src/HOL/Map.thy
author nipkow
Mon, 11 Apr 2005 12:14:23 +0200
changeset 15693 3a67e61c6e96
parent 15691 900cf45ff0a6
child 15695 f072119afa4e
permissions -rw-r--r--
tuned Map, renamed lex stuff in List.
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
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15110
diff changeset
    11
theory Map
15140
322485b816ac import -> imports
nipkow
parents: 15131
diff changeset
    12
imports List
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15110
diff changeset
    13
begin
3981
b4f93a8da835 Added the new theory Map.
nipkow
parents:
diff changeset
    14
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
    15
types ('a,'b) "~=>" = "'a => 'b option" (infixr 0)
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    16
translations (type) "a ~=> b " <= (type) "a => b option"
3981
b4f93a8da835 Added the new theory Map.
nipkow
parents:
diff changeset
    17
b4f93a8da835 Added the new theory Map.
nipkow
parents:
diff changeset
    18
consts
5300
2b1ca524ace8 defined map_upd by translation via fun_upd
oheimb
parents: 5195
diff changeset
    19
chg_map	:: "('b => 'b) => 'a => ('a ~=> 'b) => ('a ~=> 'b)"
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    20
map_add :: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)" (infixl "++" 100)
15693
3a67e61c6e96 tuned Map, renamed lex stuff in List.
nipkow
parents: 15691
diff changeset
    21
restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)" (infixl "|`"  110)
5300
2b1ca524ace8 defined map_upd by translation via fun_upd
oheimb
parents: 5195
diff changeset
    22
dom	:: "('a ~=> 'b) => 'a set"
2b1ca524ace8 defined map_upd by translation via fun_upd
oheimb
parents: 5195
diff changeset
    23
ran	:: "('a ~=> 'b) => 'b set"
2b1ca524ace8 defined map_upd by translation via fun_upd
oheimb
parents: 5195
diff changeset
    24
map_of	:: "('a * 'b)list => 'a ~=> 'b"
2b1ca524ace8 defined map_upd by translation via fun_upd
oheimb
parents: 5195
diff changeset
    25
map_upds:: "('a ~=> 'b) => 'a list => 'b list => 
14180
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    26
	    ('a ~=> 'b)"
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    27
map_upd_s::"('a ~=> 'b) => 'a set => 'b => 
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    28
	    ('a ~=> 'b)"			 ("_/'(_{|->}_/')" [900,0,0]900)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    29
map_subst::"('a ~=> 'b) => 'b => 'b => 
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    30
	    ('a ~=> 'b)"			 ("_/'(_~>_/')"    [900,0,0]900)
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
    31
map_le  :: "('a ~=> 'b) => ('a ~=> 'b) => bool" (infix "\<subseteq>\<^sub>m" 50)
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
    32
14739
86c6f272ef79 renamed `> to o_m
nipkow
parents: 14537
diff changeset
    33
syntax
86c6f272ef79 renamed `> to o_m
nipkow
parents: 14537
diff changeset
    34
  fun_map_comp :: "('b => 'c)  => ('a ~=> 'b) => ('a ~=> 'c)" (infixl "o'_m" 55)
86c6f272ef79 renamed `> to o_m
nipkow
parents: 14537
diff changeset
    35
translations
86c6f272ef79 renamed `> to o_m
nipkow
parents: 14537
diff changeset
    36
  "f o_m m" == "option_map f o m"
86c6f272ef79 renamed `> to o_m
nipkow
parents: 14537
diff changeset
    37
14180
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    38
nonterminals
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    39
  maplets maplet
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    40
5300
2b1ca524ace8 defined map_upd by translation via fun_upd
oheimb
parents: 5195
diff changeset
    41
syntax
14180
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    42
  empty	    ::  "'a ~=> 'b"
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    43
  "_maplet"  :: "['a, 'a] => maplet"             ("_ /|->/ _")
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    44
  "_maplets" :: "['a, 'a] => maplet"             ("_ /[|->]/ _")
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    45
  ""         :: "maplet => maplets"             ("_")
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    46
  "_Maplets" :: "[maplet, maplets] => maplets" ("_,/ _")
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    47
  "_MapUpd"  :: "['a ~=> 'b, maplets] => 'a ~=> 'b" ("_/'(_')" [900,0]900)
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    48
  "_Map"     :: "maplets => 'a ~=> 'b"            ("(1[_])")
3981
b4f93a8da835 Added the new theory Map.
nipkow
parents:
diff changeset
    49
12114
a8e860c86252 eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents: 10137
diff changeset
    50
syntax (xsymbols)
14739
86c6f272ef79 renamed `> to o_m
nipkow
parents: 14537
diff changeset
    51
  "~=>"     :: "[type, type] => type"    (infixr "\<rightharpoonup>" 0)
86c6f272ef79 renamed `> to o_m
nipkow
parents: 14537
diff changeset
    52
86c6f272ef79 renamed `> to o_m
nipkow
parents: 14537
diff changeset
    53
  fun_map_comp :: "('b => 'c)  => ('a ~=> 'b) => ('a ~=> 'c)" (infixl "\<circ>\<^sub>m" 55)
86c6f272ef79 renamed `> to o_m
nipkow
parents: 14537
diff changeset
    54
14180
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    55
  "_maplet"  :: "['a, 'a] => maplet"             ("_ /\<mapsto>/ _")
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    56
  "_maplets" :: "['a, 'a] => maplet"             ("_ /[\<mapsto>]/ _")
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    57
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    58
  map_upd_s  :: "('a ~=> 'b) => 'a set => 'b => ('a ~=> 'b)"
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    59
				    		 ("_/'(_/{\<mapsto>}/_')" [900,0,0]900)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    60
  map_subst :: "('a ~=> 'b) => 'b => 'b => 
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    61
	        ('a ~=> 'b)"			 ("_/'(_\<leadsto>_/')"    [900,0,0]900)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    62
 "@chg_map" :: "('a ~=> 'b) => 'a => ('b => 'b) => ('a ~=> 'b)"
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    63
					  ("_/'(_/\<mapsto>\<lambda>_. _')"  [900,0,0,0] 900)
5300
2b1ca524ace8 defined map_upd by translation via fun_upd
oheimb
parents: 5195
diff changeset
    64
15693
3a67e61c6e96 tuned Map, renamed lex stuff in List.
nipkow
parents: 15691
diff changeset
    65
syntax (latex output)
3a67e61c6e96 tuned Map, renamed lex stuff in List.
nipkow
parents: 15691
diff changeset
    66
  restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)" ("_\<restriction>\<^bsub>_\<^esub>" [111,110] 110) --"requires amssymb!"
3a67e61c6e96 tuned Map, renamed lex stuff in List.
nipkow
parents: 15691
diff changeset
    67
5300
2b1ca524ace8 defined map_upd by translation via fun_upd
oheimb
parents: 5195
diff changeset
    68
translations
13890
90611b4e0054 Made empty a translation rather than a constant.
nipkow
parents: 12919
diff changeset
    69
  "empty"    => "_K None"
90611b4e0054 Made empty a translation rather than a constant.
nipkow
parents: 12919
diff changeset
    70
  "empty"    <= "%x. None"
5300
2b1ca524ace8 defined map_upd by translation via fun_upd
oheimb
parents: 5195
diff changeset
    71
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    72
  "m(x\<mapsto>\<lambda>y. f)" == "chg_map (\<lambda>y. f) x m"
3981
b4f93a8da835 Added the new theory Map.
nipkow
parents:
diff changeset
    73
14180
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    74
  "_MapUpd m (_Maplets xy ms)"  == "_MapUpd (_MapUpd m xy) ms"
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    75
  "_MapUpd m (_maplet  x y)"    == "m(x:=Some y)"
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    76
  "_MapUpd m (_maplets x y)"    == "map_upds m x y"
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    77
  "_Map ms"                     == "_MapUpd empty ms"
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    78
  "_Map (_Maplets ms1 ms2)"     <= "_MapUpd (_Map ms1) ms2"
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    79
  "_Maplets ms1 (_Maplets ms2 ms3)" <= "_Maplets (_Maplets ms1 ms2) ms3"
d2e550609c40 Introduced new syntax for maplets x |-> y
nipkow
parents: 14134
diff changeset
    80
3981
b4f93a8da835 Added the new theory Map.
nipkow
parents:
diff changeset
    81
defs
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
    82
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
    83
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
    84
map_add_def:   "m1++m2 == %x. case m2 x of None => m1 x | Some y => Some y"
15693
3a67e61c6e96 tuned Map, renamed lex stuff in List.
nipkow
parents: 15691
diff changeset
    85
restrict_map_def: "m|`A == %x. if x : A then m x else None"
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
    86
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
    87
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
    88
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
    89
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
    90
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
    91
dom_def: "dom(m) == {a. m a ~= None}"
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
    92
ran_def: "ran(m) == {b. EX a. m a = Some b}"
3981
b4f93a8da835 Added the new theory Map.
nipkow
parents:
diff changeset
    93
14376
9fe787a90a48 Changed variable names.
nipkow
parents: 14300
diff changeset
    94
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"
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
    95
5183
89f162de39cf Adapted to new datatype package.
berghofe
parents: 3981
diff changeset
    96
primrec
89f162de39cf Adapted to new datatype package.
berghofe
parents: 3981
diff changeset
    97
  "map_of [] = empty"
5300
2b1ca524ace8 defined map_upd by translation via fun_upd
oheimb
parents: 5195
diff changeset
    98
  "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
    99
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   100
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   101
subsection {* @{term empty} *}
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   102
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   103
lemma empty_upd_none[simp]: "empty(x := None) = empty"
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   104
apply (rule ext)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   105
apply (simp (no_asm))
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   106
done
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   107
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   108
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   109
(* FIXME: what is this sum_case nonsense?? *)
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   110
lemma sum_case_empty_empty[simp]: "sum_case empty empty = empty"
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   111
apply (rule ext)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   112
apply (simp (no_asm) split add: sum.split)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   113
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   114
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   115
subsection {* @{term map_upd} *}
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   116
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   117
lemma map_upd_triv: "t k = Some x ==> t(k|->x) = t"
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   118
apply (rule ext)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   119
apply (simp (no_asm_simp))
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   120
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   121
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   122
lemma map_upd_nonempty[simp]: "t(k|->x) ~= empty"
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   123
apply safe
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   124
apply (drule_tac x = k in fun_cong)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   125
apply (simp (no_asm_use))
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   126
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   127
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   128
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
   129
by (drule fun_cong [of _ _ a], auto)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   130
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   131
lemma map_upd_Some_unfold: 
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   132
  "((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
   133
by auto
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   134
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15251
diff changeset
   135
lemma image_map_upd[simp]: "x \<notin> A \<Longrightarrow> m(x \<mapsto> y) ` A = m ` A"
eedbb8d22ca2 added lemmas
nipkow
parents: 15251
diff changeset
   136
by fastsimp
eedbb8d22ca2 added lemmas
nipkow
parents: 15251
diff changeset
   137
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   138
lemma finite_range_updI: "finite (range f) ==> finite (range (f(a|->b)))"
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   139
apply (unfold image_def)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   140
apply (simp (no_asm_use) add: full_SetCompr_eq)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   141
apply (rule finite_subset)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   142
prefer 2 apply assumption
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   143
apply auto
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   144
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   145
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   146
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   147
(* FIXME: what is this sum_case nonsense?? *)
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   148
subsection {* @{term sum_case} and @{term empty}/@{term map_upd} *}
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   149
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   150
lemma sum_case_map_upd_empty[simp]:
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   151
 "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
   152
apply (rule ext)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   153
apply (simp (no_asm) split add: sum.split)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   154
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   155
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   156
lemma sum_case_empty_map_upd[simp]:
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   157
 "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
   158
apply (rule ext)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   159
apply (simp (no_asm) split add: sum.split)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   160
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   161
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   162
lemma sum_case_map_upd_map_upd[simp]:
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   163
 "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
   164
apply (rule ext)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   165
apply (simp (no_asm) split add: sum.split)
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 chg_map} *}
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   170
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   171
lemma chg_map_new[simp]: "m a = None   ==> chg_map f a m = m"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   172
by (unfold chg_map_def, auto)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   173
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   174
lemma chg_map_upd[simp]: "m a = Some b ==> chg_map f a m = m(a|->f b)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   175
by (unfold chg_map_def, auto)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   176
14537
e95ba267e3d5 added theorem chg_map_other
oheimb
parents: 14376
diff changeset
   177
lemma chg_map_other [simp]: "a \<noteq> b \<Longrightarrow> chg_map f a m b = m b"
e95ba267e3d5 added theorem chg_map_other
oheimb
parents: 14376
diff changeset
   178
by (auto simp: chg_map_def split add: option.split)
e95ba267e3d5 added theorem chg_map_other
oheimb
parents: 14376
diff changeset
   179
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   180
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   181
subsection {* @{term map_of} *}
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   182
15304
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   183
lemma map_of_eq_None_iff:
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   184
 "(map_of xys x = None) = (x \<notin> fst ` (set xys))"
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   185
by (induct xys) simp_all
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   186
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   187
lemma map_of_is_SomeD:
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   188
 "map_of xys x = Some y \<Longrightarrow> (x,y) \<in> set xys"
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   189
apply(induct xys)
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   190
 apply simp
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   191
apply(clarsimp split:if_splits)
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   192
done
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   193
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   194
lemma map_of_eq_Some_iff[simp]:
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   195
 "distinct(map fst xys) \<Longrightarrow> (map_of xys x = Some y) = ((x,y) \<in> set xys)"
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   196
apply(induct xys)
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   197
 apply(simp)
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   198
apply(auto simp:map_of_eq_None_iff[symmetric])
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   199
done
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   200
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   201
lemma Some_eq_map_of_iff[simp]:
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   202
 "distinct(map fst xys) \<Longrightarrow> (Some y = map_of xys x) = ((x,y) \<in> set xys)"
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   203
by(auto simp del:map_of_eq_Some_iff simp add:map_of_eq_Some_iff[symmetric])
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   204
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   205
lemma [simp]: "\<lbrakk> distinct(map fst xys); (x,y) \<in> set xys \<rbrakk>
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   206
  \<Longrightarrow> map_of xys x = Some y"
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   207
apply (induct xys)
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   208
 apply simp
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   209
apply force
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   210
done
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   211
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 14739
diff changeset
   212
lemma map_of_zip_is_None[simp]:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 14739
diff changeset
   213
  "length xs = length ys \<Longrightarrow> (map_of (zip xs ys) x = None) = (x \<notin> set xs)"
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 14739
diff changeset
   214
by (induct rule:list_induct2, simp_all)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 14739
diff changeset
   215
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 14739
diff changeset
   216
lemma finite_range_map_of: "finite (range (map_of xys))"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   217
apply (induct xys)
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 14739
diff changeset
   218
apply  (simp_all (no_asm) add: image_constant)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 14739
diff changeset
   219
apply (rule finite_subset)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 14739
diff changeset
   220
prefer 2 apply assumption
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 14739
diff changeset
   221
apply auto
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 14739
diff changeset
   222
done
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 14739
diff changeset
   223
15369
paulson
parents: 15304
diff changeset
   224
lemma map_of_SomeD [rule_format]: "map_of xs k = Some y --> (k,y):set xs"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   225
by (induct "xs", auto)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   226
15369
paulson
parents: 15304
diff changeset
   227
lemma map_of_mapk_SomeI [rule_format]:
paulson
parents: 15304
diff changeset
   228
     "inj f ==> map_of t k = Some x -->  
paulson
parents: 15304
diff changeset
   229
        map_of (map (split (%k. Pair (f k))) t) (f k) = Some x"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   230
apply (induct "t")
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   231
apply  (auto simp add: inj_eq)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   232
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   233
15369
paulson
parents: 15304
diff changeset
   234
lemma weak_map_of_SomeI [rule_format]:
paulson
parents: 15304
diff changeset
   235
     "(k, x) : set l --> (\<exists>x. map_of l k = Some x)"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   236
by (induct "l", auto)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   237
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   238
lemma map_of_filter_in: 
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   239
"[| 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
   240
apply (rule mp)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   241
prefer 2 apply assumption
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   242
apply (erule thin_rl)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   243
apply (induct "xs", auto)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   244
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   245
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   246
lemma map_of_map: "map_of (map (%(a,b). (a,f b)) xs) x = option_map f (map_of xs x)"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   247
by (induct "xs", auto)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   248
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   249
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   250
subsection {* @{term option_map} related *}
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   251
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   252
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
   253
apply (rule ext)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   254
apply (simp (no_asm))
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   255
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   256
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   257
lemma option_map_o_map_upd[simp]:
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   258
 "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
   259
apply (rule ext)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   260
apply (simp (no_asm))
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   261
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   262
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   263
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   264
subsection {* @{text "++"} *}
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   265
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   266
lemma map_add_empty[simp]: "m ++ empty = m"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   267
apply (unfold map_add_def)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   268
apply (simp (no_asm))
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   269
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   270
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   271
lemma empty_map_add[simp]: "empty ++ m = m"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   272
apply (unfold map_add_def)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   273
apply (rule ext)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   274
apply (simp split add: option.split)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   275
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   276
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   277
lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   278
apply(rule ext)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   279
apply(simp add: map_add_def split:option.split)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   280
done
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   281
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   282
lemma map_add_Some_iff: 
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   283
 "((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
   284
apply (unfold map_add_def)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   285
apply (simp (no_asm) split add: option.split)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   286
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   287
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   288
lemmas map_add_SomeD = map_add_Some_iff [THEN iffD1, standard]
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   289
declare map_add_SomeD [dest!]
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   290
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   291
lemma map_add_find_right[simp]: "!!xx. n k = Some xx ==> (m ++ n) k = Some xx"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   292
by (subst map_add_Some_iff, fast)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   293
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   294
lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   295
apply (unfold map_add_def)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   296
apply (simp (no_asm) split add: option.split)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   297
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   298
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   299
lemma map_add_upd[simp]: "f ++ g(x|->y) = (f ++ g)(x|->y)"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   300
apply (unfold map_add_def)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   301
apply (rule ext, auto)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   302
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   303
14186
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   304
lemma map_add_upds[simp]: "m1 ++ (m2(xs[\<mapsto>]ys)) = (m1++m2)(xs[\<mapsto>]ys)"
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   305
by(simp add:map_upds_def)
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   306
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   307
lemma map_of_append[simp]: "map_of (xs@ys) = map_of ys ++ map_of xs"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   308
apply (unfold map_add_def)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   309
apply (induct "xs")
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   310
apply (simp (no_asm))
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   311
apply (rule ext)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   312
apply (simp (no_asm_simp) split add: option.split)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   313
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   314
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   315
declare fun_upd_apply [simp del]
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   316
lemma finite_range_map_of_map_add:
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   317
 "finite (range f) ==> finite (range (f ++ map_of l))"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   318
apply (induct "l", auto)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   319
apply (erule finite_range_updI)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   320
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   321
declare fun_upd_apply [simp]
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   322
15304
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   323
lemma inj_on_map_add_dom[iff]:
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   324
 "inj_on (m ++ m') (dom m') = inj_on m' (dom m')"
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   325
by(fastsimp simp add:map_add_def dom_def inj_on_def split:option.splits)
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   326
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   327
subsection {* @{term restrict_map} *}
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   328
15693
3a67e61c6e96 tuned Map, renamed lex stuff in List.
nipkow
parents: 15691
diff changeset
   329
lemma restrict_map_to_empty[simp]: "m|`{} = empty"
14186
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   330
by(simp add: restrict_map_def)
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   331
15693
3a67e61c6e96 tuned Map, renamed lex stuff in List.
nipkow
parents: 15691
diff changeset
   332
lemma restrict_map_empty[simp]: "empty|`D = empty"
14186
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   333
by(simp add: restrict_map_def)
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   334
15693
3a67e61c6e96 tuned Map, renamed lex stuff in List.
nipkow
parents: 15691
diff changeset
   335
lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m|`A) x = m x"
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   336
by (auto simp: restrict_map_def)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   337
15693
3a67e61c6e96 tuned Map, renamed lex stuff in List.
nipkow
parents: 15691
diff changeset
   338
lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m|`A) x = None"
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   339
by (auto simp: restrict_map_def)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   340
15693
3a67e61c6e96 tuned Map, renamed lex stuff in List.
nipkow
parents: 15691
diff changeset
   341
lemma ran_restrictD: "y \<in> ran (m|`A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y"
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   342
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
   343
15693
3a67e61c6e96 tuned Map, renamed lex stuff in List.
nipkow
parents: 15691
diff changeset
   344
lemma dom_restrict [simp]: "dom (m|`A) = dom m \<inter> A"
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   345
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
   346
15693
3a67e61c6e96 tuned Map, renamed lex stuff in List.
nipkow
parents: 15691
diff changeset
   347
lemma restrict_upd_same [simp]: "m(x\<mapsto>y)|`(-{x}) = m|`(-{x})"
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   348
by (rule ext, auto simp: restrict_map_def)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   349
15693
3a67e61c6e96 tuned Map, renamed lex stuff in List.
nipkow
parents: 15691
diff changeset
   350
lemma restrict_restrict [simp]: "m|`A|`B = m|`(A\<inter>B)"
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   351
by (rule ext, auto simp: restrict_map_def)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   352
14186
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   353
lemma restrict_fun_upd[simp]:
15693
3a67e61c6e96 tuned Map, renamed lex stuff in List.
nipkow
parents: 15691
diff changeset
   354
 "m(x := y)|`D = (if x \<in> D then (m|`(D-{x}))(x := y) else m|`D)"
14186
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   355
by(simp add: restrict_map_def expand_fun_eq)
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   356
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   357
lemma fun_upd_None_restrict[simp]:
15693
3a67e61c6e96 tuned Map, renamed lex stuff in List.
nipkow
parents: 15691
diff changeset
   358
  "(m|`D)(x := None) = (if x:D then m|`(D - {x}) else m|`D)"
14186
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   359
by(simp add: restrict_map_def expand_fun_eq)
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   360
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   361
lemma fun_upd_restrict:
15693
3a67e61c6e96 tuned Map, renamed lex stuff in List.
nipkow
parents: 15691
diff changeset
   362
 "(m|`D)(x := y) = (m|`(D-{x}))(x := y)"
14186
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   363
by(simp add: restrict_map_def expand_fun_eq)
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   364
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   365
lemma fun_upd_restrict_conv[simp]:
15693
3a67e61c6e96 tuned Map, renamed lex stuff in List.
nipkow
parents: 15691
diff changeset
   366
 "x \<in> D \<Longrightarrow> (m|`D)(x := y) = (m|`(D-{x}))(x := y)"
14186
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   367
by(simp add: restrict_map_def expand_fun_eq)
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   368
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   369
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   370
subsection {* @{term map_upds} *}
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   371
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   372
lemma map_upds_Nil1[simp]: "m([] [|->] bs) = m"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   373
by(simp add:map_upds_def)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   374
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   375
lemma map_upds_Nil2[simp]: "m(as [|->] []) = m"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   376
by(simp add:map_upds_def)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   377
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   378
lemma map_upds_Cons[simp]: "m(a#as [|->] b#bs) = (m(a|->b))(as[|->]bs)"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   379
by(simp add:map_upds_def)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   380
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14186
diff changeset
   381
lemma map_upds_append1[simp]: "\<And>ys m. size xs < size ys \<Longrightarrow>
26dfcd0ac436 Added new theorems
nipkow
parents: 14186
diff changeset
   382
  m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)"
26dfcd0ac436 Added new theorems
nipkow
parents: 14186
diff changeset
   383
apply(induct xs)
26dfcd0ac436 Added new theorems
nipkow
parents: 14186
diff changeset
   384
 apply(clarsimp simp add:neq_Nil_conv)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   385
apply (case_tac ys, simp, simp)
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14186
diff changeset
   386
done
26dfcd0ac436 Added new theorems
nipkow
parents: 14186
diff changeset
   387
26dfcd0ac436 Added new theorems
nipkow
parents: 14186
diff changeset
   388
lemma map_upds_list_update2_drop[simp]:
26dfcd0ac436 Added new theorems
nipkow
parents: 14186
diff changeset
   389
 "\<And>m ys i. \<lbrakk>size xs \<le> i; i < size ys\<rbrakk>
26dfcd0ac436 Added new theorems
nipkow
parents: 14186
diff changeset
   390
     \<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   391
apply (induct xs, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   392
apply (case_tac ys, simp)
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14186
diff changeset
   393
apply(simp split:nat.split)
26dfcd0ac436 Added new theorems
nipkow
parents: 14186
diff changeset
   394
done
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   395
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   396
lemma map_upd_upds_conv_if: "!!x y ys f.
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   397
 (f(x|->y))(xs [|->] ys) =
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   398
 (if x : set(take (length ys) xs) then f(xs [|->] ys)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   399
                                  else (f(xs [|->] ys))(x|->y))"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   400
apply (induct xs, simp)
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   401
apply(case_tac ys)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   402
 apply(auto split:split_if simp:fun_upd_twist)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   403
done
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   404
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   405
lemma map_upds_twist [simp]:
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   406
 "a ~: set as ==> m(a|->b)(as[|->]bs) = m(as[|->]bs)(a|->b)"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   407
apply(insert set_take_subset)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   408
apply (fastsimp simp add: map_upd_upds_conv_if)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   409
done
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   410
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   411
lemma map_upds_apply_nontin[simp]:
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   412
 "!!ys. x ~: set xs ==> (f(xs[|->]ys)) x = f x"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   413
apply (induct xs, simp)
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   414
apply(case_tac ys)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   415
 apply(auto simp: map_upd_upds_conv_if)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   416
done
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   417
14300
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14208
diff changeset
   418
lemma fun_upds_append_drop[simp]:
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14208
diff changeset
   419
  "!!m ys. size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)"
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14208
diff changeset
   420
apply(induct xs)
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14208
diff changeset
   421
 apply (simp)
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14208
diff changeset
   422
apply(case_tac ys)
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14208
diff changeset
   423
apply simp_all
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14208
diff changeset
   424
done
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14208
diff changeset
   425
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14208
diff changeset
   426
lemma fun_upds_append2_drop[simp]:
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14208
diff changeset
   427
  "!!m ys. size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)"
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14208
diff changeset
   428
apply(induct xs)
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14208
diff changeset
   429
 apply (simp)
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14208
diff changeset
   430
apply(case_tac ys)
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14208
diff changeset
   431
apply simp_all
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14208
diff changeset
   432
done
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14208
diff changeset
   433
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14208
diff changeset
   434
14186
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   435
lemma restrict_map_upds[simp]: "!!m ys.
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   436
 \<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk>
15693
3a67e61c6e96 tuned Map, renamed lex stuff in List.
nipkow
parents: 15691
diff changeset
   437
 \<Longrightarrow> m(xs [\<mapsto>] ys)|`D = (m|`(D - set xs))(xs [\<mapsto>] ys)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   438
apply (induct xs, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   439
apply (case_tac ys, simp)
14186
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   440
apply(simp add:Diff_insert[symmetric] insert_absorb)
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   441
apply(simp add: map_upd_upds_conv_if)
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   442
done
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   443
6d2a494e33be Added a number of thms about map restriction.
nipkow
parents: 14180
diff changeset
   444
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   445
subsection {* @{term map_upd_s} *}
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   446
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   447
lemma map_upd_s_apply [simp]: 
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   448
  "(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
   449
by (simp add: map_upd_s_def)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   450
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   451
lemma map_subst_apply [simp]: 
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   452
  "(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
   453
by (simp add: map_subst_def)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   454
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   455
subsection {* @{term dom} *}
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   456
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   457
lemma domI: "m a = Some b ==> a : dom m"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   458
by (unfold dom_def, auto)
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   459
(* declare domI [intro]? *)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   460
15369
paulson
parents: 15304
diff changeset
   461
lemma domD: "a : dom m ==> \<exists>b. m a = Some b"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   462
by (unfold dom_def, auto)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   463
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   464
lemma domIff[iff]: "(a : dom m) = (m a ~= None)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   465
by (unfold dom_def, auto)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   466
declare domIff [simp del]
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   467
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   468
lemma dom_empty[simp]: "dom empty = {}"
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   469
apply (unfold dom_def)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   470
apply (simp (no_asm))
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   471
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   472
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   473
lemma dom_fun_upd[simp]:
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   474
 "dom(f(x := y)) = (if y=None then dom f - {x} else insert x (dom f))"
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   475
by (simp add:dom_def) blast
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   476
13937
e9d57517c9b1 added a thm
nipkow
parents: 13914
diff changeset
   477
lemma dom_map_of: "dom(map_of xys) = {x. \<exists>y. (x,y) : set xys}"
e9d57517c9b1 added a thm
nipkow
parents: 13914
diff changeset
   478
apply(induct xys)
e9d57517c9b1 added a thm
nipkow
parents: 13914
diff changeset
   479
apply(auto simp del:fun_upd_apply)
e9d57517c9b1 added a thm
nipkow
parents: 13914
diff changeset
   480
done
e9d57517c9b1 added a thm
nipkow
parents: 13914
diff changeset
   481
15304
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   482
lemma dom_map_of_conv_image_fst:
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   483
  "dom(map_of xys) = fst ` (set xys)"
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   484
by(force simp: dom_map_of)
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   485
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 14739
diff changeset
   486
lemma dom_map_of_zip[simp]: "[| length xs = length ys; distinct xs |] ==>
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 14739
diff changeset
   487
  dom(map_of(zip xs ys)) = set xs"
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 14739
diff changeset
   488
by(induct rule: list_induct2, simp_all)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 14739
diff changeset
   489
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   490
lemma finite_dom_map_of: "finite (dom (map_of l))"
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   491
apply (unfold dom_def)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   492
apply (induct "l")
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   493
apply (auto simp add: insert_Collect [symmetric])
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   494
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   495
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   496
lemma dom_map_upds[simp]:
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   497
 "!!m ys. dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   498
apply (induct xs, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   499
apply (case_tac ys, auto)
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   500
done
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   501
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   502
lemma dom_map_add[simp]: "dom(m++n) = dom n Un dom m"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   503
by (unfold dom_def, auto)
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   504
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15369
diff changeset
   505
lemma dom_override_on[simp]:
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15369
diff changeset
   506
 "dom(override_on f g A) =
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15369
diff changeset
   507
 (dom f  - {a. a : A - dom g}) Un {a. a : A Int dom g}"
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15369
diff changeset
   508
by(auto simp add: dom_def override_on_def)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   509
14027
68d247b7b14b *** empty log message ***
nipkow
parents: 14026
diff changeset
   510
lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1"
68d247b7b14b *** empty log message ***
nipkow
parents: 14026
diff changeset
   511
apply(rule ext)
68d247b7b14b *** empty log message ***
nipkow
parents: 14026
diff changeset
   512
apply(fastsimp simp:map_add_def split:option.split)
68d247b7b14b *** empty log message ***
nipkow
parents: 14026
diff changeset
   513
done
68d247b7b14b *** empty log message ***
nipkow
parents: 14026
diff changeset
   514
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   515
subsection {* @{term ran} *}
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   516
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   517
lemma ranI: "m a = Some b ==> b : ran m" 
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   518
by (auto simp add: ran_def)
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   519
(* declare ranI [intro]? *)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   520
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   521
lemma ran_empty[simp]: "ran empty = {}"
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   522
apply (unfold ran_def)
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   523
apply (simp (no_asm))
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   524
done
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   525
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   526
lemma ran_map_upd[simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   527
apply (unfold ran_def, auto)
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   528
apply (subgoal_tac "~ (aa = a) ")
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   529
apply auto
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   530
done
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   531
14100
804be4c4b642 added map_image, restrict_map, some thms
oheimb
parents: 14033
diff changeset
   532
subsection {* @{text "map_le"} *}
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   533
13912
3c0a340be514 fixed document
kleing
parents: 13910
diff changeset
   534
lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g"
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   535
by(simp add:map_le_def)
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   536
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14186
diff changeset
   537
lemma [simp]: "f(x := None) \<subseteq>\<^sub>m f"
26dfcd0ac436 Added new theorems
nipkow
parents: 14186
diff changeset
   538
by(force simp add:map_le_def)
26dfcd0ac436 Added new theorems
nipkow
parents: 14186
diff changeset
   539
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   540
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
   541
by(fastsimp simp add:map_le_def)
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   542
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14186
diff changeset
   543
lemma [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)"
26dfcd0ac436 Added new theorems
nipkow
parents: 14186
diff changeset
   544
by(force simp add:map_le_def)
26dfcd0ac436 Added new theorems
nipkow
parents: 14186
diff changeset
   545
13910
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   546
lemma map_le_upds[simp]:
f9a9ef16466f Added thms
nipkow
parents: 13909
diff changeset
   547
 "!!f g bs. f \<subseteq>\<^sub>m g ==> f(as [|->] bs) \<subseteq>\<^sub>m g(as [|->] bs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   548
apply (induct as, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   549
apply (case_tac bs, auto)
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13937
diff changeset
   550
done
13908
4bdfa9f77254 Map.ML integrated into Map.thy
webertj
parents: 13890
diff changeset
   551
14033
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   552
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
   553
  by (fastsimp simp add: map_le_def dom_def)
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   554
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   555
lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f"
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   556
  by (simp add: map_le_def)
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   557
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14186
diff changeset
   558
lemma map_le_trans[trans]: "\<lbrakk> m1 \<subseteq>\<^sub>m m2; m2 \<subseteq>\<^sub>m m3\<rbrakk> \<Longrightarrow> m1 \<subseteq>\<^sub>m m3"
26dfcd0ac436 Added new theorems
nipkow
parents: 14186
diff changeset
   559
by(force simp add:map_le_def)
14033
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   560
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   561
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
   562
  apply (unfold map_le_def)
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   563
  apply (rule ext)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   564
  apply (case_tac "x \<in> dom f", simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   565
  apply (case_tac "x \<in> dom g", simp, fastsimp)
14033
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   566
done
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   567
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   568
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
   569
  by (fastsimp simp add: map_le_def)
bc723de8ec95 Added a few lemmas about map_le
webertj
parents: 14027
diff changeset
   570
15304
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   571
lemma map_le_iff_map_add_commute: "(f \<subseteq>\<^sub>m f ++ g) = (f++g = g++f)"
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   572
by(fastsimp simp add:map_add_def map_le_def expand_fun_eq split:option.splits)
3514ca74ac54 Added more lemmas
nipkow
parents: 15303
diff changeset
   573
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15251
diff changeset
   574
lemma map_add_le_mapE: "f++g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h"
eedbb8d22ca2 added lemmas
nipkow
parents: 15251
diff changeset
   575
by (fastsimp simp add: map_le_def map_add_def dom_def)
eedbb8d22ca2 added lemmas
nipkow
parents: 15251
diff changeset
   576
eedbb8d22ca2 added lemmas
nipkow
parents: 15251
diff changeset
   577
lemma map_add_le_mapI: "\<lbrakk> f \<subseteq>\<^sub>m h; g \<subseteq>\<^sub>m h; f \<subseteq>\<^sub>m f++g \<rbrakk> \<Longrightarrow> f++g \<subseteq>\<^sub>m h"
eedbb8d22ca2 added lemmas
nipkow
parents: 15251
diff changeset
   578
by (clarsimp simp add: map_le_def map_add_def dom_def split:option.splits)
eedbb8d22ca2 added lemmas
nipkow
parents: 15251
diff changeset
   579
3981
b4f93a8da835 Added the new theory Map.
nipkow
parents:
diff changeset
   580
end