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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 June-2006C, June 2006.
Process 13113 was started by mccune on cleo.thornwood,
Mon Jun 19 16:41:24 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 4.45 (+ 0.06) seconds: E.
% Length of proof is 42.
% Level of proof is 15.
% Maximum clause weight is 19.
% Given clauses 319.
]]></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 * B * A != B * A * C * A * B
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></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="23">
    <literal><![CDATA[
      (x @ y) @ z = e
    ]]></literal>
    <justification jstring="[para(16(a,1),11(a,2))].">
      <j1 rule="para" parents="16 11"/>
    </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="30">
    <literal><![CDATA[
      x ' ' * (x @ y) = y ' * x * y
    ]]></literal>
    <justification jstring="[para(10(a,1),13(a,1,2))].">
      <j1 rule="para" parents="10 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="32">
    <literal><![CDATA[
      x ' * y * x = y * (y @ x)
    ]]></literal>
    <justification jstring="[back_demod(30),demod(31(4)),flip(a)].">
      <j1 rule="back_demod" parents="30"/>
      <j2 rule="demod" parents="31"/>
      <j2 rule="flip"/>
    </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="47">
    <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="49">
    <literal><![CDATA[
      x ' * y * x * z = (x @ y ') * y * z
    ]]></literal>
    <justification jstring="[para(13(a,1),14(a,1,2,2,2)),demod(31(5))].">
      <j1 rule="para" parents="13 14"/>
      <j2 rule="demod" parents="31"/>
    </justification>
  </clause>

  <clause id="55">
    <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="59">
    <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="60">
    <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="64">
    <literal><![CDATA[
      x * y * (x * y) ' = e
    ]]></literal>
    <justification jstring="[para(55(a,1),7(a,1)),flip(a)].">
      <j1 rule="para" parents="55 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="73">
    <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="75">
    <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="88">
    <literal><![CDATA[
      x * (y * x) ' = y '
    ]]></literal>
    <justification jstring="[para(64(a,1),13(a,1,2)),demod(34(3)),flip(a)].">
      <j1 rule="para" parents="64 13"/>
      <j2 rule="demod" parents="34"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="98">
    <literal><![CDATA[
      (x * y) ' = y ' * x '
    ]]></literal>
    <justification jstring="[para(88(a,1),13(a,1,2)),flip(a)].">
      <j1 rule="para" parents="88 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="99">
    <literal><![CDATA[
      (x @ y) ' = y @ x
    ]]></literal>
    <justification jstring="[para(14(a,2),88(a,1,2,1)),demod(98(7),98(5),98(3),98(2),7(5),60(7),7(6),7(5),60(8),7(7),7(6),7(5),10(6),59(4)),flip(a)].">
      <j1 rule="para" parents="14 88"/>
      <j2 rule="demod" parents="98 98 98 98 7 60 7 7 60 7 7 7 10 59"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="109">
    <literal><![CDATA[
      x * y @ x = y @ x
    ]]></literal>
    <justification jstring="[back_demod(73),demod(98(2),7(5),10(5)),flip(a)].">
      <j1 rule="back_demod" parents="73"/>
      <j2 rule="demod" parents="98 7 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="128">
    <literal><![CDATA[
      x ' * (x @ y) * x = x @ y * x
    ]]></literal>
    <justification jstring="[back_demod(75),demod(109(3))].">
      <j1 rule="back_demod" parents="75"/>
      <j2 rule="demod" parents="109"/>
    </justification>
  </clause>

  <clause id="137">
    <literal><![CDATA[
      x ' * (y @ z) * x = y @ z
    ]]></literal>
    <justification jstring="[para(23(a,1),32(a,2,2)),demod(34(7))].">
      <j1 rule="para" parents="23 32"/>
      <j2 rule="demod" parents="34"/>
    </justification>
  </clause>

  <clause id="144">
    <literal><![CDATA[
      x @ y * x = x @ y
    ]]></literal>
    <justification jstring="[back_demod(128),demod(137(4)),flip(a)].">
      <j1 rule="back_demod" parents="128"/>
      <j2 rule="demod" parents="137"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="150">
    <literal><![CDATA[
      x ' * y @ x = y @ x
    ]]></literal>
    <justification jstring="[para(59(a,1),109(a,1,1)),flip(a)].">
      <j1 rule="para" parents="59 109"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="158">
    <literal><![CDATA[
      x ' * y @ y = y @ x
    ]]></literal>
    <justification jstring="[para(59(a,1),144(a,1,2)),demod(150(6))].">
      <j1 rule="para" parents="59 144"/>
      <j2 rule="demod" parents="150"/>
    </justification>
  </clause>

  <clause id="161">
    <literal><![CDATA[
      x * y @ y = x @ y
    ]]></literal>
    <justification jstring="[para(144(a,1),99(a,1,1)),demod(99(2)),flip(a)].">
      <j1 rule="para" parents="144 99"/>
      <j2 rule="demod" parents="99"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="162">
    <literal><![CDATA[
      x ' @ y = y @ x
    ]]></literal>
    <justification jstring="[back_demod(158),demod(161(3))].">
      <j1 rule="back_demod" parents="158"/>
      <j2 rule="demod" parents="161"/>
    </justification>
  </clause>

  <clause id="183">
    <literal><![CDATA[
      x ' @ y ' = x @ y
    ]]></literal>
    <justification jstring="[para(162(a,2),162(a,1))].">
      <j1 rule="para" parents="162 162"/>
    </justification>
  </clause>

  <clause id="343">
    <literal><![CDATA[
      x * y * (y @ x) * z = y * x * z
    ]]></literal>
    <justification jstring="[para(47(a,1),13(a,1,2)),demod(60(2))].">
      <j1 rule="para" parents="47 13"/>
      <j2 rule="demod" parents="60"/>
    </justification>
  </clause>

  <clause id="443">
    <literal><![CDATA[
      (x @ y) * y * x = x * y
    ]]></literal>
    <justification jstring="[para(9(a,1),49(a,1,2,2)),demod(60(2),34(2),183(4)),flip(a)].">
      <j1 rule="para" parents="9 49"/>
      <j2 rule="demod" parents="60 34 183"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="792">
    <literal><![CDATA[
      (x @ y) * z * y * x = z * x * y
    ]]></literal>
    <justification jstring="[para(443(a,1),47(a,1,2,2)),demod(99(2),21(6),8(7))].">
      <j1 rule="para" parents="443 47"/>
      <j2 rule="demod" parents="99 21 8"/>
    </justification>
  </clause>

  <clause id="9897">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[para(792(a,2),12(a,1,2,2)),demod(343(13)),xx(a)].">
      <j1 rule="para" parents="792 12"/>
      <j2 rule="demod" parents="343"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
