src/HOL/Library/List_Set.thy

+
+(* Author: Florian Haftmann, TU Muenchen *)
+
+header {* Relating (finite) sets and lists *}
+
+theory List_Set
+imports Main
+begin
+
+subsection {* Various additional list functions *}
+
+definition insert :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
+  "insert x xs = (if x \<in> set xs then xs else x # xs)"
+
+definition remove_all :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
+  "remove_all x xs = filter (Not o op = x) xs"
+
+
+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 = 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:
+  "Set.insert x (set xs) = set (insert x xs)"
+  by (auto simp add: insert_def)
+
+lemma remove_set:
+  "remove x (set xs) = set (remove_all x xs)"
+  by (auto simp add: remove_def remove_all_def)
+
+lemma image_set:
+  "image f (set xs) = set (remdups (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 = foldl (\<lambda>A x. Set.insert x A) A xs"
+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 minus_set:
+  "A - set xs = foldl (\<lambda>A x. remove x A) A xs"
+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 Inter_set:
+  "Inter (set (A # As)) = foldl (op \<inter>) A As"
+proof -
+  have "finite (set (A # As))" by simp
+  moreover have "fold (op \<inter>) UNIV (set (A # As)) = foldl (\<lambda>y x. x \<inter> y) UNIV (A # As)"
+    by (rule fun_left_comm_idem.fold_set, unfold_locales, auto)
+  ultimately have "Inter (set (A # As)) = foldl (op \<inter>) UNIV (A # As)"
+    by (simp only: Inter_fold_inter Int_commute)
+  then show ?thesis by simp
+qed
+
+lemma Union_set:
+  "Union (set As) = foldl (op \<union>) {} As"
+proof -
+  have "fold (op \<union>) {} (set As) = foldl (\<lambda>y x. x \<union> y) {} As"
+    by (rule fun_left_comm_idem.fold_set, unfold_locales, auto)
+  then show ?thesis
+    by (simp only: Union_fold_union finite_set Un_commute)
+qed
+
+lemma INTER_set:
+  "INTER (set (A # As)) f = foldl (\<lambda>B A. f A \<inter> B) (f A) As"
+proof -
+  have "finite (set (A # As))" by simp
+  moreover have "fold (\<lambda>A. op \<inter> (f A)) UNIV (set (A # As)) = foldl (\<lambda>B A. f A \<inter> B) UNIV (A # As)"
+    by (rule fun_left_comm_idem.fold_set, unfold_locales, auto)
+  ultimately have "INTER (set (A # As)) f = foldl (\<lambda>B A. f A \<inter> B) UNIV (A # As)"
+    by (simp only: INTER_fold_inter) 
+  then show ?thesis by simp
+qed
+
+lemma UNION_set:
+  "UNION (set As) f = foldl (\<lambda>B A. f A \<union> B) {} As"
+proof -
+  have "fold (\<lambda>A. op \<union> (f A)) {} (set As) = foldl (\<lambda>B A. f A \<union> B) {} As"
+    by (rule fun_left_comm_idem.fold_set, unfold_locales, auto)
+  then show ?thesis
+    by (simp only: UNION_fold_union finite_set)
+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)
+
+end