1.1 --- a/src/HOL/IsaMakefile Thu Jun 25 15:42:36 2009 +0200
1.2 +++ b/src/HOL/IsaMakefile Thu Jun 25 17:07:18 2009 +0200
1.3 @@ -319,7 +319,7 @@
1.4 Library/Abstract_Rat.thy \
1.5 Library/BigO.thy Library/ContNotDenum.thy Library/Efficient_Nat.thy \
1.6 Library/Euclidean_Space.thy Library/Sum_Of_Squares.thy Library/positivstellensatz.ML \
1.7 - Library/Convex_Euclidean_Space.thy \
1.8 + Library/Code_Set.thy Library/Convex_Euclidean_Space.thy \
1.9 Library/sum_of_squares.ML Library/Glbs.thy Library/normarith.ML \
1.10 Library/Executable_Set.thy Library/Infinite_Set.thy \
1.11 Library/FuncSet.thy Library/Permutations.thy Library/Determinants.thy\
1.12 @@ -329,7 +329,7 @@
1.13 Library/Fundamental_Theorem_Algebra.thy \
1.14 Library/Inner_Product.thy Library/Lattice_Syntax.thy \
1.15 Library/Legacy_GCD.thy \
1.16 - Library/Library.thy Library/List_Prefix.thy Library/State_Monad.thy \
1.17 + Library/Library.thy Library/List_Prefix.thy Library/List_Set.thy Library/State_Monad.thy \
1.18 Library/Nat_Int_Bij.thy Library/Multiset.thy Library/Permutation.thy \
1.19 Library/Primes.thy Library/Pocklington.thy Library/Quotient.thy \
1.20 Library/Quicksort.thy Library/Nat_Infinity.thy Library/Word.thy \
2.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
2.2 +++ b/src/HOL/Library/Code_Set.thy Thu Jun 25 17:07:18 2009 +0200
2.3 @@ -0,0 +1,169 @@
2.4 +
2.5 +(* Author: Florian Haftmann, TU Muenchen *)
2.6 +
2.7 +header {* Executable finite sets *}
2.8 +
2.9 +theory Code_Set
2.10 +imports List_Set
2.11 +begin
2.12 +
2.13 +lemma foldl_apply_inv:
2.14 + assumes "\<And>x. g (h x) = x"
2.15 + shows "foldl f (g s) xs = g (foldl (\<lambda>s x. h (f (g s) x)) s xs)"
2.16 + by (rule sym, induct xs arbitrary: s) (simp_all add: assms)
2.17 +
2.18 +subsection {* Lifting *}
2.19 +
2.20 +datatype 'a fset = Fset "'a set"
2.21 +
2.22 +primrec member :: "'a fset \<Rightarrow> 'a set" where
2.23 + "member (Fset A) = A"
2.24 +
2.25 +lemma Fset_member [simp]:
2.26 + "Fset (member A) = A"
2.27 + by (cases A) simp
2.28 +
2.29 +definition Set :: "'a list \<Rightarrow> 'a fset" where
2.30 + "Set xs = Fset (set xs)"
2.31 +
2.32 +lemma member_Set [simp]:
2.33 + "member (Set xs) = set xs"
2.34 + by (simp add: Set_def)
2.35 +
2.36 +code_datatype Set
2.37 +
2.38 +
2.39 +subsection {* Basic operations *}
2.40 +
2.41 +definition is_empty :: "'a fset \<Rightarrow> bool" where
2.42 + "is_empty A \<longleftrightarrow> List_Set.is_empty (member A)"
2.43 +
2.44 +lemma is_empty_Set [code]:
2.45 + "is_empty (Set xs) \<longleftrightarrow> null xs"
2.46 + by (simp add: is_empty_def is_empty_set)
2.47 +
2.48 +definition empty :: "'a fset" where
2.49 + "empty = Fset {}"
2.50 +
2.51 +lemma empty_Set [code]:
2.52 + "empty = Set []"
2.53 + by (simp add: empty_def Set_def)
2.54 +
2.55 +definition insert :: "'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset" where
2.56 + "insert x A = Fset (Set.insert x (member A))"
2.57 +
2.58 +lemma insert_Set [code]:
2.59 + "insert x (Set xs) = Set (List_Set.insert x xs)"
2.60 + by (simp add: insert_def Set_def insert_set)
2.61 +
2.62 +definition remove :: "'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset" where
2.63 + "remove x A = Fset (List_Set.remove x (member A))"
2.64 +
2.65 +lemma remove_Set [code]:
2.66 + "remove x (Set xs) = Set (remove_all x xs)"
2.67 + by (simp add: remove_def Set_def remove_set)
2.68 +
2.69 +definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset" where
2.70 + "map f A = Fset (image f (member A))"
2.71 +
2.72 +lemma map_Set [code]:
2.73 + "map f (Set xs) = Set (remdups (List.map f xs))"
2.74 + by (simp add: map_def Set_def)
2.75 +
2.76 +definition project :: "('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'a fset" where
2.77 + "project P A = Fset (List_Set.project P (member A))"
2.78 +
2.79 +lemma project_Set [code]:
2.80 + "project P (Set xs) = Set (filter P xs)"
2.81 + by (simp add: project_def Set_def project_set)
2.82 +
2.83 +definition forall :: "('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> bool" where
2.84 + "forall P A \<longleftrightarrow> Ball (member A) P"
2.85 +
2.86 +lemma forall_Set [code]:
2.87 + "forall P (Set xs) \<longleftrightarrow> list_all P xs"
2.88 + by (simp add: forall_def Set_def ball_set)
2.89 +
2.90 +definition exists :: "('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> bool" where
2.91 + "exists P A \<longleftrightarrow> Bex (member A) P"
2.92 +
2.93 +lemma exists_Set [code]:
2.94 + "exists P (Set xs) \<longleftrightarrow> list_ex P xs"
2.95 + by (simp add: exists_def Set_def bex_set)
2.96 +
2.97 +
2.98 +subsection {* Functorial operations *}
2.99 +
2.100 +definition union :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" where
2.101 + "union A B = Fset (member A \<union> member B)"
2.102 +
2.103 +lemma union_insert [code]:
2.104 + "union (Set xs) A = foldl (\<lambda>A x. insert x A) A xs"
2.105 +proof -
2.106 + have "foldl (\<lambda>A x. Set.insert x A) (member A) xs =
2.107 + member (foldl (\<lambda>A x. Fset (Set.insert x (member A))) A xs)"
2.108 + by (rule foldl_apply_inv) simp
2.109 + then show ?thesis by (simp add: union_def union_set insert_def)
2.110 +qed
2.111 +
2.112 +definition subtract :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" where
2.113 + "subtract A B = Fset (member B - member A)"
2.114 +
2.115 +lemma subtract_remove [code]:
2.116 + "subtract (Set xs) A = foldl (\<lambda>A x. remove x A) A xs"
2.117 +proof -
2.118 + have "foldl (\<lambda>A x. List_Set.remove x A) (member A) xs =
2.119 + member (foldl (\<lambda>A x. Fset (List_Set.remove x (member A))) A xs)"
2.120 + by (rule foldl_apply_inv) simp
2.121 + then show ?thesis by (simp add: subtract_def minus_set remove_def)
2.122 +qed
2.123 +
2.124 +
2.125 +subsection {* Derived operations *}
2.126 +
2.127 +lemma member_exists [code]:
2.128 + "member A y \<longleftrightarrow> exists (\<lambda>x. y = x) A"
2.129 + by (simp add: exists_def mem_def)
2.130 +
2.131 +definition subfset_eq :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" where
2.132 + "subfset_eq A B \<longleftrightarrow> member A \<subseteq> member B"
2.133 +
2.134 +lemma subfset_eq_forall [code]:
2.135 + "subfset_eq A B \<longleftrightarrow> forall (\<lambda>x. member B x) A"
2.136 + by (simp add: subfset_eq_def subset_eq forall_def mem_def)
2.137 +
2.138 +definition subfset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" where
2.139 + "subfset A B \<longleftrightarrow> member A \<subset> member B"
2.140 +
2.141 +lemma subfset_subfset_eq [code]:
2.142 + "subfset A B \<longleftrightarrow> subfset_eq A B \<and> \<not> subfset_eq B A"
2.143 + by (simp add: subfset_def subfset_eq_def subset)
2.144 +
2.145 +lemma eq_fset_subfset_eq [code]:
2.146 + "eq_class.eq A B \<longleftrightarrow> subfset_eq A B \<and> subfset_eq B A"
2.147 + by (cases A, cases B) (simp add: eq subfset_eq_def set_eq)
2.148 +
2.149 +definition inter :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" where
2.150 + "inter A B = Fset (List_Set.project (member A) (member B))"
2.151 +
2.152 +lemma inter_project [code]:
2.153 + "inter A B = project (member A) B"
2.154 + by (simp add: inter_def project_def inter)
2.155 +
2.156 +
2.157 +subsection {* Misc operations *}
2.158 +
2.159 +lemma size_fset [code]:
2.160 + "fset_size f A = 0"
2.161 + "size A = 0"
2.162 + by (cases A, simp) (cases A, simp)
2.163 +
2.164 +lemma fset_case_code [code]:
2.165 + "fset_case f A = f (member A)"
2.166 + by (cases A) simp
2.167 +
2.168 +lemma fset_rec_code [code]:
2.169 + "fset_rec f A = f (member A)"
2.170 + by (cases A) simp
2.171 +
2.172 +end
3.1 --- a/src/HOL/Library/Library.thy Thu Jun 25 15:42:36 2009 +0200
3.2 +++ b/src/HOL/Library/Library.thy Thu Jun 25 17:07:18 2009 +0200
3.3 @@ -10,6 +10,7 @@
3.4 Char_ord
3.5 Code_Char_chr
3.6 Code_Integer
3.7 + Code_Set
3.8 Coinductive_List
3.9 Commutative_Ring
3.10 Continuity
4.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
4.2 +++ b/src/HOL/Library/List_Set.thy Thu Jun 25 17:07:18 2009 +0200
4.3 @@ -0,0 +1,163 @@
4.4 +
4.5 +(* Author: Florian Haftmann, TU Muenchen *)
4.6 +
4.7 +header {* Relating (finite) sets and lists *}
4.8 +
4.9 +theory List_Set
4.10 +imports Main
4.11 +begin
4.12 +
4.13 +subsection {* Various additional list functions *}
4.14 +
4.15 +definition insert :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
4.16 + "insert x xs = (if x \<in> set xs then xs else x # xs)"
4.17 +
4.18 +definition remove_all :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
4.19 + "remove_all x xs = filter (Not o op = x) xs"
4.20 +
4.21 +
4.22 +subsection {* Various additional set functions *}
4.23 +
4.24 +definition is_empty :: "'a set \<Rightarrow> bool" where
4.25 + "is_empty A \<longleftrightarrow> A = {}"
4.26 +
4.27 +definition remove :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set" where
4.28 + "remove x A = A - {x}"
4.29 +
4.30 +lemma fun_left_comm_idem_remove:
4.31 + "fun_left_comm_idem remove"
4.32 +proof -
4.33 + have rem: "remove = (\<lambda>x A. A - {x})" by (simp add: expand_fun_eq remove_def)
4.34 + show ?thesis by (simp only: fun_left_comm_idem_remove rem)
4.35 +qed
4.36 +
4.37 +lemma minus_fold_remove:
4.38 + assumes "finite A"
4.39 + shows "B - A = fold remove B A"
4.40 +proof -
4.41 + have rem: "remove = (\<lambda>x A. A - {x})" by (simp add: expand_fun_eq remove_def)
4.42 + show ?thesis by (simp only: rem assms minus_fold_remove)
4.43 +qed
4.44 +
4.45 +definition project :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set" where
4.46 + "project P A = {a\<in>A. P a}"
4.47 +
4.48 +
4.49 +subsection {* Basic set operations *}
4.50 +
4.51 +lemma is_empty_set:
4.52 + "is_empty (set xs) \<longleftrightarrow> null xs"
4.53 + by (simp add: is_empty_def null_empty)
4.54 +
4.55 +lemma ball_set:
4.56 + "(\<forall>x\<in>set xs. P x) \<longleftrightarrow> list_all P xs"
4.57 + by (rule list_ball_code)
4.58 +
4.59 +lemma bex_set:
4.60 + "(\<exists>x\<in>set xs. P x) \<longleftrightarrow> list_ex P xs"
4.61 + by (rule list_bex_code)
4.62 +
4.63 +lemma empty_set:
4.64 + "{} = set []"
4.65 + by simp
4.66 +
4.67 +lemma insert_set:
4.68 + "Set.insert x (set xs) = set (insert x xs)"
4.69 + by (auto simp add: insert_def)
4.70 +
4.71 +lemma remove_set:
4.72 + "remove x (set xs) = set (remove_all x xs)"
4.73 + by (auto simp add: remove_def remove_all_def)
4.74 +
4.75 +lemma image_set:
4.76 + "image f (set xs) = set (remdups (map f xs))"
4.77 + by simp
4.78 +
4.79 +lemma project_set:
4.80 + "project P (set xs) = set (filter P xs)"
4.81 + by (auto simp add: project_def)
4.82 +
4.83 +
4.84 +subsection {* Functorial set operations *}
4.85 +
4.86 +lemma union_set:
4.87 + "set xs \<union> A = foldl (\<lambda>A x. Set.insert x A) A xs"
4.88 +proof -
4.89 + interpret fun_left_comm_idem Set.insert
4.90 + by (fact fun_left_comm_idem_insert)
4.91 + show ?thesis by (simp add: union_fold_insert fold_set)
4.92 +qed
4.93 +
4.94 +lemma minus_set:
4.95 + "A - set xs = foldl (\<lambda>A x. remove x A) A xs"
4.96 +proof -
4.97 + interpret fun_left_comm_idem remove
4.98 + by (fact fun_left_comm_idem_remove)
4.99 + show ?thesis
4.100 + by (simp add: minus_fold_remove [of _ A] fold_set)
4.101 +qed
4.102 +
4.103 +lemma Inter_set:
4.104 + "Inter (set (A # As)) = foldl (op \<inter>) A As"
4.105 +proof -
4.106 + have "finite (set (A # As))" by simp
4.107 + moreover have "fold (op \<inter>) UNIV (set (A # As)) = foldl (\<lambda>y x. x \<inter> y) UNIV (A # As)"
4.108 + by (rule fun_left_comm_idem.fold_set, unfold_locales, auto)
4.109 + ultimately have "Inter (set (A # As)) = foldl (op \<inter>) UNIV (A # As)"
4.110 + by (simp only: Inter_fold_inter Int_commute)
4.111 + then show ?thesis by simp
4.112 +qed
4.113 +
4.114 +lemma Union_set:
4.115 + "Union (set As) = foldl (op \<union>) {} As"
4.116 +proof -
4.117 + have "fold (op \<union>) {} (set As) = foldl (\<lambda>y x. x \<union> y) {} As"
4.118 + by (rule fun_left_comm_idem.fold_set, unfold_locales, auto)
4.119 + then show ?thesis
4.120 + by (simp only: Union_fold_union finite_set Un_commute)
4.121 +qed
4.122 +
4.123 +lemma INTER_set:
4.124 + "INTER (set (A # As)) f = foldl (\<lambda>B A. f A \<inter> B) (f A) As"
4.125 +proof -
4.126 + have "finite (set (A # As))" by simp
4.127 + moreover have "fold (\<lambda>A. op \<inter> (f A)) UNIV (set (A # As)) = foldl (\<lambda>B A. f A \<inter> B) UNIV (A # As)"
4.128 + by (rule fun_left_comm_idem.fold_set, unfold_locales, auto)
4.129 + ultimately have "INTER (set (A # As)) f = foldl (\<lambda>B A. f A \<inter> B) UNIV (A # As)"
4.130 + by (simp only: INTER_fold_inter)
4.131 + then show ?thesis by simp
4.132 +qed
4.133 +
4.134 +lemma UNION_set:
4.135 + "UNION (set As) f = foldl (\<lambda>B A. f A \<union> B) {} As"
4.136 +proof -
4.137 + have "fold (\<lambda>A. op \<union> (f A)) {} (set As) = foldl (\<lambda>B A. f A \<union> B) {} As"
4.138 + by (rule fun_left_comm_idem.fold_set, unfold_locales, auto)
4.139 + then show ?thesis
4.140 + by (simp only: UNION_fold_union finite_set)
4.141 +qed
4.142 +
4.143 +
4.144 +subsection {* Derived set operations *}
4.145 +
4.146 +lemma member:
4.147 + "a \<in> A \<longleftrightarrow> (\<exists>x\<in>A. a = x)"
4.148 + by simp
4.149 +
4.150 +lemma subset_eq:
4.151 + "A \<subseteq> B \<longleftrightarrow> (\<forall>x\<in>A. x \<in> B)"
4.152 + by (fact subset_eq)
4.153 +
4.154 +lemma subset:
4.155 + "A \<subset> B \<longleftrightarrow> A \<subseteq> B \<and> \<not> B \<subseteq> A"
4.156 + by (fact less_le_not_le)
4.157 +
4.158 +lemma set_eq:
4.159 + "A = B \<longleftrightarrow> A \<subseteq> B \<and> B \<subseteq> A"
4.160 + by (fact eq_iff)
4.161 +
4.162 +lemma inter:
4.163 + "A \<inter> B = project (\<lambda>x. x \<in> A) B"
4.164 + by (auto simp add: project_def)
4.165 +
4.166 +end
4.167 \ No newline at end of file
5.1 --- a/src/HOL/ex/Codegenerator_Candidates.thy Thu Jun 25 15:42:36 2009 +0200
5.2 +++ b/src/HOL/ex/Codegenerator_Candidates.thy Thu Jun 25 17:07:18 2009 +0200
5.3 @@ -8,6 +8,7 @@
5.4 Complex_Main
5.5 AssocList
5.6 Binomial
5.7 + Code_Set
5.8 Commutative_Ring
5.9 Enum
5.10 List_Prefix