experimental variants of Library/Cset.thy and Library/Dlist_Cset.thy defined via quotient package
1 (* Title: HOL/Quotient_Examples/List_Cset.thy
2 Author: Florian Haftmann, Alexander Krauss, TU Muenchen
5 header {* Implementation of type Cset.set based on lists. Code equations obtained via quotient lifting. *}
11 lemma [quot_respect]: "((op = ===> set_eq ===> set_eq) ===> op = ===> set_eq ===> set_eq)
13 by (simp add: fun_rel_eq)
15 lemma [quot_preserve]: "((id ---> abs_set ---> rep_set) ---> id ---> rep_set ---> abs_set) foldr = foldr"
17 by (induct_tac xa) (auto simp: Quotient_abs_rep[OF Quotient_set])
20 subsection {* Relationship to lists *}
22 (*FIXME: maybe define on sets first and then lift -> more canonical*)
23 definition coset :: "'a list \<Rightarrow> 'a Cset.set" where
24 "coset xs = Cset.uminus (Cset.set xs)"
26 code_datatype Cset.set List_Cset.coset
28 lemma member_code [code]:
29 "member x (Cset.set xs) \<longleftrightarrow> List.member xs x"
30 "member x (coset xs) \<longleftrightarrow> \<not> List.member xs x"
32 apply (lifting in_set_member)
33 by descending (simp add: in_set_member)
35 definition (in term_syntax)
36 setify :: "'a\<Colon>typerep list \<times> (unit \<Rightarrow> Code_Evaluation.term)
37 \<Rightarrow> 'a Cset.set \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
38 [code_unfold]: "setify xs = Code_Evaluation.valtermify Cset.set {\<cdot>} xs"
40 notation fcomp (infixl "\<circ>>" 60)
41 notation scomp (infixl "\<circ>\<rightarrow>" 60)
43 instantiation Cset.set :: (random) random
47 "Quickcheck.random i = Quickcheck.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (setify xs))"
53 no_notation fcomp (infixl "\<circ>>" 60)
54 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
56 subsection {* Basic operations *}
58 lemma is_empty_set [code]:
59 "Cset.is_empty (Cset.set xs) \<longleftrightarrow> List.null xs"
60 by (lifting is_empty_set)
61 hide_fact (open) is_empty_set
63 lemma empty_set [code]:
64 "Cset.empty = Cset.set []"
65 by (lifting set.simps(1)[symmetric])
66 hide_fact (open) empty_set
68 lemma UNIV_set [code]:
69 "Cset.UNIV = coset []"
70 unfolding coset_def by descending simp
71 hide_fact (open) UNIV_set
73 lemma remove_set [code]:
74 "Cset.remove x (Cset.set xs) = Cset.set (removeAll x xs)"
75 "Cset.remove x (coset xs) = coset (List.insert x xs)"
78 apply (simp add: More_Set.remove_def)
80 by (simp add: remove_set_compl)
82 lemma insert_set [code]:
83 "Cset.insert x (Cset.set xs) = Cset.set (List.insert x xs)"
84 "Cset.insert x (coset xs) = coset (removeAll x xs)"
86 apply (lifting set_insert[symmetric])
90 "Cset.map f (Cset.set xs) = Cset.set (remdups (List.map f xs))"
93 lemma filter_set [code]:
94 "Cset.filter P (Cset.set xs) = Cset.set (List.filter P xs)"
95 by descending (simp add: project_set)
97 lemma forall_set [code]:
98 "Cset.forall (Cset.set xs) P \<longleftrightarrow> list_all P xs"
99 (* FIXME: why does (lifting Ball_set_list_all) fail? *)
100 by descending (fact Ball_set_list_all)
102 lemma exists_set [code]:
103 "Cset.exists (Cset.set xs) P \<longleftrightarrow> list_ex P xs"
104 by descending (fact Bex_set_list_ex)
106 lemma card_set [code]:
107 "Cset.card (Cset.set xs) = length (remdups xs)"
108 by (lifting length_remdups_card_conv[symmetric])
110 lemma compl_set [simp, code]:
111 "Cset.uminus (Cset.set xs) = coset xs"
112 unfolding coset_def by descending simp
114 lemma compl_coset [simp, code]:
115 "Cset.uminus (coset xs) = Cset.set xs"
116 unfolding coset_def by descending simp
118 context complete_lattice
121 (* FIXME: automated lifting fails, since @{term inf} and @{term top}
122 are variables (???) *)
123 lemma Infimum_inf [code]:
124 "Infimum (Cset.set As) = foldr inf As top"
125 "Infimum (coset []) = bot"
126 unfolding Infimum_def member_code List.member_def
127 apply (simp add: mem_def Inf_set_foldr)
128 apply (simp add: Inf_UNIV[unfolded UNIV_def Collect_def])
131 lemma Supremum_sup [code]:
132 "Supremum (Cset.set As) = foldr sup As bot"
133 "Supremum (coset []) = top"
134 unfolding Supremum_def member_code List.member_def
135 apply (simp add: mem_def Sup_set_foldr)
136 apply (simp add: Sup_UNIV[unfolded UNIV_def Collect_def])
143 subsection {* Derived operations *}
145 lemma subset_eq_forall [code]:
146 "Cset.subset A B \<longleftrightarrow> Cset.forall A (\<lambda>x. member x B)"
149 lemma subset_subset_eq [code]:
150 "Cset.psubset A B \<longleftrightarrow> Cset.subset A B \<and> \<not> Cset.subset B A"
153 instantiation Cset.set :: (type) equal
157 "HOL.equal A B \<longleftrightarrow> Cset.subset A B \<and> Cset.subset B A"
161 unfolding equal_set_def
167 "HOL.equal (A :: 'a Cset.set) A \<longleftrightarrow> True"
171 subsection {* Functorial operations *}
173 lemma inter_project [code]:
174 "Cset.inter A (Cset.set xs) = Cset.set (List.filter (\<lambda>x. Cset.member x A) xs)"
175 "Cset.inter A (coset xs) = foldr Cset.remove xs A"
181 by (metis diff_eq minus_set_foldr)
183 lemma subtract_remove [code]:
184 "Cset.minus A (Cset.set xs) = foldr Cset.remove xs A"
185 "Cset.minus A (coset xs) = Cset.set (List.filter (\<lambda>x. member x A) xs)"
187 apply (lifting minus_set_foldr)
190 lemma union_insert [code]:
191 "Cset.union (Cset.set xs) A = foldr Cset.insert xs A"
192 "Cset.union (coset xs) A = coset (List.filter (\<lambda>x. \<not> member x A) xs)"
194 apply (lifting union_set_foldr)