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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>H39-H3.out</source>

<heading><![CDATA[
Prover9 (32) version November-2006, November 2006.
Process 4152 was started by mccune on cleo.thornwood,
Wed Nov 22 11:33:01 2006
The command was "/home/mccune/bin/prover9 -f lt.in H39-H3.in".
]]></heading>

<proof number="1" length="40" max_count="23">

<comments><![CDATA[
% Proof 1 at 19.26 (+ 0.14) seconds: H3.
% Length of proof is 40.
% Level of proof is 9.
% Maximum clause weight is 23.
% Given clauses 461.
]]></comments>

  <clause id="1" type="goal">
    <literal><![CDATA[
      x ^ (y v (x ^ z)) = x ^ (y v (z ^ (y v (x ^ (z v (x ^ y))))))
    ]]></literal>
    <attribute><![CDATA[
      answer(H3)
    ]]></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 (z ^ (x v u))) = x ^ (y v (z ^ (u v (x ^ z))))
    ]]></literal>
    <attribute><![CDATA[
      label(H39)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      x ^ (y v (z ^ (u v (x ^ z)))) = x ^ (y v (z ^ (x 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 (c1 ^ (c3 v (c1 ^ c2)))))) != c1 ^ (c2 v (c1 ^ c3))
    ]]></literal>
    <attribute><![CDATA[
      answer(H3)
    ]]></attribute>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      x v (y v z) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(2(a,1),3(a,1,1)),rewrite(3(2))].">
      <j1 rule="para" parents="2 3"/>
      <j2 rule="rewrite" parents="3"/>
    </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="19">
    <literal><![CDATA[
      x v ((x ^ y) v z) = x v z
    ]]></literal>
    <justification jstring="[para(7(a,1),3(a,1,1)),flip(a)].">
      <j1 rule="para" parents="7 3"/>
      <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="22">
    <literal><![CDATA[
      (x ^ y) v (x ^ (y ^ z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(5(a,1),7(a,1,2))].">
      <j1 rule="para" parents="5 7"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      x ^ x = x
    ]]></literal>
    <justification jstring="[para(7(a,1),6(a,1,2))].">
      <j1 rule="para" parents="7 6"/>
    </justification>
  </clause>

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

  <clause id="40">
    <literal><![CDATA[
      x ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(23(a,1),5(a,2,2)),rewrite(4(2))].">
      <j1 rule="para" parents="23 5"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      x v ((y ^ x) v z) = x v z
    ]]></literal>
    <justification jstring="[para(21(a,1),3(a,1,1)),flip(a)].">
      <j1 rule="para" parents="21 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="60">
    <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="76">
    <literal><![CDATA[
      (x v y) ^ (y v (x v z)) = y v x
    ]]></literal>
    <justification jstring="[para(2(a,1),16(a,1,1))].">
      <j1 rule="para" parents="2 16"/>
    </justification>
  </clause>

  <clause id="96">
    <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="103">
    <literal><![CDATA[
      (x v y) ^ ((x ^ z) v y) = (x ^ z) v y
    ]]></literal>
    <justification jstring="[para(19(a,1),15(a,1,2)),rewrite(4(4))].">
      <j1 rule="para" parents="19 15"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="130">
    <literal><![CDATA[
      (x v y) ^ ((z ^ x) v y) = (z ^ x) v y
    ]]></literal>
    <justification jstring="[para(50(a,1),15(a,1,2)),rewrite(4(4))].">
      <j1 rule="para" parents="50 15"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="166">
    <literal><![CDATA[
      (x ^ y) v (x ^ (z v y)) = x ^ (z v y)
    ]]></literal>
    <justification jstring="[para(60(a,1),21(a,1,2)),rewrite(2(4))].">
      <j1 rule="para" parents="60 21"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="196">
    <literal><![CDATA[
      (x ^ y) v (y ^ x) = x ^ y
    ]]></literal>
    <justification jstring="[para(40(a,1),22(a,1,2))].">
      <j1 rule="para" parents="40 22"/>
    </justification>
  </clause>

  <clause id="238">
    <literal><![CDATA[
      x ^ ((y ^ ((x ^ y) v z)) v u) = x ^ (u v (y ^ (x v z)))
    ]]></literal>
    <justification jstring="[para(2(a,1),25(a,1,2))].">
      <j1 rule="para" parents="2 25"/>
    </justification>
  </clause>

  <clause id="312">
    <literal><![CDATA[
      (x ^ y) v (z v (y ^ x)) = z v (y ^ x)
    ]]></literal>
    <justification jstring="[para(196(a,1),3(a,2,2)),rewrite(2(4))].">
      <j1 rule="para" parents="196 3"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="317">
    <literal><![CDATA[
      (x ^ (y ^ z)) v (y ^ x) = x ^ y
    ]]></literal>
    <justification jstring="[para(196(a,1),19(a,2)),rewrite(5(3),312(6))].">
      <j1 rule="para" parents="196 19"/>
      <j2 rule="rewrite" parents="5 312"/>
    </justification>
  </clause>

  <clause id="2170">
    <literal><![CDATA[
      x v ((x v y) ^ (x v z)) = (x v y) ^ (x v z)
    ]]></literal>
    <justification jstring="[para(6(a,1),96(a,1,1))].">
      <j1 rule="para" parents="6 96"/>
    </justification>
  </clause>

  <clause id="2340">
    <literal><![CDATA[
      x ^ ((x ^ y) v (x ^ z)) = (x ^ y) v (x ^ z)
    ]]></literal>
    <justification jstring="[para(7(a,1),103(a,1,1))].">
      <j1 rule="para" parents="7 103"/>
    </justification>
  </clause>

  <clause id="3601">
    <literal><![CDATA[
      x ^ ((y ^ z) v (y ^ x)) = x ^ y
    ]]></literal>
    <justification jstring="[para(317(a,1),130(a,1,2)),rewrite(4(5),5(5),2340(4),317(8))].">
      <j1 rule="para" parents="317 130"/>
      <j2 rule="rewrite" parents="4 5 2340 317"/>
    </justification>
  </clause>

  <clause id="3634">
    <literal><![CDATA[
      x ^ ((y ^ z) v (z ^ x)) = x ^ z
    ]]></literal>
    <justification jstring="[para(4(a,1),3601(a,1,2,1))].">
      <j1 rule="para" parents="4 3601"/>
    </justification>
  </clause>

  <clause id="5889">
    <literal><![CDATA[
      x v ((y v x) ^ (y v z)) = x v y
    ]]></literal>
    <justification jstring="[para(76(a,1),166(a,1,2)),rewrite(12(5),2(4),2170(4),76(8))].">
      <j1 rule="para" parents="76 166"/>
      <j2 rule="rewrite" parents="12 2 2170 76"/>
    </justification>
  </clause>

  <clause id="5971">
    <literal><![CDATA[
      x v ((x v y) ^ (y v z)) = x v y
    ]]></literal>
    <justification jstring="[para(2(a,1),5889(a,1,2,1))].">
      <j1 rule="para" parents="2 5889"/>
    </justification>
  </clause>

  <clause id="6095">
    <literal><![CDATA[
      x v (y ^ (x v (z ^ y))) = x v (z ^ y)
    ]]></literal>
    <justification jstring="[para(3634(a,1),5971(a,1,2)),rewrite(4(3))].">
      <j1 rule="para" parents="3634 5971"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="22936">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(H3)
    ]]></attribute>
    <justification jstring="[para(238(a,2),10(a,1,2,2)),rewrite(4(7),2(11),2340(12),2(12),50(12),6095(10)),xx(a)].">
      <j1 rule="para" parents="238 10"/>
      <j2 rule="rewrite" parents="4 2 2340 2 50 6095"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
