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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>dist-short-long.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13954 was started by mccune on cleo.thornwood,
Mon Jun 19 17:18:28 2006
The command was "/home/mccune/bin/prover9 -f dist-short-long.in".
]]></heading>

<proof number="1" length="22" max_count="27">

<comments><![CDATA[
% Proof 1 at 1.10 (+ 0.02) seconds: dist_long.
% Length of proof is 22.
% Level of proof is 6.
% Maximum clause weight is 27.
% Given clauses 56.
]]></comments>

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

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

  <clause id="6">
    <literal><![CDATA[
      f(f(c1,c4,E),f(c2,c4,E),f(c3,c4,E)) != f(f(c1,c2,c3),c4,E)
    ]]></literal>
    <attribute><![CDATA[
      answer(dist_long)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      f(x,x,y) = x
    ]]></literal>
    <attribute><![CDATA[
      label(majority)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      f(x,y,z) = f(z,x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(2a)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      f(x,y,z) = f(x,z,y)
    ]]></literal>
    <attribute><![CDATA[
      label(2b)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      f(x,y,f(z,x,u)) = f(z,x,f(x,u,y))
    ]]></literal>
    <attribute><![CDATA[
      label(associativity)
    ]]></attribute>
    <justification jstring="[copy(4),demod(8(2),8(2),8(3,R),9(3))].">
      <j1 rule="copy" parents="4"/>
      <j2 rule="demod" parents="8 8 8 9"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      f(x,f(y,z,u),f(v,z,u)) = f(z,u,f(x,y,v))
    ]]></literal>
    <attribute><![CDATA[
      label(dist_short)
    ]]></attribute>
    <justification jstring="[copy(5),demod(8(2),8(2)),flip(a)].">
      <j1 rule="copy" parents="5"/>
      <j2 rule="demod" parents="8 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      f(f(E,c1,c4),f(E,c2,c4),f(E,c3,c4)) != f(E,c4,f(c1,c2,c3))
    ]]></literal>
    <attribute><![CDATA[
      answer(dist_long)
    ]]></attribute>
    <justification jstring="[copy(6),demod(8(4),8(8),8(12),8(20),9(20))].">
      <j1 rule="copy" parents="6"/>
      <j2 rule="demod" parents="8 8 8 8 9"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      f(x,y,y) = y
    ]]></literal>
    <justification jstring="[para(8(a,1),7(a,1))].">
      <j1 rule="para" parents="8 7"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      f(x,y,f(x,z,y)) = f(x,z,y)
    ]]></literal>
    <justification jstring="[para(10(a,2),10(a,1)),demod(8(1,R),13(1),8(2),9(2)),flip(a)].">
      <j1 rule="para" parents="10 10"/>
      <j2 rule="demod" parents="8 13 8 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,z,v)) = f(y,z,f(x,u,v))
    ]]></literal>
    <justification jstring="[para(11(a,1),8(a,1)),demod(8(3),8(3),8(4),8(4),8(5,R)),flip(a)].">
      <j1 rule="para" parents="11 8"/>
      <j2 rule="demod" parents="8 8 8 8 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      f(x,f(y,z,u),f(z,u,v)) = f(z,u,f(y,v,x))
    ]]></literal>
    <justification jstring="[para(11(a,1),8(a,2)),demod(8(2),8(2),8(3),8(4),8(4))].">
      <j1 rule="para" parents="11 8"/>
      <j2 rule="demod" parents="8 8 8 8 8"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,u,v)) = f(y,u,f(x,z,v))
    ]]></literal>
    <justification jstring="[para(8(a,1),11(a,1,2)),demod(8(2),9(2),8(5,R),9(5))].">
      <j1 rule="para" parents="8 11"/>
      <j2 rule="demod" parents="8 9 8 9"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      f(x,f(y,z,u),f(v,y,z)) = f(y,z,f(v,x,u))
    ]]></literal>
    <justification jstring="[para(8(a,1),11(a,1)),demod(8(2),8(2),8(3,R),8(4))].">
      <j1 rule="para" parents="8 11"/>
      <j2 rule="demod" parents="8 8 8 8"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      f(x,f(y,z,u),f(x,v,f(z,u,w))) = f(x,v,f(z,u,f(x,y,w)))
    ]]></literal>
    <justification jstring="[para(11(a,1),10(a,2,3)),demod(8(2),8(2),8(3,R),9(3),9(5),8(7,R),9(7))].">
      <j1 rule="para" parents="11 10"/>
      <j2 rule="demod" parents="8 8 8 9 9 8 9"/>
    </justification>
  </clause>

  <clause id="479">
    <literal><![CDATA[
      f(x,y,f(z,u,f(x,v,y))) = f(x,y,f(z,v,u))
    ]]></literal>
    <justification jstring="[para(20(a,1),26(a,1,2)),demod(28(3),9(4)),flip(a)].">
      <j1 rule="para" parents="20 26"/>
      <j2 rule="demod" parents="28 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="684">
    <literal><![CDATA[
      f(x,f(y,z,f(x,u,v)),f(y,v,f(x,u,v))) = f(x,v,f(y,u,z))
    ]]></literal>
    <justification jstring="[para(27(a,1),29(a,2)),demod(8(1),9(2),8(3),8(4,R),9(4),8(6),8(7,R),9(7))].">
      <j1 rule="para" parents="27 29"/>
      <j2 rule="demod" parents="8 9 8 8 9 8 8 9"/>
    </justification>
  </clause>

  <clause id="962">
    <literal><![CDATA[
      f(x,f(y,z,f(x,u,v)),f(w,y,f(x,u,v))) = f(y,f(x,u,v),f(x,w,z))
    ]]></literal>
    <justification jstring="[para(11(a,1),32(a,2)),demod(8(3,R),9(3),32(6),9(7))].">
      <j1 rule="para" parents="11 32"/>
      <j2 rule="demod" parents="8 9 32 9"/>
    </justification>
  </clause>

  <clause id="1251">
    <literal><![CDATA[
      f(x,f(y,z,f(x,u,v)),f(y,w,f(x,u,v))) = f(y,f(x,u,v),f(x,z,w))
    ]]></literal>
    <justification jstring="[back_demod(962),demod(8(4,R),9(4),9(7))].">
      <j1 rule="back_demod" parents="962"/>
      <j2 rule="demod" parents="8 9 9"/>
    </justification>
  </clause>

  <clause id="1256">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,v,u)) = f(y,u,f(x,v,z))
    ]]></literal>
    <justification jstring="[back_demod(684),demod(1251(5),9(4))].">
      <j1 rule="back_demod" parents="684"/>
      <j2 rule="demod" parents="1251 9"/>
    </justification>
  </clause>

  <clause id="1260">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(dist_long)
    ]]></attribute>
    <justification jstring="[back_demod(12),demod(1256(13),8(9),9(9),479(10),8(6,R),9(6)),xx(a)].">
      <j1 rule="back_demod" parents="12"/>
      <j2 rule="demod" parents="1256 8 9 479 8 9"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
