author  nipkow 
Thu, 23 Mar 2006 20:03:53 +0100  
changeset 19323  ec5cd5b1804c 
parent 18576  8d98b7711e47 
child 19378  6cc9ac729eb5 
permissions  rwrr 
3981  1 
(* Title: HOL/Map.thy 
2 
ID: $Id$ 

3 
Author: Tobias Nipkow, based on a theory by David von Oheimb 

13908  4 
Copyright 19972003 TU Muenchen 
3981  5 

6 
The datatype of `maps' (written ~=>); strongly resembles maps in VDM. 

7 
*) 

8 

13914  9 
header {* Maps *} 
10 

15131  11 
theory Map 
15140  12 
imports List 
15131  13 
begin 
3981  14 

13908  15 
types ('a,'b) "~=>" = "'a => 'b option" (infixr 0) 
14100  16 
translations (type) "a ~=> b " <= (type) "a => b option" 
3981  17 

18 
consts 

14100  19 
map_add :: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)" (infixl "++" 100) 
15693  20 
restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)" (infixl "`" 110) 
5300  21 
dom :: "('a ~=> 'b) => 'a set" 
22 
ran :: "('a ~=> 'b) => 'b set" 

23 
map_of :: "('a * 'b)list => 'a ~=> 'b" 

19323  24 
map_upds:: "('a ~=> 'b) => 'a list => 'b list => ('a ~=> 'b)" 
13910  25 
map_le :: "('a ~=> 'b) => ('a ~=> 'b) => bool" (infix "\<subseteq>\<^sub>m" 50) 
26 

17391  27 
constdefs 
28 
map_comp :: "('b ~=> 'c) => ('a ~=> 'b) => ('a ~=> 'c)" (infixl "o'_m" 55) 

29 
"f o_m g == (\<lambda>k. case g k of None \<Rightarrow> None  Some v \<Rightarrow> f v)" 

14739  30 

19323  31 
syntax 
32 
empty :: "'a ~=> 'b" 

33 
translations 

34 
"empty" => "%_. None" 

35 
"empty" <= "%x. None" 

36 

14180  37 
nonterminals 
38 
maplets maplet 

39 

5300  40 
syntax 
14180  41 
"_maplet" :: "['a, 'a] => maplet" ("_ />/ _") 
42 
"_maplets" :: "['a, 'a] => maplet" ("_ /[>]/ _") 

43 
"" :: "maplet => maplets" ("_") 

44 
"_Maplets" :: "[maplet, maplets] => maplets" ("_,/ _") 

45 
"_MapUpd" :: "['a ~=> 'b, maplets] => 'a ~=> 'b" ("_/'(_')" [900,0]900) 

46 
"_Map" :: "maplets => 'a ~=> 'b" ("(1[_])") 

3981  47 

12114
a8e860c86252
eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents:
10137
diff
changeset

48 
syntax (xsymbols) 
14739  49 
"~=>" :: "[type, type] => type" (infixr "\<rightharpoonup>" 0) 
50 

17391  51 
map_comp :: "('b ~=> 'c) => ('a ~=> 'b) => ('a ~=> 'c)" (infixl "\<circ>\<^sub>m" 55) 
14739  52 

14180  53 
"_maplet" :: "['a, 'a] => maplet" ("_ /\<mapsto>/ _") 
54 
"_maplets" :: "['a, 'a] => maplet" ("_ /[\<mapsto>]/ _") 

55 

15693  56 
syntax (latex output) 
15695  57 
restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)" ("_\<restriction>\<^bsub>_\<^esub>" [111,110] 110) 
58 
"requires amssymb!" 

15693  59 

5300  60 
translations 
14180  61 
"_MapUpd m (_Maplets xy ms)" == "_MapUpd (_MapUpd m xy) ms" 
62 
"_MapUpd m (_maplet x y)" == "m(x:=Some y)" 

63 
"_MapUpd m (_maplets x y)" == "map_upds m x y" 

64 
"_Map ms" == "_MapUpd empty ms" 

65 
"_Map (_Maplets ms1 ms2)" <= "_MapUpd (_Map ms1) ms2" 

66 
"_Maplets ms1 (_Maplets ms2 ms3)" <= "_Maplets (_Maplets ms1 ms2) ms3" 

67 

3981  68 
defs 
14100  69 
map_add_def: "m1++m2 == %x. case m2 x of None => m1 x  Some y => Some y" 
15693  70 
restrict_map_def: "m`A == %x. if x : A then m x else None" 
14025  71 

72 
map_upds_def: "m(xs [>] ys) == m ++ map_of (rev(zip xs ys))" 

3981  73 

13908  74 
dom_def: "dom(m) == {a. m a ~= None}" 
14025  75 
ran_def: "ran(m) == {b. EX a. m a = Some b}" 
3981  76 

14376  77 
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  78 

5183  79 
primrec 
80 
"map_of [] = empty" 

5300  81 
"map_of (p#ps) = (map_of ps)(fst p > snd p)" 
82 

19323  83 
(* special purpose constants that should be defined somewhere else and 
84 
whose syntax is a bit odd as well: 

85 

86 
"@chg_map" :: "('a ~=> 'b) => 'a => ('b => 'b) => ('a ~=> 'b)" 

87 
("_/'(_/\<mapsto>\<lambda>_. _')" [900,0,0,0] 900) 

88 
"m(x\<mapsto>\<lambda>y. f)" == "chg_map (\<lambda>y. f) x m" 

89 

90 
map_upd_s::"('a ~=> 'b) => 'a set => 'b => 

91 
('a ~=> 'b)" ("_/'(_{>}_/')" [900,0,0]900) 

92 
map_subst::"('a ~=> 'b) => 'b => 'b => 

93 
('a ~=> 'b)" ("_/'(_~>_/')" [900,0,0]900) 

94 

95 
map_upd_s_def: "m(as{>}b) == %x. if x : as then Some b else m x" 

96 
map_subst_def: "m(a~>b) == %x. if m x = Some a then Some b else m x" 

97 

98 
map_upd_s :: "('a ~=> 'b) => 'a set => 'b => ('a ~=> 'b)" 

99 
("_/'(_/{\<mapsto>}/_')" [900,0,0]900) 

100 
map_subst :: "('a ~=> 'b) => 'b => 'b => 

101 
('a ~=> 'b)" ("_/'(_\<leadsto>_/')" [900,0,0]900) 

102 

103 

104 
subsection {* @{term [source] map_upd_s} *} 

105 

106 
lemma map_upd_s_apply [simp]: 

107 
"(m(as{>}b)) x = (if x : as then Some b else m x)" 

108 
by (simp add: map_upd_s_def) 

109 

110 
lemma map_subst_apply [simp]: 

111 
"(m(a~>b)) x = (if m x = Some a then Some b else m x)" 

112 
by (simp add: map_subst_def) 

113 

114 
*) 

13908  115 

17399
56a3a4affedc
@{term [source] ...} in subsections probably more robust;
wenzelm
parents:
17391
diff
changeset

116 
subsection {* @{term [source] empty} *} 
13908  117 

13910  118 
lemma empty_upd_none[simp]: "empty(x := None) = empty" 
13908  119 
apply (rule ext) 
120 
apply (simp (no_asm)) 

121 
done 

13910  122 

13908  123 

124 
(* FIXME: what is this sum_case nonsense?? *) 

13910  125 
lemma sum_case_empty_empty[simp]: "sum_case empty empty = empty" 
13908  126 
apply (rule ext) 
127 
apply (simp (no_asm) split add: sum.split) 

128 
done 

129 

17399
56a3a4affedc
@{term [source] ...} in subsections probably more robust;
wenzelm
parents:
17391
diff
changeset

130 
subsection {* @{term [source] map_upd} *} 
13908  131 

132 
lemma map_upd_triv: "t k = Some x ==> t(k>x) = t" 

133 
apply (rule ext) 

134 
apply (simp (no_asm_simp)) 

135 
done 

136 

13910  137 
lemma map_upd_nonempty[simp]: "t(k>x) ~= empty" 
13908  138 
apply safe 
14208  139 
apply (drule_tac x = k in fun_cong) 
13908  140 
apply (simp (no_asm_use)) 
141 
done 

142 

14100  143 
lemma map_upd_eqD1: "m(a\<mapsto>x) = n(a\<mapsto>y) \<Longrightarrow> x = y" 
144 
by (drule fun_cong [of _ _ a], auto) 

145 

146 
lemma map_upd_Some_unfold: 

147 
"((m(a>b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)" 

148 
by auto 

149 

15303  150 
lemma image_map_upd[simp]: "x \<notin> A \<Longrightarrow> m(x \<mapsto> y) ` A = m ` A" 
151 
by fastsimp 

152 

13908  153 
lemma finite_range_updI: "finite (range f) ==> finite (range (f(a>b)))" 
154 
apply (unfold image_def) 

155 
apply (simp (no_asm_use) add: full_SetCompr_eq) 

156 
apply (rule finite_subset) 

14208  157 
prefer 2 apply assumption 
13908  158 
apply auto 
159 
done 

160 

161 

162 
(* FIXME: what is this sum_case nonsense?? *) 

17399
56a3a4affedc
@{term [source] ...} in subsections probably more robust;
wenzelm
parents:
17391
diff
changeset

163 
subsection {* @{term [source] sum_case} and @{term [source] empty}/@{term [source] map_upd} *} 
13908  164 

13910  165 
lemma sum_case_map_upd_empty[simp]: 
166 
"sum_case (m(k>y)) empty = (sum_case m empty)(Inl k>y)" 

13908  167 
apply (rule ext) 
168 
apply (simp (no_asm) split add: sum.split) 

169 
done 

170 

13910  171 
lemma sum_case_empty_map_upd[simp]: 
172 
"sum_case empty (m(k>y)) = (sum_case empty m)(Inr k>y)" 

13908  173 
apply (rule ext) 
174 
apply (simp (no_asm) split add: sum.split) 

175 
done 

176 

13910  177 
lemma sum_case_map_upd_map_upd[simp]: 
178 
"sum_case (m1(k1>y1)) (m2(k2>y2)) = (sum_case (m1(k1>y1)) m2)(Inr k2>y2)" 

13908  179 
apply (rule ext) 
180 
apply (simp (no_asm) split add: sum.split) 

181 
done 

182 

183 

17399
56a3a4affedc
@{term [source] ...} in subsections probably more robust;
wenzelm
parents:
17391
diff
changeset

184 
subsection {* @{term [source] map_of} *} 
13908  185 

15304  186 
lemma map_of_eq_None_iff: 
187 
"(map_of xys x = None) = (x \<notin> fst ` (set xys))" 

188 
by (induct xys) simp_all 

189 

190 
lemma map_of_is_SomeD: 

191 
"map_of xys x = Some y \<Longrightarrow> (x,y) \<in> set xys" 

192 
apply(induct xys) 

193 
apply simp 

194 
apply(clarsimp split:if_splits) 

195 
done 

196 

197 
lemma map_of_eq_Some_iff[simp]: 

198 
"distinct(map fst xys) \<Longrightarrow> (map_of xys x = Some y) = ((x,y) \<in> set xys)" 

199 
apply(induct xys) 

200 
apply(simp) 

201 
apply(auto simp:map_of_eq_None_iff[symmetric]) 

202 
done 

203 

204 
lemma Some_eq_map_of_iff[simp]: 

205 
"distinct(map fst xys) \<Longrightarrow> (Some y = map_of xys x) = ((x,y) \<in> set xys)" 

206 
by(auto simp del:map_of_eq_Some_iff simp add:map_of_eq_Some_iff[symmetric]) 

207 

17724  208 
lemma map_of_is_SomeI [simp]: "\<lbrakk> distinct(map fst xys); (x,y) \<in> set xys \<rbrakk> 
15304  209 
\<Longrightarrow> map_of xys x = Some y" 
210 
apply (induct xys) 

211 
apply simp 

212 
apply force 

213 
done 

214 

15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
14739
diff
changeset

215 
lemma map_of_zip_is_None[simp]: 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
14739
diff
changeset

216 
"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

217 
by (induct rule:list_induct2, simp_all) 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
14739
diff
changeset

218 

78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
14739
diff
changeset

219 
lemma finite_range_map_of: "finite (range (map_of xys))" 
15251  220 
apply (induct xys) 
15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
14739
diff
changeset

221 
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

222 
apply (rule finite_subset) 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
14739
diff
changeset

223 
prefer 2 apply assumption 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
14739
diff
changeset

224 
apply auto 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
14739
diff
changeset

225 
done 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
14739
diff
changeset

226 

15369  227 
lemma map_of_SomeD [rule_format]: "map_of xs k = Some y > (k,y):set xs" 
15251  228 
by (induct "xs", auto) 
13908  229 

15369  230 
lemma map_of_mapk_SomeI [rule_format]: 
231 
"inj f ==> map_of t k = Some x > 

232 
map_of (map (split (%k. Pair (f k))) t) (f k) = Some x" 

15251  233 
apply (induct "t") 
13908  234 
apply (auto simp add: inj_eq) 
235 
done 

236 

15369  237 
lemma weak_map_of_SomeI [rule_format]: 
238 
"(k, x) : set l > (\<exists>x. map_of l k = Some x)" 

15251  239 
by (induct "l", auto) 
13908  240 

241 
lemma map_of_filter_in: 

242 
"[ map_of xs k = Some z; P k z ] ==> map_of (filter (split P) xs) k = Some z" 

243 
apply (rule mp) 

14208  244 
prefer 2 apply assumption 
13908  245 
apply (erule thin_rl) 
15251  246 
apply (induct "xs", auto) 
13908  247 
done 
248 

249 
lemma map_of_map: "map_of (map (%(a,b). (a,f b)) xs) x = option_map f (map_of xs x)" 

15251  250 
by (induct "xs", auto) 
13908  251 

252 

17399
56a3a4affedc
@{term [source] ...} in subsections probably more robust;
wenzelm
parents:
17391
diff
changeset

253 
subsection {* @{term [source] option_map} related *} 
13908  254 

13910  255 
lemma option_map_o_empty[simp]: "option_map f o empty = empty" 
13908  256 
apply (rule ext) 
257 
apply (simp (no_asm)) 

258 
done 

259 

13910  260 
lemma option_map_o_map_upd[simp]: 
261 
"option_map f o m(a>b) = (option_map f o m)(a>f b)" 

13908  262 
apply (rule ext) 
263 
apply (simp (no_asm)) 

264 
done 

265 

17399
56a3a4affedc
@{term [source] ...} in subsections probably more robust;
wenzelm
parents:
17391
diff
changeset

266 
subsection {* @{term [source] map_comp} related *} 
17391  267 

268 
lemma map_comp_empty [simp]: 

269 
"m \<circ>\<^sub>m empty = empty" 

270 
"empty \<circ>\<^sub>m m = empty" 

271 
by (auto simp add: map_comp_def intro: ext split: option.splits) 

272 

273 
lemma map_comp_simps [simp]: 

274 
"m2 k = None \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = None" 

275 
"m2 k = Some k' \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = m1 k'" 

276 
by (auto simp add: map_comp_def) 

277 

278 
lemma map_comp_Some_iff: 

279 
"((m1 \<circ>\<^sub>m m2) k = Some v) = (\<exists>k'. m2 k = Some k' \<and> m1 k' = Some v)" 

280 
by (auto simp add: map_comp_def split: option.splits) 

281 

282 
lemma map_comp_None_iff: 

283 
"((m1 \<circ>\<^sub>m m2) k = None) = (m2 k = None \<or> (\<exists>k'. m2 k = Some k' \<and> m1 k' = None)) " 

284 
by (auto simp add: map_comp_def split: option.splits) 

13908  285 

14100  286 
subsection {* @{text "++"} *} 
13908  287 

14025  288 
lemma map_add_empty[simp]: "m ++ empty = m" 
289 
apply (unfold map_add_def) 

13908  290 
apply (simp (no_asm)) 
291 
done 

292 

14025  293 
lemma empty_map_add[simp]: "empty ++ m = m" 
294 
apply (unfold map_add_def) 

13908  295 
apply (rule ext) 
296 
apply (simp split add: option.split) 

297 
done 

298 

14025  299 
lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3" 
300 
apply(rule ext) 

301 
apply(simp add: map_add_def split:option.split) 

302 
done 

303 

304 
lemma map_add_Some_iff: 

13908  305 
"((m ++ n) k = Some x) = (n k = Some x  n k = None & m k = Some x)" 
14025  306 
apply (unfold map_add_def) 
13908  307 
apply (simp (no_asm) split add: option.split) 
308 
done 

309 

14025  310 
lemmas map_add_SomeD = map_add_Some_iff [THEN iffD1, standard] 
311 
declare map_add_SomeD [dest!] 

13908  312 

14025  313 
lemma map_add_find_right[simp]: "!!xx. n k = Some xx ==> (m ++ n) k = Some xx" 
14208  314 
by (subst map_add_Some_iff, fast) 
13908  315 

14025  316 
lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)" 
317 
apply (unfold map_add_def) 

13908  318 
apply (simp (no_asm) split add: option.split) 
319 
done 

320 

14025  321 
lemma map_add_upd[simp]: "f ++ g(x>y) = (f ++ g)(x>y)" 
322 
apply (unfold map_add_def) 

14208  323 
apply (rule ext, auto) 
13908  324 
done 
325 

14186  326 
lemma map_add_upds[simp]: "m1 ++ (m2(xs[\<mapsto>]ys)) = (m1++m2)(xs[\<mapsto>]ys)" 
327 
by(simp add:map_upds_def) 

328 

14025  329 
lemma map_of_append[simp]: "map_of (xs@ys) = map_of ys ++ map_of xs" 
330 
apply (unfold map_add_def) 

15251  331 
apply (induct "xs") 
13908  332 
apply (simp (no_asm)) 
333 
apply (rule ext) 

334 
apply (simp (no_asm_simp) split add: option.split) 

335 
done 

336 

337 
declare fun_upd_apply [simp del] 

14025  338 
lemma finite_range_map_of_map_add: 
339 
"finite (range f) ==> finite (range (f ++ map_of l))" 

15251  340 
apply (induct "l", auto) 
13908  341 
apply (erule finite_range_updI) 
342 
done 

343 
declare fun_upd_apply [simp] 

344 

15304  345 
lemma inj_on_map_add_dom[iff]: 
346 
"inj_on (m ++ m') (dom m') = inj_on m' (dom m')" 

347 
by(fastsimp simp add:map_add_def dom_def inj_on_def split:option.splits) 

348 

17399
56a3a4affedc
@{term [source] ...} in subsections probably more robust;
wenzelm
parents:
17391
diff
changeset

349 
subsection {* @{term [source] restrict_map} *} 
14100  350 

15693  351 
lemma restrict_map_to_empty[simp]: "m`{} = empty" 
14186  352 
by(simp add: restrict_map_def) 
353 

15693  354 
lemma restrict_map_empty[simp]: "empty`D = empty" 
14186  355 
by(simp add: restrict_map_def) 
356 

15693  357 
lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m`A) x = m x" 
14100  358 
by (auto simp: restrict_map_def) 
359 

15693  360 
lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m`A) x = None" 
14100  361 
by (auto simp: restrict_map_def) 
362 

15693  363 
lemma ran_restrictD: "y \<in> ran (m`A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y" 
14100  364 
by (auto simp: restrict_map_def ran_def split: split_if_asm) 
365 

15693  366 
lemma dom_restrict [simp]: "dom (m`A) = dom m \<inter> A" 
14100  367 
by (auto simp: restrict_map_def dom_def split: split_if_asm) 
368 

15693  369 
lemma restrict_upd_same [simp]: "m(x\<mapsto>y)`({x}) = m`({x})" 
14100  370 
by (rule ext, auto simp: restrict_map_def) 
371 

15693  372 
lemma restrict_restrict [simp]: "m`A`B = m`(A\<inter>B)" 
14100  373 
by (rule ext, auto simp: restrict_map_def) 
374 

14186  375 
lemma restrict_fun_upd[simp]: 
15693  376 
"m(x := y)`D = (if x \<in> D then (m`(D{x}))(x := y) else m`D)" 
14186  377 
by(simp add: restrict_map_def expand_fun_eq) 
378 

379 
lemma fun_upd_None_restrict[simp]: 

15693  380 
"(m`D)(x := None) = (if x:D then m`(D  {x}) else m`D)" 
14186  381 
by(simp add: restrict_map_def expand_fun_eq) 
382 

383 
lemma fun_upd_restrict: 

15693  384 
"(m`D)(x := y) = (m`(D{x}))(x := y)" 
14186  385 
by(simp add: restrict_map_def expand_fun_eq) 
386 

387 
lemma fun_upd_restrict_conv[simp]: 

15693  388 
"x \<in> D \<Longrightarrow> (m`D)(x := y) = (m`(D{x}))(x := y)" 
14186  389 
by(simp add: restrict_map_def expand_fun_eq) 
390 

14100  391 

17399
56a3a4affedc
@{term [source] ...} in subsections probably more robust;
wenzelm
parents:
17391
diff
changeset

392 
subsection {* @{term [source] map_upds} *} 
14025  393 

394 
lemma map_upds_Nil1[simp]: "m([] [>] bs) = m" 

395 
by(simp add:map_upds_def) 

396 

397 
lemma map_upds_Nil2[simp]: "m(as [>] []) = m" 

398 
by(simp add:map_upds_def) 

399 

400 
lemma map_upds_Cons[simp]: "m(a#as [>] b#bs) = (m(a>b))(as[>]bs)" 

401 
by(simp add:map_upds_def) 

402 

14187  403 
lemma map_upds_append1[simp]: "\<And>ys m. size xs < size ys \<Longrightarrow> 
404 
m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)" 

405 
apply(induct xs) 

406 
apply(clarsimp simp add:neq_Nil_conv) 

14208  407 
apply (case_tac ys, simp, simp) 
14187  408 
done 
409 

410 
lemma map_upds_list_update2_drop[simp]: 

411 
"\<And>m ys i. \<lbrakk>size xs \<le> i; i < size ys\<rbrakk> 

412 
\<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)" 

14208  413 
apply (induct xs, simp) 
414 
apply (case_tac ys, simp) 

14187  415 
apply(simp split:nat.split) 
416 
done 

14025  417 

418 
lemma map_upd_upds_conv_if: "!!x y ys f. 

419 
(f(x>y))(xs [>] ys) = 

420 
(if x : set(take (length ys) xs) then f(xs [>] ys) 

421 
else (f(xs [>] ys))(x>y))" 

14208  422 
apply (induct xs, simp) 
14025  423 
apply(case_tac ys) 
424 
apply(auto split:split_if simp:fun_upd_twist) 

425 
done 

426 

427 
lemma map_upds_twist [simp]: 

428 
"a ~: set as ==> m(a>b)(as[>]bs) = m(as[>]bs)(a>b)" 

429 
apply(insert set_take_subset) 

430 
apply (fastsimp simp add: map_upd_upds_conv_if) 

431 
done 

432 

433 
lemma map_upds_apply_nontin[simp]: 

434 
"!!ys. x ~: set xs ==> (f(xs[>]ys)) x = f x" 

14208  435 
apply (induct xs, simp) 
14025  436 
apply(case_tac ys) 
437 
apply(auto simp: map_upd_upds_conv_if) 

438 
done 

439 

14300  440 
lemma fun_upds_append_drop[simp]: 
441 
"!!m ys. size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)" 

442 
apply(induct xs) 

443 
apply (simp) 

444 
apply(case_tac ys) 

445 
apply simp_all 

446 
done 

447 

448 
lemma fun_upds_append2_drop[simp]: 

449 
"!!m ys. size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)" 

450 
apply(induct xs) 

451 
apply (simp) 

452 
apply(case_tac ys) 

453 
apply simp_all 

454 
done 

455 

456 

14186  457 
lemma restrict_map_upds[simp]: "!!m ys. 
458 
\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk> 

15693  459 
\<Longrightarrow> m(xs [\<mapsto>] ys)`D = (m`(D  set xs))(xs [\<mapsto>] ys)" 
14208  460 
apply (induct xs, simp) 
461 
apply (case_tac ys, simp) 

14186  462 
apply(simp add:Diff_insert[symmetric] insert_absorb) 
463 
apply(simp add: map_upd_upds_conv_if) 

464 
done 

465 

466 

17399
56a3a4affedc
@{term [source] ...} in subsections probably more robust;
wenzelm
parents:
17391
diff
changeset

467 
subsection {* @{term [source] dom} *} 
13908  468 

469 
lemma domI: "m a = Some b ==> a : dom m" 

14208  470 
by (unfold dom_def, auto) 
14100  471 
(* declare domI [intro]? *) 
13908  472 

15369  473 
lemma domD: "a : dom m ==> \<exists>b. m a = Some b" 
18447  474 
apply (case_tac "m a") 
475 
apply (auto simp add: dom_def) 

476 
done 

13908  477 

13910  478 
lemma domIff[iff]: "(a : dom m) = (m a ~= None)" 
14208  479 
by (unfold dom_def, auto) 
13908  480 
declare domIff [simp del] 
481 

13910  482 
lemma dom_empty[simp]: "dom empty = {}" 
13908  483 
apply (unfold dom_def) 
484 
apply (simp (no_asm)) 

485 
done 

486 

13910  487 
lemma dom_fun_upd[simp]: 
488 
"dom(f(x := y)) = (if y=None then dom f  {x} else insert x (dom f))" 

489 
by (simp add:dom_def) blast 

13908  490 

13937  491 
lemma dom_map_of: "dom(map_of xys) = {x. \<exists>y. (x,y) : set xys}" 
492 
apply(induct xys) 

493 
apply(auto simp del:fun_upd_apply) 

494 
done 

495 

15304  496 
lemma dom_map_of_conv_image_fst: 
497 
"dom(map_of xys) = fst ` (set xys)" 

498 
by(force simp: dom_map_of) 

499 

15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
14739
diff
changeset

500 
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

501 
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

502 
by(induct rule: list_induct2, simp_all) 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
14739
diff
changeset

503 

13908  504 
lemma finite_dom_map_of: "finite (dom (map_of l))" 
505 
apply (unfold dom_def) 

15251  506 
apply (induct "l") 
13908  507 
apply (auto simp add: insert_Collect [symmetric]) 
508 
done 

509 

14025  510 
lemma dom_map_upds[simp]: 
511 
"!!m ys. dom(m(xs[>]ys)) = set(take (length ys) xs) Un dom m" 

14208  512 
apply (induct xs, simp) 
513 
apply (case_tac ys, auto) 

14025  514 
done 
13910  515 

14025  516 
lemma dom_map_add[simp]: "dom(m++n) = dom n Un dom m" 
14208  517 
by (unfold dom_def, auto) 
13910  518 

15691  519 
lemma dom_override_on[simp]: 
520 
"dom(override_on f g A) = 

521 
(dom f  {a. a : A  dom g}) Un {a. a : A Int dom g}" 

522 
by(auto simp add: dom_def override_on_def) 

13908  523 

14027  524 
lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1" 
525 
apply(rule ext) 

18576  526 
apply(force simp: map_add_def dom_def split:option.split) 
14027  527 
done 
528 

17399
56a3a4affedc
@{term [source] ...} in subsections probably more robust;
wenzelm
parents:
17391
diff
changeset

529 
subsection {* @{term [source] ran} *} 
14100  530 

531 
lemma ranI: "m a = Some b ==> b : ran m" 

532 
by (auto simp add: ran_def) 

533 
(* declare ranI [intro]? *) 

13908  534 

13910  535 
lemma ran_empty[simp]: "ran empty = {}" 
13908  536 
apply (unfold ran_def) 
537 
apply (simp (no_asm)) 

538 
done 

539 

13910  540 
lemma ran_map_upd[simp]: "m a = None ==> ran(m(a>b)) = insert b (ran m)" 
14208  541 
apply (unfold ran_def, auto) 
13908  542 
apply (subgoal_tac "~ (aa = a) ") 
543 
apply auto 

544 
done 

13910  545 

14100  546 
subsection {* @{text "map_le"} *} 
13910  547 

13912  548 
lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g" 
13910  549 
by(simp add:map_le_def) 
550 

17724  551 
lemma upd_None_map_le [simp]: "f(x := None) \<subseteq>\<^sub>m f" 
14187  552 
by(force simp add:map_le_def) 
553 

13910  554 
lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)" 
555 
by(fastsimp simp add:map_le_def) 

556 

17724  557 
lemma map_le_imp_upd_le [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)" 
14187  558 
by(force simp add:map_le_def) 
559 

13910  560 
lemma map_le_upds[simp]: 
561 
"!!f g bs. f \<subseteq>\<^sub>m g ==> f(as [>] bs) \<subseteq>\<^sub>m g(as [>] bs)" 

14208  562 
apply (induct as, simp) 
563 
apply (case_tac bs, auto) 

14025  564 
done 
13908  565 

14033  566 
lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)" 
567 
by (fastsimp simp add: map_le_def dom_def) 

568 

569 
lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f" 

570 
by (simp add: map_le_def) 

571 

14187  572 
lemma map_le_trans[trans]: "\<lbrakk> m1 \<subseteq>\<^sub>m m2; m2 \<subseteq>\<^sub>m m3\<rbrakk> \<Longrightarrow> m1 \<subseteq>\<^sub>m m3" 
18447  573 
by (auto simp add: map_le_def dom_def) 
14033  574 

575 
lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g" 

576 
apply (unfold map_le_def) 

577 
apply (rule ext) 

14208  578 
apply (case_tac "x \<in> dom f", simp) 
579 
apply (case_tac "x \<in> dom g", simp, fastsimp) 

14033  580 
done 
581 

582 
lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)" 

18576  583 
by (fastsimp simp add: map_le_def) 
14033  584 

15304  585 
lemma map_le_iff_map_add_commute: "(f \<subseteq>\<^sub>m f ++ g) = (f++g = g++f)" 
586 
by(fastsimp simp add:map_add_def map_le_def expand_fun_eq split:option.splits) 

587 

15303  588 
lemma map_add_le_mapE: "f++g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h" 
18576  589 
by (fastsimp simp add: map_le_def map_add_def dom_def) 
15303  590 

591 
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" 

592 
by (clarsimp simp add: map_le_def map_add_def dom_def split:option.splits) 

593 

3981  594 
end 