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

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13111 was started by mccune on cleo.thornwood,
Mon Jun 19 16:41:24 2006
The command was "/home/mccune/bin/prover9 -f gt.in AD.in".
]]></heading>

<proof number="1" length="60" max_count="25">

<comments><![CDATA[
% Proof 1 at 0.16 (+ 0.00) seconds: D.
% Length of proof is 60.
% Level of proof is 17.
% Maximum clause weight is 25.
% Given clauses 112.
]]></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 = u @ (v @ w)
    ]]></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="21">
    <literal><![CDATA[
      x @ (y @ z) = e
    ]]></literal>
    <justification jstring="[para(16(a,1),11(a,1,1)),demod(16(2)),flip(a)].">
      <j1 rule="para" parents="16 11"/>
      <j2 rule="demod" parents="16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="25">
    <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="27">
    <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="30">
    <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="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(30),demod(31(4)),flip(a)].">
      <j1 rule="back_demod" parents="30"/>
      <j2 rule="demod" parents="31"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      x * e = x
    ]]></literal>
    <justification jstring="[back_demod(27),demod(31(4))].">
      <j1 rule="back_demod" parents="27"/>
      <j2 rule="demod" parents="31"/>
    </justification>
  </clause>

  <clause id="48">
    <literal><![CDATA[
      x ' * y * x * z = y * (y @ x) * z
    ]]></literal>
    <justification jstring="[para(14(a,1),13(a,1,2)),demod(31(5)),flip(a)].">
      <j1 rule="para" parents="14 13"/>
      <j2 rule="demod" parents="31"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="56">
    <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="60">
    <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="61">
    <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="65">
    <literal><![CDATA[
      x * y * (x * y) ' = e
    ]]></literal>
    <justification jstring="[para(56(a,1),7(a,1)),flip(a)].">
      <j1 rule="para" parents="56 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="74">
    <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="76">
    <literal><![CDATA[
      x ' * (y * x @ y) * x = x @ y * x
    ]]></literal>
    <justification jstring="[para(14(a,1),15(a,1,2)),demod(13(3),13(8))].">
      <j1 rule="para" parents="14 15"/>
      <j2 rule="demod" parents="13 13"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      x ' * y ' * x * y * z @ u = (x @ y) * z @ u
    ]]></literal>
    <justification jstring="[para(14(a,2),15(a,2,1)),demod(15(9)),flip(a)].">
      <j1 rule="para" parents="14 15"/>
      <j2 rule="demod" parents="15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      x * (y * x) ' = y '
    ]]></literal>
    <justification jstring="[para(65(a,1),13(a,1,2)),demod(34(3)),flip(a)].">
      <j1 rule="para" parents="65 13"/>
      <j2 rule="demod" parents="34"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      (x @ y) * z * (x ' * y ' * x * y * z) ' = e
    ]]></literal>
    <justification jstring="[para(14(a,2),65(a,1,2,2,1))].">
      <j1 rule="para" parents="14 65"/>
    </justification>
  </clause>

  <clause id="100">
    <literal><![CDATA[
      (x ' * y ') ' * (x @ y) = x * y
    ]]></literal>
    <justification jstring="[para(10(a,1),25(a,1,2))].">
      <j1 rule="para" parents="10 25"/>
    </justification>
  </clause>

  <clause id="118">
    <literal><![CDATA[
      (x * y) ' = y ' * x '
    ]]></literal>
    <justification jstring="[para(89(a,1),13(a,1,2)),flip(a)].">
      <j1 rule="para" parents="89 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="119">
    <literal><![CDATA[
      (x @ y) ' = y @ x
    ]]></literal>
    <justification jstring="[para(14(a,2),89(a,1,2,1)),demod(118(7),118(5),118(3),118(2),7(5),61(7),7(6),7(5),61(8),7(7),7(6),7(5),10(6),60(4)),flip(a)].">
      <j1 rule="para" parents="14 89"/>
      <j2 rule="demod" parents="118 118 118 118 7 61 7 7 61 7 7 7 10 60"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="121">
    <literal><![CDATA[
      x ' * y ' * z * x * y * z ' = y * x @ x ' * z '
    ]]></literal>
    <justification jstring="[para(89(a,1),15(a,1,2,2,2)),demod(118(2),118(5),118(7),61(5),61(5),7(7),7(8),118(11))].">
      <j1 rule="para" parents="89 15"/>
      <j2 rule="demod" parents="118 118 118 61 61 7 7 118"/>
    </justification>
  </clause>

  <clause id="131">
    <literal><![CDATA[
      x * y * (y @ x) = y * x
    ]]></literal>
    <justification jstring="[back_demod(100),demod(118(4),61(2),61(2),7(3))].">
      <j1 rule="back_demod" parents="100"/>
      <j2 rule="demod" parents="118 61 61 7"/>
    </justification>
  </clause>

  <clause id="132">
    <literal><![CDATA[
      (x @ y) * (y @ x) = e
    ]]></literal>
    <justification jstring="[back_demod(91),demod(118(8),118(6),118(4),118(3),7(6),61(8),7(7),7(6),61(9),7(8),7(7),7(6),10(7),60(5))].">
      <j1 rule="back_demod" parents="91"/>
      <j2 rule="demod" parents="118 118 118 118 7 61 7 7 61 7 7 7 10 60"/>
    </justification>
  </clause>

  <clause id="136">
    <literal><![CDATA[
      x * y @ x = y @ x
    ]]></literal>
    <justification jstring="[back_demod(74),demod(118(2),7(5),10(5)),flip(a)].">
      <j1 rule="back_demod" parents="74"/>
      <j2 rule="demod" parents="118 7 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="153">
    <literal><![CDATA[
      x ' * (x @ y) * x = x @ y * x
    ]]></literal>
    <justification jstring="[back_demod(76),demod(136(3))].">
      <j1 rule="back_demod" parents="76"/>
      <j2 rule="demod" parents="136"/>
    </justification>
  </clause>

  <clause id="159">
    <literal><![CDATA[
      x ' * (y @ z) * x = y @ z
    ]]></literal>
    <justification jstring="[para(132(a,1),14(a,1,2,2,2)),demod(119(3),34(4),21(6),8(7))].">
      <j1 rule="para" parents="132 14"/>
      <j2 rule="demod" parents="119 34 21 8"/>
    </justification>
  </clause>

  <clause id="160">
    <literal><![CDATA[
      x @ y * x = x @ y
    ]]></literal>
    <justification jstring="[back_demod(153),demod(159(4)),flip(a)].">
      <j1 rule="back_demod" parents="153"/>
      <j2 rule="demod" parents="159"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="175">
    <literal><![CDATA[
      x * y @ x ' = y @ x '
    ]]></literal>
    <justification jstring="[para(13(a,1),136(a,1,1)),flip(a)].">
      <j1 rule="para" parents="13 136"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="177">
    <literal><![CDATA[
      x ' * y @ x = y @ x
    ]]></literal>
    <justification jstring="[para(60(a,1),136(a,1,1)),flip(a)].">
      <j1 rule="para" parents="60 136"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="182">
    <literal><![CDATA[
      x * y @ y = y @ x '
    ]]></literal>
    <justification jstring="[para(13(a,1),160(a,1,2)),demod(175(5))].">
      <j1 rule="para" parents="13 160"/>
      <j2 rule="demod" parents="175"/>
    </justification>
  </clause>

  <clause id="185">
    <literal><![CDATA[
      x ' * y @ y = y @ x
    ]]></literal>
    <justification jstring="[para(60(a,1),160(a,1,2)),demod(177(6))].">
      <j1 rule="para" parents="60 160"/>
      <j2 rule="demod" parents="177"/>
    </justification>
  </clause>

  <clause id="186">
    <literal><![CDATA[
      x * y @ y = x @ y
    ]]></literal>
    <justification jstring="[para(160(a,1),119(a,1,1)),demod(119(2)),flip(a)].">
      <j1 rule="para" parents="160 119"/>
      <j2 rule="demod" parents="119"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="189">
    <literal><![CDATA[
      x ' @ y = y @ x
    ]]></literal>
    <justification jstring="[back_demod(185),demod(186(3))].">
      <j1 rule="back_demod" parents="185"/>
      <j2 rule="demod" parents="186"/>
    </justification>
  </clause>

  <clause id="190">
    <literal><![CDATA[
      x @ y ' = y @ x
    ]]></literal>
    <justification jstring="[back_demod(182),demod(186(2)),flip(a)].">
      <j1 rule="back_demod" parents="182"/>
      <j2 rule="demod" parents="186"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="209">
    <literal><![CDATA[
      (C @ A ') * (B @ C) != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(190(a,2),12(a,1,1))].">
      <j1 rule="para" parents="190 12"/>
    </justification>
  </clause>

  <clause id="367">
    <literal><![CDATA[
      (x @ y) * z * y ' * x ' * y * x * u = z * u
    ]]></literal>
    <justification jstring="[para(14(a,2),48(a,1,2,2)),demod(119(2),21(11),8(11))].">
      <j1 rule="para" parents="14 48"/>
      <j2 rule="demod" parents="119 21 8"/>
    </justification>
  </clause>

  <clause id="372">
    <literal><![CDATA[
      (x @ y) * z = z * (x @ y)
    ]]></literal>
    <justification jstring="[para(132(a,1),48(a,1,2,2)),demod(119(2),34(3),21(4),8(5))].">
      <j1 rule="para" parents="132 48"/>
      <j2 rule="demod" parents="119 34 21 8"/>
    </justification>
  </clause>

  <clause id="404">
    <literal><![CDATA[
      x * (x @ y) * z @ y = x * y * z @ y
    ]]></literal>
    <justification jstring="[para(48(a,1),177(a,1,1))].">
      <j1 rule="para" parents="48 177"/>
    </justification>
  </clause>

  <clause id="415">
    <literal><![CDATA[
      (B @ C) * (C @ A ') != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[back_demod(209),demod(372(8,R))].">
      <j1 rule="back_demod" parents="209"/>
      <j2 rule="demod" parents="372"/>
    </justification>
  </clause>

  <clause id="417">
    <literal><![CDATA[
      B ' * C ' * B * C * (C @ A ') != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(14(a,2),415(a,1))].">
      <j1 rule="para" parents="14 415"/>
    </justification>
  </clause>

  <clause id="907">
    <literal><![CDATA[
      (x @ y) * z @ u = z @ u
    ]]></literal>
    <justification jstring="[para(77(a,1),10(a,2)),demod(118(7),118(5),118(3),118(2),7(5),61(7),7(6),7(5),61(8),7(7),7(6),7(5),10(6),7(11),7(10),7(9),7(8),7(13),367(12),10(5)),flip(a)].">
      <j1 rule="para" parents="77 10"/>
      <j2 rule="demod" parents="118 118 118 118 7 61 7 7 61 7 7 7 10 7 7 7 7 7 367 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="912">
    <literal><![CDATA[
      x ' * y * x * z @ u = y * z @ u
    ]]></literal>
    <justification jstring="[para(13(a,1),77(a,1,1,2,2,2)),demod(61(3),907(10))].">
      <j1 rule="para" parents="13 77"/>
      <j2 rule="demod" parents="61 907"/>
    </justification>
  </clause>

  <clause id="913">
    <literal><![CDATA[
      x * (x @ y) * z @ u = x * z @ u
    ]]></literal>
    <justification jstring="[para(14(a,1),77(a,1,1,2,2,2)),demod(61(3),912(7),907(12),912(9))].">
      <j1 rule="para" parents="14 77"/>
      <j2 rule="demod" parents="61 912 907 912"/>
    </justification>
  </clause>

  <clause id="941">
    <literal><![CDATA[
      x * (y @ z) @ u = x @ u
    ]]></literal>
    <justification jstring="[para(131(a,1),77(a,2,1)),demod(21(4),34(4),912(7),13(3)),flip(a)].">
      <j1 rule="para" parents="131 77"/>
      <j2 rule="demod" parents="21 34 912 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="957">
    <literal><![CDATA[
      x * y * z @ y = x * z @ y
    ]]></literal>
    <justification jstring="[back_demod(404),demod(913(4)),flip(a)].">
      <j1 rule="back_demod" parents="404"/>
      <j2 rule="demod" parents="913"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="980">
    <literal><![CDATA[
      B ' * C ' * B * A * C * A ' != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(32(a,2),417(a,1,2,2,2)),demod(61(8))].">
      <j1 rule="para" parents="32 417"/>
      <j2 rule="demod" parents="61"/>
    </justification>
  </clause>

  <clause id="1086">
    <literal><![CDATA[
      x ' * y ' * x * z * y * z ' = y @ x ' * y ' * z '
    ]]></literal>
    <justification jstring="[para(32(a,1),121(a,2,1)),demod(118(2),61(5),7(8),60(7),7(8),118(12),7(15),941(16))].">
      <j1 rule="para" parents="32 121"/>
      <j2 rule="demod" parents="118 61 7 60 7 118 7 941"/>
    </justification>
  </clause>

  <clause id="1115">
    <literal><![CDATA[
      C @ B ' * C ' * A ' != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[back_demod(980),demod(1086(14))].">
      <j1 rule="back_demod" parents="980"/>
      <j2 rule="demod" parents="1086"/>
    </justification>
  </clause>

  <clause id="1118">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(189(a,2),1115(a,1)),demod(118(9),118(6),61(3),61(4),61(6),7(5),957(7)),xx(a)].">
      <j1 rule="para" parents="189 1115"/>
      <j2 rule="demod" parents="118 118 61 61 61 7 957"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
