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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 June-2006C, June 2006.
Process 13128 was started by mccune on cleo.thornwood,
Mon Jun 19 16:41:41 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="4">
    <literal><![CDATA[
      x @ y = x ' * y ' * x * y
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      (x * y) * z = x * y * z
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      e * x = x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      x ' * x = e
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="10">
    <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="11">
    <literal><![CDATA[
      x * y * z * y * x = y * x * z * x * y
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      (A @ B) * C != C * (A @ B)
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      x ' * x * y = y
    ]]></literal>
    <justification jstring="[para(9(a,1),7(a,1,1)),demod(8(2)),flip(a)].">
      <j1 rule="para" parents="9 7"/>
      <j2 rule="demod" parents="8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      x ' * y ' * x * y * z = (x @ y) * z
    ]]></literal>
    <justification jstring="[para(10(a,1),7(a,1,1)),demod(7(7),7(6)),flip(a)].">
      <j1 rule="para" parents="10 7"/>
      <j2 rule="demod" parents="7 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      (x * y) ' * z ' * x * y * z = x * y @ z
    ]]></literal>
    <justification jstring="[para(7(a,1),10(a,1,2,2))].">
      <j1 rule="para" parents="7 10"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      x * y * z * y * x * u = y * x * z * x * y * u
    ]]></literal>
    <justification jstring="[para(11(a,1),7(a,1,1)),demod(7(5),7(4),7(3),7(2),7(9),7(8),7(7))].">
      <j1 rule="para" parents="11 7"/>
      <j2 rule="demod" parents="7 7 7 7 7 7 7"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      x * y * y * x = y * x * x * y
    ]]></literal>
    <justification jstring="[para(8(a,1),11(a,1,2,2)),demod(8(6))].">
      <j1 rule="para" parents="8 11"/>
      <j2 rule="demod" parents="8"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      x * y * (x * y) ' * y * x = y * x * e
    ]]></literal>
    <justification jstring="[para(9(a,1),11(a,1,2,2)),flip(a)].">
      <j1 rule="para" parents="9 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      x ' ' * e = x
    ]]></literal>
    <justification jstring="[para(9(a,1),13(a,1,2))].">
      <j1 rule="para" parents="9 13"/>
    </justification>
  </clause>

  <clause id="37">
    <literal><![CDATA[
      x ' ' * y = x * y
    ]]></literal>
    <justification jstring="[para(13(a,1),13(a,1,2))].">
      <j1 rule="para" parents="13 13"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      x * e = x
    ]]></literal>
    <justification jstring="[back_demod(29),demod(37(4))].">
      <j1 rule="back_demod" parents="29"/>
      <j2 rule="demod" parents="37"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      x * y * (x * y) ' * y * x = y * x
    ]]></literal>
    <justification jstring="[back_demod(25),demod(40(8))].">
      <j1 rule="back_demod" parents="25"/>
      <j2 rule="demod" parents="40"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      e ' = e
    ]]></literal>
    <justification jstring="[para(40(a,1),9(a,1))].">
      <j1 rule="para" parents="40 9"/>
    </justification>
  </clause>

  <clause id="70">
    <literal><![CDATA[
      (x * y) ' * y * x = x * y @ x
    ]]></literal>
    <justification jstring="[para(13(a,1),15(a,1,2))].">
      <j1 rule="para" parents="13 15"/>
    </justification>
  </clause>

  <clause id="76">
    <literal><![CDATA[
      x * y * (x * y @ x) = y * x
    ]]></literal>
    <justification jstring="[back_demod(42),demod(70(4))].">
      <j1 rule="back_demod" parents="42"/>
      <j2 rule="demod" parents="70"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      x * x ' = e
    ]]></literal>
    <justification jstring="[para(37(a,1),9(a,1))].">
      <j1 rule="para" parents="37 9"/>
    </justification>
  </clause>

  <clause id="81">
    <literal><![CDATA[
      x ' * y * x * y ' = x @ y '
    ]]></literal>
    <justification jstring="[para(37(a,1),10(a,1,2))].">
      <j1 rule="para" parents="37 10"/>
    </justification>
  </clause>

  <clause id="84">
    <literal><![CDATA[
      x * x ' * y = y
    ]]></literal>
    <justification jstring="[para(37(a,1),13(a,1))].">
      <j1 rule="para" parents="37 13"/>
    </justification>
  </clause>

  <clause id="88">
    <literal><![CDATA[
      x ' ' = x
    ]]></literal>
    <justification jstring="[para(37(a,1),40(a,1)),demod(40(2)),flip(a)].">
      <j1 rule="para" parents="37 40"/>
      <j2 rule="demod" parents="40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      x * y * (x * y) ' = e
    ]]></literal>
    <justification jstring="[para(79(a,1),7(a,1)),flip(a)].">
      <j1 rule="para" parents="79 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="94">
    <literal><![CDATA[
      x * y @ y ' = x * y @ x
    ]]></literal>
    <justification jstring="[para(79(a,1),15(a,1,2,2,2)),demod(88(4),40(4),70(4)),flip(a)].">
      <j1 rule="para" parents="79 15"/>
      <j2 rule="demod" parents="88 40 70"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="99">
    <literal><![CDATA[
      x ' * y ' * z * (y @ x) = y ' * x ' * z
    ]]></literal>
    <justification jstring="[para(10(a,1),19(a,1,2,2,2)),demod(13(12),9(10),40(10))].">
      <j1 rule="para" parents="10 19"/>
      <j2 rule="demod" parents="13 9 40"/>
    </justification>
  </clause>

  <clause id="123">
    <literal><![CDATA[
      x * (y * x) ' = y '
    ]]></literal>
    <justification jstring="[para(91(a,1),13(a,1,2)),demod(40(3)),flip(a)].">
      <j1 rule="para" parents="91 13"/>
      <j2 rule="demod" parents="40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="125">
    <literal><![CDATA[
      (x @ y) * z * (x ' * y ' * x * y * z) ' = e
    ]]></literal>
    <justification jstring="[para(14(a,2),91(a,1,2,2,1))].">
      <j1 rule="para" parents="14 91"/>
    </justification>
  </clause>

  <clause id="136">
    <literal><![CDATA[
      (x * y) ' = y ' * x '
    ]]></literal>
    <justification jstring="[para(123(a,1),13(a,1,2)),flip(a)].">
      <j1 rule="para" parents="123 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="137">
    <literal><![CDATA[
      (x @ y) ' = y @ x
    ]]></literal>
    <justification jstring="[para(14(a,2),123(a,1,2,1)),demod(136(7),136(5),136(3),136(2),7(5),88(7),7(6),7(5),88(8),7(7),7(6),7(5),10(6),84(4)),flip(a)].">
      <j1 rule="para" parents="14 123"/>
      <j2 rule="demod" parents="136 136 136 136 7 88 7 7 88 7 7 7 10 84"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="146">
    <literal><![CDATA[
      (x @ y) * (y @ x) = e
    ]]></literal>
    <justification jstring="[back_demod(125),demod(136(8),136(6),136(4),136(3),7(6),88(8),7(7),7(6),88(9),7(8),7(7),7(6),10(7),84(5))].">
      <j1 rule="back_demod" parents="125"/>
      <j2 rule="demod" parents="136 136 136 136 7 88 7 7 88 7 7 7 10 84"/>
    </justification>
  </clause>

  <clause id="152">
    <literal><![CDATA[
      x * y @ x = y @ x
    ]]></literal>
    <justification jstring="[back_demod(70),demod(136(2),7(5),10(5)),flip(a)].">
      <j1 rule="back_demod" parents="70"/>
      <j2 rule="demod" parents="136 7 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="165">
    <literal><![CDATA[
      x * y @ y ' = y @ x
    ]]></literal>
    <justification jstring="[back_demod(94),demod(152(5))].">
      <j1 rule="back_demod" parents="94"/>
      <j2 rule="demod" parents="152"/>
    </justification>
  </clause>

  <clause id="166">
    <literal><![CDATA[
      x * y * (y @ x) = y * x
    ]]></literal>
    <justification jstring="[back_demod(76),demod(152(2))].">
      <j1 rule="back_demod" parents="76"/>
      <j2 rule="demod" parents="152"/>
    </justification>
  </clause>

  <clause id="187">
    <literal><![CDATA[
      x * y @ x ' = y @ x '
    ]]></literal>
    <justification jstring="[para(13(a,1),152(a,1,1)),flip(a)].">
      <j1 rule="para" parents="13 152"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="191">
    <literal><![CDATA[
      x ' * y @ x = y @ x
    ]]></literal>
    <justification jstring="[para(84(a,1),152(a,1,1)),flip(a)].">
      <j1 rule="para" parents="84 152"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="193">
    <literal><![CDATA[
      x @ x * y = x @ y
    ]]></literal>
    <justification jstring="[para(152(a,1),137(a,1,1)),demod(137(2)),flip(a)].">
      <j1 rule="para" parents="152 137"/>
      <j2 rule="demod" parents="137"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="210">
    <literal><![CDATA[
      x ' @ y ' * x * y = x ' @ (x @ y)
    ]]></literal>
    <justification jstring="[para(10(a,1),193(a,1,2)),flip(a)].">
      <j1 rule="para" parents="10 193"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="226">
    <literal><![CDATA[
      x ' @ y * x = y @ x
    ]]></literal>
    <justification jstring="[para(165(a,1),137(a,1,1)),demod(137(2)),flip(a)].">
      <j1 rule="para" parents="165 137"/>
      <j2 rule="demod" parents="137"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="240">
    <literal><![CDATA[
      (x @ y) * y * x = x * y
    ]]></literal>
    <justification jstring="[para(23(a,1),14(a,1,2,2)),demod(13(6),13(4)),flip(a)].">
      <j1 rule="para" parents="23 14"/>
      <j2 rule="demod" parents="13 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="260">
    <literal><![CDATA[
      x @ y * x ' = y @ x '
    ]]></literal>
    <justification jstring="[para(88(a,1),226(a,1,1))].">
      <j1 rule="para" parents="88 226"/>
    </justification>
  </clause>

  <clause id="364">
    <literal><![CDATA[
      (x @ y) @ y * x = e
    ]]></literal>
    <justification jstring="[para(240(a,1),10(a,1,2,2)),demod(137(2),136(3),7(6),10(6),146(3)),flip(a)].">
      <j1 rule="para" parents="240 10"/>
      <j2 rule="demod" parents="137 136 7 10 146"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="366">
    <literal><![CDATA[
      (x ' @ (x @ y)) * (y @ x ') = x @ y
    ]]></literal>
    <justification jstring="[para(10(a,1),240(a,2)),demod(210(5),7(8),7(7),81(8))].">
      <j1 rule="para" parents="10 240"/>
      <j2 rule="demod" parents="210 7 7 81"/>
    </justification>
  </clause>

  <clause id="404">
    <literal><![CDATA[
      (x @ y) @ x = e
    ]]></literal>
    <justification jstring="[para(84(a,1),364(a,1,2)),demod(191(3))].">
      <j1 rule="para" parents="84 364"/>
      <j2 rule="demod" parents="191"/>
    </justification>
  </clause>

  <clause id="422">
    <literal><![CDATA[
      x @ (x @ y) = e
    ]]></literal>
    <justification jstring="[para(404(a,1),137(a,1,1)),demod(62(2)),flip(a)].">
      <j1 rule="para" parents="404 137"/>
      <j2 rule="demod" parents="62"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="425">
    <literal><![CDATA[
      (x @ y) * x = x * (x @ y)
    ]]></literal>
    <justification jstring="[para(404(a,1),166(a,1,2,2)),demod(40(3)),flip(a)].">
      <j1 rule="para" parents="404 166"/>
      <j2 rule="demod" parents="40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="462">
    <literal><![CDATA[
      (x @ y) @ x ' = e
    ]]></literal>
    <justification jstring="[para(422(a,1),165(a,2)),demod(425(2),187(4))].">
      <j1 rule="para" parents="422 165"/>
      <j2 rule="demod" parents="425 187"/>
    </justification>
  </clause>

  <clause id="478">
    <literal><![CDATA[
      x ' @ (x @ y) = e
    ]]></literal>
    <justification jstring="[para(462(a,1),137(a,1,1)),demod(62(2)),flip(a)].">
      <j1 rule="para" parents="462 137"/>
      <j2 rule="demod" parents="62"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="486">
    <literal><![CDATA[
      x @ y ' = y @ x
    ]]></literal>
    <justification jstring="[back_demod(366),demod(478(3),8(4))].">
      <j1 rule="back_demod" parents="366"/>
      <j2 rule="demod" parents="478 8"/>
    </justification>
  </clause>

  <clause id="583">
    <literal><![CDATA[
      (B @ A ') * C != C * (A @ B)
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[para(486(a,2),12(a,1,1))].">
      <j1 rule="para" parents="486 12"/>
    </justification>
  </clause>

  <clause id="591">
    <literal><![CDATA[
      x ' @ y ' = x @ y
    ]]></literal>
    <justification jstring="[para(486(a,2),152(a,1)),demod(136(2),260(4))].">
      <j1 rule="para" parents="486 152"/>
      <j2 rule="demod" parents="136 260"/>
    </justification>
  </clause>

  <clause id="656">
    <literal><![CDATA[
      (B @ A ') * C != C * (B @ A ')
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[para(486(a,2),583(a,2,2))].">
      <j1 rule="para" parents="486 583"/>
    </justification>
  </clause>

  <clause id="2416">
    <literal><![CDATA[
      (x @ y) * z = z * (x @ y)
    ]]></literal>
    <justification jstring="[para(99(a,2),14(a,1,2,2)),demod(88(2),88(2),84(7),84(5),591(5)),flip(a)].">
      <j1 rule="para" parents="99 14"/>
      <j2 rule="demod" parents="88 88 84 84 591"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2417">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[resolve(2416,a,656,a)].">
      <j1 rule="resolve" parents="2416 656"/>
    </justification>
  </clause>

</proof>

</proofs>
