src/HOL/Library/Dlist.thy
author haftmann
Mon Feb 22 15:53:18 2010 +0100 (2010-02-22)
changeset 35303 816e48d60b13
child 35688 cfe0accda6e3
permissions -rw-r--r--
added Dlist
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(* Author: Florian Haftmann, TU Muenchen *)
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header {* Lists with elements distinct as canonical example for datatype invariants *}
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theory Dlist
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imports Main Fset
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begin
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section {* Prelude *}
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text {* Without canonical argument order, higher-order things tend to get confusing quite fast: *}
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setup {* Sign.map_naming (Name_Space.add_path "List") *}
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primrec member :: "'a list \<Rightarrow> 'a \<Rightarrow> bool" where
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    "member [] y \<longleftrightarrow> False"
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  | "member (x#xs) y \<longleftrightarrow> x = y \<or> member xs y"
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lemma member_set:
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  "member = set"
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proof (rule ext)+
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  fix xs :: "'a list" and x :: 'a
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  have "member xs x \<longleftrightarrow> x \<in> set xs" by (induct xs) auto
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  then show "member xs x = set xs x" by (simp add: mem_def)
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qed
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lemma not_set_compl:
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  "Not \<circ> set xs = - set xs"
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  by (simp add: fun_Compl_def bool_Compl_def comp_def expand_fun_eq)
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primrec fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b" where
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    "fold f [] s = s"
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  | "fold f (x#xs) s = fold f xs (f x s)"
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lemma foldl_fold:
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  "foldl f s xs = List.fold (\<lambda>x s. f s x) xs s"
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  by (induct xs arbitrary: s) simp_all
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setup {* Sign.map_naming Name_Space.parent_path *}
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section {* The type of distinct lists *}
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typedef (open) 'a dlist = "{xs::'a list. distinct xs}"
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  morphisms list_of_dlist Abs_dlist
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proof
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  show "[] \<in> ?dlist" by simp
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qed
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text {* Formal, totalized constructor for @{typ "'a dlist"}: *}
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definition Dlist :: "'a list \<Rightarrow> 'a dlist" where
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  [code del]: "Dlist xs = Abs_dlist (remdups xs)"
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lemma distinct_list_of_dlist [simp]:
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  "distinct (list_of_dlist dxs)"
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  using list_of_dlist [of dxs] by simp
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lemma list_of_dlist_Dlist [simp]:
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  "list_of_dlist (Dlist xs) = remdups xs"
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  by (simp add: Dlist_def Abs_dlist_inverse)
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lemma Dlist_list_of_dlist [simp]:
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  "Dlist (list_of_dlist dxs) = dxs"
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  by (simp add: Dlist_def list_of_dlist_inverse distinct_remdups_id)
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text {* Fundamental operations: *}
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definition empty :: "'a dlist" where
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  "empty = Dlist []"
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definition insert :: "'a \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist" where
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  "insert x dxs = Dlist (List.insert x (list_of_dlist dxs))"
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definition remove :: "'a \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist" where
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  "remove x dxs = Dlist (remove1 x (list_of_dlist dxs))"
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definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a dlist \<Rightarrow> 'b dlist" where
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  "map f dxs = Dlist (remdups (List.map f (list_of_dlist dxs)))"
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definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist" where
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  "filter P dxs = Dlist (List.filter P (list_of_dlist dxs))"
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text {* Derived operations: *}
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definition null :: "'a dlist \<Rightarrow> bool" where
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  "null dxs = List.null (list_of_dlist dxs)"
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definition member :: "'a dlist \<Rightarrow> 'a \<Rightarrow> bool" where
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  "member dxs = List.member (list_of_dlist dxs)"
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definition length :: "'a dlist \<Rightarrow> nat" where
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  "length dxs = List.length (list_of_dlist dxs)"
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definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a dlist \<Rightarrow> 'b \<Rightarrow> 'b" where
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  "fold f dxs = List.fold f (list_of_dlist dxs)"
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section {* Executable version obeying invariant *}
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code_abstype Dlist list_of_dlist
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  by simp
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lemma list_of_dlist_empty [simp, code abstract]:
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  "list_of_dlist empty = []"
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  by (simp add: empty_def)
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lemma list_of_dlist_insert [simp, code abstract]:
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  "list_of_dlist (insert x dxs) = List.insert x (list_of_dlist dxs)"
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  by (simp add: insert_def)
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lemma list_of_dlist_remove [simp, code abstract]:
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  "list_of_dlist (remove x dxs) = remove1 x (list_of_dlist dxs)"
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  by (simp add: remove_def)
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lemma list_of_dlist_map [simp, code abstract]:
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  "list_of_dlist (map f dxs) = remdups (List.map f (list_of_dlist dxs))"
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  by (simp add: map_def)
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lemma list_of_dlist_filter [simp, code abstract]:
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  "list_of_dlist (filter P dxs) = List.filter P (list_of_dlist dxs)"
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  by (simp add: filter_def)
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declare null_def [code] member_def [code] length_def [code] fold_def [code] -- {* explicit is better than implicit *}
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section {* Implementation of sets by distinct lists -- canonical! *}
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definition Set :: "'a dlist \<Rightarrow> 'a fset" where
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  "Set dxs = Fset.Set (list_of_dlist dxs)"
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definition Coset :: "'a dlist \<Rightarrow> 'a fset" where
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  "Coset dxs = Fset.Coset (list_of_dlist dxs)"
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code_datatype Set Coset
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declare member_code [code del]
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declare is_empty_Set [code del]
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declare empty_Set [code del]
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declare UNIV_Set [code del]
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declare insert_Set [code del]
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declare remove_Set [code del]
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declare map_Set [code del]
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declare filter_Set [code del]
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declare forall_Set [code del]
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declare exists_Set [code del]
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declare card_Set [code del]
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declare subfset_eq_forall [code del]
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declare subfset_subfset_eq [code del]
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declare eq_fset_subfset_eq [code del]
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declare inter_project [code del]
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declare subtract_remove [code del]
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declare union_insert [code del]
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declare Infimum_inf [code del]
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declare Supremum_sup [code del]
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lemma Set_Dlist [simp]:
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  "Set (Dlist xs) = Fset (set xs)"
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  by (simp add: Set_def Fset.Set_def)
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lemma Coset_Dlist [simp]:
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  "Coset (Dlist xs) = Fset (- set xs)"
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  by (simp add: Coset_def Fset.Coset_def)
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lemma member_Set [simp]:
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  "Fset.member (Set dxs) = List.member (list_of_dlist dxs)"
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  by (simp add: Set_def member_set)
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lemma member_Coset [simp]:
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  "Fset.member (Coset dxs) = Not \<circ> List.member (list_of_dlist dxs)"
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  by (simp add: Coset_def member_set not_set_compl)
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lemma is_empty_Set [code]:
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  "Fset.is_empty (Set dxs) \<longleftrightarrow> null dxs"
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  by (simp add: null_def null_empty member_set)
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lemma bot_code [code]:
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  "bot = Set empty"
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  by (simp add: empty_def)
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lemma top_code [code]:
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  "top = Coset empty"
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  by (simp add: empty_def)
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lemma insert_code [code]:
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  "Fset.insert x (Set dxs) = Set (insert x dxs)"
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  "Fset.insert x (Coset dxs) = Coset (remove x dxs)"
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  by (simp_all add: insert_def remove_def member_set not_set_compl)
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lemma remove_code [code]:
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  "Fset.remove x (Set dxs) = Set (remove x dxs)"
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  "Fset.remove x (Coset dxs) = Coset (insert x dxs)"
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  by (auto simp add: insert_def remove_def member_set not_set_compl)
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lemma member_code [code]:
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  "Fset.member (Set dxs) = member dxs"
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  "Fset.member (Coset dxs) = Not \<circ> member dxs"
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  by (simp_all add: member_def)
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lemma map_code [code]:
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  "Fset.map f (Set dxs) = Set (map f dxs)"
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  by (simp add: member_set)
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lemma filter_code [code]:
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  "Fset.filter f (Set dxs) = Set (filter f dxs)"
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  by (simp add: member_set)
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lemma forall_Set [code]:
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  "Fset.forall P (Set xs) \<longleftrightarrow> list_all P (list_of_dlist xs)"
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  by (simp add: member_set list_all_iff)
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lemma exists_Set [code]:
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  "Fset.exists P (Set xs) \<longleftrightarrow> list_ex P (list_of_dlist xs)"
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  by (simp add: member_set list_ex_iff)
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lemma card_code [code]:
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  "Fset.card (Set dxs) = length dxs"
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  by (simp add: length_def member_set distinct_card)
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lemma foldl_list_of_dlist:
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  "foldl f s (list_of_dlist dxs) = fold (\<lambda>x s. f s x) dxs s"
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  by (simp add: foldl_fold fold_def)
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lemma inter_code [code]:
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  "inf A (Set xs) = Set (filter (Fset.member A) xs)"
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  "inf A (Coset xs) = fold Fset.remove xs A"
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  by (simp_all only: Set_def Coset_def foldl_list_of_dlist inter_project list_of_dlist_filter)
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lemma subtract_code [code]:
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  "A - Set xs = fold Fset.remove xs A"
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  "A - Coset xs = Set (filter (Fset.member A) xs)"
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  by (simp_all only: Set_def Coset_def foldl_list_of_dlist subtract_remove list_of_dlist_filter)
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lemma union_code [code]:
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  "sup (Set xs) A = fold Fset.insert xs A"
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  "sup (Coset xs) A = Coset (filter (Not \<circ> Fset.member A) xs)"
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  by (simp_all only: Set_def Coset_def foldl_list_of_dlist union_insert list_of_dlist_filter)
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context complete_lattice
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begin
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lemma Infimum_code [code]:
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  "Infimum (Set As) = fold inf As top"
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  by (simp only: Set_def Infimum_inf foldl_list_of_dlist inf.commute)
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lemma Supremum_code [code]:
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  "Supremum (Set As) = fold sup As bot"
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  by (simp only: Set_def Supremum_sup foldl_list_of_dlist sup.commute)
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end
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hide (open) const member fold empty insert remove map filter null member length fold
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end