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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>H33-H34.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13804 was started by mccune on cleo.thornwood,
Mon Jun 19 16:51:25 2006
The command was "/home/mccune/bin/prover9 -f lt.in H33-H34.in".
]]></heading>

<proof number="1" length="31" max_count="21">

<comments><![CDATA[
% Proof 1 at 1.62 (+ 0.03) seconds: H34.
% Length of proof is 31.
% Level of proof is 10.
% Maximum clause weight is 21.
% Given clauses 175.
]]></comments>

  <clause id="7">
    <literal><![CDATA[
      x ^ (y v (x ^ (z v u))) = x ^ (y v ((x v y) ^ (z v u)))
    ]]></literal>
    <attribute><![CDATA[
      label(H33)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

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

  <clause id="11">
    <literal><![CDATA[
      x ^ y = y ^ x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      (x ^ y) ^ z = x ^ (y ^ z)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

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

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

  <clause id="15">
    <literal><![CDATA[
      x ^ (y v ((x v y) ^ (z v u))) = x ^ (y v (x ^ (z v u)))
    ]]></literal>
    <attribute><![CDATA[
      label(H33)
    ]]></attribute>
    <justification jstring="[copy(7),flip(a)].">
      <j1 rule="copy" parents="7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      c1 ^ (c2 v (c3 ^ (c2 v (c4 ^ (c2 v c3))))) != c1 ^ (c2 v (c3 ^ c4))
    ]]></literal>
    <attribute><![CDATA[
      answer(H34)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      x ^ (y ^ z) = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(11(a,1),12(a,1,1)),demod(12(2))].">
      <j1 rule="para" parents="11 12"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(9(a,1),13(a,1,2))].">
      <j1 rule="para" parents="9 13"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      (x v y) ^ (x v (y v z)) = x v y
    ]]></literal>
    <justification jstring="[para(10(a,1),13(a,1,2))].">
      <j1 rule="para" parents="10 13"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      x ^ ((x v y) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(13(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="13 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),14(a,1,2))].">
      <j1 rule="para" parents="11 14"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(13(a,1),14(a,1,2))].">
      <j1 rule="para" parents="13 14"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      x ^ (y v ((x v y) ^ z)) = x ^ (y v (x ^ z))
    ]]></literal>
    <justification jstring="[para(14(a,1),15(a,1,2,2,2)),demod(14(6))].">
      <j1 rule="para" parents="14 15"/>
      <j2 rule="demod" parents="14"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(28(a,1),10(a,2,2)),demod(9(2))].">
      <j1 rule="para" parents="28 10"/>
      <j2 rule="demod" parents="9"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      x ^ ((y v x) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(19(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="19 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="60">
    <literal><![CDATA[
      x ^ (y ^ (z v x)) = y ^ x
    ]]></literal>
    <justification jstring="[para(19(a,1),18(a,1,2)),flip(a)].">
      <j1 rule="para" parents="19 18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="83">
    <literal><![CDATA[
      (x v y) ^ (y v x) = x v y
    ]]></literal>
    <justification jstring="[para(43(a,1),20(a,1,2))].">
      <j1 rule="para" parents="43 20"/>
    </justification>
  </clause>

  <clause id="96">
    <literal><![CDATA[
      (x ^ y) v ((x v z) ^ y) = (x v z) ^ y
    ]]></literal>
    <justification jstring="[para(21(a,1),25(a,1,2)),demod(9(4))].">
      <j1 rule="para" parents="21 25"/>
      <j2 rule="demod" parents="9"/>
    </justification>
  </clause>

  <clause id="120">
    <literal><![CDATA[
      (x ^ y) v ((z v x) ^ y) = (z v x) ^ y
    ]]></literal>
    <justification jstring="[para(47(a,1),25(a,1,2)),demod(9(4))].">
      <j1 rule="para" parents="47 25"/>
      <j2 rule="demod" parents="9"/>
    </justification>
  </clause>

  <clause id="203">
    <literal><![CDATA[
      (x v y) ^ (z ^ (y v x)) = z ^ (y v x)
    ]]></literal>
    <justification jstring="[para(83(a,1),12(a,2,2)),demod(11(4))].">
      <j1 rule="para" parents="83 12"/>
      <j2 rule="demod" parents="11"/>
    </justification>
  </clause>

  <clause id="206">
    <literal><![CDATA[
      (x v (y v z)) ^ (y v x) = x v y
    ]]></literal>
    <justification jstring="[para(83(a,1),21(a,2)),demod(10(3),203(6))].">
      <j1 rule="para" parents="83 21"/>
      <j2 rule="demod" parents="10 203"/>
    </justification>
  </clause>

  <clause id="1663">
    <literal><![CDATA[
      x v ((x v y) ^ (x v z)) = (x v y) ^ (x v z)
    ]]></literal>
    <justification jstring="[para(13(a,1),96(a,1,1))].">
      <j1 rule="para" parents="13 96"/>
    </justification>
  </clause>

  <clause id="2583">
    <literal><![CDATA[
      x v ((y v z) ^ (y v x)) = x v y
    ]]></literal>
    <justification jstring="[para(206(a,1),120(a,1,2)),demod(9(5),10(5),1663(4),206(8))].">
      <j1 rule="para" parents="206 120"/>
      <j2 rule="demod" parents="9 10 1663 206"/>
    </justification>
  </clause>

  <clause id="2629">
    <literal><![CDATA[
      x v ((y v z) ^ (z v x)) = x v z
    ]]></literal>
    <justification jstring="[para(9(a,1),2583(a,1,2,1))].">
      <j1 rule="para" parents="9 2583"/>
    </justification>
  </clause>

  <clause id="2695">
    <literal><![CDATA[
      x v ((y v z) ^ (x v z)) = x v z
    ]]></literal>
    <justification jstring="[para(9(a,1),2629(a,1,2,2))].">
      <j1 rule="para" parents="9 2629"/>
    </justification>
  </clause>

  <clause id="2967">
    <literal><![CDATA[
      x v (y ^ (x v (y ^ z))) = x v (y ^ z)
    ]]></literal>
    <justification jstring="[para(14(a,1),2695(a,1,2,1))].">
      <j1 rule="para" parents="14 2695"/>
    </justification>
  </clause>

  <clause id="6278">
    <literal><![CDATA[
      x ^ (y v (z ^ (y v x))) = x ^ (y v (z ^ x))
    ]]></literal>
    <justification jstring="[para(203(a,1),36(a,1,2,2)),demod(60(7))].">
      <j1 rule="para" parents="203 36"/>
      <j2 rule="demod" parents="60"/>
    </justification>
  </clause>

  <clause id="6310">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(H34)
    ]]></attribute>
    <justification jstring="[back_demod(16),demod(6278(11),11(7),2967(10)),xx(a)].">
      <j1 rule="back_demod" parents="16"/>
      <j2 rule="demod" parents="6278 11 2967"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
