(* Author: Florian Haftmann, TU Muenchen *)
header {* Relating (finite) sets and lists *}
theory More_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> List.null xs"
by (simp add: is_empty_def null_def)
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