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src/HOL/Cardinals/Order_Union.thy

author | wenzelm |

Fri Oct 27 13:50:08 2017 +0200 (23 months ago) | |

changeset 66924 | b4d4027f743b |

parent 66453 | cc19f7ca2ed6 |

child 67006 | b1278ed3cd46 |

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

more permissive;

1 (* Title: HOL/Cardinals/Order_Union.thy

2 Author: Andrei Popescu, TU Muenchen

4 The ordinal-like sum of two orders with disjoint fields

5 *)

7 section \<open>Order Union\<close>

9 theory Order_Union

10 imports HOL.Order_Relation

11 begin

13 definition Osum :: "'a rel \<Rightarrow> 'a rel \<Rightarrow> 'a rel" (infix "Osum" 60) where

14 "r Osum r' = r \<union> r' \<union> {(a, a'). a \<in> Field r \<and> a' \<in> Field r'}"

16 notation Osum (infix "\<union>o" 60)

18 lemma Field_Osum: "Field (r \<union>o r') = Field r \<union> Field r'"

19 unfolding Osum_def Field_def by blast

21 lemma Osum_wf:

22 assumes FLD: "Field r Int Field r' = {}" and

23 WF: "wf r" and WF': "wf r'"

24 shows "wf (r Osum r')"

25 unfolding wf_eq_minimal2 unfolding Field_Osum

26 proof(intro allI impI, elim conjE)

27 fix A assume *: "A \<subseteq> Field r \<union> Field r'" and **: "A \<noteq> {}"

28 obtain B where B_def: "B = A Int Field r" by blast

29 show "\<exists>a\<in>A. \<forall>a'\<in>A. (a', a) \<notin> r \<union>o r'"

30 proof(cases "B = {}")

31 assume Case1: "B \<noteq> {}"

32 hence "B \<noteq> {} \<and> B \<le> Field r" using B_def by auto

33 then obtain a where 1: "a \<in> B" and 2: "\<forall>a1 \<in> B. (a1,a) \<notin> r"

34 using WF unfolding wf_eq_minimal2 by blast

35 hence 3: "a \<in> Field r \<and> a \<notin> Field r'" using B_def FLD by auto

36 (* *)

37 have "\<forall>a1 \<in> A. (a1,a) \<notin> r Osum r'"

38 proof(intro ballI)

39 fix a1 assume **: "a1 \<in> A"

40 {assume Case11: "a1 \<in> Field r"

41 hence "(a1,a) \<notin> r" using B_def ** 2 by auto

42 moreover

43 have "(a1,a) \<notin> r'" using 3 by (auto simp add: Field_def)

44 ultimately have "(a1,a) \<notin> r Osum r'"

45 using 3 unfolding Osum_def by auto

46 }

47 moreover

48 {assume Case12: "a1 \<notin> Field r"

49 hence "(a1,a) \<notin> r" unfolding Field_def by auto

50 moreover

51 have "(a1,a) \<notin> r'" using 3 unfolding Field_def by auto

52 ultimately have "(a1,a) \<notin> r Osum r'"

53 using 3 unfolding Osum_def by auto

54 }

55 ultimately show "(a1,a) \<notin> r Osum r'" by blast

56 qed

57 thus ?thesis using 1 B_def by auto

58 next

59 assume Case2: "B = {}"

60 hence 1: "A \<noteq> {} \<and> A \<le> Field r'" using * ** B_def by auto

61 then obtain a' where 2: "a' \<in> A" and 3: "\<forall>a1' \<in> A. (a1',a') \<notin> r'"

62 using WF' unfolding wf_eq_minimal2 by blast

63 hence 4: "a' \<in> Field r' \<and> a' \<notin> Field r" using 1 FLD by blast

64 (* *)

65 have "\<forall>a1' \<in> A. (a1',a') \<notin> r Osum r'"

66 proof(unfold Osum_def, auto simp add: 3)

67 fix a1' assume "(a1', a') \<in> r"

68 thus False using 4 unfolding Field_def by blast

69 next

70 fix a1' assume "a1' \<in> A" and "a1' \<in> Field r"

71 thus False using Case2 B_def by auto

72 qed

73 thus ?thesis using 2 by blast

74 qed

75 qed

77 lemma Osum_Refl:

78 assumes FLD: "Field r Int Field r' = {}" and

79 REFL: "Refl r" and REFL': "Refl r'"

80 shows "Refl (r Osum r')"

81 using assms

82 unfolding refl_on_def Field_Osum unfolding Osum_def by blast

84 lemma Osum_trans:

85 assumes FLD: "Field r Int Field r' = {}" and

86 TRANS: "trans r" and TRANS': "trans r'"

87 shows "trans (r Osum r')"

88 proof(unfold trans_def, auto)

89 fix x y z assume *: "(x, y) \<in> r \<union>o r'" and **: "(y, z) \<in> r \<union>o r'"

90 show "(x, z) \<in> r \<union>o r'"

91 proof-

92 {assume Case1: "(x,y) \<in> r"

93 hence 1: "x \<in> Field r \<and> y \<in> Field r" unfolding Field_def by auto

94 have ?thesis

95 proof-

96 {assume Case11: "(y,z) \<in> r"

97 hence "(x,z) \<in> r" using Case1 TRANS trans_def[of r] by blast

98 hence ?thesis unfolding Osum_def by auto

99 }

100 moreover

101 {assume Case12: "(y,z) \<in> r'"

102 hence "y \<in> Field r'" unfolding Field_def by auto

103 hence False using FLD 1 by auto

104 }

105 moreover

106 {assume Case13: "z \<in> Field r'"

107 hence ?thesis using 1 unfolding Osum_def by auto

108 }

109 ultimately show ?thesis using ** unfolding Osum_def by blast

110 qed

111 }

112 moreover

113 {assume Case2: "(x,y) \<in> r'"

114 hence 2: "x \<in> Field r' \<and> y \<in> Field r'" unfolding Field_def by auto

115 have ?thesis

116 proof-

117 {assume Case21: "(y,z) \<in> r"

118 hence "y \<in> Field r" unfolding Field_def by auto

119 hence False using FLD 2 by auto

120 }

121 moreover

122 {assume Case22: "(y,z) \<in> r'"

123 hence "(x,z) \<in> r'" using Case2 TRANS' trans_def[of r'] by blast

124 hence ?thesis unfolding Osum_def by auto

125 }

126 moreover

127 {assume Case23: "y \<in> Field r"

128 hence False using FLD 2 by auto

129 }

130 ultimately show ?thesis using ** unfolding Osum_def by blast

131 qed

132 }

133 moreover

134 {assume Case3: "x \<in> Field r \<and> y \<in> Field r'"

135 have ?thesis

136 proof-

137 {assume Case31: "(y,z) \<in> r"

138 hence "y \<in> Field r" unfolding Field_def by auto

139 hence False using FLD Case3 by auto

140 }

141 moreover

142 {assume Case32: "(y,z) \<in> r'"

143 hence "z \<in> Field r'" unfolding Field_def by blast

144 hence ?thesis unfolding Osum_def using Case3 by auto

145 }

146 moreover

147 {assume Case33: "y \<in> Field r"

148 hence False using FLD Case3 by auto

149 }

150 ultimately show ?thesis using ** unfolding Osum_def by blast

151 qed

152 }

153 ultimately show ?thesis using * unfolding Osum_def by blast

154 qed

155 qed

157 lemma Osum_Preorder:

158 "\<lbrakk>Field r Int Field r' = {}; Preorder r; Preorder r'\<rbrakk> \<Longrightarrow> Preorder (r Osum r')"

159 unfolding preorder_on_def using Osum_Refl Osum_trans by blast

161 lemma Osum_antisym:

162 assumes FLD: "Field r Int Field r' = {}" and

163 AN: "antisym r" and AN': "antisym r'"

164 shows "antisym (r Osum r')"

165 proof(unfold antisym_def, auto)

166 fix x y assume *: "(x, y) \<in> r \<union>o r'" and **: "(y, x) \<in> r \<union>o r'"

167 show "x = y"

168 proof-

169 {assume Case1: "(x,y) \<in> r"

170 hence 1: "x \<in> Field r \<and> y \<in> Field r" unfolding Field_def by auto

171 have ?thesis

172 proof-

173 have "(y,x) \<in> r \<Longrightarrow> ?thesis"

174 using Case1 AN antisym_def[of r] by blast

175 moreover

176 {assume "(y,x) \<in> r'"

177 hence "y \<in> Field r'" unfolding Field_def by auto

178 hence False using FLD 1 by auto

179 }

180 moreover

181 have "x \<in> Field r' \<Longrightarrow> False" using FLD 1 by auto

182 ultimately show ?thesis using ** unfolding Osum_def by blast

183 qed

184 }

185 moreover

186 {assume Case2: "(x,y) \<in> r'"

187 hence 2: "x \<in> Field r' \<and> y \<in> Field r'" unfolding Field_def by auto

188 have ?thesis

189 proof-

190 {assume "(y,x) \<in> r"

191 hence "y \<in> Field r" unfolding Field_def by auto

192 hence False using FLD 2 by auto

193 }

194 moreover

195 have "(y,x) \<in> r' \<Longrightarrow> ?thesis"

196 using Case2 AN' antisym_def[of r'] by blast

197 moreover

198 {assume "y \<in> Field r"

199 hence False using FLD 2 by auto

200 }

201 ultimately show ?thesis using ** unfolding Osum_def by blast

202 qed

203 }

204 moreover

205 {assume Case3: "x \<in> Field r \<and> y \<in> Field r'"

206 have ?thesis

207 proof-

208 {assume "(y,x) \<in> r"

209 hence "y \<in> Field r" unfolding Field_def by auto

210 hence False using FLD Case3 by auto

211 }

212 moreover

213 {assume Case32: "(y,x) \<in> r'"

214 hence "x \<in> Field r'" unfolding Field_def by blast

215 hence False using FLD Case3 by auto

216 }

217 moreover

218 have "\<not> y \<in> Field r" using FLD Case3 by auto

219 ultimately show ?thesis using ** unfolding Osum_def by blast

220 qed

221 }

222 ultimately show ?thesis using * unfolding Osum_def by blast

223 qed

224 qed

226 lemma Osum_Partial_order:

227 "\<lbrakk>Field r Int Field r' = {}; Partial_order r; Partial_order r'\<rbrakk> \<Longrightarrow>

228 Partial_order (r Osum r')"

229 unfolding partial_order_on_def using Osum_Preorder Osum_antisym by blast

231 lemma Osum_Total:

232 assumes FLD: "Field r Int Field r' = {}" and

233 TOT: "Total r" and TOT': "Total r'"

234 shows "Total (r Osum r')"

235 using assms

236 unfolding total_on_def Field_Osum unfolding Osum_def by blast

238 lemma Osum_Linear_order:

239 "\<lbrakk>Field r Int Field r' = {}; Linear_order r; Linear_order r'\<rbrakk> \<Longrightarrow>

240 Linear_order (r Osum r')"

241 unfolding linear_order_on_def using Osum_Partial_order Osum_Total by blast

243 lemma Osum_minus_Id1:

244 assumes "r \<le> Id"

245 shows "(r Osum r') - Id \<le> (r' - Id) \<union> (Field r \<times> Field r')"

246 proof-

247 let ?Left = "(r Osum r') - Id"

248 let ?Right = "(r' - Id) \<union> (Field r \<times> Field r')"

249 {fix a::'a and b assume *: "(a,b) \<notin> Id"

250 {assume "(a,b) \<in> r"

251 with * have False using assms by auto

252 }

253 moreover

254 {assume "(a,b) \<in> r'"

255 with * have "(a,b) \<in> r' - Id" by auto

256 }

257 ultimately

258 have "(a,b) \<in> ?Left \<Longrightarrow> (a,b) \<in> ?Right"

259 unfolding Osum_def by auto

260 }

261 thus ?thesis by auto

262 qed

264 lemma Osum_minus_Id2:

265 assumes "r' \<le> Id"

266 shows "(r Osum r') - Id \<le> (r - Id) \<union> (Field r \<times> Field r')"

267 proof-

268 let ?Left = "(r Osum r') - Id"

269 let ?Right = "(r - Id) \<union> (Field r \<times> Field r')"

270 {fix a::'a and b assume *: "(a,b) \<notin> Id"

271 {assume "(a,b) \<in> r'"

272 with * have False using assms by auto

273 }

274 moreover

275 {assume "(a,b) \<in> r"

276 with * have "(a,b) \<in> r - Id" by auto

277 }

278 ultimately

279 have "(a,b) \<in> ?Left \<Longrightarrow> (a,b) \<in> ?Right"

280 unfolding Osum_def by auto

281 }

282 thus ?thesis by auto

283 qed

285 lemma Osum_minus_Id:

286 assumes TOT: "Total r" and TOT': "Total r'" and

287 NID: "\<not> (r \<le> Id)" and NID': "\<not> (r' \<le> Id)"

288 shows "(r Osum r') - Id \<le> (r - Id) Osum (r' - Id)"

289 proof-

290 {fix a a' assume *: "(a,a') \<in> (r Osum r')" and **: "a \<noteq> a'"

291 have "(a,a') \<in> (r - Id) Osum (r' - Id)"

292 proof-

293 {assume "(a,a') \<in> r \<or> (a,a') \<in> r'"

294 with ** have ?thesis unfolding Osum_def by auto

295 }

296 moreover

297 {assume "a \<in> Field r \<and> a' \<in> Field r'"

298 hence "a \<in> Field(r - Id) \<and> a' \<in> Field (r' - Id)"

299 using assms Total_Id_Field by blast

300 hence ?thesis unfolding Osum_def by auto

301 }

302 ultimately show ?thesis using * unfolding Osum_def by fast

303 qed

304 }

305 thus ?thesis by(auto simp add: Osum_def)

306 qed

308 lemma wf_Int_Times:

309 assumes "A Int B = {}"

310 shows "wf(A \<times> B)"

311 unfolding wf_def using assms by blast

313 lemma Osum_wf_Id:

314 assumes TOT: "Total r" and TOT': "Total r'" and

315 FLD: "Field r Int Field r' = {}" and

316 WF: "wf(r - Id)" and WF': "wf(r' - Id)"

317 shows "wf ((r Osum r') - Id)"

318 proof(cases "r \<le> Id \<or> r' \<le> Id")

319 assume Case1: "\<not>(r \<le> Id \<or> r' \<le> Id)"

320 have "Field(r - Id) Int Field(r' - Id) = {}"

321 using FLD mono_Field[of "r - Id" r] mono_Field[of "r' - Id" r']

322 Diff_subset[of r Id] Diff_subset[of r' Id] by blast

323 thus ?thesis

324 using Case1 Osum_minus_Id[of r r'] assms Osum_wf[of "r - Id" "r' - Id"]

325 wf_subset[of "(r - Id) \<union>o (r' - Id)" "(r Osum r') - Id"] by auto

326 next

327 have 1: "wf(Field r \<times> Field r')"

328 using FLD by (auto simp add: wf_Int_Times)

329 assume Case2: "r \<le> Id \<or> r' \<le> Id"

330 moreover

331 {assume Case21: "r \<le> Id"

332 hence "(r Osum r') - Id \<le> (r' - Id) \<union> (Field r \<times> Field r')"

333 using Osum_minus_Id1[of r r'] by simp

334 moreover

335 {have "Domain(Field r \<times> Field r') Int Range(r' - Id) = {}"

336 using FLD unfolding Field_def by blast

337 hence "wf((r' - Id) \<union> (Field r \<times> Field r'))"

338 using 1 WF' wf_Un[of "Field r \<times> Field r'" "r' - Id"]

339 by (auto simp add: Un_commute)

340 }

341 ultimately have ?thesis using wf_subset by blast

342 }

343 moreover

344 {assume Case22: "r' \<le> Id"

345 hence "(r Osum r') - Id \<le> (r - Id) \<union> (Field r \<times> Field r')"

346 using Osum_minus_Id2[of r' r] by simp

347 moreover

348 {have "Range(Field r \<times> Field r') Int Domain(r - Id) = {}"

349 using FLD unfolding Field_def by blast

350 hence "wf((r - Id) \<union> (Field r \<times> Field r'))"

351 using 1 WF wf_Un[of "r - Id" "Field r \<times> Field r'"]

352 by (auto simp add: Un_commute)

353 }

354 ultimately have ?thesis using wf_subset by blast

355 }

356 ultimately show ?thesis by blast

357 qed

359 lemma Osum_Well_order:

360 assumes FLD: "Field r Int Field r' = {}" and

361 WELL: "Well_order r" and WELL': "Well_order r'"

362 shows "Well_order (r Osum r')"

363 proof-

364 have "Total r \<and> Total r'" using WELL WELL'

365 by (auto simp add: order_on_defs)

366 thus ?thesis using assms unfolding well_order_on_def

367 using Osum_Linear_order Osum_wf_Id by blast

368 qed

370 end