<?xml version="1.0" encoding="ISO-8859-1"?>

<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>quot-np2.out</source>

<heading><![CDATA[
Prover9 (32) version 2008-05A, May 2008.
Process 12850 was started by mccune on cleo,
Wed May  7 11:34:00 2008
The command was "/home/mccune/LADR/bin/prover9 -f quot-np2.in".
]]></heading>

<proof number="1" length="26" max_count="31">

<comments><![CDATA[
% Proof 1 at 5.37 (+ 0.05) seconds.
% Length of proof is 26.
% Level of proof is 9.
% Maximum clause weight is 31.
% Given clauses 210.
]]></comments>

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

  <clause id="2" type="assumption">
    <literal><![CDATA[
      x * y != z
    ]]></literal>
    <literal><![CDATA[
      x * u != z
    ]]></literal>
    <literal><![CDATA[
      y = u
    ]]></literal>
    <attribute><![CDATA[
      label(left_cancellation_extended)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="3" type="assumption">
    <literal><![CDATA[
      x * y != z
    ]]></literal>
    <literal><![CDATA[
      u * y != z
    ]]></literal>
    <literal><![CDATA[
      x = u
    ]]></literal>
    <attribute><![CDATA[
      label(right_cancellation_extended)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="4" type="assumption">
    <literal><![CDATA[
      x * y * z * y * x = y * x * z * x * y
    ]]></literal>
    <attribute><![CDATA[
      label(nilpotent_class2)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="5" type="assumption">
    <literal><![CDATA[
      b * b0 = a * a0
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6" type="assumption">
    <literal><![CDATA[
      d * b0 = c * a0
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7" type="assumption">
    <literal><![CDATA[
      b * d0 = a * c0
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="8" type="assumption">
    <literal><![CDATA[
      d * d0 != c * c0
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

  <clause id="10">
    <literal><![CDATA[
      x * y * z != u
    ]]></literal>
    <literal><![CDATA[
      w * z != u
    ]]></literal>
    <literal><![CDATA[
      x * y = w
    ]]></literal>
    <justification jstring="[para(1(a,1),3(a,1))].">
      <j1 rule="para" parents="1 3"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      x * y * z * y * x * u = y * x * z * x * y * u
    ]]></literal>
    <justification jstring="[para(4(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="4 1"/>
      <j2 rule="rewrite" parents="1 1 1 1 1 1 1"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      b * b0 * x = a * a0 * x
    ]]></literal>
    <justification jstring="[para(5(a,1),1(a,1,1)),rewrite([1(4)]),flip(a)].">
      <j1 rule="para" parents="5 1"/>
      <j2 rule="rewrite" parents="1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      d * b0 * x = c * a0 * x
    ]]></literal>
    <justification jstring="[para(6(a,1),1(a,1,1)),rewrite([1(4)]),flip(a)].">
      <j1 rule="para" parents="6 1"/>
      <j2 rule="rewrite" parents="1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      b * d0 * x = a * c0 * x
    ]]></literal>
    <justification jstring="[para(7(a,1),1(a,1,1)),rewrite([1(4)]),flip(a)].">
      <j1 rule="para" parents="7 1"/>
      <j2 rule="rewrite" parents="1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      x * y * z * u != w
    ]]></literal>
    <literal><![CDATA[
      x * y * z * v5 != w
    ]]></literal>
    <literal><![CDATA[
      u = v5
    ]]></literal>
    <justification jstring="[para(1(a,1),9(a,1,2)),rewrite([1(6)])].">
      <j1 rule="para" parents="1 9"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      x * y * z != u
    ]]></literal>
    <literal><![CDATA[
      w * v5 * z != u
    ]]></literal>
    <literal><![CDATA[
      x * y = w * v5
    ]]></literal>
    <justification jstring="[para(1(a,1),10(b,1))].">
      <j1 rule="para" parents="1 10"/>
    </justification>
  </clause>

  <clause id="90">
    <literal><![CDATA[
      x * y * z * u * y * x * w = y * x * z * u * x * y * w
    ]]></literal>
    <justification jstring="[para(1(a,1),12(a,1,2,2)),rewrite([1(10)])].">
      <j1 rule="para" parents="1 12"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="96">
    <literal><![CDATA[
      x * y * z * y * x * u * y * x = y * x * z * y * x * u * x * y
    ]]></literal>
    <justification jstring="[para(4(a,1),12(a,1,2,2,2)),flip(a)].">
      <j1 rule="para" parents="4 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="5649">
    <literal><![CDATA[
      x * y * x * x * y = x * x * y * y * x
    ]]></literal>
    <justification jstring="[hyper(33,a,90,a,b,96,a),flip(a)].">
      <j1 rule="hyper" parents="33 90 96"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="5726">
    <literal><![CDATA[
      x * y * y * x = y * x * x * y
    ]]></literal>
    <justification jstring="[hyper(2,a,xx,b,5649,a)].">
      <j1 rule="hyper" parents="2 5649"/>
    </justification>
  </clause>

  <clause id="6150">
    <literal><![CDATA[
      a * a0 * x * x * b0 = b * x * b0 * b0 * x
    ]]></literal>
    <justification jstring="[para(5726(a,1),21(a,1,2)),flip(a)].">
      <j1 rule="para" parents="5726 21"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="6152">
    <literal><![CDATA[
      c * a0 * x * x * b0 = d * x * b0 * b0 * x
    ]]></literal>
    <justification jstring="[para(5726(a,1),25(a,1,2)),flip(a)].">
      <j1 rule="para" parents="5726 25"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="7457">
    <literal><![CDATA[
      a0 * d0 * d0 * b0 = c0 * b0 * b0 * d0
    ]]></literal>
    <justification jstring="[hyper(2,a,29,a(flip),b,6150,a),flip(a)].">
      <j1 rule="hyper" parents="2 29 6150"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="7868">
    <literal><![CDATA[
      d * d0 * b0 * b0 * d0 = c * c0 * b0 * b0 * d0
    ]]></literal>
    <justification jstring="[para(7457(a,1),6152(a,1,2)),flip(a)].">
      <j1 rule="para" parents="7457 6152"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9944">
    <literal><![CDATA[
      d * d0 = c * c0
    ]]></literal>
    <justification jstring="[hyper(65,a,xx,b,7868,a),flip(a)].">
      <j1 rule="hyper" parents="65 7868"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9945">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(9944,a,8,a)].">
      <j1 rule="resolve" parents="9944 8"/>
    </justification>
  </clause>

</proof>

</proofs>
