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(* Author: Florian Haftmann, TU Muenchen *)


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


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theory List_Set


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imports Main


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begin


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subsection {* Various additional list functions *}


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definition insert :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where


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"insert x xs = (if x \<in> set xs then xs else x # xs)"


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definition remove_all :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where


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"remove_all x xs = filter (Not o op = x) xs"


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subsection {* Various additional set functions *}


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


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


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


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"remove x A = A  {x}"


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lemma fun_left_comm_idem_remove:


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"fun_left_comm_idem remove"


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proof 


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


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show ?thesis by (simp only: fun_left_comm_idem_remove rem)


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qed


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lemma minus_fold_remove:


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assumes "finite A"


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shows "B  A = fold remove B A"


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proof 


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


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


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qed


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


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


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subsection {* Basic set operations *}


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lemma is_empty_set:


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


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by (simp add: is_empty_def null_empty)


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lemma ball_set:


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


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by (rule list_ball_code)


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lemma bex_set:


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


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by (rule list_bex_code)


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lemma empty_set:


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"{} = set []"


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by simp


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lemma insert_set:


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"Set.insert x (set xs) = set (insert x xs)"


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by (auto simp add: insert_def)


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lemma insert_set_compl:


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"Set.insert x ( set xs) =  set (List_Set.remove_all x xs)"


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by (auto simp del: mem_def simp add: remove_all_def)


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lemma remove_set:


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"remove x (set xs) = set (remove_all x xs)"


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by (auto simp add: remove_def remove_all_def)


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lemma remove_set_compl:


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"List_Set.remove x ( set xs) =  set (List_Set.insert x xs)"


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


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lemma image_set:

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

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by simp


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lemma project_set:


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


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by (auto simp add: project_def)


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subsection {* Functorial set operations *}


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lemma union_set:


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"set xs \<union> A = foldl (\<lambda>A x. Set.insert x A) A xs"


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proof 


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interpret fun_left_comm_idem Set.insert


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by (fact fun_left_comm_idem_insert)


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show ?thesis by (simp add: union_fold_insert fold_set)


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qed


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lemma minus_set:


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"A  set xs = foldl (\<lambda>A x. remove x A) A xs"


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proof 


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interpret fun_left_comm_idem remove


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by (fact fun_left_comm_idem_remove)


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show ?thesis


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


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qed


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lemma Inter_set:

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"Inter (set As) = foldl (op \<inter>) UNIV As"

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proof 

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have "fold (op \<inter>) UNIV (set As) = foldl (\<lambda>y x. x \<inter> y) UNIV As"

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by (rule fun_left_comm_idem.fold_set, unfold_locales, auto)

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then show ?thesis


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by (simp only: Inter_fold_inter finite_set Int_commute)

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qed


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lemma Union_set:


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"Union (set As) = foldl (op \<union>) {} As"


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proof 


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have "fold (op \<union>) {} (set As) = foldl (\<lambda>y x. x \<union> y) {} As"


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by (rule fun_left_comm_idem.fold_set, unfold_locales, auto)


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then show ?thesis


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by (simp only: Union_fold_union finite_set Un_commute)


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qed


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lemma INTER_set:

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"INTER (set As) f = foldl (\<lambda>B A. f A \<inter> B) UNIV As"

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proof 

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have "fold (\<lambda>A. op \<inter> (f A)) UNIV (set As) = foldl (\<lambda>B A. f A \<inter> B) UNIV As"

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by (rule fun_left_comm_idem.fold_set, unfold_locales, auto)

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then show ?thesis


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by (simp only: INTER_fold_inter finite_set)

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qed


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lemma UNION_set:


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"UNION (set As) f = foldl (\<lambda>B A. f A \<union> B) {} As"


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proof 


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have "fold (\<lambda>A. op \<union> (f A)) {} (set As) = foldl (\<lambda>B A. f A \<union> B) {} As"


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by (rule fun_left_comm_idem.fold_set, unfold_locales, auto)


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then show ?thesis


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by (simp only: UNION_fold_union finite_set)


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qed


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subsection {* Derived set operations *}


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lemma member:


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


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by simp


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lemma subset_eq:


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


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by (fact subset_eq)


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lemma subset:


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


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by (fact less_le_not_le)


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lemma set_eq:


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


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by (fact eq_iff)


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lemma inter:


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


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by (auto simp add: project_def)


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hide (open) const insert


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end 