src/HOL/Map.thy
author wenzelm
Sat Sep 30 21:39:24 2006 +0200 (2006-09-30)
changeset 20800 69c82605efcf
parent 19947 29b376397cd5
child 21210 c17fd2df4e9e
permissions -rw-r--r--
tuned specifications and proofs;
     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"
    23   "empty == %x. None"
    24 
    25 definition
    26   map_comp :: "('b ~=> 'c)  => ('a ~=> 'b) => ('a ~=> 'c)"  (infixl "o'_m" 55)
    27   "f o_m g = (\<lambda>k. case g k of None \<Rightarrow> None | Some v \<Rightarrow> f v)"
    28 
    29 const_syntax (xsymbols)
    30   map_comp  (infixl "\<circ>\<^sub>m" 55)
    31 
    32 definition
    33   map_add :: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)"  (infixl "++" 100)
    34   "m1 ++ m2 = (\<lambda>x. case m2 x of None => m1 x | Some y => Some y)"
    35 
    36   restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)"  (infixl "|`"  110)
    37   "m|`A = (\<lambda>x. if x : A then m x else None)"
    38 
    39 const_syntax (latex output)
    40   restrict_map  ("_\<restriction>\<^bsub>_\<^esub>" [111,110] 110)
    41 
    42 definition
    43   dom :: "('a ~=> 'b) => 'a set"
    44   "dom m = {a. m a ~= None}"
    45 
    46   ran :: "('a ~=> 'b) => 'b set"
    47   "ran m = {b. EX a. m a = Some b}"
    48 
    49   map_le :: "('a ~=> 'b) => ('a ~=> 'b) => bool"  (infix "\<subseteq>\<^sub>m" 50)
    50   "(m\<^isub>1 \<subseteq>\<^sub>m m\<^isub>2) = (\<forall>a \<in> dom m\<^isub>1. m\<^isub>1 a = m\<^isub>2 a)"
    51 
    52 consts
    53   map_of :: "('a * 'b) list => 'a ~=> 'b"
    54   map_upds :: "('a ~=> 'b) => 'a list => 'b list => ('a ~=> 'b)"
    55 
    56 nonterminals
    57   maplets maplet
    58 
    59 syntax
    60   "_maplet"  :: "['a, 'a] => maplet"             ("_ /|->/ _")
    61   "_maplets" :: "['a, 'a] => maplet"             ("_ /[|->]/ _")
    62   ""         :: "maplet => maplets"             ("_")
    63   "_Maplets" :: "[maplet, maplets] => maplets" ("_,/ _")
    64   "_MapUpd"  :: "['a ~=> 'b, maplets] => 'a ~=> 'b" ("_/'(_')" [900,0]900)
    65   "_Map"     :: "maplets => 'a ~=> 'b"            ("(1[_])")
    66 
    67 syntax (xsymbols)
    68   "_maplet"  :: "['a, 'a] => maplet"             ("_ /\<mapsto>/ _")
    69   "_maplets" :: "['a, 'a] => maplet"             ("_ /[\<mapsto>]/ _")
    70 
    71 translations
    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 (CONST empty) ms"
    76   "_Map (_Maplets ms1 ms2)"     <= "_MapUpd (_Map ms1) ms2"
    77   "_Maplets ms1 (_Maplets ms2 ms3)" <= "_Maplets (_Maplets ms1 ms2) ms3"
    78 
    79 primrec
    80   "map_of [] = empty"
    81   "map_of (p#ps) = (map_of ps)(fst p |-> snd p)"
    82 
    83 defs
    84   map_upds_def: "m(xs [|->] ys) == m ++ map_of (rev(zip xs ys))"
    85 
    86 (* special purpose constants that should be defined somewhere else and
    87 whose syntax is a bit odd as well:
    88 
    89  "@chg_map" :: "('a ~=> 'b) => 'a => ('b => 'b) => ('a ~=> 'b)"
    90                                           ("_/'(_/\<mapsto>\<lambda>_. _')"  [900,0,0,0] 900)
    91   "m(x\<mapsto>\<lambda>y. f)" == "chg_map (\<lambda>y. f) x m"
    92 
    93 map_upd_s::"('a ~=> 'b) => 'a set => 'b =>
    94             ('a ~=> 'b)"                         ("_/'(_{|->}_/')" [900,0,0]900)
    95 map_subst::"('a ~=> 'b) => 'b => 'b =>
    96             ('a ~=> 'b)"                         ("_/'(_~>_/')"    [900,0,0]900)
    97 
    98 map_upd_s_def: "m(as{|->}b) == %x. if x : as then Some b else m x"
    99 map_subst_def: "m(a~>b)     == %x. if m x = Some a then Some b else m x"
   100 
   101   map_upd_s  :: "('a ~=> 'b) => 'a set => 'b => ('a ~=> 'b)"
   102                                                  ("_/'(_/{\<mapsto>}/_')" [900,0,0]900)
   103   map_subst :: "('a ~=> 'b) => 'b => 'b =>
   104                 ('a ~=> 'b)"                     ("_/'(_\<leadsto>_/')"    [900,0,0]900)
   105 
   106 
   107 subsection {* @{term [source] map_upd_s} *}
   108 
   109 lemma map_upd_s_apply [simp]:
   110   "(m(as{|->}b)) x = (if x : as then Some b else m x)"
   111 by (simp add: map_upd_s_def)
   112 
   113 lemma map_subst_apply [simp]:
   114   "(m(a~>b)) x = (if m x = Some a then Some b else m x)"
   115 by (simp add: map_subst_def)
   116 
   117 *)
   118 
   119 
   120 subsection {* @{term [source] empty} *}
   121 
   122 lemma empty_upd_none [simp]: "empty(x := None) = empty"
   123   by (rule ext) simp
   124 
   125 (* FIXME: what is this sum_case nonsense?? *)
   126 lemma sum_case_empty_empty[simp]: "sum_case empty empty = empty"
   127   by (rule ext) (simp split: sum.split)
   128 
   129 
   130 subsection {* @{term [source] map_upd} *}
   131 
   132 lemma map_upd_triv: "t k = Some x ==> t(k|->x) = t"
   133   by (rule ext) simp
   134 
   135 lemma map_upd_nonempty [simp]: "t(k|->x) ~= empty"
   136 proof
   137   assume "t(k \<mapsto> x) = empty"
   138   then have "(t(k \<mapsto> x)) k = None" by simp
   139   then show False by simp
   140 qed
   141 
   142 lemma map_upd_eqD1:
   143   assumes "m(a\<mapsto>x) = n(a\<mapsto>y)"
   144   shows "x = y"
   145 proof -
   146   from prems have "(m(a\<mapsto>x)) a = (n(a\<mapsto>y)) a" by simp
   147   then show ?thesis by simp
   148 qed
   149 
   150 lemma map_upd_Some_unfold:
   151     "((m(a|->b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)"
   152   by auto
   153 
   154 lemma image_map_upd [simp]: "x \<notin> A \<Longrightarrow> m(x \<mapsto> y) ` A = m ` A"
   155   by auto
   156 
   157 lemma finite_range_updI: "finite (range f) ==> finite (range (f(a|->b)))"
   158   unfolding image_def
   159   apply (simp (no_asm_use) add: full_SetCompr_eq)
   160   apply (rule finite_subset)
   161    prefer 2 apply assumption
   162   apply auto
   163   done
   164 
   165 
   166 (* FIXME: what is this sum_case nonsense?? *)
   167 subsection {* @{term [source] sum_case} and @{term [source] empty}/@{term [source] map_upd} *}
   168 
   169 lemma sum_case_map_upd_empty [simp]:
   170     "sum_case (m(k|->y)) empty = (sum_case m empty)(Inl k|->y)"
   171   by (rule ext) (simp split: sum.split)
   172 
   173 lemma sum_case_empty_map_upd [simp]:
   174     "sum_case empty (m(k|->y)) = (sum_case empty m)(Inr k|->y)"
   175   by (rule ext) (simp split: sum.split)
   176 
   177 lemma sum_case_map_upd_map_upd [simp]:
   178     "sum_case (m1(k1|->y1)) (m2(k2|->y2)) = (sum_case (m1(k1|->y1)) m2)(Inr k2|->y2)"
   179   by (rule ext) (simp split: sum.split)
   180 
   181 
   182 subsection {* @{term [source] map_of} *}
   183 
   184 lemma map_of_eq_None_iff:
   185     "(map_of xys x = None) = (x \<notin> fst ` (set xys))"
   186   by (induct xys) simp_all
   187 
   188 lemma map_of_is_SomeD:
   189     "map_of xys x = Some y \<Longrightarrow> (x,y) \<in> set xys"
   190   apply (induct xys)
   191    apply simp
   192   apply (clarsimp split: if_splits)
   193   done
   194 
   195 lemma map_of_eq_Some_iff [simp]:
   196     "distinct(map fst xys) \<Longrightarrow> (map_of xys x = Some y) = ((x,y) \<in> set xys)"
   197   apply (induct xys)
   198    apply simp
   199   apply (auto simp: map_of_eq_None_iff [symmetric])
   200   done
   201 
   202 lemma Some_eq_map_of_iff [simp]:
   203     "distinct(map fst xys) \<Longrightarrow> (Some y = map_of xys x) = ((x,y) \<in> set xys)"
   204   by (auto simp del:map_of_eq_Some_iff simp add: map_of_eq_Some_iff [symmetric])
   205 
   206 lemma map_of_is_SomeI [simp]: "\<lbrakk> distinct(map fst xys); (x,y) \<in> set xys \<rbrakk>
   207     \<Longrightarrow> map_of xys x = Some y"
   208   apply (induct xys)
   209    apply simp
   210   apply force
   211   done
   212 
   213 lemma map_of_zip_is_None [simp]:
   214     "length xs = length ys \<Longrightarrow> (map_of (zip xs ys) x = None) = (x \<notin> set xs)"
   215   by (induct rule: list_induct2) simp_all
   216 
   217 lemma finite_range_map_of: "finite (range (map_of xys))"
   218   apply (induct xys)
   219    apply (simp_all add: image_constant)
   220   apply (rule finite_subset)
   221    prefer 2 apply assumption
   222   apply auto
   223   done
   224 
   225 lemma map_of_SomeD: "map_of xs k = Some y \<Longrightarrow> (k, y) \<in> set xs"
   226   by (induct xs) (simp, atomize (full), auto)
   227 
   228 lemma map_of_mapk_SomeI:
   229   assumes "inj f"
   230   shows "map_of t k = Some x ==>
   231     map_of (map (split (%k. Pair (f k))) t) (f k) = Some x"
   232   by (induct t) (auto simp add: `inj f` inj_eq)
   233 
   234 lemma weak_map_of_SomeI: "(k, x) : set l ==> \<exists>x. map_of l k = Some x"
   235   by (induct l) auto
   236 
   237 lemma map_of_filter_in:
   238   assumes 1: "map_of xs k = Some z"
   239     and 2: "P k z"
   240   shows "map_of (filter (split P) xs) k = Some z"
   241   using 1 by (induct xs) (insert 2, auto)
   242 
   243 lemma map_of_map: "map_of (map (%(a,b). (a,f b)) xs) x = option_map f (map_of xs x)"
   244   by (induct xs) auto
   245 
   246 
   247 subsection {* @{term [source] option_map} related *}
   248 
   249 lemma option_map_o_empty [simp]: "option_map f o empty = empty"
   250   by (rule ext) simp
   251 
   252 lemma option_map_o_map_upd [simp]:
   253     "option_map f o m(a|->b) = (option_map f o m)(a|->f b)"
   254   by (rule ext) simp
   255 
   256 
   257 subsection {* @{term [source] map_comp} related *}
   258 
   259 lemma map_comp_empty [simp]:
   260     "m \<circ>\<^sub>m empty = empty"
   261     "empty \<circ>\<^sub>m m = empty"
   262   by (auto simp add: map_comp_def intro: ext split: option.splits)
   263 
   264 lemma map_comp_simps [simp]:
   265     "m2 k = None \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = None"
   266     "m2 k = Some k' \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = m1 k'"
   267   by (auto simp add: map_comp_def)
   268 
   269 lemma map_comp_Some_iff:
   270     "((m1 \<circ>\<^sub>m m2) k = Some v) = (\<exists>k'. m2 k = Some k' \<and> m1 k' = Some v)"
   271   by (auto simp add: map_comp_def split: option.splits)
   272 
   273 lemma map_comp_None_iff:
   274     "((m1 \<circ>\<^sub>m m2) k = None) = (m2 k = None \<or> (\<exists>k'. m2 k = Some k' \<and> m1 k' = None)) "
   275   by (auto simp add: map_comp_def split: option.splits)
   276 
   277 
   278 subsection {* @{text "++"} *}
   279 
   280 lemma map_add_empty[simp]: "m ++ empty = m"
   281   unfolding map_add_def by simp
   282 
   283 lemma empty_map_add[simp]: "empty ++ m = m"
   284   unfolding map_add_def by (rule ext) (simp split: option.split)
   285 
   286 lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3"
   287   unfolding map_add_def by (rule ext) (simp add: map_add_def split: option.split)
   288 
   289 lemma map_add_Some_iff:
   290     "((m ++ n) k = Some x) = (n k = Some x | n k = None & m k = Some x)"
   291   unfolding map_add_def by (simp split: option.split)
   292 
   293 lemma map_add_SomeD [dest!]:
   294     "(m ++ n) k = Some x \<Longrightarrow> n k = Some x \<or> n k = None \<and> m k = Some x"
   295   by (rule map_add_Some_iff [THEN iffD1])
   296 
   297 lemma map_add_find_right [simp]: "!!xx. n k = Some xx ==> (m ++ n) k = Some xx"
   298   by (subst map_add_Some_iff) fast
   299 
   300 lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)"
   301   unfolding map_add_def by (simp split: option.split)
   302 
   303 lemma map_add_upd[simp]: "f ++ g(x|->y) = (f ++ g)(x|->y)"
   304   unfolding map_add_def by (rule ext) simp
   305 
   306 lemma map_add_upds[simp]: "m1 ++ (m2(xs[\<mapsto>]ys)) = (m1++m2)(xs[\<mapsto>]ys)"
   307   by (simp add: map_upds_def)
   308 
   309 lemma map_of_append[simp]: "map_of (xs @ ys) = map_of ys ++ map_of xs"
   310   unfolding map_add_def
   311   apply (induct xs)
   312    apply simp
   313   apply (rule ext)
   314   apply (simp split add: option.split)
   315   done
   316 
   317 lemma finite_range_map_of_map_add:
   318   "finite (range f) ==> finite (range (f ++ map_of l))"
   319   apply (induct l)
   320    apply (auto simp del: fun_upd_apply)
   321   apply (erule finite_range_updI)
   322   done
   323 
   324 lemma inj_on_map_add_dom [iff]:
   325     "inj_on (m ++ m') (dom m') = inj_on m' (dom m')"
   326   unfolding map_add_def dom_def inj_on_def
   327   by (fastsimp split: option.splits)
   328 
   329 
   330 subsection {* @{term [source] restrict_map} *}
   331 
   332 lemma restrict_map_to_empty [simp]: "m|`{} = empty"
   333   by (simp add: restrict_map_def)
   334 
   335 lemma restrict_map_empty [simp]: "empty|`D = empty"
   336   by (simp add: restrict_map_def)
   337 
   338 lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m|`A) x = m x"
   339   by (simp add: restrict_map_def)
   340 
   341 lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m|`A) x = None"
   342   by (simp add: restrict_map_def)
   343 
   344 lemma ran_restrictD: "y \<in> ran (m|`A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y"
   345   by (auto simp: restrict_map_def ran_def split: split_if_asm)
   346 
   347 lemma dom_restrict [simp]: "dom (m|`A) = dom m \<inter> A"
   348   by (auto simp: restrict_map_def dom_def split: split_if_asm)
   349 
   350 lemma restrict_upd_same [simp]: "m(x\<mapsto>y)|`(-{x}) = m|`(-{x})"
   351   by (rule ext) (auto simp: restrict_map_def)
   352 
   353 lemma restrict_restrict [simp]: "m|`A|`B = m|`(A\<inter>B)"
   354   by (rule ext) (auto simp: restrict_map_def)
   355 
   356 lemma restrict_fun_upd [simp]:
   357     "m(x := y)|`D = (if x \<in> D then (m|`(D-{x}))(x := y) else m|`D)"
   358   by (simp add: restrict_map_def expand_fun_eq)
   359 
   360 lemma fun_upd_None_restrict [simp]:
   361     "(m|`D)(x := None) = (if x:D then m|`(D - {x}) else m|`D)"
   362   by (simp add: restrict_map_def expand_fun_eq)
   363 
   364 lemma fun_upd_restrict: "(m|`D)(x := y) = (m|`(D-{x}))(x := y)"
   365   by (simp add: restrict_map_def expand_fun_eq)
   366 
   367 lemma fun_upd_restrict_conv [simp]:
   368     "x \<in> D \<Longrightarrow> (m|`D)(x := y) = (m|`(D-{x}))(x := y)"
   369   by (simp add: restrict_map_def expand_fun_eq)
   370 
   371 
   372 subsection {* @{term [source] map_upds} *}
   373 
   374 lemma map_upds_Nil1 [simp]: "m([] [|->] bs) = m"
   375   by (simp add: map_upds_def)
   376 
   377 lemma map_upds_Nil2 [simp]: "m(as [|->] []) = m"
   378   by (simp add:map_upds_def)
   379 
   380 lemma map_upds_Cons [simp]: "m(a#as [|->] b#bs) = (m(a|->b))(as[|->]bs)"
   381   by (simp add:map_upds_def)
   382 
   383 lemma map_upds_append1 [simp]: "\<And>ys m. size xs < size ys \<Longrightarrow>
   384     m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)"
   385   apply(induct xs)
   386    apply (clarsimp simp add: neq_Nil_conv)
   387   apply (case_tac ys)
   388    apply simp
   389   apply simp
   390   done
   391 
   392 lemma map_upds_list_update2_drop [simp]:
   393   "\<lbrakk>size xs \<le> i; i < size ys\<rbrakk>
   394     \<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)"
   395   apply (induct xs arbitrary: m ys i)
   396    apply simp
   397   apply (case_tac ys)
   398    apply simp
   399   apply (simp split: nat.split)
   400   done
   401 
   402 lemma map_upd_upds_conv_if:
   403   "(f(x|->y))(xs [|->] ys) =
   404    (if x : set(take (length ys) xs) then f(xs [|->] ys)
   405                                     else (f(xs [|->] ys))(x|->y))"
   406   apply (induct xs arbitrary: x y ys f)
   407    apply simp
   408   apply (case_tac ys)
   409    apply (auto split: split_if simp: fun_upd_twist)
   410   done
   411 
   412 lemma map_upds_twist [simp]:
   413     "a ~: set as ==> m(a|->b)(as[|->]bs) = m(as[|->]bs)(a|->b)"
   414   using set_take_subset by (fastsimp simp add: map_upd_upds_conv_if)
   415 
   416 lemma map_upds_apply_nontin [simp]:
   417     "x ~: set xs ==> (f(xs[|->]ys)) x = f x"
   418   apply (induct xs arbitrary: ys)
   419    apply simp
   420   apply (case_tac ys)
   421    apply (auto simp: map_upd_upds_conv_if)
   422   done
   423 
   424 lemma fun_upds_append_drop [simp]:
   425     "size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)"
   426   apply (induct xs arbitrary: m ys)
   427    apply simp
   428   apply (case_tac ys)
   429    apply simp_all
   430   done
   431 
   432 lemma fun_upds_append2_drop [simp]:
   433     "size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)"
   434   apply (induct xs arbitrary: m ys)
   435    apply simp
   436   apply (case_tac ys)
   437    apply simp_all
   438   done
   439 
   440 
   441 lemma restrict_map_upds[simp]:
   442   "\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk>
   443     \<Longrightarrow> m(xs [\<mapsto>] ys)|`D = (m|`(D - set xs))(xs [\<mapsto>] ys)"
   444   apply (induct xs arbitrary: m ys)
   445    apply simp
   446   apply (case_tac ys)
   447    apply simp
   448   apply (simp add: Diff_insert [symmetric] insert_absorb)
   449   apply (simp add: map_upd_upds_conv_if)
   450   done
   451 
   452 
   453 subsection {* @{term [source] dom} *}
   454 
   455 lemma domI: "m a = Some b ==> a : dom m"
   456   unfolding dom_def by simp
   457 (* declare domI [intro]? *)
   458 
   459 lemma domD: "a : dom m ==> \<exists>b. m a = Some b"
   460   by (cases "m a") (auto simp add: dom_def)
   461 
   462 lemma domIff [iff, simp del]: "(a : dom m) = (m a ~= None)"
   463   unfolding dom_def by simp
   464 
   465 lemma dom_empty [simp]: "dom empty = {}"
   466   unfolding dom_def by simp
   467 
   468 lemma dom_fun_upd [simp]:
   469     "dom(f(x := y)) = (if y=None then dom f - {x} else insert x (dom f))"
   470   unfolding dom_def by auto
   471 
   472 lemma dom_map_of: "dom(map_of xys) = {x. \<exists>y. (x,y) : set xys}"
   473   by (induct xys) (auto simp del: fun_upd_apply)
   474 
   475 lemma dom_map_of_conv_image_fst:
   476     "dom(map_of xys) = fst ` (set xys)"
   477   unfolding dom_map_of by force
   478 
   479 lemma dom_map_of_zip [simp]: "[| length xs = length ys; distinct xs |] ==>
   480     dom(map_of(zip xs ys)) = set xs"
   481   by (induct rule: list_induct2) simp_all
   482 
   483 lemma finite_dom_map_of: "finite (dom (map_of l))"
   484   unfolding dom_def
   485   by (induct l) (auto simp add: insert_Collect [symmetric])
   486 
   487 lemma dom_map_upds [simp]:
   488     "dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m"
   489   apply (induct xs arbitrary: m ys)
   490    apply simp
   491   apply (case_tac ys)
   492    apply auto
   493   done
   494 
   495 lemma dom_map_add [simp]: "dom(m++n) = dom n Un dom m"
   496   unfolding dom_def by auto
   497 
   498 lemma dom_override_on [simp]:
   499   "dom(override_on f g A) =
   500     (dom f  - {a. a : A - dom g}) Un {a. a : A Int dom g}"
   501   unfolding dom_def override_on_def by auto
   502 
   503 lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1"
   504   by (rule ext) (force simp: map_add_def dom_def split: option.split)
   505 
   506 
   507 subsection {* @{term [source] ran} *}
   508 
   509 lemma ranI: "m a = Some b ==> b : ran m"
   510   unfolding ran_def by auto
   511 (* declare ranI [intro]? *)
   512 
   513 lemma ran_empty [simp]: "ran empty = {}"
   514   unfolding ran_def by simp
   515 
   516 lemma ran_map_upd [simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)"
   517   unfolding ran_def
   518   apply auto
   519   apply (subgoal_tac "aa ~= a")
   520    apply auto
   521   done
   522 
   523 
   524 subsection {* @{text "map_le"} *}
   525 
   526 lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g"
   527   by (simp add: map_le_def)
   528 
   529 lemma upd_None_map_le [simp]: "f(x := None) \<subseteq>\<^sub>m f"
   530   by (force simp add: map_le_def)
   531 
   532 lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)"
   533   by (fastsimp simp add: map_le_def)
   534 
   535 lemma map_le_imp_upd_le [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)"
   536   by (force simp add: map_le_def)
   537 
   538 lemma map_le_upds [simp]:
   539     "f \<subseteq>\<^sub>m g ==> f(as [|->] bs) \<subseteq>\<^sub>m g(as [|->] bs)"
   540   apply (induct as arbitrary: f g bs)
   541    apply simp
   542   apply (case_tac bs)
   543    apply auto
   544   done
   545 
   546 lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)"
   547   by (fastsimp simp add: map_le_def dom_def)
   548 
   549 lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f"
   550   by (simp add: map_le_def)
   551 
   552 lemma map_le_trans[trans]: "\<lbrakk> m1 \<subseteq>\<^sub>m m2; m2 \<subseteq>\<^sub>m m3\<rbrakk> \<Longrightarrow> m1 \<subseteq>\<^sub>m m3"
   553   by (auto simp add: map_le_def dom_def)
   554 
   555 lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g"
   556   unfolding map_le_def
   557   apply (rule ext)
   558   apply (case_tac "x \<in> dom f", simp)
   559   apply (case_tac "x \<in> dom g", simp, fastsimp)
   560   done
   561 
   562 lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)"
   563   by (fastsimp simp add: map_le_def)
   564 
   565 lemma map_le_iff_map_add_commute: "(f \<subseteq>\<^sub>m f ++ g) = (f++g = g++f)"
   566   by (fastsimp simp add: map_add_def map_le_def expand_fun_eq split: option.splits)
   567 
   568 lemma map_add_le_mapE: "f++g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h"
   569   by (fastsimp simp add: map_le_def map_add_def dom_def)
   570 
   571 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"
   572   by (clarsimp simp add: map_le_def map_add_def dom_def split: option.splits)
   573 
   574 end