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<!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 August-2006A, August 2006.
Process 11391 was started by mccune on cleo.thornwood,
Sat Aug 12 21:00:08 2006
The command was "/home/mccune/bin/prover9 -f quot-np2.in".
]]></heading>

<proof number="1" length="16" max_count="23">

<comments><![CDATA[
% Proof 1 at 2.95 (+ 0.03) seconds.
% Length of proof is 16.
% Level of proof is 4.
% Maximum clause weight is 23.
% Given clauses 106.
]]></comments>

  <clause id="1">
    <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">
    <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">
    <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">
    <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">
    <literal><![CDATA[
      b * b0 = a * a0
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

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

  <clause id="8">
    <literal><![CDATA[
      d * d0 != c * c0
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </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="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="43">
    <literal><![CDATA[
      a * a0 * x * y * x * b0 = b * x * b0 * y * b0 * x
    ]]></literal>
    <justification jstring="[para(4(a,1),21(a,1,2)),flip(a)].">
      <j1 rule="para" parents="4 21"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      c * a0 * x * y * x * b0 = d * x * b0 * y * b0 * x
    ]]></literal>
    <justification jstring="[para(4(a,1),25(a,1,2)),flip(a)].">
      <j1 rule="para" parents="4 25"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3248">
    <literal><![CDATA[
      a0 * d0 * x * d0 * b0 = c0 * b0 * x * b0 * d0
    ]]></literal>
    <justification jstring="[hyper(2,a,29,a(flip),b,43,a),flip(a)].">
      <j1 rule="hyper" parents="2 29 43"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4128">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[ur(10,a,49,a(flip),c,8,a),rewrite(1(10),3248(19)),xx(a)].">
      <j1 rule="ur" parents="10 49 8"/>
      <j2 rule="rewrite" parents="1 3248"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
