src/HOL/Library/Mapping.thy
 author haftmann Wed Jul 18 20:51:21 2018 +0200 (11 months ago) changeset 68658 16cc1161ad7f parent 67399 eab6ce8368fa child 68782 8ff34c1ad580 permissions -rw-r--r--
tuned equation
```     1 (*  Title:      HOL/Library/Mapping.thy
```
```     2     Author:     Florian Haftmann and Ondrej Kuncar
```
```     3 *)
```
```     4
```
```     5 section \<open>An abstract view on maps for code generation.\<close>
```
```     6
```
```     7 theory Mapping
```
```     8 imports Main
```
```     9 begin
```
```    10
```
```    11 subsection \<open>Parametricity transfer rules\<close>
```
```    12
```
```    13 lemma map_of_foldr: "map_of xs = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) xs Map.empty"  (* FIXME move *)
```
```    14   using map_add_map_of_foldr [of Map.empty] by auto
```
```    15
```
```    16 context includes lifting_syntax
```
```    17 begin
```
```    18
```
```    19 lemma empty_parametric: "(A ===> rel_option B) Map.empty Map.empty"
```
```    20   by transfer_prover
```
```    21
```
```    22 lemma lookup_parametric: "((A ===> B) ===> A ===> B) (\<lambda>m k. m k) (\<lambda>m k. m k)"
```
```    23   by transfer_prover
```
```    24
```
```    25 lemma update_parametric:
```
```    26   assumes [transfer_rule]: "bi_unique A"
```
```    27   shows "(A ===> B ===> (A ===> rel_option B) ===> A ===> rel_option B)
```
```    28     (\<lambda>k v m. m(k \<mapsto> v)) (\<lambda>k v m. m(k \<mapsto> v))"
```
```    29   by transfer_prover
```
```    30
```
```    31 lemma delete_parametric:
```
```    32   assumes [transfer_rule]: "bi_unique A"
```
```    33   shows "(A ===> (A ===> rel_option B) ===> A ===> rel_option B)
```
```    34     (\<lambda>k m. m(k := None)) (\<lambda>k m. m(k := None))"
```
```    35   by transfer_prover
```
```    36
```
```    37 lemma is_none_parametric [transfer_rule]:
```
```    38   "(rel_option A ===> HOL.eq) Option.is_none Option.is_none"
```
```    39   by (auto simp add: Option.is_none_def rel_fun_def rel_option_iff split: option.split)
```
```    40
```
```    41 lemma dom_parametric:
```
```    42   assumes [transfer_rule]: "bi_total A"
```
```    43   shows "((A ===> rel_option B) ===> rel_set A) dom dom"
```
```    44   unfolding dom_def [abs_def] Option.is_none_def [symmetric] by transfer_prover
```
```    45
```
```    46 lemma map_of_parametric [transfer_rule]:
```
```    47   assumes [transfer_rule]: "bi_unique R1"
```
```    48   shows "(list_all2 (rel_prod R1 R2) ===> R1 ===> rel_option R2) map_of map_of"
```
```    49   unfolding map_of_def by transfer_prover
```
```    50
```
```    51 lemma map_entry_parametric [transfer_rule]:
```
```    52   assumes [transfer_rule]: "bi_unique A"
```
```    53   shows "(A ===> (B ===> B) ===> (A ===> rel_option B) ===> A ===> rel_option B)
```
```    54     (\<lambda>k f m. (case m k of None \<Rightarrow> m
```
```    55       | Some v \<Rightarrow> m (k \<mapsto> (f v)))) (\<lambda>k f m. (case m k of None \<Rightarrow> m
```
```    56       | Some v \<Rightarrow> m (k \<mapsto> (f v))))"
```
```    57   by transfer_prover
```
```    58
```
```    59 lemma tabulate_parametric:
```
```    60   assumes [transfer_rule]: "bi_unique A"
```
```    61   shows "(list_all2 A ===> (A ===> B) ===> A ===> rel_option B)
```
```    62     (\<lambda>ks f. (map_of (map (\<lambda>k. (k, f k)) ks))) (\<lambda>ks f. (map_of (map (\<lambda>k. (k, f k)) ks)))"
```
```    63   by transfer_prover
```
```    64
```
```    65 lemma bulkload_parametric:
```
```    66   "(list_all2 A ===> HOL.eq ===> rel_option A)
```
```    67     (\<lambda>xs k. if k < length xs then Some (xs ! k) else None)
```
```    68     (\<lambda>xs k. if k < length xs then Some (xs ! k) else None)"
```
```    69 proof
```
```    70   fix xs ys
```
```    71   assume "list_all2 A xs ys"
```
```    72   then show
```
```    73     "(HOL.eq ===> rel_option A)
```
```    74       (\<lambda>k. if k < length xs then Some (xs ! k) else None)
```
```    75       (\<lambda>k. if k < length ys then Some (ys ! k) else None)"
```
```    76     apply induct
```
```    77      apply auto
```
```    78     unfolding rel_fun_def
```
```    79     apply clarsimp
```
```    80     apply (case_tac xa)
```
```    81      apply (auto dest: list_all2_lengthD list_all2_nthD)
```
```    82     done
```
```    83 qed
```
```    84
```
```    85 lemma map_parametric:
```
```    86   "((A ===> B) ===> (C ===> D) ===> (B ===> rel_option C) ===> A ===> rel_option D)
```
```    87      (\<lambda>f g m. (map_option g \<circ> m \<circ> f)) (\<lambda>f g m. (map_option g \<circ> m \<circ> f))"
```
```    88   by transfer_prover
```
```    89
```
```    90 lemma combine_with_key_parametric:
```
```    91   "((A ===> B ===> B ===> B) ===> (A ===> rel_option B) ===> (A ===> rel_option B) ===>
```
```    92     (A ===> rel_option B)) (\<lambda>f m1 m2 x. combine_options (f x) (m1 x) (m2 x))
```
```    93     (\<lambda>f m1 m2 x. combine_options (f x) (m1 x) (m2 x))"
```
```    94   unfolding combine_options_def by transfer_prover
```
```    95
```
```    96 lemma combine_parametric:
```
```    97   "((B ===> B ===> B) ===> (A ===> rel_option B) ===> (A ===> rel_option B) ===>
```
```    98     (A ===> rel_option B)) (\<lambda>f m1 m2 x. combine_options f (m1 x) (m2 x))
```
```    99     (\<lambda>f m1 m2 x. combine_options f (m1 x) (m2 x))"
```
```   100   unfolding combine_options_def by transfer_prover
```
```   101
```
```   102 end
```
```   103
```
```   104
```
```   105 subsection \<open>Type definition and primitive operations\<close>
```
```   106
```
```   107 typedef ('a, 'b) mapping = "UNIV :: ('a \<rightharpoonup> 'b) set"
```
```   108   morphisms rep Mapping ..
```
```   109
```
```   110 setup_lifting type_definition_mapping
```
```   111
```
```   112 lift_definition empty :: "('a, 'b) mapping"
```
```   113   is Map.empty parametric empty_parametric .
```
```   114
```
```   115 lift_definition lookup :: "('a, 'b) mapping \<Rightarrow> 'a \<Rightarrow> 'b option"
```
```   116   is "\<lambda>m k. m k" parametric lookup_parametric .
```
```   117
```
```   118 definition "lookup_default d m k = (case Mapping.lookup m k of None \<Rightarrow> d | Some v \<Rightarrow> v)"
```
```   119
```
```   120 lemma [code abstract]:
```
```   121   "lookup (Mapping f) = f"
```
```   122   by (fact Mapping.lookup.abs_eq) (* FIXME lifting *)
```
```   123
```
```   124 lift_definition update :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
```
```   125   is "\<lambda>k v m. m(k \<mapsto> v)" parametric update_parametric .
```
```   126
```
```   127 lift_definition delete :: "'a \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
```
```   128   is "\<lambda>k m. m(k := None)" parametric delete_parametric .
```
```   129
```
```   130 lift_definition filter :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
```
```   131   is "\<lambda>P m k. case m k of None \<Rightarrow> None | Some v \<Rightarrow> if P k v then Some v else None" .
```
```   132
```
```   133 lift_definition keys :: "('a, 'b) mapping \<Rightarrow> 'a set"
```
```   134   is dom parametric dom_parametric .
```
```   135
```
```   136 lift_definition tabulate :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping"
```
```   137   is "\<lambda>ks f. (map_of (List.map (\<lambda>k. (k, f k)) ks))" parametric tabulate_parametric .
```
```   138
```
```   139 lift_definition bulkload :: "'a list \<Rightarrow> (nat, 'a) mapping"
```
```   140   is "\<lambda>xs k. if k < length xs then Some (xs ! k) else None" parametric bulkload_parametric .
```
```   141
```
```   142 lift_definition map :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('c, 'd) mapping"
```
```   143   is "\<lambda>f g m. (map_option g \<circ> m \<circ> f)" parametric map_parametric .
```
```   144
```
```   145 lift_definition map_values :: "('c \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> ('c, 'a) mapping \<Rightarrow> ('c, 'b) mapping"
```
```   146   is "\<lambda>f m x. map_option (f x) (m x)" .
```
```   147
```
```   148 lift_definition combine_with_key ::
```
```   149   "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a,'b) mapping \<Rightarrow> ('a,'b) mapping \<Rightarrow> ('a,'b) mapping"
```
```   150   is "\<lambda>f m1 m2 x. combine_options (f x) (m1 x) (m2 x)" parametric combine_with_key_parametric .
```
```   151
```
```   152 lift_definition combine ::
```
```   153   "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a,'b) mapping \<Rightarrow> ('a,'b) mapping \<Rightarrow> ('a,'b) mapping"
```
```   154   is "\<lambda>f m1 m2 x. combine_options f (m1 x) (m2 x)" parametric combine_parametric .
```
```   155
```
```   156 definition "All_mapping m P \<longleftrightarrow>
```
```   157   (\<forall>x. case Mapping.lookup m x of None \<Rightarrow> True | Some y \<Rightarrow> P x y)"
```
```   158
```
```   159 declare [[code drop: map]]
```
```   160
```
```   161
```
```   162 subsection \<open>Functorial structure\<close>
```
```   163
```
```   164 functor map: map
```
```   165   by (transfer, auto simp add: fun_eq_iff option.map_comp option.map_id)+
```
```   166
```
```   167
```
```   168 subsection \<open>Derived operations\<close>
```
```   169
```
```   170 definition ordered_keys :: "('a::linorder, 'b) mapping \<Rightarrow> 'a list"
```
```   171   where "ordered_keys m = (if finite (keys m) then sorted_list_of_set (keys m) else [])"
```
```   172
```
```   173 definition is_empty :: "('a, 'b) mapping \<Rightarrow> bool"
```
```   174   where "is_empty m \<longleftrightarrow> keys m = {}"
```
```   175
```
```   176 definition size :: "('a, 'b) mapping \<Rightarrow> nat"
```
```   177   where "size m = (if finite (keys m) then card (keys m) else 0)"
```
```   178
```
```   179 definition replace :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
```
```   180   where "replace k v m = (if k \<in> keys m then update k v m else m)"
```
```   181
```
```   182 definition default :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
```
```   183   where "default k v m = (if k \<in> keys m then m else update k v m)"
```
```   184
```
```   185 text \<open>Manual derivation of transfer rule is non-trivial\<close>
```
```   186
```
```   187 lift_definition map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" is
```
```   188   "\<lambda>k f m.
```
```   189     (case m k of
```
```   190       None \<Rightarrow> m
```
```   191     | Some v \<Rightarrow> m (k \<mapsto> (f v)))" parametric map_entry_parametric .
```
```   192
```
```   193 lemma map_entry_code [code]:
```
```   194   "map_entry k f m =
```
```   195     (case lookup m k of
```
```   196       None \<Rightarrow> m
```
```   197     | Some v \<Rightarrow> update k (f v) m)"
```
```   198   by transfer rule
```
```   199
```
```   200 definition map_default :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
```
```   201   where "map_default k v f m = map_entry k f (default k v m)"
```
```   202
```
```   203 definition of_alist :: "('k \<times> 'v) list \<Rightarrow> ('k, 'v) mapping"
```
```   204   where "of_alist xs = foldr (\<lambda>(k, v) m. update k v m) xs empty"
```
```   205
```
```   206 instantiation mapping :: (type, type) equal
```
```   207 begin
```
```   208
```
```   209 definition "HOL.equal m1 m2 \<longleftrightarrow> (\<forall>k. lookup m1 k = lookup m2 k)"
```
```   210
```
```   211 instance
```
```   212   apply standard
```
```   213   unfolding equal_mapping_def
```
```   214   apply transfer
```
```   215   apply auto
```
```   216   done
```
```   217
```
```   218 end
```
```   219
```
```   220 context includes lifting_syntax
```
```   221 begin
```
```   222
```
```   223 lemma [transfer_rule]:
```
```   224   assumes [transfer_rule]: "bi_total A"
```
```   225     and [transfer_rule]: "bi_unique B"
```
```   226   shows "(pcr_mapping A B ===> pcr_mapping A B ===> (=)) HOL.eq HOL.equal"
```
```   227   unfolding equal by transfer_prover
```
```   228
```
```   229 lemma of_alist_transfer [transfer_rule]:
```
```   230   assumes [transfer_rule]: "bi_unique R1"
```
```   231   shows "(list_all2 (rel_prod R1 R2) ===> pcr_mapping R1 R2) map_of of_alist"
```
```   232   unfolding of_alist_def [abs_def] map_of_foldr [abs_def] by transfer_prover
```
```   233
```
```   234 end
```
```   235
```
```   236
```
```   237 subsection \<open>Properties\<close>
```
```   238
```
```   239 lemma mapping_eqI: "(\<And>x. lookup m x = lookup m' x) \<Longrightarrow> m = m'"
```
```   240   by transfer (simp add: fun_eq_iff)
```
```   241
```
```   242 lemma mapping_eqI':
```
```   243   assumes "\<And>x. x \<in> Mapping.keys m \<Longrightarrow> Mapping.lookup_default d m x = Mapping.lookup_default d m' x"
```
```   244     and "Mapping.keys m = Mapping.keys m'"
```
```   245   shows "m = m'"
```
```   246 proof (intro mapping_eqI)
```
```   247   show "Mapping.lookup m x = Mapping.lookup m' x" for x
```
```   248   proof (cases "Mapping.lookup m x")
```
```   249     case None
```
```   250     then have "x \<notin> Mapping.keys m"
```
```   251       by transfer (simp add: dom_def)
```
```   252     then have "x \<notin> Mapping.keys m'"
```
```   253       by (simp add: assms)
```
```   254     then have "Mapping.lookup m' x = None"
```
```   255       by transfer (simp add: dom_def)
```
```   256     with None show ?thesis
```
```   257       by simp
```
```   258   next
```
```   259     case (Some y)
```
```   260     then have A: "x \<in> Mapping.keys m"
```
```   261       by transfer (simp add: dom_def)
```
```   262     then have "x \<in> Mapping.keys m'"
```
```   263       by (simp add: assms)
```
```   264     then have "\<exists>y'. Mapping.lookup m' x = Some y'"
```
```   265       by transfer (simp add: dom_def)
```
```   266     with Some assms(1)[OF A] show ?thesis
```
```   267       by (auto simp add: lookup_default_def)
```
```   268   qed
```
```   269 qed
```
```   270
```
```   271 lemma lookup_update: "lookup (update k v m) k = Some v"
```
```   272   by transfer simp
```
```   273
```
```   274 lemma lookup_update_neq: "k \<noteq> k' \<Longrightarrow> lookup (update k v m) k' = lookup m k'"
```
```   275   by transfer simp
```
```   276
```
```   277 lemma lookup_update': "Mapping.lookup (update k v m) k' = (if k = k' then Some v else lookup m k')"
```
```   278   by (auto simp: lookup_update lookup_update_neq)
```
```   279
```
```   280 lemma lookup_empty: "lookup empty k = None"
```
```   281   by transfer simp
```
```   282
```
```   283 lemma lookup_filter:
```
```   284   "lookup (filter P m) k =
```
```   285     (case lookup m k of
```
```   286       None \<Rightarrow> None
```
```   287     | Some v \<Rightarrow> if P k v then Some v else None)"
```
```   288   by transfer simp_all
```
```   289
```
```   290 lemma lookup_map_values: "lookup (map_values f m) k = map_option (f k) (lookup m k)"
```
```   291   by transfer simp_all
```
```   292
```
```   293 lemma lookup_default_empty: "lookup_default d empty k = d"
```
```   294   by (simp add: lookup_default_def lookup_empty)
```
```   295
```
```   296 lemma lookup_default_update: "lookup_default d (update k v m) k = v"
```
```   297   by (simp add: lookup_default_def lookup_update)
```
```   298
```
```   299 lemma lookup_default_update_neq:
```
```   300   "k \<noteq> k' \<Longrightarrow> lookup_default d (update k v m) k' = lookup_default d m k'"
```
```   301   by (simp add: lookup_default_def lookup_update_neq)
```
```   302
```
```   303 lemma lookup_default_update':
```
```   304   "lookup_default d (update k v m) k' = (if k = k' then v else lookup_default d m k')"
```
```   305   by (auto simp: lookup_default_update lookup_default_update_neq)
```
```   306
```
```   307 lemma lookup_default_filter:
```
```   308   "lookup_default d (filter P m) k =
```
```   309      (if P k (lookup_default d m k) then lookup_default d m k else d)"
```
```   310   by (simp add: lookup_default_def lookup_filter split: option.splits)
```
```   311
```
```   312 lemma lookup_default_map_values:
```
```   313   "lookup_default (f k d) (map_values f m) k = f k (lookup_default d m k)"
```
```   314   by (simp add: lookup_default_def lookup_map_values split: option.splits)
```
```   315
```
```   316 lemma lookup_combine_with_key:
```
```   317   "Mapping.lookup (combine_with_key f m1 m2) x =
```
```   318     combine_options (f x) (Mapping.lookup m1 x) (Mapping.lookup m2 x)"
```
```   319   by transfer (auto split: option.splits)
```
```   320
```
```   321 lemma combine_altdef: "combine f m1 m2 = combine_with_key (\<lambda>_. f) m1 m2"
```
```   322   by transfer' (rule refl)
```
```   323
```
```   324 lemma lookup_combine:
```
```   325   "Mapping.lookup (combine f m1 m2) x =
```
```   326      combine_options f (Mapping.lookup m1 x) (Mapping.lookup m2 x)"
```
```   327   by transfer (auto split: option.splits)
```
```   328
```
```   329 lemma lookup_default_neutral_combine_with_key:
```
```   330   assumes "\<And>x. f k d x = x" "\<And>x. f k x d = x"
```
```   331   shows "Mapping.lookup_default d (combine_with_key f m1 m2) k =
```
```   332     f k (Mapping.lookup_default d m1 k) (Mapping.lookup_default d m2 k)"
```
```   333   by (auto simp: lookup_default_def lookup_combine_with_key assms split: option.splits)
```
```   334
```
```   335 lemma lookup_default_neutral_combine:
```
```   336   assumes "\<And>x. f d x = x" "\<And>x. f x d = x"
```
```   337   shows "Mapping.lookup_default d (combine f m1 m2) x =
```
```   338     f (Mapping.lookup_default d m1 x) (Mapping.lookup_default d m2 x)"
```
```   339   by (auto simp: lookup_default_def lookup_combine assms split: option.splits)
```
```   340
```
```   341 lemma lookup_map_entry: "lookup (map_entry x f m) x = map_option f (lookup m x)"
```
```   342   by transfer (auto split: option.splits)
```
```   343
```
```   344 lemma lookup_map_entry_neq: "x \<noteq> y \<Longrightarrow> lookup (map_entry x f m) y = lookup m y"
```
```   345   by transfer (auto split: option.splits)
```
```   346
```
```   347 lemma lookup_map_entry':
```
```   348   "lookup (map_entry x f m) y =
```
```   349      (if x = y then map_option f (lookup m y) else lookup m y)"
```
```   350   by transfer (auto split: option.splits)
```
```   351
```
```   352 lemma lookup_default: "lookup (default x d m) x = Some (lookup_default d m x)"
```
```   353   unfolding lookup_default_def default_def
```
```   354   by transfer (auto split: option.splits)
```
```   355
```
```   356 lemma lookup_default_neq: "x \<noteq> y \<Longrightarrow> lookup (default x d m) y = lookup m y"
```
```   357   unfolding lookup_default_def default_def
```
```   358   by transfer (auto split: option.splits)
```
```   359
```
```   360 lemma lookup_default':
```
```   361   "lookup (default x d m) y =
```
```   362     (if x = y then Some (lookup_default d m x) else lookup m y)"
```
```   363   unfolding lookup_default_def default_def
```
```   364   by transfer (auto split: option.splits)
```
```   365
```
```   366 lemma lookup_map_default: "lookup (map_default x d f m) x = Some (f (lookup_default d m x))"
```
```   367   unfolding lookup_default_def default_def
```
```   368   by (simp add: map_default_def lookup_map_entry lookup_default lookup_default_def)
```
```   369
```
```   370 lemma lookup_map_default_neq: "x \<noteq> y \<Longrightarrow> lookup (map_default x d f m) y = lookup m y"
```
```   371   unfolding lookup_default_def default_def
```
```   372   by (simp add: map_default_def lookup_map_entry_neq lookup_default_neq)
```
```   373
```
```   374 lemma lookup_map_default':
```
```   375   "lookup (map_default x d f m) y =
```
```   376     (if x = y then Some (f (lookup_default d m x)) else lookup m y)"
```
```   377   unfolding lookup_default_def default_def
```
```   378   by (simp add: map_default_def lookup_map_entry' lookup_default' lookup_default_def)
```
```   379
```
```   380 lemma lookup_tabulate:
```
```   381   assumes "distinct xs"
```
```   382   shows "Mapping.lookup (Mapping.tabulate xs f) x = (if x \<in> set xs then Some (f x) else None)"
```
```   383   using assms by transfer (auto simp: map_of_eq_None_iff o_def dest!: map_of_SomeD)
```
```   384
```
```   385 lemma lookup_of_alist: "Mapping.lookup (Mapping.of_alist xs) k = map_of xs k"
```
```   386   by transfer simp_all
```
```   387
```
```   388 lemma keys_is_none_rep [code_unfold]: "k \<in> keys m \<longleftrightarrow> \<not> (Option.is_none (lookup m k))"
```
```   389   by transfer (auto simp add: Option.is_none_def)
```
```   390
```
```   391 lemma update_update:
```
```   392   "update k v (update k w m) = update k v m"
```
```   393   "k \<noteq> l \<Longrightarrow> update k v (update l w m) = update l w (update k v m)"
```
```   394   by (transfer; simp add: fun_upd_twist)+
```
```   395
```
```   396 lemma update_delete [simp]: "update k v (delete k m) = update k v m"
```
```   397   by transfer simp
```
```   398
```
```   399 lemma delete_update:
```
```   400   "delete k (update k v m) = delete k m"
```
```   401   "k \<noteq> l \<Longrightarrow> delete k (update l v m) = update l v (delete k m)"
```
```   402   by (transfer; simp add: fun_upd_twist)+
```
```   403
```
```   404 lemma delete_empty [simp]: "delete k empty = empty"
```
```   405   by transfer simp
```
```   406
```
```   407 lemma replace_update:
```
```   408   "k \<notin> keys m \<Longrightarrow> replace k v m = m"
```
```   409   "k \<in> keys m \<Longrightarrow> replace k v m = update k v m"
```
```   410   by (transfer; auto simp add: replace_def fun_upd_twist)+
```
```   411
```
```   412 lemma map_values_update: "map_values f (update k v m) = update k (f k v) (map_values f m)"
```
```   413   by transfer (simp_all add: fun_eq_iff)
```
```   414
```
```   415 lemma size_mono: "finite (keys m') \<Longrightarrow> keys m \<subseteq> keys m' \<Longrightarrow> size m \<le> size m'"
```
```   416   unfolding size_def by (auto intro: card_mono)
```
```   417
```
```   418 lemma size_empty [simp]: "size empty = 0"
```
```   419   unfolding size_def by transfer simp
```
```   420
```
```   421 lemma size_update:
```
```   422   "finite (keys m) \<Longrightarrow> size (update k v m) =
```
```   423     (if k \<in> keys m then size m else Suc (size m))"
```
```   424   unfolding size_def by transfer (auto simp add: insert_dom)
```
```   425
```
```   426 lemma size_delete: "size (delete k m) = (if k \<in> keys m then size m - 1 else size m)"
```
```   427   unfolding size_def by transfer simp
```
```   428
```
```   429 lemma size_tabulate [simp]: "size (tabulate ks f) = length (remdups ks)"
```
```   430   unfolding size_def by transfer (auto simp add: map_of_map_restrict card_set comp_def)
```
```   431
```
```   432 lemma keys_filter: "keys (filter P m) \<subseteq> keys m"
```
```   433   by transfer (auto split: option.splits)
```
```   434
```
```   435 lemma size_filter: "finite (keys m) \<Longrightarrow> size (filter P m) \<le> size m"
```
```   436   by (intro size_mono keys_filter)
```
```   437
```
```   438 lemma bulkload_tabulate: "bulkload xs = tabulate [0..<length xs] (nth xs)"
```
```   439   by transfer (auto simp add: map_of_map_restrict)
```
```   440
```
```   441 lemma is_empty_empty [simp]: "is_empty empty"
```
```   442   unfolding is_empty_def by transfer simp
```
```   443
```
```   444 lemma is_empty_update [simp]: "\<not> is_empty (update k v m)"
```
```   445   unfolding is_empty_def by transfer simp
```
```   446
```
```   447 lemma is_empty_delete: "is_empty (delete k m) \<longleftrightarrow> is_empty m \<or> keys m = {k}"
```
```   448   unfolding is_empty_def by transfer (auto simp del: dom_eq_empty_conv)
```
```   449
```
```   450 lemma is_empty_replace [simp]: "is_empty (replace k v m) \<longleftrightarrow> is_empty m"
```
```   451   unfolding is_empty_def replace_def by transfer auto
```
```   452
```
```   453 lemma is_empty_default [simp]: "\<not> is_empty (default k v m)"
```
```   454   unfolding is_empty_def default_def by transfer auto
```
```   455
```
```   456 lemma is_empty_map_entry [simp]: "is_empty (map_entry k f m) \<longleftrightarrow> is_empty m"
```
```   457   unfolding is_empty_def by transfer (auto split: option.split)
```
```   458
```
```   459 lemma is_empty_map_values [simp]: "is_empty (map_values f m) \<longleftrightarrow> is_empty m"
```
```   460   unfolding is_empty_def by transfer (auto simp: fun_eq_iff)
```
```   461
```
```   462 lemma is_empty_map_default [simp]: "\<not> is_empty (map_default k v f m)"
```
```   463   by (simp add: map_default_def)
```
```   464
```
```   465 lemma keys_dom_lookup: "keys m = dom (Mapping.lookup m)"
```
```   466   by transfer rule
```
```   467
```
```   468 lemma keys_empty [simp]: "keys empty = {}"
```
```   469   by transfer simp
```
```   470
```
```   471 lemma keys_update [simp]: "keys (update k v m) = insert k (keys m)"
```
```   472   by transfer simp
```
```   473
```
```   474 lemma keys_delete [simp]: "keys (delete k m) = keys m - {k}"
```
```   475   by transfer simp
```
```   476
```
```   477 lemma keys_replace [simp]: "keys (replace k v m) = keys m"
```
```   478   unfolding replace_def by transfer (simp add: insert_absorb)
```
```   479
```
```   480 lemma keys_default [simp]: "keys (default k v m) = insert k (keys m)"
```
```   481   unfolding default_def by transfer (simp add: insert_absorb)
```
```   482
```
```   483 lemma keys_map_entry [simp]: "keys (map_entry k f m) = keys m"
```
```   484   by transfer (auto split: option.split)
```
```   485
```
```   486 lemma keys_map_default [simp]: "keys (map_default k v f m) = insert k (keys m)"
```
```   487   by (simp add: map_default_def)
```
```   488
```
```   489 lemma keys_map_values [simp]: "keys (map_values f m) = keys m"
```
```   490   by transfer (simp_all add: dom_def)
```
```   491
```
```   492 lemma keys_combine_with_key [simp]:
```
```   493   "Mapping.keys (combine_with_key f m1 m2) = Mapping.keys m1 \<union> Mapping.keys m2"
```
```   494   by transfer (auto simp: dom_def combine_options_def split: option.splits)
```
```   495
```
```   496 lemma keys_combine [simp]: "Mapping.keys (combine f m1 m2) = Mapping.keys m1 \<union> Mapping.keys m2"
```
```   497   by (simp add: combine_altdef)
```
```   498
```
```   499 lemma keys_tabulate [simp]: "keys (tabulate ks f) = set ks"
```
```   500   by transfer (simp add: map_of_map_restrict o_def)
```
```   501
```
```   502 lemma keys_of_alist [simp]: "keys (of_alist xs) = set (List.map fst xs)"
```
```   503   by transfer (simp_all add: dom_map_of_conv_image_fst)
```
```   504
```
```   505 lemma keys_bulkload [simp]: "keys (bulkload xs) = {0..<length xs}"
```
```   506   by (simp add: bulkload_tabulate)
```
```   507
```
```   508 lemma distinct_ordered_keys [simp]: "distinct (ordered_keys m)"
```
```   509   by (simp add: ordered_keys_def)
```
```   510
```
```   511 lemma ordered_keys_infinite [simp]: "\<not> finite (keys m) \<Longrightarrow> ordered_keys m = []"
```
```   512   by (simp add: ordered_keys_def)
```
```   513
```
```   514 lemma ordered_keys_empty [simp]: "ordered_keys empty = []"
```
```   515   by (simp add: ordered_keys_def)
```
```   516
```
```   517 lemma ordered_keys_update [simp]:
```
```   518   "k \<in> keys m \<Longrightarrow> ordered_keys (update k v m) = ordered_keys m"
```
```   519   "finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow>
```
```   520     ordered_keys (update k v m) = insort k (ordered_keys m)"
```
```   521   by (simp_all add: ordered_keys_def)
```
```   522     (auto simp only: sorted_list_of_set_insert [symmetric] insert_absorb)
```
```   523
```
```   524 lemma ordered_keys_delete [simp]: "ordered_keys (delete k m) = remove1 k (ordered_keys m)"
```
```   525 proof (cases "finite (keys m)")
```
```   526   case False
```
```   527   then show ?thesis by simp
```
```   528 next
```
```   529   case fin: True
```
```   530   show ?thesis
```
```   531   proof (cases "k \<in> keys m")
```
```   532     case False
```
```   533     with fin have "k \<notin> set (sorted_list_of_set (keys m))"
```
```   534       by simp
```
```   535     with False show ?thesis
```
```   536       by (simp add: ordered_keys_def remove1_idem)
```
```   537   next
```
```   538     case True
```
```   539     with fin show ?thesis
```
```   540       by (simp add: ordered_keys_def sorted_list_of_set_remove)
```
```   541   qed
```
```   542 qed
```
```   543
```
```   544 lemma ordered_keys_replace [simp]: "ordered_keys (replace k v m) = ordered_keys m"
```
```   545   by (simp add: replace_def)
```
```   546
```
```   547 lemma ordered_keys_default [simp]:
```
```   548   "k \<in> keys m \<Longrightarrow> ordered_keys (default k v m) = ordered_keys m"
```
```   549   "finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> ordered_keys (default k v m) = insort k (ordered_keys m)"
```
```   550   by (simp_all add: default_def)
```
```   551
```
```   552 lemma ordered_keys_map_entry [simp]: "ordered_keys (map_entry k f m) = ordered_keys m"
```
```   553   by (simp add: ordered_keys_def)
```
```   554
```
```   555 lemma ordered_keys_map_default [simp]:
```
```   556   "k \<in> keys m \<Longrightarrow> ordered_keys (map_default k v f m) = ordered_keys m"
```
```   557   "finite (keys m) \<Longrightarrow> k \<notin> keys m \<Longrightarrow> ordered_keys (map_default k v f m) = insort k (ordered_keys m)"
```
```   558   by (simp_all add: map_default_def)
```
```   559
```
```   560 lemma ordered_keys_tabulate [simp]: "ordered_keys (tabulate ks f) = sort (remdups ks)"
```
```   561   by (simp add: ordered_keys_def sorted_list_of_set_sort_remdups)
```
```   562
```
```   563 lemma ordered_keys_bulkload [simp]: "ordered_keys (bulkload ks) = [0..<length ks]"
```
```   564   by (simp add: ordered_keys_def)
```
```   565
```
```   566 lemma tabulate_fold: "tabulate xs f = fold (\<lambda>k m. update k (f k) m) xs empty"
```
```   567 proof transfer
```
```   568   fix f :: "'a \<Rightarrow> 'b" and xs
```
```   569   have "map_of (List.map (\<lambda>k. (k, f k)) xs) = foldr (\<lambda>k m. m(k \<mapsto> f k)) xs Map.empty"
```
```   570     by (simp add: foldr_map comp_def map_of_foldr)
```
```   571   also have "foldr (\<lambda>k m. m(k \<mapsto> f k)) xs = fold (\<lambda>k m. m(k \<mapsto> f k)) xs"
```
```   572     by (rule foldr_fold) (simp add: fun_eq_iff)
```
```   573   ultimately show "map_of (List.map (\<lambda>k. (k, f k)) xs) = fold (\<lambda>k m. m(k \<mapsto> f k)) xs Map.empty"
```
```   574     by simp
```
```   575 qed
```
```   576
```
```   577 lemma All_mapping_mono:
```
```   578   "(\<And>k v. k \<in> keys m \<Longrightarrow> P k v \<Longrightarrow> Q k v) \<Longrightarrow> All_mapping m P \<Longrightarrow> All_mapping m Q"
```
```   579   unfolding All_mapping_def by transfer (auto simp: All_mapping_def dom_def split: option.splits)
```
```   580
```
```   581 lemma All_mapping_empty [simp]: "All_mapping Mapping.empty P"
```
```   582   by (auto simp: All_mapping_def lookup_empty)
```
```   583
```
```   584 lemma All_mapping_update_iff:
```
```   585   "All_mapping (Mapping.update k v m) P \<longleftrightarrow> P k v \<and> All_mapping m (\<lambda>k' v'. k = k' \<or> P k' v')"
```
```   586   unfolding All_mapping_def
```
```   587 proof safe
```
```   588   assume "\<forall>x. case Mapping.lookup (Mapping.update k v m) x of None \<Rightarrow> True | Some y \<Rightarrow> P x y"
```
```   589   then have *: "case Mapping.lookup (Mapping.update k v m) x of None \<Rightarrow> True | Some y \<Rightarrow> P x y" for x
```
```   590     by blast
```
```   591   from *[of k] show "P k v"
```
```   592     by (simp add: lookup_update)
```
```   593   show "case Mapping.lookup m x of None \<Rightarrow> True | Some v' \<Rightarrow> k = x \<or> P x v'" for x
```
```   594     using *[of x] by (auto simp add: lookup_update' split: if_splits option.splits)
```
```   595 next
```
```   596   assume "P k v"
```
```   597   assume "\<forall>x. case Mapping.lookup m x of None \<Rightarrow> True | Some v' \<Rightarrow> k = x \<or> P x v'"
```
```   598   then have A: "case Mapping.lookup m x of None \<Rightarrow> True | Some v' \<Rightarrow> k = x \<or> P x v'" for x
```
```   599     by blast
```
```   600   show "case Mapping.lookup (Mapping.update k v m) x of None \<Rightarrow> True | Some xa \<Rightarrow> P x xa" for x
```
```   601     using \<open>P k v\<close> A[of x] by (auto simp: lookup_update' split: option.splits)
```
```   602 qed
```
```   603
```
```   604 lemma All_mapping_update:
```
```   605   "P k v \<Longrightarrow> All_mapping m (\<lambda>k' v'. k = k' \<or> P k' v') \<Longrightarrow> All_mapping (Mapping.update k v m) P"
```
```   606   by (simp add: All_mapping_update_iff)
```
```   607
```
```   608 lemma All_mapping_filter_iff: "All_mapping (filter P m) Q \<longleftrightarrow> All_mapping m (\<lambda>k v. P k v \<longrightarrow> Q k v)"
```
```   609   by (auto simp: All_mapping_def lookup_filter split: option.splits)
```
```   610
```
```   611 lemma All_mapping_filter: "All_mapping m Q \<Longrightarrow> All_mapping (filter P m) Q"
```
```   612   by (auto simp: All_mapping_filter_iff intro: All_mapping_mono)
```
```   613
```
```   614 lemma All_mapping_map_values: "All_mapping (map_values f m) P \<longleftrightarrow> All_mapping m (\<lambda>k v. P k (f k v))"
```
```   615   by (auto simp: All_mapping_def lookup_map_values split: option.splits)
```
```   616
```
```   617 lemma All_mapping_tabulate: "(\<forall>x\<in>set xs. P x (f x)) \<Longrightarrow> All_mapping (Mapping.tabulate xs f) P"
```
```   618   unfolding All_mapping_def
```
```   619   apply (intro allI)
```
```   620   apply transfer
```
```   621   apply (auto split: option.split dest!: map_of_SomeD)
```
```   622   done
```
```   623
```
```   624 lemma All_mapping_alist:
```
```   625   "(\<And>k v. (k, v) \<in> set xs \<Longrightarrow> P k v) \<Longrightarrow> All_mapping (Mapping.of_alist xs) P"
```
```   626   by (auto simp: All_mapping_def lookup_of_alist dest!: map_of_SomeD split: option.splits)
```
```   627
```
```   628 lemma combine_empty [simp]: "combine f Mapping.empty y = y" "combine f y Mapping.empty = y"
```
```   629   by (transfer; force)+
```
```   630
```
```   631 lemma (in abel_semigroup) comm_monoid_set_combine: "comm_monoid_set (combine f) Mapping.empty"
```
```   632   by standard (transfer fixing: f, simp add: combine_options_ac[of f] ac_simps)+
```
```   633
```
```   634 locale combine_mapping_abel_semigroup = abel_semigroup
```
```   635 begin
```
```   636
```
```   637 sublocale combine: comm_monoid_set "combine f" Mapping.empty
```
```   638   by (rule comm_monoid_set_combine)
```
```   639
```
```   640 lemma fold_combine_code:
```
```   641   "combine.F g (set xs) = foldr (\<lambda>x. combine f (g x)) (remdups xs) Mapping.empty"
```
```   642 proof -
```
```   643   have "combine.F g (set xs) = foldr (\<lambda>x. combine f (g x)) xs Mapping.empty"
```
```   644     if "distinct xs" for xs
```
```   645     using that by (induction xs) simp_all
```
```   646   from this[of "remdups xs"] show ?thesis by simp
```
```   647 qed
```
```   648
```
```   649 lemma keys_fold_combine: "finite A \<Longrightarrow> Mapping.keys (combine.F g A) = (\<Union>x\<in>A. Mapping.keys (g x))"
```
```   650   by (induct A rule: finite_induct) simp_all
```
```   651
```
```   652 end
```
```   653
```
```   654
```
```   655 subsection \<open>Code generator setup\<close>
```
```   656
```
```   657 hide_const (open) empty is_empty rep lookup lookup_default filter update delete ordered_keys
```
```   658   keys size replace default map_entry map_default tabulate bulkload map map_values combine of_alist
```
```   659
```
```   660 end
```