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src/HOL/Library/List_Prefix.thy

author | wenzelm |

Thu Feb 11 23:00:22 2010 +0100 (2010-02-11) | |

changeset 35115 | 446c5063e4fd |

parent 30663 | 0b6aff7451b2 |

child 37474 | ce943f9edf5e |

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

modernized translations;

formal markup of @{syntax_const} and @{const_syntax};

minor tuning;

formal markup of @{syntax_const} and @{const_syntax};

minor tuning;

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

2 Author: Tobias Nipkow and Markus Wenzel, TU Muenchen

3 *)

5 header {* List prefixes and postfixes *}

7 theory List_Prefix

8 imports List Main

9 begin

11 subsection {* Prefix order on lists *}

13 instantiation list :: (type) order

14 begin

16 definition

17 prefix_def [code del]: "xs \<le> ys = (\<exists>zs. ys = xs @ zs)"

19 definition

20 strict_prefix_def [code del]: "xs < ys = (xs \<le> ys \<and> xs \<noteq> (ys::'a list))"

22 instance

23 by intro_classes (auto simp add: prefix_def strict_prefix_def)

25 end

27 lemma prefixI [intro?]: "ys = xs @ zs ==> xs \<le> ys"

28 unfolding prefix_def by blast

30 lemma prefixE [elim?]:

31 assumes "xs \<le> ys"

32 obtains zs where "ys = xs @ zs"

33 using assms unfolding prefix_def by blast

35 lemma strict_prefixI' [intro?]: "ys = xs @ z # zs ==> xs < ys"

36 unfolding strict_prefix_def prefix_def by blast

38 lemma strict_prefixE' [elim?]:

39 assumes "xs < ys"

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

41 proof -

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

43 unfolding strict_prefix_def prefix_def by blast

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

45 qed

47 lemma strict_prefixI [intro?]: "xs \<le> ys ==> xs \<noteq> ys ==> xs < (ys::'a list)"

48 unfolding strict_prefix_def by blast

50 lemma strict_prefixE [elim?]:

51 fixes xs ys :: "'a list"

52 assumes "xs < ys"

53 obtains "xs \<le> ys" and "xs \<noteq> ys"

54 using assms unfolding strict_prefix_def by blast

57 subsection {* Basic properties of prefixes *}

59 theorem Nil_prefix [iff]: "[] \<le> xs"

60 by (simp add: prefix_def)

62 theorem prefix_Nil [simp]: "(xs \<le> []) = (xs = [])"

63 by (induct xs) (simp_all add: prefix_def)

65 lemma prefix_snoc [simp]: "(xs \<le> ys @ [y]) = (xs = ys @ [y] \<or> xs \<le> ys)"

66 proof

67 assume "xs \<le> ys @ [y]"

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

69 show "xs = ys @ [y] \<or> xs \<le> ys"

70 by (metis append_Nil2 butlast_append butlast_snoc prefixI zs)

71 next

72 assume "xs = ys @ [y] \<or> xs \<le> ys"

73 then show "xs \<le> ys @ [y]"

74 by (metis order_eq_iff strict_prefixE strict_prefixI' xt1(7))

75 qed

77 lemma Cons_prefix_Cons [simp]: "(x # xs \<le> y # ys) = (x = y \<and> xs \<le> ys)"

78 by (auto simp add: prefix_def)

80 lemma same_prefix_prefix [simp]: "(xs @ ys \<le> xs @ zs) = (ys \<le> zs)"

81 by (induct xs) simp_all

83 lemma same_prefix_nil [iff]: "(xs @ ys \<le> xs) = (ys = [])"

84 by (metis append_Nil2 append_self_conv order_eq_iff prefixI)

86 lemma prefix_prefix [simp]: "xs \<le> ys ==> xs \<le> ys @ zs"

87 by (metis order_le_less_trans prefixI strict_prefixE strict_prefixI)

89 lemma append_prefixD: "xs @ ys \<le> zs \<Longrightarrow> xs \<le> zs"

90 by (auto simp add: prefix_def)

92 theorem prefix_Cons: "(xs \<le> y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> zs \<le> ys))"

93 by (cases xs) (auto simp add: prefix_def)

95 theorem prefix_append:

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

97 apply (induct zs rule: rev_induct)

98 apply force

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

100 apply (metis append_eq_appendI)

101 done

103 lemma append_one_prefix:

104 "xs \<le> ys ==> length xs < length ys ==> xs @ [ys ! length xs] \<le> ys"

105 unfolding prefix_def

106 by (metis Cons_eq_appendI append_eq_appendI append_eq_conv_conj

107 eq_Nil_appendI nth_drop')

109 theorem prefix_length_le: "xs \<le> ys ==> length xs \<le> length ys"

110 by (auto simp add: prefix_def)

112 lemma prefix_same_cases:

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

114 unfolding prefix_def by (metis append_eq_append_conv2)

116 lemma set_mono_prefix: "xs \<le> ys \<Longrightarrow> set xs \<subseteq> set ys"

117 by (auto simp add: prefix_def)

119 lemma take_is_prefix: "take n xs \<le> xs"

120 unfolding prefix_def by (metis append_take_drop_id)

122 lemma map_prefixI: "xs \<le> ys \<Longrightarrow> map f xs \<le> map f ys"

123 by (auto simp: prefix_def)

125 lemma prefix_length_less: "xs < ys \<Longrightarrow> length xs < length ys"

126 by (auto simp: strict_prefix_def prefix_def)

128 lemma strict_prefix_simps [simp]:

129 "xs < [] = False"

130 "[] < (x # xs) = True"

131 "(x # xs) < (y # ys) = (x = y \<and> xs < ys)"

132 by (simp_all add: strict_prefix_def cong: conj_cong)

134 lemma take_strict_prefix: "xs < ys \<Longrightarrow> take n xs < ys"

135 apply (induct n arbitrary: xs ys)

136 apply (case_tac ys, simp_all)[1]

137 apply (metis order_less_trans strict_prefixI take_is_prefix)

138 done

140 lemma not_prefix_cases:

141 assumes pfx: "\<not> ps \<le> ls"

142 obtains

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

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

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

146 proof (cases ps)

147 case Nil then show ?thesis using pfx by simp

148 next

149 case (Cons a as)

150 note c = `ps = a#as`

151 show ?thesis

152 proof (cases ls)

153 case Nil then show ?thesis by (metis append_Nil2 pfx c1 same_prefix_nil)

154 next

155 case (Cons x xs)

156 show ?thesis

157 proof (cases "x = a")

158 case True

159 have "\<not> as \<le> xs" using pfx c Cons True by simp

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

161 next

162 case False

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

164 qed

165 qed

166 qed

168 lemma not_prefix_induct [consumes 1, case_names Nil Neq Eq]:

169 assumes np: "\<not> ps \<le> ls"

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

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

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

173 shows "P ps ls" using np

174 proof (induct ls arbitrary: ps)

175 case Nil then show ?case

176 by (auto simp: neq_Nil_conv elim!: not_prefix_cases intro!: base)

177 next

178 case (Cons y ys)

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

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

181 by (rule not_prefix_cases) auto

182 show ?case by (metis Cons.hyps Cons_prefix_Cons npfx pv r1 r2)

183 qed

186 subsection {* Parallel lists *}

188 definition

189 parallel :: "'a list => 'a list => bool" (infixl "\<parallel>" 50) where

190 "(xs \<parallel> ys) = (\<not> xs \<le> ys \<and> \<not> ys \<le> xs)"

192 lemma parallelI [intro]: "\<not> xs \<le> ys ==> \<not> ys \<le> xs ==> xs \<parallel> ys"

193 unfolding parallel_def by blast

195 lemma parallelE [elim]:

196 assumes "xs \<parallel> ys"

197 obtains "\<not> xs \<le> ys \<and> \<not> ys \<le> xs"

198 using assms unfolding parallel_def by blast

200 theorem prefix_cases:

201 obtains "xs \<le> ys" | "ys < xs" | "xs \<parallel> ys"

202 unfolding parallel_def strict_prefix_def by blast

204 theorem parallel_decomp:

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

206 proof (induct xs rule: rev_induct)

207 case Nil

208 then have False by auto

209 then show ?case ..

210 next

211 case (snoc x xs)

212 show ?case

213 proof (rule prefix_cases)

214 assume le: "xs \<le> ys"

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

216 show ?thesis

217 proof (cases ys')

218 assume "ys' = []"

219 then show ?thesis by (metis append_Nil2 parallelE prefixI snoc.prems ys)

220 next

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

222 then show ?thesis

223 by (metis Cons_eq_appendI eq_Nil_appendI parallelE prefixI

224 same_prefix_prefix snoc.prems ys)

225 qed

226 next

227 assume "ys < xs" then have "ys \<le> xs @ [x]" by (simp add: strict_prefix_def)

228 with snoc have False by blast

229 then show ?thesis ..

230 next

231 assume "xs \<parallel> ys"

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

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

234 by blast

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

236 with neq ys show ?thesis by blast

237 qed

238 qed

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

241 apply (rule parallelI)

242 apply (erule parallelE, erule conjE,

243 induct rule: not_prefix_induct, simp+)+

244 done

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

247 by (simp add: parallel_append)

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

250 unfolding parallel_def by auto

253 subsection {* Postfix order on lists *}

255 definition

256 postfix :: "'a list => 'a list => bool" ("(_/ >>= _)" [51, 50] 50) where

257 "(xs >>= ys) = (\<exists>zs. xs = zs @ ys)"

259 lemma postfixI [intro?]: "xs = zs @ ys ==> xs >>= ys"

260 unfolding postfix_def by blast

262 lemma postfixE [elim?]:

263 assumes "xs >>= ys"

264 obtains zs where "xs = zs @ ys"

265 using assms unfolding postfix_def by blast

267 lemma postfix_refl [iff]: "xs >>= xs"

268 by (auto simp add: postfix_def)

269 lemma postfix_trans: "\<lbrakk>xs >>= ys; ys >>= zs\<rbrakk> \<Longrightarrow> xs >>= zs"

270 by (auto simp add: postfix_def)

271 lemma postfix_antisym: "\<lbrakk>xs >>= ys; ys >>= xs\<rbrakk> \<Longrightarrow> xs = ys"

272 by (auto simp add: postfix_def)

274 lemma Nil_postfix [iff]: "xs >>= []"

275 by (simp add: postfix_def)

276 lemma postfix_Nil [simp]: "([] >>= xs) = (xs = [])"

277 by (auto simp add: postfix_def)

279 lemma postfix_ConsI: "xs >>= ys \<Longrightarrow> x#xs >>= ys"

280 by (auto simp add: postfix_def)

281 lemma postfix_ConsD: "xs >>= y#ys \<Longrightarrow> xs >>= ys"

282 by (auto simp add: postfix_def)

284 lemma postfix_appendI: "xs >>= ys \<Longrightarrow> zs @ xs >>= ys"

285 by (auto simp add: postfix_def)

286 lemma postfix_appendD: "xs >>= zs @ ys \<Longrightarrow> xs >>= ys"

287 by (auto simp add: postfix_def)

289 lemma postfix_is_subset: "xs >>= ys ==> set ys \<subseteq> set xs"

290 proof -

291 assume "xs >>= ys"

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

293 then show ?thesis by (induct zs) auto

294 qed

296 lemma postfix_ConsD2: "x#xs >>= y#ys ==> xs >>= ys"

297 proof -

298 assume "x#xs >>= y#ys"

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

300 then show ?thesis

301 by (induct zs) (auto intro!: postfix_appendI postfix_ConsI)

302 qed

304 lemma postfix_to_prefix: "xs >>= ys \<longleftrightarrow> rev ys \<le> rev xs"

305 proof

306 assume "xs >>= ys"

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

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

309 then show "rev ys <= rev xs" ..

310 next

311 assume "rev ys <= rev xs"

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

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

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

315 then show "xs >>= ys" ..

316 qed

318 lemma distinct_postfix: "distinct xs \<Longrightarrow> xs >>= ys \<Longrightarrow> distinct ys"

319 by (clarsimp elim!: postfixE)

321 lemma postfix_map: "xs >>= ys \<Longrightarrow> map f xs >>= map f ys"

322 by (auto elim!: postfixE intro: postfixI)

324 lemma postfix_drop: "as >>= drop n as"

325 unfolding postfix_def

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

327 apply simp

328 done

330 lemma postfix_take: "xs >>= ys \<Longrightarrow> xs = take (length xs - length ys) xs @ ys"

331 by (clarsimp elim!: postfixE)

333 lemma parallelD1: "x \<parallel> y \<Longrightarrow> \<not> x \<le> y"

334 by blast

336 lemma parallelD2: "x \<parallel> y \<Longrightarrow> \<not> y \<le> x"

337 by blast

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

340 unfolding parallel_def by simp

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

343 unfolding parallel_def by simp

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

346 by auto

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

349 by (metis Cons_prefix_Cons parallelE parallelI)

351 lemma not_equal_is_parallel:

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

353 and len: "length xs = length ys"

354 shows "xs \<parallel> ys"

355 using len neq

356 proof (induct rule: list_induct2)

357 case Nil

358 then show ?case by simp

359 next

360 case (Cons a as b bs)

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

362 show ?case

363 proof (cases "a = b")

364 case True

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

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

367 next

368 case False

369 then show ?thesis by (rule Cons_parallelI1)

370 qed

371 qed

374 subsection {* Executable code *}

376 lemma less_eq_code [code]:

377 "([]\<Colon>'a\<Colon>{eq, ord} list) \<le> xs \<longleftrightarrow> True"

378 "(x\<Colon>'a\<Colon>{eq, ord}) # xs \<le> [] \<longleftrightarrow> False"

379 "(x\<Colon>'a\<Colon>{eq, ord}) # xs \<le> y # ys \<longleftrightarrow> x = y \<and> xs \<le> ys"

380 by simp_all

382 lemma less_code [code]:

383 "xs < ([]\<Colon>'a\<Colon>{eq, ord} list) \<longleftrightarrow> False"

384 "[] < (x\<Colon>'a\<Colon>{eq, ord})# xs \<longleftrightarrow> True"

385 "(x\<Colon>'a\<Colon>{eq, ord}) # xs < y # ys \<longleftrightarrow> x = y \<and> xs < ys"

386 unfolding strict_prefix_def by auto

388 lemmas [code] = postfix_to_prefix

390 end