src/HOL/Library/Boolean_Algebra.thy
 author wenzelm Tue Jul 12 15:45:32 2016 +0200 (2016-07-12) changeset 63462 c1fe30f2bc32 parent 61605 1bf7b186542e child 65343 0a8e30a7b10e permissions -rw-r--r--
misc tuning and modernization;
 haftmann@29629 ` 1` ```(* Title: HOL/Library/Boolean_Algebra.thy ``` haftmann@29629 ` 2` ``` Author: Brian Huffman ``` kleing@24332 ` 3` ```*) ``` kleing@24332 ` 4` wenzelm@60500 ` 5` ```section \Boolean Algebras\ ``` kleing@24332 ` 6` kleing@24332 ` 7` ```theory Boolean_Algebra ``` wenzelm@63462 ` 8` ``` imports Main ``` kleing@24332 ` 9` ```begin ``` kleing@24332 ` 10` kleing@24332 ` 11` ```locale boolean = ``` huffman@24357 ` 12` ``` fixes conj :: "'a \ 'a \ 'a" (infixr "\" 70) ``` huffman@24357 ` 13` ``` fixes disj :: "'a \ 'a \ 'a" (infixr "\" 65) ``` huffman@24357 ` 14` ``` fixes compl :: "'a \ 'a" ("\ _" [81] 80) ``` kleing@24332 ` 15` ``` fixes zero :: "'a" ("\") ``` kleing@24332 ` 16` ``` fixes one :: "'a" ("\") ``` kleing@24332 ` 17` ``` assumes conj_assoc: "(x \ y) \ z = x \ (y \ z)" ``` kleing@24332 ` 18` ``` assumes disj_assoc: "(x \ y) \ z = x \ (y \ z)" ``` kleing@24332 ` 19` ``` assumes conj_commute: "x \ y = y \ x" ``` kleing@24332 ` 20` ``` assumes disj_commute: "x \ y = y \ x" ``` kleing@24332 ` 21` ``` assumes conj_disj_distrib: "x \ (y \ z) = (x \ y) \ (x \ z)" ``` kleing@24332 ` 22` ``` assumes disj_conj_distrib: "x \ (y \ z) = (x \ y) \ (x \ z)" ``` huffman@24357 ` 23` ``` assumes conj_one_right [simp]: "x \ \ = x" ``` huffman@24357 ` 24` ``` assumes disj_zero_right [simp]: "x \ \ = x" ``` huffman@24357 ` 25` ``` assumes conj_cancel_right [simp]: "x \ \ x = \" ``` huffman@24357 ` 26` ``` assumes disj_cancel_right [simp]: "x \ \ x = \" ``` haftmann@54868 ` 27` ```begin ``` haftmann@34973 ` 28` wenzelm@61605 ` 29` ```sublocale conj: abel_semigroup conj ``` wenzelm@60855 ` 30` ``` by standard (fact conj_assoc conj_commute)+ ``` haftmann@34973 ` 31` wenzelm@61605 ` 32` ```sublocale disj: abel_semigroup disj ``` wenzelm@60855 ` 33` ``` by standard (fact disj_assoc disj_commute)+ ``` haftmann@34973 ` 34` haftmann@34973 ` 35` ```lemmas conj_left_commute = conj.left_commute ``` kleing@24332 ` 36` haftmann@34973 ` 37` ```lemmas disj_left_commute = disj.left_commute ``` haftmann@34973 ` 38` haftmann@34973 ` 39` ```lemmas conj_ac = conj.assoc conj.commute conj.left_commute ``` haftmann@34973 ` 40` ```lemmas disj_ac = disj.assoc disj.commute disj.left_commute ``` kleing@24332 ` 41` kleing@24332 ` 42` ```lemma dual: "boolean disj conj compl one zero" ``` wenzelm@63462 ` 43` ``` apply (rule boolean.intro) ``` wenzelm@63462 ` 44` ``` apply (rule disj_assoc) ``` wenzelm@63462 ` 45` ``` apply (rule conj_assoc) ``` wenzelm@63462 ` 46` ``` apply (rule disj_commute) ``` wenzelm@63462 ` 47` ``` apply (rule conj_commute) ``` wenzelm@63462 ` 48` ``` apply (rule disj_conj_distrib) ``` wenzelm@63462 ` 49` ``` apply (rule conj_disj_distrib) ``` wenzelm@63462 ` 50` ``` apply (rule disj_zero_right) ``` wenzelm@63462 ` 51` ``` apply (rule conj_one_right) ``` wenzelm@63462 ` 52` ``` apply (rule disj_cancel_right) ``` wenzelm@63462 ` 53` ``` apply (rule conj_cancel_right) ``` wenzelm@63462 ` 54` ``` done ``` kleing@24332 ` 55` wenzelm@60855 ` 56` wenzelm@60500 ` 57` ```subsection \Complement\ ``` kleing@24332 ` 58` kleing@24332 ` 59` ```lemma complement_unique: ``` kleing@24332 ` 60` ``` assumes 1: "a \ x = \" ``` kleing@24332 ` 61` ``` assumes 2: "a \ x = \" ``` kleing@24332 ` 62` ``` assumes 3: "a \ y = \" ``` kleing@24332 ` 63` ``` assumes 4: "a \ y = \" ``` kleing@24332 ` 64` ``` shows "x = y" ``` kleing@24332 ` 65` ```proof - ``` wenzelm@63462 ` 66` ``` have "(a \ x) \ (x \ y) = (a \ y) \ (x \ y)" ``` wenzelm@63462 ` 67` ``` using 1 3 by simp ``` wenzelm@63462 ` 68` ``` then have "(x \ a) \ (x \ y) = (y \ a) \ (y \ x)" ``` wenzelm@63462 ` 69` ``` using conj_commute by simp ``` wenzelm@63462 ` 70` ``` then have "x \ (a \ y) = y \ (a \ x)" ``` wenzelm@63462 ` 71` ``` using conj_disj_distrib by simp ``` wenzelm@63462 ` 72` ``` then have "x \ \ = y \ \" ``` wenzelm@63462 ` 73` ``` using 2 4 by simp ``` wenzelm@63462 ` 74` ``` then show "x = y" ``` wenzelm@63462 ` 75` ``` using conj_one_right by simp ``` kleing@24332 ` 76` ```qed ``` kleing@24332 ` 77` wenzelm@63462 ` 78` ```lemma compl_unique: "x \ y = \ \ x \ y = \ \ \ x = y" ``` wenzelm@63462 ` 79` ``` by (rule complement_unique [OF conj_cancel_right disj_cancel_right]) ``` kleing@24332 ` 80` kleing@24332 ` 81` ```lemma double_compl [simp]: "\ (\ x) = x" ``` kleing@24332 ` 82` ```proof (rule compl_unique) ``` wenzelm@63462 ` 83` ``` from conj_cancel_right show "\ x \ x = \" ``` wenzelm@63462 ` 84` ``` by (simp only: conj_commute) ``` wenzelm@63462 ` 85` ``` from disj_cancel_right show "\ x \ x = \" ``` wenzelm@63462 ` 86` ``` by (simp only: disj_commute) ``` kleing@24332 ` 87` ```qed ``` kleing@24332 ` 88` wenzelm@63462 ` 89` ```lemma compl_eq_compl_iff [simp]: "\ x = \ y \ x = y" ``` wenzelm@63462 ` 90` ``` by (rule inj_eq [OF inj_on_inverseI]) (rule double_compl) ``` kleing@24332 ` 91` wenzelm@60855 ` 92` wenzelm@60500 ` 93` ```subsection \Conjunction\ ``` kleing@24332 ` 94` huffman@24393 ` 95` ```lemma conj_absorb [simp]: "x \ x = x" ``` kleing@24332 ` 96` ```proof - ``` wenzelm@63462 ` 97` ``` have "x \ x = (x \ x) \ \" ``` wenzelm@63462 ` 98` ``` using disj_zero_right by simp ``` wenzelm@63462 ` 99` ``` also have "... = (x \ x) \ (x \ \ x)" ``` wenzelm@63462 ` 100` ``` using conj_cancel_right by simp ``` wenzelm@63462 ` 101` ``` also have "... = x \ (x \ \ x)" ``` wenzelm@63462 ` 102` ``` using conj_disj_distrib by (simp only:) ``` wenzelm@63462 ` 103` ``` also have "... = x \ \" ``` wenzelm@63462 ` 104` ``` using disj_cancel_right by simp ``` wenzelm@63462 ` 105` ``` also have "... = x" ``` wenzelm@63462 ` 106` ``` using conj_one_right by simp ``` kleing@24332 ` 107` ``` finally show ?thesis . ``` kleing@24332 ` 108` ```qed ``` kleing@24332 ` 109` kleing@24332 ` 110` ```lemma conj_zero_right [simp]: "x \ \ = \" ``` kleing@24332 ` 111` ```proof - ``` wenzelm@63462 ` 112` ``` have "x \ \ = x \ (x \ \ x)" ``` wenzelm@63462 ` 113` ``` using conj_cancel_right by simp ``` wenzelm@63462 ` 114` ``` also have "... = (x \ x) \ \ x" ``` wenzelm@63462 ` 115` ``` using conj_assoc by (simp only:) ``` wenzelm@63462 ` 116` ``` also have "... = x \ \ x" ``` wenzelm@63462 ` 117` ``` using conj_absorb by simp ``` wenzelm@63462 ` 118` ``` also have "... = \" ``` wenzelm@63462 ` 119` ``` using conj_cancel_right by simp ``` kleing@24332 ` 120` ``` finally show ?thesis . ``` kleing@24332 ` 121` ```qed ``` kleing@24332 ` 122` kleing@24332 ` 123` ```lemma compl_one [simp]: "\ \ = \" ``` wenzelm@63462 ` 124` ``` by (rule compl_unique [OF conj_zero_right disj_zero_right]) ``` kleing@24332 ` 125` kleing@24332 ` 126` ```lemma conj_zero_left [simp]: "\ \ x = \" ``` wenzelm@63462 ` 127` ``` by (subst conj_commute) (rule conj_zero_right) ``` kleing@24332 ` 128` kleing@24332 ` 129` ```lemma conj_one_left [simp]: "\ \ x = x" ``` wenzelm@63462 ` 130` ``` by (subst conj_commute) (rule conj_one_right) ``` kleing@24332 ` 131` kleing@24332 ` 132` ```lemma conj_cancel_left [simp]: "\ x \ x = \" ``` wenzelm@63462 ` 133` ``` by (subst conj_commute) (rule conj_cancel_right) ``` kleing@24332 ` 134` kleing@24332 ` 135` ```lemma conj_left_absorb [simp]: "x \ (x \ y) = x \ y" ``` wenzelm@63462 ` 136` ``` by (simp only: conj_assoc [symmetric] conj_absorb) ``` kleing@24332 ` 137` wenzelm@63462 ` 138` ```lemma conj_disj_distrib2: "(y \ z) \ x = (y \ x) \ (z \ x)" ``` wenzelm@63462 ` 139` ``` by (simp only: conj_commute conj_disj_distrib) ``` kleing@24332 ` 140` wenzelm@63462 ` 141` ```lemmas conj_disj_distribs = conj_disj_distrib conj_disj_distrib2 ``` kleing@24332 ` 142` wenzelm@60855 ` 143` wenzelm@60500 ` 144` ```subsection \Disjunction\ ``` kleing@24332 ` 145` kleing@24332 ` 146` ```lemma disj_absorb [simp]: "x \ x = x" ``` wenzelm@63462 ` 147` ``` by (rule boolean.conj_absorb [OF dual]) ``` kleing@24332 ` 148` kleing@24332 ` 149` ```lemma disj_one_right [simp]: "x \ \ = \" ``` wenzelm@63462 ` 150` ``` by (rule boolean.conj_zero_right [OF dual]) ``` kleing@24332 ` 151` kleing@24332 ` 152` ```lemma compl_zero [simp]: "\ \ = \" ``` wenzelm@63462 ` 153` ``` by (rule boolean.compl_one [OF dual]) ``` kleing@24332 ` 154` kleing@24332 ` 155` ```lemma disj_zero_left [simp]: "\ \ x = x" ``` wenzelm@63462 ` 156` ``` by (rule boolean.conj_one_left [OF dual]) ``` kleing@24332 ` 157` kleing@24332 ` 158` ```lemma disj_one_left [simp]: "\ \ x = \" ``` wenzelm@63462 ` 159` ``` by (rule boolean.conj_zero_left [OF dual]) ``` kleing@24332 ` 160` kleing@24332 ` 161` ```lemma disj_cancel_left [simp]: "\ x \ x = \" ``` wenzelm@63462 ` 162` ``` by (rule boolean.conj_cancel_left [OF dual]) ``` kleing@24332 ` 163` kleing@24332 ` 164` ```lemma disj_left_absorb [simp]: "x \ (x \ y) = x \ y" ``` wenzelm@63462 ` 165` ``` by (rule boolean.conj_left_absorb [OF dual]) ``` kleing@24332 ` 166` wenzelm@63462 ` 167` ```lemma disj_conj_distrib2: "(y \ z) \ x = (y \ x) \ (z \ x)" ``` wenzelm@63462 ` 168` ``` by (rule boolean.conj_disj_distrib2 [OF dual]) ``` kleing@24332 ` 169` wenzelm@63462 ` 170` ```lemmas disj_conj_distribs = disj_conj_distrib disj_conj_distrib2 ``` kleing@24332 ` 171` wenzelm@60855 ` 172` wenzelm@60500 ` 173` ```subsection \De Morgan's Laws\ ``` kleing@24332 ` 174` kleing@24332 ` 175` ```lemma de_Morgan_conj [simp]: "\ (x \ y) = \ x \ \ y" ``` kleing@24332 ` 176` ```proof (rule compl_unique) ``` kleing@24332 ` 177` ``` have "(x \ y) \ (\ x \ \ y) = ((x \ y) \ \ x) \ ((x \ y) \ \ y)" ``` kleing@24332 ` 178` ``` by (rule conj_disj_distrib) ``` kleing@24332 ` 179` ``` also have "... = (y \ (x \ \ x)) \ (x \ (y \ \ y))" ``` huffman@24357 ` 180` ``` by (simp only: conj_ac) ``` kleing@24332 ` 181` ``` finally show "(x \ y) \ (\ x \ \ y) = \" ``` huffman@24357 ` 182` ``` by (simp only: conj_cancel_right conj_zero_right disj_zero_right) ``` kleing@24332 ` 183` ```next ``` kleing@24332 ` 184` ``` have "(x \ y) \ (\ x \ \ y) = (x \ (\ x \ \ y)) \ (y \ (\ x \ \ y))" ``` kleing@24332 ` 185` ``` by (rule disj_conj_distrib2) ``` kleing@24332 ` 186` ``` also have "... = (\ y \ (x \ \ x)) \ (\ x \ (y \ \ y))" ``` huffman@24357 ` 187` ``` by (simp only: disj_ac) ``` kleing@24332 ` 188` ``` finally show "(x \ y) \ (\ x \ \ y) = \" ``` huffman@24357 ` 189` ``` by (simp only: disj_cancel_right disj_one_right conj_one_right) ``` kleing@24332 ` 190` ```qed ``` kleing@24332 ` 191` kleing@24332 ` 192` ```lemma de_Morgan_disj [simp]: "\ (x \ y) = \ x \ \ y" ``` wenzelm@63462 ` 193` ``` by (rule boolean.de_Morgan_conj [OF dual]) ``` kleing@24332 ` 194` kleing@24332 ` 195` ```end ``` kleing@24332 ` 196` wenzelm@60855 ` 197` wenzelm@60500 ` 198` ```subsection \Symmetric Difference\ ``` kleing@24332 ` 199` kleing@24332 ` 200` ```locale boolean_xor = boolean + ``` wenzelm@60855 ` 201` ``` fixes xor :: "'a \ 'a \ 'a" (infixr "\" 65) ``` kleing@24332 ` 202` ``` assumes xor_def: "x \ y = (x \ \ y) \ (\ x \ y)" ``` haftmann@54868 ` 203` ```begin ``` kleing@24332 ` 204` wenzelm@61605 ` 205` ```sublocale xor: abel_semigroup xor ``` wenzelm@60855 ` 206` ```proof ``` haftmann@34973 ` 207` ``` fix x y z :: 'a ``` kleing@24332 ` 208` ``` let ?t = "(x \ y \ z) \ (x \ \ y \ \ z) \ ``` kleing@24332 ` 209` ``` (\ x \ y \ \ z) \ (\ x \ \ y \ z)" ``` kleing@24332 ` 210` ``` have "?t \ (z \ x \ \ x) \ (z \ y \ \ y) = ``` kleing@24332 ` 211` ``` ?t \ (x \ y \ \ y) \ (x \ z \ \ z)" ``` huffman@24357 ` 212` ``` by (simp only: conj_cancel_right conj_zero_right) ``` wenzelm@63462 ` 213` ``` then show "(x \ y) \ z = x \ (y \ z)" ``` huffman@24357 ` 214` ``` apply (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl) ``` huffman@24357 ` 215` ``` apply (simp only: conj_disj_distribs conj_ac disj_ac) ``` kleing@24332 ` 216` ``` done ``` haftmann@34973 ` 217` ``` show "x \ y = y \ x" ``` haftmann@34973 ` 218` ``` by (simp only: xor_def conj_commute disj_commute) ``` kleing@24332 ` 219` ```qed ``` kleing@24332 ` 220` haftmann@34973 ` 221` ```lemmas xor_assoc = xor.assoc ``` haftmann@34973 ` 222` ```lemmas xor_commute = xor.commute ``` haftmann@34973 ` 223` ```lemmas xor_left_commute = xor.left_commute ``` haftmann@34973 ` 224` haftmann@34973 ` 225` ```lemmas xor_ac = xor.assoc xor.commute xor.left_commute ``` haftmann@34973 ` 226` wenzelm@63462 ` 227` ```lemma xor_def2: "x \ y = (x \ y) \ (\ x \ \ y)" ``` wenzelm@63462 ` 228` ``` by (simp only: xor_def conj_disj_distribs disj_ac conj_ac conj_cancel_right disj_zero_left) ``` kleing@24332 ` 229` kleing@24332 ` 230` ```lemma xor_zero_right [simp]: "x \ \ = x" ``` wenzelm@63462 ` 231` ``` by (simp only: xor_def compl_zero conj_one_right conj_zero_right disj_zero_right) ``` kleing@24332 ` 232` kleing@24332 ` 233` ```lemma xor_zero_left [simp]: "\ \ x = x" ``` wenzelm@63462 ` 234` ``` by (subst xor_commute) (rule xor_zero_right) ``` kleing@24332 ` 235` kleing@24332 ` 236` ```lemma xor_one_right [simp]: "x \ \ = \ x" ``` wenzelm@63462 ` 237` ``` by (simp only: xor_def compl_one conj_zero_right conj_one_right disj_zero_left) ``` kleing@24332 ` 238` kleing@24332 ` 239` ```lemma xor_one_left [simp]: "\ \ x = \ x" ``` wenzelm@63462 ` 240` ``` by (subst xor_commute) (rule xor_one_right) ``` kleing@24332 ` 241` kleing@24332 ` 242` ```lemma xor_self [simp]: "x \ x = \" ``` wenzelm@63462 ` 243` ``` by (simp only: xor_def conj_cancel_right conj_cancel_left disj_zero_right) ``` kleing@24332 ` 244` kleing@24332 ` 245` ```lemma xor_left_self [simp]: "x \ (x \ y) = y" ``` wenzelm@63462 ` 246` ``` by (simp only: xor_assoc [symmetric] xor_self xor_zero_left) ``` kleing@24332 ` 247` huffman@29996 ` 248` ```lemma xor_compl_left [simp]: "\ x \ y = \ (x \ y)" ``` wenzelm@63462 ` 249` ``` apply (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl) ``` wenzelm@63462 ` 250` ``` apply (simp only: conj_disj_distribs) ``` wenzelm@63462 ` 251` ``` apply (simp only: conj_cancel_right conj_cancel_left) ``` wenzelm@63462 ` 252` ``` apply (simp only: disj_zero_left disj_zero_right) ``` wenzelm@63462 ` 253` ``` apply (simp only: disj_ac conj_ac) ``` wenzelm@63462 ` 254` ``` done ``` kleing@24332 ` 255` huffman@29996 ` 256` ```lemma xor_compl_right [simp]: "x \ \ y = \ (x \ y)" ``` wenzelm@63462 ` 257` ``` apply (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl) ``` wenzelm@63462 ` 258` ``` apply (simp only: conj_disj_distribs) ``` wenzelm@63462 ` 259` ``` apply (simp only: conj_cancel_right conj_cancel_left) ``` wenzelm@63462 ` 260` ``` apply (simp only: disj_zero_left disj_zero_right) ``` wenzelm@63462 ` 261` ``` apply (simp only: disj_ac conj_ac) ``` wenzelm@63462 ` 262` ``` done ``` kleing@24332 ` 263` huffman@29996 ` 264` ```lemma xor_cancel_right: "x \ \ x = \" ``` wenzelm@63462 ` 265` ``` by (simp only: xor_compl_right xor_self compl_zero) ``` kleing@24332 ` 266` huffman@29996 ` 267` ```lemma xor_cancel_left: "\ x \ x = \" ``` wenzelm@63462 ` 268` ``` by (simp only: xor_compl_left xor_self compl_zero) ``` kleing@24332 ` 269` kleing@24332 ` 270` ```lemma conj_xor_distrib: "x \ (y \ z) = (x \ y) \ (x \ z)" ``` kleing@24332 ` 271` ```proof - ``` wenzelm@63462 ` 272` ``` have *: "(x \ y \ \ z) \ (x \ \ y \ z) = ``` kleing@24332 ` 273` ``` (y \ x \ \ x) \ (z \ x \ \ x) \ (x \ y \ \ z) \ (x \ \ y \ z)" ``` huffman@24357 ` 274` ``` by (simp only: conj_cancel_right conj_zero_right disj_zero_left) ``` wenzelm@63462 ` 275` ``` then show "x \ (y \ z) = (x \ y) \ (x \ z)" ``` huffman@24357 ` 276` ``` by (simp (no_asm_use) only: ``` kleing@24332 ` 277` ``` xor_def de_Morgan_disj de_Morgan_conj double_compl ``` kleing@24332 ` 278` ``` conj_disj_distribs conj_ac disj_ac) ``` kleing@24332 ` 279` ```qed ``` kleing@24332 ` 280` wenzelm@60855 ` 281` ```lemma conj_xor_distrib2: "(y \ z) \ x = (y \ x) \ (z \ x)" ``` kleing@24332 ` 282` ```proof - ``` kleing@24332 ` 283` ``` have "x \ (y \ z) = (x \ y) \ (x \ z)" ``` kleing@24332 ` 284` ``` by (rule conj_xor_distrib) ``` wenzelm@63462 ` 285` ``` then show "(y \ z) \ x = (y \ x) \ (z \ x)" ``` huffman@24357 ` 286` ``` by (simp only: conj_commute) ``` kleing@24332 ` 287` ```qed ``` kleing@24332 ` 288` wenzelm@60855 ` 289` ```lemmas conj_xor_distribs = conj_xor_distrib conj_xor_distrib2 ``` kleing@24332 ` 290` kleing@24332 ` 291` ```end ``` kleing@24332 ` 292` kleing@24332 ` 293` ```end ```