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
@@ -1,453 +1,499 @@
 (*  Author:     Johannes Hölzl, TU München
 *)
 
 section \<open>Partitions and Disjoint Sets\<close>
 
 theory Disjoint_Sets
   imports Main
 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"
   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)
 
 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 
+
 end