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<!DOCTYPE proofs SYSTEM "proof3.dtd">

<?xml-stylesheet type="text/xsl" href="proof3.xsl"?>

<proofs number_of_proofs="4">

<source>olsax-fold.out</source>

<heading><![CDATA[
Prover9 (32) version June-2007, June 2007.
Process 18409 was started by mccune on cleo,
Mon Jun 18 15:23:24 2007
The command was "prover9 -f olsax-fold.in".
]]></heading>

<proof number="1" length="58" max_count="77">

<comments><![CDATA[
% Proof 1 at 0.77 (+ 0.00) seconds: absorb.
% Length of proof is 58.
% Level of proof is 12.
% Maximum clause weight is 77.
% Given clauses 76.
]]></comments>

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

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

  <clause id="6" type="assumption">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(definition_join)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <justification jstring="[copy(6),flip(a)].">
      <j1 rule="copy" parents="6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="8" type="assumption">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(definition_meet)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      f(f(x,y),f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[copy(8),flip(a)].">
      <j1 rule="copy" parents="8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10" type="assumption">
    <literal><![CDATA[
      x' = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(definition_complementation)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      f(x,x) = x'
    ]]></literal>
    <justification jstring="[copy(10),flip(a)].">
      <j1 rule="copy" parents="10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14" type="deny">
    <literal><![CDATA[
      f(f(c4,c4),f(c4,c5)) != c4
    ]]></literal>
    <attribute><![CDATA[
      answer(absorb)
    ]]></attribute>
    <justification jstring="[deny(2)].">
      <j1 rule="deny" parents="2"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      f(c4',f(c4,c5)) != c4
    ]]></literal>
    <attribute><![CDATA[
      answer(absorb)
    ]]></attribute>
    <justification jstring="[copy(14),rewrite([11(3)])].">
      <j1 rule="copy" parents="14"/>
      <j2 rule="rewrite" parents="11"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      f(x,y)' = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(9),rewrite([11(3)])].">
      <j1 rule="back_rewrite" parents="9"/>
      <j2 rule="rewrite" parents="11"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(x',y') = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(7),rewrite([11(1),11(2)])].">
      <j1 rule="back_rewrite" parents="7"/>
      <j2 rule="rewrite" parents="11 11"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(y,f(x',x)),z))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(5),rewrite([11(5)])].">
      <j1 rule="back_rewrite" parents="5"/>
      <j2 rule="rewrite" parents="11"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      x ^ x = x''
    ]]></literal>
    <justification jstring="[para(11(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="11 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      (x v y)' = x' ^ y'
    ]]></literal>
    <justification jstring="[para(21(a,1),20(a,1,1))].">
      <j1 rule="para" parents="21 20"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      f(f(f(x',f(x,y)),z),f(x,f(f(x,f(x',x)),y))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),22(a,1,1,1,1))].">
      <j1 rule="para" parents="11 22"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      f(f(f(f(x,y),y'),z),f(y,f(f(y,f(x',x)),y))) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),22(a,1,1,1,2))].">
      <j1 rule="para" parents="11 22"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      f(f(x'',y),f(x,f(f(x,f(x',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),22(a,1,1,1)),rewrite([11(1)])].">
      <j1 rule="para" parents="11 22"/>
      <j2 rule="rewrite" parents="11"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,f(y,f(x',x)))),z),f(y,y ^ f(x',x))) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),22(a,1,2,2)),rewrite([20(11)])].">
      <j1 rule="para" parents="11 22"/>
      <j2 rule="rewrite" parents="20"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      f(f(f(x,y),f(y,z)),u) ^ f(y,f(f(y,f(x',x)),z)) = y'
    ]]></literal>
    <justification jstring="[para(22(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="22 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(x,f(f(z,x) ^ f(x,u),f(f(z,x),f(x,u)))),f(f(x,f(z',z)),u)))) = x
    ]]></literal>
    <justification jstring="[para(22(a,1),22(a,1,1,1)),rewrite([20(5)])].">
      <j1 rule="para" parents="22 22"/>
      <j2 rule="rewrite" parents="20"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      f(x' ^ f(x,y),f(x,f(f(x,f(x',x)),y))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),30(a,1,1)),rewrite([20(4)])].">
      <j1 rule="para" parents="11 30"/>
      <j2 rule="rewrite" parents="20"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(f(x,y),x' v x),z))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(30(a,1),22(a,1,1)),rewrite([21(6)])].">
      <j1 rule="para" parents="30 22"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="75">
    <literal><![CDATA[
      f(x''',f(x,f(f(x,f(x',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),32(a,1,1))].">
      <j1 rule="para" parents="11 32"/>
    </justification>
  </clause>

  <clause id="76">
    <literal><![CDATA[
      f(x'',y) ^ f(x,f(f(x,f(x',x)),x)) = x'
    ]]></literal>
    <justification jstring="[para(32(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="32 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="78">
    <literal><![CDATA[
      f(x' v y,f(x,f(f(x,f(x',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),32(a,1,1))].">
      <j1 rule="para" parents="21 32"/>
    </justification>
  </clause>

  <clause id="87">
    <literal><![CDATA[
      f(x,f(x',f(f(x',x' v x),y))) = x'
    ]]></literal>
    <justification jstring="[para(32(a,1),30(a,1,1)),rewrite([21(6)])].">
      <j1 rule="para" parents="32 30"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      x''' ^ f(x,f(f(x,f(x',x)),x)) = x'
    ]]></literal>
    <justification jstring="[para(75(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="75 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="140">
    <literal><![CDATA[
      f(x',f(x'',x)) = x''
    ]]></literal>
    <justification jstring="[para(32(a,1),87(a,1,2,2))].">
      <j1 rule="para" parents="32 87"/>
    </justification>
  </clause>

  <clause id="144">
    <literal><![CDATA[
      f(x'',x'' v x) = x'''
    ]]></literal>
    <justification jstring="[para(21(a,1),140(a,1,2))].">
      <j1 rule="para" parents="21 140"/>
    </justification>
  </clause>

  <clause id="281">
    <literal><![CDATA[
      f(x,f(f(x,y),f(x,y) ^ (x' v x))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(30(a,1),35(a,1,1)),rewrite([21(6)])].">
      <j1 rule="para" parents="30 35"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="346">
    <literal><![CDATA[
      f(x,f(x',x' v x)) = x'
    ]]></literal>
    <justification jstring="[para(281(a,1),87(a,1,2))].">
      <j1 rule="para" parents="281 87"/>
    </justification>
  </clause>

  <clause id="357">
    <literal><![CDATA[
      f(f(x,y),f(x ^ y,(x ^ y) v f(x,y))) = x ^ y
    ]]></literal>
    <justification jstring="[para(20(a,1),346(a,1,2,1)),rewrite([20(4),20(9)])].">
      <j1 rule="para" parents="20 346"/>
      <j2 rule="rewrite" parents="20 20"/>
    </justification>
  </clause>

  <clause id="380">
    <literal><![CDATA[
      f(x,y) ^ f(x,f(f(x,f(f(z,x) ^ f(x,u),f(f(z,x),f(x,u)))),f(f(x,f(z',z)),u))) = x'
    ]]></literal>
    <justification jstring="[para(22(a,1),36(a,1,1,1)),rewrite([20(5)])].">
      <j1 rule="para" parents="22 36"/>
      <j2 rule="rewrite" parents="20"/>
    </justification>
  </clause>

  <clause id="521">
    <literal><![CDATA[
      f(x''',y) ^ f(x',f(f(x',x' v x),x')) = x''
    ]]></literal>
    <justification jstring="[para(21(a,1),76(a,1,2,2,1,2))].">
      <j1 rule="para" parents="21 76"/>
    </justification>
  </clause>

  <clause id="673">
    <literal><![CDATA[
      f(x',f(f(x',f(x,y)),x)) = f(x',f(x,y))
    ]]></literal>
    <justification jstring="[para(30(a,1),62(a,1,2,2))].">
      <j1 rule="para" parents="30 62"/>
    </justification>
  </clause>

  <clause id="689">
    <literal><![CDATA[
      f(x,f(f(x,y),x' v x)) = f(x,y)
    ]]></literal>
    <justification jstring="[para(281(a,1),62(a,1,2))].">
      <j1 rule="para" parents="281 62"/>
    </justification>
  </clause>

  <clause id="770">
    <literal><![CDATA[
      f(x''',f(x,x'''' v x''')) = x
    ]]></literal>
    <justification jstring="[para(75(a,1),689(a,1,2,1)),rewrite([75(22)])].">
      <j1 rule="para" parents="75 689"/>
      <j2 rule="rewrite" parents="75"/>
    </justification>
  </clause>

  <clause id="832">
    <literal><![CDATA[
      x' ^ f(f(x',f(x,y)),x) = x' ^ f(x,y)
    ]]></literal>
    <justification jstring="[para(673(a,1),20(a,1,1)),rewrite([20(4)]),flip(a)].">
      <j1 rule="para" parents="673 20"/>
      <j2 rule="rewrite" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="834">
    <literal><![CDATA[
      f(x'',f(f(x'',x v y),x')) = f(x'',x v y)
    ]]></literal>
    <justification jstring="[para(21(a,1),673(a,1,2,1,2)),rewrite([21(14)])].">
      <j1 rule="para" parents="21 673"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="930">
    <literal><![CDATA[
      (x' ^ f(x,y))' ^ f(f((x' ^ f(x,y))',x),x' ^ f(x,y)) = (x' ^ f(x,y))' ^ x
    ]]></literal>
    <justification jstring="[para(53(a,1),832(a,1,2,1,2)),rewrite([53(27)])].">
      <j1 rule="para" parents="53 832"/>
      <j2 rule="rewrite" parents="53"/>
    </justification>
  </clause>

  <clause id="988">
    <literal><![CDATA[
      f(f(f(x,x ^ (x'''' v x''')),y),f(f(x,x'''' v x'''),f(f(f(x,x'''' v x'''),x''' v x''),f(x,x'''' v x''')))) = f(x,x'''' v x''')
    ]]></literal>
    <justification jstring="[para(770(a,1),31(a,1,1,1,1)),rewrite([20(10),21(37)])].">
      <j1 rule="para" parents="770 31"/>
      <j2 rule="rewrite" parents="20 21"/>
    </justification>
  </clause>

  <clause id="1658">
    <literal><![CDATA[
      f(x,x ^ y) = f(x,y)
    ]]></literal>
    <justification jstring="[para(43(a,1),62(a,1,2,2)),rewrite([11(3),20(2)])].">
      <j1 rule="para" parents="43 62"/>
      <j2 rule="rewrite" parents="11 20"/>
    </justification>
  </clause>

  <clause id="1758">
    <literal><![CDATA[
      f(f(f(x,x'''' v x'''),y),f(f(x,x'''' v x'''),f(f(f(x,x'''' v x'''),x''' v x''),f(x,x'''' v x''')))) = f(x,x'''' v x''')
    ]]></literal>
    <justification jstring="[back_rewrite(988),rewrite([1658(10)])].">
      <j1 rule="back_rewrite" parents="988"/>
      <j2 rule="rewrite" parents="1658"/>
    </justification>
  </clause>

  <clause id="2013">
    <literal><![CDATA[
      f(f(x'',y),x') = x
    ]]></literal>
    <justification jstring="[para(32(a,1),1658(a,2)),rewrite([76(12)])].">
      <j1 rule="para" parents="32 1658"/>
      <j2 rule="rewrite" parents="76"/>
    </justification>
  </clause>

  <clause id="2014">
    <literal><![CDATA[
      x'' v x = x
    ]]></literal>
    <justification jstring="[para(75(a,1),1658(a,2)),rewrite([89(12),21(5)])].">
      <j1 rule="para" parents="75 1658"/>
      <j2 rule="rewrite" parents="89 21"/>
    </justification>
  </clause>

  <clause id="2022">
    <literal><![CDATA[
      f(x'',x) = x'''
    ]]></literal>
    <justification jstring="[para(144(a,1),1658(a,2)),rewrite([2014(7),1658(6)])].">
      <j1 rule="para" parents="144 1658"/>
      <j2 rule="rewrite" parents="2014 1658"/>
    </justification>
  </clause>

  <clause id="2063">
    <literal><![CDATA[
      f(f(x,y),x') = x
    ]]></literal>
    <justification jstring="[para(43(a,1),1658(a,2)),rewrite([380(17)])].">
      <j1 rule="para" parents="43 1658"/>
      <j2 rule="rewrite" parents="380"/>
    </justification>
  </clause>

  <clause id="2085">
    <literal><![CDATA[
      f(x'',x v y) = x'''
    ]]></literal>
    <justification jstring="[back_rewrite(834),rewrite([2013(8),2022(3)]),flip(a)].">
      <j1 rule="back_rewrite" parents="834"/>
      <j2 rule="rewrite" parents="2013 2022"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2291">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[para(11(a,1),2063(a,1,1)),rewrite([11(3)])].">
      <j1 rule="para" parents="11 2063"/>
      <j2 rule="rewrite" parents="11"/>
    </justification>
  </clause>

  <clause id="2302">
    <literal><![CDATA[
      f(x,y') = x' v y
    ]]></literal>
    <justification jstring="[para(78(a,1),2063(a,1,1)),rewrite([27(3),2291(2),1658(3)])].">
      <j1 rule="para" parents="78 2063"/>
      <j2 rule="rewrite" parents="27 2291 1658"/>
    </justification>
  </clause>

  <clause id="2325">
    <literal><![CDATA[
      (x ^ y)' = f(x,y)
    ]]></literal>
    <justification jstring="[para(357(a,1),2063(a,1,1)),rewrite([20(3),11(3)])].">
      <j1 rule="para" parents="357 2063"/>
      <j2 rule="rewrite" parents="20 11"/>
    </justification>
  </clause>

  <clause id="2426">
    <literal><![CDATA[
      f(x,x v y) = x'
    ]]></literal>
    <justification jstring="[back_rewrite(2085),rewrite([2291(2),2291(4)])].">
      <j1 rule="back_rewrite" parents="2085"/>
      <j2 rule="rewrite" parents="2291 2291"/>
    </justification>
  </clause>

  <clause id="2451">
    <literal><![CDATA[
      f(f(x',y),x) = x'
    ]]></literal>
    <justification jstring="[back_rewrite(1758),rewrite([2291(2),2291(2),2291(2),2426(3),2291(4),2291(4),2291(4),2426(5),2291(5),2291(5),2291(5),2426(6),2291(6),2291(7),2426(7),2291(5),2291(5),2291(5),2291(5),2426(6),2302(5),2426(6),2291(4),2291(5),2291(5),2291(5),2426(6)])].">
      <j1 rule="back_rewrite" parents="1758"/>
      <j2 rule="rewrite" parents="2291 2291 2291 2426 2291 2291 2291 2426 2291 2291 2291 2426 2291 2291 2426 2291 2291 2291 2291 2426 2302 2426 2291 2291 2291 2291 2426"/>
    </justification>
  </clause>

  <clause id="2564">
    <literal><![CDATA[
      f(x',y) ^ x = x
    ]]></literal>
    <justification jstring="[back_rewrite(521),rewrite([2291(2),2426(7),2291(5),2302(5),2426(6),2291(4),2291(5)])].">
      <j1 rule="back_rewrite" parents="521"/>
      <j2 rule="rewrite" parents="2291 2426 2291 2302 2426 2291 2291"/>
    </justification>
  </clause>

  <clause id="2602">
    <literal><![CDATA[
      x ^ x = x
    ]]></literal>
    <justification jstring="[back_rewrite(25),rewrite([2291(3)])].">
      <j1 rule="back_rewrite" parents="25"/>
      <j2 rule="rewrite" parents="2291"/>
    </justification>
  </clause>

  <clause id="2842">
    <literal><![CDATA[
      f(x',f(x,y)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(930),rewrite([2325(4),2325(7),2451(7),1658(8),2602(7),2325(7),2564(7)])].">
      <j1 rule="back_rewrite" parents="930"/>
      <j2 rule="rewrite" parents="2325 2325 2451 1658 2602 2325 2564"/>
    </justification>
  </clause>

  <clause id="2843">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(absorb)
    ]]></attribute>
    <justification jstring="[resolve(2842,a,15,a)].">
      <j1 rule="resolve" parents="2842 15"/>
    </justification>
  </clause>

</proof>

<proof number="2" length="100" max_count="77">

<comments><![CDATA[
% Proof 2 at 1.78 (+ 0.01) seconds: one.
% Length of proof is 100.
% Level of proof is 20.
% Maximum clause weight is 77.
% Given clauses 137.
]]></comments>

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

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

  <clause id="6" type="assumption">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(definition_join)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <justification jstring="[copy(6),flip(a)].">
      <j1 rule="copy" parents="6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="8" type="assumption">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(definition_meet)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      f(f(x,y),f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[copy(8),flip(a)].">
      <j1 rule="copy" parents="8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10" type="assumption">
    <literal><![CDATA[
      x' = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(definition_complementation)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      f(x,x) = x'
    ]]></literal>
    <justification jstring="[copy(10),flip(a)].">
      <j1 rule="copy" parents="10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16" type="deny">
    <literal><![CDATA[
      f(c6,f(c6,c6)) != f(c7,f(c7,c7))
    ]]></literal>
    <attribute><![CDATA[
      answer(one)
    ]]></attribute>
    <justification jstring="[deny(3)].">
      <j1 rule="deny" parents="3"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(c7,c7') != f(c6,c6')
    ]]></literal>
    <attribute><![CDATA[
      answer(one)
    ]]></attribute>
    <justification jstring="[copy(16),rewrite([11(4),11(8)]),flip(a)].">
      <j1 rule="copy" parents="16"/>
      <j2 rule="rewrite" parents="11 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      f(x,y)' = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(9),rewrite([11(3)])].">
      <j1 rule="back_rewrite" parents="9"/>
      <j2 rule="rewrite" parents="11"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(x',y') = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(7),rewrite([11(1),11(2)])].">
      <j1 rule="back_rewrite" parents="7"/>
      <j2 rule="rewrite" parents="11 11"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(y,f(x',x)),z))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(5),rewrite([11(5)])].">
      <j1 rule="back_rewrite" parents="5"/>
      <j2 rule="rewrite" parents="11"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      x ^ x = x''
    ]]></literal>
    <justification jstring="[para(11(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="11 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      (x v y)' = x' ^ y'
    ]]></literal>
    <justification jstring="[para(21(a,1),20(a,1,1))].">
      <j1 rule="para" parents="21 20"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      f(x ^ y,z') = f(x,y) v z
    ]]></literal>
    <justification jstring="[para(20(a,1),21(a,1,1))].">
      <j1 rule="para" parents="20 21"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      f(x',y ^ z) = x v f(y,z)
    ]]></literal>
    <justification jstring="[para(20(a,1),21(a,1,2))].">
      <j1 rule="para" parents="20 21"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      f(f(f(x',f(x,y)),z),f(x,f(f(x,f(x',x)),y))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),22(a,1,1,1,1))].">
      <j1 rule="para" parents="11 22"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      f(f(f(f(x,y),y'),z),f(y,f(f(y,f(x',x)),y))) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),22(a,1,1,1,2))].">
      <j1 rule="para" parents="11 22"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      f(f(x'',y),f(x,f(f(x,f(x',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),22(a,1,1,1)),rewrite([11(1)])].">
      <j1 rule="para" parents="11 22"/>
      <j2 rule="rewrite" parents="11"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      f(f(f(f(x,f(x',x)),f(f(x',x),y)),z),f(f(x',x),f(x' ^ x,y))) = f(x',x)
    ]]></literal>
    <justification jstring="[para(11(a,1),22(a,1,2,2,1)),rewrite([20(13)])].">
      <j1 rule="para" parents="11 22"/>
      <j2 rule="rewrite" parents="20"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,f(y,f(x',x)))),z),f(y,y ^ f(x',x))) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),22(a,1,2,2)),rewrite([20(11)])].">
      <j1 rule="para" parents="11 22"/>
      <j2 rule="rewrite" parents="20"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      f(f(f(x,y),f(y,z)),u) ^ f(y,f(f(y,f(x',x)),z)) = y'
    ]]></literal>
    <justification jstring="[para(22(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="22 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(x,f(f(z,x) ^ f(x,u),f(f(z,x),f(x,u)))),f(f(x,f(z',z)),u)))) = x
    ]]></literal>
    <justification jstring="[para(22(a,1),22(a,1,1,1)),rewrite([20(5)])].">
      <j1 rule="para" parents="22 22"/>
      <j2 rule="rewrite" parents="20"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      f(x' ^ f(x,y),f(x,f(f(x,f(x',x)),y))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),30(a,1,1)),rewrite([20(4)])].">
      <j1 rule="para" parents="11 30"/>
      <j2 rule="rewrite" parents="20"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(f(x,y),x' v x),z))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(30(a,1),22(a,1,1)),rewrite([21(6)])].">
      <j1 rule="para" parents="30 22"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      f(f(f((x ^ y)',f(x,y) v z),u),f(x ^ y,f(f(x ^ y,(x ^ y) v f(x,y)),z'))) = x ^ y
    ]]></literal>
    <justification jstring="[para(28(a,1),30(a,1,1,1,2)),rewrite([29(12)])].">
      <j1 rule="para" parents="28 30"/>
      <j2 rule="rewrite" parents="29"/>
    </justification>
  </clause>

  <clause id="75">
    <literal><![CDATA[
      f(x''',f(x,f(f(x,f(x',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),32(a,1,1))].">
      <j1 rule="para" parents="11 32"/>
    </justification>
  </clause>

  <clause id="76">
    <literal><![CDATA[
      f(x'',y) ^ f(x,f(f(x,f(x',x)),x)) = x'
    ]]></literal>
    <justification jstring="[para(32(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="32 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="78">
    <literal><![CDATA[
      f(x' v y,f(x,f(f(x,f(x',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),32(a,1,1))].">
      <j1 rule="para" parents="21 32"/>
    </justification>
  </clause>

  <clause id="87">
    <literal><![CDATA[
      f(x,f(x',f(f(x',x' v x),y))) = x'
    ]]></literal>
    <justification jstring="[para(32(a,1),30(a,1,1)),rewrite([21(6)])].">
      <j1 rule="para" parents="32 30"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      x''' ^ f(x,f(f(x,f(x',x)),x)) = x'
    ]]></literal>
    <justification jstring="[para(75(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="75 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="97">
    <literal><![CDATA[
      (x' v y) ^ f(x,f(f(x,f(x',x)),x)) = x'
    ]]></literal>
    <justification jstring="[para(78(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="78 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="105">
    <literal><![CDATA[
      f(f(x,y) ^ y',f(y,f(f(y,f(x',x)),y))) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),31(a,1,1)),rewrite([20(4)])].">
      <j1 rule="para" parents="11 31"/>
      <j2 rule="rewrite" parents="20"/>
    </justification>
  </clause>

  <clause id="107">
    <literal><![CDATA[
      f(f(f(x,y),y'),z) ^ f(y,f(f(y,f(x',x)),y)) = y'
    ]]></literal>
    <justification jstring="[para(31(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="31 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="140">
    <literal><![CDATA[
      f(x',f(x'',x)) = x''
    ]]></literal>
    <justification jstring="[para(32(a,1),87(a,1,2,2))].">
      <j1 rule="para" parents="32 87"/>
    </justification>
  </clause>

  <clause id="144">
    <literal><![CDATA[
      f(x'',x'' v x) = x'''
    ]]></literal>
    <justification jstring="[para(21(a,1),140(a,1,2))].">
      <j1 rule="para" parents="21 140"/>
    </justification>
  </clause>

  <clause id="256">
    <literal><![CDATA[
      (x'' v y) ^ f(x',f(f(x',x' v x),x')) = x''
    ]]></literal>
    <justification jstring="[para(21(a,1),97(a,1,2,2,1,2))].">
      <j1 rule="para" parents="21 97"/>
    </justification>
  </clause>

  <clause id="281">
    <literal><![CDATA[
      f(x,f(f(x,y),f(x,y) ^ (x' v x))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(30(a,1),35(a,1,1)),rewrite([21(6)])].">
      <j1 rule="para" parents="30 35"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="346">
    <literal><![CDATA[
      f(x,f(x',x' v x)) = x'
    ]]></literal>
    <justification jstring="[para(281(a,1),87(a,1,2))].">
      <j1 rule="para" parents="281 87"/>
    </justification>
  </clause>

  <clause id="357">
    <literal><![CDATA[
      f(f(x,y),f(x ^ y,(x ^ y) v f(x,y))) = x ^ y
    ]]></literal>
    <justification jstring="[para(20(a,1),346(a,1,2,1)),rewrite([20(4),20(9)])].">
      <j1 rule="para" parents="20 346"/>
      <j2 rule="rewrite" parents="20 20"/>
    </justification>
  </clause>

  <clause id="380">
    <literal><![CDATA[
      f(x,y) ^ f(x,f(f(x,f(f(z,x) ^ f(x,u),f(f(z,x),f(x,u)))),f(f(x,f(z',z)),u))) = x'
    ]]></literal>
    <justification jstring="[para(22(a,1),36(a,1,1,1)),rewrite([20(5)])].">
      <j1 rule="para" parents="22 36"/>
      <j2 rule="rewrite" parents="20"/>
    </justification>
  </clause>

  <clause id="521">
    <literal><![CDATA[
      f(x''',y) ^ f(x',f(f(x',x' v x),x')) = x''
    ]]></literal>
    <justification jstring="[para(21(a,1),76(a,1,2,2,1,2))].">
      <j1 rule="para" parents="21 76"/>
    </justification>
  </clause>

  <clause id="673">
    <literal><![CDATA[
      f(x',f(f(x',f(x,y)),x)) = f(x',f(x,y))
    ]]></literal>
    <justification jstring="[para(30(a,1),62(a,1,2,2))].">
      <j1 rule="para" parents="30 62"/>
    </justification>
  </clause>

  <clause id="689">
    <literal><![CDATA[
      f(x,f(f(x,y),x' v x)) = f(x,y)
    ]]></literal>
    <justification jstring="[para(281(a,1),62(a,1,2))].">
      <j1 rule="para" parents="281 62"/>
    </justification>
  </clause>

  <clause id="770">
    <literal><![CDATA[
      f(x''',f(x,x'''' v x''')) = x
    ]]></literal>
    <justification jstring="[para(75(a,1),689(a,1,2,1)),rewrite([75(22)])].">
      <j1 rule="para" parents="75 689"/>
      <j2 rule="rewrite" parents="75"/>
    </justification>
  </clause>

  <clause id="832">
    <literal><![CDATA[
      x' ^ f(f(x',f(x,y)),x) = x' ^ f(x,y)
    ]]></literal>
    <justification jstring="[para(673(a,1),20(a,1,1)),rewrite([20(4)]),flip(a)].">
      <j1 rule="para" parents="673 20"/>
      <j2 rule="rewrite" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="834">
    <literal><![CDATA[
      f(x'',f(f(x'',x v y),x')) = f(x'',x v y)
    ]]></literal>
    <justification jstring="[para(21(a,1),673(a,1,2,1,2)),rewrite([21(14)])].">
      <j1 rule="para" parents="21 673"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="863">
    <literal><![CDATA[
      f((f(x,y) ^ y')',f(f((f(x,y) ^ y')',y),f(x,y) ^ y')) = f((f(x,y) ^ y')',y)
    ]]></literal>
    <justification jstring="[para(105(a,1),673(a,1,2,1,2)),rewrite([105(27)])].">
      <j1 rule="para" parents="105 673"/>
      <j2 rule="rewrite" parents="105"/>
    </justification>
  </clause>

  <clause id="930">
    <literal><![CDATA[
      (x' ^ f(x,y))' ^ f(f((x' ^ f(x,y))',x),x' ^ f(x,y)) = (x' ^ f(x,y))' ^ x
    ]]></literal>
    <justification jstring="[para(53(a,1),832(a,1,2,1,2)),rewrite([53(27)])].">
      <j1 rule="para" parents="53 832"/>
      <j2 rule="rewrite" parents="53"/>
    </justification>
  </clause>

  <clause id="988">
    <literal><![CDATA[
      f(f(f(x,x ^ (x'''' v x''')),y),f(f(x,x'''' v x'''),f(f(f(x,x'''' v x'''),x''' v x''),f(x,x'''' v x''')))) = f(x,x'''' v x''')
    ]]></literal>
    <justification jstring="[para(770(a,1),31(a,1,1,1,1)),rewrite([20(10),21(37)])].">
      <j1 rule="para" parents="770 31"/>
      <j2 rule="rewrite" parents="20 21"/>
    </justification>
  </clause>

  <clause id="1626">
    <literal><![CDATA[
      f(f(x,y),f(y,y ^ f(x',x))) = y
    ]]></literal>
    <justification jstring="[para(43(a,1),35(a,1,1))].">
      <j1 rule="para" parents="43 35"/>
    </justification>
  </clause>

  <clause id="1658">
    <literal><![CDATA[
      f(x,x ^ y) = f(x,y)
    ]]></literal>
    <justification jstring="[para(43(a,1),62(a,1,2,2)),rewrite([11(3),20(2)])].">
      <j1 rule="para" parents="43 62"/>
      <j2 rule="rewrite" parents="11 20"/>
    </justification>
  </clause>

  <clause id="1677">
    <literal><![CDATA[
      (x ^ y) ^ f(f(x ^ y,x),f(x,y)) = (x ^ y) ^ x
    ]]></literal>
    <justification jstring="[para(43(a,1),832(a,1,2,1,2)),rewrite([20(2),20(3),20(8),43(23)])].">
      <j1 rule="para" parents="43 832"/>
      <j2 rule="rewrite" parents="20 20 20 43"/>
    </justification>
  </clause>

  <clause id="1717">
    <literal><![CDATA[
      f(f(x,y),f(y,f(x',x))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1626),rewrite([1658(5)])].">
      <j1 rule="back_rewrite" parents="1626"/>
      <j2 rule="rewrite" parents="1658"/>
    </justification>
  </clause>

  <clause id="1758">
    <literal><![CDATA[
      f(f(f(x,x'''' v x'''),y),f(f(x,x'''' v x'''),f(f(f(x,x'''' v x'''),x''' v x''),f(x,x'''' v x''')))) = f(x,x'''' v x''')
    ]]></literal>
    <justification jstring="[back_rewrite(988),rewrite([1658(10)])].">
      <j1 rule="back_rewrite" parents="988"/>
      <j2 rule="rewrite" parents="1658"/>
    </justification>
  </clause>

  <clause id="2013">
    <literal><![CDATA[
      f(f(x'',y),x') = x
    ]]></literal>
    <justification jstring="[para(32(a,1),1658(a,2)),rewrite([76(12)])].">
      <j1 rule="para" parents="32 1658"/>
      <j2 rule="rewrite" parents="76"/>
    </justification>
  </clause>

  <clause id="2014">
    <literal><![CDATA[
      x'' v x = x
    ]]></literal>
    <justification jstring="[para(75(a,1),1658(a,2)),rewrite([89(12),21(5)])].">
      <j1 rule="para" parents="75 1658"/>
      <j2 rule="rewrite" parents="89 21"/>
    </justification>
  </clause>

  <clause id="2017">
    <literal><![CDATA[
      f(f(f(f(x,y),y'),z),y') = y
    ]]></literal>
    <justification jstring="[para(31(a,1),1658(a,2)),rewrite([107(14)])].">
      <j1 rule="para" parents="31 1658"/>
      <j2 rule="rewrite" parents="107"/>
    </justification>
  </clause>

  <clause id="2022">
    <literal><![CDATA[
      f(x'',x) = x'''
    ]]></literal>
    <justification jstring="[para(144(a,1),1658(a,2)),rewrite([2014(7),1658(6)])].">
      <j1 rule="para" parents="144 1658"/>
      <j2 rule="rewrite" parents="2014 1658"/>
    </justification>
  </clause>

  <clause id="2063">
    <literal><![CDATA[
      f(f(x,y),x') = x
    ]]></literal>
    <justification jstring="[para(43(a,1),1658(a,2)),rewrite([380(17)])].">
      <j1 rule="para" parents="43 1658"/>
      <j2 rule="rewrite" parents="380"/>
    </justification>
  </clause>

  <clause id="2085">
    <literal><![CDATA[
      f(x'',x v y) = x'''
    ]]></literal>
    <justification jstring="[back_rewrite(834),rewrite([2013(8),2022(3)]),flip(a)].">
      <j1 rule="back_rewrite" parents="834"/>
      <j2 rule="rewrite" parents="2013 2022"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2291">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[para(11(a,1),2063(a,1,1)),rewrite([11(3)])].">
      <j1 rule="para" parents="11 2063"/>
      <j2 rule="rewrite" parents="11"/>
    </justification>
  </clause>

  <clause id="2302">
    <literal><![CDATA[
      f(x,y') = x' v y
    ]]></literal>
    <justification jstring="[para(78(a,1),2063(a,1,1)),rewrite([27(3),2291(2),1658(3)])].">
      <j1 rule="para" parents="78 2063"/>
      <j2 rule="rewrite" parents="27 2291 1658"/>
    </justification>
  </clause>

  <clause id="2310">
    <literal><![CDATA[
      f(f((x ^ f(x',x)) v x,y),f(f(x',x),f(x' ^ x,x))) = f(x',x)
    ]]></literal>
    <justification jstring="[para(2063(a,1),34(a,1,1,1,2)),rewrite([2302(5),20(4),2291(11)])].">
      <j1 rule="para" parents="2063 34"/>
      <j2 rule="rewrite" parents="2302 20 2291"/>
    </justification>
  </clause>

  <clause id="2325">
    <literal><![CDATA[
      (x ^ y)' = f(x,y)
    ]]></literal>
    <justification jstring="[para(357(a,1),2063(a,1,1)),rewrite([20(3),11(3)])].">
      <j1 rule="para" parents="357 2063"/>
      <j2 rule="rewrite" parents="20 11"/>
    </justification>
  </clause>

  <clause id="2329">
    <literal><![CDATA[
      (x ^ y) v x = x
    ]]></literal>
    <justification jstring="[para(62(a,1),2063(a,1,1)),rewrite([2302(3),20(2)])].">
      <j1 rule="para" parents="62 2063"/>
      <j2 rule="rewrite" parents="2302 20"/>
    </justification>
  </clause>

  <clause id="2426">
    <literal><![CDATA[
      f(x,x v y) = x'
    ]]></literal>
    <justification jstring="[back_rewrite(2085),rewrite([2291(2),2291(4)])].">
      <j1 rule="back_rewrite" parents="2085"/>
      <j2 rule="rewrite" parents="2291 2291"/>
    </justification>
  </clause>

  <clause id="2451">
    <literal><![CDATA[
      f(f(x',y),x) = x'
    ]]></literal>
    <justification jstring="[back_rewrite(1758),rewrite([2291(2),2291(2),2291(2),2426(3),2291(4),2291(4),2291(4),2426(5),2291(5),2291(5),2291(5),2426(6),2291(6),2291(7),2426(7),2291(5),2291(5),2291(5),2291(5),2426(6),2302(5),2426(6),2291(4),2291(5),2291(5),2291(5),2426(6)])].">
      <j1 rule="back_rewrite" parents="1758"/>
      <j2 rule="rewrite" parents="2291 2291 2291 2426 2291 2291 2291 2426 2291 2291 2291 2426 2291 2291 2426 2291 2291 2291 2291 2426 2302 2426 2291 2291 2291 2291 2426"/>
    </justification>
  </clause>

  <clause id="2564">
    <literal><![CDATA[
      f(x',y) ^ x = x
    ]]></literal>
    <justification jstring="[back_rewrite(521),rewrite([2291(2),2426(7),2291(5),2302(5),2426(6),2291(4),2291(5)])].">
      <j1 rule="back_rewrite" parents="521"/>
      <j2 rule="rewrite" parents="2291 2426 2291 2302 2426 2291 2291"/>
    </justification>
  </clause>

  <clause id="2579">
    <literal><![CDATA[
      (x v y) ^ x = x
    ]]></literal>
    <justification jstring="[back_rewrite(256),rewrite([2291(2),2426(6),2291(4),2302(4),2426(5),2291(3),2291(4)])].">
      <j1 rule="back_rewrite" parents="256"/>
      <j2 rule="rewrite" parents="2291 2426 2291 2302 2426 2291 2291"/>
    </justification>
  </clause>

  <clause id="2602">
    <literal><![CDATA[
      x ^ x = x
    ]]></literal>
    <justification jstring="[back_rewrite(25),rewrite([2291(3)])].">
      <j1 rule="back_rewrite" parents="25"/>
      <j2 rule="rewrite" parents="2291"/>
    </justification>
  </clause>

  <clause id="2617">
    <literal><![CDATA[
      (((x ^ y) v y) ^ z) v y = y
    ]]></literal>
    <justification jstring="[back_rewrite(2017),rewrite([2302(3),20(2),2302(5),20(4)])].">
      <j1 rule="back_rewrite" parents="2017"/>
      <j2 rule="rewrite" parents="2302 20 2302 20"/>
    </justification>
  </clause>

  <clause id="2789">
    <literal><![CDATA[
      f(f(x ^ y,z),f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(67),rewrite([2325(2),2426(4),20(2),2426(8),2325(5),2302(6),20(5),2426(6),2325(4)])].">
      <j1 rule="back_rewrite" parents="67"/>
      <j2 rule="rewrite" parents="2325 2426 20 2426 2325 2302 20 2426 2325"/>
    </justification>
  </clause>

  <clause id="2794">
    <literal><![CDATA[
      c7' v c7 != c6' v c6
    ]]></literal>
    <attribute><![CDATA[
      answer(one)
    ]]></attribute>
    <justification jstring="[back_rewrite(17),rewrite([2302(4),2302(8)])].">
      <j1 rule="back_rewrite" parents="17"/>
      <j2 rule="rewrite" parents="2302 2302"/>
    </justification>
  </clause>

  <clause id="2842">
    <literal><![CDATA[
      f(x',f(x,y)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(930),rewrite([2325(4),2325(7),2451(7),1658(8),2602(7),2325(7),2564(7)])].">
      <j1 rule="back_rewrite" parents="930"/>
      <j2 rule="rewrite" parents="2325 2325 2451 1658 2602 2325 2564"/>
    </justification>
  </clause>

  <clause id="2848">
    <literal><![CDATA[
      f((x ^ y) v y,f(f((x ^ y) v y,y),f(x,y) ^ y')) = f((x ^ y) v y,y)
    ]]></literal>
    <justification jstring="[back_rewrite(863),rewrite([2325(4),2302(3),20(2),2325(6),2302(5),20(4),2325(14),2302(13),20(12)])].">
      <j1 rule="back_rewrite" parents="863"/>
      <j2 rule="rewrite" parents="2325 2302 20 2325 2302 20 2325 2302 20"/>
    </justification>
  </clause>

  <clause id="2885">
    <literal><![CDATA[
      f(f(x,y),f(f(x',x),f(x' ^ x,x))) = f(x',x)
    ]]></literal>
    <justification jstring="[back_rewrite(2310),rewrite([2329(4)])].">
      <j1 rule="back_rewrite" parents="2310"/>
      <j2 rule="rewrite" parents="2329"/>
    </justification>
  </clause>

  <clause id="2928">
    <literal><![CDATA[
      (x ^ y) ^ x = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(1677),rewrite([2789(5),2602(3)]),flip(a)].">
      <j1 rule="back_rewrite" parents="1677"/>
      <j2 rule="rewrite" parents="2789 2602"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3018">
    <literal><![CDATA[
      f(x,y ^ z) = x' v f(y,z)
    ]]></literal>
    <justification jstring="[para(2291(a,1),29(a,1,1))].">
      <j1 rule="para" parents="2291 29"/>
    </justification>
  </clause>

  <clause id="3035">
    <literal><![CDATA[
      f((x ^ y) v y,y) = f(x,y) ^ y'
    ]]></literal>
    <justification jstring="[back_rewrite(2848),rewrite([3018(9),20(6),2302(8),20(7),2329(8),11(5),27(3),2325(2)]),flip(a)].">
      <j1 rule="back_rewrite" parents="2848"/>
      <j2 rule="rewrite" parents="3018 20 2302 20 2329 11 27 2325"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3327">
    <literal><![CDATA[
      f(f(x,y),f(x ^ y,z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(2325(a,1),2842(a,1,1))].">
      <j1 rule="para" parents="2325 2842"/>
    </justification>
  </clause>

  <clause id="3376">
    <literal><![CDATA[
      (x ^ y) v f(x',x) = f(x',x)
    ]]></literal>
    <justification jstring="[back_rewrite(2885),rewrite([3327(7),3018(4),20(2)])].">
      <j1 rule="back_rewrite" parents="2885"/>
      <j2 rule="rewrite" parents="3327 3018 20"/>
    </justification>
  </clause>

  <clause id="3465">
    <literal><![CDATA[
      f(x,y) = x' v y'
    ]]></literal>
    <justification jstring="[para(2291(a,1),2302(a,1,2))].">
      <j1 rule="para" parents="2291 2302"/>
    </justification>
  </clause>

  <clause id="3553">
    <literal><![CDATA[
      (x ^ y) v (x v x') = x v x'
    ]]></literal>
    <justification jstring="[back_rewrite(3376),rewrite([3465(3),2291(3),3465(6),2291(6)])].">
      <j1 rule="back_rewrite" parents="3376"/>
      <j2 rule="rewrite" parents="3465 2291 3465 2291"/>
    </justification>
  </clause>

  <clause id="3801">
    <literal><![CDATA[
      ((x' v y') ^ y') v y' = (x' v y') ^ y'
    ]]></literal>
    <justification jstring="[back_rewrite(3035),rewrite([3465(3),27(3),2325(2),3465(1),3465(8)])].">
      <j1 rule="back_rewrite" parents="3035"/>
      <j2 rule="rewrite" parents="3465 27 2325 3465 3465"/>
    </justification>
  </clause>

  <clause id="3999">
    <literal><![CDATA[
      (x ^ y)' = x' v y'
    ]]></literal>
    <justification jstring="[back_rewrite(2325),rewrite([3465(3)])].">
      <j1 rule="back_rewrite" parents="2325"/>
      <j2 rule="rewrite" parents="3465"/>
    </justification>
  </clause>

  <clause id="4068">
    <literal><![CDATA[
      (x ^ y) v (y ^ (x v x')) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1717),rewrite([3465(1),3465(5),2291(5),3465(6),27(7),2291(7),3465(8),27(4),2291(2),2291(2),27(6),2291(3),3999(4),2291(3)])].">
      <j1 rule="back_rewrite" parents="1717"/>
      <j2 rule="rewrite" parents="3465 3465 2291 3465 27 2291 3465 27 2291 2291 27 2291 3999 2291"/>
    </justification>
  </clause>

  <clause id="4263">
    <literal><![CDATA[
      (x ^ y) v y = y
    ]]></literal>
    <justification jstring="[para(2579(a,1),2617(a,1,1))].">
      <j1 rule="para" parents="2579 2617"/>
    </justification>
  </clause>

  <clause id="4302">
    <literal><![CDATA[
      (x' v y') ^ y' = y'
    ]]></literal>
    <justification jstring="[back_rewrite(3801),rewrite([4263(7)]),flip(a)].">
      <j1 rule="back_rewrite" parents="3801"/>
      <j2 rule="rewrite" parents="4263"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4394">
    <literal><![CDATA[
      (x' v y) ^ y = y
    ]]></literal>
    <justification jstring="[para(2291(a,1),4302(a,1,1,2)),rewrite([2291(4),2291(5)])].">
      <j1 rule="para" parents="2291 4302"/>
      <j2 rule="rewrite" parents="2291 2291"/>
    </justification>
  </clause>

  <clause id="4409">
    <literal><![CDATA[
      (x v y) ^ y = y
    ]]></literal>
    <justification jstring="[para(2291(a,1),4394(a,1,1,1))].">
      <j1 rule="para" parents="2291 4394"/>
    </justification>
  </clause>

  <clause id="4427">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(4409(a,1),2928(a,1,1)),rewrite([4409(4)])].">
      <j1 rule="para" parents="4409 2928"/>
      <j2 rule="rewrite" parents="4409"/>
    </justification>
  </clause>

  <clause id="4586">
    <literal><![CDATA[
      (x v x') v y = x v x'
    ]]></literal>
    <justification jstring="[para(2579(a,1),4068(a,1,2)),rewrite([3553(7)]),flip(a)].">
      <j1 rule="para" parents="2579 4068"/>
      <j2 rule="rewrite" parents="3553"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4603">
    <literal><![CDATA[
      (x v x') ^ y = y
    ]]></literal>
    <justification jstring="[para(4586(a,1),4409(a,1,1))].">
      <j1 rule="para" parents="4586 4409"/>
    </justification>
  </clause>

  <clause id="4604">
    <literal><![CDATA[
      x ^ (y v y') = x
    ]]></literal>
    <justification jstring="[para(4586(a,1),4427(a,1,2))].">
      <j1 rule="para" parents="4586 4427"/>
    </justification>
  </clause>

  <clause id="4729">
    <literal><![CDATA[
      x v x' = y v y'
    ]]></literal>
    <justification jstring="[para(4604(a,1),4603(a,1))].">
      <j1 rule="para" parents="4604 4603"/>
    </justification>
  </clause>

  <clause id="4781">
    <literal><![CDATA[
      x v x' = c_0
    ]]></literal>
    <justification jstring="[new_symbol(4729)].">
      <j1 rule="new_symbol" parents="4729"/>
    </justification>
  </clause>

  <clause id="4855">
    <literal><![CDATA[
      x' v x = c_0
    ]]></literal>
    <justification jstring="[para(2291(a,1),4781(a,1,2))].">
      <j1 rule="para" parents="2291 4781"/>
    </justification>
  </clause>

  <clause id="4942">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(one)
    ]]></attribute>
    <justification jstring="[back_rewrite(2794),rewrite([4855(4),4855(5)]),xx(a)].">
      <j1 rule="back_rewrite" parents="2794"/>
      <j2 rule="rewrite" parents="4855 4855"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

<proof number="3" length="189" max_count="77">

<comments><![CDATA[
% Proof 3 at 16.42 (+ 0.09) seconds: combined.
% Length of proof is 189.
% Level of proof is 36.
% Maximum clause weight is 77.
% Given clauses 648.
]]></comments>

  <clause id="4" type="goal">
    <literal><![CDATA[
      f(x,f(f(y,z),f(y,z))) = f(y,f(f(x,z),f(x,z))) & f(f(x,x),f(x,y)) = x & f(x,f(x,x)) = f(y,f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      answer(combined)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

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

  <clause id="6" type="assumption">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(definition_join)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <justification jstring="[copy(6),flip(a)].">
      <j1 rule="copy" parents="6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="8" type="assumption">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(definition_meet)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      f(f(x,y),f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[copy(8),flip(a)].">
      <j1 rule="copy" parents="8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10" type="assumption">
    <literal><![CDATA[
      x' = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(definition_complementation)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      f(x,x) = x'
    ]]></literal>
    <justification jstring="[copy(10),flip(a)].">
      <j1 rule="copy" parents="10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18" type="deny">
    <literal><![CDATA[
      f(c8,f(f(c9,c10),f(c9,c10))) != f(c9,f(f(c8,c10),f(c8,c10)))
    ]]></literal>
    <literal><![CDATA[
      f(f(c8,c8),f(c8,c9)) != c8
    ]]></literal>
    <literal><![CDATA[
      f(c8,f(c8,c8)) != f(c9,f(c9,c9))
    ]]></literal>
    <attribute><![CDATA[
      answer(combined)
    ]]></attribute>
    <justification jstring="[deny(4)].">
      <j1 rule="deny" parents="4"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(c9,f(c8,c10)') != f(c8,f(c9,c10)')
    ]]></literal>
    <literal><![CDATA[
      f(c8',f(c8,c9)) != c8
    ]]></literal>
    <literal><![CDATA[
      f(c9,c9') != f(c8,c8')
    ]]></literal>
    <attribute><![CDATA[
      answer(combined)
    ]]></attribute>
    <justification jstring="[copy(18),rewrite([11(8),11(14),11(16),11(25),11(29)]),flip(a),flip(c)].">
      <j1 rule="copy" parents="18"/>
      <j2 rule="rewrite" parents="11 11 11 11 11"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      f(x,y)' = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(9),rewrite([11(3)])].">
      <j1 rule="back_rewrite" parents="9"/>
      <j2 rule="rewrite" parents="11"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(x',y') = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(7),rewrite([11(1),11(2)])].">
      <j1 rule="back_rewrite" parents="7"/>
      <j2 rule="rewrite" parents="11 11"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(y,f(x',x)),z))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(5),rewrite([11(5)])].">
      <j1 rule="back_rewrite" parents="5"/>
      <j2 rule="rewrite" parents="11"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      f(c9,c8 ^ c10) != f(c8,c9 ^ c10)
    ]]></literal>
    <literal><![CDATA[
      f(c8',f(c8,c9)) != c8
    ]]></literal>
    <literal><![CDATA[
      f(c9,c9') != f(c8,c8')
    ]]></literal>
    <attribute><![CDATA[
      answer(combined)
    ]]></attribute>
    <justification jstring="[back_rewrite(19),rewrite([20(5),20(10)])].">
      <j1 rule="back_rewrite" parents="19"/>
      <j2 rule="rewrite" parents="20 20"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      x ^ x = x''
    ]]></literal>
    <justification jstring="[para(11(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="11 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      (x v y)' = x' ^ y'
    ]]></literal>
    <justification jstring="[para(21(a,1),20(a,1,1))].">
      <j1 rule="para" parents="21 20"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      f(x ^ y,z') = f(x,y) v z
    ]]></literal>
    <justification jstring="[para(20(a,1),21(a,1,1))].">
      <j1 rule="para" parents="20 21"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      f(x',y ^ z) = x v f(y,z)
    ]]></literal>
    <justification jstring="[para(20(a,1),21(a,1,2))].">
      <j1 rule="para" parents="20 21"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      f(f(f(x',f(x,y)),z),f(x,f(f(x,f(x',x)),y))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),22(a,1,1,1,1))].">
      <j1 rule="para" parents="11 22"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      f(f(f(f(x,y),y'),z),f(y,f(f(y,f(x',x)),y))) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),22(a,1,1,1,2))].">
      <j1 rule="para" parents="11 22"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      f(f(x'',y),f(x,f(f(x,f(x',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),22(a,1,1,1)),rewrite([11(1)])].">
      <j1 rule="para" parents="11 22"/>
      <j2 rule="rewrite" parents="11"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      f(f(x,y) ^ f(y,z),f(y,f(f(y,f(x',x)),z))) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),22(a,1,1)),rewrite([20(4)])].">
      <j1 rule="para" parents="11 22"/>
      <j2 rule="rewrite" parents="20"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      f(f(f(f(x,f(x',x)),f(f(x',x),y)),z),f(f(x',x),f(x' ^ x,y))) = f(x',x)
    ]]></literal>
    <justification jstring="[para(11(a,1),22(a,1,2,2,1)),rewrite([20(13)])].">
      <j1 rule="para" parents="11 22"/>
      <j2 rule="rewrite" parents="20"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,f(y,f(x',x)))),z),f(y,y ^ f(x',x))) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),22(a,1,2,2)),rewrite([20(11)])].">
      <j1 rule="para" parents="11 22"/>
      <j2 rule="rewrite" parents="20"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      f(f(f(x,y),f(y,z)),u) ^ f(y,f(f(y,f(x',x)),z)) = y'
    ]]></literal>
    <justification jstring="[para(22(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="22 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      f(f(f(x v y,f(y',z)),u),f(y',f(f(y',x' v x),z))) = y'
    ]]></literal>
    <justification jstring="[para(21(a,1),22(a,1,1,1,1)),rewrite([21(11)])].">
      <j1 rule="para" parents="21 22"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(x,f(f(z,x) ^ f(x,u),f(f(z,x),f(x,u)))),f(f(x,f(z',z)),u)))) = x
    ]]></literal>
    <justification jstring="[para(22(a,1),22(a,1,1,1)),rewrite([20(5)])].">
      <j1 rule="para" parents="22 22"/>
      <j2 rule="rewrite" parents="20"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      f(x' ^ f(x,y),f(x,f(f(x,f(x',x)),y))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),30(a,1,1)),rewrite([20(4)])].">
      <j1 rule="para" parents="11 30"/>
      <j2 rule="rewrite" parents="20"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(f(x,y),x' v x),z))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(30(a,1),22(a,1,1)),rewrite([21(6)])].">
      <j1 rule="para" parents="30 22"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      f(f(f((x ^ y)',f(x,y) v z),u),f(x ^ y,f(f(x ^ y,(x ^ y) v f(x,y)),z'))) = x ^ y
    ]]></literal>
    <justification jstring="[para(28(a,1),30(a,1,1,1,2)),rewrite([29(12)])].">
      <j1 rule="para" parents="28 30"/>
      <j2 rule="rewrite" parents="29"/>
    </justification>
  </clause>

  <clause id="75">
    <literal><![CDATA[
      f(x''',f(x,f(f(x,f(x',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),32(a,1,1))].">
      <j1 rule="para" parents="11 32"/>
    </justification>
  </clause>

  <clause id="76">
    <literal><![CDATA[
      f(x'',y) ^ f(x,f(f(x,f(x',x)),x)) = x'
    ]]></literal>
    <justification jstring="[para(32(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="32 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="78">
    <literal><![CDATA[
      f(x' v y,f(x,f(f(x,f(x',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),32(a,1,1))].">
      <j1 rule="para" parents="21 32"/>
    </justification>
  </clause>

  <clause id="87">
    <literal><![CDATA[
      f(x,f(x',f(f(x',x' v x),y))) = x'
    ]]></literal>
    <justification jstring="[para(32(a,1),30(a,1,1)),rewrite([21(6)])].">
      <j1 rule="para" parents="32 30"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      x''' ^ f(x,f(f(x,f(x',x)),x)) = x'
    ]]></literal>
    <justification jstring="[para(75(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="75 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="97">
    <literal><![CDATA[
      (x' v y) ^ f(x,f(f(x,f(x',x)),x)) = x'
    ]]></literal>
    <justification jstring="[para(78(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="78 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="105">
    <literal><![CDATA[
      f(f(x,y) ^ y',f(y,f(f(y,f(x',x)),y))) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),31(a,1,1)),rewrite([20(4)])].">
      <j1 rule="para" parents="11 31"/>
      <j2 rule="rewrite" parents="20"/>
    </justification>
  </clause>

  <clause id="107">
    <literal><![CDATA[
      f(f(f(x,y),y'),z) ^ f(y,f(f(y,f(x',x)),y)) = y'
    ]]></literal>
    <justification jstring="[para(31(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="31 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="132">
    <literal><![CDATA[
      x ^ f(x',f(f(x',x' v x),y)) = x''
    ]]></literal>
    <justification jstring="[para(87(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="87 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="140">
    <literal><![CDATA[
      f(x',f(x'',x)) = x''
    ]]></literal>
    <justification jstring="[para(32(a,1),87(a,1,2,2))].">
      <j1 rule="para" parents="32 87"/>
    </justification>
  </clause>

  <clause id="144">
    <literal><![CDATA[
      f(x'',x'' v x) = x'''
    ]]></literal>
    <justification jstring="[para(21(a,1),140(a,1,2))].">
      <j1 rule="para" parents="21 140"/>
    </justification>
  </clause>

  <clause id="166">
    <literal><![CDATA[
      (f(x,y) ^ f(y,z)) ^ f(y,f(f(y,f(x',x)),z)) = y'
    ]]></literal>
    <justification jstring="[para(33(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="33 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="256">
    <literal><![CDATA[
      (x'' v y) ^ f(x',f(f(x',x' v x),x')) = x''
    ]]></literal>
    <justification jstring="[para(21(a,1),97(a,1,2,2,1,2))].">
      <j1 rule="para" parents="21 97"/>
    </justification>
  </clause>

  <clause id="281">
    <literal><![CDATA[
      f(x,f(f(x,y),f(x,y) ^ (x' v x))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(30(a,1),35(a,1,1)),rewrite([21(6)])].">
      <j1 rule="para" parents="30 35"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="346">
    <literal><![CDATA[
      f(x,f(x',x' v x)) = x'
    ]]></literal>
    <justification jstring="[para(281(a,1),87(a,1,2))].">
      <j1 rule="para" parents="281 87"/>
    </justification>
  </clause>

  <clause id="357">
    <literal><![CDATA[
      f(f(x,y),f(x ^ y,(x ^ y) v f(x,y))) = x ^ y
    ]]></literal>
    <justification jstring="[para(20(a,1),346(a,1,2,1)),rewrite([20(4),20(9)])].">
      <j1 rule="para" parents="20 346"/>
      <j2 rule="rewrite" parents="20 20"/>
    </justification>
  </clause>

  <clause id="380">
    <literal><![CDATA[
      f(x,y) ^ f(x,f(f(x,f(f(z,x) ^ f(x,u),f(f(z,x),f(x,u)))),f(f(x,f(z',z)),u))) = x'
    ]]></literal>
    <justification jstring="[para(22(a,1),36(a,1,1,1)),rewrite([20(5)])].">
      <j1 rule="para" parents="22 36"/>
      <j2 rule="rewrite" parents="20"/>
    </justification>
  </clause>

  <clause id="392">
    <literal><![CDATA[
      x ^ f(f(x,y),f(f(f(x,y),x' v x),z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(30(a,1),36(a,1,1)),rewrite([21(6),20(10)])].">
      <j1 rule="para" parents="30 36"/>
      <j2 rule="rewrite" parents="21 20"/>
    </justification>
  </clause>

  <clause id="521">
    <literal><![CDATA[
      f(x''',y) ^ f(x',f(f(x',x' v x),x')) = x''
    ]]></literal>
    <justification jstring="[para(21(a,1),76(a,1,2,2,1,2))].">
      <j1 rule="para" parents="21 76"/>
    </justification>
  </clause>

  <clause id="673">
    <literal><![CDATA[
      f(x',f(f(x',f(x,y)),x)) = f(x',f(x,y))
    ]]></literal>
    <justification jstring="[para(30(a,1),62(a,1,2,2))].">
      <j1 rule="para" parents="30 62"/>
    </justification>
  </clause>

  <clause id="679">
    <literal><![CDATA[
      f(f(x,y),f(f(f(x,y),y'),y)) = f(f(x,y),y')
    ]]></literal>
    <justification jstring="[para(31(a,1),62(a,1,2,2))].">
      <j1 rule="para" parents="31 62"/>
    </justification>
  </clause>

  <clause id="689">
    <literal><![CDATA[
      f(x,f(f(x,y),x' v x)) = f(x,y)
    ]]></literal>
    <justification jstring="[para(281(a,1),62(a,1,2))].">
      <j1 rule="para" parents="281 62"/>
    </justification>
  </clause>

  <clause id="770">
    <literal><![CDATA[
      f(x''',f(x,x'''' v x''')) = x
    ]]></literal>
    <justification jstring="[para(75(a,1),689(a,1,2,1)),rewrite([75(22)])].">
      <j1 rule="para" parents="75 689"/>
      <j2 rule="rewrite" parents="75"/>
    </justification>
  </clause>

  <clause id="832">
    <literal><![CDATA[
      x' ^ f(f(x',f(x,y)),x) = x' ^ f(x,y)
    ]]></literal>
    <justification jstring="[para(673(a,1),20(a,1,1)),rewrite([20(4)]),flip(a)].">
      <j1 rule="para" parents="673 20"/>
      <j2 rule="rewrite" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="834">
    <literal><![CDATA[
      f(x'',f(f(x'',x v y),x')) = f(x'',x v y)
    ]]></literal>
    <justification jstring="[para(21(a,1),673(a,1,2,1,2)),rewrite([21(14)])].">
      <j1 rule="para" parents="21 673"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="863">
    <literal><![CDATA[
      f((f(x,y) ^ y')',f(f((f(x,y) ^ y')',y),f(x,y) ^ y')) = f((f(x,y) ^ y')',y)
    ]]></literal>
    <justification jstring="[para(105(a,1),673(a,1,2,1,2)),rewrite([105(27)])].">
      <j1 rule="para" parents="105 673"/>
      <j2 rule="rewrite" parents="105"/>
    </justification>
  </clause>

  <clause id="920">
    <literal><![CDATA[
      x'' ^ f(f(x'',x v y),x') = x'' ^ (x v y)
    ]]></literal>
    <justification jstring="[para(21(a,1),832(a,1,2,1,2)),rewrite([21(14)])].">
      <j1 rule="para" parents="21 832"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="930">
    <literal><![CDATA[
      (x' ^ f(x,y))' ^ f(f((x' ^ f(x,y))',x),x' ^ f(x,y)) = (x' ^ f(x,y))' ^ x
    ]]></literal>
    <justification jstring="[para(53(a,1),832(a,1,2,1,2)),rewrite([53(27)])].">
      <j1 rule="para" parents="53 832"/>
      <j2 rule="rewrite" parents="53"/>
    </justification>
  </clause>

  <clause id="934">
    <literal><![CDATA[
      f(x,y) ^ f(f(f(x,y),y'),y) = f(x,y) ^ y'
    ]]></literal>
    <justification jstring="[para(679(a,1),20(a,1,1)),rewrite([20(4)]),flip(a)].">
      <j1 rule="para" parents="679 20"/>
      <j2 rule="rewrite" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="988">
    <literal><![CDATA[
      f(f(f(x,x ^ (x'''' v x''')),y),f(f(x,x'''' v x'''),f(f(f(x,x'''' v x'''),x''' v x''),f(x,x'''' v x''')))) = f(x,x'''' v x''')
    ]]></literal>
    <justification jstring="[para(770(a,1),31(a,1,1,1,1)),rewrite([20(10),21(37)])].">
      <j1 rule="para" parents="770 31"/>
      <j2 rule="rewrite" parents="20 21"/>
    </justification>
  </clause>

  <clause id="1140">
    <literal><![CDATA[
      x'' ^ f(x' v f(x'',x),f(x'',x)) = x'' ^ (x'' ^ x)
    ]]></literal>
    <justification jstring="[para(140(a,1),934(a,1,1)),rewrite([140(7),20(8),29(8),140(17),20(18)])].">
      <j1 rule="para" parents="140 934"/>
      <j2 rule="rewrite" parents="140 20 29 140 20"/>
    </justification>
  </clause>

  <clause id="1606">
    <literal><![CDATA[
      f(x,x'') = x'
    ]]></literal>
    <justification jstring="[para(43(a,1),87(a,1,2,2)),rewrite([11(3)])].">
      <j1 rule="para" parents="43 87"/>
      <j2 rule="rewrite" parents="11"/>
    </justification>
  </clause>

  <clause id="1625">
    <literal><![CDATA[
      x ^ x'' = x''
    ]]></literal>
    <justification jstring="[para(43(a,1),132(a,1,2,2)),rewrite([11(3)])].">
      <j1 rule="para" parents="43 132"/>
      <j2 rule="rewrite" parents="11"/>
    </justification>
  </clause>

  <clause id="1626">
    <literal><![CDATA[
      f(f(x,y),f(y,y ^ f(x',x))) = y
    ]]></literal>
    <justification jstring="[para(43(a,1),35(a,1,1))].">
      <j1 rule="para" parents="43 35"/>
    </justification>
  </clause>

  <clause id="1658">
    <literal><![CDATA[
      f(x,x ^ y) = f(x,y)
    ]]></literal>
    <justification jstring="[para(43(a,1),62(a,1,2,2)),rewrite([11(3),20(2)])].">
      <j1 rule="para" parents="43 62"/>
      <j2 rule="rewrite" parents="11 20"/>
    </justification>
  </clause>

  <clause id="1677">
    <literal><![CDATA[
      (x ^ y) ^ f(f(x ^ y,x),f(x,y)) = (x ^ y) ^ x
    ]]></literal>
    <justification jstring="[para(43(a,1),832(a,1,2,1,2)),rewrite([20(2),20(3),20(8),43(23)])].">
      <j1 rule="para" parents="43 832"/>
      <j2 rule="rewrite" parents="20 20 20 43"/>
    </justification>
  </clause>

  <clause id="1704">
    <literal><![CDATA[
      x ^ (x ^ y) = x ^ y
    ]]></literal>
    <justification jstring="[para(43(a,1),392(a,1,2,2)),rewrite([11(3),20(2)])].">
      <j1 rule="para" parents="43 392"/>
      <j2 rule="rewrite" parents="11 20"/>
    </justification>
  </clause>

  <clause id="1717">
    <literal><![CDATA[
      f(f(x,y),f(y,f(x',x))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1626),rewrite([1658(5)])].">
      <j1 rule="back_rewrite" parents="1626"/>
      <j2 rule="rewrite" parents="1658"/>
    </justification>
  </clause>

  <clause id="1758">
    <literal><![CDATA[
      f(f(f(x,x'''' v x'''),y),f(f(x,x'''' v x'''),f(f(f(x,x'''' v x'''),x''' v x''),f(x,x'''' v x''')))) = f(x,x'''' v x''')
    ]]></literal>
    <justification jstring="[back_rewrite(988),rewrite([1658(10)])].">
      <j1 rule="back_rewrite" parents="988"/>
      <j2 rule="rewrite" parents="1658"/>
    </justification>
  </clause>

  <clause id="1903">
    <literal><![CDATA[
      x'' ^ f(x' v f(x'',x),f(x'',x)) = x'' ^ x
    ]]></literal>
    <justification jstring="[back_rewrite(1140),rewrite([1704(18)])].">
      <j1 rule="back_rewrite" parents="1140"/>
      <j2 rule="rewrite" parents="1704"/>
    </justification>
  </clause>

  <clause id="1943">
    <literal><![CDATA[
      x v x'' = x''
    ]]></literal>
    <justification jstring="[para(1606(a,1),21(a,1)),flip(a)].">
      <j1 rule="para" parents="1606 21"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2013">
    <literal><![CDATA[
      f(f(x'',y),x') = x
    ]]></literal>
    <justification jstring="[para(32(a,1),1658(a,2)),rewrite([76(12)])].">
      <j1 rule="para" parents="32 1658"/>
      <j2 rule="rewrite" parents="76"/>
    </justification>
  </clause>

  <clause id="2014">
    <literal><![CDATA[
      x'' v x = x
    ]]></literal>
    <justification jstring="[para(75(a,1),1658(a,2)),rewrite([89(12),21(5)])].">
      <j1 rule="para" parents="75 1658"/>
      <j2 rule="rewrite" parents="89 21"/>
    </justification>
  </clause>

  <clause id="2017">
    <literal><![CDATA[
      f(f(f(f(x,y),y'),z),y') = y
    ]]></literal>
    <justification jstring="[para(31(a,1),1658(a,2)),rewrite([107(14)])].">
      <j1 rule="para" parents="31 1658"/>
      <j2 rule="rewrite" parents="107"/>
    </justification>
  </clause>

  <clause id="2021">
    <literal><![CDATA[
      f(f(x,y),f(y,z)) v y = y
    ]]></literal>
    <justification jstring="[para(33(a,1),1658(a,2)),rewrite([166(12),28(5)])].">
      <j1 rule="para" parents="33 1658"/>
      <j2 rule="rewrite" parents="166 28"/>
    </justification>
  </clause>

  <clause id="2022">
    <literal><![CDATA[
      f(x'',x) = x'''
    ]]></literal>
    <justification jstring="[para(144(a,1),1658(a,2)),rewrite([2014(7),1658(6)])].">
      <j1 rule="para" parents="144 1658"/>
      <j2 rule="rewrite" parents="2014 1658"/>
    </justification>
  </clause>

  <clause id="2063">
    <literal><![CDATA[
      f(f(x,y),x') = x
    ]]></literal>
    <justification jstring="[para(43(a,1),1658(a,2)),rewrite([380(17)])].">
      <j1 rule="para" parents="43 1658"/>
      <j2 rule="rewrite" parents="380"/>
    </justification>
  </clause>

  <clause id="2064">
    <literal><![CDATA[
      x'' ^ x = x''''
    ]]></literal>
    <justification jstring="[back_rewrite(1903),rewrite([2022(6),1943(7),2022(8),11(9),1625(7)]),flip(a)].">
      <j1 rule="back_rewrite" parents="1903"/>
      <j2 rule="rewrite" parents="2022 1943 2022 11 1625"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2083">
    <literal><![CDATA[
      x'' ^ (x v y) = x''''
    ]]></literal>
    <justification jstring="[back_rewrite(920),rewrite([2013(8),2064(3)]),flip(a)].">
      <j1 rule="back_rewrite" parents="920"/>
      <j2 rule="rewrite" parents="2013 2064"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2085">
    <literal><![CDATA[
      f(x'',x v y) = x'''
    ]]></literal>
    <justification jstring="[back_rewrite(834),rewrite([2013(8),2022(3)]),flip(a)].">
      <j1 rule="back_rewrite" parents="834"/>
      <j2 rule="rewrite" parents="2013 2022"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2291">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[para(11(a,1),2063(a,1,1)),rewrite([11(3)])].">
      <j1 rule="para" parents="11 2063"/>
      <j2 rule="rewrite" parents="11"/>
    </justification>
  </clause>

  <clause id="2292">
    <literal><![CDATA[
      f(x,y) ^ x' = x'
    ]]></literal>
    <justification jstring="[para(2063(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="2063 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2294">
    <literal><![CDATA[
      f(x v y,x) = x'
    ]]></literal>
    <justification jstring="[para(21(a,1),2063(a,1,1)),rewrite([2291(3)])].">
      <j1 rule="para" parents="21 2063"/>
      <j2 rule="rewrite" parents="2291"/>
    </justification>
  </clause>

  <clause id="2302">
    <literal><![CDATA[
      f(x,y') = x' v y
    ]]></literal>
    <justification jstring="[para(78(a,1),2063(a,1,1)),rewrite([27(3),2291(2),1658(3)])].">
      <j1 rule="para" parents="78 2063"/>
      <j2 rule="rewrite" parents="27 2291 1658"/>
    </justification>
  </clause>

  <clause id="2310">
    <literal><![CDATA[
      f(f((x ^ f(x',x)) v x,y),f(f(x',x),f(x' ^ x,x))) = f(x',x)
    ]]></literal>
    <justification jstring="[para(2063(a,1),34(a,1,1,1,2)),rewrite([2302(5),20(4),2291(11)])].">
      <j1 rule="para" parents="2063 34"/>
      <j2 rule="rewrite" parents="2302 20 2291"/>
    </justification>
  </clause>

  <clause id="2325">
    <literal><![CDATA[
      (x ^ y)' = f(x,y)
    ]]></literal>
    <justification jstring="[para(357(a,1),2063(a,1,1)),rewrite([20(3),11(3)])].">
      <j1 rule="para" parents="357 2063"/>
      <j2 rule="rewrite" parents="20 11"/>
    </justification>
  </clause>

  <clause id="2329">
    <literal><![CDATA[
      (x ^ y) v x = x
    ]]></literal>
    <justification jstring="[para(62(a,1),2063(a,1,1)),rewrite([2302(3),20(2)])].">
      <j1 rule="para" parents="62 2063"/>
      <j2 rule="rewrite" parents="2302 20"/>
    </justification>
  </clause>

  <clause id="2426">
    <literal><![CDATA[
      f(x,x v y) = x'
    ]]></literal>
    <justification jstring="[back_rewrite(2085),rewrite([2291(2),2291(4)])].">
      <j1 rule="back_rewrite" parents="2085"/>
      <j2 rule="rewrite" parents="2291 2291"/>
    </justification>
  </clause>

  <clause id="2427">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[back_rewrite(2083),rewrite([2291(2),2291(4),2291(4)])].">
      <j1 rule="back_rewrite" parents="2083"/>
      <j2 rule="rewrite" parents="2291 2291 2291"/>
    </justification>
  </clause>

  <clause id="2451">
    <literal><![CDATA[
      f(f(x',y),x) = x'
    ]]></literal>
    <justification jstring="[back_rewrite(1758),rewrite([2291(2),2291(2),2291(2),2426(3),2291(4),2291(4),2291(4),2426(5),2291(5),2291(5),2291(5),2426(6),2291(6),2291(7),2426(7),2291(5),2291(5),2291(5),2291(5),2426(6),2302(5),2426(6),2291(4),2291(5),2291(5),2291(5),2426(6)])].">
      <j1 rule="back_rewrite" parents="1758"/>
      <j2 rule="rewrite" parents="2291 2291 2291 2426 2291 2291 2291 2426 2291 2291 2291 2426 2291 2291 2426 2291 2291 2291 2291 2426 2302 2426 2291 2291 2291 2291 2426"/>
    </justification>
  </clause>

  <clause id="2564">
    <literal><![CDATA[
      f(x',y) ^ x = x
    ]]></literal>
    <justification jstring="[back_rewrite(521),rewrite([2291(2),2426(7),2291(5),2302(5),2426(6),2291(4),2291(5)])].">
      <j1 rule="back_rewrite" parents="521"/>
      <j2 rule="rewrite" parents="2291 2426 2291 2302 2426 2291 2291"/>
    </justification>
  </clause>

  <clause id="2579">
    <literal><![CDATA[
      (x v y) ^ x = x
    ]]></literal>
    <justification jstring="[back_rewrite(256),rewrite([2291(2),2426(6),2291(4),2302(4),2426(5),2291(3),2291(4)])].">
      <j1 rule="back_rewrite" parents="256"/>
      <j2 rule="rewrite" parents="2291 2426 2291 2302 2426 2291 2291"/>
    </justification>
  </clause>

  <clause id="2602">
    <literal><![CDATA[
      x ^ x = x
    ]]></literal>
    <justification jstring="[back_rewrite(25),rewrite([2291(3)])].">
      <j1 rule="back_rewrite" parents="25"/>
      <j2 rule="rewrite" parents="2291"/>
    </justification>
  </clause>

  <clause id="2617">
    <literal><![CDATA[
      (((x ^ y) v y) ^ z) v y = y
    ]]></literal>
    <justification jstring="[back_rewrite(2017),rewrite([2302(3),20(2),2302(5),20(4)])].">
      <j1 rule="back_rewrite" parents="2017"/>
      <j2 rule="rewrite" parents="2302 20 2302 20"/>
    </justification>
  </clause>

  <clause id="2789">
    <literal><![CDATA[
      f(f(x ^ y,z),f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(67),rewrite([2325(2),2426(4),20(2),2426(8),2325(5),2302(6),20(5),2426(6),2325(4)])].">
      <j1 rule="back_rewrite" parents="67"/>
      <j2 rule="rewrite" parents="2325 2426 20 2426 2325 2302 20 2426 2325"/>
    </justification>
  </clause>

  <clause id="2793">
    <literal><![CDATA[
      f(c9,c8 ^ c10) != f(c8,c9 ^ c10)
    ]]></literal>
    <literal><![CDATA[
      f(c8',f(c8,c9)) != c8
    ]]></literal>
    <literal><![CDATA[
      c9' v c9 != c8' v c8
    ]]></literal>
    <attribute><![CDATA[
      answer(combined)
    ]]></attribute>
    <justification jstring="[back_rewrite(23),rewrite([2302(23),2302(27)])].">
      <j1 rule="back_rewrite" parents="23"/>
      <j2 rule="rewrite" parents="2302 2302"/>
    </justification>
  </clause>

  <clause id="2842">
    <literal><![CDATA[
      f(x',f(x,y)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(930),rewrite([2325(4),2325(7),2451(7),1658(8),2602(7),2325(7),2564(7)])].">
      <j1 rule="back_rewrite" parents="930"/>
      <j2 rule="rewrite" parents="2325 2325 2451 1658 2602 2325 2564"/>
    </justification>
  </clause>

  <clause id="2848">
    <literal><![CDATA[
      f((x ^ y) v y,f(f((x ^ y) v y,y),f(x,y) ^ y')) = f((x ^ y) v y,y)
    ]]></literal>
    <justification jstring="[back_rewrite(863),rewrite([2325(4),2302(3),20(2),2325(6),2302(5),20(4),2325(14),2302(13),20(12)])].">
      <j1 rule="back_rewrite" parents="863"/>
      <j2 rule="rewrite" parents="2325 2302 20 2325 2302 20 2325 2302 20"/>
    </justification>
  </clause>

  <clause id="2885">
    <literal><![CDATA[
      f(f(x,y),f(f(x',x),f(x' ^ x,x))) = f(x',x)
    ]]></literal>
    <justification jstring="[back_rewrite(2310),rewrite([2329(4)])].">
      <j1 rule="back_rewrite" parents="2310"/>
      <j2 rule="rewrite" parents="2329"/>
    </justification>
  </clause>

  <clause id="2928">
    <literal><![CDATA[
      (x ^ y) ^ x = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(1677),rewrite([2789(5),2602(3)]),flip(a)].">
      <j1 rule="back_rewrite" parents="1677"/>
      <j2 rule="rewrite" parents="2789 2602"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2938">
    <literal><![CDATA[
      f(c9,c8 ^ c10) != f(c8,c9 ^ c10)
    ]]></literal>
    <literal><![CDATA[
      c9' v c9 != c8' v c8
    ]]></literal>
    <attribute><![CDATA[
      answer(combined)
    ]]></attribute>
    <justification jstring="[back_rewrite(2793),rewrite([2842(17)]),xx(b)].">
      <j1 rule="back_rewrite" parents="2793"/>
      <j2 rule="rewrite" parents="2842"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="3018">
    <literal><![CDATA[
      f(x,y ^ z) = x' v f(y,z)
    ]]></literal>
    <justification jstring="[para(2291(a,1),29(a,1,1))].">
      <j1 rule="para" parents="2291 29"/>
    </justification>
  </clause>

  <clause id="3029">
    <literal><![CDATA[
      c9' v f(c8,c10) != c8' v f(c9,c10)
    ]]></literal>
    <literal><![CDATA[
      c9' v c9 != c8' v c8
    ]]></literal>
    <attribute><![CDATA[
      answer(combined)
    ]]></attribute>
    <justification jstring="[back_rewrite(2938),rewrite([3018(5),3018(11)])].">
      <j1 rule="back_rewrite" parents="2938"/>
      <j2 rule="rewrite" parents="3018 3018"/>
    </justification>
  </clause>

  <clause id="3035">
    <literal><![CDATA[
      f((x ^ y) v y,y) = f(x,y) ^ y'
    ]]></literal>
    <justification jstring="[back_rewrite(2848),rewrite([3018(9),20(6),2302(8),20(7),2329(8),11(5),27(3),2325(2)]),flip(a)].">
      <j1 rule="back_rewrite" parents="2848"/>
      <j2 rule="rewrite" parents="3018 20 2302 20 2329 11 27 2325"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3192">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(2579(a,1),2329(a,1,1))].">
      <j1 rule="para" parents="2579 2329"/>
    </justification>
  </clause>

  <clause id="3205">
    <literal><![CDATA[
      f(f(x' ^ y,z),f(x',f(f(x',(x' ^ y) v f(x',y)),y))) = x'
    ]]></literal>
    <justification jstring="[para(2294(a,1),38(a,1,1,1)),rewrite([20(3),20(8)])].">
      <j1 rule="para" parents="2294 38"/>
      <j2 rule="rewrite" parents="20 20"/>
    </justification>
  </clause>

  <clause id="3325">
    <literal><![CDATA[
      (x v y) v x = x v y
    ]]></literal>
    <justification jstring="[para(2294(a,1),2842(a,1,2)),rewrite([27(2),2302(5),2325(4),2302(3),2291(2)])].">
      <j1 rule="para" parents="2294 2842"/>
      <j2 rule="rewrite" parents="27 2302 2325 2302 2291"/>
    </justification>
  </clause>

  <clause id="3327">
    <literal><![CDATA[
      f(f(x,y),f(x ^ y,z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(2325(a,1),2842(a,1,1))].">
      <j1 rule="para" parents="2325 2842"/>
    </justification>
  </clause>

  <clause id="3376">
    <literal><![CDATA[
      (x ^ y) v f(x',x) = f(x',x)
    ]]></literal>
    <justification jstring="[back_rewrite(2885),rewrite([3327(7),3018(4),20(2)])].">
      <j1 rule="back_rewrite" parents="2885"/>
      <j2 rule="rewrite" parents="3327 3018 20"/>
    </justification>
  </clause>

  <clause id="3428">
    <literal><![CDATA[
      x ^ f(f(y,x),f(x,z)) = f(f(y,x),f(x,z))
    ]]></literal>
    <justification jstring="[para(33(a,1),2292(a,1,1)),rewrite([2325(4),2325(8)])].">
      <j1 rule="para" parents="33 2292"/>
      <j2 rule="rewrite" parents="2325 2325"/>
    </justification>
  </clause>

  <clause id="3465">
    <literal><![CDATA[
      f(x,y) = x' v y'
    ]]></literal>
    <justification jstring="[para(2291(a,1),2302(a,1,2))].">
      <j1 rule="para" parents="2291 2302"/>
    </justification>
  </clause>

  <clause id="3510">
    <literal><![CDATA[
      x ^ ((y ^ x) v (x ^ z)) = (y ^ x) v (x ^ z)
    ]]></literal>
    <justification jstring="[back_rewrite(3428),rewrite([3465(1),3465(4),3465(7),27(4),2291(2),2291(2),27(5),2291(3),2291(3),3465(5),3465(8),3465(11),27(8),2291(6),2291(6),27(9),2291(7),2291(7)])].">
      <j1 rule="back_rewrite" parents="3428"/>
      <j2 rule="rewrite" parents="3465 3465 3465 27 2291 2291 27 2291 2291 3465 3465 3465 27 2291 2291 27 2291 2291"/>
    </justification>
  </clause>

  <clause id="3553">
    <literal><![CDATA[
      (x ^ y) v (x v x') = x v x'
    ]]></literal>
    <justification jstring="[back_rewrite(3376),rewrite([3465(3),2291(3),3465(6),2291(6)])].">
      <j1 rule="back_rewrite" parents="3376"/>
      <j2 rule="rewrite" parents="3465 2291 3465 2291"/>
    </justification>
  </clause>

  <clause id="3683">
    <literal><![CDATA[
      ((x' ^ y) ^ z) v (x' ^ ((x' ^ ((x' ^ y) v (x v y'))) v y')) = x'
    ]]></literal>
    <justification jstring="[back_rewrite(3205),rewrite([3465(3),2325(3),3465(2),2291(2),3465(10),2291(10),3465(12),2291(7),27(11),2325(8),3465(7),2291(7),27(10),2291(10),3465(12),27(12),2325(12),3465(11),27(9),2291(9),2325(11),3465(10),2291(10),3465(15),2291(6),27(14),2325(12),3465(11),2291(6),27(10),2325(7),3465(6),2291(6),27(9),2291(9),2291(12),3465(13),27(5),27(3),2291(3),2291(4),27(12),2325(12),3465(11),27(11),2325(11),3465(10),27(8),2291(8),2325(10),3465(9),2291(9)])].">
      <j1 rule="back_rewrite" parents="3205"/>
      <j2 rule="rewrite" parents="3465 2325 3465 2291 3465 2291 3465 2291 27 2325 3465 2291 27 2291 3465 27 2325 3465 27 2291 2325 3465 2291 3465 2291 27 2325 3465 2291 27 2325 3465 2291 27 2291 2291 3465 27 27 2291 2291 27 2325 3465 27 2325 3465 27 2291 2325 3465 2291"/>
    </justification>
  </clause>

  <clause id="3801">
    <literal><![CDATA[
      ((x' v y') ^ y') v y' = (x' v y') ^ y'
    ]]></literal>
    <justification jstring="[back_rewrite(3035),rewrite([3465(3),27(3),2325(2),3465(1),3465(8)])].">
      <j1 rule="back_rewrite" parents="3035"/>
      <j2 rule="rewrite" parents="3465 27 2325 3465 3465"/>
    </justification>
  </clause>

  <clause id="3806">
    <literal><![CDATA[
      c9' v (c8' v c10') != c8' v (c9' v c10')
    ]]></literal>
    <literal><![CDATA[
      c9' v c9 != c8' v c8
    ]]></literal>
    <attribute><![CDATA[
      answer(combined)
    ]]></attribute>
    <justification jstring="[back_rewrite(3029),rewrite([3465(5),3465(13)])].">
      <j1 rule="back_rewrite" parents="3029"/>
      <j2 rule="rewrite" parents="3465 3465"/>
    </justification>
  </clause>

  <clause id="3999">
    <literal><![CDATA[
      (x ^ y)' = x' v y'
    ]]></literal>
    <justification jstring="[back_rewrite(2325),rewrite([3465(3)])].">
      <j1 rule="back_rewrite" parents="2325"/>
      <j2 rule="rewrite" parents="3465"/>
    </justification>
  </clause>

  <clause id="4034">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v y = y
    ]]></literal>
    <justification jstring="[back_rewrite(2021),rewrite([3465(1),3465(4),3465(7),27(4),2291(2),2291(2),27(5),2291(3),2291(3)])].">
      <j1 rule="back_rewrite" parents="2021"/>
      <j2 rule="rewrite" parents="3465 3465 3465 27 2291 2291 27 2291 2291"/>
    </justification>
  </clause>

  <clause id="4068">
    <literal><![CDATA[
      (x ^ y) v (y ^ (x v x')) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1717),rewrite([3465(1),3465(5),2291(5),3465(6),27(7),2291(7),3465(8),27(4),2291(2),2291(2),27(6),2291(3),3999(4),2291(3)])].">
      <j1 rule="back_rewrite" parents="1717"/>
      <j2 rule="rewrite" parents="3465 3465 2291 3465 27 2291 3465 27 2291 2291 27 2291 3999 2291"/>
    </justification>
  </clause>

  <clause id="4263">
    <literal><![CDATA[
      (x ^ y) v y = y
    ]]></literal>
    <justification jstring="[para(2579(a,1),2617(a,1,1))].">
      <j1 rule="para" parents="2579 2617"/>
    </justification>
  </clause>

  <clause id="4302">
    <literal><![CDATA[
      (x' v y') ^ y' = y'
    ]]></literal>
    <justification jstring="[back_rewrite(3801),rewrite([4263(7)]),flip(a)].">
      <j1 rule="back_rewrite" parents="3801"/>
      <j2 rule="rewrite" parents="4263"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4380">
    <literal><![CDATA[
      ((x ^ (y v z)) v y) v (y v z) = y v z
    ]]></literal>
    <justification jstring="[para(2579(a,1),4034(a,1,1,2))].">
      <j1 rule="para" parents="2579 4034"/>
    </justification>
  </clause>

  <clause id="4381">
    <literal><![CDATA[
      ((x ^ y) v (x ^ z)) v x = x
    ]]></literal>
    <justification jstring="[para(2928(a,1),4034(a,1,1,1))].">
      <j1 rule="para" parents="2928 4034"/>
    </justification>
  </clause>

  <clause id="4382">
    <literal><![CDATA[
      x v ((y ^ x) v (x ^ z)) = x
    ]]></literal>
    <justification jstring="[para(4034(a,1),3325(a,1,1)),rewrite([4034(8)])].">
      <j1 rule="para" parents="4034 3325"/>
      <j2 rule="rewrite" parents="4034"/>
    </justification>
  </clause>

  <clause id="4394">
    <literal><![CDATA[
      (x' v y) ^ y = y
    ]]></literal>
    <justification jstring="[para(2291(a,1),4302(a,1,1,2)),rewrite([2291(4),2291(5)])].">
      <j1 rule="para" parents="2291 4302"/>
      <j2 rule="rewrite" parents="2291 2291"/>
    </justification>
  </clause>

  <clause id="4409">
    <literal><![CDATA[
      (x v y) ^ y = y
    ]]></literal>
    <justification jstring="[para(2291(a,1),4394(a,1,1,1))].">
      <j1 rule="para" parents="2291 4394"/>
    </justification>
  </clause>

  <clause id="4425">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(4409(a,1),2329(a,1,1))].">
      <j1 rule="para" parents="4409 2329"/>
    </justification>
  </clause>

  <clause id="4427">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(4409(a,1),2928(a,1,1)),rewrite([4409(4)])].">
      <j1 rule="para" parents="4409 2928"/>
      <j2 rule="rewrite" parents="4409"/>
    </justification>
  </clause>

  <clause id="4428">
    <literal><![CDATA[
      ((x ^ (y v z)) v z) v (y v z) = y v z
    ]]></literal>
    <justification jstring="[para(4409(a,1),4034(a,1,1,2))].">
      <j1 rule="para" parents="4409 4034"/>
    </justification>
  </clause>

  <clause id="4475">
    <literal><![CDATA[
      (((x v y) ^ z) v x) v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(2579(a,1),4381(a,1,1,2))].">
      <j1 rule="para" parents="2579 4381"/>
    </justification>
  </clause>

  <clause id="4510">
    <literal><![CDATA[
      (x v y) v (y v ((x v y) ^ z)) = x v y
    ]]></literal>
    <justification jstring="[para(4427(a,1),4382(a,1,2,1))].">
      <j1 rule="para" parents="4427 4382"/>
    </justification>
  </clause>

  <clause id="4586">
    <literal><![CDATA[
      (x v x') v y = x v x'
    ]]></literal>
    <justification jstring="[para(2579(a,1),4068(a,1,2)),rewrite([3553(7)]),flip(a)].">
      <j1 rule="para" parents="2579 4068"/>
      <j2 rule="rewrite" parents="3553"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4603">
    <literal><![CDATA[
      (x v x') ^ y = y
    ]]></literal>
    <justification jstring="[para(4586(a,1),4409(a,1,1))].">
      <j1 rule="para" parents="4586 4409"/>
    </justification>
  </clause>

  <clause id="4604">
    <literal><![CDATA[
      x ^ (y v y') = x
    ]]></literal>
    <justification jstring="[para(4586(a,1),4427(a,1,2))].">
      <j1 rule="para" parents="4586 4427"/>
    </justification>
  </clause>

  <clause id="4729">
    <literal><![CDATA[
      x v x' = y v y'
    ]]></literal>
    <justification jstring="[para(4604(a,1),4603(a,1))].">
      <j1 rule="para" parents="4604 4603"/>
    </justification>
  </clause>

  <clause id="4781">
    <literal><![CDATA[
      x v x' = c_0
    ]]></literal>
    <justification jstring="[new_symbol(4729)].">
      <j1 rule="new_symbol" parents="4729"/>
    </justification>
  </clause>

  <clause id="4783">
    <literal><![CDATA[
      x ^ c_0 = x
    ]]></literal>
    <justification jstring="[back_rewrite(4604),rewrite([4781(2)])].">
      <j1 rule="back_rewrite" parents="4604"/>
      <j2 rule="rewrite" parents="4781"/>
    </justification>
  </clause>

  <clause id="4855">
    <literal><![CDATA[
      x' v x = c_0
    ]]></literal>
    <justification jstring="[para(2291(a,1),4781(a,1,2))].">
      <j1 rule="para" parents="2291 4781"/>
    </justification>
  </clause>

  <clause id="4856">
    <literal><![CDATA[
      (x ^ y) v (x' v y') = c_0
    ]]></literal>
    <justification jstring="[para(3999(a,1),4781(a,1,2))].">
      <j1 rule="para" parents="3999 4781"/>
    </justification>
  </clause>

  <clause id="4941">
    <literal><![CDATA[
      c9' v (c8' v c10') != c8' v (c9' v c10')
    ]]></literal>
    <attribute><![CDATA[
      answer(combined)
    ]]></attribute>
    <justification jstring="[back_rewrite(3806),rewrite([4855(21),4855(22)]),xx(b)].">
      <j1 rule="back_rewrite" parents="3806"/>
      <j2 rule="rewrite" parents="4855 4855"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="5069">
    <literal><![CDATA[
      (x v y) ^ (x v ((x v y) ^ z)) = x v ((x v y) ^ z)
    ]]></literal>
    <justification jstring="[para(2427(a,1),3510(a,1,2,1)),rewrite([2427(7)])].">
      <j1 rule="para" parents="2427 3510"/>
      <j2 rule="rewrite" parents="2427"/>
    </justification>
  </clause>

  <clause id="5094">
    <literal><![CDATA[
      (x' ^ y) v (x v y') = c_0
    ]]></literal>
    <justification jstring="[para(2291(a,1),4856(a,1,2,1))].">
      <j1 rule="para" parents="2291 4856"/>
    </justification>
  </clause>

  <clause id="5100">
    <literal><![CDATA[
      ((x' ^ y) ^ z) v x' = x'
    ]]></literal>
    <justification jstring="[back_rewrite(3683),rewrite([5094(10),4783(7),2427(8)])].">
      <j1 rule="back_rewrite" parents="3683"/>
      <j2 rule="rewrite" parents="5094 4783 2427"/>
    </justification>
  </clause>

  <clause id="5291">
    <literal><![CDATA[
      ((x v y') v z') ^ x = x
    ]]></literal>
    <justification jstring="[para(5100(a,1),27(a,1,1)),rewrite([2291(2),3999(4),3999(3),2291(2),2291(6)]),flip(a)].">
      <j1 rule="para" parents="5100 27"/>
      <j2 rule="rewrite" parents="2291 3999 3999 2291 2291"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="5292">
    <literal><![CDATA[
      ((x ^ y) ^ z) v x = x
    ]]></literal>
    <justification jstring="[para(2291(a,1),5100(a,1,1,1,1)),rewrite([2291(4),2291(5)])].">
      <j1 rule="para" parents="2291 5100"/>
      <j2 rule="rewrite" parents="2291 2291"/>
    </justification>
  </clause>

  <clause id="5301">
    <literal><![CDATA[
      (x ^ y) v (x v z) = x v z
    ]]></literal>
    <justification jstring="[para(2579(a,1),5292(a,1,1,1))].">
      <j1 rule="para" parents="2579 5292"/>
    </justification>
  </clause>

  <clause id="5448">
    <literal><![CDATA[
      ((x v y) v z') ^ x = x
    ]]></literal>
    <justification jstring="[para(2291(a,1),5291(a,1,1,1,2))].">
      <j1 rule="para" parents="2291 5291"/>
    </justification>
  </clause>

  <clause id="5477">
    <literal><![CDATA[
      ((x v y) v z) ^ x = x
    ]]></literal>
    <justification jstring="[para(2291(a,1),5448(a,1,1,2))].">
      <j1 rule="para" parents="2291 5448"/>
    </justification>
  </clause>

  <clause id="5550">
    <literal><![CDATA[
      x ^ ((x v y) v z) = x
    ]]></literal>
    <justification jstring="[para(5477(a,1),2928(a,1,1)),rewrite([5477(6)])].">
      <j1 rule="para" parents="5477 2928"/>
      <j2 rule="rewrite" parents="5477"/>
    </justification>
  </clause>

  <clause id="5554">
    <literal><![CDATA[
      ((x v y) v z) ^ y = y
    ]]></literal>
    <justification jstring="[para(4425(a,1),5477(a,1,1,1))].">
      <j1 rule="para" parents="4425 5477"/>
    </justification>
  </clause>

  <clause id="5555">
    <literal><![CDATA[
      (x v (y v z)) ^ y = y
    ]]></literal>
    <justification jstring="[para(4425(a,1),5477(a,1,1))].">
      <j1 rule="para" parents="4425 5477"/>
    </justification>
  </clause>

  <clause id="5581">
    <literal><![CDATA[
      x ^ ((y v x) v z) = x
    ]]></literal>
    <justification jstring="[para(4425(a,1),5550(a,1,2,1))].">
      <j1 rule="para" parents="4425 5550"/>
    </justification>
  </clause>

  <clause id="5582">
    <literal><![CDATA[
      x ^ (y v (x v z)) = x
    ]]></literal>
    <justification jstring="[para(4425(a,1),5550(a,1,2))].">
      <j1 rule="para" parents="4425 5550"/>
    </justification>
  </clause>

  <clause id="5594">
    <literal><![CDATA[
      x v ((y v x) v z) = (y v x) v z
    ]]></literal>
    <justification jstring="[para(5554(a,1),2329(a,1,1))].">
      <j1 rule="para" parents="5554 2329"/>
    </justification>
  </clause>

  <clause id="5597">
    <literal><![CDATA[
      (x v (y v z)) ^ z = z
    ]]></literal>
    <justification jstring="[para(4425(a,1),5554(a,1,1))].">
      <j1 rule="para" parents="4425 5554"/>
    </justification>
  </clause>

  <clause id="5614">
    <literal><![CDATA[
      (x v y) ^ (z ^ y) = z ^ y
    ]]></literal>
    <justification jstring="[para(4263(a,1),5555(a,1,1,2))].">
      <j1 rule="para" parents="4263 5555"/>
    </justification>
  </clause>

  <clause id="5663">
    <literal><![CDATA[
      x ^ (y v (z v x)) = x
    ]]></literal>
    <justification jstring="[para(4425(a,1),5581(a,1,2))].">
      <j1 rule="para" parents="4425 5581"/>
    </justification>
  </clause>

  <clause id="5668">
    <literal><![CDATA[
      (x ^ y) ^ (z v x) = x ^ y
    ]]></literal>
    <justification jstring="[para(2329(a,1),5582(a,1,2,2))].">
      <j1 rule="para" parents="2329 5582"/>
    </justification>
  </clause>

  <clause id="5669">
    <literal><![CDATA[
      (x ^ y) ^ (z v y) = x ^ y
    ]]></literal>
    <justification jstring="[para(4263(a,1),5582(a,1,2,2))].">
      <j1 rule="para" parents="4263 5582"/>
    </justification>
  </clause>

  <clause id="5681">
    <literal><![CDATA[
      x v (y v (z v x)) = y v (z v x)
    ]]></literal>
    <justification jstring="[para(5597(a,1),2329(a,1,1))].">
      <j1 rule="para" parents="5597 2329"/>
    </justification>
  </clause>

  <clause id="5738">
    <literal><![CDATA[
      (x v (y v z)) ^ (y ^ u) = y ^ u
    ]]></literal>
    <justification jstring="[para(5301(a,1),5555(a,1,1,2))].">
      <j1 rule="para" parents="5301 5555"/>
    </justification>
  </clause>

  <clause id="6034">
    <literal><![CDATA[
      x ^ (y v (z v (u v x))) = x
    ]]></literal>
    <justification jstring="[para(5597(a,1),5668(a,1,1)),rewrite([5597(7)])].">
      <j1 rule="para" parents="5597 5668"/>
      <j2 rule="rewrite" parents="5597"/>
    </justification>
  </clause>

  <clause id="9482">
    <literal><![CDATA[
      (x v y) v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(4427(a,1),4380(a,1,1,1))].">
      <j1 rule="para" parents="4427 4380"/>
    </justification>
  </clause>

  <clause id="9493">
    <literal><![CDATA[
      (x v y) v (y v (x v z)) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(5582(a,1),4380(a,1,1,1))].">
      <j1 rule="para" parents="5582 4380"/>
    </justification>
  </clause>

  <clause id="9494">
    <literal><![CDATA[
      (x v y) v (y v (z v x)) = y v (z v x)
    ]]></literal>
    <justification jstring="[para(5663(a,1),4380(a,1,1,1))].">
      <j1 rule="para" parents="5663 4380"/>
    </justification>
  </clause>

  <clause id="9497">
    <literal><![CDATA[
      ((x ^ y) v z) v (z v x) = z v x
    ]]></literal>
    <justification jstring="[para(5668(a,1),4380(a,1,1,1))].">
      <j1 rule="para" parents="5668 4380"/>
    </justification>
  </clause>

  <clause id="9498">
    <literal><![CDATA[
      ((x ^ y) v z) v (z v y) = z v y
    ]]></literal>
    <justification jstring="[para(5669(a,1),4380(a,1,1,1))].">
      <j1 rule="para" parents="5669 4380"/>
    </justification>
  </clause>

  <clause id="9520">
    <literal><![CDATA[
      (x v y) v (y v (z v (u v x))) = y v (z v (u v x))
    ]]></literal>
    <justification jstring="[para(6034(a,1),4380(a,1,1,1))].">
      <j1 rule="para" parents="6034 4380"/>
    </justification>
  </clause>

  <clause id="9606">
    <literal><![CDATA[
      (x v y) v ((z v x) v y) = (z v x) v y
    ]]></literal>
    <justification jstring="[para(5581(a,1),4428(a,1,1,1))].">
      <j1 rule="para" parents="5581 4428"/>
    </justification>
  </clause>

  <clause id="10380">
    <literal><![CDATA[
      ((x ^ y) v z) v (z v (x v u)) = z v (x v u)
    ]]></literal>
    <justification jstring="[para(5738(a,1),4475(a,1,1,1))].">
      <j1 rule="para" parents="5738 4475"/>
    </justification>
  </clause>

  <clause id="10433">
    <literal><![CDATA[
      (x v y) ^ (y v x) = x v y
    ]]></literal>
    <justification jstring="[para(9482(a,1),2427(a,1,2))].">
      <j1 rule="para" parents="9482 2427"/>
    </justification>
  </clause>

  <clause id="10434">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[para(9482(a,1),2579(a,1,1)),rewrite([10433(3)])].">
      <j1 rule="para" parents="9482 2579"/>
      <j2 rule="rewrite" parents="10433"/>
    </justification>
  </clause>

  <clause id="10650">
    <literal><![CDATA[
      (x v y) v ((z ^ y) v x) = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(9498),rewrite([10434(4)])].">
      <j1 rule="back_rewrite" parents="9498"/>
      <j2 rule="rewrite" parents="10434"/>
    </justification>
  </clause>

  <clause id="10651">
    <literal><![CDATA[
      (x v y) v ((y ^ z) v x) = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(9497),rewrite([10434(4)])].">
      <j1 rule="back_rewrite" parents="9497"/>
      <j2 rule="rewrite" parents="10434"/>
    </justification>
  </clause>

  <clause id="13361">
    <literal><![CDATA[
      x v (y v ((y v x) ^ z)) = y v x
    ]]></literal>
    <justification jstring="[para(5069(a,1),4510(a,1,2,2)),rewrite([9493(6)])].">
      <j1 rule="para" parents="5069 4510"/>
      <j2 rule="rewrite" parents="9493"/>
    </justification>
  </clause>

  <clause id="24203">
    <literal><![CDATA[
      ((x ^ y) v z) v (u v (z v y)) = u v (z v y)
    ]]></literal>
    <justification jstring="[para(10650(a,1),5681(a,1,2,2)),rewrite([10650(9)])].">
      <j1 rule="para" parents="10650 5681"/>
      <j2 rule="rewrite" parents="10650"/>
    </justification>
  </clause>

  <clause id="24303">
    <literal><![CDATA[
      ((x ^ y) v z) v ((z v x) v u) = (z v x) v u
    ]]></literal>
    <justification jstring="[para(10651(a,1),5594(a,1,2,1)),rewrite([10651(9)])].">
      <j1 rule="para" parents="10651 5594"/>
      <j2 rule="rewrite" parents="10651"/>
    </justification>
  </clause>

  <clause id="25999">
    <literal><![CDATA[
      x v (y v (z ^ x)) = y v x
    ]]></literal>
    <justification jstring="[para(5614(a,1),13361(a,1,2,2))].">
      <j1 rule="para" parents="5614 13361"/>
    </justification>
  </clause>

  <clause id="26091">
    <literal><![CDATA[
      (x v y) v (z v x) = z v (x v y)
    ]]></literal>
    <justification jstring="[para(2427(a,1),25999(a,1,2,2))].">
      <j1 rule="para" parents="2427 25999"/>
    </justification>
  </clause>

  <clause id="26092">
    <literal><![CDATA[
      (x v y) v (z v y) = z v (x v y)
    ]]></literal>
    <justification jstring="[para(4427(a,1),25999(a,1,2,2))].">
      <j1 rule="para" parents="4427 25999"/>
    </justification>
  </clause>

  <clause id="26107">
    <literal><![CDATA[
      x v ((y ^ x) v z) = z v x
    ]]></literal>
    <justification jstring="[para(10434(a,1),25999(a,1,2))].">
      <j1 rule="para" parents="10434 25999"/>
    </justification>
  </clause>

  <clause id="26176">
    <literal><![CDATA[
      (x v y) v (y v z) = (x v y) v z
    ]]></literal>
    <justification jstring="[back_rewrite(9606),rewrite([26092(4)])].">
      <j1 rule="back_rewrite" parents="9606"/>
      <j2 rule="rewrite" parents="26092"/>
    </justification>
  </clause>

  <clause id="26200">
    <literal><![CDATA[
      ((x ^ y) v z) v (x v u) = z v (x v u)
    ]]></literal>
    <justification jstring="[back_rewrite(10380),rewrite([26176(5)])].">
      <j1 rule="back_rewrite" parents="10380"/>
      <j2 rule="rewrite" parents="26176"/>
    </justification>
  </clause>

  <clause id="26201">
    <literal><![CDATA[
      (x v y) v (z v (u v x)) = y v (z v (u v x))
    ]]></literal>
    <justification jstring="[back_rewrite(9520),rewrite([26176(5)])].">
      <j1 rule="back_rewrite" parents="9520"/>
      <j2 rule="rewrite" parents="26176"/>
    </justification>
  </clause>

  <clause id="26205">
    <literal><![CDATA[
      x v (y v z) = z v (x v y)
    ]]></literal>
    <justification jstring="[back_rewrite(9494),rewrite([26176(4),26091(3)])].">
      <j1 rule="back_rewrite" parents="9494"/>
      <j2 rule="rewrite" parents="26176 26091"/>
    </justification>
  </clause>

  <clause id="26542">
    <literal><![CDATA[
      x v (y v ((x ^ z) v u)) = u v (x v y)
    ]]></literal>
    <justification jstring="[back_rewrite(26200),rewrite([26205(4,R)])].">
      <j1 rule="back_rewrite" parents="26200"/>
      <j2 rule="rewrite" parents="26205"/>
    </justification>
  </clause>

  <clause id="26606">
    <literal><![CDATA[
      (x v y) v z = x v (y v z)
    ]]></literal>
    <justification jstring="[back_rewrite(24303),rewrite([26205(5,R),26201(5),26542(4)]),flip(a)].">
      <j1 rule="back_rewrite" parents="24303"/>
      <j2 rule="rewrite" parents="26205 26201 26542"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="26612">
    <literal><![CDATA[
      x v (y v z) = y v (x v z)
    ]]></literal>
    <justification jstring="[back_rewrite(24203),rewrite([26205(5),26606(4),26606(5),26107(4),26606(2),3192(3)])].">
      <j1 rule="back_rewrite" parents="24203"/>
      <j2 rule="rewrite" parents="26205 26606 26606 26107 26606 3192"/>
    </justification>
  </clause>

  <clause id="26613">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(combined)
    ]]></attribute>
    <justification jstring="[resolve(26612,a,4941,a)].">
      <j1 rule="resolve" parents="26612 4941"/>
    </justification>
  </clause>

</proof>

<proof number="4" length="185" max_count="77">

<comments><![CDATA[
% Proof 4 at 16.42 (+ 0.10) seconds: assoc.
% Length of proof is 185.
% Level of proof is 36.
% Maximum clause weight is 77.
% Given clauses 648.
]]></comments>

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

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

  <clause id="6" type="assumption">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(definition_join)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <justification jstring="[copy(6),flip(a)].">
      <j1 rule="copy" parents="6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="8" type="assumption">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(definition_meet)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      f(f(x,y),f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[copy(8),flip(a)].">
      <j1 rule="copy" parents="8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10" type="assumption">
    <literal><![CDATA[
      x' = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(definition_complementation)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      f(x,x) = x'
    ]]></literal>
    <justification jstring="[copy(10),flip(a)].">
      <j1 rule="copy" parents="10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="12" type="deny">
    <literal><![CDATA[
      f(c1,f(f(c2,c3),f(c2,c3))) != f(c2,f(f(c1,c3),f(c1,c3)))
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc)
    ]]></attribute>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      f(c2,f(c1,c3)') != f(c1,f(c2,c3)')
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc)
    ]]></attribute>
    <justification jstring="[copy(12),rewrite([11(8),11(14)]),flip(a)].">
      <j1 rule="copy" parents="12"/>
      <j2 rule="rewrite" parents="11 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      f(x,y)' = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(9),rewrite([11(3)])].">
      <j1 rule="back_rewrite" parents="9"/>
      <j2 rule="rewrite" parents="11"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(x',y') = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(7),rewrite([11(1),11(2)])].">
      <j1 rule="back_rewrite" parents="7"/>
      <j2 rule="rewrite" parents="11 11"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(y,f(x',x)),z))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(5),rewrite([11(5)])].">
      <j1 rule="back_rewrite" parents="5"/>
      <j2 rule="rewrite" parents="11"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      f(c2,c1 ^ c3) != f(c1,c2 ^ c3)
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc)
    ]]></attribute>
    <justification jstring="[back_rewrite(13),rewrite([20(5),20(10)])].">
      <j1 rule="back_rewrite" parents="13"/>
      <j2 rule="rewrite" parents="20 20"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      x ^ x = x''
    ]]></literal>
    <justification jstring="[para(11(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="11 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      (x v y)' = x' ^ y'
    ]]></literal>
    <justification jstring="[para(21(a,1),20(a,1,1))].">
      <j1 rule="para" parents="21 20"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      f(x ^ y,z') = f(x,y) v z
    ]]></literal>
    <justification jstring="[para(20(a,1),21(a,1,1))].">
      <j1 rule="para" parents="20 21"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      f(x',y ^ z) = x v f(y,z)
    ]]></literal>
    <justification jstring="[para(20(a,1),21(a,1,2))].">
      <j1 rule="para" parents="20 21"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      f(f(f(x',f(x,y)),z),f(x,f(f(x,f(x',x)),y))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),22(a,1,1,1,1))].">
      <j1 rule="para" parents="11 22"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      f(f(f(f(x,y),y'),z),f(y,f(f(y,f(x',x)),y))) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),22(a,1,1,1,2))].">
      <j1 rule="para" parents="11 22"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      f(f(x'',y),f(x,f(f(x,f(x',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),22(a,1,1,1)),rewrite([11(1)])].">
      <j1 rule="para" parents="11 22"/>
      <j2 rule="rewrite" parents="11"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      f(f(x,y) ^ f(y,z),f(y,f(f(y,f(x',x)),z))) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),22(a,1,1)),rewrite([20(4)])].">
      <j1 rule="para" parents="11 22"/>
      <j2 rule="rewrite" parents="20"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      f(f(f(f(x,f(x',x)),f(f(x',x),y)),z),f(f(x',x),f(x' ^ x,y))) = f(x',x)
    ]]></literal>
    <justification jstring="[para(11(a,1),22(a,1,2,2,1)),rewrite([20(13)])].">
      <j1 rule="para" parents="11 22"/>
      <j2 rule="rewrite" parents="20"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,f(y,f(x',x)))),z),f(y,y ^ f(x',x))) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),22(a,1,2,2)),rewrite([20(11)])].">
      <j1 rule="para" parents="11 22"/>
      <j2 rule="rewrite" parents="20"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      f(f(f(x,y),f(y,z)),u) ^ f(y,f(f(y,f(x',x)),z)) = y'
    ]]></literal>
    <justification jstring="[para(22(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="22 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      f(f(f(x v y,f(y',z)),u),f(y',f(f(y',x' v x),z))) = y'
    ]]></literal>
    <justification jstring="[para(21(a,1),22(a,1,1,1,1)),rewrite([21(11)])].">
      <j1 rule="para" parents="21 22"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(x,f(f(z,x) ^ f(x,u),f(f(z,x),f(x,u)))),f(f(x,f(z',z)),u)))) = x
    ]]></literal>
    <justification jstring="[para(22(a,1),22(a,1,1,1)),rewrite([20(5)])].">
      <j1 rule="para" parents="22 22"/>
      <j2 rule="rewrite" parents="20"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      f(x' ^ f(x,y),f(x,f(f(x,f(x',x)),y))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),30(a,1,1)),rewrite([20(4)])].">
      <j1 rule="para" parents="11 30"/>
      <j2 rule="rewrite" parents="20"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(f(x,y),x' v x),z))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(30(a,1),22(a,1,1)),rewrite([21(6)])].">
      <j1 rule="para" parents="30 22"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      f(f(f((x ^ y)',f(x,y) v z),u),f(x ^ y,f(f(x ^ y,(x ^ y) v f(x,y)),z'))) = x ^ y
    ]]></literal>
    <justification jstring="[para(28(a,1),30(a,1,1,1,2)),rewrite([29(12)])].">
      <j1 rule="para" parents="28 30"/>
      <j2 rule="rewrite" parents="29"/>
    </justification>
  </clause>

  <clause id="75">
    <literal><![CDATA[
      f(x''',f(x,f(f(x,f(x',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),32(a,1,1))].">
      <j1 rule="para" parents="11 32"/>
    </justification>
  </clause>

  <clause id="76">
    <literal><![CDATA[
      f(x'',y) ^ f(x,f(f(x,f(x',x)),x)) = x'
    ]]></literal>
    <justification jstring="[para(32(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="32 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="78">
    <literal><![CDATA[
      f(x' v y,f(x,f(f(x,f(x',x)),x))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),32(a,1,1))].">
      <j1 rule="para" parents="21 32"/>
    </justification>
  </clause>

  <clause id="87">
    <literal><![CDATA[
      f(x,f(x',f(f(x',x' v x),y))) = x'
    ]]></literal>
    <justification jstring="[para(32(a,1),30(a,1,1)),rewrite([21(6)])].">
      <j1 rule="para" parents="32 30"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      x''' ^ f(x,f(f(x,f(x',x)),x)) = x'
    ]]></literal>
    <justification jstring="[para(75(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="75 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="97">
    <literal><![CDATA[
      (x' v y) ^ f(x,f(f(x,f(x',x)),x)) = x'
    ]]></literal>
    <justification jstring="[para(78(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="78 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="105">
    <literal><![CDATA[
      f(f(x,y) ^ y',f(y,f(f(y,f(x',x)),y))) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),31(a,1,1)),rewrite([20(4)])].">
      <j1 rule="para" parents="11 31"/>
      <j2 rule="rewrite" parents="20"/>
    </justification>
  </clause>

  <clause id="107">
    <literal><![CDATA[
      f(f(f(x,y),y'),z) ^ f(y,f(f(y,f(x',x)),y)) = y'
    ]]></literal>
    <justification jstring="[para(31(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="31 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="132">
    <literal><![CDATA[
      x ^ f(x',f(f(x',x' v x),y)) = x''
    ]]></literal>
    <justification jstring="[para(87(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="87 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="140">
    <literal><![CDATA[
      f(x',f(x'',x)) = x''
    ]]></literal>
    <justification jstring="[para(32(a,1),87(a,1,2,2))].">
      <j1 rule="para" parents="32 87"/>
    </justification>
  </clause>

  <clause id="144">
    <literal><![CDATA[
      f(x'',x'' v x) = x'''
    ]]></literal>
    <justification jstring="[para(21(a,1),140(a,1,2))].">
      <j1 rule="para" parents="21 140"/>
    </justification>
  </clause>

  <clause id="166">
    <literal><![CDATA[
      (f(x,y) ^ f(y,z)) ^ f(y,f(f(y,f(x',x)),z)) = y'
    ]]></literal>
    <justification jstring="[para(33(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="33 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="256">
    <literal><![CDATA[
      (x'' v y) ^ f(x',f(f(x',x' v x),x')) = x''
    ]]></literal>
    <justification jstring="[para(21(a,1),97(a,1,2,2,1,2))].">
      <j1 rule="para" parents="21 97"/>
    </justification>
  </clause>

  <clause id="281">
    <literal><![CDATA[
      f(x,f(f(x,y),f(x,y) ^ (x' v x))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(30(a,1),35(a,1,1)),rewrite([21(6)])].">
      <j1 rule="para" parents="30 35"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="346">
    <literal><![CDATA[
      f(x,f(x',x' v x)) = x'
    ]]></literal>
    <justification jstring="[para(281(a,1),87(a,1,2))].">
      <j1 rule="para" parents="281 87"/>
    </justification>
  </clause>

  <clause id="357">
    <literal><![CDATA[
      f(f(x,y),f(x ^ y,(x ^ y) v f(x,y))) = x ^ y
    ]]></literal>
    <justification jstring="[para(20(a,1),346(a,1,2,1)),rewrite([20(4),20(9)])].">
      <j1 rule="para" parents="20 346"/>
      <j2 rule="rewrite" parents="20 20"/>
    </justification>
  </clause>

  <clause id="380">
    <literal><![CDATA[
      f(x,y) ^ f(x,f(f(x,f(f(z,x) ^ f(x,u),f(f(z,x),f(x,u)))),f(f(x,f(z',z)),u))) = x'
    ]]></literal>
    <justification jstring="[para(22(a,1),36(a,1,1,1)),rewrite([20(5)])].">
      <j1 rule="para" parents="22 36"/>
      <j2 rule="rewrite" parents="20"/>
    </justification>
  </clause>

  <clause id="392">
    <literal><![CDATA[
      x ^ f(f(x,y),f(f(f(x,y),x' v x),z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(30(a,1),36(a,1,1)),rewrite([21(6),20(10)])].">
      <j1 rule="para" parents="30 36"/>
      <j2 rule="rewrite" parents="21 20"/>
    </justification>
  </clause>

  <clause id="521">
    <literal><![CDATA[
      f(x''',y) ^ f(x',f(f(x',x' v x),x')) = x''
    ]]></literal>
    <justification jstring="[para(21(a,1),76(a,1,2,2,1,2))].">
      <j1 rule="para" parents="21 76"/>
    </justification>
  </clause>

  <clause id="673">
    <literal><![CDATA[
      f(x',f(f(x',f(x,y)),x)) = f(x',f(x,y))
    ]]></literal>
    <justification jstring="[para(30(a,1),62(a,1,2,2))].">
      <j1 rule="para" parents="30 62"/>
    </justification>
  </clause>

  <clause id="679">
    <literal><![CDATA[
      f(f(x,y),f(f(f(x,y),y'),y)) = f(f(x,y),y')
    ]]></literal>
    <justification jstring="[para(31(a,1),62(a,1,2,2))].">
      <j1 rule="para" parents="31 62"/>
    </justification>
  </clause>

  <clause id="689">
    <literal><![CDATA[
      f(x,f(f(x,y),x' v x)) = f(x,y)
    ]]></literal>
    <justification jstring="[para(281(a,1),62(a,1,2))].">
      <j1 rule="para" parents="281 62"/>
    </justification>
  </clause>

  <clause id="770">
    <literal><![CDATA[
      f(x''',f(x,x'''' v x''')) = x
    ]]></literal>
    <justification jstring="[para(75(a,1),689(a,1,2,1)),rewrite([75(22)])].">
      <j1 rule="para" parents="75 689"/>
      <j2 rule="rewrite" parents="75"/>
    </justification>
  </clause>

  <clause id="832">
    <literal><![CDATA[
      x' ^ f(f(x',f(x,y)),x) = x' ^ f(x,y)
    ]]></literal>
    <justification jstring="[para(673(a,1),20(a,1,1)),rewrite([20(4)]),flip(a)].">
      <j1 rule="para" parents="673 20"/>
      <j2 rule="rewrite" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="834">
    <literal><![CDATA[
      f(x'',f(f(x'',x v y),x')) = f(x'',x v y)
    ]]></literal>
    <justification jstring="[para(21(a,1),673(a,1,2,1,2)),rewrite([21(14)])].">
      <j1 rule="para" parents="21 673"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="863">
    <literal><![CDATA[
      f((f(x,y) ^ y')',f(f((f(x,y) ^ y')',y),f(x,y) ^ y')) = f((f(x,y) ^ y')',y)
    ]]></literal>
    <justification jstring="[para(105(a,1),673(a,1,2,1,2)),rewrite([105(27)])].">
      <j1 rule="para" parents="105 673"/>
      <j2 rule="rewrite" parents="105"/>
    </justification>
  </clause>

  <clause id="920">
    <literal><![CDATA[
      x'' ^ f(f(x'',x v y),x') = x'' ^ (x v y)
    ]]></literal>
    <justification jstring="[para(21(a,1),832(a,1,2,1,2)),rewrite([21(14)])].">
      <j1 rule="para" parents="21 832"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="930">
    <literal><![CDATA[
      (x' ^ f(x,y))' ^ f(f((x' ^ f(x,y))',x),x' ^ f(x,y)) = (x' ^ f(x,y))' ^ x
    ]]></literal>
    <justification jstring="[para(53(a,1),832(a,1,2,1,2)),rewrite([53(27)])].">
      <j1 rule="para" parents="53 832"/>
      <j2 rule="rewrite" parents="53"/>
    </justification>
  </clause>

  <clause id="934">
    <literal><![CDATA[
      f(x,y) ^ f(f(f(x,y),y'),y) = f(x,y) ^ y'
    ]]></literal>
    <justification jstring="[para(679(a,1),20(a,1,1)),rewrite([20(4)]),flip(a)].">
      <j1 rule="para" parents="679 20"/>
      <j2 rule="rewrite" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="988">
    <literal><![CDATA[
      f(f(f(x,x ^ (x'''' v x''')),y),f(f(x,x'''' v x'''),f(f(f(x,x'''' v x'''),x''' v x''),f(x,x'''' v x''')))) = f(x,x'''' v x''')
    ]]></literal>
    <justification jstring="[para(770(a,1),31(a,1,1,1,1)),rewrite([20(10),21(37)])].">
      <j1 rule="para" parents="770 31"/>
      <j2 rule="rewrite" parents="20 21"/>
    </justification>
  </clause>

  <clause id="1140">
    <literal><![CDATA[
      x'' ^ f(x' v f(x'',x),f(x'',x)) = x'' ^ (x'' ^ x)
    ]]></literal>
    <justification jstring="[para(140(a,1),934(a,1,1)),rewrite([140(7),20(8),29(8),140(17),20(18)])].">
      <j1 rule="para" parents="140 934"/>
      <j2 rule="rewrite" parents="140 20 29 140 20"/>
    </justification>
  </clause>

  <clause id="1606">
    <literal><![CDATA[
      f(x,x'') = x'
    ]]></literal>
    <justification jstring="[para(43(a,1),87(a,1,2,2)),rewrite([11(3)])].">
      <j1 rule="para" parents="43 87"/>
      <j2 rule="rewrite" parents="11"/>
    </justification>
  </clause>

  <clause id="1625">
    <literal><![CDATA[
      x ^ x'' = x''
    ]]></literal>
    <justification jstring="[para(43(a,1),132(a,1,2,2)),rewrite([11(3)])].">
      <j1 rule="para" parents="43 132"/>
      <j2 rule="rewrite" parents="11"/>
    </justification>
  </clause>

  <clause id="1626">
    <literal><![CDATA[
      f(f(x,y),f(y,y ^ f(x',x))) = y
    ]]></literal>
    <justification jstring="[para(43(a,1),35(a,1,1))].">
      <j1 rule="para" parents="43 35"/>
    </justification>
  </clause>

  <clause id="1658">
    <literal><![CDATA[
      f(x,x ^ y) = f(x,y)
    ]]></literal>
    <justification jstring="[para(43(a,1),62(a,1,2,2)),rewrite([11(3),20(2)])].">
      <j1 rule="para" parents="43 62"/>
      <j2 rule="rewrite" parents="11 20"/>
    </justification>
  </clause>

  <clause id="1677">
    <literal><![CDATA[
      (x ^ y) ^ f(f(x ^ y,x),f(x,y)) = (x ^ y) ^ x
    ]]></literal>
    <justification jstring="[para(43(a,1),832(a,1,2,1,2)),rewrite([20(2),20(3),20(8),43(23)])].">
      <j1 rule="para" parents="43 832"/>
      <j2 rule="rewrite" parents="20 20 20 43"/>
    </justification>
  </clause>

  <clause id="1704">
    <literal><![CDATA[
      x ^ (x ^ y) = x ^ y
    ]]></literal>
    <justification jstring="[para(43(a,1),392(a,1,2,2)),rewrite([11(3),20(2)])].">
      <j1 rule="para" parents="43 392"/>
      <j2 rule="rewrite" parents="11 20"/>
    </justification>
  </clause>

  <clause id="1717">
    <literal><![CDATA[
      f(f(x,y),f(y,f(x',x))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1626),rewrite([1658(5)])].">
      <j1 rule="back_rewrite" parents="1626"/>
      <j2 rule="rewrite" parents="1658"/>
    </justification>
  </clause>

  <clause id="1758">
    <literal><![CDATA[
      f(f(f(x,x'''' v x'''),y),f(f(x,x'''' v x'''),f(f(f(x,x'''' v x'''),x''' v x''),f(x,x'''' v x''')))) = f(x,x'''' v x''')
    ]]></literal>
    <justification jstring="[back_rewrite(988),rewrite([1658(10)])].">
      <j1 rule="back_rewrite" parents="988"/>
      <j2 rule="rewrite" parents="1658"/>
    </justification>
  </clause>

  <clause id="1903">
    <literal><![CDATA[
      x'' ^ f(x' v f(x'',x),f(x'',x)) = x'' ^ x
    ]]></literal>
    <justification jstring="[back_rewrite(1140),rewrite([1704(18)])].">
      <j1 rule="back_rewrite" parents="1140"/>
      <j2 rule="rewrite" parents="1704"/>
    </justification>
  </clause>

  <clause id="1943">
    <literal><![CDATA[
      x v x'' = x''
    ]]></literal>
    <justification jstring="[para(1606(a,1),21(a,1)),flip(a)].">
      <j1 rule="para" parents="1606 21"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2013">
    <literal><![CDATA[
      f(f(x'',y),x') = x
    ]]></literal>
    <justification jstring="[para(32(a,1),1658(a,2)),rewrite([76(12)])].">
      <j1 rule="para" parents="32 1658"/>
      <j2 rule="rewrite" parents="76"/>
    </justification>
  </clause>

  <clause id="2014">
    <literal><![CDATA[
      x'' v x = x
    ]]></literal>
    <justification jstring="[para(75(a,1),1658(a,2)),rewrite([89(12),21(5)])].">
      <j1 rule="para" parents="75 1658"/>
      <j2 rule="rewrite" parents="89 21"/>
    </justification>
  </clause>

  <clause id="2017">
    <literal><![CDATA[
      f(f(f(f(x,y),y'),z),y') = y
    ]]></literal>
    <justification jstring="[para(31(a,1),1658(a,2)),rewrite([107(14)])].">
      <j1 rule="para" parents="31 1658"/>
      <j2 rule="rewrite" parents="107"/>
    </justification>
  </clause>

  <clause id="2021">
    <literal><![CDATA[
      f(f(x,y),f(y,z)) v y = y
    ]]></literal>
    <justification jstring="[para(33(a,1),1658(a,2)),rewrite([166(12),28(5)])].">
      <j1 rule="para" parents="33 1658"/>
      <j2 rule="rewrite" parents="166 28"/>
    </justification>
  </clause>

  <clause id="2022">
    <literal><![CDATA[
      f(x'',x) = x'''
    ]]></literal>
    <justification jstring="[para(144(a,1),1658(a,2)),rewrite([2014(7),1658(6)])].">
      <j1 rule="para" parents="144 1658"/>
      <j2 rule="rewrite" parents="2014 1658"/>
    </justification>
  </clause>

  <clause id="2063">
    <literal><![CDATA[
      f(f(x,y),x') = x
    ]]></literal>
    <justification jstring="[para(43(a,1),1658(a,2)),rewrite([380(17)])].">
      <j1 rule="para" parents="43 1658"/>
      <j2 rule="rewrite" parents="380"/>
    </justification>
  </clause>

  <clause id="2064">
    <literal><![CDATA[
      x'' ^ x = x''''
    ]]></literal>
    <justification jstring="[back_rewrite(1903),rewrite([2022(6),1943(7),2022(8),11(9),1625(7)]),flip(a)].">
      <j1 rule="back_rewrite" parents="1903"/>
      <j2 rule="rewrite" parents="2022 1943 2022 11 1625"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2083">
    <literal><![CDATA[
      x'' ^ (x v y) = x''''
    ]]></literal>
    <justification jstring="[back_rewrite(920),rewrite([2013(8),2064(3)]),flip(a)].">
      <j1 rule="back_rewrite" parents="920"/>
      <j2 rule="rewrite" parents="2013 2064"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2085">
    <literal><![CDATA[
      f(x'',x v y) = x'''
    ]]></literal>
    <justification jstring="[back_rewrite(834),rewrite([2013(8),2022(3)]),flip(a)].">
      <j1 rule="back_rewrite" parents="834"/>
      <j2 rule="rewrite" parents="2013 2022"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2291">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[para(11(a,1),2063(a,1,1)),rewrite([11(3)])].">
      <j1 rule="para" parents="11 2063"/>
      <j2 rule="rewrite" parents="11"/>
    </justification>
  </clause>

  <clause id="2292">
    <literal><![CDATA[
      f(x,y) ^ x' = x'
    ]]></literal>
    <justification jstring="[para(2063(a,1),20(a,1,1)),flip(a)].">
      <j1 rule="para" parents="2063 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2294">
    <literal><![CDATA[
      f(x v y,x) = x'
    ]]></literal>
    <justification jstring="[para(21(a,1),2063(a,1,1)),rewrite([2291(3)])].">
      <j1 rule="para" parents="21 2063"/>
      <j2 rule="rewrite" parents="2291"/>
    </justification>
  </clause>

  <clause id="2302">
    <literal><![CDATA[
      f(x,y') = x' v y
    ]]></literal>
    <justification jstring="[para(78(a,1),2063(a,1,1)),rewrite([27(3),2291(2),1658(3)])].">
      <j1 rule="para" parents="78 2063"/>
      <j2 rule="rewrite" parents="27 2291 1658"/>
    </justification>
  </clause>

  <clause id="2310">
    <literal><![CDATA[
      f(f((x ^ f(x',x)) v x,y),f(f(x',x),f(x' ^ x,x))) = f(x',x)
    ]]></literal>
    <justification jstring="[para(2063(a,1),34(a,1,1,1,2)),rewrite([2302(5),20(4),2291(11)])].">
      <j1 rule="para" parents="2063 34"/>
      <j2 rule="rewrite" parents="2302 20 2291"/>
    </justification>
  </clause>

  <clause id="2325">
    <literal><![CDATA[
      (x ^ y)' = f(x,y)
    ]]></literal>
    <justification jstring="[para(357(a,1),2063(a,1,1)),rewrite([20(3),11(3)])].">
      <j1 rule="para" parents="357 2063"/>
      <j2 rule="rewrite" parents="20 11"/>
    </justification>
  </clause>

  <clause id="2329">
    <literal><![CDATA[
      (x ^ y) v x = x
    ]]></literal>
    <justification jstring="[para(62(a,1),2063(a,1,1)),rewrite([2302(3),20(2)])].">
      <j1 rule="para" parents="62 2063"/>
      <j2 rule="rewrite" parents="2302 20"/>
    </justification>
  </clause>

  <clause id="2426">
    <literal><![CDATA[
      f(x,x v y) = x'
    ]]></literal>
    <justification jstring="[back_rewrite(2085),rewrite([2291(2),2291(4)])].">
      <j1 rule="back_rewrite" parents="2085"/>
      <j2 rule="rewrite" parents="2291 2291"/>
    </justification>
  </clause>

  <clause id="2427">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[back_rewrite(2083),rewrite([2291(2),2291(4),2291(4)])].">
      <j1 rule="back_rewrite" parents="2083"/>
      <j2 rule="rewrite" parents="2291 2291 2291"/>
    </justification>
  </clause>

  <clause id="2451">
    <literal><![CDATA[
      f(f(x',y),x) = x'
    ]]></literal>
    <justification jstring="[back_rewrite(1758),rewrite([2291(2),2291(2),2291(2),2426(3),2291(4),2291(4),2291(4),2426(5),2291(5),2291(5),2291(5),2426(6),2291(6),2291(7),2426(7),2291(5),2291(5),2291(5),2291(5),2426(6),2302(5),2426(6),2291(4),2291(5),2291(5),2291(5),2426(6)])].">
      <j1 rule="back_rewrite" parents="1758"/>
      <j2 rule="rewrite" parents="2291 2291 2291 2426 2291 2291 2291 2426 2291 2291 2291 2426 2291 2291 2426 2291 2291 2291 2291 2426 2302 2426 2291 2291 2291 2291 2426"/>
    </justification>
  </clause>

  <clause id="2564">
    <literal><![CDATA[
      f(x',y) ^ x = x
    ]]></literal>
    <justification jstring="[back_rewrite(521),rewrite([2291(2),2426(7),2291(5),2302(5),2426(6),2291(4),2291(5)])].">
      <j1 rule="back_rewrite" parents="521"/>
      <j2 rule="rewrite" parents="2291 2426 2291 2302 2426 2291 2291"/>
    </justification>
  </clause>

  <clause id="2579">
    <literal><![CDATA[
      (x v y) ^ x = x
    ]]></literal>
    <justi