--- a/src/HOL/Library/Cset.thy Sat Aug 27 09:02:25 2011 +0200
+++ b/src/HOL/Library/Cset.thy Sat Aug 27 09:44:45 2011 +0200
@@ -152,6 +152,10 @@
"set xs = Set (List.set xs)"
hide_const (open) set
+definition coset :: "'a list \<Rightarrow> 'a Cset.set" where
+ "coset xs = Set (- List.set xs)"
+hide_const (open) coset
+
text {* conversion from @{typ "'a Predicate.pred"} *}
definition pred_of_cset :: "'a Cset.set \<Rightarrow> 'a Predicate.pred" where
@@ -200,6 +204,21 @@
by (simp add: set_def)
hide_fact (open) set_def
+lemma member_set [simp]:
+ "member (Cset.set xs) = (\<lambda>x. x \<in> set xs)"
+ by (simp add: fun_eq_iff member_def)
+hide_fact (open) member_set
+
+lemma set_of_coset [simp]:
+ "set_of (Cset.coset xs) = - set xs"
+ by (simp add: coset_def)
+hide_fact (open) coset_def
+
+lemma member_coset [simp]:
+ "member (Cset.coset xs) = (\<lambda>x. x \<in> - set xs)"
+ by (simp add: fun_eq_iff member_def)
+hide_fact (open) member_coset
+
lemma set_simps [simp]:
"Cset.set [] = Cset.empty"
"Cset.set (x # xs) = insert x (Cset.set xs)"
@@ -268,6 +287,82 @@
"single a = insert a Cset.empty"
by (simp add: Cset.single_def)
+lemma compl_set [simp]:
+ "- Cset.set xs = Cset.coset xs"
+ by (simp add: Cset.set_def Cset.coset_def)
+
+lemma compl_coset [simp]:
+ "- Cset.coset xs = Cset.set xs"
+ by (simp add: Cset.set_def Cset.coset_def)
+
+lemma member_cset_of:
+ "member = set_of"
+ by (rule ext)+ (simp add: member_def mem_def)
+
+lemma inter_project:
+ "inf A (Cset.set xs) = Cset.set (List.filter (Cset.member A) xs)"
+ "inf A (Cset.coset xs) = foldr Cset.remove xs A"
+proof -
+ show "inf A (Cset.set xs) = Cset.set (List.filter (member A) xs)"
+ by (simp add: inter project_def Cset.set_def member_def)
+ have *: "\<And>x::'a. Cset.remove = (\<lambda>x. Set \<circ> More_Set.remove x \<circ> member)"
+ by (simp add: fun_eq_iff More_Set.remove_def member_cset_of)
+ have "member \<circ> fold (\<lambda>x. Set \<circ> More_Set.remove x \<circ> member) xs =
+ fold More_Set.remove xs \<circ> member"
+ by (rule fold_commute) (simp add: fun_eq_iff mem_def)
+ then have "fold More_Set.remove xs (member A) =
+ member (fold (\<lambda>x. Set \<circ> More_Set.remove x \<circ> member) xs A)"
+ by (simp add: fun_eq_iff)
+ then have "inf A (Cset.coset xs) = fold Cset.remove xs A"
+ by (simp add: Diff_eq [symmetric] minus_set * member_cset_of)
+ moreover have "\<And>x y :: 'a. Cset.remove y \<circ> Cset.remove x = Cset.remove x \<circ> Cset.remove y"
+ by (auto simp add: More_Set.remove_def * member_cset_of)
+ ultimately show "inf A (Cset.coset xs) = foldr Cset.remove xs A"
+ by (simp add: foldr_fold)
+qed
+
+lemma subtract_remove:
+ "A - Cset.set xs = foldr Cset.remove xs A"
+ "A - Cset.coset xs = Cset.set (List.filter (member A) xs)"
+ by (simp_all only: diff_eq compl_set compl_coset inter_project)
+
+lemma union_insert:
+ "sup (Cset.set xs) A = foldr Cset.insert xs A"
+ "sup (Cset.coset xs) A = Cset.coset (List.filter (Not \<circ> member A) xs)"
+proof -
+ have *: "\<And>x::'a. Cset.insert = (\<lambda>x. Set \<circ> Set.insert x \<circ> member)"
+ by (simp add: fun_eq_iff member_cset_of)
+ have "member \<circ> fold (\<lambda>x. Set \<circ> Set.insert x \<circ> member) xs =
+ fold Set.insert xs \<circ> member"
+ by (rule fold_commute) (simp add: fun_eq_iff mem_def)
+ then have "fold Set.insert xs (member A) =
+ member (fold (\<lambda>x. Set \<circ> Set.insert x \<circ> member) xs A)"
+ by (simp add: fun_eq_iff)
+ then have "sup (Cset.set xs) A = fold Cset.insert xs A"
+ by (simp add: union_set * member_cset_of)
+ moreover have "\<And>x y :: 'a. Cset.insert y \<circ> Cset.insert x = Cset.insert x \<circ> Cset.insert y"
+ by (auto simp add: * member_cset_of)
+ ultimately show "sup (Cset.set xs) A = foldr Cset.insert xs A"
+ by (simp add: foldr_fold)
+ show "sup (Cset.coset xs) A = Cset.coset (List.filter (Not \<circ> member A) xs)"
+ by (auto simp add: Cset.coset_def member_cset_of mem_def)
+qed
+
+context complete_lattice
+begin
+
+lemma Infimum_inf:
+ "Infimum (Cset.set As) = foldr inf As top"
+ "Infimum (Cset.coset []) = bot"
+ by (simp_all add: Inf_set_foldr)
+
+lemma Supremum_sup:
+ "Supremum (Cset.set As) = foldr sup As bot"
+ "Supremum (Cset.coset []) = top"
+ by (simp_all add: Sup_set_foldr)
+
+end
+
lemma of_pred_code [code]:
"of_pred (Predicate.Seq f) = (case f () of
Predicate.Empty \<Rightarrow> Cset.empty
@@ -287,6 +382,22 @@
apply simp_all
done
+lemma bind_set:
+ "Cset.bind (Cset.set xs) f = fold (sup \<circ> f) xs (Cset.set [])"
+ by (simp add: Cset.bind_def SUPR_set_fold)
+hide_fact (open) bind_set
+
+lemma pred_of_cset_set:
+ "pred_of_cset (Cset.set xs) = foldr sup (List.map Predicate.single xs) bot"
+proof -
+ have "pred_of_cset (Cset.set xs) = Predicate.Pred (\<lambda>x. x \<in> set xs)"
+ by (simp add: Cset.pred_of_cset_def member_set)
+ moreover have "foldr sup (List.map Predicate.single xs) bot = \<dots>"
+ by (induct xs) (auto simp add: bot_pred_def intro: pred_eqI, simp add: mem_def)
+ ultimately show ?thesis by simp
+qed
+hide_fact (open) pred_of_cset_set
+
no_notation bind (infixl "\<guillemotright>=" 70)
hide_const (open) is_empty insert remove map filter forall exists card