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src/HOL/Library/Sublist.thy

author | wenzelm |

Wed Sep 12 13:42:28 2012 +0200 (2012-09-12) | |

changeset 49322 | fbb320d02420 |

parent 49107 | ec34e9df0514 |

child 50516 | ed6b40d15d1c |

permissions | -rw-r--r-- |

tuned headers;

1 (* Title: HOL/Library/Sublist.thy

2 Author: Tobias Nipkow and Markus Wenzel, TU Muenchen

3 Author: Christian Sternagel, JAIST

4 *)

6 header {* List prefixes, suffixes, and embedding*}

8 theory Sublist

9 imports Main

10 begin

12 subsection {* Prefix order on lists *}

14 definition prefixeq :: "'a list => 'a list => bool"

15 where "prefixeq xs ys \<longleftrightarrow> (\<exists>zs. ys = xs @ zs)"

17 definition prefix :: "'a list => 'a list => bool"

18 where "prefix xs ys \<longleftrightarrow> prefixeq xs ys \<and> xs \<noteq> ys"

20 interpretation prefix_order: order prefixeq prefix

21 by default (auto simp: prefixeq_def prefix_def)

23 interpretation prefix_bot: bot prefixeq prefix Nil

24 by default (simp add: prefixeq_def)

26 lemma prefixeqI [intro?]: "ys = xs @ zs ==> prefixeq xs ys"

27 unfolding prefixeq_def by blast

29 lemma prefixeqE [elim?]:

30 assumes "prefixeq xs ys"

31 obtains zs where "ys = xs @ zs"

32 using assms unfolding prefixeq_def by blast

34 lemma prefixI' [intro?]: "ys = xs @ z # zs ==> prefix xs ys"

35 unfolding prefix_def prefixeq_def by blast

37 lemma prefixE' [elim?]:

38 assumes "prefix xs ys"

39 obtains z zs where "ys = xs @ z # zs"

40 proof -

41 from `prefix xs ys` obtain us where "ys = xs @ us" and "xs \<noteq> ys"

42 unfolding prefix_def prefixeq_def by blast

43 with that show ?thesis by (auto simp add: neq_Nil_conv)

44 qed

46 lemma prefixI [intro?]: "prefixeq xs ys ==> xs \<noteq> ys ==> prefix xs ys"

47 unfolding prefix_def by blast

49 lemma prefixE [elim?]:

50 fixes xs ys :: "'a list"

51 assumes "prefix xs ys"

52 obtains "prefixeq xs ys" and "xs \<noteq> ys"

53 using assms unfolding prefix_def by blast

56 subsection {* Basic properties of prefixes *}

58 theorem Nil_prefixeq [iff]: "prefixeq [] xs"

59 by (simp add: prefixeq_def)

61 theorem prefixeq_Nil [simp]: "(prefixeq xs []) = (xs = [])"

62 by (induct xs) (simp_all add: prefixeq_def)

64 lemma prefixeq_snoc [simp]: "prefixeq xs (ys @ [y]) \<longleftrightarrow> xs = ys @ [y] \<or> prefixeq xs ys"

65 proof

66 assume "prefixeq xs (ys @ [y])"

67 then obtain zs where zs: "ys @ [y] = xs @ zs" ..

68 show "xs = ys @ [y] \<or> prefixeq xs ys"

69 by (metis append_Nil2 butlast_append butlast_snoc prefixeqI zs)

70 next

71 assume "xs = ys @ [y] \<or> prefixeq xs ys"

72 then show "prefixeq xs (ys @ [y])"

73 by (metis prefix_order.eq_iff prefix_order.order_trans prefixeqI)

74 qed

76 lemma Cons_prefixeq_Cons [simp]: "prefixeq (x # xs) (y # ys) = (x = y \<and> prefixeq xs ys)"

77 by (auto simp add: prefixeq_def)

79 lemma prefixeq_code [code]:

80 "prefixeq [] xs \<longleftrightarrow> True"

81 "prefixeq (x # xs) [] \<longleftrightarrow> False"

82 "prefixeq (x # xs) (y # ys) \<longleftrightarrow> x = y \<and> prefixeq xs ys"

83 by simp_all

85 lemma same_prefixeq_prefixeq [simp]: "prefixeq (xs @ ys) (xs @ zs) = prefixeq ys zs"

86 by (induct xs) simp_all

88 lemma same_prefixeq_nil [iff]: "prefixeq (xs @ ys) xs = (ys = [])"

89 by (metis append_Nil2 append_self_conv prefix_order.eq_iff prefixeqI)

91 lemma prefixeq_prefixeq [simp]: "prefixeq xs ys ==> prefixeq xs (ys @ zs)"

92 by (metis prefix_order.le_less_trans prefixeqI prefixE prefixI)

94 lemma append_prefixeqD: "prefixeq (xs @ ys) zs \<Longrightarrow> prefixeq xs zs"

95 by (auto simp add: prefixeq_def)

97 theorem prefixeq_Cons: "prefixeq xs (y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> prefixeq zs ys))"

98 by (cases xs) (auto simp add: prefixeq_def)

100 theorem prefixeq_append:

101 "prefixeq xs (ys @ zs) = (prefixeq xs ys \<or> (\<exists>us. xs = ys @ us \<and> prefixeq us zs))"

102 apply (induct zs rule: rev_induct)

103 apply force

104 apply (simp del: append_assoc add: append_assoc [symmetric])

105 apply (metis append_eq_appendI)

106 done

108 lemma append_one_prefixeq:

109 "prefixeq xs ys ==> length xs < length ys ==> prefixeq (xs @ [ys ! length xs]) ys"

110 unfolding prefixeq_def

111 by (metis Cons_eq_appendI append_eq_appendI append_eq_conv_conj

112 eq_Nil_appendI nth_drop')

114 theorem prefixeq_length_le: "prefixeq xs ys ==> length xs \<le> length ys"

115 by (auto simp add: prefixeq_def)

117 lemma prefixeq_same_cases:

118 "prefixeq (xs\<^isub>1::'a list) ys \<Longrightarrow> prefixeq xs\<^isub>2 ys \<Longrightarrow> prefixeq xs\<^isub>1 xs\<^isub>2 \<or> prefixeq xs\<^isub>2 xs\<^isub>1"

119 unfolding prefixeq_def by (metis append_eq_append_conv2)

121 lemma set_mono_prefixeq: "prefixeq xs ys \<Longrightarrow> set xs \<subseteq> set ys"

122 by (auto simp add: prefixeq_def)

124 lemma take_is_prefixeq: "prefixeq (take n xs) xs"

125 unfolding prefixeq_def by (metis append_take_drop_id)

127 lemma map_prefixeqI: "prefixeq xs ys \<Longrightarrow> prefixeq (map f xs) (map f ys)"

128 by (auto simp: prefixeq_def)

130 lemma prefixeq_length_less: "prefix xs ys \<Longrightarrow> length xs < length ys"

131 by (auto simp: prefix_def prefixeq_def)

133 lemma prefix_simps [simp, code]:

134 "prefix xs [] \<longleftrightarrow> False"

135 "prefix [] (x # xs) \<longleftrightarrow> True"

136 "prefix (x # xs) (y # ys) \<longleftrightarrow> x = y \<and> prefix xs ys"

137 by (simp_all add: prefix_def cong: conj_cong)

139 lemma take_prefix: "prefix xs ys \<Longrightarrow> prefix (take n xs) ys"

140 apply (induct n arbitrary: xs ys)

141 apply (case_tac ys, simp_all)[1]

142 apply (metis prefix_order.less_trans prefixI take_is_prefixeq)

143 done

145 lemma not_prefixeq_cases:

146 assumes pfx: "\<not> prefixeq ps ls"

147 obtains

148 (c1) "ps \<noteq> []" and "ls = []"

149 | (c2) a as x xs where "ps = a#as" and "ls = x#xs" and "x = a" and "\<not> prefixeq as xs"

150 | (c3) a as x xs where "ps = a#as" and "ls = x#xs" and "x \<noteq> a"

151 proof (cases ps)

152 case Nil

153 then show ?thesis using pfx by simp

154 next

155 case (Cons a as)

156 note c = `ps = a#as`

157 show ?thesis

158 proof (cases ls)

159 case Nil then show ?thesis by (metis append_Nil2 pfx c1 same_prefixeq_nil)

160 next

161 case (Cons x xs)

162 show ?thesis

163 proof (cases "x = a")

164 case True

165 have "\<not> prefixeq as xs" using pfx c Cons True by simp

166 with c Cons True show ?thesis by (rule c2)

167 next

168 case False

169 with c Cons show ?thesis by (rule c3)

170 qed

171 qed

172 qed

174 lemma not_prefixeq_induct [consumes 1, case_names Nil Neq Eq]:

175 assumes np: "\<not> prefixeq ps ls"

176 and base: "\<And>x xs. P (x#xs) []"

177 and r1: "\<And>x xs y ys. x \<noteq> y \<Longrightarrow> P (x#xs) (y#ys)"

178 and r2: "\<And>x xs y ys. \<lbrakk> x = y; \<not> prefixeq xs ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys)"

179 shows "P ps ls" using np

180 proof (induct ls arbitrary: ps)

181 case Nil then show ?case

182 by (auto simp: neq_Nil_conv elim!: not_prefixeq_cases intro!: base)

183 next

184 case (Cons y ys)

185 then have npfx: "\<not> prefixeq ps (y # ys)" by simp

186 then obtain x xs where pv: "ps = x # xs"

187 by (rule not_prefixeq_cases) auto

188 show ?case by (metis Cons.hyps Cons_prefixeq_Cons npfx pv r1 r2)

189 qed

192 subsection {* Parallel lists *}

194 definition parallel :: "'a list => 'a list => bool" (infixl "\<parallel>" 50)

195 where "(xs \<parallel> ys) = (\<not> prefixeq xs ys \<and> \<not> prefixeq ys xs)"

197 lemma parallelI [intro]: "\<not> prefixeq xs ys ==> \<not> prefixeq ys xs ==> xs \<parallel> ys"

198 unfolding parallel_def by blast

200 lemma parallelE [elim]:

201 assumes "xs \<parallel> ys"

202 obtains "\<not> prefixeq xs ys \<and> \<not> prefixeq ys xs"

203 using assms unfolding parallel_def by blast

205 theorem prefixeq_cases:

206 obtains "prefixeq xs ys" | "prefix ys xs" | "xs \<parallel> ys"

207 unfolding parallel_def prefix_def by blast

209 theorem parallel_decomp:

210 "xs \<parallel> ys ==> \<exists>as b bs c cs. b \<noteq> c \<and> xs = as @ b # bs \<and> ys = as @ c # cs"

211 proof (induct xs rule: rev_induct)

212 case Nil

213 then have False by auto

214 then show ?case ..

215 next

216 case (snoc x xs)

217 show ?case

218 proof (rule prefixeq_cases)

219 assume le: "prefixeq xs ys"

220 then obtain ys' where ys: "ys = xs @ ys'" ..

221 show ?thesis

222 proof (cases ys')

223 assume "ys' = []"

224 then show ?thesis by (metis append_Nil2 parallelE prefixeqI snoc.prems ys)

225 next

226 fix c cs assume ys': "ys' = c # cs"

227 then show ?thesis

228 by (metis Cons_eq_appendI eq_Nil_appendI parallelE prefixeqI

229 same_prefixeq_prefixeq snoc.prems ys)

230 qed

231 next

232 assume "prefix ys xs"

233 then have "prefixeq ys (xs @ [x])" by (simp add: prefix_def)

234 with snoc have False by blast

235 then show ?thesis ..

236 next

237 assume "xs \<parallel> ys"

238 with snoc obtain as b bs c cs where neq: "(b::'a) \<noteq> c"

239 and xs: "xs = as @ b # bs" and ys: "ys = as @ c # cs"

240 by blast

241 from xs have "xs @ [x] = as @ b # (bs @ [x])" by simp

242 with neq ys show ?thesis by blast

243 qed

244 qed

246 lemma parallel_append: "a \<parallel> b \<Longrightarrow> a @ c \<parallel> b @ d"

247 apply (rule parallelI)

248 apply (erule parallelE, erule conjE,

249 induct rule: not_prefixeq_induct, simp+)+

250 done

252 lemma parallel_appendI: "xs \<parallel> ys \<Longrightarrow> x = xs @ xs' \<Longrightarrow> y = ys @ ys' \<Longrightarrow> x \<parallel> y"

253 by (simp add: parallel_append)

255 lemma parallel_commute: "a \<parallel> b \<longleftrightarrow> b \<parallel> a"

256 unfolding parallel_def by auto

259 subsection {* Suffix order on lists *}

261 definition suffixeq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"

262 where "suffixeq xs ys = (\<exists>zs. ys = zs @ xs)"

264 definition suffix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"

265 where "suffix xs ys \<longleftrightarrow> (\<exists>us. ys = us @ xs \<and> us \<noteq> [])"

267 lemma suffix_imp_suffixeq:

268 "suffix xs ys \<Longrightarrow> suffixeq xs ys"

269 by (auto simp: suffixeq_def suffix_def)

271 lemma suffixeqI [intro?]: "ys = zs @ xs ==> suffixeq xs ys"

272 unfolding suffixeq_def by blast

274 lemma suffixeqE [elim?]:

275 assumes "suffixeq xs ys"

276 obtains zs where "ys = zs @ xs"

277 using assms unfolding suffixeq_def by blast

279 lemma suffixeq_refl [iff]: "suffixeq xs xs"

280 by (auto simp add: suffixeq_def)

281 lemma suffix_trans:

282 "suffix xs ys \<Longrightarrow> suffix ys zs \<Longrightarrow> suffix xs zs"

283 by (auto simp: suffix_def)

284 lemma suffixeq_trans: "\<lbrakk>suffixeq xs ys; suffixeq ys zs\<rbrakk> \<Longrightarrow> suffixeq xs zs"

285 by (auto simp add: suffixeq_def)

286 lemma suffixeq_antisym: "\<lbrakk>suffixeq xs ys; suffixeq ys xs\<rbrakk> \<Longrightarrow> xs = ys"

287 by (auto simp add: suffixeq_def)

289 lemma suffixeq_tl [simp]: "suffixeq (tl xs) xs"

290 by (induct xs) (auto simp: suffixeq_def)

292 lemma suffix_tl [simp]: "xs \<noteq> [] \<Longrightarrow> suffix (tl xs) xs"

293 by (induct xs) (auto simp: suffix_def)

295 lemma Nil_suffixeq [iff]: "suffixeq [] xs"

296 by (simp add: suffixeq_def)

297 lemma suffixeq_Nil [simp]: "(suffixeq xs []) = (xs = [])"

298 by (auto simp add: suffixeq_def)

300 lemma suffixeq_ConsI: "suffixeq xs ys \<Longrightarrow> suffixeq xs (y # ys)"

301 by (auto simp add: suffixeq_def)

302 lemma suffixeq_ConsD: "suffixeq (x # xs) ys \<Longrightarrow> suffixeq xs ys"

303 by (auto simp add: suffixeq_def)

305 lemma suffixeq_appendI: "suffixeq xs ys \<Longrightarrow> suffixeq xs (zs @ ys)"

306 by (auto simp add: suffixeq_def)

307 lemma suffixeq_appendD: "suffixeq (zs @ xs) ys \<Longrightarrow> suffixeq xs ys"

308 by (auto simp add: suffixeq_def)

310 lemma suffix_set_subset:

311 "suffix xs ys \<Longrightarrow> set xs \<subseteq> set ys" by (auto simp: suffix_def)

313 lemma suffixeq_set_subset:

314 "suffixeq xs ys \<Longrightarrow> set xs \<subseteq> set ys" by (auto simp: suffixeq_def)

316 lemma suffixeq_ConsD2: "suffixeq (x # xs) (y # ys) \<Longrightarrow> suffixeq xs ys"

317 proof -

318 assume "suffixeq (x # xs) (y # ys)"

319 then obtain zs where "y # ys = zs @ x # xs" ..

320 then show ?thesis

321 by (induct zs) (auto intro!: suffixeq_appendI suffixeq_ConsI)

322 qed

324 lemma suffixeq_to_prefixeq [code]: "suffixeq xs ys \<longleftrightarrow> prefixeq (rev xs) (rev ys)"

325 proof

326 assume "suffixeq xs ys"

327 then obtain zs where "ys = zs @ xs" ..

328 then have "rev ys = rev xs @ rev zs" by simp

329 then show "prefixeq (rev xs) (rev ys)" ..

330 next

331 assume "prefixeq (rev xs) (rev ys)"

332 then obtain zs where "rev ys = rev xs @ zs" ..

333 then have "rev (rev ys) = rev zs @ rev (rev xs)" by simp

334 then have "ys = rev zs @ xs" by simp

335 then show "suffixeq xs ys" ..

336 qed

338 lemma distinct_suffixeq: "distinct ys \<Longrightarrow> suffixeq xs ys \<Longrightarrow> distinct xs"

339 by (clarsimp elim!: suffixeqE)

341 lemma suffixeq_map: "suffixeq xs ys \<Longrightarrow> suffixeq (map f xs) (map f ys)"

342 by (auto elim!: suffixeqE intro: suffixeqI)

344 lemma suffixeq_drop: "suffixeq (drop n as) as"

345 unfolding suffixeq_def

346 apply (rule exI [where x = "take n as"])

347 apply simp

348 done

350 lemma suffixeq_take: "suffixeq xs ys \<Longrightarrow> ys = take (length ys - length xs) ys @ xs"

351 by (auto elim!: suffixeqE)

353 lemma suffixeq_suffix_reflclp_conv: "suffixeq = suffix\<^sup>=\<^sup>="

354 proof (intro ext iffI)

355 fix xs ys :: "'a list"

356 assume "suffixeq xs ys"

357 show "suffix\<^sup>=\<^sup>= xs ys"

358 proof

359 assume "xs \<noteq> ys"

360 with `suffixeq xs ys` show "suffix xs ys"

361 by (auto simp: suffixeq_def suffix_def)

362 qed

363 next

364 fix xs ys :: "'a list"

365 assume "suffix\<^sup>=\<^sup>= xs ys"

366 then show "suffixeq xs ys"

367 proof

368 assume "suffix xs ys" then show "suffixeq xs ys"

369 by (rule suffix_imp_suffixeq)

370 next

371 assume "xs = ys" then show "suffixeq xs ys"

372 by (auto simp: suffixeq_def)

373 qed

374 qed

376 lemma parallelD1: "x \<parallel> y \<Longrightarrow> \<not> prefixeq x y"

377 by blast

379 lemma parallelD2: "x \<parallel> y \<Longrightarrow> \<not> prefixeq y x"

380 by blast

382 lemma parallel_Nil1 [simp]: "\<not> x \<parallel> []"

383 unfolding parallel_def by simp

385 lemma parallel_Nil2 [simp]: "\<not> [] \<parallel> x"

386 unfolding parallel_def by simp

388 lemma Cons_parallelI1: "a \<noteq> b \<Longrightarrow> a # as \<parallel> b # bs"

389 by auto

391 lemma Cons_parallelI2: "\<lbrakk> a = b; as \<parallel> bs \<rbrakk> \<Longrightarrow> a # as \<parallel> b # bs"

392 by (metis Cons_prefixeq_Cons parallelE parallelI)

394 lemma not_equal_is_parallel:

395 assumes neq: "xs \<noteq> ys"

396 and len: "length xs = length ys"

397 shows "xs \<parallel> ys"

398 using len neq

399 proof (induct rule: list_induct2)

400 case Nil

401 then show ?case by simp

402 next

403 case (Cons a as b bs)

404 have ih: "as \<noteq> bs \<Longrightarrow> as \<parallel> bs" by fact

405 show ?case

406 proof (cases "a = b")

407 case True

408 then have "as \<noteq> bs" using Cons by simp

409 then show ?thesis by (rule Cons_parallelI2 [OF True ih])

410 next

411 case False

412 then show ?thesis by (rule Cons_parallelI1)

413 qed

414 qed

416 lemma suffix_reflclp_conv: "suffix\<^sup>=\<^sup>= = suffixeq"

417 by (intro ext) (auto simp: suffixeq_def suffix_def)

419 lemma suffix_lists: "suffix xs ys \<Longrightarrow> ys \<in> lists A \<Longrightarrow> xs \<in> lists A"

420 unfolding suffix_def by auto

423 subsection {* Embedding on lists *}

425 inductive emb :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"

426 for P :: "('a \<Rightarrow> 'a \<Rightarrow> bool)"

427 where

428 emb_Nil [intro, simp]: "emb P [] ys"

429 | emb_Cons [intro] : "emb P xs ys \<Longrightarrow> emb P xs (y#ys)"

430 | emb_Cons2 [intro]: "P x y \<Longrightarrow> emb P xs ys \<Longrightarrow> emb P (x#xs) (y#ys)"

432 lemma emb_Nil2 [simp]:

433 assumes "emb P xs []" shows "xs = []"

434 using assms by (cases rule: emb.cases) auto

436 lemma emb_Cons_Nil [simp]: "emb P (x#xs) [] = False"

437 proof -

438 { assume "emb P (x#xs) []"

439 from emb_Nil2 [OF this] have False by simp

440 } moreover {

441 assume False

442 then have "emb P (x#xs) []" by simp

443 } ultimately show ?thesis by blast

444 qed

446 lemma emb_append2 [intro]: "emb P xs ys \<Longrightarrow> emb P xs (zs @ ys)"

447 by (induct zs) auto

449 lemma emb_prefix [intro]:

450 assumes "emb P xs ys" shows "emb P xs (ys @ zs)"

451 using assms

452 by (induct arbitrary: zs) auto

454 lemma emb_ConsD:

455 assumes "emb P (x#xs) ys"

456 shows "\<exists>us v vs. ys = us @ v # vs \<and> P x v \<and> emb P xs vs"

457 using assms

458 proof (induct x \<equiv> "x # xs" ys arbitrary: x xs)

459 case emb_Cons

460 then show ?case by (metis append_Cons)

461 next

462 case (emb_Cons2 x y xs ys)

463 then show ?case by (cases xs) (auto, blast+)

464 qed

466 lemma emb_appendD:

467 assumes "emb P (xs @ ys) zs"

468 shows "\<exists>us vs. zs = us @ vs \<and> emb P xs us \<and> emb P ys vs"

469 using assms

470 proof (induction xs arbitrary: ys zs)

471 case Nil then show ?case by auto

472 next

473 case (Cons x xs)

474 then obtain us v vs where "zs = us @ v # vs"

475 and "P x v" and "emb P (xs @ ys) vs" by (auto dest: emb_ConsD)

476 with Cons show ?case by (metis append_Cons append_assoc emb_Cons2 emb_append2)

477 qed

479 lemma emb_suffix:

480 assumes "emb P xs ys" and "suffix ys zs"

481 shows "emb P xs zs"

482 using assms(2) and emb_append2 [OF assms(1)] by (auto simp: suffix_def)

484 lemma emb_suffixeq:

485 assumes "emb P xs ys" and "suffixeq ys zs"

486 shows "emb P xs zs"

487 using assms and emb_suffix unfolding suffixeq_suffix_reflclp_conv by auto

489 lemma emb_length: "emb P xs ys \<Longrightarrow> length xs \<le> length ys"

490 by (induct rule: emb.induct) auto

492 (*FIXME: move*)

493 definition transp_on :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"

494 where "transp_on P A \<longleftrightarrow> (\<forall>a\<in>A. \<forall>b\<in>A. \<forall>c\<in>A. P a b \<and> P b c \<longrightarrow> P a c)"

495 lemma transp_onI [Pure.intro]:

496 "(\<And>a b c. \<lbrakk>a \<in> A; b \<in> A; c \<in> A; P a b; P b c\<rbrakk> \<Longrightarrow> P a c) \<Longrightarrow> transp_on P A"

497 unfolding transp_on_def by blast

499 lemma transp_on_emb:

500 assumes "transp_on P A"

501 shows "transp_on (emb P) (lists A)"

502 proof

503 fix xs ys zs

504 assume "emb P xs ys" and "emb P ys zs"

505 and "xs \<in> lists A" and "ys \<in> lists A" and "zs \<in> lists A"

506 then show "emb P xs zs"

507 proof (induction arbitrary: zs)

508 case emb_Nil show ?case by blast

509 next

510 case (emb_Cons xs ys y)

511 from emb_ConsD [OF `emb P (y#ys) zs`] obtain us v vs

512 where zs: "zs = us @ v # vs" and "P y v" and "emb P ys vs" by blast

513 then have "emb P ys (v#vs)" by blast

514 then have "emb P ys zs" unfolding zs by (rule emb_append2)

515 from emb_Cons.IH [OF this] and emb_Cons.prems show ?case by simp

516 next

517 case (emb_Cons2 x y xs ys)

518 from emb_ConsD [OF `emb P (y#ys) zs`] obtain us v vs

519 where zs: "zs = us @ v # vs" and "P y v" and "emb P ys vs" by blast

520 with emb_Cons2 have "emb P xs vs" by simp

521 moreover have "P x v"

522 proof -

523 from zs and `zs \<in> lists A` have "v \<in> A" by auto

524 moreover have "x \<in> A" and "y \<in> A" using emb_Cons2 by simp_all

525 ultimately show ?thesis using `P x y` and `P y v` and assms

526 unfolding transp_on_def by blast

527 qed

528 ultimately have "emb P (x#xs) (v#vs)" by blast

529 then show ?case unfolding zs by (rule emb_append2)

530 qed

531 qed

534 subsection {* Sublists (special case of embedding) *}

536 abbreviation sub :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"

537 where "sub xs ys \<equiv> emb (op =) xs ys"

539 lemma sub_Cons2: "sub xs ys \<Longrightarrow> sub (x#xs) (x#ys)" by auto

541 lemma sub_same_length:

542 assumes "sub xs ys" and "length xs = length ys" shows "xs = ys"

543 using assms by (induct) (auto dest: emb_length)

545 lemma not_sub_length [simp]: "length ys < length xs \<Longrightarrow> \<not> sub xs ys"

546 by (metis emb_length linorder_not_less)

548 lemma [code]:

549 "emb P [] ys \<longleftrightarrow> True"

550 "emb P (x#xs) [] \<longleftrightarrow> False"

551 by (simp_all)

553 lemma sub_Cons': "sub (x#xs) ys \<Longrightarrow> sub xs ys"

554 by (induct xs) (auto dest: emb_ConsD)

556 lemma sub_Cons2':

557 assumes "sub (x#xs) (x#ys)" shows "sub xs ys"

558 using assms by (cases) (rule sub_Cons')

560 lemma sub_Cons2_neq:

561 assumes "sub (x#xs) (y#ys)"

562 shows "x \<noteq> y \<Longrightarrow> sub (x#xs) ys"

563 using assms by (cases) auto

565 lemma sub_Cons2_iff [simp, code]:

566 "sub (x#xs) (y#ys) = (if x = y then sub xs ys else sub (x#xs) ys)"

567 by (metis emb_Cons emb_Cons2 [of "op =", OF refl] sub_Cons2' sub_Cons2_neq)

569 lemma sub_append': "sub (zs @ xs) (zs @ ys) \<longleftrightarrow> sub xs ys"

570 by (induct zs) simp_all

572 lemma sub_refl [simp, intro!]: "sub xs xs" by (induct xs) simp_all

574 lemma sub_antisym:

575 assumes "sub xs ys" and "sub ys xs"

576 shows "xs = ys"

577 using assms

578 proof (induct)

579 case emb_Nil

580 from emb_Nil2 [OF this] show ?case by simp

581 next

582 case emb_Cons2

583 then show ?case by simp

584 next

585 case emb_Cons

586 then show ?case

587 by (metis sub_Cons' emb_length Suc_length_conv Suc_n_not_le_n)

588 qed

590 lemma transp_on_sub: "transp_on sub UNIV"

591 proof -

592 have "transp_on (op =) UNIV" by (simp add: transp_on_def)

593 from transp_on_emb [OF this] show ?thesis by simp

594 qed

596 lemma sub_trans: "sub xs ys \<Longrightarrow> sub ys zs \<Longrightarrow> sub xs zs"

597 using transp_on_sub [unfolded transp_on_def] by blast

599 lemma sub_append_le_same_iff: "sub (xs @ ys) ys \<longleftrightarrow> xs = []"

600 by (auto dest: emb_length)

602 lemma emb_append_mono:

603 "\<lbrakk> emb P xs xs'; emb P ys ys' \<rbrakk> \<Longrightarrow> emb P (xs@ys) (xs'@ys')"

604 apply (induct rule: emb.induct)

605 apply (metis eq_Nil_appendI emb_append2)

606 apply (metis append_Cons emb_Cons)

607 apply (metis append_Cons emb_Cons2)

608 done

611 subsection {* Appending elements *}

613 lemma sub_append [simp]:

614 "sub (xs @ zs) (ys @ zs) \<longleftrightarrow> sub xs ys" (is "?l = ?r")

615 proof

616 { fix xs' ys' xs ys zs :: "'a list" assume "sub xs' ys'"

617 then have "xs' = xs @ zs & ys' = ys @ zs \<longrightarrow> sub xs ys"

618 proof (induct arbitrary: xs ys zs)

619 case emb_Nil show ?case by simp

620 next

621 case (emb_Cons xs' ys' x)

622 { assume "ys=[]" then have ?case using emb_Cons(1) by auto }

623 moreover

624 { fix us assume "ys = x#us"

625 then have ?case using emb_Cons(2) by(simp add: emb.emb_Cons) }

626 ultimately show ?case by (auto simp:Cons_eq_append_conv)

627 next

628 case (emb_Cons2 x y xs' ys')

629 { assume "xs=[]" then have ?case using emb_Cons2(1) by auto }

630 moreover

631 { fix us vs assume "xs=x#us" "ys=x#vs" then have ?case using emb_Cons2 by auto}

632 moreover

633 { fix us assume "xs=x#us" "ys=[]" then have ?case using emb_Cons2(2) by bestsimp }

634 ultimately show ?case using `x = y` by (auto simp: Cons_eq_append_conv)

635 qed }

636 moreover assume ?l

637 ultimately show ?r by blast

638 next

639 assume ?r then show ?l by (metis emb_append_mono sub_refl)

640 qed

642 lemma sub_drop_many: "sub xs ys \<Longrightarrow> sub xs (zs @ ys)"

643 by (induct zs) auto

645 lemma sub_rev_drop_many: "sub xs ys \<Longrightarrow> sub xs (ys @ zs)"

646 by (metis append_Nil2 emb_Nil emb_append_mono)

649 subsection {* Relation to standard list operations *}

651 lemma sub_map:

652 assumes "sub xs ys" shows "sub (map f xs) (map f ys)"

653 using assms by (induct) auto

655 lemma sub_filter_left [simp]: "sub (filter P xs) xs"

656 by (induct xs) auto

658 lemma sub_filter [simp]:

659 assumes "sub xs ys" shows "sub (filter P xs) (filter P ys)"

660 using assms by (induct) auto

662 lemma "sub xs ys \<longleftrightarrow> (\<exists>N. xs = sublist ys N)" (is "?L = ?R")

663 proof

664 assume ?L

665 then show ?R

666 proof (induct)

667 case emb_Nil show ?case by (metis sublist_empty)

668 next

669 case (emb_Cons xs ys x)

670 then obtain N where "xs = sublist ys N" by blast

671 then have "xs = sublist (x#ys) (Suc ` N)"

672 by (clarsimp simp add:sublist_Cons inj_image_mem_iff)

673 then show ?case by blast

674 next

675 case (emb_Cons2 x y xs ys)

676 then obtain N where "xs = sublist ys N" by blast

677 then have "x#xs = sublist (x#ys) (insert 0 (Suc ` N))"

678 by (clarsimp simp add:sublist_Cons inj_image_mem_iff)

679 then show ?case unfolding `x = y` by blast

680 qed

681 next

682 assume ?R

683 then obtain N where "xs = sublist ys N" ..

684 moreover have "sub (sublist ys N) ys"

685 proof (induct ys arbitrary: N)

686 case Nil show ?case by simp

687 next

688 case Cons then show ?case by (auto simp: sublist_Cons)

689 qed

690 ultimately show ?L by simp

691 qed

693 end