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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

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

<heading><![CDATA[
Prover9 (32) version August-2006A, August 2006.
Process 11348 was started by mccune on cleo.thornwood,
Sat Aug 12 20:59:46 2006
The command was "/home/mccune/bin/prover9 -f gt.in AE.in".
]]></heading>

<proof number="1" length="42" max_count="19">

<comments><![CDATA[
% Proof 1 at 4.44 (+ 0.03) seconds: E.
% Length of proof is 42.
% Level of proof is 15.
% Maximum clause weight is 19.
% Given clauses 319.
]]></comments>

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

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

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

  <clause id="4">
    <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">
    <literal><![CDATA[
      (x @ y) @ z = u @ (v @ w)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7">
    <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="11">
    <literal><![CDATA[
      e @ x = e
    ]]></literal>
    <justification jstring="[para(2(a,1),5(a,1,2,2)),rewrite(3(4),3(4)),flip(a)].">
      <j1 rule="para" parents="2 5"/>
      <j2 rule="rewrite" parents="3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      x @ (y @ z) = e
    ]]></literal>
    <justification jstring="[para(11(a,1),6(a,1,1)),rewrite(11(2)),flip(a)].">
      <j1 rule="para" parents="11 6"/>
      <j2 rule="rewrite" parents="11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      (x @ y) @ z = e
    ]]></literal>
    <justification jstring="[para(11(a,1),6(a,2))].">
      <j1 rule="para" parents="11 6"/>
    </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="26">
    <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="27">
    <literal><![CDATA[
      x' * y * x = y * (y @ x)
    ]]></literal>
    <justification jstring="[back_rewrite(25),rewrite(26(4)),flip(a)].">
      <j1 rule="back_rewrite" parents="25"/>
      <j2 rule="rewrite" parents="26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      x * e = x
    ]]></literal>
    <justification jstring="[back_rewrite(22),rewrite(26(4))].">
      <j1 rule="back_rewrite" parents="22"/>
      <j2 rule="rewrite" parents="26"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      x' * y * x * z = y * (y @ x) * z
    ]]></literal>
    <justification jstring="[para(9(a,1),8(a,1,2)),rewrite(26(5)),flip(a)].">
      <j1 rule="para" parents="9 8"/>
      <j2 rule="rewrite" parents="26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      x' * y * x * z = (x @ y') * y * z
    ]]></literal>
    <justification jstring="[para(8(a,1),9(a,1,2,2,2)),rewrite(26(5))].">
      <j1 rule="para" parents="8 9"/>
      <j2 rule="rewrite" parents="26"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      x * x' = e
    ]]></literal>
    <justification jstring="[para(26(a,1),3(a,1))].">
      <j1 rule="para" parents="26 3"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      x * x' * y = y
    ]]></literal>
    <justification jstring="[para(26(a,1),8(a,1))].">
      <j1 rule="para" parents="26 8"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[para(26(a,1),29(a,1)),rewrite(29(2)),flip(a)].">
      <j1 rule="para" parents="26 29"/>
      <j2 rule="rewrite" parents="29"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      x * y * (x * y)' = e
    ]]></literal>
    <justification jstring="[para(50(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="50 1"/>
      <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 = x @ y * x
    ]]></literal>
    <justification jstring="[para(9(a,1),10(a,1,2)),rewrite(8(3),8(8))].">
      <j1 rule="para" parents="9 10"/>
      <j2 rule="rewrite" parents="8 8"/>
    </justification>
  </clause>

  <clause id="83">
    <literal><![CDATA[
      x * (y * x)' = y'
    ]]></literal>
    <justification jstring="[para(59(a,1),8(a,1,2)),rewrite(29(3)),flip(a)].">
      <j1 rule="para" parents="59 8"/>
      <j2 rule="rewrite" parents="29"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="93">
    <literal><![CDATA[
      (x * y)' = y' * x'
    ]]></literal>
    <justification jstring="[para(83(a,1),8(a,1,2)),flip(a)].">
      <j1 rule="para" parents="83 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="94">
    <literal><![CDATA[
      (x @ y)' = y @ x
    ]]></literal>
    <justification jstring="[para(9(a,2),83(a,1,2,1)),rewrite(93(7),93(5),93(3),93(2),1(5),55(7),1(6),1(5),55(8),1(7),1(6),1(5),5(6),54(4)),flip(a)].">
      <j1 rule="para" parents="9 83"/>
      <j2 rule="rewrite" parents="93 93 93 93 1 55 1 1 55 1 1 1 5 54"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="104">
    <literal><![CDATA[
      x * y @ x = y @ x
    ]]></literal>
    <justification jstring="[back_rewrite(68),rewrite(93(2),1(5),5(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="68"/>
      <j2 rule="rewrite" parents="93 1 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="123">
    <literal><![CDATA[
      x' * (x @ y) * x = x @ y * x
    ]]></literal>
    <justification jstring="[back_rewrite(70),rewrite(104(3))].">
      <j1 rule="back_rewrite" parents="70"/>
      <j2 rule="rewrite" parents="104"/>
    </justification>
  </clause>

  <clause id="132">
    <literal><![CDATA[
      x' * (y @ z) * x = y @ z
    ]]></literal>
    <justification jstring="[para(18(a,1),27(a,2,2)),rewrite(29(7))].">
      <j1 rule="para" parents="18 27"/>
      <j2 rule="rewrite" parents="29"/>
    </justification>
  </clause>

  <clause id="139">
    <literal><![CDATA[
      x @ y * x = x @ y
    ]]></literal>
    <justification jstring="[back_rewrite(123),rewrite(132(4)),flip(a)].">
      <j1 rule="back_rewrite" parents="123"/>
      <j2 rule="rewrite" parents="132"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="145">
    <literal><![CDATA[
      x' * y @ x = y @ x
    ]]></literal>
    <justification jstring="[para(54(a,1),104(a,1,1)),flip(a)].">
      <j1 rule="para" parents="54 104"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="153">
    <literal><![CDATA[
      x' * y @ y = y @ x
    ]]></literal>
    <justification jstring="[para(54(a,1),139(a,1,2)),rewrite(145(6))].">
      <j1 rule="para" parents="54 139"/>
      <j2 rule="rewrite" parents="145"/>
    </justification>
  </clause>

  <clause id="156">
    <literal><![CDATA[
      x * y @ y = x @ y
    ]]></literal>
    <justification jstring="[para(139(a,1),94(a,1,1)),rewrite(94(2)),flip(a)].">
      <j1 rule="para" parents="139 94"/>
      <j2 rule="rewrite" parents="94"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="157">
    <literal><![CDATA[
      x' @ y = y @ x
    ]]></literal>
    <justification jstring="[back_rewrite(153),rewrite(156(3))].">
      <j1 rule="back_rewrite" parents="153"/>
      <j2 rule="rewrite" parents="156"/>
    </justification>
  </clause>

  <clause id="178">
    <literal><![CDATA[
      x' @ y' = x @ y
    ]]></literal>
    <justification jstring="[para(157(a,2),157(a,1))].">
      <j1 rule="para" parents="157 157"/>
    </justification>
  </clause>

  <clause id="338">
    <literal><![CDATA[
      x * y * (y @ x) * z = y * x * z
    ]]></literal>
    <justification jstring="[para(42(a,1),8(a,1,2)),rewrite(55(2))].">
      <j1 rule="para" parents="42 8"/>
      <j2 rule="rewrite" parents="55"/>
    </justification>
  </clause>

  <clause id="438">
    <literal><![CDATA[
      (x @ y) * y * x = x * y
    ]]></literal>
    <justification jstring="[para(3(a,1),44(a,1,2,2)),rewrite(55(2),29(2),178(4)),flip(a)].">
      <j1 rule="para" parents="3 44"/>
      <j2 rule="rewrite" parents="55 29 178"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="787">
    <literal><![CDATA[
      (x @ y) * z * y * x = z * x * y
    ]]></literal>
    <justification jstring="[para(438(a,1),42(a,1,2,2)),rewrite(94(2),16(6),2(7))].">
      <j1 rule="para" parents="438 42"/>
      <j2 rule="rewrite" parents="94 16 2"/>
    </justification>
  </clause>

  <clause id="9892">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[para(787(a,2),7(a,1,2,2)),rewrite(338(13)),xx(a)].">
      <j1 rule="para" parents="787 7"/>
      <j2 rule="rewrite" parents="338"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
