2 (* Author: Florian Haftmann, TU Muenchen *)
4 header {* implementation of Cset.sets based on lists *}
10 declare mem_def [simp]
11 declare Cset.set_code [code del]
13 definition coset :: "'a list \<Rightarrow> 'a Cset.set" where
14 "coset xs = Set (- set xs)"
15 hide_const (open) coset
17 lemma member_coset [simp]:
18 "member (List_Cset.coset xs) = - set xs"
19 by (simp add: coset_def)
20 hide_fact (open) member_coset
22 code_datatype Cset.set List_Cset.coset
24 lemma member_code [code]:
25 "member (Cset.set xs) = List.member xs"
26 "member (List_Cset.coset xs) = Not \<circ> List.member xs"
27 by (simp_all add: fun_eq_iff member_def fun_Compl_def bool_Compl_def)
29 lemma member_image_UNIV [simp]:
30 "member ` UNIV = UNIV"
32 have "\<And>A \<Colon> 'a set. \<exists>B \<Colon> 'a Cset.set. A = member B"
35 show "A = member (Set A)" by simp
37 then show ?thesis by (simp add: image_def)
40 definition (in term_syntax)
41 setify :: "'a\<Colon>typerep list \<times> (unit \<Rightarrow> Code_Evaluation.term)
42 \<Rightarrow> 'a Cset.set \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
43 [code_unfold]: "setify xs = Code_Evaluation.valtermify Cset.set {\<cdot>} xs"
45 notation fcomp (infixl "\<circ>>" 60)
46 notation scomp (infixl "\<circ>\<rightarrow>" 60)
48 instantiation Cset.set :: (random) random
52 "Quickcheck.random i = Quickcheck.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (setify xs))"
58 no_notation fcomp (infixl "\<circ>>" 60)
59 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
61 subsection {* Basic operations *}
63 lemma is_empty_set [code]:
64 "Cset.is_empty (Cset.set xs) \<longleftrightarrow> List.null xs"
65 by (simp add: is_empty_set null_def)
66 hide_fact (open) is_empty_set
68 lemma empty_set [code]:
69 "Cset.empty = Cset.set []"
70 by (simp add: set_def)
71 hide_fact (open) empty_set
73 lemma UNIV_set [code]:
74 "top = List_Cset.coset []"
75 by (simp add: coset_def)
76 hide_fact (open) UNIV_set
78 lemma remove_set [code]:
79 "Cset.remove x (Cset.set xs) = Cset.set (removeAll x xs)"
80 "Cset.remove x (List_Cset.coset xs) = List_Cset.coset (List.insert x xs)"
81 by (simp_all add: Cset.set_def coset_def)
82 (metis List.set_insert More_Set.remove_def remove_set_compl)
84 lemma insert_set [code]:
85 "Cset.insert x (Cset.set xs) = Cset.set (List.insert x xs)"
86 "Cset.insert x (List_Cset.coset xs) = List_Cset.coset (removeAll x xs)"
87 by (simp_all add: Cset.set_def coset_def)
90 "Cset.map f (Cset.set xs) = Cset.set (remdups (List.map f xs))"
91 by (simp add: Cset.set_def)
93 lemma filter_set [code]:
94 "Cset.filter P (Cset.set xs) = Cset.set (List.filter P xs)"
95 by (simp add: Cset.set_def project_set)
97 lemma forall_set [code]:
98 "Cset.forall P (Cset.set xs) \<longleftrightarrow> list_all P xs"
99 by (simp add: Cset.set_def list_all_iff)
101 lemma exists_set [code]:
102 "Cset.exists P (Cset.set xs) \<longleftrightarrow> list_ex P xs"
103 by (simp add: Cset.set_def list_ex_iff)
105 lemma card_set [code]:
106 "Cset.card (Cset.set xs) = length (remdups xs)"
108 have "Finite_Set.card (set (remdups xs)) = length (remdups xs)"
109 by (rule distinct_card) simp
110 then show ?thesis by (simp add: Cset.set_def)
113 lemma compl_set [simp, code]:
114 "- Cset.set xs = List_Cset.coset xs"
115 by (simp add: Cset.set_def coset_def)
117 lemma compl_coset [simp, code]:
118 "- List_Cset.coset xs = Cset.set xs"
119 by (simp add: Cset.set_def coset_def)
121 context complete_lattice
124 lemma Infimum_inf [code]:
125 "Infimum (Cset.set As) = foldr inf As top"
126 "Infimum (List_Cset.coset []) = bot"
127 by (simp_all add: Inf_set_foldr Inf_UNIV)
129 lemma Supremum_sup [code]:
130 "Supremum (Cset.set As) = foldr sup As bot"
131 "Supremum (List_Cset.coset []) = top"
132 by (simp_all add: Sup_set_foldr Sup_UNIV)
136 declare single_code [code del]
137 lemma single_set [code]:
138 "Cset.single a = Cset.set [a]"
139 by(simp add: Cset.single_code)
140 hide_fact (open) single_set
142 lemma bind_set [code]:
143 "Cset.bind (Cset.set xs) f = foldl (\<lambda>A x. sup A (f x)) (Cset.set []) xs"
145 show "foldl (\<lambda>A x. sup A (f x)) (Cset.set []) xs = Cset.bind (Cset.set xs) f"
146 by(induct xs rule: rev_induct)(auto simp add: Cset.bind_def Cset.set_def)
148 hide_fact (open) bind_set
150 lemma pred_of_cset_set [code]:
151 "pred_of_cset (Cset.set xs) = foldr sup (map Predicate.single xs) bot"
153 have "pred_of_cset (Cset.set xs) = Predicate.Pred (\<lambda>x. x \<in> set xs)"
154 by(auto simp add: Cset.pred_of_cset_def mem_def)
155 moreover have "foldr sup (map Predicate.single xs) bot = \<dots>"
156 by(induct xs)(auto simp add: bot_pred_def simp del: mem_def intro: pred_eqI, simp)
157 ultimately show ?thesis by(simp)
159 hide_fact (open) pred_of_cset_set
161 subsection {* Derived operations *}
163 lemma subset_eq_forall [code]:
164 "A \<le> B \<longleftrightarrow> Cset.forall (member B) A"
165 by (simp add: subset_eq)
167 lemma subset_subset_eq [code]:
168 "A < B \<longleftrightarrow> A \<le> B \<and> \<not> B \<le> (A :: 'a Cset.set)"
169 by (fact less_le_not_le)
171 instantiation Cset.set :: (type) equal
175 "HOL.equal A B \<longleftrightarrow> A \<le> B \<and> B \<le> (A :: 'a Cset.set)"
178 qed (simp add: equal_set_def set_eq [symmetric] Cset.set_eq_iff)
183 "HOL.equal (A :: 'a Cset.set) A \<longleftrightarrow> True"
187 subsection {* Functorial operations *}
189 lemma inter_project [code]:
190 "inf A (Cset.set xs) = Cset.set (List.filter (Cset.member A) xs)"
191 "inf A (List_Cset.coset xs) = foldr Cset.remove xs A"
193 show "inf A (Cset.set xs) = Cset.set (List.filter (member A) xs)"
194 by (simp add: inter project_def Cset.set_def)
195 have *: "\<And>x::'a. Cset.remove = (\<lambda>x. Set \<circ> More_Set.remove x \<circ> member)"
196 by (simp add: fun_eq_iff More_Set.remove_def)
197 have "member \<circ> fold (\<lambda>x. Set \<circ> More_Set.remove x \<circ> member) xs =
198 fold More_Set.remove xs \<circ> member"
199 by (rule fold_commute) (simp add: fun_eq_iff)
200 then have "fold More_Set.remove xs (member A) =
201 member (fold (\<lambda>x. Set \<circ> More_Set.remove x \<circ> member) xs A)"
202 by (simp add: fun_eq_iff)
203 then have "inf A (List_Cset.coset xs) = fold Cset.remove xs A"
204 by (simp add: Diff_eq [symmetric] minus_set *)
205 moreover have "\<And>x y :: 'a. Cset.remove y \<circ> Cset.remove x = Cset.remove x \<circ> Cset.remove y"
206 by (auto simp add: More_Set.remove_def * intro: ext)
207 ultimately show "inf A (List_Cset.coset xs) = foldr Cset.remove xs A"
208 by (simp add: foldr_fold)
211 lemma subtract_remove [code]:
212 "A - Cset.set xs = foldr Cset.remove xs A"
213 "A - List_Cset.coset xs = Cset.set (List.filter (member A) xs)"
214 by (simp_all only: diff_eq compl_set compl_coset inter_project)
216 lemma union_insert [code]:
217 "sup (Cset.set xs) A = foldr Cset.insert xs A"
218 "sup (List_Cset.coset xs) A = List_Cset.coset (List.filter (Not \<circ> member A) xs)"
220 have *: "\<And>x::'a. Cset.insert = (\<lambda>x. Set \<circ> Set.insert x \<circ> member)"
221 by (simp add: fun_eq_iff)
222 have "member \<circ> fold (\<lambda>x. Set \<circ> Set.insert x \<circ> member) xs =
223 fold Set.insert xs \<circ> member"
224 by (rule fold_commute) (simp add: fun_eq_iff)
225 then have "fold Set.insert xs (member A) =
226 member (fold (\<lambda>x. Set \<circ> Set.insert x \<circ> member) xs A)"
227 by (simp add: fun_eq_iff)
228 then have "sup (Cset.set xs) A = fold Cset.insert xs A"
229 by (simp add: union_set *)
230 moreover have "\<And>x y :: 'a. Cset.insert y \<circ> Cset.insert x = Cset.insert x \<circ> Cset.insert y"
231 by (auto simp add: * intro: ext)
232 ultimately show "sup (Cset.set xs) A = foldr Cset.insert xs A"
233 by (simp add: foldr_fold)
234 show "sup (List_Cset.coset xs) A = List_Cset.coset (List.filter (Not \<circ> member A) xs)"
235 by (auto simp add: coset_def)
238 declare mem_def[simp del]