--- a/src/HOL/Library/List_Set.thy Thu May 20 16:35:54 2010 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,137 +0,0 @@
-
-(* 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
\ No newline at end of file