src/HOL/Library/List_Set.thy
 author haftmann Tue, 06 Oct 2009 15:59:12 +0200 changeset 32880 b8bee63c7202 parent 31851 c04f8c51d0ab child 33939 fcb50b497763 permissions -rw-r--r--
sets and cosets
```

(* 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"

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)"

lemma insert_set_compl:
"Set.insert x (- set xs) = - set (List_Set.remove_all x xs)"
by (auto simp del: mem_def simp add: remove_all_def)

lemma remove_set:
"remove x (set xs) = set (remove_all x xs)"
by (auto simp add: remove_def remove_all_def)

lemma remove_set_compl:
"List_Set.remove x (- set xs) = - set (List_Set.insert x xs)"
by (auto simp del: mem_def simp add: remove_def List_Set.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)"

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 As) = foldl (op \<inter>) UNIV As"
proof -
have "fold (op \<inter>) UNIV (set As) = foldl (\<lambda>y x. x \<inter> y) UNIV As"
by (rule fun_left_comm_idem.fold_set, unfold_locales, auto)
then show ?thesis
by (simp only: Inter_fold_inter finite_set Int_commute)
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 As) f = foldl (\<lambda>B A. f A \<inter> B) UNIV As"
proof -
have "fold (\<lambda>A. op \<inter> (f A)) UNIV (set As) = foldl (\<lambda>B A. f A \<inter> B) UNIV As"
by (rule fun_left_comm_idem.fold_set, unfold_locales, auto)
then show ?thesis
by (simp only: INTER_fold_inter finite_set)
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"