<?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 August-2006A, August 2006.
Process 11421 was started by mccune on cleo.thornwood,
Sat Aug 12 21:01:05 2006
The command was "/home/mccune/bin/prover9 -f omlsax2.in".
]]></heading>

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

<comments><![CDATA[
% Proof 1 at 0.11 (+ 0.00) seconds: B_SS.
% Length of proof is 66.
% Level of proof is 19.
% Maximum clause weight is 38.
% Given clauses 74.
]]></comments>

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

  <clause id="4">
    <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">
    <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">
    <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">
    <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">
    <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="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="44">
    <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="45">
    <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="47">
    <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="51">
    <literal><![CDATA[
      x''' ^ f(x,f(f(x,f(x',x)),x)) = x'
    ]]></literal>
    <justification jstring="[para(44(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="44 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="57">
    <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="68">
    <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="69">
    <literal><![CDATA[
      (x' v y) ^ f(x,f(f(x,f(x',x)),x)) = x'
    ]]></literal>
    <justification jstring="[para(47(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="47 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="75">
    <literal><![CDATA[
      x ^ f(x',f(f(y,f(x'',y)),y)) = x''
    ]]></literal>
    <justification jstring="[para(68(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="68 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="109">
    <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="117">
    <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(45(12),27(18))].">
      <j1 rule="para" parents="27 36"/>
      <j2 rule="rewrite" parents="45 27"/>
    </justification>
  </clause>

  <clause id="118">
    <literal><![CDATA[
      f(x''',f(x,f(f(y,f(x',y)),y))) = x
    ]]></literal>
    <justification jstring="[para(44(a,1),36(a,1,2,1)),rewrite(51(12),44(18))].">
      <j1 rule="para" parents="44 36"/>
      <j2 rule="rewrite" parents="51 44"/>
    </justification>
  </clause>

  <clause id="120">
    <literal><![CDATA[
      f(x' v y,f(x,f(f(z,f(x',z)),z))) = x
    ]]></literal>
    <justification jstring="[para(47(a,1),36(a,1,2,1)),rewrite(69(10),47(16))].">
      <j1 rule="para" parents="47 36"/>
      <j2 rule="rewrite" parents="69 47"/>
    </justification>
  </clause>

  <clause id="135">
    <literal><![CDATA[
      f(x''',f(x,f(x v x',x'))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),118(a,1,2,2,1,2)),rewrite(17(7))].">
      <j1 rule="para" parents="10 118"/>
      <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(118(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="118 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="148">
    <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),75(a,1,2,2,1,2))].">
      <j1 rule="para" parents="135 75"/>
    </justification>
  </clause>

  <clause id="174">
    <literal><![CDATA[
      f(f(x'',y),f(x,f(f(z',x v z),z'))) = x
    ]]></literal>
    <justification jstring="[para(17(a,1),117(a,1,2,2,1,2))].">
      <j1 rule="para" parents="17 117"/>
    </justification>
  </clause>

  <clause id="220">
    <literal><![CDATA[
      (x' ^ f(x,y)) ^ f(x,f(f(y,f(x',y)),y)) = x'
    ]]></literal>
    <justification jstring="[para(57(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="57 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="229">
    <literal><![CDATA[
      f(x' ^ f(x,y),f(x,f(f(z,f(x',z)),z))) = x
    ]]></literal>
    <justification jstring="[para(57(a,1),36(a,1,2,1)),rewrite(220(12),57(18))].">
      <j1 rule="para" parents="57 36"/>
      <j2 rule="rewrite" parents="220 57"/>
    </justification>
  </clause>

  <clause id="284">
    <literal><![CDATA[
      f(x' ^ f(x,y),f(x,f(x v x',x'))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),229(a,1,2,2,1,2)),rewrite(17(7))].">
      <j1 rule="para" parents="10 229"/>
      <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(284(a,1),36(a,1,2,2,1,2))].">
      <j1 rule="para" parents="284 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(284(a,1),109(a,1,2,2,1,2))].">
      <j1 rule="para" parents="284 109"/>
    </justification>
  </clause>

  <clause id="341">
    <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="351">
    <literal><![CDATA[
      x' ^ f(f(x',f(x,y)),x) = x' ^ f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(311),rewrite(341(16))].">
      <j1 rule="back_rewrite" parents="311"/>
      <j2 rule="rewrite" parents="341"/>
    </justification>
  </clause>

  <clause id="352">
    <literal><![CDATA[
      f(x',f(f(x',f(x,y)),x)) = f(x',f(x,y))
    ]]></literal>
    <justification jstring="[back_rewrite(307),rewrite(341(16))].">
      <j1 rule="back_rewrite" parents="307"/>
      <j2 rule="rewrite" parents="341"/>
    </justification>
  </clause>

  <clause id="360">
    <literal><![CDATA[
      x' ^ f(x'',x) = x'''
    ]]></literal>
    <justification jstring="[back_rewrite(148),rewrite(341(15))].">
      <j1 rule="back_rewrite" parents="148"/>
      <j2 rule="rewrite" parents="341"/>
    </justification>
  </clause>

  <clause id="380">
    <literal><![CDATA[
      x'' ^ (x'' v x) = x''''
    ]]></literal>
    <justification jstring="[para(17(a,1),360(a,1,2))].">
      <j1 rule="para" parents="17 360"/>
    </justification>
  </clause>

  <clause id="393">
    <literal><![CDATA[
      f(x',f(x,f(x v x',x'))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),341(a,1,1))].">
      <j1 rule="para" parents="10 341"/>
    </justification>
  </clause>

  <clause id="414">
    <literal><![CDATA[
      f(x,x'') = x'
    ]]></literal>
    <justification jstring="[para(393(a,1),68(a,1,2,2,1,2)),rewrite(341(22),10(3))].">
      <j1 rule="para" parents="393 68"/>
      <j2 rule="rewrite" parents="341 10"/>
    </justification>
  </clause>

  <clause id="415">
    <literal><![CDATA[
      x ^ x'' = x''
    ]]></literal>
    <justification jstring="[para(393(a,1),75(a,1,2,2,1,2)),rewrite(341(22),10(3))].">
      <j1 rule="para" parents="393 75"/>
      <j2 rule="rewrite" parents="341 10"/>
    </justification>
  </clause>

  <clause id="417">
    <literal><![CDATA[
      x'' v x = x
    ]]></literal>
    <justification jstring="[para(393(a,1),118(a,1,2,2,1,2)),rewrite(341(15),10(4),17(5))].">
      <j1 rule="para" parents="393 118"/>
      <j2 rule="rewrite" parents="341 10 17"/>
    </justification>
  </clause>

  <clause id="418">
    <literal><![CDATA[
      f(x' v y,x') = x
    ]]></literal>
    <justification jstring="[para(393(a,1),120(a,1,2,2,1,2)),rewrite(341(14),10(3))].">
      <j1 rule="para" parents="393 120"/>
      <j2 rule="rewrite" parents="341 10"/>
    </justification>
  </clause>

  <clause id="420">
    <literal><![CDATA[
      x''' ^ x' = x'
    ]]></literal>
    <justification jstring="[para(393(a,1),136(a,1,2,2,1,2)),rewrite(341(15),10(4))].">
      <j1 rule="para" parents="393 136"/>
      <j2 rule="rewrite" parents="341 10"/>
    </justification>
  </clause>

  <clause id="424">
    <literal><![CDATA[
      f(x',f(x,f(f(y',x v y),y'))) = x
    ]]></literal>
    <justification jstring="[para(393(a,1),174(a,1,1))].">
      <j1 rule="para" parents="393 174"/>
    </justification>
  </clause>

  <clause id="439">
    <literal><![CDATA[
      x'' ^ x = x''''
    ]]></literal>
    <justification jstring="[back_rewrite(380),rewrite(417(5))].">
      <j1 rule="back_rewrite" parents="380"/>
      <j2 rule="rewrite" parents="417"/>
    </justification>
  </clause>

  <clause id="443">
    <literal><![CDATA[
      x''''' = x'
    ]]></literal>
    <justification jstring="[back_rewrite(420),rewrite(439(5))].">
      <j1 rule="back_rewrite" parents="420"/>
      <j2 rule="rewrite" parents="439"/>
    </justification>
  </clause>

  <clause id="522">
    <literal><![CDATA[
      x'''' = x
    ]]></literal>
    <justification jstring="[para(443(a,1),414(a,1,2)),rewrite(17(5),417(3)),flip(a)].">
      <j1 rule="para" parents="443 414"/>
      <j2 rule="rewrite" parents="17 417"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="533">
    <literal><![CDATA[
      x'' ^ x = x
    ]]></literal>
    <justification jstring="[back_rewrite(439),rewrite(522(7))].">
      <j1 rule="back_rewrite" parents="439"/>
      <j2 rule="rewrite" parents="522"/>
    </justification>
  </clause>

  <clause id="535">
    <literal><![CDATA[
      f(x,y) = (x ^ y)'''
    ]]></literal>
    <justification jstring="[para(16(a,1),522(a,1,1,1,1)),flip(a)].">
      <j1 rule="para" parents="16 522"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="536">
    <literal><![CDATA[
      (x ^ y')''' = x''' v y
    ]]></literal>
    <justification jstring="[para(522(a,1),17(a,1,1)),rewrite(535(2))].">
      <j1 rule="para" parents="522 17"/>
      <j2 rule="rewrite" parents="535"/>
    </justification>
  </clause>

  <clause id="537">
    <literal><![CDATA[
      (x' ^ y)''' = x v y'''
    ]]></literal>
    <justification jstring="[para(522(a,1),17(a,1,2)),rewrite(535(2))].">
      <j1 rule="para" parents="522 17"/>
      <j2 rule="rewrite" parents="535"/>
    </justification>
  </clause>

  <clause id="584">
    <literal><![CDATA[
      ((x v y) ^ x)''' = x'''
    ]]></literal>
    <justification jstring="[para(522(a,1),418(a,1,1,1)),rewrite(522(5),535(2))].">
      <j1 rule="para" parents="522 418"/>
      <j2 rule="rewrite" parents="522 535"/>
    </justification>
  </clause>

  <clause id="611">
    <literal><![CDATA[
      x v (x ^ ((y' ^ (x v y))'' v y))'' = x
    ]]></literal>
    <justification jstring="[back_rewrite(424),rewrite(535(4),537(7),22(3),535(9),536(12),22(8),536(8),522(6),535(8),535(12),536(15),522(4))].">
      <j1 rule="back_rewrite" parents="424"/>
      <j2 rule="rewrite" parents="535 537 22 535 536 22 536 522 535 535 536 522"/>
    </justification>
  </clause>

  <clause id="625">
    <literal><![CDATA[
      x v (x ^ y)'' = x''
    ]]></literal>
    <justification jstring="[back_rewrite(352),rewrite(535(3),535(7),536(10),522(5),535(6),584(9),535(5),415(5),522(4),535(4),535(8),536(11),522(6)),flip(a)].">
      <j1 rule="back_rewrite" parents="352"/>
      <j2 rule="rewrite" parents="535 535 536 522 535 584 535 415 522 535 535 536 522"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="626">
    <literal><![CDATA[
      x' ^ (x ^ y)''' = x'''
    ]]></literal>
    <justification jstring="[back_rewrite(351),rewrite(535(3),535(7),536(10),522(5),625(5),535(4),533(4),415(5),535(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="351"/>
      <j2 rule="rewrite" parents="535 535 536 522 625 535 533 415 535"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="697">
    <literal><![CDATA[
      c4'' != c4
    ]]></literal>
    <attribute><![CDATA[
      answer(B_SS)
    ]]></attribute>
    <justification jstring="[back_rewrite(14),rewrite(535(5),535(9),626(9),522(5))].">
      <j1 rule="back_rewrite" parents="14"/>
      <j2 rule="rewrite" parents="535 535 626 522"/>
    </justification>
  </clause>

  <clause id="699">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[back_rewrite(611),rewrite(625(10))].">
      <j1 rule="back_rewrite" parents="611"/>
      <j2 rule="rewrite" parents="625"/>
    </justification>
  </clause>

  <clause id="700">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(B_SS)
    ]]></attribute>
    <justification jstring="[resolve(699,a,697,a)].">
      <j1 rule="resolve" parents="699 697"/>
    </justification>
  </clause>

</proof>

<proof number="2" length="112" max_count="38">

<comments><![CDATA[
% Proof 2 at 0.35 (+ 0.00) seconds: OM_SS.
% Length of proof is 112.
% Level of proof is 28.
% Maximum clause weight is 38.
% Given clauses 203.
]]></comments>

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

  <clause id="4">
    <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">
    <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">
    <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">
    <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">
    <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="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="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="26">
    <literal><![CDATA[
      f(f(f(f(x,y),y'),z),f(y,f(f(y,f(y',y)),y))) = y
    ]]></literal>
    <justification jstring="[para(10(a,1),18(a,1,1,1,2))].">
      <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="44">
    <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="45">
    <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="47">
    <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="51">
    <literal><![CDATA[
      x''' ^ f(x,f(f(x,f(x',x)),x)) = x'
    ]]></literal>
    <justification jstring="[para(44(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="44 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="57">
    <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="68">
    <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="69">
    <literal><![CDATA[
      (x' v y) ^ f(x,f(f(x,f(x',x)),x)) = x'
    ]]></literal>
    <justification jstring="[para(47(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="47 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="75">
    <literal><![CDATA[
      x ^ f(x',f(f(y,f(x'',y)),y)) = x''
    ]]></literal>
    <justification jstring="[para(68(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="68 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      f(f(x,y) ^ y',f(y,f(f(y,f(y',y)),y))) = y
    ]]></literal>
    <justification jstring="[para(10(a,1),26(a,1,1)),rewrite(16(4))].">
      <j1 rule="para" parents="10 26"/>
      <j2 rule="rewrite" parents="16"/>
    </justification>
  </clause>

  <clause id="90">
    <literal><![CDATA[
      f(f(f(x,y),y'),z) ^ f(y,f(f(y,f(y',y)),y)) = y'
    ]]></literal>
    <justification jstring="[para(26(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="26 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="109">
    <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="117">
    <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(45(12),27(18))].">
      <j1 rule="para" parents="27 36"/>
      <j2 rule="rewrite" parents="45 27"/>
    </justification>
  </clause>

  <clause id="118">
    <literal><![CDATA[
      f(x''',f(x,f(f(y,f(x',y)),y))) = x
    ]]></literal>
    <justification jstring="[para(44(a,1),36(a,1,2,1)),rewrite(51(12),44(18))].">
      <j1 rule="para" parents="44 36"/>
      <j2 rule="rewrite" parents="51 44"/>
    </justification>
  </clause>

  <clause id="120">
    <literal><![CDATA[
      f(x' v y,f(x,f(f(z,f(x',z)),z))) = x
    ]]></literal>
    <justification jstring="[para(47(a,1),36(a,1,2,1)),rewrite(69(10),47(16))].">
      <j1 rule="para" parents="47 36"/>
      <j2 rule="rewrite" parents="69 47"/>
    </justification>
  </clause>

  <clause id="121">
    <literal><![CDATA[
      f(f(f(f(x,y),y'),z),f(y,f(f(u,f(y',u)),u))) = y
    ]]></literal>
    <justification jstring="[para(26(a,1),36(a,1,2,1)),rewrite(90(14),26(20))].">
      <j1 rule="para" parents="26 36"/>
      <j2 rule="rewrite" parents="90 26"/>
    </justification>
  </clause>

  <clause id="123">
    <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="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(123(12),28(18))].">
      <j1 rule="para" parents="28 36"/>
      <j2 rule="rewrite" parents="123 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),118(a,1,2,2,1,2)),rewrite(17(7))].">
      <j1 rule="para" parents="10 118"/>
      <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(118(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="118 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="148">
    <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),75(a,1,2,2,1,2))].">
      <j1 rule="para" parents="135 75"/>
    </justification>
  </clause>

  <clause id="151">
    <literal><![CDATA[
      (x' v y) ^ f(x,f(f(z,f(x',z)),z)) = x'
    ]]></literal>
    <justification jstring="[para(120(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="120 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="174">
    <literal><![CDATA[
      f(f(x'',y),f(x,f(f(z',x v z),z'))) = x
    ]]></literal>
    <justification jstring="[para(17(a,1),117(a,1,2,2,1,2))].">
      <j1 rule="para" parents="17 117"/>
    </justification>
  </clause>

  <clause id="220">
    <literal><![CDATA[
      (x' ^ f(x,y)) ^ f(x,f(f(y,f(x',y)),y)) = x'
    ]]></literal>
    <justification jstring="[para(57(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="57 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="229">
    <literal><![CDATA[
      f(x' ^ f(x,y),f(x,f(f(z,f(x',z)),z))) = x
    ]]></literal>
    <justification jstring="[para(57(a,1),36(a,1,2,1)),rewrite(220(12),57(18))].">
      <j1 rule="para" parents="57 36"/>
      <j2 rule="rewrite" parents="220 57"/>
    </justification>
  </clause>

  <clause id="231">
    <literal><![CDATA[
      (f(x,y) ^ y') ^ f(y,f(f(y,f(y',y)),y)) = y'
    ]]></literal>
    <justification jstring="[para(89(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="89 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="238">
    <literal><![CDATA[
      f(f(x,y) ^ y',f(y,f(f(z,f(y',z)),z))) = y
    ]]></literal>
    <justification jstring="[para(89(a,1),36(a,1,2,1)),rewrite(231(12),89(18))].">
      <j1 rule="para" parents="89 36"/>
      <j2 rule="rewrite" parents="231 89"/>
    </justification>
  </clause>

  <clause id="284">
    <literal><![CDATA[
      f(x' ^ f(x,y),f(x,f(x v x',x'))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),229(a,1,2,2,1,2)),rewrite(17(7))].">
      <j1 rule="para" parents="10 229"/>
      <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(284(a,1),36(a,1,2,2,1,2))].">
      <j1 rule="para" parents="284 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(284(a,1),109(a,1,2,2,1,2))].">
      <j1 rule="para" parents="284 109"/>
    </justification>
  </clause>

  <clause id="341">
    <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="351">
    <literal><![CDATA[
      x' ^ f(f(x',f(x,y)),x) = x' ^ f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(311),rewrite(341(16))].">
      <j1 rule="back_rewrite" parents="311"/>
      <j2 rule="rewrite" parents="341"/>
    </justification>
  </clause>

  <clause id="352">
    <literal><![CDATA[
      f(x',f(f(x',f(x,y)),x)) = f(x',f(x,y))
    ]]></literal>
    <justification jstring="[back_rewrite(307),rewrite(341(16))].">
      <j1 rule="back_rewrite" parents="307"/>
      <j2 rule="rewrite" parents="341"/>
    </justification>
  </clause>

  <clause id="360">
    <literal><![CDATA[
      x' ^ f(x'',x) = x'''
    ]]></literal>
    <justification jstring="[back_rewrite(148),rewrite(341(15))].">
      <j1 rule="back_rewrite" parents="148"/>
      <j2 rule="rewrite" parents="341"/>
    </justification>
  </clause>

  <clause id="380">
    <literal><![CDATA[
      x'' ^ (x'' v x) = x''''
    ]]></literal>
    <justification jstring="[para(17(a,1),360(a,1,2))].">
      <j1 rule="para" parents="17 360"/>
    </justification>
  </clause>

  <clause id="393">
    <literal><![CDATA[
      f(x',f(x,f(x v x',x'))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),341(a,1,1))].">
      <j1 rule="para" parents="10 341"/>
    </justification>
  </clause>

  <clause id="414">
    <literal><![CDATA[
      f(x,x'') = x'
    ]]></literal>
    <justification jstring="[para(393(a,1),68(a,1,2,2,1,2)),rewrite(341(22),10(3))].">
      <j1 rule="para" parents="393 68"/>
      <j2 rule="rewrite" parents="341 10"/>
    </justification>
  </clause>

  <clause id="415">
    <literal><![CDATA[
      x ^ x'' = x''
    ]]></literal>
    <justification jstring="[para(393(a,1),75(a,1,2,2,1,2)),rewrite(341(22),10(3))].">
      <j1 rule="para" parents="393 75"/>
      <j2 rule="rewrite" parents="341 10"/>
    </justification>
  </clause>

  <clause id="417">
    <literal><![CDATA[
      x'' v x = x
    ]]></literal>
    <justification jstring="[para(393(a,1),118(a,1,2,2,1,2)),rewrite(341(15),10(4),17(5))].">
      <j1 rule="para" parents="393 118"/>
      <j2 rule="rewrite" parents="341 10 17"/>
    </justification>
  </clause>

  <clause id="418">
    <literal><![CDATA[
      f(x' v y,x') = x
    ]]></literal>
    <justification jstring="[para(393(a,1),120(a,1,2,2,1,2)),rewrite(341(14),10(3))].">
      <j1 rule="para" parents="393 120"/>
      <j2 rule="rewrite" parents="341 10"/>
    </justification>
  </clause>

  <clause id="420">
    <literal><![CDATA[
      x''' ^ x' = x'
    ]]></literal>
    <justification jstring="[para(393(a,1),136(a,1,2,2,1,2)),rewrite(341(15),10(4))].">
      <j1 rule="para" parents="393 136"/>
      <j2 rule="rewrite" parents="341 10"/>
    </justification>
  </clause>

  <clause id="421">
    <literal><![CDATA[
      (x' v y) ^ x' = x'
    ]]></literal>
    <justification jstring="[para(393(a,1),151(a,1,2,2,1,2)),rewrite(341(14),10(3))].">
      <j1 rule="para" parents="393 151"/>
      <j2 rule="rewrite" parents="341 10"/>
    </justification>
  </clause>

  <clause id="424">
    <literal><![CDATA[
      f(x',f(x,f(f(y',x v y),y'))) = x
    ]]></literal>
    <justification jstring="[para(393(a,1),174(a,1,1))].">
      <j1 rule="para" parents="393 174"/>
    </justification>
  </clause>

  <clause id="428">
    <literal><![CDATA[
      f(f(x,y),y') v y = y
    ]]></literal>
    <justification jstring="[para(393(a,1),238(a,1,2,2,1,2)),rewrite(341(15),10(4),23(5))].">
      <j1 rule="para" parents="393 238"/>
      <j2 rule="rewrite" parents="341 10 23"/>
    </justification>
  </clause>

  <clause id="439">
    <literal><![CDATA[
      x'' ^ x = x''''
    ]]></literal>
    <justification jstring="[back_rewrite(380),rewrite(417(5))].">
      <j1 rule="back_rewrite" parents="380"/>
      <j2 rule="rewrite" parents="417"/>
    </justification>
  </clause>

  <clause id="443">
    <literal><![CDATA[
      x''''' = x'
    ]]></literal>
    <justification jstring="[back_rewrite(420),rewrite(439(5))].">
      <j1 rule="back_rewrite" parents="420"/>
      <j2 rule="rewrite" parents="439"/>
    </justification>
  </clause>

  <clause id="522">
    <literal><![CDATA[
      x'''' = x
    ]]></literal>
    <justification jstring="[para(443(a,1),414(a,1,2)),rewrite(17(5),417(3)),flip(a)].">
      <j1 rule="para" parents="443 414"/>
      <j2 rule="rewrite" parents="17 417"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="533">
    <literal><![CDATA[
      x'' ^ x = x
    ]]></literal>
    <justification jstring="[back_rewrite(439),rewrite(522(7))].">
      <j1 rule="back_rewrite" parents="439"/>
      <j2 rule="rewrite" parents="522"/>
    </justification>
  </clause>

  <clause id="535">
    <literal><![CDATA[
      f(x,y) = (x ^ y)'''
    ]]></literal>
    <justification jstring="[para(16(a,1),522(a,1,1,1,1)),flip(a)].">
      <j1 rule="para" parents="16 522"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="536">
    <literal><![CDATA[
      (x ^ y')''' = x''' v y
    ]]></literal>
    <justification jstring="[para(522(a,1),17(a,1,1)),rewrite(535(2))].">
      <j1 rule="para" parents="522 17"/>
      <j2 rule="rewrite" parents="535"/>
    </justification>
  </clause>

  <clause id="537">
    <literal><![CDATA[
      (x' ^ y)''' = x v y'''
    ]]></literal>
    <justification jstring="[para(522(a,1),17(a,1,2)),rewrite(535(2))].">
      <j1 rule="para" parents="522 17"/>
      <j2 rule="rewrite" parents="535"/>
    </justification>
  </clause>

  <clause id="584">
    <literal><![CDATA[
      ((x v y) ^ x)''' = x'''
    ]]></literal>
    <justification jstring="[para(522(a,1),418(a,1,1,1)),rewrite(522(5),535(2))].">
      <j1 rule="para" parents="522 418"/>
      <j2 rule="rewrite" parents="522 535"/>
    </justification>
  </clause>

  <clause id="609">
    <literal><![CDATA[
      ((x ^ y)'' v y) v y = y
    ]]></literal>
    <justification jstring="[back_rewrite(428),rewrite(535(1),535(6),536(9),522(5))].">
      <j1 rule="back_rewrite" parents="428"/>
      <j2 rule="rewrite" parents="535 535 536 522"/>
    </justification>
  </clause>

  <clause id="611">
    <literal><![CDATA[
      x v (x ^ ((y' ^ (x v y))'' v y))'' = x
    ]]></literal>
    <justification jstring="[back_rewrite(424),rewrite(535(4),537(7),22(3),535(9),536(12),22(8),536(8),522(6),535(8),535(12),536(15),522(4))].">
      <j1 rule="back_rewrite" parents="424"/>
      <j2 rule="rewrite" parents="535 537 22 535 536 22 536 522 535 535 536 522"/>
    </justification>
  </clause>

  <clause id="625">
    <literal><![CDATA[
      x v (x ^ y)'' = x''
    ]]></literal>
    <justification jstring="[back_rewrite(352),rewrite(535(3),535(7),536(10),522(5),535(6),584(9),535(5),415(5),522(4),535(4),535(8),536(11),522(6)),flip(a)].">
      <j1 rule="back_rewrite" parents="352"/>
      <j2 rule="rewrite" parents="535 535 536 522 535 584 535 415 522 535 535 536 522"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="626">
    <literal><![CDATA[
      x' ^ (x ^ y)''' = x'''
    ]]></literal>
    <justification jstring="[back_rewrite(351),rewrite(535(3),535(7),536(10),522(5),625(5),535(4),533(4),415(5),535(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="351"/>
      <j2 rule="rewrite" parents="535 535 536 522 625 535 533 415 535"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="680">
    <literal><![CDATA[
      ((x ^ y)'' v (y ^ z)'') v (y ^ ((u ^ (y v u'''))'' v u'''))'' = y
    ]]></literal>
    <justification jstring="[back_rewrite(134),rewrite(535(1),535(5),535(11),537(14),535(14),535(18),537(21),535(21),535(25),536(28),536(12),522(5))].">
      <j1 rule="back_rewrite" parents="134"/>
      <j2 rule="rewrite" parents="535 535 535 537 535 535 537 535 535 536 536 522"/>
    </justification>
  </clause>

  <clause id="686">
    <literal><![CDATA[
      (((x ^ y)'' v y) ^ z)'' v (y ^ ((u ^ (y v u'''))'' v u'''))'' = y
    ]]></literal>
    <justification jstring="[back_rewrite(121),rewrite(535(1),535(6),536(9),522(5),535(5),535(10),537(13),535(13),535(17),537(20),535(20),535(24),536(27),522(9))].">
      <j1 rule="back_rewrite" parents="121"/>
      <j2 rule="rewrite" parents="535 535 536 522 535 535 537 535 535 537 535 535 536 522"/>
    </justification>
  </clause>

  <clause id="696">
    <literal><![CDATA[
      (c6 ^ (c6''' v (c6 ^ c7)''))''' != (c6 ^ c7)'''
    ]]></literal>
    <attribute><![CDATA[
      answer(OM_SS)
    ]]></attribute>
    <justification jstring="[back_rewrite(15),rewrite(535(5),535(9),536(12),535(12),535(18))].">
      <j1 rule="back_rewrite" parents="15"/>
      <j2 rule="rewrite" parents="535 535 536 535 535"/>
    </justification>
  </clause>

  <clause id="699">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[back_rewrite(611),rewrite(625(10))].">
      <j1 rule="back_rewrite" parents="611"/>
      <j2 rule="rewrite" parents="625"/>
    </justification>
  </clause>

  <clause id="745">
    <literal><![CDATA[
      (c6 ^ (c6' v (c6 ^ c7)))' != (c6 ^ c7)'
    ]]></literal>
    <attribute><![CDATA[
      answer(OM_SS)
    ]]></attribute>
    <justification jstring="[back_rewrite(696),rewrite(699(4),699(8),699(10),699(14))].">
      <j1 rule="back_rewrite" parents="696"/>
      <j2 rule="rewrite" parents="699 699 699 699"/>
    </justification>
  </clause>

  <clause id="753">
    <literal><![CDATA[
      (((x ^ y) v y) ^ z) v (y ^ ((u ^ (y v u')) v u')) = y
    ]]></literal>
    <justification jstring="[back_rewrite(686),rewrite(699(3),699(5),699(5),699(8),699(8),699(11))].">
      <j1 rule="back_rewrite" parents="686"/>
      <j2 rule="rewrite" parents="699 699 699 699 699 699"/>
    </justification>
  </clause>

  <clause id="758">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v (y ^ ((u ^ (y v u')) v u')) = y
    ]]></literal>
    <justification jstring="[back_rewrite(680),rewrite(699(3),699(4),699(5),699(8),699(8),699(11))].">
      <j1 rule="back_rewrite" parents="680"/>
      <j2 rule="rewrite" parents="699 699 699 699 699 699"/>
    </justification>
  </clause>

  <clause id="789">
    <literal><![CDATA[
      x' ^ (x ^ y)' = x'
    ]]></literal>
    <justification jstring="[back_rewrite(626),rewrite(699(4),699(6))].">
      <j1 rule="back_rewrite" parents="626"/>
      <j2 rule="rewrite" parents="699 699"/>
    </justification>
  </clause>

  <clause id="790">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[back_rewrite(625),rewrite(699(3),699(4))].">
      <j1 rule="back_rewrite" parents="625"/>
      <j2 rule="rewrite" parents="699 699"/>
    </justification>
  </clause>

  <clause id="797">
    <literal><![CDATA[
      ((x ^ y) v y) v y = y
    ]]></literal>
    <justification jstring="[back_rewrite(609),rewrite(699(3))].">
      <j1 rule="back_rewrite" parents="609"/>
      <j2 rule="rewrite" parents="699"/>
    </justification>
  </clause>

  <clause id="817">
    <literal><![CDATA[
      (x' ^ y)' = x v y'
    ]]></literal>
    <justification jstring="[back_rewrite(537),rewrite(699(4),699(5))].">
      <j1 rule="back_rewrite" parents="537"/>
      <j2 rule="rewrite" parents="699 699"/>
    </justification>
  </clause>

  <clause id="823">
    <literal><![CDATA[
      (x v y) ^ x = x
    ]]></literal>
    <justification jstring="[para(699(a,1),421(a,1,1,1)),rewrite(699(3),699(4))].">
      <j1 rule="para" parents="699 421"/>
      <j2 rule="rewrite" parents="699 699"/>
    </justification>
  </clause>

  <clause id="829">
    <literal><![CDATA[
      (x v y) v x = x v y
    ]]></literal>
    <justification jstring="[para(823(a,1),790(a,1,2))].">
      <j1 rule="para" parents="823 790"/>
    </justification>
  </clause>

  <clause id="866">
    <literal><![CDATA[
      x ^ (x v y') = x
    ]]></literal>
    <justification jstring="[para(699(a,1),789(a,1,1)),rewrite(817(3),699(5))].">
      <j1 rule="para" parents="699 789"/>
      <j2 rule="rewrite" parents="817 699"/>
    </justification>
  </clause>

  <clause id="871">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[para(699(a,1),866(a,1,2,2))].">
      <j1 rule="para" parents="699 866"/>
    </justification>
  </clause>

  <clause id="887">
    <literal><![CDATA[
      (x ^ y)' = x' v y'
    ]]></literal>
    <justification jstring="[para(699(a,1),817(a,1,1,1))].">
      <j1 rule="para" parents="699 817"/>
    </justification>
  </clause>

  <clause id="914">
    <literal><![CDATA[
      c6' v (c6 ^ (c6' v c7')) != c6' v c7'
    ]]></literal>
    <attribute><![CDATA[
      answer(OM_SS)
    ]]></attribute>
    <justification jstring="[back_rewrite(745),rewrite(887(9),22(9),699(5),887(7),887(14))].">
      <j1 rule="back_rewrite" parents="745"/>
      <j2 rule="rewrite" parents="887 22 699 887 887"/>
    </justification>
  </clause>

  <clause id="924">
    <literal><![CDATA[
      (x ^ y) v (y ^ ((z ^ (y v z')) v z')) = y
    ]]></literal>
    <justification jstring="[para(823(a,1),753(a,1,1))].">
      <j1 rule="para" parents="823 753"/>
    </justification>
  </clause>

  <clause id="925">
    <literal><![CDATA[
      x ^ ((y ^ (x v y')) v y') = x
    ]]></literal>
    <justification jstring="[para(753(a,1),797(a,1,1)),rewrite(790(7)),flip(a)].">
      <j1 rule="para" parents="753 797"/>
      <j2 rule="rewrite" parents="790"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="929">
    <literal><![CDATA[
      (((x ^ y) v y) ^ z) ^ y = ((x ^ y) v y) ^ z
    ]]></literal>
    <justification jstring="[para(753(a,1),871(a,1,2))].">
      <j1 rule="para" parents="753 871"/>
    </justification>
  </clause>

  <clause id="934">
    <literal><![CDATA[
      (x ^ y) v y = y
    ]]></literal>
    <justification jstring="[back_rewrite(924),rewrite(925(7))].">
      <j1 rule="back_rewrite" parents="924"/>
      <j2 rule="rewrite" parents="925"/>
    </justification>
  </clause>

  <clause id="936">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v y = y
    ]]></literal>
    <justification jstring="[back_rewrite(758),rewrite(925(9))].">
      <j1 rule="back_rewrite" parents="758"/>
      <j2 rule="rewrite" parents="925"/>
    </justification>
  </clause>

  <clause id="938">
    <literal><![CDATA[
      (x ^ y) ^ x = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(929),rewrite(934(2),934(4))].">
      <j1 rule="back_rewrite" parents="929"/>
      <j2 rule="rewrite" parents="934 934"/>
    </justification>
  </clause>

  <clause id="943">
    <literal><![CDATA[
      (x' v y') ^ y' = y'
    ]]></literal>
    <justification jstring="[para(934(a,1),22(a,1,1)),rewrite(887(3)),flip(a)].">
      <j1 rule="para" parents="934 22"/>
      <j2 rule="rewrite" parents="887"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="996">
    <literal><![CDATA[
      x ^ ((y ^ x) v (x ^ z)) = (y ^ x) v (x ^ z)
    ]]></literal>
    <justification jstring="[para(936(a,1),823(a,1,1))].">
      <j1 rule="para" parents="936 823"/>
    </justification>
  </clause>

  <clause id="997">
    <literal><![CDATA[
      ((x ^ (y v z)) v y) v (y v z) = y v z
    ]]></literal>
    <justification jstring="[para(823(a,1),936(a,1,1,2))].">
      <j1 rule="para" parents="823 936"/>
    </justification>
  </clause>

  <clause id="1013">
    <literal><![CDATA[
      (x' v y) ^ y = y
    ]]></literal>
    <justification jstring="[para(699(a,1),943(a,1,1,2)),rewrite(699(4),699(5))].">
      <j1 rule="para" parents="699 943"/>
      <j2 rule="rewrite" parents="699 699"/>
    </justification>
  </clause>

  <clause id="1024">
    <literal><![CDATA[
      (x v y) ^ y = y
    ]]></literal>
    <justification jstring="[para(699(a,1),1013(a,1,1,1))].">
      <j1 rule="para" parents="699 1013"/>
    </justification>
  </clause>

  <clause id="1032">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(1024(a,1),938(a,1,1)),rewrite(1024(4))].">
      <j1 rule="para" parents="1024 938"/>
      <j2 rule="rewrite" parents="1024"/>
    </justification>
  </clause>

  <clause id="1168">
    <literal><![CDATA[
      (x' v y) ^ ((x ^ (x' v y)) v x') = x' v y
    ]]></literal>
    <justification jstring="[para(829(a,1),925(a,1,2,1,2))].">
      <j1 rule="para" parents="829 925"/>
    </justification>
  </clause>

  <clause id="1448">
    <literal><![CDATA[
      (x v y) ^ ((z ^ (x v y)) v x) = (z ^ (x v y)) v x
    ]]></literal>
    <justification jstring="[para(823(a,1),996(a,1,2,2)),rewrite(823(9))].">
      <j1 rule="para" parents="823 996"/>
      <j2 rule="rewrite" parents="823"/>
    </justification>
  </clause>

  <clause id="1477">
    <literal><![CDATA[
      (x ^ (x' v y)) v x' = x' v y
    ]]></literal>
    <justification jstring="[back_rewrite(1168),rewrite(1448(8))].">
      <j1 rule="back_rewrite" parents="1168"/>
      <j2 rule="rewrite" parents="1448"/>
    </justification>
  </clause>

  <clause id="1573">
    <literal><![CDATA[
      (x v y) v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(1032(a,1),997(a,1,1,1))].">
      <j1 rule="para" parents="1032 997"/>
    </justification>
  </clause>

  <clause id="2291">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[para(1573(a,1),829(a,1,1)),rewrite(1573(3),1573(4))].">
      <j1 rule="para" parents="1573 829"/>
      <j2 rule="rewrite" parents="1573 1573"/>
    </justification>
  </clause>

  <clause id="2467">
    <literal><![CDATA[
      x' v (x ^ (x' v y)) = x' v y
    ]]></literal>
    <justification jstring="[back_rewrite(1477),rewrite(2291(5))].">
      <j1 rule="back_rewrite" parents="1477"/>
      <j2 rule="rewrite" parents="2291"/>
    </justification>
  </clause>

  <clause id="2468">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(OM_SS)
    ]]></attribute>
    <justification jstring="[resolve(2467,a,914,a)].">
      <j1 rule="resolve" parents="2467 914"/>
    </justification>
  </clause>

</proof>

<proof number="3" length="175" max_count="38">

<comments><![CDATA[
% Proof 3 at 9.00 (+ 0.10) seconds: A_SS.
% Length of proof is 175.
% Level of proof is 37.
% Maximum clause weight is 38.
% Given clauses 649.
]]></comments>

  <clause id="1">
    <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>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="4">
    <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">
    <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">
    <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">
    <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">
    <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="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="26">
    <literal><![CDATA[
      f(f(f(f(x,y),y'),z),f(y,f(f(y,f(y',y)),y))) = y
    ]]></literal>
    <justification jstring="[para(10(a,1),18(a,1,1,1,2))].">
      <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="34">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z')),u),f(y,f(f(z',y v z),z'))) = y
    ]]></literal>
    <justification jstring="[para(17(a,1),18(a,1,2,2,1,2))].">
      <j1 rule="para" parents="17 18"/>
    </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="44">
    <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="45">
    <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="47">
    <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="51">
    <literal><![CDATA[
      x''' ^ f(x,f(f(x,f(x',x)),x)) = x'
    ]]></literal>
    <justification jstring="[para(44(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="44 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="57">
    <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="68">
    <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="69">
    <literal><![CDATA[
      (x' v y) ^ f(x,f(f(x,f(x',x)),x)) = x'
    ]]></literal>
    <justification jstring="[para(47(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="47 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="75">
    <literal><![CDATA[
      x ^ f(x',f(f(y,f(x'',y)),y)) = x''
    ]]></literal>
    <justification jstring="[para(68(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="68 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      f(f(x,y) ^ y',f(y,f(f(y,f(y',y)),y))) = y
    ]]></literal>
    <justification jstring="[para(10(a,1),26(a,1,1)),rewrite(16(4))].">
      <j1 rule="para" parents="10 26"/>
      <j2 rule="rewrite" parents="16"/>
    </justification>
  </clause>

  <clause id="90">
    <literal><![CDATA[
      f(f(f(x,y),y'),z) ^ f(y,f(f(y,f(y',y)),y)) = y'
    ]]></literal>
    <justification jstring="[para(26(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="26 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="109">
    <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="117">
    <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(45(12),27(18))].">
      <j1 rule="para" parents="27 36"/>
      <j2 rule="rewrite" parents="45 27"/>
    </justification>
  </clause>

  <clause id="118">
    <literal><![CDATA[
      f(x''',f(x,f(f(y,f(x',y)),y))) = x
    ]]></literal>
    <justification jstring="[para(44(a,1),36(a,1,2,1)),rewrite(51(12),44(18))].">
      <j1 rule="para" parents="44 36"/>
      <j2 rule="rewrite" parents="51 44"/>
    </justification>
  </clause>

  <clause id="120">
    <literal><![CDATA[
      f(x' v y,f(x,f(f(z,f(x',z)),z))) = x
    ]]></literal>
    <justification jstring="[para(47(a,1),36(a,1,2,1)),rewrite(69(10),47(16))].">
      <j1 rule="para" parents="47 36"/>
      <j2 rule="rewrite" parents="69 47"/>
    </justification>
  </clause>

  <clause id="121">
    <literal><![CDATA[
      f(f(f(f(x,y),y'),z),f(y,f(f(u,f(y',u)),u))) = y
    ]]></literal>
    <justification jstring="[para(26(a,1),36(a,1,2,1)),rewrite(90(14),26(20))].">
      <j1 rule="para" parents="26 36"/>
      <j2 rule="rewrite" parents="90 26"/>
    </justification>
  </clause>

  <clause id="123">
    <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="127">
    <literal><![CDATA[
      f(f(x,y) ^ f(y,z'),f(y,f(f(z',y v z),z'))) = y
    ]]></literal>
    <justification jstring="[para(17(a,1),28(a,1,2,2,1,2))].">
      <j1 rule="para" parents="17 28"/>
    </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(123(12),28(18))].">
      <j1 rule="para" parents="28 36"/>
      <j2 rule="rewrite" parents="123 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),118(a,1,2,2,1,2)),rewrite(17(7))].">
      <j1 rule="para" parents="10 118"/>
      <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(118(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="118 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="148">
    <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),75(a,1,2,2,1,2))].">
      <j1 rule="para" parents="135 75"/>
    </justification>
  </clause>

  <clause id="151">
    <literal><![CDATA[
      (x' v y) ^ f(x,f(f(z,f(x',z)),z)) = x'
    ]]></literal>
    <justification jstring="[para(120(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="120 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="174">
    <literal><![CDATA[
      f(f(x'',y),f(x,f(f(z',x v z),z'))) = x
    ]]></literal>
    <justification jstring="[para(17(a,1),117(a,1,2,2,1,2))].">
      <j1 rule="para" parents="17 117"/>
    </justification>
  </clause>

  <clause id="208">
    <literal><![CDATA[
      (x' v y) ^ f(x,f(f(z',x v z),z')) = x'
    ]]></literal>
    <justification jstring="[para(17(a,1),151(a,1,2,2,1,2))].">
      <j1 rule="para" parents="17 151"/>
    </justification>
  </clause>

  <clause id="220">
    <literal><![CDATA[
      (x' ^ f(x,y)) ^ f(x,f(f(y,f(x',y)),y)) = x'
    ]]></literal>
    <justification jstring="[para(57(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="57 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="229">
    <literal><![CDATA[
      f(x' ^ f(x,y),f(x,f(f(z,f(x',z)),z))) = x
    ]]></literal>
    <justification jstring="[para(57(a,1),36(a,1,2,1)),rewrite(220(12),57(18))].">
      <j1 rule="para" parents="57 36"/>
      <j2 rule="rewrite" parents="220 57"/>
    </justification>
  </clause>

  <clause id="231">
    <literal><![CDATA[
      (f(x,y) ^ y') ^ f(y,f(f(y,f(y',y)),y)) = y'
    ]]></literal>
    <justification jstring="[para(89(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="89 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="238">
    <literal><![CDATA[
      f(f(x,y) ^ y',f(y,f(f(z,f(y',z)),z))) = y
    ]]></literal>
    <justification jstring="[para(89(a,1),36(a,1,2,1)),rewrite(231(12),89(18))].">
      <j1 rule="para" parents="89 36"/>
      <j2 rule="rewrite" parents="231 89"/>
    </justification>
  </clause>

  <clause id="284">
    <literal><![CDATA[
      f(x' ^ f(x,y),f(x,f(x v x',x'))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),229(a,1,2,2,1,2)),rewrite(17(7))].">
      <j1 rule="para" parents="10 229"/>
      <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(284(a,1),36(a,1,2,2,1,2))].">
      <j1 rule="para" parents="284 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(284(a,1),109(a,1,2,2,1,2))].">
      <j1 rule="para" parents="284 109"/>
    </justification>
  </clause>

  <clause id="341">
    <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="351">
    <literal><![CDATA[
      x' ^ f(f(x',f(x,y)),x) = x' ^ f(x,y)
    ]]></literal>
    <justification jstring="[back_rewrite(311),rewrite(341(16))].">
      <j1 rule="back_rewrite" parents="311"/>
      <j2 rule="rewrite" parents="341"/>
    </justification>
  </clause>

  <clause id="352">
    <literal><![CDATA[
      f(x',f(f(x',f(x,y)),x)) = f(x',f(x,y))
    ]]></literal>
    <justification jstring="[back_rewrite(307),rewrite(341(16))].">
      <j1 rule="back_rewrite" parents="307"/>
      <j2 rule="rewrite" parents="341"/>
    </justification>
  </clause>

  <clause id="360">
    <literal><![CDATA[
      x' ^ f(x'',x) = x'''
    ]]></literal>
    <justification jstring="[back_rewrite(148),rewrite(341(15))].">
      <j1 rule="back_rewrite" parents="148"/>
      <j2 rule="rewrite" parents="341"/>
    </justification>
  </clause>

  <clause id="380">
    <literal><![CDATA[
      x'' ^ (x'' v x) = x''''
    ]]></literal>
    <justification jstring="[para(17(a,1),360(a,1,2))].">
      <j1 rule="para" parents="17 360"/>
    </justification>
  </clause>

  <clause id="393">
    <literal><![CDATA[
      f(x',f(x,f(x v x',x'))) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),341(a,1,1))].">
      <j1 rule="para" parents="10 341"/>
    </justification>
  </clause>

  <clause id="414">
    <literal><![CDATA[
      f(x,x'') = x'
    ]]></literal>
    <justification jstring="[para(393(a,1),68(a,1,2,2,1,2)),rewrite(341(22),10(3))].">
      <j1 rule="para" parents="393 68"/>
      <j2 rule="rewrite" parents="341 10"/>
    </justification>
  </clause>

  <clause id="415">
    <literal><![CDATA[
      x ^ x'' = x''
    ]]></literal>
    <justification jstring="[para(393(a,1),75(a,1,2,2,1,2)),rewrite(341(22),10(3))].">
      <j1 rule="para" parents="393 75"/>
      <j2 rule="rewrite" parents="341 10"/>
    </justification>
  </clause>

  <clause id="417">
    <literal><![CDATA[
      x'' v x = x
    ]]></literal>
    <justification jstring="[para(393(a,1),118(a,1,2,2,1,2)),rewrite(341(15),10(4),17(5))].">
      <j1 rule="para" parents="393 118"/>
      <j2 rule="rewrite" parents="341 10 17"/>
    </justification>
  </clause>

  <clause id="418">
    <literal><![CDATA[
      f(x' v y,x') = x
    ]]></literal>
    <justification jstring="[para(393(a,1),120(a,1,2,2,1,2)),rewrite(341(14),10(3))].">
      <j1 rule="para" parents="393 120"/>
      <j2 rule="rewrite" parents="341 10"/>
    </justification>
  </clause>

  <clause id="420">
    <literal><![CDATA[
      x''' ^ x' = x'
    ]]></literal>
    <justification jstring="[para(393(a,1),136(a,1,2,2,1,2)),rewrite(341(15),10(4))].">
      <j1 rule="para" parents="393 136"/>
      <j2 rule="rewrite" parents="341 10"/>
    </justification>
  </clause>

  <clause id="421">
    <literal><![CDATA[
      (x' v y) ^ x' = x'
    ]]></literal>
    <justification jstring="[para(393(a,1),151(a,1,2,2,1,2)),rewrite(341(14),10(3))].">
      <j1 rule="para" parents="393 151"/>
      <j2 rule="rewrite" parents="341 10"/>
    </justification>
  </clause>

  <clause id="424">
    <literal><![CDATA[
      f(x',f(x,f(f(y',x v y),y'))) = x
    ]]></literal>
    <justification jstring="[para(393(a,1),174(a,1,1))].">
      <j1 rule="para" parents="393 174"/>
    </justification>
  </clause>

  <clause id="428">
    <literal><![CDATA[
      f(f(x,y),y') v y = y
    ]]></literal>
    <justification jstring="[para(393(a,1),238(a,1,2,2,1,2)),rewrite(341(15),10(4),23(5))].">
      <j1 rule="para" parents="393 238"/>
      <j2 rule="rewrite" parents="341 10 23"/>
    </justification>
  </clause>

  <clause id="439">
    <literal><![CDATA[
      x'' ^ x = x''''
    ]]></literal>
    <justification jstring="[back_rewrite(380),rewrite(417(5))].">
      <j1 rule="back_rewrite" parents="380"/>
      <j2 rule="rewrite" parents="417"/>
    </justification>
  </clause>

  <clause id="443">
    <literal><![CDATA[
      x''''' = x'
    ]]></literal>
    <justification jstring="[back_rewrite(420),rewrite(439(5))].">
      <j1 rule="back_rewrite" parents="420"/>
      <j2 rule="rewrite" parents="439"/>
    </justification>
  </clause>

  <clause id="522">
    <literal><![CDATA[
      x'''' = x
    ]]></literal>
    <justification jstring="[para(443(a,1),414(a,1,2)),rewrite(17(5),417(3)),flip(a)].">
      <j1 rule="para" parents="443 414"/>
      <j2 rule="rewrite" parents="17 417"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="533">
    <literal><![CDATA[
      x'' ^ x = x
    ]]></literal>
    <justification jstring="[back_rewrite(439),rewrite(522(7))].">
      <j1 rule="back_rewrite" parents="439"/>
      <j2 rule="rewrite" parents="522"/>
    </justification>
  </clause>

  <clause id="535">
    <literal><![CDATA[
      f(x,y) = (x ^ y)'''
    ]]></literal>
    <justification jstring="[para(16(a,1),522(a,1,1,1,1)),flip(a)].">
      <j1 rule="para" parents="16 522"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="536">
    <literal><![CDATA[
      (x ^ y')''' = x''' v y
    ]]></literal>
    <justification jstring="[para(522(a,1),17(a,1,1)),rewrite(535(2))].">
      <j1 rule="para" parents="522 17"/>
      <j2 rule="rewrite" parents="535"/>
    </justification>
  </clause>

  <clause id="537">
    <literal><![CDATA[
      (x' ^ y)''' = x v y'''
    ]]></literal>
    <justification jstring="[para(522(a,1),17(a,1,2)),rewrite(535(2))].">
      <j1 rule="para" parents="522 17"/>
      <j2 rule="rewrite" parents="535"/>
    </justification>
  </clause>

  <clause id="539">
    <literal><![CDATA[
      (x ^ (y ^ z))''' = x''' v (y ^ z)'''
    ]]></literal>
    <justification jstring="[para(522(a,1),24(a,1,1)),rewrite(535(2),535(9))].">
      <j1 rule="para" parents="522 24"/>
      <j2 rule="rewrite" parents="535 535"/>
    </justification>
  </clause>

  <clause id="563">
    <literal><![CDATA[
      (x v y) ^ (x'' v ((z''' v (x ^ z)'') ^ z)'') = x
    ]]></literal>
    <justification jstring="[para(522(a,1),151(a,1,1,1)),rewrite(522(8),535(5),535(9),536(12),535(12),535(16),536(19),522(5),522(19))].">
      <j1 rule="para" parents="522 151"/>
      <j2 rule="rewrite" parents="522 535 535 536 535 535 536 522 522"/>
    </justification>
  </clause>

  <clause id="576">
    <literal><![CDATA[
      (x v y) ^ (x'' v ((z v (x ^ z')'') ^ z')'') = x
    ]]></literal>
    <justification jstring="[para(522(a,1),208(a,1,1,1)),rewrite(535(10),537(13),22(9),522(8),535(11),536(14),22(10),536(10),535(14),537(17),22(13),537(12),22(8),522(7),522(18))].">
      <j1 rule="para" parents="522 208"/>
      <j2 rule="rewrite" parents="535 537 22 522 535 536 22 536 535 537 22 537 22 522 522"/>
    </justification>
  </clause>

  <clause id="578">
    <literal><![CDATA[
      (((x ^ y)'' v (y ^ z)'') ^ u)'' v (y ^ ((z ^ (y v z'''))'' v z'''))'' = y
    ]]></literal>
    <justification jstring="[para(522(a,1),34(a,1,1,1,2,2)),rewrite(535(1),535(5),535(9),536(12),522(5),535(8),522(15),535(16),522(23),535(20),537(23),535(23),535(27),536(30),522(12))].">
      <j1 rule="para" parents="522 34"/>
      <j2 rule="rewrite" parents="535 535 535 536 522 535 522 535 522 535 537 535 535 536 522"/>
    </justification>
  </clause>

  <clause id="584">
    <literal><![CDATA[
      ((x v y) ^ x)''' = x'''
    ]]></literal>
    <justification jstring="[para(522(a,1),418(a,1,1,1)),rewrite(522(5),535(2))].">
      <j1 rule="para" parents="522 418"/>
      <j2 rule="rewrite" parents="522 535"/>
    </justification>
  </clause>

  <clause id="609">
    <literal><![CDATA[
      ((x ^ y)'' v y) v y = y
    ]]></literal>
    <justification jstring="[back_rewrite(428),rewrite(535(1),535(6),536(9),522(5))].">
      <j1 rule="back_rewrite" parents="428"/>
      <j2 rule="rewrite" parents="535 535 536 522"/>
    </justification>
  </clause>

  <clause id="611">
    <literal><![CDATA[
      x v (x ^ ((y' ^ (x v y))'' v y))'' = x
    ]]></literal>
    <justification jstring="[back_rewrite(424),rewrite(535(4),537(7),22(3),535(9),536(12),22(8),536(8),522(6),535(8),535(12),536(15),522(4))].">
      <j1 rule="back_rewrite" parents="424"/>
      <j2 rule="rewrite" parents="535 537 22 535 536 22 536 522 535 535 536 522"/>
    </justification>
  </clause>

  <clause id="625">
    <literal><![CDATA[
      x v (x ^ y)'' = x''
    ]]></literal>
    <justification jstring="[back_rewrite(352),rewrite(535(3),535(7),536(10),522(5),535(6),584(9),535(5),415(5),522(4),535(4),535(8),536(11),522(6)),flip(a)].">
      <j1 rule="back_rewrite" parents="352"/>
      <j2 rule="rewrite" parents="535 535 536 522 535 584 535 415 522 535 535 536 522"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="626">
    <literal><![CDATA[
      x' ^ (x ^ y)''' = x'''
    ]]></literal>
    <justification jstring="[back_rewrite(351),rewrite(535(3),535(7),536(10),522(5),625(5),535(4),533(4),415(5),535(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="351"/>
      <j2 rule="rewrite" parents="535 535 536 522 625 535 533 415 535"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="680">
    <literal><![CDATA[
      ((x ^ y)'' v (y ^ z)'') v (y ^ ((u ^ (y v u'''))'' v u'''))'' = y
    ]]></literal>
    <justification jstring="[back_rewrite(134),rewrite(535(1),535(5),535(11),537(14),535(14),535(18),537(21),535(21),535(25),536(28),536(12),522(5))].">
      <j1 rule="back_rewrite" parents="134"/>
      <j2 rule="rewrite" parents="535 535 535 537 535 535 537 535 535 536 536 522"/>
    </justification>
  </clause>

  <clause id="681">
    <literal><![CDATA[
      ((x ^ y)'' v (y ^ z')'') v (y ^ ((z' ^ (y v z))'' v z))'' = y
    ]]></literal>
    <justification jstring="[back_rewrite(127),rewrite(535(1),535(6),536(9),535(12),537(15),22(11),535(17),536(20),22(16),536(16),522(14),535(16),535(20),536(23),537(12),22(8),522(7))].">
      <j1 rule="back_rewrite" parents="127"/>
      <j2 rule="rewrite" parents="535 535 536 535 537 22 535 536 22 536 522 535 535 536 537 22 522"/>
    </justification>
  </clause>

  <clause id="686">
    <literal><![CDATA[
      (((x ^ y)'' v y) ^ z)'' v (y ^ ((u ^ (y v u'''))'' v u'''))'' = y
    ]]></literal>
    <justification jstring="[back_rewrite(121),rewrite(535(1),535(6),536(9),522(5),535(5),535(10),537(13),535(13),535(17),537(20),535(20),535(24),536(27),522(9))].">
      <j1 rule="back_rewrite" parents="121"/>
      <j2 rule="rewrite" parents="535 535 536 522 535 535 537 535 535 537 535 535 536 522"/>
    </justification>
  </clause>

  <clause id="695">
    <literal><![CDATA[
      c2''' v (c1 ^ c3)''' != c1''' v (c2 ^ c3)'''
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <justification jstring="[back_rewrite(19),rewrite(535(5),539(8),535(16),539(19))].">
      <j1 rule="back_rewrite" parents="19"/>
      <j2 rule="rewrite" parents="535 539 535 539"/>
    </justification>
  </clause>

  <clause id="699">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[back_rewrite(611),rewrite(625(10))].">
      <j1 rule="back_rewrite" parents="611"/>
      <j2 rule="rewrite" parents="625"/>
    </justification>
  </clause>

  <clause id="746">
    <literal><![CDATA[
      c2' v (c1 ^ c3)' != c1' v (c2 ^ c3)'
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <justification jstring="[back_rewrite(695),rewrite(699(3),699(7),699(10),699(14))].">
      <j1 rule="back_rewrite" parents="695"/>
      <j2 rule="rewrite" parents="699 699 699 699"/>
    </justification>
  </clause>

  <clause id="753">
    <literal><![CDATA[
      (((x ^ y) v y) ^ z) v (y ^ ((u ^ (y v u')) v u')) = y
    ]]></literal>
    <justification jstring="[back_rewrite(686),rewrite(699(3),699(5),699(5),699(8),699(8),699(11))].">
      <j1 rule="back_rewrite" parents="686"/>
      <j2 rule="rewrite" parents="699 699 699 699 699 699"/>
    </justification>
  </clause>

  <clause id="757">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z')) v (y ^ ((z' ^ (y v z)) v z)) = y
    ]]></literal>
    <justification jstring="[back_rewrite(681),rewrite(699(3),699(5),699(9),699(11))].">
      <j1 rule="back_rewrite" parents="681"/>
      <j2 rule="rewrite" parents="699 699 699 699"/>
    </justification>
  </clause>

  <clause id="758">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v (y ^ ((u ^ (y v u')) v u')) = y
    ]]></literal>
    <justification jstring="[back_rewrite(680),rewrite(699(3),699(4),699(5),699(8),699(8),699(11))].">
      <j1 rule="back_rewrite" parents="680"/>
      <j2 rule="rewrite" parents="699 699 699 699 699 699"/>
    </justification>
  </clause>

  <clause id="789">
    <literal><![CDATA[
      x' ^ (x ^ y)' = x'
    ]]></literal>
    <justification jstring="[back_rewrite(626),rewrite(699(4),699(6))].">
      <j1 rule="back_rewrite" parents="626"/>
      <j2 rule="rewrite" parents="699 699"/>
    </justification>
  </clause>

  <clause id="790">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[back_rewrite(625),rewrite(699(3),699(4))].">
      <j1 rule="back_rewrite" parents="625"/>
      <j2 rule="rewrite" parents="699 699"/>
    </justification>
  </clause>

  <clause id="797">
    <literal><![CDATA[
      ((x ^ y) v y) v y = y
    ]]></literal>
    <justification jstring="[back_rewrite(609),rewrite(699(3))].">
      <j1 rule="back_rewrite" parents="609"/>
      <j2 rule="rewrite" parents="699"/>
    </justification>
  </clause>

  <clause id="805">
    <literal><![CDATA[
      (((x ^ y) v (y ^ z)) ^ u) v (y ^ ((z ^ (y v z')) v z')) = y
    ]]></literal>
    <justification jstring="[back_rewrite(578),rewrite(699(3),699(4),699(6),699(6),699(9),699(9),699(12))].">
      <j1 rule="back_rewrite" parents="578"/>
      <j2 rule="rewrite" parents="699 699 699 699 699 699 699"/>
    </justification>
  </clause>

  <clause id="807">
    <literal><![CDATA[
      (x v y) ^ (x v ((z v (x ^ z')) ^ z')) = x
    ]]></literal>
    <justification jstring="[back_rewrite(576),rewrite(699(3),699(5),699(8))].">
      <j1 rule="back_rewrite" parents="576"/>
      <j2 rule="rewrite" parents="699 699 699"/>
    </justification>
  </clause>

  <clause id="812">
    <literal><![CDATA[
      (x v y) ^ (x v ((z' v (x ^ z)) ^ z)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(563),rewrite(699(3),699(3),699(5),699(7))].">
      <j1 rule="back_rewrite" parents="563"/>
      <j2 rule="rewrite" parents="699 699 699 699"/>
    </justification>
  </clause>

  <clause id="817">
    <literal><![CDATA[
      (x' ^ y)' = x v y'
    ]]></literal>
    <justification jstring="[back_rewrite(537),rewrite(699(4),699(5))].">
      <j1 rule="back_rewrite" parents="537"/>
      <j2 rule="rewrite" parents="699 699"/>
    </justification>
  </clause>

  <clause id="821">
    <literal><![CDATA[
      x ^ x = x
    ]]></literal>
    <justification jstring="[back_rewrite(20),rewrite(699(3))].">
      <j1 rule="back_rewrite" parents="20"/>
      <j2 rule="rewrite" parents="699"/>
    </justification>
  </clause>

  <clause id="823">
    <literal><![CDATA[
      (x v y) ^ x = x
    ]]></literal>
    <justification jstring="[para(699(a,1),421(a,1,1,1)),rewrite(699(3),699(4))].">
      <j1 rule="para" parents="699 421"/>
      <j2 rule="rewrite" parents="699 699"/>
    </justification>
  </clause>

  <clause id="829">
    <literal><![CDATA[
      (x v y) v x = x v y
    ]]></literal>
    <justification jstring="[para(823(a,1),790(a,1,2))].">
      <j1 rule="para" parents="823 790"/>
    </justification>
  </clause>

  <clause id="866">
    <literal><![CDATA[
      x ^ (x v y') = x
    ]]></literal>
    <justification jstring="[para(699(a,1),789(a,1,1)),rewrite(817(3),699(5))].">
      <j1 rule="para" parents="699 789"/>
      <j2 rule="rewrite" parents="817 699"/>
    </justification>
  </clause>

  <clause id="871">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[para(699(a,1),866(a,1,2,2))].">
      <j1 rule="para" parents="699 866"/>
    </justification>
  </clause>

  <clause id="887">
    <literal><![CDATA[
      (x ^ y)' = x' v y'
    ]]></literal>
    <justification jstring="[para(699(a,1),817(a,1,1,1))].">
      <j1 rule="para" parents="699 817"/>
    </justification>
  </clause>

  <clause id="913">
    <literal><![CDATA[
      c2' v (c1' v c3') != c1' v (c2' v c3')
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <justification jstring="[back_rewrite(746),rewrite(887(6),887(14))].">
      <j1 rule="back_rewrite" parents="746"/>
      <j2 rule="rewrite" parents="887 887"/>
    </justification>
  </clause>

  <clause id="924">
    <literal><![CDATA[
      (x ^ y) v (y ^ ((z ^ (y v z')) v z')) = y
    ]]></literal>
    <justification jstring="[para(823(a,1),753(a,1,1))].">
      <j1 rule="para" parents="823 753"/>
    </justification>
  </clause>

  <clause id="925">
    <literal><![CDATA[
      x ^ ((y ^ (x v y')) v y') = x
    ]]></literal>
    <justification jstring="[para(753(a,1),797(a,1,1)),rewrite(790(7)),flip(a)].">
      <j1 rule="para" parents="753 797"/>
      <j2 rule="rewrite" parents="790"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="929">
    <literal><![CDATA[
      (((x ^ y) v y) ^ z) ^ y = ((x ^ y) v y) ^ z
    ]]></literal>
    <justification jstring="[para(753(a,1),871(a,1,2))].">
      <j1 rule="para" parents="753 871"/>
    </justification>
  </clause>

  <clause id="933">
    <literal><![CDATA[
      (x ^ y) v x = x
    ]]></literal>
    <justification jstring="[para(753(a,1),753(a,1,1,1)),rewrite(925(7),925(7),925(7),925(8))].">
      <j1 rule="para" parents="753 753"/>
      <j2 rule="rewrite" parents="925 925 925 925"/>
    </justification>
  </clause>

  <clause id="934">
    <literal><![CDATA[
      (x ^ y) v y = y
    ]]></literal>
    <justification jstring="[back_rewrite(924),rewrite(925(7))].">
      <j1 rule="back_rewrite" parents="924"/>
      <j2 rule="rewrite" parents="925"/>
    </justification>
  </clause>

  <clause id="935">
    <literal><![CDATA[
      (((x ^ y) v (y ^ z)) ^ u) v y = y
    ]]></literal>
    <justification jstring="[back_rewrite(805),rewrite(925(10))].">
      <j1 rule="back_rewrite" parents="805"/>
      <j2 rule="rewrite" parents="925"/>
    </justification>
  </clause>

  <clause id="936">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v y = y
    ]]></literal>
    <justification jstring="[back_rewrite(758),rewrite(925(9))].">
      <j1 rule="back_rewrite" parents="758"/>
      <j2 rule="rewrite" parents="925"/>
    </justification>
  </clause>

  <clause id="938">
    <literal><![CDATA[
      (x ^ y) ^ x = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(929),rewrite(934(2),934(4))].">
      <j1 rule="back_rewrite" parents="929"/>
      <j2 rule="rewrite" parents="934 934"/>
    </justification>
  </clause>

  <clause id="943">
    <literal><![CDATA[
      (x' v y') ^ y' = y'
    ]]></literal>
    <justification jstring="[para(934(a,1),22(a,1,1)),rewrite(887(3)),flip(a)].">
      <j1 rule="para" parents="934 22"/>
      <j2 rule="rewrite" parents="887"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="945">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(934(a,1),829(a,1,1)),rewrite(934(4))].">
      <j1 rule="para" parents="934 829"/>
      <j2 rule="rewrite" parents="934"/>
    </justification>
  </clause>

  <clause id="951">
    <literal><![CDATA[
      ((x' v y') ^ (y' v z)) ^ (y' v ((z v (y' ^ z')) ^ z')) = y'
    ]]></literal>
    <justification jstring="[para(757(a,1),22(a,1,1)),rewrite(22(6),887(3),887(7),699(7),887(13),22(13),887(12),699(10),22(10)),flip(a)].">
      <j1 rule="para" parents="757 22"/>
      <j2 rule="rewrite" parents="22 887 887 699 887 22 887 699 22"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="996">
    <literal><![CDATA[
      x ^ ((y ^ x) v (x ^ z)) = (y ^ x) v (x ^ z)
    ]]></literal>
    <justification jstring="[para(936(a,1),823(a,1,1))].">
      <j1 rule="para" parents="936 823"/>
    </justification>
  </clause>

  <clause id="997">
    <literal><![CDATA[
      ((x ^ (y v z)) v y) v (y v z) = y v z
    ]]></literal>
    <justification jstring="[para(823(a,1),936(a,1,1,2))].">
      <j1 rule="para" parents="823 936"/>
    </justification>
  </clause>

  <clause id="998">
    <literal><![CDATA[
      x v ((y ^ x) v (x ^ z)) = x
    ]]></literal>
    <justification jstring="[para(936(a,1),829(a,1,1)),rewrite(936(8))].">
      <j1 rule="para" parents="936 829"/>
      <j2 rule="rewrite" parents="936"/>
    </justification>
  </clause>

  <clause id="1013">
    <literal><![CDATA[
      (x' v y) ^ y = y
    ]]></literal>
    <justification jstring="[para(699(a,1),943(a,1,1,2)),rewrite(699(4),699(5))].">
      <j1 rule="para" parents="699 943"/>
      <j2 rule="rewrite" parents="699 699"/>
    </justification>
  </clause>

  <clause id="1024">
    <literal><![CDATA[
      (x v y) ^ y = y
    ]]></literal>
    <justification jstring="[para(699(a,1),1013(a,1,1,1))].">
      <j1 rule="para" parents="699 1013"/>
    </justification>
  </clause>

  <clause id="1031">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(1024(a,1),933(a,1,1))].">
      <j1 rule="para" parents="1024 933"/>
    </justification>
  </clause>

  <clause id="1032">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(1024(a,1),938(a,1,1)),rewrite(1024(4))].">
      <j1 rule="para" parents="1024 938"/>
      <j2 rule="rewrite" parents="1024"/>
    </justification>
  </clause>

  <clause id="1034">
    <literal><![CDATA[
      ((x ^ (y v z)) v z) v (y v z) = y v z
    ]]></literal>
    <justification jstring="[para(1024(a,1),936(a,1,1,2))].">
      <j1 rule="para" parents="1024 936"/>
    </justification>
  </clause>

  <clause id="1037">
    <literal><![CDATA[
      x v ((y v (x ^ y')) ^ y') = x
    ]]></literal>
    <justification jstring="[para(807(a,1),821(a,1)),flip(a)].">
      <j1 rule="para" parents="807 821"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1041">
    <literal><![CDATA[
      ((x' v y') ^ (y' v z)) ^ y' = y'
    ]]></literal>
    <justification jstring="[back_rewrite(951),rewrite(1037(14))].">
      <j1 rule="back_rewrite" parents="951"/>
      <j2 rule="rewrite" parents="1037"/>
    </justification>
  </clause>

  <clause id="1051">
    <literal><![CDATA[
      (x v y) v (y v ((x v y) ^ z)) = x v y
    ]]></literal>
    <justification jstring="[para(1032(a,1),998(a,1,2,1))].">
      <j1 rule="para" parents="1032 998"/>
    </justification>
  </clause>

  <clause id="1067">
    <literal><![CDATA[
      x v ((y' v (x ^ y)) ^ y) = x
    ]]></literal>
    <justification jstring="[para(812(a,1),821(a,1)),flip(a)].">
      <j1 rule="para" parents="812 821"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1126">
    <literal><![CDATA[
      ((x' v (y ^ x)) ^ x) v y = y
    ]]></literal>
    <justification jstring="[para(1067(a,1),1031(a,1,2)),rewrite(1067(10))].">
      <j1 rule="para" parents="1067 1031"/>
      <j2 rule="rewrite" parents="1067"/>
    </justification>
  </clause>

  <clause id="1142">
    <literal><![CDATA[
      (((x' ^ y') v x) ^ (x v y)) v x = x
    ]]></literal>
    <justification jstring="[para(871(a,1),1126(a,1,1,1,2)),rewrite(22(2))].">
      <j1 rule="para" parents="871 1126"/>
      <j2 rule="rewrite" parents="22"/>
    </justification>
  </clause>

  <clause id="1146">
    <literal><![CDATA[
      (((x' ^ y') v y) ^ (x v y)) v y = y
    ]]></literal>
    <justification jstring="[para(1032(a,1),1126(a,1,1,1,2)),rewrite(22(2))].">
      <j1 rule="para" parents="1032 1126"/>
      <j2 rule="rewrite" parents="22"/>
    </justification>
  </clause>

  <clause id="1188">
    <literal><![CDATA[
      ((x ^ y) ^ z) v x = x
    ]]></literal>
    <justification jstring="[para(933(a,1),935(a,1,1,1))].">
      <j1 rule="para" parents="933 935"/>
    </justification>
  </clause>

  <clause id="1200">
    <literal><![CDATA[
      (x ^ y) v (x v z) = x v z
    ]]></literal>
    <justification jstring="[para(823(a,1),1188(a,1,1,1))].">
      <j1 rule="para" parents="823 1188"/>
    </justification>
  </clause>

  <clause id="1204">
    <literal><![CDATA[
      (x ^ y) v (z v x) = z v x
    ]]></literal>
    <justification jstring="[para(1024(a,1),1188(a,1,1,1))].">
      <j1 rule="para" parents="1024 1188"/>
    </justification>
  </clause>

  <clause id="1278">
    <literal><![CDATA[
      (x v y) ^ (x ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(1200(a,1),823(a,1,1))].">
      <j1 rule="para" parents="1200 823"/>
    </justification>
  </clause>

  <clause id="1292">
    <literal><![CDATA[
      (x v y) ^ (y ^ z) = y ^ z
    ]]></literal>
    <justification jstring="[para(1204(a,1),823(a,1,1))].">
      <j1 rule="para" parents="1204 823"/>
    </justification>
  </clause>

  <clause id="1383">
    <literal><![CDATA[
      ((x v y) v z) ^ x = x
    ]]></literal>
    <justification jstring="[para(823(a,1),1278(a,1,2)),rewrite(823(5))].">
      <j1 rule="para" parents="823 1278"/>
      <j2 rule="rewrite" parents="823"/>
    </justification>
  </clause>

  <clause id="1385">
    <literal><![CDATA[
      ((x v y) v z) ^ y = y
    ]]></literal>
    <justification jstring="[para(1024(a,1),1278(a,1,2)),rewrite(1024(5))].">
      <j1 rule="para" parents="1024 1278"/>
      <j2 rule="rewrite" parents="1024"/>
    </justification>
  </clause>

  <clause id="1416">
    <literal><![CDATA[
      x ^ ((x v y) v z) = x
    ]]></literal>
    <justification jstring="[para(1383(a,1),938(a,1,1)),rewrite(1383(6))].">
      <j1 rule="para" parents="1383 938"/>
      <j2 rule="rewrite" parents="1383"/>
    </justification>
  </clause>

  <clause id="1449">
    <literal><![CDATA[
      (x v y) ^ (x v ((x v y) ^ z)) = x v ((x v y) ^ z)
    ]]></literal>
    <justification jstring="[para(871(a,1),996(a,1,2,1)),rewrite(871(7))].">
      <j1 rule="para" parents="871 996"/>
      <j2 rule="rewrite" parents="871"/>
    </justification>
  </clause>

  <clause id="1489">
    <literal><![CDATA[
      x ^ ((y v x) v z) = x
    ]]></literal>
    <justification jstring="[para(1385(a,1),938(a,1,1)),rewrite(1385(6))].">
      <j1 rule="para" parents="1385 938"/>
      <j2 rule="rewrite" parents="1385"/>
    </justification>
  </clause>

  <clause id="1513">
    <literal><![CDATA[
      x ^ (y v (x v z)) = x
    ]]></literal>
    <justification jstring="[para(1031(a,1),1416(a,1,2))].">
      <j1 rule="para" parents="1031 1416"/>
    </justification>
  </clause>

  <clause id="1565">
    <literal><![CDATA[
      x ^ (y v (z v x)) = x
    ]]></literal>
    <justification jstring="[para(1031(a,1),1489(a,1,2))].">
      <j1 rule="para" parents="1031 1489"/>
    </justification>
  </clause>

  <clause id="1573">
    <literal><![CDATA[
      (x v y) v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(1032(a,1),997(a,1,1,1))].">
      <j1 rule="para" parents="1032 997"/>
    </justification>
  </clause>

  <clause id="1635">
    <literal><![CDATA[
      (x v y) v (y v (x v z)) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(1513(a,1),997(a,1,1,1))].">
      <j1 rule="para" parents="1513 997"/>
    </justification>
  </clause>

  <clause id="1641">
    <literal><![CDATA[
      (x v y) v (y v (z v x)) = y v (z v x)
    ]]></literal>
    <justification jstring="[para(1565(a,1),997(a,1,1,1))].">
      <j1 rule="para" parents="1565 997"/>
    </justification>
  </clause>

  <clause id="1939">
    <literal><![CDATA[
      (x v y) v ((z v x) v y) = (z v x) v y
    ]]></literal>
    <justification jstring="[para(1489(a,1),1034(a,1,1,1))].">
      <j1 rule="para" parents="1489 1034"/>
    </justification>
  </clause>

  <clause id="2291">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[para(1573(a,1),829(a,1,1)),rewrite(1573(3),1573(4))].">
      <j1 rule="para" parents="1573 829"/>
      <j2 rule="rewrite" parents="1573 1573"/>
    </justification>
  </clause>

  <clause id="2499">
    <literal><![CDATA[
      x v ((x v (y' ^ x')) ^ (y v x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1146),rewrite(2291(4),2291(7))].">
      <j1 rule="back_rewrite" parents="1146"/>
      <j2 rule="rewrite" parents="2291 2291"/>
    </justification>
  </clause>

  <clause id="2500">
    <literal><![CDATA[
      x v ((x v (x' ^ y')) ^ (x v y)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1142),rewrite(2291(4),2291(7))].">
      <j1 rule="back_rewrite" parents="1142"/>
      <j2 rule="rewrite" parents="2291 2291"/>
    </justification>
  </clause>

  <clause id="2515">
    <literal><![CDATA[
      x' ^ y' = y' ^ x'
    ]]></literal>
    <justification jstring="[para(2291(a,1),22(a,1,1)),rewrite(22(2))].">
      <j1 rule="para" parents="2291 22"/>
      <j2 rule="rewrite" parents="22"/>
    </justification>
  </clause>

  <clause id="2536">
    <literal><![CDATA[
      ((x' v y) ^ (y v z)) ^ y = y
    ]]></literal>
    <justification jstring="[para(699(a,1),1041(a,1,1,1,2)),rewrite(699(4),699(6),699(7))].">
      <j1 rule="para" parents="699 1041"/>
      <j2 rule="rewrite" parents="699 699 699"/>
    </justification>
  </clause>

  <clause id="2869">
    <literal><![CDATA[
      x' ^ y = y ^ x'
    ]]></literal>
    <justification jstring="[para(699(a,1),2515(a,1,1)),rewrite(699(5)),flip(a)].">
      <j1 rule="para" parents="699 2515"/>
      <j2 rule="rewrite" parents="699"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2926">
    <literal><![CDATA[
      x ^ y = y ^ x
    ]]></literal>
    <justification jstring="[para(699(a,1),2869(a,1,1)),rewrite(699(3))].">
      <j1 rule="para" parents="699 2869"/>
      <j2 rule="rewrite" parents="699"/>
    </justification>
  </clause>

  <clause id="2981">
    <literal><![CDATA[
      x ^ ((y' v x) ^ (x v z)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(2536),rewrite(2926(5))].">
      <j1 rule="back_rewrite" parents="2536"/>
      <j2 rule="rewrite" parents="2926"/>
    </justification>
  </clause>

  <clause id="2988">
    <literal><![CDATA[
      x v ((x v y) ^ (x v (x' ^ y'))) = x
    ]]></literal>
    <justification jstring="[back_rewrite(2500),rewrite(2926(6))].">
      <j1 rule="back_rewrite" parents="2500"/>
      <j2 rule="rewrite" parents="2926"/>
    </justification>
  </clause>

  <clause id="2989">
    <literal><![CDATA[
      x v ((y v x) ^ (x v (y' ^ x'))) = x
    ]]></literal>
    <justification jstring="[back_rewrite(2499),rewrite(2926(6))].">
      <j1 rule="back_rewrite" parents="2499"/>
      <j2 rule="rewrite" parents="2926"/>
    </justification>
  </clause>

  <clause id="4201">
    <literal><![CDATA[
      x ^ ((y v x) ^ (x v z)) = x
    ]]></literal>
    <justification jstring="[para(699(a,1),2981(a,1,2,1,1))].">
      <j1 rule="para" parents="699 2981"/>
    </justification>
  </clause>

  <clause id="4252">
    <literal><![CDATA[
      x v ((y v x) ^ (x v z)) = (y v x) ^ (x v z)
    ]]></literal>
    <justification jstring="[para(4201(a,1),945(a,1,2)),rewrite(2291(4))].">
      <j1 rule="para" parents="4201 945"/>
      <j2 rule="rewrite" parents="2291"/>
    </justification>
  </clause>

  <clause id="4291">
    <literal><![CDATA[
      x ^ ((x v y) ^ (x v z)) = x
    ]]></literal>
    <justification jstring="[para(2291(a,1),4201(a,1,2,1))].">
      <j1 rule="para" parents="2291 4201"/>
    </justification>
  </clause>

  <clause id="4307">
    <literal><![CDATA[
      (x v y) ^ (y v (x' ^ y')) = y
    ]]></literal>
    <justification jstring="[back_rewrite(2989),rewrite(4252(7))].">
      <j1 rule="back_rewrite" parents="2989"/>
      <j2 rule="rewrite" parents="4252"/>
    </justification>
  </clause>

  <clause id="4371">
    <literal><![CDATA[
      x v ((x v y) ^ (x v z)) = (x v y) ^ (x v z)
    ]]></literal>
    <justification jstring="[para(4291(a,1),945(a,1,2)),rewrite(2291(4))].">
      <j1 rule="para" parents="4291 945"/>
      <j2 rule="rewrite" parents="2291"/>
    </justification>
  </clause>

  <clause id="4406">
    <literal><![CDATA[
      (x v y) ^ (x v (x' ^ y')) = x
    ]]></literal>
    <justification jstring="[back_rewrite(2988),rewrite(4371(7))].">
      <j1 rule="back_rewrite" parents="2988"/>
      <j2 rule="rewrite" parents="4371"/>
    </justification>
  </clause>

  <clause id="12688">
    <literal><![CDATA[
      x v (y v ((y v x) ^ z)) = y v x
    ]]></literal>
    <justification jstring="[para(1449(a,1),1051(a,1,2,2)),rewrite(1635(6))].">
      <j1 rule="para" parents="1449 1051"/>
      <j2 rule="rewrite" parents="1635"/>
    </justification>
  </clause>

  <clause id="13852">
    <literal><![CDATA[
      x v (y v (x ^ z)) = y v x
    ]]></literal>
    <justification jstring="[para(1292(a,1),12688(a,1,2,2))].">
      <j1 rule="para" parents="1292 12688"/>
    </justification>
  </clause>

  <clause id="13946">
    <literal><![CDATA[
      (x v y) v (z v y) = z v (x v y)
    ]]></literal>
    <justification jstring="[para(4307(a,1),13852(a,1,2,2))].">
      <j1 rule="para" parents="4307 13852"/>
    </justification>
  </clause>

  <clause id="13947">
    <literal><![CDATA[
      (x v y) v (z v x) = z v (x v y)
    ]]></literal>
    <justification jstring="[para(4406(a,1),13852(a,1,2,2))].">
      <j1 rule="para" parents="4406 13852"/>
    </justification>
  </clause>

  <clause id="13965">
    <literal><![CDATA[
      (x v y) v (y v z) = (x v y) v z
    ]]></literal>
    <justification jstring="[back_rewrite(1939),rewrite(13946(4))].">
      <j1 rule="back_rewrite" parents="1939"/>
      <j2 rule="rewrite" parents="13946"/>
    </justification>
  </clause>

  <clause id="13979">
    <literal><![CDATA[
      x v (y v z) = z v (x v y)
    ]]></literal>
    <justification jstring="[back_rewrite(1641),rewrite(13965(4),13947(3))].">
      <j1 rule="back_rewrite" parents="1641"/>
      <j2 rule="rewrite" parents="13965 13947"/>
    </justification>
  </clause>

  <clause id="14423">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <justification jstring="[back_rewrite(913),rewrite(13979(8,R),2291(7)),xx(a)].">
      <j1 rule="back_rewrite" parents="913"/>
      <j2 rule="rewrite" parents="13979 2291"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
