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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>pair-def.out</source>

<heading><![CDATA[
Prover9 (32) version August-2006A, August 2006.
Process 11623 was started by mccune on cleo.thornwood,
Sat Aug 12 21:15:02 2006
The command was "/home/mccune/bin/prover9 -f pair-def.in".
]]></heading>

<proof number="1" length="96" max_count="43">

<comments><![CDATA[
% Proof 1 at 6.00 (+ 0.04) seconds: "Sheffer".
% Length of proof is 96.
% Level of proof is 30.
% Maximum clause weight is 43.
% Given clauses 246.
]]></comments>

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

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

  <clause id="3">
    <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="4">
    <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(a,f(b,c)),f(a,f(b,c))) != f(f(a,f(b,b)),f(a,f(c,c)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[copy(3),rewrite(1(25),1(30)),flip(c)].">
      <j1 rule="copy" parents="3"/>
      <j2 rule="rewrite" parents="1 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="5">
    <literal><![CDATA[
      x' = f(x,x)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6">
    <literal><![CDATA[
      f(x,x) = x'
    ]]></literal>
    <justification jstring="[copy(5),flip(a)].">
      <j1 rule="copy" parents="5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      a'' != a
    ]]></literal>
    <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(4),rewrite(6(3),6(5),6(5),6(10),6(14),6(25),6(24),6(28)),flip(c)].">
      <j1 rule="back_rewrite" parents="4"/>
      <j2 rule="rewrite" parents="6 6 6 6 6 6 6 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

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

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

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

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

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

  <clause id="14">
    <literal><![CDATA[
      f(f(x,y),f(x,y')) = x
    ]]></literal>
    <justification jstring="[para(6(a,1),2(a,1,2,2))].">
      <j1 rule="para" parents="6 2"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      f(x',f(x,f(y,x))) = x
    ]]></literal>
    <justification jstring="[para(1(a,1),13(a,1,2,2))].">
      <j1 rule="para" parents="1 13"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      f(x,f(x,x')) = x'
    ]]></literal>
    <justification jstring="[para(13(a,1),2(a,1,2)),rewrite(1(2),1(3))].">
      <j1 rule="para" parents="13 2"/>
      <j2 rule="rewrite" parents="1 1"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(y,x)) = x
    ]]></literal>
    <justification jstring="[para(8(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="8 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      f(f(x,y),f(y,f(z,x))) = y
    ]]></literal>
    <justification jstring="[para(1(a,1),8(a,1,2,2))].">
      <j1 rule="para" parents="1 8"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      f(x,f(f(y,x),f(f(x,f(y,z)),u))) = f(y,x)
    ]]></literal>
    <justification jstring="[para(8(a,1),2(a,1,1))].">
      <j1 rule="para" parents="8 2"/>
    </justification>
  </clause>

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

  <clause id="28">
    <literal><![CDATA[
      f(f(x,y),f(y,x')) = y
    ]]></literal>
    <justification jstring="[para(6(a,1),8(a,1,2,2))].">
      <j1 rule="para" parents="6 8"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      f(f(x',y),f(y,x)) = y
    ]]></literal>
    <justification jstring="[para(13(a,1),8(a,1,2,2))].">
      <j1 rule="para" parents="13 8"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[para(18(a,1),2(a,1,2)),rewrite(6(1),6(3))].">
      <j1 rule="para" parents="18 2"/>
      <j2 rule="rewrite" parents="6 6"/>
    </justification>
  </clause>

  <clause id="37">
    <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(7),rewrite(35(3)),xx(a)].">
      <j1 rule="back_rewrite" parents="7"/>
      <j2 rule="rewrite" parents="35"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      f(f(x,y),f(f(z,y),x)) = x
    ]]></literal>
    <justification jstring="[para(1(a,1),9(a,1,2))].">
      <j1 rule="para" parents="1 9"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(x,f(z,y)),u))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(9(a,1),2(a,1,1))].">
      <j1 rule="para" parents="9 2"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      f(x,f(f(y,x),f(y,z))) = f(y,x)
    ]]></literal>
    <justification jstring="[para(8(a,1),9(a,1,2)),rewrite(1(4))].">
      <j1 rule="para" parents="8 9"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      f(x,f(f(x,y),f(z,y))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(9(a,1),9(a,1,2)),rewrite(1(4))].">
      <j1 rule="para" parents="9 9"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      f(x,f(y,x)') = f(y,x)
    ]]></literal>
    <justification jstring="[para(8(a,1),16(a,1,2)),rewrite(1(3))].">
      <j1 rule="para" parents="8 16"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="76">
    <literal><![CDATA[
      f(f(x,y'),f(y,x)) = x
    ]]></literal>
    <justification jstring="[para(28(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="28 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      f(f(x,y),f(x',y)) = y
    ]]></literal>
    <justification jstring="[para(1(a,1),28(a,1,2))].">
      <j1 rule="para" parents="1 28"/>
    </justification>
  </clause>

  <clause id="99">
    <literal><![CDATA[
      f(x,f(f(y,z),f(x,y))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(10(a,1),8(a,1,2)),rewrite(1(4))].">
      <j1 rule="para" parents="10 8"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="100">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(z,x)) = x
    ]]></literal>
    <justification jstring="[para(8(a,1),10(a,1,2,1))].">
      <j1 rule="para" parents="8 10"/>
    </justification>
  </clause>

  <clause id="101">
    <literal><![CDATA[
      f(x,f(y,f(x,f(y,z)))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(8(a,1),10(a,1,2)),rewrite(1(3),1(4))].">
      <j1 rule="para" parents="8 10"/>
      <j2 rule="rewrite" parents="1 1"/>
    </justification>
  </clause>

  <clause id="117">
    <literal><![CDATA[
      f(x',f(x,y)) = x
    ]]></literal>
    <justification jstring="[para(31(a,1),69(a,1,2,1)),rewrite(1(3),31(7))].">
      <j1 rule="para" parents="31 69"/>
      <j2 rule="rewrite" parents="1 31"/>
    </justification>
  </clause>

  <clause id="121">
    <literal><![CDATA[
      f(x',f(y,x)) = x
    ]]></literal>
    <justification jstring="[para(1(a,1),117(a,1,2))].">
      <j1 rule="para" parents="1 117"/>
    </justification>
  </clause>

  <clause id="124">
    <literal><![CDATA[
      f(x,f(x',y)) = x'
    ]]></literal>
    <justification jstring="[para(35(a,1),117(a,1,1))].">
      <j1 rule="para" parents="35 117"/>
    </justification>
  </clause>

  <clause id="126">
    <literal><![CDATA[
      f(x,f(y,x')) = x'
    ]]></literal>
    <justification jstring="[para(121(a,1),8(a,1,2)),rewrite(1(3))].">
      <j1 rule="para" parents="121 8"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="164">
    <literal><![CDATA[
      f(x,f(f(y,x)',f(y',x))) = f(y',x)
    ]]></literal>
    <justification jstring="[para(77(a,1),28(a,1,1)),rewrite(1(5))].">
      <j1 rule="para" parents="77 28"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="175">
    <literal><![CDATA[
      f(x,f(f(y,z),f(y,x))) = f(y,x)
    ]]></literal>
    <justification jstring="[para(8(a,1),22(a,1,2)),rewrite(1(4))].">
      <j1 rule="para" parents="8 22"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="314">
    <literal><![CDATA[
      f(x,f(y,f(x,y))) = x'
    ]]></literal>
    <justification jstring="[para(100(a,1),26(a,1,2)),rewrite(6(1)),flip(a)].">
      <j1 rule="para" parents="100 26"/>
      <j2 rule="rewrite" parents="6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="316">
    <literal><![CDATA[
      f(x,f(y,f(y,x))) = x'
    ]]></literal>
    <justification jstring="[para(1(a,1),314(a,1,2,2))].">
      <j1 rule="para" parents="1 314"/>
    </justification>
  </clause>

  <clause id="330">
    <literal><![CDATA[
      f(x,f(f(y,x),f(y,x)')) = x'
    ]]></literal>
    <justification jstring="[para(69(a,1),314(a,1,2,2)),rewrite(1(4))].">
      <j1 rule="para" parents="69 314"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="366">
    <literal><![CDATA[
      f(f(x,x'),f(x',y)) = f(x',y)'
    ]]></literal>
    <justification jstring="[para(124(a,1),316(a,1,2,2)),rewrite(1(5))].">
      <j1 rule="para" parents="124 316"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="488">
    <literal><![CDATA[
      f(f(x,y'),f(x,f(f(x,y),z))) = x
    ]]></literal>
    <justification jstring="[para(14(a,1),49(a,1,2,1)),rewrite(14(10))].">
      <j1 rule="para" parents="14 49"/>
      <j2 rule="rewrite" parents="14"/>
    </justification>
  </clause>

  <clause id="498">
    <literal><![CDATA[
      f(f(f(x,y),f(x,z)),f(f(x,y),f(f(f(f(x,y),f(x,z)),f(y,u)),v))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(49(a,1),24(a,1,2,1)),rewrite(49(16))].">
      <j1 rule="para" parents="49 24"/>
      <j2 rule="rewrite" parents="49"/>
    </justification>
  </clause>

  <clause id="517">
    <literal><![CDATA[
      f(f(x,y),f(x,f(z,f(x,y')))) = x
    ]]></literal>
    <justification jstring="[para(14(a,1),53(a,1,2,1)),rewrite(14(10))].">
      <j1 rule="para" parents="14 53"/>
      <j2 rule="rewrite" parents="14"/>
    </justification>
  </clause>

  <clause id="527">
    <literal><![CDATA[
      f(f(x,y'),f(x,f(z,f(y,x)))) = x
    ]]></literal>
    <justification jstring="[para(76(a,1),53(a,1,2,1)),rewrite(76(10))].">
      <j1 rule="para" parents="76 53"/>
      <j2 rule="rewrite" parents="76"/>
    </justification>
  </clause>

  <clause id="593">
    <literal><![CDATA[
      f(x,f(x',y)') = f(x,x')
    ]]></literal>
    <justification jstring="[para(126(a,1),42(a,1,2,2,1)),rewrite(366(5))].">
      <j1 rule="para" parents="126 42"/>
      <j2 rule="rewrite" parents="366"/>
    </justification>
  </clause>

  <clause id="635">
    <literal><![CDATA[
      f(x,f(y,x')') = f(x,x')
    ]]></literal>
    <justification jstring="[para(1(a,1),593(a,1,2,1))].">
      <j1 rule="para" parents="1 593"/>
    </justification>
  </clause>

  <clause id="638">
    <literal><![CDATA[
      f(x',f(x,y)') = f(x,x')
    ]]></literal>
    <justification jstring="[para(35(a,1),593(a,1,2,1,1)),rewrite(35(7),1(6))].">
      <j1 rule="para" parents="35 593"/>
      <j2 rule="rewrite" parents="35 1"/>
    </justification>
  </clause>

  <clause id="669">
    <literal><![CDATA[
      f(x',f(y,x)') = f(x,x')
    ]]></literal>
    <justification jstring="[para(35(a,1),635(a,1,2,1,2)),rewrite(35(7),1(6))].">
      <j1 rule="para" parents="35 635"/>
      <j2 rule="rewrite" parents="35 1"/>
    </justification>
  </clause>

  <clause id="702">
    <literal><![CDATA[
      f(f(x,y),f(x,y)') = f(x,x')
    ]]></literal>
    <justification jstring="[para(2(a,1),638(a,1,2,1)),rewrite(1(4),638(4)),flip(a)].">
      <j1 rule="para" parents="2 638"/>
      <j2 rule="rewrite" parents="1 638"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="704">
    <literal><![CDATA[
      f(x,x') = f(y,y')
    ]]></literal>
    <justification jstring="[para(8(a,1),638(a,1,2,1)),rewrite(1(4),669(4),702(6))].">
      <j1 rule="para" parents="8 638"/>
      <j2 rule="rewrite" parents="1 669 702"/>
    </justification>
  </clause>

  <clause id="723">
    <literal><![CDATA[
      f(f(x,y)',f(x',z)') = f(x,x')
    ]]></literal>
    <justification jstring="[para(638(a,1),49(a,1,2,1)),rewrite(366(7),638(10))].">
      <j1 rule="para" parents="638 49"/>
      <j2 rule="rewrite" parents="366 638"/>
    </justification>
  </clause>

  <clause id="736">
    <literal><![CDATA[
      f(x,f(y,y')) = x'
    ]]></literal>
    <justification jstring="[back_rewrite(330),rewrite(702(4))].">
      <j1 rule="back_rewrite" parents="330"/>
      <j2 rule="rewrite" parents="702"/>
    </justification>
  </clause>

  <clause id="737">
    <literal><![CDATA[
      f(x,x') = c_0
    ]]></literal>
    <justification jstring="[new_symbol(704)].">
      <j1 rule="new_symbol" parents="704"/>
    </justification>
  </clause>

  <clause id="747">
    <literal><![CDATA[
      f(a,c_0) != 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(37),rewrite(737(5))].">
      <j1 rule="back_rewrite" parents="37"/>
      <j2 rule="rewrite" parents="737"/>
    </justification>
  </clause>

  <clause id="748">
    <literal><![CDATA[
      f(x,c_0) = x'
    ]]></literal>
    <justification jstring="[back_rewrite(736),rewrite(737(2))].">
      <j1 rule="back_rewrite" parents="736"/>
      <j2 rule="rewrite" parents="737"/>
    </justification>
  </clause>

  <clause id="755">
    <literal><![CDATA[
      f(f(x,y)',f(x',z)') = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(723),rewrite(737(8))].">
      <j1 rule="back_rewrite" parents="723"/>
      <j2 rule="rewrite" parents="737"/>
    </justification>
  </clause>

  <clause id="781">
    <literal><![CDATA[
      f(x',f(y,x)') = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(669),rewrite(737(6))].">
      <j1 rule="back_rewrite" parents="669"/>
      <j2 rule="rewrite" parents="737"/>
    </justification>
  </clause>

  <clause id="799">
    <literal><![CDATA[
      f(f(a,b'),f(a,c')) != f(a,f(b,c))'
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(747),rewrite(748(3)),xx(a)].">
      <j1 rule="back_rewrite" parents="747"/>
      <j2 rule="rewrite" parents="748"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="820">
    <literal><![CDATA[
      f(c_0,x) = x'
    ]]></literal>
    <justification jstring="[para(737(a,1),8(a,1,1)),rewrite(117(4))].">
      <j1 rule="para" parents="737 8"/>
      <j2 rule="rewrite" parents="117"/>
    </justification>
  </clause>

  <clause id="957">
    <literal><![CDATA[
      f(x,f(y,f(x,y))') = c_0
    ]]></literal>
    <justification jstring="[para(314(a,1),781(a,1,2,1)),rewrite(35(5),1(4))].">
      <j1 rule="para" parents="314 781"/>
      <j2 rule="rewrite" parents="35 1"/>
    </justification>
  </clause>

  <clause id="1043">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(x,f(y,x))') = c_0
    ]]></literal>
    <justification jstring="[para(21(a,1),957(a,1,2,1,2)),rewrite(1(4))].">
      <j1 rule="para" parents="21 957"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="1056">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(x,f(z,x))') = c_0
    ]]></literal>
    <justification jstring="[para(100(a,1),957(a,1,2,1,2)),rewrite(1(4))].">
      <j1 rule="para" parents="100 957"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="1206">
    <literal><![CDATA[
      f(f(x,y)',f(x',z)) = f(x,y)
    ]]></literal>
    <justification jstring="[para(755(a,1),314(a,1,2,2)),rewrite(1(7),820(7),35(6),35(8))].">
      <j1 rule="para" parents="755 314"/>
      <j2 rule="rewrite" parents="1 820 35 35"/>
    </justification>
  </clause>

  <clause id="1218">
    <literal><![CDATA[
      f(x',y) = f(y,f(x,y))
    ]]></literal>
    <justification jstring="[back_rewrite(164),rewrite(1206(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="164"/>
      <j2 rule="rewrite" parents="1206"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1278">
    <literal><![CDATA[
      f(x,f(y,x)) = f(x,y')
    ]]></literal>
    <justification jstring="[para(1218(a,1),1(a,1))].">
      <j1 rule="para" parents="1218 1"/>
    </justification>
  </clause>

  <clause id="1290">
    <literal><![CDATA[
      f(f(x,y)',f(f(y,z),x)) = f(x,f(y,z)')
    ]]></literal>
    <justification jstring="[para(10(a,1),1218(a,2,2)),rewrite(1(8),1278(8))].">
      <j1 rule="para" parents="10 1218"/>
      <j2 rule="rewrite" parents="1 1278"/>
    </justification>
  </clause>

  <clause id="1299">
    <literal><![CDATA[
      f(x,f(x,y)) = f(x,y')
    ]]></literal>
    <justification jstring="[para(1218(a,2),12(a,1,2,2)),rewrite(1278(3),35(2),1278(4))].">
      <j1 rule="para" parents="1218 12"/>
      <j2 rule="rewrite" parents="1278 35 1278"/>
    </justification>
  </clause>

  <clause id="1317">
    <literal><![CDATA[
      f(f(x,y)',f(f(z,y),x)) = f(x,f(z,y)')
    ]]></literal>
    <justification jstring="[para(41(a,1),1218(a,2,2)),rewrite(1(8),1278(8))].">
      <j1 rule="para" parents="41 1218"/>
      <j2 rule="rewrite" parents="1 1278"/>
    </justification>
  </clause>

  <clause id="1362">
    <literal><![CDATA[
      f(f(x,y')',f(x,f(z,y))) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(1056),rewrite(1278(4),1(6))].">
      <j1 rule="back_rewrite" parents="1056"/>
      <j2 rule="rewrite" parents="1278 1"/>
    </justification>
  </clause>

  <clause id="1365">
    <literal><![CDATA[
      f(f(x,y')',f(x,f(y,z))) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(1043),rewrite(1278(4),1(6))].">
      <j1 rule="back_rewrite" parents="1043"/>
      <j2 rule="rewrite" parents="1278 1"/>
    </justification>
  </clause>

  <clause id="2498">
    <literal><![CDATA[
      f(f(x',y)',f(y,f(z,x))) = c_0
    ]]></literal>
    <justification jstring="[para(1(a,1),1362(a,1,1,1))].">
      <j1 rule="para" parents="1 1362"/>
    </justification>
  </clause>

  <clause id="2563">
    <literal><![CDATA[
      f(f(x,y)',f(x,f(y',z))) = c_0
    ]]></literal>
    <justification jstring="[para(35(a,1),1365(a,1,1,1,2))].">
      <j1 rule="para" parents="35 1365"/>
    </justification>
  </clause>

  <clause id="3006">
    <literal><![CDATA[
      f(x,f(f(x,y),f(x,z))) = f(f(x,y),f(x,z))'
    ]]></literal>
    <justification jstring="[para(101(a,1),175(a,1,2)),rewrite(6(7)),flip(a)].">
      <j1 rule="para" parents="101 175"/>
      <j2 rule="rewrite" parents="6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3030">
    <literal><![CDATA[
      f(x,f(f(y,z)',f(x,y))') = c_0
    ]]></literal>
    <justification jstring="[para(2(a,1),2498(a,1,2)),rewrite(1(6))].">
      <j1 rule="para" parents="2 2498"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="3045">
    <literal><![CDATA[
      f(f(f(x,y)',z)',f(z,y)) = c_0
    ]]></literal>
    <justification jstring="[para(121(a,1),2498(a,1,2,2))].">
      <j1 rule="para" parents="121 2498"/>
    </justification>
  </clause>

  <clause id="4063">
    <literal><![CDATA[
      f(f(x,y),f(x,f(y',z))) = f(x,f(y',z))'
    ]]></literal>
    <justification jstring="[para(2563(a,1),1278(a,1,2)),rewrite(1(5),820(5),35(10),1(9)),flip(a)].">
      <j1 rule="para" parents="2563 1278"/>
      <j2 rule="rewrite" parents="1 820 35 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4483">
    <literal><![CDATA[
      f(x,f(f(y,z)',f(x,y))) = x'
    ]]></literal>
    <justification jstring="[para(3030(a,1),1299(a,1,2)),rewrite(748(2),35(7)),flip(a)].">
      <j1 rule="para" parents="3030 1299"/>
      <j2 rule="rewrite" parents="748 35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="5014">
    <literal><![CDATA[
      f(f(f(x,y)',z)',f(z,y)') = f(f(x,y)',z)
    ]]></literal>
    <justification jstring="[para(3045(a,1),1299(a,1,2)),rewrite(1(6),820(6),35(5)),flip(a)].">
      <j1 rule="para" parents="3045 1299"/>
      <j2 rule="rewrite" parents="1 820 35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="7307">
    <literal><![CDATA[
      f(f(x,y)',f(z,x)) = f(f(x,y)',z')
    ]]></literal>
    <justification jstring="[para(4483(a,1),101(a,1,2)),flip(a)].">
      <j1 rule="para" parents="4483 101"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="7372">
    <literal><![CDATA[
      f(f(x,y)',f(z,y)') = f(x,f(z,y)')
    ]]></literal>
    <justification jstring="[back_rewrite(1317),rewrite(7307(5))].">
      <j1 rule="back_rewrite" parents="1317"/>
      <j2 rule="rewrite" parents="7307"/>
    </justification>
  </clause>

  <clause id="7373">
    <literal><![CDATA[
      f(f(x,y)',f(y,z)') = f(x,f(y,z)')
    ]]></literal>
    <justification jstring="[back_rewrite(1290),rewrite(7307(5))].">
      <j1 rule="back_rewrite" parents="1290"/>
      <j2 rule="rewrite" parents="7307"/>
    </justification>
  </clause>

  <clause id="7376">
    <literal><![CDATA[
      f(f(x,y)',z) = f(x,f(z,y)')
    ]]></literal>
    <justification jstring="[back_rewrite(5014),rewrite(7373(7),7372(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="5014"/>
      <j2 rule="rewrite" parents="7373 7372"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9255">
    <literal><![CDATA[
      f(f(x,y)',z) = f(x,f(y,z)')
    ]]></literal>
    <justification jstring="[para(1(a,1),7376(a,2,2,1))].">
      <j1 rule="para" parents="1 7376"/>
    </justification>
  </clause>

  <clause id="9439">
    <literal><![CDATA[
      f(x,f(y,f(x,z)')) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(7376(a,1),1278(a,1,2)),rewrite(35(7))].">
      <j1 rule="para" parents="7376 1278"/>
      <j2 rule="rewrite" parents="35"/>
    </justification>
  </clause>

  <clause id="9443">
    <literal><![CDATA[
      f(x,f(y,f(x,z))') = f(x,f(y,z')')
    ]]></literal>
    <justification jstring="[para(7376(a,1),1299(a,1)),rewrite(9255(3),1(4),9439(4),9255(8))].">
      <j1 rule="para" parents="7376 1299"/>
      <j2 rule="rewrite" parents="9255 1 9439 9255"/>
    </justification>
  </clause>

  <clause id="11722">
    <literal><![CDATA[
      f(x,f(f(x,y),z)') = f(x,f(y',z)')
    ]]></literal>
    <justification jstring="[para(1299(a,1),9255(a,1,1,1)),rewrite(9255(4)),flip(a)].">
      <j1 rule="para" parents="1299 9255"/>
      <j2 rule="rewrite" parents="9255"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="12506">
    <literal><![CDATA[
      f(x,f(f(x,y),z)) = f(x,f(y',z))
    ]]></literal>
    <justification jstring="[para(488(a,1),10(a,1,2)),rewrite(1(4),1299(4),11722(4),1(5),1299(5),35(4)),flip(a)].">
      <j1 rule="para" parents="488 10"/>
      <j2 rule="rewrite" parents="1 1299 11722 1 1299 35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="12742">
    <literal><![CDATA[
      f(f(x,y),f(x,z))' = f(x,f(y',f(x,z)))
    ]]></literal>
    <justification jstring="[back_rewrite(3006),rewrite(12506(4)),flip(a)].">
      <j1 rule="back_rewrite" parents="3006"/>
      <j2 rule="rewrite" parents="12506"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="12941">
    <literal><![CDATA[
      f(x,f(y,f(x,z'))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(517(a,1),10(a,1,2)),rewrite(1(5),1299(5),9443(5),35(2),1(4),1299(4),35(3)),flip(a)].">
      <j1 rule="para" parents="517 10"/>
      <j2 rule="rewrite" parents="1 1299 9443 35 1 1299 35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13081">
    <literal><![CDATA[
      f(x,f(y,f(x,z))) = f(x,f(y,z'))
    ]]></literal>
    <justification jstring="[para(527(a,1),99(a,1,2,1)),rewrite(12941(4))].">
      <j1 rule="para" parents="527 99"/>
      <j2 rule="rewrite" parents="12941"/>
    </justification>
  </clause>

  <clause id="13217">
    <literal><![CDATA[
      f(f(x,y),f(x,z))' = f(x,f(y',z'))
    ]]></literal>
    <justification jstring="[back_rewrite(12742),rewrite(13081(8))].">
      <j1 rule="back_rewrite" parents="12742"/>
      <j2 rule="rewrite" parents="13081"/>
    </justification>
  </clause>

  <clause id="14026">
    <literal><![CDATA[
      f(f(x,y),f(x,z)) = f(x,f(y',z'))'
    ]]></literal>
    <justification jstring="[para(498(a,1),117(a,1,2)),rewrite(13217(4),1(6),4063(6)),flip(a)].">
      <j1 rule="para" parents="498 117"/>
      <j2 rule="rewrite" parents="13217 1 4063"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14432">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(799),rewrite(14026(9),35(4),35(5)),xx(a)].">
      <j1 rule="back_rewrite" parents="799"/>
      <j2 rule="rewrite" parents="14026 35 35"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
