<?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>EA-gt.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13124 was started by mccune on cleo.thornwood,
Mon Jun 19 16:41:34 2006
The command was "/home/mccune/bin/prover9 -f gt.in EA.in".
]]></heading>

<proof number="1" length="69" max_count="20">

<comments><![CDATA[
% Proof 1 at 0.33 (+ 0.00) seconds: A.
% Length of proof is 69.
% Level of proof is 19.
% Maximum clause weight is 20.
% Given clauses 132.
]]></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 * y * x = y * x * z * x * y
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      (A @ B) @ C != D @ (F @ G)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></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="17">
    <literal><![CDATA[
      x ' @ x = e
    ]]></literal>
    <justification jstring="[para(9(a,1),10(a,1,2,2)),demod(13(6)),flip(a)].">
      <j1 rule="para" parents="9 10"/>
      <j2 rule="demod" parents="13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      x * y * y * x = y * x * x * y
    ]]></literal>
    <justification jstring="[para(8(a,1),11(a,1,2,2)),demod(8(6))].">
      <j1 rule="para" parents="8 11"/>
      <j2 rule="demod" parents="8"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      x * y * (x * y) ' * y * x = y * x * e
    ]]></literal>
    <justification jstring="[para(9(a,1),11(a,1,2,2)),flip(a)].">
      <j1 rule="para" parents="9 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

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

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

  <clause id="37">
    <literal><![CDATA[
      x ' ' * y = x * y
    ]]></literal>
    <justification jstring="[para(13(a,1),13(a,1,2))].">
      <j1 rule="para" parents="13 13"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      x ' * y * x = y * (y @ x)
    ]]></literal>
    <justification jstring="[back_demod(32),demod(37(4)),flip(a)].">
      <j1 rule="back_demod" parents="32"/>
      <j2 rule="demod" parents="37"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      x * e = x
    ]]></literal>
    <justification jstring="[back_demod(29),demod(37(4))].">
      <j1 rule="back_demod" parents="29"/>
      <j2 rule="demod" parents="37"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      x * y * (x * y) ' * y * x = y * x
    ]]></literal>
    <justification jstring="[back_demod(25),demod(40(8))].">
      <j1 rule="back_demod" parents="25"/>
      <j2 rule="demod" parents="40"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      (x @ y) * z * y * x = z * x * y
    ]]></literal>
    <justification jstring="[para(11(a,1),14(a,1,2,2)),demod(13(7),13(5)),flip(a)].">
      <j1 rule="para" parents="11 14"/>
      <j2 rule="demod" parents="13 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      e ' = e
    ]]></literal>
    <justification jstring="[para(40(a,1),9(a,1))].">
      <j1 rule="para" parents="40 9"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      x @ e = e
    ]]></literal>
    <justification jstring="[para(40(a,1),10(a,1,2,2)),demod(61(3),8(3),9(2)),flip(a)].">
      <j1 rule="para" parents="40 10"/>
      <j2 rule="demod" parents="61 8 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="63">
    <literal><![CDATA[
      x * x ' = e
    ]]></literal>
    <justification jstring="[para(37(a,1),9(a,1))].">
      <j1 rule="para" parents="37 9"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      x ' * y * x * y ' = x @ y '
    ]]></literal>
    <justification jstring="[para(37(a,1),10(a,1,2))].">
      <j1 rule="para" parents="37 10"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      x * x ' * y = y
    ]]></literal>
    <justification jstring="[para(37(a,1),13(a,1))].">
      <j1 rule="para" parents="37 13"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      x ' ' = x
    ]]></literal>
    <justification jstring="[para(37(a,1),40(a,1)),demod(40(2)),flip(a)].">
      <j1 rule="para" parents="37 40"/>
      <j2 rule="demod" parents="40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      (x * y ') ' * y ' * x = x * y ' @ y
    ]]></literal>
    <justification jstring="[para(9(a,1),15(a,1,2,2,2)),demod(40(6))].">
      <j1 rule="para" parents="9 15"/>
      <j2 rule="demod" parents="40"/>
    </justification>
  </clause>

  <clause id="79">
    <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 ' @ y = x * y ' @ x
    ]]></literal>
    <justification jstring="[back_demod(74),demod(79(6)),flip(a)].">
      <j1 rule="back_demod" parents="74"/>
      <j2 rule="demod" parents="79"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="85">
    <literal><![CDATA[
      x * y * (x * y @ x) = y * x
    ]]></literal>
    <justification jstring="[back_demod(42),demod(79(4))].">
      <j1 rule="back_demod" parents="42"/>
      <j2 rule="demod" parents="79"/>
    </justification>
  </clause>

  <clause id="86">
    <literal><![CDATA[
      x @ x ' = e
    ]]></literal>
    <justification jstring="[para(72(a,1),17(a,1,1))].">
      <j1 rule="para" parents="72 17"/>
    </justification>
  </clause>

  <clause id="88">
    <literal><![CDATA[
      x * y * (x * y) ' = e
    ]]></literal>
    <justification jstring="[para(63(a,1),7(a,1)),flip(a)].">
      <j1 rule="para" parents="63 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      x * y @ y ' = x * y @ x
    ]]></literal>
    <justification jstring="[para(63(a,1),15(a,1,2,2,2)),demod(72(4),40(4),79(4)),flip(a)].">
      <j1 rule="para" parents="63 15"/>
      <j2 rule="demod" parents="72 40 79"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="122">
    <literal><![CDATA[
      x * (y * x) ' = y '
    ]]></literal>
    <justification jstring="[para(88(a,1),13(a,1,2)),demod(40(3)),flip(a)].">
      <j1 rule="para" parents="88 13"/>
      <j2 rule="demod" parents="40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="124">
    <literal><![CDATA[
      (x @ y) * z * (x ' * y ' * x * y * z) ' = e
    ]]></literal>
    <justification jstring="[para(14(a,2),88(a,1,2,2,1))].">
      <j1 rule="para" parents="14 88"/>
    </justification>
  </clause>

  <clause id="135">
    <literal><![CDATA[
      (x * y) ' = y ' * x '
    ]]></literal>
    <justification jstring="[para(122(a,1),13(a,1,2)),flip(a)].">
      <j1 rule="para" parents="122 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="136">
    <literal><![CDATA[
      (x @ y) ' = y @ x
    ]]></literal>
    <justification jstring="[para(14(a,2),122(a,1,2,1)),demod(135(7),135(5),135(3),135(2),7(5),72(7),7(6),7(5),72(8),7(7),7(6),7(5),10(6),68(4)),flip(a)].">
      <j1 rule="para" parents="14 122"/>
      <j2 rule="demod" parents="135 135 135 135 7 72 7 7 72 7 7 7 10 68"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="145">
    <literal><![CDATA[
      (x @ y) * (y @ x) = e
    ]]></literal>
    <justification jstring="[back_demod(124),demod(135(8),135(6),135(4),135(3),7(6),72(8),7(7),7(6),72(9),7(8),7(7),7(6),10(7),68(5))].">
      <j1 rule="back_demod" parents="124"/>
      <j2 rule="demod" parents="135 135 135 135 7 72 7 7 72 7 7 7 10 68"/>
    </justification>
  </clause>

  <clause id="151">
    <literal><![CDATA[
      x * y @ x = y @ x
    ]]></literal>
    <justification jstring="[back_demod(79),demod(135(2),7(5),10(5)),flip(a)].">
      <j1 rule="back_demod" parents="79"/>
      <j2 rule="demod" parents="135 7 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="164">
    <literal><![CDATA[
      x * y @ y ' = y @ x
    ]]></literal>
    <justification jstring="[back_demod(91),demod(151(5))].">
      <j1 rule="back_demod" parents="91"/>
      <j2 rule="demod" parents="151"/>
    </justification>
  </clause>

  <clause id="165">
    <literal><![CDATA[
      x * y * (y @ x) = y * x
    ]]></literal>
    <justification jstring="[back_demod(85),demod(151(2))].">
      <j1 rule="back_demod" parents="85"/>
      <j2 rule="demod" parents="151"/>
    </justification>
  </clause>

  <clause id="166">
    <literal><![CDATA[
      x * y ' @ y = y ' @ x
    ]]></literal>
    <justification jstring="[back_demod(84),demod(151(6))].">
      <j1 rule="back_demod" parents="84"/>
      <j2 rule="demod" parents="151"/>
    </justification>
  </clause>

  <clause id="186">
    <literal><![CDATA[
      x * y @ x ' = y @ x '
    ]]></literal>
    <justification jstring="[para(13(a,1),151(a,1,1)),flip(a)].">
      <j1 rule="para" parents="13 151"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="190">
    <literal><![CDATA[
      x ' * y @ x = y @ x
    ]]></literal>
    <justification jstring="[para(68(a,1),151(a,1,1)),flip(a)].">
      <j1 rule="para" parents="68 151"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="191">
    <literal><![CDATA[
      x @ x * y = x @ y
    ]]></literal>
    <justification jstring="[para(151(a,1),136(a,1,1)),demod(136(2)),flip(a)].">
      <j1 rule="para" parents="151 136"/>
      <j2 rule="demod" parents="136"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="194">
    <literal><![CDATA[
      x ' @ y ' * x * y = x ' @ (x @ y)
    ]]></literal>
    <justification jstring="[para(10(a,1),191(a,1,2)),flip(a)].">
      <j1 rule="para" parents="10 191"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="224">
    <literal><![CDATA[
      x ' @ y * x = y @ x
    ]]></literal>
    <justification jstring="[para(164(a,1),136(a,1,1)),demod(136(2)),flip(a)].">
      <j1 rule="para" parents="164 136"/>
      <j2 rule="demod" parents="136"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="245">
    <literal><![CDATA[
      (x @ y) * y * x = x * y
    ]]></literal>
    <justification jstring="[para(23(a,1),14(a,1,2,2)),demod(13(6),13(4)),flip(a)].">
      <j1 rule="para" parents="23 14"/>
      <j2 rule="demod" parents="13 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="362">
    <literal><![CDATA[
      (x @ y) @ y * x = e
    ]]></literal>
    <justification jstring="[para(245(a,1),10(a,1,2,2)),demod(136(2),135(3),7(6),10(6),145(3)),flip(a)].">
      <j1 rule="para" parents="245 10"/>
      <j2 rule="demod" parents="136 135 7 10 145"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="364">
    <literal><![CDATA[
      (x ' @ (x @ y)) * (y @ x ') = x @ y
    ]]></literal>
    <justification jstring="[para(10(a,1),245(a,2)),demod(194(5),7(8),7(7),65(8))].">
      <j1 rule="para" parents="10 245"/>
      <j2 rule="demod" parents="194 7 7 65"/>
    </justification>
  </clause>

  <clause id="402">
    <literal><![CDATA[
      (x @ y) @ x = e
    ]]></literal>
    <justification jstring="[para(68(a,1),362(a,1,2)),demod(190(3))].">
      <j1 rule="para" parents="68 362"/>
      <j2 rule="demod" parents="190"/>
    </justification>
  </clause>

  <clause id="420">
    <literal><![CDATA[
      x @ (x @ y) = e
    ]]></literal>
    <justification jstring="[para(402(a,1),136(a,1,1)),demod(61(2)),flip(a)].">
      <j1 rule="para" parents="402 136"/>
      <j2 rule="demod" parents="61"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="423">
    <literal><![CDATA[
      (x @ y) * x = x * (x @ y)
    ]]></literal>
    <justification jstring="[para(402(a,1),165(a,1,2,2)),demod(40(3)),flip(a)].">
      <j1 rule="para" parents="402 165"/>
      <j2 rule="demod" parents="40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="437">
    <literal><![CDATA[
      (x @ y) @ x ' = e
    ]]></literal>
    <justification jstring="[para(420(a,1),164(a,2)),demod(423(2),186(4))].">
      <j1 rule="para" parents="420 164"/>
      <j2 rule="demod" parents="423 186"/>
    </justification>
  </clause>

  <clause id="476">
    <literal><![CDATA[
      x ' @ (x @ y) = e
    ]]></literal>
    <justification jstring="[para(437(a,1),136(a,1,1)),demod(61(2)),flip(a)].">
      <j1 rule="para" parents="437 136"/>
      <j2 rule="demod" parents="61"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="484">
    <literal><![CDATA[
      x @ y ' = y @ x
    ]]></literal>
    <justification jstring="[back_demod(364),demod(476(3),8(4))].">
      <j1 rule="back_demod" parents="364"/>
      <j2 rule="demod" parents="476 8"/>
    </justification>
  </clause>

  <clause id="584">
    <literal><![CDATA[
      (F @ G) @ D ' != (A @ B) @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(484(a,2),12(a,2)),flip(a)].">
      <j1 rule="para" parents="484 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="587">
    <literal><![CDATA[
      (x @ y) @ z = z @ (y @ x)
    ]]></literal>
    <justification jstring="[para(136(a,1),484(a,1,2)),flip(a)].">
      <j1 rule="para" parents="136 484"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="598">
    <literal><![CDATA[
      x ' @ y = x @ y '
    ]]></literal>
    <justification jstring="[para(484(a,2),166(a,1)),demod(135(3),72(2),191(3)),flip(a)].">
      <j1 rule="para" parents="484 166"/>
      <j2 rule="demod" parents="135 72 191"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="606">
    <literal><![CDATA[
      (F @ G) @ D ' != C @ (B @ A)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[back_demod(584),demod(587(11))].">
      <j1 rule="back_demod" parents="584"/>
      <j2 rule="demod" parents="587"/>
    </justification>
  </clause>

  <clause id="658">
    <literal><![CDATA[
      D ' @ (F ' @ G) != C @ (B @ A)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(484(a,2),606(a,1,1)),demod(587(7))].">
      <j1 rule="para" parents="484 606"/>
      <j2 rule="demod" parents="587"/>
    </justification>
  </clause>

  <clause id="811">
    <literal><![CDATA[
      D ' @ (F ' @ G) != C @ (A ' @ B)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(484(a,2),658(a,2,2)),demod(598(12,R))].">
      <j1 rule="para" parents="484 658"/>
      <j2 rule="demod" parents="598"/>
    </justification>
  </clause>

  <clause id="900">
    <literal><![CDATA[
      (x @ y) * z * y * (y @ x) = z * y
    ]]></literal>
    <justification jstring="[para(38(a,1),54(a,1,2,2)),demod(164(3),7(8),63(7),40(7))].">
      <j1 rule="para" parents="38 54"/>
      <j2 rule="demod" parents="164 7 63 40"/>
    </justification>
  </clause>

  <clause id="1178">
    <literal><![CDATA[
      D ' @ (F ' @ G) != (A ' @ B) @ C '
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(484(a,2),811(a,2))].">
      <j1 rule="para" parents="484 811"/>
    </justification>
  </clause>

  <clause id="2109">
    <literal><![CDATA[
      (x @ (y @ z)) * y = y
    ]]></literal>
    <justification jstring="[para(423(a,1),14(a,1,2,2,2)),demod(136(3),900(6),13(3)),flip(a)].">
      <j1 rule="para" parents="423 14"/>
      <j2 rule="demod" parents="136 900 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2242">
    <literal><![CDATA[
      x @ ((x @ y) @ z) = (x @ y) @ z
    ]]></literal>
    <justification jstring="[para(2109(a,1),10(a,1,2,2)),demod(136(3),9(4),40(4),587(5)),flip(a)].">
      <j1 rule="para" parents="2109 10"/>
      <j2 rule="demod" parents="136 9 40 587"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2252">
    <literal><![CDATA[
      (x @ y) @ z = e
    ]]></literal>
    <justification jstring="[para(2109(a,1),224(a,1,2)),demod(598(2),86(2),587(4),2242(4)),flip(a)].">
      <j1 rule="para" parents="2109 224"/>
      <j2 rule="demod" parents="598 86 587 2242"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2261">
    <literal><![CDATA[
      x @ (y @ z) = e
    ]]></literal>
    <justification jstring="[para(2109(a,1),38(a,1,2)),demod(9(2),587(6),2252(5),62(5),40(5)),flip(a)].">
      <j1 rule="para" parents="2109 38"/>
      <j2 rule="demod" parents="9 587 2252 62 40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2289">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[back_demod(1178),demod(2261(7),2252(8)),xx(a)].">
      <j1 rule="back_demod" parents="1178"/>
      <j2 rule="demod" parents="2261 2252"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
