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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

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

<heading><![CDATA[
Prover9 (32) version November-2006, November 2006.
Process 3875 was started by mccune on cleo.thornwood,
Wed Nov 22 11:24:24 2006
The command was "/home/mccune/bin/prover9 -f gt.in NE.in".
]]></heading>

<proof number="1" length="29" max_count="19">

<comments><![CDATA[
% Proof 1 at 4.25 (+ 0.05) seconds: E.
% Length of proof is 29.
% Level of proof is 13.
% Maximum clause weight is 19.
% Given clauses 302.
]]></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 * B * C * B * A != B * A * C * A * B
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></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="14">
    <literal><![CDATA[
      (x @ y) * z * u = z * (x @ y) * u
    ]]></literal>
    <justification jstring="[para(6(a,1),1(a,1,1)),rewrite(1(3)),flip(a)].">
      <j1 rule="para" parents="6 1"/>
      <j2 rule="rewrite" parents="1"/>
      <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="27">
    <literal><![CDATA[
      x' * y * x = y * (y @ x)
    ]]></literal>
    <justification jstring="[para(5(a,1),8(a,1,2)),rewrite(24(2)),flip(a)].">
      <j1 rule="para" parents="5 8"/>
      <j2 rule="rewrite" parents="24"/>
      <j2 rule="flip"/>
    </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="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="78">
    <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="82">
    <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="88">
    <literal><![CDATA[
      (x * y)' * y * x = y @ x * y
    ]]></literal>
    <justification jstring="[back_rewrite(78),rewrite(82(6))].">
      <j1 rule="back_rewrite" parents="78"/>
      <j2 rule="rewrite" parents="82"/>
    </justification>
  </clause>

  <clause id="94">
    <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="107">
    <literal><![CDATA[
      (x * y)' = y' * x'
    ]]></literal>
    <justification jstring="[para(94(a,1),8(a,1,2)),flip(a)].">
      <j1 rule="para" parents="94 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="119">
    <literal><![CDATA[
      x @ y * x = x @ y
    ]]></literal>
    <justification jstring="[back_rewrite(88),rewrite(107(2),1(5),5(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="88"/>
      <j2 rule="rewrite" parents="107 1 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="146">
    <literal><![CDATA[
      x * y @ x = y @ x
    ]]></literal>
    <justification jstring="[back_rewrite(82),rewrite(119(4))].">
      <j1 rule="back_rewrite" parents="82"/>
      <j2 rule="rewrite" parents="119"/>
    </justification>
  </clause>

  <clause id="298">
    <literal><![CDATA[
      x * y * (y @ x) = y * x
    ]]></literal>
    <justification jstring="[para(27(a,1),8(a,1,2)),rewrite(24(2))].">
      <j1 rule="para" parents="27 8"/>
      <j2 rule="rewrite" parents="24"/>
    </justification>
  </clause>

  <clause id="308">
    <literal><![CDATA[
      (x @ y) * y * x = x * y
    ]]></literal>
    <justification jstring="[para(146(a,1),27(a,2,2)),rewrite(1(3),8(4),6(4,R)),flip(a)].">
      <j1 rule="para" parents="146 27"/>
      <j2 rule="rewrite" parents="1 8 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="601">
    <literal><![CDATA[
      x * y * (y @ x) * z = y * x * z
    ]]></literal>
    <justification jstring="[para(298(a,1),1(a,1,1)),rewrite(1(2),1(5)),flip(a)].">
      <j1 rule="para" parents="298 1"/>
      <j2 rule="rewrite" parents="1 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="625">
    <literal><![CDATA[
      (x @ y) * z * y * x = z * x * y
    ]]></literal>
    <justification jstring="[para(308(a,1),14(a,2,2))].">
      <j1 rule="para" parents="308 14"/>
    </justification>
  </clause>

  <clause id="9544">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[para(625(a,2),7(a,1,2,2)),rewrite(601(13)),xx(a)].">
      <j1 rule="para" parents="625 7"/>
      <j2 rule="rewrite" parents="601"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
