<?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="6">

<source>mol-hints.out</source>

<heading><![CDATA[
Prover9 (32) version 22-May-2007, May 2007.
Process 26999 was started by mccune on cleo,
Tue May 22 14:44:36 2007
The command was "/home/mccune/bin/prover9 -f mol-hints.in".
]]></heading>

<proof number="1" length="28" max_count="41">

<comments><![CDATA[
% Proof 1 at 0.20 (+ 0.01) seconds: absorb.
% Length of proof is 28.
% Level of proof is 14.
% Maximum clause weight is 1009.
% Given clauses 44.
]]></comments>

  <clause id="3" type="goal">
    <literal><![CDATA[
      f(f(x,x),f(x,y)) = x
    ]]></literal>
    <attribute><![CDATA[
      answer(absorb)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="7" type="assumption">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(y,u)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(MOL_sax)
    ]]></attribute>
    <attribute><![CDATA[
      label(492)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="10" type="deny">
    <literal><![CDATA[
      f(f(c5,c5),f(c5,c6)) != c5
    ]]></literal>
    <attribute><![CDATA[
      answer(absorb)
    ]]></attribute>
    <justification jstring="[deny(3)].">
      <j1 rule="deny" parents="3"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),u),f(f(f(f(f(f(y,z),x),u),u),f(y,z)),z))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(499)
    ]]></attribute>
    <justification jstring="[para(7(a,1),7(a,1,2,2,2))].">
      <j1 rule="para" parents="7 7"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(f(f(y,y),u),f(f(f(f(f(y,v),u),u),y),f(y,w)))))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(500)
    ]]></attribute>
    <justification jstring="[para(7(a,1),15(a,1,1,2)),rewrite([7(12),7(12),7(14),7(17),7(30)])].">
      <j1 rule="para" parents="7 15"/>
      <j2 rule="rewrite" parents="7 7 7 7 7"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(508A)
    ]]></attribute>
    <justification jstring="[para(7(a,1),16(a,1,2,2))].">
      <j1 rule="para" parents="7 16"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      f(f(x,y),f(y,y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(508B)
    ]]></attribute>
    <justification jstring="[para(7(a,1),17(a,1,2,1))].">
      <j1 rule="para" parents="7 17"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(f(f(f(y,x),f(y,y)),f(y,y)),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(503)
    ]]></attribute>
    <justification jstring="[para(18(a,1),7(a,1,2,1))].">
      <j1 rule="para" parents="18 7"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),f(x,x)),f(f(f(x,f(x,x)),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(583A)
    ]]></attribute>
    <justification jstring="[para(18(a,1),7(a,1,2,2,1,1,1))].">
      <j1 rule="para" parents="18 7"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),f(x,x)),f(f(f(x,f(x,x)),f(y,z)),z))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(586A)
    ]]></attribute>
    <justification jstring="[para(18(a,1),15(a,1,2,2,1,1,1))].">
      <j1 rule="para" parents="18 15"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      f(f(x,f(y,y)),y) = f(y,y)
    ]]></literal>
    <attribute><![CDATA[
      label(508)
    ]]></attribute>
    <justification jstring="[para(18(a,1),15(a,1,2,2,1)),rewrite([18(5),18(5)])].">
      <j1 rule="para" parents="18 15"/>
      <j2 rule="rewrite" parents="18 18"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      f(f(f(x,x),y),x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(510B)
    ]]></attribute>
    <justification jstring="[para(17(a,1),18(a,1,1)),rewrite([17(7)]),flip(a)].">
      <j1 rule="para" parents="17 18"/>
      <j2 rule="rewrite" parents="17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(y,y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(511)
    ]]></attribute>
    <justification jstring="[back_rewrite(19),rewrite([24(7)])].">
      <j1 rule="back_rewrite" parents="19"/>
      <j2 rule="rewrite" parents="24"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      f(f(x,x),f(f(f(x,x),x),f(f(f(x,x),x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(512)
    ]]></attribute>
    <justification jstring="[para(26(a,1),7(a,1,2,2,1,1))].">
      <j1 rule="para" parents="26 7"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      f(f(x,y),f(x,x)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(513)
    ]]></attribute>
    <justification jstring="[para(18(a,1),26(a,1,1,1)),rewrite([18(6)])].">
      <j1 rule="para" parents="18 26"/>
      <j2 rule="rewrite" parents="18"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),f(f(y,x),f(y,x))),f(f(f(y,x),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(514)
    ]]></attribute>
    <justification jstring="[para(29(a,1),7(a,1,2,2,1,1))].">
      <j1 rule="para" parents="29 7"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      f(x,f(f(y,x),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(515)
    ]]></attribute>
    <justification jstring="[para(7(a,1),29(a,1,1))].">
      <j1 rule="para" parents="7 29"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      f(f(x,f(f(x,x),f(x,y))),f(x,f(f(x,x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(519A)
    ]]></attribute>
    <justification jstring="[para(27(a,1),26(a,1)),flip(a)].">
      <j1 rule="para" parents="27 26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      f(f(x,x),f(f(f(x,x),x),f(f(f(x,x),x),f(y,x)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(520)
    ]]></attribute>
    <justification jstring="[para(31(a,1),28(a,1,2,2,2))].">
      <j1 rule="para" parents="31 28"/>
    </justification>
  </clause>

  <clause id="37">
    <literal><![CDATA[
      f(x,f(f(x,x),f(x,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(521B)
    ]]></attribute>
    <justification jstring="[para(35(a,1),18(a,1,1)),rewrite([35(9)]),flip(a)].">
      <j1 rule="para" parents="35 18"/>
      <j2 rule="rewrite" parents="35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      f(x,f(f(x,f(x,x)),f(f(x,f(x,x)),f(y,f(x,x))))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(526)
    ]]></attribute>
    <justification jstring="[para(18(a,1),36(a,1,1)),rewrite([29(3),29(5)])].">
      <j1 rule="para" parents="18 36"/>
      <j2 rule="rewrite" parents="29 29"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(f(x,f(x,x)),f(y,f(x,x)))),x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(527)
    ]]></attribute>
    <justification jstring="[para(42(a,1),31(a,1,2,1)),rewrite([42(18),29(11),42(18)])].">
      <j1 rule="para" parents="42 31"/>
      <j2 rule="rewrite" parents="42 29 42"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),f(x,x)),f(f(f(x,f(x,x)),f(y,z)),y))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(528)
    ]]></attribute>
    <justification jstring="[para(29(a,1),21(a,1,2,2,2))].">
      <j1 rule="para" parents="29 21"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(f(x,x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(529)
    ]]></attribute>
    <justification jstring="[para(43(a,1),21(a,1,2,2,1)),rewrite([24(3),31(8)])].">
      <j1 rule="para" parents="43 21"/>
      <j2 rule="rewrite" parents="24 31"/>
    </justification>
  </clause>

  <clause id="48">
    <literal><![CDATA[
      f(f(f(x,x),x),f(f(x,f(x,x)),f(f(x,x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(532)
    ]]></attribute>
    <justification jstring="[para(24(a,1),30(a,1,2,2,1)),rewrite([31(9)])].">
      <j1 rule="para" parents="24 30"/>
      <j2 rule="rewrite" parents="31"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(x,x),f(x,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(534F)
    ]]></attribute>
    <justification jstring="[para(45(a,1),23(a,1,1)),rewrite([29(16),29(18),48(23),37(18),18(16)]),flip(a)].">
      <j1 rule="para" parents="45 23"/>
      <j2 rule="rewrite" parents="29 29 48 37 18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      f(f(x,x),f(x,y)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(525)
    ]]></attribute>
    <justification jstring="[para(37(a,1),44(a,1,1)),rewrite([50(16),29(13),24(11),29(3)]),flip(a)].">
      <j1 rule="para" parents="37 44"/>
      <j2 rule="rewrite" parents="50 29 24 29"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="56">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(absorb)
    ]]></attribute>
    <justification jstring="[resolve(55,a,10,a)].">
      <j1 rule="resolve" parents="55 10"/>
    </justification>
  </clause>

</proof>

<proof number="2" length="63" max_count="41">

<comments><![CDATA[
% Proof 2 at 0.38 (+ 0.01) seconds: one.
% Length of proof is 63.
% Level of proof is 24.
% Maximum clause weight is 1009.
% Given clauses 69.
]]></comments>

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

  <clause id="7" type="assumption">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(y,u)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(MOL_sax)
    ]]></attribute>
    <attribute><![CDATA[
      label(492)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="9" type="deny">
    <literal><![CDATA[
      f(c4,f(c4,c4)) != f(c3,f(c3,c3))
    ]]></literal>
    <attribute><![CDATA[
      answer(one)
    ]]></attribute>
    <justification jstring="[deny(2)].">
      <j1 rule="deny" parents="2"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),u),f(f(f(f(f(f(y,z),x),u),u),f(y,z)),z))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(499)
    ]]></attribute>
    <justification jstring="[para(7(a,1),7(a,1,2,2,2))].">
      <j1 rule="para" parents="7 7"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(f(f(y,y),u),f(f(f(f(f(y,v),u),u),y),f(y,w)))))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(500)
    ]]></attribute>
    <justification jstring="[para(7(a,1),15(a,1,1,2)),rewrite([7(12),7(12),7(14),7(17),7(30)])].">
      <j1 rule="para" parents="7 15"/>
      <j2 rule="rewrite" parents="7 7 7 7 7"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(508A)
    ]]></attribute>
    <justification jstring="[para(7(a,1),16(a,1,2,2))].">
      <j1 rule="para" parents="7 16"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      f(f(x,y),f(y,y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(508B)
    ]]></attribute>
    <justification jstring="[para(7(a,1),17(a,1,2,1))].">
      <j1 rule="para" parents="7 17"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(f(f(f(y,x),f(y,y)),f(y,y)),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(503)
    ]]></attribute>
    <justification jstring="[para(18(a,1),7(a,1,2,1))].">
      <j1 rule="para" parents="18 7"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      f(f(f(x,x),f(y,x)),f(f(f(f(y,x),f(y,x)),z),f(f(f(f(x,z),z),f(y,x)),f(f(y,x),u)))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(504)
    ]]></attribute>
    <justification jstring="[para(18(a,1),7(a,1,2,2,1,1,1,1))].">
      <j1 rule="para" parents="18 7"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),f(x,x)),f(f(f(x,f(x,x)),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(583A)
    ]]></attribute>
    <justification jstring="[para(18(a,1),7(a,1,2,2,1,1,1))].">
      <j1 rule="para" parents="18 7"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),f(x,x)),f(f(f(x,f(x,x)),f(y,z)),z))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(586A)
    ]]></attribute>
    <justification jstring="[para(18(a,1),15(a,1,2,2,1,1,1))].">
      <j1 rule="para" parents="18 15"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      f(f(x,f(y,y)),y) = f(y,y)
    ]]></literal>
    <attribute><![CDATA[
      label(508)
    ]]></attribute>
    <justification jstring="[para(18(a,1),15(a,1,2,2,1)),rewrite([18(5),18(5)])].">
      <j1 rule="para" parents="18 15"/>
      <j2 rule="rewrite" parents="18 18"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      f(f(f(x,x),y),x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(510B)
    ]]></attribute>
    <justification jstring="[para(17(a,1),18(a,1,1)),rewrite([17(7)]),flip(a)].">
      <j1 rule="para" parents="17 18"/>
      <j2 rule="rewrite" parents="17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(y,y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(511)
    ]]></attribute>
    <justification jstring="[back_rewrite(19),rewrite([24(7)])].">
      <j1 rule="back_rewrite" parents="19"/>
      <j2 rule="rewrite" parents="24"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      f(f(x,x),f(f(f(x,x),x),f(f(f(x,x),x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(512)
    ]]></attribute>
    <justification jstring="[para(26(a,1),7(a,1,2,2,1,1))].">
      <j1 rule="para" parents="26 7"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      f(f(x,y),f(x,x)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(513)
    ]]></attribute>
    <justification jstring="[para(18(a,1),26(a,1,1,1)),rewrite([18(6)])].">
      <j1 rule="para" parents="18 26"/>
      <j2 rule="rewrite" parents="18"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),f(f(y,x),f(y,x))),f(f(f(y,x),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(514)
    ]]></attribute>
    <justification jstring="[para(29(a,1),7(a,1,2,2,1,1))].">
      <j1 rule="para" parents="29 7"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      f(x,f(f(y,x),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(515)
    ]]></attribute>
    <justification jstring="[para(7(a,1),29(a,1,1))].">
      <j1 rule="para" parents="7 29"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(u,y)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(517)
    ]]></attribute>
    <justification jstring="[para(31(a,1),7(a,1,2,2,2))].">
      <j1 rule="para" parents="31 7"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      f(f(x,f(f(x,x),f(x,y))),f(x,f(f(x,x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(519A)
    ]]></attribute>
    <justification jstring="[para(27(a,1),26(a,1)),flip(a)].">
      <j1 rule="para" parents="27 26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      f(f(x,x),f(f(f(x,x),x),f(f(f(x,x),x),f(y,x)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(520)
    ]]></attribute>
    <justification jstring="[para(31(a,1),28(a,1,2,2,2))].">
      <j1 rule="para" parents="31 28"/>
    </justification>
  </clause>

  <clause id="37">
    <literal><![CDATA[
      f(x,f(f(x,x),f(x,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(521B)
    ]]></attribute>
    <justification jstring="[para(35(a,1),18(a,1,1)),rewrite([35(9)]),flip(a)].">
      <j1 rule="para" parents="35 18"/>
      <j2 rule="rewrite" parents="35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      f(f(x,y),f(f(f(x,y),f(x,y)),y)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(522)
    ]]></attribute>
    <justification jstring="[para(7(a,1),37(a,1,2,2))].">
      <j1 rule="para" parents="7 37"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      f(f(x,x),f(x,f(f(x,x),y))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(523)
    ]]></attribute>
    <justification jstring="[para(18(a,1),37(a,1,2,1)),rewrite([29(8)])].">
      <j1 rule="para" parents="18 37"/>
      <j2 rule="rewrite" parents="29"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      f(f(x,y),f(f(f(x,y),f(x,y)),x)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(524)
    ]]></attribute>
    <justification jstring="[para(29(a,1),37(a,1,2,2))].">
      <j1 rule="para" parents="29 37"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      f(f(x,x),f(x,f(y,f(x,x)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(525)
    ]]></attribute>
    <justification jstring="[para(31(a,1),39(a,1,2,2))].">
      <j1 rule="para" parents="31 39"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      f(x,f(f(x,f(x,x)),f(f(x,f(x,x)),f(y,f(x,x))))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(526)
    ]]></attribute>
    <justification jstring="[para(18(a,1),36(a,1,1)),rewrite([29(3),29(5)])].">
      <j1 rule="para" parents="18 36"/>
      <j2 rule="rewrite" parents="29 29"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(f(x,f(x,x)),f(y,f(x,x)))),x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(527)
    ]]></attribute>
    <justification jstring="[para(42(a,1),31(a,1,2,1)),rewrite([42(18),29(11),42(18)])].">
      <j1 rule="para" parents="42 31"/>
      <j2 rule="rewrite" parents="42 29 42"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),f(x,x)),f(f(f(x,f(x,x)),f(y,z)),y))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(528)
    ]]></attribute>
    <justification jstring="[para(29(a,1),21(a,1,2,2,2))].">
      <j1 rule="para" parents="29 21"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(f(x,x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(529)
    ]]></attribute>
    <justification jstring="[para(43(a,1),21(a,1,2,2,1)),rewrite([24(3),31(8)])].">
      <j1 rule="para" parents="43 21"/>
      <j2 rule="rewrite" parents="24 31"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      f(x,f(f(f(x,x),x),f(x,f(f(x,x),y)))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(530)
    ]]></attribute>
    <justification jstring="[para(18(a,1),45(a,1,1)),rewrite([29(4),29(5)])].">
      <j1 rule="para" parents="18 45"/>
      <j2 rule="rewrite" parents="29 29"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(f(x,x),x),f(x,f(f(x,x),y)))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(531)
    ]]></attribute>
    <justification jstring="[para(18(a,1),30(a,1,2,2,1)),rewrite([29(5),31(8)])].">
      <j1 rule="para" parents="18 30"/>
      <j2 rule="rewrite" parents="29 31"/>
    </justification>
  </clause>

  <clause id="48">
    <literal><![CDATA[
      f(f(f(x,x),x),f(f(x,f(x,x)),f(f(x,x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(532)
    ]]></attribute>
    <justification jstring="[para(24(a,1),30(a,1,2,2,1)),rewrite([31(9)])].">
      <j1 rule="para" parents="24 30"/>
      <j2 rule="rewrite" parents="31"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(x,x),f(x,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(534F)
    ]]></attribute>
    <justification jstring="[para(45(a,1),23(a,1,1)),rewrite([29(16),29(18),48(23),37(18),18(16)]),flip(a)].">
      <j1 rule="para" parents="45 23"/>
      <j2 rule="rewrite" parents="29 29 48 37 18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="51">
    <literal><![CDATA[
      f(f(f(x,x),x),f(x,f(f(x,x),y))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(535E)
    ]]></attribute>
    <justification jstring="[para(46(a,1),23(a,1,1)),rewrite([47(25),39(21),24(17),29(3)]),flip(a)].">
      <j1 rule="para" parents="46 23"/>
      <j2 rule="rewrite" parents="47 39 24 29"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      f(f(f(x,x),x),f(x,f(y,f(x,x)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(536)
    ]]></attribute>
    <justification jstring="[para(31(a,1),51(a,1,2,2))].">
      <j1 rule="para" parents="31 51"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      f(f(f(x,y),f(f(x,y),f(x,y))),f(f(f(x,y),f(x,y)),y)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(537)
    ]]></attribute>
    <justification jstring="[para(7(a,1),50(a,1,2,2))].">
      <j1 rule="para" parents="7 50"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      f(f(f(x,y),f(f(x,y),f(x,y))),f(f(f(x,y),f(x,y)),x)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(538)
    ]]></attribute>
    <justification jstring="[para(29(a,1),50(a,1,2,2))].">
      <j1 rule="para" parents="29 50"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      f(f(x,x),f(x,y)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(525)
    ]]></attribute>
    <justification jstring="[para(37(a,1),44(a,1,1)),rewrite([50(16),29(13),24(11),29(3)]),flip(a)].">
      <j1 rule="para" parents="37 44"/>
      <j2 rule="rewrite" parents="50 29 24 29"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="57">
    <literal><![CDATA[
      f(x,f(f(x,x),y)) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(541E)
    ]]></attribute>
    <justification jstring="[para(39(a,1),44(a,1,1)),rewrite([55(10),55(12),51(14),18(10)]),flip(a)].">
      <j1 rule="para" parents="39 44"/>
      <j2 rule="rewrite" parents="55 55 51 18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="58">
    <literal><![CDATA[
      f(x,f(y,f(x,x))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(542E)
    ]]></attribute>
    <justification jstring="[para(41(a,1),44(a,1,1)),rewrite([55(10),55(12),52(14),18(10)]),flip(a)].">
      <j1 rule="para" parents="41 44"/>
      <j2 rule="rewrite" parents="55 55 52 18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      f(f(f(x,y),f(x,y)),y) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(543E)
    ]]></attribute>
    <justification jstring="[para(38(a,1),44(a,1,1)),rewrite([53(26),55(23),24(18),55(7)]),flip(a)].">
      <j1 rule="para" parents="38 44"/>
      <j2 rule="rewrite" parents="53 55 24 55"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="60">
    <literal><![CDATA[
      f(f(f(x,y),f(x,y)),x) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(544E)
    ]]></attribute>
    <justification jstring="[para(40(a,1),44(a,1,1)),rewrite([54(26),55(23),24(18),55(7)]),flip(a)].">
      <j1 rule="para" parents="40 44"/>
      <j2 rule="rewrite" parents="54 55 24 55"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(f(f(f(y,x),f(y,z)),f(y,z)),y),f(u,y)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(546)
    ]]></attribute>
    <justification jstring="[para(57(a,1),33(a,1,2,1)),rewrite([55(4),55(5),55(7)])].">
      <j1 rule="para" parents="57 33"/>
      <j2 rule="rewrite" parents="55 55 55"/>
    </justification>
  </clause>

  <clause id="63">
    <literal><![CDATA[
      f(f(x,x),f(y,x)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(548)
    ]]></attribute>
    <justification jstring="[para(18(a,1),58(a,1,2,2)),rewrite([55(6)])].">
      <j1 rule="para" parents="18 58"/>
      <j2 rule="rewrite" parents="55"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      f(x,f(f(f(f(y,x),f(y,x)),z),f(f(f(f(x,z),z),f(y,x)),f(f(y,x),u)))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(550)
    ]]></attribute>
    <justification jstring="[back_rewrite(20),rewrite([63(3)])].">
      <j1 rule="back_rewrite" parents="20"/>
      <j2 rule="rewrite" parents="63"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      f(f(x,y),f(f(f(f(y,x),f(y,z)),f(y,z)),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(552)
    ]]></attribute>
    <justification jstring="[para(31(a,1),62(a,1,2))].">
      <j1 rule="para" parents="31 62"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      f(x,f(f(f(x,f(f(y,x),z)),f(f(y,x),z)),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(553)
    ]]></attribute>
    <justification jstring="[para(18(a,1),67(a,1,2,1,1,1)),rewrite([63(3)])].">
      <j1 rule="para" parents="18 67"/>
      <j2 rule="rewrite" parents="63"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      f(x,f(f(f(x,x),x),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(557)
    ]]></attribute>
    <justification jstring="[para(7(a,1),68(a,1,2,1,1,2)),rewrite([7(12)])].">
      <j1 rule="para" parents="7 68"/>
      <j2 rule="rewrite" parents="7"/>
    </justification>
  </clause>

  <clause id="73">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(y,f(x,x)))) = f(y,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(558)
    ]]></attribute>
    <justification jstring="[para(24(a,1),68(a,1,2,1,1,2)),rewrite([55(4),24(4)])].">
      <j1 rule="para" parents="24 68"/>
      <j2 rule="rewrite" parents="55 24"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      f(x,f(f(f(x,y),y),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(559)
    ]]></attribute>
    <justification jstring="[para(29(a,1),68(a,1,2,1,1,2)),rewrite([29(4)])].">
      <j1 rule="para" parents="29 68"/>
      <j2 rule="rewrite" parents="29"/>
    </justification>
  </clause>

  <clause id="75">
    <literal><![CDATA[
      f(x,f(f(f(x,x),x),f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(560)
    ]]></attribute>
    <justification jstring="[para(60(a,1),72(a,1,2,2)),rewrite([60(9)])].">
      <j1 rule="para" parents="60 72"/>
      <j2 rule="rewrite" parents="60"/>
    </justification>
  </clause>

  <clause id="76">
    <literal><![CDATA[
      f(f(x,x),x) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(561)
    ]]></attribute>
    <justification jstring="[para(26(a,1),74(a,1,2,1)),rewrite([55(5)])].">
      <j1 rule="para" parents="26 74"/>
      <j2 rule="rewrite" parents="55"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      f(x,f(f(x,f(x,x)),f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(562)
    ]]></attribute>
    <justification jstring="[back_rewrite(75),rewrite([76(2)])].">
      <j1 rule="back_rewrite" parents="75"/>
      <j2 rule="rewrite" parents="76"/>
    </justification>
  </clause>

  <clause id="78">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(f(x,x),y))) = f(f(x,x),y)
    ]]></literal>
    <attribute><![CDATA[
      label(563)
    ]]></attribute>
    <justification jstring="[para(18(a,1),77(a,1,2,1,2)),rewrite([76(3)])].">
      <j1 rule="para" parents="18 77"/>
      <j2 rule="rewrite" parents="76"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      f(x,f(f(y,x),f(f(f(f(x,y),y),f(y,x)),f(f(y,x),z)))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(564)
    ]]></attribute>
    <justification jstring="[para(60(a,1),65(a,1,2,1))].">
      <j1 rule="para" parents="60 65"/>
    </justification>
  </clause>

  <clause id="80">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(x,f(f(x,f(x,x)),y)))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(565)
    ]]></attribute>
    <justification jstring="[para(26(a,1),79(a,1,2,2,1,1)),rewrite([55(7)])].">
      <j1 rule="para" parents="26 79"/>
      <j2 rule="rewrite" parents="55"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(x,f(x,x)),f(x,f(x,x)))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(567D)
    ]]></attribute>
    <justification jstring="[para(43(a,1),80(a,1,2,2,2)),rewrite([29(17),43(14),59(7)]),flip(a)].">
      <j1 rule="para" parents="43 80"/>
      <j2 rule="rewrite" parents="29 43 59"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="83">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(x,y))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(568)
    ]]></attribute>
    <justification jstring="[para(77(a,1),80(a,1,2,2))].">
      <j1 rule="para" parents="77 80"/>
    </justification>
  </clause>

  <clause id="84">
    <literal><![CDATA[
      f(x,f(f(y,f(y,y)),f(y,f(y,y)))) = f(y,f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(570C)
    ]]></attribute>
    <justification jstring="[para(82(a,1),73(a,1,2,1)),rewrite([58(14),55(11)]),flip(a)].">
      <j1 rule="para" parents="82 73"/>
      <j2 rule="rewrite" parents="58 55"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="90">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(y,f(y,y)))) = f(y,f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(576)
    ]]></attribute>
    <justification jstring="[para(84(a,1),78(a,1,2,2)),rewrite([84(14)])].">
      <j1 rule="para" parents="84 78"/>
      <j2 rule="rewrite" parents="84"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      f(x,f(x,x)) = f(y,f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(577)
    ]]></attribute>
    <justification jstring="[para(84(a,1),83(a,1,2,2)),rewrite([90(7)])].">
      <j1 rule="para" parents="84 83"/>
      <j2 rule="rewrite" parents="90"/>
    </justification>
  </clause>

  <clause id="92">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(one)
    ]]></attribute>
    <justification jstring="[resolve(91,a,9,a)].">
      <j1 rule="resolve" parents="91 9"/>
    </justification>
  </clause>

</proof>

<proof number="3" length="59" max_count="41">

<comments><![CDATA[
% Proof 3 at 0.41 (+ 0.01) seconds: oml.
% Length of proof is 59.
% Level of proof is 25.
% Maximum clause weight is 1009.
% Given clauses 72.
]]></comments>

  <clause id="4" type="goal">
    <literal><![CDATA[
      f(x,f(x,f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      answer(oml)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="7" type="assumption">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(y,u)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(MOL_sax)
    ]]></attribute>
    <attribute><![CDATA[
      label(492)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11" type="deny">
    <literal><![CDATA[
      f(c7,f(c7,f(c7,c8))) != f(c7,c8)
    ]]></literal>
    <attribute><![CDATA[
      answer(oml)
    ]]></attribute>
    <justification jstring="[deny(4)].">
      <j1 rule="deny" parents="4"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),u),f(f(f(f(f(f(y,z),x),u),u),f(y,z)),z))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(499)
    ]]></attribute>
    <justification jstring="[para(7(a,1),7(a,1,2,2,2))].">
      <j1 rule="para" parents="7 7"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(f(f(y,y),u),f(f(f(f(f(y,v),u),u),y),f(y,w)))))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(500)
    ]]></attribute>
    <justification jstring="[para(7(a,1),15(a,1,1,2)),rewrite([7(12),7(12),7(14),7(17),7(30)])].">
      <j1 rule="para" parents="7 15"/>
      <j2 rule="rewrite" parents="7 7 7 7 7"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(508A)
    ]]></attribute>
    <justification jstring="[para(7(a,1),16(a,1,2,2))].">
      <j1 rule="para" parents="7 16"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      f(f(x,y),f(y,y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(508B)
    ]]></attribute>
    <justification jstring="[para(7(a,1),17(a,1,2,1))].">
      <j1 rule="para" parents="7 17"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(f(f(f(y,x),f(y,y)),f(y,y)),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(503)
    ]]></attribute>
    <justification jstring="[para(18(a,1),7(a,1,2,1))].">
      <j1 rule="para" parents="18 7"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      f(f(f(x,x),f(y,x)),f(f(f(f(y,x),f(y,x)),z),f(f(f(f(x,z),z),f(y,x)),f(f(y,x),u)))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(504)
    ]]></attribute>
    <justification jstring="[para(18(a,1),7(a,1,2,2,1,1,1,1))].">
      <j1 rule="para" parents="18 7"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),f(x,x)),f(f(f(x,f(x,x)),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(583A)
    ]]></attribute>
    <justification jstring="[para(18(a,1),7(a,1,2,2,1,1,1))].">
      <j1 rule="para" parents="18 7"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),f(x,x)),f(f(f(x,f(x,x)),f(y,z)),z))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(586A)
    ]]></attribute>
    <justification jstring="[para(18(a,1),15(a,1,2,2,1,1,1))].">
      <j1 rule="para" parents="18 15"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      f(f(x,f(y,y)),y) = f(y,y)
    ]]></literal>
    <attribute><![CDATA[
      label(508)
    ]]></attribute>
    <justification jstring="[para(18(a,1),15(a,1,2,2,1)),rewrite([18(5),18(5)])].">
      <j1 rule="para" parents="18 15"/>
      <j2 rule="rewrite" parents="18 18"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      f(f(f(x,x),y),x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(510B)
    ]]></attribute>
    <justification jstring="[para(17(a,1),18(a,1,1)),rewrite([17(7)]),flip(a)].">
      <j1 rule="para" parents="17 18"/>
      <j2 rule="rewrite" parents="17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(y,y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(511)
    ]]></attribute>
    <justification jstring="[back_rewrite(19),rewrite([24(7)])].">
      <j1 rule="back_rewrite" parents="19"/>
      <j2 rule="rewrite" parents="24"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      f(f(x,x),f(f(f(x,x),x),f(f(f(x,x),x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(512)
    ]]></attribute>
    <justification jstring="[para(26(a,1),7(a,1,2,2,1,1))].">
      <j1 rule="para" parents="26 7"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      f(f(x,y),f(x,x)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(513)
    ]]></attribute>
    <justification jstring="[para(18(a,1),26(a,1,1,1)),rewrite([18(6)])].">
      <j1 rule="para" parents="18 26"/>
      <j2 rule="rewrite" parents="18"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),f(f(y,x),f(y,x))),f(f(f(y,x),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(514)
    ]]></attribute>
    <justification jstring="[para(29(a,1),7(a,1,2,2,1,1))].">
      <j1 rule="para" parents="29 7"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      f(x,f(f(y,x),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(515)
    ]]></attribute>
    <justification jstring="[para(7(a,1),29(a,1,1))].">
      <j1 rule="para" parents="7 29"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(u,y)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(517)
    ]]></attribute>
    <justification jstring="[para(31(a,1),7(a,1,2,2,2))].">
      <j1 rule="para" parents="31 7"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      f(f(x,f(f(x,x),f(x,y))),f(x,f(f(x,x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(519A)
    ]]></attribute>
    <justification jstring="[para(27(a,1),26(a,1)),flip(a)].">
      <j1 rule="para" parents="27 26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      f(f(x,x),f(f(f(x,x),x),f(f(f(x,x),x),f(y,x)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(520)
    ]]></attribute>
    <justification jstring="[para(31(a,1),28(a,1,2,2,2))].">
      <j1 rule="para" parents="31 28"/>
    </justification>
  </clause>

  <clause id="37">
    <literal><![CDATA[
      f(x,f(f(x,x),f(x,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(521B)
    ]]></attribute>
    <justification jstring="[para(35(a,1),18(a,1,1)),rewrite([35(9)]),flip(a)].">
      <j1 rule="para" parents="35 18"/>
      <j2 rule="rewrite" parents="35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      f(f(x,y),f(f(f(x,y),f(x,y)),y)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(522)
    ]]></attribute>
    <justification jstring="[para(7(a,1),37(a,1,2,2))].">
      <j1 rule="para" parents="7 37"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      f(f(x,x),f(x,f(f(x,x),y))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(523)
    ]]></attribute>
    <justification jstring="[para(18(a,1),37(a,1,2,1)),rewrite([29(8)])].">
      <j1 rule="para" parents="18 37"/>
      <j2 rule="rewrite" parents="29"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      f(f(x,y),f(f(f(x,y),f(x,y)),x)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(524)
    ]]></attribute>
    <justification jstring="[para(29(a,1),37(a,1,2,2))].">
      <j1 rule="para" parents="29 37"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      f(f(x,x),f(x,f(y,f(x,x)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(525)
    ]]></attribute>
    <justification jstring="[para(31(a,1),39(a,1,2,2))].">
      <j1 rule="para" parents="31 39"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      f(x,f(f(x,f(x,x)),f(f(x,f(x,x)),f(y,f(x,x))))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(526)
    ]]></attribute>
    <justification jstring="[para(18(a,1),36(a,1,1)),rewrite([29(3),29(5)])].">
      <j1 rule="para" parents="18 36"/>
      <j2 rule="rewrite" parents="29 29"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(f(x,f(x,x)),f(y,f(x,x)))),x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(527)
    ]]></attribute>
    <justification jstring="[para(42(a,1),31(a,1,2,1)),rewrite([42(18),29(11),42(18)])].">
      <j1 rule="para" parents="42 31"/>
      <j2 rule="rewrite" parents="42 29 42"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),f(x,x)),f(f(f(x,f(x,x)),f(y,z)),y))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(528)
    ]]></attribute>
    <justification jstring="[para(29(a,1),21(a,1,2,2,2))].">
      <j1 rule="para" parents="29 21"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(f(x,x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(529)
    ]]></attribute>
    <justification jstring="[para(43(a,1),21(a,1,2,2,1)),rewrite([24(3),31(8)])].">
      <j1 rule="para" parents="43 21"/>
      <j2 rule="rewrite" parents="24 31"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      f(x,f(f(f(x,x),x),f(x,f(f(x,x),y)))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(530)
    ]]></attribute>
    <justification jstring="[para(18(a,1),45(a,1,1)),rewrite([29(4),29(5)])].">
      <j1 rule="para" parents="18 45"/>
      <j2 rule="rewrite" parents="29 29"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(f(x,x),x),f(x,f(f(x,x),y)))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(531)
    ]]></attribute>
    <justification jstring="[para(18(a,1),30(a,1,2,2,1)),rewrite([29(5),31(8)])].">
      <j1 rule="para" parents="18 30"/>
      <j2 rule="rewrite" parents="29 31"/>
    </justification>
  </clause>

  <clause id="48">
    <literal><![CDATA[
      f(f(f(x,x),x),f(f(x,f(x,x)),f(f(x,x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(532)
    ]]></attribute>
    <justification jstring="[para(24(a,1),30(a,1,2,2,1)),rewrite([31(9)])].">
      <j1 rule="para" parents="24 30"/>
      <j2 rule="rewrite" parents="31"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(x,x),f(x,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(534F)
    ]]></attribute>
    <justification jstring="[para(45(a,1),23(a,1,1)),rewrite([29(16),29(18),48(23),37(18),18(16)]),flip(a)].">
      <j1 rule="para" parents="45 23"/>
      <j2 rule="rewrite" parents="29 29 48 37 18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="51">
    <literal><![CDATA[
      f(f(f(x,x),x),f(x,f(f(x,x),y))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(535E)
    ]]></attribute>
    <justification jstring="[para(46(a,1),23(a,1,1)),rewrite([47(25),39(21),24(17),29(3)]),flip(a)].">
      <j1 rule="para" parents="46 23"/>
      <j2 rule="rewrite" parents="47 39 24 29"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      f(f(f(x,x),x),f(x,f(y,f(x,x)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(536)
    ]]></attribute>
    <justification jstring="[para(31(a,1),51(a,1,2,2))].">
      <j1 rule="para" parents="31 51"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      f(f(f(x,y),f(f(x,y),f(x,y))),f(f(f(x,y),f(x,y)),y)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(537)
    ]]></attribute>
    <justification jstring="[para(7(a,1),50(a,1,2,2))].">
      <j1 rule="para" parents="7 50"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      f(f(f(x,y),f(f(x,y),f(x,y))),f(f(f(x,y),f(x,y)),x)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(538)
    ]]></attribute>
    <justification jstring="[para(29(a,1),50(a,1,2,2))].">
      <j1 rule="para" parents="29 50"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      f(f(x,x),f(x,y)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(525)
    ]]></attribute>
    <justification jstring="[para(37(a,1),44(a,1,1)),rewrite([50(16),29(13),24(11),29(3)]),flip(a)].">
      <j1 rule="para" parents="37 44"/>
      <j2 rule="rewrite" parents="50 29 24 29"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="57">
    <literal><![CDATA[
      f(x,f(f(x,x),y)) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(541E)
    ]]></attribute>
    <justification jstring="[para(39(a,1),44(a,1,1)),rewrite([55(10),55(12),51(14),18(10)]),flip(a)].">
      <j1 rule="para" parents="39 44"/>
      <j2 rule="rewrite" parents="55 55 51 18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="58">
    <literal><![CDATA[
      f(x,f(y,f(x,x))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(542E)
    ]]></attribute>
    <justification jstring="[para(41(a,1),44(a,1,1)),rewrite([55(10),55(12),52(14),18(10)]),flip(a)].">
      <j1 rule="para" parents="41 44"/>
      <j2 rule="rewrite" parents="55 55 52 18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      f(f(f(x,y),f(x,y)),y) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(543E)
    ]]></attribute>
    <justification jstring="[para(38(a,1),44(a,1,1)),rewrite([53(26),55(23),24(18),55(7)]),flip(a)].">
      <j1 rule="para" parents="38 44"/>
      <j2 rule="rewrite" parents="53 55 24 55"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="60">
    <literal><![CDATA[
      f(f(f(x,y),f(x,y)),x) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(544E)
    ]]></attribute>
    <justification jstring="[para(40(a,1),44(a,1,1)),rewrite([54(26),55(23),24(18),55(7)]),flip(a)].">
      <j1 rule="para" parents="40 44"/>
      <j2 rule="rewrite" parents="54 55 24 55"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(f(f(f(y,x),f(y,z)),f(y,z)),y),f(u,y)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(546)
    ]]></attribute>
    <justification jstring="[para(57(a,1),33(a,1,2,1)),rewrite([55(4),55(5),55(7)])].">
      <j1 rule="para" parents="57 33"/>
      <j2 rule="rewrite" parents="55 55 55"/>
    </justification>
  </clause>

  <clause id="63">
    <literal><![CDATA[
      f(f(x,x),f(y,x)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(548)
    ]]></attribute>
    <justification jstring="[para(18(a,1),58(a,1,2,2)),rewrite([55(6)])].">
      <j1 rule="para" parents="18 58"/>
      <j2 rule="rewrite" parents="55"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      f(x,f(f(f(f(y,x),f(y,x)),z),f(f(f(f(x,z),z),f(y,x)),f(f(y,x),u)))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(550)
    ]]></attribute>
    <justification jstring="[back_rewrite(20),rewrite([63(3)])].">
      <j1 rule="back_rewrite" parents="20"/>
      <j2 rule="rewrite" parents="63"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      f(f(x,y),f(f(f(f(y,x),f(y,z)),f(y,z)),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(552)
    ]]></attribute>
    <justification jstring="[para(31(a,1),62(a,1,2))].">
      <j1 rule="para" parents="31 62"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      f(x,f(f(f(x,f(f(y,x),z)),f(f(y,x),z)),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(553)
    ]]></attribute>
    <justification jstring="[para(18(a,1),67(a,1,2,1,1,1)),rewrite([63(3)])].">
      <j1 rule="para" parents="18 67"/>
      <j2 rule="rewrite" parents="63"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      f(x,f(f(f(x,f(f(x,y),z)),f(f(x,y),z)),f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(554)
    ]]></attribute>
    <justification jstring="[para(29(a,1),67(a,1,2,1,1,1)),rewrite([55(3)])].">
      <j1 rule="para" parents="29 67"/>
      <j2 rule="rewrite" parents="55"/>
    </justification>
  </clause>

  <clause id="73">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(y,f(x,x)))) = f(y,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(558)
    ]]></attribute>
    <justification jstring="[para(24(a,1),68(a,1,2,1,1,2)),rewrite([55(4),24(4)])].">
      <j1 rule="para" parents="24 68"/>
      <j2 rule="rewrite" parents="55 24"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      f(x,f(f(y,x),f(f(f(f(x,y),y),f(y,x)),f(f(y,x),z)))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(564)
    ]]></attribute>
    <justification jstring="[para(60(a,1),65(a,1,2,1))].">
      <j1 rule="para" parents="60 65"/>
    </justification>
  </clause>

  <clause id="80">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(x,f(f(x,f(x,x)),y)))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(565)
    ]]></attribute>
    <justification jstring="[para(26(a,1),79(a,1,2,2,1,1)),rewrite([55(7)])].">
      <j1 rule="para" parents="26 79"/>
      <j2 rule="rewrite" parents="55"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(x,f(x,x)),f(x,f(x,x)))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(567D)
    ]]></attribute>
    <justification jstring="[para(43(a,1),80(a,1,2,2,2)),rewrite([29(17),43(14),59(7)]),flip(a)].">
      <j1 rule="para" parents="43 80"/>
      <j2 rule="rewrite" parents="29 43 59"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="84">
    <literal><![CDATA[
      f(x,f(f(y,f(y,y)),f(y,f(y,y)))) = f(y,f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(570C)
    ]]></attribute>
    <justification jstring="[para(82(a,1),73(a,1,2,1)),rewrite([58(14),55(11)]),flip(a)].">
      <j1 rule="para" parents="82 73"/>
      <j2 rule="rewrite" parents="58 55"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="88">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(y,y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(574)
    ]]></attribute>
    <justification jstring="[para(84(a,1),29(a,1,1))].">
      <j1 rule="para" parents="84 29"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      f(x,f(y,f(y,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(575)
    ]]></attribute>
    <justification jstring="[para(84(a,1),57(a,1,2))].">
      <j1 rule="para" parents="84 57"/>
    </justification>
  </clause>

  <clause id="104">
    <literal><![CDATA[
      f(f(x,x),f(y,f(y,y))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(594)
    ]]></attribute>
    <justification jstring="[para(88(a,1),60(a,1,1,1)),rewrite([88(4),88(8)])].">
      <j1 rule="para" parents="88 60"/>
      <j2 rule="rewrite" parents="88 88"/>
    </justification>
  </clause>

  <clause id="108">
    <literal><![CDATA[
      f(x,f(x,f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(599)
    ]]></attribute>
    <justification jstring="[para(89(a,1),69(a,1,2,1,1)),rewrite([104(7)])].">
      <j1 rule="para" parents="89 69"/>
      <j2 rule="rewrite" parents="104"/>
    </justification>
  </clause>

  <clause id="109">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(oml)
    ]]></attribute>
    <justification jstring="[resolve(108,a,11,a)].">
      <j1 rule="resolve" parents="108 11"/>
    </justification>
  </clause>

</proof>

<proof number="4" length="97" max_count="41">

<comments><![CDATA[
% Proof 4 at 0.73 (+ 0.01) seconds: comm.
% Length of proof is 97.
% Level of proof is 38.
% Maximum clause weight is 1009.
% Given clauses 117.
]]></comments>

  <clause id="1" type="goal">
    <literal><![CDATA[
      f(x,y) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      answer(comm)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="7" type="assumption">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(y,u)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(MOL_sax)
    ]]></attribute>
    <attribute><![CDATA[
      label(492)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="8" type="deny">
    <literal><![CDATA[
      f(c2,c1) != f(c1,c2)
    ]]></literal>
    <attribute><![CDATA[
      answer(comm)
    ]]></attribute>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),u),f(f(f(f(f(f(y,z),x),u),u),f(y,z)),z))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(499)
    ]]></attribute>
    <justification jstring="[para(7(a,1),7(a,1,2,2,2))].">
      <j1 rule="para" parents="7 7"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(f(f(y,y),u),f(f(f(f(f(y,v),u),u),y),f(y,w)))))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(500)
    ]]></attribute>
    <justification jstring="[para(7(a,1),15(a,1,1,2)),rewrite([7(12),7(12),7(14),7(17),7(30)])].">
      <j1 rule="para" parents="7 15"/>
      <j2 rule="rewrite" parents="7 7 7 7 7"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(508A)
    ]]></attribute>
    <justification jstring="[para(7(a,1),16(a,1,2,2))].">
      <j1 rule="para" parents="7 16"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      f(f(x,y),f(y,y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(508B)
    ]]></attribute>
    <justification jstring="[para(7(a,1),17(a,1,2,1))].">
      <j1 rule="para" parents="7 17"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(f(f(f(y,x),f(y,y)),f(y,y)),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(503)
    ]]></attribute>
    <justification jstring="[para(18(a,1),7(a,1,2,1))].">
      <j1 rule="para" parents="18 7"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      f(f(f(x,x),f(y,x)),f(f(f(f(y,x),f(y,x)),z),f(f(f(f(x,z),z),f(y,x)),f(f(y,x),u)))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(504)
    ]]></attribute>
    <justification jstring="[para(18(a,1),7(a,1,2,2,1,1,1,1))].">
      <j1 rule="para" parents="18 7"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),f(x,x)),f(f(f(x,f(x,x)),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(583A)
    ]]></attribute>
    <justification jstring="[para(18(a,1),7(a,1,2,2,1,1,1))].">
      <j1 rule="para" parents="18 7"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      f(f(f(x,x),f(y,x)),f(f(f(f(y,x),f(y,x)),z),f(f(f(f(x,z),z),f(y,x)),x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(506)
    ]]></attribute>
    <justification jstring="[para(18(a,1),15(a,1,2,2,1,1,1,1))].">
      <j1 rule="para" parents="18 15"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),f(x,x)),f(f(f(x,f(x,x)),f(y,z)),z))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(586A)
    ]]></attribute>
    <justification jstring="[para(18(a,1),15(a,1,2,2,1,1,1))].">
      <j1 rule="para" parents="18 15"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      f(f(x,f(y,y)),y) = f(y,y)
    ]]></literal>
    <attribute><![CDATA[
      label(508)
    ]]></attribute>
    <justification jstring="[para(18(a,1),15(a,1,2,2,1)),rewrite([18(5),18(5)])].">
      <j1 rule="para" parents="18 15"/>
      <j2 rule="rewrite" parents="18 18"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      f(f(f(x,x),y),x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(510B)
    ]]></attribute>
    <justification jstring="[para(17(a,1),18(a,1,1)),rewrite([17(7)]),flip(a)].">
      <j1 rule="para" parents="17 18"/>
      <j2 rule="rewrite" parents="17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(y,y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(511)
    ]]></attribute>
    <justification jstring="[back_rewrite(19),rewrite([24(7)])].">
      <j1 rule="back_rewrite" parents="19"/>
      <j2 rule="rewrite" parents="24"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      f(f(x,x),f(f(f(x,x),x),f(f(f(x,x),x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(512)
    ]]></attribute>
    <justification jstring="[para(26(a,1),7(a,1,2,2,1,1))].">
      <j1 rule="para" parents="26 7"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      f(f(x,y),f(x,x)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(513)
    ]]></attribute>
    <justification jstring="[para(18(a,1),26(a,1,1,1)),rewrite([18(6)])].">
      <j1 rule="para" parents="18 26"/>
      <j2 rule="rewrite" parents="18"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),f(f(y,x),f(y,x))),f(f(f(y,x),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(514)
    ]]></attribute>
    <justification jstring="[para(29(a,1),7(a,1,2,2,1,1))].">
      <j1 rule="para" parents="29 7"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      f(x,f(f(y,x),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(515)
    ]]></attribute>
    <justification jstring="[para(7(a,1),29(a,1,1))].">
      <j1 rule="para" parents="7 29"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      f(x,f(f(x,y),f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(516)
    ]]></attribute>
    <justification jstring="[para(29(a,1),29(a,1,1))].">
      <j1 rule="para" parents="29 29"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(u,y)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(517)
    ]]></attribute>
    <justification jstring="[para(31(a,1),7(a,1,2,2,2))].">
      <j1 rule="para" parents="31 7"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      f(f(f(f(x,x),y),f(f(f(f(f(x,z),y),y),x),f(x,u))),f(x,x)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(518)
    ]]></attribute>
    <justification jstring="[para(7(a,1),31(a,1,2,1)),rewrite([7(20),7(22)])].">
      <j1 rule="para" parents="7 31"/>
      <j2 rule="rewrite" parents="7 7"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      f(f(x,f(f(x,x),f(x,y))),f(x,f(f(x,x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(519A)
    ]]></attribute>
    <justification jstring="[para(27(a,1),26(a,1)),flip(a)].">
      <j1 rule="para" parents="27 26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      f(f(x,x),f(f(f(x,x),x),f(f(f(x,x),x),f(y,x)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(520)
    ]]></attribute>
    <justification jstring="[para(31(a,1),28(a,1,2,2,2))].">
      <j1 rule="para" parents="31 28"/>
    </justification>
  </clause>

  <clause id="37">
    <literal><![CDATA[
      f(x,f(f(x,x),f(x,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(521B)
    ]]></attribute>
    <justification jstring="[para(35(a,1),18(a,1,1)),rewrite([35(9)]),flip(a)].">
      <j1 rule="para" parents="35 18"/>
      <j2 rule="rewrite" parents="35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      f(f(x,y),f(f(f(x,y),f(x,y)),y)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(522)
    ]]></attribute>
    <justification jstring="[para(7(a,1),37(a,1,2,2))].">
      <j1 rule="para" parents="7 37"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      f(f(x,x),f(x,f(f(x,x),y))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(523)
    ]]></attribute>
    <justification jstring="[para(18(a,1),37(a,1,2,1)),rewrite([29(8)])].">
      <j1 rule="para" parents="18 37"/>
      <j2 rule="rewrite" parents="29"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      f(f(x,y),f(f(f(x,y),f(x,y)),x)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(524)
    ]]></attribute>
    <justification jstring="[para(29(a,1),37(a,1,2,2))].">
      <j1 rule="para" parents="29 37"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      f(f(x,x),f(x,f(y,f(x,x)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(525)
    ]]></attribute>
    <justification jstring="[para(31(a,1),39(a,1,2,2))].">
      <j1 rule="para" parents="31 39"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      f(x,f(f(x,f(x,x)),f(f(x,f(x,x)),f(y,f(x,x))))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(526)
    ]]></attribute>
    <justification jstring="[para(18(a,1),36(a,1,1)),rewrite([29(3),29(5)])].">
      <j1 rule="para" parents="18 36"/>
      <j2 rule="rewrite" parents="29 29"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(f(x,f(x,x)),f(y,f(x,x)))),x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(527)
    ]]></attribute>
    <justification jstring="[para(42(a,1),31(a,1,2,1)),rewrite([42(18),29(11),42(18)])].">
      <j1 rule="para" parents="42 31"/>
      <j2 rule="rewrite" parents="42 29 42"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),f(x,x)),f(f(f(x,f(x,x)),f(y,z)),y))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(528)
    ]]></attribute>
    <justification jstring="[para(29(a,1),21(a,1,2,2,2))].">
      <j1 rule="para" parents="29 21"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(f(x,x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(529)
    ]]></attribute>
    <justification jstring="[para(43(a,1),21(a,1,2,2,1)),rewrite([24(3),31(8)])].">
      <j1 rule="para" parents="43 21"/>
      <j2 rule="rewrite" parents="24 31"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      f(x,f(f(f(x,x),x),f(x,f(f(x,x),y)))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(530)
    ]]></attribute>
    <justification jstring="[para(18(a,1),45(a,1,1)),rewrite([29(4),29(5)])].">
      <j1 rule="para" parents="18 45"/>
      <j2 rule="rewrite" parents="29 29"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(f(x,x),x),f(x,f(f(x,x),y)))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(531)
    ]]></attribute>
    <justification jstring="[para(18(a,1),30(a,1,2,2,1)),rewrite([29(5),31(8)])].">
      <j1 rule="para" parents="18 30"/>
      <j2 rule="rewrite" parents="29 31"/>
    </justification>
  </clause>

  <clause id="48">
    <literal><![CDATA[
      f(f(f(x,x),x),f(f(x,f(x,x)),f(f(x,x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(532)
    ]]></attribute>
    <justification jstring="[para(24(a,1),30(a,1,2,2,1)),rewrite([31(9)])].">
      <j1 rule="para" parents="24 30"/>
      <j2 rule="rewrite" parents="31"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      f(f(f(x,x),f(y,x)),f(f(f(f(y,x),f(y,x)),f(x,x)),f(f(x,f(y,x)),x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(533)
    ]]></attribute>
    <justification jstring="[para(29(a,1),22(a,1,2,2,1,1))].">
      <j1 rule="para" parents="29 22"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(x,x),f(x,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(534F)
    ]]></attribute>
    <justification jstring="[para(45(a,1),23(a,1,1)),rewrite([29(16),29(18),48(23),37(18),18(16)]),flip(a)].">
      <j1 rule="para" parents="45 23"/>
      <j2 rule="rewrite" parents="29 29 48 37 18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="51">
    <literal><![CDATA[
      f(f(f(x,x),x),f(x,f(f(x,x),y))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(535E)
    ]]></attribute>
    <justification jstring="[para(46(a,1),23(a,1,1)),rewrite([47(25),39(21),24(17),29(3)]),flip(a)].">
      <j1 rule="para" parents="46 23"/>
      <j2 rule="rewrite" parents="47 39 24 29"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      f(f(f(x,x),x),f(x,f(y,f(x,x)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(536)
    ]]></attribute>
    <justification jstring="[para(31(a,1),51(a,1,2,2))].">
      <j1 rule="para" parents="31 51"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      f(f(f(x,y),f(f(x,y),f(x,y))),f(f(f(x,y),f(x,y)),y)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(537)
    ]]></attribute>
    <justification jstring="[para(7(a,1),50(a,1,2,2))].">
      <j1 rule="para" parents="7 50"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      f(f(f(x,y),f(f(x,y),f(x,y))),f(f(f(x,y),f(x,y)),x)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(538)
    ]]></attribute>
    <justification jstring="[para(29(a,1),50(a,1,2,2))].">
      <j1 rule="para" parents="29 50"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      f(f(x,x),f(x,y)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(525)
    ]]></attribute>
    <justification jstring="[para(37(a,1),44(a,1,1)),rewrite([50(16),29(13),24(11),29(3)]),flip(a)].">
      <j1 rule="para" parents="37 44"/>
      <j2 rule="rewrite" parents="50 29 24 29"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="57">
    <literal><![CDATA[
      f(x,f(f(x,x),y)) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(541E)
    ]]></attribute>
    <justification jstring="[para(39(a,1),44(a,1,1)),rewrite([55(10),55(12),51(14),18(10)]),flip(a)].">
      <j1 rule="para" parents="39 44"/>
      <j2 rule="rewrite" parents="55 55 51 18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="58">
    <literal><![CDATA[
      f(x,f(y,f(x,x))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(542E)
    ]]></attribute>
    <justification jstring="[para(41(a,1),44(a,1,1)),rewrite([55(10),55(12),52(14),18(10)]),flip(a)].">
      <j1 rule="para" parents="41 44"/>
      <j2 rule="rewrite" parents="55 55 52 18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      f(f(f(x,y),f(x,y)),y) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(543E)
    ]]></attribute>
    <justification jstring="[para(38(a,1),44(a,1,1)),rewrite([53(26),55(23),24(18),55(7)]),flip(a)].">
      <j1 rule="para" parents="38 44"/>
      <j2 rule="rewrite" parents="53 55 24 55"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="60">
    <literal><![CDATA[
      f(f(f(x,y),f(x,y)),x) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(544E)
    ]]></attribute>
    <justification jstring="[para(40(a,1),44(a,1,1)),rewrite([54(26),55(23),24(18),55(7)]),flip(a)].">
      <j1 rule="para" parents="40 44"/>
      <j2 rule="rewrite" parents="54 55 24 55"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(f(f(f(y,x),f(y,z)),f(y,z)),y),f(u,y)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(546)
    ]]></attribute>
    <justification jstring="[para(57(a,1),33(a,1,2,1)),rewrite([55(4),55(5),55(7)])].">
      <j1 rule="para" parents="57 33"/>
      <j2 rule="rewrite" parents="55 55 55"/>
    </justification>
  </clause>

  <clause id="63">
    <literal><![CDATA[
      f(f(x,x),f(y,x)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(548)
    ]]></attribute>
    <justification jstring="[para(18(a,1),58(a,1,2,2)),rewrite([55(6)])].">
      <j1 rule="para" parents="18 58"/>
      <j2 rule="rewrite" parents="55"/>
    </justification>
  </clause>

  <clause id="64">
    <literal><![CDATA[
      f(x,f(f(f(f(y,x),f(y,x)),f(x,x)),f(f(x,f(y,x)),x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(549)
    ]]></attribute>
    <justification jstring="[back_rewrite(49),rewrite([63(3)])].">
      <j1 rule="back_rewrite" parents="49"/>
      <j2 rule="rewrite" parents="63"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      f(x,f(f(f(f(y,x),f(y,x)),z),f(f(f(f(x,z),z),f(y,x)),f(f(y,x),u)))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(550)
    ]]></attribute>
    <justification jstring="[back_rewrite(20),rewrite([63(3)])].">
      <j1 rule="back_rewrite" parents="20"/>
      <j2 rule="rewrite" parents="63"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      f(f(x,y),f(f(f(f(y,x),f(y,z)),f(y,z)),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(552)
    ]]></attribute>
    <justification jstring="[para(31(a,1),62(a,1,2))].">
      <j1 rule="para" parents="31 62"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      f(x,f(f(f(x,f(f(y,x),z)),f(f(y,x),z)),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(553)
    ]]></attribute>
    <justification jstring="[para(18(a,1),67(a,1,2,1,1,1)),rewrite([63(3)])].">
      <j1 rule="para" parents="18 67"/>
      <j2 rule="rewrite" parents="63"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      f(x,f(f(f(x,f(f(x,y),z)),f(f(x,y),z)),f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(554)
    ]]></attribute>
    <justification jstring="[para(29(a,1),67(a,1,2,1,1,1)),rewrite([55(3)])].">
      <j1 rule="para" parents="29 67"/>
      <j2 rule="rewrite" parents="55"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      f(x,f(f(f(x,x),x),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(557)
    ]]></attribute>
    <justification jstring="[para(7(a,1),68(a,1,2,1,1,2)),rewrite([7(12)])].">
      <j1 rule="para" parents="7 68"/>
      <j2 rule="rewrite" parents="7"/>
    </justification>
  </clause>

  <clause id="73">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(y,f(x,x)))) = f(y,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(558)
    ]]></attribute>
    <justification jstring="[para(24(a,1),68(a,1,2,1,1,2)),rewrite([55(4),24(4)])].">
      <j1 rule="para" parents="24 68"/>
      <j2 rule="rewrite" parents="55 24"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      f(x,f(f(f(x,y),y),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(559)
    ]]></attribute>
    <justification jstring="[para(29(a,1),68(a,1,2,1,1,2)),rewrite([29(4)])].">
      <j1 rule="para" parents="29 68"/>
      <j2 rule="rewrite" parents="29"/>
    </justification>
  </clause>

  <clause id="75">
    <literal><![CDATA[
      f(x,f(f(f(x,x),x),f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(560)
    ]]></attribute>
    <justification jstring="[para(60(a,1),72(a,1,2,2)),rewrite([60(9)])].">
      <j1 rule="para" parents="60 72"/>
      <j2 rule="rewrite" parents="60"/>
    </justification>
  </clause>

  <clause id="76">
    <literal><![CDATA[
      f(f(x,x),x) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(561)
    ]]></attribute>
    <justification jstring="[para(26(a,1),74(a,1,2,1)),rewrite([55(5)])].">
      <j1 rule="para" parents="26 74"/>
      <j2 rule="rewrite" parents="55"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      f(x,f(f(x,f(x,x)),f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(562)
    ]]></attribute>
    <justification jstring="[back_rewrite(75),rewrite([76(2)])].">
      <j1 rule="back_rewrite" parents="75"/>
      <j2 rule="rewrite" parents="76"/>
    </justification>
  </clause>

  <clause id="78">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(f(x,x),y))) = f(f(x,x),y)
    ]]></literal>
    <attribute><![CDATA[
      label(563)
    ]]></attribute>
    <justification jstring="[para(18(a,1),77(a,1,2,1,2)),rewrite([76(3)])].">
      <j1 rule="para" parents="18 77"/>
      <j2 rule="rewrite" parents="76"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      f(x,f(f(y,x),f(f(f(f(x,y),y),f(y,x)),f(f(y,x),z)))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(564)
    ]]></attribute>
    <justification jstring="[para(60(a,1),65(a,1,2,1))].">
      <j1 rule="para" parents="60 65"/>
    </justification>
  </clause>

  <clause id="80">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(x,f(f(x,f(x,x)),y)))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(565)
    ]]></attribute>
    <justification jstring="[para(26(a,1),79(a,1,2,2,1,1)),rewrite([55(7)])].">
      <j1 rule="para" parents="26 79"/>
      <j2 rule="rewrite" parents="55"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(x,f(x,x)),f(x,f(x,x)))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(567D)
    ]]></attribute>
    <justification jstring="[para(43(a,1),80(a,1,2,2,2)),rewrite([29(17),43(14),59(7)]),flip(a)].">
      <j1 rule="para" parents="43 80"/>
      <j2 rule="rewrite" parents="29 43 59"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="83">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(x,y))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(568)
    ]]></attribute>
    <justification jstring="[para(77(a,1),80(a,1,2,2))].">
      <j1 rule="para" parents="77 80"/>
    </justification>
  </clause>

  <clause id="84">
    <literal><![CDATA[
      f(x,f(f(y,f(y,y)),f(y,f(y,y)))) = f(y,f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(570C)
    ]]></attribute>
    <justification jstring="[para(82(a,1),73(a,1,2,1)),rewrite([58(14),55(11)]),flip(a)].">
      <j1 rule="para" parents="82 73"/>
      <j2 rule="rewrite" parents="58 55"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="87">
    <literal><![CDATA[
      f(f(x,f(x,x)),y) = f(y,y)
    ]]></literal>
    <attribute><![CDATA[
      label(573)
    ]]></attribute>
    <justification jstring="[para(84(a,1),26(a,1,1))].">
      <j1 rule="para" parents="84 26"/>
    </justification>
  </clause>

  <clause id="88">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(y,y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(574)
    ]]></attribute>
    <justification jstring="[para(84(a,1),29(a,1,1))].">
      <j1 rule="para" parents="84 29"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      f(x,f(y,f(y,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(575)
    ]]></attribute>
    <justification jstring="[para(84(a,1),57(a,1,2))].">
      <j1 rule="para" parents="84 57"/>
    </justification>
  </clause>

  <clause id="90">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(y,f(y,y)))) = f(y,f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(576)
    ]]></attribute>
    <justification jstring="[para(84(a,1),78(a,1,2,2)),rewrite([84(14)])].">
      <j1 rule="para" parents="84 78"/>
      <j2 rule="rewrite" parents="84"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      f(x,f(x,x)) = f(y,f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(577)
    ]]></attribute>
    <justification jstring="[para(84(a,1),83(a,1,2,2)),rewrite([90(7)])].">
      <j1 rule="para" parents="84 83"/>
      <j2 rule="rewrite" parents="90"/>
    </justification>
  </clause>

  <clause id="95">
    <literal><![CDATA[
      f(x,f(f(y,f(y,y)),f(z,x))) = f(z,x)
    ]]></literal>
    <attribute><![CDATA[
      label(581)
    ]]></attribute>
    <justification jstring="[para(87(a,2),31(a,1,2))].">
      <j1 rule="para" parents="87 31"/>
    </justification>
  </clause>

  <clause id="96">
    <literal><![CDATA[
      f(x,f(f(y,f(y,y)),f(x,z))) = f(x,z)
    ]]></literal>
    <attribute><![CDATA[
      label(582)
    ]]></attribute>
    <justification jstring="[para(87(a,2),32(a,1,2))].">
      <j1 rule="para" parents="87 32"/>
    </justification>
  </clause>

  <clause id="97">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),y),f(y,z)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(583)
    ]]></attribute>
    <justification jstring="[para(87(a,1),21(a,1,2,1,2)),rewrite([84(10),84(13),96(12)])].">
      <j1 rule="para" parents="87 21"/>
      <j2 rule="rewrite" parents="84 84 96"/>
    </justification>
  </clause>

  <clause id="101">
    <literal><![CDATA[
      f(f(x,x),f(f(x,y),f(x,y))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(590)
    ]]></attribute>
    <justification jstring="[para(87(a,1),83(a,1,2))].">
      <j1 rule="para" parents="87 83"/>
    </justification>
  </clause>

  <clause id="104">
    <literal><![CDATA[
      f(f(x,x),f(y,f(y,y))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(594)
    ]]></attribute>
    <justification jstring="[para(88(a,1),60(a,1,1,1)),rewrite([88(4),88(8)])].">
      <j1 rule="para" parents="88 60"/>
      <j2 rule="rewrite" parents="88 88"/>
    </justification>
  </clause>

  <clause id="105">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(y,f(y,y)),z)) = z
    ]]></literal>
    <attribute><![CDATA[
      label(595)
    ]]></attribute>
    <justification jstring="[para(87(a,2),88(a,1,2))].">
      <j1 rule="para" parents="87 88"/>
    </justification>
  </clause>

  <clause id="106">
    <literal><![CDATA[
      f(f(x,y),f(f(y,x),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(597)
    ]]></attribute>
    <justification jstring="[para(89(a,1),67(a,1,2,1,1)),rewrite([104(7)])].">
      <j1 rule="para" parents="89 67"/>
      <j2 rule="rewrite" parents="104"/>
    </justification>
  </clause>

  <clause id="107">
    <literal><![CDATA[
      f(x,f(x,f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(598)
    ]]></attribute>
    <justification jstring="[para(89(a,1),68(a,1,2,1,1)),rewrite([104(7)])].">
      <j1 rule="para" parents="89 68"/>
      <j2 rule="rewrite" parents="104"/>
    </justification>
  </clause>

  <clause id="108">
    <literal><![CDATA[
      f(x,f(x,f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(599)
    ]]></attribute>
    <justification jstring="[para(89(a,1),69(a,1,2,1,1)),rewrite([104(7)])].">
      <j1 rule="para" parents="89 69"/>
      <j2 rule="rewrite" parents="104"/>
    </justification>
  </clause>

  <clause id="114">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(f(f(f(f(y,z),y),y),y),f(y,u))),f(y,y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(607)
    ]]></attribute>
    <justification jstring="[para(91(a,1),34(a,1,1,1)),rewrite([55(6),55(7)])].">
      <j1 rule="para" parents="91 34"/>
      <j2 rule="rewrite" parents="55 55"/>
    </justification>
  </clause>

  <clause id="119">
    <literal><![CDATA[
      f(f(x,y),f(f(f(x,f(x,y)),x),f(x,f(x,y)))) = f(x,f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(616)
    ]]></attribute>
    <justification jstring="[para(108(a,1),106(a,1,1))].">
      <j1 rule="para" parents="108 106"/>
    </justification>
  </clause>

  <clause id="121">
    <literal><![CDATA[
      f(f(f(x,y),x),f(x,y)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(620)
    ]]></attribute>
    <justification jstring="[para(97(a,1),64(a,1,2,1,1,1)),rewrite([97(6),101(6),97(9),95(9),97(9)])].">
      <j1 rule="para" parents="97 64"/>
      <j2 rule="rewrite" parents="97 101 97 95 97"/>
    </justification>
  </clause>

  <clause id="122">
    <literal><![CDATA[
      f(f(x,y),x) = f(x,f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(623)
    ]]></attribute>
    <justification jstring="[back_rewrite(119),rewrite([121(7)])].">
      <j1 rule="back_rewrite" parents="119"/>
      <j2 rule="rewrite" parents="121"/>
    </justification>
  </clause>

  <clause id="125">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(f(y,f(y,z)),f(y,u))),f(y,y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(626)
    ]]></attribute>
    <justification jstring="[back_rewrite(114),rewrite([122(4),122(5),108(5),122(4)])].">
      <j1 rule="back_rewrite" parents="114"/>
      <j2 rule="rewrite" parents="122 122 108 122"/>
    </justification>
  </clause>

  <clause id="141">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(f(y,z),f(z,u))),f(z,z)) = z
    ]]></literal>
    <attribute><![CDATA[
      label(644)
    ]]></attribute>
    <justification jstring="[para(107(a,1),125(a,1,1,2,1))].">
      <j1 rule="para" parents="107 125"/>
    </justification>
  </clause>

  <clause id="142">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(f(y,z),f(u,z))),f(z,z)) = z
    ]]></literal>
    <attribute><![CDATA[
      label(646)
    ]]></attribute>
    <justification jstring="[para(31(a,1),141(a,1,1,2,2))].">
      <j1 rule="para" parents="31 141"/>
    </justification>
  </clause>

  <clause id="144">
    <literal><![CDATA[
      f(f(f(x,y),f(z,y)),y) = f(f(u,f(u,u)),f(f(x,y),f(z,y)))
    ]]></literal>
    <attribute><![CDATA[
      label(649)
    ]]></attribute>
    <justification jstring="[para(142(a,1),97(a,1,2)),rewrite([105(9)])].">
      <j1 rule="para" parents="142 97"/>
      <j2 rule="rewrite" parents="105"/>
    </justification>
  </clause>

  <clause id="154">
    <literal><![CDATA[
      f(f(f(x,y),f(z,y)),y) = f(y,f(f(x,y),f(z,y)))
    ]]></literal>
    <attribute><![CDATA[
      label(664B)
    ]]></attribute>
    <justification jstring="[para(144(a,1),31(a,1,2,1)),rewrite([97(11)]),flip(a)].">
      <j1 rule="para" parents="144 31"/>
      <j2 rule="rewrite" parents="97"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="156">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,y),f(x,y)))) = f(f(z,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(666)
    ]]></attribute>
    <justification jstring="[para(144(a,2),95(a,1,2)),rewrite([154(5)])].">
      <j1 rule="para" parents="144 95"/>
      <j2 rule="rewrite" parents="154"/>
    </justification>
  </clause>

  <clause id="158">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(y,z),f(x,y)))) = f(f(y,z),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(668)
    ]]></attribute>
    <justification jstring="[para(60(a,1),156(a,1,2,2,1)),rewrite([60(10)])].">
      <j1 rule="para" parents="60 156"/>
      <j2 rule="rewrite" parents="60"/>
    </justification>
  </clause>

  <clause id="159">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(z,f(x,y)),x))) = f(f(z,f(x,y)),x)
    ]]></literal>
    <attribute><![CDATA[
      label(669)
    ]]></attribute>
    <justification jstring="[para(97(a,1),156(a,1,1)),rewrite([97(8),97(13)])].">
      <j1 rule="para" parents="97 156"/>
      <j2 rule="rewrite" parents="97 97"/>
    </justification>
  </clause>

  <clause id="160">
    <literal><![CDATA[
      f(x,f(f(x,y),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(670)
    ]]></attribute>
    <justification jstring="[para(63(a,1),159(a,1,2,2,1)),rewrite([63(7)])].">
      <j1 rule="para" parents="63 159"/>
      <j2 rule="rewrite" parents="63"/>
    </justification>
  </clause>

  <clause id="161">
    <literal><![CDATA[
      f(f(x,y),f(y,x)) = f(f(y,x),f(y,x))
    ]]></literal>
    <attribute><![CDATA[
      label(671A)
    ]]></attribute>
    <justification jstring="[para(160(a,1),158(a,1,2)),flip(a)].">
      <j1 rule="para" parents="160 158"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="162">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(z,y),f(y,z))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(672)
    ]]></attribute>
    <justification jstring="[para(161(a,2),18(a,1,2))].">
      <j1 rule="para" parents="161 18"/>
    </justification>
  </clause>

  <clause id="164">
    <literal><![CDATA[
      f(x,y) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(674B)
    ]]></attribute>
    <justification jstring="[para(161(a,1),161(a,1,1)),rewrite([55(7),162(8)])].">
      <j1 rule="para" parents="161 161"/>
      <j2 rule="rewrite" parents="55 162"/>
    </justification>
  </clause>

  <clause id="165">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(comm)
    ]]></attribute>
    <justification jstring="[resolve(164,a,8,a)].">
      <j1 rule="resolve" parents="164 8"/>
    </justification>
  </clause>

</proof>

<proof number="5" length="169" max_count="41">

<comments><![CDATA[
% Proof 5 at 2.45 (+ 0.01) seconds: assoc.
% Length of proof is 169.
% Level of proof is 63.
% Maximum clause weight is 1009.
% Given clauses 213.
]]></comments>

  <clause id="5" type="goal">
    <literal><![CDATA[
      f(x,f(f(y,z),f(y,z))) = f(z,f(f(y,x),f(y,x)))
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="7" type="assumption">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(y,u)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(MOL_sax)
    ]]></attribute>
    <attribute><![CDATA[
      label(492)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="12" type="deny">
    <literal><![CDATA[
      f(c9,f(f(c10,c11),f(c10,c11))) != f(c11,f(f(c10,c9),f(c10,c9)))
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc)
    ]]></attribute>
    <justification jstring="[deny(5)].">
      <j1 rule="deny" parents="5"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),u),f(f(f(f(f(f(y,z),x),u),u),f(y,z)),z))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(499)
    ]]></attribute>
    <justification jstring="[para(7(a,1),7(a,1,2,2,2))].">
      <j1 rule="para" parents="7 7"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(f(f(y,y),u),f(f(f(f(f(y,v),u),u),y),f(y,w)))))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(500)
    ]]></attribute>
    <justification jstring="[para(7(a,1),15(a,1,1,2)),rewrite([7(12),7(12),7(14),7(17),7(30)])].">
      <j1 rule="para" parents="7 15"/>
      <j2 rule="rewrite" parents="7 7 7 7 7"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(508A)
    ]]></attribute>
    <justification jstring="[para(7(a,1),16(a,1,2,2))].">
      <j1 rule="para" parents="7 16"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      f(f(x,y),f(y,y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(508B)
    ]]></attribute>
    <justification jstring="[para(7(a,1),17(a,1,2,1))].">
      <j1 rule="para" parents="7 17"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(f(f(f(y,x),f(y,y)),f(y,y)),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(503)
    ]]></attribute>
    <justification jstring="[para(18(a,1),7(a,1,2,1))].">
      <j1 rule="para" parents="18 7"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      f(f(f(x,x),f(y,x)),f(f(f(f(y,x),f(y,x)),z),f(f(f(f(x,z),z),f(y,x)),f(f(y,x),u)))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(504)
    ]]></attribute>
    <justification jstring="[para(18(a,1),7(a,1,2,2,1,1,1,1))].">
      <j1 rule="para" parents="18 7"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),f(x,x)),f(f(f(x,f(x,x)),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(583A)
    ]]></attribute>
    <justification jstring="[para(18(a,1),7(a,1,2,2,1,1,1))].">
      <j1 rule="para" parents="18 7"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      f(f(f(x,x),f(y,x)),f(f(f(f(y,x),f(y,x)),z),f(f(f(f(x,z),z),f(y,x)),x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(506)
    ]]></attribute>
    <justification jstring="[para(18(a,1),15(a,1,2,2,1,1,1,1))].">
      <j1 rule="para" parents="18 15"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),f(x,x)),f(f(f(x,f(x,x)),f(y,z)),z))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(586A)
    ]]></attribute>
    <justification jstring="[para(18(a,1),15(a,1,2,2,1,1,1))].">
      <j1 rule="para" parents="18 15"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      f(f(x,f(y,y)),y) = f(y,y)
    ]]></literal>
    <attribute><![CDATA[
      label(508)
    ]]></attribute>
    <justification jstring="[para(18(a,1),15(a,1,2,2,1)),rewrite([18(5),18(5)])].">
      <j1 rule="para" parents="18 15"/>
      <j2 rule="rewrite" parents="18 18"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      f(f(f(x,x),y),x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(510B)
    ]]></attribute>
    <justification jstring="[para(17(a,1),18(a,1,1)),rewrite([17(7)]),flip(a)].">
      <j1 rule="para" parents="17 18"/>
      <j2 rule="rewrite" parents="17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(y,y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(511)
    ]]></attribute>
    <justification jstring="[back_rewrite(19),rewrite([24(7)])].">
      <j1 rule="back_rewrite" parents="19"/>
      <j2 rule="rewrite" parents="24"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      f(f(x,x),f(f(f(x,x),x),f(f(f(x,x),x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(512)
    ]]></attribute>
    <justification jstring="[para(26(a,1),7(a,1,2,2,1,1))].">
      <j1 rule="para" parents="26 7"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      f(f(x,y),f(x,x)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(513)
    ]]></attribute>
    <justification jstring="[para(18(a,1),26(a,1,1,1)),rewrite([18(6)])].">
      <j1 rule="para" parents="18 26"/>
      <j2 rule="rewrite" parents="18"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),f(f(y,x),f(y,x))),f(f(f(y,x),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(514)
    ]]></attribute>
    <justification jstring="[para(29(a,1),7(a,1,2,2,1,1))].">
      <j1 rule="para" parents="29 7"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      f(x,f(f(y,x),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(515)
    ]]></attribute>
    <justification jstring="[para(7(a,1),29(a,1,1))].">
      <j1 rule="para" parents="7 29"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      f(x,f(f(x,y),f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(516)
    ]]></attribute>
    <justification jstring="[para(29(a,1),29(a,1,1))].">
      <j1 rule="para" parents="29 29"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(u,y)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(517)
    ]]></attribute>
    <justification jstring="[para(31(a,1),7(a,1,2,2,2))].">
      <j1 rule="para" parents="31 7"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      f(f(f(f(x,x),y),f(f(f(f(f(x,z),y),y),x),f(x,u))),f(x,x)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(518)
    ]]></attribute>
    <justification jstring="[para(7(a,1),31(a,1,2,1)),rewrite([7(20),7(22)])].">
      <j1 rule="para" parents="7 31"/>
      <j2 rule="rewrite" parents="7 7"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      f(f(x,f(f(x,x),f(x,y))),f(x,f(f(x,x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(519A)
    ]]></attribute>
    <justification jstring="[para(27(a,1),26(a,1)),flip(a)].">
      <j1 rule="para" parents="27 26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      f(f(x,x),f(f(f(x,x),x),f(f(f(x,x),x),f(y,x)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(520)
    ]]></attribute>
    <justification jstring="[para(31(a,1),28(a,1,2,2,2))].">
      <j1 rule="para" parents="31 28"/>
    </justification>
  </clause>

  <clause id="37">
    <literal><![CDATA[
      f(x,f(f(x,x),f(x,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(521B)
    ]]></attribute>
    <justification jstring="[para(35(a,1),18(a,1,1)),rewrite([35(9)]),flip(a)].">
      <j1 rule="para" parents="35 18"/>
      <j2 rule="rewrite" parents="35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      f(f(x,y),f(f(f(x,y),f(x,y)),y)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(522)
    ]]></attribute>
    <justification jstring="[para(7(a,1),37(a,1,2,2))].">
      <j1 rule="para" parents="7 37"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      f(f(x,x),f(x,f(f(x,x),y))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(523)
    ]]></attribute>
    <justification jstring="[para(18(a,1),37(a,1,2,1)),rewrite([29(8)])].">
      <j1 rule="para" parents="18 37"/>
      <j2 rule="rewrite" parents="29"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      f(f(x,y),f(f(f(x,y),f(x,y)),x)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(524)
    ]]></attribute>
    <justification jstring="[para(29(a,1),37(a,1,2,2))].">
      <j1 rule="para" parents="29 37"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      f(f(x,x),f(x,f(y,f(x,x)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(525)
    ]]></attribute>
    <justification jstring="[para(31(a,1),39(a,1,2,2))].">
      <j1 rule="para" parents="31 39"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      f(x,f(f(x,f(x,x)),f(f(x,f(x,x)),f(y,f(x,x))))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(526)
    ]]></attribute>
    <justification jstring="[para(18(a,1),36(a,1,1)),rewrite([29(3),29(5)])].">
      <j1 rule="para" parents="18 36"/>
      <j2 rule="rewrite" parents="29 29"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(f(x,f(x,x)),f(y,f(x,x)))),x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(527)
    ]]></attribute>
    <justification jstring="[para(42(a,1),31(a,1,2,1)),rewrite([42(18),29(11),42(18)])].">
      <j1 rule="para" parents="42 31"/>
      <j2 rule="rewrite" parents="42 29 42"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),f(x,x)),f(f(f(x,f(x,x)),f(y,z)),y))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(528)
    ]]></attribute>
    <justification jstring="[para(29(a,1),21(a,1,2,2,2))].">
      <j1 rule="para" parents="29 21"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(f(x,x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(529)
    ]]></attribute>
    <justification jstring="[para(43(a,1),21(a,1,2,2,1)),rewrite([24(3),31(8)])].">
      <j1 rule="para" parents="43 21"/>
      <j2 rule="rewrite" parents="24 31"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      f(x,f(f(f(x,x),x),f(x,f(f(x,x),y)))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(530)
    ]]></attribute>
    <justification jstring="[para(18(a,1),45(a,1,1)),rewrite([29(4),29(5)])].">
      <j1 rule="para" parents="18 45"/>
      <j2 rule="rewrite" parents="29 29"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(f(x,x),x),f(x,f(f(x,x),y)))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(531)
    ]]></attribute>
    <justification jstring="[para(18(a,1),30(a,1,2,2,1)),rewrite([29(5),31(8)])].">
      <j1 rule="para" parents="18 30"/>
      <j2 rule="rewrite" parents="29 31"/>
    </justification>
  </clause>

  <clause id="48">
    <literal><![CDATA[
      f(f(f(x,x),x),f(f(x,f(x,x)),f(f(x,x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(532)
    ]]></attribute>
    <justification jstring="[para(24(a,1),30(a,1,2,2,1)),rewrite([31(9)])].">
      <j1 rule="para" parents="24 30"/>
      <j2 rule="rewrite" parents="31"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      f(f(f(x,x),f(y,x)),f(f(f(f(y,x),f(y,x)),f(x,x)),f(f(x,f(y,x)),x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(533)
    ]]></attribute>
    <justification jstring="[para(29(a,1),22(a,1,2,2,1,1))].">
      <j1 rule="para" parents="29 22"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(x,x),f(x,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(534F)
    ]]></attribute>
    <justification jstring="[para(45(a,1),23(a,1,1)),rewrite([29(16),29(18),48(23),37(18),18(16)]),flip(a)].">
      <j1 rule="para" parents="45 23"/>
      <j2 rule="rewrite" parents="29 29 48 37 18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="51">
    <literal><![CDATA[
      f(f(f(x,x),x),f(x,f(f(x,x),y))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(535E)
    ]]></attribute>
    <justification jstring="[para(46(a,1),23(a,1,1)),rewrite([47(25),39(21),24(17),29(3)]),flip(a)].">
      <j1 rule="para" parents="46 23"/>
      <j2 rule="rewrite" parents="47 39 24 29"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      f(f(f(x,x),x),f(x,f(y,f(x,x)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(536)
    ]]></attribute>
    <justification jstring="[para(31(a,1),51(a,1,2,2))].">
      <j1 rule="para" parents="31 51"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      f(f(f(x,y),f(f(x,y),f(x,y))),f(f(f(x,y),f(x,y)),y)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(537)
    ]]></attribute>
    <justification jstring="[para(7(a,1),50(a,1,2,2))].">
      <j1 rule="para" parents="7 50"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      f(f(f(x,y),f(f(x,y),f(x,y))),f(f(f(x,y),f(x,y)),x)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(538)
    ]]></attribute>
    <justification jstring="[para(29(a,1),50(a,1,2,2))].">
      <j1 rule="para" parents="29 50"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      f(f(x,x),f(x,y)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(525)
    ]]></attribute>
    <justification jstring="[para(37(a,1),44(a,1,1)),rewrite([50(16),29(13),24(11),29(3)]),flip(a)].">
      <j1 rule="para" parents="37 44"/>
      <j2 rule="rewrite" parents="50 29 24 29"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="57">
    <literal><![CDATA[
      f(x,f(f(x,x),y)) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(541E)
    ]]></attribute>
    <justification jstring="[para(39(a,1),44(a,1,1)),rewrite([55(10),55(12),51(14),18(10)]),flip(a)].">
      <j1 rule="para" parents="39 44"/>
      <j2 rule="rewrite" parents="55 55 51 18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="58">
    <literal><![CDATA[
      f(x,f(y,f(x,x))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(542E)
    ]]></attribute>
    <justification jstring="[para(41(a,1),44(a,1,1)),rewrite([55(10),55(12),52(14),18(10)]),flip(a)].">
      <j1 rule="para" parents="41 44"/>
      <j2 rule="rewrite" parents="55 55 52 18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      f(f(f(x,y),f(x,y)),y) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(543E)
    ]]></attribute>
    <justification jstring="[para(38(a,1),44(a,1,1)),rewrite([53(26),55(23),24(18),55(7)]),flip(a)].">
      <j1 rule="para" parents="38 44"/>
      <j2 rule="rewrite" parents="53 55 24 55"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="60">
    <literal><![CDATA[
      f(f(f(x,y),f(x,y)),x) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(544E)
    ]]></attribute>
    <justification jstring="[para(40(a,1),44(a,1,1)),rewrite([54(26),55(23),24(18),55(7)]),flip(a)].">
      <j1 rule="para" parents="40 44"/>
      <j2 rule="rewrite" parents="54 55 24 55"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(f(f(f(y,x),f(y,z)),f(y,z)),y),f(u,y)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(546)
    ]]></attribute>
    <justification jstring="[para(57(a,1),33(a,1,2,1)),rewrite([55(4),55(5),55(7)])].">
      <j1 rule="para" parents="57 33"/>
      <j2 rule="rewrite" parents="55 55 55"/>
    </justification>
  </clause>

  <clause id="63">
    <literal><![CDATA[
      f(f(x,x),f(y,x)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(548)
    ]]></attribute>
    <justification jstring="[para(18(a,1),58(a,1,2,2)),rewrite([55(6)])].">
      <j1 rule="para" parents="18 58"/>
      <j2 rule="rewrite" parents="55"/>
    </justification>
  </clause>

  <clause id="64">
    <literal><![CDATA[
      f(x,f(f(f(f(y,x),f(y,x)),f(x,x)),f(f(x,f(y,x)),x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(549)
    ]]></attribute>
    <justification jstring="[back_rewrite(49),rewrite([63(3)])].">
      <j1 rule="back_rewrite" parents="49"/>
      <j2 rule="rewrite" parents="63"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      f(x,f(f(f(f(y,x),f(y,x)),z),f(f(f(f(x,z),z),f(y,x)),f(f(y,x),u)))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(550)
    ]]></attribute>
    <justification jstring="[back_rewrite(20),rewrite([63(3)])].">
      <j1 rule="back_rewrite" parents="20"/>
      <j2 rule="rewrite" parents="63"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(x,y),z),z),y),f(y,u)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(551)
    ]]></attribute>
    <justification jstring="[para(59(a,1),7(a,1,1)),rewrite([31(7)])].">
      <j1 rule="para" parents="59 7"/>
      <j2 rule="rewrite" parents="31"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      f(f(x,y),f(f(f(f(y,x),f(y,z)),f(y,z)),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(552)
    ]]></attribute>
    <justification jstring="[para(31(a,1),62(a,1,2))].">
      <j1 rule="para" parents="31 62"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      f(x,f(f(f(x,f(f(y,x),z)),f(f(y,x),z)),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(553)
    ]]></attribute>
    <justification jstring="[para(18(a,1),67(a,1,2,1,1,1)),rewrite([63(3)])].">
      <j1 rule="para" parents="18 67"/>
      <j2 rule="rewrite" parents="63"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      f(x,f(f(f(x,f(f(x,y),z)),f(f(x,y),z)),f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(554)
    ]]></attribute>
    <justification jstring="[para(29(a,1),67(a,1,2,1,1,1)),rewrite([55(3)])].">
      <j1 rule="para" parents="29 67"/>
      <j2 rule="rewrite" parents="55"/>
    </justification>
  </clause>

  <clause id="70">
    <literal><![CDATA[
      f(f(x,y),f(f(f(f(x,y),f(y,z)),f(y,z)),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(555)
    ]]></attribute>
    <justification jstring="[para(31(a,1),67(a,1,2,1,1,1)),rewrite([59(4)])].">
      <j1 rule="para" parents="31 67"/>
      <j2 rule="rewrite" parents="59"/>
    </justification>
  </clause>

  <clause id="71">
    <literal><![CDATA[
      f(f(x,y),f(f(f(f(x,y),f(z,y)),f(z,y)),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(556)
    ]]></attribute>
    <justification jstring="[para(31(a,1),70(a,1,2,1,1,2)),rewrite([31(8)])].">
      <j1 rule="para" parents="31 70"/>
      <j2 rule="rewrite" parents="31"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      f(x,f(f(f(x,x),x),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(557)
    ]]></attribute>
    <justification jstring="[para(7(a,1),68(a,1,2,1,1,2)),rewrite([7(12)])].">
      <j1 rule="para" parents="7 68"/>
      <j2 rule="rewrite" parents="7"/>
    </justification>
  </clause>

  <clause id="73">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(y,f(x,x)))) = f(y,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(558)
    ]]></attribute>
    <justification jstring="[para(24(a,1),68(a,1,2,1,1,2)),rewrite([55(4),24(4)])].">
      <j1 rule="para" parents="24 68"/>
      <j2 rule="rewrite" parents="55 24"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      f(x,f(f(f(x,y),y),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(559)
    ]]></attribute>
    <justification jstring="[para(29(a,1),68(a,1,2,1,1,2)),rewrite([29(4)])].">
      <j1 rule="para" parents="29 68"/>
      <j2 rule="rewrite" parents="29"/>
    </justification>
  </clause>

  <clause id="75">
    <literal><![CDATA[
      f(x,f(f(f(x,x),x),f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(560)
    ]]></attribute>
    <justification jstring="[para(60(a,1),72(a,1,2,2)),rewrite([60(9)])].">
      <j1 rule="para" parents="60 72"/>
      <j2 rule="rewrite" parents="60"/>
    </justification>
  </clause>

  <clause id="76">
    <literal><![CDATA[
      f(f(x,x),x) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(561)
    ]]></attribute>
    <justification jstring="[para(26(a,1),74(a,1,2,1)),rewrite([55(5)])].">
      <j1 rule="para" parents="26 74"/>
      <j2 rule="rewrite" parents="55"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      f(x,f(f(x,f(x,x)),f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(562)
    ]]></attribute>
    <justification jstring="[back_rewrite(75),rewrite([76(2)])].">
      <j1 rule="back_rewrite" parents="75"/>
      <j2 rule="rewrite" parents="76"/>
    </justification>
  </clause>

  <clause id="78">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(f(x,x),y))) = f(f(x,x),y)
    ]]></literal>
    <attribute><![CDATA[
      label(563)
    ]]></attribute>
    <justification jstring="[para(18(a,1),77(a,1,2,1,2)),rewrite([76(3)])].">
      <j1 rule="para" parents="18 77"/>
      <j2 rule="rewrite" parents="76"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      f(x,f(f(y,x),f(f(f(f(x,y),y),f(y,x)),f(f(y,x),z)))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(564)
    ]]></attribute>
    <justification jstring="[para(60(a,1),65(a,1,2,1))].">
      <j1 rule="para" parents="60 65"/>
    </justification>
  </clause>

  <clause id="80">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(x,f(f(x,f(x,x)),y)))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(565)
    ]]></attribute>
    <justification jstring="[para(26(a,1),79(a,1,2,2,1,1)),rewrite([55(7)])].">
      <j1 rule="para" parents="26 79"/>
      <j2 rule="rewrite" parents="55"/>
    </justification>
  </clause>

  <clause id="81">
    <literal><![CDATA[
      f(x,f(f(y,x),f(f(f(f(x,y),y),f(y,x)),f(z,f(y,x))))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(566)
    ]]></attribute>
    <justification jstring="[para(31(a,1),79(a,1,2,2,2))].">
      <j1 rule="para" parents="31 79"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(x,f(x,x)),f(x,f(x,x)))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(567D)
    ]]></attribute>
    <justification jstring="[para(43(a,1),80(a,1,2,2,2)),rewrite([29(17),43(14),59(7)]),flip(a)].">
      <j1 rule="para" parents="43 80"/>
      <j2 rule="rewrite" parents="29 43 59"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="83">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(x,y))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(568)
    ]]></attribute>
    <justification jstring="[para(77(a,1),80(a,1,2,2))].">
      <j1 rule="para" parents="77 80"/>
    </justification>
  </clause>

  <clause id="84">
    <literal><![CDATA[
      f(x,f(f(y,f(y,y)),f(y,f(y,y)))) = f(y,f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(570C)
    ]]></attribute>
    <justification jstring="[para(82(a,1),73(a,1,2,1)),rewrite([58(14),55(11)]),flip(a)].">
      <j1 rule="para" parents="82 73"/>
      <j2 rule="rewrite" parents="58 55"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="87">
    <literal><![CDATA[
      f(f(x,f(x,x)),y) = f(y,y)
    ]]></literal>
    <attribute><![CDATA[
      label(573)
    ]]></attribute>
    <justification jstring="[para(84(a,1),26(a,1,1))].">
      <j1 rule="para" parents="84 26"/>
    </justification>
  </clause>

  <clause id="88">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(y,y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(574)
    ]]></attribute>
    <justification jstring="[para(84(a,1),29(a,1,1))].">
      <j1 rule="para" parents="84 29"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      f(x,f(y,f(y,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(575)
    ]]></attribute>
    <justification jstring="[para(84(a,1),57(a,1,2))].">
      <j1 rule="para" parents="84 57"/>
    </justification>
  </clause>

  <clause id="90">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(y,f(y,y)))) = f(y,f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(576)
    ]]></attribute>
    <justification jstring="[para(84(a,1),78(a,1,2,2)),rewrite([84(14)])].">
      <j1 rule="para" parents="84 78"/>
      <j2 rule="rewrite" parents="84"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      f(x,f(x,x)) = f(y,f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(577)
    ]]></attribute>
    <justification jstring="[para(84(a,1),83(a,1,2,2)),rewrite([90(7)])].">
      <j1 rule="para" parents="84 83"/>
      <j2 rule="rewrite" parents="90"/>
    </justification>
  </clause>

  <clause id="93">
    <literal><![CDATA[
      f(f(x,y),f(f(z,f(z,z)),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(579)
    ]]></attribute>
    <justification jstring="[para(87(a,2),18(a,1,2))].">
      <j1 rule="para" parents="87 18"/>
    </justification>
  </clause>

  <clause id="95">
    <literal><![CDATA[
      f(x,f(f(y,f(y,y)),f(z,x))) = f(z,x)
    ]]></literal>
    <attribute><![CDATA[
      label(581)
    ]]></attribute>
    <justification jstring="[para(87(a,2),31(a,1,2))].">
      <j1 rule="para" parents="87 31"/>
    </justification>
  </clause>

  <clause id="96">
    <literal><![CDATA[
      f(x,f(f(y,f(y,y)),f(x,z))) = f(x,z)
    ]]></literal>
    <attribute><![CDATA[
      label(582)
    ]]></attribute>
    <justification jstring="[para(87(a,2),32(a,1,2))].">
      <j1 rule="para" parents="87 32"/>
    </justification>
  </clause>

  <clause id="97">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),y),f(y,z)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(583)
    ]]></attribute>
    <justification jstring="[para(87(a,1),21(a,1,2,1,2)),rewrite([84(10),84(13),96(12)])].">
      <j1 rule="para" parents="87 21"/>
      <j2 rule="rewrite" parents="84 84 96"/>
    </justification>
  </clause>

  <clause id="98">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(u,f(u,u)),f(f(f(f(y,x),z),z),y)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(585)
    ]]></attribute>
    <justification jstring="[para(87(a,2),33(a,1,2,2))].">
      <j1 rule="para" parents="87 33"/>
    </justification>
  </clause>

  <clause id="99">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(y,z)),z) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(586)
    ]]></attribute>
    <justification jstring="[para(87(a,1),23(a,1,2,1,2)),rewrite([84(13),84(14),96(13)])].">
      <j1 rule="para" parents="87 23"/>
      <j2 rule="rewrite" parents="84 84 96"/>
    </justification>
  </clause>

  <clause id="100">
    <literal><![CDATA[
      f(x,f(f(y,f(y,y)),z)) = f(x,f(z,z))
    ]]></literal>
    <attribute><![CDATA[
      label(589B)
    ]]></attribute>
    <justification jstring="[para(87(a,2),73(a,2,2)),rewrite([95(7)]),flip(a)].">
      <j1 rule="para" parents="87 73"/>
      <j2 rule="rewrite" parents="95"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="101">
    <literal><![CDATA[
      f(f(x,x),f(f(x,y),f(x,y))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(590)
    ]]></attribute>
    <justification jstring="[para(87(a,1),83(a,1,2))].">
      <j1 rule="para" parents="87 83"/>
    </justification>
  </clause>

  <clause id="104">
    <literal><![CDATA[
      f(f(x,x),f(y,f(y,y))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(594)
    ]]></attribute>
    <justification jstring="[para(88(a,1),60(a,1,1,1)),rewrite([88(4),88(8)])].">
      <j1 rule="para" parents="88 60"/>
      <j2 rule="rewrite" parents="88 88"/>
    </justification>
  </clause>

  <clause id="105">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(y,f(y,y)),z)) = z
    ]]></literal>
    <attribute><![CDATA[
      label(595)
    ]]></attribute>
    <justification jstring="[para(87(a,2),88(a,1,2))].">
      <j1 rule="para" parents="87 88"/>
    </justification>
  </clause>

  <clause id="106">
    <literal><![CDATA[
      f(f(x,y),f(f(y,x),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(597)
    ]]></attribute>
    <justification jstring="[para(89(a,1),67(a,1,2,1,1)),rewrite([104(7)])].">
      <j1 rule="para" parents="89 67"/>
      <j2 rule="rewrite" parents="104"/>
    </justification>
  </clause>

  <clause id="107">
    <literal><![CDATA[
      f(x,f(x,f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(598)
    ]]></attribute>
    <justification jstring="[para(89(a,1),68(a,1,2,1,1)),rewrite([104(7)])].">
      <j1 rule="para" parents="89 68"/>
      <j2 rule="rewrite" parents="104"/>
    </justification>
  </clause>

  <clause id="108">
    <literal><![CDATA[
      f(x,f(x,f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(599)
    ]]></attribute>
    <justification jstring="[para(89(a,1),69(a,1,2,1,1)),rewrite([104(7)])].">
      <j1 rule="para" parents="89 69"/>
      <j2 rule="rewrite" parents="104"/>
    </justification>
  </clause>

  <clause id="110">
    <literal><![CDATA[
      f(x,f(f(y,x),f(f(x,f(y,x)),f(f(y,x),z)))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(601)
    ]]></attribute>
    <justification jstring="[para(89(a,1),65(a,1,2,2,1,1,1)),rewrite([104(6),104(5)])].">
      <j1 rule="para" parents="89 65"/>
      <j2 rule="rewrite" parents="104 104"/>
    </justification>
  </clause>

  <clause id="113">
    <literal><![CDATA[
      f(f(x,y),f(f(z,f(z,z)),f(f(f(f(f(y,x),y),y),y),f(u,y)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(606)
    ]]></attribute>
    <justification jstring="[para(91(a,1),33(a,1,2,1)),rewrite([55(7),55(8)])].">
      <j1 rule="para" parents="91 33"/>
      <j2 rule="rewrite" parents="55 55"/>
    </justification>
  </clause>

  <clause id="114">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(f(f(f(f(y,z),y),y),y),f(y,u))),f(y,y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(607)
    ]]></attribute>
    <justification jstring="[para(91(a,1),34(a,1,1,1)),rewrite([55(6),55(7)])].">
      <j1 rule="para" parents="91 34"/>
      <j2 rule="rewrite" parents="55 55"/>
    </justification>
  </clause>

  <clause id="116">
    <literal><![CDATA[
      f(f(x,f(y,f(y,y))),f(z,f(z,z))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(613)
    ]]></attribute>
    <justification jstring="[para(89(a,2),104(a,1,1))].">
      <j1 rule="para" parents="89 104"/>
    </justification>
  </clause>

  <clause id="119">
    <literal><![CDATA[
      f(f(x,y),f(f(f(x,f(x,y)),x),f(x,f(x,y)))) = f(x,f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(616)
    ]]></attribute>
    <justification jstring="[para(108(a,1),106(a,1,1))].">
      <j1 rule="para" parents="108 106"/>
    </justification>
  </clause>

  <clause id="121">
    <literal><![CDATA[
      f(f(f(x,y),x),f(x,y)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(620)
    ]]></attribute>
    <justification jstring="[para(97(a,1),64(a,1,2,1,1,1)),rewrite([97(6),101(6),97(9),95(9),97(9)])].">
      <j1 rule="para" parents="97 64"/>
      <j2 rule="rewrite" parents="97 101 97 95 97"/>
    </justification>
  </clause>

  <clause id="122">
    <literal><![CDATA[
      f(f(x,y),x) = f(x,f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(623)
    ]]></attribute>
    <justification jstring="[back_rewrite(119),rewrite([121(7)])].">
      <j1 rule="back_rewrite" parents="119"/>
      <j2 rule="rewrite" parents="121"/>
    </justification>
  </clause>

  <clause id="125">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(f(y,f(y,z)),f(y,u))),f(y,y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(626)
    ]]></attribute>
    <justification jstring="[back_rewrite(114),rewrite([122(4),122(5),108(5),122(4)])].">
      <j1 rule="back_rewrite" parents="114"/>
      <j2 rule="rewrite" parents="122 122 108 122"/>
    </justification>
  </clause>

  <clause id="126">
    <literal><![CDATA[
      f(f(x,y),f(f(z,f(z,z)),f(f(y,f(y,x)),f(u,y)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(627)
    ]]></attribute>
    <justification jstring="[back_rewrite(113),rewrite([122(5),122(6),108(6),122(5)])].">
      <j1 rule="back_rewrite" parents="113"/>
      <j2 rule="rewrite" parents="122 122 108 122"/>
    </justification>
  </clause>

  <clause id="127">
    <literal><![CDATA[
      f(f(x,y),y) = f(y,f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(628B)
    ]]></attribute>
    <justification jstring="[para(7(a,1),122(a,1,1)),rewrite([7(14)]),flip(a)].">
      <j1 rule="para" parents="7 122"/>
      <j2 rule="rewrite" parents="7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="128">
    <literal><![CDATA[
      f(f(x,y),f(y,f(x,y))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(629)
    ]]></attribute>
    <justification jstring="[para(122(a,1),71(a,1,2,1,1)),rewrite([127(2),127(5),122(8),107(8),122(5),107(5),127(4)])].">
      <j1 rule="para" parents="122 71"/>
      <j2 rule="rewrite" parents="127 127 122 107 122 107 127"/>
    </justification>
  </clause>

  <clause id="129">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),x),f(f(y,f(x,y)),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(630)
    ]]></attribute>
    <justification jstring="[para(122(a,1),66(a,1,2,2,1,1,1)),rewrite([122(6),108(6),127(5)])].">
      <j1 rule="para" parents="122 66"/>
      <j2 rule="rewrite" parents="122 108 127"/>
    </justification>
  </clause>

  <clause id="130">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(u,f(u,u)),f(f(z,f(f(y,x),z)),y)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(632)
    ]]></attribute>
    <justification jstring="[back_rewrite(98),rewrite([127(8)])].">
      <j1 rule="back_rewrite" parents="98"/>
      <j2 rule="rewrite" parents="127"/>
    </justification>
  </clause>

  <clause id="131">
    <literal><![CDATA[
      f(x,f(f(y,x),f(f(f(y,f(x,y)),f(y,x)),f(z,f(y,x))))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(633)
    ]]></attribute>
    <justification jstring="[back_rewrite(81),rewrite([127(3)])].">
      <j1 rule="back_rewrite" parents="81"/>
      <j2 rule="rewrite" parents="127"/>
    </justification>
  </clause>

  <clause id="134">
    <literal><![CDATA[
      f(f(f(x,y),f(x,y)),f(x,x)) = f(f(x,y),f(f(x,y),f(x,y)))
    ]]></literal>
    <attribute><![CDATA[
      label(637)
    ]]></attribute>
    <justification jstring="[para(29(a,1),101(a,1,2,1)),rewrite([29(6)])].">
      <j1 rule="para" parents="29 101"/>
      <j2 rule="rewrite" parents="29"/>
    </justification>
  </clause>

  <clause id="139">
    <literal><![CDATA[
      f(f(x,f(y,x)),f(x,f(f(y,x),f(x,z)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(640)
    ]]></attribute>
    <justification jstring="[para(128(a,1),110(a,1,2,1)),rewrite([128(8),122(5),107(5),128(7),128(11)])].">
      <j1 rule="para" parents="128 110"/>
      <j2 rule="rewrite" parents="128 122 107 128 128"/>
    </justification>
  </clause>

  <clause id="141">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(f(y,z),f(z,u))),f(z,z)) = z
    ]]></literal>
    <attribute><![CDATA[
      label(644)
    ]]></attribute>
    <justification jstring="[para(107(a,1),125(a,1,1,2,1))].">
      <j1 rule="para" parents="107 125"/>
    </justification>
  </clause>

  <clause id="142">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(f(y,z),f(u,z))),f(z,z)) = z
    ]]></literal>
    <attribute><![CDATA[
      label(646)
    ]]></attribute>
    <justification jstring="[para(31(a,1),141(a,1,1,2,2))].">
      <j1 rule="para" parents="31 141"/>
    </justification>
  </clause>

  <clause id="144">
    <literal><![CDATA[
      f(f(f(x,y),f(z,y)),y) = f(f(u,f(u,u)),f(f(x,y),f(z,y)))
    ]]></literal>
    <attribute><![CDATA[
      label(649)
    ]]></attribute>
    <justification jstring="[para(142(a,1),97(a,1,2)),rewrite([105(9)])].">
      <j1 rule="para" parents="142 97"/>
      <j2 rule="rewrite" parents="105"/>
    </justification>
  </clause>

  <clause id="147">
    <literal><![CDATA[
      f(f(x,f(x,y)),f(y,x)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(652A)
    ]]></attribute>
    <justification jstring="[para(126(a,1),95(a,1)),flip(a)].">
      <j1 rule="para" parents="126 95"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="149">
    <literal><![CDATA[
      f(f(x,x),f(f(y,x),f(x,z))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(658)
    ]]></attribute>
    <justification jstring="[para(141(a,1),134(a,1,1,1)),rewrite([141(8),93(14),141(13),141(13),141(13)])].">
      <j1 rule="para" parents="141 134"/>
      <j2 rule="rewrite" parents="141 93 141 141 141"/>
    </justification>
  </clause>

  <clause id="151">
    <literal><![CDATA[
      f(f(x,x),f(f(y,x),f(z,f(z,z)))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(660)
    ]]></attribute>
    <justification jstring="[para(91(a,1),149(a,1,2,2))].">
      <j1 rule="para" parents="91 149"/>
    </justification>
  </clause>

  <clause id="154">
    <literal><![CDATA[
      f(f(f(x,y),f(z,y)),y) = f(y,f(f(x,y),f(z,y)))
    ]]></literal>
    <attribute><![CDATA[
      label(664B)
    ]]></attribute>
    <justification jstring="[para(144(a,1),31(a,1,2,1)),rewrite([97(11)]),flip(a)].">
      <j1 rule="para" parents="144 31"/>
      <j2 rule="rewrite" parents="97"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="156">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,y),f(x,y)))) = f(f(z,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(666)
    ]]></attribute>
    <justification jstring="[para(144(a,2),95(a,1,2)),rewrite([154(5)])].">
      <j1 rule="para" parents="144 95"/>
      <j2 rule="rewrite" parents="154"/>
    </justification>
  </clause>

  <clause id="157">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(z,x),f(x,y)))) = f(f(z,x),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(667)
    ]]></attribute>
    <justification jstring="[para(60(a,1),156(a,1,1)),rewrite([60(6),60(11)])].">
      <j1 rule="para" parents="60 156"/>
      <j2 rule="rewrite" parents="60 60"/>
    </justification>
  </clause>

  <clause id="158">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(y,z),f(x,y)))) = f(f(y,z),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(668)
    ]]></attribute>
    <justification jstring="[para(60(a,1),156(a,1,2,2,1)),rewrite([60(10)])].">
      <j1 rule="para" parents="60 156"/>
      <j2 rule="rewrite" parents="60"/>
    </justification>
  </clause>

  <clause id="159">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(z,f(x,y)),x))) = f(f(z,f(x,y)),x)
    ]]></literal>
    <attribute><![CDATA[
      label(669)
    ]]></attribute>
    <justification jstring="[para(97(a,1),156(a,1,1)),rewrite([97(8),97(13)])].">
      <j1 rule="para" parents="97 156"/>
      <j2 rule="rewrite" parents="97 97"/>
    </justification>
  </clause>

  <clause id="160">
    <literal><![CDATA[
      f(x,f(f(x,y),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(670)
    ]]></attribute>
    <justification jstring="[para(63(a,1),159(a,1,2,2,1)),rewrite([63(7)])].">
      <j1 rule="para" parents="63 159"/>
      <j2 rule="rewrite" parents="63"/>
    </justification>
  </clause>

  <clause id="161">
    <literal><![CDATA[
      f(f(x,y),f(y,x)) = f(f(y,x),f(y,x))
    ]]></literal>
    <attribute><![CDATA[
      label(671A)
    ]]></attribute>
    <justification jstring="[para(160(a,1),158(a,1,2)),flip(a)].">
      <j1 rule="para" parents="160 158"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="162">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(z,y),f(y,z))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(672)
    ]]></attribute>
    <justification jstring="[para(161(a,2),18(a,1,2))].">
      <j1 rule="para" parents="161 18"/>
    </justification>
  </clause>

  <clause id="163">
    <literal><![CDATA[
      f(f(x,f(y,x)),f(x,y)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(673)
    ]]></attribute>
    <justification jstring="[para(161(a,1),139(a,1,2,2)),rewrite([32(6)])].">
      <j1 rule="para" parents="161 139"/>
      <j2 rule="rewrite" parent