src/HOL/Map.thy
author haftmann
Fri Apr 20 11:21:42 2007 +0200 (2007-04-20)
changeset 22744 5cbe966d67a2
parent 22230 bdec4a82f385
child 24331 76f7a8c6e842
permissions -rw-r--r--
Isar definitions are now added explicitly to code theorem table
     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 syntax (xsymbols)
    19   "~=>" :: "[type, type] => type"  (infixr "\<rightharpoonup>" 0)
    20 
    21 abbreviation
    22   empty :: "'a ~=> 'b" where
    23   "empty == %x. None"
    24 
    25 definition
    26   map_comp :: "('b ~=> 'c)  => ('a ~=> 'b) => ('a ~=> 'c)"  (infixl "o'_m" 55) where
    27   "f o_m g = (\<lambda>k. case g k of None \<Rightarrow> None | Some v \<Rightarrow> f v)"
    28 
    29 notation (xsymbols)
    30   map_comp  (infixl "\<circ>\<^sub>m" 55)
    31 
    32 definition
    33   map_add :: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)"  (infixl "++" 100) where
    34   "m1 ++ m2 = (\<lambda>x. case m2 x of None => m1 x | Some y => Some y)"
    35 
    36 definition
    37   restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)"  (infixl "|`"  110) where
    38   "m|`A = (\<lambda>x. if x : A then m x else None)"
    39 
    40 notation (latex output)
    41   restrict_map  ("_\<restriction>\<^bsub>_\<^esub>" [111,110] 110)
    42 
    43 definition
    44   dom :: "('a ~=> 'b) => 'a set" where
    45   "dom m = {a. m a ~= None}"
    46 
    47 definition
    48   ran :: "('a ~=> 'b) => 'b set" where
    49   "ran m = {b. EX a. m a = Some b}"
    50 
    51 definition
    52   map_le :: "('a ~=> 'b) => ('a ~=> 'b) => bool"  (infix "\<subseteq>\<^sub>m" 50) where
    53   "(m\<^isub>1 \<subseteq>\<^sub>m m\<^isub>2) = (\<forall>a \<in> dom m\<^isub>1. m\<^isub>1 a = m\<^isub>2 a)"
    54 
    55 consts
    56   map_of :: "('a * 'b) list => 'a ~=> 'b"
    57   map_upds :: "('a ~=> 'b) => 'a list => 'b list => ('a ~=> 'b)"
    58 
    59 nonterminals
    60   maplets maplet
    61 
    62 syntax
    63   "_maplet"  :: "['a, 'a] => maplet"             ("_ /|->/ _")
    64   "_maplets" :: "['a, 'a] => maplet"             ("_ /[|->]/ _")
    65   ""         :: "maplet => maplets"             ("_")
    66   "_Maplets" :: "[maplet, maplets] => maplets" ("_,/ _")
    67   "_MapUpd"  :: "['a ~=> 'b, maplets] => 'a ~=> 'b" ("_/'(_')" [900,0]900)
    68   "_Map"     :: "maplets => 'a ~=> 'b"            ("(1[_])")
    69 
    70 syntax (xsymbols)
    71   "_maplet"  :: "['a, 'a] => maplet"             ("_ /\<mapsto>/ _")
    72   "_maplets" :: "['a, 'a] => maplet"             ("_ /[\<mapsto>]/ _")
    73 
    74 translations
    75   "_MapUpd m (_Maplets xy ms)"  == "_MapUpd (_MapUpd m xy) ms"
    76   "_MapUpd m (_maplet  x y)"    == "m(x:=Some y)"
    77   "_MapUpd m (_maplets x y)"    == "map_upds m x y"
    78   "_Map ms"                     == "_MapUpd (CONST empty) ms"
    79   "_Map (_Maplets ms1 ms2)"     <= "_MapUpd (_Map ms1) ms2"
    80   "_Maplets ms1 (_Maplets ms2 ms3)" <= "_Maplets (_Maplets ms1 ms2) ms3"
    81 
    82 primrec
    83   "map_of [] = empty"
    84   "map_of (p#ps) = (map_of ps)(fst p |-> snd p)"
    85 
    86 defs
    87   map_upds_def [code func]: "m(xs [|->] ys) == m ++ map_of (rev(zip xs ys))"
    88 
    89 
    90 subsection {* @{term [source] empty} *}
    91 
    92 lemma empty_upd_none [simp]: "empty(x := None) = empty"
    93   by (rule ext) simp
    94 
    95 
    96 subsection {* @{term [source] map_upd} *}
    97 
    98 lemma map_upd_triv: "t k = Some x ==> t(k|->x) = t"
    99   by (rule ext) simp
   100 
   101 lemma map_upd_nonempty [simp]: "t(k|->x) ~= empty"
   102 proof
   103   assume "t(k \<mapsto> x) = empty"
   104   then have "(t(k \<mapsto> x)) k = None" by simp
   105   then show False by simp
   106 qed
   107 
   108 lemma map_upd_eqD1:
   109   assumes "m(a\<mapsto>x) = n(a\<mapsto>y)"
   110   shows "x = y"
   111 proof -
   112   from prems have "(m(a\<mapsto>x)) a = (n(a\<mapsto>y)) a" by simp
   113   then show ?thesis by simp
   114 qed
   115 
   116 lemma map_upd_Some_unfold:
   117     "((m(a|->b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)"
   118   by auto
   119 
   120 lemma image_map_upd [simp]: "x \<notin> A \<Longrightarrow> m(x \<mapsto> y) ` A = m ` A"
   121   by auto
   122 
   123 lemma finite_range_updI: "finite (range f) ==> finite (range (f(a|->b)))"
   124   unfolding image_def
   125   apply (simp (no_asm_use) add: full_SetCompr_eq)
   126   apply (rule finite_subset)
   127    prefer 2 apply assumption
   128   apply auto
   129   done
   130 
   131 
   132 subsection {* @{term [source] map_of} *}
   133 
   134 lemma map_of_eq_None_iff:
   135     "(map_of xys x = None) = (x \<notin> fst ` (set xys))"
   136   by (induct xys) simp_all
   137 
   138 lemma map_of_is_SomeD:
   139     "map_of xys x = Some y \<Longrightarrow> (x,y) \<in> set xys"
   140   apply (induct xys)
   141    apply simp
   142   apply (clarsimp split: if_splits)
   143   done
   144 
   145 lemma map_of_eq_Some_iff [simp]:
   146     "distinct(map fst xys) \<Longrightarrow> (map_of xys x = Some y) = ((x,y) \<in> set xys)"
   147   apply (induct xys)
   148    apply simp
   149   apply (auto simp: map_of_eq_None_iff [symmetric])
   150   done
   151 
   152 lemma Some_eq_map_of_iff [simp]:
   153     "distinct(map fst xys) \<Longrightarrow> (Some y = map_of xys x) = ((x,y) \<in> set xys)"
   154   by (auto simp del:map_of_eq_Some_iff simp add: map_of_eq_Some_iff [symmetric])
   155 
   156 lemma map_of_is_SomeI [simp]: "\<lbrakk> distinct(map fst xys); (x,y) \<in> set xys \<rbrakk>
   157     \<Longrightarrow> map_of xys x = Some y"
   158   apply (induct xys)
   159    apply simp
   160   apply force
   161   done
   162 
   163 lemma map_of_zip_is_None [simp]:
   164     "length xs = length ys \<Longrightarrow> (map_of (zip xs ys) x = None) = (x \<notin> set xs)"
   165   by (induct rule: list_induct2) simp_all
   166 
   167 lemma finite_range_map_of: "finite (range (map_of xys))"
   168   apply (induct xys)
   169    apply (simp_all add: image_constant)
   170   apply (rule finite_subset)
   171    prefer 2 apply assumption
   172   apply auto
   173   done
   174 
   175 lemma map_of_SomeD: "map_of xs k = Some y \<Longrightarrow> (k, y) \<in> set xs"
   176   by (induct xs) (simp, atomize (full), auto)
   177 
   178 lemma map_of_mapk_SomeI:
   179   assumes "inj f"
   180   shows "map_of t k = Some x ==>
   181     map_of (map (split (%k. Pair (f k))) t) (f k) = Some x"
   182   by (induct t) (auto simp add: `inj f` inj_eq)
   183 
   184 lemma weak_map_of_SomeI: "(k, x) : set l ==> \<exists>x. map_of l k = Some x"
   185   by (induct l) auto
   186 
   187 lemma map_of_filter_in:
   188   assumes 1: "map_of xs k = Some z"
   189     and 2: "P k z"
   190   shows "map_of (filter (split P) xs) k = Some z"
   191   using 1 by (induct xs) (insert 2, auto)
   192 
   193 lemma map_of_map: "map_of (map (%(a,b). (a,f b)) xs) x = option_map f (map_of xs x)"
   194   by (induct xs) auto
   195 
   196 
   197 subsection {* @{term [source] option_map} related *}
   198 
   199 lemma option_map_o_empty [simp]: "option_map f o empty = empty"
   200   by (rule ext) simp
   201 
   202 lemma option_map_o_map_upd [simp]:
   203     "option_map f o m(a|->b) = (option_map f o m)(a|->f b)"
   204   by (rule ext) simp
   205 
   206 
   207 subsection {* @{term [source] map_comp} related *}
   208 
   209 lemma map_comp_empty [simp]:
   210     "m \<circ>\<^sub>m empty = empty"
   211     "empty \<circ>\<^sub>m m = empty"
   212   by (auto simp add: map_comp_def intro: ext split: option.splits)
   213 
   214 lemma map_comp_simps [simp]:
   215     "m2 k = None \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = None"
   216     "m2 k = Some k' \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = m1 k'"
   217   by (auto simp add: map_comp_def)
   218 
   219 lemma map_comp_Some_iff:
   220     "((m1 \<circ>\<^sub>m m2) k = Some v) = (\<exists>k'. m2 k = Some k' \<and> m1 k' = Some v)"
   221   by (auto simp add: map_comp_def split: option.splits)
   222 
   223 lemma map_comp_None_iff:
   224     "((m1 \<circ>\<^sub>m m2) k = None) = (m2 k = None \<or> (\<exists>k'. m2 k = Some k' \<and> m1 k' = None)) "
   225   by (auto simp add: map_comp_def split: option.splits)
   226 
   227 
   228 subsection {* @{text "++"} *}
   229 
   230 lemma map_add_empty[simp]: "m ++ empty = m"
   231   unfolding map_add_def by simp
   232 
   233 lemma empty_map_add[simp]: "empty ++ m = m"
   234   unfolding map_add_def by (rule ext) (simp split: option.split)
   235 
   236 lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3"
   237   unfolding map_add_def by (rule ext) (simp add: map_add_def split: option.split)
   238 
   239 lemma map_add_Some_iff:
   240     "((m ++ n) k = Some x) = (n k = Some x | n k = None & m k = Some x)"
   241   unfolding map_add_def by (simp split: option.split)
   242 
   243 lemma map_add_SomeD [dest!]:
   244     "(m ++ n) k = Some x \<Longrightarrow> n k = Some x \<or> n k = None \<and> m k = Some x"
   245   by (rule map_add_Some_iff [THEN iffD1])
   246 
   247 lemma map_add_find_right [simp]: "!!xx. n k = Some xx ==> (m ++ n) k = Some xx"
   248   by (subst map_add_Some_iff) fast
   249 
   250 lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)"
   251   unfolding map_add_def by (simp split: option.split)
   252 
   253 lemma map_add_upd[simp]: "f ++ g(x|->y) = (f ++ g)(x|->y)"
   254   unfolding map_add_def by (rule ext) simp
   255 
   256 lemma map_add_upds[simp]: "m1 ++ (m2(xs[\<mapsto>]ys)) = (m1++m2)(xs[\<mapsto>]ys)"
   257   by (simp add: map_upds_def)
   258 
   259 lemma map_of_append[simp]: "map_of (xs @ ys) = map_of ys ++ map_of xs"
   260   unfolding map_add_def
   261   apply (induct xs)
   262    apply simp
   263   apply (rule ext)
   264   apply (simp split add: option.split)
   265   done
   266 
   267 lemma finite_range_map_of_map_add:
   268   "finite (range f) ==> finite (range (f ++ map_of l))"
   269   apply (induct l)
   270    apply (auto simp del: fun_upd_apply)
   271   apply (erule finite_range_updI)
   272   done
   273 
   274 lemma inj_on_map_add_dom [iff]:
   275     "inj_on (m ++ m') (dom m') = inj_on m' (dom m')"
   276   unfolding map_add_def dom_def inj_on_def
   277   by (fastsimp split: option.splits)
   278 
   279 
   280 subsection {* @{term [source] restrict_map} *}
   281 
   282 lemma restrict_map_to_empty [simp]: "m|`{} = empty"
   283   by (simp add: restrict_map_def)
   284 
   285 lemma restrict_map_empty [simp]: "empty|`D = empty"
   286   by (simp add: restrict_map_def)
   287 
   288 lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m|`A) x = m x"
   289   by (simp add: restrict_map_def)
   290 
   291 lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m|`A) x = None"
   292   by (simp add: restrict_map_def)
   293 
   294 lemma ran_restrictD: "y \<in> ran (m|`A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y"
   295   by (auto simp: restrict_map_def ran_def split: split_if_asm)
   296 
   297 lemma dom_restrict [simp]: "dom (m|`A) = dom m \<inter> A"
   298   by (auto simp: restrict_map_def dom_def split: split_if_asm)
   299 
   300 lemma restrict_upd_same [simp]: "m(x\<mapsto>y)|`(-{x}) = m|`(-{x})"
   301   by (rule ext) (auto simp: restrict_map_def)
   302 
   303 lemma restrict_restrict [simp]: "m|`A|`B = m|`(A\<inter>B)"
   304   by (rule ext) (auto simp: restrict_map_def)
   305 
   306 lemma restrict_fun_upd [simp]:
   307     "m(x := y)|`D = (if x \<in> D then (m|`(D-{x}))(x := y) else m|`D)"
   308   by (simp add: restrict_map_def expand_fun_eq)
   309 
   310 lemma fun_upd_None_restrict [simp]:
   311     "(m|`D)(x := None) = (if x:D then m|`(D - {x}) else m|`D)"
   312   by (simp add: restrict_map_def expand_fun_eq)
   313 
   314 lemma fun_upd_restrict: "(m|`D)(x := y) = (m|`(D-{x}))(x := y)"
   315   by (simp add: restrict_map_def expand_fun_eq)
   316 
   317 lemma fun_upd_restrict_conv [simp]:
   318     "x \<in> D \<Longrightarrow> (m|`D)(x := y) = (m|`(D-{x}))(x := y)"
   319   by (simp add: restrict_map_def expand_fun_eq)
   320 
   321 
   322 subsection {* @{term [source] map_upds} *}
   323 
   324 lemma map_upds_Nil1 [simp]: "m([] [|->] bs) = m"
   325   by (simp add: map_upds_def)
   326 
   327 lemma map_upds_Nil2 [simp]: "m(as [|->] []) = m"
   328   by (simp add:map_upds_def)
   329 
   330 lemma map_upds_Cons [simp]: "m(a#as [|->] b#bs) = (m(a|->b))(as[|->]bs)"
   331   by (simp add:map_upds_def)
   332 
   333 lemma map_upds_append1 [simp]: "\<And>ys m. size xs < size ys \<Longrightarrow>
   334     m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)"
   335   apply(induct xs)
   336    apply (clarsimp simp add: neq_Nil_conv)
   337   apply (case_tac ys)
   338    apply simp
   339   apply simp
   340   done
   341 
   342 lemma map_upds_list_update2_drop [simp]:
   343   "\<lbrakk>size xs \<le> i; i < size ys\<rbrakk>
   344     \<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)"
   345   apply (induct xs arbitrary: m ys i)
   346    apply simp
   347   apply (case_tac ys)
   348    apply simp
   349   apply (simp split: nat.split)
   350   done
   351 
   352 lemma map_upd_upds_conv_if:
   353   "(f(x|->y))(xs [|->] ys) =
   354    (if x : set(take (length ys) xs) then f(xs [|->] ys)
   355                                     else (f(xs [|->] ys))(x|->y))"
   356   apply (induct xs arbitrary: x y ys f)
   357    apply simp
   358   apply (case_tac ys)
   359    apply (auto split: split_if simp: fun_upd_twist)
   360   done
   361 
   362 lemma map_upds_twist [simp]:
   363     "a ~: set as ==> m(a|->b)(as[|->]bs) = m(as[|->]bs)(a|->b)"
   364   using set_take_subset by (fastsimp simp add: map_upd_upds_conv_if)
   365 
   366 lemma map_upds_apply_nontin [simp]:
   367     "x ~: set xs ==> (f(xs[|->]ys)) x = f x"
   368   apply (induct xs arbitrary: ys)
   369    apply simp
   370   apply (case_tac ys)
   371    apply (auto simp: map_upd_upds_conv_if)
   372   done
   373 
   374 lemma fun_upds_append_drop [simp]:
   375     "size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)"
   376   apply (induct xs arbitrary: m ys)
   377    apply simp
   378   apply (case_tac ys)
   379    apply simp_all
   380   done
   381 
   382 lemma fun_upds_append2_drop [simp]:
   383     "size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)"
   384   apply (induct xs arbitrary: m ys)
   385    apply simp
   386   apply (case_tac ys)
   387    apply simp_all
   388   done
   389 
   390 
   391 lemma restrict_map_upds[simp]:
   392   "\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk>
   393     \<Longrightarrow> m(xs [\<mapsto>] ys)|`D = (m|`(D - set xs))(xs [\<mapsto>] ys)"
   394   apply (induct xs arbitrary: m ys)
   395    apply simp
   396   apply (case_tac ys)
   397    apply 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 [source] dom} *}
   404 
   405 lemma domI: "m a = Some b ==> a : dom m"
   406   unfolding dom_def by simp
   407 (* declare domI [intro]? *)
   408 
   409 lemma domD: "a : dom m ==> \<exists>b. m a = Some b"
   410   by (cases "m a") (auto simp add: dom_def)
   411 
   412 lemma domIff [iff, simp del]: "(a : dom m) = (m a ~= None)"
   413   unfolding dom_def by simp
   414 
   415 lemma dom_empty [simp]: "dom empty = {}"
   416   unfolding dom_def by simp
   417 
   418 lemma dom_fun_upd [simp]:
   419     "dom(f(x := y)) = (if y=None then dom f - {x} else insert x (dom f))"
   420   unfolding dom_def by auto
   421 
   422 lemma dom_map_of: "dom(map_of xys) = {x. \<exists>y. (x,y) : set xys}"
   423   by (induct xys) (auto simp del: fun_upd_apply)
   424 
   425 lemma dom_map_of_conv_image_fst:
   426     "dom(map_of xys) = fst ` (set xys)"
   427   unfolding dom_map_of by force
   428 
   429 lemma dom_map_of_zip [simp]: "[| length xs = length ys; distinct xs |] ==>
   430     dom(map_of(zip xs ys)) = set xs"
   431   by (induct rule: list_induct2) simp_all
   432 
   433 lemma finite_dom_map_of: "finite (dom (map_of l))"
   434   unfolding dom_def
   435   by (induct l) (auto simp add: insert_Collect [symmetric])
   436 
   437 lemma dom_map_upds [simp]:
   438     "dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m"
   439   apply (induct xs arbitrary: m ys)
   440    apply simp
   441   apply (case_tac ys)
   442    apply auto
   443   done
   444 
   445 lemma dom_map_add [simp]: "dom(m++n) = dom n Un dom m"
   446   unfolding dom_def by auto
   447 
   448 lemma dom_override_on [simp]:
   449   "dom(override_on f g A) =
   450     (dom f  - {a. a : A - dom g}) Un {a. a : A Int dom g}"
   451   unfolding dom_def override_on_def by auto
   452 
   453 lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1"
   454   by (rule ext) (force simp: map_add_def dom_def split: option.split)
   455 
   456 (* Due to John Matthews - could be rephrased with dom *)
   457 lemma finite_map_freshness:
   458   "finite (dom (f :: 'a \<rightharpoonup> 'b)) \<Longrightarrow> \<not> finite (UNIV :: 'a set) \<Longrightarrow>
   459    \<exists>x. f x = None"
   460 by(bestsimp dest:ex_new_if_finite)
   461 
   462 subsection {* @{term [source] ran} *}
   463 
   464 lemma ranI: "m a = Some b ==> b : ran m"
   465   unfolding ran_def by auto
   466 (* declare ranI [intro]? *)
   467 
   468 lemma ran_empty [simp]: "ran empty = {}"
   469   unfolding ran_def by simp
   470 
   471 lemma ran_map_upd [simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)"
   472   unfolding ran_def
   473   apply auto
   474   apply (subgoal_tac "aa ~= a")
   475    apply auto
   476   done
   477 
   478 
   479 subsection {* @{text "map_le"} *}
   480 
   481 lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g"
   482   by (simp add: map_le_def)
   483 
   484 lemma upd_None_map_le [simp]: "f(x := None) \<subseteq>\<^sub>m f"
   485   by (force simp add: map_le_def)
   486 
   487 lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)"
   488   by (fastsimp simp add: map_le_def)
   489 
   490 lemma map_le_imp_upd_le [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)"
   491   by (force simp add: map_le_def)
   492 
   493 lemma map_le_upds [simp]:
   494     "f \<subseteq>\<^sub>m g ==> f(as [|->] bs) \<subseteq>\<^sub>m g(as [|->] bs)"
   495   apply (induct as arbitrary: f g bs)
   496    apply simp
   497   apply (case_tac bs)
   498    apply auto
   499   done
   500 
   501 lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)"
   502   by (fastsimp simp add: map_le_def dom_def)
   503 
   504 lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f"
   505   by (simp add: map_le_def)
   506 
   507 lemma map_le_trans[trans]: "\<lbrakk> m1 \<subseteq>\<^sub>m m2; m2 \<subseteq>\<^sub>m m3\<rbrakk> \<Longrightarrow> m1 \<subseteq>\<^sub>m m3"
   508   by (auto simp add: map_le_def dom_def)
   509 
   510 lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g"
   511   unfolding map_le_def
   512   apply (rule ext)
   513   apply (case_tac "x \<in> dom f", simp)
   514   apply (case_tac "x \<in> dom g", simp, fastsimp)
   515   done
   516 
   517 lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)"
   518   by (fastsimp simp add: map_le_def)
   519 
   520 lemma map_le_iff_map_add_commute: "(f \<subseteq>\<^sub>m f ++ g) = (f++g = g++f)"
   521   by (fastsimp simp add: map_add_def map_le_def expand_fun_eq split: option.splits)
   522 
   523 lemma map_add_le_mapE: "f++g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h"
   524   by (fastsimp simp add: map_le_def map_add_def dom_def)
   525 
   526 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"
   527   by (clarsimp simp add: map_le_def map_add_def dom_def split: option.splits)
   528 
   529 end