--- a/src/HOL/Library/List_Set.thy Thu May 20 16:35:53 2010 +0200
+++ b/src/HOL/Library/List_Set.thy Thu May 20 16:35:54 2010 +0200
@@ -4,7 +4,7 @@
header {* Relating (finite) sets and lists *}
theory List_Set
-imports Main
+imports Main More_List
begin
subsection {* Various additional set functions *}
@@ -24,7 +24,7 @@
lemma minus_fold_remove:
assumes "finite A"
- shows "B - A = fold remove B 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)
@@ -72,15 +72,23 @@
subsection {* Functorial set operations *}
lemma union_set:
- "set xs \<union> A = foldl (\<lambda>A x. Set.insert x A) A xs"
+ "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 = foldl (\<lambda>A x. remove x A) A xs"
+ "A - set xs = fold remove xs A"
proof -
interpret fun_left_comm_idem remove
by (fact fun_left_comm_idem_remove)
@@ -88,6 +96,14 @@
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 *}
@@ -111,4 +127,11 @@
"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
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