<?xml version="1.0" encoding="ISO-8859-1"?>

<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="14">

<source>oml6.out</source>

<heading><![CDATA[
Prover9 (32) version 2009-02A, February 2009.
Process 11188 was started by mccune on cleo,
Wed Feb 25 09:32:09 2009
The command was "/home/mccune/bin/prover9 -f oml6.in".
]]></heading>

<proof number="1" length="17" max_count="37">

<comments><![CDATA[
% Proof 1 at 0.20 (+ 0.02) seconds: cc.
% Length of proof is 17.
% Level of proof is 6.
% Maximum clause weight is 37.
% Given clauses 74.

]]></comments>

  <clause id="8" type="goal">
    <literal><![CDATA[
      c(c(x)) = x
    ]]></literal>
    <attribute><![CDATA[
      answer(cc)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="15" type="assumption">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="16" type="assumption">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <justification jstring="[copy(16),flip(a)].">
      <j1 rule="copy" parents="16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20" type="assumption">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(x,x) = c(x)
    ]]></literal>
    <justification jstring="[copy(20),flip(a)].">
      <j1 rule="copy" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="31" type="deny">
    <literal><![CDATA[
      c(c(c14)) != c14
    ]]></literal>
    <attribute><![CDATA[
      answer(cc)
    ]]></attribute>
    <justification jstring="[deny(8)].">
      <j1 rule="deny" parents="8"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      f(c(x),c(y)) = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(17),rewrite([21(1),21(2)])].">
      <j1 rule="back_rewrite" parents="17"/>
      <j2 rule="rewrite" parents="21 21"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(15),rewrite([21(5)])].">
      <j1 rule="back_rewrite" parents="15"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      c(c(x)) = x v x
    ]]></literal>
    <justification jstring="[para(41(a,1),21(a,1)),flip(a)].">
      <j1 rule="para" parents="41 21"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="51">
    <literal><![CDATA[
      c14 v c14 != c14
    ]]></literal>
    <attribute><![CDATA[
      answer(cc)
    ]]></attribute>
    <justification jstring="[back_rewrite(31),rewrite([45(3)])].">
      <j1 rule="back_rewrite" parents="31"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      f(f(f(c(x),f(x,y)),z),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1,1))].">
      <j1 rule="para" parents="21 42"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(f(f(z,f(c(x),z)),z),f(c(x),f(f(z,f(c(x),z)),z))),f(f(z,f(c(x),z)),z)))) = x
    ]]></literal>
    <justification jstring="[para(42(a,1),42(a,1,1,1))].">
      <j1 rule="para" parents="42 42"/>
    </justification>
  </clause>

  <clause id="1623">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(62(a,1),42(a,1,1))].">
      <j1 rule="para" parents="62 42"/>
    </justification>
  </clause>

  <clause id="1625">
    <literal><![CDATA[
      f(c(x),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(62(a,1),52(a,1,1))].">
      <j1 rule="para" parents="62 52"/>
    </justification>
  </clause>

  <clause id="1739">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(1625(a,1),1625(a,1,2,2,1,2)),rewrite([1623(13),21(2),21(3),45(2)])].">
      <j1 rule="para" parents="1625 1625"/>
      <j2 rule="rewrite" parents="1623 21 21 45"/>
    </justification>
  </clause>

  <clause id="1740">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(cc)
    ]]></attribute>
    <justification jstring="[resolve(1739,a,51,a)].">
      <j1 rule="resolve" parents="1739 51"/>
    </justification>
  </clause>

</proof>

<proof number="2" length="22" max_count="37">

<comments><![CDATA[
% Proof 2 at 0.20 (+ 0.02) seconds: idempotence_meet.
% Length of proof is 22.
% Level of proof is 6.
% Maximum clause weight is 37.
% Given clauses 74.

]]></comments>

  <clause id="7" type="goal">
    <literal><![CDATA[
      x ^ x = x
    ]]></literal>
    <attribute><![CDATA[
      answer(idempotence_meet)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="15" type="assumption">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="16" type="assumption">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <justification jstring="[copy(16),flip(a)].">
      <j1 rule="copy" parents="16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18" type="assumption">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_meet)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(f(x,y),f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[copy(18),flip(a)].">
      <j1 rule="copy" parents="18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20" type="assumption">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(x,x) = c(x)
    ]]></literal>
    <justification jstring="[copy(20),flip(a)].">
      <j1 rule="copy" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="30" type="deny">
    <literal><![CDATA[
      c13 ^ c13 != c13
    ]]></literal>
    <attribute><![CDATA[
      answer(idempotence_meet)
    ]]></attribute>
    <justification jstring="[deny(7)].">
      <j1 rule="deny" parents="7"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      c(f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(19),rewrite([21(3)])].">
      <j1 rule="back_rewrite" parents="19"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      f(c(x),c(y)) = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(17),rewrite([21(1),21(2)])].">
      <j1 rule="back_rewrite" parents="17"/>
      <j2 rule="rewrite" parents="21 21"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(15),rewrite([21(5)])].">
      <j1 rule="back_rewrite" parents="15"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      x ^ x = c(c(x))
    ]]></literal>
    <justification jstring="[para(21(a,1),40(a,1,1)),flip(a)].">
      <j1 rule="para" parents="21 40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      c(c(c13)) != c13
    ]]></literal>
    <attribute><![CDATA[
      answer(idempotence_meet)
    ]]></attribute>
    <justification jstring="[back_rewrite(30),rewrite([43(3)])].">
      <j1 rule="back_rewrite" parents="30"/>
      <j2 rule="rewrite" parents="43"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      c(c(x)) = x v x
    ]]></literal>
    <justification jstring="[para(41(a,1),21(a,1)),flip(a)].">
      <j1 rule="para" parents="41 21"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      c13 v c13 != c13
    ]]></literal>
    <attribute><![CDATA[
      answer(idempotence_meet)
    ]]></attribute>
    <justification jstring="[back_rewrite(44),rewrite([45(3)])].">
      <j1 rule="back_rewrite" parents="44"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      f(f(f(c(x),f(x,y)),z),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1,1))].">
      <j1 rule="para" parents="21 42"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(f(f(z,f(c(x),z)),z),f(c(x),f(f(z,f(c(x),z)),z))),f(f(z,f(c(x),z)),z)))) = x
    ]]></literal>
    <justification jstring="[para(42(a,1),42(a,1,1,1))].">
      <j1 rule="para" parents="42 42"/>
    </justification>
  </clause>

  <clause id="1623">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(62(a,1),42(a,1,1))].">
      <j1 rule="para" parents="62 42"/>
    </justification>
  </clause>

  <clause id="1625">
    <literal><![CDATA[
      f(c(x),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(62(a,1),52(a,1,1))].">
      <j1 rule="para" parents="62 52"/>
    </justification>
  </clause>

  <clause id="1739">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(1625(a,1),1625(a,1,2,2,1,2)),rewrite([1623(13),21(2),21(3),45(2)])].">
      <j1 rule="para" parents="1625 1625"/>
      <j2 rule="rewrite" parents="1623 21 21 45"/>
    </justification>
  </clause>

  <clause id="1741">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(idempotence_meet)
    ]]></attribute>
    <justification jstring="[resolve(1739,a,49,a)].">
      <j1 rule="resolve" parents="1739 49"/>
    </justification>
  </clause>

</proof>

<proof number="3" length="16" max_count="37">

<comments><![CDATA[
% Proof 3 at 0.20 (+ 0.02) seconds: idempotence_join.
% Length of proof is 16.
% Level of proof is 6.
% Maximum clause weight is 37.
% Given clauses 74.

]]></comments>

  <clause id="6" type="goal">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <attribute><![CDATA[
      answer(idempotence_join)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="15" type="assumption">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="16" type="assumption">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <justification jstring="[copy(16),flip(a)].">
      <j1 rule="copy" parents="16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20" type="assumption">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(x,x) = c(x)
    ]]></literal>
    <justification jstring="[copy(20),flip(a)].">
      <j1 rule="copy" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="29" type="deny">
    <literal><![CDATA[
      c12 v c12 != c12
    ]]></literal>
    <attribute><![CDATA[
      answer(idempotence_join)
    ]]></attribute>
    <justification jstring="[deny(6)].">
      <j1 rule="deny" parents="6"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      f(c(x),c(y)) = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(17),rewrite([21(1),21(2)])].">
      <j1 rule="back_rewrite" parents="17"/>
      <j2 rule="rewrite" parents="21 21"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(15),rewrite([21(5)])].">
      <j1 rule="back_rewrite" parents="15"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      c(c(x)) = x v x
    ]]></literal>
    <justification jstring="[para(41(a,1),21(a,1)),flip(a)].">
      <j1 rule="para" parents="41 21"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      f(f(f(c(x),f(x,y)),z),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1,1))].">
      <j1 rule="para" parents="21 42"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(f(f(z,f(c(x),z)),z),f(c(x),f(f(z,f(c(x),z)),z))),f(f(z,f(c(x),z)),z)))) = x
    ]]></literal>
    <justification jstring="[para(42(a,1),42(a,1,1,1))].">
      <j1 rule="para" parents="42 42"/>
    </justification>
  </clause>

  <clause id="1623">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(62(a,1),42(a,1,1))].">
      <j1 rule="para" parents="62 42"/>
    </justification>
  </clause>

  <clause id="1625">
    <literal><![CDATA[
      f(c(x),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(62(a,1),52(a,1,1))].">
      <j1 rule="para" parents="62 52"/>
    </justification>
  </clause>

  <clause id="1739">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(1625(a,1),1625(a,1,2,2,1,2)),rewrite([1623(13),21(2),21(3),45(2)])].">
      <j1 rule="para" parents="1625 1625"/>
      <j2 rule="rewrite" parents="1623 21 21 45"/>
    </justification>
  </clause>

  <clause id="1742">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(idempotence_join)
    ]]></attribute>
    <justification jstring="[resolve(1739,a,29,a)].">
      <j1 rule="resolve" parents="1739 29"/>
    </justification>
  </clause>

</proof>

<proof number="4" length="24" max_count="37">

<comments><![CDATA[
% Proof 4 at 0.29 (+ 0.02) seconds: DM.
% Length of proof is 24.
% Level of proof is 7.
% Maximum clause weight is 37.
% Given clauses 74.

]]></comments>

  <clause id="4" type="goal">
    <literal><![CDATA[
      x ^ y = c(c(x) v c(y))
    ]]></literal>
    <attribute><![CDATA[
      answer(DM)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="15" type="assumption">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="16" type="assumption">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <justification jstring="[copy(16),flip(a)].">
      <j1 rule="copy" parents="16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18" type="assumption">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_meet)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(f(x,y),f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[copy(18),flip(a)].">
      <j1 rule="copy" parents="18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20" type="assumption">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(x,x) = c(x)
    ]]></literal>
    <justification jstring="[copy(20),flip(a)].">
      <j1 rule="copy" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="26" type="deny">
    <literal><![CDATA[
      c(c(c8) v c(c9)) != c8 ^ c9
    ]]></literal>
    <attribute><![CDATA[
      answer(DM)
    ]]></attribute>
    <justification jstring="[deny(4)].">
      <j1 rule="deny" parents="4"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      c8 ^ c9 != c(c(c8) v c(c9))
    ]]></literal>
    <attribute><![CDATA[
      answer(DM)
    ]]></attribute>
    <justification jstring="[copy(26),flip(a)].">
      <j1 rule="copy" parents="26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      c(f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(19),rewrite([21(3)])].">
      <j1 rule="back_rewrite" parents="19"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      f(c(x),c(y)) = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(17),rewrite([21(1),21(2)])].">
      <j1 rule="back_rewrite" parents="17"/>
      <j2 rule="rewrite" parents="21 21"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(15),rewrite([21(5)])].">
      <j1 rule="back_rewrite" parents="15"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      c(c(x)) = x v x
    ]]></literal>
    <justification jstring="[para(41(a,1),21(a,1)),flip(a)].">
      <j1 rule="para" parents="41 21"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      c(x) ^ c(y) = c(x v y)
    ]]></literal>
    <justification jstring="[para(41(a,1),40(a,1,1)),flip(a)].">
      <j1 rule="para" parents="41 40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      f(f(f(c(x),f(x,y)),z),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1,1))].">
      <j1 rule="para" parents="21 42"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(f(f(z,f(c(x),z)),z),f(c(x),f(f(z,f(c(x),z)),z))),f(f(z,f(c(x),z)),z)))) = x
    ]]></literal>
    <justification jstring="[para(42(a,1),42(a,1,1,1))].">
      <j1 rule="para" parents="42 42"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      (x v x) ^ c(y) = c(c(x) v y)
    ]]></literal>
    <justification jstring="[para(45(a,1),46(a,1,1))].">
      <j1 rule="para" parents="45 46"/>
    </justification>
  </clause>

  <clause id="95">
    <literal><![CDATA[
      (x v x) ^ (y v y) = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[para(45(a,1),72(a,1,2))].">
      <j1 rule="para" parents="45 72"/>
    </justification>
  </clause>

  <clause id="1623">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(62(a,1),42(a,1,1))].">
      <j1 rule="para" parents="62 42"/>
    </justification>
  </clause>

  <clause id="1625">
    <literal><![CDATA[
      f(c(x),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(62(a,1),52(a,1,1))].">
      <j1 rule="para" parents="62 52"/>
    </justification>
  </clause>

  <clause id="1739">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(1625(a,1),1625(a,1,2,2,1,2)),rewrite([1623(13),21(2),21(3),45(2)])].">
      <j1 rule="para" parents="1625 1625"/>
      <j2 rule="rewrite" parents="1623 21 21 45"/>
    </justification>
  </clause>

  <clause id="2374">
    <literal><![CDATA[
      x ^ y = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[back_rewrite(95),rewrite([1739(1),1739(1)])].">
      <j1 rule="back_rewrite" parents="95"/>
      <j2 rule="rewrite" parents="1739 1739"/>
    </justification>
  </clause>

  <clause id="2375">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(DM)
    ]]></attribute>
    <justification jstring="[resolve(2374,a,27,a)].">
      <j1 rule="resolve" parents="2374 27"/>
    </justification>
  </clause>

</proof>

<proof number="5" length="19" max_count="37">

<comments><![CDATA[
% Proof 5 at 0.29 (+ 0.02) seconds: DEF_SS.
% Length of proof is 19.
% Level of proof is 7.
% Maximum clause weight is 37.
% Given clauses 74.

]]></comments>

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

  <clause id="15" type="assumption">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="16" type="assumption">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <justification jstring="[copy(16),flip(a)].">
      <j1 rule="copy" parents="16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20" type="assumption">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(x,x) = c(x)
    ]]></literal>
    <justification jstring="[copy(20),flip(a)].">
      <j1 rule="copy" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="28" type="deny">
    <literal><![CDATA[
      f(c10,c11) != c(c10) v c(c11)
    ]]></literal>
    <attribute><![CDATA[
      answer(DEF_SS)
    ]]></attribute>
    <justification jstring="[deny(5)].">
      <j1 rule="deny" parents="5"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      f(c(x),c(y)) = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(17),rewrite([21(1),21(2)])].">
      <j1 rule="back_rewrite" parents="17"/>
      <j2 rule="rewrite" parents="21 21"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(15),rewrite([21(5)])].">
      <j1 rule="back_rewrite" parents="15"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      c(c(x)) = x v x
    ]]></literal>
    <justification jstring="[para(41(a,1),21(a,1)),flip(a)].">
      <j1 rule="para" parents="41 21"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      f(f(f(c(x),f(x,y)),z),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1,1))].">
      <j1 rule="para" parents="21 42"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(f(f(z,f(c(x),z)),z),f(c(x),f(f(z,f(c(x),z)),z))),f(f(z,f(c(x),z)),z)))) = x
    ]]></literal>
    <justification jstring="[para(42(a,1),42(a,1,1,1))].">
      <j1 rule="para" parents="42 42"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      f(x v x,c(y)) = c(x) v y
    ]]></literal>
    <justification jstring="[para(45(a,1),41(a,1,1))].">
      <j1 rule="para" parents="45 41"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      f(x v x,y v y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[para(45(a,1),65(a,1,2))].">
      <j1 rule="para" parents="45 65"/>
    </justification>
  </clause>

  <clause id="1623">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(62(a,1),42(a,1,1))].">
      <j1 rule="para" parents="62 42"/>
    </justification>
  </clause>

  <clause id="1625">
    <literal><![CDATA[
      f(c(x),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(62(a,1),52(a,1,1))].">
      <j1 rule="para" parents="62 52"/>
    </justification>
  </clause>

  <clause id="1739">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(1625(a,1),1625(a,1,2,2,1,2)),rewrite([1623(13),21(2),21(3),45(2)])].">
      <j1 rule="para" parents="1625 1625"/>
      <j2 rule="rewrite" parents="1623 21 21 45"/>
    </justification>
  </clause>

  <clause id="2384">
    <literal><![CDATA[
      f(x,y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(77),rewrite([1739(1),1739(1)])].">
      <j1 rule="back_rewrite" parents="77"/>
      <j2 rule="rewrite" parents="1739 1739"/>
    </justification>
  </clause>

  <clause id="2385">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(DEF_SS)
    ]]></attribute>
    <justification jstring="[resolve(2384,a,28,a)].">
      <j1 rule="resolve" parents="2384 28"/>
    </justification>
  </clause>

</proof>

<proof number="6" length="57" max_count="37">

<comments><![CDATA[
% Proof 6 at 0.46 (+ 0.02) seconds: B1.
% Length of proof is 57.
% Level of proof is 15.
% Maximum clause weight is 37.
% Given clauses 97.

]]></comments>

  <clause id="1" type="goal">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <attribute><![CDATA[
      answer(B1)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="15" type="assumption">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="16" type="assumption">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <justification jstring="[copy(16),flip(a)].">
      <j1 rule="copy" parents="16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18" type="assumption">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_meet)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(f(x,y),f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[copy(18),flip(a)].">
      <j1 rule="copy" parents="18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20" type="assumption">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(x,x) = c(x)
    ]]></literal>
    <justification jstring="[copy(20),flip(a)].">
      <j1 rule="copy" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22" type="deny">
    <literal><![CDATA[
      c1 v (c1 ^ c2) != c1
    ]]></literal>
    <attribute><![CDATA[
      answer(B1)
    ]]></attribute>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      c(f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(19),rewrite([21(3)])].">
      <j1 rule="back_rewrite" parents="19"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      f(c(x),c(y)) = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(17),rewrite([21(1),21(2)])].">
      <j1 rule="back_rewrite" parents="17"/>
      <j2 rule="rewrite" parents="21 21"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(15),rewrite([21(5)])].">
      <j1 rule="back_rewrite" parents="15"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      x ^ x = c(c(x))
    ]]></literal>
    <justification jstring="[para(21(a,1),40(a,1,1)),flip(a)].">
      <j1 rule="para" parents="21 40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      c(c(x)) = x v x
    ]]></literal>
    <justification jstring="[para(41(a,1),21(a,1)),flip(a)].">
      <j1 rule="para" parents="41 21"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      c(x) ^ c(y) = c(x v y)
    ]]></literal>
    <justification jstring="[para(41(a,1),40(a,1,1)),flip(a)].">
      <j1 rule="para" parents="41 40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      x ^ x = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(43),rewrite([45(3)])].">
      <j1 rule="back_rewrite" parents="43"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      f(f(f(c(x),f(x,y)),z),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1,1))].">
      <j1 rule="para" parents="21 42"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      f(f(f(f(x,y),c(y)),z),f(y,f(f(y,f(c(y),y)),y))) = y
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1,2))].">
      <j1 rule="para" parents="21 42"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      f(f(x v x,y),f(x,f(f(x,f(c(x),x)),x))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1)),rewrite([21(1),45(2)])].">
      <j1 rule="para" parents="21 42"/>
      <j2 rule="rewrite" parents="21 45"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      f(f(x,y) ^ f(y,z),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1)),rewrite([40(4)])].">
      <j1 rule="para" parents="21 42"/>
      <j2 rule="rewrite" parents="40"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      f(f(f(x v y,f(c(y),z)),u),f(c(y),f(f(z,f(y v y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[para(41(a,1),42(a,1,1,1,1)),rewrite([45(8)])].">
      <j1 rule="para" parents="41 42"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(f(f(z,f(c(x),z)),z),f(c(x),f(f(z,f(c(x),z)),z))),f(f(z,f(c(x),z)),z)))) = x
    ]]></literal>
    <justification jstring="[para(42(a,1),42(a,1,1,1))].">
      <j1 rule="para" parents="42 42"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      f(x v x,c(y)) = c(x) v y
    ]]></literal>
    <justification jstring="[para(45(a,1),41(a,1,1))].">
      <j1 rule="para" parents="45 41"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      (x v x) ^ c(y) = c(c(x) v y)
    ]]></literal>
    <justification jstring="[para(45(a,1),46(a,1,1))].">
      <j1 rule="para" parents="45 46"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      f(x v x,y v y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[para(45(a,1),65(a,1,2))].">
      <j1 rule="para" parents="45 65"/>
    </justification>
  </clause>

  <clause id="95">
    <literal><![CDATA[
      (x v x) ^ (y v y) = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[para(45(a,1),72(a,1,2))].">
      <j1 rule="para" parents="45 72"/>
    </justification>
  </clause>

  <clause id="158">
    <literal><![CDATA[
      f(x,f(c(x),f(f(y,f(x v x,y)),y))) = c(x)
    ]]></literal>
    <justification jstring="[para(53(a,1),42(a,1,1)),rewrite([45(3)])].">
      <j1 rule="para" parents="53 42"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="203">
    <literal><![CDATA[
      f(c(x) v y,f(x,f(f(x,f(c(x),x)),x))) = x
    ]]></literal>
    <justification jstring="[para(65(a,1),54(a,1,1))].">
      <j1 rule="para" parents="65 54"/>
    </justification>
  </clause>

  <clause id="776">
    <literal><![CDATA[
      f((x v x) v y,f(c(x),f(f(c(x),c(x) v x),c(x)))) = c(x)
    ]]></literal>
    <justification jstring="[para(41(a,1),203(a,1,2,2,1,2)),rewrite([45(2)])].">
      <j1 rule="para" parents="41 203"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="785">
    <literal><![CDATA[
      f(c(x),f(x v x,f(f(f(x,f(f(x,f(c(x),x)),x)),x),f(x,f(f(x,f(c(x),x)),x))))) = x v x
    ]]></literal>
    <justification jstring="[para(203(a,1),158(a,1,2,2,1,2)),rewrite([45(3),45(18)])].">
      <j1 rule="para" parents="203 158"/>
      <j2 rule="rewrite" parents="45 45"/>
    </justification>
  </clause>

  <clause id="1623">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(62(a,1),42(a,1,1))].">
      <j1 rule="para" parents="62 42"/>
    </justification>
  </clause>

  <clause id="1625">
    <literal><![CDATA[
      f(c(x),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(62(a,1),52(a,1,1))].">
      <j1 rule="para" parents="62 52"/>
    </justification>
  </clause>

  <clause id="1628">
    <literal><![CDATA[
      f(x v y,f(c(y),f(f(z,f(y v y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[para(62(a,1),59(a,1,1))].">
      <j1 rule="para" parents="62 59"/>
    </justification>
  </clause>

  <clause id="1630">
    <literal><![CDATA[
      f(c(x),f(x v x,x)) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(785),rewrite([1623(14)])].">
      <j1 rule="back_rewrite" parents="785"/>
      <j2 rule="rewrite" parents="1623"/>
    </justification>
  </clause>

  <clause id="1647">
    <literal><![CDATA[
      f((x v x) ^ f(f(x v x,x),y),f(f(x v x,x),f(f(y,f((x v x) ^ x,y)),y))) = f(x v x,x)
    ]]></literal>
    <justification jstring="[para(1630(a,1),55(a,1,1,1)),rewrite([40(10)])].">
      <j1 rule="para" parents="1630 55"/>
      <j2 rule="rewrite" parents="40"/>
    </justification>
  </clause>

  <clause id="1693">
    <literal><![CDATA[
      f(c(x),f(x,f(f(c(y),x v y),c(y)))) = x
    ]]></literal>
    <justification jstring="[para(41(a,1),1625(a,1,2,2,1,2))].">
      <j1 rule="para" parents="41 1625"/>
    </justification>
  </clause>

  <clause id="1739">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(1625(a,1),1625(a,1,2,2,1,2)),rewrite([1623(13),21(2),21(3),45(2)])].">
      <j1 rule="para" parents="1625 1625"/>
      <j2 rule="rewrite" parents="1623 21 21 45"/>
    </justification>
  </clause>

  <clause id="1787">
    <literal><![CDATA[
      f(x ^ f(c(x),y),f(c(x),f(f(y,f(x,y)),y))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(1647),rewrite([1739(1),1739(1),21(1),1739(4),21(4),1739(5),50(5),1739(5),1739(10),21(10)])].">
      <j1 rule="back_rewrite" parents="1647"/>
      <j2 rule="rewrite" parents="1739 1739 21 1739 21 1739 50 1739 1739 21"/>
    </justification>
  </clause>

  <clause id="1798">
    <literal><![CDATA[
      f(x v y,f(c(y),f(f(z,f(y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1628),rewrite([1739(3)])].">
      <j1 rule="back_rewrite" parents="1628"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="2130">
    <literal><![CDATA[
      f(x v y,f(c(x),f(f(c(x),c(x) v x),c(x)))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(776),rewrite([1739(1)])].">
      <j1 rule="back_rewrite" parents="776"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="2374">
    <literal><![CDATA[
      x ^ y = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[back_rewrite(95),rewrite([1739(1),1739(1)])].">
      <j1 rule="back_rewrite" parents="95"/>
      <j2 rule="rewrite" parents="1739 1739"/>
    </justification>
  </clause>

  <clause id="2384">
    <literal><![CDATA[
      f(x,y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(77),rewrite([1739(1),1739(1)])].">
      <j1 rule="back_rewrite" parents="77"/>
      <j2 rule="rewrite" parents="1739 1739"/>
    </justification>
  </clause>

  <clause id="2392">
    <literal><![CDATA[
      c(c(x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(45),rewrite([1739(3)])].">
      <j1 rule="back_rewrite" parents="45"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="2410">
    <literal><![CDATA[
      (c(x) v c(x v c(y))) v c(x v c(c(c(y) v c(c(x) v c(y))) v c(y))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(1787),rewrite([2384(2),2392(2),2374(3),2384(8),2384(11),2384(14),2384(17),2392(8),2384(18),2392(7)])].">
      <j1 rule="back_rewrite" parents="1787"/>
      <j2 rule="rewrite" parents="2384 2392 2374 2384 2384 2384 2384 2392 2384 2392"/>
    </justification>
  </clause>

  <clause id="2441">
    <literal><![CDATA[
      c1 v c(c(c1) v c(c2)) != c1
    ]]></literal>
    <attribute><![CDATA[
      answer(B1)
    ]]></attribute>
    <justification jstring="[back_rewrite(22),rewrite([2374(4)])].">
      <j1 rule="back_rewrite" parents="22"/>
      <j2 rule="rewrite" parents="2374"/>
    </justification>
  </clause>

  <clause id="2450">
    <literal><![CDATA[
      c(x v y) v c(x v c(c(x v c(c(x) v x)) v x)) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(2130),rewrite([2384(6),2392(4),2384(8),2392(9),2384(9),2392(3),2384(10)])].">
      <j1 rule="back_rewrite" parents="2130"/>
      <j2 rule="rewrite" parents="2384 2392 2384 2392 2384 2392 2384"/>
    </justification>
  </clause>

  <clause id="2454">
    <literal><![CDATA[
      c(x v y) v c(y v c(c(c(z) v c(c(y) v c(z))) v c(z))) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1798),rewrite([2384(3),2384(6),2384(9),2384(12),2392(3),2384(13)])].">
      <j1 rule="back_rewrite" parents="1798"/>
      <j2 rule="rewrite" parents="2384 2384 2384 2384 2392 2384"/>
    </justification>
  </clause>

  <clause id="2465">
    <literal><![CDATA[
      x v c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1693),rewrite([2384(4),2392(3),2384(6),2392(7),2384(7),2384(10),2392(2)])].">
      <j1 rule="back_rewrite" parents="1693"/>
      <j2 rule="rewrite" parents="2384 2392 2384 2392 2384 2384 2392"/>
    </justification>
  </clause>

  <clause id="2479">
    <literal><![CDATA[
      c(x v c(c(x v c(c(x) v x)) v x)) = c(x)
    ]]></literal>
    <justification jstring="[para(2450(a,1),1739(a,1)),flip(a)].">
      <j1 rule="para" parents="2450 1739"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2481">
    <literal><![CDATA[
      c(x v y) v c(x) = c(x)
    ]]></literal>
    <justification jstring="[para(2450(a,1),2384(a,2)),rewrite([2384(10),2479(11)])].">
      <j1 rule="para" parents="2450 2384"/>
      <j2 rule="rewrite" parents="2384 2479"/>
    </justification>
  </clause>

  <clause id="2486">
    <literal><![CDATA[
      c(c(x) v y) v x = x
    ]]></literal>
    <justification jstring="[para(2392(a,1),2481(a,1,2)),rewrite([2392(6)])].">
      <j1 rule="para" parents="2392 2481"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="2487">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2481(a,1,1,1)),rewrite([2392(2),2392(3),2392(5)])].">
      <j1 rule="para" parents="2481 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2511">
    <literal><![CDATA[
      (c(x) v c(x v y)) v c(x v c(c(y v c(c(x) v y)) v y)) = c(x)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2410(a,1,1,2,1,2)),rewrite([2392(6),2392(7),2392(11)])].">
      <j1 rule="para" parents="2392 2410"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2513">
    <literal><![CDATA[
      (c(x) v c(x v c(y))) v c(x) = c(x) v c(x v c(y))
    ]]></literal>
    <justification jstring="[para(2410(a,1),2465(a,1,2,1,2,1,1,1,2,1)),rewrite([2392(25),2454(38),2392(13),2486(12)])].">
      <j1 rule="para" parents="2410 2465"/>
      <j2 rule="rewrite" parents="2392 2454 2392 2486"/>
    </justification>
  </clause>

  <clause id="2517">
    <literal><![CDATA[
      c(x) v c(x v c(y)) = c(x)
    ]]></literal>
    <justification jstring="[para(2410(a,1),2487(a,1,2)),rewrite([2513(7),2511(23)])].">
      <j1 rule="para" parents="2410 2487"/>
      <j2 rule="rewrite" parents="2513 2511"/>
    </justification>
  </clause>

  <clause id="2526">
    <literal><![CDATA[
      x v c(c(x) v c(y)) = x
    ]]></literal>
    <justification jstring="[para(2392(a,1),2517(a,1,1)),rewrite([2392(7)])].">
      <j1 rule="para" parents="2392 2517"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="2527">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(B1)
    ]]></attribute>
    <justification jstring="[resolve(2526,a,2441,a)].">
      <j1 rule="resolve" parents="2526 2441"/>
    </justification>
  </clause>

</proof>

<proof number="7" length="111" max_count="37">

<comments><![CDATA[
% Proof 7 at 0.84 (+ 0.03) seconds: 1.
% Length of proof is 111.
% Level of proof is 27.
% Maximum clause weight is 37.
% Given clauses 207.

]]></comments>

  <clause id="11" type="goal">
    <literal><![CDATA[
      x v c(x) = y v c(y)
    ]]></literal>
    <attribute><![CDATA[
      answer(1)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="15" type="assumption">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="16" type="assumption">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <justification jstring="[copy(16),flip(a)].">
      <j1 rule="copy" parents="16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18" type="assumption">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_meet)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(f(x,y),f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[copy(18),flip(a)].">
      <j1 rule="copy" parents="18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20" type="assumption">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(x,x) = c(x)
    ]]></literal>
    <justification jstring="[copy(20),flip(a)].">
      <j1 rule="copy" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="34" type="deny">
    <literal><![CDATA[
      c19 v c(c19) != c20 v c(c20)
    ]]></literal>
    <attribute><![CDATA[
      answer(1)
    ]]></attribute>
    <justification jstring="[deny(11)].">
      <j1 rule="deny" parents="11"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      c20 v c(c20) != c19 v c(c19)
    ]]></literal>
    <attribute><![CDATA[
      answer(1)
    ]]></attribute>
    <justification jstring="[copy(34),flip(a)].">
      <j1 rule="copy" parents="34"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      c(f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(19),rewrite([21(3)])].">
      <j1 rule="back_rewrite" parents="19"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      f(c(x),c(y)) = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(17),rewrite([21(1),21(2)])].">
      <j1 rule="back_rewrite" parents="17"/>
      <j2 rule="rewrite" parents="21 21"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(15),rewrite([21(5)])].">
      <j1 rule="back_rewrite" parents="15"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      x ^ x = c(c(x))
    ]]></literal>
    <justification jstring="[para(21(a,1),40(a,1,1)),flip(a)].">
      <j1 rule="para" parents="21 40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      c(c(x)) = x v x
    ]]></literal>
    <justification jstring="[para(41(a,1),21(a,1)),flip(a)].">
      <j1 rule="para" parents="41 21"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      c(x) ^ c(y) = c(x v y)
    ]]></literal>
    <justification jstring="[para(41(a,1),40(a,1,1)),flip(a)].">
      <j1 rule="para" parents="41 40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      x ^ x = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(43),rewrite([45(3)])].">
      <j1 rule="back_rewrite" parents="43"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      f(f(f(c(x),f(x,y)),z),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1,1))].">
      <j1 rule="para" parents="21 42"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      f(f(f(f(x,y),c(y)),z),f(y,f(f(y,f(c(y),y)),y))) = y
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1,2))].">
      <j1 rule="para" parents="21 42"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      f(f(x v x,y),f(x,f(f(x,f(c(x),x)),x))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1)),rewrite([21(1),45(2)])].">
      <j1 rule="para" parents="21 42"/>
      <j2 rule="rewrite" parents="21 45"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      f(f(x,y) ^ f(y,z),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1)),rewrite([40(4)])].">
      <j1 rule="para" parents="21 42"/>
      <j2 rule="rewrite" parents="40"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      f(f(f(x v y,f(c(y),z)),u),f(c(y),f(f(z,f(y v y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[para(41(a,1),42(a,1,1,1,1)),rewrite([45(8)])].">
      <j1 rule="para" parents="41 42"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,c(z))),u),f(y,f(f(c(z),y v z),c(z)))) = y
    ]]></literal>
    <justification jstring="[para(41(a,1),42(a,1,2,2,1,2))].">
      <j1 rule="para" parents="41 42"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(f(f(z,f(c(x),z)),z),f(c(x),f(f(z,f(c(x),z)),z))),f(f(z,f(c(x),z)),z)))) = x
    ]]></literal>
    <justification jstring="[para(42(a,1),42(a,1,1,1))].">
      <j1 rule="para" parents="42 42"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      f(x v x,c(y)) = c(x) v y
    ]]></literal>
    <justification jstring="[para(45(a,1),41(a,1,1))].">
      <j1 rule="para" parents="45 41"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      f(c(x),y v y) = x v c(y)
    ]]></literal>
    <justification jstring="[para(45(a,1),41(a,1,2))].">
      <j1 rule="para" parents="45 41"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      c(x v x) = c(x) v c(x)
    ]]></literal>
    <justification jstring="[para(45(a,1),45(a,1,1))].">
      <j1 rule="para" parents="45 45"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      (x v x) ^ c(y) = c(c(x) v y)
    ]]></literal>
    <justification jstring="[para(45(a,1),46(a,1,1))].">
      <j1 rule="para" parents="45 46"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      f(x v x,y v y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[para(45(a,1),65(a,1,2))].">
      <j1 rule="para" parents="45 65"/>
    </justification>
  </clause>

  <clause id="95">
    <literal><![CDATA[
      (x v x) ^ (y v y) = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[para(45(a,1),72(a,1,2))].">
      <j1 rule="para" parents="45 72"/>
    </justification>
  </clause>

  <clause id="158">
    <literal><![CDATA[
      f(x,f(c(x),f(f(y,f(x v x,y)),y))) = c(x)
    ]]></literal>
    <justification jstring="[para(53(a,1),42(a,1,1)),rewrite([45(3)])].">
      <j1 rule="para" parents="53 42"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="203">
    <literal><![CDATA[
      f(c(x) v y,f(x,f(f(x,f(c(x),x)),x))) = x
    ]]></literal>
    <justification jstring="[para(65(a,1),54(a,1,1))].">
      <j1 rule="para" parents="65 54"/>
    </justification>
  </clause>

  <clause id="304">
    <literal><![CDATA[
      f(f(x,y v y) ^ (c(y) v z),f(y v y,f(f(c(z),(y v y) v z),c(z)))) = y v y
    ]]></literal>
    <justification jstring="[para(65(a,1),55(a,1,1,2)),rewrite([68(9),65(12),45(9)])].">
      <j1 rule="para" parents="65 55"/>
      <j2 rule="rewrite" parents="68 65 45"/>
    </justification>
  </clause>

  <clause id="776">
    <literal><![CDATA[
      f((x v x) v y,f(c(x),f(f(c(x),c(x) v x),c(x)))) = c(x)
    ]]></literal>
    <justification jstring="[para(41(a,1),203(a,1,2,2,1,2)),rewrite([45(2)])].">
      <j1 rule="para" parents="41 203"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="785">
    <literal><![CDATA[
      f(c(x),f(x v x,f(f(f(x,f(f(x,f(c(x),x)),x)),x),f(x,f(f(x,f(c(x),x)),x))))) = x v x
    ]]></literal>
    <justification jstring="[para(203(a,1),158(a,1,2,2,1,2)),rewrite([45(3),45(18)])].">
      <j1 rule="para" parents="203 158"/>
      <j2 rule="rewrite" parents="45 45"/>
    </justification>
  </clause>

  <clause id="1263">
    <literal><![CDATA[
      f(f(f(x v c(y),c(y) v z),u),f(y v y,f(f(c(z),(y v y) v z),c(z)))) = y v y
    ]]></literal>
    <justification jstring="[para(66(a,1),61(a,1,1,1,1)),rewrite([65(5)])].">
      <j1 rule="para" parents="66 61"/>
      <j2 rule="rewrite" parents="65"/>
    </justification>
  </clause>

  <clause id="1623">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(62(a,1),42(a,1,1))].">
      <j1 rule="para" parents="62 42"/>
    </justification>
  </clause>

  <clause id="1625">
    <literal><![CDATA[
      f(c(x),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(62(a,1),52(a,1,1))].">
      <j1 rule="para" parents="62 52"/>
    </justification>
  </clause>

  <clause id="1628">
    <literal><![CDATA[
      f(x v y,f(c(y),f(f(z,f(y v y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[para(62(a,1),59(a,1,1))].">
      <j1 rule="para" parents="62 59"/>
    </justification>
  </clause>

  <clause id="1630">
    <literal><![CDATA[
      f(c(x),f(x v x,x)) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(785),rewrite([1623(14)])].">
      <j1 rule="back_rewrite" parents="785"/>
      <j2 rule="rewrite" parents="1623"/>
    </justification>
  </clause>

  <clause id="1647">
    <literal><![CDATA[
      f((x v x) ^ f(f(x v x,x),y),f(f(x v x,x),f(f(y,f((x v x) ^ x,y)),y))) = f(x v x,x)
    ]]></literal>
    <justification jstring="[para(1630(a,1),55(a,1,1,1)),rewrite([40(10)])].">
      <j1 rule="para" parents="1630 55"/>
      <j2 rule="rewrite" parents="40"/>
    </justification>
  </clause>

  <clause id="1648">
    <literal><![CDATA[
      f(f(x,c(y)) ^ (y v y),f(c(y),f(f(f(y v y,y),f(y v y,f(y v y,y))),f(y v y,y)))) = c(y)
    ]]></literal>
    <justification jstring="[para(1630(a,1),55(a,1,1,2)),rewrite([45(9)])].">
      <j1 rule="para" parents="1630 55"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="1693">
    <literal><![CDATA[
      f(c(x),f(x,f(f(c(y),x v y),c(y)))) = x
    ]]></literal>
    <justification jstring="[para(41(a,1),1625(a,1,2,2,1,2))].">
      <j1 rule="para" parents="41 1625"/>
    </justification>
  </clause>

  <clause id="1739">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(1625(a,1),1625(a,1,2,2,1,2)),rewrite([1623(13),21(2),21(3),45(2)])].">
      <j1 rule="para" parents="1625 1625"/>
      <j2 rule="rewrite" parents="1623 21 21 45"/>
    </justification>
  </clause>

  <clause id="1786">
    <literal><![CDATA[
      f(f(x,c(y)) ^ y,f(c(y),f(f(c(y),f(y,c(y))),c(y)))) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1648),rewrite([1739(3),1739(5),21(5),1739(6),1739(6),21(6),1739(9),21(9)])].">
      <j1 rule="back_rewrite" parents="1648"/>
      <j2 rule="rewrite" parents="1739 1739 21 1739 1739 21 1739 21"/>
    </justification>
  </clause>

  <clause id="1787">
    <literal><![CDATA[
      f(x ^ f(c(x),y),f(c(x),f(f(y,f(x,y)),y))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(1647),rewrite([1739(1),1739(1),21(1),1739(4),21(4),1739(5),50(5),1739(5),1739(10),21(10)])].">
      <j1 rule="back_rewrite" parents="1647"/>
      <j2 rule="rewrite" parents="1739 1739 21 1739 21 1739 50 1739 1739 21"/>
    </justification>
  </clause>

  <clause id="1798">
    <literal><![CDATA[
      f(x v y,f(c(y),f(f(z,f(y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1628),rewrite([1739(3)])].">
      <j1 rule="back_rewrite" parents="1628"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="1958">
    <literal><![CDATA[
      f(f(f(x v c(y),c(y) v z),u),f(y,f(f(c(z),y v z),c(z)))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1263),rewrite([1739(7),1739(8),1739(14)])].">
      <j1 rule="back_rewrite" parents="1263"/>
      <j2 rule="rewrite" parents="1739 1739 1739"/>
    </justification>
  </clause>

  <clause id="2130">
    <literal><![CDATA[
      f(x v y,f(c(x),f(f(c(x),c(x) v x),c(x)))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(776),rewrite([1739(1)])].">
      <j1 rule="back_rewrite" parents="776"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="2302">
    <literal><![CDATA[
      f(f(x,y) ^ (c(y) v z),f(y,f(f(c(z),y v z),c(z)))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(304),rewrite([1739(1),1739(5),1739(6),1739(12)])].">
      <j1 rule="back_rewrite" parents="304"/>
      <j2 rule="rewrite" parents="1739 1739 1739 1739"/>
    </justification>
  </clause>

  <clause id="2374">
    <literal><![CDATA[
      x ^ y = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[back_rewrite(95),rewrite([1739(1),1739(1)])].">
      <j1 rule="back_rewrite" parents="95"/>
      <j2 rule="rewrite" parents="1739 1739"/>
    </justification>
  </clause>

  <clause id="2384">
    <literal><![CDATA[
      f(x,y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(77),rewrite([1739(1),1739(1)])].">
      <j1 rule="back_rewrite" parents="77"/>
      <j2 rule="rewrite" parents="1739 1739"/>
    </justification>
  </clause>

  <clause id="2392">
    <literal><![CDATA[
      c(c(x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(45),rewrite([1739(3)])].">
      <j1 rule="back_rewrite" parents="45"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="2400">
    <literal><![CDATA[
      (c(c(x) v c(y)) v c(c(y) v z)) v c(c(y) v c(c(z v c(y v z)) v z)) = y
    ]]></literal>
    <justification jstring="[back_rewrite(2302),rewrite([2384(1),2374(6),2384(12),2392(11),2384(14),2392(15),2384(15),2384(18),2392(10)])].">
      <j1 rule="back_rewrite" parents="2302"/>
      <j2 rule="rewrite" parents="2384 2374 2384 2392 2384 2392 2384 2384 2392"/>
    </justification>
  </clause>

  <clause id="2410">
    <literal><![CDATA[
      (c(x) v c(x v c(y))) v c(x v c(c(c(y) v c(c(x) v c(y))) v c(y))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(1787),rewrite([2384(2),2392(2),2374(3),2384(8),2384(11),2384(14),2384(17),2392(8),2384(18),2392(7)])].">
      <j1 rule="back_rewrite" parents="1787"/>
      <j2 rule="rewrite" parents="2384 2392 2374 2384 2384 2384 2384 2392 2384 2392"/>
    </justification>
  </clause>

  <clause id="2411">
    <literal><![CDATA[
      (c(c(x) v y) v c(y)) v c(y v c(c(y v c(c(y) v y)) v y)) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1786),rewrite([2384(2),2392(3),2374(3),2384(10),2392(11),2384(11),2392(9),2384(13),2392(14),2384(14),2392(8),2384(15),2392(7)])].">
      <j1 rule="back_rewrite" parents="1786"/>
      <j2 rule="rewrite" parents="2384 2392 2374 2384 2392 2384 2392 2384 2392 2384 2392 2384 2392"/>
    </justification>
  </clause>

  <clause id="2450">
    <literal><![CDATA[
      c(x v y) v c(x v c(c(x v c(c(x) v x)) v x)) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(2130),rewrite([2384(6),2392(4),2384(8),2392(9),2384(9),2392(3),2384(10)])].">
      <j1 rule="back_rewrite" parents="2130"/>
      <j2 rule="rewrite" parents="2384 2392 2384 2392 2384 2392 2384"/>
    </justification>
  </clause>

  <clause id="2453">
    <literal><![CDATA[
      c(c(c(x v c(y)) v c(c(y) v z)) v c(u)) v c(c(y) v c(c(z v c(y v z)) v z)) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1958),rewrite([2384(5),2384(8),2384(13),2392(12),2384(15),2392(16),2384(16),2384(19)])].">
      <j1 rule="back_rewrite" parents="1958"/>
      <j2 rule="rewrite" parents="2384 2384 2384 2392 2384 2392 2384 2384"/>
    </justification>
  </clause>

  <clause id="2454">
    <literal><![CDATA[
      c(x v y) v c(y v c(c(c(z) v c(c(y) v c(z))) v c(z))) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1798),rewrite([2384(3),2384(6),2384(9),2384(12),2392(3),2384(13)])].">
      <j1 rule="back_rewrite" parents="1798"/>
      <j2 rule="rewrite" parents="2384 2384 2384 2384 2392 2384"/>
    </justification>
  </clause>

  <clause id="2465">
    <literal><![CDATA[
      x v c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1693),rewrite([2384(4),2392(3),2384(6),2392(7),2384(7),2384(10),2392(2)])].">
      <j1 rule="back_rewrite" parents="1693"/>
      <j2 rule="rewrite" parents="2384 2392 2384 2392 2384 2384 2392"/>
    </justification>
  </clause>

  <clause id="2479">
    <literal><![CDATA[
      c(x v c(c(x v c(c(x) v x)) v x)) = c(x)
    ]]></literal>
    <justification jstring="[para(2450(a,1),1739(a,1)),flip(a)].">
      <j1 rule="para" parents="2450 1739"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2481">
    <literal><![CDATA[
      c(x v y) v c(x) = c(x)
    ]]></literal>
    <justification jstring="[para(2450(a,1),2384(a,2)),rewrite([2384(10),2479(11)])].">
      <j1 rule="para" parents="2450 2384"/>
      <j2 rule="rewrite" parents="2384 2479"/>
    </justification>
  </clause>

  <clause id="2485">
    <literal><![CDATA[
      (c(c(x) v y) v c(y)) v c(y) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(2411),rewrite([2479(14)])].">
      <j1 rule="back_rewrite" parents="2411"/>
      <j2 rule="rewrite" parents="2479"/>
    </justification>
  </clause>

  <clause id="2486">
    <literal><![CDATA[
      c(c(x) v y) v x = x
    ]]></literal>
    <justification jstring="[para(2392(a,1),2481(a,1,2)),rewrite([2392(6)])].">
      <j1 rule="para" parents="2392 2481"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="2487">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2481(a,1,1,1)),rewrite([2392(2),2392(3),2392(5)])].">
      <j1 rule="para" parents="2481 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2492">
    <literal><![CDATA[
      c(x) v c(c(c(y) v c(x)) v c(c(x) v z)) = c(c(c(y) v c(x)) v c(c(x) v z))
    ]]></literal>
    <justification jstring="[para(2400(a,1),2481(a,1,1,1))].">
      <j1 rule="para" parents="2400 2481"/>
    </justification>
  </clause>

  <clause id="2500">
    <literal><![CDATA[
      (c(c(x) v c(y)) v y) v y = y
    ]]></literal>
    <justification jstring="[para(2392(a,1),2485(a,1,1,2)),rewrite([2392(7),2392(8)])].">
      <j1 rule="para" parents="2392 2485"/>
      <j2 rule="rewrite" parents="2392 2392"/>
    </justification>
  </clause>

  <clause id="2507">
    <literal><![CDATA[
      (c(x v c(y)) v y) v y = y
    ]]></literal>
    <justification jstring="[para(2392(a,1),2500(a,1,1,1,1,1))].">
      <j1 rule="para" parents="2392 2500"/>
    </justification>
  </clause>

  <clause id="2511">
    <literal><![CDATA[
      (c(x) v c(x v y)) v c(x v c(c(y v c(c(x) v y)) v y)) = c(x)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2410(a,1,1,2,1,2)),rewrite([2392(6),2392(7),2392(11)])].">
      <j1 rule="para" parents="2392 2410"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2513">
    <literal><![CDATA[
      (c(x) v c(x v c(y))) v c(x) = c(x) v c(x v c(y))
    ]]></literal>
    <justification jstring="[para(2410(a,1),2465(a,1,2,1,2,1,1,1,2,1)),rewrite([2392(25),2454(38),2392(13),2486(12)])].">
      <j1 rule="para" parents="2410 2465"/>
      <j2 rule="rewrite" parents="2392 2454 2392 2486"/>
    </justification>
  </clause>

  <clause id="2517">
    <literal><![CDATA[
      c(x) v c(x v c(y)) = c(x)
    ]]></literal>
    <justification jstring="[para(2410(a,1),2487(a,1,2)),rewrite([2513(7),2511(23)])].">
      <j1 rule="para" parents="2410 2487"/>
      <j2 rule="rewrite" parents="2513 2511"/>
    </justification>
  </clause>

  <clause id="2528">
    <literal><![CDATA[
      c(x) v c(x v y) = c(x)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2517(a,1,2,1,2))].">
      <j1 rule="para" parents="2392 2517"/>
    </justification>
  </clause>

  <clause id="2529">
    <literal><![CDATA[
      (x v y) v x = x v y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2517(a,1,2,1)),rewrite([2392(3),2392(3),2392(5)])].">
      <j1 rule="para" parents="2481 2517"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2533">
    <literal><![CDATA[
      x v c(c(x) v y) = x
    ]]></literal>
    <justification jstring="[para(2486(a,1),2529(a,1,1)),rewrite([2486(8)])].">
      <j1 rule="para" parents="2486 2529"/>
      <j2 rule="rewrite" parents="2486"/>
    </justification>
  </clause>

  <clause id="2552">
    <literal><![CDATA[
      c((c(x) v y) v c(z)) v c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[para(2481(a,1),2453(a,1,1,1,1,1)),rewrite([2392(4)])].">
      <j1 rule="para" parents="2481 2453"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="2553">
    <literal><![CDATA[
      c(x v c(y)) v c(c(y) v c(c(z v c(y v z)) v z)) = y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2453(a,1,1,1)),rewrite([2392(4)])].">
      <j1 rule="para" parents="2481 2453"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="2560">
    <literal><![CDATA[
      c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[para(2453(a,1),2507(a,1,1)),rewrite([2533(10)]),flip(a)].">
      <j1 rule="para" parents="2453 2507"/>
      <j2 rule="rewrite" parents="2533"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2574">
    <literal><![CDATA[
      c(x v c(y)) v y = y
    ]]></literal>
    <justification jstring="[back_rewrite(2553),rewrite([2560(12)])].">
      <j1 rule="back_rewrite" parents="2553"/>
      <j2 rule="rewrite" parents="2560"/>
    </justification>
  </clause>

  <clause id="2575">
    <literal><![CDATA[
      c((c(x) v y) v c(z)) v x = x
    ]]></literal>
    <justification jstring="[back_rewrite(2552),rewrite([2560(14)])].">
      <j1 rule="back_rewrite" parents="2552"/>
      <j2 rule="rewrite" parents="2560"/>
    </justification>
  </clause>

  <clause id="2583">
    <literal><![CDATA[
      c(x v y) v c(y) = c(y)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2574(a,1,1,1,2))].">
      <j1 rule="para" parents="2392 2574"/>
    </justification>
  </clause>

  <clause id="2585">
    <literal><![CDATA[
      x v c(y v c(x)) = x
    ]]></literal>
    <justification jstring="[para(2574(a,1),2529(a,1,1)),rewrite([2574(8)])].">
      <j1 rule="para" parents="2574 2529"/>
      <j2 rule="rewrite" parents="2574"/>
    </justification>
  </clause>

  <clause id="2591">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(2583(a,1),2481(a,1,1,1)),rewrite([2392(2),2392(3),2392(5)])].">
      <j1 rule="para" parents="2583 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2599">
    <literal><![CDATA[
      c((c(x) v y) v z) v x = x
    ]]></literal>
    <justification jstring="[para(2392(a,1),2575(a,1,1,1,2))].">
      <j1 rule="para" parents="2392 2575"/>
    </justification>
  </clause>

  <clause id="2612">
    <literal><![CDATA[
      c(c(x) v y) v (x v z) = x v z
    ]]></literal>
    <justification jstring="[para(2481(a,1),2599(a,1,1,1,1))].">
      <j1 rule="para" parents="2481 2599"/>
    </justification>
  </clause>

  <clause id="2614">
    <literal><![CDATA[
      x v c((c(x) v y) v z) = x
    ]]></literal>
    <justification jstring="[para(2599(a,1),2529(a,1,1)),rewrite([2599(10)])].">
      <j1 rule="para" parents="2599 2529"/>
      <j2 rule="rewrite" parents="2599"/>
    </justification>
  </clause>

  <clause id="2618">
    <literal><![CDATA[
      c((x v c(y)) v z) v y = y
    ]]></literal>
    <justification jstring="[para(2591(a,1),2599(a,1,1,1,1))].">
      <j1 rule="para" parents="2591 2599"/>
    </justification>
  </clause>

  <clause id="2619">
    <literal><![CDATA[
      c(x v (c(y) v z)) v y = y
    ]]></literal>
    <justification jstring="[para(2591(a,1),2599(a,1,1,1))].">
      <j1 rule="para" parents="2591 2599"/>
    </justification>
  </clause>

  <clause id="2645">
    <literal><![CDATA[
      (x v y) v c(c(x) v z) = x v y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2614(a,1,2,1,1))].">
      <j1 rule="para" parents="2481 2614"/>
    </justification>
  </clause>

  <clause id="2649">
    <literal><![CDATA[
      x v c((y v c(x)) v z) = x
    ]]></literal>
    <justification jstring="[para(2591(a,1),2614(a,1,2,1,1))].">
      <j1 rule="para" parents="2591 2614"/>
    </justification>
  </clause>

  <clause id="2650">
    <literal><![CDATA[
      x v c(y v (c(x) v z)) = x
    ]]></literal>
    <justification jstring="[para(2591(a,1),2614(a,1,2,1))].">
      <j1 rule="para" parents="2591 2614"/>
    </justification>
  </clause>

  <clause id="2656">
    <literal><![CDATA[
      c(x v (y v c(z))) v z = z
    ]]></literal>
    <justification jstring="[para(2591(a,1),2618(a,1,1,1))].">
      <j1 rule="para" parents="2591 2618"/>
    </justification>
  </clause>

  <clause id="2660">
    <literal><![CDATA[
      c(x v (y v z)) v c(y) = c(y)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2619(a,1,1,1,2,1))].">
      <j1 rule="para" parents="2392 2619"/>
    </justification>
  </clause>

  <clause id="2674">
    <literal><![CDATA[
      x v c(y v (z v c(x))) = x
    ]]></literal>
    <justification jstring="[para(2591(a,1),2649(a,1,2,1))].">
      <j1 rule="para" parents="2591 2649"/>
    </justification>
  </clause>

  <clause id="2675">
    <literal><![CDATA[
      x v c(c(c(y) v x) v c(x v z)) = c(c(c(y) v x) v c(x v z))
    ]]></literal>
    <justification jstring="[para(2392(a,1),2492(a,1,1)),rewrite([2392(3),2392(5),2392(11),2392(13)])].">
      <j1 rule="para" parents="2392 2492"/>
      <j2 rule="rewrite" parents="2392 2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2698">
    <literal><![CDATA[
      (x v y) v c(z v c(x)) = x v y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2650(a,1,2,1,2))].">
      <j1 rule="para" parents="2481 2650"/>
    </justification>
  </clause>

  <clause id="2706">
    <literal><![CDATA[
      c(x v (y v z)) v c(z) = c(z)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2656(a,1,1,1,2,2))].">
      <j1 rule="para" parents="2392 2656"/>
    </justification>
  </clause>

  <clause id="2737">
    <literal><![CDATA[
      x v (y v (x v z)) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(2660(a,1),2481(a,1,1,1)),rewrite([2392(2),2392(4),2392(7)])].">
      <j1 rule="para" parents="2660 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2746">
    <literal><![CDATA[
      c(c(x v y) v x) v c(x v y) = c(x)
    ]]></literal>
    <justification jstring="[para(2528(a,1),2560(a,1,1,2,1,1,1,2,1)),rewrite([2392(2),2392(4),2675(9),2392(9)])].">
      <j1 rule="para" parents="2528 2560"/>
      <j2 rule="rewrite" parents="2392 2392 2675 2392"/>
    </justification>
  </clause>

  <clause id="2765">
    <literal><![CDATA[
      (x v (y v z)) v z = x v (y v z)
    ]]></literal>
    <justification jstring="[para(2706(a,1),2533(a,1,2,1)),rewrite([2392(4)])].">
      <j1 rule="para" parents="2706 2533"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="2781">
    <literal><![CDATA[
      c((x v (y v c(z))) v u) v z = z
    ]]></literal>
    <justification jstring="[para(2656(a,1),2612(a,1,2)),rewrite([2392(5),2656(11)])].">
      <j1 rule="para" parents="2656 2612"/>
      <j2 rule="rewrite" parents="2392 2656"/>
    </justification>
  </clause>

  <clause id="2842">
    <literal><![CDATA[
      x v c((y v (z v c(x))) v u) = x
    ]]></literal>
    <justification jstring="[para(2656(a,1),2645(a,1,1)),rewrite([2392(5),2656(11)])].">
      <j1 rule="para" parents="2656 2645"/>
      <j2 rule="rewrite" parents="2392 2656"/>
    </justification>
  </clause>

  <clause id="4044">
    <literal><![CDATA[
      c(x v c(x v y)) v x = x v y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2746(a,1,1,1,1,1)),rewrite([2392(2),2481(8),2392(6),2392(8)])].">
      <j1 rule="para" parents="2481 2746"/>
      <j2 rule="rewrite" parents="2392 2481 2392 2392"/>
    </justification>
  </clause>

  <clause id="4089">
    <literal><![CDATA[
      c(x) v x = x v c(x)
    ]]></literal>
    <justification jstring="[para(2585(a,1),4044(a,1,1,1))].">
      <j1 rule="para" parents="2585 4044"/>
    </justification>
  </clause>

  <clause id="4091">
    <literal><![CDATA[
      x v (y v c(x)) = x v c(x)
    ]]></literal>
    <justification jstring="[para(2674(a,1),4044(a,1,1,1)),rewrite([4089(2)]),flip(a)].">
      <j1 rule="para" parents="2674 4044"/>
      <j2 rule="rewrite" parents="4089"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4094">
    <literal><![CDATA[
      (x v y) v c(x v y) = (x v y) v c(x)
    ]]></literal>
    <justification jstring="[para(2698(a,1),4044(a,1,1,1)),rewrite([4089(4)])].">
      <j1 rule="para" parents="2698 4044"/>
      <j2 rule="rewrite" parents="4089"/>
    </justification>
  </clause>

  <clause id="4130">
    <literal><![CDATA[
      (x v y) v c(x) = x v c(x)
    ]]></literal>
    <justification jstring="[para(4089(a,1),2737(a,1,2)),rewrite([4094(4),4091(4),4089(6),4094(6)]),flip(a)].">
      <j1 rule="para" parents="4089 2737"/>
      <j2 rule="rewrite" parents="4094 4091 4089 4094"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4132">
    <literal><![CDATA[
      (x v c(x)) v y = x v c(x)
    ]]></literal>
    <justification jstring="[para(4089(a,1),2765(a,1,1)),rewrite([4094(4),4130(3),4089(7),4094(7),4130(6)])].">
      <j1 rule="para" parents="4089 2765"/>
      <j2 rule="rewrite" parents="4094 4130 4089 4094 4130"/>
    </justification>
  </clause>

  <clause id="4133">
    <literal><![CDATA[
      c(x v c(x)) v y = y
    ]]></literal>
    <justification jstring="[para(4089(a,1),2781(a,1,1,1,1)),rewrite([4094(6),4130(4),4132(3)])].">
      <j1 rule="para" parents="4089 2781"/>
      <j2 rule="rewrite" parents="4094 4130 4132"/>
    </justification>
  </clause>

  <clause id="4134">
    <literal><![CDATA[
      x v c(y v c(y)) = x
    ]]></literal>
    <justification jstring="[para(4089(a,1),2842(a,1,2,1,1)),rewrite([4094(6),4130(4),4132(3)])].">
      <j1 rule="para" parents="4089 2842"/>
      <j2 rule="rewrite" parents="4094 4130 4132"/>
    </justification>
  </clause>

  <clause id="4207">
    <literal><![CDATA[
      x v c(x) = y v c(y)
    ]]></literal>
    <justification jstring="[para(4133(a,1),2746(a,1,1,1,1,1)),rewrite([4134(5),2392(2),4133(4),2392(6)])].">
      <j1 rule="para" parents="4133 2746"/>
      <j2 rule="rewrite" parents="4134 2392 4133 2392"/>
    </justification>
  </clause>

  <clause id="4208">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(1)
    ]]></attribute>
    <justification jstring="[resolve(4207,a,35,a)].">
      <j1 rule="resolve" parents="4207 35"/>
    </justification>
  </clause>

</proof>

<proof number="8" length="113" max_count="37">

<comments><![CDATA[
% Proof 8 at 0.85 (+ 0.03) seconds: 0.
% Length of proof is 113.
% Level of proof is 27.
% Maximum clause weight is 37.
% Given clauses 208.

]]></comments>

  <clause id="12" type="goal">
    <literal><![CDATA[
      x ^ c(x) = y ^ c(y)
    ]]></literal>
    <attribute><![CDATA[
      answer(0)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="15" type="assumption">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="16" type="assumption">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <justification jstring="[copy(16),flip(a)].">
      <j1 rule="copy" parents="16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18" type="assumption">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_meet)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(f(x,y),f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[copy(18),flip(a)].">
      <j1 rule="copy" parents="18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20" type="assumption">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(x,x) = c(x)
    ]]></literal>
    <justification jstring="[copy(20),flip(a)].">
      <j1 rule="copy" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="36" type="deny">
    <literal><![CDATA[
      c21 ^ c(c21) != c22 ^ c(c22)
    ]]></literal>
    <attribute><![CDATA[
      answer(0)
    ]]></attribute>
    <justification jstring="[deny(12)].">
      <j1 rule="deny" parents="12"/>
    </justification>
  </clause>

  <clause id="37">
    <literal><![CDATA[
      c22 ^ c(c22) != c21 ^ c(c21)
    ]]></literal>
    <attribute><![CDATA[
      answer(0)
    ]]></attribute>
    <justification jstring="[copy(36),flip(a)].">
      <j1 rule="copy" parents="36"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      c(f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(19),rewrite([21(3)])].">
      <j1 rule="back_rewrite" parents="19"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      f(c(x),c(y)) = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(17),rewrite([21(1),21(2)])].">
      <j1 rule="back_rewrite" parents="17"/>
      <j2 rule="rewrite" parents="21 21"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(15),rewrite([21(5)])].">
      <j1 rule="back_rewrite" parents="15"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      x ^ x = c(c(x))
    ]]></literal>
    <justification jstring="[para(21(a,1),40(a,1,1)),flip(a)].">
      <j1 rule="para" parents="21 40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      c(c(x)) = x v x
    ]]></literal>
    <justification jstring="[para(41(a,1),21(a,1)),flip(a)].">
      <j1 rule="para" parents="41 21"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      c(x) ^ c(y) = c(x v y)
    ]]></literal>
    <justification jstring="[para(41(a,1),40(a,1,1)),flip(a)].">
      <j1 rule="para" parents="41 40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      x ^ x = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(43),rewrite([45(3)])].">
      <j1 rule="back_rewrite" parents="43"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      f(f(f(c(x),f(x,y)),z),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1,1))].">
      <j1 rule="para" parents="21 42"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      f(f(f(f(x,y),c(y)),z),f(y,f(f(y,f(c(y),y)),y))) = y
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1,2))].">
      <j1 rule="para" parents="21 42"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      f(f(x v x,y),f(x,f(f(x,f(c(x),x)),x))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1)),rewrite([21(1),45(2)])].">
      <j1 rule="para" parents="21 42"/>
      <j2 rule="rewrite" parents="21 45"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      f(f(x,y) ^ f(y,z),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1)),rewrite([40(4)])].">
      <j1 rule="para" parents="21 42"/>
      <j2 rule="rewrite" parents="40"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      f(f(f(x v y,f(c(y),z)),u),f(c(y),f(f(z,f(y v y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[para(41(a,1),42(a,1,1,1,1)),rewrite([45(8)])].">
      <j1 rule="para" parents="41 42"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,c(z))),u),f(y,f(f(c(z),y v z),c(z)))) = y
    ]]></literal>
    <justification jstring="[para(41(a,1),42(a,1,2,2,1,2))].">
      <j1 rule="para" parents="41 42"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(f(f(z,f(c(x),z)),z),f(c(x),f(f(z,f(c(x),z)),z))),f(f(z,f(c(x),z)),z)))) = x
    ]]></literal>
    <justification jstring="[para(42(a,1),42(a,1,1,1))].">
      <j1 rule="para" parents="42 42"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      f(x v x,c(y)) = c(x) v y
    ]]></literal>
    <justification jstring="[para(45(a,1),41(a,1,1))].">
      <j1 rule="para" parents="45 41"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      f(c(x),y v y) = x v c(y)
    ]]></literal>
    <justification jstring="[para(45(a,1),41(a,1,2))].">
      <j1 rule="para" parents="45 41"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      c(x v x) = c(x) v c(x)
    ]]></literal>
    <justification jstring="[para(45(a,1),45(a,1,1))].">
      <j1 rule="para" parents="45 45"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      (x v x) ^ c(y) = c(c(x) v y)
    ]]></literal>
    <justification jstring="[para(45(a,1),46(a,1,1))].">
      <j1 rule="para" parents="45 46"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      f(x v x,y v y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[para(45(a,1),65(a,1,2))].">
      <j1 rule="para" parents="45 65"/>
    </justification>
  </clause>

  <clause id="95">
    <literal><![CDATA[
      (x v x) ^ (y v y) = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[para(45(a,1),72(a,1,2))].">
      <j1 rule="para" parents="45 72"/>
    </justification>
  </clause>

  <clause id="158">
    <literal><![CDATA[
      f(x,f(c(x),f(f(y,f(x v x,y)),y))) = c(x)
    ]]></literal>
    <justification jstring="[para(53(a,1),42(a,1,1)),rewrite([45(3)])].">
      <j1 rule="para" parents="53 42"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="203">
    <literal><![CDATA[
      f(c(x) v y,f(x,f(f(x,f(c(x),x)),x))) = x
    ]]></literal>
    <justification jstring="[para(65(a,1),54(a,1,1))].">
      <j1 rule="para" parents="65 54"/>
    </justification>
  </clause>

  <clause id="304">
    <literal><![CDATA[
      f(f(x,y v y) ^ (c(y) v z),f(y v y,f(f(c(z),(y v y) v z),c(z)))) = y v y
    ]]></literal>
    <justification jstring="[para(65(a,1),55(a,1,1,2)),rewrite([68(9),65(12),45(9)])].">
      <j1 rule="para" parents="65 55"/>
      <j2 rule="rewrite" parents="68 65 45"/>
    </justification>
  </clause>

  <clause id="776">
    <literal><![CDATA[
      f((x v x) v y,f(c(x),f(f(c(x),c(x) v x),c(x)))) = c(x)
    ]]></literal>
    <justification jstring="[para(41(a,1),203(a,1,2,2,1,2)),rewrite([45(2)])].">
      <j1 rule="para" parents="41 203"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="785">
    <literal><![CDATA[
      f(c(x),f(x v x,f(f(f(x,f(f(x,f(c(x),x)),x)),x),f(x,f(f(x,f(c(x),x)),x))))) = x v x
    ]]></literal>
    <justification jstring="[para(203(a,1),158(a,1,2,2,1,2)),rewrite([45(3),45(18)])].">
      <j1 rule="para" parents="203 158"/>
      <j2 rule="rewrite" parents="45 45"/>
    </justification>
  </clause>

  <clause id="1263">
    <literal><![CDATA[
      f(f(f(x v c(y),c(y) v z),u),f(y v y,f(f(c(z),(y v y) v z),c(z)))) = y v y
    ]]></literal>
    <justification jstring="[para(66(a,1),61(a,1,1,1,1)),rewrite([65(5)])].">
      <j1 rule="para" parents="66 61"/>
      <j2 rule="rewrite" parents="65"/>
    </justification>
  </clause>

  <clause id="1623">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(62(a,1),42(a,1,1))].">
      <j1 rule="para" parents="62 42"/>
    </justification>
  </clause>

  <clause id="1625">
    <literal><![CDATA[
      f(c(x),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(62(a,1),52(a,1,1))].">
      <j1 rule="para" parents="62 52"/>
    </justification>
  </clause>

  <clause id="1628">
    <literal><![CDATA[
      f(x v y,f(c(y),f(f(z,f(y v y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[para(62(a,1),59(a,1,1))].">
      <j1 rule="para" parents="62 59"/>
    </justification>
  </clause>

  <clause id="1630">
    <literal><![CDATA[
      f(c(x),f(x v x,x)) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(785),rewrite([1623(14)])].">
      <j1 rule="back_rewrite" parents="785"/>
      <j2 rule="rewrite" parents="1623"/>
    </justification>
  </clause>

  <clause id="1647">
    <literal><![CDATA[
      f((x v x) ^ f(f(x v x,x),y),f(f(x v x,x),f(f(y,f((x v x) ^ x,y)),y))) = f(x v x,x)
    ]]></literal>
    <justification jstring="[para(1630(a,1),55(a,1,1,1)),rewrite([40(10)])].">
      <j1 rule="para" parents="1630 55"/>
      <j2 rule="rewrite" parents="40"/>
    </justification>
  </clause>

  <clause id="1648">
    <literal><![CDATA[
      f(f(x,c(y)) ^ (y v y),f(c(y),f(f(f(y v y,y),f(y v y,f(y v y,y))),f(y v y,y)))) = c(y)
    ]]></literal>
    <justification jstring="[para(1630(a,1),55(a,1,1,2)),rewrite([45(9)])].">
      <j1 rule="para" parents="1630 55"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="1693">
    <literal><![CDATA[
      f(c(x),f(x,f(f(c(y),x v y),c(y)))) = x
    ]]></literal>
    <justification jstring="[para(41(a,1),1625(a,1,2,2,1,2))].">
      <j1 rule="para" parents="41 1625"/>
    </justification>
  </clause>

  <clause id="1739">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(1625(a,1),1625(a,1,2,2,1,2)),rewrite([1623(13),21(2),21(3),45(2)])].">
      <j1 rule="para" parents="1625 1625"/>
      <j2 rule="rewrite" parents="1623 21 21 45"/>
    </justification>
  </clause>

  <clause id="1786">
    <literal><![CDATA[
      f(f(x,c(y)) ^ y,f(c(y),f(f(c(y),f(y,c(y))),c(y)))) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1648),rewrite([1739(3),1739(5),21(5),1739(6),1739(6),21(6),1739(9),21(9)])].">
      <j1 rule="back_rewrite" parents="1648"/>
      <j2 rule="rewrite" parents="1739 1739 21 1739 1739 21 1739 21"/>
    </justification>
  </clause>

  <clause id="1787">
    <literal><![CDATA[
      f(x ^ f(c(x),y),f(c(x),f(f(y,f(x,y)),y))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(1647),rewrite([1739(1),1739(1),21(1),1739(4),21(4),1739(5),50(5),1739(5),1739(10),21(10)])].">
      <j1 rule="back_rewrite" parents="1647"/>
      <j2 rule="rewrite" parents="1739 1739 21 1739 21 1739 50 1739 1739 21"/>
    </justification>
  </clause>

  <clause id="1798">
    <literal><![CDATA[
      f(x v y,f(c(y),f(f(z,f(y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1628),rewrite([1739(3)])].">
      <j1 rule="back_rewrite" parents="1628"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="1958">
    <literal><![CDATA[
      f(f(f(x v c(y),c(y) v z),u),f(y,f(f(c(z),y v z),c(z)))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1263),rewrite([1739(7),1739(8),1739(14)])].">
      <j1 rule="back_rewrite" parents="1263"/>
      <j2 rule="rewrite" parents="1739 1739 1739"/>
    </justification>
  </clause>

  <clause id="2130">
    <literal><![CDATA[
      f(x v y,f(c(x),f(f(c(x),c(x) v x),c(x)))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(776),rewrite([1739(1)])].">
      <j1 rule="back_rewrite" parents="776"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="2302">
    <literal><![CDATA[
      f(f(x,y) ^ (c(y) v z),f(y,f(f(c(z),y v z),c(z)))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(304),rewrite([1739(1),1739(5),1739(6),1739(12)])].">
      <j1 rule="back_rewrite" parents="304"/>
      <j2 rule="rewrite" parents="1739 1739 1739 1739"/>
    </justification>
  </clause>

  <clause id="2374">
    <literal><![CDATA[
      x ^ y = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[back_rewrite(95),rewrite([1739(1),1739(1)])].">
      <j1 rule="back_rewrite" parents="95"/>
      <j2 rule="rewrite" parents="1739 1739"/>
    </justification>
  </clause>

  <clause id="2384">
    <literal><![CDATA[
      f(x,y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(77),rewrite([1739(1),1739(1)])].">
      <j1 rule="back_rewrite" parents="77"/>
      <j2 rule="rewrite" parents="1739 1739"/>
    </justification>
  </clause>

  <clause id="2392">
    <literal><![CDATA[
      c(c(x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(45),rewrite([1739(3)])].">
      <j1 rule="back_rewrite" parents="45"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="2400">
    <literal><![CDATA[
      (c(c(x) v c(y)) v c(c(y) v z)) v c(c(y) v c(c(z v c(y v z)) v z)) = y
    ]]></literal>
    <justification jstring="[back_rewrite(2302),rewrite([2384(1),2374(6),2384(12),2392(11),2384(14),2392(15),2384(15),2384(18),2392(10)])].">
      <j1 rule="back_rewrite" parents="2302"/>
      <j2 rule="rewrite" parents="2384 2374 2384 2392 2384 2392 2384 2384 2392"/>
    </justification>
  </clause>

  <clause id="2410">
    <literal><![CDATA[
      (c(x) v c(x v c(y))) v c(x v c(c(c(y) v c(c(x) v c(y))) v c(y))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(1787),rewrite([2384(2),2392(2),2374(3),2384(8),2384(11),2384(14),2384(17),2392(8),2384(18),2392(7)])].">
      <j1 rule="back_rewrite" parents="1787"/>
      <j2 rule="rewrite" parents="2384 2392 2374 2384 2384 2384 2384 2392 2384 2392"/>
    </justification>
  </clause>

  <clause id="2411">
    <literal><![CDATA[
      (c(c(x) v y) v c(y)) v c(y v c(c(y v c(c(y) v y)) v y)) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1786),rewrite([2384(2),2392(3),2374(3),2384(10),2392(11),2384(11),2392(9),2384(13),2392(14),2384(14),2392(8),2384(15),2392(7)])].">
      <j1 rule="back_rewrite" parents="1786"/>
      <j2 rule="rewrite" parents="2384 2392 2374 2384 2392 2384 2392 2384 2392 2384 2392 2384 2392"/>
    </justification>
  </clause>

  <clause id="2438">
    <literal><![CDATA[
      c(c(c22) v c22) != c(c(c21) v c21)
    ]]></literal>
    <attribute><![CDATA[
      answer(0)
    ]]></attribute>
    <justification jstring="[back_rewrite(37),rewrite([2374(4),2392(5),2374(9),2392(10)])].">
      <j1 rule="back_rewrite" parents="37"/>
      <j2 rule="rewrite" parents="2374 2392 2374 2392"/>
    </justification>
  </clause>

  <clause id="2450">
    <literal><![CDATA[
      c(x v y) v c(x v c(c(x v c(c(x) v x)) v x)) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(2130),rewrite([2384(6),2392(4),2384(8),2392(9),2384(9),2392(3),2384(10)])].">
      <j1 rule="back_rewrite" parents="2130"/>
      <j2 rule="rewrite" parents="2384 2392 2384 2392 2384 2392 2384"/>
    </justification>
  </clause>

  <clause id="2453">
    <literal><![CDATA[
      c(c(c(x v c(y)) v c(c(y) v z)) v c(u)) v c(c(y) v c(c(z v c(y v z)) v z)) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1958),rewrite([2384(5),2384(8),2384(13),2392(12),2384(15),2392(16),2384(16),2384(19)])].">
      <j1 rule="back_rewrite" parents="1958"/>
      <j2 rule="rewrite" parents="2384 2384 2384 2392 2384 2392 2384 2384"/>
    </justification>
  </clause>

  <clause id="2454">
    <literal><![CDATA[
      c(x v y) v c(y v c(c(c(z) v c(c(y) v c(z))) v c(z))) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1798),rewrite([2384(3),2384(6),2384(9),2384(12),2392(3),2384(13)])].">
      <j1 rule="back_rewrite" parents="1798"/>
      <j2 rule="rewrite" parents="2384 2384 2384 2384 2392 2384"/>
    </justification>
  </clause>

  <clause id="2465">
    <literal><![CDATA[
      x v c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1693),rewrite([2384(4),2392(3),2384(6),2392(7),2384(7),2384(10),2392(2)])].">
      <j1 rule="back_rewrite" parents="1693"/>
      <j2 rule="rewrite" parents="2384 2392 2384 2392 2384 2384 2392"/>
    </justification>
  </clause>

  <clause id="2479">
    <literal><![CDATA[
      c(x v c(c(x v c(c(x) v x)) v x)) = c(x)
    ]]></literal>
    <justification jstring="[para(2450(a,1),1739(a,1)),flip(a)].">
      <j1 rule="para" parents="2450 1739"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2481">
    <literal><![CDATA[
      c(x v y) v c(x) = c(x)
    ]]></literal>
    <justification jstring="[para(2450(a,1),2384(a,2)),rewrite([2384(10),2479(11)])].">
      <j1 rule="para" parents="2450 2384"/>
      <j2 rule="rewrite" parents="2384 2479"/>
    </justification>
  </clause>

  <clause id="2485">
    <literal><![CDATA[
      (c(c(x) v y) v c(y)) v c(y) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(2411),rewrite([2479(14)])].">
      <j1 rule="back_rewrite" parents="2411"/>
      <j2 rule="rewrite" parents="2479"/>
    </justification>
  </clause>

  <clause id="2486">
    <literal><![CDATA[
      c(c(x) v y) v x = x
    ]]></literal>
    <justification jstring="[para(2392(a,1),2481(a,1,2)),rewrite([2392(6)])].">
      <j1 rule="para" parents="2392 2481"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="2487">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2481(a,1,1,1)),rewrite([2392(2),2392(3),2392(5)])].">
      <j1 rule="para" parents="2481 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2492">
    <literal><![CDATA[
      c(x) v c(c(c(y) v c(x)) v c(c(x) v z)) = c(c(c(y) v c(x)) v c(c(x) v z))
    ]]></literal>
    <justification jstring="[para(2400(a,1),2481(a,1,1,1))].">
      <j1 rule="para" parents="2400 2481"/>
    </justification>
  </clause>

  <clause id="2500">
    <literal><![CDATA[
      (c(c(x) v c(y)) v y) v y = y
    ]]></literal>
    <justification jstring="[para(2392(a,1),2485(a,1,1,2)),rewrite([2392(7),2392(8)])].">
      <j1 rule="para" parents="2392 2485"/>
      <j2 rule="rewrite" parents="2392 2392"/>
    </justification>
  </clause>

  <clause id="2507">
    <literal><![CDATA[
      (c(x v c(y)) v y) v y = y
    ]]></literal>
    <justification jstring="[para(2392(a,1),2500(a,1,1,1,1,1))].">
      <j1 rule="para" parents="2392 2500"/>
    </justification>
  </clause>

  <clause id="2511">
    <literal><![CDATA[
      (c(x) v c(x v y)) v c(x v c(c(y v c(c(x) v y)) v y)) = c(x)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2410(a,1,1,2,1,2)),rewrite([2392(6),2392(7),2392(11)])].">
      <j1 rule="para" parents="2392 2410"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2513">
    <literal><![CDATA[
      (c(x) v c(x v c(y))) v c(x) = c(x) v c(x v c(y))
    ]]></literal>
    <justification jstring="[para(2410(a,1),2465(a,1,2,1,2,1,1,1,2,1)),rewrite([2392(25),2454(38),2392(13),2486(12)])].">
      <j1 rule="para" parents="2410 2465"/>
      <j2 rule="rewrite" parents="2392 2454 2392 2486"/>
    </justification>
  </clause>

  <clause id="2517">
    <literal><![CDATA[
      c(x) v c(x v c(y)) = c(x)
    ]]></literal>
    <justification jstring="[para(2410(a,1),2487(a,1,2)),rewrite([2513(7),2511(23)])].">
      <j1 rule="para" parents="2410 2487"/>
      <j2 rule="rewrite" parents="2513 2511"/>
    </justification>
  </clause>

  <clause id="2528">
    <literal><![CDATA[
      c(x) v c(x v y) = c(x)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2517(a,1,2,1,2))].">
      <j1 rule="para" parents="2392 2517"/>
    </justification>
  </clause>

  <clause id="2529">
    <literal><![CDATA[
      (x v y) v x = x v y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2517(a,1,2,1)),rewrite([2392(3),2392(3),2392(5)])].">
      <j1 rule="para" parents="2481 2517"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2533">
    <literal><![CDATA[
      x v c(c(x) v y) = x
    ]]></literal>
    <justification jstring="[para(2486(a,1),2529(a,1,1)),rewrite([2486(8)])].">
      <j1 rule="para" parents="2486 2529"/>
      <j2 rule="rewrite" parents="2486"/>
    </justification>
  </clause>

  <clause id="2552">
    <literal><![CDATA[
      c((c(x) v y) v c(z)) v c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[para(2481(a,1),2453(a,1,1,1,1,1)),rewrite([2392(4)])].">
      <j1 rule="para" parents="2481 2453"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="2553">
    <literal><![CDATA[
      c(x v c(y)) v c(c(y) v c(c(z v c(y v z)) v z)) = y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2453(a,1,1,1)),rewrite([2392(4)])].">
      <j1 rule="para" parents="2481 2453"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="2560">
    <literal><![CDATA[
      c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[para(2453(a,1),2507(a,1,1)),rewrite([2533(10)]),flip(a)].">
      <j1 rule="para" parents="2453 2507"/>
      <j2 rule="rewrite" parents="2533"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2574">
    <literal><![CDATA[
      c(x v c(y)) v y = y
    ]]></literal>
    <justification jstring="[back_rewrite(2553),rewrite([2560(12)])].">
      <j1 rule="back_rewrite" parents="2553"/>
      <j2 rule="rewrite" parents="2560"/>
    </justification>
  </clause>

  <clause id="2575">
    <literal><![CDATA[
      c((c(x) v y) v c(z)) v x = x
    ]]></literal>
    <justification jstring="[back_rewrite(2552),rewrite([2560(14)])].">
      <j1 rule="back_rewrite" parents="2552"/>
      <j2 rule="rewrite" parents="2560"/>
    </justification>
  </clause>

  <clause id="2583">
    <literal><![CDATA[
      c(x v y) v c(y) = c(y)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2574(a,1,1,1,2))].">
      <j1 rule="para" parents="2392 2574"/>
    </justification>
  </clause>

  <clause id="2585">
    <literal><![CDATA[
      x v c(y v c(x)) = x
    ]]></literal>
    <justification jstring="[para(2574(a,1),2529(a,1,1)),rewrite([2574(8)])].">
      <j1 rule="para" parents="2574 2529"/>
      <j2 rule="rewrite" parents="2574"/>
    </justification>
  </clause>

  <clause id="2591">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(2583(a,1),2481(a,1,1,1)),rewrite([2392(2),2392(3),2392(5)])].">
      <j1 rule="para" parents="2583 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2599">
    <literal><![CDATA[
      c((c(x) v y) v z) v x = x
    ]]></literal>
    <justification jstring="[para(2392(a,1),2575(a,1,1,1,2))].">
      <j1 rule="para" parents="2392 2575"/>
    </justification>
  </clause>

  <clause id="2612">
    <literal><![CDATA[
      c(c(x) v y) v (x v z) = x v z
    ]]></literal>
    <justification jstring="[para(2481(a,1),2599(a,1,1,1,1))].">
      <j1 rule="para" parents="2481 2599"/>
    </justification>
  </clause>

  <clause id="2614">
    <literal><![CDATA[
      x v c((c(x) v y) v z) = x
    ]]></literal>
    <justification jstring="[para(2599(a,1),2529(a,1,1)),rewrite([2599(10)])].">
      <j1 rule="para" parents="2599 2529"/>
      <j2 rule="rewrite" parents="2599"/>
    </justification>
  </clause>

  <clause id="2618">
    <literal><![CDATA[
      c((x v c(y)) v z) v y = y
    ]]></literal>
    <justification jstring="[para(2591(a,1),2599(a,1,1,1,1))].">
      <j1 rule="para" parents="2591 2599"/>
    </justification>
  </clause>

  <clause id="2619">
    <literal><![CDATA[
      c(x v (c(y) v z)) v y = y
    ]]></literal>
    <justification jstring="[para(2591(a,1),2599(a,1,1,1))].">
      <j1 rule="para" parents="2591 2599"/>
    </justification>
  </clause>

  <clause id="2645">
    <literal><![CDATA[
      (x v y) v c(c(x) v z) = x v y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2614(a,1,2,1,1))].">
      <j1 rule="para" parents="2481 2614"/>
    </justification>
  </clause>

  <clause id="2649">
    <literal><![CDATA[
      x v c((y v c(x)) v z) = x
    ]]></literal>
    <justification jstring="[para(2591(a,1),2614(a,1,2,1,1))].">
      <j1 rule="para" parents="2591 2614"/>
    </justification>
  </clause>

  <clause id="2650">
    <literal><![CDATA[
      x v c(y v (c(x) v z)) = x
    ]]></literal>
    <justification jstring="[para(2591(a,1),2614(a,1,2,1))].">
      <j1 rule="para" parents="2591 2614"/>
    </justification>
  </clause>

  <clause id="2656">
    <literal><![CDATA[
      c(x v (y v c(z))) v z = z
    ]]></literal>
    <justification jstring="[para(2591(a,1),2618(a,1,1,1))].">
      <j1 rule="para" parents="2591 2618"/>
    </justification>
  </clause>

  <clause id="2660">
    <literal><![CDATA[
      c(x v (y v z)) v c(y) = c(y)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2619(a,1,1,1,2,1))].">
      <j1 rule="para" parents="2392 2619"/>
    </justification>
  </clause>

  <clause id="2674">
    <literal><![CDATA[
      x v c(y v (z v c(x))) = x
    ]]></literal>
    <justification jstring="[para(2591(a,1),2649(a,1,2,1))].">
      <j1 rule="para" parents="2591 2649"/>
    </justification>
  </clause>

  <clause id="2675">
    <literal><![CDATA[
      x v c(c(c(y) v x) v c(x v z)) = c(c(c(y) v x) v c(x v z))
    ]]></literal>
    <justification jstring="[para(2392(a,1),2492(a,1,1)),rewrite([2392(3),2392(5),2392(11),2392(13)])].">
      <j1 rule="para" parents="2392 2492"/>
      <j2 rule="rewrite" parents="2392 2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2698">
    <literal><![CDATA[
      (x v y) v c(z v c(x)) = x v y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2650(a,1,2,1,2))].">
      <j1 rule="para" parents="2481 2650"/>
    </justification>
  </clause>

  <clause id="2706">
    <literal><![CDATA[
      c(x v (y v z)) v c(z) = c(z)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2656(a,1,1,1,2,2))].">
      <j1 rule="para" parents="2392 2656"/>
    </justification>
  </clause>

  <clause id="2737">
    <literal><![CDATA[
      x v (y v (x v z)) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(2660(a,1),2481(a,1,1,1)),rewrite([2392(2),2392(4),2392(7)])].">
      <j1 rule="para" parents="2660 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2746">
    <literal><![CDATA[
      c(c(x v y) v x) v c(x v y) = c(x)
    ]]></literal>
    <justification jstring="[para(2528(a,1),2560(a,1,1,2,1,1,1,2,1)),rewrite([2392(2),2392(4),2675(9),2392(9)])].">
      <j1 rule="para" parents="2528 2560"/>
      <j2 rule="rewrite" parents="2392 2392 2675 2392"/>
    </justification>
  </clause>

  <clause id="2765">
    <literal><![CDATA[
      (x v (y v z)) v z = x v (y v z)
    ]]></literal>
    <justification jstring="[para(2706(a,1),2533(a,1,2,1)),rewrite([2392(4)])].">
      <j1 rule="para" parents="2706 2533"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="2781">
    <literal><![CDATA[
      c((x v (y v c(z))) v u) v z = z
    ]]></literal>
    <justification jstring="[para(2656(a,1),2612(a,1,2)),rewrite([2392(5),2656(11)])].">
      <j1 rule="para" parents="2656 2612"/>
      <j2 rule="rewrite" parents="2392 2656"/>
    </justification>
  </clause>

  <clause id="2842">
    <literal><![CDATA[
      x v c((y v (z v c(x))) v u) = x
    ]]></literal>
    <justification jstring="[para(2656(a,1),2645(a,1,1)),rewrite([2392(5),2656(11)])].">
      <j1 rule="para" parents="2656 2645"/>
      <j2 rule="rewrite" parents="2392 2656"/>
    </justification>
  </clause>

  <clause id="4044">
    <literal><![CDATA[
      c(x v c(x v y)) v x = x v y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2746(a,1,1,1,1,1)),rewrite([2392(2),2481(8),2392(6),2392(8)])].">
      <j1 rule="para" parents="2481 2746"/>
      <j2 rule="rewrite" parents="2392 2481 2392 2392"/>
    </justification>
  </clause>

  <clause id="4089">
    <literal><![CDATA[
      c(x) v x = x v c(x)
    ]]></literal>
    <justification jstring="[para(2585(a,1),4044(a,1,1,1))].">
      <j1 rule="para" parents="2585 4044"/>
    </justification>
  </clause>

  <clause id="4091">
    <literal><![CDATA[
      x v (y v c(x)) = x v c(x)
    ]]></literal>
    <justification jstring="[para(2674(a,1),4044(a,1,1,1)),rewrite([4089(2)]),flip(a)].">
      <j1 rule="para" parents="2674 4044"/>
      <j2 rule="rewrite" parents="4089"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4094">
    <literal><![CDATA[
      (x v y) v c(x v y) = (x v y) v c(x)
    ]]></literal>
    <justification jstring="[para(2698(a,1),4044(a,1,1,1)),rewrite([4089(4)])].">
      <j1 rule="para" parents="2698 4044"/>
      <j2 rule="rewrite" parents="4089"/>
    </justification>
  </clause>

  <clause id="4129">
    <literal><![CDATA[
      c(c22 v c(c22)) != c(c21 v c(c21))
    ]]></literal>
    <attribute><![CDATA[
      answer(0)
    ]]></attribute>
    <justification jstring="[back_rewrite(2438),rewrite([4089(4),4089(9)])].">
      <j1 rule="back_rewrite" parents="2438"/>
      <j2 rule="rewrite" parents="4089 4089"/>
    </justification>
  </clause>

  <clause id="4130">
    <literal><![CDATA[
      (x v y) v c(x) = x v c(x)
    ]]></literal>
    <justification jstring="[para(4089(a,1),2737(a,1,2)),rewrite([4094(4),4091(4),4089(6),4094(6)]),flip(a)].">
      <j1 rule="para" parents="4089 2737"/>
      <j2 rule="rewrite" parents="4094 4091 4089 4094"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4132">
    <literal><![CDATA[
      (x v c(x)) v y = x v c(x)
    ]]></literal>
    <justification jstring="[para(4089(a,1),2765(a,1,1)),rewrite([4094(4),4130(3),4089(7),4094(7),4130(6)])].">
      <j1 rule="para" parents="4089 2765"/>
      <j2 rule="rewrite" parents="4094 4130 4089 4094 4130"/>
    </justification>
  </clause>

  <clause id="4133">
    <literal><![CDATA[
      c(x v c(x)) v y = y
    ]]></literal>
    <justification jstring="[para(4089(a,1),2781(a,1,1,1,1)),rewrite([4094(6),4130(4),4132(3)])].">
      <j1 rule="para" parents="4089 2781"/>
      <j2 rule="rewrite" parents="4094 4130 4132"/>
    </justification>
  </clause>

  <clause id="4134">
    <literal><![CDATA[
      x v c(y v c(y)) = x
    ]]></literal>
    <justification jstring="[para(4089(a,1),2842(a,1,2,1,1)),rewrite([4094(6),4130(4),4132(3)])].">
      <j1 rule="para" parents="4089 2842"/>
      <j2 rule="rewrite" parents="4094 4130 4132"/>
    </justification>
  </clause>

  <clause id="4209">
    <literal><![CDATA[
      c(x v c(x)) = c(y v c(y))
    ]]></literal>
    <justification jstring="[para(4134(a,1),4133(a,1))].">
      <j1 rule="para" parents="4134 4133"/>
    </justification>
  </clause>

  <clause id="4210">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(0)
    ]]></attribute>
    <justification jstring="[resolve(4209,a,4129,a)].">
      <j1 rule="resolve" parents="4209 4129"/>
    </justification>
  </clause>

</proof>

<proof number="9" length="82" max_count="37">

<comments><![CDATA[
% Proof 9 at 1.40 (+ 0.03) seconds: commutativity_meet.
% Length of proof is 82.
% Level of proof is 23.
% Maximum clause weight is 37.
% Given clauses 267.

]]></comments>

  <clause id="10" type="goal">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <attribute><![CDATA[
      answer(commutativity_meet)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="15" type="assumption">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="16" type="assumption">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <justification jstring="[copy(16),flip(a)].">
      <j1 rule="copy" parents="16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18" type="assumption">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_meet)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(f(x,y),f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[copy(18),flip(a)].">
      <j1 rule="copy" parents="18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20" type="assumption">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(x,x) = c(x)
    ]]></literal>
    <justification jstring="[copy(20),flip(a)].">
      <j1 rule="copy" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="33" type="deny">
    <literal><![CDATA[
      c18 v c17 != c17 v c18
    ]]></literal>
    <attribute><![CDATA[
      answer(commutativity_meet)
    ]]></attribute>
    <justification jstring="[deny(10)].">
      <j1 rule="deny" parents="10"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      c(f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(19),rewrite([21(3)])].">
      <j1 rule="back_rewrite" parents="19"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      f(c(x),c(y)) = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(17),rewrite([21(1),21(2)])].">
      <j1 rule="back_rewrite" parents="17"/>
      <j2 rule="rewrite" parents="21 21"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(15),rewrite([21(5)])].">
      <j1 rule="back_rewrite" parents="15"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      x ^ x = c(c(x))
    ]]></literal>
    <justification jstring="[para(21(a,1),40(a,1,1)),flip(a)].">
      <j1 rule="para" parents="21 40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      c(c(x)) = x v x
    ]]></literal>
    <justification jstring="[para(41(a,1),21(a,1)),flip(a)].">
      <j1 rule="para" parents="41 21"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      c(x) ^ c(y) = c(x v y)
    ]]></literal>
    <justification jstring="[para(41(a,1),40(a,1,1)),flip(a)].">
      <j1 rule="para" parents="41 40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      x ^ x = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(43),rewrite([45(3)])].">
      <j1 rule="back_rewrite" parents="43"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      f(f(f(c(x),f(x,y)),z),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1,1))].">
      <j1 rule="para" parents="21 42"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      f(f(f(f(x,y),c(y)),z),f(y,f(f(y,f(c(y),y)),y))) = y
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1,2))].">
      <j1 rule="para" parents="21 42"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      f(f(x v x,y),f(x,f(f(x,f(c(x),x)),x))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1)),rewrite([21(1),45(2)])].">
      <j1 rule="para" parents="21 42"/>
      <j2 rule="rewrite" parents="21 45"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      f(f(x,y) ^ f(y,z),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1)),rewrite([40(4)])].">
      <j1 rule="para" parents="21 42"/>
      <j2 rule="rewrite" parents="40"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      f(f(f(x v y,f(c(y),z)),u),f(c(y),f(f(z,f(y v y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[para(41(a,1),42(a,1,1,1,1)),rewrite([45(8)])].">
      <j1 rule="para" parents="41 42"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,c(z))),u),f(y,f(f(c(z),y v z),c(z)))) = y
    ]]></literal>
    <justification jstring="[para(41(a,1),42(a,1,2,2,1,2))].">
      <j1 rule="para" parents="41 42"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(f(f(z,f(c(x),z)),z),f(c(x),f(f(z,f(c(x),z)),z))),f(f(z,f(c(x),z)),z)))) = x
    ]]></literal>
    <justification jstring="[para(42(a,1),42(a,1,1,1))].">
      <j1 rule="para" parents="42 42"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      f(x v x,c(y)) = c(x) v y
    ]]></literal>
    <justification jstring="[para(45(a,1),41(a,1,1))].">
      <j1 rule="para" parents="45 41"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      f(c(x),y v y) = x v c(y)
    ]]></literal>
    <justification jstring="[para(45(a,1),41(a,1,2))].">
      <j1 rule="para" parents="45 41"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      c(x v x) = c(x) v c(x)
    ]]></literal>
    <justification jstring="[para(45(a,1),45(a,1,1))].">
      <j1 rule="para" parents="45 45"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      (x v x) ^ c(y) = c(c(x) v y)
    ]]></literal>
    <justification jstring="[para(45(a,1),46(a,1,1))].">
      <j1 rule="para" parents="45 46"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      f(x v x,y v y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[para(45(a,1),65(a,1,2))].">
      <j1 rule="para" parents="45 65"/>
    </justification>
  </clause>

  <clause id="95">
    <literal><![CDATA[
      (x v x) ^ (y v y) = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[para(45(a,1),72(a,1,2))].">
      <j1 rule="para" parents="45 72"/>
    </justification>
  </clause>

  <clause id="158">
    <literal><![CDATA[
      f(x,f(c(x),f(f(y,f(x v x,y)),y))) = c(x)
    ]]></literal>
    <justification jstring="[para(53(a,1),42(a,1,1)),rewrite([45(3)])].">
      <j1 rule="para" parents="53 42"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="203">
    <literal><![CDATA[
      f(c(x) v y,f(x,f(f(x,f(c(x),x)),x))) = x
    ]]></literal>
    <justification jstring="[para(65(a,1),54(a,1,1))].">
      <j1 rule="para" parents="65 54"/>
    </justification>
  </clause>

  <clause id="304">
    <literal><![CDATA[
      f(f(x,y v y) ^ (c(y) v z),f(y v y,f(f(c(z),(y v y) v z),c(z)))) = y v y
    ]]></literal>
    <justification jstring="[para(65(a,1),55(a,1,1,2)),rewrite([68(9),65(12),45(9)])].">
      <j1 rule="para" parents="65 55"/>
      <j2 rule="rewrite" parents="68 65 45"/>
    </justification>
  </clause>

  <clause id="776">
    <literal><![CDATA[
      f((x v x) v y,f(c(x),f(f(c(x),c(x) v x),c(x)))) = c(x)
    ]]></literal>
    <justification jstring="[para(41(a,1),203(a,1,2,2,1,2)),rewrite([45(2)])].">
      <j1 rule="para" parents="41 203"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="785">
    <literal><![CDATA[
      f(c(x),f(x v x,f(f(f(x,f(f(x,f(c(x),x)),x)),x),f(x,f(f(x,f(c(x),x)),x))))) = x v x
    ]]></literal>
    <justification jstring="[para(203(a,1),158(a,1,2,2,1,2)),rewrite([45(3),45(18)])].">
      <j1 rule="para" parents="203 158"/>
      <j2 rule="rewrite" parents="45 45"/>
    </justification>
  </clause>

  <clause id="1263">
    <literal><![CDATA[
      f(f(f(x v c(y),c(y) v z),u),f(y v y,f(f(c(z),(y v y) v z),c(z)))) = y v y
    ]]></literal>
    <justification jstring="[para(66(a,1),61(a,1,1,1,1)),rewrite([65(5)])].">
      <j1 rule="para" parents="66 61"/>
      <j2 rule="rewrite" parents="65"/>
    </justification>
  </clause>

  <clause id="1623">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(62(a,1),42(a,1,1))].">
      <j1 rule="para" parents="62 42"/>
    </justification>
  </clause>

  <clause id="1625">
    <literal><![CDATA[
      f(c(x),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(62(a,1),52(a,1,1))].">
      <j1 rule="para" parents="62 52"/>
    </justification>
  </clause>

  <clause id="1628">
    <literal><![CDATA[
      f(x v y,f(c(y),f(f(z,f(y v y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[para(62(a,1),59(a,1,1))].">
      <j1 rule="para" parents="62 59"/>
    </justification>
  </clause>

  <clause id="1630">
    <literal><![CDATA[
      f(c(x),f(x v x,x)) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(785),rewrite([1623(14)])].">
      <j1 rule="back_rewrite" parents="785"/>
      <j2 rule="rewrite" parents="1623"/>
    </justification>
  </clause>

  <clause id="1647">
    <literal><![CDATA[
      f((x v x) ^ f(f(x v x,x),y),f(f(x v x,x),f(f(y,f((x v x) ^ x,y)),y))) = f(x v x,x)
    ]]></literal>
    <justification jstring="[para(1630(a,1),55(a,1,1,1)),rewrite([40(10)])].">
      <j1 rule="para" parents="1630 55"/>
      <j2 rule="rewrite" parents="40"/>
    </justification>
  </clause>

  <clause id="1648">
    <literal><![CDATA[
      f(f(x,c(y)) ^ (y v y),f(c(y),f(f(f(y v y,y),f(y v y,f(y v y,y))),f(y v y,y)))) = c(y)
    ]]></literal>
    <justification jstring="[para(1630(a,1),55(a,1,1,2)),rewrite([45(9)])].">
      <j1 rule="para" parents="1630 55"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="1693">
    <literal><![CDATA[
      f(c(x),f(x,f(f(c(y),x v y),c(y)))) = x
    ]]></literal>
    <justification jstring="[para(41(a,1),1625(a,1,2,2,1,2))].">
      <j1 rule="para" parents="41 1625"/>
    </justification>
  </clause>

  <clause id="1739">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(1625(a,1),1625(a,1,2,2,1,2)),rewrite([1623(13),21(2),21(3),45(2)])].">
      <j1 rule="para" parents="1625 1625"/>
      <j2 rule="rewrite" parents="1623 21 21 45"/>
    </justification>
  </clause>

  <clause id="1786">
    <literal><![CDATA[
      f(f(x,c(y)) ^ y,f(c(y),f(f(c(y),f(y,c(y))),c(y)))) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1648),rewrite([1739(3),1739(5),21(5),1739(6),1739(6),21(6),1739(9),21(9)])].">
      <j1 rule="back_rewrite" parents="1648"/>
      <j2 rule="rewrite" parents="1739 1739 21 1739 1739 21 1739 21"/>
    </justification>
  </clause>

  <clause id="1787">
    <literal><![CDATA[
      f(x ^ f(c(x),y),f(c(x),f(f(y,f(x,y)),y))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(1647),rewrite([1739(1),1739(1),21(1),1739(4),21(4),1739(5),50(5),1739(5),1739(10),21(10)])].">
      <j1 rule="back_rewrite" parents="1647"/>
      <j2 rule="rewrite" parents="1739 1739 21 1739 21 1739 50 1739 1739 21"/>
    </justification>
  </clause>

  <clause id="1798">
    <literal><![CDATA[
      f(x v y,f(c(y),f(f(z,f(y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1628),rewrite([1739(3)])].">
      <j1 rule="back_rewrite" parents="1628"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="1958">
    <literal><![CDATA[
      f(f(f(x v c(y),c(y) v z),u),f(y,f(f(c(z),y v z),c(z)))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1263),rewrite([1739(7),1739(8),1739(14)])].">
      <j1 rule="back_rewrite" parents="1263"/>
      <j2 rule="rewrite" parents="1739 1739 1739"/>
    </justification>
  </clause>

  <clause id="2130">
    <literal><![CDATA[
      f(x v y,f(c(x),f(f(c(x),c(x) v x),c(x)))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(776),rewrite([1739(1)])].">
      <j1 rule="back_rewrite" parents="776"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="2302">
    <literal><![CDATA[
      f(f(x,y) ^ (c(y) v z),f(y,f(f(c(z),y v z),c(z)))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(304),rewrite([1739(1),1739(5),1739(6),1739(12)])].">
      <j1 rule="back_rewrite" parents="304"/>
      <j2 rule="rewrite" parents="1739 1739 1739 1739"/>
    </justification>
  </clause>

  <clause id="2374">
    <literal><![CDATA[
      x ^ y = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[back_rewrite(95),rewrite([1739(1),1739(1)])].">
      <j1 rule="back_rewrite" parents="95"/>
      <j2 rule="rewrite" parents="1739 1739"/>
    </justification>
  </clause>

  <clause id="2384">
    <literal><![CDATA[
      f(x,y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(77),rewrite([1739(1),1739(1)])].">
      <j1 rule="back_rewrite" parents="77"/>
      <j2 rule="rewrite" parents="1739 1739"/>
    </justification>
  </clause>

  <clause id="2392">
    <literal><![CDATA[
      c(c(x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(45),rewrite([1739(3)])].">
      <j1 rule="back_rewrite" parents="45"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="2400">
    <literal><![CDATA[
      (c(c(x) v c(y)) v c(c(y) v z)) v c(c(y) v c(c(z v c(y v z)) v z)) = y
    ]]></literal>
    <justification jstring="[back_rewrite(2302),rewrite([2384(1),2374(6),2384(12),2392(11),2384(14),2392(15),2384(15),2384(18),2392(10)])].">
      <j1 rule="back_rewrite" parents="2302"/>
      <j2 rule="rewrite" parents="2384 2374 2384 2392 2384 2392 2384 2384 2392"/>
    </justification>
  </clause>

  <clause id="2410">
    <literal><![CDATA[
      (c(x) v c(x v c(y))) v c(x v c(c(c(y) v c(c(x) v c(y))) v c(y))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(1787),rewrite([2384(2),2392(2),2374(3),2384(8),2384(11),2384(14),2384(17),2392(8),2384(18),2392(7)])].">
      <j1 rule="back_rewrite" parents="1787"/>
      <j2 rule="rewrite" parents="2384 2392 2374 2384 2384 2384 2384 2392 2384 2392"/>
    </justification>
  </clause>

  <clause id="2411">
    <literal><![CDATA[
      (c(c(x) v y) v c(y)) v c(y v c(c(y v c(c(y) v y)) v y)) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1786),rewrite([2384(2),2392(3),2374(3),2384(10),2392(11),2384(11),2392(9),2384(13),2392(14),2384(14),2392(8),2384(15),2392(7)])].">
      <j1 rule="back_rewrite" parents="1786"/>
      <j2 rule="rewrite" parents="2384 2392 2374 2384 2392 2384 2392 2384 2392 2384 2392 2384 2392"/>
    </justification>
  </clause>

  <clause id="2450">
    <literal><![CDATA[
      c(x v y) v c(x v c(c(x v c(c(x) v x)) v x)) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(2130),rewrite([2384(6),2392(4),2384(8),2392(9),2384(9),2392(3),2384(10)])].">
      <j1 rule="back_rewrite" parents="2130"/>
      <j2 rule="rewrite" parents="2384 2392 2384 2392 2384 2392 2384"/>
    </justification>
  </clause>

  <clause id="2453">
    <literal><![CDATA[
      c(c(c(x v c(y)) v c(c(y) v z)) v c(u)) v c(c(y) v c(c(z v c(y v z)) v z)) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1958),rewrite([2384(5),2384(8),2384(13),2392(12),2384(15),2392(16),2384(16),2384(19)])].">
      <j1 rule="back_rewrite" parents="1958"/>
      <j2 rule="rewrite" parents="2384 2384 2384 2392 2384 2392 2384 2384"/>
    </justification>
  </clause>

  <clause id="2454">
    <literal><![CDATA[
      c(x v y) v c(y v c(c(c(z) v c(c(y) v c(z))) v c(z))) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1798),rewrite([2384(3),2384(6),2384(9),2384(12),2392(3),2384(13)])].">
      <j1 rule="back_rewrite" parents="1798"/>
      <j2 rule="rewrite" parents="2384 2384 2384 2384 2392 2384"/>
    </justification>
  </clause>

  <clause id="2465">
    <literal><![CDATA[
      x v c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1693),rewrite([2384(4),2392(3),2384(6),2392(7),2384(7),2384(10),2392(2)])].">
      <j1 rule="back_rewrite" parents="1693"/>
      <j2 rule="rewrite" parents="2384 2392 2384 2392 2384 2384 2392"/>
    </justification>
  </clause>

  <clause id="2479">
    <literal><![CDATA[
      c(x v c(c(x v c(c(x) v x)) v x)) = c(x)
    ]]></literal>
    <justification jstring="[para(2450(a,1),1739(a,1)),flip(a)].">
      <j1 rule="para" parents="2450 1739"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2481">
    <literal><![CDATA[
      c(x v y) v c(x) = c(x)
    ]]></literal>
    <justification jstring="[para(2450(a,1),2384(a,2)),rewrite([2384(10),2479(11)])].">
      <j1 rule="para" parents="2450 2384"/>
      <j2 rule="rewrite" parents="2384 2479"/>
    </justification>
  </clause>

  <clause id="2485">
    <literal><![CDATA[
      (c(c(x) v y) v c(y)) v c(y) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(2411),rewrite([2479(14)])].">
      <j1 rule="back_rewrite" parents="2411"/>
      <j2 rule="rewrite" parents="2479"/>
    </justification>
  </clause>

  <clause id="2486">
    <literal><![CDATA[
      c(c(x) v y) v x = x
    ]]></literal>
    <justification jstring="[para(2392(a,1),2481(a,1,2)),rewrite([2392(6)])].">
      <j1 rule="para" parents="2392 2481"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="2487">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2481(a,1,1,1)),rewrite([2392(2),2392(3),2392(5)])].">
      <j1 rule="para" parents="2481 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2492">
    <literal><![CDATA[
      c(x) v c(c(c(y) v c(x)) v c(c(x) v z)) = c(c(c(y) v c(x)) v c(c(x) v z))
    ]]></literal>
    <justification jstring="[para(2400(a,1),2481(a,1,1,1))].">
      <j1 rule="para" parents="2400 2481"/>
    </justification>
  </clause>

  <clause id="2500">
    <literal><![CDATA[
      (c(c(x) v c(y)) v y) v y = y
    ]]></literal>
    <justification jstring="[para(2392(a,1),2485(a,1,1,2)),rewrite([2392(7),2392(8)])].">
      <j1 rule="para" parents="2392 2485"/>
      <j2 rule="rewrite" parents="2392 2392"/>
    </justification>
  </clause>

  <clause id="2507">
    <literal><![CDATA[
      (c(x v c(y)) v y) v y = y
    ]]></literal>
    <justification jstring="[para(2392(a,1),2500(a,1,1,1,1,1))].">
      <j1 rule="para" parents="2392 2500"/>
    </justification>
  </clause>

  <clause id="2511">
    <literal><![CDATA[
      (c(x) v c(x v y)) v c(x v c(c(y v c(c(x) v y)) v y)) = c(x)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2410(a,1,1,2,1,2)),rewrite([2392(6),2392(7),2392(11)])].">
      <j1 rule="para" parents="2392 2410"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2513">
    <literal><![CDATA[
      (c(x) v c(x v c(y))) v c(x) = c(x) v c(x v c(y))
    ]]></literal>
    <justification jstring="[para(2410(a,1),2465(a,1,2,1,2,1,1,1,2,1)),rewrite([2392(25),2454(38),2392(13),2486(12)])].">
      <j1 rule="para" parents="2410 2465"/>
      <j2 rule="rewrite" parents="2392 2454 2392 2486"/>
    </justification>
  </clause>

  <clause id="2517">
    <literal><![CDATA[
      c(x) v c(x v c(y)) = c(x)
    ]]></literal>
    <justification jstring="[para(2410(a,1),2487(a,1,2)),rewrite([2513(7),2511(23)])].">
      <j1 rule="para" parents="2410 2487"/>
      <j2 rule="rewrite" parents="2513 2511"/>
    </justification>
  </clause>

  <clause id="2529">
    <literal><![CDATA[
      (x v y) v x = x v y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2517(a,1,2,1)),rewrite([2392(3),2392(3),2392(5)])].">
      <j1 rule="para" parents="2481 2517"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2533">
    <literal><![CDATA[
      x v c(c(x) v y) = x
    ]]></literal>
    <justification jstring="[para(2486(a,1),2529(a,1,1)),rewrite([2486(8)])].">
      <j1 rule="para" parents="2486 2529"/>
      <j2 rule="rewrite" parents="2486"/>
    </justification>
  </clause>

  <clause id="2553">
    <literal><![CDATA[
      c(x v c(y)) v c(c(y) v c(c(z v c(y v z)) v z)) = y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2453(a,1,1,1)),rewrite([2392(4)])].">
      <j1 rule="para" parents="2481 2453"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="2560">
    <literal><![CDATA[
      c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[para(2453(a,1),2507(a,1,1)),rewrite([2533(10)]),flip(a)].">
      <j1 rule="para" parents="2453 2507"/>
      <j2 rule="rewrite" parents="2533"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2574">
    <literal><![CDATA[
      c(x v c(y)) v y = y
    ]]></literal>
    <justification jstring="[back_rewrite(2553),rewrite([2560(12)])].">
      <j1 rule="back_rewrite" parents="2553"/>
      <j2 rule="rewrite" parents="2560"/>
    </justification>
  </clause>

  <clause id="2585">
    <literal><![CDATA[
      x v c(y v c(x)) = x
    ]]></literal>
    <justification jstring="[para(2574(a,1),2529(a,1,1)),rewrite([2574(8)])].">
      <j1 rule="para" parents="2574 2529"/>
      <j2 rule="rewrite" parents="2574"/>
    </justification>
  </clause>

  <clause id="2590">
    <literal><![CDATA[
      c(x) v c(y v x) = c(x)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2585(a,1,2,1,2))].">
      <j1 rule="para" parents="2392 2585"/>
    </justification>
  </clause>

  <clause id="2677">
    <literal><![CDATA[
      c(x v y) v c(c(c(z) v c(x v y)) v x) = c(c(c(z) v c(x v y)) v x)
    ]]></literal>
    <justification jstring="[para(2481(a,1),2492(a,1,2,1,2,1)),rewrite([2392(9),2481(19),2392(17)])].">
      <j1 rule="para" parents="2481 2492"/>
      <j2 rule="rewrite" parents="2392 2481 2392"/>
    </justification>
  </clause>

  <clause id="5628">
    <literal><![CDATA[
      c(x v y) v c(y v x) = c(y v x)
    ]]></literal>
    <justification jstring="[para(2585(a,1),2677(a,1,2,1,1,1)),rewrite([2392(2),2392(4),2392(8),2590(9),2392(7)])].">
      <j1 rule="para" parents="2585 2677"/>
      <j2 rule="rewrite" parents="2392 2392 2392 2590 2392"/>
    </justification>
  </clause>

  <clause id="5749">
    <literal><![CDATA[
      (x v y) v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(5628(a,1),2481(a,1,1,1)),rewrite([2392(3),2392(4),2392(6)])].">
      <j1 rule="para" parents="5628 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="5751">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[para(5628(a,1),2533(a,1,2,1)),rewrite([2392(4),5749(3)])].">
      <j1 rule="para" parents="5628 2533"/>
      <j2 rule="rewrite" parents="2392 5749"/>
    </justification>
  </clause>

  <clause id="5752">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(commutativity_meet)
    ]]></attribute>
    <justification jstring="[resolve(5751,a,33,a)].">
      <j1 rule="resolve" parents="5751 33"/>
    </justification>
  </clause>

</proof>

<proof number="10" length="82" max_count="37">

<comments><![CDATA[
% Proof 10 at 1.40 (+ 0.03) seconds: commutativity_join.
% Length of proof is 82.
% Level of proof is 23.
% Maximum clause weight is 37.
% Given clauses 267.

]]></comments>

  <clause id="9" type="goal">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <attribute><![CDATA[
      answer(commutativity_join)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="15" type="assumption">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="16" type="assumption">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <justification jstring="[copy(16),flip(a)].">
      <j1 rule="copy" parents="16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18" type="assumption">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_meet)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(f(x,y),f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[copy(18),flip(a)].">
      <j1 rule="copy" parents="18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20" type="assumption">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(x,x) = c(x)
    ]]></literal>
    <justification jstring="[copy(20),flip(a)].">
      <j1 rule="copy" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="32" type="deny">
    <literal><![CDATA[
      c16 v c15 != c15 v c16
    ]]></literal>
    <attribute><![CDATA[
      answer(commutativity_join)
    ]]></attribute>
    <justification jstring="[deny(9)].">
      <j1 rule="deny" parents="9"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      c(f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(19),rewrite([21(3)])].">
      <j1 rule="back_rewrite" parents="19"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      f(c(x),c(y)) = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(17),rewrite([21(1),21(2)])].">
      <j1 rule="back_rewrite" parents="17"/>
      <j2 rule="rewrite" parents="21 21"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(15),rewrite([21(5)])].">
      <j1 rule="back_rewrite" parents="15"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      x ^ x = c(c(x))
    ]]></literal>
    <justification jstring="[para(21(a,1),40(a,1,1)),flip(a)].">
      <j1 rule="para" parents="21 40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      c(c(x)) = x v x
    ]]></literal>
    <justification jstring="[para(41(a,1),21(a,1)),flip(a)].">
      <j1 rule="para" parents="41 21"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      c(x) ^ c(y) = c(x v y)
    ]]></literal>
    <justification jstring="[para(41(a,1),40(a,1,1)),flip(a)].">
      <j1 rule="para" parents="41 40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      x ^ x = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(43),rewrite([45(3)])].">
      <j1 rule="back_rewrite" parents="43"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      f(f(f(c(x),f(x,y)),z),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1,1))].">
      <j1 rule="para" parents="21 42"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      f(f(f(f(x,y),c(y)),z),f(y,f(f(y,f(c(y),y)),y))) = y
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1,2))].">
      <j1 rule="para" parents="21 42"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      f(f(x v x,y),f(x,f(f(x,f(c(x),x)),x))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1)),rewrite([21(1),45(2)])].">
      <j1 rule="para" parents="21 42"/>
      <j2 rule="rewrite" parents="21 45"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      f(f(x,y) ^ f(y,z),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1)),rewrite([40(4)])].">
      <j1 rule="para" parents="21 42"/>
      <j2 rule="rewrite" parents="40"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      f(f(f(x v y,f(c(y),z)),u),f(c(y),f(f(z,f(y v y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[para(41(a,1),42(a,1,1,1,1)),rewrite([45(8)])].">
      <j1 rule="para" parents="41 42"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,c(z))),u),f(y,f(f(c(z),y v z),c(z)))) = y
    ]]></literal>
    <justification jstring="[para(41(a,1),42(a,1,2,2,1,2))].">
      <j1 rule="para" parents="41 42"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(f(f(z,f(c(x),z)),z),f(c(x),f(f(z,f(c(x),z)),z))),f(f(z,f(c(x),z)),z)))) = x
    ]]></literal>
    <justification jstring="[para(42(a,1),42(a,1,1,1))].">
      <j1 rule="para" parents="42 42"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      f(x v x,c(y)) = c(x) v y
    ]]></literal>
    <justification jstring="[para(45(a,1),41(a,1,1))].">
      <j1 rule="para" parents="45 41"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      f(c(x),y v y) = x v c(y)
    ]]></literal>
    <justification jstring="[para(45(a,1),41(a,1,2))].">
      <j1 rule="para" parents="45 41"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      c(x v x) = c(x) v c(x)
    ]]></literal>
    <justification jstring="[para(45(a,1),45(a,1,1))].">
      <j1 rule="para" parents="45 45"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      (x v x) ^ c(y) = c(c(x) v y)
    ]]></literal>
    <justification jstring="[para(45(a,1),46(a,1,1))].">
      <j1 rule="para" parents="45 46"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      f(x v x,y v y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[para(45(a,1),65(a,1,2))].">
      <j1 rule="para" parents="45 65"/>
    </justification>
  </clause>

  <clause id="95">
    <literal><![CDATA[
      (x v x) ^ (y v y) = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[para(45(a,1),72(a,1,2))].">
      <j1 rule="para" parents="45 72"/>
    </justification>
  </clause>

  <clause id="158">
    <literal><![CDATA[
      f(x,f(c(x),f(f(y,f(x v x,y)),y))) = c(x)
    ]]></literal>
    <justification jstring="[para(53(a,1),42(a,1,1)),rewrite([45(3)])].">
      <j1 rule="para" parents="53 42"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="203">
    <literal><![CDATA[
      f(c(x) v y,f(x,f(f(x,f(c(x),x)),x))) = x
    ]]></literal>
    <justification jstring="[para(65(a,1),54(a,1,1))].">
      <j1 rule="para" parents="65 54"/>
    </justification>
  </clause>

  <clause id="304">
    <literal><![CDATA[
      f(f(x,y v y) ^ (c(y) v z),f(y v y,f(f(c(z),(y v y) v z),c(z)))) = y v y
    ]]></literal>
    <justification jstring="[para(65(a,1),55(a,1,1,2)),rewrite([68(9),65(12),45(9)])].">
      <j1 rule="para" parents="65 55"/>
      <j2 rule="rewrite" parents="68 65 45"/>
    </justification>
  </clause>

  <clause id="776">
    <literal><![CDATA[
      f((x v x) v y,f(c(x),f(f(c(x),c(x) v x),c(x)))) = c(x)
    ]]></literal>
    <justification jstring="[para(41(a,1),203(a,1,2,2,1,2)),rewrite([45(2)])].">
      <j1 rule="para" parents="41 203"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="785">
    <literal><![CDATA[
      f(c(x),f(x v x,f(f(f(x,f(f(x,f(c(x),x)),x)),x),f(x,f(f(x,f(c(x),x)),x))))) = x v x
    ]]></literal>
    <justification jstring="[para(203(a,1),158(a,1,2,2,1,2)),rewrite([45(3),45(18)])].">
      <j1 rule="para" parents="203 158"/>
      <j2 rule="rewrite" parents="45 45"/>
    </justification>
  </clause>

  <clause id="1263">
    <literal><![CDATA[
      f(f(f(x v c(y),c(y) v z),u),f(y v y,f(f(c(z),(y v y) v z),c(z)))) = y v y
    ]]></literal>
    <justification jstring="[para(66(a,1),61(a,1,1,1,1)),rewrite([65(5)])].">
      <j1 rule="para" parents="66 61"/>
      <j2 rule="rewrite" parents="65"/>
    </justification>
  </clause>

  <clause id="1623">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(62(a,1),42(a,1,1))].">
      <j1 rule="para" parents="62 42"/>
    </justification>
  </clause>

  <clause id="1625">
    <literal><![CDATA[
      f(c(x),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(62(a,1),52(a,1,1))].">
      <j1 rule="para" parents="62 52"/>
    </justification>
  </clause>

  <clause id="1628">
    <literal><![CDATA[
      f(x v y,f(c(y),f(f(z,f(y v y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[para(62(a,1),59(a,1,1))].">
      <j1 rule="para" parents="62 59"/>
    </justification>
  </clause>

  <clause id="1630">
    <literal><![CDATA[
      f(c(x),f(x v x,x)) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(785),rewrite([1623(14)])].">
      <j1 rule="back_rewrite" parents="785"/>
      <j2 rule="rewrite" parents="1623"/>
    </justification>
  </clause>

  <clause id="1647">
    <literal><![CDATA[
      f((x v x) ^ f(f(x v x,x),y),f(f(x v x,x),f(f(y,f((x v x) ^ x,y)),y))) = f(x v x,x)
    ]]></literal>
    <justification jstring="[para(1630(a,1),55(a,1,1,1)),rewrite([40(10)])].">
      <j1 rule="para" parents="1630 55"/>
      <j2 rule="rewrite" parents="40"/>
    </justification>
  </clause>

  <clause id="1648">
    <literal><![CDATA[
      f(f(x,c(y)) ^ (y v y),f(c(y),f(f(f(y v y,y),f(y v y,f(y v y,y))),f(y v y,y)))) = c(y)
    ]]></literal>
    <justification jstring="[para(1630(a,1),55(a,1,1,2)),rewrite([45(9)])].">
      <j1 rule="para" parents="1630 55"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="1693">
    <literal><![CDATA[
      f(c(x),f(x,f(f(c(y),x v y),c(y)))) = x
    ]]></literal>
    <justification jstring="[para(41(a,1),1625(a,1,2,2,1,2))].">
      <j1 rule="para" parents="41 1625"/>
    </justification>
  </clause>

  <clause id="1739">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(1625(a,1),1625(a,1,2,2,1,2)),rewrite([1623(13),21(2),21(3),45(2)])].">
      <j1 rule="para" parents="1625 1625"/>
      <j2 rule="rewrite" parents="1623 21 21 45"/>
    </justification>
  </clause>

  <clause id="1786">
    <literal><![CDATA[
      f(f(x,c(y)) ^ y,f(c(y),f(f(c(y),f(y,c(y))),c(y)))) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1648),rewrite([1739(3),1739(5),21(5),1739(6),1739(6),21(6),1739(9),21(9)])].">
      <j1 rule="back_rewrite" parents="1648"/>
      <j2 rule="rewrite" parents="1739 1739 21 1739 1739 21 1739 21"/>
    </justification>
  </clause>

  <clause id="1787">
    <literal><![CDATA[
      f(x ^ f(c(x),y),f(c(x),f(f(y,f(x,y)),y))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(1647),rewrite([1739(1),1739(1),21(1),1739(4),21(4),1739(5),50(5),1739(5),1739(10),21(10)])].">
      <j1 rule="back_rewrite" parents="1647"/>
      <j2 rule="rewrite" parents="1739 1739 21 1739 21 1739 50 1739 1739 21"/>
    </justification>
  </clause>

  <clause id="1798">
    <literal><![CDATA[
      f(x v y,f(c(y),f(f(z,f(y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1628),rewrite([1739(3)])].">
      <j1 rule="back_rewrite" parents="1628"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="1958">
    <literal><![CDATA[
      f(f(f(x v c(y),c(y) v z),u),f(y,f(f(c(z),y v z),c(z)))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1263),rewrite([1739(7),1739(8),1739(14)])].">
      <j1 rule="back_rewrite" parents="1263"/>
      <j2 rule="rewrite" parents="1739 1739 1739"/>
    </justification>
  </clause>

  <clause id="2130">
    <literal><![CDATA[
      f(x v y,f(c(x),f(f(c(x),c(x) v x),c(x)))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(776),rewrite([1739(1)])].">
      <j1 rule="back_rewrite" parents="776"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="2302">
    <literal><![CDATA[
      f(f(x,y) ^ (c(y) v z),f(y,f(f(c(z),y v z),c(z)))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(304),rewrite([1739(1),1739(5),1739(6),1739(12)])].">
      <j1 rule="back_rewrite" parents="304"/>
      <j2 rule="rewrite" parents="1739 1739 1739 1739"/>
    </justification>
  </clause>

  <clause id="2374">
    <literal><![CDATA[
      x ^ y = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[back_rewrite(95),rewrite([1739(1),1739(1)])].">
      <j1 rule="back_rewrite" parents="95"/>
      <j2 rule="rewrite" parents="1739 1739"/>
    </justification>
  </clause>

  <clause id="2384">
    <literal><![CDATA[
      f(x,y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(77),rewrite([1739(1),1739(1)])].">
      <j1 rule="back_rewrite" parents="77"/>
      <j2 rule="rewrite" parents="1739 1739"/>
    </justification>
  </clause>

  <clause id="2392">
    <literal><![CDATA[
      c(c(x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(45),rewrite([1739(3)])].">
      <j1 rule="back_rewrite" parents="45"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="2400">
    <literal><![CDATA[
      (c(c(x) v c(y)) v c(c(y) v z)) v c(c(y) v c(c(z v c(y v z)) v z)) = y
    ]]></literal>
    <justification jstring="[back_rewrite(2302),rewrite([2384(1),2374(6),2384(12),2392(11),2384(14),2392(15),2384(15),2384(18),2392(10)])].">
      <j1 rule="back_rewrite" parents="2302"/>
      <j2 rule="rewrite" parents="2384 2374 2384 2392 2384 2392 2384 2384 2392"/>
    </justification>
  </clause>

  <clause id="2410">
    <literal><![CDATA[
      (c(x) v c(x v c(y))) v c(x v c(c(c(y) v c(c(x) v c(y))) v c(y))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(1787),rewrite([2384(2),2392(2),2374(3),2384(8),2384(11),2384(14),2384(17),2392(8),2384(18),2392(7)])].">
      <j1 rule="back_rewrite" parents="1787"/>
      <j2 rule="rewrite" parents="2384 2392 2374 2384 2384 2384 2384 2392 2384 2392"/>
    </justification>
  </clause>

  <clause id="2411">
    <literal><![CDATA[
      (c(c(x) v y) v c(y)) v c(y v c(c(y v c(c(y) v y)) v y)) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1786),rewrite([2384(2),2392(3),2374(3),2384(10),2392(11),2384(11),2392(9),2384(13),2392(14),2384(14),2392(8),2384(15),2392(7)])].">
      <j1 rule="back_rewrite" parents="1786"/>
      <j2 rule="rewrite" parents="2384 2392 2374 2384 2392 2384 2392 2384 2392 2384 2392 2384 2392"/>
    </justification>
  </clause>

  <clause id="2450">
    <literal><![CDATA[
      c(x v y) v c(x v c(c(x v c(c(x) v x)) v x)) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(2130),rewrite([2384(6),2392(4),2384(8),2392(9),2384(9),2392(3),2384(10)])].">
      <j1 rule="back_rewrite" parents="2130"/>
      <j2 rule="rewrite" parents="2384 2392 2384 2392 2384 2392 2384"/>
    </justification>
  </clause>

  <clause id="2453">
    <literal><![CDATA[
      c(c(c(x v c(y)) v c(c(y) v z)) v c(u)) v c(c(y) v c(c(z v c(y v z)) v z)) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1958),rewrite([2384(5),2384(8),2384(13),2392(12),2384(15),2392(16),2384(16),2384(19)])].">
      <j1 rule="back_rewrite" parents="1958"/>
      <j2 rule="rewrite" parents="2384 2384 2384 2392 2384 2392 2384 2384"/>
    </justification>
  </clause>

  <clause id="2454">
    <literal><![CDATA[
      c(x v y) v c(y v c(c(c(z) v c(c(y) v c(z))) v c(z))) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1798),rewrite([2384(3),2384(6),2384(9),2384(12),2392(3),2384(13)])].">
      <j1 rule="back_rewrite" parents="1798"/>
      <j2 rule="rewrite" parents="2384 2384 2384 2384 2392 2384"/>
    </justification>
  </clause>

  <clause id="2465">
    <literal><![CDATA[
      x v c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1693),rewrite([2384(4),2392(3),2384(6),2392(7),2384(7),2384(10),2392(2)])].">
      <j1 rule="back_rewrite" parents="1693"/>
      <j2 rule="rewrite" parents="2384 2392 2384 2392 2384 2384 2392"/>
    </justification>
  </clause>

  <clause id="2479">
    <literal><![CDATA[
      c(x v c(c(x v c(c(x) v x)) v x)) = c(x)
    ]]></literal>
    <justification jstring="[para(2450(a,1),1739(a,1)),flip(a)].">
      <j1 rule="para" parents="2450 1739"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2481">
    <literal><![CDATA[
      c(x v y) v c(x) = c(x)
    ]]></literal>
    <justification jstring="[para(2450(a,1),2384(a,2)),rewrite([2384(10),2479(11)])].">
      <j1 rule="para" parents="2450 2384"/>
      <j2 rule="rewrite" parents="2384 2479"/>
    </justification>
  </clause>

  <clause id="2485">
    <literal><![CDATA[
      (c(c(x) v y) v c(y)) v c(y) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(2411),rewrite([2479(14)])].">
      <j1 rule="back_rewrite" parents="2411"/>
      <j2 rule="rewrite" parents="2479"/>
    </justification>
  </clause>

  <clause id="2486">
    <literal><![CDATA[
      c(c(x) v y) v x = x
    ]]></literal>
    <justification jstring="[para(2392(a,1),2481(a,1,2)),rewrite([2392(6)])].">
      <j1 rule="para" parents="2392 2481"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="2487">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2481(a,1,1,1)),rewrite([2392(2),2392(3),2392(5)])].">
      <j1 rule="para" parents="2481 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2492">
    <literal><![CDATA[
      c(x) v c(c(c(y) v c(x)) v c(c(x) v z)) = c(c(c(y) v c(x)) v c(c(x) v z))
    ]]></literal>
    <justification jstring="[para(2400(a,1),2481(a,1,1,1))].">
      <j1 rule="para" parents="2400 2481"/>
    </justification>
  </clause>

  <clause id="2500">
    <literal><![CDATA[
      (c(c(x) v c(y)) v y) v y = y
    ]]></literal>
    <justification jstring="[para(2392(a,1),2485(a,1,1,2)),rewrite([2392(7),2392(8)])].">
      <j1 rule="para" parents="2392 2485"/>
      <j2 rule="rewrite" parents="2392 2392"/>
    </justification>
  </clause>

  <clause id="2507">
    <literal><![CDATA[
      (c(x v c(y)) v y) v y = y
    ]]></literal>
    <justification jstring="[para(2392(a,1),2500(a,1,1,1,1,1))].">
      <j1 rule="para" parents="2392 2500"/>
    </justification>
  </clause>

  <clause id="2511">
    <literal><![CDATA[
      (c(x) v c(x v y)) v c(x v c(c(y v c(c(x) v y)) v y)) = c(x)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2410(a,1,1,2,1,2)),rewrite([2392(6),2392(7),2392(11)])].">
      <j1 rule="para" parents="2392 2410"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2513">
    <literal><![CDATA[
      (c(x) v c(x v c(y))) v c(x) = c(x) v c(x v c(y))
    ]]></literal>
    <justification jstring="[para(2410(a,1),2465(a,1,2,1,2,1,1,1,2,1)),rewrite([2392(25),2454(38),2392(13),2486(12)])].">
      <j1 rule="para" parents="2410 2465"/>
      <j2 rule="rewrite" parents="2392 2454 2392 2486"/>
    </justification>
  </clause>

  <clause id="2517">
    <literal><![CDATA[
      c(x) v c(x v c(y)) = c(x)
    ]]></literal>
    <justification jstring="[para(2410(a,1),2487(a,1,2)),rewrite([2513(7),2511(23)])].">
      <j1 rule="para" parents="2410 2487"/>
      <j2 rule="rewrite" parents="2513 2511"/>
    </justification>
  </clause>

  <clause id="2529">
    <literal><![CDATA[
      (x v y) v x = x v y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2517(a,1,2,1)),rewrite([2392(3),2392(3),2392(5)])].">
      <j1 rule="para" parents="2481 2517"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2533">
    <literal><![CDATA[
      x v c(c(x) v y) = x
    ]]></literal>
    <justification jstring="[para(2486(a,1),2529(a,1,1)),rewrite([2486(8)])].">
      <j1 rule="para" parents="2486 2529"/>
      <j2 rule="rewrite" parents="2486"/>
    </justification>
  </clause>

  <clause id="2553">
    <literal><![CDATA[
      c(x v c(y)) v c(c(y) v c(c(z v c(y v z)) v z)) = y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2453(a,1,1,1)),rewrite([2392(4)])].">
      <j1 rule="para" parents="2481 2453"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="2560">
    <literal><![CDATA[
      c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[para(2453(a,1),2507(a,1,1)),rewrite([2533(10)]),flip(a)].">
      <j1 rule="para" parents="2453 2507"/>
      <j2 rule="rewrite" parents="2533"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2574">
    <literal><![CDATA[
      c(x v c(y)) v y = y
    ]]></literal>
    <justification jstring="[back_rewrite(2553),rewrite([2560(12)])].">
      <j1 rule="back_rewrite" parents="2553"/>
      <j2 rule="rewrite" parents="2560"/>
    </justification>
  </clause>

  <clause id="2585">
    <literal><![CDATA[
      x v c(y v c(x)) = x
    ]]></literal>
    <justification jstring="[para(2574(a,1),2529(a,1,1)),rewrite([2574(8)])].">
      <j1 rule="para" parents="2574 2529"/>
      <j2 rule="rewrite" parents="2574"/>
    </justification>
  </clause>

  <clause id="2590">
    <literal><![CDATA[
      c(x) v c(y v x) = c(x)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2585(a,1,2,1,2))].">
      <j1 rule="para" parents="2392 2585"/>
    </justification>
  </clause>

  <clause id="2677">
    <literal><![CDATA[
      c(x v y) v c(c(c(z) v c(x v y)) v x) = c(c(c(z) v c(x v y)) v x)
    ]]></literal>
    <justification jstring="[para(2481(a,1),2492(a,1,2,1,2,1)),rewrite([2392(9),2481(19),2392(17)])].">
      <j1 rule="para" parents="2481 2492"/>
      <j2 rule="rewrite" parents="2392 2481 2392"/>
    </justification>
  </clause>

  <clause id="5628">
    <literal><![CDATA[
      c(x v y) v c(y v x) = c(y v x)
    ]]></literal>
    <justification jstring="[para(2585(a,1),2677(a,1,2,1,1,1)),rewrite([2392(2),2392(4),2392(8),2590(9),2392(7)])].">
      <j1 rule="para" parents="2585 2677"/>
      <j2 rule="rewrite" parents="2392 2392 2392 2590 2392"/>
    </justification>
  </clause>

  <clause id="5749">
    <literal><![CDATA[
      (x v y) v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(5628(a,1),2481(a,1,1,1)),rewrite([2392(3),2392(4),2392(6)])].">
      <j1 rule="para" parents="5628 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="5751">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[para(5628(a,1),2533(a,1,2,1)),rewrite([2392(4),5749(3)])].">
      <j1 rule="para" parents="5628 2533"/>
      <j2 rule="rewrite" parents="2392 5749"/>
    </justification>
  </clause>

  <clause id="5753">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(commutativity_join)
    ]]></attribute>
    <justification jstring="[resolve(5751,a,32,a)].">
      <j1 rule="resolve" parents="5751 32"/>
    </justification>
  </clause>

</proof>

<proof number="11" length="88" max_count="37">

<comments><![CDATA[
% Proof 11 at 1.44 (+ 0.03) seconds: OM.
% Length of proof is 88.
% Level of proof is 24.
% Maximum clause weight is 37.
% Given clauses 267.

]]></comments>

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

  <clause id="15" type="assumption">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="16" type="assumption">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <justification jstring="[copy(16),flip(a)].">
      <j1 rule="copy" parents="16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18" type="assumption">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_meet)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(f(x,y),f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[copy(18),flip(a)].">
      <j1 rule="copy" parents="18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20" type="assumption">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(x,x) = c(x)
    ]]></literal>
    <justification jstring="[copy(20),flip(a)].">
      <j1 rule="copy" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23" type="deny">
    <literal><![CDATA[
      c3 v (c(c3) ^ (c3 v c4)) != c3 v c4
    ]]></literal>
    <attribute><![CDATA[
      answer(OM)
    ]]></attribute>
    <justification jstring="[deny(2)].">
      <j1 rule="deny" parents="2"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      c(f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(19),rewrite([21(3)])].">
      <j1 rule="back_rewrite" parents="19"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      f(c(x),c(y)) = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(17),rewrite([21(1),21(2)])].">
      <j1 rule="back_rewrite" parents="17"/>
      <j2 rule="rewrite" parents="21 21"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(15),rewrite([21(5)])].">
      <j1 rule="back_rewrite" parents="15"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      x ^ x = c(c(x))
    ]]></literal>
    <justification jstring="[para(21(a,1),40(a,1,1)),flip(a)].">
      <j1 rule="para" parents="21 40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      c(c(x)) = x v x
    ]]></literal>
    <justification jstring="[para(41(a,1),21(a,1)),flip(a)].">
      <j1 rule="para" parents="41 21"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      c(x) ^ c(y) = c(x v y)
    ]]></literal>
    <justification jstring="[para(41(a,1),40(a,1,1)),flip(a)].">
      <j1 rule="para" parents="41 40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      x ^ x = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(43),rewrite([45(3)])].">
      <j1 rule="back_rewrite" parents="43"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      f(f(f(c(x),f(x,y)),z),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1,1))].">
      <j1 rule="para" parents="21 42"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      f(f(f(f(x,y),c(y)),z),f(y,f(f(y,f(c(y),y)),y))) = y
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1,2))].">
      <j1 rule="para" parents="21 42"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      f(f(x v x,y),f(x,f(f(x,f(c(x),x)),x))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1)),rewrite([21(1),45(2)])].">
      <j1 rule="para" parents="21 42"/>
      <j2 rule="rewrite" parents="21 45"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      f(f(x,y) ^ f(y,z),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1)),rewrite([40(4)])].">
      <j1 rule="para" parents="21 42"/>
      <j2 rule="rewrite" parents="40"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      f(f(f(x v y,f(c(y),z)),u),f(c(y),f(f(z,f(y v y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[para(41(a,1),42(a,1,1,1,1)),rewrite([45(8)])].">
      <j1 rule="para" parents="41 42"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,c(z))),u),f(y,f(f(c(z),y v z),c(z)))) = y
    ]]></literal>
    <justification jstring="[para(41(a,1),42(a,1,2,2,1,2))].">
      <j1 rule="para" parents="41 42"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(f(f(z,f(c(x),z)),z),f(c(x),f(f(z,f(c(x),z)),z))),f(f(z,f(c(x),z)),z)))) = x
    ]]></literal>
    <justification jstring="[para(42(a,1),42(a,1,1,1))].">
      <j1 rule="para" parents="42 42"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      f(x v x,c(y)) = c(x) v y
    ]]></literal>
    <justification jstring="[para(45(a,1),41(a,1,1))].">
      <j1 rule="para" parents="45 41"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      f(c(x),y v y) = x v c(y)
    ]]></literal>
    <justification jstring="[para(45(a,1),41(a,1,2))].">
      <j1 rule="para" parents="45 41"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      c(x v x) = c(x) v c(x)
    ]]></literal>
    <justification jstring="[para(45(a,1),45(a,1,1))].">
      <j1 rule="para" parents="45 45"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      (x v x) ^ c(y) = c(c(x) v y)
    ]]></literal>
    <justification jstring="[para(45(a,1),46(a,1,1))].">
      <j1 rule="para" parents="45 46"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      f(x v x,y v y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[para(45(a,1),65(a,1,2))].">
      <j1 rule="para" parents="45 65"/>
    </justification>
  </clause>

  <clause id="95">
    <literal><![CDATA[
      (x v x) ^ (y v y) = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[para(45(a,1),72(a,1,2))].">
      <j1 rule="para" parents="45 72"/>
    </justification>
  </clause>

  <clause id="158">
    <literal><![CDATA[
      f(x,f(c(x),f(f(y,f(x v x,y)),y))) = c(x)
    ]]></literal>
    <justification jstring="[para(53(a,1),42(a,1,1)),rewrite([45(3)])].">
      <j1 rule="para" parents="53 42"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="203">
    <literal><![CDATA[
      f(c(x) v y,f(x,f(f(x,f(c(x),x)),x))) = x
    ]]></literal>
    <justification jstring="[para(65(a,1),54(a,1,1))].">
      <j1 rule="para" parents="65 54"/>
    </justification>
  </clause>

  <clause id="304">
    <literal><![CDATA[
      f(f(x,y v y) ^ (c(y) v z),f(y v y,f(f(c(z),(y v y) v z),c(z)))) = y v y
    ]]></literal>
    <justification jstring="[para(65(a,1),55(a,1,1,2)),rewrite([68(9),65(12),45(9)])].">
      <j1 rule="para" parents="65 55"/>
      <j2 rule="rewrite" parents="68 65 45"/>
    </justification>
  </clause>

  <clause id="776">
    <literal><![CDATA[
      f((x v x) v y,f(c(x),f(f(c(x),c(x) v x),c(x)))) = c(x)
    ]]></literal>
    <justification jstring="[para(41(a,1),203(a,1,2,2,1,2)),rewrite([45(2)])].">
      <j1 rule="para" parents="41 203"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="785">
    <literal><![CDATA[
      f(c(x),f(x v x,f(f(f(x,f(f(x,f(c(x),x)),x)),x),f(x,f(f(x,f(c(x),x)),x))))) = x v x
    ]]></literal>
    <justification jstring="[para(203(a,1),158(a,1,2,2,1,2)),rewrite([45(3),45(18)])].">
      <j1 rule="para" parents="203 158"/>
      <j2 rule="rewrite" parents="45 45"/>
    </justification>
  </clause>

  <clause id="1263">
    <literal><![CDATA[
      f(f(f(x v c(y),c(y) v z),u),f(y v y,f(f(c(z),(y v y) v z),c(z)))) = y v y
    ]]></literal>
    <justification jstring="[para(66(a,1),61(a,1,1,1,1)),rewrite([65(5)])].">
      <j1 rule="para" parents="66 61"/>
      <j2 rule="rewrite" parents="65"/>
    </justification>
  </clause>

  <clause id="1623">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(62(a,1),42(a,1,1))].">
      <j1 rule="para" parents="62 42"/>
    </justification>
  </clause>

  <clause id="1625">
    <literal><![CDATA[
      f(c(x),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(62(a,1),52(a,1,1))].">
      <j1 rule="para" parents="62 52"/>
    </justification>
  </clause>

  <clause id="1628">
    <literal><![CDATA[
      f(x v y,f(c(y),f(f(z,f(y v y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[para(62(a,1),59(a,1,1))].">
      <j1 rule="para" parents="62 59"/>
    </justification>
  </clause>

  <clause id="1630">
    <literal><![CDATA[
      f(c(x),f(x v x,x)) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(785),rewrite([1623(14)])].">
      <j1 rule="back_rewrite" parents="785"/>
      <j2 rule="rewrite" parents="1623"/>
    </justification>
  </clause>

  <clause id="1647">
    <literal><![CDATA[
      f((x v x) ^ f(f(x v x,x),y),f(f(x v x,x),f(f(y,f((x v x) ^ x,y)),y))) = f(x v x,x)
    ]]></literal>
    <justification jstring="[para(1630(a,1),55(a,1,1,1)),rewrite([40(10)])].">
      <j1 rule="para" parents="1630 55"/>
      <j2 rule="rewrite" parents="40"/>
    </justification>
  </clause>

  <clause id="1648">
    <literal><![CDATA[
      f(f(x,c(y)) ^ (y v y),f(c(y),f(f(f(y v y,y),f(y v y,f(y v y,y))),f(y v y,y)))) = c(y)
    ]]></literal>
    <justification jstring="[para(1630(a,1),55(a,1,1,2)),rewrite([45(9)])].">
      <j1 rule="para" parents="1630 55"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="1693">
    <literal><![CDATA[
      f(c(x),f(x,f(f(c(y),x v y),c(y)))) = x
    ]]></literal>
    <justification jstring="[para(41(a,1),1625(a,1,2,2,1,2))].">
      <j1 rule="para" parents="41 1625"/>
    </justification>
  </clause>

  <clause id="1739">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(1625(a,1),1625(a,1,2,2,1,2)),rewrite([1623(13),21(2),21(3),45(2)])].">
      <j1 rule="para" parents="1625 1625"/>
      <j2 rule="rewrite" parents="1623 21 21 45"/>
    </justification>
  </clause>

  <clause id="1786">
    <literal><![CDATA[
      f(f(x,c(y)) ^ y,f(c(y),f(f(c(y),f(y,c(y))),c(y)))) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1648),rewrite([1739(3),1739(5),21(5),1739(6),1739(6),21(6),1739(9),21(9)])].">
      <j1 rule="back_rewrite" parents="1648"/>
      <j2 rule="rewrite" parents="1739 1739 21 1739 1739 21 1739 21"/>
    </justification>
  </clause>

  <clause id="1787">
    <literal><![CDATA[
      f(x ^ f(c(x),y),f(c(x),f(f(y,f(x,y)),y))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(1647),rewrite([1739(1),1739(1),21(1),1739(4),21(4),1739(5),50(5),1739(5),1739(10),21(10)])].">
      <j1 rule="back_rewrite" parents="1647"/>
      <j2 rule="rewrite" parents="1739 1739 21 1739 21 1739 50 1739 1739 21"/>
    </justification>
  </clause>

  <clause id="1798">
    <literal><![CDATA[
      f(x v y,f(c(y),f(f(z,f(y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1628),rewrite([1739(3)])].">
      <j1 rule="back_rewrite" parents="1628"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="1958">
    <literal><![CDATA[
      f(f(f(x v c(y),c(y) v z),u),f(y,f(f(c(z),y v z),c(z)))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1263),rewrite([1739(7),1739(8),1739(14)])].">
      <j1 rule="back_rewrite" parents="1263"/>
      <j2 rule="rewrite" parents="1739 1739 1739"/>
    </justification>
  </clause>

  <clause id="2130">
    <literal><![CDATA[
      f(x v y,f(c(x),f(f(c(x),c(x) v x),c(x)))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(776),rewrite([1739(1)])].">
      <j1 rule="back_rewrite" parents="776"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="2302">
    <literal><![CDATA[
      f(f(x,y) ^ (c(y) v z),f(y,f(f(c(z),y v z),c(z)))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(304),rewrite([1739(1),1739(5),1739(6),1739(12)])].">
      <j1 rule="back_rewrite" parents="304"/>
      <j2 rule="rewrite" parents="1739 1739 1739 1739"/>
    </justification>
  </clause>

  <clause id="2374">
    <literal><![CDATA[
      x ^ y = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[back_rewrite(95),rewrite([1739(1),1739(1)])].">
      <j1 rule="back_rewrite" parents="95"/>
      <j2 rule="rewrite" parents="1739 1739"/>
    </justification>
  </clause>

  <clause id="2384">
    <literal><![CDATA[
      f(x,y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(77),rewrite([1739(1),1739(1)])].">
      <j1 rule="back_rewrite" parents="77"/>
      <j2 rule="rewrite" parents="1739 1739"/>
    </justification>
  </clause>

  <clause id="2392">
    <literal><![CDATA[
      c(c(x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(45),rewrite([1739(3)])].">
      <j1 rule="back_rewrite" parents="45"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="2400">
    <literal><![CDATA[
      (c(c(x) v c(y)) v c(c(y) v z)) v c(c(y) v c(c(z v c(y v z)) v z)) = y
    ]]></literal>
    <justification jstring="[back_rewrite(2302),rewrite([2384(1),2374(6),2384(12),2392(11),2384(14),2392(15),2384(15),2384(18),2392(10)])].">
      <j1 rule="back_rewrite" parents="2302"/>
      <j2 rule="rewrite" parents="2384 2374 2384 2392 2384 2392 2384 2384 2392"/>
    </justification>
  </clause>

  <clause id="2410">
    <literal><![CDATA[
      (c(x) v c(x v c(y))) v c(x v c(c(c(y) v c(c(x) v c(y))) v c(y))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(1787),rewrite([2384(2),2392(2),2374(3),2384(8),2384(11),2384(14),2384(17),2392(8),2384(18),2392(7)])].">
      <j1 rule="back_rewrite" parents="1787"/>
      <j2 rule="rewrite" parents="2384 2392 2374 2384 2384 2384 2384 2392 2384 2392"/>
    </justification>
  </clause>

  <clause id="2411">
    <literal><![CDATA[
      (c(c(x) v y) v c(y)) v c(y v c(c(y v c(c(y) v y)) v y)) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1786),rewrite([2384(2),2392(3),2374(3),2384(10),2392(11),2384(11),2392(9),2384(13),2392(14),2384(14),2392(8),2384(15),2392(7)])].">
      <j1 rule="back_rewrite" parents="1786"/>
      <j2 rule="rewrite" parents="2384 2392 2374 2384 2392 2384 2392 2384 2392 2384 2392 2384 2392"/>
    </justification>
  </clause>

  <clause id="2440">
    <literal><![CDATA[
      c3 v c(c3 v c(c3 v c4)) != c3 v c4
    ]]></literal>
    <attribute><![CDATA[
      answer(OM)
    ]]></attribute>
    <justification jstring="[back_rewrite(23),rewrite([2374(7),2392(4)])].">
      <j1 rule="back_rewrite" parents="23"/>
      <j2 rule="rewrite" parents="2374 2392"/>
    </justification>
  </clause>

  <clause id="2450">
    <literal><![CDATA[
      c(x v y) v c(x v c(c(x v c(c(x) v x)) v x)) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(2130),rewrite([2384(6),2392(4),2384(8),2392(9),2384(9),2392(3),2384(10)])].">
      <j1 rule="back_rewrite" parents="2130"/>
      <j2 rule="rewrite" parents="2384 2392 2384 2392 2384 2392 2384"/>
    </justification>
  </clause>

  <clause id="2453">
    <literal><![CDATA[
      c(c(c(x v c(y)) v c(c(y) v z)) v c(u)) v c(c(y) v c(c(z v c(y v z)) v z)) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1958),rewrite([2384(5),2384(8),2384(13),2392(12),2384(15),2392(16),2384(16),2384(19)])].">
      <j1 rule="back_rewrite" parents="1958"/>
      <j2 rule="rewrite" parents="2384 2384 2384 2392 2384 2392 2384 2384"/>
    </justification>
  </clause>

  <clause id="2454">
    <literal><![CDATA[
      c(x v y) v c(y v c(c(c(z) v c(c(y) v c(z))) v c(z))) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1798),rewrite([2384(3),2384(6),2384(9),2384(12),2392(3),2384(13)])].">
      <j1 rule="back_rewrite" parents="1798"/>
      <j2 rule="rewrite" parents="2384 2384 2384 2384 2392 2384"/>
    </justification>
  </clause>

  <clause id="2465">
    <literal><![CDATA[
      x v c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1693),rewrite([2384(4),2392(3),2384(6),2392(7),2384(7),2384(10),2392(2)])].">
      <j1 rule="back_rewrite" parents="1693"/>
      <j2 rule="rewrite" parents="2384 2392 2384 2392 2384 2384 2392"/>
    </justification>
  </clause>

  <clause id="2479">
    <literal><![CDATA[
      c(x v c(c(x v c(c(x) v x)) v x)) = c(x)
    ]]></literal>
    <justification jstring="[para(2450(a,1),1739(a,1)),flip(a)].">
      <j1 rule="para" parents="2450 1739"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2481">
    <literal><![CDATA[
      c(x v y) v c(x) = c(x)
    ]]></literal>
    <justification jstring="[para(2450(a,1),2384(a,2)),rewrite([2384(10),2479(11)])].">
      <j1 rule="para" parents="2450 2384"/>
      <j2 rule="rewrite" parents="2384 2479"/>
    </justification>
  </clause>

  <clause id="2485">
    <literal><![CDATA[
      (c(c(x) v y) v c(y)) v c(y) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(2411),rewrite([2479(14)])].">
      <j1 rule="back_rewrite" parents="2411"/>
      <j2 rule="rewrite" parents="2479"/>
    </justification>
  </clause>

  <clause id="2486">
    <literal><![CDATA[
      c(c(x) v y) v x = x
    ]]></literal>
    <justification jstring="[para(2392(a,1),2481(a,1,2)),rewrite([2392(6)])].">
      <j1 rule="para" parents="2392 2481"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="2487">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2481(a,1,1,1)),rewrite([2392(2),2392(3),2392(5)])].">
      <j1 rule="para" parents="2481 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2492">
    <literal><![CDATA[
      c(x) v c(c(c(y) v c(x)) v c(c(x) v z)) = c(c(c(y) v c(x)) v c(c(x) v z))
    ]]></literal>
    <justification jstring="[para(2400(a,1),2481(a,1,1,1))].">
      <j1 rule="para" parents="2400 2481"/>
    </justification>
  </clause>

  <clause id="2500">
    <literal><![CDATA[
      (c(c(x) v c(y)) v y) v y = y
    ]]></literal>
    <justification jstring="[para(2392(a,1),2485(a,1,1,2)),rewrite([2392(7),2392(8)])].">
      <j1 rule="para" parents="2392 2485"/>
      <j2 rule="rewrite" parents="2392 2392"/>
    </justification>
  </clause>

  <clause id="2507">
    <literal><![CDATA[
      (c(x v c(y)) v y) v y = y
    ]]></literal>
    <justification jstring="[para(2392(a,1),2500(a,1,1,1,1,1))].">
      <j1 rule="para" parents="2392 2500"/>
    </justification>
  </clause>

  <clause id="2511">
    <literal><![CDATA[
      (c(x) v c(x v y)) v c(x v c(c(y v c(c(x) v y)) v y)) = c(x)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2410(a,1,1,2,1,2)),rewrite([2392(6),2392(7),2392(11)])].">
      <j1 rule="para" parents="2392 2410"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2513">
    <literal><![CDATA[
      (c(x) v c(x v c(y))) v c(x) = c(x) v c(x v c(y))
    ]]></literal>
    <justification jstring="[para(2410(a,1),2465(a,1,2,1,2,1,1,1,2,1)),rewrite([2392(25),2454(38),2392(13),2486(12)])].">
      <j1 rule="para" parents="2410 2465"/>
      <j2 rule="rewrite" parents="2392 2454 2392 2486"/>
    </justification>
  </clause>

  <clause id="2517">
    <literal><![CDATA[
      c(x) v c(x v c(y)) = c(x)
    ]]></literal>
    <justification jstring="[para(2410(a,1),2487(a,1,2)),rewrite([2513(7),2511(23)])].">
      <j1 rule="para" parents="2410 2487"/>
      <j2 rule="rewrite" parents="2513 2511"/>
    </justification>
  </clause>

  <clause id="2528">
    <literal><![CDATA[
      c(x) v c(x v y) = c(x)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2517(a,1,2,1,2))].">
      <j1 rule="para" parents="2392 2517"/>
    </justification>
  </clause>

  <clause id="2529">
    <literal><![CDATA[
      (x v y) v x = x v y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2517(a,1,2,1)),rewrite([2392(3),2392(3),2392(5)])].">
      <j1 rule="para" parents="2481 2517"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2533">
    <literal><![CDATA[
      x v c(c(x) v y) = x
    ]]></literal>
    <justification jstring="[para(2486(a,1),2529(a,1,1)),rewrite([2486(8)])].">
      <j1 rule="para" parents="2486 2529"/>
      <j2 rule="rewrite" parents="2486"/>
    </justification>
  </clause>

  <clause id="2553">
    <literal><![CDATA[
      c(x v c(y)) v c(c(y) v c(c(z v c(y v z)) v z)) = y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2453(a,1,1,1)),rewrite([2392(4)])].">
      <j1 rule="para" parents="2481 2453"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="2560">
    <literal><![CDATA[
      c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[para(2453(a,1),2507(a,1,1)),rewrite([2533(10)]),flip(a)].">
      <j1 rule="para" parents="2453 2507"/>
      <j2 rule="rewrite" parents="2533"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2574">
    <literal><![CDATA[
      c(x v c(y)) v y = y
    ]]></literal>
    <justification jstring="[back_rewrite(2553),rewrite([2560(12)])].">
      <j1 rule="back_rewrite" parents="2553"/>
      <j2 rule="rewrite" parents="2560"/>
    </justification>
  </clause>

  <clause id="2585">
    <literal><![CDATA[
      x v c(y v c(x)) = x
    ]]></literal>
    <justification jstring="[para(2574(a,1),2529(a,1,1)),rewrite([2574(8)])].">
      <j1 rule="para" parents="2574 2529"/>
      <j2 rule="rewrite" parents="2574"/>
    </justification>
  </clause>

  <clause id="2590">
    <literal><![CDATA[
      c(x) v c(y v x) = c(x)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2585(a,1,2,1,2))].">
      <j1 rule="para" parents="2392 2585"/>
    </justification>
  </clause>

  <clause id="2675">
    <literal><![CDATA[
      x v c(c(c(y) v x) v c(x v z)) = c(c(c(y) v x) v c(x v z))
    ]]></literal>
    <justification jstring="[para(2392(a,1),2492(a,1,1)),rewrite([2392(3),2392(5),2392(11),2392(13)])].">
      <j1 rule="para" parents="2392 2492"/>
      <j2 rule="rewrite" parents="2392 2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2677">
    <literal><![CDATA[
      c(x v y) v c(c(c(z) v c(x v y)) v x) = c(c(c(z) v c(x v y)) v x)
    ]]></literal>
    <justification jstring="[para(2481(a,1),2492(a,1,2,1,2,1)),rewrite([2392(9),2481(19),2392(17)])].">
      <j1 rule="para" parents="2481 2492"/>
      <j2 rule="rewrite" parents="2392 2481 2392"/>
    </justification>
  </clause>

  <clause id="2746">
    <literal><![CDATA[
      c(c(x v y) v x) v c(x v y) = c(x)
    ]]></literal>
    <justification jstring="[para(2528(a,1),2560(a,1,1,2,1,1,1,2,1)),rewrite([2392(2),2392(4),2675(9),2392(9)])].">
      <j1 rule="para" parents="2528 2560"/>
      <j2 rule="rewrite" parents="2392 2392 2675 2392"/>
    </justification>
  </clause>

  <clause id="4088">
    <literal><![CDATA[
      x v c(c(x v y) v x) = x v y
    ]]></literal>
    <justification jstring="[para(2746(a,1),2746(a,1,1,1)),rewrite([2392(2),2392(8)])].">
      <j1 rule="para" parents="2746 2746"/>
      <j2 rule="rewrite" parents="2392 2392"/>
    </justification>
  </clause>

  <clause id="5628">
    <literal><![CDATA[
      c(x v y) v c(y v x) = c(y v x)
    ]]></literal>
    <justification jstring="[para(2585(a,1),2677(a,1,2,1,1,1)),rewrite([2392(2),2392(4),2392(8),2590(9),2392(7)])].">
      <j1 rule="para" parents="2585 2677"/>
      <j2 rule="rewrite" parents="2392 2392 2392 2590 2392"/>
    </justification>
  </clause>

  <clause id="5749">
    <literal><![CDATA[
      (x v y) v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(5628(a,1),2481(a,1,1,1)),rewrite([2392(3),2392(4),2392(6)])].">
      <j1 rule="para" parents="5628 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="5751">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[para(5628(a,1),2533(a,1,2,1)),rewrite([2392(4),5749(3)])].">
      <j1 rule="para" parents="5628 2533"/>
      <j2 rule="rewrite" parents="2392 5749"/>
    </justification>
  </clause>

  <clause id="5872">
    <literal><![CDATA[
      x v c(x v c(x v y)) = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(4088),rewrite([5751(3)])].">
      <j1 rule="back_rewrite" parents="4088"/>
      <j2 rule="rewrite" parents="5751"/>
    </justification>
  </clause>

  <clause id="5873">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(OM)
    ]]></attribute>
    <justification jstring="[resolve(5872,a,2440,a)].">
      <j1 rule="resolve" parents="5872 2440"/>
    </justification>
  </clause>

</proof>

<proof number="12" length="111" max_count="37">

<comments><![CDATA[
% Proof 12 at 6.95 (+ 0.08) seconds: AJ.
% Length of proof is 111.
% Level of proof is 29.
% Maximum clause weight is 37.
% Given clauses 444.

]]></comments>

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

  <clause id="15" type="assumption">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="16" type="assumption">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <justification jstring="[copy(16),flip(a)].">
      <j1 rule="copy" parents="16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18" type="assumption">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_meet)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(f(x,y),f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[copy(18),flip(a)].">
      <j1 rule="copy" parents="18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20" type="assumption">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(x,x) = c(x)
    ]]></literal>
    <justification jstring="[copy(20),flip(a)].">
      <j1 rule="copy" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="24" type="deny">
    <literal><![CDATA[
      c5 v (c6 v c7) != c6 v (c5 v c7)
    ]]></literal>
    <attribute><![CDATA[
      answer(AJ)
    ]]></attribute>
    <justification jstring="[deny(3)].">
      <j1 rule="deny" parents="3"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      c6 v (c5 v c7) != c5 v (c6 v c7)
    ]]></literal>
    <attribute><![CDATA[
      answer(AJ)
    ]]></attribute>
    <justification jstring="[copy(24),flip(a)].">
      <j1 rule="copy" parents="24"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      c(f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(19),rewrite([21(3)])].">
      <j1 rule="back_rewrite" parents="19"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      f(c(x),c(y)) = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(17),rewrite([21(1),21(2)])].">
      <j1 rule="back_rewrite" parents="17"/>
      <j2 rule="rewrite" parents="21 21"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(15),rewrite([21(5)])].">
      <j1 rule="back_rewrite" parents="15"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      x ^ x = c(c(x))
    ]]></literal>
    <justification jstring="[para(21(a,1),40(a,1,1)),flip(a)].">
      <j1 rule="para" parents="21 40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      c(c(x)) = x v x
    ]]></literal>
    <justification jstring="[para(41(a,1),21(a,1)),flip(a)].">
      <j1 rule="para" parents="41 21"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      c(x) ^ c(y) = c(x v y)
    ]]></literal>
    <justification jstring="[para(41(a,1),40(a,1,1)),flip(a)].">
      <j1 rule="para" parents="41 40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      x ^ x = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(43),rewrite([45(3)])].">
      <j1 rule="back_rewrite" parents="43"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      f(f(f(c(x),f(x,y)),z),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1,1))].">
      <j1 rule="para" parents="21 42"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      f(f(f(f(x,y),c(y)),z),f(y,f(f(y,f(c(y),y)),y))) = y
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1,2))].">
      <j1 rule="para" parents="21 42"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      f(f(x v x,y),f(x,f(f(x,f(c(x),x)),x))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1)),rewrite([21(1),45(2)])].">
      <j1 rule="para" parents="21 42"/>
      <j2 rule="rewrite" parents="21 45"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      f(f(x,y) ^ f(y,z),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1)),rewrite([40(4)])].">
      <j1 rule="para" parents="21 42"/>
      <j2 rule="rewrite" parents="40"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      f(f(f(x v y,f(c(y),z)),u),f(c(y),f(f(z,f(y v y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[para(41(a,1),42(a,1,1,1,1)),rewrite([45(8)])].">
      <j1 rule="para" parents="41 42"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,c(z))),u),f(y,f(f(c(z),y v z),c(z)))) = y
    ]]></literal>
    <justification jstring="[para(41(a,1),42(a,1,2,2,1,2))].">
      <j1 rule="para" parents="41 42"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(f(f(z,f(c(x),z)),z),f(c(x),f(f(z,f(c(x),z)),z))),f(f(z,f(c(x),z)),z)))) = x
    ]]></literal>
    <justification jstring="[para(42(a,1),42(a,1,1,1))].">
      <j1 rule="para" parents="42 42"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      f(x v x,c(y)) = c(x) v y
    ]]></literal>
    <justification jstring="[para(45(a,1),41(a,1,1))].">
      <j1 rule="para" parents="45 41"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      f(c(x),y v y) = x v c(y)
    ]]></literal>
    <justification jstring="[para(45(a,1),41(a,1,2))].">
      <j1 rule="para" parents="45 41"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      c(x v x) = c(x) v c(x)
    ]]></literal>
    <justification jstring="[para(45(a,1),45(a,1,1))].">
      <j1 rule="para" parents="45 45"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      (x v x) ^ c(y) = c(c(x) v y)
    ]]></literal>
    <justification jstring="[para(45(a,1),46(a,1,1))].">
      <j1 rule="para" parents="45 46"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      f(x v x,y v y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[para(45(a,1),65(a,1,2))].">
      <j1 rule="para" parents="45 65"/>
    </justification>
  </clause>

  <clause id="95">
    <literal><![CDATA[
      (x v x) ^ (y v y) = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[para(45(a,1),72(a,1,2))].">
      <j1 rule="para" parents="45 72"/>
    </justification>
  </clause>

  <clause id="158">
    <literal><![CDATA[
      f(x,f(c(x),f(f(y,f(x v x,y)),y))) = c(x)
    ]]></literal>
    <justification jstring="[para(53(a,1),42(a,1,1)),rewrite([45(3)])].">
      <j1 rule="para" parents="53 42"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="203">
    <literal><![CDATA[
      f(c(x) v y,f(x,f(f(x,f(c(x),x)),x))) = x
    ]]></literal>
    <justification jstring="[para(65(a,1),54(a,1,1))].">
      <j1 rule="para" parents="65 54"/>
    </justification>
  </clause>

  <clause id="304">
    <literal><![CDATA[
      f(f(x,y v y) ^ (c(y) v z),f(y v y,f(f(c(z),(y v y) v z),c(z)))) = y v y
    ]]></literal>
    <justification jstring="[para(65(a,1),55(a,1,1,2)),rewrite([68(9),65(12),45(9)])].">
      <j1 rule="para" parents="65 55"/>
      <j2 rule="rewrite" parents="68 65 45"/>
    </justification>
  </clause>

  <clause id="776">
    <literal><![CDATA[
      f((x v x) v y,f(c(x),f(f(c(x),c(x) v x),c(x)))) = c(x)
    ]]></literal>
    <justification jstring="[para(41(a,1),203(a,1,2,2,1,2)),rewrite([45(2)])].">
      <j1 rule="para" parents="41 203"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="785">
    <literal><![CDATA[
      f(c(x),f(x v x,f(f(f(x,f(f(x,f(c(x),x)),x)),x),f(x,f(f(x,f(c(x),x)),x))))) = x v x
    ]]></literal>
    <justification jstring="[para(203(a,1),158(a,1,2,2,1,2)),rewrite([45(3),45(18)])].">
      <j1 rule="para" parents="203 158"/>
      <j2 rule="rewrite" parents="45 45"/>
    </justification>
  </clause>

  <clause id="1263">
    <literal><![CDATA[
      f(f(f(x v c(y),c(y) v z),u),f(y v y,f(f(c(z),(y v y) v z),c(z)))) = y v y
    ]]></literal>
    <justification jstring="[para(66(a,1),61(a,1,1,1,1)),rewrite([65(5)])].">
      <j1 rule="para" parents="66 61"/>
      <j2 rule="rewrite" parents="65"/>
    </justification>
  </clause>

  <clause id="1623">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(62(a,1),42(a,1,1))].">
      <j1 rule="para" parents="62 42"/>
    </justification>
  </clause>

  <clause id="1625">
    <literal><![CDATA[
      f(c(x),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(62(a,1),52(a,1,1))].">
      <j1 rule="para" parents="62 52"/>
    </justification>
  </clause>

  <clause id="1628">
    <literal><![CDATA[
      f(x v y,f(c(y),f(f(z,f(y v y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[para(62(a,1),59(a,1,1))].">
      <j1 rule="para" parents="62 59"/>
    </justification>
  </clause>

  <clause id="1630">
    <literal><![CDATA[
      f(c(x),f(x v x,x)) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(785),rewrite([1623(14)])].">
      <j1 rule="back_rewrite" parents="785"/>
      <j2 rule="rewrite" parents="1623"/>
    </justification>
  </clause>

  <clause id="1647">
    <literal><![CDATA[
      f((x v x) ^ f(f(x v x,x),y),f(f(x v x,x),f(f(y,f((x v x) ^ x,y)),y))) = f(x v x,x)
    ]]></literal>
    <justification jstring="[para(1630(a,1),55(a,1,1,1)),rewrite([40(10)])].">
      <j1 rule="para" parents="1630 55"/>
      <j2 rule="rewrite" parents="40"/>
    </justification>
  </clause>

  <clause id="1648">
    <literal><![CDATA[
      f(f(x,c(y)) ^ (y v y),f(c(y),f(f(f(y v y,y),f(y v y,f(y v y,y))),f(y v y,y)))) = c(y)
    ]]></literal>
    <justification jstring="[para(1630(a,1),55(a,1,1,2)),rewrite([45(9)])].">
      <j1 rule="para" parents="1630 55"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="1693">
    <literal><![CDATA[
      f(c(x),f(x,f(f(c(y),x v y),c(y)))) = x
    ]]></literal>
    <justification jstring="[para(41(a,1),1625(a,1,2,2,1,2))].">
      <j1 rule="para" parents="41 1625"/>
    </justification>
  </clause>

  <clause id="1739">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(1625(a,1),1625(a,1,2,2,1,2)),rewrite([1623(13),21(2),21(3),45(2)])].">
      <j1 rule="para" parents="1625 1625"/>
      <j2 rule="rewrite" parents="1623 21 21 45"/>
    </justification>
  </clause>

  <clause id="1786">
    <literal><![CDATA[
      f(f(x,c(y)) ^ y,f(c(y),f(f(c(y),f(y,c(y))),c(y)))) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1648),rewrite([1739(3),1739(5),21(5),1739(6),1739(6),21(6),1739(9),21(9)])].">
      <j1 rule="back_rewrite" parents="1648"/>
      <j2 rule="rewrite" parents="1739 1739 21 1739 1739 21 1739 21"/>
    </justification>
  </clause>

  <clause id="1787">
    <literal><![CDATA[
      f(x ^ f(c(x),y),f(c(x),f(f(y,f(x,y)),y))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(1647),rewrite([1739(1),1739(1),21(1),1739(4),21(4),1739(5),50(5),1739(5),1739(10),21(10)])].">
      <j1 rule="back_rewrite" parents="1647"/>
      <j2 rule="rewrite" parents="1739 1739 21 1739 21 1739 50 1739 1739 21"/>
    </justification>
  </clause>

  <clause id="1798">
    <literal><![CDATA[
      f(x v y,f(c(y),f(f(z,f(y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1628),rewrite([1739(3)])].">
      <j1 rule="back_rewrite" parents="1628"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="1958">
    <literal><![CDATA[
      f(f(f(x v c(y),c(y) v z),u),f(y,f(f(c(z),y v z),c(z)))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1263),rewrite([1739(7),1739(8),1739(14)])].">
      <j1 rule="back_rewrite" parents="1263"/>
      <j2 rule="rewrite" parents="1739 1739 1739"/>
    </justification>
  </clause>

  <clause id="2130">
    <literal><![CDATA[
      f(x v y,f(c(x),f(f(c(x),c(x) v x),c(x)))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(776),rewrite([1739(1)])].">
      <j1 rule="back_rewrite" parents="776"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="2302">
    <literal><![CDATA[
      f(f(x,y) ^ (c(y) v z),f(y,f(f(c(z),y v z),c(z)))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(304),rewrite([1739(1),1739(5),1739(6),1739(12)])].">
      <j1 rule="back_rewrite" parents="304"/>
      <j2 rule="rewrite" parents="1739 1739 1739 1739"/>
    </justification>
  </clause>

  <clause id="2374">
    <literal><![CDATA[
      x ^ y = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[back_rewrite(95),rewrite([1739(1),1739(1)])].">
      <j1 rule="back_rewrite" parents="95"/>
      <j2 rule="rewrite" parents="1739 1739"/>
    </justification>
  </clause>

  <clause id="2384">
    <literal><![CDATA[
      f(x,y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(77),rewrite([1739(1),1739(1)])].">
      <j1 rule="back_rewrite" parents="77"/>
      <j2 rule="rewrite" parents="1739 1739"/>
    </justification>
  </clause>

  <clause id="2392">
    <literal><![CDATA[
      c(c(x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(45),rewrite([1739(3)])].">
      <j1 rule="back_rewrite" parents="45"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="2400">
    <literal><![CDATA[
      (c(c(x) v c(y)) v c(c(y) v z)) v c(c(y) v c(c(z v c(y v z)) v z)) = y
    ]]></literal>
    <justification jstring="[back_rewrite(2302),rewrite([2384(1),2374(6),2384(12),2392(11),2384(14),2392(15),2384(15),2384(18),2392(10)])].">
      <j1 rule="back_rewrite" parents="2302"/>
      <j2 rule="rewrite" parents="2384 2374 2384 2392 2384 2392 2384 2384 2392"/>
    </justification>
  </clause>

  <clause id="2410">
    <literal><![CDATA[
      (c(x) v c(x v c(y))) v c(x v c(c(c(y) v c(c(x) v c(y))) v c(y))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(1787),rewrite([2384(2),2392(2),2374(3),2384(8),2384(11),2384(14),2384(17),2392(8),2384(18),2392(7)])].">
      <j1 rule="back_rewrite" parents="1787"/>
      <j2 rule="rewrite" parents="2384 2392 2374 2384 2384 2384 2384 2392 2384 2392"/>
    </justification>
  </clause>

  <clause id="2411">
    <literal><![CDATA[
      (c(c(x) v y) v c(y)) v c(y v c(c(y v c(c(y) v y)) v y)) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1786),rewrite([2384(2),2392(3),2374(3),2384(10),2392(11),2384(11),2392(9),2384(13),2392(14),2384(14),2392(8),2384(15),2392(7)])].">
      <j1 rule="back_rewrite" parents="1786"/>
      <j2 rule="rewrite" parents="2384 2392 2374 2384 2392 2384 2392 2384 2392 2384 2392 2384 2392"/>
    </justification>
  </clause>

  <clause id="2450">
    <literal><![CDATA[
      c(x v y) v c(x v c(c(x v c(c(x) v x)) v x)) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(2130),rewrite([2384(6),2392(4),2384(8),2392(9),2384(9),2392(3),2384(10)])].">
      <j1 rule="back_rewrite" parents="2130"/>
      <j2 rule="rewrite" parents="2384 2392 2384 2392 2384 2392 2384"/>
    </justification>
  </clause>

  <clause id="2453">
    <literal><![CDATA[
      c(c(c(x v c(y)) v c(c(y) v z)) v c(u)) v c(c(y) v c(c(z v c(y v z)) v z)) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1958),rewrite([2384(5),2384(8),2384(13),2392(12),2384(15),2392(16),2384(16),2384(19)])].">
      <j1 rule="back_rewrite" parents="1958"/>
      <j2 rule="rewrite" parents="2384 2384 2384 2392 2384 2392 2384 2384"/>
    </justification>
  </clause>

  <clause id="2454">
    <literal><![CDATA[
      c(x v y) v c(y v c(c(c(z) v c(c(y) v c(z))) v c(z))) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1798),rewrite([2384(3),2384(6),2384(9),2384(12),2392(3),2384(13)])].">
      <j1 rule="back_rewrite" parents="1798"/>
      <j2 rule="rewrite" parents="2384 2384 2384 2384 2392 2384"/>
    </justification>
  </clause>

  <clause id="2465">
    <literal><![CDATA[
      x v c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1693),rewrite([2384(4),2392(3),2384(6),2392(7),2384(7),2384(10),2392(2)])].">
      <j1 rule="back_rewrite" parents="1693"/>
      <j2 rule="rewrite" parents="2384 2392 2384 2392 2384 2384 2392"/>
    </justification>
  </clause>

  <clause id="2479">
    <literal><![CDATA[
      c(x v c(c(x v c(c(x) v x)) v x)) = c(x)
    ]]></literal>
    <justification jstring="[para(2450(a,1),1739(a,1)),flip(a)].">
      <j1 rule="para" parents="2450 1739"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2481">
    <literal><![CDATA[
      c(x v y) v c(x) = c(x)
    ]]></literal>
    <justification jstring="[para(2450(a,1),2384(a,2)),rewrite([2384(10),2479(11)])].">
      <j1 rule="para" parents="2450 2384"/>
      <j2 rule="rewrite" parents="2384 2479"/>
    </justification>
  </clause>

  <clause id="2485">
    <literal><![CDATA[
      (c(c(x) v y) v c(y)) v c(y) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(2411),rewrite([2479(14)])].">
      <j1 rule="back_rewrite" parents="2411"/>
      <j2 rule="rewrite" parents="2479"/>
    </justification>
  </clause>

  <clause id="2486">
    <literal><![CDATA[
      c(c(x) v y) v x = x
    ]]></literal>
    <justification jstring="[para(2392(a,1),2481(a,1,2)),rewrite([2392(6)])].">
      <j1 rule="para" parents="2392 2481"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="2487">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2481(a,1,1,1)),rewrite([2392(2),2392(3),2392(5)])].">
      <j1 rule="para" parents="2481 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2492">
    <literal><![CDATA[
      c(x) v c(c(c(y) v c(x)) v c(c(x) v z)) = c(c(c(y) v c(x)) v c(c(x) v z))
    ]]></literal>
    <justification jstring="[para(2400(a,1),2481(a,1,1,1))].">
      <j1 rule="para" parents="2400 2481"/>
    </justification>
  </clause>

  <clause id="2500">
    <literal><![CDATA[
      (c(c(x) v c(y)) v y) v y = y
    ]]></literal>
    <justification jstring="[para(2392(a,1),2485(a,1,1,2)),rewrite([2392(7),2392(8)])].">
      <j1 rule="para" parents="2392 2485"/>
      <j2 rule="rewrite" parents="2392 2392"/>
    </justification>
  </clause>

  <clause id="2507">
    <literal><![CDATA[
      (c(x v c(y)) v y) v y = y
    ]]></literal>
    <justification jstring="[para(2392(a,1),2500(a,1,1,1,1,1))].">
      <j1 rule="para" parents="2392 2500"/>
    </justification>
  </clause>

  <clause id="2511">
    <literal><![CDATA[
      (c(x) v c(x v y)) v c(x v c(c(y v c(c(x) v y)) v y)) = c(x)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2410(a,1,1,2,1,2)),rewrite([2392(6),2392(7),2392(11)])].">
      <j1 rule="para" parents="2392 2410"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2513">
    <literal><![CDATA[
      (c(x) v c(x v c(y))) v c(x) = c(x) v c(x v c(y))
    ]]></literal>
    <justification jstring="[para(2410(a,1),2465(a,1,2,1,2,1,1,1,2,1)),rewrite([2392(25),2454(38),2392(13),2486(12)])].">
      <j1 rule="para" parents="2410 2465"/>
      <j2 rule="rewrite" parents="2392 2454 2392 2486"/>
    </justification>
  </clause>

  <clause id="2517">
    <literal><![CDATA[
      c(x) v c(x v c(y)) = c(x)
    ]]></literal>
    <justification jstring="[para(2410(a,1),2487(a,1,2)),rewrite([2513(7),2511(23)])].">
      <j1 rule="para" parents="2410 2487"/>
      <j2 rule="rewrite" parents="2513 2511"/>
    </justification>
  </clause>

  <clause id="2529">
    <literal><![CDATA[
      (x v y) v x = x v y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2517(a,1,2,1)),rewrite([2392(3),2392(3),2392(5)])].">
      <j1 rule="para" parents="2481 2517"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2533">
    <literal><![CDATA[
      x v c(c(x) v y) = x
    ]]></literal>
    <justification jstring="[para(2486(a,1),2529(a,1,1)),rewrite([2486(8)])].">
      <j1 rule="para" parents="2486 2529"/>
      <j2 rule="rewrite" parents="2486"/>
    </justification>
  </clause>

  <clause id="2552">
    <literal><![CDATA[
      c((c(x) v y) v c(z)) v c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[para(2481(a,1),2453(a,1,1,1,1,1)),rewrite([2392(4)])].">
      <j1 rule="para" parents="2481 2453"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="2553">
    <literal><![CDATA[
      c(x v c(y)) v c(c(y) v c(c(z v c(y v z)) v z)) = y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2453(a,1,1,1)),rewrite([2392(4)])].">
      <j1 rule="para" parents="2481 2453"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="2560">
    <literal><![CDATA[
      c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[para(2453(a,1),2507(a,1,1)),rewrite([2533(10)]),flip(a)].">
      <j1 rule="para" parents="2453 2507"/>
      <j2 rule="rewrite" parents="2533"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2574">
    <literal><![CDATA[
      c(x v c(y)) v y = y
    ]]></literal>
    <justification jstring="[back_rewrite(2553),rewrite([2560(12)])].">
      <j1 rule="back_rewrite" parents="2553"/>
      <j2 rule="rewrite" parents="2560"/>
    </justification>
  </clause>

  <clause id="2575">
    <literal><![CDATA[
      c((c(x) v y) v c(z)) v x = x
    ]]></literal>
    <justification jstring="[back_rewrite(2552),rewrite([2560(14)])].">
      <j1 rule="back_rewrite" parents="2552"/>
      <j2 rule="rewrite" parents="2560"/>
    </justification>
  </clause>

  <clause id="2583">
    <literal><![CDATA[
      c(x v y) v c(y) = c(y)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2574(a,1,1,1,2))].">
      <j1 rule="para" parents="2392 2574"/>
    </justification>
  </clause>

  <clause id="2585">
    <literal><![CDATA[
      x v c(y v c(x)) = x
    ]]></literal>
    <justification jstring="[para(2574(a,1),2529(a,1,1)),rewrite([2574(8)])].">
      <j1 rule="para" parents="2574 2529"/>
      <j2 rule="rewrite" parents="2574"/>
    </justification>
  </clause>

  <clause id="2590">
    <literal><![CDATA[
      c(x) v c(y v x) = c(x)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2585(a,1,2,1,2))].">
      <j1 rule="para" parents="2392 2585"/>
    </justification>
  </clause>

  <clause id="2591">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(2583(a,1),2481(a,1,1,1)),rewrite([2392(2),2392(3),2392(5)])].">
      <j1 rule="para" parents="2583 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2599">
    <literal><![CDATA[
      c((c(x) v y) v z) v x = x
    ]]></literal>
    <justification jstring="[para(2392(a,1),2575(a,1,1,1,2))].">
      <j1 rule="para" parents="2392 2575"/>
    </justification>
  </clause>

  <clause id="2612">
    <literal><![CDATA[
      c(c(x) v y) v (x v z) = x v z
    ]]></literal>
    <justification jstring="[para(2481(a,1),2599(a,1,1,1,1))].">
      <j1 rule="para" parents="2481 2599"/>
    </justification>
  </clause>

  <clause id="2614">
    <literal><![CDATA[
      x v c((c(x) v y) v z) = x
    ]]></literal>
    <justification jstring="[para(2599(a,1),2529(a,1,1)),rewrite([2599(10)])].">
      <j1 rule="para" parents="2599 2529"/>
      <j2 rule="rewrite" parents="2599"/>
    </justification>
  </clause>

  <clause id="2617">
    <literal><![CDATA[
      c(c(x) v y) v (z v x) = z v x
    ]]></literal>
    <justification jstring="[para(2583(a,1),2599(a,1,1,1,1))].">
      <j1 rule="para" parents="2583 2599"/>
    </justification>
  </clause>

  <clause id="2618">
    <literal><![CDATA[
      c((x v c(y)) v z) v y = y
    ]]></literal>
    <justification jstring="[para(2591(a,1),2599(a,1,1,1,1))].">
      <j1 rule="para" parents="2591 2599"/>
    </justification>
  </clause>

  <clause id="2619">
    <literal><![CDATA[
      c(x v (c(y) v z)) v y = y
    ]]></literal>
    <justification jstring="[para(2591(a,1),2599(a,1,1,1))].">
      <j1 rule="para" parents="2591 2599"/>
    </justification>
  </clause>

  <clause id="2650">
    <literal><![CDATA[
      x v c(y v (c(x) v z)) = x
    ]]></literal>
    <justification jstring="[para(2591(a,1),2614(a,1,2,1))].">
      <j1 rule="para" parents="2591 2614"/>
    </justification>
  </clause>

  <clause id="2656">
    <literal><![CDATA[
      c(x v (y v c(z))) v z = z
    ]]></literal>
    <justification jstring="[para(2591(a,1),2618(a,1,1,1))].">
      <j1 rule="para" parents="2591 2618"/>
    </justification>
  </clause>

  <clause id="2660">
    <literal><![CDATA[
      c(x v (y v z)) v c(y) = c(y)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2619(a,1,1,1,2,1))].">
      <j1 rule="para" parents="2392 2619"/>
    </justification>
  </clause>

  <clause id="2677">
    <literal><![CDATA[
      c(x v y) v c(c(c(z) v c(x v y)) v x) = c(c(c(z) v c(x v y)) v x)
    ]]></literal>
    <justification jstring="[para(2481(a,1),2492(a,1,2,1,2,1)),rewrite([2392(9),2481(19),2392(17)])].">
      <j1 rule="para" parents="2481 2492"/>
      <j2 rule="rewrite" parents="2392 2481 2392"/>
    </justification>
  </clause>

  <clause id="2697">
    <literal><![CDATA[
      c(x) v c(y v (x v z)) = c(x)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2650(a,1,2,1,2,1))].">
      <j1 rule="para" parents="2392 2650"/>
    </justification>
  </clause>

  <clause id="2706">
    <literal><![CDATA[
      c(x v (y v z)) v c(z) = c(z)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2656(a,1,1,1,2,2))].">
      <j1 rule="para" parents="2392 2656"/>
    </justification>
  </clause>

  <clause id="2737">
    <literal><![CDATA[
      x v (y v (x v z)) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(2660(a,1),2481(a,1,1,1)),rewrite([2392(2),2392(4),2392(7)])].">
      <j1 rule="para" parents="2660 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2764">
    <literal><![CDATA[
      x v (y v (z v x)) = y v (z v x)
    ]]></literal>
    <justification jstring="[para(2706(a,1),2481(a,1,1,1)),rewrite([2392(2),2392(4),2392(7)])].">
      <j1 rule="para" parents="2706 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="5628">
    <literal><![CDATA[
      c(x v y) v c(y v x) = c(y v x)
    ]]></literal>
    <justification jstring="[para(2585(a,1),2677(a,1,2,1,1,1)),rewrite([2392(2),2392(4),2392(8),2590(9),2392(7)])].">
      <j1 rule="para" parents="2585 2677"/>
      <j2 rule="rewrite" parents="2392 2392 2392 2590 2392"/>
    </justification>
  </clause>

  <clause id="5636">
    <literal><![CDATA[
      c(x v (y v z)) v c(y v x) = c(y v x)
    ]]></literal>
    <justification jstring="[para(2650(a,1),2677(a,1,2,1,1,1)),rewrite([2392(2),2392(5),2392(9),2697(11),2392(8)])].">
      <j1 rule="para" parents="2650 2677"/>
      <j2 rule="rewrite" parents="2392 2392 2392 2697 2392"/>
    </justification>
  </clause>

  <clause id="5749">
    <literal><![CDATA[
      (x v y) v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(5628(a,1),2481(a,1,1,1)),rewrite([2392(3),2392(4),2392(6)])].">
      <j1 rule="para" parents="5628 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="5751">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[para(5628(a,1),2533(a,1,2,1)),rewrite([2392(4),5749(3)])].">
      <j1 rule="para" parents="5628 2533"/>
      <j2 rule="rewrite" parents="2392 5749"/>
    </justification>
  </clause>

  <clause id="5754">
    <literal><![CDATA[
      (x v y) v c(c(y v x) v z) = x v y
    ]]></literal>
    <justification jstring="[para(5628(a,1),2599(a,1,1,1,1)),rewrite([5751(6)])].">
      <j1 rule="para" parents="5628 2599"/>
      <j2 rule="rewrite" parents="5751"/>
    </justification>
  </clause>

  <clause id="12548">
    <literal><![CDATA[
      (x v (y v z)) v (y v x) = x v (y v z)
    ]]></literal>
    <justification jstring="[para(5636(a,1),2533(a,1,2,1)),rewrite([2392(5)])].">
      <j1 rule="para" parents="5636 2533"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="12684">
    <literal><![CDATA[
      (x v y) v ((x v z) v y) = (x v z) v y
    ]]></literal>
    <justification jstring="[para(5636(a,1),5754(a,1,2,1)),rewrite([2392(5),5751(4)])].">
      <j1 rule="para" parents="5636 5754"/>
      <j2 rule="rewrite" parents="2392 5751"/>
    </justification>
  </clause>

  <clause id="12718">
    <literal><![CDATA[
      c(c(x v y) v z) v (y v (x v u)) = y v (x v u)
    ]]></literal>
    <justification jstring="[para(12548(a,1),2617(a,1,2)),rewrite([12548(11)])].">
      <j1 rule="para" parents="12548 2617"/>
      <j2 rule="rewrite" parents="12548"/>
    </justification>
  </clause>

  <clause id="12728">
    <literal><![CDATA[
      (x v y) v (x v (y v z)) = x v (y v z)
    ]]></literal>
    <justification jstring="[para(2737(a,1),12548(a,1,1)),rewrite([5751(4),2737(7)])].">
      <j1 rule="para" parents="2737 12548"/>
      <j2 rule="rewrite" parents="5751 2737"/>
    </justification>
  </clause>

  <clause id="12730">
    <literal><![CDATA[
      (x v y) v (x v (z v y)) = x v (z v y)
    ]]></literal>
    <justification jstring="[para(2764(a,1),12548(a,1,1)),rewrite([5751(4),2764(7)])].">
      <j1 rule="para" parents="2764 12548"/>
      <j2 rule="rewrite" parents="5751 2764"/>
    </justification>
  </clause>

  <clause id="13101">
    <literal><![CDATA[
      (x v y) v (x v z) = (x v y) v z
    ]]></literal>
    <justification jstring="[para(12684(a,1),12548(a,1,1)),rewrite([12730(6),12684(7)])].">
      <j1 rule="para" parents="12684 12548"/>
      <j2 rule="rewrite" parents="12730 12684"/>
    </justification>
  </clause>

  <clause id="14172">
    <literal><![CDATA[
      (x v y) v (y v z) = x v (y v z)
    ]]></literal>
    <justification jstring="[back_rewrite(12728),rewrite([13101(4)])].">
      <j1 rule="back_rewrite" parents="12728"/>
      <j2 rule="rewrite" parents="13101"/>
    </justification>
  </clause>

  <clause id="14240">
    <literal><![CDATA[
      (x v y) v z = x v (y v z)
    ]]></literal>
    <justification jstring="[para(2591(a,1),13101(a,1,1)),rewrite([14172(3),2591(4)]),flip(a)].">
      <j1 rule="para" parents="2591 13101"/>
      <j2 rule="rewrite" parents="14172 2591"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14251">
    <literal><![CDATA[
      x v (y v z) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(13101(a,1),2612(a,2)),rewrite([14240(7),2737(7),12718(7),14240(4)])].">
      <j1 rule="para" parents="13101 2612"/>
      <j2 rule="rewrite" parents="14240 2737 12718 14240"/>
    </justification>
  </clause>

  <clause id="14252">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(AJ)
    ]]></attribute>
    <justification jstring="[resolve(14251,a,25,a)].">
      <j1 rule="resolve" parents="14251 25"/>
    </justification>
  </clause>

</proof>

<proof number="13" length="107" max_count="37">

<comments><![CDATA[
% Proof 13 at 7.94 (+ 0.08) seconds: assoc_meet.
% Length of proof is 107.
% Level of proof is 28.
% Maximum clause weight is 37.
% Given clauses 444.

]]></comments>

  <clause id="14" type="goal">
    <literal><![CDATA[
      (x ^ y) ^ z = x ^ (y ^ z)
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc_meet)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="15" type="assumption">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="16" type="assumption">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <justification jstring="[copy(16),flip(a)].">
      <j1 rule="copy" parents="16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18" type="assumption">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_meet)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(f(x,y),f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[copy(18),flip(a)].">
      <j1 rule="copy" parents="18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20" type="assumption">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(x,x) = c(x)
    ]]></literal>
    <justification jstring="[copy(20),flip(a)].">
      <j1 rule="copy" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="39" type="deny">
    <literal><![CDATA[
      (c26 ^ c27) ^ c28 != c26 ^ (c27 ^ c28)
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc_meet)
    ]]></attribute>
    <justification jstring="[deny(14)].">
      <j1 rule="deny" parents="14"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      c(f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(19),rewrite([21(3)])].">
      <j1 rule="back_rewrite" parents="19"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      f(c(x),c(y)) = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(17),rewrite([21(1),21(2)])].">
      <j1 rule="back_rewrite" parents="17"/>
      <j2 rule="rewrite" parents="21 21"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(15),rewrite([21(5)])].">
      <j1 rule="back_rewrite" parents="15"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      x ^ x = c(c(x))
    ]]></literal>
    <justification jstring="[para(21(a,1),40(a,1,1)),flip(a)].">
      <j1 rule="para" parents="21 40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      c(c(x)) = x v x
    ]]></literal>
    <justification jstring="[para(41(a,1),21(a,1)),flip(a)].">
      <j1 rule="para" parents="41 21"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      c(x) ^ c(y) = c(x v y)
    ]]></literal>
    <justification jstring="[para(41(a,1),40(a,1,1)),flip(a)].">
      <j1 rule="para" parents="41 40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      x ^ x = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(43),rewrite([45(3)])].">
      <j1 rule="back_rewrite" parents="43"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      f(f(f(c(x),f(x,y)),z),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1,1))].">
      <j1 rule="para" parents="21 42"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      f(f(f(f(x,y),c(y)),z),f(y,f(f(y,f(c(y),y)),y))) = y
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1,2))].">
      <j1 rule="para" parents="21 42"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      f(f(x v x,y),f(x,f(f(x,f(c(x),x)),x))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1)),rewrite([21(1),45(2)])].">
      <j1 rule="para" parents="21 42"/>
      <j2 rule="rewrite" parents="21 45"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      f(f(x,y) ^ f(y,z),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1)),rewrite([40(4)])].">
      <j1 rule="para" parents="21 42"/>
      <j2 rule="rewrite" parents="40"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      f(f(f(x v y,f(c(y),z)),u),f(c(y),f(f(z,f(y v y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[para(41(a,1),42(a,1,1,1,1)),rewrite([45(8)])].">
      <j1 rule="para" parents="41 42"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,c(z))),u),f(y,f(f(c(z),y v z),c(z)))) = y
    ]]></literal>
    <justification jstring="[para(41(a,1),42(a,1,2,2,1,2))].">
      <j1 rule="para" parents="41 42"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(f(f(z,f(c(x),z)),z),f(c(x),f(f(z,f(c(x),z)),z))),f(f(z,f(c(x),z)),z)))) = x
    ]]></literal>
    <justification jstring="[para(42(a,1),42(a,1,1,1))].">
      <j1 rule="para" parents="42 42"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      f(x v x,c(y)) = c(x) v y
    ]]></literal>
    <justification jstring="[para(45(a,1),41(a,1,1))].">
      <j1 rule="para" parents="45 41"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      f(c(x),y v y) = x v c(y)
    ]]></literal>
    <justification jstring="[para(45(a,1),41(a,1,2))].">
      <j1 rule="para" parents="45 41"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      c(x v x) = c(x) v c(x)
    ]]></literal>
    <justification jstring="[para(45(a,1),45(a,1,1))].">
      <j1 rule="para" parents="45 45"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      (x v x) ^ c(y) = c(c(x) v y)
    ]]></literal>
    <justification jstring="[para(45(a,1),46(a,1,1))].">
      <j1 rule="para" parents="45 46"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      f(x v x,y v y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[para(45(a,1),65(a,1,2))].">
      <j1 rule="para" parents="45 65"/>
    </justification>
  </clause>

  <clause id="95">
    <literal><![CDATA[
      (x v x) ^ (y v y) = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[para(45(a,1),72(a,1,2))].">
      <j1 rule="para" parents="45 72"/>
    </justification>
  </clause>

  <clause id="158">
    <literal><![CDATA[
      f(x,f(c(x),f(f(y,f(x v x,y)),y))) = c(x)
    ]]></literal>
    <justification jstring="[para(53(a,1),42(a,1,1)),rewrite([45(3)])].">
      <j1 rule="para" parents="53 42"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="203">
    <literal><![CDATA[
      f(c(x) v y,f(x,f(f(x,f(c(x),x)),x))) = x
    ]]></literal>
    <justification jstring="[para(65(a,1),54(a,1,1))].">
      <j1 rule="para" parents="65 54"/>
    </justification>
  </clause>

  <clause id="304">
    <literal><![CDATA[
      f(f(x,y v y) ^ (c(y) v z),f(y v y,f(f(c(z),(y v y) v z),c(z)))) = y v y
    ]]></literal>
    <justification jstring="[para(65(a,1),55(a,1,1,2)),rewrite([68(9),65(12),45(9)])].">
      <j1 rule="para" parents="65 55"/>
      <j2 rule="rewrite" parents="68 65 45"/>
    </justification>
  </clause>

  <clause id="776">
    <literal><![CDATA[
      f((x v x) v y,f(c(x),f(f(c(x),c(x) v x),c(x)))) = c(x)
    ]]></literal>
    <justification jstring="[para(41(a,1),203(a,1,2,2,1,2)),rewrite([45(2)])].">
      <j1 rule="para" parents="41 203"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="785">
    <literal><![CDATA[
      f(c(x),f(x v x,f(f(f(x,f(f(x,f(c(x),x)),x)),x),f(x,f(f(x,f(c(x),x)),x))))) = x v x
    ]]></literal>
    <justification jstring="[para(203(a,1),158(a,1,2,2,1,2)),rewrite([45(3),45(18)])].">
      <j1 rule="para" parents="203 158"/>
      <j2 rule="rewrite" parents="45 45"/>
    </justification>
  </clause>

  <clause id="1263">
    <literal><![CDATA[
      f(f(f(x v c(y),c(y) v z),u),f(y v y,f(f(c(z),(y v y) v z),c(z)))) = y v y
    ]]></literal>
    <justification jstring="[para(66(a,1),61(a,1,1,1,1)),rewrite([65(5)])].">
      <j1 rule="para" parents="66 61"/>
      <j2 rule="rewrite" parents="65"/>
    </justification>
  </clause>

  <clause id="1623">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(62(a,1),42(a,1,1))].">
      <j1 rule="para" parents="62 42"/>
    </justification>
  </clause>

  <clause id="1625">
    <literal><![CDATA[
      f(c(x),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(62(a,1),52(a,1,1))].">
      <j1 rule="para" parents="62 52"/>
    </justification>
  </clause>

  <clause id="1628">
    <literal><![CDATA[
      f(x v y,f(c(y),f(f(z,f(y v y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[para(62(a,1),59(a,1,1))].">
      <j1 rule="para" parents="62 59"/>
    </justification>
  </clause>

  <clause id="1630">
    <literal><![CDATA[
      f(c(x),f(x v x,x)) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(785),rewrite([1623(14)])].">
      <j1 rule="back_rewrite" parents="785"/>
      <j2 rule="rewrite" parents="1623"/>
    </justification>
  </clause>

  <clause id="1647">
    <literal><![CDATA[
      f((x v x) ^ f(f(x v x,x),y),f(f(x v x,x),f(f(y,f((x v x) ^ x,y)),y))) = f(x v x,x)
    ]]></literal>
    <justification jstring="[para(1630(a,1),55(a,1,1,1)),rewrite([40(10)])].">
      <j1 rule="para" parents="1630 55"/>
      <j2 rule="rewrite" parents="40"/>
    </justification>
  </clause>

  <clause id="1648">
    <literal><![CDATA[
      f(f(x,c(y)) ^ (y v y),f(c(y),f(f(f(y v y,y),f(y v y,f(y v y,y))),f(y v y,y)))) = c(y)
    ]]></literal>
    <justification jstring="[para(1630(a,1),55(a,1,1,2)),rewrite([45(9)])].">
      <j1 rule="para" parents="1630 55"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="1693">
    <literal><![CDATA[
      f(c(x),f(x,f(f(c(y),x v y),c(y)))) = x
    ]]></literal>
    <justification jstring="[para(41(a,1),1625(a,1,2,2,1,2))].">
      <j1 rule="para" parents="41 1625"/>
    </justification>
  </clause>

  <clause id="1739">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(1625(a,1),1625(a,1,2,2,1,2)),rewrite([1623(13),21(2),21(3),45(2)])].">
      <j1 rule="para" parents="1625 1625"/>
      <j2 rule="rewrite" parents="1623 21 21 45"/>
    </justification>
  </clause>

  <clause id="1786">
    <literal><![CDATA[
      f(f(x,c(y)) ^ y,f(c(y),f(f(c(y),f(y,c(y))),c(y)))) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1648),rewrite([1739(3),1739(5),21(5),1739(6),1739(6),21(6),1739(9),21(9)])].">
      <j1 rule="back_rewrite" parents="1648"/>
      <j2 rule="rewrite" parents="1739 1739 21 1739 1739 21 1739 21"/>
    </justification>
  </clause>

  <clause id="1787">
    <literal><![CDATA[
      f(x ^ f(c(x),y),f(c(x),f(f(y,f(x,y)),y))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(1647),rewrite([1739(1),1739(1),21(1),1739(4),21(4),1739(5),50(5),1739(5),1739(10),21(10)])].">
      <j1 rule="back_rewrite" parents="1647"/>
      <j2 rule="rewrite" parents="1739 1739 21 1739 21 1739 50 1739 1739 21"/>
    </justification>
  </clause>

  <clause id="1798">
    <literal><![CDATA[
      f(x v y,f(c(y),f(f(z,f(y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1628),rewrite([1739(3)])].">
      <j1 rule="back_rewrite" parents="1628"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="1958">
    <literal><![CDATA[
      f(f(f(x v c(y),c(y) v z),u),f(y,f(f(c(z),y v z),c(z)))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1263),rewrite([1739(7),1739(8),1739(14)])].">
      <j1 rule="back_rewrite" parents="1263"/>
      <j2 rule="rewrite" parents="1739 1739 1739"/>
    </justification>
  </clause>

  <clause id="2130">
    <literal><![CDATA[
      f(x v y,f(c(x),f(f(c(x),c(x) v x),c(x)))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(776),rewrite([1739(1)])].">
      <j1 rule="back_rewrite" parents="776"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="2302">
    <literal><![CDATA[
      f(f(x,y) ^ (c(y) v z),f(y,f(f(c(z),y v z),c(z)))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(304),rewrite([1739(1),1739(5),1739(6),1739(12)])].">
      <j1 rule="back_rewrite" parents="304"/>
      <j2 rule="rewrite" parents="1739 1739 1739 1739"/>
    </justification>
  </clause>

  <clause id="2374">
    <literal><![CDATA[
      x ^ y = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[back_rewrite(95),rewrite([1739(1),1739(1)])].">
      <j1 rule="back_rewrite" parents="95"/>
      <j2 rule="rewrite" parents="1739 1739"/>
    </justification>
  </clause>

  <clause id="2384">
    <literal><![CDATA[
      f(x,y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(77),rewrite([1739(1),1739(1)])].">
      <j1 rule="back_rewrite" parents="77"/>
      <j2 rule="rewrite" parents="1739 1739"/>
    </justification>
  </clause>

  <clause id="2392">
    <literal><![CDATA[
      c(c(x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(45),rewrite([1739(3)])].">
      <j1 rule="back_rewrite" parents="45"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="2400">
    <literal><![CDATA[
      (c(c(x) v c(y)) v c(c(y) v z)) v c(c(y) v c(c(z v c(y v z)) v z)) = y
    ]]></literal>
    <justification jstring="[back_rewrite(2302),rewrite([2384(1),2374(6),2384(12),2392(11),2384(14),2392(15),2384(15),2384(18),2392(10)])].">
      <j1 rule="back_rewrite" parents="2302"/>
      <j2 rule="rewrite" parents="2384 2374 2384 2392 2384 2392 2384 2384 2392"/>
    </justification>
  </clause>

  <clause id="2410">
    <literal><![CDATA[
      (c(x) v c(x v c(y))) v c(x v c(c(c(y) v c(c(x) v c(y))) v c(y))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(1787),rewrite([2384(2),2392(2),2374(3),2384(8),2384(11),2384(14),2384(17),2392(8),2384(18),2392(7)])].">
      <j1 rule="back_rewrite" parents="1787"/>
      <j2 rule="rewrite" parents="2384 2392 2374 2384 2384 2384 2384 2392 2384 2392"/>
    </justification>
  </clause>

  <clause id="2411">
    <literal><![CDATA[
      (c(c(x) v y) v c(y)) v c(y v c(c(y v c(c(y) v y)) v y)) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1786),rewrite([2384(2),2392(3),2374(3),2384(10),2392(11),2384(11),2392(9),2384(13),2392(14),2384(14),2392(8),2384(15),2392(7)])].">
      <j1 rule="back_rewrite" parents="1786"/>
      <j2 rule="rewrite" parents="2384 2392 2374 2384 2392 2384 2392 2384 2392 2384 2392 2384 2392"/>
    </justification>
  </clause>

  <clause id="2437">
    <literal><![CDATA[
      c((c(c26) v c(c27)) v c(c28)) != c(c(c26) v (c(c27) v c(c28)))
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc_meet)
    ]]></attribute>
    <justification jstring="[back_rewrite(39),rewrite([2374(3),2374(8),2392(7),2374(13),2374(17),2392(18)])].">
      <j1 rule="back_rewrite" parents="39"/>
      <j2 rule="rewrite" parents="2374 2374 2392 2374 2374 2392"/>
    </justification>
  </clause>

  <clause id="2450">
    <literal><![CDATA[
      c(x v y) v c(x v c(c(x v c(c(x) v x)) v x)) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(2130),rewrite([2384(6),2392(4),2384(8),2392(9),2384(9),2392(3),2384(10)])].">
      <j1 rule="back_rewrite" parents="2130"/>
      <j2 rule="rewrite" parents="2384 2392 2384 2392 2384 2392 2384"/>
    </justification>
  </clause>

  <clause id="2453">
    <literal><![CDATA[
      c(c(c(x v c(y)) v c(c(y) v z)) v c(u)) v c(c(y) v c(c(z v c(y v z)) v z)) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1958),rewrite([2384(5),2384(8),2384(13),2392(12),2384(15),2392(16),2384(16),2384(19)])].">
      <j1 rule="back_rewrite" parents="1958"/>
      <j2 rule="rewrite" parents="2384 2384 2384 2392 2384 2392 2384 2384"/>
    </justification>
  </clause>

  <clause id="2454">
    <literal><![CDATA[
      c(x v y) v c(y v c(c(c(z) v c(c(y) v c(z))) v c(z))) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1798),rewrite([2384(3),2384(6),2384(9),2384(12),2392(3),2384(13)])].">
      <j1 rule="back_rewrite" parents="1798"/>
      <j2 rule="rewrite" parents="2384 2384 2384 2384 2392 2384"/>
    </justification>
  </clause>

  <clause id="2465">
    <literal><![CDATA[
      x v c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1693),rewrite([2384(4),2392(3),2384(6),2392(7),2384(7),2384(10),2392(2)])].">
      <j1 rule="back_rewrite" parents="1693"/>
      <j2 rule="rewrite" parents="2384 2392 2384 2392 2384 2384 2392"/>
    </justification>
  </clause>

  <clause id="2479">
    <literal><![CDATA[
      c(x v c(c(x v c(c(x) v x)) v x)) = c(x)
    ]]></literal>
    <justification jstring="[para(2450(a,1),1739(a,1)),flip(a)].">
      <j1 rule="para" parents="2450 1739"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2481">
    <literal><![CDATA[
      c(x v y) v c(x) = c(x)
    ]]></literal>
    <justification jstring="[para(2450(a,1),2384(a,2)),rewrite([2384(10),2479(11)])].">
      <j1 rule="para" parents="2450 2384"/>
      <j2 rule="rewrite" parents="2384 2479"/>
    </justification>
  </clause>

  <clause id="2485">
    <literal><![CDATA[
      (c(c(x) v y) v c(y)) v c(y) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(2411),rewrite([2479(14)])].">
      <j1 rule="back_rewrite" parents="2411"/>
      <j2 rule="rewrite" parents="2479"/>
    </justification>
  </clause>

  <clause id="2486">
    <literal><![CDATA[
      c(c(x) v y) v x = x
    ]]></literal>
    <justification jstring="[para(2392(a,1),2481(a,1,2)),rewrite([2392(6)])].">
      <j1 rule="para" parents="2392 2481"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="2487">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2481(a,1,1,1)),rewrite([2392(2),2392(3),2392(5)])].">
      <j1 rule="para" parents="2481 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2492">
    <literal><![CDATA[
      c(x) v c(c(c(y) v c(x)) v c(c(x) v z)) = c(c(c(y) v c(x)) v c(c(x) v z))
    ]]></literal>
    <justification jstring="[para(2400(a,1),2481(a,1,1,1))].">
      <j1 rule="para" parents="2400 2481"/>
    </justification>
  </clause>

  <clause id="2500">
    <literal><![CDATA[
      (c(c(x) v c(y)) v y) v y = y
    ]]></literal>
    <justification jstring="[para(2392(a,1),2485(a,1,1,2)),rewrite([2392(7),2392(8)])].">
      <j1 rule="para" parents="2392 2485"/>
      <j2 rule="rewrite" parents="2392 2392"/>
    </justification>
  </clause>

  <clause id="2507">
    <literal><![CDATA[
      (c(x v c(y)) v y) v y = y
    ]]></literal>
    <justification jstring="[para(2392(a,1),2500(a,1,1,1,1,1))].">
      <j1 rule="para" parents="2392 2500"/>
    </justification>
  </clause>

  <clause id="2511">
    <literal><![CDATA[
      (c(x) v c(x v y)) v c(x v c(c(y v c(c(x) v y)) v y)) = c(x)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2410(a,1,1,2,1,2)),rewrite([2392(6),2392(7),2392(11)])].">
      <j1 rule="para" parents="2392 2410"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2513">
    <literal><![CDATA[
      (c(x) v c(x v c(y))) v c(x) = c(x) v c(x v c(y))
    ]]></literal>
    <justification jstring="[para(2410(a,1),2465(a,1,2,1,2,1,1,1,2,1)),rewrite([2392(25),2454(38),2392(13),2486(12)])].">
      <j1 rule="para" parents="2410 2465"/>
      <j2 rule="rewrite" parents="2392 2454 2392 2486"/>
    </justification>
  </clause>

  <clause id="2517">
    <literal><![CDATA[
      c(x) v c(x v c(y)) = c(x)
    ]]></literal>
    <justification jstring="[para(2410(a,1),2487(a,1,2)),rewrite([2513(7),2511(23)])].">
      <j1 rule="para" parents="2410 2487"/>
      <j2 rule="rewrite" parents="2513 2511"/>
    </justification>
  </clause>

  <clause id="2529">
    <literal><![CDATA[
      (x v y) v x = x v y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2517(a,1,2,1)),rewrite([2392(3),2392(3),2392(5)])].">
      <j1 rule="para" parents="2481 2517"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2533">
    <literal><![CDATA[
      x v c(c(x) v y) = x
    ]]></literal>
    <justification jstring="[para(2486(a,1),2529(a,1,1)),rewrite([2486(8)])].">
      <j1 rule="para" parents="2486 2529"/>
      <j2 rule="rewrite" parents="2486"/>
    </justification>
  </clause>

  <clause id="2552">
    <literal><![CDATA[
      c((c(x) v y) v c(z)) v c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[para(2481(a,1),2453(a,1,1,1,1,1)),rewrite([2392(4)])].">
      <j1 rule="para" parents="2481 2453"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="2553">
    <literal><![CDATA[
      c(x v c(y)) v c(c(y) v c(c(z v c(y v z)) v z)) = y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2453(a,1,1,1)),rewrite([2392(4)])].">
      <j1 rule="para" parents="2481 2453"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="2560">
    <literal><![CDATA[
      c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[para(2453(a,1),2507(a,1,1)),rewrite([2533(10)]),flip(a)].">
      <j1 rule="para" parents="2453 2507"/>
      <j2 rule="rewrite" parents="2533"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2574">
    <literal><![CDATA[
      c(x v c(y)) v y = y
    ]]></literal>
    <justification jstring="[back_rewrite(2553),rewrite([2560(12)])].">
      <j1 rule="back_rewrite" parents="2553"/>
      <j2 rule="rewrite" parents="2560"/>
    </justification>
  </clause>

  <clause id="2575">
    <literal><![CDATA[
      c((c(x) v y) v c(z)) v x = x
    ]]></literal>
    <justification jstring="[back_rewrite(2552),rewrite([2560(14)])].">
      <j1 rule="back_rewrite" parents="2552"/>
      <j2 rule="rewrite" parents="2560"/>
    </justification>
  </clause>

  <clause id="2583">
    <literal><![CDATA[
      c(x v y) v c(y) = c(y)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2574(a,1,1,1,2))].">
      <j1 rule="para" parents="2392 2574"/>
    </justification>
  </clause>

  <clause id="2585">
    <literal><![CDATA[
      x v c(y v c(x)) = x
    ]]></literal>
    <justification jstring="[para(2574(a,1),2529(a,1,1)),rewrite([2574(8)])].">
      <j1 rule="para" parents="2574 2529"/>
      <j2 rule="rewrite" parents="2574"/>
    </justification>
  </clause>

  <clause id="2590">
    <literal><![CDATA[
      c(x) v c(y v x) = c(x)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2585(a,1,2,1,2))].">
      <j1 rule="para" parents="2392 2585"/>
    </justification>
  </clause>

  <clause id="2591">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(2583(a,1),2481(a,1,1,1)),rewrite([2392(2),2392(3),2392(5)])].">
      <j1 rule="para" parents="2583 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2599">
    <literal><![CDATA[
      c((c(x) v y) v z) v x = x
    ]]></literal>
    <justification jstring="[para(2392(a,1),2575(a,1,1,1,2))].">
      <j1 rule="para" parents="2392 2575"/>
    </justification>
  </clause>

  <clause id="2614">
    <literal><![CDATA[
      x v c((c(x) v y) v z) = x
    ]]></literal>
    <justification jstring="[para(2599(a,1),2529(a,1,1)),rewrite([2599(10)])].">
      <j1 rule="para" parents="2599 2529"/>
      <j2 rule="rewrite" parents="2599"/>
    </justification>
  </clause>

  <clause id="2618">
    <literal><![CDATA[
      c((x v c(y)) v z) v y = y
    ]]></literal>
    <justification jstring="[para(2591(a,1),2599(a,1,1,1,1))].">
      <j1 rule="para" parents="2591 2599"/>
    </justification>
  </clause>

  <clause id="2619">
    <literal><![CDATA[
      c(x v (c(y) v z)) v y = y
    ]]></literal>
    <justification jstring="[para(2591(a,1),2599(a,1,1,1))].">
      <j1 rule="para" parents="2591 2599"/>
    </justification>
  </clause>

  <clause id="2650">
    <literal><![CDATA[
      x v c(y v (c(x) v z)) = x
    ]]></literal>
    <justification jstring="[para(2591(a,1),2614(a,1,2,1))].">
      <j1 rule="para" parents="2591 2614"/>
    </justification>
  </clause>

  <clause id="2656">
    <literal><![CDATA[
      c(x v (y v c(z))) v z = z
    ]]></literal>
    <justification jstring="[para(2591(a,1),2618(a,1,1,1))].">
      <j1 rule="para" parents="2591 2618"/>
    </justification>
  </clause>

  <clause id="2660">
    <literal><![CDATA[
      c(x v (y v z)) v c(y) = c(y)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2619(a,1,1,1,2,1))].">
      <j1 rule="para" parents="2392 2619"/>
    </justification>
  </clause>

  <clause id="2677">
    <literal><![CDATA[
      c(x v y) v c(c(c(z) v c(x v y)) v x) = c(c(c(z) v c(x v y)) v x)
    ]]></literal>
    <justification jstring="[para(2481(a,1),2492(a,1,2,1,2,1)),rewrite([2392(9),2481(19),2392(17)])].">
      <j1 rule="para" parents="2481 2492"/>
      <j2 rule="rewrite" parents="2392 2481 2392"/>
    </justification>
  </clause>

  <clause id="2697">
    <literal><![CDATA[
      c(x) v c(y v (x v z)) = c(x)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2650(a,1,2,1,2,1))].">
      <j1 rule="para" parents="2392 2650"/>
    </justification>
  </clause>

  <clause id="2706">
    <literal><![CDATA[
      c(x v (y v z)) v c(z) = c(z)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2656(a,1,1,1,2,2))].">
      <j1 rule="para" parents="2392 2656"/>
    </justification>
  </clause>

  <clause id="2737">
    <literal><![CDATA[
      x v (y v (x v z)) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(2660(a,1),2481(a,1,1,1)),rewrite([2392(2),2392(4),2392(7)])].">
      <j1 rule="para" parents="2660 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2764">
    <literal><![CDATA[
      x v (y v (z v x)) = y v (z v x)
    ]]></literal>
    <justification jstring="[para(2706(a,1),2481(a,1,1,1)),rewrite([2392(2),2392(4),2392(7)])].">
      <j1 rule="para" parents="2706 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="5628">
    <literal><![CDATA[
      c(x v y) v c(y v x) = c(y v x)
    ]]></literal>
    <justification jstring="[para(2585(a,1),2677(a,1,2,1,1,1)),rewrite([2392(2),2392(4),2392(8),2590(9),2392(7)])].">
      <j1 rule="para" parents="2585 2677"/>
      <j2 rule="rewrite" parents="2392 2392 2392 2590 2392"/>
    </justification>
  </clause>

  <clause id="5636">
    <literal><![CDATA[
      c(x v (y v z)) v c(y v x) = c(y v x)
    ]]></literal>
    <justification jstring="[para(2650(a,1),2677(a,1,2,1,1,1)),rewrite([2392(2),2392(5),2392(9),2697(11),2392(8)])].">
      <j1 rule="para" parents="2650 2677"/>
      <j2 rule="rewrite" parents="2392 2392 2392 2697 2392"/>
    </justification>
  </clause>

  <clause id="5749">
    <literal><![CDATA[
      (x v y) v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(5628(a,1),2481(a,1,1,1)),rewrite([2392(3),2392(4),2392(6)])].">
      <j1 rule="para" parents="5628 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="5751">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[para(5628(a,1),2533(a,1,2,1)),rewrite([2392(4),5749(3)])].">
      <j1 rule="para" parents="5628 2533"/>
      <j2 rule="rewrite" parents="2392 5749"/>
    </justification>
  </clause>

  <clause id="5754">
    <literal><![CDATA[
      (x v y) v c(c(y v x) v z) = x v y
    ]]></literal>
    <justification jstring="[para(5628(a,1),2599(a,1,1,1,1)),rewrite([5751(6)])].">
      <j1 rule="para" parents="5628 2599"/>
      <j2 rule="rewrite" parents="5751"/>
    </justification>
  </clause>

  <clause id="12548">
    <literal><![CDATA[
      (x v (y v z)) v (y v x) = x v (y v z)
    ]]></literal>
    <justification jstring="[para(5636(a,1),2533(a,1,2,1)),rewrite([2392(5)])].">
      <j1 rule="para" parents="5636 2533"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="12684">
    <literal><![CDATA[
      (x v y) v ((x v z) v y) = (x v z) v y
    ]]></literal>
    <justification jstring="[para(5636(a,1),5754(a,1,2,1)),rewrite([2392(5),5751(4)])].">
      <j1 rule="para" parents="5636 5754"/>
      <j2 rule="rewrite" parents="2392 5751"/>
    </justification>
  </clause>

  <clause id="12728">
    <literal><![CDATA[
      (x v y) v (x v (y v z)) = x v (y v z)
    ]]></literal>
    <justification jstring="[para(2737(a,1),12548(a,1,1)),rewrite([5751(4),2737(7)])].">
      <j1 rule="para" parents="2737 12548"/>
      <j2 rule="rewrite" parents="5751 2737"/>
    </justification>
  </clause>

  <clause id="12730">
    <literal><![CDATA[
      (x v y) v (x v (z v y)) = x v (z v y)
    ]]></literal>
    <justification jstring="[para(2764(a,1),12548(a,1,1)),rewrite([5751(4),2764(7)])].">
      <j1 rule="para" parents="2764 12548"/>
      <j2 rule="rewrite" parents="5751 2764"/>
    </justification>
  </clause>

  <clause id="13101">
    <literal><![CDATA[
      (x v y) v (x v z) = (x v y) v z
    ]]></literal>
    <justification jstring="[para(12684(a,1),12548(a,1,1)),rewrite([12730(6),12684(7)])].">
      <j1 rule="para" parents="12684 12548"/>
      <j2 rule="rewrite" parents="12730 12684"/>
    </justification>
  </clause>

  <clause id="14172">
    <literal><![CDATA[
      (x v y) v (y v z) = x v (y v z)
    ]]></literal>
    <justification jstring="[back_rewrite(12728),rewrite([13101(4)])].">
      <j1 rule="back_rewrite" parents="12728"/>
      <j2 rule="rewrite" parents="13101"/>
    </justification>
  </clause>

  <clause id="14240">
    <literal><![CDATA[
      (x v y) v z = x v (y v z)
    ]]></literal>
    <justification jstring="[para(2591(a,1),13101(a,1,1)),rewrite([14172(3),2591(4)]),flip(a)].">
      <j1 rule="para" parents="2591 13101"/>
      <j2 rule="rewrite" parents="14172 2591"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="15060">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc_meet)
    ]]></attribute>
    <justification jstring="[back_rewrite(2437),rewrite([14240(8)]),xx(a)].">
      <j1 rule="back_rewrite" parents="2437"/>
      <j2 rule="rewrite" parents="14240"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

<proof number="14" length="111" max_count="37">

<comments><![CDATA[
% Proof 14 at 7.95 (+ 0.08) seconds: assoc_join.
% Length of proof is 111.
% Level of proof is 29.
% Maximum clause weight is 37.
% Given clauses 444.

]]></comments>

  <clause id="13" type="goal">
    <literal><![CDATA[
      (x v y) v z = x v (y v z)
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc_join)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="15" type="assumption">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="16" type="assumption">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <justification jstring="[copy(16),flip(a)].">
      <j1 rule="copy" parents="16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18" type="assumption">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_meet)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(f(x,y),f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[copy(18),flip(a)].">
      <j1 rule="copy" parents="18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20" type="assumption">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(x,x) = c(x)
    ]]></literal>
    <justification jstring="[copy(20),flip(a)].">
      <j1 rule="copy" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="38" type="deny">
    <literal><![CDATA[
      (c23 v c24) v c25 != c23 v (c24 v c25)
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc_join)
    ]]></attribute>
    <justification jstring="[deny(13)].">
      <j1 rule="deny" parents="13"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      c(f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(19),rewrite([21(3)])].">
      <j1 rule="back_rewrite" parents="19"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      f(c(x),c(y)) = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(17),rewrite([21(1),21(2)])].">
      <j1 rule="back_rewrite" parents="17"/>
      <j2 rule="rewrite" parents="21 21"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(15),rewrite([21(5)])].">
      <j1 rule="back_rewrite" parents="15"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      x ^ x = c(c(x))
    ]]></literal>
    <justification jstring="[para(21(a,1),40(a,1,1)),flip(a)].">
      <j1 rule="para" parents="21 40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      c(c(x)) = x v x
    ]]></literal>
    <justification jstring="[para(41(a,1),21(a,1)),flip(a)].">
      <j1 rule="para" parents="41 21"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      c(x) ^ c(y) = c(x v y)
    ]]></literal>
    <justification jstring="[para(41(a,1),40(a,1,1)),flip(a)].">
      <j1 rule="para" parents="41 40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      x ^ x = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(43),rewrite([45(3)])].">
      <j1 rule="back_rewrite" parents="43"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      f(f(f(c(x),f(x,y)),z),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1,1))].">
      <j1 rule="para" parents="21 42"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      f(f(f(f(x,y),c(y)),z),f(y,f(f(y,f(c(y),y)),y))) = y
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1,2))].">
      <j1 rule="para" parents="21 42"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      f(f(x v x,y),f(x,f(f(x,f(c(x),x)),x))) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1,1)),rewrite([21(1),45(2)])].">
      <j1 rule="para" parents="21 42"/>
      <j2 rule="rewrite" parents="21 45"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      f(f(x,y) ^ f(y,z),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(21(a,1),42(a,1,1)),rewrite([40(4)])].">
      <j1 rule="para" parents="21 42"/>
      <j2 rule="rewrite" parents="40"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      f(f(f(x v y,f(c(y),z)),u),f(c(y),f(f(z,f(y v y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[para(41(a,1),42(a,1,1,1,1)),rewrite([45(8)])].">
      <j1 rule="para" parents="41 42"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,c(z))),u),f(y,f(f(c(z),y v z),c(z)))) = y
    ]]></literal>
    <justification jstring="[para(41(a,1),42(a,1,2,2,1,2))].">
      <j1 rule="para" parents="41 42"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(f(f(z,f(c(x),z)),z),f(c(x),f(f(z,f(c(x),z)),z))),f(f(z,f(c(x),z)),z)))) = x
    ]]></literal>
    <justification jstring="[para(42(a,1),42(a,1,1,1))].">
      <j1 rule="para" parents="42 42"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      f(x v x,c(y)) = c(x) v y
    ]]></literal>
    <justification jstring="[para(45(a,1),41(a,1,1))].">
      <j1 rule="para" parents="45 41"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      f(c(x),y v y) = x v c(y)
    ]]></literal>
    <justification jstring="[para(45(a,1),41(a,1,2))].">
      <j1 rule="para" parents="45 41"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      c(x v x) = c(x) v c(x)
    ]]></literal>
    <justification jstring="[para(45(a,1),45(a,1,1))].">
      <j1 rule="para" parents="45 45"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      (x v x) ^ c(y) = c(c(x) v y)
    ]]></literal>
    <justification jstring="[para(45(a,1),46(a,1,1))].">
      <j1 rule="para" parents="45 46"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      f(x v x,y v y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[para(45(a,1),65(a,1,2))].">
      <j1 rule="para" parents="45 65"/>
    </justification>
  </clause>

  <clause id="95">
    <literal><![CDATA[
      (x v x) ^ (y v y) = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[para(45(a,1),72(a,1,2))].">
      <j1 rule="para" parents="45 72"/>
    </justification>
  </clause>

  <clause id="158">
    <literal><![CDATA[
      f(x,f(c(x),f(f(y,f(x v x,y)),y))) = c(x)
    ]]></literal>
    <justification jstring="[para(53(a,1),42(a,1,1)),rewrite([45(3)])].">
      <j1 rule="para" parents="53 42"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="203">
    <literal><![CDATA[
      f(c(x) v y,f(x,f(f(x,f(c(x),x)),x))) = x
    ]]></literal>
    <justification jstring="[para(65(a,1),54(a,1,1))].">
      <j1 rule="para" parents="65 54"/>
    </justification>
  </clause>

  <clause id="304">
    <literal><![CDATA[
      f(f(x,y v y) ^ (c(y) v z),f(y v y,f(f(c(z),(y v y) v z),c(z)))) = y v y
    ]]></literal>
    <justification jstring="[para(65(a,1),55(a,1,1,2)),rewrite([68(9),65(12),45(9)])].">
      <j1 rule="para" parents="65 55"/>
      <j2 rule="rewrite" parents="68 65 45"/>
    </justification>
  </clause>

  <clause id="776">
    <literal><![CDATA[
      f((x v x) v y,f(c(x),f(f(c(x),c(x) v x),c(x)))) = c(x)
    ]]></literal>
    <justification jstring="[para(41(a,1),203(a,1,2,2,1,2)),rewrite([45(2)])].">
      <j1 rule="para" parents="41 203"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="785">
    <literal><![CDATA[
      f(c(x),f(x v x,f(f(f(x,f(f(x,f(c(x),x)),x)),x),f(x,f(f(x,f(c(x),x)),x))))) = x v x
    ]]></literal>
    <justification jstring="[para(203(a,1),158(a,1,2,2,1,2)),rewrite([45(3),45(18)])].">
      <j1 rule="para" parents="203 158"/>
      <j2 rule="rewrite" parents="45 45"/>
    </justification>
  </clause>

  <clause id="1263">
    <literal><![CDATA[
      f(f(f(x v c(y),c(y) v z),u),f(y v y,f(f(c(z),(y v y) v z),c(z)))) = y v y
    ]]></literal>
    <justification jstring="[para(66(a,1),61(a,1,1,1,1)),rewrite([65(5)])].">
      <j1 rule="para" parents="66 61"/>
      <j2 rule="rewrite" parents="65"/>
    </justification>
  </clause>

  <clause id="1623">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(62(a,1),42(a,1,1))].">
      <j1 rule="para" parents="62 42"/>
    </justification>
  </clause>

  <clause id="1625">
    <literal><![CDATA[
      f(c(x),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(62(a,1),52(a,1,1))].">
      <j1 rule="para" parents="62 52"/>
    </justification>
  </clause>

  <clause id="1628">
    <literal><![CDATA[
      f(x v y,f(c(y),f(f(z,f(y v y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[para(62(a,1),59(a,1,1))].">
      <j1 rule="para" parents="62 59"/>
    </justification>
  </clause>

  <clause id="1630">
    <literal><![CDATA[
      f(c(x),f(x v x,x)) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(785),rewrite([1623(14)])].">
      <j1 rule="back_rewrite" parents="785"/>
      <j2 rule="rewrite" parents="1623"/>
    </justification>
  </clause>

  <clause id="1647">
    <literal><![CDATA[
      f((x v x) ^ f(f(x v x,x),y),f(f(x v x,x),f(f(y,f((x v x) ^ x,y)),y))) = f(x v x,x)
    ]]></literal>
    <justification jstring="[para(1630(a,1),55(a,1,1,1)),rewrite([40(10)])].">
      <j1 rule="para" parents="1630 55"/>
      <j2 rule="rewrite" parents="40"/>
    </justification>
  </clause>

  <clause id="1648">
    <literal><![CDATA[
      f(f(x,c(y)) ^ (y v y),f(c(y),f(f(f(y v y,y),f(y v y,f(y v y,y))),f(y v y,y)))) = c(y)
    ]]></literal>
    <justification jstring="[para(1630(a,1),55(a,1,1,2)),rewrite([45(9)])].">
      <j1 rule="para" parents="1630 55"/>
      <j2 rule="rewrite" parents="45"/>
    </justification>
  </clause>

  <clause id="1693">
    <literal><![CDATA[
      f(c(x),f(x,f(f(c(y),x v y),c(y)))) = x
    ]]></literal>
    <justification jstring="[para(41(a,1),1625(a,1,2,2,1,2))].">
      <j1 rule="para" parents="41 1625"/>
    </justification>
  </clause>

  <clause id="1739">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(1625(a,1),1625(a,1,2,2,1,2)),rewrite([1623(13),21(2),21(3),45(2)])].">
      <j1 rule="para" parents="1625 1625"/>
      <j2 rule="rewrite" parents="1623 21 21 45"/>
    </justification>
  </clause>

  <clause id="1786">
    <literal><![CDATA[
      f(f(x,c(y)) ^ y,f(c(y),f(f(c(y),f(y,c(y))),c(y)))) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1648),rewrite([1739(3),1739(5),21(5),1739(6),1739(6),21(6),1739(9),21(9)])].">
      <j1 rule="back_rewrite" parents="1648"/>
      <j2 rule="rewrite" parents="1739 1739 21 1739 1739 21 1739 21"/>
    </justification>
  </clause>

  <clause id="1787">
    <literal><![CDATA[
      f(x ^ f(c(x),y),f(c(x),f(f(y,f(x,y)),y))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(1647),rewrite([1739(1),1739(1),21(1),1739(4),21(4),1739(5),50(5),1739(5),1739(10),21(10)])].">
      <j1 rule="back_rewrite" parents="1647"/>
      <j2 rule="rewrite" parents="1739 1739 21 1739 21 1739 50 1739 1739 21"/>
    </justification>
  </clause>

  <clause id="1798">
    <literal><![CDATA[
      f(x v y,f(c(y),f(f(z,f(y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1628),rewrite([1739(3)])].">
      <j1 rule="back_rewrite" parents="1628"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="1958">
    <literal><![CDATA[
      f(f(f(x v c(y),c(y) v z),u),f(y,f(f(c(z),y v z),c(z)))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1263),rewrite([1739(7),1739(8),1739(14)])].">
      <j1 rule="back_rewrite" parents="1263"/>
      <j2 rule="rewrite" parents="1739 1739 1739"/>
    </justification>
  </clause>

  <clause id="2130">
    <literal><![CDATA[
      f(x v y,f(c(x),f(f(c(x),c(x) v x),c(x)))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(776),rewrite([1739(1)])].">
      <j1 rule="back_rewrite" parents="776"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="2302">
    <literal><![CDATA[
      f(f(x,y) ^ (c(y) v z),f(y,f(f(c(z),y v z),c(z)))) = y
    ]]></literal>
    <justification jstring="[back_rewrite(304),rewrite([1739(1),1739(5),1739(6),1739(12)])].">
      <j1 rule="back_rewrite" parents="304"/>
      <j2 rule="rewrite" parents="1739 1739 1739 1739"/>
    </justification>
  </clause>

  <clause id="2374">
    <literal><![CDATA[
      x ^ y = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[back_rewrite(95),rewrite([1739(1),1739(1)])].">
      <j1 rule="back_rewrite" parents="95"/>
      <j2 rule="rewrite" parents="1739 1739"/>
    </justification>
  </clause>

  <clause id="2384">
    <literal><![CDATA[
      f(x,y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(77),rewrite([1739(1),1739(1)])].">
      <j1 rule="back_rewrite" parents="77"/>
      <j2 rule="rewrite" parents="1739 1739"/>
    </justification>
  </clause>

  <clause id="2392">
    <literal><![CDATA[
      c(c(x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(45),rewrite([1739(3)])].">
      <j1 rule="back_rewrite" parents="45"/>
      <j2 rule="rewrite" parents="1739"/>
    </justification>
  </clause>

  <clause id="2400">
    <literal><![CDATA[
      (c(c(x) v c(y)) v c(c(y) v z)) v c(c(y) v c(c(z v c(y v z)) v z)) = y
    ]]></literal>
    <justification jstring="[back_rewrite(2302),rewrite([2384(1),2374(6),2384(12),2392(11),2384(14),2392(15),2384(15),2384(18),2392(10)])].">
      <j1 rule="back_rewrite" parents="2302"/>
      <j2 rule="rewrite" parents="2384 2374 2384 2392 2384 2392 2384 2384 2392"/>
    </justification>
  </clause>

  <clause id="2410">
    <literal><![CDATA[
      (c(x) v c(x v c(y))) v c(x v c(c(c(y) v c(c(x) v c(y))) v c(y))) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(1787),rewrite([2384(2),2392(2),2374(3),2384(8),2384(11),2384(14),2384(17),2392(8),2384(18),2392(7)])].">
      <j1 rule="back_rewrite" parents="1787"/>
      <j2 rule="rewrite" parents="2384 2392 2374 2384 2384 2384 2384 2392 2384 2392"/>
    </justification>
  </clause>

  <clause id="2411">
    <literal><![CDATA[
      (c(c(x) v y) v c(y)) v c(y v c(c(y v c(c(y) v y)) v y)) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1786),rewrite([2384(2),2392(3),2374(3),2384(10),2392(11),2384(11),2392(9),2384(13),2392(14),2384(14),2392(8),2384(15),2392(7)])].">
      <j1 rule="back_rewrite" parents="1786"/>
      <j2 rule="rewrite" parents="2384 2392 2374 2384 2392 2384 2392 2384 2392 2384 2392 2384 2392"/>
    </justification>
  </clause>

  <clause id="2450">
    <literal><![CDATA[
      c(x v y) v c(x v c(c(x v c(c(x) v x)) v x)) = c(x)
    ]]></literal>
    <justification jstring="[back_rewrite(2130),rewrite([2384(6),2392(4),2384(8),2392(9),2384(9),2392(3),2384(10)])].">
      <j1 rule="back_rewrite" parents="2130"/>
      <j2 rule="rewrite" parents="2384 2392 2384 2392 2384 2392 2384"/>
    </justification>
  </clause>

  <clause id="2453">
    <literal><![CDATA[
      c(c(c(x v c(y)) v c(c(y) v z)) v c(u)) v c(c(y) v c(c(z v c(y v z)) v z)) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1958),rewrite([2384(5),2384(8),2384(13),2392(12),2384(15),2392(16),2384(16),2384(19)])].">
      <j1 rule="back_rewrite" parents="1958"/>
      <j2 rule="rewrite" parents="2384 2384 2384 2392 2384 2392 2384 2384"/>
    </justification>
  </clause>

  <clause id="2454">
    <literal><![CDATA[
      c(x v y) v c(y v c(c(c(z) v c(c(y) v c(z))) v c(z))) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(1798),rewrite([2384(3),2384(6),2384(9),2384(12),2392(3),2384(13)])].">
      <j1 rule="back_rewrite" parents="1798"/>
      <j2 rule="rewrite" parents="2384 2384 2384 2384 2392 2384"/>
    </justification>
  </clause>

  <clause id="2465">
    <literal><![CDATA[
      x v c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1693),rewrite([2384(4),2392(3),2384(6),2392(7),2384(7),2384(10),2392(2)])].">
      <j1 rule="back_rewrite" parents="1693"/>
      <j2 rule="rewrite" parents="2384 2392 2384 2392 2384 2384 2392"/>
    </justification>
  </clause>

  <clause id="2479">
    <literal><![CDATA[
      c(x v c(c(x v c(c(x) v x)) v x)) = c(x)
    ]]></literal>
    <justification jstring="[para(2450(a,1),1739(a,1)),flip(a)].">
      <j1 rule="para" parents="2450 1739"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2481">
    <literal><![CDATA[
      c(x v y) v c(x) = c(x)
    ]]></literal>
    <justification jstring="[para(2450(a,1),2384(a,2)),rewrite([2384(10),2479(11)])].">
      <j1 rule="para" parents="2450 2384"/>
      <j2 rule="rewrite" parents="2384 2479"/>
    </justification>
  </clause>

  <clause id="2485">
    <literal><![CDATA[
      (c(c(x) v y) v c(y)) v c(y) = c(y)
    ]]></literal>
    <justification jstring="[back_rewrite(2411),rewrite([2479(14)])].">
      <j1 rule="back_rewrite" parents="2411"/>
      <j2 rule="rewrite" parents="2479"/>
    </justification>
  </clause>

  <clause id="2486">
    <literal><![CDATA[
      c(c(x) v y) v x = x
    ]]></literal>
    <justification jstring="[para(2392(a,1),2481(a,1,2)),rewrite([2392(6)])].">
      <j1 rule="para" parents="2392 2481"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="2487">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2481(a,1,1,1)),rewrite([2392(2),2392(3),2392(5)])].">
      <j1 rule="para" parents="2481 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2492">
    <literal><![CDATA[
      c(x) v c(c(c(y) v c(x)) v c(c(x) v z)) = c(c(c(y) v c(x)) v c(c(x) v z))
    ]]></literal>
    <justification jstring="[para(2400(a,1),2481(a,1,1,1))].">
      <j1 rule="para" parents="2400 2481"/>
    </justification>
  </clause>

  <clause id="2500">
    <literal><![CDATA[
      (c(c(x) v c(y)) v y) v y = y
    ]]></literal>
    <justification jstring="[para(2392(a,1),2485(a,1,1,2)),rewrite([2392(7),2392(8)])].">
      <j1 rule="para" parents="2392 2485"/>
      <j2 rule="rewrite" parents="2392 2392"/>
    </justification>
  </clause>

  <clause id="2507">
    <literal><![CDATA[
      (c(x v c(y)) v y) v y = y
    ]]></literal>
    <justification jstring="[para(2392(a,1),2500(a,1,1,1,1,1))].">
      <j1 rule="para" parents="2392 2500"/>
    </justification>
  </clause>

  <clause id="2511">
    <literal><![CDATA[
      (c(x) v c(x v y)) v c(x v c(c(y v c(c(x) v y)) v y)) = c(x)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2410(a,1,1,2,1,2)),rewrite([2392(6),2392(7),2392(11)])].">
      <j1 rule="para" parents="2392 2410"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2513">
    <literal><![CDATA[
      (c(x) v c(x v c(y))) v c(x) = c(x) v c(x v c(y))
    ]]></literal>
    <justification jstring="[para(2410(a,1),2465(a,1,2,1,2,1,1,1,2,1)),rewrite([2392(25),2454(38),2392(13),2486(12)])].">
      <j1 rule="para" parents="2410 2465"/>
      <j2 rule="rewrite" parents="2392 2454 2392 2486"/>
    </justification>
  </clause>

  <clause id="2517">
    <literal><![CDATA[
      c(x) v c(x v c(y)) = c(x)
    ]]></literal>
    <justification jstring="[para(2410(a,1),2487(a,1,2)),rewrite([2513(7),2511(23)])].">
      <j1 rule="para" parents="2410 2487"/>
      <j2 rule="rewrite" parents="2513 2511"/>
    </justification>
  </clause>

  <clause id="2529">
    <literal><![CDATA[
      (x v y) v x = x v y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2517(a,1,2,1)),rewrite([2392(3),2392(3),2392(5)])].">
      <j1 rule="para" parents="2481 2517"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2533">
    <literal><![CDATA[
      x v c(c(x) v y) = x
    ]]></literal>
    <justification jstring="[para(2486(a,1),2529(a,1,1)),rewrite([2486(8)])].">
      <j1 rule="para" parents="2486 2529"/>
      <j2 rule="rewrite" parents="2486"/>
    </justification>
  </clause>

  <clause id="2552">
    <literal><![CDATA[
      c((c(x) v y) v c(z)) v c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[para(2481(a,1),2453(a,1,1,1,1,1)),rewrite([2392(4)])].">
      <j1 rule="para" parents="2481 2453"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="2553">
    <literal><![CDATA[
      c(x v c(y)) v c(c(y) v c(c(z v c(y v z)) v z)) = y
    ]]></literal>
    <justification jstring="[para(2481(a,1),2453(a,1,1,1)),rewrite([2392(4)])].">
      <j1 rule="para" parents="2481 2453"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="2560">
    <literal><![CDATA[
      c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[para(2453(a,1),2507(a,1,1)),rewrite([2533(10)]),flip(a)].">
      <j1 rule="para" parents="2453 2507"/>
      <j2 rule="rewrite" parents="2533"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2574">
    <literal><![CDATA[
      c(x v c(y)) v y = y
    ]]></literal>
    <justification jstring="[back_rewrite(2553),rewrite([2560(12)])].">
      <j1 rule="back_rewrite" parents="2553"/>
      <j2 rule="rewrite" parents="2560"/>
    </justification>
  </clause>

  <clause id="2575">
    <literal><![CDATA[
      c((c(x) v y) v c(z)) v x = x
    ]]></literal>
    <justification jstring="[back_rewrite(2552),rewrite([2560(14)])].">
      <j1 rule="back_rewrite" parents="2552"/>
      <j2 rule="rewrite" parents="2560"/>
    </justification>
  </clause>

  <clause id="2583">
    <literal><![CDATA[
      c(x v y) v c(y) = c(y)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2574(a,1,1,1,2))].">
      <j1 rule="para" parents="2392 2574"/>
    </justification>
  </clause>

  <clause id="2585">
    <literal><![CDATA[
      x v c(y v c(x)) = x
    ]]></literal>
    <justification jstring="[para(2574(a,1),2529(a,1,1)),rewrite([2574(8)])].">
      <j1 rule="para" parents="2574 2529"/>
      <j2 rule="rewrite" parents="2574"/>
    </justification>
  </clause>

  <clause id="2590">
    <literal><![CDATA[
      c(x) v c(y v x) = c(x)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2585(a,1,2,1,2))].">
      <j1 rule="para" parents="2392 2585"/>
    </justification>
  </clause>

  <clause id="2591">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(2583(a,1),2481(a,1,1,1)),rewrite([2392(2),2392(3),2392(5)])].">
      <j1 rule="para" parents="2583 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2599">
    <literal><![CDATA[
      c((c(x) v y) v z) v x = x
    ]]></literal>
    <justification jstring="[para(2392(a,1),2575(a,1,1,1,2))].">
      <j1 rule="para" parents="2392 2575"/>
    </justification>
  </clause>

  <clause id="2612">
    <literal><![CDATA[
      c(c(x) v y) v (x v z) = x v z
    ]]></literal>
    <justification jstring="[para(2481(a,1),2599(a,1,1,1,1))].">
      <j1 rule="para" parents="2481 2599"/>
    </justification>
  </clause>

  <clause id="2614">
    <literal><![CDATA[
      x v c((c(x) v y) v z) = x
    ]]></literal>
    <justification jstring="[para(2599(a,1),2529(a,1,1)),rewrite([2599(10)])].">
      <j1 rule="para" parents="2599 2529"/>
      <j2 rule="rewrite" parents="2599"/>
    </justification>
  </clause>

  <clause id="2617">
    <literal><![CDATA[
      c(c(x) v y) v (z v x) = z v x
    ]]></literal>
    <justification jstring="[para(2583(a,1),2599(a,1,1,1,1))].">
      <j1 rule="para" parents="2583 2599"/>
    </justification>
  </clause>

  <clause id="2618">
    <literal><![CDATA[
      c((x v c(y)) v z) v y = y
    ]]></literal>
    <justification jstring="[para(2591(a,1),2599(a,1,1,1,1))].">
      <j1 rule="para" parents="2591 2599"/>
    </justification>
  </clause>

  <clause id="2619">
    <literal><![CDATA[
      c(x v (c(y) v z)) v y = y
    ]]></literal>
    <justification jstring="[para(2591(a,1),2599(a,1,1,1))].">
      <j1 rule="para" parents="2591 2599"/>
    </justification>
  </clause>

  <clause id="2650">
    <literal><![CDATA[
      x v c(y v (c(x) v z)) = x
    ]]></literal>
    <justification jstring="[para(2591(a,1),2614(a,1,2,1))].">
      <j1 rule="para" parents="2591 2614"/>
    </justification>
  </clause>

  <clause id="2656">
    <literal><![CDATA[
      c(x v (y v c(z))) v z = z
    ]]></literal>
    <justification jstring="[para(2591(a,1),2618(a,1,1,1))].">
      <j1 rule="para" parents="2591 2618"/>
    </justification>
  </clause>

  <clause id="2660">
    <literal><![CDATA[
      c(x v (y v z)) v c(y) = c(y)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2619(a,1,1,1,2,1))].">
      <j1 rule="para" parents="2392 2619"/>
    </justification>
  </clause>

  <clause id="2677">
    <literal><![CDATA[
      c(x v y) v c(c(c(z) v c(x v y)) v x) = c(c(c(z) v c(x v y)) v x)
    ]]></literal>
    <justification jstring="[para(2481(a,1),2492(a,1,2,1,2,1)),rewrite([2392(9),2481(19),2392(17)])].">
      <j1 rule="para" parents="2481 2492"/>
      <j2 rule="rewrite" parents="2392 2481 2392"/>
    </justification>
  </clause>

  <clause id="2697">
    <literal><![CDATA[
      c(x) v c(y v (x v z)) = c(x)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2650(a,1,2,1,2,1))].">
      <j1 rule="para" parents="2392 2650"/>
    </justification>
  </clause>

  <clause id="2706">
    <literal><![CDATA[
      c(x v (y v z)) v c(z) = c(z)
    ]]></literal>
    <justification jstring="[para(2392(a,1),2656(a,1,1,1,2,2))].">
      <j1 rule="para" parents="2392 2656"/>
    </justification>
  </clause>

  <clause id="2737">
    <literal><![CDATA[
      x v (y v (x v z)) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(2660(a,1),2481(a,1,1,1)),rewrite([2392(2),2392(4),2392(7)])].">
      <j1 rule="para" parents="2660 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="2764">
    <literal><![CDATA[
      x v (y v (z v x)) = y v (z v x)
    ]]></literal>
    <justification jstring="[para(2706(a,1),2481(a,1,1,1)),rewrite([2392(2),2392(4),2392(7)])].">
      <j1 rule="para" parents="2706 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="5628">
    <literal><![CDATA[
      c(x v y) v c(y v x) = c(y v x)
    ]]></literal>
    <justification jstring="[para(2585(a,1),2677(a,1,2,1,1,1)),rewrite([2392(2),2392(4),2392(8),2590(9),2392(7)])].">
      <j1 rule="para" parents="2585 2677"/>
      <j2 rule="rewrite" parents="2392 2392 2392 2590 2392"/>
    </justification>
  </clause>

  <clause id="5636">
    <literal><![CDATA[
      c(x v (y v z)) v c(y v x) = c(y v x)
    ]]></literal>
    <justification jstring="[para(2650(a,1),2677(a,1,2,1,1,1)),rewrite([2392(2),2392(5),2392(9),2697(11),2392(8)])].">
      <j1 rule="para" parents="2650 2677"/>
      <j2 rule="rewrite" parents="2392 2392 2392 2697 2392"/>
    </justification>
  </clause>

  <clause id="5749">
    <literal><![CDATA[
      (x v y) v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(5628(a,1),2481(a,1,1,1)),rewrite([2392(3),2392(4),2392(6)])].">
      <j1 rule="para" parents="5628 2481"/>
      <j2 rule="rewrite" parents="2392 2392 2392"/>
    </justification>
  </clause>

  <clause id="5751">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[para(5628(a,1),2533(a,1,2,1)),rewrite([2392(4),5749(3)])].">
      <j1 rule="para" parents="5628 2533"/>
      <j2 rule="rewrite" parents="2392 5749"/>
    </justification>
  </clause>

  <clause id="5754">
    <literal><![CDATA[
      (x v y) v c(c(y v x) v z) = x v y
    ]]></literal>
    <justification jstring="[para(5628(a,1),2599(a,1,1,1,1)),rewrite([5751(6)])].">
      <j1 rule="para" parents="5628 2599"/>
      <j2 rule="rewrite" parents="5751"/>
    </justification>
  </clause>

  <clause id="6160">
    <literal><![CDATA[
      c25 v (c23 v c24) != c23 v (c24 v c25)
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc_join)
    ]]></attribute>
    <justification jstring="[back_rewrite(38),rewrite([5751(5)])].">
      <j1 rule="back_rewrite" parents="38"/>
      <j2 rule="rewrite" parents="5751"/>
    </justification>
  </clause>

  <clause id="12548">
    <literal><![CDATA[
      (x v (y v z)) v (y v x) = x v (y v z)
    ]]></literal>
    <justification jstring="[para(5636(a,1),2533(a,1,2,1)),rewrite([2392(5)])].">
      <j1 rule="para" parents="5636 2533"/>
      <j2 rule="rewrite" parents="2392"/>
    </justification>
  </clause>

  <clause id="12684">
    <literal><![CDATA[
      (x v y) v ((x v z) v y) = (x v z) v y
    ]]></literal>
    <justification jstring="[para(5636(a,1),5754(a,1,2,1)),rewrite([2392(5),5751(4)])].">
      <j1 rule="para" parents="5636 5754"/>
      <j2 rule="rewrite" parents="2392 5751"/>
    </justification>
  </clause>

  <clause id="12718">
    <literal><![CDATA[
      c(c(x v y) v z) v (y v (x v u)) = y v (x v u)
    ]]></literal>
    <justification jstring="[para(12548(a,1),2617(a,1,2)),rewrite([12548(11)])].">
      <j1 rule="para" parents="12548 2617"/>
      <j2 rule="rewrite" parents="12548"/>
    </justification>
  </clause>

  <clause id="12728">
    <literal><![CDATA[
      (x v y) v (x v (y v z)) = x v (y v z)
    ]]></literal>
    <justification jstring="[para(2737(a,1),12548(a,1,1)),rewrite([5751(4),2737(7)])].">
      <j1 rule="para" parents="2737 12548"/>
      <j2 rule="rewrite" parents="5751 2737"/>
    </justification>
  </clause>

  <clause id="12730">
    <literal><![CDATA[
      (x v y) v (x v (z v y)) = x v (z v y)
    ]]></literal>
    <justification jstring="[para(2764(a,1),12548(a,1,1)),rewrite([5751(4),2764(7)])].">
      <j1 rule="para" parents="2764 12548"/>
      <j2 rule="rewrite" parents="5751 2764"/>
    </justification>
  </clause>

  <clause id="13101">
    <literal><![CDATA[
      (x v y) v (x v z) = (x v y) v z
    ]]></literal>
    <justification jstring="[para(12684(a,1),12548(a,1,1)),rewrite([12730(6),12684(7)])].">
      <j1 rule="para" parents="12684 12548"/>
      <j2 rule="rewrite" parents="12730 12684"/>
    </justification>
  </clause>

  <clause id="14172">
    <literal><![CDATA[
      (x v y) v (y v z) = x v (y v z)
    ]]></literal>
    <justification jstring="[back_rewrite(12728),rewrite([13101(4)])].">
      <j1 rule="back_rewrite" parents="12728"/>
      <j2 rule="rewrite" parents="13101"/>
    </justification>
  </clause>

  <clause id="14240">
    <literal><![CDATA[
      (x v y) v z = x v (y v z)
    ]]></literal>
    <justification jstring="[para(2591(a,1),13101(a,1,1)),rewrite([14172(3),2591(4)]),flip(a)].">
      <j1 rule="para" parents="2591 13101"/>
      <j2 rule="rewrite" parents="14172 2591"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14251">
    <literal><![CDATA[
      x v (y v z) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(13101(a,1),2612(a,2)),rewrite([14240(7),2737(7),12718(7),14240(4)])].">
      <j1 rule="para" parents="13101 2612"/>
      <j2 rule="rewrite" parents="14240 2737 12718 14240"/>
    </justification>
  </clause>

  <clause id="15086">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc_join)
    ]]></attribute>
    <justification jstring="[back_rewrite(6160),rewrite([14251(5),5751(4)]),xx(a)].">
      <j1 rule="back_rewrite" parents="6160"/>
      <j2 rule="rewrite" parents="14251 5751"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
