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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

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

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13115 was started by mccune on cleo.thornwood,
Mon Jun 19 16:41:29 2006
The command was "/home/mccune/bin/prover9 -f gt.in AN.in".
]]></heading>

<proof number="1" length="43" max_count="20">

<comments><![CDATA[
% Proof 1 at 0.03 (+ 0.01) seconds: N.
% Length of proof is 43.
% Level of proof is 14.
% Maximum clause weight is 20.
% Given clauses 56.
]]></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 = u @ (v @ w)
    ]]></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="16">
    <literal><![CDATA[
      e @ x = e
    ]]></literal>
    <justification jstring="[para(8(a,1),10(a,1,2,2)),demod(9(4),9(4)),flip(a)].">
      <j1 rule="para" parents="8 10"/>
      <j2 rule="demod" parents="9 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      x @ (y @ z) = e
    ]]></literal>
    <justification jstring="[para(16(a,1),11(a,1,1)),demod(16(2)),flip(a)].">
      <j1 rule="para" parents="16 11"/>
      <j2 rule="demod" parents="16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="27">
    <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="31">
    <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="34">
    <literal><![CDATA[
      x * e = x
    ]]></literal>
    <justification jstring="[back_demod(27),demod(31(4))].">
      <j1 rule="back_demod" parents="27"/>
      <j2 rule="demod" parents="31"/>
    </justification>
  </clause>

  <clause id="48">
    <literal><![CDATA[
      x ' * y * x * z = y * (y @ x) * z
    ]]></literal>
    <justification jstring="[para(14(a,1),13(a,1,2)),demod(31(5)),flip(a)].">
      <j1 rule="para" parents="14 13"/>
      <j2 rule="demod" parents="31"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="56">
    <literal><![CDATA[
      x * x ' = e
    ]]></literal>
    <justification jstring="[para(31(a,1),9(a,1))].">
      <j1 rule="para" parents="31 9"/>
    </justification>
  </clause>

  <clause id="60">
    <literal><![CDATA[
      x * x ' * y = y
    ]]></literal>
    <justification jstring="[para(31(a,1),13(a,1))].">
      <j1 rule="para" parents="31 13"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      x ' ' = x
    ]]></literal>
    <justification jstring="[para(31(a,1),34(a,1)),demod(34(2)),flip(a)].">
      <j1 rule="para" parents="31 34"/>
      <j2 rule="demod" parents="34"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      x * y * (x * y) ' = e
    ]]></literal>
    <justification jstring="[para(56(a,1),7(a,1)),flip(a)].">
      <j1 rule="para" parents="56 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="74">
    <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 = x @ y * x
    ]]></literal>
    <justification jstring="[para(14(a,1),15(a,1,2)),demod(13(3),13(8))].">
      <j1 rule="para" parents="14 15"/>
      <j2 rule="demod" parents="13 13"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      x * (y * x) ' = y '
    ]]></literal>
    <justification jstring="[para(65(a,1),13(a,1,2)),demod(34(3)),flip(a)].">
      <j1 rule="para" parents="65 13"/>
      <j2 rule="demod" parents="34"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      (x @ y) * z * (x ' * y ' * x * y * z) ' = e
    ]]></literal>
    <justification jstring="[para(14(a,2),65(a,1,2,2,1))].">
      <j1 rule="para" parents="14 65"/>
    </justification>
  </clause>

  <clause id="118">
    <literal><![CDATA[
      (x * y) ' = y ' * x '
    ]]></literal>
    <justification jstring="[para(89(a,1),13(a,1,2)),flip(a)].">
      <j1 rule="para" parents="89 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="119">
    <literal><![CDATA[
      (x @ y) ' = y @ x
    ]]></literal>
    <justification jstring="[para(14(a,2),89(a,1,2,1)),demod(118(7),118(5),118(3),118(2),7(5),61(7),7(6),7(5),61(8),7(7),7(6),7(5),10(6),60(4)),flip(a)].">
      <j1 rule="para" parents="14 89"/>
      <j2 rule="demod" parents="118 118 118 118 7 61 7 7 61 7 7 7 10 60"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="132">
    <literal><![CDATA[
      (x @ y) * (y @ x) = e
    ]]></literal>
    <justification jstring="[back_demod(91),demod(118(8),118(6),118(4),118(3),7(6),61(8),7(7),7(6),61(9),7(8),7(7),7(6),10(7),60(5))].">
      <j1 rule="back_demod" parents="91"/>
      <j2 rule="demod" parents="118 118 118 118 7 61 7 7 61 7 7 7 10 60"/>
    </justification>
  </clause>

  <clause id="136">
    <literal><![CDATA[
      x * y @ x = y @ x
    ]]></literal>
    <justification jstring="[back_demod(74),demod(118(2),7(5),10(5)),flip(a)].">
      <j1 rule="back_demod" parents="74"/>
      <j2 rule="demod" parents="118 7 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="153">
    <literal><![CDATA[
      x ' * (x @ y) * x = x @ y * x
    ]]></literal>
    <justification jstring="[back_demod(76),demod(136(3))].">
      <j1 rule="back_demod" parents="76"/>
      <j2 rule="demod" parents="136"/>
    </justification>
  </clause>

  <clause id="159">
    <literal><![CDATA[
      x ' * (y @ z) * x = y @ z
    ]]></literal>
    <justification jstring="[para(132(a,1),14(a,1,2,2,2)),demod(119(3),34(4),21(6),8(7))].">
      <j1 rule="para" parents="132 14"/>
      <j2 rule="demod" parents="119 34 21 8"/>
    </justification>
  </clause>

  <clause id="160">
    <literal><![CDATA[
      x @ y * x = x @ y
    ]]></literal>
    <justification jstring="[back_demod(153),demod(159(4)),flip(a)].">
      <j1 rule="back_demod" parents="153"/>
      <j2 rule="demod" parents="159"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="175">
    <literal><![CDATA[
      x * y @ x ' = y @ x '
    ]]></literal>
    <justification jstring="[para(13(a,1),136(a,1,1)),flip(a)].">
      <j1 rule="para" parents="13 136"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="177">
    <literal><![CDATA[
      x ' * y @ x = y @ x
    ]]></literal>
    <justification jstring="[para(60(a,1),136(a,1,1)),flip(a)].">
      <j1 rule="para" parents="60 136"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="182">
    <literal><![CDATA[
      x * y @ y = y @ x '
    ]]></literal>
    <justification jstring="[para(13(a,1),160(a,1,2)),demod(175(5))].">
      <j1 rule="para" parents="13 160"/>
      <j2 rule="demod" parents="175"/>
    </justification>
  </clause>

  <clause id="185">
    <literal><![CDATA[
      x ' * y @ y = y @ x
    ]]></literal>
    <justification jstring="[para(60(a,1),160(a,1,2)),demod(177(6))].">
      <j1 rule="para" parents="60 160"/>
      <j2 rule="demod" parents="177"/>
    </justification>
  </clause>

  <clause id="186">
    <literal><![CDATA[
      x * y @ y = x @ y
    ]]></literal>
    <justification jstring="[para(160(a,1),119(a,1,1)),demod(119(2)),flip(a)].">
      <j1 rule="para" parents="160 119"/>
      <j2 rule="demod" parents="119"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="189">
    <literal><![CDATA[
      x ' @ y = y @ x
    ]]></literal>
    <justification jstring="[back_demod(185),demod(186(3))].">
      <j1 rule="back_demod" parents="185"/>
      <j2 rule="demod" parents="186"/>
    </justification>
  </clause>

  <clause id="190">
    <literal><![CDATA[
      x @ y ' = y @ x
    ]]></literal>
    <justification jstring="[back_demod(182),demod(186(2)),flip(a)].">
      <j1 rule="back_demod" parents="182"/>
      <j2 rule="demod" parents="186"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="207">
    <literal><![CDATA[
      (B @ A ') * C != C * (A @ B)
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[para(190(a,2),12(a,1,1))].">
      <j1 rule="para" parents="190 12"/>
    </justification>
  </clause>

  <clause id="216">
    <literal><![CDATA[
      x ' @ y = x @ y '
    ]]></literal>
    <justification jstring="[para(190(a,1),189(a,1)),flip(a)].">
      <j1 rule="para" parents="190 189"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="232">
    <literal><![CDATA[
      (B @ A ') * C != C * (B @ A ')
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[para(189(a,2),207(a,2,2)),demod(216(11))].">
      <j1 rule="para" parents="189 207"/>
      <j2 rule="demod" parents="216"/>
    </justification>
  </clause>

  <clause id="372">
    <literal><![CDATA[
      (x @ y) * z = z * (x @ y)
    ]]></literal>
    <justification jstring="[para(132(a,1),48(a,1,2,2)),demod(119(2),34(3),21(4),8(5))].">
      <j1 rule="para" parents="132 48"/>
      <j2 rule="demod" parents="119 34 21 8"/>
    </justification>
  </clause>

  <clause id="373">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[resolve(372,a,232,a)].">
      <j1 rule="resolve" parents="372 232"/>
    </justification>
  </clause>

</proof>

</proofs>
