author  nipkow 
Sat, 17 Oct 2009 13:46:39 +0200  
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parent 32740  9dd0a2f83429 
child 32988  d1d4d7a08a66 
permissions  rwrr 
1475  1 
(* Title: HOL/Fun.thy 
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Author: Tobias Nipkow, Cambridge University Computer Laboratory 

923  3 
Copyright 1994 University of Cambridge 
18154  4 
*) 
923  5 

18154  6 
header {* Notions about functions *} 
923  7 

15510  8 
theory Fun 
32139  9 
imports Complete_Lattice 
32554  10 
uses ("Tools/transfer.ML") 
15131  11 
begin 
2912  12 

26147  13 
text{*As a simplification rule, it replaces all function equalities by 
14 
firstorder equalities.*} 

15 
lemma expand_fun_eq: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)" 

16 
apply (rule iffI) 

17 
apply (simp (no_asm_simp)) 

18 
apply (rule ext) 

19 
apply (simp (no_asm_simp)) 

20 
done 

5305  21 

26147  22 
lemma apply_inverse: 
26357  23 
"f x = u \<Longrightarrow> (\<And>x. P x \<Longrightarrow> g (f x) = x) \<Longrightarrow> P x \<Longrightarrow> x = g u" 
26147  24 
by auto 
2912  25 

12258  26 

26147  27 
subsection {* The Identity Function @{text id} *} 
6171  28 

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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  33 

26147  34 
lemma id_apply [simp]: "id x = x" 
35 
by (simp add: id_def) 

36 

37 
lemma image_ident [simp]: "(%x. x) ` Y = Y" 

38 
by blast 

39 

40 
lemma image_id [simp]: "id ` Y = Y" 

41 
by (simp add: id_def) 

42 

43 
lemma vimage_ident [simp]: "(%x. x) ` Y = Y" 

44 
by blast 

45 

46 
lemma vimage_id [simp]: "id ` A = A" 

47 
by (simp add: id_def) 

48 

49 

50 
subsection {* The Composition Operator @{text "f \<circ> g"} *} 

51 

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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))" 
11123  56 

21210  57 
notation (xsymbols) 
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tuned concrete syntax  abbreviation/const_syntax;
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comp (infixl "\<circ>" 55) 
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21210  60 
notation (HTML output) 
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comp (infixl "\<circ>" 55) 
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13585  63 
text{*compatibility*} 
64 
lemmas o_def = comp_def 

2912  65 

13585  66 
lemma o_apply [simp]: "(f o g) x = f (g x)" 
67 
by (simp add: comp_def) 

68 

69 
lemma o_assoc: "f o (g o h) = f o g o h" 

70 
by (simp add: comp_def) 

71 

72 
lemma id_o [simp]: "id o g = g" 

73 
by (simp add: comp_def) 

74 

75 
lemma o_id [simp]: "f o id = f" 

76 
by (simp add: comp_def) 

77 

78 
lemma image_compose: "(f o g) ` r = f`(g`r)" 

79 
by (simp add: comp_def, blast) 

80 

81 
lemma UN_o: "UNION A (g o f) = UNION (f`A) g" 

82 
by (unfold comp_def, blast) 

83 

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subsection {* The Forward Composition Operator @{text fcomp} *} 
26357  86 

87 
definition 

88 
fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o>" 60) 

89 
where 

90 
"f o> g = (\<lambda>x. g (f x))" 

91 

92 
lemma fcomp_apply: "(f o> g) x = g (f x)" 

93 
by (simp add: fcomp_def) 

94 

95 
lemma fcomp_assoc: "(f o> g) o> h = f o> (g o> h)" 

96 
by (simp add: fcomp_def) 

97 

98 
lemma id_fcomp [simp]: "id o> g = g" 

99 
by (simp add: fcomp_def) 

100 

101 
lemma fcomp_id [simp]: "f o> id = f" 

102 
by (simp add: fcomp_def) 

103 

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code_const fcomp 
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(Eval infixl 1 "#>") 
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no_notation fcomp (infixl "o>" 60) 
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26357  109 

26147  110 
subsection {* Injectivity and Surjectivity *} 
111 

112 
constdefs 

113 
inj_on :: "['a => 'b, 'a set] => bool"  "injective" 

114 
"inj_on f A == ! x:A. ! y:A. f(x)=f(y) > x=y" 

115 

116 
text{*A common special case: functions injective over the entire domain type.*} 

117 

118 
abbreviation 

119 
"inj f == inj_on f UNIV" 

13585  120 

26147  121 
definition 
122 
bij_betw :: "('a => 'b) => 'a set => 'b set => bool" where  "bijective" 

28562  123 
[code del]: "bij_betw f A B \<longleftrightarrow> inj_on f A & f ` A = B" 
26147  124 

125 
constdefs 

126 
surj :: "('a => 'b) => bool" (*surjective*) 

127 
"surj f == ! y. ? x. y=f(x)" 

13585  128 

26147  129 
bij :: "('a => 'b) => bool" (*bijective*) 
130 
"bij f == inj f & surj f" 

131 

132 
lemma injI: 

133 
assumes "\<And>x y. f x = f y \<Longrightarrow> x = y" 

134 
shows "inj f" 

135 
using assms unfolding inj_on_def by auto 

13585  136 

31775  137 
text{*For Proofs in @{text "Tools/Datatype/datatype_rep_proofs"}*} 
13585  138 
lemma datatype_injI: 
139 
"(!! x. ALL y. f(x) = f(y) > x=y) ==> inj(f)" 

140 
by (simp add: inj_on_def) 

141 

13637  142 
theorem range_ex1_eq: "inj f \<Longrightarrow> b : range f = (EX! x. b = f x)" 
143 
by (unfold inj_on_def, blast) 

144 

13585  145 
lemma injD: "[ inj(f); f(x) = f(y) ] ==> x=y" 
146 
by (simp add: inj_on_def) 

147 

148 
(*Useful with the simplifier*) 

149 
lemma inj_eq: "inj(f) ==> (f(x) = f(y)) = (x=y)" 

150 
by (force simp add: inj_on_def) 

151 

26147  152 
lemma inj_on_id[simp]: "inj_on id A" 
153 
by (simp add: inj_on_def) 

13585  154 

26147  155 
lemma inj_on_id2[simp]: "inj_on (%x. x) A" 
156 
by (simp add: inj_on_def) 

157 

158 
lemma surj_id[simp]: "surj id" 

159 
by (simp add: surj_def) 

160 

161 
lemma bij_id[simp]: "bij id" 

162 
by (simp add: bij_def inj_on_id surj_id) 

13585  163 

164 
lemma inj_onI: 

165 
"(!! x y. [ x:A; y:A; f(x) = f(y) ] ==> x=y) ==> inj_on f A" 

166 
by (simp add: inj_on_def) 

167 

168 
lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A" 

169 
by (auto dest: arg_cong [of concl: g] simp add: inj_on_def) 

170 

171 
lemma inj_onD: "[ inj_on f A; f(x)=f(y); x:A; y:A ] ==> x=y" 

172 
by (unfold inj_on_def, blast) 

173 

174 
lemma inj_on_iff: "[ inj_on f A; x:A; y:A ] ==> (f(x)=f(y)) = (x=y)" 

175 
by (blast dest!: inj_onD) 

176 

177 
lemma comp_inj_on: 

178 
"[ inj_on f A; inj_on g (f`A) ] ==> inj_on (g o f) A" 

179 
by (simp add: comp_def inj_on_def) 

180 

15303  181 
lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)" 
182 
apply(simp add:inj_on_def image_def) 

183 
apply blast 

184 
done 

185 

15439  186 
lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y); 
187 
inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A" 

188 
apply(unfold inj_on_def) 

189 
apply blast 

190 
done 

191 

13585  192 
lemma inj_on_contraD: "[ inj_on f A; ~x=y; x:A; y:A ] ==> ~ f(x)=f(y)" 
193 
by (unfold inj_on_def, blast) 

12258  194 

13585  195 
lemma inj_singleton: "inj (%s. {s})" 
196 
by (simp add: inj_on_def) 

197 

15111  198 
lemma inj_on_empty[iff]: "inj_on f {}" 
199 
by(simp add: inj_on_def) 

200 

15303  201 
lemma subset_inj_on: "[ inj_on f B; A <= B ] ==> inj_on f A" 
13585  202 
by (unfold inj_on_def, blast) 
203 

15111  204 
lemma inj_on_Un: 
205 
"inj_on f (A Un B) = 

206 
(inj_on f A & inj_on f B & f`(AB) Int f`(BA) = {})" 

207 
apply(unfold inj_on_def) 

208 
apply (blast intro:sym) 

209 
done 

210 

211 
lemma inj_on_insert[iff]: 

212 
"inj_on f (insert a A) = (inj_on f A & f a ~: f`(A{a}))" 

213 
apply(unfold inj_on_def) 

214 
apply (blast intro:sym) 

215 
done 

216 

217 
lemma inj_on_diff: "inj_on f A ==> inj_on f (AB)" 

218 
apply(unfold inj_on_def) 

219 
apply (blast) 

220 
done 

221 

13585  222 
lemma surjI: "(!! x. g(f x) = x) ==> surj g" 
223 
apply (simp add: surj_def) 

224 
apply (blast intro: sym) 

225 
done 

226 

227 
lemma surj_range: "surj f ==> range f = UNIV" 

228 
by (auto simp add: surj_def) 

229 

230 
lemma surjD: "surj f ==> EX x. y = f x" 

231 
by (simp add: surj_def) 

232 

233 
lemma surjE: "surj f ==> (!!x. y = f x ==> C) ==> C" 

234 
by (simp add: surj_def, blast) 

235 

236 
lemma comp_surj: "[ surj f; surj g ] ==> surj (g o f)" 

237 
apply (simp add: comp_def surj_def, clarify) 

238 
apply (drule_tac x = y in spec, clarify) 

239 
apply (drule_tac x = x in spec, blast) 

240 
done 

241 

242 
lemma bijI: "[ inj f; surj f ] ==> bij f" 

243 
by (simp add: bij_def) 

244 

245 
lemma bij_is_inj: "bij f ==> inj f" 

246 
by (simp add: bij_def) 

247 

248 
lemma bij_is_surj: "bij f ==> surj f" 

249 
by (simp add: bij_def) 

250 

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lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A" 
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by (simp add: bij_betw_def) 
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32337  254 
lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)" 
255 
by(fastsimp intro: comp_inj_on comp_surj simp: bij_def surj_range) 

256 

31438  257 
lemma bij_betw_trans: 
258 
"bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C" 

259 
by(auto simp add:bij_betw_def comp_inj_on) 

260 

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lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A" 
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proof  
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have i: "inj_on f A" and s: "f ` A = B" 
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using assms by(auto simp:bij_betw_def) 
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let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)" 
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{ fix a b assume P: "?P b a" 
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hence ex1: "\<exists>a. ?P b a" using s unfolding image_def by blast 
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hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i]) 
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hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp 
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} note g = this 
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have "inj_on ?g B" 
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proof(rule inj_onI) 
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fix x y assume "x:B" "y:B" "?g x = ?g y" 
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from s `x:B` obtain a1 where a1: "?P x a1" unfolding image_def by blast 
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from s `y:B` obtain a2 where a2: "?P y a2" unfolding image_def by blast 
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from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp 
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qed 
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moreover have "?g ` B = A" 
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proof(auto simp:image_def) 
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fix b assume "b:B" 
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with s obtain a where P: "?P b a" unfolding image_def by blast 
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thus "?g b \<in> A" using g[OF P] by auto 
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next 
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fix a assume "a:A" 
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then obtain b where P: "?P b a" using s unfolding image_def by blast 
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then have "b:B" using s unfolding image_def by blast 
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with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast 
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qed 
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ultimately show ?thesis by(auto simp:bij_betw_def) 
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qed 
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13585  292 
lemma surj_image_vimage_eq: "surj f ==> f ` (f ` A) = A" 
293 
by (simp add: surj_range) 

294 

295 
lemma inj_vimage_image_eq: "inj f ==> f ` (f ` A) = A" 

296 
by (simp add: inj_on_def, blast) 

297 

298 
lemma vimage_subsetD: "surj f ==> f ` B <= A ==> B <= f ` A" 

299 
apply (unfold surj_def) 

300 
apply (blast intro: sym) 

301 
done 

302 

303 
lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f ` B <= A" 

304 
by (unfold inj_on_def, blast) 

305 

306 
lemma vimage_subset_eq: "bij f ==> (f ` B <= A) = (B <= f ` A)" 

307 
apply (unfold bij_def) 

308 
apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD) 

309 
done 

310 

31438  311 
lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B" 
312 
by(blast dest: inj_onD) 

313 

13585  314 
lemma inj_on_image_Int: 
315 
"[ inj_on f C; A<=C; B<=C ] ==> f`(A Int B) = f`A Int f`B" 

316 
apply (simp add: inj_on_def, blast) 

317 
done 

318 

319 
lemma inj_on_image_set_diff: 

320 
"[ inj_on f C; A<=C; B<=C ] ==> f`(AB) = f`A  f`B" 

321 
apply (simp add: inj_on_def, blast) 

322 
done 

323 

324 
lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B" 

325 
by (simp add: inj_on_def, blast) 

326 

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

328 
by (simp add: inj_on_def, blast) 

329 

330 
lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)" 

331 
by (blast dest: injD) 

332 

333 
lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)" 

334 
by (simp add: inj_on_def, blast) 

335 

336 
lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)" 

337 
by (blast dest: injD) 

338 

339 
(*injectivity's required. Lefttoright inclusion holds even if A is empty*) 

340 
lemma image_INT: 

341 
"[ inj_on f C; ALL x:A. B x <= C; j:A ] 

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

343 
apply (simp add: inj_on_def, blast) 

344 
done 

345 

346 
(*Compare with image_INT: no use of inj_on, and if f is surjective then 

347 
it doesn't matter whether A is empty*) 

348 
lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)" 

349 
apply (simp add: bij_def) 

350 
apply (simp add: inj_on_def surj_def, blast) 

351 
done 

352 

353 
lemma surj_Compl_image_subset: "surj f ==> (f`A) <= f`(A)" 

354 
by (auto simp add: surj_def) 

355 

356 
lemma inj_image_Compl_subset: "inj f ==> f`(A) <= (f`A)" 

357 
by (auto simp add: inj_on_def) 

5852  358 

13585  359 
lemma bij_image_Compl_eq: "bij f ==> f`(A) = (f`A)" 
360 
apply (simp add: bij_def) 

361 
apply (rule equalityI) 

362 
apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset) 

363 
done 

364 

365 

366 
subsection{*Function Updating*} 

367 

26147  368 
constdefs 
369 
fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" 

370 
"fun_upd f a b == % x. if x=a then b else f x" 

371 

372 
nonterminals 

373 
updbinds updbind 

374 
syntax 

375 
"_updbind" :: "['a, 'a] => updbind" ("(2_ :=/ _)") 

376 
"" :: "updbind => updbinds" ("_") 

377 
"_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _") 

378 
"_Update" :: "['a, updbinds] => 'a" ("_/'((_)')" [1000,0] 900) 

379 

380 
translations 

381 
"_Update f (_updbinds b bs)" == "_Update (_Update f b) bs" 

382 
"f(x:=y)" == "fun_upd f x y" 

383 

384 
(* Hint: to define the sum of two functions (or maps), use sum_case. 

385 
A nice infix syntax could be defined (in Datatype.thy or below) by 

386 
consts 

387 
fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80) 

388 
translations 

389 
"fun_sum" == sum_case 

390 
*) 

391 

13585  392 
lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)" 
393 
apply (simp add: fun_upd_def, safe) 

394 
apply (erule subst) 

395 
apply (rule_tac [2] ext, auto) 

396 
done 

397 

398 
(* f x = y ==> f(x:=y) = f *) 

399 
lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard] 

400 

401 
(* f(x := f x) = f *) 

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402 
lemmas fun_upd_triv = refl [THEN fun_upd_idem] 
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403 
declare fun_upd_triv [iff] 
13585  404 

405 
lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)" 

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406 
by (simp add: fun_upd_def) 
13585  407 

408 
(* fun_upd_apply supersedes these two, but they are useful 

409 
if fun_upd_apply is intentionally removed from the simpset *) 

410 
lemma fun_upd_same: "(f(x:=y)) x = y" 

411 
by simp 

412 

413 
lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z" 

414 
by simp 

415 

416 
lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)" 

417 
by (simp add: expand_fun_eq) 

418 

419 
lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)" 

420 
by (rule ext, auto) 

421 

15303  422 
lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A" 
423 
by(fastsimp simp:inj_on_def image_def) 

424 

15510  425 
lemma fun_upd_image: 
426 
"f(x:=y) ` A = (if x \<in> A then insert y (f ` (A{x})) else f ` A)" 

427 
by auto 

428 

31080  429 
lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)" 
430 
by(auto intro: ext) 

431 

26147  432 

433 
subsection {* @{text override_on} *} 

434 

435 
definition 

436 
override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b" 

437 
where 

438 
"override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)" 

13910  439 

15691  440 
lemma override_on_emptyset[simp]: "override_on f g {} = f" 
441 
by(simp add:override_on_def) 

13910  442 

15691  443 
lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a" 
444 
by(simp add:override_on_def) 

13910  445 

15691  446 
lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a" 
447 
by(simp add:override_on_def) 

13910  448 

26147  449 

450 
subsection {* @{text swap} *} 

15510  451 

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452 
definition 
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453 
swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)" 
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454 
where 
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455 
"swap a b f = f (a := f b, b:= f a)" 
15510  456 

457 
lemma swap_self: "swap a a f = f" 

15691  458 
by (simp add: swap_def) 
15510  459 

460 
lemma swap_commute: "swap a b f = swap b a f" 

461 
by (rule ext, simp add: fun_upd_def swap_def) 

462 

463 
lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f" 

464 
by (rule ext, simp add: fun_upd_def swap_def) 

465 

466 
lemma inj_on_imp_inj_on_swap: 

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467 
"[inj_on f A; a \<in> A; b \<in> A] ==> inj_on (swap a b f) A" 
15510  468 
by (simp add: inj_on_def swap_def, blast) 
469 

470 
lemma inj_on_swap_iff [simp]: 

471 
assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A = inj_on f A" 

472 
proof 

473 
assume "inj_on (swap a b f) A" 

474 
with A have "inj_on (swap a b (swap a b f)) A" 

17589  475 
by (iprover intro: inj_on_imp_inj_on_swap) 
15510  476 
thus "inj_on f A" by simp 
477 
next 

478 
assume "inj_on f A" 

27165  479 
with A show "inj_on (swap a b f) A" by(iprover intro: inj_on_imp_inj_on_swap) 
15510  480 
qed 
481 

482 
lemma surj_imp_surj_swap: "surj f ==> surj (swap a b f)" 

483 
apply (simp add: surj_def swap_def, clarify) 

27125  484 
apply (case_tac "y = f b", blast) 
485 
apply (case_tac "y = f a", auto) 

15510  486 
done 
487 

488 
lemma surj_swap_iff [simp]: "surj (swap a b f) = surj f" 

489 
proof 

490 
assume "surj (swap a b f)" 

491 
hence "surj (swap a b (swap a b f))" by (rule surj_imp_surj_swap) 

492 
thus "surj f" by simp 

493 
next 

494 
assume "surj f" 

495 
thus "surj (swap a b f)" by (rule surj_imp_surj_swap) 

496 
qed 

497 

498 
lemma bij_swap_iff: "bij (swap a b f) = bij f" 

499 
by (simp add: bij_def) 

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500 

27188  501 
hide (open) const swap 
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502 

31949  503 

504 
subsection {* Inversion of injective functions *} 

505 

506 
definition inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where 

507 
"inv f y = (THE x. f x = y)" 

508 

509 
lemma inv_f_f: 

510 
assumes "inj f" 

511 
shows "inv f (f x) = x" 

512 
proof  

513 
from assms have "(THE x'. f x' = f x) = (THE x'. x' = x)" 

514 
by (simp only: inj_eq) 

515 
also have "... = x" by (rule the_eq_trivial) 

516 
finally show ?thesis by (unfold inv_def) 

517 
qed 

518 

519 
lemma f_inv_f: 

520 
assumes "inj f" 

521 
and "y \<in> range f" 

522 
shows "f (inv f y) = y" 

523 
proof (unfold inv_def) 

524 
from `y \<in> range f` obtain x where "y = f x" .. 

525 
then have "f x = y" .. 

526 
then show "f (THE x. f x = y) = y" 

527 
proof (rule theI) 

528 
fix x' assume "f x' = y" 

529 
with `f x = y` have "f x' = f x" by simp 

530 
with `inj f` show "x' = x" by (rule injD) 

531 
qed 

532 
qed 

533 

534 
hide (open) const inv 

535 

32961  536 
definition the_inv_onto :: "'a set => ('a => 'b) => ('b => 'a)" where 
537 
"the_inv_onto A f == %x. THE y. y : A & f y = x" 

538 

539 
lemma the_inv_onto_f_f: 

540 
"[ inj_on f A; x : A ] ==> the_inv_onto A f (f x) = x" 

541 
apply (simp add: the_inv_onto_def inj_on_def) 

542 
apply (blast intro: the_equality) 

543 
done 

544 

545 
lemma f_the_inv_onto_f: 

546 
"inj_on f A ==> y : f`A ==> f (the_inv_onto A f y) = y" 

547 
apply (simp add: the_inv_onto_def) 

548 
apply (rule the1I2) 

549 
apply(blast dest: inj_onD) 

550 
apply blast 

551 
done 

552 

553 
lemma the_inv_onto_into: 

554 
"[ inj_on f A; x : f ` A; A <= B ] ==> the_inv_onto A f x : B" 

555 
apply (simp add: the_inv_onto_def) 

556 
apply (rule the1I2) 

557 
apply(blast dest: inj_onD) 

558 
apply blast 

559 
done 

560 

561 
lemma the_inv_onto_onto[simp]: 

562 
"inj_on f A ==> the_inv_onto A f ` (f ` A) = A" 

563 
by (fast intro:the_inv_onto_into the_inv_onto_f_f[symmetric]) 

564 

565 
lemma the_inv_onto_f_eq: 

566 
"[ inj_on f A; f x = y; x : A ] ==> the_inv_onto A f y = x" 

567 
apply (erule subst) 

568 
apply (erule the_inv_onto_f_f, assumption) 

569 
done 

570 

571 
lemma the_inv_onto_comp: 

572 
"[ inj_on f (g ` A); inj_on g A; x : f ` g ` A ] ==> 

573 
the_inv_onto A (f o g) x = (the_inv_onto A g o the_inv_onto (g ` A) f) x" 

574 
apply (rule the_inv_onto_f_eq) 

575 
apply (fast intro: comp_inj_on) 

576 
apply (simp add: f_the_inv_onto_f the_inv_onto_into) 

577 
apply (simp add: the_inv_onto_into) 

578 
done 

579 

580 
lemma inj_on_the_inv_onto: 

581 
"inj_on f A \<Longrightarrow> inj_on (the_inv_onto A f) (f ` A)" 

582 
by (auto intro: inj_onI simp: image_def the_inv_onto_f_f) 

583 

584 
lemma bij_betw_the_inv_onto: 

585 
"bij_betw f A B \<Longrightarrow> bij_betw (the_inv_onto A f) B A" 

586 
by (auto simp add: bij_betw_def inj_on_the_inv_onto the_inv_onto_into) 

587 

31949  588 

22845  589 
subsection {* Proof tool setup *} 
590 

591 
text {* simplifies terms of the form 

592 
f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *} 

593 

24017  594 
simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ => 
22845  595 
let 
596 
fun gen_fun_upd NONE T _ _ = NONE 

24017  597 
 gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) $ f $ x $ y) 
22845  598 
fun dest_fun_T1 (Type (_, T :: Ts)) = T 
599 
fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) = 

600 
let 

601 
fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) = 

602 
if v aconv x then SOME g else gen_fun_upd (find g) T v w 

603 
 find t = NONE 

604 
in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end 

24017  605 

606 
fun proc ss ct = 

607 
let 

608 
val ctxt = Simplifier.the_context ss 

609 
val t = Thm.term_of ct 

610 
in 

611 
case find_double t of 

612 
(T, NONE) => NONE 

613 
 (T, SOME rhs) => 

27330  614 
SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs)) 
24017  615 
(fn _ => 
616 
rtac eq_reflection 1 THEN 

617 
rtac ext 1 THEN 

618 
simp_tac (Simplifier.inherit_context ss @{simpset}) 1)) 

619 
end 

620 
in proc end 

22845  621 
*} 
622 

623 

32554  624 
subsection {* Generic transfer procedure *} 
625 

626 
definition TransferMorphism:: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> bool" 

627 
where "TransferMorphism a B \<longleftrightarrow> True" 

628 

629 
use "Tools/transfer.ML" 

630 

631 
setup Transfer.setup 

632 

633 

21870  634 
subsection {* Code generator setup *} 
635 

25886
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636 
types_code 
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637 
"fun" ("(_ >/ _)") 
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638 
attach (term_of) {* 
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639 
fun term_of_fun_type _ aT _ bT _ = Free ("<function>", aT > bT); 
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640 
*} 
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641 
attach (test) {* 
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642 
fun gen_fun_type aF aT bG bT i = 
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643 
let 
32740  644 
val tab = Unsynchronized.ref []; 
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645 
fun mk_upd (x, (_, y)) t = Const ("Fun.fun_upd", 
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646 
(aT > bT) > aT > bT > aT > bT) $ t $ aF x $ y () 
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647 
in 
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648 
(fn x => 
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649 
case AList.lookup op = (!tab) x of 
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650 
NONE => 
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651 
let val p as (y, _) = bG i 
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652 
in (tab := (x, p) :: !tab; y) end 
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653 
 SOME (y, _) => y, 
28711  654 
fn () => Basics.fold mk_upd (!tab) (Const ("HOL.undefined", aT > bT))) 
25886
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655 
end; 
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656 
*} 
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657 

21870  658 
code_const "op \<circ>" 
659 
(SML infixl 5 "o") 

660 
(Haskell infixr 9 ".") 

661 

21906  662 
code_const "id" 
663 
(Haskell "id") 

664 

2912  665 
end 