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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>2basis.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13903 was started by mccune on cleo.thornwood,
Mon Jun 19 17:06:22 2006
The command was "/home/mccune/bin/prover9 -f 2basis.in".
]]></heading>

<proof number="1" length="30" max_count="27">

<comments><![CDATA[
% Proof 1 at 0.07 (+ 0.01) seconds: 2a_2b.
% Length of proof is 30.
% Level of proof is 15.
% Maximum clause weight is 27.
% Given clauses 34.
]]></comments>

  <clause id="3">
    <literal><![CDATA[
      f(A,B,C) != f(C,A,B)
    ]]></literal>
    <literal><![CDATA[
      f(A,B,C) != f(A,C,B)
    ]]></literal>
    <attribute><![CDATA[
      answer(2a_2b)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      f(x,x,y) = x
    ]]></literal>
    <attribute><![CDATA[
      label(majority)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="5">
    <literal><![CDATA[
      f(f(x,y,z),u,y) = f(y,x,f(y,u,z))
    ]]></literal>
    <attribute><![CDATA[
      label(associativity2)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="6">
    <literal><![CDATA[
      f(C,A,B) != f(A,B,C)
    ]]></literal>
    <literal><![CDATA[
      f(A,C,B) != f(A,B,C)
    ]]></literal>
    <attribute><![CDATA[
      answer(2a_2b)
    ]]></attribute>
    <justification jstring="[copy(3),flip(a),flip(b)].">
      <j1 rule="copy" parents="3"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      f(x,y,f(x,f(y,x,z),z)) = f(y,x,z)
    ]]></literal>
    <justification jstring="[para(5(a,1),4(a,1))].">
      <j1 rule="para" parents="5 4"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      f(x,y,x) = x
    ]]></literal>
    <justification jstring="[para(4(a,1),5(a,1,1)),demod(4(3))].">
      <j1 rule="para" parents="4 5"/>
      <j2 rule="demod" parents="4"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      f(x,y,f(x,z,y)) = f(y,z,x)
    ]]></literal>
    <justification jstring="[para(8(a,1),5(a,1,1)),flip(a)].">
      <j1 rule="para" parents="8 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      f(x,y,y) = y
    ]]></literal>
    <justification jstring="[para(4(a,1),10(a,1,3)),demod(8(1)),flip(a)].">
      <j1 rule="para" parents="4 10"/>
      <j2 rule="demod" parents="8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      f(x,y,f(x,z,f(y,u,x))) = f(f(x,u,y),z,x)
    ]]></literal>
    <justification jstring="[para(10(a,1),5(a,1,1)),flip(a)].">
      <j1 rule="para" parents="10 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      f(f(x,y,z),y,f(y,x,f(y,u,z))) = f(y,u,f(x,y,z))
    ]]></literal>
    <justification jstring="[para(5(a,1),10(a,1,3))].">
      <j1 rule="para" parents="5 10"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      f(x,f(x,y,z),f(z,y,x)) = f(f(x,y,z),z,x)
    ]]></literal>
    <justification jstring="[para(10(a,1),10(a,1,3))].">
      <j1 rule="para" parents="10 10"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(x,y,f(f(x,z,y),y,x)) = f(x,z,y)
    ]]></literal>
    <justification jstring="[para(10(a,1),7(a,1,3,2)),demod(14(3),10(5))].">
      <j1 rule="para" parents="10 7"/>
      <j2 rule="demod" parents="14 10"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(f(x,y,z),y,f(f(y,x,f(y,u,z)),y,f(x,y,z))) = f(y,x,f(y,u,z))
    ]]></literal>
    <justification jstring="[para(5(a,1),19(a,1,3,1)),demod(5(8))].">
      <j1 rule="para" parents="5 19"/>
      <j2 rule="demod" parents="5"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      f(f(x,y,z),x,z) = f(z,y,x)
    ]]></literal>
    <justification jstring="[para(19(a,1),12(a,1,3)),demod(11(3),10(3)),flip(a)].">
      <j1 rule="para" parents="19 12"/>
      <j2 rule="demod" parents="11 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      f(f(x,y,z),z,f(z,y,x)) = f(z,x,f(x,y,z))
    ]]></literal>
    <justification jstring="[para(34(a,1),10(a,1,3))].">
      <j1 rule="para" parents="34 10"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      f(f(x,y,z),z,x) = f(x,z,f(z,y,x))
    ]]></literal>
    <justification jstring="[para(10(a,1),34(a,1,1)),demod(38(3)),flip(a)].">
      <j1 rule="para" parents="10 34"/>
      <j2 rule="demod" parents="38"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      f(x,y,f(y,z,x)) = f(x,z,y)
    ]]></literal>
    <justification jstring="[para(19(a,1),34(a,2)),demod(39(2),39(5),34(6),12(3),39(2))].">
      <j1 rule="para" parents="19 34"/>
      <j2 rule="demod" parents="39 39 34 12 39"/>
    </justification>
  </clause>

  <clause id="51">
    <literal><![CDATA[
      f(x,y,z) = f(z,y,x)
    ]]></literal>
    <justification jstring="[back_demod(19),demod(39(2),44(2),10(2))].">
      <j1 rule="back_demod" parents="19"/>
      <j2 rule="demod" parents="39 44 10"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      f(f(x,y,z),z,f(z,y,x)) = f(z,y,x)
    ]]></literal>
    <justification jstring="[back_demod(38),demod(44(5))].">
      <j1 rule="back_demod" parents="38"/>
      <j2 rule="demod" parents="44"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      f(x,y,f(z,x,u)) = f(x,z,f(x,y,u))
    ]]></literal>
    <justification jstring="[back_demod(21),demod(51(5),13(5),44(4),51(2))].">
      <j1 rule="back_demod" parents="21"/>
      <j2 rule="demod" parents="51 13 44 51"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      f(B,A,C) != f(A,B,C)
    ]]></literal>
    <literal><![CDATA[
      f(A,C,B) != f(A,B,C)
    ]]></literal>
    <attribute><![CDATA[
      answer(2a_2b)
    ]]></attribute>
    <justification jstring="[back_demod(6),demod(51(4))].">
      <j1 rule="back_demod" parents="6"/>
      <j2 rule="demod" parents="51"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      f(f(x,y,z),y,f(y,x,z)) = f(y,z,f(x,y,z))
    ]]></literal>
    <justification jstring="[para(11(a,1),13(a,1,3,3))].">
      <j1 rule="para" parents="11 13"/>
    </justification>
  </clause>

  <clause id="78">
    <literal><![CDATA[
      f(x,y,f(z,y,x)) = f(x,y,z)
    ]]></literal>
    <justification jstring="[para(11(a,1),13(a,1,3)),demod(51(2),44(2),53(3),51(3),44(3)),flip(a)].">
      <j1 rule="para" parents="11 13"/>
      <j2 rule="demod" parents="51 44 53 51 44"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      f(x,y,f(y,x,z)) = f(y,x,z)
    ]]></literal>
    <justification jstring="[para(7(a,1),78(a,2)),demod(51(3),7(3))].">
      <j1 rule="para" parents="7 78"/>
      <j2 rule="demod" parents="51 7"/>
    </justification>
  </clause>

  <clause id="93">
    <literal><![CDATA[
      f(x,y,f(x,y,z)) = f(x,y,z)
    ]]></literal>
    <justification jstring="[para(51(a,1),78(a,1,3))].">
      <j1 rule="para" parents="51 78"/>
    </justification>
  </clause>

  <clause id="121">
    <literal><![CDATA[
      f(x,y,f(z,x,y)) = f(z,x,y)
    ]]></literal>
    <justification jstring="[para(89(a,1),13(a,2)),demod(93(3),77(3))].">
      <j1 rule="para" parents="89 13"/>
      <j2 rule="demod" parents="93 77"/>
    </justification>
  </clause>

  <clause id="219">
    <literal><![CDATA[
      f(x,y,z) = f(y,x,z)
    ]]></literal>
    <justification jstring="[para(11(a,1),67(a,2,3)),demod(121(2))].">
      <j1 rule="para" parents="11 67"/>
      <j2 rule="demod" parents="121"/>
    </justification>
  </clause>

  <clause id="310">
    <literal><![CDATA[
      f(A,C,B) != f(A,B,C)
    ]]></literal>
    <attribute><![CDATA[
      answer(2a_2b)
    ]]></attribute>
    <justification jstring="[back_demod(72),demod(219(4)),xx(a)].">
      <j1 rule="back_demod" parents="72"/>
      <j2 rule="demod" parents="219"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="326">
    <literal><![CDATA[
      f(x,y,z) = f(x,z,y)
    ]]></literal>
    <justification jstring="[para(219(a,1),44(a,1,3)),demod(78(2))].">
      <j1 rule="para" parents="219 44"/>
      <j2 rule="demod" parents="78"/>
    </justification>
  </clause>

  <clause id="327">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(2a_2b)
    ]]></attribute>
    <justification jstring="[resolve(326,a,310,a)].">
      <j1 rule="resolve" parents="326 310"/>
    </justification>
  </clause>

</proof>

</proofs>
