<?xml version="1.0" encoding="ISO-8859-1"?>

<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>sh1.out</source>

<heading><![CDATA[
Prover9 (32) version 2009-02A, February 2009.
Process 11744 was started by mccune on cleo,
Wed Feb 25 09:39:24 2009
The command was "/home/mccune/bin/prover9 -f sh1.in".
]]></heading>

<proof number="1" length="168" max_count="87">

<comments><![CDATA[
% Proof 1 at 544.09 (+ 1.73) seconds: "Sheffer".
% Length of proof is 168.
% Level of proof is 30.
% Maximum clause weight is 87.
% Given clauses 1327.

]]></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="32">
    <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(f(f(x,f(y,z)),u),u)))) = f(z,f(f(x,z),z))
    ]]></literal>
    <justification jstring="[para(2(a,1),15(a,1,1,2,1))].">
      <j1 rule="para" parents="2 15"/>
    </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(w,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),f(f(z,f(f(x,z),z)),f(w,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="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="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="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="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="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="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="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="340">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(x,f(z,y)),u))) = f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(32),rewrite([298(6),269(3),269(6),269(9)])].">
      <j1 rule="back_rewrite" parents="32"/>
      <j2 rule="rewrite" parents="298 269 269 269"/>
    </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="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="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="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="474">
    <literal><![CDATA[
      f(f(x,x),f(f(y,f(x,y)),f(x,z))) = f(y,f(x,y))
    ]]></literal>
    <justification jstring="[para(427(a,1),453(a,1,1))].">
      <j1 rule="para" parents="427 453"/>
    </justification>
  </clause>

  <clause id="475">
    <literal><![CDATA[
      f(x,f(f(x,f(y,z)),f(f(y,x),u))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(453(a,1),453(a,1,1))].">
      <j1 rule="para" parents="453 453"/>
    </justification>
  </clause>

  <clause id="505">
    <literal><![CDATA[
      f(x,f(f(f(x,x),y),f(x,z))) = f(x,z)
    ]]></literal>
    <justification jstring="[para(324(a,1),454(a,1,1))].">
      <j1 rule="para" parents="324 454"/>
    </justification>
  </clause>

  <clause id="518">
    <literal><![CDATA[
      f(x,f(f(y,z),f(x,z))) = f(x,z)
    ]]></literal>
    <justification jstring="[para(353(a,1),455(a,1,2)),rewrite([421(4)])].">
      <j1 rule="para" parents="353 455"/>
      <j2 rule="rewrite" parents="421"/>
    </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="648">
    <literal><![CDATA[
      f(f(f(x,f(y,x)),z),f(f(y,y),z)) = z
    ]]></literal>
    <justification jstring="[para(427(a,1),519(a,1,2,1))].">
      <j1 rule="para" parents="427 519"/>
    </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="800">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(z,y),f(x,y)))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(518(a,1),427(a,1,2,2)),rewrite([421(5)])].">
      <j1 rule="para" parents="518 427"/>
      <j2 rule="rewrite" parents="421"/>
    </justification>
  </clause>

  <clause id="804">
    <literal><![CDATA[
      f(f(f(x,y),z),f(x,f(f(u,z),f(f(x,y),z)))) = f(f(u,z),f(f(x,y),z))
    ]]></literal>
    <justification jstring="[para(518(a,1),454(a,1,2)),rewrite([421(5),421(8)])].">
      <j1 rule="para" parents="518 454"/>
      <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="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="1091">
    <literal><![CDATA[
      f(f(x,f(y,f(x,f(z,u)))),f(u,x)) = x
    ]]></literal>
    <justification jstring="[para(531(a,1),330(a,1,2))].">
      <j1 rule="para" parents="531 330"/>
    </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="1547">
    <literal><![CDATA[
      f(x,f(f(y,y),x)) = f(x,y)
    ]]></literal>
    <justification jstring="[para(421(a,1),1415(a,1,2))].">
      <j1 rule="para" parents="421 1415"/>
    </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="1663">
    <literal><![CDATA[
      f(f(f(f(x,x),y),z),f(z,f(y,x))) = z
    ]]></literal>
    <justification jstring="[para(1422(a,2),466(a,1,1,1))].">
      <j1 rule="para" parents="1422 466"/>
    </justification>
  </clause>

  <clause id="1898">
    <literal><![CDATA[
      f(f(x,f(f(y,y),z)),f(x,f(z,y))) = x
    ]]></literal>
    <justification jstring="[para(1547(a,1),353(a,1,2,2))].">
      <j1 rule="para" parents="1547 353"/>
    </justification>
  </clause>

  <clause id="2565">
    <literal><![CDATA[
      f(x,f(f(y,f(x,y)),f(x,z))) = f(x,z)
    ]]></literal>
    <justification jstring="[para(1647(a,1),505(a,1,2,1))].">
      <j1 rule="para" parents="1647 505"/>
    </justification>
  </clause>

  <clause id="2820">
    <literal><![CDATA[
      f(f(x,x),f(y,f(f(x,f(x,x)),f(z,x)))) = x
    ]]></literal>
    <justification jstring="[para(304(a,1),533(a,1,2)),rewrite([280(3),280(5),421(3)]),flip(a)].">
      <j1 rule="para" parents="304 533"/>
      <j2 rule="rewrite" parents="280 280 421"/>
      <j2 rule="flip"/>
    </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="11509">
    <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="11510">
    <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,11509)].">
      <j1 rule="back_unit_del" parents="1398"/>
      <j2 rule="unit_del" parents="11509"/>
    </justification>
  </clause>

  <clause id="11513">
    <literal><![CDATA[
      f(f(x,x),f(y,f(y,y))) = x
    ]]></literal>
    <justification jstring="[para(11509(a,2),163(a,1))].">
      <j1 rule="para" parents="11509 163"/>
    </justification>
  </clause>

  <clause id="11534">
    <literal><![CDATA[
      f(f(x,x),f(f(x,y),f(z,f(z,z)))) = f(z,f(z,z))
    ]]></literal>
    <justification jstring="[para(11509(a,1),472(a,1,1))].">
      <j1 rule="para" parents="11509 472"/>
    </justification>
  </clause>

  <clause id="11539">
    <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(11509(a,2),304(a,1,2,2,2)),rewrite([421(2),421(5)])].">
      <j1 rule="para" parents="11509 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(11509(a,2),307(a,1,2)),rewrite([421(3),11534(7),421(4)])].">
      <j1 rule="para" parents="11509 307"/>
      <j2 rule="rewrite" parents="421 11534 421"/>
    </justification>
  </clause>

  <clause id="11559">
    <literal><![CDATA[
      f(x,f(f(y,f(y,y)),f(y,f(y,y)))) = f(x,f(x,x))
    ]]></literal>
    <justification jstring="[para(11509(a,1),1380(a,2,2))].">
      <j1 rule="para" parents="11509 1380"/>
    </justification>
  </clause>

  <clause id="11561">
    <literal><![CDATA[
      f(x,f(x,f(y,f(z,f(z,z))))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(11509(a,2),1415(a,1,2,2))].">
      <j1 rule="para" parents="11509 1415"/>
    </justification>
  </clause>

  <clause id="11562">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(x,f(x,x)),y)) = y
    ]]></literal>
    <justification jstring="[para(11509(a,1),1422(a,1)),rewrite([280(3)]),flip(a)].">
      <j1 rule="para" parents="11509 1422"/>
      <j2 rule="rewrite" parents="280"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11565">
    <literal><![CDATA[
      f(x,f(y,f(z,f(z,z)))) = f(x,f(y,x))
    ]]></literal>
    <justification jstring="[para(11509(a,2),1423(a,1,2))].">
      <j1 rule="para" parents="11509 1423"/>
    </justification>
  </clause>

  <clause id="11568">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(y,y)) = y
    ]]></literal>
    <justification jstring="[para(11509(a,1),1647(a,2,2)),rewrite([11513(4)]),flip(a)].">
      <j1 rule="para" parents="11509 1647"/>
      <j2 rule="rewrite" parents="11513"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11569">
    <literal><![CDATA[
      f(f(x,f(y,f(y,y))),z) = f(z,f(x,z))
    ]]></literal>
    <justification jstring="[para(11509(a,2),1647(a,1,1))].">
      <j1 rule="para" parents="11509 1647"/>
    </justification>
  </clause>

  <clause id="11601">
    <literal><![CDATA[
      f(x,f(f(y,f(x,y)),f(z,f(z,z)))) = f(z,f(z,z))
    ]]></literal>
    <justification jstring="[para(11509(a,1),648(a,1,2)),rewrite([280(8),421(6)])].">
      <j1 rule="para" parents="11509 648"/>
      <j2 rule="rewrite" parents="280 421"/>
    </justification>
  </clause>

  <clause id="11669">
    <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="11727">
    <literal><![CDATA[
      f(x,f(c_0,f(y,f(x,y)))) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(11601),rewrite([11669(4),421(4),11669(7)])].">
      <j1 rule="back_rewrite" parents="11601"/>
      <j2 rule="rewrite" parents="11669 421 11669"/>
    </justification>
  </clause>

  <clause id="11755">
    <literal><![CDATA[
      f(f(x,c_0),y) = f(y,f(x,y))
    ]]></literal>
    <justification jstring="[back_rewrite(11569),rewrite([11669(2)])].">
      <j1 rule="back_rewrite" parents="11569"/>
      <j2 rule="rewrite" parents="11669"/>
    </justification>
  </clause>

  <clause id="11756">
    <literal><![CDATA[
      f(c_0,f(x,x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(11568),rewrite([11669(2)])].">
      <j1 rule="back_rewrite" parents="11568"/>
      <j2 rule="rewrite" parents="11669"/>
    </justification>
  </clause>

  <clause id="11759">
    <literal><![CDATA[
      f(x,f(y,c_0)) = f(x,f(y,x))
    ]]></literal>
    <justification jstring="[back_rewrite(11565),rewrite([11669(2)])].">
      <j1 rule="back_rewrite" parents="11565"/>
      <j2 rule="rewrite" parents="11669"/>
    </justification>
  </clause>

  <clause id="11762">
    <literal><![CDATA[
      f(c_0,f(c_0,x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(11562),rewrite([11669(2),11669(3)])].">
      <j1 rule="back_rewrite" parents="11562"/>
      <j2 rule="rewrite" parents="11669 11669"/>
    </justification>
  </clause>

  <clause id="11763">
    <literal><![CDATA[
      f(x,f(x,f(y,c_0))) = f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(11561),rewrite([11669(2)])].">
      <j1 rule="back_rewrite" parents="11561"/>
      <j2 rule="rewrite" parents="11669"/>
    </justification>
  </clause>

  <clause id="11765">
    <literal><![CDATA[
      f(x,f(c_0,c_0)) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(11559),rewrite([11669(2),11669(3),11669(6)])].">
      <j1 rule="back_rewrite" parents="11559"/>
      <j2 rule="rewrite" parents="11669 11669 11669"/>
    </justification>
  </clause>

  <clause id="11782">
    <literal><![CDATA[
      f(c_0,x) = f(x,x)
    ]]></literal>
    <justification jstring="[back_rewrite(11539),rewrite([11669(2),11669(4),11669(5),11765(6),421(4),11756(4)])].">
      <j1 rule="back_rewrite" parents="11539"/>
      <j2 rule="rewrite" parents="11669 11669 11669 11765 421 11756"/>
    </justification>
  </clause>

  <clause id="11871">
    <literal><![CDATA[
      f(f(x,x),f(y,f(c_0,f(z,x)))) = x
    ]]></literal>
    <justification jstring="[back_rewrite(2820),rewrite([11669(3)])].">
      <j1 rule="back_rewrite" parents="2820"/>
      <j2 rule="rewrite" parents="11669"/>
    </justification>
  </clause>

  <clause id="12007">
    <literal><![CDATA[
      f(x,f(f(y,x),f(c_0,f(y,z)))) = f(x,x)
    ]]></literal>
    <justification jstring="[back_rewrite(886),rewrite([11759(5,R),421(4)])].">
      <j1 rule="back_rewrite" parents="886"/>
      <j2 rule="rewrite" parents="11759 421"/>
    </justification>
  </clause>

  <clause id="12008">
    <literal><![CDATA[
      f(x,f(f(x,y),f(c_0,f(z,y)))) = f(x,x)
    ]]></literal>
    <justification jstring="[back_rewrite(800),rewrite([11759(5,R),421(4)])].">
      <j1 rule="back_rewrite" parents="800"/>
      <j2 rule="rewrite" parents="11759 421"/>
    </justification>
  </clause>

  <clause id="12331">
    <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(11510),rewrite([11782(11,R)]),flip(a)].">
      <j1 rule="back_rewrite" parents="11510"/>
      <j2 rule="rewrite" parents="11782"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13857">
    <literal><![CDATA[
      f(f(c_0,x),f(f(c_0,f(y,x)),z)) = x
    ]]></literal>
    <justification jstring="[para(340(a,1),11762(a,1,2)),rewrite([11762(4)]),flip(a)].">
      <j1 rule="para" parents="340 11762"/>
      <j2 rule="rewrite" parents="11762"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13883">
    <literal><![CDATA[
      f(x,f(c_0,y)) = f(x,f(x,y))
    ]]></literal>
    <justification jstring="[para(11782(a,2),1380(a,1,2))].">
      <j1 rule="para" parents="11782 1380"/>
    </justification>
  </clause>

  <clause id="13884">
    <literal><![CDATA[
      f(x,f(x,f(c_0,y))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(11782(a,2),1415(a,1,2,2))].">
      <j1 rule="para" parents="11782 1415"/>
    </justification>
  </clause>

  <clause id="13886">
    <literal><![CDATA[
      f(x,f(c_0,y)) = f(x,f(y,x))
    ]]></literal>
    <justification jstring="[para(11782(a,2),1423(a,1,2))].">
      <j1 rule="para" parents="11782 1423"/>
    </justification>
  </clause>

  <clause id="14975">
    <literal><![CDATA[
      f(f(x,y),f(c_0,f(y,f(c_0,f(x,z))))) = c_0
    ]]></literal>
    <justification jstring="[para(472(a,1),11727(a,1,2,2,2)),rewrite([421(5),13886(5,R)])].">
      <j1 rule="para" parents="472 11727"/>
      <j2 rule="rewrite" parents="421 13886"/>
    </justification>
  </clause>

  <clause id="16966">
    <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),11763(a,1,2)),rewrite([421(8),11762(8),421(11)]),flip(a)].">
      <j1 rule="para" parents="1089 11763"/>
      <j2 rule="rewrite" parents="421 11762 421"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16968">
    <literal><![CDATA[
      f(x,f(c_0,f(y,f(c_0,f(z,x))))) = f(y,f(c_0,f(z,x)))
    ]]></literal>
    <justification jstring="[para(1091(a,1),11763(a,1,2)),rewrite([421(8),11762(8),421(11)]),flip(a)].">
      <j1 rule="para" parents="1091 11763"/>
      <j2 rule="rewrite" parents="421 11762 421"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20848">
    <literal><![CDATA[
      f(x,f(y,f(x,f(f(y,x),z)))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(1089(a,1),475(a,1,2)),flip(a)].">
      <j1 rule="para" parents="1089 475"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21026">
    <literal><![CDATA[
      f(f(x,x),f(y,f(y,f(z,x)))) = x
    ]]></literal>
    <justification jstring="[para(13883(a,1),11871(a,1,2))].">
      <j1 rule="para" parents="13883 11871"/>
    </justification>
  </clause>

  <clause id="21274">
    <literal><![CDATA[
      f(f(x,y),f(f(f(c_0,f(z,x)),u),y)) = y
    ]]></literal>
    <justification jstring="[para(13857(a,1),519(a,1,2,1)),rewrite([421(7)])].">
      <j1 rule="para" parents="13857 519"/>
      <j2 rule="rewrite" parents="421"/>
    </justification>
  </clause>

  <clause id="21704">
    <literal><![CDATA[
      f(f(x,x),f(c_0,f(y,f(y,f(z,x))))) = c_0
    ]]></literal>
    <justification jstring="[para(21026(a,1),1380(a,2,2)),rewrite([11782(8,R),421(9),11669(9)])].">
      <j1 rule="para" parents="21026 1380"/>
      <j2 rule="rewrite" parents="11782 421 11669"/>
    </justification>
  </clause>

  <clause id="24376">
    <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([11782(5,R),11756(5),421(5),533(5)])].">
      <j1 rule="para" parents="392 1658"/>
      <j2 rule="rewrite" parents="11782 11756 421 533"/>
    </justification>
  </clause>

  <clause id="24525">
    <literal><![CDATA[
      f(f(f(x,f(y,x)),z),f(z,f(y,u))) = z
    ]]></literal>
    <justification jstring="[para(474(a,1),1663(a,1,1,1)),rewrite([421(8),2565(8)])].">
      <j1 rule="para" parents="474 1663"/>
      <j2 rule="rewrite" parents="421 2565"/>
    </justification>
  </clause>

  <clause id="25368">
    <literal><![CDATA[
      f(f(x,f(y,f(z,y))),f(x,f(z,u))) = x
    ]]></literal>
    <justification jstring="[para(474(a,1),1898(a,1,1,2)),rewrite([421(8),2565(8)])].">
      <j1 rule="para" parents="474 1898"/>
      <j2 rule="rewrite" parents="421 2565"/>
    </justification>
  </clause>

  <clause id="26193">
    <literal><![CDATA[
      f(x,f(y,f(f(f(y,y),z),x))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(163(a,1),2946(a,1,2,1))].">
      <j1 rule="para" parents="163 2946"/>
    </justification>
  </clause>

  <clause id="29010">
    <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="31233">
    <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(12007(a,1),10652(a,1,2,2)),rewrite([264(6),11782(10,R)]),flip(a)].">
      <j1 rule="para" parents="12007 10652"/>
      <j2 rule="rewrite" parents="264 11782"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="31460">
    <literal><![CDATA[
      f(f(c_0,f(x,y)),f(c_0,f(z,y))) = f(f(c_0,f(x,y)),z)
    ]]></literal>
    <justification jstring="[para(12008(a,1),10652(a,1,2,2)),rewrite([324(6),11782(10,R)]),flip(a)].">
      <j1 rule="para" parents="12008 10652"/>
      <j2 rule="rewrite" parents="324 11782"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="51689">
    <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([11782(5,R),421(6)]),flip(a)].">
      <j1 rule="para" parents="598 549"/>
      <j2 rule="rewrite" parents="11782 421"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="69048">
    <literal><![CDATA[
      f(x,f(y,f(f(x,y),z))) = f(x,f(y,y))
    ]]></literal>
    <justification jstring="[para(20848(a,1),441(a,1,2)),flip(a)].">
      <j1 rule="para" parents="20848 441"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="87490">
    <literal><![CDATA[
      f(f(c_0,f(x,f(c_0,f(y,z)))),f(y,x)) = c_0
    ]]></literal>
    <justification jstring="[para(14975(a,1),21704(a,1,2,2,2)),rewrite([11782(9,R),421(10),11762(11)])].">
      <j1 rule="para" parents="14975 21704"/>
      <j2 rule="rewrite" parents="11782 421 11762"/>
    </justification>
  </clause>

  <clause id="90853">
    <literal><![CDATA[
      f(f(x,x),f(c_0,f(f(x,y),z))) = f(f(x,x),z)
    ]]></literal>
    <justification jstring="[para(24376(a,1),10652(a,1,2)),rewrite([11782(8,R)]),flip(a)].">
      <j1 rule="para" parents="24376 10652"/>
      <j2 rule="rewrite" parents="11782"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="92607">
    <literal><![CDATA[
      f(x,f(y,f(c_0,f(f(x,y),z)))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(26193(a,1),584(a,1)),rewrite([11782(6,R),421(6),421(7)]),flip(a)].">
      <j1 rule="para" parents="26193 584"/>
      <j2 rule="rewrite" parents="11782 421 421"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="94473">
    <literal><![CDATA[
      f(x,f(c_0,f(f(y,f(x,y)),z))) = f(x,z)
    ]]></literal>
    <justification jstring="[para(24525(a,1),29010(a,1,2)),rewrite([11782(8,R)]),flip(a)].">
      <j1 rule="para" parents="24525 29010"/>
      <j2 rule="rewrite" parents="11782"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="94474">
    <literal><![CDATA[
      f(x,f(c_0,f(y,f(z,f(x,z))))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(25368(a,1),29010(a,1,2)),rewrite([11782(8,R)]),flip(a)].">
      <j1 rule="para" parents="25368 29010"/>
      <j2 rule="rewrite" parents="11782"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="108306">
    <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(21274(a,1),69048(a,1,2)),rewrite([11782(10,R)]),flip(a)].">
      <j1 rule="para" parents="21274 69048"/>
      <j2 rule="rewrite" parents="11782"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="115332">
    <literal><![CDATA[
      f(f(c_0,f(x,y)),z) = f(y,f(c_0,f(x,z)))
    ]]></literal>
    <justification jstring="[para(87490(a,1),1422(a,2,2)),rewrite([11782(3,R),108306(10),31233(7),421(12),11762(12)])].">
      <j1 rule="para" parents="87490 1422"/>
      <j2 rule="rewrite" parents="11782 108306 31233 421 11762"/>
    </justification>
  </clause>

  <clause id="115453">
    <literal><![CDATA[
      f(x,f(c_0,f(y,z))) = f(y,f(c_0,f(x,z)))
    ]]></literal>
    <justification jstring="[back_rewrite(108306),rewrite([115332(7),16966(7),115332(8)])].">
      <j1 rule="back_rewrite" parents="108306"/>
      <j2 rule="rewrite" parents="115332 16966 115332"/>
    </justification>
  </clause>

  <clause id="115812">
    <literal><![CDATA[
      f(x,f(c_0,f(y,z))) = f(z,f(c_0,f(x,y)))
    ]]></literal>
    <justification jstring="[back_rewrite(31460),rewrite([115332(7),16968(7),115332(8)])].">
      <j1 rule="back_rewrite" parents="31460"/>
      <j2 rule="rewrite" parents="115332 16968 115332"/>
    </justification>
  </clause>

  <clause id="121312">
    <literal><![CDATA[
      f(x,f(f(y,y),z)) = f(x,f(f(y,x),z))
    ]]></literal>
    <justification jstring="[para(92607(a,1),29010(a,1,2,2)),rewrite([421(6),90853(6),11782(12,R),11762(9)])].">
      <j1 rule="para" parents="92607 29010"/>
      <j2 rule="rewrite" parents="421 90853 11782 11762"/>
    </justification>
  </clause>

  <clause id="122506">
    <literal><![CDATA[
      f(x,f(y,f(f(z,y),f(u,f(x,u))))) = f(x,f(z,y))
    ]]></literal>
    <justification jstring="[para(549(a,1),94473(a,1,2,2)),rewrite([94474(7)]),flip(a)].">
      <j1 rule="para" parents="549 94473"/>
      <j2 rule="rewrite" parents="94474"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="122508">
    <literal><![CDATA[
      f(x,f(y,f(z,f(x,z)))) = f(x,f(y,c_0))
    ]]></literal>
    <justification jstring="[para(11755(a,2),94473(a,1,2,2)),rewrite([122506(8)]),flip(a)].">
      <j1 rule="para" parents="11755 94473"/>
      <j2 rule="rewrite" parents="122506"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="124905">
    <literal><![CDATA[
      f(x,f(y,f(c_0,f(x,z)))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(115453(a,1),1380(a,2,2)),rewrite([11782(7,R),11762(5)]),flip(a)].">
      <j1 rule="para" parents="115453 1380"/>
      <j2 rule="rewrite" parents="11782 11762"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="125394">
    <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(115812(a,1),307(a,1,2)),rewrite([11782(3,R),11782(7,R),115332(8),421(9),124905(9),115332(9),124905(7),11782(9,R),115332(10)])].">
      <j1 rule="para" parents="115812 307"/>
      <j2 rule="rewrite" parents="11782 11782 115332 421 124905 115332 124905 11782 115332"/>
    </justification>
  </clause>

  <clause id="127473">
    <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(121312(a,2),692(a,1,2)),rewrite([421(2),11782(5,R),115332(7),124905(8),51689(6),421(7)]),flip(a)].">
      <j1 rule="para" parents="121312 692"/>
      <j2 rule="rewrite" parents="421 11782 115332 124905 51689 421"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="127575">
    <literal><![CDATA[
      f(c_0,f(c1,f(c2,f(c1,f(c1,c3))))) != f(c_0,f(c1,f(c2,c3)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(12331),rewrite([127473(11)])].">
      <j1 rule="back_rewrite" parents="12331"/>
      <j2 rule="rewrite" parents="127473"/>
    </justification>
  </clause>

  <clause id="127838">
    <literal><![CDATA[
      f(x,f(y,f(x,f(x,z)))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(122508(a,1),804(a,1,2)),rewrite([1423(4,R),324(3),421(5),125394(6),13884(5),1423(9,R),324(8),421(6)]),flip(a)].">
      <j1 rule="para" parents="122508 804"/>
      <j2 rule="rewrite" parents="1423 324 421 125394 13884 1423 324 421"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="127865">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(127575),rewrite([127838(10)]),xx(a)].">
      <j1 rule="back_rewrite" parents="127575"/>
      <j2 rule="rewrite" parents="127838"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
