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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>cancel2.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13012 was started by mccune on cleo.thornwood,
Mon Jun 19 16:40:06 2006
The command was "/home/mccune/bin/prover9 -f cancel2.in".
]]></heading>

<proof number="1" length="25" max_count="15">

<comments><![CDATA[
% Proof 1 at 0.03 (+ 0.00) seconds.
% Length of proof is 25.
% Level of proof is 11.
% Maximum clause weight is 15.
% Given clauses 28.
]]></comments>

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

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

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

  <clause id="10">
    <literal><![CDATA[
      ((x * x) * x) * x = e
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      c3 * c2 = c1 * c2
    ]]></literal>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      c3 != c1
    ]]></literal>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      e * e = e
    ]]></literal>
    <justification jstring="[para(7(a,1),7(a,1,1)),demod(8(5)),flip(a)].">
      <j1 rule="para" parents="7 7"/>
      <j2 rule="demod" parents="8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      ((x * y) * x) * (e * y) = x * y
    ]]></literal>
    <justification jstring="[para(9(a,1),7(a,1))].">
      <j1 rule="para" parents="9 7"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      (x * y) * ((x * z) * u) = z * (y * u)
    ]]></literal>
    <justification jstring="[para(8(a,1),9(a,1,1)),flip(a)].">
      <j1 rule="para" parents="8 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      ((x * x) * x) * e = x
    ]]></literal>
    <justification jstring="[para(10(a,1),8(a,1,2))].">
      <j1 rule="para" parents="10 8"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      (x * y) * (e * (y * x)) = x
    ]]></literal>
    <justification jstring="[para(8(a,1),14(a,1,1,1)),demod(8(7))].">
      <j1 rule="para" parents="8 14"/>
      <j2 rule="demod" parents="8"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      (x * y) * x = e * (y * x)
    ]]></literal>
    <justification jstring="[para(41(a,1),8(a,1,2))].">
      <j1 rule="para" parents="41 8"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      x * (e * (x * x)) = e * (x * x)
    ]]></literal>
    <justification jstring="[para(10(a,1),41(a,1,1)),demod(67(4),8(8),67(6))].">
      <j1 rule="para" parents="10 41"/>
      <j2 rule="demod" parents="67 8 67"/>
    </justification>
  </clause>

  <clause id="73">
    <literal><![CDATA[
      e * ((x * x) * e) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),41(a,1,2,2)),demod(67(2),72(4),13(6),67(5))].">
      <j1 rule="para" parents="10 41"/>
      <j2 rule="demod" parents="67 72 13 67"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      (c1 * c2) * (e * (c2 * c3)) = c3
    ]]></literal>
    <justification jstring="[para(11(a,1),41(a,1,1))].">
      <j1 rule="para" parents="11 41"/>
    </justification>
  </clause>

  <clause id="76">
    <literal><![CDATA[
      (x * x) * (e * x) = e
    ]]></literal>
    <justification jstring="[para(22(a,1),41(a,1,2,2)),demod(67(3),8(5))].">
      <j1 rule="para" parents="22 41"/>
      <j2 rule="demod" parents="67 8"/>
    </justification>
  </clause>

  <clause id="96">
    <literal><![CDATA[
      (x * x) * e = e * x
    ]]></literal>
    <justification jstring="[para(73(a,1),8(a,1,2)),flip(a)].">
      <j1 rule="para" parents="73 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="212">
    <literal><![CDATA[
      (c1 * e) * c3 = c3
    ]]></literal>
    <justification jstring="[para(74(a,1),9(a,1)),demod(8(9)),flip(a)].">
      <j1 rule="para" parents="74 9"/>
      <j2 rule="demod" parents="8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="219">
    <literal><![CDATA[
      (c3 * (c1 * e)) * (e * c3) = c3
    ]]></literal>
    <justification jstring="[para(212(a,1),41(a,1,2,2))].">
      <j1 rule="para" parents="212 41"/>
    </justification>
  </clause>

  <clause id="254">
    <literal><![CDATA[
      (x * y) * e = x * (y * (e * x))
    ]]></literal>
    <justification jstring="[para(76(a,1),18(a,1,2))].">
      <j1 rule="para" parents="76 18"/>
    </justification>
  </clause>

  <clause id="279">
    <literal><![CDATA[
      (x * y) * (e * x) = x * (y * e)
    ]]></literal>
    <justification jstring="[para(96(a,1),18(a,1,2))].">
      <j1 rule="para" parents="96 18"/>
    </justification>
  </clause>

  <clause id="358">
    <literal><![CDATA[
      c3 * (c1 * c1) = c3
    ]]></literal>
    <justification jstring="[back_demod(219),demod(279(9),254(6),8(7))].">
      <j1 rule="back_demod" parents="219"/>
      <j2 rule="demod" parents="279 254 8"/>
    </justification>
  </clause>

  <clause id="361">
    <literal><![CDATA[
      c3 * c3 = c1 * c1
    ]]></literal>
    <justification jstring="[para(358(a,1),8(a,1,2))].">
      <j1 rule="para" parents="358 8"/>
    </justification>
  </clause>

  <clause id="376">
    <literal><![CDATA[
      c3 = c1
    ]]></literal>
    <justification jstring="[para(361(a,1),41(a,1,1)),demod(361(7),41(9)),flip(a)].">
      <j1 rule="para" parents="361 41"/>
      <j2 rule="demod" parents="361 41"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="377">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(376,a,12,a)].">
      <j1 rule="resolve" parents="376 12"/>
    </justification>
  </clause>

</proof>

</proofs>
