src/HOL/Map.thy
author haftmann
Sat Mar 06 15:31:31 2010 +0100 (2010-03-06)
changeset 35619 b5f6481772f3
parent 35607 896f01fe825b
child 39198 f967a16dfcdd
permissions -rw-r--r--
lemma restrict_complement_singleton_eq
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@35565
    14
types ('a,'b) "map" = "'a => 'b option" (infixr "~=>" 0)
wenzelm@35427
    15
translations (type) "'a ~=> 'b" <= (type) "'a => 'b option"
nipkow@3981
    16
wenzelm@35427
    17
type_notation (xsymbols)
haftmann@35565
    18
  "map" (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
nipkow@14180
    54
nonterminals
nipkow@14180
    55
  maplets maplet
nipkow@14180
    56
oheimb@5300
    57
syntax
nipkow@14180
    58
  "_maplet"  :: "['a, 'a] => maplet"             ("_ /|->/ _")
nipkow@14180
    59
  "_maplets" :: "['a, 'a] => maplet"             ("_ /[|->]/ _")
nipkow@14180
    60
  ""         :: "maplet => maplets"             ("_")
nipkow@14180
    61
  "_Maplets" :: "[maplet, maplets] => maplets" ("_,/ _")
nipkow@14180
    62
  "_MapUpd"  :: "['a ~=> 'b, maplets] => 'a ~=> 'b" ("_/'(_')" [900,0]900)
nipkow@14180
    63
  "_Map"     :: "maplets => 'a ~=> 'b"            ("(1[_])")
nipkow@3981
    64
wenzelm@12114
    65
syntax (xsymbols)
nipkow@14180
    66
  "_maplet"  :: "['a, 'a] => maplet"             ("_ /\<mapsto>/ _")
nipkow@14180
    67
  "_maplets" :: "['a, 'a] => maplet"             ("_ /[\<mapsto>]/ _")
nipkow@14180
    68
oheimb@5300
    69
translations
nipkow@14180
    70
  "_MapUpd m (_Maplets xy ms)"  == "_MapUpd (_MapUpd m xy) ms"
wenzelm@35115
    71
  "_MapUpd m (_maplet  x y)"    == "m(x := CONST Some y)"
wenzelm@19947
    72
  "_Map ms"                     == "_MapUpd (CONST empty) ms"
nipkow@14180
    73
  "_Map (_Maplets ms1 ms2)"     <= "_MapUpd (_Map ms1) ms2"
nipkow@14180
    74
  "_Maplets ms1 (_Maplets ms2 ms3)" <= "_Maplets (_Maplets ms1 ms2) ms3"
nipkow@14180
    75
berghofe@5183
    76
primrec
haftmann@34941
    77
  map_of :: "('a \<times> 'b) list \<Rightarrow> 'a \<rightharpoonup> 'b" where
haftmann@34941
    78
    "map_of [] = empty"
haftmann@34941
    79
  | "map_of (p # ps) = (map_of ps)(fst p \<mapsto> snd p)"
oheimb@5300
    80
haftmann@34941
    81
definition
haftmann@34941
    82
  map_upds :: "('a \<rightharpoonup> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> 'a \<rightharpoonup> 'b" where
haftmann@34941
    83
  "map_upds m xs ys = m ++ map_of (rev (zip xs ys))"
haftmann@34941
    84
haftmann@34941
    85
translations
haftmann@34941
    86
  "_MapUpd m (_maplets x y)"    == "CONST map_upds m x y"
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
wenzelm@17399
    94
subsection {* @{term [source] empty} *}
webertj@13908
    95
wenzelm@20800
    96
lemma empty_upd_none [simp]: "empty(x := None) = empty"
nipkow@24331
    97
by (rule ext) simp
webertj@13908
    98
webertj@13908
    99
wenzelm@17399
   100
subsection {* @{term [source] map_upd} *}
webertj@13908
   101
webertj@13908
   102
lemma map_upd_triv: "t k = Some x ==> t(k|->x) = t"
nipkow@24331
   103
by (rule ext) simp
webertj@13908
   104
wenzelm@20800
   105
lemma map_upd_nonempty [simp]: "t(k|->x) ~= empty"
wenzelm@20800
   106
proof
wenzelm@20800
   107
  assume "t(k \<mapsto> x) = empty"
wenzelm@20800
   108
  then have "(t(k \<mapsto> x)) k = None" by simp
wenzelm@20800
   109
  then show False by simp
wenzelm@20800
   110
qed
webertj@13908
   111
wenzelm@20800
   112
lemma map_upd_eqD1:
wenzelm@20800
   113
  assumes "m(a\<mapsto>x) = n(a\<mapsto>y)"
wenzelm@20800
   114
  shows "x = y"
wenzelm@20800
   115
proof -
wenzelm@20800
   116
  from prems have "(m(a\<mapsto>x)) a = (n(a\<mapsto>y)) a" by simp
wenzelm@20800
   117
  then show ?thesis by simp
wenzelm@20800
   118
qed
oheimb@14100
   119
wenzelm@20800
   120
lemma map_upd_Some_unfold:
nipkow@24331
   121
  "((m(a|->b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)"
nipkow@24331
   122
by auto
oheimb@14100
   123
wenzelm@20800
   124
lemma image_map_upd [simp]: "x \<notin> A \<Longrightarrow> m(x \<mapsto> y) ` A = m ` A"
nipkow@24331
   125
by auto
nipkow@15303
   126
webertj@13908
   127
lemma finite_range_updI: "finite (range f) ==> finite (range (f(a|->b)))"
nipkow@24331
   128
unfolding image_def
nipkow@24331
   129
apply (simp (no_asm_use) add:full_SetCompr_eq)
nipkow@24331
   130
apply (rule finite_subset)
nipkow@24331
   131
 prefer 2 apply assumption
nipkow@24331
   132
apply (auto)
nipkow@24331
   133
done
webertj@13908
   134
webertj@13908
   135
wenzelm@17399
   136
subsection {* @{term [source] map_of} *}
webertj@13908
   137
nipkow@15304
   138
lemma map_of_eq_None_iff:
nipkow@24331
   139
  "(map_of xys x = None) = (x \<notin> fst ` (set xys))"
nipkow@24331
   140
by (induct xys) simp_all
nipkow@15304
   141
nipkow@24331
   142
lemma map_of_is_SomeD: "map_of xys x = Some y \<Longrightarrow> (x,y) \<in> set xys"
nipkow@24331
   143
apply (induct xys)
nipkow@24331
   144
 apply simp
nipkow@24331
   145
apply (clarsimp split: if_splits)
nipkow@24331
   146
done
nipkow@15304
   147
wenzelm@20800
   148
lemma map_of_eq_Some_iff [simp]:
nipkow@24331
   149
  "distinct(map fst xys) \<Longrightarrow> (map_of xys x = Some y) = ((x,y) \<in> set xys)"
nipkow@24331
   150
apply (induct xys)
nipkow@24331
   151
 apply simp
nipkow@24331
   152
apply (auto simp: map_of_eq_None_iff [symmetric])
nipkow@24331
   153
done
nipkow@15304
   154
wenzelm@20800
   155
lemma Some_eq_map_of_iff [simp]:
nipkow@24331
   156
  "distinct(map fst xys) \<Longrightarrow> (Some y = map_of xys x) = ((x,y) \<in> set xys)"
nipkow@24331
   157
by (auto simp del:map_of_eq_Some_iff simp add: map_of_eq_Some_iff [symmetric])
nipkow@15304
   158
paulson@17724
   159
lemma map_of_is_SomeI [simp]: "\<lbrakk> distinct(map fst xys); (x,y) \<in> set xys \<rbrakk>
wenzelm@20800
   160
    \<Longrightarrow> map_of xys x = Some y"
nipkow@24331
   161
apply (induct xys)
nipkow@24331
   162
 apply simp
nipkow@24331
   163
apply force
nipkow@24331
   164
done
nipkow@15304
   165
wenzelm@20800
   166
lemma map_of_zip_is_None [simp]:
nipkow@24331
   167
  "length xs = length ys \<Longrightarrow> (map_of (zip xs ys) x = None) = (x \<notin> set xs)"
nipkow@24331
   168
by (induct rule: list_induct2) simp_all
nipkow@15110
   169
haftmann@26443
   170
lemma map_of_zip_is_Some:
haftmann@26443
   171
  assumes "length xs = length ys"
haftmann@26443
   172
  shows "x \<in> set xs \<longleftrightarrow> (\<exists>y. map_of (zip xs ys) x = Some y)"
haftmann@26443
   173
using assms by (induct rule: list_induct2) simp_all
haftmann@26443
   174
haftmann@26443
   175
lemma map_of_zip_upd:
haftmann@26443
   176
  fixes x :: 'a and xs :: "'a list" and ys zs :: "'b list"
haftmann@26443
   177
  assumes "length ys = length xs"
haftmann@26443
   178
    and "length zs = length xs"
haftmann@26443
   179
    and "x \<notin> set xs"
haftmann@26443
   180
    and "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)"
haftmann@26443
   181
  shows "map_of (zip xs ys) = map_of (zip xs zs)"
haftmann@26443
   182
proof
haftmann@26443
   183
  fix x' :: 'a
haftmann@26443
   184
  show "map_of (zip xs ys) x' = map_of (zip xs zs) x'"
haftmann@26443
   185
  proof (cases "x = x'")
haftmann@26443
   186
    case True
haftmann@26443
   187
    from assms True map_of_zip_is_None [of xs ys x']
haftmann@26443
   188
      have "map_of (zip xs ys) x' = None" by simp
haftmann@26443
   189
    moreover from assms True map_of_zip_is_None [of xs zs x']
haftmann@26443
   190
      have "map_of (zip xs zs) x' = None" by simp
haftmann@26443
   191
    ultimately show ?thesis by simp
haftmann@26443
   192
  next
haftmann@26443
   193
    case False from assms
haftmann@26443
   194
      have "(map_of (zip xs ys)(x \<mapsto> y)) x' = (map_of (zip xs zs)(x \<mapsto> z)) x'" by auto
haftmann@26443
   195
    with False show ?thesis by simp
haftmann@26443
   196
  qed
haftmann@26443
   197
qed
haftmann@26443
   198
haftmann@26443
   199
lemma map_of_zip_inject:
haftmann@26443
   200
  assumes "length ys = length xs"
haftmann@26443
   201
    and "length zs = length xs"
haftmann@26443
   202
    and dist: "distinct xs"
haftmann@26443
   203
    and map_of: "map_of (zip xs ys) = map_of (zip xs zs)"
haftmann@26443
   204
  shows "ys = zs"
haftmann@26443
   205
using assms(1) assms(2)[symmetric] using dist map_of proof (induct ys xs zs rule: list_induct3)
haftmann@26443
   206
  case Nil show ?case by simp
haftmann@26443
   207
next
haftmann@26443
   208
  case (Cons y ys x xs z zs)
haftmann@26443
   209
  from `map_of (zip (x#xs) (y#ys)) = map_of (zip (x#xs) (z#zs))`
haftmann@26443
   210
    have map_of: "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)" by simp
haftmann@26443
   211
  from Cons have "length ys = length xs" and "length zs = length xs"
haftmann@26443
   212
    and "x \<notin> set xs" by simp_all
haftmann@26443
   213
  then have "map_of (zip xs ys) = map_of (zip xs zs)" using map_of by (rule map_of_zip_upd)
haftmann@26443
   214
  with Cons.hyps `distinct (x # xs)` have "ys = zs" by simp
haftmann@26443
   215
  moreover from map_of have "y = z" by (rule map_upd_eqD1)
haftmann@26443
   216
  ultimately show ?case by simp
haftmann@26443
   217
qed
haftmann@26443
   218
haftmann@33635
   219
lemma map_of_zip_map:
haftmann@33635
   220
  "map_of (zip xs (map f xs)) = (\<lambda>x. if x \<in> set xs then Some (f x) else None)"
haftmann@33635
   221
  by (induct xs) (simp_all add: expand_fun_eq)
haftmann@33635
   222
nipkow@15110
   223
lemma finite_range_map_of: "finite (range (map_of xys))"
nipkow@24331
   224
apply (induct xys)
nipkow@24331
   225
 apply (simp_all add: image_constant)
nipkow@24331
   226
apply (rule finite_subset)
nipkow@24331
   227
 prefer 2 apply assumption
nipkow@24331
   228
apply auto
nipkow@24331
   229
done
nipkow@15110
   230
wenzelm@20800
   231
lemma map_of_SomeD: "map_of xs k = Some y \<Longrightarrow> (k, y) \<in> set xs"
nipkow@24331
   232
by (induct xs) (simp, atomize (full), auto)
webertj@13908
   233
wenzelm@20800
   234
lemma map_of_mapk_SomeI:
nipkow@24331
   235
  "inj f ==> map_of t k = Some x ==>
nipkow@24331
   236
   map_of (map (split (%k. Pair (f k))) t) (f k) = Some x"
nipkow@24331
   237
by (induct t) (auto simp add: inj_eq)
webertj@13908
   238
wenzelm@20800
   239
lemma weak_map_of_SomeI: "(k, x) : set l ==> \<exists>x. map_of l k = Some x"
nipkow@24331
   240
by (induct l) auto
webertj@13908
   241
wenzelm@20800
   242
lemma map_of_filter_in:
nipkow@24331
   243
  "map_of xs k = Some z \<Longrightarrow> P k z \<Longrightarrow> map_of (filter (split P) xs) k = Some z"
nipkow@24331
   244
by (induct xs) auto
webertj@13908
   245
haftmann@35607
   246
lemma map_of_map:
haftmann@35607
   247
  "map_of (map (\<lambda>(k, v). (k, f v)) xs) = Option.map f \<circ> map_of xs"
haftmann@35607
   248
  by (induct xs) (auto simp add: expand_fun_eq)
haftmann@35607
   249
haftmann@35607
   250
lemma dom_option_map:
haftmann@35607
   251
  "dom (\<lambda>k. Option.map (f k) (m k)) = dom m"
haftmann@35607
   252
  by (simp add: dom_def)
webertj@13908
   253
webertj@13908
   254
nipkow@30235
   255
subsection {* @{const Option.map} related *}
webertj@13908
   256
nipkow@30235
   257
lemma option_map_o_empty [simp]: "Option.map f o empty = empty"
nipkow@24331
   258
by (rule ext) simp
webertj@13908
   259
wenzelm@20800
   260
lemma option_map_o_map_upd [simp]:
nipkow@30235
   261
  "Option.map f o m(a|->b) = (Option.map f o m)(a|->f b)"
nipkow@24331
   262
by (rule ext) simp
wenzelm@20800
   263
webertj@13908
   264
wenzelm@17399
   265
subsection {* @{term [source] map_comp} related *}
schirmer@17391
   266
wenzelm@20800
   267
lemma map_comp_empty [simp]:
nipkow@24331
   268
  "m \<circ>\<^sub>m empty = empty"
nipkow@24331
   269
  "empty \<circ>\<^sub>m m = empty"
nipkow@24331
   270
by (auto simp add: map_comp_def intro: ext split: option.splits)
schirmer@17391
   271
wenzelm@20800
   272
lemma map_comp_simps [simp]:
nipkow@24331
   273
  "m2 k = None \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = None"
nipkow@24331
   274
  "m2 k = Some k' \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = m1 k'"
nipkow@24331
   275
by (auto simp add: map_comp_def)
schirmer@17391
   276
schirmer@17391
   277
lemma map_comp_Some_iff:
nipkow@24331
   278
  "((m1 \<circ>\<^sub>m m2) k = Some v) = (\<exists>k'. m2 k = Some k' \<and> m1 k' = Some v)"
nipkow@24331
   279
by (auto simp add: map_comp_def split: option.splits)
schirmer@17391
   280
schirmer@17391
   281
lemma map_comp_None_iff:
nipkow@24331
   282
  "((m1 \<circ>\<^sub>m m2) k = None) = (m2 k = None \<or> (\<exists>k'. m2 k = Some k' \<and> m1 k' = None)) "
nipkow@24331
   283
by (auto simp add: map_comp_def split: option.splits)
webertj@13908
   284
wenzelm@20800
   285
oheimb@14100
   286
subsection {* @{text "++"} *}
webertj@13908
   287
nipkow@14025
   288
lemma map_add_empty[simp]: "m ++ empty = m"
nipkow@24331
   289
by(simp add: map_add_def)
webertj@13908
   290
nipkow@14025
   291
lemma empty_map_add[simp]: "empty ++ m = m"
nipkow@24331
   292
by (rule ext) (simp add: map_add_def split: option.split)
webertj@13908
   293
nipkow@14025
   294
lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3"
nipkow@24331
   295
by (rule ext) (simp add: map_add_def split: option.split)
wenzelm@20800
   296
wenzelm@20800
   297
lemma map_add_Some_iff:
nipkow@24331
   298
  "((m ++ n) k = Some x) = (n k = Some x | n k = None & m k = Some x)"
nipkow@24331
   299
by (simp add: map_add_def split: option.split)
nipkow@14025
   300
wenzelm@20800
   301
lemma map_add_SomeD [dest!]:
nipkow@24331
   302
  "(m ++ n) k = Some x \<Longrightarrow> n k = Some x \<or> n k = None \<and> m k = Some x"
nipkow@24331
   303
by (rule map_add_Some_iff [THEN iffD1])
webertj@13908
   304
wenzelm@20800
   305
lemma map_add_find_right [simp]: "!!xx. n k = Some xx ==> (m ++ n) k = Some xx"
nipkow@24331
   306
by (subst map_add_Some_iff) fast
webertj@13908
   307
nipkow@14025
   308
lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)"
nipkow@24331
   309
by (simp add: map_add_def split: option.split)
webertj@13908
   310
nipkow@14025
   311
lemma map_add_upd[simp]: "f ++ g(x|->y) = (f ++ g)(x|->y)"
nipkow@24331
   312
by (rule ext) (simp add: map_add_def)
webertj@13908
   313
nipkow@14186
   314
lemma map_add_upds[simp]: "m1 ++ (m2(xs[\<mapsto>]ys)) = (m1++m2)(xs[\<mapsto>]ys)"
nipkow@24331
   315
by (simp add: map_upds_def)
nipkow@14186
   316
krauss@32236
   317
lemma map_add_upd_left: "m\<notin>dom e2 \<Longrightarrow> e1(m \<mapsto> u1) ++ e2 = (e1 ++ e2)(m \<mapsto> u1)"
krauss@32236
   318
by (rule ext) (auto simp: map_add_def dom_def split: option.split)
krauss@32236
   319
wenzelm@20800
   320
lemma map_of_append[simp]: "map_of (xs @ ys) = map_of ys ++ map_of xs"
nipkow@24331
   321
unfolding map_add_def
nipkow@24331
   322
apply (induct xs)
nipkow@24331
   323
 apply simp
nipkow@24331
   324
apply (rule ext)
nipkow@24331
   325
apply (simp split add: option.split)
nipkow@24331
   326
done
webertj@13908
   327
nipkow@14025
   328
lemma finite_range_map_of_map_add:
wenzelm@20800
   329
  "finite (range f) ==> finite (range (f ++ map_of l))"
nipkow@24331
   330
apply (induct l)
nipkow@24331
   331
 apply (auto simp del: fun_upd_apply)
nipkow@24331
   332
apply (erule finite_range_updI)
nipkow@24331
   333
done
webertj@13908
   334
wenzelm@20800
   335
lemma inj_on_map_add_dom [iff]:
nipkow@24331
   336
  "inj_on (m ++ m') (dom m') = inj_on m' (dom m')"
nipkow@24331
   337
by (fastsimp simp: map_add_def dom_def inj_on_def split: option.splits)
wenzelm@20800
   338
haftmann@34979
   339
lemma map_upds_fold_map_upd:
haftmann@35552
   340
  "m(ks[\<mapsto>]vs) = foldl (\<lambda>m (k, v). m(k \<mapsto> v)) m (zip ks vs)"
haftmann@34979
   341
unfolding map_upds_def proof (rule sym, rule zip_obtain_same_length)
haftmann@34979
   342
  fix ks :: "'a list" and vs :: "'b list"
haftmann@34979
   343
  assume "length ks = length vs"
haftmann@35552
   344
  then show "foldl (\<lambda>m (k, v). m(k\<mapsto>v)) m (zip ks vs) = m ++ map_of (rev (zip ks vs))"
haftmann@35552
   345
    by(induct arbitrary: m rule: list_induct2) simp_all
haftmann@34979
   346
qed
haftmann@34979
   347
haftmann@34979
   348
lemma map_add_map_of_foldr:
haftmann@34979
   349
  "m ++ map_of ps = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) ps m"
haftmann@34979
   350
  by (induct ps) (auto simp add: expand_fun_eq map_add_def)
haftmann@34979
   351
nipkow@15304
   352
wenzelm@17399
   353
subsection {* @{term [source] restrict_map} *}
oheimb@14100
   354
wenzelm@20800
   355
lemma restrict_map_to_empty [simp]: "m|`{} = empty"
nipkow@24331
   356
by (simp add: restrict_map_def)
nipkow@14186
   357
haftmann@31380
   358
lemma restrict_map_insert: "f |` (insert a A) = (f |` A)(a := f a)"
haftmann@31380
   359
by (auto simp add: restrict_map_def intro: ext)
haftmann@31380
   360
wenzelm@20800
   361
lemma restrict_map_empty [simp]: "empty|`D = empty"
nipkow@24331
   362
by (simp add: restrict_map_def)
nipkow@14186
   363
nipkow@15693
   364
lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m|`A) x = m x"
nipkow@24331
   365
by (simp add: restrict_map_def)
oheimb@14100
   366
nipkow@15693
   367
lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m|`A) x = None"
nipkow@24331
   368
by (simp add: restrict_map_def)
oheimb@14100
   369
nipkow@15693
   370
lemma ran_restrictD: "y \<in> ran (m|`A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y"
nipkow@24331
   371
by (auto simp: restrict_map_def ran_def split: split_if_asm)
oheimb@14100
   372
nipkow@15693
   373
lemma dom_restrict [simp]: "dom (m|`A) = dom m \<inter> A"
nipkow@24331
   374
by (auto simp: restrict_map_def dom_def split: split_if_asm)
oheimb@14100
   375
nipkow@15693
   376
lemma restrict_upd_same [simp]: "m(x\<mapsto>y)|`(-{x}) = m|`(-{x})"
nipkow@24331
   377
by (rule ext) (auto simp: restrict_map_def)
oheimb@14100
   378
nipkow@15693
   379
lemma restrict_restrict [simp]: "m|`A|`B = m|`(A\<inter>B)"
nipkow@24331
   380
by (rule ext) (auto simp: restrict_map_def)
oheimb@14100
   381
wenzelm@20800
   382
lemma restrict_fun_upd [simp]:
nipkow@24331
   383
  "m(x := y)|`D = (if x \<in> D then (m|`(D-{x}))(x := y) else m|`D)"
nipkow@24331
   384
by (simp add: restrict_map_def expand_fun_eq)
nipkow@14186
   385
wenzelm@20800
   386
lemma fun_upd_None_restrict [simp]:
nipkow@24331
   387
  "(m|`D)(x := None) = (if x:D then m|`(D - {x}) else m|`D)"
nipkow@24331
   388
by (simp add: restrict_map_def expand_fun_eq)
nipkow@14186
   389
wenzelm@20800
   390
lemma fun_upd_restrict: "(m|`D)(x := y) = (m|`(D-{x}))(x := y)"
nipkow@24331
   391
by (simp add: restrict_map_def expand_fun_eq)
nipkow@14186
   392
wenzelm@20800
   393
lemma fun_upd_restrict_conv [simp]:
nipkow@24331
   394
  "x \<in> D \<Longrightarrow> (m|`D)(x := y) = (m|`(D-{x}))(x := y)"
nipkow@24331
   395
by (simp add: restrict_map_def expand_fun_eq)
nipkow@14186
   396
haftmann@35159
   397
lemma map_of_map_restrict:
haftmann@35159
   398
  "map_of (map (\<lambda>k. (k, f k)) ks) = (Some \<circ> f) |` set ks"
haftmann@35159
   399
  by (induct ks) (simp_all add: expand_fun_eq restrict_map_insert)
haftmann@35159
   400
haftmann@35619
   401
lemma restrict_complement_singleton_eq:
haftmann@35619
   402
  "f |` (- {x}) = f(x := None)"
haftmann@35619
   403
  by (simp add: restrict_map_def expand_fun_eq)
haftmann@35619
   404
oheimb@14100
   405
wenzelm@17399
   406
subsection {* @{term [source] map_upds} *}
nipkow@14025
   407
wenzelm@20800
   408
lemma map_upds_Nil1 [simp]: "m([] [|->] bs) = m"
nipkow@24331
   409
by (simp add: map_upds_def)
nipkow@14025
   410
wenzelm@20800
   411
lemma map_upds_Nil2 [simp]: "m(as [|->] []) = m"
nipkow@24331
   412
by (simp add:map_upds_def)
wenzelm@20800
   413
wenzelm@20800
   414
lemma map_upds_Cons [simp]: "m(a#as [|->] b#bs) = (m(a|->b))(as[|->]bs)"
nipkow@24331
   415
by (simp add:map_upds_def)
nipkow@14025
   416
wenzelm@20800
   417
lemma map_upds_append1 [simp]: "\<And>ys m. size xs < size ys \<Longrightarrow>
nipkow@24331
   418
  m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)"
nipkow@24331
   419
apply(induct xs)
nipkow@24331
   420
 apply (clarsimp simp add: neq_Nil_conv)
nipkow@24331
   421
apply (case_tac ys)
nipkow@24331
   422
 apply simp
nipkow@24331
   423
apply simp
nipkow@24331
   424
done
nipkow@14187
   425
wenzelm@20800
   426
lemma map_upds_list_update2_drop [simp]:
wenzelm@20800
   427
  "\<lbrakk>size xs \<le> i; i < size ys\<rbrakk>
wenzelm@20800
   428
    \<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)"
nipkow@24331
   429
apply (induct xs arbitrary: m ys i)
nipkow@24331
   430
 apply simp
nipkow@24331
   431
apply (case_tac ys)
nipkow@24331
   432
 apply simp
nipkow@24331
   433
apply (simp split: nat.split)
nipkow@24331
   434
done
nipkow@14025
   435
wenzelm@20800
   436
lemma map_upd_upds_conv_if:
wenzelm@20800
   437
  "(f(x|->y))(xs [|->] ys) =
wenzelm@20800
   438
   (if x : set(take (length ys) xs) then f(xs [|->] ys)
wenzelm@20800
   439
                                    else (f(xs [|->] ys))(x|->y))"
nipkow@24331
   440
apply (induct xs arbitrary: x y ys f)
nipkow@24331
   441
 apply simp
nipkow@24331
   442
apply (case_tac ys)
nipkow@24331
   443
 apply (auto split: split_if simp: fun_upd_twist)
nipkow@24331
   444
done
nipkow@14025
   445
nipkow@14025
   446
lemma map_upds_twist [simp]:
nipkow@24331
   447
  "a ~: set as ==> m(a|->b)(as[|->]bs) = m(as[|->]bs)(a|->b)"
nipkow@24331
   448
using set_take_subset by (fastsimp simp add: map_upd_upds_conv_if)
nipkow@14025
   449
wenzelm@20800
   450
lemma map_upds_apply_nontin [simp]:
nipkow@24331
   451
  "x ~: set xs ==> (f(xs[|->]ys)) x = f x"
nipkow@24331
   452
apply (induct xs arbitrary: ys)
nipkow@24331
   453
 apply simp
nipkow@24331
   454
apply (case_tac ys)
nipkow@24331
   455
 apply (auto simp: map_upd_upds_conv_if)
nipkow@24331
   456
done
nipkow@14025
   457
wenzelm@20800
   458
lemma fun_upds_append_drop [simp]:
nipkow@24331
   459
  "size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)"
nipkow@24331
   460
apply (induct xs arbitrary: m ys)
nipkow@24331
   461
 apply simp
nipkow@24331
   462
apply (case_tac ys)
nipkow@24331
   463
 apply simp_all
nipkow@24331
   464
done
nipkow@14300
   465
wenzelm@20800
   466
lemma fun_upds_append2_drop [simp]:
nipkow@24331
   467
  "size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)"
nipkow@24331
   468
apply (induct xs arbitrary: m ys)
nipkow@24331
   469
 apply simp
nipkow@24331
   470
apply (case_tac ys)
nipkow@24331
   471
 apply simp_all
nipkow@24331
   472
done
nipkow@14300
   473
nipkow@14300
   474
wenzelm@20800
   475
lemma restrict_map_upds[simp]:
wenzelm@20800
   476
  "\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk>
wenzelm@20800
   477
    \<Longrightarrow> m(xs [\<mapsto>] ys)|`D = (m|`(D - set xs))(xs [\<mapsto>] ys)"
nipkow@24331
   478
apply (induct xs arbitrary: m ys)
nipkow@24331
   479
 apply simp
nipkow@24331
   480
apply (case_tac ys)
nipkow@24331
   481
 apply simp
nipkow@24331
   482
apply (simp add: Diff_insert [symmetric] insert_absorb)
nipkow@24331
   483
apply (simp add: map_upd_upds_conv_if)
nipkow@24331
   484
done
nipkow@14186
   485
nipkow@14186
   486
wenzelm@17399
   487
subsection {* @{term [source] dom} *}
webertj@13908
   488
nipkow@31080
   489
lemma dom_eq_empty_conv [simp]: "dom f = {} \<longleftrightarrow> f = empty"
nipkow@31080
   490
by(auto intro!:ext simp: dom_def)
nipkow@31080
   491
webertj@13908
   492
lemma domI: "m a = Some b ==> a : dom m"
nipkow@24331
   493
by(simp add:dom_def)
oheimb@14100
   494
(* declare domI [intro]? *)
webertj@13908
   495
paulson@15369
   496
lemma domD: "a : dom m ==> \<exists>b. m a = Some b"
nipkow@24331
   497
by (cases "m a") (auto simp add: dom_def)
webertj@13908
   498
wenzelm@20800
   499
lemma domIff [iff, simp del]: "(a : dom m) = (m a ~= None)"
nipkow@24331
   500
by(simp add:dom_def)
webertj@13908
   501
wenzelm@20800
   502
lemma dom_empty [simp]: "dom empty = {}"
nipkow@24331
   503
by(simp add:dom_def)
webertj@13908
   504
wenzelm@20800
   505
lemma dom_fun_upd [simp]:
nipkow@24331
   506
  "dom(f(x := y)) = (if y=None then dom f - {x} else insert x (dom f))"
nipkow@24331
   507
by(auto simp add:dom_def)
webertj@13908
   508
haftmann@34979
   509
lemma dom_if:
haftmann@34979
   510
  "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@34979
   511
  by (auto split: if_splits)
nipkow@13937
   512
nipkow@15304
   513
lemma dom_map_of_conv_image_fst:
haftmann@34979
   514
  "dom (map_of xys) = fst ` set xys"
haftmann@34979
   515
  by (induct xys) (auto simp add: dom_if)
nipkow@15304
   516
wenzelm@20800
   517
lemma dom_map_of_zip [simp]: "[| length xs = length ys; distinct xs |] ==>
nipkow@24331
   518
  dom(map_of(zip xs ys)) = set xs"
nipkow@24331
   519
by (induct rule: list_induct2) simp_all
nipkow@15110
   520
webertj@13908
   521
lemma finite_dom_map_of: "finite (dom (map_of l))"
nipkow@24331
   522
by (induct l) (auto simp add: dom_def insert_Collect [symmetric])
webertj@13908
   523
wenzelm@20800
   524
lemma dom_map_upds [simp]:
nipkow@24331
   525
  "dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m"
nipkow@24331
   526
apply (induct xs arbitrary: m ys)
nipkow@24331
   527
 apply simp
nipkow@24331
   528
apply (case_tac ys)
nipkow@24331
   529
 apply auto
nipkow@24331
   530
done
nipkow@13910
   531
wenzelm@20800
   532
lemma dom_map_add [simp]: "dom(m++n) = dom n Un dom m"
nipkow@24331
   533
by(auto simp:dom_def)
nipkow@13910
   534
wenzelm@20800
   535
lemma dom_override_on [simp]:
wenzelm@20800
   536
  "dom(override_on f g A) =
wenzelm@20800
   537
    (dom f  - {a. a : A - dom g}) Un {a. a : A Int dom g}"
nipkow@24331
   538
by(auto simp: dom_def override_on_def)
webertj@13908
   539
nipkow@14027
   540
lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1"
nipkow@24331
   541
by (rule ext) (force simp: map_add_def dom_def split: option.split)
wenzelm@20800
   542
krauss@32236
   543
lemma map_add_dom_app_simps:
krauss@32236
   544
  "\<lbrakk> m\<in>dom l2 \<rbrakk> \<Longrightarrow> (l1++l2) m = l2 m"
krauss@32236
   545
  "\<lbrakk> m\<notin>dom l1 \<rbrakk> \<Longrightarrow> (l1++l2) m = l2 m"
krauss@32236
   546
  "\<lbrakk> m\<notin>dom l2 \<rbrakk> \<Longrightarrow> (l1++l2) m = l1 m"
krauss@32236
   547
by (auto simp add: map_add_def split: option.split_asm)
krauss@32236
   548
haftmann@29622
   549
lemma dom_const [simp]:
haftmann@35159
   550
  "dom (\<lambda>x. Some (f x)) = UNIV"
haftmann@29622
   551
  by auto
haftmann@29622
   552
nipkow@22230
   553
(* Due to John Matthews - could be rephrased with dom *)
nipkow@22230
   554
lemma finite_map_freshness:
nipkow@22230
   555
  "finite (dom (f :: 'a \<rightharpoonup> 'b)) \<Longrightarrow> \<not> finite (UNIV :: 'a set) \<Longrightarrow>
nipkow@22230
   556
   \<exists>x. f x = None"
nipkow@22230
   557
by(bestsimp dest:ex_new_if_finite)
nipkow@14027
   558
haftmann@28790
   559
lemma dom_minus:
haftmann@28790
   560
  "f x = None \<Longrightarrow> dom f - insert x A = dom f - A"
haftmann@28790
   561
  unfolding dom_def by simp
haftmann@28790
   562
haftmann@28790
   563
lemma insert_dom:
haftmann@28790
   564
  "f x = Some y \<Longrightarrow> insert x (dom f) = dom f"
haftmann@28790
   565
  unfolding dom_def by auto
haftmann@28790
   566
haftmann@35607
   567
lemma map_of_map_keys:
haftmann@35607
   568
  "set xs = dom m \<Longrightarrow> map_of (map (\<lambda>k. (k, the (m k))) xs) = m"
haftmann@35607
   569
  by (rule ext) (auto simp add: map_of_map_restrict restrict_map_def)
haftmann@35607
   570
haftmann@28790
   571
wenzelm@17399
   572
subsection {* @{term [source] ran} *}
oheimb@14100
   573
wenzelm@20800
   574
lemma ranI: "m a = Some b ==> b : ran m"
nipkow@24331
   575
by(auto simp: ran_def)
oheimb@14100
   576
(* declare ranI [intro]? *)
webertj@13908
   577
wenzelm@20800
   578
lemma ran_empty [simp]: "ran empty = {}"
nipkow@24331
   579
by(auto simp: ran_def)
webertj@13908
   580
wenzelm@20800
   581
lemma ran_map_upd [simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)"
nipkow@24331
   582
unfolding ran_def
nipkow@24331
   583
apply auto
nipkow@24331
   584
apply (subgoal_tac "aa ~= a")
nipkow@24331
   585
 apply auto
nipkow@24331
   586
done
wenzelm@20800
   587
haftmann@34979
   588
lemma ran_distinct: 
haftmann@34979
   589
  assumes dist: "distinct (map fst al)" 
haftmann@34979
   590
  shows "ran (map_of al) = snd ` set al"
haftmann@34979
   591
using assms proof (induct al)
haftmann@34979
   592
  case Nil then show ?case by simp
haftmann@34979
   593
next
haftmann@34979
   594
  case (Cons kv al)
haftmann@34979
   595
  then have "ran (map_of al) = snd ` set al" by simp
haftmann@34979
   596
  moreover from Cons.prems have "map_of al (fst kv) = None"
haftmann@34979
   597
    by (simp add: map_of_eq_None_iff)
haftmann@34979
   598
  ultimately show ?case by (simp only: map_of.simps ran_map_upd) simp
haftmann@34979
   599
qed
haftmann@34979
   600
nipkow@13910
   601
oheimb@14100
   602
subsection {* @{text "map_le"} *}
nipkow@13910
   603
kleing@13912
   604
lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g"
nipkow@24331
   605
by (simp add: map_le_def)
nipkow@13910
   606
paulson@17724
   607
lemma upd_None_map_le [simp]: "f(x := None) \<subseteq>\<^sub>m f"
nipkow@24331
   608
by (force simp add: map_le_def)
nipkow@14187
   609
nipkow@13910
   610
lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)"
nipkow@24331
   611
by (fastsimp simp add: map_le_def)
nipkow@13910
   612
paulson@17724
   613
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
   614
by (force simp add: map_le_def)
nipkow@14187
   615
wenzelm@20800
   616
lemma map_le_upds [simp]:
nipkow@24331
   617
  "f \<subseteq>\<^sub>m g ==> f(as [|->] bs) \<subseteq>\<^sub>m g(as [|->] bs)"
nipkow@24331
   618
apply (induct as arbitrary: f g bs)
nipkow@24331
   619
 apply simp
nipkow@24331
   620
apply (case_tac bs)
nipkow@24331
   621
 apply auto
nipkow@24331
   622
done
webertj@13908
   623
webertj@14033
   624
lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)"
nipkow@24331
   625
by (fastsimp simp add: map_le_def dom_def)
webertj@14033
   626
webertj@14033
   627
lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f"
nipkow@24331
   628
by (simp add: map_le_def)
webertj@14033
   629
nipkow@14187
   630
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
   631
by (auto simp add: map_le_def dom_def)
webertj@14033
   632
webertj@14033
   633
lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g"
nipkow@24331
   634
unfolding map_le_def
nipkow@24331
   635
apply (rule ext)
nipkow@24331
   636
apply (case_tac "x \<in> dom f", simp)
nipkow@24331
   637
apply (case_tac "x \<in> dom g", simp, fastsimp)
nipkow@24331
   638
done
webertj@14033
   639
webertj@14033
   640
lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)"
nipkow@24331
   641
by (fastsimp simp add: map_le_def)
webertj@14033
   642
nipkow@15304
   643
lemma map_le_iff_map_add_commute: "(f \<subseteq>\<^sub>m f ++ g) = (f++g = g++f)"
nipkow@24331
   644
by(fastsimp simp: map_add_def map_le_def expand_fun_eq split: option.splits)
nipkow@15304
   645
nipkow@15303
   646
lemma map_add_le_mapE: "f++g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h"
nipkow@24331
   647
by (fastsimp simp add: map_le_def map_add_def dom_def)
nipkow@15303
   648
nipkow@15303
   649
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
   650
by (clarsimp simp add: map_le_def map_add_def dom_def split: option.splits)
nipkow@15303
   651
nipkow@31080
   652
lemma dom_eq_singleton_conv: "dom f = {x} \<longleftrightarrow> (\<exists>v. f = [x \<mapsto> v])"
nipkow@31080
   653
proof(rule iffI)
nipkow@31080
   654
  assume "\<exists>v. f = [x \<mapsto> v]"
nipkow@31080
   655
  thus "dom f = {x}" by(auto split: split_if_asm)
nipkow@31080
   656
next
nipkow@31080
   657
  assume "dom f = {x}"
nipkow@31080
   658
  then obtain v where "f x = Some v" by auto
nipkow@31080
   659
  hence "[x \<mapsto> v] \<subseteq>\<^sub>m f" by(auto simp add: map_le_def)
nipkow@31080
   660
  moreover have "f \<subseteq>\<^sub>m [x \<mapsto> v]" using `dom f = {x}` `f x = Some v`
nipkow@31080
   661
    by(auto simp add: map_le_def)
nipkow@31080
   662
  ultimately have "f = [x \<mapsto> v]" by-(rule map_le_antisym)
nipkow@31080
   663
  thus "\<exists>v. f = [x \<mapsto> v]" by blast
nipkow@31080
   664
qed
nipkow@31080
   665
haftmann@35565
   666
haftmann@35565
   667
subsection {* Various *}
haftmann@35565
   668
haftmann@35565
   669
lemma set_map_of_compr:
haftmann@35565
   670
  assumes distinct: "distinct (map fst xs)"
haftmann@35565
   671
  shows "set xs = {(k, v). map_of xs k = Some v}"
haftmann@35565
   672
using assms proof (induct xs)
haftmann@35565
   673
  case Nil then show ?case by simp
haftmann@35565
   674
next
haftmann@35565
   675
  case (Cons x xs)
haftmann@35565
   676
  obtain k v where "x = (k, v)" by (cases x) blast
haftmann@35565
   677
  with Cons.prems have "k \<notin> dom (map_of xs)"
haftmann@35565
   678
    by (simp add: dom_map_of_conv_image_fst)
haftmann@35565
   679
  then have *: "insert (k, v) {(k, v). map_of xs k = Some v} =
haftmann@35565
   680
    {(k', v'). (map_of xs(k \<mapsto> v)) k' = Some v'}"
haftmann@35565
   681
    by (auto split: if_splits)
haftmann@35565
   682
  from Cons have "set xs = {(k, v). map_of xs k = Some v}" by simp
haftmann@35565
   683
  with * `x = (k, v)` show ?case by simp
haftmann@35565
   684
qed
haftmann@35565
   685
haftmann@35565
   686
lemma map_of_inject_set:
haftmann@35565
   687
  assumes distinct: "distinct (map fst xs)" "distinct (map fst ys)"
haftmann@35565
   688
  shows "map_of xs = map_of ys \<longleftrightarrow> set xs = set ys" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@35565
   689
proof
haftmann@35565
   690
  assume ?lhs
haftmann@35565
   691
  moreover from `distinct (map fst xs)` have "set xs = {(k, v). map_of xs k = Some v}"
haftmann@35565
   692
    by (rule set_map_of_compr)
haftmann@35565
   693
  moreover from `distinct (map fst ys)` have "set ys = {(k, v). map_of ys k = Some v}"
haftmann@35565
   694
    by (rule set_map_of_compr)
haftmann@35565
   695
  ultimately show ?rhs by simp
haftmann@35565
   696
next
haftmann@35565
   697
  assume ?rhs show ?lhs proof
haftmann@35565
   698
    fix k
haftmann@35565
   699
    show "map_of xs k = map_of ys k" proof (cases "map_of xs k")
haftmann@35565
   700
      case None
haftmann@35565
   701
      moreover with `?rhs` have "map_of ys k = None"
haftmann@35565
   702
        by (simp add: map_of_eq_None_iff)
haftmann@35565
   703
      ultimately show ?thesis by simp
haftmann@35565
   704
    next
haftmann@35565
   705
      case (Some v)
haftmann@35565
   706
      moreover with distinct `?rhs` have "map_of ys k = Some v"
haftmann@35565
   707
        by simp
haftmann@35565
   708
      ultimately show ?thesis by simp
haftmann@35565
   709
    qed
haftmann@35565
   710
  qed
haftmann@35565
   711
qed
haftmann@35565
   712
nipkow@3981
   713
end