src/HOL/Library/Boolean_Algebra.thy
 author bulwahn Tue Apr 05 09:38:28 2011 +0200 (2011-04-05) changeset 42231 bc1891226d00 parent 34973 ae634fad947e child 54868 bab6cade3cc5 permissions -rw-r--r--
 haftmann@29629 ` 1` ```(* Title: HOL/Library/Boolean_Algebra.thy ``` haftmann@29629 ` 2` ``` Author: Brian Huffman ``` kleing@24332 ` 3` ```*) ``` kleing@24332 ` 4` kleing@24332 ` 5` ```header {* Boolean Algebras *} ``` kleing@24332 ` 6` kleing@24332 ` 7` ```theory Boolean_Algebra ``` haftmann@30663 ` 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@34973 ` 27` haftmann@34973 ` 28` ```sublocale boolean < conj!: abel_semigroup conj proof ``` haftmann@34973 ` 29` ```qed (fact conj_assoc conj_commute)+ ``` haftmann@34973 ` 30` haftmann@34973 ` 31` ```sublocale boolean < disj!: abel_semigroup disj proof ``` haftmann@34973 ` 32` ```qed (fact disj_assoc disj_commute)+ ``` haftmann@34973 ` 33` haftmann@34973 ` 34` ```context boolean ``` kleing@24332 ` 35` ```begin ``` kleing@24332 ` 36` haftmann@34973 ` 37` ```lemmas conj_left_commute = conj.left_commute ``` kleing@24332 ` 38` haftmann@34973 ` 39` ```lemmas disj_left_commute = disj.left_commute ``` haftmann@34973 ` 40` haftmann@34973 ` 41` ```lemmas conj_ac = conj.assoc conj.commute conj.left_commute ``` haftmann@34973 ` 42` ```lemmas disj_ac = disj.assoc disj.commute disj.left_commute ``` kleing@24332 ` 43` kleing@24332 ` 44` ```lemma dual: "boolean disj conj compl one zero" ``` kleing@24332 ` 45` ```apply (rule boolean.intro) ``` kleing@24332 ` 46` ```apply (rule disj_assoc) ``` kleing@24332 ` 47` ```apply (rule conj_assoc) ``` kleing@24332 ` 48` ```apply (rule disj_commute) ``` kleing@24332 ` 49` ```apply (rule conj_commute) ``` kleing@24332 ` 50` ```apply (rule disj_conj_distrib) ``` kleing@24332 ` 51` ```apply (rule conj_disj_distrib) ``` kleing@24332 ` 52` ```apply (rule disj_zero_right) ``` kleing@24332 ` 53` ```apply (rule conj_one_right) ``` kleing@24332 ` 54` ```apply (rule disj_cancel_right) ``` kleing@24332 ` 55` ```apply (rule conj_cancel_right) ``` kleing@24332 ` 56` ```done ``` kleing@24332 ` 57` huffman@24357 ` 58` ```subsection {* Complement *} ``` kleing@24332 ` 59` kleing@24332 ` 60` ```lemma complement_unique: ``` kleing@24332 ` 61` ``` assumes 1: "a \ x = \" ``` kleing@24332 ` 62` ``` assumes 2: "a \ x = \" ``` kleing@24332 ` 63` ``` assumes 3: "a \ y = \" ``` kleing@24332 ` 64` ``` assumes 4: "a \ y = \" ``` kleing@24332 ` 65` ``` shows "x = y" ``` kleing@24332 ` 66` ```proof - ``` kleing@24332 ` 67` ``` have "(a \ x) \ (x \ y) = (a \ y) \ (x \ y)" using 1 3 by simp ``` kleing@24332 ` 68` ``` hence "(x \ a) \ (x \ y) = (y \ a) \ (y \ x)" using conj_commute by simp ``` kleing@24332 ` 69` ``` hence "x \ (a \ y) = y \ (a \ x)" using conj_disj_distrib by simp ``` kleing@24332 ` 70` ``` hence "x \ \ = y \ \" using 2 4 by simp ``` kleing@24332 ` 71` ``` thus "x = y" using conj_one_right by simp ``` kleing@24332 ` 72` ```qed ``` kleing@24332 ` 73` huffman@24357 ` 74` ```lemma compl_unique: "\x \ y = \; x \ y = \\ \ \ x = y" ``` kleing@24332 ` 75` ```by (rule complement_unique [OF conj_cancel_right disj_cancel_right]) ``` kleing@24332 ` 76` kleing@24332 ` 77` ```lemma double_compl [simp]: "\ (\ x) = x" ``` kleing@24332 ` 78` ```proof (rule compl_unique) ``` huffman@24357 ` 79` ``` from conj_cancel_right show "\ x \ x = \" by (simp only: conj_commute) ``` huffman@24357 ` 80` ``` from disj_cancel_right show "\ x \ x = \" by (simp only: disj_commute) ``` kleing@24332 ` 81` ```qed ``` kleing@24332 ` 82` kleing@24332 ` 83` ```lemma compl_eq_compl_iff [simp]: "(\ x = \ y) = (x = y)" ``` kleing@24332 ` 84` ```by (rule inj_eq [OF inj_on_inverseI], rule double_compl) ``` kleing@24332 ` 85` huffman@24357 ` 86` ```subsection {* Conjunction *} ``` kleing@24332 ` 87` huffman@24393 ` 88` ```lemma conj_absorb [simp]: "x \ x = x" ``` kleing@24332 ` 89` ```proof - ``` kleing@24332 ` 90` ``` have "x \ x = (x \ x) \ \" using disj_zero_right by simp ``` kleing@24332 ` 91` ``` also have "... = (x \ x) \ (x \ \ x)" using conj_cancel_right by simp ``` huffman@24357 ` 92` ``` also have "... = x \ (x \ \ x)" using conj_disj_distrib by (simp only:) ``` kleing@24332 ` 93` ``` also have "... = x \ \" using disj_cancel_right by simp ``` kleing@24332 ` 94` ``` also have "... = x" using conj_one_right by simp ``` kleing@24332 ` 95` ``` finally show ?thesis . ``` kleing@24332 ` 96` ```qed ``` kleing@24332 ` 97` kleing@24332 ` 98` ```lemma conj_zero_right [simp]: "x \ \ = \" ``` kleing@24332 ` 99` ```proof - ``` kleing@24332 ` 100` ``` have "x \ \ = x \ (x \ \ x)" using conj_cancel_right by simp ``` huffman@24393 ` 101` ``` also have "... = (x \ x) \ \ x" using conj_assoc by (simp only:) ``` kleing@24332 ` 102` ``` also have "... = x \ \ x" using conj_absorb by simp ``` kleing@24332 ` 103` ``` also have "... = \" using conj_cancel_right by simp ``` kleing@24332 ` 104` ``` finally show ?thesis . ``` kleing@24332 ` 105` ```qed ``` kleing@24332 ` 106` kleing@24332 ` 107` ```lemma compl_one [simp]: "\ \ = \" ``` kleing@24332 ` 108` ```by (rule compl_unique [OF conj_zero_right disj_zero_right]) ``` kleing@24332 ` 109` kleing@24332 ` 110` ```lemma conj_zero_left [simp]: "\ \ x = \" ``` kleing@24332 ` 111` ```by (subst conj_commute) (rule conj_zero_right) ``` kleing@24332 ` 112` kleing@24332 ` 113` ```lemma conj_one_left [simp]: "\ \ x = x" ``` kleing@24332 ` 114` ```by (subst conj_commute) (rule conj_one_right) ``` kleing@24332 ` 115` kleing@24332 ` 116` ```lemma conj_cancel_left [simp]: "\ x \ x = \" ``` kleing@24332 ` 117` ```by (subst conj_commute) (rule conj_cancel_right) ``` kleing@24332 ` 118` kleing@24332 ` 119` ```lemma conj_left_absorb [simp]: "x \ (x \ y) = x \ y" ``` huffman@24357 ` 120` ```by (simp only: conj_assoc [symmetric] conj_absorb) ``` kleing@24332 ` 121` kleing@24332 ` 122` ```lemma conj_disj_distrib2: ``` kleing@24332 ` 123` ``` "(y \ z) \ x = (y \ x) \ (z \ x)" ``` huffman@24357 ` 124` ```by (simp only: conj_commute conj_disj_distrib) ``` kleing@24332 ` 125` kleing@24332 ` 126` ```lemmas conj_disj_distribs = ``` kleing@24332 ` 127` ``` conj_disj_distrib conj_disj_distrib2 ``` kleing@24332 ` 128` huffman@24357 ` 129` ```subsection {* Disjunction *} ``` kleing@24332 ` 130` kleing@24332 ` 131` ```lemma disj_absorb [simp]: "x \ x = x" ``` kleing@24332 ` 132` ```by (rule boolean.conj_absorb [OF dual]) ``` kleing@24332 ` 133` kleing@24332 ` 134` ```lemma disj_one_right [simp]: "x \ \ = \" ``` kleing@24332 ` 135` ```by (rule boolean.conj_zero_right [OF dual]) ``` kleing@24332 ` 136` kleing@24332 ` 137` ```lemma compl_zero [simp]: "\ \ = \" ``` kleing@24332 ` 138` ```by (rule boolean.compl_one [OF dual]) ``` kleing@24332 ` 139` kleing@24332 ` 140` ```lemma disj_zero_left [simp]: "\ \ x = x" ``` kleing@24332 ` 141` ```by (rule boolean.conj_one_left [OF dual]) ``` kleing@24332 ` 142` kleing@24332 ` 143` ```lemma disj_one_left [simp]: "\ \ x = \" ``` kleing@24332 ` 144` ```by (rule boolean.conj_zero_left [OF dual]) ``` kleing@24332 ` 145` kleing@24332 ` 146` ```lemma disj_cancel_left [simp]: "\ x \ x = \" ``` kleing@24332 ` 147` ```by (rule boolean.conj_cancel_left [OF dual]) ``` kleing@24332 ` 148` kleing@24332 ` 149` ```lemma disj_left_absorb [simp]: "x \ (x \ y) = x \ y" ``` kleing@24332 ` 150` ```by (rule boolean.conj_left_absorb [OF dual]) ``` kleing@24332 ` 151` kleing@24332 ` 152` ```lemma disj_conj_distrib2: ``` kleing@24332 ` 153` ``` "(y \ z) \ x = (y \ x) \ (z \ x)" ``` kleing@24332 ` 154` ```by (rule boolean.conj_disj_distrib2 [OF dual]) ``` kleing@24332 ` 155` kleing@24332 ` 156` ```lemmas disj_conj_distribs = ``` kleing@24332 ` 157` ``` disj_conj_distrib disj_conj_distrib2 ``` kleing@24332 ` 158` huffman@24357 ` 159` ```subsection {* De Morgan's Laws *} ``` kleing@24332 ` 160` kleing@24332 ` 161` ```lemma de_Morgan_conj [simp]: "\ (x \ y) = \ x \ \ y" ``` kleing@24332 ` 162` ```proof (rule compl_unique) ``` kleing@24332 ` 163` ``` have "(x \ y) \ (\ x \ \ y) = ((x \ y) \ \ x) \ ((x \ y) \ \ y)" ``` kleing@24332 ` 164` ``` by (rule conj_disj_distrib) ``` kleing@24332 ` 165` ``` also have "... = (y \ (x \ \ x)) \ (x \ (y \ \ y))" ``` huffman@24357 ` 166` ``` by (simp only: conj_ac) ``` kleing@24332 ` 167` ``` finally show "(x \ y) \ (\ x \ \ y) = \" ``` huffman@24357 ` 168` ``` by (simp only: conj_cancel_right conj_zero_right disj_zero_right) ``` kleing@24332 ` 169` ```next ``` kleing@24332 ` 170` ``` have "(x \ y) \ (\ x \ \ y) = (x \ (\ x \ \ y)) \ (y \ (\ x \ \ y))" ``` kleing@24332 ` 171` ``` by (rule disj_conj_distrib2) ``` kleing@24332 ` 172` ``` also have "... = (\ y \ (x \ \ x)) \ (\ x \ (y \ \ y))" ``` huffman@24357 ` 173` ``` by (simp only: disj_ac) ``` kleing@24332 ` 174` ``` finally show "(x \ y) \ (\ x \ \ y) = \" ``` huffman@24357 ` 175` ``` by (simp only: disj_cancel_right disj_one_right conj_one_right) ``` kleing@24332 ` 176` ```qed ``` kleing@24332 ` 177` kleing@24332 ` 178` ```lemma de_Morgan_disj [simp]: "\ (x \ y) = \ x \ \ y" ``` kleing@24332 ` 179` ```by (rule boolean.de_Morgan_conj [OF dual]) ``` kleing@24332 ` 180` kleing@24332 ` 181` ```end ``` kleing@24332 ` 182` huffman@24357 ` 183` ```subsection {* Symmetric Difference *} ``` kleing@24332 ` 184` kleing@24332 ` 185` ```locale boolean_xor = boolean + ``` kleing@24332 ` 186` ``` fixes xor :: "'a => 'a => 'a" (infixr "\" 65) ``` kleing@24332 ` 187` ``` assumes xor_def: "x \ y = (x \ \ y) \ (\ x \ y)" ``` kleing@24332 ` 188` haftmann@34973 ` 189` ```sublocale boolean_xor < xor!: abel_semigroup xor proof ``` haftmann@34973 ` 190` ``` fix x y z :: 'a ``` kleing@24332 ` 191` ``` let ?t = "(x \ y \ z) \ (x \ \ y \ \ z) \ ``` kleing@24332 ` 192` ``` (\ x \ y \ \ z) \ (\ x \ \ y \ z)" ``` kleing@24332 ` 193` ``` have "?t \ (z \ x \ \ x) \ (z \ y \ \ y) = ``` kleing@24332 ` 194` ``` ?t \ (x \ y \ \ y) \ (x \ z \ \ z)" ``` huffman@24357 ` 195` ``` by (simp only: conj_cancel_right conj_zero_right) ``` kleing@24332 ` 196` ``` thus "(x \ y) \ z = x \ (y \ z)" ``` huffman@24357 ` 197` ``` apply (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl) ``` huffman@24357 ` 198` ``` apply (simp only: conj_disj_distribs conj_ac disj_ac) ``` kleing@24332 ` 199` ``` done ``` haftmann@34973 ` 200` ``` show "x \ y = y \ x" ``` haftmann@34973 ` 201` ``` by (simp only: xor_def conj_commute disj_commute) ``` kleing@24332 ` 202` ```qed ``` kleing@24332 ` 203` haftmann@34973 ` 204` ```context boolean_xor ``` haftmann@34973 ` 205` ```begin ``` haftmann@34973 ` 206` haftmann@34973 ` 207` ```lemmas xor_assoc = xor.assoc ``` haftmann@34973 ` 208` ```lemmas xor_commute = xor.commute ``` haftmann@34973 ` 209` ```lemmas xor_left_commute = xor.left_commute ``` haftmann@34973 ` 210` haftmann@34973 ` 211` ```lemmas xor_ac = xor.assoc xor.commute xor.left_commute ``` haftmann@34973 ` 212` haftmann@34973 ` 213` ```lemma xor_def2: ``` haftmann@34973 ` 214` ``` "x \ y = (x \ y) \ (\ x \ \ y)" ``` haftmann@34973 ` 215` ```by (simp only: xor_def conj_disj_distribs ``` haftmann@34973 ` 216` ``` disj_ac conj_ac conj_cancel_right disj_zero_left) ``` kleing@24332 ` 217` kleing@24332 ` 218` ```lemma xor_zero_right [simp]: "x \ \ = x" ``` huffman@24357 ` 219` ```by (simp only: xor_def compl_zero conj_one_right conj_zero_right disj_zero_right) ``` kleing@24332 ` 220` kleing@24332 ` 221` ```lemma xor_zero_left [simp]: "\ \ x = x" ``` kleing@24332 ` 222` ```by (subst xor_commute) (rule xor_zero_right) ``` kleing@24332 ` 223` kleing@24332 ` 224` ```lemma xor_one_right [simp]: "x \ \ = \ x" ``` huffman@24357 ` 225` ```by (simp only: xor_def compl_one conj_zero_right conj_one_right disj_zero_left) ``` kleing@24332 ` 226` kleing@24332 ` 227` ```lemma xor_one_left [simp]: "\ \ x = \ x" ``` kleing@24332 ` 228` ```by (subst xor_commute) (rule xor_one_right) ``` kleing@24332 ` 229` kleing@24332 ` 230` ```lemma xor_self [simp]: "x \ x = \" ``` huffman@24357 ` 231` ```by (simp only: xor_def conj_cancel_right conj_cancel_left disj_zero_right) ``` kleing@24332 ` 232` kleing@24332 ` 233` ```lemma xor_left_self [simp]: "x \ (x \ y) = y" ``` huffman@24357 ` 234` ```by (simp only: xor_assoc [symmetric] xor_self xor_zero_left) ``` kleing@24332 ` 235` huffman@29996 ` 236` ```lemma xor_compl_left [simp]: "\ x \ y = \ (x \ y)" ``` huffman@24357 ` 237` ```apply (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl) ``` huffman@24357 ` 238` ```apply (simp only: conj_disj_distribs) ``` huffman@24357 ` 239` ```apply (simp only: conj_cancel_right conj_cancel_left) ``` huffman@24357 ` 240` ```apply (simp only: disj_zero_left disj_zero_right) ``` huffman@24357 ` 241` ```apply (simp only: disj_ac conj_ac) ``` kleing@24332 ` 242` ```done ``` kleing@24332 ` 243` huffman@29996 ` 244` ```lemma xor_compl_right [simp]: "x \ \ y = \ (x \ y)" ``` huffman@24357 ` 245` ```apply (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl) ``` huffman@24357 ` 246` ```apply (simp only: conj_disj_distribs) ``` huffman@24357 ` 247` ```apply (simp only: conj_cancel_right conj_cancel_left) ``` huffman@24357 ` 248` ```apply (simp only: disj_zero_left disj_zero_right) ``` huffman@24357 ` 249` ```apply (simp only: disj_ac conj_ac) ``` kleing@24332 ` 250` ```done ``` kleing@24332 ` 251` huffman@29996 ` 252` ```lemma xor_cancel_right: "x \ \ x = \" ``` huffman@24357 ` 253` ```by (simp only: xor_compl_right xor_self compl_zero) ``` kleing@24332 ` 254` huffman@29996 ` 255` ```lemma xor_cancel_left: "\ x \ x = \" ``` huffman@29996 ` 256` ```by (simp only: xor_compl_left xor_self compl_zero) ``` kleing@24332 ` 257` kleing@24332 ` 258` ```lemma conj_xor_distrib: "x \ (y \ z) = (x \ y) \ (x \ z)" ``` kleing@24332 ` 259` ```proof - ``` kleing@24332 ` 260` ``` have "(x \ y \ \ z) \ (x \ \ y \ z) = ``` kleing@24332 ` 261` ``` (y \ x \ \ x) \ (z \ x \ \ x) \ (x \ y \ \ z) \ (x \ \ y \ z)" ``` huffman@24357 ` 262` ``` by (simp only: conj_cancel_right conj_zero_right disj_zero_left) ``` kleing@24332 ` 263` ``` thus "x \ (y \ z) = (x \ y) \ (x \ z)" ``` huffman@24357 ` 264` ``` by (simp (no_asm_use) only: ``` kleing@24332 ` 265` ``` xor_def de_Morgan_disj de_Morgan_conj double_compl ``` kleing@24332 ` 266` ``` conj_disj_distribs conj_ac disj_ac) ``` kleing@24332 ` 267` ```qed ``` kleing@24332 ` 268` kleing@24332 ` 269` ```lemma conj_xor_distrib2: ``` kleing@24332 ` 270` ``` "(y \ z) \ x = (y \ x) \ (z \ x)" ``` kleing@24332 ` 271` ```proof - ``` kleing@24332 ` 272` ``` have "x \ (y \ z) = (x \ y) \ (x \ z)" ``` kleing@24332 ` 273` ``` by (rule conj_xor_distrib) ``` kleing@24332 ` 274` ``` thus "(y \ z) \ x = (y \ x) \ (z \ x)" ``` huffman@24357 ` 275` ``` by (simp only: conj_commute) ``` kleing@24332 ` 276` ```qed ``` kleing@24332 ` 277` kleing@24332 ` 278` ```lemmas conj_xor_distribs = ``` kleing@24332 ` 279` ``` conj_xor_distrib conj_xor_distrib2 ``` kleing@24332 ` 280` kleing@24332 ` 281` ```end ``` kleing@24332 ` 282` kleing@24332 ` 283` ```end ```