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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 June-2006C, June 2006.
Process 13805 was started by mccune on cleo.thornwood,
Mon Jun 19 16:51:26 2006
The command was "/home/mccune/bin/prover9 -f lt.in H39-H3.in".
]]></heading>

<proof number="1" length="39" max_count="23">

<comments><![CDATA[
% Proof 1 at 14.30 (+ 0.11) seconds: H3.
% Length of proof is 39.
% Level of proof is 9.
% Maximum clause weight is 23.
% Given clauses 461.
]]></comments>

  <clause id="7">
    <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="[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 (z ^ (u v (x ^ z)))) = x ^ (y v (z ^ (x v u)))
    ]]></literal>
    <attribute><![CDATA[
      label(H39)
    ]]></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 (c1 ^ (c3 v (c1 ^ c2)))))) != c1 ^ (c2 v (c1 ^ c3))
    ]]></literal>
    <attribute><![CDATA[
      answer(H3)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      x v (y v z) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(9(a,1),10(a,1,1)),demod(10(2))].">
      <j1 rule="para" parents="9 10"/>
      <j2 rule="demod" parents="10"/>
    </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="23">
    <literal><![CDATA[
      x v ((x ^ y) v z) = x v z
    ]]></literal>
    <justification jstring="[para(14(a,1),10(a,1,1)),flip(a)].">
      <j1 rule="para" parents="14 10"/>
      <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="26">
    <literal><![CDATA[
      (x ^ y) v (x ^ (y ^ z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(12(a,1),14(a,1,2))].">
      <j1 rule="para" parents="12 14"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      x ^ x = x
    ]]></literal>
    <justification jstring="[para(14(a,1),13(a,1,2))].">
      <j1 rule="para" parents="14 13"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      x ^ (y v (z ^ ((x ^ z) v u))) = x ^ (y v (z ^ (x v u)))
    ]]></literal>
    <justification jstring="[para(9(a,1),15(a,1,2,2,2))].">
      <j1 rule="para" parents="9 15"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      x ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(27(a,1),12(a,2,2)),demod(11(2))].">
      <j1 rule="para" parents="27 12"/>
      <j2 rule="demod" parents="11"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      x v ((y ^ x) v z) = x v z
    ]]></literal>
    <justification jstring="[para(25(a,1),10(a,1,1)),flip(a)].">
      <j1 rule="para" parents="25 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="64">
    <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="80">
    <literal><![CDATA[
      (x v y) ^ (y v (x v z)) = y v x
    ]]></literal>
    <justification jstring="[para(9(a,1),20(a,1,1))].">
      <j1 rule="para" parents="9 20"/>
    </justification>
  </clause>

  <clause id="100">
    <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="107">
    <literal><![CDATA[
      (x v y) ^ ((x ^ z) v y) = (x ^ z) v y
    ]]></literal>
    <justification jstring="[para(23(a,1),19(a,1,2)),demod(11(4))].">
      <j1 rule="para" parents="23 19"/>
      <j2 rule="demod" parents="11"/>
    </justification>
  </clause>

  <clause id="134">
    <literal><![CDATA[
      (x v y) ^ ((z ^ x) v y) = (z ^ x) v y
    ]]></literal>
    <justification jstring="[para(54(a,1),19(a,1,2)),demod(11(4))].">
      <j1 rule="para" parents="54 19"/>
      <j2 rule="demod" parents="11"/>
    </justification>
  </clause>

  <clause id="170">
    <literal><![CDATA[
      (x ^ y) v (x ^ (z v y)) = x ^ (z v y)
    ]]></literal>
    <justification jstring="[para(64(a,1),25(a,1,2)),demod(9(4))].">
      <j1 rule="para" parents="64 25"/>
      <j2 rule="demod" parents="9"/>
    </justification>
  </clause>

  <clause id="200">
    <literal><![CDATA[
      (x ^ y) v (y ^ x) = x ^ y
    ]]></literal>
    <justification jstring="[para(44(a,1),26(a,1,2))].">
      <j1 rule="para" parents="44 26"/>
    </justification>
  </clause>

  <clause id="242">
    <literal><![CDATA[
      x ^ ((y ^ ((x ^ y) v z)) v u) = x ^ (u v (y ^ (x v z)))
    ]]></literal>
    <justification jstring="[para(9(a,1),29(a,1,2))].">
      <j1 rule="para" parents="9 29"/>
    </justification>
  </clause>

  <clause id="316">
    <literal><![CDATA[
      (x ^ y) v (z v (y ^ x)) = z v (y ^ x)
    ]]></literal>
    <justification jstring="[para(200(a,1),10(a,2,2)),demod(9(4))].">
      <j1 rule="para" parents="200 10"/>
      <j2 rule="demod" parents="9"/>
    </justification>
  </clause>

  <clause id="321">
    <literal><![CDATA[
      (x ^ (y ^ z)) v (y ^ x) = x ^ y
    ]]></literal>
    <justification jstring="[para(200(a,1),23(a,2)),demod(12(3),316(6))].">
      <j1 rule="para" parents="200 23"/>
      <j2 rule="demod" parents="12 316"/>
    </justification>
  </clause>

  <clause id="2174">
    <literal><![CDATA[
      x v ((x v y) ^ (x v z)) = (x v y) ^ (x v z)
    ]]></literal>
    <justification jstring="[para(13(a,1),100(a,1,1))].">
      <j1 rule="para" parents="13 100"/>
    </justification>
  </clause>

  <clause id="2344">
    <literal><![CDATA[
      x ^ ((x ^ y) v (x ^ z)) = (x ^ y) v (x ^ z)
    ]]></literal>
    <justification jstring="[para(14(a,1),107(a,1,1))].">
      <j1 rule="para" parents="14 107"/>
    </justification>
  </clause>

  <clause id="3605">
    <literal><![CDATA[
      x ^ ((y ^ z) v (y ^ x)) = x ^ y
    ]]></literal>
    <justification jstring="[para(321(a,1),134(a,1,2)),demod(11(5),12(5),2344(4),321(8))].">
      <j1 rule="para" parents="321 134"/>
      <j2 rule="demod" parents="11 12 2344 321"/>
    </justification>
  </clause>

  <clause id="3638">
    <literal><![CDATA[
      x ^ ((y ^ z) v (z ^ x)) = x ^ z
    ]]></literal>
    <justification jstring="[para(11(a,1),3605(a,1,2,1))].">
      <j1 rule="para" parents="11 3605"/>
    </justification>
  </clause>

  <clause id="5892">
    <literal><![CDATA[
      x v ((y v x) ^ (y v z)) = x v y
    ]]></literal>
    <justification jstring="[para(80(a,1),170(a,1,2)),demod(17(5),9(4),2174(4),80(8))].">
      <j1 rule="para" parents="80 170"/>
      <j2 rule="demod" parents="17 9 2174 80"/>
    </justification>
  </clause>

  <clause id="5972">
    <literal><![CDATA[
      x v ((x v y) ^ (y v z)) = x v y
    ]]></literal>
    <justification jstring="[para(9(a,1),5892(a,1,2,1))].">
      <j1 rule="para" parents="9 5892"/>
    </justification>
  </clause>

  <clause id="6096">
    <literal><![CDATA[
      x v (y ^ (x v (z ^ y))) = x v (z ^ y)
    ]]></literal>
    <justification jstring="[para(3638(a,1),5972(a,1,2)),demod(11(3))].">
      <j1 rule="para" parents="3638 5972"/>
      <j2 rule="demod" parents="11"/>
    </justification>
  </clause>

  <clause id="12709">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(H3)
    ]]></attribute>
    <justification jstring="[para(242(a,2),16(a,1,2,2)),demod(11(7),9(11),2344(12),9(12),54(12),6096(10)),xx(a)].">
      <j1 rule="para" parents="242 16"/>
      <j2 rule="demod" parents="11 9 2344 9 54 6096"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
