src/HOL/Map.thy
author nipkow
Wed Aug 04 19:10:45 2004 +0200 (2004-08-04)
changeset 15110 78b5636eabc7
parent 14739 86c6f272ef79
child 15131 c69542757a4d
permissions -rw-r--r--
Added a number of new thms and the new function remove1
     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 = List:
    12 
    13 types ('a,'b) "~=>" = "'a => 'b option" (infixr 0)
    14 translations (type) "a ~=> b " <= (type) "a => b option"
    15 
    16 consts
    17 chg_map	:: "('b => 'b) => 'a => ('a ~=> 'b) => ('a ~=> 'b)"
    18 map_add :: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)" (infixl "++" 100)
    19 restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)" ("_|'__" [90, 91] 90)
    20 dom	:: "('a ~=> 'b) => 'a set"
    21 ran	:: "('a ~=> 'b) => 'b set"
    22 map_of	:: "('a * 'b)list => 'a ~=> 'b"
    23 map_upds:: "('a ~=> 'b) => 'a list => 'b list => 
    24 	    ('a ~=> 'b)"
    25 map_upd_s::"('a ~=> 'b) => 'a set => 'b => 
    26 	    ('a ~=> 'b)"			 ("_/'(_{|->}_/')" [900,0,0]900)
    27 map_subst::"('a ~=> 'b) => 'b => 'b => 
    28 	    ('a ~=> 'b)"			 ("_/'(_~>_/')"    [900,0,0]900)
    29 map_le  :: "('a ~=> 'b) => ('a ~=> 'b) => bool" (infix "\<subseteq>\<^sub>m" 50)
    30 
    31 syntax
    32   fun_map_comp :: "('b => 'c)  => ('a ~=> 'b) => ('a ~=> 'c)" (infixl "o'_m" 55)
    33 translations
    34   "f o_m m" == "option_map f o m"
    35 
    36 nonterminals
    37   maplets maplet
    38 
    39 syntax
    40   empty	    ::  "'a ~=> 'b"
    41   "_maplet"  :: "['a, 'a] => maplet"             ("_ /|->/ _")
    42   "_maplets" :: "['a, 'a] => maplet"             ("_ /[|->]/ _")
    43   ""         :: "maplet => maplets"             ("_")
    44   "_Maplets" :: "[maplet, maplets] => maplets" ("_,/ _")
    45   "_MapUpd"  :: "['a ~=> 'b, maplets] => 'a ~=> 'b" ("_/'(_')" [900,0]900)
    46   "_Map"     :: "maplets => 'a ~=> 'b"            ("(1[_])")
    47 
    48 syntax (xsymbols)
    49   "~=>"     :: "[type, type] => type"    (infixr "\<rightharpoonup>" 0)
    50 
    51   fun_map_comp :: "('b => 'c)  => ('a ~=> 'b) => ('a ~=> 'c)" (infixl "\<circ>\<^sub>m" 55)
    52 
    53   "_maplet"  :: "['a, 'a] => maplet"             ("_ /\<mapsto>/ _")
    54   "_maplets" :: "['a, 'a] => maplet"             ("_ /[\<mapsto>]/ _")
    55 
    56   restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)" ("_\<lfloor>_" [90, 91] 90)
    57   map_upd_s  :: "('a ~=> 'b) => 'a set => 'b => ('a ~=> 'b)"
    58 				    		 ("_/'(_/{\<mapsto>}/_')" [900,0,0]900)
    59   map_subst :: "('a ~=> 'b) => 'b => 'b => 
    60 	        ('a ~=> 'b)"			 ("_/'(_\<leadsto>_/')"    [900,0,0]900)
    61  "@chg_map" :: "('a ~=> 'b) => 'a => ('b => 'b) => ('a ~=> 'b)"
    62 					  ("_/'(_/\<mapsto>\<lambda>_. _')"  [900,0,0,0] 900)
    63 
    64 translations
    65   "empty"    => "_K None"
    66   "empty"    <= "%x. None"
    67 
    68   "m(x\<mapsto>\<lambda>y. f)" == "chg_map (\<lambda>y. f) x m"
    69 
    70   "_MapUpd m (_Maplets xy ms)"  == "_MapUpd (_MapUpd m xy) ms"
    71   "_MapUpd m (_maplet  x y)"    == "m(x:=Some y)"
    72   "_MapUpd m (_maplets x y)"    == "map_upds m x y"
    73   "_Map ms"                     == "_MapUpd empty ms"
    74   "_Map (_Maplets ms1 ms2)"     <= "_MapUpd (_Map ms1) ms2"
    75   "_Maplets ms1 (_Maplets ms2 ms3)" <= "_Maplets (_Maplets ms1 ms2) ms3"
    76 
    77 defs
    78 chg_map_def:  "chg_map f a m == case m a of None => m | Some b => m(a|->f b)"
    79 
    80 map_add_def:   "m1++m2 == %x. case m2 x of None => m1 x | Some y => Some y"
    81 restrict_map_def: "m|_A == %x. if x : A then m x else None"
    82 
    83 map_upds_def: "m(xs [|->] ys) == m ++ map_of (rev(zip xs ys))"
    84 map_upd_s_def: "m(as{|->}b) == %x. if x : as then Some b else m x"
    85 map_subst_def: "m(a~>b)     == %x. if m x = Some a then Some b else m x"
    86 
    87 dom_def: "dom(m) == {a. m a ~= None}"
    88 ran_def: "ran(m) == {b. EX a. m a = Some b}"
    89 
    90 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"
    91 
    92 primrec
    93   "map_of [] = empty"
    94   "map_of (p#ps) = (map_of ps)(fst p |-> snd p)"
    95 
    96 
    97 subsection {* @{term empty} *}
    98 
    99 lemma empty_upd_none[simp]: "empty(x := None) = empty"
   100 apply (rule ext)
   101 apply (simp (no_asm))
   102 done
   103 
   104 
   105 (* FIXME: what is this sum_case nonsense?? *)
   106 lemma sum_case_empty_empty[simp]: "sum_case empty empty = empty"
   107 apply (rule ext)
   108 apply (simp (no_asm) split add: sum.split)
   109 done
   110 
   111 subsection {* @{term map_upd} *}
   112 
   113 lemma map_upd_triv: "t k = Some x ==> t(k|->x) = t"
   114 apply (rule ext)
   115 apply (simp (no_asm_simp))
   116 done
   117 
   118 lemma map_upd_nonempty[simp]: "t(k|->x) ~= empty"
   119 apply safe
   120 apply (drule_tac x = k in fun_cong)
   121 apply (simp (no_asm_use))
   122 done
   123 
   124 lemma map_upd_eqD1: "m(a\<mapsto>x) = n(a\<mapsto>y) \<Longrightarrow> x = y"
   125 by (drule fun_cong [of _ _ a], auto)
   126 
   127 lemma map_upd_Some_unfold: 
   128   "((m(a|->b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)"
   129 by auto
   130 
   131 lemma finite_range_updI: "finite (range f) ==> finite (range (f(a|->b)))"
   132 apply (unfold image_def)
   133 apply (simp (no_asm_use) add: full_SetCompr_eq)
   134 apply (rule finite_subset)
   135 prefer 2 apply assumption
   136 apply auto
   137 done
   138 
   139 
   140 (* FIXME: what is this sum_case nonsense?? *)
   141 subsection {* @{term sum_case} and @{term empty}/@{term map_upd} *}
   142 
   143 lemma sum_case_map_upd_empty[simp]:
   144  "sum_case (m(k|->y)) empty =  (sum_case m empty)(Inl k|->y)"
   145 apply (rule ext)
   146 apply (simp (no_asm) split add: sum.split)
   147 done
   148 
   149 lemma sum_case_empty_map_upd[simp]:
   150  "sum_case empty (m(k|->y)) =  (sum_case empty m)(Inr k|->y)"
   151 apply (rule ext)
   152 apply (simp (no_asm) split add: sum.split)
   153 done
   154 
   155 lemma sum_case_map_upd_map_upd[simp]:
   156  "sum_case (m1(k1|->y1)) (m2(k2|->y2)) = (sum_case (m1(k1|->y1)) m2)(Inr k2|->y2)"
   157 apply (rule ext)
   158 apply (simp (no_asm) split add: sum.split)
   159 done
   160 
   161 
   162 subsection {* @{term chg_map} *}
   163 
   164 lemma chg_map_new[simp]: "m a = None   ==> chg_map f a m = m"
   165 by (unfold chg_map_def, auto)
   166 
   167 lemma chg_map_upd[simp]: "m a = Some b ==> chg_map f a m = m(a|->f b)"
   168 by (unfold chg_map_def, auto)
   169 
   170 lemma chg_map_other [simp]: "a \<noteq> b \<Longrightarrow> chg_map f a m b = m b"
   171 by (auto simp: chg_map_def split add: option.split)
   172 
   173 
   174 subsection {* @{term map_of} *}
   175 
   176 lemma map_of_zip_is_None[simp]:
   177   "length xs = length ys \<Longrightarrow> (map_of (zip xs ys) x = None) = (x \<notin> set xs)"
   178 by (induct rule:list_induct2, simp_all)
   179 
   180 lemma finite_range_map_of: "finite (range (map_of xys))"
   181 apply (induct_tac xys)
   182 apply  (simp_all (no_asm) add: image_constant)
   183 apply (rule finite_subset)
   184 prefer 2 apply assumption
   185 apply auto
   186 done
   187 
   188 lemma map_of_SomeD [rule_format (no_asm)]: "map_of xs k = Some y --> (k,y):set xs"
   189 by (induct_tac "xs", auto)
   190 
   191 lemma map_of_mapk_SomeI [rule_format (no_asm)]: "inj f ==> map_of t k = Some x -->  
   192    map_of (map (split (%k. Pair (f k))) t) (f k) = Some x"
   193 apply (induct_tac "t")
   194 apply  (auto simp add: inj_eq)
   195 done
   196 
   197 lemma weak_map_of_SomeI [rule_format (no_asm)]: "(k, x) : set l --> (? x. map_of l k = Some x)"
   198 by (induct_tac "l", auto)
   199 
   200 lemma map_of_filter_in: 
   201 "[| map_of xs k = Some z; P k z |] ==> map_of (filter (split P) xs) k = Some z"
   202 apply (rule mp)
   203 prefer 2 apply assumption
   204 apply (erule thin_rl)
   205 apply (induct_tac "xs", auto)
   206 done
   207 
   208 lemma map_of_map: "map_of (map (%(a,b). (a,f b)) xs) x = option_map f (map_of xs x)"
   209 by (induct_tac "xs", auto)
   210 
   211 
   212 subsection {* @{term option_map} related *}
   213 
   214 lemma option_map_o_empty[simp]: "option_map f o empty = empty"
   215 apply (rule ext)
   216 apply (simp (no_asm))
   217 done
   218 
   219 lemma option_map_o_map_upd[simp]:
   220  "option_map f o m(a|->b) = (option_map f o m)(a|->f b)"
   221 apply (rule ext)
   222 apply (simp (no_asm))
   223 done
   224 
   225 
   226 subsection {* @{text "++"} *}
   227 
   228 lemma map_add_empty[simp]: "m ++ empty = m"
   229 apply (unfold map_add_def)
   230 apply (simp (no_asm))
   231 done
   232 
   233 lemma empty_map_add[simp]: "empty ++ m = m"
   234 apply (unfold map_add_def)
   235 apply (rule ext)
   236 apply (simp split add: option.split)
   237 done
   238 
   239 lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3"
   240 apply(rule ext)
   241 apply(simp add: map_add_def split:option.split)
   242 done
   243 
   244 lemma map_add_Some_iff: 
   245  "((m ++ n) k = Some x) = (n k = Some x | n k = None & m k = Some x)"
   246 apply (unfold map_add_def)
   247 apply (simp (no_asm) split add: option.split)
   248 done
   249 
   250 lemmas map_add_SomeD = map_add_Some_iff [THEN iffD1, standard]
   251 declare map_add_SomeD [dest!]
   252 
   253 lemma map_add_find_right[simp]: "!!xx. n k = Some xx ==> (m ++ n) k = Some xx"
   254 by (subst map_add_Some_iff, fast)
   255 
   256 lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)"
   257 apply (unfold map_add_def)
   258 apply (simp (no_asm) split add: option.split)
   259 done
   260 
   261 lemma map_add_upd[simp]: "f ++ g(x|->y) = (f ++ g)(x|->y)"
   262 apply (unfold map_add_def)
   263 apply (rule ext, auto)
   264 done
   265 
   266 lemma map_add_upds[simp]: "m1 ++ (m2(xs[\<mapsto>]ys)) = (m1++m2)(xs[\<mapsto>]ys)"
   267 by(simp add:map_upds_def)
   268 
   269 lemma map_of_append[simp]: "map_of (xs@ys) = map_of ys ++ map_of xs"
   270 apply (unfold map_add_def)
   271 apply (induct_tac "xs")
   272 apply (simp (no_asm))
   273 apply (rule ext)
   274 apply (simp (no_asm_simp) split add: option.split)
   275 done
   276 
   277 declare fun_upd_apply [simp del]
   278 lemma finite_range_map_of_map_add:
   279  "finite (range f) ==> finite (range (f ++ map_of l))"
   280 apply (induct_tac "l", auto)
   281 apply (erule finite_range_updI)
   282 done
   283 declare fun_upd_apply [simp]
   284 
   285 subsection {* @{term restrict_map} *}
   286 
   287 lemma restrict_map_to_empty[simp]: "m\<lfloor>{} = empty"
   288 by(simp add: restrict_map_def)
   289 
   290 lemma restrict_map_empty[simp]: "empty\<lfloor>D = empty"
   291 by(simp add: restrict_map_def)
   292 
   293 lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m\<lfloor>A) x = m x"
   294 by (auto simp: restrict_map_def)
   295 
   296 lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m\<lfloor>A) x = None"
   297 by (auto simp: restrict_map_def)
   298 
   299 lemma ran_restrictD: "y \<in> ran (m\<lfloor>A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y"
   300 by (auto simp: restrict_map_def ran_def split: split_if_asm)
   301 
   302 lemma dom_restrict [simp]: "dom (m\<lfloor>A) = dom m \<inter> A"
   303 by (auto simp: restrict_map_def dom_def split: split_if_asm)
   304 
   305 lemma restrict_upd_same [simp]: "m(x\<mapsto>y)\<lfloor>(-{x}) = m\<lfloor>(-{x})"
   306 by (rule ext, auto simp: restrict_map_def)
   307 
   308 lemma restrict_restrict [simp]: "m\<lfloor>A\<lfloor>B = m\<lfloor>(A\<inter>B)"
   309 by (rule ext, auto simp: restrict_map_def)
   310 
   311 lemma restrict_fun_upd[simp]:
   312  "m(x := y)\<lfloor>D = (if x \<in> D then (m\<lfloor>(D-{x}))(x := y) else m\<lfloor>D)"
   313 by(simp add: restrict_map_def expand_fun_eq)
   314 
   315 lemma fun_upd_None_restrict[simp]:
   316   "(m\<lfloor>D)(x := None) = (if x:D then m\<lfloor>(D - {x}) else m\<lfloor>D)"
   317 by(simp add: restrict_map_def expand_fun_eq)
   318 
   319 lemma fun_upd_restrict:
   320  "(m\<lfloor>D)(x := y) = (m\<lfloor>(D-{x}))(x := y)"
   321 by(simp add: restrict_map_def expand_fun_eq)
   322 
   323 lemma fun_upd_restrict_conv[simp]:
   324  "x \<in> D \<Longrightarrow> (m\<lfloor>D)(x := y) = (m\<lfloor>(D-{x}))(x := y)"
   325 by(simp add: restrict_map_def expand_fun_eq)
   326 
   327 
   328 subsection {* @{term map_upds} *}
   329 
   330 lemma map_upds_Nil1[simp]: "m([] [|->] bs) = m"
   331 by(simp add:map_upds_def)
   332 
   333 lemma map_upds_Nil2[simp]: "m(as [|->] []) = m"
   334 by(simp add:map_upds_def)
   335 
   336 lemma map_upds_Cons[simp]: "m(a#as [|->] b#bs) = (m(a|->b))(as[|->]bs)"
   337 by(simp add:map_upds_def)
   338 
   339 lemma map_upds_append1[simp]: "\<And>ys m. size xs < size ys \<Longrightarrow>
   340   m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)"
   341 apply(induct xs)
   342  apply(clarsimp simp add:neq_Nil_conv)
   343 apply (case_tac ys, simp, simp)
   344 done
   345 
   346 lemma map_upds_list_update2_drop[simp]:
   347  "\<And>m ys i. \<lbrakk>size xs \<le> i; i < size ys\<rbrakk>
   348      \<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)"
   349 apply (induct xs, simp)
   350 apply (case_tac ys, simp)
   351 apply(simp split:nat.split)
   352 done
   353 
   354 lemma map_upd_upds_conv_if: "!!x y ys f.
   355  (f(x|->y))(xs [|->] ys) =
   356  (if x : set(take (length ys) xs) then f(xs [|->] ys)
   357                                   else (f(xs [|->] ys))(x|->y))"
   358 apply (induct xs, simp)
   359 apply(case_tac ys)
   360  apply(auto split:split_if simp:fun_upd_twist)
   361 done
   362 
   363 lemma map_upds_twist [simp]:
   364  "a ~: set as ==> m(a|->b)(as[|->]bs) = m(as[|->]bs)(a|->b)"
   365 apply(insert set_take_subset)
   366 apply (fastsimp simp add: map_upd_upds_conv_if)
   367 done
   368 
   369 lemma map_upds_apply_nontin[simp]:
   370  "!!ys. x ~: set xs ==> (f(xs[|->]ys)) x = f x"
   371 apply (induct xs, simp)
   372 apply(case_tac ys)
   373  apply(auto simp: map_upd_upds_conv_if)
   374 done
   375 
   376 lemma fun_upds_append_drop[simp]:
   377   "!!m ys. size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)"
   378 apply(induct xs)
   379  apply (simp)
   380 apply(case_tac ys)
   381 apply simp_all
   382 done
   383 
   384 lemma fun_upds_append2_drop[simp]:
   385   "!!m ys. size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)"
   386 apply(induct xs)
   387  apply (simp)
   388 apply(case_tac ys)
   389 apply simp_all
   390 done
   391 
   392 
   393 lemma restrict_map_upds[simp]: "!!m ys.
   394  \<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk>
   395  \<Longrightarrow> m(xs [\<mapsto>] ys)\<lfloor>D = (m\<lfloor>(D - set xs))(xs [\<mapsto>] ys)"
   396 apply (induct xs, simp)
   397 apply (case_tac ys, simp)
   398 apply(simp add:Diff_insert[symmetric] insert_absorb)
   399 apply(simp add: map_upd_upds_conv_if)
   400 done
   401 
   402 
   403 subsection {* @{term map_upd_s} *}
   404 
   405 lemma map_upd_s_apply [simp]: 
   406   "(m(as{|->}b)) x = (if x : as then Some b else m x)"
   407 by (simp add: map_upd_s_def)
   408 
   409 lemma map_subst_apply [simp]: 
   410   "(m(a~>b)) x = (if m x = Some a then Some b else m x)" 
   411 by (simp add: map_subst_def)
   412 
   413 subsection {* @{term dom} *}
   414 
   415 lemma domI: "m a = Some b ==> a : dom m"
   416 by (unfold dom_def, auto)
   417 (* declare domI [intro]? *)
   418 
   419 lemma domD: "a : dom m ==> ? b. m a = Some b"
   420 by (unfold dom_def, auto)
   421 
   422 lemma domIff[iff]: "(a : dom m) = (m a ~= None)"
   423 by (unfold dom_def, auto)
   424 declare domIff [simp del]
   425 
   426 lemma dom_empty[simp]: "dom empty = {}"
   427 apply (unfold dom_def)
   428 apply (simp (no_asm))
   429 done
   430 
   431 lemma dom_fun_upd[simp]:
   432  "dom(f(x := y)) = (if y=None then dom f - {x} else insert x (dom f))"
   433 by (simp add:dom_def) blast
   434 
   435 lemma dom_map_of: "dom(map_of xys) = {x. \<exists>y. (x,y) : set xys}"
   436 apply(induct xys)
   437 apply(auto simp del:fun_upd_apply)
   438 done
   439 
   440 lemma dom_map_of_zip[simp]: "[| length xs = length ys; distinct xs |] ==>
   441   dom(map_of(zip xs ys)) = set xs"
   442 by(induct rule: list_induct2, simp_all)
   443 
   444 lemma finite_dom_map_of: "finite (dom (map_of l))"
   445 apply (unfold dom_def)
   446 apply (induct_tac "l")
   447 apply (auto simp add: insert_Collect [symmetric])
   448 done
   449 
   450 lemma dom_map_upds[simp]:
   451  "!!m ys. dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m"
   452 apply (induct xs, simp)
   453 apply (case_tac ys, auto)
   454 done
   455 
   456 lemma dom_map_add[simp]: "dom(m++n) = dom n Un dom m"
   457 by (unfold dom_def, auto)
   458 
   459 lemma dom_overwrite[simp]:
   460  "dom(f(g|A)) = (dom f  - {a. a : A - dom g}) Un {a. a : A Int dom g}"
   461 by(auto simp add: dom_def overwrite_def)
   462 
   463 lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1"
   464 apply(rule ext)
   465 apply(fastsimp simp:map_add_def split:option.split)
   466 done
   467 
   468 subsection {* @{term ran} *}
   469 
   470 lemma ranI: "m a = Some b ==> b : ran m" 
   471 by (auto simp add: ran_def)
   472 (* declare ranI [intro]? *)
   473 
   474 lemma ran_empty[simp]: "ran empty = {}"
   475 apply (unfold ran_def)
   476 apply (simp (no_asm))
   477 done
   478 
   479 lemma ran_map_upd[simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)"
   480 apply (unfold ran_def, auto)
   481 apply (subgoal_tac "~ (aa = a) ")
   482 apply auto
   483 done
   484 
   485 subsection {* @{text "map_le"} *}
   486 
   487 lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g"
   488 by(simp add:map_le_def)
   489 
   490 lemma [simp]: "f(x := None) \<subseteq>\<^sub>m f"
   491 by(force simp add:map_le_def)
   492 
   493 lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)"
   494 by(fastsimp simp add:map_le_def)
   495 
   496 lemma [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)"
   497 by(force simp add:map_le_def)
   498 
   499 lemma map_le_upds[simp]:
   500  "!!f g bs. f \<subseteq>\<^sub>m g ==> f(as [|->] bs) \<subseteq>\<^sub>m g(as [|->] bs)"
   501 apply (induct as, simp)
   502 apply (case_tac bs, auto)
   503 done
   504 
   505 lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)"
   506   by (fastsimp simp add: map_le_def dom_def)
   507 
   508 lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f"
   509   by (simp add: map_le_def)
   510 
   511 lemma map_le_trans[trans]: "\<lbrakk> m1 \<subseteq>\<^sub>m m2; m2 \<subseteq>\<^sub>m m3\<rbrakk> \<Longrightarrow> m1 \<subseteq>\<^sub>m m3"
   512 by(force simp add:map_le_def)
   513 
   514 lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g"
   515   apply (unfold map_le_def)
   516   apply (rule ext)
   517   apply (case_tac "x \<in> dom f", simp)
   518   apply (case_tac "x \<in> dom g", simp, fastsimp)
   519 done
   520 
   521 lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)"
   522   by (fastsimp simp add: map_le_def)
   523 
   524 end