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<!DOCTYPE proofs SYSTEM "proof3.dtd">

<?xml-stylesheet type="text/xsl" href="proof3.xsl"?>

<proofs number_of_proofs="1">

<source>H42.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13671 was started by mccune on cleo.thornwood,
Mon Jun 19 16:48:36 2006
The command was "/home/mccune/bin/prover9 -f lt.in uc.in H42.in".
]]></heading>

<proof number="1" length="46" max_count="23">

<comments><![CDATA[
% Proof 1 at 68.66 (+ 0.95) seconds.
% Length of proof is 46.
% Level of proof is 11.
% Maximum clause weight is 23.
% Given clauses 1571.
]]></comments>

  <clause id="10">
    <literal><![CDATA[
      x ^ (y v (z ^ (x v u))) = x ^ (y v (z ^ (y v (u v (x ^ z)))))
    ]]></literal>
    <attribute><![CDATA[
      label(H42)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      c1 ' != c1 ' v c2 '
    ]]></literal>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      (x v y) v z = x v (y v z)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      x ^ y = y ^ x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      (x ^ y) ^ z = x ^ (y ^ z)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      x v x ' = 1
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x ^ x ' = 0
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      x v y != 1
    ]]></literal>
    <literal><![CDATA[
      x ^ y != 0
    ]]></literal>
    <literal><![CDATA[
      x ' = y
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      x ^ (y v (z ^ (y v (u v (x ^ z))))) = x ^ (y v (z ^ (x v u)))
    ]]></literal>
    <attribute><![CDATA[
      label(H42)
    ]]></attribute>
    <justification jstring="[copy(10),flip(a)].">
      <j1 rule="copy" parents="10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      c1 ^ c2 = c1
    ]]></literal>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      c1 ' v c2 ' != c1 '
    ]]></literal>
    <justification jstring="[copy(12),flip(a)].">
      <j1 rule="copy" parents="12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      x ^ (y ^ z) = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(15(a,1),16(a,1,1)),demod(16(2))].">
      <j1 rule="para" parents="15 16"/>
      <j2 rule="demod" parents="16"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      x ^ ((x v y) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(17(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="17 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      x v ((x ^ y) v z) = x v z
    ]]></literal>
    <justification jstring="[para(18(a,1),14(a,1,1)),flip(a)].">
      <j1 rule="para" parents="18 14"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(15(a,1),18(a,1,2))].">
      <j1 rule="para" parents="15 18"/>
    </justification>
  </clause>

  <clause id="37">
    <literal><![CDATA[
      x v (x ' v y) = 1 v y
    ]]></literal>
    <justification jstring="[para(19(a,1),14(a,1,1)),flip(a)].">
      <j1 rule="para" parents="19 14"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      x ^ 1 = x
    ]]></literal>
    <justification jstring="[para(19(a,1),17(a,1,2))].">
      <j1 rule="para" parents="19 17"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      x ^ (x ' ^ y) = 0 ^ y
    ]]></literal>
    <justification jstring="[para(20(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="20 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      x v 0 = x
    ]]></literal>
    <justification jstring="[para(20(a,1),18(a,1,2))].">
      <j1 rule="para" parents="20 18"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      x ^ (y v (z ^ (y v ((x ^ z) v u)))) = x ^ (y v (z ^ (x v u)))
    ]]></literal>
    <justification jstring="[para(13(a,1),22(a,1,2,2,2,2))].">
      <j1 rule="para" parents="13 22"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      c1 ^ (c2 ^ x) = c1 ^ x
    ]]></literal>
    <justification jstring="[para(23(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="23 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="78">
    <literal><![CDATA[
      1 ^ x = x
    ]]></literal>
    <justification jstring="[para(39(a,1),15(a,1)),flip(a)].">
      <j1 rule="para" parents="39 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="84">
    <literal><![CDATA[
      0 v x = x
    ]]></literal>
    <justification jstring="[para(42(a,1),13(a,1)),flip(a)].">
      <j1 rule="para" parents="42 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="87">
    <literal><![CDATA[
      1 v x = 1
    ]]></literal>
    <justification jstring="[para(78(a,1),17(a,1))].">
      <j1 rule="para" parents="78 17"/>
    </justification>
  </clause>

  <clause id="88">
    <literal><![CDATA[
      x v (x ' v y) = 1
    ]]></literal>
    <justification jstring="[back_demod(37),demod(87(5))].">
      <j1 rule="back_demod" parents="37"/>
      <j2 rule="demod" parents="87"/>
    </justification>
  </clause>

  <clause id="90">
    <literal><![CDATA[
      0 ^ x = 0
    ]]></literal>
    <justification jstring="[para(84(a,1),17(a,1,2))].">
      <j1 rule="para" parents="84 17"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      x ^ (x ' ^ y) = 0
    ]]></literal>
    <justification jstring="[back_demod(40),demod(90(5))].">
      <j1 rule="back_demod" parents="40"/>
      <j2 rule="demod" parents="90"/>
    </justification>
  </clause>

  <clause id="97">
    <literal><![CDATA[
      x v 1 = 1
    ]]></literal>
    <justification jstring="[para(87(a,1),13(a,1)),flip(a)].">
      <j1 rule="para" parents="87 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="100">
    <literal><![CDATA[
      x ^ 0 = 0
    ]]></literal>
    <justification jstring="[para(90(a,1),15(a,1)),flip(a)].">
      <j1 rule="para" parents="90 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="113">
    <literal><![CDATA[
      c1 v c2 = c2
    ]]></literal>
    <justification jstring="[para(23(a,1),33(a,1,2)),demod(13(3))].">
      <j1 rule="para" parents="23 33"/>
      <j2 rule="demod" parents="13"/>
    </justification>
  </clause>

  <clause id="131">
    <literal><![CDATA[
      c1 ^ (c1 ' v c2 ') != 0
    ]]></literal>
    <justification jstring="[ur(21,a,88,a,c,24,a(flip))].">
      <j1 rule="ur" parents="21 88 24"/>
    </justification>
  </clause>

  <clause id="142">
    <literal><![CDATA[
      x ^ (x v y) ' = 0
    ]]></literal>
    <justification jstring="[para(20(a,1),29(a,1,2)),demod(100(2)),flip(a)].">
      <j1 rule="para" parents="20 29"/>
      <j2 rule="demod" parents="100"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="157">
    <literal><![CDATA[
      c1 ^ c2 ' = 0
    ]]></literal>
    <justification jstring="[para(113(a,1),142(a,1,2,1))].">
      <j1 rule="para" parents="113 142"/>
    </justification>
  </clause>

  <clause id="184">
    <literal><![CDATA[
      (c1 ^ x) v (c2 ^ x) = c2 ^ x
    ]]></literal>
    <justification jstring="[para(66(a,1),33(a,1,2)),demod(13(5))].">
      <j1 rule="para" parents="66 33"/>
      <j2 rule="demod" parents="13"/>
    </justification>
  </clause>

  <clause id="191">
    <literal><![CDATA[
      x v (x ^ y) ' = 1
    ]]></literal>
    <justification jstring="[para(19(a,1),31(a,1,2)),demod(97(2)),flip(a)].">
      <j1 rule="para" parents="19 31"/>
      <j2 rule="demod" parents="97"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="224">
    <literal><![CDATA[
      x v (y ^ x) ' = 1
    ]]></literal>
    <justification jstring="[para(15(a,1),191(a,1,2,1))].">
      <j1 rule="para" parents="15 191"/>
    </justification>
  </clause>

  <clause id="237">
    <literal><![CDATA[
      (c2 ^ x) v (c1 ^ x) ' = 1
    ]]></literal>
    <justification jstring="[para(66(a,1),224(a,1,2,1))].">
      <j1 rule="para" parents="66 224"/>
    </justification>
  </clause>

  <clause id="580">
    <literal><![CDATA[
      c1 ^ (x v (c2 ' ^ (c1 v y))) = c1 ^ (x v (c2 ' ^ (x v y)))
    ]]></literal>
    <justification jstring="[para(157(a,1),49(a,1,2,2,2,2,1)),demod(84(5)),flip(a)].">
      <j1 rule="para" parents="157 49"/>
      <j2 rule="demod" parents="84"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1330">
    <literal><![CDATA[
      (c2 ^ (c1 v x)) v c1 ' = 1
    ]]></literal>
    <justification jstring="[para(17(a,1),237(a,1,2,1))].">
      <j1 rule="para" parents="17 237"/>
    </justification>
  </clause>

  <clause id="2204">
    <literal><![CDATA[
      c1 ' v (c2 ^ (c1 v x)) = 1
    ]]></literal>
    <justification jstring="[para(1330(a,1),13(a,1)),flip(a)].">
      <j1 rule="para" parents="1330 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2850">
    <literal><![CDATA[
      c1 v (c2 ^ (c1 v x)) = c2 ^ (c1 v x)
    ]]></literal>
    <justification jstring="[para(17(a,1),184(a,1,1))].">
      <j1 rule="para" parents="17 184"/>
    </justification>
  </clause>

  <clause id="14906">
    <literal><![CDATA[
      c1 ^ (c1 ' v c2 ') = 0
    ]]></literal>
    <justification jstring="[para(2204(a,1),580(a,2,2,2,2)),demod(2850(11),26(10),91(10),13(5),84(5),20(4),15(8),78(8)),flip(a)].">
      <j1 rule="para" parents="2204 580"/>
      <j2 rule="demod" parents="2850 26 91 13 84 20 15 78"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14907">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(14906,a,131,a)].">
      <j1 rule="resolve" parents="14906 131"/>
    </justification>
  </clause>

</proof>

</proofs>
