author | huffman |
Tue, 21 Aug 2007 20:53:37 +0200 | |
changeset 24393 | b9718519f3d2 |
parent 24357 | d42cf77da51f |
child 25283 | c532fd8445a2 |
permissions | -rw-r--r-- |
24332
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boolean algebras as locales and numbers as types by Brian Huffman
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1 |
(* |
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2 |
ID: $Id$ |
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3 |
Author: Brian Huffman |
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4 |
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5 |
Boolean algebras as locales. |
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boolean algebras as locales and numbers as types by Brian Huffman
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6 |
*) |
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7 |
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8 |
header {* Boolean Algebras *} |
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9 |
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boolean algebras as locales and numbers as types by Brian Huffman
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10 |
theory Boolean_Algebra |
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11 |
imports Main |
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12 |
begin |
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13 |
|
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14 |
locale boolean = |
24357 | 15 |
fixes conj :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "\<sqinter>" 70) |
16 |
fixes disj :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "\<squnion>" 65) |
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17 |
fixes compl :: "'a \<Rightarrow> 'a" ("\<sim> _" [81] 80) |
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24332
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18 |
fixes zero :: "'a" ("\<zero>") |
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19 |
fixes one :: "'a" ("\<one>") |
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20 |
assumes conj_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)" |
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21 |
assumes disj_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)" |
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22 |
assumes conj_commute: "x \<sqinter> y = y \<sqinter> x" |
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23 |
assumes disj_commute: "x \<squnion> y = y \<squnion> x" |
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24 |
assumes conj_disj_distrib: "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" |
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25 |
assumes disj_conj_distrib: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)" |
24357 | 26 |
assumes conj_one_right [simp]: "x \<sqinter> \<one> = x" |
27 |
assumes disj_zero_right [simp]: "x \<squnion> \<zero> = x" |
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28 |
assumes conj_cancel_right [simp]: "x \<sqinter> \<sim> x = \<zero>" |
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29 |
assumes disj_cancel_right [simp]: "x \<squnion> \<sim> x = \<one>" |
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30 |
begin |
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boolean algebras as locales and numbers as types by Brian Huffman
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31 |
|
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32 |
lemmas disj_ac = |
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33 |
disj_assoc disj_commute |
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34 |
mk_left_commute [of "disj", OF disj_assoc disj_commute] |
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35 |
|
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36 |
lemmas conj_ac = |
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37 |
conj_assoc conj_commute |
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38 |
mk_left_commute [of "conj", OF conj_assoc conj_commute] |
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39 |
|
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40 |
lemma dual: "boolean disj conj compl one zero" |
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41 |
apply (rule boolean.intro) |
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42 |
apply (rule disj_assoc) |
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43 |
apply (rule conj_assoc) |
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44 |
apply (rule disj_commute) |
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45 |
apply (rule conj_commute) |
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46 |
apply (rule disj_conj_distrib) |
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47 |
apply (rule conj_disj_distrib) |
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48 |
apply (rule disj_zero_right) |
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49 |
apply (rule conj_one_right) |
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boolean algebras as locales and numbers as types by Brian Huffman
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50 |
apply (rule disj_cancel_right) |
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boolean algebras as locales and numbers as types by Brian Huffman
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parents:
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51 |
apply (rule conj_cancel_right) |
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boolean algebras as locales and numbers as types by Brian Huffman
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52 |
done |
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53 |
|
24357 | 54 |
subsection {* Complement *} |
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55 |
|
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boolean algebras as locales and numbers as types by Brian Huffman
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56 |
lemma complement_unique: |
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57 |
assumes 1: "a \<sqinter> x = \<zero>" |
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58 |
assumes 2: "a \<squnion> x = \<one>" |
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boolean algebras as locales and numbers as types by Brian Huffman
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59 |
assumes 3: "a \<sqinter> y = \<zero>" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
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parents:
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60 |
assumes 4: "a \<squnion> y = \<one>" |
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boolean algebras as locales and numbers as types by Brian Huffman
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61 |
shows "x = y" |
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boolean algebras as locales and numbers as types by Brian Huffman
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parents:
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62 |
proof - |
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boolean algebras as locales and numbers as types by Brian Huffman
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parents:
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63 |
have "(a \<sqinter> x) \<squnion> (x \<sqinter> y) = (a \<sqinter> y) \<squnion> (x \<sqinter> y)" using 1 3 by simp |
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boolean algebras as locales and numbers as types by Brian Huffman
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64 |
hence "(x \<sqinter> a) \<squnion> (x \<sqinter> y) = (y \<sqinter> a) \<squnion> (y \<sqinter> x)" using conj_commute by simp |
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boolean algebras as locales and numbers as types by Brian Huffman
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parents:
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65 |
hence "x \<sqinter> (a \<squnion> y) = y \<sqinter> (a \<squnion> x)" using conj_disj_distrib by simp |
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boolean algebras as locales and numbers as types by Brian Huffman
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parents:
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66 |
hence "x \<sqinter> \<one> = y \<sqinter> \<one>" using 2 4 by simp |
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boolean algebras as locales and numbers as types by Brian Huffman
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parents:
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67 |
thus "x = y" using conj_one_right by simp |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
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68 |
qed |
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boolean algebras as locales and numbers as types by Brian Huffman
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parents:
diff
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69 |
|
24357 | 70 |
lemma compl_unique: "\<lbrakk>x \<sqinter> y = \<zero>; x \<squnion> y = \<one>\<rbrakk> \<Longrightarrow> \<sim> x = y" |
24332
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71 |
by (rule complement_unique [OF conj_cancel_right disj_cancel_right]) |
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boolean algebras as locales and numbers as types by Brian Huffman
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parents:
diff
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72 |
|
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boolean algebras as locales and numbers as types by Brian Huffman
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parents:
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73 |
lemma double_compl [simp]: "\<sim> (\<sim> x) = x" |
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boolean algebras as locales and numbers as types by Brian Huffman
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parents:
diff
changeset
|
74 |
proof (rule compl_unique) |
24357 | 75 |
from conj_cancel_right show "\<sim> x \<sqinter> x = \<zero>" by (simp only: conj_commute) |
76 |
from disj_cancel_right show "\<sim> x \<squnion> x = \<one>" by (simp only: disj_commute) |
|
24332
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boolean algebras as locales and numbers as types by Brian Huffman
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parents:
diff
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77 |
qed |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
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parents:
diff
changeset
|
78 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
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parents:
diff
changeset
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79 |
lemma compl_eq_compl_iff [simp]: "(\<sim> x = \<sim> y) = (x = y)" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
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parents:
diff
changeset
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80 |
by (rule inj_eq [OF inj_on_inverseI], rule double_compl) |
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boolean algebras as locales and numbers as types by Brian Huffman
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parents:
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81 |
|
24357 | 82 |
subsection {* Conjunction *} |
24332
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boolean algebras as locales and numbers as types by Brian Huffman
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parents:
diff
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83 |
|
24393 | 84 |
lemma conj_absorb [simp]: "x \<sqinter> x = x" |
24332
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boolean algebras as locales and numbers as types by Brian Huffman
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parents:
diff
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85 |
proof - |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
86 |
have "x \<sqinter> x = (x \<sqinter> x) \<squnion> \<zero>" using disj_zero_right by simp |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
87 |
also have "... = (x \<sqinter> x) \<squnion> (x \<sqinter> \<sim> x)" using conj_cancel_right by simp |
24357 | 88 |
also have "... = x \<sqinter> (x \<squnion> \<sim> x)" using conj_disj_distrib by (simp only:) |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
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89 |
also have "... = x \<sqinter> \<one>" using disj_cancel_right by simp |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
90 |
also have "... = x" using conj_one_right by simp |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
91 |
finally show ?thesis . |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
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parents:
diff
changeset
|
92 |
qed |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
93 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
94 |
lemma conj_zero_right [simp]: "x \<sqinter> \<zero> = \<zero>" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
95 |
proof - |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
96 |
have "x \<sqinter> \<zero> = x \<sqinter> (x \<sqinter> \<sim> x)" using conj_cancel_right by simp |
24393 | 97 |
also have "... = (x \<sqinter> x) \<sqinter> \<sim> x" using conj_assoc by (simp only:) |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
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98 |
also have "... = x \<sqinter> \<sim> x" using conj_absorb by simp |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
99 |
also have "... = \<zero>" using conj_cancel_right by simp |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
100 |
finally show ?thesis . |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
101 |
qed |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
102 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
103 |
lemma compl_one [simp]: "\<sim> \<one> = \<zero>" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
104 |
by (rule compl_unique [OF conj_zero_right disj_zero_right]) |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
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parents:
diff
changeset
|
105 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
106 |
lemma conj_zero_left [simp]: "\<zero> \<sqinter> x = \<zero>" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
107 |
by (subst conj_commute) (rule conj_zero_right) |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
108 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
109 |
lemma conj_one_left [simp]: "\<one> \<sqinter> x = x" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
110 |
by (subst conj_commute) (rule conj_one_right) |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
111 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
112 |
lemma conj_cancel_left [simp]: "\<sim> x \<sqinter> x = \<zero>" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
113 |
by (subst conj_commute) (rule conj_cancel_right) |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
114 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
115 |
lemma conj_left_absorb [simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y" |
24357 | 116 |
by (simp only: conj_assoc [symmetric] conj_absorb) |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
117 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
118 |
lemma conj_disj_distrib2: |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
119 |
"(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)" |
24357 | 120 |
by (simp only: conj_commute conj_disj_distrib) |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
121 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
122 |
lemmas conj_disj_distribs = |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
123 |
conj_disj_distrib conj_disj_distrib2 |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
124 |
|
24357 | 125 |
subsection {* Disjunction *} |
24332
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boolean algebras as locales and numbers as types by Brian Huffman
kleing
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diff
changeset
|
126 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
127 |
lemma disj_absorb [simp]: "x \<squnion> x = x" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
128 |
by (rule boolean.conj_absorb [OF dual]) |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
129 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
130 |
lemma disj_one_right [simp]: "x \<squnion> \<one> = \<one>" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
131 |
by (rule boolean.conj_zero_right [OF dual]) |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
132 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
133 |
lemma compl_zero [simp]: "\<sim> \<zero> = \<one>" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
134 |
by (rule boolean.compl_one [OF dual]) |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
135 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
136 |
lemma disj_zero_left [simp]: "\<zero> \<squnion> x = x" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
137 |
by (rule boolean.conj_one_left [OF dual]) |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
138 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
139 |
lemma disj_one_left [simp]: "\<one> \<squnion> x = \<one>" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
140 |
by (rule boolean.conj_zero_left [OF dual]) |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
141 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
142 |
lemma disj_cancel_left [simp]: "\<sim> x \<squnion> x = \<one>" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
143 |
by (rule boolean.conj_cancel_left [OF dual]) |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
144 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
145 |
lemma disj_left_absorb [simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
146 |
by (rule boolean.conj_left_absorb [OF dual]) |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
147 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
148 |
lemma disj_conj_distrib2: |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
149 |
"(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
150 |
by (rule boolean.conj_disj_distrib2 [OF dual]) |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
151 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
152 |
lemmas disj_conj_distribs = |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
153 |
disj_conj_distrib disj_conj_distrib2 |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
154 |
|
24357 | 155 |
subsection {* De Morgan's Laws *} |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
156 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
157 |
lemma de_Morgan_conj [simp]: "\<sim> (x \<sqinter> y) = \<sim> x \<squnion> \<sim> y" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
158 |
proof (rule compl_unique) |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
159 |
have "(x \<sqinter> y) \<sqinter> (\<sim> x \<squnion> \<sim> y) = ((x \<sqinter> y) \<sqinter> \<sim> x) \<squnion> ((x \<sqinter> y) \<sqinter> \<sim> y)" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
160 |
by (rule conj_disj_distrib) |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
161 |
also have "... = (y \<sqinter> (x \<sqinter> \<sim> x)) \<squnion> (x \<sqinter> (y \<sqinter> \<sim> y))" |
24357 | 162 |
by (simp only: conj_ac) |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
163 |
finally show "(x \<sqinter> y) \<sqinter> (\<sim> x \<squnion> \<sim> y) = \<zero>" |
24357 | 164 |
by (simp only: conj_cancel_right conj_zero_right disj_zero_right) |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
165 |
next |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
166 |
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))" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
167 |
by (rule disj_conj_distrib2) |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
168 |
also have "... = (\<sim> y \<squnion> (x \<squnion> \<sim> x)) \<sqinter> (\<sim> x \<squnion> (y \<squnion> \<sim> y))" |
24357 | 169 |
by (simp only: disj_ac) |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
170 |
finally show "(x \<sqinter> y) \<squnion> (\<sim> x \<squnion> \<sim> y) = \<one>" |
24357 | 171 |
by (simp only: disj_cancel_right disj_one_right conj_one_right) |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
172 |
qed |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
173 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
174 |
lemma de_Morgan_disj [simp]: "\<sim> (x \<squnion> y) = \<sim> x \<sqinter> \<sim> y" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
175 |
by (rule boolean.de_Morgan_conj [OF dual]) |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
176 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
177 |
end |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
178 |
|
24357 | 179 |
subsection {* Symmetric Difference *} |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
180 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
181 |
locale boolean_xor = boolean + |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
182 |
fixes xor :: "'a => 'a => 'a" (infixr "\<oplus>" 65) |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
183 |
assumes xor_def: "x \<oplus> y = (x \<sqinter> \<sim> y) \<squnion> (\<sim> x \<sqinter> y)" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
184 |
begin |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
185 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
186 |
lemma xor_def2: |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
187 |
"x \<oplus> y = (x \<squnion> y) \<sqinter> (\<sim> x \<squnion> \<sim> y)" |
24357 | 188 |
by (simp only: xor_def conj_disj_distribs |
189 |
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
|
190 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
191 |
lemma xor_commute: "x \<oplus> y = y \<oplus> x" |
24357 | 192 |
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
|
193 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
194 |
lemma xor_assoc: "(x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
195 |
proof - |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
196 |
let ?t = "(x \<sqinter> y \<sqinter> z) \<squnion> (x \<sqinter> \<sim> y \<sqinter> \<sim> z) \<squnion> |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
197 |
(\<sim> x \<sqinter> y \<sqinter> \<sim> z) \<squnion> (\<sim> x \<sqinter> \<sim> y \<sqinter> z)" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
198 |
have "?t \<squnion> (z \<sqinter> x \<sqinter> \<sim> x) \<squnion> (z \<sqinter> y \<sqinter> \<sim> y) = |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
199 |
?t \<squnion> (x \<sqinter> y \<sqinter> \<sim> y) \<squnion> (x \<sqinter> z \<sqinter> \<sim> z)" |
24357 | 200 |
by (simp only: conj_cancel_right conj_zero_right) |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
201 |
thus "(x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)" |
24357 | 202 |
apply (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl) |
203 |
apply (simp only: conj_disj_distribs conj_ac disj_ac) |
|
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
204 |
done |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
205 |
qed |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
206 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
207 |
lemmas xor_ac = |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
208 |
xor_assoc xor_commute |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
209 |
mk_left_commute [of "xor", OF xor_assoc xor_commute] |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
210 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
211 |
lemma xor_zero_right [simp]: "x \<oplus> \<zero> = x" |
24357 | 212 |
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
|
213 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
214 |
lemma xor_zero_left [simp]: "\<zero> \<oplus> x = x" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
215 |
by (subst xor_commute) (rule xor_zero_right) |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
216 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
217 |
lemma xor_one_right [simp]: "x \<oplus> \<one> = \<sim> x" |
24357 | 218 |
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
|
219 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
220 |
lemma xor_one_left [simp]: "\<one> \<oplus> x = \<sim> x" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
221 |
by (subst xor_commute) (rule xor_one_right) |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
222 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
223 |
lemma xor_self [simp]: "x \<oplus> x = \<zero>" |
24357 | 224 |
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
|
225 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
226 |
lemma xor_left_self [simp]: "x \<oplus> (x \<oplus> y) = y" |
24357 | 227 |
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
|
228 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
229 |
lemma xor_compl_left: "\<sim> x \<oplus> y = \<sim> (x \<oplus> y)" |
24357 | 230 |
apply (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl) |
231 |
apply (simp only: conj_disj_distribs) |
|
232 |
apply (simp only: conj_cancel_right conj_cancel_left) |
|
233 |
apply (simp only: disj_zero_left disj_zero_right) |
|
234 |
apply (simp only: disj_ac conj_ac) |
|
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
235 |
done |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
236 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
237 |
lemma xor_compl_right: "x \<oplus> \<sim> y = \<sim> (x \<oplus> y)" |
24357 | 238 |
apply (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl) |
239 |
apply (simp only: conj_disj_distribs) |
|
240 |
apply (simp only: conj_cancel_right conj_cancel_left) |
|
241 |
apply (simp only: disj_zero_left disj_zero_right) |
|
242 |
apply (simp only: disj_ac conj_ac) |
|
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
243 |
done |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
244 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
245 |
lemma xor_cancel_right [simp]: "x \<oplus> \<sim> x = \<one>" |
24357 | 246 |
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
|
247 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
248 |
lemma xor_cancel_left [simp]: "\<sim> x \<oplus> x = \<one>" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
249 |
by (subst xor_commute) (rule xor_cancel_right) |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
250 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
251 |
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
|
252 |
proof - |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
253 |
have "(x \<sqinter> y \<sqinter> \<sim> z) \<squnion> (x \<sqinter> \<sim> y \<sqinter> z) = |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
254 |
(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 | 255 |
by (simp only: conj_cancel_right conj_zero_right disj_zero_left) |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
256 |
thus "x \<sqinter> (y \<oplus> z) = (x \<sqinter> y) \<oplus> (x \<sqinter> z)" |
24357 | 257 |
by (simp (no_asm_use) only: |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
258 |
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
|
259 |
conj_disj_distribs conj_ac disj_ac) |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
260 |
qed |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
261 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
262 |
lemma conj_xor_distrib2: |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
263 |
"(y \<oplus> z) \<sqinter> x = (y \<sqinter> x) \<oplus> (z \<sqinter> x)" |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
264 |
proof - |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
265 |
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
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by (rule conj_xor_distrib) |
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thus "(y \<oplus> z) \<sqinter> x = (y \<sqinter> x) \<oplus> (z \<sqinter> x)" |
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by (simp only: conj_commute) |
24332
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269 |
qed |
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270 |
|
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271 |
lemmas conj_xor_distribs = |
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conj_xor_distrib conj_xor_distrib2 |
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273 |
|
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274 |
end |
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|
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276 |
end |