src/HOL/Library/Boolean_Algebra.thy
 author wenzelm Wed Jun 17 11:03:05 2015 +0200 (2015-06-17) changeset 60500 903bb1495239 parent 58881 b9556a055632 child 60855 6449ae4b85f9 permissions -rw-r--r--
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 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 ``` 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@54868 ` 27` ```begin ``` haftmann@34973 ` 28` haftmann@54868 ` 29` ```sublocale conj!: abel_semigroup conj proof ``` haftmann@34973 ` 30` ```qed (fact conj_assoc conj_commute)+ ``` haftmann@34973 ` 31` haftmann@54868 ` 32` ```sublocale disj!: abel_semigroup disj proof ``` haftmann@34973 ` 33` ```qed (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" ``` kleing@24332 ` 43` ```apply (rule boolean.intro) ``` kleing@24332 ` 44` ```apply (rule disj_assoc) ``` kleing@24332 ` 45` ```apply (rule conj_assoc) ``` kleing@24332 ` 46` ```apply (rule disj_commute) ``` kleing@24332 ` 47` ```apply (rule conj_commute) ``` kleing@24332 ` 48` ```apply (rule disj_conj_distrib) ``` kleing@24332 ` 49` ```apply (rule conj_disj_distrib) ``` kleing@24332 ` 50` ```apply (rule disj_zero_right) ``` kleing@24332 ` 51` ```apply (rule conj_one_right) ``` kleing@24332 ` 52` ```apply (rule disj_cancel_right) ``` kleing@24332 ` 53` ```apply (rule conj_cancel_right) ``` kleing@24332 ` 54` ```done ``` kleing@24332 ` 55` wenzelm@60500 ` 56` ```subsection \Complement\ ``` kleing@24332 ` 57` kleing@24332 ` 58` ```lemma complement_unique: ``` kleing@24332 ` 59` ``` assumes 1: "a \ x = \" ``` kleing@24332 ` 60` ``` assumes 2: "a \ x = \" ``` kleing@24332 ` 61` ``` assumes 3: "a \ y = \" ``` kleing@24332 ` 62` ``` assumes 4: "a \ y = \" ``` kleing@24332 ` 63` ``` shows "x = y" ``` kleing@24332 ` 64` ```proof - ``` kleing@24332 ` 65` ``` have "(a \ x) \ (x \ y) = (a \ y) \ (x \ y)" using 1 3 by simp ``` kleing@24332 ` 66` ``` hence "(x \ a) \ (x \ y) = (y \ a) \ (y \ x)" using conj_commute by simp ``` kleing@24332 ` 67` ``` hence "x \ (a \ y) = y \ (a \ x)" using conj_disj_distrib by simp ``` kleing@24332 ` 68` ``` hence "x \ \ = y \ \" using 2 4 by simp ``` kleing@24332 ` 69` ``` thus "x = y" using conj_one_right by simp ``` kleing@24332 ` 70` ```qed ``` kleing@24332 ` 71` huffman@24357 ` 72` ```lemma compl_unique: "\x \ y = \; x \ y = \\ \ \ x = y" ``` kleing@24332 ` 73` ```by (rule complement_unique [OF conj_cancel_right disj_cancel_right]) ``` kleing@24332 ` 74` kleing@24332 ` 75` ```lemma double_compl [simp]: "\ (\ x) = x" ``` kleing@24332 ` 76` ```proof (rule compl_unique) ``` huffman@24357 ` 77` ``` from conj_cancel_right show "\ x \ x = \" by (simp only: conj_commute) ``` huffman@24357 ` 78` ``` from disj_cancel_right show "\ x \ x = \" by (simp only: disj_commute) ``` kleing@24332 ` 79` ```qed ``` kleing@24332 ` 80` kleing@24332 ` 81` ```lemma compl_eq_compl_iff [simp]: "(\ x = \ y) = (x = y)" ``` kleing@24332 ` 82` ```by (rule inj_eq [OF inj_on_inverseI], rule double_compl) ``` kleing@24332 ` 83` wenzelm@60500 ` 84` ```subsection \Conjunction\ ``` kleing@24332 ` 85` huffman@24393 ` 86` ```lemma conj_absorb [simp]: "x \ x = x" ``` kleing@24332 ` 87` ```proof - ``` kleing@24332 ` 88` ``` have "x \ x = (x \ x) \ \" using disj_zero_right by simp ``` kleing@24332 ` 89` ``` also have "... = (x \ x) \ (x \ \ x)" using conj_cancel_right by simp ``` huffman@24357 ` 90` ``` also have "... = x \ (x \ \ x)" using conj_disj_distrib by (simp only:) ``` kleing@24332 ` 91` ``` also have "... = x \ \" using disj_cancel_right by simp ``` kleing@24332 ` 92` ``` also have "... = x" using conj_one_right by simp ``` kleing@24332 ` 93` ``` finally show ?thesis . ``` kleing@24332 ` 94` ```qed ``` kleing@24332 ` 95` kleing@24332 ` 96` ```lemma conj_zero_right [simp]: "x \ \ = \" ``` kleing@24332 ` 97` ```proof - ``` kleing@24332 ` 98` ``` have "x \ \ = x \ (x \ \ x)" using conj_cancel_right by simp ``` huffman@24393 ` 99` ``` also have "... = (x \ x) \ \ x" using conj_assoc by (simp only:) ``` kleing@24332 ` 100` ``` also have "... = x \ \ x" using conj_absorb by simp ``` kleing@24332 ` 101` ``` also have "... = \" using conj_cancel_right by simp ``` kleing@24332 ` 102` ``` finally show ?thesis . ``` kleing@24332 ` 103` ```qed ``` kleing@24332 ` 104` kleing@24332 ` 105` ```lemma compl_one [simp]: "\ \ = \" ``` kleing@24332 ` 106` ```by (rule compl_unique [OF conj_zero_right disj_zero_right]) ``` kleing@24332 ` 107` kleing@24332 ` 108` ```lemma conj_zero_left [simp]: "\ \ x = \" ``` kleing@24332 ` 109` ```by (subst conj_commute) (rule conj_zero_right) ``` kleing@24332 ` 110` kleing@24332 ` 111` ```lemma conj_one_left [simp]: "\ \ x = x" ``` kleing@24332 ` 112` ```by (subst conj_commute) (rule conj_one_right) ``` kleing@24332 ` 113` kleing@24332 ` 114` ```lemma conj_cancel_left [simp]: "\ x \ x = \" ``` kleing@24332 ` 115` ```by (subst conj_commute) (rule conj_cancel_right) ``` kleing@24332 ` 116` kleing@24332 ` 117` ```lemma conj_left_absorb [simp]: "x \ (x \ y) = x \ y" ``` huffman@24357 ` 118` ```by (simp only: conj_assoc [symmetric] conj_absorb) ``` kleing@24332 ` 119` kleing@24332 ` 120` ```lemma conj_disj_distrib2: ``` kleing@24332 ` 121` ``` "(y \ z) \ x = (y \ x) \ (z \ x)" ``` huffman@24357 ` 122` ```by (simp only: conj_commute conj_disj_distrib) ``` kleing@24332 ` 123` kleing@24332 ` 124` ```lemmas conj_disj_distribs = ``` kleing@24332 ` 125` ``` conj_disj_distrib conj_disj_distrib2 ``` kleing@24332 ` 126` wenzelm@60500 ` 127` ```subsection \Disjunction\ ``` kleing@24332 ` 128` kleing@24332 ` 129` ```lemma disj_absorb [simp]: "x \ x = x" ``` kleing@24332 ` 130` ```by (rule boolean.conj_absorb [OF dual]) ``` kleing@24332 ` 131` kleing@24332 ` 132` ```lemma disj_one_right [simp]: "x \ \ = \" ``` kleing@24332 ` 133` ```by (rule boolean.conj_zero_right [OF dual]) ``` kleing@24332 ` 134` kleing@24332 ` 135` ```lemma compl_zero [simp]: "\ \ = \" ``` kleing@24332 ` 136` ```by (rule boolean.compl_one [OF dual]) ``` kleing@24332 ` 137` kleing@24332 ` 138` ```lemma disj_zero_left [simp]: "\ \ x = x" ``` kleing@24332 ` 139` ```by (rule boolean.conj_one_left [OF dual]) ``` kleing@24332 ` 140` kleing@24332 ` 141` ```lemma disj_one_left [simp]: "\ \ x = \" ``` kleing@24332 ` 142` ```by (rule boolean.conj_zero_left [OF dual]) ``` kleing@24332 ` 143` kleing@24332 ` 144` ```lemma disj_cancel_left [simp]: "\ x \ x = \" ``` kleing@24332 ` 145` ```by (rule boolean.conj_cancel_left [OF dual]) ``` kleing@24332 ` 146` kleing@24332 ` 147` ```lemma disj_left_absorb [simp]: "x \ (x \ y) = x \ y" ``` kleing@24332 ` 148` ```by (rule boolean.conj_left_absorb [OF dual]) ``` kleing@24332 ` 149` kleing@24332 ` 150` ```lemma disj_conj_distrib2: ``` kleing@24332 ` 151` ``` "(y \ z) \ x = (y \ x) \ (z \ x)" ``` kleing@24332 ` 152` ```by (rule boolean.conj_disj_distrib2 [OF dual]) ``` kleing@24332 ` 153` kleing@24332 ` 154` ```lemmas disj_conj_distribs = ``` kleing@24332 ` 155` ``` disj_conj_distrib disj_conj_distrib2 ``` kleing@24332 ` 156` wenzelm@60500 ` 157` ```subsection \De Morgan's Laws\ ``` kleing@24332 ` 158` kleing@24332 ` 159` ```lemma de_Morgan_conj [simp]: "\ (x \ y) = \ x \ \ y" ``` kleing@24332 ` 160` ```proof (rule compl_unique) ``` kleing@24332 ` 161` ``` have "(x \ y) \ (\ x \ \ y) = ((x \ y) \ \ x) \ ((x \ y) \ \ y)" ``` kleing@24332 ` 162` ``` by (rule conj_disj_distrib) ``` kleing@24332 ` 163` ``` also have "... = (y \ (x \ \ x)) \ (x \ (y \ \ y))" ``` huffman@24357 ` 164` ``` by (simp only: conj_ac) ``` kleing@24332 ` 165` ``` finally show "(x \ y) \ (\ x \ \ y) = \" ``` huffman@24357 ` 166` ``` by (simp only: conj_cancel_right conj_zero_right disj_zero_right) ``` kleing@24332 ` 167` ```next ``` kleing@24332 ` 168` ``` have "(x \ y) \ (\ x \ \ y) = (x \ (\ x \ \ y)) \ (y \ (\ x \ \ y))" ``` kleing@24332 ` 169` ``` by (rule disj_conj_distrib2) ``` kleing@24332 ` 170` ``` also have "... = (\ y \ (x \ \ x)) \ (\ x \ (y \ \ y))" ``` huffman@24357 ` 171` ``` by (simp only: disj_ac) ``` kleing@24332 ` 172` ``` finally show "(x \ y) \ (\ x \ \ y) = \" ``` huffman@24357 ` 173` ``` by (simp only: disj_cancel_right disj_one_right conj_one_right) ``` kleing@24332 ` 174` ```qed ``` kleing@24332 ` 175` kleing@24332 ` 176` ```lemma de_Morgan_disj [simp]: "\ (x \ y) = \ x \ \ y" ``` kleing@24332 ` 177` ```by (rule boolean.de_Morgan_conj [OF dual]) ``` kleing@24332 ` 178` kleing@24332 ` 179` ```end ``` kleing@24332 ` 180` wenzelm@60500 ` 181` ```subsection \Symmetric Difference\ ``` kleing@24332 ` 182` kleing@24332 ` 183` ```locale boolean_xor = boolean + ``` kleing@24332 ` 184` ``` fixes xor :: "'a => 'a => 'a" (infixr "\" 65) ``` kleing@24332 ` 185` ``` assumes xor_def: "x \ y = (x \ \ y) \ (\ x \ y)" ``` haftmann@54868 ` 186` ```begin ``` kleing@24332 ` 187` haftmann@54868 ` 188` ```sublocale xor!: abel_semigroup xor proof ``` haftmann@34973 ` 189` ``` fix x y z :: 'a ``` kleing@24332 ` 190` ``` let ?t = "(x \ y \ z) \ (x \ \ y \ \ z) \ ``` kleing@24332 ` 191` ``` (\ x \ y \ \ z) \ (\ x \ \ y \ z)" ``` kleing@24332 ` 192` ``` have "?t \ (z \ x \ \ x) \ (z \ y \ \ y) = ``` kleing@24332 ` 193` ``` ?t \ (x \ y \ \ y) \ (x \ z \ \ z)" ``` huffman@24357 ` 194` ``` by (simp only: conj_cancel_right conj_zero_right) ``` kleing@24332 ` 195` ``` thus "(x \ y) \ z = x \ (y \ z)" ``` huffman@24357 ` 196` ``` apply (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl) ``` huffman@24357 ` 197` ``` apply (simp only: conj_disj_distribs conj_ac disj_ac) ``` kleing@24332 ` 198` ``` done ``` haftmann@34973 ` 199` ``` show "x \ y = y \ x" ``` haftmann@34973 ` 200` ``` by (simp only: xor_def conj_commute disj_commute) ``` kleing@24332 ` 201` ```qed ``` kleing@24332 ` 202` haftmann@34973 ` 203` ```lemmas xor_assoc = xor.assoc ``` haftmann@34973 ` 204` ```lemmas xor_commute = xor.commute ``` haftmann@34973 ` 205` ```lemmas xor_left_commute = xor.left_commute ``` haftmann@34973 ` 206` haftmann@34973 ` 207` ```lemmas xor_ac = xor.assoc xor.commute xor.left_commute ``` haftmann@34973 ` 208` haftmann@34973 ` 209` ```lemma xor_def2: ``` haftmann@34973 ` 210` ``` "x \ y = (x \ y) \ (\ x \ \ y)" ``` haftmann@34973 ` 211` ```by (simp only: xor_def conj_disj_distribs ``` haftmann@34973 ` 212` ``` disj_ac conj_ac conj_cancel_right disj_zero_left) ``` kleing@24332 ` 213` kleing@24332 ` 214` ```lemma xor_zero_right [simp]: "x \ \ = x" ``` huffman@24357 ` 215` ```by (simp only: xor_def compl_zero conj_one_right conj_zero_right disj_zero_right) ``` kleing@24332 ` 216` kleing@24332 ` 217` ```lemma xor_zero_left [simp]: "\ \ x = x" ``` kleing@24332 ` 218` ```by (subst xor_commute) (rule xor_zero_right) ``` kleing@24332 ` 219` kleing@24332 ` 220` ```lemma xor_one_right [simp]: "x \ \ = \ x" ``` huffman@24357 ` 221` ```by (simp only: xor_def compl_one conj_zero_right conj_one_right disj_zero_left) ``` kleing@24332 ` 222` kleing@24332 ` 223` ```lemma xor_one_left [simp]: "\ \ x = \ x" ``` kleing@24332 ` 224` ```by (subst xor_commute) (rule xor_one_right) ``` kleing@24332 ` 225` kleing@24332 ` 226` ```lemma xor_self [simp]: "x \ x = \" ``` huffman@24357 ` 227` ```by (simp only: xor_def conj_cancel_right conj_cancel_left disj_zero_right) ``` kleing@24332 ` 228` kleing@24332 ` 229` ```lemma xor_left_self [simp]: "x \ (x \ y) = y" ``` huffman@24357 ` 230` ```by (simp only: xor_assoc [symmetric] xor_self xor_zero_left) ``` kleing@24332 ` 231` huffman@29996 ` 232` ```lemma xor_compl_left [simp]: "\ x \ y = \ (x \ y)" ``` huffman@24357 ` 233` ```apply (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl) ``` huffman@24357 ` 234` ```apply (simp only: conj_disj_distribs) ``` huffman@24357 ` 235` ```apply (simp only: conj_cancel_right conj_cancel_left) ``` huffman@24357 ` 236` ```apply (simp only: disj_zero_left disj_zero_right) ``` huffman@24357 ` 237` ```apply (simp only: disj_ac conj_ac) ``` kleing@24332 ` 238` ```done ``` kleing@24332 ` 239` huffman@29996 ` 240` ```lemma xor_compl_right [simp]: "x \ \ y = \ (x \ y)" ``` huffman@24357 ` 241` ```apply (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl) ``` huffman@24357 ` 242` ```apply (simp only: conj_disj_distribs) ``` huffman@24357 ` 243` ```apply (simp only: conj_cancel_right conj_cancel_left) ``` huffman@24357 ` 244` ```apply (simp only: disj_zero_left disj_zero_right) ``` huffman@24357 ` 245` ```apply (simp only: disj_ac conj_ac) ``` kleing@24332 ` 246` ```done ``` kleing@24332 ` 247` huffman@29996 ` 248` ```lemma xor_cancel_right: "x \ \ x = \" ``` huffman@24357 ` 249` ```by (simp only: xor_compl_right xor_self compl_zero) ``` kleing@24332 ` 250` huffman@29996 ` 251` ```lemma xor_cancel_left: "\ x \ x = \" ``` huffman@29996 ` 252` ```by (simp only: xor_compl_left xor_self compl_zero) ``` kleing@24332 ` 253` kleing@24332 ` 254` ```lemma conj_xor_distrib: "x \ (y \ z) = (x \ y) \ (x \ z)" ``` kleing@24332 ` 255` ```proof - ``` kleing@24332 ` 256` ``` have "(x \ y \ \ z) \ (x \ \ y \ z) = ``` kleing@24332 ` 257` ``` (y \ x \ \ x) \ (z \ x \ \ x) \ (x \ y \ \ z) \ (x \ \ y \ z)" ``` huffman@24357 ` 258` ``` by (simp only: conj_cancel_right conj_zero_right disj_zero_left) ``` kleing@24332 ` 259` ``` thus "x \ (y \ z) = (x \ y) \ (x \ z)" ``` huffman@24357 ` 260` ``` by (simp (no_asm_use) only: ``` kleing@24332 ` 261` ``` xor_def de_Morgan_disj de_Morgan_conj double_compl ``` kleing@24332 ` 262` ``` conj_disj_distribs conj_ac disj_ac) ``` kleing@24332 ` 263` ```qed ``` kleing@24332 ` 264` kleing@24332 ` 265` ```lemma conj_xor_distrib2: ``` kleing@24332 ` 266` ``` "(y \ z) \ x = (y \ x) \ (z \ x)" ``` kleing@24332 ` 267` ```proof - ``` kleing@24332 ` 268` ``` have "x \ (y \ z) = (x \ y) \ (x \ z)" ``` kleing@24332 ` 269` ``` by (rule conj_xor_distrib) ``` kleing@24332 ` 270` ``` thus "(y \ z) \ x = (y \ x) \ (z \ x)" ``` huffman@24357 ` 271` ``` by (simp only: conj_commute) ``` kleing@24332 ` 272` ```qed ``` kleing@24332 ` 273` kleing@24332 ` 274` ```lemmas conj_xor_distribs = ``` kleing@24332 ` 275` ``` conj_xor_distrib conj_xor_distrib2 ``` kleing@24332 ` 276` kleing@24332 ` 277` ```end ``` kleing@24332 ` 278` kleing@24332 ` 279` ```end ```