<?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-base.out</source>

<heading><![CDATA[
Prover9 (32) version 2009-02A, February 2009.
Process 12225 was started by mccune on cleo,
Wed Feb 25 10:11:19 2009
The command was "/home/mccune/bin/prover9 -f BA2.in".
]]></heading>

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

<comments><![CDATA[
% Proof 1 at 94.11 (+ 0.12) seconds: "Sheffer".
% Length of proof is 75.
% Level of proof is 24.
% Maximum clause weight is 43.
% Given clauses 315.

]]></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>
    <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>
    <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(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>
    <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>
    <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="14">
    <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="16">
    <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([14(7)]),xx(a)].">
      <j1 rule="back_rewrite" parents="5"/>
      <j2 rule="rewrite" parents="14"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(x,f(f(x,x),f(y,f(x,z)))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(14(a,1),3(a,1,1)),rewrite([2(3)])].">
      <j1 rule="para" parents="14 3"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(x,f(f(x,y),f(z,f(x,x)))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(14(a,1),6(a,1,1)),rewrite([2(3)])].">
      <j1 rule="para" parents="14 6"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(x,f(y,f(x,x))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(14(a,1),14(a,1,1)),rewrite([2(2)])].">
      <j1 rule="para" parents="14 14"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="22">
    <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="23">
    <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="24">
    <literal><![CDATA[
      f(x,f(f(y,x),f(y,z))) = f(y,x)
    ]]></literal>
    <justification jstring="[para(6(a,1),9(a,1,2)),rewrite([2(4)])].">
      <j1 rule="para" parents="6 9"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="33">
    <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="35">
    <literal><![CDATA[
      f(f(x,x),f(f(x,y),f(z,f(x,x)))) = f(z,f(x,x))
    ]]></literal>
    <justification jstring="[para(21(a,1),6(a,1,1)),rewrite([2(5)])].">
      <j1 rule="para" parents="21 6"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      f(f(x,y),f(y,y)) = y
    ]]></literal>
    <justification jstring="[para(21(a,1),6(a,1,2))].">
      <j1 rule="para" parents="21 6"/>
    </justification>
  </clause>

  <clause id="37">
    <literal><![CDATA[
      f(f(x,y),f(y,f(z,f(f(x,y),f(u,f(y,y)))))) = y
    ]]></literal>
    <justification jstring="[para(36(a,1),7(a,1,2,1)),rewrite([2(4),36(11)])].">
      <j1 rule="para" parents="36 7"/>
      <j2 rule="rewrite" parents="2 36"/>
    </justification>
  </clause>

  <clause id="48">
    <literal><![CDATA[
      f(f(x,f(y,f(x,z))),f(x,f(z,u))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),9(a,1,2))].">
      <j1 rule="para" parents="11 9"/>
    </justification>
  </clause>

  <clause id="56">
    <literal><![CDATA[
      f(x,f(y,f(x,f(y,z)))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(3(a,1),22(a,1,2,2)),rewrite([2(3)])].">
      <j1 rule="para" parents="3 22"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      f(x,f(y,f(x,f(z,y)))) = f(x,f(z,y))
    ]]></literal>
    <justification jstring="[para(6(a,1),22(a,1,2,2)),rewrite([2(3)])].">
      <j1 rule="para" parents="6 22"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="73">
    <literal><![CDATA[
      f(x,f(f(x,y),f(z,y))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(2(a,1),33(a,1,2,1)),rewrite([2(2),2(3),2(5)])].">
      <j1 rule="para" parents="2 33"/>
      <j2 rule="rewrite" parents="2 2 2"/>
    </justification>
  </clause>

  <clause id="87">
    <literal><![CDATA[
      f(x,f(f(x,x),f(y,z))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(11(a,1),17(a,1,2))].">
      <j1 rule="para" parents="11 17"/>
    </justification>
  </clause>

  <clause id="136">
    <literal><![CDATA[
      f(x,f(y,f(x,y))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(48(a,1),73(a,1,2)),flip(a)].">
      <j1 rule="para" parents="48 73"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="145">
    <literal><![CDATA[
      f(x,f(y,f(y,x))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(136(a,1),2(a,2)),rewrite([2(1),2(3)])].">
      <j1 rule="para" parents="136 2"/>
      <j2 rule="rewrite" parents="2 2"/>
    </justification>
  </clause>

  <clause id="165">
    <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>
    <justification jstring="[para(3(a,1),145(a,1,2,2)),rewrite([2(4),2(5)]),flip(a)].">
      <j1 rule="para" parents="3 145"/>
      <j2 rule="rewrite" parents="2 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="212">
    <literal><![CDATA[
      f(x,f(f(x,f(y,z)),f(y,f(x,f(y,z))))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(56(a,1),136(a,1,2,2)),rewrite([2(6)])].">
      <j1 rule="para" parents="56 136"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="213">
    <literal><![CDATA[
      f(x,f(y,y)) = f(x,f(x,y))
    ]]></literal>
    <justification jstring="[para(136(a,1),56(a,1,2)),rewrite([2(3)])].">
      <j1 rule="para" parents="136 56"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="222">
    <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(16),rewrite([213(27),213(32)])].">
      <j1 rule="back_rewrite" parents="16"/>
      <j2 rule="rewrite" parents="213 213"/>
    </justification>
  </clause>

  <clause id="223">
    <literal><![CDATA[
      f(x,f(y,y)) = f(x,f(y,x))
    ]]></literal>
    <justification jstring="[para(213(a,1),2(a,2)),rewrite([2(2),2(3)])].">
      <j1 rule="para" parents="213 2"/>
      <j2 rule="rewrite" parents="2 2"/>
    </justification>
  </clause>

  <clause id="226">
    <literal><![CDATA[
      f(x,f(x,f(y,y))) = f(y,x)
    ]]></literal>
    <justification jstring="[para(213(a,1),9(a,1)),rewrite([2(2),223(2,R),2(4),223(4,R),2(6),6(6),2(3)])].">
      <j1 rule="para" parents="213 9"/>
      <j2 rule="rewrite" parents="2 223 2 223 2 6 2"/>
    </justification>
  </clause>

  <clause id="244">
    <literal><![CDATA[
      f(x,f(f(y,y),f(x,f(y,z)))) = f(x,x)
    ]]></literal>
    <justification jstring="[back_rewrite(212),rewrite([223(6,R),2(4)])].">
      <j1 rule="back_rewrite" parents="212"/>
      <j2 rule="rewrite" parents="223 2"/>
    </justification>
  </clause>

  <clause id="275">
    <literal><![CDATA[
      f(x,f(y,f(y,f(z,z)))) = f(x,f(z,y))
    ]]></literal>
    <justification jstring="[para(226(a,1),11(a,1,2,1,2)),rewrite([23(6)]),flip(a)].">
      <j1 rule="para" parents="226 11"/>
      <j2 rule="rewrite" parents="23"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="287">
    <literal><![CDATA[
      f(f(x,f(y,y)),f(f(y,z),f(y,z))) = f(y,z)
    ]]></literal>
    <justification jstring="[para(226(a,2),19(a,1)),rewrite([2(10),35(10),2(7),223(7,R)])].">
      <j1 rule="para" parents="226 19"/>
      <j2 rule="rewrite" parents="2 35 2 223"/>
    </justification>
  </clause>

  <clause id="400">
    <literal><![CDATA[
      f(x,f(y,f(x,f(z,f(x,y))))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(48(a,1),59(a,1,2)),flip(a)].">
      <j1 rule="para" parents="48 59"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="623">
    <literal><![CDATA[
      f(x,f(y,f(x,f(z,f(y,y))))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(14(a,1),244(a,1,2,1)),rewrite([2(2)])].">
      <j1 rule="para" parents="14 244"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="642">
    <literal><![CDATA[
      f(x,f(y,f(y,y))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(244(a,1),59(a,1)),flip(a)].">
      <j1 rule="para" parents="244 59"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="652">
    <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(222),unit_del(a,642)].">
      <j1 rule="back_unit_del" parents="222"/>
      <j2 rule="unit_del" parents="642"/>
    </justification>
  </clause>

  <clause id="657">
    <literal><![CDATA[
      f(f(x,x),f(y,f(y,y))) = x
    ]]></literal>
    <justification jstring="[para(642(a,2),14(a,1))].">
      <j1 rule="para" parents="642 14"/>
    </justification>
  </clause>

  <clause id="665">
    <literal><![CDATA[
      f(x,f(x,x)) = f(y,f(y,y))
    ]]></literal>
    <justification jstring="[para(642(a,1),36(a,1,1)),rewrite([223(7),2(5),657(5),2(2)])].">
      <j1 rule="para" parents="642 36"/>
      <j2 rule="rewrite" parents="223 2 657 2"/>
    </justification>
  </clause>

  <clause id="671">
    <literal><![CDATA[
      f(x,f(f(x,y),f(z,f(z,z)))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(642(a,2),24(a,1,2)),rewrite([2(1),2(6)])].">
      <j1 rule="para" parents="642 24"/>
      <j2 rule="rewrite" parents="2 2"/>
    </justification>
  </clause>

  <clause id="714">
    <literal><![CDATA[
      f(x,f(x,x)) = c_0
    ]]></literal>
    <justification jstring="[new_symbol(665)].">
      <j1 rule="new_symbol" parents="665"/>
    </justification>
  </clause>

  <clause id="748">
    <literal><![CDATA[
      f(x,f(c_0,f(x,y))) = f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(671),rewrite([714(3),2(3)])].">
      <j1 rule="back_rewrite" parents="671"/>
      <j2 rule="rewrite" parents="714 2"/>
    </justification>
  </clause>

  <clause id="756">
    <literal><![CDATA[
      f(c_0,f(x,x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(657),rewrite([714(3),2(3)])].">
      <j1 rule="back_rewrite" parents="657"/>
      <j2 rule="rewrite" parents="714 2"/>
    </justification>
  </clause>

  <clause id="763">
    <literal><![CDATA[
      f(x,c_0) = f(x,x)
    ]]></literal>
    <justification jstring="[back_rewrite(642),rewrite([714(2)])].">
      <j1 rule="back_rewrite" parents="642"/>
      <j2 rule="rewrite" parents="714"/>
    </justification>
  </clause>

  <clause id="778">
    <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(652),rewrite([763(11,R),2(7)]),flip(a)].">
      <j1 rule="back_rewrite" parents="652"/>
      <j2 rule="rewrite" parents="763 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="796">
    <literal><![CDATA[
      f(f(x,f(x,y)),f(x,f(y,z))) = f(c_0,f(x,f(y,z)))
    ]]></literal>
    <justification jstring="[back_rewrite(165),rewrite([763(5,R),2(4)]),flip(a)].">
      <j1 rule="back_rewrite" parents="165"/>
      <j2 rule="rewrite" parents="763 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="815">
    <literal><![CDATA[
      f(f(x,f(x,y)),f(c_0,f(y,z))) = f(y,z)
    ]]></literal>
    <justification jstring="[back_rewrite(287),rewrite([223(2),2(1),763(5,R),2(5)])].">
      <j1 rule="back_rewrite" parents="287"/>
      <j2 rule="rewrite" parents="223 2 763 2"/>
    </justification>
  </clause>

  <clause id="943">
    <literal><![CDATA[
      f(f(x,f(y,y)),f(c_0,f(y,z))) = f(y,z)
    ]]></literal>
    <justification jstring="[para(35(a,1),223(a,2,2)),rewrite([763(7,R),2(7),756(7),2(5),19(5),2(8),223(8,R),763(6,R),2(6)]),flip(a)].">
      <j1 rule="para" parents="35 223"/>
      <j2 rule="rewrite" parents="763 2 756 2 19 2 223 763 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="970">
    <literal><![CDATA[
      f(c_0,f(c_0,f(x,y))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(11(a,1),748(a,1,2)),rewrite([763(7,R),2(5),2(10),763(10),2(11),943(11)])].">
      <j1 rule="para" parents="11 748"/>
      <j2 rule="rewrite" parents="763 2 2 763 2 943"/>
    </justification>
  </clause>

  <clause id="1032">
    <literal><![CDATA[
      f(x,f(f(x,y),f(c_0,f(z,y)))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(37(a,1),244(a,1,2,2,2)),rewrite([763(3,R),2(3),2(5)])].">
      <j1 rule="para" parents="37 244"/>
      <j2 rule="rewrite" parents="763 2 2"/>
    </justification>
  </clause>

  <clause id="1519">
    <literal><![CDATA[
      f(x,f(c_0,f(y,z))) = f(x,f(x,f(y,z)))
    ]]></literal>
    <justification jstring="[para(275(a,1),223(a,1,2)),rewrite([223(2),2(1),8(3),2(2),763(3,R),2(3),223(6),2(5),8(7),2(6)])].">
      <j1 rule="para" parents="275 223"/>
      <j2 rule="rewrite" parents="223 2 8 2 763 2 223 2 8 2"/>
    </justification>
  </clause>

  <clause id="1819">
    <literal><![CDATA[
      f(x,f(y,f(z,f(x,y)))) = f(x,f(x,y))
    ]]></literal>
    <justification jstring="[para(400(a,1),56(a,1,2)),rewrite([223(2),2(1),2(3)]),flip(a)].">
      <j1 rule="para" parents="400 56"/>
      <j2 rule="rewrite" parents="223 2 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1906">
    <literal><![CDATA[
      f(x,f(y,f(z,f(x,x)))) = f(x,f(x,y))
    ]]></literal>
    <justification jstring="[para(623(a,1),56(a,1,2)),rewrite([223(2),2(1)]),flip(a)].">
      <j1 rule="para" parents="623 56"/>
      <j2 rule="rewrite" parents="223 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2257">
    <literal><![CDATA[
      f(f(x,f(x,y)),f(c_0,f(z,f(x,f(y,u))))) = f(z,f(x,f(y,u)))
    ]]></literal>
    <justification jstring="[para(3(a,1),815(a,1,1,2)),rewrite([2(2),2(6),2(11)])].">
      <j1 rule="para" parents="3 815"/>
      <j2 rule="rewrite" parents="2 2 2"/>
    </justification>
  </clause>

  <clause id="2771">
    <literal><![CDATA[
      f(f(x,x),f(y,f(c_0,f(z,x)))) = x
    ]]></literal>
    <justification jstring="[para(6(a,1),943(a,1,2,2)),rewrite([763(3,R),2(3),2(6),763(6),2(6),6(10)])].">
      <j1 rule="para" parents="6 943"/>
      <j2 rule="rewrite" parents="763 2 2 763 2 6"/>
    </justification>
  </clause>

  <clause id="2780">
    <literal><![CDATA[
      f(f(x,y),f(x,f(z,f(c_0,f(u,y))))) = x
    ]]></literal>
    <justification jstring="[para(2771(a,1),9(a,1,2,2)),rewrite([2(7)])].">
      <j1 rule="para" parents="2771 9"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="2786">
    <literal><![CDATA[
      f(f(x,y),f(c_0,f(f(x,x),f(z,u)))) = f(f(x,x),f(z,u))
    ]]></literal>
    <justification jstring="[para(87(a,1),2771(a,1,2,2,2)),rewrite([763(7,R),2(5),756(8),2(6),2(7)])].">
      <j1 rule="para" parents="87 2771"/>
      <j2 rule="rewrite" parents="763 2 756 2 2"/>
    </justification>
  </clause>

  <clause id="2984">
    <literal><![CDATA[
      f(x,f(f(x,f(y,z)),f(z,z))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(73(a,1),1032(a,1,2,2)),rewrite([2(4),763(4)])].">
      <j1 rule="para" parents="73 1032"/>
      <j2 rule="rewrite" parents="2 763"/>
    </justification>
  </clause>

  <clause id="4444">
    <literal><![CDATA[
      f(x,f(c_0,f(y,f(z,f(x,y))))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(1819(a,1),1519(a,2,2)),rewrite([8(9)])].">
      <j1 rule="para" parents="1819 1519"/>
      <j2 rule="rewrite" parents="8"/>
    </justification>
  </clause>

  <clause id="6406">
    <literal><![CDATA[
      f(f(c_0,f(x,y)),f(c_0,f(y,z))) = f(z,f(c_0,f(x,y)))
    ]]></literal>
    <justification jstring="[para(2780(a,1),1906(a,1,2)),rewrite([2(4),2(11),2(12),223(13,R),763(10,R),2(10)]),flip(a)].">
      <j1 rule="para" parents="2780 1906"/>
      <j2 rule="rewrite" parents="2 2 2 223 763 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="7453">
    <literal><![CDATA[
      f(x,f(c_0,f(y,f(z,f(y,x))))) = f(y,x)
    ]]></literal>
    <justification jstring="[para(4444(a,1),2(a,2)),rewrite([2(2),2(6),2(7)])].">
      <j1 rule="para" parents="4444 2"/>
      <j2 rule="rewrite" parents="2 2 2"/>
    </justification>
  </clause>

  <clause id="7555">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(z,z)) = f(f(x,x),f(z,z))
    ]]></literal>
    <justification jstring="[para(2984(a,1),4444(a,1,2,2,2)),rewrite([2(6),2786(8)]),flip(a)].">
      <j1 rule="para" parents="2984 4444"/>
      <j2 rule="rewrite" parents="2 2786"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18323">
    <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),7555(a,1,1)),rewrite([763(3,R),2(3),763(7,R),2(7),763(10,R),2(10)]),flip(a)].">
      <j1 rule="para" parents="3 7555"/>
      <j2 rule="rewrite" parents="763 2 763 2 763 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18743">
    <literal><![CDATA[
      f(x,f(c_0,f(y,z))) = f(z,f(c_0,f(x,y)))
    ]]></literal>
    <justification jstring="[back_rewrite(6406),rewrite([18323(7)])].">
      <j1 rule="back_rewrite" parents="6406"/>
      <j2 rule="rewrite" parents="18323"/>
    </justification>
  </clause>

  <clause id="18986">
    <literal><![CDATA[
      f(x,f(y,f(c_0,f(x,z)))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(18743(a,2),1519(a,2,2)),rewrite([970(5),2(4)]),flip(a)].">
      <j1 rule="para" parents="18743 1519"/>
      <j2 rule="rewrite" parents="970 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21298">
    <literal><![CDATA[
      f(x,f(y,f(x,f(x,z)))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(8(a,1),18986(a,1,2,2,2)),rewrite([18986(5)]),flip(a)].">
      <j1 rule="para" parents="8 18986"/>
      <j2 rule="rewrite" parents="18986"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23790">
    <literal><![CDATA[
      f(x,f(y,f(z,x))) = f(x,f(y,f(y,z)))
    ]]></literal>
    <justification jstring="[para(21298(a,1),7453(a,1,2,2,2)),rewrite([2(4),2257(8)])].">
      <j1 rule="para" parents="21298 7453"/>
      <j2 rule="rewrite" parents="2 2257"/>
    </justification>
  </clause>

  <clause id="24821">
    <literal><![CDATA[
      f(f(x,f(x,y)),f(x,f(x,z))) = f(c_0,f(x,f(y,z)))
    ]]></literal>
    <justification jstring="[para(21298(a,1),23790(a,1,2)),rewrite([2(3),796(5)]),flip(a)].">
      <j1 rule="para" parents="21298 23790"/>
      <j2 rule="rewrite" parents="2 796"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="24822">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[resolve(24821,a,778,a)].">
      <j1 rule="resolve" parents="24821 778"/>
    </justification>
  </clause>

</proof>

</proofs>
