1.1 --- a/src/HOL/Library/Order_Union.thy Thu Dec 05 17:52:12 2013 +0100
1.2 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000
1.3 @@ -1,376 +0,0 @@
1.4 -(* Title: HOL/Library/Order_Union.thy
1.5 - Author: Andrei Popescu, TU Muenchen
1.6 -
1.7 -The ordinal-like sum of two orders with disjoint fields
1.8 -*)
1.9 -
1.10 -header {* Order Union *}
1.11 -
1.12 -theory Order_Union
1.13 -imports "~~/src/HOL/Cardinals/Wellfounded_More_Base"
1.14 -begin
1.15 -
1.16 -definition Osum :: "'a rel \<Rightarrow> 'a rel \<Rightarrow> 'a rel" (infix "Osum" 60) where
1.17 - "r Osum r' = r \<union> r' \<union> {(a, a'). a \<in> Field r \<and> a' \<in> Field r'}"
1.18 -
1.19 -notation Osum (infix "\<union>o" 60)
1.20 -
1.21 -lemma Field_Osum: "Field (r \<union>o r') = Field r \<union> Field r'"
1.22 - unfolding Osum_def Field_def by blast
1.23 -
1.24 -lemma Osum_wf:
1.25 -assumes FLD: "Field r Int Field r' = {}" and
1.26 - WF: "wf r" and WF': "wf r'"
1.27 -shows "wf (r Osum r')"
1.28 -unfolding wf_eq_minimal2 unfolding Field_Osum
1.29 -proof(intro allI impI, elim conjE)
1.30 - fix A assume *: "A \<subseteq> Field r \<union> Field r'" and **: "A \<noteq> {}"
1.31 - obtain B where B_def: "B = A Int Field r" by blast
1.32 - show "\<exists>a\<in>A. \<forall>a'\<in>A. (a', a) \<notin> r \<union>o r'"
1.33 - proof(cases "B = {}")
1.34 - assume Case1: "B \<noteq> {}"
1.35 - hence "B \<noteq> {} \<and> B \<le> Field r" using B_def by auto
1.36 - then obtain a where 1: "a \<in> B" and 2: "\<forall>a1 \<in> B. (a1,a) \<notin> r"
1.37 - using WF unfolding wf_eq_minimal2 by blast
1.38 - hence 3: "a \<in> Field r \<and> a \<notin> Field r'" using B_def FLD by auto
1.39 - (* *)
1.40 - have "\<forall>a1 \<in> A. (a1,a) \<notin> r Osum r'"
1.41 - proof(intro ballI)
1.42 - fix a1 assume **: "a1 \<in> A"
1.43 - {assume Case11: "a1 \<in> Field r"
1.44 - hence "(a1,a) \<notin> r" using B_def ** 2 by auto
1.45 - moreover
1.46 - have "(a1,a) \<notin> r'" using 3 by (auto simp add: Field_def)
1.47 - ultimately have "(a1,a) \<notin> r Osum r'"
1.48 - using 3 unfolding Osum_def by auto
1.49 - }
1.50 - moreover
1.51 - {assume Case12: "a1 \<notin> Field r"
1.52 - hence "(a1,a) \<notin> r" unfolding Field_def by auto
1.53 - moreover
1.54 - have "(a1,a) \<notin> r'" using 3 unfolding Field_def by auto
1.55 - ultimately have "(a1,a) \<notin> r Osum r'"
1.56 - using 3 unfolding Osum_def by auto
1.57 - }
1.58 - ultimately show "(a1,a) \<notin> r Osum r'" by blast
1.59 - qed
1.60 - thus ?thesis using 1 B_def by auto
1.61 - next
1.62 - assume Case2: "B = {}"
1.63 - hence 1: "A \<noteq> {} \<and> A \<le> Field r'" using * ** B_def by auto
1.64 - then obtain a' where 2: "a' \<in> A" and 3: "\<forall>a1' \<in> A. (a1',a') \<notin> r'"
1.65 - using WF' unfolding wf_eq_minimal2 by blast
1.66 - hence 4: "a' \<in> Field r' \<and> a' \<notin> Field r" using 1 FLD by blast
1.67 - (* *)
1.68 - have "\<forall>a1' \<in> A. (a1',a') \<notin> r Osum r'"
1.69 - proof(unfold Osum_def, auto simp add: 3)
1.70 - fix a1' assume "(a1', a') \<in> r"
1.71 - thus False using 4 unfolding Field_def by blast
1.72 - next
1.73 - fix a1' assume "a1' \<in> A" and "a1' \<in> Field r"
1.74 - thus False using Case2 B_def by auto
1.75 - qed
1.76 - thus ?thesis using 2 by blast
1.77 - qed
1.78 -qed
1.79 -
1.80 -lemma Osum_Refl:
1.81 -assumes FLD: "Field r Int Field r' = {}" and
1.82 - REFL: "Refl r" and REFL': "Refl r'"
1.83 -shows "Refl (r Osum r')"
1.84 -using assms
1.85 -unfolding refl_on_def Field_Osum unfolding Osum_def by blast
1.86 -
1.87 -lemma Osum_trans:
1.88 -assumes FLD: "Field r Int Field r' = {}" and
1.89 - TRANS: "trans r" and TRANS': "trans r'"
1.90 -shows "trans (r Osum r')"
1.91 -proof(unfold trans_def, auto)
1.92 - fix x y z assume *: "(x, y) \<in> r \<union>o r'" and **: "(y, z) \<in> r \<union>o r'"
1.93 - show "(x, z) \<in> r \<union>o r'"
1.94 - proof-
1.95 - {assume Case1: "(x,y) \<in> r"
1.96 - hence 1: "x \<in> Field r \<and> y \<in> Field r" unfolding Field_def by auto
1.97 - have ?thesis
1.98 - proof-
1.99 - {assume Case11: "(y,z) \<in> r"
1.100 - hence "(x,z) \<in> r" using Case1 TRANS trans_def[of r] by blast
1.101 - hence ?thesis unfolding Osum_def by auto
1.102 - }
1.103 - moreover
1.104 - {assume Case12: "(y,z) \<in> r'"
1.105 - hence "y \<in> Field r'" unfolding Field_def by auto
1.106 - hence False using FLD 1 by auto
1.107 - }
1.108 - moreover
1.109 - {assume Case13: "z \<in> Field r'"
1.110 - hence ?thesis using 1 unfolding Osum_def by auto
1.111 - }
1.112 - ultimately show ?thesis using ** unfolding Osum_def by blast
1.113 - qed
1.114 - }
1.115 - moreover
1.116 - {assume Case2: "(x,y) \<in> r'"
1.117 - hence 2: "x \<in> Field r' \<and> y \<in> Field r'" unfolding Field_def by auto
1.118 - have ?thesis
1.119 - proof-
1.120 - {assume Case21: "(y,z) \<in> r"
1.121 - hence "y \<in> Field r" unfolding Field_def by auto
1.122 - hence False using FLD 2 by auto
1.123 - }
1.124 - moreover
1.125 - {assume Case22: "(y,z) \<in> r'"
1.126 - hence "(x,z) \<in> r'" using Case2 TRANS' trans_def[of r'] by blast
1.127 - hence ?thesis unfolding Osum_def by auto
1.128 - }
1.129 - moreover
1.130 - {assume Case23: "y \<in> Field r"
1.131 - hence False using FLD 2 by auto
1.132 - }
1.133 - ultimately show ?thesis using ** unfolding Osum_def by blast
1.134 - qed
1.135 - }
1.136 - moreover
1.137 - {assume Case3: "x \<in> Field r \<and> y \<in> Field r'"
1.138 - have ?thesis
1.139 - proof-
1.140 - {assume Case31: "(y,z) \<in> r"
1.141 - hence "y \<in> Field r" unfolding Field_def by auto
1.142 - hence False using FLD Case3 by auto
1.143 - }
1.144 - moreover
1.145 - {assume Case32: "(y,z) \<in> r'"
1.146 - hence "z \<in> Field r'" unfolding Field_def by blast
1.147 - hence ?thesis unfolding Osum_def using Case3 by auto
1.148 - }
1.149 - moreover
1.150 - {assume Case33: "y \<in> Field r"
1.151 - hence False using FLD Case3 by auto
1.152 - }
1.153 - ultimately show ?thesis using ** unfolding Osum_def by blast
1.154 - qed
1.155 - }
1.156 - ultimately show ?thesis using * unfolding Osum_def by blast
1.157 - qed
1.158 -qed
1.159 -
1.160 -lemma Osum_Preorder:
1.161 -"\<lbrakk>Field r Int Field r' = {}; Preorder r; Preorder r'\<rbrakk> \<Longrightarrow> Preorder (r Osum r')"
1.162 -unfolding preorder_on_def using Osum_Refl Osum_trans by blast
1.163 -
1.164 -lemma Osum_antisym:
1.165 -assumes FLD: "Field r Int Field r' = {}" and
1.166 - AN: "antisym r" and AN': "antisym r'"
1.167 -shows "antisym (r Osum r')"
1.168 -proof(unfold antisym_def, auto)
1.169 - fix x y assume *: "(x, y) \<in> r \<union>o r'" and **: "(y, x) \<in> r \<union>o r'"
1.170 - show "x = y"
1.171 - proof-
1.172 - {assume Case1: "(x,y) \<in> r"
1.173 - hence 1: "x \<in> Field r \<and> y \<in> Field r" unfolding Field_def by auto
1.174 - have ?thesis
1.175 - proof-
1.176 - have "(y,x) \<in> r \<Longrightarrow> ?thesis"
1.177 - using Case1 AN antisym_def[of r] by blast
1.178 - moreover
1.179 - {assume "(y,x) \<in> r'"
1.180 - hence "y \<in> Field r'" unfolding Field_def by auto
1.181 - hence False using FLD 1 by auto
1.182 - }
1.183 - moreover
1.184 - have "x \<in> Field r' \<Longrightarrow> False" using FLD 1 by auto
1.185 - ultimately show ?thesis using ** unfolding Osum_def by blast
1.186 - qed
1.187 - }
1.188 - moreover
1.189 - {assume Case2: "(x,y) \<in> r'"
1.190 - hence 2: "x \<in> Field r' \<and> y \<in> Field r'" unfolding Field_def by auto
1.191 - have ?thesis
1.192 - proof-
1.193 - {assume "(y,x) \<in> r"
1.194 - hence "y \<in> Field r" unfolding Field_def by auto
1.195 - hence False using FLD 2 by auto
1.196 - }
1.197 - moreover
1.198 - have "(y,x) \<in> r' \<Longrightarrow> ?thesis"
1.199 - using Case2 AN' antisym_def[of r'] by blast
1.200 - moreover
1.201 - {assume "y \<in> Field r"
1.202 - hence False using FLD 2 by auto
1.203 - }
1.204 - ultimately show ?thesis using ** unfolding Osum_def by blast
1.205 - qed
1.206 - }
1.207 - moreover
1.208 - {assume Case3: "x \<in> Field r \<and> y \<in> Field r'"
1.209 - have ?thesis
1.210 - proof-
1.211 - {assume "(y,x) \<in> r"
1.212 - hence "y \<in> Field r" unfolding Field_def by auto
1.213 - hence False using FLD Case3 by auto
1.214 - }
1.215 - moreover
1.216 - {assume Case32: "(y,x) \<in> r'"
1.217 - hence "x \<in> Field r'" unfolding Field_def by blast
1.218 - hence False using FLD Case3 by auto
1.219 - }
1.220 - moreover
1.221 - have "\<not> y \<in> Field r" using FLD Case3 by auto
1.222 - ultimately show ?thesis using ** unfolding Osum_def by blast
1.223 - qed
1.224 - }
1.225 - ultimately show ?thesis using * unfolding Osum_def by blast
1.226 - qed
1.227 -qed
1.228 -
1.229 -lemma Osum_Partial_order:
1.230 -"\<lbrakk>Field r Int Field r' = {}; Partial_order r; Partial_order r'\<rbrakk> \<Longrightarrow>
1.231 - Partial_order (r Osum r')"
1.232 -unfolding partial_order_on_def using Osum_Preorder Osum_antisym by blast
1.233 -
1.234 -lemma Osum_Total:
1.235 -assumes FLD: "Field r Int Field r' = {}" and
1.236 - TOT: "Total r" and TOT': "Total r'"
1.237 -shows "Total (r Osum r')"
1.238 -using assms
1.239 -unfolding total_on_def Field_Osum unfolding Osum_def by blast
1.240 -
1.241 -lemma Osum_Linear_order:
1.242 -"\<lbrakk>Field r Int Field r' = {}; Linear_order r; Linear_order r'\<rbrakk> \<Longrightarrow>
1.243 - Linear_order (r Osum r')"
1.244 -unfolding linear_order_on_def using Osum_Partial_order Osum_Total by blast
1.245 -
1.246 -lemma Osum_minus_Id1:
1.247 -assumes "r \<le> Id"
1.248 -shows "(r Osum r') - Id \<le> (r' - Id) \<union> (Field r \<times> Field r')"
1.249 -proof-
1.250 - let ?Left = "(r Osum r') - Id"
1.251 - let ?Right = "(r' - Id) \<union> (Field r \<times> Field r')"
1.252 - {fix a::'a and b assume *: "(a,b) \<notin> Id"
1.253 - {assume "(a,b) \<in> r"
1.254 - with * have False using assms by auto
1.255 - }
1.256 - moreover
1.257 - {assume "(a,b) \<in> r'"
1.258 - with * have "(a,b) \<in> r' - Id" by auto
1.259 - }
1.260 - ultimately
1.261 - have "(a,b) \<in> ?Left \<Longrightarrow> (a,b) \<in> ?Right"
1.262 - unfolding Osum_def by auto
1.263 - }
1.264 - thus ?thesis by auto
1.265 -qed
1.266 -
1.267 -lemma Osum_minus_Id2:
1.268 -assumes "r' \<le> Id"
1.269 -shows "(r Osum r') - Id \<le> (r - Id) \<union> (Field r \<times> Field r')"
1.270 -proof-
1.271 - let ?Left = "(r Osum r') - Id"
1.272 - let ?Right = "(r - Id) \<union> (Field r \<times> Field r')"
1.273 - {fix a::'a and b assume *: "(a,b) \<notin> Id"
1.274 - {assume "(a,b) \<in> r'"
1.275 - with * have False using assms by auto
1.276 - }
1.277 - moreover
1.278 - {assume "(a,b) \<in> r"
1.279 - with * have "(a,b) \<in> r - Id" by auto
1.280 - }
1.281 - ultimately
1.282 - have "(a,b) \<in> ?Left \<Longrightarrow> (a,b) \<in> ?Right"
1.283 - unfolding Osum_def by auto
1.284 - }
1.285 - thus ?thesis by auto
1.286 -qed
1.287 -
1.288 -lemma Osum_minus_Id:
1.289 -assumes TOT: "Total r" and TOT': "Total r'" and
1.290 - NID: "\<not> (r \<le> Id)" and NID': "\<not> (r' \<le> Id)"
1.291 -shows "(r Osum r') - Id \<le> (r - Id) Osum (r' - Id)"
1.292 -proof-
1.293 - {fix a a' assume *: "(a,a') \<in> (r Osum r')" and **: "a \<noteq> a'"
1.294 - have "(a,a') \<in> (r - Id) Osum (r' - Id)"
1.295 - proof-
1.296 - {assume "(a,a') \<in> r \<or> (a,a') \<in> r'"
1.297 - with ** have ?thesis unfolding Osum_def by auto
1.298 - }
1.299 - moreover
1.300 - {assume "a \<in> Field r \<and> a' \<in> Field r'"
1.301 - hence "a \<in> Field(r - Id) \<and> a' \<in> Field (r' - Id)"
1.302 - using assms Total_Id_Field by blast
1.303 - hence ?thesis unfolding Osum_def by auto
1.304 - }
1.305 - ultimately show ?thesis using * unfolding Osum_def by blast
1.306 - qed
1.307 - }
1.308 - thus ?thesis by(auto simp add: Osum_def)
1.309 -qed
1.310 -
1.311 -lemma wf_Int_Times:
1.312 -assumes "A Int B = {}"
1.313 -shows "wf(A \<times> B)"
1.314 -proof(unfold wf_def, auto)
1.315 - fix P x
1.316 - assume *: "\<forall>x. (\<forall>y. y \<in> A \<and> x \<in> B \<longrightarrow> P y) \<longrightarrow> P x"
1.317 - moreover have "\<forall>y \<in> A. P y" using assms * by blast
1.318 - ultimately show "P x" using * by (case_tac "x \<in> B", auto)
1.319 -qed
1.320 -
1.321 -lemma Osum_wf_Id:
1.322 -assumes TOT: "Total r" and TOT': "Total r'" and
1.323 - FLD: "Field r Int Field r' = {}" and
1.324 - WF: "wf(r - Id)" and WF': "wf(r' - Id)"
1.325 -shows "wf ((r Osum r') - Id)"
1.326 -proof(cases "r \<le> Id \<or> r' \<le> Id")
1.327 - assume Case1: "\<not>(r \<le> Id \<or> r' \<le> Id)"
1.328 - have "Field(r - Id) Int Field(r' - Id) = {}"
1.329 - using FLD mono_Field[of "r - Id" r] mono_Field[of "r' - Id" r']
1.330 - Diff_subset[of r Id] Diff_subset[of r' Id] by blast
1.331 - thus ?thesis
1.332 - using Case1 Osum_minus_Id[of r r'] assms Osum_wf[of "r - Id" "r' - Id"]
1.333 - wf_subset[of "(r - Id) \<union>o (r' - Id)" "(r Osum r') - Id"] by auto
1.334 -next
1.335 - have 1: "wf(Field r \<times> Field r')"
1.336 - using FLD by (auto simp add: wf_Int_Times)
1.337 - assume Case2: "r \<le> Id \<or> r' \<le> Id"
1.338 - moreover
1.339 - {assume Case21: "r \<le> Id"
1.340 - hence "(r Osum r') - Id \<le> (r' - Id) \<union> (Field r \<times> Field r')"
1.341 - using Osum_minus_Id1[of r r'] by simp
1.342 - moreover
1.343 - {have "Domain(Field r \<times> Field r') Int Range(r' - Id) = {}"
1.344 - using FLD unfolding Field_def by blast
1.345 - hence "wf((r' - Id) \<union> (Field r \<times> Field r'))"
1.346 - using 1 WF' wf_Un[of "Field r \<times> Field r'" "r' - Id"]
1.347 - by (auto simp add: Un_commute)
1.348 - }
1.349 - ultimately have ?thesis by (auto simp add: wf_subset)
1.350 - }
1.351 - moreover
1.352 - {assume Case22: "r' \<le> Id"
1.353 - hence "(r Osum r') - Id \<le> (r - Id) \<union> (Field r \<times> Field r')"
1.354 - using Osum_minus_Id2[of r' r] by simp
1.355 - moreover
1.356 - {have "Range(Field r \<times> Field r') Int Domain(r - Id) = {}"
1.357 - using FLD unfolding Field_def by blast
1.358 - hence "wf((r - Id) \<union> (Field r \<times> Field r'))"
1.359 - using 1 WF wf_Un[of "r - Id" "Field r \<times> Field r'"]
1.360 - by (auto simp add: Un_commute)
1.361 - }
1.362 - ultimately have ?thesis by (auto simp add: wf_subset)
1.363 - }
1.364 - ultimately show ?thesis by blast
1.365 -qed
1.366 -
1.367 -lemma Osum_Well_order:
1.368 -assumes FLD: "Field r Int Field r' = {}" and
1.369 - WELL: "Well_order r" and WELL': "Well_order r'"
1.370 -shows "Well_order (r Osum r')"
1.371 -proof-
1.372 - have "Total r \<and> Total r'" using WELL WELL'
1.373 - by (auto simp add: order_on_defs)
1.374 - thus ?thesis using assms unfolding well_order_on_def
1.375 - using Osum_Linear_order Osum_wf_Id by blast
1.376 -qed
1.377 -
1.378 -end
1.379 -