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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>pair.out</source>

<heading><![CDATA[
Prover9 (32) version 22-May-2007, May 2007.
Process 27933 was started by mccune on cleo,
Tue May 22 14:58:16 2007
The command was "/home/mccune/bin/prover9 -f pair.in".
]]></heading>

<proof number="1" length="125" max_count="45">

<comments><![CDATA[
% Proof 1 at 463.59 (+ 1.42) seconds: "Sheffer".
% Length of proof is 125.
% Level of proof is 19.
% Maximum clause weight is 43.
% Given clauses 1336.
]]></comments>

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

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

  <clause id="3" type="assumption">
    <literal><![CDATA[
      f(f(x,y),f(x,f(y,z))) = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="4" type="deny">
    <literal><![CDATA[
      f(f(c1,c1),f(c1,c1)) != c1
    ]]></literal>
    <literal><![CDATA[
      f(c1,f(c2,f(c2,c2))) != f(c1,c1)
    ]]></literal>
    <literal><![CDATA[
      f(f(f(c2,c2),c1),f(f(c3,c3),c1)) != f(f(c1,f(c2,c3)),f(c1,f(c2,c3)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

  <clause id="5">
    <literal><![CDATA[
      f(f(c1,c1),f(c1,c1)) != c1
    ]]></literal>
    <literal><![CDATA[
      f(c1,f(c2,f(c2,c2))) != f(c1,c1)
    ]]></literal>
    <literal><![CDATA[
      f(f(c1,f(c2,c3)),f(c1,f(c2,c3))) != f(f(c1,f(c2,c2)),f(c1,f(c3,c3)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[copy(4),rewrite([2(25),2(30)]),flip(c)].">
      <j1 rule="copy" parents="4"/>
      <j2 rule="rewrite" parents="2 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="6">
    <literal><![CDATA[
      f(f(x,y),f(y,f(x,z))) = y
    ]]></literal>
    <justification jstring="[para(2(a,1),3(a,1,1))].">
      <j1 rule="para" parents="2 3"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      f(f(x,y),f(x,f(z,y))) = x
    ]]></literal>
    <justification jstring="[para(2(a,1),3(a,1,2,2))].">
      <j1 rule="para" parents="2 3"/>
    </justification>
  </clause>

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

  <clause id="9">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(x,f(y,z)),u))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(3(a,1),3(a,1,1))].">
      <j1 rule="para" parents="3 3"/>
    </justification>
  </clause>

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

  <clause id="11">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(y,x)) = x
    ]]></literal>
    <justification jstring="[para(6(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="6 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      f(f(x,y),f(y,f(z,x))) = y
    ]]></literal>
    <justification jstring="[para(2(a,1),6(a,1,2,2))].">
      <j1 rule="para" parents="2 6"/>
    </justification>
  </clause>

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

  <clause id="14">
    <literal><![CDATA[
      f(x,f(f(y,x),f(f(x,f(y,z)),u))) = f(y,x)
    ]]></literal>
    <justification jstring="[para(6(a,1),3(a,1,1))].">
      <j1 rule="para" parents="6 3"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      f(x,f(f(x,f(y,z)),f(f(x,y),u))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(3(a,1),6(a,1,1))].">
      <j1 rule="para" parents="3 6"/>
    </justification>
  </clause>

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

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

  <clause id="21">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(x,f(z,y)),u))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(7(a,1),3(a,1,1))].">
      <j1 rule="para" parents="7 3"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      f(x,f(f(x,f(y,z)),f(f(x,z),u))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(7(a,1),6(a,1,1))].">
      <j1 rule="para" parents="7 6"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      f(x,f(f(y,x),f(y,z))) = f(y,x)
    ]]></literal>
    <justification jstring="[para(6(a,1),7(a,1,2)),rewrite([2(4)])].">
      <j1 rule="para" parents="6 7"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      f(x,f(f(x,y),f(z,y))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(7(a,1),7(a,1,2)),rewrite([2(4)])].">
      <j1 rule="para" parents="7 7"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      f(x,f(f(y,z),f(x,y))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(8(a,1),6(a,1,2)),rewrite([2(4)])].">
      <j1 rule="para" parents="8 6"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(z,x)) = x
    ]]></literal>
    <justification jstring="[para(6(a,1),8(a,1,2,1))].">
      <j1 rule="para" parents="6 8"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      f(x,f(y,f(x,f(y,z)))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(6(a,1),8(a,1,2)),rewrite([2(3),2(4)])].">
      <j1 rule="para" parents="6 8"/>
      <j2 rule="rewrite" parents="2 2"/>
    </justification>
  </clause>

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

  <clause id="39">
    <literal><![CDATA[
      f(c1,f(c2,f(c2,c2))) != f(c1,c1)
    ]]></literal>
    <literal><![CDATA[
      f(f(c1,f(c2,c3)),f(c1,f(c2,c3))) != f(f(c1,f(c2,c2)),f(c1,f(c3,c3)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(5),rewrite([36(7)]),xx(a)].">
      <j1 rule="back_rewrite" parents="5"/>
      <j2 rule="rewrite" parents="36"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      f(f(x,y),f(x,x)) = x
    ]]></literal>
    <justification jstring="[para(36(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="36 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      f(f(x,x),f(y,x)) = x
    ]]></literal>
    <justification jstring="[para(2(a,1),36(a,1,2))].">
      <j1 rule="para" parents="2 36"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      f(x,f(f(x,x),y)) = f(x,x)
    ]]></literal>
    <justification jstring="[para(36(a,1),7(a,1,2)),rewrite([2(3)])].">
      <j1 rule="para" parents="36 7"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      f(f(f(x,y),z),f(x,f(f(f(x,y),z),f(f(f(x,y),f(z,u)),v)))) = f(f(f(x,y),z),f(f(f(x,y),f(z,u)),v))
    ]]></literal>
    <justification jstring="[para(9(a,1),8(a,1,2)),rewrite([2(8),2(11)])].">
      <j1 rule="para" parents="9 8"/>
      <j2 rule="rewrite" parents="2 2"/>
    </justification>
  </clause>

  <clause id="57">
    <literal><![CDATA[
      f(f(x,y),f(y,y)) = y
    ]]></literal>
    <justification jstring="[para(2(a,1),40(a,1,1))].">
      <j1 rule="para" parents="2 40"/>
    </justification>
  </clause>

  <clause id="58">
    <literal><![CDATA[
      f(x,f(f(x,y),f(z,f(x,x)))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(40(a,1),7(a,1,1))].">
      <j1 rule="para" parents="40 7"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(x,f(y,f(x,x))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(41(a,1),6(a,1,2)),rewrite([2(3)])].">
      <j1 rule="para" parents="41 6"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="64">
    <literal><![CDATA[
      f(x,f(f(y,x),f(z,f(x,x)))) = f(y,x)
    ]]></literal>
    <justification jstring="[para(57(a,1),7(a,1,1))].">
      <j1 rule="para" parents="57 7"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      f(x,f(f(f(y,x),z),f(x,f(y,u)))) = f(x,f(y,u))
    ]]></literal>
    <justification jstring="[para(11(a,1),8(a,1,1))].">
      <j1 rule="para" parents="11 8"/>
    </justification>
  </clause>

  <clause id="71">
    <literal><![CDATA[
      f(f(x,y),f(f(z,x),y)) = y
    ]]></literal>
    <justification jstring="[para(2(a,1),12(a,1,2))].">
      <j1 rule="para" parents="2 12"/>
    </justification>
  </clause>

  <clause id="80">
    <literal><![CDATA[
      f(x,f(y,f(x,f(z,y)))) = f(x,f(z,y))
    ]]></literal>
    <justification jstring="[para(12(a,1),8(a,1,2)),rewrite([2(3),2(4)])].">
      <j1 rule="para" parents="12 8"/>
      <j2 rule="rewrite" parents="2 2"/>
    </justification>
  </clause>

  <clause id="87">
    <literal><![CDATA[
      f(x,f(f(f(x,y),z),f(x,f(y,u)))) = f(x,f(y,u))
    ]]></literal>
    <justification jstring="[para(3(a,1),13(a,1,1))].">
      <j1 rule="para" parents="3 13"/>
    </justification>
  </clause>

  <clause id="90">
    <literal><![CDATA[
      f(x,f(f(f(x,y),z),f(x,f(u,y)))) = f(x,f(u,y))
    ]]></literal>
    <justification jstring="[para(7(a,1),13(a,1,1))].">
      <j1 rule="para" parents="7 13"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      f(x,f(y,f(f(y,z),x))) = f(f(y,z),x)
    ]]></literal>
    <justification jstring="[para(13(a,1),8(a,1,2)),rewrite([2(3),2(4)])].">
      <j1 rule="para" parents="13 8"/>
      <j2 rule="rewrite" parents="2 2"/>
    </justification>
  </clause>

  <clause id="97">
    <literal><![CDATA[
      f(x,f(f(f(y,x),z),f(x,f(u,y)))) = f(x,f(u,y))
    ]]></literal>
    <justification jstring="[para(12(a,1),13(a,1,1))].">
      <j1 rule="para" parents="12 13"/>
    </justification>
  </clause>

  <clause id="107">
    <literal><![CDATA[
      f(x,f(y,f(f(z,y),x))) = f(f(z,y),x)
    ]]></literal>
    <justification jstring="[para(19(a,1),7(a,1,2)),rewrite([2(3),2(4)])].">
      <j1 rule="para" parents="19 7"/>
      <j2 rule="rewrite" parents="2 2"/>
    </justification>
  </clause>

  <clause id="143">
    <literal><![CDATA[
      f(x,f(f(y,f(z,x)),f(x,f(u,z)))) = f(x,f(u,z))
    ]]></literal>
    <justification jstring="[para(32(a,1),19(a,1,2)),rewrite([2(6)])].">
      <j1 rule="para" parents="32 19"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="151">
    <literal><![CDATA[
      f(f(x,x),f(f(x,y),f(z,f(x,x)))) = f(z,f(x,x))
    ]]></literal>
    <justification jstring="[para(62(a,1),13(a,1,1))].">
      <j1 rule="para" parents="62 13"/>
    </justification>
  </clause>

  <clause id="156">
    <literal><![CDATA[
      f(f(x,f(f(x,y),z)),f(x,f(y,u))) = x
    ]]></literal>
    <justification jstring="[para(16(a,1),7(a,1,2))].">
      <j1 rule="para" parents="16 7"/>
    </justification>
  </clause>

  <clause id="184">
    <literal><![CDATA[
      f(x,f(y,f(x,y))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(32(a,1),16(a,1,2)),flip(a)].">
      <j1 rule="para" parents="32 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="201">
    <literal><![CDATA[
      f(f(x,x),f(f(y,x),f(z,f(x,x)))) = f(z,f(x,x))
    ]]></literal>
    <justification jstring="[para(62(a,1),71(a,1,1))].">
      <j1 rule="para" parents="62 71"/>
    </justification>
  </clause>

  <clause id="239">
    <literal><![CDATA[
      f(f(f(x,f(y,x)),z),f(f(y,y),z)) = z
    ]]></literal>
    <justification jstring="[para(184(a,1),71(a,1,2,1))].">
      <j1 rule="para" parents="184 71"/>
    </justification>
  </clause>

  <clause id="295">
    <literal><![CDATA[
      f(f(x,f(f(y,x),z)),f(x,f(y,u))) = x
    ]]></literal>
    <justification jstring="[para(18(a,1),7(a,1,2))].">
      <j1 rule="para" parents="18 7"/>
    </justification>
  </clause>

  <clause id="325">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(y,f(f(x,f(y,z)),f(x,u)))) = f(f(x,f(y,z)),f(x,u))
    ]]></literal>
    <justification jstring="[para(26(a,1),8(a,1,2)),rewrite([2(5),2(8)])].">
      <j1 rule="para" parents="26 8"/>
      <j2 rule="rewrite" parents="2 2"/>
    </justification>
  </clause>

  <clause id="355">
    <literal><![CDATA[
      f(f(x,y),f(y,f(z,f(y,f(u,x))))) = y
    ]]></literal>
    <justification jstring="[para(12(a,1),28(a,1,2,1)),rewrite([12(10)])].">
      <j1 rule="para" parents="12 28"/>
      <j2 rule="rewrite" parents="12"/>
    </justification>
  </clause>

  <clause id="365">
    <literal><![CDATA[
      f(x,f(f(x,f(f(f(x,y),f(z,y)),u)),f(f(x,y),v))) = f(x,f(f(f(x,y),f(z,y)),u))
    ]]></literal>
    <justification jstring="[para(28(a,1),16(a,1,2,2,1))].">
      <j1 rule="para" parents="28 16"/>
    </justification>
  </clause>

  <clause id="369">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(x,y),f(z,y)))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(28(a,1),184(a,1,2,2)),rewrite([2(5)])].">
      <j1 rule="para" parents="28 184"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="387">
    <literal><![CDATA[
      f(x,f(f(x,f(f(f(y,z),f(x,y)),u)),f(f(x,y),v))) = f(x,f(f(f(y,z),f(x,y)),u))
    ]]></literal>
    <justification jstring="[para(31(a,1),16(a,1,2,2,1))].">
      <j1 rule="para" parents="31 16"/>
    </justification>
  </clause>

  <clause id="569">
    <literal><![CDATA[
      f(x,f(f(x,f(y,z)),f(y,f(x,f(y,z))))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(33(a,1),184(a,1,2,2)),rewrite([2(6)])].">
      <j1 rule="para" parents="33 184"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="570">
    <literal><![CDATA[
      f(x,f(y,y)) = f(x,f(y,x))
    ]]></literal>
    <justification jstring="[para(184(a,1),33(a,1,2))].">
      <j1 rule="para" parents="184 33"/>
    </justification>
  </clause>

  <clause id="582">
    <literal><![CDATA[
      f(x,f(f(x,f(y,z)),f(y,y))) = f(x,x)
    ]]></literal>
    <justification jstring="[back_rewrite(569),rewrite([570(6,R)])].">
      <j1 rule="back_rewrite" parents="569"/>
      <j2 rule="rewrite" parents="570"/>
    </justification>
  </clause>

  <clause id="583">
    <literal><![CDATA[
      f(c1,f(c2,f(c2,c2))) != f(c1,c1)
    ]]></literal>
    <literal><![CDATA[
      f(f(c1,f(c2,c3)),f(c1,f(c2,c3))) != f(f(c1,f(c1,c2)),f(c1,f(c1,c3)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(39),rewrite([570(27),2(26),570(32),2(31)])].">
      <j1 rule="back_rewrite" parents="39"/>
      <j2 rule="rewrite" parents="570 2 570 2"/>
    </justification>
  </clause>

  <clause id="637">
    <literal><![CDATA[
      f(f(x,x),y) = f(y,f(x,y))
    ]]></literal>
    <justification jstring="[para(570(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="570 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="638">
    <literal><![CDATA[
      f(x,f(y,y)) = f(x,f(x,y))
    ]]></literal>
    <justification jstring="[para(2(a,1),570(a,2,2))].">
      <j1 rule="para" parents="2 570"/>
    </justification>
  </clause>

  <clause id="656">
    <literal><![CDATA[
      f(f(x,x),f(f(x,y),f(x,y))) = f(x,f(x,x))
    ]]></literal>
    <justification jstring="[para(40(a,1),570(a,2,2)),rewrite([2(7)])].">
      <j1 rule="para" parents="40 570"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="658">
    <literal><![CDATA[
      f(f(x,x),f(f(y,x),f(y,x))) = f(x,f(x,x))
    ]]></literal>
    <justification jstring="[para(57(a,1),570(a,2,2)),rewrite([2(7)])].">
      <j1 rule="para" parents="57 570"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="697">
    <literal><![CDATA[
      f(f(f(x,y),z),f(f(y,f(x,x)),z)) = z
    ]]></literal>
    <justification jstring="[para(570(a,2),71(a,1,2,1))].">
      <j1 rule="para" parents="570 71"/>
    </justification>
  </clause>

  <clause id="741">
    <literal><![CDATA[
      f(f(x,x),y) = f(y,f(y,x))
    ]]></literal>
    <justification jstring="[para(2(a,1),637(a,2,2))].">
      <j1 rule="para" parents="2 637"/>
    </justification>
  </clause>

  <clause id="1951">
    <literal><![CDATA[
      f(x,f(y,f(x,f(f(x,y),z)))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(156(a,1),80(a,1,2)),flip(a)].">
      <j1 rule="para" parents="156 80"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="5175">
    <literal><![CDATA[
      f(x,f(y,f(x,f(f(y,x),z)))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(295(a,1),80(a,1,2)),flip(a)].">
      <j1 rule="para" parents="295 80"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="5880">
    <literal><![CDATA[
      f(f(x,y),f(f(x,y),f(x,y))) = f(x,f(x,x))
    ]]></literal>
    <justification jstring="[para(14(a,1),52(a,1,2,2)),rewrite([369(9),2(5),656(5),36(16),26(14)]),flip(a)].">
      <j1 rule="para" parents="14 52"/>
      <j2 rule="rewrite" parents="369 2 656 36 26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9092">
    <literal><![CDATA[
      f(x,f(y,f(x,f(f(y,y),z)))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(36(a,1),582(a,1,2,2)),rewrite([2(4)])].">
      <j1 rule="para" parents="36 582"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="9101">
    <literal><![CDATA[
      f(x,f(y,f(y,y))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(14(a,1),582(a,1,2,1)),rewrite([5880(5)])].">
      <j1 rule="para" parents="14 582"/>
      <j2 rule="rewrite" parents="5880"/>
    </justification>
  </clause>

  <clause id="9201">
    <literal><![CDATA[
      f(f(c1,f(c2,c3)),f(c1,f(c2,c3))) != f(f(c1,f(c1,c2)),f(c1,f(c1,c3)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_unit_del(583),unit_del(a,9101)].">
      <j1 rule="back_unit_del" parents="583"/>
      <j2 rule="unit_del" parents="9101"/>
    </justification>
  </clause>

  <clause id="9287">
    <literal><![CDATA[
      f(f(x,f(x,x)),y) = f(y,y)
    ]]></literal>
    <justification jstring="[para(9101(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="9101 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9296">
    <literal><![CDATA[
      f(f(x,x),f(y,f(y,y))) = x
    ]]></literal>
    <justification jstring="[para(9101(a,2),36(a,1))].">
      <j1 rule="para" parents="9101 36"/>
    </justification>
  </clause>

  <clause id="9330">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(y,y)) = y
    ]]></literal>
    <justification jstring="[para(9101(a,1),637(a,2,2)),rewrite([9296(4)]),flip(a)].">
      <j1 rule="para" parents="9101 637"/>
      <j2 rule="rewrite" parents="9296"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9337">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(x,f(x,x)),y)) = y
    ]]></literal>
    <justification jstring="[para(9101(a,1),741(a,1)),rewrite([36(3)]),flip(a)].">
      <j1 rule="para" parents="9101 741"/>
      <j2 rule="rewrite" parents="36"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9338">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(x,f(x,x))),y) = f(y,f(y,y))
    ]]></literal>
    <justification jstring="[para(9101(a,1),741(a,2,2))].">
      <j1 rule="para" parents="9101 741"/>
    </justification>
  </clause>

  <clause id="9354">
    <literal><![CDATA[
      f(x,f(x,x)) = f(y,f(y,y))
    ]]></literal>
    <justification jstring="[para(9101(a,1),107(a,1,2)),rewrite([2(5),658(5),5880(7)])].">
      <j1 rule="para" parents="9101 107"/>
      <j2 rule="rewrite" parents="2 658 5880"/>
    </justification>
  </clause>

  <clause id="9377">
    <literal><![CDATA[
      f(x,f(f(y,f(x,y)),f(z,f(z,z)))) = f(z,f(z,z))
    ]]></literal>
    <justification jstring="[para(9101(a,1),239(a,1,2)),rewrite([36(8),2(6)])].">
      <j1 rule="para" parents="9101 239"/>
      <j2 rule="rewrite" parents="36 2"/>
    </justification>
  </clause>

  <clause id="9469">
    <literal><![CDATA[
      f(f(c1,f(c2,c3)),f(x,f(x,x))) != f(f(c1,f(c1,c2)),f(c1,f(c1,c3)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[para(9101(a,2),9201(a,1))].">
      <j1 rule="para" parents="9101 9201"/>
    </justification>
  </clause>

  <clause id="9472">
    <literal><![CDATA[
      f(x,f(x,x)) = c_0
    ]]></literal>
    <justification jstring="[new_symbol(9354)].">
      <j1 rule="new_symbol" parents="9354"/>
    </justification>
  </clause>

  <clause id="9629">
    <literal><![CDATA[
      f(f(c1,f(c1,c2)),f(c1,f(c1,c3))) != f(c_0,f(c1,f(c2,c3)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(9469),rewrite([9472(7),2(7)]),flip(a)].">
      <j1 rule="back_rewrite" parents="9469"/>
      <j2 rule="rewrite" parents="9472 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9713">
    <literal><![CDATA[
      f(x,f(c_0,f(y,f(x,y)))) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(9377),rewrite([9472(4),2(4),9472(7)])].">
      <j1 rule="back_rewrite" parents="9377"/>
      <j2 rule="rewrite" parents="9472 2 9472"/>
    </justification>
  </clause>

  <clause id="9749">
    <literal><![CDATA[
      f(f(c_0,c_0),x) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(9338),rewrite([9472(2),9472(3),9472(6)])].">
      <j1 rule="back_rewrite" parents="9338"/>
      <j2 rule="rewrite" parents="9472 9472 9472"/>
    </justification>
  </clause>

  <clause id="9750">
    <literal><![CDATA[
      f(c_0,f(c_0,x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(9337),rewrite([9472(2),9472(3)])].">
      <j1 rule="back_rewrite" parents="9337"/>
      <j2 rule="rewrite" parents="9472 9472"/>
    </justification>
  </clause>

  <clause id="9755">
    <literal><![CDATA[
      f(c_0,f(x,x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(9330),rewrite([9472(2)])].">
      <j1 rule="back_rewrite" parents="9330"/>
      <j2 rule="rewrite" parents="9472"/>
    </justification>
  </clause>

  <clause id="9778">
    <literal><![CDATA[
      f(c_0,x) = f(x,x)
    ]]></literal>
    <justification jstring="[back_rewrite(9287),rewrite([9472(2)])].">
      <j1 rule="back_rewrite" parents="9287"/>
      <j2 rule="rewrite" parents="9472"/>
    </justification>
  </clause>

  <clause id="11085">
    <literal><![CDATA[
      f(f(x,y),f(c_0,x)) = x
    ]]></literal>
    <justification jstring="[para(9472(a,1),8(a,1,2,1))].">
      <j1 rule="para" parents="9472 8"/>
    </justification>
  </clause>

  <clause id="11118">
    <literal><![CDATA[
      f(f(c_0,x),f(f(c_0,f(x,y)),z)) = x
    ]]></literal>
    <justification jstring="[para(9749(a,1),52(a,1,1,1)),rewrite([9749(9),9749(11),9749(13),2(4),9750(4),9749(4),9749(6)]),flip(a)].">
      <j1 rule="para" parents="9749 52"/>
      <j2 rule="rewrite" parents="9749 9749 9749 2 9750 9749 9749"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11125">
    <literal><![CDATA[
      f(x,f(f(c_0,x),y)) = f(c_0,x)
    ]]></literal>
    <justification jstring="[para(9750(a,1),26(a,1,2,2))].">
      <j1 rule="para" parents="9750 26"/>
    </justification>
  </clause>

  <clause id="11127">
    <literal><![CDATA[
      f(f(c_0,x),f(f(c_0,f(y,x)),z)) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),9750(a,1,2)),rewrite([9750(4)]),flip(a)].">
      <j1 rule="para" parents="21 9750"/>
      <j2 rule="rewrite" parents="9750"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11130">
    <literal><![CDATA[
      f(f(c_0,f(x,y)),f(f(c_0,y),z)) = f(x,y)
    ]]></literal>
    <justification jstring="[para(24(a,1),9750(a,1,2)),rewrite([9750(5)]),flip(a)].">
      <j1 rule="para" parents="24 9750"/>
      <j2 rule="rewrite" parents="9750"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11229">
    <literal><![CDATA[
      f(x,f(c_0,y)) = f(x,f(y,x))
    ]]></literal>
    <justification jstring="[para(9778(a,2),570(a,1,2))].">
      <j1 rule="para" parents="9778 570"/>
    </justification>
  </clause>

  <clause id="11231">
    <literal><![CDATA[
      f(x,f(c_0,y)) = f(x,f(x,y))
    ]]></literal>
    <justification jstring="[para(9778(a,2),638(a,1,2))].">
      <j1 rule="para" parents="9778 638"/>
    </justification>
  </clause>

  <clause id="11233">
    <literal><![CDATA[
      f(f(c_0,x),y) = f(y,f(y,x))
    ]]></literal>
    <justification jstring="[para(9778(a,2),741(a,1,1))].">
      <j1 rule="para" parents="9778 741"/>
    </justification>
  </clause>

  <clause id="11242">
    <literal><![CDATA[
      f(f(c_0,x),y) = f(y,f(x,x))
    ]]></literal>
    <justification jstring="[para(9778(a,2),91(a,2,1)),rewrite([45(3)]),flip(a)].">
      <j1 rule="para" parents="9778 91"/>
      <j2 rule="rewrite" parents="45"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="12244">
    <literal><![CDATA[
      f(x,f(y,f(x,f(z,f(y,x))))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(355(a,1),68(a,1,2)),flip(a)].">
      <j1 rule="para" parents="355 68"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="12738">
    <literal><![CDATA[
      f(f(x,y),f(c_0,f(y,f(c_0,f(x,z))))) = c_0
    ]]></literal>
    <justification jstring="[para(13(a,1),9713(a,1,2,2,2)),rewrite([2(5),11229(5,R)])].">
      <j1 rule="para" parents="13 9713"/>
      <j2 rule="rewrite" parents="2 11229"/>
    </justification>
  </clause>

  <clause id="18798">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(c_0,f(x,z)),u))) = y
    ]]></literal>
    <justification jstring="[para(11118(a,1),19(a,1,1,1))].">
      <j1 rule="para" parents="11118 19"/>
    </justification>
  </clause>

  <clause id="18869">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(c_0,f(z,x)),u))) = y
    ]]></literal>
    <justification jstring="[para(11127(a,1),19(a,1,1,1))].">
      <j1 rule="para" parents="11127 19"/>
    </justification>
  </clause>

  <clause id="19511">
    <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(87(a,1),107(a,1,2)),rewrite([9778(5,R),2(6)]),flip(a)].">
      <j1 rule="para" parents="87 107"/>
      <j2 rule="rewrite" parents="9778 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22055">
    <literal><![CDATA[
      f(f(f(x,y),z),f(f(u,f(x,x)),z)) = z
    ]]></literal>
    <justification jstring="[para(58(a,1),697(a,1,1,1)),rewrite([2(8),151(8)])].">
      <j1 rule="para" parents="58 697"/>
      <j2 rule="rewrite" parents="2 151"/>
    </justification>
  </clause>

  <clause id="22057">
    <literal><![CDATA[
      f(f(f(x,y),z),f(f(u,f(y,y)),z)) = z
    ]]></literal>
    <justification jstring="[para(64(a,1),697(a,1,1,1)),rewrite([2(8),201(8)])].">
      <j1 rule="para" parents="64 697"/>
      <j2 rule="rewrite" parents="2 201"/>
    </justification>
  </clause>

  <clause id="24987">
    <literal><![CDATA[
      f(x,f(y,f(f(y,x),z))) = f(x,f(y,y))
    ]]></literal>
    <justification jstring="[para(1951(a,1),33(a,1,2)),flip(a)].">
      <j1 rule="para" parents="1951 33"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="26362">
    <literal><![CDATA[
      f(x,f(y,f(f(x,y),z))) = f(x,f(y,y))
    ]]></literal>
    <justification jstring="[para(5175(a,1),33(a,1,2)),flip(a)].">
      <j1 rule="para" parents="5175 33"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="26615">
    <literal><![CDATA[
      f(x,f(y,f(c_0,f(f(x,y),z)))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(9092(a,1),90(a,1)),rewrite([9778(6,R),2(6)]),flip(a)].">
      <j1 rule="para" parents="9092 90"/>
      <j2 rule="rewrite" parents="9778 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="26616">
    <literal><![CDATA[
      f(x,f(y,f(c_0,f(f(y,x),z)))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(9092(a,1),97(a,1)),rewrite([9778(6,R),2(6)]),flip(a)].">
      <j1 rule="para" parents="9092 97"/>
      <j2 rule="rewrite" parents="9778 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="27939">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(c_0,z),u),x)) = x
    ]]></literal>
    <justification jstring="[para(11130(a,1),32(a,1,1,2))].">
      <j1 rule="para" parents="11130 32"/>
    </justification>
  </clause>

  <clause id="28705">
    <literal><![CDATA[
      f(x,f(y,f(z,f(x,y)))) = f(x,f(y,y))
    ]]></literal>
    <justification jstring="[para(12244(a,1),33(a,1,2)),flip(a)].">
      <j1 rule="para" parents="12244 33"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="28848">
    <literal><![CDATA[
      f(x,f(y,f(c_0,f(z,f(y,x))))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(143(a,1),9092(a,1)),rewrite([9778(5,R),2(5)])].">
      <j1 rule="para" parents="143 9092"/>
      <j2 rule="rewrite" parents="9778 2"/>
    </justification>
  </clause>

  <clause id="30029">
    <literal><![CDATA[
      f(f(c_0,f(x,f(c_0,f(y,z)))),f(y,x)) = c_0
    ]]></literal>
    <justification jstring="[para(12738(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="12738 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="38718">
    <literal><![CDATA[
      f(f(x,x),f(c_0,f(f(x,y),z))) = f(f(x,x),z)
    ]]></literal>
    <justification jstring="[para(22055(a,1),24987(a,1,2)),rewrite([9778(8,R)]),flip(a)].">
      <j1 rule="para" parents="22055 24987"/>
      <j2 rule="rewrite" parents="9778"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="38719">
    <literal><![CDATA[
      f(f(x,x),f(c_0,f(f(y,x),z))) = f(f(x,x),z)
    ]]></literal>
    <justification jstring="[para(22057(a,1),24987(a,1,2)),rewrite([9778(8,R)]),flip(a)].">
      <j1 rule="para" parents="22057 24987"/>
      <j2 rule="rewrite" parents="9778"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="39333">
    <literal><![CDATA[
      f(f(c_0,x),f(c_0,f(y,f(z,x)))) = f(f(c_0,x),y)
    ]]></literal>
    <justification jstring="[para(27939(a,1),26362(a,1,2)),rewrite([9778(10,R)]),flip(a)].">
      <j1 rule="para" parents="27939 26362"/>
      <j2 rule="rewrite" parents="9778"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="39512">
    <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(18798(a,1),28705(a,1,2)),rewrite([9778(10,R)]),flip(a)].">
      <j1 rule="para" parents="18798 28705"/>
      <j2 rule="rewrite" parents="9778"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="39514">
    <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(18869(a,1),28705(a,1,2)),rewrite([9778(10,R)]),flip(a)].">
      <j1 rule="para" parents="18869 28705"/>
      <j2 rule="rewrite" parents="9778"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="39523">
    <literal><![CDATA[
      f(x,f(f(y,y),z)) = f(x,f(f(y,x),z))
    ]]></literal>
    <justification jstring="[para(26615(a,1),28705(a,1,2,2)),rewrite([2(6),38718(6),9778(12,R),9750(9)])].">
      <j1 rule="para" parents="26615 28705"/>
      <j2 rule="rewrite" parents="2 38718 9778 9750"/>
    </justification>
  </clause>

  <clause id="39524">
    <literal><![CDATA[
      f(x,f(f(y,y),z)) = f(x,f(f(x,y),z))
    ]]></literal>
    <justification jstring="[para(26616(a,1),28705(a,1,2,2)),rewrite([2(6),38719(6),9778(12,R),9750(9)])].">
      <j1 rule="para" parents="26616 28705"/>
      <j2 rule="rewrite" parents="2 38719 9778 9750"/>
    </justification>
  </clause>

  <clause id="39569">
    <literal><![CDATA[
      f(x,f(f(c_0,y),z)) = f(x,f(z,f(x,y)))
    ]]></literal>
    <justification jstring="[para(28848(a,1),28705(a,1,2,2)),rewrite([11242(6,R),39333(7),9778(13,R),9750(10)])].">
      <j1 rule="para" parents="28848 28705"/>
      <j2 rule="rewrite" parents="11242 39333 9778 9750"/>
    </justification>
  </clause>

  <clause id="39836">
    <literal><![CDATA[
      f(f(c_0,f(x,y)),z) = f(y,f(c_0,f(x,z)))
    ]]></literal>
    <justification jstring="[para(30029(a,1),741(a,2,2)),rewrite([9778(3,R),39514(10),39512(7),2(12),9750(12)])].">
      <j1 rule="para" parents="30029 741"/>
      <j2 rule="rewrite" parents="9778 39514 39512 2 9750"/>
    </justification>
  </clause>

  <clause id="41346">
    <literal><![CDATA[
      f(x,f(c_0,f(y,z))) = f(z,f(z,f(x,y)))
    ]]></literal>
    <justification jstring="[para(387(a,1),11233(a,1,1)),rewrite([11085(5),39836(4),11085(9),39836(11),11125(9),9750(8)])].">
      <j1 rule="para" parents="387 11233"/>
      <j2 rule="rewrite" parents="11085 39836 11085 39836 11125 9750"/>
    </justification>
  </clause>

  <clause id="47092">
    <literal><![CDATA[
      f(x,f(f(x,f(x,y)),z)) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(741(a,1),39523(a,2,2,1)),rewrite([9778(3,R),9755(3)]),flip(a)].">
      <j1 rule="para" parents="741 39523"/>
      <j2 rule="rewrite" parents="9778 9755"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="47235">
    <literal><![CDATA[
      f(x,f(y,f(c_0,f(x,z)))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(39523(a,1),365(a,1,2,1)),rewrite([2(2),47092(4),16(6),9778(5,R),39836(6)]),flip(a)].">
      <j1 rule="para" parents="39523 365"/>
      <j2 rule="rewrite" parents="2 47092 16 9778 39836"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="47476">
    <literal><![CDATA[
      f(f(x,f(x,y)),f(x,z)) = f(c_0,f(x,f(y,f(x,z))))
    ]]></literal>
    <justification jstring="[para(39524(a,2),325(a,1,2)),rewrite([9778(5,R),39836(7),47235(8),19511(6)]),flip(a)].">
      <j1 rule="para" parents="39524 325"/>
      <j2 rule="rewrite" parents="9778 39836 47235 19511"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="47575">
    <literal><![CDATA[
      f(c_0,f(c1,f(c2,f(c1,f(c1,c3))))) != f(c_0,f(c1,f(c2,c3)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(9629),rewrite([47476(11)])].">
      <j1 rule="back_rewrite" parents="9629"/>
      <j2 rule="rewrite" parents="47476"/>
    </justification>
  </clause>

  <clause id="48028">
    <literal><![CDATA[
      f(c_0,f(c1,f(c2,f(c2,f(c1,c3))))) != f(c_0,f(c1,f(c2,c3)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[para(39569(a,2),47575(a,1,2)),rewrite([2(9),11231(9)])].">
      <j1 rule="para" parents="39569 47575"/>
      <j2 rule="rewrite" parents="2 11231"/>
    </justification>
  </clause>

  <clause id="48744">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[para(41346(a,2),48028(a,1,2,2)),rewrite([2(7),11231(9),10(10)]),xx(a)].">
      <j1 rule="para" parents="41346 48028"/>
      <j2 rule="rewrite" parents="2 11231 10"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
