(* 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 (Complete_Lattice.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 (Complete_Lattice.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