src/HOL/Map.thy
author haftmann
Fri Jan 25 14:54:41 2008 +0100 (2008-01-25)
changeset 25965 05df64f786a4
parent 25670 497474b69c86
child 26443 cae9fa186541
permissions -rw-r--r--
improved code theorem setup
nipkow@3981
     1
(*  Title:      HOL/Map.thy
nipkow@3981
     2
    ID:         $Id$
nipkow@3981
     3
    Author:     Tobias Nipkow, based on a theory by David von Oheimb
webertj@13908
     4
    Copyright   1997-2003 TU Muenchen
nipkow@3981
     5
nipkow@3981
     6
The datatype of `maps' (written ~=>); strongly resembles maps in VDM.
nipkow@3981
     7
*)
nipkow@3981
     8
nipkow@13914
     9
header {* Maps *}
nipkow@13914
    10
nipkow@15131
    11
theory Map
nipkow@15140
    12
imports List
nipkow@15131
    13
begin
nipkow@3981
    14
haftmann@25490
    15
types ('a,'b) "~=>" = "'a => 'b option"  (infixr 0)
oheimb@14100
    16
translations (type) "a ~=> b " <= (type) "a => b option"
nipkow@3981
    17
wenzelm@19656
    18
syntax (xsymbols)
haftmann@25490
    19
  "~=>" :: "[type, type] => type"  (infixr "\<rightharpoonup>" 0)
wenzelm@19656
    20
nipkow@19378
    21
abbreviation
wenzelm@21404
    22
  empty :: "'a ~=> 'b" where
nipkow@19378
    23
  "empty == %x. None"
nipkow@19378
    24
wenzelm@19656
    25
definition
haftmann@25670
    26
  map_comp :: "('b ~=> 'c) => ('a ~=> 'b) => ('a ~=> 'c)"  (infixl "o'_m" 55) where
wenzelm@20800
    27
  "f o_m g = (\<lambda>k. case g k of None \<Rightarrow> None | Some v \<Rightarrow> f v)"
nipkow@19378
    28
wenzelm@21210
    29
notation (xsymbols)
wenzelm@19656
    30
  map_comp  (infixl "\<circ>\<^sub>m" 55)
wenzelm@19656
    31
wenzelm@20800
    32
definition
wenzelm@21404
    33
  map_add :: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)"  (infixl "++" 100) where
wenzelm@20800
    34
  "m1 ++ m2 = (\<lambda>x. case m2 x of None => m1 x | Some y => Some y)"
wenzelm@20800
    35
wenzelm@21404
    36
definition
wenzelm@21404
    37
  restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)"  (infixl "|`"  110) where
wenzelm@20800
    38
  "m|`A = (\<lambda>x. if x : A then m x else None)"
nipkow@13910
    39
wenzelm@21210
    40
notation (latex output)
wenzelm@19656
    41
  restrict_map  ("_\<restriction>\<^bsub>_\<^esub>" [111,110] 110)
wenzelm@19656
    42
wenzelm@20800
    43
definition
wenzelm@21404
    44
  dom :: "('a ~=> 'b) => 'a set" where
wenzelm@20800
    45
  "dom m = {a. m a ~= None}"
wenzelm@20800
    46
wenzelm@21404
    47
definition
wenzelm@21404
    48
  ran :: "('a ~=> 'b) => 'b set" where
wenzelm@20800
    49
  "ran m = {b. EX a. m a = Some b}"
wenzelm@20800
    50
wenzelm@21404
    51
definition
wenzelm@21404
    52
  map_le :: "('a ~=> 'b) => ('a ~=> 'b) => bool"  (infix "\<subseteq>\<^sub>m" 50) where
wenzelm@20800
    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)"
wenzelm@20800
    54
wenzelm@20800
    55
consts
wenzelm@20800
    56
  map_of :: "('a * 'b) list => 'a ~=> 'b"
wenzelm@20800
    57
  map_upds :: "('a ~=> 'b) => 'a list => 'b list => ('a ~=> 'b)"
wenzelm@20800
    58
nipkow@14180
    59
nonterminals
nipkow@14180
    60
  maplets maplet
nipkow@14180
    61
oheimb@5300
    62
syntax
nipkow@14180
    63
  "_maplet"  :: "['a, 'a] => maplet"             ("_ /|->/ _")
nipkow@14180
    64
  "_maplets" :: "['a, 'a] => maplet"             ("_ /[|->]/ _")
nipkow@14180
    65
  ""         :: "maplet => maplets"             ("_")
nipkow@14180
    66
  "_Maplets" :: "[maplet, maplets] => maplets" ("_,/ _")
nipkow@14180
    67
  "_MapUpd"  :: "['a ~=> 'b, maplets] => 'a ~=> 'b" ("_/'(_')" [900,0]900)
nipkow@14180
    68
  "_Map"     :: "maplets => 'a ~=> 'b"            ("(1[_])")
nipkow@3981
    69
wenzelm@12114
    70
syntax (xsymbols)
nipkow@14180
    71
  "_maplet"  :: "['a, 'a] => maplet"             ("_ /\<mapsto>/ _")
nipkow@14180
    72
  "_maplets" :: "['a, 'a] => maplet"             ("_ /[\<mapsto>]/ _")
nipkow@14180
    73
oheimb@5300
    74
translations
nipkow@14180
    75
  "_MapUpd m (_Maplets xy ms)"  == "_MapUpd (_MapUpd m xy) ms"
nipkow@14180
    76
  "_MapUpd m (_maplet  x y)"    == "m(x:=Some y)"
nipkow@14180
    77
  "_MapUpd m (_maplets x y)"    == "map_upds m x y"
wenzelm@19947
    78
  "_Map ms"                     == "_MapUpd (CONST empty) ms"
nipkow@14180
    79
  "_Map (_Maplets ms1 ms2)"     <= "_MapUpd (_Map ms1) ms2"
nipkow@14180
    80
  "_Maplets ms1 (_Maplets ms2 ms3)" <= "_Maplets (_Maplets ms1 ms2) ms3"
nipkow@14180
    81
berghofe@5183
    82
primrec
berghofe@5183
    83
  "map_of [] = empty"
oheimb@5300
    84
  "map_of (p#ps) = (map_of ps)(fst p |-> snd p)"
oheimb@5300
    85
haftmann@25965
    86
declare map_of.simps [code del]
haftmann@25965
    87
haftmann@25965
    88
lemma map_of_Cons_code [code]: 
haftmann@25965
    89
  "map_of [] k = None"
haftmann@25965
    90
  "map_of ((l, v) # ps) k = (if l = k then Some v else map_of ps k)"
haftmann@25965
    91
  by simp_all
haftmann@25965
    92
wenzelm@20800
    93
defs
haftmann@22744
    94
  map_upds_def [code func]: "m(xs [|->] ys) == m ++ map_of (rev(zip xs ys))"
wenzelm@20800
    95
wenzelm@20800
    96
wenzelm@17399
    97
subsection {* @{term [source] empty} *}
webertj@13908
    98
wenzelm@20800
    99
lemma empty_upd_none [simp]: "empty(x := None) = empty"
nipkow@24331
   100
by (rule ext) simp
webertj@13908
   101
webertj@13908
   102
wenzelm@17399
   103
subsection {* @{term [source] map_upd} *}
webertj@13908
   104
webertj@13908
   105
lemma map_upd_triv: "t k = Some x ==> t(k|->x) = t"
nipkow@24331
   106
by (rule ext) simp
webertj@13908
   107
wenzelm@20800
   108
lemma map_upd_nonempty [simp]: "t(k|->x) ~= empty"
wenzelm@20800
   109
proof
wenzelm@20800
   110
  assume "t(k \<mapsto> x) = empty"
wenzelm@20800
   111
  then have "(t(k \<mapsto> x)) k = None" by simp
wenzelm@20800
   112
  then show False by simp
wenzelm@20800
   113
qed
webertj@13908
   114
wenzelm@20800
   115
lemma map_upd_eqD1:
wenzelm@20800
   116
  assumes "m(a\<mapsto>x) = n(a\<mapsto>y)"
wenzelm@20800
   117
  shows "x = y"
wenzelm@20800
   118
proof -
wenzelm@20800
   119
  from prems have "(m(a\<mapsto>x)) a = (n(a\<mapsto>y)) a" by simp
wenzelm@20800
   120
  then show ?thesis by simp
wenzelm@20800
   121
qed
oheimb@14100
   122
wenzelm@20800
   123
lemma map_upd_Some_unfold:
nipkow@24331
   124
  "((m(a|->b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)"
nipkow@24331
   125
by auto
oheimb@14100
   126
wenzelm@20800
   127
lemma image_map_upd [simp]: "x \<notin> A \<Longrightarrow> m(x \<mapsto> y) ` A = m ` A"
nipkow@24331
   128
by auto
nipkow@15303
   129
webertj@13908
   130
lemma finite_range_updI: "finite (range f) ==> finite (range (f(a|->b)))"
nipkow@24331
   131
unfolding image_def
nipkow@24331
   132
apply (simp (no_asm_use) add:full_SetCompr_eq)
nipkow@24331
   133
apply (rule finite_subset)
nipkow@24331
   134
 prefer 2 apply assumption
nipkow@24331
   135
apply (auto)
nipkow@24331
   136
done
webertj@13908
   137
webertj@13908
   138
wenzelm@17399
   139
subsection {* @{term [source] map_of} *}
webertj@13908
   140
nipkow@15304
   141
lemma map_of_eq_None_iff:
nipkow@24331
   142
  "(map_of xys x = None) = (x \<notin> fst ` (set xys))"
nipkow@24331
   143
by (induct xys) simp_all
nipkow@15304
   144
nipkow@24331
   145
lemma map_of_is_SomeD: "map_of xys x = Some y \<Longrightarrow> (x,y) \<in> set xys"
nipkow@24331
   146
apply (induct xys)
nipkow@24331
   147
 apply simp
nipkow@24331
   148
apply (clarsimp split: if_splits)
nipkow@24331
   149
done
nipkow@15304
   150
wenzelm@20800
   151
lemma map_of_eq_Some_iff [simp]:
nipkow@24331
   152
  "distinct(map fst xys) \<Longrightarrow> (map_of xys x = Some y) = ((x,y) \<in> set xys)"
nipkow@24331
   153
apply (induct xys)
nipkow@24331
   154
 apply simp
nipkow@24331
   155
apply (auto simp: map_of_eq_None_iff [symmetric])
nipkow@24331
   156
done
nipkow@15304
   157
wenzelm@20800
   158
lemma Some_eq_map_of_iff [simp]:
nipkow@24331
   159
  "distinct(map fst xys) \<Longrightarrow> (Some y = map_of xys x) = ((x,y) \<in> set xys)"
nipkow@24331
   160
by (auto simp del:map_of_eq_Some_iff simp add: map_of_eq_Some_iff [symmetric])
nipkow@15304
   161
paulson@17724
   162
lemma map_of_is_SomeI [simp]: "\<lbrakk> distinct(map fst xys); (x,y) \<in> set xys \<rbrakk>
wenzelm@20800
   163
    \<Longrightarrow> map_of xys x = Some y"
nipkow@24331
   164
apply (induct xys)
nipkow@24331
   165
 apply simp
nipkow@24331
   166
apply force
nipkow@24331
   167
done
nipkow@15304
   168
wenzelm@20800
   169
lemma map_of_zip_is_None [simp]:
nipkow@24331
   170
  "length xs = length ys \<Longrightarrow> (map_of (zip xs ys) x = None) = (x \<notin> set xs)"
nipkow@24331
   171
by (induct rule: list_induct2) simp_all
nipkow@15110
   172
nipkow@15110
   173
lemma finite_range_map_of: "finite (range (map_of xys))"
nipkow@24331
   174
apply (induct xys)
nipkow@24331
   175
 apply (simp_all add: image_constant)
nipkow@24331
   176
apply (rule finite_subset)
nipkow@24331
   177
 prefer 2 apply assumption
nipkow@24331
   178
apply auto
nipkow@24331
   179
done
nipkow@15110
   180
wenzelm@20800
   181
lemma map_of_SomeD: "map_of xs k = Some y \<Longrightarrow> (k, y) \<in> set xs"
nipkow@24331
   182
by (induct xs) (simp, atomize (full), auto)
webertj@13908
   183
wenzelm@20800
   184
lemma map_of_mapk_SomeI:
nipkow@24331
   185
  "inj f ==> map_of t k = Some x ==>
nipkow@24331
   186
   map_of (map (split (%k. Pair (f k))) t) (f k) = Some x"
nipkow@24331
   187
by (induct t) (auto simp add: inj_eq)
webertj@13908
   188
wenzelm@20800
   189
lemma weak_map_of_SomeI: "(k, x) : set l ==> \<exists>x. map_of l k = Some x"
nipkow@24331
   190
by (induct l) auto
webertj@13908
   191
wenzelm@20800
   192
lemma map_of_filter_in:
nipkow@24331
   193
  "map_of xs k = Some z \<Longrightarrow> P k z \<Longrightarrow> map_of (filter (split P) xs) k = Some z"
nipkow@24331
   194
by (induct xs) auto
webertj@13908
   195
webertj@13908
   196
lemma map_of_map: "map_of (map (%(a,b). (a,f b)) xs) x = option_map f (map_of xs x)"
nipkow@24331
   197
by (induct xs) auto
webertj@13908
   198
webertj@13908
   199
wenzelm@17399
   200
subsection {* @{term [source] option_map} related *}
webertj@13908
   201
wenzelm@20800
   202
lemma option_map_o_empty [simp]: "option_map f o empty = empty"
nipkow@24331
   203
by (rule ext) simp
webertj@13908
   204
wenzelm@20800
   205
lemma option_map_o_map_upd [simp]:
nipkow@24331
   206
  "option_map f o m(a|->b) = (option_map f o m)(a|->f b)"
nipkow@24331
   207
by (rule ext) simp
wenzelm@20800
   208
webertj@13908
   209
wenzelm@17399
   210
subsection {* @{term [source] map_comp} related *}
schirmer@17391
   211
wenzelm@20800
   212
lemma map_comp_empty [simp]:
nipkow@24331
   213
  "m \<circ>\<^sub>m empty = empty"
nipkow@24331
   214
  "empty \<circ>\<^sub>m m = empty"
nipkow@24331
   215
by (auto simp add: map_comp_def intro: ext split: option.splits)
schirmer@17391
   216
wenzelm@20800
   217
lemma map_comp_simps [simp]:
nipkow@24331
   218
  "m2 k = None \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = None"
nipkow@24331
   219
  "m2 k = Some k' \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = m1 k'"
nipkow@24331
   220
by (auto simp add: map_comp_def)
schirmer@17391
   221
schirmer@17391
   222
lemma map_comp_Some_iff:
nipkow@24331
   223
  "((m1 \<circ>\<^sub>m m2) k = Some v) = (\<exists>k'. m2 k = Some k' \<and> m1 k' = Some v)"
nipkow@24331
   224
by (auto simp add: map_comp_def split: option.splits)
schirmer@17391
   225
schirmer@17391
   226
lemma map_comp_None_iff:
nipkow@24331
   227
  "((m1 \<circ>\<^sub>m m2) k = None) = (m2 k = None \<or> (\<exists>k'. m2 k = Some k' \<and> m1 k' = None)) "
nipkow@24331
   228
by (auto simp add: map_comp_def split: option.splits)
webertj@13908
   229
wenzelm@20800
   230
oheimb@14100
   231
subsection {* @{text "++"} *}
webertj@13908
   232
nipkow@14025
   233
lemma map_add_empty[simp]: "m ++ empty = m"
nipkow@24331
   234
by(simp add: map_add_def)
webertj@13908
   235
nipkow@14025
   236
lemma empty_map_add[simp]: "empty ++ m = m"
nipkow@24331
   237
by (rule ext) (simp add: map_add_def split: option.split)
webertj@13908
   238
nipkow@14025
   239
lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3"
nipkow@24331
   240
by (rule ext) (simp add: map_add_def split: option.split)
wenzelm@20800
   241
wenzelm@20800
   242
lemma map_add_Some_iff:
nipkow@24331
   243
  "((m ++ n) k = Some x) = (n k = Some x | n k = None & m k = Some x)"
nipkow@24331
   244
by (simp add: map_add_def split: option.split)
nipkow@14025
   245
wenzelm@20800
   246
lemma map_add_SomeD [dest!]:
nipkow@24331
   247
  "(m ++ n) k = Some x \<Longrightarrow> n k = Some x \<or> n k = None \<and> m k = Some x"
nipkow@24331
   248
by (rule map_add_Some_iff [THEN iffD1])
webertj@13908
   249
wenzelm@20800
   250
lemma map_add_find_right [simp]: "!!xx. n k = Some xx ==> (m ++ n) k = Some xx"
nipkow@24331
   251
by (subst map_add_Some_iff) fast
webertj@13908
   252
nipkow@14025
   253
lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)"
nipkow@24331
   254
by (simp add: map_add_def split: option.split)
webertj@13908
   255
nipkow@14025
   256
lemma map_add_upd[simp]: "f ++ g(x|->y) = (f ++ g)(x|->y)"
nipkow@24331
   257
by (rule ext) (simp add: map_add_def)
webertj@13908
   258
nipkow@14186
   259
lemma map_add_upds[simp]: "m1 ++ (m2(xs[\<mapsto>]ys)) = (m1++m2)(xs[\<mapsto>]ys)"
nipkow@24331
   260
by (simp add: map_upds_def)
nipkow@14186
   261
wenzelm@20800
   262
lemma map_of_append[simp]: "map_of (xs @ ys) = map_of ys ++ map_of xs"
nipkow@24331
   263
unfolding map_add_def
nipkow@24331
   264
apply (induct xs)
nipkow@24331
   265
 apply simp
nipkow@24331
   266
apply (rule ext)
nipkow@24331
   267
apply (simp split add: option.split)
nipkow@24331
   268
done
webertj@13908
   269
nipkow@14025
   270
lemma finite_range_map_of_map_add:
wenzelm@20800
   271
  "finite (range f) ==> finite (range (f ++ map_of l))"
nipkow@24331
   272
apply (induct l)
nipkow@24331
   273
 apply (auto simp del: fun_upd_apply)
nipkow@24331
   274
apply (erule finite_range_updI)
nipkow@24331
   275
done
webertj@13908
   276
wenzelm@20800
   277
lemma inj_on_map_add_dom [iff]:
nipkow@24331
   278
  "inj_on (m ++ m') (dom m') = inj_on m' (dom m')"
nipkow@24331
   279
by (fastsimp simp: map_add_def dom_def inj_on_def split: option.splits)
wenzelm@20800
   280
nipkow@15304
   281
wenzelm@17399
   282
subsection {* @{term [source] restrict_map} *}
oheimb@14100
   283
wenzelm@20800
   284
lemma restrict_map_to_empty [simp]: "m|`{} = empty"
nipkow@24331
   285
by (simp add: restrict_map_def)
nipkow@14186
   286
wenzelm@20800
   287
lemma restrict_map_empty [simp]: "empty|`D = empty"
nipkow@24331
   288
by (simp add: restrict_map_def)
nipkow@14186
   289
nipkow@15693
   290
lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m|`A) x = m x"
nipkow@24331
   291
by (simp add: restrict_map_def)
oheimb@14100
   292
nipkow@15693
   293
lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m|`A) x = None"
nipkow@24331
   294
by (simp add: restrict_map_def)
oheimb@14100
   295
nipkow@15693
   296
lemma ran_restrictD: "y \<in> ran (m|`A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y"
nipkow@24331
   297
by (auto simp: restrict_map_def ran_def split: split_if_asm)
oheimb@14100
   298
nipkow@15693
   299
lemma dom_restrict [simp]: "dom (m|`A) = dom m \<inter> A"
nipkow@24331
   300
by (auto simp: restrict_map_def dom_def split: split_if_asm)
oheimb@14100
   301
nipkow@15693
   302
lemma restrict_upd_same [simp]: "m(x\<mapsto>y)|`(-{x}) = m|`(-{x})"
nipkow@24331
   303
by (rule ext) (auto simp: restrict_map_def)
oheimb@14100
   304
nipkow@15693
   305
lemma restrict_restrict [simp]: "m|`A|`B = m|`(A\<inter>B)"
nipkow@24331
   306
by (rule ext) (auto simp: restrict_map_def)
oheimb@14100
   307
wenzelm@20800
   308
lemma restrict_fun_upd [simp]:
nipkow@24331
   309
  "m(x := y)|`D = (if x \<in> D then (m|`(D-{x}))(x := y) else m|`D)"
nipkow@24331
   310
by (simp add: restrict_map_def expand_fun_eq)
nipkow@14186
   311
wenzelm@20800
   312
lemma fun_upd_None_restrict [simp]:
nipkow@24331
   313
  "(m|`D)(x := None) = (if x:D then m|`(D - {x}) else m|`D)"
nipkow@24331
   314
by (simp add: restrict_map_def expand_fun_eq)
nipkow@14186
   315
wenzelm@20800
   316
lemma fun_upd_restrict: "(m|`D)(x := y) = (m|`(D-{x}))(x := y)"
nipkow@24331
   317
by (simp add: restrict_map_def expand_fun_eq)
nipkow@14186
   318
wenzelm@20800
   319
lemma fun_upd_restrict_conv [simp]:
nipkow@24331
   320
  "x \<in> D \<Longrightarrow> (m|`D)(x := y) = (m|`(D-{x}))(x := y)"
nipkow@24331
   321
by (simp add: restrict_map_def expand_fun_eq)
nipkow@14186
   322
oheimb@14100
   323
wenzelm@17399
   324
subsection {* @{term [source] map_upds} *}
nipkow@14025
   325
wenzelm@20800
   326
lemma map_upds_Nil1 [simp]: "m([] [|->] bs) = m"
nipkow@24331
   327
by (simp add: map_upds_def)
nipkow@14025
   328
wenzelm@20800
   329
lemma map_upds_Nil2 [simp]: "m(as [|->] []) = m"
nipkow@24331
   330
by (simp add:map_upds_def)
wenzelm@20800
   331
wenzelm@20800
   332
lemma map_upds_Cons [simp]: "m(a#as [|->] b#bs) = (m(a|->b))(as[|->]bs)"
nipkow@24331
   333
by (simp add:map_upds_def)
nipkow@14025
   334
wenzelm@20800
   335
lemma map_upds_append1 [simp]: "\<And>ys m. size xs < size ys \<Longrightarrow>
nipkow@24331
   336
  m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)"
nipkow@24331
   337
apply(induct xs)
nipkow@24331
   338
 apply (clarsimp simp add: neq_Nil_conv)
nipkow@24331
   339
apply (case_tac ys)
nipkow@24331
   340
 apply simp
nipkow@24331
   341
apply simp
nipkow@24331
   342
done
nipkow@14187
   343
wenzelm@20800
   344
lemma map_upds_list_update2_drop [simp]:
wenzelm@20800
   345
  "\<lbrakk>size xs \<le> i; i < size ys\<rbrakk>
wenzelm@20800
   346
    \<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)"
nipkow@24331
   347
apply (induct xs arbitrary: m ys i)
nipkow@24331
   348
 apply simp
nipkow@24331
   349
apply (case_tac ys)
nipkow@24331
   350
 apply simp
nipkow@24331
   351
apply (simp split: nat.split)
nipkow@24331
   352
done
nipkow@14025
   353
wenzelm@20800
   354
lemma map_upd_upds_conv_if:
wenzelm@20800
   355
  "(f(x|->y))(xs [|->] ys) =
wenzelm@20800
   356
   (if x : set(take (length ys) xs) then f(xs [|->] ys)
wenzelm@20800
   357
                                    else (f(xs [|->] ys))(x|->y))"
nipkow@24331
   358
apply (induct xs arbitrary: x y ys f)
nipkow@24331
   359
 apply simp
nipkow@24331
   360
apply (case_tac ys)
nipkow@24331
   361
 apply (auto split: split_if simp: fun_upd_twist)
nipkow@24331
   362
done
nipkow@14025
   363
nipkow@14025
   364
lemma map_upds_twist [simp]:
nipkow@24331
   365
  "a ~: set as ==> m(a|->b)(as[|->]bs) = m(as[|->]bs)(a|->b)"
nipkow@24331
   366
using set_take_subset by (fastsimp simp add: map_upd_upds_conv_if)
nipkow@14025
   367
wenzelm@20800
   368
lemma map_upds_apply_nontin [simp]:
nipkow@24331
   369
  "x ~: set xs ==> (f(xs[|->]ys)) x = f x"
nipkow@24331
   370
apply (induct xs arbitrary: ys)
nipkow@24331
   371
 apply simp
nipkow@24331
   372
apply (case_tac ys)
nipkow@24331
   373
 apply (auto simp: map_upd_upds_conv_if)
nipkow@24331
   374
done
nipkow@14025
   375
wenzelm@20800
   376
lemma fun_upds_append_drop [simp]:
nipkow@24331
   377
  "size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)"
nipkow@24331
   378
apply (induct xs arbitrary: m ys)
nipkow@24331
   379
 apply simp
nipkow@24331
   380
apply (case_tac ys)
nipkow@24331
   381
 apply simp_all
nipkow@24331
   382
done
nipkow@14300
   383
wenzelm@20800
   384
lemma fun_upds_append2_drop [simp]:
nipkow@24331
   385
  "size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)"
nipkow@24331
   386
apply (induct xs arbitrary: m ys)
nipkow@24331
   387
 apply simp
nipkow@24331
   388
apply (case_tac ys)
nipkow@24331
   389
 apply simp_all
nipkow@24331
   390
done
nipkow@14300
   391
nipkow@14300
   392
wenzelm@20800
   393
lemma restrict_map_upds[simp]:
wenzelm@20800
   394
  "\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk>
wenzelm@20800
   395
    \<Longrightarrow> m(xs [\<mapsto>] ys)|`D = (m|`(D - set xs))(xs [\<mapsto>] ys)"
nipkow@24331
   396
apply (induct xs arbitrary: m ys)
nipkow@24331
   397
 apply simp
nipkow@24331
   398
apply (case_tac ys)
nipkow@24331
   399
 apply simp
nipkow@24331
   400
apply (simp add: Diff_insert [symmetric] insert_absorb)
nipkow@24331
   401
apply (simp add: map_upd_upds_conv_if)
nipkow@24331
   402
done
nipkow@14186
   403
nipkow@14186
   404
wenzelm@17399
   405
subsection {* @{term [source] dom} *}
webertj@13908
   406
webertj@13908
   407
lemma domI: "m a = Some b ==> a : dom m"
nipkow@24331
   408
by(simp add:dom_def)
oheimb@14100
   409
(* declare domI [intro]? *)
webertj@13908
   410
paulson@15369
   411
lemma domD: "a : dom m ==> \<exists>b. m a = Some b"
nipkow@24331
   412
by (cases "m a") (auto simp add: dom_def)
webertj@13908
   413
wenzelm@20800
   414
lemma domIff [iff, simp del]: "(a : dom m) = (m a ~= None)"
nipkow@24331
   415
by(simp add:dom_def)
webertj@13908
   416
wenzelm@20800
   417
lemma dom_empty [simp]: "dom empty = {}"
nipkow@24331
   418
by(simp add:dom_def)
webertj@13908
   419
wenzelm@20800
   420
lemma dom_fun_upd [simp]:
nipkow@24331
   421
  "dom(f(x := y)) = (if y=None then dom f - {x} else insert x (dom f))"
nipkow@24331
   422
by(auto simp add:dom_def)
webertj@13908
   423
nipkow@13937
   424
lemma dom_map_of: "dom(map_of xys) = {x. \<exists>y. (x,y) : set xys}"
nipkow@24331
   425
by (induct xys) (auto simp del: fun_upd_apply)
nipkow@13937
   426
nipkow@15304
   427
lemma dom_map_of_conv_image_fst:
nipkow@24331
   428
  "dom(map_of xys) = fst ` (set xys)"
nipkow@24331
   429
by(force simp: dom_map_of)
nipkow@15304
   430
wenzelm@20800
   431
lemma dom_map_of_zip [simp]: "[| length xs = length ys; distinct xs |] ==>
nipkow@24331
   432
  dom(map_of(zip xs ys)) = set xs"
nipkow@24331
   433
by (induct rule: list_induct2) simp_all
nipkow@15110
   434
webertj@13908
   435
lemma finite_dom_map_of: "finite (dom (map_of l))"
nipkow@24331
   436
by (induct l) (auto simp add: dom_def insert_Collect [symmetric])
webertj@13908
   437
wenzelm@20800
   438
lemma dom_map_upds [simp]:
nipkow@24331
   439
  "dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m"
nipkow@24331
   440
apply (induct xs arbitrary: m ys)
nipkow@24331
   441
 apply simp
nipkow@24331
   442
apply (case_tac ys)
nipkow@24331
   443
 apply auto
nipkow@24331
   444
done
nipkow@13910
   445
wenzelm@20800
   446
lemma dom_map_add [simp]: "dom(m++n) = dom n Un dom m"
nipkow@24331
   447
by(auto simp:dom_def)
nipkow@13910
   448
wenzelm@20800
   449
lemma dom_override_on [simp]:
wenzelm@20800
   450
  "dom(override_on f g A) =
wenzelm@20800
   451
    (dom f  - {a. a : A - dom g}) Un {a. a : A Int dom g}"
nipkow@24331
   452
by(auto simp: dom_def override_on_def)
webertj@13908
   453
nipkow@14027
   454
lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1"
nipkow@24331
   455
by (rule ext) (force simp: map_add_def dom_def split: option.split)
wenzelm@20800
   456
nipkow@22230
   457
(* Due to John Matthews - could be rephrased with dom *)
nipkow@22230
   458
lemma finite_map_freshness:
nipkow@22230
   459
  "finite (dom (f :: 'a \<rightharpoonup> 'b)) \<Longrightarrow> \<not> finite (UNIV :: 'a set) \<Longrightarrow>
nipkow@22230
   460
   \<exists>x. f x = None"
nipkow@22230
   461
by(bestsimp dest:ex_new_if_finite)
nipkow@14027
   462
wenzelm@17399
   463
subsection {* @{term [source] ran} *}
oheimb@14100
   464
wenzelm@20800
   465
lemma ranI: "m a = Some b ==> b : ran m"
nipkow@24331
   466
by(auto simp: ran_def)
oheimb@14100
   467
(* declare ranI [intro]? *)
webertj@13908
   468
wenzelm@20800
   469
lemma ran_empty [simp]: "ran empty = {}"
nipkow@24331
   470
by(auto simp: ran_def)
webertj@13908
   471
wenzelm@20800
   472
lemma ran_map_upd [simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)"
nipkow@24331
   473
unfolding ran_def
nipkow@24331
   474
apply auto
nipkow@24331
   475
apply (subgoal_tac "aa ~= a")
nipkow@24331
   476
 apply auto
nipkow@24331
   477
done
wenzelm@20800
   478
nipkow@13910
   479
oheimb@14100
   480
subsection {* @{text "map_le"} *}
nipkow@13910
   481
kleing@13912
   482
lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g"
nipkow@24331
   483
by (simp add: map_le_def)
nipkow@13910
   484
paulson@17724
   485
lemma upd_None_map_le [simp]: "f(x := None) \<subseteq>\<^sub>m f"
nipkow@24331
   486
by (force simp add: map_le_def)
nipkow@14187
   487
nipkow@13910
   488
lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)"
nipkow@24331
   489
by (fastsimp simp add: map_le_def)
nipkow@13910
   490
paulson@17724
   491
lemma map_le_imp_upd_le [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)"
nipkow@24331
   492
by (force simp add: map_le_def)
nipkow@14187
   493
wenzelm@20800
   494
lemma map_le_upds [simp]:
nipkow@24331
   495
  "f \<subseteq>\<^sub>m g ==> f(as [|->] bs) \<subseteq>\<^sub>m g(as [|->] bs)"
nipkow@24331
   496
apply (induct as arbitrary: f g bs)
nipkow@24331
   497
 apply simp
nipkow@24331
   498
apply (case_tac bs)
nipkow@24331
   499
 apply auto
nipkow@24331
   500
done
webertj@13908
   501
webertj@14033
   502
lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)"
nipkow@24331
   503
by (fastsimp simp add: map_le_def dom_def)
webertj@14033
   504
webertj@14033
   505
lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f"
nipkow@24331
   506
by (simp add: map_le_def)
webertj@14033
   507
nipkow@14187
   508
lemma map_le_trans[trans]: "\<lbrakk> m1 \<subseteq>\<^sub>m m2; m2 \<subseteq>\<^sub>m m3\<rbrakk> \<Longrightarrow> m1 \<subseteq>\<^sub>m m3"
nipkow@24331
   509
by (auto simp add: map_le_def dom_def)
webertj@14033
   510
webertj@14033
   511
lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g"
nipkow@24331
   512
unfolding map_le_def
nipkow@24331
   513
apply (rule ext)
nipkow@24331
   514
apply (case_tac "x \<in> dom f", simp)
nipkow@24331
   515
apply (case_tac "x \<in> dom g", simp, fastsimp)
nipkow@24331
   516
done
webertj@14033
   517
webertj@14033
   518
lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)"
nipkow@24331
   519
by (fastsimp simp add: map_le_def)
webertj@14033
   520
nipkow@15304
   521
lemma map_le_iff_map_add_commute: "(f \<subseteq>\<^sub>m f ++ g) = (f++g = g++f)"
nipkow@24331
   522
by(fastsimp simp: map_add_def map_le_def expand_fun_eq split: option.splits)
nipkow@15304
   523
nipkow@15303
   524
lemma map_add_le_mapE: "f++g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h"
nipkow@24331
   525
by (fastsimp simp add: map_le_def map_add_def dom_def)
nipkow@15303
   526
nipkow@15303
   527
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"
nipkow@24331
   528
by (clarsimp simp add: map_le_def map_add_def dom_def split: option.splits)
nipkow@15303
   529
nipkow@3981
   530
end