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

<heading><![CDATA[
Prover9 (32) version August-2006A, August 2006.
Process 11748 was started by mccune on cleo.thornwood,
Sat Aug 12 21:31:40 2006
The command was "/home/mccune/bin/prover9 -f dist-long-short.in".
]]></heading>

<proof number="1" length="91" max_count="54">

<comments><![CDATA[
% Proof 1 at 48.08 (+ 0.06) seconds: dist_short.
% Length of proof is 91.
% Level of proof is 15.
% Maximum clause weight is 54.
% Given clauses 187.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      f(f(x,y,z),u,v) = f(x,f(y,u,v),f(z,u,v))
    ]]></literal>
    <attribute><![CDATA[
      answer(dist_short)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

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

  <clause id="3">
    <literal><![CDATA[
      f(x,y,z) = f(z,x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(2a)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      f(x,y,z) = f(x,z,y)
    ]]></literal>
    <attribute><![CDATA[
      label(2b)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

  <clause id="6">
    <literal><![CDATA[
      f(x,y,f(x,z,u)) = f(x,z,f(x,y,u))
    ]]></literal>
    <justification jstring="[copy(5),rewrite(3(2),3(2),3(3,R),4(3)),rewrite(3(1,R),4(1),4(3),3(4,R),4(4))].">
      <j1 rule="copy" parents="5"/>
      <j2 rule="rewrite" parents="3 3 3 4"/>
      <j2 rule="rewrite" parents="3 4 4 3 4"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      f(f(x,y,z),u,v) = f(f(x,u,v),f(y,u,v),f(z,u,v))
    ]]></literal>
    <attribute><![CDATA[
      label(dist_long)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      f(f(x,y,z),f(y,z,u),f(y,z,v)) = f(y,z,f(x,u,v))
    ]]></literal>
    <justification jstring="[copy(7),rewrite(3(2),3(2)),flip(a),rewrite(3(2),3(2),3(3),3(3))].">
      <j1 rule="copy" parents="7"/>
      <j2 rule="rewrite" parents="3 3"/>
      <j2 rule="flip"/>
      <j2 rule="rewrite" parents="3 3 3 3"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      f(f(c1,c2,c3),c4,c5) != f(c1,f(c2,c4,c5),f(c3,c4,c5))
    ]]></literal>
    <attribute><![CDATA[
      answer(dist_short)
    ]]></attribute>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      f(c1,f(c2,c4,c5),f(c3,c4,c5)) != f(c4,c5,f(c1,c2,c3))
    ]]></literal>
    <attribute><![CDATA[
      answer(dist_short)
    ]]></attribute>
    <justification jstring="[copy(9),rewrite(3(7),3(7)),flip(a)].">
      <j1 rule="copy" parents="9"/>
      <j2 rule="rewrite" parents="3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

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

  <clause id="12">
    <literal><![CDATA[
      f(x,y,f(z,x,u)) = f(z,x,f(x,y,u))
    ]]></literal>
    <justification jstring="[para(6(a,1),3(a,2)),rewrite(4(2),3(3,R),4(3)),flip(a)].">
      <j1 rule="para" parents="6 3"/>
      <j2 rule="rewrite" parents="4 3 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      f(x,y,f(x,z,u)) = f(x,u,f(x,y,z))
    ]]></literal>
    <justification jstring="[para(3(a,1),6(a,1,3)),rewrite(3(1,R),4(1))].">
      <j1 rule="para" parents="3 6"/>
      <j2 rule="rewrite" parents="3 4"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      f(x,y,f(x,z,f(x,u,v))) = f(x,u,f(x,y,f(x,z,v)))
    ]]></literal>
    <justification jstring="[para(6(a,1),6(a,1,3))].">
      <j1 rule="para" parents="6 6"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      f(x,f(x,y,z),f(x,y,u)) = f(x,y,f(x,z,u))
    ]]></literal>
    <justification jstring="[para(2(a,1),8(a,1,1))].">
      <j1 rule="para" parents="2 8"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      f(f(x,y,z),f(x,y,u),f(x,y,v)) = f(x,y,f(u,v,z))
    ]]></literal>
    <justification jstring="[para(8(a,1),3(a,1)),rewrite(3(4),3(4)),flip(a)].">
      <j1 rule="para" parents="8 3"/>
      <j2 rule="rewrite" parents="3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(f(x,y,z),f(x,z,u),f(x,z,v)) = f(x,z,f(y,u,v))
    ]]></literal>
    <justification jstring="[para(3(a,1),8(a,1,1)),rewrite(3(2,R),4(2),3(3,R),4(3),3(6,R),4(6))].">
      <j1 rule="para" parents="3 8"/>
      <j2 rule="rewrite" parents="3 4 3 4 3 4"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      f(f(x,y,z),f(y,z,u),f(v,f(y,z,w),f(x,y,z))) = f(v,f(x,y,z),f(y,z,f(x,u,w)))
    ]]></literal>
    <justification jstring="[para(8(a,1),6(a,1,3)),rewrite(3(4,R),4(4),3(9,R)),flip(a)].">
      <j1 rule="para" parents="8 6"/>
      <j2 rule="rewrite" parents="3 4 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      f(f(x,y,z),f(y,u,f(y,z,v)),f(y,z,w)) = f(y,z,f(x,w,f(y,u,v)))
    ]]></literal>
    <justification jstring="[para(6(a,1),8(a,1,2)),rewrite(4(7))].">
      <j1 rule="para" parents="6 8"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      f(x,y,f(z,u,f(z,v,w))) = f(x,y,f(z,v,f(z,u,w)))
    ]]></literal>
    <justification jstring="[para(6(a,1),8(a,2,3)),rewrite(8(5))].">
      <j1 rule="para" parents="6 8"/>
      <j2 rule="rewrite" parents="8"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      f(f(x,f(y,z,u),f(z,u,v)),f(z,u,f(y,v,w)),f(v6,f(y,z,u),f(z,u,v))) = f(f(y,z,u),f(z,u,v),f(x,v6,f(z,u,w)))
    ]]></literal>
    <justification jstring="[para(8(a,1),8(a,1,2)),rewrite(3(8),4(13))].">
      <j1 rule="para" parents="8 8"/>
      <j2 rule="rewrite" parents="3 4"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      f(x,y,f(x,z,y)) = f(x,z,y)
    ]]></literal>
    <justification jstring="[para(11(a,1),6(a,1,3)),flip(a)].">
      <j1 rule="para" parents="11 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      f(x,y,f(x,y,z)) = f(x,y,z)
    ]]></literal>
    <justification jstring="[para(11(a,1),6(a,1)),rewrite(4(3),28(3)),flip(a)].">
      <j1 rule="para" parents="11 6"/>
      <j2 rule="rewrite" parents="4 28"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      f(x,f(y,x,z),f(u,y,x)) = f(y,x,f(u,x,z))
    ]]></literal>
    <justification jstring="[para(11(a,1),8(a,1,2)),rewrite(3(3,R))].">
      <j1 rule="para" parents="11 8"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      f(f(x,y,z),f(z,u,f(x,y,z)),f(z,v,f(x,y,z))) = f(z,f(x,y,z),f(x,u,v))
    ]]></literal>
    <justification jstring="[para(28(a,1),8(a,1,1)),rewrite(4(3),4(5))].">
      <j1 rule="para" parents="28 8"/>
      <j2 rule="rewrite" parents="4 4"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      f(x,y,f(z,u,f(x,y,v))) = f(x,y,f(z,v,u))
    ]]></literal>
    <justification jstring="[para(28(a,1),8(a,1,2)),rewrite(4(2),8(4),4(3),4(4)),flip(a)].">
      <j1 rule="para" parents="28 8"/>
      <j2 rule="rewrite" parents="4 8 4 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      f(x,y,f(z,y,u)) = f(z,y,f(x,y,u))
    ]]></literal>
    <justification jstring="[para(12(a,1),3(a,2)),rewrite(4(2),3(3,R),4(3))].">
      <j1 rule="para" parents="12 3"/>
      <j2 rule="rewrite" parents="4 3 4"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      f(x,y,f(z,x,u)) = f(z,x,f(x,u,y))
    ]]></literal>
    <justification jstring="[para(3(a,1),12(a,1)),rewrite(3(2),3(2),4(3))].">
      <j1 rule="para" parents="3 12"/>
      <j2 rule="rewrite" parents="3 3 4"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      f(f(x,y,f(y,z,u)),f(z,v,f(x,y,u)),f(z,w,f(x,y,u))) = f(z,f(x,y,u),f(y,v,w))
    ]]></literal>
    <justification jstring="[para(12(a,1),8(a,1,1)),rewrite(4(4),4(6))].">
      <j1 rule="para" parents="12 8"/>
      <j2 rule="rewrite" parents="4 4"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      f(x,f(x,y,z),f(u,x,y)) = f(u,x,f(x,y,z))
    ]]></literal>
    <justification jstring="[para(12(a,1),8(a,2)),rewrite(3(2),2(2),3(3,R))].">
      <j1 rule="para" parents="12 8"/>
      <j2 rule="rewrite" parents="3 2 3"/>
    </justification>
  </clause>

  <clause id="48">
    <literal><![CDATA[
      f(f(x,y,z),f(y,z,u),f(v,f(x,y,z),f(y,z,w))) = f(v,f(x,y,z),f(y,z,f(x,u,w)))
    ]]></literal>
    <justification jstring="[para(8(a,1),12(a,2,3))].">
      <j1 rule="para" parents="8 12"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(x,y,f(z,u,f(y,z,v))) = f(y,z,f(x,y,f(z,u,v)))
    ]]></literal>
    <justification jstring="[para(6(a,1),38(a,1,3)),rewrite(3(1,R),4(1),3(6,R),4(6))].">
      <j1 rule="para" parents="6 38"/>
      <j2 rule="rewrite" parents="3 4 3 4"/>
    </justification>
  </clause>

  <clause id="94">
    <literal><![CDATA[
      f(x,f(x,y,z),f(y,u,f(x,y,v))) = f(x,y,f(x,z,f(y,u,v)))
    ]]></literal>
    <justification jstring="[para(12(a,2),15(a,1,3))].">
      <j1 rule="para" parents="12 15"/>
    </justification>
  </clause>

  <clause id="115">
    <literal><![CDATA[
      f(x,f(x,y,z),f(x,u,v)) = f(x,y,f(x,u,f(x,v,z)))
    ]]></literal>
    <justification jstring="[para(14(a,1),13(a,1)),flip(a)].">
      <j1 rule="para" parents="14 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="141">
    <literal><![CDATA[
      f(x,f(y,x,z),f(y,x,u)) = f(y,x,f(x,u,z))
    ]]></literal>
    <justification jstring="[para(30(a,1),3(a,1)),rewrite(3(1,R),4(1),3(3),3(3),3(5,R)),flip(a)].">
      <j1 rule="para" parents="30 3"/>
      <j2 rule="rewrite" parents="3 4 3 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="148">
    <literal><![CDATA[
      f(x,f(y,x,z),f(y,u,f(y,x,z))) = f(y,x,z)
    ]]></literal>
    <justification jstring="[para(29(a,1),30(a,1,3)),rewrite(4(3),3(5),2(5),4(4)),flip(a)].">
      <j1 rule="para" parents="29 30"/>
      <j2 rule="rewrite" parents="4 3 2 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="149">
    <literal><![CDATA[
      f(x,f(y,x,z),f(u,f(y,x,v),f(y,x,z))) = f(u,f(y,x,z),f(y,x,f(x,z,v)))
    ]]></literal>
    <justification jstring="[para(30(a,1),12(a,1,3)),rewrite(3(2),3(2),3(4,R),4(4),3(7),3(7),3(8,R)),flip(a)].">
      <j1 rule="para" parents="30 12"/>
      <j2 rule="rewrite" parents="3 3 3 4 3 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="151">
    <literal><![CDATA[
      f(f(x,y,z),f(x,u,f(x,y,z)),f(x,y,f(x,z,v))) = f(x,f(x,y,z),f(u,v,f(x,y,z)))
    ]]></literal>
    <justification jstring="[para(12(a,2),30(a,1,3)),rewrite(4(3),3(4),3(4),3(9))].">
      <j1 rule="para" parents="12 30"/>
      <j2 rule="rewrite" parents="4 3 3 3"/>
    </justification>
  </clause>

  <clause id="198">
    <literal><![CDATA[
      f(x,y,f(z,u,f(v,w,f(x,y,v6)))) = f(x,y,f(z,u,f(v,v6,w)))
    ]]></literal>
    <justification jstring="[para(36(a,1),8(a,1,2)),rewrite(8(5),4(2),4(6)),flip(a)].">
      <j1 rule="para" parents="36 8"/>
      <j2 rule="rewrite" parents="8 4 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="213">
    <literal><![CDATA[
      f(x,y,f(y,z,f(x,z,u))) = f(x,y,z)
    ]]></literal>
    <justification jstring="[para(30(a,1),36(a,1,3)),rewrite(2(5))].">
      <j1 rule="para" parents="30 36"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="218">
    <literal><![CDATA[
      f(f(x,y,z),f(x,u,f(x,y,v)),f(x,y,w)) = f(x,y,f(z,w,f(x,u,v)))
    ]]></literal>
    <justification jstring="[para(6(a,1),16(a,1,2)),rewrite(3(7),4(7))].">
      <j1 rule="para" parents="6 16"/>
      <j2 rule="rewrite" parents="3 4"/>
    </justification>
  </clause>

  <clause id="226">
    <literal><![CDATA[
      f(x,y,f(z,u,f(x,v,y))) = f(x,y,f(v,z,u))
    ]]></literal>
    <justification jstring="[para(28(a,1),16(a,1,1)),rewrite(17(4)),flip(a)].">
      <j1 rule="para" parents="28 16"/>
      <j2 rule="rewrite" parents="17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="249">
    <literal><![CDATA[
      f(x,f(x,y,z),f(u,v,f(x,y,z))) = f(x,y,f(x,z,f(z,u,v)))
    ]]></literal>
    <justification jstring="[back_rewrite(151),rewrite(218(6),4(2),141(3)),flip(a)].">
      <j1 rule="back_rewrite" parents="151"/>
      <j2 rule="rewrite" parents="218 4 141"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="253">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,z,f(y,u,v))) = f(y,z,f(y,u,f(u,x,v)))
    ]]></literal>
    <justification jstring="[back_rewrite(149),rewrite(3(1,R),4(1),3(2,R),4(2),3(3,R),4(3),249(5),198(4),4(1),3(4,R),4(4),3(6,R),4(6)),flip(a)].">
      <j1 rule="back_rewrite" parents="149"/>
      <j2 rule="rewrite" parents="3 4 3 4 3 4 249 198 4 3 4 3 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="254">
    <literal><![CDATA[
      f(x,y,f(x,z,f(y,z,u))) = f(x,y,z)
    ]]></literal>
    <justification jstring="[back_rewrite(148),rewrite(3(1,R),4(1),3(2,R),4(2),94(4),4(1),3(4,R),4(4))].">
      <j1 rule="back_rewrite" parents="148"/>
      <j2 rule="rewrite" parents="3 4 3 4 94 4 3 4"/>
    </justification>
  </clause>

  <clause id="258">
    <literal><![CDATA[
      f(x,y,f(x,z,f(z,y,u))) = f(x,z,y)
    ]]></literal>
    <justification jstring="[para(213(a,1),3(a,2)),rewrite(3(1,R),4(1),4(3),3(4),3(4))].">
      <j1 rule="para" parents="213 3"/>
      <j2 rule="rewrite" parents="3 4 4 3 3"/>
    </justification>
  </clause>

  <clause id="271">
    <literal><![CDATA[
      f(x,f(y,z,u),f(x,y,z)) = f(x,y,z)
    ]]></literal>
    <justification jstring="[para(213(a,1),39(a,2)),rewrite(3(2,R),4(2),3(4,R),4(4))].">
      <j1 rule="para" parents="213 39"/>
      <j2 rule="rewrite" parents="3 4 3 4"/>
    </justification>
  </clause>

  <clause id="285">
    <literal><![CDATA[
      f(x,y,f(x,f(y,z,u),f(y,v,f(y,z,u)))) = f(y,z,f(x,y,u))
    ]]></literal>
    <justification jstring="[para(12(a,2),254(a,2)),rewrite(4(3))].">
      <j1 rule="para" parents="12 254"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="288">
    <literal><![CDATA[
      f(x,y,f(z,x,u)) = f(x,u,f(z,x,y))
    ]]></literal>
    <justification jstring="[para(39(a,2),254(a,2)),rewrite(4(3),285(5))].">
      <j1 rule="para" parents="39 254"/>
      <j2 rule="rewrite" parents="4 285"/>
    </justification>
  </clause>

  <clause id="294">
    <literal><![CDATA[
      f(x,y,f(z,x,f(z,y,u))) = f(z,x,y)
    ]]></literal>
    <justification jstring="[para(213(a,1),254(a,2)),rewrite(4(5),285(7))].">
      <j1 rule="para" parents="213 254"/>
      <j2 rule="rewrite" parents="4 285"/>
    </justification>
  </clause>

  <clause id="298">
    <literal><![CDATA[
      f(x,f(y,z,u),f(x,z,f(z,v,f(y,z,u)))) = f(y,z,f(x,z,u))
    ]]></literal>
    <justification jstring="[para(38(a,1),258(a,2)),rewrite(4(3))].">
      <j1 rule="para" parents="38 258"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="305">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,z,x)) = f(y,z,x)
    ]]></literal>
    <justification jstring="[para(271(a,1),3(a,2)),rewrite(3(2),3(2),3(3),3(4),3(4))].">
      <j1 rule="para" parents="271 3"/>
      <j2 rule="rewrite" parents="3 3 3 3 3"/>
    </justification>
  </clause>

  <clause id="306">
    <literal><![CDATA[
      f(x,f(y,z,u),f(x,z,u)) = f(x,z,u)
    ]]></literal>
    <justification jstring="[para(3(a,1),271(a,1,2))].">
      <j1 rule="para" parents="3 271"/>
    </justification>
  </clause>

  <clause id="320">
    <literal><![CDATA[
      f(x,f(y,z,f(y,u,v)),f(x,y,v)) = f(x,y,v)
    ]]></literal>
    <justification jstring="[para(13(a,2),271(a,1,2))].">
      <j1 rule="para" parents="13 271"/>
    </justification>
  </clause>

  <clause id="346">
    <literal><![CDATA[
      f(x,f(y,z,u),f(x,v,f(y,z,v))) = f(x,v,f(y,z,u))
    ]]></literal>
    <justification jstring="[para(271(a,1),258(a,1,3,3)),rewrite(3(2),3(2))].">
      <j1 rule="para" parents="271 258"/>
      <j2 rule="rewrite" parents="3 3"/>
    </justification>
  </clause>

  <clause id="367">
    <literal><![CDATA[
      f(f(x,y,f(x,z,u)),f(x,v,f(x,y,z)),f(x,w,f(x,y,z))) = f(x,f(x,y,z),f(u,v,w))
    ]]></literal>
    <justification jstring="[para(13(a,2),17(a,1,1)),rewrite(4(4),4(6))].">
      <j1 rule="para" parents="13 17"/>
      <j2 rule="rewrite" parents="4 4"/>
    </justification>
  </clause>

  <clause id="393">
    <literal><![CDATA[
      f(x,y,f(z,u,x)) = f(z,x,f(u,x,y))
    ]]></literal>
    <justification jstring="[para(288(a,1),3(a,2)),rewrite(4(2),3(3)),flip(a)].">
      <j1 rule="para" parents="288 3"/>
      <j2 rule="rewrite" parents="4 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="394">
    <literal><![CDATA[
      f(x,y,f(z,x,y)) = f(z,x,y)
    ]]></literal>
    <justification jstring="[para(11(a,1),288(a,1)),rewrite(29(3)),flip(a)].">
      <j1 rule="para" parents="11 288"/>
      <j2 rule="rewrite" parents="29"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="410">
    <literal><![CDATA[
      f(x,y,f(z,u,f(v,x,y))) = f(x,y,f(v,z,u))
    ]]></literal>
    <justification jstring="[para(394(a,1),16(a,1,1)),rewrite(8(4)),flip(a)].">
      <j1 rule="para" parents="394 16"/>
      <j2 rule="rewrite" parents="8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="427">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,x,z)) = f(y,x,z)
    ]]></literal>
    <justification jstring="[para(294(a,1),288(a,1)),flip(a)].">
      <j1 rule="para" parents="294 288"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="429">
    <literal><![CDATA[
      f(x,y,f(z,u,x)) = f(u,x,f(z,x,y))
    ]]></literal>
    <justification jstring="[para(288(a,1),294(a,2)),rewrite(4(3),3(4,R),4(4),298(5),3(3)),flip(a)].">
      <j1 rule="para" parents="288 294"/>
      <j2 rule="rewrite" parents="4 3 4 298 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="446">
    <literal><![CDATA[
      f(x,f(y,z,x),f(u,v,f(y,z,w))) = f(x,f(y,z,x),f(y,u,v))
    ]]></literal>
    <justification jstring="[para(305(a,1),17(a,1,1)),rewrite(4(3),4(5),35(6),3(6),3(6)),flip(a)].">
      <j1 rule="para" parents="305 17"/>
      <j2 rule="rewrite" parents="4 4 35 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="462">
    <literal><![CDATA[
      f(f(x,y,z),f(y,z,u),f(x,y,f(y,z,v))) = f(y,z,f(y,f(x,y,z),f(x,u,v)))
    ]]></literal>
    <justification jstring="[para(6(a,1),18(a,2)),rewrite(45(5))].">
      <j1 rule="para" parents="6 18"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="490">
    <literal><![CDATA[
      f(x,y,f(x,f(z,x,y),f(z,u,v))) = f(x,f(z,u,v),f(z,x,y))
    ]]></literal>
    <justification jstring="[para(13(a,1),18(a,2)),rewrite(45(5),462(5),4(7),394(7))].">
      <j1 rule="para" parents="13 18"/>
      <j2 rule="rewrite" parents="45 462 4 394"/>
    </justification>
  </clause>

  <clause id="497">
    <literal><![CDATA[
      f(x,f(y,x,z),f(x,z,f(y,u,v))) = f(x,f(y,u,v),f(y,x,z))
    ]]></literal>
    <justification jstring="[para(39(a,1),18(a,1,3)),rewrite(29(4),462(5),490(4)),flip(a)].">
      <j1 rule="para" parents="39 18"/>
      <j2 rule="rewrite" parents="29 462 490"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="586">
    <literal><![CDATA[
      f(x,f(y,z,u),f(z,u,x)) = f(z,u,x)
    ]]></literal>
    <justification jstring="[para(306(a,1),3(a,2)),rewrite(3(2),3(2),3(3),3(4),3(4))].">
      <j1 rule="para" parents="306 3"/>
      <j2 rule="rewrite" parents="3 3 3 3 3"/>
    </justification>
  </clause>

  <clause id="594">
    <literal><![CDATA[
      f(x,y,f(x,z,f(u,y,z))) = f(x,y,z)
    ]]></literal>
    <justification jstring="[para(306(a,1),8(a,2)),rewrite(2(2),4(2),4(4),115(5),11(3))].">
      <j1 rule="para" parents="306 8"/>
      <j2 rule="rewrite" parents="2 4 4 115 11"/>
    </justification>
  </clause>

  <clause id="713">
    <literal><![CDATA[
      f(x,y,f(z,u,y)) = f(z,y,f(x,u,y))
    ]]></literal>
    <justification jstring="[para(393(a,1),3(a,2)),rewrite(4(2),3(3))].">
      <j1 rule="para" parents="393 3"/>
      <j2 rule="rewrite" parents="4 3"/>
    </justification>
  </clause>

  <clause id="715">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,v,f(y,z,u))) = f(y,z,f(y,u,f(u,v,x)))
    ]]></literal>
    <justification jstring="[para(12(a,2),393(a,1,3)),rewrite(3(2),3(2),3(4,R),4(4),253(4),4(6)),flip(a)].">
      <j1 rule="para" parents="12 393"/>
      <j2 rule="rewrite" parents="3 3 3 4 253 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="738">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,u,f(z,u,v))) = f(u,f(y,z,u),f(y,x,v))
    ]]></literal>
    <justification jstring="[para(393(a,1),393(a,1,3)),rewrite(3(4,R),4(4),3(7),446(8))].">
      <j1 rule="para" parents="393 393"/>
      <j2 rule="rewrite" parents="3 4 3 446"/>
    </justification>
  </clause>

  <clause id="739">
    <literal><![CDATA[
      f(x,f(y,z,u),f(x,y,v)) = f(x,y,f(x,v,f(y,z,u)))
    ]]></literal>
    <justification jstring="[back_rewrite(497),rewrite(3(1,R),4(1),115(4),4(2),29(3),3(5,R),4(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="497"/>
      <j2 rule="rewrite" parents="3 4 115 4 29 3 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="758">
    <literal><![CDATA[
      f(x,y,f(x,z,f(y,u,f(y,v,z)))) = f(x,y,z)
    ]]></literal>
    <justification jstring="[back_rewrite(320),rewrite(739(4))].">
      <j1 rule="back_rewrite" parents="320"/>
      <j2 rule="rewrite" parents="739"/>
    </justification>
  </clause>

  <clause id="925">
    <literal><![CDATA[
      f(f(x,y,z),f(y,z,u),f(u,v,f(y,z,u))) = f(y,z,u)
    ]]></literal>
    <justification jstring="[para(306(a,1),427(a,1,3)),rewrite(3(2),3(2),4(3),3(4),3(4),4(5),3(7),3(7),586(8))].">
      <j1 rule="para" parents="306 427"/>
      <j2 rule="rewrite" parents="3 3 4 3 3 4 3 3 586"/>
    </justification>
  </clause>

  <clause id="948">
    <literal><![CDATA[
      f(x,f(y,z,u),f(u,v,f(y,z,u))) = f(u,f(y,z,u),f(y,v,x))
    ]]></literal>
    <justification jstring="[para(393(a,1),429(a,1,3)),rewrite(3(4,R),4(4),738(4),4(6)),flip(a)].">
      <j1 rule="para" parents="393 429"/>
      <j2 rule="rewrite" parents="3 4 738 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="959">
    <literal><![CDATA[
      f(x,f(y,z,x),f(y,u,f(v,y,z))) = f(y,z,x)
    ]]></literal>
    <justification jstring="[back_rewrite(925),rewrite(948(5))].">
      <j1 rule="back_rewrite" parents="925"/>
      <j2 rule="rewrite" parents="948"/>
    </justification>
  </clause>

  <clause id="1351">
    <literal><![CDATA[
      f(x,f(y,z,u),f(v,y,f(y,z,u))) = f(y,z,f(y,u,f(v,u,x)))
    ]]></literal>
    <justification jstring="[para(12(a,2),713(a,1,3)),rewrite(3(2),3(2),253(4),3(1,R),4(1)),flip(a)].">
      <j1 rule="para" parents="12 713"/>
      <j2 rule="rewrite" parents="3 3 253 3 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1352">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,z,f(v,y,u))) = f(y,z,f(y,u,f(v,u,x)))
    ]]></literal>
    <justification jstring="[para(12(a,2),713(a,2,3)),rewrite(3(3,R),4(3),715(4),3(1)),flip(a)].">
      <j1 rule="para" parents="12 713"/>
      <j2 rule="rewrite" parents="3 4 715 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1432">
    <literal><![CDATA[
      f(f(x,y,z),f(x,u,f(x,y,v)),f(x,w,f(x,y,v6))) = f(x,y,f(z,f(x,w,v6),f(x,u,v)))
    ]]></literal>
    <justification jstring="[para(20(a,1),22(a,1)),rewrite(3(5),3(5)),flip(a)].">
      <j1 rule="para" parents="20 22"/>
      <j2 rule="rewrite" parents="3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1443">
    <literal><![CDATA[
      f(x,f(x,y,z),f(u,v,w)) = f(x,y,f(x,z,f(u,v,w)))
    ]]></literal>
    <justification jstring="[back_rewrite(367),rewrite(1432(7),4(2),4(3),16(4),3(1),4(1)),flip(a)].">
      <j1 rule="back_rewrite" parents="367"/>
      <j2 rule="rewrite" parents="1432 4 4 16 3 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2694">
    <literal><![CDATA[
      f(x,f(y,z,u),f(z,u,f(x,y,v))) = f(x,f(y,z,u),f(z,u,v))
    ]]></literal>
    <justification jstring="[para(410(a,1),226(a,1,3)),rewrite(3(2,R),4(2),3(6),3(6))].">
      <j1 rule="para" parents="410 226"/>
      <j2 rule="rewrite" parents="3 4 3 3"/>
    </justification>
  </clause>

  <clause id="3865">
    <literal><![CDATA[
      f(x,f(y,z,f(u,v,w)),f(x,f(y,v6,f(u,v,w)),f(v7,f(y,z,f(u,v,w)),f(y,f(u,v,w),f(z,v,v6))))) = f(x,f(y,z,f(u,v,w)),f(y,v6,f(u,v,w)))
    ]]></literal>
    <justification jstring="[para(41(a,1),959(a,1,3,3)),rewrite(3(5),3(9,R),4(9),3(11,R),4(11),1443(12),3(17))].">
      <j1 rule="para" parents="41 959"/>
      <j2 rule="rewrite" parents="3 3 4 3 4 1443 3"/>
    </justification>
  </clause>

  <clause id="4983">
    <literal><![CDATA[
      f(x,f(y,z,u),f(z,u,f(y,v,x))) = f(z,u,f(y,v,x))
    ]]></literal>
    <justification jstring="[para(586(a,1),48(a,1,3)),rewrite(8(4)),flip(a)].">
      <j1 rule="para" parents="586 48"/>
      <j2 rule="rewrite" parents="8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="7393">
    <literal><![CDATA[
      f(x,f(y,x,z),f(u,x,z)) = f(y,x,f(u,x,z))
    ]]></literal>
    <justification jstring="[para(586(a,1),62(a,2,3)),rewrite(3(3),4(3),410(4),3(2,R),4(2),2694(4),3(2),2(2),4(2),3(4),3(5,R)),flip(a)].">
      <j1 rule="para" parents="586 62"/>
      <j2 rule="rewrite" parents="3 4 410 3 4 2694 3 2 4 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="8192">
    <literal><![CDATA[
      f(x,f(y,z,u),f(z,u,f(z,v,f(x,y,v)))) = f(x,f(y,z,u),f(z,u,v))
    ]]></literal>
    <justification jstring="[para(7393(a,1),594(a,1,3,3)),rewrite(3(2),3(2),3(3),3(3),1351(5),3(2),3(7),3(7))].">
      <j1 rule="para" parents="7393 594"/>
      <j2 rule="rewrite" parents="3 3 3 3 1351 3 3 3"/>
    </justification>
  </clause>

  <clause id="11728">
    <literal><![CDATA[
      f(x,f(y,z,u),f(z,v,f(z,w,f(x,y,w)))) = f(x,f(z,v,w),f(y,z,u))
    ]]></literal>
    <justification jstring="[para(12(a,2),346(a,1,3,3)),rewrite(1352(5),3(2))].">
      <j1 rule="para" parents="12 346"/>
      <j2 rule="rewrite" parents="1352 3"/>
    </justification>
  </clause>

  <clause id="11769">
    <literal><![CDATA[
      f(x,f(y,z,u),f(v,y,z)) = f(y,z,f(v,u,x))
    ]]></literal>
    <justification jstring="[para(346(a,1),24(a,2)),rewrite(2(4),48(7),4(3),29(3),4983(4),3(5,R)),flip(a)].">
      <j1 rule="para" parents="346 24"/>
      <j2 rule="rewrite" parents="2 48 4 29 4983 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11840">
    <literal><![CDATA[
      f(x,f(y,z,u),f(z,u,v)) = f(z,u,f(x,y,v))
    ]]></literal>
    <justification jstring="[back_rewrite(8192),rewrite(11728(5),11769(3),3(1)),flip(a)].">
      <j1 rule="back_rewrite" parents="8192"/>
      <j2 rule="rewrite" parents="11728 11769 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="12331">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,u,v)) = f(y,u,f(x,z,v))
    ]]></literal>
    <justification jstring="[para(3(a,2),11769(a,1,2)),rewrite(3(2),4(2),3(4),4(4),3(5,R),4(5))].">
      <j1 rule="para" parents="3 11769"/>
      <j2 rule="rewrite" parents="3 4 3 4 3 4"/>
    </justification>
  </clause>

  <clause id="12557">
    <literal><![CDATA[
      f(x,f(y,z,f(u,v,w)),f(y,v6,f(u,v,w))) = f(y,f(u,v,w),f(x,z,v6))
    ]]></literal>
    <justification jstring="[back_rewrite(3865),rewrite(12331(10),3(7,R),4(7),12331(9),12331(8),758(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="3865"/>
      <j2 rule="rewrite" parents="12331 3 4 12331 12331 758"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13185">
    <literal><![CDATA[
      f(x,f(y,z,u),f(v,z,u)) = f(z,u,f(v,x,y))
    ]]></literal>
    <justification jstring="[para(11840(a,1),11769(a,2)),rewrite(4(2),3(4,R),4(4),12557(5),4(2),3(4))].">
      <j1 rule="para" parents="11840 11769"/>
      <j2 rule="rewrite" parents="4 3 4 12557 4 3"/>
    </justification>
  </clause>

  <clause id="13263">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(dist_short)
    ]]></attribute>
    <justification jstring="[back_rewrite(10),rewrite(13185(10),3(6),3(6)),xx(a)].">
      <j1 rule="back_rewrite" parents="10"/>
      <j2 rule="rewrite" parents="13185 3 3"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
