<?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="3">

<source>olsax2.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13580 was started by mccune on cleo.thornwood,
Mon Jun 19 16:42:53 2006
The command was "/home/mccune/bin/prover9 -f olsax2.in".
]]></heading>

<proof number="1" length="44" max_count="36">

<comments><![CDATA[
% Proof 1 at 0.23 (+ 0.01) seconds: B_SS.
% Length of proof is 44.
% Level of proof is 14.
% Maximum clause weight is 36.
% Given clauses 74.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(y,f(f(x,x),x)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OL_Sh)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

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

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

  <clause id="4">
    <literal><![CDATA[
      x ' = f(x,x)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="6">
    <literal><![CDATA[
      f(f(c4,c4),f(c4,c5)) != c4
    ]]></literal>
    <attribute><![CDATA[
      answer(B_SS)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(y,f(f(x,x),x)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OL_Sh)
    ]]></attribute>
    <justification jstring="[copy(1)].">
      <j1 rule="copy" parents="1"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <justification jstring="[copy(2),flip(a)].">
      <j1 rule="copy" parents="2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

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

  <clause id="11">
    <literal><![CDATA[
      f(x,x) = x '
    ]]></literal>
    <justification jstring="[copy(4),flip(a)].">
      <j1 rule="copy" parents="4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      f(x,y) ' = x ^ y
    ]]></literal>
    <justification jstring="[back_demod(10),demod(11(3))].">
      <j1 rule="back_demod" parents="10"/>
      <j2 rule="demod" parents="11"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      f(x ',y ') = x v y
    ]]></literal>
    <justification jstring="[back_demod(9),demod(11(1),11(2))].">
      <j1 rule="back_demod" parents="9"/>
      <j2 rule="demod" parents="11 11"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(y,f(x ',x)),z))) = y
    ]]></literal>
    <justification jstring="[back_demod(8),demod(11(5))].">
      <j1 rule="back_demod" parents="8"/>
      <j2 rule="demod" parents="11"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      f(c4 ',f(c4,c5)) != c4
    ]]></literal>
    <attribute><![CDATA[
      answer(B_SS)
    ]]></attribute>
    <justification jstring="[copy(6),demod(11(3))].">
      <j1 rule="copy" parents="6"/>
      <j2 rule="demod" parents="11"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      x ^ x = x ' '
    ]]></literal>
    <justification jstring="[para(11(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="11 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      f(x ',y ^ z) = x v f(y,z)
    ]]></literal>
    <justification jstring="[para(12(a,1),13(a,1,2))].">
      <j1 rule="para" parents="12 13"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      f(f(f(x ',f(x,y)),z),f(x,f(f(x,f(x ',x)),y))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),14(a,1,1,1,1))].">
      <j1 rule="para" parents="11 14"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      f(f(x ' ',y),f(x,f(f(x,f(x ',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),14(a,1,1,1)),demod(11(1))].">
      <j1 rule="para" parents="11 14"/>
      <j2 rule="demod" parents="11"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      f(f(f(x,y),f(y,z)),u) ^ f(y,f(f(y,f(x ',x)),z)) = y '
    ]]></literal>
    <justification jstring="[para(14(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="14 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(x,f(f(z,x) ^ f(x,u),f(f(z,x),f(x,u)))),f(f(x,f(z ',z)),u)))) = x
    ]]></literal>
    <justification jstring="[para(14(a,1),14(a,1,1,1)),demod(12(5))].">
      <j1 rule="para" parents="14 14"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(f(x,y),f(z ^ x,f(z,x))),u))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(14(a,1),14(a,1,1)),demod(12(4))].">
      <j1 rule="para" parents="14 14"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="174">
    <literal><![CDATA[
      f(x ' ' ',f(x,f(f(x,f(x ',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),25(a,1,1))].">
      <j1 rule="para" parents="11 25"/>
    </justification>
  </clause>

  <clause id="183">
    <literal><![CDATA[
      f(x,f(x ',f(f(x ',x ' v x),y))) = x '
    ]]></literal>
    <justification jstring="[para(25(a,1),23(a,1,1)),demod(13(6))].">
      <j1 rule="para" parents="25 23"/>
      <j2 rule="demod" parents="13"/>
    </justification>
  </clause>

  <clause id="210">
    <literal><![CDATA[
      x ' ' ' ^ f(x,f(f(x,f(x ',x)),x)) = x '
    ]]></literal>
    <justification jstring="[para(174(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="174 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="229">
    <literal><![CDATA[
      f(x ',f(x ' ',x)) = x ' '
    ]]></literal>
    <justification jstring="[para(25(a,1),183(a,1,2,2))].">
      <j1 rule="para" parents="25 183"/>
    </justification>
  </clause>

  <clause id="233">
    <literal><![CDATA[
      f(x ' ',x ' ' v x) = x ' ' '
    ]]></literal>
    <justification jstring="[para(13(a,1),229(a,1,2))].">
      <j1 rule="para" parents="13 229"/>
    </justification>
  </clause>

  <clause id="326">
    <literal><![CDATA[
      f(x,y) ^ f(x,f(f(x,f(f(z,x) ^ f(x,u),f(f(z,x),f(x,u)))),f(f(x,f(z ',z)),u))) = x '
    ]]></literal>
    <justification jstring="[para(14(a,1),29(a,1,1,1)),demod(12(5))].">
      <j1 rule="para" parents="14 29"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="737">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(y,f(x ',x)),z))) = y
    ]]></literal>
    <justification jstring="[para(34(a,1),14(a,1,1))].">
      <j1 rule="para" parents="34 14"/>
    </justification>
  </clause>

  <clause id="748">
    <literal><![CDATA[
      f(x,y) ^ f(y,f(f(y,f(x ',x)),z)) = y '
    ]]></literal>
    <justification jstring="[para(34(a,1),29(a,1,1))].">
      <j1 rule="para" parents="34 29"/>
    </justification>
  </clause>

  <clause id="838">
    <literal><![CDATA[
      f(x ',f(f(x ',f(x,y)),x)) = f(x ',f(x,y))
    ]]></literal>
    <justification jstring="[para(23(a,1),35(a,1,2,2))].">
      <j1 rule="para" parents="23 35"/>
    </justification>
  </clause>

  <clause id="864">
    <literal><![CDATA[
      f(x,x ^ y) = f(x,y)
    ]]></literal>
    <justification jstring="[para(34(a,1),35(a,1,2,2)),demod(11(3),12(2))].">
      <j1 rule="para" parents="34 35"/>
      <j2 rule="demod" parents="11 12"/>
    </justification>
  </clause>

  <clause id="985">
    <literal><![CDATA[
      x ' ' v x = x
    ]]></literal>
    <justification jstring="[para(174(a,1),864(a,2)),demod(210(12),13(5))].">
      <j1 rule="para" parents="174 864"/>
      <j2 rule="demod" parents="210 13"/>
    </justification>
  </clause>

  <clause id="990">
    <literal><![CDATA[
      f(x ' ',x) = x ' ' '
    ]]></literal>
    <justification jstring="[para(233(a,1),864(a,2)),demod(985(7),864(6))].">
      <j1 rule="para" parents="233 864"/>
      <j2 rule="demod" parents="985 864"/>
    </justification>
  </clause>

  <clause id="1011">
    <literal><![CDATA[
      f(f(x,y),x ') = x
    ]]></literal>
    <justification jstring="[para(34(a,1),864(a,2)),demod(326(17))].">
      <j1 rule="para" parents="34 864"/>
      <j2 rule="demod" parents="326"/>
    </justification>
  </clause>

  <clause id="1055">
    <literal><![CDATA[
      f(x,y) = (x ^ y) ' ' '
    ]]></literal>
    <justification jstring="[para(985(a,1),22(a,2)),demod(12(2),990(5)),flip(a)].">
      <j1 rule="para" parents="985 22"/>
      <j2 rule="demod" parents="12 990"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1070">
    <literal><![CDATA[
      ((x ^ y) ' ' ' ^ x ') ' ' ' = x
    ]]></literal>
    <justification jstring="[back_demod(1011),demod(1055(1),1055(6))].">
      <j1 rule="back_demod" parents="1011"/>
      <j2 rule="demod" parents="1055 1055"/>
    </justification>
  </clause>

  <clause id="1102">
    <literal><![CDATA[
      (x ' ^ ((x ' ^ (x ^ y) ' ' ') ' ' ' ^ x) ' ' ') ' ' ' = (x ' ^ (x ^ y) ' ' ') ' ' '
    ]]></literal>
    <justification jstring="[back_demod(838),demod(1055(3),1055(7),1055(11),1055(15),1055(20),1055(24))].">
      <j1 rule="back_demod" parents="838"/>
      <j2 rule="demod" parents="1055 1055 1055 1055 1055 1055"/>
    </justification>
  </clause>

  <clause id="1121">
    <literal><![CDATA[
      (x ^ y) ' ' ' ^ (y ^ ((y ^ (x ' ^ x) ' ' ') ' ' ' ^ z) ' ' ') ' ' ' = y '
    ]]></literal>
    <justification jstring="[back_demod(748),demod(1055(1),1055(6),1055(10),1055(14),1055(18))].">
      <j1 rule="back_demod" parents="748"/>
      <j2 rule="demod" parents="1055 1055 1055 1055 1055"/>
    </justification>
  </clause>

  <clause id="1123">
    <literal><![CDATA[
      x ' ' ' ' = x
    ]]></literal>
    <justification jstring="[back_demod(737),demod(1055(1),1055(6),1055(10),1055(14),1055(18),1055(22),1121(22))].">
      <j1 rule="back_demod" parents="737"/>
      <j2 rule="demod" parents="1055 1055 1055 1055 1055 1055 1121"/>
    </justification>
  </clause>

  <clause id="1137">
    <literal><![CDATA[
      (c4 ' ^ (c4 ^ c5) ' ' ') ' ' ' != c4
    ]]></literal>
    <attribute><![CDATA[
      answer(B_SS)
    ]]></attribute>
    <justification jstring="[back_demod(16),demod(1055(5),1055(9))].">
      <j1 rule="back_demod" parents="16"/>
      <j2 rule="demod" parents="1055 1055"/>
    </justification>
  </clause>

  <clause id="1144">
    <literal><![CDATA[
      x ' ' = x
    ]]></literal>
    <justification jstring="[para(18(a,1),1070(a,1,1,1,1,1,1,1,1)),demod(1123(4),18(3),1123(4))].">
      <j1 rule="para" parents="18 1070"/>
      <j2 rule="demod" parents="1123 18 1123"/>
    </justification>
  </clause>

  <clause id="1148">
    <literal><![CDATA[
      ((x ' ^ y) ' ^ x) ' = x '
    ]]></literal>
    <justification jstring="[para(1123(a,1),1070(a,1,1,1,1,2)),demod(1144(2),1144(4),1144(6),1144(7))].">
      <j1 rule="para" parents="1123 1070"/>
      <j2 rule="demod" parents="1144 1144 1144 1144"/>
    </justification>
  </clause>

  <clause id="1152">
    <literal><![CDATA[
      (c4 ' ^ (c4 ^ c5) ') ' != c4
    ]]></literal>
    <attribute><![CDATA[
      answer(B_SS)
    ]]></attribute>
    <justification jstring="[back_demod(1137),demod(1144(7),1144(9))].">
      <j1 rule="back_demod" parents="1137"/>
      <j2 rule="demod" parents="1144 1144"/>
    </justification>
  </clause>

  <clause id="1164">
    <literal><![CDATA[
      (x ' ^ (x ^ y) ') ' = x
    ]]></literal>
    <justification jstring="[back_demod(1102),demod(1144(5),1144(7),1148(8),1144(3),18(3),1144(2),1144(2),1144(2),1144(4),1144(6)),flip(a)].">
      <j1 rule="back_demod" parents="1102"/>
      <j2 rule="demod" parents="1144 1144 1148 1144 18 1144 1144 1144 1144 1144"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1165">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(B_SS)
    ]]></attribute>
    <justification jstring="[resolve(1164,a,1152,a)].">
      <j1 rule="resolve" parents="1164 1152"/>
    </justification>
  </clause>

</proof>

<proof number="2" length="87" max_count="39">

<comments><![CDATA[
% Proof 2 at 0.38 (+ 0.01) seconds: ONE_SS.
% Length of proof is 87.
% Level of proof is 24.
% Maximum clause weight is 39.
% Given clauses 144.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(y,f(f(x,x),x)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OL_Sh)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

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

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

  <clause id="4">
    <literal><![CDATA[
      x ' = f(x,x)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      f(c7,f(c7,c7)) != f(c6,f(c6,c6))
    ]]></literal>
    <attribute><![CDATA[
      answer(ONE_SS)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(y,f(f(x,x),x)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OL_Sh)
    ]]></attribute>
    <justification jstring="[copy(1)].">
      <j1 rule="copy" parents="1"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <justification jstring="[copy(2),flip(a)].">
      <j1 rule="copy" parents="2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

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

  <clause id="11">
    <literal><![CDATA[
      f(x,x) = x '
    ]]></literal>
    <justification jstring="[copy(4),flip(a)].">
      <j1 rule="copy" parents="4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      f(x,y) ' = x ^ y
    ]]></literal>
    <justification jstring="[back_demod(10),demod(11(3))].">
      <j1 rule="back_demod" parents="10"/>
      <j2 rule="demod" parents="11"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      f(x ',y ') = x v y
    ]]></literal>
    <justification jstring="[back_demod(9),demod(11(1),11(2))].">
      <j1 rule="back_demod" parents="9"/>
      <j2 rule="demod" parents="11 11"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(y,f(x ',x)),z))) = y
    ]]></literal>
    <justification jstring="[back_demod(8),demod(11(5))].">
      <j1 rule="back_demod" parents="8"/>
      <j2 rule="demod" parents="11"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(c7,c7 ') != f(c6,c6 ')
    ]]></literal>
    <attribute><![CDATA[
      answer(ONE_SS)
    ]]></attribute>
    <justification jstring="[copy(7),demod(11(4),11(8))].">
      <j1 rule="copy" parents="7"/>
      <j2 rule="demod" parents="11 11"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      x ^ x = x ' '
    ]]></literal>
    <justification jstring="[para(11(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="11 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      f(x ',y ^ z) = x v f(y,z)
    ]]></literal>
    <justification jstring="[para(12(a,1),13(a,1,2))].">
      <j1 rule="para" parents="12 13"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      f(f(f(x ',f(x,y)),z),f(x,f(f(x,f(x ',x)),y))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),14(a,1,1,1,1))].">
      <j1 rule="para" parents="11 14"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      f(f(x ' ',y),f(x,f(f(x,f(x ',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),14(a,1,1,1)),demod(11(1))].">
      <j1 rule="para" parents="11 14"/>
      <j2 rule="demod" parents="11"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      f(f(x,y) ^ f(y,z),f(y,f(f(y,f(x ',x)),z))) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),14(a,1,1)),demod(12(4))].">
      <j1 rule="para" parents="11 14"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      f(f(f(x,y),f(y,z)),u) ^ f(y,f(f(y,f(x ',x)),z)) = y '
    ]]></literal>
    <justification jstring="[para(14(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="14 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(x,f(f(z,x) ^ f(x,u),f(f(z,x),f(x,u)))),f(f(x,f(z ',z)),u)))) = x
    ]]></literal>
    <justification jstring="[para(14(a,1),14(a,1,1,1)),demod(12(5))].">
      <j1 rule="para" parents="14 14"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(f(x,y),f(z ^ x,f(z,x))),u))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(14(a,1),14(a,1,1)),demod(12(4))].">
      <j1 rule="para" parents="14 14"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="174">
    <literal><![CDATA[
      f(x ' ' ',f(x,f(f(x,f(x ',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),25(a,1,1))].">
      <j1 rule="para" parents="11 25"/>
    </justification>
  </clause>

  <clause id="183">
    <literal><![CDATA[
      f(x,f(x ',f(f(x ',x ' v x),y))) = x '
    ]]></literal>
    <justification jstring="[para(25(a,1),23(a,1,1)),demod(13(6))].">
      <j1 rule="para" parents="25 23"/>
      <j2 rule="demod" parents="13"/>
    </justification>
  </clause>

  <clause id="187">
    <literal><![CDATA[
      (f(x,y) ^ f(y,z)) ^ f(y,f(f(y,f(x ',x)),z)) = y '
    ]]></literal>
    <justification jstring="[para(26(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="26 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="210">
    <literal><![CDATA[
      x ' ' ' ^ f(x,f(f(x,f(x ',x)),x)) = x '
    ]]></literal>
    <justification jstring="[para(174(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="174 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="223">
    <literal><![CDATA[
      x ^ f(x ',f(f(x ',x ' v x),y)) = x ' '
    ]]></literal>
    <justification jstring="[para(183(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="183 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="229">
    <literal><![CDATA[
      f(x ',f(x ' ',x)) = x ' '
    ]]></literal>
    <justification jstring="[para(25(a,1),183(a,1,2,2))].">
      <j1 rule="para" parents="25 183"/>
    </justification>
  </clause>

  <clause id="233">
    <literal><![CDATA[
      f(x ' ',x ' ' v x) = x ' ' '
    ]]></literal>
    <justification jstring="[para(13(a,1),229(a,1,2))].">
      <j1 rule="para" parents="13 229"/>
    </justification>
  </clause>

  <clause id="326">
    <literal><![CDATA[
      f(x,y) ^ f(x,f(f(x,f(f(z,x) ^ f(x,u),f(f(z,x),f(x,u)))),f(f(x,f(z ',z)),u))) = x '
    ]]></literal>
    <justification jstring="[para(14(a,1),29(a,1,1,1)),demod(12(5))].">
      <j1 rule="para" parents="14 29"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="737">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(y,f(x ',x)),z))) = y
    ]]></literal>
    <justification jstring="[para(34(a,1),14(a,1,1))].">
      <j1 rule="para" parents="34 14"/>
    </justification>
  </clause>

  <clause id="748">
    <literal><![CDATA[
      f(x,y) ^ f(y,f(f(y,f(x ',x)),z)) = y '
    ]]></literal>
    <justification jstring="[para(34(a,1),29(a,1,1))].">
      <j1 rule="para" parents="34 29"/>
    </justification>
  </clause>

  <clause id="749">
    <literal><![CDATA[
      x ^ x ' ' = x ' '
    ]]></literal>
    <justification jstring="[para(34(a,1),223(a,1,2,2)),demod(11(3))].">
      <j1 rule="para" parents="34 223"/>
      <j2 rule="demod" parents="11"/>
    </justification>
  </clause>

  <clause id="838">
    <literal><![CDATA[
      f(x ',f(f(x ',f(x,y)),x)) = f(x ',f(x,y))
    ]]></literal>
    <justification jstring="[para(23(a,1),35(a,1,2,2))].">
      <j1 rule="para" parents="23 35"/>
    </justification>
  </clause>

  <clause id="864">
    <literal><![CDATA[
      f(x,x ^ y) = f(x,y)
    ]]></literal>
    <justification jstring="[para(34(a,1),35(a,1,2,2)),demod(11(3),12(2))].">
      <j1 rule="para" parents="34 35"/>
      <j2 rule="demod" parents="11 12"/>
    </justification>
  </clause>

  <clause id="966">
    <literal><![CDATA[
      x ^ (x ^ y) = x ^ y
    ]]></literal>
    <justification jstring="[para(864(a,1),12(a,1,1)),demod(12(2)),flip(a)].">
      <j1 rule="para" parents="864 12"/>
      <j2 rule="demod" parents="12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="970">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),y ') = y
    ]]></literal>
    <justification jstring="[para(14(a,1),864(a,2)),demod(29(14))].">
      <j1 rule="para" parents="14 864"/>
      <j2 rule="demod" parents="29"/>
    </justification>
  </clause>

  <clause id="985">
    <literal><![CDATA[
      x ' ' v x = x
    ]]></literal>
    <justification jstring="[para(174(a,1),864(a,2)),demod(210(12),13(5))].">
      <j1 rule="para" parents="174 864"/>
      <j2 rule="demod" parents="210 13"/>
    </justification>
  </clause>

  <clause id="990">
    <literal><![CDATA[
      f(x ' ',x) = x ' ' '
    ]]></literal>
    <justification jstring="[para(233(a,1),864(a,2)),demod(985(7),864(6))].">
      <j1 rule="para" parents="233 864"/>
      <j2 rule="demod" parents="985 864"/>
    </justification>
  </clause>

  <clause id="1011">
    <literal><![CDATA[
      f(f(x,y),x ') = x
    ]]></literal>
    <justification jstring="[para(34(a,1),864(a,2)),demod(326(17))].">
      <j1 rule="para" parents="34 864"/>
      <j2 rule="demod" parents="326"/>
    </justification>
  </clause>

  <clause id="1055">
    <literal><![CDATA[
      f(x,y) = (x ^ y) ' ' '
    ]]></literal>
    <justification jstring="[para(985(a,1),22(a,2)),demod(12(2),990(5)),flip(a)].">
      <j1 rule="para" parents="985 22"/>
      <j2 rule="demod" parents="12 990"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1070">
    <literal><![CDATA[
      ((x ^ y) ' ' ' ^ x ') ' ' ' = x
    ]]></literal>
    <justification jstring="[back_demod(1011),demod(1055(1),1055(6))].">
      <j1 rule="back_demod" parents="1011"/>
      <j2 rule="demod" parents="1055 1055"/>
    </justification>
  </clause>

  <clause id="1085">
    <literal><![CDATA[
      ((((x ^ y) ' ' ' ^ (y ^ z) ' ' ') ' ' ' ^ u) ' ' ' ^ y ') ' ' ' = y
    ]]></literal>
    <justification jstring="[back_demod(970),demod(1055(1),1055(5),1055(9),1055(13),1055(18))].">
      <j1 rule="back_demod" parents="970"/>
      <j2 rule="demod" parents="1055 1055 1055 1055 1055"/>
    </justification>
  </clause>

  <clause id="1102">
    <literal><![CDATA[
      (x ' ^ ((x ' ^ (x ^ y) ' ' ') ' ' ' ^ x) ' ' ') ' ' ' = (x ' ^ (x ^ y) ' ' ') ' ' '
    ]]></literal>
    <justification jstring="[back_demod(838),demod(1055(3),1055(7),1055(11),1055(15),1055(20),1055(24))].">
      <j1 rule="back_demod" parents="838"/>
      <j2 rule="demod" parents="1055 1055 1055 1055 1055 1055"/>
    </justification>
  </clause>

  <clause id="1121">
    <literal><![CDATA[
      (x ^ y) ' ' ' ^ (y ^ ((y ^ (x ' ^ x) ' ' ') ' ' ' ^ z) ' ' ') ' ' ' = y '
    ]]></literal>
    <justification jstring="[back_demod(748),demod(1055(1),1055(6),1055(10),1055(14),1055(18))].">
      <j1 rule="back_demod" parents="748"/>
      <j2 rule="demod" parents="1055 1055 1055 1055 1055"/>
    </justification>
  </clause>

  <clause id="1123">
    <literal><![CDATA[
      x ' ' ' ' = x
    ]]></literal>
    <justification jstring="[back_demod(737),demod(1055(1),1055(6),1055(10),1055(14),1055(18),1055(22),1121(22))].">
      <j1 rule="back_demod" parents="737"/>
      <j2 rule="demod" parents="1055 1055 1055 1055 1055 1055 1121"/>
    </justification>
  </clause>

  <clause id="1134">
    <literal><![CDATA[
      ((x ^ y) ' ' ' ^ (y ^ z) ' ' ') ^ (y ^ ((y ^ (x ' ^ x) ' ' ') ' ' ' ^ z) ' ' ') ' ' ' = y '
    ]]></literal>
    <justification jstring="[back_demod(187),demod(1055(1),1055(5),1055(11),1055(15),1055(19),1055(23))].">
      <j1 rule="back_demod" parents="187"/>
      <j2 rule="demod" parents="1055 1055 1055 1055 1055 1055"/>
    </justification>
  </clause>

  <clause id="1136">
    <literal><![CDATA[
      (c7 ^ c7 ') ' ' ' != (c6 ^ c6 ') ' ' '
    ]]></literal>
    <attribute><![CDATA[
      answer(ONE_SS)
    ]]></attribute>
    <justification jstring="[back_demod(17),demod(1055(4),1055(11))].">
      <j1 rule="back_demod" parents="17"/>
      <j2 rule="demod" parents="1055 1055"/>
    </justification>
  </clause>

  <clause id="1144">
    <literal><![CDATA[
      x ' ' = x
    ]]></literal>
    <justification jstring="[para(18(a,1),1070(a,1,1,1,1,1,1,1,1)),demod(1123(4),18(3),1123(4))].">
      <j1 rule="para" parents="18 1070"/>
      <j2 rule="demod" parents="1123 18 1123"/>
    </justification>
  </clause>

  <clause id="1147">
    <literal><![CDATA[
      (x ^ y) ' ^ x ' = x '
    ]]></literal>
    <justification jstring="[para(1070(a,1),749(a,2,1)),demod(1144(3),1144(6),1144(7),1144(10),1144(10),18(9),1144(6))].">
      <j1 rule="para" parents="1070 749"/>
      <j2 rule="demod" parents="1144 1144 1144 1144 1144 18 1144"/>
    </justification>
  </clause>

  <clause id="1148">
    <literal><![CDATA[
      ((x ' ^ y) ' ^ x) ' = x '
    ]]></literal>
    <justification jstring="[para(1123(a,1),1070(a,1,1,1,1,2)),demod(1144(2),1144(4),1144(6),1144(7))].">
      <j1 rule="para" parents="1123 1070"/>
      <j2 rule="demod" parents="1144 1144 1144 1144"/>
    </justification>
  </clause>

  <clause id="1153">
    <literal><![CDATA[
      (c7 ^ c7 ') ' != (c6 ^ c6 ') '
    ]]></literal>
    <attribute><![CDATA[
      answer(ONE_SS)
    ]]></attribute>
    <justification jstring="[back_demod(1136),demod(1144(6),1144(11))].">
      <j1 rule="back_demod" parents="1136"/>
      <j2 rule="demod" parents="1144 1144"/>
    </justification>
  </clause>

  <clause id="1155">
    <literal><![CDATA[
      ((x ^ y) ' ^ (y ^ z) ') ^ (y ^ ((y ^ (x ' ^ x) ') ' ^ z) ') ' = y '
    ]]></literal>
    <justification jstring="[back_demod(1134),demod(1144(3),1144(5),1144(9),1144(11),1144(13),1144(15))].">
      <j1 rule="back_demod" parents="1134"/>
      <j2 rule="demod" parents="1144 1144 1144 1144 1144 1144"/>
    </justification>
  </clause>

  <clause id="1163">
    <literal><![CDATA[
      (x ^ y) ' ^ (y ^ ((y ^ (x ' ^ x) ') ' ^ z) ') ' = y '
    ]]></literal>
    <justification jstring="[back_demod(1121),demod(1144(3),1144(6),1144(8),1144(10),1144(12))].">
      <j1 rule="back_demod" parents="1121"/>
      <j2 rule="demod" parents="1144 1144 1144 1144 1144"/>
    </justification>
  </clause>

  <clause id="1164">
    <literal><![CDATA[
      (x ' ^ (x ^ y) ') ' = x
    ]]></literal>
    <justification jstring="[back_demod(1102),demod(1144(5),1144(7),1148(8),1144(3),18(3),1144(2),1144(2),1144(2),1144(4),1144(6)),flip(a)].">
      <j1 rule="back_demod" parents="1102"/>
      <j2 rule="demod" parents="1144 1144 1148 1144 18 1144 1144 1144 1144 1144"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1174">
    <literal><![CDATA[
      ((((x ^ y) ' ^ (y ^ z) ') ' ^ u) ' ^ y ') ' = y
    ]]></literal>
    <justification jstring="[back_demod(1085),demod(1144(3),1144(5),1144(7),1144(9),1144(12))].">
      <j1 rule="back_demod" parents="1085"/>
      <j2 rule="demod" parents="1144 1144 1144 1144 1144"/>
    </justification>
  </clause>

  <clause id="1182">
    <literal><![CDATA[
      x ^ x = x
    ]]></literal>
    <justification jstring="[back_demod(18),demod(1144(3))].">
      <j1 rule="back_demod" parents="18"/>
      <j2 rule="demod" parents="1144"/>
    </justification>
  </clause>

  <clause id="1183">
    <literal><![CDATA[
      (x ' ^ y) ' ^ x = x
    ]]></literal>
    <justification jstring="[para(1144(a,1),1147(a,1,2)),demod(1144(6))].">
      <j1 rule="para" parents="1144 1147"/>
      <j2 rule="demod" parents="1144"/>
    </justification>
  </clause>

  <clause id="1184">
    <literal><![CDATA[
      x ' ^ (x ^ y) ' = x '
    ]]></literal>
    <justification jstring="[para(1164(a,1),1144(a,1,1)),flip(a)].">
      <j1 rule="para" parents="1164 1144"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1191">
    <literal><![CDATA[
      x ^ ((y ^ x) ' ^ (x ^ z) ') ' = ((y ^ x) ' ^ (x ^ z) ') '
    ]]></literal>
    <justification jstring="[para(1155(a,1),1147(a,1,1,1)),demod(1144(2))].">
      <j1 rule="para" parents="1155 1147"/>
      <j2 rule="demod" parents="1144"/>
    </justification>
  </clause>

  <clause id="1193">
    <literal><![CDATA[
      ((x ^ (y ^ z) ') ' ^ y) ^ ((y ^ z) ' ^ (((y ^ z) ' ^ (x ' ^ x) ') ' ^ y ') ') ' = y ^ z
    ]]></literal>
    <justification jstring="[para(1147(a,1),1155(a,1,1,2,1)),demod(1144(6),1144(23))].">
      <j1 rule="para" parents="1147 1155"/>
      <j2 rule="demod" parents="1144 1144"/>
    </justification>
  </clause>

  <clause id="1199">
    <literal><![CDATA[
      x ^ (x ' ^ y) ' = x
    ]]></literal>
    <justification jstring="[para(1144(a,1),1184(a,1,1)),demod(1144(6))].">
      <j1 rule="para" parents="1144 1184"/>
      <j2 rule="demod" parents="1144"/>
    </justification>
  </clause>

  <clause id="1200">
    <literal><![CDATA[
      (x ^ y) ^ x = x ^ y
    ]]></literal>
    <justification jstring="[para(1147(a,1),1184(a,1,2,1)),demod(1144(3),1144(3),1144(5))].">
      <j1 rule="para" parents="1147 1184"/>
      <j2 rule="demod" parents="1144 1144 1144"/>
    </justification>
  </clause>

  <clause id="1204">
    <literal><![CDATA[
      ((x ^ y) ' ^ (y ^ z) ') ' ^ y = ((x ^ y) ' ^ (y ^ z) ') '
    ]]></literal>
    <justification jstring="[para(1155(a,1),1184(a,1,2,1)),demod(1144(8))].">
      <j1 rule="para" parents="1155 1184"/>
      <j2 rule="demod" parents="1144"/>
    </justification>
  </clause>

  <clause id="1251">
    <literal><![CDATA[
      (x ^ y) ' ^ y ' = y '
    ]]></literal>
    <justification jstring="[para(1163(a,1),966(a,1,2)),demod(1163(16))].">
      <j1 rule="para" parents="1163 966"/>
      <j2 rule="demod" parents="1163"/>
    </justification>
  </clause>

  <clause id="1253">
    <literal><![CDATA[
      x ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(1163(a,1),1147(a,1,1,1)),demod(1144(2),1144(3),1144(5))].">
      <j1 rule="para" parents="1163 1147"/>
      <j2 rule="demod" parents="1144 1144 1144"/>
    </justification>
  </clause>

  <clause id="1256">
    <literal><![CDATA[
      (x ^ y ') ' ^ y = y
    ]]></literal>
    <justification jstring="[para(1183(a,1),1163(a,1,2,1,2,1)),demod(1182(6),1144(5),1144(6))].">
      <j1 rule="para" parents="1183 1163"/>
      <j2 rule="demod" parents="1182 1144 1144"/>
    </justification>
  </clause>

  <clause id="1264">
    <literal><![CDATA[
      x ^ (y ^ x ') ' = x
    ]]></literal>
    <justification jstring="[para(1253(a,1),1199(a,1,2,1))].">
      <j1 rule="para" parents="1253 1199"/>
    </justification>
  </clause>

  <clause id="1273">
    <literal><![CDATA[
      (x ^ (y ^ (x ' ^ x)) ') ' ^ ((y ^ (x ' ^ x)) ' ^ ((x ' ^ x) ^ z) ') ' = y ^ (x ' ^ x)
    ]]></literal>
    <justification jstring="[para(1256(a,1),1163(a,1,2,1,2,1,1,1)),demod(1144(4),1144(10),1144(14),1144(21),1144(22))].">
      <j1 rule="para" parents="1256 1163"/>
      <j2 rule="demod" parents="1144 1144 1144 1144 1144"/>
    </justification>
  </clause>

  <clause id="1347">
    <literal><![CDATA[
      (((x ^ y) ^ z) ' ^ x ') ' = x
    ]]></literal>
    <justification jstring="[para(1147(a,1),1174(a,1,1,1,1,1,1)),demod(1144(3))].">
      <j1 rule="para" parents="1147 1174"/>
      <j2 rule="demod" parents="1144"/>
    </justification>
  </clause>

  <clause id="1364">
    <literal><![CDATA[
      ((x ^ y) ^ z) ' ^ x ' = x '
    ]]></literal>
    <justification jstring="[para(1347(a,1),1144(a,1,1)),flip(a)].">
      <j1 rule="para" parents="1347 1144"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1366">
    <literal><![CDATA[
      x ^ ((x ^ y) ^ z) = (x ^ y) ^ z
    ]]></literal>
    <justification jstring="[para(1347(a,1),1147(a,1,1)),demod(1144(4),1144(7))].">
      <j1 rule="para" parents="1347 1147"/>
      <j2 rule="demod" parents="1144 1144"/>
    </justification>
  </clause>

  <clause id="1399">
    <literal><![CDATA[
      ((x ' ^ y) ^ z) ' ^ x = x
    ]]></literal>
    <justification jstring="[para(1144(a,1),1364(a,1,2)),demod(1144(7))].">
      <j1 rule="para" parents="1144 1364"/>
      <j2 rule="demod" parents="1144"/>
    </justification>
  </clause>

  <clause id="1413">
    <literal><![CDATA[
      x ^ ((x ' ^ y) ^ z) ' = x
    ]]></literal>
    <justification jstring="[para(1399(a,1),1200(a,1,1)),demod(1399(10))].">
      <j1 rule="para" parents="1399 1200"/>
      <j2 rule="demod" parents="1399"/>
    </justification>
  </clause>

  <clause id="1428">
    <literal><![CDATA[
      (x ^ y) ^ (y ' ^ z) ' = x ^ y
    ]]></literal>
    <justification jstring="[para(1163(a,1),1413(a,1,2,1,1))].">
      <j1 rule="para" parents="1163 1413"/>
    </justification>
  </clause>

  <clause id="1430">
    <literal><![CDATA[
      x ^ (y ^ (x ' ^ z)) ' = x
    ]]></literal>
    <justification jstring="[para(1253(a,1),1413(a,1,2,1))].">
      <j1 rule="para" parents="1253 1413"/>
    </justification>
  </clause>

  <clause id="1436">
    <literal><![CDATA[
      x ' ^ ((y ^ (x ' ^ x)) ' ^ ((x ' ^ x) ^ z) ') ' = y ^ (x ' ^ x)
    ]]></literal>
    <justification jstring="[back_demod(1273),demod(1430(5))].">
      <j1 rule="back_demod" parents="1273"/>
      <j2 rule="demod" parents="1430"/>
    </justification>
  </clause>

  <clause id="1667">
    <literal><![CDATA[
      x ^ ((y ^ (x ^ z)) ' ^ ((x ^ z) ^ u) ') ' = ((y ^ (x ^ z)) ' ^ ((x ^ z) ^ u) ') '
    ]]></literal>
    <justification jstring="[para(1191(a,1),1366(a,1,2)),demod(1191(19))].">
      <j1 rule="para" parents="1191 1366"/>
      <j2 rule="demod" parents="1191"/>
    </justification>
  </clause>

  <clause id="1672">
    <literal><![CDATA[
      ((x ^ (y ' ^ y)) ' ^ ((y ' ^ y) ^ z) ') ' = x ^ (y ' ^ y)
    ]]></literal>
    <justification jstring="[back_demod(1436),demod(1667(12))].">
      <j1 rule="back_demod" parents="1436"/>
      <j2 rule="demod" parents="1667"/>
    </justification>
  </clause>

  <clause id="1697">
    <literal><![CDATA[
      (x ' ^ x) ^ y = x ' ^ x
    ]]></literal>
    <justification jstring="[para(1147(a,1),1193(a,1,2,1,2,1,1,1)),demod(1413(5),966(4),1144(10),1428(12),1147(10),1144(6),1182(5)),flip(a)].">
      <j1 rule="para" parents="1147 1193"/>
      <j2 rule="demod" parents="1413 966 1144 1428 1147 1144 1182"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1731">
    <literal><![CDATA[
      x ^ (y ' ^ y) = y ' ^ y
    ]]></literal>
    <justification jstring="[back_demod(1672),demod(1697(7),1251(8),1144(4)),flip(a)].">
      <j1 rule="back_demod" parents="1672"/>
      <j2 rule="demod" parents="1697 1251 1144"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1739">
    <literal><![CDATA[
      (x ' ^ x) ' ^ y = y
    ]]></literal>
    <justification jstring="[para(1697(a,1),1256(a,1,1,1))].">
      <j1 rule="para" parents="1697 1256"/>
    </justification>
  </clause>

  <clause id="1740">
    <literal><![CDATA[
      x ^ (y ' ^ y) ' = x
    ]]></literal>
    <justification jstring="[para(1697(a,1),1264(a,1,2,1))].">
      <j1 rule="para" parents="1697 1264"/>
    </justification>
  </clause>

  <clause id="1750">
    <literal><![CDATA[
      (x ' ^ x) ' = (y ' ^ y) '
    ]]></literal>
    <justification jstring="[para(1740(a,1),1739(a,1))].">
      <j1 rule="para" parents="1740 1739"/>
    </justification>
  </clause>

  <clause id="1753">
    <literal><![CDATA[
      (x ' ^ x) ' = c_0
    ]]></literal>
    <justification jstring="[new_symbol(1750)].">
      <j1 rule="new_symbol" parents="1750"/>
    </justification>
  </clause>

  <clause id="1794">
    <literal><![CDATA[
      x ' ^ x = c_0 '
    ]]></literal>
    <justification jstring="[para(1697(a,1),1204(a,1,1,1,2,1)),demod(1731(3),1753(3),1753(4),1182(3),1731(5),1731(5),1753(5),1697(6),1753(6),1182(5))].">
      <j1 rule="para" parents="1697 1204"/>
      <j2 rule="demod" parents="1731 1753 1753 1182 1731 1731 1753 1697 1753 1182"/>
    </justification>
  </clause>

  <clause id="1799">
    <literal><![CDATA[
      x ^ x ' = c_0 '
    ]]></literal>
    <justification jstring="[para(1144(a,1),1794(a,1,1))].">
      <j1 rule="para" parents="1144 1794"/>
    </justification>
  </clause>

  <clause id="1800">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(ONE_SS)
    ]]></attribute>
    <justification jstring="[back_demod(1153),demod(1799(4),1144(3),1799(5),1144(4)),xx(a)].">
      <j1 rule="back_demod" parents="1153"/>
      <j2 rule="demod" parents="1799 1144 1799 1144"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

<proof number="3" length="112" max_count="39">

<comments><![CDATA[
% Proof 3 at 3.05 (+ 0.04) seconds: A_SS.
% Length of proof is 112.
% Level of proof is 28.
% Maximum clause weight is 39.
% Given clauses 312.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(y,f(f(x,x),x)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OL_Sh)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

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

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

  <clause id="4">
    <literal><![CDATA[
      x ' = f(x,x)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="5">
    <literal><![CDATA[
      f(c2,f(f(c1,c3),f(c1,c3))) != f(c1,f(f(c2,c3),f(c2,c3)))
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(y,f(f(x,x),x)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OL_Sh)
    ]]></attribute>
    <justification jstring="[copy(1)].">
      <j1 rule="copy" parents="1"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <justification jstring="[copy(2),flip(a)].">
      <j1 rule="copy" parents="2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

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

  <clause id="11">
    <literal><![CDATA[
      f(x,x) = x '
    ]]></literal>
    <justification jstring="[copy(4),flip(a)].">
      <j1 rule="copy" parents="4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      f(x,y) ' = x ^ y
    ]]></literal>
    <justification jstring="[back_demod(10),demod(11(3))].">
      <j1 rule="back_demod" parents="10"/>
      <j2 rule="demod" parents="11"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      f(x ',y ') = x v y
    ]]></literal>
    <justification jstring="[back_demod(9),demod(11(1),11(2))].">
      <j1 rule="back_demod" parents="9"/>
      <j2 rule="demod" parents="11 11"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(y,f(x ',x)),z))) = y
    ]]></literal>
    <justification jstring="[back_demod(8),demod(11(5))].">
      <j1 rule="back_demod" parents="8"/>
      <j2 rule="demod" parents="11"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      f(c2,c1 ^ c3) != f(c1,c2 ^ c3)
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <justification jstring="[copy(5),demod(11(8),12(5),11(13),12(10))].">
      <j1 rule="copy" parents="5"/>
      <j2 rule="demod" parents="11 12 11 12"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      x ^ x = x ' '
    ]]></literal>
    <justification jstring="[para(11(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="11 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      (x v y) ' = x ' ^ y '
    ]]></literal>
    <justification jstring="[para(13(a,1),12(a,1,1))].">
      <j1 rule="para" parents="13 12"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      f(x ',y ^ z) = x v f(y,z)
    ]]></literal>
    <justification jstring="[para(12(a,1),13(a,1,2))].">
      <j1 rule="para" parents="12 13"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      f(f(f(x ',f(x,y)),z),f(x,f(f(x,f(x ',x)),y))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),14(a,1,1,1,1))].">
      <j1 rule="para" parents="11 14"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      f(f(x ' ',y),f(x,f(f(x,f(x ',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),14(a,1,1,1)),demod(11(1))].">
      <j1 rule="para" parents="11 14"/>
      <j2 rule="demod" parents="11"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      f(f(x,y) ^ f(y,z),f(y,f(f(y,f(x ',x)),z))) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),14(a,1,1)),demod(12(4))].">
      <j1 rule="para" parents="11 14"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,f(y,f(x ',x)))),z),f(y,y ^ f(x ',x))) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),14(a,1,2,2)),demod(12(11))].">
      <j1 rule="para" parents="11 14"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      f(f(f(x,y),f(y,z)),u) ^ f(y,f(f(y,f(x ',x)),z)) = y '
    ]]></literal>
    <justification jstring="[para(14(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="14 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(x,f(f(z,x) ^ f(x,u),f(f(z,x),f(x,u)))),f(f(x,f(z ',z)),u)))) = x
    ]]></literal>
    <justification jstring="[para(14(a,1),14(a,1,1,1)),demod(12(5))].">
      <j1 rule="para" parents="14 14"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(f(x,y),f(z ^ x,f(z,x))),u))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(14(a,1),14(a,1,1)),demod(12(4))].">
      <j1 rule="para" parents="14 14"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      f(x ',y ' ^ z ') = x v (y v z)
    ]]></literal>
    <justification jstring="[para(20(a,1),13(a,1,2))].">
      <j1 rule="para" parents="20 13"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      f(x ' ^ y ',z ^ u) = (x v y) v f(z,u)
    ]]></literal>
    <justification jstring="[para(20(a,1),22(a,1,1))].">
      <j1 rule="para" parents="20 22"/>
    </justification>
  </clause>

  <clause id="174">
    <literal><![CDATA[
      f(x ' ' ',f(x,f(f(x,f(x ',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),25(a,1,1))].">
      <j1 rule="para" parents="11 25"/>
    </justification>
  </clause>

  <clause id="183">
    <literal><![CDATA[
      f(x,f(x ',f(f(x ',x ' v x),y))) = x '
    ]]></literal>
    <justification jstring="[para(25(a,1),23(a,1,1)),demod(13(6))].">
      <j1 rule="para" parents="25 23"/>
      <j2 rule="demod" parents="13"/>
    </justification>
  </clause>

  <clause id="187">
    <literal><![CDATA[
      (f(x,y) ^ f(y,z)) ^ f(y,f(f(y,f(x ',x)),z)) = y '
    ]]></literal>
    <justification jstring="[para(26(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="26 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="210">
    <literal><![CDATA[
      x ' ' ' ^ f(x,f(f(x,f(x ',x)),x)) = x '
    ]]></literal>
    <justification jstring="[para(174(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="174 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="223">
    <literal><![CDATA[
      x ^ f(x ',f(f(x ',x ' v x),y)) = x ' '
    ]]></literal>
    <justification jstring="[para(183(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="183 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="229">
    <literal><![CDATA[
      f(x ',f(x ' ',x)) = x ' '
    ]]></literal>
    <justification jstring="[para(25(a,1),183(a,1,2,2))].">
      <j1 rule="para" parents="25 183"/>
    </justification>
  </clause>

  <clause id="231">
    <literal><![CDATA[
      x ' ^ f(x ' ',x) = x ' ' '
    ]]></literal>
    <justification jstring="[para(229(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="229 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="233">
    <literal><![CDATA[
      f(x ' ',x ' ' v x) = x ' ' '
    ]]></literal>
    <justification jstring="[para(13(a,1),229(a,1,2))].">
      <j1 rule="para" parents="13 229"/>
    </justification>
  </clause>

  <clause id="254">
    <literal><![CDATA[
      x ' ' ^ (x ' ' v x) = x ' ' ' '
    ]]></literal>
    <justification jstring="[para(13(a,1),231(a,1,2))].">
      <j1 rule="para" parents="13 231"/>
    </justification>
  </clause>

  <clause id="299">
    <literal><![CDATA[
      f(x,f(f(x,y),f(x,y) ^ (x ' v x))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(23(a,1),28(a,1,1)),demod(13(6))].">
      <j1 rule="para" parents="23 28"/>
      <j2 rule="demod" parents="13"/>
    </justification>
  </clause>

  <clause id="326">
    <literal><![CDATA[
      f(x,y) ^ f(x,f(f(x,f(f(z,x) ^ f(x,u),f(f(z,x),f(x,u)))),f(f(x,f(z ',z)),u))) = x '
    ]]></literal>
    <justification jstring="[para(14(a,1),29(a,1,1,1)),demod(12(5))].">
      <j1 rule="para" parents="14 29"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="394">
    <literal><![CDATA[
      x ^ f(f(x,y),f(x,y) ^ (x ' v x)) = x ^ y
    ]]></literal>
    <justification jstring="[para(299(a,1),12(a,1,1)),demod(12(2)),flip(a)].">
      <j1 rule="para" parents="299 12"/>
      <j2 rule="demod" parents="12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="737">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(y,f(x ',x)),z))) = y
    ]]></literal>
    <justification jstring="[para(34(a,1),14(a,1,1))].">
      <j1 rule="para" parents="34 14"/>
    </justification>
  </clause>

  <clause id="748">
    <literal><![CDATA[
      f(x,y) ^ f(y,f(f(y,f(x ',x)),z)) = y '
    ]]></literal>
    <justification jstring="[para(34(a,1),29(a,1,1))].">
      <j1 rule="para" parents="34 29"/>
    </justification>
  </clause>

  <clause id="749">
    <literal><![CDATA[
      x ^ x ' ' = x ' '
    ]]></literal>
    <justification jstring="[para(34(a,1),223(a,1,2,2)),demod(11(3))].">
      <j1 rule="para" parents="34 223"/>
      <j2 rule="demod" parents="11"/>
    </justification>
  </clause>

  <clause id="755">
    <literal><![CDATA[
      f(x,y) ^ f(x,x ^ ((x ^ y) v f(x,y))) = x '
    ]]></literal>
    <justification jstring="[para(34(a,1),394(a,1,2,1)),demod(34(17),12(3),326(23))].">
      <j1 rule="para" parents="34 394"/>
      <j2 rule="demod" parents="34 12 326"/>
    </justification>
  </clause>

  <clause id="838">
    <literal><![CDATA[
      f(x ',f(f(x ',f(x,y)),x)) = f(x ',f(x,y))
    ]]></literal>
    <justification jstring="[para(23(a,1),35(a,1,2,2))].">
      <j1 rule="para" parents="23 35"/>
    </justification>
  </clause>

  <clause id="864">
    <literal><![CDATA[
      f(x,x ^ y) = f(x,y)
    ]]></literal>
    <justification jstring="[para(34(a,1),35(a,1,2,2)),demod(11(3),12(2))].">
      <j1 rule="para" parents="34 35"/>
      <j2 rule="demod" parents="11 12"/>
    </justification>
  </clause>

  <clause id="879">
    <literal><![CDATA[
      f(x,y) ^ f(x,(x ^ y) v f(x,y)) = x '
    ]]></literal>
    <justification jstring="[back_demod(755),demod(864(6))].">
      <j1 rule="back_demod" parents="755"/>
      <j2 rule="demod" parents="864"/>
    </justification>
  </clause>

  <clause id="966">
    <literal><![CDATA[
      x ^ (x ^ y) = x ^ y
    ]]></literal>
    <justification jstring="[para(864(a,1),12(a,1,1)),demod(12(2)),flip(a)].">
      <j1 rule="para" parents="864 12"/>
      <j2 rule="demod" parents="12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="970">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),y ') = y
    ]]></literal>
    <justification jstring="[para(14(a,1),864(a,2)),demod(29(14))].">
      <j1 rule="para" parents="14 864"/>
      <j2 rule="demod" parents="29"/>
    </justification>
  </clause>

  <clause id="985">
    <literal><![CDATA[
      x ' ' v x = x
    ]]></literal>
    <justification jstring="[para(174(a,1),864(a,2)),demod(210(12),13(5))].">
      <j1 rule="para" parents="174 864"/>
      <j2 rule="demod" parents="210 13"/>
    </justification>
  </clause>

  <clause id="990">
    <literal><![CDATA[
      f(x ' ',x) = x ' ' '
    ]]></literal>
    <justification jstring="[para(233(a,1),864(a,2)),demod(985(7),864(6))].">
      <j1 rule="para" parents="233 864"/>
      <j2 rule="demod" parents="985 864"/>
    </justification>
  </clause>

  <clause id="1011">
    <literal><![CDATA[
      f(f(x,y),x ') = x
    ]]></literal>
    <justification jstring="[para(34(a,1),864(a,2)),demod(326(17))].">
      <j1 rule="para" parents="34 864"/>
      <j2 rule="demod" parents="326"/>
    </justification>
  </clause>

  <clause id="1041">
    <literal><![CDATA[
      x ' ' ^ x = x ' ' ' '
    ]]></literal>
    <justification jstring="[back_demod(254),demod(985(5))].">
      <j1 rule="back_demod" parents="254"/>
      <j2 rule="demod" parents="985"/>
    </justification>
  </clause>

  <clause id="1053">
    <literal><![CDATA[
      x ' ' ' ' ' = x '
    ]]></literal>
    <justification jstring="[para(985(a,1),20(a,1,1)),demod(1041(6)),flip(a)].">
      <j1 rule="para" parents="985 20"/>
      <j2 rule="demod" parents="1041"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1055">
    <literal><![CDATA[
      f(x,y) = (x ^ y) ' ' '
    ]]></literal>
    <justification jstring="[para(985(a,1),22(a,2)),demod(12(2),990(5)),flip(a)].">
      <j1 rule="para" parents="985 22"/>
      <j2 rule="demod" parents="12 990"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1056">
    <literal><![CDATA[
      x v y = (x ' ^ y ') ' ' '
    ]]></literal>
    <justification jstring="[para(985(a,1),36(a,2,2)),demod(1041(6),1053(6),1055(3)),flip(a)].">
      <j1 rule="para" parents="985 36"/>
      <j2 rule="demod" parents="1041 1053 1055"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1058">
    <literal><![CDATA[
      (x ' ^ (y ^ z) ' ' ' ') ' ' ' = (x ' ^ (y ^ z)) ' ' '
    ]]></literal>
    <justification jstring="[para(985(a,1),45(a,2,1)),demod(1041(5),1053(5),1055(3),1055(7),1056(11)),flip(a)].">
      <j1 rule="para" parents="985 45"/>
      <j2 rule="demod" parents="1041 1053 1055 1055 1056"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1070">
    <literal><![CDATA[
      ((x ^ y) ' ' ' ^ x ') ' ' ' = x
    ]]></literal>
    <justification jstring="[back_demod(1011),demod(1055(1),1055(6))].">
      <j1 rule="back_demod" parents="1011"/>
      <j2 rule="demod" parents="1055 1055"/>
    </justification>
  </clause>

  <clause id="1085">
    <literal><![CDATA[
      ((((x ^ y) ' ' ' ^ (y ^ z) ' ' ') ' ' ' ^ u) ' ' ' ^ y ') ' ' ' = y
    ]]></literal>
    <justification jstring="[back_demod(970),demod(1055(1),1055(5),1055(9),1055(13),1055(18))].">
      <j1 rule="back_demod" parents="970"/>
      <j2 rule="demod" parents="1055 1055 1055 1055 1055"/>
    </justification>
  </clause>

  <clause id="1097">
    <literal><![CDATA[
      (x ^ y) ' ' ' ^ (x ^ ((x ^ y) ' ^ (x ^ y)) ' ' ') ' ' ' = x '
    ]]></literal>
    <justification jstring="[back_demod(879),demod(1055(1),1055(6),1056(10),1058(15),1055(12))].">
      <j1 rule="back_demod" parents="879"/>
      <j2 rule="demod" parents="1055 1055 1056 1058 1055"/>
    </justification>
  </clause>

  <clause id="1102">
    <literal><![CDATA[
      (x ' ^ ((x ' ^ (x ^ y) ' ' ') ' ' ' ^ x) ' ' ') ' ' ' = (x ' ^ (x ^ y) ' ' ') ' ' '
    ]]></literal>
    <justification jstring="[back_demod(838),demod(1055(3),1055(7),1055(11),1055(15),1055(20),1055(24))].">
      <j1 rule="back_demod" parents="838"/>
      <j2 rule="demod" parents="1055 1055 1055 1055 1055 1055"/>
    </justification>
  </clause>

  <clause id="1121">
    <literal><![CDATA[
      (x ^ y) ' ' ' ^ (y ^ ((y ^ (x ' ^ x) ' ' ') ' ' ' ^ z) ' ' ') ' ' ' = y '
    ]]></literal>
    <justification jstring="[back_demod(748),demod(1055(1),1055(6),1055(10),1055(14),1055(18))].">
      <j1 rule="back_demod" parents="748"/>
      <j2 rule="demod" parents="1055 1055 1055 1055 1055"/>
    </justification>
  </clause>

  <clause id="1123">
    <literal><![CDATA[
      x ' ' ' ' = x
    ]]></literal>
    <justification jstring="[back_demod(737),demod(1055(1),1055(6),1055(10),1055(14),1055(18),1055(22),1121(22))].">
      <j1 rule="back_demod" parents="737"/>
      <j2 rule="demod" parents="1055 1055 1055 1055 1055 1055 1121"/>
    </justification>
  </clause>

  <clause id="1134">
    <literal><![CDATA[
      ((x ^ y) ' ' ' ^ (y ^ z) ' ' ') ^ (y ^ ((y ^ (x ' ^ x) ' ' ') ' ' ' ^ z) ' ' ') ' ' ' = y '
    ]]></literal>
    <justification jstring="[back_demod(187),demod(1055(1),1055(5),1055(11),1055(15),1055(19),1055(23))].">
      <j1 rule="back_demod" parents="187"/>
      <j2 rule="demod" parents="1055 1055 1055 1055 1055 1055"/>
    </justification>
  </clause>

  <clause id="1138">
    <literal><![CDATA[
      (c2 ^ (c1 ^ c3)) ' ' ' != (c1 ^ (c2 ^ c3)) ' ' '
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <justification jstring="[back_demod(15),demod(1055(5),1055(13))].">
      <j1 rule="back_demod" parents="15"/>
      <j2 rule="demod" parents="1055 1055"/>
    </justification>
  </clause>

  <clause id="1144">
    <literal><![CDATA[
      x ' ' = x
    ]]></literal>
    <justification jstring="[para(18(a,1),1070(a,1,1,1,1,1,1,1,1)),demod(1123(4),18(3),1123(4))].">
      <j1 rule="para" parents="18 1070"/>
      <j2 rule="demod" parents="1123 18 1123"/>
    </justification>
  </clause>

  <clause id="1147">
    <literal><![CDATA[
      (x ^ y) ' ^ x ' = x '
    ]]></literal>
    <justification jstring="[para(1070(a,1),749(a,2,1)),demod(1144(3),1144(6),1144(7),1144(10),1144(10),18(9),1144(6))].">
      <j1 rule="para" parents="1070 749"/>
      <j2 rule="demod" parents="1144 1144 1144 1144 1144 18 1144"/>
    </justification>
  </clause>

  <clause id="1148">
    <literal><![CDATA[
      ((x ' ^ y) ' ^ x) ' = x '
    ]]></literal>
    <justification jstring="[para(1123(a,1),1070(a,1,1,1,1,2)),demod(1144(2),1144(4),1144(6),1144(7))].">
      <j1 rule="para" parents="1123 1070"/>
      <j2 rule="demod" parents="1144 1144 1144 1144"/>
    </justification>
  </clause>

  <clause id="1151">
    <literal><![CDATA[
      (c2 ^ (c1 ^ c3)) ' != (c1 ^ (c2 ^ c3)) '
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <justification jstring="[back_demod(1138),demod(1144(7),1144(13))].">
      <j1 rule="back_demod" parents="1138"/>
      <j2 rule="demod" parents="1144 1144"/>
    </justification>
  </clause>

  <clause id="1155">
    <literal><![CDATA[
      ((x ^ y) ' ^ (y ^ z) ') ^ (y ^ ((y ^ (x ' ^ x) ') ' ^ z) ') ' = y '
    ]]></literal>
    <justification jstring="[back_demod(1134),demod(1144(3),1144(5),1144(9),1144(11),1144(13),1144(15))].">
      <j1 rule="back_demod" parents="1134"/>
      <j2 rule="demod" parents="1144 1144 1144 1144 1144 1144"/>
    </justification>
  </clause>

  <clause id="1163">
    <literal><![CDATA[
      (x ^ y) ' ^ (y ^ ((y ^ (x ' ^ x) ') ' ^ z) ') ' = y '
    ]]></literal>
    <justification jstring="[back_demod(1121),demod(1144(3),1144(6),1144(8),1144(10),1144(12))].">
      <j1 rule="back_demod" parents="1121"/>
      <j2 rule="demod" parents="1144 1144 1144 1144 1144"/>
    </justification>
  </clause>

  <clause id="1164">
    <literal><![CDATA[
      (x ' ^ (x ^ y) ') ' = x
    ]]></literal>
    <justification jstring="[back_demod(1102),demod(1144(5),1144(7),1148(8),1144(3),18(3),1144(2),1144(2),1144(2),1144(4),1144(6)),flip(a)].">
      <j1 rule="back_demod" parents="1102"/>
      <j2 rule="demod" parents="1144 1144 1148 1144 18 1144 1144 1144 1144 1144"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1170">
    <literal><![CDATA[
      (x ^ y) ' ^ (x ^ ((x ^ y) ' ^ (x ^ y)) ') ' = x '
    ]]></literal>
    <justification jstring="[back_demod(1097),demod(1144(3),1144(8),1144(10))].">
      <j1 rule="back_demod" parents="1097"/>
      <j2 rule="demod" parents="1144 1144 1144"/>
    </justification>
  </clause>

  <clause id="1174">
    <literal><![CDATA[
      ((((x ^ y) ' ^ (y ^ z) ') ' ^ u) ' ^ y ') ' = y
    ]]></literal>
    <justification jstring="[back_demod(1085),demod(1144(3),1144(5),1144(7),1144(9),1144(12))].">
      <j1 rule="back_demod" parents="1085"/>
      <j2 rule="demod" parents="1144 1144 1144 1144 1144"/>
    </justification>
  </clause>

  <clause id="1182">
    <literal><![CDATA[
      x ^ x = x
    ]]></literal>
    <justification jstring="[back_demod(18),demod(1144(3))].">
      <j1 rule="back_demod" parents="18"/>
      <j2 rule="demod" parents="1144"/>
    </justification>
  </clause>

  <clause id="1184">
    <literal><![CDATA[
      x ' ^ (x ^ y) ' = x '
    ]]></literal>
    <justification jstring="[para(1164(a,1),1144(a,1,1)),flip(a)].">
      <j1 rule="para" parents="1164 1144"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1191">
    <literal><![CDATA[
      x ^ ((y ^ x) ' ^ (x ^ z) ') ' = ((y ^ x) ' ^ (x ^ z) ') '
    ]]></literal>
    <justification jstring="[para(1155(a,1),1147(a,1,1,1)),demod(1144(2))].">
      <j1 rule="para" parents="1155 1147"/>
      <j2 rule="demod" parents="1144"/>
    </justification>
  </clause>

  <clause id="1193">
    <literal><![CDATA[
      ((x ^ (y ^ z) ') ' ^ y) ^ ((y ^ z) ' ^ (((y ^ z) ' ^ (x ' ^ x) ') ' ^ y ') ') ' = y ^ z
    ]]></literal>
    <justification jstring="[para(1147(a,1),1155(a,1,1,2,1)),demod(1144(6),1144(23))].">
      <j1 rule="para" parents="1147 1155"/>
      <j2 rule="demod" parents="1144 1144"/>
    </justification>
  </clause>

  <clause id="1199">
    <literal><![CDATA[
      x ^ (x ' ^ y) ' = x
    ]]></literal>
    <justification jstring="[para(1144(a,1),1184(a,1,1)),demod(1144(6))].">
      <j1 rule="para" parents="1144 1184"/>
      <j2 rule="demod" parents="1144"/>
    </justification>
  </clause>

  <clause id="1200">
    <literal><![CDATA[
      (x ^ y) ^ x = x ^ y
    ]]></literal>
    <justification jstring="[para(1147(a,1),1184(a,1,2,1)),demod(1144(3),1144(3),1144(5))].">
      <j1 rule="para" parents="1147 1184"/>
      <j2 rule="demod" parents="1144 1144 1144"/>
    </justification>
  </clause>

  <clause id="1209">
    <literal><![CDATA[
      x ' ^ ((y ^ x) ' ^ (x ^ z) ') = x '
    ]]></literal>
    <justification jstring="[para(1155(a,1),1200(a,1,1)),demod(1155(22))].">
      <j1 rule="para" parents="1155 1200"/>
      <j2 rule="demod" parents="1155"/>
    </justification>
  </clause>

  <clause id="1253">
    <literal><![CDATA[
      x ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(1163(a,1),1147(a,1,1,1)),demod(1144(2),1144(3),1144(5))].">
      <j1 rule="para" parents="1163 1147"/>
      <j2 rule="demod" parents="1144 1144 1144"/>
    </justification>
  </clause>

  <clause id="1258">
    <literal><![CDATA[
      x ' ^ (y ^ x) ' = x '
    ]]></literal>
    <justification jstring="[para(1163(a,1),1200(a,1,1)),demod(1163(16))].">
      <j1 rule="para" parents="1163 1200"/>
      <j2 rule="demod" parents="1163"/>
    </justification>
  </clause>

  <clause id="1264">
    <literal><![CDATA[
      x ^ (y ^ x ') ' = x
    ]]></literal>
    <justification jstring="[para(1253(a,1),1199(a,1,2,1))].">
      <j1 rule="para" parents="1253 1199"/>
    </justification>
  </clause>

  <clause id="1268">
    <literal><![CDATA[
      x ' ^ ((y ^ x) ' ^ (z ^ x) ') = x '
    ]]></literal>
    <justification jstring="[para(1253(a,1),1209(a,1,2,2,1))].">
      <j1 rule="para" parents="1253 1209"/>
    </justification>
  </clause>

  <clause id="1310">
    <literal><![CDATA[
      ((x ^ y) ' ^ (y ^ y ') ') ' = x ^ y
    ]]></literal>
    <justification jstring="[para(1163(a,1),1170(a,1,1,1)),demod(1144(2),1163(14),1144(4),1163(14),1191(8),1144(10))].">
      <j1 rule="para" parents="1163 1170"/>
      <j2 rule="demod" parents="1144 1163 1144 1163 1191 1144"/>
    </justification>
  </clause>

  <clause id="1336">
    <literal><![CDATA[
      (x ^ y) ^ (x ^ (z ^ (x ^ y) ') ') = x ^ y
    ]]></literal>
    <justification jstring="[para(1184(a,1),1268(a,1,2,1,1)),demod(1144(3),1144(3),1144(10))].">
      <j1 rule="para" parents="1184 1268"/>
      <j2 rule="demod" parents="1144 1144 1144"/>
    </justification>
  </clause>

  <clause id="1347">
    <literal><![CDATA[
      (((x ^ y) ^ z) ' ^ x ') ' = x
    ]]></literal>
    <justification jstring="[para(1147(a,1),1174(a,1,1,1,1,1,1)),demod(1144(3))].">
      <j1 rule="para" parents="1147 1174"/>
      <j2 rule="demod" parents="1144"/>
    </justification>
  </clause>

  <clause id="1364">
    <literal><![CDATA[
      ((x ^ y) ^ z) ' ^ x ' = x '
    ]]></literal>
    <justification jstring="[para(1347(a,1),1144(a,1,1)),flip(a)].">
      <j1 rule="para" parents="1347 1144"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1399">
    <literal><![CDATA[
      ((x ' ^ y) ^ z) ' ^ x = x
    ]]></literal>
    <justification jstring="[para(1144(a,1),1364(a,1,2)),demod(1144(7))].">
      <j1 rule="para" parents="1144 1364"/>
      <j2 rule="demod" parents="1144"/>
    </justification>
  </clause>

  <clause id="1413">
    <literal><![CDATA[
      x ^ ((x ' ^ y) ^ z) ' = x
    ]]></literal>
    <justification jstring="[para(1399(a,1),1200(a,1,1)),demod(1399(10))].">
      <j1 rule="para" parents="1399 1200"/>
      <j2 rule="demod" parents="1399"/>
    </justification>
  </clause>

  <clause id="1428">
    <literal><![CDATA[
      (x ^ y) ^ (y ' ^ z) ' = x ^ y
    ]]></literal>
    <justification jstring="[para(1163(a,1),1413(a,1,2,1,1))].">
      <j1 rule="para" parents="1163 1413"/>
    </justification>
  </clause>

  <clause id="1429">
    <literal><![CDATA[
      x ^ ((y ^ x ') ^ z) ' = x
    ]]></literal>
    <justification jstring="[para(1253(a,1),1413(a,1,2,1,1))].">
      <j1 rule="para" parents="1253 1413"/>
    </justification>
  </clause>

  <clause id="1430">
    <literal><![CDATA[
      x ^ (y ^ (x ' ^ z)) ' = x
    ]]></literal>
    <justification jstring="[para(1253(a,1),1413(a,1,2,1))].">
      <j1 rule="para" parents="1253 1413"/>
    </justification>
  </clause>

  <clause id="1466">
    <literal><![CDATA[
      x ^ (y ^ (z ^ x ')) ' = x
    ]]></literal>
    <justification jstring="[para(1253(a,1),1429(a,1,2,1))].">
      <j1 rule="para" parents="1253 1429"/>
    </justification>
  </clause>

  <clause id="1471">
    <literal><![CDATA[
      x ' ^ (y ^ (x ^ z)) ' = x '
    ]]></literal>
    <justification jstring="[para(1144(a,1),1430(a,1,2,1,2,1))].">
      <j1 rule="para" parents="1144 1430"/>
    </justification>
  </clause>

  <clause id="1499">
    <literal><![CDATA[
      x ' ^ (y ^ (z ^ x)) ' = x '
    ]]></literal>
    <justification jstring="[para(1144(a,1),1466(a,1,2,1,2,2))].">
      <j1 rule="para" parents="1144 1466"/>
    </justification>
  </clause>

  <clause id="1697">
    <literal><![CDATA[
      (x ' ^ x) ^ y = x ' ^ x
    ]]></literal>
    <justification jstring="[para(1147(a,1),1193(a,1,2,1,2,1,1,1)),demod(1413(5),966(4),1144(10),1428(12),1147(10),1144(6),1182(5)),flip(a)].">
      <j1 rule="para" parents="1147 1193"/>
      <j2 rule="demod" parents="1413 966 1144 1428 1147 1144 1182"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1703">
    <literal><![CDATA[
      (x ^ y) ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(1258(a,1),1193(a,1,1,1,1)),demod(1144(2),1144(7),1310(10),1184(8),1144(4))].">
      <j1 rule="para" parents="1258 1193"/>
      <j2 rule="demod" parents="1144 1144 1310 1184 1144"/>
    </justification>
  </clause>

  <clause id="1726">
    <literal><![CDATA[
      (x ^ y) ^ ((y ^ (x ^ z)) ' ^ (((y ^ (x ^ z)) ' ^ (x ^ x ') ') ' ^ y ') ') ' = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(1471(a,1),1193(a,1,1,1,1)),demod(1144(2),1144(9))].">
      <j1 rule="para" parents="1471 1193"/>
      <j2 rule="demod" parents="1144 1144"/>
    </justification>
  </clause>

  <clause id="1728">
    <literal><![CDATA[
      (x ^ y) ^ ((y ^ (z ^ x)) ' ^ (((y ^ (z ^ x)) ' ^ (x ^ x ') ') ' ^ y ') ') ' = y ^ (z ^ x)
    ]]></literal>
    <justification jstring="[para(1499(a,1),1193(a,1,1,1,1)),demod(1144(2),1144(9))].">
      <j1 rule="para" parents="1499 1193"/>
      <j2 rule="demod" parents="1144 1144"/>
    </justification>
  </clause>

  <clause id="1740">
    <literal><![CDATA[
      x ^ (y ' ^ y) ' = x
    ]]></literal>
    <justification jstring="[para(1697(a,1),1264(a,1,2,1))].">
      <j1 rule="para" parents="1697 1264"/>
    </justification>
  </clause>

  <clause id="1749">
    <literal><![CDATA[
      x ^ (y ^ y ') ' = x
    ]]></literal>
    <justification jstring="[para(1144(a,1),1740(a,1,2,1,1))].">
      <j1 rule="para" parents="1144 1740"/>
    </justification>
  </clause>

  <clause id="1751">
    <literal><![CDATA[
      (x ^ y) ^ (y ^ (z ^ x)) = y ^ (z ^ x)
    ]]></literal>
    <justification jstring="[back_demod(1728),demod(1749(11),1144(8),1184(10),1144(5))].">
      <j1 rule="back_demod" parents="1728"/>
      <j2 rule="demod" parents="1749 1144 1184 1144"/>
    </justification>
  </clause>

  <clause id="1752">
    <literal><![CDATA[
      (x ^ y) ^ (y ^ (x ^ z)) = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[back_demod(1726),demod(1749(11),1144(8),1184(10),1144(5))].">
      <j1 rule="back_demod" parents="1726"/>
      <j2 rule="demod" parents="1749 1144 1184 1144"/>
    </justification>
  </clause>

  <clause id="1812">
    <literal><![CDATA[
      x ^ y = y ^ x
    ]]></literal>
    <justification jstring="[para(1703(a,1),1200(a,1,1)),demod(1703(3),1703(4))].">
      <j1 rule="para" parents="1703 1200"/>
      <j2 rule="demod" parents="1703 1703"/>
    </justification>
  </clause>

  <clause id="2690">
    <literal><![CDATA[
      (x ^ y) ^ (x ^ (z ^ y)) = x ^ (z ^ y)
    ]]></literal>
    <justification jstring="[para(1499(a,1),1336(a,1,2,2,1)),demod(1144(4),1812(4))].">
      <j1 rule="para" parents="1499 1336"/>
      <j2 rule="demod" parents="1144 1812"/>
    </justification>
  </clause>

  <clause id="5056">
    <literal><![CDATA[
      (x ^ (y ^ z)) ^ (z ^ x) = x ^ (y ^ z)
    ]]></literal>
    <justification jstring="[para(1751(a,1),1703(a,1,2)),demod(1812(7),966(7),1751(8))].">
      <j1 rule="para" parents="1751 1703"/>
      <j2 rule="demod" parents="1812 966 1751"/>
    </justification>
  </clause>

  <clause id="5187">
    <literal><![CDATA[
      (x ^ y) ^ (y ^ z) = z ^ (x ^ y)
    ]]></literal>
    <justification jstring="[para(1751(a,1),1751(a,1,2)),demod(5056(6),1751(7))].">
      <j1 rule="para" parents="1751 1751"/>
      <j2 rule="demod" parents="5056 1751"/>
    </justification>
  </clause>

  <clause id="5481">
    <literal><![CDATA[
      (x ^ y) ^ (y ^ z) = (x ^ y) ^ z
    ]]></literal>
    <justification jstring="[para(5187(a,2),1703(a,1,1)),demod(1812(6),2690(6))].">
      <j1 rule="para" parents="5187 1703"/>
      <j2 rule="demod" parents="1812 2690"/>
    </justification>
  </clause>

  <clause id="5483">
    <literal><![CDATA[
      (x ^ y) ^ (x ^ z) = z ^ (y ^ x)
    ]]></literal>
    <justification jstring="[para(1812(a,1),5187(a,1,1))].">
      <j1 rule="para" parents="1812 5187"/>
    </justification>
  </clause>

  <clause id="5584">
    <literal><![CDATA[
      (x ^ y) ^ (x ^ z) = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[back_demod(1752),demod(5481(4))].">
      <j1 rule="back_demod" parents="1752"/>
      <j2 rule="demod" parents="5481"/>
    </justification>
  </clause>

  <clause id="5585">
    <literal><![CDATA[
      x ^ (y ^ z) = z ^ (x ^ y)
    ]]></literal>
    <justification jstring="[back_demod(5483),demod(5584(3))].">
      <j1 rule="back_demod" parents="5483"/>
      <j2 rule="demod" parents="5584"/>
    </justification>
  </clause>

  <clause id="5721">
    <literal><![CDATA[
      x ^ (y ^ z) = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(1812(a,1),5585(a,1,2))].">
      <j1 rule="para" parents="1812 5585"/>
    </justification>
  </clause>

  <clause id="6489">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <justification jstring="[back_demod(1151),demod(5721(5)),xx(a)].">
      <j1 rule="back_demod" parents="1151"/>
      <j2 rule="demod" parents="5721"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
