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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 June-2006C, June 2006.
Process 13132 was started by mccune on cleo.thornwood,
Mon Jun 19 16:41:41 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 0.10 (+ 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="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 @ C) * (B @ C) != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></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="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="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="39">
    <literal><![CDATA[
      x * x ' * y = y
    ]]></literal>
    <justification jstring="[para(29(a,1),13(a,1,1))].">
      <j1 rule="para" parents="29 13"/>
    </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="53">
    <literal><![CDATA[
      x ' * y ' * x * (z @ u) * y = (x @ y) * (z @ u)
    ]]></literal>
    <justification jstring="[para(11(a,2),14(a,1,2,2,2))].">
      <j1 rule="para" parents="11 14"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      x ' * y * x * z = y * (y @ x) * z
    ]]></literal>
    <justification jstring="[para(14(a,1),13(a,1,2)),demod(29(2)),flip(a)].">
      <j1 rule="para" parents="14 13"/>
      <j2 rule="demod" parents="29"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="80">
    <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="84">
    <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="89">
    <literal><![CDATA[
      (x * y) ' * y * x = y @ x * y
    ]]></literal>
    <justification jstring="[back_demod(80),demod(84(6))].">
      <j1 rule="back_demod" parents="80"/>
      <j2 rule="demod" parents="84"/>
    </justification>
  </clause>

  <clause id="100">
    <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="113">
    <literal><![CDATA[
      (x * y) ' = y ' * x '
    ]]></literal>
    <justification jstring="[para(100(a,1),13(a,1,2)),flip(a)].">
      <j1 rule="para" parents="100 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="114">
    <literal><![CDATA[
      (x @ y) ' = y @ x
    ]]></literal>
    <justification jstring="[para(14(a,2),100(a,1,2,1)),demod(113(7),113(5),113(3),113(2),7(5),29(7),7(6),7(5),29(8),7(7),7(6),7(5),10(6),39(4)),flip(a)].">
      <j1 rule="para" parents="14 100"/>
      <j2 rule="demod" parents="113 113 113 113 7 29 7 7 29 7 7 7 10 39"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="126">
    <literal><![CDATA[
      x @ y * x = x @ y
    ]]></literal>
    <justification jstring="[back_demod(89),demod(113(2),7(5),10(5)),flip(a)].">
      <j1 rule="back_demod" parents="89"/>
      <j2 rule="demod" parents="113 7 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="152">
    <literal><![CDATA[
      x * y @ x = y @ x
    ]]></literal>
    <justification jstring="[back_demod(84),demod(126(4))].">
      <j1 rule="back_demod" parents="84"/>
      <j2 rule="demod" parents="126"/>
    </justification>
  </clause>

  <clause id="173">
    <literal><![CDATA[
      x * y @ x ' = x * y @ y
    ]]></literal>
    <justification jstring="[para(13(a,1),126(a,1,2)),flip(a)].">
      <j1 rule="para" parents="13 126"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="180">
    <literal><![CDATA[
      x * y @ y = x @ y
    ]]></literal>
    <justification jstring="[para(126(a,1),114(a,1,1)),demod(114(2)),flip(a)].">
      <j1 rule="para" parents="126 114"/>
      <j2 rule="demod" parents="114"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="182">
    <literal><![CDATA[
      x * y @ x ' = x @ y
    ]]></literal>
    <justification jstring="[back_demod(173),demod(180(5))].">
      <j1 rule="back_demod" parents="173"/>
      <j2 rule="demod" parents="180"/>
    </justification>
  </clause>

  <clause id="184">
    <literal><![CDATA[
      x @ y ' = y @ x
    ]]></literal>
    <justification jstring="[para(13(a,1),152(a,1,1)),demod(182(5))].">
      <j1 rule="para" parents="13 152"/>
      <j2 rule="demod" parents="182"/>
    </justification>
  </clause>

  <clause id="213">
    <literal><![CDATA[
      (B @ C) * (C @ A ') != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(184(a,2),12(a,1,1)),demod(11(8,R))].">
      <j1 rule="para" parents="184 12"/>
      <j2 rule="demod" parents="11"/>
    </justification>
  </clause>

  <clause id="273">
    <literal><![CDATA[
      (C @ A ') * (C @ B ') != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(184(a,2),213(a,1,1)),demod(11(9,R))].">
      <j1 rule="para" parents="184 213"/>
      <j2 rule="demod" parents="11"/>
    </justification>
  </clause>

  <clause id="333">
    <literal><![CDATA[
      C @ B ' * A ' != (C @ A ') * (C @ B ')
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(184(a,2),273(a,2)),demod(113(14)),flip(a)].">
      <j1 rule="para" parents="184 273"/>
      <j2 rule="demod" parents="113"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="337">
    <literal><![CDATA[
      x ' * y ' * z ' * x * z * y = x @ z * y
    ]]></literal>
    <justification jstring="[para(113(a,1),10(a,1,2,1)),demod(7(7))].">
      <j1 rule="para" parents="113 10"/>
      <j2 rule="demod" parents="7"/>
    </justification>
  </clause>

  <clause id="786">
    <literal><![CDATA[
      x @ y * z = (x @ z) * (x @ y)
    ]]></literal>
    <justification jstring="[para(55(a,2),53(a,1,2,2)),demod(337(8))].">
      <j1 rule="para" parents="55 53"/>
      <j2 rule="demod" parents="337"/>
    </justification>
  </clause>

  <clause id="787">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[resolve(786,a,333,a)].">
      <j1 rule="resolve" parents="786 333"/>
    </justification>
  </clause>

</proof>

</proofs>
