--- a/src/HOL/Library/Fset.thy Thu May 20 16:35:53 2010 +0200
+++ b/src/HOL/Library/Fset.thy Thu May 20 16:35:54 2010 +0200
@@ -4,7 +4,7 @@
header {* Executable finite sets *}
theory Fset
-imports List_Set
+imports List_Set More_List
begin
declare mem_def [simp]
@@ -41,9 +41,9 @@
code_datatype Set Coset
lemma member_code [code]:
- "member (Set xs) y \<longleftrightarrow> List.member y xs"
- "member (Coset xs) y \<longleftrightarrow> \<not> List.member y xs"
- by (simp_all add: mem_iff fun_Compl_def bool_Compl_def)
+ "member (Set xs) = List.member xs"
+ "member (Coset xs) = Not \<circ> List.member xs"
+ by (simp_all add: expand_fun_eq mem_iff fun_Compl_def bool_Compl_def)
lemma member_image_UNIV [simp]:
"member ` UNIV = UNIV"
@@ -105,6 +105,7 @@
end
+
subsection {* Basic operations *}
definition is_empty :: "'a fset \<Rightarrow> bool" where
@@ -128,7 +129,7 @@
lemma insert_Set [code]:
"insert x (Set xs) = Set (List.insert x xs)"
"insert x (Coset xs) = Coset (removeAll x xs)"
- by (simp_all add: Set_def Coset_def set_insert)
+ by (simp_all add: Set_def Coset_def)
definition remove :: "'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset" where
[simp]: "remove x A = Fset (List_Set.remove x (member A))"
@@ -175,9 +176,17 @@
proof -
have "Finite_Set.card (set (remdups xs)) = length (remdups xs)"
by (rule distinct_card) simp
- then show ?thesis by (simp add: Set_def card_def)
+ then show ?thesis by (simp add: Set_def)
qed
+lemma compl_Set [simp, code]:
+ "- Set xs = Coset xs"
+ by (simp add: Set_def Coset_def)
+
+lemma compl_Coset [simp, code]:
+ "- Coset xs = Set xs"
+ by (simp add: Set_def Coset_def)
+
subsection {* Derived operations *}
@@ -198,39 +207,49 @@
lemma inter_project [code]:
"inf A (Set xs) = Set (List.filter (member A) xs)"
- "inf A (Coset xs) = foldl (\<lambda>A x. remove x A) A xs"
+ "inf A (Coset xs) = foldr remove xs A"
proof -
show "inf A (Set xs) = Set (List.filter (member A) xs)"
by (simp add: inter project_def Set_def)
- have "foldl (\<lambda>A x. List_Set.remove x A) (member A) xs =
- member (foldl (\<lambda>A x. Fset (List_Set.remove x (member A))) A xs)"
- by (rule foldl_apply) (simp add: expand_fun_eq)
- then show "inf A (Coset xs) = foldl (\<lambda>A x. remove x A) A xs"
- by (simp add: Diff_eq [symmetric] minus_set)
+ have *: "\<And>x::'a. remove = (\<lambda>x. Fset \<circ> List_Set.remove x \<circ> member)"
+ by (simp add: expand_fun_eq)
+ have "member \<circ> fold (\<lambda>x. Fset \<circ> List_Set.remove x \<circ> member) xs =
+ fold List_Set.remove xs \<circ> member"
+ by (rule fold_apply) (simp add: expand_fun_eq)
+ then have "fold List_Set.remove xs (member A) =
+ member (fold (\<lambda>x. Fset \<circ> List_Set.remove x \<circ> member) xs A)"
+ by (simp add: expand_fun_eq)
+ then have "inf A (Coset xs) = fold remove xs A"
+ by (simp add: Diff_eq [symmetric] minus_set *)
+ moreover have "\<And>x y :: 'a. Fset.remove y \<circ> Fset.remove x = Fset.remove x \<circ> Fset.remove y"
+ by (auto simp add: List_Set.remove_def * intro: ext)
+ ultimately show "inf A (Coset xs) = foldr remove xs A"
+ by (simp add: foldr_fold)
qed
lemma subtract_remove [code]:
- "A - Set xs = foldl (\<lambda>A x. remove x A) A xs"
+ "A - Set xs = foldr remove xs A"
"A - Coset xs = Set (List.filter (member A) xs)"
-proof -
- have "foldl (\<lambda>A x. List_Set.remove x A) (member A) xs =
- member (foldl (\<lambda>A x. Fset (List_Set.remove x (member A))) A xs)"
- by (rule foldl_apply) (simp add: expand_fun_eq)
- then show "A - Set xs = foldl (\<lambda>A x. remove x A) A xs"
- by (simp add: minus_set)
- show "A - Coset xs = Set (List.filter (member A) xs)"
- by (auto simp add: Coset_def Set_def)
-qed
+ by (simp_all only: diff_eq compl_Set compl_Coset inter_project)
lemma union_insert [code]:
- "sup (Set xs) A = foldl (\<lambda>A x. insert x A) A xs"
+ "sup (Set xs) A = foldr insert xs A"
"sup (Coset xs) A = Coset (List.filter (Not \<circ> member A) xs)"
proof -
- have "foldl (\<lambda>A x. Set.insert x A) (member A) xs =
- member (foldl (\<lambda>A x. Fset (Set.insert x (member A))) A xs)"
- by (rule foldl_apply) (simp add: expand_fun_eq)
- then show "sup (Set xs) A = foldl (\<lambda>A x. insert x A) A xs"
- by (simp add: union_set)
+ have *: "\<And>x::'a. insert = (\<lambda>x. Fset \<circ> Set.insert x \<circ> member)"
+ by (simp add: expand_fun_eq)
+ have "member \<circ> fold (\<lambda>x. Fset \<circ> Set.insert x \<circ> member) xs =
+ fold Set.insert xs \<circ> member"
+ by (rule fold_apply) (simp add: expand_fun_eq)
+ then have "fold Set.insert xs (member A) =
+ member (fold (\<lambda>x. Fset \<circ> Set.insert x \<circ> member) xs A)"
+ by (simp add: expand_fun_eq)
+ then have "sup (Set xs) A = fold insert xs A"
+ by (simp add: union_set *)
+ moreover have "\<And>x y :: 'a. Fset.insert y \<circ> Fset.insert x = Fset.insert x \<circ> Fset.insert y"
+ by (auto simp add: * intro: ext)
+ ultimately show "sup (Set xs) A = foldr insert xs A"
+ by (simp add: foldr_fold)
show "sup (Coset xs) A = Coset (List.filter (Not \<circ> member A) xs)"
by (auto simp add: Coset_def)
qed
@@ -242,17 +261,17 @@
[simp]: "Infimum A = Inf (member A)"
lemma Infimum_inf [code]:
- "Infimum (Set As) = foldl inf top As"
+ "Infimum (Set As) = foldr inf As top"
"Infimum (Coset []) = bot"
- by (simp_all add: Inf_set_fold Inf_UNIV)
+ by (simp_all add: Inf_set_foldr Inf_UNIV)
definition Supremum :: "'a fset \<Rightarrow> 'a" where
[simp]: "Supremum A = Sup (member A)"
lemma Supremum_sup [code]:
- "Supremum (Set As) = foldl sup bot As"
+ "Supremum (Set As) = foldr sup As bot"
"Supremum (Coset []) = top"
- by (simp_all add: Sup_set_fold Sup_UNIV)
+ by (simp_all add: Sup_set_foldr Sup_UNIV)
end