author  haftmann 
Sun, 06 May 2007 21:50:17 +0200  
changeset 22845  5f9138bcb3d7 
parent 22744  5cbe966d67a2 
child 22886  cdff6ef76009 
permissions  rwrr 
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(* Title: HOL/Fun.thy 
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ID: $Id$ 
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Author: Tobias Nipkow, Cambridge University Computer Laboratory 
923  4 
Copyright 1994 University of Cambridge 
18154  5 
*) 
923  6 

18154  7 
header {* Notions about functions *} 
923  8 

15510  9 
theory Fun 
21870  10 
imports Set Code_Generator 
15131  11 
begin 
2912  12 

13585  13 
constdefs 
14 
fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" 

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[code func]: "fun_upd f a b == % x. if x=a then b else f x" 
6171  16 

9141  17 
nonterminals 
18 
updbinds updbind 

5305  19 
syntax 
13585  20 
"_updbind" :: "['a, 'a] => updbind" ("(2_ :=/ _)") 
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"" :: "updbind => updbinds" ("_") 

22 
"_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _") 

23 
"_Update" :: "['a, updbinds] => 'a" ("_/'((_)')" [1000,0] 900) 

5305  24 

25 
translations 

26 
"_Update f (_updbinds b bs)" == "_Update (_Update f b) bs" 

27 
"f(x:=y)" == "fun_upd f x y" 

2912  28 

9340  29 
(* Hint: to define the sum of two functions (or maps), use sum_case. 
30 
A nice infix syntax could be defined (in Datatype.thy or below) by 

31 
consts 

32 
fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80) 

33 
translations 

13585  34 
"fun_sum" == sum_case 
9340  35 
*) 
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definition 
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override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b" 
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where 
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"override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)" 
6171  41 

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definition 
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id :: "'a \<Rightarrow> 'a" 
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where 
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"id = (\<lambda>x. x)" 
13910  46 

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definition 
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comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o" 55) 
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where 
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"f o g = (\<lambda>x. f (g x))" 
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21210  52 
notation (xsymbols) 
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tuned concrete syntax  abbreviation/const_syntax;
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comp (infixl "\<circ>" 55) 
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tuned concrete syntax  abbreviation/const_syntax;
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notation (HTML output) 
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tuned concrete syntax  abbreviation/const_syntax;
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comp (infixl "\<circ>" 55) 
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13585  58 
text{*compatibility*} 
59 
lemmas o_def = comp_def 

2912  60 

13585  61 
constdefs 
62 
inj_on :: "['a => 'b, 'a set] => bool" (*injective*) 

19363  63 
"inj_on f A == ! x:A. ! y:A. f(x)=f(y) > x=y" 
6171  64 

13585  65 
text{*A common special case: functions injective over the entire domain type.*} 
19323  66 

19363  67 
abbreviation 
68 
"inj f == inj_on f UNIV" 

5852  69 

7374  70 
constdefs 
13585  71 
surj :: "('a => 'b) => bool" (*surjective*) 
19363  72 
"surj f == ! y. ? x. y=f(x)" 
12258  73 

13585  74 
bij :: "('a => 'b) => bool" (*bijective*) 
19363  75 
"bij f == inj f & surj f" 
12258  76 

7374  77 

13585  78 

79 
text{*As a simplification rule, it replaces all function equalities by 

80 
firstorder equalities.*} 

21327  81 
lemma expand_fun_eq: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)" 
13585  82 
apply (rule iffI) 
83 
apply (simp (no_asm_simp)) 

21327  84 
apply (rule ext) 
85 
apply (simp (no_asm_simp)) 

13585  86 
done 
87 

88 
lemma apply_inverse: 

89 
"[ f(x)=u; !!x. P(x) ==> g(f(x)) = x; P(x) ] ==> x=g(u)" 

90 
by auto 

91 

92 

93 
text{*The Identity Function: @{term id}*} 

94 
lemma id_apply [simp]: "id x = x" 

95 
by (simp add: id_def) 

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lemma inj_on_id[simp]: "inj_on id A" 
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by (simp add: inj_on_def) 
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lemma inj_on_id2[simp]: "inj_on (%x. x) A" 
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by (simp add: inj_on_def) 
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lemma surj_id[simp]: "surj id" 
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by (simp add: surj_def) 
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lemma bij_id[simp]: "bij id" 
15510  107 
by (simp add: bij_def inj_on_id surj_id) 
108 

109 

13585  110 

111 
subsection{*The Composition Operator: @{term "f \<circ> g"}*} 

112 

113 
lemma o_apply [simp]: "(f o g) x = f (g x)" 

114 
by (simp add: comp_def) 

115 

116 
lemma o_assoc: "f o (g o h) = f o g o h" 

117 
by (simp add: comp_def) 

118 

119 
lemma id_o [simp]: "id o g = g" 

120 
by (simp add: comp_def) 

121 

122 
lemma o_id [simp]: "f o id = f" 

123 
by (simp add: comp_def) 

124 

125 
lemma image_compose: "(f o g) ` r = f`(g`r)" 

126 
by (simp add: comp_def, blast) 

127 

128 
lemma image_eq_UN: "f`A = (UN x:A. {f x})" 

129 
by blast 

130 

131 
lemma UN_o: "UNION A (g o f) = UNION (f`A) g" 

132 
by (unfold comp_def, blast) 

133 

134 

135 
subsection{*The Injectivity Predicate, @{term inj}*} 

136 

137 
text{*NB: @{term inj} now just translates to @{term inj_on}*} 

138 

139 

140 
text{*For Proofs in @{text "Tools/datatype_rep_proofs"}*} 

141 
lemma datatype_injI: 

142 
"(!! x. ALL y. f(x) = f(y) > x=y) ==> inj(f)" 

143 
by (simp add: inj_on_def) 

144 

13637  145 
theorem range_ex1_eq: "inj f \<Longrightarrow> b : range f = (EX! x. b = f x)" 
146 
by (unfold inj_on_def, blast) 

147 

13585  148 
lemma injD: "[ inj(f); f(x) = f(y) ] ==> x=y" 
149 
by (simp add: inj_on_def) 

150 

151 
(*Useful with the simplifier*) 

152 
lemma inj_eq: "inj(f) ==> (f(x) = f(y)) = (x=y)" 

153 
by (force simp add: inj_on_def) 

154 

155 

156 
subsection{*The Predicate @{term inj_on}: Injectivity On A Restricted Domain*} 

157 

158 
lemma inj_onI: 

159 
"(!! x y. [ x:A; y:A; f(x) = f(y) ] ==> x=y) ==> inj_on f A" 

160 
by (simp add: inj_on_def) 

161 

162 
lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A" 

163 
by (auto dest: arg_cong [of concl: g] simp add: inj_on_def) 

164 

165 
lemma inj_onD: "[ inj_on f A; f(x)=f(y); x:A; y:A ] ==> x=y" 

166 
by (unfold inj_on_def, blast) 

167 

168 
lemma inj_on_iff: "[ inj_on f A; x:A; y:A ] ==> (f(x)=f(y)) = (x=y)" 

169 
by (blast dest!: inj_onD) 

170 

171 
lemma comp_inj_on: 

172 
"[ inj_on f A; inj_on g (f`A) ] ==> inj_on (g o f) A" 

173 
by (simp add: comp_def inj_on_def) 

174 

15303  175 
lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)" 
176 
apply(simp add:inj_on_def image_def) 

177 
apply blast 

178 
done 

179 

15439  180 
lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y); 
181 
inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A" 

182 
apply(unfold inj_on_def) 

183 
apply blast 

184 
done 

185 

13585  186 
lemma inj_on_contraD: "[ inj_on f A; ~x=y; x:A; y:A ] ==> ~ f(x)=f(y)" 
187 
by (unfold inj_on_def, blast) 

12258  188 

13585  189 
lemma inj_singleton: "inj (%s. {s})" 
190 
by (simp add: inj_on_def) 

191 

15111  192 
lemma inj_on_empty[iff]: "inj_on f {}" 
193 
by(simp add: inj_on_def) 

194 

15303  195 
lemma subset_inj_on: "[ inj_on f B; A <= B ] ==> inj_on f A" 
13585  196 
by (unfold inj_on_def, blast) 
197 

15111  198 
lemma inj_on_Un: 
199 
"inj_on f (A Un B) = 

200 
(inj_on f A & inj_on f B & f`(AB) Int f`(BA) = {})" 

201 
apply(unfold inj_on_def) 

202 
apply (blast intro:sym) 

203 
done 

204 

205 
lemma inj_on_insert[iff]: 

206 
"inj_on f (insert a A) = (inj_on f A & f a ~: f`(A{a}))" 

207 
apply(unfold inj_on_def) 

208 
apply (blast intro:sym) 

209 
done 

210 

211 
lemma inj_on_diff: "inj_on f A ==> inj_on f (AB)" 

212 
apply(unfold inj_on_def) 

213 
apply (blast) 

214 
done 

215 

13585  216 

217 
subsection{*The Predicate @{term surj}: Surjectivity*} 

218 

219 
lemma surjI: "(!! x. g(f x) = x) ==> surj g" 

220 
apply (simp add: surj_def) 

221 
apply (blast intro: sym) 

222 
done 

223 

224 
lemma surj_range: "surj f ==> range f = UNIV" 

225 
by (auto simp add: surj_def) 

226 

227 
lemma surjD: "surj f ==> EX x. y = f x" 

228 
by (simp add: surj_def) 

229 

230 
lemma surjE: "surj f ==> (!!x. y = f x ==> C) ==> C" 

231 
by (simp add: surj_def, blast) 

232 

233 
lemma comp_surj: "[ surj f; surj g ] ==> surj (g o f)" 

234 
apply (simp add: comp_def surj_def, clarify) 

235 
apply (drule_tac x = y in spec, clarify) 

236 
apply (drule_tac x = x in spec, blast) 

237 
done 

238 

239 

240 

241 
subsection{*The Predicate @{term bij}: Bijectivity*} 

242 

243 
lemma bijI: "[ inj f; surj f ] ==> bij f" 

244 
by (simp add: bij_def) 

245 

246 
lemma bij_is_inj: "bij f ==> inj f" 

247 
by (simp add: bij_def) 

248 

249 
lemma bij_is_surj: "bij f ==> surj f" 

250 
by (simp add: bij_def) 

251 

252 

253 
subsection{*Facts About the Identity Function*} 

5852  254 

13585  255 
text{*We seem to need both the @{term id} forms and the @{term "\<lambda>x. x"} 
256 
forms. The latter can arise by rewriting, while @{term id} may be used 

257 
explicitly.*} 

258 

259 
lemma image_ident [simp]: "(%x. x) ` Y = Y" 

260 
by blast 

261 

262 
lemma image_id [simp]: "id ` Y = Y" 

263 
by (simp add: id_def) 

264 

265 
lemma vimage_ident [simp]: "(%x. x) ` Y = Y" 

266 
by blast 

267 

268 
lemma vimage_id [simp]: "id ` A = A" 

269 
by (simp add: id_def) 

270 

271 
lemma vimage_image_eq: "f ` (f ` A) = {y. EX x:A. f x = f y}" 

272 
by (blast intro: sym) 

273 

274 
lemma image_vimage_subset: "f ` (f ` A) <= A" 

275 
by blast 

276 

277 
lemma image_vimage_eq [simp]: "f ` (f ` A) = A Int range f" 

278 
by blast 

279 

280 
lemma surj_image_vimage_eq: "surj f ==> f ` (f ` A) = A" 

281 
by (simp add: surj_range) 

282 

283 
lemma inj_vimage_image_eq: "inj f ==> f ` (f ` A) = A" 

284 
by (simp add: inj_on_def, blast) 

285 

286 
lemma vimage_subsetD: "surj f ==> f ` B <= A ==> B <= f ` A" 

287 
apply (unfold surj_def) 

288 
apply (blast intro: sym) 

289 
done 

290 

291 
lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f ` B <= A" 

292 
by (unfold inj_on_def, blast) 

293 

294 
lemma vimage_subset_eq: "bij f ==> (f ` B <= A) = (B <= f ` A)" 

295 
apply (unfold bij_def) 

296 
apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD) 

297 
done 

298 

299 
lemma image_Int_subset: "f`(A Int B) <= f`A Int f`B" 

300 
by blast 

301 

302 
lemma image_diff_subset: "f`A  f`B <= f`(A  B)" 

303 
by blast 

5852  304 

13585  305 
lemma inj_on_image_Int: 
306 
"[ inj_on f C; A<=C; B<=C ] ==> f`(A Int B) = f`A Int f`B" 

307 
apply (simp add: inj_on_def, blast) 

308 
done 

309 

310 
lemma inj_on_image_set_diff: 

311 
"[ inj_on f C; A<=C; B<=C ] ==> f`(AB) = f`A  f`B" 

312 
apply (simp add: inj_on_def, blast) 

313 
done 

314 

315 
lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B" 

316 
by (simp add: inj_on_def, blast) 

317 

318 
lemma image_set_diff: "inj f ==> f`(AB) = f`A  f`B" 

319 
by (simp add: inj_on_def, blast) 

320 

321 
lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)" 

322 
by (blast dest: injD) 

323 

324 
lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)" 

325 
by (simp add: inj_on_def, blast) 

326 

327 
lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)" 

328 
by (blast dest: injD) 

329 

330 
lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (B x)))" 

331 
by blast 

332 

333 
(*injectivity's required. Lefttoright inclusion holds even if A is empty*) 

334 
lemma image_INT: 

335 
"[ inj_on f C; ALL x:A. B x <= C; j:A ] 

336 
==> f ` (INTER A B) = (INT x:A. f ` B x)" 

337 
apply (simp add: inj_on_def, blast) 

338 
done 

339 

340 
(*Compare with image_INT: no use of inj_on, and if f is surjective then 

341 
it doesn't matter whether A is empty*) 

342 
lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)" 

343 
apply (simp add: bij_def) 

344 
apply (simp add: inj_on_def surj_def, blast) 

345 
done 

346 

347 
lemma surj_Compl_image_subset: "surj f ==> (f`A) <= f`(A)" 

348 
by (auto simp add: surj_def) 

349 

350 
lemma inj_image_Compl_subset: "inj f ==> f`(A) <= (f`A)" 

351 
by (auto simp add: inj_on_def) 

5852  352 

13585  353 
lemma bij_image_Compl_eq: "bij f ==> f`(A) = (f`A)" 
354 
apply (simp add: bij_def) 

355 
apply (rule equalityI) 

356 
apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset) 

357 
done 

358 

359 

360 
subsection{*Function Updating*} 

361 

362 
lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)" 

363 
apply (simp add: fun_upd_def, safe) 

364 
apply (erule subst) 

365 
apply (rule_tac [2] ext, auto) 

366 
done 

367 

368 
(* f x = y ==> f(x:=y) = f *) 

369 
lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard] 

370 

371 
(* f(x := f x) = f *) 

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lemmas fun_upd_triv = refl [THEN fun_upd_idem] 
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declare fun_upd_triv [iff] 
13585  374 

375 
lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)" 

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by (simp add: fun_upd_def) 
13585  377 

378 
(* fun_upd_apply supersedes these two, but they are useful 

379 
if fun_upd_apply is intentionally removed from the simpset *) 

380 
lemma fun_upd_same: "(f(x:=y)) x = y" 

381 
by simp 

382 

383 
lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z" 

384 
by simp 

385 

386 
lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)" 

387 
by (simp add: expand_fun_eq) 

388 

389 
lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)" 

390 
by (rule ext, auto) 

391 

15303  392 
lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A" 
393 
by(fastsimp simp:inj_on_def image_def) 

394 

15510  395 
lemma fun_upd_image: 
396 
"f(x:=y) ` A = (if x \<in> A then insert y (f ` (A{x})) else f ` A)" 

397 
by auto 

398 

15691  399 
subsection{* @{text override_on} *} 
13910  400 

15691  401 
lemma override_on_emptyset[simp]: "override_on f g {} = f" 
402 
by(simp add:override_on_def) 

13910  403 

15691  404 
lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a" 
405 
by(simp add:override_on_def) 

13910  406 

15691  407 
lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a" 
408 
by(simp add:override_on_def) 

13910  409 

15510  410 
subsection{* swap *} 
411 

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definition 
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swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)" 
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"swap a b f = f (a := f b, b:= f a)" 
15510  416 

417 
lemma swap_self: "swap a a f = f" 

15691  418 
by (simp add: swap_def) 
15510  419 

420 
lemma swap_commute: "swap a b f = swap b a f" 

421 
by (rule ext, simp add: fun_upd_def swap_def) 

422 

423 
lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f" 

424 
by (rule ext, simp add: fun_upd_def swap_def) 

425 

426 
lemma inj_on_imp_inj_on_swap: 

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"[inj_on f A; a \<in> A; b \<in> A] ==> inj_on (swap a b f) A" 
15510  428 
by (simp add: inj_on_def swap_def, blast) 
429 

430 
lemma inj_on_swap_iff [simp]: 

431 
assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A = inj_on f A" 

432 
proof 

433 
assume "inj_on (swap a b f) A" 

434 
with A have "inj_on (swap a b (swap a b f)) A" 

17589  435 
by (iprover intro: inj_on_imp_inj_on_swap) 
15510  436 
thus "inj_on f A" by simp 
437 
next 

438 
assume "inj_on f A" 

17589  439 
with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap) 
15510  440 
qed 
441 

442 
lemma surj_imp_surj_swap: "surj f ==> surj (swap a b f)" 

443 
apply (simp add: surj_def swap_def, clarify) 

444 
apply (rule_tac P = "y = f b" in case_split_thm, blast) 

445 
apply (rule_tac P = "y = f a" in case_split_thm, auto) 

446 
{*We don't yet have @{text case_tac}*} 

447 
done 

448 

449 
lemma surj_swap_iff [simp]: "surj (swap a b f) = surj f" 

450 
proof 

451 
assume "surj (swap a b f)" 

452 
hence "surj (swap a b (swap a b f))" by (rule surj_imp_surj_swap) 

453 
thus "surj f" by simp 

454 
next 

455 
assume "surj f" 

456 
thus "surj (swap a b f)" by (rule surj_imp_surj_swap) 

457 
qed 

458 

459 
lemma bij_swap_iff: "bij (swap a b f) = bij f" 

460 
by (simp add: bij_def) 

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22453  463 
subsection {* Order and lattice on functions *} 
464 

465 
instance "fun" :: (type, ord) ord 

466 
le_fun_def: "f \<le> g \<equiv> \<forall>x. f x \<le> g x" 

467 
less_fun_def: "f < g \<equiv> f \<le> g \<and> f \<noteq> g" .. 

468 

22845  469 
lemmas [code func del] = le_fun_def less_fun_def 
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instance "fun" :: (type, order) order 
22845  472 
by default 
473 
(auto simp add: le_fun_def less_fun_def expand_fun_eq 

474 
intro: order_trans order_antisym) 

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lemma le_funI: "(\<And>x. f x \<le> g x) \<Longrightarrow> f \<le> g" 
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unfolding le_fun_def by simp 
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lemma le_funE: "f \<le> g \<Longrightarrow> (f x \<le> g x \<Longrightarrow> P) \<Longrightarrow> P" 
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unfolding le_fun_def by simp 
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lemma le_funD: "f \<le> g \<Longrightarrow> f x \<le> g x" 
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unfolding le_fun_def by simp 
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22453  485 
text {* 
486 
Handy introduction and elimination rules for @{text "\<le>"} 

487 
on unary and binary predicates 

488 
*} 

489 

490 
lemma predicate1I [Pure.intro!, intro!]: 

491 
assumes PQ: "\<And>x. P x \<Longrightarrow> Q x" 

492 
shows "P \<le> Q" 

493 
apply (rule le_funI) 

494 
apply (rule le_boolI) 

495 
apply (rule PQ) 

496 
apply assumption 

497 
done 

498 

499 
lemma predicate1D [Pure.dest, dest]: "P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x" 

500 
apply (erule le_funE) 

501 
apply (erule le_boolE) 

502 
apply assumption+ 

503 
done 

504 

505 
lemma predicate2I [Pure.intro!, intro!]: 

506 
assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y" 

507 
shows "P \<le> Q" 

508 
apply (rule le_funI)+ 

509 
apply (rule le_boolI) 

510 
apply (rule PQ) 

511 
apply assumption 

512 
done 

513 

514 
lemma predicate2D [Pure.dest, dest]: "P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y" 

515 
apply (erule le_funE)+ 

516 
apply (erule le_boolE) 

517 
apply assumption+ 

518 
done 

519 

520 
lemma rev_predicate1D: "P x ==> P <= Q ==> Q x" 

521 
by (rule predicate1D) 

522 

523 
lemma rev_predicate2D: "P x y ==> P <= Q ==> Q x y" 

524 
by (rule predicate2D) 

525 

526 
instance "fun" :: (type, lattice) lattice 

527 
inf_fun_eq: "inf f g \<equiv> (\<lambda>x. inf (f x) (g x))" 

528 
sup_fun_eq: "sup f g \<equiv> (\<lambda>x. sup (f x) (g x))" 

529 
apply intro_classes 

530 
unfolding inf_fun_eq sup_fun_eq 

531 
apply (auto intro: le_funI) 

532 
apply (rule le_funI) 

533 
apply (auto dest: le_funD) 

534 
apply (rule le_funI) 

535 
apply (auto dest: le_funD) 

536 
done 

537 

22845  538 
lemmas [code func del] = inf_fun_eq sup_fun_eq 
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22453  540 
instance "fun" :: (type, distrib_lattice) distrib_lattice 
541 
by default (auto simp add: inf_fun_eq sup_fun_eq sup_inf_distrib1) 

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22845  544 
subsection {* Proof tool setup *} 
545 

546 
text {* simplifies terms of the form 

547 
f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *} 

548 

549 
ML {* 

550 
let 

551 
fun gen_fun_upd NONE T _ _ = NONE 

552 
 gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd},T) $ f $ x $ y) 

553 
fun dest_fun_T1 (Type (_, T :: Ts)) = T 

554 
fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) = 

555 
let 

556 
fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) = 

557 
if v aconv x then SOME g else gen_fun_upd (find g) T v w 

558 
 find t = NONE 

559 
in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end 

560 
fun fun_upd_prover ss = 

561 
rtac eq_reflection 1 THEN rtac ext 1 THEN 

562 
simp_tac (Simplifier.inherit_context ss @{simpset}) 1 

563 
val fun_upd2_simproc = 

564 
Simplifier.simproc @{theory} 

565 
"fun_upd2" ["f(v := w, x := y)"] 

566 
(fn _ => fn ss => fn t => 

567 
case find_double t of (T, NONE) => NONE 

568 
 (T, SOME rhs) => 

569 
SOME (Goal.prove (Simplifier.the_context ss) [] [] 

570 
(Term.equals T $ t $ rhs) (K (fun_upd_prover ss)))) 

571 
in 

572 
Addsimprocs [fun_upd2_simproc] 

573 
end; 

574 
*} 

575 

576 

21870  577 
subsection {* Code generator setup *} 
578 

579 
code_const "op \<circ>" 

580 
(SML infixl 5 "o") 

581 
(Haskell infixr 9 ".") 

582 

21906  583 
code_const "id" 
584 
(Haskell "id") 

585 

21870  586 

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subsection {* ML legacy bindings *} 
15510  588 

22845  589 
ML {* 
590 
val set_cs = claset() delrules [equalityI] 

591 
*} 

5852  592 

22845  593 
ML {* 
594 
val id_apply = @{thm id_apply} 

595 
val id_def = @{thm id_def} 

596 
val o_apply = @{thm o_apply} 

597 
val o_assoc = @{thm o_assoc} 

598 
val o_def = @{thm o_def} 

599 
val injD = @{thm injD} 

600 
val datatype_injI = @{thm datatype_injI} 

601 
val range_ex1_eq = @{thm range_ex1_eq} 

602 
val expand_fun_eq = @{thm expand_fun_eq} 

13585  603 
*} 
5852  604 

2912  605 
end 