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<!DOCTYPE proofs SYSTEM "proof3.dtd">

<?xml-stylesheet type="text/xsl" href="proof3.xsl"?>

<proofs number_of_proofs="1">

<source>rp-ident.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 12987 was started by mccune on cleo.thornwood,
Mon Jun 19 16:39:41 2006
The command was "/home/mccune/bin/prover9 -f rp-ident.in".
]]></heading>

<proof number="1" length="46" max_count="23">

<comments><![CDATA[
% Proof 1 at 0.96 (+ 0.00) seconds.
% Length of proof is 46.
% Level of proof is 19.
% Maximum clause weight is 23.
% Given clauses 97.
]]></comments>

  <clause id="6">
    <literal><![CDATA[
      (x * e) * x = x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      x * (x * y) = y
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      (x * y) * (z * u) = (x * z) * (y * u)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      ((x * x) * x) * x = e
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      ((c4 * c2) * c3) * c1 != ((c1 * c2) * c3) * c4
    ]]></literal>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      e * e = e
    ]]></literal>
    <justification jstring="[para(6(a,1),6(a,1,1)),demod(7(5)),flip(a)].">
      <j1 rule="para" parents="6 6"/>
      <j2 rule="demod" parents="7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      ((x * e) * y) * (x * z) = x * (y * z)
    ]]></literal>
    <justification jstring="[para(6(a,1),8(a,1,1)),flip(a)].">
      <j1 rule="para" parents="6 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      (x * (y * e)) * (z * y) = (x * z) * y
    ]]></literal>
    <justification jstring="[para(6(a,1),8(a,1,2)),flip(a)].">
      <j1 rule="para" parents="6 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      (x * y) * ((x * z) * u) = z * (y * u)
    ]]></literal>
    <justification jstring="[para(7(a,1),8(a,1,1)),flip(a)].">
      <j1 rule="para" parents="7 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      (x * y) * (z * (y * u)) = (x * z) * u
    ]]></literal>
    <justification jstring="[para(7(a,1),8(a,1,2)),flip(a)].">
      <j1 rule="para" parents="7 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      ((x * x) * x) * e = x
    ]]></literal>
    <justification jstring="[para(9(a,1),7(a,1,2))].">
      <j1 rule="para" parents="9 7"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      (e * x) * (e * y) = e * (x * y)
    ]]></literal>
    <justification jstring="[para(11(a,1),8(a,1,1)),flip(a)].">
      <j1 rule="para" parents="11 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      (x * e) * (y * e) = (x * y) * e
    ]]></literal>
    <justification jstring="[para(11(a,1),8(a,1,2)),flip(a)].">
      <j1 rule="para" parents="11 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      x * ((x * x) * x) = (x * x) * x
    ]]></literal>
    <justification jstring="[para(20(a,1),6(a,1,1))].">
      <j1 rule="para" parents="20 6"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      e * (x * (e * y)) = (e * x) * y
    ]]></literal>
    <justification jstring="[para(7(a,1),26(a,1,2)),flip(a)].">
      <j1 rule="para" parents="7 26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      ((x * e) * y) * z = x * (y * (x * z))
    ]]></literal>
    <justification jstring="[para(7(a,1),13(a,1,2))].">
      <j1 rule="para" parents="7 13"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      ((x * e) * y) * ((z * e) * u) = ((x * z) * e) * (y * u)
    ]]></literal>
    <justification jstring="[para(27(a,1),8(a,1,1)),flip(a)].">
      <j1 rule="para" parents="27 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="83">
    <literal><![CDATA[
      x * ((x * e) * e) = x
    ]]></literal>
    <justification jstring="[para(13(a,1),28(a,2)),demod(27(7),27(8),20(6),6(3)),flip(a)].">
      <j1 rule="para" parents="13 28"/>
      <j2 rule="demod" parents="27 27 20 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="84">
    <literal><![CDATA[
      (x * e) * e = x * x
    ]]></literal>
    <justification jstring="[para(83(a,1),7(a,1,2)),flip(a)].">
      <j1 rule="para" parents="83 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="88">
    <literal><![CDATA[
      x * ((((x * e) * e) * e) * y) = (x * e) * (x * y)
    ]]></literal>
    <justification jstring="[para(83(a,1),13(a,1,1)),flip(a)].">
      <j1 rule="para" parents="83 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      (x * e) * (x * x) = e
    ]]></literal>
    <justification jstring="[para(84(a,1),7(a,1,2))].">
      <j1 rule="para" parents="84 7"/>
    </justification>
  </clause>

  <clause id="101">
    <literal><![CDATA[
      (x * x) * (x * y) = x * (e * y)
    ]]></literal>
    <justification jstring="[para(84(a,1),13(a,1,1))].">
      <j1 rule="para" parents="84 13"/>
    </justification>
  </clause>

  <clause id="104">
    <literal><![CDATA[
      ((x * x) * y) * e = x * (y * x)
    ]]></literal>
    <justification jstring="[para(84(a,2),13(a,1,2)),demod(66(8),27(6))].">
      <j1 rule="para" parents="84 13"/>
      <j2 rule="demod" parents="66 27"/>
    </justification>
  </clause>

  <clause id="107">
    <literal><![CDATA[
      ((x * e) * y) * e = (x * x) * (y * e)
    ]]></literal>
    <justification jstring="[para(84(a,1),27(a,1,1)),flip(a)].">
      <j1 rule="para" parents="84 27"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="118">
    <literal><![CDATA[
      (x * y) * x = e * (y * x)
    ]]></literal>
    <justification jstring="[para(7(a,1),14(a,1,1)),flip(a)].">
      <j1 rule="para" parents="7 14"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="144">
    <literal><![CDATA[
      (x * x) * (e * x) = e
    ]]></literal>
    <justification jstring="[para(91(a,1),8(a,1)),flip(a)].">
      <j1 rule="para" parents="91 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="150">
    <literal><![CDATA[
      (x * x) * e = e * x
    ]]></literal>
    <justification jstring="[para(144(a,1),7(a,1,2))].">
      <j1 rule="para" parents="144 7"/>
    </justification>
  </clause>

  <clause id="164">
    <literal><![CDATA[
      (x * x) * (y * e) = (e * y) * x
    ]]></literal>
    <justification jstring="[para(150(a,1),13(a,1,2)),demod(150(3),118(6),40(5)),flip(a)].">
      <j1 rule="para" parents="150 13"/>
      <j2 rule="demod" parents="150 118 40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="168">
    <literal><![CDATA[
      e * (x * e) = x * (e * x)
    ]]></literal>
    <justification jstring="[para(150(a,1),84(a,1,1)),demod(118(4),101(7))].">
      <j1 rule="para" parents="150 84"/>
      <j2 rule="demod" parents="118 101"/>
    </justification>
  </clause>

  <clause id="172">
    <literal><![CDATA[
      ((x * e) * y) * e = (e * y) * x
    ]]></literal>
    <justification jstring="[back_demod(107),demod(164(9))].">
      <j1 rule="back_demod" parents="107"/>
      <j2 rule="demod" parents="164"/>
    </justification>
  </clause>

  <clause id="176">
    <literal><![CDATA[
      (x * e) * (x * y) = x * ((e * x) * y)
    ]]></literal>
    <justification jstring="[back_demod(88),demod(172(6),11(3)),flip(a)].">
      <j1 rule="back_demod" parents="88"/>
      <j2 rule="demod" parents="172 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="182">
    <literal><![CDATA[
      x * ((e * x) * x) = e
    ]]></literal>
    <justification jstring="[back_demod(91),demod(176(4))].">
      <j1 rule="back_demod" parents="91"/>
      <j2 rule="demod" parents="176"/>
    </justification>
  </clause>

  <clause id="193">
    <literal><![CDATA[
      x * (y * ((z * x) * u)) = (z * y) * u
    ]]></literal>
    <justification jstring="[para(7(a,1),16(a,1,2)),flip(a)].">
      <j1 rule="para" parents="7 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="194">
    <literal><![CDATA[
      ((x * y) * z) * (((x * u) * v) * w) = (u * (y * v)) * (z * w)
    ]]></literal>
    <justification jstring="[para(16(a,1),8(a,1,1)),flip(a)].">
      <j1 rule="para" parents="16 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="214">
    <literal><![CDATA[
      x * (y * (e * x)) = (x * y) * e
    ]]></literal>
    <justification jstring="[para(144(a,1),16(a,1,2)),flip(a)].">
      <j1 rule="para" parents="144 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="224">
    <literal><![CDATA[
      (e * x) * x = x * e
    ]]></literal>
    <justification jstring="[para(182(a,1),7(a,1,2)),flip(a)].">
      <j1 rule="para" parents="182 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="297">
    <literal><![CDATA[
      (x * (y * z)) * (u * v) = z * ((y * (x * u)) * v)
    ]]></literal>
    <justification jstring="[para(17(a,1),16(a,1,1)),demod(194(6))].">
      <j1 rule="para" parents="17 16"/>
      <j2 rule="demod" parents="194"/>
    </justification>
  </clause>

  <clause id="301">
    <literal><![CDATA[
      (x * e) * (y * (x * z)) = ((e * x) * y) * z
    ]]></literal>
    <justification jstring="[para(224(a,1),17(a,1,1))].">
      <j1 rule="para" parents="224 17"/>
    </justification>
  </clause>

  <clause id="466">
    <literal><![CDATA[
      x * (y * x) = y * (x * y)
    ]]></literal>
    <justification jstring="[para(104(a,1),224(a,2)),demod(297(7),168(5,R),150(4),7(4))].">
      <j1 rule="para" parents="104 224"/>
      <j2 rule="demod" parents="297 168 150 7"/>
    </justification>
  </clause>

  <clause id="505">
    <literal><![CDATA[
      x * (y * (x * y)) = y * x
    ]]></literal>
    <justification jstring="[para(466(a,1),7(a,1,2))].">
      <j1 rule="para" parents="466 7"/>
    </justification>
  </clause>

  <clause id="680">
    <literal><![CDATA[
      ((x * e) * y) * y = y * x
    ]]></literal>
    <justification jstring="[para(505(a,1),13(a,2)),demod(7(5))].">
      <j1 rule="para" parents="505 13"/>
      <j2 rule="demod" parents="7"/>
    </justification>
  </clause>

  <clause id="848">
    <literal><![CDATA[
      ((x * (y * x)) * z) * z = z * ((x * x) * y)
    ]]></literal>
    <justification jstring="[para(104(a,1),680(a,1,1,1))].">
      <j1 rule="para" parents="104 680"/>
    </justification>
  </clause>

  <clause id="1009">
    <literal><![CDATA[
      x * ((x * y) * e) = y * (e * x)
    ]]></literal>
    <justification jstring="[para(224(a,1),50(a,2,2)),demod(301(6),118(4),848(6),11(3)),flip(a)].">
      <j1 rule="para" parents="224 50"/>
      <j2 rule="demod" parents="301 118 848 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1517">
    <literal><![CDATA[
      ((x * e) * y) * (z * e) = e * (z * ((e * y) * x))
    ]]></literal>
    <justification jstring="[para(172(a,1),13(a,1,2)),demod(172(5),118(8)),flip(a)].">
      <j1 rule="para" parents="172 13"/>
      <j2 rule="demod" parents="172 118"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3834">
    <literal><![CDATA[
      ((x * y) * z) * u = ((u * y) * z) * x
    ]]></literal>
    <justification jstring="[para(1009(a,1),66(a,2,2)),demod(27(9),1517(8),297(7),214(5),193(7),17(10))].">
      <j1 rule="para" parents="1009 66"/>
      <j2 rule="demod" parents="27 1517 297 214 193 17"/>
    </justification>
  </clause>

  <clause id="3835">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(3834,a,10,a)].">
      <j1 rule="resolve" parents="3834 10"/>
    </justification>
  </clause>

</proof>

</proofs>
