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

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13117 was started by mccune on cleo.thornwood,
Mon Jun 19 16:41:29 2006
The command was "/home/mccune/bin/prover9 -f gt.in DA.in".
]]></heading>

<proof number="1" length="63" max_count="25">

<comments><![CDATA[
% Proof 1 at 0.20 (+ 0.00) seconds: A.
% Length of proof is 63.
% Level of proof is 17.
% Maximum clause weight is 25.
% Given clauses 141.
]]></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 != 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="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="19">
    <literal><![CDATA[
      (x * y @ z) * u = (x @ z) * (y @ z) * u
    ]]></literal>
    <justification jstring="[para(11(a,1),7(a,1,1))].">
      <j1 rule="para" parents="11 7"/>
    </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="29">
    <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="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="32">
    <literal><![CDATA[
      x ' * y * x = y * (y @ x)
    ]]></literal>
    <justification jstring="[back_demod(29),demod(31(4)),flip(a)].">
      <j1 rule="back_demod" parents="29"/>
      <j2 rule="demod" parents="31"/>
      <j2 rule="flip"/>
    </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="59">
    <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="63">
    <literal><![CDATA[
      x * x ' * y = y
    ]]></literal>
    <justification jstring="[para(31(a,1),13(a,1))].">
      <j1 rule="para" parents="31 13"/>
    </justification>
  </clause>

  <clause id="67">
    <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="69">
    <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="72">
    <literal><![CDATA[
      x ' * y * z * x = y * z * (y * z @ x)
    ]]></literal>
    <justification jstring="[para(15(a,1),13(a,1,2)),demod(67(3),7(4)),flip(a)].">
      <j1 rule="para" parents="15 13"/>
      <j2 rule="demod" parents="67 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="74">
    <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="77">
    <literal><![CDATA[
      x ' @ y = y @ x
    ]]></literal>
    <justification jstring="[back_demod(69),demod(74(6))].">
      <j1 rule="back_demod" parents="69"/>
      <j2 rule="demod" parents="74"/>
    </justification>
  </clause>

  <clause id="78">
    <literal><![CDATA[
      x @ x ' = e
    ]]></literal>
    <justification jstring="[para(67(a,1),17(a,1,1))].">
      <j1 rule="para" parents="67 17"/>
    </justification>
  </clause>

  <clause id="83">
    <literal><![CDATA[
      x * y @ y ' = y @ x
    ]]></literal>
    <justification jstring="[para(59(a,1),15(a,1,2,2,2)),demod(67(4),34(4),74(4)),flip(a)].">
      <j1 rule="para" parents="59 15"/>
      <j2 rule="demod" parents="67 34 74"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="85">
    <literal><![CDATA[
      x @ y ' = y @ x
    ]]></literal>
    <justification jstring="[para(78(a,1),11(a,1,2)),demod(34(4),83(5))].">
      <j1 rule="para" parents="78 11"/>
      <j2 rule="demod" parents="34 83"/>
    </justification>
  </clause>

  <clause id="87">
    <literal><![CDATA[
      x ' * y ' * x * z = (x @ y) * y ' * z
    ]]></literal>
    <justification jstring="[para(63(a,1),14(a,1,2,2,2))].">
      <j1 rule="para" parents="63 14"/>
    </justification>
  </clause>

  <clause id="103">
    <literal><![CDATA[
      (B ' @ A) @ C != D @ (F @ G)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(77(a,2),12(a,1,1))].">
      <j1 rule="para" parents="77 12"/>
    </justification>
  </clause>

  <clause id="114">
    <literal><![CDATA[
      x ' @ y ' = x @ y
    ]]></literal>
    <justification jstring="[para(77(a,2),77(a,1))].">
      <j1 rule="para" parents="77 77"/>
    </justification>
  </clause>

  <clause id="136">
    <literal><![CDATA[
      x ' @ y = x @ y '
    ]]></literal>
    <justification jstring="[para(85(a,1),77(a,1)),flip(a)].">
      <j1 rule="para" parents="85 77"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="139">
    <literal><![CDATA[
      (F @ G) @ D ' != (B @ A ') @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(85(a,2),103(a,2)),demod(136(4)),flip(a)].">
      <j1 rule="para" parents="85 103"/>
      <j2 rule="demod" parents="136"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="144">
    <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="196">
    <literal><![CDATA[
      (x @ y) @ x = e
    ]]></literal>
    <justification jstring="[para(10(a,1),22(a,1,1)),demod(144(6),9(4),16(4))].">
      <j1 rule="para" parents="10 22"/>
      <j2 rule="demod" parents="144 9 16"/>
    </justification>
  </clause>

  <clause id="207">
    <literal><![CDATA[
      x ' @ (x @ y) = e
    ]]></literal>
    <justification jstring="[para(196(a,1),77(a,2))].">
      <j1 rule="para" parents="196 77"/>
    </justification>
  </clause>

  <clause id="242">
    <literal><![CDATA[
      (x * y) ' * x = y '
    ]]></literal>
    <justification jstring="[para(59(a,1),24(a,1,2,2)),demod(34(4))].">
      <j1 rule="para" parents="59 24"/>
      <j2 rule="demod" parents="34"/>
    </justification>
  </clause>

  <clause id="255">
    <literal><![CDATA[
      x @ (x @ y) = e
    ]]></literal>
    <justification jstring="[para(114(a,1),207(a,1,2)),demod(67(2))].">
      <j1 rule="para" parents="114 207"/>
      <j2 rule="demod" parents="67"/>
    </justification>
  </clause>

  <clause id="272">
    <literal><![CDATA[
      (x @ y) ' = x ' @ y
    ]]></literal>
    <justification jstring="[para(59(a,1),30(a,1,2,1)),demod(16(4),34(4))].">
      <j1 rule="para" parents="59 30"/>
      <j2 rule="demod" parents="16 34"/>
    </justification>
  </clause>

  <clause id="310">
    <literal><![CDATA[
      (F @ G) @ D ' != C @ (B @ A)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(85(a,2),139(a,2)),demod(272(12),114(12))].">
      <j1 rule="para" parents="85 139"/>
      <j2 rule="demod" parents="272 114"/>
    </justification>
  </clause>

  <clause id="312">
    <literal><![CDATA[
      D @ (F @ G) != C @ (B @ A)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(77(a,2),310(a,1)),demod(67(3))].">
      <j1 rule="para" parents="77 310"/>
      <j2 rule="demod" parents="67"/>
    </justification>
  </clause>

  <clause id="318">
    <literal><![CDATA[
      x * y @ (y @ z) = x @ (y @ z)
    ]]></literal>
    <justification jstring="[para(255(a,1),11(a,1,2)),demod(34(4)),flip(a)].">
      <j1 rule="para" parents="255 11"/>
      <j2 rule="demod" parents="34"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="336">
    <literal><![CDATA[
      (x * y) ' * (y ' @ z) * x * z ' * y * z = x @ (y @ z)
    ]]></literal>
    <justification jstring="[para(32(a,2),15(a,1,2,2,2)),demod(272(4),318(13))].">
      <j1 rule="para" parents="32 15"/>
      <j2 rule="demod" parents="272 318"/>
    </justification>
  </clause>

  <clause id="351">
    <literal><![CDATA[
      D @ (G @ F ') != C @ (B @ A)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(77(a,2),312(a,1,2)),demod(136(5))].">
      <j1 rule="para" parents="77 312"/>
      <j2 rule="demod" parents="136"/>
    </justification>
  </clause>

  <clause id="355">
    <literal><![CDATA[
      (G @ F) @ D != C @ (B @ A)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(77(a,2),351(a,1)),demod(272(5),114(5))].">
      <j1 rule="para" parents="77 351"/>
      <j2 rule="demod" parents="272 114"/>
    </justification>
  </clause>

  <clause id="377">
    <literal><![CDATA[
      D ' @ (G @ F) != C @ (B @ A)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(77(a,2),355(a,1))].">
      <j1 rule="para" parents="77 355"/>
    </justification>
  </clause>

  <clause id="456">
    <literal><![CDATA[
      D ' @ (G @ F) != (B @ A ') @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(77(a,2),377(a,2)),demod(272(10),136(10))].">
      <j1 rule="para" parents="77 377"/>
      <j2 rule="demod" parents="272 136"/>
    </justification>
  </clause>

  <clause id="595">
    <literal><![CDATA[
      (x * y) ' = y ' * x '
    ]]></literal>
    <justification jstring="[para(13(a,1),242(a,1,1,1)),flip(a)].">
      <j1 rule="para" parents="13 242"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="635">
    <literal><![CDATA[
      x ' * y ' * (x ' @ z) * y * z ' * x * z = y @ (x @ z)
    ]]></literal>
    <justification jstring="[back_demod(336),demod(595(2),7(11))].">
      <j1 rule="back_demod" parents="336"/>
      <j2 rule="demod" parents="595 7"/>
    </justification>
  </clause>

  <clause id="1188">
    <literal><![CDATA[
      D ' @ (G @ F) != C ' @ (B @ A ')
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(77(a,2),456(a,2))].">
      <j1 rule="para" parents="77 456"/>
    </justification>
  </clause>

  <clause id="1437">
    <literal><![CDATA[
      (x ' @ y) * (z ' @ y) * y ' * z * x * y = z * x
    ]]></literal>
    <justification jstring="[para(72(a,2),72(a,1,2)),demod(272(3),595(2),19(9),255(14),34(12))].">
      <j1 rule="para" parents="72 72"/>
      <j2 rule="demod" parents="272 595 19 255 34"/>
    </justification>
  </clause>

  <clause id="1581">
    <literal><![CDATA[
      (x ' @ y) * y ' * x * z = x * y ' * z
    ]]></literal>
    <justification jstring="[para(13(a,1),87(a,1,2,2)),demod(67(2)),flip(a)].">
      <j1 rule="para" parents="13 87"/>
      <j2 rule="demod" parents="67"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1641">
    <literal><![CDATA[
      (x ' @ y) * z * y ' * x * y = z * x
    ]]></literal>
    <justification jstring="[back_demod(1437),demod(1581(9))].">
      <j1 rule="back_demod" parents="1437"/>
      <j2 rule="demod" parents="1581"/>
    </justification>
  </clause>

  <clause id="1643">
    <literal><![CDATA[
      x @ (y @ z) = e
    ]]></literal>
    <justification jstring="[back_demod(635),demod(1641(9),13(4),9(2)),flip(a)].">
      <j1 rule="back_demod" parents="635"/>
      <j2 rule="demod" parents="1641 13 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1670">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[back_demod(1188),demod(1643(6),1643(8)),xx(a)].">
      <j1 rule="back_demod" parents="1188"/>
      <j2 rule="demod" parents="1643 1643"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
