src/HOL/Map.thy
author nipkow
Wed, 03 Sep 2003 18:20:57 +0200
changeset 14180 d2e550609c40
parent 14134 0fdf5708c7a8
child 14186 6d2a494e33be
permissions -rw-r--r--
Introduced new syntax for maplets x |-> y
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
3981
b4f93a8da835 Added the new theory Map.
nipkow
parents:
diff changeset
     1
(*  Title:      HOL/Map.thy
b4f93a8da835 Added the new theory Map.
nipkow
parents:
diff changeset
     2
    ID:         $Id$
b4f93a8da835 Added the new theory Map.
nipkow
parents:
diff changeset
     3
    Author:     Tobias Nipkow, based on a theory by David von Oheimb
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
     4
    Copyright   1997-2003 TU Muenchen
3981
b4f93a8da835 Added the new theory Map.
nipkow
parents:
diff changeset
     5
b4f93a8da835 Added the new theory Map.
nipkow
parents:
diff changeset
     6
The datatype of `maps' (written ~=>); strongly resembles maps in VDM.
b4f93a8da835 Added the new theory Map.
nipkow
parents:
diff changeset
     7
*)
b4f93a8da835 Added the new theory Map.
nipkow
parents:
diff changeset
     8
13914
nipkow
parents: 13912
diff changeset
     9
header {* Maps *}
nipkow
parents: 13912
diff changeset
    10
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
    11
theory Map = List:
3981
b4f93a8da835 Added the new theory Map.
nipkow
parents:
diff changeset
    12
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
    13
types ('a,'b) "~=>" = "'a => 'b option" (infixr 0)
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    14
translations (type) "a ~=> b " <= (type) "a => b option"
3981
b4f93a8da835 Added the new theory Map.
nipkow
parents:
diff changeset
    15
b4f93a8da835 Added the new theory Map.
nipkow
parents:
diff changeset
    16
consts
5300
2b1ca524ace8 defined map_upd by translation via fun_upd
oheimb
parents: 5195
diff changeset
    17
chg_map	:: "('b => 'b) => 'a => ('a ~=> 'b) => ('a ~=> 'b)"
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    18
map_add :: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)" (infixl "++" 100)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    19
map_image::"('b => 'c)  => ('a ~=> 'b) => ('a ~=> 'c)" (infixr "`>" 90)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    20
restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)" ("_|'__" [90, 91] 90)
5300
2b1ca524ace8 defined map_upd by translation via fun_upd
oheimb
parents: 5195
diff changeset
    21
dom	:: "('a ~=> 'b) => 'a set"
2b1ca524ace8 defined map_upd by translation via fun_upd
oheimb
parents: 5195
diff changeset
    22
ran	:: "('a ~=> 'b) => 'b set"
2b1ca524ace8 defined map_upd by translation via fun_upd
oheimb
parents: 5195
diff changeset
    23
map_of	:: "('a * 'b)list => 'a ~=> 'b"
2b1ca524ace8 defined map_upd by translation via fun_upd
oheimb
parents: 5195
diff changeset
    24
map_upds:: "('a ~=> 'b) => 'a list => 'b list => 
14180
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    25
	    ('a ~=> 'b)"
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    26
map_upd_s::"('a ~=> 'b) => 'a set => 'b => 
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    27
	    ('a ~=> 'b)"			 ("_/'(_{|->}_/')" [900,0,0]900)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    28
map_subst::"('a ~=> 'b) => 'b => 'b => 
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    29
	    ('a ~=> 'b)"			 ("_/'(_~>_/')"    [900,0,0]900)
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
    30
map_le  :: "('a ~=> 'b) => ('a ~=> 'b) => bool" (infix "\<subseteq>\<^sub>m" 50)
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
    31
14180
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    32
nonterminals
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    33
  maplets maplet
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    34
5300
2b1ca524ace8 defined map_upd by translation via fun_upd
oheimb
parents: 5195
diff changeset
    35
syntax
14180
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    36
  empty	    ::  "'a ~=> 'b"
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    37
  "_maplet"  :: "['a, 'a] => maplet"             ("_ /|->/ _")
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    38
  "_maplets" :: "['a, 'a] => maplet"             ("_ /[|->]/ _")
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    39
  ""         :: "maplet => maplets"             ("_")
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    40
  "_Maplets" :: "[maplet, maplets] => maplets" ("_,/ _")
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    41
  "_MapUpd"  :: "['a ~=> 'b, maplets] => 'a ~=> 'b" ("_/'(_')" [900,0]900)
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    42
  "_Map"     :: "maplets => 'a ~=> 'b"            ("(1[_])")
3981
b4f93a8da835 Added the new theory Map.
nipkow
parents:
diff changeset
    43
12114
a8e860c86252 eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents: 10137
diff changeset
    44
syntax (xsymbols)
14180
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    45
  "_maplet"  :: "['a, 'a] => maplet"             ("_ /\<mapsto>/ _")
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    46
  "_maplets" :: "['a, 'a] => maplet"             ("_ /[\<mapsto>]/ _")
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    47
14134
0fdf5708c7a8 Replaced \<leadsto> by \<rightharpoonup>
nipkow
parents: 14100
diff changeset
    48
  "~=>"     :: "[type, type] => type"    (infixr "\<rightharpoonup>" 0)
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    49
  restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)" ("_\<lfloor>_" [90, 91] 90)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    50
  map_upd_s  :: "('a ~=> 'b) => 'a set => 'b => ('a ~=> 'b)"
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    51
				    		 ("_/'(_/{\<mapsto>}/_')" [900,0,0]900)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    52
  map_subst :: "('a ~=> 'b) => 'b => 'b => 
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    53
	        ('a ~=> 'b)"			 ("_/'(_\<leadsto>_/')"    [900,0,0]900)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    54
 "@chg_map" :: "('a ~=> 'b) => 'a => ('b => 'b) => ('a ~=> 'b)"
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    55
					  ("_/'(_/\<mapsto>\<lambda>_. _')"  [900,0,0,0] 900)
5300
2b1ca524ace8 defined map_upd by translation via fun_upd
oheimb
parents: 5195
diff changeset
    56
2b1ca524ace8 defined map_upd by translation via fun_upd
oheimb
parents: 5195
diff changeset
    57
translations
13890
90611b4e0054 Made empty a translation rather than a constant.
nipkow
parents: 12919
diff changeset
    58
  "empty"    => "_K None"
90611b4e0054 Made empty a translation rather than a constant.
nipkow
parents: 12919
diff changeset
    59
  "empty"    <= "%x. None"
5300
2b1ca524ace8 defined map_upd by translation via fun_upd
oheimb
parents: 5195
diff changeset
    60
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    61
  "m(x\<mapsto>\<lambda>y. f)" == "chg_map (\<lambda>y. f) x m"
3981
b4f93a8da835 Added the new theory Map.
nipkow
parents:
diff changeset
    62
14180
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    63
  "_MapUpd m (_Maplets xy ms)"  == "_MapUpd (_MapUpd m xy) ms"
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    64
  "_MapUpd m (_maplet  x y)"    == "m(x:=Some y)"
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    65
  "_MapUpd m (_maplets x y)"    == "map_upds m x y"
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    66
  "_Map ms"                     == "_MapUpd empty ms"
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    67
  "_Map (_Maplets ms1 ms2)"     <= "_MapUpd (_Map ms1) ms2"
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    68
  "_Maplets ms1 (_Maplets ms2 ms3)" <= "_Maplets (_Maplets ms1 ms2) ms3"
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    69
3981
b4f93a8da835 Added the new theory Map.
nipkow
parents:
diff changeset
    70
defs
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
    71
chg_map_def:  "chg_map f a m == case m a of None => m | Some b => m(a|->f b)"
3981
b4f93a8da835 Added the new theory Map.
nipkow
parents:
diff changeset
    72
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    73
map_add_def:   "m1++m2 == %x. case m2 x of None => m1 x | Some y => Some y"
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    74
map_image_def: "f`>m == option_map f o m"
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    75
restrict_map_def: "m|_A == %x. if x : A then m x else None"
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
    76
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
    77
map_upds_def: "m(xs [|->] ys) == m ++ map_of (rev(zip xs ys))"
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    78
map_upd_s_def: "m(as{|->}b) == %x. if x : as then Some b else m x"
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    79
map_subst_def: "m(a~>b)     == %x. if m x = Some a then Some b else m x"
3981
b4f93a8da835 Added the new theory Map.
nipkow
parents:
diff changeset
    80
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
    81
dom_def: "dom(m) == {a. m a ~= None}"
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
    82
ran_def: "ran(m) == {b. EX a. m a = Some b}"
3981
b4f93a8da835 Added the new theory Map.
nipkow
parents:
diff changeset
    83
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
    84
map_le_def: "m1 \<subseteq>\<^sub>m m2  ==  ALL a : dom m1. m1 a = m2 a"
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
    85
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 3981
diff changeset
    86
primrec
89f162de39cf Adapted to new datatype package.
berghofe
parents: 3981
diff changeset
    87
  "map_of [] = empty"
5300
2b1ca524ace8 defined map_upd by translation via fun_upd
oheimb
parents: 5195
diff changeset
    88
  "map_of (p#ps) = (map_of ps)(fst p |-> snd p)"
2b1ca524ace8 defined map_upd by translation via fun_upd
oheimb
parents: 5195
diff changeset
    89
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
    90
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    91
subsection {* @{term empty} *}
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
    92
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
    93
lemma empty_upd_none[simp]: "empty(x := None) = empty"
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
    94
apply (rule ext)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
    95
apply (simp (no_asm))
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
    96
done
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
    97
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
    98
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
    99
(* FIXME: what is this sum_case nonsense?? *)
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   100
lemma sum_case_empty_empty[simp]: "sum_case empty empty = empty"
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   101
apply (rule ext)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   102
apply (simp (no_asm) split add: sum.split)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   103
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   104
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   105
subsection {* @{term map_upd} *}
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   106
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   107
lemma map_upd_triv: "t k = Some x ==> t(k|->x) = t"
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   108
apply (rule ext)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   109
apply (simp (no_asm_simp))
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   110
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   111
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   112
lemma map_upd_nonempty[simp]: "t(k|->x) ~= empty"
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   113
apply safe
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   114
apply (drule_tac x = "k" in fun_cong)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   115
apply (simp (no_asm_use))
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   116
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   117
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   118
lemma map_upd_eqD1: "m(a\<mapsto>x) = n(a\<mapsto>y) \<Longrightarrow> x = y"
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   119
by (drule fun_cong [of _ _ a], auto)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   120
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   121
lemma map_upd_Some_unfold: 
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   122
  "((m(a|->b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)"
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   123
by auto
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   124
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   125
lemma finite_range_updI: "finite (range f) ==> finite (range (f(a|->b)))"
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   126
apply (unfold image_def)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   127
apply (simp (no_asm_use) add: full_SetCompr_eq)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   128
apply (rule finite_subset)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   129
prefer 2 apply (assumption)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   130
apply auto
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   131
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   132
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   133
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   134
(* FIXME: what is this sum_case nonsense?? *)
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   135
subsection {* @{term sum_case} and @{term empty}/@{term map_upd} *}
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   136
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   137
lemma sum_case_map_upd_empty[simp]:
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   138
 "sum_case (m(k|->y)) empty =  (sum_case m empty)(Inl k|->y)"
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   139
apply (rule ext)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   140
apply (simp (no_asm) split add: sum.split)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   141
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   142
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   143
lemma sum_case_empty_map_upd[simp]:
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   144
 "sum_case empty (m(k|->y)) =  (sum_case empty m)(Inr k|->y)"
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   145
apply (rule ext)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   146
apply (simp (no_asm) split add: sum.split)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   147
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   148
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   149
lemma sum_case_map_upd_map_upd[simp]:
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   150
 "sum_case (m1(k1|->y1)) (m2(k2|->y2)) = (sum_case (m1(k1|->y1)) m2)(Inr k2|->y2)"
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   151
apply (rule ext)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   152
apply (simp (no_asm) split add: sum.split)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   153
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   154
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   155
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   156
subsection {* @{term chg_map} *}
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   157
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   158
lemma chg_map_new[simp]: "m a = None   ==> chg_map f a m = m"
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   159
apply (unfold chg_map_def)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   160
apply auto
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   161
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   162
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   163
lemma chg_map_upd[simp]: "m a = Some b ==> chg_map f a m = m(a|->f b)"
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   164
apply (unfold chg_map_def)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   165
apply auto
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   166
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   167
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   168
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   169
subsection {* @{term map_of} *}
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   170
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   171
lemma map_of_SomeD [rule_format (no_asm)]: "map_of xs k = Some y --> (k,y):set xs"
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   172
apply (induct_tac "xs")
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   173
apply  auto
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   174
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   175
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   176
lemma map_of_mapk_SomeI [rule_format (no_asm)]: "inj f ==> map_of t k = Some x -->  
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   177
   map_of (map (split (%k. Pair (f k))) t) (f k) = Some x"
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   178
apply (induct_tac "t")
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   179
apply  (auto simp add: inj_eq)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   180
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   181
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   182
lemma weak_map_of_SomeI [rule_format (no_asm)]: "(k, x) : set l --> (? x. map_of l k = Some x)"
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   183
apply (induct_tac "l")
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   184
apply  auto
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   185
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   186
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   187
lemma map_of_filter_in: 
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   188
"[| map_of xs k = Some z; P k z |] ==> map_of (filter (split P) xs) k = Some z"
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   189
apply (rule mp)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   190
prefer 2 apply (assumption)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   191
apply (erule thin_rl)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   192
apply (induct_tac "xs")
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   193
apply  auto
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   194
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   195
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   196
lemma finite_range_map_of: "finite (range (map_of l))"
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   197
apply (induct_tac "l")
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   198
apply  (simp_all (no_asm) add: image_constant)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   199
apply (rule finite_subset)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   200
prefer 2 apply (assumption)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   201
apply auto
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   202
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   203
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   204
lemma map_of_map: "map_of (map (%(a,b). (a,f b)) xs) x = option_map f (map_of xs x)"
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   205
apply (induct_tac "xs")
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   206
apply auto
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   207
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   208
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   209
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   210
subsection {* @{term option_map} related *}
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   211
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   212
lemma option_map_o_empty[simp]: "option_map f o empty = empty"
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   213
apply (rule ext)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   214
apply (simp (no_asm))
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   215
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   216
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   217
lemma option_map_o_map_upd[simp]:
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   218
 "option_map f o m(a|->b) = (option_map f o m)(a|->f b)"
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   219
apply (rule ext)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   220
apply (simp (no_asm))
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   221
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   222
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   223
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   224
subsection {* @{text "++"} *}
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   225
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   226
lemma map_add_empty[simp]: "m ++ empty = m"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   227
apply (unfold map_add_def)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   228
apply (simp (no_asm))
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   229
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   230
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   231
lemma empty_map_add[simp]: "empty ++ m = m"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   232
apply (unfold map_add_def)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   233
apply (rule ext)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   234
apply (simp split add: option.split)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   235
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   236
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   237
lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   238
apply(rule ext)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   239
apply(simp add: map_add_def split:option.split)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   240
done
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   241
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   242
lemma map_add_Some_iff: 
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   243
 "((m ++ n) k = Some x) = (n k = Some x | n k = None & m k = Some x)"
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   244
apply (unfold map_add_def)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   245
apply (simp (no_asm) split add: option.split)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   246
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   247
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   248
lemmas map_add_SomeD = map_add_Some_iff [THEN iffD1, standard]
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   249
declare map_add_SomeD [dest!]
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   250
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   251
lemma map_add_find_right[simp]: "!!xx. n k = Some xx ==> (m ++ n) k = Some xx"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   252
apply (subst map_add_Some_iff)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   253
apply fast
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   254
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   255
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   256
lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   257
apply (unfold map_add_def)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   258
apply (simp (no_asm) split add: option.split)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   259
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   260
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   261
lemma map_add_upd[simp]: "f ++ g(x|->y) = (f ++ g)(x|->y)"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   262
apply (unfold map_add_def)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   263
apply (rule ext)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   264
apply auto
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   265
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   266
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   267
lemma map_of_append[simp]: "map_of (xs@ys) = map_of ys ++ map_of xs"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   268
apply (unfold map_add_def)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   269
apply (induct_tac "xs")
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   270
apply (simp (no_asm))
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   271
apply (rule ext)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   272
apply (simp (no_asm_simp) split add: option.split)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   273
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   274
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   275
declare fun_upd_apply [simp del]
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   276
lemma finite_range_map_of_map_add:
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   277
 "finite (range f) ==> finite (range (f ++ map_of l))"
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   278
apply (induct_tac "l")
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   279
apply  auto
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   280
apply (erule finite_range_updI)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   281
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   282
declare fun_upd_apply [simp]
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   283
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   284
subsection {* @{term map_image} *}
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   285
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   286
lemma map_image_empty [simp]: "f`>empty = empty" 
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   287
by (auto simp: map_image_def empty_def)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   288
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   289
lemma map_image_upd [simp]: "f`>m(a|->b) = (f`>m)(a|->f b)" 
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   290
apply (auto simp: map_image_def fun_upd_def)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   291
by (rule ext, auto)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   292
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   293
subsection {* @{term restrict_map} *}
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   294
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   295
lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m\<lfloor>A) x = m x"
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   296
by (auto simp: restrict_map_def)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   297
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   298
lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m\<lfloor>A) x = None"
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   299
by (auto simp: restrict_map_def)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   300
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   301
lemma ran_restrictD: "y \<in> ran (m\<lfloor>A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y"
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   302
by (auto simp: restrict_map_def ran_def split: split_if_asm)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   303
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   304
lemma dom_valF_restrict [simp]: "dom (m\<lfloor>A) = dom m \<inter> A"
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   305
by (auto simp: restrict_map_def dom_def split: split_if_asm)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   306
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   307
lemma restrict_upd_same [simp]: "m(x\<mapsto>y)\<lfloor>(-{x}) = m\<lfloor>(-{x})"
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   308
by (rule ext, auto simp: restrict_map_def)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   309
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   310
lemma restrict_restrict [simp]: "m\<lfloor>A\<lfloor>B = m\<lfloor>(A\<inter>B)"
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   311
by (rule ext, auto simp: restrict_map_def)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   312
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   313
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   314
subsection {* @{term map_upds} *}
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   315
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   316
lemma map_upds_Nil1[simp]: "m([] [|->] bs) = m"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   317
by(simp add:map_upds_def)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   318
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   319
lemma map_upds_Nil2[simp]: "m(as [|->] []) = m"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   320
by(simp add:map_upds_def)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   321
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   322
lemma map_upds_Cons[simp]: "m(a#as [|->] b#bs) = (m(a|->b))(as[|->]bs)"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   323
by(simp add:map_upds_def)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   324
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   325
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   326
lemma map_upd_upds_conv_if: "!!x y ys f.
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   327
 (f(x|->y))(xs [|->] ys) =
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   328
 (if x : set(take (length ys) xs) then f(xs [|->] ys)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   329
                                  else (f(xs [|->] ys))(x|->y))"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   330
apply(induct xs)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   331
 apply simp
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   332
apply(case_tac ys)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   333
 apply(auto split:split_if simp:fun_upd_twist)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   334
done
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   335
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   336
lemma map_upds_twist [simp]:
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   337
 "a ~: set as ==> m(a|->b)(as[|->]bs) = m(as[|->]bs)(a|->b)"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   338
apply(insert set_take_subset)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   339
apply (fastsimp simp add: map_upd_upds_conv_if)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   340
done
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   341
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   342
lemma map_upds_apply_nontin[simp]:
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   343
 "!!ys. x ~: set xs ==> (f(xs[|->]ys)) x = f x"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   344
apply(induct xs)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   345
 apply simp
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   346
apply(case_tac ys)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   347
 apply(auto simp: map_upd_upds_conv_if)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   348
done
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   349
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   350
subsection {* @{term map_upd_s} *}
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   351
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   352
lemma map_upd_s_apply [simp]: 
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   353
  "(m(as{|->}b)) x = (if x : as then Some b else m x)"
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   354
by (simp add: map_upd_s_def)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   355
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   356
lemma map_subst_apply [simp]: 
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   357
  "(m(a~>b)) x = (if m x = Some a then Some b else m x)" 
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   358
by (simp add: map_subst_def)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   359
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   360
subsection {* @{term dom} *}
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   361
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   362
lemma domI: "m a = Some b ==> a : dom m"
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   363
apply (unfold dom_def)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   364
apply auto
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   365
done
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   366
(* declare domI [intro]? *)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   367
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   368
lemma domD: "a : dom m ==> ? b. m a = Some b"
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   369
apply (unfold dom_def)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   370
apply auto
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   371
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   372
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   373
lemma domIff[iff]: "(a : dom m) = (m a ~= None)"
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   374
apply (unfold dom_def)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   375
apply auto
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   376
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   377
declare domIff [simp del]
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   378
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   379
lemma dom_empty[simp]: "dom empty = {}"
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   380
apply (unfold dom_def)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   381
apply (simp (no_asm))
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   382
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   383
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   384
lemma dom_fun_upd[simp]:
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   385
 "dom(f(x := y)) = (if y=None then dom f - {x} else insert x (dom f))"
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   386
by (simp add:dom_def) blast
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   387
13937
e9d57517c9b1 added a thm
nipkow
parents: 13914
diff changeset
   388
lemma dom_map_of: "dom(map_of xys) = {x. \<exists>y. (x,y) : set xys}"
e9d57517c9b1 added a thm
nipkow
parents: 13914
diff changeset
   389
apply(induct xys)
e9d57517c9b1 added a thm
nipkow
parents: 13914
diff changeset
   390
apply(auto simp del:fun_upd_apply)
e9d57517c9b1 added a thm
nipkow
parents: 13914
diff changeset
   391
done
e9d57517c9b1 added a thm
nipkow
parents: 13914
diff changeset
   392
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   393
lemma finite_dom_map_of: "finite (dom (map_of l))"
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   394
apply (unfold dom_def)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   395
apply (induct_tac "l")
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   396
apply (auto simp add: insert_Collect [symmetric])
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   397
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   398
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   399
lemma dom_map_upds[simp]:
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   400
 "!!m ys. dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   401
apply(induct xs)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   402
 apply simp
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   403
apply(case_tac ys)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   404
 apply auto
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   405
done
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   406
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   407
lemma dom_map_add[simp]: "dom(m++n) = dom n Un dom m"
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   408
apply (unfold dom_def)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   409
apply auto
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   410
done
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   411
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   412
lemma dom_overwrite[simp]:
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   413
 "dom(f(g|A)) = (dom f  - {a. a : A - dom g}) Un {a. a : A Int dom g}"
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   414
by(auto simp add: dom_def overwrite_def)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   415
14027
68d247b7b14b *** empty log message ***
nipkow
parents: 14026
diff changeset
   416
lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1"
68d247b7b14b *** empty log message ***
nipkow
parents: 14026
diff changeset
   417
apply(rule ext)
68d247b7b14b *** empty log message ***
nipkow
parents: 14026
diff changeset
   418
apply(fastsimp simp:map_add_def split:option.split)
68d247b7b14b *** empty log message ***
nipkow
parents: 14026
diff changeset
   419
done
68d247b7b14b *** empty log message ***
nipkow
parents: 14026
diff changeset
   420
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   421
subsection {* @{term ran} *}
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   422
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   423
lemma ranI: "m a = Some b ==> b : ran m" 
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   424
by (auto simp add: ran_def)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   425
(* declare ranI [intro]? *)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   426
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   427
lemma ran_empty[simp]: "ran empty = {}"
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   428
apply (unfold ran_def)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   429
apply (simp (no_asm))
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   430
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   431
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   432
lemma ran_map_upd[simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)"
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   433
apply (unfold ran_def)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   434
apply auto
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   435
apply (subgoal_tac "~ (aa = a) ")
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   436
apply auto
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   437
done
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   438
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   439
subsection {* @{text "map_le"} *}
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   440
13912
3c0a340be514 fixed document
kleing
parents: 13910
diff changeset
   441
lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g"
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   442
by(simp add:map_le_def)
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   443
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   444
lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)"
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   445
by(fastsimp simp add:map_le_def)
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   446
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   447
lemma map_le_upds[simp]:
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   448
 "!!f g bs. f \<subseteq>\<^sub>m g ==> f(as [|->] bs) \<subseteq>\<^sub>m g(as [|->] bs)"
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   449
apply(induct as)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   450
 apply simp
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   451
apply(case_tac bs)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   452
 apply auto
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   453
done
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   454
14033
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   455
lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)"
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   456
  by (fastsimp simp add: map_le_def dom_def)
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   457
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   458
lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f"
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   459
  by (simp add: map_le_def)
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   460
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   461
lemma map_le_trans: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m h \<rbrakk> \<Longrightarrow> f \<subseteq>\<^sub>m h"
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   462
  apply (clarsimp simp add: map_le_def)
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   463
  apply (drule_tac x="a" in bspec, fastsimp)+
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   464
  apply assumption
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   465
done
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   466
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   467
lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g"
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   468
  apply (unfold map_le_def)
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   469
  apply (rule ext)
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   470
  apply (case_tac "x \<in> dom f")
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   471
    apply simp
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   472
  apply (case_tac "x \<in> dom g")
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   473
    apply simp
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   474
  apply fastsimp
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   475
done
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   476
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   477
lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)"
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   478
  by (fastsimp simp add: map_le_def)
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   479
3981
b4f93a8da835 Added the new theory Map.
nipkow
parents:
diff changeset
   480
end