author | haftmann |
Sat, 20 Apr 2019 18:02:22 +0000 | |
changeset 70188 | e626bffe28bc |
parent 70187 | 2082287357e6 |
child 70189 | 6d2effbbf8d4 |
permissions | -rw-r--r-- |
29629 | 1 |
(* Title: HOL/Library/Boolean_Algebra.thy |
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Author: Brian Huffman |
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*) |
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section \<open>Boolean Algebras\<close> |
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theory Boolean_Algebra |
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imports Main |
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begin |
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locale boolean_algebra = conj: abel_semigroup "(\<sqinter>)" + disj: abel_semigroup "(\<squnion>)" |
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for conj :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "\<sqinter>" 70) |
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and disj :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "\<squnion>" 65) + |
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fixes compl :: "'a \<Rightarrow> 'a" ("\<sim> _" [81] 80) |
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and zero :: "'a" ("\<zero>") |
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and one :: "'a" ("\<one>") |
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assumes conj_disj_distrib: "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" |
65343 | 18 |
and disj_conj_distrib: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)" |
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and conj_one_right: "x \<sqinter> \<one> = x" |
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and disj_zero_right: "x \<squnion> \<zero> = x" |
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65343 | 21 |
and conj_cancel_right [simp]: "x \<sqinter> \<sim> x = \<zero>" |
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and disj_cancel_right [simp]: "x \<squnion> \<sim> x = \<one>" |
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begin |
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sublocale conj: semilattice_neutr "(\<sqinter>)" "\<one>" |
26 |
proof |
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show "x \<sqinter> \<one> = x" for x |
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by (fact conj_one_right) |
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show "x \<sqinter> x = x" for x |
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proof - |
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have "x \<sqinter> x = (x \<sqinter> x) \<squnion> \<zero>" |
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by (simp add: disj_zero_right) |
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also have "\<dots> = (x \<sqinter> x) \<squnion> (x \<sqinter> \<sim> x)" |
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by simp |
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also have "\<dots> = x \<sqinter> (x \<squnion> \<sim> x)" |
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by (simp only: conj_disj_distrib) |
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also have "\<dots> = x \<sqinter> \<one>" |
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by simp |
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also have "\<dots> = x" |
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by (simp add: conj_one_right) |
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finally show ?thesis . |
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qed |
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qed |
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70188 | 45 |
sublocale disj: semilattice_neutr "(\<squnion>)" "\<zero>" |
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proof |
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show "x \<squnion> \<zero> = x" for x |
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by (fact disj_zero_right) |
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show "x \<squnion> x = x" for x |
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proof - |
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have "x \<squnion> x = (x \<squnion> x) \<sqinter> \<one>" |
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by simp |
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also have "\<dots> = (x \<squnion> x) \<sqinter> (x \<squnion> \<sim> x)" |
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by simp |
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also have "\<dots> = x \<squnion> (x \<sqinter> \<sim> x)" |
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by (simp only: disj_conj_distrib) |
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also have "\<dots> = x \<squnion> \<zero>" |
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by simp |
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also have "\<dots> = x" |
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by (simp add: disj_zero_right) |
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finally show ?thesis . |
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qed |
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qed |
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60855 | 65 |
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subsection \<open>Complement\<close> |
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67 |
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lemma complement_unique: |
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assumes 1: "a \<sqinter> x = \<zero>" |
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assumes 2: "a \<squnion> x = \<one>" |
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assumes 3: "a \<sqinter> y = \<zero>" |
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assumes 4: "a \<squnion> y = \<one>" |
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shows "x = y" |
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proof - |
65343 | 75 |
from 1 3 have "(a \<sqinter> x) \<squnion> (x \<sqinter> y) = (a \<sqinter> y) \<squnion> (x \<sqinter> y)" |
76 |
by simp |
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then have "(x \<sqinter> a) \<squnion> (x \<sqinter> y) = (y \<sqinter> a) \<squnion> (y \<sqinter> x)" |
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by (simp add: ac_simps) |
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then have "x \<sqinter> (a \<squnion> y) = y \<sqinter> (a \<squnion> x)" |
65343 | 80 |
by (simp add: conj_disj_distrib) |
81 |
with 2 4 have "x \<sqinter> \<one> = y \<sqinter> \<one>" |
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by simp |
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63462 | 83 |
then show "x = y" |
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by simp |
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qed |
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63462 | 87 |
lemma compl_unique: "x \<sqinter> y = \<zero> \<Longrightarrow> x \<squnion> y = \<one> \<Longrightarrow> \<sim> x = y" |
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by (rule complement_unique [OF conj_cancel_right disj_cancel_right]) |
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89 |
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lemma double_compl [simp]: "\<sim> (\<sim> x) = x" |
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proof (rule compl_unique) |
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show "\<sim> x \<sqinter> x = \<zero>" |
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by (simp only: conj_cancel_right conj.commute) |
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show "\<sim> x \<squnion> x = \<one>" |
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by (simp only: disj_cancel_right disj.commute) |
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qed |
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lemma compl_eq_compl_iff [simp]: "\<sim> x = \<sim> y \<longleftrightarrow> x = y" |
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by (rule inj_eq [OF inj_on_inverseI]) (rule double_compl) |
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100 |
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subsection \<open>Conjunction\<close> |
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103 |
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lemma conj_zero_right [simp]: "x \<sqinter> \<zero> = \<zero>" |
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105 |
proof - |
65343 | 106 |
from conj_cancel_right have "x \<sqinter> \<zero> = x \<sqinter> (x \<sqinter> \<sim> x)" |
107 |
by simp |
|
108 |
also from conj_assoc have "\<dots> = (x \<sqinter> x) \<sqinter> \<sim> x" |
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70188 | 109 |
by (simp only: ac_simps) |
65343 | 110 |
also from conj_absorb have "\<dots> = x \<sqinter> \<sim> x" |
111 |
by simp |
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also have "\<dots> = \<zero>" |
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by simp |
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finally show ?thesis . |
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115 |
qed |
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116 |
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lemma compl_one [simp]: "\<sim> \<one> = \<zero>" |
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by (rule compl_unique [OF conj_zero_right disj_zero_right]) |
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119 |
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120 |
lemma conj_zero_left [simp]: "\<zero> \<sqinter> x = \<zero>" |
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by (subst conj.commute) (rule conj_zero_right) |
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122 |
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lemma conj_cancel_left [simp]: "\<sim> x \<sqinter> x = \<zero>" |
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by (subst conj.commute) (rule conj_cancel_right) |
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125 |
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63462 | 126 |
lemma conj_disj_distrib2: "(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)" |
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by (simp only: conj.commute conj_disj_distrib) |
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lemmas conj_disj_distribs = conj_disj_distrib conj_disj_distrib2 |
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70188 | 131 |
lemma conj_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)" |
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by (fact ac_simps) |
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lemma conj_commute: "x \<sqinter> y = y \<sqinter> x" |
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by (fact ac_simps) |
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lemmas conj_left_commute = conj.left_commute |
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lemmas conj_ac = conj.assoc conj.commute conj.left_commute |
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lemma conj_one_left: "\<one> \<sqinter> x = x" |
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by (fact conj.left_neutral) |
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lemma conj_left_absorb: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y" |
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by (fact conj.left_idem) |
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lemma conj_absorb: "x \<sqinter> x = x" |
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by (fact conj.idem) |
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subsection \<open>Disjunction\<close> |
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70188 | 152 |
interpretation dual: boolean_algebra "(\<squnion>)" "(\<sqinter>)" compl \<one> \<zero> |
153 |
apply standard |
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apply (rule disj_conj_distrib) |
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apply (rule conj_disj_distrib) |
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apply simp_all |
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done |
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159 |
lemma compl_zero [simp]: "\<sim> \<zero> = \<one>" |
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by (fact dual.compl_one) |
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lemma disj_one_right [simp]: "x \<squnion> \<one> = \<one>" |
70188 | 163 |
by (fact dual.conj_zero_right) |
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164 |
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lemma disj_one_left [simp]: "\<one> \<squnion> x = \<one>" |
70188 | 166 |
by (fact dual.conj_zero_left) |
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167 |
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lemma disj_cancel_left [simp]: "\<sim> x \<squnion> x = \<one>" |
70188 | 169 |
by (rule dual.conj_cancel_left) |
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170 |
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63462 | 171 |
lemma disj_conj_distrib2: "(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)" |
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by (rule dual.conj_disj_distrib2) |
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173 |
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63462 | 174 |
lemmas disj_conj_distribs = disj_conj_distrib disj_conj_distrib2 |
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175 |
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70188 | 176 |
lemma disj_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)" |
177 |
by (fact ac_simps) |
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178 |
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179 |
lemma disj_commute: "x \<squnion> y = y \<squnion> x" |
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by (fact ac_simps) |
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182 |
lemmas disj_left_commute = disj.left_commute |
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184 |
lemmas disj_ac = disj.assoc disj.commute disj.left_commute |
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186 |
lemma disj_zero_left: "\<zero> \<squnion> x = x" |
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by (fact disj.left_neutral) |
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189 |
lemma disj_left_absorb: "x \<squnion> (x \<squnion> y) = x \<squnion> y" |
|
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by (fact disj.left_idem) |
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191 |
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192 |
lemma disj_absorb: "x \<squnion> x = x" |
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193 |
by (fact disj.idem) |
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194 |
||
60855 | 195 |
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60500 | 196 |
subsection \<open>De Morgan's Laws\<close> |
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197 |
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198 |
lemma de_Morgan_conj [simp]: "\<sim> (x \<sqinter> y) = \<sim> x \<squnion> \<sim> y" |
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199 |
proof (rule compl_unique) |
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200 |
have "(x \<sqinter> y) \<sqinter> (\<sim> x \<squnion> \<sim> y) = ((x \<sqinter> y) \<sqinter> \<sim> x) \<squnion> ((x \<sqinter> y) \<sqinter> \<sim> y)" |
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201 |
by (rule conj_disj_distrib) |
65343 | 202 |
also have "\<dots> = (y \<sqinter> (x \<sqinter> \<sim> x)) \<squnion> (x \<sqinter> (y \<sqinter> \<sim> y))" |
24357 | 203 |
by (simp only: conj_ac) |
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204 |
finally show "(x \<sqinter> y) \<sqinter> (\<sim> x \<squnion> \<sim> y) = \<zero>" |
24357 | 205 |
by (simp only: conj_cancel_right conj_zero_right disj_zero_right) |
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206 |
next |
e3a2b75b1cf9
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207 |
have "(x \<sqinter> y) \<squnion> (\<sim> x \<squnion> \<sim> y) = (x \<squnion> (\<sim> x \<squnion> \<sim> y)) \<sqinter> (y \<squnion> (\<sim> x \<squnion> \<sim> y))" |
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|
208 |
by (rule disj_conj_distrib2) |
65343 | 209 |
also have "\<dots> = (\<sim> y \<squnion> (x \<squnion> \<sim> x)) \<sqinter> (\<sim> x \<squnion> (y \<squnion> \<sim> y))" |
24357 | 210 |
by (simp only: disj_ac) |
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211 |
finally show "(x \<sqinter> y) \<squnion> (\<sim> x \<squnion> \<sim> y) = \<one>" |
24357 | 212 |
by (simp only: disj_cancel_right disj_one_right conj_one_right) |
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213 |
qed |
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214 |
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215 |
lemma de_Morgan_disj [simp]: "\<sim> (x \<squnion> y) = \<sim> x \<sqinter> \<sim> y" |
70188 | 216 |
using dual.boolean_algebra_axioms by (rule boolean_algebra.de_Morgan_conj) |
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217 |
|
60855 | 218 |
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60500 | 219 |
subsection \<open>Symmetric Difference\<close> |
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220 |
|
70187
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221 |
definition xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "\<oplus>" 65) |
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222 |
where "x \<oplus> y = (x \<sqinter> \<sim> y) \<squnion> (\<sim> x \<sqinter> y)" |
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223 |
|
61605 | 224 |
sublocale xor: abel_semigroup xor |
60855 | 225 |
proof |
34973
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
30663
diff
changeset
|
226 |
fix x y z :: 'a |
65343 | 227 |
let ?t = "(x \<sqinter> y \<sqinter> z) \<squnion> (x \<sqinter> \<sim> y \<sqinter> \<sim> z) \<squnion> (\<sim> x \<sqinter> y \<sqinter> \<sim> z) \<squnion> (\<sim> x \<sqinter> \<sim> y \<sqinter> z)" |
228 |
have "?t \<squnion> (z \<sqinter> x \<sqinter> \<sim> x) \<squnion> (z \<sqinter> y \<sqinter> \<sim> y) = ?t \<squnion> (x \<sqinter> y \<sqinter> \<sim> y) \<squnion> (x \<sqinter> z \<sqinter> \<sim> z)" |
|
24357 | 229 |
by (simp only: conj_cancel_right conj_zero_right) |
63462 | 230 |
then show "(x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)" |
65343 | 231 |
by (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl) |
232 |
(simp only: conj_disj_distribs conj_ac disj_ac) |
|
34973
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
30663
diff
changeset
|
233 |
show "x \<oplus> y = y \<oplus> x" |
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
30663
diff
changeset
|
234 |
by (simp only: xor_def conj_commute disj_commute) |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
235 |
qed |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
236 |
|
34973
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
30663
diff
changeset
|
237 |
lemmas xor_assoc = xor.assoc |
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
30663
diff
changeset
|
238 |
lemmas xor_commute = xor.commute |
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
30663
diff
changeset
|
239 |
lemmas xor_left_commute = xor.left_commute |
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
30663
diff
changeset
|
240 |
|
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
30663
diff
changeset
|
241 |
lemmas xor_ac = xor.assoc xor.commute xor.left_commute |
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
30663
diff
changeset
|
242 |
|
63462 | 243 |
lemma xor_def2: "x \<oplus> y = (x \<squnion> y) \<sqinter> (\<sim> x \<squnion> \<sim> y)" |
244 |
by (simp only: xor_def conj_disj_distribs disj_ac conj_ac conj_cancel_right disj_zero_left) |
|
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
245 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
246 |
lemma xor_zero_right [simp]: "x \<oplus> \<zero> = x" |
63462 | 247 |
by (simp only: xor_def compl_zero conj_one_right conj_zero_right disj_zero_right) |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
248 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
249 |
lemma xor_zero_left [simp]: "\<zero> \<oplus> x = x" |
63462 | 250 |
by (subst xor_commute) (rule xor_zero_right) |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
251 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
252 |
lemma xor_one_right [simp]: "x \<oplus> \<one> = \<sim> x" |
63462 | 253 |
by (simp only: xor_def compl_one conj_zero_right conj_one_right disj_zero_left) |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
254 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
255 |
lemma xor_one_left [simp]: "\<one> \<oplus> x = \<sim> x" |
63462 | 256 |
by (subst xor_commute) (rule xor_one_right) |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
257 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
258 |
lemma xor_self [simp]: "x \<oplus> x = \<zero>" |
63462 | 259 |
by (simp only: xor_def conj_cancel_right conj_cancel_left disj_zero_right) |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
260 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
261 |
lemma xor_left_self [simp]: "x \<oplus> (x \<oplus> y) = y" |
63462 | 262 |
by (simp only: xor_assoc [symmetric] xor_self xor_zero_left) |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
263 |
|
29996 | 264 |
lemma xor_compl_left [simp]: "\<sim> x \<oplus> y = \<sim> (x \<oplus> y)" |
63462 | 265 |
apply (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl) |
266 |
apply (simp only: conj_disj_distribs) |
|
267 |
apply (simp only: conj_cancel_right conj_cancel_left) |
|
268 |
apply (simp only: disj_zero_left disj_zero_right) |
|
269 |
apply (simp only: disj_ac conj_ac) |
|
270 |
done |
|
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
271 |
|
29996 | 272 |
lemma xor_compl_right [simp]: "x \<oplus> \<sim> y = \<sim> (x \<oplus> y)" |
63462 | 273 |
apply (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl) |
274 |
apply (simp only: conj_disj_distribs) |
|
275 |
apply (simp only: conj_cancel_right conj_cancel_left) |
|
276 |
apply (simp only: disj_zero_left disj_zero_right) |
|
277 |
apply (simp only: disj_ac conj_ac) |
|
278 |
done |
|
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
279 |
|
29996 | 280 |
lemma xor_cancel_right: "x \<oplus> \<sim> x = \<one>" |
63462 | 281 |
by (simp only: xor_compl_right xor_self compl_zero) |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
282 |
|
29996 | 283 |
lemma xor_cancel_left: "\<sim> x \<oplus> x = \<one>" |
63462 | 284 |
by (simp only: xor_compl_left xor_self compl_zero) |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
285 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
286 |
lemma conj_xor_distrib: "x \<sqinter> (y \<oplus> z) = (x \<sqinter> y) \<oplus> (x \<sqinter> z)" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
287 |
proof - |
63462 | 288 |
have *: "(x \<sqinter> y \<sqinter> \<sim> z) \<squnion> (x \<sqinter> \<sim> y \<sqinter> z) = |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
289 |
(y \<sqinter> x \<sqinter> \<sim> x) \<squnion> (z \<sqinter> x \<sqinter> \<sim> x) \<squnion> (x \<sqinter> y \<sqinter> \<sim> z) \<squnion> (x \<sqinter> \<sim> y \<sqinter> z)" |
24357 | 290 |
by (simp only: conj_cancel_right conj_zero_right disj_zero_left) |
63462 | 291 |
then show "x \<sqinter> (y \<oplus> z) = (x \<sqinter> y) \<oplus> (x \<sqinter> z)" |
24357 | 292 |
by (simp (no_asm_use) only: |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
293 |
xor_def de_Morgan_disj de_Morgan_conj double_compl |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
294 |
conj_disj_distribs conj_ac disj_ac) |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
295 |
qed |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
296 |
|
60855 | 297 |
lemma conj_xor_distrib2: "(y \<oplus> z) \<sqinter> x = (y \<sqinter> x) \<oplus> (z \<sqinter> x)" |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
298 |
proof - |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
299 |
have "x \<sqinter> (y \<oplus> z) = (x \<sqinter> y) \<oplus> (x \<sqinter> z)" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
300 |
by (rule conj_xor_distrib) |
63462 | 301 |
then show "(y \<oplus> z) \<sqinter> x = (y \<sqinter> x) \<oplus> (z \<sqinter> x)" |
24357 | 302 |
by (simp only: conj_commute) |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
303 |
qed |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
304 |
|
60855 | 305 |
lemmas conj_xor_distribs = conj_xor_distrib conj_xor_distrib2 |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
306 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
307 |
end |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
308 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
309 |
end |