<?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 November-2006, November 2006.
Process 3295 was started by mccune on cleo.thornwood,
Wed Nov 22 11:21:52 2006
The command was "/home/mccune/bin/prover9 -f gtsax.in".
]]></heading>

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

<comments><![CDATA[
% Proof 1 at 0.01 (+ 0.00) seconds: inverse.
% Length of proof is 12.
% Level of proof is 7.
% Maximum clause weight is 29.
% Given clauses 10.
]]></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="13">
    <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(13(a,1),4(a,1,2,1,2))].">
      <j1 rule="para" parents="13 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="111">
    <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="125">
    <literal><![CDATA[
      x * x' = y * y'
    ]]></literal>
    <justification jstring="[para(111(a,1),4(a,1,2,1))].">
      <j1 rule="para" parents="111 4"/>
    </justification>
  </clause>

  <clause id="126">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(inverse)
    ]]></attribute>
    <justification jstring="[resolve(125,a,7,a)].">
      <j1 rule="resolve" parents="125 7"/>
    </justification>
  </clause>

</proof>

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

<comments><![CDATA[
% Proof 2 at 0.21 (+ 0.02) seconds: identity.
% Length of proof is 34.
% Level of proof is 18.
% Maximum clause weight is 34.
% Given clauses 39.
]]></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 * 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="15">
    <literal><![CDATA[
      (x * (((y * y') * (z * x)') * u))' = v * ((((w * w') * z') * (v6 * v6')) * (u * v))'
    ]]></literal>
    <justification jstring="[para(4(a,1),10(a,1,2,1,1,1,2,1)),flip(a)].">
      <j1 rule="para" parents="4 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      (x * x') * (y * (z * ((u * (v * v')) * (y * z))'))' = u
    ]]></literal>
    <justification jstring="[para(13(a,1),4(a,1,2,1,2))].">
      <j1 rule="para" parents="13 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="111">
    <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="125">
    <literal><![CDATA[
      x * x' = y * y'
    ]]></literal>
    <justification jstring="[para(111(a,1),4(a,1,2,1))].">
      <j1 rule="para" parents="111 4"/>
    </justification>
  </clause>

  <clause id="129">
    <literal><![CDATA[
      (x * (((y * y') * (z * x)') * ((u * u') * ((v * v') * w)')))' = w * z
    ]]></literal>
    <justification jstring="[para(4(a,1),111(a,1,2)),flip(a)].">
      <j1 rule="para" parents="4 111"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="202">
    <literal><![CDATA[
      x * (y' * ((z * z') * x))' = y
    ]]></literal>
    <justification jstring="[para(125(a,1),4(a,1,2,1,2,1))].">
      <j1 rule="para" parents="125 4"/>
    </justification>
  </clause>

  <clause id="203">
    <literal><![CDATA[
      c6 * (x * x') != c6
    ]]></literal>
    <attribute><![CDATA[
      answer(identity)
    ]]></attribute>
    <justification jstring="[para(125(a,1),8(a,1,2))].">
      <j1 rule="para" parents="125 8"/>
    </justification>
  </clause>

  <clause id="205">
    <literal><![CDATA[
      x * (((y * y') * (z * z')) * (u * x))' = u'
    ]]></literal>
    <justification jstring="[para(125(a,1),10(a,1,2,1,1,1))].">
      <j1 rule="para" parents="125 10"/>
    </justification>
  </clause>

  <clause id="218">
    <literal><![CDATA[
      (x * x')' * ((y * y') * z) = z
    ]]></literal>
    <justification jstring="[para(125(a,1),111(a,1,2,1))].">
      <j1 rule="para" parents="125 111"/>
    </justification>
  </clause>

  <clause id="253">
    <literal><![CDATA[
      (x * x')' * (y' * (z * z'))' = y
    ]]></literal>
    <justification jstring="[para(125(a,1),202(a,1,2,1,2))].">
      <j1 rule="para" parents="125 202"/>
    </justification>
  </clause>

  <clause id="278">
    <literal><![CDATA[
      (x * x')' = (y * y')' * (z * z')
    ]]></literal>
    <justification jstring="[para(125(a,1),218(a,1,2)),flip(a)].">
      <j1 rule="para" parents="125 218"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="407">
    <literal><![CDATA[
      (x * x')' = (y * y')'
    ]]></literal>
    <justification jstring="[para(278(a,2),218(a,1))].">
      <j1 rule="para" parents="278 218"/>
    </justification>
  </clause>

  <clause id="440">
    <literal><![CDATA[
      (x * x') * (y * y')' = z * z'
    ]]></literal>
    <justification jstring="[para(407(a,1),125(a,1,2))].">
      <j1 rule="para" parents="407 125"/>
    </justification>
  </clause>

  <clause id="561">
    <literal><![CDATA[
      ((x * x')' * y) * (z' * y)' = z
    ]]></literal>
    <justification jstring="[para(440(a,2),11(a,1,1,1,2,2,2,1)),rewrite(129(18))].">
      <j1 rule="para" parents="440 11"/>
      <j2 rule="rewrite" parents="129"/>
    </justification>
  </clause>

  <clause id="621">
    <literal><![CDATA[
      x * (y' * (x' * (z * z'))')' = y
    ]]></literal>
    <justification jstring="[para(253(a,1),561(a,1,1))].">
      <j1 rule="para" parents="253 561"/>
    </justification>
  </clause>

  <clause id="635">
    <literal><![CDATA[
      ((x * x')' * y'') * ((z * z') * (u * u')')' = y
    ]]></literal>
    <justification jstring="[para(440(a,2),561(a,1,2,1))].">
      <j1 rule="para" parents="440 561"/>
    </justification>
  </clause>

  <clause id="691">
    <literal><![CDATA[
      (x * x') * y' = (z * z') * y'
    ]]></literal>
    <justification jstring="[para(15(a,2),39(a,1,2,1,2)),rewrite(4(11))].">
      <j1 rule="para" parents="15 39"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="837">
    <literal><![CDATA[
      ((x * x') * ((y * y') * z')') * (u * u') = z''
    ]]></literal>
    <justification jstring="[para(691(a,1),13(a,1,1,2,1)),rewrite(205(21))].">
      <j1 rule="para" parents="691 13"/>
      <j2 rule="rewrite" parents="205"/>
    </justification>
  </clause>

  <clause id="1044">
    <literal><![CDATA[
      (x * x') * (y * z')' = z * y'
    ]]></literal>
    <justification jstring="[para(4(a,1),621(a,1,2,1)),flip(a)].">
      <j1 rule="para" parents="4 621"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1070">
    <literal><![CDATA[
      (x * x')' * y'' = y
    ]]></literal>
    <justification jstring="[para(218(a,1),621(a,1,2,1,2,1)),rewrite(1044(10))].">
      <j1 rule="para" parents="218 621"/>
      <j2 rule="rewrite" parents="1044"/>
    </justification>
  </clause>

  <clause id="1114">
    <literal><![CDATA[
      x'' = (x * (y * y')') * (z * z')
    ]]></literal>
    <justification jstring="[back_rewrite(837),rewrite(1044(8)),flip(a)].">
      <j1 rule="back_rewrite" parents="837"/>
      <j2 rule="rewrite" parents="1044"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1122">
    <literal><![CDATA[
      x * (y * y')' = x
    ]]></literal>
    <justification jstring="[back_rewrite(635),rewrite(1070(6),1044(6))].">
      <j1 rule="back_rewrite" parents="635"/>
      <j2 rule="rewrite" parents="1070 1044"/>
    </justification>
  </clause>

  <clause id="1167">
    <literal><![CDATA[
      x'' = x * (y * y')
    ]]></literal>
    <justification jstring="[back_rewrite(1114),rewrite(1122(6))].">
      <j1 rule="back_rewrite" parents="1114"/>
      <j2 rule="rewrite" parents="1122"/>
    </justification>
  </clause>

  <clause id="1508">
    <literal><![CDATA[
      c6'' != c6
    ]]></literal>
    <attribute><![CDATA[
      answer(identity)
    ]]></attribute>
    <justification jstring="[para(1167(a,2),203(a,1))].">
      <j1 rule="para" parents="1167 203"/>
    </justification>
  </clause>

  <clause id="1511">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[para(1167(a,2),202(a,1,2,1)),rewrite(1070(8))].">
      <j1 rule="para" parents="1167 202"/>
      <j2 rule="rewrite" parents="1070"/>
    </justification>
  </clause>

  <clause id="1512">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(identity)
    ]]></attribute>
    <justification jstring="[resolve(1511,a,1508,a)].">
      <j1 rule="resolve" parents="1511 1508"/>
    </justification>
  </clause>

</proof>

<proof number="3" length="67" max_count="66">

<comments><![CDATA[
% Proof 3 at 0.29 (+ 0.02) seconds: associativity.
% Length of proof is 67.
% Level of proof is 27.
% Maximum clause weight is 66.
% Given clauses 39.
]]></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 * 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="14">
    <literal><![CDATA[
      ((((x * x') * (y * z)') * (u * u')) * (y * ((v * v') * (w * v6)')))' * (v6 * z)' = w
    ]]></literal>
    <justification jstring="[para(10(a,1),4(a,1,2,1,2))].">
      <j1 rule="para" parents="10 4"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      (x * (((y * y') * (z * x)') * u))' = v * ((((w * w') * z') * (v6 * v6')) * (u * v))'
    ]]></literal>
    <justification jstring="[para(4(a,1),10(a,1,2,1,1,1,2,1)),flip(a)].">
      <j1 rule="para" parents="4 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      (x * (((y * y') * (z * x)') * u))' * ((((v * v') * (u * w)') * (v6 * v6')) * z)' = w
    ]]></literal>
    <justification jstring="[para(4(a,1),10(a,1,2,1,2))].">
      <j1 rule="para" parents="4 10"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      ((((x * x') * (y * z)') * (u * u')) * (y * v))' = w * ((((v6 * v6') * z') * (v7 * v7')) * (v * w))'
    ]]></literal>
    <justification jstring="[para(10(a,1),10(a,1,2,1,1,1,2,1)),flip(a)].">
      <j1 rule="para" parents="10 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      (x * x') * (y * (z * ((u * (v * v')) * (y * z))'))' = u
    ]]></literal>
    <justification jstring="[para(13(a,1),4(a,1,2,1,2))].">
      <j1 rule="para" parents="13 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="111">
    <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="125">
    <literal><![CDATA[
      x * x' = y * y'
    ]]></literal>
    <justification jstring="[para(111(a,1),4(a,1,2,1))].">
      <j1 rule="para" parents="111 4"/>
    </justification>
  </clause>

  <clause id="129">
    <literal><![CDATA[
      (x * (((y * y') * (z * x)') * ((u * u') * ((v * v') * w)')))' = w * z
    ]]></literal>
    <justification jstring="[para(4(a,1),111(a,1,2)),flip(a)].">
      <j1 rule="para" parents="4 111"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="202">
    <literal><![CDATA[
      x * (y' * ((z * z') * x))' = y
    ]]></literal>
    <justification jstring="[para(125(a,1),4(a,1,2,1,2,1))].">
      <j1 rule="para" parents="125 4"/>
    </justification>
  </clause>

  <clause id="205">
    <literal><![CDATA[
      x * (((y * y') * (z * z')) * (u * x))' = u'
    ]]></literal>
    <justification jstring="[para(125(a,1),10(a,1,2,1,1,1))].">
      <j1 rule="para" parents="125 10"/>
    </justification>
  </clause>

  <clause id="218">
    <literal><![CDATA[
      (x * x')' * ((y * y') * z) = z
    ]]></literal>
    <justification jstring="[para(125(a,1),111(a,1,2,1))].">
      <j1 rule="para" parents="125 111"/>
    </justification>
  </clause>

  <clause id="253">
    <literal><![CDATA[
      (x * x')' * (y' * (z * z'))' = y
    ]]></literal>
    <justification jstring="[para(125(a,1),202(a,1,2,1,2))].">
      <j1 rule="para" parents="125 202"/>
    </justification>
  </clause>

  <clause id="278">
    <literal><![CDATA[
      (x * x')' = (y * y')' * (z * z')
    ]]></literal>
    <justification jstring="[para(125(a,1),218(a,1,2)),flip(a)].">
      <j1 rule="para" parents="125 218"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="390">
    <literal><![CDATA[
      (x * x') * ((y * (z * z')) * (u * u')')'' = y
    ]]></literal>
    <justification jstring="[para(278(a,2),39(a,1,2,1,2,2,1,2)),rewrite(218(17))].">
      <j1 rule="para" parents="278 39"/>
      <j2 rule="rewrite" parents="218"/>
    </justification>
  </clause>

  <clause id="407">
    <literal><![CDATA[
      (x * x')' = (y * y')'
    ]]></literal>
    <justification jstring="[para(278(a,2),218(a,1))].">
      <j1 rule="para" parents="278 218"/>
    </justification>
  </clause>

  <clause id="440">
    <literal><![CDATA[
      (x * x') * (y * y')' = z * z'
    ]]></literal>
    <justification jstring="[para(407(a,1),125(a,1,2))].">
      <j1 rule="para" parents="407 125"/>
    </justification>
  </clause>

  <clause id="561">
    <literal><![CDATA[
      ((x * x')' * y) * (z' * y)' = z
    ]]></literal>
    <justification jstring="[para(440(a,2),11(a,1,1,1,2,2,2,1)),rewrite(129(18))].">
      <j1 rule="para" parents="440 11"/>
      <j2 rule="rewrite" parents="129"/>
    </justification>
  </clause>

  <clause id="578">
    <literal><![CDATA[
      x * ((y * y')'' * (z * x))' = z'
    ]]></literal>
    <justification jstring="[para(561(a,1),4(a,1,2,1,2,1))].">
      <j1 rule="para" parents="561 4"/>
    </justification>
  </clause>

  <clause id="621">
    <literal><![CDATA[
      x * (y' * (x' * (z * z'))')' = y
    ]]></literal>
    <justification jstring="[para(253(a,1),561(a,1,1))].">
      <j1 rule="para" parents="253 561"/>
    </justification>
  </clause>

  <clause id="635">
    <literal><![CDATA[
      ((x * x')' * y'') * ((z * z') * (u * u')')' = y
    ]]></literal>
    <justification jstring="[para(440(a,2),561(a,1,2,1))].">
      <j1 rule="para" parents="440 561"/>
    </justification>
  </clause>

  <clause id="691">
    <literal><![CDATA[
      (x * x') * y' = (z * z') * y'
    ]]></literal>
    <justification jstring="[para(15(a,2),39(a,1,2,1,2)),rewrite(4(11))].">
      <j1 rule="para" parents="15 39"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="837">
    <literal><![CDATA[
      ((x * x') * ((y * y') * z')') * (u * u') = z''
    ]]></literal>
    <justification jstring="[para(691(a,1),13(a,1,1,2,1)),rewrite(205(21))].">
      <j1 rule="para" parents="691 13"/>
      <j2 rule="rewrite" parents="205"/>
    </justification>
  </clause>

  <clause id="859">
    <literal><![CDATA[
      ((x * x')'' * (y * ((z * z') * (u * v)')))' = w * ((v * y')' * (((v6 * v6') * u') * w))'
    ]]></literal>
    <justification jstring="[para(578(a,1),9(a,1,2,1,1,1,2)),flip(a)].">
      <j1 rule="para" parents="578 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1044">
    <literal><![CDATA[
      (x * x') * (y * z')' = z * y'
    ]]></literal>
    <justification jstring="[para(4(a,1),621(a,1,2,1)),flip(a)].">
      <j1 rule="para" parents="4 621"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1061">
    <literal><![CDATA[
      (x' * (((y * y') * ((z * z') * u)')' * (v * v'))')' = u * x
    ]]></literal>
    <justification jstring="[para(621(a,1),111(a,1,2)),flip(a)].">
      <j1 rule="para" parents="621 111"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1070">
    <literal><![CDATA[
      (x * x')' * y'' = y
    ]]></literal>
    <justification jstring="[para(218(a,1),621(a,1,2,1,2,1)),rewrite(1044(10))].">
      <j1 rule="para" parents="218 621"/>
      <j2 rule="rewrite" parents="1044"/>
    </justification>
  </clause>

  <clause id="1114">
    <literal><![CDATA[
      x'' = (x * (y * y')') * (z * z')
    ]]></literal>
    <justification jstring="[back_rewrite(837),rewrite(1044(8)),flip(a)].">
      <j1 rule="back_rewrite" parents="837"/>
      <j2 rule="rewrite" parents="1044"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1122">
    <literal><![CDATA[
      x * (y * y')' = x
    ]]></literal>
    <justification jstring="[back_rewrite(635),rewrite(1070(6),1044(6))].">
      <j1 rule="back_rewrite" parents="635"/>
      <j2 rule="rewrite" parents="1070 1044"/>
    </justification>
  </clause>

  <clause id="1167">
    <literal><![CDATA[
      x'' = x * (y * y')
    ]]></literal>
    <justification jstring="[back_rewrite(1114),rewrite(1122(6))].">
      <j1 rule="back_rewrite" parents="1114"/>
      <j2 rule="rewrite" parents="1122"/>
    </justification>
  </clause>

  <clause id="1189">
    <literal><![CDATA[
      (x * x') * (y * (z * z'))'' = y
    ]]></literal>
    <justification jstring="[back_rewrite(390),rewrite(1122(9))].">
      <j1 rule="back_rewrite" parents="390"/>
      <j2 rule="rewrite" parents="1122"/>
    </justification>
  </clause>

  <clause id="1358">
    <literal><![CDATA[
      ((((x * x') * (((((y * y') * (z * u)') * (v * v')) * (z * ((w * w') * (v6 * v7)')))' * v8)') * (v9 * v9')) * v6)' = v10 * ((((v11 * v11') * v8') * (v12 * v12')) * ((v7 * u)' * v10))'
    ]]></literal>
    <justification jstring="[para(14(a,1),17(a,1,1,2))].">
      <j1 rule="para" parents="14 17"/>
    </justification>
  </clause>

  <clause id="1371">
    <literal><![CDATA[
      (x * ((((y * y') * (z * u)') * (v * v')) * (z * w))')' = v6 * ((((v7 * v7') * v8') * (v9 * v9')) * (((((v10 * v10') * u') * (v11 * v11')) * (w * ((v12 * v12') * (v8 * x)')))' * v6))'
    ]]></literal>
    <justification jstring="[para(17(a,2),15(a,1,1,2))].">
      <j1 rule="para" parents="17 15"/>
    </justification>
  </clause>

  <clause id="1398">
    <literal><![CDATA[
      ((((x * x') * (y * z)') * (u * u')) * v')' = w * ((((v6 * v6') * z') * (v7 * v7')) * (((v8 * v8')'' * (v * y))' * w))'
    ]]></literal>
    <justification jstring="[para(578(a,1),17(a,1,1,2))].">
      <j1 rule="para" parents="578 17"/>
    </justification>
  </clause>

  <clause id="1412">
    <literal><![CDATA[
      ((((x * x') * y') * (z * z')) * ((u * (((v * v') * (w * u)') * v6))' * v7))' = v8 * ((((v9 * v9') * ((((v10 * v10') * (v6 * y)') * (v11 * v11')) * w)'') * (v12 * v12')) * (v7 * v8))'
    ]]></literal>
    <justification jstring="[para(16(a,1),17(a,1,1,1,1,2,1))].">
      <j1 rule="para" parents="16 17"/>
    </justification>
  </clause>

  <clause id="1418">
    <literal><![CDATA[
      ((((x * x') * y') * (z * z')) * (u * v))' = w * ((((v6 * v6') * (y' * (u' * (v7 * v7'))')'') * (v8 * v8')) * (v * w))'
    ]]></literal>
    <justification jstring="[para(621(a,1),17(a,1,1,1,1,2,1))].">
      <j1 rule="para" parents="621 17"/>
    </justification>
  </clause>

  <clause id="1468">
    <literal><![CDATA[
      x' * (((y * y') * (x * z)') * (u * u'))''' = z
    ]]></literal>
    <justification jstring="[para(1167(a,2),10(a,1,2,1))].">
      <j1 rule="para" parents="1167 10"/>
    </justification>
  </clause>

  <clause id="1472">
    <literal><![CDATA[
      x * (((y * y') * (((z * z') * u''') * v))' * (((w * w') * u') * x))' = v
    ]]></literal>
    <justification jstring="[para(1167(a,2),9(a,1,2,1,1,1,2,1,2,1))].">
      <j1 rule="para" parents="1167 9"/>
    </justification>
  </clause>

  <clause id="1474">
    <literal><![CDATA[
      ((x * x') * (y * z)')' = u * (z''' * (((v * v') * y') * u))'
    ]]></literal>
    <justification jstring="[para(1167(a,2),9(a,1,2,1,1,1)),flip(a)].">
      <j1 rule="para" parents="1167 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1480">
    <literal><![CDATA[
      ((x * x') * (y * z)')'' = u * ((y * (v * v')) * (z * u))'
    ]]></literal>
    <justification jstring="[para(1167(a,2),13(a,1))].">
      <j1 rule="para" parents="1167 13"/>
    </justification>
  </clause>

  <clause id="1482">
    <literal><![CDATA[
      ((x * x') * (y * z)') * (u * u') = v * (y'' * (z * v))'
    ]]></literal>
    <justification jstring="[para(1167(a,2),13(a,2,2,1,1))].">
      <j1 rule="para" parents="1167 13"/>
    </justification>
  </clause>

  <clause id="1484">
    <literal><![CDATA[
      x' * (y * (z * z'))''' = ((u * u') * (y * x)') * (v * v')
    ]]></literal>
    <justification jstring="[para(1167(a,2),13(a,2,2,1)),flip(a)].">
      <j1 rule="para" parents="1167 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1511">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[para(1167(a,2),202(a,1,2,1)),rewrite(1070(8))].">
      <j1 rule="para" parents="1167 202"/>
      <j2 rule="rewrite" parents="1070"/>
    </justification>
  </clause>

  <clause id="1539">
    <literal><![CDATA[
      x * (y * y') = x
    ]]></literal>
    <justification jstring="[para(407(a,1),1167(a,2,2,2)),rewrite(1511(2),1122(6)),flip(a)].">
      <j1 rule="para" parents="407 1167"/>
      <j2 rule="rewrite" parents="1511 1122"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1566">
    <literal><![CDATA[
      (x * x') * ((y * y') * z)' = z'
    ]]></literal>
    <justification jstring="[para(1167(a,2),578(a,1,2,1,2)),rewrite(1511(6),1511(6))].">
      <j1 rule="para" parents="1167 578"/>
      <j2 rule="rewrite" parents="1511 1511"/>
    </justification>
  </clause>

  <clause id="1604">
    <literal><![CDATA[
      (x' * y')' = y * x
    ]]></literal>
    <justification jstring="[back_rewrite(1061),rewrite(1566(8),1511(3),1539(4))].">
      <j1 rule="back_rewrite" parents="1061"/>
      <j2 rule="rewrite" parents="1566 1511 1539"/>
    </justification>
  </clause>

  <clause id="1624">
    <literal><![CDATA[
      (x * x') * (y * z)' = z' * y'
    ]]></literal>
    <justification jstring="[back_rewrite(1484),rewrite(1539(4),1511(3),1539(11)),flip(a)].">
      <j1 rule="back_rewrite" parents="1484"/>
      <j2 rule="rewrite" parents="1539 1511 1539"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1625">
    <literal><![CDATA[
      x * (y * (z * x))' = z' * y'
    ]]></literal>
    <justification jstring="[back_rewrite(1482),rewrite(1624(5),1539(6),1511(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="1482"/>
      <j2 rule="rewrite" parents="1624 1539 1511"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1626">
    <literal><![CDATA[
      (x * y)' = y' * x'
    ]]></literal>
    <justification jstring="[back_rewrite(1480),rewrite(1624(5),1604(4),1539(5),1625(6))].">
      <j1 rule="back_rewrite" parents="1480"/>
      <j2 rule="rewrite" parents="1624 1604 1539 1625"/>
    </justification>
  </clause>

  <clause id="1628">
    <literal><![CDATA[
      x * ((x' * y) * z) = y * z
    ]]></literal>
    <justification jstring="[back_rewrite(1474),rewrite(1626(4),1626(7),1626(4),1511(2),1511(2),1626(4),1511(3),1539(4),1511(3),1626(9),1626(7),1626(7),1511(4),1626(5),1511(4),1539(5),1511(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="1474"/>
      <j2 rule="rewrite" parents="1626 1626 1626 1511 1511 1626 1511 1539 1511 1626 1626 1626 1511 1626 1511 1539 1511"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1630">
    <literal><![CDATA[
      x * (x' * y) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1472),rewrite(1511(6),1626(9),1626(6),1626(6),1511(3),1626(4),1511(3),1539(4),1626(5),1511(4),1539(5),1626(9),1626(6),1626(6),1511(3),1626(4),1511(3),1539(4),1626(5),1511(5),1628(6))].">
      <j1 rule="back_rewrite" parents="1472"/>
      <j2 rule="rewrite" parents="1511 1626 1626 1626 1511 1626 1511 1539 1626 1511 1539 1626 1626 1626 1511 1626 1511 1539 1626 1511 1628"/>
    </justification>
  </clause>

  <clause id="1632">
    <literal><![CDATA[
      x' * (x * y) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1468),rewrite(1626(5),1539(10),1626(8),1626(5),1511(3),1511(3),1626(5),1511(4),1539(5),1626(3),1626(5),1511(3),1511(3))].">
      <j1 rule="back_rewrite" parents="1468"/>
      <j2 rule="rewrite" parents="1626 1539 1626 1626 1511 1511 1626 1511 1539 1626 1626 1511 1511"/>
    </justification>
  </clause>

  <clause id="1637">
    <literal><![CDATA[
      (x' * y') * z = x' * (y' * z)
    ]]></literal>
    <justification jstring="[back_rewrite(1418),rewrite(1539(7),1626(7),1626(2),1626(8),1511(5),1626(6),1511(5),1539(6),1539(11),1511(9),1626(9),1511(9),1626(9),1511(9),1539(12),1626(12),1626(6),1626(13),1626(10),1511(10),1626(12),1511(11),1539(12),1628(11))].">
      <j1 rule="back_rewrite" parents="1418"/>
      <j2 rule="rewrite" parents="1539 1626 1626 1626 1511 1626 1511 1539 1539 1511 1626 1511 1626 1511 1539 1626 1626 1626 1626 1511 1626 1511 1539 1628"/>
    </justification>
  </clause>

  <clause id="1638">
    <literal><![CDATA[
      (x' * (y' * z)) * u = x' * ((y' * z) * u)
    ]]></literal>
    <justification jstring="[back_rewrite(1412),rewrite(1539(7),1626(8),1626(13),1626(12),1626(12),1626(9),1511(7),1511(7),1626(9),1511(8),1539(9),1626(12),1626(7),1626(7),1511(3),1626(5),1626(3),1511(6),1637(5),1630(6),1626(9),1511(6),1626(7),1511(6),1539(7),1626(11),1539(16),1626(15),1626(15),1626(12),1511(10),1511(10),1626(12),1511(11),1539(12),1626(11),1626(9),1511(12),1637(11),1539(15),1626(15),1626(7),1626(16),1626(13),1626(11),1511(11),1511(12),1626(14),1511(13),1539(14),1637(12),1630(13))].">
      <j1 rule="back_rewrite" parents="1412"/>
      <j2 rule="rewrite" parents="1539 1626 1626 1626 1626 1626 1511 1511 1626 1511 1539 1626 1626 1626 1511 1626 1626 1511 1637 1630 1626 1511 1626 1511 1539 1626 1539 1626 1626 1626 1511 1511 1626 1511 1539 1626 1626 1511 1637 1539 1626 1626 1626 1626 1626 1511 1511 1626 1511 1539 1637 1630"/>
    </justification>
  </clause>

  <clause id="1642">
    <literal><![CDATA[
      x * (((x' * y) * z) * u) = y * (z * u)
    ]]></literal>
    <justification jstring="[back_rewrite(1398),rewrite(1626(4),1539(9),1626(9),1511(2),1626(7),1626(4),1511(2),1511(2),1626(4),1511(3),1539(4),1539(9),1626(9),1511(8),1626(9),1511(8),1626(11),1626(8),1626(12),1511(11),1539(12),1637(10),1626(12),1626(7),1626(5),1511(5),1511(6),1626(10),1511(7),1626(8),1511(7),1539(8)),flip(a)].">
      <j1 rule="back_rewrite" parents="1398"/>
      <j2 rule="rewrite" parents="1626 1539 1626 1511 1626 1626 1511 1511 1626 1511 1539 1539 1626 1511 1626 1511 1626 1626 1626 1511 1539 1637 1626 1626 1626 1511 1511 1626 1511 1626 1511 1539"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1656">
    <literal><![CDATA[
      (x * x') * y = y
    ]]></literal>
    <justification jstring="[back_rewrite(1189),rewrite(1539(5),1511(4))].">
      <j1 rule="back_rewrite" parents="1189"/>
      <j2 rule="rewrite" parents="1539 1511"/>
    </justification>
  </clause>

  <clause id="1672">
    <literal><![CDATA[
      (x * y) * z' = x * (y * z')
    ]]></literal>
    <justification jstring="[back_rewrite(859),rewrite(1626(3),1511(2),1626(3),1511(2),1626(6),1656(8),1656(7),1626(5),1626(4),1511(2),1511(2),1626(6),1511(5),1656(9),1626(9),1626(6),1511(6),1626(8),1511(7),1628(9))].">
      <j1 rule="back_rewrite" parents="859"/>
      <j2 rule="rewrite" parents="1626 1511 1626 1511 1626 1656 1656 1626 1626 1511 1511 1626 1511 1656 1626 1626 1511 1626 1511 1628"/>
    </justification>
  </clause>

  <clause id="1700">
    <literal><![CDATA[
      (x' * (y * (z' * u'))) * u = x' * (y * z')
    ]]></literal>
    <justification jstring="[back_rewrite(1371),rewrite(1626(4),1656(6),1539(6),1637(5),1632(4),1626(3),1511(3),1626(4),1626(3),1511(3),1672(4),1672(8),1630(8),1539(8),1672(9),1630(9),1539(9),1626(10),1656(12),1626(12),1626(10),1626(9),1511(7),1511(7),1672(8),1511(10),1626(12),1626(10),1626(10),1626(10),1626(9),1511(8),1672(10),1511(14),1638(13),1630(14)),flip(a)].">
      <j1 rule="back_rewrite" parents="1371"/>
      <j2 rule="rewrite" parents="1626 1656 1539 1637 1632 1626 1511 1626 1626 1511 1672 1672 1630 1539 1672 1630 1539 1626 1656 1626 1626 1626 1511 1511 1672 1511 1626 1626 1626 1626 1626 1511 1672 1511 1638 1630"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1701">
    <literal><![CDATA[
      (x * y) * z = x * (y * z)
    ]]></literal>
    <justification jstring="[back_rewrite(1358),rewrite(1626(6),1656(8),1539(8),1626(9),1656(11),1637(10),1632(9),1626(8),1626(6),1511(4),1511(4),1511(5),1626(6),1626(6),1626(6),1656(10),1539(10),1700(8),1626(6),1626(4),1511(2),1511(2),1511(3),1672(6),1630(6),1539(6),1626(5),1637(7),1626(9),1626(7),1626(5),1511(5),1511(6),1511(7),1642(7))].">
      <j1 rule="back_rewrite" parents="1358"/>
      <j2 rule="rewrite" parents="1626 1656 1539 1626 1656 1637 1632 1626 1626 1511 1511 1511 1626 1626 1626 1656 1539 1700 1626 1626 1511 1511 1511 1672 1630 1539 1626 1637 1626 1626 1626 1511 1511 1511 1642"/>
    </justification>
  </clause>

  <clause id="1702">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(associativity)
    ]]></attribute>
    <justification jstring="[resolve(1701,a,5,a)].">
      <j1 rule="resolve" parents="1701 5"/>
    </justification>
  </clause>

</proof>

</proofs>
