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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>sh1.out</source>

<heading><![CDATA[
Prover9 (32) version 22-May-2007, May 2007.
Process 27705 was started by mccune on cleo,
Tue May 22 14:52:02 2007
The command was "/home/mccune/bin/prover9 -f sh1.in".
]]></heading>

<proof number="1" length="163" max_count="87">

<comments><![CDATA[
% Proof 1 at 346.46 (+ 1.15) seconds: "Sheffer".
% Length of proof is 163.
% Level of proof is 28.
% Maximum clause weight is 87.
% Given clauses 1209.
]]></comments>

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

  <clause id="2" type="assumption">
    <literal><![CDATA[
      f(f(x,f(f(y,x),x)),f(y,f(z,x))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(Sh_1)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="3" type="deny">
    <literal><![CDATA[
      f(f(c1,c1),f(c1,c1)) != c1
    ]]></literal>
    <literal><![CDATA[
      f(c1,f(c2,f(c2,c2))) != f(c1,c1)
    ]]></literal>
    <literal><![CDATA[
      f(f(f(c2,c2),c1),f(f(c3,c3),c1)) != f(f(c1,f(c2,c3)),f(c1,f(c2,c3)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

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

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

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

  <clause id="13">
    <literal><![CDATA[
      f(f(f(x,y),f(f(z,f(x,y)),f(x,y))),f(z,x)) = z
    ]]></literal>
    <justification jstring="[para(5(a,1),2(a,1,2,2))].">
      <j1 rule="para" parents="5 2"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y
    ]]></literal>
    <justification jstring="[para(2(a,1),5(a,1,1,1)),rewrite([2(6),2(7)])].">
      <j1 rule="para" parents="2 5"/>
      <j2 rule="rewrite" parents="2 2"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      f(f(x,f(f(f(y,f(f(y,y),y)),x),x)),y) = f(y,f(f(y,y),y))
    ]]></literal>
    <justification jstring="[para(2(a,1),15(a,1,2))].">
      <j1 rule="para" parents="2 15"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      f(x,f(f(x,x),x)) = f(x,x)
    ]]></literal>
    <justification jstring="[para(15(a,1),4(a,1,2)),rewrite([15(8)]),flip(a)].">
      <j1 rule="para" parents="15 4"/>
      <j2 rule="rewrite" parents="15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      f(f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(u,f(f(x,f(y,z)),f(x,f(x,f(y,z))))),f(f(x,f(y,z)),f(x,f(x,f(y,z)))))),f(u,f(f(f(z,f(f(x,z),z)),f(v,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(f(z,f(f(x,z),z)),f(v,f(x,f(y,z)))))))) = u
    ]]></literal>
    <justification jstring="[para(4(a,1),15(a,1,2,2,2,1))].">
      <j1 rule="para" parents="4 15"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      f(f(x,f(f(f(f(y,y),f(y,f(y,y))),x),x)),f(y,y)) = f(f(y,y),f(y,f(y,y)))
    ]]></literal>
    <justification jstring="[para(4(a,1),15(a,1,2)),rewrite([36(3),36(4),36(10),36(12),36(13)])].">
      <j1 rule="para" parents="4 15"/>
      <j2 rule="rewrite" parents="36 36 36 36 36"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      f(f(f(x,f(y,f(f(z,y),y))),f(x,f(x,f(y,f(f(z,y),y))))),f(f(z,f(f(x,z),z)),f(u,f(f(f(x,f(y,f(f(z,y),y))),u),u)))) = f(z,f(f(x,z),z))
    ]]></literal>
    <justification jstring="[para(15(a,1),15(a,1,1,2,1))].">
      <j1 rule="para" parents="15 15"/>
    </justification>
  </clause>

  <clause id="48">
    <literal><![CDATA[
      f(f(x,f(f(f(y,y),x),x)),y) = f(y,y)
    ]]></literal>
    <justification jstring="[back_rewrite(33),rewrite([36(3),36(8)])].">
      <j1 rule="back_rewrite" parents="33"/>
      <j2 rule="rewrite" parents="36 36"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      f(f(x,x),f(x,f(y,x))) = x
    ]]></literal>
    <justification jstring="[para(36(a,1),2(a,1,1))].">
      <j1 rule="para" parents="36 2"/>
    </justification>
  </clause>

  <clause id="51">
    <literal><![CDATA[
      f(f(x,x),f(x,x)) = x
    ]]></literal>
    <justification jstring="[para(36(a,1),2(a,1,2)),rewrite([36(3)])].">
      <j1 rule="para" parents="36 2"/>
      <j2 rule="rewrite" parents="36"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      f(f(f(x,y),f(x,y)),f(f(x,y),x)) = f(x,y)
    ]]></literal>
    <justification jstring="[para(36(a,1),13(a,1,1))].">
      <j1 rule="para" parents="36 13"/>
    </justification>
  </clause>

  <clause id="60">
    <literal><![CDATA[
      f(f(x,f(f(y,x),x)),f(y,y)) = y
    ]]></literal>
    <justification jstring="[back_rewrite(39),rewrite([50(4),50(9)])].">
      <j1 rule="back_rewrite" parents="39"/>
      <j2 rule="rewrite" parents="50 50"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      f(c1,f(c2,f(c2,c2))) != f(c1,c1)
    ]]></literal>
    <literal><![CDATA[
      f(f(f(c2,c2),c1),f(f(c3,c3),c1)) != f(f(c1,f(c2,c3)),f(c1,f(c2,c3)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(3),rewrite([51(7)]),xx(a)].">
      <j1 rule="back_rewrite" parents="3"/>
      <j2 rule="rewrite" parents="51"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      f(f(f(x,f(y,z)),f(f(f(f(y,z),f(y,f(y,z))),f(x,f(y,z))),f(x,f(y,z)))),f(f(z,u),f(f(y,f(z,u)),f(z,u)))) = f(f(y,z),f(y,f(y,z)))
    ]]></literal>
    <justification jstring="[para(13(a,1),6(a,1,1,2,1,1,2,1)),rewrite([13(29)])].">
      <j1 rule="para" parents="13 6"/>
      <j2 rule="rewrite" parents="13"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      f(f(x,f(f(f(f(y,z),f(f(f(z,f(f(f(u,f(f(x,u),u)),z),z)),f(y,z)),f(y,z))),x),x)),f(z,f(f(f(u,f(f(x,u),u)),z),z))) = f(f(y,z),f(f(f(z,f(f(f(u,f(f(x,u),u)),z),z)),f(y,z)),f(y,z)))
    ]]></literal>
    <justification jstring="[para(6(a,1),15(a,1,2))].">
      <j1 rule="para" parents="6 15"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      f(f(f(f(x,f(f(y,x),x)),f(z,f(y,f(u,x)))),f(f(x,f(f(y,x),x)),f(f(x,f(f(y,x),x)),f(z,f(y,f(u,x)))))),f(f(f(y,f(u,x)),f(y,f(y,f(u,x)))),f(f(y,f(u,x)),f(y,f(y,f(u,x)))))) = f(f(y,f(u,x)),f(y,f(y,f(u,x))))
    ]]></literal>
    <justification jstring="[para(4(a,1),60(a,1,1,2,1))].">
      <j1 rule="para" parents="4 60"/>
    </justification>
  </clause>

  <clause id="94">
    <literal><![CDATA[
      f(f(f(x,f(f(y,x),x)),f(f(z,f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),f(z,y)) = z
    ]]></literal>
    <justification jstring="[para(60(a,1),15(a,1,2,2,2,1)),rewrite([50(16)])].">
      <j1 rule="para" parents="60 15"/>
      <j2 rule="rewrite" parents="50"/>
    </justification>
  </clause>

  <clause id="122">
    <literal><![CDATA[
      f(f(x,x),f(f(y,f(f(f(x,x),y),y)),f(z,x))) = f(y,f(f(f(x,x),y),y))
    ]]></literal>
    <justification jstring="[para(48(a,1),2(a,1,1,2,1)),rewrite([36(3)])].">
      <j1 rule="para" parents="48 2"/>
      <j2 rule="rewrite" parents="36"/>
    </justification>
  </clause>

  <clause id="125">
    <literal><![CDATA[
      f(f(f(f(x,f(f(f(y,y),x),x)),f(z,y)),f(f(x,f(f(f(y,y),x),x)),f(f(x,f(f(f(y,y),x),x)),f(z,y)))),f(f(y,y),f(u,f(f(x,f(f(f(y,y),x),x)),f(z,y))))) = f(y,y)
    ]]></literal>
    <justification jstring="[para(48(a,1),4(a,1,2,1,2,1)),rewrite([36(21),48(33),36(31)])].">
      <j1 rule="para" parents="48 4"/>
      <j2 rule="rewrite" parents="36 48 36"/>
    </justification>
  </clause>

  <clause id="131">
    <literal><![CDATA[
      f(x,f(f(f(f(y,x),f(y,x)),x),x)) = f(y,x)
    ]]></literal>
    <justification jstring="[para(48(a,1),6(a,1,1,2,1)),rewrite([36(7),51(7)]),flip(a)].">
      <j1 rule="para" parents="48 6"/>
      <j2 rule="rewrite" parents="36 51"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="163">
    <literal><![CDATA[
      f(f(x,x),f(y,x)) = x
    ]]></literal>
    <justification jstring="[para(131(a,1),2(a,1,2)),rewrite([36(3)])].">
      <j1 rule="para" parents="131 2"/>
      <j2 rule="rewrite" parents="36"/>
    </justification>
  </clause>

  <clause id="171">
    <literal><![CDATA[
      f(f(f(x,y),f(f(z,f(x,y)),f(x,y))),f(z,y)) = z
    ]]></literal>
    <justification jstring="[para(131(a,1),13(a,1,1,1)),rewrite([131(7),131(9)])].">
      <j1 rule="para" parents="131 13"/>
      <j2 rule="rewrite" parents="131 131"/>
    </justification>
  </clause>

  <clause id="175">
    <literal><![CDATA[
      f(f(x,y),f(f(f(x,x),f(x,y)),f(x,y))) = x
    ]]></literal>
    <justification jstring="[para(13(a,1),131(a,1,2,1,1,1)),rewrite([13(9),13(15)])].">
      <j1 rule="para" parents="13 131"/>
      <j2 rule="rewrite" parents="13 13"/>
    </justification>
  </clause>

  <clause id="184">
    <literal><![CDATA[
      f(f(f(f(f(f(x,y),f(x,y)),y),y),f(f(x,y),f(f(f(f(x,y),f(x,y)),y),y))),f(y,y)) = y
    ]]></literal>
    <justification jstring="[para(131(a,1),60(a,1,1,2,1))].">
      <j1 rule="para" parents="131 60"/>
    </justification>
  </clause>

  <clause id="190">
    <literal><![CDATA[
      f(f(f(x,y),f(x,y)),f(f(x,y),y)) = f(x,y)
    ]]></literal>
    <justification jstring="[para(131(a,1),55(a,1,1,1)),rewrite([131(7),131(9),131(12)])].">
      <j1 rule="para" parents="131 55"/>
      <j2 rule="rewrite" parents="131 131 131"/>
    </justification>
  </clause>

  <clause id="192">
    <literal><![CDATA[
      f(f(f(f(f(x,y),f(x,y)),y),y),f(f(f(f(x,y),f(x,y)),f(f(f(f(x,y),f(x,y)),y),y)),f(f(f(f(x,y),f(x,y)),y),y))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(131(a,1),131(a,1,2,1,1,1)),rewrite([131(12),131(27)])].">
      <j1 rule="para" parents="131 131"/>
      <j2 rule="rewrite" parents="131 131"/>
    </justification>
  </clause>

  <clause id="238">
    <literal><![CDATA[
      f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(y,z))) = f(z,f(f(x,z),z))
    ]]></literal>
    <justification jstring="[para(163(a,1),4(a,1,2,2))].">
      <j1 rule="para" parents="163 4"/>
    </justification>
  </clause>

  <clause id="239">
    <literal><![CDATA[
      f(f(f(x,y),f(y,f(x,y))),f(f(y,y),x)) = f(y,y)
    ]]></literal>
    <justification jstring="[para(163(a,1),13(a,1,1,2,1))].">
      <j1 rule="para" parents="163 13"/>
    </justification>
  </clause>

  <clause id="243">
    <literal><![CDATA[
      f(f(x,f(f(f(f(y,x),f(f(z,f(y,x)),f(y,x))),x),x)),z) = f(f(y,x),f(f(z,f(y,x)),f(y,x)))
    ]]></literal>
    <justification jstring="[para(163(a,1),6(a,1,1,1)),rewrite([163(9),163(10)])].">
      <j1 rule="para" parents="163 6"/>
      <j2 rule="rewrite" parents="163 163"/>
    </justification>
  </clause>

  <clause id="244">
    <literal><![CDATA[
      f(f(x,x),f(y,f(f(x,y),y))) = x
    ]]></literal>
    <justification jstring="[para(6(a,1),163(a,1,2))].">
      <j1 rule="para" parents="6 163"/>
    </justification>
  </clause>

  <clause id="250">
    <literal><![CDATA[
      f(f(x,y),f(y,f(x,y))) = y
    ]]></literal>
    <justification jstring="[para(163(a,1),131(a,1,2,1,1,1)),rewrite([163(4),163(4),163(7)])].">
      <j1 rule="para" parents="163 131"/>
      <j2 rule="rewrite" parents="163 163 163"/>
    </justification>
  </clause>

  <clause id="256">
    <literal><![CDATA[
      f(f(f(x,y),f(x,y)),y) = f(x,y)
    ]]></literal>
    <justification jstring="[para(163(a,1),163(a,1,2))].">
      <j1 rule="para" parents="163 163"/>
    </justification>
  </clause>

  <clause id="258">
    <literal><![CDATA[
      f(x,f(f(x,x),y)) = f(x,x)
    ]]></literal>
    <justification jstring="[back_rewrite(239),rewrite([250(4)])].">
      <j1 rule="back_rewrite" parents="239"/>
      <j2 rule="rewrite" parents="250"/>
    </justification>
  </clause>

  <clause id="262">
    <literal><![CDATA[
      f(f(f(x,y),y),f(f(x,y),f(f(x,y),y))) = f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(192),rewrite([256(4),256(9),190(8),256(7)])].">
      <j1 rule="back_rewrite" parents="192"/>
      <j2 rule="rewrite" parents="256 256 190 256"/>
    </justification>
  </clause>

  <clause id="264">
    <literal><![CDATA[
      f(f(x,y),f(y,y)) = y
    ]]></literal>
    <justification jstring="[back_rewrite(184),rewrite([256(4),256(7),262(7)])].">
      <j1 rule="back_rewrite" parents="184"/>
      <j2 rule="rewrite" parents="256 256 262"/>
    </justification>
  </clause>

  <clause id="269">
    <literal><![CDATA[
      f(x,f(f(y,x),x)) = f(y,x)
    ]]></literal>
    <justification jstring="[back_rewrite(131),rewrite([256(4)])].">
      <j1 rule="back_rewrite" parents="131"/>
      <j2 rule="rewrite" parents="256"/>
    </justification>
  </clause>

  <clause id="280">
    <literal><![CDATA[
      f(f(x,x),f(x,y)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(244),rewrite([269(4)])].">
      <j1 rule="back_rewrite" parents="244"/>
      <j2 rule="rewrite" parents="269"/>
    </justification>
  </clause>

  <clause id="281">
    <literal><![CDATA[
      f(f(f(x,f(y,z)),z),x) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[back_rewrite(243),rewrite([269(6),269(5),269(10)])].">
      <j1 rule="back_rewrite" parents="243"/>
      <j2 rule="rewrite" parents="269 269 269"/>
    </justification>
  </clause>

  <clause id="282">
    <literal><![CDATA[
      f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(x,z),f(y,z))) = f(x,z)
    ]]></literal>
    <justification jstring="[back_rewrite(238),rewrite([269(9),269(13)])].">
      <j1 rule="back_rewrite" parents="238"/>
      <j2 rule="rewrite" parents="269 269"/>
    </justification>
  </clause>

  <clause id="298">
    <literal><![CDATA[
      f(f(x,y),f(x,f(x,y))) = x
    ]]></literal>
    <justification jstring="[back_rewrite(175),rewrite([280(4)])].">
      <j1 rule="back_rewrite" parents="175"/>
      <j2 rule="rewrite" parents="280"/>
    </justification>
  </clause>

  <clause id="299">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(x,z)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(171),rewrite([269(6)])].">
      <j1 rule="back_rewrite" parents="171"/>
      <j2 rule="rewrite" parents="269"/>
    </justification>
  </clause>

  <clause id="304">
    <literal><![CDATA[
      f(f(f(x,x),y),f(f(x,x),f(z,f(f(f(x,x),y),f(u,x))))) = f(x,x)
    ]]></literal>
    <justification jstring="[back_rewrite(125),rewrite([269(4),269(8),269(10),298(12),269(7)])].">
      <j1 rule="back_rewrite" parents="125"/>
      <j2 rule="rewrite" parents="269 269 269 298 269"/>
    </justification>
  </clause>

  <clause id="307">
    <literal><![CDATA[
      f(f(x,x),f(f(f(x,x),y),f(z,x))) = f(f(x,x),y)
    ]]></literal>
    <justification jstring="[back_rewrite(122),rewrite([269(5),269(10)])].">
      <j1 rule="back_rewrite" parents="122"/>
      <j2 rule="rewrite" parents="269 269"/>
    </justification>
  </clause>

  <clause id="321">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(x,y)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(94),rewrite([269(3),269(4),269(6),269(6)])].">
      <j1 rule="back_rewrite" parents="94"/>
      <j2 rule="rewrite" parents="269 269 269 269"/>
    </justification>
  </clause>

  <clause id="324">
    <literal><![CDATA[
      f(f(x,y),f(x,x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(89),rewrite([269(3),269(8),269(9),298(13),298(7),298(7),298(9)])].">
      <j1 rule="back_rewrite" parents="89"/>
      <j2 rule="rewrite" parents="269 269 269 298 298 298 298"/>
    </justification>
  </clause>

  <clause id="329">
    <literal><![CDATA[
      f(f(f(f(f(x,y),z),f(u,z)),x),f(f(x,y),z)) = f(f(f(x,y),z),f(u,z))
    ]]></literal>
    <justification jstring="[back_rewrite(68),rewrite([269(4),269(5),269(8),269(7),269(8),269(9),269(12),269(13),269(16)])].">
      <j1 rule="back_rewrite" parents="68"/>
      <j2 rule="rewrite" parents="269 269 269 269 269 269 269 269 269"/>
    </justification>
  </clause>

  <clause id="330">
    <literal><![CDATA[
      f(f(x,f(y,f(x,z))),f(x,f(z,u))) = x
    ]]></literal>
    <justification jstring="[back_rewrite(65),rewrite([298(6),269(9),269(9),298(10)])].">
      <j1 rule="back_rewrite" parents="65"/>
      <j2 rule="rewrite" parents="298 269 269 298"/>
    </justification>
  </clause>

  <clause id="334">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(x,f(y,z)),u))) = f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(45),rewrite([269(3),269(5),298(6),269(3),269(4),269(6),269(9)])].">
      <j1 rule="back_rewrite" parents="45"/>
      <j2 rule="rewrite" parents="269 269 298 269 269 269 269"/>
    </justification>
  </clause>

  <clause id="338">
    <literal><![CDATA[
      f(f(x,y),f(x,f(y,z))) = x
    ]]></literal>
    <justification jstring="[back_rewrite(38),rewrite([298(6),298(6),298(7),269(3),269(4),269(9),269(10),298(14)])].">
      <j1 rule="back_rewrite" parents="38"/>
      <j2 rule="rewrite" parents="298 298 298 269 269 269 269 298"/>
    </justification>
  </clause>

  <clause id="353">
    <literal><![CDATA[
      f(f(x,y),f(x,f(z,y))) = x
    ]]></literal>
    <justification jstring="[back_rewrite(2),rewrite([269(3)])].">
      <j1 rule="back_rewrite" parents="2"/>
      <j2 rule="rewrite" parents="269"/>
    </justification>
  </clause>

  <clause id="356">
    <literal><![CDATA[
      f(x,f(f(x,y),f(z,y))) = f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(282),rewrite([298(6)])].">
      <j1 rule="back_rewrite" parents="282"/>
      <j2 rule="rewrite" parents="298"/>
    </justification>
  </clause>

  <clause id="387">
    <literal><![CDATA[
      f(x,f(f(x,f(y,z)),y)) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(321(a,1),299(a,1,1))].">
      <j1 rule="para" parents="321 299"/>
    </justification>
  </clause>

  <clause id="392">
    <literal><![CDATA[
      f(x,f(f(y,x),f(f(x,x),z))) = f(y,x)
    ]]></literal>
    <justification jstring="[para(264(a,1),338(a,1,1))].">
      <j1 rule="para" parents="264 338"/>
    </justification>
  </clause>

  <clause id="399">
    <literal><![CDATA[
      f(x,f(f(x,y),f(y,z))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(338(a,1),299(a,1,1))].">
      <j1 rule="para" parents="338 299"/>
    </justification>
  </clause>

  <clause id="401">
    <literal><![CDATA[
      f(x,f(f(x,f(y,z)),f(f(x,z),u))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(299(a,1),338(a,1,1))].">
      <j1 rule="para" parents="299 338"/>
    </justification>
  </clause>

  <clause id="421">
    <literal><![CDATA[
      f(x,y) = f(y,x)
    ]]></literal>
    <justification jstring="[para(264(a,1),281(a,1,1,1,2)),rewrite([264(3),264(4)])].">
      <j1 rule="para" parents="264 281"/>
      <j2 rule="rewrite" parents="264 264"/>
    </justification>
  </clause>

  <clause id="427">
    <literal><![CDATA[
      f(x,f(y,f(x,y))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(321(a,1),281(a,1,1)),flip(a)].">
      <j1 rule="para" parents="321 281"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="429">
    <literal><![CDATA[
      f(x,f(y,f(x,f(z,y)))) = f(x,f(z,y))
    ]]></literal>
    <justification jstring="[para(281(a,1),256(a,2)),rewrite([421(3),421(4),421(7),421(8),421(10),356(10)])].">
      <j1 rule="para" parents="281 256"/>
      <j2 rule="rewrite" parents="421 421 421 421 421 356"/>
    </justification>
  </clause>

  <clause id="441">
    <literal><![CDATA[
      f(x,f(y,f(x,f(y,z)))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[back_rewrite(387),rewrite([421(3)])].">
      <j1 rule="back_rewrite" parents="387"/>
      <j2 rule="rewrite" parents="421"/>
    </justification>
  </clause>

  <clause id="444">
    <literal><![CDATA[
      f(f(f(x,y),z),f(x,f(f(f(x,y),z),f(u,z)))) = f(f(f(x,y),z),f(u,z))
    ]]></literal>
    <justification jstring="[back_rewrite(329),rewrite([421(5),421(8)])].">
      <j1 rule="back_rewrite" parents="329"/>
      <j2 rule="rewrite" parents="421 421"/>
    </justification>
  </clause>

  <clause id="451">
    <literal><![CDATA[
      f(c1,f(c2,f(c2,c2))) != f(c1,c1)
    ]]></literal>
    <literal><![CDATA[
      f(f(c1,f(c2,c3)),f(c1,f(c2,c3))) != f(f(c1,f(c2,c2)),f(c1,f(c3,c3)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(61),rewrite([421(16),421(21)]),flip(b)].">
      <j1 rule="back_rewrite" parents="61"/>
      <j2 rule="rewrite" parents="421 421"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="453">
    <literal><![CDATA[
      f(f(x,y),f(y,f(x,z))) = y
    ]]></literal>
    <justification jstring="[para(421(a,1),338(a,1,1))].">
      <j1 rule="para" parents="421 338"/>
    </justification>
  </clause>

  <clause id="454">
    <literal><![CDATA[
      f(f(x,y),f(f(y,z),x)) = x
    ]]></literal>
    <justification jstring="[para(421(a,1),338(a,1,2))].">
      <j1 rule="para" parents="421 338"/>
    </justification>
  </clause>

  <clause id="455">
    <literal><![CDATA[
      f(f(x,y),f(y,f(z,x))) = y
    ]]></literal>
    <justification jstring="[para(421(a,1),353(a,1,1))].">
      <j1 rule="para" parents="421 353"/>
    </justification>
  </clause>

  <clause id="456">
    <literal><![CDATA[
      f(f(x,y),f(f(z,y),x)) = x
    ]]></literal>
    <justification jstring="[para(421(a,1),353(a,1,2))].">
      <j1 rule="para" parents="421 353"/>
    </justification>
  </clause>

  <clause id="464">
    <literal><![CDATA[
      f(x,f(y,f(y,x))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(421(a,1),427(a,1,2,2))].">
      <j1 rule="para" parents="421 427"/>
    </justification>
  </clause>

  <clause id="466">
    <literal><![CDATA[
      f(f(f(x,y),z),f(z,y)) = z
    ]]></literal>
    <justification jstring="[para(264(a,1),453(a,1,2,2))].">
      <j1 rule="para" parents="264 453"/>
    </justification>
  </clause>

  <clause id="467">
    <literal><![CDATA[
      f(f(f(x,y),z),f(z,x)) = z
    ]]></literal>
    <justification jstring="[para(324(a,1),453(a,1,2,2))].">
      <j1 rule="para" parents="324 453"/>
    </justification>
  </clause>

  <clause id="470">
    <literal><![CDATA[
      f(x,f(f(y,x),f(y,z))) = f(y,x)
    ]]></literal>
    <justification jstring="[para(453(a,1),353(a,1,2)),rewrite([421(4)])].">
      <j1 rule="para" parents="453 353"/>
      <j2 rule="rewrite" parents="421"/>
    </justification>
  </clause>

  <clause id="472">
    <literal><![CDATA[
      f(f(x,y),f(f(x,z),y)) = y
    ]]></literal>
    <justification jstring="[para(421(a,1),453(a,1,2))].">
      <j1 rule="para" parents="421 453"/>
    </justification>
  </clause>

  <clause id="504">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(z,x)) = x
    ]]></literal>
    <justification jstring="[para(264(a,1),454(a,1,2,1))].">
      <j1 rule="para" parents="264 454"/>
    </justification>
  </clause>

  <clause id="519">
    <literal><![CDATA[
      f(f(x,y),f(f(z,x),y)) = y
    ]]></literal>
    <justification jstring="[para(421(a,1),455(a,1,2))].">
      <j1 rule="para" parents="421 455"/>
    </justification>
  </clause>

  <clause id="525">
    <literal><![CDATA[
      f(x,f(f(y,z),f(y,x))) = f(y,x)
    ]]></literal>
    <justification jstring="[para(453(a,1),455(a,1,2)),rewrite([421(4)])].">
      <j1 rule="para" parents="453 455"/>
      <j2 rule="rewrite" parents="421"/>
    </justification>
  </clause>

  <clause id="531">
    <literal><![CDATA[
      f(x,f(f(y,z),f(z,x))) = f(z,x)
    ]]></literal>
    <justification jstring="[para(455(a,1),455(a,1,2)),rewrite([421(4)])].">
      <j1 rule="para" parents="455 455"/>
      <j2 rule="rewrite" parents="421"/>
    </justification>
  </clause>

  <clause id="533">
    <literal><![CDATA[
      f(x,f(f(y,f(x,x)),f(x,z))) = f(x,z)
    ]]></literal>
    <justification jstring="[para(324(a,1),456(a,1,1))].">
      <j1 rule="para" parents="324 456"/>
    </justification>
  </clause>

  <clause id="545">
    <literal><![CDATA[
      f(x,f(y,f(f(y,z),x))) = f(f(y,z),x)
    ]]></literal>
    <justification jstring="[para(454(a,1),456(a,1,2)),rewrite([421(3),421(4)])].">
      <j1 rule="para" parents="454 456"/>
      <j2 rule="rewrite" parents="421 421"/>
    </justification>
  </clause>

  <clause id="549">
    <literal><![CDATA[
      f(x,f(y,f(f(z,y),x))) = f(f(z,y),x)
    ]]></literal>
    <justification jstring="[para(456(a,1),456(a,1,2)),rewrite([421(3),421(4)])].">
      <j1 rule="para" parents="456 456"/>
      <j2 rule="rewrite" parents="421 421"/>
    </justification>
  </clause>

  <clause id="584">
    <literal><![CDATA[
      f(x,f(f(f(x,y),z),f(f(u,y),x))) = f(f(u,y),x)
    ]]></literal>
    <justification jstring="[para(466(a,1),454(a,1,1))].">
      <j1 rule="para" parents="466 454"/>
    </justification>
  </clause>

  <clause id="598">
    <literal><![CDATA[
      f(x,f(f(f(x,y),z),f(x,f(y,u)))) = f(x,f(y,u))
    ]]></literal>
    <justification jstring="[para(338(a,1),472(a,1,1))].">
      <j1 rule="para" parents="338 472"/>
    </justification>
  </clause>

  <clause id="639">
    <literal><![CDATA[
      f(f(x,f(y,y)),f(f(z,f(z,y)),x)) = x
    ]]></literal>
    <justification jstring="[para(464(a,1),504(a,1,1,2))].">
      <j1 rule="para" parents="464 504"/>
    </justification>
  </clause>

  <clause id="656">
    <literal><![CDATA[
      f(f(f(x,f(x,y)),z),f(f(y,y),z)) = z
    ]]></literal>
    <justification jstring="[para(464(a,1),519(a,1,2,1))].">
      <j1 rule="para" parents="464 519"/>
    </justification>
  </clause>

  <clause id="692">
    <literal><![CDATA[
      f(f(f(x,y),z),f(x,f(f(f(x,y),z),f(z,u)))) = f(f(f(x,y),z),f(z,u))
    ]]></literal>
    <justification jstring="[para(399(a,1),454(a,1,2)),rewrite([421(5),421(8)])].">
      <j1 rule="para" parents="399 454"/>
      <j2 rule="rewrite" parents="421 421"/>
    </justification>
  </clause>

  <clause id="886">
    <literal><![CDATA[
      f(x,f(f(y,x),f(f(y,z),f(y,x)))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(525(a,1),427(a,1,2,2)),rewrite([421(5)])].">
      <j1 rule="para" parents="525 427"/>
      <j2 rule="rewrite" parents="421"/>
    </justification>
  </clause>

  <clause id="912">
    <literal><![CDATA[
      f(x,f(f(y,x),f(f(z,y),f(y,x)))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(531(a,1),427(a,1,2,2)),rewrite([421(5)])].">
      <j1 rule="para" parents="531 427"/>
      <j2 rule="rewrite" parents="421"/>
    </justification>
  </clause>

  <clause id="1043">
    <literal><![CDATA[
      f(f(x,f(y,f(z,x))),f(x,f(z,u))) = x
    ]]></literal>
    <justification jstring="[para(421(a,1),330(a,1,1,2,2))].">
      <j1 rule="para" parents="421 330"/>
    </justification>
  </clause>

  <clause id="1089">
    <literal><![CDATA[
      f(f(x,f(y,f(x,f(z,u)))),f(z,x)) = x
    ]]></literal>
    <justification jstring="[para(525(a,1),330(a,1,2))].">
      <j1 rule="para" parents="525 330"/>
    </justification>
  </clause>

  <clause id="1380">
    <literal><![CDATA[
      f(x,f(y,y)) = f(x,f(x,y))
    ]]></literal>
    <justification jstring="[para(464(a,1),429(a,1,2))].">
      <j1 rule="para" parents="464 429"/>
    </justification>
  </clause>

  <clause id="1394">
    <literal><![CDATA[
      f(x,f(y,f(x,f(z,f(x,y))))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(330(a,1),429(a,1,2)),flip(a)].">
      <j1 rule="para" parents="330 429"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1398">
    <literal><![CDATA[
      f(c1,f(c2,f(c2,c2))) != f(c1,c1)
    ]]></literal>
    <literal><![CDATA[
      f(f(c1,f(c2,c3)),f(c1,f(c2,c3))) != f(f(c1,f(c1,c2)),f(c1,f(c1,c3)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(451),rewrite([1380(27),1380(32)])].">
      <j1 rule="back_rewrite" parents="451"/>
      <j2 rule="rewrite" parents="1380 1380"/>
    </justification>
  </clause>

  <clause id="1415">
    <literal><![CDATA[
      f(x,f(x,f(y,y))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(163(a,1),1380(a,1,2)),flip(a)].">
      <j1 rule="para" parents="163 1380"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1421">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(x,f(y,f(z,z)))) = x
    ]]></literal>
    <justification jstring="[para(1380(a,2),353(a,1,2,2))].">
      <j1 rule="para" parents="1380 353"/>
    </justification>
  </clause>

  <clause id="1422">
    <literal><![CDATA[
      f(f(x,x),y) = f(y,f(y,x))
    ]]></literal>
    <justification jstring="[para(1380(a,1),421(a,1)),flip(a)].">
      <j1 rule="para" parents="1380 421"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1423">
    <literal><![CDATA[
      f(x,f(y,y)) = f(x,f(y,x))
    ]]></literal>
    <justification jstring="[para(421(a,1),1380(a,2,2))].">
      <j1 rule="para" parents="421 1380"/>
    </justification>
  </clause>

  <clause id="1647">
    <literal><![CDATA[
      f(f(x,x),y) = f(y,f(x,y))
    ]]></literal>
    <justification jstring="[para(421(a,1),1422(a,2,2))].">
      <j1 rule="para" parents="421 1422"/>
    </justification>
  </clause>

  <clause id="1658">
    <literal><![CDATA[
      f(f(f(x,y),z),f(z,f(f(y,y),x))) = z
    ]]></literal>
    <justification jstring="[para(1422(a,2),455(a,1,2,2))].">
      <j1 rule="para" parents="1422 455"/>
    </justification>
  </clause>

  <clause id="2946">
    <literal><![CDATA[
      f(x,f(f(y,y),f(f(y,z),x))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(545(a,1),427(a,1,2,2)),rewrite([421(6),1423(6,R),421(4)])].">
      <j1 rule="para" parents="545 427"/>
      <j2 rule="rewrite" parents="421 1423 421"/>
    </justification>
  </clause>

  <clause id="9584">
    <literal><![CDATA[
      f(x,f(y,f(x,f(z,f(y,x))))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(1043(a,1),429(a,1,2)),flip(a)].">
      <j1 rule="para" parents="1043 429"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10652">
    <literal><![CDATA[
      f(x,f(y,f(z,f(y,x)))) = f(x,f(y,y))
    ]]></literal>
    <justification jstring="[para(1394(a,1),441(a,1,2)),flip(a)].">
      <j1 rule="para" parents="1394 441"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11507">
    <literal><![CDATA[
      f(x,f(y,f(y,y))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(1421(a,1),1421(a,1,2)),rewrite([421(2),421(6),353(6),421(3)]),flip(a)].">
      <j1 rule="para" parents="1421 1421"/>
      <j2 rule="rewrite" parents="421 421 353 421"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11508">
    <literal><![CDATA[
      f(f(c1,f(c2,c3)),f(c1,f(c2,c3))) != f(f(c1,f(c1,c2)),f(c1,f(c1,c3)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_unit_del(1398),unit_del(a,11507)].">
      <j1 rule="back_unit_del" parents="1398"/>
      <j2 rule="unit_del" parents="11507"/>
    </justification>
  </clause>

  <clause id="11511">
    <literal><![CDATA[
      f(f(x,x),f(y,f(y,y))) = x
    ]]></literal>
    <justification jstring="[para(11507(a,2),163(a,1))].">
      <j1 rule="para" parents="11507 163"/>
    </justification>
  </clause>

  <clause id="11513">
    <literal><![CDATA[
      f(f(x,y),f(y,f(z,f(z,z)))) = y
    ]]></literal>
    <justification jstring="[para(11507(a,2),264(a,1,2))].">
      <j1 rule="para" parents="11507 264"/>
    </justification>
  </clause>

  <clause id="11533">
    <literal><![CDATA[
      f(f(x,x),f(f(x,y),f(z,f(z,z)))) = f(z,f(z,z))
    ]]></literal>
    <justification jstring="[para(11507(a,1),472(a,1,1))].">
      <j1 rule="para" parents="11507 472"/>
    </justification>
  </clause>

  <clause id="11538">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(x,x),f(y,f(f(x,f(x,x)),f(z,f(z,z)))))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(11507(a,2),304(a,1,2,2,2)),rewrite([421(2),421(5)])].">
      <j1 rule="para" parents="11507 304"/>
      <j2 rule="rewrite" parents="421 421"/>
    </justification>
  </clause>

  <clause id="11548">
    <literal><![CDATA[
      f(x,f(x,x)) = f(y,f(y,y))
    ]]></literal>
    <justification jstring="[para(11507(a,2),307(a,1,2)),rewrite([421(3),11533(7),421(4)])].">
      <j1 rule="para" parents="11507 307"/>
      <j2 rule="rewrite" parents="421 11533 421"/>
    </justification>
  </clause>

  <clause id="11549">
    <literal><![CDATA[
      f(x,f(f(y,x),f(z,f(z,z)))) = f(y,x)
    ]]></literal>
    <justification jstring="[para(11507(a,2),470(a,1,2))].">
      <j1 rule="para" parents="11507 470"/>
    </justification>
  </clause>

  <clause id="11565">
    <literal><![CDATA[
      f(x,f(f(y,f(y,y)),f(y,f(y,y)))) = f(x,f(x,x))
    ]]></literal>
    <justification jstring="[para(11507(a,1),1380(a,2,2))].">
      <j1 rule="para" parents="11507 1380"/>
    </justification>
  </clause>

  <clause id="11567">
    <literal><![CDATA[
      f(x,f(x,f(y,f(z,f(z,z))))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(11507(a,2),1415(a,1,2,2))].">
      <j1 rule="para" parents="11507 1415"/>
    </justification>
  </clause>

  <clause id="11569">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(x,f(x,x)),y)) = y
    ]]></literal>
    <justification jstring="[para(11507(a,1),1422(a,1)),rewrite([280(3)]),flip(a)].">
      <j1 rule="para" parents="11507 1422"/>
      <j2 rule="rewrite" parents="280"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11572">
    <literal><![CDATA[
      f(x,f(y,f(z,f(z,z)))) = f(x,f(y,x))
    ]]></literal>
    <justification jstring="[para(11507(a,2),1423(a,1,2))].">
      <j1 rule="para" parents="11507 1423"/>
    </justification>
  </clause>

  <clause id="11575">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(y,y)) = y
    ]]></literal>
    <justification jstring="[para(11507(a,1),1647(a,2,2)),rewrite([11511(4)]),flip(a)].">
      <j1 rule="para" parents="11507 1647"/>
      <j2 rule="rewrite" parents="11511"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11611">
    <literal><![CDATA[
      f(f(x,f(y,f(z,f(z,z)))),f(f(u,f(u,y)),x)) = x
    ]]></literal>
    <justification jstring="[para(11507(a,2),639(a,1,1,2))].">
      <j1 rule="para" parents="11507 639"/>
    </justification>
  </clause>

  <clause id="11615">
    <literal><![CDATA[
      f(x,f(f(y,f(y,x)),f(z,f(z,z)))) = f(z,f(z,z))
    ]]></literal>
    <justification jstring="[para(11507(a,1),656(a,1,2)),rewrite([280(8),421(6)])].">
      <j1 rule="para" parents="11507 656"/>
      <j2 rule="rewrite" parents="280 421"/>
    </justification>
  </clause>

  <clause id="11683">
    <literal><![CDATA[
      f(x,f(x,x)) = c_0
    ]]></literal>
    <justification jstring="[new_symbol(11548)].">
      <j1 rule="new_symbol" parents="11548"/>
    </justification>
  </clause>

  <clause id="11741">
    <literal><![CDATA[
      f(x,f(c_0,f(y,f(y,x)))) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(11615),rewrite([11683(4),421(4),11683(7)])].">
      <j1 rule="back_rewrite" parents="11615"/>
      <j2 rule="rewrite" parents="11683 421 11683"/>
    </justification>
  </clause>

  <clause id="11745">
    <literal><![CDATA[
      f(f(x,f(y,c_0)),f(f(z,f(z,y)),x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(11611),rewrite([11683(2)])].">
      <j1 rule="back_rewrite" parents="11611"/>
      <j2 rule="rewrite" parents="11683"/>
    </justification>
  </clause>

  <clause id="11777">
    <literal><![CDATA[
      f(c_0,f(x,x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(11575),rewrite([11683(2)])].">
      <j1 rule="back_rewrite" parents="11575"/>
      <j2 rule="rewrite" parents="11683"/>
    </justification>
  </clause>

  <clause id="11780">
    <literal><![CDATA[
      f(x,f(y,c_0)) = f(x,f(y,x))
    ]]></literal>
    <justification jstring="[back_rewrite(11572),rewrite([11683(2)])].">
      <j1 rule="back_rewrite" parents="11572"/>
      <j2 rule="rewrite" parents="11683"/>
    </justification>
  </clause>

  <clause id="11783">
    <literal><![CDATA[
      f(c_0,f(c_0,x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(11569),rewrite([11683(2),11683(3)])].">
      <j1 rule="back_rewrite" parents="11569"/>
      <j2 rule="rewrite" parents="11683 11683"/>
    </justification>
  </clause>

  <clause id="11785">
    <literal><![CDATA[
      f(x,f(x,f(y,c_0))) = f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(11567),rewrite([11683(2)])].">
      <j1 rule="back_rewrite" parents="11567"/>
      <j2 rule="rewrite" parents="11683"/>
    </justification>
  </clause>

  <clause id="11787">
    <literal><![CDATA[
      f(x,f(c_0,c_0)) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(11565),rewrite([11683(2),11683(3),11683(6)])].">
      <j1 rule="back_rewrite" parents="11565"/>
      <j2 rule="rewrite" parents="11683 11683 11683"/>
    </justification>
  </clause>

  <clause id="11800">
    <literal><![CDATA[
      f(x,f(c_0,f(y,x))) = f(y,x)
    ]]></literal>
    <justification jstring="[back_rewrite(11549),rewrite([11683(3),421(3)])].">
      <j1 rule="back_rewrite" parents="11549"/>
      <j2 rule="rewrite" parents="11683 421"/>
    </justification>
  </clause>

  <clause id="11808">
    <literal><![CDATA[
      f(c_0,x) = f(x,x)
    ]]></literal>
    <justification jstring="[back_rewrite(11538),rewrite([11683(2),11683(4),11683(5),11787(6),421(4),11777(4)])].">
      <j1 rule="back_rewrite" parents="11538"/>
      <j2 rule="rewrite" parents="11683 11683 11683 11787 421 11777"/>
    </justification>
  </clause>

  <clause id="11824">
    <literal><![CDATA[
      f(f(x,y),f(y,c_0)) = y
    ]]></literal>
    <justification jstring="[back_rewrite(11513),rewrite([11683(3)])].">
      <j1 rule="back_rewrite" parents="11513"/>
      <j2 rule="rewrite" parents="11683"/>
    </justification>
  </clause>

  <clause id="12087">
    <literal><![CDATA[
      f(x,f(f(y,x),f(c_0,f(z,y)))) = f(x,x)
    ]]></literal>
    <justification jstring="[back_rewrite(912),rewrite([11780(5,R),421(4)])].">
      <j1 rule="back_rewrite" parents="912"/>
      <j2 rule="rewrite" parents="11780 421"/>
    </justification>
  </clause>

  <clause id="12088">
    <literal><![CDATA[
      f(x,f(f(y,x),f(c_0,f(y,z)))) = f(x,x)
    ]]></literal>
    <justification jstring="[back_rewrite(886),rewrite([11780(5,R),421(4)])].">
      <j1 rule="back_rewrite" parents="886"/>
      <j2 rule="rewrite" parents="11780 421"/>
    </justification>
  </clause>

  <clause id="12508">
    <literal><![CDATA[
      f(f(c1,f(c1,c2)),f(c1,f(c1,c3))) != f(c_0,f(c1,f(c2,c3)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(11508),rewrite([11808(11,R)]),flip(a)].">
      <j1 rule="back_rewrite" parents="11508"/>
      <j2 rule="rewrite" parents="11808"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14132">
    <literal><![CDATA[
      f(f(c_0,x),f(f(c_0,f(x,y)),z)) = x
    ]]></literal>
    <justification jstring="[para(334(a,1),11783(a,1,2)),rewrite([11783(4)]),flip(a)].">
      <j1 rule="para" parents="334 11783"/>
      <j2 rule="rewrite" parents="11783"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14145">
    <literal><![CDATA[
      f(f(c_0,f(x,y)),f(f(c_0,y),z)) = f(x,y)
    ]]></literal>
    <justification jstring="[para(401(a,1),11783(a,1,2)),rewrite([11783(5)]),flip(a)].">
      <j1 rule="para" parents="401 11783"/>
      <j2 rule="rewrite" parents="11783"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14162">
    <literal><![CDATA[
      f(x,f(x,f(c_0,y))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(11808(a,2),1415(a,1,2,2))].">
      <j1 rule="para" parents="11808 1415"/>
    </justification>
  </clause>

  <clause id="14163">
    <literal><![CDATA[
      f(f(c_0,x),y) = f(y,f(y,x))
    ]]></literal>
    <justification jstring="[para(11808(a,2),1422(a,1,1))].">
      <j1 rule="para" parents="11808 1422"/>
    </justification>
  </clause>

  <clause id="14164">
    <literal><![CDATA[
      f(x,f(c_0,y)) = f(x,f(y,x))
    ]]></literal>
    <justification jstring="[para(11808(a,2),1423(a,1,2))].">
      <j1 rule="para" parents="11808 1423"/>
    </justification>
  </clause>

  <clause id="14176">
    <literal><![CDATA[
      f(f(c_0,x),y) = f(y,f(x,x))
    ]]></literal>
    <justification jstring="[para(11808(a,2),545(a,2,1)),rewrite([258(3)]),flip(a)].">
      <j1 rule="para" parents="11808 545"/>
      <j2 rule="rewrite" parents="258"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14992">
    <literal><![CDATA[
      f(f(x,y),f(c_0,f(x,f(c_0,f(y,z))))) = c_0
    ]]></literal>
    <justification jstring="[para(467(a,1),11741(a,1,2,2,2)),rewrite([421(5),14164(5,R)])].">
      <j1 rule="para" parents="467 11741"/>
      <j2 rule="rewrite" parents="421 14164"/>
    </justification>
  </clause>

  <clause id="15062">
    <literal><![CDATA[
      f(f(f(x,f(x,y)),z),f(c_0,f(z,y))) = c_0
    ]]></literal>
    <justification jstring="[para(639(a,1),11741(a,1,2,2,2)),rewrite([421(7),1415(7)])].">
      <j1 rule="para" parents="639 11741"/>
      <j2 rule="rewrite" parents="421 1415"/>
    </justification>
  </clause>

  <clause id="17171">
    <literal><![CDATA[
      f(x,f(c_0,f(y,f(c_0,f(x,z))))) = f(y,f(c_0,f(x,z)))
    ]]></literal>
    <justification jstring="[para(1089(a,1),11785(a,1,2)),rewrite([421(8),11783(8),421(11)]),flip(a)].">
      <j1 rule="para" parents="1089 11785"/>
      <j2 rule="rewrite" parents="421 11783 421"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="17607">
    <literal><![CDATA[
      f(f(f(c_0,x),y),f(c_0,f(z,x))) = f(z,x)
    ]]></literal>
    <justification jstring="[para(401(a,1),11800(a,1,2)),rewrite([14145(14)])].">
      <j1 rule="para" parents="401 11800"/>
      <j2 rule="rewrite" parents="14145"/>
    </justification>
  </clause>

  <clause id="21469">
    <literal><![CDATA[
      f(x,f(f(c_0,f(y,f(x,y))),z)) = f(x,x)
    ]]></literal>
    <justification jstring="[para(1647(a,1),14132(a,1,2,1,2)),rewrite([11777(3)])].">
      <j1 rule="para" parents="1647 14132"/>
      <j2 rule="rewrite" parents="11777"/>
    </justification>
  </clause>

  <clause id="24762">
    <literal><![CDATA[
      f(f(f(x,y),z),f(z,f(u,f(x,x)))) = z
    ]]></literal>
    <justification jstring="[para(392(a,1),1658(a,1,2,2)),rewrite([11808(5,R),11777(5),421(5),533(5)])].">
      <j1 rule="para" parents="392 1658"/>
      <j2 rule="rewrite" parents="11808 11777 421 533"/>
    </justification>
  </clause>

  <clause id="26303">
    <literal><![CDATA[
      f(x,f(y,f(f(z,f(z,y)),x))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(1422(a,1),2946(a,1,2,2,1)),rewrite([11808(3,R),11777(3)])].">
      <j1 rule="para" parents="1422 2946"/>
      <j2 rule="rewrite" parents="11808 11777"/>
    </justification>
  </clause>

  <clause id="27586">
    <literal><![CDATA[
      f(x,f(y,f(z,f(x,y)))) = f(x,f(y,y))
    ]]></literal>
    <justification jstring="[para(9584(a,1),441(a,1,2)),flip(a)].">
      <j1 rule="para" parents="9584 441"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="28475">
    <literal><![CDATA[
      f(f(c_0,f(x,y)),f(c_0,f(y,z))) = f(f(c_0,f(x,y)),z)
    ]]></literal>
    <justification jstring="[para(12087(a,1),10652(a,1,2,2)),rewrite([264(6),11808(10,R)]),flip(a)].">
      <j1 rule="para" parents="12087 10652"/>
      <j2 rule="rewrite" parents="264 11808"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="28533">
    <literal><![CDATA[
      f(f(c_0,f(x,y)),f(c_0,f(x,z))) = f(f(c_0,f(x,y)),z)
    ]]></literal>
    <justification jstring="[para(12088(a,1),10652(a,1,2,2)),rewrite([264(6),11808(10,R)]),flip(a)].">
      <j1 rule="para" parents="12088 10652"/>
      <j2 rule="rewrite" parents="264 11808"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="30231">
    <literal><![CDATA[
      f(f(c_0,f(x,y)),z) = f(x,f(c_0,f(y,z)))
    ]]></literal>
    <justification jstring="[para(14992(a,1),464(a,1,2,2)),rewrite([421(9),421(10),28533(10),28475(7),11808(17,R),11783(12)])].">
      <j1 rule="para" parents="14992 464"/>
      <j2 rule="rewrite" parents="421 421 28533 28475 11808 11783"/>
    </justification>
  </clause>

  <clause id="30496">
    <literal><![CDATA[
      f(x,f(c_0,f(y,z))) = f(y,f(c_0,f(x,z)))
    ]]></literal>
    <justification jstring="[back_rewrite(28533),rewrite([30231(7),17171(7),30231(8)])].">
      <j1 rule="back_rewrite" parents="28533"/>
      <j2 rule="rewrite" parents="30231 17171 30231"/>
    </justification>
  </clause>

  <clause id="30816">
    <literal><![CDATA[
      f(x,f(y,f(c_0,f(f(x,y),z)))) = f(x,x)
    ]]></literal>
    <justification jstring="[back_rewrite(21469),rewrite([30231(5)])].">
      <j1 rule="back_rewrite" parents="21469"/>
      <j2 rule="rewrite" parents="30231"/>
    </justification>
  </clause>

  <clause id="31904">
    <literal><![CDATA[
      f(c_0,f(x,f(f(y,f(y,x)),z))) = f(c_0,f(x,f(c_0,z)))
    ]]></literal>
    <justification jstring="[para(15062(a,1),11745(a,1,2)),rewrite([421(5),30231(9),11824(7),421(5),421(10)]),flip(a)].">
      <j1 rule="para" parents="15062 11745"/>
      <j2 rule="rewrite" parents="421 30231 11824 421 421"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="32481">
    <literal><![CDATA[
      f(x,f(c_0,f(y,z))) = f(z,f(z,f(x,y)))
    ]]></literal>
    <justification jstring="[para(584(a,1),14163(a,1,1)),rewrite([421(3),30231(4),421(10),17607(11)])].">
      <j1 rule="para" parents="584 14163"/>
      <j2 rule="rewrite" parents="421 30231 421 17607"/>
    </justification>
  </clause>

  <clause id="33782">
    <literal><![CDATA[
      f(f(x,f(x,y)),f(x,f(y,z))) = f(c_0,f(x,f(y,z)))
    ]]></literal>
    <justification jstring="[para(598(a,1),549(a,1,2)),rewrite([11808(5,R),421(6)]),flip(a)].">
      <j1 rule="para" parents="598 549"/>
      <j2 rule="rewrite" parents="11808 421"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="38529">
    <literal><![CDATA[
      f(f(x,x),f(c_0,f(f(x,y),z))) = f(f(x,x),z)
    ]]></literal>
    <justification jstring="[para(24762(a,1),10652(a,1,2)),rewrite([11808(8,R)]),flip(a)].">
      <j1 rule="para" parents="24762 10652"/>
      <j2 rule="rewrite" parents="11808"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="39061">
    <literal><![CDATA[
      f(x,f(f(y,f(y,x)),z)) = f(x,f(c_0,z))
    ]]></literal>
    <justification jstring="[para(26303(a,1),163(a,1,2)),rewrite([11808(9,R),31904(6),14176(7,R),11800(8)]),flip(a)].">
      <j1 rule="para" parents="26303 163"/>
      <j2 rule="rewrite" parents="11808 31904 14176 11800"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="42603">
    <literal><![CDATA[
      f(x,f(y,f(c_0,f(x,z)))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(30496(a,1),1380(a,2,2)),rewrite([11808(7,R),11783(5)]),flip(a)].">
      <j1 rule="para" parents="30496 1380"/>
      <j2 rule="rewrite" parents="11808 11783"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="42966">
    <literal><![CDATA[
      f(x,f(f(y,y),z)) = f(x,f(f(y,x),z))
    ]]></literal>
    <justification jstring="[para(30816(a,1),27586(a,1,2,2)),rewrite([421(6),38529(6),11808(12,R),11783(9)])].">
      <j1 rule="para" parents="30816 27586"/>
      <j2 rule="rewrite" parents="421 38529 11808 11783"/>
    </justification>
  </clause>

  <clause id="43790">
    <literal><![CDATA[
      f(x,f(c_0,f(y,f(x,f(x,z))))) = f(x,f(c_0,f(y,z)))
    ]]></literal>
    <justification jstring="[para(32481(a,2),307(a,1,2)),rewrite([11808(3,R),11808(7,R),30231(8),42603(9),30231(9),42603(7),11808(9,R),30231(10)])].">
      <j1 rule="para" parents="32481 307"/>
      <j2 rule="rewrite" parents="11808 11808 30231 42603 30231 42603 11808 30231"/>
    </justification>
  </clause>

  <clause id="44163">
    <literal><![CDATA[
      f(x,f(y,f(x,f(x,z)))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(39061(a,1),444(a,1,2)),rewrite([421(3),1423(4,R),324(3),421(3),43790(6),14162(5),421(5),1423(6,R),324(5),421(4)]),flip(a)].">
      <j1 rule="para" parents="39061 444"/>
      <j2 rule="rewrite" parents="421 1423 324 421 43790 14162 421 1423 324 421"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="46136">
    <literal><![CDATA[
      f(f(x,f(x,y)),f(x,z)) = f(c_0,f(x,f(y,f(x,z))))
    ]]></literal>
    <justification jstring="[para(42966(a,2),692(a,1,2)),rewrite([421(2),11808(5,R),30231(7),14162(8),33782(6),421(7)]),flip(a)].">
      <j1 rule="para" parents="42966 692"/>
      <j2 rule="rewrite" parents="421 11808 30231 14162 33782 421"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="46227">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(12508),rewrite([46136(11),44163(10)]),xx(a)].">
      <j1 rule="back_rewrite" parents="12508"/>
      <j2 rule="rewrite" parents="46136 44163"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
