src/HOL/Map.thy
 author hoelzl Tue Jan 18 21:37:23 2011 +0100 (2011-01-18) changeset 41654 32fe42892983 parent 41229 d797baa3d57c child 41550 efa734d9b221 permissions -rw-r--r--
Gauge measure removed
```     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' (written ~=>); strongly resembles maps in VDM.
```
```     6 *)
```
```     7
```
```     8 header {* Maps *}
```
```     9
```
```    10 theory Map
```
```    11 imports List
```
```    12 begin
```
```    13
```
```    14 types ('a,'b) "map" = "'a => 'b option" (infixr "~=>" 0)
```
```    15
```
```    16 type_notation (xsymbols)
```
```    17   "map" (infixr "\<rightharpoonup>" 0)
```
```    18
```
```    19 abbreviation
```
```    20   empty :: "'a ~=> 'b" where
```
```    21   "empty == %x. None"
```
```    22
```
```    23 definition
```
```    24   map_comp :: "('b ~=> 'c) => ('a ~=> 'b) => ('a ~=> 'c)"  (infixl "o'_m" 55) where
```
```    25   "f o_m g = (\<lambda>k. case g k of None \<Rightarrow> None | Some v \<Rightarrow> f v)"
```
```    26
```
```    27 notation (xsymbols)
```
```    28   map_comp  (infixl "\<circ>\<^sub>m" 55)
```
```    29
```
```    30 definition
```
```    31   map_add :: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)"  (infixl "++" 100) where
```
```    32   "m1 ++ m2 = (\<lambda>x. case m2 x of None => m1 x | Some y => Some y)"
```
```    33
```
```    34 definition
```
```    35   restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)"  (infixl "|`"  110) where
```
```    36   "m|`A = (\<lambda>x. if x : A then m x else None)"
```
```    37
```
```    38 notation (latex output)
```
```    39   restrict_map  ("_\<restriction>\<^bsub>_\<^esub>" [111,110] 110)
```
```    40
```
```    41 definition
```
```    42   dom :: "('a ~=> 'b) => 'a set" where
```
```    43   "dom m = {a. m a ~= None}"
```
```    44
```
```    45 definition
```
```    46   ran :: "('a ~=> 'b) => 'b set" where
```
```    47   "ran m = {b. EX a. m a = Some b}"
```
```    48
```
```    49 definition
```
```    50   map_le :: "('a ~=> 'b) => ('a ~=> 'b) => bool"  (infix "\<subseteq>\<^sub>m" 50) where
```
```    51   "(m\<^isub>1 \<subseteq>\<^sub>m m\<^isub>2) = (\<forall>a \<in> dom m\<^isub>1. m\<^isub>1 a = m\<^isub>2 a)"
```
```    52
```
```    53 nonterminal maplets and maplet
```
```    54
```
```    55 syntax
```
```    56   "_maplet"  :: "['a, 'a] => maplet"             ("_ /|->/ _")
```
```    57   "_maplets" :: "['a, 'a] => maplet"             ("_ /[|->]/ _")
```
```    58   ""         :: "maplet => maplets"             ("_")
```
```    59   "_Maplets" :: "[maplet, maplets] => maplets" ("_,/ _")
```
```    60   "_MapUpd"  :: "['a ~=> 'b, maplets] => 'a ~=> 'b" ("_/'(_')" [900,0]900)
```
```    61   "_Map"     :: "maplets => 'a ~=> 'b"            ("(1[_])")
```
```    62
```
```    63 syntax (xsymbols)
```
```    64   "_maplet"  :: "['a, 'a] => maplet"             ("_ /\<mapsto>/ _")
```
```    65   "_maplets" :: "['a, 'a] => maplet"             ("_ /[\<mapsto>]/ _")
```
```    66
```
```    67 translations
```
```    68   "_MapUpd m (_Maplets xy ms)"  == "_MapUpd (_MapUpd m xy) ms"
```
```    69   "_MapUpd m (_maplet  x y)"    == "m(x := CONST Some y)"
```
```    70   "_Map ms"                     == "_MapUpd (CONST empty) ms"
```
```    71   "_Map (_Maplets ms1 ms2)"     <= "_MapUpd (_Map ms1) ms2"
```
```    72   "_Maplets ms1 (_Maplets ms2 ms3)" <= "_Maplets (_Maplets ms1 ms2) ms3"
```
```    73
```
```    74 primrec
```
```    75   map_of :: "('a \<times> 'b) list \<Rightarrow> 'a \<rightharpoonup> 'b" where
```
```    76     "map_of [] = empty"
```
```    77   | "map_of (p # ps) = (map_of ps)(fst p \<mapsto> snd p)"
```
```    78
```
```    79 definition
```
```    80   map_upds :: "('a \<rightharpoonup> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> 'a \<rightharpoonup> 'b" where
```
```    81   "map_upds m xs ys = m ++ map_of (rev (zip xs ys))"
```
```    82
```
```    83 translations
```
```    84   "_MapUpd m (_maplets x y)"    == "CONST map_upds m x y"
```
```    85
```
```    86 lemma map_of_Cons_code [code]:
```
```    87   "map_of [] k = None"
```
```    88   "map_of ((l, v) # ps) k = (if l = k then Some v else map_of ps k)"
```
```    89   by simp_all
```
```    90
```
```    91
```
```    92 subsection {* @{term [source] empty} *}
```
```    93
```
```    94 lemma empty_upd_none [simp]: "empty(x := None) = empty"
```
```    95 by (rule ext) simp
```
```    96
```
```    97
```
```    98 subsection {* @{term [source] map_upd} *}
```
```    99
```
```   100 lemma map_upd_triv: "t k = Some x ==> t(k|->x) = t"
```
```   101 by (rule ext) simp
```
```   102
```
```   103 lemma map_upd_nonempty [simp]: "t(k|->x) ~= empty"
```
```   104 proof
```
```   105   assume "t(k \<mapsto> x) = empty"
```
```   106   then have "(t(k \<mapsto> x)) k = None" by simp
```
```   107   then show False by simp
```
```   108 qed
```
```   109
```
```   110 lemma map_upd_eqD1:
```
```   111   assumes "m(a\<mapsto>x) = n(a\<mapsto>y)"
```
```   112   shows "x = y"
```
```   113 proof -
```
```   114   from prems have "(m(a\<mapsto>x)) a = (n(a\<mapsto>y)) a" by simp
```
```   115   then show ?thesis by simp
```
```   116 qed
```
```   117
```
```   118 lemma map_upd_Some_unfold:
```
```   119   "((m(a|->b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)"
```
```   120 by auto
```
```   121
```
```   122 lemma image_map_upd [simp]: "x \<notin> A \<Longrightarrow> m(x \<mapsto> y) ` A = m ` A"
```
```   123 by auto
```
```   124
```
```   125 lemma finite_range_updI: "finite (range f) ==> finite (range (f(a|->b)))"
```
```   126 unfolding image_def
```
```   127 apply (simp (no_asm_use) add:full_SetCompr_eq)
```
```   128 apply (rule finite_subset)
```
```   129  prefer 2 apply assumption
```
```   130 apply (auto)
```
```   131 done
```
```   132
```
```   133
```
```   134 subsection {* @{term [source] map_of} *}
```
```   135
```
```   136 lemma map_of_eq_None_iff:
```
```   137   "(map_of xys x = None) = (x \<notin> fst ` (set xys))"
```
```   138 by (induct xys) simp_all
```
```   139
```
```   140 lemma map_of_is_SomeD: "map_of xys x = Some y \<Longrightarrow> (x,y) \<in> set xys"
```
```   141 apply (induct xys)
```
```   142  apply simp
```
```   143 apply (clarsimp split: if_splits)
```
```   144 done
```
```   145
```
```   146 lemma map_of_eq_Some_iff [simp]:
```
```   147   "distinct(map fst xys) \<Longrightarrow> (map_of xys x = Some y) = ((x,y) \<in> set xys)"
```
```   148 apply (induct xys)
```
```   149  apply simp
```
```   150 apply (auto simp: map_of_eq_None_iff [symmetric])
```
```   151 done
```
```   152
```
```   153 lemma Some_eq_map_of_iff [simp]:
```
```   154   "distinct(map fst xys) \<Longrightarrow> (Some y = map_of xys x) = ((x,y) \<in> set xys)"
```
```   155 by (auto simp del:map_of_eq_Some_iff simp add: map_of_eq_Some_iff [symmetric])
```
```   156
```
```   157 lemma map_of_is_SomeI [simp]: "\<lbrakk> distinct(map fst xys); (x,y) \<in> set xys \<rbrakk>
```
```   158     \<Longrightarrow> map_of xys x = Some y"
```
```   159 apply (induct xys)
```
```   160  apply simp
```
```   161 apply force
```
```   162 done
```
```   163
```
```   164 lemma map_of_zip_is_None [simp]:
```
```   165   "length xs = length ys \<Longrightarrow> (map_of (zip xs ys) x = None) = (x \<notin> set xs)"
```
```   166 by (induct rule: list_induct2) simp_all
```
```   167
```
```   168 lemma map_of_zip_is_Some:
```
```   169   assumes "length xs = length ys"
```
```   170   shows "x \<in> set xs \<longleftrightarrow> (\<exists>y. map_of (zip xs ys) x = Some y)"
```
```   171 using assms by (induct rule: list_induct2) simp_all
```
```   172
```
```   173 lemma map_of_zip_upd:
```
```   174   fixes x :: 'a and xs :: "'a list" and ys zs :: "'b list"
```
```   175   assumes "length ys = length xs"
```
```   176     and "length zs = length xs"
```
```   177     and "x \<notin> set xs"
```
```   178     and "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)"
```
```   179   shows "map_of (zip xs ys) = map_of (zip xs zs)"
```
```   180 proof
```
```   181   fix x' :: 'a
```
```   182   show "map_of (zip xs ys) x' = map_of (zip xs zs) x'"
```
```   183   proof (cases "x = x'")
```
```   184     case True
```
```   185     from assms True map_of_zip_is_None [of xs ys x']
```
```   186       have "map_of (zip xs ys) x' = None" by simp
```
```   187     moreover from assms True map_of_zip_is_None [of xs zs x']
```
```   188       have "map_of (zip xs zs) x' = None" by simp
```
```   189     ultimately show ?thesis by simp
```
```   190   next
```
```   191     case False from assms
```
```   192       have "(map_of (zip xs ys)(x \<mapsto> y)) x' = (map_of (zip xs zs)(x \<mapsto> z)) x'" by auto
```
```   193     with False show ?thesis by simp
```
```   194   qed
```
```   195 qed
```
```   196
```
```   197 lemma map_of_zip_inject:
```
```   198   assumes "length ys = length xs"
```
```   199     and "length zs = length xs"
```
```   200     and dist: "distinct xs"
```
```   201     and map_of: "map_of (zip xs ys) = map_of (zip xs zs)"
```
```   202   shows "ys = zs"
```
```   203 using assms(1) assms(2)[symmetric] using dist map_of proof (induct ys xs zs rule: list_induct3)
```
```   204   case Nil show ?case by simp
```
```   205 next
```
```   206   case (Cons y ys x xs z zs)
```
```   207   from `map_of (zip (x#xs) (y#ys)) = map_of (zip (x#xs) (z#zs))`
```
```   208     have map_of: "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)" by simp
```
```   209   from Cons have "length ys = length xs" and "length zs = length xs"
```
```   210     and "x \<notin> set xs" by simp_all
```
```   211   then have "map_of (zip xs ys) = map_of (zip xs zs)" using map_of by (rule map_of_zip_upd)
```
```   212   with Cons.hyps `distinct (x # xs)` have "ys = zs" by simp
```
```   213   moreover from map_of have "y = z" by (rule map_upd_eqD1)
```
```   214   ultimately show ?case by simp
```
```   215 qed
```
```   216
```
```   217 lemma map_of_zip_map:
```
```   218   "map_of (zip xs (map f xs)) = (\<lambda>x. if x \<in> set xs then Some (f x) else None)"
```
```   219   by (induct xs) (simp_all add: fun_eq_iff)
```
```   220
```
```   221 lemma finite_range_map_of: "finite (range (map_of xys))"
```
```   222 apply (induct xys)
```
```   223  apply (simp_all add: image_constant)
```
```   224 apply (rule finite_subset)
```
```   225  prefer 2 apply assumption
```
```   226 apply auto
```
```   227 done
```
```   228
```
```   229 lemma map_of_SomeD: "map_of xs k = Some y \<Longrightarrow> (k, y) \<in> set xs"
```
```   230 by (induct xs) (simp, atomize (full), auto)
```
```   231
```
```   232 lemma map_of_mapk_SomeI:
```
```   233   "inj f ==> map_of t k = Some x ==>
```
```   234    map_of (map (split (%k. Pair (f k))) t) (f k) = Some x"
```
```   235 by (induct t) (auto simp add: inj_eq)
```
```   236
```
```   237 lemma weak_map_of_SomeI: "(k, x) : set l ==> \<exists>x. map_of l k = Some x"
```
```   238 by (induct l) auto
```
```   239
```
```   240 lemma map_of_filter_in:
```
```   241   "map_of xs k = Some z \<Longrightarrow> P k z \<Longrightarrow> map_of (filter (split P) xs) k = Some z"
```
```   242 by (induct xs) auto
```
```   243
```
```   244 lemma map_of_map:
```
```   245   "map_of (map (\<lambda>(k, v). (k, f v)) xs) = Option.map f \<circ> map_of xs"
```
```   246   by (induct xs) (auto simp add: fun_eq_iff)
```
```   247
```
```   248 lemma dom_option_map:
```
```   249   "dom (\<lambda>k. Option.map (f k) (m k)) = dom m"
```
```   250   by (simp add: dom_def)
```
```   251
```
```   252
```
```   253 subsection {* @{const Option.map} related *}
```
```   254
```
```   255 lemma option_map_o_empty [simp]: "Option.map f o empty = empty"
```
```   256 by (rule ext) simp
```
```   257
```
```   258 lemma option_map_o_map_upd [simp]:
```
```   259   "Option.map f o m(a|->b) = (Option.map f o m)(a|->f b)"
```
```   260 by (rule ext) simp
```
```   261
```
```   262
```
```   263 subsection {* @{term [source] map_comp} related *}
```
```   264
```
```   265 lemma map_comp_empty [simp]:
```
```   266   "m \<circ>\<^sub>m empty = empty"
```
```   267   "empty \<circ>\<^sub>m m = empty"
```
```   268 by (auto simp add: map_comp_def intro: ext split: option.splits)
```
```   269
```
```   270 lemma map_comp_simps [simp]:
```
```   271   "m2 k = None \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = None"
```
```   272   "m2 k = Some k' \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = m1 k'"
```
```   273 by (auto simp add: map_comp_def)
```
```   274
```
```   275 lemma map_comp_Some_iff:
```
```   276   "((m1 \<circ>\<^sub>m m2) k = Some v) = (\<exists>k'. m2 k = Some k' \<and> m1 k' = Some v)"
```
```   277 by (auto simp add: map_comp_def split: option.splits)
```
```   278
```
```   279 lemma map_comp_None_iff:
```
```   280   "((m1 \<circ>\<^sub>m m2) k = None) = (m2 k = None \<or> (\<exists>k'. m2 k = Some k' \<and> m1 k' = None)) "
```
```   281 by (auto simp add: map_comp_def split: option.splits)
```
```   282
```
```   283
```
```   284 subsection {* @{text "++"} *}
```
```   285
```
```   286 lemma map_add_empty[simp]: "m ++ empty = m"
```
```   287 by(simp add: map_add_def)
```
```   288
```
```   289 lemma empty_map_add[simp]: "empty ++ m = m"
```
```   290 by (rule ext) (simp add: map_add_def split: option.split)
```
```   291
```
```   292 lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3"
```
```   293 by (rule ext) (simp add: map_add_def split: option.split)
```
```   294
```
```   295 lemma map_add_Some_iff:
```
```   296   "((m ++ n) k = Some x) = (n k = Some x | n k = None & m k = Some x)"
```
```   297 by (simp add: map_add_def split: option.split)
```
```   298
```
```   299 lemma map_add_SomeD [dest!]:
```
```   300   "(m ++ n) k = Some x \<Longrightarrow> n k = Some x \<or> n k = None \<and> m k = Some x"
```
```   301 by (rule map_add_Some_iff [THEN iffD1])
```
```   302
```
```   303 lemma map_add_find_right [simp]: "!!xx. n k = Some xx ==> (m ++ n) k = Some xx"
```
```   304 by (subst map_add_Some_iff) fast
```
```   305
```
```   306 lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)"
```
```   307 by (simp add: map_add_def split: option.split)
```
```   308
```
```   309 lemma map_add_upd[simp]: "f ++ g(x|->y) = (f ++ g)(x|->y)"
```
```   310 by (rule ext) (simp add: map_add_def)
```
```   311
```
```   312 lemma map_add_upds[simp]: "m1 ++ (m2(xs[\<mapsto>]ys)) = (m1++m2)(xs[\<mapsto>]ys)"
```
```   313 by (simp add: map_upds_def)
```
```   314
```
```   315 lemma map_add_upd_left: "m\<notin>dom e2 \<Longrightarrow> e1(m \<mapsto> u1) ++ e2 = (e1 ++ e2)(m \<mapsto> u1)"
```
```   316 by (rule ext) (auto simp: map_add_def dom_def split: option.split)
```
```   317
```
```   318 lemma map_of_append[simp]: "map_of (xs @ ys) = map_of ys ++ map_of xs"
```
```   319 unfolding map_add_def
```
```   320 apply (induct xs)
```
```   321  apply simp
```
```   322 apply (rule ext)
```
```   323 apply (simp split add: option.split)
```
```   324 done
```
```   325
```
```   326 lemma finite_range_map_of_map_add:
```
```   327   "finite (range f) ==> finite (range (f ++ map_of l))"
```
```   328 apply (induct l)
```
```   329  apply (auto simp del: fun_upd_apply)
```
```   330 apply (erule finite_range_updI)
```
```   331 done
```
```   332
```
```   333 lemma inj_on_map_add_dom [iff]:
```
```   334   "inj_on (m ++ m') (dom m') = inj_on m' (dom m')"
```
```   335 by (fastsimp simp: map_add_def dom_def inj_on_def split: option.splits)
```
```   336
```
```   337 lemma map_upds_fold_map_upd:
```
```   338   "m(ks[\<mapsto>]vs) = foldl (\<lambda>m (k, v). m(k \<mapsto> v)) m (zip ks vs)"
```
```   339 unfolding map_upds_def proof (rule sym, rule zip_obtain_same_length)
```
```   340   fix ks :: "'a list" and vs :: "'b list"
```
```   341   assume "length ks = length vs"
```
```   342   then show "foldl (\<lambda>m (k, v). m(k\<mapsto>v)) m (zip ks vs) = m ++ map_of (rev (zip ks vs))"
```
```   343     by(induct arbitrary: m rule: list_induct2) simp_all
```
```   344 qed
```
```   345
```
```   346 lemma map_add_map_of_foldr:
```
```   347   "m ++ map_of ps = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) ps m"
```
```   348   by (induct ps) (auto simp add: fun_eq_iff map_add_def)
```
```   349
```
```   350
```
```   351 subsection {* @{term [source] restrict_map} *}
```
```   352
```
```   353 lemma restrict_map_to_empty [simp]: "m|`{} = empty"
```
```   354 by (simp add: restrict_map_def)
```
```   355
```
```   356 lemma restrict_map_insert: "f |` (insert a A) = (f |` A)(a := f a)"
```
```   357 by (auto simp add: restrict_map_def intro: ext)
```
```   358
```
```   359 lemma restrict_map_empty [simp]: "empty|`D = empty"
```
```   360 by (simp add: restrict_map_def)
```
```   361
```
```   362 lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m|`A) x = m x"
```
```   363 by (simp add: restrict_map_def)
```
```   364
```
```   365 lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m|`A) x = None"
```
```   366 by (simp add: restrict_map_def)
```
```   367
```
```   368 lemma ran_restrictD: "y \<in> ran (m|`A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y"
```
```   369 by (auto simp: restrict_map_def ran_def split: split_if_asm)
```
```   370
```
```   371 lemma dom_restrict [simp]: "dom (m|`A) = dom m \<inter> A"
```
```   372 by (auto simp: restrict_map_def dom_def split: split_if_asm)
```
```   373
```
```   374 lemma restrict_upd_same [simp]: "m(x\<mapsto>y)|`(-{x}) = m|`(-{x})"
```
```   375 by (rule ext) (auto simp: restrict_map_def)
```
```   376
```
```   377 lemma restrict_restrict [simp]: "m|`A|`B = m|`(A\<inter>B)"
```
```   378 by (rule ext) (auto simp: restrict_map_def)
```
```   379
```
```   380 lemma restrict_fun_upd [simp]:
```
```   381   "m(x := y)|`D = (if x \<in> D then (m|`(D-{x}))(x := y) else m|`D)"
```
```   382 by (simp add: restrict_map_def fun_eq_iff)
```
```   383
```
```   384 lemma fun_upd_None_restrict [simp]:
```
```   385   "(m|`D)(x := None) = (if x:D then m|`(D - {x}) else m|`D)"
```
```   386 by (simp add: restrict_map_def fun_eq_iff)
```
```   387
```
```   388 lemma fun_upd_restrict: "(m|`D)(x := y) = (m|`(D-{x}))(x := y)"
```
```   389 by (simp add: restrict_map_def fun_eq_iff)
```
```   390
```
```   391 lemma fun_upd_restrict_conv [simp]:
```
```   392   "x \<in> D \<Longrightarrow> (m|`D)(x := y) = (m|`(D-{x}))(x := y)"
```
```   393 by (simp add: restrict_map_def fun_eq_iff)
```
```   394
```
```   395 lemma map_of_map_restrict:
```
```   396   "map_of (map (\<lambda>k. (k, f k)) ks) = (Some \<circ> f) |` set ks"
```
```   397   by (induct ks) (simp_all add: fun_eq_iff restrict_map_insert)
```
```   398
```
```   399 lemma restrict_complement_singleton_eq:
```
```   400   "f |` (- {x}) = f(x := None)"
```
```   401   by (simp add: restrict_map_def fun_eq_iff)
```
```   402
```
```   403
```
```   404 subsection {* @{term [source] map_upds} *}
```
```   405
```
```   406 lemma map_upds_Nil1 [simp]: "m([] [|->] bs) = m"
```
```   407 by (simp add: map_upds_def)
```
```   408
```
```   409 lemma map_upds_Nil2 [simp]: "m(as [|->] []) = m"
```
```   410 by (simp add:map_upds_def)
```
```   411
```
```   412 lemma map_upds_Cons [simp]: "m(a#as [|->] b#bs) = (m(a|->b))(as[|->]bs)"
```
```   413 by (simp add:map_upds_def)
```
```   414
```
```   415 lemma map_upds_append1 [simp]: "\<And>ys m. size xs < size ys \<Longrightarrow>
```
```   416   m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)"
```
```   417 apply(induct xs)
```
```   418  apply (clarsimp simp add: neq_Nil_conv)
```
```   419 apply (case_tac ys)
```
```   420  apply simp
```
```   421 apply simp
```
```   422 done
```
```   423
```
```   424 lemma map_upds_list_update2_drop [simp]:
```
```   425   "\<lbrakk>size xs \<le> i; i < size ys\<rbrakk>
```
```   426     \<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)"
```
```   427 apply (induct xs arbitrary: m ys i)
```
```   428  apply simp
```
```   429 apply (case_tac ys)
```
```   430  apply simp
```
```   431 apply (simp split: nat.split)
```
```   432 done
```
```   433
```
```   434 lemma map_upd_upds_conv_if:
```
```   435   "(f(x|->y))(xs [|->] ys) =
```
```   436    (if x : set(take (length ys) xs) then f(xs [|->] ys)
```
```   437                                     else (f(xs [|->] ys))(x|->y))"
```
```   438 apply (induct xs arbitrary: x y ys f)
```
```   439  apply simp
```
```   440 apply (case_tac ys)
```
```   441  apply (auto split: split_if simp: fun_upd_twist)
```
```   442 done
```
```   443
```
```   444 lemma map_upds_twist [simp]:
```
```   445   "a ~: set as ==> m(a|->b)(as[|->]bs) = m(as[|->]bs)(a|->b)"
```
```   446 using set_take_subset by (fastsimp simp add: map_upd_upds_conv_if)
```
```   447
```
```   448 lemma map_upds_apply_nontin [simp]:
```
```   449   "x ~: set xs ==> (f(xs[|->]ys)) x = f x"
```
```   450 apply (induct xs arbitrary: ys)
```
```   451  apply simp
```
```   452 apply (case_tac ys)
```
```   453  apply (auto simp: map_upd_upds_conv_if)
```
```   454 done
```
```   455
```
```   456 lemma fun_upds_append_drop [simp]:
```
```   457   "size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)"
```
```   458 apply (induct xs arbitrary: m ys)
```
```   459  apply simp
```
```   460 apply (case_tac ys)
```
```   461  apply simp_all
```
```   462 done
```
```   463
```
```   464 lemma fun_upds_append2_drop [simp]:
```
```   465   "size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)"
```
```   466 apply (induct xs arbitrary: m ys)
```
```   467  apply simp
```
```   468 apply (case_tac ys)
```
```   469  apply simp_all
```
```   470 done
```
```   471
```
```   472
```
```   473 lemma restrict_map_upds[simp]:
```
```   474   "\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk>
```
```   475     \<Longrightarrow> m(xs [\<mapsto>] ys)|`D = (m|`(D - set xs))(xs [\<mapsto>] ys)"
```
```   476 apply (induct xs arbitrary: m ys)
```
```   477  apply simp
```
```   478 apply (case_tac ys)
```
```   479  apply simp
```
```   480 apply (simp add: Diff_insert [symmetric] insert_absorb)
```
```   481 apply (simp add: map_upd_upds_conv_if)
```
```   482 done
```
```   483
```
```   484
```
```   485 subsection {* @{term [source] dom} *}
```
```   486
```
```   487 lemma dom_eq_empty_conv [simp]: "dom f = {} \<longleftrightarrow> f = empty"
```
```   488 by(auto intro!:ext simp: dom_def)
```
```   489
```
```   490 lemma domI: "m a = Some b ==> a : dom m"
```
```   491 by(simp add:dom_def)
```
```   492 (* declare domI [intro]? *)
```
```   493
```
```   494 lemma domD: "a : dom m ==> \<exists>b. m a = Some b"
```
```   495 by (cases "m a") (auto simp add: dom_def)
```
```   496
```
```   497 lemma domIff [iff, simp del]: "(a : dom m) = (m a ~= None)"
```
```   498 by(simp add:dom_def)
```
```   499
```
```   500 lemma dom_empty [simp]: "dom empty = {}"
```
```   501 by(simp add:dom_def)
```
```   502
```
```   503 lemma dom_fun_upd [simp]:
```
```   504   "dom(f(x := y)) = (if y=None then dom f - {x} else insert x (dom f))"
```
```   505 by(auto simp add:dom_def)
```
```   506
```
```   507 lemma dom_if:
```
```   508   "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}"
```
```   509   by (auto split: if_splits)
```
```   510
```
```   511 lemma dom_map_of_conv_image_fst:
```
```   512   "dom (map_of xys) = fst ` set xys"
```
```   513   by (induct xys) (auto simp add: dom_if)
```
```   514
```
```   515 lemma dom_map_of_zip [simp]: "[| length xs = length ys; distinct xs |] ==>
```
```   516   dom(map_of(zip xs ys)) = set xs"
```
```   517 by (induct rule: list_induct2) simp_all
```
```   518
```
```   519 lemma finite_dom_map_of: "finite (dom (map_of l))"
```
```   520 by (induct l) (auto simp add: dom_def insert_Collect [symmetric])
```
```   521
```
```   522 lemma dom_map_upds [simp]:
```
```   523   "dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m"
```
```   524 apply (induct xs arbitrary: m ys)
```
```   525  apply simp
```
```   526 apply (case_tac ys)
```
```   527  apply auto
```
```   528 done
```
```   529
```
```   530 lemma dom_map_add [simp]: "dom(m++n) = dom n Un dom m"
```
```   531 by(auto simp:dom_def)
```
```   532
```
```   533 lemma dom_override_on [simp]:
```
```   534   "dom(override_on f g A) =
```
```   535     (dom f  - {a. a : A - dom g}) Un {a. a : A Int dom g}"
```
```   536 by(auto simp: dom_def override_on_def)
```
```   537
```
```   538 lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1"
```
```   539 by (rule ext) (force simp: map_add_def dom_def split: option.split)
```
```   540
```
```   541 lemma map_add_dom_app_simps:
```
```   542   "\<lbrakk> m\<in>dom l2 \<rbrakk> \<Longrightarrow> (l1++l2) m = l2 m"
```
```   543   "\<lbrakk> m\<notin>dom l1 \<rbrakk> \<Longrightarrow> (l1++l2) m = l2 m"
```
```   544   "\<lbrakk> m\<notin>dom l2 \<rbrakk> \<Longrightarrow> (l1++l2) m = l1 m"
```
```   545 by (auto simp add: map_add_def split: option.split_asm)
```
```   546
```
```   547 lemma dom_const [simp]:
```
```   548   "dom (\<lambda>x. Some (f x)) = UNIV"
```
```   549   by auto
```
```   550
```
```   551 (* Due to John Matthews - could be rephrased with dom *)
```
```   552 lemma finite_map_freshness:
```
```   553   "finite (dom (f :: 'a \<rightharpoonup> 'b)) \<Longrightarrow> \<not> finite (UNIV :: 'a set) \<Longrightarrow>
```
```   554    \<exists>x. f x = None"
```
```   555 by(bestsimp dest:ex_new_if_finite)
```
```   556
```
```   557 lemma dom_minus:
```
```   558   "f x = None \<Longrightarrow> dom f - insert x A = dom f - A"
```
```   559   unfolding dom_def by simp
```
```   560
```
```   561 lemma insert_dom:
```
```   562   "f x = Some y \<Longrightarrow> insert x (dom f) = dom f"
```
```   563   unfolding dom_def by auto
```
```   564
```
```   565 lemma map_of_map_keys:
```
```   566   "set xs = dom m \<Longrightarrow> map_of (map (\<lambda>k. (k, the (m k))) xs) = m"
```
```   567   by (rule ext) (auto simp add: map_of_map_restrict restrict_map_def)
```
```   568
```
```   569 lemma map_of_eqI:
```
```   570   assumes set_eq: "set (map fst xs) = set (map fst ys)"
```
```   571   assumes map_eq: "\<forall>k\<in>set (map fst xs). map_of xs k = map_of ys k"
```
```   572   shows "map_of xs = map_of ys"
```
```   573 proof (rule ext)
```
```   574   fix k show "map_of xs k = map_of ys k"
```
```   575   proof (cases "map_of xs k")
```
```   576     case None then have "k \<notin> set (map fst xs)" by (simp add: map_of_eq_None_iff)
```
```   577     with set_eq have "k \<notin> set (map fst ys)" by simp
```
```   578     then have "map_of ys k = None" by (simp add: map_of_eq_None_iff)
```
```   579     with None show ?thesis by simp
```
```   580   next
```
```   581     case (Some v) then have "k \<in> set (map fst xs)" by (auto simp add: dom_map_of_conv_image_fst [symmetric])
```
```   582     with map_eq show ?thesis by auto
```
```   583   qed
```
```   584 qed
```
```   585
```
```   586 lemma map_of_eq_dom:
```
```   587   assumes "map_of xs = map_of ys"
```
```   588   shows "fst ` set xs = fst ` set ys"
```
```   589 proof -
```
```   590   from assms have "dom (map_of xs) = dom (map_of ys)" by simp
```
```   591   then show ?thesis by (simp add: dom_map_of_conv_image_fst)
```
```   592 qed
```
```   593
```
```   594
```
```   595 subsection {* @{term [source] ran} *}
```
```   596
```
```   597 lemma ranI: "m a = Some b ==> b : ran m"
```
```   598 by(auto simp: ran_def)
```
```   599 (* declare ranI [intro]? *)
```
```   600
```
```   601 lemma ran_empty [simp]: "ran empty = {}"
```
```   602 by(auto simp: ran_def)
```
```   603
```
```   604 lemma ran_map_upd [simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)"
```
```   605 unfolding ran_def
```
```   606 apply auto
```
```   607 apply (subgoal_tac "aa ~= a")
```
```   608  apply auto
```
```   609 done
```
```   610
```
```   611 lemma ran_distinct:
```
```   612   assumes dist: "distinct (map fst al)"
```
```   613   shows "ran (map_of al) = snd ` set al"
```
```   614 using assms proof (induct al)
```
```   615   case Nil then show ?case by simp
```
```   616 next
```
```   617   case (Cons kv al)
```
```   618   then have "ran (map_of al) = snd ` set al" by simp
```
```   619   moreover from Cons.prems have "map_of al (fst kv) = None"
```
```   620     by (simp add: map_of_eq_None_iff)
```
```   621   ultimately show ?case by (simp only: map_of.simps ran_map_upd) simp
```
```   622 qed
```
```   623
```
```   624
```
```   625 subsection {* @{text "map_le"} *}
```
```   626
```
```   627 lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g"
```
```   628 by (simp add: map_le_def)
```
```   629
```
```   630 lemma upd_None_map_le [simp]: "f(x := None) \<subseteq>\<^sub>m f"
```
```   631 by (force simp add: map_le_def)
```
```   632
```
```   633 lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)"
```
```   634 by (fastsimp simp add: map_le_def)
```
```   635
```
```   636 lemma map_le_imp_upd_le [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)"
```
```   637 by (force simp add: map_le_def)
```
```   638
```
```   639 lemma map_le_upds [simp]:
```
```   640   "f \<subseteq>\<^sub>m g ==> f(as [|->] bs) \<subseteq>\<^sub>m g(as [|->] bs)"
```
```   641 apply (induct as arbitrary: f g bs)
```
```   642  apply simp
```
```   643 apply (case_tac bs)
```
```   644  apply auto
```
```   645 done
```
```   646
```
```   647 lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)"
```
```   648 by (fastsimp simp add: map_le_def dom_def)
```
```   649
```
```   650 lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f"
```
```   651 by (simp add: map_le_def)
```
```   652
```
```   653 lemma map_le_trans[trans]: "\<lbrakk> m1 \<subseteq>\<^sub>m m2; m2 \<subseteq>\<^sub>m m3\<rbrakk> \<Longrightarrow> m1 \<subseteq>\<^sub>m m3"
```
```   654 by (auto simp add: map_le_def dom_def)
```
```   655
```
```   656 lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g"
```
```   657 unfolding map_le_def
```
```   658 apply (rule ext)
```
```   659 apply (case_tac "x \<in> dom f", simp)
```
```   660 apply (case_tac "x \<in> dom g", simp, fastsimp)
```
```   661 done
```
```   662
```
```   663 lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)"
```
```   664 by (fastsimp simp add: map_le_def)
```
```   665
```
```   666 lemma map_le_iff_map_add_commute: "(f \<subseteq>\<^sub>m f ++ g) = (f++g = g++f)"
```
```   667 by(fastsimp simp: map_add_def map_le_def fun_eq_iff split: option.splits)
```
```   668
```
```   669 lemma map_add_le_mapE: "f++g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h"
```
```   670 by (fastsimp simp add: map_le_def map_add_def dom_def)
```
```   671
```
```   672 lemma map_add_le_mapI: "\<lbrakk> f \<subseteq>\<^sub>m h; g \<subseteq>\<^sub>m h; f \<subseteq>\<^sub>m f++g \<rbrakk> \<Longrightarrow> f++g \<subseteq>\<^sub>m h"
```
```   673 by (clarsimp simp add: map_le_def map_add_def dom_def split: option.splits)
```
```   674
```
```   675 lemma dom_eq_singleton_conv: "dom f = {x} \<longleftrightarrow> (\<exists>v. f = [x \<mapsto> v])"
```
```   676 proof(rule iffI)
```
```   677   assume "\<exists>v. f = [x \<mapsto> v]"
```
```   678   thus "dom f = {x}" by(auto split: split_if_asm)
```
```   679 next
```
```   680   assume "dom f = {x}"
```
```   681   then obtain v where "f x = Some v" by auto
```
```   682   hence "[x \<mapsto> v] \<subseteq>\<^sub>m f" by(auto simp add: map_le_def)
```
```   683   moreover have "f \<subseteq>\<^sub>m [x \<mapsto> v]" using `dom f = {x}` `f x = Some v`
```
```   684     by(auto simp add: map_le_def)
```
```   685   ultimately have "f = [x \<mapsto> v]" by-(rule map_le_antisym)
```
```   686   thus "\<exists>v. f = [x \<mapsto> v]" by blast
```
```   687 qed
```
```   688
```
```   689
```
```   690 subsection {* Various *}
```
```   691
```
```   692 lemma set_map_of_compr:
```
```   693   assumes distinct: "distinct (map fst xs)"
```
```   694   shows "set xs = {(k, v). map_of xs k = Some v}"
```
```   695 using assms proof (induct xs)
```
```   696   case Nil then show ?case by simp
```
```   697 next
```
```   698   case (Cons x xs)
```
```   699   obtain k v where "x = (k, v)" by (cases x) blast
```
```   700   with Cons.prems have "k \<notin> dom (map_of xs)"
```
```   701     by (simp add: dom_map_of_conv_image_fst)
```
```   702   then have *: "insert (k, v) {(k, v). map_of xs k = Some v} =
```
```   703     {(k', v'). (map_of xs(k \<mapsto> v)) k' = Some v'}"
```
```   704     by (auto split: if_splits)
```
```   705   from Cons have "set xs = {(k, v). map_of xs k = Some v}" by simp
```
```   706   with * `x = (k, v)` show ?case by simp
```
```   707 qed
```
```   708
```
```   709 lemma map_of_inject_set:
```
```   710   assumes distinct: "distinct (map fst xs)" "distinct (map fst ys)"
```
```   711   shows "map_of xs = map_of ys \<longleftrightarrow> set xs = set ys" (is "?lhs \<longleftrightarrow> ?rhs")
```
```   712 proof
```
```   713   assume ?lhs
```
```   714   moreover from `distinct (map fst xs)` have "set xs = {(k, v). map_of xs k = Some v}"
```
```   715     by (rule set_map_of_compr)
```
```   716   moreover from `distinct (map fst ys)` have "set ys = {(k, v). map_of ys k = Some v}"
```
```   717     by (rule set_map_of_compr)
```
```   718   ultimately show ?rhs by simp
```
```   719 next
```
```   720   assume ?rhs show ?lhs proof
```
```   721     fix k
```
```   722     show "map_of xs k = map_of ys k" proof (cases "map_of xs k")
```
```   723       case None
```
```   724       moreover with `?rhs` have "map_of ys k = None"
```
```   725         by (simp add: map_of_eq_None_iff)
```
```   726       ultimately show ?thesis by simp
```
```   727     next
```
```   728       case (Some v)
```
```   729       moreover with distinct `?rhs` have "map_of ys k = Some v"
```
```   730         by simp
```
```   731       ultimately show ?thesis by simp
```
```   732     qed
```
```   733   qed
```
```   734 qed
```
```   735
```
```   736 end
```