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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 August-2006A, August 2006.
Process 11536 was started by mccune on cleo.thornwood,
Sat Aug 12 21:08:20 2006
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 1.58 (+ 0.02) seconds: H34.
% Length of proof is 32.
% Level of proof is 10.
% Maximum clause weight is 21.
% Given clauses 175.
]]></comments>

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

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

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

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

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

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

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

  <clause id="8">
    <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">
    <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="39">
    <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="56">
    <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="79">
    <literal><![CDATA[
      (x v y) ^ (y v x) = x v y
    ]]></literal>
    <justification jstring="[para(39(a,1),16(a,1,2))].">
      <j1 rule="para" parents="39 16"/>
    </justification>
  </clause>

  <clause id="92">
    <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="116">
    <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="199">
    <literal><![CDATA[
      (x v y) ^ (z ^ (y v x)) = z ^ (y v x)
    ]]></literal>
    <justification jstring="[para(79(a,1),5(a,2,2)),rewrite(4(4))].">
      <j1 rule="para" parents="79 5"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="202">
    <literal><![CDATA[
      (x v (y v z)) ^ (y v x) = x v y
    ]]></literal>
    <justification jstring="[para(79(a,1),17(a,2)),rewrite(3(3),199(6))].">
      <j1 rule="para" parents="79 17"/>
      <j2 rule="rewrite" parents="3 199"/>
    </justification>
  </clause>

  <clause id="1659">
    <literal><![CDATA[
      x v ((x v y) ^ (x v z)) = (x v y) ^ (x v z)
    ]]></literal>
    <justification jstring="[para(6(a,1),92(a,1,1))].">
      <j1 rule="para" parents="6 92"/>
    </justification>
  </clause>

  <clause id="2579">
    <literal><![CDATA[
      x v ((y v z) ^ (y v x)) = x v y
    ]]></literal>
    <justification jstring="[para(202(a,1),116(a,1,2)),rewrite(2(5),3(5),1659(4),202(8))].">
      <j1 rule="para" parents="202 116"/>
      <j2 rule="rewrite" parents="2 3 1659 202"/>
    </justification>
  </clause>

  <clause id="2625">
    <literal><![CDATA[
      x v ((y v z) ^ (z v x)) = x v z
    ]]></literal>
    <justification jstring="[para(2(a,1),2579(a,1,2,1))].">
      <j1 rule="para" parents="2 2579"/>
    </justification>
  </clause>

  <clause id="2691">
    <literal><![CDATA[
      x v ((y v z) ^ (x v z)) = x v z
    ]]></literal>
    <justification jstring="[para(2(a,1),2625(a,1,2,2))].">
      <j1 rule="para" parents="2 2625"/>
    </justification>
  </clause>

  <clause id="2963">
    <literal><![CDATA[
      x v (y ^ (x v (y ^ z))) = x v (y ^ z)
    ]]></literal>
    <justification jstring="[para(7(a,1),2691(a,1,2,1))].">
      <j1 rule="para" parents="7 2691"/>
    </justification>
  </clause>

  <clause id="6274">
    <literal><![CDATA[
      x ^ (y v (z ^ (y v x))) = x ^ (y v (z ^ x))
    ]]></literal>
    <justification jstring="[para(199(a,1),32(a,1,2,2)),rewrite(56(7))].">
      <j1 rule="para" parents="199 32"/>
      <j2 rule="rewrite" parents="56"/>
    </justification>
  </clause>

  <clause id="6306">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(H34)
    ]]></attribute>
    <justification jstring="[back_rewrite(10),rewrite(6274(11),4(7),2963(10)),xx(a)].">
      <j1 rule="back_rewrite" parents="10"/>
      <j2 rule="rewrite" parents="6274 4 2963"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
