<?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-xy3a.out</source>

<heading><![CDATA[
Prover9 (32) version 2008-05A, May 2008.
Process 12857 was started by mccune on cleo,
Wed May  7 11:34:06 2008
The command was "/home/mccune/LADR/bin/prover9 -f quot-xy3a.in".
]]></heading>

<proof number="1" length="23" max_count="33">

<comments><![CDATA[
% Proof 1 at 2.60 (+ 0.03) seconds.
% Length of proof is 23.
% Level of proof is 7.
% Maximum clause weight is 33.
% Given clauses 217.
]]></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 * 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" 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="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[
      x * y * y * y * z * z * z != u
    ]]></literal>
    <literal><![CDATA[
      x * y * w != u
    ]]></literal>
    <literal><![CDATA[
      z * y * z * y * z = w
    ]]></literal>
    <justification jstring="[para(4(a,1),9(a,1,2))].">
      <j1 rule="para" parents="4 9"/>
    </justification>
  </clause>

  <clause id="70">
    <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="116">
    <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="735">
    <literal><![CDATA[
      d0 * d0 * x * x * x = x * d0 * x * d0 * x
    ]]></literal>
    <justification jstring="[hyper(35,a,30,a,b,30,a),flip(a)].">
      <j1 rule="hyper" parents="35 30 30"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="747">
    <literal><![CDATA[
      d * d0 * d0 * b0 * b0 * b0 = c * a0 * d0 * b0 * d0 * b0
    ]]></literal>
    <justification jstring="[para(735(a,2),26(a,1,2))].">
      <j1 rule="para" parents="735 26"/>
    </justification>
  </clause>

  <clause id="1704">
    <literal><![CDATA[
      a0 * d0 * b0 * d0 * b0 = c0 * d0 * b0 * b0 * b0
    ]]></literal>
    <justification jstring="[hyper(2,a,30,a(flip),b,116,a),flip(a)].">
      <j1 rule="hyper" parents="2 30 116"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1746">
    <literal><![CDATA[
      d * d0 * d0 * b0 * b0 * b0 = c * c0 * d0 * b0 * b0 * b0
    ]]></literal>
    <justification jstring="[back_rewrite(747),rewrite([1704(21)])].">
      <j1 rule="back_rewrite" parents="747"/>
      <j2 rule="rewrite" parents="1704"/>
    </justification>
  </clause>

  <clause id="3486">
    <literal><![CDATA[
      d * d0 = c * c0
    ]]></literal>
    <justification jstring="[hyper(70,a,xx,b,1746,a),flip(a)].">
      <j1 rule="hyper" parents="70 1746"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3487">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(3486,a,8,a)].">
      <j1 rule="resolve" parents="3486 8"/>
    </justification>
  </clause>

</proof>

</proofs>
