src/HOL/Library/Boolean_Algebra.thy
 author wenzelm Mon Dec 28 01:28:28 2015 +0100 (2015-12-28) changeset 61945 1135b8de26c3 parent 61605 1bf7b186542e child 63462 c1fe30f2bc32 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` 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" ``` 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@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 - ``` kleing@24332 ` 66` ``` have "(a \ x) \ (x \ y) = (a \ y) \ (x \ y)" using 1 3 by simp ``` kleing@24332 ` 67` ``` hence "(x \ a) \ (x \ y) = (y \ a) \ (y \ x)" using conj_commute by simp ``` kleing@24332 ` 68` ``` hence "x \ (a \ y) = y \ (a \ x)" using conj_disj_distrib by simp ``` kleing@24332 ` 69` ``` hence "x \ \ = y \ \" using 2 4 by simp ``` kleing@24332 ` 70` ``` thus "x = y" using conj_one_right by simp ``` kleing@24332 ` 71` ```qed ``` kleing@24332 ` 72` huffman@24357 ` 73` ```lemma compl_unique: "\x \ y = \; x \ y = \\ \ \ x = y" ``` kleing@24332 ` 74` ```by (rule complement_unique [OF conj_cancel_right disj_cancel_right]) ``` kleing@24332 ` 75` kleing@24332 ` 76` ```lemma double_compl [simp]: "\ (\ x) = x" ``` kleing@24332 ` 77` ```proof (rule compl_unique) ``` huffman@24357 ` 78` ``` from conj_cancel_right show "\ x \ x = \" by (simp only: conj_commute) ``` huffman@24357 ` 79` ``` from disj_cancel_right show "\ x \ x = \" by (simp only: disj_commute) ``` kleing@24332 ` 80` ```qed ``` kleing@24332 ` 81` kleing@24332 ` 82` ```lemma compl_eq_compl_iff [simp]: "(\ x = \ y) = (x = y)" ``` kleing@24332 ` 83` ```by (rule inj_eq [OF inj_on_inverseI], rule double_compl) ``` kleing@24332 ` 84` wenzelm@60855 ` 85` wenzelm@60500 ` 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: ``` wenzelm@60855 ` 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` wenzelm@60855 ` 129` wenzelm@60500 ` 130` ```subsection \Disjunction\ ``` kleing@24332 ` 131` kleing@24332 ` 132` ```lemma disj_absorb [simp]: "x \ x = x" ``` kleing@24332 ` 133` ```by (rule boolean.conj_absorb [OF dual]) ``` kleing@24332 ` 134` kleing@24332 ` 135` ```lemma disj_one_right [simp]: "x \ \ = \" ``` kleing@24332 ` 136` ```by (rule boolean.conj_zero_right [OF dual]) ``` kleing@24332 ` 137` kleing@24332 ` 138` ```lemma compl_zero [simp]: "\ \ = \" ``` kleing@24332 ` 139` ```by (rule boolean.compl_one [OF dual]) ``` kleing@24332 ` 140` kleing@24332 ` 141` ```lemma disj_zero_left [simp]: "\ \ x = x" ``` kleing@24332 ` 142` ```by (rule boolean.conj_one_left [OF dual]) ``` kleing@24332 ` 143` kleing@24332 ` 144` ```lemma disj_one_left [simp]: "\ \ x = \" ``` kleing@24332 ` 145` ```by (rule boolean.conj_zero_left [OF dual]) ``` kleing@24332 ` 146` kleing@24332 ` 147` ```lemma disj_cancel_left [simp]: "\ x \ x = \" ``` kleing@24332 ` 148` ```by (rule boolean.conj_cancel_left [OF dual]) ``` kleing@24332 ` 149` kleing@24332 ` 150` ```lemma disj_left_absorb [simp]: "x \ (x \ y) = x \ y" ``` kleing@24332 ` 151` ```by (rule boolean.conj_left_absorb [OF dual]) ``` kleing@24332 ` 152` kleing@24332 ` 153` ```lemma disj_conj_distrib2: ``` kleing@24332 ` 154` ``` "(y \ z) \ x = (y \ x) \ (z \ x)" ``` kleing@24332 ` 155` ```by (rule boolean.conj_disj_distrib2 [OF dual]) ``` kleing@24332 ` 156` kleing@24332 ` 157` ```lemmas disj_conj_distribs = ``` kleing@24332 ` 158` ``` disj_conj_distrib disj_conj_distrib2 ``` kleing@24332 ` 159` wenzelm@60855 ` 160` wenzelm@60500 ` 161` ```subsection \De Morgan's Laws\ ``` kleing@24332 ` 162` kleing@24332 ` 163` ```lemma de_Morgan_conj [simp]: "\ (x \ y) = \ x \ \ y" ``` kleing@24332 ` 164` ```proof (rule compl_unique) ``` kleing@24332 ` 165` ``` have "(x \ y) \ (\ x \ \ y) = ((x \ y) \ \ x) \ ((x \ y) \ \ y)" ``` kleing@24332 ` 166` ``` by (rule conj_disj_distrib) ``` kleing@24332 ` 167` ``` also have "... = (y \ (x \ \ x)) \ (x \ (y \ \ y))" ``` huffman@24357 ` 168` ``` by (simp only: conj_ac) ``` kleing@24332 ` 169` ``` finally show "(x \ y) \ (\ x \ \ y) = \" ``` huffman@24357 ` 170` ``` by (simp only: conj_cancel_right conj_zero_right disj_zero_right) ``` kleing@24332 ` 171` ```next ``` kleing@24332 ` 172` ``` have "(x \ y) \ (\ x \ \ y) = (x \ (\ x \ \ y)) \ (y \ (\ x \ \ y))" ``` kleing@24332 ` 173` ``` by (rule disj_conj_distrib2) ``` kleing@24332 ` 174` ``` also have "... = (\ y \ (x \ \ x)) \ (\ x \ (y \ \ y))" ``` huffman@24357 ` 175` ``` by (simp only: disj_ac) ``` kleing@24332 ` 176` ``` finally show "(x \ y) \ (\ x \ \ y) = \" ``` huffman@24357 ` 177` ``` by (simp only: disj_cancel_right disj_one_right conj_one_right) ``` kleing@24332 ` 178` ```qed ``` kleing@24332 ` 179` kleing@24332 ` 180` ```lemma de_Morgan_disj [simp]: "\ (x \ y) = \ x \ \ y" ``` kleing@24332 ` 181` ```by (rule boolean.de_Morgan_conj [OF dual]) ``` kleing@24332 ` 182` kleing@24332 ` 183` ```end ``` kleing@24332 ` 184` wenzelm@60855 ` 185` wenzelm@60500 ` 186` ```subsection \Symmetric Difference\ ``` kleing@24332 ` 187` kleing@24332 ` 188` ```locale boolean_xor = boolean + ``` wenzelm@60855 ` 189` ``` fixes xor :: "'a \ 'a \ 'a" (infixr "\" 65) ``` kleing@24332 ` 190` ``` assumes xor_def: "x \ y = (x \ \ y) \ (\ x \ y)" ``` haftmann@54868 ` 191` ```begin ``` kleing@24332 ` 192` wenzelm@61605 ` 193` ```sublocale xor: abel_semigroup xor ``` wenzelm@60855 ` 194` ```proof ``` haftmann@34973 ` 195` ``` fix x y z :: 'a ``` kleing@24332 ` 196` ``` let ?t = "(x \ y \ z) \ (x \ \ y \ \ z) \ ``` kleing@24332 ` 197` ``` (\ x \ y \ \ z) \ (\ x \ \ y \ z)" ``` kleing@24332 ` 198` ``` have "?t \ (z \ x \ \ x) \ (z \ y \ \ y) = ``` kleing@24332 ` 199` ``` ?t \ (x \ y \ \ y) \ (x \ z \ \ z)" ``` huffman@24357 ` 200` ``` by (simp only: conj_cancel_right conj_zero_right) ``` kleing@24332 ` 201` ``` thus "(x \ y) \ z = x \ (y \ z)" ``` huffman@24357 ` 202` ``` apply (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl) ``` huffman@24357 ` 203` ``` apply (simp only: conj_disj_distribs conj_ac disj_ac) ``` kleing@24332 ` 204` ``` done ``` haftmann@34973 ` 205` ``` show "x \ y = y \ x" ``` haftmann@34973 ` 206` ``` by (simp only: xor_def conj_commute disj_commute) ``` kleing@24332 ` 207` ```qed ``` kleing@24332 ` 208` haftmann@34973 ` 209` ```lemmas xor_assoc = xor.assoc ``` haftmann@34973 ` 210` ```lemmas xor_commute = xor.commute ``` haftmann@34973 ` 211` ```lemmas xor_left_commute = xor.left_commute ``` haftmann@34973 ` 212` haftmann@34973 ` 213` ```lemmas xor_ac = xor.assoc xor.commute xor.left_commute ``` haftmann@34973 ` 214` haftmann@34973 ` 215` ```lemma xor_def2: ``` haftmann@34973 ` 216` ``` "x \ y = (x \ y) \ (\ x \ \ y)" ``` haftmann@34973 ` 217` ```by (simp only: xor_def conj_disj_distribs ``` haftmann@34973 ` 218` ``` disj_ac conj_ac conj_cancel_right disj_zero_left) ``` kleing@24332 ` 219` kleing@24332 ` 220` ```lemma xor_zero_right [simp]: "x \ \ = x" ``` huffman@24357 ` 221` ```by (simp only: xor_def compl_zero conj_one_right conj_zero_right disj_zero_right) ``` kleing@24332 ` 222` kleing@24332 ` 223` ```lemma xor_zero_left [simp]: "\ \ x = x" ``` kleing@24332 ` 224` ```by (subst xor_commute) (rule xor_zero_right) ``` kleing@24332 ` 225` kleing@24332 ` 226` ```lemma xor_one_right [simp]: "x \ \ = \ x" ``` huffman@24357 ` 227` ```by (simp only: xor_def compl_one conj_zero_right conj_one_right disj_zero_left) ``` kleing@24332 ` 228` kleing@24332 ` 229` ```lemma xor_one_left [simp]: "\ \ x = \ x" ``` kleing@24332 ` 230` ```by (subst xor_commute) (rule xor_one_right) ``` kleing@24332 ` 231` kleing@24332 ` 232` ```lemma xor_self [simp]: "x \ x = \" ``` huffman@24357 ` 233` ```by (simp only: xor_def conj_cancel_right conj_cancel_left disj_zero_right) ``` kleing@24332 ` 234` kleing@24332 ` 235` ```lemma xor_left_self [simp]: "x \ (x \ y) = y" ``` huffman@24357 ` 236` ```by (simp only: xor_assoc [symmetric] xor_self xor_zero_left) ``` kleing@24332 ` 237` huffman@29996 ` 238` ```lemma xor_compl_left [simp]: "\ x \ y = \ (x \ y)" ``` huffman@24357 ` 239` ```apply (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl) ``` huffman@24357 ` 240` ```apply (simp only: conj_disj_distribs) ``` huffman@24357 ` 241` ```apply (simp only: conj_cancel_right conj_cancel_left) ``` huffman@24357 ` 242` ```apply (simp only: disj_zero_left disj_zero_right) ``` huffman@24357 ` 243` ```apply (simp only: disj_ac conj_ac) ``` kleing@24332 ` 244` ```done ``` kleing@24332 ` 245` huffman@29996 ` 246` ```lemma xor_compl_right [simp]: "x \ \ y = \ (x \ y)" ``` huffman@24357 ` 247` ```apply (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl) ``` huffman@24357 ` 248` ```apply (simp only: conj_disj_distribs) ``` huffman@24357 ` 249` ```apply (simp only: conj_cancel_right conj_cancel_left) ``` huffman@24357 ` 250` ```apply (simp only: disj_zero_left disj_zero_right) ``` huffman@24357 ` 251` ```apply (simp only: disj_ac conj_ac) ``` kleing@24332 ` 252` ```done ``` kleing@24332 ` 253` huffman@29996 ` 254` ```lemma xor_cancel_right: "x \ \ x = \" ``` huffman@24357 ` 255` ```by (simp only: xor_compl_right xor_self compl_zero) ``` kleing@24332 ` 256` huffman@29996 ` 257` ```lemma xor_cancel_left: "\ x \ x = \" ``` huffman@29996 ` 258` ```by (simp only: xor_compl_left xor_self compl_zero) ``` kleing@24332 ` 259` kleing@24332 ` 260` ```lemma conj_xor_distrib: "x \ (y \ z) = (x \ y) \ (x \ z)" ``` kleing@24332 ` 261` ```proof - ``` kleing@24332 ` 262` ``` have "(x \ y \ \ z) \ (x \ \ y \ z) = ``` kleing@24332 ` 263` ``` (y \ x \ \ x) \ (z \ x \ \ x) \ (x \ y \ \ z) \ (x \ \ y \ z)" ``` huffman@24357 ` 264` ``` by (simp only: conj_cancel_right conj_zero_right disj_zero_left) ``` kleing@24332 ` 265` ``` thus "x \ (y \ z) = (x \ y) \ (x \ z)" ``` huffman@24357 ` 266` ``` by (simp (no_asm_use) only: ``` kleing@24332 ` 267` ``` xor_def de_Morgan_disj de_Morgan_conj double_compl ``` kleing@24332 ` 268` ``` conj_disj_distribs conj_ac disj_ac) ``` kleing@24332 ` 269` ```qed ``` kleing@24332 ` 270` wenzelm@60855 ` 271` ```lemma conj_xor_distrib2: "(y \ z) \ x = (y \ x) \ (z \ x)" ``` kleing@24332 ` 272` ```proof - ``` kleing@24332 ` 273` ``` have "x \ (y \ z) = (x \ y) \ (x \ z)" ``` kleing@24332 ` 274` ``` by (rule conj_xor_distrib) ``` kleing@24332 ` 275` ``` thus "(y \ z) \ x = (y \ x) \ (z \ x)" ``` huffman@24357 ` 276` ``` by (simp only: conj_commute) ``` kleing@24332 ` 277` ```qed ``` kleing@24332 ` 278` wenzelm@60855 ` 279` ```lemmas conj_xor_distribs = conj_xor_distrib conj_xor_distrib2 ``` kleing@24332 ` 280` kleing@24332 ` 281` ```end ``` kleing@24332 ` 282` kleing@24332 ` 283` ```end ```