<?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>BA2.out</source>

<heading><![CDATA[
Prover9 (32) version November-2006, November 2006.
Process 4412 was started by mccune on cleo.thornwood,
Wed Nov 22 12:04:01 2006
The command was "/home/mccune/bin/prover9 -f BA2.in BA2-interp.outx".
]]></heading>

<proof number="1" length="75" max_count="45">

<comments><![CDATA[
% Proof 1 at 18.35 (+ 0.04) seconds: "Sheffer".
% Length of proof is 75.
% Level of proof is 24.
% Maximum clause weight is 43.
% Given clauses 176.
]]></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(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="2" type="assumption">
    <literal><![CDATA[
      f(x,y) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(commutativity)
    ]]></attribute>
    <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>
    <attribute><![CDATA[
      label(Veroff_2)
    ]]></attribute>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="4" type="deny">
    <literal><![CDATA[
      f(f(c4,c4),f(c4,c4)) != c4
    ]]></literal>
    <literal><![CDATA[
      f(c4,f(c5,f(c5,c5))) != f(c4,c4)
    ]]></literal>
    <literal><![CDATA[
      f(f(f(c5,c5),c4),f(f(c6,c6),c4)) != f(f(c4,f(c5,c6)),f(c4,f(c5,c6)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

  <clause id="5">
    <literal><![CDATA[
      f(f(c4,c4),f(c4,c4)) != c4
    ]]></literal>
    <literal><![CDATA[
      f(c4,f(c5,f(c5,c5))) != f(c4,c4)
    ]]></literal>
    <literal><![CDATA[
      f(f(c4,f(c5,c6)),f(c4,f(c5,c6))) != f(f(c4,f(c5,c5)),f(c4,f(c6,c6)))
    ]]></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>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(2(a,1),3(a,1,1))].">
      <j1 rule="para" parents="2 3"/>
    </justification>
  </clause>

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

  <clause id="8">
    <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="9">
    <literal><![CDATA[
      f(f(x,y),f(x,f(z,y))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(6(a,1),3(a,1,2,2)),rewrite(2(4))].">
      <j1 rule="para" parents="6 3"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      f(x,f(f(x,f(y,z)),f(u,f(x,y)))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(3(a,1),6(a,1,1)),rewrite(2(4))].">
      <j1 rule="para" parents="3 6"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      f(f(x,y),f(y,f(z,x))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(6(a,1),6(a,1,2,2)),rewrite(2(2),2(3),2(4))].">
      <j1 rule="para" parents="6 6"/>
      <j2 rule="rewrite" parents="2 2 2"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      f(x,f(f(x,y),f(y,z))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(3(a,1),9(a,1,2)),rewrite(2(4))].">
      <j1 rule="para" parents="3 9"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      f(x,f(f(x,f(y,z)),f(u,f(x,z)))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(9(a,1),6(a,1,1)),rewrite(2(4))].">
      <j1 rule="para" parents="9 6"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      f(x,f(f(y,z),f(y,x))) = f(y,x)
    ]]></literal>
    <justification jstring="[para(3(a,1),12(a,1,2)),rewrite(2(2),2(4),2(5))].">
      <j1 rule="para" parents="3 12"/>
      <j2 rule="rewrite" parents="2 2 2"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(f(x,x),f(x,y)) = x
    ]]></literal>
    <justification jstring="[para(8(a,1),3(a,1,2))].">
      <j1 rule="para" parents="8 3"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      f(c4,f(c5,f(c5,c5))) != f(c4,c4)
    ]]></literal>
    <literal><![CDATA[
      f(f(c4,f(c5,c6)),f(c4,f(c5,c6))) != f(f(c4,f(c5,c5)),f(c4,f(c6,c6)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(5),rewrite(19(7)),xx(a)].">
      <j1 rule="back_rewrite" parents="5"/>
      <j2 rule="rewrite" parents="19"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      f(f(x,y),f(y,f(z,f(f(x,y),f(u,f(y,f(x,v))))))) = y
    ]]></literal>
    <justification jstring="[para(6(a,1),7(a,1,2,1)),rewrite(2(5),6(13))].">
      <j1 rule="para" parents="6 7"/>
      <j2 rule="rewrite" parents="2 6"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      f(x,f(f(x,y),f(z,f(x,x)))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(19(a,1),6(a,1,1)),rewrite(2(3))].">
      <j1 rule="para" parents="19 6"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      f(x,f(y,f(x,x))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(19(a,1),9(a,1,2)),rewrite(2(2),2(3))].">
      <j1 rule="para" parents="19 9"/>
      <j2 rule="rewrite" parents="2 2"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      f(f(x,x),f(f(x,y),f(z,f(x,x)))) = f(z,f(x,x))
    ]]></literal>
    <justification jstring="[para(30(a,1),6(a,1,1)),rewrite(2(5))].">
      <j1 rule="para" parents="30 6"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      f(f(x,y),f(y,y)) = y
    ]]></literal>
    <justification jstring="[para(30(a,1),6(a,1,2))].">
      <j1 rule="para" parents="30 6"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      f(x,f(y,f(x,f(y,z)))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(3(a,1),13(a,1,2,2)),rewrite(2(3))].">
      <j1 rule="para" parents="3 13"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      f(x,f(f(x,y),f(z,y))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(2(a,1),16(a,1,2,1)),rewrite(2(2),2(3),2(5))].">
      <j1 rule="para" parents="2 16"/>
      <j2 rule="rewrite" parents="2 2 2"/>
    </justification>
  </clause>

  <clause id="117">
    <literal><![CDATA[
      f(x,f(y,f(x,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(9(a,1),36(a,1,2)),rewrite(2(3)),flip(a)].">
      <j1 rule="para" parents="9 36"/>
      <j2 rule="rewrite" parents="2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="122">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(x,f(y,z))) = f(f(x,f(x,y)),f(x,f(y,z)))
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(11(a,1),36(a,1,2)),rewrite(2(10))].">
      <j1 rule="para" parents="11 36"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="153">
    <literal><![CDATA[
      f(f(x,f(y,f(y,z))),f(x,f(z,z))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(117(a,1),9(a,1,2,2)),rewrite(2(1))].">
      <j1 rule="para" parents="117 9"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="169">
    <literal><![CDATA[
      f(x,f(y,y)) = f(x,f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(117(a,1),36(a,1,2)),rewrite(2(3))].">
      <j1 rule="para" parents="117 36"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="170">
    <literal><![CDATA[
      f(x,f(f(x,f(y,z)),f(y,f(x,f(y,z))))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(36(a,1),117(a,1,2,2)),rewrite(2(6))].">
      <j1 rule="para" parents="36 117"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="173">
    <literal><![CDATA[
      f(f(x,f(x,y)),f(x,f(z,f(z,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[back_rewrite(153),rewrite(169(5),2(6))].">
      <j1 rule="back_rewrite" parents="153"/>
      <j2 rule="rewrite" parents="169 2"/>
    </justification>
  </clause>

  <clause id="175">
    <literal><![CDATA[
      f(c4,f(c5,f(c5,c5))) != f(c4,c4)
    ]]></literal>
    <literal><![CDATA[
      f(f(c4,f(c5,c6)),f(c4,f(c5,c6))) != f(f(c4,f(c4,c5)),f(c4,f(c4,c6)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(20),rewrite(169(27),169(32))].">
      <j1 rule="back_rewrite" parents="20"/>
      <j2 rule="rewrite" parents="169 169"/>
    </justification>
  </clause>

  <clause id="176">
    <literal><![CDATA[
      f(x,f(y,y)) = f(x,f(y,x))
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(169(a,1),2(a,2)),rewrite(2(2),2(3))].">
      <j1 rule="para" parents="169 2"/>
      <j2 rule="rewrite" parents="2 2"/>
    </justification>
  </clause>

  <clause id="177">
    <literal><![CDATA[
      f(f(x,f(x,y)),f(x,f(z,f(y,y)))) = x
    ]]></literal>
    <justification jstring="[para(169(a,1),3(a,1,1)),rewrite(2(4))].">
      <j1 rule="para" parents="169 3"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="179">
    <literal><![CDATA[
      f(x,f(x,f(y,y))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(169(a,1),9(a,1)),rewrite(2(2),176(2,R),2(4),176(4,R),2(6),6(6),2(3))].">
      <j1 rule="para" parents="169 9"/>
      <j2 rule="rewrite" parents="2 176 2 176 2 6 2"/>
    </justification>
  </clause>

  <clause id="197">
    <literal><![CDATA[
      f(x,f(f(y,y),f(x,f(y,z)))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[back_rewrite(170),rewrite(176(6,R),2(4))].">
      <j1 rule="back_rewrite" parents="170"/>
      <j2 rule="rewrite" parents="176 2"/>
    </justification>
  </clause>

  <clause id="201">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(x,f(z,f(y,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(176(a,2),9(a,1,2,2))].">
      <j1 rule="para" parents="176 9"/>
    </justification>
  </clause>

  <clause id="231">
    <literal><![CDATA[
      f(x,f(y,f(y,f(z,z)))) = f(x,f(z,y))
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(179(a,1),11(a,1,2,1,2)),rewrite(14(6)),flip(a)].">
      <j1 rule="para" parents="179 11"/>
      <j2 rule="rewrite" parents="14"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="247">
    <literal><![CDATA[
      f(f(x,f(y,y)),f(f(y,z),f(y,z))) = f(y,z)
    ]]></literal>
    <justification jstring="[para(179(a,2),29(a,1)),rewrite(2(10),32(10),2(7),176(7,R))].">
      <j1 rule="para" parents="179 29"/>
      <j2 rule="rewrite" parents="2 32 2 176"/>
    </justification>
  </clause>

  <clause id="359">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(x,f(y,f(y,f(z,f(x,x)))))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(30(a,1),173(a,1,1,2))].">
      <j1 rule="para" parents="30 173"/>
    </justification>
  </clause>

  <clause id="410">
    <literal><![CDATA[
      f(f(x,x),f(y,f(x,z))) = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(197(a,1),36(a,1,2)),flip(a)].">
      <j1 rule="para" parents="197 36"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="452">
    <literal><![CDATA[
      f(x,f(y,f(y,y))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(201(a,1),201(a,1,2)),rewrite(2(5),176(5),2(4),2(6),177(6)),flip(a)].">
      <j1 rule="para" parents="201 201"/>
      <j2 rule="rewrite" parents="2 176 2 2 177"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="453">
    <literal><![CDATA[
      f(f(c4,f(c5,c6)),f(c4,f(c5,c6))) != f(f(c4,f(c4,c5)),f(c4,f(c4,c6)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_unit_del(175),unit_del(a,452)].">
      <j1 rule="back_unit_del" parents="175"/>
      <j2 rule="unit_del" parents="452"/>
    </justification>
  </clause>

  <clause id="463">
    <literal><![CDATA[
      f(f(x,x),f(y,f(y,y))) = x
    ]]></literal>
    <justification jstring="[para(452(a,2),19(a,1))].">
      <j1 rule="para" parents="452 19"/>
    </justification>
  </clause>

  <clause id="466">
    <literal><![CDATA[
      f(x,f(x,x)) = f(y,f(y,y))
    ]]></literal>
    <justification jstring="[para(452(a,1),33(a,1,1)),rewrite(176(7),2(5),463(5),2(2))].">
      <j1 rule="para" parents="452 33"/>
      <j2 rule="rewrite" parents="176 2 463 2"/>
    </justification>
  </clause>

  <clause id="504">
    <literal><![CDATA[
      f(x,f(x,x)) = c_0
    ]]></literal>
    <justification jstring="[new_symbol(466)].">
      <j1 rule="new_symbol" parents="466"/>
    </justification>
  </clause>

  <clause id="535">
    <literal><![CDATA[
      f(c_0,f(x,x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(463),rewrite(504(3),2(3))].">
      <j1 rule="back_rewrite" parents="463"/>
      <j2 rule="rewrite" parents="504 2"/>
    </justification>
  </clause>

  <clause id="541">
    <literal><![CDATA[
      f(x,c_0) = f(x,x)
    ]]></literal>
    <justification jstring="[back_rewrite(452),rewrite(504(2))].">
      <j1 rule="back_rewrite" parents="452"/>
      <j2 rule="rewrite" parents="504"/>
    </justification>
  </clause>

  <clause id="546">
    <literal><![CDATA[
      f(c_0,f(x,f(y,f(y,f(z,f(x,x)))))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[back_rewrite(359),rewrite(504(2))].">
      <j1 rule="back_rewrite" parents="359"/>
      <j2 rule="rewrite" parents="504"/>
    </justification>
  </clause>

  <clause id="551">
    <literal><![CDATA[
      f(f(c4,f(c4,c5)),f(c4,f(c4,c6))) != f(c_0,f(c4,f(c5,c6)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(453),rewrite(541(11,R),2(7)),flip(a)].">
      <j1 rule="back_rewrite" parents="453"/>
      <j2 rule="rewrite" parents="541 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="572">
    <literal><![CDATA[
      f(f(x,f(x,y)),f(c_0,f(y,z))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[back_rewrite(247),rewrite(176(2),2(1),541(5,R),2(5))].">
      <j1 rule="back_rewrite" parents="247"/>
      <j2 rule="rewrite" parents="176 2 541 2"/>
    </justification>
  </clause>

  <clause id="590">
    <literal><![CDATA[
      f(f(x,f(x,y)),f(x,f(y,z))) = f(c_0,f(x,f(y,z)))
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[back_rewrite(122),rewrite(541(5,R),2(4)),flip(a)].">
      <j1 rule="back_rewrite" parents="122"/>
      <j2 rule="rewrite" parents="541 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="606">
    <literal><![CDATA[
      f(x,f(y,f(y,x))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(19(a,1),23(a,1,2)),rewrite(176(2),2(1),2(3))].">
      <j1 rule="para" parents="19 23"/>
      <j2 rule="rewrite" parents="176 2 2"/>
    </justification>
  </clause>

  <clause id="636">
    <literal><![CDATA[
      f(x,f(f(x,y),f(c_0,f(z,y)))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(23(a,1),197(a,1,2,2,2)),rewrite(541(3,R),2(3),2(5))].">
      <j1 rule="para" parents="23 197"/>
      <j2 rule="rewrite" parents="541 2 2"/>
    </justification>
  </clause>

  <clause id="760">
    <literal><![CDATA[
      f(x,f(x,f(c_0,f(y,z)))) = f(x,f(y,z))
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(231(a,1),179(a,1,2,2)),rewrite(176(2),2(1),8(3),2(2),541(3,R),2(3),176(7),2(6),8(8),2(7))].">
      <j1 rule="para" parents="231 179"/>
      <j2 rule="rewrite" parents="176 2 8 2 541 2 176 2 8 2"/>
    </justification>
  </clause>

  <clause id="830">
    <literal><![CDATA[
      f(c_0,f(c_0,f(x,y))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(3(a,1),546(a,1,2,2,2,2)),rewrite(606(6),541(4,R),2(4))].">
      <j1 rule="para" parents="3 546"/>
      <j2 rule="rewrite" parents="606 541 2"/>
    </justification>
  </clause>

  <clause id="832">
    <literal><![CDATA[
      f(x,f(c_0,f(y,f(y,f(z,f(z,x)))))) = c_0
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(546(a,1),9(a,1,2)),rewrite(176(3),2(2),2(7))].">
      <j1 rule="para" parents="546 9"/>
      <j2 rule="rewrite" parents="176 2 2"/>
    </justification>
  </clause>

  <clause id="895">
    <literal><![CDATA[
      f(f(x,y),f(c_0,f(z,f(z,x)))) = f(z,f(z,x))
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(572(a,1),3(a,1,2)),rewrite(2(4),2(6))].">
      <j1 rule="para" parents="572 3"/>
      <j2 rule="rewrite" parents="2 2"/>
    </justification>
  </clause>

  <clause id="899">
    <literal><![CDATA[
      f(f(x,f(x,f(y,z))),f(z,z)) = z
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(6(a,1),572(a,1,2,2)),rewrite(2(5),541(5),6(9))].">
      <j1 rule="para" parents="6 572"/>
      <j2 rule="rewrite" parents="2 541 6"/>
    </justification>
  </clause>

  <clause id="992">
    <literal><![CDATA[
      f(x,f(y,f(y,f(z,f(z,x))))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(19(a,1),899(a,1,2)),rewrite(176(2),2(1),2(5))].">
      <j1 rule="para" parents="19 899"/>
      <j2 rule="rewrite" parents="176 2 2"/>
    </justification>
  </clause>

  <clause id="1054">
    <literal><![CDATA[
      f(x,f(f(x,f(y,z)),f(z,z))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(66(a,1),636(a,1,2,2)),rewrite(2(4),541(4))].">
      <j1 rule="para" parents="66 636"/>
      <j2 rule="rewrite" parents="2 541"/>
    </justification>
  </clause>

  <clause id="1443">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(c_0,f(x,f(x,y)))) = c_0
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(7(a,1),832(a,1,2,2,2))].">
      <j1 rule="para" parents="7 832"/>
    </justification>
  </clause>

  <clause id="1593">
    <literal><![CDATA[
      f(f(x,x),f(f(x,y),f(f(x,f(z,u)),f(u,u)))) = f(f(x,f(z,u)),f(u,u))
    ]]></literal>
    <justification jstring="[para(1054(a,1),6(a,1,1)),rewrite(2(7))].">
      <j1 rule="para" parents="1054 6"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="1900">
    <literal><![CDATA[
      f(f(x,f(y,f(z,u))),f(c_0,f(y,f(y,z)))) = f(y,f(y,z))
    ]]></literal>
    <justification jstring="[para(1443(a,1),992(a,1,2,2,2,2)),rewrite(2(8),760(10),2(8),541(17,R),2(14),830(14))].">
      <j1 rule="para" parents="1443 992"/>
      <j2 rule="rewrite" parents="2 760 2 541 2 830"/>
    </justification>
  </clause>

  <clause id="2962">
    <literal><![CDATA[
      f(f(x,x),f(f(x,y),f(f(x,y),f(z,z)))) = f(f(x,x),f(z,z))
    ]]></literal>
    <justification jstring="[para(179(a,2),410(a,1,2))].">
      <j1 rule="para" parents="179 410"/>
    </justification>
  </clause>

  <clause id="2991">
    <literal><![CDATA[
      f(f(x,x),f(f(x,y),f(z,u))) = f(f(x,x),f(c_0,f(z,u)))
    ]]></literal>
    <justification jstring="[para(231(a,1),410(a,2,2)),rewrite(176(3),2(2),8(4),2(4),176(8),2(7),8(9),2(8),541(9,R),2(9))].">
      <j1 rule="para" parents="231 410"/>
      <j2 rule="rewrite" parents="176 2 8 2 176 2 8 2 541 2"/>
    </justification>
  </clause>

  <clause id="3110">
    <literal><![CDATA[
      f(f(x,x),f(c_0,f(f(x,y),f(z,z)))) = f(f(x,x),f(z,z))
    ]]></literal>
    <justification jstring="[back_rewrite(2962),rewrite(2991(7))].">
      <j1 rule="back_rewrite" parents="2962"/>
      <j2 rule="rewrite" parents="2991"/>
    </justification>
  </clause>

  <clause id="3132">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(z,z)) = f(f(x,x),f(z,z))
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[back_rewrite(1593),rewrite(2991(8),3110(8)),flip(a)].">
      <j1 rule="back_rewrite" parents="1593"/>
      <j2 rule="rewrite" parents="2991 3110"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4091">
    <literal><![CDATA[
      f(f(x,x),f(c_0,f(y,f(x,z)))) = f(y,f(y,x))
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(410(a,1),895(a,1,2,2,2)),rewrite(2(3),2(8),176(9,R),541(8,R),2(8),535(8),2(6),176(6),2(5),1900(8),2991(8)),flip(a)].">
      <j1 rule="para" parents="410 895"/>
      <j2 rule="rewrite" parents="2 2 176 541 2 535 2 176 2 1900 2991"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="5826">
    <literal><![CDATA[
      f(f(c_0,f(x,y)),f(c_0,f(y,z))) = f(x,f(c_0,f(y,z)))
    ]]></literal>
    <justification jstring="[para(3(a,1),3132(a,1,1)),rewrite(541(3,R),2(3),541(7,R),2(7),541(10,R),2(10)),flip(a)].">
      <j1 rule="para" parents="3 3132"/>
      <j2 rule="rewrite" parents="541 2 541 2 541 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="6340">
    <literal><![CDATA[
      f(x,f(c_0,f(y,z))) = f(z,f(z,f(x,y)))
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(6(a,1),4091(a,1,2,2,2)),rewrite(541(3,R),2(3),2(5),5826(7))].">
      <j1 rule="para" parents="6 4091"/>
      <j2 rule="rewrite" parents="541 2 2 5826"/>
    </justification>
  </clause>

  <clause id="6714">
    <literal><![CDATA[
      f(x,f(y,f(c_0,f(x,z)))) = f(x,f(y,z))
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(6340(a,2),8(a,1,2)),rewrite(2(2))].">
      <j1 rule="para" parents="6340 8"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="8820">
    <literal><![CDATA[
      f(x,f(y,f(x,f(x,z)))) = f(x,f(y,z))
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(8(a,1),6714(a,1,2,2,2)),rewrite(6714(5)),flip(a)].">
      <j1 rule="para" parents="8 6714"/>
      <j2 rule="rewrite" parents="6714"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="8827">
    <literal><![CDATA[
      f(x,f(y,f(y,z))) = f(x,f(y,f(x,z)))
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(176(a,1),6714(a,1,2,2,2)),rewrite(2(2),6714(6),176(5),2(4)),flip(a)].">
      <j1 rule="para" parents="176 6714"/>
      <j2 rule="rewrite" parents="2 6714 176 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10686">
    <literal><![CDATA[
      f(x,f(y,f(z,x))) = f(x,f(y,f(y,z)))
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(8827(a,1),2(a,2)),rewrite(2(3),2(4)),flip(a)].">
      <j1 rule="para" parents="8827 2"/>
      <j2 rule="rewrite" parents="2 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11795">
    <literal><![CDATA[
      f(f(x,f(x,y)),f(x,f(x,z))) = f(c_0,f(x,f(y,z)))
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(8820(a,1),10686(a,1,2)),rewrite(2(3),590(5)),flip(a)].">
      <j1 rule="para" parents="8820 10686"/>
      <j2 rule="rewrite" parents="2 590"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11796">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[resolve(11795,a,551,a)].">
      <j1 rule="resolve" parents="11795 551"/>
    </justification>
  </clause>

</proof>

</proofs>
