--- a/src/HOL/Library/Cset.thy Tue Jun 07 11:11:01 2011 +0200
+++ b/src/HOL/Library/Cset.thy Tue Jun 07 11:12:05 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