author | nipkow |
Wed, 13 Feb 2019 07:48:42 +0100 | |
changeset 69801 | a99a0f5474c5 |
parent 65343 | 0a8e30a7b10e |
child 70186 | 18e94864fd0f |
permissions | -rw-r--r-- |
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(* 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 = |
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fixes 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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and 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_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)" |
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and disj_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)" |
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and conj_commute: "x \<sqinter> y = y \<sqinter> x" |
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and disj_commute: "x \<squnion> y = y \<squnion> x" |
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and conj_disj_distrib: "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" |
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and disj_conj_distrib: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)" |
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and conj_one_right [simp]: "x \<sqinter> \<one> = x" |
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and disj_zero_right [simp]: "x \<squnion> \<zero> = x" |
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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: abel_semigroup conj |
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by standard (fact conj_assoc conj_commute)+ |
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sublocale disj: abel_semigroup disj |
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by standard (fact disj_assoc disj_commute)+ |
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lemmas conj_left_commute = conj.left_commute |
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lemmas disj_left_commute = disj.left_commute |
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lemmas conj_ac = conj.assoc conj.commute conj.left_commute |
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lemmas disj_ac = disj.assoc disj.commute disj.left_commute |
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lemma dual: "boolean disj conj compl one zero" |
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apply (rule boolean.intro) |
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apply (rule disj_assoc) |
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apply (rule conj_assoc) |
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apply (rule disj_commute) |
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apply (rule conj_commute) |
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apply (rule disj_conj_distrib) |
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apply (rule conj_disj_distrib) |
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apply (rule disj_zero_right) |
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apply (rule conj_one_right) |
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apply (rule disj_cancel_right) |
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apply (rule conj_cancel_right) |
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done |
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subsection \<open>Complement\<close> |
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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 - |
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from 1 3 have "(a \<sqinter> x) \<squnion> (x \<sqinter> y) = (a \<sqinter> y) \<squnion> (x \<sqinter> y)" |
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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: conj_commute) |
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then have "x \<sqinter> (a \<squnion> y) = y \<sqinter> (a \<squnion> x)" |
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by (simp add: conj_disj_distrib) |
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with 2 4 have "x \<sqinter> \<one> = y \<sqinter> \<one>" |
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by simp |
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then show "x = y" |
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by simp |
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qed |
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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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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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90 |
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subsection \<open>Conjunction\<close> |
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lemma conj_absorb [simp]: "x \<sqinter> x = 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 |
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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 |
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finally show ?thesis . |
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qed |
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108 |
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lemma conj_zero_right [simp]: "x \<sqinter> \<zero> = \<zero>" |
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proof - |
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from conj_cancel_right have "x \<sqinter> \<zero> = x \<sqinter> (x \<sqinter> \<sim> x)" |
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by simp |
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also from conj_assoc have "\<dots> = (x \<sqinter> x) \<sqinter> \<sim> x" |
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by (simp only:) |
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also from conj_absorb have "\<dots> = x \<sqinter> \<sim> x" |
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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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qed |
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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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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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lemma conj_one_left [simp]: "\<one> \<sqinter> x = x" |
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by (subst conj_commute) (rule conj_one_right) |
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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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lemma conj_left_absorb [simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y" |
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by (simp only: conj_assoc [symmetric] conj_absorb) |
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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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141 |
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subsection \<open>Disjunction\<close> |
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lemma disj_absorb [simp]: "x \<squnion> x = x" |
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by (rule boolean.conj_absorb [OF dual]) |
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147 |
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lemma disj_one_right [simp]: "x \<squnion> \<one> = \<one>" |
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by (rule boolean.conj_zero_right [OF dual]) |
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lemma compl_zero [simp]: "\<sim> \<zero> = \<one>" |
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by (rule boolean.compl_one [OF dual]) |
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153 |
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lemma disj_zero_left [simp]: "\<zero> \<squnion> x = x" |
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by (rule boolean.conj_one_left [OF dual]) |
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156 |
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lemma disj_one_left [simp]: "\<one> \<squnion> x = \<one>" |
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by (rule boolean.conj_zero_left [OF dual]) |
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160 |
lemma disj_cancel_left [simp]: "\<sim> x \<squnion> x = \<one>" |
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by (rule boolean.conj_cancel_left [OF dual]) |
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162 |
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163 |
lemma disj_left_absorb [simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y" |
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by (rule boolean.conj_left_absorb [OF dual]) |
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165 |
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63462 | 166 |
lemma disj_conj_distrib2: "(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)" |
167 |
by (rule boolean.conj_disj_distrib2 [OF dual]) |
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168 |
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lemmas disj_conj_distribs = disj_conj_distrib disj_conj_distrib2 |
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170 |
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subsection \<open>De Morgan's Laws\<close> |
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173 |
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174 |
lemma de_Morgan_conj [simp]: "\<sim> (x \<sqinter> y) = \<sim> x \<squnion> \<sim> y" |
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175 |
proof (rule compl_unique) |
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176 |
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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177 |
by (rule conj_disj_distrib) |
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also have "\<dots> = (y \<sqinter> (x \<sqinter> \<sim> x)) \<squnion> (x \<sqinter> (y \<sqinter> \<sim> y))" |
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by (simp only: conj_ac) |
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180 |
finally show "(x \<sqinter> y) \<sqinter> (\<sim> x \<squnion> \<sim> y) = \<zero>" |
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by (simp only: conj_cancel_right conj_zero_right disj_zero_right) |
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182 |
next |
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183 |
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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184 |
by (rule disj_conj_distrib2) |
65343 | 185 |
also have "\<dots> = (\<sim> y \<squnion> (x \<squnion> \<sim> x)) \<sqinter> (\<sim> x \<squnion> (y \<squnion> \<sim> y))" |
24357 | 186 |
by (simp only: disj_ac) |
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kleing
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|
187 |
finally show "(x \<sqinter> y) \<squnion> (\<sim> x \<squnion> \<sim> y) = \<one>" |
24357 | 188 |
by (simp only: disj_cancel_right disj_one_right conj_one_right) |
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|
189 |
qed |
e3a2b75b1cf9
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kleing
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diff
changeset
|
190 |
|
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changeset
|
191 |
lemma de_Morgan_disj [simp]: "\<sim> (x \<squnion> y) = \<sim> x \<sqinter> \<sim> y" |
63462 | 192 |
by (rule boolean.de_Morgan_conj [OF dual]) |
24332
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kleing
parents:
diff
changeset
|
193 |
|
e3a2b75b1cf9
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changeset
|
194 |
end |
e3a2b75b1cf9
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|
195 |
|
60855 | 196 |
|
60500 | 197 |
subsection \<open>Symmetric Difference\<close> |
24332
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|
198 |
|
e3a2b75b1cf9
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199 |
locale boolean_xor = boolean + |
60855 | 200 |
fixes xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "\<oplus>" 65) |
24332
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|
201 |
assumes xor_def: "x \<oplus> y = (x \<sqinter> \<sim> y) \<squnion> (\<sim> x \<sqinter> y)" |
54868 | 202 |
begin |
24332
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changeset
|
203 |
|
61605 | 204 |
sublocale xor: abel_semigroup xor |
60855 | 205 |
proof |
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diff
changeset
|
206 |
fix x y z :: 'a |
65343 | 207 |
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)" |
208 |
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 | 209 |
by (simp only: conj_cancel_right conj_zero_right) |
63462 | 210 |
then show "(x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)" |
65343 | 211 |
by (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl) |
212 |
(simp only: conj_disj_distribs conj_ac disj_ac) |
|
34973
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haftmann
parents:
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diff
changeset
|
213 |
show "x \<oplus> y = y \<oplus> x" |
ae634fad947e
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haftmann
parents:
30663
diff
changeset
|
214 |
by (simp only: xor_def conj_commute disj_commute) |
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|
215 |
qed |
e3a2b75b1cf9
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kleing
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diff
changeset
|
216 |
|
34973
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haftmann
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changeset
|
217 |
lemmas xor_assoc = xor.assoc |
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haftmann
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diff
changeset
|
218 |
lemmas xor_commute = xor.commute |
ae634fad947e
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haftmann
parents:
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changeset
|
219 |
lemmas xor_left_commute = xor.left_commute |
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haftmann
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diff
changeset
|
220 |
|
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haftmann
parents:
30663
diff
changeset
|
221 |
lemmas xor_ac = xor.assoc xor.commute xor.left_commute |
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haftmann
parents:
30663
diff
changeset
|
222 |
|
63462 | 223 |
lemma xor_def2: "x \<oplus> y = (x \<squnion> y) \<sqinter> (\<sim> x \<squnion> \<sim> y)" |
224 |
by (simp only: xor_def conj_disj_distribs disj_ac conj_ac conj_cancel_right disj_zero_left) |
|
24332
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changeset
|
225 |
|
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|
226 |
lemma xor_zero_right [simp]: "x \<oplus> \<zero> = x" |
63462 | 227 |
by (simp only: xor_def compl_zero conj_one_right conj_zero_right disj_zero_right) |
24332
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changeset
|
228 |
|
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|
229 |
lemma xor_zero_left [simp]: "\<zero> \<oplus> x = x" |
63462 | 230 |
by (subst xor_commute) (rule xor_zero_right) |
24332
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changeset
|
231 |
|
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changeset
|
232 |
lemma xor_one_right [simp]: "x \<oplus> \<one> = \<sim> x" |
63462 | 233 |
by (simp only: xor_def compl_one conj_zero_right conj_one_right disj_zero_left) |
24332
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parents:
diff
changeset
|
234 |
|
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changeset
|
235 |
lemma xor_one_left [simp]: "\<one> \<oplus> x = \<sim> x" |
63462 | 236 |
by (subst xor_commute) (rule xor_one_right) |
24332
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parents:
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changeset
|
237 |
|
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changeset
|
238 |
lemma xor_self [simp]: "x \<oplus> x = \<zero>" |
63462 | 239 |
by (simp only: xor_def conj_cancel_right conj_cancel_left disj_zero_right) |
24332
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changeset
|
240 |
|
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diff
changeset
|
241 |
lemma xor_left_self [simp]: "x \<oplus> (x \<oplus> y) = y" |
63462 | 242 |
by (simp only: xor_assoc [symmetric] xor_self xor_zero_left) |
24332
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changeset
|
243 |
|
29996 | 244 |
lemma xor_compl_left [simp]: "\<sim> x \<oplus> y = \<sim> (x \<oplus> y)" |
63462 | 245 |
apply (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl) |
246 |
apply (simp only: conj_disj_distribs) |
|
247 |
apply (simp only: conj_cancel_right conj_cancel_left) |
|
248 |
apply (simp only: disj_zero_left disj_zero_right) |
|
249 |
apply (simp only: disj_ac conj_ac) |
|
250 |
done |
|
24332
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changeset
|
251 |
|
29996 | 252 |
lemma xor_compl_right [simp]: "x \<oplus> \<sim> y = \<sim> (x \<oplus> y)" |
63462 | 253 |
apply (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl) |
254 |
apply (simp only: conj_disj_distribs) |
|
255 |
apply (simp only: conj_cancel_right conj_cancel_left) |
|
256 |
apply (simp only: disj_zero_left disj_zero_right) |
|
257 |
apply (simp only: disj_ac conj_ac) |
|
258 |
done |
|
24332
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diff
changeset
|
259 |
|
29996 | 260 |
lemma xor_cancel_right: "x \<oplus> \<sim> x = \<one>" |
63462 | 261 |
by (simp only: xor_compl_right xor_self compl_zero) |
24332
e3a2b75b1cf9
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kleing
parents:
diff
changeset
|
262 |
|
29996 | 263 |
lemma xor_cancel_left: "\<sim> x \<oplus> x = \<one>" |
63462 | 264 |
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
|
265 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
266 |
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
|
267 |
proof - |
63462 | 268 |
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
|
269 |
(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 | 270 |
by (simp only: conj_cancel_right conj_zero_right disj_zero_left) |
63462 | 271 |
then show "x \<sqinter> (y \<oplus> z) = (x \<sqinter> y) \<oplus> (x \<sqinter> z)" |
24357 | 272 |
by (simp (no_asm_use) only: |
24332
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kleing
parents:
diff
changeset
|
273 |
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
|
274 |
conj_disj_distribs conj_ac disj_ac) |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
275 |
qed |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
276 |
|
60855 | 277 |
lemma conj_xor_distrib2: "(y \<oplus> z) \<sqinter> x = (y \<sqinter> x) \<oplus> (z \<sqinter> x)" |
24332
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kleing
parents:
diff
changeset
|
278 |
proof - |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
279 |
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
|
280 |
by (rule conj_xor_distrib) |
63462 | 281 |
then show "(y \<oplus> z) \<sqinter> x = (y \<sqinter> x) \<oplus> (z \<sqinter> x)" |
24357 | 282 |
by (simp only: conj_commute) |
24332
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boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
283 |
qed |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
284 |
|
60855 | 285 |
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
|
286 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
287 |
end |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
288 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
289 |
end |