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

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13122 was started by mccune on cleo.thornwood,
Mon Jun 19 16:41:34 2006
The command was "/home/mccune/bin/prover9 -f gt.in DN.in".
]]></heading>

<proof number="1" length="59" max_count="24">

<comments><![CDATA[
% Proof 1 at 0.14 (+ 0.02) seconds: N.
% Length of proof is 59.
% Level of proof is 14.
% Maximum clause weight is 24.
% Given clauses 107.
]]></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 * z @ y
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      (A @ B) * C != C * (A @ B)
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></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="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="20">
    <literal><![CDATA[
      (x @ y) ' * (z @ y) ' * (x * z @ y) = (x @ y) @ (z @ y)
    ]]></literal>
    <justification jstring="[para(11(a,1),10(a,1,2,2))].">
      <j1 rule="para" parents="11 10"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      (x @ y) * e = x * e @ y
    ]]></literal>
    <justification jstring="[para(16(a,1),11(a,1,2))].">
      <j1 rule="para" parents="16 11"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      x ' * y @ x = y @ x
    ]]></literal>
    <justification jstring="[para(17(a,1),11(a,1,1)),demod(8(3)),flip(a)].">
      <j1 rule="para" parents="17 11"/>
      <j2 rule="demod" parents="8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      x * y ' @ y = x * e @ y
    ]]></literal>
    <justification jstring="[para(17(a,1),11(a,1,2)),demod(21(3)),flip(a)].">
      <j1 rule="para" parents="17 11"/>
      <j2 rule="demod" parents="21"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      (x * y) ' * x * y * z = z
    ]]></literal>
    <justification jstring="[para(7(a,1),13(a,1,2))].">
      <j1 rule="para" parents="7 13"/>
    </justification>
  </clause>

  <clause id="26">
    <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="28">
    <literal><![CDATA[
      x @ x = e
    ]]></literal>
    <justification jstring="[para(13(a,1),10(a,1,2)),demod(9(2)),flip(a)].">
      <j1 rule="para" parents="13 10"/>
      <j2 rule="demod" parents="9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      (x @ y) ' * (x * z @ y) = z @ y
    ]]></literal>
    <justification jstring="[para(11(a,1),13(a,1,2))].">
      <j1 rule="para" parents="11 13"/>
    </justification>
  </clause>

  <clause id="31">
    <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="34">
    <literal><![CDATA[
      x * e = x
    ]]></literal>
    <justification jstring="[back_demod(26),demod(31(4))].">
      <j1 rule="back_demod" parents="26"/>
      <j2 rule="demod" parents="31"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      x * y ' @ y = x @ y
    ]]></literal>
    <justification jstring="[back_demod(23),demod(34(5))].">
      <j1 rule="back_demod" parents="23"/>
      <j2 rule="demod" parents="34"/>
    </justification>
  </clause>

  <clause id="37">
    <literal><![CDATA[
      x * y @ x = y @ x
    ]]></literal>
    <justification jstring="[para(28(a,1),11(a,1,1)),demod(8(3)),flip(a)].">
      <j1 rule="para" parents="28 11"/>
      <j2 rule="demod" parents="8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      x ' * y * x * z = (x @ y ') * y * z
    ]]></literal>
    <justification jstring="[para(13(a,1),14(a,1,2,2,2)),demod(31(5))].">
      <j1 rule="para" parents="13 14"/>
      <j2 rule="demod" parents="31"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      (x * y ') ' * y ' * x = x @ y
    ]]></literal>
    <justification jstring="[para(9(a,1),15(a,1,2,2,2)),demod(34(6),36(9))].">
      <j1 rule="para" parents="9 15"/>
      <j2 rule="demod" parents="34 36"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      (x * y) ' * y * x = y @ x
    ]]></literal>
    <justification jstring="[para(13(a,1),15(a,1,2)),demod(37(6))].">
      <j1 rule="para" parents="13 15"/>
      <j2 rule="demod" parents="37"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      x ' @ y = y @ x
    ]]></literal>
    <justification jstring="[back_demod(61),demod(66(6))].">
      <j1 rule="back_demod" parents="61"/>
      <j2 rule="demod" parents="66"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      (B ' @ A) * C != C * (A @ B)
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[para(69(a,2),12(a,1,1))].">
      <j1 rule="para" parents="69 12"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      x ' @ y ' = x @ y
    ]]></literal>
    <justification jstring="[para(69(a,2),69(a,1))].">
      <j1 rule="para" parents="69 69"/>
    </justification>
  </clause>

  <clause id="85">
    <literal><![CDATA[
      x * x ' = e
    ]]></literal>
    <justification jstring="[para(31(a,1),9(a,1))].">
      <j1 rule="para" parents="31 9"/>
    </justification>
  </clause>

  <clause id="93">
    <literal><![CDATA[
      x ' ' = x
    ]]></literal>
    <justification jstring="[para(31(a,1),34(a,1)),demod(34(2)),flip(a)].">
      <j1 rule="para" parents="31 34"/>
      <j2 rule="demod" parents="34"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="95">
    <literal><![CDATA[
      x @ x ' = e
    ]]></literal>
    <justification jstring="[para(93(a,1),17(a,1,1))].">
      <j1 rule="para" parents="93 17"/>
    </justification>
  </clause>

  <clause id="96">
    <literal><![CDATA[
      x @ y ' = y @ x
    ]]></literal>
    <justification jstring="[para(93(a,1),69(a,1,1)),flip(a)].">
      <j1 rule="para" parents="93 69"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="110">
    <literal><![CDATA[
      x * y @ y ' = y @ x
    ]]></literal>
    <justification jstring="[para(85(a,1),15(a,1,2,2,2)),demod(93(4),34(4),66(4)),flip(a)].">
      <j1 rule="para" parents="85 15"/>
      <j2 rule="demod" parents="93 34 66"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="126">
    <literal><![CDATA[
      (x ' @ y) @ (z @ x) = (y @ x) @ (z @ x)
    ]]></literal>
    <justification jstring="[para(69(a,2),20(a,2,1)),demod(20(8)),flip(a)].">
      <j1 rule="para" parents="69 20"/>
      <j2 rule="demod" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="138">
    <literal><![CDATA[
      (x @ y) * (y @ z ') = x * z @ y
    ]]></literal>
    <justification jstring="[para(96(a,2),11(a,1,2))].">
      <j1 rule="para" parents="96 11"/>
    </justification>
  </clause>

  <clause id="145">
    <literal><![CDATA[
      x ' @ y = x @ y '
    ]]></literal>
    <justification jstring="[para(96(a,1),69(a,1)),flip(a)].">
      <j1 rule="para" parents="96 69"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="146">
    <literal><![CDATA[
      (B @ A ') * C != C * (B @ A ')
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[para(96(a,2),74(a,2,2)),demod(145(4))].">
      <j1 rule="para" parents="96 74"/>
      <j2 rule="demod" parents="145"/>
    </justification>
  </clause>

  <clause id="153">
    <literal><![CDATA[
      (x @ y) ' * (z @ y) ' * (y @ (x * z) ') = (x @ y) @ (z @ y)
    ]]></literal>
    <justification jstring="[para(96(a,2),20(a,1,2,2))].">
      <j1 rule="para" parents="96 20"/>
    </justification>
  </clause>

  <clause id="161">
    <literal><![CDATA[
      x ' * y * x @ y = (y @ x) @ y
    ]]></literal>
    <justification jstring="[para(10(a,1),22(a,1,1)),flip(a)].">
      <j1 rule="para" parents="10 22"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="164">
    <literal><![CDATA[
      x ' * y * x * z @ y = (y @ x) * z @ y
    ]]></literal>
    <justification jstring="[para(14(a,1),22(a,1,1)),flip(a)].">
      <j1 rule="para" parents="14 22"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="171">
    <literal><![CDATA[
      x * y * z @ y = x * z @ y
    ]]></literal>
    <justification jstring="[para(37(a,1),11(a,1,2)),demod(11(3)),flip(a)].">
      <j1 rule="para" parents="37 11"/>
      <j2 rule="demod" parents="11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="180">
    <literal><![CDATA[
      (x @ y) * z @ x = z @ x
    ]]></literal>
    <justification jstring="[back_demod(164),demod(171(5),13(3)),flip(a)].">
      <j1 rule="back_demod" parents="164"/>
      <j2 rule="demod" parents="171 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="181">
    <literal><![CDATA[
      (x @ y) @ x = e
    ]]></literal>
    <justification jstring="[back_demod(161),demod(171(4),9(2),16(2)),flip(a)].">
      <j1 rule="back_demod" parents="161"/>
      <j2 rule="demod" parents="171 9 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="188">
    <literal><![CDATA[
      x ' @ (x @ y) = e
    ]]></literal>
    <justification jstring="[para(181(a,1),69(a,2))].">
      <j1 rule="para" parents="181 69"/>
    </justification>
  </clause>

  <clause id="214">
    <literal><![CDATA[
      (x * y) ' * x = y '
    ]]></literal>
    <justification jstring="[para(85(a,1),24(a,1,2,2)),demod(34(4))].">
      <j1 rule="para" parents="85 24"/>
      <j2 rule="demod" parents="34"/>
    </justification>
  </clause>

  <clause id="247">
    <literal><![CDATA[
      (x @ y) ' = x ' @ y
    ]]></literal>
    <justification jstring="[para(85(a,1),30(a,1,2,1)),demod(16(4),34(4))].">
      <j1 rule="para" parents="85 30"/>
      <j2 rule="demod" parents="16 34"/>
    </justification>
  </clause>

  <clause id="249">
    <literal><![CDATA[
      (x @ y) * (y @ x) = e
    ]]></literal>
    <justification jstring="[para(95(a,1),30(a,2)),demod(247(3),79(3),110(4))].">
      <j1 rule="para" parents="95 30"/>
      <j2 rule="demod" parents="247 79 110"/>
    </justification>
  </clause>

  <clause id="269">
    <literal><![CDATA[
      (x @ y) @ (z @ y) = (x @ z) @ y
    ]]></literal>
    <justification jstring="[back_demod(153),demod(247(2),247(4),138(8),11(7),10(5)),flip(a)].">
      <j1 rule="back_demod" parents="153"/>
      <j2 rule="demod" parents="247 247 138 11 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="283">
    <literal><![CDATA[
      x ' * y @ (x @ z) = y @ (x @ z)
    ]]></literal>
    <justification jstring="[para(188(a,1),11(a,1,1)),demod(8(4)),flip(a)].">
      <j1 rule="para" parents="188 11"/>
      <j2 rule="demod" parents="8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="445">
    <literal><![CDATA[
      (x * y) ' = y ' * x '
    ]]></literal>
    <justification jstring="[para(13(a,1),214(a,1,1,1)),flip(a)].">
      <j1 rule="para" parents="13 214"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="502">
    <literal><![CDATA[
      (x @ y ') * (x @ y) = e
    ]]></literal>
    <justification jstring="[para(69(a,1),249(a,1,2))].">
      <j1 rule="para" parents="69 249"/>
    </justification>
  </clause>

  <clause id="1179">
    <literal><![CDATA[
      ((x @ y ') @ z ') * z * (x @ y) = (x @ y) * z
    ]]></literal>
    <justification jstring="[para(502(a,1),55(a,1,2,2)),demod(247(3),79(3),34(3)),flip(a)].">
      <j1 rule="para" parents="502 55"/>
      <j2 rule="demod" parents="247 79 34"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1427">
    <literal><![CDATA[
      (x ' @ y) @ (z @ x) = e
    ]]></literal>
    <justification jstring="[para(180(a,1),188(a,1,2)),demod(445(3),247(3),283(6))].">
      <j1 rule="para" parents="180 188"/>
      <j2 rule="demod" parents="445 247 283"/>
    </justification>
  </clause>

  <clause id="1453">
    <literal><![CDATA[
      (x @ y) @ (z @ y) = e
    ]]></literal>
    <justification jstring="[back_demod(126),demod(1427(4)),flip(a)].">
      <j1 rule="back_demod" parents="126"/>
      <j2 rule="demod" parents="1427"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1457">
    <literal><![CDATA[
      (x @ y) @ z = e
    ]]></literal>
    <justification jstring="[back_demod(269),demod(1453(3)),flip(a)].">
      <j1 rule="back_demod" parents="269"/>
      <j2 rule="demod" parents="1453"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1470">
    <literal><![CDATA[
      (x @ y) * z = z * (x @ y)
    ]]></literal>
    <justification jstring="[back_demod(1179),demod(1457(4),8(4)),flip(a)].">
      <j1 rule="back_demod" parents="1179"/>
      <j2 rule="demod" parents="1457 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1471">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[resolve(1470,a,146,a)].">
      <j1 rule="resolve" parents="1470 146"/>
    </justification>
  </clause>

</proof>

</proofs>
