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