<?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-def.out</source>

<heading><![CDATA[
Prover9 (32) version August-2006A, August 2006.
Process 11621 was started by mccune on cleo.thornwood,
Sat Aug 12 21:14:54 2006
The command was "/home/mccune/bin/prover9 -f sh1-def.in".
]]></heading>

<proof number="1" length="178" max_count="95">

<comments><![CDATA[
% Proof 1 at 8.36 (+ 0.05) seconds: "Sheffer".
% Length of proof is 178.
% Level of proof is 42.
% Maximum clause weight is 95.
% Given clauses 274.
]]></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[
      x' = f(x,x)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      f(x,x) = x'
    ]]></literal>
    <justification jstring="[copy(3),flip(a)].">
      <j1 rule="copy" parents="3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="5">
    <literal><![CDATA[
      a'' != a
    ]]></literal>
    <literal><![CDATA[
      f(a,f(b,b')) != a'
    ]]></literal>
    <literal><![CDATA[
      f(f(b',a),f(c',a)) != f(a,f(b,c))'
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(2),rewrite(4(3),4(5),4(5),4(10),4(14),4(17),4(21),4(34))].">
      <j1 rule="back_rewrite" parents="2"/>
      <j2 rule="rewrite" parents="4 4 4 4 4 4 4 4"/>
    </justification>
  </clause>

  <clause id="6">
    <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="7">
    <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="8">
    <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="9">
    <literal><![CDATA[
      f(f(x,f(x',x)),f(x,f(y,x))) = x
    ]]></literal>
    <justification jstring="[para(4(a,1),1(a,1,1,2,1))].">
      <j1 rule="para" parents="4 1"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      f(f(x,f(f(y,x),x)),f(y,x')) = y
    ]]></literal>
    <justification jstring="[para(4(a,1),1(a,1,2,2))].">
      <j1 rule="para" parents="4 1"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      f(x,f(x',x))' = x
    ]]></literal>
    <justification jstring="[para(4(a,1),1(a,1)),rewrite(4(1))].">
      <j1 rule="para" parents="4 1"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="18">
    <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(v,f(f(x,f(f(y,x),x)),f(z,f(y,f(u,x))))))) = f(f(y,f(u,x)),f(y,f(y,f(u,x))))
    ]]></literal>
    <justification jstring="[para(6(a,1),1(a,1,1,2,1))].">
      <j1 rule="para" parents="6 1"/>
    </justification>
  </clause>

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

  <clause id="22">
    <literal><![CDATA[
      f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(z,f(f(x,z),z)),x)) = f(z,f(f(x,z),z))
    ]]></literal>
    <justification jstring="[para(1(a,1),6(a,1,2,2))].">
      <j1 rule="para" parents="1 6"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      f(f(x',f(f(f(x,f(f(y,x),x)),x'),x')),y) = f(x,f(f(y,x),x))
    ]]></literal>
    <justification jstring="[para(10(a,1),1(a,1,2))].">
      <j1 rule="para" parents="10 1"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      f(f(x,f(f(y,f(f(x,y),y)),x)),f(f(y',f(f(f(y,f(f(x,y),y)),y'),y')),f(z,x))) = f(y',f(f(f(y,f(f(x,y),y)),y'),y'))
    ]]></literal>
    <justification jstring="[para(10(a,1),6(a,1,1,1)),rewrite(10(9),10(20))].">
      <j1 rule="para" parents="10 6"/>
      <j2 rule="rewrite" parents="10 10"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      f(f(f(x,y),f(f(z,f(x,y)),f(x,y))),f(z,x)) = z
    ]]></literal>
    <justification jstring="[para(7(a,1),1(a,1,2,2))].">
      <j1 rule="para" parents="7 1"/>
    </justification>
  </clause>

  <clause id="67">
    <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),7(a,1,1,1)),rewrite(1(6),1(7))].">
      <j1 rule="para" parents="1 7"/>
      <j2 rule="rewrite" parents="1 1"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      f(f(f(x,y),f(f(x,y)',f(x,y))),f(f(x,y),x)) = f(x,y)
    ]]></literal>
    <justification jstring="[para(7(a,1),9(a,1,2,2))].">
      <j1 rule="para" parents="7 9"/>
    </justification>
  </clause>

  <clause id="70">
    <literal><![CDATA[
      f(f(x,f(f(y,x),x)),f(y,f(x,f(x',x)))) = y
    ]]></literal>
    <justification jstring="[para(9(a,1),7(a,1,1,1)),rewrite(9(6),9(7))].">
      <j1 rule="para" parents="9 7"/>
      <j2 rule="rewrite" parents="9 9"/>
    </justification>
  </clause>

  <clause id="88">
    <literal><![CDATA[
      f(f(x,f(x',x)),f(x,f(x,f(x',x)))) = x
    ]]></literal>
    <justification jstring="[para(4(a,1),70(a,1,1,2,1))].">
      <j1 rule="para" parents="4 70"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      f(x,f(x',x)) = x'
    ]]></literal>
    <justification jstring="[para(70(a,1),6(a,1,2)),rewrite(88(8),4(1),4(2)),flip(a)].">
      <j1 rule="para" parents="70 6"/>
      <j2 rule="rewrite" parents="88 4 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="92">
    <literal><![CDATA[
      f(f(x',f(f(f(x',f(x,x')),x'),x')),x') = f(x',f(x,x'))
    ]]></literal>
    <justification jstring="[para(6(a,1),70(a,1,2)),rewrite(4(1),91(3),4(2),91(4),4(3),91(5),4(6),91(8),4(8),91(10),4(11),91(13),4(13),91(15),4(14),91(16))].">
      <j1 rule="para" parents="6 70"/>
      <j2 rule="rewrite" parents="4 91 4 91 4 91 4 91 4 91 4 91 4 91 4 91"/>
    </justification>
  </clause>

  <clause id="96">
    <literal><![CDATA[
      f(x',f(x,x')) = x
    ]]></literal>
    <justification jstring="[back_rewrite(88),rewrite(91(3),91(4))].">
      <j1 rule="back_rewrite" parents="88"/>
      <j2 rule="rewrite" parents="91 91"/>
    </justification>
  </clause>

  <clause id="100">
    <literal><![CDATA[
      f(f(x,y)',f(f(x,y),x)) = f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(69),rewrite(91(6))].">
      <j1 rule="back_rewrite" parents="69"/>
      <j2 rule="rewrite" parents="91"/>
    </justification>
  </clause>

  <clause id="112">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[back_rewrite(12),rewrite(91(3))].">
      <j1 rule="back_rewrite" parents="12"/>
      <j2 rule="rewrite" parents="91"/>
    </justification>
  </clause>

  <clause id="114">
    <literal><![CDATA[
      f(f(x',f(f(x,x'),x')),x') = x
    ]]></literal>
    <justification jstring="[back_rewrite(92),rewrite(96(5),96(12))].">
      <j1 rule="back_rewrite" parents="92"/>
      <j2 rule="rewrite" parents="96 96"/>
    </justification>
  </clause>

  <clause id="122">
    <literal><![CDATA[
      f(a,f(b,b')) != a'
    ]]></literal>
    <literal><![CDATA[
      f(f(b',a),f(c',a)) != f(a,f(b,c))'
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(5),rewrite(112(3)),xx(a)].">
      <j1 rule="back_rewrite" parents="5"/>
      <j2 rule="rewrite" parents="112"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="123">
    <literal><![CDATA[
      f(f(x',f(f(y,x'),x')),f(y,x)) = y
    ]]></literal>
    <justification jstring="[para(112(a,1),10(a,1,2,2))].">
      <j1 rule="para" parents="112 10"/>
    </justification>
  </clause>

  <clause id="140">
    <literal><![CDATA[
      f(f(f(x,f(f(y,x),x)),f(f(f(y,f(f(z,y),y)),f(x,f(f(y,x),x))),f(x,f(f(y,x),x)))),z) = f(y,f(f(z,y),y))
    ]]></literal>
    <justification jstring="[para(8(a,1),8(a,1,1,1)),rewrite(8(16),8(20))].">
      <j1 rule="para" parents="8 8"/>
      <j2 rule="rewrite" parents="8 8"/>
    </justification>
  </clause>

  <clause id="199">
    <literal><![CDATA[
      f(f(x,f(f(f(x',f(f(y,x'),x')),x),x)),y) = f(x',f(f(y,x'),x'))
    ]]></literal>
    <justification jstring="[para(114(a,1),8(a,1,1,1)),rewrite(114(14),114(15))].">
      <j1 rule="para" parents="114 8"/>
      <j2 rule="rewrite" parents="114 114"/>
    </justification>
  </clause>

  <clause id="231">
    <literal><![CDATA[
      f(f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(f(f(f(z,f(f(x,z),z)),f(f(u,f(z,f(f(x,z),z))),f(z,f(f(x,z),z)))),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)))))),u) = f(f(z,f(f(x,z),z)),f(f(u,f(z,f(f(x,z),z))),f(z,f(f(x,z),z))))
    ]]></literal>
    <justification jstring="[para(19(a,1),8(a,1,1,1)),rewrite(19(40),19(47))].">
      <j1 rule="para" parents="19 8"/>
      <j2 rule="rewrite" parents="19 19"/>
    </justification>
  </clause>

  <clause id="280">
    <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(18(a,1),65(a,1,1,1)),rewrite(18(41),18(48))].">
      <j1 rule="para" parents="18 65"/>
      <j2 rule="rewrite" parents="18 18"/>
    </justification>
  </clause>

  <clause id="296">
    <literal><![CDATA[
      f(f(x,f(f(y',x),x)),y) = y'
    ]]></literal>
    <justification jstring="[para(1(a,1),67(a,1,2)),rewrite(4(1),91(3),4(6),91(8))].">
      <j1 rule="para" parents="1 67"/>
      <j2 rule="rewrite" parents="4 91 4 91"/>
    </justification>
  </clause>

  <clause id="297">
    <literal><![CDATA[
      f(f(x,f(f(y,f(f(x,y),y)),x)),f(f(f(z,f(f(y,z),z)),f(f(f(y,f(f(x,y),y)),f(z,f(f(y,z),z))),f(z,f(f(y,z),z)))),f(u,x))) = f(f(z,f(f(y,z),z)),f(f(f(y,f(f(x,y),y)),f(z,f(f(y,z),z))),f(z,f(f(y,z),z))))
    ]]></literal>
    <justification jstring="[para(67(a,1),6(a,1,1,1)),rewrite(67(11),67(28))].">
      <j1 rule="para" parents="67 6"/>
      <j2 rule="rewrite" parents="67 67"/>
    </justification>
  </clause>

  <clause id="298">
    <literal><![CDATA[
      f(f(x,f(f(y,x),x)),y') = y
    ]]></literal>
    <justification jstring="[para(6(a,1),67(a,1,2)),rewrite(4(1),91(3),4(2),91(4),96(4),4(4),91(6),4(6),91(8),4(7),91(9),96(9))].">
      <j1 rule="para" parents="6 67"/>
      <j2 rule="rewrite" parents="4 91 4 91 96 4 91 4 91 4 91 96"/>
    </justification>
  </clause>

  <clause id="299">
    <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(x,f(y,f(f(z,y),y)))')) = f(z,f(f(x,z),z))
    ]]></literal>
    <justification jstring="[para(67(a,1),10(a,1,1,2,1))].">
      <j1 rule="para" parents="67 10"/>
    </justification>
  </clause>

  <clause id="329">
    <literal><![CDATA[
      f(f(f(x,y),f(f(f(y,f(f(z,f(f(y,z),z)),y)),f(x,y)),f(x,y))),f(f(u,f(f(z,u),u)),f(f(f(z,f(f(y,z),z)),f(u,f(f(z,u),u))),f(u,f(f(z,u),u))))) = f(y,f(f(z,f(f(y,z),z)),y))
    ]]></literal>
    <justification jstring="[para(67(a,1),19(a,1,1,1,2)),rewrite(67(9),67(12),67(14),67(16),67(35),67(38))].">
      <j1 rule="para" parents="67 19"/>
      <j2 rule="rewrite" parents="67 67 67 67 67 67"/>
    </justification>
  </clause>

  <clause id="337">
    <literal><![CDATA[
      f(f(f(x,y),f(x,f(x,y))),f(f(f(y,z),f(f(x,f(y,z)),f(y,z))),f(u,f(f(f(x,y),u),u)))) = f(f(y,z),f(f(x,f(y,z)),f(y,z)))
    ]]></literal>
    <justification jstring="[para(65(a,1),67(a,1,1,2,1))].">
      <j1 rule="para" parents="65 67"/>
    </justification>
  </clause>

  <clause id="354">
    <literal><![CDATA[
      f(x,f(f(y,f(f(x,y),y)),x)) = f(y,f(f(x,y),y))
    ]]></literal>
    <justification jstring="[para(298(a,1),10(a,1,1,2,1)),rewrite(96(4),112(5))].">
      <j1 rule="para" parents="298 10"/>
      <j2 rule="rewrite" parents="96 112"/>
    </justification>
  </clause>

  <clause id="375">
    <literal><![CDATA[
      f(x,f(y,f(f(x,y),y))') = f(y,f(f(x,y),y))
    ]]></literal>
    <justification jstring="[para(298(a,1),298(a,1,1,2,1)),rewrite(96(4))].">
      <j1 rule="para" parents="298 298"/>
      <j2 rule="rewrite" parents="96"/>
    </justification>
  </clause>

  <clause id="380">
    <literal><![CDATA[
      f(f(f(x,y),f(f(f(z,f(f(y,z),z)),f(x,y)),f(x,y))),f(f(u,f(f(z,u),u)),f(f(f(z,f(f(y,z),z)),f(u,f(f(z,u),u))),f(u,f(f(z,u),u))))) = f(z,f(f(y,z),z))
    ]]></literal>
    <justification jstring="[back_rewrite(329),rewrite(354(6),354(30))].">
      <j1 rule="back_rewrite" parents="329"/>
      <j2 rule="rewrite" parents="354 354"/>
    </justification>
  </clause>

  <clause id="382">
    <literal><![CDATA[
      f(f(x,f(f(y,x),x)),f(f(f(z,f(f(x,z),z)),f(f(f(x,f(f(y,x),x)),f(z,f(f(x,z),z))),f(z,f(f(x,z),z)))),f(u,y))) = f(f(z,f(f(x,z),z)),f(f(f(x,f(f(y,x),x)),f(z,f(f(x,z),z))),f(z,f(f(x,z),z))))
    ]]></literal>
    <justification jstring="[back_rewrite(297),rewrite(354(5))].">
      <j1 rule="back_rewrite" parents="297"/>
      <j2 rule="rewrite" parents="354"/>
    </justification>
  </clause>

  <clause id="391">
    <literal><![CDATA[
      f(f(x,f(f(y,x),x)),f(f(x',f(f(f(x,f(f(y,x),x)),x'),x')),f(z,y))) = f(x',f(f(f(x,f(f(y,x),x)),x'),x'))
    ]]></literal>
    <justification jstring="[back_rewrite(43),rewrite(354(5))].">
      <j1 rule="back_rewrite" parents="43"/>
      <j2 rule="rewrite" parents="354"/>
    </justification>
  </clause>

  <clause id="406">
    <literal><![CDATA[
      f(x',f(f(y,f(f(x',y),y)),f(z,x))) = f(y,f(f(x',y),y))
    ]]></literal>
    <justification jstring="[para(296(a,1),1(a,1,1,2,1)),rewrite(91(3))].">
      <j1 rule="para" parents="296 1"/>
      <j2 rule="rewrite" parents="91"/>
    </justification>
  </clause>

  <clause id="407">
    <literal><![CDATA[
      f(f(x,f(f(f(y,f(f(f(z,x)',y),y)),x),x)),f(z,x)') = f(y,f(f(f(z,x)',y),y))
    ]]></literal>
    <justification jstring="[para(296(a,1),1(a,1,2))].">
      <j1 rule="para" parents="296 1"/>
    </justification>
  </clause>

  <clause id="409">
    <literal><![CDATA[
      f(f(f(f(x,f(f(y',x),x)),f(z,y)),f(f(x,f(f(y',x),x)),f(f(x,f(f(y',x),x)),f(z,y)))),f(y',f(u,f(f(x,f(f(y',x),x)),f(z,y))))) = y'
    ]]></literal>
    <justification jstring="[para(296(a,1),6(a,1,2,1,2,1)),rewrite(91(21),296(33),91(31))].">
      <j1 rule="para" parents="296 6"/>
      <j2 rule="rewrite" parents="91 296 91"/>
    </justification>
  </clause>

  <clause id="411">
    <literal><![CDATA[
      f(x,f(f(f(y,x)',x),x)) = f(y,x)
    ]]></literal>
    <justification jstring="[para(296(a,1),8(a,1,1,2,1)),rewrite(91(6),4(5),112(3)),flip(a)].">
      <j1 rule="para" parents="296 8"/>
      <j2 rule="rewrite" parents="91 4 112"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="416">
    <literal><![CDATA[
      f(f(x,f(f(f(y,f(f(f(z,f(f(x,z),z))',y),y)),x),x)),f(z,f(f(x,z),z))') = f(y,f(f(f(z,f(f(x,z),z))',y),y))
    ]]></literal>
    <justification jstring="[para(296(a,1),67(a,1,2))].">
      <j1 rule="para" parents="296 67"/>
    </justification>
  </clause>

  <clause id="485">
    <literal><![CDATA[
      f(x',f(y,x)) = x
    ]]></literal>
    <justification jstring="[para(411(a,1),1(a,1,2)),rewrite(4(1),91(3))].">
      <j1 rule="para" parents="411 1"/>
      <j2 rule="rewrite" parents="4 91"/>
    </justification>
  </clause>

  <clause id="492">
    <literal><![CDATA[
      f(x,f(y,x')) = x'
    ]]></literal>
    <justification jstring="[para(411(a,1),6(a,1,2)),rewrite(4(1),91(3),4(2),91(4),96(4),4(1),91(3),4(4),91(6))].">
      <j1 rule="para" parents="411 6"/>
      <j2 rule="rewrite" parents="4 91 4 91 96 4 91 4 91"/>
    </justification>
  </clause>

  <clause id="496">
    <literal><![CDATA[
      f(f(x',x),x) = x'
    ]]></literal>
    <justification jstring="[para(91(a,1),411(a,1,2,1,1,1)),rewrite(112(4),91(5),485(6),91(6))].">
      <j1 rule="para" parents="91 411"/>
      <j2 rule="rewrite" parents="112 91 485 91"/>
    </justification>
  </clause>

  <clause id="500">
    <literal><![CDATA[
      f(f(x,x'),x') = x
    ]]></literal>
    <justification jstring="[para(96(a,1),411(a,1,2,1,1,1)),rewrite(96(6),492(5),96(8))].">
      <j1 rule="para" parents="96 411"/>
      <j2 rule="rewrite" parents="96 492 96"/>
    </justification>
  </clause>

  <clause id="505">
    <literal><![CDATA[
      f(f(x,y)',f(f(x,y),y)) = f(x,y)
    ]]></literal>
    <justification jstring="[para(411(a,1),100(a,1,1,1)),rewrite(411(7),411(10))].">
      <j1 rule="para" parents="411 100"/>
      <j2 rule="rewrite" parents="411 411"/>
    </justification>
  </clause>

  <clause id="511">
    <literal><![CDATA[
      f(f(x,y),f(f(x',f(x,y)),f(x,y))) = x
    ]]></literal>
    <justification jstring="[para(123(a,1),411(a,1,2,1,1,1)),rewrite(123(15))].">
      <j1 rule="para" parents="123 411"/>
      <j2 rule="rewrite" parents="123"/>
    </justification>
  </clause>

  <clause id="524">
    <literal><![CDATA[
      f(f(f(f(f(x,y)',y),y),f(f(x,y),f(f(f(x,y)',y),y))),y') = y
    ]]></literal>
    <justification jstring="[para(411(a,1),298(a,1,1,2,1))].">
      <j1 rule="para" parents="411 298"/>
    </justification>
  </clause>

  <clause id="525">
    <literal><![CDATA[
      f(f(f(f(f(x,y')',y'),y'),f(f(x,y'),f(f(f(x,y')',y'),y'))),y) = y'
    ]]></literal>
    <justification jstring="[para(411(a,1),296(a,1,1,2,1))].">
      <j1 rule="para" parents="411 296"/>
    </justification>
  </clause>

  <clause id="532">
    <literal><![CDATA[
      f(f(f(x,f(y,f(f(f(z,x)',x),x))),f(x,f(x,f(y,f(f(f(z,x)',x),x))))),f(f(f(f(f(z,x)',x),x),f(f(z,x),f(f(f(z,x)',x),x))),x)) = f(f(f(f(z,x)',x),x),f(f(z,x),f(f(f(z,x)',x),x)))
    ]]></literal>
    <justification jstring="[para(411(a,1),22(a,1,2,1,2,1)),rewrite(411(36))].">
      <j1 rule="para" parents="411 22"/>
      <j2 rule="rewrite" parents="411"/>
    </justification>
  </clause>

  <clause id="536">
    <literal><![CDATA[
      f(f(f(f(x,y)',y),y),f(f(f(x,y)',f(f(f(x,y)',y),y)),f(f(f(x,y)',y),y))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(411(a,1),411(a,1,2,1,1,1)),rewrite(411(22))].">
      <j1 rule="para" parents="411 411"/>
      <j2 rule="rewrite" parents="411"/>
    </justification>
  </clause>

  <clause id="591">
    <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(485(a,1),6(a,1,2,2))].">
      <j1 rule="para" parents="485 6"/>
    </justification>
  </clause>

  <clause id="595">
    <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(485(a,1),8(a,1,1,1)),rewrite(485(9),485(10))].">
      <j1 rule="para" parents="485 8"/>
      <j2 rule="rewrite" parents="485 485"/>
    </justification>
  </clause>

  <clause id="596">
    <literal><![CDATA[
      f(x',f(y,f(f(x,y),y))) = x
    ]]></literal>
    <justification jstring="[para(8(a,1),485(a,1,2))].">
      <j1 rule="para" parents="8 485"/>
    </justification>
  </clause>

  <clause id="603">
    <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(u,z),f(f(y,f(u,z)),f(u,z)))) = f(f(y,z),f(y,f(y,z)))
    ]]></literal>
    <justification jstring="[para(485(a,1),19(a,1,1,1,2,2)),rewrite(485(5),485(6),485(9),485(12),485(23),485(24))].">
      <j1 rule="para" parents="485 19"/>
      <j2 rule="rewrite" parents="485 485 485 485 485 485"/>
    </justification>
  </clause>

  <clause id="605">
    <literal><![CDATA[
      f(f(f(x,y),f(f(f(f(z,f(x,y)),f(z,f(z,f(x,y)))),f(x,y)),f(x,y))),f(y,f(f(z,y),y))) = f(f(z,f(x,y)),f(z,f(z,f(x,y))))
    ]]></literal>
    <justification jstring="[para(485(a,1),19(a,1,1,1)),rewrite(485(12),485(14))].">
      <j1 rule="para" parents="485 19"/>
      <j2 rule="rewrite" parents="485 485"/>
    </justification>
  </clause>

  <clause id="614">
    <literal><![CDATA[
      f(x',x) = f(x,x')
    ]]></literal>
    <justification jstring="[para(485(a,1),22(a,1,1,1)),rewrite(485(4),91(3),496(4),500(5),496(5))].">
      <j1 rule="para" parents="485 22"/>
      <j2 rule="rewrite" parents="485 91 496 500 496"/>
    </justification>
  </clause>

  <clause id="618">
    <literal><![CDATA[
      f(f(x,y)',y) = f(x,y)
    ]]></literal>
    <justification jstring="[para(485(a,1),485(a,1,2))].">
      <j1 rule="para" parents="485 485"/>
    </justification>
  </clause>

  <clause id="643">
    <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(536),rewrite(618(3),618(7),505(7),618(6))].">
      <j1 rule="back_rewrite" parents="536"/>
      <j2 rule="rewrite" parents="618 618 505 618"/>
    </justification>
  </clause>

  <clause id="644">
    <literal><![CDATA[
      f(f(f(x,f(y,f(f(z,x),x))),f(x,f(x,f(y,f(f(z,x),x))))),f(f(z,x),x)) = f(z,x)
    ]]></literal>
    <justification jstring="[back_rewrite(532),rewrite(618(3),618(7),618(13),618(16),643(17),618(16),618(19),643(20))].">
      <j1 rule="back_rewrite" parents="532"/>
      <j2 rule="rewrite" parents="618 618 618 618 643 618 618 643"/>
    </justification>
  </clause>

  <clause id="645">
    <literal><![CDATA[
      f(f(x,y'),y) = y'
    ]]></literal>
    <justification jstring="[back_rewrite(525),rewrite(618(5),618(11),643(12))].">
      <j1 rule="back_rewrite" parents="525"/>
      <j2 rule="rewrite" parents="618 618 643"/>
    </justification>
  </clause>

  <clause id="646">
    <literal><![CDATA[
      f(f(x,y),y') = y
    ]]></literal>
    <justification jstring="[back_rewrite(524),rewrite(618(3),618(6),643(7))].">
      <j1 rule="back_rewrite" parents="524"/>
      <j2 rule="rewrite" parents="618 618 643"/>
    </justification>
  </clause>

  <clause id="654">
    <literal><![CDATA[
      f(x,f(f(y,x),x)) = f(y,x)
    ]]></literal>
    <justification jstring="[back_rewrite(411),rewrite(618(3))].">
      <j1 rule="back_rewrite" parents="411"/>
      <j2 rule="rewrite" parents="618"/>
    </justification>
  </clause>

  <clause id="666">
    <literal><![CDATA[
      f(f(f(f(x,f(y,z)),f(x,f(x,f(y,z)))),f(y,z)),f(x,z)) = f(f(x,f(y,z)),f(x,f(x,f(y,z))))
    ]]></literal>
    <justification jstring="[back_rewrite(605),rewrite(654(12),654(11))].">
      <j1 rule="back_rewrite" parents="605"/>
      <j2 rule="rewrite" parents="654 654"/>
    </justification>
  </clause>

  <clause id="667">
    <literal><![CDATA[
      f(f(f(f(x,y),f(x,f(x,y))),f(z,f(x,y))),f(x,f(u,y))) = f(f(x,y),f(x,f(x,y)))
    ]]></literal>
    <justification jstring="[back_rewrite(603),rewrite(654(13),654(13))].">
      <j1 rule="back_rewrite" parents="603"/>
      <j2 rule="rewrite" parents="654 654"/>
    </justification>
  </clause>

  <clause id="671">
    <literal><![CDATA[
      f(x',f(x,y)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(596),rewrite(654(4))].">
      <j1 rule="back_rewrite" parents="596"/>
      <j2 rule="rewrite" parents="654"/>
    </justification>
  </clause>

  <clause id="672">
    <literal><![CDATA[
      f(f(f(x,f(y,z)),z),x) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[back_rewrite(595),rewrite(654(6),654(5),654(10))].">
      <j1 rule="back_rewrite" parents="595"/>
      <j2 rule="rewrite" parents="654 654 654"/>
    </justification>
  </clause>

  <clause id="673">
    <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(591),rewrite(654(9),654(13))].">
      <j1 rule="back_rewrite" parents="591"/>
      <j2 rule="rewrite" parents="654 654"/>
    </justification>
  </clause>

  <clause id="688">
    <literal><![CDATA[
      f(f(x,y),f(x,f(x,y))) = x
    ]]></literal>
    <justification jstring="[back_rewrite(511),rewrite(671(4))].">
      <j1 rule="back_rewrite" parents="511"/>
      <j2 rule="rewrite" parents="671"/>
    </justification>
  </clause>

  <clause id="692">
    <literal><![CDATA[
      f(f(f(f(x,y)',z),x),f(x,y)') = f(f(x,y)',z)
    ]]></literal>
    <justification jstring="[back_rewrite(416),rewrite(654(3),654(5),654(6),654(7),654(10),654(12))].">
      <j1 rule="back_rewrite" parents="416"/>
      <j2 rule="rewrite" parents="654 654 654 654 654 654"/>
    </justification>
  </clause>

  <clause id="697">
    <literal><![CDATA[
      f(f(x',y),f(x',f(z,f(f(x',y),f(u,x))))) = x'
    ]]></literal>
    <justification jstring="[back_rewrite(409),rewrite(654(4),654(8),654(10),688(12),654(7))].">
      <j1 rule="back_rewrite" parents="409"/>
      <j2 rule="rewrite" parents="654 654 654 688 654"/>
    </justification>
  </clause>

  <clause id="699">
    <literal><![CDATA[
      f(f(f(f(x,y)',z),y),f(x,y)') = f(f(x,y)',z)
    ]]></literal>
    <justification jstring="[back_rewrite(407),rewrite(654(5),654(6),654(12))].">
      <j1 rule="back_rewrite" parents="407"/>
      <j2 rule="rewrite" parents="654 654 654"/>
    </justification>
  </clause>

  <clause id="700">
    <literal><![CDATA[
      f(x',f(f(x',y),f(z,x))) = f(x',y)
    ]]></literal>
    <justification jstring="[back_rewrite(406),rewrite(654(5),654(10))].">
      <j1 rule="back_rewrite" parents="406"/>
      <j2 rule="rewrite" parents="654 654"/>
    </justification>
  </clause>

  <clause id="702">
    <literal><![CDATA[
      f(f(x,y),f(y,f(z,x))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(391),rewrite(654(3),654(5),646(5),671(5),654(8),646(8),671(8))].">
      <j1 rule="back_rewrite" parents="391"/>
      <j2 rule="rewrite" parents="654 654 646 671 654 646 671"/>
    </justification>
  </clause>

  <clause id="710">
    <literal><![CDATA[
      f(f(x,y),f(f(f(x,y),f(y,z)),f(u,x))) = f(f(x,y),f(y,z))
    ]]></literal>
    <justification jstring="[back_rewrite(382),rewrite(654(3),654(4),654(5),654(6),654(8),654(8),654(10),654(11),654(12),654(14),654(14))].">
      <j1 rule="back_rewrite" parents="382"/>
      <j2 rule="rewrite" parents="654 654 654 654 654 654 654 654 654 654 654"/>
    </justification>
  </clause>

  <clause id="712">
    <literal><![CDATA[
      f(f(f(x,y),f(z,x)),f(f(x,y),f(y,u))) = f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(380),rewrite(654(4),654(7),654(6),654(7),654(8),654(10),654(10),654(10))].">
      <j1 rule="back_rewrite" parents="380"/>
      <j2 rule="rewrite" parents="654 654 654 654 654 654 654 654"/>
    </justification>
  </clause>

  <clause id="714">
    <literal><![CDATA[
      f(x,f(x,y)') = f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(375),rewrite(654(3),654(6))].">
      <j1 rule="back_rewrite" parents="375"/>
      <j2 rule="rewrite" parents="654 654"/>
    </justification>
  </clause>

  <clause id="727">
    <literal><![CDATA[
      f(x,f(f(x,f(y,z)),f(f(x,y),u))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[back_rewrite(337),rewrite(688(4),654(6),654(6),654(12))].">
      <j1 rule="back_rewrite" parents="337"/>
      <j2 rule="rewrite" parents="688 654 654 654"/>
    </justification>
  </clause>

  <clause id="735">
    <literal><![CDATA[
      f(x,f(f(x,y),f(x,f(y,z))')) = f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(299),rewrite(654(3),654(5),688(6),654(3),654(4),654(9))].">
      <j1 rule="back_rewrite" parents="299"/>
      <j2 rule="rewrite" parents="654 654 688 654 654 654"/>
    </justification>
  </clause>

  <clause id="740">
    <literal><![CDATA[
      f(f(x,y),f(x,f(y,z))) = x
    ]]></literal>
    <justification jstring="[back_rewrite(280),rewrite(688(6),688(6),688(7),654(3),654(4),654(9),654(10),688(14))].">
      <j1 rule="back_rewrite" parents="280"/>
      <j2 rule="rewrite" parents="688 688 688 654 654 654 654 688"/>
    </justification>
  </clause>

  <clause id="749">
    <literal><![CDATA[
      f(f(f(x,f(y,z)),y),x) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[back_rewrite(231),rewrite(688(6),654(3),654(4),654(6),654(6),688(8),688(9),654(5),654(7),654(8),654(10),654(10))].">
      <j1 rule="back_rewrite" parents="231"/>
      <j2 rule="rewrite" parents="688 654 654 654 654 688 688 654 654 654 654 654"/>
    </justification>
  </clause>

  <clause id="759">
    <literal><![CDATA[
      f(x',y) = f(y,x')
    ]]></literal>
    <justification jstring="[back_rewrite(199),rewrite(654(6),645(3),614(2),492(3),654(8))].">
      <j1 rule="back_rewrite" parents="199"/>
      <j2 rule="rewrite" parents="654 645 614 492 654"/>
    </justification>
  </clause>

  <clause id="765">
    <literal><![CDATA[
      f(f(f(x,y),f(y,z)),x) = f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(140),rewrite(654(3),654(4),654(5),654(7),654(7),654(7))].">
      <j1 rule="back_rewrite" parents="140"/>
      <j2 rule="rewrite" parents="654 654 654 654 654 654"/>
    </justification>
  </clause>

  <clause id="771">
    <literal><![CDATA[
      f(f(x',f(x',f(y,x'))),f(y,x)) = y
    ]]></literal>
    <justification jstring="[back_rewrite(123),rewrite(759(5,R))].">
      <j1 rule="back_rewrite" parents="123"/>
      <j2 rule="rewrite" parents="759"/>
    </justification>
  </clause>

  <clause id="776">
    <literal><![CDATA[
      f(x,y) = f(y,x)
    ]]></literal>
    <justification jstring="[back_rewrite(38),rewrite(654(4),759(4,R),485(4),671(4),654(4))].">
      <j1 rule="back_rewrite" parents="38"/>
      <j2 rule="rewrite" parents="654 759 485 671 654"/>
    </justification>
  </clause>

  <clause id="790">
    <literal><![CDATA[
      f(x,f(f(x,y),f(z,y))) = f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(673),rewrite(688(6))].">
      <j1 rule="back_rewrite" parents="673"/>
      <j2 rule="rewrite" parents="688"/>
    </justification>
  </clause>

  <clause id="794">
    <literal><![CDATA[
      f(f(x,f(y,f(x,z))),f(x,f(u,z))) = x
    ]]></literal>
    <justification jstring="[back_rewrite(667),rewrite(688(4),688(10))].">
      <j1 rule="back_rewrite" parents="667"/>
      <j2 rule="rewrite" parents="688 688"/>
    </justification>
  </clause>

  <clause id="795">
    <literal><![CDATA[
      f(f(x,y),f(x,f(z,y))) = x
    ]]></literal>
    <justification jstring="[back_rewrite(666),rewrite(688(6),776(4),688(10))].">
      <j1 rule="back_rewrite" parents="666"/>
      <j2 rule="rewrite" parents="688 776 688"/>
    </justification>
  </clause>

  <clause id="798">
    <literal><![CDATA[
      f(x,f(x,f(y,x))) = f(y,x)
    ]]></literal>
    <justification jstring="[back_rewrite(644),rewrite(776(2),776(6),688(10),776(2))].">
      <j1 rule="back_rewrite" parents="644"/>
      <j2 rule="rewrite" parents="776 776 688 776"/>
    </justification>
  </clause>

  <clause id="803">
    <literal><![CDATA[
      f(f(x,y)',f(y,f(f(x,y)',z))) = f(f(x,y)',z)
    ]]></literal>
    <justification jstring="[back_rewrite(699),rewrite(776(4),776(7))].">
      <j1 rule="back_rewrite" parents="699"/>
      <j2 rule="rewrite" parents="776 776"/>
    </justification>
  </clause>

  <clause id="804">
    <literal><![CDATA[
      f(f(x,y)',f(x,f(f(x,y)',z))) = f(f(x,y)',z)
    ]]></literal>
    <justification jstring="[back_rewrite(692),rewrite(776(4),776(7))].">
      <j1 rule="back_rewrite" parents="692"/>
      <j2 rule="rewrite" parents="776 776"/>
    </justification>
  </clause>

  <clause id="806">
    <literal><![CDATA[
      f(a,f(b,b')) != a'
    ]]></literal>
    <literal><![CDATA[
      f(f(a,b'),f(a,c')) != f(a,f(b,c))'
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(122),rewrite(776(13),776(17))].">
      <j1 rule="back_rewrite" parents="122"/>
      <j2 rule="rewrite" parents="776 776"/>
    </justification>
  </clause>

  <clause id="807">
    <literal><![CDATA[
      f(f(x,y),f(x,y')) = x
    ]]></literal>
    <justification jstring="[back_rewrite(771),rewrite(798(6),776(4))].">
      <j1 rule="back_rewrite" parents="771"/>
      <j2 rule="rewrite" parents="798 776"/>
    </justification>
  </clause>

  <clause id="811">
    <literal><![CDATA[
      f(x,f(f(x,y),f(y,z))) = f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(765),rewrite(776(4))].">
      <j1 rule="back_rewrite" parents="765"/>
      <j2 rule="rewrite" parents="776"/>
    </justification>
  </clause>

  <clause id="814">
    <literal><![CDATA[
      f(x,f(y,f(x,f(y,z)))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[back_rewrite(749),rewrite(776(3),776(4))].">
      <j1 rule="back_rewrite" parents="749"/>
      <j2 rule="rewrite" parents="776 776"/>
    </justification>
  </clause>

  <clause id="827">
    <literal><![CDATA[
      f(x,f(y,f(x,f(z,y)))) = f(x,f(z,y))
    ]]></literal>
    <justification jstring="[back_rewrite(672),rewrite(776(3),776(4))].">
      <j1 rule="back_rewrite" parents="672"/>
      <j2 rule="rewrite" parents="776 776"/>
    </justification>
  </clause>

  <clause id="835">
    <literal><![CDATA[
      f(f(x,y),f(y,x')) = y
    ]]></literal>
    <justification jstring="[para(776(a,1),807(a,1,1))].">
      <j1 rule="para" parents="776 807"/>
    </justification>
  </clause>

  <clause id="836">
    <literal><![CDATA[
      f(f(x,y),f(y',x)) = x
    ]]></literal>
    <justification jstring="[para(776(a,1),807(a,1,2))].">
      <j1 rule="para" parents="776 807"/>
    </justification>
  </clause>

  <clause id="838">
    <literal><![CDATA[
      f(f(x,y'),f(y,x)) = x
    ]]></literal>
    <justification jstring="[para(835(a,1),776(a,1)),flip(a)].">
      <j1 rule="para" parents="835 776"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="850">
    <literal><![CDATA[
      f(x,f(f(y,x)',f(x,y'))) = f(x,y')
    ]]></literal>
    <justification jstring="[para(838(a,1),836(a,1,1))].">
      <j1 rule="para" parents="838 836"/>
    </justification>
  </clause>

  <clause id="853">
    <literal><![CDATA[
      f(x,f(f(y,x),f(z,x'))) = f(y,x)
    ]]></literal>
    <justification jstring="[para(485(a,1),702(a,1,1))].">
      <j1 rule="para" parents="485 702"/>
    </justification>
  </clause>

  <clause id="855">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(z,x)) = x
    ]]></literal>
    <justification jstring="[para(702(a,1),776(a,1)),flip(a)].">
      <j1 rule="para" parents="702 776"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="856">
    <literal><![CDATA[
      f(f(x,y),f(y,f(x,z))) = y
    ]]></literal>
    <justification jstring="[para(776(a,1),702(a,1,2,2))].">
      <j1 rule="para" parents="776 702"/>
    </justification>
  </clause>

  <clause id="857">
    <literal><![CDATA[
      f(f(x,y),f(f(z,x),y)) = y
    ]]></literal>
    <justification jstring="[para(776(a,1),702(a,1,2))].">
      <j1 rule="para" parents="776 702"/>
    </justification>
  </clause>

  <clause id="888">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(x,y)',z))) = x
    ]]></literal>
    <justification jstring="[para(714(a,1),740(a,1,1))].">
      <j1 rule="para" parents="714 740"/>
    </justification>
  </clause>

  <clause id="928">
    <literal><![CDATA[
      f(x,f(f(y,x),f(x',z))) = f(y,x)
    ]]></literal>
    <justification jstring="[para(485(a,1),856(a,1,1))].">
      <j1 rule="para" parents="485 856"/>
    </justification>
  </clause>

  <clause id="929">
    <literal><![CDATA[
      f(f(x,y),f(f(x,z),y)) = y
    ]]></literal>
    <justification jstring="[para(776(a,1),856(a,1,2))].">
      <j1 rule="para" parents="776 856"/>
    </justification>
  </clause>

  <clause id="947">
    <literal><![CDATA[
      f(x,f(f(y,x'),f(z,x))) = f(z,x)
    ]]></literal>
    <justification jstring="[para(485(a,1),857(a,1,1))].">
      <j1 rule="para" parents="485 857"/>
    </justification>
  </clause>

  <clause id="948">
    <literal><![CDATA[
      f(x',f(f(y,x),f(z,x'))) = f(z,x')
    ]]></literal>
    <justification jstring="[para(492(a,1),857(a,1,1))].">
      <j1 rule="para" parents="492 857"/>
    </justification>
  </clause>

  <clause id="949">
    <literal><![CDATA[
      f(x,f(f(y,x'),f(x,z))) = f(x,z)
    ]]></literal>
    <justification jstring="[para(671(a,1),857(a,1,1))].">
      <j1 rule="para" parents="671 857"/>
    </justification>
  </clause>

  <clause id="1189">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(x,y),z))) = f(x,f(f(x,y),z))'
    ]]></literal>
    <justification jstring="[para(740(a,1),710(a,1,2)),rewrite(776(3),776(6),4(7),776(7),776(9)),flip(a)].">
      <j1 rule="para" parents="740 710"/>
      <j2 rule="rewrite" parents="776 776 4 776 776"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1533">
    <literal><![CDATA[
      f(f(x,y),f(y,x)) = f(x,y)'
    ]]></literal>
    <justification jstring="[para(712(a,1),790(a,1,2)),rewrite(4(3)),flip(a)].">
      <j1 rule="para" parents="712 790"/>
      <j2 rule="rewrite" parents="4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1551">
    <literal><![CDATA[
      f(f(x,y)',f(y,x)') = f(x,y)
    ]]></literal>
    <justification jstring="[para(1533(a,1),485(a,1,2))].">
      <j1 rule="para" parents="1533 485"/>
    </justification>
  </clause>

  <clause id="1902">
    <literal><![CDATA[
      f(x',f(y,f(f(x,x'),f(z,x)))) = x
    ]]></literal>
    <justification jstring="[para(697(a,1),949(a,1,2)),rewrite(4(3),112(2),112(4),776(3)),flip(a)].">
      <j1 rule="para" parents="697 949"/>
      <j2 rule="rewrite" parents="4 112 112 776"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1923">
    <literal><![CDATA[
      f(x,f(f(x,x')',y)) = x'
    ]]></literal>
    <justification jstring="[para(888(a,1),949(a,1,2)),rewrite(4(1)),flip(a)].">
      <j1 rule="para" parents="888 949"/>
      <j2 rule="rewrite" parents="4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1926">
    <literal><![CDATA[
      f(x',f(f(x,x')',y)) = x
    ]]></literal>
    <justification jstring="[para(112(a,1),1923(a,1,2,1,1,2)),rewrite(776(3),112(8))].">
      <j1 rule="para" parents="112 1923"/>
      <j2 rule="rewrite" parents="776 112"/>
    </justification>
  </clause>

  <clause id="1928">
    <literal><![CDATA[
      f(f(x,x')',y) = f(x,x')
    ]]></literal>
    <justification jstring="[para(1923(a,1),835(a,1,1)),rewrite(776(7),1926(7),776(2)),flip(a)].">
      <j1 rule="para" parents="1923 835"/>
      <j2 rule="rewrite" parents="776 1926 776"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2013">
    <literal><![CDATA[
      f(x,f(y,y')) = x'
    ]]></literal>
    <justification jstring="[para(1928(a,1),492(a,1,2))].">
      <j1 rule="para" parents="1928 492"/>
    </justification>
  </clause>

  <clause id="2016">
    <literal><![CDATA[
      f(x,x') = f(y,y')
    ]]></literal>
    <justification jstring="[para(1928(a,1),798(a,1,2,2)),rewrite(2013(3),1928(6))].">
      <j1 rule="para" parents="1928 798"/>
      <j2 rule="rewrite" parents="2013 1928"/>
    </justification>
  </clause>

  <clause id="2024">
    <literal><![CDATA[
      f(f(x,x'),y) = y'
    ]]></literal>
    <justification jstring="[para(1928(a,1),928(a,1,2,2)),rewrite(2013(5),2013(6),112(4),2013(6))].">
      <j1 rule="para" parents="1928 928"/>
      <j2 rule="rewrite" parents="2013 2013 112 2013"/>
    </justification>
  </clause>

  <clause id="2026">
    <literal><![CDATA[
      f(x,f(y,x')') = f(z,z')
    ]]></literal>
    <justification jstring="[para(1928(a,1),947(a,1,2,2)),rewrite(2013(5),1928(8))].">
      <j1 rule="para" parents="1928 947"/>
      <j2 rule="rewrite" parents="2013 1928"/>
    </justification>
  </clause>

  <clause id="2027">
    <literal><![CDATA[
      f(f(a,b'),f(a,c')) != f(a,f(b,c))'
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(806),rewrite(2013(6)),xx(a)].">
      <j1 rule="back_rewrite" parents="806"/>
      <j2 rule="rewrite" parents="2013"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="2028">
    <literal><![CDATA[
      f(x,x') = c_0
    ]]></literal>
    <justification jstring="[new_symbol(2016)].">
      <j1 rule="new_symbol" parents="2016"/>
    </justification>
  </clause>

  <clause id="2039">
    <literal><![CDATA[
      f(x',f(y,f(c_0,f(z,x)))) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1902),rewrite(2028(3))].">
      <j1 rule="back_rewrite" parents="1902"/>
      <j2 rule="rewrite" parents="2028"/>
    </justification>
  </clause>

  <clause id="2040">
    <literal><![CDATA[
      f(x,f(y,x')') = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(2026),rewrite(2028(6))].">
      <j1 rule="back_rewrite" parents="2026"/>
      <j2 rule="rewrite" parents="2028"/>
    </justification>
  </clause>

  <clause id="2042">
    <literal><![CDATA[
      f(c_0,x) = x'
    ]]></literal>
    <justification jstring="[back_rewrite(2024),rewrite(2028(2))].">
      <j1 rule="back_rewrite" parents="2024"/>
      <j2 rule="rewrite" parents="2028"/>
    </justification>
  </clause>

  <clause id="2043">
    <literal><![CDATA[
      f(x,c_0) = x'
    ]]></literal>
    <justification jstring="[back_rewrite(2013),rewrite(2028(2))].">
      <j1 rule="back_rewrite" parents="2013"/>
      <j2 rule="rewrite" parents="2028"/>
    </justification>
  </clause>

  <clause id="2046">
    <literal><![CDATA[
      f(x',f(y,f(z,x)')) = x
    ]]></literal>
    <justification jstring="[back_rewrite(2039),rewrite(2042(4))].">
      <j1 rule="back_rewrite" parents="2039"/>
      <j2 rule="rewrite" parents="2042"/>
    </justification>
  </clause>

  <clause id="2049">
    <literal><![CDATA[
      f(x',f(y,x)') = c_0
    ]]></literal>
    <justification jstring="[para(2028(a,1),700(a,1,2,1)),rewrite(2042(4),112(7),776(6),2028(6))].">
      <j1 rule="para" parents="2028 700"/>
      <j2 rule="rewrite" parents="2042 112 776 2028"/>
    </justification>
  </clause>

  <clause id="2080">
    <literal><![CDATA[
      f(x,f(y,f(x,y))) = x'
    ]]></literal>
    <justification jstring="[para(855(a,1),727(a,1,2)),rewrite(4(1)),flip(a)].">
      <j1 rule="para" parents="855 727"/>
      <j2 rule="rewrite" parents="4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2230">
    <literal><![CDATA[
      f(f(x,y),f(y,x)') = c_0
    ]]></literal>
    <justification jstring="[para(1551(a,1),2040(a,1,2,1))].">
      <j1 rule="para" parents="1551 2040"/>
    </justification>
  </clause>

  <clause id="2360">
    <literal><![CDATA[
      f(x,f(y,f(y,x))) = x'
    ]]></literal>
    <justification jstring="[para(776(a,1),2080(a,1,2,2))].">
      <j1 rule="para" parents="776 2080"/>
    </justification>
  </clause>

  <clause id="2451">
    <literal><![CDATA[
      f(f(x,y),f(f(y,x)',z)') = c_0
    ]]></literal>
    <justification jstring="[para(2230(a,1),811(a,1,2,1)),rewrite(2042(6),2230(10))].">
      <j1 rule="para" parents="2230 811"/>
      <j2 rule="rewrite" parents="2042 2230"/>
    </justification>
  </clause>

  <clause id="2534">
    <literal><![CDATA[
      f(x,f(y,f(y,x))') = c_0
    ]]></literal>
    <justification jstring="[para(2360(a,1),2049(a,1,2,1)),rewrite(112(5),776(4))].">
      <j1 rule="para" parents="2360 2049"/>
      <j2 rule="rewrite" parents="112 776"/>
    </justification>
  </clause>

  <clause id="2700">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(x,f(x,z))') = c_0
    ]]></literal>
    <justification jstring="[para(795(a,1),2534(a,1,2,1,2)),rewrite(776(4))].">
      <j1 rule="para" parents="795 2534"/>
      <j2 rule="rewrite" parents="776"/>
    </justification>
  </clause>

  <clause id="2753">
    <literal><![CDATA[
      f(f(x,y)',f(z,y')) = f(x,y)
    ]]></literal>
    <justification jstring="[para(485(a,1),2046(a,1,2,2,1))].">
      <j1 rule="para" parents="485 2046"/>
    </justification>
  </clause>

  <clause id="2757">
    <literal><![CDATA[
      f(f(x,y)',f(z,x')) = f(x,y)
    ]]></literal>
    <justification jstring="[para(671(a,1),2046(a,1,2,2,1))].">
      <j1 rule="para" parents="671 2046"/>
    </justification>
  </clause>

  <clause id="2789">
    <literal><![CDATA[
      f(x,f(x,y)) = f(x,y')
    ]]></literal>
    <justification jstring="[para(2046(a,1),735(a,1,2,2,1,2)),rewrite(776(5),2753(5))].">
      <j1 rule="para" parents="2046 735"/>
      <j2 rule="rewrite" parents="776 2753"/>
    </justification>
  </clause>

  <clause id="2792">
    <literal><![CDATA[
      f(x,f(y,x)) = f(x,y')
    ]]></literal>
    <justification jstring="[back_rewrite(850),rewrite(2757(5))].">
      <j1 rule="back_rewrite" parents="850"/>
      <j2 rule="rewrite" parents="2757"/>
    </justification>
  </clause>

  <clause id="2815">
    <literal><![CDATA[
      f(f(x,y')',f(x,f(z,y))) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(2700),rewrite(2789(4),776(6))].">
      <j1 rule="back_rewrite" parents="2700"/>
      <j2 rule="rewrite" parents="2789 776"/>
    </justification>
  </clause>

  <clause id="3157">
    <literal><![CDATA[
      f(f(x,y)',f(f(x,z),y)) = f(y,f(x,z)')
    ]]></literal>
    <justification jstring="[para(929(a,1),2792(a,1,2)),rewrite(776(3),2792(3),776(8)),flip(a)].">
      <j1 rule="para" parents="929 2792"/>
      <j2 rule="rewrite" parents="776 2792 776"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3182">
    <literal><![CDATA[
      f(f(x,y),f(z,y')') = f(z,y')
    ]]></literal>
    <justification jstring="[para(853(a,1),2792(a,1,2)),rewrite(776(6),2789(6),776(11),948(11))].">
      <j1 rule="para" parents="853 2792"/>
      <j2 rule="rewrite" parents="776 2789 776 948"/>
    </justification>
  </clause>

  <clause id="4310">
    <literal><![CDATA[
      f(f(x,y)',f(f(y,x)',z)') = f(f(y,x)',z)
    ]]></literal>
    <justification jstring="[para(2451(a,1),2792(a,1,2)),rewrite(776(6),2042(6),112(5),776(10)),flip(a)].">
      <j1 rule="para" parents="2451 2792"/>
      <j2 rule="rewrite" parents="776 2042 112 776"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4528">
    <literal><![CDATA[
      f(f(x',y)',f(y,f(z,x))) = c_0
    ]]></literal>
    <justification jstring="[para(776(a,1),2815(a,1,1,1))].">
      <j1 rule="para" parents="776 2815"/>
    </justification>
  </clause>

  <clause id="5211">
    <literal><![CDATA[
      f(x,f(f(y,z)',f(z,x))') = c_0
    ]]></literal>
    <justification jstring="[para(702(a,1),4528(a,1,2)),rewrite(776(6))].">
      <j1 rule="para" parents="702 4528"/>
      <j2 rule="rewrite" parents="776"/>
    </justification>
  </clause>

  <clause id="5215">
    <literal><![CDATA[
      f(x,f(f(y,z)',f(x,z))') = c_0
    ]]></literal>
    <justification jstring="[para(795(a,1),4528(a,1,2)),rewrite(776(6))].">
      <j1 rule="para" parents="795 4528"/>
      <j2 rule="rewrite" parents="776"/>
    </justification>
  </clause>

  <clause id="5219">
    <literal><![CDATA[
      f(x,f(f(y,z)',f(y,x))') = c_0
    ]]></literal>
    <justification jstring="[para(856(a,1),4528(a,1,2)),rewrite(776(6))].">
      <j1 rule="para" parents="856 4528"/>
      <j2 rule="rewrite" parents="776"/>
    </justification>
  </clause>

  <clause id="6633">
    <literal><![CDATA[
      f(x,f(f(x,f(y,z))',f(x,f(u,f(x,z))))) = f(x,f(u,f(x,z)))
    ]]></literal>
    <justification jstring="[para(794(a,1),836(a,1,1))].">
      <j1 rule="para" parents="794 836"/>
    </justification>
  </clause>

  <clause id="7590">
    <literal><![CDATA[
      f(x,f(f(y,z)',f(z,x))) = x'
    ]]></literal>
    <justification jstring="[para(5211(a,1),2789(a,1,2)),rewrite(2043(2),112(7)),flip(a)].">
      <j1 rule="para" parents="5211 2789"/>
      <j2 rule="rewrite" parents="2043 112"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="7856">
    <literal><![CDATA[
      f(x,f(f(y,z)',f(x,z))) = x'
    ]]></literal>
    <justification jstring="[para(5215(a,1),2789(a,1,2)),rewrite(2043(2),112(7)),flip(a)].">
      <j1 rule="para" parents="5215 2789"/>
      <j2 rule="rewrite" parents="2043 112"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="7956">
    <literal><![CDATA[
      f(x,f(f(y,z)',f(y,x))) = x'
    ]]></literal>
    <justification jstring="[para(5219(a,1),2789(a,1,2)),rewrite(2043(2),112(7)),flip(a)].">
      <j1 rule="para" parents="5219 2789"/>
      <j2 rule="rewrite" parents="2043 112"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="8045">
    <literal><![CDATA[
      f(f(x,y)',f(x,f(f(x,y)',z))') = f(f(x,y)',z')
    ]]></literal>
    <justification jstring="[para(804(a,1),2789(a,1,2)),rewrite(2789(6)),flip(a)].">
      <j1 rule="para" parents="804 2789"/>
      <j2 rule="rewrite" parents="2789"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10329">
    <literal><![CDATA[
      f(f(x,y)',f(z,y)) = f(f(x,y)',z')
    ]]></literal>
    <justification jstring="[para(7856(a,1),814(a,1,2)),flip(a)].">
      <j1 rule="para" parents="7856 814"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10344">
    <literal><![CDATA[
      f(f(x,y)',f(x,z)') = f(y,f(x,z)')
    ]]></literal>
    <justification jstring="[back_rewrite(3157),rewrite(10329(5))].">
      <j1 rule="back_rewrite" parents="3157"/>
      <j2 rule="rewrite" parents="10329"/>
    </justification>
  </clause>

  <clause id="10348">
    <literal><![CDATA[
      f(f(x,y)',z') = f(y,f(x,f(f(x,y)',z))')
    ]]></literal>
    <justification jstring="[back_rewrite(8045),rewrite(10344(8)),flip(a)].">
      <j1 rule="back_rewrite" parents="8045"/>
      <j2 rule="rewrite" parents="10344"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10358">
    <literal><![CDATA[
      f(x,f(y,f(f(y,x)',f(y,z)))') = f(x,f(y,z)')
    ]]></literal>
    <justification jstring="[back_rewrite(10344),rewrite(10348(5))].">
      <j1 rule="back_rewrite" parents="10344"/>
      <j2 rule="rewrite" parents="10348"/>
    </justification>
  </clause>

  <clause id="10510">
    <literal><![CDATA[
      f(x,f(y,f(f(y,x)',f(f(x,y)',z)))') = f(f(x,y)',z)
    ]]></literal>
    <justification jstring="[back_rewrite(4310),rewrite(10348(7))].">
      <j1 rule="back_rewrite" parents="4310"/>
      <j2 rule="rewrite" parents="10348"/>
    </justification>
  </clause>

  <clause id="13929">
    <literal><![CDATA[
      f(f(x,y)',f(y,z)) = f(y,f(x,f(f(x,y)',z))')
    ]]></literal>
    <justification jstring="[para(7590(a,1),827(a,1,2)),rewrite(10348(4)),flip(a)].">
      <j1 rule="para" parents="7590 827"/>
      <j2 rule="rewrite" parents="10348"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13933">
    <literal><![CDATA[
      f(f(x,y)',f(x,z)) = f(y,f(x,f(f(x,y)',z))')
    ]]></literal>
    <justification jstring="[para(7956(a,1),827(a,1,2)),rewrite(10348(4)),flip(a)].">
      <j1 rule="para" parents="7956 827"/>
      <j2 rule="rewrite" parents="10348"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14080">
    <literal><![CDATA[
      f(x,f(y,f(x,f(y,f(f(y,x)',z))'))') = f(f(y,x)',z)
    ]]></literal>
    <justification jstring="[back_rewrite(803),rewrite(13929(7),2789(6),10348(4))].">
      <j1 rule="back_rewrite" parents="803"/>
      <j2 rule="rewrite" parents="13929 2789 10348"/>
    </justification>
  </clause>

  <clause id="14093">
    <literal><![CDATA[
      f(f(x,y)',z) = f(y,f(x,z)')
    ]]></literal>
    <justification jstring="[back_rewrite(10358),rewrite(13933(4),14080(9))].">
      <j1 rule="back_rewrite" parents="10358"/>
      <j2 rule="rewrite" parents="13933 14080"/>
    </justification>
  </clause>

  <clause id="14114">
    <literal><![CDATA[
      f(x,f(f(y,z),f(x,f(u,f(x,z))')')) = f(x,f(u,f(x,z)))
    ]]></literal>
    <justification jstring="[back_rewrite(6633),rewrite(14093(7),2789(5))].">
      <j1 rule="back_rewrite" parents="6633"/>
      <j2 rule="rewrite" parents="14093 2789"/>
    </justification>
  </clause>

  <clause id="14271">
    <literal><![CDATA[
      f(x,f(y,f(x,f(y,z)'))') = f(x,f(y,z')')
    ]]></literal>
    <justification jstring="[back_rewrite(13933),rewrite(14093(4),2789(2),14093(7)),flip(a)].">
      <j1 rule="back_rewrite" parents="13933"/>
      <j2 rule="rewrite" parents="14093 2789 14093"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14272">
    <literal><![CDATA[
      f(x,f(y,f(x,z))') = f(x,f(y,z')')
    ]]></literal>
    <justification jstring="[back_rewrite(13929),rewrite(14093(4),14093(7),14271(10))].">
      <j1 rule="back_rewrite" parents="13929"/>
      <j2 rule="rewrite" parents="14093 14093 14271"/>
    </justification>
  </clause>

  <clause id="14862">
    <literal><![CDATA[
      f(x,f(y,z)') = f(y,f(x,z)')
    ]]></literal>
    <justification jstring="[back_rewrite(10510),rewrite(14093(5),14093(6),2789(4),112(3),14272(4),14272(7),112(4),2789(3),112(2),14093(6))].">
      <j1 rule="back_rewrite" parents="10510"/>
      <j2 rule="rewrite" parents="14093 14093 2789 112 14272 14272 112 2789 112 14093"/>
    </justification>
  </clause>

  <clause id="15273">
    <literal><![CDATA[
      f(x,f(f(y,z),f(x,f(u,z')')')) = f(x,f(u,f(x,z)))
    ]]></literal>
    <justification jstring="[back_rewrite(14114),rewrite(14272(5))].">
      <j1 rule="back_rewrite" parents="14114"/>
      <j2 rule="rewrite" parents="14272"/>
    </justification>
  </clause>

  <clause id="16175">
    <literal><![CDATA[
      f(x,f(y,f(x,z)')) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(14862(a,1),2789(a,1,2)),rewrite(112(7))].">
      <j1 rule="para" parents="14862 2789"/>
      <j2 rule="rewrite" parents="112"/>
    </justification>
  </clause>

  <clause id="16253">
    <literal><![CDATA[
      f(x,f(y,f(x,z))) = f(x,f(y,z'))
    ]]></literal>
    <justification jstring="[back_rewrite(15273),rewrite(16175(8),3182(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="15273"/>
      <j2 rule="rewrite" parents="16175 3182"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="17320">
    <literal><![CDATA[
      f(f(x,y),f(x,z')) = f(x,f(f(x,y),z))'
    ]]></literal>
    <justification jstring="[back_rewrite(1189),rewrite(16253(5))].">
      <j1 rule="back_rewrite" parents="1189"/>
      <j2 rule="rewrite" parents="16253"/>
    </justification>
  </clause>

  <clause id="17670">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(2027),rewrite(17320(9),776(7),16253(8),112(5),776(4)),xx(a)].">
      <j1 rule="back_rewrite" parents="2027"/>
      <j2 rule="rewrite" parents="17320 776 16253 112 776"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
