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(* Author: Florian Haftmann, TU Muenchen *)
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header {* Executable finite sets *}
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31849
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theory Fset
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imports List_Set
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begin
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declare mem_def [simp]
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subsection {* Lifting *}
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datatype 'a fset = Fset "'a set"
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primrec member :: "'a fset \<Rightarrow> 'a set" where
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"member (Fset A) = A"
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lemma Fset_member [simp]:
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"Fset (member A) = A"
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by (cases A) simp
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definition Set :: "'a list \<Rightarrow> 'a fset" where
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"Set xs = Fset (set xs)"
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lemma member_Set [simp]:
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"member (Set xs) = set xs"
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by (simp add: Set_def)
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code_datatype Set
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subsection {* Basic operations *}
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definition is_empty :: "'a fset \<Rightarrow> bool" where
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[simp]: "is_empty A \<longleftrightarrow> List_Set.is_empty (member A)"
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lemma is_empty_Set [code]:
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"is_empty (Set xs) \<longleftrightarrow> null xs"
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by (simp add: is_empty_set)
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definition empty :: "'a fset" where
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[simp]: "empty = Fset {}"
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lemma empty_Set [code]:
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"empty = Set []"
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by (simp add: Set_def)
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definition insert :: "'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset" where
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[simp]: "insert x A = Fset (Set.insert x (member A))"
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lemma insert_Set [code]:
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"insert x (Set xs) = Set (List_Set.insert x xs)"
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by (simp add: Set_def insert_set)
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definition remove :: "'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset" where
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[simp]: "remove x A = Fset (List_Set.remove x (member A))"
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lemma remove_Set [code]:
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"remove x (Set xs) = Set (remove_all x xs)"
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by (simp add: Set_def remove_set)
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definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset" where
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[simp]: "map f A = Fset (image f (member A))"
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lemma map_Set [code]:
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"map f (Set xs) = Set (remdups (List.map f xs))"
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by (simp add: Set_def)
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definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'a fset" where
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[simp]: "filter P A = Fset (List_Set.project P (member A))"
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lemma filter_Set [code]:
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"filter P (Set xs) = Set (List.filter P xs)"
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by (simp add: Set_def project_set)
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definition forall :: "('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> bool" where
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[simp]: "forall P A \<longleftrightarrow> Ball (member A) P"
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lemma forall_Set [code]:
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"forall P (Set xs) \<longleftrightarrow> list_all P xs"
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by (simp add: Set_def ball_set)
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definition exists :: "('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> bool" where
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[simp]: "exists P A \<longleftrightarrow> Bex (member A) P"
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lemma exists_Set [code]:
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"exists P (Set xs) \<longleftrightarrow> list_ex P xs"
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by (simp add: Set_def bex_set)
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definition card :: "'a fset \<Rightarrow> nat" where
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[simp]: "card A = Finite_Set.card (member A)"
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lemma card_Set [code]:
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"card (Set xs) = length (remdups xs)"
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proof -
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have "Finite_Set.card (set (remdups xs)) = length (remdups xs)"
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by (rule distinct_card) simp
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then show ?thesis by (simp add: Set_def card_def)
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qed
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subsection {* Derived operations *}
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lemma member_exists [code]:
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"member A y \<longleftrightarrow> exists (\<lambda>x. y = x) A"
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by simp
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definition subfset_eq :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" where
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[simp]: "subfset_eq A B \<longleftrightarrow> member A \<subseteq> member B"
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lemma subfset_eq_forall [code]:
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"subfset_eq A B \<longleftrightarrow> forall (\<lambda>x. member B x) A"
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by (simp add: subset_eq)
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definition subfset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" where
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[simp]: "subfset A B \<longleftrightarrow> member A \<subset> member B"
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lemma subfset_subfset_eq [code]:
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"subfset A B \<longleftrightarrow> subfset_eq A B \<and> \<not> subfset_eq B A"
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by (simp add: subset)
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lemma eq_fset_subfset_eq [code]:
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"eq_class.eq A B \<longleftrightarrow> subfset_eq A B \<and> subfset_eq B A"
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by (cases A, cases B) (simp add: eq set_eq)
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definition inter :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" where
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[simp]: "inter A B = Fset (project (member A) (member B))"
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lemma inter_project [code]:
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"inter A B = filter (member A) B"
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by (simp add: inter)
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subsection {* Functorial operations *}
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definition union :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" where
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[simp]: "union A B = Fset (member A \<union> member B)"
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lemma union_insert [code]:
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"union (Set xs) A = foldl (\<lambda>A x. insert x A) A xs"
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proof -
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have "foldl (\<lambda>A x. Set.insert x A) (member A) xs =
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member (foldl (\<lambda>A x. Fset (Set.insert x (member A))) A xs)"
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by (rule foldl_apply_inv) simp
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then show ?thesis by (simp add: union_set)
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qed
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definition subtract :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" where
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[simp]: "subtract A B = Fset (member B - member A)"
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lemma subtract_remove [code]:
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"subtract (Set xs) A = foldl (\<lambda>A x. remove x A) A xs"
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proof -
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have "foldl (\<lambda>A x. List_Set.remove x A) (member A) xs =
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member (foldl (\<lambda>A x. Fset (List_Set.remove x (member A))) A xs)"
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by (rule foldl_apply_inv) simp
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then show ?thesis by (simp add: minus_set)
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qed
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definition Inter :: "'a fset fset \<Rightarrow> 'a fset" where
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[simp]: "Inter A = Fset (Complete_Lattice.Inter (member ` member A))"
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lemma Inter_inter [code]:
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"Inter (Set (A # As)) = foldl inter A As"
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proof -
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note Inter_image_eq [simp del] set_map [simp del] set.simps [simp del]
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have "foldl (op \<inter>) (member A) (List.map member As) =
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member (foldl (\<lambda>B A. Fset (member B \<inter> A)) A (List.map member As))"
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by (rule foldl_apply_inv) simp
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then show ?thesis
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by (simp add: Inter_set image_set inter_def_raw inter foldl_map)
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qed
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definition Union :: "'a fset fset \<Rightarrow> 'a fset" where
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[simp]: "Union A = Fset (Complete_Lattice.Union (member ` member A))"
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lemma Union_union [code]:
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"Union (Set As) = foldl union empty As"
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proof -
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note Union_image_eq [simp del] set_map [simp del]
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have "foldl (op \<union>) (member empty) (List.map member As) =
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member (foldl (\<lambda>B A. Fset (member B \<union> A)) empty (List.map member As))"
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by (rule foldl_apply_inv) simp
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then show ?thesis
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by (simp add: Union_set image_set union_def_raw foldl_map)
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qed
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subsection {* Misc operations *}
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lemma size_fset [code]:
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"fset_size f A = 0"
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"size A = 0"
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by (cases A, simp) (cases A, simp)
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lemma fset_case_code [code]:
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"fset_case f A = f (member A)"
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by (cases A) simp
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lemma fset_rec_code [code]:
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"fset_rec f A = f (member A)"
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by (cases A) simp
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subsection {* Simplified simprules *}
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lemma is_empty_simp [simp]:
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"is_empty A \<longleftrightarrow> member A = {}"
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by (simp add: List_Set.is_empty_def)
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declare is_empty_def [simp del]
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lemma remove_simp [simp]:
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"remove x A = Fset (member A - {x})"
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by (simp add: List_Set.remove_def)
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declare remove_def [simp del]
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lemma filter_simp [simp]:
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"filter P A = Fset {x \<in> member A. P x}"
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by (simp add: List_Set.project_def)
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declare filter_def [simp del]
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lemma inter_simp [simp]:
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"inter A B = Fset (member A \<inter> member B)"
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by (simp add: inter)
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declare inter_def [simp del]
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declare mem_def [simp del]
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hide (open) const is_empty empty insert remove map filter forall exists card
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subfset_eq subfset inter union subtract Inter Union
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end
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