src/HOL/Map.thy
author oheimb
Mon Apr 12 19:54:09 2004 +0200 (2004-04-12)
changeset 14537 e95ba267e3d5
parent 14376 9fe787a90a48
child 14739 86c6f272ef79
permissions -rw-r--r--
added theorem chg_map_other
nipkow@3981
     1
(*  Title:      HOL/Map.thy
nipkow@3981
     2
    ID:         $Id$
nipkow@3981
     3
    Author:     Tobias Nipkow, based on a theory by David von Oheimb
webertj@13908
     4
    Copyright   1997-2003 TU Muenchen
nipkow@3981
     5
nipkow@3981
     6
The datatype of `maps' (written ~=>); strongly resembles maps in VDM.
nipkow@3981
     7
*)
nipkow@3981
     8
nipkow@13914
     9
header {* Maps *}
nipkow@13914
    10
webertj@13908
    11
theory Map = List:
nipkow@3981
    12
webertj@13908
    13
types ('a,'b) "~=>" = "'a => 'b option" (infixr 0)
oheimb@14100
    14
translations (type) "a ~=> b " <= (type) "a => b option"
nipkow@3981
    15
nipkow@3981
    16
consts
oheimb@5300
    17
chg_map	:: "('b => 'b) => 'a => ('a ~=> 'b) => ('a ~=> 'b)"
oheimb@14100
    18
map_add :: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)" (infixl "++" 100)
oheimb@14100
    19
map_image::"('b => 'c)  => ('a ~=> 'b) => ('a ~=> 'c)" (infixr "`>" 90)
oheimb@14100
    20
restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)" ("_|'__" [90, 91] 90)
oheimb@5300
    21
dom	:: "('a ~=> 'b) => 'a set"
oheimb@5300
    22
ran	:: "('a ~=> 'b) => 'b set"
oheimb@5300
    23
map_of	:: "('a * 'b)list => 'a ~=> 'b"
oheimb@5300
    24
map_upds:: "('a ~=> 'b) => 'a list => 'b list => 
nipkow@14180
    25
	    ('a ~=> 'b)"
oheimb@14100
    26
map_upd_s::"('a ~=> 'b) => 'a set => 'b => 
oheimb@14100
    27
	    ('a ~=> 'b)"			 ("_/'(_{|->}_/')" [900,0,0]900)
oheimb@14100
    28
map_subst::"('a ~=> 'b) => 'b => 'b => 
oheimb@14100
    29
	    ('a ~=> 'b)"			 ("_/'(_~>_/')"    [900,0,0]900)
nipkow@13910
    30
map_le  :: "('a ~=> 'b) => ('a ~=> 'b) => bool" (infix "\<subseteq>\<^sub>m" 50)
nipkow@13910
    31
nipkow@14180
    32
nonterminals
nipkow@14180
    33
  maplets maplet
nipkow@14180
    34
oheimb@5300
    35
syntax
nipkow@14180
    36
  empty	    ::  "'a ~=> 'b"
nipkow@14180
    37
  "_maplet"  :: "['a, 'a] => maplet"             ("_ /|->/ _")
nipkow@14180
    38
  "_maplets" :: "['a, 'a] => maplet"             ("_ /[|->]/ _")
nipkow@14180
    39
  ""         :: "maplet => maplets"             ("_")
nipkow@14180
    40
  "_Maplets" :: "[maplet, maplets] => maplets" ("_,/ _")
nipkow@14180
    41
  "_MapUpd"  :: "['a ~=> 'b, maplets] => 'a ~=> 'b" ("_/'(_')" [900,0]900)
nipkow@14180
    42
  "_Map"     :: "maplets => 'a ~=> 'b"            ("(1[_])")
nipkow@3981
    43
wenzelm@12114
    44
syntax (xsymbols)
nipkow@14180
    45
  "_maplet"  :: "['a, 'a] => maplet"             ("_ /\<mapsto>/ _")
nipkow@14180
    46
  "_maplets" :: "['a, 'a] => maplet"             ("_ /[\<mapsto>]/ _")
nipkow@14180
    47
nipkow@14134
    48
  "~=>"     :: "[type, type] => type"    (infixr "\<rightharpoonup>" 0)
oheimb@14100
    49
  restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)" ("_\<lfloor>_" [90, 91] 90)
oheimb@14100
    50
  map_upd_s  :: "('a ~=> 'b) => 'a set => 'b => ('a ~=> 'b)"
oheimb@14100
    51
				    		 ("_/'(_/{\<mapsto>}/_')" [900,0,0]900)
oheimb@14100
    52
  map_subst :: "('a ~=> 'b) => 'b => 'b => 
oheimb@14100
    53
	        ('a ~=> 'b)"			 ("_/'(_\<leadsto>_/')"    [900,0,0]900)
oheimb@14100
    54
 "@chg_map" :: "('a ~=> 'b) => 'a => ('b => 'b) => ('a ~=> 'b)"
oheimb@14100
    55
					  ("_/'(_/\<mapsto>\<lambda>_. _')"  [900,0,0,0] 900)
oheimb@5300
    56
oheimb@5300
    57
translations
nipkow@13890
    58
  "empty"    => "_K None"
nipkow@13890
    59
  "empty"    <= "%x. None"
oheimb@5300
    60
oheimb@14100
    61
  "m(x\<mapsto>\<lambda>y. f)" == "chg_map (\<lambda>y. f) x m"
nipkow@3981
    62
nipkow@14180
    63
  "_MapUpd m (_Maplets xy ms)"  == "_MapUpd (_MapUpd m xy) ms"
nipkow@14180
    64
  "_MapUpd m (_maplet  x y)"    == "m(x:=Some y)"
nipkow@14180
    65
  "_MapUpd m (_maplets x y)"    == "map_upds m x y"
nipkow@14180
    66
  "_Map ms"                     == "_MapUpd empty ms"
nipkow@14180
    67
  "_Map (_Maplets ms1 ms2)"     <= "_MapUpd (_Map ms1) ms2"
nipkow@14180
    68
  "_Maplets ms1 (_Maplets ms2 ms3)" <= "_Maplets (_Maplets ms1 ms2) ms3"
nipkow@14180
    69
nipkow@3981
    70
defs
webertj@13908
    71
chg_map_def:  "chg_map f a m == case m a of None => m | Some b => m(a|->f b)"
nipkow@3981
    72
oheimb@14100
    73
map_add_def:   "m1++m2 == %x. case m2 x of None => m1 x | Some y => Some y"
oheimb@14100
    74
map_image_def: "f`>m == option_map f o m"
oheimb@14100
    75
restrict_map_def: "m|_A == %x. if x : A then m x else None"
nipkow@14025
    76
nipkow@14025
    77
map_upds_def: "m(xs [|->] ys) == m ++ map_of (rev(zip xs ys))"
oheimb@14100
    78
map_upd_s_def: "m(as{|->}b) == %x. if x : as then Some b else m x"
oheimb@14100
    79
map_subst_def: "m(a~>b)     == %x. if m x = Some a then Some b else m x"
nipkow@3981
    80
webertj@13908
    81
dom_def: "dom(m) == {a. m a ~= None}"
nipkow@14025
    82
ran_def: "ran(m) == {b. EX a. m a = Some b}"
nipkow@3981
    83
nipkow@14376
    84
map_le_def: "m\<^isub>1 \<subseteq>\<^sub>m m\<^isub>2  ==  ALL a : dom m\<^isub>1. m\<^isub>1 a = m\<^isub>2 a"
nipkow@13910
    85
berghofe@5183
    86
primrec
berghofe@5183
    87
  "map_of [] = empty"
oheimb@5300
    88
  "map_of (p#ps) = (map_of ps)(fst p |-> snd p)"
oheimb@5300
    89
webertj@13908
    90
oheimb@14100
    91
subsection {* @{term empty} *}
webertj@13908
    92
nipkow@13910
    93
lemma empty_upd_none[simp]: "empty(x := None) = empty"
webertj@13908
    94
apply (rule ext)
webertj@13908
    95
apply (simp (no_asm))
webertj@13908
    96
done
nipkow@13910
    97
webertj@13908
    98
webertj@13908
    99
(* FIXME: what is this sum_case nonsense?? *)
nipkow@13910
   100
lemma sum_case_empty_empty[simp]: "sum_case empty empty = empty"
webertj@13908
   101
apply (rule ext)
webertj@13908
   102
apply (simp (no_asm) split add: sum.split)
webertj@13908
   103
done
webertj@13908
   104
oheimb@14100
   105
subsection {* @{term map_upd} *}
webertj@13908
   106
webertj@13908
   107
lemma map_upd_triv: "t k = Some x ==> t(k|->x) = t"
webertj@13908
   108
apply (rule ext)
webertj@13908
   109
apply (simp (no_asm_simp))
webertj@13908
   110
done
webertj@13908
   111
nipkow@13910
   112
lemma map_upd_nonempty[simp]: "t(k|->x) ~= empty"
webertj@13908
   113
apply safe
paulson@14208
   114
apply (drule_tac x = k in fun_cong)
webertj@13908
   115
apply (simp (no_asm_use))
webertj@13908
   116
done
webertj@13908
   117
oheimb@14100
   118
lemma map_upd_eqD1: "m(a\<mapsto>x) = n(a\<mapsto>y) \<Longrightarrow> x = y"
oheimb@14100
   119
by (drule fun_cong [of _ _ a], auto)
oheimb@14100
   120
oheimb@14100
   121
lemma map_upd_Some_unfold: 
oheimb@14100
   122
  "((m(a|->b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)"
oheimb@14100
   123
by auto
oheimb@14100
   124
webertj@13908
   125
lemma finite_range_updI: "finite (range f) ==> finite (range (f(a|->b)))"
webertj@13908
   126
apply (unfold image_def)
webertj@13908
   127
apply (simp (no_asm_use) add: full_SetCompr_eq)
webertj@13908
   128
apply (rule finite_subset)
paulson@14208
   129
prefer 2 apply assumption
webertj@13908
   130
apply auto
webertj@13908
   131
done
webertj@13908
   132
webertj@13908
   133
webertj@13908
   134
(* FIXME: what is this sum_case nonsense?? *)
oheimb@14100
   135
subsection {* @{term sum_case} and @{term empty}/@{term map_upd} *}
webertj@13908
   136
nipkow@13910
   137
lemma sum_case_map_upd_empty[simp]:
nipkow@13910
   138
 "sum_case (m(k|->y)) empty =  (sum_case m empty)(Inl k|->y)"
webertj@13908
   139
apply (rule ext)
webertj@13908
   140
apply (simp (no_asm) split add: sum.split)
webertj@13908
   141
done
webertj@13908
   142
nipkow@13910
   143
lemma sum_case_empty_map_upd[simp]:
nipkow@13910
   144
 "sum_case empty (m(k|->y)) =  (sum_case empty m)(Inr k|->y)"
webertj@13908
   145
apply (rule ext)
webertj@13908
   146
apply (simp (no_asm) split add: sum.split)
webertj@13908
   147
done
webertj@13908
   148
nipkow@13910
   149
lemma sum_case_map_upd_map_upd[simp]:
nipkow@13910
   150
 "sum_case (m1(k1|->y1)) (m2(k2|->y2)) = (sum_case (m1(k1|->y1)) m2)(Inr k2|->y2)"
webertj@13908
   151
apply (rule ext)
webertj@13908
   152
apply (simp (no_asm) split add: sum.split)
webertj@13908
   153
done
webertj@13908
   154
webertj@13908
   155
oheimb@14100
   156
subsection {* @{term chg_map} *}
webertj@13908
   157
nipkow@13910
   158
lemma chg_map_new[simp]: "m a = None   ==> chg_map f a m = m"
paulson@14208
   159
by (unfold chg_map_def, auto)
webertj@13908
   160
nipkow@13910
   161
lemma chg_map_upd[simp]: "m a = Some b ==> chg_map f a m = m(a|->f b)"
paulson@14208
   162
by (unfold chg_map_def, auto)
webertj@13908
   163
oheimb@14537
   164
lemma chg_map_other [simp]: "a \<noteq> b \<Longrightarrow> chg_map f a m b = m b"
oheimb@14537
   165
by (auto simp: chg_map_def split add: option.split)
oheimb@14537
   166
webertj@13908
   167
oheimb@14100
   168
subsection {* @{term map_of} *}
webertj@13908
   169
webertj@13908
   170
lemma map_of_SomeD [rule_format (no_asm)]: "map_of xs k = Some y --> (k,y):set xs"
paulson@14208
   171
by (induct_tac "xs", auto)
webertj@13908
   172
webertj@13908
   173
lemma map_of_mapk_SomeI [rule_format (no_asm)]: "inj f ==> map_of t k = Some x -->  
webertj@13908
   174
   map_of (map (split (%k. Pair (f k))) t) (f k) = Some x"
webertj@13908
   175
apply (induct_tac "t")
webertj@13908
   176
apply  (auto simp add: inj_eq)
webertj@13908
   177
done
webertj@13908
   178
webertj@13908
   179
lemma weak_map_of_SomeI [rule_format (no_asm)]: "(k, x) : set l --> (? x. map_of l k = Some x)"
paulson@14208
   180
by (induct_tac "l", auto)
webertj@13908
   181
webertj@13908
   182
lemma map_of_filter_in: 
webertj@13908
   183
"[| map_of xs k = Some z; P k z |] ==> map_of (filter (split P) xs) k = Some z"
webertj@13908
   184
apply (rule mp)
paulson@14208
   185
prefer 2 apply assumption
webertj@13908
   186
apply (erule thin_rl)
paulson@14208
   187
apply (induct_tac "xs", auto)
webertj@13908
   188
done
webertj@13908
   189
webertj@13908
   190
lemma finite_range_map_of: "finite (range (map_of l))"
webertj@13908
   191
apply (induct_tac "l")
webertj@13908
   192
apply  (simp_all (no_asm) add: image_constant)
webertj@13908
   193
apply (rule finite_subset)
paulson@14208
   194
prefer 2 apply assumption
webertj@13908
   195
apply auto
webertj@13908
   196
done
webertj@13908
   197
webertj@13908
   198
lemma map_of_map: "map_of (map (%(a,b). (a,f b)) xs) x = option_map f (map_of xs x)"
paulson@14208
   199
by (induct_tac "xs", auto)
webertj@13908
   200
webertj@13908
   201
oheimb@14100
   202
subsection {* @{term option_map} related *}
webertj@13908
   203
nipkow@13910
   204
lemma option_map_o_empty[simp]: "option_map f o empty = empty"
webertj@13908
   205
apply (rule ext)
webertj@13908
   206
apply (simp (no_asm))
webertj@13908
   207
done
webertj@13908
   208
nipkow@13910
   209
lemma option_map_o_map_upd[simp]:
nipkow@13910
   210
 "option_map f o m(a|->b) = (option_map f o m)(a|->f b)"
webertj@13908
   211
apply (rule ext)
webertj@13908
   212
apply (simp (no_asm))
webertj@13908
   213
done
webertj@13908
   214
webertj@13908
   215
oheimb@14100
   216
subsection {* @{text "++"} *}
webertj@13908
   217
nipkow@14025
   218
lemma map_add_empty[simp]: "m ++ empty = m"
nipkow@14025
   219
apply (unfold map_add_def)
webertj@13908
   220
apply (simp (no_asm))
webertj@13908
   221
done
webertj@13908
   222
nipkow@14025
   223
lemma empty_map_add[simp]: "empty ++ m = m"
nipkow@14025
   224
apply (unfold map_add_def)
webertj@13908
   225
apply (rule ext)
webertj@13908
   226
apply (simp split add: option.split)
webertj@13908
   227
done
webertj@13908
   228
nipkow@14025
   229
lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3"
nipkow@14025
   230
apply(rule ext)
nipkow@14025
   231
apply(simp add: map_add_def split:option.split)
nipkow@14025
   232
done
nipkow@14025
   233
nipkow@14025
   234
lemma map_add_Some_iff: 
webertj@13908
   235
 "((m ++ n) k = Some x) = (n k = Some x | n k = None & m k = Some x)"
nipkow@14025
   236
apply (unfold map_add_def)
webertj@13908
   237
apply (simp (no_asm) split add: option.split)
webertj@13908
   238
done
webertj@13908
   239
nipkow@14025
   240
lemmas map_add_SomeD = map_add_Some_iff [THEN iffD1, standard]
nipkow@14025
   241
declare map_add_SomeD [dest!]
webertj@13908
   242
nipkow@14025
   243
lemma map_add_find_right[simp]: "!!xx. n k = Some xx ==> (m ++ n) k = Some xx"
paulson@14208
   244
by (subst map_add_Some_iff, fast)
webertj@13908
   245
nipkow@14025
   246
lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)"
nipkow@14025
   247
apply (unfold map_add_def)
webertj@13908
   248
apply (simp (no_asm) split add: option.split)
webertj@13908
   249
done
webertj@13908
   250
nipkow@14025
   251
lemma map_add_upd[simp]: "f ++ g(x|->y) = (f ++ g)(x|->y)"
nipkow@14025
   252
apply (unfold map_add_def)
paulson@14208
   253
apply (rule ext, auto)
webertj@13908
   254
done
webertj@13908
   255
nipkow@14186
   256
lemma map_add_upds[simp]: "m1 ++ (m2(xs[\<mapsto>]ys)) = (m1++m2)(xs[\<mapsto>]ys)"
nipkow@14186
   257
by(simp add:map_upds_def)
nipkow@14186
   258
nipkow@14025
   259
lemma map_of_append[simp]: "map_of (xs@ys) = map_of ys ++ map_of xs"
nipkow@14025
   260
apply (unfold map_add_def)
webertj@13908
   261
apply (induct_tac "xs")
webertj@13908
   262
apply (simp (no_asm))
webertj@13908
   263
apply (rule ext)
webertj@13908
   264
apply (simp (no_asm_simp) split add: option.split)
webertj@13908
   265
done
webertj@13908
   266
webertj@13908
   267
declare fun_upd_apply [simp del]
nipkow@14025
   268
lemma finite_range_map_of_map_add:
nipkow@14025
   269
 "finite (range f) ==> finite (range (f ++ map_of l))"
paulson@14208
   270
apply (induct_tac "l", auto)
webertj@13908
   271
apply (erule finite_range_updI)
webertj@13908
   272
done
webertj@13908
   273
declare fun_upd_apply [simp]
webertj@13908
   274
oheimb@14100
   275
subsection {* @{term map_image} *}
webertj@13908
   276
oheimb@14100
   277
lemma map_image_empty [simp]: "f`>empty = empty" 
oheimb@14100
   278
by (auto simp: map_image_def empty_def)
oheimb@14100
   279
oheimb@14100
   280
lemma map_image_upd [simp]: "f`>m(a|->b) = (f`>m)(a|->f b)" 
oheimb@14100
   281
apply (auto simp: map_image_def fun_upd_def)
oheimb@14100
   282
by (rule ext, auto)
oheimb@14100
   283
oheimb@14100
   284
subsection {* @{term restrict_map} *}
oheimb@14100
   285
nipkow@14186
   286
lemma restrict_map_to_empty[simp]: "m\<lfloor>{} = empty"
nipkow@14186
   287
by(simp add: restrict_map_def)
nipkow@14186
   288
nipkow@14186
   289
lemma restrict_map_empty[simp]: "empty\<lfloor>D = empty"
nipkow@14186
   290
by(simp add: restrict_map_def)
nipkow@14186
   291
oheimb@14100
   292
lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m\<lfloor>A) x = m x"
oheimb@14100
   293
by (auto simp: restrict_map_def)
oheimb@14100
   294
oheimb@14100
   295
lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m\<lfloor>A) x = None"
oheimb@14100
   296
by (auto simp: restrict_map_def)
oheimb@14100
   297
oheimb@14100
   298
lemma ran_restrictD: "y \<in> ran (m\<lfloor>A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y"
oheimb@14100
   299
by (auto simp: restrict_map_def ran_def split: split_if_asm)
oheimb@14100
   300
nipkow@14186
   301
lemma dom_restrict [simp]: "dom (m\<lfloor>A) = dom m \<inter> A"
oheimb@14100
   302
by (auto simp: restrict_map_def dom_def split: split_if_asm)
oheimb@14100
   303
oheimb@14100
   304
lemma restrict_upd_same [simp]: "m(x\<mapsto>y)\<lfloor>(-{x}) = m\<lfloor>(-{x})"
oheimb@14100
   305
by (rule ext, auto simp: restrict_map_def)
oheimb@14100
   306
oheimb@14100
   307
lemma restrict_restrict [simp]: "m\<lfloor>A\<lfloor>B = m\<lfloor>(A\<inter>B)"
oheimb@14100
   308
by (rule ext, auto simp: restrict_map_def)
oheimb@14100
   309
nipkow@14186
   310
lemma restrict_fun_upd[simp]:
nipkow@14186
   311
 "m(x := y)\<lfloor>D = (if x \<in> D then (m\<lfloor>(D-{x}))(x := y) else m\<lfloor>D)"
nipkow@14186
   312
by(simp add: restrict_map_def expand_fun_eq)
nipkow@14186
   313
nipkow@14186
   314
lemma fun_upd_None_restrict[simp]:
nipkow@14186
   315
  "(m\<lfloor>D)(x := None) = (if x:D then m\<lfloor>(D - {x}) else m\<lfloor>D)"
nipkow@14186
   316
by(simp add: restrict_map_def expand_fun_eq)
nipkow@14186
   317
nipkow@14186
   318
lemma fun_upd_restrict:
nipkow@14186
   319
 "(m\<lfloor>D)(x := y) = (m\<lfloor>(D-{x}))(x := y)"
nipkow@14186
   320
by(simp add: restrict_map_def expand_fun_eq)
nipkow@14186
   321
nipkow@14186
   322
lemma fun_upd_restrict_conv[simp]:
nipkow@14186
   323
 "x \<in> D \<Longrightarrow> (m\<lfloor>D)(x := y) = (m\<lfloor>(D-{x}))(x := y)"
nipkow@14186
   324
by(simp add: restrict_map_def expand_fun_eq)
nipkow@14186
   325
oheimb@14100
   326
oheimb@14100
   327
subsection {* @{term map_upds} *}
nipkow@14025
   328
nipkow@14025
   329
lemma map_upds_Nil1[simp]: "m([] [|->] bs) = m"
nipkow@14025
   330
by(simp add:map_upds_def)
nipkow@14025
   331
nipkow@14025
   332
lemma map_upds_Nil2[simp]: "m(as [|->] []) = m"
nipkow@14025
   333
by(simp add:map_upds_def)
nipkow@14025
   334
nipkow@14025
   335
lemma map_upds_Cons[simp]: "m(a#as [|->] b#bs) = (m(a|->b))(as[|->]bs)"
nipkow@14025
   336
by(simp add:map_upds_def)
nipkow@14025
   337
nipkow@14187
   338
lemma map_upds_append1[simp]: "\<And>ys m. size xs < size ys \<Longrightarrow>
nipkow@14187
   339
  m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)"
nipkow@14187
   340
apply(induct xs)
nipkow@14187
   341
 apply(clarsimp simp add:neq_Nil_conv)
paulson@14208
   342
apply (case_tac ys, simp, simp)
nipkow@14187
   343
done
nipkow@14187
   344
nipkow@14187
   345
lemma map_upds_list_update2_drop[simp]:
nipkow@14187
   346
 "\<And>m ys i. \<lbrakk>size xs \<le> i; i < size ys\<rbrakk>
nipkow@14187
   347
     \<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)"
paulson@14208
   348
apply (induct xs, simp)
paulson@14208
   349
apply (case_tac ys, simp)
nipkow@14187
   350
apply(simp split:nat.split)
nipkow@14187
   351
done
nipkow@14025
   352
nipkow@14025
   353
lemma map_upd_upds_conv_if: "!!x y ys f.
nipkow@14025
   354
 (f(x|->y))(xs [|->] ys) =
nipkow@14025
   355
 (if x : set(take (length ys) xs) then f(xs [|->] ys)
nipkow@14025
   356
                                  else (f(xs [|->] ys))(x|->y))"
paulson@14208
   357
apply (induct xs, simp)
nipkow@14025
   358
apply(case_tac ys)
nipkow@14025
   359
 apply(auto split:split_if simp:fun_upd_twist)
nipkow@14025
   360
done
nipkow@14025
   361
nipkow@14025
   362
lemma map_upds_twist [simp]:
nipkow@14025
   363
 "a ~: set as ==> m(a|->b)(as[|->]bs) = m(as[|->]bs)(a|->b)"
nipkow@14025
   364
apply(insert set_take_subset)
nipkow@14025
   365
apply (fastsimp simp add: map_upd_upds_conv_if)
nipkow@14025
   366
done
nipkow@14025
   367
nipkow@14025
   368
lemma map_upds_apply_nontin[simp]:
nipkow@14025
   369
 "!!ys. x ~: set xs ==> (f(xs[|->]ys)) x = f x"
paulson@14208
   370
apply (induct xs, simp)
nipkow@14025
   371
apply(case_tac ys)
nipkow@14025
   372
 apply(auto simp: map_upd_upds_conv_if)
nipkow@14025
   373
done
nipkow@14025
   374
nipkow@14300
   375
lemma fun_upds_append_drop[simp]:
nipkow@14300
   376
  "!!m ys. size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)"
nipkow@14300
   377
apply(induct xs)
nipkow@14300
   378
 apply (simp)
nipkow@14300
   379
apply(case_tac ys)
nipkow@14300
   380
apply simp_all
nipkow@14300
   381
done
nipkow@14300
   382
nipkow@14300
   383
lemma fun_upds_append2_drop[simp]:
nipkow@14300
   384
  "!!m ys. size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)"
nipkow@14300
   385
apply(induct xs)
nipkow@14300
   386
 apply (simp)
nipkow@14300
   387
apply(case_tac ys)
nipkow@14300
   388
apply simp_all
nipkow@14300
   389
done
nipkow@14300
   390
nipkow@14300
   391
nipkow@14186
   392
lemma restrict_map_upds[simp]: "!!m ys.
nipkow@14186
   393
 \<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk>
nipkow@14186
   394
 \<Longrightarrow> m(xs [\<mapsto>] ys)\<lfloor>D = (m\<lfloor>(D - set xs))(xs [\<mapsto>] ys)"
paulson@14208
   395
apply (induct xs, simp)
paulson@14208
   396
apply (case_tac ys, simp)
nipkow@14186
   397
apply(simp add:Diff_insert[symmetric] insert_absorb)
nipkow@14186
   398
apply(simp add: map_upd_upds_conv_if)
nipkow@14186
   399
done
nipkow@14186
   400
nipkow@14186
   401
oheimb@14100
   402
subsection {* @{term map_upd_s} *}
oheimb@14100
   403
oheimb@14100
   404
lemma map_upd_s_apply [simp]: 
oheimb@14100
   405
  "(m(as{|->}b)) x = (if x : as then Some b else m x)"
oheimb@14100
   406
by (simp add: map_upd_s_def)
oheimb@14100
   407
oheimb@14100
   408
lemma map_subst_apply [simp]: 
oheimb@14100
   409
  "(m(a~>b)) x = (if m x = Some a then Some b else m x)" 
oheimb@14100
   410
by (simp add: map_subst_def)
oheimb@14100
   411
oheimb@14100
   412
subsection {* @{term dom} *}
webertj@13908
   413
webertj@13908
   414
lemma domI: "m a = Some b ==> a : dom m"
paulson@14208
   415
by (unfold dom_def, auto)
oheimb@14100
   416
(* declare domI [intro]? *)
webertj@13908
   417
webertj@13908
   418
lemma domD: "a : dom m ==> ? b. m a = Some b"
paulson@14208
   419
by (unfold dom_def, auto)
webertj@13908
   420
nipkow@13910
   421
lemma domIff[iff]: "(a : dom m) = (m a ~= None)"
paulson@14208
   422
by (unfold dom_def, auto)
webertj@13908
   423
declare domIff [simp del]
webertj@13908
   424
nipkow@13910
   425
lemma dom_empty[simp]: "dom empty = {}"
webertj@13908
   426
apply (unfold dom_def)
webertj@13908
   427
apply (simp (no_asm))
webertj@13908
   428
done
webertj@13908
   429
nipkow@13910
   430
lemma dom_fun_upd[simp]:
nipkow@13910
   431
 "dom(f(x := y)) = (if y=None then dom f - {x} else insert x (dom f))"
nipkow@13910
   432
by (simp add:dom_def) blast
webertj@13908
   433
nipkow@13937
   434
lemma dom_map_of: "dom(map_of xys) = {x. \<exists>y. (x,y) : set xys}"
nipkow@13937
   435
apply(induct xys)
nipkow@13937
   436
apply(auto simp del:fun_upd_apply)
nipkow@13937
   437
done
nipkow@13937
   438
webertj@13908
   439
lemma finite_dom_map_of: "finite (dom (map_of l))"
webertj@13908
   440
apply (unfold dom_def)
webertj@13908
   441
apply (induct_tac "l")
webertj@13908
   442
apply (auto simp add: insert_Collect [symmetric])
webertj@13908
   443
done
webertj@13908
   444
nipkow@14025
   445
lemma dom_map_upds[simp]:
nipkow@14025
   446
 "!!m ys. dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m"
paulson@14208
   447
apply (induct xs, simp)
paulson@14208
   448
apply (case_tac ys, auto)
nipkow@14025
   449
done
nipkow@13910
   450
nipkow@14025
   451
lemma dom_map_add[simp]: "dom(m++n) = dom n Un dom m"
paulson@14208
   452
by (unfold dom_def, auto)
nipkow@13910
   453
nipkow@13910
   454
lemma dom_overwrite[simp]:
nipkow@13910
   455
 "dom(f(g|A)) = (dom f  - {a. a : A - dom g}) Un {a. a : A Int dom g}"
nipkow@13910
   456
by(auto simp add: dom_def overwrite_def)
webertj@13908
   457
nipkow@14027
   458
lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1"
nipkow@14027
   459
apply(rule ext)
nipkow@14027
   460
apply(fastsimp simp:map_add_def split:option.split)
nipkow@14027
   461
done
nipkow@14027
   462
oheimb@14100
   463
subsection {* @{term ran} *}
oheimb@14100
   464
oheimb@14100
   465
lemma ranI: "m a = Some b ==> b : ran m" 
oheimb@14100
   466
by (auto simp add: ran_def)
oheimb@14100
   467
(* declare ranI [intro]? *)
webertj@13908
   468
nipkow@13910
   469
lemma ran_empty[simp]: "ran empty = {}"
webertj@13908
   470
apply (unfold ran_def)
webertj@13908
   471
apply (simp (no_asm))
webertj@13908
   472
done
webertj@13908
   473
nipkow@13910
   474
lemma ran_map_upd[simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)"
paulson@14208
   475
apply (unfold ran_def, auto)
webertj@13908
   476
apply (subgoal_tac "~ (aa = a) ")
webertj@13908
   477
apply auto
webertj@13908
   478
done
nipkow@13910
   479
oheimb@14100
   480
subsection {* @{text "map_le"} *}
nipkow@13910
   481
kleing@13912
   482
lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g"
nipkow@13910
   483
by(simp add:map_le_def)
nipkow@13910
   484
nipkow@14187
   485
lemma [simp]: "f(x := None) \<subseteq>\<^sub>m f"
nipkow@14187
   486
by(force simp add:map_le_def)
nipkow@14187
   487
nipkow@13910
   488
lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)"
nipkow@13910
   489
by(fastsimp simp add:map_le_def)
nipkow@13910
   490
nipkow@14187
   491
lemma [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)"
nipkow@14187
   492
by(force simp add:map_le_def)
nipkow@14187
   493
nipkow@13910
   494
lemma map_le_upds[simp]:
nipkow@13910
   495
 "!!f g bs. f \<subseteq>\<^sub>m g ==> f(as [|->] bs) \<subseteq>\<^sub>m g(as [|->] bs)"
paulson@14208
   496
apply (induct as, simp)
paulson@14208
   497
apply (case_tac bs, auto)
nipkow@14025
   498
done
webertj@13908
   499
webertj@14033
   500
lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)"
webertj@14033
   501
  by (fastsimp simp add: map_le_def dom_def)
webertj@14033
   502
webertj@14033
   503
lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f"
webertj@14033
   504
  by (simp add: map_le_def)
webertj@14033
   505
nipkow@14187
   506
lemma map_le_trans[trans]: "\<lbrakk> m1 \<subseteq>\<^sub>m m2; m2 \<subseteq>\<^sub>m m3\<rbrakk> \<Longrightarrow> m1 \<subseteq>\<^sub>m m3"
nipkow@14187
   507
by(force simp add:map_le_def)
webertj@14033
   508
webertj@14033
   509
lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g"
webertj@14033
   510
  apply (unfold map_le_def)
webertj@14033
   511
  apply (rule ext)
paulson@14208
   512
  apply (case_tac "x \<in> dom f", simp)
paulson@14208
   513
  apply (case_tac "x \<in> dom g", simp, fastsimp)
webertj@14033
   514
done
webertj@14033
   515
webertj@14033
   516
lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)"
webertj@14033
   517
  by (fastsimp simp add: map_le_def)
webertj@14033
   518
nipkow@3981
   519
end