarrow> c \<in> B"
   by (simp add: less_eq_set_def le_fun_def)
   \<comment> \<open>Rule in Modus Ponens style.\<close>
 
 lemma rev_subsetD [intro?,no_atp]: "c \<in> A \<Longrightarrow> A \<subseteq> B \<Longrightarrow> c \<in> B"
   \<comment> \<open>The same, with reversed premises for use with @{method erule} -- cf. @{thm rev_mp}.\<close>
   by (rule subsetD)
 
 lemma subsetCE [elim,no_atp]: "A \<subseteq> B \<Longrightarrow> (c \<notin> A \<Longrightarrow> P) \<Longrightarrow> (c \<in> B \<Longrightarrow> P) \<Longrightarrow> P"
   \<comment> \<open>Classical elimination rule.\<close>
   by (auto simp add: less_eq_set_def le_fun_def)
 
 lemma subset_eq: "A \<subseteq> B \<longleftrightarrow> (\<forall>x\<in>A. x \<in> B)"
   by blast
 
 lemma contra_subsetD [no_atp]: "A \<subseteq> B \<Longrightarrow> c \<notin> B \<Longrightarrow> c \<notin> A"
   by blast
 
 lemma subset_refl: "A \<subseteq> A"
   by (fact order_refl) (* already [iff] *)
 
 lemma subset_trans: "A \<subseteq> B \<Longrightarrow> B \<subseteq> C \<Longrightarrow> A \<subseteq> C"
   by (fact order_trans)
 
 lemma subset_not_subset_eq [code]: "A \<subset> B \<longleftrightarrow> A \<subseteq> B \<and> \<not> B \<subseteq> A"
   by (fact less_le_not_le)
 
 lemma eq_mem_trans: "a = b \<Longrightarrow> b \<in> A \<Longrightarrow> a \<in> A"
   by simp
 
 lemmas basic_trans_rules [trans] =
   order_trans_rules rev_subsetD subsetD eq_mem_trans
 
 
 subsubsection \<open>Equality\<close>
 
 lemma subset_antisym [intro!]: "A \<subseteq> B \<Longrightarrow> B \<subseteq> A \<Longrightarrow> A = B"
   \<comment> \<open>Anti-symmetry of the subset relation.\<close>
   by (iprover intro: set_eqI subsetD)
 
 text \<open>\<^medskip> Equality rules from ZF set theory -- are they appropriate here?\<close>
 
 lemma equalityD1: "A = B \<Longrightarrow> A \<subseteq> B"
   by simp
 
 lemma equalityD2: "A = B \<Longrightarrow> B \<subseteq> A"
   by simp
 
 text \<open>
   \<^medskip>
   Be careful when adding this to the claset as \<open>subset_empty\<close> is in the
   simpset: \<^prop>\<open>A = {}\<close> goes to \<^prop>\<open>{} \<subseteq> A\<close> and \<^prop>\<open>A \<subseteq> {}\<close>
   and then back to \<^prop>\<open>A = {}\<close>!
 \<close>
 
 lemma equalityE: "A = B \<Longrightarrow> (A \<subseteq> B \<Longrightarrow> B \<subseteq> A \<Longrightarrow> P) \<Longrightarrow> P"
   by simp
 
 lemma equalityCE [elim]: "A = B \<Longrightarrow> (c \<in> A \<Longrightarrow> c \<in> B \<Longrightarrow> P) \<Longrightarrow> (c \<notin> A \<Longrightarrow> c \<notin> B \<Longrightarrow> P) \<Longrightarrow> P"
   by blast
 
 lemma eqset_imp_iff: "A = B \<Longrightarrow> x \<in> A \<longleftrightarrow> x \<in> B"
   by simp
 
 lemma eqelem_imp_iff: "x = y \<Longrightarrow> x \<in> A \<longleftrightarrow> y \<in> A"
   by simp
 
 
 subsubsection \<open>The empty set\<close>
 
 lemma empty_def: "{} = {x. False}"
   by (simp add: bot_set_def bot_fun_def)
 
 lemma empty_iff [simp]: "c \<in> {} \<longleftrightarrow> False"
   by (simp add: empty_def)
 
 lemma emptyE [elim!]: "a \<in> {} \<Longrightarrow> P"
   by simp
 
 lemma empty_subsetI [iff]: "{} \<subseteq> A"
   \<comment> \<open>One effect is to delete the ASSUMPTION \<^prop>\<open>{} \<subseteq> A\<close>\<close>
   by blast
 
 lemma equals0I: "(\<And>y. y \<in> A \<Longrightarrow> False) \<Longrightarrow> A = {}"
   by blast
 
 lemma equals0D: "A = {} \<Longrightarrow> a \<notin> A"
   \<comment> \<open>Use for reasoning about disjointness: \<open>A \<inter> B = {}\<close>\<close>
   by blast
 
 lemma ball_empty [simp]: "Ball {} P \<longleftrightarrow> True"
   by (simp add: Ball_def)
 
 lemma bex_empty [simp]: "Bex {} P \<longleftrightarrow> False"
   by (simp add: Bex_def)
 
 
 subsubsection \<open>The universal set -- UNIV\<close>
 
 abbreviation UNIV :: "'a set"
   where "UNIV \<equiv> top"
 
 lemma UNIV_def: "UNIV = {x. True}"
   by (simp add: top_set_def top_fun_def)
 
 lemma UNIV_I [simp]: "x \<in> UNIV"
   by (simp add: UNIV_def)
 
 declare UNIV_I [intro]  \<comment> \<open>unsafe makes it less likely to cause problems\<close>
 
 lemma UNIV_witness [intro?]: "\<exists>x. x \<in> UNIV"
   by simp
 
 lemma subset_UNIV: "A \<subseteq> UNIV"
   by (fact top_greatest) (* already simp *)
 
 text \<open>
   \<^medskip>
   Eta-contracting these two rules (to remove \<open>P\<close>) causes them
   to be ignored because of their interaction with congruence rules.
 \<close>
 
 lemma ball_UNIV [simp]: "Ball UNIV P \<longleftrightarrow> All P"
   by (simp add: Ball_def)
 
 lemma bex_UNIV [simp]: "Bex UNIV P \<longleftrightarrow> Ex P"
   by (simp add: Bex_def)
 
 lemma UNIV_eq_I: "(\<And>x. x \<in> A) \<Longrightarrow> UNIV = A"
   by auto
 
 lemma UNIV_not_empty [iff]: "UNIV \<noteq> {}"
   by (blast elim: equalityE)
 
 lemma empty_not_UNIV[simp]: "{} \<noteq> UNIV"
   by blast
 
 
 subsubsection \<open>The Powerset operator -- Pow\<close>
 
 definition Pow :: "'a set \<Rightarrow> 'a set set"
   where Pow_def: "Pow A = {B. B \<subseteq> A}"
 
 lemma Pow_iff [iff]: "A \<in> Pow B \<longleftrightarrow> A \<subseteq> B"
   by (simp add: Pow_def)
 
 lemma PowI: "A \<subseteq> B \<Longrightarrow> A \<in> Pow B"
   by (simp add: Pow_def)
 
 lemma PowD: "A \<in> Pow B \<Longrightarrow> A \<subseteq> B"
   by (simp add: Pow_def)
 
 lemma Pow_bottom: "{} \<in> Pow B"
   by simp
 
 lemma Pow_top: "A \<in> Pow A"
   by simp
 
 lemma Pow_not_empty: "Pow A \<noteq> {}"
   using Pow_top by blast
 
 
 subsubsection \<open>Set complement\<close>
 
 lemma Compl_iff [simp]: "c \<in> - A \<longleftrightarrow> c \<notin> A"
   by (simp add: fun_Compl_def uminus_set_def)
 
 lemma ComplI [intro!]: "(c \<in> A \<Longrightarrow> False) \<Longrightarrow> c \<in> - A"
   by (simp add: fun_Compl_def uminus_set_def) blast
 
 text \<open>
   \<^medskip>
   This form, with negated conclusion, works well with the Classical prover.
   Negated assumptions behave like formulae on the right side of the
   notional turnstile \dots
 \<close>
 
 lemma ComplD [dest!]: "c \<in> - A \<Longrightarrow> c \<notin> A"
   by simp
 
 lemmas ComplE = ComplD [elim_format]
 
 lemma Compl_eq: "- A = {x. \<not> x \<in> A}"
   by blast
 
 
 subsubsection \<open>Binary intersection\<close>
 
 abbreviation inter :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "\<inter>" 70)
   where "(\<inter>) \<equiv> inf"
 
 notation (ASCII)
   inter  (infixl "Int" 70)
 
 lemma Int_def: "A \<inter> B = {x. x \<in> A \<and> x \<in> B}"
   by (simp add: inf_set_def inf_fun_def)
 
 lemma Int_iff [simp]: "c \<in> A \<inter> B \<longleftrightarrow> c \<in> A \<and> c \<in> B"
   unfolding Int_def by blast
 
 lemma IntI [intro!]: "c \<in> A \<Longrightarrow> c \<in> B \<Longrightarrow> c \<in> A \<inter> B"
   by simp
 
 lemma IntD1: "c \<in> A \<inter> B \<Longrightarrow> c \<in> A"
   by simp
 
 lemma IntD2: "c \<in> A \<inter> B \<Longrightarrow> c \<in> B"
   by simp
 
 lemma IntE [elim!]: "c \<in> A \<inter> B \<Longrightarrow> (c \<in> A \<Longrightarrow> c \<in> B \<Longrightarrow> P) \<Longrightarrow> P"
   by simp
 
 lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B"
   by (fact mono_inf)
 
 
 subsubsection \<open>Binary union\<close>
 
 abbreviation union :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "\<union>" 65)
   where "union \<equiv> sup"
 
 notation (ASCII)
   union  (infixl "Un" 65)
 
 lemma Un_def: "A \<union> B = {x. x \<in> A \<or> x \<in> B}"
   by (simp add: sup_set_def sup_fun_def)
 
 lemma Un_iff [simp]: "c \<in> A \<union> B \<longleftrightarrow> c \<in> A \<or> c \<in> B"
   unfolding Un_def by blast
 
 lemma UnI1 [elim?]: "c \<in> A \<Longrightarrow> c \<in> A \<union> B"
   by simp
 
 lemma UnI2 [elim?]: "c \<in> B \<Longrightarrow> c \<in> A \<union> B"
   by simp
 
 text \<open>\<^medskip> Classical introduction rule: no commitment to \<open>A\<close> vs. \<open>B\<close>.\<close>
 lemma UnCI [intro!]: "(c \<notin> B \<Longrightarrow> c \<in> A) \<Longrightarrow> c \<in> A \<union> B"
   by auto
 
 lemma UnE [elim!]: "c \<in> A \<union> B \<Longrightarrow> (c \<in> A \<Longrightarrow> P) \<Longrightarrow> (c \<in> B \<Longrightarrow> P) \<Longrightarrow> P"
   unfolding Un_def by blast
 
 lemma insert_def: "insert a B = {x. x = a} \<union> B"
   by (simp add: insert_compr Un_def)
 
 lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)"
   by (fact mono_sup)
 
 
 subsubsection \<open>Set difference\<close>
 
 lemma Diff_iff [simp]: "c \<in> A - B \<longleftrightarrow> c \<in> A \<and> c \<notin> B"
   by (simp add: minus_set_def fun_diff_def)
 
 lemma DiffI [intro!]: "c \<in> A \<Longrightarrow> c \<notin> B \<Longrightarrow> c \<in> A - B"
   by simp
 
 lemma DiffD1: "c \<in> A - B \<Longrightarrow> c \<in> A"
   by simp
 
 lemma DiffD2: "c \<in> A - B \<Longrightarrow> c \<in> B \<Longrightarrow> P"
   by simp
 
 lemma DiffE [elim!]: "c \<in> A - B \<Longrightarrow> (c \<in> A \<Longrightarrow> c \<notin> B \<Longrightarrow> P) \<Longrightarrow> P"
   by simp
 
 lemma set_diff_eq: "A - B = {x. x \<in> A \<and> x \<notin> B}"
   by blast
 
 lemma Compl_eq_Diff_UNIV: "- A = (UNIV - A)"
   by blast
 
 
 subsubsection \<open>Augmenting a set -- \<^const>\<open>insert\<close>\<close>
 
 lemma insert_iff [simp]: "a \<in> insert b A \<longleftrightarrow> a = b \<or> a \<in> A"
   unfolding insert_def by blast
 
 lemma insertI1: "a \<in> insert a B"
   by simp
 
 lemma insertI2: "a \<in> B \<Longrightarrow> a \<in> insert b B"
   by simp
 
 lemma insertE [elim!]: "a \<in> insert b A \<Longrightarrow> (a = b \<Longrightarrow> P) \<Longrightarrow> (a \<in> A \<Longrightarrow> P) \<Longrightarrow> P"
   unfolding insert_def by blast
 
 lemma insertCI [intro!]: "(a \<notin> B \<Longrightarrow> a = b) \<Longrightarrow> a \<in> insert b B"
   \<comment> \<open>Classical introduction rule.\<close>
   by auto
 
 lemma subset_insert_iff: "A \<subseteq> insert x B \<longleftrightarrow> (if x \<in> A then A - {x} \<subseteq> B else A \<subseteq> B)"
   by auto
 
 lemma set_insert:
   assumes "x \<in> A"
   obtains B where "A = insert x B" and "x \<notin> B"
 proof
   show "A = insert x (A - {x})" using assms by blast
   show "x \<notin> A - {x}" by blast
 qed
 
 lemma insert_ident: "x \<notin> A \<Longrightarrow> x \<notin> B \<Longrightarrow> insert x A = insert x B \<longleftrightarrow> A = B"
   by auto
 
 lemma insert_eq_iff:
   assumes "a \<notin> A" "b \<notin> B"
   shows "insert a A = insert b B \<longleftrightarrow>
     (if a = b then A = B else \<exists>C. A = insert b C \<and> b \<notin> C \<and> B = insert a C \<and> a \<notin> C)"
     (is "?L \<longleftrightarrow> ?R")
 proof
   show ?R if ?L
   proof (cases "a = b")
     case True
     with assms \<open>?L\<close> show ?R
       by (simp add: insert_ident)
   next
     case False
     let ?C = "A - {b}"
     have "A = insert b ?C \<and> b \<notin> ?C \<and> B = insert a ?C \<and> a \<notin> ?C"
       using assms \<open>?L\<close> \<open>a \<noteq> b\<close> by auto
     then show ?R using \<open>a \<noteq> b\<close> by auto
   qed
   show ?L if ?R
     using that by (auto split: if_splits)
 qed
 
 lemma insert_UNIV: "insert x UNIV = UNIV"
   by auto
 
 
 subsubsection \<open>Singletons, using insert\<close>
 
 lemma singletonI [intro!]: "a \<in> {a}"
   \<comment> \<open>Redundant? But unlike \<open>insertCI\<close>, it proves the subgoal immediately!\<close>
   by (rule insertI1)
 
 lemma singletonD [dest!]: "b \<in> {a} \<Longrightarrow> b = a"
   by blast
 
 lemmas singletonE = singletonD [elim_format]
 
 lemma singleton_iff: "b \<in> {a} \<longleftrightarrow> b = a"
   by blast
 
 lemma singleton_inject [dest!]: "{a} = {b} \<Longrightarrow> a = b"
   by blast
 
 lemma singleton_insert_inj_eq [iff]: "{b} = insert a A \<longleftrightarrow> a = b \<and> A \<subseteq> {b}"
   by blast
 
 lemma singleton_insert_inj_eq' [iff]: "insert a A = {b} \<longleftrightarrow> a = b \<and> A \<subseteq> {b}"
   by blast
 
 lemma subset_singletonD: "A \<subseteq> {x} \<Longrightarrow> A = {} \<or> A = {x}"
   by fast
 
 lemma subset_singleton_iff: "X \<subseteq> {a} \<longleftrightarrow> X = {} \<or> X = {a}"
   by blast
 
 lemma subset_singleton_iff_Uniq: "(\<exists>a. A \<subseteq> {a}) \<longleftrightarrow> (\<exists>\<^sub>\<le>\<^sub>1x. x \<in> A)"
   unfolding Uniq_def by blast
 
 lemma singleton_conv [simp]: "{x. x = a} = {a}"
   by blast
 
 lemma singleton_conv2 [simp]: "{x. a = x} = {a}"
   by blast
 
 lemma Diff_single_insert: "A - {x} \<subseteq> B \<Longrightarrow> A \<subseteq> insert x B"
   by blast
 
 lemma subset_Diff_insert: "A \<subseteq> B - insert x C \<longleftrightarrow> A \<subseteq> B - C \<and> x \<notin> A"
   by blast
 
 lemma doubleton_eq_iff: "{a, b} = {c, d} \<longleftrightarrow> a = c \<and> b = d \<or> a = d \<and> b = c"
   by (blast elim: equalityE)
 
 lemma Un_singleton_iff: "A \<union> B = {x} \<longleftrightarrow> A = {} \<and> B = {x} \<or> A = {x} \<and> B = {} \<or> A = {x} \<and> B = {x}"
   by auto
 
 lemma singleton_Un_iff: "{x} = A \<union> B \<longleftrightarrow> A = {} \<and> B = {x} \<or> A = {x} \<and> B = {} \<or> A = {x} \<and> B = {x}"
   by auto
 
 
 subsubsection \<open>Image of a set under a function\<close>
 
 text \<open>Frequently \<open>b\<close> does not have the syntactic form of \<open>f x\<close>.\<close>
 
 definition image :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set"    (infixr "`" 90)
   where "f ` A = {y. \<exists>x\<in>A. y = f x}"
 
 lemma image_eqI [simp, intro]: "b = f x \<Longrightarrow> x \<in> A \<Longrightarrow> b \<in> f ` A"
   unfolding image_def by blast
 
 lemma imageI: "x \<in> A \<Longrightarrow> f x \<in> f ` A"
   by (rule image_eqI) (rule refl)
 
 lemma rev_image_eqI: "x \<in> A \<Longrightarrow> b = f x \<Longrightarrow> b \<in> f ` A"
   \<comment> \<open>This version's more effective when we already have the required \<open>x\<close>.\<close>
   by (rule image_eqI)
 
 lemma imageE [elim!]:
   assumes "b \<in> (\<lambda>x. f x) ` A"  \<comment> \<open>The eta-expansion gives variable-name preservation.\<close>
   obtains x where "b = f x" and "x \<in> A"
   using assms unfolding image_def by blast
 
 lemma Compr_image_eq: "{x \<in> f ` A. P x} = f ` {x \<in> A. P (f x)}"
   by auto
 
 lemma image_Un: "f ` (A \<union> B) = f ` A \<union> f ` B"
   by blast
 
 lemma image_iff: "z \<in> f ` A \<longleftrightarrow> (\<exists>x\<in>A. z = f x)"
   by blast
 
 lemma image_subsetI: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B) \<Longrightarrow> f ` A \<subseteq> B"
   \<comment> \<open>Replaces the three steps \<open>subsetI\<close>, \<open>imageE\<close>,
     \<open>hypsubst\<close>, but breaks too many existing proofs.\<close>
   by blast
 
 lemma image_subset_iff: "f ` A \<subseteq> B \<longleftrightarrow> (\<forall>x\<in>A. f x \<in> B)"
   \<comment> \<open>This rewrite rule would confuse users if made default.\<close>
   by blast
 
 lemma subset_imageE:
   assumes "B \<subseteq> f ` A"
   obtains C where "C \<subseteq> A" and "B = f ` C"
 proof -
   from assms have "B = f ` {a \<in> A. f a \<in> B}" by fast
   moreover have "{a \<in> A. f a \<in> B} \<subseteq> A" by blast
   ultimately show thesis by (blast intro: that)
 qed
 
 lemma subset_image_iff: "B \<subseteq> f ` A \<longleftrightarrow> (\<exists>AA\<subseteq>A. B = f ` AA)"
   by (blast elim: subset_imageE)
 
 lemma image_ident [simp]: "(\<lambda>x. x) ` Y = Y"
   by blast
 
 lemma image_empty [simp]: "f ` {} = {}"
   by blast
 
 lemma image_insert [simp]: "f ` insert a B = insert (f a) (f ` B)"
   by blast
 
 lemma image_constant: "x \<in> A \<Longrightarrow> (\<lambda>x. c) ` A = {c}"
   by auto
 
 lemma image_constant_conv: "(\<lambda>x. c) ` A = (if A = {} then {} else {c})"
   by auto
 
 lemma image_image: "f ` (g ` A) = (\<lambda>x. f (g x)) ` A"
   by blast
 
 lemma insert_image [simp]: "x \<in> A \<Longrightarrow> insert (f x) (f ` A) = f ` A"
   by blast
 
 lemma image_is_empty [iff]: "f ` A = {} \<longleftrightarrow> A = {}"
   by blast
 
 lemma empty_is_image [iff]: "{} = f ` A \<longleftrightarrow> A = {}"
   by blast
 
 lemma image_Collect: "f ` {x. P x} = {f x | x. P x}"
   \<comment> \<open>NOT suitable as a default simp rule: the RHS isn't simpler than the LHS,
       with its implicit quantifier and conjunction.  Also image enjoys better
       equational properties than does the RHS.\<close>
   by blast
 
 lemma if_image_distrib [simp]:
   "(\<lambda>x. if P x then f x else g x) ` S = f ` (S \<inter> {x. P x}) \<union> g ` (S \<inter> {x. \<not> P x})"
   by auto
 
 lemma image_cong:
   "f ` M = g ` N" if "M = N" "\<And>x. x \<in> N \<Longrightarrow> f x = g x"
   using that by (simp add: image_def)
 
 lemma image_cong_simp [cong]:
   "f ` M = g ` N" if "M = N" "\<And>x. x \<in> N =simp=> f x = g x"
   using that image_cong [of M N f g] by (simp add: simp_implies_def)
 
 lemma image_Int_subset: "f ` (A \<inter> B) \<subseteq> f ` A \<inter> f ` B"
   by blast
 
 lemma image_diff_subset: "f ` A - f ` B \<subseteq> f ` (A - B)"
   by blast
 
 lemma Setcompr_eq_image: "{f x |x. x \<in> A} = f ` A"
   by blast
 
 lemma setcompr_eq_image: "{f x |x. P x} = f ` {x. P x}"
   by auto
 
 lemma ball_imageD: "\<forall>x\<in>f ` A. P x \<Longrightarrow> \<forall>x\<in>A. P (f x)"
   by simp
 
 lemma bex_imageD: "\<exists>x\<in>f ` A. P x \<Longrightarrow> \<exists>x\<in>A. P (f x)"
   by auto
 
 lemma image_add_0 [simp]: "(+) (0::'a::comm_monoid_add) ` S = S"
   by auto
 
 
 text \<open>\<^medskip> Range of a function -- just an abbreviation for image!\<close>
 
 abbreviation range :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b set"  \<comment> \<open>of function\<close>
   where "range f \<equiv> f ` UNIV"
 
 lemma range_eqI: "b = f x \<Longrightarrow> b \<in> range f"
   by simp
 
 lemma rangeI: "f x \<in> range f"
   by simp
 
 lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) \<Longrightarrow> (\<And>x. b = f x \<Longrightarrow> P) \<Longrightarrow> P"
   by (rule imageE)
 
+lemma range_subsetD: "range f \<subseteq> B \<Longrightarrow> f i \<in> B"
+  by blast
+
 lemma full_SetCompr_eq: "{u. \<exists>x. u = f x} = range f"
   by auto
 
 lemma range_composition: "range (\<lambda>x. f (g x)) = f ` range g"
   by auto
 
 lemma range_constant [simp]: "range (\<lambda>_. x) = {x}"
   by (simp add: image_constant)
 
 lemma range_eq_singletonD: "range f = {a} \<Longrightarrow> f x = a"
   by auto
 
 
 subsubsection \<open>Some rules with \<open>if\<close>\<close>
 
 text \<open>Elimination of \<open>{x. \<dots> \<and> x = t \<and> \<dots>}\<close>.\<close>
 
 lemma Collect_conv_if: "{x. x = a \<and> P x} = (if P a then {a} else {})"
   by auto
 
 lemma Collect_conv_if2: "{x. a = x \<and> P x} = (if P a then {a} else {})"
   by auto
 
 text \<open>
   Rewrite rules for boolean case-splitting: faster than \<open>if_split [split]\<close>.
 \<close>
 
 lemma if_split_eq1: "(if Q then x else y) = b \<longleftrightarrow> (Q \<longrightarrow> x = b) \<and> (\<not> Q \<longrightarrow> y = b)"
   by (rule if_split)
 
 lemma if_split_eq2: "a = (if Q then x else y) \<longleftrightarrow> (Q \<longrightarrow> a = x) \<and> (\<not> Q \<longrightarrow> a = y)"
   by (rule if_split)
 
 text \<open>
   Split ifs on either side of the membership relation.
   Not for \<open>[simp]\<close> -- can cause goals to blow up!
 \<close>
 
 lemma if_split_mem1: "(if Q then x else y) \<in> b \<longleftrightarrow> (Q \<longrightarrow> x \<in> b) \<and> (\<not> Q \<longrightarrow> y \<in> b)"
   by (rule if_split)
 
 lemma if_split_mem2: "(a \<in> (if Q then x else y)) \<longleftrightarrow> (Q \<longrightarrow> a \<in> x) \<and> (\<not> Q \<longrightarrow> a \<in> y)"
   by (rule if_split [where P = "\<lambda>S. a \<in> S"])
 
 lemmas split_ifs = if_bool_eq_conj if_split_eq1 if_split_eq2 if_split_mem1 if_split_mem2
 
 (*Would like to add these, but the existing code only searches for the
   outer-level constant, which in this case is just Set.member; we instead need
   to use term-nets to associate patterns with rules.  Also, if a rule fails to
   apply, then the formula should be kept.
   [("uminus", Compl_iff RS iffD1), ("minus", [Diff_iff RS iffD1]),
    ("Int", [IntD1,IntD2]),
    ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
  *)
 
 
 subsection \<open>Further operations and lemmas\<close>
 
 subsubsection \<open>The ``proper subset'' relation\<close>
 
 lemma psubsetI [intro!]: "A \<subseteq> B \<Longrightarrow> A \<noteq> B \<Longrightarrow> A \<subset> B"
   unfolding less_le by blast
 
 lemma psubsetE [elim!]: "A \<subset> B \<Longrightarrow> (A \<subseteq> B \<Longrightarrow> \<not> B \<subseteq> A \<Longrightarrow> R) \<Longrightarrow> R"
   unfolding less_le by blast
 
 lemma psubset_insert_iff:
   "A \<subset> insert x B \<longleftrightarrow> (if x \<in> B then A \<subset> B else if x \<in> A then A - {x} \<subset> B else A \<subseteq> B)"
   by (auto simp add: less_le subset_insert_iff)
 
 lemma psubset_eq: "A \<subset> B \<longleftrightarrow> A \<subseteq> B \<and> A \<noteq> B"
   by (simp only: less_le)
 
 lemma psubset_imp_subset: "A \<subset> B \<Longrightarrow> A \<subseteq> B"
   by (simp add: psubset_eq)
 
 lemma psubset_trans: "A \<subset> B \<Longrightarrow> B \<subset> C \<Longrightarrow> A \<subset> C"
   unfolding less_le by (auto dest: subset_antisym)
 
 lemma psubsetD: "A \<subset> B \<Longrightarrow> c \<in> A \<Longrightarrow> c \<in> B"
   unfolding less_le by (auto dest: subsetD)
 
 lemma psubset_subset_trans: "A \<subset> B \<Longrightarrow> B \<subseteq> C \<Longrightarrow> A \<subset> C"
   by (auto simp add: psubset_eq)
 
 lemma subset_psubset_trans: "A \<subseteq> B \<Longrightarrow> B \<subset> C \<Longrightarrow> A \<subset> C"
   by (auto simp add: psubset_eq)
 
 lemma psubset_imp_ex_mem: "A \<subset> B \<Longrightarrow> \<exists>b. b \<in> B - A"
   unfolding less_le by blast
 
 lemma atomize_ball: "(\<And>x. x \<in> A \<Longrightarrow> P x) \<equiv> Trueprop (\<forall>x\<in>A. P x)"
   by (simp only: Ball_def atomize_all atomize_imp)
 
 lemmas [symmetric, rulify] = atomize_ball
   and [symmetric, defn] = atomize_ball
 
 lemma image_Pow_mono: "f ` A \<subseteq> B \<Longrightarrow> image f ` Pow A \<subseteq> Pow B"
   by blast
 
 lemma image_Pow_surj: "f ` A = B \<Longrightarrow> image f ` Pow A = Pow B"
   by (blast elim: subset_imageE)
 
 
 subsubsection \<open>Derived rules involving subsets.\<close>
 
 text \<open>\<open>insert\<close>.\<close>
 
 lemma subset_insertI: "B \<subseteq> insert a B"
   by (rule subsetI) (erule insertI2)
 
 lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B"
   by blast
 
 lemma subset_insert: "x \<notin> A \<Longrightarrow> A \<subseteq> insert x B \<longleftrightarrow> A \<subseteq> B"
   by blast
 
 
 text \<open>\<^medskip> Finite Union -- the least upper bound of two sets.\<close>
 
 lemma Un_upper1: "A \<subseteq> A \<union> B"
   by (fact sup_ge1)
 
 lemma Un_upper2: "B \<subseteq> A \<union> B"
   by (fact sup_ge2)
 
 lemma Un_least: "A \<subseteq> C \<Longrightarrow> B \<subseteq> C \<Longrightarrow> A \<union> B \<subseteq> C"
   by (fact sup_least)
 
 
 text \<open>\<^medskip> Finite Intersection -- the greatest lower bound of two sets.\<close>
 
 lemma Int_lower1: "A \<inter> B \<subseteq> A"
   by (fact inf_le1)
 
 lemma Int_lower2: "A \<inter> B \<subseteq> B"
   by (fact inf_le2)
 
 lemma Int_greatest: "C \<subseteq> A \<Longrightarrow> C \<subseteq> B \<Longrightarrow> C \<subseteq> A \<inter> B"
   by (fact inf_greatest)
 
 
 text \<open>\<^medskip> Set difference.\<close>
 
 lemma Diff_subset[simp]: "A - B \<subseteq> A"
   by blast
 
 lemma Diff_subset_conv: "A - B \<subseteq> C \<longleftrightarrow> A \<subseteq> B \<union> C"
   by blast
 
 
 subsubsection \<open>Equalities involving union, intersection, inclusion, etc.\<close>
 
 text \<open>\<open>{}\<close>.\<close>
 
 lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})"
   \<comment> \<open>supersedes \<open>Collect_False_empty\<close>\<close>
   by auto
 
 lemma subset_empty [simp]: "A \<subseteq> {} \<longleftrightarrow> A = {}"
   by (fact bot_unique)
 
 lemma not_psubset_empty [iff]: "\<not> (A < {})"
   by (fact not_less_bot) (* FIXME: already simp *)
 
 lemma Collect_subset [simp]: "{x\<in>A. P x} \<subseteq> A" by auto
 
 lemma Collect_empty_eq [simp]: "Collect P = {} \<longleftrightarrow> (\<forall>x. \<not> P x)"
   by blast
 
 lemma empty_Collect_eq [simp]: "{} = Collect P \<longleftrightarrow> (\<forall>x. \<not> P x)"
   by blast
 
 lemma Collect_neg_eq: "{x. \<not> P x} = - {x. P x}"
   by blast
 
 lemma Collect_disj_eq: "{x. P x \<or> Q x} = {x. P x} \<union> {x. Q x}"
   by blast
 
 lemma Collect_imp_eq: "{x. P x \<longrightarrow> Q x} = - {x. P x} \<union> {x. Q x}"
   by blast
 
 lemma Collect_conj_eq: "{x. P x \<and> Q x} = {x. P x} \<inter> {x. Q x}"
   by blast
 
 lemma Collect_mono_iff: "Collect P \<subseteq> Collect Q \<longleftrightarrow> (\<forall>x. P x \<longrightarrow> Q x)"
   by blast
 
 
 text \<open>\<^medskip> \<open>insert\<close>.\<close>
 
 lemma insert_is_Un: "insert a A = {a} \<union> A"
   \<comment> \<open>NOT SUITABLE FOR REWRITING since \<open>{a} \<equiv> insert a {}\<close>\<close>
   by blast
 
 lemma insert_not_empty [simp]: "insert a A \<noteq> {}"
   and empty_not_insert [simp]: "{} \<noteq> insert a A"
   by blast+
 
 lemma insert_absorb: "a \<in> A \<Longrightarrow> insert a A = A"
   \<comment> \<open>\<open>[simp]\<close> causes recursive calls when there are nested inserts\<close>
   \<comment> \<open>with \<^emph>\<open>quadratic\<close> running time\<close>
   by blast
 
 lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A"
   by blast
 
 lemma insert_commute: "insert x (insert y A) = insert y (insert x A)"
   by blast
 
 lemma insert_subset [simp]: "insert x A \<subseteq> B \<longleftrightarrow> x \<in> B \<and> A \<subseteq> B"
   by blast
 
 lemma mk_disjoint_insert: "a \<in> A \<Longrightarrow> \<exists>B. A = insert a B \<and> a \<notin> B"
   \<comment> \<open>use new \<open>B\<close> rather than \<open>A - {a}\<close> to avoid infinite unfolding\<close>
   by (rule exI [where x = "A - {a}"]) blast
 
 lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a \<longrightarrow> P u}"
   by auto
 
 lemma insert_inter_insert [simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)"
   by blast
 
 lemma insert_disjoint [simp]:
   "insert a A \<inter> B = {} \<longleftrightarrow> a \<notin> B \<and> A \<inter> B = {}"
   "{} = insert a A \<inter> B \<longleftrightarrow> a \<notin> B \<and> {} = A \<inter> B"
   by auto
 
 lemma disjoint_insert [simp]:
   "B \<inter> insert a A = {} \<longleftrightarrow> a \<notin> B \<and> B \<inter> A = {}"
   "{} = A \<inter> insert b B \<longleftrightarrow> b \<notin> A \<and> {} = A \<inter> B"
   by auto
 
 
 text \<open>\<^medskip> \<open>Int\<close>\<close>
 
 lemma Int_absorb: "A \<inter> A = A"
   by (fact inf_idem) (* already simp *)
 
 lemma Int_left_absorb: "A \<inter> (A \<inter> B) = A \<inter> B"
   by (fact inf_left_idem)
 
 lemma Int_commute: "A \<inter> B = B \<inter> A"
   by (fact inf_commute)
 
 lemma Int_left_commute: "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"
   by (fact inf_left_commute)
 
 lemma Int_assoc: "(A \<inter> B) \<inter> C = A \<inter> (B \<inter> C)"
   by (fact inf_assoc)
 
 lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute
   \<comment> \<open>Intersection is an AC-operator\<close>
 
 lemma Int_absorb1: "B \<subseteq> A \<Longrightarrow> A \<inter> B = B"
   by (fact inf_absorb2)
 
 lemma Int_absorb2: "A \<subseteq> B \<Longrightarrow> A \<inter> B = A"
   by (fact inf_absorb1)
 
 lemma Int_empty_left: "{} \<inter> B = {}"
   by (fact inf_bot_left) (* already simp *)
 
 lemma Int_empty_right: "A \<inter> {} = {}"
   by (fact inf_bot_right) (* already simp *)
 
 lemma disjoint_eq_subset_Compl: "A \<inter> B = {} \<longleftrightarrow> A \<subseteq> - B"
   by blast
 
 lemma disjoint_iff: "A \<inter> B = {} \<longleftrightarrow> (\<forall>x. x\<in>A \<longrightarrow> x \<notin> B)"
   by blast
 
 lemma disjoint_iff_not_equal: "A \<inter> B = {} \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"
   by blast
 
 lemma Int_UNIV_left: "UNIV \<inter> B = B"
   by (fact inf_top_left) (* already simp *)
 
 lemma Int_UNIV_right: "A \<inter> UNIV = A"
   by (fact inf_top_right) (* already simp *)
 
 lemma Int_Un_distrib: "A \<inter> (B \<union> C) = (A \<inter> B) \<union> (A \<inter> C)"
   by (fact inf_sup_distrib1)
 
 lemma Int_Un_distrib2: "(B \<union> C) \<inter> A = (B \<inter> A) \<union> (C \<inter> A)"
   by (fact inf_sup_distrib2)
 
 lemma Int_UNIV [simp]: "A \<inter> B = UNIV \<longleftrightarrow> A = UNIV \<and> B = UNIV"
   by (fact inf_eq_top_iff) (* already simp *)
 
 lemma Int_subset_iff [simp]: "C \<subseteq> A \<inter> B \<longleftrightarrow> C \<subseteq> A \<and> C \<subseteq> B"
   by (fact le_inf_iff)
 
 lemma Int_Collect: "x \<in> A \<inter> {x. P x} \<longleftrightarrow> x \<in> A \<and> P x"
   by blast
 
 
 text \<open>\<^medskip> \<open>Un\<close>.\<close>
 
 lemma Un_absorb: "A \<union> A = A"
   by (fact sup_idem) (* already simp *)
 
 lemma Un_left_absorb: "A \<union> (A \<union> B) = A \<union> B"
   by (fact sup_left_idem)
 
 lemma Un_commute: "A \<union> B = B \<union> A"
   by (fact sup_commute)
 
 lemma Un_left_commute: "A \<union> (B \<union> C) = B \<union> (A \<union> C)"
   by (fact sup_left_commute)
 
 lemma Un_assoc: "(A \<union> B) \<union> C = A \<union> (B \<union> C)"
   by (fact sup_assoc)
 
 lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute
   \<comment> \<open>Union is an AC-operator\<close>
 
 lemma Un_absorb1: "A \<subseteq> B \<Longrightarrow> A \<union> B = B"
   by (fact sup_absorb2)
 
 lemma Un_absorb2: "B \<subseteq> A \<Longrightarrow> A \<union> B = A"
   by (fact sup_absorb1)
 
 lemma Un_empty_left: "{} \<union> B = B"
   by (fact sup_bot_left) (* already simp *)
 
 lemma Un_empty_right: "A \<union> {} = A"
   by (fact sup_bot_right) (* already simp *)
 
 lemma Un_UNIV_left: "UNIV \<union> B = UNIV"
   by (fact sup_top_left) (* already simp *)
 
 lemma Un_UNIV_right: "A \<union> UNIV = UNIV"
   by (fact sup_top_right) (* already simp *)
 
 lemma Un_insert_left [simp]: "(insert a B) \<union> C = insert a (B \<union> C)"
   by blast
 
 lemma Un_insert_right [simp]: "A \<union> (insert a B) = insert a (A \<union> B)"
   by blast
 
 lemma Int_insert_left: "(insert a B) \<inter> C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)"
   by auto
 
 lemma Int_insert_left_if0 [simp]: "a \<notin> C \<Longrightarrow> (insert a B) \<inter> C = B \<inter> C"
   by auto
 
 lemma Int_insert_left_if1 [simp]: "a \<in> C \<Longrightarrow> (insert a B) \<inter> C = insert a (B \<inter> C)"
   by auto
 
 lemma Int_insert_right: "A \<inter> (insert a B) = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)"
   by auto
 
 lemma Int_insert_right_if0 [simp]: "a \<notin> A \<Longrightarrow> A \<inter> (insert a B) = A \<inter> B"
   by auto
 
 lemma Int_insert_right_if1 [simp]: "a \<in> A \<Longrightarrow> A \<inter> (insert a B) = insert a (A \<inter> B)"
   by auto
 
 lemma Un_Int_distrib: "A \<union> (B \<inter> C) = (A \<union> B) \<inter> (A \<union> C)"
   by (fact sup_inf_distrib1)
 
 lemma Un_Int_distrib2: "(B \<inter> C) \<union> A = (B \<union> A) \<inter> (C \<union> A)"
   by (fact sup_inf_distrib2)
 
 lemma Un_Int_crazy: "(A \<inter> B) \<union> (B \<inter> C) \<union> (C \<inter> A) = (A \<union> B) \<inter> (B \<union> C) \<inter> (C \<union> A)"
   by blast
 
 lemma subset_Un_eq: "A \<subseteq> B \<longleftrightarrow> A \<union> B = B"
   by (fact le_iff_sup)
 
 lemma Un_empty [iff]: "A \<union> B = {} \<longleftrightarrow> A = {} \<and> B = {}"
   by (fact sup_eq_bot_iff) (* FIXME: already simp *)
 
 lemma Un_subset_iff [simp]: "A \<union> B \<subseteq> C \<longleftrightarrow> A \<subseteq> C \<and> B \<subseteq> C"
   by (fact le_sup_iff)
 
 lemma Un_Diff_Int: "(A - B) \<union> (A \<inter> B) = A"
   by blast
 
 lemma Diff_Int2: "A \<inter> C - B \<inter> C = A \<inter> C - B"
   by blast
 
 lemma subset_UnE: 
   assumes "C \<subseteq> A \<union> B"
   obtains A' B' where "A' \<subseteq> A" "B' \<subseteq> B" "C = A' \<union> B'"
 proof
   show "C \<inter> A \<subseteq> A" "C \<inter> B \<subseteq> B" "C = (C \<inter> A) \<union> (C \<inter> B)"
     using assms by blast+
 qed
 
 lemma Un_Int_eq [simp]: "(S \<union> T) \<inter> S = S" "(S \<union> T) \<inter> T = T" "S \<inter> (S \<union> T) = S" "T \<inter> (S \<union> T) = T"
   by auto
 
 lemma Int_Un_eq [simp]: "(S \<inter> T) \<union> S = S" "(S \<inter> T) \<union> T = T" "S \<union> (S \<inter> T) = S" "T \<union> (S \<inter> T) = T"
   by auto
 
 text \<open>\<^medskip> Set complement\<close>
 
 lemma Compl_disjoint [simp]: "A \<inter> - A = {}"
   by (fact inf_compl_bot)
 
 lemma Compl_disjoint2 [simp]: "- A \<inter> A = {}"
   by (fact compl_inf_bot)
 
 lemma Compl_partition: "A \<union> - A = UNIV"
   by (fact sup_compl_top)
 
 lemma Compl_partition2: "- A \<union> A = UNIV"
   by (fact compl_sup_top)
 
 lemma double_complement: "- (-A) = A" for A :: "'a set"
   by (fact double_compl) (* already simp *)
 
 lemma Compl_Un: "- (A \<union> B) = (- A) \<inter> (- B)"
   by (fact compl_sup) (* already simp *)
 
 lemma Compl_Int: "- (A \<inter> B) = (- A) \<union> (- B)"
   by (fact compl_inf) (* already simp *)
 
 lemma subset_Compl_self_eq: "A \<subseteq> - A \<longleftrightarrow> A = {}"
   by blast
 
 lemma Un_Int_assoc_eq: "(A \<inter> B) \<union> C = A \<inter> (B \<union> C) \<longleftrightarrow> C \<subseteq> A"
   \<comment> \<open>Halmos, Naive Set Theory, page 16.\<close>
   by blast
 
 lemma Compl_UNIV_eq: "- UNIV = {}"
   by (fact compl_top_eq) (* already simp *)
 
 lemma Compl_empty_eq: "- {} = UNIV"
   by (fact compl_bot_eq) (* already simp *)
 
 lemma Compl_subset_Compl_iff [iff]: "- A \<subseteq> - B \<longleftrightarrow> B \<subseteq> A"
   by (fact compl_le_compl_iff) (* FIXME: already simp *)
 
 lemma Compl_eq_Compl_iff [iff]: "- A = - B \<longleftrightarrow> A = B"
   for A B :: "'a set"
   by (fact compl_eq_compl_iff) (* FIXME: already simp *)
 
 lemma Compl_insert: "- insert x A = (- A) - {x}"
   by blast
 
 text \<open>\<^medskip> Bounded quantifiers.
 
   The following are not added to the default simpset because
   (a) they duplicate the body and (b) there are no similar rules for \<open>Int\<close>.
 \<close>
 
 lemma ball_Un: "(\<forall>x \<in> A \<union> B. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x) \<and> (\<forall>x\<in>B. P x)"
   by blast
 
 lemma bex_Un: "(\<exists>x \<in> A \<union> B. P x) \<longleftrightarrow> (\<exists>x\<in>A. P x) \<or> (\<exists>x\<in>B. P x)"
   by blast
 
 
 text \<open>\<^medskip> Set difference.\<close>
 
 lemma Diff_eq: "A - B = A \<inter> (- B)"
   by blast
 
 lemma Diff_eq_empty_iff [simp]: "A - B = {} \<longleftrightarrow> A \<subseteq> B"
   by blast
 
 lemma Diff_cancel [simp]: "A - A = {}"
   by blast
 
 lemma Diff_idemp [simp]: "(A - B) - B = A - B"
   for A B :: "'a set"
   by blast
 
 lemma Diff_triv: "A \<inter> B = {} \<Longrightarrow> A - B = A"
   by (blast elim: equalityE)
 
 lemma empty_Diff [simp]: "{} - A = {}"
   by blast
 
 lemma Diff_empty [simp]: "A - {} = A"
   by blast
 
 lemma Diff_UNIV [simp]: "A - UNIV = {}"
   by blast
 
 lemma Diff_insert0 [simp]: "x \<notin> A \<Longrightarrow> A - insert x B = A - B"
   by blast
 
 lemma Diff_insert: "A - insert a B = A - B - {a}"
   \<comment> \<open>NOT SUITABLE FOR REWRITING since \<open>{a} \<equiv> insert a 0\<close>\<close>
   by blast
 
 lemma Diff_insert2: "A - insert a B = A - {a} - B"
   \<comment> \<open>NOT SUITABLE FOR REWRITING since \<open>{a} \<equiv> insert a 0\<close>\<close>
   by blast
 
 lemma insert_Diff_if: "insert x A - B = (if x \<in> B then A - B else insert x (A - B))"
   by auto
 
 lemma insert_Diff1 [simp]: "x \<in> B \<Longrightarrow> insert x A - B = A - B"
   by blast
 
 lemma insert_Diff_single[simp]: "insert a (A - {a}) = insert a A"
   by blast
 
 lemma insert_Diff: "a \<in> A \<Longrightarrow> insert a (A - {a}) = A"
   by blast
 
 lemma Diff_insert_absorb: "x \<notin> A \<Longrightarrow> (insert x A) - {x} = A"
   by auto
 
 lemma Diff_disjoint [simp]: "A \<inter> (B - A) = {}"
   by blast
 
 lemma Diff_partition: "A \<subseteq> B \<Longrightarrow> A \<union> (B - A) = B"
   by blast
 
 lemma double_diff: "A \<subseteq> B \<Longrightarrow> B \<subseteq> C \<Longrightarrow> B - (C - A) = A"
   by blast
 
 lemma Un_Diff_cancel [simp]: "A \<union> (B - A) = A \<union> B"
   by blast
 
 lemma Un_Diff_cancel2 [simp]: "(B - A) \<union> A = B \<union> A"
   by blast
 
 lemma Diff_Un: "A - (B \<union> C) = (A - B) \<inter> (A - C)"
   by blast
 
 lemma Diff_Int: "A - (B \<inter> C) = (A - B) \<union> (A - C)"
   by blast
 
 lemma Diff_Diff_Int: "A - (A - B) = A \<inter> B"
   by blast
 
 lemma Un_Diff: "(A \<union> B) - C = (A - C) \<union> (B - C)"
   by blast
 
 lemma Int_Diff: "(A \<inter> B) - C = A \<inter> (B - C)"
   by blast
 
 lemma Diff_Int_distrib: "C \<inter> (A - B) = (C \<inter> A) - (C \<inter> B)"
   by blast
 
 lemma Diff_Int_distrib2: "(A - B) \<inter> C = (A \<inter> C) - (B \<inter> C)"
   by blast
 
 lemma Diff_Compl [simp]: "A - (- B) = A \<inter> B"
   by auto
 
 lemma Compl_Diff_eq [simp]: "- (A - B) = - A \<union> B"
   by blast
 
 lemma subset_Compl_singleton [simp]: "A \<subseteq> - {b} \<longleftrightarrow> b \<notin> A"
   by blast
 
 text \<open>\<^medskip> Quantification over type \<^typ>\<open>bool\<close>.\<close>
 
 lemma bool_induct: "P True \<Longrightarrow> P False \<Longrightarrow> P x"
   by (cases x) auto
 
 lemma all_bool_eq: "(\<forall>b. P b) \<longleftrightarrow> P True \<and> P False"
   by (auto intro: bool_induct)
 
 lemma bool_contrapos: "P x \<Longrightarrow> \<not> P False \<Longrightarrow> P True"
   by (cases x) auto
 
 lemma ex_bool_eq: "(\<exists>b. P b) \<longleftrightarrow> P True \<or> P False"
   by (auto intro: bool_contrapos)
 
 lemma UNIV_bool: "UNIV = {False, True}"
   by (auto intro: bool_induct)
 
 text \<open>\<^medskip> \<open>Pow\<close>\<close>
 
 lemma Pow_empty [simp]: "Pow {} = {{}}"
   by (auto simp add: Pow_def)
 
 lemma Pow_singleton_iff [simp]: "Pow X = {Y} \<longleftrightarrow> X = {} \<and> Y = {}"
   by blast  (* somewhat slow *)
 
 lemma Pow_insert: "Pow (insert a A) = Pow A \<union> (insert a ` Pow A)"
   by (blast intro: image_eqI [where ?x = "u - {a}" for u])
 
 lemma Pow_Compl: "Pow (- A) = {- B | B. A \<in> Pow B}"
   by (blast intro: exI [where ?x = "- u" for u])
 
 lemma Pow_UNIV [simp]: "Pow UNIV = UNIV"
   by blast
 
 lemma Un_Pow_subset: "Pow A \<union> Pow B \<subseteq> Pow (A \<union> B)"
   by blast
 
 lemma Pow_Int_eq [simp]: "Pow (A \<inter> B) = Pow A \<inter> Pow B"
   by blast
 
 
 text \<open>\<^medskip> Miscellany.\<close>
 
+lemma Int_Diff_disjoint: "A \<inter> B \<inter> (A - B) = {}"
+  by blast
+
+lemma Int_Diff_Un: "A \<inter> B \<union> (A - B) = A"
+  by blast
+
 lemma set_eq_subset: "A = B \<longleftrightarrow> A \<subseteq> B \<and> B \<subseteq> A"
   by blast
 
 lemma subset_iff: "A \<subseteq> B \<longleftrightarrow> (\<forall>t. t \<in> A \<longrightarrow> t \<in> B)"
   by blast
 
 lemma subset_iff_psubset_eq: "A \<subseteq> B \<longleftrightarrow> A \<subset> B \<or> A = B"
   unfolding less_le by blast
 
 lemma all_not_in_conv [simp]: "(\<forall>x. x \<notin> A) \<longleftrightarrow> A = {}"
   by blast
 
 lemma ex_in_conv: "(\<exists>x. x \<in> A) \<longleftrightarrow> A \<noteq> {}"
   by blast
 
 lemma ball_simps [simp, no_atp]:
   "\<And>A P Q. (\<forall>x\<in>A. P x \<or> Q) \<longleftrightarrow> ((\<forall>x\<in>A. P x) \<or> Q)"
   "\<And>A P Q. (\<forall>x\<in>A. P \<or> Q x) \<longleftrightarrow> (P \<or> (\<forall>x\<in>A. Q x))"
   "\<And>A P Q. (\<forall>x\<in>A. P \<longrightarrow> Q x) \<longleftrightarrow> (P \<longrightarrow> (\<forall>x\<in>A. Q x))"
   "\<And>A P Q. (\<forall>x\<in>A. P x \<longrightarrow> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<longrightarrow> Q)"
   "\<And>P. (\<forall>x\<in>{}. P x) \<longleftrightarrow> True"
   "\<And>P. (\<forall>x\<in>UNIV. P x) \<longleftrightarrow> (\<forall>x. P x)"
   "\<And>a B P. (\<forall>x\<in>insert a B. P x) \<longleftrightarrow> (P a \<and> (\<forall>x\<in>B. P x))"
   "\<And>P Q. (\<forall>x\<in>Collect Q. P x) \<longleftrightarrow> (\<forall>x. Q x \<longrightarrow> P x)"
   "\<And>A P f. (\<forall>x\<in>f`A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P (f x))"
   "\<And>A P. (\<not> (\<forall>x\<in>A. P x)) \<longleftrightarrow> (\<exists>x\<in>A. \<not> P x)"
   by auto
 
 lemma bex_simps [simp, no_atp]:
   "\<And>A P Q. (\<exists>x\<in>A. P x \<and> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<and> Q)"
   "\<And>A P Q. (\<exists>x\<in>A. P \<and> Q x) \<longleftrightarrow> (P \<and> (\<exists>x\<in>A. Q x))"
   "\<And>P. (\<exists>x\<in>{}. P x) \<longleftrightarrow> False"
   "\<And>P. (\<exists>x\<in>UNIV. P x) \<longleftrightarrow> (\<exists>x. P x)"
   "\<And>a B P. (\<exists>x\<in>insert a B. P x) \<longleftrightarrow> (P a \<or> (\<exists>x\<in>B. P x))"
   "\<And>P Q. (\<exists>x\<in>Collect Q. P x) \<longleftrightarrow> (\<exists>x. Q x \<and> P x)"
   "\<And>A P f. (\<exists>x\<in>f`A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P (f x))"
   "\<And>A P. (\<not>(\<exists>x\<in>A. P x)) \<longleftrightarrow> (\<forall>x\<in>A. \<not> P x)"
   by auto
 
 lemma ex_image_cong_iff [simp, no_atp]:
   "(\<exists>x. x\<in>f`A) \<longleftrightarrow> A \<noteq> {}" "(\<exists>x. x\<in>f`A \<and> P x) \<longleftrightarrow> (\<exists>x\<in>A. P (f x))"
   by auto
 
 subsubsection \<open>Monotonicity of various operations\<close>
 
 lemma image_mono: "A \<subseteq> B \<Longrightarrow> f ` A \<subseteq> f ` B"
   by blast
 
 lemma Pow_mono: "A \<subseteq> B \<Longrightarrow> Pow A \<subseteq> Pow B"
   by blast
 
 lemma insert_mono: "C \<subseteq> D \<Longrightarrow> insert a C \<subseteq> insert a D"
   by blast
 
 lemma Un_mono: "A \<subseteq> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> A \<union> B \<subseteq> C \<union> D"
   by (fact sup_mono)
 
 lemma Int_mono: "A \<subseteq> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> A \<inter> B \<subseteq> C \<inter> D"
   by (fact inf_mono)
 
 lemma Diff_mono: "A \<subseteq> C \<Longrightarrow> D \<subseteq> B \<Longrightarrow> A - B \<subseteq> C - D"
   by blast
 
 lemma Compl_anti_mono: "A \<subseteq> B \<Longrightarrow> - B \<subseteq> - A"
   by (fact compl_mono)
 
 text \<open>\<^medskip> Monotonicity of implications.\<close>
 
 lemma in_mono: "A \<subseteq> B \<Longrightarrow> x \<in> A \<longrightarrow> x \<in> B"
   by (rule impI) (erule subsetD)
 
 lemma conj_mono: "P1 \<longrightarrow> Q1 \<Longrightarrow> P2 \<longrightarrow> Q2 \<Longrightarrow> (P1 \<and> P2) \<longrightarrow> (Q1 \<and> Q2)"
   by iprover
 
 lemma disj_mono: "P1 \<longrightarrow> Q1 \<Longrightarrow> P2 \<longrightarrow> Q2 \<Longrightarrow> (P1 \<or> P2) \<longrightarrow> (Q1 \<or> Q2)"
   by iprover
 
 lemma imp_mono: "Q1 \<longrightarrow> P1 \<Longrightarrow> P2 \<longrightarrow> Q2 \<Longrightarrow> (P1 \<longrightarrow> P2) \<longrightarrow> (Q1 \<longrightarrow> Q2)"
   by iprover
 
 lemma imp_refl: "P \<longrightarrow> P" ..
 
 lemma not_mono: "Q \<longrightarrow> P \<Longrightarrow> \<not> P \<longrightarrow> \<not> Q"
   by iprover
 
 lemma ex_mono: "(\<And>x. P x \<longrightarrow> Q x) \<Longrightarrow> (\<exists>x. P x) \<longrightarrow> (\<exists>x. Q x)"
   by iprover
 
 lemma all_mono: "(\<And>x. P x \<longrightarrow> Q x) \<Longrightarrow> (\<forall>x. P x) \<longrightarrow> (\<forall>x. Q x)"
   by iprover
 
 lemma Collect_mono: "(\<And>x. P x \<longrightarrow> Q x) \<Longrightarrow> Collect P \<subseteq> Collect Q"
   by blast
 
 lemma Int_Collect_mono: "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> P x \<longrightarrow> Q x) \<Longrightarrow> A \<inter> Collect P \<subseteq> B \<inter> Collect Q"
   by blast
 
 lemmas basic_monos =
   subset_refl imp_refl disj_mono conj_mono ex_mono Collect_mono in_mono
 
 lemma eq_to_mono: "a = b \<Longrightarrow> c = d \<Longrightarrow> b \<longrightarrow> d \<Longrightarrow> a \<longrightarrow> c"
   by iprover
 
 
 subsubsection \<open>Inverse image of a function\<close>
 
 definition vimage :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b set \<Rightarrow> 'a set"  (infixr "-`" 90)
   where "f -` B \<equiv> {x. f x \<in> B}"
 
 lemma vimage_eq [simp]: "a \<in> f -` B \<longleftrightarrow> f a \<in> B"
   unfolding vimage_def by blast
 
 lemma vimage_singleton_eq: "a \<in> f -` {b} \<longleftrightarrow> f a = b"
   by simp
 
 lemma vimageI [intro]: "f a = b \<Longrightarrow> b \<in> B \<Longrightarrow> a \<in> f -` B"
   unfolding vimage_def by blast
 
 lemma vimageI2: "f a \<in> A \<Longrightarrow> a \<in> f -` A"
   unfolding vimage_def by fast
 
 lemma vimageE [elim!]: "a \<in> f -` B \<Longrightarrow> (\<And>x. f a = x \<Longrightarrow> x \<in> B \<Longrightarrow> P) \<Longrightarrow> P"
   unfolding vimage_def by blast
 
 lemma vimageD: "a \<in> f -` A \<Longrightarrow> f a \<in> A"
   unfolding vimage_def by fast
 
 lemma vimage_empty [simp]: "f -` {} = {}"
   by blast
 
 lemma vimage_Compl: "f -` (- A) = - (f -` A)"
   by blast
 
 lemma vimage_Un [simp]: "f -` (A \<union> B) = (f -` A) \<union> (f -` B)"
   by blast
 
 lemma vimage_Int [simp]: "f -` (A \<inter> B) = (f -` A) \<inter> (f -` B)"
   by fast
 
 lemma vimage_Collect_eq [simp]: "f -` Collect P = {y. P (f y)}"
   by blast
 
 lemma vimage_Collect: "(\<And>x. P (f x) = Q x) \<Longrightarrow> f -` (Collect P) = Collect Q"
   by blast
 
 lemma vimage_insert: "f -` (insert a B) = (f -` {a}) \<union> (f -` B)"
   \<comment> \<open>NOT suitable for rewriting because of the recurrence of \<open>{a}\<close>.\<close>
   by blast
 
 lemma vimage_Diff: "f -` (A - B) = (f -` A) - (f -` B)"
   by blast
 
 lemma vimage_UNIV [simp]: "f -` UNIV = UNIV"
   by blast
 
 lemma vimage_mono: "A \<subseteq> B \<Longrightarrow> f -` A \<subseteq> f -` B"
   \<comment> \<open>monotonicity\<close>
   by blast
 
 lemma vimage_image_eq: "f -` (f ` A) = {y. \<exists>x\<in>A. f x = f y}"
   by (blast intro: sym)
 
 lemma image_vimage_subset: "f ` (f -` A) \<subseteq> A"
   by blast
 
 lemma image_vimage_eq [simp]: "f ` (f -` A) = A \<inter> range f"
   by blast
 
 lemma image_subset_iff_subset_vimage: "f ` A \<subseteq> B \<longleftrightarrow> A \<subseteq> f -` B"
   by blast
 
 lemma subset_vimage_iff: "A \<subseteq> f -` B \<longleftrightarrow> (\<forall>x\<in>A. f x \<in> B)"
   by auto
 
 lemma vimage_const [simp]: "((\<lambda>x. c) -` A) = (if c \<in> A then UNIV else {})"
   by auto
 
 lemma vimage_if [simp]: "((\<lambda>x. if x \<in> B then c else d) -` A) =
    (if c \<in> A then (if d \<in> A then UNIV else B)
     else if d \<in> A then - B else {})"
   by (auto simp add: vimage_def)
 
 lemma vimage_inter_cong: "(\<And> w. w \<in> S \<Longrightarrow> f w = g w) \<Longrightarrow> f -` y \<inter> S = g -` y \<inter> S"
   by auto
 
 lemma vimage_ident [simp]: "(\<lambda>x. x) -` Y = Y"
   by blast
 
 
 subsubsection \<open>Singleton sets\<close>
 
 definition is_singleton :: "'a set \<Rightarrow> bool"
   where "is_singleton A \<longleftrightarrow> (\<exists>x. A = {x})"
 
 lemma is_singletonI [simp, intro!]: "is_singleton {x}"
   unfolding is_singleton_def by simp
 
 lemma is_singletonI': "A \<noteq> {} \<Longrightarrow> (\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x = y) \<Longrightarrow> is_singleton A"
   unfolding is_singleton_def by blast
 
 lemma is_singletonE: "is_singleton A \<Longrightarrow> (\<And>x. A = {x} \<Longrightarrow> P) \<Longrightarrow> P"
   unfolding is_singleton_def by blast
 
 
 subsubsection \<open>Getting the contents of a singleton set\<close>
 
 definition the_elem :: "'a set \<Rightarrow> 'a"
   where "the_elem X = (THE x. X = {x})"
 
 lemma the_elem_eq [simp]: "the_elem {x} = x"
   by (simp add: the_elem_def)
 
 lemma is_singleton_the_elem: "is_singleton A \<longleftrightarrow> A = {the_elem A}"
   by (auto simp: is_singleton_def)
 
 lemma the_elem_image_unique:
   assumes "A \<noteq> {}"
     and *: "\<And>y. y \<in> A \<Longrightarrow> f y = f x"
   shows "the_elem (f ` A) = f x"
   unfolding the_elem_def
 proof (rule the1_equality)
   from \<open>A \<noteq> {}\<close> obtain y where "y \<in> A" by auto
   with * have "f x = f y" by simp
   with \<open>y \<in> A\<close> have "f x \<in> f ` A" by blast
   with * show "f ` A = {f x}" by auto
   then show "\<exists>!x. f ` A = {x}" by auto
 qed
 
 
 subsubsection \<open>Least value operator\<close>
 
 lemma Least_mono: "mono f \<Longrightarrow> \<exists>x\<in>S. \<forall>y\<in>S. x \<le> y \<Longrightarrow> (LEAST y. y \<in> f ` S) = f (LEAST x. x \<in> S)"
   for f :: "'a::order \<Rightarrow> 'b::order"
   \<comment> \<open>Courtesy of Stephan Merz\<close>
   apply clarify
   apply (erule_tac P = "\<lambda>x. x \<in> S" in LeastI2_order)
    apply fast
   apply (rule LeastI2_order)
     apply (auto elim: monoD intro!: order_antisym)
   done
 
 
 subsubsection \<open>Monad operation\<close>
 
 definition bind :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set"
   where "bind A f = {x. \<exists>B \<in> f`A. x \<in> B}"
 
 hide_const (open) bind
 
 lemma bind_bind: "Set.bind (Set.bind A B) C = Set.bind A (\<lambda>x. Set.bind (B x) C)"
   for A :: "'a set"
   by (auto simp: bind_def)
 
 lemma empty_bind [simp]: "Set.bind {} f = {}"
   by (simp add: bind_def)
 
 lemma nonempty_bind_const: "A \<noteq> {} \<Longrightarrow> Set.bind A (\<lambda>_. B) = B"
   by (auto simp: bind_def)
 
 lemma bind_const: "Set.bind A (\<lambda>_. B) = (if A = {} then {} else B)"
   by (auto simp: bind_def)
 
 lemma bind_singleton_conv_image: "Set.bind A (\<lambda>x. {f x}) = f ` A"
   by (auto simp: bind_def)
 
 
 subsubsection \<open>Operations for execution\<close>
 
 definition is_empty :: "'a set \<Rightarrow> bool"
   where [code_abbrev]: "is_empty A \<longleftrightarrow> A = {}"
 
 hide_const (open) is_empty
 
 definition remove :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set"
   where [code_abbrev]: "remove x A = A - {x}"
 
 hide_const (open) remove
 
 lemma member_remove [simp]: "x \<in> Set.remove y A \<longleftrightarrow> x \<in> A \<and> x \<noteq> y"
   by (simp add: remove_def)
 
 definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set"
   where [code_abbrev]: "filter P A = {a \<in> A. P a}"
 
 hide_const (open) filter
 
 lemma member_filter [simp]: "x \<in> Set.filter P A \<longleftrightarrow> x \<in> A \<and> P x"
   by (simp add: filter_def)
 
 instantiation set :: (equal) equal
 begin
 
 definition "HOL.equal A B \<longleftrightarrow> A \<subseteq> B \<and> B \<subseteq> A"
 
 instance by standard (auto simp add: equal_set_def)
 
 end
 
 
 text \<open>Misc\<close>
 
 definition pairwise :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
   where "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y \<in> S. x \<noteq> y \<longrightarrow> R x y)"
 
 lemma pairwise_alt: "pairwise R S \<longleftrightarrow> (\<forall>x\<in>S. \<forall>y\<in>S-{x}. R x y)"
 by (auto simp add: pairwise_def)
 
 lemma pairwise_trivial [simp]: "pairwise (\<lambda>i j. j \<noteq> i) I"
   by (auto simp: pairwise_def)
 
 lemma pairwiseI [intro?]:
   "pairwise R S" if "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x \<noteq> y \<Longrightarrow> R x y"
   using that by (simp add: pairwise_def)
 
 lemma pairwiseD:
   "R x y" and "R y x"
   if "pairwise R S" "x \<in> S" and "y \<in> S" and "x \<noteq> y"
   using that by (simp_all add: pairwise_def)
 
 lemma pairwise_empty [simp]: "pairwise P {}"
   by (simp add: pairwise_def)
 
 lemma pairwise_singleton [simp]: "pairwise P {A}"
   by (simp add: pairwise_def)
 
 lemma pairwise_insert:
   "pairwise r (insert x s) \<longleftrightarrow> (\<forall>y. y \<in> s \<and> y \<noteq> x \<longrightarrow> r x y \<and> r y x) \<and> pairwise r s"
   by (force simp: pairwise_def)
 
 lemma pairwise_subset: "pairwise P S \<Longrightarrow> T \<subseteq> S \<Longrightarrow> pairwise P T"
   by (force simp: pairwise_def)
 
 lemma pairwise_mono: "\<lbrakk>pairwise P A; \<And>x y. P x y \<Longrightarrow> Q x y; B \<subseteq> A\<rbrakk> \<Longrightarrow> pairwise Q B"
   by (fastforce simp: pairwise_def)
 
 lemma pairwise_imageI:
   "pairwise P (f ` A)"
   if "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> f x \<noteq> f y \<Longrightarrow> P (f x) (f y)"
   using that by (auto intro: pairwiseI)
 
 lemma pairwise_image: "pairwise r (f ` s) \<longleftrightarrow> pairwise (\<lambda>x y. (f x \<noteq> f y) \<longrightarrow> r (f x) (f y)) s"
   by (force simp: pairwise_def)
 
 definition disjnt :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
   where "disjnt A B \<longleftrightarrow> A \<inter> B = {}"
 
 lemma disjnt_self_iff_empty [simp]: "disjnt S S \<longleftrightarrow> S = {}"
   by (auto simp: disjnt_def)
 
 lemma disjnt_iff: "disjnt A B \<longleftrightarrow> (\<forall>x. \<not> (x \<in> A \<and> x \<in> B))"
   by (force simp: disjnt_def)
 
 lemma disjnt_sym: "disjnt A B \<Longrightarrow> disjnt B A"
   using disjnt_iff by blast
 
 lemma disjnt_empty1 [simp]: "disjnt {} A" and disjnt_empty2 [simp]: "disjnt A {}"
   by (auto simp: disjnt_def)
 
 lemma disjnt_insert1 [simp]: "disjnt (insert a X) Y \<longleftrightarrow> a \<notin> Y \<and> disjnt X Y"
   by (simp add: disjnt_def)
 
 lemma disjnt_insert2 [simp]: "disjnt Y (insert a X) \<longleftrightarrow> a \<notin> Y \<and> disjnt Y X"
   by (simp add: disjnt_def)
 
 lemma disjnt_subset1 : "\<lbrakk>disjnt X Y; Z \<subseteq> X\<rbrakk> \<Longrightarrow> disjnt Z Y"
   by (auto simp: disjnt_def)
 
 lemma disjnt_subset2 : "\<lbrakk>disjnt X Y; Z \<subseteq> Y\<rbrakk> \<Longrightarrow> disjnt X Z"
   by (auto simp: disjnt_def)
 
 lemma disjnt_Un1 [simp]: "disjnt (A \<union> B) C \<longleftrightarrow> disjnt A C \<and> disjnt B C"
   by (auto simp: disjnt_def)
 
 lemma disjnt_Un2 [simp]: "disjnt C (A \<union> B) \<longleftrightarrow> disjnt C A \<and> disjnt C B"
   by (auto simp: disjnt_def)
 
 lemma disjoint_image_subset: "\<lbrakk>pairwise disjnt \<A>; \<And>X. X \<in> \<A> \<Longrightarrow> f X \<subseteq> X\<rbrakk> \<Longrightarrow> pairwise disjnt (f `\<A>)"
   unfolding disjnt_def pairwise_def by fast
 
 lemma pairwise_disjnt_iff: "pairwise disjnt \<A> \<longleftrightarrow> (\<forall>x. \<exists>\<^sub>\<le>\<^sub>1 X. X \<in> \<A> \<and> x \<in> X)"
   by (auto simp: Uniq_def disjnt_iff pairwise_def)
 
 lemma disjnt_insert: \<^marker>\<open>contributor \<open>Lars Hupel\<close>\<close>
   \<open>disjnt (insert x M) N\<close> if \<open>x \<notin> N\<close> \<open>disjnt M N\<close>
   using that by (simp add: disjnt_def)
 
 lemma Int_emptyI: "(\<And>x. x \<in> A \<Longrightarrow> x \<in> B \<Longrightarrow> False) \<Longrightarrow> A \<inter> B = {}"
   by blast
 
 lemma in_image_insert_iff:
   assumes "\<And>C. C \<in> B \<Longrightarrow> x \<notin> C"
   shows "A \<in> insert x ` B \<longleftrightarrow> x \<in> A \<and> A - {x} \<in> B" (is "?P \<longleftrightarrow> ?Q")
 proof
   assume ?P then show ?Q
     using assms by auto
 next
   assume ?Q
   then have "x \<in> A" and "A - {x} \<in> B"
     by simp_all
   from \<open>A - {x} \<in> B\<close> have "insert x (A - {x}) \<in> insert x ` B"
     by (rule imageI)
   also from \<open>x \<in> A\<close>
   have "insert x (A - {x}) = A"
     by auto
   finally show ?P .
 qed
 
 hide_const (open) member not_member
 
 lemmas equalityI = subset_antisym
 lemmas set_mp = subsetD
 lemmas set_rev_mp = rev_subsetD
 
 ML \<open>
 val Ball_def = @{thm Ball_def}
 val Bex_def = @{thm Bex_def}
 val CollectD = @{thm CollectD}
 val CollectE = @{thm CollectE}
 val CollectI = @{thm CollectI}
 val Collect_conj_eq = @{thm Collect_conj_eq}
 val Collect_mem_eq = @{thm Collect_mem_eq}
 val IntD1 = @{thm IntD1}
 val IntD2 = @{thm IntD2}
 val IntE = @{thm IntE}
 val IntI = @{thm IntI}
 val Int_Collect = @{thm Int_Collect}
 val UNIV_I = @{thm UNIV_I}
 val UNIV_witness = @{thm UNIV_witness}
 val UnE = @{thm UnE}
 val UnI1 = @{thm UnI1}
 val UnI2 = @{thm UnI2}
 val ballE = @{thm ballE}
 val ballI = @{thm ballI}
 val bexCI = @{thm bexCI}
 val bexE = @{thm bexE}
 val bexI = @{thm bexI}
 val bex_triv = @{thm bex_triv}
 val bspec = @{thm bspec}
 val contra_subsetD = @{thm contra_subsetD}
 val equalityCE = @{thm equalityCE}
 val equalityD1 = @{thm equalityD1}
 val equalityD2 = @{thm equalityD2}
 val equalityE = @{thm equalityE}
 val equalityI = @{thm equalityI}
 val imageE = @{thm imageE}
 val imageI = @{thm imageI}
 val image_Un = @{thm image_Un}
 val image_insert = @{thm image_insert}
 val insert_commute = @{thm insert_commute}
 val insert_iff = @{thm insert_iff}
 val mem_Collect_eq = @{thm mem_Collect_eq}
 val rangeE = @{thm rangeE}
 val rangeI = @{thm rangeI}
 val range_eqI = @{thm range_eqI}
 val subsetCE = @{thm subsetCE}
 val subsetD = @{thm subsetD}
 val subsetI = @{thm subsetI}
 val subset_refl = @{thm subset_refl}
 val subset_trans = @{thm subset_trans}
 val vimageD = @{thm vimageD}
 val vimageE = @{thm vimageE}
 val vimageI = @{thm vimageI}
 val vimageI2 = @{thm vimageI2}
 val vimage_Collect = @{thm vimage_Collect}
 val vimage_Int = @{thm vimage_Int}
 val vimage_Un = @{thm vimage_Un}
 \<close>
 
 end