<?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="3">

<source>gtsax.out</source>

<heading><![CDATA[
Prover9 (32) version April-2007, April 2007.
Process 26424 was started by mccune on cleo,
Fri Apr 13 09:13:51 2007
The command was "/home/mccune/bin/prover9 -f gtsax.in".
]]></heading>

<proof number="1" length="12" max_count="29">

<comments><![CDATA[
% Proof 1 at 0.02 (+ 0.00) seconds: inverse.
% Length of proof is 12.
% Level of proof is 7.
% Maximum clause weight is 29.
% Given clauses 11.
]]></comments>

  <clause id="2" type="goal">
    <literal><![CDATA[
      x * x' = y * y'
    ]]></literal>
    <attribute><![CDATA[
      label(inverse)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="4" type="assumption">
    <literal><![CDATA[
      x * (y * (((z * z') * (u * y)') * x))' = u
    ]]></literal>
    <attribute><![CDATA[
      label(gt_sax)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6" type="deny">
    <literal><![CDATA[
      c4 * c4' != c5 * c5'
    ]]></literal>
    <attribute><![CDATA[
      label(inverse)
    ]]></attribute>
    <attribute><![CDATA[
      answer(inverse)
    ]]></attribute>
    <justification jstring="[deny(2)].">
      <j1 rule="deny" parents="2"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      c5 * c5' != c4 * c4'
    ]]></literal>
    <attribute><![CDATA[
      answer(inverse)
    ]]></attribute>
    <justification jstring="[copy(6),flip(a)].">
      <j1 rule="copy" parents="6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

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

  <clause id="10">
    <literal><![CDATA[
      x * ((((y * y') * (z * u)') * (v * v')) * (z * x))' = u
    ]]></literal>
    <justification jstring="[para(4(a,1),4(a,1,2,1,2,1))].">
      <j1 rule="para" parents="4 4"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      ((x * x') * (y * z)') * (u * u') = v * ((y * (w * w')) * (z * v))'
    ]]></literal>
    <justification jstring="[para(10(a,1),4(a,1,2,1,2,1)),flip(a)].">
      <j1 rule="para" parents="10 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      (x * x') * (y * (z * ((u * (v * v')) * (y * z))'))' = u
    ]]></literal>
    <justification jstring="[para(24(a,1),4(a,1,2,1,2))].">
      <j1 rule="para" parents="24 4"/>
    </justification>
  </clause>

  <clause id="101">
    <literal><![CDATA[
      (x * (y * ((z * (u * u')) * (x * y))'))' * (v' * z)' = v
    ]]></literal>
    <justification jstring="[para(39(a,1),4(a,1,2,1,2))].">
      <j1 rule="para" parents="39 4"/>
    </justification>
  </clause>

  <clause id="107">
    <literal><![CDATA[
      x * (((y * y') * ((z * z') * x)') * u) = u
    ]]></literal>
    <justification jstring="[para(39(a,1),9(a,1,2,1,2)),rewrite(101(22))].">
      <j1 rule="para" parents="39 9"/>
      <j2 rule="rewrite" parents="101"/>
    </justification>
  </clause>

  <clause id="160">
    <literal><![CDATA[
      x * x' = y * y'
    ]]></literal>
    <justification jstring="[para(107(a,1),4(a,1,2,1))].">
      <j1 rule="para" parents="107 4"/>
    </justification>
  </clause>

  <clause id="161">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(inverse)
    ]]></attribute>
    <justification jstring="[resolve(160,a,7,a)].">
      <j1 rule="resolve" parents="160 7"/>
    </justification>
  </clause>

</proof>

<proof number="2" length="33" max_count="34">

<comments><![CDATA[
% Proof 2 at 0.18 (+ 0.00) seconds: identity.
% Length of proof is 33.
% Level of proof is 18.
% Maximum clause weight is 34.
% Given clauses 37.
]]></comments>

  <clause id="3" type="goal">
    <literal><![CDATA[
      x * (y * y') = x
    ]]></literal>
    <attribute><![CDATA[
      label(identity)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="4" type="assumption">
    <literal><![CDATA[
      x * (y * (((z * z') * (u * y)') * x))' = u
    ]]></literal>
    <attribute><![CDATA[
      label(gt_sax)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="8" type="deny">
    <literal><![CDATA[
      c6 * (c7 * c7') != c6
    ]]></literal>
    <attribute><![CDATA[
      label(identity)
    ]]></attribute>
    <attribute><![CDATA[
      answer(identity)
    ]]></attribute>
    <justification jstring="[deny(3)].">
      <j1 rule="deny" parents="3"/>
    </justification>
  </clause>

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

  <clause id="10">
    <literal><![CDATA[
      x * ((((y * y') * (z * u)') * (v * v')) * (z * x))' = u
    ]]></literal>
    <justification jstring="[para(4(a,1),4(a,1,2,1,2,1))].">
      <j1 rule="para" parents="4 4"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      (x * (((y * y') * (z * x)') * ((u * u') * (v * w)')))' * (w * z)' = v
    ]]></literal>
    <justification jstring="[para(4(a,1),4(a,1,2,1,2))].">
      <j1 rule="para" parents="4 4"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      (x * (((y * y') * (z * x)') * u))' = v * ((((w * w') * z') * (v6 * v6')) * (u * v))'
    ]]></literal>
    <justification jstring="[para(9(a,1),4(a,1,2,1,2,1)),flip(a)].">
      <j1 rule="para" parents="9 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      ((x * x') * (y * z)') * (u * u') = v * ((y * (w * w')) * (z * v))'
    ]]></literal>
    <justification jstring="[para(10(a,1),4(a,1,2,1,2,1)),flip(a)].">
      <j1 rule="para" parents="10 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      (x * x') * (y * (z * ((u * (v * v')) * (y * z))'))' = u
    ]]></literal>
    <justification jstring="[para(24(a,1),4(a,1,2,1,2))].">
      <j1 rule="para" parents="24 4"/>
    </justification>
  </clause>

  <clause id="101">
    <literal><![CDATA[
      (x * (y * ((z * (u * u')) * (x * y))'))' * (v' * z)' = v
    ]]></literal>
    <justification jstring="[para(39(a,1),4(a,1,2,1,2))].">
      <j1 rule="para" parents="39 4"/>
    </justification>
  </clause>

  <clause id="107">
    <literal><![CDATA[
      x * (((y * y') * ((z * z') * x)') * u) = u
    ]]></literal>
    <justification jstring="[para(39(a,1),9(a,1,2,1,2)),rewrite(101(22))].">
      <j1 rule="para" parents="39 9"/>
      <j2 rule="rewrite" parents="101"/>
    </justification>
  </clause>

  <clause id="160">
    <literal><![CDATA[
      x * x' = y * y'
    ]]></literal>
    <justification jstring="[para(107(a,1),4(a,1,2,1))].">
      <j1 rule="para" parents="107 4"/>
    </justification>
  </clause>

  <clause id="164">
    <literal><![CDATA[
      (x * (((y * y') * (z * x)') * ((u * u') * ((v * v') * w)')))' = w * z
    ]]></literal>
    <justification jstring="[para(4(a,1),107(a,1,2)),flip(a)].">
      <j1 rule="para" parents="4 107"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="203">
    <literal><![CDATA[
      x * (y' * ((z * z') * x))' = y
    ]]></literal>
    <justification jstring="[para(160(a,1),4(a,1,2,1,2,1))].">
      <j1 rule="para" parents="160 4"/>
    </justification>
  </clause>

  <clause id="204">
    <literal><![CDATA[
      c6 * (x * x') != c6
    ]]></literal>
    <attribute><![CDATA[
      answer(identity)
    ]]></attribute>
    <justification jstring="[para(160(a,1),8(a,1,2))].">
      <j1 rule="para" parents="160 8"/>
    </justification>
  </clause>

  <clause id="210">
    <literal><![CDATA[
      x * (((y * y') * (z * z')) * (u * x))' = u'
    ]]></literal>
    <justification jstring="[para(160(a,1),10(a,1,2,1,1,1))].">
      <j1 rule="para" parents="160 10"/>
    </justification>
  </clause>

  <clause id="224">
    <literal><![CDATA[
      (x * x')' * ((y * y') * z) = z
    ]]></literal>
    <justification jstring="[para(160(a,1),107(a,1,2,1))].">
      <j1 rule="para" parents="160 107"/>
    </justification>
  </clause>

  <clause id="253">
    <literal><![CDATA[
      (x * x')' * (y' * (z * z'))' = y
    ]]></literal>
    <justification jstring="[para(160(a,1),203(a,1,2,1,2))].">
      <j1 rule="para" parents="160 203"/>
    </justification>
  </clause>

  <clause id="278">
    <literal><![CDATA[
      (x * x')' = (y * y')' * (z * z')
    ]]></literal>
    <justification jstring="[para(160(a,1),224(a,1,2)),flip(a)].">
      <j1 rule="para" parents="160 224"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="419">
    <literal><![CDATA[
      (x * x')' = (y * y')'
    ]]></literal>
    <justification jstring="[para(278(a,2),224(a,1))].">
      <j1 rule="para" parents="278 224"/>
    </justification>
  </clause>

  <clause id="453">
    <literal><![CDATA[
      (x * x') * (y * y')' = z * z'
    ]]></literal>
    <justification jstring="[para(419(a,1),160(a,1,2))].">
      <j1 rule="para" parents="419 160"/>
    </justification>
  </clause>

  <clause id="485">
    <literal><![CDATA[
      ((x * x')' * y) * (z' * y)' = z
    ]]></literal>
    <justification jstring="[para(453(a,2),11(a,1,1,1,2,2,2,1)),rewrite(164(18))].">
      <j1 rule="para" parents="453 11"/>
      <j2 rule="rewrite" parents="164"/>
    </justification>
  </clause>

  <clause id="555">
    <literal><![CDATA[
      (x * x') * y' = (z * z') * y'
    ]]></literal>
    <justification jstring="[para(13(a,2),39(a,1,2,1,2)),rewrite(4(11))].">
      <j1 rule="para" parents="13 39"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="734">
    <literal><![CDATA[
      x * (y' * (x' * (z * z'))')' = y
    ]]></literal>
    <justification jstring="[para(253(a,1),485(a,1,1))].">
      <j1 rule="para" parents="253 485"/>
    </justification>
  </clause>

  <clause id="742">
    <literal><![CDATA[
      ((x * x')' * y'') * ((z * z') * (u * u')')' = y
    ]]></literal>
    <justification jstring="[para(453(a,2),485(a,1,2,1))].">
      <j1 rule="para" parents="453 485"/>
    </justification>
  </clause>

  <clause id="757">
    <literal><![CDATA[
      ((x * x') * ((y * y') * z')') * (u * u') = z''
    ]]></literal>
    <justification jstring="[para(555(a,1),24(a,1,1,2,1)),rewrite(210(21))].">
      <j1 rule="para" parents="555 24"/>
      <j2 rule="rewrite" parents="210"/>
    </justification>
  </clause>

  <clause id="833">
    <literal><![CDATA[
      (x * x') * (y * z')' = z * y'
    ]]></literal>
    <justification jstring="[para(4(a,1),734(a,1,2,1)),flip(a)].">
      <j1 rule="para" parents="4 734"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="859">
    <literal><![CDATA[
      (x * x')' * y'' = y
    ]]></literal>
    <justification jstring="[para(224(a,1),734(a,1,2,1,2,1)),rewrite(833(10))].">
      <j1 rule="para" parents="224 734"/>
      <j2 rule="rewrite" parents="833"/>
    </justification>
  </clause>

  <clause id="886">
    <literal><![CDATA[
      x'' = (x * (y * y')') * (z * z')
    ]]></literal>
    <justification jstring="[back_rewrite(757),rewrite(833(8)),flip(a)].">
      <j1 rule="back_rewrite" parents="757"/>
      <j2 rule="rewrite" parents="833"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="889">
    <literal><![CDATA[
      x * (y * y')' = x
    ]]></literal>
    <justification jstring="[back_rewrite(742),rewrite(859(6),833(6))].">
      <j1 rule="back_rewrite" parents="742"/>
      <j2 rule="rewrite" parents="859 833"/>
    </justification>
  </clause>

  <clause id="920">
    <literal><![CDATA[
      x'' = x * (y * y')
    ]]></literal>
    <justification jstring="[back_rewrite(886),rewrite(889(6))].">
      <j1 rule="back_rewrite" parents="886"/>
      <j2 rule="rewrite" parents="889"/>
    </justification>
  </clause>

  <clause id="1236">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[para(920(a,2),203(a,1,2,1)),rewrite(859(8))].">
      <j1 rule="para" parents="920 203"/>
      <j2 rule="rewrite" parents="859"/>
    </justification>
  </clause>

  <clause id="1253">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(identity)
    ]]></attribute>
    <justification jstring="[para(920(a,2),204(a,1)),rewrite(1236(3)),xx(a)].">
      <j1 rule="para" parents="920 204"/>
      <j2 rule="rewrite" parents="1236"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

<proof number="3" length="48" max_count="34">

<comments><![CDATA[
% Proof 3 at 0.31 (+ 0.00) seconds: associativity.
% Length of proof is 48.
% Level of proof is 24.
% Maximum clause weight is 34.
% Given clauses 47.
]]></comments>

  <clause id="1" type="goal">
    <literal><![CDATA[
      (x * y) * z = x * (y * z)
    ]]></literal>
    <attribute><![CDATA[
      label(associativity)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="4" type="assumption">
    <literal><![CDATA[
      x * (y * (((z * z') * (u * y)') * x))' = u
    ]]></literal>
    <attribute><![CDATA[
      label(gt_sax)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="5" type="deny">
    <literal><![CDATA[
      (c1 * c2) * c3 != c1 * (c2 * c3)
    ]]></literal>
    <attribute><![CDATA[
      label(associativity)
    ]]></attribute>
    <attribute><![CDATA[
      answer(associativity)
    ]]></attribute>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

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

  <clause id="10">
    <literal><![CDATA[
      x * ((((y * y') * (z * u)') * (v * v')) * (z * x))' = u
    ]]></literal>
    <justification jstring="[para(4(a,1),4(a,1,2,1,2,1))].">
      <j1 rule="para" parents="4 4"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      (x * (((y * y') * (z * x)') * ((u * u') * (v * w)')))' * (w * z)' = v
    ]]></literal>
    <justification jstring="[para(4(a,1),4(a,1,2,1,2))].">
      <j1 rule="para" parents="4 4"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      (x * (((y * y') * (z * x)') * u))' = v * ((((w * w') * z') * (v6 * v6')) * (u * v))'
    ]]></literal>
    <justification jstring="[para(9(a,1),4(a,1,2,1,2,1)),flip(a)].">
      <j1 rule="para" parents="9 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      ((x * x') * (y * z)') * (u * u') = v * ((y * (w * w')) * (z * v))'
    ]]></literal>
    <justification jstring="[para(10(a,1),4(a,1,2,1,2,1)),flip(a)].">
      <j1 rule="para" parents="10 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      (x * x') * (y * (z * ((u * (v * v')) * (y * z))'))' = u
    ]]></literal>
    <justification jstring="[para(24(a,1),4(a,1,2,1,2))].">
      <j1 rule="para" parents="24 4"/>
    </justification>
  </clause>

  <clause id="101">
    <literal><![CDATA[
      (x * (y * ((z * (u * u')) * (x * y))'))' * (v' * z)' = v
    ]]></literal>
    <justification jstring="[para(39(a,1),4(a,1,2,1,2))].">
      <j1 rule="para" parents="39 4"/>
    </justification>
  </clause>

  <clause id="107">
    <literal><![CDATA[
      x * (((y * y') * ((z * z') * x)') * u) = u
    ]]></literal>
    <justification jstring="[para(39(a,1),9(a,1,2,1,2)),rewrite(101(22))].">
      <j1 rule="para" parents="39 9"/>
      <j2 rule="rewrite" parents="101"/>
    </justification>
  </clause>

  <clause id="160">
    <literal><![CDATA[
      x * x' = y * y'
    ]]></literal>
    <justification jstring="[para(107(a,1),4(a,1,2,1))].">
      <j1 rule="para" parents="107 4"/>
    </justification>
  </clause>

  <clause id="164">
    <literal><![CDATA[
      (x * (((y * y') * (z * x)') * ((u * u') * ((v * v') * w)')))' = w * z
    ]]></literal>
    <justification jstring="[para(4(a,1),107(a,1,2)),flip(a)].">
      <j1 rule="para" parents="4 107"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="203">
    <literal><![CDATA[
      x * (y' * ((z * z') * x))' = y
    ]]></literal>
    <justification jstring="[para(160(a,1),4(a,1,2,1,2,1))].">
      <j1 rule="para" parents="160 4"/>
    </justification>
  </clause>

  <clause id="210">
    <literal><![CDATA[
      x * (((y * y') * (z * z')) * (u * x))' = u'
    ]]></literal>
    <justification jstring="[para(160(a,1),10(a,1,2,1,1,1))].">
      <j1 rule="para" parents="160 10"/>
    </justification>
  </clause>

  <clause id="224">
    <literal><![CDATA[
      (x * x')' * ((y * y') * z) = z
    ]]></literal>
    <justification jstring="[para(160(a,1),107(a,1,2,1))].">
      <j1 rule="para" parents="160 107"/>
    </justification>
  </clause>

  <clause id="253">
    <literal><![CDATA[
      (x * x')' * (y' * (z * z'))' = y
    ]]></literal>
    <justification jstring="[para(160(a,1),203(a,1,2,1,2))].">
      <j1 rule="para" parents="160 203"/>
    </justification>
  </clause>

  <clause id="278">
    <literal><![CDATA[
      (x * x')' = (y * y')' * (z * z')
    ]]></literal>
    <justification jstring="[para(160(a,1),224(a,1,2)),flip(a)].">
      <j1 rule="para" parents="160 224"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="419">
    <literal><![CDATA[
      (x * x')' = (y * y')'
    ]]></literal>
    <justification jstring="[para(278(a,2),224(a,1))].">
      <j1 rule="para" parents="278 224"/>
    </justification>
  </clause>

  <clause id="453">
    <literal><![CDATA[
      (x * x') * (y * y')' = z * z'
    ]]></literal>
    <justification jstring="[para(419(a,1),160(a,1,2))].">
      <j1 rule="para" parents="419 160"/>
    </justification>
  </clause>

  <clause id="485">
    <literal><![CDATA[
      ((x * x')' * y) * (z' * y)' = z
    ]]></literal>
    <justification jstring="[para(453(a,2),11(a,1,1,1,2,2,2,1)),rewrite(164(18))].">
      <j1 rule="para" parents="453 11"/>
      <j2 rule="rewrite" parents="164"/>
    </justification>
  </clause>

  <clause id="555">
    <literal><![CDATA[
      (x * x') * y' = (z * z') * y'
    ]]></literal>
    <justification jstring="[para(13(a,2),39(a,1,2,1,2)),rewrite(4(11))].">
      <j1 rule="para" parents="13 39"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="691">
    <literal><![CDATA[
      x * ((y * y')'' * (z * x))' = z'
    ]]></literal>
    <justification jstring="[para(485(a,1),4(a,1,2,1,2,1))].">
      <j1 rule="para" parents="485 4"/>
    </justification>
  </clause>

  <clause id="734">
    <literal><![CDATA[
      x * (y' * (x' * (z * z'))')' = y
    ]]></literal>
    <justification jstring="[para(253(a,1),485(a,1,1))].">
      <j1 rule="para" parents="253 485"/>
    </justification>
  </clause>

  <clause id="742">
    <literal><![CDATA[
      ((x * x')' * y'') * ((z * z') * (u * u')')' = y
    ]]></literal>
    <justification jstring="[para(453(a,2),485(a,1,2,1))].">
      <j1 rule="para" parents="453 485"/>
    </justification>
  </clause>

  <clause id="757">
    <literal><![CDATA[
      ((x * x') * ((y * y') * z')') * (u * u') = z''
    ]]></literal>
    <justification jstring="[para(555(a,1),24(a,1,1,2,1)),rewrite(210(21))].">
      <j1 rule="para" parents="555 24"/>
      <j2 rule="rewrite" parents="210"/>
    </justification>
  </clause>

  <clause id="833">
    <literal><![CDATA[
      (x * x') * (y * z')' = z * y'
    ]]></literal>
    <justification jstring="[para(4(a,1),734(a,1,2,1)),flip(a)].">
      <j1 rule="para" parents="4 734"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="855">
    <literal><![CDATA[
      (x' * (((y * y') * ((z * z') * u)')' * (v * v'))')' = u * x
    ]]></literal>
    <justification jstring="[para(734(a,1),107(a,1,2)),flip(a)].">
      <j1 rule="para" parents="734 107"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="859">
    <literal><![CDATA[
      (x * x')' * y'' = y
    ]]></literal>
    <justification jstring="[para(224(a,1),734(a,1,2,1,2,1)),rewrite(833(10))].">
      <j1 rule="para" parents="224 734"/>
      <j2 rule="rewrite" parents="833"/>
    </justification>
  </clause>

  <clause id="886">
    <literal><![CDATA[
      x'' = (x * (y * y')') * (z * z')
    ]]></literal>
    <justification jstring="[back_rewrite(757),rewrite(833(8)),flip(a)].">
      <j1 rule="back_rewrite" parents="757"/>
      <j2 rule="rewrite" parents="833"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="889">
    <literal><![CDATA[
      x * (y * y')' = x
    ]]></literal>
    <justification jstring="[back_rewrite(742),rewrite(859(6),833(6))].">
      <j1 rule="back_rewrite" parents="742"/>
      <j2 rule="rewrite" parents="859 833"/>
    </justification>
  </clause>

  <clause id="920">
    <literal><![CDATA[
      x'' = x * (y * y')
    ]]></literal>
    <justification jstring="[back_rewrite(886),rewrite(889(6))].">
      <j1 rule="back_rewrite" parents="886"/>
      <j2 rule="rewrite" parents="889"/>
    </justification>
  </clause>

  <clause id="1195">
    <literal><![CDATA[
      ((x * x') * (y * z)')' = u * (z''' * (((v * v') * y') * u))'
    ]]></literal>
    <justification jstring="[para(920(a,2),9(a,1,2,1,1,1)),flip(a)].">
      <j1 rule="para" parents="920 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1203">
    <literal><![CDATA[
      x' * (((y * y') * (x * z)') * (u * u'))''' = z
    ]]></literal>
    <justification jstring="[para(920(a,2),10(a,1,2,1))].">
      <j1 rule="para" parents="920 10"/>
    </justification>
  </clause>

  <clause id="1207">
    <literal><![CDATA[
      ((x * x') * (y * z)')'' = u * ((y * (v * v')) * (z * u))'
    ]]></literal>
    <justification jstring="[para(920(a,2),24(a,1))].">
      <j1 rule="para" parents="920 24"/>
    </justification>
  </clause>

  <clause id="1209">
    <literal><![CDATA[
      ((x * x') * (y * z)') * (u * u') = v * (y'' * (z * v))'
    ]]></literal>
    <justification jstring="[para(920(a,2),24(a,2,2,1,1))].">
      <j1 rule="para" parents="920 24"/>
    </justification>
  </clause>

  <clause id="1211">
    <literal><![CDATA[
      x' * (y * (z * z'))''' = ((u * u') * (y * x)') * (v * v')
    ]]></literal>
    <justification jstring="[para(920(a,2),24(a,2,2,1)),flip(a)].">
      <j1 rule="para" parents="920 24"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1236">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[para(920(a,2),203(a,1,2,1)),rewrite(859(8))].">
      <j1 rule="para" parents="920 203"/>
      <j2 rule="rewrite" parents="859"/>
    </justification>
  </clause>

  <clause id="1267">
    <literal><![CDATA[
      x * (y * y') = x
    ]]></literal>
    <justification jstring="[para(419(a,1),920(a,2,2,2)),rewrite(1236(2),889(6)),flip(a)].">
      <j1 rule="para" parents="419 920"/>
      <j2 rule="rewrite" parents="1236 889"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1288">
    <literal><![CDATA[
      (x * x') * ((y * y') * z)' = z'
    ]]></literal>
    <justification jstring="[para(920(a,2),691(a,1,2,1,2)),rewrite(1236(6),1236(6))].">
      <j1 rule="para" parents="920 691"/>
      <j2 rule="rewrite" parents="1236 1236"/>
    </justification>
  </clause>

  <clause id="1327">
    <literal><![CDATA[
      (x' * y')' = y * x
    ]]></literal>
    <justification jstring="[back_rewrite(855),rewrite(1288(8),1236(3),1267(4))].">
      <j1 rule="back_rewrite" parents="855"/>
      <j2 rule="rewrite" parents="1288 1236 1267"/>
    </justification>
  </clause>

  <clause id="1346">
    <literal><![CDATA[
      (x * x') * (y * z)' = z' * y'
    ]]></literal>
    <justification jstring="[back_rewrite(1211),rewrite(1267(4),1236(3),1267(11)),flip(a)].">
      <j1 rule="back_rewrite" parents="1211"/>
      <j2 rule="rewrite" parents="1267 1236 1267"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1347">
    <literal><![CDATA[
      x * (y * (z * x))' = z' * y'
    ]]></literal>
    <justification jstring="[back_rewrite(1209),rewrite(1346(5),1267(6),1236(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="1209"/>
      <j2 rule="rewrite" parents="1346 1267 1236"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1348">
    <literal><![CDATA[
      (x * y)' = y' * x'
    ]]></literal>
    <justification jstring="[back_rewrite(1207),rewrite(1346(5),1327(4),1267(5),1347(6))].">
      <j1 rule="back_rewrite" parents="1207"/>
      <j2 rule="rewrite" parents="1346 1327 1267 1347"/>
    </justification>
  </clause>

  <clause id="1349">
    <literal><![CDATA[
      x' * (x * y) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1203),rewrite(1348(5),1267(10),1348(8),1348(5),1236(3),1236(3),1348(5),1236(4),1267(5),1348(3),1348(5),1236(3),1236(3))].">
      <j1 rule="back_rewrite" parents="1203"/>
      <j2 rule="rewrite" parents="1348 1267 1348 1348 1236 1236 1348 1236 1267 1348 1348 1236 1236"/>
    </justification>
  </clause>

  <clause id="1354">
    <literal><![CDATA[
      x * ((x' * y) * z) = y * z
    ]]></literal>
    <justification jstring="[back_rewrite(1195),rewrite(1348(4),1348(7),1348(4),1236(2),1236(2),1348(4),1236(3),1267(4),1236(3),1348(9),1348(7),1348(7),1236(4),1348(5),1236(4),1267(5),1236(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="1195"/>
      <j2 rule="rewrite" parents="1348 1348 1348 1236 1236 1348 1236 1267 1236 1348 1348 1348 1236 1348 1236 1267 1236"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1586">
    <literal><![CDATA[
      (x * y) * z = x * (y * z)
    ]]></literal>
    <justification jstring="[para(1349(a,1),1354(a,1,2,1)),flip(a)].">
      <j1 rule="para" parents="1349 1354"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1587">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(associativity)
    ]]></attribute>
    <justification jstring="[resolve(1586,a,5,a)].">
      <j1 rule="resolve" parents="1586 5"/>
    </justification>
  </clause>

</proof>

</proofs>
