src/HOL/Map.thy
author haftmann
Fri Nov 27 08:41:10 2009 +0100 (2009-11-27)
changeset 33963 977b94b64905
parent 33635 dcaada178c6f
child 34941 156925dd67af
permissions -rw-r--r--
renamed former datatype.ML to datatype_data.ML; datatype.ML provides uniform view on datatype.ML and datatype_rep_proofs.ML
nipkow@3981
     1
(*  Title:      HOL/Map.thy
nipkow@3981
     2
    Author:     Tobias Nipkow, based on a theory by David von Oheimb
webertj@13908
     3
    Copyright   1997-2003 TU Muenchen
nipkow@3981
     4
nipkow@3981
     5
The datatype of `maps' (written ~=>); strongly resembles maps in VDM.
nipkow@3981
     6
*)
nipkow@3981
     7
nipkow@13914
     8
header {* Maps *}
nipkow@13914
     9
nipkow@15131
    10
theory Map
nipkow@15140
    11
imports List
nipkow@15131
    12
begin
nipkow@3981
    13
haftmann@30935
    14
types ('a,'b) "~=>" = "'a => 'b option"  (infixr "~=>" 0)
oheimb@14100
    15
translations (type) "a ~=> b " <= (type) "a => b option"
nipkow@3981
    16
wenzelm@19656
    17
syntax (xsymbols)
haftmann@25490
    18
  "~=>" :: "[type, type] => type"  (infixr "\<rightharpoonup>" 0)
wenzelm@19656
    19
nipkow@19378
    20
abbreviation
wenzelm@21404
    21
  empty :: "'a ~=> 'b" where
nipkow@19378
    22
  "empty == %x. None"
nipkow@19378
    23
wenzelm@19656
    24
definition
haftmann@25670
    25
  map_comp :: "('b ~=> 'c) => ('a ~=> 'b) => ('a ~=> 'c)"  (infixl "o'_m" 55) where
wenzelm@20800
    26
  "f o_m g = (\<lambda>k. case g k of None \<Rightarrow> None | Some v \<Rightarrow> f v)"
nipkow@19378
    27
wenzelm@21210
    28
notation (xsymbols)
wenzelm@19656
    29
  map_comp  (infixl "\<circ>\<^sub>m" 55)
wenzelm@19656
    30
wenzelm@20800
    31
definition
wenzelm@21404
    32
  map_add :: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)"  (infixl "++" 100) where
wenzelm@20800
    33
  "m1 ++ m2 = (\<lambda>x. case m2 x of None => m1 x | Some y => Some y)"
wenzelm@20800
    34
wenzelm@21404
    35
definition
wenzelm@21404
    36
  restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)"  (infixl "|`"  110) where
wenzelm@20800
    37
  "m|`A = (\<lambda>x. if x : A then m x else None)"
nipkow@13910
    38
wenzelm@21210
    39
notation (latex output)
wenzelm@19656
    40
  restrict_map  ("_\<restriction>\<^bsub>_\<^esub>" [111,110] 110)
wenzelm@19656
    41
wenzelm@20800
    42
definition
wenzelm@21404
    43
  dom :: "('a ~=> 'b) => 'a set" where
wenzelm@20800
    44
  "dom m = {a. m a ~= None}"
wenzelm@20800
    45
wenzelm@21404
    46
definition
wenzelm@21404
    47
  ran :: "('a ~=> 'b) => 'b set" where
wenzelm@20800
    48
  "ran m = {b. EX a. m a = Some b}"
wenzelm@20800
    49
wenzelm@21404
    50
definition
wenzelm@21404
    51
  map_le :: "('a ~=> 'b) => ('a ~=> 'b) => bool"  (infix "\<subseteq>\<^sub>m" 50) where
wenzelm@20800
    52
  "(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
    53
wenzelm@20800
    54
consts
wenzelm@20800
    55
  map_of :: "('a * 'b) list => 'a ~=> 'b"
wenzelm@20800
    56
  map_upds :: "('a ~=> 'b) => 'a list => 'b list => ('a ~=> 'b)"
wenzelm@20800
    57
nipkow@14180
    58
nonterminals
nipkow@14180
    59
  maplets maplet
nipkow@14180
    60
oheimb@5300
    61
syntax
nipkow@14180
    62
  "_maplet"  :: "['a, 'a] => maplet"             ("_ /|->/ _")
nipkow@14180
    63
  "_maplets" :: "['a, 'a] => maplet"             ("_ /[|->]/ _")
nipkow@14180
    64
  ""         :: "maplet => maplets"             ("_")
nipkow@14180
    65
  "_Maplets" :: "[maplet, maplets] => maplets" ("_,/ _")
nipkow@14180
    66
  "_MapUpd"  :: "['a ~=> 'b, maplets] => 'a ~=> 'b" ("_/'(_')" [900,0]900)
nipkow@14180
    67
  "_Map"     :: "maplets => 'a ~=> 'b"            ("(1[_])")
nipkow@3981
    68
wenzelm@12114
    69
syntax (xsymbols)
nipkow@14180
    70
  "_maplet"  :: "['a, 'a] => maplet"             ("_ /\<mapsto>/ _")
nipkow@14180
    71
  "_maplets" :: "['a, 'a] => maplet"             ("_ /[\<mapsto>]/ _")
nipkow@14180
    72
oheimb@5300
    73
translations
nipkow@14180
    74
  "_MapUpd m (_Maplets xy ms)"  == "_MapUpd (_MapUpd m xy) ms"
nipkow@14180
    75
  "_MapUpd m (_maplet  x y)"    == "m(x:=Some y)"
nipkow@14180
    76
  "_MapUpd m (_maplets x y)"    == "map_upds m x y"
wenzelm@19947
    77
  "_Map ms"                     == "_MapUpd (CONST empty) ms"
nipkow@14180
    78
  "_Map (_Maplets ms1 ms2)"     <= "_MapUpd (_Map ms1) ms2"
nipkow@14180
    79
  "_Maplets ms1 (_Maplets ms2 ms3)" <= "_Maplets (_Maplets ms1 ms2) ms3"
nipkow@14180
    80
berghofe@5183
    81
primrec
berghofe@5183
    82
  "map_of [] = empty"
oheimb@5300
    83
  "map_of (p#ps) = (map_of ps)(fst p |-> snd p)"
oheimb@5300
    84
haftmann@25965
    85
declare map_of.simps [code del]
haftmann@25965
    86
haftmann@25965
    87
lemma map_of_Cons_code [code]: 
haftmann@25965
    88
  "map_of [] k = None"
haftmann@25965
    89
  "map_of ((l, v) # ps) k = (if l = k then Some v else map_of ps k)"
haftmann@25965
    90
  by simp_all
haftmann@25965
    91
wenzelm@20800
    92
defs
haftmann@28562
    93
  map_upds_def [code]: "m(xs [|->] ys) == m ++ map_of (rev(zip xs ys))"
wenzelm@20800
    94
wenzelm@20800
    95
wenzelm@17399
    96
subsection {* @{term [source] empty} *}
webertj@13908
    97
wenzelm@20800
    98
lemma empty_upd_none [simp]: "empty(x := None) = empty"
nipkow@24331
    99
by (rule ext) simp
webertj@13908
   100
webertj@13908
   101
wenzelm@17399
   102
subsection {* @{term [source] map_upd} *}
webertj@13908
   103
webertj@13908
   104
lemma map_upd_triv: "t k = Some x ==> t(k|->x) = t"
nipkow@24331
   105
by (rule ext) simp
webertj@13908
   106
wenzelm@20800
   107
lemma map_upd_nonempty [simp]: "t(k|->x) ~= empty"
wenzelm@20800
   108
proof
wenzelm@20800
   109
  assume "t(k \<mapsto> x) = empty"
wenzelm@20800
   110
  then have "(t(k \<mapsto> x)) k = None" by simp
wenzelm@20800
   111
  then show False by simp
wenzelm@20800
   112
qed
webertj@13908
   113
wenzelm@20800
   114
lemma map_upd_eqD1:
wenzelm@20800
   115
  assumes "m(a\<mapsto>x) = n(a\<mapsto>y)"
wenzelm@20800
   116
  shows "x = y"
wenzelm@20800
   117
proof -
wenzelm@20800
   118
  from prems have "(m(a\<mapsto>x)) a = (n(a\<mapsto>y)) a" by simp
wenzelm@20800
   119
  then show ?thesis by simp
wenzelm@20800
   120
qed
oheimb@14100
   121
wenzelm@20800
   122
lemma map_upd_Some_unfold:
nipkow@24331
   123
  "((m(a|->b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)"
nipkow@24331
   124
by auto
oheimb@14100
   125
wenzelm@20800
   126
lemma image_map_upd [simp]: "x \<notin> A \<Longrightarrow> m(x \<mapsto> y) ` A = m ` A"
nipkow@24331
   127
by auto
nipkow@15303
   128
webertj@13908
   129
lemma finite_range_updI: "finite (range f) ==> finite (range (f(a|->b)))"
nipkow@24331
   130
unfolding image_def
nipkow@24331
   131
apply (simp (no_asm_use) add:full_SetCompr_eq)
nipkow@24331
   132
apply (rule finite_subset)
nipkow@24331
   133
 prefer 2 apply assumption
nipkow@24331
   134
apply (auto)
nipkow@24331
   135
done
webertj@13908
   136
webertj@13908
   137
wenzelm@17399
   138
subsection {* @{term [source] map_of} *}
webertj@13908
   139
nipkow@15304
   140
lemma map_of_eq_None_iff:
nipkow@24331
   141
  "(map_of xys x = None) = (x \<notin> fst ` (set xys))"
nipkow@24331
   142
by (induct xys) simp_all
nipkow@15304
   143
nipkow@24331
   144
lemma map_of_is_SomeD: "map_of xys x = Some y \<Longrightarrow> (x,y) \<in> set xys"
nipkow@24331
   145
apply (induct xys)
nipkow@24331
   146
 apply simp
nipkow@24331
   147
apply (clarsimp split: if_splits)
nipkow@24331
   148
done
nipkow@15304
   149
wenzelm@20800
   150
lemma map_of_eq_Some_iff [simp]:
nipkow@24331
   151
  "distinct(map fst xys) \<Longrightarrow> (map_of xys x = Some y) = ((x,y) \<in> set xys)"
nipkow@24331
   152
apply (induct xys)
nipkow@24331
   153
 apply simp
nipkow@24331
   154
apply (auto simp: map_of_eq_None_iff [symmetric])
nipkow@24331
   155
done
nipkow@15304
   156
wenzelm@20800
   157
lemma Some_eq_map_of_iff [simp]:
nipkow@24331
   158
  "distinct(map fst xys) \<Longrightarrow> (Some y = map_of xys x) = ((x,y) \<in> set xys)"
nipkow@24331
   159
by (auto simp del:map_of_eq_Some_iff simp add: map_of_eq_Some_iff [symmetric])
nipkow@15304
   160
paulson@17724
   161
lemma map_of_is_SomeI [simp]: "\<lbrakk> distinct(map fst xys); (x,y) \<in> set xys \<rbrakk>
wenzelm@20800
   162
    \<Longrightarrow> map_of xys x = Some y"
nipkow@24331
   163
apply (induct xys)
nipkow@24331
   164
 apply simp
nipkow@24331
   165
apply force
nipkow@24331
   166
done
nipkow@15304
   167
wenzelm@20800
   168
lemma map_of_zip_is_None [simp]:
nipkow@24331
   169
  "length xs = length ys \<Longrightarrow> (map_of (zip xs ys) x = None) = (x \<notin> set xs)"
nipkow@24331
   170
by (induct rule: list_induct2) simp_all
nipkow@15110
   171
haftmann@26443
   172
lemma map_of_zip_is_Some:
haftmann@26443
   173
  assumes "length xs = length ys"
haftmann@26443
   174
  shows "x \<in> set xs \<longleftrightarrow> (\<exists>y. map_of (zip xs ys) x = Some y)"
haftmann@26443
   175
using assms by (induct rule: list_induct2) simp_all
haftmann@26443
   176
haftmann@26443
   177
lemma map_of_zip_upd:
haftmann@26443
   178
  fixes x :: 'a and xs :: "'a list" and ys zs :: "'b list"
haftmann@26443
   179
  assumes "length ys = length xs"
haftmann@26443
   180
    and "length zs = length xs"
haftmann@26443
   181
    and "x \<notin> set xs"
haftmann@26443
   182
    and "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)"
haftmann@26443
   183
  shows "map_of (zip xs ys) = map_of (zip xs zs)"
haftmann@26443
   184
proof
haftmann@26443
   185
  fix x' :: 'a
haftmann@26443
   186
  show "map_of (zip xs ys) x' = map_of (zip xs zs) x'"
haftmann@26443
   187
  proof (cases "x = x'")
haftmann@26443
   188
    case True
haftmann@26443
   189
    from assms True map_of_zip_is_None [of xs ys x']
haftmann@26443
   190
      have "map_of (zip xs ys) x' = None" by simp
haftmann@26443
   191
    moreover from assms True map_of_zip_is_None [of xs zs x']
haftmann@26443
   192
      have "map_of (zip xs zs) x' = None" by simp
haftmann@26443
   193
    ultimately show ?thesis by simp
haftmann@26443
   194
  next
haftmann@26443
   195
    case False from assms
haftmann@26443
   196
      have "(map_of (zip xs ys)(x \<mapsto> y)) x' = (map_of (zip xs zs)(x \<mapsto> z)) x'" by auto
haftmann@26443
   197
    with False show ?thesis by simp
haftmann@26443
   198
  qed
haftmann@26443
   199
qed
haftmann@26443
   200
haftmann@26443
   201
lemma map_of_zip_inject:
haftmann@26443
   202
  assumes "length ys = length xs"
haftmann@26443
   203
    and "length zs = length xs"
haftmann@26443
   204
    and dist: "distinct xs"
haftmann@26443
   205
    and map_of: "map_of (zip xs ys) = map_of (zip xs zs)"
haftmann@26443
   206
  shows "ys = zs"
haftmann@26443
   207
using assms(1) assms(2)[symmetric] using dist map_of proof (induct ys xs zs rule: list_induct3)
haftmann@26443
   208
  case Nil show ?case by simp
haftmann@26443
   209
next
haftmann@26443
   210
  case (Cons y ys x xs z zs)
haftmann@26443
   211
  from `map_of (zip (x#xs) (y#ys)) = map_of (zip (x#xs) (z#zs))`
haftmann@26443
   212
    have map_of: "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)" by simp
haftmann@26443
   213
  from Cons have "length ys = length xs" and "length zs = length xs"
haftmann@26443
   214
    and "x \<notin> set xs" by simp_all
haftmann@26443
   215
  then have "map_of (zip xs ys) = map_of (zip xs zs)" using map_of by (rule map_of_zip_upd)
haftmann@26443
   216
  with Cons.hyps `distinct (x # xs)` have "ys = zs" by simp
haftmann@26443
   217
  moreover from map_of have "y = z" by (rule map_upd_eqD1)
haftmann@26443
   218
  ultimately show ?case by simp
haftmann@26443
   219
qed
haftmann@26443
   220
haftmann@33635
   221
lemma map_of_zip_map:
haftmann@33635
   222
  "map_of (zip xs (map f xs)) = (\<lambda>x. if x \<in> set xs then Some (f x) else None)"
haftmann@33635
   223
  by (induct xs) (simp_all add: expand_fun_eq)
haftmann@33635
   224
nipkow@15110
   225
lemma finite_range_map_of: "finite (range (map_of xys))"
nipkow@24331
   226
apply (induct xys)
nipkow@24331
   227
 apply (simp_all add: image_constant)
nipkow@24331
   228
apply (rule finite_subset)
nipkow@24331
   229
 prefer 2 apply assumption
nipkow@24331
   230
apply auto
nipkow@24331
   231
done
nipkow@15110
   232
wenzelm@20800
   233
lemma map_of_SomeD: "map_of xs k = Some y \<Longrightarrow> (k, y) \<in> set xs"
nipkow@24331
   234
by (induct xs) (simp, atomize (full), auto)
webertj@13908
   235
wenzelm@20800
   236
lemma map_of_mapk_SomeI:
nipkow@24331
   237
  "inj f ==> map_of t k = Some x ==>
nipkow@24331
   238
   map_of (map (split (%k. Pair (f k))) t) (f k) = Some x"
nipkow@24331
   239
by (induct t) (auto simp add: inj_eq)
webertj@13908
   240
wenzelm@20800
   241
lemma weak_map_of_SomeI: "(k, x) : set l ==> \<exists>x. map_of l k = Some x"
nipkow@24331
   242
by (induct l) auto
webertj@13908
   243
wenzelm@20800
   244
lemma map_of_filter_in:
nipkow@24331
   245
  "map_of xs k = Some z \<Longrightarrow> P k z \<Longrightarrow> map_of (filter (split P) xs) k = Some z"
nipkow@24331
   246
by (induct xs) auto
webertj@13908
   247
nipkow@30235
   248
lemma map_of_map: "map_of (map (%(a,b). (a,f b)) xs) x = Option.map f (map_of xs x)"
nipkow@24331
   249
by (induct xs) auto
webertj@13908
   250
webertj@13908
   251
nipkow@30235
   252
subsection {* @{const Option.map} related *}
webertj@13908
   253
nipkow@30235
   254
lemma option_map_o_empty [simp]: "Option.map f o empty = empty"
nipkow@24331
   255
by (rule ext) simp
webertj@13908
   256
wenzelm@20800
   257
lemma option_map_o_map_upd [simp]:
nipkow@30235
   258
  "Option.map f o m(a|->b) = (Option.map f o m)(a|->f b)"
nipkow@24331
   259
by (rule ext) simp
wenzelm@20800
   260
webertj@13908
   261
wenzelm@17399
   262
subsection {* @{term [source] map_comp} related *}
schirmer@17391
   263
wenzelm@20800
   264
lemma map_comp_empty [simp]:
nipkow@24331
   265
  "m \<circ>\<^sub>m empty = empty"
nipkow@24331
   266
  "empty \<circ>\<^sub>m m = empty"
nipkow@24331
   267
by (auto simp add: map_comp_def intro: ext split: option.splits)
schirmer@17391
   268
wenzelm@20800
   269
lemma map_comp_simps [simp]:
nipkow@24331
   270
  "m2 k = None \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = None"
nipkow@24331
   271
  "m2 k = Some k' \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = m1 k'"
nipkow@24331
   272
by (auto simp add: map_comp_def)
schirmer@17391
   273
schirmer@17391
   274
lemma map_comp_Some_iff:
nipkow@24331
   275
  "((m1 \<circ>\<^sub>m m2) k = Some v) = (\<exists>k'. m2 k = Some k' \<and> m1 k' = Some v)"
nipkow@24331
   276
by (auto simp add: map_comp_def split: option.splits)
schirmer@17391
   277
schirmer@17391
   278
lemma map_comp_None_iff:
nipkow@24331
   279
  "((m1 \<circ>\<^sub>m m2) k = None) = (m2 k = None \<or> (\<exists>k'. m2 k = Some k' \<and> m1 k' = None)) "
nipkow@24331
   280
by (auto simp add: map_comp_def split: option.splits)
webertj@13908
   281
wenzelm@20800
   282
oheimb@14100
   283
subsection {* @{text "++"} *}
webertj@13908
   284
nipkow@14025
   285
lemma map_add_empty[simp]: "m ++ empty = m"
nipkow@24331
   286
by(simp add: map_add_def)
webertj@13908
   287
nipkow@14025
   288
lemma empty_map_add[simp]: "empty ++ m = m"
nipkow@24331
   289
by (rule ext) (simp add: map_add_def split: option.split)
webertj@13908
   290
nipkow@14025
   291
lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3"
nipkow@24331
   292
by (rule ext) (simp add: map_add_def split: option.split)
wenzelm@20800
   293
wenzelm@20800
   294
lemma map_add_Some_iff:
nipkow@24331
   295
  "((m ++ n) k = Some x) = (n k = Some x | n k = None & m k = Some x)"
nipkow@24331
   296
by (simp add: map_add_def split: option.split)
nipkow@14025
   297
wenzelm@20800
   298
lemma map_add_SomeD [dest!]:
nipkow@24331
   299
  "(m ++ n) k = Some x \<Longrightarrow> n k = Some x \<or> n k = None \<and> m k = Some x"
nipkow@24331
   300
by (rule map_add_Some_iff [THEN iffD1])
webertj@13908
   301
wenzelm@20800
   302
lemma map_add_find_right [simp]: "!!xx. n k = Some xx ==> (m ++ n) k = Some xx"
nipkow@24331
   303
by (subst map_add_Some_iff) fast
webertj@13908
   304
nipkow@14025
   305
lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)"
nipkow@24331
   306
by (simp add: map_add_def split: option.split)
webertj@13908
   307
nipkow@14025
   308
lemma map_add_upd[simp]: "f ++ g(x|->y) = (f ++ g)(x|->y)"
nipkow@24331
   309
by (rule ext) (simp add: map_add_def)
webertj@13908
   310
nipkow@14186
   311
lemma map_add_upds[simp]: "m1 ++ (m2(xs[\<mapsto>]ys)) = (m1++m2)(xs[\<mapsto>]ys)"
nipkow@24331
   312
by (simp add: map_upds_def)
nipkow@14186
   313
krauss@32236
   314
lemma map_add_upd_left: "m\<notin>dom e2 \<Longrightarrow> e1(m \<mapsto> u1) ++ e2 = (e1 ++ e2)(m \<mapsto> u1)"
krauss@32236
   315
by (rule ext) (auto simp: map_add_def dom_def split: option.split)
krauss@32236
   316
wenzelm@20800
   317
lemma map_of_append[simp]: "map_of (xs @ ys) = map_of ys ++ map_of xs"
nipkow@24331
   318
unfolding map_add_def
nipkow@24331
   319
apply (induct xs)
nipkow@24331
   320
 apply simp
nipkow@24331
   321
apply (rule ext)
nipkow@24331
   322
apply (simp split add: option.split)
nipkow@24331
   323
done
webertj@13908
   324
nipkow@14025
   325
lemma finite_range_map_of_map_add:
wenzelm@20800
   326
  "finite (range f) ==> finite (range (f ++ map_of l))"
nipkow@24331
   327
apply (induct l)
nipkow@24331
   328
 apply (auto simp del: fun_upd_apply)
nipkow@24331
   329
apply (erule finite_range_updI)
nipkow@24331
   330
done
webertj@13908
   331
wenzelm@20800
   332
lemma inj_on_map_add_dom [iff]:
nipkow@24331
   333
  "inj_on (m ++ m') (dom m') = inj_on m' (dom m')"
nipkow@24331
   334
by (fastsimp simp: map_add_def dom_def inj_on_def split: option.splits)
wenzelm@20800
   335
nipkow@15304
   336
wenzelm@17399
   337
subsection {* @{term [source] restrict_map} *}
oheimb@14100
   338
wenzelm@20800
   339
lemma restrict_map_to_empty [simp]: "m|`{} = empty"
nipkow@24331
   340
by (simp add: restrict_map_def)
nipkow@14186
   341
haftmann@31380
   342
lemma restrict_map_insert: "f |` (insert a A) = (f |` A)(a := f a)"
haftmann@31380
   343
by (auto simp add: restrict_map_def intro: ext)
haftmann@31380
   344
wenzelm@20800
   345
lemma restrict_map_empty [simp]: "empty|`D = empty"
nipkow@24331
   346
by (simp add: restrict_map_def)
nipkow@14186
   347
nipkow@15693
   348
lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m|`A) x = m x"
nipkow@24331
   349
by (simp add: restrict_map_def)
oheimb@14100
   350
nipkow@15693
   351
lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m|`A) x = None"
nipkow@24331
   352
by (simp add: restrict_map_def)
oheimb@14100
   353
nipkow@15693
   354
lemma ran_restrictD: "y \<in> ran (m|`A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y"
nipkow@24331
   355
by (auto simp: restrict_map_def ran_def split: split_if_asm)
oheimb@14100
   356
nipkow@15693
   357
lemma dom_restrict [simp]: "dom (m|`A) = dom m \<inter> A"
nipkow@24331
   358
by (auto simp: restrict_map_def dom_def split: split_if_asm)
oheimb@14100
   359
nipkow@15693
   360
lemma restrict_upd_same [simp]: "m(x\<mapsto>y)|`(-{x}) = m|`(-{x})"
nipkow@24331
   361
by (rule ext) (auto simp: restrict_map_def)
oheimb@14100
   362
nipkow@15693
   363
lemma restrict_restrict [simp]: "m|`A|`B = m|`(A\<inter>B)"
nipkow@24331
   364
by (rule ext) (auto simp: restrict_map_def)
oheimb@14100
   365
wenzelm@20800
   366
lemma restrict_fun_upd [simp]:
nipkow@24331
   367
  "m(x := y)|`D = (if x \<in> D then (m|`(D-{x}))(x := y) else m|`D)"
nipkow@24331
   368
by (simp add: restrict_map_def expand_fun_eq)
nipkow@14186
   369
wenzelm@20800
   370
lemma fun_upd_None_restrict [simp]:
nipkow@24331
   371
  "(m|`D)(x := None) = (if x:D then m|`(D - {x}) else m|`D)"
nipkow@24331
   372
by (simp add: restrict_map_def expand_fun_eq)
nipkow@14186
   373
wenzelm@20800
   374
lemma fun_upd_restrict: "(m|`D)(x := y) = (m|`(D-{x}))(x := y)"
nipkow@24331
   375
by (simp add: restrict_map_def expand_fun_eq)
nipkow@14186
   376
wenzelm@20800
   377
lemma fun_upd_restrict_conv [simp]:
nipkow@24331
   378
  "x \<in> D \<Longrightarrow> (m|`D)(x := y) = (m|`(D-{x}))(x := y)"
nipkow@24331
   379
by (simp add: restrict_map_def expand_fun_eq)
nipkow@14186
   380
oheimb@14100
   381
wenzelm@17399
   382
subsection {* @{term [source] map_upds} *}
nipkow@14025
   383
wenzelm@20800
   384
lemma map_upds_Nil1 [simp]: "m([] [|->] bs) = m"
nipkow@24331
   385
by (simp add: map_upds_def)
nipkow@14025
   386
wenzelm@20800
   387
lemma map_upds_Nil2 [simp]: "m(as [|->] []) = m"
nipkow@24331
   388
by (simp add:map_upds_def)
wenzelm@20800
   389
wenzelm@20800
   390
lemma map_upds_Cons [simp]: "m(a#as [|->] b#bs) = (m(a|->b))(as[|->]bs)"
nipkow@24331
   391
by (simp add:map_upds_def)
nipkow@14025
   392
wenzelm@20800
   393
lemma map_upds_append1 [simp]: "\<And>ys m. size xs < size ys \<Longrightarrow>
nipkow@24331
   394
  m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)"
nipkow@24331
   395
apply(induct xs)
nipkow@24331
   396
 apply (clarsimp simp add: neq_Nil_conv)
nipkow@24331
   397
apply (case_tac ys)
nipkow@24331
   398
 apply simp
nipkow@24331
   399
apply simp
nipkow@24331
   400
done
nipkow@14187
   401
wenzelm@20800
   402
lemma map_upds_list_update2_drop [simp]:
wenzelm@20800
   403
  "\<lbrakk>size xs \<le> i; i < size ys\<rbrakk>
wenzelm@20800
   404
    \<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)"
nipkow@24331
   405
apply (induct xs arbitrary: m ys i)
nipkow@24331
   406
 apply simp
nipkow@24331
   407
apply (case_tac ys)
nipkow@24331
   408
 apply simp
nipkow@24331
   409
apply (simp split: nat.split)
nipkow@24331
   410
done
nipkow@14025
   411
wenzelm@20800
   412
lemma map_upd_upds_conv_if:
wenzelm@20800
   413
  "(f(x|->y))(xs [|->] ys) =
wenzelm@20800
   414
   (if x : set(take (length ys) xs) then f(xs [|->] ys)
wenzelm@20800
   415
                                    else (f(xs [|->] ys))(x|->y))"
nipkow@24331
   416
apply (induct xs arbitrary: x y ys f)
nipkow@24331
   417
 apply simp
nipkow@24331
   418
apply (case_tac ys)
nipkow@24331
   419
 apply (auto split: split_if simp: fun_upd_twist)
nipkow@24331
   420
done
nipkow@14025
   421
nipkow@14025
   422
lemma map_upds_twist [simp]:
nipkow@24331
   423
  "a ~: set as ==> m(a|->b)(as[|->]bs) = m(as[|->]bs)(a|->b)"
nipkow@24331
   424
using set_take_subset by (fastsimp simp add: map_upd_upds_conv_if)
nipkow@14025
   425
wenzelm@20800
   426
lemma map_upds_apply_nontin [simp]:
nipkow@24331
   427
  "x ~: set xs ==> (f(xs[|->]ys)) x = f x"
nipkow@24331
   428
apply (induct xs arbitrary: ys)
nipkow@24331
   429
 apply simp
nipkow@24331
   430
apply (case_tac ys)
nipkow@24331
   431
 apply (auto simp: map_upd_upds_conv_if)
nipkow@24331
   432
done
nipkow@14025
   433
wenzelm@20800
   434
lemma fun_upds_append_drop [simp]:
nipkow@24331
   435
  "size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)"
nipkow@24331
   436
apply (induct xs arbitrary: m ys)
nipkow@24331
   437
 apply simp
nipkow@24331
   438
apply (case_tac ys)
nipkow@24331
   439
 apply simp_all
nipkow@24331
   440
done
nipkow@14300
   441
wenzelm@20800
   442
lemma fun_upds_append2_drop [simp]:
nipkow@24331
   443
  "size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)"
nipkow@24331
   444
apply (induct xs arbitrary: m ys)
nipkow@24331
   445
 apply simp
nipkow@24331
   446
apply (case_tac ys)
nipkow@24331
   447
 apply simp_all
nipkow@24331
   448
done
nipkow@14300
   449
nipkow@14300
   450
wenzelm@20800
   451
lemma restrict_map_upds[simp]:
wenzelm@20800
   452
  "\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk>
wenzelm@20800
   453
    \<Longrightarrow> m(xs [\<mapsto>] ys)|`D = (m|`(D - set xs))(xs [\<mapsto>] ys)"
nipkow@24331
   454
apply (induct xs arbitrary: m ys)
nipkow@24331
   455
 apply simp
nipkow@24331
   456
apply (case_tac ys)
nipkow@24331
   457
 apply simp
nipkow@24331
   458
apply (simp add: Diff_insert [symmetric] insert_absorb)
nipkow@24331
   459
apply (simp add: map_upd_upds_conv_if)
nipkow@24331
   460
done
nipkow@14186
   461
nipkow@14186
   462
wenzelm@17399
   463
subsection {* @{term [source] dom} *}
webertj@13908
   464
nipkow@31080
   465
lemma dom_eq_empty_conv [simp]: "dom f = {} \<longleftrightarrow> f = empty"
nipkow@31080
   466
by(auto intro!:ext simp: dom_def)
nipkow@31080
   467
webertj@13908
   468
lemma domI: "m a = Some b ==> a : dom m"
nipkow@24331
   469
by(simp add:dom_def)
oheimb@14100
   470
(* declare domI [intro]? *)
webertj@13908
   471
paulson@15369
   472
lemma domD: "a : dom m ==> \<exists>b. m a = Some b"
nipkow@24331
   473
by (cases "m a") (auto simp add: dom_def)
webertj@13908
   474
wenzelm@20800
   475
lemma domIff [iff, simp del]: "(a : dom m) = (m a ~= None)"
nipkow@24331
   476
by(simp add:dom_def)
webertj@13908
   477
wenzelm@20800
   478
lemma dom_empty [simp]: "dom empty = {}"
nipkow@24331
   479
by(simp add:dom_def)
webertj@13908
   480
wenzelm@20800
   481
lemma dom_fun_upd [simp]:
nipkow@24331
   482
  "dom(f(x := y)) = (if y=None then dom f - {x} else insert x (dom f))"
nipkow@24331
   483
by(auto simp add:dom_def)
webertj@13908
   484
nipkow@13937
   485
lemma dom_map_of: "dom(map_of xys) = {x. \<exists>y. (x,y) : set xys}"
nipkow@24331
   486
by (induct xys) (auto simp del: fun_upd_apply)
nipkow@13937
   487
nipkow@15304
   488
lemma dom_map_of_conv_image_fst:
nipkow@24331
   489
  "dom(map_of xys) = fst ` (set xys)"
nipkow@24331
   490
by(force simp: dom_map_of)
nipkow@15304
   491
wenzelm@20800
   492
lemma dom_map_of_zip [simp]: "[| length xs = length ys; distinct xs |] ==>
nipkow@24331
   493
  dom(map_of(zip xs ys)) = set xs"
nipkow@24331
   494
by (induct rule: list_induct2) simp_all
nipkow@15110
   495
webertj@13908
   496
lemma finite_dom_map_of: "finite (dom (map_of l))"
nipkow@24331
   497
by (induct l) (auto simp add: dom_def insert_Collect [symmetric])
webertj@13908
   498
wenzelm@20800
   499
lemma dom_map_upds [simp]:
nipkow@24331
   500
  "dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m"
nipkow@24331
   501
apply (induct xs arbitrary: m ys)
nipkow@24331
   502
 apply simp
nipkow@24331
   503
apply (case_tac ys)
nipkow@24331
   504
 apply auto
nipkow@24331
   505
done
nipkow@13910
   506
wenzelm@20800
   507
lemma dom_map_add [simp]: "dom(m++n) = dom n Un dom m"
nipkow@24331
   508
by(auto simp:dom_def)
nipkow@13910
   509
wenzelm@20800
   510
lemma dom_override_on [simp]:
wenzelm@20800
   511
  "dom(override_on f g A) =
wenzelm@20800
   512
    (dom f  - {a. a : A - dom g}) Un {a. a : A Int dom g}"
nipkow@24331
   513
by(auto simp: dom_def override_on_def)
webertj@13908
   514
nipkow@14027
   515
lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1"
nipkow@24331
   516
by (rule ext) (force simp: map_add_def dom_def split: option.split)
wenzelm@20800
   517
krauss@32236
   518
lemma map_add_dom_app_simps:
krauss@32236
   519
  "\<lbrakk> m\<in>dom l2 \<rbrakk> \<Longrightarrow> (l1++l2) m = l2 m"
krauss@32236
   520
  "\<lbrakk> m\<notin>dom l1 \<rbrakk> \<Longrightarrow> (l1++l2) m = l2 m"
krauss@32236
   521
  "\<lbrakk> m\<notin>dom l2 \<rbrakk> \<Longrightarrow> (l1++l2) m = l1 m"
krauss@32236
   522
by (auto simp add: map_add_def split: option.split_asm)
krauss@32236
   523
haftmann@29622
   524
lemma dom_const [simp]:
haftmann@29622
   525
  "dom (\<lambda>x. Some y) = UNIV"
haftmann@29622
   526
  by auto
haftmann@29622
   527
haftmann@29622
   528
lemma dom_if:
haftmann@29622
   529
  "dom (\<lambda>x. if P x then f x else g x) = dom f \<inter> {x. P x} \<union> dom g \<inter> {x. \<not> P x}"
haftmann@29622
   530
  by (auto split: if_splits)
haftmann@29622
   531
haftmann@29622
   532
nipkow@22230
   533
(* Due to John Matthews - could be rephrased with dom *)
nipkow@22230
   534
lemma finite_map_freshness:
nipkow@22230
   535
  "finite (dom (f :: 'a \<rightharpoonup> 'b)) \<Longrightarrow> \<not> finite (UNIV :: 'a set) \<Longrightarrow>
nipkow@22230
   536
   \<exists>x. f x = None"
nipkow@22230
   537
by(bestsimp dest:ex_new_if_finite)
nipkow@14027
   538
haftmann@28790
   539
lemma dom_minus:
haftmann@28790
   540
  "f x = None \<Longrightarrow> dom f - insert x A = dom f - A"
haftmann@28790
   541
  unfolding dom_def by simp
haftmann@28790
   542
haftmann@28790
   543
lemma insert_dom:
haftmann@28790
   544
  "f x = Some y \<Longrightarrow> insert x (dom f) = dom f"
haftmann@28790
   545
  unfolding dom_def by auto
haftmann@28790
   546
haftmann@28790
   547
wenzelm@17399
   548
subsection {* @{term [source] ran} *}
oheimb@14100
   549
wenzelm@20800
   550
lemma ranI: "m a = Some b ==> b : ran m"
nipkow@24331
   551
by(auto simp: ran_def)
oheimb@14100
   552
(* declare ranI [intro]? *)
webertj@13908
   553
wenzelm@20800
   554
lemma ran_empty [simp]: "ran empty = {}"
nipkow@24331
   555
by(auto simp: ran_def)
webertj@13908
   556
wenzelm@20800
   557
lemma ran_map_upd [simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)"
nipkow@24331
   558
unfolding ran_def
nipkow@24331
   559
apply auto
nipkow@24331
   560
apply (subgoal_tac "aa ~= a")
nipkow@24331
   561
 apply auto
nipkow@24331
   562
done
wenzelm@20800
   563
nipkow@13910
   564
oheimb@14100
   565
subsection {* @{text "map_le"} *}
nipkow@13910
   566
kleing@13912
   567
lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g"
nipkow@24331
   568
by (simp add: map_le_def)
nipkow@13910
   569
paulson@17724
   570
lemma upd_None_map_le [simp]: "f(x := None) \<subseteq>\<^sub>m f"
nipkow@24331
   571
by (force simp add: map_le_def)
nipkow@14187
   572
nipkow@13910
   573
lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)"
nipkow@24331
   574
by (fastsimp simp add: map_le_def)
nipkow@13910
   575
paulson@17724
   576
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
   577
by (force simp add: map_le_def)
nipkow@14187
   578
wenzelm@20800
   579
lemma map_le_upds [simp]:
nipkow@24331
   580
  "f \<subseteq>\<^sub>m g ==> f(as [|->] bs) \<subseteq>\<^sub>m g(as [|->] bs)"
nipkow@24331
   581
apply (induct as arbitrary: f g bs)
nipkow@24331
   582
 apply simp
nipkow@24331
   583
apply (case_tac bs)
nipkow@24331
   584
 apply auto
nipkow@24331
   585
done
webertj@13908
   586
webertj@14033
   587
lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)"
nipkow@24331
   588
by (fastsimp simp add: map_le_def dom_def)
webertj@14033
   589
webertj@14033
   590
lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f"
nipkow@24331
   591
by (simp add: map_le_def)
webertj@14033
   592
nipkow@14187
   593
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
   594
by (auto simp add: map_le_def dom_def)
webertj@14033
   595
webertj@14033
   596
lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g"
nipkow@24331
   597
unfolding map_le_def
nipkow@24331
   598
apply (rule ext)
nipkow@24331
   599
apply (case_tac "x \<in> dom f", simp)
nipkow@24331
   600
apply (case_tac "x \<in> dom g", simp, fastsimp)
nipkow@24331
   601
done
webertj@14033
   602
webertj@14033
   603
lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)"
nipkow@24331
   604
by (fastsimp simp add: map_le_def)
webertj@14033
   605
nipkow@15304
   606
lemma map_le_iff_map_add_commute: "(f \<subseteq>\<^sub>m f ++ g) = (f++g = g++f)"
nipkow@24331
   607
by(fastsimp simp: map_add_def map_le_def expand_fun_eq split: option.splits)
nipkow@15304
   608
nipkow@15303
   609
lemma map_add_le_mapE: "f++g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h"
nipkow@24331
   610
by (fastsimp simp add: map_le_def map_add_def dom_def)
nipkow@15303
   611
nipkow@15303
   612
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
   613
by (clarsimp simp add: map_le_def map_add_def dom_def split: option.splits)
nipkow@15303
   614
nipkow@31080
   615
nipkow@31080
   616
lemma dom_eq_singleton_conv: "dom f = {x} \<longleftrightarrow> (\<exists>v. f = [x \<mapsto> v])"
nipkow@31080
   617
proof(rule iffI)
nipkow@31080
   618
  assume "\<exists>v. f = [x \<mapsto> v]"
nipkow@31080
   619
  thus "dom f = {x}" by(auto split: split_if_asm)
nipkow@31080
   620
next
nipkow@31080
   621
  assume "dom f = {x}"
nipkow@31080
   622
  then obtain v where "f x = Some v" by auto
nipkow@31080
   623
  hence "[x \<mapsto> v] \<subseteq>\<^sub>m f" by(auto simp add: map_le_def)
nipkow@31080
   624
  moreover have "f \<subseteq>\<^sub>m [x \<mapsto> v]" using `dom f = {x}` `f x = Some v`
nipkow@31080
   625
    by(auto simp add: map_le_def)
nipkow@31080
   626
  ultimately have "f = [x \<mapsto> v]" by-(rule map_le_antisym)
nipkow@31080
   627
  thus "\<exists>v. f = [x \<mapsto> v]" by blast
nipkow@31080
   628
qed
nipkow@31080
   629
nipkow@3981
   630
end