<?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-xy3b.out</source>

<heading><![CDATA[
Prover9 (32) version 22-May-2007, May 2007.
Process 27471 was started by mccune on cleo,
Tue May 22 14:46:13 2007
The command was "/home/mccune/bin/prover9 -f quot-xy3b.in".
]]></heading>

<proof number="1" length="25" max_count="31">

<comments><![CDATA[
% Proof 1 at 26.63 (+ 0.04) seconds.
% Length of proof is 25.
% Level of proof is 7.
% Maximum clause weight is 31.
% Given clauses 815.
]]></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 = 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" 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="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 * x * y * x * y * z = y * x * y * x * y * x * z
    ]]></literal>
    <justification jstring="[para(4(a,1),1(a,1,1)),rewrite([1(6),1(5),1(4),1(3),1(2),1(11),1(10),1(9),1(8)])].">
      <j1 rule="para" parents="4 1"/>
      <j2 rule="rewrite" parents="1 1 1 1 1 1 1 1 1"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      x * y * z * y * z * y * z = x * z * y * z * y * z * y
    ]]></literal>
    <justification jstring="[para(4(a,1),1(a,2,2)),rewrite([1(6)])].">
      <j1 rule="para" parents="4 1"/>
      <j2 rule="rewrite" parents="1"/>
    </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="39">
    <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="47">
    <literal><![CDATA[
      a * c0 * x != y
    ]]></literal>
    <literal><![CDATA[
      b * z != y
    ]]></literal>
    <literal><![CDATA[
      d0 * x = z
    ]]></literal>
    <justification jstring="[para(26(a,1),2(a,1))].">
      <j1 rule="para" parents="26 2"/>
    </justification>
  </clause>

  <clause id="57">
    <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="122">
    <literal><![CDATA[
      a * c0 * x * d0 * x * d0 * x * y = b * x * d0 * x * d0 * x * d0 * y
    ]]></literal>
    <justification jstring="[para(12(a,1),26(a,1,2)),flip(a)].">
      <j1 rule="para" parents="12 26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="179">
    <literal><![CDATA[
      x * y * z * u * z * u * z * u = x * y * u * z * u * z * u * z
    ]]></literal>
    <justification jstring="[para(14(a,1),1(a,1)),rewrite([1(7)])].">
      <j1 rule="para" parents="14 1"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="359">
    <literal><![CDATA[
      a0 * d0 * b0 * d0 * b0 * d0 = c0 * b0 * d0 * b0 * d0 * b0
    ]]></literal>
    <justification jstring="[hyper(2,a,26,a(flip),b,39,a(flip)),flip(a)].">
      <j1 rule="hyper" parents="2 26 39"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="534">
    <literal><![CDATA[
      a0 * d0 * b0 * d0 * b0 * d0 * x = c0 * b0 * d0 * b0 * d0 * b0 * x
    ]]></literal>
    <justification jstring="[para(359(a,1),1(a,1,1)),rewrite([1(12),1(11),1(10),1(9),1(8),1(23),1(22),1(21),1(20)]),flip(a)].">
      <j1 rule="para" parents="359 1"/>
      <j2 rule="rewrite" parents="1 1 1 1 1 1 1 1 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2077">
    <literal><![CDATA[
      d0 * d0 * x * d0 * x * d0 * x = x * d0 * x * d0 * x * d0 * d0
    ]]></literal>
    <justification jstring="[hyper(47,a,179,a,b,122,a(flip))].">
      <j1 rule="hyper" parents="47 179 122"/>
    </justification>
  </clause>

  <clause id="2082">
    <literal><![CDATA[
      d * d0 * d0 * b0 * d0 * b0 * d0 * b0 = c * c0 * b0 * d0 * b0 * d0 * b0 * d0
    ]]></literal>
    <justification jstring="[para(2077(a,2),22(a,1,2)),rewrite([534(29)])].">
      <j1 rule="para" parents="2077 22"/>
      <j2 rule="rewrite" parents="534"/>
    </justification>
  </clause>

  <clause id="2507">
    <literal><![CDATA[
      d * d0 = c * c0
    ]]></literal>
    <justification jstring="[hyper(57,a,179,a,b,2082,a),flip(a)].">
      <j1 rule="hyper" parents="57 179 2082"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2508">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(2507,a,8,a)].">
      <j1 rule="resolve" parents="2507 8"/>
    </justification>
  </clause>

</proof>

</proofs>
