<?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 November-2006, November 2006.
Process 3899 was started by mccune on cleo.thornwood,
Wed Nov 22 11:24:28 2006
The command was "/home/mccune/bin/prover9 -f quot-np2.in".
]]></heading>

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

<comments><![CDATA[
% Proof 1 at 5.11 (+ 0.04) seconds.
% Length of proof is 26.
% Level of proof is 9.
% Maximum clause weight is 31.
% Given clauses 208.
]]></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 * v != u
    ]]></literal>
    <literal><![CDATA[
      z = v
    ]]></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[
      v * z != u
    ]]></literal>
    <literal><![CDATA[
      x * y = v
    ]]></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="51">
    <literal><![CDATA[
      x * y * z * u != v
    ]]></literal>
    <literal><![CDATA[
      x * y * z * w != v
    ]]></literal>
    <literal><![CDATA[
      u = w
    ]]></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="70">
    <literal><![CDATA[
      x * y * z != u
    ]]></literal>
    <literal><![CDATA[
      v * w * z != u
    ]]></literal>
    <literal><![CDATA[
      x * y = v * w
    ]]></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 * v = y * x * z * u * x * y * v
    ]]></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="5664">
    <literal><![CDATA[
      x * y * x * x * y = x * x * y * y * x
    ]]></literal>
    <justification jstring="[hyper(51,a,90,a,b,96,a),flip(a)].">
      <j1 rule="hyper" parents="51 90 96"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="5741">
    <literal><![CDATA[
      x * y * y * x = y * x * x * y
    ]]></literal>
    <justification jstring="[hyper(2,a,xx,b,5664,a)].">
      <j1 rule="hyper" parents="2 5664"/>
    </justification>
  </clause>

  <clause id="6172">
    <literal><![CDATA[
      a * a0 * x * x * b0 = b * x * b0 * b0 * x
    ]]></literal>
    <justification jstring="[para(5741(a,1),21(a,1,2)),flip(a)].">
      <j1 rule="para" parents="5741 21"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="6174">
    <literal><![CDATA[
      c * a0 * x * x * b0 = d * x * b0 * b0 * x
    ]]></literal>
    <justification jstring="[para(5741(a,1),25(a,1,2)),flip(a)].">
      <j1 rule="para" parents="5741 25"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="7663">
    <literal><![CDATA[
      a0 * d0 * d0 * b0 = c0 * b0 * b0 * d0
    ]]></literal>
    <justification jstring="[hyper(2,a,29,a(flip),b,6172,a),flip(a)].">
      <j1 rule="hyper" parents="2 29 6172"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="8170">
    <literal><![CDATA[
      d * d0 * b0 * b0 * d0 = c * c0 * b0 * b0 * d0
    ]]></literal>
    <justification jstring="[para(7663(a,1),6174(a,1,2)),flip(a)].">
      <j1 rule="para" parents="7663 6174"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10288">
    <literal><![CDATA[
      d * d0 = c * c0
    ]]></literal>
    <justification jstring="[hyper(70,a,xx,b,8170,a),flip(a)].">
      <j1 rule="hyper" parents="70 8170"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10289">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(10288,a,8,a)].">
      <j1 rule="resolve" parents="10288 8"/>
    </justification>
  </clause>

</proof>

</proofs>
