author | haftmann |
Sun, 11 Apr 2010 16:51:07 +0200 | |
changeset 36112 | 7fa17a225852 |
parent 35688 | cfe0accda6e3 |
child 36176 | 3fe7e97ccca8 |
permissions | -rw-r--r-- |
35303 | 1 |
(* 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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36112
7fa17a225852
user interface for abstract datatypes is an attribute, not a command
haftmann
parents:
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changeset
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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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36112
7fa17a225852
user interface for abstract datatypes is an attribute, not a command
haftmann
parents:
35688
diff
changeset
|
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lemma Dlist_list_of_dlist [simp, code abstype]: |
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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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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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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 |