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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>dist-both.out</source>

<heading><![CDATA[
Prover9 (32) version August-2006A, August 2006.
Process 11678 was started by mccune on cleo.thornwood,
Sat Aug 12 21:22:12 2006
The command was "/home/mccune/bin/prover9 -f dist-both.in".
]]></heading>

<proof number="1" length="175" max_count="39">

<comments><![CDATA[
% Proof 1 at 564.19 (+ 1.15) seconds: dist_both.
% Length of proof is 175.
% Level of proof is 33.
% Maximum clause weight is 44.
% Given clauses 809.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      f(x,x,y) = x
    ]]></literal>
    <attribute><![CDATA[
      label(majority)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="2">
    <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="3">
    <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="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="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

  <clause id="6">
    <literal><![CDATA[
      f(f(A,B,C),D,E) != f(f(A,D,E),f(B,D,E),f(C,D,E))
    ]]></literal>
    <literal><![CDATA[
      f(f(A,B,C),D,E) != f(A,f(B,D,E),f(C,D,E))
    ]]></literal>
    <attribute><![CDATA[
      answer(dist_both)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      f(f(A,D,E),f(B,D,E),f(C,D,E)) != f(D,E,f(A,B,C))
    ]]></literal>
    <literal><![CDATA[
      f(A,f(B,D,E),f(C,D,E)) != f(D,E,f(A,B,C))
    ]]></literal>
    <attribute><![CDATA[
      answer(dist_both)
    ]]></attribute>
    <justification jstring="[copy(6),rewrite(2(7),2(7),2(28),2(28)),flip(a),flip(b)].">
      <j1 rule="copy" parents="6"/>
      <j2 rule="rewrite" parents="2 2 2 2"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

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

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

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

  <clause id="11">
    <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(5(a,1),5(a,1,3))].">
      <j1 rule="para" parents="5 5"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      f(x,y,f(x,z,y)) = f(x,z,y)
    ]]></literal>
    <justification jstring="[para(8(a,1),5(a,1,3)),flip(a)].">
      <j1 rule="para" parents="8 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      f(x,y,f(x,y,z)) = f(x,y,z)
    ]]></literal>
    <justification jstring="[para(8(a,1),5(a,1)),rewrite(3(3),12(3)),flip(a)].">
      <j1 rule="para" parents="8 5"/>
      <j2 rule="rewrite" parents="3 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      f(x,y,f(x,z,f(x,u,y))) = f(x,z,f(x,u,y))
    ]]></literal>
    <justification jstring="[para(12(a,1),5(a,1,3)),flip(a)].">
      <j1 rule="para" parents="12 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      f(x,y,f(x,z,f(x,y,u))) = f(x,z,f(x,y,u))
    ]]></literal>
    <justification jstring="[para(12(a,1),5(a,2,3)),rewrite(3(1),3(4))].">
      <j1 rule="para" parents="12 5"/>
      <j2 rule="rewrite" parents="3 3"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(x,y,f(z,y,u)) = f(z,y,f(x,y,u))
    ]]></literal>
    <justification jstring="[para(9(a,1),2(a,2)),rewrite(3(2),2(3,R),3(3))].">
      <j1 rule="para" parents="9 2"/>
      <j2 rule="rewrite" parents="3 2 3"/>
    </justification>
  </clause>

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

  <clause id="19">
    <literal><![CDATA[
      f(x,y,f(z,u,f(x,z,v))) = f(x,z,f(x,y,f(z,u,v)))
    ]]></literal>
    <justification jstring="[para(9(a,2),5(a,1,3))].">
      <j1 rule="para" parents="9 5"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,z,f(x,y,u))) = f(x,y,f(y,z,u))
    ]]></literal>
    <justification jstring="[para(9(a,2),12(a,1,3))].">
      <j1 rule="para" parents="9 12"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      f(x,y,f(y,z,f(x,y,u))) = f(x,y,f(y,z,u))
    ]]></literal>
    <justification jstring="[para(9(a,2),13(a,1,3))].">
      <j1 rule="para" parents="9 13"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      f(x,y,f(z,u,f(x,u,v))) = f(x,u,f(x,y,f(z,u,v)))
    ]]></literal>
    <justification jstring="[para(9(a,1),9(a,1,3)),rewrite(2(1,R),3(1),2(6,R),3(6))].">
      <j1 rule="para" parents="9 9"/>
      <j2 rule="rewrite" parents="2 3 2 3"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      f(x,y,f(x,z,f(x,u,v))) = f(x,v,f(x,y,f(x,z,u)))
    ]]></literal>
    <justification jstring="[para(10(a,2),5(a,1,3))].">
      <j1 rule="para" parents="10 5"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      f(x,y,f(x,z,f(x,u,v))) = f(x,u,f(x,y,f(x,v,z)))
    ]]></literal>
    <justification jstring="[para(10(a,1),5(a,2,3)),flip(a)].">
      <j1 rule="para" parents="10 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      f(x,f(y,z,u),f(x,v,y)) = f(x,v,f(y,z,f(x,y,u)))
    ]]></literal>
    <justification jstring="[para(9(a,2),10(a,1,3)),flip(a)].">
      <j1 rule="para" parents="9 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      f(x,y,f(z,x,f(x,u,v))) = f(z,x,f(x,u,f(x,y,v)))
    ]]></literal>
    <justification jstring="[para(11(a,1),2(a,2)),rewrite(3(3),2(5,R),3(5)),flip(a)].">
      <j1 rule="para" parents="11 2"/>
      <j2 rule="rewrite" parents="3 2 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      f(x,y,f(y,z,f(y,u,v))) = f(y,z,f(y,u,f(x,y,v)))
    ]]></literal>
    <justification jstring="[para(11(a,2),2(a,2)),rewrite(3(3),2(4,R),3(4))].">
      <j1 rule="para" parents="11 2"/>
      <j2 rule="rewrite" parents="3 2 3"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      f(x,y,f(x,z,f(u,v,f(x,u,w)))) = f(x,u,f(x,y,f(x,z,f(u,v,w))))
    ]]></literal>
    <justification jstring="[para(9(a,2),11(a,1,3,3))].">
      <j1 rule="para" parents="9 11"/>
    </justification>
  </clause>

  <clause id="50">
    <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(11(a,1),10(a,1)),flip(a)].">
      <j1 rule="para" parents="11 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="63">
    <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(5(a,1),17(a,1,3)),rewrite(2(1,R),3(1),2(6,R),3(6))].">
      <j1 rule="para" parents="5 17"/>
      <j2 rule="rewrite" parents="2 3 2 3"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      f(x,y,f(z,x,f(x,y,u))) = f(z,x,f(x,y,u))
    ]]></literal>
    <justification jstring="[para(12(a,1),17(a,1,3)),rewrite(2(1),2(3),2(5,R),3(5)),flip(a)].">
      <j1 rule="para" parents="12 17"/>
      <j2 rule="rewrite" parents="2 2 2 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      f(x,y,f(z,u,f(y,u,v))) = f(y,u,f(x,y,f(z,u,v)))
    ]]></literal>
    <justification jstring="[para(9(a,1),17(a,1,3)),rewrite(2(1,R),3(1),2(6,R),3(6))].">
      <j1 rule="para" parents="9 17"/>
      <j2 rule="rewrite" parents="2 3 2 3"/>
    </justification>
  </clause>

  <clause id="75">
    <literal><![CDATA[
      f(x,y,f(z,u,f(x,z,v))) = f(x,z,f(x,y,f(z,v,u)))
    ]]></literal>
    <justification jstring="[para(18(a,2),5(a,2,3)),flip(a)].">
      <j1 rule="para" parents="18 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="88">
    <literal><![CDATA[
      f(x,y,f(x,z,f(y,u,f(x,y,v)))) = f(x,z,f(y,u,f(x,y,v)))
    ]]></literal>
    <justification jstring="[para(14(a,1),9(a,1,3)),rewrite(2(1),2(4),2(7,R),3(7)),flip(a)].">
      <j1 rule="para" parents="14 9"/>
      <j2 rule="rewrite" parents="2 2 2 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="96">
    <literal><![CDATA[
      f(x,y,f(y,z,f(x,u,f(x,y,v)))) = f(y,z,f(x,u,f(x,y,v)))
    ]]></literal>
    <justification jstring="[para(14(a,1),17(a,2,3)),rewrite(3(1),2(3,R),3(3),3(5),2(7,R),3(7))].">
      <j1 rule="para" parents="14 17"/>
      <j2 rule="rewrite" parents="3 2 3 3 2 3"/>
    </justification>
  </clause>

  <clause id="100">
    <literal><![CDATA[
      f(x,y,f(z,u,f(x,z,v))) = f(x,y,f(x,z,f(z,v,u)))
    ]]></literal>
    <justification jstring="[para(18(a,2),14(a,2,3)),rewrite(14(5)),flip(a)].">
      <j1 rule="para" parents="18 14"/>
      <j2 rule="rewrite" parents="14"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="115">
    <literal><![CDATA[
      f(x,y,f(z,y,f(x,y,u))) = f(z,y,f(x,y,u))
    ]]></literal>
    <justification jstring="[para(66(a,1),2(a,2)),rewrite(2(1,R),3(1),3(3),2(4,R),3(4))].">
      <j1 rule="para" parents="66 2"/>
      <j2 rule="rewrite" parents="2 3 3 2 3"/>
    </justification>
  </clause>

  <clause id="125">
    <literal><![CDATA[
      f(x,y,f(z,u,f(x,z,u))) = f(x,y,f(x,z,u))
    ]]></literal>
    <justification jstring="[para(8(a,1),19(a,1,3)),rewrite(3(4),88(6)),flip(a)].">
      <j1 rule="para" parents="8 19"/>
      <j2 rule="rewrite" parents="3 88"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="126">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,z,f(y,x,u))) = f(y,z,f(y,x,u))
    ]]></literal>
    <justification jstring="[para(8(a,1),19(a,1)),rewrite(2(1,R),3(1),2(3,R),3(3),3(6),2(7,R),3(7),96(7)),flip(a)].">
      <j1 rule="para" parents="8 19"/>
      <j2 rule="rewrite" parents="2 3 2 3 3 2 3 96"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="140">
    <literal><![CDATA[
      f(x,y,f(z,y,f(z,x,u))) = f(z,x,f(z,y,f(x,y,u)))
    ]]></literal>
    <justification jstring="[para(19(a,1),17(a,1)),flip(a)].">
      <j1 rule="para" parents="19 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="156">
    <literal><![CDATA[
      f(x,y,f(z,y,f(x,z,u))) = f(z,y,f(x,z,f(x,y,u)))
    ]]></literal>
    <justification jstring="[para(19(a,1),115(a,1,3)),rewrite(2(2,R),3(2),2(3,R),3(3),88(4),2(4,R),3(4),2(6,R),3(6))].">
      <j1 rule="para" parents="19 115"/>
      <j2 rule="rewrite" parents="2 3 2 3 88 2 3 2 3"/>
    </justification>
  </clause>

  <clause id="162">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,z,f(y,u,x))) = f(y,x,f(y,z,u))
    ]]></literal>
    <justification jstring="[para(21(a,1),2(a,2)),rewrite(2(2),2(2),2(4),2(6,R),3(6))].">
      <j1 rule="para" parents="21 2"/>
      <j2 rule="rewrite" parents="2 2 2 2 3"/>
    </justification>
  </clause>

  <clause id="163">
    <literal><![CDATA[
      f(x,f(y,z,u),f(z,u,f(x,y,z))) = f(x,z,f(y,z,u))
    ]]></literal>
    <justification jstring="[para(2(a,1),21(a,1,2)),rewrite(3(2),2(5))].">
      <j1 rule="para" parents="2 21"/>
      <j2 rule="rewrite" parents="3 2"/>
    </justification>
  </clause>

  <clause id="194">
    <literal><![CDATA[
      f(x,f(y,z,f(u,v,f(x,u,w))),f(y,z,f(x,u,f(x,y,f(u,v,w))))) = f(x,y,f(y,z,f(u,v,f(x,u,w))))
    ]]></literal>
    <justification jstring="[para(19(a,1),21(a,1,3,3))].">
      <j1 rule="para" parents="19 21"/>
    </justification>
  </clause>

  <clause id="198">
    <literal><![CDATA[
      f(x,y,f(z,x,y)) = f(z,x,y)
    ]]></literal>
    <justification jstring="[para(125(a,1),8(a,1)),rewrite(3(4),125(4),8(3)),flip(a)].">
      <j1 rule="para" parents="125 8"/>
      <j2 rule="rewrite" parents="3 125 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="208">
    <literal><![CDATA[
      f(x,y,f(z,x,f(x,u,y))) = f(z,x,f(x,u,y))
    ]]></literal>
    <justification jstring="[para(198(a,1),17(a,1,3)),rewrite(2(1),2(3),2(5,R),3(5)),flip(a)].">
      <j1 rule="para" parents="198 17"/>
      <j2 rule="rewrite" parents="2 2 2 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="213">
    <literal><![CDATA[
      f(x,y,f(z,u,f(u,y,v))) = f(u,y,f(x,y,f(z,u,v)))
    ]]></literal>
    <justification jstring="[para(23(a,1),2(a,2)),rewrite(2(1,R),3(1),3(3),2(5,R),3(5),2(6,R),3(6))].">
      <j1 rule="para" parents="23 2"/>
      <j2 rule="rewrite" parents="2 3 3 2 3 2 3"/>
    </justification>
  </clause>

  <clause id="234">
    <literal><![CDATA[
      f(x,y,f(z,u,f(x,z,y))) = f(x,z,f(z,y,f(x,y,u)))
    ]]></literal>
    <justification jstring="[para(18(a,2),23(a,2,3)),flip(a)].">
      <j1 rule="para" parents="18 23"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="256">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,z,f(x,z,f(x,y,u)))) = f(x,z,f(x,y,f(y,z,u)))
    ]]></literal>
    <justification jstring="[para(21(a,1),23(a,2,3))].">
      <j1 rule="para" parents="21 23"/>
    </justification>
  </clause>

  <clause id="259">
    <literal><![CDATA[
      f(x,y,f(z,x,f(z,y,u))) = f(z,x,f(z,y,f(x,y,u)))
    ]]></literal>
    <justification jstring="[para(23(a,1),208(a,2)),rewrite(2(4,R),3(4),14(5),2(5,R),3(5)),flip(a)].">
      <j1 rule="para" parents="23 208"/>
      <j2 rule="rewrite" parents="2 3 14 2 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="314">
    <literal><![CDATA[
      f(x,f(y,z,u),f(v,y,x)) = f(v,x,f(y,z,f(y,x,u)))
    ]]></literal>
    <justification jstring="[para(31(a,2),2(a,2)),rewrite(2(1,R),3(1),3(3),2(5),2(5)),flip(a)].">
      <j1 rule="para" parents="31 2"/>
      <j2 rule="rewrite" parents="2 3 3 2 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="316">
    <literal><![CDATA[
      f(x,f(y,z,u),f(x,u,v)) = f(x,v,f(y,u,f(x,z,u)))
    ]]></literal>
    <justification jstring="[para(2(a,2),31(a,1,2)),rewrite(3(2),3(4),2(5,R),3(5))].">
      <j1 rule="para" parents="2 31"/>
      <j2 rule="rewrite" parents="3 3 2 3"/>
    </justification>
  </clause>

  <clause id="326">
    <literal><![CDATA[
      f(x,y,f(z,f(x,y,u),f(x,v,f(x,y,u)))) = f(x,f(x,y,u),f(z,v,f(x,y,u)))
    ]]></literal>
    <justification jstring="[para(13(a,1),31(a,1,3)),rewrite(2(2),2(2),3(4),3(7),2(8,R),3(8)),flip(a)].">
      <j1 rule="para" parents="13 31"/>
      <j2 rule="rewrite" parents="2 2 3 3 2 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="362">
    <literal><![CDATA[
      f(x,f(y,x,z),f(y,x,u)) = f(y,x,f(x,u,z))
    ]]></literal>
    <justification jstring="[para(18(a,2),31(a,2)),rewrite(8(2),3(2)),flip(a)].">
      <j1 rule="para" parents="18 31"/>
      <j2 rule="rewrite" parents="8 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="381">
    <literal><![CDATA[
      f(x,y,f(z,f(x,y,u),f(x,y,z))) = f(y,z,f(x,y,f(x,z,u)))
    ]]></literal>
    <justification jstring="[para(31(a,2),115(a,1,3)),rewrite(2(2),3(2),2(5,R),3(5),2(7,R),3(7))].">
      <j1 rule="para" parents="31 115"/>
      <j2 rule="rewrite" parents="2 3 2 3 2 3"/>
    </justification>
  </clause>

  <clause id="387">
    <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="[para(31(a,2),19(a,1)),rewrite(3(2))].">
      <j1 rule="para" parents="31 19"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="432">
    <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(5(a,1),362(a,1,3)),rewrite(2(1,R),3(1),2(2,R),3(2),3(6),2(7,R),3(7))].">
      <j1 rule="para" parents="5 362"/>
      <j2 rule="rewrite" parents="2 3 2 3 3 2 3"/>
    </justification>
  </clause>

  <clause id="496">
    <literal><![CDATA[
      f(x,y,f(x,z,f(u,x,y))) = f(u,x,f(x,z,y))
    ]]></literal>
    <justification jstring="[para(12(a,1),34(a,1,3)),flip(a)].">
      <j1 rule="para" parents="12 34"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="541">
    <literal><![CDATA[
      f(x,y,f(z,y,f(z,u,f(x,z,v)))) = f(x,z,f(z,y,f(x,y,f(z,u,v))))
    ]]></literal>
    <justification jstring="[para(34(a,1),23(a,2,3)),flip(a)].">
      <j1 rule="para" parents="34 23"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="567">
    <literal><![CDATA[
      f(x,f(y,x,z),f(x,u,z)) = f(y,x,f(x,u,z))
    ]]></literal>
    <justification jstring="[para(496(a,1),10(a,1)),rewrite(3(4)),flip(a)].">
      <j1 rule="para" parents="496 10"/>
      <j2 rule="rewrite" parents="3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="595">
    <literal><![CDATA[
      f(x,f(y,x,z),f(x,z,u)) = f(y,x,f(x,z,u))
    ]]></literal>
    <justification jstring="[para(567(a,1),2(a,2)),rewrite(3(2),2(3),3(4))].">
      <j1 rule="para" parents="567 2"/>
      <j2 rule="rewrite" parents="3 2 3"/>
    </justification>
  </clause>

  <clause id="631">
    <literal><![CDATA[
      f(x,f(x,y,z),f(u,x,y)) = f(u,x,f(x,y,z))
    ]]></literal>
    <justification jstring="[para(595(a,1),3(a,1)),flip(a)].">
      <j1 rule="para" parents="595 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="632">
    <literal><![CDATA[
      f(x,f(x,y,z),f(u,v,f(x,y,z))) = f(x,z,f(x,y,f(u,v,f(x,y,z))))
    ]]></literal>
    <justification jstring="[para(595(a,1),19(a,2)),rewrite(2(1),2(1),2(2),2(2),3(3),2(4,R),3(4),326(5),2(5),2(5),2(6),2(6),2(8,R),3(8))].">
      <j1 rule="para" parents="595 19"/>
      <j2 rule="rewrite" parents="2 2 2 2 3 2 3 326 2 2 2 2 2 3"/>
    </justification>
  </clause>

  <clause id="778">
    <literal><![CDATA[
      f(x,y,f(x,z,f(y,z,u))) = f(x,y,z)
    ]]></literal>
    <justification jstring="[para(631(a,1),33(a,1,3)),rewrite(2(4,R),8(4),2(3,R),8(3),2(5),3(5),2(6,R),3(6),432(6),3(3),50(6),3(4),13(4),13(5)),flip(a)].">
      <j1 rule="para" parents="631 33"/>
      <j2 rule="rewrite" parents="2 8 2 8 2 3 2 3 432 3 50 3 13 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="784">
    <literal><![CDATA[
      f(x,y,f(z,y,f(x,z,u))) = f(x,z,y)
    ]]></literal>
    <justification jstring="[back_rewrite(259),rewrite(2(1,R),3(1),2(2,R),3(2),778(3),2(3,R),3(3),2(4,R),3(4)),flip(a)].">
      <j1 rule="back_rewrite" parents="259"/>
      <j2 rule="rewrite" parents="2 3 2 3 778 2 3 2 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="786">
    <literal><![CDATA[
      f(x,y,f(y,z,f(x,z,u))) = f(x,y,z)
    ]]></literal>
    <justification jstring="[back_rewrite(140),rewrite(2(1,R),3(1),2(2,R),3(2),2(5,R),3(5),2(6,R),3(6),784(6))].">
      <j1 rule="back_rewrite" parents="140"/>
      <j2 rule="rewrite" parents="2 3 2 3 2 3 2 3 784"/>
    </justification>
  </clause>

  <clause id="787">
    <literal><![CDATA[
      f(x,y,f(z,y,f(z,x,u))) = f(z,x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(156),rewrite(2(2,R),3(2),786(3),2(4,R),3(4)),flip(a)].">
      <j1 rule="back_rewrite" parents="156"/>
      <j2 rule="rewrite" parents="2 3 786 2 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="788">
    <literal><![CDATA[
      f(x,y,f(y,z,f(z,u,f(x,z,v)))) = f(x,y,z)
    ]]></literal>
    <justification jstring="[back_rewrite(541),rewrite(2(3,R),3(3),2(7,R),3(7),784(8))].">
      <j1 rule="back_rewrite" parents="541"/>
      <j2 rule="rewrite" parents="2 3 2 3 784"/>
    </justification>
  </clause>

  <clause id="791">
    <literal><![CDATA[
      f(x,y,f(z,u,f(x,y,z))) = f(x,y,z)
    ]]></literal>
    <justification jstring="[back_rewrite(234),rewrite(3(1),2(5,R),3(5),784(6))].">
      <j1 rule="back_rewrite" parents="234"/>
      <j2 rule="rewrite" parents="3 2 3 784"/>
    </justification>
  </clause>

  <clause id="793">
    <literal><![CDATA[
      f(x,y,f(x,z,f(z,y,u))) = f(x,z,y)
    ]]></literal>
    <justification jstring="[back_rewrite(256),rewrite(787(4),387(3),778(3)),flip(a)].">
      <j1 rule="back_rewrite" parents="256"/>
      <j2 rule="rewrite" parents="787 387 778"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="795">
    <literal><![CDATA[
      f(x,y,f(z,x,f(z,y,u))) = f(z,x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(381),rewrite(791(4)),flip(a)].">
      <j1 rule="back_rewrite" parents="381"/>
      <j2 rule="rewrite" parents="791"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="801">
    <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(9(a,2),778(a,2)),rewrite(3(3))].">
      <j1 rule="para" parents="9 778"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="808">
    <literal><![CDATA[
      f(x,y,f(z,x,u)) = f(x,u,f(z,x,y))
    ]]></literal>
    <justification jstring="[para(18(a,2),778(a,2)),rewrite(3(3),801(5))].">
      <j1 rule="para" parents="18 778"/>
      <j2 rule="rewrite" parents="3 801"/>
    </justification>
  </clause>

  <clause id="824">
    <literal><![CDATA[
      f(x,f(y,z,u),f(x,v,f(x,y,z))) = f(x,v,f(x,y,z))
    ]]></literal>
    <justification jstring="[para(778(a,1),25(a,1,3)),flip(a)].">
      <j1 rule="para" parents="778 25"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="840">
    <literal><![CDATA[
      f(x,y,f(z,u,x)) = f(z,x,f(u,x,y))
    ]]></literal>
    <justification jstring="[para(808(a,1),2(a,2)),rewrite(3(2),2(3)),flip(a)].">
      <j1 rule="para" parents="808 2"/>
      <j2 rule="rewrite" parents="3 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="844">
    <literal><![CDATA[
      f(f(x,y,z),f(x,y,f(x,z,u)),f(u,v,f(x,y,z))) = f(v,f(x,y,z),f(x,u,f(x,y,z)))
    ]]></literal>
    <justification jstring="[para(21(a,1),808(a,2,3)),rewrite(2(2),2(2),3(5),2(9,R),3(9),2(10,R),3(10))].">
      <j1 rule="para" parents="21 808"/>
      <j2 rule="rewrite" parents="2 2 3 2 3 2 3"/>
    </justification>
  </clause>

  <clause id="864">
    <literal><![CDATA[
      f(x,y,f(z,u,y)) = f(z,y,f(x,u,y))
    ]]></literal>
    <justification jstring="[para(840(a,1),2(a,2)),rewrite(3(2),2(3))].">
      <j1 rule="para" parents="840 2"/>
      <j2 rule="rewrite" parents="3 2"/>
    </justification>
  </clause>

  <clause id="889">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,z,x)) = f(y,z,x)
    ]]></literal>
    <justification jstring="[para(840(a,2),31(a,2)),rewrite(3(2),387(3),778(3)),flip(a)].">
      <j1 rule="para" parents="840 31"/>
      <j2 rule="rewrite" parents="3 387 778"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="892">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,x,z)) = f(y,x,z)
    ]]></literal>
    <justification jstring="[para(31(a,2),840(a,2)),rewrite(2(2),387(3),778(3),2(2,R),3(2),2(3,R),3(3)),flip(a)].">
      <j1 rule="para" parents="31 840"/>
      <j2 rule="rewrite" parents="2 387 778 2 3 2 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="957">
    <literal><![CDATA[
      f(x,y,f(z,u,f(x,u,y))) = f(x,u,y)
    ]]></literal>
    <justification jstring="[para(13(a,1),864(a,1,3)),rewrite(8(3),632(5),115(4)),flip(a)].">
      <j1 rule="para" parents="13 864"/>
      <j2 rule="rewrite" parents="8 632 115"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="963">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,v,f(y,z,u))) = f(v,f(y,z,u),f(y,u,f(x,y,z)))
    ]]></literal>
    <justification jstring="[para(18(a,2),864(a,2,3)),rewrite(2(3,R),3(3))].">
      <j1 rule="para" parents="18 864"/>
      <j2 rule="rewrite" parents="2 3"/>
    </justification>
  </clause>

  <clause id="997">
    <literal><![CDATA[
      f(x,f(y,z,u),f(x,z,u)) = f(x,z,u)
    ]]></literal>
    <justification jstring="[para(2(a,1),889(a,1,2)),rewrite(2(2),2(4))].">
      <j1 rule="para" parents="2 889"/>
      <j2 rule="rewrite" parents="2 2"/>
    </justification>
  </clause>

  <clause id="998">
    <literal><![CDATA[
      f(x,y,f(x,z,f(y,u,z))) = f(x,y,z)
    ]]></literal>
    <justification jstring="[para(2(a,2),889(a,1,2)),rewrite(2(2),3(2),387(3),2(4),3(4))].">
      <j1 rule="para" parents="2 889"/>
      <j2 rule="rewrite" parents="2 3 387 2 3"/>
    </justification>
  </clause>

  <clause id="1004">
    <literal><![CDATA[
      f(x,f(y,z,u),f(v,f(y,z,x),f(y,z,u))) = f(v,f(y,z,x),f(y,z,u))
    ]]></literal>
    <justification jstring="[para(889(a,1),9(a,1,3)),rewrite(2(3,R),2(7,R)),flip(a)].">
      <j1 rule="para" parents="889 9"/>
      <j2 rule="rewrite" parents="2 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1005">
    <literal><![CDATA[
      f(x,f(y,z,f(z,u,v)),f(x,z,u)) = f(x,z,u)
    ]]></literal>
    <justification jstring="[para(9(a,1),889(a,1,2)),rewrite(2(3),2(5))].">
      <j1 rule="para" parents="9 889"/>
      <j2 rule="rewrite" parents="2 2"/>
    </justification>
  </clause>

  <clause id="1014">
    <literal><![CDATA[
      f(x,f(y,z,f(z,u,v)),f(x,z,v)) = f(x,z,v)
    ]]></literal>
    <justification jstring="[para(18(a,1),889(a,1,2)),rewrite(2(3),2(5))].">
      <j1 rule="para" parents="18 889"/>
      <j2 rule="rewrite" parents="2 2"/>
    </justification>
  </clause>

  <clause id="1042">
    <literal><![CDATA[
      f(f(x,y,z),f(x,y,u),f(u,v,f(x,y,z))) = f(v,f(x,y,u),f(x,y,z))
    ]]></literal>
    <justification jstring="[para(889(a,1),362(a,1,2)),rewrite(3(4),2(9,R),1004(10))].">
      <j1 rule="para" parents="889 362"/>
      <j2 rule="rewrite" parents="3 2 1004"/>
    </justification>
  </clause>

  <clause id="1059">
    <literal><![CDATA[
      f(x,y,f(x,f(z,u,v),f(y,z,u))) = f(x,y,f(z,u,v))
    ]]></literal>
    <justification jstring="[para(889(a,1),778(a,1,3,3)),rewrite(2(2))].">
      <j1 rule="para" parents="889 778"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="1067">
    <literal><![CDATA[
      f(x,f(y,z,u),f(x,u,f(y,z,v))) = f(x,u,f(y,z,v))
    ]]></literal>
    <justification jstring="[para(889(a,1),889(a,1,2)),rewrite(2(3),2(6))].">
      <j1 rule="para" parents="889 889"/>
      <j2 rule="rewrite" parents="2 2"/>
    </justification>
  </clause>

  <clause id="1068">
    <literal><![CDATA[
      f(x,f(y,x,z),f(y,z,u)) = f(y,x,z)
    ]]></literal>
    <justification jstring="[para(892(a,1),3(a,1)),flip(a)].">
      <j1 rule="para" parents="892 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1084">
    <literal><![CDATA[
      f(x,f(y,z,u),f(v,x,y)) = f(v,x,f(y,z,f(x,y,u)))
    ]]></literal>
    <justification jstring="[para(31(a,2),892(a,2)),rewrite(3(3),2(6,R),3(6),387(7),998(7),2(5,R),3(5)),flip(a)].">
      <j1 rule="para" parents="31 892"/>
      <j2 rule="rewrite" parents="3 2 3 387 998 2 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1099">
    <literal><![CDATA[
      f(x,f(y,z,u),f(z,u,x)) = f(z,u,x)
    ]]></literal>
    <justification jstring="[para(997(a,1),2(a,2)),rewrite(2(2),2(2),2(3),2(4),2(4))].">
      <j1 rule="para" parents="997 2"/>
      <j2 rule="rewrite" parents="2 2 2 2 2"/>
    </justification>
  </clause>

  <clause id="1100">
    <literal><![CDATA[
      f(x,f(y,z,u),f(x,v,f(x,z,u))) = f(x,v,f(x,z,u))
    ]]></literal>
    <justification jstring="[para(997(a,1),5(a,2,3))].">
      <j1 rule="para" parents="997 5"/>
    </justification>
  </clause>

  <clause id="1127">
    <literal><![CDATA[
      f(x,f(y,z,u),f(v,f(x,z,u),f(y,z,u))) = f(v,f(x,z,u),f(y,z,u))
    ]]></literal>
    <justification jstring="[para(997(a,1),21(a,1,3,3)),rewrite(2(3,R),2(6,R),8(7),2(7,R)),flip(a)].">
      <j1 rule="para" parents="997 21"/>
      <j2 rule="rewrite" parents="2 2 8 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1136">
    <literal><![CDATA[
      f(x,f(y,z,u),f(x,u,f(x,z,v))) = f(x,v,f(x,z,u))
    ]]></literal>
    <justification jstring="[para(997(a,1),26(a,2,3)),rewrite(3(2))].">
      <j1 rule="para" parents="997 26"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="1162">
    <literal><![CDATA[
      f(x,f(y,z,f(z,u,v)),f(x,f(y,z,u),f(z,u,v))) = f(x,f(y,z,u),f(z,u,v))
    ]]></literal>
    <justification jstring="[para(595(a,1),997(a,1,2))].">
      <j1 rule="para" parents="595 997"/>
    </justification>
  </clause>

  <clause id="1502">
    <literal><![CDATA[
      f(x,y,f(y,f(z,u,f(x,u,v)),f(x,w,f(z,u,f(x,u,v))))) = f(x,u,f(x,y,f(z,u,v)))
    ]]></literal>
    <justification jstring="[para(23(a,1),786(a,2)),rewrite(3(5))].">
      <j1 rule="para" parents="23 786"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="1508">
    <literal><![CDATA[
      f(x,y,f(y,f(z,u,v),f(x,z,u))) = f(x,y,f(z,u,v))
    ]]></literal>
    <justification jstring="[para(889(a,1),786(a,1,3,3)),rewrite(2(2))].">
      <j1 rule="para" parents="889 786"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="1510">
    <literal><![CDATA[
      f(x,y,f(y,f(z,u,v),f(x,u,v))) = f(x,y,f(z,u,v))
    ]]></literal>
    <justification jstring="[para(997(a,1),786(a,1,3,3))].">
      <j1 rule="para" parents="997 786"/>
    </justification>
  </clause>

  <clause id="1650">
    <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(17(a,1),793(a,2)),rewrite(3(3))].">
      <j1 rule="para" parents="17 793"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="1661">
    <literal><![CDATA[
      f(x,y,f(z,u,x)) = f(u,x,f(z,x,y))
    ]]></literal>
    <justification jstring="[para(840(a,2),793(a,2)),rewrite(3(3),1650(5)),flip(a)].">
      <j1 rule="para" parents="840 793"/>
      <j2 rule="rewrite" parents="3 1650"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1662">
    <literal><![CDATA[
      f(x,f(y,z,u),f(x,u,f(u,v,f(y,z,u)))) = f(y,u,f(x,z,u))
    ]]></literal>
    <justification jstring="[para(864(a,1),793(a,2)),rewrite(3(3))].">
      <j1 rule="para" parents="864 793"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="1664">
    <literal><![CDATA[
      f(x,f(y,z,u),f(x,v,f(z,u,v))) = f(x,v,f(y,z,u))
    ]]></literal>
    <justification jstring="[para(997(a,1),793(a,1,3,3)),rewrite(2(2),2(2))].">
      <j1 rule="para" parents="997 793"/>
      <j2 rule="rewrite" parents="2 2"/>
    </justification>
  </clause>

  <clause id="1668">
    <literal><![CDATA[
      f(x,y,f(z,u,y)) = f(u,y,f(x,z,y))
    ]]></literal>
    <justification jstring="[para(1661(a,1),2(a,2)),rewrite(3(2),2(3))].">
      <j1 rule="para" parents="1661 2"/>
      <j2 rule="rewrite" parents="3 2"/>
    </justification>
  </clause>

  <clause id="1754">
    <literal><![CDATA[
      f(x,y,f(z,u,y)) = f(u,y,f(z,x,y))
    ]]></literal>
    <justification jstring="[para(1668(a,1),793(a,2)),rewrite(3(3),1662(5))].">
      <j1 rule="para" parents="1668 793"/>
      <j2 rule="rewrite" parents="3 1662"/>
    </justification>
  </clause>

  <clause id="1771">
    <literal><![CDATA[
      f(f(x,y,z),f(x,z,u),f(y,v,f(x,z,u))) = f(v,f(x,z,u),f(x,y,z))
    ]]></literal>
    <justification jstring="[para(1068(a,1),1754(a,2,3))].">
      <j1 rule="para" parents="1068 1754"/>
    </justification>
  </clause>

  <clause id="1803">
    <literal><![CDATA[
      f(x,y,f(z,u,f(x,u,f(y,u,v)))) = f(x,y,u)
    ]]></literal>
    <justification jstring="[para(34(a,2),795(a,1,3)),rewrite(2(1,R),3(1),2(2,R),3(2),2(5),2(5))].">
      <j1 rule="para" parents="34 795"/>
      <j2 rule="rewrite" parents="2 3 2 3 2 2"/>
    </justification>
  </clause>

  <clause id="1804">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,z,f(z,u,x))) = f(z,x,f(y,z,u))
    ]]></literal>
    <justification jstring="[para(567(a,1),795(a,1,3)),rewrite(3(2),2(4,R),3(4),3(6))].">
      <j1 rule="para" parents="567 795"/>
      <j2 rule="rewrite" parents="3 2 3 3"/>
    </justification>
  </clause>

  <clause id="1848">
    <literal><![CDATA[
      f(x,y,f(z,u,f(u,x,y))) = f(u,x,y)
    ]]></literal>
    <justification jstring="[para(2(a,1),957(a,1)),rewrite(2(1,R),3(1),2(3),2(3),2(4,R),3(4))].">
      <j1 rule="para" parents="2 957"/>
      <j2 rule="rewrite" parents="2 3 2 2 2 3"/>
    </justification>
  </clause>

  <clause id="2513">
    <literal><![CDATA[
      f(x,f(y,z,f(x,y,u)),f(x,v,w)) = f(x,y,f(x,v,f(x,w,f(y,z,u))))
    ]]></literal>
    <justification jstring="[para(47(a,1),10(a,1)),flip(a)].">
      <j1 rule="para" parents="47 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2745">
    <literal><![CDATA[
      f(x,f(y,z,f(u,z,v)),f(x,u,z)) = f(x,u,z)
    ]]></literal>
    <justification jstring="[para(1005(a,1),14(a,2)),rewrite(2(1,R),3(1),3(3),15(5),3(5))].">
      <j1 rule="para" parents="1005 14"/>
      <j2 rule="rewrite" parents="2 3 3 15 3"/>
    </justification>
  </clause>

  <clause id="3618">
    <literal><![CDATA[
      f(x,f(y,x,z),f(u,y,f(y,x,v))) = f(x,z,f(u,y,f(y,x,v)))
    ]]></literal>
    <justification jstring="[para(2745(a,1),795(a,1,3)),rewrite(2(1,R),3(1),2(3),3(3),2(4,R),2(5,R),3(5),2(7))].">
      <j1 rule="para" parents="2745 795"/>
      <j2 rule="rewrite" parents="2 3 2 3 2 2 3 2"/>
    </justification>
  </clause>

  <clause id="3828">
    <literal><![CDATA[
      f(x,y,f(y,f(x,z,u),f(v,f(x,z,u),f(x,w,f(x,z,u))))) = f(x,z,f(x,y,u))
    ]]></literal>
    <justification jstring="[para(5(a,1),788(a,2)),rewrite(3(4),2(5,R),3(5))].">
      <j1 rule="para" parents="5 788"/>
      <j2 rule="rewrite" parents="3 2 3"/>
    </justification>
  </clause>

  <clause id="3859">
    <literal><![CDATA[
      f(x,y,f(x,z,f(u,y,z))) = f(x,y,z)
    ]]></literal>
    <justification jstring="[para(997(a,1),788(a,2)),rewrite(3(6),2(7,R),3(7),3828(9),3(2))].">
      <j1 rule="para" parents="997 788"/>
      <j2 rule="rewrite" parents="3 2 3 3828 3"/>
    </justification>
  </clause>

  <clause id="3885">
    <literal><![CDATA[
      f(x,y,f(x,z,f(u,y,f(y,v,z)))) = f(x,y,z)
    ]]></literal>
    <justification jstring="[para(1014(a,1),788(a,2)),rewrite(3(8),2(9,R),3(9),3828(11),3(3))].">
      <j1 rule="para" parents="1014 788"/>
      <j2 rule="rewrite" parents="3 2 3 3828 3"/>
    </justification>
  </clause>

  <clause id="5045">
    <literal><![CDATA[
      f(x,y,f(x,z,f(u,y,f(z,y,v)))) = f(x,z,y)
    ]]></literal>
    <justification jstring="[para(1803(a,1),786(a,2)),rewrite(3(7),1502(9))].">
      <j1 rule="para" parents="1803 786"/>
      <j2 rule="rewrite" parents="3 1502"/>
    </justification>
  </clause>

  <clause id="6497">
    <literal><![CDATA[
      f(x,y,f(y,z,f(u,x,y))) = f(y,z,f(u,x,y))
    ]]></literal>
    <justification jstring="[para(1099(a,1),5045(a,1,3)),rewrite(2(2),2(2),2(4),2(4),2(6,R),3(6),198(6),2(5),2(5),2(7),3(7),12(7))].">
      <j1 rule="para" parents="1099 5045"/>
      <j2 rule="rewrite" parents="2 2 2 2 2 3 198 2 2 2 3 12"/>
    </justification>
  </clause>

  <clause id="6530">
    <literal><![CDATA[
      f(x,y,f(y,z,f(x,u,z))) = f(x,y,z)
    ]]></literal>
    <justification jstring="[para(1099(a,1),6497(a,2)),rewrite(2(1),2(2),2(2),22(4),3(2),2(4),2(4))].">
      <j1 rule="para" parents="1099 6497"/>
      <j2 rule="rewrite" parents="2 2 2 22 3 2 2"/>
    </justification>
  </clause>

  <clause id="6743">
    <literal><![CDATA[
      f(x,f(x,y,z),f(u,y,z)) = f(x,y,z)
    ]]></literal>
    <justification jstring="[para(997(a,1),63(a,2,3)),rewrite(2(3),2(3),3859(4),997(3),2(4,R)),flip(a)].">
      <j1 rule="para" parents="997 63"/>
      <j2 rule="rewrite" parents="2 2 3859 997 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="6803">
    <literal><![CDATA[
      f(x,f(x,y,z),f(u,y,f(y,v,z))) = f(x,y,z)
    ]]></literal>
    <justification jstring="[para(1014(a,1),63(a,2,3)),rewrite(2(5),2(5),3885(6),1014(4),2(5,R)),flip(a)].">
      <j1 rule="para" parents="1014 63"/>
      <j2 rule="rewrite" parents="2 2 3885 1014 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="6918">
    <literal><![CDATA[
      f(f(x,y,z),f(u,x,y),f(v,z,f(u,x,y))) = f(v,f(x,y,z),f(u,x,y))
    ]]></literal>
    <justification jstring="[para(6743(a,1),1661(a,1,3)),rewrite(2(2),2(2),2(3,R),2(4),2(4),2(7)),flip(a)].">
      <j1 rule="para" parents="6743 1661"/>
      <j2 rule="rewrite" parents="2 2 2 2 2 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="7077">
    <literal><![CDATA[
      f(x,f(y,z,x),f(u,y,f(y,v,z))) = f(y,z,x)
    ]]></literal>
    <justification jstring="[para(2(a,1),6803(a,1)),rewrite(2(3),2(3),2(4,R),2(5),2(5))].">
      <j1 rule="para" parents="2 6803"/>
      <j2 rule="rewrite" parents="2 2 2 2 2"/>
    </justification>
  </clause>

  <clause id="7539">
    <literal><![CDATA[
      f(x,f(y,z,x),f(z,x,u)) = f(y,x,f(z,x,u))
    ]]></literal>
    <justification jstring="[para(889(a,1),67(a,2,3)),rewrite(2(3),3(3),115(4),12(4),2(4),2(5,R)),flip(a)].">
      <j1 rule="para" parents="889 67"/>
      <j2 rule="rewrite" parents="2 3 115 12 2 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="7547">
    <literal><![CDATA[
      f(x,f(y,x,z),f(u,y,x)) = f(x,z,f(u,y,x))
    ]]></literal>
    <justification jstring="[para(67(a,1),1099(a,1)),rewrite(2(3),2(3),2(6),3(6),6918(6),2(5,R),3(5),12(5),2(5),2(5),6497(6))].">
      <j1 rule="para" parents="67 1099"/>
      <j2 rule="rewrite" parents="2 2 2 3 6918 2 3 12 2 2 6497"/>
    </justification>
  </clause>

  <clause id="7657">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,v,f(y,z,u))) = f(y,u,f(y,z,f(v,x,f(y,z,u))))
    ]]></literal>
    <justification jstring="[para(5(a,1),7539(a,1,2)),rewrite(3(2),3(5),844(6),3(7),632(8))].">
      <j1 rule="para" parents="5 7539"/>
      <j2 rule="rewrite" parents="3 3 844 3 632"/>
    </justification>
  </clause>

  <clause id="9558">
    <literal><![CDATA[
      f(x,y,f(z,f(x,u,v),f(x,z,v))) = f(x,y,f(x,z,v))
    ]]></literal>
    <justification jstring="[para(6530(a,1),100(a,2,3))].">
      <j1 rule="para" parents="6530 100"/>
    </justification>
  </clause>

  <clause id="10211">
    <literal><![CDATA[
      f(x,y,f(z,y,f(x,u,f(x,v,y)))) = f(z,y,f(x,u,f(x,v,y)))
    ]]></literal>
    <justification jstring="[para(6497(a,1),126(a,1,2)),rewrite(2(1),2(3),2(5,R),3(5),2(6,R),3(6),957(7),2(4),2(6,R),3(6),2(7,R),3(7)),flip(a)].">
      <j1 rule="para" parents="6497 126"/>
      <j2 rule="rewrite" parents="2 2 2 3 2 3 957 2 2 3 2 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10268">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,u,x)) = f(y,u,x)
    ]]></literal>
    <justification jstring="[para(998(a,1),162(a,1,3)),rewrite(2(2),2(2),2(3),2(3),8(4),2(3),2(3)),flip(a)].">
      <j1 rule="para" parents="998 162"/>
      <j2 rule="rewrite" parents="2 2 2 2 8 2 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10473">
    <literal><![CDATA[
      f(x,f(y,z,u),f(z,u,f(y,z,x))) = f(z,x,f(y,z,u))
    ]]></literal>
    <justification jstring="[para(163(a,1),2(a,2)),rewrite(2(2),2(2),2(4),2(6,R),3(6))].">
      <j1 rule="para" parents="163 2"/>
      <j2 rule="rewrite" parents="2 2 2 2 3"/>
    </justification>
  </clause>

  <clause id="10969">
    <literal><![CDATA[
      f(x,f(y,x,z),f(z,u,v)) = f(y,x,f(z,u,f(x,z,v)))
    ]]></literal>
    <justification jstring="[para(314(a,1),3(a,1)),rewrite(2(1,R),3(1),3(4)),flip(a)].">
      <j1 rule="para" parents="314 3"/>
      <j2 rule="rewrite" parents="2 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11010">
    <literal><![CDATA[
      f(f(x,y,z),f(u,v,w),f(x,y,z)) = f(x,y,z)
    ]]></literal>
    <justification jstring="[para(314(a,1),47(a,2,3)),rewrite(2(5),2(5),12(5),13(5),2(5,R),8(5),3(9),1(11),2(7,R),8(7))].">
      <j1 rule="para" parents="314 47"/>
      <j2 rule="rewrite" parents="2 2 12 13 2 8 3 1 2 8"/>
    </justification>
  </clause>

  <clause id="11017">
    <literal><![CDATA[
      f(x,f(y,z,u),f(z,x,u)) = f(z,x,u)
    ]]></literal>
    <justification jstring="[para(314(a,2),6497(a,2)),rewrite(2(4,R),3(4),2(5,R),3(5),3618(5),1848(4),8(3),2(2,R),3(2),2(4),2(4),13(4)),flip(a)].">
      <j1 rule="para" parents="314 6497"/>
      <j2 rule="rewrite" parents="2 3 2 3 3618 1848 8 2 3 2 2 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11086">
    <literal><![CDATA[
      f(f(x,y,z),f(u,v,w),f(x,y,f(x,z,f(u,v,w)))) = f(x,y,f(x,z,f(u,v,w)))
    ]]></literal>
    <justification jstring="[para(11010(a,1),63(a,1)),rewrite(2(2),2(2),2(6),2(6),2(10),3(10),2(11,R),3(11),10211(11)),flip(a)].">
      <j1 rule="para" parents="11010 63"/>
      <j2 rule="rewrite" parents="2 2 2 2 2 3 2 3 10211"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11087">
    <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="[para(11010(a,1),63(a,2,3)),rewrite(2(4),2(4),11086(6),2(6,R)),flip(a)].">
      <j1 rule="para" parents="11010 63"/>
      <j2 rule="rewrite" parents="2 2 11086 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11201">
    <literal><![CDATA[
      f(x,f(y,z,f(z,u,v)),f(x,w,f(z,u,v))) = f(x,w,f(y,f(z,u,v),f(z,v,f(x,z,u))))
    ]]></literal>
    <justification jstring="[para(18(a,2),316(a,2,3,3)),rewrite(3(4))].">
      <j1 rule="para" parents="18 316"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="11208">
    <literal><![CDATA[
      f(x,f(y,z,u),f(x,y,z)) = f(x,y,z)
    ]]></literal>
    <justification jstring="[para(316(a,2),22(a,2)),rewrite(3(1),795(3),13(2),3(2),3(3)),flip(a)].">
      <j1 rule="para" parents="316 22"/>
      <j2 rule="rewrite" parents="3 795 13 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11218">
    <literal><![CDATA[
      f(x,f(y,z,u),f(x,y,u)) = f(x,y,u)
    ]]></literal>
    <justification jstring="[para(316(a,2),208(a,1,3)),rewrite(2(3),9558(5),6530(6),2(4,R),3(4))].">
      <j1 rule="para" parents="316 208"/>
      <j2 rule="rewrite" parents="2 9558 6530 2 3"/>
    </justification>
  </clause>

  <clause id="11219">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,x,u)) = f(y,x,u)
    ]]></literal>
    <justification jstring="[para(316(a,2),208(a,2)),rewrite(6530(4),2(2,R),3(2),11218(3),2(3)),flip(a)].">
      <j1 rule="para" parents="316 208"/>
      <j2 rule="rewrite" parents="6530 2 3 11218 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11249">
    <literal><![CDATA[
      f(x,f(y,z,u),f(x,z,u)) = f(x,z,u)
    ]]></literal>
    <justification jstring="[para(316(a,2),864(a,1)),rewrite(12(3),12(6),8(6))].">
      <j1 rule="para" parents="316 864"/>
      <j2 rule="rewrite" parents="12 12 8"/>
    </justification>
  </clause>

  <clause id="11370">
    <literal><![CDATA[
      f(x,f(y,z,u),f(x,y,f(z,v,u))) = f(x,y,f(z,v,u))
    ]]></literal>
    <justification jstring="[para(316(a,2),7077(a,1,3)),rewrite(2(2),2(4),11219(5),3(4),2(6))].">
      <j1 rule="para" parents="316 7077"/>
      <j2 rule="rewrite" parents="2 2 11219 3 2"/>
    </justification>
  </clause>

  <clause id="11372">
    <literal><![CDATA[
      f(x,y,f(z,u,f(y,v,u))) = f(y,u,f(x,y,f(z,v,u)))
    ]]></literal>
    <justification jstring="[para(316(a,2),67(a,1,3)),rewrite(8(2),3(2),12(5))].">
      <j1 rule="para" parents="316 67"/>
      <j2 rule="rewrite" parents="8 3 12"/>
    </justification>
  </clause>

  <clause id="12815">
    <literal><![CDATA[
      f(x,f(y,z,u),f(v,x,f(v,y,z))) = f(v,x,f(y,z,u))
    ]]></literal>
    <justification jstring="[para(824(a,1),213(a,2,3)),rewrite(1508(5),12(4)),flip(a)].">
      <j1 rule="para" parents="824 213"/>
      <j2 rule="rewrite" parents="1508 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13038">
    <literal><![CDATA[
      f(x,y,f(x,f(y,z,u),f(z,v,f(x,y,u)))) = f(x,y,f(z,v,f(x,y,u)))
    ]]></literal>
    <justification jstring="[para(19(a,1),1059(a,1,3)),rewrite(3(2),3(4),13(6),3(7))].">
      <j1 rule="para" parents="19 1059"/>
      <j2 rule="rewrite" parents="3 3 13 3"/>
    </justification>
  </clause>

  <clause id="13105">
    <literal><![CDATA[
      f(x,f(y,z,u),f(z,v,f(x,y,v))) = f(x,v,f(y,z,u))
    ]]></literal>
    <justification jstring="[para(1059(a,1),75(a,2)),rewrite(3(2),2(3,R),3(3))].">
      <j1 rule="para" parents="1059 75"/>
      <j2 rule="rewrite" parents="3 2 3"/>
    </justification>
  </clause>

  <clause id="13451">
    <literal><![CDATA[
      f(x,y,f(x,z,f(x,f(y,u,v),f(u,v,w)))) = f(x,z,f(x,y,f(u,v,w)))
    ]]></literal>
    <justification jstring="[para(1067(a,1),11(a,1,3)),rewrite(2(4)),flip(a)].">
      <j1 rule="para" parents="1067 11"/>
      <j2 rule="rewrite" parents="2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13517">
    <literal><![CDATA[
      f(x,f(y,z,u),f(x,y,f(x,v,w))) = f(x,y,f(x,v,f(x,w,f(y,z,u))))
    ]]></literal>
    <justification jstring="[para(1084(a,2),7547(a,1,2)),rewrite(2(2),2(2),2(4),3(4),11087(5),11087(4),13(3),2(7),3(7),2513(8))].">
      <j1 rule="para" parents="1084 7547"/>
      <j2 rule="rewrite" parents="2 2 2 3 11087 11087 13 2 3 2513"/>
    </justification>
  </clause>

  <clause id="13654">
    <literal><![CDATA[
      f(x,f(y,z,u),f(v,x,f(v,z,u))) = f(v,x,f(y,z,u))
    ]]></literal>
    <justification jstring="[para(1100(a,1),213(a,2,3)),rewrite(1510(5),12(4)),flip(a)].">
      <j1 rule="para" parents="1100 213"/>
      <j2 rule="rewrite" parents="1510 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13663">
    <literal><![CDATA[
      f(x,f(y,z,u),f(u,x,f(z,x,v))) = f(x,v,f(z,u,x))
    ]]></literal>
    <justification jstring="[para(1136(a,1),2(a,2)),rewrite(2(2,R),3(2),2(3,R),3(3),2(4),2(5),2(5))].">
      <j1 rule="para" parents="1136 2"/>
      <j2 rule="rewrite" parents="2 3 2 3 2 2 2"/>
    </justification>
  </clause>

  <clause id="15803">
    <literal><![CDATA[
      f(f(x,y,z),f(u,x,y),f(u,v,f(x,y,z))) = f(v,f(u,x,y),f(x,y,z))
    ]]></literal>
    <justification jstring="[para(11208(a,1),10473(a,1,3,3)),rewrite(3(3),2(6,R),11017(7),3(7)),flip(a)].">
      <j1 rule="para" parents="11208 10473"/>
      <j2 rule="rewrite" parents="3 2 11017 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="15975">
    <literal><![CDATA[
      f(x,y,f(x,z,f(u,v,f(x,z,y)))) = f(x,y,f(z,u,f(x,z,v)))
    ]]></literal>
    <justification jstring="[para(10969(a,1),1664(a,2)),rewrite(2(2),2(2),2(3),2(3),11087(5),13517(6),3(4),13451(6),2(7,R),3(7))].">
      <j1 rule="para" parents="10969 1664"/>
      <j2 rule="rewrite" parents="2 2 2 2 11087 13517 3 13451 2 3"/>
    </justification>
  </clause>

  <clause id="15991">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,v,f(y,z,u))) = f(y,u,f(x,z,f(y,z,v)))
    ]]></literal>
    <justification jstring="[back_rewrite(7657),rewrite(2(6,R),3(6),15975(8),2(6,R),3(6))].">
      <j1 rule="back_rewrite" parents="7657"/>
      <j2 rule="rewrite" parents="2 3 15975 2 3"/>
    </justification>
  </clause>

  <clause id="16037">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,u,f(v,y,z))) = f(y,u,f(v,z,f(y,z,x)))
    ]]></literal>
    <justification jstring="[back_rewrite(963),rewrite(15991(4)),flip(a)].">
      <j1 rule="back_rewrite" parents="963"/>
      <j2 rule="rewrite" parents="15991"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16078">
    <literal><![CDATA[
      f(x,f(y,z,f(z,u,v)),f(x,w,f(z,u,v))) = f(x,w,f(z,v,f(x,u,f(y,z,u))))
    ]]></literal>
    <justification jstring="[back_rewrite(11201),rewrite(16037(9),2(6))].">
      <j1 rule="back_rewrite" parents="11201"/>
      <j2 rule="rewrite" parents="16037 2"/>
    </justification>
  </clause>

  <clause id="16103">
    <literal><![CDATA[
      f(x,f(y,z,u),f(z,v,f(x,u,f(y,z,u)))) = f(x,f(y,z,u),f(z,u,v))
    ]]></literal>
    <justification jstring="[back_rewrite(1162),rewrite(16078(6))].">
      <j1 rule="back_rewrite" parents="1162"/>
      <j2 rule="rewrite" parents="16078"/>
    </justification>
  </clause>

  <clause id="17083">
    <literal><![CDATA[
      f(f(x,y,z),f(y,z,u),f(v,x,f(y,z,u))) = f(v,f(y,z,u),f(x,y,z))
    ]]></literal>
    <justification jstring="[para(11370(a,1),11372(a,2,3)),rewrite(3(2),3(4),2(5,R),11208(5),12(5),3(5),3(6)),flip(a)].">
      <j1 rule="para" parents="11370 11372"/>
      <j2 rule="rewrite" parents="3 3 2 11208 12 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18010">
    <literal><![CDATA[
      f(f(x,y,z),f(x,y,u),f(x,y,v)) = f(v,f(x,y,z),f(x,y,u))
    ]]></literal>
    <justification jstring="[para(11208(a,1),12815(a,1,3)),rewrite(2(3),2(3))].">
      <j1 rule="para" parents="11208 12815"/>
      <j2 rule="rewrite" parents="2 2"/>
    </justification>
  </clause>

  <clause id="18011">
    <literal><![CDATA[
      f(f(x,y,z),f(x,z,u),f(x,z,v)) = f(v,f(x,y,z),f(x,z,u))
    ]]></literal>
    <justification jstring="[para(11218(a,1),12815(a,1,3)),rewrite(2(3),2(3))].">
      <j1 rule="para" parents="11218 12815"/>
      <j2 rule="rewrite" parents="2 2"/>
    </justification>
  </clause>

  <clause id="18012">
    <literal><![CDATA[
      f(f(x,y,z),f(y,z,u),f(y,z,v)) = f(v,f(x,y,z),f(y,z,u))
    ]]></literal>
    <justification jstring="[para(11249(a,1),12815(a,1,3)),rewrite(2(3),2(3))].">
      <j1 rule="para" parents="11249 12815"/>
      <j2 rule="rewrite" parents="2 2"/>
    </justification>
  </clause>

  <clause id="18024">
    <literal><![CDATA[
      f(f(x,y,z),f(u,x,y),f(u,x,z)) = f(x,z,f(u,x,y))
    ]]></literal>
    <justification jstring="[para(1804(a,1),12815(a,2)),rewrite(2(5),2(5),889(6),3(5),18010(5))].">
      <j1 rule="para" parents="1804 12815"/>
      <j2 rule="rewrite" parents="2 2 889 3 18010"/>
    </justification>
  </clause>

  <clause id="18052">
    <literal><![CDATA[
      f(f(x,y,z),f(x,y,u),f(y,z,u)) = f(y,u,f(x,y,z))
    ]]></literal>
    <justification jstring="[para(18024(a,1),2(a,2))].">
      <j1 rule="para" parents="18024 2"/>
    </justification>
  </clause>

  <clause id="18054">
    <literal><![CDATA[
      f(f(x,y,z),f(x,z,u),f(y,z,u)) = f(y,z,f(x,z,u))
    ]]></literal>
    <justification jstring="[para(2(a,2),18024(a,1,1)),rewrite(2(2),3(2),2(3),3(3),2(5),3(5),2(6,R),3(6))].">
      <j1 rule="para" parents="2 18024"/>
      <j2 rule="rewrite" parents="2 3 2 3 2 3 2 3"/>
    </justification>
  </clause>

  <clause id="18290">
    <literal><![CDATA[
      f(f(x,y,z),f(x,u,y),f(u,y,z)) = f(u,y,f(x,y,z))
    ]]></literal>
    <justification jstring="[para(18054(a,1),12(a,1,3)),rewrite(18012(5),18054(8))].">
      <j1 rule="para" parents="18054 12"/>
      <j2 rule="rewrite" parents="18012 18054"/>
    </justification>
  </clause>

  <clause id="18433">
    <literal><![CDATA[
      f(x,f(y,z,u),f(z,v,f(y,x,v))) = f(y,f(z,x,v),f(x,v,f(y,z,u)))
    ]]></literal>
    <justification jstring="[para(18290(a,1),194(a,1,2)),rewrite(2(2,R),3(2),2(5,R),3(5),2(6),2(6),2(8,R),3(8),3(10),13038(11),2(10,R),3(10),17083(10),3(7),16078(8),16103(6),2(6,R),3(6),2(8,R),3(8),2(9),2(9),2(12,R),18052(12),16037(10),2(6)),flip(a)].">
      <j1 rule="para" parents="18290 194"/>
      <j2 rule="rewrite" parents="2 3 2 3 2 2 2 3 3 13038 2 3 17083 3 16078 16103 2 3 2 3 2 2 2 18052 16037 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18586">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,u,f(x,z,v))) = f(y,u,f(x,z,v))
    ]]></literal>
    <justification jstring="[back_rewrite(18433),rewrite(2(2,R),3(2),13105(4),2(3,R),3(3)),flip(a)].">
      <j1 rule="back_rewrite" parents="18433"/>
      <j2 rule="rewrite" parents="2 3 13105 2 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18804">
    <literal><![CDATA[
      f(f(x,y,z),f(u,v,w),f(w,f(x,y,z),f(x,y,v))) = f(w,f(x,y,v),f(x,y,z))
    ]]></literal>
    <justification jstring="[para(889(a,1),13663(a,1,3,3)),rewrite(1042(11))].">
      <j1 rule="para" parents="889 13663"/>
      <j2 rule="rewrite" parents="1042"/>
    </justification>
  </clause>

  <clause id="21103">
    <literal><![CDATA[
      f(f(x,y,z),f(u,x,v),f(u,v,y)) = f(u,v,f(x,y,z))
    ]]></literal>
    <justification jstring="[para(18586(a,1),12815(a,2)),rewrite(2(2,R),3(2),2(5),2(5),10268(6),3(5),18011(5),2(5,R),3(5))].">
      <j1 rule="para" parents="18586 12815"/>
      <j2 rule="rewrite" parents="2 3 2 2 10268 3 18011 2 3"/>
    </justification>
  </clause>

  <clause id="21122">
    <literal><![CDATA[
      f(f(x,y,z),f(x,z,u),f(y,u,v)) = f(x,z,f(y,u,v))
    ]]></literal>
    <justification jstring="[para(21103(a,1),2(a,2))].">
      <j1 rule="para" parents="21103 2"/>
    </justification>
  </clause>

  <clause id="21132">
    <literal><![CDATA[
      f(x,f(y,z,u),f(z,u,v)) = f(z,u,f(y,x,f(z,u,v)))
    ]]></literal>
    <justification jstring="[para(13(a,1),21103(a,1,3)),rewrite(3(2),2(3,R),3(3),2(5),3(5),15803(5),3(5))].">
      <j1 rule="para" parents="13 21103"/>
      <j2 rule="rewrite" parents="3 2 3 2 3 15803 3"/>
    </justification>
  </clause>

  <clause id="21249">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,v,z)) = f(y,z,f(v,x,f(y,z,u)))
    ]]></literal>
    <justification jstring="[para(12(a,1),21122(a,1,2)),rewrite(3(2),3(3),3(4),1771(5),3(4),3(5))].">
      <j1 rule="para" parents="12 21122"/>
      <j2 rule="rewrite" parents="3 3 3 1771 3 3"/>
    </justification>
  </clause>

  <clause id="21283">
    <literal><![CDATA[
      f(x,f(y,z,u),f(y,z,v)) = f(y,z,f(v,u,x))
    ]]></literal>
    <justification jstring="[para(21122(a,1),13654(a,2)),rewrite(2(2,R),3(2),3(3),3(5),2(6),2(6),1042(7),18804(6),2(4,R),3(4))].">
      <j1 rule="para" parents="21122 13654"/>
      <j2 rule="rewrite" parents="2 3 3 3 2 2 1042 18804 2 3"/>
    </justification>
  </clause>

  <clause id="21304">
    <literal><![CDATA[
      f(x,y,f(z,u,f(x,y,v))) = f(x,y,f(z,v,u))
    ]]></literal>
    <justification jstring="[back_rewrite(21249),rewrite(3(2),21283(3),2(1),3(1),2(4,R),3(4)),flip(a)].">
      <j1 rule="back_rewrite" parents="21249"/>
      <j2 rule="rewrite" parents="3 21283 2 3 2 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21343">
    <literal><![CDATA[
      f(x,f(y,z,u),f(z,u,v)) = f(z,u,f(v,x,f(y,z,u)))
    ]]></literal>
    <justification jstring="[back_rewrite(18012),rewrite(21283(4),2(2,R),3(2)),flip(a)].">
      <j1 rule="back_rewrite" parents="18012"/>
      <j2 rule="rewrite" parents="21283 2 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21370">
    <literal><![CDATA[
      f(x,f(y,z,u),f(z,u,v)) = f(z,u,f(x,y,v))
    ]]></literal>
    <justification jstring="[back_rewrite(21132),rewrite(2(5,R),3(5),21304(6),3(4))].">
      <j1 rule="back_rewrite" parents="21132"/>
      <j2 rule="rewrite" parents="2 3 21304 3"/>
    </justification>
  </clause>

  <clause id="21439">
    <literal><![CDATA[
      f(x,y,f(z,u,f(v,x,y))) = f(x,y,f(z,v,u))
    ]]></literal>
    <justification jstring="[back_rewrite(21343),rewrite(21370(3),2(4,R),3(4)),flip(a)].">
      <j1 rule="back_rewrite" parents="21343"/>
      <j2 rule="rewrite" parents="21370 2 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22153">
    <literal><![CDATA[
      f(x,f(y,z,u),f(v,z,u)) = f(z,u,f(y,v,x))
    ]]></literal>
    <justification jstring="[para(21122(a,1),21370(a,2)),rewrite(2(1,R),3(1),2(2),11249(3),2(2,R),3(2),2(3),2(4),1127(5))].">
      <j1 rule="para" parents="21122 21370"/>
      <j2 rule="rewrite" parents="2 3 2 11249 2 3 2 2 1127"/>
    </justification>
  </clause>

  <clause id="22178">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(dist_both)
    ]]></attribute>
    <justification jstring="[back_rewrite(7),rewrite(22153(13),21439(10),2(6,R),3(6),22153(25),2(21)),xx(a),xx(b)].">
      <j1 rule="back_rewrite" parents="7"/>
      <j2 rule="rewrite" parents="22153 21439 2 3 22153 2"/>
      <j2 rule="xx"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
