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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>NE-gt.out</source>

<heading><![CDATA[
Prover9 (32) version March-2007, March 2007.
Process 21023 was started by mccune on cleo,
Mon Mar 19 17:02:11 2007
The command was "/home/mccune/bin/prover9 -f gt.in NE.in".
]]></heading>

<proof number="1" length="31" max_count="19">

<comments><![CDATA[
% Proof 1 at 4.37 (+ 0.04) seconds: E.
% Length of proof is 31.
% Level of proof is 11.
% Maximum clause weight is 19.
% Given clauses 304.
]]></comments>

  <clause id="1" type="assumption">
    <literal><![CDATA[
      (x * y) * z = x * y * z
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

  <clause id="3" type="assumption">
    <literal><![CDATA[
      x' * x = e
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

  <clause id="5">
    <literal><![CDATA[
      x' * y' * x * y = x @ y
    ]]></literal>
    <justification jstring="[copy(4),flip(a)].">
      <j1 rule="copy" parents="4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

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

  <clause id="7" type="assumption">
    <literal><![CDATA[
      A * B * C * B * A != B * A * C * A * B
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      x' * x * y = y
    ]]></literal>
    <justification jstring="[para(3(a,1),1(a,1,1)),rewrite(2(2)),flip(a)].">
      <j1 rule="para" parents="3 1"/>
      <j2 rule="rewrite" parents="2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      x' * y' * x * y * z = (x @ y) * z
    ]]></literal>
    <justification jstring="[para(5(a,1),1(a,1,1)),rewrite(1(7),1(6)),flip(a)].">
      <j1 rule="para" parents="5 1"/>
      <j2 rule="rewrite" parents="1 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      (x * y)' * z' * x * y * z = x * y @ z
    ]]></literal>
    <justification jstring="[para(1(a,1),5(a,1,2,2))].">
      <j1 rule="para" parents="1 5"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      (x @ y) * z * u = z * (x @ y) * u
    ]]></literal>
    <justification jstring="[para(6(a,1),1(a,1,1)),rewrite(1(3)),flip(a)].">
      <j1 rule="para" parents="6 1"/>
      <j2 rule="rewrite" parents="1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      x'' * e = x
    ]]></literal>
    <justification jstring="[para(3(a,1),8(a,1,2))].">
      <j1 rule="para" parents="3 8"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      x'' * (x @ y) = y' * x * y
    ]]></literal>
    <justification jstring="[para(5(a,1),8(a,1,2))].">
      <j1 rule="para" parents="5 8"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      x'' * y = x * y
    ]]></literal>
    <justification jstring="[para(8(a,1),8(a,1,2))].">
      <j1 rule="para" parents="8 8"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      x' * y * x = y * (y @ x)
    ]]></literal>
    <justification jstring="[back_rewrite(25),rewrite(28(4)),flip(a)].">
      <j1 rule="back_rewrite" parents="25"/>
      <j2 rule="rewrite" parents="28"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      x * e = x
    ]]></literal>
    <justification jstring="[back_rewrite(22),rewrite(28(4))].">
      <j1 rule="back_rewrite" parents="22"/>
      <j2 rule="rewrite" parents="28"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      x' * y * x * z = y * (y @ x) * z
    ]]></literal>
    <justification jstring="[para(9(a,1),8(a,1,2)),rewrite(28(5)),flip(a)].">
      <j1 rule="para" parents="9 8"/>
      <j2 rule="rewrite" parents="28"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      (x * y)' * y * x = x * y @ x
    ]]></literal>
    <justification jstring="[para(8(a,1),10(a,1,2))].">
      <j1 rule="para" parents="8 10"/>
    </justification>
  </clause>

  <clause id="70">
    <literal><![CDATA[
      x * y @ x = y @ x * y
    ]]></literal>
    <justification jstring="[para(9(a,1),10(a,1,2)),rewrite(8(3),6(4),8(5),8(5))].">
      <j1 rule="para" parents="9 10"/>
      <j2 rule="rewrite" parents="8 6 8 8"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      (x * y)' * y * x = y @ x * y
    ]]></literal>
    <justification jstring="[back_rewrite(68),rewrite(70(6))].">
      <j1 rule="back_rewrite" parents="68"/>
      <j2 rule="rewrite" parents="70"/>
    </justification>
  </clause>

  <clause id="76">
    <literal><![CDATA[
      x * x' = e
    ]]></literal>
    <justification jstring="[para(28(a,1),3(a,1))].">
      <j1 rule="para" parents="28 3"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[para(28(a,1),31(a,1)),rewrite(31(2)),flip(a)].">
      <j1 rule="para" parents="28 31"/>
      <j2 rule="rewrite" parents="31"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="87">
    <literal><![CDATA[
      x * y * (x * y)' = e
    ]]></literal>
    <justification jstring="[para(76(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="76 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="122">
    <literal><![CDATA[
      x * (y * x)' = y'
    ]]></literal>
    <justification jstring="[para(87(a,1),8(a,1,2)),rewrite(31(3)),flip(a)].">
      <j1 rule="para" parents="87 8"/>
      <j2 rule="rewrite" parents="31"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="137">
    <literal><![CDATA[
      (x * y)' = y' * x'
    ]]></literal>
    <justification jstring="[para(122(a,1),8(a,1,2)),flip(a)].">
      <j1 rule="para" parents="122 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="153">
    <literal><![CDATA[
      x @ y * x = x @ y
    ]]></literal>
    <justification jstring="[back_rewrite(74),rewrite(137(2),1(5),5(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="74"/>
      <j2 rule="rewrite" parents="137 1 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="174">
    <literal><![CDATA[
      x * y @ x = y @ x
    ]]></literal>
    <justification jstring="[back_rewrite(70),rewrite(153(4))].">
      <j1 rule="back_rewrite" parents="70"/>
      <j2 rule="rewrite" parents="153"/>
    </justification>
  </clause>

  <clause id="323">
    <literal><![CDATA[
      (x @ y) * y * x = x * y
    ]]></literal>
    <justification jstring="[para(174(a,1),29(a,2,2)),rewrite(1(3),8(4),6(4,R)),flip(a)].">
      <j1 rule="para" parents="174 29"/>
      <j2 rule="rewrite" parents="1 8 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="661">
    <literal><![CDATA[
      x * y * (y @ x) * z = y * x * z
    ]]></literal>
    <justification jstring="[para(47(a,1),8(a,1,2)),rewrite(82(2))].">
      <j1 rule="para" parents="47 8"/>
      <j2 rule="rewrite" parents="82"/>
    </justification>
  </clause>

  <clause id="779">
    <literal><![CDATA[
      (x @ y) * z * y * x = z * x * y
    ]]></literal>
    <justification jstring="[para(323(a,1),14(a,2,2))].">
      <j1 rule="para" parents="323 14"/>
    </justification>
  </clause>

  <clause id="8540">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[para(779(a,2),7(a,1,2,2)),rewrite(661(13)),xx(a)].">
      <j1 rule="para" parents="779 7"/>
      <j2 rule="rewrite" parents="661"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
