1 (* Author: Florian Haftmann, TU Muenchen *)
3 header {* An abstract view on maps for code generation. *}
9 subsection {* Type definition and primitive operations *}
11 datatype ('a, 'b) mapping = Mapping "'a \<rightharpoonup> 'b"
13 definition empty :: "('a, 'b) mapping" where
14 "empty = Mapping (\<lambda>_. None)"
16 primrec lookup :: "('a, 'b) mapping \<Rightarrow> 'a \<rightharpoonup> 'b" where
17 "lookup (Mapping f) = f"
19 primrec update :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
20 "update k v (Mapping f) = Mapping (f (k \<mapsto> v))"
22 primrec delete :: "'a \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
23 "delete k (Mapping f) = Mapping (f (k := None))"
26 subsection {* Derived operations *}
28 definition keys :: "('a, 'b) mapping \<Rightarrow> 'a set" where
29 "keys m = dom (lookup m)"
31 definition ordered_keys :: "('a\<Colon>linorder, 'b) mapping \<Rightarrow> 'a list" where
32 "ordered_keys m = (if finite (keys m) then sorted_list_of_set (keys m) else [])"
34 definition is_empty :: "('a, 'b) mapping \<Rightarrow> bool" where
35 "is_empty m \<longleftrightarrow> keys m = {}"
37 definition size :: "('a, 'b) mapping \<Rightarrow> nat" where
38 "size m = (if finite (keys m) then card (keys m) else 0)"
40 definition replace :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
41 "replace k v m = (if k \<in> keys m then update k v m else m)"
43 definition default :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
44 "default k v m = (if k \<in> keys m then m else update k v m)"
46 definition map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
47 "map_entry k f m = (case lookup m k of None \<Rightarrow> m
48 | Some v \<Rightarrow> update k (f v) m)"
50 definition map_default :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
51 "map_default k v f m = map_entry k f (default k v m)"
53 definition tabulate :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping" where
54 "tabulate ks f = Mapping (map_of (map (\<lambda>k. (k, f k)) ks))"
56 definition bulkload :: "'a list \<Rightarrow> (nat, 'a) mapping" where
57 "bulkload xs = Mapping (\<lambda>k. if k < length xs then Some (xs ! k) else None)"
60 subsection {* Properties *}
62 lemma lookup_inject [simp]:
63 "lookup m = lookup n \<longleftrightarrow> m = n"
64 by (cases m, cases n) simp
67 assumes "lookup m = lookup n"
71 lemma keys_is_none_lookup [code_inline]:
72 "k \<in> keys m \<longleftrightarrow> \<not> (Option.is_none (lookup m k))"
73 by (auto simp add: keys_def is_none_def)
75 lemma lookup_empty [simp]:
76 "lookup empty = Map.empty"
77 by (simp add: empty_def)
79 lemma lookup_update [simp]:
80 "lookup (update k v m) = (lookup m) (k \<mapsto> v)"
83 lemma lookup_delete [simp]:
84 "lookup (delete k m) = (lookup m) (k := None)"
87 lemma lookup_map_entry [simp]:
88 "lookup (map_entry k f m) = (lookup m) (k := Option.map f (lookup m k))"
89 by (cases "lookup m k") (simp_all add: map_entry_def expand_fun_eq)
91 lemma lookup_tabulate [simp]:
92 "lookup (tabulate ks f) = (Some o f) |` set ks"
93 by (induct ks) (auto simp add: tabulate_def restrict_map_def expand_fun_eq)
95 lemma lookup_bulkload [simp]:
96 "lookup (bulkload xs) = (\<lambda>k. if k < length xs then Some (xs ! k) else None)"
97 by (simp add: bulkload_def)
100 "update k v (update k w m) = update k v m"
101 "k \<noteq> l \<Longrightarrow> update k v (update l w m) = update l w (update k v m)"
102 by (rule mapping_eqI, simp add: fun_upd_twist)+
104 lemma update_delete [simp]:
105 "update k v (delete k m) = update k v m"
106 by (rule mapping_eqI) simp
109 "delete k (update k v m) = delete k m"
110 "k \<noteq> l \<Longrightarrow> delete k (update l v m) = update l v (delete k m)"
111 by (rule mapping_eqI, simp add: fun_upd_twist)+
113 lemma delete_empty [simp]:
114 "delete k empty = empty"
115 by (rule mapping_eqI) simp
117 lemma replace_update:
118 "k \<notin> keys m \<Longrightarrow> replace k v m = m"
119 "k \<in> keys m \<Longrightarrow> replace k v m = update k v m"
120 by (rule mapping_eqI) (auto simp add: replace_def fun_upd_twist)+
122 lemma size_empty [simp]:
124 by (simp add: size_def keys_def)
127 "finite (keys m) \<Longrightarrow> size (update k v m) =
128 (if k \<in> keys m then size m else Suc (size m))"
129 by (auto simp add: size_def insert_dom keys_def)
132 "size (delete k m) = (if k \<in> keys m then size m - 1 else size m)"
133 by (simp add: size_def keys_def)
135 lemma size_tabulate [simp]:
136 "size (tabulate ks f) = length (remdups ks)"
137 by (simp add: size_def distinct_card [of "remdups ks", symmetric] comp_def keys_def)
139 lemma bulkload_tabulate:
140 "bulkload xs = tabulate [0..<length xs] (nth xs)"
141 by (rule mapping_eqI) (simp add: expand_fun_eq)
143 lemma is_empty_empty: (*FIXME*)
144 "is_empty m \<longleftrightarrow> m = Mapping Map.empty"
145 by (cases m) (simp add: is_empty_def keys_def)
147 lemma is_empty_empty' [simp]:
149 by (simp add: is_empty_empty empty_def)
151 lemma is_empty_update [simp]:
152 "\<not> is_empty (update k v m)"
153 by (cases m) (simp add: is_empty_empty)
155 lemma is_empty_delete:
156 "is_empty (delete k m) \<longleftrightarrow> is_empty m \<or> keys m = {k}"
157 by (cases m) (auto simp add: is_empty_empty keys_def dom_eq_empty_conv [symmetric] simp del: dom_eq_empty_conv)
159 lemma is_empty_replace [simp]:
160 "is_empty (replace k v m) \<longleftrightarrow> is_empty m"
161 by (auto simp add: replace_def) (simp add: is_empty_def)
163 lemma is_empty_default [simp]:
164 "\<not> is_empty (default k v m)"
165 by (auto simp add: default_def) (simp add: is_empty_def)
167 lemma is_empty_map_entry [simp]:
168 "is_empty (map_entry k f m) \<longleftrightarrow> is_empty m"
169 by (cases "lookup m k")
170 (auto simp add: map_entry_def, simp add: is_empty_empty)
172 lemma is_empty_map_default [simp]:
173 "\<not> is_empty (map_default k v f m)"
174 by (simp add: map_default_def)
176 lemma keys_empty [simp]:
178 by (simp add: keys_def)
180 lemma keys_update [simp]:
181 "keys (update k v m) = insert k (keys m)"
182 by (simp add: keys_def)
184 lemma keys_delete [simp]:
185 "keys (delete k m) = keys m - {k}"
186 by (simp add: keys_def)
188 lemma keys_replace [simp]:
189 "keys (replace k v m) = keys m"
190 by (auto simp add: keys_def replace_def)
192 lemma keys_default [simp]:
193 "keys (default k v m) = insert k (keys m)"
194 by (auto simp add: keys_def default_def)
196 lemma keys_map_entry [simp]:
197 "keys (map_entry k f m) = keys m"
198 by (auto simp add: keys_def)
200 lemma keys_map_default [simp]:
201 "keys (map_default k v f m) = insert k (keys m)"
202 by (simp add: map_default_def)
204 lemma keys_tabulate [simp]:
205 "keys (tabulate ks f) = set ks"
206 by (simp add: tabulate_def keys_def map_of_map_restrict o_def)
208 lemma keys_bulkload [simp]:
209 "keys (bulkload xs) = {0..<length xs}"
210 by (simp add: keys_tabulate bulkload_tabulate)
212 lemma distinct_ordered_keys [simp]:
213 "distinct (ordered_keys m)"
214 by (simp add: ordered_keys_def)
216 lemma ordered_keys_infinite [simp]:
217 "\<not> finite (keys m) \<Longrightarrow> ordered_keys m = []"
218 by (simp add: ordered_keys_def)
220 lemma ordered_keys_empty [simp]:
221 "ordered_keys empty = []"
222 by (simp add: ordered_keys_def)
224 lemma ordered_keys_update [simp]:
225 "k \<in> keys m \<Longrightarrow> ordered_keys (update k v m) = ordered_keys m"
226 "finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> ordered_keys (update k v m) = insort k (ordered_keys m)"
227 by (simp_all add: ordered_keys_def) (auto simp only: sorted_list_of_set_insert [symmetric] insert_absorb)
229 lemma ordered_keys_delete [simp]:
230 "ordered_keys (delete k m) = remove1 k (ordered_keys m)"
231 proof (cases "finite (keys m)")
232 case False then show ?thesis by simp
234 case True note fin = True
236 proof (cases "k \<in> keys m")
237 case False with fin have "k \<notin> set (sorted_list_of_set (keys m))" by simp
238 with False show ?thesis by (simp add: ordered_keys_def remove1_idem)
240 case True with fin show ?thesis by (simp add: ordered_keys_def sorted_list_of_set_remove)
244 lemma ordered_keys_replace [simp]:
245 "ordered_keys (replace k v m) = ordered_keys m"
246 by (simp add: replace_def)
248 lemma ordered_keys_default [simp]:
249 "k \<in> keys m \<Longrightarrow> ordered_keys (default k v m) = ordered_keys m"
250 "finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> ordered_keys (default k v m) = insort k (ordered_keys m)"
251 by (simp_all add: default_def)
253 lemma ordered_keys_map_entry [simp]:
254 "ordered_keys (map_entry k f m) = ordered_keys m"
255 by (simp add: ordered_keys_def)
257 lemma ordered_keys_map_default [simp]:
258 "k \<in> keys m \<Longrightarrow> ordered_keys (map_default k v f m) = ordered_keys m"
259 "finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> ordered_keys (map_default k v f m) = insort k (ordered_keys m)"
260 by (simp_all add: map_default_def)
262 lemma ordered_keys_tabulate [simp]:
263 "ordered_keys (tabulate ks f) = sort (remdups ks)"
264 by (simp add: ordered_keys_def sorted_list_of_set_sort_remdups)
266 lemma ordered_keys_bulkload [simp]:
267 "ordered_keys (bulkload ks) = [0..<length ks]"
268 by (simp add: ordered_keys_def)
271 subsection {* Some technical code lemmas *}
274 "mapping_case f m = f (Mapping.lookup m)"
278 "mapping_rec f m = f (Mapping.lookup m)"
282 "Nat.size (m :: (_, _) mapping) = 0"
286 "mapping_size f g m = 0"
290 hide_const (open) empty is_empty lookup update delete ordered_keys keys size replace tabulate bulkload