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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

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

<heading><![CDATA[
Prover9 (32) version August-2006A, August 2006.
Process 11363 was started by mccune on cleo.thornwood,
Sat Aug 12 21:00:03 2006
The command was "/home/mccune/bin/prover9 -f gt.in EN.in".
]]></heading>

<proof number="1" length="55" max_count="23">

<comments><![CDATA[
% Proof 1 at 0.37 (+ 0.01) seconds: N.
% Length of proof is 55.
% Level of proof is 16.
% Maximum clause weight is 23.
% Given clauses 151.
]]></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 * y * x = y * x * z * x * y
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      (A @ B) * C != C * (A @ B)
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></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 * y * x * u = y * x * z * x * y * u
    ]]></literal>
    <justification jstring="[para(6(a,1),1(a,1,1)),rewrite(1(5),1(4),1(3),1(2),1(9),1(8),1(7))].">
      <j1 rule="para" parents="6 1"/>
      <j2 rule="rewrite" parents="1 1 1 1 1 1 1"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      x * y * y * x = y * x * x * y
    ]]></literal>
    <justification jstring="[para(2(a,1),6(a,1,2,2)),rewrite(2(6))].">
      <j1 rule="para" parents="2 6"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x * y * (x * y)' * y * x = y * x * e
    ]]></literal>
    <justification jstring="[para(3(a,1),6(a,1,2,2)),flip(a)].">
      <j1 rule="para" parents="3 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="24">
    <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="32">
    <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="35">
    <literal><![CDATA[
      x * e = x
    ]]></literal>
    <justification jstring="[back_rewrite(24),rewrite(32(4))].">
      <j1 rule="back_rewrite" parents="24"/>
      <j2 rule="rewrite" parents="32"/>
    </justification>
  </clause>

  <clause id="37">
    <literal><![CDATA[
      x * y * (x * y)' * y * x = y * x
    ]]></literal>
    <justification jstring="[back_rewrite(20),rewrite(35(8))].">
      <j1 rule="back_rewrite" parents="20"/>
      <j2 rule="rewrite" parents="35"/>
    </justification>
  </clause>

  <clause id="57">
    <literal><![CDATA[
      e' = e
    ]]></literal>
    <justification jstring="[para(35(a,1),3(a,1))].">
      <j1 rule="para" parents="35 3"/>
    </justification>
  </clause>

  <clause id="65">
    <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="71">
    <literal><![CDATA[
      x * y * (x * y @ x) = y * x
    ]]></literal>
    <justification jstring="[back_rewrite(37),rewrite(65(4))].">
      <j1 rule="back_rewrite" parents="37"/>
      <j2 rule="rewrite" parents="65"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      x * x' = e
    ]]></literal>
    <justification jstring="[para(32(a,1),3(a,1))].">
      <j1 rule="para" parents="32 3"/>
    </justification>
  </clause>

  <clause id="76">
    <literal><![CDATA[
      x' * y * x * y' = x @ y'
    ]]></literal>
    <justification jstring="[para(32(a,1),5(a,1,2))].">
      <j1 rule="para" parents="32 5"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      x * x' * y = y
    ]]></literal>
    <justification jstring="[para(32(a,1),8(a,1))].">
      <j1 rule="para" parents="32 8"/>
    </justification>
  </clause>

  <clause id="83">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[para(32(a,1),35(a,1)),rewrite(35(2)),flip(a)].">
      <j1 rule="para" parents="32 35"/>
      <j2 rule="rewrite" parents="35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="86">
    <literal><![CDATA[
      x * y * (x * y)' = e
    ]]></literal>
    <justification jstring="[para(74(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="74 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      x * y @ y' = x * y @ x
    ]]></literal>
    <justification jstring="[para(74(a,1),10(a,1,2,2,2)),rewrite(83(4),35(4),65(4)),flip(a)].">
      <j1 rule="para" parents="74 10"/>
      <j2 rule="rewrite" parents="83 35 65"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="94">
    <literal><![CDATA[
      x' * y' * z * (y @ x) = y' * x' * z
    ]]></literal>
    <justification jstring="[para(5(a,1),14(a,1,2,2,2)),rewrite(8(12),3(10),35(10))].">
      <j1 rule="para" parents="5 14"/>
      <j2 rule="rewrite" parents="8 3 35"/>
    </justification>
  </clause>

  <clause id="118">
    <literal><![CDATA[
      x * (y * x)' = y'
    ]]></literal>
    <justification jstring="[para(86(a,1),8(a,1,2)),rewrite(35(3)),flip(a)].">
      <j1 rule="para" parents="86 8"/>
      <j2 rule="rewrite" parents="35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="120">
    <literal><![CDATA[
      (x @ y) * z * (x' * y' * x * y * z)' = e
    ]]></literal>
    <justification jstring="[para(9(a,2),86(a,1,2,2,1))].">
      <j1 rule="para" parents="9 86"/>
    </justification>
  </clause>

  <clause id="131">
    <literal><![CDATA[
      (x * y)' = y' * x'
    ]]></literal>
    <justification jstring="[para(118(a,1),8(a,1,2)),flip(a)].">
      <j1 rule="para" parents="118 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="132">
    <literal><![CDATA[
      (x @ y)' = y @ x
    ]]></literal>
    <justification jstring="[para(9(a,2),118(a,1,2,1)),rewrite(131(7),131(5),131(3),131(2),1(5),83(7),1(6),1(5),83(8),1(7),1(6),1(5),5(6),79(4)),flip(a)].">
      <j1 rule="para" parents="9 118"/>
      <j2 rule="rewrite" parents="131 131 131 131 1 83 1 1 83 1 1 1 5 79"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="141">
    <literal><![CDATA[
      (x @ y) * (y @ x) = e
    ]]></literal>
    <justification jstring="[back_rewrite(120),rewrite(131(8),131(6),131(4),131(3),1(6),83(8),1(7),1(6),83(9),1(8),1(7),1(6),5(7),79(5))].">
      <j1 rule="back_rewrite" parents="120"/>
      <j2 rule="rewrite" parents="131 131 131 131 1 83 1 1 83 1 1 1 5 79"/>
    </justification>
  </clause>

  <clause id="147">
    <literal><![CDATA[
      x * y @ x = y @ x
    ]]></literal>
    <justification jstring="[back_rewrite(65),rewrite(131(2),1(5),5(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="65"/>
      <j2 rule="rewrite" parents="131 1 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="160">
    <literal><![CDATA[
      x * y @ y' = y @ x
    ]]></literal>
    <justification jstring="[back_rewrite(89),rewrite(147(5))].">
      <j1 rule="back_rewrite" parents="89"/>
      <j2 rule="rewrite" parents="147"/>
    </justification>
  </clause>

  <clause id="161">
    <literal><![CDATA[
      x * y * (y @ x) = y * x
    ]]></literal>
    <justification jstring="[back_rewrite(71),rewrite(147(2))].">
      <j1 rule="back_rewrite" parents="71"/>
      <j2 rule="rewrite" parents="147"/>
    </justification>
  </clause>

  <clause id="182">
    <literal><![CDATA[
      x * y @ x' = y @ x'
    ]]></literal>
    <justification jstring="[para(8(a,1),147(a,1,1)),flip(a)].">
      <j1 rule="para" parents="8 147"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="186">
    <literal><![CDATA[
      x' * y @ x = y @ x
    ]]></literal>
    <justification jstring="[para(79(a,1),147(a,1,1)),flip(a)].">
      <j1 rule="para" parents="79 147"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="188">
    <literal><![CDATA[
      x @ x * y = x @ y
    ]]></literal>
    <justification jstring="[para(147(a,1),132(a,1,1)),rewrite(132(2)),flip(a)].">
      <j1 rule="para" parents="147 132"/>
      <j2 rule="rewrite" parents="132"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="205">
    <literal><![CDATA[
      x' @ y' * x * y = x' @ (x @ y)
    ]]></literal>
    <justification jstring="[para(5(a,1),188(a,1,2)),flip(a)].">
      <j1 rule="para" parents="5 188"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="221">
    <literal><![CDATA[
      x' @ y * x = y @ x
    ]]></literal>
    <justification jstring="[para(160(a,1),132(a,1,1)),rewrite(132(2)),flip(a)].">
      <j1 rule="para" parents="160 132"/>
      <j2 rule="rewrite" parents="132"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="235">
    <literal><![CDATA[
      (x @ y) * y * x = x * y
    ]]></literal>
    <justification jstring="[para(18(a,1),9(a,1,2,2)),rewrite(8(6),8(4)),flip(a)].">
      <j1 rule="para" parents="18 9"/>
      <j2 rule="rewrite" parents="8 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="255">
    <literal><![CDATA[
      x @ y * x' = y @ x'
    ]]></literal>
    <justification jstring="[para(83(a,1),221(a,1,1))].">
      <j1 rule="para" parents="83 221"/>
    </justification>
  </clause>

  <clause id="359">
    <literal><![CDATA[
      (x @ y) @ y * x = e
    ]]></literal>
    <justification jstring="[para(235(a,1),5(a,1,2,2)),rewrite(132(2),131(3),1(6),5(6),141(3)),flip(a)].">
      <j1 rule="para" parents="235 5"/>
      <j2 rule="rewrite" parents="132 131 1 5 141"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="361">
    <literal><![CDATA[
      (x' @ (x @ y)) * (y @ x') = x @ y
    ]]></literal>
    <justification jstring="[para(5(a,1),235(a,2)),rewrite(205(5),1(8),1(7),76(8))].">
      <j1 rule="para" parents="5 235"/>
      <j2 rule="rewrite" parents="205 1 1 76"/>
    </justification>
  </clause>

  <clause id="399">
    <literal><![CDATA[
      (x @ y) @ x = e
    ]]></literal>
    <justification jstring="[para(79(a,1),359(a,1,2)),rewrite(186(3))].">
      <j1 rule="para" parents="79 359"/>
      <j2 rule="rewrite" parents="186"/>
    </justification>
  </clause>

  <clause id="417">
    <literal><![CDATA[
      x @ (x @ y) = e
    ]]></literal>
    <justification jstring="[para(399(a,1),132(a,1,1)),rewrite(57(2)),flip(a)].">
      <j1 rule="para" parents="399 132"/>
      <j2 rule="rewrite" parents="57"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="420">
    <literal><![CDATA[
      (x @ y) * x = x * (x @ y)
    ]]></literal>
    <justification jstring="[para(399(a,1),161(a,1,2,2)),rewrite(35(3)),flip(a)].">
      <j1 rule="para" parents="399 161"/>
      <j2 rule="rewrite" parents="35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="457">
    <literal><![CDATA[
      (x @ y) @ x' = e
    ]]></literal>
    <justification jstring="[para(417(a,1),160(a,2)),rewrite(420(2),182(4))].">
      <j1 rule="para" parents="417 160"/>
      <j2 rule="rewrite" parents="420 182"/>
    </justification>
  </clause>

  <clause id="473">
    <literal><![CDATA[
      x' @ (x @ y) = e
    ]]></literal>
    <justification jstring="[para(457(a,1),132(a,1,1)),rewrite(57(2)),flip(a)].">
      <j1 rule="para" parents="457 132"/>
      <j2 rule="rewrite" parents="57"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="481">
    <literal><![CDATA[
      x @ y' = y @ x
    ]]></literal>
    <justification jstring="[back_rewrite(361),rewrite(473(3),2(4))].">
      <j1 rule="back_rewrite" parents="361"/>
      <j2 rule="rewrite" parents="473 2"/>
    </justification>
  </clause>

  <clause id="578">
    <literal><![CDATA[
      (B @ A') * C != C * (A @ B)
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[para(481(a,2),7(a,1,1))].">
      <j1 rule="para" parents="481 7"/>
    </justification>
  </clause>

  <clause id="586">
    <literal><![CDATA[
      x' @ y' = x @ y
    ]]></literal>
    <justification jstring="[para(481(a,2),147(a,1)),rewrite(131(2),255(4))].">
      <j1 rule="para" parents="481 147"/>
      <j2 rule="rewrite" parents="131 255"/>
    </justification>
  </clause>

  <clause id="651">
    <literal><![CDATA[
      (B @ A') * C != C * (B @ A')
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[para(481(a,2),578(a,2,2))].">
      <j1 rule="para" parents="481 578"/>
    </justification>
  </clause>

  <clause id="2411">
    <literal><![CDATA[
      (x @ y) * z = z * (x @ y)
    ]]></literal>
    <justification jstring="[para(94(a,2),9(a,1,2,2)),rewrite(83(2),83(2),79(7),79(5),586(5)),flip(a)].">
      <j1 rule="para" parents="94 9"/>
      <j2 rule="rewrite" parents="83 83 79 79 586"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2412">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[resolve(2411,a,651,a)].">
      <j1 rule="resolve" parents="2411 651"/>
    </justification>
  </clause>

</proof>

</proofs>
