(* Author: Florian Haftmann, TU Muenchen *)

 header {* Relating (finite) sets and lists *}

 theory List_Set

 imports Main More_List

 begin

 subsection {* Various additional set functions *}

 definition is_empty :: "'a set \<Rightarrow> bool" where

   "is_empty A \<longleftrightarrow> A = {}"

 definition remove :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set" where

   "remove x A = A - {x}"

 lemma fun_left_comm_idem_remove:

   "fun_left_comm_idem remove"

 proof -

   have rem: "remove = (\<lambda>x A. A - {x})" by (simp add: expand_fun_eq remove_def)

   show ?thesis by (simp only: fun_left_comm_idem_remove rem)

 qed

 lemma minus_fold_remove:

   assumes "finite A"

   shows "B - A = Finite_Set.fold remove B A"

 proof -

   have rem: "remove = (\<lambda>x A. A - {x})" by (simp add: expand_fun_eq remove_def)

   show ?thesis by (simp only: rem assms minus_fold_remove)

 qed

 definition project :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set" where

   "project P A = {a\<in>A. P a}"

 subsection {* Basic set operations *}

 lemma is_empty_set:

   "is_empty (set xs) \<longleftrightarrow> null xs"

   by (simp add: is_empty_def null_empty)

 lemma ball_set:

   "(\<forall>x\<in>set xs. P x) \<longleftrightarrow> list_all P xs"

   by (rule list_ball_code)

 lemma bex_set:

   "(\<exists>x\<in>set xs. P x) \<longleftrightarrow> list_ex P xs"

   by (rule list_bex_code)

 lemma empty_set:

   "{} = set []"

   by simp

 lemma insert_set_compl:

   "insert x (- set xs) = - set (removeAll x xs)"

   by auto

 lemma remove_set_compl:

   "remove x (- set xs) = - set (List.insert x xs)"

   by (auto simp del: mem_def simp add: remove_def List.insert_def)

 lemma image_set:

   "image f (set xs) = set (map f xs)"

   by simp

 lemma project_set:

   "project P (set xs) = set (filter P xs)"

   by (auto simp add: project_def)

 subsection {* Functorial set operations *}

 lemma union_set:

   "set xs \<union> A = fold Set.insert xs A"

 proof -

   interpret fun_left_comm_idem Set.insert

     by (fact fun_left_comm_idem_insert)

   show ?thesis by (simp add: union_fold_insert fold_set)

 qed

 lemma union_set_foldr:

   "set xs \<union> A = foldr Set.insert xs A"

 proof -

   have "\<And>x y :: 'a. insert y \<circ> insert x = insert x \<circ> insert y"

     by (auto intro: ext)

   then show ?thesis by (simp add: union_set foldr_fold)

 qed

 lemma minus_set:

   "A - set xs = fold remove xs A"

 proof -

   interpret fun_left_comm_idem remove

     by (fact fun_left_comm_idem_remove)

   show ?thesis

     by (simp add: minus_fold_remove [of _ A] fold_set)

 qed

 lemma minus_set_foldr:

   "A - set xs = foldr remove xs A"

 proof -

   have "\<And>x y :: 'a. remove y \<circ> remove x = remove x \<circ> remove y"

     by (auto simp add: remove_def intro: ext)

   then show ?thesis by (simp add: minus_set foldr_fold)

 qed

 subsection {* Derived set operations *}

 lemma member:

   "a \<in> A \<longleftrightarrow> (\<exists>x\<in>A. a = x)"

   by simp

 lemma subset_eq:

   "A \<subseteq> B \<longleftrightarrow> (\<forall>x\<in>A. x \<in> B)"

   by (fact subset_eq)

 lemma subset:

   "A \<subset> B \<longleftrightarrow> A \<subseteq> B \<and> \<not> B \<subseteq> A"

   by (fact less_le_not_le)

 lemma set_eq:

   "A = B \<longleftrightarrow> A \<subseteq> B \<and> B \<subseteq> A"

   by (fact eq_iff)

 lemma inter:

   "A \<inter> B = project (\<lambda>x. x \<in> A) B"

   by (auto simp add: project_def)

 subsection {* Various lemmas *}

 lemma not_set_compl:

   "Not \<circ> set xs = - set xs"

   by (simp add: fun_Compl_def bool_Compl_def comp_def expand_fun_eq)

 end