<?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 April-2007, April 2007.
Process 28060 was started by mccune on cleo,
Fri Apr 13 09:56:39 2007
The command was "/home/mccune/bin/prover9 -f dist-long-short.in".
]]></heading>

<proof number="1" length="110" max_count="45">

<comments><![CDATA[
% Proof 1 at 78.24 (+ 0.11) seconds: dist_short.
% Length of proof is 110.
% Level of proof is 19.
% Maximum clause weight is 45.
% Given clauses 185.
]]></comments>

  <clause id="1" type="goal">
    <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>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

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

  <clause id="3" type="assumption">
    <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" type="assumption">
    <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" type="assumption">
    <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" type="assumption">
    <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" type="deny">
    <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="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="19">
    <literal><![CDATA[
      f(f(x,y,f(x,z,u)),f(z,v,f(x,y,u)),f(z,w,f(x,y,u))) = f(z,f(x,y,u),f(x,v,w))
    ]]></literal>
    <justification jstring="[para(6(a,1),8(a,1,1)),rewrite(4(4),4(6))].">
      <j1 rule="para" parents="6 8"/>
      <j2 rule="rewrite" parents="4 4"/>
    </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="21">
    <literal><![CDATA[
      f(f(x,y,z),f(y,z,u),f(y,v,f(y,z,w))) = f(y,z,f(x,u,f(y,v,w)))
    ]]></literal>
    <justification jstring="[para(6(a,1),8(a,1,3))].">
      <j1 rule="para" parents="6 8"/>
    </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="31">
    <literal><![CDATA[
      f(x,f(y,z,x),f(z,x,u)) = f(z,x,f(y,x,u))
    ]]></literal>
    <justification jstring="[para(11(a,1),8(a,1,3)),rewrite(3(3),4(4))].">
      <j1 rule="para" parents="11 8"/>
      <j2 rule="rewrite" parents="3 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="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="83">
    <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="96">
    <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="98">
    <literal><![CDATA[
      f(f(x,y,f(y,z,u)),f(y,v,f(x,y,u)),f(y,w,f(x,y,u))) = f(y,f(x,y,u),f(z,v,w))
    ]]></literal>
    <justification jstring="[para(38(a,1),8(a,1,1)),rewrite(3(1,R),4(1),4(4),4(6))].">
      <j1 rule="para" parents="38 8"/>
      <j2 rule="rewrite" parents="3 4 4 4"/>
    </justification>
  </clause>

  <clause id="125">
    <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="135">
    <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="139">
    <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="149">
    <literal><![CDATA[
      f(x,f(y,x,z),f(u,f(y,x,z),f(y,x,v))) = f(u,f(y,x,z),f(y,x,f(x,z,v)))
    ]]></literal>
    <justification jstring="[para(30(a,1),38(a,1,3)),rewrite(3(2),3(2),3(7),3(7)),flip(a)].">
      <j1 rule="para" parents="30 38"/>
      <j2 rule="rewrite" parents="3 3 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="153">
    <literal><![CDATA[
      f(f(x,y,z),f(x,y,f(y,z,u)),f(y,v,f(x,y,z))) = f(v,f(x,y,z),f(x,y,f(y,z,u)))
    ]]></literal>
    <justification jstring="[para(30(a,1),30(a,1,2)),rewrite(3(2),3(2),3(5,R),4(5),3(9),3(9),149(11))].">
      <j1 rule="para" parents="30 30"/>
      <j2 rule="rewrite" parents="3 3 3 4 3 3 149"/>
    </justification>
  </clause>

  <clause id="154">
    <literal><![CDATA[
      f(f(x,y,z),f(u,x,y),f(y,v,f(u,x,y))) = f(v,f(u,x,y),f(x,y,f(u,y,z)))
    ]]></literal>
    <justification jstring="[para(30(a,1),30(a,1,3)),rewrite(3(4),4(7),18(7),4(3),28(3),4(8)),flip(a)].">
      <j1 rule="para" parents="30 30"/>
      <j2 rule="rewrite" parents="3 4 18 4 28 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="157">
    <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="165">
    <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="169">
    <literal><![CDATA[
      f(f(x,y,z),f(x,y,u),f(y,v,f(x,y,w))) = f(x,y,f(z,u,f(y,v,w)))
    ]]></literal>
    <justification jstring="[para(12(a,2),16(a,1,3)),rewrite(3(7))].">
      <j1 rule="para" parents="12 16"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="178">
    <literal><![CDATA[
      f(x,f(y,x,z),f(u,v,f(y,x,w))) = f(x,f(y,x,z),f(w,u,v))
    ]]></literal>
    <justification jstring="[para(30(a,1),16(a,1,1)),rewrite(3(1,R),4(1),4(4),4(6),98(7),3(5),3(5)),flip(a)].">
      <j1 rule="para" parents="30 16"/>
      <j2 rule="rewrite" parents="3 4 4 4 98 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="183">
    <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(139),rewrite(157(6),4(2),125(3)),flip(a)].">
      <j1 rule="back_rewrite" parents="139"/>
      <j2 rule="rewrite" parents="157 4 125"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="185">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,z,f(z,u,v))) = f(y,z,f(z,u,f(u,v,x)))
    ]]></literal>
    <justification jstring="[back_rewrite(153),rewrite(169(6),4(2),125(3),4(1)),flip(a)].">
      <j1 rule="back_rewrite" parents="153"/>
      <j2 rule="rewrite" parents="169 4 125 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="187">
    <literal><![CDATA[
      f(x,y,f(x,z,f(y,z,u))) = f(x,y,z)
    ]]></literal>
    <justification jstring="[back_rewrite(135),rewrite(3(1,R),4(1),3(2,R),4(2),183(4),3(1,R),4(1),3(4,R),4(4))].">
      <j1 rule="back_rewrite" parents="135"/>
      <j2 rule="rewrite" parents="3 4 3 4 183 3 4 3 4"/>
    </justification>
  </clause>

  <clause id="189">
    <literal><![CDATA[
      f(x,y,f(y,z,f(x,z,u))) = f(x,y,z)
    ]]></literal>
    <justification jstring="[para(187(a,1),3(a,2)),rewrite(4(3),3(4,R),4(4))].">
      <j1 rule="para" parents="187 3"/>
      <j2 rule="rewrite" parents="4 3 4"/>
    </justification>
  </clause>

  <clause id="190">
    <literal><![CDATA[
      f(x,y,f(x,z,f(z,y,u))) = f(x,z,y)
    ]]></literal>
    <justification jstring="[para(3(a,1),187(a,1)),rewrite(3(1,R),4(1),3(3),3(3),4(4))].">
      <j1 rule="para" parents="3 187"/>
      <j2 rule="rewrite" parents="3 4 3 3 4"/>
    </justification>
  </clause>

  <clause id="198">
    <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),187(a,2)),rewrite(4(3))].">
      <j1 rule="para" parents="12 187"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="200">
    <literal><![CDATA[
      f(x,f(y,z,u),f(x,y,z)) = f(x,y,z)
    ]]></literal>
    <justification jstring="[para(187(a,1),13(a,1)),flip(a)].">
      <j1 rule="para" parents="187 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="204">
    <literal><![CDATA[
      f(x,y,f(z,x,u)) = f(x,u,f(z,x,y))
    ]]></literal>
    <justification jstring="[para(39(a,2),187(a,2)),rewrite(4(3),198(5))].">
      <j1 rule="para" parents="39 187"/>
      <j2 rule="rewrite" parents="4 198"/>
    </justification>
  </clause>

  <clause id="226">
    <literal><![CDATA[
      f(x,y,f(z,x,f(z,y,u))) = f(z,x,y)
    ]]></literal>
    <justification jstring="[para(189(a,1),187(a,2)),rewrite(4(5),198(7))].">
      <j1 rule="para" parents="189 187"/>
      <j2 rule="rewrite" parents="4 198"/>
    </justification>
  </clause>

  <clause id="233">
    <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),190(a,2)),rewrite(4(3))].">
      <j1 rule="para" parents="38 190"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="236">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,z,x)) = f(y,z,x)
    ]]></literal>
    <justification jstring="[para(200(a,1),3(a,2)),rewrite(3(2),3(2),3(3),3(4),3(4))].">
      <j1 rule="para" parents="200 3"/>
      <j2 rule="rewrite" parents="3 3 3 3 3"/>
    </justification>
  </clause>

  <clause id="237">
    <literal><![CDATA[
      f(x,f(y,z,u),f(x,z,u)) = f(x,z,u)
    ]]></literal>
    <justification jstring="[para(3(a,1),200(a,1,2))].">
      <j1 rule="para" parents="3 200"/>
    </justification>
  </clause>

  <clause id="239">
    <literal><![CDATA[
      f(x,f(x,y,z),f(y,z,u)) = f(x,y,z)
    ]]></literal>
    <justification jstring="[para(200(a,1),4(a,1)),flip(a)].">
      <j1 rule="para" parents="200 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="249">
    <literal><![CDATA[
      f(x,f(y,z,f(z,u,v)),f(x,z,u)) = f(x,z,u)
    ]]></literal>
    <justification jstring="[para(12(a,1),200(a,1,2))].">
      <j1 rule="para" parents="12 200"/>
    </justification>
  </clause>

  <clause id="275">
    <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(200(a,1),190(a,1,3,3)),rewrite(3(2),3(2))].">
      <j1 rule="para" parents="200 190"/>
      <j2 rule="rewrite" parents="3 3"/>
    </justification>
  </clause>

  <clause id="295">
    <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="320">
    <literal><![CDATA[
      f(x,y,f(z,u,x)) = f(z,x,f(u,x,y))
    ]]></literal>
    <justification jstring="[para(204(a,1),3(a,2)),rewrite(4(2),3(3)),flip(a)].">
      <j1 rule="para" parents="204 3"/>
      <j2 rule="rewrite" parents="4 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="321">
    <literal><![CDATA[
      f(x,y,f(z,x,y)) = f(z,x,y)
    ]]></literal>
    <justification jstring="[para(11(a,1),204(a,1)),rewrite(29(3)),flip(a)].">
      <j1 rule="para" parents="11 204"/>
      <j2 rule="rewrite" parents="29"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="335">
    <literal><![CDATA[
      f(x,y,f(z,u,f(v,x,y))) = f(x,y,f(v,z,u))
    ]]></literal>
    <justification jstring="[para(321(a,1),16(a,1,1)),rewrite(8(4)),flip(a)].">
      <j1 rule="para" parents="321 16"/>
      <j2 rule="rewrite" parents="8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="353">
    <literal><![CDATA[
      f(x,y,f(z,u,x)) = f(u,x,f(z,x,y))
    ]]></literal>
    <justification jstring="[para(204(a,1),226(a,2)),rewrite(4(3),3(4,R),4(4),233(5),3(3)),flip(a)].">
      <j1 rule="para" parents="204 226"/>
      <j2 rule="rewrite" parents="4 3 4 233 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="370">
    <literal><![CDATA[
      f(f(x,y,z),f(y,z,u),f(x,y,z)) = f(x,y,z)
    ]]></literal>
    <justification jstring="[para(321(a,1),236(a,1,3)),rewrite(321(6))].">
      <j1 rule="para" parents="321 236"/>
      <j2 rule="rewrite" parents="321"/>
    </justification>
  </clause>

  <clause id="403">
    <literal><![CDATA[
      f(x,f(y,x,z),f(z,u,f(x,z,v))) = f(y,x,f(x,z,f(z,u,v)))
    ]]></literal>
    <justification jstring="[para(12(a,1),18(a,2,3)),rewrite(3(2),2(2),3(5,R),4(5),178(5),185(8),4(5))].">
      <j1 rule="para" parents="12 18"/>
      <j2 rule="rewrite" parents="3 2 3 4 178 185 4"/>
    </justification>
  </clause>

  <clause id="469">
    <literal><![CDATA[
      f(x,f(y,z,u),f(z,v,f(y,z,u))) = f(y,z,f(z,u,f(u,x,v)))
    ]]></literal>
    <justification jstring="[para(204(a,1),18(a,2,3)),rewrite(3(2),2(2),3(5,R),4(5),178(5),403(4)),flip(a)].">
      <j1 rule="para" parents="204 18"/>
      <j2 rule="rewrite" parents="3 2 3 4 178 403"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="497">
    <literal><![CDATA[
      f(x,f(y,z,u),f(z,u,x)) = f(z,u,x)
    ]]></literal>
    <justification jstring="[para(237(a,1),3(a,2)),rewrite(3(2),3(2),3(3),3(4),3(4))].">
      <j1 rule="para" parents="237 3"/>
      <j2 rule="rewrite" parents="3 3 3 3 3"/>
    </justification>
  </clause>

  <clause id="499">
    <literal><![CDATA[
      f(x,y,f(x,z,f(y,u,z))) = f(x,y,z)
    ]]></literal>
    <justification jstring="[para(237(a,1),6(a,2)),rewrite(3(1,R),4(1),4(2))].">
      <j1 rule="para" parents="237 6"/>
      <j2 rule="rewrite" parents="3 4 4"/>
    </justification>
  </clause>

  <clause id="505">
    <literal><![CDATA[
      f(x,y,f(x,z,f(u,y,z))) = f(x,y,z)
    ]]></literal>
    <justification jstring="[para(237(a,1),8(a,2)),rewrite(2(2),4(2),4(4),83(5),11(3))].">
      <j1 rule="para" parents="237 8"/>
      <j2 rule="rewrite" parents="2 4 4 83 11"/>
    </justification>
  </clause>

  <clause id="513">
    <literal><![CDATA[
      f(x,f(y,z,u),f(x,z,f(x,u,v))) = f(x,v,f(x,z,u))
    ]]></literal>
    <justification jstring="[para(237(a,1),14(a,2,3)),rewrite(4(2))].">
      <j1 rule="para" parents="237 14"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="528">
    <literal><![CDATA[
      f(x,f(y,z,f(u,v,y)),f(x,v,y)) = f(x,v,y)
    ]]></literal>
    <justification jstring="[para(237(a,1),187(a,2)),rewrite(3(5),183(6),513(6))].">
      <j1 rule="para" parents="237 187"/>
      <j2 rule="rewrite" parents="3 183 513"/>
    </justification>
  </clause>

  <clause id="591">
    <literal><![CDATA[
      f(f(x,y,z),f(u,f(x,y,z),f(y,z,v)),f(w,f(x,y,z),f(y,z,v))) = f(f(x,y,z),f(y,z,v),f(x,u,w))
    ]]></literal>
    <justification jstring="[para(239(a,1),8(a,1,1)),rewrite(3(4),3(7))].">
      <j1 rule="para" parents="239 8"/>
      <j2 rule="rewrite" parents="3 3"/>
    </justification>
  </clause>

  <clause id="619">
    <literal><![CDATA[
      f(x,y,f(z,u,y)) = f(z,y,f(x,u,y))
    ]]></literal>
    <justification jstring="[para(320(a,1),3(a,2)),rewrite(4(2),3(3))].">
      <j1 rule="para" parents="320 3"/>
      <j2 rule="rewrite" parents="4 3"/>
    </justification>
  </clause>

  <clause id="620">
    <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="[para(6(a,1),320(a,1,3)),rewrite(4(2),3(4,R),4(4),3(7),183(8))].">
      <j1 rule="para" parents="6 320"/>
      <j2 rule="rewrite" parents="4 3 4 3 183"/>
    </justification>
  </clause>

  <clause id="622">
    <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),320(a,1,3)),rewrite(3(2),3(2),3(4,R),4(4),620(4),4(6)),flip(a)].">
      <j1 rule="para" parents="12 320"/>
      <j2 rule="rewrite" parents="3 3 3 4 620 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="639">
    <literal><![CDATA[
      f(f(x,y,z),f(y,z,u),f(y,v,f(x,y,z))) = f(x,y,f(y,z,f(z,u,v)))
    ]]></literal>
    <justification jstring="[para(320(a,1),18(a,2,3)),rewrite(3(3),2(3),3(4,R),4(4),3(7),3(7),185(9))].">
      <j1 rule="para" parents="320 18"/>
      <j2 rule="rewrite" parents="3 2 3 4 3 3 185"/>
    </justification>
  </clause>

  <clause id="756">
    <literal><![CDATA[
      f(x,y,f(y,z,f(z,u,f(x,y,v)))) = f(y,f(x,y,z),f(x,v,u))
    ]]></literal>
    <justification jstring="[para(204(a,1),19(a,1,2)),rewrite(29(2),639(6),4(2))].">
      <j1 rule="para" parents="204 19"/>
      <j2 rule="rewrite" parents="29 639 4"/>
    </justification>
  </clause>

  <clause id="791">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,z,f(z,v,u))) = f(z,f(y,z,u),f(y,v,x))
    ]]></literal>
    <justification jstring="[back_rewrite(154),rewrite(3(2),3(2),3(3),3(3),469(5),4(2),756(4),3(4),3(4),3(5),3(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="154"/>
      <j2 rule="rewrite" parents="3 3 3 3 469 4 756 3 3 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="832">
    <literal><![CDATA[
      f(x,y,f(z,u,y)) = f(u,y,f(x,z,y))
    ]]></literal>
    <justification jstring="[para(353(a,1),3(a,2)),rewrite(4(2),3(3))].">
      <j1 rule="para" parents="353 3"/>
      <j2 rule="rewrite" parents="4 3"/>
    </justification>
  </clause>

  <clause id="1143">
    <literal><![CDATA[
      f(f(x,y,z),f(y,z,u),f(v,w,f(x,y,z))) = f(f(x,y,z),f(y,z,u),f(v,x,w))
    ]]></literal>
    <justification jstring="[para(18(a,1),21(a,1,3)),rewrite(3(6),4(9),2(11),3(8),591(8),3(3,R),4(3),370(10)),flip(a)].">
      <j1 rule="para" parents="18 21"/>
      <j2 rule="rewrite" parents="3 4 2 3 591 3 4 370"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1188">
    <literal><![CDATA[
      f(f(x,y,z),f(y,z,u),f(x,y,v)) = f(x,y,f(y,z,f(z,u,v)))
    ]]></literal>
    <justification jstring="[back_rewrite(639),rewrite(1143(5),3(3,R),4(3))].">
      <j1 rule="back_rewrite" parents="639"/>
      <j2 rule="rewrite" parents="1143 3 4"/>
    </justification>
  </clause>

  <clause id="1233">
    <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),619(a,1,3)),rewrite(3(2),3(2),620(4),3(1,R),4(1)),flip(a)].">
      <j1 rule="para" parents="12 619"/>
      <j2 rule="rewrite" parents="3 3 620 3 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1234">
    <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),619(a,2,3)),rewrite(3(3,R),4(3),622(4),3(1)),flip(a)].">
      <j1 rule="para" parents="12 619"/>
      <j2 rule="rewrite" parents="3 4 622 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1308">
    <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="1318">
    <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(295),rewrite(1308(7),4(2),4(3),16(4),3(1),4(1)),flip(a)].">
      <j1 rule="back_rewrite" parents="295"/>
      <j2 rule="rewrite" parents="1308 4 4 16 3 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1375">
    <literal><![CDATA[
      f(x,y,f(x,f(y,z,u),f(y,u,v))) = f(y,f(x,y,u),f(x,z,v))
    ]]></literal>
    <justification jstring="[para(505(a,1),832(a,2,3)),rewrite(3(1),3(3),1318(6),165(5),3(5),4(7),4(8),791(8))].">
      <j1 rule="para" parents="505 832"/>
      <j2 rule="rewrite" parents="3 3 1318 165 3 4 4 791"/>
    </justification>
  </clause>

  <clause id="1711">
    <literal><![CDATA[
      f(x,f(y,x,z),f(y,u,v)) = f(x,z,f(y,x,f(y,u,v)))
    ]]></literal>
    <justification jstring="[para(31(a,1),21(a,2,3)),rewrite(3(2),4(2),3(5,R),4(5),1188(6),36(4),756(4),3(4,R),4(4),3(5,R),4(5))].">
      <j1 rule="para" parents="31 21"/>
      <j2 rule="rewrite" parents="3 4 3 4 1188 36 756 3 4 3 4"/>
    </justification>
  </clause>

  <clause id="1735">
    <literal><![CDATA[
      f(x,y,f(x,f(y,z,u),f(y,u,v))) = f(y,u,f(x,y,f(x,z,v)))
    ]]></literal>
    <justification jstring="[back_rewrite(1375),rewrite(1711(7))].">
      <j1 rule="back_rewrite" parents="1375"/>
      <j2 rule="rewrite" parents="1711"/>
    </justification>
  </clause>

  <clause id="2254">
    <literal><![CDATA[
      f(x,f(y,z,f(z,u,v)),f(z,u,x)) = f(z,u,x)
    ]]></literal>
    <justification jstring="[para(249(a,1),3(a,2)),rewrite(3(3),3(3),3(4),3(5),3(5))].">
      <j1 rule="para" parents="249 3"/>
      <j2 rule="rewrite" parents="3 3 3 3 3"/>
    </justification>
  </clause>

  <clause id="2670">
    <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(335(a,1),165(a,1,3)),rewrite(3(2,R),4(2),3(6),3(6))].">
      <j1 rule="para" parents="335 165"/>
      <j2 rule="rewrite" parents="3 4 3 3"/>
    </justification>
  </clause>

  <clause id="2835">
    <literal><![CDATA[
      f(x,f(y,z,u),f(x,y,u)) = f(x,y,u)
    ]]></literal>
    <justification jstring="[para(2(a,1),528(a,1,2,3)),rewrite(4(2),4(4))].">
      <j1 rule="para" parents="2 528"/>
      <j2 rule="rewrite" parents="4 4"/>
    </justification>
  </clause>

  <clause id="3388">
    <literal><![CDATA[
      f(x,f(y,z,f(z,x,u)),f(x,v,f(y,z,u))) = f(x,f(y,z,u),f(z,x,v))
    ]]></literal>
    <justification jstring="[para(2(a,1),41(a,1,3)),rewrite(3(5),4(7))].">
      <j1 rule="para" parents="2 41"/>
      <j2 rule="rewrite" parents="3 4"/>
    </justification>
  </clause>

  <clause id="3513">
    <literal><![CDATA[
      f(x,f(y,z,f(u,v,w)),f(x,f(y,v6,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),505(a,1,3,3)),rewrite(3(6,R),4(6))].">
      <j1 rule="para" parents="41 505"/>
      <j2 rule="rewrite" parents="3 4"/>
    </justification>
  </clause>

  <clause id="3604">
    <literal><![CDATA[
      f(x,y,f(z,x,f(z,u,y))) = f(z,x,y)
    ]]></literal>
    <justification jstring="[para(528(a,1),41(a,2)),rewrite(3(1),2(2),11(1),3(1),3(4),3(6,R),4(6),29(6),4(6),28(6),3(4))].">
      <j1 rule="para" parents="528 41"/>
      <j2 rule="rewrite" parents="3 2 11 3 3 3 4 29 4 28 3"/>
    </justification>
  </clause>

  <clause id="3656">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,z,f(z,v,u))) = f(z,u,f(y,z,f(y,v,x)))
    ]]></literal>
    <justification jstring="[para(3604(a,1),619(a,2,3)),rewrite(3(2,R),4(2),3(4,R),4(4),3(5,R),4(5),1318(6),165(5),1735(4),3(5,R),4(5),3(6,R),4(6),4(7)),flip(a)].">
      <j1 rule="para" parents="3604 619"/>
      <j2 rule="rewrite" parents="3 4 3 4 3 4 1318 165 1735 3 4 3 4 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4823">
    <literal><![CDATA[
      f(x,f(y,z,f(z,u,v)),f(z,x,u)) = f(z,x,u)
    ]]></literal>
    <justification jstring="[para(2254(a,1),41(a,2)),rewrite(2(9),3(7),3388(7),4(5))].">
      <j1 rule="para" parents="2254 41"/>
      <j2 rule="rewrite" parents="2 3 3388 4"/>
    </justification>
  </clause>

  <clause id="5050">
    <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(497(a,1),48(a,1,3)),rewrite(8(4)),flip(a)].">
      <j1 rule="para" parents="497 48"/>
      <j2 rule="rewrite" parents="8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="6664">
    <literal><![CDATA[
      f(x,y,f(z,y,f(z,u,f(v,z,x)))) = f(z,x,y)
    ]]></literal>
    <justification jstring="[para(4823(a,1),41(a,2)),rewrite(2(6),3(7,R),1318(7),3656(6),11(2),1318(3))].">
      <j1 rule="para" parents="4823 41"/>
      <j2 rule="rewrite" parents="2 3 1318 3656 11 1318"/>
    </justification>
  </clause>

  <clause id="7732">
    <literal><![CDATA[
      f(x,f(y,z,f(x,u,v)),f(y,v,f(x,u,v))) = f(x,v,f(y,z,u))
    ]]></literal>
    <justification jstring="[para(2835(a,1),6664(a,1,3)),rewrite(3(1),3(3),3(4,R),4(4),3(6),3(8,R),4(8),165(8),3(6),3(6))].">
      <j1 rule="para" parents="2835 6664"/>
      <j2 rule="rewrite" parents="3 3 3 4 3 3 4 165 3 3"/>
    </justification>
  </clause>

  <clause id="9590">
    <literal><![CDATA[
      f(x,f(y,x,z),f(u,x,z)) = f(y,x,f(u,x,z))
    ]]></literal>
    <justification jstring="[para(497(a,1),96(a,2,3)),rewrite(3(3),4(3),335(4),3(2,R),4(2),2670(4),3(2),2(2),4(2),3(4),3(5,R)),flip(a)].">
      <j1 rule="para" parents="497 96"/>
      <j2 rule="rewrite" parents="3 4 335 3 4 2670 3 2 4 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9859">
    <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(9590(a,1),505(a,1,3,3)),rewrite(3(2),3(2),3(3),3(3),1233(5),3(2),3(7),3(7))].">
      <j1 rule="para" parents="9590 505"/>
      <j2 rule="rewrite" parents="3 3 3 3 1233 3 3 3"/>
    </justification>
  </clause>

  <clause id="15767">
    <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),275(a,1,3,3)),rewrite(1234(5),3(2))].">
      <j1 rule="para" parents="12 275"/>
      <j2 rule="rewrite" parents="1234 3"/>
    </justification>
  </clause>

  <clause id="15825">
    <literal><![CDATA[
      f(x,f(y,z,u),f(v,y,z)) = f(y,z,f(v,u,x))
    ]]></literal>
    <justification jstring="[para(275(a,1),24(a,2)),rewrite(2(4),48(7),4(3),29(3),5050(4),3(5,R)),flip(a)].">
      <j1 rule="para" parents="275 24"/>
      <j2 rule="rewrite" parents="2 48 4 29 5050 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16006">
    <literal><![CDATA[
      f(x,f(y,z,u),f(z,u,v)) = f(z,u,f(x,y,v))
    ]]></literal>
    <justification jstring="[back_rewrite(9859),rewrite(15767(5),15825(3),3(1)),flip(a)].">
      <j1 rule="back_rewrite" parents="9859"/>
      <j2 rule="rewrite" parents="15767 15825 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18711">
    <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),15825(a,1,2)),rewrite(3(2),4(2),3(4),4(4),3(5,R),4(5))].">
      <j1 rule="para" parents="3 15825"/>
      <j2 rule="rewrite" parents="3 4 3 4 3 4"/>
    </justification>
  </clause>

  <clause id="19118">
    <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(3513),rewrite(18711(8),18711(7),499(4)),flip(a)].">
      <j1 rule="back_rewrite" parents="3513"/>
      <j2 rule="rewrite" parents="18711 18711 499"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="19770">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,v,u)) = f(y,u,f(x,v,z))
    ]]></literal>
    <justification jstring="[back_rewrite(7732),rewrite(19118(5))].">
      <j1 rule="back_rewrite" parents="7732"/>
      <j2 rule="rewrite" parents="19118"/>
    </justification>
  </clause>

  <clause id="20100">
    <literal><![CDATA[
      f(x,f(y,z,u),f(v,z,u)) = f(z,u,f(v,x,y))
    ]]></literal>
    <justification jstring="[para(16006(a,1),15825(a,2)),rewrite(4(2),3(4,R),4(4),19770(5),3(4))].">
      <j1 rule="para" parents="16006 15825"/>
      <j2 rule="rewrite" parents="4 3 4 19770 3"/>
    </justification>
  </clause>

  <clause id="20445">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(dist_short)
    ]]></attribute>
    <justification jstring="[back_rewrite(10),rewrite(20100(10),3(6),3(6)),xx(a)].">
      <j1 rule="back_rewrite" parents="10"/>
      <j2 rule="rewrite" parents="20100 3 3"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
