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

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13119 was started by mccune on cleo.thornwood,
Mon Jun 19 16:41:29 2006
The command was "/home/mccune/bin/prover9 -f gt.in DE.in".
]]></heading>

<proof number="1" length="72" max_count="24">

<comments><![CDATA[
% Proof 1 at 4.76 (+ 0.08) seconds: E.
% Length of proof is 72.
% Level of proof is 16.
% Maximum clause weight is 24.
% Given clauses 368.
]]></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 * B * A != B * A * C * A * B
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></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="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="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="38">
    <literal><![CDATA[
      x * y @ y = x @ y
    ]]></literal>
    <justification jstring="[para(28(a,1),11(a,1,2)),demod(34(3)),flip(a)].">
      <j1 rule="para" parents="28 11"/>
      <j2 rule="demod" parents="34"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="48">
    <literal><![CDATA[
      (x ' @ y) * x = x * (y @ x)
    ]]></literal>
    <justification jstring="[para(10(a,1),14(a,1,2)),demod(31(4)),flip(a)].">
      <j1 rule="para" parents="10 14"/>
      <j2 rule="demod" parents="31"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      x ' * y ' * x * y * (z @ y) = x * z @ y
    ]]></literal>
    <justification jstring="[para(14(a,2),11(a,1))].">
      <j1 rule="para" parents="14 11"/>
    </justification>
  </clause>

  <clause id="54">
    <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="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="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="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="110">
    <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="117">
    <literal><![CDATA[
      (x @ y) * (y @ z ') = x * z @ y
    ]]></literal>
    <justification jstring="[para(85(a,2),11(a,1,2))].">
      <j1 rule="para" parents="85 11"/>
    </justification>
  </clause>

  <clause id="130">
    <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="137">
    <literal><![CDATA[
      x ' @ x * y = y @ x
    ]]></literal>
    <justification jstring="[para(37(a,1),77(a,2))].">
      <j1 rule="para" parents="37 77"/>
    </justification>
  </clause>

  <clause id="144">
    <literal><![CDATA[
      x @ y * x = x @ y
    ]]></literal>
    <justification jstring="[para(13(a,1),38(a,1,1)),demod(137(5))].">
      <j1 rule="para" parents="13 38"/>
      <j2 rule="demod" parents="137"/>
    </justification>
  </clause>

  <clause id="160">
    <literal><![CDATA[
      (x ' @ y) ' * (z @ x) ' * (y * z @ x) = (y @ x) @ (z @ x)
    ]]></literal>
    <justification jstring="[para(77(a,2),20(a,1,1,1))].">
      <j1 rule="para" parents="77 20"/>
    </justification>
  </clause>

  <clause id="169">
    <literal><![CDATA[
      (x @ y) ' * (z @ y) ' * (y @ (x * z) ') = (x @ y) @ (z @ y)
    ]]></literal>
    <justification jstring="[para(85(a,2),20(a,1,2,2))].">
      <j1 rule="para" parents="85 20"/>
    </justification>
  </clause>

  <clause id="203">
    <literal><![CDATA[
      (x @ y) @ x = e
    ]]></literal>
    <justification jstring="[para(10(a,1),22(a,1,1)),demod(130(6),9(4),16(4))].">
      <j1 rule="para" parents="10 22"/>
      <j2 rule="demod" parents="130 9 16"/>
    </justification>
  </clause>

  <clause id="222">
    <literal><![CDATA[
      (x ' * y ') ' * (x @ y) * z = x * y * z
    ]]></literal>
    <justification jstring="[para(14(a,1),24(a,1,2))].">
      <j1 rule="para" parents="14 24"/>
    </justification>
  </clause>

  <clause id="227">
    <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="237">
    <literal><![CDATA[
      (x @ y) @ y = e
    ]]></literal>
    <justification jstring="[para(85(a,1),203(a,1,1))].">
      <j1 rule="para" parents="85 203"/>
    </justification>
  </clause>

  <clause id="250">
    <literal><![CDATA[
      x * (y @ z) @ z = x @ z
    ]]></literal>
    <justification jstring="[para(237(a,1),11(a,1,2)),demod(34(3)),flip(a)].">
      <j1 rule="para" parents="237 11"/>
      <j2 rule="demod" parents="34"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="277">
    <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="278">
    <literal><![CDATA[
      (x @ y) * (y @ x) = e
    ]]></literal>
    <justification jstring="[para(78(a,1),30(a,2)),demod(277(3),110(3),83(4))].">
      <j1 rule="para" parents="78 30"/>
      <j2 rule="demod" parents="277 110 83"/>
    </justification>
  </clause>

  <clause id="292">
    <literal><![CDATA[
      (x @ y) * (z ' * y * z @ x) = (y @ x) @ (z @ x)
    ]]></literal>
    <justification jstring="[back_demod(160),demod(277(3),67(2),277(3),11(6))].">
      <j1 rule="back_demod" parents="160"/>
      <j2 rule="demod" parents="277 67 277 11"/>
    </justification>
  </clause>

  <clause id="305">
    <literal><![CDATA[
      (x @ y) @ (z @ y) = (x @ z) @ y
    ]]></literal>
    <justification jstring="[back_demod(169),demod(277(2),277(4),117(8),11(7),10(5)),flip(a)].">
      <j1 rule="back_demod" parents="169"/>
      <j2 rule="demod" parents="277 277 117 11 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="340">
    <literal><![CDATA[
      x * y * (y @ x) = y * x
    ]]></literal>
    <justification jstring="[para(32(a,1),13(a,1,2)),demod(67(2))].">
      <j1 rule="para" parents="32 13"/>
      <j2 rule="demod" parents="67"/>
    </justification>
  </clause>

  <clause id="350">
    <literal><![CDATA[
      x * (y @ x) = y * x * y '
    ]]></literal>
    <justification jstring="[para(85(a,1),32(a,2,2)),demod(67(2)),flip(a)].">
      <j1 rule="para" parents="85 32"/>
      <j2 rule="demod" parents="67"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="402">
    <literal><![CDATA[
      (x * y) ' = y ' * x '
    ]]></literal>
    <justification jstring="[para(13(a,1),227(a,1,1,1)),flip(a)].">
      <j1 rule="para" parents="13 227"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="421">
    <literal><![CDATA[
      x * y * (y @ x) * z = y * x * z
    ]]></literal>
    <justification jstring="[back_demod(222),demod(402(4),67(2),67(2),7(4))].">
      <j1 rule="back_demod" parents="222"/>
      <j2 rule="demod" parents="402 67 67 7"/>
    </justification>
  </clause>

  <clause id="515">
    <literal><![CDATA[
      (x @ y ') * (x @ y) = e
    ]]></literal>
    <justification jstring="[para(77(a,1),278(a,1,2))].">
      <j1 rule="para" parents="77 278"/>
    </justification>
  </clause>

  <clause id="741">
    <literal><![CDATA[
      (x @ y) * y = y * (x @ y)
    ]]></literal>
    <justification jstring="[para(77(a,1),48(a,1,1))].">
      <j1 rule="para" parents="77 48"/>
    </justification>
  </clause>

  <clause id="1241">
    <literal><![CDATA[
      ((x @ y ') @ z ') * z * (x @ y) = (x @ y) * z
    ]]></literal>
    <justification jstring="[para(515(a,1),54(a,1,2,2)),demod(277(3),110(3),34(3)),flip(a)].">
      <j1 rule="para" parents="515 54"/>
      <j2 rule="demod" parents="277 110 34"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1921">
    <literal><![CDATA[
      x ' * y * x @ z = y @ z
    ]]></literal>
    <justification jstring="[para(250(a,1),10(a,2)),demod(402(3),277(2),7(8),741(7),7(10),49(9),11(5))].">
      <j1 rule="para" parents="250 10"/>
      <j2 rule="demod" parents="402 277 7 741 7 49 11"/>
    </justification>
  </clause>

  <clause id="1985">
    <literal><![CDATA[
      (x @ y) @ (z @ y) = e
    ]]></literal>
    <justification jstring="[back_demod(292),demod(1921(5),278(3)),flip(a)].">
      <j1 rule="back_demod" parents="292"/>
      <j2 rule="demod" parents="1921 278"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1991">
    <literal><![CDATA[
      (x @ y) @ z = e
    ]]></literal>
    <justification jstring="[back_demod(305),demod(1985(3)),flip(a)].">
      <j1 rule="back_demod" parents="305"/>
      <j2 rule="demod" parents="1985"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2003">
    <literal><![CDATA[
      (x @ y) * z = z * (x @ y)
    ]]></literal>
    <justification jstring="[back_demod(1241),demod(1991(4),8(4)),flip(a)].">
      <j1 rule="back_demod" parents="1241"/>
      <j2 rule="demod" parents="1991 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2197">
    <literal><![CDATA[
      x * (y @ z) @ u = x @ u
    ]]></literal>
    <justification jstring="[para(1991(a,1),11(a,1,2)),demod(34(3)),flip(a)].">
      <j1 rule="para" parents="1991 11"/>
      <j2 rule="demod" parents="34"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2202">
    <literal><![CDATA[
      x * y @ z = y * x @ z
    ]]></literal>
    <justification jstring="[para(340(a,1),11(a,2,1)),demod(2197(4),11(3))].">
      <j1 rule="para" parents="340 11"/>
      <j2 rule="demod" parents="2197 11"/>
    </justification>
  </clause>

  <clause id="2604">
    <literal><![CDATA[
      (x @ y) * y * x = x * y
    ]]></literal>
    <justification jstring="[para(144(a,1),350(a,1,2)),demod(2003(3,R),7(6),59(5),34(5))].">
      <j1 rule="para" parents="144 350"/>
      <j2 rule="demod" parents="2003 7 59 34"/>
    </justification>
  </clause>

  <clause id="2642">
    <literal><![CDATA[
      (x @ y) * y * x * z = x * y * z
    ]]></literal>
    <justification jstring="[para(2604(a,1),7(a,1,1)),demod(7(2),7(5)),flip(a)].">
      <j1 rule="para" parents="2604 7"/>
      <j2 rule="demod" parents="7 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2672">
    <literal><![CDATA[
      (x @ y) * z * y * x = z * x * y
    ]]></literal>
    <justification jstring="[para(2202(a,1),2604(a,1,1)),demod(19(5),2642(5),7(6))].">
      <j1 rule="para" parents="2202 2604"/>
      <j2 rule="demod" parents="19 2642 7"/>
    </justification>
  </clause>

  <clause id="10653">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[para(2672(a,2),12(a,1,2,2)),demod(421(13)),xx(a)].">
      <j1 rule="para" parents="2672 12"/>
      <j2 rule="demod" parents="421"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
