src/HOL/Map.thy
author haftmann
Sat Nov 11 18:41:08 2017 +0000 (17 months ago)
changeset 67051 e7e54a0b9197
parent 66584 acb02fa48ef3
child 67091 1393c2340eec
permissions -rw-r--r--
dedicated definition for coprimality
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
wenzelm@60838
     5
The datatype of "maps"; strongly resembles maps in VDM.
nipkow@3981
     6
*)
nipkow@3981
     7
wenzelm@60758
     8
section \<open>Maps\<close>
nipkow@13914
     9
nipkow@15131
    10
theory Map
nipkow@15140
    11
imports List
nipkow@15131
    12
begin
nipkow@3981
    13
wenzelm@61069
    14
type_synonym ('a, 'b) "map" = "'a \<Rightarrow> 'b option" (infixr "\<rightharpoonup>" 0)
wenzelm@19656
    15
nipkow@19378
    16
abbreviation
wenzelm@60838
    17
  empty :: "'a \<rightharpoonup> 'b" where
wenzelm@60839
    18
  "empty \<equiv> \<lambda>x. None"
nipkow@19378
    19
wenzelm@19656
    20
definition
wenzelm@61069
    21
  map_comp :: "('b \<rightharpoonup> 'c) \<Rightarrow> ('a \<rightharpoonup> 'b) \<Rightarrow> ('a \<rightharpoonup> 'c)"  (infixl "\<circ>\<^sub>m" 55) where
wenzelm@61069
    22
  "f \<circ>\<^sub>m g = (\<lambda>k. case g k of None \<Rightarrow> None | Some v \<Rightarrow> f v)"
wenzelm@19656
    23
wenzelm@20800
    24
definition
wenzelm@60839
    25
  map_add :: "('a \<rightharpoonup> 'b) \<Rightarrow> ('a \<rightharpoonup> 'b) \<Rightarrow> ('a \<rightharpoonup> 'b)"  (infixl "++" 100) where
wenzelm@60839
    26
  "m1 ++ m2 = (\<lambda>x. case m2 x of None \<Rightarrow> m1 x | Some y \<Rightarrow> Some y)"
wenzelm@20800
    27
wenzelm@21404
    28
definition
wenzelm@60839
    29
  restrict_map :: "('a \<rightharpoonup> 'b) \<Rightarrow> 'a set \<Rightarrow> ('a \<rightharpoonup> 'b)"  (infixl "|`"  110) where
wenzelm@60839
    30
  "m|`A = (\<lambda>x. if x \<in> A then m x else None)"
nipkow@13910
    31
wenzelm@21210
    32
notation (latex output)
wenzelm@19656
    33
  restrict_map  ("_\<restriction>\<^bsub>_\<^esub>" [111,110] 110)
wenzelm@19656
    34
wenzelm@20800
    35
definition
wenzelm@60839
    36
  dom :: "('a \<rightharpoonup> 'b) \<Rightarrow> 'a set" where
wenzelm@60839
    37
  "dom m = {a. m a \<noteq> None}"
wenzelm@20800
    38
wenzelm@21404
    39
definition
wenzelm@60839
    40
  ran :: "('a \<rightharpoonup> 'b) \<Rightarrow> 'b set" where
wenzelm@60839
    41
  "ran m = {b. \<exists>a. m a = Some b}"
wenzelm@20800
    42
wenzelm@21404
    43
definition
wenzelm@60839
    44
  map_le :: "('a \<rightharpoonup> 'b) \<Rightarrow> ('a \<rightharpoonup> 'b) \<Rightarrow> bool"  (infix "\<subseteq>\<^sub>m" 50) where
wenzelm@60839
    45
  "(m\<^sub>1 \<subseteq>\<^sub>m m\<^sub>2) \<longleftrightarrow> (\<forall>a \<in> dom m\<^sub>1. m\<^sub>1 a = m\<^sub>2 a)"
wenzelm@20800
    46
wenzelm@41229
    47
nonterminal maplets and maplet
nipkow@14180
    48
oheimb@5300
    49
syntax
wenzelm@61955
    50
  "_maplet"  :: "['a, 'a] \<Rightarrow> maplet"             ("_ /\<mapsto>/ _")
wenzelm@61955
    51
  "_maplets" :: "['a, 'a] \<Rightarrow> maplet"             ("_ /[\<mapsto>]/ _")
wenzelm@60839
    52
  ""         :: "maplet \<Rightarrow> maplets"             ("_")
wenzelm@60839
    53
  "_Maplets" :: "[maplet, maplets] \<Rightarrow> maplets" ("_,/ _")
wenzelm@61955
    54
  "_MapUpd"  :: "['a \<rightharpoonup> 'b, maplets] \<Rightarrow> 'a \<rightharpoonup> 'b" ("_/'(_')" [900, 0] 900)
wenzelm@60839
    55
  "_Map"     :: "maplets \<Rightarrow> 'a \<rightharpoonup> 'b"            ("(1[_])")
nipkow@3981
    56
wenzelm@61955
    57
syntax (ASCII)
wenzelm@61955
    58
  "_maplet"  :: "['a, 'a] \<Rightarrow> maplet"             ("_ /|->/ _")
wenzelm@61955
    59
  "_maplets" :: "['a, 'a] \<Rightarrow> maplet"             ("_ /[|->]/ _")
nipkow@14180
    60
oheimb@5300
    61
translations
wenzelm@60839
    62
  "_MapUpd m (_Maplets xy ms)"  \<rightleftharpoons> "_MapUpd (_MapUpd m xy) ms"
wenzelm@60839
    63
  "_MapUpd m (_maplet  x y)"    \<rightleftharpoons> "m(x := CONST Some y)"
wenzelm@60839
    64
  "_Map ms"                     \<rightleftharpoons> "_MapUpd (CONST empty) ms"
wenzelm@60839
    65
  "_Map (_Maplets ms1 ms2)"     \<leftharpoondown> "_MapUpd (_Map ms1) ms2"
wenzelm@60839
    66
  "_Maplets ms1 (_Maplets ms2 ms3)" \<leftharpoondown> "_Maplets (_Maplets ms1 ms2) ms3"
nipkow@14180
    67
wenzelm@61955
    68
primrec map_of :: "('a \<times> 'b) list \<Rightarrow> 'a \<rightharpoonup> 'b"
wenzelm@61955
    69
where
wenzelm@61955
    70
  "map_of [] = empty"
wenzelm@61955
    71
| "map_of (p # ps) = (map_of ps)(fst p \<mapsto> snd p)"
oheimb@5300
    72
wenzelm@61955
    73
definition map_upds :: "('a \<rightharpoonup> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> 'a \<rightharpoonup> 'b"
wenzelm@61955
    74
  where "map_upds m xs ys = m ++ map_of (rev (zip xs ys))"
haftmann@34941
    75
translations
wenzelm@60839
    76
  "_MapUpd m (_maplets x y)" \<rightleftharpoons> "CONST map_upds m x y"
haftmann@25965
    77
wenzelm@60839
    78
lemma map_of_Cons_code [code]:
haftmann@25965
    79
  "map_of [] k = None"
haftmann@25965
    80
  "map_of ((l, v) # ps) k = (if l = k then Some v else map_of ps k)"
haftmann@25965
    81
  by simp_all
haftmann@25965
    82
wenzelm@20800
    83
wenzelm@60758
    84
subsection \<open>@{term [source] empty}\<close>
webertj@13908
    85
wenzelm@20800
    86
lemma empty_upd_none [simp]: "empty(x := None) = empty"
wenzelm@60839
    87
  by (rule ext) simp
webertj@13908
    88
webertj@13908
    89
wenzelm@60758
    90
subsection \<open>@{term [source] map_upd}\<close>
webertj@13908
    91
wenzelm@60839
    92
lemma map_upd_triv: "t k = Some x \<Longrightarrow> t(k\<mapsto>x) = t"
wenzelm@60839
    93
  by (rule ext) simp
webertj@13908
    94
wenzelm@60839
    95
lemma map_upd_nonempty [simp]: "t(k\<mapsto>x) \<noteq> empty"
wenzelm@20800
    96
proof
wenzelm@20800
    97
  assume "t(k \<mapsto> x) = empty"
wenzelm@20800
    98
  then have "(t(k \<mapsto> x)) k = None" by simp
wenzelm@20800
    99
  then show False by simp
wenzelm@20800
   100
qed
webertj@13908
   101
wenzelm@20800
   102
lemma map_upd_eqD1:
wenzelm@20800
   103
  assumes "m(a\<mapsto>x) = n(a\<mapsto>y)"
wenzelm@20800
   104
  shows "x = y"
wenzelm@20800
   105
proof -
wenzelm@41550
   106
  from assms have "(m(a\<mapsto>x)) a = (n(a\<mapsto>y)) a" by simp
wenzelm@20800
   107
  then show ?thesis by simp
wenzelm@20800
   108
qed
oheimb@14100
   109
wenzelm@20800
   110
lemma map_upd_Some_unfold:
wenzelm@60838
   111
  "((m(a\<mapsto>b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)"
nipkow@24331
   112
by auto
oheimb@14100
   113
wenzelm@20800
   114
lemma image_map_upd [simp]: "x \<notin> A \<Longrightarrow> m(x \<mapsto> y) ` A = m ` A"
nipkow@24331
   115
by auto
nipkow@15303
   116
wenzelm@60839
   117
lemma finite_range_updI: "finite (range f) \<Longrightarrow> finite (range (f(a\<mapsto>b)))"
nipkow@24331
   118
unfolding image_def
nipkow@24331
   119
apply (simp (no_asm_use) add:full_SetCompr_eq)
nipkow@24331
   120
apply (rule finite_subset)
nipkow@24331
   121
 prefer 2 apply assumption
nipkow@24331
   122
apply (auto)
nipkow@24331
   123
done
webertj@13908
   124
webertj@13908
   125
wenzelm@60758
   126
subsection \<open>@{term [source] map_of}\<close>
webertj@13908
   127
nipkow@15304
   128
lemma map_of_eq_None_iff:
nipkow@24331
   129
  "(map_of xys x = None) = (x \<notin> fst ` (set xys))"
nipkow@24331
   130
by (induct xys) simp_all
nipkow@15304
   131
wenzelm@20800
   132
lemma map_of_eq_Some_iff [simp]:
nipkow@24331
   133
  "distinct(map fst xys) \<Longrightarrow> (map_of xys x = Some y) = ((x,y) \<in> set xys)"
nipkow@24331
   134
apply (induct xys)
nipkow@24331
   135
 apply simp
nipkow@24331
   136
apply (auto simp: map_of_eq_None_iff [symmetric])
nipkow@24331
   137
done
nipkow@15304
   138
wenzelm@20800
   139
lemma Some_eq_map_of_iff [simp]:
nipkow@24331
   140
  "distinct(map fst xys) \<Longrightarrow> (Some y = map_of xys x) = ((x,y) \<in> set xys)"
wenzelm@60839
   141
by (auto simp del: map_of_eq_Some_iff simp: map_of_eq_Some_iff [symmetric])
nipkow@15304
   142
paulson@17724
   143
lemma map_of_is_SomeI [simp]: "\<lbrakk> distinct(map fst xys); (x,y) \<in> set xys \<rbrakk>
wenzelm@20800
   144
    \<Longrightarrow> map_of xys x = Some y"
nipkow@24331
   145
apply (induct xys)
nipkow@24331
   146
 apply simp
nipkow@24331
   147
apply force
nipkow@24331
   148
done
nipkow@15304
   149
wenzelm@20800
   150
lemma map_of_zip_is_None [simp]:
nipkow@24331
   151
  "length xs = length ys \<Longrightarrow> (map_of (zip xs ys) x = None) = (x \<notin> set xs)"
nipkow@24331
   152
by (induct rule: list_induct2) simp_all
nipkow@15110
   153
haftmann@26443
   154
lemma map_of_zip_is_Some:
haftmann@26443
   155
  assumes "length xs = length ys"
haftmann@26443
   156
  shows "x \<in> set xs \<longleftrightarrow> (\<exists>y. map_of (zip xs ys) x = Some y)"
haftmann@26443
   157
using assms by (induct rule: list_induct2) simp_all
haftmann@26443
   158
haftmann@26443
   159
lemma map_of_zip_upd:
haftmann@26443
   160
  fixes x :: 'a and xs :: "'a list" and ys zs :: "'b list"
haftmann@26443
   161
  assumes "length ys = length xs"
haftmann@26443
   162
    and "length zs = length xs"
haftmann@26443
   163
    and "x \<notin> set xs"
haftmann@26443
   164
    and "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)"
haftmann@26443
   165
  shows "map_of (zip xs ys) = map_of (zip xs zs)"
haftmann@26443
   166
proof
haftmann@26443
   167
  fix x' :: 'a
haftmann@26443
   168
  show "map_of (zip xs ys) x' = map_of (zip xs zs) x'"
haftmann@26443
   169
  proof (cases "x = x'")
haftmann@26443
   170
    case True
haftmann@26443
   171
    from assms True map_of_zip_is_None [of xs ys x']
haftmann@26443
   172
      have "map_of (zip xs ys) x' = None" by simp
haftmann@26443
   173
    moreover from assms True map_of_zip_is_None [of xs zs x']
haftmann@26443
   174
      have "map_of (zip xs zs) x' = None" by simp
haftmann@26443
   175
    ultimately show ?thesis by simp
haftmann@26443
   176
  next
haftmann@26443
   177
    case False from assms
haftmann@26443
   178
      have "(map_of (zip xs ys)(x \<mapsto> y)) x' = (map_of (zip xs zs)(x \<mapsto> z)) x'" by auto
haftmann@26443
   179
    with False show ?thesis by simp
haftmann@26443
   180
  qed
haftmann@26443
   181
qed
haftmann@26443
   182
haftmann@26443
   183
lemma map_of_zip_inject:
haftmann@26443
   184
  assumes "length ys = length xs"
haftmann@26443
   185
    and "length zs = length xs"
haftmann@26443
   186
    and dist: "distinct xs"
haftmann@26443
   187
    and map_of: "map_of (zip xs ys) = map_of (zip xs zs)"
haftmann@26443
   188
  shows "ys = zs"
wenzelm@60839
   189
  using assms(1) assms(2)[symmetric]
wenzelm@60839
   190
  using dist map_of
wenzelm@60839
   191
proof (induct ys xs zs rule: list_induct3)
haftmann@26443
   192
  case Nil show ?case by simp
haftmann@26443
   193
next
haftmann@26443
   194
  case (Cons y ys x xs z zs)
wenzelm@60758
   195
  from \<open>map_of (zip (x#xs) (y#ys)) = map_of (zip (x#xs) (z#zs))\<close>
haftmann@26443
   196
    have map_of: "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)" by simp
haftmann@26443
   197
  from Cons have "length ys = length xs" and "length zs = length xs"
haftmann@26443
   198
    and "x \<notin> set xs" by simp_all
haftmann@26443
   199
  then have "map_of (zip xs ys) = map_of (zip xs zs)" using map_of by (rule map_of_zip_upd)
wenzelm@60758
   200
  with Cons.hyps \<open>distinct (x # xs)\<close> have "ys = zs" by simp
haftmann@26443
   201
  moreover from map_of have "y = z" by (rule map_upd_eqD1)
haftmann@26443
   202
  ultimately show ?case by simp
haftmann@26443
   203
qed
haftmann@26443
   204
bulwahn@66584
   205
lemma map_of_zip_nth:
bulwahn@66584
   206
  assumes "length xs = length ys"
bulwahn@66584
   207
  assumes "distinct xs"
bulwahn@66584
   208
  assumes "i < length ys"
bulwahn@66584
   209
  shows "map_of (zip xs ys) (xs ! i) = Some (ys ! i)"
bulwahn@66584
   210
using assms proof (induct arbitrary: i rule: list_induct2)
bulwahn@66584
   211
  case Nil
bulwahn@66584
   212
  then show ?case by simp
bulwahn@66584
   213
next
bulwahn@66584
   214
  case (Cons x xs y ys)
bulwahn@66584
   215
  then show ?case
bulwahn@66584
   216
    using less_Suc_eq_0_disj by auto
bulwahn@66584
   217
qed
bulwahn@66584
   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)"
nipkow@39302
   221
  by (induct xs) (simp_all add: fun_eq_iff)
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"
wenzelm@60841
   232
  by (induct xs) (auto split: if_splits)
webertj@13908
   233
wenzelm@20800
   234
lemma map_of_mapk_SomeI:
wenzelm@60839
   235
  "inj f \<Longrightarrow> map_of t k = Some x \<Longrightarrow>
haftmann@61032
   236
   map_of (map (case_prod (\<lambda>k. Pair (f k))) t) (f k) = Some x"
wenzelm@60839
   237
by (induct t) (auto simp: inj_eq)
webertj@13908
   238
wenzelm@60839
   239
lemma weak_map_of_SomeI: "(k, x) \<in> set l \<Longrightarrow> \<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:
haftmann@61032
   243
  "map_of xs k = Some z \<Longrightarrow> P k z \<Longrightarrow> map_of (filter (case_prod P) xs) k = Some z"
nipkow@24331
   244
by (induct xs) auto
webertj@13908
   245
haftmann@35607
   246
lemma map_of_map:
blanchet@55466
   247
  "map_of (map (\<lambda>(k, v). (k, f v)) xs) = map_option f \<circ> map_of xs"
wenzelm@60839
   248
  by (induct xs) (auto simp: fun_eq_iff)
haftmann@35607
   249
blanchet@55466
   250
lemma dom_map_option:
blanchet@55466
   251
  "dom (\<lambda>k. map_option (f k) (m k)) = dom m"
haftmann@35607
   252
  by (simp add: dom_def)
webertj@13908
   253
haftmann@56545
   254
lemma dom_map_option_comp [simp]:
haftmann@56545
   255
  "dom (map_option g \<circ> m) = dom m"
haftmann@56545
   256
  using dom_map_option [of "\<lambda>_. g" m] by (simp add: comp_def)
haftmann@56545
   257
webertj@13908
   258
wenzelm@60758
   259
subsection \<open>@{const map_option} related\<close>
webertj@13908
   260
blanchet@55466
   261
lemma map_option_o_empty [simp]: "map_option f o empty = empty"
nipkow@24331
   262
by (rule ext) simp
webertj@13908
   263
blanchet@55466
   264
lemma map_option_o_map_upd [simp]:
wenzelm@60838
   265
  "map_option f o m(a\<mapsto>b) = (map_option f o m)(a\<mapsto>f b)"
nipkow@24331
   266
by (rule ext) simp
wenzelm@20800
   267
webertj@13908
   268
wenzelm@60758
   269
subsection \<open>@{term [source] map_comp} related\<close>
schirmer@17391
   270
wenzelm@20800
   271
lemma map_comp_empty [simp]:
nipkow@24331
   272
  "m \<circ>\<^sub>m empty = empty"
nipkow@24331
   273
  "empty \<circ>\<^sub>m m = empty"
wenzelm@60839
   274
by (auto simp: map_comp_def split: option.splits)
schirmer@17391
   275
wenzelm@20800
   276
lemma map_comp_simps [simp]:
nipkow@24331
   277
  "m2 k = None \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = None"
nipkow@24331
   278
  "m2 k = Some k' \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = m1 k'"
wenzelm@60839
   279
by (auto simp: map_comp_def)
schirmer@17391
   280
schirmer@17391
   281
lemma map_comp_Some_iff:
nipkow@24331
   282
  "((m1 \<circ>\<^sub>m m2) k = Some v) = (\<exists>k'. m2 k = Some k' \<and> m1 k' = Some v)"
wenzelm@60839
   283
by (auto simp: map_comp_def split: option.splits)
schirmer@17391
   284
schirmer@17391
   285
lemma map_comp_None_iff:
nipkow@24331
   286
  "((m1 \<circ>\<^sub>m m2) k = None) = (m2 k = None \<or> (\<exists>k'. m2 k = Some k' \<and> m1 k' = None)) "
wenzelm@60839
   287
by (auto simp: map_comp_def split: option.splits)
webertj@13908
   288
wenzelm@20800
   289
wenzelm@61799
   290
subsection \<open>\<open>++\<close>\<close>
webertj@13908
   291
nipkow@14025
   292
lemma map_add_empty[simp]: "m ++ empty = m"
nipkow@24331
   293
by(simp add: map_add_def)
webertj@13908
   294
nipkow@14025
   295
lemma empty_map_add[simp]: "empty ++ m = m"
nipkow@24331
   296
by (rule ext) (simp add: map_add_def split: option.split)
webertj@13908
   297
nipkow@14025
   298
lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3"
nipkow@24331
   299
by (rule ext) (simp add: map_add_def split: option.split)
wenzelm@20800
   300
wenzelm@20800
   301
lemma map_add_Some_iff:
nipkow@24331
   302
  "((m ++ n) k = Some x) = (n k = Some x | n k = None & m k = Some x)"
nipkow@24331
   303
by (simp add: map_add_def split: option.split)
nipkow@14025
   304
wenzelm@20800
   305
lemma map_add_SomeD [dest!]:
nipkow@24331
   306
  "(m ++ n) k = Some x \<Longrightarrow> n k = Some x \<or> n k = None \<and> m k = Some x"
nipkow@24331
   307
by (rule map_add_Some_iff [THEN iffD1])
webertj@13908
   308
wenzelm@60839
   309
lemma map_add_find_right [simp]: "n k = Some xx \<Longrightarrow> (m ++ n) k = Some xx"
nipkow@24331
   310
by (subst map_add_Some_iff) fast
webertj@13908
   311
nipkow@14025
   312
lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)"
nipkow@24331
   313
by (simp add: map_add_def split: option.split)
webertj@13908
   314
wenzelm@60838
   315
lemma map_add_upd[simp]: "f ++ g(x\<mapsto>y) = (f ++ g)(x\<mapsto>y)"
nipkow@24331
   316
by (rule ext) (simp add: map_add_def)
webertj@13908
   317
nipkow@14186
   318
lemma map_add_upds[simp]: "m1 ++ (m2(xs[\<mapsto>]ys)) = (m1++m2)(xs[\<mapsto>]ys)"
nipkow@24331
   319
by (simp add: map_upds_def)
nipkow@14186
   320
krauss@32236
   321
lemma map_add_upd_left: "m\<notin>dom e2 \<Longrightarrow> e1(m \<mapsto> u1) ++ e2 = (e1 ++ e2)(m \<mapsto> u1)"
krauss@32236
   322
by (rule ext) (auto simp: map_add_def dom_def split: option.split)
krauss@32236
   323
wenzelm@20800
   324
lemma map_of_append[simp]: "map_of (xs @ ys) = map_of ys ++ map_of xs"
nipkow@24331
   325
unfolding map_add_def
nipkow@24331
   326
apply (induct xs)
nipkow@24331
   327
 apply simp
nipkow@24331
   328
apply (rule ext)
nipkow@63648
   329
apply (simp split: option.split)
nipkow@24331
   330
done
webertj@13908
   331
nipkow@14025
   332
lemma finite_range_map_of_map_add:
wenzelm@60839
   333
  "finite (range f) \<Longrightarrow> finite (range (f ++ map_of l))"
nipkow@24331
   334
apply (induct l)
nipkow@24331
   335
 apply (auto simp del: fun_upd_apply)
nipkow@24331
   336
apply (erule finite_range_updI)
nipkow@24331
   337
done
webertj@13908
   338
wenzelm@20800
   339
lemma inj_on_map_add_dom [iff]:
nipkow@24331
   340
  "inj_on (m ++ m') (dom m') = inj_on m' (dom m')"
nipkow@44890
   341
by (fastforce simp: map_add_def dom_def inj_on_def split: option.splits)
wenzelm@20800
   342
haftmann@34979
   343
lemma map_upds_fold_map_upd:
haftmann@35552
   344
  "m(ks[\<mapsto>]vs) = foldl (\<lambda>m (k, v). m(k \<mapsto> v)) m (zip ks vs)"
haftmann@34979
   345
unfolding map_upds_def proof (rule sym, rule zip_obtain_same_length)
haftmann@34979
   346
  fix ks :: "'a list" and vs :: "'b list"
haftmann@34979
   347
  assume "length ks = length vs"
haftmann@35552
   348
  then show "foldl (\<lambda>m (k, v). m(k\<mapsto>v)) m (zip ks vs) = m ++ map_of (rev (zip ks vs))"
haftmann@35552
   349
    by(induct arbitrary: m rule: list_induct2) simp_all
haftmann@34979
   350
qed
haftmann@34979
   351
haftmann@34979
   352
lemma map_add_map_of_foldr:
haftmann@34979
   353
  "m ++ map_of ps = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) ps m"
wenzelm@60839
   354
  by (induct ps) (auto simp: fun_eq_iff map_add_def)
haftmann@34979
   355
nipkow@15304
   356
wenzelm@60758
   357
subsection \<open>@{term [source] restrict_map}\<close>
oheimb@14100
   358
wenzelm@20800
   359
lemma restrict_map_to_empty [simp]: "m|`{} = empty"
nipkow@24331
   360
by (simp add: restrict_map_def)
nipkow@14186
   361
haftmann@31380
   362
lemma restrict_map_insert: "f |` (insert a A) = (f |` A)(a := f a)"
wenzelm@60839
   363
by (auto simp: restrict_map_def)
haftmann@31380
   364
wenzelm@20800
   365
lemma restrict_map_empty [simp]: "empty|`D = empty"
nipkow@24331
   366
by (simp add: restrict_map_def)
nipkow@14186
   367
nipkow@15693
   368
lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m|`A) x = m x"
nipkow@24331
   369
by (simp add: restrict_map_def)
oheimb@14100
   370
nipkow@15693
   371
lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m|`A) x = None"
nipkow@24331
   372
by (simp add: restrict_map_def)
oheimb@14100
   373
nipkow@15693
   374
lemma ran_restrictD: "y \<in> ran (m|`A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y"
nipkow@62390
   375
by (auto simp: restrict_map_def ran_def split: if_split_asm)
oheimb@14100
   376
nipkow@15693
   377
lemma dom_restrict [simp]: "dom (m|`A) = dom m \<inter> A"
nipkow@62390
   378
by (auto simp: restrict_map_def dom_def split: if_split_asm)
oheimb@14100
   379
nipkow@15693
   380
lemma restrict_upd_same [simp]: "m(x\<mapsto>y)|`(-{x}) = m|`(-{x})"
nipkow@24331
   381
by (rule ext) (auto simp: restrict_map_def)
oheimb@14100
   382
nipkow@15693
   383
lemma restrict_restrict [simp]: "m|`A|`B = m|`(A\<inter>B)"
nipkow@24331
   384
by (rule ext) (auto simp: restrict_map_def)
oheimb@14100
   385
wenzelm@20800
   386
lemma restrict_fun_upd [simp]:
nipkow@24331
   387
  "m(x := y)|`D = (if x \<in> D then (m|`(D-{x}))(x := y) else m|`D)"
nipkow@39302
   388
by (simp add: restrict_map_def fun_eq_iff)
nipkow@14186
   389
wenzelm@20800
   390
lemma fun_upd_None_restrict [simp]:
wenzelm@60839
   391
  "(m|`D)(x := None) = (if x \<in> D then m|`(D - {x}) else m|`D)"
nipkow@39302
   392
by (simp add: restrict_map_def fun_eq_iff)
nipkow@14186
   393
wenzelm@20800
   394
lemma fun_upd_restrict: "(m|`D)(x := y) = (m|`(D-{x}))(x := y)"
nipkow@39302
   395
by (simp add: restrict_map_def fun_eq_iff)
nipkow@14186
   396
wenzelm@20800
   397
lemma fun_upd_restrict_conv [simp]:
nipkow@24331
   398
  "x \<in> D \<Longrightarrow> (m|`D)(x := y) = (m|`(D-{x}))(x := y)"
nipkow@39302
   399
by (simp add: restrict_map_def fun_eq_iff)
nipkow@14186
   400
haftmann@35159
   401
lemma map_of_map_restrict:
haftmann@35159
   402
  "map_of (map (\<lambda>k. (k, f k)) ks) = (Some \<circ> f) |` set ks"
nipkow@39302
   403
  by (induct ks) (simp_all add: fun_eq_iff restrict_map_insert)
haftmann@35159
   404
haftmann@35619
   405
lemma restrict_complement_singleton_eq:
haftmann@35619
   406
  "f |` (- {x}) = f(x := None)"
nipkow@39302
   407
  by (simp add: restrict_map_def fun_eq_iff)
haftmann@35619
   408
oheimb@14100
   409
wenzelm@60758
   410
subsection \<open>@{term [source] map_upds}\<close>
nipkow@14025
   411
wenzelm@60838
   412
lemma map_upds_Nil1 [simp]: "m([] [\<mapsto>] bs) = m"
nipkow@24331
   413
by (simp add: map_upds_def)
nipkow@14025
   414
wenzelm@60838
   415
lemma map_upds_Nil2 [simp]: "m(as [\<mapsto>] []) = m"
nipkow@24331
   416
by (simp add:map_upds_def)
wenzelm@20800
   417
wenzelm@60838
   418
lemma map_upds_Cons [simp]: "m(a#as [\<mapsto>] b#bs) = (m(a\<mapsto>b))(as[\<mapsto>]bs)"
nipkow@24331
   419
by (simp add:map_upds_def)
nipkow@14025
   420
wenzelm@60839
   421
lemma map_upds_append1 [simp]: "size xs < size ys \<Longrightarrow>
nipkow@24331
   422
  m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)"
wenzelm@60839
   423
apply(induct xs arbitrary: ys m)
nipkow@24331
   424
 apply (clarsimp simp add: neq_Nil_conv)
nipkow@24331
   425
apply (case_tac ys)
nipkow@24331
   426
 apply simp
nipkow@24331
   427
apply simp
nipkow@24331
   428
done
nipkow@14187
   429
wenzelm@20800
   430
lemma map_upds_list_update2_drop [simp]:
bulwahn@46588
   431
  "size xs \<le> i \<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)"
nipkow@24331
   432
apply (induct xs arbitrary: m ys i)
nipkow@24331
   433
 apply simp
nipkow@24331
   434
apply (case_tac ys)
nipkow@24331
   435
 apply simp
nipkow@24331
   436
apply (simp split: nat.split)
nipkow@24331
   437
done
nipkow@14025
   438
wenzelm@20800
   439
lemma map_upd_upds_conv_if:
wenzelm@60838
   440
  "(f(x\<mapsto>y))(xs [\<mapsto>] ys) =
wenzelm@60839
   441
   (if x \<in> set(take (length ys) xs) then f(xs [\<mapsto>] ys)
wenzelm@60838
   442
                                    else (f(xs [\<mapsto>] ys))(x\<mapsto>y))"
nipkow@24331
   443
apply (induct xs arbitrary: x y ys f)
nipkow@24331
   444
 apply simp
nipkow@24331
   445
apply (case_tac ys)
nipkow@62390
   446
 apply (auto split: if_split simp: fun_upd_twist)
nipkow@24331
   447
done
nipkow@14025
   448
nipkow@14025
   449
lemma map_upds_twist [simp]:
wenzelm@60839
   450
  "a \<notin> set as \<Longrightarrow> m(a\<mapsto>b)(as[\<mapsto>]bs) = m(as[\<mapsto>]bs)(a\<mapsto>b)"
nipkow@44890
   451
using set_take_subset by (fastforce simp add: map_upd_upds_conv_if)
nipkow@14025
   452
wenzelm@20800
   453
lemma map_upds_apply_nontin [simp]:
wenzelm@60839
   454
  "x \<notin> set xs \<Longrightarrow> (f(xs[\<mapsto>]ys)) x = f x"
nipkow@24331
   455
apply (induct xs arbitrary: ys)
nipkow@24331
   456
 apply simp
nipkow@24331
   457
apply (case_tac ys)
nipkow@24331
   458
 apply (auto simp: map_upd_upds_conv_if)
nipkow@24331
   459
done
nipkow@14025
   460
wenzelm@20800
   461
lemma fun_upds_append_drop [simp]:
nipkow@24331
   462
  "size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)"
nipkow@24331
   463
apply (induct xs arbitrary: m ys)
nipkow@24331
   464
 apply simp
nipkow@24331
   465
apply (case_tac ys)
nipkow@24331
   466
 apply simp_all
nipkow@24331
   467
done
nipkow@14300
   468
wenzelm@20800
   469
lemma fun_upds_append2_drop [simp]:
nipkow@24331
   470
  "size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)"
nipkow@24331
   471
apply (induct xs arbitrary: m ys)
nipkow@24331
   472
 apply simp
nipkow@24331
   473
apply (case_tac ys)
nipkow@24331
   474
 apply simp_all
nipkow@24331
   475
done
nipkow@14300
   476
nipkow@14300
   477
wenzelm@20800
   478
lemma restrict_map_upds[simp]:
wenzelm@20800
   479
  "\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk>
wenzelm@20800
   480
    \<Longrightarrow> m(xs [\<mapsto>] ys)|`D = (m|`(D - set xs))(xs [\<mapsto>] ys)"
nipkow@24331
   481
apply (induct xs arbitrary: m ys)
nipkow@24331
   482
 apply simp
nipkow@24331
   483
apply (case_tac ys)
nipkow@24331
   484
 apply simp
nipkow@24331
   485
apply (simp add: Diff_insert [symmetric] insert_absorb)
nipkow@24331
   486
apply (simp add: map_upd_upds_conv_if)
nipkow@24331
   487
done
nipkow@14186
   488
nipkow@14186
   489
wenzelm@60758
   490
subsection \<open>@{term [source] dom}\<close>
webertj@13908
   491
nipkow@31080
   492
lemma dom_eq_empty_conv [simp]: "dom f = {} \<longleftrightarrow> f = empty"
huffman@44921
   493
  by (auto simp: dom_def)
nipkow@31080
   494
wenzelm@60839
   495
lemma domI: "m a = Some b \<Longrightarrow> a \<in> dom m"
wenzelm@60839
   496
  by (simp add: dom_def)
oheimb@14100
   497
(* declare domI [intro]? *)
webertj@13908
   498
wenzelm@60839
   499
lemma domD: "a \<in> dom m \<Longrightarrow> \<exists>b. m a = Some b"
wenzelm@60839
   500
  by (cases "m a") (auto simp add: dom_def)
webertj@13908
   501
haftmann@66010
   502
lemma domIff [iff, simp del, code_unfold]: "a \<in> dom m \<longleftrightarrow> m a \<noteq> None"
wenzelm@60839
   503
  by (simp add: dom_def)
webertj@13908
   504
wenzelm@20800
   505
lemma dom_empty [simp]: "dom empty = {}"
wenzelm@60839
   506
  by (simp add: dom_def)
webertj@13908
   507
wenzelm@20800
   508
lemma dom_fun_upd [simp]:
wenzelm@60839
   509
  "dom(f(x := y)) = (if y = None then dom f - {x} else insert x (dom f))"
wenzelm@60839
   510
  by (auto simp: dom_def)
webertj@13908
   511
haftmann@34979
   512
lemma dom_if:
haftmann@34979
   513
  "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
   514
  by (auto split: if_splits)
nipkow@13937
   515
nipkow@15304
   516
lemma dom_map_of_conv_image_fst:
haftmann@34979
   517
  "dom (map_of xys) = fst ` set xys"
haftmann@34979
   518
  by (induct xys) (auto simp add: dom_if)
nipkow@15304
   519
wenzelm@60839
   520
lemma dom_map_of_zip [simp]: "length xs = length ys \<Longrightarrow> dom (map_of (zip xs ys)) = set xs"
wenzelm@60839
   521
  by (induct rule: list_induct2) (auto simp: dom_if)
nipkow@15110
   522
webertj@13908
   523
lemma finite_dom_map_of: "finite (dom (map_of l))"
wenzelm@60839
   524
  by (induct l) (auto simp: dom_def insert_Collect [symmetric])
webertj@13908
   525
wenzelm@20800
   526
lemma dom_map_upds [simp]:
wenzelm@60839
   527
  "dom(m(xs[\<mapsto>]ys)) = set(take (length ys) xs) \<union> dom m"
nipkow@24331
   528
apply (induct xs arbitrary: m ys)
nipkow@24331
   529
 apply simp
nipkow@24331
   530
apply (case_tac ys)
nipkow@24331
   531
 apply auto
nipkow@24331
   532
done
nipkow@13910
   533
wenzelm@60839
   534
lemma dom_map_add [simp]: "dom (m ++ n) = dom n \<union> dom m"
wenzelm@60839
   535
  by (auto simp: dom_def)
nipkow@13910
   536
wenzelm@20800
   537
lemma dom_override_on [simp]:
wenzelm@60839
   538
  "dom (override_on f g A) =
wenzelm@60839
   539
    (dom f  - {a. a \<in> A - dom g}) \<union> {a. a \<in> A \<inter> dom g}"
wenzelm@60839
   540
  by (auto simp: dom_def override_on_def)
webertj@13908
   541
wenzelm@60839
   542
lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1 ++ m2 = m2 ++ m1"
wenzelm@60839
   543
  by (rule ext) (force simp: map_add_def dom_def split: option.split)
wenzelm@20800
   544
krauss@32236
   545
lemma map_add_dom_app_simps:
wenzelm@60839
   546
  "m \<in> dom l2 \<Longrightarrow> (l1 ++ l2) m = l2 m"
wenzelm@60839
   547
  "m \<notin> dom l1 \<Longrightarrow> (l1 ++ l2) m = l2 m"
wenzelm@60839
   548
  "m \<notin> dom l2 \<Longrightarrow> (l1 ++ l2) m = l1 m"
wenzelm@60839
   549
  by (auto simp add: map_add_def split: option.split_asm)
krauss@32236
   550
haftmann@29622
   551
lemma dom_const [simp]:
haftmann@35159
   552
  "dom (\<lambda>x. Some (f x)) = UNIV"
haftmann@29622
   553
  by auto
haftmann@29622
   554
nipkow@22230
   555
(* Due to John Matthews - could be rephrased with dom *)
nipkow@22230
   556
lemma finite_map_freshness:
nipkow@22230
   557
  "finite (dom (f :: 'a \<rightharpoonup> 'b)) \<Longrightarrow> \<not> finite (UNIV :: 'a set) \<Longrightarrow>
nipkow@22230
   558
   \<exists>x. f x = None"
wenzelm@60839
   559
  by (bestsimp dest: ex_new_if_finite)
nipkow@14027
   560
haftmann@28790
   561
lemma dom_minus:
haftmann@28790
   562
  "f x = None \<Longrightarrow> dom f - insert x A = dom f - A"
haftmann@28790
   563
  unfolding dom_def by simp
haftmann@28790
   564
haftmann@28790
   565
lemma insert_dom:
haftmann@28790
   566
  "f x = Some y \<Longrightarrow> insert x (dom f) = dom f"
haftmann@28790
   567
  unfolding dom_def by auto
haftmann@28790
   568
haftmann@35607
   569
lemma map_of_map_keys:
haftmann@35607
   570
  "set xs = dom m \<Longrightarrow> map_of (map (\<lambda>k. (k, the (m k))) xs) = m"
haftmann@35607
   571
  by (rule ext) (auto simp add: map_of_map_restrict restrict_map_def)
haftmann@35607
   572
haftmann@39379
   573
lemma map_of_eqI:
haftmann@39379
   574
  assumes set_eq: "set (map fst xs) = set (map fst ys)"
haftmann@39379
   575
  assumes map_eq: "\<forall>k\<in>set (map fst xs). map_of xs k = map_of ys k"
haftmann@39379
   576
  shows "map_of xs = map_of ys"
haftmann@39379
   577
proof (rule ext)
haftmann@39379
   578
  fix k show "map_of xs k = map_of ys k"
haftmann@39379
   579
  proof (cases "map_of xs k")
wenzelm@60839
   580
    case None
wenzelm@60839
   581
    then have "k \<notin> set (map fst xs)" by (simp add: map_of_eq_None_iff)
haftmann@39379
   582
    with set_eq have "k \<notin> set (map fst ys)" by simp
haftmann@39379
   583
    then have "map_of ys k = None" by (simp add: map_of_eq_None_iff)
haftmann@39379
   584
    with None show ?thesis by simp
haftmann@39379
   585
  next
wenzelm@60839
   586
    case (Some v)
wenzelm@60839
   587
    then have "k \<in> set (map fst xs)" by (auto simp add: dom_map_of_conv_image_fst [symmetric])
haftmann@39379
   588
    with map_eq show ?thesis by auto
haftmann@39379
   589
  qed
haftmann@39379
   590
qed
haftmann@39379
   591
haftmann@39379
   592
lemma map_of_eq_dom:
haftmann@39379
   593
  assumes "map_of xs = map_of ys"
haftmann@39379
   594
  shows "fst ` set xs = fst ` set ys"
haftmann@39379
   595
proof -
haftmann@39379
   596
  from assms have "dom (map_of xs) = dom (map_of ys)" by simp
haftmann@39379
   597
  then show ?thesis by (simp add: dom_map_of_conv_image_fst)
haftmann@39379
   598
qed
haftmann@39379
   599
nipkow@53820
   600
lemma finite_set_of_finite_maps:
wenzelm@60839
   601
  assumes "finite A" "finite B"
wenzelm@60839
   602
  shows "finite {m. dom m = A \<and> ran m \<subseteq> B}" (is "finite ?S")
nipkow@53820
   603
proof -
nipkow@53820
   604
  let ?S' = "{m. \<forall>x. (x \<in> A \<longrightarrow> m x \<in> Some ` B) \<and> (x \<notin> A \<longrightarrow> m x = None)}"
nipkow@53820
   605
  have "?S = ?S'"
nipkow@53820
   606
  proof
wenzelm@60839
   607
    show "?S \<subseteq> ?S'" by (auto simp: dom_def ran_def image_def)
nipkow@53820
   608
    show "?S' \<subseteq> ?S"
nipkow@53820
   609
    proof
nipkow@53820
   610
      fix m assume "m \<in> ?S'"
nipkow@53820
   611
      hence 1: "dom m = A" by force
wenzelm@60839
   612
      hence 2: "ran m \<subseteq> B" using \<open>m \<in> ?S'\<close> by (auto simp: dom_def ran_def)
nipkow@53820
   613
      from 1 2 show "m \<in> ?S" by blast
nipkow@53820
   614
    qed
nipkow@53820
   615
  qed
nipkow@53820
   616
  with assms show ?thesis by(simp add: finite_set_of_finite_funs)
nipkow@53820
   617
qed
haftmann@28790
   618
wenzelm@60839
   619
wenzelm@60758
   620
subsection \<open>@{term [source] ran}\<close>
oheimb@14100
   621
wenzelm@60839
   622
lemma ranI: "m a = Some b \<Longrightarrow> b \<in> ran m"
wenzelm@60839
   623
  by (auto simp: ran_def)
oheimb@14100
   624
(* declare ranI [intro]? *)
webertj@13908
   625
wenzelm@20800
   626
lemma ran_empty [simp]: "ran empty = {}"
wenzelm@60839
   627
  by (auto simp: ran_def)
webertj@13908
   628
wenzelm@60839
   629
lemma ran_map_upd [simp]: "m a = None \<Longrightarrow> ran(m(a\<mapsto>b)) = insert b (ran m)"
wenzelm@60839
   630
  unfolding ran_def
nipkow@24331
   631
apply auto
wenzelm@60839
   632
apply (subgoal_tac "aa \<noteq> a")
nipkow@24331
   633
 apply auto
nipkow@24331
   634
done
wenzelm@20800
   635
bulwahn@66583
   636
lemma ran_map_add:
bulwahn@66583
   637
  assumes "dom m1 \<inter> dom m2 = {}"
bulwahn@66583
   638
  shows "ran (m1 ++ m2) = ran m1 \<union> ran m2"
bulwahn@66583
   639
proof
bulwahn@66583
   640
  show "ran (m1 ++ m2) \<subseteq> ran m1 \<union> ran m2"
bulwahn@66583
   641
    unfolding ran_def by auto
bulwahn@66583
   642
next
bulwahn@66583
   643
  show "ran m1 \<union> ran m2 \<subseteq> ran (m1 ++ m2)"
bulwahn@66583
   644
  proof -
bulwahn@66583
   645
    have "(m1 ++ m2) x = Some y" if "m1 x = Some y" for x y
bulwahn@66583
   646
      using assms map_add_comm that by fastforce
bulwahn@66583
   647
    moreover have "(m1 ++ m2) x = Some y" if "m2 x = Some y" for x y
bulwahn@66583
   648
      using assms that by auto
bulwahn@66583
   649
    ultimately show ?thesis
bulwahn@66583
   650
      unfolding ran_def by blast
bulwahn@66583
   651
  qed
bulwahn@66583
   652
qed
bulwahn@66583
   653
bulwahn@66583
   654
lemma finite_ran:
bulwahn@66583
   655
  assumes "finite (dom p)"
bulwahn@66583
   656
  shows "finite (ran p)"
bulwahn@66583
   657
proof -
bulwahn@66583
   658
  have "ran p = (\<lambda>x. the (p x)) ` dom p"
bulwahn@66583
   659
    unfolding ran_def by force
bulwahn@66583
   660
  from this \<open>finite (dom p)\<close> show ?thesis by auto
bulwahn@66583
   661
qed
bulwahn@66583
   662
wenzelm@60839
   663
lemma ran_distinct:
wenzelm@60839
   664
  assumes dist: "distinct (map fst al)"
haftmann@34979
   665
  shows "ran (map_of al) = snd ` set al"
wenzelm@60839
   666
  using assms
wenzelm@60839
   667
proof (induct al)
wenzelm@60839
   668
  case Nil
wenzelm@60839
   669
  then show ?case by simp
haftmann@34979
   670
next
haftmann@34979
   671
  case (Cons kv al)
haftmann@34979
   672
  then have "ran (map_of al) = snd ` set al" by simp
haftmann@34979
   673
  moreover from Cons.prems have "map_of al (fst kv) = None"
haftmann@34979
   674
    by (simp add: map_of_eq_None_iff)
haftmann@34979
   675
  ultimately show ?case by (simp only: map_of.simps ran_map_upd) simp
haftmann@34979
   676
qed
haftmann@34979
   677
bulwahn@66584
   678
lemma ran_map_of_zip:
bulwahn@66584
   679
  assumes "length xs = length ys" "distinct xs"
bulwahn@66584
   680
  shows "ran (map_of (zip xs ys)) = set ys"
bulwahn@66584
   681
using assms by (simp add: ran_distinct set_map[symmetric])
bulwahn@66584
   682
Andreas@60057
   683
lemma ran_map_option: "ran (\<lambda>x. map_option f (m x)) = f ` ran m"
wenzelm@60839
   684
  by (auto simp add: ran_def)
wenzelm@60839
   685
nipkow@13910
   686
wenzelm@61799
   687
subsection \<open>\<open>map_le\<close>\<close>
nipkow@13910
   688
kleing@13912
   689
lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g"
wenzelm@60839
   690
  by (simp add: map_le_def)
nipkow@13910
   691
paulson@17724
   692
lemma upd_None_map_le [simp]: "f(x := None) \<subseteq>\<^sub>m f"
wenzelm@60839
   693
  by (force simp add: map_le_def)
nipkow@14187
   694
nipkow@13910
   695
lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)"
wenzelm@60839
   696
  by (fastforce simp add: map_le_def)
nipkow@13910
   697
paulson@17724
   698
lemma map_le_imp_upd_le [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)"
wenzelm@60839
   699
  by (force simp add: map_le_def)
nipkow@14187
   700
wenzelm@20800
   701
lemma map_le_upds [simp]:
wenzelm@60839
   702
  "f \<subseteq>\<^sub>m g \<Longrightarrow> f(as [\<mapsto>] bs) \<subseteq>\<^sub>m g(as [\<mapsto>] bs)"
nipkow@24331
   703
apply (induct as arbitrary: f g bs)
nipkow@24331
   704
 apply simp
nipkow@24331
   705
apply (case_tac bs)
nipkow@24331
   706
 apply auto
nipkow@24331
   707
done
webertj@13908
   708
webertj@14033
   709
lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)"
wenzelm@60839
   710
  by (fastforce simp add: map_le_def dom_def)
webertj@14033
   711
webertj@14033
   712
lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f"
wenzelm@60839
   713
  by (simp add: map_le_def)
webertj@14033
   714
nipkow@14187
   715
lemma map_le_trans[trans]: "\<lbrakk> m1 \<subseteq>\<^sub>m m2; m2 \<subseteq>\<^sub>m m3\<rbrakk> \<Longrightarrow> m1 \<subseteq>\<^sub>m m3"
wenzelm@60839
   716
  by (auto simp add: map_le_def dom_def)
webertj@14033
   717
webertj@14033
   718
lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g"
nipkow@24331
   719
unfolding map_le_def
nipkow@24331
   720
apply (rule ext)
nipkow@24331
   721
apply (case_tac "x \<in> dom f", simp)
nipkow@44890
   722
apply (case_tac "x \<in> dom g", simp, fastforce)
nipkow@24331
   723
done
webertj@14033
   724
wenzelm@60839
   725
lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m g ++ f"
wenzelm@60839
   726
  by (fastforce simp: map_le_def)
webertj@14033
   727
wenzelm@60839
   728
lemma map_le_iff_map_add_commute: "f \<subseteq>\<^sub>m f ++ g \<longleftrightarrow> f ++ g = g ++ f"
wenzelm@60839
   729
  by (fastforce simp: map_add_def map_le_def fun_eq_iff split: option.splits)
nipkow@15304
   730
wenzelm@60839
   731
lemma map_add_le_mapE: "f ++ g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h"
wenzelm@60839
   732
  by (fastforce simp: map_le_def map_add_def dom_def)
nipkow@15303
   733
wenzelm@60839
   734
lemma map_add_le_mapI: "\<lbrakk> f \<subseteq>\<^sub>m h; g \<subseteq>\<^sub>m h \<rbrakk> \<Longrightarrow> f ++ g \<subseteq>\<^sub>m h"
wenzelm@60839
   735
  by (auto simp: map_le_def map_add_def dom_def split: option.splits)
nipkow@15303
   736
nipkow@63828
   737
lemma map_add_subsumed1: "f \<subseteq>\<^sub>m g \<Longrightarrow> f++g = g"
nipkow@63828
   738
by (simp add: map_add_le_mapI map_le_antisym)
nipkow@63828
   739
nipkow@63828
   740
lemma map_add_subsumed2: "f \<subseteq>\<^sub>m g \<Longrightarrow> g++f = g"
nipkow@63828
   741
by (metis map_add_subsumed1 map_le_iff_map_add_commute)
nipkow@63828
   742
nipkow@31080
   743
lemma dom_eq_singleton_conv: "dom f = {x} \<longleftrightarrow> (\<exists>v. f = [x \<mapsto> v])"
wenzelm@63834
   744
  (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@63834
   745
proof
wenzelm@63834
   746
  assume ?rhs
wenzelm@63834
   747
  then show ?lhs by (auto split: if_split_asm)
nipkow@31080
   748
next
wenzelm@63834
   749
  assume ?lhs
wenzelm@63834
   750
  then obtain v where v: "f x = Some v" by auto
wenzelm@63834
   751
  show ?rhs
wenzelm@63834
   752
  proof
wenzelm@63834
   753
    show "f = [x \<mapsto> v]"
wenzelm@63834
   754
    proof (rule map_le_antisym)
wenzelm@63834
   755
      show "[x \<mapsto> v] \<subseteq>\<^sub>m f"
wenzelm@63834
   756
        using v by (auto simp add: map_le_def)
wenzelm@63834
   757
      show "f \<subseteq>\<^sub>m [x \<mapsto> v]"
wenzelm@63834
   758
        using \<open>dom f = {x}\<close> \<open>f x = Some v\<close> by (auto simp add: map_le_def)
wenzelm@63834
   759
    qed
wenzelm@63834
   760
  qed
nipkow@31080
   761
qed
nipkow@31080
   762
haftmann@35565
   763
wenzelm@60758
   764
subsection \<open>Various\<close>
haftmann@35565
   765
haftmann@35565
   766
lemma set_map_of_compr:
haftmann@35565
   767
  assumes distinct: "distinct (map fst xs)"
haftmann@35565
   768
  shows "set xs = {(k, v). map_of xs k = Some v}"
wenzelm@60839
   769
  using assms
wenzelm@60839
   770
proof (induct xs)
wenzelm@60839
   771
  case Nil
wenzelm@60839
   772
  then show ?case by simp
haftmann@35565
   773
next
haftmann@35565
   774
  case (Cons x xs)
haftmann@35565
   775
  obtain k v where "x = (k, v)" by (cases x) blast
haftmann@35565
   776
  with Cons.prems have "k \<notin> dom (map_of xs)"
haftmann@35565
   777
    by (simp add: dom_map_of_conv_image_fst)
haftmann@35565
   778
  then have *: "insert (k, v) {(k, v). map_of xs k = Some v} =
haftmann@35565
   779
    {(k', v'). (map_of xs(k \<mapsto> v)) k' = Some v'}"
haftmann@35565
   780
    by (auto split: if_splits)
haftmann@35565
   781
  from Cons have "set xs = {(k, v). map_of xs k = Some v}" by simp
wenzelm@60758
   782
  with * \<open>x = (k, v)\<close> show ?case by simp
haftmann@35565
   783
qed
haftmann@35565
   784
haftmann@67051
   785
lemma eq_key_imp_eq_value:
haftmann@67051
   786
  "v1 = v2"
haftmann@67051
   787
  if "distinct (map fst xs)" "(k, v1) \<in> set xs" "(k, v2) \<in> set xs"
haftmann@67051
   788
proof -
haftmann@67051
   789
  from that have "inj_on fst (set xs)"
haftmann@67051
   790
    by (simp add: distinct_map)
haftmann@67051
   791
  moreover have "fst (k, v1) = fst (k, v2)"
haftmann@67051
   792
    by simp
haftmann@67051
   793
  ultimately have "(k, v1) = (k, v2)"
haftmann@67051
   794
    by (rule inj_onD) (fact that)+
haftmann@67051
   795
  then show ?thesis
haftmann@67051
   796
    by simp
haftmann@67051
   797
qed
haftmann@67051
   798
haftmann@35565
   799
lemma map_of_inject_set:
haftmann@35565
   800
  assumes distinct: "distinct (map fst xs)" "distinct (map fst ys)"
haftmann@35565
   801
  shows "map_of xs = map_of ys \<longleftrightarrow> set xs = set ys" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@35565
   802
proof
haftmann@35565
   803
  assume ?lhs
wenzelm@60758
   804
  moreover from \<open>distinct (map fst xs)\<close> have "set xs = {(k, v). map_of xs k = Some v}"
haftmann@35565
   805
    by (rule set_map_of_compr)
wenzelm@60758
   806
  moreover from \<open>distinct (map fst ys)\<close> have "set ys = {(k, v). map_of ys k = Some v}"
haftmann@35565
   807
    by (rule set_map_of_compr)
haftmann@35565
   808
  ultimately show ?rhs by simp
haftmann@35565
   809
next
wenzelm@53374
   810
  assume ?rhs show ?lhs
wenzelm@53374
   811
  proof
haftmann@35565
   812
    fix k
wenzelm@60839
   813
    show "map_of xs k = map_of ys k"
wenzelm@60839
   814
    proof (cases "map_of xs k")
haftmann@35565
   815
      case None
wenzelm@60758
   816
      with \<open>?rhs\<close> have "map_of ys k = None"
haftmann@35565
   817
        by (simp add: map_of_eq_None_iff)
wenzelm@53374
   818
      with None show ?thesis by simp
haftmann@35565
   819
    next
haftmann@35565
   820
      case (Some v)
wenzelm@60758
   821
      with distinct \<open>?rhs\<close> have "map_of ys k = Some v"
haftmann@35565
   822
        by simp
wenzelm@53374
   823
      with Some show ?thesis by simp
haftmann@35565
   824
    qed
haftmann@35565
   825
  qed
haftmann@35565
   826
qed
haftmann@35565
   827
nipkow@3981
   828
end