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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>quot-xy3a.out</source>

<heading><![CDATA[
Prover9 (32) version August-2006A, August 2006.
Process 11392 was started by mccune on cleo.thornwood,
Sat Aug 12 21:00:11 2006
The command was "/home/mccune/bin/prover9 -f quot-xy3a.in".
]]></heading>

<proof number="1" length="21" max_count="33">

<comments><![CDATA[
% Proof 1 at 46.77 (+ 0.08) seconds.
% Length of proof is 21.
% Level of proof is 6.
% Maximum clause weight is 33.
% Given clauses 275.
]]></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 = x * x * x * y * y * y
    ]]></literal>
    <attribute><![CDATA[
      label("(xy)^3 = x^3y^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="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="11">
    <literal><![CDATA[
      x * y * x * y * x = y * y * x * x * x
    ]]></literal>
    <justification jstring="[hyper(2,a,xx,b,4,a),flip(a)].">
      <j1 rule="hyper" parents="2 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22">
    <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="26">
    <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="30">
    <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="35">
    <literal><![CDATA[
      d * d0 * x != c * c0 * x
    ]]></literal>
    <justification jstring="[ur(3,b,1,a,c,8,a),rewrite(1(4))].">
      <j1 rule="ur" parents="3 1 8"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="56">
    <literal><![CDATA[
      x * y * y * y * z * z * z != u
    ]]></literal>
    <literal><![CDATA[
      x * y * v != u
    ]]></literal>
    <literal><![CDATA[
      z * y * z * y * z = v
    ]]></literal>
    <justification jstring="[para(4(a,1),9(a,1,2))].">
      <j1 rule="para" parents="4 9"/>
    </justification>
  </clause>

  <clause id="209">
    <literal><![CDATA[
      a * a0 * x * b0 * x * b0 = b * x * x * b0 * b0 * b0
    ]]></literal>
    <justification jstring="[para(11(a,1),22(a,1,2)),flip(a)].">
      <j1 rule="para" parents="11 22"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2708">
    <literal><![CDATA[
      d0 * d0 * x * x * x = x * d0 * x * d0 * x
    ]]></literal>
    <justification jstring="[hyper(56,a,30,a,b,30,a),flip(a)].">
      <j1 rule="hyper" parents="56 30 30"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3222">
    <literal><![CDATA[
      d * d0 * d0 * b0 * b0 * b0 = c * a0 * d0 * b0 * d0 * b0
    ]]></literal>
    <justification jstring="[para(2708(a,2),26(a,1,2))].">
      <j1 rule="para" parents="2708 26"/>
    </justification>
  </clause>

  <clause id="6352">
    <literal><![CDATA[
      a0 * d0 * b0 * d0 * b0 = c0 * d0 * b0 * b0 * b0
    ]]></literal>
    <justification jstring="[hyper(2,a,30,a(flip),b,209,a),flip(a)].">
      <j1 rule="hyper" parents="2 30 209"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="6374">
    <literal><![CDATA[
      d * d0 * d0 * b0 * b0 * b0 = c * c0 * d0 * b0 * b0 * b0
    ]]></literal>
    <justification jstring="[back_rewrite(3222),rewrite(6352(21))].">
      <j1 rule="back_rewrite" parents="3222"/>
      <j2 rule="rewrite" parents="6352"/>
    </justification>
  </clause>

  <clause id="6375">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(6374,a,35,a)].">
      <j1 rule="resolve" parents="6374 35"/>
    </justification>
  </clause>

</proof>

</proofs>
