<?xml version="1.0" encoding="ISO-8859-1"?>

<!DOCTYPE proofs SYSTEM "proof3.dtd">

<?xml-stylesheet type="text/xsl" href="proof3.xsl"?>

<proofs number_of_proofs="1">

<source>ND-gt.out</source>

<heading><![CDATA[
Prover9 (32) version August-2006A, August 2006.
Process 11367 was started by mccune on cleo.thornwood,
Sat Aug 12 21:00:03 2006
The command was "/home/mccune/bin/prover9 -f gt.in ND.in".
]]></heading>

<proof number="1" length="36" max_count="21">

<comments><![CDATA[
% Proof 1 at 0.09 (+ 0.00) seconds: D.
% Length of proof is 36.
% Level of proof is 16.
% Maximum clause weight is 21.
% Given clauses 86.
]]></comments>

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

  <clause id="2">
    <literal><![CDATA[
      e * x = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="3">
    <literal><![CDATA[
      x' * x = e
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      x @ y = x' * y' * x * y
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="5">
    <literal><![CDATA[
      x' * y' * x * y = x @ y
    ]]></literal>
    <justification jstring="[copy(4),flip(a)].">
      <j1 rule="copy" parents="4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="6">
    <literal><![CDATA[
      (x @ y) * z = z * (x @ y)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      (A @ C) * (B @ C) != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      x' * x * y = y
    ]]></literal>
    <justification jstring="[para(3(a,1),1(a,1,1)),rewrite(2(2)),flip(a)].">
      <j1 rule="para" parents="3 1"/>
      <j2 rule="rewrite" parents="2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      x' * y' * x * y * z = (x @ y) * z
    ]]></literal>
    <justification jstring="[para(5(a,1),1(a,1,1)),rewrite(1(7),1(6)),flip(a)].">
      <j1 rule="para" parents="5 1"/>
      <j2 rule="rewrite" parents="1 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      (x * y)' * z' * x * y * z = x * y @ z
    ]]></literal>
    <justification jstring="[para(1(a,1),5(a,1,2,2))].">
      <j1 rule="para" parents="1 5"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      e @ x = e
    ]]></literal>
    <justification jstring="[para(2(a,1),5(a,1,2,2)),rewrite(3(4),3(4)),flip(a)].">
      <j1 rule="para" parents="2 5"/>
      <j2 rule="rewrite" parents="3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x * e = x
    ]]></literal>
    <justification jstring="[para(11(a,1),6(a,1,1)),rewrite(2(2),11(2)),flip(a)].">
      <j1 rule="para" parents="11 6"/>
      <j2 rule="rewrite" parents="2 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[para(3(a,1),8(a,1,2)),rewrite(20(4))].">
      <j1 rule="para" parents="3 8"/>
      <j2 rule="rewrite" parents="20"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      x * x' = e
    ]]></literal>
    <justification jstring="[para(24(a,1),3(a,1,1))].">
      <j1 rule="para" parents="24 3"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      x * x' * y = y
    ]]></literal>
    <justification jstring="[para(24(a,1),8(a,1,1))].">
      <j1 rule="para" parents="24 8"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      x * y * (x * y)' = e
    ]]></literal>
    <justification jstring="[para(31(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="31 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="48">
    <literal><![CDATA[
      x' * y' * x * (z @ u) * y = (x @ y) * (z @ u)
    ]]></literal>
    <justification jstring="[para(6(a,2),9(a,1,2,2,2))].">
      <j1 rule="para" parents="6 9"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      x' * y * x * z = y * (y @ x) * z
    ]]></literal>
    <justification jstring="[para(9(a,1),8(a,1,2)),rewrite(24(2)),flip(a)].">
      <j1 rule="para" parents="9 8"/>
      <j2 rule="rewrite" parents="24"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="75">
    <literal><![CDATA[
      (x * y)' * y * x = x * y @ x
    ]]></literal>
    <justification jstring="[para(8(a,1),10(a,1,2))].">
      <j1 rule="para" parents="8 10"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      x * y @ x = y @ x * y
    ]]></literal>
    <justification jstring="[para(9(a,1),10(a,1,2)),rewrite(8(3),6(4),8(5),8(5))].">
      <j1 rule="para" parents="9 10"/>
      <j2 rule="rewrite" parents="8 6 8 8"/>
    </justification>
  </clause>

  <clause id="84">
    <literal><![CDATA[
      (x * y)' * y * x = y @ x * y
    ]]></literal>
    <justification jstring="[back_rewrite(75),rewrite(79(6))].">
      <j1 rule="back_rewrite" parents="75"/>
      <j2 rule="rewrite" parents="79"/>
    </justification>
  </clause>

  <clause id="95">
    <literal><![CDATA[
      x * (y * x)' = y'
    ]]></literal>
    <justification jstring="[para(36(a,1),8(a,1,2)),rewrite(20(3)),flip(a)].">
      <j1 rule="para" parents="36 8"/>
      <j2 rule="rewrite" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="108">
    <literal><![CDATA[
      (x * y)' = y' * x'
    ]]></literal>
    <justification jstring="[para(95(a,1),8(a,1,2)),flip(a)].">
      <j1 rule="para" parents="95 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="109">
    <literal><![CDATA[
      (x @ y)' = y @ x
    ]]></literal>
    <justification jstring="[para(9(a,2),95(a,1,2,1)),rewrite(108(7),108(5),108(3),108(2),1(5),24(7),1(6),1(5),24(8),1(7),1(6),1(5),5(6),34(4)),flip(a)].">
      <j1 rule="para" parents="9 95"/>
      <j2 rule="rewrite" parents="108 108 108 108 1 24 1 1 24 1 1 1 5 34"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="121">
    <literal><![CDATA[
      x @ y * x = x @ y
    ]]></literal>
    <justification jstring="[back_rewrite(84),rewrite(108(2),1(5),5(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="84"/>
      <j2 rule="rewrite" parents="108 1 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="147">
    <literal><![CDATA[
      x * y @ x = y @ x
    ]]></literal>
    <justification jstring="[back_rewrite(79),rewrite(121(4))].">
      <j1 rule="back_rewrite" parents="79"/>
      <j2 rule="rewrite" parents="121"/>
    </justification>
  </clause>

  <clause id="168">
    <literal><![CDATA[
      x * y @ x' = x * y @ y
    ]]></literal>
    <justification jstring="[para(8(a,1),121(a,1,2)),flip(a)].">
      <j1 rule="para" parents="8 121"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="175">
    <literal><![CDATA[
      x * y @ y = x @ y
    ]]></literal>
    <justification jstring="[para(121(a,1),109(a,1,1)),rewrite(109(2)),flip(a)].">
      <j1 rule="para" parents="121 109"/>
      <j2 rule="rewrite" parents="109"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="177">
    <literal><![CDATA[
      x * y @ x' = x @ y
    ]]></literal>
    <justification jstring="[back_rewrite(168),rewrite(175(5))].">
      <j1 rule="back_rewrite" parents="168"/>
      <j2 rule="rewrite" parents="175"/>
    </justification>
  </clause>

  <clause id="179">
    <literal><![CDATA[
      x @ y' = y @ x
    ]]></literal>
    <justification jstring="[para(8(a,1),147(a,1,1)),rewrite(177(5))].">
      <j1 rule="para" parents="8 147"/>
      <j2 rule="rewrite" parents="177"/>
    </justification>
  </clause>

  <clause id="208">
    <literal><![CDATA[
      (B @ C) * (C @ A') != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(179(a,2),7(a,1,1)),rewrite(6(8,R))].">
      <j1 rule="para" parents="179 7"/>
      <j2 rule="rewrite" parents="6"/>
    </justification>
  </clause>

  <clause id="268">
    <literal><![CDATA[
      (C @ A') * (C @ B') != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(179(a,2),208(a,1,1)),rewrite(6(9,R))].">
      <j1 rule="para" parents="179 208"/>
      <j2 rule="rewrite" parents="6"/>
    </justification>
  </clause>

  <clause id="328">
    <literal><![CDATA[
      C @ B' * A' != (C @ A') * (C @ B')
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(179(a,2),268(a,2)),rewrite(108(14)),flip(a)].">
      <j1 rule="para" parents="179 268"/>
      <j2 rule="rewrite" parents="108"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="332">
    <literal><![CDATA[
      x' * y' * z' * x * z * y = x @ z * y
    ]]></literal>
    <justification jstring="[para(108(a,1),5(a,1,2,1)),rewrite(1(7))].">
      <j1 rule="para" parents="108 5"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="781">
    <literal><![CDATA[
      x @ y * z = (x @ z) * (x @ y)
    ]]></literal>
    <justification jstring="[para(50(a,2),48(a,1,2,2)),rewrite(332(8))].">
      <j1 rule="para" parents="50 48"/>
      <j2 rule="rewrite" parents="332"/>
    </justification>
  </clause>

  <clause id="782">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[resolve(781,a,328,a)].">
      <j1 rule="resolve" parents="781 328"/>
    </justification>
  </clause>

</proof>

</proofs>
