src/HOL/Library/Mapping.thy
 author wenzelm Sun Nov 02 17:20:45 2014 +0100 (2014-11-02) changeset 58881 b9556a055632 parent 56545 8f1e7596deb7 child 59485 792272e6ee6b permissions -rw-r--r--
```     1 (*  Title:      HOL/Library/Mapping.thy
```
```     2     Author:     Florian Haftmann and Ondrej Kuncar
```
```     3 *)
```
```     4
```
```     5 section {* An abstract view on maps for code generation. *}
```
```     6
```
```     7 theory Mapping
```
```     8 imports Main
```
```     9 begin
```
```    10
```
```    11 subsection {* Parametricity transfer rules *}
```
```    12
```
```    13 lemma map_of_foldr: -- {* FIXME move *}
```
```    14   "map_of xs = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) xs Map.empty"
```
```    15   using map_add_map_of_foldr [of Map.empty] by auto
```
```    16
```
```    17 context
```
```    18 begin
```
```    19
```
```    20 interpretation lifting_syntax .
```
```    21
```
```    22 lemma empty_parametric:
```
```    23   "(A ===> rel_option B) Map.empty Map.empty"
```
```    24   by transfer_prover
```
```    25
```
```    26 lemma lookup_parametric: "((A ===> B) ===> A ===> B) (\<lambda>m k. m k) (\<lambda>m k. m k)"
```
```    27   by transfer_prover
```
```    28
```
```    29 lemma update_parametric:
```
```    30   assumes [transfer_rule]: "bi_unique A"
```
```    31   shows "(A ===> B ===> (A ===> rel_option B) ===> A ===> rel_option B)
```
```    32     (\<lambda>k v m. m(k \<mapsto> v)) (\<lambda>k v m. m(k \<mapsto> v))"
```
```    33   by transfer_prover
```
```    34
```
```    35 lemma delete_parametric:
```
```    36   assumes [transfer_rule]: "bi_unique A"
```
```    37   shows "(A ===> (A ===> rel_option B) ===> A ===> rel_option B)
```
```    38     (\<lambda>k m. m(k := None)) (\<lambda>k m. m(k := None))"
```
```    39   by transfer_prover
```
```    40
```
```    41 lemma is_none_parametric [transfer_rule]:
```
```    42   "(rel_option A ===> HOL.eq) Option.is_none Option.is_none"
```
```    43   by (auto simp add: is_none_def rel_fun_def rel_option_iff split: option.split)
```
```    44
```
```    45 lemma dom_parametric:
```
```    46   assumes [transfer_rule]: "bi_total A"
```
```    47   shows "((A ===> rel_option B) ===> rel_set A) dom dom"
```
```    48   unfolding dom_def [abs_def] is_none_def [symmetric] by transfer_prover
```
```    49
```
```    50 lemma map_of_parametric [transfer_rule]:
```
```    51   assumes [transfer_rule]: "bi_unique R1"
```
```    52   shows "(list_all2 (rel_prod R1 R2) ===> R1 ===> rel_option R2) map_of map_of"
```
```    53   unfolding map_of_def by transfer_prover
```
```    54
```
```    55 lemma map_entry_parametric [transfer_rule]:
```
```    56   assumes [transfer_rule]: "bi_unique A"
```
```    57   shows "(A ===> (B ===> B) ===> (A ===> rel_option B) ===> A ===> rel_option B)
```
```    58     (\<lambda>k f m. (case m k of None \<Rightarrow> m
```
```    59       | Some v \<Rightarrow> m (k \<mapsto> (f v)))) (\<lambda>k f m. (case m k of None \<Rightarrow> m
```
```    60       | Some v \<Rightarrow> m (k \<mapsto> (f v))))"
```
```    61   by transfer_prover
```
```    62
```
```    63 lemma tabulate_parametric:
```
```    64   assumes [transfer_rule]: "bi_unique A"
```
```    65   shows "(list_all2 A ===> (A ===> B) ===> A ===> rel_option B)
```
```    66     (\<lambda>ks f. (map_of (map (\<lambda>k. (k, f k)) ks))) (\<lambda>ks f. (map_of (map (\<lambda>k. (k, f k)) ks)))"
```
```    67   by transfer_prover
```
```    68
```
```    69 lemma bulkload_parametric:
```
```    70   "(list_all2 A ===> HOL.eq ===> rel_option A)
```
```    71     (\<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)"
```
```    72 proof
```
```    73   fix xs ys
```
```    74   assume "list_all2 A xs ys"
```
```    75   then show "(HOL.eq ===> rel_option A)
```
```    76     (\<lambda>k. if k < length xs then Some (xs ! k) else None)
```
```    77     (\<lambda>k. if k < length ys then Some (ys ! k) else None)"
```
```    78     apply induct
```
```    79     apply auto
```
```    80     unfolding rel_fun_def
```
```    81     apply clarsimp
```
```    82     apply (case_tac xa)
```
```    83     apply (auto dest: list_all2_lengthD list_all2_nthD)
```
```    84     done
```
```    85 qed
```
```    86
```
```    87 lemma map_parametric:
```
```    88   "((A ===> B) ===> (C ===> D) ===> (B ===> rel_option C) ===> A ===> rel_option D)
```
```    89      (\<lambda>f g m. (map_option g \<circ> m \<circ> f)) (\<lambda>f g m. (map_option g \<circ> m \<circ> f))"
```
```    90   by transfer_prover
```
```    91
```
```    92 end
```
```    93
```
```    94
```
```    95 subsection {* Type definition and primitive operations *}
```
```    96
```
```    97 typedef ('a, 'b) mapping = "UNIV :: ('a \<rightharpoonup> 'b) set"
```
```    98   morphisms rep Mapping
```
```    99   ..
```
```   100
```
```   101 setup_lifting (no_code) type_definition_mapping
```
```   102
```
```   103 lift_definition empty :: "('a, 'b) mapping"
```
```   104   is Map.empty parametric empty_parametric .
```
```   105
```
```   106 lift_definition lookup :: "('a, 'b) mapping \<Rightarrow> 'a \<Rightarrow> 'b option"
```
```   107   is "\<lambda>m k. m k" parametric lookup_parametric .
```
```   108
```
```   109 lift_definition update :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
```
```   110   is "\<lambda>k v m. m(k \<mapsto> v)" parametric update_parametric .
```
```   111
```
```   112 lift_definition delete :: "'a \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
```
```   113   is "\<lambda>k m. m(k := None)" parametric delete_parametric .
```
```   114
```
```   115 lift_definition keys :: "('a, 'b) mapping \<Rightarrow> 'a set"
```
```   116   is dom parametric dom_parametric .
```
```   117
```
```   118 lift_definition tabulate :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping"
```
```   119   is "\<lambda>ks f. (map_of (List.map (\<lambda>k. (k, f k)) ks))" parametric tabulate_parametric .
```
```   120
```
```   121 lift_definition bulkload :: "'a list \<Rightarrow> (nat, 'a) mapping"
```
```   122   is "\<lambda>xs k. if k < length xs then Some (xs ! k) else None" parametric bulkload_parametric .
```
```   123
```
```   124 lift_definition map :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('c, 'd) mapping"
```
```   125   is "\<lambda>f g m. (map_option g \<circ> m \<circ> f)" parametric map_parametric .
```
```   126
```
```   127
```
```   128 subsection {* Functorial structure *}
```
```   129
```
```   130 functor map: map
```
```   131   by (transfer, auto simp add: fun_eq_iff option.map_comp option.map_id)+
```
```   132
```
```   133
```
```   134 subsection {* Derived operations *}
```
```   135
```
```   136 definition ordered_keys :: "('a\<Colon>linorder, 'b) mapping \<Rightarrow> 'a list"
```
```   137 where
```
```   138   "ordered_keys m = (if finite (keys m) then sorted_list_of_set (keys m) else [])"
```
```   139
```
```   140 definition is_empty :: "('a, 'b) mapping \<Rightarrow> bool"
```
```   141 where
```
```   142   "is_empty m \<longleftrightarrow> keys m = {}"
```
```   143
```
```   144 definition size :: "('a, 'b) mapping \<Rightarrow> nat"
```
```   145 where
```
```   146   "size m = (if finite (keys m) then card (keys m) else 0)"
```
```   147
```
```   148 definition replace :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
```
```   149 where
```
```   150   "replace k v m = (if k \<in> keys m then update k v m else m)"
```
```   151
```
```   152 definition default :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
```
```   153 where
```
```   154   "default k v m = (if k \<in> keys m then m else update k v m)"
```
```   155
```
```   156 text {* Manual derivation of transfer rule is non-trivial *}
```
```   157
```
```   158 lift_definition map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" is
```
```   159   "\<lambda>k f m. (case m k of None \<Rightarrow> m
```
```   160     | Some v \<Rightarrow> m (k \<mapsto> (f v)))" parametric map_entry_parametric .
```
```   161
```
```   162 lemma map_entry_code [code]:
```
```   163   "map_entry k f m = (case lookup m k of None \<Rightarrow> m
```
```   164     | Some v \<Rightarrow> update k (f v) m)"
```
```   165   by transfer rule
```
```   166
```
```   167 definition map_default :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
```
```   168 where
```
```   169   "map_default k v f m = map_entry k f (default k v m)"
```
```   170
```
```   171 definition of_alist :: "('k \<times> 'v) list \<Rightarrow> ('k, 'v) mapping"
```
```   172 where
```
```   173   "of_alist xs = foldr (\<lambda>(k, v) m. update k v m) xs empty"
```
```   174
```
```   175 instantiation mapping :: (type, type) equal
```
```   176 begin
```
```   177
```
```   178 definition
```
```   179   "HOL.equal m1 m2 \<longleftrightarrow> (\<forall>k. lookup m1 k = lookup m2 k)"
```
```   180
```
```   181 instance proof
```
```   182 qed (unfold equal_mapping_def, transfer, auto)
```
```   183
```
```   184 end
```
```   185
```
```   186 context
```
```   187 begin
```
```   188
```
```   189 interpretation lifting_syntax .
```
```   190
```
```   191 lemma [transfer_rule]:
```
```   192   assumes [transfer_rule]: "bi_total A"
```
```   193   assumes [transfer_rule]: "bi_unique B"
```
```   194   shows "(pcr_mapping A B ===> pcr_mapping A B ===> op=) HOL.eq HOL.equal"
```
```   195   by (unfold equal) transfer_prover
```
```   196
```
```   197 lemma of_alist_transfer [transfer_rule]:
```
```   198   assumes [transfer_rule]: "bi_unique R1"
```
```   199   shows "(list_all2 (rel_prod R1 R2) ===> pcr_mapping R1 R2) map_of of_alist"
```
```   200   unfolding of_alist_def [abs_def] map_of_foldr [abs_def] by transfer_prover
```
```   201
```
```   202 end
```
```   203
```
```   204
```
```   205 subsection {* Properties *}
```
```   206
```
```   207 lemma lookup_update:
```
```   208   "lookup (update k v m) k = Some v"
```
```   209   by transfer simp
```
```   210
```
```   211 lemma lookup_update_neq:
```
```   212   "k \<noteq> k' \<Longrightarrow> lookup (update k v m) k' = lookup m k'"
```
```   213   by transfer simp
```
```   214
```
```   215 lemma lookup_empty:
```
```   216   "lookup empty k = None"
```
```   217   by transfer simp
```
```   218
```
```   219 lemma keys_is_none_rep [code_unfold]:
```
```   220   "k \<in> keys m \<longleftrightarrow> \<not> (Option.is_none (lookup m k))"
```
```   221   by transfer (auto simp add: is_none_def)
```
```   222
```
```   223 lemma update_update:
```
```   224   "update k v (update k w m) = update k v m"
```
```   225   "k \<noteq> l \<Longrightarrow> update k v (update l w m) = update l w (update k v m)"
```
```   226   by (transfer, simp add: fun_upd_twist)+
```
```   227
```
```   228 lemma update_delete [simp]:
```
```   229   "update k v (delete k m) = update k v m"
```
```   230   by transfer simp
```
```   231
```
```   232 lemma delete_update:
```
```   233   "delete k (update k v m) = delete k m"
```
```   234   "k \<noteq> l \<Longrightarrow> delete k (update l v m) = update l v (delete k m)"
```
```   235   by (transfer, simp add: fun_upd_twist)+
```
```   236
```
```   237 lemma delete_empty [simp]:
```
```   238   "delete k empty = empty"
```
```   239   by transfer simp
```
```   240
```
```   241 lemma replace_update:
```
```   242   "k \<notin> keys m \<Longrightarrow> replace k v m = m"
```
```   243   "k \<in> keys m \<Longrightarrow> replace k v m = update k v m"
```
```   244   by (transfer, auto simp add: replace_def fun_upd_twist)+
```
```   245
```
```   246 lemma size_empty [simp]:
```
```   247   "size empty = 0"
```
```   248   unfolding size_def by transfer simp
```
```   249
```
```   250 lemma size_update:
```
```   251   "finite (keys m) \<Longrightarrow> size (update k v m) =
```
```   252     (if k \<in> keys m then size m else Suc (size m))"
```
```   253   unfolding size_def by transfer (auto simp add: insert_dom)
```
```   254
```
```   255 lemma size_delete:
```
```   256   "size (delete k m) = (if k \<in> keys m then size m - 1 else size m)"
```
```   257   unfolding size_def by transfer simp
```
```   258
```
```   259 lemma size_tabulate [simp]:
```
```   260   "size (tabulate ks f) = length (remdups ks)"
```
```   261   unfolding size_def by transfer (auto simp add: map_of_map_restrict  card_set comp_def)
```
```   262
```
```   263 lemma bulkload_tabulate:
```
```   264   "bulkload xs = tabulate [0..<length xs] (nth xs)"
```
```   265   by transfer (auto simp add: map_of_map_restrict)
```
```   266
```
```   267 lemma is_empty_empty [simp]:
```
```   268   "is_empty empty"
```
```   269   unfolding is_empty_def by transfer simp
```
```   270
```
```   271 lemma is_empty_update [simp]:
```
```   272   "\<not> is_empty (update k v m)"
```
```   273   unfolding is_empty_def by transfer simp
```
```   274
```
```   275 lemma is_empty_delete:
```
```   276   "is_empty (delete k m) \<longleftrightarrow> is_empty m \<or> keys m = {k}"
```
```   277   unfolding is_empty_def by transfer (auto simp del: dom_eq_empty_conv)
```
```   278
```
```   279 lemma is_empty_replace [simp]:
```
```   280   "is_empty (replace k v m) \<longleftrightarrow> is_empty m"
```
```   281   unfolding is_empty_def replace_def by transfer auto
```
```   282
```
```   283 lemma is_empty_default [simp]:
```
```   284   "\<not> is_empty (default k v m)"
```
```   285   unfolding is_empty_def default_def by transfer auto
```
```   286
```
```   287 lemma is_empty_map_entry [simp]:
```
```   288   "is_empty (map_entry k f m) \<longleftrightarrow> is_empty m"
```
```   289   unfolding is_empty_def by transfer (auto split: option.split)
```
```   290
```
```   291 lemma is_empty_map_default [simp]:
```
```   292   "\<not> is_empty (map_default k v f m)"
```
```   293   by (simp add: map_default_def)
```
```   294
```
```   295 lemma keys_dom_lookup:
```
```   296   "keys m = dom (Mapping.lookup m)"
```
```   297   by transfer rule
```
```   298
```
```   299 lemma keys_empty [simp]:
```
```   300   "keys empty = {}"
```
```   301   by transfer simp
```
```   302
```
```   303 lemma keys_update [simp]:
```
```   304   "keys (update k v m) = insert k (keys m)"
```
```   305   by transfer simp
```
```   306
```
```   307 lemma keys_delete [simp]:
```
```   308   "keys (delete k m) = keys m - {k}"
```
```   309   by transfer simp
```
```   310
```
```   311 lemma keys_replace [simp]:
```
```   312   "keys (replace k v m) = keys m"
```
```   313   unfolding replace_def by transfer (simp add: insert_absorb)
```
```   314
```
```   315 lemma keys_default [simp]:
```
```   316   "keys (default k v m) = insert k (keys m)"
```
```   317   unfolding default_def by transfer (simp add: insert_absorb)
```
```   318
```
```   319 lemma keys_map_entry [simp]:
```
```   320   "keys (map_entry k f m) = keys m"
```
```   321   by transfer (auto split: option.split)
```
```   322
```
```   323 lemma keys_map_default [simp]:
```
```   324   "keys (map_default k v f m) = insert k (keys m)"
```
```   325   by (simp add: map_default_def)
```
```   326
```
```   327 lemma keys_tabulate [simp]:
```
```   328   "keys (tabulate ks f) = set ks"
```
```   329   by transfer (simp add: map_of_map_restrict o_def)
```
```   330
```
```   331 lemma keys_bulkload [simp]:
```
```   332   "keys (bulkload xs) = {0..<length xs}"
```
```   333   by (simp add: bulkload_tabulate)
```
```   334
```
```   335 lemma distinct_ordered_keys [simp]:
```
```   336   "distinct (ordered_keys m)"
```
```   337   by (simp add: ordered_keys_def)
```
```   338
```
```   339 lemma ordered_keys_infinite [simp]:
```
```   340   "\<not> finite (keys m) \<Longrightarrow> ordered_keys m = []"
```
```   341   by (simp add: ordered_keys_def)
```
```   342
```
```   343 lemma ordered_keys_empty [simp]:
```
```   344   "ordered_keys empty = []"
```
```   345   by (simp add: ordered_keys_def)
```
```   346
```
```   347 lemma ordered_keys_update [simp]:
```
```   348   "k \<in> keys m \<Longrightarrow> ordered_keys (update k v m) = ordered_keys m"
```
```   349   "finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> ordered_keys (update k v m) = insort k (ordered_keys m)"
```
```   350   by (simp_all add: ordered_keys_def) (auto simp only: sorted_list_of_set_insert [symmetric] insert_absorb)
```
```   351
```
```   352 lemma ordered_keys_delete [simp]:
```
```   353   "ordered_keys (delete k m) = remove1 k (ordered_keys m)"
```
```   354 proof (cases "finite (keys m)")
```
```   355   case False then show ?thesis by simp
```
```   356 next
```
```   357   case True note fin = True
```
```   358   show ?thesis
```
```   359   proof (cases "k \<in> keys m")
```
```   360     case False with fin have "k \<notin> set (sorted_list_of_set (keys m))" by simp
```
```   361     with False show ?thesis by (simp add: ordered_keys_def remove1_idem)
```
```   362   next
```
```   363     case True with fin show ?thesis by (simp add: ordered_keys_def sorted_list_of_set_remove)
```
```   364   qed
```
```   365 qed
```
```   366
```
```   367 lemma ordered_keys_replace [simp]:
```
```   368   "ordered_keys (replace k v m) = ordered_keys m"
```
```   369   by (simp add: replace_def)
```
```   370
```
```   371 lemma ordered_keys_default [simp]:
```
```   372   "k \<in> keys m \<Longrightarrow> ordered_keys (default k v m) = ordered_keys m"
```
```   373   "finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> ordered_keys (default k v m) = insort k (ordered_keys m)"
```
```   374   by (simp_all add: default_def)
```
```   375
```
```   376 lemma ordered_keys_map_entry [simp]:
```
```   377   "ordered_keys (map_entry k f m) = ordered_keys m"
```
```   378   by (simp add: ordered_keys_def)
```
```   379
```
```   380 lemma ordered_keys_map_default [simp]:
```
```   381   "k \<in> keys m \<Longrightarrow> ordered_keys (map_default k v f m) = ordered_keys m"
```
```   382   "finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> ordered_keys (map_default k v f m) = insort k (ordered_keys m)"
```
```   383   by (simp_all add: map_default_def)
```
```   384
```
```   385 lemma ordered_keys_tabulate [simp]:
```
```   386   "ordered_keys (tabulate ks f) = sort (remdups ks)"
```
```   387   by (simp add: ordered_keys_def sorted_list_of_set_sort_remdups)
```
```   388
```
```   389 lemma ordered_keys_bulkload [simp]:
```
```   390   "ordered_keys (bulkload ks) = [0..<length ks]"
```
```   391   by (simp add: ordered_keys_def)
```
```   392
```
```   393 lemma tabulate_fold:
```
```   394   "tabulate xs f = fold (\<lambda>k m. update k (f k) m) xs empty"
```
```   395 proof transfer
```
```   396   fix f :: "'a \<Rightarrow> 'b" and xs
```
```   397   have "map_of (List.map (\<lambda>k. (k, f k)) xs) = foldr (\<lambda>k m. m(k \<mapsto> f k)) xs Map.empty"
```
```   398     by (simp add: foldr_map comp_def map_of_foldr)
```
```   399   also have "foldr (\<lambda>k m. m(k \<mapsto> f k)) xs = fold (\<lambda>k m. m(k \<mapsto> f k)) xs"
```
```   400     by (rule foldr_fold) (simp add: fun_eq_iff)
```
```   401   ultimately show "map_of (List.map (\<lambda>k. (k, f k)) xs) = fold (\<lambda>k m. m(k \<mapsto> f k)) xs Map.empty"
```
```   402     by simp
```
```   403 qed
```
```   404
```
```   405
```
```   406 subsection {* Code generator setup *}
```
```   407
```
```   408 code_datatype empty update
```
```   409
```
```   410 hide_const (open) empty is_empty rep lookup update delete ordered_keys keys size
```
```   411   replace default map_entry map_default tabulate bulkload map of_alist
```
```   412
```
```   413 end
```