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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 June-2006C, June 2006.
Process 13134 was started by mccune on cleo.thornwood,
Mon Jun 19 16:41:41 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 4.32 (+ 0.06) seconds: E.
% Length of proof is 29.
% Level of proof is 13.
% Maximum clause weight is 19.
% Given clauses 302.
]]></comments>

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

  <clause id="7">
    <literal><![CDATA[
      (x * y) * z = x * y * z
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      e * x = x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      x ' * x = e
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="10">
    <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="11">
    <literal><![CDATA[
      (x @ y) * z = z * (x @ y)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      A * B * C * B * A != B * A * C * A * B
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      x ' * x * y = y
    ]]></literal>
    <justification jstring="[para(9(a,1),7(a,1,1)),demod(8(2)),flip(a)].">
      <j1 rule="para" parents="9 7"/>
      <j2 rule="demod" parents="8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      x ' * y ' * x * y * z = (x @ y) * z
    ]]></literal>
    <justification jstring="[para(10(a,1),7(a,1,1)),demod(7(7),7(6)),flip(a)].">
      <j1 rule="para" parents="10 7"/>
      <j2 rule="demod" parents="7 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      (x * y) ' * z ' * x * y * z = x * y @ z
    ]]></literal>
    <justification jstring="[para(7(a,1),10(a,1,2,2))].">
      <j1 rule="para" parents="7 10"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      e @ x = e
    ]]></literal>
    <justification jstring="[para(8(a,1),10(a,1,2,2)),demod(9(4),9(4)),flip(a)].">
      <j1 rule="para" parents="8 10"/>
      <j2 rule="demod" parents="9 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      (x @ y) * z * u = z * (x @ y) * u
    ]]></literal>
    <justification jstring="[para(11(a,1),7(a,1,1)),demod(7(3)),flip(a)].">
      <j1 rule="para" parents="11 7"/>
      <j2 rule="demod" parents="7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      x * e = x
    ]]></literal>
    <justification jstring="[para(16(a,1),11(a,1,1)),demod(8(2),16(2)),flip(a)].">
      <j1 rule="para" parents="16 11"/>
      <j2 rule="demod" parents="8 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      x ' ' = x
    ]]></literal>
    <justification jstring="[para(9(a,1),13(a,1,2)),demod(25(4))].">
      <j1 rule="para" parents="9 13"/>
      <j2 rule="demod" parents="25"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      x ' * y * x = y * (y @ x)
    ]]></literal>
    <justification jstring="[para(10(a,1),13(a,1,2)),demod(29(2)),flip(a)].">
      <j1 rule="para" parents="10 13"/>
      <j2 rule="demod" parents="29"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      x * x ' = e
    ]]></literal>
    <justification jstring="[para(29(a,1),9(a,1,1))].">
      <j1 rule="para" parents="29 9"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      x * y * (x * y) ' = e
    ]]></literal>
    <justification jstring="[para(36(a,1),7(a,1)),flip(a)].">
      <j1 rule="para" parents="36 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="83">
    <literal><![CDATA[
      (x * y) ' * y * x = x * y @ x
    ]]></literal>
    <justification jstring="[para(13(a,1),15(a,1,2))].">
      <j1 rule="para" parents="13 15"/>
    </justification>
  </clause>

  <clause id="87">
    <literal><![CDATA[
      x * y @ x = y @ x * y
    ]]></literal>
    <justification jstring="[para(14(a,1),15(a,1,2)),demod(13(3),11(4),13(5),13(5))].">
      <j1 rule="para" parents="14 15"/>
      <j2 rule="demod" parents="13 11 13 13"/>
    </justification>
  </clause>

  <clause id="93">
    <literal><![CDATA[
      (x * y) ' * y * x = y @ x * y
    ]]></literal>
    <justification jstring="[back_demod(83),demod(87(6))].">
      <j1 rule="back_demod" parents="83"/>
      <j2 rule="demod" parents="87"/>
    </justification>
  </clause>

  <clause id="99">
    <literal><![CDATA[
      x * (y * x) ' = y '
    ]]></literal>
    <justification jstring="[para(41(a,1),13(a,1,2)),demod(25(3)),flip(a)].">
      <j1 rule="para" parents="41 13"/>
      <j2 rule="demod" parents="25"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="112">
    <literal><![CDATA[
      (x * y) ' = y ' * x '
    ]]></literal>
    <justification jstring="[para(99(a,1),13(a,1,2)),flip(a)].">
      <j1 rule="para" parents="99 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="124">
    <literal><![CDATA[
      x @ y * x = x @ y
    ]]></literal>
    <justification jstring="[back_demod(93),demod(112(2),7(5),10(5)),flip(a)].">
      <j1 rule="back_demod" parents="93"/>
      <j2 rule="demod" parents="112 7 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="151">
    <literal><![CDATA[
      x * y @ x = y @ x
    ]]></literal>
    <justification jstring="[back_demod(87),demod(124(4))].">
      <j1 rule="back_demod" parents="87"/>
      <j2 rule="demod" parents="124"/>
    </justification>
  </clause>

  <clause id="303">
    <literal><![CDATA[
      x * y * (y @ x) = y * x
    ]]></literal>
    <justification jstring="[para(32(a,1),13(a,1,2)),demod(29(2))].">
      <j1 rule="para" parents="32 13"/>
      <j2 rule="demod" parents="29"/>
    </justification>
  </clause>

  <clause id="313">
    <literal><![CDATA[
      (x @ y) * y * x = x * y
    ]]></literal>
    <justification jstring="[para(151(a,1),32(a,2,2)),demod(7(3),13(4),11(4,R)),flip(a)].">
      <j1 rule="para" parents="151 32"/>
      <j2 rule="demod" parents="7 13 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="606">
    <literal><![CDATA[
      x * y * (y @ x) * z = y * x * z
    ]]></literal>
    <justification jstring="[para(303(a,1),7(a,1,1)),demod(7(2),7(5)),flip(a)].">
      <j1 rule="para" parents="303 7"/>
      <j2 rule="demod" parents="7 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="630">
    <literal><![CDATA[
      (x @ y) * z * y * x = z * x * y
    ]]></literal>
    <justification jstring="[para(313(a,1),19(a,2,2))].">
      <j1 rule="para" parents="313 19"/>
    </justification>
  </clause>

  <clause id="9096">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[para(630(a,2),12(a,1,2,2)),demod(606(13)),xx(a)].">
      <j1 rule="para" parents="630 12"/>
      <j2 rule="demod" parents="606"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
