<?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 August-2006A, August 2006.
Process 10640 was started by mccune on cleo.thornwood,
Sat Aug 12 20:57:37 2006
The command was "/home/mccune/bin/prover9 -f gtsax.in".
]]></heading>

<proof number="1" length="11" max_count="29">

<comments><![CDATA[
% Proof 1 at 0.01 (+ 0.00) seconds: inverse.
% Length of proof is 11.
% Level of proof is 7.
% Maximum clause weight is 29.
% Given clauses 10.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      x * (y * (((z * z') * (u * y)') * x))' = u
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="3">
    <literal><![CDATA[
      A * A' != B * B'
    ]]></literal>
    <attribute><![CDATA[
      answer(inverse)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      B * B' != A * A'
    ]]></literal>
    <attribute><![CDATA[
      answer(inverse)
    ]]></attribute>
    <justification jstring="[copy(3),flip(a)].">
      <j1 rule="copy" parents="3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="6">
    <literal><![CDATA[
      x * ((y * (((z * z') * (u * y)') * v))' * (((w * w') * u') * x))' = v
    ]]></literal>
    <justification jstring="[para(1(a,1),1(a,1,2,1,2,1,2,1))].">
      <j1 rule="para" parents="1 1"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      x * ((((y * y') * (z * u)') * (v * v')) * (z * x))' = u
    ]]></literal>
    <justification jstring="[para(1(a,1),1(a,1,2,1,2,1))].">
      <j1 rule="para" parents="1 1"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      ((x * x') * (y * z)') * (u * u') = v * ((y * (w * w')) * (z * v))'
    ]]></literal>
    <justification jstring="[para(7(a,1),1(a,1,2,1,2,1)),flip(a)].">
      <j1 rule="para" parents="7 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      (x * x') * (y * (z * ((u * (v * v')) * (y * z))'))' = u
    ]]></literal>
    <justification jstring="[para(10(a,1),1(a,1,2,1,2))].">
      <j1 rule="para" parents="10 1"/>
    </justification>
  </clause>

  <clause id="98">
    <literal><![CDATA[
      (x * (y * ((z * (u * u')) * (x * y))'))' * (v' * z)' = v
    ]]></literal>
    <justification jstring="[para(36(a,1),1(a,1,2,1,2))].">
      <j1 rule="para" parents="36 1"/>
    </justification>
  </clause>

  <clause id="108">
    <literal><![CDATA[
      x * (((y * y') * ((z * z') * x)') * u) = u
    ]]></literal>
    <justification jstring="[para(36(a,1),6(a,1,2,1,2)),rewrite(98(22))].">
      <j1 rule="para" parents="36 6"/>
      <j2 rule="rewrite" parents="98"/>
    </justification>
  </clause>

  <clause id="122">
    <literal><![CDATA[
      x * x' = y * y'
    ]]></literal>
    <justification jstring="[para(108(a,1),1(a,1,2,1))].">
      <j1 rule="para" parents="108 1"/>
    </justification>
  </clause>

  <clause id="123">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(inverse)
    ]]></attribute>
    <justification jstring="[resolve(122,a,4,a)].">
      <j1 rule="resolve" parents="122 4"/>
    </justification>
  </clause>

</proof>

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

<comments><![CDATA[
% Proof 2 at 0.23 (+ 0.01) seconds: identity.
% Length of proof is 33.
% Level of proof is 18.
% Maximum clause weight is 34.
% Given clauses 39.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      x * (y * (((z * z') * (u * y)') * x))' = u
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="5">
    <literal><![CDATA[
      A * (B * B') != A
    ]]></literal>
    <attribute><![CDATA[
      answer(identity)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6">
    <literal><![CDATA[
      x * ((y * (((z * z') * (u * y)') * v))' * (((w * w') * u') * x))' = v
    ]]></literal>
    <justification jstring="[para(1(a,1),1(a,1,2,1,2,1,2,1))].">
      <j1 rule="para" parents="1 1"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      x * ((((y * y') * (z * u)') * (v * v')) * (z * x))' = u
    ]]></literal>
    <justification jstring="[para(1(a,1),1(a,1,2,1,2,1))].">
      <j1 rule="para" parents="1 1"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      (x * (((y * y') * (z * x)') * ((u * u') * (v * w)')))' * (w * z)' = v
    ]]></literal>
    <justification jstring="[para(1(a,1),1(a,1,2,1,2))].">
      <j1 rule="para" parents="1 1"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      ((x * x') * (y * z)') * (u * u') = v * ((y * (w * w')) * (z * v))'
    ]]></literal>
    <justification jstring="[para(7(a,1),1(a,1,2,1,2,1)),flip(a)].">
      <j1 rule="para" parents="7 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      (x * (((y * y') * (z * x)') * u))' = v * ((((w * w') * z') * (v6 * v6')) * (u * v))'
    ]]></literal>
    <justification jstring="[para(1(a,1),7(a,1,2,1,1,1,2,1)),flip(a)].">
      <j1 rule="para" parents="1 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      (x * x') * (y * (z * ((u * (v * v')) * (y * z))'))' = u
    ]]></literal>
    <justification jstring="[para(10(a,1),1(a,1,2,1,2))].">
      <j1 rule="para" parents="10 1"/>
    </justification>
  </clause>

  <clause id="98">
    <literal><![CDATA[
      (x * (y * ((z * (u * u')) * (x * y))'))' * (v' * z)' = v
    ]]></literal>
    <justification jstring="[para(36(a,1),1(a,1,2,1,2))].">
      <j1 rule="para" parents="36 1"/>
    </justification>
  </clause>

  <clause id="108">
    <literal><![CDATA[
      x * (((y * y') * ((z * z') * x)') * u) = u
    ]]></literal>
    <justification jstring="[para(36(a,1),6(a,1,2,1,2)),rewrite(98(22))].">
      <j1 rule="para" parents="36 6"/>
      <j2 rule="rewrite" parents="98"/>
    </justification>
  </clause>

  <clause id="122">
    <literal><![CDATA[
      x * x' = y * y'
    ]]></literal>
    <justification jstring="[para(108(a,1),1(a,1,2,1))].">
      <j1 rule="para" parents="108 1"/>
    </justification>
  </clause>

  <clause id="126">
    <literal><![CDATA[
      (x * (((y * y') * (z * x)') * ((u * u') * ((v * v') * w)')))' = w * z
    ]]></literal>
    <justification jstring="[para(1(a,1),108(a,1,2)),flip(a)].">
      <j1 rule="para" parents="1 108"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="199">
    <literal><![CDATA[
      x * (y' * ((z * z') * x))' = y
    ]]></literal>
    <justification jstring="[para(122(a,1),1(a,1,2,1,2,1))].">
      <j1 rule="para" parents="122 1"/>
    </justification>
  </clause>

  <clause id="200">
    <literal><![CDATA[
      A * (x * x') != A
    ]]></literal>
    <attribute><![CDATA[
      answer(identity)
    ]]></attribute>
    <justification jstring="[para(122(a,1),5(a,1,2))].">
      <j1 rule="para" parents="122 5"/>
    </justification>
  </clause>

  <clause id="202">
    <literal><![CDATA[
      x * (((y * y') * (z * z')) * (u * x))' = u'
    ]]></literal>
    <justification jstring="[para(122(a,1),7(a,1,2,1,1,1))].">
      <j1 rule="para" parents="122 7"/>
    </justification>
  </clause>

  <clause id="215">
    <literal><![CDATA[
      (x * x')' * ((y * y') * z) = z
    ]]></literal>
    <justification jstring="[para(122(a,1),108(a,1,2,1))].">
      <j1 rule="para" parents="122 108"/>
    </justification>
  </clause>

  <clause id="250">
    <literal><![CDATA[
      (x * x')' * (y' * (z * z'))' = y
    ]]></literal>
    <justification jstring="[para(122(a,1),199(a,1,2,1,2))].">
      <j1 rule="para" parents="122 199"/>
    </justification>
  </clause>

  <clause id="275">
    <literal><![CDATA[
      (x * x')' = (y * y')' * (z * z')
    ]]></literal>
    <justification jstring="[para(122(a,1),215(a,1,2)),flip(a)].">
      <j1 rule="para" parents="122 215"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="404">
    <literal><![CDATA[
      (x * x')' = (y * y')'
    ]]></literal>
    <justification jstring="[para(275(a,2),215(a,1))].">
      <j1 rule="para" parents="275 215"/>
    </justification>
  </clause>

  <clause id="437">
    <literal><![CDATA[
      (x * x') * (y * y')' = z * z'
    ]]></literal>
    <justification jstring="[para(404(a,1),122(a,1,2))].">
      <j1 rule="para" parents="404 122"/>
    </justification>
  </clause>

  <clause id="558">
    <literal><![CDATA[
      ((x * x')' * y) * (z' * y)' = z
    ]]></literal>
    <justification jstring="[para(437(a,2),8(a,1,1,1,2,2,2,1)),rewrite(126(18))].">
      <j1 rule="para" parents="437 8"/>
      <j2 rule="rewrite" parents="126"/>
    </justification>
  </clause>

  <clause id="618">
    <literal><![CDATA[
      x * (y' * (x' * (z * z'))')' = y
    ]]></literal>
    <justification jstring="[para(250(a,1),558(a,1,1))].">
      <j1 rule="para" parents="250 558"/>
    </justification>
  </clause>

  <clause id="632">
    <literal><![CDATA[
      ((x * x')' * y'') * ((z * z') * (u * u')')' = y
    ]]></literal>
    <justification jstring="[para(437(a,2),558(a,1,2,1))].">
      <j1 rule="para" parents="437 558"/>
    </justification>
  </clause>

  <clause id="688">
    <literal><![CDATA[
      (x * x') * y' = (z * z') * y'
    ]]></literal>
    <justification jstring="[para(12(a,2),36(a,1,2,1,2)),rewrite(1(11))].">
      <j1 rule="para" parents="12 36"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="834">
    <literal><![CDATA[
      ((x * x') * ((y * y') * z')') * (u * u') = z''
    ]]></literal>
    <justification jstring="[para(688(a,1),10(a,1,1,2,1)),rewrite(202(21))].">
      <j1 rule="para" parents="688 10"/>
      <j2 rule="rewrite" parents="202"/>
    </justification>
  </clause>

  <clause id="1041">
    <literal><![CDATA[
      (x * x') * (y * z')' = z * y'
    ]]></literal>
    <justification jstring="[para(1(a,1),618(a,1,2,1)),flip(a)].">
      <j1 rule="para" parents="1 618"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1067">
    <literal><![CDATA[
      (x * x')' * y'' = y
    ]]></literal>
    <justification jstring="[para(215(a,1),618(a,1,2,1,2,1)),rewrite(1041(10))].">
      <j1 rule="para" parents="215 618"/>
      <j2 rule="rewrite" parents="1041"/>
    </justification>
  </clause>

  <clause id="1111">
    <literal><![CDATA[
      x'' = (x * (y * y')') * (z * z')
    ]]></literal>
    <justification jstring="[back_rewrite(834),rewrite(1041(8)),flip(a)].">
      <j1 rule="back_rewrite" parents="834"/>
      <j2 rule="rewrite" parents="1041"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1119">
    <literal><![CDATA[
      x * (y * y')' = x
    ]]></literal>
    <justification jstring="[back_rewrite(632),rewrite(1067(6),1041(6))].">
      <j1 rule="back_rewrite" parents="632"/>
      <j2 rule="rewrite" parents="1067 1041"/>
    </justification>
  </clause>

  <clause id="1164">
    <literal><![CDATA[
      x'' = x * (y * y')
    ]]></literal>
    <justification jstring="[back_rewrite(1111),rewrite(1119(6))].">
      <j1 rule="back_rewrite" parents="1111"/>
      <j2 rule="rewrite" parents="1119"/>
    </justification>
  </clause>

  <clause id="1505">
    <literal><![CDATA[
      A'' != A
    ]]></literal>
    <attribute><![CDATA[
      answer(identity)
    ]]></attribute>
    <justification jstring="[para(1164(a,2),200(a,1))].">
      <j1 rule="para" parents="1164 200"/>
    </justification>
  </clause>

  <clause id="1508">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[para(1164(a,2),199(a,1,2,1)),rewrite(1067(8))].">
      <j1 rule="para" parents="1164 199"/>
      <j2 rule="rewrite" parents="1067"/>
    </justification>
  </clause>

  <clause id="1509">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(identity)
    ]]></attribute>
    <justification jstring="[resolve(1508,a,1505,a)].">
      <j1 rule="resolve" parents="1508 1505"/>
    </justification>
  </clause>

</proof>

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

<comments><![CDATA[
% Proof 3 at 0.31 (+ 0.01) seconds: associativity.
% Length of proof is 66.
% Level of proof is 27.
% Maximum clause weight is 66.
% Given clauses 39.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      x * (y * (((z * z') * (u * y)') * x))' = u
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="2">
    <literal><![CDATA[
      (A * B) * C != A * (B * C)
    ]]></literal>
    <attribute><![CDATA[
      answer(associativity)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6">
    <literal><![CDATA[
      x * ((y * (((z * z') * (u * y)') * v))' * (((w * w') * u') * x))' = v
    ]]></literal>
    <justification jstring="[para(1(a,1),1(a,1,2,1,2,1,2,1))].">
      <j1 rule="para" parents="1 1"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      x * ((((y * y') * (z * u)') * (v * v')) * (z * x))' = u
    ]]></literal>
    <justification jstring="[para(1(a,1),1(a,1,2,1,2,1))].">
      <j1 rule="para" parents="1 1"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      (x * (((y * y') * (z * x)') * ((u * u') * (v * w)')))' * (w * z)' = v
    ]]></literal>
    <justification jstring="[para(1(a,1),1(a,1,2,1,2))].">
      <j1 rule="para" parents="1 1"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      ((x * x') * (y * z)') * (u * u') = v * ((y * (w * w')) * (z * v))'
    ]]></literal>
    <justification jstring="[para(7(a,1),1(a,1,2,1,2,1)),flip(a)].">
      <j1 rule="para" parents="7 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      ((((x * x') * (y * z)') * (u * u')) * (y * ((v * v') * (w * v6)')))' * (v6 * z)' = w
    ]]></literal>
    <justification jstring="[para(7(a,1),1(a,1,2,1,2))].">
      <j1 rule="para" parents="7 1"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      (x * (((y * y') * (z * x)') * u))' = v * ((((w * w') * z') * (v6 * v6')) * (u * v))'
    ]]></literal>
    <justification jstring="[para(1(a,1),7(a,1,2,1,1,1,2,1)),flip(a)].">
      <j1 rule="para" parents="1 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      (x * (((y * y') * (z * x)') * u))' * ((((v * v') * (u * w)') * (v6 * v6')) * z)' = w
    ]]></literal>
    <justification jstring="[para(1(a,1),7(a,1,2,1,2))].">
      <j1 rule="para" parents="1 7"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      ((((x * x') * (y * z)') * (u * u')) * (y * v))' = w * ((((v6 * v6') * z') * (v7 * v7')) * (v * w))'
    ]]></literal>
    <justification jstring="[para(7(a,1),7(a,1,2,1,1,1,2,1)),flip(a)].">
      <j1 rule="para" parents="7 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      (x * x') * (y * (z * ((u * (v * v')) * (y * z))'))' = u
    ]]></literal>
    <justification jstring="[para(10(a,1),1(a,1,2,1,2))].">
      <j1 rule="para" parents="10 1"/>
    </justification>
  </clause>

  <clause id="98">
    <literal><![CDATA[
      (x * (y * ((z * (u * u')) * (x * y))'))' * (v' * z)' = v
    ]]></literal>
    <justification jstring="[para(36(a,1),1(a,1,2,1,2))].">
      <j1 rule="para" parents="36 1"/>
    </justification>
  </clause>

  <clause id="108">
    <literal><![CDATA[
      x * (((y * y') * ((z * z') * x)') * u) = u
    ]]></literal>
    <justification jstring="[para(36(a,1),6(a,1,2,1,2)),rewrite(98(22))].">
      <j1 rule="para" parents="36 6"/>
      <j2 rule="rewrite" parents="98"/>
    </justification>
  </clause>

  <clause id="122">
    <literal><![CDATA[
      x * x' = y * y'
    ]]></literal>
    <justification jstring="[para(108(a,1),1(a,1,2,1))].">
      <j1 rule="para" parents="108 1"/>
    </justification>
  </clause>

  <clause id="126">
    <literal><![CDATA[
      (x * (((y * y') * (z * x)') * ((u * u') * ((v * v') * w)')))' = w * z
    ]]></literal>
    <justification jstring="[para(1(a,1),108(a,1,2)),flip(a)].">
      <j1 rule="para" parents="1 108"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="199">
    <literal><![CDATA[
      x * (y' * ((z * z') * x))' = y
    ]]></literal>
    <justification jstring="[para(122(a,1),1(a,1,2,1,2,1))].">
      <j1 rule="para" parents="122 1"/>
    </justification>
  </clause>

  <clause id="202">
    <literal><![CDATA[
      x * (((y * y') * (z * z')) * (u * x))' = u'
    ]]></literal>
    <justification jstring="[para(122(a,1),7(a,1,2,1,1,1))].">
      <j1 rule="para" parents="122 7"/>
    </justification>
  </clause>

  <clause id="215">
    <literal><![CDATA[
      (x * x')' * ((y * y') * z) = z
    ]]></literal>
    <justification jstring="[para(122(a,1),108(a,1,2,1))].">
      <j1 rule="para" parents="122 108"/>
    </justification>
  </clause>

  <clause id="250">
    <literal><![CDATA[
      (x * x')' * (y' * (z * z'))' = y
    ]]></literal>
    <justification jstring="[para(122(a,1),199(a,1,2,1,2))].">
      <j1 rule="para" parents="122 199"/>
    </justification>
  </clause>

  <clause id="275">
    <literal><![CDATA[
      (x * x')' = (y * y')' * (z * z')
    ]]></literal>
    <justification jstring="[para(122(a,1),215(a,1,2)),flip(a)].">
      <j1 rule="para" parents="122 215"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="387">
    <literal><![CDATA[
      (x * x') * ((y * (z * z')) * (u * u')')'' = y
    ]]></literal>
    <justification jstring="[para(275(a,2),36(a,1,2,1,2,2,1,2)),rewrite(215(17))].">
      <j1 rule="para" parents="275 36"/>
      <j2 rule="rewrite" parents="215"/>
    </justification>
  </clause>

  <clause id="404">
    <literal><![CDATA[
      (x * x')' = (y * y')'
    ]]></literal>
    <justification jstring="[para(275(a,2),215(a,1))].">
      <j1 rule="para" parents="275 215"/>
    </justification>
  </clause>

  <clause id="437">
    <literal><![CDATA[
      (x * x') * (y * y')' = z * z'
    ]]></literal>
    <justification jstring="[para(404(a,1),122(a,1,2))].">
      <j1 rule="para" parents="404 122"/>
    </justification>
  </clause>

  <clause id="558">
    <literal><![CDATA[
      ((x * x')' * y) * (z' * y)' = z
    ]]></literal>
    <justification jstring="[para(437(a,2),8(a,1,1,1,2,2,2,1)),rewrite(126(18))].">
      <j1 rule="para" parents="437 8"/>
      <j2 rule="rewrite" parents="126"/>
    </justification>
  </clause>

  <clause id="575">
    <literal><![CDATA[
      x * ((y * y')'' * (z * x))' = z'
    ]]></literal>
    <justification jstring="[para(558(a,1),1(a,1,2,1,2,1))].">
      <j1 rule="para" parents="558 1"/>
    </justification>
  </clause>

  <clause id="618">
    <literal><![CDATA[
      x * (y' * (x' * (z * z'))')' = y
    ]]></literal>
    <justification jstring="[para(250(a,1),558(a,1,1))].">
      <j1 rule="para" parents="250 558"/>
    </justification>
  </clause>

  <clause id="632">
    <literal><![CDATA[
      ((x * x')' * y'') * ((z * z') * (u * u')')' = y
    ]]></literal>
    <justification jstring="[para(437(a,2),558(a,1,2,1))].">
      <j1 rule="para" parents="437 558"/>
    </justification>
  </clause>

  <clause id="688">
    <literal><![CDATA[
      (x * x') * y' = (z * z') * y'
    ]]></literal>
    <justification jstring="[para(12(a,2),36(a,1,2,1,2)),rewrite(1(11))].">
      <j1 rule="para" parents="12 36"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="834">
    <literal><![CDATA[
      ((x * x') * ((y * y') * z')') * (u * u') = z''
    ]]></literal>
    <justification jstring="[para(688(a,1),10(a,1,1,2,1)),rewrite(202(21))].">
      <j1 rule="para" parents="688 10"/>
      <j2 rule="rewrite" parents="202"/>
    </justification>
  </clause>

  <clause id="856">
    <literal><![CDATA[
      ((x * x')'' * (y * ((z * z') * (u * v)')))' = w * ((v * y')' * (((v6 * v6') * u') * w))'
    ]]></literal>
    <justification jstring="[para(575(a,1),6(a,1,2,1,1,1,2)),flip(a)].">
      <j1 rule="para" parents="575 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1041">
    <literal><![CDATA[
      (x * x') * (y * z')' = z * y'
    ]]></literal>
    <justification jstring="[para(1(a,1),618(a,1,2,1)),flip(a)].">
      <j1 rule="para" parents="1 618"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1058">
    <literal><![CDATA[
      (x' * (((y * y') * ((z * z') * u)')' * (v * v'))')' = u * x
    ]]></literal>
    <justification jstring="[para(618(a,1),108(a,1,2)),flip(a)].">
      <j1 rule="para" parents="618 108"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1067">
    <literal><![CDATA[
      (x * x')' * y'' = y
    ]]></literal>
    <justification jstring="[para(215(a,1),618(a,1,2,1,2,1)),rewrite(1041(10))].">
      <j1 rule="para" parents="215 618"/>
      <j2 rule="rewrite" parents="1041"/>
    </justification>
  </clause>

  <clause id="1111">
    <literal><![CDATA[
      x'' = (x * (y * y')') * (z * z')
    ]]></literal>
    <justification jstring="[back_rewrite(834),rewrite(1041(8)),flip(a)].">
      <j1 rule="back_rewrite" parents="834"/>
      <j2 rule="rewrite" parents="1041"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1119">
    <literal><![CDATA[
      x * (y * y')' = x
    ]]></literal>
    <justification jstring="[back_rewrite(632),rewrite(1067(6),1041(6))].">
      <j1 rule="back_rewrite" parents="632"/>
      <j2 rule="rewrite" parents="1067 1041"/>
    </justification>
  </clause>

  <clause id="1164">
    <literal><![CDATA[
      x'' = x * (y * y')
    ]]></literal>
    <justification jstring="[back_rewrite(1111),rewrite(1119(6))].">
      <j1 rule="back_rewrite" parents="1111"/>
      <j2 rule="rewrite" parents="1119"/>
    </justification>
  </clause>

  <clause id="1186">
    <literal><![CDATA[
      (x * x') * (y * (z * z'))'' = y
    ]]></literal>
    <justification jstring="[back_rewrite(387),rewrite(1119(9))].">
      <j1 rule="back_rewrite" parents="387"/>
      <j2 rule="rewrite" parents="1119"/>
    </justification>
  </clause>

  <clause id="1355">
    <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(11(a,1),14(a,1,1,2))].">
      <j1 rule="para" parents="11 14"/>
    </justification>
  </clause>

  <clause id="1368">
    <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(14(a,2),12(a,1,1,2))].">
      <j1 rule="para" parents="14 12"/>
    </justification>
  </clause>

  <clause id="1395">
    <literal><![CDATA[
      ((((x * x') * (y * z)') * (u * u')) * v')' = w * ((((v6 * v6') * z') * (v7 * v7')) * (((v8 * v8')'' * (v * y))' * w))'
    ]]></literal>
    <justification jstring="[para(575(a,1),14(a,1,1,2))].">
      <j1 rule="para" parents="575 14"/>
    </justification>
  </clause>

  <clause id="1409">
    <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(13(a,1),14(a,1,1,1,1,2,1))].">
      <j1 rule="para" parents="13 14"/>
    </justification>
  </clause>

  <clause id="1415">
    <literal><![CDATA[
      ((((x * x') * y') * (z * z')) * (u * v))' = w * ((((v6 * v6') * (y' * (u' * (v7 * v7'))')'') * (v8 * v8')) * (v * w))'
    ]]></literal>
    <justification jstring="[para(618(a,1),14(a,1,1,1,1,2,1))].">
      <j1 rule="para" parents="618 14"/>
    </justification>
  </clause>

  <clause id="1465">
    <literal><![CDATA[
      x' * (((y * y') * (x * z)') * (u * u'))''' = z
    ]]></literal>
    <justification jstring="[para(1164(a,2),7(a,1,2,1))].">
      <j1 rule="para" parents="1164 7"/>
    </justification>
  </clause>

  <clause id="1469">
    <literal><![CDATA[
      x * (((y * y') * (((z * z') * u''') * v))' * (((w * w') * u') * x))' = v
    ]]></literal>
    <justification jstring="[para(1164(a,2),6(a,1,2,1,1,1,2,1,2,1))].">
      <j1 rule="para" parents="1164 6"/>
    </justification>
  </clause>

  <clause id="1471">
    <literal><![CDATA[
      ((x * x') * (y * z)')' = u * (z''' * (((v * v') * y') * u))'
    ]]></literal>
    <justification jstring="[para(1164(a,2),6(a,1,2,1,1,1)),flip(a)].">
      <j1 rule="para" parents="1164 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1477">
    <literal><![CDATA[
      ((x * x') * (y * z)')'' = u * ((y * (v * v')) * (z * u))'
    ]]></literal>
    <justification jstring="[para(1164(a,2),10(a,1))].">
      <j1 rule="para" parents="1164 10"/>
    </justification>
  </clause>

  <clause id="1479">
    <literal><![CDATA[
      ((x * x') * (y * z)') * (u * u') = v * (y'' * (z * v))'
    ]]></literal>
    <justification jstring="[para(1164(a,2),10(a,2,2,1,1))].">
      <j1 rule="para" parents="1164 10"/>
    </justification>
  </clause>

  <clause id="1481">
    <literal><![CDATA[
      x' * (y * (z * z'))''' = ((u * u') * (y * x)') * (v * v')
    ]]></literal>
    <justification jstring="[para(1164(a,2),10(a,2,2,1)),flip(a)].">
      <j1 rule="para" parents="1164 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1508">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[para(1164(a,2),199(a,1,2,1)),rewrite(1067(8))].">
      <j1 rule="para" parents="1164 199"/>
      <j2 rule="rewrite" parents="1067"/>
    </justification>
  </clause>

  <clause id="1536">
    <literal><![CDATA[
      x * (y * y') = x
    ]]></literal>
    <justification jstring="[para(404(a,1),1164(a,2,2,2)),rewrite(1508(2),1119(6)),flip(a)].">
      <j1 rule="para" parents="404 1164"/>
      <j2 rule="rewrite" parents="1508 1119"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1563">
    <literal><![CDATA[
      (x * x') * ((y * y') * z)' = z'
    ]]></literal>
    <justification jstring="[para(1164(a,2),575(a,1,2,1,2)),rewrite(1508(6),1508(6))].">
      <j1 rule="para" parents="1164 575"/>
      <j2 rule="rewrite" parents="1508 1508"/>
    </justification>
  </clause>

  <clause id="1601">
    <literal><![CDATA[
      (x' * y')' = y * x
    ]]></literal>
    <justification jstring="[back_rewrite(1058),rewrite(1563(8),1508(3),1536(4))].">
      <j1 rule="back_rewrite" parents="1058"/>
      <j2 rule="rewrite" parents="1563 1508 1536"/>
    </justification>
  </clause>

  <clause id="1621">
    <literal><![CDATA[
      (x * x') * (y * z)' = z' * y'
    ]]></literal>
    <justification jstring="[back_rewrite(1481),rewrite(1536(4),1508(3),1536(11)),flip(a)].">
      <j1 rule="back_rewrite" parents="1481"/>
      <j2 rule="rewrite" parents="1536 1508 1536"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1622">
    <literal><![CDATA[
      x * (y * (z * x))' = z' * y'
    ]]></literal>
    <justification jstring="[back_rewrite(1479),rewrite(1621(5),1536(6),1508(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="1479"/>
      <j2 rule="rewrite" parents="1621 1536 1508"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1623">
    <literal><![CDATA[
      (x * y)' = y' * x'
    ]]></literal>
    <justification jstring="[back_rewrite(1477),rewrite(1621(5),1601(4),1536(5),1622(6))].">
      <j1 rule="back_rewrite" parents="1477"/>
      <j2 rule="rewrite" parents="1621 1601 1536 1622"/>
    </justification>
  </clause>

  <clause id="1625">
    <literal><![CDATA[
      x * ((x' * y) * z) = y * z
    ]]></literal>
    <justification jstring="[back_rewrite(1471),rewrite(1623(4),1623(7),1623(4),1508(2),1508(2),1623(4),1508(3),1536(4),1508(3),1623(9),1623(7),1623(7),1508(4),1623(5),1508(4),1536(5),1508(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="1471"/>
      <j2 rule="rewrite" parents="1623 1623 1623 1508 1508 1623 1508 1536 1508 1623 1623 1623 1508 1623 1508 1536 1508"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1627">
    <literal><![CDATA[
      x * (x' * y) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1469),rewrite(1508(6),1623(9),1623(6),1623(6),1508(3),1623(4),1508(3),1536(4),1623(5),1508(4),1536(5),1623(9),1623(6),1623(6),1508(3),1623(4),1508(3),1536(4),1623(5),1508(5),1625(6))].">
      <j1 rule="back_rewrite" parents="1469"/>
      <j2 rule="rewrite" parents="1508 1623 1623 1623 1508 1623 1508 1536 1623 1508 1536 1623 1623 1623 1508 1623 1508 1536 1623 1508 1625"/>
    </justification>
  </clause>

  <clause id="1629">
    <literal><![CDATA[
      x' * (x * y) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1465),rewrite(1623(5),1536(10),1623(8),1623(5),1508(3),1508(3),1623(5),1508(4),1536(5),1623(3),1623(5),1508(3),1508(3))].">
      <j1 rule="back_rewrite" parents="1465"/>
      <j2 rule="rewrite" parents="1623 1536 1623 1623 1508 1508 1623 1508 1536 1623 1623 1508 1508"/>
    </justification>
  </clause>

  <clause id="1634">
    <literal><![CDATA[
      (x' * y') * z = x' * (y' * z)
    ]]></literal>
    <justification jstring="[back_rewrite(1415),rewrite(1536(7),1623(7),1623(2),1623(8),1508(5),1623(6),1508(5),1536(6),1536(11),1508(9),1623(9),1508(9),1623(9),1508(9),1536(12),1623(12),1623(6),1623(13),1623(10),1508(10),1623(12),1508(11),1536(12),1625(11))].">
      <j1 rule="back_rewrite" parents="1415"/>
      <j2 rule="rewrite" parents="1536 1623 1623 1623 1508 1623 1508 1536 1536 1508 1623 1508 1623 1508 1536 1623 1623 1623 1623 1508 1623 1508 1536 1625"/>
    </justification>
  </clause>

  <clause id="1635">
    <literal><![CDATA[
      (x' * (y' * z)) * u = x' * ((y' * z) * u)
    ]]></literal>
    <justification jstring="[back_rewrite(1409),rewrite(1536(7),1623(8),1623(13),1623(12),1623(12),1623(9),1508(7),1508(7),1623(9),1508(8),1536(9),1623(12),1623(7),1623(7),1508(3),1623(5),1623(3),1508(6),1634(5),1627(6),1623(9),1508(6),1623(7),1508(6),1536(7),1623(11),1536(16),1623(15),1623(15),1623(12),1508(10),1508(10),1623(12),1508(11),1536(12),1623(11),1623(9),1508(12),1634(11),1536(15),1623(15),1623(7),1623(16),1623(13),1623(11),1508(11),1508(12),1623(14),1508(13),1536(14),1634(12),1627(13))].">
      <j1 rule="back_rewrite" parents="1409"/>
      <j2 rule="rewrite" parents="1536 1623 1623 1623 1623 1623 1508 1508 1623 1508 1536 1623 1623 1623 1508 1623 1623 1508 1634 1627 1623 1508 1623 1508 1536 1623 1536 1623 1623 1623 1508 1508 1623 1508 1536 1623 1623 1508 1634 1536 1623 1623 1623 1623 1623 1508 1508 1623 1508 1536 1634 1627"/>
    </justification>
  </clause>

  <clause id="1639">
    <literal><![CDATA[
      x * (((x' * y) * z) * u) = y * (z * u)
    ]]></literal>
    <justification jstring="[back_rewrite(1395),rewrite(1623(4),1536(9),1623(9),1508(2),1623(7),1623(4),1508(2),1508(2),1623(4),1508(3),1536(4),1536(9),1623(9),1508(8),1623(9),1508(8),1623(11),1623(8),1623(12),1508(11),1536(12),1634(10),1623(12),1623(7),1623(5),1508(5),1508(6),1623(10),1508(7),1623(8),1508(7),1536(8)),flip(a)].">
      <j1 rule="back_rewrite" parents="1395"/>
      <j2 rule="rewrite" parents="1623 1536 1623 1508 1623 1623 1508 1508 1623 1508 1536 1536 1623 1508 1623 1508 1623 1623 1623 1508 1536 1634 1623 1623 1623 1508 1508 1623 1508 1623 1508 1536"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1653">
    <literal><![CDATA[
      (x * x') * y = y
    ]]></literal>
    <justification jstring="[back_rewrite(1186),rewrite(1536(5),1508(4))].">
      <j1 rule="back_rewrite" parents="1186"/>
      <j2 rule="rewrite" parents="1536 1508"/>
    </justification>
  </clause>

  <clause id="1669">
    <literal><![CDATA[
      (x * y) * z' = x * (y * z')
    ]]></literal>
    <justification jstring="[back_rewrite(856),rewrite(1623(3),1508(2),1623(3),1508(2),1623(6),1653(8),1653(7),1623(5),1623(4),1508(2),1508(2),1623(6),1508(5),1653(9),1623(9),1623(6),1508(6),1623(8),1508(7),1625(9))].">
      <j1 rule="back_rewrite" parents="856"/>
      <j2 rule="rewrite" parents="1623 1508 1623 1508 1623 1653 1653 1623 1623 1508 1508 1623 1508 1653 1623 1623 1508 1623 1508 1625"/>
    </justification>
  </clause>

  <clause id="1697">
    <literal><![CDATA[
      (x' * (y * (z' * u'))) * u = x' * (y * z')
    ]]></literal>
    <justification jstring="[back_rewrite(1368),rewrite(1623(4),1653(6),1536(6),1634(5),1629(4),1623(3),1508(3),1623(4),1623(3),1508(3),1669(4),1669(8),1627(8),1536(8),1669(9),1627(9),1536(9),1623(10),1653(12),1623(12),1623(10),1623(9),1508(7),1508(7),1669(8),1508(10),1623(12),1623(10),1623(10),1623(10),1623(9),1508(8),1669(10),1508(14),1635(13),1627(14)),flip(a)].">
      <j1 rule="back_rewrite" parents="1368"/>
      <j2 rule="rewrite" parents="1623 1653 1536 1634 1629 1623 1508 1623 1623 1508 1669 1669 1627 1536 1669 1627 1536 1623 1653 1623 1623 1623 1508 1508 1669 1508 1623 1623 1623 1623 1623 1508 1669 1508 1635 1627"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1698">
    <literal><![CDATA[
      (x * y) * z = x * (y * z)
    ]]></literal>
    <justification jstring="[back_rewrite(1355),rewrite(1623(6),1653(8),1536(8),1623(9),1653(11),1634(10),1629(9),1623(8),1623(6),1508(4),1508(4),1508(5),1623(6),1623(6),1623(6),1653(10),1536(10),1697(8),1623(6),1623(4),1508(2),1508(2),1508(3),1669(6),1627(6),1536(6),1623(5),1634(7),1623(9),1623(7),1623(5),1508(5),1508(6),1508(7),1639(7))].">
      <j1 rule="back_rewrite" parents="1355"/>
      <j2 rule="rewrite" parents="1623 1653 1536 1623 1653 1634 1629 1623 1623 1508 1508 1508 1623 1623 1623 1653 1536 1697 1623 1623 1508 1508 1508 1669 1627 1536 1623 1634 1623 1623 1623 1508 1508 1508 1639"/>
    </justification>
  </clause>

  <clause id="1699">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(associativity)
    ]]></attribute>
    <justification jstring="[resolve(1698,a,2,a)].">
      <j1 rule="resolve" parents="1698 2"/>
    </justification>
  </clause>

</proof>

</proofs>
