# HG changeset patch
# User bulwahn
# Date 1307437925 -7200
# Node ID 93b1183e43e548107c90af993c28ab187dede129
# Parent  da47097bd589f029b81980a038078df4d9d98715
splitting Cset into Cset and List_Cset

diff -r da47097bd589 -r 93b1183e43e5 src/HOL/IsaMakefile
--- a/src/HOL/IsaMakefile	Tue Jun 07 11:11:01 2011 +0200
+++ b/src/HOL/IsaMakefile	Tue Jun 07 11:12:05 2011 +0200
@@ -453,7 +453,7 @@
   Library/Indicator_Function.thy Library/Infinite_Set.thy		\
   Library/Inner_Product.thy Library/Kleene_Algebra.thy			\
   Library/LaTeXsugar.thy Library/Lattice_Algebras.thy			\
-  Library/Lattice_Syntax.thy Library/Library.thy			\
+  Library/Lattice_Syntax.thy Library/Library.thy Library/List_Cset.thy 	\
   Library/List_Prefix.thy Library/List_lexord.thy Library/Mapping.thy	\
   Library/Monad_Syntax.thy Library/More_List.thy Library/More_Set.thy	\
   Library/Multiset.thy Library/Nat_Bijection.thy			\
diff -r da47097bd589 -r 93b1183e43e5 src/HOL/Library/Cset.thy
--- a/src/HOL/Library/Cset.thy	Tue Jun 07 11:11:01 2011 +0200
+++ b/src/HOL/Library/Cset.thy	Tue Jun 07 11:12:05 2011 +0200
@@ -35,66 +35,6 @@
   by (simp add: Cset.set_eq_iff)
 hide_fact (open) set_eqI
 
-declare mem_def [simp]
-
-definition set :: "'a list \<Rightarrow> 'a Cset.set" where
-  "set xs = Set (List.set xs)"
-hide_const (open) set
-
-lemma member_set [simp]:
-  "member (Cset.set xs) = set xs"
-  by (simp add: set_def)
-hide_fact (open) member_set
-
-definition coset :: "'a list \<Rightarrow> 'a Cset.set" where
-  "coset xs = Set (- set xs)"
-hide_const (open) coset
-
-lemma member_coset [simp]:
-  "member (Cset.coset xs) = - set xs"
-  by (simp add: coset_def)
-hide_fact (open) member_coset
-
-code_datatype Cset.set Cset.coset
-
-lemma member_code [code]:
-  "member (Cset.set xs) = List.member xs"
-  "member (Cset.coset xs) = Not \<circ> List.member xs"
-  by (simp_all add: fun_eq_iff member_def fun_Compl_def bool_Compl_def)
-
-lemma member_image_UNIV [simp]:
-  "member ` UNIV = UNIV"
-proof -
-  have "\<And>A \<Colon> 'a set. \<exists>B \<Colon> 'a Cset.set. A = member B"
-  proof
-    fix A :: "'a set"
-    show "A = member (Set A)" by simp
-  qed
-  then show ?thesis by (simp add: image_def)
-qed
-
-definition (in term_syntax)
-  setify :: "'a\<Colon>typerep list \<times> (unit \<Rightarrow> Code_Evaluation.term)
-    \<Rightarrow> 'a Cset.set \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
-  [code_unfold]: "setify xs = Code_Evaluation.valtermify Cset.set {\<cdot>} xs"
-
-notation fcomp (infixl "\<circ>>" 60)
-notation scomp (infixl "\<circ>\<rightarrow>" 60)
-
-instantiation Cset.set :: (random) random
-begin
-
-definition
-  "Quickcheck.random i = Quickcheck.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (setify xs))"
-
-instance ..
-
-end
-
-no_notation fcomp (infixl "\<circ>>" 60)
-no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
-
-
 subsection {* Lattice instantiation *}
 
 instantiation Cset.set :: (type) boolean_algebra
@@ -149,185 +89,39 @@
 definition is_empty :: "'a Cset.set \<Rightarrow> bool" where
   [simp]: "is_empty A \<longleftrightarrow> More_Set.is_empty (member A)"
 
-lemma is_empty_set [code]:
-  "is_empty (Cset.set xs) \<longleftrightarrow> List.null xs"
-  by (simp add: is_empty_set)
-hide_fact (open) is_empty_set
-
-lemma empty_set [code]:
-  "bot = Cset.set []"
-  by (simp add: set_def)
-hide_fact (open) empty_set
-
-lemma UNIV_set [code]:
-  "top = Cset.coset []"
-  by (simp add: coset_def)
-hide_fact (open) UNIV_set
-
 definition insert :: "'a \<Rightarrow> 'a Cset.set \<Rightarrow> 'a Cset.set" where
   [simp]: "insert x A = Set (Set.insert x (member A))"
 
-lemma insert_set [code]:
-  "insert x (Cset.set xs) = Cset.set (List.insert x xs)"
-  "insert x (Cset.coset xs) = Cset.coset (removeAll x xs)"
-  by (simp_all add: set_def coset_def)
-
 definition remove :: "'a \<Rightarrow> 'a Cset.set \<Rightarrow> 'a Cset.set" where
   [simp]: "remove x A = Set (More_Set.remove x (member A))"
 
-lemma remove_set [code]:
-  "remove x (Cset.set xs) = Cset.set (removeAll x xs)"
-  "remove x (Cset.coset xs) = Cset.coset (List.insert x xs)"
-  by (simp_all add: set_def coset_def remove_set_compl)
-    (simp add: More_Set.remove_def)
-
 definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a Cset.set \<Rightarrow> 'b Cset.set" where
   [simp]: "map f A = Set (image f (member A))"
 
-lemma map_set [code]:
-  "map f (Cset.set xs) = Cset.set (remdups (List.map f xs))"
-  by (simp add: set_def)
-
 enriched_type map: map
   by (simp_all add: fun_eq_iff image_compose)
 
 definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a Cset.set \<Rightarrow> 'a Cset.set" where
   [simp]: "filter P A = Set (More_Set.project P (member A))"
 
-lemma filter_set [code]:
-  "filter P (Cset.set xs) = Cset.set (List.filter P xs)"
-  by (simp add: set_def project_set)
-
 definition forall :: "('a \<Rightarrow> bool) \<Rightarrow> 'a Cset.set \<Rightarrow> bool" where
   [simp]: "forall P A \<longleftrightarrow> Ball (member A) P"
 
-lemma forall_set [code]:
-  "forall P (Cset.set xs) \<longleftrightarrow> list_all P xs"
-  by (simp add: set_def list_all_iff)
-
 definition exists :: "('a \<Rightarrow> bool) \<Rightarrow> 'a Cset.set \<Rightarrow> bool" where
   [simp]: "exists P A \<longleftrightarrow> Bex (member A) P"
 
-lemma exists_set [code]:
-  "exists P (Cset.set xs) \<longleftrightarrow> list_ex P xs"
-  by (simp add: set_def list_ex_iff)
-
 definition card :: "'a Cset.set \<Rightarrow> nat" where
   [simp]: "card A = Finite_Set.card (member A)"
-
-lemma card_set [code]:
-  "card (Cset.set xs) = length (remdups xs)"
-proof -
-  have "Finite_Set.card (set (remdups xs)) = length (remdups xs)"
-    by (rule distinct_card) simp
-  then show ?thesis by (simp add: set_def)
-qed
-
-lemma compl_set [simp, code]:
-  "- Cset.set xs = Cset.coset xs"
-  by (simp add: set_def coset_def)
-
-lemma compl_coset [simp, code]:
-  "- Cset.coset xs = Cset.set xs"
-  by (simp add: set_def coset_def)
-
-
-subsection {* Derived operations *}
-
-lemma subset_eq_forall [code]:
-  "A \<le> B \<longleftrightarrow> forall (member B) A"
-  by (simp add: subset_eq)
-
-lemma subset_subset_eq [code]:
-  "A < B \<longleftrightarrow> A \<le> B \<and> \<not> B \<le> (A :: 'a Cset.set)"
-  by (fact less_le_not_le)
-
-instantiation Cset.set :: (type) equal
-begin
-
-definition [code]:
-  "HOL.equal A B \<longleftrightarrow> A \<le> B \<and> B \<le> (A :: 'a Cset.set)"
-
-instance proof
-qed (simp add: equal_set_def set_eq [symmetric] Cset.set_eq_iff)
-
-end
-
-lemma [code nbe]:
-  "HOL.equal (A :: 'a Cset.set) A \<longleftrightarrow> True"
-  by (fact equal_refl)
-
-
-subsection {* Functorial operations *}
-
-lemma inter_project [code]:
-  "inf A (Cset.set xs) = Cset.set (List.filter (member A) xs)"
-  "inf A (Cset.coset xs) = foldr remove xs A"
-proof -
-  show "inf A (Cset.set xs) = Cset.set (List.filter (member A) xs)"
-    by (simp add: inter project_def set_def)
-  have *: "\<And>x::'a. remove = (\<lambda>x. Set \<circ> More_Set.remove x \<circ> member)"
-    by (simp add: fun_eq_iff)
-  have "member \<circ> fold (\<lambda>x. Set \<circ> More_Set.remove x \<circ> member) xs =
-    fold More_Set.remove xs \<circ> member"
-    by (rule fold_commute) (simp add: fun_eq_iff)
-  then have "fold More_Set.remove xs (member A) = 
-    member (fold (\<lambda>x. Set \<circ> More_Set.remove x \<circ> member) xs A)"
-    by (simp add: fun_eq_iff)
-  then have "inf A (Cset.coset xs) = fold remove xs A"
-    by (simp add: Diff_eq [symmetric] minus_set *)
-  moreover have "\<And>x y :: 'a. Cset.remove y \<circ> Cset.remove x = Cset.remove x \<circ> Cset.remove y"
-    by (auto simp add: More_Set.remove_def * intro: ext)
-  ultimately show "inf A (Cset.coset xs) = foldr remove xs A"
-    by (simp add: foldr_fold)
-qed
-
-lemma subtract_remove [code]:
-  "A - Cset.set xs = foldr remove xs A"
-  "A - Cset.coset xs = Cset.set (List.filter (member A) xs)"
-  by (simp_all only: diff_eq compl_set compl_coset inter_project)
-
-lemma union_insert [code]:
-  "sup (Cset.set xs) A = foldr insert xs A"
-  "sup (Cset.coset xs) A = Cset.coset (List.filter (Not \<circ> member A) xs)"
-proof -
-  have *: "\<And>x::'a. insert = (\<lambda>x. Set \<circ> Set.insert x \<circ> member)"
-    by (simp add: fun_eq_iff)
-  have "member \<circ> fold (\<lambda>x. Set \<circ> Set.insert x \<circ> member) xs =
-    fold Set.insert xs \<circ> member"
-    by (rule fold_commute) (simp add: fun_eq_iff)
-  then have "fold Set.insert xs (member A) =
-    member (fold (\<lambda>x. Set \<circ> Set.insert x \<circ> member) xs A)"
-    by (simp add: fun_eq_iff)
-  then have "sup (Cset.set xs) A = fold insert xs A"
-    by (simp add: union_set *)
-  moreover have "\<And>x y :: 'a. Cset.insert y \<circ> Cset.insert x = Cset.insert x \<circ> Cset.insert y"
-    by (auto simp add: * intro: ext)
-  ultimately show "sup (Cset.set xs) A = foldr insert xs A"
-    by (simp add: foldr_fold)
-  show "sup (Cset.coset xs) A = Cset.coset (List.filter (Not \<circ> member A) xs)"
-    by (auto simp add: coset_def)
-qed
-
+  
 context complete_lattice
 begin
 
 definition Infimum :: "'a Cset.set \<Rightarrow> 'a" where
   [simp]: "Infimum A = Inf (member A)"
 
-lemma Infimum_inf [code]:
-  "Infimum (Cset.set As) = foldr inf As top"
-  "Infimum (Cset.coset []) = bot"
-  by (simp_all add: Inf_set_foldr Inf_UNIV)
-
 definition Supremum :: "'a Cset.set \<Rightarrow> 'a" where
   [simp]: "Supremum A = Sup (member A)"
 
-lemma Supremum_sup [code]:
-  "Supremum (Cset.set As) = foldr sup As bot"
-  "Supremum (Cset.coset []) = top"
-  by (simp_all add: Sup_set_foldr Sup_UNIV)
-
 end
 
 
@@ -351,7 +145,7 @@
 declare mem_def [simp del]
 
 
-hide_const (open) setify is_empty insert remove map filter forall exists card
+hide_const (open) is_empty insert remove map filter forall exists card
   Inter Union
 
 end
diff -r da47097bd589 -r 93b1183e43e5 src/HOL/Library/Dlist_Cset.thy
--- a/src/HOL/Library/Dlist_Cset.thy	Tue Jun 07 11:11:01 2011 +0200
+++ b/src/HOL/Library/Dlist_Cset.thy	Tue Jun 07 11:12:05 2011 +0200
@@ -3,21 +3,21 @@
 header {* Canonical implementation of sets by distinct lists *}
 
 theory Dlist_Cset
-imports Dlist Cset
+imports Dlist List_Cset
 begin
 
 definition Set :: "'a dlist \<Rightarrow> 'a Cset.set" where
-  "Set dxs = Cset.set (list_of_dlist dxs)"
+  "Set dxs = List_Cset.set (list_of_dlist dxs)"
 
 definition Coset :: "'a dlist \<Rightarrow> 'a Cset.set" where
-  "Coset dxs = Cset.coset (list_of_dlist dxs)"
+  "Coset dxs = List_Cset.coset (list_of_dlist dxs)"
 
 code_datatype Set Coset
 
 declare member_code [code del]
-declare Cset.is_empty_set [code del]
-declare Cset.empty_set [code del]
-declare Cset.UNIV_set [code del]
+declare List_Cset.is_empty_set [code del]
+declare List_Cset.empty_set [code del]
+declare List_Cset.UNIV_set [code del]
 declare insert_set [code del]
 declare remove_set [code del]
 declare compl_set [code del]
@@ -50,11 +50,11 @@
   by (simp add: Coset_def member_set not_set_compl)
 
 lemma Set_dlist_of_list [code]:
-  "Cset.set xs = Set (dlist_of_list xs)"
+  "List_Cset.set xs = Set (dlist_of_list xs)"
   by (rule Cset.set_eqI) simp
 
 lemma Coset_dlist_of_list [code]:
-  "Cset.coset xs = Coset (dlist_of_list xs)"
+  "List_Cset.coset xs = Coset (dlist_of_list xs)"
   by (rule Cset.set_eqI) simp
 
 lemma is_empty_Set [code]:
diff -r da47097bd589 -r 93b1183e43e5 src/HOL/Library/Library.thy
--- a/src/HOL/Library/Library.thy	Tue Jun 07 11:11:01 2011 +0200
+++ b/src/HOL/Library/Library.thy	Tue Jun 07 11:12:05 2011 +0200
@@ -29,6 +29,7 @@
   Lattice_Algebras
   Lattice_Syntax
   ListVector
+  List_Cset
   Kleene_Algebra
   Mapping
   Monad_Syntax
diff -r da47097bd589 -r 93b1183e43e5 src/HOL/Library/List_Cset.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/List_Cset.thy	Tue Jun 07 11:12:05 2011 +0200
@@ -0,0 +1,222 @@
+
+(* Author: Florian Haftmann, TU Muenchen *)
+
+header {* implementation of Cset.sets based on lists *}
+
+theory List_Cset
+imports Cset
+begin
+
+declare mem_def [simp]
+
+definition set :: "'a list \<Rightarrow> 'a Cset.set" where
+  "set xs = Set (List.set xs)"
+hide_const (open) set
+
+lemma member_set [simp]:
+  "member (List_Cset.set xs) = set xs"
+  by (simp add: set_def)
+hide_fact (open) member_set
+
+definition coset :: "'a list \<Rightarrow> 'a Cset.set" where
+  "coset xs = Set (- set xs)"
+hide_const (open) coset
+
+lemma member_coset [simp]:
+  "member (List_Cset.coset xs) = - set xs"
+  by (simp add: coset_def)
+hide_fact (open) member_coset
+
+code_datatype List_Cset.set List_Cset.coset
+
+lemma member_code [code]:
+  "member (List_Cset.set xs) = List.member xs"
+  "member (List_Cset.coset xs) = Not \<circ> List.member xs"
+  by (simp_all add: fun_eq_iff member_def fun_Compl_def bool_Compl_def)
+
+lemma member_image_UNIV [simp]:
+  "member ` UNIV = UNIV"
+proof -
+  have "\<And>A \<Colon> 'a set. \<exists>B \<Colon> 'a Cset.set. A = member B"
+  proof
+    fix A :: "'a set"
+    show "A = member (Set A)" by simp
+  qed
+  then show ?thesis by (simp add: image_def)
+qed
+
+definition (in term_syntax)
+  setify :: "'a\<Colon>typerep list \<times> (unit \<Rightarrow> Code_Evaluation.term)
+    \<Rightarrow> 'a Cset.set \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
+  [code_unfold]: "setify xs = Code_Evaluation.valtermify List_Cset.set {\<cdot>} xs"
+
+notation fcomp (infixl "\<circ>>" 60)
+notation scomp (infixl "\<circ>\<rightarrow>" 60)
+
+instantiation Cset.set :: (random) random
+begin
+
+definition
+  "Quickcheck.random i = Quickcheck.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (setify xs))"
+
+instance ..
+
+end
+
+no_notation fcomp (infixl "\<circ>>" 60)
+no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
+
+subsection {* Basic operations *}
+
+lemma is_empty_set [code]:
+  "Cset.is_empty (List_Cset.set xs) \<longleftrightarrow> List.null xs"
+  by (simp add: is_empty_set null_def)
+hide_fact (open) is_empty_set
+
+lemma empty_set [code]:
+  "bot = List_Cset.set []"
+  by (simp add: set_def)
+hide_fact (open) empty_set
+
+lemma UNIV_set [code]:
+  "top = List_Cset.coset []"
+  by (simp add: coset_def)
+hide_fact (open) UNIV_set
+
+lemma remove_set [code]:
+  "Cset.remove x (List_Cset.set xs) = List_Cset.set (removeAll x xs)"
+  "Cset.remove x (List_Cset.coset xs) = List_Cset.coset (List.insert x xs)"
+by (simp_all add: set_def coset_def)
+  (metis List.set_insert More_Set.remove_def remove_set_compl)
+
+lemma insert_set [code]:
+  "Cset.insert x (List_Cset.set xs) = List_Cset.set (List.insert x xs)"
+  "Cset.insert x (List_Cset.coset xs) = List_Cset.coset (removeAll x xs)"
+  by (simp_all add: set_def coset_def)
+
+lemma map_set [code]:
+  "Cset.map f (List_Cset.set xs) = List_Cset.set (remdups (List.map f xs))"
+  by (simp add: set_def)
+  
+lemma filter_set [code]:
+  "Cset.filter P (List_Cset.set xs) = List_Cset.set (List.filter P xs)"
+  by (simp add: set_def project_set)
+
+lemma forall_set [code]:
+  "Cset.forall P (List_Cset.set xs) \<longleftrightarrow> list_all P xs"
+  by (simp add: set_def list_all_iff)
+
+lemma exists_set [code]:
+  "Cset.exists P (List_Cset.set xs) \<longleftrightarrow> list_ex P xs"
+  by (simp add: set_def list_ex_iff)
+
+lemma card_set [code]:
+  "Cset.card (List_Cset.set xs) = length (remdups xs)"
+proof -
+  have "Finite_Set.card (set (remdups xs)) = length (remdups xs)"
+    by (rule distinct_card) simp
+  then show ?thesis by (simp add: set_def)
+qed
+
+lemma compl_set [simp, code]:
+  "- List_Cset.set xs = List_Cset.coset xs"
+  by (simp add: set_def coset_def)
+
+lemma compl_coset [simp, code]:
+  "- List_Cset.coset xs = List_Cset.set xs"
+  by (simp add: set_def coset_def)
+
+context complete_lattice
+begin
+
+lemma Infimum_inf [code]:
+  "Infimum (List_Cset.set As) = foldr inf As top"
+  "Infimum (List_Cset.coset []) = bot"
+  by (simp_all add: Inf_set_foldr Inf_UNIV)
+
+lemma Supremum_sup [code]:
+  "Supremum (List_Cset.set As) = foldr sup As bot"
+  "Supremum (List_Cset.coset []) = top"
+  by (simp_all add: Sup_set_foldr Sup_UNIV)
+
+end
+
+
+subsection {* Derived operations *}
+
+lemma subset_eq_forall [code]:
+  "A \<le> B \<longleftrightarrow> Cset.forall (member B) A"
+  by (simp add: subset_eq)
+
+lemma subset_subset_eq [code]:
+  "A < B \<longleftrightarrow> A \<le> B \<and> \<not> B \<le> (A :: 'a Cset.set)"
+  by (fact less_le_not_le)
+
+instantiation Cset.set :: (type) equal
+begin
+
+definition [code]:
+  "HOL.equal A B \<longleftrightarrow> A \<le> B \<and> B \<le> (A :: 'a Cset.set)"
+
+instance proof
+qed (simp add: equal_set_def set_eq [symmetric] Cset.set_eq_iff)
+
+end
+
+lemma [code nbe]:
+  "HOL.equal (A :: 'a Cset.set) A \<longleftrightarrow> True"
+  by (fact equal_refl)
+
+
+subsection {* Functorial operations *}
+
+lemma inter_project [code]:
+  "inf A (List_Cset.set xs) = List_Cset.set (List.filter (Cset.member A) xs)"
+  "inf A (List_Cset.coset xs) = foldr Cset.remove xs A"
+proof -
+  show "inf A (List_Cset.set xs) = List_Cset.set (List.filter (member A) xs)"
+    by (simp add: inter project_def set_def)
+  have *: "\<And>x::'a. Cset.remove = (\<lambda>x. Set \<circ> More_Set.remove x \<circ> member)"
+    by (simp add: fun_eq_iff More_Set.remove_def)
+  have "member \<circ> fold (\<lambda>x. Set \<circ> More_Set.remove x \<circ> member) xs =
+    fold More_Set.remove xs \<circ> member"
+    by (rule fold_commute) (simp add: fun_eq_iff)
+  then have "fold More_Set.remove xs (member A) = 
+    member (fold (\<lambda>x. Set \<circ> More_Set.remove x \<circ> member) xs A)"
+    by (simp add: fun_eq_iff)
+  then have "inf A (List_Cset.coset xs) = fold Cset.remove xs A"
+    by (simp add: Diff_eq [symmetric] minus_set *)
+  moreover have "\<And>x y :: 'a. Cset.remove y \<circ> Cset.remove x = Cset.remove x \<circ> Cset.remove y"
+    by (auto simp add: More_Set.remove_def * intro: ext)
+  ultimately show "inf A (List_Cset.coset xs) = foldr Cset.remove xs A"
+    by (simp add: foldr_fold)
+qed
+
+lemma subtract_remove [code]:
+  "A - List_Cset.set xs = foldr Cset.remove xs A"
+  "A - List_Cset.coset xs = List_Cset.set (List.filter (member A) xs)"
+  by (simp_all only: diff_eq compl_set compl_coset inter_project)
+
+lemma union_insert [code]:
+  "sup (List_Cset.set xs) A = foldr Cset.insert xs A"
+  "sup (List_Cset.coset xs) A = List_Cset.coset (List.filter (Not \<circ> member A) xs)"
+proof -
+  have *: "\<And>x::'a. Cset.insert = (\<lambda>x. Set \<circ> Set.insert x \<circ> member)"
+    by (simp add: fun_eq_iff)
+  have "member \<circ> fold (\<lambda>x. Set \<circ> Set.insert x \<circ> member) xs =
+    fold Set.insert xs \<circ> member"
+    by (rule fold_commute) (simp add: fun_eq_iff)
+  then have "fold Set.insert xs (member A) =
+    member (fold (\<lambda>x. Set \<circ> Set.insert x \<circ> member) xs A)"
+    by (simp add: fun_eq_iff)
+  then have "sup (List_Cset.set xs) A = fold Cset.insert xs A"
+    by (simp add: union_set *)
+  moreover have "\<And>x y :: 'a. Cset.insert y \<circ> Cset.insert x = Cset.insert x \<circ> Cset.insert y"
+    by (auto simp add: * intro: ext)
+  ultimately show "sup (List_Cset.set xs) A = foldr Cset.insert xs A"
+    by (simp add: foldr_fold)
+  show "sup (List_Cset.coset xs) A = List_Cset.coset (List.filter (Not \<circ> member A) xs)"
+    by (auto simp add: coset_def)
+qed
+
+end
\ No newline at end of file