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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>quot-xy3b.out</source>

<heading><![CDATA[
Prover9 (32) version August-2006A, August 2006.
Process 11396 was started by mccune on cleo.thornwood,
Sat Aug 12 21:00:58 2006
The command was "/home/mccune/bin/prover9 -f quot-xy3b.in".
]]></heading>

<proof number="1" length="16" max_count="27">

<comments><![CDATA[
% Proof 1 at 1.40 (+ 0.01) seconds.
% Length of proof is 16.
% Level of proof is 4.
% Maximum clause weight is 27.
% Given clauses 81.
]]></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 * x * y * x * y = y * x * y * x * y * x
    ]]></literal>
    <attribute><![CDATA[
      label("(xy)^3 = (yx)^3")
    ]]></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="18">
    <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="22">
    <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="26">
    <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="38">
    <literal><![CDATA[
      b * x * b0 * x * b0 * x * b0 = a * a0 * x * b0 * x * b0 * x
    ]]></literal>
    <justification jstring="[para(4(a,1),18(a,1,2))].">
      <j1 rule="para" parents="4 18"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      d * x * b0 * x * b0 * x * b0 = c * a0 * x * b0 * x * b0 * x
    ]]></literal>
    <justification jstring="[para(4(a,1),22(a,1,2))].">
      <j1 rule="para" parents="4 22"/>
    </justification>
  </clause>

  <clause id="1118">
    <literal><![CDATA[
      a0 * d0 * b0 * d0 * b0 * d0 = c0 * b0 * d0 * b0 * d0 * b0
    ]]></literal>
    <justification jstring="[hyper(2,a,26,a(flip),b,38,a(flip)),flip(a)].">
      <j1 rule="hyper" parents="2 26 38"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1338">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[ur(10,a,42,a,c,8,a),rewrite(1(13),1118(25)),xx(a)].">
      <j1 rule="ur" parents="10 42 8"/>
      <j2 rule="rewrite" parents="1 1118"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
