# HG changeset patch
# User bulwahn
# Date 1307438656 -7200
# Node ID 3c58977e0911b0f96cb25280ce133b57d0f2460e
# Parent  93b1183e43e548107c90af993c28ab187dede129# Parent  9d717505595f24b20bb0fe0c4e4799ba23297c9a
merged; manually merged IsaMakefile

diff -r 9d717505595f -r 3c58977e0911 src/HOL/IsaMakefile
--- a/src/HOL/IsaMakefile	Tue Jun 07 11:05:09 2011 +0200
+++ b/src/HOL/IsaMakefile	Tue Jun 07 11:24:16 2011 +0200
@@ -453,7 +453,7 @@
   Library/Indicator_Function.thy Library/Infinite_Set.thy		\
   Library/Inner_Product.thy Library/Kleene_Algebra.thy			\
   Library/LaTeXsugar.thy Library/Lattice_Algebras.thy			\
-  Library/Lattice_Syntax.thy Library/Library.thy			\
+  Library/Lattice_Syntax.thy Library/Library.thy Library/List_Cset.thy 	\
   Library/List_Prefix.thy Library/List_lexord.thy Library/Mapping.thy	\
   Library/Monad_Syntax.thy Library/More_List.thy Library/More_Set.thy	\
   Library/Multiset.thy Library/Nat_Bijection.thy			\
@@ -1047,7 +1047,7 @@
 $(LOG)/HOL-ex.gz: $(OUT)/HOL Decision_Procs/Commutative_Ring.thy	\
   Number_Theory/Primes.thy ex/Abstract_NAT.thy ex/Antiquote.thy		\
   ex/ATP_Export.thy ex/Arith_Examples.thy ex/Arithmetic_Series_Complex.thy \
-  ex/BT.thy	ex/BinEx.thy ex/Binary.thy ex/Birthday_Paradoxon.thy			\
+  ex/BT.thy	ex/BinEx.thy ex/Binary.thy ex/Birthday_Paradox.thy			\
   ex/CASC_Setup.thy ex/CTL.thy ex/Case_Product.thy			\
   ex/Chinese.thy ex/Classical.thy ex/CodegenSML_Test.thy		\
   ex/Coercion_Examples.thy ex/Coherent.thy ex/Dedekind_Real.thy		\
diff -r 9d717505595f -r 3c58977e0911 src/HOL/Library/Cset.thy
--- a/src/HOL/Library/Cset.thy	Tue Jun 07 11:05:09 2011 +0200
+++ b/src/HOL/Library/Cset.thy	Tue Jun 07 11:24:16 2011 +0200
@@ -35,66 +35,6 @@
   by (simp add: Cset.set_eq_iff)
 hide_fact (open) set_eqI
 
-declare mem_def [simp]
-
-definition set :: "'a list \<Rightarrow> 'a Cset.set" where
-  "set xs = Set (List.set xs)"
-hide_const (open) set
-
-lemma member_set [simp]:
-  "member (Cset.set xs) = set xs"
-  by (simp add: set_def)
-hide_fact (open) member_set
-
-definition coset :: "'a list \<Rightarrow> 'a Cset.set" where
-  "coset xs = Set (- set xs)"
-hide_const (open) coset
-
-lemma member_coset [simp]:
-  "member (Cset.coset xs) = - set xs"
-  by (simp add: coset_def)
-hide_fact (open) member_coset
-
-code_datatype Cset.set Cset.coset
-
-lemma member_code [code]:
-  "member (Cset.set xs) = List.member xs"
-  "member (Cset.coset xs) = Not \<circ> List.member xs"
-  by (simp_all add: fun_eq_iff member_def fun_Compl_def bool_Compl_def)
-
-lemma member_image_UNIV [simp]:
-  "member ` UNIV = UNIV"
-proof -
-  have "\<And>A \<Colon> 'a set. \<exists>B \<Colon> 'a Cset.set. A = member B"
-  proof
-    fix A :: "'a set"
-    show "A = member (Set A)" by simp
-  qed
-  then show ?thesis by (simp add: image_def)
-qed
-
-definition (in term_syntax)
-  setify :: "'a\<Colon>typerep list \<times> (unit \<Rightarrow> Code_Evaluation.term)
-    \<Rightarrow> 'a Cset.set \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
-  [code_unfold]: "setify xs = Code_Evaluation.valtermify Cset.set {\<cdot>} xs"
-
-notation fcomp (infixl "\<circ>>" 60)
-notation scomp (infixl "\<circ>\<rightarrow>" 60)
-
-instantiation Cset.set :: (random) random
-begin
-
-definition
-  "Quickcheck.random i = Quickcheck.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (setify xs))"
-
-instance ..
-
-end
-
-no_notation fcomp (infixl "\<circ>>" 60)
-no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
-
-
 subsection {* Lattice instantiation *}
 
 instantiation Cset.set :: (type) boolean_algebra
@@ -149,185 +89,39 @@
 definition is_empty :: "'a Cset.set \<Rightarrow> bool" where
   [simp]: "is_empty A \<longleftrightarrow> More_Set.is_empty (member A)"
 
-lemma is_empty_set [code]:
-  "is_empty (Cset.set xs) \<longleftrightarrow> List.null xs"
-  by (simp add: is_empty_set)
-hide_fact (open) is_empty_set
-
-lemma empty_set [code]:
-  "bot = Cset.set []"
-  by (simp add: set_def)
-hide_fact (open) empty_set
-
-lemma UNIV_set [code]:
-  "top = Cset.coset []"
-  by (simp add: coset_def)
-hide_fact (open) UNIV_set
-
 definition insert :: "'a \<Rightarrow> 'a Cset.set \<Rightarrow> 'a Cset.set" where
   [simp]: "insert x A = Set (Set.insert x (member A))"
 
-lemma insert_set [code]:
-  "insert x (Cset.set xs) = Cset.set (List.insert x xs)"
-  "insert x (Cset.coset xs) = Cset.coset (removeAll x xs)"
-  by (simp_all add: set_def coset_def)
-
 definition remove :: "'a \<Rightarrow> 'a Cset.set \<Rightarrow> 'a Cset.set" where
   [simp]: "remove x A = Set (More_Set.remove x (member A))"
 
-lemma remove_set [code]:
-  "remove x (Cset.set xs) = Cset.set (removeAll x xs)"
-  "remove x (Cset.coset xs) = Cset.coset (List.insert x xs)"
-  by (simp_all add: set_def coset_def remove_set_compl)
-    (simp add: More_Set.remove_def)
-
 definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a Cset.set \<Rightarrow> 'b Cset.set" where
   [simp]: "map f A = Set (image f (member A))"
 
-lemma map_set [code]:
-  "map f (Cset.set xs) = Cset.set (remdups (List.map f xs))"
-  by (simp add: set_def)
-
 enriched_type map: map
   by (simp_all add: fun_eq_iff image_compose)
 
 definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a Cset.set \<Rightarrow> 'a Cset.set" where
   [simp]: "filter P A = Set (More_Set.project P (member A))"
 
-lemma filter_set [code]:
-  "filter P (Cset.set xs) = Cset.set (List.filter P xs)"
-  by (simp add: set_def project_set)
-
 definition forall :: "('a \<Rightarrow> bool) \<Rightarrow> 'a Cset.set \<Rightarrow> bool" where
   [simp]: "forall P A \<longleftrightarrow> Ball (member A) P"
 
-lemma forall_set [code]:
-  "forall P (Cset.set xs) \<longleftrightarrow> list_all P xs"
-  by (simp add: set_def list_all_iff)
-
 definition exists :: "('a \<Rightarrow> bool) \<Rightarrow> 'a Cset.set \<Rightarrow> bool" where
   [simp]: "exists P A \<longleftrightarrow> Bex (member A) P"
 
-lemma exists_set [code]:
-  "exists P (Cset.set xs) \<longleftrightarrow> list_ex P xs"
-  by (simp add: set_def list_ex_iff)
-
 definition card :: "'a Cset.set \<Rightarrow> nat" where
   [simp]: "card A = Finite_Set.card (member A)"
-
-lemma card_set [code]:
-  "card (Cset.set xs) = length (remdups xs)"
-proof -
-  have "Finite_Set.card (set (remdups xs)) = length (remdups xs)"
-    by (rule distinct_card) simp
-  then show ?thesis by (simp add: set_def)
-qed
-
-lemma compl_set [simp, code]:
-  "- Cset.set xs = Cset.coset xs"
-  by (simp add: set_def coset_def)
-
-lemma compl_coset [simp, code]:
-  "- Cset.coset xs = Cset.set xs"
-  by (simp add: set_def coset_def)
-
-
-subsection {* Derived operations *}
-
-lemma subset_eq_forall [code]:
-  "A \<le> B \<longleftrightarrow> forall (member B) A"
-  by (simp add: subset_eq)
-
-lemma subset_subset_eq [code]:
-  "A < B \<longleftrightarrow> A \<le> B \<and> \<not> B \<le> (A :: 'a Cset.set)"
-  by (fact less_le_not_le)
-
-instantiation Cset.set :: (type) equal
-begin
-
-definition [code]:
-  "HOL.equal A B \<longleftrightarrow> A \<le> B \<and> B \<le> (A :: 'a Cset.set)"
-
-instance proof
-qed (simp add: equal_set_def set_eq [symmetric] Cset.set_eq_iff)
-
-end
-
-lemma [code nbe]:
-  "HOL.equal (A :: 'a Cset.set) A \<longleftrightarrow> True"
-  by (fact equal_refl)
-
-
-subsection {* Functorial operations *}
-
-lemma inter_project [code]:
-  "inf A (Cset.set xs) = Cset.set (List.filter (member A) xs)"
-  "inf A (Cset.coset xs) = foldr remove xs A"
-proof -
-  show "inf A (Cset.set xs) = Cset.set (List.filter (member A) xs)"
-    by (simp add: inter project_def set_def)
-  have *: "\<And>x::'a. remove = (\<lambda>x. Set \<circ> More_Set.remove x \<circ> member)"
-    by (simp add: fun_eq_iff)
-  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)
-  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 remove xs A"
-    by (simp add: Diff_eq [symmetric] minus_set *)
-  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 * intro: ext)
-  ultimately show "inf A (Cset.coset xs) = foldr remove xs A"
-    by (simp add: foldr_fold)
-qed
-
-lemma subtract_remove [code]:
-  "A - Cset.set xs = foldr 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 [code]:
-  "sup (Cset.set xs) A = foldr insert xs A"
-  "sup (Cset.coset xs) A = Cset.coset (List.filter (Not \<circ> member A) xs)"
-proof -
-  have *: "\<And>x::'a. insert = (\<lambda>x. Set \<circ> Set.insert x \<circ> member)"
-    by (simp add: fun_eq_iff)
-  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)
-  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 insert xs A"
-    by (simp add: union_set *)
-  moreover have "\<And>x y :: 'a. Cset.insert y \<circ> Cset.insert x = Cset.insert x \<circ> Cset.insert y"
-    by (auto simp add: * intro: ext)
-  ultimately show "sup (Cset.set xs) A = foldr 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: coset_def)
-qed
-
+  
 context complete_lattice
 begin
 
 definition Infimum :: "'a Cset.set \<Rightarrow> 'a" where
   [simp]: "Infimum A = Inf (member A)"
 
-lemma Infimum_inf [code]:
-  "Infimum (Cset.set As) = foldr inf As top"
-  "Infimum (Cset.coset []) = bot"
-  by (simp_all add: Inf_set_foldr Inf_UNIV)
-
 definition Supremum :: "'a Cset.set \<Rightarrow> 'a" where
   [simp]: "Supremum A = Sup (member A)"
 
-lemma Supremum_sup [code]:
-  "Supremum (Cset.set As) = foldr sup As bot"
-  "Supremum (Cset.coset []) = top"
-  by (simp_all add: Sup_set_foldr Sup_UNIV)
-
 end
 
 
@@ -351,7 +145,7 @@
 declare mem_def [simp del]
 
 
-hide_const (open) setify is_empty insert remove map filter forall exists card
+hide_const (open) is_empty insert remove map filter forall exists card
   Inter Union
 
 end
diff -r 9d717505595f -r 3c58977e0911 src/HOL/Library/Dlist_Cset.thy
--- a/src/HOL/Library/Dlist_Cset.thy	Tue Jun 07 11:05:09 2011 +0200
+++ b/src/HOL/Library/Dlist_Cset.thy	Tue Jun 07 11:24:16 2011 +0200
@@ -3,21 +3,21 @@
 header {* Canonical implementation of sets by distinct lists *}
 
 theory Dlist_Cset
-imports Dlist Cset
+imports Dlist List_Cset
 begin
 
 definition Set :: "'a dlist \<Rightarrow> 'a Cset.set" where
-  "Set dxs = Cset.set (list_of_dlist dxs)"
+  "Set dxs = List_Cset.set (list_of_dlist dxs)"
 
 definition Coset :: "'a dlist \<Rightarrow> 'a Cset.set" where
-  "Coset dxs = Cset.coset (list_of_dlist dxs)"
+  "Coset dxs = List_Cset.coset (list_of_dlist dxs)"
 
 code_datatype Set Coset
 
 declare member_code [code del]
-declare Cset.is_empty_set [code del]
-declare Cset.empty_set [code del]
-declare Cset.UNIV_set [code del]
+declare List_Cset.is_empty_set [code del]
+declare List_Cset.empty_set [code del]
+declare List_Cset.UNIV_set [code del]
 declare insert_set [code del]
 declare remove_set [code del]
 declare compl_set [code del]
@@ -50,11 +50,11 @@
   by (simp add: Coset_def member_set not_set_compl)
 
 lemma Set_dlist_of_list [code]:
-  "Cset.set xs = Set (dlist_of_list xs)"
+  "List_Cset.set xs = Set (dlist_of_list xs)"
   by (rule Cset.set_eqI) simp
 
 lemma Coset_dlist_of_list [code]:
-  "Cset.coset xs = Coset (dlist_of_list xs)"
+  "List_Cset.coset xs = Coset (dlist_of_list xs)"
   by (rule Cset.set_eqI) simp
 
 lemma is_empty_Set [code]:
diff -r 9d717505595f -r 3c58977e0911 src/HOL/Library/Library.thy
--- a/src/HOL/Library/Library.thy	Tue Jun 07 11:05:09 2011 +0200
+++ b/src/HOL/Library/Library.thy	Tue Jun 07 11:24:16 2011 +0200
@@ -29,6 +29,7 @@
   Lattice_Algebras
   Lattice_Syntax
   ListVector
+  List_Cset
   Kleene_Algebra
   Mapping
   Monad_Syntax
diff -r 9d717505595f -r 3c58977e0911 src/HOL/Library/List_Cset.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/List_Cset.thy	Tue Jun 07 11:24:16 2011 +0200
@@ -0,0 +1,222 @@
+
+(* Author: Florian Haftmann, TU Muenchen *)
+
+header {* implementation of Cset.sets based on lists *}
+
+theory List_Cset
+imports Cset
+begin
+
+declare mem_def [simp]
+
+definition set :: "'a list \<Rightarrow> 'a Cset.set" where
+  "set xs = Set (List.set xs)"
+hide_const (open) set
+
+lemma member_set [simp]:
+  "member (List_Cset.set xs) = set xs"
+  by (simp add: set_def)
+hide_fact (open) member_set
+
+definition coset :: "'a list \<Rightarrow> 'a Cset.set" where
+  "coset xs = Set (- set xs)"
+hide_const (open) coset
+
+lemma member_coset [simp]:
+  "member (List_Cset.coset xs) = - set xs"
+  by (simp add: coset_def)
+hide_fact (open) member_coset
+
+code_datatype List_Cset.set List_Cset.coset
+
+lemma member_code [code]:
+  "member (List_Cset.set xs) = List.member xs"
+  "member (List_Cset.coset xs) = Not \<circ> List.member xs"
+  by (simp_all add: fun_eq_iff member_def fun_Compl_def bool_Compl_def)
+
+lemma member_image_UNIV [simp]:
+  "member ` UNIV = UNIV"
+proof -
+  have "\<And>A \<Colon> 'a set. \<exists>B \<Colon> 'a Cset.set. A = member B"
+  proof
+    fix A :: "'a set"
+    show "A = member (Set A)" by simp
+  qed
+  then show ?thesis by (simp add: image_def)
+qed
+
+definition (in term_syntax)
+  setify :: "'a\<Colon>typerep list \<times> (unit \<Rightarrow> Code_Evaluation.term)
+    \<Rightarrow> 'a Cset.set \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
+  [code_unfold]: "setify xs = Code_Evaluation.valtermify List_Cset.set {\<cdot>} xs"
+
+notation fcomp (infixl "\<circ>>" 60)
+notation scomp (infixl "\<circ>\<rightarrow>" 60)
+
+instantiation Cset.set :: (random) random
+begin
+
+definition
+  "Quickcheck.random i = Quickcheck.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (setify xs))"
+
+instance ..
+
+end
+
+no_notation fcomp (infixl "\<circ>>" 60)
+no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
+
+subsection {* Basic operations *}
+
+lemma is_empty_set [code]:
+  "Cset.is_empty (List_Cset.set xs) \<longleftrightarrow> List.null xs"
+  by (simp add: is_empty_set null_def)
+hide_fact (open) is_empty_set
+
+lemma empty_set [code]:
+  "bot = List_Cset.set []"
+  by (simp add: set_def)
+hide_fact (open) empty_set
+
+lemma UNIV_set [code]:
+  "top = List_Cset.coset []"
+  by (simp add: coset_def)
+hide_fact (open) UNIV_set
+
+lemma remove_set [code]:
+  "Cset.remove x (List_Cset.set xs) = List_Cset.set (removeAll x xs)"
+  "Cset.remove x (List_Cset.coset xs) = List_Cset.coset (List.insert x xs)"
+by (simp_all add: set_def coset_def)
+  (metis List.set_insert More_Set.remove_def remove_set_compl)
+
+lemma insert_set [code]:
+  "Cset.insert x (List_Cset.set xs) = List_Cset.set (List.insert x xs)"
+  "Cset.insert x (List_Cset.coset xs) = List_Cset.coset (removeAll x xs)"
+  by (simp_all add: set_def coset_def)
+
+lemma map_set [code]:
+  "Cset.map f (List_Cset.set xs) = List_Cset.set (remdups (List.map f xs))"
+  by (simp add: set_def)
+  
+lemma filter_set [code]:
+  "Cset.filter P (List_Cset.set xs) = List_Cset.set (List.filter P xs)"
+  by (simp add: set_def project_set)
+
+lemma forall_set [code]:
+  "Cset.forall P (List_Cset.set xs) \<longleftrightarrow> list_all P xs"
+  by (simp add: set_def list_all_iff)
+
+lemma exists_set [code]:
+  "Cset.exists P (List_Cset.set xs) \<longleftrightarrow> list_ex P xs"
+  by (simp add: set_def list_ex_iff)
+
+lemma card_set [code]:
+  "Cset.card (List_Cset.set xs) = length (remdups xs)"
+proof -
+  have "Finite_Set.card (set (remdups xs)) = length (remdups xs)"
+    by (rule distinct_card) simp
+  then show ?thesis by (simp add: set_def)
+qed
+
+lemma compl_set [simp, code]:
+  "- List_Cset.set xs = List_Cset.coset xs"
+  by (simp add: set_def coset_def)
+
+lemma compl_coset [simp, code]:
+  "- List_Cset.coset xs = List_Cset.set xs"
+  by (simp add: set_def coset_def)
+
+context complete_lattice
+begin
+
+lemma Infimum_inf [code]:
+  "Infimum (List_Cset.set As) = foldr inf As top"
+  "Infimum (List_Cset.coset []) = bot"
+  by (simp_all add: Inf_set_foldr Inf_UNIV)
+
+lemma Supremum_sup [code]:
+  "Supremum (List_Cset.set As) = foldr sup As bot"
+  "Supremum (List_Cset.coset []) = top"
+  by (simp_all add: Sup_set_foldr Sup_UNIV)
+
+end
+
+
+subsection {* Derived operations *}
+
+lemma subset_eq_forall [code]:
+  "A \<le> B \<longleftrightarrow> Cset.forall (member B) A"
+  by (simp add: subset_eq)
+
+lemma subset_subset_eq [code]:
+  "A < B \<longleftrightarrow> A \<le> B \<and> \<not> B \<le> (A :: 'a Cset.set)"
+  by (fact less_le_not_le)
+
+instantiation Cset.set :: (type) equal
+begin
+
+definition [code]:
+  "HOL.equal A B \<longleftrightarrow> A \<le> B \<and> B \<le> (A :: 'a Cset.set)"
+
+instance proof
+qed (simp add: equal_set_def set_eq [symmetric] Cset.set_eq_iff)
+
+end
+
+lemma [code nbe]:
+  "HOL.equal (A :: 'a Cset.set) A \<longleftrightarrow> True"
+  by (fact equal_refl)
+
+
+subsection {* Functorial operations *}
+
+lemma inter_project [code]:
+  "inf A (List_Cset.set xs) = List_Cset.set (List.filter (Cset.member A) xs)"
+  "inf A (List_Cset.coset xs) = foldr Cset.remove xs A"
+proof -
+  show "inf A (List_Cset.set xs) = List_Cset.set (List.filter (member A) xs)"
+    by (simp add: inter project_def set_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)
+  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)
+  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 (List_Cset.coset xs) = fold Cset.remove xs A"
+    by (simp add: Diff_eq [symmetric] minus_set *)
+  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 * intro: ext)
+  ultimately show "inf A (List_Cset.coset xs) = foldr Cset.remove xs A"
+    by (simp add: foldr_fold)
+qed
+
+lemma subtract_remove [code]:
+  "A - List_Cset.set xs = foldr Cset.remove xs A"
+  "A - List_Cset.coset xs = List_Cset.set (List.filter (member A) xs)"
+  by (simp_all only: diff_eq compl_set compl_coset inter_project)
+
+lemma union_insert [code]:
+  "sup (List_Cset.set xs) A = foldr Cset.insert xs A"
+  "sup (List_Cset.coset xs) A = List_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)
+  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)
+  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 (List_Cset.set xs) A = fold Cset.insert xs A"
+    by (simp add: union_set *)
+  moreover have "\<And>x y :: 'a. Cset.insert y \<circ> Cset.insert x = Cset.insert x \<circ> Cset.insert y"
+    by (auto simp add: * intro: ext)
+  ultimately show "sup (List_Cset.set xs) A = foldr Cset.insert xs A"
+    by (simp add: foldr_fold)
+  show "sup (List_Cset.coset xs) A = List_Cset.coset (List.filter (Not \<circ> member A) xs)"
+    by (auto simp add: coset_def)
+qed
+
+end
\ No newline at end of file
diff -r 9d717505595f -r 3c58977e0911 src/HOL/Library/Quickcheck_Narrowing.thy
--- a/src/HOL/Library/Quickcheck_Narrowing.thy	Tue Jun 07 11:05:09 2011 +0200
+++ b/src/HOL/Library/Quickcheck_Narrowing.thy	Tue Jun 07 11:24:16 2011 +0200
@@ -5,6 +5,8 @@
 theory Quickcheck_Narrowing
 imports Main "~~/src/HOL/Library/Code_Char"
 uses
+  ("~~/src/HOL/Tools/Quickcheck/PNF_Narrowing_Engine.hs")
+  ("~~/src/HOL/Tools/Quickcheck/Narrowing_Engine.hs")
   ("~~/src/HOL/Tools/Quickcheck/narrowing_generators.ML")
 begin
 
@@ -454,6 +456,17 @@
 
 end
 
+datatype property = Universal narrowing_type "(narrowing_term => property)" "narrowing_term => Code_Evaluation.term" | Existential narrowing_type "(narrowing_term => property)" "narrowing_term => Code_Evaluation.term" | Property bool
+
+(* FIXME: hard-wired maximal depth of 100 here *)
+fun exists :: "('a :: {narrowing, partial_term_of} => property) => property"
+where
+  "exists f = (case narrowing (100 :: code_int) of C ty cs => Existential ty (\<lambda> t. f (conv cs t)) (partial_term_of (TYPE('a))))"
+
+fun "all" :: "('a :: {narrowing, partial_term_of} => property) => property"
+where
+  "all f = (case narrowing (100 :: code_int) of C ty cs => Universal ty (\<lambda>t. f (conv cs t)) (partial_term_of (TYPE('a))))"
+
 subsubsection {* class @{text is_testable} *}
 
 text {* The class @{text is_testable} ensures that all necessary type instances are generated. *}
@@ -492,13 +505,15 @@
 hide_const (open) Constant eval_cfun
 
 subsubsection {* Setting up the counterexample generator *}
-  
+
+setup {* Thy_Load.provide_file (Path.explode ("~~/src/HOL/Tools/Quickcheck/PNF_Narrowing_Engine.hs")) *}
+setup {* Thy_Load.provide_file (Path.explode ("~~/src/HOL/Tools/Quickcheck/Narrowing_Engine.hs")) *}
 use "~~/src/HOL/Tools/Quickcheck/narrowing_generators.ML"
 
 setup {* Narrowing_Generators.setup *}
 
 hide_type (open) code_int narrowing_type narrowing_term cons
 hide_const (open) int_of of_int nth error toEnum map_index split_At empty
-  C cons conv nonEmpty "apply" sum cons1 cons2 ensure_testable
+  C cons conv nonEmpty "apply" sum cons1 cons2 ensure_testable all exists
 
 end
\ No newline at end of file
diff -r 9d717505595f -r 3c58977e0911 src/HOL/Tools/Quickcheck/narrowing_generators.ML
--- a/src/HOL/Tools/Quickcheck/narrowing_generators.ML	Tue Jun 07 11:05:09 2011 +0200
+++ b/src/HOL/Tools/Quickcheck/narrowing_generators.ML	Tue Jun 07 11:24:16 2011 +0200
@@ -6,10 +6,15 @@
 
 signature NARROWING_GENERATORS =
 sig
-  val test_term: Proof.context -> bool * bool -> term * term list -> Quickcheck.result
-  val put_counterexample: (unit -> term list option) -> Proof.context -> Proof.context
+  val allow_existentials : bool Config.T
   val finite_functions : bool Config.T
   val overlord : bool Config.T
+  val test_term: Proof.context -> bool * bool -> term * term list -> Quickcheck.result
+  datatype counterexample = Universal_Counterexample of (term * counterexample)
+    | Existential_Counterexample of (term * counterexample) list
+    | Empty_Assignment
+  val put_counterexample: (unit -> term list option) -> Proof.context -> Proof.context
+  val put_existential_counterexample : (unit -> counterexample option) -> Proof.context -> Proof.context
   val setup: theory -> theory
 end;
 
@@ -18,6 +23,7 @@
 
 (* configurations *)
 
+val allow_existentials = Attrib.setup_config_bool @{binding quickcheck_allow_existentials} (K true)
 val finite_functions = Attrib.setup_config_bool @{binding quickcheck_finite_functions} (K true)
 val overlord = Attrib.setup_config_bool @{binding quickcheck_narrowing_overlord} (K false)
 
@@ -165,18 +171,25 @@
   in
     eqs
   end
-  
+    
+fun contains_recursive_type_under_function_types xs =
+  exists (fn (_, (_, _, cs)) => cs |> exists (snd #> exists (fn dT =>
+    (case Datatype_Aux.strip_dtyp dT of (_ :: _, Datatype.DtRec _) => true | _ => false)))) xs
+
 fun instantiate_narrowing_datatype config descr vs tycos prfx (names, auxnames) (Ts, Us) thy =
   let
     val _ = Datatype_Aux.message config "Creating narrowing generators ...";
     val narrowingsN = map (prefix (narrowingN ^ "_")) (names @ auxnames);
   in
-    thy
-    |> Class.instantiation (tycos, vs, @{sort narrowing})
-    |> Quickcheck_Common.define_functions
-      (fn narrowings => mk_equations descr vs tycos narrowings (Ts, Us), NONE)
-      prfx [] narrowingsN (map narrowingT (Ts @ Us))
-    |> Class.prove_instantiation_exit (K (Class.intro_classes_tac []))
+    if not (contains_recursive_type_under_function_types descr) then
+      thy
+      |> Class.instantiation (tycos, vs, @{sort narrowing})
+      |> Quickcheck_Common.define_functions
+        (fn narrowings => mk_equations descr vs tycos narrowings (Ts, Us), NONE)
+        prfx [] narrowingsN (map narrowingT (Ts @ Us))
+      |> Class.prove_instantiation_exit (K (Class.intro_classes_tac []))
+    else
+      thy
   end;
 
 (* testing framework *)
@@ -186,6 +199,7 @@
 (** invocation of Haskell interpreter **)
 
 val narrowing_engine = File.read (Path.explode "~~/src/HOL/Tools/Quickcheck/Narrowing_Engine.hs")
+val pnf_narrowing_engine = File.read (Path.explode "~~/src/HOL/Tools/Quickcheck/PNF_Narrowing_Engine.hs")
 
 fun exec verbose code =
   ML_Context.exec (fn () => Secure.use_text ML_Env.local_context (0, "generated code") verbose code)
@@ -196,7 +210,7 @@
     val _ = Isabelle_System.mkdirs path;
   in Exn.release (Exn.capture f path) end;
   
-fun value ctxt (get, put, put_ml) (code, value_name) =
+fun value contains_existentials ctxt (get, put, put_ml) (code, value_name) =
   let
     fun message s = if Config.get ctxt Quickcheck.quiet then () else Output.urgent_message s
     val tmp_prefix = "Quickcheck_Narrowing"
@@ -216,16 +230,14 @@
         val code' = prefix "module Code where {\n\ndata Typerep = Typerep String [Typerep];\n"
           (unprefix "module Code where {" code)
         val _ = File.write code_file code'
-        val _ = File.write narrowing_engine_file narrowing_engine
+        val _ = File.write narrowing_engine_file
+          (if contains_existentials then pnf_narrowing_engine else narrowing_engine)
         val _ = File.write main_file main
         val executable = File.shell_path (Path.append in_path (Path.basic "isabelle_quickcheck_narrowing"))
         val cmd = "exec \"$ISABELLE_GHC\" -fglasgow-exts " ^
           (space_implode " " (map File.shell_path [code_file, narrowing_engine_file, main_file])) ^
           " -o " ^ executable ^ ";"
-        val _ = if bash cmd <> 0 then
-          error "Compilation failed!"
-        else
-          ()
+        val _ = if bash cmd <> 0 then error "Compilation with GHC failed" else ()
         fun with_size k =
           if k > Config.get ctxt Quickcheck.size then
             NONE
@@ -252,10 +264,10 @@
       end
   in with_tmp_dir tmp_prefix run end;
 
-fun dynamic_value_strict cookie thy postproc t =
+fun dynamic_value_strict contains_existentials cookie thy postproc t =
   let
     val ctxt = Proof_Context.init_global thy
-    fun evaluator naming program ((_, vs_ty), t) deps = Exn.interruptible_capture (value ctxt cookie)
+    fun evaluator naming program ((_, vs_ty), t) deps = Exn.interruptible_capture (value contains_existentials ctxt cookie)
       (Code_Target.evaluator thy target naming program deps (vs_ty, t));    
   in Exn.release (Code_Thingol.dynamic_value thy (Exn.map_result o postproc) evaluator t) end;
 
@@ -267,11 +279,29 @@
   fun init _ () = error "Counterexample"
 )
 
+datatype counterexample = Universal_Counterexample of (term * counterexample)
+  | Existential_Counterexample of (term * counterexample) list
+  | Empty_Assignment
+  
+fun map_counterexample f Empty_Assignment = Empty_Assignment
+  | map_counterexample f (Universal_Counterexample (t, c)) =
+      Universal_Counterexample (f t, map_counterexample f c)
+  | map_counterexample f (Existential_Counterexample cs) =
+      Existential_Counterexample (map (fn (t, c) => (f t, map_counterexample f c)) cs)
+
+structure Existential_Counterexample = Proof_Data
+(
+  type T = unit -> counterexample option
+  fun init _ () = error "Counterexample"
+)
+
+val put_existential_counterexample = Existential_Counterexample.put
+
 val put_counterexample = Counterexample.put
 
-fun finitize_functions t =
+fun finitize_functions (xTs, t) =
   let
-    val ((names, Ts), t') = apfst split_list (strip_abs t)
+    val (names, boundTs) = split_list xTs
     fun mk_eval_ffun dT rT =
       Const (@{const_name "Quickcheck_Narrowing.eval_ffun"}, 
         Type (@{type_name "Quickcheck_Narrowing.ffun"}, [dT, rT]) --> dT --> rT)
@@ -290,27 +320,97 @@
             Type (@{type_name "Quickcheck_Narrowing.ffun"}, [dT, rT']))
       end
       | eval_function T = (I, T)
-    val (tt, Ts') = split_list (map eval_function Ts)
-    val t'' = subst_bounds (map2 (fn f => fn x => f x) (rev tt) (map_index (Bound o fst) Ts), t')
+    val (tt, boundTs') = split_list (map eval_function boundTs)
+    val t' = subst_bounds (map2 (fn f => fn x => f x) (rev tt) (map_index (Bound o fst) boundTs), t)
   in
-    list_abs (names ~~ Ts', t'')
+    (names ~~ boundTs', t')
   end
 
 (** tester **)
+
+val rewrs =
+    map (swap o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of)
+      (@{thms all_simps} @ @{thms ex_simps})
+    @ map (HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of)
+        [@{thm iff_conv_conj_imp}, @{thm not_ex}, @{thm not_all}]
+
+fun make_pnf_term thy t = Pattern.rewrite_term thy rewrs [] t
+
+fun strip_quantifiers (Const (@{const_name Ex}, _) $ Abs (x, T, t)) =
+    apfst (cons (@{const_name Ex}, (x, T))) (strip_quantifiers t)
+  | strip_quantifiers (Const (@{const_name All}, _) $ Abs (x, T, t)) =
+    apfst (cons (@{const_name All}, (x, T))) (strip_quantifiers t)
+  | strip_quantifiers t = ([], t)
+
+fun contains_existentials t = exists (fn (Q, _) => Q = @{const_name Ex}) (fst (strip_quantifiers t))
+
+fun mk_property qs t =
+  let
+    fun enclose (@{const_name Ex}, (x, T)) t =
+        Const (@{const_name Quickcheck_Narrowing.exists}, (T --> @{typ property}) --> @{typ property})
+          $ Abs (x, T, t)
+      | enclose (@{const_name All}, (x, T)) t =
+        Const (@{const_name Quickcheck_Narrowing.all}, (T --> @{typ property}) --> @{typ property})
+          $ Abs (x, T, t)
+  in
+    fold_rev enclose qs (@{term Quickcheck_Narrowing.Property} $
+      (list_comb (t , map Bound (((length qs) - 1) downto 0))))
+  end
+
+fun mk_case_term ctxt p ((@{const_name Ex}, (x, T)) :: qs') (Existential_Counterexample cs) =
+    fst (Datatype.make_case ctxt Datatype_Case.Quiet [] (Free (x, T)) (map (fn (t, c) =>
+      (t, mk_case_term ctxt (p - 1) qs' c)) cs))
+  | mk_case_term ctxt p ((@{const_name All}, (x, T)) :: qs') (Universal_Counterexample (t, c)) =
+    if p = 0 then t else mk_case_term ctxt (p - 1) qs' c
+
+fun mk_terms ctxt qs result =
+  let
+    val
+      ps = filter (fn (_, (@{const_name All}, _)) => true | _ => false) (map_index I qs)
+    in
+      map (fn (p, (_, (x, T))) => (x, mk_case_term ctxt p qs result)) ps
+    end
   
-fun test_term  ctxt (limit_time, is_interactive) (t, eval_terms) =
+fun test_term ctxt (limit_time, is_interactive) (t, eval_terms) =
   let
     val thy = Proof_Context.theory_of ctxt
-    val t' = list_abs_free (Term.add_frees t [], t)
-    val t'' = if Config.get ctxt finite_functions then finitize_functions t' else t'
-    fun ensure_testable t =
-      Const (@{const_name Quickcheck_Narrowing.ensure_testable}, fastype_of t --> fastype_of t) $ t
-    val result = dynamic_value_strict
-      (Counterexample.get, Counterexample.put, "Narrowing_Generators.put_counterexample")
-      thy (Option.map o map) (ensure_testable t'')
+    val t' = fold_rev (fn (x, T) => fn t => HOLogic.mk_all (x, T, t)) (Term.add_frees t []) t
+    val pnf_t = make_pnf_term thy t'
   in
-    Quickcheck.Result {counterexample = Option.map ((curry (op ~~)) (Term.add_free_names t [])) result,
-      evaluation_terms = Option.map (K []) result, timings = [], reports = []}
+    if Config.get ctxt allow_existentials andalso contains_existentials pnf_t then
+      let
+        fun wrap f (qs, t) =
+          let val (qs1, qs2) = split_list qs in
+          apfst (map2 pair qs1) (f (qs2, t)) end
+        val finitize = if Config.get ctxt finite_functions then wrap finitize_functions else I
+        val (qs, prop_t) = finitize (strip_quantifiers pnf_t)
+        val prop_term = fold_rev (fn (_, (x, T)) => fn t => Abs (x, T, t)) qs prop_t
+        val ((prop_def, _), ctxt') = Local_Theory.define ((Binding.conceal (Binding.name "test_property"), NoSyn),
+          ((Binding.conceal Binding.empty, [Code.add_default_eqn_attrib]), prop_term)) ctxt
+        val (prop_def', thy') = Local_Theory.exit_result_global Morphism.term (prop_def, ctxt') 
+        val result = dynamic_value_strict true
+          (Existential_Counterexample.get, Existential_Counterexample.put,
+            "Narrowing_Generators.put_existential_counterexample")
+          thy' (Option.map o map_counterexample) (mk_property qs prop_def')
+        val result' = Option.map (mk_terms ctxt' (fst (strip_quantifiers pnf_t))) result
+      in
+        Quickcheck.Result {counterexample = result', evaluation_terms = Option.map (K []) result,
+          timings = [], reports = []}
+      end
+    else
+      let
+        val t' = Term.list_abs_free (Term.add_frees t [], t)
+        fun wrap f t = list_abs (f (strip_abs t))
+        val finitize = if Config.get ctxt finite_functions then wrap finitize_functions else I
+        fun ensure_testable t =
+          Const (@{const_name Quickcheck_Narrowing.ensure_testable}, fastype_of t --> fastype_of t) $ t
+        val result = dynamic_value_strict false
+          (Counterexample.get, Counterexample.put, "Narrowing_Generators.put_counterexample")
+          thy (Option.map o map) (ensure_testable (finitize t'))
+      in
+        Quickcheck.Result {counterexample = Option.map ((curry (op ~~)) (Term.add_free_names t [])) result,
+          evaluation_terms = Option.map (K []) result, timings = [], reports = []}
+      end
   end;
 
 fun test_goals ctxt (limit_time, is_interactive) insts goals =
diff -r 9d717505595f -r 3c58977e0911 src/HOL/ex/Birthday_Paradox.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ex/Birthday_Paradox.thy	Tue Jun 07 11:24:16 2011 +0200
@@ -0,0 +1,101 @@
+(*  Title: HOL/ex/Birthday_Paradox.thy
+    Author: Lukas Bulwahn, TU Muenchen, 2007
+*)
+
+header {* A Formulation of the Birthday Paradox *}
+
+theory Birthday_Paradox
+imports Main "~~/src/HOL/Fact" "~~/src/HOL/Library/FuncSet"
+begin
+
+section {* Cardinality *}
+
+lemma card_product_dependent:
+  assumes "finite S"
+  assumes "\<forall>x \<in> S. finite (T x)" 
+  shows "card {(x, y). x \<in> S \<and> y \<in> T x} = (\<Sum>x \<in> S. card (T x))"
+proof -
+  note `finite S`
+  moreover
+  have "{(x, y). x \<in> S \<and> y \<in> T x} = (UN x : S. Pair x ` T x)" by auto
+  moreover
+  from `\<forall>x \<in> S. finite (T x)` have "ALL x:S. finite (Pair x ` T x)" by auto
+  moreover
+  have " ALL i:S. ALL j:S. i ~= j --> Pair i ` T i Int Pair j ` T j = {}" by auto
+  moreover  
+  ultimately have "card {(x, y). x \<in> S \<and> y \<in> T x} = (SUM i:S. card (Pair i ` T i))"
+    by (auto, subst card_UN_disjoint) auto
+  also have "... = (SUM x:S. card (T x))"
+    by (subst card_image) (auto intro: inj_onI)
+  finally show ?thesis by auto
+qed
+
+lemma card_extensional_funcset_inj_on:
+  assumes "finite S" "finite T" "card S \<le> card T"
+  shows "card {f \<in> extensional_funcset S T. inj_on f S} = fact (card T) div (fact (card T - card S))"
+using assms
+proof (induct S arbitrary: T rule: finite_induct)
+  case empty
+  from this show ?case by (simp add: Collect_conv_if extensional_funcset_empty_domain)
+next
+  case (insert x S)
+  { fix x
+    from `finite T` have "finite (T - {x})" by auto
+    from `finite S` this have "finite (extensional_funcset S (T - {x}))"
+      by (rule finite_extensional_funcset)
+    moreover
+    have "{f : extensional_funcset S (T - {x}). inj_on f S} \<subseteq> (extensional_funcset S (T - {x}))" by auto    
+    ultimately have "finite {f : extensional_funcset S (T - {x}). inj_on f S}"
+      by (auto intro: finite_subset)
+  } note finite_delete = this
+  from insert have hyps: "\<forall>y \<in> T. card ({g. g \<in> extensional_funcset S (T - {y}) \<and> inj_on g S}) = fact (card T - 1) div fact ((card T - 1) - card S)"(is "\<forall> _ \<in> T. _ = ?k") by auto
+  from extensional_funcset_extend_domain_inj_on_eq[OF `x \<notin> S`]
+  have "card {f. f : extensional_funcset (insert x S) T & inj_on f (insert x S)} =
+    card ((%(y, g). g(x := y)) ` {(y, g). y : T & g : extensional_funcset S (T - {y}) & inj_on g S})"
+    by metis
+  also from extensional_funcset_extend_domain_inj_onI[OF `x \<notin> S`, of T] have "... =  card {(y, g). y : T & g : extensional_funcset S (T - {y}) & inj_on g S}"
+    by (simp add: card_image)
+  also have "card {(y, g). y \<in> T \<and> g \<in> extensional_funcset S (T - {y}) \<and> inj_on g S} =
+    card {(y, g). y \<in> T \<and> g \<in> {f \<in> extensional_funcset S (T - {y}). inj_on f S}}" by auto
+  also from `finite T` finite_delete have "... = (\<Sum>y \<in> T. card {g. g \<in> extensional_funcset S (T - {y}) \<and>  inj_on g S})"
+    by (subst card_product_dependent) auto
+  also from hyps have "... = (card T) * ?k"
+    by auto
+  also have "... = card T * fact (card T - 1) div fact (card T - card (insert x S))"
+    using insert unfolding div_mult1_eq[of "card T" "fact (card T - 1)"]
+    by (simp add: fact_mod)
+  also have "... = fact (card T) div fact (card T - card (insert x S))"
+    using insert by (simp add: fact_reduce_nat[of "card T"])
+  finally show ?case .
+qed
+
+lemma card_extensional_funcset_not_inj_on:
+  assumes "finite S" "finite T" "card S \<le> card T"
+  shows "card {f \<in> extensional_funcset S T. \<not> inj_on f S} = (card T) ^ (card S) - (fact (card T)) div (fact (card T - card S))"
+proof -
+  have subset: "{f : extensional_funcset S T. inj_on f S} <= extensional_funcset S T" by auto
+  from finite_subset[OF subset] assms have finite: "finite {f : extensional_funcset S T. inj_on f S}"
+    by (auto intro!: finite_extensional_funcset)
+  have "{f \<in> extensional_funcset S T. \<not> inj_on f S} = extensional_funcset S T - {f \<in> extensional_funcset S T. inj_on f S}" by auto 
+  from assms this finite subset show ?thesis
+    by (simp add: card_Diff_subset card_extensional_funcset card_extensional_funcset_inj_on)
+qed
+
+lemma setprod_upto_nat_unfold:
+  "setprod f {m..(n::nat)} = (if n < m then 1 else (if n = 0 then f 0 else f n * setprod f {m..(n - 1)}))"
+  by auto (auto simp add: gr0_conv_Suc atLeastAtMostSuc_conv)
+
+section {* Birthday paradox *}
+
+lemma birthday_paradox:
+  assumes "card S = 23" "card T = 365"
+  shows "2 * card {f \<in> extensional_funcset S T. \<not> inj_on f S} \<ge> card (extensional_funcset S T)"
+proof -
+  from `card S = 23` `card T = 365` have "finite S" "finite T" "card S <= card T" by (auto intro: card_ge_0_finite)
+  from assms show ?thesis
+    using card_extensional_funcset[OF `finite S`, of T]
+      card_extensional_funcset_not_inj_on[OF `finite S` `finite T` `card S <= card T`]
+    by (simp add: fact_div_fact setprod_upto_nat_unfold)
+qed
+
+end
diff -r 9d717505595f -r 3c58977e0911 src/HOL/ex/Birthday_Paradoxon.thy
--- a/src/HOL/ex/Birthday_Paradoxon.thy	Tue Jun 07 11:05:09 2011 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,101 +0,0 @@
-(*  Title: HOL/ex/Birthday_Paradoxon.thy
-    Author: Lukas Bulwahn, TU Muenchen, 2007
-*)
-
-header {* A Formulation of the Birthday Paradoxon *}
-
-theory Birthday_Paradoxon
-imports Main "~~/src/HOL/Fact" "~~/src/HOL/Library/FuncSet"
-begin
-
-section {* Cardinality *}
-
-lemma card_product_dependent:
-  assumes "finite S"
-  assumes "\<forall>x \<in> S. finite (T x)" 
-  shows "card {(x, y). x \<in> S \<and> y \<in> T x} = (\<Sum>x \<in> S. card (T x))"
-proof -
-  note `finite S`
-  moreover
-  have "{(x, y). x \<in> S \<and> y \<in> T x} = (UN x : S. Pair x ` T x)" by auto
-  moreover
-  from `\<forall>x \<in> S. finite (T x)` have "ALL x:S. finite (Pair x ` T x)" by auto
-  moreover
-  have " ALL i:S. ALL j:S. i ~= j --> Pair i ` T i Int Pair j ` T j = {}" by auto
-  moreover  
-  ultimately have "card {(x, y). x \<in> S \<and> y \<in> T x} = (SUM i:S. card (Pair i ` T i))"
-    by (auto, subst card_UN_disjoint) auto
-  also have "... = (SUM x:S. card (T x))"
-    by (subst card_image) (auto intro: inj_onI)
-  finally show ?thesis by auto
-qed
-
-lemma card_extensional_funcset_inj_on:
-  assumes "finite S" "finite T" "card S \<le> card T"
-  shows "card {f \<in> extensional_funcset S T. inj_on f S} = fact (card T) div (fact (card T - card S))"
-using assms
-proof (induct S arbitrary: T rule: finite_induct)
-  case empty
-  from this show ?case by (simp add: Collect_conv_if extensional_funcset_empty_domain)
-next
-  case (insert x S)
-  { fix x
-    from `finite T` have "finite (T - {x})" by auto
-    from `finite S` this have "finite (extensional_funcset S (T - {x}))"
-      by (rule finite_extensional_funcset)
-    moreover
-    have "{f : extensional_funcset S (T - {x}). inj_on f S} \<subseteq> (extensional_funcset S (T - {x}))" by auto    
-    ultimately have "finite {f : extensional_funcset S (T - {x}). inj_on f S}"
-      by (auto intro: finite_subset)
-  } note finite_delete = this
-  from insert have hyps: "\<forall>y \<in> T. card ({g. g \<in> extensional_funcset S (T - {y}) \<and> inj_on g S}) = fact (card T - 1) div fact ((card T - 1) - card S)"(is "\<forall> _ \<in> T. _ = ?k") by auto
-  from extensional_funcset_extend_domain_inj_on_eq[OF `x \<notin> S`]
-  have "card {f. f : extensional_funcset (insert x S) T & inj_on f (insert x S)} =
-    card ((%(y, g). g(x := y)) ` {(y, g). y : T & g : extensional_funcset S (T - {y}) & inj_on g S})"
-    by metis
-  also from extensional_funcset_extend_domain_inj_onI[OF `x \<notin> S`, of T] have "... =  card {(y, g). y : T & g : extensional_funcset S (T - {y}) & inj_on g S}"
-    by (simp add: card_image)
-  also have "card {(y, g). y \<in> T \<and> g \<in> extensional_funcset S (T - {y}) \<and> inj_on g S} =
-    card {(y, g). y \<in> T \<and> g \<in> {f \<in> extensional_funcset S (T - {y}). inj_on f S}}" by auto
-  also from `finite T` finite_delete have "... = (\<Sum>y \<in> T. card {g. g \<in> extensional_funcset S (T - {y}) \<and>  inj_on g S})"
-    by (subst card_product_dependent) auto
-  also from hyps have "... = (card T) * ?k"
-    by auto
-  also have "... = card T * fact (card T - 1) div fact (card T - card (insert x S))"
-    using insert unfolding div_mult1_eq[of "card T" "fact (card T - 1)"]
-    by (simp add: fact_mod)
-  also have "... = fact (card T) div fact (card T - card (insert x S))"
-    using insert by (simp add: fact_reduce_nat[of "card T"])
-  finally show ?case .
-qed
-
-lemma card_extensional_funcset_not_inj_on:
-  assumes "finite S" "finite T" "card S \<le> card T"
-  shows "card {f \<in> extensional_funcset S T. \<not> inj_on f S} = (card T) ^ (card S) - (fact (card T)) div (fact (card T - card S))"
-proof -
-  have subset: "{f : extensional_funcset S T. inj_on f S} <= extensional_funcset S T" by auto
-  from finite_subset[OF subset] assms have finite: "finite {f : extensional_funcset S T. inj_on f S}"
-    by (auto intro!: finite_extensional_funcset)
-  have "{f \<in> extensional_funcset S T. \<not> inj_on f S} = extensional_funcset S T - {f \<in> extensional_funcset S T. inj_on f S}" by auto 
-  from assms this finite subset show ?thesis
-    by (simp add: card_Diff_subset card_extensional_funcset card_extensional_funcset_inj_on)
-qed
-
-lemma setprod_upto_nat_unfold:
-  "setprod f {m..(n::nat)} = (if n < m then 1 else (if n = 0 then f 0 else f n * setprod f {m..(n - 1)}))"
-  by auto (auto simp add: gr0_conv_Suc atLeastAtMostSuc_conv)
-
-section {* Birthday paradoxon *}
-
-lemma birthday_paradoxon:
-  assumes "card S = 23" "card T = 365"
-  shows "2 * card {f \<in> extensional_funcset S T. \<not> inj_on f S} \<ge> card (extensional_funcset S T)"
-proof -
-  from `card S = 23` `card T = 365` have "finite S" "finite T" "card S <= card T" by (auto intro: card_ge_0_finite)
-  from assms show ?thesis
-    using card_extensional_funcset[OF `finite S`, of T]
-      card_extensional_funcset_not_inj_on[OF `finite S` `finite T` `card S <= card T`]
-    by (simp add: fact_div_fact setprod_upto_nat_unfold)
-qed
-
-end
diff -r 9d717505595f -r 3c58977e0911 src/HOL/ex/Quickcheck_Narrowing_Examples.thy
--- a/src/HOL/ex/Quickcheck_Narrowing_Examples.thy	Tue Jun 07 11:05:09 2011 +0200
+++ b/src/HOL/ex/Quickcheck_Narrowing_Examples.thy	Tue Jun 07 11:24:16 2011 +0200
@@ -23,14 +23,56 @@
 oops
 *)
 
+(*declare [[quickcheck_narrowing_overlord]]*)
+subsection {* Simple examples with existentials *}
+
+lemma
+  "\<exists> y :: nat. \<forall> x. x = y"
+quickcheck[tester = narrowing, finite_types = false, expect = counterexample]
+oops
+
+lemma
+  "x > 1 --> (\<exists>y :: nat. x < y & y <= 1)"
+quickcheck[tester = narrowing, finite_types = false, expect = counterexample]
+oops
+
+lemma
+  "x > 2 --> (\<exists>y :: nat. x < y & y <= 2)"
+quickcheck[tester = narrowing, finite_types = false, size = 10]
+oops
+
+lemma
+  "\<forall> x. \<exists> y :: nat. x > 3 --> (y < x & y > 3)"
+quickcheck[tester = narrowing, finite_types = false, size = 7]
+oops
+
+text {* A false conjecture derived from an partial copy-n-paste of @{thm not_distinct_decomp} *}
+lemma
+  "~ distinct ws ==> EX xs ys zs y. ws = xs @ [y] @ ys @ [y]"
+quickcheck[tester = narrowing, finite_types = false, default_type = nat, expect = counterexample]
+oops
+
+text {* A false conjecture derived from theorems @{thm split_list_first} and @{thm split_list_last} *}  
+lemma
+  "x : set xs ==> EX ys zs. xs = ys @ x # zs & x ~: set zs & x ~: set ys"
+quickcheck[tester = narrowing, finite_types = false, default_type = nat, expect = counterexample]
+oops
+
+text {* A false conjecture derived from @{thm map_eq_Cons_conv} with a typo *}
+lemma
+  "(map f xs = y # ys) = (EX z zs. xs = z' # zs & f z = y & map f zs = ys)"
+quickcheck[tester = narrowing, finite_types = false, default_type = nat, expect = counterexample]
+oops
+
 subsection {* Simple list examples *}
 
 lemma "rev xs = xs"
-  quickcheck[tester = narrowing, finite_types = false, default_type = nat, expect = counterexample]
+quickcheck[tester = narrowing, finite_types = false, default_type = nat, expect = counterexample]
 oops
 
+(* FIXME: integer has strange representation! *)
 lemma "rev xs = xs"
-  quickcheck[tester = narrowing, finite_types = false, default_type = int, expect = counterexample]
+quickcheck[tester = narrowing, finite_types = false, default_type = int, expect = counterexample]
 oops
 (*
 lemma "rev xs = xs"
@@ -39,30 +81,29 @@
 *)
 subsection {* Simple examples with functions *}
 
-declare [[quickcheck_finite_functions]]
-(*
 lemma "map f xs = map g xs"
-  quickcheck[tester = narrowing, finite_types = true, expect = counterexample]
+  quickcheck[tester = narrowing, finite_types = false, expect = counterexample]
 oops
 
 lemma "map f xs = map f ys ==> xs = ys"
-  quickcheck[tester = narrowing, finite_types = true, expect = counterexample]
+  quickcheck[tester = narrowing, finite_types = false, expect = counterexample]
 oops
 
 lemma
   "list_all2 P (rev xs) (rev ys) = list_all2 P xs (rev ys)"
-  quickcheck[tester = narrowing, expect = counterexample]
+  quickcheck[tester = narrowing, finite_types = false, expect = counterexample]
 oops
 
 lemma "map f xs = F f xs"
-  quickcheck[tester = narrowing, finite_types = true, expect = counterexample]
-oops
-*)
-lemma "map f xs = F f xs"
   quickcheck[tester = narrowing, finite_types = false, expect = counterexample]
 oops
 
-(* requires some work...
+lemma "map f xs = F f xs"
+  quickcheck[tester = narrowing, finite_types = false, expect = counterexample]
+oops
+
+(* requires some work...*)
+(*
 lemma "R O S = S O R"
   quickcheck[tester = narrowing, size = 2]
 oops
diff -r 9d717505595f -r 3c58977e0911 src/HOL/ex/ROOT.ML
--- a/src/HOL/ex/ROOT.ML	Tue Jun 07 11:05:09 2011 +0200
+++ b/src/HOL/ex/ROOT.ML	Tue Jun 07 11:24:16 2011 +0200
@@ -73,7 +73,7 @@
   "Gauge_Integration",
   "Dedekind_Real",
   "Quicksort",
-  "Birthday_Paradoxon",
+  "Birthday_Paradox",
   "List_to_Set_Comprehension_Examples",
   "Set_Algebras"
 ];