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<!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 April-2007, April 2007.
Process 26958 was started by mccune on cleo,
Fri Apr 13 09:16:48 2007
The command was "/home/mccune/bin/prover9 -f gt.in ND.in".
]]></heading>

<proof number="1" length="37" max_count="21">

<comments><![CDATA[
% Proof 1 at 0.10 (+ 0.00) seconds: D.
% Length of proof is 37.
% Level of proof is 14.
% Maximum clause weight is 21.
% Given clauses 103.
]]></comments>

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

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

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

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

  <clause id="7" type="assumption">
    <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="22">
    <literal><![CDATA[
      x'' * e = x
    ]]></literal>
    <justification jstring="[para(3(a,1),8(a,1,2))].">
      <j1 rule="para" parents="3 8"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      x'' * y = x * y
    ]]></literal>
    <justification jstring="[para(8(a,1),8(a,1,2))].">
      <j1 rule="para" parents="8 8"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      x * e = x
    ]]></literal>
    <justification jstring="[back_rewrite(22),rewrite(28(4))].">
      <j1 rule="back_rewrite" parents="22"/>
      <j2 rule="rewrite" parents="28"/>
    </justification>
  </clause>

  <clause id="46">
    <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="48">
    <literal><![CDATA[
      x' * y * x * z = y * (y @ x) * z
    ]]></literal>
    <justification jstring="[para(9(a,1),8(a,1,2)),rewrite(28(5)),flip(a)].">
      <j1 rule="para" parents="9 8"/>
      <j2 rule="rewrite" parents="28"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="69">
    <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="71">
    <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="75">
    <literal><![CDATA[
      (x * y)' * y * x = y @ x * y
    ]]></literal>
    <justification jstring="[back_rewrite(69),rewrite(71(6))].">
      <j1 rule="back_rewrite" parents="69"/>
      <j2 rule="rewrite" parents="71"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      x * x' = e
    ]]></literal>
    <justification jstring="[para(28(a,1),3(a,1))].">
      <j1 rule="para" parents="28 3"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      x * x' * y = y
    ]]></literal>
    <justification jstring="[para(28(a,1),8(a,1))].">
      <j1 rule="para" parents="28 8"/>
    </justification>
  </clause>

  <clause id="83">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[para(28(a,1),31(a,1)),rewrite(31(2)),flip(a)].">
      <j1 rule="para" parents="28 31"/>
      <j2 rule="rewrite" parents="31"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="88">
    <literal><![CDATA[
      x * y * (x * y)' = e
    ]]></literal>
    <justification jstring="[para(77(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="77 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="123">
    <literal><![CDATA[
      x * (y * x)' = y'
    ]]></literal>
    <justification jstring="[para(88(a,1),8(a,1,2)),rewrite(31(3)),flip(a)].">
      <j1 rule="para" parents="88 8"/>
      <j2 rule="rewrite" parents="31"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="138">
    <literal><![CDATA[
      (x * y)' = y' * x'
    ]]></literal>
    <justification jstring="[para(123(a,1),8(a,1,2)),flip(a)].">
      <j1 rule="para" parents="123 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="139">
    <literal><![CDATA[
      (x @ y)' = y @ x
    ]]></literal>
    <justification jstring="[para(9(a,2),123(a,1,2,1)),rewrite(138(7),138(5),138(3),138(2),1(5),83(7),1(6),1(5),83(8),1(7),1(6),1(5),5(6),82(4)),flip(a)].">
      <j1 rule="para" parents="9 123"/>
      <j2 rule="rewrite" parents="138 138 138 138 1 83 1 1 83 1 1 1 5 82"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="154">
    <literal><![CDATA[
      x @ y * x = x @ y
    ]]></literal>
    <justification jstring="[back_rewrite(75),rewrite(138(2),1(5),5(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="75"/>
      <j2 rule="rewrite" parents="138 1 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="175">
    <literal><![CDATA[
      x * y @ x = y @ x
    ]]></literal>
    <justification jstring="[back_rewrite(71),rewrite(154(4))].">
      <j1 rule="back_rewrite" parents="71"/>
      <j2 rule="rewrite" parents="154"/>
    </justification>
  </clause>

  <clause id="210">
    <literal><![CDATA[
      x * y @ x' = x * y @ y
    ]]></literal>
    <justification jstring="[para(8(a,1),154(a,1,2)),flip(a)].">
      <j1 rule="para" parents="8 154"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="217">
    <literal><![CDATA[
      x * y @ y = x @ y
    ]]></literal>
    <justification jstring="[para(154(a,1),139(a,1,1)),rewrite(139(2)),flip(a)].">
      <j1 rule="para" parents="154 139"/>
      <j2 rule="rewrite" parents="139"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="219">
    <literal><![CDATA[
      x * y @ x' = x @ y
    ]]></literal>
    <justification jstring="[back_rewrite(210),rewrite(217(5))].">
      <j1 rule="back_rewrite" parents="210"/>
      <j2 rule="rewrite" parents="217"/>
    </justification>
  </clause>

  <clause id="221">
    <literal><![CDATA[
      x @ y' = y @ x
    ]]></literal>
    <justification jstring="[para(8(a,1),175(a,1,1)),rewrite(219(5))].">
      <j1 rule="para" parents="8 175"/>
      <j2 rule="rewrite" parents="219"/>
    </justification>
  </clause>

  <clause id="240">
    <literal><![CDATA[
      (B @ C) * (C @ A') != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(221(a,2),7(a,1,1)),rewrite(6(8,R))].">
      <j1 rule="para" parents="221 7"/>
      <j2 rule="rewrite" parents="6"/>
    </justification>
  </clause>

  <clause id="292">
    <literal><![CDATA[
      (C @ A') * (C @ B') != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(221(a,2),240(a,1,1)),rewrite(6(9,R))].">
      <j1 rule="para" parents="221 240"/>
      <j2 rule="rewrite" parents="6"/>
    </justification>
  </clause>

  <clause id="312">
    <literal><![CDATA[
      x' * y' * z' * x * z * y = x @ z * y
    ]]></literal>
    <justification jstring="[para(138(a,1),5(a,1,2,1)),rewrite(1(7))].">
      <j1 rule="para" parents="138 5"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="357">
    <literal><![CDATA[
      C @ B' * A' != (C @ A') * (C @ B')
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(221(a,2),292(a,2)),rewrite(138(14)),flip(a)].">
      <j1 rule="para" parents="221 292"/>
      <j2 rule="rewrite" parents="138"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="821">
    <literal><![CDATA[
      x @ y * z = (x @ z) * (x @ y)
    ]]></literal>
    <justification jstring="[para(48(a,2),46(a,1,2,2)),rewrite(312(8))].">
      <j1 rule="para" parents="48 46"/>
      <j2 rule="rewrite" parents="312"/>
    </justification>
  </clause>

  <clause id="822">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[resolve(821,a,357,a)].">
      <j1 rule="resolve" parents="821 357"/>
    </justification>
  </clause>

</proof>

</proofs>
