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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 22-May-2007, May 2007.
Process 27688 was started by mccune on cleo,
Tue May 22 14:51:30 2007
The command was "/home/mccune/bin/prover9 -f lt.in H33-H34.in".
]]></heading>

<proof number="1" length="32" max_count="21">

<comments><![CDATA[
% Proof 1 at 1.51 (+ 0.01) seconds: H34.
% Length of proof is 32.
% Level of proof is 10.
% Maximum clause weight is 21.
% Given clauses 173.
]]></comments>

  <clause id="1" type="goal">
    <literal><![CDATA[
      x ^ (y v (z ^ u)) = x ^ (y v (z ^ (y v (u ^ (y v z)))))
    ]]></literal>
    <attribute><![CDATA[
      answer(H34)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="2" type="assumption">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="3" type="assumption">
    <literal><![CDATA[
      (x v y) v z = x v (y v z)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="4" type="assumption">
    <literal><![CDATA[
      x ^ y = y ^ x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="5" type="assumption">
    <literal><![CDATA[
      (x ^ y) ^ z = x ^ (y ^ z)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6" type="assumption">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7" type="assumption">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="8" type="assumption">
    <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="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      x ^ (y v ((x v y) ^ (z v u))) = x ^ (y v (x ^ (z v u)))
    ]]></literal>
    <justification jstring="[copy(8),flip(a)].">
      <j1 rule="copy" parents="8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10" type="deny">
    <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="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      x ^ (y ^ z) = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(4(a,1),5(a,1,1)),rewrite([5(2)])].">
      <j1 rule="para" parents="4 5"/>
      <j2 rule="rewrite" parents="5"/>
    </justification>
  </clause>

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

  <clause id="16">
    <literal><![CDATA[
      (x v y) ^ (x v (y v z)) = x v y
    ]]></literal>
    <justification jstring="[para(3(a,1),6(a,1,2))].">
      <j1 rule="para" parents="3 6"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      x ^ ((x v y) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(6(a,1),5(a,1,1)),flip(a)].">
      <j1 rule="para" parents="6 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(4(a,1),7(a,1,2))].">
      <j1 rule="para" parents="4 7"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(6(a,1),7(a,1,2))].">
      <j1 rule="para" parents="6 7"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      x ^ (y v ((x v y) ^ z)) = x ^ (y v (x ^ z))
    ]]></literal>
    <justification jstring="[para(7(a,1),9(a,1,2,2,2)),rewrite([7(6)])].">
      <j1 rule="para" parents="7 9"/>
      <j2 rule="rewrite" parents="7"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(24(a,1),3(a,2,2)),rewrite([2(2)])].">
      <j1 rule="para" parents="24 3"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      x ^ ((y v x) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(15(a,1),5(a,1,1)),flip(a)].">
      <j1 rule="para" parents="15 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      x ^ (y ^ (z v x)) = y ^ x
    ]]></literal>
    <justification jstring="[para(15(a,1),14(a,1,2)),flip(a)].">
      <j1 rule="para" parents="15 14"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="73">
    <literal><![CDATA[
      (x v y) ^ (y v x) = x v y
    ]]></literal>
    <justification jstring="[para(41(a,1),16(a,1,2))].">
      <j1 rule="para" parents="41 16"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      (x ^ y) v ((x v z) ^ y) = (x v z) ^ y
    ]]></literal>
    <justification jstring="[para(17(a,1),21(a,1,2)),rewrite([2(4)])].">
      <j1 rule="para" parents="17 21"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="108">
    <literal><![CDATA[
      (x ^ y) v ((z v x) ^ y) = (z v x) ^ y
    ]]></literal>
    <justification jstring="[para(43(a,1),21(a,1,2)),rewrite([2(4)])].">
      <j1 rule="para" parents="43 21"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="178">
    <literal><![CDATA[
      (x v y) ^ (z ^ (y v x)) = z ^ (y v x)
    ]]></literal>
    <justification jstring="[para(73(a,1),5(a,2,2)),rewrite([4(4)])].">
      <j1 rule="para" parents="73 5"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="181">
    <literal><![CDATA[
      (x v (y v z)) ^ (y v x) = x v y
    ]]></literal>
    <justification jstring="[para(73(a,1),17(a,2)),rewrite([3(3),178(6)])].">
      <j1 rule="para" parents="73 17"/>
      <j2 rule="rewrite" parents="3 178"/>
    </justification>
  </clause>

  <clause id="1738">
    <literal><![CDATA[
      x v ((x v y) ^ (x v z)) = (x v y) ^ (x v z)
    ]]></literal>
    <justification jstring="[para(6(a,1),91(a,1,1))].">
      <j1 rule="para" parents="6 91"/>
    </justification>
  </clause>

  <clause id="2066">
    <literal><![CDATA[
      x v ((y v z) ^ (y v x)) = x v y
    ]]></literal>
    <justification jstring="[para(181(a,1),108(a,1,2)),rewrite([2(5),3(5),1738(4),181(8)])].">
      <j1 rule="para" parents="181 108"/>
      <j2 rule="rewrite" parents="2 3 1738 181"/>
    </justification>
  </clause>

  <clause id="2097">
    <literal><![CDATA[
      x v ((y v z) ^ (z v x)) = x v z
    ]]></literal>
    <justification jstring="[para(2(a,1),2066(a,1,2,1))].">
      <j1 rule="para" parents="2 2066"/>
    </justification>
  </clause>

  <clause id="2221">
    <literal><![CDATA[
      x v ((y v z) ^ (x v z)) = x v z
    ]]></literal>
    <justification jstring="[para(2(a,1),2097(a,1,2,2))].">
      <j1 rule="para" parents="2 2097"/>
    </justification>
  </clause>

  <clause id="2390">
    <literal><![CDATA[
      x v (y ^ (x v (y ^ z))) = x v (y ^ z)
    ]]></literal>
    <justification jstring="[para(7(a,1),2221(a,1,2,1))].">
      <j1 rule="para" parents="7 2221"/>
    </justification>
  </clause>

  <clause id="6236">
    <literal><![CDATA[
      x ^ (y v (z ^ (y v x))) = x ^ (y v (z ^ x))
    ]]></literal>
    <justification jstring="[para(178(a,1),32(a,1,2,2)),rewrite([52(7)])].">
      <j1 rule="para" parents="178 32"/>
      <j2 rule="rewrite" parents="52"/>
    </justification>
  </clause>

  <clause id="6272">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(H34)
    ]]></attribute>
    <justification jstring="[back_rewrite(10),rewrite([6236(11),4(7),2390(10)]),xx(a)].">
      <j1 rule="back_rewrite" parents="10"/>
      <j2 rule="rewrite" parents="6236 4 2390"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
