src/HOL/Map.thy
 author paulson Tue Apr 25 16:39:54 2017 +0100 (2017-04-25) changeset 65578 e4997c181cce parent 63834 6a757f36997e child 66010 2f7d39285a1a permissions -rw-r--r--
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
```     1 (*  Title:      HOL/Map.thy
```
```     2     Author:     Tobias Nipkow, based on a theory by David von Oheimb
```
```     3     Copyright   1997-2003 TU Muenchen
```
```     4
```
```     5 The datatype of "maps"; strongly resembles maps in VDM.
```
```     6 *)
```
```     7
```
```     8 section \<open>Maps\<close>
```
```     9
```
```    10 theory Map
```
```    11 imports List
```
```    12 begin
```
```    13
```
```    14 type_synonym ('a, 'b) "map" = "'a \<Rightarrow> 'b option" (infixr "\<rightharpoonup>" 0)
```
```    15
```
```    16 abbreviation
```
```    17   empty :: "'a \<rightharpoonup> 'b" where
```
```    18   "empty \<equiv> \<lambda>x. None"
```
```    19
```
```    20 definition
```
```    21   map_comp :: "('b \<rightharpoonup> 'c) \<Rightarrow> ('a \<rightharpoonup> 'b) \<Rightarrow> ('a \<rightharpoonup> 'c)"  (infixl "\<circ>\<^sub>m" 55) where
```
```    22   "f \<circ>\<^sub>m g = (\<lambda>k. case g k of None \<Rightarrow> None | Some v \<Rightarrow> f v)"
```
```    23
```
```    24 definition
```
```    25   map_add :: "('a \<rightharpoonup> 'b) \<Rightarrow> ('a \<rightharpoonup> 'b) \<Rightarrow> ('a \<rightharpoonup> 'b)"  (infixl "++" 100) where
```
```    26   "m1 ++ m2 = (\<lambda>x. case m2 x of None \<Rightarrow> m1 x | Some y \<Rightarrow> Some y)"
```
```    27
```
```    28 definition
```
```    29   restrict_map :: "('a \<rightharpoonup> 'b) \<Rightarrow> 'a set \<Rightarrow> ('a \<rightharpoonup> 'b)"  (infixl "|`"  110) where
```
```    30   "m|`A = (\<lambda>x. if x \<in> A then m x else None)"
```
```    31
```
```    32 notation (latex output)
```
```    33   restrict_map  ("_\<restriction>\<^bsub>_\<^esub>" [111,110] 110)
```
```    34
```
```    35 definition
```
```    36   dom :: "('a \<rightharpoonup> 'b) \<Rightarrow> 'a set" where
```
```    37   "dom m = {a. m a \<noteq> None}"
```
```    38
```
```    39 definition
```
```    40   ran :: "('a \<rightharpoonup> 'b) \<Rightarrow> 'b set" where
```
```    41   "ran m = {b. \<exists>a. m a = Some b}"
```
```    42
```
```    43 definition
```
```    44   map_le :: "('a \<rightharpoonup> 'b) \<Rightarrow> ('a \<rightharpoonup> 'b) \<Rightarrow> bool"  (infix "\<subseteq>\<^sub>m" 50) where
```
```    45   "(m\<^sub>1 \<subseteq>\<^sub>m m\<^sub>2) \<longleftrightarrow> (\<forall>a \<in> dom m\<^sub>1. m\<^sub>1 a = m\<^sub>2 a)"
```
```    46
```
```    47 nonterminal maplets and maplet
```
```    48
```
```    49 syntax
```
```    50   "_maplet"  :: "['a, 'a] \<Rightarrow> maplet"             ("_ /\<mapsto>/ _")
```
```    51   "_maplets" :: "['a, 'a] \<Rightarrow> maplet"             ("_ /[\<mapsto>]/ _")
```
```    52   ""         :: "maplet \<Rightarrow> maplets"             ("_")
```
```    53   "_Maplets" :: "[maplet, maplets] \<Rightarrow> maplets" ("_,/ _")
```
```    54   "_MapUpd"  :: "['a \<rightharpoonup> 'b, maplets] \<Rightarrow> 'a \<rightharpoonup> 'b" ("_/'(_')" [900, 0] 900)
```
```    55   "_Map"     :: "maplets \<Rightarrow> 'a \<rightharpoonup> 'b"            ("(1[_])")
```
```    56
```
```    57 syntax (ASCII)
```
```    58   "_maplet"  :: "['a, 'a] \<Rightarrow> maplet"             ("_ /|->/ _")
```
```    59   "_maplets" :: "['a, 'a] \<Rightarrow> maplet"             ("_ /[|->]/ _")
```
```    60
```
```    61 translations
```
```    62   "_MapUpd m (_Maplets xy ms)"  \<rightleftharpoons> "_MapUpd (_MapUpd m xy) ms"
```
```    63   "_MapUpd m (_maplet  x y)"    \<rightleftharpoons> "m(x := CONST Some y)"
```
```    64   "_Map ms"                     \<rightleftharpoons> "_MapUpd (CONST empty) ms"
```
```    65   "_Map (_Maplets ms1 ms2)"     \<leftharpoondown> "_MapUpd (_Map ms1) ms2"
```
```    66   "_Maplets ms1 (_Maplets ms2 ms3)" \<leftharpoondown> "_Maplets (_Maplets ms1 ms2) ms3"
```
```    67
```
```    68 primrec map_of :: "('a \<times> 'b) list \<Rightarrow> 'a \<rightharpoonup> 'b"
```
```    69 where
```
```    70   "map_of [] = empty"
```
```    71 | "map_of (p # ps) = (map_of ps)(fst p \<mapsto> snd p)"
```
```    72
```
```    73 definition map_upds :: "('a \<rightharpoonup> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> 'a \<rightharpoonup> 'b"
```
```    74   where "map_upds m xs ys = m ++ map_of (rev (zip xs ys))"
```
```    75 translations
```
```    76   "_MapUpd m (_maplets x y)" \<rightleftharpoons> "CONST map_upds m x y"
```
```    77
```
```    78 lemma map_of_Cons_code [code]:
```
```    79   "map_of [] k = None"
```
```    80   "map_of ((l, v) # ps) k = (if l = k then Some v else map_of ps k)"
```
```    81   by simp_all
```
```    82
```
```    83
```
```    84 subsection \<open>@{term [source] empty}\<close>
```
```    85
```
```    86 lemma empty_upd_none [simp]: "empty(x := None) = empty"
```
```    87   by (rule ext) simp
```
```    88
```
```    89
```
```    90 subsection \<open>@{term [source] map_upd}\<close>
```
```    91
```
```    92 lemma map_upd_triv: "t k = Some x \<Longrightarrow> t(k\<mapsto>x) = t"
```
```    93   by (rule ext) simp
```
```    94
```
```    95 lemma map_upd_nonempty [simp]: "t(k\<mapsto>x) \<noteq> empty"
```
```    96 proof
```
```    97   assume "t(k \<mapsto> x) = empty"
```
```    98   then have "(t(k \<mapsto> x)) k = None" by simp
```
```    99   then show False by simp
```
```   100 qed
```
```   101
```
```   102 lemma map_upd_eqD1:
```
```   103   assumes "m(a\<mapsto>x) = n(a\<mapsto>y)"
```
```   104   shows "x = y"
```
```   105 proof -
```
```   106   from assms have "(m(a\<mapsto>x)) a = (n(a\<mapsto>y)) a" by simp
```
```   107   then show ?thesis by simp
```
```   108 qed
```
```   109
```
```   110 lemma map_upd_Some_unfold:
```
```   111   "((m(a\<mapsto>b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)"
```
```   112 by auto
```
```   113
```
```   114 lemma image_map_upd [simp]: "x \<notin> A \<Longrightarrow> m(x \<mapsto> y) ` A = m ` A"
```
```   115 by auto
```
```   116
```
```   117 lemma finite_range_updI: "finite (range f) \<Longrightarrow> finite (range (f(a\<mapsto>b)))"
```
```   118 unfolding image_def
```
```   119 apply (simp (no_asm_use) add:full_SetCompr_eq)
```
```   120 apply (rule finite_subset)
```
```   121  prefer 2 apply assumption
```
```   122 apply (auto)
```
```   123 done
```
```   124
```
```   125
```
```   126 subsection \<open>@{term [source] map_of}\<close>
```
```   127
```
```   128 lemma map_of_eq_None_iff:
```
```   129   "(map_of xys x = None) = (x \<notin> fst ` (set xys))"
```
```   130 by (induct xys) simp_all
```
```   131
```
```   132 lemma map_of_eq_Some_iff [simp]:
```
```   133   "distinct(map fst xys) \<Longrightarrow> (map_of xys x = Some y) = ((x,y) \<in> set xys)"
```
```   134 apply (induct xys)
```
```   135  apply simp
```
```   136 apply (auto simp: map_of_eq_None_iff [symmetric])
```
```   137 done
```
```   138
```
```   139 lemma Some_eq_map_of_iff [simp]:
```
```   140   "distinct(map fst xys) \<Longrightarrow> (Some y = map_of xys x) = ((x,y) \<in> set xys)"
```
```   141 by (auto simp del: map_of_eq_Some_iff simp: map_of_eq_Some_iff [symmetric])
```
```   142
```
```   143 lemma map_of_is_SomeI [simp]: "\<lbrakk> distinct(map fst xys); (x,y) \<in> set xys \<rbrakk>
```
```   144     \<Longrightarrow> map_of xys x = Some y"
```
```   145 apply (induct xys)
```
```   146  apply simp
```
```   147 apply force
```
```   148 done
```
```   149
```
```   150 lemma map_of_zip_is_None [simp]:
```
```   151   "length xs = length ys \<Longrightarrow> (map_of (zip xs ys) x = None) = (x \<notin> set xs)"
```
```   152 by (induct rule: list_induct2) simp_all
```
```   153
```
```   154 lemma map_of_zip_is_Some:
```
```   155   assumes "length xs = length ys"
```
```   156   shows "x \<in> set xs \<longleftrightarrow> (\<exists>y. map_of (zip xs ys) x = Some y)"
```
```   157 using assms by (induct rule: list_induct2) simp_all
```
```   158
```
```   159 lemma map_of_zip_upd:
```
```   160   fixes x :: 'a and xs :: "'a list" and ys zs :: "'b list"
```
```   161   assumes "length ys = length xs"
```
```   162     and "length zs = length xs"
```
```   163     and "x \<notin> set xs"
```
```   164     and "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)"
```
```   165   shows "map_of (zip xs ys) = map_of (zip xs zs)"
```
```   166 proof
```
```   167   fix x' :: 'a
```
```   168   show "map_of (zip xs ys) x' = map_of (zip xs zs) x'"
```
```   169   proof (cases "x = x'")
```
```   170     case True
```
```   171     from assms True map_of_zip_is_None [of xs ys x']
```
```   172       have "map_of (zip xs ys) x' = None" by simp
```
```   173     moreover from assms True map_of_zip_is_None [of xs zs x']
```
```   174       have "map_of (zip xs zs) x' = None" by simp
```
```   175     ultimately show ?thesis by simp
```
```   176   next
```
```   177     case False from assms
```
```   178       have "(map_of (zip xs ys)(x \<mapsto> y)) x' = (map_of (zip xs zs)(x \<mapsto> z)) x'" by auto
```
```   179     with False show ?thesis by simp
```
```   180   qed
```
```   181 qed
```
```   182
```
```   183 lemma map_of_zip_inject:
```
```   184   assumes "length ys = length xs"
```
```   185     and "length zs = length xs"
```
```   186     and dist: "distinct xs"
```
```   187     and map_of: "map_of (zip xs ys) = map_of (zip xs zs)"
```
```   188   shows "ys = zs"
```
```   189   using assms(1) assms(2)[symmetric]
```
```   190   using dist map_of
```
```   191 proof (induct ys xs zs rule: list_induct3)
```
```   192   case Nil show ?case by simp
```
```   193 next
```
```   194   case (Cons y ys x xs z zs)
```
```   195   from \<open>map_of (zip (x#xs) (y#ys)) = map_of (zip (x#xs) (z#zs))\<close>
```
```   196     have map_of: "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)" by simp
```
```   197   from Cons have "length ys = length xs" and "length zs = length xs"
```
```   198     and "x \<notin> set xs" by simp_all
```
```   199   then have "map_of (zip xs ys) = map_of (zip xs zs)" using map_of by (rule map_of_zip_upd)
```
```   200   with Cons.hyps \<open>distinct (x # xs)\<close> have "ys = zs" by simp
```
```   201   moreover from map_of have "y = z" by (rule map_upd_eqD1)
```
```   202   ultimately show ?case by simp
```
```   203 qed
```
```   204
```
```   205 lemma map_of_zip_map:
```
```   206   "map_of (zip xs (map f xs)) = (\<lambda>x. if x \<in> set xs then Some (f x) else None)"
```
```   207   by (induct xs) (simp_all add: fun_eq_iff)
```
```   208
```
```   209 lemma finite_range_map_of: "finite (range (map_of xys))"
```
```   210 apply (induct xys)
```
```   211  apply (simp_all add: image_constant)
```
```   212 apply (rule finite_subset)
```
```   213  prefer 2 apply assumption
```
```   214 apply auto
```
```   215 done
```
```   216
```
```   217 lemma map_of_SomeD: "map_of xs k = Some y \<Longrightarrow> (k, y) \<in> set xs"
```
```   218   by (induct xs) (auto split: if_splits)
```
```   219
```
```   220 lemma map_of_mapk_SomeI:
```
```   221   "inj f \<Longrightarrow> map_of t k = Some x \<Longrightarrow>
```
```   222    map_of (map (case_prod (\<lambda>k. Pair (f k))) t) (f k) = Some x"
```
```   223 by (induct t) (auto simp: inj_eq)
```
```   224
```
```   225 lemma weak_map_of_SomeI: "(k, x) \<in> set l \<Longrightarrow> \<exists>x. map_of l k = Some x"
```
```   226 by (induct l) auto
```
```   227
```
```   228 lemma map_of_filter_in:
```
```   229   "map_of xs k = Some z \<Longrightarrow> P k z \<Longrightarrow> map_of (filter (case_prod P) xs) k = Some z"
```
```   230 by (induct xs) auto
```
```   231
```
```   232 lemma map_of_map:
```
```   233   "map_of (map (\<lambda>(k, v). (k, f v)) xs) = map_option f \<circ> map_of xs"
```
```   234   by (induct xs) (auto simp: fun_eq_iff)
```
```   235
```
```   236 lemma dom_map_option:
```
```   237   "dom (\<lambda>k. map_option (f k) (m k)) = dom m"
```
```   238   by (simp add: dom_def)
```
```   239
```
```   240 lemma dom_map_option_comp [simp]:
```
```   241   "dom (map_option g \<circ> m) = dom m"
```
```   242   using dom_map_option [of "\<lambda>_. g" m] by (simp add: comp_def)
```
```   243
```
```   244
```
```   245 subsection \<open>@{const map_option} related\<close>
```
```   246
```
```   247 lemma map_option_o_empty [simp]: "map_option f o empty = empty"
```
```   248 by (rule ext) simp
```
```   249
```
```   250 lemma map_option_o_map_upd [simp]:
```
```   251   "map_option f o m(a\<mapsto>b) = (map_option f o m)(a\<mapsto>f b)"
```
```   252 by (rule ext) simp
```
```   253
```
```   254
```
```   255 subsection \<open>@{term [source] map_comp} related\<close>
```
```   256
```
```   257 lemma map_comp_empty [simp]:
```
```   258   "m \<circ>\<^sub>m empty = empty"
```
```   259   "empty \<circ>\<^sub>m m = empty"
```
```   260 by (auto simp: map_comp_def split: option.splits)
```
```   261
```
```   262 lemma map_comp_simps [simp]:
```
```   263   "m2 k = None \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = None"
```
```   264   "m2 k = Some k' \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = m1 k'"
```
```   265 by (auto simp: map_comp_def)
```
```   266
```
```   267 lemma map_comp_Some_iff:
```
```   268   "((m1 \<circ>\<^sub>m m2) k = Some v) = (\<exists>k'. m2 k = Some k' \<and> m1 k' = Some v)"
```
```   269 by (auto simp: map_comp_def split: option.splits)
```
```   270
```
```   271 lemma map_comp_None_iff:
```
```   272   "((m1 \<circ>\<^sub>m m2) k = None) = (m2 k = None \<or> (\<exists>k'. m2 k = Some k' \<and> m1 k' = None)) "
```
```   273 by (auto simp: map_comp_def split: option.splits)
```
```   274
```
```   275
```
```   276 subsection \<open>\<open>++\<close>\<close>
```
```   277
```
```   278 lemma map_add_empty[simp]: "m ++ empty = m"
```
```   279 by(simp add: map_add_def)
```
```   280
```
```   281 lemma empty_map_add[simp]: "empty ++ m = m"
```
```   282 by (rule ext) (simp add: map_add_def split: option.split)
```
```   283
```
```   284 lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3"
```
```   285 by (rule ext) (simp add: map_add_def split: option.split)
```
```   286
```
```   287 lemma map_add_Some_iff:
```
```   288   "((m ++ n) k = Some x) = (n k = Some x | n k = None & m k = Some x)"
```
```   289 by (simp add: map_add_def split: option.split)
```
```   290
```
```   291 lemma map_add_SomeD [dest!]:
```
```   292   "(m ++ n) k = Some x \<Longrightarrow> n k = Some x \<or> n k = None \<and> m k = Some x"
```
```   293 by (rule map_add_Some_iff [THEN iffD1])
```
```   294
```
```   295 lemma map_add_find_right [simp]: "n k = Some xx \<Longrightarrow> (m ++ n) k = Some xx"
```
```   296 by (subst map_add_Some_iff) fast
```
```   297
```
```   298 lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)"
```
```   299 by (simp add: map_add_def split: option.split)
```
```   300
```
```   301 lemma map_add_upd[simp]: "f ++ g(x\<mapsto>y) = (f ++ g)(x\<mapsto>y)"
```
```   302 by (rule ext) (simp add: map_add_def)
```
```   303
```
```   304 lemma map_add_upds[simp]: "m1 ++ (m2(xs[\<mapsto>]ys)) = (m1++m2)(xs[\<mapsto>]ys)"
```
```   305 by (simp add: map_upds_def)
```
```   306
```
```   307 lemma map_add_upd_left: "m\<notin>dom e2 \<Longrightarrow> e1(m \<mapsto> u1) ++ e2 = (e1 ++ e2)(m \<mapsto> u1)"
```
```   308 by (rule ext) (auto simp: map_add_def dom_def split: option.split)
```
```   309
```
```   310 lemma map_of_append[simp]: "map_of (xs @ ys) = map_of ys ++ map_of xs"
```
```   311 unfolding map_add_def
```
```   312 apply (induct xs)
```
```   313  apply simp
```
```   314 apply (rule ext)
```
```   315 apply (simp split: option.split)
```
```   316 done
```
```   317
```
```   318 lemma finite_range_map_of_map_add:
```
```   319   "finite (range f) \<Longrightarrow> finite (range (f ++ map_of l))"
```
```   320 apply (induct l)
```
```   321  apply (auto simp del: fun_upd_apply)
```
```   322 apply (erule finite_range_updI)
```
```   323 done
```
```   324
```
```   325 lemma inj_on_map_add_dom [iff]:
```
```   326   "inj_on (m ++ m') (dom m') = inj_on m' (dom m')"
```
```   327 by (fastforce simp: map_add_def dom_def inj_on_def split: option.splits)
```
```   328
```
```   329 lemma map_upds_fold_map_upd:
```
```   330   "m(ks[\<mapsto>]vs) = foldl (\<lambda>m (k, v). m(k \<mapsto> v)) m (zip ks vs)"
```
```   331 unfolding map_upds_def proof (rule sym, rule zip_obtain_same_length)
```
```   332   fix ks :: "'a list" and vs :: "'b list"
```
```   333   assume "length ks = length vs"
```
```   334   then show "foldl (\<lambda>m (k, v). m(k\<mapsto>v)) m (zip ks vs) = m ++ map_of (rev (zip ks vs))"
```
```   335     by(induct arbitrary: m rule: list_induct2) simp_all
```
```   336 qed
```
```   337
```
```   338 lemma map_add_map_of_foldr:
```
```   339   "m ++ map_of ps = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) ps m"
```
```   340   by (induct ps) (auto simp: fun_eq_iff map_add_def)
```
```   341
```
```   342
```
```   343 subsection \<open>@{term [source] restrict_map}\<close>
```
```   344
```
```   345 lemma restrict_map_to_empty [simp]: "m|`{} = empty"
```
```   346 by (simp add: restrict_map_def)
```
```   347
```
```   348 lemma restrict_map_insert: "f |` (insert a A) = (f |` A)(a := f a)"
```
```   349 by (auto simp: restrict_map_def)
```
```   350
```
```   351 lemma restrict_map_empty [simp]: "empty|`D = empty"
```
```   352 by (simp add: restrict_map_def)
```
```   353
```
```   354 lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m|`A) x = m x"
```
```   355 by (simp add: restrict_map_def)
```
```   356
```
```   357 lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m|`A) x = None"
```
```   358 by (simp add: restrict_map_def)
```
```   359
```
```   360 lemma ran_restrictD: "y \<in> ran (m|`A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y"
```
```   361 by (auto simp: restrict_map_def ran_def split: if_split_asm)
```
```   362
```
```   363 lemma dom_restrict [simp]: "dom (m|`A) = dom m \<inter> A"
```
```   364 by (auto simp: restrict_map_def dom_def split: if_split_asm)
```
```   365
```
```   366 lemma restrict_upd_same [simp]: "m(x\<mapsto>y)|`(-{x}) = m|`(-{x})"
```
```   367 by (rule ext) (auto simp: restrict_map_def)
```
```   368
```
```   369 lemma restrict_restrict [simp]: "m|`A|`B = m|`(A\<inter>B)"
```
```   370 by (rule ext) (auto simp: restrict_map_def)
```
```   371
```
```   372 lemma restrict_fun_upd [simp]:
```
```   373   "m(x := y)|`D = (if x \<in> D then (m|`(D-{x}))(x := y) else m|`D)"
```
```   374 by (simp add: restrict_map_def fun_eq_iff)
```
```   375
```
```   376 lemma fun_upd_None_restrict [simp]:
```
```   377   "(m|`D)(x := None) = (if x \<in> D then m|`(D - {x}) else m|`D)"
```
```   378 by (simp add: restrict_map_def fun_eq_iff)
```
```   379
```
```   380 lemma fun_upd_restrict: "(m|`D)(x := y) = (m|`(D-{x}))(x := y)"
```
```   381 by (simp add: restrict_map_def fun_eq_iff)
```
```   382
```
```   383 lemma fun_upd_restrict_conv [simp]:
```
```   384   "x \<in> D \<Longrightarrow> (m|`D)(x := y) = (m|`(D-{x}))(x := y)"
```
```   385 by (simp add: restrict_map_def fun_eq_iff)
```
```   386
```
```   387 lemma map_of_map_restrict:
```
```   388   "map_of (map (\<lambda>k. (k, f k)) ks) = (Some \<circ> f) |` set ks"
```
```   389   by (induct ks) (simp_all add: fun_eq_iff restrict_map_insert)
```
```   390
```
```   391 lemma restrict_complement_singleton_eq:
```
```   392   "f |` (- {x}) = f(x := None)"
```
```   393   by (simp add: restrict_map_def fun_eq_iff)
```
```   394
```
```   395
```
```   396 subsection \<open>@{term [source] map_upds}\<close>
```
```   397
```
```   398 lemma map_upds_Nil1 [simp]: "m([] [\<mapsto>] bs) = m"
```
```   399 by (simp add: map_upds_def)
```
```   400
```
```   401 lemma map_upds_Nil2 [simp]: "m(as [\<mapsto>] []) = m"
```
```   402 by (simp add:map_upds_def)
```
```   403
```
```   404 lemma map_upds_Cons [simp]: "m(a#as [\<mapsto>] b#bs) = (m(a\<mapsto>b))(as[\<mapsto>]bs)"
```
```   405 by (simp add:map_upds_def)
```
```   406
```
```   407 lemma map_upds_append1 [simp]: "size xs < size ys \<Longrightarrow>
```
```   408   m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)"
```
```   409 apply(induct xs arbitrary: ys m)
```
```   410  apply (clarsimp simp add: neq_Nil_conv)
```
```   411 apply (case_tac ys)
```
```   412  apply simp
```
```   413 apply simp
```
```   414 done
```
```   415
```
```   416 lemma map_upds_list_update2_drop [simp]:
```
```   417   "size xs \<le> i \<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)"
```
```   418 apply (induct xs arbitrary: m ys i)
```
```   419  apply simp
```
```   420 apply (case_tac ys)
```
```   421  apply simp
```
```   422 apply (simp split: nat.split)
```
```   423 done
```
```   424
```
```   425 lemma map_upd_upds_conv_if:
```
```   426   "(f(x\<mapsto>y))(xs [\<mapsto>] ys) =
```
```   427    (if x \<in> set(take (length ys) xs) then f(xs [\<mapsto>] ys)
```
```   428                                     else (f(xs [\<mapsto>] ys))(x\<mapsto>y))"
```
```   429 apply (induct xs arbitrary: x y ys f)
```
```   430  apply simp
```
```   431 apply (case_tac ys)
```
```   432  apply (auto split: if_split simp: fun_upd_twist)
```
```   433 done
```
```   434
```
```   435 lemma map_upds_twist [simp]:
```
```   436   "a \<notin> set as \<Longrightarrow> m(a\<mapsto>b)(as[\<mapsto>]bs) = m(as[\<mapsto>]bs)(a\<mapsto>b)"
```
```   437 using set_take_subset by (fastforce simp add: map_upd_upds_conv_if)
```
```   438
```
```   439 lemma map_upds_apply_nontin [simp]:
```
```   440   "x \<notin> set xs \<Longrightarrow> (f(xs[\<mapsto>]ys)) x = f x"
```
```   441 apply (induct xs arbitrary: ys)
```
```   442  apply simp
```
```   443 apply (case_tac ys)
```
```   444  apply (auto simp: map_upd_upds_conv_if)
```
```   445 done
```
```   446
```
```   447 lemma fun_upds_append_drop [simp]:
```
```   448   "size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)"
```
```   449 apply (induct xs arbitrary: m ys)
```
```   450  apply simp
```
```   451 apply (case_tac ys)
```
```   452  apply simp_all
```
```   453 done
```
```   454
```
```   455 lemma fun_upds_append2_drop [simp]:
```
```   456   "size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)"
```
```   457 apply (induct xs arbitrary: m ys)
```
```   458  apply simp
```
```   459 apply (case_tac ys)
```
```   460  apply simp_all
```
```   461 done
```
```   462
```
```   463
```
```   464 lemma restrict_map_upds[simp]:
```
```   465   "\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk>
```
```   466     \<Longrightarrow> m(xs [\<mapsto>] ys)|`D = (m|`(D - set xs))(xs [\<mapsto>] ys)"
```
```   467 apply (induct xs arbitrary: m ys)
```
```   468  apply simp
```
```   469 apply (case_tac ys)
```
```   470  apply simp
```
```   471 apply (simp add: Diff_insert [symmetric] insert_absorb)
```
```   472 apply (simp add: map_upd_upds_conv_if)
```
```   473 done
```
```   474
```
```   475
```
```   476 subsection \<open>@{term [source] dom}\<close>
```
```   477
```
```   478 lemma dom_eq_empty_conv [simp]: "dom f = {} \<longleftrightarrow> f = empty"
```
```   479   by (auto simp: dom_def)
```
```   480
```
```   481 lemma domI: "m a = Some b \<Longrightarrow> a \<in> dom m"
```
```   482   by (simp add: dom_def)
```
```   483 (* declare domI [intro]? *)
```
```   484
```
```   485 lemma domD: "a \<in> dom m \<Longrightarrow> \<exists>b. m a = Some b"
```
```   486   by (cases "m a") (auto simp add: dom_def)
```
```   487
```
```   488 lemma domIff [iff, simp del]: "a \<in> dom m \<longleftrightarrow> m a \<noteq> None"
```
```   489   by (simp add: dom_def)
```
```   490
```
```   491 lemma dom_empty [simp]: "dom empty = {}"
```
```   492   by (simp add: dom_def)
```
```   493
```
```   494 lemma dom_fun_upd [simp]:
```
```   495   "dom(f(x := y)) = (if y = None then dom f - {x} else insert x (dom f))"
```
```   496   by (auto simp: dom_def)
```
```   497
```
```   498 lemma dom_if:
```
```   499   "dom (\<lambda>x. if P x then f x else g x) = dom f \<inter> {x. P x} \<union> dom g \<inter> {x. \<not> P x}"
```
```   500   by (auto split: if_splits)
```
```   501
```
```   502 lemma dom_map_of_conv_image_fst:
```
```   503   "dom (map_of xys) = fst ` set xys"
```
```   504   by (induct xys) (auto simp add: dom_if)
```
```   505
```
```   506 lemma dom_map_of_zip [simp]: "length xs = length ys \<Longrightarrow> dom (map_of (zip xs ys)) = set xs"
```
```   507   by (induct rule: list_induct2) (auto simp: dom_if)
```
```   508
```
```   509 lemma finite_dom_map_of: "finite (dom (map_of l))"
```
```   510   by (induct l) (auto simp: dom_def insert_Collect [symmetric])
```
```   511
```
```   512 lemma dom_map_upds [simp]:
```
```   513   "dom(m(xs[\<mapsto>]ys)) = set(take (length ys) xs) \<union> dom m"
```
```   514 apply (induct xs arbitrary: m ys)
```
```   515  apply simp
```
```   516 apply (case_tac ys)
```
```   517  apply auto
```
```   518 done
```
```   519
```
```   520 lemma dom_map_add [simp]: "dom (m ++ n) = dom n \<union> dom m"
```
```   521   by (auto simp: dom_def)
```
```   522
```
```   523 lemma dom_override_on [simp]:
```
```   524   "dom (override_on f g A) =
```
```   525     (dom f  - {a. a \<in> A - dom g}) \<union> {a. a \<in> A \<inter> dom g}"
```
```   526   by (auto simp: dom_def override_on_def)
```
```   527
```
```   528 lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1 ++ m2 = m2 ++ m1"
```
```   529   by (rule ext) (force simp: map_add_def dom_def split: option.split)
```
```   530
```
```   531 lemma map_add_dom_app_simps:
```
```   532   "m \<in> dom l2 \<Longrightarrow> (l1 ++ l2) m = l2 m"
```
```   533   "m \<notin> dom l1 \<Longrightarrow> (l1 ++ l2) m = l2 m"
```
```   534   "m \<notin> dom l2 \<Longrightarrow> (l1 ++ l2) m = l1 m"
```
```   535   by (auto simp add: map_add_def split: option.split_asm)
```
```   536
```
```   537 lemma dom_const [simp]:
```
```   538   "dom (\<lambda>x. Some (f x)) = UNIV"
```
```   539   by auto
```
```   540
```
```   541 (* Due to John Matthews - could be rephrased with dom *)
```
```   542 lemma finite_map_freshness:
```
```   543   "finite (dom (f :: 'a \<rightharpoonup> 'b)) \<Longrightarrow> \<not> finite (UNIV :: 'a set) \<Longrightarrow>
```
```   544    \<exists>x. f x = None"
```
```   545   by (bestsimp dest: ex_new_if_finite)
```
```   546
```
```   547 lemma dom_minus:
```
```   548   "f x = None \<Longrightarrow> dom f - insert x A = dom f - A"
```
```   549   unfolding dom_def by simp
```
```   550
```
```   551 lemma insert_dom:
```
```   552   "f x = Some y \<Longrightarrow> insert x (dom f) = dom f"
```
```   553   unfolding dom_def by auto
```
```   554
```
```   555 lemma map_of_map_keys:
```
```   556   "set xs = dom m \<Longrightarrow> map_of (map (\<lambda>k. (k, the (m k))) xs) = m"
```
```   557   by (rule ext) (auto simp add: map_of_map_restrict restrict_map_def)
```
```   558
```
```   559 lemma map_of_eqI:
```
```   560   assumes set_eq: "set (map fst xs) = set (map fst ys)"
```
```   561   assumes map_eq: "\<forall>k\<in>set (map fst xs). map_of xs k = map_of ys k"
```
```   562   shows "map_of xs = map_of ys"
```
```   563 proof (rule ext)
```
```   564   fix k show "map_of xs k = map_of ys k"
```
```   565   proof (cases "map_of xs k")
```
```   566     case None
```
```   567     then have "k \<notin> set (map fst xs)" by (simp add: map_of_eq_None_iff)
```
```   568     with set_eq have "k \<notin> set (map fst ys)" by simp
```
```   569     then have "map_of ys k = None" by (simp add: map_of_eq_None_iff)
```
```   570     with None show ?thesis by simp
```
```   571   next
```
```   572     case (Some v)
```
```   573     then have "k \<in> set (map fst xs)" by (auto simp add: dom_map_of_conv_image_fst [symmetric])
```
```   574     with map_eq show ?thesis by auto
```
```   575   qed
```
```   576 qed
```
```   577
```
```   578 lemma map_of_eq_dom:
```
```   579   assumes "map_of xs = map_of ys"
```
```   580   shows "fst ` set xs = fst ` set ys"
```
```   581 proof -
```
```   582   from assms have "dom (map_of xs) = dom (map_of ys)" by simp
```
```   583   then show ?thesis by (simp add: dom_map_of_conv_image_fst)
```
```   584 qed
```
```   585
```
```   586 lemma finite_set_of_finite_maps:
```
```   587   assumes "finite A" "finite B"
```
```   588   shows "finite {m. dom m = A \<and> ran m \<subseteq> B}" (is "finite ?S")
```
```   589 proof -
```
```   590   let ?S' = "{m. \<forall>x. (x \<in> A \<longrightarrow> m x \<in> Some ` B) \<and> (x \<notin> A \<longrightarrow> m x = None)}"
```
```   591   have "?S = ?S'"
```
```   592   proof
```
```   593     show "?S \<subseteq> ?S'" by (auto simp: dom_def ran_def image_def)
```
```   594     show "?S' \<subseteq> ?S"
```
```   595     proof
```
```   596       fix m assume "m \<in> ?S'"
```
```   597       hence 1: "dom m = A" by force
```
```   598       hence 2: "ran m \<subseteq> B" using \<open>m \<in> ?S'\<close> by (auto simp: dom_def ran_def)
```
```   599       from 1 2 show "m \<in> ?S" by blast
```
```   600     qed
```
```   601   qed
```
```   602   with assms show ?thesis by(simp add: finite_set_of_finite_funs)
```
```   603 qed
```
```   604
```
```   605
```
```   606 subsection \<open>@{term [source] ran}\<close>
```
```   607
```
```   608 lemma ranI: "m a = Some b \<Longrightarrow> b \<in> ran m"
```
```   609   by (auto simp: ran_def)
```
```   610 (* declare ranI [intro]? *)
```
```   611
```
```   612 lemma ran_empty [simp]: "ran empty = {}"
```
```   613   by (auto simp: ran_def)
```
```   614
```
```   615 lemma ran_map_upd [simp]: "m a = None \<Longrightarrow> ran(m(a\<mapsto>b)) = insert b (ran m)"
```
```   616   unfolding ran_def
```
```   617 apply auto
```
```   618 apply (subgoal_tac "aa \<noteq> a")
```
```   619  apply auto
```
```   620 done
```
```   621
```
```   622 lemma ran_distinct:
```
```   623   assumes dist: "distinct (map fst al)"
```
```   624   shows "ran (map_of al) = snd ` set al"
```
```   625   using assms
```
```   626 proof (induct al)
```
```   627   case Nil
```
```   628   then show ?case by simp
```
```   629 next
```
```   630   case (Cons kv al)
```
```   631   then have "ran (map_of al) = snd ` set al" by simp
```
```   632   moreover from Cons.prems have "map_of al (fst kv) = None"
```
```   633     by (simp add: map_of_eq_None_iff)
```
```   634   ultimately show ?case by (simp only: map_of.simps ran_map_upd) simp
```
```   635 qed
```
```   636
```
```   637 lemma ran_map_option: "ran (\<lambda>x. map_option f (m x)) = f ` ran m"
```
```   638   by (auto simp add: ran_def)
```
```   639
```
```   640
```
```   641 subsection \<open>\<open>map_le\<close>\<close>
```
```   642
```
```   643 lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g"
```
```   644   by (simp add: map_le_def)
```
```   645
```
```   646 lemma upd_None_map_le [simp]: "f(x := None) \<subseteq>\<^sub>m f"
```
```   647   by (force simp add: map_le_def)
```
```   648
```
```   649 lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)"
```
```   650   by (fastforce simp add: map_le_def)
```
```   651
```
```   652 lemma map_le_imp_upd_le [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)"
```
```   653   by (force simp add: map_le_def)
```
```   654
```
```   655 lemma map_le_upds [simp]:
```
```   656   "f \<subseteq>\<^sub>m g \<Longrightarrow> f(as [\<mapsto>] bs) \<subseteq>\<^sub>m g(as [\<mapsto>] bs)"
```
```   657 apply (induct as arbitrary: f g bs)
```
```   658  apply simp
```
```   659 apply (case_tac bs)
```
```   660  apply auto
```
```   661 done
```
```   662
```
```   663 lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)"
```
```   664   by (fastforce simp add: map_le_def dom_def)
```
```   665
```
```   666 lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f"
```
```   667   by (simp add: map_le_def)
```
```   668
```
```   669 lemma map_le_trans[trans]: "\<lbrakk> m1 \<subseteq>\<^sub>m m2; m2 \<subseteq>\<^sub>m m3\<rbrakk> \<Longrightarrow> m1 \<subseteq>\<^sub>m m3"
```
```   670   by (auto simp add: map_le_def dom_def)
```
```   671
```
```   672 lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g"
```
```   673 unfolding map_le_def
```
```   674 apply (rule ext)
```
```   675 apply (case_tac "x \<in> dom f", simp)
```
```   676 apply (case_tac "x \<in> dom g", simp, fastforce)
```
```   677 done
```
```   678
```
```   679 lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m g ++ f"
```
```   680   by (fastforce simp: map_le_def)
```
```   681
```
```   682 lemma map_le_iff_map_add_commute: "f \<subseteq>\<^sub>m f ++ g \<longleftrightarrow> f ++ g = g ++ f"
```
```   683   by (fastforce simp: map_add_def map_le_def fun_eq_iff split: option.splits)
```
```   684
```
```   685 lemma map_add_le_mapE: "f ++ g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h"
```
```   686   by (fastforce simp: map_le_def map_add_def dom_def)
```
```   687
```
```   688 lemma map_add_le_mapI: "\<lbrakk> f \<subseteq>\<^sub>m h; g \<subseteq>\<^sub>m h \<rbrakk> \<Longrightarrow> f ++ g \<subseteq>\<^sub>m h"
```
```   689   by (auto simp: map_le_def map_add_def dom_def split: option.splits)
```
```   690
```
```   691 lemma map_add_subsumed1: "f \<subseteq>\<^sub>m g \<Longrightarrow> f++g = g"
```
```   692 by (simp add: map_add_le_mapI map_le_antisym)
```
```   693
```
```   694 lemma map_add_subsumed2: "f \<subseteq>\<^sub>m g \<Longrightarrow> g++f = g"
```
```   695 by (metis map_add_subsumed1 map_le_iff_map_add_commute)
```
```   696
```
```   697 lemma dom_eq_singleton_conv: "dom f = {x} \<longleftrightarrow> (\<exists>v. f = [x \<mapsto> v])"
```
```   698   (is "?lhs \<longleftrightarrow> ?rhs")
```
```   699 proof
```
```   700   assume ?rhs
```
```   701   then show ?lhs by (auto split: if_split_asm)
```
```   702 next
```
```   703   assume ?lhs
```
```   704   then obtain v where v: "f x = Some v" by auto
```
```   705   show ?rhs
```
```   706   proof
```
```   707     show "f = [x \<mapsto> v]"
```
```   708     proof (rule map_le_antisym)
```
```   709       show "[x \<mapsto> v] \<subseteq>\<^sub>m f"
```
```   710         using v by (auto simp add: map_le_def)
```
```   711       show "f \<subseteq>\<^sub>m [x \<mapsto> v]"
```
```   712         using \<open>dom f = {x}\<close> \<open>f x = Some v\<close> by (auto simp add: map_le_def)
```
```   713     qed
```
```   714   qed
```
```   715 qed
```
```   716
```
```   717
```
```   718 subsection \<open>Various\<close>
```
```   719
```
```   720 lemma set_map_of_compr:
```
```   721   assumes distinct: "distinct (map fst xs)"
```
```   722   shows "set xs = {(k, v). map_of xs k = Some v}"
```
```   723   using assms
```
```   724 proof (induct xs)
```
```   725   case Nil
```
```   726   then show ?case by simp
```
```   727 next
```
```   728   case (Cons x xs)
```
```   729   obtain k v where "x = (k, v)" by (cases x) blast
```
```   730   with Cons.prems have "k \<notin> dom (map_of xs)"
```
```   731     by (simp add: dom_map_of_conv_image_fst)
```
```   732   then have *: "insert (k, v) {(k, v). map_of xs k = Some v} =
```
```   733     {(k', v'). (map_of xs(k \<mapsto> v)) k' = Some v'}"
```
```   734     by (auto split: if_splits)
```
```   735   from Cons have "set xs = {(k, v). map_of xs k = Some v}" by simp
```
```   736   with * \<open>x = (k, v)\<close> show ?case by simp
```
```   737 qed
```
```   738
```
```   739 lemma map_of_inject_set:
```
```   740   assumes distinct: "distinct (map fst xs)" "distinct (map fst ys)"
```
```   741   shows "map_of xs = map_of ys \<longleftrightarrow> set xs = set ys" (is "?lhs \<longleftrightarrow> ?rhs")
```
```   742 proof
```
```   743   assume ?lhs
```
```   744   moreover from \<open>distinct (map fst xs)\<close> have "set xs = {(k, v). map_of xs k = Some v}"
```
```   745     by (rule set_map_of_compr)
```
```   746   moreover from \<open>distinct (map fst ys)\<close> have "set ys = {(k, v). map_of ys k = Some v}"
```
```   747     by (rule set_map_of_compr)
```
```   748   ultimately show ?rhs by simp
```
```   749 next
```
```   750   assume ?rhs show ?lhs
```
```   751   proof
```
```   752     fix k
```
```   753     show "map_of xs k = map_of ys k"
```
```   754     proof (cases "map_of xs k")
```
```   755       case None
```
```   756       with \<open>?rhs\<close> have "map_of ys k = None"
```
```   757         by (simp add: map_of_eq_None_iff)
```
```   758       with None show ?thesis by simp
```
```   759     next
```
```   760       case (Some v)
```
```   761       with distinct \<open>?rhs\<close> have "map_of ys k = Some v"
```
```   762         by simp
```
```   763       with Some show ?thesis by simp
```
```   764     qed
```
```   765   qed
```
```   766 qed
```
```   767
```
```   768 end
```