<?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 2009-02A, February 2009.
Process 11656 was started by mccune on cleo,
Wed Feb 25 09:35:32 2009
The command was "/home/mccune/bin/prover9 -f omlsax2.in".
]]></heading>

<proof number="1" length="77" max_count="38">

<comments><![CDATA[
% Proof 1 at 0.13 (+ 0.00) seconds: B_SS.
% Length of proof is 77.
% Level of proof is 20.
% Maximum clause weight is 38.
% Given clauses 71.

]]></comments>

  <clause id="2" type="goal">
    <literal><![CDATA[
      f(f(x,x),f(x,y)) = x
    ]]></literal>
    <attribute><![CDATA[
      answer(B_SS)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="4" type="assumption">
    <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="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="5" type="assumption">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(definition_join)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

  <clause id="7" type="assumption">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(definition_meet)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      f(f(x,y),f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[copy(7),flip(a)].">
      <j1 rule="copy" parents="7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9" type="assumption">
    <literal><![CDATA[
      x' = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(definition_complementation)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      f(x,x) = x'
    ]]></literal>
    <justification jstring="[copy(9),flip(a)].">
      <j1 rule="copy" parents="9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13" type="deny">
    <literal><![CDATA[
      f(f(c4,c4),f(c4,c5)) != c4
    ]]></literal>
    <attribute><![CDATA[
      answer(B_SS)
    ]]></attribute>
    <justification jstring="[deny(2)].">
      <j1 rule="deny" parents="2"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      f(c4',f(c4,c5)) != c4
    ]]></literal>
    <attribute><![CDATA[
      answer(B_SS)
    ]]></attribute>
    <justification jstring="[copy(13),rewrite([10(3)])].">
      <j1 rule="copy" parents="13"/>
      <j2 rule="rewrite" parents="10"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      f(x,y)' = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(8),rewrite([10(3)])].">
      <j1 rule="back_rewrite" parents="8"/>
      <j2 rule="rewrite" parents="10"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(x',y') = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(6),rewrite([10(1),10(2)])].">
      <j1 rule="back_rewrite" parents="6"/>
      <j2 rule="rewrite" parents="10 10"/>
    </justification>
  </clause>

  <clause id="18">
    <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_rewrite(4),rewrite([10(5)])].">
      <j1 rule="back_rewrite" parents="4"/>
      <j2 rule="rewrite" parents="10"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      (x v y)' = x' ^ y'
    ]]></literal>
    <justification jstring="[para(17(a,1),16(a,1,1))].">
      <j1 rule="para" parents="17 16"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      f(x ^ y,z') = f(x,y) v z
    ]]></literal>
    <justification jstring="[para(16(a,1),17(a,1,1))].">
      <j1 rule="para" parents="16 17"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      f(x',y ^ z) = x v f(y,z)
    ]]></literal>
    <justification jstring="[para(16(a,1),17(a,1,2))].">
      <j1 rule="para" parents="16 17"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      f(f(f(x',f(x,y)),z),f(x,f(f(y,f(x',y)),y))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),18(a,1,1,1,1))].">
      <j1 rule="para" parents="10 18"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      f(f(x'',y),f(x,f(f(x,f(x',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),18(a,1,1,1)),rewrite([10(1)])].">
      <j1 rule="para" parents="10 18"/>
      <j2 rule="rewrite" parents="10"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,y')),z),f(y,f(y v y',y'))) = y
    ]]></literal>
    <justification jstring="[para(10(a,1),18(a,1,2,2,1,2)),rewrite([17(9)])].">
      <j1 rule="para" parents="10 18"/>
      <j2 rule="rewrite" parents="17"/>
    </justification>
  </clause>

  <clause id="35">
    <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(18(a,1),18(a,1,1,1))].">
      <j1 rule="para" parents="18 18"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(z,f(x ^ y,z)),z))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(18(a,1),18(a,1,1)),rewrite([16(3)])].">
      <j1 rule="para" parents="18 18"/>
      <j2 rule="rewrite" parents="16"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      f(x' ^ f(x,y),f(x,f(f(y,f(x',y)),y))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),25(a,1,1)),rewrite([16(4)])].">
      <j1 rule="para" parents="10 25"/>
      <j2 rule="rewrite" parents="16"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      f(x''',f(x,f(f(x,f(x',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),27(a,1,1))].">
      <j1 rule="para" parents="10 27"/>
    </justification>
  </clause>

  <clause id="58">
    <literal><![CDATA[
      f(x' v y,f(x,f(f(x,f(x',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(17(a,1),27(a,1,1))].">
      <j1 rule="para" parents="17 27"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(x,f(x',f(f(y,f(x'',y)),y))) = x'
    ]]></literal>
    <justification jstring="[para(27(a,1),25(a,1,1))].">
      <j1 rule="para" parents="27 25"/>
    </justification>
  </clause>

  <clause id="63">
    <literal><![CDATA[
      x''' ^ f(x,f(f(x,f(x',x)),x)) = x'
    ]]></literal>
    <justification jstring="[para(55(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="55 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      (x' v y) ^ f(x,f(f(x,f(x',x)),x)) = x'
    ]]></literal>
    <justification jstring="[para(58(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="58 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="83">
    <literal><![CDATA[
      x ^ f(x',f(f(y,f(x'',y)),y)) = x''
    ]]></literal>
    <justification jstring="[para(62(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="62 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="85">
    <literal><![CDATA[
      f(x,f(x',f(f(y',x' v y),y'))) = x'
    ]]></literal>
    <justification jstring="[para(17(a,1),62(a,1,2,2,1,2))].">
      <j1 rule="para" parents="17 62"/>
    </justification>
  </clause>

  <clause id="121">
    <literal><![CDATA[
      x ^ f(f(x,y),f(f(z,f(x ^ y,z)),z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(36(a,1),16(a,1,1)),rewrite([16(2)]),flip(a)].">
      <j1 rule="para" parents="36 16"/>
      <j2 rule="rewrite" parents="16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="131">
    <literal><![CDATA[
      f(x''',f(x,f(f(y,f(x',y)),y))) = x
    ]]></literal>
    <justification jstring="[para(55(a,1),36(a,1,2,1)),rewrite([63(12),55(18)])].">
      <j1 rule="para" parents="55 36"/>
      <j2 rule="rewrite" parents="63 55"/>
    </justification>
  </clause>

  <clause id="132">
    <literal><![CDATA[
      f(x' v y,f(x,f(f(z,f(x',z)),z))) = x
    ]]></literal>
    <justification jstring="[para(58(a,1),36(a,1,2,1)),rewrite([69(10),58(16)])].">
      <j1 rule="para" parents="58 36"/>
      <j2 rule="rewrite" parents="69 58"/>
    </justification>
  </clause>

  <clause id="135">
    <literal><![CDATA[
      f(x''',f(x,f(x v x',x'))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),131(a,1,2,2,1,2)),rewrite([17(7)])].">
      <j1 rule="para" parents="10 131"/>
      <j2 rule="rewrite" parents="17"/>
    </justification>
  </clause>

  <clause id="136">
    <literal><![CDATA[
      x''' ^ f(x,f(f(y,f(x',y)),y)) = x'
    ]]></literal>
    <justification jstring="[para(131(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="131 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="154">
    <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(135(a,1),83(a,1,2,2,1,2))].">
      <j1 rule="para" parents="135 83"/>
    </justification>
  </clause>

  <clause id="224">
    <literal><![CDATA[
      (x' ^ f(x,y)) ^ f(x,f(f(y,f(x',y)),y)) = x'
    ]]></literal>
    <justification jstring="[para(43(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="43 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="233">
    <literal><![CDATA[
      f(x' ^ f(x,y),f(x,f(f(z,f(x',z)),z))) = x
    ]]></literal>
    <justification jstring="[para(43(a,1),36(a,1,2,1)),rewrite([224(12),43(18)])].">
      <j1 rule="para" parents="43 36"/>
      <j2 rule="rewrite" parents="224 43"/>
    </justification>
  </clause>

  <clause id="287">
    <literal><![CDATA[
      f(x' ^ f(x,y),f(x,f(x v x',x'))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),233(a,1,2,2,1,2)),rewrite([17(7)])].">
      <j1 rule="para" parents="10 233"/>
      <j2 rule="rewrite" parents="17"/>
    </justification>
  </clause>

  <clause id="307">
    <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(287(a,1),36(a,1,2,2,1,2))].">
      <j1 rule="para" parents="287 36"/>
    </justification>
  </clause>

  <clause id="311">
    <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(287(a,1),121(a,1,2,2,1,2))].">
      <j1 rule="para" parents="287 121"/>
    </justification>
  </clause>

  <clause id="317">
    <literal><![CDATA[
      f(f(x,y),f(y,f(y v y',y'))) = y
    ]]></literal>
    <justification jstring="[para(35(a,1),29(a,1,1))].">
      <j1 rule="para" parents="35 29"/>
    </justification>
  </clause>

  <clause id="323">
    <literal><![CDATA[
      x' ^ f(f(x',f(x,y)),x) = x' ^ f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(311),rewrite([317(16)])].">
      <j1 rule="back_rewrite" parents="311"/>
      <j2 rule="rewrite" parents="317"/>
    </justification>
  </clause>

  <clause id="324">
    <literal><![CDATA[
      f(x',f(f(x',f(x,y)),x)) = f(x',f(x,y))
    ]]></literal>
    <justification jstring="[back_rewrite(307),rewrite([317(16)])].">
      <j1 rule="back_rewrite" parents="307"/>
      <j2 rule="rewrite" parents="317"/>
    </justification>
  </clause>

  <clause id="332">
    <literal><![CDATA[
      x' ^ f(x'',x) = x'''
    ]]></literal>
    <justification jstring="[back_rewrite(154),rewrite([317(15)])].">
      <j1 rule="back_rewrite" parents="154"/>
      <j2 rule="rewrite" parents="317"/>
    </justification>
  </clause>

  <clause id="349">
    <literal><![CDATA[
      x'' ^ (x'' v x) = x''''
    ]]></literal>
    <justification jstring="[para(17(a,1),332(a,1,2))].">
      <j1 rule="para" parents="17 332"/>
    </justification>
  </clause>

  <clause id="360">
    <literal><![CDATA[
      f(x',f(x,f(x v x',x'))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),317(a,1,1))].">
      <j1 rule="para" parents="10 317"/>
    </justification>
  </clause>

  <clause id="377">
    <literal><![CDATA[
      x' ^ f(x,f(x v x',x')) = x'
    ]]></literal>
    <justification jstring="[para(360(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="360 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="378">
    <literal><![CDATA[
      f(x ^ y,f(f(x,y),f(f(x,y) v (x ^ y),x ^ y))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(16(a,1),360(a,1,1)),rewrite([16(5),16(7)])].">
      <j1 rule="para" parents="16 360"/>
      <j2 rule="rewrite" parents="16 16"/>
    </justification>
  </clause>

  <clause id="379">
    <literal><![CDATA[
      f(x' ^ y',f(x v y,f((x v y) v (x' ^ y'),x' ^ y'))) = x v y
    ]]></literal>
    <justification jstring="[para(22(a,1),360(a,1,1)),rewrite([22(7),22(11)])].">
      <j1 rule="para" parents="22 360"/>
      <j2 rule="rewrite" parents="22 22"/>
    </justification>
  </clause>

  <clause id="380">
    <literal><![CDATA[
      f(x,x'') = x'
    ]]></literal>
    <justification jstring="[para(360(a,1),62(a,1,2,2,1,2)),rewrite([317(22),10(3)])].">
      <j1 rule="para" parents="360 62"/>
      <j2 rule="rewrite" parents="317 10"/>
    </justification>
  </clause>

  <clause id="381">
    <literal><![CDATA[
      x ^ x'' = x''
    ]]></literal>
    <justification jstring="[para(360(a,1),83(a,1,2,2,1,2)),rewrite([317(22),10(3)])].">
      <j1 rule="para" parents="360 83"/>
      <j2 rule="rewrite" parents="317 10"/>
    </justification>
  </clause>

  <clause id="383">
    <literal><![CDATA[
      x'' v x = x
    ]]></literal>
    <justification jstring="[para(360(a,1),131(a,1,2,2,1,2)),rewrite([317(15),10(4),17(5)])].">
      <j1 rule="para" parents="360 131"/>
      <j2 rule="rewrite" parents="317 10 17"/>
    </justification>
  </clause>

  <clause id="384">
    <literal><![CDATA[
      f(x' v y,x') = x
    ]]></literal>
    <justification jstring="[para(360(a,1),132(a,1,2,2,1,2)),rewrite([317(14),10(3)])].">
      <j1 rule="para" parents="360 132"/>
      <j2 rule="rewrite" parents="317 10"/>
    </justification>
  </clause>

  <clause id="386">
    <literal><![CDATA[
      x''' ^ x' = x'
    ]]></literal>
    <justification jstring="[para(360(a,1),136(a,1,2,2,1,2)),rewrite([317(15),10(4)])].">
      <j1 rule="para" parents="360 136"/>
      <j2 rule="rewrite" parents="317 10"/>
    </justification>
  </clause>

  <clause id="388">
    <literal><![CDATA[
      x' ^ f(x,f(f(y,f(x',y)),y)) = x'
    ]]></literal>
    <justification jstring="[para(360(a,1),121(a,1,2,1)),rewrite([377(8),377(14)])].">
      <j1 rule="para" parents="360 121"/>
      <j2 rule="rewrite" parents="377 377"/>
    </justification>
  </clause>

  <clause id="402">
    <literal><![CDATA[
      x'' ^ x = x''''
    ]]></literal>
    <justification jstring="[back_rewrite(349),rewrite([383(5)])].">
      <j1 rule="back_rewrite" parents="349"/>
      <j2 rule="rewrite" parents="383"/>
    </justification>
  </clause>

  <clause id="406">
    <literal><![CDATA[
      x''''' = x'
    ]]></literal>
    <justification jstring="[back_rewrite(386),rewrite([402(5)])].">
      <j1 rule="back_rewrite" parents="386"/>
      <j2 rule="rewrite" parents="402"/>
    </justification>
  </clause>

  <clause id="428">
    <literal><![CDATA[
      x v x'' = x''
    ]]></literal>
    <justification jstring="[para(380(a,1),17(a,1)),flip(a)].">
      <j1 rule="para" parents="380 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="481">
    <literal><![CDATA[
      x'''' = x
    ]]></literal>
    <justification jstring="[para(406(a,1),380(a,1,2)),rewrite([17(5),383(3)]),flip(a)].">
      <j1 rule="para" parents="406 380"/>
      <j2 rule="rewrite" parents="17 383"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="494">
    <literal><![CDATA[
      (x ^ y)''' = f(x,y)
    ]]></literal>
    <justification jstring="[para(16(a,1),481(a,1,1,1,1))].">
      <j1 rule="para" parents="16 481"/>
    </justification>
  </clause>

  <clause id="495">
    <literal><![CDATA[
      f(x,y') = x''' v y
    ]]></literal>
    <justification jstring="[para(481(a,1),17(a,1,1))].">
      <j1 rule="para" parents="481 17"/>
    </justification>
  </clause>

  <clause id="496">
    <literal><![CDATA[
      f(x',y) = x v y'''
    ]]></literal>
    <justification jstring="[para(481(a,1),17(a,1,2))].">
      <j1 rule="para" parents="481 17"/>
    </justification>
  </clause>

  <clause id="497">
    <literal><![CDATA[
      f(x ^ y,z) = f(x,y) v z'''
    ]]></literal>
    <justification jstring="[para(481(a,1),23(a,1,2))].">
      <j1 rule="para" parents="481 23"/>
    </justification>
  </clause>

  <clause id="498">
    <literal><![CDATA[
      f(x,y ^ z) = x''' v f(y,z)
    ]]></literal>
    <justification jstring="[para(481(a,1),24(a,1,1))].">
      <j1 rule="para" parents="481 24"/>
    </justification>
  </clause>

  <clause id="505">
    <literal><![CDATA[
      x'' v (x ^ ((y' ^ (x v y))'' v y))'' = x
    ]]></literal>
    <justification jstring="[para(481(a,1),85(a,1,2,1)),rewrite([481(8),496(6),22(5),495(11),22(10),494(10),495(7),481(8),496(11),16(10),481(16)])].">
      <j1 rule="para" parents="481 85"/>
      <j2 rule="rewrite" parents="481 496 22 495 22 494 495 481 496 16 481"/>
    </justification>
  </clause>

  <clause id="543">
    <literal><![CDATA[
      f(x v y,x) = x'''
    ]]></literal>
    <justification jstring="[para(481(a,1),384(a,1,1,1)),rewrite([481(5)])].">
      <j1 rule="para" parents="481 384"/>
      <j2 rule="rewrite" parents="481"/>
    </justification>
  </clause>

  <clause id="646">
    <literal><![CDATA[
      x' ^ f(x,f(f(y,x v y'''),y)) = x'
    ]]></literal>
    <justification jstring="[back_rewrite(388),rewrite([496(3)])].">
      <j1 rule="back_rewrite" parents="388"/>
      <j2 rule="rewrite" parents="496"/>
    </justification>
  </clause>

  <clause id="650">
    <literal><![CDATA[
      x v (x ^ y)'' = x''
    ]]></literal>
    <justification jstring="[back_rewrite(324),rewrite([496(4),16(3),543(6),495(5),481(4),428(3),496(5),16(4)]),flip(a)].">
      <j1 rule="back_rewrite" parents="324"/>
      <j2 rule="rewrite" parents="496 16 543 495 481 428 496 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="651">
    <literal><![CDATA[
      x' ^ f(x,y) = x'''
    ]]></literal>
    <justification jstring="[back_rewrite(323),rewrite([496(4),16(3),650(5),496(4),428(6),381(5)]),flip(a)].">
      <j1 rule="back_rewrite" parents="323"/>
      <j2 rule="rewrite" parents="496 16 650 496 428 381"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="699">
    <literal><![CDATA[
      c4'' != c4
    ]]></literal>
    <attribute><![CDATA[
      answer(B_SS)
    ]]></attribute>
    <justification jstring="[back_rewrite(14),rewrite([496(6),16(5),650(7)])].">
      <j1 rule="back_rewrite" parents="14"/>
      <j2 rule="rewrite" parents="496 16 650"/>
    </justification>
  </clause>

  <clause id="701">
    <literal><![CDATA[
      (x' ^ y')' = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(379),rewrite([498(13),22(10),22(6),495(17),481(18),497(18),495(3),481(4),16(16),650(18),22(2)])].">
      <j1 rule="back_rewrite" parents="379"/>
      <j2 rule="rewrite" parents="498 22 22 495 481 497 495 481 16 650 22"/>
    </justification>
  </clause>

  <clause id="702">
    <literal><![CDATA[
      (x ^ y)' = f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(378),rewrite([498(7),22(6),16(4),497(12),16(12),650(14),16(2)])].">
      <j1 rule="back_rewrite" parents="378"/>
      <j2 rule="rewrite" parents="498 22 16 497 16 650 16"/>
    </justification>
  </clause>

  <clause id="778">
    <literal><![CDATA[
      x''' = x'
    ]]></literal>
    <justification jstring="[back_rewrite(646),rewrite([651(9)])].">
      <j1 rule="back_rewrite" parents="646"/>
      <j2 rule="rewrite" parents="651"/>
    </justification>
  </clause>

  <clause id="816">
    <literal><![CDATA[
      x'' v y = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(701),rewrite([702(4),495(3),778(3)])].">
      <j1 rule="back_rewrite" parents="701"/>
      <j2 rule="rewrite" parents="702 495 778"/>
    </justification>
  </clause>

  <clause id="842">
    <literal><![CDATA[
      x v (x ^ y) = x''
    ]]></literal>
    <justification jstring="[back_rewrite(650),rewrite([702(2),16(2)])].">
      <j1 rule="back_rewrite" parents="650"/>
      <j2 rule="rewrite" parents="702 16"/>
    </justification>
  </clause>

  <clause id="910">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[back_rewrite(505),rewrite([702(6),496(5),22(4),702(6),495(5),778(5),816(5),22(4),22(7),702(7),495(6),778(6),816(6),702(8),16(8),816(8),842(6)])].">
      <j1 rule="back_rewrite" parents="505"/>
      <j2 rule="rewrite" parents="702 496 22 702 495 778 816 22 22 702 495 778 816 702 16 816 842"/>
    </justification>
  </clause>

  <clause id="911">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(B_SS)
    ]]></attribute>
    <justification jstring="[resolve(910,a,699,a)].">
      <j1 rule="resolve" parents="910 699"/>
    </justification>
  </clause>

</proof>

<proof number="2" length="135" max_count="38">

<comments><![CDATA[
% Proof 2 at 0.87 (+ 0.02) seconds: OM_SS.
% Length of proof is 135.
% Level of proof is 31.
% Maximum clause weight is 38.
% Given clauses 323.

]]></comments>

  <clause id="3" type="goal">
    <literal><![CDATA[
      f(x,f(x,f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      answer(OM_SS)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="4" type="assumption">
    <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="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="5" type="assumption">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(definition_join)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

  <clause id="7" type="assumption">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(definition_meet)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      f(f(x,y),f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[copy(7),flip(a)].">
      <j1 rule="copy" parents="7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9" type="assumption">
    <literal><![CDATA[
      x' = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(definition_complementation)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      f(x,x) = x'
    ]]></literal>
    <justification jstring="[copy(9),flip(a)].">
      <j1 rule="copy" parents="9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="15" type="deny">
    <literal><![CDATA[
      f(c6,f(c6,f(c6,c7))) != f(c6,c7)
    ]]></literal>
    <attribute><![CDATA[
      answer(OM_SS)
    ]]></attribute>
    <justification jstring="[deny(3)].">
      <j1 rule="deny" parents="3"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      f(x,y)' = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(8),rewrite([10(3)])].">
      <j1 rule="back_rewrite" parents="8"/>
      <j2 rule="rewrite" parents="10"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(x',y') = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(6),rewrite([10(1),10(2)])].">
      <j1 rule="back_rewrite" parents="6"/>
      <j2 rule="rewrite" parents="10 10"/>
    </justification>
  </clause>

  <clause id="18">
    <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_rewrite(4),rewrite([10(5)])].">
      <j1 rule="back_rewrite" parents="4"/>
      <j2 rule="rewrite" parents="10"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x ^ x = x''
    ]]></literal>
    <justification jstring="[para(10(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="10 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      x v x = x''
    ]]></literal>
    <justification jstring="[para(17(a,1),10(a,1))].">
      <j1 rule="para" parents="17 10"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      (x v y)' = x' ^ y'
    ]]></literal>
    <justification jstring="[para(17(a,1),16(a,1,1))].">
      <j1 rule="para" parents="17 16"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      f(x ^ y,z') = f(x,y) v z
    ]]></literal>
    <justification jstring="[para(16(a,1),17(a,1,1))].">
      <j1 rule="para" parents="16 17"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      f(x',y ^ z) = x v f(y,z)
    ]]></literal>
    <justification jstring="[para(16(a,1),17(a,1,2))].">
      <j1 rule="para" parents="16 17"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      f(f(f(x',f(x,y)),z),f(x,f(f(y,f(x',y)),y))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),18(a,1,1,1,1))].">
      <j1 rule="para" parents="10 18"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      f(f(x'',y),f(x,f(f(x,f(x',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),18(a,1,1,1)),rewrite([10(1)])].">
      <j1 rule="para" parents="10 18"/>
      <j2 rule="rewrite" parents="10"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      f(f(x,y) ^ f(y,z),f(y,f(f(z,f(y',z)),z))) = y
    ]]></literal>
    <justification jstring="[para(10(a,1),18(a,1,1)),rewrite([16(4)])].">
      <j1 rule="para" parents="10 18"/>
      <j2 rule="rewrite" parents="16"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,y')),z),f(y,f(y v y',y'))) = y
    ]]></literal>
    <justification jstring="[para(10(a,1),18(a,1,2,2,1,2)),rewrite([17(9)])].">
      <j1 rule="para" parents="10 18"/>
      <j2 rule="rewrite" parents="17"/>
    </justification>
  </clause>

  <clause id="35">
    <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(18(a,1),18(a,1,1,1))].">
      <j1 rule="para" parents="18 18"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(z,f(x ^ y,z)),z))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(18(a,1),18(a,1,1)),rewrite([16(3)])].">
      <j1 rule="para" parents="18 18"/>
      <j2 rule="rewrite" parents="16"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      f(x' ^ f(x,y),f(x,f(f(y,f(x',y)),y))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),25(a,1,1)),rewrite([16(4)])].">
      <j1 rule="para" parents="10 25"/>
      <j2 rule="rewrite" parents="16"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      f(x''',f(x,f(f(x,f(x',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),27(a,1,1))].">
      <j1 rule="para" parents="10 27"/>
    </justification>
  </clause>

  <clause id="56">
    <literal><![CDATA[
      f(x'',y) ^ f(x,f(f(x,f(x',x)),x)) = x'
    ]]></literal>
    <justification jstring="[para(27(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="27 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="58">
    <literal><![CDATA[
      f(x' v y,f(x,f(f(x,f(x',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(17(a,1),27(a,1,1))].">
      <j1 rule="para" parents="17 27"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(x,f(x',f(f(y,f(x'',y)),y))) = x'
    ]]></literal>
    <justification jstring="[para(27(a,1),25(a,1,1))].">
      <j1 rule="para" parents="27 25"/>
    </justification>
  </clause>

  <clause id="63">
    <literal><![CDATA[
      x''' ^ f(x,f(f(x,f(x',x)),x)) = x'
    ]]></literal>
    <justification jstring="[para(55(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="55 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      (x' v y) ^ f(x,f(f(x,f(x',x)),x)) = x'
    ]]></literal>
    <justification jstring="[para(58(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="58 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="83">
    <literal><![CDATA[
      x ^ f(x',f(f(y,f(x'',y)),y)) = x''
    ]]></literal>
    <justification jstring="[para(62(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="62 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="85">
    <literal><![CDATA[
      f(x,f(x',f(f(y',x' v y),y'))) = x'
    ]]></literal>
    <justification jstring="[para(17(a,1),62(a,1,2,2,1,2))].">
      <j1 rule="para" parents="17 62"/>
    </justification>
  </clause>

  <clause id="101">
    <literal><![CDATA[
      (f(x,y) ^ f(y,z)) ^ f(y,f(f(z,f(y',z)),z)) = y'
    ]]></literal>
    <justification jstring="[para(28(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="28 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="121">
    <literal><![CDATA[
      x ^ f(f(x,y),f(f(z,f(x ^ y,z)),z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(36(a,1),16(a,1,1)),rewrite([16(2)]),flip(a)].">
      <j1 rule="para" parents="36 16"/>
      <j2 rule="rewrite" parents="16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="130">
    <literal><![CDATA[
      f(f(x'',y),f(x,f(f(z,f(x',z)),z))) = x
    ]]></literal>
    <justification jstring="[para(27(a,1),36(a,1,2,1)),rewrite([56(12),27(18)])].">
      <j1 rule="para" parents="27 36"/>
      <j2 rule="rewrite" parents="56 27"/>
    </justification>
  </clause>

  <clause id="131">
    <literal><![CDATA[
      f(x''',f(x,f(f(y,f(x',y)),y))) = x
    ]]></literal>
    <justification jstring="[para(55(a,1),36(a,1,2,1)),rewrite([63(12),55(18)])].">
      <j1 rule="para" parents="55 36"/>
      <j2 rule="rewrite" parents="63 55"/>
    </justification>
  </clause>

  <clause id="132">
    <literal><![CDATA[
      f(x' v y,f(x,f(f(z,f(x',z)),z))) = x
    ]]></literal>
    <justification jstring="[para(58(a,1),36(a,1,2,1)),rewrite([69(10),58(16)])].">
      <j1 rule="para" parents="58 36"/>
      <j2 rule="rewrite" parents="69 58"/>
    </justification>
  </clause>

  <clause id="134">
    <literal><![CDATA[
      f(f(x,y) ^ f(y,z),f(y,f(f(u,f(y',u)),u))) = y
    ]]></literal>
    <justification jstring="[para(28(a,1),36(a,1,2,1)),rewrite([101(12),28(18)])].">
      <j1 rule="para" parents="28 36"/>
      <j2 rule="rewrite" parents="101 28"/>
    </justification>
  </clause>

  <clause id="135">
    <literal><![CDATA[
      f(x''',f(x,f(x v x',x'))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),131(a,1,2,2,1,2)),rewrite([17(7)])].">
      <j1 rule="para" parents="10 131"/>
      <j2 rule="rewrite" parents="17"/>
    </justification>
  </clause>

  <clause id="136">
    <literal><![CDATA[
      x''' ^ f(x,f(f(y,f(x',y)),y)) = x'
    ]]></literal>
    <justification jstring="[para(131(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="131 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="154">
    <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(135(a,1),83(a,1,2,2,1,2))].">
      <j1 rule="para" parents="135 83"/>
    </justification>
  </clause>

  <clause id="157">
    <literal><![CDATA[
      (x' v y) ^ f(x,f(f(z,f(x',z)),z)) = x'
    ]]></literal>
    <justification jstring="[para(132(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="132 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="185">
    <literal><![CDATA[
      f(f(x'',y),f(x,f(f(z',x v z),z'))) = x
    ]]></literal>
    <justification jstring="[para(17(a,1),130(a,1,2,2,1,2))].">
      <j1 rule="para" parents="17 130"/>
    </justification>
  </clause>

  <clause id="209">
    <literal><![CDATA[
      (x' v y) ^ f(x,f(f(z',x v z),z')) = x'
    ]]></literal>
    <justification jstring="[para(17(a,1),157(a,1,2,2,1,2))].">
      <j1 rule="para" parents="17 157"/>
    </justification>
  </clause>

  <clause id="224">
    <literal><![CDATA[
      (x' ^ f(x,y)) ^ f(x,f(f(y,f(x',y)),y)) = x'
    ]]></literal>
    <justification jstring="[para(43(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="43 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="233">
    <literal><![CDATA[
      f(x' ^ f(x,y),f(x,f(f(z,f(x',z)),z))) = x
    ]]></literal>
    <justification jstring="[para(43(a,1),36(a,1,2,1)),rewrite([224(12),43(18)])].">
      <j1 rule="para" parents="43 36"/>
      <j2 rule="rewrite" parents="224 43"/>
    </justification>
  </clause>

  <clause id="287">
    <literal><![CDATA[
      f(x' ^ f(x,y),f(x,f(x v x',x'))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),233(a,1,2,2,1,2)),rewrite([17(7)])].">
      <j1 rule="para" parents="10 233"/>
      <j2 rule="rewrite" parents="17"/>
    </justification>
  </clause>

  <clause id="307">
    <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(287(a,1),36(a,1,2,2,1,2))].">
      <j1 rule="para" parents="287 36"/>
    </justification>
  </clause>

  <clause id="311">
    <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(287(a,1),121(a,1,2,2,1,2))].">
      <j1 rule="para" parents="287 121"/>
    </justification>
  </clause>

  <clause id="317">
    <literal><![CDATA[
      f(f(x,y),f(y,f(y v y',y'))) = y
    ]]></literal>
    <justification jstring="[para(35(a,1),29(a,1,1))].">
      <j1 rule="para" parents="35 29"/>
    </justification>
  </clause>

  <clause id="323">
    <literal><![CDATA[
      x' ^ f(f(x',f(x,y)),x) = x' ^ f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(311),rewrite([317(16)])].">
      <j1 rule="back_rewrite" parents="311"/>
      <j2 rule="rewrite" parents="317"/>
    </justification>
  </clause>

  <clause id="324">
    <literal><![CDATA[
      f(x',f(f(x',f(x,y)),x)) = f(x',f(x,y))
    ]]></literal>
    <justification jstring="[back_rewrite(307),rewrite([317(16)])].">
      <j1 rule="back_rewrite" parents="307"/>
      <j2 rule="rewrite" parents="317"/>
    </justification>
  </clause>

  <clause id="332">
    <literal><![CDATA[
      x' ^ f(x'',x) = x'''
    ]]></literal>
    <justification jstring="[back_rewrite(154),rewrite([317(15)])].">
      <j1 rule="back_rewrite" parents="154"/>
      <j2 rule="rewrite" parents="317"/>
    </justification>
  </clause>

  <clause id="349">
    <literal><![CDATA[
      x'' ^ (x'' v x) = x''''
    ]]></literal>
    <justification jstring="[para(17(a,1),332(a,1,2))].">
      <j1 rule="para" parents="17 332"/>
    </justification>
  </clause>

  <clause id="360">
    <literal><![CDATA[
      f(x',f(x,f(x v x',x'))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),317(a,1,1))].">
      <j1 rule="para" parents="10 317"/>
    </justification>
  </clause>

  <clause id="361">
    <literal><![CDATA[
      f(x,y) ^ f(y,f(y v y',y')) = y'
    ]]></literal>
    <justification jstring="[para(317(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="317 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="377">
    <literal><![CDATA[
      x' ^ f(x,f(x v x',x')) = x'
    ]]></literal>
    <justification jstring="[para(360(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="360 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="378">
    <literal><![CDATA[
      f(x ^ y,f(f(x,y),f(f(x,y) v (x ^ y),x ^ y))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(16(a,1),360(a,1,1)),rewrite([16(5),16(7)])].">
      <j1 rule="para" parents="16 360"/>
      <j2 rule="rewrite" parents="16 16"/>
    </justification>
  </clause>

  <clause id="379">
    <literal><![CDATA[
      f(x' ^ y',f(x v y,f((x v y) v (x' ^ y'),x' ^ y'))) = x v y
    ]]></literal>
    <justification jstring="[para(22(a,1),360(a,1,1)),rewrite([22(7),22(11)])].">
      <j1 rule="para" parents="22 360"/>
      <j2 rule="rewrite" parents="22 22"/>
    </justification>
  </clause>

  <clause id="380">
    <literal><![CDATA[
      f(x,x'') = x'
    ]]></literal>
    <justification jstring="[para(360(a,1),62(a,1,2,2,1,2)),rewrite([317(22),10(3)])].">
      <j1 rule="para" parents="360 62"/>
      <j2 rule="rewrite" parents="317 10"/>
    </justification>
  </clause>

  <clause id="381">
    <literal><![CDATA[
      x ^ x'' = x''
    ]]></literal>
    <justification jstring="[para(360(a,1),83(a,1,2,2,1,2)),rewrite([317(22),10(3)])].">
      <j1 rule="para" parents="360 83"/>
      <j2 rule="rewrite" parents="317 10"/>
    </justification>
  </clause>

  <clause id="383">
    <literal><![CDATA[
      x'' v x = x
    ]]></literal>
    <justification jstring="[para(360(a,1),131(a,1,2,2,1,2)),rewrite([317(15),10(4),17(5)])].">
      <j1 rule="para" parents="360 131"/>
      <j2 rule="rewrite" parents="317 10 17"/>
    </justification>
  </clause>

  <clause id="384">
    <literal><![CDATA[
      f(x' v y,x') = x
    ]]></literal>
    <justification jstring="[para(360(a,1),132(a,1,2,2,1,2)),rewrite([317(14),10(3)])].">
      <j1 rule="para" parents="360 132"/>
      <j2 rule="rewrite" parents="317 10"/>
    </justification>
  </clause>

  <clause id="385">
    <literal><![CDATA[
      f(f(x'',y),x') = x
    ]]></literal>
    <justification jstring="[para(360(a,1),130(a,1,2,2,1,2)),rewrite([317(15),10(4)])].">
      <j1 rule="para" parents="360 130"/>
      <j2 rule="rewrite" parents="317 10"/>
    </justification>
  </clause>

  <clause id="386">
    <literal><![CDATA[
      x''' ^ x' = x'
    ]]></literal>
    <justification jstring="[para(360(a,1),136(a,1,2,2,1,2)),rewrite([317(15),10(4)])].">
      <j1 rule="para" parents="360 136"/>
      <j2 rule="rewrite" parents="317 10"/>
    </justification>
  </clause>

  <clause id="387">
    <literal><![CDATA[
      (x' v y) ^ x' = x'
    ]]></literal>
    <justification jstring="[para(360(a,1),157(a,1,2,2,1,2)),rewrite([317(14),10(3)])].">
      <j1 rule="para" parents="360 157"/>
      <j2 rule="rewrite" parents="317 10"/>
    </justification>
  </clause>

  <clause id="388">
    <literal><![CDATA[
      x' ^ f(x,f(f(y,f(x',y)),y)) = x'
    ]]></literal>
    <justification jstring="[para(360(a,1),121(a,1,2,1)),rewrite([377(8),377(14)])].">
      <j1 rule="para" parents="360 121"/>
      <j2 rule="rewrite" parents="377 377"/>
    </justification>
  </clause>

  <clause id="402">
    <literal><![CDATA[
      x'' ^ x = x''''
    ]]></literal>
    <justification jstring="[back_rewrite(349),rewrite([383(5)])].">
      <j1 rule="back_rewrite" parents="349"/>
      <j2 rule="rewrite" parents="383"/>
    </justification>
  </clause>

  <clause id="406">
    <literal><![CDATA[
      x''''' = x'
    ]]></literal>
    <justification jstring="[back_rewrite(386),rewrite([402(5)])].">
      <j1 rule="back_rewrite" parents="386"/>
      <j2 rule="rewrite" parents="402"/>
    </justification>
  </clause>

  <clause id="423">
    <literal><![CDATA[
      (x''' v y) ^ f(x'',f(f(x',x),x')) = x'''
    ]]></literal>
    <justification jstring="[para(383(a,1),209(a,1,2,2,1,2))].">
      <j1 rule="para" parents="383 209"/>
    </justification>
  </clause>

  <clause id="428">
    <literal><![CDATA[
      x v x'' = x''
    ]]></literal>
    <justification jstring="[para(380(a,1),17(a,1)),flip(a)].">
      <j1 rule="para" parents="380 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="481">
    <literal><![CDATA[
      x'''' = x
    ]]></literal>
    <justification jstring="[para(406(a,1),380(a,1,2)),rewrite([17(5),383(3)]),flip(a)].">
      <j1 rule="para" parents="406 380"/>
      <j2 rule="rewrite" parents="17 383"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="494">
    <literal><![CDATA[
      (x ^ y)''' = f(x,y)
    ]]></literal>
    <justification jstring="[para(16(a,1),481(a,1,1,1,1))].">
      <j1 rule="para" parents="16 481"/>
    </justification>
  </clause>

  <clause id="495">
    <literal><![CDATA[
      f(x,y') = x''' v y
    ]]></literal>
    <justification jstring="[para(481(a,1),17(a,1,1))].">
      <j1 rule="para" parents="481 17"/>
    </justification>
  </clause>

  <clause id="496">
    <literal><![CDATA[
      f(x',y) = x v y'''
    ]]></literal>
    <justification jstring="[para(481(a,1),17(a,1,2))].">
      <j1 rule="para" parents="481 17"/>
    </justification>
  </clause>

  <clause id="497">
    <literal><![CDATA[
      f(x ^ y,z) = f(x,y) v z'''
    ]]></literal>
    <justification jstring="[para(481(a,1),23(a,1,2))].">
      <j1 rule="para" parents="481 23"/>
    </justification>
  </clause>

  <clause id="498">
    <literal><![CDATA[
      f(x,y ^ z) = x''' v f(y,z)
    ]]></literal>
    <justification jstring="[para(481(a,1),24(a,1,1))].">
      <j1 rule="para" parents="481 24"/>
    </justification>
  </clause>

  <clause id="505">
    <literal><![CDATA[
      x'' v (x ^ ((y' ^ (x v y))'' v y))'' = x
    ]]></literal>
    <justification jstring="[para(481(a,1),85(a,1,2,1)),rewrite([481(8),496(6),22(5),495(11),22(10),494(10),495(7),481(8),496(11),16(10),481(16)])].">
      <j1 rule="para" parents="481 85"/>
      <j2 rule="rewrite" parents="481 496 22 495 22 494 495 481 496 16 481"/>
    </justification>
  </clause>

  <clause id="533">
    <literal><![CDATA[
      f(f(x,y),x' v ((z v (x''' ^ z')'') ^ z')'') = x''
    ]]></literal>
    <justification jstring="[para(481(a,1),185(a,1,1,1)),rewrite([496(8),22(7),495(13),22(12),494(12),495(9),481(8),496(12),22(11),494(10),496(7),22(6)])].">
      <j1 rule="para" parents="481 185"/>
      <j2 rule="rewrite" parents="496 22 495 22 494 495 481 496 22 494 496 22"/>
    </justification>
  </clause>

  <clause id="543">
    <literal><![CDATA[
      f(x v y,x) = x'''
    ]]></literal>
    <justification jstring="[para(481(a,1),384(a,1,1,1)),rewrite([481(5)])].">
      <j1 rule="para" parents="481 384"/>
      <j2 rule="rewrite" parents="481"/>
    </justification>
  </clause>

  <clause id="555">
    <literal><![CDATA[
      (x''' v y) ^ (x' v ((x v x''') ^ x')'') = x'''
    ]]></literal>
    <justification jstring="[back_rewrite(423),rewrite([496(8),495(12),22(11),481(11),496(12),22(11),494(10),496(7)])].">
      <j1 rule="back_rewrite" parents="423"/>
      <j2 rule="rewrite" parents="496 495 22 481 496 22 494 496"/>
    </justification>
  </clause>

  <clause id="572">
    <literal><![CDATA[
      (x'' ^ y)'' v x = x
    ]]></literal>
    <justification jstring="[back_rewrite(385),rewrite([496(3),495(7),22(6),481(6)])].">
      <j1 rule="back_rewrite" parents="385"/>
      <j2 rule="rewrite" parents="496 495 22 481"/>
    </justification>
  </clause>

  <clause id="579">
    <literal><![CDATA[
      f(x,y) ^ f(y,(y' ^ y'')'' v y) = y'
    ]]></literal>
    <justification jstring="[back_rewrite(361),rewrite([495(5),22(4)])].">
      <j1 rule="back_rewrite" parents="361"/>
      <j2 rule="rewrite" parents="495 22"/>
    </justification>
  </clause>

  <clause id="646">
    <literal><![CDATA[
      x' ^ f(x,f(f(y,x v y'''),y)) = x'
    ]]></literal>
    <justification jstring="[back_rewrite(388),rewrite([496(3)])].">
      <j1 rule="back_rewrite" parents="388"/>
      <j2 rule="rewrite" parents="496"/>
    </justification>
  </clause>

  <clause id="650">
    <literal><![CDATA[
      x v (x ^ y)'' = x''
    ]]></literal>
    <justification jstring="[back_rewrite(324),rewrite([496(4),16(3),543(6),495(5),481(4),428(3),496(5),16(4)]),flip(a)].">
      <j1 rule="back_rewrite" parents="324"/>
      <j2 rule="rewrite" parents="496 16 543 495 481 428 496 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="651">
    <literal><![CDATA[
      x' ^ f(x,y) = x'''
    ]]></literal>
    <justification jstring="[back_rewrite(323),rewrite([496(4),16(3),650(5),496(4),428(6),381(5)]),flip(a)].">
      <j1 rule="back_rewrite" parents="323"/>
      <j2 rule="rewrite" parents="496 16 650 496 428 381"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="682">
    <literal><![CDATA[
      f(f(x,y),f(y,z)) v (y ^ f(f(u,y v u'''),u))'' = y
    ]]></literal>
    <justification jstring="[back_rewrite(134),rewrite([496(5),497(11),16(11)])].">
      <j1 rule="back_rewrite" parents="134"/>
      <j2 rule="rewrite" parents="496 497 16"/>
    </justification>
  </clause>

  <clause id="701">
    <literal><![CDATA[
      (x' ^ y')' = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(379),rewrite([498(13),22(10),22(6),495(17),481(18),497(18),495(3),481(4),16(16),650(18),22(2)])].">
      <j1 rule="back_rewrite" parents="379"/>
      <j2 rule="rewrite" parents="498 22 22 495 481 497 495 481 16 650 22"/>
    </justification>
  </clause>

  <clause id="702">
    <literal><![CDATA[
      (x ^ y)' = f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(378),rewrite([498(7),22(6),16(4),497(12),16(12),650(14),16(2)])].">
      <j1 rule="back_rewrite" parents="378"/>
      <j2 rule="rewrite" parents="498 22 16 497 16 650 16"/>
    </justification>
  </clause>

  <clause id="778">
    <literal><![CDATA[
      x''' = x'
    ]]></literal>
    <justification jstring="[back_rewrite(646),rewrite([651(9)])].">
      <j1 rule="back_rewrite" parents="646"/>
      <j2 rule="rewrite" parents="651"/>
    </justification>
  </clause>

  <clause id="802">
    <literal><![CDATA[
      f(x,y) ^ f(y,(y' ^ y'') v y) = y'
    ]]></literal>
    <justification jstring="[back_rewrite(579),rewrite([702(6),495(5),778(4),22(6),778(4)])].">
      <j1 rule="back_rewrite" parents="579"/>
      <j2 rule="rewrite" parents="702 495 778 22 778"/>
    </justification>
  </clause>

  <clause id="811">
    <literal><![CDATA[
      f(f(x,y),x' v ((z v (x' ^ z')) ^ z')) = x''
    ]]></literal>
    <justification jstring="[back_rewrite(533),rewrite([778(5),702(6),495(5),778(5),22(6),778(5),702(9),495(8),22(7),702(7),495(6),778(6),702(8),496(7),22(6),778(5),702(6),495(5),778(5),22(6),778(5),22(7),702(7),495(6),778(6),22(9),702(8),496(7),22(6),778(5),702(6),495(5),778(5),22(6),778(5)])].">
      <j1 rule="back_rewrite" parents="533"/>
      <j2 rule="rewrite" parents="778 702 495 778 22 778 702 495 22 702 495 778 702 496 22 778 702 495 778 22 778 22 702 495 778 22 702 496 22 778 702 495 778 22 778"/>
    </justification>
  </clause>

  <clause id="816">
    <literal><![CDATA[
      x'' v y = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(701),rewrite([702(4),495(3),778(3)])].">
      <j1 rule="back_rewrite" parents="701"/>
      <j2 rule="rewrite" parents="702 495 778"/>
    </justification>
  </clause>

  <clause id="830">
    <literal><![CDATA[
      f(f(x,y),f(y,z)) v (y ^ f(f(u,y v u'),u)) = y
    ]]></literal>
    <justification jstring="[back_rewrite(682),rewrite([778(6),702(9),16(9)])].">
      <j1 rule="back_rewrite" parents="682"/>
      <j2 rule="rewrite" parents="778 702 16"/>
    </justification>
  </clause>

  <clause id="842">
    <literal><![CDATA[
      x v (x ^ y) = x''
    ]]></literal>
    <justification jstring="[back_rewrite(650),rewrite([702(2),16(2)])].">
      <j1 rule="back_rewrite" parents="650"/>
      <j2 rule="rewrite" parents="702 16"/>
    </justification>
  </clause>

  <clause id="874">
    <literal><![CDATA[
      (x'' ^ y'') v x = x
    ]]></literal>
    <justification jstring="[back_rewrite(572),rewrite([702(4),496(3),778(4),22(4)])].">
      <j1 rule="back_rewrite" parents="572"/>
      <j2 rule="rewrite" parents="702 496 778 22"/>
    </justification>
  </clause>

  <clause id="883">
    <literal><![CDATA[
      (x' v y) ^ (x' v ((x v x') ^ x')) = x'
    ]]></literal>
    <justification jstring="[back_rewrite(555),rewrite([778(3),778(6),702(8),495(7),22(6),702(8),495(7),778(6),816(7),22(6),22(9),702(8),495(7),778(6),816(7),778(12)])].">
      <j1 rule="back_rewrite" parents="555"/>
      <j2 rule="rewrite" parents="778 778 702 495 22 702 495 778 816 22 22 702 495 778 816 778"/>
    </justification>
  </clause>

  <clause id="910">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[back_rewrite(505),rewrite([702(6),496(5),22(4),702(6),495(5),778(5),816(5),22(4),22(7),702(7),495(6),778(6),816(6),702(8),16(8),816(8),842(6)])].">
      <j1 rule="back_rewrite" parents="505"/>
      <j2 rule="rewrite" parents="702 496 22 702 495 778 816 22 22 702 495 778 816 702 16 816 842"/>
    </justification>
  </clause>

  <clause id="946">
    <literal><![CDATA[
      x' ^ f(x,y) = x'
    ]]></literal>
    <justification jstring="[back_rewrite(651),rewrite([910(5)])].">
      <j1 rule="back_rewrite" parents="651"/>
      <j2 rule="rewrite" parents="910"/>
    </justification>
  </clause>

  <clause id="954">
    <literal><![CDATA[
      f(x',y) = x v y'
    ]]></literal>
    <justification jstring="[back_rewrite(496),rewrite([910(4)])].">
      <j1 rule="back_rewrite" parents="496"/>
      <j2 rule="rewrite" parents="910"/>
    </justification>
  </clause>

  <clause id="979">
    <literal><![CDATA[
      (x ^ y) v x = x
    ]]></literal>
    <justification jstring="[back_rewrite(874),rewrite([910(2),910(2)])].">
      <j1 rule="back_rewrite" parents="874"/>
      <j2 rule="rewrite" parents="910 910"/>
    </justification>
  </clause>

  <clause id="985">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[back_rewrite(842),rewrite([910(4)])].">
      <j1 rule="back_rewrite" parents="842"/>
      <j2 rule="rewrite" parents="910"/>
    </justification>
  </clause>

  <clause id="994">
    <literal><![CDATA[
      f(f(x,y),x' v ((z v (x' ^ z')) ^ z')) = x
    ]]></literal>
    <justification jstring="[back_rewrite(811),rewrite([910(12)])].">
      <j1 rule="back_rewrite" parents="811"/>
      <j2 rule="rewrite" parents="910"/>
    </justification>
  </clause>

  <clause id="999">
    <literal><![CDATA[
      f(x,y) ^ f(y,(y' ^ y) v y) = y'
    ]]></literal>
    <justification jstring="[back_rewrite(802),rewrite([910(4)])].">
      <j1 rule="back_rewrite" parents="802"/>
      <j2 rule="rewrite" parents="910"/>
    </justification>
  </clause>

  <clause id="1007">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[back_rewrite(21),rewrite([910(3)])].">
      <j1 rule="back_rewrite" parents="21"/>
      <j2 rule="rewrite" parents="910"/>
    </justification>
  </clause>

  <clause id="1008">
    <literal><![CDATA[
      x ^ x = x
    ]]></literal>
    <justification jstring="[back_rewrite(20),rewrite([910(3)])].">
      <j1 rule="back_rewrite" parents="20"/>
      <j2 rule="rewrite" parents="910"/>
    </justification>
  </clause>

  <clause id="1010">
    <literal><![CDATA[
      (x v y) ^ x = x
    ]]></literal>
    <justification jstring="[para(910(a,1),387(a,1,1,1)),rewrite([910(3),910(4)])].">
      <j1 rule="para" parents="910 387"/>
      <j2 rule="rewrite" parents="910 910"/>
    </justification>
  </clause>

  <clause id="1018">
    <literal><![CDATA[
      x ^ (x v y') = x
    ]]></literal>
    <justification jstring="[para(910(a,1),946(a,1,1)),rewrite([954(2),910(5)])].">
      <j1 rule="para" parents="910 946"/>
      <j2 rule="rewrite" parents="954 910"/>
    </justification>
  </clause>

  <clause id="1021">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[para(910(a,1),1018(a,1,2,2))].">
      <j1 rule="para" parents="910 1018"/>
    </justification>
  </clause>

  <clause id="1041">
    <literal><![CDATA[
      (x ^ y) ^ x = x ^ y
    ]]></literal>
    <justification jstring="[para(979(a,1),1021(a,1,2))].">
      <j1 rule="para" parents="979 1021"/>
    </justification>
  </clause>

  <clause id="1046">
    <literal><![CDATA[
      f(x,y) = x' v y'
    ]]></literal>
    <justification jstring="[para(910(a,1),954(a,1,1))].">
      <j1 rule="para" parents="910 954"/>
    </justification>
  </clause>

  <clause id="1063">
    <literal><![CDATA[
      (x' v y') ^ (y' v ((y v y') ^ y')) = y'
    ]]></literal>
    <justification jstring="[back_rewrite(999),rewrite([1046(1),1046(7),22(8),702(7),1046(6),910(6)])].">
      <j1 rule="back_rewrite" parents="999"/>
      <j2 rule="rewrite" parents="1046 1046 22 702 1046 910"/>
    </justification>
  </clause>

  <clause id="1066">
    <literal><![CDATA[
      (x ^ y) v (x ^ ((z' ^ (x v z)) v z)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(994),rewrite([1046(1),1046(12),22(4),910(2),910(2),22(10),910(3),702(8),1046(7),22(6),702(6),1046(5),910(4),910(4),910(6)])].">
      <j1 rule="back_rewrite" parents="994"/>
      <j2 rule="rewrite" parents="1046 1046 22 910 910 22 910 702 1046 22 702 1046 910 910 910"/>
    </justification>
  </clause>

  <clause id="1106">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v (y ^ ((u ^ (y v u')) v u')) = y
    ]]></literal>
    <justification jstring="[back_rewrite(830),rewrite([1046(1),1046(4),1046(7),22(4),910(2),910(2),22(5),910(3),910(3),1046(6),22(7),910(7),1046(8),22(8),910(5),702(6),1046(5),910(5)])].">
      <j1 rule="back_rewrite" parents="830"/>
      <j2 rule="rewrite" parents="1046 1046 1046 22 910 910 22 910 910 1046 22 910 1046 22 910 702 1046 910"/>
    </justification>
  </clause>

  <clause id="1121">
    <literal><![CDATA[
      (x ^ y)' = x' v y'
    ]]></literal>
    <justification jstring="[back_rewrite(702),rewrite([1046(3)])].">
      <j1 rule="back_rewrite" parents="702"/>
      <j2 rule="rewrite" parents="1046"/>
    </justification>
  </clause>

  <clause id="1122">
    <literal><![CDATA[
      c6' v (c6 ^ (c6' v c7')) != c6' v c7'
    ]]></literal>
    <attribute><![CDATA[
      answer(OM_SS)
    ]]></attribute>
    <justification jstring="[back_rewrite(15),rewrite([1046(5),1046(8),22(9),910(6),910(7),1046(8),22(9),910(5),1121(7),1046(13)])].">
      <j1 rule="back_rewrite" parents="15"/>
      <j2 rule="rewrite" parents="1046 1046 22 910 910 1046 22 910 1121 1046"/>
    </justification>
  </clause>

  <clause id="1127">
    <literal><![CDATA[
      x' v ((x v x') ^ x') = x'
    ]]></literal>
    <justification jstring="[para(883(a,1),1008(a,1)),flip(a)].">
      <j1 rule="para" parents="883 1008"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1128">
    <literal><![CDATA[
      (x' v y') ^ y' = y'
    ]]></literal>
    <justification jstring="[back_rewrite(1063),rewrite([1127(9)])].">
      <j1 rule="back_rewrite" parents="1063"/>
      <j2 rule="rewrite" parents="1127"/>
    </justification>
  </clause>

  <clause id="1153">
    <literal><![CDATA[
      (x' v y) ^ y = y
    ]]></literal>
    <justification jstring="[para(910(a,1),1128(a,1,1,2)),rewrite([910(4),910(5)])].">
      <j1 rule="para" parents="910 1128"/>
      <j2 rule="rewrite" parents="910 910"/>
    </justification>
  </clause>

  <clause id="1155">
    <literal><![CDATA[
      (x v y) ^ y = y
    ]]></literal>
    <justification jstring="[para(910(a,1),1153(a,1,1,1))].">
      <j1 rule="para" parents="910 1153"/>
    </justification>
  </clause>

  <clause id="1168">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(1155(a,1),1041(a,1,1)),rewrite([1155(4)])].">
      <j1 rule="para" parents="1155 1041"/>
      <j2 rule="rewrite" parents="1155"/>
    </justification>
  </clause>

  <clause id="1249">
    <literal><![CDATA[
      x ^ ((y' ^ (x v y)) v y) = x
    ]]></literal>
    <justification jstring="[para(1066(a,1),1007(a,1)),flip(a)].">
      <j1 rule="para" parents="1066 1007"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1263">
    <literal><![CDATA[
      x ^ ((y ^ (x v y')) v y') = x
    ]]></literal>
    <justification jstring="[para(910(a,1),1249(a,1,2,1,1))].">
      <j1 rule="para" parents="910 1249"/>
    </justification>
  </clause>

  <clause id="1264">
    <literal><![CDATA[
      x ^ (((x' v y') ^ x) v (x ^ y)) = x
    ]]></literal>
    <justification jstring="[para(985(a,1),1249(a,1,2,1,2)),rewrite([1121(2)])].">
      <j1 rule="para" parents="985 1249"/>
      <j2 rule="rewrite" parents="1121"/>
    </justification>
  </clause>

  <clause id="1280">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v y = y
    ]]></literal>
    <justification jstring="[back_rewrite(1106),rewrite([1263(9)])].">
      <j1 rule="back_rewrite" parents="1106"/>
      <j2 rule="rewrite" parents="1263"/>
    </justification>
  </clause>

  <clause id="1282">
    <literal><![CDATA[
      x ^ ((y ^ x) v (x ^ z)) = (y ^ x) v (x ^ z)
    ]]></literal>
    <justification jstring="[para(1280(a,1),1010(a,1,1))].">
      <j1 rule="para" parents="1280 1010"/>
    </justification>
  </clause>

  <clause id="1283">
    <literal><![CDATA[
      ((x ^ (y v z)) v y) v (y v z) = y v z
    ]]></literal>
    <justification jstring="[para(1010(a,1),1280(a,1,1,2))].">
      <j1 rule="para" parents="1010 1280"/>
    </justification>
  </clause>

  <clause id="1292">
    <literal><![CDATA[
      ((x' v y') ^ x) v (x ^ y) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1264),rewrite([1282(7)])].">
      <j1 rule="back_rewrite" parents="1264"/>
      <j2 rule="rewrite" parents="1282"/>
    </justification>
  </clause>

  <clause id="3361">
    <literal><![CDATA[
      (x' ^ (x v y)) v x = x v y
    ]]></literal>
    <justification jstring="[para(1010(a,1),1292(a,1,2)),rewrite([22(2),979(5)])].">
      <j1 rule="para" parents="1010 1292"/>
      <j2 rule="rewrite" parents="22 979"/>
    </justification>
  </clause>

  <clause id="3402">
    <literal><![CDATA[
      (x ^ (x' v y)) v x' = x' v y
    ]]></literal>
    <justification jstring="[para(910(a,1),3361(a,1,1,1))].">
      <j1 rule="para" parents="910 3361"/>
    </justification>
  </clause>

  <clause id="4076">
    <literal><![CDATA[
      (x v y) v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(1168(a,1),1283(a,1,1,1))].">
      <j1 rule="para" parents="1168 1283"/>
    </justification>
  </clause>

  <clause id="4469">
    <literal><![CDATA[
      (x v y) ^ (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(4076(a,1),1010(a,1,1))].">
      <j1 rule="para" parents="4076 1010"/>
    </justification>
  </clause>

  <clause id="4470">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[para(4076(a,1),1021(a,1,2)),rewrite([4469(3)])].">
      <j1 rule="para" parents="4076 1021"/>
      <j2 rule="rewrite" parents="4469"/>
    </justification>
  </clause>

  <clause id="4695">
    <literal><![CDATA[
      x' v (x ^ (x' v y)) = x' v y
    ]]></literal>
    <justification jstring="[back_rewrite(3402),rewrite([4470(5)])].">
      <j1 rule="back_rewrite" parents="3402"/>
      <j2 rule="rewrite" parents="4470"/>
    </justification>
  </clause>

  <clause id="4696">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(OM_SS)
    ]]></attribute>
    <justification jstring="[resolve(4695,a,1122,a)].">
      <j1 rule="resolve" parents="4695 1122"/>
    </justification>
  </clause>

</proof>

<proof number="3" length="177" max_count="38">

<comments><![CDATA[
% Proof 3 at 40.33 (+ 0.34) seconds: A_SS.
% Length of proof is 177.
% Level of proof is 39.
% Maximum clause weight is 38.
% Given clauses 1120.

]]></comments>

  <clause id="1" type="goal">
    <literal><![CDATA[
      f(x,f(f(y,z),f(y,z))) = f(y,f(f(x,z),f(x,z)))
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="4" type="assumption">
    <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="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="5" type="assumption">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(definition_join)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

  <clause id="7" type="assumption">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(definition_meet)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      f(f(x,y),f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[copy(7),flip(a)].">
      <j1 rule="copy" parents="7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9" type="assumption">
    <literal><![CDATA[
      x' = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(definition_complementation)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      f(x,x) = x'
    ]]></literal>
    <justification jstring="[copy(9),flip(a)].">
      <j1 rule="copy" parents="9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11" type="deny">
    <literal><![CDATA[
      f(c1,f(f(c2,c3),f(c2,c3))) != f(c2,f(f(c1,c3),f(c1,c3)))
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      f(c2,f(c1,c3)') != f(c1,f(c2,c3)')
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <justification jstring="[copy(11),rewrite([10(8),10(14)]),flip(a)].">
      <j1 rule="copy" parents="11"/>
      <j2 rule="rewrite" parents="10 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      f(x,y)' = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(8),rewrite([10(3)])].">
      <j1 rule="back_rewrite" parents="8"/>
      <j2 rule="rewrite" parents="10"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(x',y') = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(6),rewrite([10(1),10(2)])].">
      <j1 rule="back_rewrite" parents="6"/>
      <j2 rule="rewrite" parents="10 10"/>
    </justification>
  </clause>

  <clause id="18">
    <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_rewrite(4),rewrite([10(5)])].">
      <j1 rule="back_rewrite" parents="4"/>
      <j2 rule="rewrite" parents="10"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(c2,c1 ^ c3) != f(c1,c2 ^ c3)
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <justification jstring="[back_rewrite(12),rewrite([16(5),16(10)])].">
      <j1 rule="back_rewrite" parents="12"/>
      <j2 rule="rewrite" parents="16 16"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x ^ x = x''
    ]]></literal>
    <justification jstring="[para(10(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="10 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      x v x = x''
    ]]></literal>
    <justification jstring="[para(17(a,1),10(a,1))].">
      <j1 rule="para" parents="17 10"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      (x v y)' = x' ^ y'
    ]]></literal>
    <justification jstring="[para(17(a,1),16(a,1,1))].">
      <j1 rule="para" parents="17 16"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      f(x ^ y,z') = f(x,y) v z
    ]]></literal>
    <justification jstring="[para(16(a,1),17(a,1,1))].">
      <j1 rule="para" parents="16 17"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      f(x',y ^ z) = x v f(y,z)
    ]]></literal>
    <justification jstring="[para(16(a,1),17(a,1,2))].">
      <j1 rule="para" parents="16 17"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      f(f(f(x',f(x,y)),z),f(x,f(f(y,f(x',y)),y))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),18(a,1,1,1,1))].">
      <j1 rule="para" parents="10 18"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      f(f(x'',y),f(x,f(f(x,f(x',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),18(a,1,1,1)),rewrite([10(1)])].">
      <j1 rule="para" parents="10 18"/>
      <j2 rule="rewrite" parents="10"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      f(f(x,y) ^ f(y,z),f(y,f(f(z,f(y',z)),z))) = y
    ]]></literal>
    <justification jstring="[para(10(a,1),18(a,1,1)),rewrite([16(4)])].">
      <j1 rule="para" parents="10 18"/>
      <j2 rule="rewrite" parents="16"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,y')),z),f(y,f(y v y',y'))) = y
    ]]></literal>
    <justification jstring="[para(10(a,1),18(a,1,2,2,1,2)),rewrite([17(9)])].">
      <j1 rule="para" parents="10 18"/>
      <j2 rule="rewrite" parents="17"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      f(f(f(x,y),f(y,z)),u) ^ f(y,f(f(z,f(y',z)),z)) = y'
    ]]></literal>
    <justification jstring="[para(18(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="18 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="35">
    <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(18(a,1),18(a,1,1,1))].">
      <j1 rule="para" parents="18 18"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(z,f(x ^ y,z)),z))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(18(a,1),18(a,1,1)),rewrite([16(3)])].">
      <j1 rule="para" parents="18 18"/>
      <j2 rule="rewrite" parents="16"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      f(x' ^ f(x,y),f(x,f(f(y,f(x',y)),y))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),25(a,1,1)),rewrite([16(4)])].">
      <j1 rule="para" parents="10 25"/>
      <j2 rule="rewrite" parents="16"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      f(x''',f(x,f(f(x,f(x',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),27(a,1,1))].">
      <j1 rule="para" parents="10 27"/>
    </justification>
  </clause>

  <clause id="56">
    <literal><![CDATA[
      f(x'',y) ^ f(x,f(f(x,f(x',x)),x)) = x'
    ]]></literal>
    <justification jstring="[para(27(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="27 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="58">
    <literal><![CDATA[
      f(x' v y,f(x,f(f(x,f(x',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(17(a,1),27(a,1,1))].">
      <j1 rule="para" parents="17 27"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(x,f(x',f(f(y,f(x'',y)),y))) = x'
    ]]></literal>
    <justification jstring="[para(27(a,1),25(a,1,1))].">
      <j1 rule="para" parents="27 25"/>
    </justification>
  </clause>

  <clause id="63">
    <literal><![CDATA[
      x''' ^ f(x,f(f(x,f(x',x)),x)) = x'
    ]]></literal>
    <justification jstring="[para(55(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="55 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      (x' v y) ^ f(x,f(f(x,f(x',x)),x)) = x'
    ]]></literal>
    <justification jstring="[para(58(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="58 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="83">
    <literal><![CDATA[
      x ^ f(x',f(f(y,f(x'',y)),y)) = x''
    ]]></literal>
    <justification jstring="[para(62(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="62 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="85">
    <literal><![CDATA[
      f(x,f(x',f(f(y',x' v y),y'))) = x'
    ]]></literal>
    <justification jstring="[para(17(a,1),62(a,1,2,2,1,2))].">
      <j1 rule="para" parents="17 62"/>
    </justification>
  </clause>

  <clause id="101">
    <literal><![CDATA[
      (f(x,y) ^ f(y,z)) ^ f(y,f(f(z,f(y',z)),z)) = y'
    ]]></literal>
    <justification jstring="[para(28(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="28 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="121">
    <literal><![CDATA[
      x ^ f(f(x,y),f(f(z,f(x ^ y,z)),z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(36(a,1),16(a,1,1)),rewrite([16(2)]),flip(a)].">
      <j1 rule="para" parents="36 16"/>
      <j2 rule="rewrite" parents="16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="123">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(w,f(y',w)),w))) = y
    ]]></literal>
    <justification jstring="[para(18(a,1),36(a,1,2,1)),rewrite([30(14),18(20)])].">
      <j1 rule="para" parents="18 36"/>
      <j2 rule="rewrite" parents="30 18"/>
    </justification>
  </clause>

  <clause id="130">
    <literal><![CDATA[
      f(f(x'',y),f(x,f(f(z,f(x',z)),z))) = x
    ]]></literal>
    <justification jstring="[para(27(a,1),36(a,1,2,1)),rewrite([56(12),27(18)])].">
      <j1 rule="para" parents="27 36"/>
      <j2 rule="rewrite" parents="56 27"/>
    </justification>
  </clause>

  <clause id="131">
    <literal><![CDATA[
      f(x''',f(x,f(f(y,f(x',y)),y))) = x
    ]]></literal>
    <justification jstring="[para(55(a,1),36(a,1,2,1)),rewrite([63(12),55(18)])].">
      <j1 rule="para" parents="55 36"/>
      <j2 rule="rewrite" parents="63 55"/>
    </justification>
  </clause>

  <clause id="132">
    <literal><![CDATA[
      f(x' v y,f(x,f(f(z,f(x',z)),z))) = x
    ]]></literal>
    <justification jstring="[para(58(a,1),36(a,1,2,1)),rewrite([69(10),58(16)])].">
      <j1 rule="para" parents="58 36"/>
      <j2 rule="rewrite" parents="69 58"/>
    </justification>
  </clause>

  <clause id="134">
    <literal><![CDATA[
      f(f(x,y) ^ f(y,z),f(y,f(f(u,f(y',u)),u))) = y
    ]]></literal>
    <justification jstring="[para(28(a,1),36(a,1,2,1)),rewrite([101(12),28(18)])].">
      <j1 rule="para" parents="28 36"/>
      <j2 rule="rewrite" parents="101 28"/>
    </justification>
  </clause>

  <clause id="135">
    <literal><![CDATA[
      f(x''',f(x,f(x v x',x'))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),131(a,1,2,2,1,2)),rewrite([17(7)])].">
      <j1 rule="para" parents="10 131"/>
      <j2 rule="rewrite" parents="17"/>
    </justification>
  </clause>

  <clause id="136">
    <literal><![CDATA[
      x''' ^ f(x,f(f(y,f(x',y)),y)) = x'
    ]]></literal>
    <justification jstring="[para(131(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="131 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="154">
    <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(135(a,1),83(a,1,2,2,1,2))].">
      <j1 rule="para" parents="135 83"/>
    </justification>
  </clause>

  <clause id="157">
    <literal><![CDATA[
      (x' v y) ^ f(x,f(f(z,f(x',z)),z)) = x'
    ]]></literal>
    <justification jstring="[para(132(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="132 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="171">
    <literal><![CDATA[
      f(f(f(x,y'),y v z),u) ^ f(y',f(f(z',y' v z),z')) = y''
    ]]></literal>
    <justification jstring="[para(17(a,1),30(a,1,1,1,2)),rewrite([17(11)])].">
      <j1 rule="para" parents="17 30"/>
      <j2 rule="rewrite" parents="17"/>
    </justification>
  </clause>

  <clause id="185">
    <literal><![CDATA[
      f(f(x'',y),f(x,f(f(z',x v z),z'))) = x
    ]]></literal>
    <justification jstring="[para(17(a,1),130(a,1,2,2,1,2))].">
      <j1 rule="para" parents="17 130"/>
    </justification>
  </clause>

  <clause id="209">
    <literal><![CDATA[
      (x' v y) ^ f(x,f(f(z',x v z),z')) = x'
    ]]></literal>
    <justification jstring="[para(17(a,1),157(a,1,2,2,1,2))].">
      <j1 rule="para" parents="17 157"/>
    </justification>
  </clause>

  <clause id="224">
    <literal><![CDATA[
      (x' ^ f(x,y)) ^ f(x,f(f(y,f(x',y)),y)) = x'
    ]]></literal>
    <justification jstring="[para(43(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="43 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="233">
    <literal><![CDATA[
      f(x' ^ f(x,y),f(x,f(f(z,f(x',z)),z))) = x
    ]]></literal>
    <justification jstring="[para(43(a,1),36(a,1,2,1)),rewrite([224(12),43(18)])].">
      <j1 rule="para" parents="43 36"/>
      <j2 rule="rewrite" parents="224 43"/>
    </justification>
  </clause>

  <clause id="287">
    <literal><![CDATA[
      f(x' ^ f(x,y),f(x,f(x v x',x'))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),233(a,1,2,2,1,2)),rewrite([17(7)])].">
      <j1 rule="para" parents="10 233"/>
      <j2 rule="rewrite" parents="17"/>
    </justification>
  </clause>

  <clause id="307">
    <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(287(a,1),36(a,1,2,2,1,2))].">
      <j1 rule="para" parents="287 36"/>
    </justification>
  </clause>

  <clause id="311">
    <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(287(a,1),121(a,1,2,2,1,2))].">
      <j1 rule="para" parents="287 121"/>
    </justification>
  </clause>

  <clause id="317">
    <literal><![CDATA[
      f(f(x,y),f(y,f(y v y',y'))) = y
    ]]></literal>
    <justification jstring="[para(35(a,1),29(a,1,1))].">
      <j1 rule="para" parents="35 29"/>
    </justification>
  </clause>

  <clause id="323">
    <literal><![CDATA[
      x' ^ f(f(x',f(x,y)),x) = x' ^ f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(311),rewrite([317(16)])].">
      <j1 rule="back_rewrite" parents="311"/>
      <j2 rule="rewrite" parents="317"/>
    </justification>
  </clause>

  <clause id="324">
    <literal><![CDATA[
      f(x',f(f(x',f(x,y)),x)) = f(x',f(x,y))
    ]]></literal>
    <justification jstring="[back_rewrite(307),rewrite([317(16)])].">
      <j1 rule="back_rewrite" parents="307"/>
      <j2 rule="rewrite" parents="317"/>
    </justification>
  </clause>

  <clause id="332">
    <literal><![CDATA[
      x' ^ f(x'',x) = x'''
    ]]></literal>
    <justification jstring="[back_rewrite(154),rewrite([317(15)])].">
      <j1 rule="back_rewrite" parents="154"/>
      <j2 rule="rewrite" parents="317"/>
    </justification>
  </clause>

  <clause id="349">
    <literal><![CDATA[
      x'' ^ (x'' v x) = x''''
    ]]></literal>
    <justification jstring="[para(17(a,1),332(a,1,2))].">
      <j1 rule="para" parents="17 332"/>
    </justification>
  </clause>

  <clause id="360">
    <literal><![CDATA[
      f(x',f(x,f(x v x',x'))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),317(a,1,1))].">
      <j1 rule="para" parents="10 317"/>
    </justification>
  </clause>

  <clause id="361">
    <literal><![CDATA[
      f(x,y) ^ f(y,f(y v y',y')) = y'
    ]]></literal>
    <justification jstring="[para(317(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="317 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="377">
    <literal><![CDATA[
      x' ^ f(x,f(x v x',x')) = x'
    ]]></literal>
    <justification jstring="[para(360(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="360 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="378">
    <literal><![CDATA[
      f(x ^ y,f(f(x,y),f(f(x,y) v (x ^ y),x ^ y))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(16(a,1),360(a,1,1)),rewrite([16(5),16(7)])].">
      <j1 rule="para" parents="16 360"/>
      <j2 rule="rewrite" parents="16 16"/>
    </justification>
  </clause>

  <clause id="379">
    <literal><![CDATA[
      f(x' ^ y',f(x v y,f((x v y) v (x' ^ y'),x' ^ y'))) = x v y
    ]]></literal>
    <justification jstring="[para(22(a,1),360(a,1,1)),rewrite([22(7),22(11)])].">
      <j1 rule="para" parents="22 360"/>
      <j2 rule="rewrite" parents="22 22"/>
    </justification>
  </clause>

  <clause id="380">
    <literal><![CDATA[
      f(x,x'') = x'
    ]]></literal>
    <justification jstring="[para(360(a,1),62(a,1,2,2,1,2)),rewrite([317(22),10(3)])].">
      <j1 rule="para" parents="360 62"/>
      <j2 rule="rewrite" parents="317 10"/>
    </justification>
  </clause>

  <clause id="381">
    <literal><![CDATA[
      x ^ x'' = x''
    ]]></literal>
    <justification jstring="[para(360(a,1),83(a,1,2,2,1,2)),rewrite([317(22),10(3)])].">
      <j1 rule="para" parents="360 83"/>
      <j2 rule="rewrite" parents="317 10"/>
    </justification>
  </clause>

  <clause id="383">
    <literal><![CDATA[
      x'' v x = x
    ]]></literal>
    <justification jstring="[para(360(a,1),131(a,1,2,2,1,2)),rewrite([317(15),10(4),17(5)])].">
      <j1 rule="para" parents="360 131"/>
      <j2 rule="rewrite" parents="317 10 17"/>
    </justification>
  </clause>

  <clause id="384">
    <literal><![CDATA[
      f(x' v y,x') = x
    ]]></literal>
    <justification jstring="[para(360(a,1),132(a,1,2,2,1,2)),rewrite([317(14),10(3)])].">
      <j1 rule="para" parents="360 132"/>
      <j2 rule="rewrite" parents="317 10"/>
    </justification>
  </clause>

  <clause id="385">
    <literal><![CDATA[
      f(f(x'',y),x') = x
    ]]></literal>
    <justification jstring="[para(360(a,1),130(a,1,2,2,1,2)),rewrite([317(15),10(4)])].">
      <j1 rule="para" parents="360 130"/>
      <j2 rule="rewrite" parents="317 10"/>
    </justification>
  </clause>

  <clause id="386">
    <literal><![CDATA[
      x''' ^ x' = x'
    ]]></literal>
    <justification jstring="[para(360(a,1),136(a,1,2,2,1,2)),rewrite([317(15),10(4)])].">
      <j1 rule="para" parents="360 136"/>
      <j2 rule="rewrite" parents="317 10"/>
    </justification>
  </clause>

  <clause id="387">
    <literal><![CDATA[
      (x' v y) ^ x' = x'
    ]]></literal>
    <justification jstring="[para(360(a,1),157(a,1,2,2,1,2)),rewrite([317(14),10(3)])].">
      <j1 rule="para" parents="360 157"/>
      <j2 rule="rewrite" parents="317 10"/>
    </justification>
  </clause>

  <clause id="388">
    <literal><![CDATA[
      x' ^ f(x,f(f(y,f(x',y)),y)) = x'
    ]]></literal>
    <justification jstring="[para(360(a,1),121(a,1,2,1)),rewrite([377(8),377(14)])].">
      <j1 rule="para" parents="360 121"/>
      <j2 rule="rewrite" parents="377 377"/>
    </justification>
  </clause>

  <clause id="402">
    <literal><![CDATA[
      x'' ^ x = x''''
    ]]></literal>
    <justification jstring="[back_rewrite(349),rewrite([383(5)])].">
      <j1 rule="back_rewrite" parents="349"/>
      <j2 rule="rewrite" parents="383"/>
    </justification>
  </clause>

  <clause id="406">
    <literal><![CDATA[
      x''''' = x'
    ]]></literal>
    <justification jstring="[back_rewrite(386),rewrite([402(5)])].">
      <j1 rule="back_rewrite" parents="386"/>
      <j2 rule="rewrite" parents="402"/>
    </justification>
  </clause>

  <clause id="423">
    <literal><![CDATA[
      (x''' v y) ^ f(x'',f(f(x',x),x')) = x'''
    ]]></literal>
    <justification jstring="[para(383(a,1),209(a,1,2,2,1,2))].">
      <j1 rule="para" parents="383 209"/>
    </justification>
  </clause>

  <clause id="428">
    <literal><![CDATA[
      x v x'' = x''
    ]]></literal>
    <justification jstring="[para(380(a,1),17(a,1)),flip(a)].">
      <j1 rule="para" parents="380 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="481">
    <literal><![CDATA[
      x'''' = x
    ]]></literal>
    <justification jstring="[para(406(a,1),380(a,1,2)),rewrite([17(5),383(3)]),flip(a)].">
      <j1 rule="para" parents="406 380"/>
      <j2 rule="rewrite" parents="17 383"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="494">
    <literal><![CDATA[
      (x ^ y)''' = f(x,y)
    ]]></literal>
    <justification jstring="[para(16(a,1),481(a,1,1,1,1))].">
      <j1 rule="para" parents="16 481"/>
    </justification>
  </clause>

  <clause id="495">
    <literal><![CDATA[
      f(x,y') = x''' v y
    ]]></literal>
    <justification jstring="[para(481(a,1),17(a,1,1))].">
      <j1 rule="para" parents="481 17"/>
    </justification>
  </clause>

  <clause id="496">
    <literal><![CDATA[
      f(x',y) = x v y'''
    ]]></literal>
    <justification jstring="[para(481(a,1),17(a,1,2))].">
      <j1 rule="para" parents="481 17"/>
    </justification>
  </clause>

  <clause id="497">
    <literal><![CDATA[
      f(x ^ y,z) = f(x,y) v z'''
    ]]></literal>
    <justification jstring="[para(481(a,1),23(a,1,2))].">
      <j1 rule="para" parents="481 23"/>
    </justification>
  </clause>

  <clause id="498">
    <literal><![CDATA[
      f(x,y ^ z) = x''' v f(y,z)
    ]]></literal>
    <justification jstring="[para(481(a,1),24(a,1,1))].">
      <j1 rule="para" parents="481 24"/>
    </justification>
  </clause>

  <clause id="505">
    <literal><![CDATA[
      x'' v (x ^ ((y' ^ (x v y))'' v y))'' = x
    ]]></literal>
    <justification jstring="[para(481(a,1),85(a,1,2,1)),rewrite([481(8),496(6),22(5),495(11),22(10),494(10),495(7),481(8),496(11),16(10),481(16)])].">
      <j1 rule="para" parents="481 85"/>
      <j2 rule="rewrite" parents="481 496 22 495 22 494 495 481 496 16 481"/>
    </justification>
  </clause>

  <clause id="533">
    <literal><![CDATA[
      f(f(x,y),x' v ((z v (x''' ^ z')'') ^ z')'') = x''
    ]]></literal>
    <justification jstring="[para(481(a,1),185(a,1,1,1)),rewrite([496(8),22(7),495(13),22(12),494(12),495(9),481(8),496(12),22(11),494(10),496(7),22(6)])].">
      <j1 rule="para" parents="481 185"/>
      <j2 rule="rewrite" parents="496 22 495 22 494 495 481 496 22 494 496 22"/>
    </justification>
  </clause>

  <clause id="537">
    <literal><![CDATA[
      (x v y) ^ (x'' v ((z v (x ^ z')'') ^ z')'') = x
    ]]></literal>
    <justification jstring="[para(481(a,1),209(a,1,1,1)),rewrite([496(10),22(9),481(8),495(11),22(10),494(10),495(7),496(14),22(13),494(12),496(9),22(8),481(7),481(18)])].">
      <j1 rule="para" parents="481 209"/>
      <j2 rule="rewrite" parents="496 22 481 495 22 494 495 496 22 494 496 22 481 481"/>
    </justification>
  </clause>

  <clause id="543">
    <literal><![CDATA[
      f(x v y,x) = x'''
    ]]></literal>
    <justification jstring="[para(481(a,1),384(a,1,1,1)),rewrite([481(5)])].">
      <j1 rule="para" parents="481 384"/>
      <j2 rule="rewrite" parents="481"/>
    </justification>
  </clause>

  <clause id="555">
    <literal><![CDATA[
      (x''' v y) ^ (x' v ((x v x''') ^ x')'') = x'''
    ]]></literal>
    <justification jstring="[back_rewrite(423),rewrite([496(8),495(12),22(11),481(11),496(12),22(11),494(10),496(7)])].">
      <j1 rule="back_rewrite" parents="423"/>
      <j2 rule="rewrite" parents="496 495 22 481 496 22 494 496"/>
    </justification>
  </clause>

  <clause id="572">
    <literal><![CDATA[
      (x'' ^ y)'' v x = x
    ]]></literal>
    <justification jstring="[back_rewrite(385),rewrite([496(3),495(7),22(6),481(6)])].">
      <j1 rule="back_rewrite" parents="385"/>
      <j2 rule="rewrite" parents="496 495 22 481"/>
    </justification>
  </clause>

  <clause id="579">
    <literal><![CDATA[
      f(x,y) ^ f(y,(y' ^ y'')'' v y) = y'
    ]]></literal>
    <justification jstring="[back_rewrite(361),rewrite([495(5),22(4)])].">
      <j1 rule="back_rewrite" parents="361"/>
      <j2 rule="rewrite" parents="495 22"/>
    </justification>
  </clause>

  <clause id="616">
    <literal><![CDATA[
      f(f(x''' v y,y v z),u) ^ (y v ((z v (y'' ^ z')'') ^ z')'') = y''
    ]]></literal>
    <justification jstring="[back_rewrite(171),rewrite([495(2),496(12),22(11),495(17),22(16),494(16),495(13),481(13),496(16),22(15),494(14),496(11),22(10)])].">
      <j1 rule="back_rewrite" parents="171"/>
      <j2 rule="rewrite" parents="495 496 22 495 22 494 495 481 496 22 494 496 22"/>
    </justification>
  </clause>

  <clause id="646">
    <literal><![CDATA[
      x' ^ f(x,f(f(y,x v y'''),y)) = x'
    ]]></literal>
    <justification jstring="[back_rewrite(388),rewrite([496(3)])].">
      <j1 rule="back_rewrite" parents="388"/>
      <j2 rule="rewrite" parents="496"/>
    </justification>
  </clause>

  <clause id="650">
    <literal><![CDATA[
      x v (x ^ y)'' = x''
    ]]></literal>
    <justification jstring="[back_rewrite(324),rewrite([496(4),16(3),543(6),495(5),481(4),428(3),496(5),16(4)]),flip(a)].">
      <j1 rule="back_rewrite" parents="324"/>
      <j2 rule="rewrite" parents="496 16 543 495 481 428 496 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="651">
    <literal><![CDATA[
      x' ^ f(x,y) = x'''
    ]]></literal>
    <justification jstring="[back_rewrite(323),rewrite([496(4),16(3),650(5),496(4),428(6),381(5)]),flip(a)].">
      <j1 rule="back_rewrite" parents="323"/>
      <j2 rule="rewrite" parents="496 16 650 496 428 381"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="682">
    <literal><![CDATA[
      f(f(x,y),f(y,z)) v (y ^ f(f(u,y v u'''),u))'' = y
    ]]></literal>
    <justification jstring="[back_rewrite(134),rewrite([496(5),497(11),16(11)])].">
      <j1 rule="back_rewrite" parents="134"/>
      <j2 rule="rewrite" parents="496 497 16"/>
    </justification>
  </clause>

  <clause id="684">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(w,y v w'''),w))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(123),rewrite([496(6)])].">
      <j1 rule="back_rewrite" parents="123"/>
      <j2 rule="rewrite" parents="496"/>
    </justification>
  </clause>

  <clause id="701">
    <literal><![CDATA[
      (x' ^ y')' = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(379),rewrite([498(13),22(10),22(6),495(17),481(18),497(18),495(3),481(4),16(16),650(18),22(2)])].">
      <j1 rule="back_rewrite" parents="379"/>
      <j2 rule="rewrite" parents="498 22 22 495 481 497 495 481 16 650 22"/>
    </justification>
  </clause>

  <clause id="702">
    <literal><![CDATA[
      (x ^ y)' = f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(378),rewrite([498(7),22(6),16(4),497(12),16(12),650(14),16(2)])].">
      <j1 rule="back_rewrite" parents="378"/>
      <j2 rule="rewrite" parents="498 22 16 497 16 650 16"/>
    </justification>
  </clause>

  <clause id="764">
    <literal><![CDATA[
      c2''' v f(c1,c3) != c1''' v f(c2,c3)
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <justification jstring="[back_rewrite(19),rewrite([498(5),498(13)])].">
      <j1 rule="back_rewrite" parents="19"/>
      <j2 rule="rewrite" parents="498 498"/>
    </justification>
  </clause>

  <clause id="778">
    <literal><![CDATA[
      x''' = x'
    ]]></literal>
    <justification jstring="[back_rewrite(646),rewrite([651(9)])].">
      <j1 rule="back_rewrite" parents="646"/>
      <j2 rule="rewrite" parents="651"/>
    </justification>
  </clause>

  <clause id="793">
    <literal><![CDATA[
      f(f(x' v y,y v z),u) ^ (y v ((z v (y'' ^ z')) ^ z')) = y''
    ]]></literal>
    <justification jstring="[back_rewrite(616),rewrite([778(3),702(10),495(9),778(8),778(8),22(8),702(13),495(12),22(11),702(11),495(10),778(9),778(9),702(10),496(9),22(8),702(10),495(9),778(8),778(8),22(8),22(11),702(11),495(10),778(9),778(9),22(11),702(10),496(9),22(8),702(10),495(9),778(8),778(8),22(8)])].">
      <j1 rule="back_rewrite" parents="616"/>
      <j2 rule="rewrite" parents="778 702 495 778 778 22 702 495 22 702 495 778 778 702 496 22 702 495 778 778 22 22 702 495 778 778 22 702 496 22 702 495 778 778 22"/>
    </justification>
  </clause>

  <clause id="802">
    <literal><![CDATA[
      f(x,y) ^ f(y,(y' ^ y'') v y) = y'
    ]]></literal>
    <justification jstring="[back_rewrite(579),rewrite([702(6),495(5),778(4),22(6),778(4)])].">
      <j1 rule="back_rewrite" parents="579"/>
      <j2 rule="rewrite" parents="702 495 778 22 778"/>
    </justification>
  </clause>

  <clause id="811">
    <literal><![CDATA[
      f(f(x,y),x' v ((z v (x' ^ z')) ^ z')) = x''
    ]]></literal>
    <justification jstring="[back_rewrite(533),rewrite([778(5),702(6),495(5),778(5),22(6),778(5),702(9),495(8),22(7),702(7),495(6),778(6),702(8),496(7),22(6),778(5),702(6),495(5),778(5),22(6),778(5),22(7),702(7),495(6),778(6),22(9),702(8),496(7),22(6),778(5),702(6),495(5),778(5),22(6),778(5)])].">
      <j1 rule="back_rewrite" parents="533"/>
      <j2 rule="rewrite" parents="778 702 495 778 22 778 702 495 22 702 495 778 702 496 22 778 702 495 778 22 778 22 702 495 778 22 702 496 22 778 702 495 778 22 778"/>
    </justification>
  </clause>

  <clause id="816">
    <literal><![CDATA[
      x'' v y = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(701),rewrite([702(4),495(3),778(3)])].">
      <j1 rule="back_rewrite" parents="701"/>
      <j2 rule="rewrite" parents="702 495 778"/>
    </justification>
  </clause>

  <clause id="830">
    <literal><![CDATA[
      f(f(x,y),f(y,z)) v (y ^ f(f(u,y v u'),u)) = y
    ]]></literal>
    <justification jstring="[back_rewrite(682),rewrite([778(6),702(9),16(9)])].">
      <j1 rule="back_rewrite" parents="682"/>
      <j2 rule="rewrite" parents="778 702 16"/>
    </justification>
  </clause>

  <clause id="842">
    <literal><![CDATA[
      x v (x ^ y) = x''
    ]]></literal>
    <justification jstring="[back_rewrite(650),rewrite([702(2),16(2)])].">
      <j1 rule="back_rewrite" parents="650"/>
      <j2 rule="rewrite" parents="702 16"/>
    </justification>
  </clause>

  <clause id="874">
    <literal><![CDATA[
      (x'' ^ y'') v x = x
    ]]></literal>
    <justification jstring="[back_rewrite(572),rewrite([702(4),496(3),778(4),22(4)])].">
      <j1 rule="back_rewrite" parents="572"/>
      <j2 rule="rewrite" parents="702 496 778 22"/>
    </justification>
  </clause>

  <clause id="883">
    <literal><![CDATA[
      (x' v y) ^ (x' v ((x v x') ^ x')) = x'
    ]]></literal>
    <justification jstring="[back_rewrite(555),rewrite([778(3),778(6),702(8),495(7),22(6),702(8),495(7),778(6),816(7),22(6),22(9),702(8),495(7),778(6),816(7),778(12)])].">
      <j1 rule="back_rewrite" parents="555"/>
      <j2 rule="rewrite" parents="778 778 702 495 22 702 495 778 816 22 22 702 495 778 816 778"/>
    </justification>
  </clause>

  <clause id="892">
    <literal><![CDATA[
      (x v y) ^ (x v ((z v (x'' ^ z')) ^ z')) = x
    ]]></literal>
    <justification jstring="[back_rewrite(537),rewrite([702(6),495(5),778(6),22(6),702(11),495(10),22(9),702(9),495(8),778(7),778(7),702(8),496(7),22(6),702(8),495(7),778(6),778(6),22(6),22(9),702(9),495(8),778(7),778(7),22(9),702(8),496(7),22(6),702(8),495(7),778(6),778(6),22(6),816(11)])].">
      <j1 rule="back_rewrite" parents="537"/>
      <j2 rule="rewrite" parents="702 495 778 22 702 495 22 702 495 778 778 702 496 22 702 495 778 778 22 22 702 495 778 778 22 702 496 22 702 495 778 778 22 816"/>
    </justification>
  </clause>

  <clause id="910">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[back_rewrite(505),rewrite([702(6),496(5),22(4),702(6),495(5),778(5),816(5),22(4),22(7),702(7),495(6),778(6),816(6),702(8),16(8),816(8),842(6)])].">
      <j1 rule="back_rewrite" parents="505"/>
      <j2 rule="rewrite" parents="702 496 22 702 495 778 816 22 22 702 495 778 816 702 16 816 842"/>
    </justification>
  </clause>

  <clause id="916">
    <literal><![CDATA[
      c2' v f(c1,c3) != c1' v f(c2,c3)
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <justification jstring="[back_rewrite(764),rewrite([910(3),910(9)])].">
      <j1 rule="back_rewrite" parents="764"/>
      <j2 rule="rewrite" parents="910 910"/>
    </justification>
  </clause>

  <clause id="941">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(w,y v w'),w))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(684),rewrite([910(6)])].">
      <j1 rule="back_rewrite" parents="684"/>
      <j2 rule="rewrite" parents="910"/>
    </justification>
  </clause>

  <clause id="946">
    <literal><![CDATA[
      x' ^ f(x,y) = x'
    ]]></literal>
    <justification jstring="[back_rewrite(651),rewrite([910(5)])].">
      <j1 rule="back_rewrite" parents="651"/>
      <j2 rule="rewrite" parents="910"/>
    </justification>
  </clause>

  <clause id="954">
    <literal><![CDATA[
      f(x',y) = x v y'
    ]]></literal>
    <justification jstring="[back_rewrite(496),rewrite([910(4)])].">
      <j1 rule="back_rewrite" parents="496"/>
      <j2 rule="rewrite" parents="910"/>
    </justification>
  </clause>

  <clause id="973">
    <literal><![CDATA[
      (x v y) ^ (x v ((z v (x ^ z')) ^ z')) = x
    ]]></literal>
    <justification jstring="[back_rewrite(892),rewrite([910(3)])].">
      <j1 rule="back_rewrite" parents="892"/>
      <j2 rule="rewrite" parents="910"/>
    </justification>
  </clause>

  <clause id="979">
    <literal><![CDATA[
      (x ^ y) v x = x
    ]]></literal>
    <justification jstring="[back_rewrite(874),rewrite([910(2),910(2)])].">
      <j1 rule="back_rewrite" parents="874"/>
      <j2 rule="rewrite" parents="910 910"/>
    </justification>
  </clause>

  <clause id="985">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[back_rewrite(842),rewrite([910(4)])].">
      <j1 rule="back_rewrite" parents="842"/>
      <j2 rule="rewrite" parents="910"/>
    </justification>
  </clause>

  <clause id="994">
    <literal><![CDATA[
      f(f(x,y),x' v ((z v (x' ^ z')) ^ z')) = x
    ]]></literal>
    <justification jstring="[back_rewrite(811),rewrite([910(12)])].">
      <j1 rule="back_rewrite" parents="811"/>
      <j2 rule="rewrite" parents="910"/>
    </justification>
  </clause>

  <clause id="999">
    <literal><![CDATA[
      f(x,y) ^ f(y,(y' ^ y) v y) = y'
    ]]></literal>
    <justification jstring="[back_rewrite(802),rewrite([910(4)])].">
      <j1 rule="back_rewrite" parents="802"/>
      <j2 rule="rewrite" parents="910"/>
    </justification>
  </clause>

  <clause id="1002">
    <literal><![CDATA[
      f(f(x' v y,y v z),u) ^ (y v ((z v (y ^ z')) ^ z')) = y
    ]]></literal>
    <justification jstring="[back_rewrite(793),rewrite([910(7),910(14)])].">
      <j1 rule="back_rewrite" parents="793"/>
      <j2 rule="rewrite" parents="910 910"/>
    </justification>
  </clause>

  <clause id="1007">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[back_rewrite(21),rewrite([910(3)])].">
      <j1 rule="back_rewrite" parents="21"/>
      <j2 rule="rewrite" parents="910"/>
    </justification>
  </clause>

  <clause id="1008">
    <literal><![CDATA[
      x ^ x = x
    ]]></literal>
    <justification jstring="[back_rewrite(20),rewrite([910(3)])].">
      <j1 rule="back_rewrite" parents="20"/>
      <j2 rule="rewrite" parents="910"/>
    </justification>
  </clause>

  <clause id="1010">
    <literal><![CDATA[
      (x v y) ^ x = x
    ]]></literal>
    <justification jstring="[para(910(a,1),387(a,1,1,1)),rewrite([910(3),910(4)])].">
      <j1 rule="para" parents="910 387"/>
      <j2 rule="rewrite" parents="910 910"/>
    </justification>
  </clause>

  <clause id="1012">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(1010(a,1),979(a,1,1))].">
      <j1 rule="para" parents="1010 979"/>
    </justification>
  </clause>

  <clause id="1014">
    <literal><![CDATA[
      (x v y) v x = x v y
    ]]></literal>
    <justification jstring="[para(1010(a,1),985(a,1,2))].">
      <j1 rule="para" parents="1010 985"/>
    </justification>
  </clause>

  <clause id="1018">
    <literal><![CDATA[
      x ^ (x v y') = x
    ]]></literal>
    <justification jstring="[para(910(a,1),946(a,1,1)),rewrite([954(2),910(5)])].">
      <j1 rule="para" parents="910 946"/>
      <j2 rule="rewrite" parents="954 910"/>
    </justification>
  </clause>

  <clause id="1021">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[para(910(a,1),1018(a,1,2,2))].">
      <j1 rule="para" parents="910 1018"/>
    </justification>
  </clause>

  <clause id="1041">
    <literal><![CDATA[
      (x ^ y) ^ x = x ^ y
    ]]></literal>
    <justification jstring="[para(979(a,1),1021(a,1,2))].">
      <j1 rule="para" parents="979 1021"/>
    </justification>
  </clause>

  <clause id="1046">
    <literal><![CDATA[
      f(x,y) = x' v y'
    ]]></literal>
    <justification jstring="[para(910(a,1),954(a,1,1))].">
      <j1 rule="para" parents="910 954"/>
    </justification>
  </clause>

  <clause id="1060">
    <literal><![CDATA[
      (((x' v y) ^ (y v z)) v u') ^ (y v ((z v (y ^ z')) ^ z')) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1002),rewrite([1046(4),22(3),910(2),22(4),1046(7),22(7),702(3),1046(2),910(3),702(6),1046(5),910(4),910(4)])].">
      <j1 rule="back_rewrite" parents="1002"/>
      <j2 rule="rewrite" parents="1046 22 910 22 1046 22 702 1046 910 702 1046 910 910"/>
    </justification>
  </clause>

  <clause id="1063">
    <literal><![CDATA[
      (x' v y') ^ (y' v ((y v y') ^ y')) = y'
    ]]></literal>
    <justification jstring="[back_rewrite(999),rewrite([1046(1),1046(7),22(8),702(7),1046(6),910(6)])].">
      <j1 rule="back_rewrite" parents="999"/>
      <j2 rule="rewrite" parents="1046 1046 22 702 1046 910"/>
    </justification>
  </clause>

  <clause id="1066">
    <literal><![CDATA[
      (x ^ y) v (x ^ ((z' ^ (x v z)) v z)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(994),rewrite([1046(1),1046(12),22(4),910(2),910(2),22(10),910(3),702(8),1046(7),22(6),702(6),1046(5),910(4),910(4),910(6)])].">
      <j1 rule="back_rewrite" parents="994"/>
      <j2 rule="rewrite" parents="1046 1046 22 910 910 22 910 702 1046 22 702 1046 910 910 910"/>
    </justification>
  </clause>

  <clause id="1079">
    <literal><![CDATA[
      (((x ^ y) v (y ^ z)) ^ u) v (y ^ ((w ^ (y v w')) v w')) = y
    ]]></literal>
    <justification jstring="[back_rewrite(941),rewrite([1046(1),1046(4),1046(7),22(4),910(2),910(2),22(5),910(3),910(3),1046(4),22(4),702(2),1046(1),702(5),1046(4),1046(12),22(13),910(13),1046(14),22(14),910(11),702(12),1046(11),910(11),1046(15),22(16),702(14),1046(13),22(14),910(14),910(16),1046(17),22(10),702(8),1046(7),22(4),910(2),910(2),22(5),910(3),910(3),910(5),22(12),910(6),702(10),1046(9),22(9),910(6),702(7),1046(6),910(6)])].">
      <j1 rule="back_rewrite" parents="941"/>
      <j2 rule="rewrite" parents="1046 1046 1046 22 910 910 22 910 910 1046 22 702 1046 702 1046 1046 22 910 1046 22 910 702 1046 910 1046 22 702 1046 22 910 910 1046 22 702 1046 22 910 910 22 910 910 910 22 910 702 1046 22 910 702 1046 910"/>
    </justification>
  </clause>

  <clause id="1092">
    <literal><![CDATA[
      c2' v (c1' v c3') != c1' v (c2' v c3')
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <justification jstring="[back_rewrite(916),rewrite([1046(5),1046(13)])].">
      <j1 rule="back_rewrite" parents="916"/>
      <j2 rule="rewrite" parents="1046 1046"/>
    </justification>
  </clause>

  <clause id="1106">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v (y ^ ((u ^ (y v u')) v u')) = y
    ]]></literal>
    <justification jstring="[back_rewrite(830),rewrite([1046(1),1046(4),1046(7),22(4),910(2),910(2),22(5),910(3),910(3),1046(6),22(7),910(7),1046(8),22(8),910(5),702(6),1046(5),910(5)])].">
      <j1 rule="back_rewrite" parents="830"/>
      <j2 rule="rewrite" parents="1046 1046 1046 22 910 910 22 910 910 1046 22 910 1046 22 910 702 1046 910"/>
    </justification>
  </clause>

  <clause id="1127">
    <literal><![CDATA[
      x' v ((x v x') ^ x') = x'
    ]]></literal>
    <justification jstring="[para(883(a,1),1008(a,1)),flip(a)].">
      <j1 rule="para" parents="883 1008"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1128">
    <literal><![CDATA[
      (x' v y') ^ y' = y'
    ]]></literal>
    <justification jstring="[back_rewrite(1063),rewrite([1127(9)])].">
      <j1 rule="back_rewrite" parents="1063"/>
      <j2 rule="rewrite" parents="1127"/>
    </justification>
  </clause>

  <clause id="1144">
    <literal><![CDATA[
      x v ((y v (x ^ y')) ^ y') = x
    ]]></literal>
    <justification jstring="[para(973(a,1),1008(a,1)),flip(a)].">
      <j1 rule="para" parents="973 1008"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1148">
    <literal><![CDATA[
      (((x' v y) ^ (y v z)) v u') ^ y = y
    ]]></literal>
    <justification jstring="[back_rewrite(1060),rewrite([1144(12)])].">
      <j1 rule="back_rewrite" parents="1060"/>
      <j2 rule="rewrite" parents="1144"/>
    </justification>
  </clause>

  <clause id="1153">
    <literal><![CDATA[
      (x' v y) ^ y = y
    ]]></literal>
    <justification jstring="[para(910(a,1),1128(a,1,1,2)),rewrite([910(4),910(5)])].">
      <j1 rule="para" parents="910 1128"/>
      <j2 rule="rewrite" parents="910 910"/>
    </justification>
  </clause>

  <clause id="1155">
    <literal><![CDATA[
      (x v y) ^ y = y
    ]]></literal>
    <justification jstring="[para(910(a,1),1153(a,1,1,1))].">
      <j1 rule="para" parents="910 1153"/>
    </justification>
  </clause>

  <clause id="1166">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(1155(a,1),979(a,1,1))].">
      <j1 rule="para" parents="1155 979"/>
    </justification>
  </clause>

  <clause id="1168">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(1155(a,1),1041(a,1,1)),rewrite([1155(4)])].">
      <j1 rule="para" parents="1155 1041"/>
      <j2 rule="rewrite" parents="1155"/>
    </justification>
  </clause>

  <clause id="1249">
    <literal><![CDATA[
      x ^ ((y' ^ (x v y)) v y) = x
    ]]></literal>
    <justification jstring="[para(1066(a,1),1007(a,1)),flip(a)].">
      <j1 rule="para" parents="1066 1007"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1263">
    <literal><![CDATA[
      x ^ ((y ^ (x v y')) v y') = x
    ]]></literal>
    <justification jstring="[para(910(a,1),1249(a,1,2,1,1))].">
      <j1 rule="para" parents="910 1249"/>
    </justification>
  </clause>

  <clause id="1280">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v y = y
    ]]></literal>
    <justification jstring="[back_rewrite(1106),rewrite([1263(9)])].">
      <j1 rule="back_rewrite" parents="1106"/>
      <j2 rule="rewrite" parents="1263"/>
    </justification>
  </clause>

  <clause id="1281">
    <literal><![CDATA[
      (((x ^ y) v (y ^ z)) ^ u) v y = y
    ]]></literal>
    <justification jstring="[back_rewrite(1079),rewrite([1263(10)])].">
      <j1 rule="back_rewrite" parents="1079"/>
      <j2 rule="rewrite" parents="1263"/>
    </justification>
  </clause>

  <clause id="1283">
    <literal><![CDATA[
      ((x ^ (y v z)) v y) v (y v z) = y v z
    ]]></literal>
    <justification jstring="[para(1010(a,1),1280(a,1,1,2))].">
      <j1 rule="para" parents="1010 1280"/>
    </justification>
  </clause>

  <clause id="1285">
    <literal><![CDATA[
      (x v ((x v y) ^ z)) v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(1021(a,1),1280(a,1,1,1))].">
      <j1 rule="para" parents="1021 1280"/>
    </justification>
  </clause>

  <clause id="1380">
    <literal><![CDATA[
      ((x ^ y) ^ z) v x = x
    ]]></literal>
    <justification jstring="[para(979(a,1),1281(a,1,1,1))].">
      <j1 rule="para" parents="979 1281"/>
    </justification>
  </clause>

  <clause id="1403">
    <literal><![CDATA[
      (x ^ y) v (x v z) = x v z
    ]]></literal>
    <justification jstring="[para(1010(a,1),1380(a,1,1,1))].">
      <j1 rule="para" parents="1010 1380"/>
    </justification>
  </clause>

  <clause id="1405">
    <literal><![CDATA[
      x v ((x ^ y) ^ z) = x
    ]]></literal>
    <justification jstring="[para(1380(a,1),1014(a,1,1)),rewrite([1380(6)])].">
      <j1 rule="para" parents="1380 1014"/>
      <j2 rule="rewrite" parents="1380"/>
    </justification>
  </clause>

  <clause id="1416">
    <literal><![CDATA[
      (x v y) v (x ^ z) = x v y
    ]]></literal>
    <justification jstring="[para(1010(a,1),1405(a,1,2,1))].">
      <j1 rule="para" parents="1010 1405"/>
    </justification>
  </clause>

  <clause id="1444">
    <literal><![CDATA[
      ((x' v y) v z') ^ y = y
    ]]></literal>
    <justification jstring="[para(1168(a,1),1148(a,1,1,1))].">
      <j1 rule="para" parents="1168 1148"/>
    </justification>
  </clause>

  <clause id="1508">
    <literal><![CDATA[
      x v ((x v y) v z) = (x v y) v z
    ]]></literal>
    <justification jstring="[para(1010(a,1),1403(a,1,1))].">
      <j1 rule="para" parents="1010 1403"/>
    </justification>
  </clause>

  <clause id="1546">
    <literal><![CDATA[
      (x v y) v ((x ^ z) ^ u) = x v y
    ]]></literal>
    <justification jstring="[para(1403(a,1),1416(a,1,1)),rewrite([1403(7)])].">
      <j1 rule="para" parents="1403 1416"/>
      <j2 rule="rewrite" parents="1403"/>
    </justification>
  </clause>

  <clause id="1592">
    <literal><![CDATA[
      ((x v y) v z') ^ y = y
    ]]></literal>
    <justification jstring="[para(910(a,1),1444(a,1,1,1,1))].">
      <j1 rule="para" parents="910 1444"/>
    </justification>
  </clause>

  <clause id="1623">
    <literal><![CDATA[
      ((x v y) v z) ^ y = y
    ]]></literal>
    <justification jstring="[para(910(a,1),1592(a,1,1,2))].">
      <j1 rule="para" parents="910 1592"/>
    </justification>
  </clause>

  <clause id="1671">
    <literal><![CDATA[
      ((x v y) v z) ^ x = x
    ]]></literal>
    <justification jstring="[para(1014(a,1),1623(a,1,1,1))].">
      <j1 rule="para" parents="1014 1623"/>
    </justification>
  </clause>

  <clause id="1672">
    <literal><![CDATA[
      x ^ ((y v x) v z) = x
    ]]></literal>
    <justification jstring="[para(1623(a,1),1041(a,1,1)),rewrite([1623(6)])].">
      <j1 rule="para" parents="1623 1041"/>
      <j2 rule="rewrite" parents="1623"/>
    </justification>
  </clause>

  <clause id="1703">
    <literal><![CDATA[
      x ^ ((x v y) v z) = x
    ]]></literal>
    <justification jstring="[para(1671(a,1),1041(a,1,1)),rewrite([1671(6)])].">
      <j1 rule="para" parents="1671 1041"/>
      <j2 rule="rewrite" parents="1671"/>
    </justification>
  </clause>

  <clause id="1704">
    <literal><![CDATA[
      (x v (y v z)) ^ y = y
    ]]></literal>
    <justification jstring="[para(1166(a,1),1671(a,1,1))].">
      <j1 rule="para" parents="1166 1671"/>
    </justification>
  </clause>

  <clause id="1725">
    <literal><![CDATA[
      x ^ (y v (z v x)) = x
    ]]></literal>
    <justification jstring="[para(1166(a,1),1672(a,1,2))].">
      <j1 rule="para" parents="1166 1672"/>
    </justification>
  </clause>

  <clause id="1787">
    <literal><![CDATA[
      x ^ (y v (x v z)) = x
    ]]></literal>
    <justification jstring="[para(1166(a,1),1703(a,1,2))].">
      <j1 rule="para" parents="1166 1703"/>
    </justification>
  </clause>

  <clause id="4076">
    <literal><![CDATA[
      (x v y) v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(1168(a,1),1283(a,1,1,1))].">
      <j1 rule="para" parents="1168 1283"/>
    </justification>
  </clause>

  <clause id="4090">
    <literal><![CDATA[
      (x v y) v (y v (z v x)) = y v (z v x)
    ]]></literal>
    <justification jstring="[para(1725(a,1),1283(a,1,1,1))].">
      <j1 rule="para" parents="1725 1283"/>
    </justification>
  </clause>

  <clause id="4092">
    <literal><![CDATA[
      (x v y) v (y v (x v z)) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(1787(a,1),1283(a,1,1,1))].">
      <j1 rule="para" parents="1787 1283"/>
    </justification>
  </clause>

  <clause id="4262">
    <literal><![CDATA[
      (x v y) v (x v (y v z)) = x v (y v z)
    ]]></literal>
    <justification jstring="[para(1704(a,1),1285(a,1,1,2))].">
      <j1 rule="para" parents="1704 1285"/>
    </justification>
  </clause>

  <clause id="4469">
    <literal><![CDATA[
      (x v y) ^ (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(4076(a,1),1010(a,1,1))].">
      <j1 rule="para" parents="4076 1010"/>
    </justification>
  </clause>

  <clause id="4470">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[para(4076(a,1),1021(a,1,2)),rewrite([4469(3)])].">
      <j1 rule="para" parents="4076 1021"/>
      <j2 rule="rewrite" parents="4469"/>
    </justification>
  </clause>

  <clause id="30891">
    <literal><![CDATA[
      (x v y) v (y v z) = z v (x v y)
    ]]></literal>
    <justification jstring="[para(4090(a,1),4090(a,1,2)),rewrite([4470(6),4090(6),4090(7)])].">
      <j1 rule="para" parents="4090 4090"/>
      <j2 rule="rewrite" parents="4470 4090 4090"/>
    </justification>
  </clause>

  <clause id="30906">
    <literal><![CDATA[
      (x v y) v (x v z) = z v (x v y)
    ]]></literal>
    <justification jstring="[back_rewrite(4092),rewrite([30891(4)])].">
      <j1 rule="back_rewrite" parents="4092"/>
      <j2 rule="rewrite" parents="30891"/>
    </justification>
  </clause>

  <clause id="30917">
    <literal><![CDATA[
      (x v y) v (z v x) = z v (x v y)
    ]]></literal>
    <justification jstring="[back_rewrite(4262),rewrite([30906(4)])].">
      <j1 rule="back_rewrite" parents="4262"/>
      <j2 rule="rewrite" parents="30906"/>
    </justification>
  </clause>

  <clause id="31726">
    <literal><![CDATA[
      x v (y v z) = z v (x v y)
    ]]></literal>
    <justification jstring="[para(1012(a,1),30891(a,1,2)),rewrite([30891(3),30917(5)]),flip(a)].">
      <j1 rule="para" parents="1012 30891"/>
      <j2 rule="rewrite" parents="30891 30917"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="31733">
    <literal><![CDATA[
      x v (((x ^ y) ^ z) v u) = u v x
    ]]></literal>
    <justification jstring="[para(1405(a,1),30891(a,1,1)),rewrite([1405(7)])].">
      <j1 rule="para" parents="1405 30891"/>
      <j2 rule="rewrite" parents="1405"/>
    </justification>
  </clause>

  <clause id="31763">
    <literal><![CDATA[
      (x v y) v z = y v (z v x)
    ]]></literal>
    <justification jstring="[para(30891(a,1),4470(a,1)),rewrite([31726(5),1508(5)]),flip(a)].">
      <j1 rule="para" parents="30891 4470"/>
      <j2 rule="rewrite" parents="31726 1508"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="31783">
    <literal><![CDATA[
      x v (y v z) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(1546(a,1),30891(a,1,1)),rewrite([31763(5),4470(4),31733(4),31763(6),4470(5),1405(5)])].">
      <j1 rule="para" parents="1546 30891"/>
      <j2 rule="rewrite" parents="31763 4470 31733 31763 4470 1405"/>
    </justification>
  </clause>

  <clause id="31784">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <justification jstring="[resolve(31783,a,1092,a)].">
      <j1 rule="resolve" parents="31783 1092"/>
    </justification>
  </clause>

</proof>

</proofs>
