src/HOL/Cardinals/Order_Union.thy
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 blanchet@54545 ` 1` ```(* Title: HOL/Cardinals/Order_Union.thy ``` popescua@52184 ` 2` ``` Author: Andrei Popescu, TU Muenchen ``` popescua@52184 ` 3` popescua@52199 ` 4` ```The ordinal-like sum of two orders with disjoint fields ``` popescua@52184 ` 5` ```*) ``` popescua@52184 ` 6` wenzelm@63167 ` 7` ```section \Order Union\ ``` popescua@52184 ` 8` popescua@52184 ` 9` ```theory Order_Union ``` wenzelm@67006 ` 10` ```imports Main ``` popescua@52184 ` 11` ```begin ``` popescua@52184 ` 12` popescua@52184 ` 13` ```definition Osum :: "'a rel \ 'a rel \ 'a rel" (infix "Osum" 60) where ``` popescua@52184 ` 14` ``` "r Osum r' = r \ r' \ {(a, a'). a \ Field r \ a' \ Field r'}" ``` popescua@52184 ` 15` wenzelm@52191 ` 16` ```notation Osum (infix "\o" 60) ``` popescua@52184 ` 17` popescua@52184 ` 18` ```lemma Field_Osum: "Field (r \o r') = Field r \ Field r'" ``` popescua@52184 ` 19` ``` unfolding Osum_def Field_def by blast ``` popescua@52184 ` 20` popescua@52184 ` 21` ```lemma Osum_wf: ``` popescua@52184 ` 22` ```assumes FLD: "Field r Int Field r' = {}" and ``` popescua@52184 ` 23` ``` WF: "wf r" and WF': "wf r'" ``` popescua@52184 ` 24` ```shows "wf (r Osum r')" ``` popescua@52184 ` 25` ```unfolding wf_eq_minimal2 unfolding Field_Osum ``` popescua@52184 ` 26` ```proof(intro allI impI, elim conjE) ``` popescua@52184 ` 27` ``` fix A assume *: "A \ Field r \ Field r'" and **: "A \ {}" ``` popescua@52184 ` 28` ``` obtain B where B_def: "B = A Int Field r" by blast ``` popescua@52184 ` 29` ``` show "\a\A. \a'\A. (a', a) \ r \o r'" ``` popescua@52184 ` 30` ``` proof(cases "B = {}") ``` popescua@52184 ` 31` ``` assume Case1: "B \ {}" ``` popescua@52184 ` 32` ``` hence "B \ {} \ B \ Field r" using B_def by auto ``` popescua@52184 ` 33` ``` then obtain a where 1: "a \ B" and 2: "\a1 \ B. (a1,a) \ r" ``` blanchet@55021 ` 34` ``` using WF unfolding wf_eq_minimal2 by blast ``` popescua@52184 ` 35` ``` hence 3: "a \ Field r \ a \ Field r'" using B_def FLD by auto ``` popescua@52184 ` 36` ``` (* *) ``` popescua@52184 ` 37` ``` have "\a1 \ A. (a1,a) \ r Osum r'" ``` popescua@52184 ` 38` ``` proof(intro ballI) ``` popescua@52184 ` 39` ``` fix a1 assume **: "a1 \ A" ``` popescua@52184 ` 40` ``` {assume Case11: "a1 \ Field r" ``` popescua@52184 ` 41` ``` hence "(a1,a) \ r" using B_def ** 2 by auto ``` popescua@52184 ` 42` ``` moreover ``` popescua@52184 ` 43` ``` have "(a1,a) \ r'" using 3 by (auto simp add: Field_def) ``` popescua@52184 ` 44` ``` ultimately have "(a1,a) \ r Osum r'" ``` popescua@52184 ` 45` ``` using 3 unfolding Osum_def by auto ``` popescua@52184 ` 46` ``` } ``` popescua@52184 ` 47` ``` moreover ``` popescua@52184 ` 48` ``` {assume Case12: "a1 \ Field r" ``` popescua@52184 ` 49` ``` hence "(a1,a) \ r" unfolding Field_def by auto ``` popescua@52184 ` 50` ``` moreover ``` popescua@52184 ` 51` ``` have "(a1,a) \ r'" using 3 unfolding Field_def by auto ``` popescua@52184 ` 52` ``` ultimately have "(a1,a) \ r Osum r'" ``` popescua@52184 ` 53` ``` using 3 unfolding Osum_def by auto ``` popescua@52184 ` 54` ``` } ``` popescua@52184 ` 55` ``` ultimately show "(a1,a) \ r Osum r'" by blast ``` popescua@52184 ` 56` ``` qed ``` popescua@52184 ` 57` ``` thus ?thesis using 1 B_def by auto ``` popescua@52184 ` 58` ``` next ``` popescua@52184 ` 59` ``` assume Case2: "B = {}" ``` popescua@52184 ` 60` ``` hence 1: "A \ {} \ A \ Field r'" using * ** B_def by auto ``` popescua@52184 ` 61` ``` then obtain a' where 2: "a' \ A" and 3: "\a1' \ A. (a1',a') \ r'" ``` blanchet@55021 ` 62` ``` using WF' unfolding wf_eq_minimal2 by blast ``` popescua@52184 ` 63` ``` hence 4: "a' \ Field r' \ a' \ Field r" using 1 FLD by blast ``` popescua@52184 ` 64` ``` (* *) ``` popescua@52184 ` 65` ``` have "\a1' \ A. (a1',a') \ r Osum r'" ``` popescua@52184 ` 66` ``` proof(unfold Osum_def, auto simp add: 3) ``` popescua@52184 ` 67` ``` fix a1' assume "(a1', a') \ r" ``` popescua@52184 ` 68` ``` thus False using 4 unfolding Field_def by blast ``` popescua@52184 ` 69` ``` next ``` popescua@52184 ` 70` ``` fix a1' assume "a1' \ A" and "a1' \ Field r" ``` popescua@52184 ` 71` ``` thus False using Case2 B_def by auto ``` popescua@52184 ` 72` ``` qed ``` popescua@52184 ` 73` ``` thus ?thesis using 2 by blast ``` popescua@52184 ` 74` ``` qed ``` popescua@52184 ` 75` ```qed ``` popescua@52184 ` 76` popescua@52184 ` 77` ```lemma Osum_Refl: ``` popescua@52184 ` 78` ```assumes FLD: "Field r Int Field r' = {}" and ``` popescua@52184 ` 79` ``` REFL: "Refl r" and REFL': "Refl r'" ``` popescua@52184 ` 80` ```shows "Refl (r Osum r')" ``` blanchet@58127 ` 81` ```using assms ``` popescua@52184 ` 82` ```unfolding refl_on_def Field_Osum unfolding Osum_def by blast ``` popescua@52184 ` 83` popescua@52184 ` 84` ```lemma Osum_trans: ``` popescua@52184 ` 85` ```assumes FLD: "Field r Int Field r' = {}" and ``` popescua@52184 ` 86` ``` TRANS: "trans r" and TRANS': "trans r'" ``` popescua@52184 ` 87` ```shows "trans (r Osum r')" ``` popescua@52184 ` 88` ```proof(unfold trans_def, auto) ``` popescua@52184 ` 89` ``` fix x y z assume *: "(x, y) \ r \o r'" and **: "(y, z) \ r \o r'" ``` popescua@52184 ` 90` ``` show "(x, z) \ r \o r'" ``` popescua@52184 ` 91` ``` proof- ``` popescua@52184 ` 92` ``` {assume Case1: "(x,y) \ r" ``` popescua@52184 ` 93` ``` hence 1: "x \ Field r \ y \ Field r" unfolding Field_def by auto ``` popescua@52184 ` 94` ``` have ?thesis ``` popescua@52184 ` 95` ``` proof- ``` popescua@52184 ` 96` ``` {assume Case11: "(y,z) \ r" ``` popescua@52184 ` 97` ``` hence "(x,z) \ r" using Case1 TRANS trans_def[of r] by blast ``` popescua@52184 ` 98` ``` hence ?thesis unfolding Osum_def by auto ``` popescua@52184 ` 99` ``` } ``` popescua@52184 ` 100` ``` moreover ``` popescua@52184 ` 101` ``` {assume Case12: "(y,z) \ r'" ``` popescua@52184 ` 102` ``` hence "y \ Field r'" unfolding Field_def by auto ``` popescua@52184 ` 103` ``` hence False using FLD 1 by auto ``` popescua@52184 ` 104` ``` } ``` popescua@52184 ` 105` ``` moreover ``` popescua@52184 ` 106` ``` {assume Case13: "z \ Field r'" ``` popescua@52184 ` 107` ``` hence ?thesis using 1 unfolding Osum_def by auto ``` popescua@52184 ` 108` ``` } ``` popescua@52184 ` 109` ``` ultimately show ?thesis using ** unfolding Osum_def by blast ``` popescua@52184 ` 110` ``` qed ``` popescua@52184 ` 111` ``` } ``` popescua@52184 ` 112` ``` moreover ``` popescua@52184 ` 113` ``` {assume Case2: "(x,y) \ r'" ``` popescua@52184 ` 114` ``` hence 2: "x \ Field r' \ y \ Field r'" unfolding Field_def by auto ``` popescua@52184 ` 115` ``` have ?thesis ``` popescua@52184 ` 116` ``` proof- ``` popescua@52184 ` 117` ``` {assume Case21: "(y,z) \ r" ``` popescua@52184 ` 118` ``` hence "y \ Field r" unfolding Field_def by auto ``` popescua@52184 ` 119` ``` hence False using FLD 2 by auto ``` popescua@52184 ` 120` ``` } ``` popescua@52184 ` 121` ``` moreover ``` popescua@52184 ` 122` ``` {assume Case22: "(y,z) \ r'" ``` popescua@52184 ` 123` ``` hence "(x,z) \ r'" using Case2 TRANS' trans_def[of r'] by blast ``` popescua@52184 ` 124` ``` hence ?thesis unfolding Osum_def by auto ``` popescua@52184 ` 125` ``` } ``` popescua@52184 ` 126` ``` moreover ``` popescua@52184 ` 127` ``` {assume Case23: "y \ Field r" ``` popescua@52184 ` 128` ``` hence False using FLD 2 by auto ``` popescua@52184 ` 129` ``` } ``` popescua@52184 ` 130` ``` ultimately show ?thesis using ** unfolding Osum_def by blast ``` popescua@52184 ` 131` ``` qed ``` popescua@52184 ` 132` ``` } ``` popescua@52184 ` 133` ``` moreover ``` popescua@52184 ` 134` ``` {assume Case3: "x \ Field r \ y \ Field r'" ``` popescua@52184 ` 135` ``` have ?thesis ``` popescua@52184 ` 136` ``` proof- ``` popescua@52184 ` 137` ``` {assume Case31: "(y,z) \ r" ``` popescua@52184 ` 138` ``` hence "y \ Field r" unfolding Field_def by auto ``` popescua@52184 ` 139` ``` hence False using FLD Case3 by auto ``` popescua@52184 ` 140` ``` } ``` popescua@52184 ` 141` ``` moreover ``` popescua@52184 ` 142` ``` {assume Case32: "(y,z) \ r'" ``` popescua@52184 ` 143` ``` hence "z \ Field r'" unfolding Field_def by blast ``` popescua@52184 ` 144` ``` hence ?thesis unfolding Osum_def using Case3 by auto ``` popescua@52184 ` 145` ``` } ``` popescua@52184 ` 146` ``` moreover ``` popescua@52184 ` 147` ``` {assume Case33: "y \ Field r" ``` popescua@52184 ` 148` ``` hence False using FLD Case3 by auto ``` popescua@52184 ` 149` ``` } ``` popescua@52184 ` 150` ``` ultimately show ?thesis using ** unfolding Osum_def by blast ``` popescua@52184 ` 151` ``` qed ``` popescua@52184 ` 152` ``` } ``` popescua@52184 ` 153` ``` ultimately show ?thesis using * unfolding Osum_def by blast ``` popescua@52184 ` 154` ``` qed ``` popescua@52184 ` 155` ```qed ``` popescua@52184 ` 156` popescua@52184 ` 157` ```lemma Osum_Preorder: ``` popescua@52184 ` 158` ```"\Field r Int Field r' = {}; Preorder r; Preorder r'\ \ Preorder (r Osum r')" ``` popescua@52184 ` 159` ```unfolding preorder_on_def using Osum_Refl Osum_trans by blast ``` popescua@52184 ` 160` popescua@52184 ` 161` ```lemma Osum_antisym: ``` popescua@52184 ` 162` ```assumes FLD: "Field r Int Field r' = {}" and ``` popescua@52184 ` 163` ``` AN: "antisym r" and AN': "antisym r'" ``` popescua@52184 ` 164` ```shows "antisym (r Osum r')" ``` popescua@52184 ` 165` ```proof(unfold antisym_def, auto) ``` popescua@52184 ` 166` ``` fix x y assume *: "(x, y) \ r \o r'" and **: "(y, x) \ r \o r'" ``` popescua@52184 ` 167` ``` show "x = y" ``` popescua@52184 ` 168` ``` proof- ``` popescua@52184 ` 169` ``` {assume Case1: "(x,y) \ r" ``` popescua@52184 ` 170` ``` hence 1: "x \ Field r \ y \ Field r" unfolding Field_def by auto ``` popescua@52184 ` 171` ``` have ?thesis ``` popescua@52184 ` 172` ``` proof- ``` popescua@52184 ` 173` ``` have "(y,x) \ r \ ?thesis" ``` popescua@52184 ` 174` ``` using Case1 AN antisym_def[of r] by blast ``` popescua@52184 ` 175` ``` moreover ``` popescua@52184 ` 176` ``` {assume "(y,x) \ r'" ``` popescua@52184 ` 177` ``` hence "y \ Field r'" unfolding Field_def by auto ``` popescua@52184 ` 178` ``` hence False using FLD 1 by auto ``` popescua@52184 ` 179` ``` } ``` popescua@52184 ` 180` ``` moreover ``` popescua@52184 ` 181` ``` have "x \ Field r' \ False" using FLD 1 by auto ``` popescua@52184 ` 182` ``` ultimately show ?thesis using ** unfolding Osum_def by blast ``` popescua@52184 ` 183` ``` qed ``` popescua@52184 ` 184` ``` } ``` popescua@52184 ` 185` ``` moreover ``` popescua@52184 ` 186` ``` {assume Case2: "(x,y) \ r'" ``` popescua@52184 ` 187` ``` hence 2: "x \ Field r' \ y \ Field r'" unfolding Field_def by auto ``` popescua@52184 ` 188` ``` have ?thesis ``` popescua@52184 ` 189` ``` proof- ``` popescua@52184 ` 190` ``` {assume "(y,x) \ r" ``` popescua@52184 ` 191` ``` hence "y \ Field r" unfolding Field_def by auto ``` popescua@52184 ` 192` ``` hence False using FLD 2 by auto ``` popescua@52184 ` 193` ``` } ``` popescua@52184 ` 194` ``` moreover ``` popescua@52184 ` 195` ``` have "(y,x) \ r' \ ?thesis" ``` popescua@52184 ` 196` ``` using Case2 AN' antisym_def[of r'] by blast ``` popescua@52184 ` 197` ``` moreover ``` popescua@52184 ` 198` ``` {assume "y \ Field r" ``` popescua@52184 ` 199` ``` hence False using FLD 2 by auto ``` popescua@52184 ` 200` ``` } ``` popescua@52184 ` 201` ``` ultimately show ?thesis using ** unfolding Osum_def by blast ``` popescua@52184 ` 202` ``` qed ``` popescua@52184 ` 203` ``` } ``` popescua@52184 ` 204` ``` moreover ``` popescua@52184 ` 205` ``` {assume Case3: "x \ Field r \ y \ Field r'" ``` popescua@52184 ` 206` ``` have ?thesis ``` popescua@52184 ` 207` ``` proof- ``` popescua@52184 ` 208` ``` {assume "(y,x) \ r" ``` popescua@52184 ` 209` ``` hence "y \ Field r" unfolding Field_def by auto ``` popescua@52184 ` 210` ``` hence False using FLD Case3 by auto ``` popescua@52184 ` 211` ``` } ``` popescua@52184 ` 212` ``` moreover ``` popescua@52184 ` 213` ``` {assume Case32: "(y,x) \ r'" ``` popescua@52184 ` 214` ``` hence "x \ Field r'" unfolding Field_def by blast ``` popescua@52184 ` 215` ``` hence False using FLD Case3 by auto ``` popescua@52184 ` 216` ``` } ``` popescua@52184 ` 217` ``` moreover ``` popescua@52184 ` 218` ``` have "\ y \ Field r" using FLD Case3 by auto ``` popescua@52184 ` 219` ``` ultimately show ?thesis using ** unfolding Osum_def by blast ``` popescua@52184 ` 220` ``` qed ``` popescua@52184 ` 221` ``` } ``` popescua@52184 ` 222` ``` ultimately show ?thesis using * unfolding Osum_def by blast ``` popescua@52184 ` 223` ``` qed ``` popescua@52184 ` 224` ```qed ``` popescua@52184 ` 225` popescua@52184 ` 226` ```lemma Osum_Partial_order: ``` popescua@52184 ` 227` ```"\Field r Int Field r' = {}; Partial_order r; Partial_order r'\ \ ``` popescua@52184 ` 228` ``` Partial_order (r Osum r')" ``` popescua@52184 ` 229` ```unfolding partial_order_on_def using Osum_Preorder Osum_antisym by blast ``` popescua@52184 ` 230` popescua@52184 ` 231` ```lemma Osum_Total: ``` popescua@52184 ` 232` ```assumes FLD: "Field r Int Field r' = {}" and ``` popescua@52184 ` 233` ``` TOT: "Total r" and TOT': "Total r'" ``` popescua@52184 ` 234` ```shows "Total (r Osum r')" ``` popescua@52184 ` 235` ```using assms ``` popescua@52184 ` 236` ```unfolding total_on_def Field_Osum unfolding Osum_def by blast ``` popescua@52184 ` 237` popescua@52184 ` 238` ```lemma Osum_Linear_order: ``` popescua@52184 ` 239` ```"\Field r Int Field r' = {}; Linear_order r; Linear_order r'\ \ ``` popescua@52184 ` 240` ``` Linear_order (r Osum r')" ``` popescua@52184 ` 241` ```unfolding linear_order_on_def using Osum_Partial_order Osum_Total by blast ``` popescua@52184 ` 242` popescua@52184 ` 243` ```lemma Osum_minus_Id1: ``` popescua@52184 ` 244` ```assumes "r \ Id" ``` popescua@52184 ` 245` ```shows "(r Osum r') - Id \ (r' - Id) \ (Field r \ Field r')" ``` popescua@52184 ` 246` ```proof- ``` popescua@52184 ` 247` ``` let ?Left = "(r Osum r') - Id" ``` popescua@52184 ` 248` ``` let ?Right = "(r' - Id) \ (Field r \ Field r')" ``` popescua@52184 ` 249` ``` {fix a::'a and b assume *: "(a,b) \ Id" ``` popescua@52184 ` 250` ``` {assume "(a,b) \ r" ``` popescua@52184 ` 251` ``` with * have False using assms by auto ``` popescua@52184 ` 252` ``` } ``` popescua@52184 ` 253` ``` moreover ``` popescua@52184 ` 254` ``` {assume "(a,b) \ r'" ``` popescua@52184 ` 255` ``` with * have "(a,b) \ r' - Id" by auto ``` popescua@52184 ` 256` ``` } ``` popescua@52184 ` 257` ``` ultimately ``` popescua@52184 ` 258` ``` have "(a,b) \ ?Left \ (a,b) \ ?Right" ``` popescua@52184 ` 259` ``` unfolding Osum_def by auto ``` popescua@52184 ` 260` ``` } ``` popescua@52184 ` 261` ``` thus ?thesis by auto ``` popescua@52184 ` 262` ```qed ``` popescua@52184 ` 263` popescua@52184 ` 264` ```lemma Osum_minus_Id2: ``` popescua@52184 ` 265` ```assumes "r' \ Id" ``` popescua@52184 ` 266` ```shows "(r Osum r') - Id \ (r - Id) \ (Field r \ Field r')" ``` popescua@52184 ` 267` ```proof- ``` popescua@52184 ` 268` ``` let ?Left = "(r Osum r') - Id" ``` popescua@52184 ` 269` ``` let ?Right = "(r - Id) \ (Field r \ Field r')" ``` popescua@52184 ` 270` ``` {fix a::'a and b assume *: "(a,b) \ Id" ``` popescua@52184 ` 271` ``` {assume "(a,b) \ r'" ``` popescua@52184 ` 272` ``` with * have False using assms by auto ``` popescua@52184 ` 273` ``` } ``` popescua@52184 ` 274` ``` moreover ``` popescua@52184 ` 275` ``` {assume "(a,b) \ r" ``` popescua@52184 ` 276` ``` with * have "(a,b) \ r - Id" by auto ``` popescua@52184 ` 277` ``` } ``` popescua@52184 ` 278` ``` ultimately ``` popescua@52184 ` 279` ``` have "(a,b) \ ?Left \ (a,b) \ ?Right" ``` popescua@52184 ` 280` ``` unfolding Osum_def by auto ``` popescua@52184 ` 281` ``` } ``` popescua@52184 ` 282` ``` thus ?thesis by auto ``` popescua@52184 ` 283` ```qed ``` popescua@52184 ` 284` popescua@52184 ` 285` ```lemma Osum_minus_Id: ``` popescua@52184 ` 286` ```assumes TOT: "Total r" and TOT': "Total r'" and ``` popescua@52184 ` 287` ``` NID: "\ (r \ Id)" and NID': "\ (r' \ Id)" ``` popescua@52184 ` 288` ```shows "(r Osum r') - Id \ (r - Id) Osum (r' - Id)" ``` popescua@52184 ` 289` ```proof- ``` popescua@52184 ` 290` ``` {fix a a' assume *: "(a,a') \ (r Osum r')" and **: "a \ a'" ``` popescua@52184 ` 291` ``` have "(a,a') \ (r - Id) Osum (r' - Id)" ``` popescua@52184 ` 292` ``` proof- ``` popescua@52184 ` 293` ``` {assume "(a,a') \ r \ (a,a') \ r'" ``` popescua@52184 ` 294` ``` with ** have ?thesis unfolding Osum_def by auto ``` popescua@52184 ` 295` ``` } ``` popescua@52184 ` 296` ``` moreover ``` popescua@52184 ` 297` ``` {assume "a \ Field r \ a' \ Field r'" ``` popescua@52184 ` 298` ``` hence "a \ Field(r - Id) \ a' \ Field (r' - Id)" ``` popescua@52184 ` 299` ``` using assms Total_Id_Field by blast ``` popescua@52184 ` 300` ``` hence ?thesis unfolding Osum_def by auto ``` popescua@52184 ` 301` ``` } ``` blanchet@54482 ` 302` ``` ultimately show ?thesis using * unfolding Osum_def by fast ``` popescua@52184 ` 303` ``` qed ``` popescua@52184 ` 304` ``` } ``` popescua@52184 ` 305` ``` thus ?thesis by(auto simp add: Osum_def) ``` popescua@52184 ` 306` ```qed ``` popescua@52184 ` 307` popescua@52184 ` 308` ```lemma wf_Int_Times: ``` popescua@52184 ` 309` ```assumes "A Int B = {}" ``` popescua@52184 ` 310` ```shows "wf(A \ B)" ``` blanchet@54482 ` 311` ```unfolding wf_def using assms by blast ``` popescua@52184 ` 312` popescua@52184 ` 313` ```lemma Osum_wf_Id: ``` popescua@52184 ` 314` ```assumes TOT: "Total r" and TOT': "Total r'" and ``` popescua@52184 ` 315` ``` FLD: "Field r Int Field r' = {}" and ``` popescua@52184 ` 316` ``` WF: "wf(r - Id)" and WF': "wf(r' - Id)" ``` popescua@52184 ` 317` ```shows "wf ((r Osum r') - Id)" ``` popescua@52184 ` 318` ```proof(cases "r \ Id \ r' \ Id") ``` popescua@52184 ` 319` ``` assume Case1: "\(r \ Id \ r' \ Id)" ``` popescua@52184 ` 320` ``` have "Field(r - Id) Int Field(r' - Id) = {}" ``` popescua@52184 ` 321` ``` using FLD mono_Field[of "r - Id" r] mono_Field[of "r' - Id" r'] ``` popescua@52184 ` 322` ``` Diff_subset[of r Id] Diff_subset[of r' Id] by blast ``` popescua@52184 ` 323` ``` thus ?thesis ``` popescua@52184 ` 324` ``` using Case1 Osum_minus_Id[of r r'] assms Osum_wf[of "r - Id" "r' - Id"] ``` popescua@52184 ` 325` ``` wf_subset[of "(r - Id) \o (r' - Id)" "(r Osum r') - Id"] by auto ``` popescua@52184 ` 326` ```next ``` popescua@52184 ` 327` ``` have 1: "wf(Field r \ Field r')" ``` popescua@52184 ` 328` ``` using FLD by (auto simp add: wf_Int_Times) ``` popescua@52184 ` 329` ``` assume Case2: "r \ Id \ r' \ Id" ``` popescua@52184 ` 330` ``` moreover ``` popescua@52184 ` 331` ``` {assume Case21: "r \ Id" ``` popescua@52184 ` 332` ``` hence "(r Osum r') - Id \ (r' - Id) \ (Field r \ Field r')" ``` popescua@52184 ` 333` ``` using Osum_minus_Id1[of r r'] by simp ``` popescua@52184 ` 334` ``` moreover ``` popescua@52184 ` 335` ``` {have "Domain(Field r \ Field r') Int Range(r' - Id) = {}" ``` popescua@52184 ` 336` ``` using FLD unfolding Field_def by blast ``` popescua@52184 ` 337` ``` hence "wf((r' - Id) \ (Field r \ Field r'))" ``` popescua@52184 ` 338` ``` using 1 WF' wf_Un[of "Field r \ Field r'" "r' - Id"] ``` popescua@52184 ` 339` ``` by (auto simp add: Un_commute) ``` popescua@52184 ` 340` ``` } ``` blanchet@55021 ` 341` ``` ultimately have ?thesis using wf_subset by blast ``` popescua@52184 ` 342` ``` } ``` popescua@52184 ` 343` ``` moreover ``` popescua@52184 ` 344` ``` {assume Case22: "r' \ Id" ``` popescua@52184 ` 345` ``` hence "(r Osum r') - Id \ (r - Id) \ (Field r \ Field r')" ``` popescua@52184 ` 346` ``` using Osum_minus_Id2[of r' r] by simp ``` popescua@52184 ` 347` ``` moreover ``` popescua@52184 ` 348` ``` {have "Range(Field r \ Field r') Int Domain(r - Id) = {}" ``` popescua@52184 ` 349` ``` using FLD unfolding Field_def by blast ``` popescua@52184 ` 350` ``` hence "wf((r - Id) \ (Field r \ Field r'))" ``` popescua@52184 ` 351` ``` using 1 WF wf_Un[of "r - Id" "Field r \ Field r'"] ``` popescua@52184 ` 352` ``` by (auto simp add: Un_commute) ``` popescua@52184 ` 353` ``` } ``` blanchet@55021 ` 354` ``` ultimately have ?thesis using wf_subset by blast ``` popescua@52184 ` 355` ``` } ``` popescua@52184 ` 356` ``` ultimately show ?thesis by blast ``` popescua@52184 ` 357` ```qed ``` popescua@52184 ` 358` popescua@52184 ` 359` ```lemma Osum_Well_order: ``` popescua@52184 ` 360` ```assumes FLD: "Field r Int Field r' = {}" and ``` popescua@52184 ` 361` ``` WELL: "Well_order r" and WELL': "Well_order r'" ``` popescua@52184 ` 362` ```shows "Well_order (r Osum r')" ``` popescua@52184 ` 363` ```proof- ``` popescua@52184 ` 364` ``` have "Total r \ Total r'" using WELL WELL' ``` popescua@52184 ` 365` ``` by (auto simp add: order_on_defs) ``` popescua@52184 ` 366` ``` thus ?thesis using assms unfolding well_order_on_def ``` popescua@52184 ` 367` ``` using Osum_Linear_order Osum_wf_Id by blast ``` popescua@52184 ` 368` ```qed ``` popescua@52184 ` 369` popescua@52184 ` 370` ```end ```