<?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 August-2006A, August 2006.
Process 11562 was started by mccune on cleo.thornwood,
Sat Aug 12 21:08:40 2006
The command was "/home/mccune/bin/prover9 -f sh1.in".
]]></heading>

<proof number="1" length="139" max_count="87">

<comments><![CDATA[
% Proof 1 at 368.49 (+ 1.84) seconds: "Sheffer".
% Length of proof is 139.
% Level of proof is 27.
% Maximum clause weight is 87.
% Given clauses 1264.
]]></comments>

  <clause id="1">
    <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="2">
    <literal><![CDATA[
      f(f(a,a),f(a,a)) != a
    ]]></literal>
    <literal><![CDATA[
      f(a,f(b,f(b,b))) != f(a,a)
    ]]></literal>
    <literal><![CDATA[
      f(f(f(b,b),a),f(f(c,c),a)) != f(f(a,f(b,c)),f(a,f(b,c)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

  <clause id="4">
    <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(1(a,1),1(a,1,2,2))].">
      <j1 rule="para" parents="1 1"/>
    </justification>
  </clause>

  <clause id="5">
    <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(1(a,1),1(a,1,2))].">
      <j1 rule="para" parents="1 1"/>
    </justification>
  </clause>

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

  <clause id="9">
    <literal><![CDATA[
      f(f(x,f(f(y,x),x)),f(y,f(z,f(f(x,z),z)))) = y
    ]]></literal>
    <justification jstring="[para(1(a,1),4(a,1,1,1)),rewrite(1(6),1(7))].">
      <j1 rule="para" parents="1 4"/>
      <j2 rule="rewrite" parents="1 1"/>
    </justification>
  </clause>

  <clause id="23">
    <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(1(a,1),9(a,1,2))].">
      <j1 rule="para" parents="1 9"/>
    </justification>
  </clause>

  <clause id="29">
    <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(9(a,1),9(a,1,1,2,1))].">
      <j1 rule="para" parents="9 9"/>
    </justification>
  </clause>

  <clause id="39">
    <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(3(a,1),9(a,1,2,2,2,1))].">
      <j1 rule="para" parents="3 9"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      f(f(x,f(f(y,x),x)),f(y,f(f(y,y),y))) = y
    ]]></literal>
    <justification jstring="[para(3(a,1),9(a,1,2)),rewrite(9(8),9(15))].">
      <j1 rule="para" parents="3 9"/>
      <j2 rule="rewrite" parents="9 9"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      f(x,f(f(x,x),x)) = f(x,x)
    ]]></literal>
    <justification jstring="[para(9(a,1),3(a,1,2)),rewrite(9(8)),flip(a)].">
      <j1 rule="para" parents="9 3"/>
      <j2 rule="rewrite" parents="9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      f(f(x,f(f(y,x),x)),f(y,y)) = y
    ]]></literal>
    <justification jstring="[back_rewrite(40),rewrite(43(6))].">
      <j1 rule="back_rewrite" parents="40"/>
      <j2 rule="rewrite" parents="43"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      f(f(x,f(f(f(y,y),x),x)),y) = f(y,y)
    ]]></literal>
    <justification jstring="[back_rewrite(23),rewrite(43(3),43(8))].">
      <j1 rule="back_rewrite" parents="23"/>
      <j2 rule="rewrite" parents="43 43"/>
    </justification>
  </clause>

  <clause id="48">
    <literal><![CDATA[
      f(f(x,x),f(x,f(y,x))) = x
    ]]></literal>
    <justification jstring="[para(43(a,1),1(a,1,1))].">
      <j1 rule="para" parents="43 1"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      f(f(x,x),f(x,x)) = x
    ]]></literal>
    <justification jstring="[para(43(a,1),1(a,1,2)),rewrite(43(3))].">
      <j1 rule="para" parents="43 1"/>
      <j2 rule="rewrite" parents="43"/>
    </justification>
  </clause>

  <clause id="51">
    <literal><![CDATA[
      f(f(f(x,y),f(x,y)),f(f(x,y),x)) = f(x,y)
    ]]></literal>
    <justification jstring="[para(43(a,1),7(a,1,1))].">
      <j1 rule="para" parents="43 7"/>
    </justification>
  </clause>

  <clause id="57">
    <literal><![CDATA[
      f(a,f(b,f(b,b))) != f(a,a)
    ]]></literal>
    <literal><![CDATA[
      f(f(f(b,b),a),f(f(c,c),a)) != f(f(a,f(b,c)),f(a,f(b,c)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(2),rewrite(49(7)),xx(a)].">
      <j1 rule="back_rewrite" parents="2"/>
      <j2 rule="rewrite" parents="49"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="71">
    <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(7(a,1),5(a,1,1,2,1,1,2,1)),rewrite(7(29))].">
      <j1 rule="para" parents="7 5"/>
      <j2 rule="rewrite" parents="7"/>
    </justification>
  </clause>

  <clause id="81">
    <literal><![CDATA[
      f(f(x,f(f(f(y,f(f(x,y),y)),x),x)),x) = f(y,f(f(x,y),y))
    ]]></literal>
    <justification jstring="[para(45(a,1),1(a,1,2))].">
      <j1 rule="para" parents="45 1"/>
    </justification>
  </clause>

  <clause id="87">
    <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(45(a,1),9(a,1,2,2,2,1)),rewrite(48(16))].">
      <j1 rule="para" parents="45 9"/>
      <j2 rule="rewrite" parents="48"/>
    </justification>
  </clause>

  <clause id="91">
    <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(3(a,1),45(a,1,1,2,1))].">
      <j1 rule="para" parents="3 45"/>
    </justification>
  </clause>

  <clause id="109">
    <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(46(a,1),1(a,1,1,2,1)),rewrite(43(3))].">
      <j1 rule="para" parents="46 1"/>
      <j2 rule="rewrite" parents="43"/>
    </justification>
  </clause>

  <clause id="116">
    <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(46(a,1),3(a,1,2,1,2,1)),rewrite(43(21),46(33),43(31))].">
      <j1 rule="para" parents="46 3"/>
      <j2 rule="rewrite" parents="43 46 43"/>
    </justification>
  </clause>

  <clause id="118">
    <literal><![CDATA[
      f(x,f(f(f(f(y,x),f(y,x)),x),x)) = f(y,x)
    ]]></literal>
    <justification jstring="[para(46(a,1),5(a,1,1,2,1)),rewrite(43(7),49(7)),flip(a)].">
      <j1 rule="para" parents="46 5"/>
      <j2 rule="rewrite" parents="43 49"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="169">
    <literal><![CDATA[
      f(f(x,x),f(y,x)) = x
    ]]></literal>
    <justification jstring="[para(118(a,1),1(a,1,2)),rewrite(43(3))].">
      <j1 rule="para" parents="118 1"/>
      <j2 rule="rewrite" parents="43"/>
    </justification>
  </clause>

  <clause id="172">
    <literal><![CDATA[
      f(f(f(x,y),f(f(z,f(x,y)),f(x,y))),f(z,y)) = z
    ]]></literal>
    <justification jstring="[para(118(a,1),7(a,1,1,1)),rewrite(118(7),118(9))].">
      <j1 rule="para" parents="118 7"/>
      <j2 rule="rewrite" parents="118 118"/>
    </justification>
  </clause>

  <clause id="176">
    <literal><![CDATA[
      f(f(x,y),f(f(f(x,x),f(x,y)),f(x,y))) = x
    ]]></literal>
    <justification jstring="[para(7(a,1),118(a,1,2,1,1,1)),rewrite(7(9),7(15))].">
      <j1 rule="para" parents="7 118"/>
      <j2 rule="rewrite" parents="7 7"/>
    </justification>
  </clause>

  <clause id="190">
    <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(118(a,1),45(a,1,1,2,1))].">
      <j1 rule="para" parents="118 45"/>
    </justification>
  </clause>

  <clause id="199">
    <literal><![CDATA[
      f(f(f(x,y),f(x,y)),f(f(x,y),y)) = f(x,y)
    ]]></literal>
    <justification jstring="[para(118(a,1),51(a,1,1,1)),rewrite(118(7),118(9),118(12))].">
      <j1 rule="para" parents="118 51"/>
      <j2 rule="rewrite" parents="118 118 118"/>
    </justification>
  </clause>

  <clause id="201">
    <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(118(a,1),118(a,1,2,1,1,1)),rewrite(118(12),118(27))].">
      <j1 rule="para" parents="118 118"/>
      <j2 rule="rewrite" parents="118 118"/>
    </justification>
  </clause>

  <clause id="235">
    <literal><![CDATA[
      f(f(f(x,y),f(y,f(x,y))),f(f(y,y),x)) = f(y,y)
    ]]></literal>
    <justification jstring="[para(169(a,1),7(a,1,1,2,1))].">
      <j1 rule="para" parents="169 7"/>
    </justification>
  </clause>

  <clause id="242">
    <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(169(a,1),3(a,1,2,2))].">
      <j1 rule="para" parents="169 3"/>
    </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(169(a,1),5(a,1,1,1)),rewrite(169(9),169(10))].">
      <j1 rule="para" parents="169 5"/>
      <j2 rule="rewrite" parents="169 169"/>
    </justification>
  </clause>

  <clause id="245">
    <literal><![CDATA[
      f(f(x,x),f(y,f(f(x,y),y))) = x
    ]]></literal>
    <justification jstring="[para(5(a,1),169(a,1,2))].">
      <j1 rule="para" parents="5 169"/>
    </justification>
  </clause>

  <clause id="250">
    <literal><![CDATA[
      f(f(x,y),f(y,f(x,y))) = y
    ]]></literal>
    <justification jstring="[para(169(a,1),118(a,1,2,1,1,1)),rewrite(169(4),169(4),169(7))].">
      <j1 rule="para" parents="169 118"/>
      <j2 rule="rewrite" parents="169 169 169"/>
    </justification>
  </clause>

  <clause id="252">
    <literal><![CDATA[
      f(f(f(x,y),f(x,y)),y) = f(x,y)
    ]]></literal>
    <justification jstring="[para(169(a,1),169(a,1,2))].">
      <j1 rule="para" parents="169 169"/>
    </justification>
  </clause>

  <clause id="261">
    <literal><![CDATA[
      f(x,f(f(x,x),y)) = f(x,x)
    ]]></literal>
    <justification jstring="[back_rewrite(235),rewrite(250(4))].">
      <j1 rule="back_rewrite" parents="235"/>
      <j2 rule="rewrite" parents="250"/>
    </justification>
  </clause>

  <clause id="263">
    <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(201),rewrite(252(4),252(9),199(8),252(7))].">
      <j1 rule="back_rewrite" parents="201"/>
      <j2 rule="rewrite" parents="252 252 199 252"/>
    </justification>
  </clause>

  <clause id="266">
    <literal><![CDATA[
      f(f(x,y),f(y,y)) = y
    ]]></literal>
    <justification jstring="[back_rewrite(190),rewrite(252(4),252(7),263(7))].">
      <j1 rule="back_rewrite" parents="190"/>
      <j2 rule="rewrite" parents="252 252 263"/>
    </justification>
  </clause>

  <clause id="271">
    <literal><![CDATA[
      f(x,f(f(y,x),x)) = f(y,x)
    ]]></literal>
    <justification jstring="[back_rewrite(118),rewrite(252(4))].">
      <j1 rule="back_rewrite" parents="118"/>
      <j2 rule="rewrite" parents="252"/>
    </justification>
  </clause>

  <clause id="277">
    <literal><![CDATA[
      f(f(x,x),f(x,y)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(245),rewrite(271(4))].">
      <j1 rule="back_rewrite" parents="245"/>
      <j2 rule="rewrite" parents="271"/>
    </justification>
  </clause>

  <clause id="278">
    <literal><![CDATA[
      f(f(f(x,f(y,z)),z),x) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[back_rewrite(243),rewrite(271(6),271(5),271(10))].">
      <j1 rule="back_rewrite" parents="243"/>
      <j2 rule="rewrite" parents="271 271 271"/>
    </justification>
  </clause>

  <clause id="279">
    <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(242),rewrite(271(9),271(13))].">
      <j1 rule="back_rewrite" parents="242"/>
      <j2 rule="rewrite" parents="271 271"/>
    </justification>
  </clause>

  <clause id="291">
    <literal><![CDATA[
      f(f(x,y),f(x,f(x,y))) = x
    ]]></literal>
    <justification jstring="[back_rewrite(176),rewrite(277(4))].">
      <j1 rule="back_rewrite" parents="176"/>
      <j2 rule="rewrite" parents="277"/>
    </justification>
  </clause>

  <clause id="292">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(x,z)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(172),rewrite(271(6))].">
      <j1 rule="back_rewrite" parents="172"/>
      <j2 rule="rewrite" parents="271"/>
    </justification>
  </clause>

  <clause id="294">
    <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(116),rewrite(271(4),271(8),271(10),291(12),271(7))].">
      <j1 rule="back_rewrite" parents="116"/>
      <j2 rule="rewrite" parents="271 271 271 291 271"/>
    </justification>
  </clause>

  <clause id="301">
    <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(109),rewrite(271(5),271(10))].">
      <j1 rule="back_rewrite" parents="109"/>
      <j2 rule="rewrite" parents="271 271"/>
    </justification>
  </clause>

  <clause id="310">
    <literal><![CDATA[
      f(f(x,y),f(x,x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(91),rewrite(271(3),271(8),271(9),291(13),291(7),291(7),291(9))].">
      <j1 rule="back_rewrite" parents="91"/>
      <j2 rule="rewrite" parents="271 271 271 291 291 291 291"/>
    </justification>
  </clause>

  <clause id="313">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(x,y)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(87),rewrite(271(3),271(4),271(6),271(6))].">
      <j1 rule="back_rewrite" parents="87"/>
      <j2 rule="rewrite" parents="271 271 271 271"/>
    </justification>
  </clause>

  <clause id="316">
    <literal><![CDATA[
      f(f(f(x,y),x),x) = f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(81),rewrite(271(3),271(4),271(6))].">
      <j1 rule="back_rewrite" parents="81"/>
      <j2 rule="rewrite" parents="271 271 271"/>
    </justification>
  </clause>

  <clause id="320">
    <literal><![CDATA[
      f(f(x,f(y,f(x,z))),f(x,f(z,u))) = x
    ]]></literal>
    <justification jstring="[back_rewrite(71),rewrite(291(6),271(9),271(9),291(10))].">
      <j1 rule="back_rewrite" parents="71"/>
      <j2 rule="rewrite" parents="291 271 271 291"/>
    </justification>
  </clause>

  <clause id="324">
    <literal><![CDATA[
      f(f(x,y),f(x,f(y,z))) = x
    ]]></literal>
    <justification jstring="[back_rewrite(39),rewrite(291(6),291(6),291(7),271(3),271(4),271(9),271(10),291(14))].">
      <j1 rule="back_rewrite" parents="39"/>
      <j2 rule="rewrite" parents="291 291 291 271 271 271 271 291"/>
    </justification>
  </clause>

  <clause id="330">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(x,f(y,z)),u))) = f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(29),rewrite(271(3),271(5),291(6),271(3),271(4),271(6),271(9))].">
      <j1 rule="back_rewrite" parents="29"/>
      <j2 rule="rewrite" parents="271 271 291 271 271 271 271"/>
    </justification>
  </clause>

  <clause id="343">
    <literal><![CDATA[
      f(f(x,y),f(x,f(z,y))) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1),rewrite(271(3))].">
      <j1 rule="back_rewrite" parents="1"/>
      <j2 rule="rewrite" parents="271"/>
    </justification>
  </clause>

  <clause id="347">
    <literal><![CDATA[
      f(x,f(f(x,y),f(z,y))) = f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(279),rewrite(291(6))].">
      <j1 rule="back_rewrite" parents="279"/>
      <j2 rule="rewrite" parents="291"/>
    </justification>
  </clause>

  <clause id="360">
    <literal><![CDATA[
      f(f(x,y),x) = f(x,f(x,y))
    ]]></literal>
    <justification jstring="[para(292(a,1),271(a,1,2))].">
      <j1 rule="para" parents="292 271"/>
    </justification>
  </clause>

  <clause id="368">
    <literal><![CDATA[
      f(x,f(x,f(x,y))) = f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(316),rewrite(360(2),360(3))].">
      <j1 rule="back_rewrite" parents="316"/>
      <j2 rule="rewrite" parents="360 360"/>
    </justification>
  </clause>

  <clause id="375">
    <literal><![CDATA[
      f(x,f(f(x,f(y,z)),y)) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(313(a,1),292(a,1,1))].">
      <j1 rule="para" parents="313 292"/>
    </justification>
  </clause>

  <clause id="421">
    <literal><![CDATA[
      f(x,y) = f(y,x)
    ]]></literal>
    <justification jstring="[para(266(a,1),278(a,1,1,1,2)),rewrite(266(3),266(4))].">
      <j1 rule="para" parents="266 278"/>
      <j2 rule="rewrite" parents="266 266"/>
    </justification>
  </clause>

  <clause id="424">
    <literal><![CDATA[
      f(x,f(y,f(x,f(z,y)))) = f(x,f(z,y))
    ]]></literal>
    <justification jstring="[para(278(a,1),252(a,2)),rewrite(421(3),421(4),421(7),421(8),421(10),347(10))].">
      <j1 rule="para" parents="278 252"/>
      <j2 rule="rewrite" parents="421 421 421 421 421 347"/>
    </justification>
  </clause>

  <clause id="429">
    <literal><![CDATA[
      f(x,f(y,f(x,y))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(313(a,1),278(a,1,1)),flip(a)].">
      <j1 rule="para" parents="313 278"/>
      <j2 rule="flip"/>
    </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(375),rewrite(421(3))].">
      <j1 rule="back_rewrite" parents="375"/>
      <j2 rule="rewrite" parents="421"/>
    </justification>
  </clause>

  <clause id="449">
    <literal><![CDATA[
      f(a,f(b,f(b,b))) != f(a,a)
    ]]></literal>
    <literal><![CDATA[
      f(f(a,f(b,c)),f(a,f(b,c))) != f(f(a,f(b,b)),f(a,f(c,c)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(57),rewrite(421(16),421(21)),flip(b)].">
      <j1 rule="back_rewrite" parents="57"/>
      <j2 rule="rewrite" parents="421 421"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="451">
    <literal><![CDATA[
      f(f(x,y),f(y,f(x,z))) = y
    ]]></literal>
    <justification jstring="[para(421(a,1),324(a,1,1))].">
      <j1 rule="para" parents="421 324"/>
    </justification>
  </clause>

  <clause id="452">
    <literal><![CDATA[
      f(f(x,y),f(f(y,z),x)) = x
    ]]></literal>
    <justification jstring="[para(421(a,1),324(a,1,2))].">
      <j1 rule="para" parents="421 324"/>
    </justification>
  </clause>

  <clause id="453">
    <literal><![CDATA[
      f(f(x,y),f(y,f(z,x))) = y
    ]]></literal>
    <justification jstring="[para(421(a,1),343(a,1,1))].">
      <j1 rule="para" parents="421 343"/>
    </justification>
  </clause>

  <clause id="454">
    <literal><![CDATA[
      f(f(x,y),f(f(z,y),x)) = x
    ]]></literal>
    <justification jstring="[para(421(a,1),343(a,1,2))].">
      <j1 rule="para" parents="421 343"/>
    </justification>
  </clause>

  <clause id="462">
    <literal><![CDATA[
      f(x,f(y,f(y,x))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(421(a,1),429(a,1,2,2))].">
      <j1 rule="para" parents="421 429"/>
    </justification>
  </clause>

  <clause id="465">
    <literal><![CDATA[
      f(f(f(x,y),z),f(z,x)) = z
    ]]></literal>
    <justification jstring="[para(310(a,1),451(a,1,2,2))].">
      <j1 rule="para" parents="310 451"/>
    </justification>
  </clause>

  <clause id="470">
    <literal><![CDATA[
      f(f(x,y),f(f(x,z),y)) = y
    ]]></literal>
    <justification jstring="[para(421(a,1),451(a,1,2))].">
      <j1 rule="para" parents="421 451"/>
    </justification>
  </clause>

  <clause id="504">
    <literal><![CDATA[
      f(f(x,y),f(f(z,x),y)) = y
    ]]></literal>
    <justification jstring="[para(421(a,1),453(a,1,2))].">
      <j1 rule="para" parents="421 453"/>
    </justification>
  </clause>

  <clause id="510">
    <literal><![CDATA[
      f(x,f(f(y,z),f(y,x))) = f(y,x)
    ]]></literal>
    <justification jstring="[para(451(a,1),453(a,1,2)),rewrite(421(4))].">
      <j1 rule="para" parents="451 453"/>
      <j2 rule="rewrite" parents="421"/>
    </justification>
  </clause>

  <clause id="514">
    <literal><![CDATA[
      f(x,f(f(y,z),f(z,x))) = f(z,x)
    ]]></literal>
    <justification jstring="[para(453(a,1),453(a,1,2)),rewrite(421(4))].">
      <j1 rule="para" parents="453 453"/>
      <j2 rule="rewrite" parents="421"/>
    </justification>
  </clause>

  <clause id="527">
    <literal><![CDATA[
      f(x,f(y,f(f(y,z),x))) = f(f(y,z),x)
    ]]></literal>
    <justification jstring="[para(452(a,1),454(a,1,2)),rewrite(421(3),421(4))].">
      <j1 rule="para" parents="452 454"/>
      <j2 rule="rewrite" parents="421 421"/>
    </justification>
  </clause>

  <clause id="531">
    <literal><![CDATA[
      f(x,f(y,f(f(z,y),x))) = f(f(z,y),x)
    ]]></literal>
    <justification jstring="[para(454(a,1),454(a,1,2)),rewrite(421(3),421(4))].">
      <j1 rule="para" parents="454 454"/>
      <j2 rule="rewrite" parents="421 421"/>
    </justification>
  </clause>

  <clause id="577">
    <literal><![CDATA[
      f(x,f(f(f(x,y),z),f(x,f(y,u)))) = f(x,f(y,u))
    ]]></literal>
    <justification jstring="[para(324(a,1),470(a,1,1))].">
      <j1 rule="para" parents="324 470"/>
    </justification>
  </clause>

  <clause id="616">
    <literal><![CDATA[
      f(f(f(x,f(x,y)),z),f(f(y,y),z)) = z
    ]]></literal>
    <justification jstring="[para(462(a,1),504(a,1,2,1))].">
      <j1 rule="para" parents="462 504"/>
    </justification>
  </clause>

  <clause id="851">
    <literal><![CDATA[
      f(x,f(f(y,x),f(f(y,z),f(y,x)))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(510(a,1),429(a,1,2,2)),rewrite(421(5))].">
      <j1 rule="para" parents="510 429"/>
      <j2 rule="rewrite" parents="421"/>
    </justification>
  </clause>

  <clause id="880">
    <literal><![CDATA[
      f(x,f(f(y,x),f(f(z,y),f(y,x)))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(514(a,1),429(a,1,2,2)),rewrite(421(5))].">
      <j1 rule="para" parents="514 429"/>
      <j2 rule="rewrite" parents="421"/>
    </justification>
  </clause>

  <clause id="1004">
    <literal><![CDATA[
      f(f(x,f(y,f(z,x))),f(x,f(z,u))) = x
    ]]></literal>
    <justification jstring="[para(421(a,1),320(a,1,1,2,2))].">
      <j1 rule="para" parents="421 320"/>
    </justification>
  </clause>

  <clause id="1005">
    <literal><![CDATA[
      f(f(x,f(f(x,y),z)),f(x,f(y,u))) = x
    ]]></literal>
    <justification jstring="[para(421(a,1),320(a,1,1,2))].">
      <j1 rule="para" parents="421 320"/>
    </justification>
  </clause>

  <clause id="1051">
    <literal><![CDATA[
      f(f(x,f(y,f(x,f(z,u)))),f(z,x)) = x
    ]]></literal>
    <justification jstring="[para(510(a,1),320(a,1,2))].">
      <j1 rule="para" parents="510 320"/>
    </justification>
  </clause>

  <clause id="1141">
    <literal><![CDATA[
      f(x,f(y,y)) = f(x,f(x,y))
    ]]></literal>
    <justification jstring="[para(462(a,1),424(a,1,2))].">
      <j1 rule="para" parents="462 424"/>
    </justification>
  </clause>

  <clause id="1156">
    <literal><![CDATA[
      f(x,f(y,f(x,f(z,f(x,y))))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(320(a,1),424(a,1,2)),flip(a)].">
      <j1 rule="para" parents="320 424"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1157">
    <literal><![CDATA[
      f(a,f(b,f(b,b))) != f(a,a)
    ]]></literal>
    <literal><![CDATA[
      f(f(a,f(b,c)),f(a,f(b,c))) != f(f(a,f(a,b)),f(a,f(a,c)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(449),rewrite(1141(27),1141(32))].">
      <j1 rule="back_rewrite" parents="449"/>
      <j2 rule="rewrite" parents="1141 1141"/>
    </justification>
  </clause>

  <clause id="1227">
    <literal><![CDATA[
      f(x,f(x,f(y,y))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(169(a,1),1141(a,1,2)),flip(a)].">
      <j1 rule="para" parents="169 1141"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1233">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(x,f(y,f(z,z)))) = x
    ]]></literal>
    <justification jstring="[para(1141(a,2),343(a,1,2,2))].">
      <j1 rule="para" parents="1141 343"/>
    </justification>
  </clause>

  <clause id="1234">
    <literal><![CDATA[
      f(f(x,x),y) = f(y,f(y,x))
    ]]></literal>
    <justification jstring="[para(1141(a,1),421(a,1)),flip(a)].">
      <j1 rule="para" parents="1141 421"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1235">
    <literal><![CDATA[
      f(x,f(y,y)) = f(x,f(y,x))
    ]]></literal>
    <justification jstring="[para(421(a,1),1141(a,2,2))].">
      <j1 rule="para" parents="421 1141"/>
    </justification>
  </clause>

  <clause id="1373">
    <literal><![CDATA[
      f(f(x,x),y) = f(y,f(x,y))
    ]]></literal>
    <justification jstring="[para(421(a,1),1234(a,2,2))].">
      <j1 rule="para" parents="421 1234"/>
    </justification>
  </clause>

  <clause id="8208">
    <literal><![CDATA[
      f(x,f(y,f(x,f(z,f(y,x))))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(1004(a,1),424(a,1,2)),flip(a)].">
      <j1 rule="para" parents="1004 424"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="8556">
    <literal><![CDATA[
      f(x,f(y,f(z,f(y,x)))) = f(x,f(y,y))
    ]]></literal>
    <justification jstring="[para(1156(a,1),441(a,1,2)),flip(a)].">
      <j1 rule="para" parents="1156 441"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="8902">
    <literal><![CDATA[
      f(x,f(y,f(y,y))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(1233(a,1),1233(a,1,2)),rewrite(421(2),421(6),343(6),421(3)),flip(a)].">
      <j1 rule="para" parents="1233 1233"/>
      <j2 rule="rewrite" parents="421 421 343 421"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="8903">
    <literal><![CDATA[
      f(f(a,f(b,c)),f(a,f(b,c))) != f(f(a,f(a,b)),f(a,f(a,c)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_unit_del(1157),unit_del(a,8902)].">
      <j1 rule="back_unit_del" parents="1157"/>
      <j2 rule="unit_del" parents="8902"/>
    </justification>
  </clause>

  <clause id="8906">
    <literal><![CDATA[
      f(f(x,x),f(y,f(y,y))) = x
    ]]></literal>
    <justification jstring="[para(8902(a,2),169(a,1))].">
      <j1 rule="para" parents="8902 169"/>
    </justification>
  </clause>

  <clause id="8918">
    <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(8902(a,2),294(a,1,2,2,2)),rewrite(421(2),421(5))].">
      <j1 rule="para" parents="8902 294"/>
      <j2 rule="rewrite" parents="421 421"/>
    </justification>
  </clause>

  <clause id="8929">
    <literal><![CDATA[
      f(f(x,x),f(f(x,y),f(z,f(z,z)))) = f(z,f(z,z))
    ]]></literal>
    <justification jstring="[para(8902(a,1),470(a,1,1))].">
      <j1 rule="para" parents="8902 470"/>
    </justification>
  </clause>

  <clause id="8933">
    <literal><![CDATA[
      f(x,f(x,x)) = f(y,f(y,y))
    ]]></literal>
    <justification jstring="[para(8902(a,2),301(a,1,2)),rewrite(421(3),8929(7),421(4))].">
      <j1 rule="para" parents="8902 301"/>
      <j2 rule="rewrite" parents="421 8929 421"/>
    </justification>
  </clause>

  <clause id="8951">
    <literal><![CDATA[
      f(x,f(f(y,f(y,y)),f(y,f(y,y)))) = f(x,f(x,x))
    ]]></literal>
    <justification jstring="[para(8902(a,1),1141(a,2,2))].">
      <j1 rule="para" parents="8902 1141"/>
    </justification>
  </clause>

  <clause id="8952">
    <literal><![CDATA[
      f(x,f(x,f(y,f(z,f(z,z))))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(8902(a,2),1227(a,1,2,2))].">
      <j1 rule="para" parents="8902 1227"/>
    </justification>
  </clause>

  <clause id="8953">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(x,f(x,x)),y)) = y
    ]]></literal>
    <justification jstring="[para(8902(a,1),1234(a,1)),rewrite(277(3)),flip(a)].">
      <j1 rule="para" parents="8902 1234"/>
      <j2 rule="rewrite" parents="277"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="8958">
    <literal><![CDATA[
      f(x,f(y,f(z,f(z,z)))) = f(x,f(y,x))
    ]]></literal>
    <justification jstring="[para(8902(a,2),1235(a,1,2))].">
      <j1 rule="para" parents="8902 1235"/>
    </justification>
  </clause>

  <clause id="8962">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(y,y)) = y
    ]]></literal>
    <justification jstring="[para(8902(a,1),1373(a,2,2)),rewrite(8906(4)),flip(a)].">
      <j1 rule="para" parents="8902 1373"/>
      <j2 rule="rewrite" parents="8906"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="8997">
    <literal><![CDATA[
      f(x,f(f(y,f(y,x)),f(z,f(z,z)))) = f(z,f(z,z))
    ]]></literal>
    <justification jstring="[para(8902(a,1),616(a,1,2)),rewrite(277(8),421(6))].">
      <j1 rule="para" parents="8902 616"/>
      <j2 rule="rewrite" parents="277 421"/>
    </justification>
  </clause>

  <clause id="9061">
    <literal><![CDATA[
      f(f(a,f(b,c)),f(x,f(x,x))) != f(f(a,f(a,b)),f(a,f(a,c)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[para(8902(a,2),8903(a,1))].">
      <j1 rule="para" parents="8902 8903"/>
    </justification>
  </clause>

  <clause id="9064">
    <literal><![CDATA[
      f(x,f(x,x)) = c_0
    ]]></literal>
    <justification jstring="[new_symbol(8933)].">
      <j1 rule="new_symbol" parents="8933"/>
    </justification>
  </clause>

  <clause id="9066">
    <literal><![CDATA[
      f(f(a,f(a,b)),f(a,f(a,c))) != f(c_0,f(a,f(b,c)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(9061),rewrite(9064(7),421(7)),flip(a)].">
      <j1 rule="back_rewrite" parents="9061"/>
      <j2 rule="rewrite" parents="9064 421"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9122">
    <literal><![CDATA[
      f(x,f(c_0,f(y,f(y,x)))) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(8997),rewrite(9064(4),421(4),9064(7))].">
      <j1 rule="back_rewrite" parents="8997"/>
      <j2 rule="rewrite" parents="9064 421 9064"/>
    </justification>
  </clause>

  <clause id="9154">
    <literal><![CDATA[
      f(c_0,f(x,x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(8962),rewrite(9064(2))].">
      <j1 rule="back_rewrite" parents="8962"/>
      <j2 rule="rewrite" parents="9064"/>
    </justification>
  </clause>

  <clause id="9158">
    <literal><![CDATA[
      f(x,f(y,c_0)) = f(x,f(y,x))
    ]]></literal>
    <justification jstring="[back_rewrite(8958),rewrite(9064(2))].">
      <j1 rule="back_rewrite" parents="8958"/>
      <j2 rule="rewrite" parents="9064"/>
    </justification>
  </clause>

  <clause id="9162">
    <literal><![CDATA[
      f(c_0,f(c_0,x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(8953),rewrite(9064(2),9064(3))].">
      <j1 rule="back_rewrite" parents="8953"/>
      <j2 rule="rewrite" parents="9064 9064"/>
    </justification>
  </clause>

  <clause id="9163">
    <literal><![CDATA[
      f(x,f(x,f(y,c_0))) = f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(8952),rewrite(9064(2))].">
      <j1 rule="back_rewrite" parents="8952"/>
      <j2 rule="rewrite" parents="9064"/>
    </justification>
  </clause>

  <clause id="9164">
    <literal><![CDATA[
      f(x,f(c_0,c_0)) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(8951),rewrite(9064(2),9064(3),9064(6))].">
      <j1 rule="back_rewrite" parents="8951"/>
      <j2 rule="rewrite" parents="9064 9064 9064"/>
    </justification>
  </clause>

  <clause id="9188">
    <literal><![CDATA[
      f(c_0,x) = f(x,x)
    ]]></literal>
    <justification jstring="[back_rewrite(8918),rewrite(9064(2),9064(4),9064(5),9164(6),421(4),9154(4))].">
      <j1 rule="back_rewrite" parents="8918"/>
      <j2 rule="rewrite" parents="9064 9064 9064 9164 421 9154"/>
    </justification>
  </clause>

  <clause id="9414">
    <literal><![CDATA[
      f(x,f(f(y,x),f(c_0,f(z,y)))) = f(x,x)
    ]]></literal>
    <justification jstring="[back_rewrite(880),rewrite(9158(5,R),421(4))].">
      <j1 rule="back_rewrite" parents="880"/>
      <j2 rule="rewrite" parents="9158 421"/>
    </justification>
  </clause>

  <clause id="9415">
    <literal><![CDATA[
      f(x,f(f(y,x),f(c_0,f(y,z)))) = f(x,x)
    ]]></literal>
    <justification jstring="[back_rewrite(851),rewrite(9158(5,R),421(4))].">
      <j1 rule="back_rewrite" parents="851"/>
      <j2 rule="rewrite" parents="9158 421"/>
    </justification>
  </clause>

  <clause id="10700">
    <literal><![CDATA[
      f(f(c_0,x),f(f(c_0,f(x,y)),z)) = x
    ]]></literal>
    <justification jstring="[para(330(a,1),9162(a,1,2)),rewrite(9162(4)),flip(a)].">
      <j1 rule="para" parents="330 9162"/>
      <j2 rule="rewrite" parents="9162"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10728">
    <literal><![CDATA[
      f(x,f(x,f(c_0,y))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(9188(a,2),1227(a,1,2,2))].">
      <j1 rule="para" parents="9188 1227"/>
    </justification>
  </clause>

  <clause id="10729">
    <literal><![CDATA[
      f(f(c_0,x),y) = f(y,f(y,x))
    ]]></literal>
    <justification jstring="[para(9188(a,2),1234(a,1,1))].">
      <j1 rule="para" parents="9188 1234"/>
    </justification>
  </clause>

  <clause id="10731">
    <literal><![CDATA[
      f(x,f(c_0,y)) = f(x,f(y,x))
    ]]></literal>
    <justification jstring="[para(9188(a,2),1235(a,1,2))].">
      <j1 rule="para" parents="9188 1235"/>
    </justification>
  </clause>

  <clause id="10749">
    <literal><![CDATA[
      f(f(c_0,x),y) = f(y,f(x,x))
    ]]></literal>
    <justification jstring="[para(9188(a,2),527(a,2,1)),rewrite(261(3)),flip(a)].">
      <j1 rule="para" parents="9188 527"/>
      <j2 rule="rewrite" parents="261"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11431">
    <literal><![CDATA[
      f(f(x,y),f(c_0,f(x,f(c_0,f(y,z))))) = c_0
    ]]></literal>
    <justification jstring="[para(465(a,1),9122(a,1,2,2,2)),rewrite(421(5),10731(5,R))].">
      <j1 rule="para" parents="465 9122"/>
      <j2 rule="rewrite" parents="421 10731"/>
    </justification>
  </clause>

  <clause id="12917">
    <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(1051(a,1),9163(a,1,2)),rewrite(421(8),9162(8),421(11)),flip(a)].">
      <j1 rule="para" parents="1051 9163"/>
      <j2 rule="rewrite" parents="421 9162 421"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14066">
    <literal><![CDATA[
      f(f(c_0,f(f(c_0,x),y)),f(x,z)) = f(f(c_0,x),y)
    ]]></literal>
    <justification jstring="[para(1005(a,1),10728(a,1,2)),rewrite(421(7),9162(7)),flip(a)].">
      <j1 rule="para" parents="1005 10728"/>
      <j2 rule="rewrite" parents="421 9162"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16897">
    <literal><![CDATA[
      f(x,f(y,f(z,f(x,y)))) = f(x,f(y,y))
    ]]></literal>
    <justification jstring="[para(8208(a,1),441(a,1,2)),flip(a)].">
      <j1 rule="para" parents="8208 441"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16928">
    <literal><![CDATA[
      f(x,f(f(c_0,f(y,z)),f(x,y))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(10700(a,1),8208(a,1,2,2,2))].">
      <j1 rule="para" parents="10700 8208"/>
    </justification>
  </clause>

  <clause id="17472">
    <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(9414(a,1),8556(a,1,2,2)),rewrite(266(6),9188(10,R)),flip(a)].">
      <j1 rule="para" parents="9414 8556"/>
      <j2 rule="rewrite" parents="266 9188"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="17512">
    <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(9415(a,1),8556(a,1,2,2)),rewrite(266(6),9188(10,R)),flip(a)].">
      <j1 rule="para" parents="9415 8556"/>
      <j2 rule="rewrite" parents="266 9188"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18218">
    <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(577(a,1),531(a,1,2)),rewrite(9188(5,R),421(6)),flip(a)].">
      <j1 rule="para" parents="577 531"/>
      <j2 rule="rewrite" parents="9188 421"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18708">
    <literal><![CDATA[
      f(f(c_0,f(x,y)),z) = f(x,f(c_0,f(y,z)))
    ]]></literal>
    <justification jstring="[para(11431(a,1),462(a,1,2,2)),rewrite(421(9),421(10),17512(10),17472(7),9188(17,R),9162(12))].">
      <j1 rule="para" parents="11431 462"/>
      <j2 rule="rewrite" parents="421 421 17512 17472 9188 9162"/>
    </justification>
  </clause>

  <clause id="18881">
    <literal><![CDATA[
      f(x,f(c_0,f(y,z))) = f(y,f(c_0,f(x,z)))
    ]]></literal>
    <justification jstring="[back_rewrite(17512),rewrite(18708(7),12917(7),18708(8))].">
      <j1 rule="back_rewrite" parents="17512"/>
      <j2 rule="rewrite" parents="18708 12917 18708"/>
    </justification>
  </clause>

  <clause id="18916">
    <literal><![CDATA[
      f(x,f(y,f(c_0,f(z,f(x,y))))) = f(x,x)
    ]]></literal>
    <justification jstring="[back_rewrite(16928),rewrite(18708(5))].">
      <j1 rule="back_rewrite" parents="16928"/>
      <j2 rule="rewrite" parents="18708"/>
    </justification>
  </clause>

  <clause id="19104">
    <literal><![CDATA[
      f(f(c_0,x),f(c_0,f(y,f(x,z)))) = f(f(c_0,x),y)
    ]]></literal>
    <justification jstring="[back_rewrite(14066),rewrite(18708(7))].">
      <j1 rule="back_rewrite" parents="14066"/>
      <j2 rule="rewrite" parents="18708"/>
    </justification>
  </clause>

  <clause id="21726">
    <literal><![CDATA[
      f(x,f(y,f(c_0,f(x,z)))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(18881(a,1),1141(a,2,2)),rewrite(9188(7,R),9162(5)),flip(a)].">
      <j1 rule="para" parents="18881 1141"/>
      <j2 rule="rewrite" parents="9188 9162"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21736">
    <literal><![CDATA[
      f(x,f(f(c_0,y),z)) = f(x,f(z,f(y,x)))
    ]]></literal>
    <justification jstring="[para(18916(a,1),16897(a,1,2,2)),rewrite(10749(6,R),19104(7),9188(13,R),9162(10))].">
      <j1 rule="para" parents="18916 16897"/>
      <j2 rule="rewrite" parents="10749 19104 9188 9162"/>
    </justification>
  </clause>

  <clause id="22470">
    <literal><![CDATA[
      f(x,f(y,f(x,f(x,z)))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(368(a,1),21726(a,1,2,2,2)),rewrite(21726(5)),flip(a)].">
      <j1 rule="para" parents="368 21726"/>
      <j2 rule="rewrite" parents="21726"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22565">
    <literal><![CDATA[
      f(x,f(y,f(z,x))) = f(x,f(y,f(y,z)))
    ]]></literal>
    <justification jstring="[para(10729(a,1),21736(a,1,2)),flip(a)].">
      <j1 rule="para" parents="10729 21736"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22989">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[para(22565(a,2),9066(a,1)),rewrite(22470(14),421(9),18218(11)),xx(a)].">
      <j1 rule="para" parents="22565 9066"/>
      <j2 rule="rewrite" parents="22470 421 18218"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
