src/HOL/Map.thy
author paulson
Fri Dec 03 15:27:47 2004 +0100 (2004-12-03)
changeset 15369 090b16d6c6e0
parent 15304 3514ca74ac54
child 15691 900cf45ff0a6
permissions -rw-r--r--
tidied
     1 (*  Title:      HOL/Map.thy
     2     ID:         $Id$
     3     Author:     Tobias Nipkow, based on a theory by David von Oheimb
     4     Copyright   1997-2003 TU Muenchen
     5 
     6 The datatype of `maps' (written ~=>); strongly resembles maps in VDM.
     7 *)
     8 
     9 header {* Maps *}
    10 
    11 theory Map
    12 imports List
    13 begin
    14 
    15 types ('a,'b) "~=>" = "'a => 'b option" (infixr 0)
    16 translations (type) "a ~=> b " <= (type) "a => b option"
    17 
    18 consts
    19 chg_map	:: "('b => 'b) => 'a => ('a ~=> 'b) => ('a ~=> 'b)"
    20 map_add :: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)" (infixl "++" 100)
    21 restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)" ("_|'__" [90, 91] 90)
    22 dom	:: "('a ~=> 'b) => 'a set"
    23 ran	:: "('a ~=> 'b) => 'b set"
    24 map_of	:: "('a * 'b)list => 'a ~=> 'b"
    25 map_upds:: "('a ~=> 'b) => 'a list => 'b list => 
    26 	    ('a ~=> 'b)"
    27 map_upd_s::"('a ~=> 'b) => 'a set => 'b => 
    28 	    ('a ~=> 'b)"			 ("_/'(_{|->}_/')" [900,0,0]900)
    29 map_subst::"('a ~=> 'b) => 'b => 'b => 
    30 	    ('a ~=> 'b)"			 ("_/'(_~>_/')"    [900,0,0]900)
    31 map_le  :: "('a ~=> 'b) => ('a ~=> 'b) => bool" (infix "\<subseteq>\<^sub>m" 50)
    32 
    33 syntax
    34   fun_map_comp :: "('b => 'c)  => ('a ~=> 'b) => ('a ~=> 'c)" (infixl "o'_m" 55)
    35 translations
    36   "f o_m m" == "option_map f o m"
    37 
    38 nonterminals
    39   maplets maplet
    40 
    41 syntax
    42   empty	    ::  "'a ~=> 'b"
    43   "_maplet"  :: "['a, 'a] => maplet"             ("_ /|->/ _")
    44   "_maplets" :: "['a, 'a] => maplet"             ("_ /[|->]/ _")
    45   ""         :: "maplet => maplets"             ("_")
    46   "_Maplets" :: "[maplet, maplets] => maplets" ("_,/ _")
    47   "_MapUpd"  :: "['a ~=> 'b, maplets] => 'a ~=> 'b" ("_/'(_')" [900,0]900)
    48   "_Map"     :: "maplets => 'a ~=> 'b"            ("(1[_])")
    49 
    50 syntax (xsymbols)
    51   "~=>"     :: "[type, type] => type"    (infixr "\<rightharpoonup>" 0)
    52 
    53   fun_map_comp :: "('b => 'c)  => ('a ~=> 'b) => ('a ~=> 'c)" (infixl "\<circ>\<^sub>m" 55)
    54 
    55   "_maplet"  :: "['a, 'a] => maplet"             ("_ /\<mapsto>/ _")
    56   "_maplets" :: "['a, 'a] => maplet"             ("_ /[\<mapsto>]/ _")
    57 
    58   restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)" ("_\<lfloor>_" [90, 91] 90)
    59   map_upd_s  :: "('a ~=> 'b) => 'a set => 'b => ('a ~=> 'b)"
    60 				    		 ("_/'(_/{\<mapsto>}/_')" [900,0,0]900)
    61   map_subst :: "('a ~=> 'b) => 'b => 'b => 
    62 	        ('a ~=> 'b)"			 ("_/'(_\<leadsto>_/')"    [900,0,0]900)
    63  "@chg_map" :: "('a ~=> 'b) => 'a => ('b => 'b) => ('a ~=> 'b)"
    64 					  ("_/'(_/\<mapsto>\<lambda>_. _')"  [900,0,0,0] 900)
    65 
    66 translations
    67   "empty"    => "_K None"
    68   "empty"    <= "%x. None"
    69 
    70   "m(x\<mapsto>\<lambda>y. f)" == "chg_map (\<lambda>y. f) x m"
    71 
    72   "_MapUpd m (_Maplets xy ms)"  == "_MapUpd (_MapUpd m xy) ms"
    73   "_MapUpd m (_maplet  x y)"    == "m(x:=Some y)"
    74   "_MapUpd m (_maplets x y)"    == "map_upds m x y"
    75   "_Map ms"                     == "_MapUpd empty ms"
    76   "_Map (_Maplets ms1 ms2)"     <= "_MapUpd (_Map ms1) ms2"
    77   "_Maplets ms1 (_Maplets ms2 ms3)" <= "_Maplets (_Maplets ms1 ms2) ms3"
    78 
    79 defs
    80 chg_map_def:  "chg_map f a m == case m a of None => m | Some b => m(a|->f b)"
    81 
    82 map_add_def:   "m1++m2 == %x. case m2 x of None => m1 x | Some y => Some y"
    83 restrict_map_def: "m|_A == %x. if x : A then m x else None"
    84 
    85 map_upds_def: "m(xs [|->] ys) == m ++ map_of (rev(zip xs ys))"
    86 map_upd_s_def: "m(as{|->}b) == %x. if x : as then Some b else m x"
    87 map_subst_def: "m(a~>b)     == %x. if m x = Some a then Some b else m x"
    88 
    89 dom_def: "dom(m) == {a. m a ~= None}"
    90 ran_def: "ran(m) == {b. EX a. m a = Some b}"
    91 
    92 map_le_def: "m\<^isub>1 \<subseteq>\<^sub>m m\<^isub>2  ==  ALL a : dom m\<^isub>1. m\<^isub>1 a = m\<^isub>2 a"
    93 
    94 primrec
    95   "map_of [] = empty"
    96   "map_of (p#ps) = (map_of ps)(fst p |-> snd p)"
    97 
    98 
    99 subsection {* @{term empty} *}
   100 
   101 lemma empty_upd_none[simp]: "empty(x := None) = empty"
   102 apply (rule ext)
   103 apply (simp (no_asm))
   104 done
   105 
   106 
   107 (* FIXME: what is this sum_case nonsense?? *)
   108 lemma sum_case_empty_empty[simp]: "sum_case empty empty = empty"
   109 apply (rule ext)
   110 apply (simp (no_asm) split add: sum.split)
   111 done
   112 
   113 subsection {* @{term map_upd} *}
   114 
   115 lemma map_upd_triv: "t k = Some x ==> t(k|->x) = t"
   116 apply (rule ext)
   117 apply (simp (no_asm_simp))
   118 done
   119 
   120 lemma map_upd_nonempty[simp]: "t(k|->x) ~= empty"
   121 apply safe
   122 apply (drule_tac x = k in fun_cong)
   123 apply (simp (no_asm_use))
   124 done
   125 
   126 lemma map_upd_eqD1: "m(a\<mapsto>x) = n(a\<mapsto>y) \<Longrightarrow> x = y"
   127 by (drule fun_cong [of _ _ a], auto)
   128 
   129 lemma map_upd_Some_unfold: 
   130   "((m(a|->b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)"
   131 by auto
   132 
   133 lemma image_map_upd[simp]: "x \<notin> A \<Longrightarrow> m(x \<mapsto> y) ` A = m ` A"
   134 by fastsimp
   135 
   136 lemma finite_range_updI: "finite (range f) ==> finite (range (f(a|->b)))"
   137 apply (unfold image_def)
   138 apply (simp (no_asm_use) add: full_SetCompr_eq)
   139 apply (rule finite_subset)
   140 prefer 2 apply assumption
   141 apply auto
   142 done
   143 
   144 
   145 (* FIXME: what is this sum_case nonsense?? *)
   146 subsection {* @{term sum_case} and @{term empty}/@{term map_upd} *}
   147 
   148 lemma sum_case_map_upd_empty[simp]:
   149  "sum_case (m(k|->y)) empty =  (sum_case m empty)(Inl k|->y)"
   150 apply (rule ext)
   151 apply (simp (no_asm) split add: sum.split)
   152 done
   153 
   154 lemma sum_case_empty_map_upd[simp]:
   155  "sum_case empty (m(k|->y)) =  (sum_case empty m)(Inr k|->y)"
   156 apply (rule ext)
   157 apply (simp (no_asm) split add: sum.split)
   158 done
   159 
   160 lemma sum_case_map_upd_map_upd[simp]:
   161  "sum_case (m1(k1|->y1)) (m2(k2|->y2)) = (sum_case (m1(k1|->y1)) m2)(Inr k2|->y2)"
   162 apply (rule ext)
   163 apply (simp (no_asm) split add: sum.split)
   164 done
   165 
   166 
   167 subsection {* @{term chg_map} *}
   168 
   169 lemma chg_map_new[simp]: "m a = None   ==> chg_map f a m = m"
   170 by (unfold chg_map_def, auto)
   171 
   172 lemma chg_map_upd[simp]: "m a = Some b ==> chg_map f a m = m(a|->f b)"
   173 by (unfold chg_map_def, auto)
   174 
   175 lemma chg_map_other [simp]: "a \<noteq> b \<Longrightarrow> chg_map f a m b = m b"
   176 by (auto simp: chg_map_def split add: option.split)
   177 
   178 
   179 subsection {* @{term map_of} *}
   180 
   181 lemma map_of_eq_None_iff:
   182  "(map_of xys x = None) = (x \<notin> fst ` (set xys))"
   183 by (induct xys) simp_all
   184 
   185 lemma map_of_is_SomeD:
   186  "map_of xys x = Some y \<Longrightarrow> (x,y) \<in> set xys"
   187 apply(induct xys)
   188  apply simp
   189 apply(clarsimp split:if_splits)
   190 done
   191 
   192 lemma map_of_eq_Some_iff[simp]:
   193  "distinct(map fst xys) \<Longrightarrow> (map_of xys x = Some y) = ((x,y) \<in> set xys)"
   194 apply(induct xys)
   195  apply(simp)
   196 apply(auto simp:map_of_eq_None_iff[symmetric])
   197 done
   198 
   199 lemma Some_eq_map_of_iff[simp]:
   200  "distinct(map fst xys) \<Longrightarrow> (Some y = map_of xys x) = ((x,y) \<in> set xys)"
   201 by(auto simp del:map_of_eq_Some_iff simp add:map_of_eq_Some_iff[symmetric])
   202 
   203 lemma [simp]: "\<lbrakk> distinct(map fst xys); (x,y) \<in> set xys \<rbrakk>
   204   \<Longrightarrow> map_of xys x = Some y"
   205 apply (induct xys)
   206  apply simp
   207 apply force
   208 done
   209 
   210 lemma map_of_zip_is_None[simp]:
   211   "length xs = length ys \<Longrightarrow> (map_of (zip xs ys) x = None) = (x \<notin> set xs)"
   212 by (induct rule:list_induct2, simp_all)
   213 
   214 lemma finite_range_map_of: "finite (range (map_of xys))"
   215 apply (induct xys)
   216 apply  (simp_all (no_asm) add: image_constant)
   217 apply (rule finite_subset)
   218 prefer 2 apply assumption
   219 apply auto
   220 done
   221 
   222 lemma map_of_SomeD [rule_format]: "map_of xs k = Some y --> (k,y):set xs"
   223 by (induct "xs", auto)
   224 
   225 lemma map_of_mapk_SomeI [rule_format]:
   226      "inj f ==> map_of t k = Some x -->  
   227         map_of (map (split (%k. Pair (f k))) t) (f k) = Some x"
   228 apply (induct "t")
   229 apply  (auto simp add: inj_eq)
   230 done
   231 
   232 lemma weak_map_of_SomeI [rule_format]:
   233      "(k, x) : set l --> (\<exists>x. map_of l k = Some x)"
   234 by (induct "l", auto)
   235 
   236 lemma map_of_filter_in: 
   237 "[| map_of xs k = Some z; P k z |] ==> map_of (filter (split P) xs) k = Some z"
   238 apply (rule mp)
   239 prefer 2 apply assumption
   240 apply (erule thin_rl)
   241 apply (induct "xs", auto)
   242 done
   243 
   244 lemma map_of_map: "map_of (map (%(a,b). (a,f b)) xs) x = option_map f (map_of xs x)"
   245 by (induct "xs", auto)
   246 
   247 
   248 subsection {* @{term option_map} related *}
   249 
   250 lemma option_map_o_empty[simp]: "option_map f o empty = empty"
   251 apply (rule ext)
   252 apply (simp (no_asm))
   253 done
   254 
   255 lemma option_map_o_map_upd[simp]:
   256  "option_map f o m(a|->b) = (option_map f o m)(a|->f b)"
   257 apply (rule ext)
   258 apply (simp (no_asm))
   259 done
   260 
   261 
   262 subsection {* @{text "++"} *}
   263 
   264 lemma map_add_empty[simp]: "m ++ empty = m"
   265 apply (unfold map_add_def)
   266 apply (simp (no_asm))
   267 done
   268 
   269 lemma empty_map_add[simp]: "empty ++ m = m"
   270 apply (unfold map_add_def)
   271 apply (rule ext)
   272 apply (simp split add: option.split)
   273 done
   274 
   275 lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3"
   276 apply(rule ext)
   277 apply(simp add: map_add_def split:option.split)
   278 done
   279 
   280 lemma map_add_Some_iff: 
   281  "((m ++ n) k = Some x) = (n k = Some x | n k = None & m k = Some x)"
   282 apply (unfold map_add_def)
   283 apply (simp (no_asm) split add: option.split)
   284 done
   285 
   286 lemmas map_add_SomeD = map_add_Some_iff [THEN iffD1, standard]
   287 declare map_add_SomeD [dest!]
   288 
   289 lemma map_add_find_right[simp]: "!!xx. n k = Some xx ==> (m ++ n) k = Some xx"
   290 by (subst map_add_Some_iff, fast)
   291 
   292 lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)"
   293 apply (unfold map_add_def)
   294 apply (simp (no_asm) split add: option.split)
   295 done
   296 
   297 lemma map_add_upd[simp]: "f ++ g(x|->y) = (f ++ g)(x|->y)"
   298 apply (unfold map_add_def)
   299 apply (rule ext, auto)
   300 done
   301 
   302 lemma map_add_upds[simp]: "m1 ++ (m2(xs[\<mapsto>]ys)) = (m1++m2)(xs[\<mapsto>]ys)"
   303 by(simp add:map_upds_def)
   304 
   305 lemma map_of_append[simp]: "map_of (xs@ys) = map_of ys ++ map_of xs"
   306 apply (unfold map_add_def)
   307 apply (induct "xs")
   308 apply (simp (no_asm))
   309 apply (rule ext)
   310 apply (simp (no_asm_simp) split add: option.split)
   311 done
   312 
   313 declare fun_upd_apply [simp del]
   314 lemma finite_range_map_of_map_add:
   315  "finite (range f) ==> finite (range (f ++ map_of l))"
   316 apply (induct "l", auto)
   317 apply (erule finite_range_updI)
   318 done
   319 declare fun_upd_apply [simp]
   320 
   321 lemma inj_on_map_add_dom[iff]:
   322  "inj_on (m ++ m') (dom m') = inj_on m' (dom m')"
   323 by(fastsimp simp add:map_add_def dom_def inj_on_def split:option.splits)
   324 
   325 subsection {* @{term restrict_map} *}
   326 
   327 lemma restrict_map_to_empty[simp]: "m\<lfloor>{} = empty"
   328 by(simp add: restrict_map_def)
   329 
   330 lemma restrict_map_empty[simp]: "empty\<lfloor>D = empty"
   331 by(simp add: restrict_map_def)
   332 
   333 lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m\<lfloor>A) x = m x"
   334 by (auto simp: restrict_map_def)
   335 
   336 lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m\<lfloor>A) x = None"
   337 by (auto simp: restrict_map_def)
   338 
   339 lemma ran_restrictD: "y \<in> ran (m\<lfloor>A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y"
   340 by (auto simp: restrict_map_def ran_def split: split_if_asm)
   341 
   342 lemma dom_restrict [simp]: "dom (m\<lfloor>A) = dom m \<inter> A"
   343 by (auto simp: restrict_map_def dom_def split: split_if_asm)
   344 
   345 lemma restrict_upd_same [simp]: "m(x\<mapsto>y)\<lfloor>(-{x}) = m\<lfloor>(-{x})"
   346 by (rule ext, auto simp: restrict_map_def)
   347 
   348 lemma restrict_restrict [simp]: "m\<lfloor>A\<lfloor>B = m\<lfloor>(A\<inter>B)"
   349 by (rule ext, auto simp: restrict_map_def)
   350 
   351 lemma restrict_fun_upd[simp]:
   352  "m(x := y)\<lfloor>D = (if x \<in> D then (m\<lfloor>(D-{x}))(x := y) else m\<lfloor>D)"
   353 by(simp add: restrict_map_def expand_fun_eq)
   354 
   355 lemma fun_upd_None_restrict[simp]:
   356   "(m\<lfloor>D)(x := None) = (if x:D then m\<lfloor>(D - {x}) else m\<lfloor>D)"
   357 by(simp add: restrict_map_def expand_fun_eq)
   358 
   359 lemma fun_upd_restrict:
   360  "(m\<lfloor>D)(x := y) = (m\<lfloor>(D-{x}))(x := y)"
   361 by(simp add: restrict_map_def expand_fun_eq)
   362 
   363 lemma fun_upd_restrict_conv[simp]:
   364  "x \<in> D \<Longrightarrow> (m\<lfloor>D)(x := y) = (m\<lfloor>(D-{x}))(x := y)"
   365 by(simp add: restrict_map_def expand_fun_eq)
   366 
   367 
   368 subsection {* @{term map_upds} *}
   369 
   370 lemma map_upds_Nil1[simp]: "m([] [|->] bs) = m"
   371 by(simp add:map_upds_def)
   372 
   373 lemma map_upds_Nil2[simp]: "m(as [|->] []) = m"
   374 by(simp add:map_upds_def)
   375 
   376 lemma map_upds_Cons[simp]: "m(a#as [|->] b#bs) = (m(a|->b))(as[|->]bs)"
   377 by(simp add:map_upds_def)
   378 
   379 lemma map_upds_append1[simp]: "\<And>ys m. size xs < size ys \<Longrightarrow>
   380   m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)"
   381 apply(induct xs)
   382  apply(clarsimp simp add:neq_Nil_conv)
   383 apply (case_tac ys, simp, simp)
   384 done
   385 
   386 lemma map_upds_list_update2_drop[simp]:
   387  "\<And>m ys i. \<lbrakk>size xs \<le> i; i < size ys\<rbrakk>
   388      \<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)"
   389 apply (induct xs, simp)
   390 apply (case_tac ys, simp)
   391 apply(simp split:nat.split)
   392 done
   393 
   394 lemma map_upd_upds_conv_if: "!!x y ys f.
   395  (f(x|->y))(xs [|->] ys) =
   396  (if x : set(take (length ys) xs) then f(xs [|->] ys)
   397                                   else (f(xs [|->] ys))(x|->y))"
   398 apply (induct xs, simp)
   399 apply(case_tac ys)
   400  apply(auto split:split_if simp:fun_upd_twist)
   401 done
   402 
   403 lemma map_upds_twist [simp]:
   404  "a ~: set as ==> m(a|->b)(as[|->]bs) = m(as[|->]bs)(a|->b)"
   405 apply(insert set_take_subset)
   406 apply (fastsimp simp add: map_upd_upds_conv_if)
   407 done
   408 
   409 lemma map_upds_apply_nontin[simp]:
   410  "!!ys. x ~: set xs ==> (f(xs[|->]ys)) x = f x"
   411 apply (induct xs, simp)
   412 apply(case_tac ys)
   413  apply(auto simp: map_upd_upds_conv_if)
   414 done
   415 
   416 lemma fun_upds_append_drop[simp]:
   417   "!!m ys. size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)"
   418 apply(induct xs)
   419  apply (simp)
   420 apply(case_tac ys)
   421 apply simp_all
   422 done
   423 
   424 lemma fun_upds_append2_drop[simp]:
   425   "!!m ys. size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)"
   426 apply(induct xs)
   427  apply (simp)
   428 apply(case_tac ys)
   429 apply simp_all
   430 done
   431 
   432 
   433 lemma restrict_map_upds[simp]: "!!m ys.
   434  \<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk>
   435  \<Longrightarrow> m(xs [\<mapsto>] ys)\<lfloor>D = (m\<lfloor>(D - set xs))(xs [\<mapsto>] ys)"
   436 apply (induct xs, simp)
   437 apply (case_tac ys, simp)
   438 apply(simp add:Diff_insert[symmetric] insert_absorb)
   439 apply(simp add: map_upd_upds_conv_if)
   440 done
   441 
   442 
   443 subsection {* @{term map_upd_s} *}
   444 
   445 lemma map_upd_s_apply [simp]: 
   446   "(m(as{|->}b)) x = (if x : as then Some b else m x)"
   447 by (simp add: map_upd_s_def)
   448 
   449 lemma map_subst_apply [simp]: 
   450   "(m(a~>b)) x = (if m x = Some a then Some b else m x)" 
   451 by (simp add: map_subst_def)
   452 
   453 subsection {* @{term dom} *}
   454 
   455 lemma domI: "m a = Some b ==> a : dom m"
   456 by (unfold dom_def, auto)
   457 (* declare domI [intro]? *)
   458 
   459 lemma domD: "a : dom m ==> \<exists>b. m a = Some b"
   460 by (unfold dom_def, auto)
   461 
   462 lemma domIff[iff]: "(a : dom m) = (m a ~= None)"
   463 by (unfold dom_def, auto)
   464 declare domIff [simp del]
   465 
   466 lemma dom_empty[simp]: "dom empty = {}"
   467 apply (unfold dom_def)
   468 apply (simp (no_asm))
   469 done
   470 
   471 lemma dom_fun_upd[simp]:
   472  "dom(f(x := y)) = (if y=None then dom f - {x} else insert x (dom f))"
   473 by (simp add:dom_def) blast
   474 
   475 lemma dom_map_of: "dom(map_of xys) = {x. \<exists>y. (x,y) : set xys}"
   476 apply(induct xys)
   477 apply(auto simp del:fun_upd_apply)
   478 done
   479 
   480 lemma dom_map_of_conv_image_fst:
   481   "dom(map_of xys) = fst ` (set xys)"
   482 by(force simp: dom_map_of)
   483 
   484 lemma dom_map_of_zip[simp]: "[| length xs = length ys; distinct xs |] ==>
   485   dom(map_of(zip xs ys)) = set xs"
   486 by(induct rule: list_induct2, simp_all)
   487 
   488 lemma finite_dom_map_of: "finite (dom (map_of l))"
   489 apply (unfold dom_def)
   490 apply (induct "l")
   491 apply (auto simp add: insert_Collect [symmetric])
   492 done
   493 
   494 lemma dom_map_upds[simp]:
   495  "!!m ys. dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m"
   496 apply (induct xs, simp)
   497 apply (case_tac ys, auto)
   498 done
   499 
   500 lemma dom_map_add[simp]: "dom(m++n) = dom n Un dom m"
   501 by (unfold dom_def, auto)
   502 
   503 lemma dom_overwrite[simp]:
   504  "dom(f(g|A)) = (dom f  - {a. a : A - dom g}) Un {a. a : A Int dom g}"
   505 by(auto simp add: dom_def overwrite_def)
   506 
   507 lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1"
   508 apply(rule ext)
   509 apply(fastsimp simp:map_add_def split:option.split)
   510 done
   511 
   512 subsection {* @{term ran} *}
   513 
   514 lemma ranI: "m a = Some b ==> b : ran m" 
   515 by (auto simp add: ran_def)
   516 (* declare ranI [intro]? *)
   517 
   518 lemma ran_empty[simp]: "ran empty = {}"
   519 apply (unfold ran_def)
   520 apply (simp (no_asm))
   521 done
   522 
   523 lemma ran_map_upd[simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)"
   524 apply (unfold ran_def, auto)
   525 apply (subgoal_tac "~ (aa = a) ")
   526 apply auto
   527 done
   528 
   529 subsection {* @{text "map_le"} *}
   530 
   531 lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g"
   532 by(simp add:map_le_def)
   533 
   534 lemma [simp]: "f(x := None) \<subseteq>\<^sub>m f"
   535 by(force simp add:map_le_def)
   536 
   537 lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)"
   538 by(fastsimp simp add:map_le_def)
   539 
   540 lemma [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)"
   541 by(force simp add:map_le_def)
   542 
   543 lemma map_le_upds[simp]:
   544  "!!f g bs. f \<subseteq>\<^sub>m g ==> f(as [|->] bs) \<subseteq>\<^sub>m g(as [|->] bs)"
   545 apply (induct as, simp)
   546 apply (case_tac bs, auto)
   547 done
   548 
   549 lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)"
   550   by (fastsimp simp add: map_le_def dom_def)
   551 
   552 lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f"
   553   by (simp add: map_le_def)
   554 
   555 lemma map_le_trans[trans]: "\<lbrakk> m1 \<subseteq>\<^sub>m m2; m2 \<subseteq>\<^sub>m m3\<rbrakk> \<Longrightarrow> m1 \<subseteq>\<^sub>m m3"
   556 by(force simp add:map_le_def)
   557 
   558 lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g"
   559   apply (unfold map_le_def)
   560   apply (rule ext)
   561   apply (case_tac "x \<in> dom f", simp)
   562   apply (case_tac "x \<in> dom g", simp, fastsimp)
   563 done
   564 
   565 lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)"
   566   by (fastsimp simp add: map_le_def)
   567 
   568 lemma map_le_iff_map_add_commute: "(f \<subseteq>\<^sub>m f ++ g) = (f++g = g++f)"
   569 by(fastsimp simp add:map_add_def map_le_def expand_fun_eq split:option.splits)
   570 
   571 lemma map_add_le_mapE: "f++g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h"
   572 by (fastsimp simp add: map_le_def map_add_def dom_def)
   573 
   574 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"
   575 by (clarsimp simp add: map_le_def map_add_def dom_def split:option.splits)
   576 
   577 end