<?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>omlsax2.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13581 was started by mccune on cleo.thornwood,
Mon Jun 19 16:42:56 2006
The command was "/home/mccune/bin/prover9 -f omlsax2.in".
]]></heading>

<proof number="1" length="66" max_count="38">

<comments><![CDATA[
% Proof 1 at 0.10 (+ 0.01) seconds: B_SS.
% Length of proof is 66.
% Level of proof is 19.
% Maximum clause weight is 38.
% Given clauses 69.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_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(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_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(z,f(y ',z)),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="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="23">
    <literal><![CDATA[
      f(f(f(x ',f(x,y)),z),f(x,f(f(y,f(x ',y)),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="27">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,y ')),z),f(y,f(y v y ',y '))) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),14(a,1,2,2,1,2)),demod(13(9))].">
      <j1 rule="para" parents="11 14"/>
      <j2 rule="demod" parents="13"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(f(f(z,f(x ',z)),z),f(x ',f(f(z,f(x ',z)),z))),f(f(z,f(x ',z)),z)))) = x
    ]]></literal>
    <justification jstring="[para(14(a,1),14(a,1,1,1))].">
      <j1 rule="para" parents="14 14"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(z,f(x ^ y,z)),z))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(14(a,1),14(a,1,1)),demod(12(3))].">
      <j1 rule="para" parents="14 14"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      f(x ' ^ f(x,y),f(x,f(f(y,f(x ',y)),y))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),23(a,1,1)),demod(12(4))].">
      <j1 rule="para" parents="11 23"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="53">
    <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="54">
    <literal><![CDATA[
      f(x ' ',y) ^ f(x,f(f(x,f(x ',x)),x)) = x '
    ]]></literal>
    <justification jstring="[para(25(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="25 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="56">
    <literal><![CDATA[
      f(x ' v y,f(x,f(f(x,f(x ',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(13(a,1),25(a,1,1))].">
      <j1 rule="para" parents="13 25"/>
    </justification>
  </clause>

  <clause id="60">
    <literal><![CDATA[
      f(x,f(x ',f(f(y,f(x ' ',y)),y))) = x '
    ]]></literal>
    <justification jstring="[para(25(a,1),23(a,1,1))].">
      <j1 rule="para" parents="25 23"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      x ' ' ' ^ f(x,f(f(x,f(x ',x)),x)) = x '
    ]]></literal>
    <justification jstring="[para(53(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="53 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      (x ' v y) ^ f(x,f(f(x,f(x ',x)),x)) = x '
    ]]></literal>
    <justification jstring="[para(56(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="56 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="73">
    <literal><![CDATA[
      x ^ f(x ',f(f(y,f(x ' ',y)),y)) = x ' '
    ]]></literal>
    <justification jstring="[para(60(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="60 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="119">
    <literal><![CDATA[
      x ^ f(f(x,y),f(f(z,f(x ^ y,z)),z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(34(a,1),12(a,1,1)),demod(12(2)),flip(a)].">
      <j1 rule="para" parents="34 12"/>
      <j2 rule="demod" parents="12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="128">
    <literal><![CDATA[
      f(f(x ' ',y),f(x,f(f(z,f(x ',z)),z))) = x
    ]]></literal>
    <justification jstring="[para(25(a,1),34(a,1,2,1)),demod(54(12),25(18))].">
      <j1 rule="para" parents="25 34"/>
      <j2 rule="demod" parents="54 25"/>
    </justification>
  </clause>

  <clause id="129">
    <literal><![CDATA[
      f(x ' ' ',f(x,f(f(y,f(x ',y)),y))) = x
    ]]></literal>
    <justification jstring="[para(53(a,1),34(a,1,2,1)),demod(61(12),53(18))].">
      <j1 rule="para" parents="53 34"/>
      <j2 rule="demod" parents="61 53"/>
    </justification>
  </clause>

  <clause id="130">
    <literal><![CDATA[
      f(x ' v y,f(x,f(f(z,f(x ',z)),z))) = x
    ]]></literal>
    <justification jstring="[para(56(a,1),34(a,1,2,1)),demod(67(10),56(16))].">
      <j1 rule="para" parents="56 34"/>
      <j2 rule="demod" parents="67 56"/>
    </justification>
  </clause>

  <clause id="133">
    <literal><![CDATA[
      f(x ' ' ',f(x,f(x v x ',x '))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),129(a,1,2,2,1,2)),demod(13(7))].">
      <j1 rule="para" parents="11 129"/>
      <j2 rule="demod" parents="13"/>
    </justification>
  </clause>

  <clause id="134">
    <literal><![CDATA[
      x ' ' ' ^ f(x,f(f(y,f(x ',y)),y)) = x '
    ]]></literal>
    <justification jstring="[para(129(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="129 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="146">
    <literal><![CDATA[
      x ' ^ f(x ' ',f(f(f(x,f(x v x ',x ')),x),f(x,f(x v x ',x ')))) = x ' ' '
    ]]></literal>
    <justification jstring="[para(133(a,1),73(a,1,2,2,1,2))].">
      <j1 rule="para" parents="133 73"/>
    </justification>
  </clause>

  <clause id="183">
    <literal><![CDATA[
      f(f(x ' ',y),f(x,f(f(z ',x v z),z '))) = x
    ]]></literal>
    <justification jstring="[para(13(a,1),128(a,1,2,2,1,2))].">
      <j1 rule="para" parents="13 128"/>
    </justification>
  </clause>

  <clause id="218">
    <literal><![CDATA[
      (x ' ^ f(x,y)) ^ f(x,f(f(y,f(x ',y)),y)) = x '
    ]]></literal>
    <justification jstring="[para(42(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="42 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="227">
    <literal><![CDATA[
      f(x ' ^ f(x,y),f(x,f(f(z,f(x ',z)),z))) = x
    ]]></literal>
    <justification jstring="[para(42(a,1),34(a,1,2,1)),demod(218(12),42(18))].">
      <j1 rule="para" parents="42 34"/>
      <j2 rule="demod" parents="218 42"/>
    </justification>
  </clause>

  <clause id="285">
    <literal><![CDATA[
      f(x ' ^ f(x,y),f(x,f(x v x ',x '))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),227(a,1,2,2,1,2)),demod(13(7))].">
      <j1 rule="para" parents="11 227"/>
      <j2 rule="demod" parents="13"/>
    </justification>
  </clause>

  <clause id="305">
    <literal><![CDATA[
      f(x ',f(f(x ',f(x,y)),f(f(f(x,f(x v x ',x ')),x),f(x,f(x v x ',x '))))) = f(x ',f(x,y))
    ]]></literal>
    <justification jstring="[para(285(a,1),34(a,1,2,2,1,2))].">
      <j1 rule="para" parents="285 34"/>
    </justification>
  </clause>

  <clause id="309">
    <literal><![CDATA[
      x ' ^ f(f(x ',f(x,y)),f(f(f(x,f(x v x ',x ')),x),f(x,f(x v x ',x ')))) = x ' ^ f(x,y)
    ]]></literal>
    <justification jstring="[para(285(a,1),119(a,1,2,2,1,2))].">
      <j1 rule="para" parents="285 119"/>
    </justification>
  </clause>

  <clause id="325">
    <literal><![CDATA[
      f(f(x,y),f(y,f(y v y ',y '))) = y
    ]]></literal>
    <justification jstring="[para(33(a,1),27(a,1,1))].">
      <j1 rule="para" parents="33 27"/>
    </justification>
  </clause>

  <clause id="332">
    <literal><![CDATA[
      x ' ^ f(f(x ',f(x,y)),x) = x ' ^ f(x,y)
    ]]></literal>
    <justification jstring="[back_demod(309),demod(325(16))].">
      <j1 rule="back_demod" parents="309"/>
      <j2 rule="demod" parents="325"/>
    </justification>
  </clause>

  <clause id="333">
    <literal><![CDATA[
      f(x ',f(f(x ',f(x,y)),x)) = f(x ',f(x,y))
    ]]></literal>
    <justification jstring="[back_demod(305),demod(325(16))].">
      <j1 rule="back_demod" parents="305"/>
      <j2 rule="demod" parents="325"/>
    </justification>
  </clause>

  <clause id="341">
    <literal><![CDATA[
      x ' ^ f(x ' ',x) = x ' ' '
    ]]></literal>
    <justification jstring="[back_demod(146),demod(325(15))].">
      <j1 rule="back_demod" parents="146"/>
      <j2 rule="demod" parents="325"/>
    </justification>
  </clause>

  <clause id="360">
    <literal><![CDATA[
      x ' ' ^ (x ' ' v x) = x ' ' ' '
    ]]></literal>
    <justification jstring="[para(13(a,1),341(a,1,2))].">
      <j1 rule="para" parents="13 341"/>
    </justification>
  </clause>

  <clause id="373">
    <literal><![CDATA[
      f(x ',f(x,f(x v x ',x '))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),325(a,1,1))].">
      <j1 rule="para" parents="11 325"/>
    </justification>
  </clause>

  <clause id="393">
    <literal><![CDATA[
      f(x,x ' ') = x '
    ]]></literal>
    <justification jstring="[para(373(a,1),60(a,1,2,2,1,2)),demod(325(22),11(3))].">
      <j1 rule="para" parents="373 60"/>
      <j2 rule="demod" parents="325 11"/>
    </justification>
  </clause>

  <clause id="394">
    <literal><![CDATA[
      x ^ x ' ' = x ' '
    ]]></literal>
    <justification jstring="[para(373(a,1),73(a,1,2,2,1,2)),demod(325(22),11(3))].">
      <j1 rule="para" parents="373 73"/>
      <j2 rule="demod" parents="325 11"/>
    </justification>
  </clause>

  <clause id="396">
    <literal><![CDATA[
      x ' ' v x = x
    ]]></literal>
    <justification jstring="[para(373(a,1),129(a,1,2,2,1,2)),demod(325(15),11(4),13(5))].">
      <j1 rule="para" parents="373 129"/>
      <j2 rule="demod" parents="325 11 13"/>
    </justification>
  </clause>

  <clause id="397">
    <literal><![CDATA[
      f(x ' v y,x ') = x
    ]]></literal>
    <justification jstring="[para(373(a,1),130(a,1,2,2,1,2)),demod(325(14),11(3))].">
      <j1 rule="para" parents="373 130"/>
      <j2 rule="demod" parents="325 11"/>
    </justification>
  </clause>

  <clause id="399">
    <literal><![CDATA[
      x ' ' ' ^ x ' = x '
    ]]></literal>
    <justification jstring="[para(373(a,1),134(a,1,2,2,1,2)),demod(325(15),11(4))].">
      <j1 rule="para" parents="373 134"/>
      <j2 rule="demod" parents="325 11"/>
    </justification>
  </clause>

  <clause id="403">
    <literal><![CDATA[
      f(x ',f(x,f(f(y ',x v y),y '))) = x
    ]]></literal>
    <justification jstring="[para(373(a,1),183(a,1,1))].">
      <j1 rule="para" parents="373 183"/>
    </justification>
  </clause>

  <clause id="417">
    <literal><![CDATA[
      x ' ' ^ x = x ' ' ' '
    ]]></literal>
    <justification jstring="[back_demod(360),demod(396(5))].">
      <j1 rule="back_demod" parents="360"/>
      <j2 rule="demod" parents="396"/>
    </justification>
  </clause>

  <clause id="421">
    <literal><![CDATA[
      x ' ' ' ' ' = x '
    ]]></literal>
    <justification jstring="[back_demod(399),demod(417(5))].">
      <j1 rule="back_demod" parents="399"/>
      <j2 rule="demod" parents="417"/>
    </justification>
  </clause>

  <clause id="498">
    <literal><![CDATA[
      x ' ' ' ' = x
    ]]></literal>
    <justification jstring="[para(421(a,1),393(a,1,2)),demod(13(5),396(3)),flip(a)].">
      <j1 rule="para" parents="421 393"/>
      <j2 rule="demod" parents="13 396"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="509">
    <literal><![CDATA[
      x ' ' ^ x = x
    ]]></literal>
    <justification jstring="[back_demod(417),demod(498(7))].">
      <j1 rule="back_demod" parents="417"/>
      <j2 rule="demod" parents="498"/>
    </justification>
  </clause>

  <clause id="511">
    <literal><![CDATA[
      f(x,y) = (x ^ y) ' ' '
    ]]></literal>
    <justification jstring="[para(12(a,1),498(a,1,1,1,1)),flip(a)].">
      <j1 rule="para" parents="12 498"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="512">
    <literal><![CDATA[
      (x ^ y ') ' ' ' = x ' ' ' v y
    ]]></literal>
    <justification jstring="[para(498(a,1),13(a,1,1)),demod(511(2))].">
      <j1 rule="para" parents="498 13"/>
      <j2 rule="demod" parents="511"/>
    </justification>
  </clause>

  <clause id="513">
    <literal><![CDATA[
      (x ' ^ y) ' ' ' = x v y ' ' '
    ]]></literal>
    <justification jstring="[para(498(a,1),13(a,1,2)),demod(511(2))].">
      <j1 rule="para" parents="498 13"/>
      <j2 rule="demod" parents="511"/>
    </justification>
  </clause>

  <clause id="558">
    <literal><![CDATA[
      ((x v y) ^ x) ' ' ' = x ' ' '
    ]]></literal>
    <justification jstring="[para(498(a,1),397(a,1,1,1)),demod(498(5),511(2))].">
      <j1 rule="para" parents="498 397"/>
      <j2 rule="demod" parents="498 511"/>
    </justification>
  </clause>

  <clause id="585">
    <literal><![CDATA[
      x v (x ^ ((y ' ^ (x v y)) ' ' v y)) ' ' = x
    ]]></literal>
    <justification jstring="[back_demod(403),demod(511(4),513(7),20(3),511(9),512(12),20(8),512(8),498(6),511(8),511(12),512(15),498(4))].">
      <j1 rule="back_demod" parents="403"/>
      <j2 rule="demod" parents="511 513 20 511 512 20 512 498 511 511 512 498"/>
    </justification>
  </clause>

  <clause id="598">
    <literal><![CDATA[
      x v (x ^ y) ' ' = x ' '
    ]]></literal>
    <justification jstring="[back_demod(333),demod(511(3),511(7),512(10),498(5),511(6),558(9),511(5),394(5),498(4),511(4),511(8),512(11),498(6)),flip(a)].">
      <j1 rule="back_demod" parents="333"/>
      <j2 rule="demod" parents="511 511 512 498 511 558 511 394 498 511 511 512 498"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="599">
    <literal><![CDATA[
      x ' ^ (x ^ y) ' ' ' = x ' ' '
    ]]></literal>
    <justification jstring="[back_demod(332),demod(511(3),511(7),512(10),498(5),598(5),511(4),509(4),394(5),511(5)),flip(a)].">
      <j1 rule="back_demod" parents="332"/>
      <j2 rule="demod" parents="511 511 512 498 598 511 509 394 511"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="663">
    <literal><![CDATA[
      c4 ' ' != c4
    ]]></literal>
    <attribute><![CDATA[
      answer(B_SS)
    ]]></attribute>
    <justification jstring="[back_demod(16),demod(511(5),511(9),599(9),498(5))].">
      <j1 rule="back_demod" parents="16"/>
      <j2 rule="demod" parents="511 511 599 498"/>
    </justification>
  </clause>

  <clause id="666">
    <literal><![CDATA[
      x ' ' = x
    ]]></literal>
    <justification jstring="[back_demod(585),demod(598(10))].">
      <j1 rule="back_demod" parents="585"/>
      <j2 rule="demod" parents="598"/>
    </justification>
  </clause>

  <clause id="667">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(B_SS)
    ]]></attribute>
    <justification jstring="[resolve(666,a,663,a)].">
      <j1 rule="resolve" parents="666 663"/>
    </justification>
  </clause>

</proof>

<proof number="2" length="116" max_count="38">

<comments><![CDATA[
% Proof 2 at 0.32 (+ 0.01) seconds: OM_SS.
% Length of proof is 116.
% Level of proof is 28.
% Maximum clause weight is 38.
% Given clauses 191.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_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="8">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_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(z,f(y ',z)),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(c6,f(c6,f(c6,c7))) != f(c6,c7)
    ]]></literal>
    <attribute><![CDATA[
      answer(OM_SS)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </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="21">
    <literal><![CDATA[
      f(x ^ y,z ') = f(x,y) v z
    ]]></literal>
    <justification jstring="[para(12(a,1),13(a,1,1))].">
      <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(y,f(x ',y)),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="24">
    <literal><![CDATA[
      f(f(f(f(x,y),y '),z),f(y,f(f(y,f(y ',y)),y))) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),14(a,1,1,1,2))].">
      <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(z,f(y ',z)),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="27">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,y ')),z),f(y,f(y v y ',y '))) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),14(a,1,2,2,1,2)),demod(13(9))].">
      <j1 rule="para" parents="11 14"/>
      <j2 rule="demod" parents="13"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(f(f(z,f(x ',z)),z),f(x ',f(f(z,f(x ',z)),z))),f(f(z,f(x ',z)),z)))) = x
    ]]></literal>
    <justification jstring="[para(14(a,1),14(a,1,1,1))].">
      <j1 rule="para" parents="14 14"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(z,f(x ^ y,z)),z))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(14(a,1),14(a,1,1)),demod(12(3))].">
      <j1 rule="para" parents="14 14"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      f(x ' ^ f(x,y),f(x,f(f(y,f(x ',y)),y))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),23(a,1,1)),demod(12(4))].">
      <j1 rule="para" parents="11 23"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="53">
    <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="54">
    <literal><![CDATA[
      f(x ' ',y) ^ f(x,f(f(x,f(x ',x)),x)) = x '
    ]]></literal>
    <justification jstring="[para(25(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="25 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="56">
    <literal><![CDATA[
      f(x ' v y,f(x,f(f(x,f(x ',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(13(a,1),25(a,1,1))].">
      <j1 rule="para" parents="13 25"/>
    </justification>
  </clause>

  <clause id="60">
    <literal><![CDATA[
      f(x,f(x ',f(f(y,f(x ' ',y)),y))) = x '
    ]]></literal>
    <justification jstring="[para(25(a,1),23(a,1,1))].">
      <j1 rule="para" parents="25 23"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      x ' ' ' ^ f(x,f(f(x,f(x ',x)),x)) = x '
    ]]></literal>
    <justification jstring="[para(53(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="53 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      (x ' v y) ^ f(x,f(f(x,f(x ',x)),x)) = x '
    ]]></literal>
    <justification jstring="[para(56(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="56 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="73">
    <literal><![CDATA[
      x ^ f(x ',f(f(y,f(x ' ',y)),y)) = x ' '
    ]]></literal>
    <justification jstring="[para(60(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="60 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      f(f(x,y) ^ y ',f(y,f(f(y,f(y ',y)),y))) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),24(a,1,1)),demod(12(4))].">
      <j1 rule="para" parents="11 24"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="80">
    <literal><![CDATA[
      f(f(f(x,y),y '),z) ^ f(y,f(f(y,f(y ',y)),y)) = y '
    ]]></literal>
    <justification jstring="[para(24(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="24 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="104">
    <literal><![CDATA[
      (f(x,y) ^ f(y,z)) ^ f(y,f(f(z,f(y ',z)),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="119">
    <literal><![CDATA[
      x ^ f(f(x,y),f(f(z,f(x ^ y,z)),z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(34(a,1),12(a,1,1)),demod(12(2)),flip(a)].">
      <j1 rule="para" parents="34 12"/>
      <j2 rule="demod" parents="12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="128">
    <literal><![CDATA[
      f(f(x ' ',y),f(x,f(f(z,f(x ',z)),z))) = x
    ]]></literal>
    <justification jstring="[para(25(a,1),34(a,1,2,1)),demod(54(12),25(18))].">
      <j1 rule="para" parents="25 34"/>
      <j2 rule="demod" parents="54 25"/>
    </justification>
  </clause>

  <clause id="129">
    <literal><![CDATA[
      f(x ' ' ',f(x,f(f(y,f(x ',y)),y))) = x
    ]]></literal>
    <justification jstring="[para(53(a,1),34(a,1,2,1)),demod(61(12),53(18))].">
      <j1 rule="para" parents="53 34"/>
      <j2 rule="demod" parents="61 53"/>
    </justification>
  </clause>

  <clause id="130">
    <literal><![CDATA[
      f(x ' v y,f(x,f(f(z,f(x ',z)),z))) = x
    ]]></literal>
    <justification jstring="[para(56(a,1),34(a,1,2,1)),demod(67(10),56(16))].">
      <j1 rule="para" parents="56 34"/>
      <j2 rule="demod" parents="67 56"/>
    </justification>
  </clause>

  <clause id="131">
    <literal><![CDATA[
      f(f(f(f(x,y),y '),z),f(y,f(f(u,f(y ',u)),u))) = y
    ]]></literal>
    <justification jstring="[para(24(a,1),34(a,1,2,1)),demod(80(14),24(20))].">
      <j1 rule="para" parents="24 34"/>
      <j2 rule="demod" parents="80 24"/>
    </justification>
  </clause>

  <clause id="132">
    <literal><![CDATA[
      f(f(x,y) ^ f(y,z),f(y,f(f(u,f(y ',u)),u))) = y
    ]]></literal>
    <justification jstring="[para(26(a,1),34(a,1,2,1)),demod(104(12),26(18))].">
      <j1 rule="para" parents="26 34"/>
      <j2 rule="demod" parents="104 26"/>
    </justification>
  </clause>

  <clause id="133">
    <literal><![CDATA[
      f(x ' ' ',f(x,f(x v x ',x '))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),129(a,1,2,2,1,2)),demod(13(7))].">
      <j1 rule="para" parents="11 129"/>
      <j2 rule="demod" parents="13"/>
    </justification>
  </clause>

  <clause id="134">
    <literal><![CDATA[
      x ' ' ' ^ f(x,f(f(y,f(x ',y)),y)) = x '
    ]]></literal>
    <justification jstring="[para(129(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="129 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="146">
    <literal><![CDATA[
      x ' ^ f(x ' ',f(f(f(x,f(x v x ',x ')),x),f(x,f(x v x ',x ')))) = x ' ' '
    ]]></literal>
    <justification jstring="[para(133(a,1),73(a,1,2,2,1,2))].">
      <j1 rule="para" parents="133 73"/>
    </justification>
  </clause>

  <clause id="155">
    <literal><![CDATA[
      (x ' v y) ^ f(x,f(f(z,f(x ',z)),z)) = x '
    ]]></literal>
    <justification jstring="[para(130(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="130 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="183">
    <literal><![CDATA[
      f(f(x ' ',y),f(x,f(f(z ',x v z),z '))) = x
    ]]></literal>
    <justification jstring="[para(13(a,1),128(a,1,2,2,1,2))].">
      <j1 rule="para" parents="13 128"/>
    </justification>
  </clause>

  <clause id="218">
    <literal><![CDATA[
      (x ' ^ f(x,y)) ^ f(x,f(f(y,f(x ',y)),y)) = x '
    ]]></literal>
    <justification jstring="[para(42(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="42 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="227">
    <literal><![CDATA[
      f(x ' ^ f(x,y),f(x,f(f(z,f(x ',z)),z))) = x
    ]]></literal>
    <justification jstring="[para(42(a,1),34(a,1,2,1)),demod(218(12),42(18))].">
      <j1 rule="para" parents="42 34"/>
      <j2 rule="demod" parents="218 42"/>
    </justification>
  </clause>

  <clause id="233">
    <literal><![CDATA[
      (f(x,y) ^ y ') ^ f(y,f(f(y,f(y ',y)),y)) = y '
    ]]></literal>
    <justification jstring="[para(79(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="79 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="240">
    <literal><![CDATA[
      f(f(x,y) ^ y ',f(y,f(f(z,f(y ',z)),z))) = y
    ]]></literal>
    <justification jstring="[para(79(a,1),34(a,1,2,1)),demod(233(12),79(18))].">
      <j1 rule="para" parents="79 34"/>
      <j2 rule="demod" parents="233 79"/>
    </justification>
  </clause>

  <clause id="285">
    <literal><![CDATA[
      f(x ' ^ f(x,y),f(x,f(x v x ',x '))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),227(a,1,2,2,1,2)),demod(13(7))].">
      <j1 rule="para" parents="11 227"/>
      <j2 rule="demod" parents="13"/>
    </justification>
  </clause>

  <clause id="305">
    <literal><![CDATA[
      f(x ',f(f(x ',f(x,y)),f(f(f(x,f(x v x ',x ')),x),f(x,f(x v x ',x '))))) = f(x ',f(x,y))
    ]]></literal>
    <justification jstring="[para(285(a,1),34(a,1,2,2,1,2))].">
      <j1 rule="para" parents="285 34"/>
    </justification>
  </clause>

  <clause id="309">
    <literal><![CDATA[
      x ' ^ f(f(x ',f(x,y)),f(f(f(x,f(x v x ',x ')),x),f(x,f(x v x ',x ')))) = x ' ^ f(x,y)
    ]]></literal>
    <justification jstring="[para(285(a,1),119(a,1,2,2,1,2))].">
      <j1 rule="para" parents="285 119"/>
    </justification>
  </clause>

  <clause id="325">
    <literal><![CDATA[
      f(f(x,y),f(y,f(y v y ',y '))) = y
    ]]></literal>
    <justification jstring="[para(33(a,1),27(a,1,1))].">
      <j1 rule="para" parents="33 27"/>
    </justification>
  </clause>

  <clause id="332">
    <literal><![CDATA[
      x ' ^ f(f(x ',f(x,y)),x) = x ' ^ f(x,y)
    ]]></literal>
    <justification jstring="[back_demod(309),demod(325(16))].">
      <j1 rule="back_demod" parents="309"/>
      <j2 rule="demod" parents="325"/>
    </justification>
  </clause>

  <clause id="333">
    <literal><![CDATA[
      f(x ',f(f(x ',f(x,y)),x)) = f(x ',f(x,y))
    ]]></literal>
    <justification jstring="[back_demod(305),demod(325(16))].">
      <j1 rule="back_demod" parents="305"/>
      <j2 rule="demod" parents="325"/>
    </justification>
  </clause>

  <clause id="341">
    <literal><![CDATA[
      x ' ^ f(x ' ',x) = x ' ' '
    ]]></literal>
    <justification jstring="[back_demod(146),demod(325(15))].">
      <j1 rule="back_demod" parents="146"/>
      <j2 rule="demod" parents="325"/>
    </justification>
  </clause>

  <clause id="360">
    <literal><![CDATA[
      x ' ' ^ (x ' ' v x) = x ' ' ' '
    ]]></literal>
    <justification jstring="[para(13(a,1),341(a,1,2))].">
      <j1 rule="para" parents="13 341"/>
    </justification>
  </clause>

  <clause id="373">
    <literal><![CDATA[
      f(x ',f(x,f(x v x ',x '))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),325(a,1,1))].">
      <j1 rule="para" parents="11 325"/>
    </justification>
  </clause>

  <clause id="393">
    <literal><![CDATA[
      f(x,x ' ') = x '
    ]]></literal>
    <justification jstring="[para(373(a,1),60(a,1,2,2,1,2)),demod(325(22),11(3))].">
      <j1 rule="para" parents="373 60"/>
      <j2 rule="demod" parents="325 11"/>
    </justification>
  </clause>

  <clause id="394">
    <literal><![CDATA[
      x ^ x ' ' = x ' '
    ]]></literal>
    <justification jstring="[para(373(a,1),73(a,1,2,2,1,2)),demod(325(22),11(3))].">
      <j1 rule="para" parents="373 73"/>
      <j2 rule="demod" parents="325 11"/>
    </justification>
  </clause>

  <clause id="396">
    <literal><![CDATA[
      x ' ' v x = x
    ]]></literal>
    <justification jstring="[para(373(a,1),129(a,1,2,2,1,2)),demod(325(15),11(4),13(5))].">
      <j1 rule="para" parents="373 129"/>
      <j2 rule="demod" parents="325 11 13"/>
    </justification>
  </clause>

  <clause id="397">
    <literal><![CDATA[
      f(x ' v y,x ') = x
    ]]></literal>
    <justification jstring="[para(373(a,1),130(a,1,2,2,1,2)),demod(325(14),11(3))].">
      <j1 rule="para" parents="373 130"/>
      <j2 rule="demod" parents="325 11"/>
    </justification>
  </clause>

  <clause id="399">
    <literal><![CDATA[
      x ' ' ' ^ x ' = x '
    ]]></literal>
    <justification jstring="[para(373(a,1),134(a,1,2,2,1,2)),demod(325(15),11(4))].">
      <j1 rule="para" parents="373 134"/>
      <j2 rule="demod" parents="325 11"/>
    </justification>
  </clause>

  <clause id="400">
    <literal><![CDATA[
      (x ' v y) ^ x ' = x '
    ]]></literal>
    <justification jstring="[para(373(a,1),155(a,1,2,2,1,2)),demod(325(14),11(3))].">
      <j1 rule="para" parents="373 155"/>
      <j2 rule="demod" parents="325 11"/>
    </justification>
  </clause>

  <clause id="403">
    <literal><![CDATA[
      f(x ',f(x,f(f(y ',x v y),y '))) = x
    ]]></literal>
    <justification jstring="[para(373(a,1),183(a,1,1))].">
      <j1 rule="para" parents="373 183"/>
    </justification>
  </clause>

  <clause id="407">
    <literal><![CDATA[
      f(f(x,y),y ') v y = y
    ]]></literal>
    <justification jstring="[para(373(a,1),240(a,1,2,2,1,2)),demod(325(15),11(4),21(5))].">
      <j1 rule="para" parents="373 240"/>
      <j2 rule="demod" parents="325 11 21"/>
    </justification>
  </clause>

  <clause id="417">
    <literal><![CDATA[
      x ' ' ^ x = x ' ' ' '
    ]]></literal>
    <justification jstring="[back_demod(360),demod(396(5))].">
      <j1 rule="back_demod" parents="360"/>
      <j2 rule="demod" parents="396"/>
    </justification>
  </clause>

  <clause id="421">
    <literal><![CDATA[
      x ' ' ' ' ' = x '
    ]]></literal>
    <justification jstring="[back_demod(399),demod(417(5))].">
      <j1 rule="back_demod" parents="399"/>
      <j2 rule="demod" parents="417"/>
    </justification>
  </clause>

  <clause id="498">
    <literal><![CDATA[
      x ' ' ' ' = x
    ]]></literal>
    <justification jstring="[para(421(a,1),393(a,1,2)),demod(13(5),396(3)),flip(a)].">
      <j1 rule="para" parents="421 393"/>
      <j2 rule="demod" parents="13 396"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="509">
    <literal><![CDATA[
      x ' ' ^ x = x
    ]]></literal>
    <justification jstring="[back_demod(417),demod(498(7))].">
      <j1 rule="back_demod" parents="417"/>
      <j2 rule="demod" parents="498"/>
    </justification>
  </clause>

  <clause id="511">
    <literal><![CDATA[
      f(x,y) = (x ^ y) ' ' '
    ]]></literal>
    <justification jstring="[para(12(a,1),498(a,1,1,1,1)),flip(a)].">
      <j1 rule="para" parents="12 498"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="512">
    <literal><![CDATA[
      (x ^ y ') ' ' ' = x ' ' ' v y
    ]]></literal>
    <justification jstring="[para(498(a,1),13(a,1,1)),demod(511(2))].">
      <j1 rule="para" parents="498 13"/>
      <j2 rule="demod" parents="511"/>
    </justification>
  </clause>

  <clause id="513">
    <literal><![CDATA[
      (x ' ^ y) ' ' ' = x v y ' ' '
    ]]></literal>
    <justification jstring="[para(498(a,1),13(a,1,2)),demod(511(2))].">
      <j1 rule="para" parents="498 13"/>
      <j2 rule="demod" parents="511"/>
    </justification>
  </clause>

  <clause id="558">
    <literal><![CDATA[
      ((x v y) ^ x) ' ' ' = x ' ' '
    ]]></literal>
    <justification jstring="[para(498(a,1),397(a,1,1,1)),demod(498(5),511(2))].">
      <j1 rule="para" parents="498 397"/>
      <j2 rule="demod" parents="498 511"/>
    </justification>
  </clause>

  <clause id="583">
    <literal><![CDATA[
      ((x ^ y) ' ' v y) v y = y
    ]]></literal>
    <justification jstring="[back_demod(407),demod(511(1),511(6),512(9),498(5))].">
      <j1 rule="back_demod" parents="407"/>
      <j2 rule="demod" parents="511 511 512 498"/>
    </justification>
  </clause>

  <clause id="585">
    <literal><![CDATA[
      x v (x ^ ((y ' ^ (x v y)) ' ' v y)) ' ' = x
    ]]></literal>
    <justification jstring="[back_demod(403),demod(511(4),513(7),20(3),511(9),512(12),20(8),512(8),498(6),511(8),511(12),512(15),498(4))].">
      <j1 rule="back_demod" parents="403"/>
      <j2 rule="demod" parents="511 513 20 511 512 20 512 498 511 511 512 498"/>
    </justification>
  </clause>

  <clause id="598">
    <literal><![CDATA[
      x v (x ^ y) ' ' = x ' '
    ]]></literal>
    <justification jstring="[back_demod(333),demod(511(3),511(7),512(10),498(5),511(6),558(9),511(5),394(5),498(4),511(4),511(8),512(11),498(6)),flip(a)].">
      <j1 rule="back_demod" parents="333"/>
      <j2 rule="demod" parents="511 511 512 498 511 558 511 394 498 511 511 512 498"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="599">
    <literal><![CDATA[
      x ' ^ (x ^ y) ' ' ' = x ' ' '
    ]]></literal>
    <justification jstring="[back_demod(332),demod(511(3),511(7),512(10),498(5),598(5),511(4),509(4),394(5),511(5)),flip(a)].">
      <j1 rule="back_demod" parents="332"/>
      <j2 rule="demod" parents="511 511 512 498 598 511 509 394 511"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="647">
    <literal><![CDATA[
      ((x ^ y) ' ' v (y ^ z) ' ') v (y ^ ((u ^ (y v u ' ' ')) ' ' v u ' ' ')) ' ' = y
    ]]></literal>
    <justification jstring="[back_demod(132),demod(511(1),511(5),511(11),513(14),511(14),511(18),513(21),511(21),511(25),512(28),512(12),498(5))].">
      <j1 rule="back_demod" parents="132"/>
      <j2 rule="demod" parents="511 511 511 513 511 511 513 511 511 512 512 498"/>
    </justification>
  </clause>

  <clause id="648">
    <literal><![CDATA[
      (((x ^ y) ' ' v y) ^ z) ' ' v (y ^ ((u ^ (y v u ' ' ')) ' ' v u ' ' ')) ' ' = y
    ]]></literal>
    <justification jstring="[back_demod(131),demod(511(1),511(6),512(9),498(5),511(5),511(10),513(13),511(13),511(17),513(20),511(20),511(24),512(27),498(9))].">
      <j1 rule="back_demod" parents="131"/>
      <j2 rule="demod" parents="511 511 512 498 511 511 513 511 511 513 511 511 512 498"/>
    </justification>
  </clause>

  <clause id="662">
    <literal><![CDATA[
      (c6 ^ (c6 ' ' ' v (c6 ^ c7) ' ')) ' ' ' != (c6 ^ c7) ' ' '
    ]]></literal>
    <attribute><![CDATA[
      answer(OM_SS)
    ]]></attribute>
    <justification jstring="[back_demod(17),demod(511(5),511(9),512(12),511(12),511(18))].">
      <j1 rule="back_demod" parents="17"/>
      <j2 rule="demod" parents="511 511 512 511 511"/>
    </justification>
  </clause>

  <clause id="666">
    <literal><![CDATA[
      x ' ' = x
    ]]></literal>
    <justification jstring="[back_demod(585),demod(598(10))].">
      <j1 rule="back_demod" parents="585"/>
      <j2 rule="demod" parents="598"/>
    </justification>
  </clause>

  <clause id="707">
    <literal><![CDATA[
      (c6 ^ (c6 ' v (c6 ^ c7))) ' != (c6 ^ c7) '
    ]]></literal>
    <attribute><![CDATA[
      answer(OM_SS)
    ]]></attribute>
    <justification jstring="[back_demod(662),demod(666(4),666(8),666(10),666(14))].">
      <j1 rule="back_demod" parents="662"/>
      <j2 rule="demod" parents="666 666 666 666"/>
    </justification>
  </clause>

  <clause id="718">
    <literal><![CDATA[
      (((x ^ y) v y) ^ z) v (y ^ ((u ^ (y v u ')) v u ')) = y
    ]]></literal>
    <justification jstring="[back_demod(648),demod(666(3),666(5),666(5),666(8),666(8),666(11))].">
      <j1 rule="back_demod" parents="648"/>
      <j2 rule="demod" parents="666 666 666 666 666 666"/>
    </justification>
  </clause>

  <clause id="719">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v (y ^ ((u ^ (y v u ')) v u ')) = y
    ]]></literal>
    <justification jstring="[back_demod(647),demod(666(3),666(4),666(5),666(8),666(8),666(11))].">
      <j1 rule="back_demod" parents="647"/>
      <j2 rule="demod" parents="666 666 666 666 666 666"/>
    </justification>
  </clause>

  <clause id="747">
    <literal><![CDATA[
      x ' ^ (x ^ y) ' = x '
    ]]></literal>
    <justification jstring="[back_demod(599),demod(666(4),666(6))].">
      <j1 rule="back_demod" parents="599"/>
      <j2 rule="demod" parents="666 666"/>
    </justification>
  </clause>

  <clause id="748">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[back_demod(598),demod(666(3),666(4))].">
      <j1 rule="back_demod" parents="598"/>
      <j2 rule="demod" parents="666 666"/>
    </justification>
  </clause>

  <clause id="754">
    <literal><![CDATA[
      ((x ^ y) v y) v y = y
    ]]></literal>
    <justification jstring="[back_demod(583),demod(666(3))].">
      <j1 rule="back_demod" parents="583"/>
      <j2 rule="demod" parents="666"/>
    </justification>
  </clause>

  <clause id="774">
    <literal><![CDATA[
      (x ' ^ y) ' = x v y '
    ]]></literal>
    <justification jstring="[back_demod(513),demod(666(4),666(5))].">
      <j1 rule="back_demod" parents="513"/>
      <j2 rule="demod" parents="666 666"/>
    </justification>
  </clause>

  <clause id="780">
    <literal><![CDATA[
      (x v y) ^ x = x
    ]]></literal>
    <justification jstring="[para(666(a,1),400(a,1,1,1)),demod(666(3),666(4))].">
      <j1 rule="para" parents="666 400"/>
      <j2 rule="demod" parents="666 666"/>
    </justification>
  </clause>

  <clause id="781">
    <literal><![CDATA[
      (x v y) v x = x v y
    ]]></literal>
    <justification jstring="[para(780(a,1),748(a,1,2))].">
      <j1 rule="para" parents="780 748"/>
    </justification>
  </clause>

  <clause id="823">
    <literal><![CDATA[
      x ^ (x v y ') = x
    ]]></literal>
    <justification jstring="[para(666(a,1),747(a,1,1)),demod(774(3),666(5))].">
      <j1 rule="para" parents="666 747"/>
      <j2 rule="demod" parents="774 666"/>
    </justification>
  </clause>

  <clause id="828">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[para(666(a,1),823(a,1,2,2))].">
      <j1 rule="para" parents="666 823"/>
    </justification>
  </clause>

  <clause id="836">
    <literal><![CDATA[
      (x ^ y) ' = x ' v y '
    ]]></literal>
    <justification jstring="[para(666(a,1),774(a,1,1,1))].">
      <j1 rule="para" parents="666 774"/>
    </justification>
  </clause>

  <clause id="858">
    <literal><![CDATA[
      c6 ' v (c6 ^ (c6 ' v c7 ')) != c6 ' v c7 '
    ]]></literal>
    <attribute><![CDATA[
      answer(OM_SS)
    ]]></attribute>
    <justification jstring="[back_demod(707),demod(836(9),20(9),666(5),836(7),836(14))].">
      <j1 rule="back_demod" parents="707"/>
      <j2 rule="demod" parents="836 20 666 836 836"/>
    </justification>
  </clause>

  <clause id="867">
    <literal><![CDATA[
      (x ' ^ y ') v x ' = x '
    ]]></literal>
    <justification jstring="[para(780(a,1),836(a,1,1)),demod(20(3)),flip(a)].">
      <j1 rule="para" parents="780 836"/>
      <j2 rule="demod" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="870">
    <literal><![CDATA[
      (x ^ y ') v x = x
    ]]></literal>
    <justification jstring="[para(666(a,1),867(a,1,1,1)),demod(666(4),666(5))].">
      <j1 rule="para" parents="666 867"/>
      <j2 rule="demod" parents="666 666"/>
    </justification>
  </clause>

  <clause id="874">
    <literal><![CDATA[
      (x ^ y) v x = x
    ]]></literal>
    <justification jstring="[para(666(a,1),870(a,1,1,2))].">
      <j1 rule="para" parents="666 870"/>
    </justification>
  </clause>

  <clause id="892">
    <literal><![CDATA[
      (x ^ y) ^ x = x ^ y
    ]]></literal>
    <justification jstring="[para(874(a,1),828(a,1,2))].">
      <j1 rule="para" parents="874 828"/>
    </justification>
  </clause>

  <clause id="903">
    <literal><![CDATA[
      (x ^ y) v (y ^ ((z ^ (y v z ')) v z ')) = y
    ]]></literal>
    <justification jstring="[para(780(a,1),718(a,1,1))].">
      <j1 rule="para" parents="780 718"/>
    </justification>
  </clause>

  <clause id="904">
    <literal><![CDATA[
      x ^ ((y ^ (x v y ')) v y ') = x
    ]]></literal>
    <justification jstring="[para(718(a,1),754(a,1,1)),demod(748(7)),flip(a)].">
      <j1 rule="para" parents="718 754"/>
      <j2 rule="demod" parents="748"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="912">
    <literal><![CDATA[
      (x ^ y) v y = y
    ]]></literal>
    <justification jstring="[back_demod(903),demod(904(7))].">
      <j1 rule="back_demod" parents="903"/>
      <j2 rule="demod" parents="904"/>
    </justification>
  </clause>

  <clause id="914">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v y = y
    ]]></literal>
    <justification jstring="[back_demod(719),demod(904(9))].">
      <j1 rule="back_demod" parents="719"/>
      <j2 rule="demod" parents="904"/>
    </justification>
  </clause>

  <clause id="918">
    <literal><![CDATA[
      (x ' v y ') ^ y ' = y '
    ]]></literal>
    <justification jstring="[para(912(a,1),20(a,1,1)),demod(836(3)),flip(a)].">
      <j1 rule="para" parents="912 20"/>
      <j2 rule="demod" parents="836"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="934">
    <literal><![CDATA[
      x ^ ((y ^ x) v (x ^ z)) = (y ^ x) v (x ^ z)
    ]]></literal>
    <justification jstring="[para(914(a,1),780(a,1,1))].">
      <j1 rule="para" parents="914 780"/>
    </justification>
  </clause>

  <clause id="935">
    <literal><![CDATA[
      ((x ^ (y v z)) v y) v (y v z) = y v z
    ]]></literal>
    <justification jstring="[para(780(a,1),914(a,1,1,2))].">
      <j1 rule="para" parents="780 914"/>
    </justification>
  </clause>

  <clause id="936">
    <literal><![CDATA[
      x v ((y ^ x) v (x ^ z)) = x
    ]]></literal>
    <justification jstring="[para(914(a,1),781(a,1,1)),demod(914(8))].">
      <j1 rule="para" parents="914 781"/>
      <j2 rule="demod" parents="914"/>
    </justification>
  </clause>

  <clause id="945">
    <literal><![CDATA[
      (x ' v y) ^ y = y
    ]]></literal>
    <justification jstring="[para(666(a,1),918(a,1,1,2)),demod(666(4),666(5))].">
      <j1 rule="para" parents="666 918"/>
      <j2 rule="demod" parents="666 666"/>
    </justification>
  </clause>

  <clause id="952">
    <literal><![CDATA[
      (x v y) ^ y = y
    ]]></literal>
    <justification jstring="[para(666(a,1),945(a,1,1,1))].">
      <j1 rule="para" parents="666 945"/>
    </justification>
  </clause>

  <clause id="961">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(952(a,1),892(a,1,1)),demod(952(4))].">
      <j1 rule="para" parents="952 892"/>
      <j2 rule="demod" parents="952"/>
    </justification>
  </clause>

  <clause id="975">
    <literal><![CDATA[
      (x v y) v ((z ^ (x v y)) v x) = x v y
    ]]></literal>
    <justification jstring="[para(780(a,1),936(a,1,2,2))].">
      <j1 rule="para" parents="780 936"/>
    </justification>
  </clause>

  <clause id="1084">
    <literal><![CDATA[
      (x ' v y) ^ ((x ^ (x ' v y)) v x ') = x ' v y
    ]]></literal>
    <justification jstring="[para(781(a,1),904(a,1,2,1,2))].">
      <j1 rule="para" parents="781 904"/>
    </justification>
  </clause>

  <clause id="1427">
    <literal><![CDATA[
      (x v y) ^ ((z ^ (x v y)) v x) = (z ^ (x v y)) v x
    ]]></literal>
    <justification jstring="[para(780(a,1),934(a,1,2,2)),demod(780(9))].">
      <j1 rule="para" parents="780 934"/>
      <j2 rule="demod" parents="780"/>
    </justification>
  </clause>

  <clause id="1458">
    <literal><![CDATA[
      (x ^ (x ' v y)) v x ' = x ' v y
    ]]></literal>
    <justification jstring="[back_demod(1084),demod(1427(8))].">
      <j1 rule="back_demod" parents="1084"/>
      <j2 rule="demod" parents="1427"/>
    </justification>
  </clause>

  <clause id="1547">
    <literal><![CDATA[
      (x v y) v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(961(a,1),935(a,1,1,1))].">
      <j1 rule="para" parents="961 935"/>
    </justification>
  </clause>

  <clause id="2241">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[para(961(a,1),975(a,1,2,1)),demod(1547(3))].">
      <j1 rule="para" parents="961 975"/>
      <j2 rule="demod" parents="1547"/>
    </justification>
  </clause>

  <clause id="2397">
    <literal><![CDATA[
      x ' v (x ^ (x ' v y)) = x ' v y
    ]]></literal>
    <justification jstring="[back_demod(1458),demod(2241(5))].">
      <j1 rule="back_demod" parents="1458"/>
      <j2 rule="demod" parents="2241"/>
    </justification>
  </clause>

  <clause id="2398">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(OM_SS)
    ]]></attribute>
    <justification jstring="[resolve(2397,a,858,a)].">
      <j1 rule="resolve" parents="2397 858"/>
    </justification>
  </clause>

</proof>

<proof number="3" length="168" max_count="38">

<comments><![CDATA[
% Proof 3 at 9.76 (+ 0.10) seconds: A_SS.
% Length of proof is 168.
% Level of proof is 35.
% Maximum clause weight is 38.
% Given clauses 648.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_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(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_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(z,f(y ',z)),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="21">
    <literal><![CDATA[
      f(x ^ y,z ') = f(x,y) v z
    ]]></literal>
    <justification jstring="[para(12(a,1),13(a,1,1))].">
      <j1 rule="para" parents="12 13"/>
    </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(y,f(x ',y)),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="24">
    <literal><![CDATA[
      f(f(f(f(x,y),y '),z),f(y,f(f(y,f(y ',y)),y))) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),14(a,1,1,1,2))].">
      <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(z,f(y ',z)),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="27">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,y ')),z),f(y,f(y v y ',y '))) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),14(a,1,2,2,1,2)),demod(13(9))].">
      <j1 rule="para" parents="11 14"/>
      <j2 rule="demod" parents="13"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z ')),u),f(y,f(f(z ',y v z),z '))) = y
    ]]></literal>
    <justification jstring="[para(13(a,1),14(a,1,2,2,1,2))].">
      <j1 rule="para" parents="13 14"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(f(f(z,f(x ',z)),z),f(x ',f(f(z,f(x ',z)),z))),f(f(z,f(x ',z)),z)))) = x
    ]]></literal>
    <justification jstring="[para(14(a,1),14(a,1,1,1))].">
      <j1 rule="para" parents="14 14"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(z,f(x ^ y,z)),z))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(14(a,1),14(a,1,1)),demod(12(3))].">
      <j1 rule="para" parents="14 14"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      f(x ' ^ f(x,y),f(x,f(f(y,f(x ',y)),y))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),23(a,1,1)),demod(12(4))].">
      <j1 rule="para" parents="11 23"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="53">
    <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="54">
    <literal><![CDATA[
      f(x ' ',y) ^ f(x,f(f(x,f(x ',x)),x)) = x '
    ]]></literal>
    <justification jstring="[para(25(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="25 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="56">
    <literal><![CDATA[
      f(x ' v y,f(x,f(f(x,f(x ',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(13(a,1),25(a,1,1))].">
      <j1 rule="para" parents="13 25"/>
    </justification>
  </clause>

  <clause id="60">
    <literal><![CDATA[
      f(x,f(x ',f(f(y,f(x ' ',y)),y))) = x '
    ]]></literal>
    <justification jstring="[para(25(a,1),23(a,1,1))].">
      <j1 rule="para" parents="25 23"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      x ' ' ' ^ f(x,f(f(x,f(x ',x)),x)) = x '
    ]]></literal>
    <justification jstring="[para(53(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="53 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      (x ' v y) ^ f(x,f(f(x,f(x ',x)),x)) = x '
    ]]></literal>
    <justification jstring="[para(56(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="56 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="73">
    <literal><![CDATA[
      x ^ f(x ',f(f(y,f(x ' ',y)),y)) = x ' '
    ]]></literal>
    <justification jstring="[para(60(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="60 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      f(f(x,y) ^ y ',f(y,f(f(y,f(y ',y)),y))) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),24(a,1,1)),demod(12(4))].">
      <j1 rule="para" parents="11 24"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="80">
    <literal><![CDATA[
      f(f(f(x,y),y '),z) ^ f(y,f(f(y,f(y ',y)),y)) = y '
    ]]></literal>
    <justification jstring="[para(24(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="24 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="104">
    <literal><![CDATA[
      (f(x,y) ^ f(y,z)) ^ f(y,f(f(z,f(y ',z)),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="119">
    <literal><![CDATA[
      x ^ f(f(x,y),f(f(z,f(x ^ y,z)),z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(34(a,1),12(a,1,1)),demod(12(2)),flip(a)].">
      <j1 rule="para" parents="34 12"/>
      <j2 rule="demod" parents="12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="128">
    <literal><![CDATA[
      f(f(x ' ',y),f(x,f(f(z,f(x ',z)),z))) = x
    ]]></literal>
    <justification jstring="[para(25(a,1),34(a,1,2,1)),demod(54(12),25(18))].">
      <j1 rule="para" parents="25 34"/>
      <j2 rule="demod" parents="54 25"/>
    </justification>
  </clause>

  <clause id="129">
    <literal><![CDATA[
      f(x ' ' ',f(x,f(f(y,f(x ',y)),y))) = x
    ]]></literal>
    <justification jstring="[para(53(a,1),34(a,1,2,1)),demod(61(12),53(18))].">
      <j1 rule="para" parents="53 34"/>
      <j2 rule="demod" parents="61 53"/>
    </justification>
  </clause>

  <clause id="130">
    <literal><![CDATA[
      f(x ' v y,f(x,f(f(z,f(x ',z)),z))) = x
    ]]></literal>
    <justification jstring="[para(56(a,1),34(a,1,2,1)),demod(67(10),56(16))].">
      <j1 rule="para" parents="56 34"/>
      <j2 rule="demod" parents="67 56"/>
    </justification>
  </clause>

  <clause id="131">
    <literal><![CDATA[
      f(f(f(f(x,y),y '),z),f(y,f(f(u,f(y ',u)),u))) = y
    ]]></literal>
    <justification jstring="[para(24(a,1),34(a,1,2,1)),demod(80(14),24(20))].">
      <j1 rule="para" parents="24 34"/>
      <j2 rule="demod" parents="80 24"/>
    </justification>
  </clause>

  <clause id="132">
    <literal><![CDATA[
      f(f(x,y) ^ f(y,z),f(y,f(f(u,f(y ',u)),u))) = y
    ]]></literal>
    <justification jstring="[para(26(a,1),34(a,1,2,1)),demod(104(12),26(18))].">
      <j1 rule="para" parents="26 34"/>
      <j2 rule="demod" parents="104 26"/>
    </justification>
  </clause>

  <clause id="133">
    <literal><![CDATA[
      f(x ' ' ',f(x,f(x v x ',x '))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),129(a,1,2,2,1,2)),demod(13(7))].">
      <j1 rule="para" parents="11 129"/>
      <j2 rule="demod" parents="13"/>
    </justification>
  </clause>

  <clause id="134">
    <literal><![CDATA[
      x ' ' ' ^ f(x,f(f(y,f(x ',y)),y)) = x '
    ]]></literal>
    <justification jstring="[para(129(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="129 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="146">
    <literal><![CDATA[
      x ' ^ f(x ' ',f(f(f(x,f(x v x ',x ')),x),f(x,f(x v x ',x ')))) = x ' ' '
    ]]></literal>
    <justification jstring="[para(133(a,1),73(a,1,2,2,1,2))].">
      <j1 rule="para" parents="133 73"/>
    </justification>
  </clause>

  <clause id="155">
    <literal><![CDATA[
      (x ' v y) ^ f(x,f(f(z,f(x ',z)),z)) = x '
    ]]></literal>
    <justification jstring="[para(130(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="130 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="183">
    <literal><![CDATA[
      f(f(x ' ',y),f(x,f(f(z ',x v z),z '))) = x
    ]]></literal>
    <justification jstring="[para(13(a,1),128(a,1,2,2,1,2))].">
      <j1 rule="para" parents="13 128"/>
    </justification>
  </clause>

  <clause id="218">
    <literal><![CDATA[
      (x ' ^ f(x,y)) ^ f(x,f(f(y,f(x ',y)),y)) = x '
    ]]></literal>
    <justification jstring="[para(42(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="42 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="227">
    <literal><![CDATA[
      f(x ' ^ f(x,y),f(x,f(f(z,f(x ',z)),z))) = x
    ]]></literal>
    <justification jstring="[para(42(a,1),34(a,1,2,1)),demod(218(12),42(18))].">
      <j1 rule="para" parents="42 34"/>
      <j2 rule="demod" parents="218 42"/>
    </justification>
  </clause>

  <clause id="233">
    <literal><![CDATA[
      (f(x,y) ^ y ') ^ f(y,f(f(y,f(y ',y)),y)) = y '
    ]]></literal>
    <justification jstring="[para(79(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="79 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="240">
    <literal><![CDATA[
      f(f(x,y) ^ y ',f(y,f(f(z,f(y ',z)),z))) = y
    ]]></literal>
    <justification jstring="[para(79(a,1),34(a,1,2,1)),demod(233(12),79(18))].">
      <j1 rule="para" parents="79 34"/>
      <j2 rule="demod" parents="233 79"/>
    </justification>
  </clause>

  <clause id="285">
    <literal><![CDATA[
      f(x ' ^ f(x,y),f(x,f(x v x ',x '))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),227(a,1,2,2,1,2)),demod(13(7))].">
      <j1 rule="para" parents="11 227"/>
      <j2 rule="demod" parents="13"/>
    </justification>
  </clause>

  <clause id="305">
    <literal><![CDATA[
      f(x ',f(f(x ',f(x,y)),f(f(f(x,f(x v x ',x ')),x),f(x,f(x v x ',x '))))) = f(x ',f(x,y))
    ]]></literal>
    <justification jstring="[para(285(a,1),34(a,1,2,2,1,2))].">
      <j1 rule="para" parents="285 34"/>
    </justification>
  </clause>

  <clause id="309">
    <literal><![CDATA[
      x ' ^ f(f(x ',f(x,y)),f(f(f(x,f(x v x ',x ')),x),f(x,f(x v x ',x ')))) = x ' ^ f(x,y)
    ]]></literal>
    <justification jstring="[para(285(a,1),119(a,1,2,2,1,2))].">
      <j1 rule="para" parents="285 119"/>
    </justification>
  </clause>

  <clause id="325">
    <literal><![CDATA[
      f(f(x,y),f(y,f(y v y ',y '))) = y
    ]]></literal>
    <justification jstring="[para(33(a,1),27(a,1,1))].">
      <j1 rule="para" parents="33 27"/>
    </justification>
  </clause>

  <clause id="332">
    <literal><![CDATA[
      x ' ^ f(f(x ',f(x,y)),x) = x ' ^ f(x,y)
    ]]></literal>
    <justification jstring="[back_demod(309),demod(325(16))].">
      <j1 rule="back_demod" parents="309"/>
      <j2 rule="demod" parents="325"/>
    </justification>
  </clause>

  <clause id="333">
    <literal><![CDATA[
      f(x ',f(f(x ',f(x,y)),x)) = f(x ',f(x,y))
    ]]></literal>
    <justification jstring="[back_demod(305),demod(325(16))].">
      <j1 rule="back_demod" parents="305"/>
      <j2 rule="demod" parents="325"/>
    </justification>
  </clause>

  <clause id="341">
    <literal><![CDATA[
      x ' ^ f(x ' ',x) = x ' ' '
    ]]></literal>
    <justification jstring="[back_demod(146),demod(325(15))].">
      <j1 rule="back_demod" parents="146"/>
      <j2 rule="demod" parents="325"/>
    </justification>
  </clause>

  <clause id="360">
    <literal><![CDATA[
      x ' ' ^ (x ' ' v x) = x ' ' ' '
    ]]></literal>
    <justification jstring="[para(13(a,1),341(a,1,2))].">
      <j1 rule="para" parents="13 341"/>
    </justification>
  </clause>

  <clause id="373">
    <literal><![CDATA[
      f(x ',f(x,f(x v x ',x '))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),325(a,1,1))].">
      <j1 rule="para" parents="11 325"/>
    </justification>
  </clause>

  <clause id="393">
    <literal><![CDATA[
      f(x,x ' ') = x '
    ]]></literal>
    <justification jstring="[para(373(a,1),60(a,1,2,2,1,2)),demod(325(22),11(3))].">
      <j1 rule="para" parents="373 60"/>
      <j2 rule="demod" parents="325 11"/>
    </justification>
  </clause>

  <clause id="394">
    <literal><![CDATA[
      x ^ x ' ' = x ' '
    ]]></literal>
    <justification jstring="[para(373(a,1),73(a,1,2,2,1,2)),demod(325(22),11(3))].">
      <j1 rule="para" parents="373 73"/>
      <j2 rule="demod" parents="325 11"/>
    </justification>
  </clause>

  <clause id="396">
    <literal><![CDATA[
      x ' ' v x = x
    ]]></literal>
    <justification jstring="[para(373(a,1),129(a,1,2,2,1,2)),demod(325(15),11(4),13(5))].">
      <j1 rule="para" parents="373 129"/>
      <j2 rule="demod" parents="325 11 13"/>
    </justification>
  </clause>

  <clause id="397">
    <literal><![CDATA[
      f(x ' v y,x ') = x
    ]]></literal>
    <justification jstring="[para(373(a,1),130(a,1,2,2,1,2)),demod(325(14),11(3))].">
      <j1 rule="para" parents="373 130"/>
      <j2 rule="demod" parents="325 11"/>
    </justification>
  </clause>

  <clause id="399">
    <literal><![CDATA[
      x ' ' ' ^ x ' = x '
    ]]></literal>
    <justification jstring="[para(373(a,1),134(a,1,2,2,1,2)),demod(325(15),11(4))].">
      <j1 rule="para" parents="373 134"/>
      <j2 rule="demod" parents="325 11"/>
    </justification>
  </clause>

  <clause id="400">
    <literal><![CDATA[
      (x ' v y) ^ x ' = x '
    ]]></literal>
    <justification jstring="[para(373(a,1),155(a,1,2,2,1,2)),demod(325(14),11(3))].">
      <j1 rule="para" parents="373 155"/>
      <j2 rule="demod" parents="325 11"/>
    </justification>
  </clause>

  <clause id="403">
    <literal><![CDATA[
      f(x ',f(x,f(f(y ',x v y),y '))) = x
    ]]></literal>
    <justification jstring="[para(373(a,1),183(a,1,1))].">
      <j1 rule="para" parents="373 183"/>
    </justification>
  </clause>

  <clause id="407">
    <literal><![CDATA[
      f(f(x,y),y ') v y = y
    ]]></literal>
    <justification jstring="[para(373(a,1),240(a,1,2,2,1,2)),demod(325(15),11(4),21(5))].">
      <j1 rule="para" parents="373 240"/>
      <j2 rule="demod" parents="325 11 21"/>
    </justification>
  </clause>

  <clause id="417">
    <literal><![CDATA[
      x ' ' ^ x = x ' ' ' '
    ]]></literal>
    <justification jstring="[back_demod(360),demod(396(5))].">
      <j1 rule="back_demod" parents="360"/>
      <j2 rule="demod" parents="396"/>
    </justification>
  </clause>

  <clause id="421">
    <literal><![CDATA[
      x ' ' ' ' ' = x '
    ]]></literal>
    <justification jstring="[back_demod(399),demod(417(5))].">
      <j1 rule="back_demod" parents="399"/>
      <j2 rule="demod" parents="417"/>
    </justification>
  </clause>

  <clause id="498">
    <literal><![CDATA[
      x ' ' ' ' = x
    ]]></literal>
    <justification jstring="[para(421(a,1),393(a,1,2)),demod(13(5),396(3)),flip(a)].">
      <j1 rule="para" parents="421 393"/>
      <j2 rule="demod" parents="13 396"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="509">
    <literal><![CDATA[
      x ' ' ^ x = x
    ]]></literal>
    <justification jstring="[back_demod(417),demod(498(7))].">
      <j1 rule="back_demod" parents="417"/>
      <j2 rule="demod" parents="498"/>
    </justification>
  </clause>

  <clause id="511">
    <literal><![CDATA[
      f(x,y) = (x ^ y) ' ' '
    ]]></literal>
    <justification jstring="[para(12(a,1),498(a,1,1,1,1)),flip(a)].">
      <j1 rule="para" parents="12 498"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="512">
    <literal><![CDATA[
      (x ^ y ') ' ' ' = x ' ' ' v y
    ]]></literal>
    <justification jstring="[para(498(a,1),13(a,1,1)),demod(511(2))].">
      <j1 rule="para" parents="498 13"/>
      <j2 rule="demod" parents="511"/>
    </justification>
  </clause>

  <clause id="513">
    <literal><![CDATA[
      (x ' ^ y) ' ' ' = x v y ' ' '
    ]]></literal>
    <justification jstring="[para(498(a,1),13(a,1,2)),demod(511(2))].">
      <j1 rule="para" parents="498 13"/>
      <j2 rule="demod" parents="511"/>
    </justification>
  </clause>

  <clause id="515">
    <literal><![CDATA[
      (x ^ (y ^ z)) ' ' ' = x ' ' ' v (y ^ z) ' ' '
    ]]></literal>
    <justification jstring="[para(498(a,1),22(a,1,1)),demod(511(2),511(9))].">
      <j1 rule="para" parents="498 22"/>
      <j2 rule="demod" parents="511 511"/>
    </justification>
  </clause>

  <clause id="539">
    <literal><![CDATA[
      (x v y) ^ (x ' ' v ((z ' ' ' v (x ^ z) ' ') ^ z) ' ') = x
    ]]></literal>
    <justification jstring="[para(498(a,1),155(a,1,1,1)),demod(498(8),511(5),511(9),512(12),511(12),511(16),512(19),498(5),498(19))].">
      <j1 rule="para" parents="498 155"/>
      <j2 rule="demod" parents="498 511 511 512 511 511 512 498 498"/>
    </justification>
  </clause>

  <clause id="554">
    <literal><![CDATA[
      (((x ^ y) ' ' v (y ^ z) ' ') ^ u) ' ' v (y ^ ((z ^ (y v z ' ' ')) ' ' v z ' ' ')) ' ' = y
    ]]></literal>
    <justification jstring="[para(498(a,1),32(a,1,1,1,2,2)),demod(511(1),511(5),511(9),512(12),498(5),511(8),498(15),511(16),498(23),511(20),513(23),511(23),511(27),512(30),498(12))].">
      <j1 rule="para" parents="498 32"/>
      <j2 rule="demod" parents="511 511 511 512 498 511 498 511 498 511 513 511 511 512 498"/>
    </justification>
  </clause>

  <clause id="558">
    <literal><![CDATA[
      ((x v y) ^ x) ' ' ' = x ' ' '
    ]]></literal>
    <justification jstring="[para(498(a,1),397(a,1,1,1)),demod(498(5),511(2))].">
      <j1 rule="para" parents="498 397"/>
      <j2 rule="demod" parents="498 511"/>
    </justification>
  </clause>

  <clause id="583">
    <literal><![CDATA[
      ((x ^ y) ' ' v y) v y = y
    ]]></literal>
    <justification jstring="[back_demod(407),demod(511(1),511(6),512(9),498(5))].">
      <j1 rule="back_demod" parents="407"/>
      <j2 rule="demod" parents="511 511 512 498"/>
    </justification>
  </clause>

  <clause id="585">
    <literal><![CDATA[
      x v (x ^ ((y ' ^ (x v y)) ' ' v y)) ' ' = x
    ]]></literal>
    <justification jstring="[back_demod(403),demod(511(4),513(7),20(3),511(9),512(12),20(8),512(8),498(6),511(8),511(12),512(15),498(4))].">
      <j1 rule="back_demod" parents="403"/>
      <j2 rule="demod" parents="511 513 20 511 512 20 512 498 511 511 512 498"/>
    </justification>
  </clause>

  <clause id="598">
    <literal><![CDATA[
      x v (x ^ y) ' ' = x ' '
    ]]></literal>
    <justification jstring="[back_demod(333),demod(511(3),511(7),512(10),498(5),511(6),558(9),511(5),394(5),498(4),511(4),511(8),512(11),498(6)),flip(a)].">
      <j1 rule="back_demod" parents="333"/>
      <j2 rule="demod" parents="511 511 512 498 511 558 511 394 498 511 511 512 498"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="599">
    <literal><![CDATA[
      x ' ^ (x ^ y) ' ' ' = x ' ' '
    ]]></literal>
    <justification jstring="[back_demod(332),demod(511(3),511(7),512(10),498(5),598(5),511(4),509(4),394(5),511(5)),flip(a)].">
      <j1 rule="back_demod" parents="332"/>
      <j2 rule="demod" parents="511 511 512 498 598 511 509 394 511"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="647">
    <literal><![CDATA[
      ((x ^ y) ' ' v (y ^ z) ' ') v (y ^ ((u ^ (y v u ' ' ')) ' ' v u ' ' ')) ' ' = y
    ]]></literal>
    <justification jstring="[back_demod(132),demod(511(1),511(5),511(11),513(14),511(14),511(18),513(21),511(21),511(25),512(28),512(12),498(5))].">
      <j1 rule="back_demod" parents="132"/>
      <j2 rule="demod" parents="511 511 511 513 511 511 513 511 511 512 512 498"/>
    </justification>
  </clause>

  <clause id="648">
    <literal><![CDATA[
      (((x ^ y) ' ' v y) ^ z) ' ' v (y ^ ((u ^ (y v u ' ' ')) ' ' v u ' ' ')) ' ' = y
    ]]></literal>
    <justification jstring="[back_demod(131),demod(511(1),511(6),512(9),498(5),511(5),511(10),513(13),511(13),511(17),513(20),511(20),511(24),512(27),498(9))].">
      <j1 rule="back_demod" parents="131"/>
      <j2 rule="demod" parents="511 511 512 498 511 511 513 511 511 513 511 511 512 498"/>
    </justification>
  </clause>

  <clause id="664">
    <literal><![CDATA[
      c2 ' ' ' v (c1 ^ c3) ' ' ' != c1 ' ' ' v (c2 ^ c3) ' ' '
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <justification jstring="[back_demod(15),demod(511(5),515(8),511(16),515(19))].">
      <j1 rule="back_demod" parents="15"/>
      <j2 rule="demod" parents="511 515 511 515"/>
    </justification>
  </clause>

  <clause id="666">
    <literal><![CDATA[
      x ' ' = x
    ]]></literal>
    <justification jstring="[back_demod(585),demod(598(10))].">
      <j1 rule="back_demod" parents="585"/>
      <j2 rule="demod" parents="598"/>
    </justification>
  </clause>

  <clause id="706">
    <literal><![CDATA[
      c2 ' v (c1 ^ c3) ' != c1 ' v (c2 ^ c3) '
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <justification jstring="[back_demod(664),demod(666(3),666(7),666(10),666(14))].">
      <j1 rule="back_demod" parents="664"/>
      <j2 rule="demod" parents="666 666 666 666"/>
    </justification>
  </clause>

  <clause id="718">
    <literal><![CDATA[
      (((x ^ y) v y) ^ z) v (y ^ ((u ^ (y v u ')) v u ')) = y
    ]]></literal>
    <justification jstring="[back_demod(648),demod(666(3),666(5),666(5),666(8),666(8),666(11))].">
      <j1 rule="back_demod" parents="648"/>
      <j2 rule="demod" parents="666 666 666 666 666 666"/>
    </justification>
  </clause>

  <clause id="719">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v (y ^ ((u ^ (y v u ')) v u ')) = y
    ]]></literal>
    <justification jstring="[back_demod(647),demod(666(3),666(4),666(5),666(8),666(8),666(11))].">
      <j1 rule="back_demod" parents="647"/>
      <j2 rule="demod" parents="666 666 666 666 666 666"/>
    </justification>
  </clause>

  <clause id="747">
    <literal><![CDATA[
      x ' ^ (x ^ y) ' = x '
    ]]></literal>
    <justification jstring="[back_demod(599),demod(666(4),666(6))].">
      <j1 rule="back_demod" parents="599"/>
      <j2 rule="demod" parents="666 666"/>
    </justification>
  </clause>

  <clause id="748">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[back_demod(598),demod(666(3),666(4))].">
      <j1 rule="back_demod" parents="598"/>
      <j2 rule="demod" parents="666 666"/>
    </justification>
  </clause>

  <clause id="754">
    <literal><![CDATA[
      ((x ^ y) v y) v y = y
    ]]></literal>
    <justification jstring="[back_demod(583),demod(666(3))].">
      <j1 rule="back_demod" parents="583"/>
      <j2 rule="demod" parents="666"/>
    </justification>
  </clause>

  <clause id="762">
    <literal><![CDATA[
      (((x ^ y) v (y ^ z)) ^ u) v (y ^ ((z ^ (y v z ')) v z ')) = y
    ]]></literal>
    <justification jstring="[back_demod(554),demod(666(3),666(4),666(6),666(6),666(9),666(9),666(12))].">
      <j1 rule="back_demod" parents="554"/>
      <j2 rule="demod" parents="666 666 666 666 666 666 666"/>
    </justification>
  </clause>

  <clause id="769">
    <literal><![CDATA[
      (x v y) ^ (x v ((z ' v (x ^ z)) ^ z)) = x
    ]]></literal>
    <justification jstring="[back_demod(539),demod(666(3),666(3),666(5),666(7))].">
      <j1 rule="back_demod" parents="539"/>
      <j2 rule="demod" parents="666 666 666 666"/>
    </justification>
  </clause>

  <clause id="774">
    <literal><![CDATA[
      (x ' ^ y) ' = x v y '
    ]]></literal>
    <justification jstring="[back_demod(513),demod(666(4),666(5))].">
      <j1 rule="back_demod" parents="513"/>
      <j2 rule="demod" parents="666 666"/>
    </justification>
  </clause>

  <clause id="778">
    <literal><![CDATA[
      x ^ x = x
    ]]></literal>
    <justification jstring="[back_demod(18),demod(666(3))].">
      <j1 rule="back_demod" parents="18"/>
      <j2 rule="demod" parents="666"/>
    </justification>
  </clause>

  <clause id="780">
    <literal><![CDATA[
      (x v y) ^ x = x
    ]]></literal>
    <justification jstring="[para(666(a,1),400(a,1,1,1)),demod(666(3),666(4))].">
      <j1 rule="para" parents="666 400"/>
      <j2 rule="demod" parents="666 666"/>
    </justification>
  </clause>

  <clause id="781">
    <literal><![CDATA[
      (x v y) v x = x v y
    ]]></literal>
    <justification jstring="[para(780(a,1),748(a,1,2))].">
      <j1 rule="para" parents="780 748"/>
    </justification>
  </clause>

  <clause id="823">
    <literal><![CDATA[
      x ^ (x v y ') = x
    ]]></literal>
    <justification jstring="[para(666(a,1),747(a,1,1)),demod(774(3),666(5))].">
      <j1 rule="para" parents="666 747"/>
      <j2 rule="demod" parents="774 666"/>
    </justification>
  </clause>

  <clause id="828">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[para(666(a,1),823(a,1,2,2))].">
      <j1 rule="para" parents="666 823"/>
    </justification>
  </clause>

  <clause id="836">
    <literal><![CDATA[
      (x ^ y) ' = x ' v y '
    ]]></literal>
    <justification jstring="[para(666(a,1),774(a,1,1,1))].">
      <j1 rule="para" parents="666 774"/>
    </justification>
  </clause>

  <clause id="859">
    <literal><![CDATA[
      c2 ' v (c1 ' v c3 ') != c1 ' v (c2 ' v c3 ')
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <justification jstring="[back_demod(706),demod(836(6),836(14))].">
      <j1 rule="back_demod" parents="706"/>
      <j2 rule="demod" parents="836 836"/>
    </justification>
  </clause>

  <clause id="867">
    <literal><![CDATA[
      (x ' ^ y ') v x ' = x '
    ]]></literal>
    <justification jstring="[para(780(a,1),836(a,1,1)),demod(20(3)),flip(a)].">
      <j1 rule="para" parents="780 836"/>
      <j2 rule="demod" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="870">
    <literal><![CDATA[
      (x ^ y ') v x = x
    ]]></literal>
    <justification jstring="[para(666(a,1),867(a,1,1,1)),demod(666(4),666(5))].">
      <j1 rule="para" parents="666 867"/>
      <j2 rule="demod" parents="666 666"/>
    </justification>
  </clause>

  <clause id="874">
    <literal><![CDATA[
      (x ^ y) v x = x
    ]]></literal>
    <justification jstring="[para(666(a,1),870(a,1,1,2))].">
      <j1 rule="para" parents="666 870"/>
    </justification>
  </clause>

  <clause id="892">
    <literal><![CDATA[
      (x ^ y) ^ x = x ^ y
    ]]></literal>
    <justification jstring="[para(874(a,1),828(a,1,2))].">
      <j1 rule="para" parents="874 828"/>
    </justification>
  </clause>

  <clause id="903">
    <literal><![CDATA[
      (x ^ y) v (y ^ ((z ^ (y v z ')) v z ')) = y
    ]]></literal>
    <justification jstring="[para(780(a,1),718(a,1,1))].">
      <j1 rule="para" parents="780 718"/>
    </justification>
  </clause>

  <clause id="904">
    <literal><![CDATA[
      x ^ ((y ^ (x v y ')) v y ') = x
    ]]></literal>
    <justification jstring="[para(718(a,1),754(a,1,1)),demod(748(7)),flip(a)].">
      <j1 rule="para" parents="718 754"/>
      <j2 rule="demod" parents="748"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="912">
    <literal><![CDATA[
      (x ^ y) v y = y
    ]]></literal>
    <justification jstring="[back_demod(903),demod(904(7))].">
      <j1 rule="back_demod" parents="903"/>
      <j2 rule="demod" parents="904"/>
    </justification>
  </clause>

  <clause id="913">
    <literal><![CDATA[
      (((x ^ y) v (y ^ z)) ^ u) v y = y
    ]]></literal>
    <justification jstring="[back_demod(762),demod(904(10))].">
      <j1 rule="back_demod" parents="762"/>
      <j2 rule="demod" parents="904"/>
    </justification>
  </clause>

  <clause id="914">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v y = y
    ]]></literal>
    <justification jstring="[back_demod(719),demod(904(9))].">
      <j1 rule="back_demod" parents="719"/>
      <j2 rule="demod" parents="904"/>
    </justification>
  </clause>

  <clause id="918">
    <literal><![CDATA[
      (x ' v y ') ^ y ' = y '
    ]]></literal>
    <justification jstring="[para(912(a,1),20(a,1,1)),demod(836(3)),flip(a)].">
      <j1 rule="para" parents="912 20"/>
      <j2 rule="demod" parents="836"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="920">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(912(a,1),781(a,1,1)),demod(912(4))].">
      <j1 rule="para" parents="912 781"/>
      <j2 rule="demod" parents="912"/>
    </justification>
  </clause>

  <clause id="934">
    <literal><![CDATA[
      x ^ ((y ^ x) v (x ^ z)) = (y ^ x) v (x ^ z)
    ]]></literal>
    <justification jstring="[para(914(a,1),780(a,1,1))].">
      <j1 rule="para" parents="914 780"/>
    </justification>
  </clause>

  <clause id="935">
    <literal><![CDATA[
      ((x ^ (y v z)) v y) v (y v z) = y v z
    ]]></literal>
    <justification jstring="[para(780(a,1),914(a,1,1,2))].">
      <j1 rule="para" parents="780 914"/>
    </justification>
  </clause>

  <clause id="936">
    <literal><![CDATA[
      x v ((y ^ x) v (x ^ z)) = x
    ]]></literal>
    <justification jstring="[para(914(a,1),781(a,1,1)),demod(914(8))].">
      <j1 rule="para" parents="914 781"/>
      <j2 rule="demod" parents="914"/>
    </justification>
  </clause>

  <clause id="945">
    <literal><![CDATA[
      (x ' v y) ^ y = y
    ]]></literal>
    <justification jstring="[para(666(a,1),918(a,1,1,2)),demod(666(4),666(5))].">
      <j1 rule="para" parents="666 918"/>
      <j2 rule="demod" parents="666 666"/>
    </justification>
  </clause>

  <clause id="952">
    <literal><![CDATA[
      (x v y) ^ y = y
    ]]></literal>
    <justification jstring="[para(666(a,1),945(a,1,1,1))].">
      <j1 rule="para" parents="666 945"/>
    </justification>
  </clause>

  <clause id="960">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(952(a,1),874(a,1,1))].">
      <j1 rule="para" parents="952 874"/>
    </justification>
  </clause>

  <clause id="961">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(952(a,1),892(a,1,1)),demod(952(4))].">
      <j1 rule="para" parents="952 892"/>
      <j2 rule="demod" parents="952"/>
    </justification>
  </clause>

  <clause id="963">
    <literal><![CDATA[
      ((x ^ (y v z)) v z) v (y v z) = y v z
    ]]></literal>
    <justification jstring="[para(952(a,1),914(a,1,1,2))].">
      <j1 rule="para" parents="952 914"/>
    </justification>
  </clause>

  <clause id="974">
    <literal><![CDATA[
      x ' ^ ((y ' v x ') ^ (x ' v z ')) = x '
    ]]></literal>
    <justification jstring="[para(936(a,1),20(a,1,1)),demod(20(6),836(4),836(7)),flip(a)].">
      <j1 rule="para" parents="936 20"/>
      <j2 rule="demod" parents="20 836 836"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="975">
    <literal><![CDATA[
      (x v y) v ((z ^ (x v y)) v x) = x v y
    ]]></literal>
    <justification jstring="[para(780(a,1),936(a,1,2,2))].">
      <j1 rule="para" parents="780 936"/>
    </justification>
  </clause>

  <clause id="980">
    <literal><![CDATA[
      (x v y) v (y v ((x v y) ^ z)) = x v y
    ]]></literal>
    <justification jstring="[para(961(a,1),936(a,1,2,1))].">
      <j1 rule="para" parents="961 936"/>
    </justification>
  </clause>

  <clause id="1056">
    <literal><![CDATA[
      x v ((y ' v (x ^ y)) ^ y) = x
    ]]></literal>
    <justification jstring="[para(769(a,1),778(a,1)),flip(a)].">
      <j1 rule="para" parents="769 778"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1071">
    <literal><![CDATA[
      ((x ' v (y ^ x)) ^ x) v y = y
    ]]></literal>
    <justification jstring="[para(1056(a,1),960(a,1,2)),demod(1056(10))].">
      <j1 rule="para" parents="1056 960"/>
      <j2 rule="demod" parents="1056"/>
    </justification>
  </clause>

  <clause id="1077">
    <literal><![CDATA[
      (((x ' ^ y ') v x) ^ (x v y)) v x = x
    ]]></literal>
    <justification jstring="[para(828(a,1),1071(a,1,1,1,2)),demod(20(2))].">
      <j1 rule="para" parents="828 1071"/>
      <j2 rule="demod" parents="20"/>
    </justification>
  </clause>

  <clause id="1081">
    <literal><![CDATA[
      (((x ' ^ y ') v y) ^ (x v y)) v y = y
    ]]></literal>
    <justification jstring="[para(961(a,1),1071(a,1,1,1,2)),demod(20(2))].">
      <j1 rule="para" parents="961 1071"/>
      <j2 rule="demod" parents="20"/>
    </justification>
  </clause>

  <clause id="1104">
    <literal><![CDATA[
      ((x ^ y) ^ z) v x = x
    ]]></literal>
    <justification jstring="[para(874(a,1),913(a,1,1,1))].">
      <j1 rule="para" parents="874 913"/>
    </justification>
  </clause>

  <clause id="1127">
    <literal><![CDATA[
      (x ^ y) v (x v z) = x v z
    ]]></literal>
    <justification jstring="[para(780(a,1),1104(a,1,1,1))].">
      <j1 rule="para" parents="780 1104"/>
    </justification>
  </clause>

  <clause id="1131">
    <literal><![CDATA[
      (x ^ y) v (z v x) = z v x
    ]]></literal>
    <justification jstring="[para(952(a,1),1104(a,1,1,1))].">
      <j1 rule="para" parents="952 1104"/>
    </justification>
  </clause>

  <clause id="1214">
    <literal><![CDATA[
      (x v y) ^ (x ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(1127(a,1),780(a,1,1))].">
      <j1 rule="para" parents="1127 780"/>
    </justification>
  </clause>

  <clause id="1228">
    <literal><![CDATA[
      (x v y) ^ (y ^ z) = y ^ z
    ]]></literal>
    <justification jstring="[para(1131(a,1),780(a,1,1))].">
      <j1 rule="para" parents="1131 780"/>
    </justification>
  </clause>

  <clause id="1321">
    <literal><![CDATA[
      ((x v y) v z) ^ x = x
    ]]></literal>
    <justification jstring="[para(780(a,1),1214(a,1,2)),demod(780(5))].">
      <j1 rule="para" parents="780 1214"/>
      <j2 rule="demod" parents="780"/>
    </justification>
  </clause>

  <clause id="1323">
    <literal><![CDATA[
      ((x v y) v z) ^ y = y
    ]]></literal>
    <justification jstring="[para(952(a,1),1214(a,1,2)),demod(952(5))].">
      <j1 rule="para" parents="952 1214"/>
      <j2 rule="demod" parents="952"/>
    </justification>
  </clause>

  <clause id="1353">
    <literal><![CDATA[
      x ^ ((x v y) v z) = x
    ]]></literal>
    <justification jstring="[para(1321(a,1),892(a,1,1)),demod(1321(6))].">
      <j1 rule="para" parents="1321 892"/>
      <j2 rule="demod" parents="1321"/>
    </justification>
  </clause>

  <clause id="1384">
    <literal><![CDATA[
      x ^ ((y v x) v z) = x
    ]]></literal>
    <justification jstring="[para(1323(a,1),892(a,1,1)),demod(1323(6))].">
      <j1 rule="para" parents="1323 892"/>
      <j2 rule="demod" parents="1323"/>
    </justification>
  </clause>

  <clause id="1407">
    <literal><![CDATA[
      x ^ (y v (x v z)) = x
    ]]></literal>
    <justification jstring="[para(960(a,1),1353(a,1,2))].">
      <j1 rule="para" parents="960 1353"/>
    </justification>
  </clause>

  <clause id="1428">
    <literal><![CDATA[
      (x v y) ^ (x v ((x v y) ^ z)) = x v ((x v y) ^ z)
    ]]></literal>
    <justification jstring="[para(828(a,1),934(a,1,2,1)),demod(828(7))].">
      <j1 rule="para" parents="828 934"/>
      <j2 rule="demod" parents="828"/>
    </justification>
  </clause>

  <clause id="1501">
    <literal><![CDATA[
      x ^ (y v (z v x)) = x
    ]]></literal>
    <justification jstring="[para(960(a,1),1384(a,1,2))].">
      <j1 rule="para" parents="960 1384"/>
    </justification>
  </clause>

  <clause id="1547">
    <literal><![CDATA[
      (x v y) v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(961(a,1),935(a,1,1,1))].">
      <j1 rule="para" parents="961 935"/>
    </justification>
  </clause>

  <clause id="1568">
    <literal><![CDATA[
      (x v y) v (y v (x v z)) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(1407(a,1),935(a,1,1,1))].">
      <j1 rule="para" parents="1407 935"/>
    </justification>
  </clause>

  <clause id="1574">
    <literal><![CDATA[
      (x v y) v (y v (z v x)) = y v (z v x)
    ]]></literal>
    <justification jstring="[para(1501(a,1),935(a,1,1,1))].">
      <j1 rule="para" parents="1501 935"/>
    </justification>
  </clause>

  <clause id="1788">
    <literal><![CDATA[
      (x v y) v ((z v x) v y) = (z v x) v y
    ]]></literal>
    <justification jstring="[para(1384(a,1),963(a,1,1,1))].">
      <j1 rule="para" parents="1384 963"/>
    </justification>
  </clause>

  <clause id="2017">
    <literal><![CDATA[
      x ^ ((y ' v x) ^ (x v z ')) = x
    ]]></literal>
    <justification jstring="[para(666(a,1),974(a,1,1)),demod(666(3),666(4),666(8))].">
      <j1 rule="para" parents="666 974"/>
      <j2 rule="demod" parents="666 666 666"/>
    </justification>
  </clause>

  <clause id="2241">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[para(961(a,1),975(a,1,2,1)),demod(1547(3))].">
      <j1 rule="para" parents="961 975"/>
      <j2 rule="demod" parents="1547"/>
    </justification>
  </clause>

  <clause id="2441">
    <literal><![CDATA[
      x v ((x v (y ' ^ x ')) ^ (y v x)) = x
    ]]></literal>
    <justification jstring="[back_demod(1081),demod(2241(4),2241(7))].">
      <j1 rule="back_demod" parents="1081"/>
      <j2 rule="demod" parents="2241 2241"/>
    </justification>
  </clause>

  <clause id="2442">
    <literal><![CDATA[
      x v ((x v (x ' ^ y ')) ^ (x v y)) = x
    ]]></literal>
    <justification jstring="[back_demod(1077),demod(2241(4),2241(7))].">
      <j1 rule="back_demod" parents="1077"/>
      <j2 rule="demod" parents="2241 2241"/>
    </justification>
  </clause>

  <clause id="2456">
    <literal><![CDATA[
      x ' ^ y ' = y ' ^ x '
    ]]></literal>
    <justification jstring="[para(2241(a,1),20(a,1,1)),demod(20(2))].">
      <j1 rule="para" parents="2241 20"/>
      <j2 rule="demod" parents="20"/>
    </justification>
  </clause>

  <clause id="2746">
    <literal><![CDATA[
      x ' ^ y = y ^ x '
    ]]></literal>
    <justification jstring="[para(666(a,1),2456(a,1,1)),demod(666(5)),flip(a)].">
      <j1 rule="para" parents="666 2456"/>
      <j2 rule="demod" parents="666"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2789">
    <literal><![CDATA[
      x ^ y = y ^ x
    ]]></literal>
    <justification jstring="[para(666(a,1),2746(a,1,1)),demod(666(3))].">
      <j1 rule="para" parents="666 2746"/>
      <j2 rule="demod" parents="666"/>
    </justification>
  </clause>

  <clause id="2845">
    <literal><![CDATA[
      x v ((x v y) ^ (x v (x ' ^ y '))) = x
    ]]></literal>
    <justification jstring="[back_demod(2442),demod(2789(6))].">
      <j1 rule="back_demod" parents="2442"/>
      <j2 rule="demod" parents="2789"/>
    </justification>
  </clause>

  <clause id="2846">
    <literal><![CDATA[
      x v ((y v x) ^ (x v (y ' ^ x '))) = x
    ]]></literal>
    <justification jstring="[back_demod(2441),demod(2789(6))].">
      <j1 rule="back_demod" parents="2441"/>
      <j2 rule="demod" parents="2789"/>
    </justification>
  </clause>

  <clause id="8061">
    <literal><![CDATA[
      x ^ ((y v x) ^ (x v z ')) = x
    ]]></literal>
    <justification jstring="[para(666(a,1),2017(a,1,2,1,1))].">
      <j1 rule="para" parents="666 2017"/>
    </justification>
  </clause>

  <clause id="8106">
    <literal><![CDATA[
      x ^ ((y v x) ^ (x v z)) = x
    ]]></literal>
    <justification jstring="[para(666(a,1),8061(a,1,2,2,2))].">
      <j1 rule="para" parents="666 8061"/>
    </justification>
  </clause>

  <clause id="8175">
    <literal><![CDATA[
      x v ((y v x) ^ (x v z)) = (y v x) ^ (x v z)
    ]]></literal>
    <justification jstring="[para(8106(a,1),920(a,1,2)),demod(2241(4))].">
      <j1 rule="para" parents="8106 920"/>
      <j2 rule="demod" parents="2241"/>
    </justification>
  </clause>

  <clause id="8214">
    <literal><![CDATA[
      x ^ ((x v y) ^ (x v z)) = x
    ]]></literal>
    <justification jstring="[para(2241(a,1),8106(a,1,2,1))].">
      <j1 rule="para" parents="2241 8106"/>
    </justification>
  </clause>

  <clause id="8259">
    <literal><![CDATA[
      (x v y) ^ (y v (x ' ^ y ')) = y
    ]]></literal>
    <justification jstring="[back_demod(2846),demod(8175(7))].">
      <j1 rule="back_demod" parents="2846"/>
      <j2 rule="demod" parents="8175"/>
    </justification>
  </clause>

  <clause id="8374">
    <literal><![CDATA[
      x v ((x v y) ^ (x v z)) = (x v y) ^ (x v z)
    ]]></literal>
    <justification jstring="[para(8214(a,1),920(a,1,2)),demod(2241(4))].">
      <j1 rule="para" parents="8214 920"/>
      <j2 rule="demod" parents="2241"/>
    </justification>
  </clause>

  <clause id="8433">
    <literal><![CDATA[
      (x v y) ^ (x v (x ' ^ y ')) = x
    ]]></literal>
    <justification jstring="[back_demod(2845),demod(8374(7))].">
      <j1 rule="back_demod" parents="2845"/>
      <j2 rule="demod" parents="8374"/>
    </justification>
  </clause>

  <clause id="13951">
    <literal><![CDATA[
      x v (y v ((y v x) ^ z)) = y v x
    ]]></literal>
    <justification jstring="[para(1428(a,1),980(a,1,2,2)),demod(1568(6))].">
      <j1 rule="para" parents="1428 980"/>
      <j2 rule="demod" parents="1568"/>
    </justification>
  </clause>

  <clause id="14068">
    <literal><![CDATA[
      x v (y v (x ^ z)) = y v x
    ]]></literal>
    <justification jstring="[para(1228(a,1),13951(a,1,2,2))].">
      <j1 rule="para" parents="1228 13951"/>
    </justification>
  </clause>

  <clause id="14152">
    <literal><![CDATA[
      (x v y) v (z v y) = z v (x v y)
    ]]></literal>
    <justification jstring="[para(8259(a,1),14068(a,1,2,2))].">
      <j1 rule="para" parents="8259 14068"/>
    </justification>
  </clause>

  <clause id="14153">
    <literal><![CDATA[
      (x v y) v (z v x) = z v (x v y)
    ]]></literal>
    <justification jstring="[para(8433(a,1),14068(a,1,2,2))].">
      <j1 rule="para" parents="8433 14068"/>
    </justification>
  </clause>

  <clause id="14166">
    <literal><![CDATA[
      (x v y) v (y v z) = (x v y) v z
    ]]></literal>
    <justification jstring="[back_demod(1788),demod(14152(4))].">
      <j1 rule="back_demod" parents="1788"/>
      <j2 rule="demod" parents="14152"/>
    </justification>
  </clause>

  <clause id="14181">
    <literal><![CDATA[
      x v (y v z) = z v (x v y)
    ]]></literal>
    <justification jstring="[back_demod(1574),demod(14166(4),14153(3))].">
      <j1 rule="back_demod" parents="1574"/>
      <j2 rule="demod" parents="14166 14153"/>
    </justification>
  </clause>

  <clause id="14713">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <justification jstring="[back_demod(859),demod(14181(8,R),2241(7)),xx(a)].">
      <j1 rule="back_demod" parents="859"/>
      <j2 rule="demod" parents="14181 2241"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
