1 (* Title: HOL/Library/Mapping.thy
2 Author: Florian Haftmann and Ondrej Kuncar
5 header {* An abstract view on maps for code generation. *}
11 subsection {* Parametricity transfer rules *}
15 interpretation lifting_syntax .
17 lemma empty_transfer: "(A ===> option_rel B) Map.empty Map.empty" by transfer_prover
19 lemma lookup_transfer: "((A ===> B) ===> A ===> B) (\<lambda>m k. m k) (\<lambda>m k. m k)" by transfer_prover
21 lemma update_transfer:
22 assumes [transfer_rule]: "bi_unique A"
23 shows "(A ===> B ===> (A ===> option_rel B) ===> A ===> option_rel B)
24 (\<lambda>k v m. m(k \<mapsto> v)) (\<lambda>k v m. m(k \<mapsto> v))"
27 lemma delete_transfer:
28 assumes [transfer_rule]: "bi_unique A"
29 shows "(A ===> (A ===> option_rel B) ===> A ===> option_rel B)
30 (\<lambda>k m. m(k := None)) (\<lambda>k m. m(k := None))"
33 definition equal_None :: "'a option \<Rightarrow> bool" where "equal_None x \<equiv> x = None"
35 lemma [transfer_rule]: "(option_rel A ===> op=) equal_None equal_None"
36 unfolding fun_rel_def option_rel_def equal_None_def by (auto split: option.split)
39 assumes [transfer_rule]: "bi_total A"
40 shows "((A ===> option_rel B) ===> set_rel A) dom dom"
41 unfolding dom_def[abs_def] equal_None_def[symmetric]
44 lemma map_of_transfer [transfer_rule]:
45 assumes [transfer_rule]: "bi_unique R1"
46 shows "(list_all2 (prod_rel R1 R2) ===> R1 ===> option_rel R2) map_of map_of"
47 unfolding map_of_def by transfer_prover
49 lemma tabulate_transfer:
50 assumes [transfer_rule]: "bi_unique A"
51 shows "(list_all2 A ===> (A ===> B) ===> A ===> option_rel B)
52 (\<lambda>ks f. (map_of (List.map (\<lambda>k. (k, f k)) ks))) (\<lambda>ks f. (map_of (List.map (\<lambda>k. (k, f k)) ks)))"
55 lemma bulkload_transfer:
56 "(list_all2 A ===> op= ===> option_rel A)
57 (\<lambda>xs k. if k < length xs then Some (xs ! k) else None) (\<lambda>xs k. if k < length xs then Some (xs ! k) else None)"
60 apply (erule list_all2_induct)
64 by (auto dest: list_all2_lengthD list_all2_nthD)
67 "((A ===> B) ===> (C ===> D) ===> (B ===> option_rel C) ===> A ===> option_rel D)
68 (\<lambda>f g m. (Option.map g \<circ> m \<circ> f)) (\<lambda>f g m. (Option.map g \<circ> m \<circ> f))"
71 lemma map_entry_transfer:
72 assumes [transfer_rule]: "bi_unique A"
73 shows "(A ===> (B ===> B) ===> (A ===> option_rel B) ===> A ===> option_rel B)
74 (\<lambda>k f m. (case m k of None \<Rightarrow> m
75 | Some v \<Rightarrow> m (k \<mapsto> (f v)))) (\<lambda>k f m. (case m k of None \<Rightarrow> m
76 | Some v \<Rightarrow> m (k \<mapsto> (f v))))"
81 subsection {* Type definition and primitive operations *}
83 typedef ('a, 'b) mapping = "UNIV :: ('a \<rightharpoonup> 'b) set"
84 morphisms rep Mapping ..
86 setup_lifting(no_code) type_definition_mapping
88 lift_definition empty :: "('a, 'b) mapping" is Map.empty parametric empty_transfer .
90 lift_definition lookup :: "('a, 'b) mapping \<Rightarrow> 'a \<Rightarrow> 'b option" is "\<lambda>m k. m k"
91 parametric lookup_transfer .
93 lift_definition update :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" is "\<lambda>k v m. m(k \<mapsto> v)"
94 parametric update_transfer .
96 lift_definition delete :: "'a \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" is "\<lambda>k m. m(k := None)"
97 parametric delete_transfer .
99 lift_definition keys :: "('a, 'b) mapping \<Rightarrow> 'a set" is dom parametric dom_transfer .
101 lift_definition tabulate :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping" is
102 "\<lambda>ks f. (map_of (List.map (\<lambda>k. (k, f k)) ks))" parametric tabulate_transfer .
104 lift_definition bulkload :: "'a list \<Rightarrow> (nat, 'a) mapping" is
105 "\<lambda>xs k. if k < length xs then Some (xs ! k) else None" parametric bulkload_transfer .
107 lift_definition map :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('c, 'd) mapping" is
108 "\<lambda>f g m. (Option.map g \<circ> m \<circ> f)" parametric map_transfer .
111 subsection {* Functorial structure *}
113 enriched_type map: map
114 by (transfer, auto simp add: fun_eq_iff Option.map.compositionality Option.map.id)+
117 subsection {* Derived operations *}
119 definition ordered_keys :: "('a\<Colon>linorder, 'b) mapping \<Rightarrow> 'a list" where
120 "ordered_keys m = (if finite (keys m) then sorted_list_of_set (keys m) else [])"
122 definition is_empty :: "('a, 'b) mapping \<Rightarrow> bool" where
123 "is_empty m \<longleftrightarrow> keys m = {}"
125 definition size :: "('a, 'b) mapping \<Rightarrow> nat" where
126 "size m = (if finite (keys m) then card (keys m) else 0)"
128 definition replace :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
129 "replace k v m = (if k \<in> keys m then update k v m else m)"
131 definition default :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
132 "default k v m = (if k \<in> keys m then m else update k v m)"
134 lift_definition map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" is
135 "\<lambda>k f m. (case m k of None \<Rightarrow> m
136 | Some v \<Rightarrow> m (k \<mapsto> (f v)))" parametric map_entry_transfer .
138 lemma map_entry_code [code]: "map_entry k f m = (case lookup m k of None \<Rightarrow> m
139 | Some v \<Rightarrow> update k (f v) m)"
142 definition map_default :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
143 "map_default k v f m = map_entry k f (default k v m)"
145 lift_definition assoc_list_to_mapping :: "('k \<times> 'v) list \<Rightarrow> ('k, 'v) mapping"
146 is map_of parametric map_of_transfer .
148 lemma assoc_list_to_mapping_code [code]:
149 "assoc_list_to_mapping xs = foldr (\<lambda>(k, v) m. update k v m) xs empty"
150 by transfer(simp add: map_add_map_of_foldr[symmetric])
152 instantiation mapping :: (type, type) equal
156 "HOL.equal m1 m2 \<longleftrightarrow> (\<forall>k. lookup m1 k = lookup m2 k)"
159 qed (unfold equal_mapping_def, transfer, auto)
165 interpretation lifting_syntax .
167 lemma [transfer_rule]:
168 assumes [transfer_rule]: "bi_total A"
169 assumes [transfer_rule]: "bi_unique B"
170 shows "(pcr_mapping A B ===> pcr_mapping A B ===> op=) HOL.eq HOL.equal"
171 by (unfold equal) transfer_prover
175 subsection {* Properties *}
177 lemma lookup_update: "lookup (update k v m) k = Some v"
180 lemma lookup_update_neq: "k \<noteq> k' \<Longrightarrow> lookup (update k v m) k' = lookup m k'"
183 lemma lookup_empty: "lookup empty k = None"
186 lemma keys_is_none_rep [code_unfold]:
187 "k \<in> keys m \<longleftrightarrow> \<not> (Option.is_none (lookup m k))"
188 by transfer (auto simp add: is_none_def)
190 lemma tabulate_alt_def:
191 "map_of (List.map (\<lambda>k. (k, f k)) ks) = (Some o f) |` set ks"
192 by (induct ks) (auto simp add: tabulate_def restrict_map_def)
195 "update k v (update k w m) = update k v m"
196 "k \<noteq> l \<Longrightarrow> update k v (update l w m) = update l w (update k v m)"
197 by (transfer, simp add: fun_upd_twist)+
199 lemma update_delete [simp]:
200 "update k v (delete k m) = update k v m"
204 "delete k (update k v m) = delete k m"
205 "k \<noteq> l \<Longrightarrow> delete k (update l v m) = update l v (delete k m)"
206 by (transfer, simp add: fun_upd_twist)+
208 lemma delete_empty [simp]:
209 "delete k empty = empty"
212 lemma replace_update:
213 "k \<notin> keys m \<Longrightarrow> replace k v m = m"
214 "k \<in> keys m \<Longrightarrow> replace k v m = update k v m"
215 by (transfer, auto simp add: replace_def fun_upd_twist)+
217 lemma size_empty [simp]:
219 unfolding size_def by transfer simp
222 "finite (keys m) \<Longrightarrow> size (update k v m) =
223 (if k \<in> keys m then size m else Suc (size m))"
224 unfolding size_def by transfer (auto simp add: insert_dom)
227 "size (delete k m) = (if k \<in> keys m then size m - 1 else size m)"
228 unfolding size_def by transfer simp
230 lemma size_tabulate [simp]:
231 "size (tabulate ks f) = length (remdups ks)"
232 unfolding size_def by transfer (auto simp add: tabulate_alt_def card_set comp_def)
234 lemma bulkload_tabulate:
235 "bulkload xs = tabulate [0..<length xs] (nth xs)"
236 by transfer (auto simp add: tabulate_alt_def)
238 lemma is_empty_empty [simp]:
240 unfolding is_empty_def by transfer simp
242 lemma is_empty_update [simp]:
243 "\<not> is_empty (update k v m)"
244 unfolding is_empty_def by transfer simp
246 lemma is_empty_delete:
247 "is_empty (delete k m) \<longleftrightarrow> is_empty m \<or> keys m = {k}"
248 unfolding is_empty_def by transfer (auto simp del: dom_eq_empty_conv)
250 lemma is_empty_replace [simp]:
251 "is_empty (replace k v m) \<longleftrightarrow> is_empty m"
252 unfolding is_empty_def replace_def by transfer auto
254 lemma is_empty_default [simp]:
255 "\<not> is_empty (default k v m)"
256 unfolding is_empty_def default_def by transfer auto
258 lemma is_empty_map_entry [simp]:
259 "is_empty (map_entry k f m) \<longleftrightarrow> is_empty m"
260 unfolding is_empty_def
261 apply transfer by (case_tac "m k") auto
263 lemma is_empty_map_default [simp]:
264 "\<not> is_empty (map_default k v f m)"
265 by (simp add: map_default_def)
267 lemma keys_empty [simp]:
271 lemma keys_update [simp]:
272 "keys (update k v m) = insert k (keys m)"
275 lemma keys_delete [simp]:
276 "keys (delete k m) = keys m - {k}"
279 lemma keys_replace [simp]:
280 "keys (replace k v m) = keys m"
281 unfolding replace_def by transfer (simp add: insert_absorb)
283 lemma keys_default [simp]:
284 "keys (default k v m) = insert k (keys m)"
285 unfolding default_def by transfer (simp add: insert_absorb)
287 lemma keys_map_entry [simp]:
288 "keys (map_entry k f m) = keys m"
289 apply transfer by (case_tac "m k") auto
291 lemma keys_map_default [simp]:
292 "keys (map_default k v f m) = insert k (keys m)"
293 by (simp add: map_default_def)
295 lemma keys_tabulate [simp]:
296 "keys (tabulate ks f) = set ks"
297 by transfer (simp add: map_of_map_restrict o_def)
299 lemma keys_bulkload [simp]:
300 "keys (bulkload xs) = {0..<length xs}"
301 by (simp add: keys_tabulate bulkload_tabulate)
303 lemma distinct_ordered_keys [simp]:
304 "distinct (ordered_keys m)"
305 by (simp add: ordered_keys_def)
307 lemma ordered_keys_infinite [simp]:
308 "\<not> finite (keys m) \<Longrightarrow> ordered_keys m = []"
309 by (simp add: ordered_keys_def)
311 lemma ordered_keys_empty [simp]:
312 "ordered_keys empty = []"
313 by (simp add: ordered_keys_def)
315 lemma ordered_keys_update [simp]:
316 "k \<in> keys m \<Longrightarrow> ordered_keys (update k v m) = ordered_keys m"
317 "finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> ordered_keys (update k v m) = insort k (ordered_keys m)"
318 by (simp_all add: ordered_keys_def) (auto simp only: sorted_list_of_set_insert [symmetric] insert_absorb)
320 lemma ordered_keys_delete [simp]:
321 "ordered_keys (delete k m) = remove1 k (ordered_keys m)"
322 proof (cases "finite (keys m)")
323 case False then show ?thesis by simp
325 case True note fin = True
327 proof (cases "k \<in> keys m")
328 case False with fin have "k \<notin> set (sorted_list_of_set (keys m))" by simp
329 with False show ?thesis by (simp add: ordered_keys_def remove1_idem)
331 case True with fin show ?thesis by (simp add: ordered_keys_def sorted_list_of_set_remove)
335 lemma ordered_keys_replace [simp]:
336 "ordered_keys (replace k v m) = ordered_keys m"
337 by (simp add: replace_def)
339 lemma ordered_keys_default [simp]:
340 "k \<in> keys m \<Longrightarrow> ordered_keys (default k v m) = ordered_keys m"
341 "finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> ordered_keys (default k v m) = insort k (ordered_keys m)"
342 by (simp_all add: default_def)
344 lemma ordered_keys_map_entry [simp]:
345 "ordered_keys (map_entry k f m) = ordered_keys m"
346 by (simp add: ordered_keys_def)
348 lemma ordered_keys_map_default [simp]:
349 "k \<in> keys m \<Longrightarrow> ordered_keys (map_default k v f m) = ordered_keys m"
350 "finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> ordered_keys (map_default k v f m) = insort k (ordered_keys m)"
351 by (simp_all add: map_default_def)
353 lemma ordered_keys_tabulate [simp]:
354 "ordered_keys (tabulate ks f) = sort (remdups ks)"
355 by (simp add: ordered_keys_def sorted_list_of_set_sort_remdups)
357 lemma ordered_keys_bulkload [simp]:
358 "ordered_keys (bulkload ks) = [0..<length ks]"
359 by (simp add: ordered_keys_def)
362 subsection {* Code generator setup *}
364 code_datatype empty update
366 hide_const (open) empty is_empty rep lookup update delete ordered_keys keys size
367 replace default map_entry map_default tabulate bulkload map