<?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 June-2006C, June 2006.
Process 13067 was started by mccune on cleo.thornwood,
Mon Jun 19 16:40:53 2006
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 (+ 0.01) seconds: cc.
% Length of proof is 17.
% Level of proof is 6.
% Maximum clause weight is 37.
% Given clauses 59.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="2">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[copy(1)].">
      <j1 rule="copy" parents="1"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[copy(2),flip(a)].">
      <j1 rule="copy" parents="2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      f(x,x) = c(x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[copy(4),flip(a)].">
      <j1 rule="copy" parents="4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      f(c(x),c(y)) = x v y
    ]]></literal>
    <justification jstring="[back_demod(20),demod(22(1),22(2))].">
      <j1 rule="back_demod" parents="20"/>
      <j2 rule="demod" parents="22 22"/>
    </justification>
  </clause>

  <clause id="25">
    <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_demod(19),demod(22(5))].">
      <j1 rule="back_demod" parents="19"/>
      <j2 rule="demod" parents="22"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      c(c(c14)) != c14
    ]]></literal>
    <attribute><![CDATA[
      answer(cc)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      c(c(x)) = x v x
    ]]></literal>
    <justification jstring="[para(24(a,1),22(a,1)),flip(a)].">
      <j1 rule="para" parents="24 22"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="48">
    <literal><![CDATA[
      c14 v c14 != c14
    ]]></literal>
    <attribute><![CDATA[
      answer(cc)
    ]]></attribute>
    <justification jstring="[back_demod(33),demod(42(3))].">
      <j1 rule="back_demod" parents="33"/>
      <j2 rule="demod" parents="42"/>
    </justification>
  </clause>

  <clause id="49">
    <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(22(a,1),25(a,1,1,1,1))].">
      <j1 rule="para" parents="22 25"/>
    </justification>
  </clause>

  <clause id="59">
    <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(25(a,1),25(a,1,1,1))].">
      <j1 rule="para" parents="25 25"/>
    </justification>
  </clause>

  <clause id="1641">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(59(a,1),25(a,1,1))].">
      <j1 rule="para" parents="59 25"/>
    </justification>
  </clause>

  <clause id="1643">
    <literal><![CDATA[
      f(c(x),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(59(a,1),49(a,1,1))].">
      <j1 rule="para" parents="59 49"/>
    </justification>
  </clause>

  <clause id="1757">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(1643(a,1),1643(a,1,2,2,1,2)),demod(1641(13),22(2),22(3),42(2))].">
      <j1 rule="para" parents="1643 1643"/>
      <j2 rule="demod" parents="1641 22 22 42"/>
    </justification>
  </clause>

  <clause id="1758">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(cc)
    ]]></attribute>
    <justification jstring="[resolve(1757,a,48,a)].">
      <j1 rule="resolve" parents="1757 48"/>
    </justification>
  </clause>

</proof>

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

<comments><![CDATA[
% Proof 2 at 0.20 (+ 0.01) seconds: idempotence_meet.
% Length of proof is 22.
% Level of proof is 6.
% Maximum clause weight is 37.
% Given clauses 59.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="2">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="3">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_meet)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[copy(1)].">
      <j1 rule="copy" parents="1"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[copy(2),flip(a)].">
      <j1 rule="copy" parents="2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(f(x,y),f(x,y)) = x ^ y
    ]]></literal>
    <attribute><![CDATA[
      label(Def_meet)
    ]]></attribute>
    <justification jstring="[copy(3),flip(a)].">
      <j1 rule="copy" parents="3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      f(x,x) = c(x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[copy(4),flip(a)].">
      <j1 rule="copy" parents="4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      c(f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[back_demod(21),demod(22(3))].">
      <j1 rule="back_demod" parents="21"/>
      <j2 rule="demod" parents="22"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      f(c(x),c(y)) = x v y
    ]]></literal>
    <justification jstring="[back_demod(20),demod(22(1),22(2))].">
      <j1 rule="back_demod" parents="20"/>
      <j2 rule="demod" parents="22 22"/>
    </justification>
  </clause>

  <clause id="25">
    <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_demod(19),demod(22(5))].">
      <j1 rule="back_demod" parents="19"/>
      <j2 rule="demod" parents="22"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      c13 ^ c13 != c13
    ]]></literal>
    <attribute><![CDATA[
      answer(idempotence_meet)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      x ^ x = c(c(x))
    ]]></literal>
    <justification jstring="[para(22(a,1),23(a,1,1)),flip(a)].">
      <j1 rule="para" parents="22 23"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      c(c(c13)) != c13
    ]]></literal>
    <attribute><![CDATA[
      answer(idempotence_meet)
    ]]></attribute>
    <justification jstring="[back_demod(32),demod(40(3))].">
      <j1 rule="back_demod" parents="32"/>
      <j2 rule="demod" parents="40"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      c(c(x)) = x v x
    ]]></literal>
    <justification jstring="[para(24(a,1),22(a,1)),flip(a)].">
      <j1 rule="para" parents="24 22"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      c13 v c13 != c13
    ]]></literal>
    <attribute><![CDATA[
      answer(idempotence_meet)
    ]]></attribute>
    <justification jstring="[back_demod(41),demod(42(3))].">
      <j1 rule="back_demod" parents="41"/>
      <j2 rule="demod" parents="42"/>
    </justification>
  </clause>

  <clause id="49">
    <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(22(a,1),25(a,1,1,1,1))].">
      <j1 rule="para" parents="22 25"/>
    </justification>
  </clause>

  <clause id="59">
    <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(25(a,1),25(a,1,1,1))].">
      <j1 rule="para" parents="25 25"/>
    </justification>
  </clause>

  <clause id="1641">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(59(a,1),25(a,1,1))].">
      <j1 rule="para" parents="59 25"/>
    </justification>
  </clause>

  <clause id="1643">
    <literal><![CDATA[
      f(c(x),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(59(a,1),49(a,1,1))].">
      <j1 rule="para" parents="59 49"/>
    </justification>
  </clause>

  <clause id="1757">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(1643(a,1),1643(a,1,2,2,1,2)),demod(1641(13),22(2),22(3),42(2))].">
      <j1 rule="para" parents="1643 1643"/>
      <j2 rule="demod" parents="1641 22 22 42"/>
    </justification>
  </clause>

  <clause id="1759">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(idempotence_meet)
    ]]></attribute>
    <justification jstring="[resolve(1757,a,46,a)].">
      <j1 rule="resolve" parents="1757 46"/>
    </justification>
  </clause>

</proof>

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

<comments><![CDATA[
% Proof 3 at 0.20 (+ 0.01) seconds: idempotence_join.
% Length of proof is 16.
% Level of proof is 6.
% Maximum clause weight is 37.
% Given clauses 59.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="2">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[copy(1)].">
      <j1 rule="copy" parents="1"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[copy(2),flip(a)].">
      <j1 rule="copy" parents="2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      f(x,x) = c(x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[copy(4),flip(a)].">
      <j1 rule="copy" parents="4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      f(c(x),c(y)) = x v y
    ]]></literal>
    <justification jstring="[back_demod(20),demod(22(1),22(2))].">
      <j1 rule="back_demod" parents="20"/>
      <j2 rule="demod" parents="22 22"/>
    </justification>
  </clause>

  <clause id="25">
    <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_demod(19),demod(22(5))].">
      <j1 rule="back_demod" parents="19"/>
      <j2 rule="demod" parents="22"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      c12 v c12 != c12
    ]]></literal>
    <attribute><![CDATA[
      answer(idempotence_join)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      c(c(x)) = x v x
    ]]></literal>
    <justification jstring="[para(24(a,1),22(a,1)),flip(a)].">
      <j1 rule="para" parents="24 22"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="49">
    <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(22(a,1),25(a,1,1,1,1))].">
      <j1 rule="para" parents="22 25"/>
    </justification>
  </clause>

  <clause id="59">
    <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(25(a,1),25(a,1,1,1))].">
      <j1 rule="para" parents="25 25"/>
    </justification>
  </clause>

  <clause id="1641">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(59(a,1),25(a,1,1))].">
      <j1 rule="para" parents="59 25"/>
    </justification>
  </clause>

  <clause id="1643">
    <literal><![CDATA[
      f(c(x),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(59(a,1),49(a,1,1))].">
      <j1 rule="para" parents="59 49"/>
    </justification>
  </clause>

  <clause id="1757">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(1643(a,1),1643(a,1,2,2,1,2)),demod(1641(13),22(2),22(3),42(2))].">
      <j1 rule="para" parents="1643 1643"/>
      <j2 rule="demod" parents="1641 22 22 42"/>
    </justification>
  </clause>

  <clause id="1760">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(idempotence_join)
    ]]></attribute>
    <justification jstring="[resolve(1757,a,31,a)].">
      <j1 rule="resolve" parents="1757 31"/>
    </justification>
  </clause>

</proof>

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

<comments><![CDATA[
% Proof 4 at 0.28 (+ 0.01) seconds: DM.
% Length of proof is 24.
% Level of proof is 7.
% Maximum clause weight is 37.
% Given clauses 59.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="2">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="3">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_meet)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      c(c(c8) v c(c9)) != c8 ^ c9
    ]]></literal>
    <attribute><![CDATA[
      answer(DM)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[copy(1)].">
      <j1 rule="copy" parents="1"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[copy(2),flip(a)].">
      <j1 rule="copy" parents="2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(f(x,y),f(x,y)) = x ^ y
    ]]></literal>
    <attribute><![CDATA[
      label(Def_meet)
    ]]></attribute>
    <justification jstring="[copy(3),flip(a)].">
      <j1 rule="copy" parents="3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      f(x,x) = c(x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[copy(4),flip(a)].">
      <j1 rule="copy" parents="4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      c(f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[back_demod(21),demod(22(3))].">
      <j1 rule="back_demod" parents="21"/>
      <j2 rule="demod" parents="22"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      f(c(x),c(y)) = x v y
    ]]></literal>
    <justification jstring="[back_demod(20),demod(22(1),22(2))].">
      <j1 rule="back_demod" parents="20"/>
      <j2 rule="demod" parents="22 22"/>
    </justification>
  </clause>

  <clause id="25">
    <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_demod(19),demod(22(5))].">
      <j1 rule="back_demod" parents="19"/>
      <j2 rule="demod" parents="22"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      c8 ^ c9 != c(c(c8) v c(c9))
    ]]></literal>
    <attribute><![CDATA[
      answer(DM)
    ]]></attribute>
    <justification jstring="[copy(8),flip(a)].">
      <j1 rule="copy" parents="8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      c(c(x)) = x v x
    ]]></literal>
    <justification jstring="[para(24(a,1),22(a,1)),flip(a)].">
      <j1 rule="para" parents="24 22"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      c(x) ^ c(y) = c(x v y)
    ]]></literal>
    <justification jstring="[para(24(a,1),23(a,1,1)),flip(a)].">
      <j1 rule="para" parents="24 23"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="49">
    <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(22(a,1),25(a,1,1,1,1))].">
      <j1 rule="para" parents="22 25"/>
    </justification>
  </clause>

  <clause id="59">
    <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(25(a,1),25(a,1,1,1))].">
      <j1 rule="para" parents="25 25"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      (x v x) ^ c(y) = c(c(x) v y)
    ]]></literal>
    <justification jstring="[para(42(a,1),43(a,1,1))].">
      <j1 rule="para" parents="42 43"/>
    </justification>
  </clause>

  <clause id="92">
    <literal><![CDATA[
      (x v x) ^ (y v y) = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[para(42(a,1),69(a,1,2))].">
      <j1 rule="para" parents="42 69"/>
    </justification>
  </clause>

  <clause id="1641">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(59(a,1),25(a,1,1))].">
      <j1 rule="para" parents="59 25"/>
    </justification>
  </clause>

  <clause id="1643">
    <literal><![CDATA[
      f(c(x),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(59(a,1),49(a,1,1))].">
      <j1 rule="para" parents="59 49"/>
    </justification>
  </clause>

  <clause id="1757">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(1643(a,1),1643(a,1,2,2,1,2)),demod(1641(13),22(2),22(3),42(2))].">
      <j1 rule="para" parents="1643 1643"/>
      <j2 rule="demod" parents="1641 22 22 42"/>
    </justification>
  </clause>

  <clause id="2409">
    <literal><![CDATA[
      x ^ y = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[back_demod(92),demod(1757(1),1757(1))].">
      <j1 rule="back_demod" parents="92"/>
      <j2 rule="demod" parents="1757 1757"/>
    </justification>
  </clause>

  <clause id="2410">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(DM)
    ]]></attribute>
    <justification jstring="[resolve(2409,a,29,a)].">
      <j1 rule="resolve" parents="2409 29"/>
    </justification>
  </clause>

</proof>

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

<comments><![CDATA[
% Proof 5 at 0.28 (+ 0.01) seconds: DEF_SS.
% Length of proof is 19.
% Level of proof is 7.
% Maximum clause weight is 37.
% Given clauses 59.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="2">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[copy(1)].">
      <j1 rule="copy" parents="1"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[copy(2),flip(a)].">
      <j1 rule="copy" parents="2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      f(x,x) = c(x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[copy(4),flip(a)].">
      <j1 rule="copy" parents="4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      f(c(x),c(y)) = x v y
    ]]></literal>
    <justification jstring="[back_demod(20),demod(22(1),22(2))].">
      <j1 rule="back_demod" parents="20"/>
      <j2 rule="demod" parents="22 22"/>
    </justification>
  </clause>

  <clause id="25">
    <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_demod(19),demod(22(5))].">
      <j1 rule="back_demod" parents="19"/>
      <j2 rule="demod" parents="22"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      f(c10,c11) != c(c10) v c(c11)
    ]]></literal>
    <attribute><![CDATA[
      answer(DEF_SS)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      c(c(x)) = x v x
    ]]></literal>
    <justification jstring="[para(24(a,1),22(a,1)),flip(a)].">
      <j1 rule="para" parents="24 22"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="49">
    <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(22(a,1),25(a,1,1,1,1))].">
      <j1 rule="para" parents="22 25"/>
    </justification>
  </clause>

  <clause id="59">
    <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(25(a,1),25(a,1,1,1))].">
      <j1 rule="para" parents="25 25"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(x v x,c(y)) = c(x) v y
    ]]></literal>
    <justification jstring="[para(42(a,1),24(a,1,1))].">
      <j1 rule="para" parents="42 24"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      f(x v x,y v y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[para(42(a,1),62(a,1,2))].">
      <j1 rule="para" parents="42 62"/>
    </justification>
  </clause>

  <clause id="1641">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(59(a,1),25(a,1,1))].">
      <j1 rule="para" parents="59 25"/>
    </justification>
  </clause>

  <clause id="1643">
    <literal><![CDATA[
      f(c(x),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(59(a,1),49(a,1,1))].">
      <j1 rule="para" parents="59 49"/>
    </justification>
  </clause>

  <clause id="1757">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(1643(a,1),1643(a,1,2,2,1,2)),demod(1641(13),22(2),22(3),42(2))].">
      <j1 rule="para" parents="1643 1643"/>
      <j2 rule="demod" parents="1641 22 22 42"/>
    </justification>
  </clause>

  <clause id="2419">
    <literal><![CDATA[
      f(x,y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[back_demod(74),demod(1757(1),1757(1))].">
      <j1 rule="back_demod" parents="74"/>
      <j2 rule="demod" parents="1757 1757"/>
    </justification>
  </clause>

  <clause id="2420">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(DEF_SS)
    ]]></attribute>
    <justification jstring="[resolve(2419,a,30,a)].">
      <j1 rule="resolve" parents="2419 30"/>
    </justification>
  </clause>

</proof>

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

<comments><![CDATA[
% Proof 6 at 0.44 (+ 0.01) seconds: B1.
% Length of proof is 57.
% Level of proof is 15.
% Maximum clause weight is 37.
% Given clauses 78.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="2">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="3">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_meet)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[copy(1)].">
      <j1 rule="copy" parents="1"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[copy(2),flip(a)].">
      <j1 rule="copy" parents="2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(f(x,y),f(x,y)) = x ^ y
    ]]></literal>
    <attribute><![CDATA[
      label(Def_meet)
    ]]></attribute>
    <justification jstring="[copy(3),flip(a)].">
      <j1 rule="copy" parents="3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      f(x,x) = c(x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[copy(4),flip(a)].">
      <j1 rule="copy" parents="4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      c(f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[back_demod(21),demod(22(3))].">
      <j1 rule="back_demod" parents="21"/>
      <j2 rule="demod" parents="22"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      f(c(x),c(y)) = x v y
    ]]></literal>
    <justification jstring="[back_demod(20),demod(22(1),22(2))].">
      <j1 rule="back_demod" parents="20"/>
      <j2 rule="demod" parents="22 22"/>
    </justification>
  </clause>

  <clause id="25">
    <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_demod(19),demod(22(5))].">
      <j1 rule="back_demod" parents="19"/>
      <j2 rule="demod" parents="22"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      c1 v (c1 ^ c2) != c1
    ]]></literal>
    <attribute><![CDATA[
      answer(B1)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      x ^ x = c(c(x))
    ]]></literal>
    <justification jstring="[para(22(a,1),23(a,1,1)),flip(a)].">
      <j1 rule="para" parents="22 23"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      c(c(x)) = x v x
    ]]></literal>
    <justification jstring="[para(24(a,1),22(a,1)),flip(a)].">
      <j1 rule="para" parents="24 22"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      c(x) ^ c(y) = c(x v y)
    ]]></literal>
    <justification jstring="[para(24(a,1),23(a,1,1)),flip(a)].">
      <j1 rule="para" parents="24 23"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      x ^ x = x v x
    ]]></literal>
    <justification jstring="[back_demod(40),demod(42(3))].">
      <j1 rule="back_demod" parents="40"/>
      <j2 rule="demod" parents="42"/>
    </justification>
  </clause>

  <clause id="49">
    <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(22(a,1),25(a,1,1,1,1))].">
      <j1 rule="para" parents="22 25"/>
    </justification>
  </clause>

  <clause id="50">
    <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(22(a,1),25(a,1,1,1,2))].">
      <j1 rule="para" parents="22 25"/>
    </justification>
  </clause>

  <clause id="51">
    <literal><![CDATA[
      f(f(x v x,y),f(x,f(f(x,f(c(x),x)),x))) = x
    ]]></literal>
    <justification jstring="[para(22(a,1),25(a,1,1,1)),demod(22(1),42(2))].">
      <j1 rule="para" parents="22 25"/>
      <j2 rule="demod" parents="22 42"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      f(f(x,y) ^ f(y,z),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(22(a,1),25(a,1,1)),demod(23(4))].">
      <j1 rule="para" parents="22 25"/>
      <j2 rule="demod" parents="23"/>
    </justification>
  </clause>

  <clause id="56">
    <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(24(a,1),25(a,1,1,1,1)),demod(42(8))].">
      <j1 rule="para" parents="24 25"/>
      <j2 rule="demod" parents="42"/>
    </justification>
  </clause>

  <clause id="59">
    <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(25(a,1),25(a,1,1,1))].">
      <j1 rule="para" parents="25 25"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(x v x,c(y)) = c(x) v y
    ]]></literal>
    <justification jstring="[para(42(a,1),24(a,1,1))].">
      <j1 rule="para" parents="42 24"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      (x v x) ^ c(y) = c(c(x) v y)
    ]]></literal>
    <justification jstring="[para(42(a,1),43(a,1,1))].">
      <j1 rule="para" parents="42 43"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      f(x v x,y v y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[para(42(a,1),62(a,1,2))].">
      <j1 rule="para" parents="42 62"/>
    </justification>
  </clause>

  <clause id="92">
    <literal><![CDATA[
      (x v x) ^ (y v y) = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[para(42(a,1),69(a,1,2))].">
      <j1 rule="para" parents="42 69"/>
    </justification>
  </clause>

  <clause id="167">
    <literal><![CDATA[
      f(x,f(c(x),f(f(y,f(x v x,y)),y))) = c(x)
    ]]></literal>
    <justification jstring="[para(50(a,1),25(a,1,1)),demod(42(3))].">
      <j1 rule="para" parents="50 25"/>
      <j2 rule="demod" parents="42"/>
    </justification>
  </clause>

  <clause id="225">
    <literal><![CDATA[
      f(c(x) v y,f(x,f(f(x,f(c(x),x)),x))) = x
    ]]></literal>
    <justification jstring="[para(62(a,1),51(a,1,1))].">
      <j1 rule="para" parents="62 51"/>
    </justification>
  </clause>

  <clause id="773">
    <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(24(a,1),225(a,1,2,2,1,2)),demod(42(2))].">
      <j1 rule="para" parents="24 225"/>
      <j2 rule="demod" parents="42"/>
    </justification>
  </clause>

  <clause id="782">
    <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(225(a,1),167(a,1,2,2,1,2)),demod(42(3),42(18))].">
      <j1 rule="para" parents="225 167"/>
      <j2 rule="demod" parents="42 42"/>
    </justification>
  </clause>

  <clause id="1641">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(59(a,1),25(a,1,1))].">
      <j1 rule="para" parents="59 25"/>
    </justification>
  </clause>

  <clause id="1643">
    <literal><![CDATA[
      f(c(x),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(59(a,1),49(a,1,1))].">
      <j1 rule="para" parents="59 49"/>
    </justification>
  </clause>

  <clause id="1646">
    <literal><![CDATA[
      f(x v y,f(c(y),f(f(z,f(y v y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[para(59(a,1),56(a,1,1))].">
      <j1 rule="para" parents="59 56"/>
    </justification>
  </clause>

  <clause id="1649">
    <literal><![CDATA[
      f(c(x),f(x v x,x)) = x v x
    ]]></literal>
    <justification jstring="[back_demod(782),demod(1641(14))].">
      <j1 rule="back_demod" parents="782"/>
      <j2 rule="demod" parents="1641"/>
    </justification>
  </clause>

  <clause id="1665">
    <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(1649(a,1),52(a,1,1,1)),demod(23(10))].">
      <j1 rule="para" parents="1649 52"/>
      <j2 rule="demod" parents="23"/>
    </justification>
  </clause>

  <clause id="1711">
    <literal><![CDATA[
      f(c(x),f(x,f(f(c(y),x v y),c(y)))) = x
    ]]></literal>
    <justification jstring="[para(24(a,1),1643(a,1,2,2,1,2))].">
      <j1 rule="para" parents="24 1643"/>
    </justification>
  </clause>

  <clause id="1757">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(1643(a,1),1643(a,1,2,2,1,2)),demod(1641(13),22(2),22(3),42(2))].">
      <j1 rule="para" parents="1643 1643"/>
      <j2 rule="demod" parents="1641 22 22 42"/>
    </justification>
  </clause>

  <clause id="1805">
    <literal><![CDATA[
      f(x ^ f(c(x),y),f(c(x),f(f(y,f(x,y)),y))) = c(x)
    ]]></literal>
    <justification jstring="[back_demod(1665),demod(1757(1),1757(1),22(1),1757(4),22(4),1757(5),47(5),1757(5),1757(10),22(10))].">
      <j1 rule="back_demod" parents="1665"/>
      <j2 rule="demod" parents="1757 1757 22 1757 22 1757 47 1757 1757 22"/>
    </justification>
  </clause>

  <clause id="1816">
    <literal><![CDATA[
      f(x v y,f(c(y),f(f(z,f(y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[back_demod(1646),demod(1757(3))].">
      <j1 rule="back_demod" parents="1646"/>
      <j2 rule="demod" parents="1757"/>
    </justification>
  </clause>

  <clause id="2166">
    <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_demod(773),demod(1757(1))].">
      <j1 rule="back_demod" parents="773"/>
      <j2 rule="demod" parents="1757"/>
    </justification>
  </clause>

  <clause id="2409">
    <literal><![CDATA[
      x ^ y = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[back_demod(92),demod(1757(1),1757(1))].">
      <j1 rule="back_demod" parents="92"/>
      <j2 rule="demod" parents="1757 1757"/>
    </justification>
  </clause>

  <clause id="2419">
    <literal><![CDATA[
      f(x,y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[back_demod(74),demod(1757(1),1757(1))].">
      <j1 rule="back_demod" parents="74"/>
      <j2 rule="demod" parents="1757 1757"/>
    </justification>
  </clause>

  <clause id="2425">
    <literal><![CDATA[
      c(c(x)) = x
    ]]></literal>
    <justification jstring="[back_demod(42),demod(1757(3))].">
      <j1 rule="back_demod" parents="42"/>
      <j2 rule="demod" parents="1757"/>
    </justification>
  </clause>

  <clause id="2447">
    <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_demod(1805),demod(2419(2),2425(2),2409(3),2419(8),2419(11),2419(14),2419(17),2425(8),2419(18),2425(7))].">
      <j1 rule="back_demod" parents="1805"/>
      <j2 rule="demod" parents="2419 2425 2409 2419 2419 2419 2419 2425 2419 2425"/>
    </justification>
  </clause>

  <clause id="2477">
    <literal><![CDATA[
      c1 v c(c(c1) v c(c2)) != c1
    ]]></literal>
    <attribute><![CDATA[
      answer(B1)
    ]]></attribute>
    <justification jstring="[back_demod(26),demod(2409(4))].">
      <j1 rule="back_demod" parents="26"/>
      <j2 rule="demod" parents="2409"/>
    </justification>
  </clause>

  <clause id="2486">
    <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_demod(2166),demod(2419(6),2425(4),2419(8),2425(9),2419(9),2425(3),2419(10))].">
      <j1 rule="back_demod" parents="2166"/>
      <j2 rule="demod" parents="2419 2425 2419 2425 2419 2425 2419"/>
    </justification>
  </clause>

  <clause id="2490">
    <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_demod(1816),demod(2419(3),2419(6),2419(9),2419(12),2425(3),2419(13))].">
      <j1 rule="back_demod" parents="1816"/>
      <j2 rule="demod" parents="2419 2419 2419 2419 2425 2419"/>
    </justification>
  </clause>

  <clause id="2501">
    <literal><![CDATA[
      x v c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[back_demod(1711),demod(2419(4),2425(3),2419(6),2425(7),2419(7),2419(10),2425(2))].">
      <j1 rule="back_demod" parents="1711"/>
      <j2 rule="demod" parents="2419 2425 2419 2425 2419 2419 2425"/>
    </justification>
  </clause>

  <clause id="2514">
    <literal><![CDATA[
      c(x v c(c(x v c(c(x) v x)) v x)) = c(x)
    ]]></literal>
    <justification jstring="[para(2486(a,1),1757(a,1)),flip(a)].">
      <j1 rule="para" parents="2486 1757"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2516">
    <literal><![CDATA[
      c(x v y) v c(x) = c(x)
    ]]></literal>
    <justification jstring="[para(2486(a,1),2419(a,2)),demod(2419(10),2514(11))].">
      <j1 rule="para" parents="2486 2419"/>
      <j2 rule="demod" parents="2419 2514"/>
    </justification>
  </clause>

  <clause id="2521">
    <literal><![CDATA[
      c(c(x) v y) v x = x
    ]]></literal>
    <justification jstring="[para(2425(a,1),2516(a,1,2)),demod(2425(6))].">
      <j1 rule="para" parents="2425 2516"/>
      <j2 rule="demod" parents="2425"/>
    </justification>
  </clause>

  <clause id="2522">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(2516(a,1),2516(a,1,1,1)),demod(2425(2),2425(3),2425(5))].">
      <j1 rule="para" parents="2516 2516"/>
      <j2 rule="demod" parents="2425 2425 2425"/>
    </justification>
  </clause>

  <clause id="2546">
    <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(2425(a,1),2447(a,1,1,2,1,2)),demod(2425(6),2425(7),2425(11))].">
      <j1 rule="para" parents="2425 2447"/>
      <j2 rule="demod" parents="2425 2425 2425"/>
    </justification>
  </clause>

  <clause id="2548">
    <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(2447(a,1),2501(a,1,2,1,2,1,1,1,2,1)),demod(2425(25),2490(38),2425(13),2521(12))].">
      <j1 rule="para" parents="2447 2501"/>
      <j2 rule="demod" parents="2425 2490 2425 2521"/>
    </justification>
  </clause>

  <clause id="2552">
    <literal><![CDATA[
      c(x) v c(x v c(y)) = c(x)
    ]]></literal>
    <justification jstring="[para(2447(a,1),2522(a,1,2)),demod(2548(7),2546(23))].">
      <j1 rule="para" parents="2447 2522"/>
      <j2 rule="demod" parents="2548 2546"/>
    </justification>
  </clause>

  <clause id="2561">
    <literal><![CDATA[
      x v c(c(x) v c(y)) = x
    ]]></literal>
    <justification jstring="[para(2425(a,1),2552(a,1,1)),demod(2425(7))].">
      <j1 rule="para" parents="2425 2552"/>
      <j2 rule="demod" parents="2425"/>
    </justification>
  </clause>

  <clause id="2562">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(B1)
    ]]></attribute>
    <justification jstring="[resolve(2561,a,2477,a)].">
      <j1 rule="resolve" parents="2561 2477"/>
    </justification>
  </clause>

</proof>

<proof number="7" length="110" max_count="37">

<comments><![CDATA[
% Proof 7 at 0.80 (+ 0.02) seconds: 1.
% Length of proof is 110.
% Level of proof is 27.
% Maximum clause weight is 37.
% Given clauses 187.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="2">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="3">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_meet)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[copy(1)].">
      <j1 rule="copy" parents="1"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[copy(2),flip(a)].">
      <j1 rule="copy" parents="2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(f(x,y),f(x,y)) = x ^ y
    ]]></literal>
    <attribute><![CDATA[
      label(Def_meet)
    ]]></attribute>
    <justification jstring="[copy(3),flip(a)].">
      <j1 rule="copy" parents="3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      f(x,x) = c(x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[copy(4),flip(a)].">
      <j1 rule="copy" parents="4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      c(f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[back_demod(21),demod(22(3))].">
      <j1 rule="back_demod" parents="21"/>
      <j2 rule="demod" parents="22"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      f(c(x),c(y)) = x v y
    ]]></literal>
    <justification jstring="[back_demod(20),demod(22(1),22(2))].">
      <j1 rule="back_demod" parents="20"/>
      <j2 rule="demod" parents="22 22"/>
    </justification>
  </clause>

  <clause id="25">
    <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_demod(19),demod(22(5))].">
      <j1 rule="back_demod" parents="19"/>
      <j2 rule="demod" parents="22"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      c20 v c(c20) != c19 v c(c19)
    ]]></literal>
    <attribute><![CDATA[
      answer(1)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      x ^ x = c(c(x))
    ]]></literal>
    <justification jstring="[para(22(a,1),23(a,1,1)),flip(a)].">
      <j1 rule="para" parents="22 23"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      c(c(x)) = x v x
    ]]></literal>
    <justification jstring="[para(24(a,1),22(a,1)),flip(a)].">
      <j1 rule="para" parents="24 22"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      c(x) ^ c(y) = c(x v y)
    ]]></literal>
    <justification jstring="[para(24(a,1),23(a,1,1)),flip(a)].">
      <j1 rule="para" parents="24 23"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      x ^ x = x v x
    ]]></literal>
    <justification jstring="[back_demod(40),demod(42(3))].">
      <j1 rule="back_demod" parents="40"/>
      <j2 rule="demod" parents="42"/>
    </justification>
  </clause>

  <clause id="49">
    <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(22(a,1),25(a,1,1,1,1))].">
      <j1 rule="para" parents="22 25"/>
    </justification>
  </clause>

  <clause id="50">
    <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(22(a,1),25(a,1,1,1,2))].">
      <j1 rule="para" parents="22 25"/>
    </justification>
  </clause>

  <clause id="51">
    <literal><![CDATA[
      f(f(x v x,y),f(x,f(f(x,f(c(x),x)),x))) = x
    ]]></literal>
    <justification jstring="[para(22(a,1),25(a,1,1,1)),demod(22(1),42(2))].">
      <j1 rule="para" parents="22 25"/>
      <j2 rule="demod" parents="22 42"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      f(f(x,y) ^ f(y,z),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(22(a,1),25(a,1,1)),demod(23(4))].">
      <j1 rule="para" parents="22 25"/>
      <j2 rule="demod" parents="23"/>
    </justification>
  </clause>

  <clause id="56">
    <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(24(a,1),25(a,1,1,1,1)),demod(42(8))].">
      <j1 rule="para" parents="24 25"/>
      <j2 rule="demod" parents="42"/>
    </justification>
  </clause>

  <clause id="58">
    <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(24(a,1),25(a,1,2,2,1,2))].">
      <j1 rule="para" parents="24 25"/>
    </justification>
  </clause>

  <clause id="59">
    <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(25(a,1),25(a,1,1,1))].">
      <j1 rule="para" parents="25 25"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(x v x,c(y)) = c(x) v y
    ]]></literal>
    <justification jstring="[para(42(a,1),24(a,1,1))].">
      <j1 rule="para" parents="42 24"/>
    </justification>
  </clause>

  <clause id="63">
    <literal><![CDATA[
      f(c(x),y v y) = x v c(y)
    ]]></literal>
    <justification jstring="[para(42(a,1),24(a,1,2))].">
      <j1 rule="para" parents="42 24"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      c(x v x) = c(x) v c(x)
    ]]></literal>
    <justification jstring="[para(42(a,1),42(a,1,1))].">
      <j1 rule="para" parents="42 42"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      (x v x) ^ c(y) = c(c(x) v y)
    ]]></literal>
    <justification jstring="[para(42(a,1),43(a,1,1))].">
      <j1 rule="para" parents="42 43"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      f(x v x,y v y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[para(42(a,1),62(a,1,2))].">
      <j1 rule="para" parents="42 62"/>
    </justification>
  </clause>

  <clause id="92">
    <literal><![CDATA[
      (x v x) ^ (y v y) = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[para(42(a,1),69(a,1,2))].">
      <j1 rule="para" parents="42 69"/>
    </justification>
  </clause>

  <clause id="167">
    <literal><![CDATA[
      f(x,f(c(x),f(f(y,f(x v x,y)),y))) = c(x)
    ]]></literal>
    <justification jstring="[para(50(a,1),25(a,1,1)),demod(42(3))].">
      <j1 rule="para" parents="50 25"/>
      <j2 rule="demod" parents="42"/>
    </justification>
  </clause>

  <clause id="225">
    <literal><![CDATA[
      f(c(x) v y,f(x,f(f(x,f(c(x),x)),x))) = x
    ]]></literal>
    <justification jstring="[para(62(a,1),51(a,1,1))].">
      <j1 rule="para" parents="62 51"/>
    </justification>
  </clause>

  <clause id="318">
    <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(62(a,1),52(a,1,1,2)),demod(65(9),62(12),42(9))].">
      <j1 rule="para" parents="62 52"/>
      <j2 rule="demod" parents="65 62 42"/>
    </justification>
  </clause>

  <clause id="773">
    <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(24(a,1),225(a,1,2,2,1,2)),demod(42(2))].">
      <j1 rule="para" parents="24 225"/>
      <j2 rule="demod" parents="42"/>
    </justification>
  </clause>

  <clause id="782">
    <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(225(a,1),167(a,1,2,2,1,2)),demod(42(3),42(18))].">
      <j1 rule="para" parents="225 167"/>
      <j2 rule="demod" parents="42 42"/>
    </justification>
  </clause>

  <clause id="1350">
    <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(63(a,1),58(a,1,1,1,1)),demod(62(5))].">
      <j1 rule="para" parents="63 58"/>
      <j2 rule="demod" parents="62"/>
    </justification>
  </clause>

  <clause id="1641">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(59(a,1),25(a,1,1))].">
      <j1 rule="para" parents="59 25"/>
    </justification>
  </clause>

  <clause id="1643">
    <literal><![CDATA[
      f(c(x),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(59(a,1),49(a,1,1))].">
      <j1 rule="para" parents="59 49"/>
    </justification>
  </clause>

  <clause id="1646">
    <literal><![CDATA[
      f(x v y,f(c(y),f(f(z,f(y v y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[para(59(a,1),56(a,1,1))].">
      <j1 rule="para" parents="59 56"/>
    </justification>
  </clause>

  <clause id="1649">
    <literal><![CDATA[
      f(c(x),f(x v x,x)) = x v x
    ]]></literal>
    <justification jstring="[back_demod(782),demod(1641(14))].">
      <j1 rule="back_demod" parents="782"/>
      <j2 rule="demod" parents="1641"/>
    </justification>
  </clause>

  <clause id="1665">
    <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(1649(a,1),52(a,1,1,1)),demod(23(10))].">
      <j1 rule="para" parents="1649 52"/>
      <j2 rule="demod" parents="23"/>
    </justification>
  </clause>

  <clause id="1666">
    <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(1649(a,1),52(a,1,1,2)),demod(42(9))].">
      <j1 rule="para" parents="1649 52"/>
      <j2 rule="demod" parents="42"/>
    </justification>
  </clause>

  <clause id="1711">
    <literal><![CDATA[
      f(c(x),f(x,f(f(c(y),x v y),c(y)))) = x
    ]]></literal>
    <justification jstring="[para(24(a,1),1643(a,1,2,2,1,2))].">
      <j1 rule="para" parents="24 1643"/>
    </justification>
  </clause>

  <clause id="1757">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(1643(a,1),1643(a,1,2,2,1,2)),demod(1641(13),22(2),22(3),42(2))].">
      <j1 rule="para" parents="1643 1643"/>
      <j2 rule="demod" parents="1641 22 22 42"/>
    </justification>
  </clause>

  <clause id="1804">
    <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_demod(1666),demod(1757(3),1757(5),22(5),1757(6),1757(6),22(6),1757(9),22(9))].">
      <j1 rule="back_demod" parents="1666"/>
      <j2 rule="demod" parents="1757 1757 22 1757 1757 22 1757 22"/>
    </justification>
  </clause>

  <clause id="1805">
    <literal><![CDATA[
      f(x ^ f(c(x),y),f(c(x),f(f(y,f(x,y)),y))) = c(x)
    ]]></literal>
    <justification jstring="[back_demod(1665),demod(1757(1),1757(1),22(1),1757(4),22(4),1757(5),47(5),1757(5),1757(10),22(10))].">
      <j1 rule="back_demod" parents="1665"/>
      <j2 rule="demod" parents="1757 1757 22 1757 22 1757 47 1757 1757 22"/>
    </justification>
  </clause>

  <clause id="1816">
    <literal><![CDATA[
      f(x v y,f(c(y),f(f(z,f(y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[back_demod(1646),demod(1757(3))].">
      <j1 rule="back_demod" parents="1646"/>
      <j2 rule="demod" parents="1757"/>
    </justification>
  </clause>

  <clause id="1969">
    <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_demod(1350),demod(1757(7),1757(8),1757(14))].">
      <j1 rule="back_demod" parents="1350"/>
      <j2 rule="demod" parents="1757 1757 1757"/>
    </justification>
  </clause>

  <clause id="2166">
    <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_demod(773),demod(1757(1))].">
      <j1 rule="back_demod" parents="773"/>
      <j2 rule="demod" parents="1757"/>
    </justification>
  </clause>

  <clause id="2332">
    <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_demod(318),demod(1757(1),1757(5),1757(6),1757(12))].">
      <j1 rule="back_demod" parents="318"/>
      <j2 rule="demod" parents="1757 1757 1757 1757"/>
    </justification>
  </clause>

  <clause id="2409">
    <literal><![CDATA[
      x ^ y = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[back_demod(92),demod(1757(1),1757(1))].">
      <j1 rule="back_demod" parents="92"/>
      <j2 rule="demod" parents="1757 1757"/>
    </justification>
  </clause>

  <clause id="2419">
    <literal><![CDATA[
      f(x,y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[back_demod(74),demod(1757(1),1757(1))].">
      <j1 rule="back_demod" parents="74"/>
      <j2 rule="demod" parents="1757 1757"/>
    </justification>
  </clause>

  <clause id="2425">
    <literal><![CDATA[
      c(c(x)) = x
    ]]></literal>
    <justification jstring="[back_demod(42),demod(1757(3))].">
      <j1 rule="back_demod" parents="42"/>
      <j2 rule="demod" parents="1757"/>
    </justification>
  </clause>

  <clause id="2434">
    <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_demod(2332),demod(2419(1),2409(6),2419(12),2425(11),2419(14),2425(15),2419(15),2419(18),2425(10))].">
      <j1 rule="back_demod" parents="2332"/>
      <j2 rule="demod" parents="2419 2409 2419 2425 2419 2425 2419 2419 2425"/>
    </justification>
  </clause>

  <clause id="2447">
    <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_demod(1805),demod(2419(2),2425(2),2409(3),2419(8),2419(11),2419(14),2419(17),2425(8),2419(18),2425(7))].">
      <j1 rule="back_demod" parents="1805"/>
      <j2 rule="demod" parents="2419 2425 2409 2419 2419 2419 2419 2425 2419 2425"/>
    </justification>
  </clause>

  <clause id="2448">
    <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_demod(1804),demod(2419(2),2425(3),2409(3),2419(10),2425(11),2419(11),2425(9),2419(13),2425(14),2419(14),2425(8),2419(15),2425(7))].">
      <j1 rule="back_demod" parents="1804"/>
      <j2 rule="demod" parents="2419 2425 2409 2419 2425 2419 2425 2419 2425 2419 2425 2419 2425"/>
    </justification>
  </clause>

  <clause id="2486">
    <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_demod(2166),demod(2419(6),2425(4),2419(8),2425(9),2419(9),2425(3),2419(10))].">
      <j1 rule="back_demod" parents="2166"/>
      <j2 rule="demod" parents="2419 2425 2419 2425 2419 2425 2419"/>
    </justification>
  </clause>

  <clause id="2489">
    <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_demod(1969),demod(2419(5),2419(8),2419(13),2425(12),2419(15),2425(16),2419(16),2419(19))].">
      <j1 rule="back_demod" parents="1969"/>
      <j2 rule="demod" parents="2419 2419 2419 2425 2419 2425 2419 2419"/>
    </justification>
  </clause>

  <clause id="2490">
    <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_demod(1816),demod(2419(3),2419(6),2419(9),2419(12),2425(3),2419(13))].">
      <j1 rule="back_demod" parents="1816"/>
      <j2 rule="demod" parents="2419 2419 2419 2419 2425 2419"/>
    </justification>
  </clause>

  <clause id="2501">
    <literal><![CDATA[
      x v c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[back_demod(1711),demod(2419(4),2425(3),2419(6),2425(7),2419(7),2419(10),2425(2))].">
      <j1 rule="back_demod" parents="1711"/>
      <j2 rule="demod" parents="2419 2425 2419 2425 2419 2419 2425"/>
    </justification>
  </clause>

  <clause id="2514">
    <literal><![CDATA[
      c(x v c(c(x v c(c(x) v x)) v x)) = c(x)
    ]]></literal>
    <justification jstring="[para(2486(a,1),1757(a,1)),flip(a)].">
      <j1 rule="para" parents="2486 1757"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2516">
    <literal><![CDATA[
      c(x v y) v c(x) = c(x)
    ]]></literal>
    <justification jstring="[para(2486(a,1),2419(a,2)),demod(2419(10),2514(11))].">
      <j1 rule="para" parents="2486 2419"/>
      <j2 rule="demod" parents="2419 2514"/>
    </justification>
  </clause>

  <clause id="2520">
    <literal><![CDATA[
      (c(c(x) v y) v c(y)) v c(y) = c(y)
    ]]></literal>
    <justification jstring="[back_demod(2448),demod(2514(14))].">
      <j1 rule="back_demod" parents="2448"/>
      <j2 rule="demod" parents="2514"/>
    </justification>
  </clause>

  <clause id="2521">
    <literal><![CDATA[
      c(c(x) v y) v x = x
    ]]></literal>
    <justification jstring="[para(2425(a,1),2516(a,1,2)),demod(2425(6))].">
      <j1 rule="para" parents="2425 2516"/>
      <j2 rule="demod" parents="2425"/>
    </justification>
  </clause>

  <clause id="2522">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(2516(a,1),2516(a,1,1,1)),demod(2425(2),2425(3),2425(5))].">
      <j1 rule="para" parents="2516 2516"/>
      <j2 rule="demod" parents="2425 2425 2425"/>
    </justification>
  </clause>

  <clause id="2527">
    <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(2434(a,1),2516(a,1,1,1))].">
      <j1 rule="para" parents="2434 2516"/>
    </justification>
  </clause>

  <clause id="2535">
    <literal><![CDATA[
      (c(c(x) v c(y)) v y) v y = y
    ]]></literal>
    <justification jstring="[para(2425(a,1),2520(a,1,1,2)),demod(2425(7),2425(8))].">
      <j1 rule="para" parents="2425 2520"/>
      <j2 rule="demod" parents="2425 2425"/>
    </justification>
  </clause>

  <clause id="2542">
    <literal><![CDATA[
      (c(x v c(y)) v y) v y = y
    ]]></literal>
    <justification jstring="[para(2425(a,1),2535(a,1,1,1,1,1))].">
      <j1 rule="para" parents="2425 2535"/>
    </justification>
  </clause>

  <clause id="2546">
    <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(2425(a,1),2447(a,1,1,2,1,2)),demod(2425(6),2425(7),2425(11))].">
      <j1 rule="para" parents="2425 2447"/>
      <j2 rule="demod" parents="2425 2425 2425"/>
    </justification>
  </clause>

  <clause id="2548">
    <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(2447(a,1),2501(a,1,2,1,2,1,1,1,2,1)),demod(2425(25),2490(38),2425(13),2521(12))].">
      <j1 rule="para" parents="2447 2501"/>
      <j2 rule="demod" parents="2425 2490 2425 2521"/>
    </justification>
  </clause>

  <clause id="2552">
    <literal><![CDATA[
      c(x) v c(x v c(y)) = c(x)
    ]]></literal>
    <justification jstring="[para(2447(a,1),2522(a,1,2)),demod(2548(7),2546(23))].">
      <j1 rule="para" parents="2447 2522"/>
      <j2 rule="demod" parents="2548 2546"/>
    </justification>
  </clause>

  <clause id="2563">
    <literal><![CDATA[
      c(x) v c(x v y) = c(x)
    ]]></literal>
    <justification jstring="[para(2425(a,1),2552(a,1,2,1,2))].">
      <j1 rule="para" parents="2425 2552"/>
    </justification>
  </clause>

  <clause id="2564">
    <literal><![CDATA[
      (x v y) v x = x v y
    ]]></literal>
    <justification jstring="[para(2516(a,1),2552(a,1,2,1)),demod(2425(3),2425(3),2425(5))].">
      <j1 rule="para" parents="2516 2552"/>
      <j2 rule="demod" parents="2425 2425 2425"/>
    </justification>
  </clause>

  <clause id="2567">
    <literal><![CDATA[
      x v c(c(x) v y) = x
    ]]></literal>
    <justification jstring="[para(2521(a,1),2564(a,1,1)),demod(2521(8))].">
      <j1 rule="para" parents="2521 2564"/>
      <j2 rule="demod" parents="2521"/>
    </justification>
  </clause>

  <clause id="2586">
    <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(2516(a,1),2489(a,1,1,1,1,1)),demod(2425(4))].">
      <j1 rule="para" parents="2516 2489"/>
      <j2 rule="demod" parents="2425"/>
    </justification>
  </clause>

  <clause id="2587">
    <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(2516(a,1),2489(a,1,1,1)),demod(2425(4))].">
      <j1 rule="para" parents="2516 2489"/>
      <j2 rule="demod" parents="2425"/>
    </justification>
  </clause>

  <clause id="2594">
    <literal><![CDATA[
      c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[para(2489(a,1),2542(a,1,1)),demod(2567(10)),flip(a)].">
      <j1 rule="para" parents="2489 2542"/>
      <j2 rule="demod" parents="2567"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2608">
    <literal><![CDATA[
      c(x v c(y)) v y = y
    ]]></literal>
    <justification jstring="[back_demod(2587),demod(2594(12))].">
      <j1 rule="back_demod" parents="2587"/>
      <j2 rule="demod" parents="2594"/>
    </justification>
  </clause>

  <clause id="2609">
    <literal><![CDATA[
      c((c(x) v y) v c(z)) v x = x
    ]]></literal>
    <justification jstring="[back_demod(2586),demod(2594(14))].">
      <j1 rule="back_demod" parents="2586"/>
      <j2 rule="demod" parents="2594"/>
    </justification>
  </clause>

  <clause id="2617">
    <literal><![CDATA[
      c(x v y) v c(y) = c(y)
    ]]></literal>
    <justification jstring="[para(2425(a,1),2608(a,1,1,1,2))].">
      <j1 rule="para" parents="2425 2608"/>
    </justification>
  </clause>

  <clause id="2619">
    <literal><![CDATA[
      x v c(y v c(x)) = x
    ]]></literal>
    <justification jstring="[para(2608(a,1),2564(a,1,1)),demod(2608(8))].">
      <j1 rule="para" parents="2608 2564"/>
      <j2 rule="demod" parents="2608"/>
    </justification>
  </clause>

  <clause id="2625">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(2617(a,1),2516(a,1,1,1)),demod(2425(2),2425(3),2425(5))].">
      <j1 rule="para" parents="2617 2516"/>
      <j2 rule="demod" parents="2425 2425 2425"/>
    </justification>
  </clause>

  <clause id="2633">
    <literal><![CDATA[
      c((c(x) v y) v z) v x = x
    ]]></literal>
    <justification jstring="[para(2425(a,1),2609(a,1,1,1,2))].">
      <j1 rule="para" parents="2425 2609"/>
    </justification>
  </clause>

  <clause id="2646">
    <literal><![CDATA[
      c(c(x) v y) v (x v z) = x v z
    ]]></literal>
    <justification jstring="[para(2516(a,1),2633(a,1,1,1,1))].">
      <j1 rule="para" parents="2516 2633"/>
    </justification>
  </clause>

  <clause id="2648">
    <literal><![CDATA[
      x v c((c(x) v y) v z) = x
    ]]></literal>
    <justification jstring="[para(2633(a,1),2564(a,1,1)),demod(2633(10))].">
      <j1 rule="para" parents="2633 2564"/>
      <j2 rule="demod" parents="2633"/>
    </justification>
  </clause>

  <clause id="2652">
    <literal><![CDATA[
      c((x v c(y)) v z) v y = y
    ]]></literal>
    <justification jstring="[para(2625(a,1),2633(a,1,1,1,1))].">
      <j1 rule="para" parents="2625 2633"/>
    </justification>
  </clause>

  <clause id="2653">
    <literal><![CDATA[
      c(x v (c(y) v z)) v y = y
    ]]></literal>
    <justification jstring="[para(2625(a,1),2633(a,1,1,1))].">
      <j1 rule="para" parents="2625 2633"/>
    </justification>
  </clause>

  <clause id="2679">
    <literal><![CDATA[
      (x v y) v c(c(x) v z) = x v y
    ]]></literal>
    <justification jstring="[para(2516(a,1),2648(a,1,2,1,1))].">
      <j1 rule="para" parents="2516 2648"/>
    </justification>
  </clause>

  <clause id="2683">
    <literal><![CDATA[
      x v c((y v c(x)) v z) = x
    ]]></literal>
    <justification jstring="[para(2625(a,1),2648(a,1,2,1,1))].">
      <j1 rule="para" parents="2625 2648"/>
    </justification>
  </clause>

  <clause id="2684">
    <literal><![CDATA[
      x v c(y v (c(x) v z)) = x
    ]]></literal>
    <justification jstring="[para(2625(a,1),2648(a,1,2,1))].">
      <j1 rule="para" parents="2625 2648"/>
    </justification>
  </clause>

  <clause id="2690">
    <literal><![CDATA[
      c(x v (y v c(z))) v z = z
    ]]></literal>
    <justification jstring="[para(2625(a,1),2652(a,1,1,1))].">
      <j1 rule="para" parents="2625 2652"/>
    </justification>
  </clause>

  <clause id="2694">
    <literal><![CDATA[
      c(x v (y v z)) v c(y) = c(y)
    ]]></literal>
    <justification jstring="[para(2425(a,1),2653(a,1,1,1,2,1))].">
      <j1 rule="para" parents="2425 2653"/>
    </justification>
  </clause>

  <clause id="2708">
    <literal><![CDATA[
      x v c(y v (z v c(x))) = x
    ]]></literal>
    <justification jstring="[para(2625(a,1),2683(a,1,2,1))].">
      <j1 rule="para" parents="2625 2683"/>
    </justification>
  </clause>

  <clause id="2709">
    <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(2425(a,1),2527(a,1,1)),demod(2425(3),2425(5),2425(11),2425(13))].">
      <j1 rule="para" parents="2425 2527"/>
      <j2 rule="demod" parents="2425 2425 2425 2425"/>
    </justification>
  </clause>

  <clause id="2732">
    <literal><![CDATA[
      (x v y) v c(z v c(x)) = x v y
    ]]></literal>
    <justification jstring="[para(2516(a,1),2684(a,1,2,1,2))].">
      <j1 rule="para" parents="2516 2684"/>
    </justification>
  </clause>

  <clause id="2740">
    <literal><![CDATA[
      c(x v (y v z)) v c(z) = c(z)
    ]]></literal>
    <justification jstring="[para(2425(a,1),2690(a,1,1,1,2,2))].">
      <j1 rule="para" parents="2425 2690"/>
    </justification>
  </clause>

  <clause id="2771">
    <literal><![CDATA[
      x v (y v (x v z)) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(2694(a,1),2516(a,1,1,1)),demod(2425(2),2425(4),2425(7))].">
      <j1 rule="para" parents="2694 2516"/>
      <j2 rule="demod" parents="2425 2425 2425"/>
    </justification>
  </clause>

  <clause id="2780">
    <literal><![CDATA[
      c(c(x v y) v x) v c(x v y) = c(x)
    ]]></literal>
    <justification jstring="[para(2563(a,1),2594(a,1,1,2,1,1,1,2,1)),demod(2425(2),2425(4),2709(9),2425(9))].">
      <j1 rule="para" parents="2563 2594"/>
      <j2 rule="demod" parents="2425 2425 2709 2425"/>
    </justification>
  </clause>

  <clause id="2799">
    <literal><![CDATA[
      (x v (y v z)) v z = x v (y v z)
    ]]></literal>
    <justification jstring="[para(2740(a,1),2567(a,1,2,1)),demod(2425(4))].">
      <j1 rule="para" parents="2740 2567"/>
      <j2 rule="demod" parents="2425"/>
    </justification>
  </clause>

  <clause id="2815">
    <literal><![CDATA[
      c((x v (y v c(z))) v u) v z = z
    ]]></literal>
    <justification jstring="[para(2690(a,1),2646(a,1,2)),demod(2425(5),2690(11))].">
      <j1 rule="para" parents="2690 2646"/>
      <j2 rule="demod" parents="2425 2690"/>
    </justification>
  </clause>

  <clause id="2876">
    <literal><![CDATA[
      x v c((y v (z v c(x))) v u) = x
    ]]></literal>
    <justification jstring="[para(2690(a,1),2679(a,1,1)),demod(2425(5),2690(11))].">
      <j1 rule="para" parents="2690 2679"/>
      <j2 rule="demod" parents="2425 2690"/>
    </justification>
  </clause>

  <clause id="4078">
    <literal><![CDATA[
      c(x v c(x v y)) v x = x v y
    ]]></literal>
    <justification jstring="[para(2516(a,1),2780(a,1,1,1,1,1)),demod(2425(2),2516(8),2425(6),2425(8))].">
      <j1 rule="para" parents="2516 2780"/>
      <j2 rule="demod" parents="2425 2516 2425 2425"/>
    </justification>
  </clause>

  <clause id="4123">
    <literal><![CDATA[
      c(x) v x = x v c(x)
    ]]></literal>
    <justification jstring="[para(2619(a,1),4078(a,1,1,1))].">
      <j1 rule="para" parents="2619 4078"/>
    </justification>
  </clause>

  <clause id="4125">
    <literal><![CDATA[
      x v (y v c(x)) = x v c(x)
    ]]></literal>
    <justification jstring="[para(2708(a,1),4078(a,1,1,1)),demod(4123(2)),flip(a)].">
      <j1 rule="para" parents="2708 4078"/>
      <j2 rule="demod" parents="4123"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4128">
    <literal><![CDATA[
      (x v y) v c(x v y) = (x v y) v c(x)
    ]]></literal>
    <justification jstring="[para(2732(a,1),4078(a,1,1,1)),demod(4123(4))].">
      <j1 rule="para" parents="2732 4078"/>
      <j2 rule="demod" parents="4123"/>
    </justification>
  </clause>

  <clause id="4164">
    <literal><![CDATA[
      (x v y) v c(x) = x v c(x)
    ]]></literal>
    <justification jstring="[para(4123(a,1),2771(a,1,2)),demod(4128(4),4125(4),4123(6),4128(6)),flip(a)].">
      <j1 rule="para" parents="4123 2771"/>
      <j2 rule="demod" parents="4128 4125 4123 4128"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4166">
    <literal><![CDATA[
      (x v c(x)) v y = x v c(x)
    ]]></literal>
    <justification jstring="[para(4123(a,1),2799(a,1,1)),demod(4128(4),4164(3),4123(7),4128(7),4164(6))].">
      <j1 rule="para" parents="4123 2799"/>
      <j2 rule="demod" parents="4128 4164 4123 4128 4164"/>
    </justification>
  </clause>

  <clause id="4167">
    <literal><![CDATA[
      c(x v c(x)) v y = y
    ]]></literal>
    <justification jstring="[para(4123(a,1),2815(a,1,1,1,1)),demod(4128(6),4164(4),4166(3))].">
      <j1 rule="para" parents="4123 2815"/>
      <j2 rule="demod" parents="4128 4164 4166"/>
    </justification>
  </clause>

  <clause id="4168">
    <literal><![CDATA[
      x v c(y v c(y)) = x
    ]]></literal>
    <justification jstring="[para(4123(a,1),2876(a,1,2,1,1)),demod(4128(6),4164(4),4166(3))].">
      <j1 rule="para" parents="4123 2876"/>
      <j2 rule="demod" parents="4128 4164 4166"/>
    </justification>
  </clause>

  <clause id="4241">
    <literal><![CDATA[
      x v c(x) = y v c(y)
    ]]></literal>
    <justification jstring="[para(4167(a,1),2780(a,1,1,1,1,1)),demod(4168(5),2425(2),4167(4),2425(6))].">
      <j1 rule="para" parents="4167 2780"/>
      <j2 rule="demod" parents="4168 2425 4167 2425"/>
    </justification>
  </clause>

  <clause id="4242">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(1)
    ]]></attribute>
    <justification jstring="[resolve(4241,a,36,a)].">
      <j1 rule="resolve" parents="4241 36"/>
    </justification>
  </clause>

</proof>

<proof number="8" length="112" max_count="37">

<comments><![CDATA[
% Proof 8 at 0.80 (+ 0.02) seconds: 0.
% Length of proof is 112.
% Level of proof is 27.
% Maximum clause weight is 37.
% Given clauses 188.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="2">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="3">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_meet)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[copy(1)].">
      <j1 rule="copy" parents="1"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[copy(2),flip(a)].">
      <j1 rule="copy" parents="2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(f(x,y),f(x,y)) = x ^ y
    ]]></literal>
    <attribute><![CDATA[
      label(Def_meet)
    ]]></attribute>
    <justification jstring="[copy(3),flip(a)].">
      <j1 rule="copy" parents="3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      f(x,x) = c(x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[copy(4),flip(a)].">
      <j1 rule="copy" parents="4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      c(f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[back_demod(21),demod(22(3))].">
      <j1 rule="back_demod" parents="21"/>
      <j2 rule="demod" parents="22"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      f(c(x),c(y)) = x v y
    ]]></literal>
    <justification jstring="[back_demod(20),demod(22(1),22(2))].">
      <j1 rule="back_demod" parents="20"/>
      <j2 rule="demod" parents="22 22"/>
    </justification>
  </clause>

  <clause id="25">
    <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_demod(19),demod(22(5))].">
      <j1 rule="back_demod" parents="19"/>
      <j2 rule="demod" parents="22"/>
    </justification>
  </clause>

  <clause id="37">
    <literal><![CDATA[
      c22 ^ c(c22) != c21 ^ c(c21)
    ]]></literal>
    <attribute><![CDATA[
      answer(0)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      x ^ x = c(c(x))
    ]]></literal>
    <justification jstring="[para(22(a,1),23(a,1,1)),flip(a)].">
      <j1 rule="para" parents="22 23"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      c(c(x)) = x v x
    ]]></literal>
    <justification jstring="[para(24(a,1),22(a,1)),flip(a)].">
      <j1 rule="para" parents="24 22"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      c(x) ^ c(y) = c(x v y)
    ]]></literal>
    <justification jstring="[para(24(a,1),23(a,1,1)),flip(a)].">
      <j1 rule="para" parents="24 23"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      x ^ x = x v x
    ]]></literal>
    <justification jstring="[back_demod(40),demod(42(3))].">
      <j1 rule="back_demod" parents="40"/>
      <j2 rule="demod" parents="42"/>
    </justification>
  </clause>

  <clause id="49">
    <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(22(a,1),25(a,1,1,1,1))].">
      <j1 rule="para" parents="22 25"/>
    </justification>
  </clause>

  <clause id="50">
    <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(22(a,1),25(a,1,1,1,2))].">
      <j1 rule="para" parents="22 25"/>
    </justification>
  </clause>

  <clause id="51">
    <literal><![CDATA[
      f(f(x v x,y),f(x,f(f(x,f(c(x),x)),x))) = x
    ]]></literal>
    <justification jstring="[para(22(a,1),25(a,1,1,1)),demod(22(1),42(2))].">
      <j1 rule="para" parents="22 25"/>
      <j2 rule="demod" parents="22 42"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      f(f(x,y) ^ f(y,z),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(22(a,1),25(a,1,1)),demod(23(4))].">
      <j1 rule="para" parents="22 25"/>
      <j2 rule="demod" parents="23"/>
    </justification>
  </clause>

  <clause id="56">
    <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(24(a,1),25(a,1,1,1,1)),demod(42(8))].">
      <j1 rule="para" parents="24 25"/>
      <j2 rule="demod" parents="42"/>
    </justification>
  </clause>

  <clause id="58">
    <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(24(a,1),25(a,1,2,2,1,2))].">
      <j1 rule="para" parents="24 25"/>
    </justification>
  </clause>

  <clause id="59">
    <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(25(a,1),25(a,1,1,1))].">
      <j1 rule="para" parents="25 25"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(x v x,c(y)) = c(x) v y
    ]]></literal>
    <justification jstring="[para(42(a,1),24(a,1,1))].">
      <j1 rule="para" parents="42 24"/>
    </justification>
  </clause>

  <clause id="63">
    <literal><![CDATA[
      f(c(x),y v y) = x v c(y)
    ]]></literal>
    <justification jstring="[para(42(a,1),24(a,1,2))].">
      <j1 rule="para" parents="42 24"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      c(x v x) = c(x) v c(x)
    ]]></literal>
    <justification jstring="[para(42(a,1),42(a,1,1))].">
      <j1 rule="para" parents="42 42"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      (x v x) ^ c(y) = c(c(x) v y)
    ]]></literal>
    <justification jstring="[para(42(a,1),43(a,1,1))].">
      <j1 rule="para" parents="42 43"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      f(x v x,y v y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[para(42(a,1),62(a,1,2))].">
      <j1 rule="para" parents="42 62"/>
    </justification>
  </clause>

  <clause id="92">
    <literal><![CDATA[
      (x v x) ^ (y v y) = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[para(42(a,1),69(a,1,2))].">
      <j1 rule="para" parents="42 69"/>
    </justification>
  </clause>

  <clause id="167">
    <literal><![CDATA[
      f(x,f(c(x),f(f(y,f(x v x,y)),y))) = c(x)
    ]]></literal>
    <justification jstring="[para(50(a,1),25(a,1,1)),demod(42(3))].">
      <j1 rule="para" parents="50 25"/>
      <j2 rule="demod" parents="42"/>
    </justification>
  </clause>

  <clause id="225">
    <literal><![CDATA[
      f(c(x) v y,f(x,f(f(x,f(c(x),x)),x))) = x
    ]]></literal>
    <justification jstring="[para(62(a,1),51(a,1,1))].">
      <j1 rule="para" parents="62 51"/>
    </justification>
  </clause>

  <clause id="318">
    <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(62(a,1),52(a,1,1,2)),demod(65(9),62(12),42(9))].">
      <j1 rule="para" parents="62 52"/>
      <j2 rule="demod" parents="65 62 42"/>
    </justification>
  </clause>

  <clause id="773">
    <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(24(a,1),225(a,1,2,2,1,2)),demod(42(2))].">
      <j1 rule="para" parents="24 225"/>
      <j2 rule="demod" parents="42"/>
    </justification>
  </clause>

  <clause id="782">
    <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(225(a,1),167(a,1,2,2,1,2)),demod(42(3),42(18))].">
      <j1 rule="para" parents="225 167"/>
      <j2 rule="demod" parents="42 42"/>
    </justification>
  </clause>

  <clause id="1350">
    <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(63(a,1),58(a,1,1,1,1)),demod(62(5))].">
      <j1 rule="para" parents="63 58"/>
      <j2 rule="demod" parents="62"/>
    </justification>
  </clause>

  <clause id="1641">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(59(a,1),25(a,1,1))].">
      <j1 rule="para" parents="59 25"/>
    </justification>
  </clause>

  <clause id="1643">
    <literal><![CDATA[
      f(c(x),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(59(a,1),49(a,1,1))].">
      <j1 rule="para" parents="59 49"/>
    </justification>
  </clause>

  <clause id="1646">
    <literal><![CDATA[
      f(x v y,f(c(y),f(f(z,f(y v y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[para(59(a,1),56(a,1,1))].">
      <j1 rule="para" parents="59 56"/>
    </justification>
  </clause>

  <clause id="1649">
    <literal><![CDATA[
      f(c(x),f(x v x,x)) = x v x
    ]]></literal>
    <justification jstring="[back_demod(782),demod(1641(14))].">
      <j1 rule="back_demod" parents="782"/>
      <j2 rule="demod" parents="1641"/>
    </justification>
  </clause>

  <clause id="1665">
    <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(1649(a,1),52(a,1,1,1)),demod(23(10))].">
      <j1 rule="para" parents="1649 52"/>
      <j2 rule="demod" parents="23"/>
    </justification>
  </clause>

  <clause id="1666">
    <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(1649(a,1),52(a,1,1,2)),demod(42(9))].">
      <j1 rule="para" parents="1649 52"/>
      <j2 rule="demod" parents="42"/>
    </justification>
  </clause>

  <clause id="1711">
    <literal><![CDATA[
      f(c(x),f(x,f(f(c(y),x v y),c(y)))) = x
    ]]></literal>
    <justification jstring="[para(24(a,1),1643(a,1,2,2,1,2))].">
      <j1 rule="para" parents="24 1643"/>
    </justification>
  </clause>

  <clause id="1757">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(1643(a,1),1643(a,1,2,2,1,2)),demod(1641(13),22(2),22(3),42(2))].">
      <j1 rule="para" parents="1643 1643"/>
      <j2 rule="demod" parents="1641 22 22 42"/>
    </justification>
  </clause>

  <clause id="1804">
    <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_demod(1666),demod(1757(3),1757(5),22(5),1757(6),1757(6),22(6),1757(9),22(9))].">
      <j1 rule="back_demod" parents="1666"/>
      <j2 rule="demod" parents="1757 1757 22 1757 1757 22 1757 22"/>
    </justification>
  </clause>

  <clause id="1805">
    <literal><![CDATA[
      f(x ^ f(c(x),y),f(c(x),f(f(y,f(x,y)),y))) = c(x)
    ]]></literal>
    <justification jstring="[back_demod(1665),demod(1757(1),1757(1),22(1),1757(4),22(4),1757(5),47(5),1757(5),1757(10),22(10))].">
      <j1 rule="back_demod" parents="1665"/>
      <j2 rule="demod" parents="1757 1757 22 1757 22 1757 47 1757 1757 22"/>
    </justification>
  </clause>

  <clause id="1816">
    <literal><![CDATA[
      f(x v y,f(c(y),f(f(z,f(y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[back_demod(1646),demod(1757(3))].">
      <j1 rule="back_demod" parents="1646"/>
      <j2 rule="demod" parents="1757"/>
    </justification>
  </clause>

  <clause id="1969">
    <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_demod(1350),demod(1757(7),1757(8),1757(14))].">
      <j1 rule="back_demod" parents="1350"/>
      <j2 rule="demod" parents="1757 1757 1757"/>
    </justification>
  </clause>

  <clause id="2166">
    <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_demod(773),demod(1757(1))].">
      <j1 rule="back_demod" parents="773"/>
      <j2 rule="demod" parents="1757"/>
    </justification>
  </clause>

  <clause id="2332">
    <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_demod(318),demod(1757(1),1757(5),1757(6),1757(12))].">
      <j1 rule="back_demod" parents="318"/>
      <j2 rule="demod" parents="1757 1757 1757 1757"/>
    </justification>
  </clause>

  <clause id="2409">
    <literal><![CDATA[
      x ^ y = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[back_demod(92),demod(1757(1),1757(1))].">
      <j1 rule="back_demod" parents="92"/>
      <j2 rule="demod" parents="1757 1757"/>
    </justification>
  </clause>

  <clause id="2419">
    <literal><![CDATA[
      f(x,y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[back_demod(74),demod(1757(1),1757(1))].">
      <j1 rule="back_demod" parents="74"/>
      <j2 rule="demod" parents="1757 1757"/>
    </justification>
  </clause>

  <clause id="2425">
    <literal><![CDATA[
      c(c(x)) = x
    ]]></literal>
    <justification jstring="[back_demod(42),demod(1757(3))].">
      <j1 rule="back_demod" parents="42"/>
      <j2 rule="demod" parents="1757"/>
    </justification>
  </clause>

  <clause id="2434">
    <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_demod(2332),demod(2419(1),2409(6),2419(12),2425(11),2419(14),2425(15),2419(15),2419(18),2425(10))].">
      <j1 rule="back_demod" parents="2332"/>
      <j2 rule="demod" parents="2419 2409 2419 2425 2419 2425 2419 2419 2425"/>
    </justification>
  </clause>

  <clause id="2447">
    <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_demod(1805),demod(2419(2),2425(2),2409(3),2419(8),2419(11),2419(14),2419(17),2425(8),2419(18),2425(7))].">
      <j1 rule="back_demod" parents="1805"/>
      <j2 rule="demod" parents="2419 2425 2409 2419 2419 2419 2419 2425 2419 2425"/>
    </justification>
  </clause>

  <clause id="2448">
    <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_demod(1804),demod(2419(2),2425(3),2409(3),2419(10),2425(11),2419(11),2425(9),2419(13),2425(14),2419(14),2425(8),2419(15),2425(7))].">
      <j1 rule="back_demod" parents="1804"/>
      <j2 rule="demod" parents="2419 2425 2409 2419 2425 2419 2425 2419 2425 2419 2425 2419 2425"/>
    </justification>
  </clause>

  <clause id="2475">
    <literal><![CDATA[
      c(c(c22) v c22) != c(c(c21) v c21)
    ]]></literal>
    <attribute><![CDATA[
      answer(0)
    ]]></attribute>
    <justification jstring="[back_demod(37),demod(2409(4),2425(5),2409(9),2425(10))].">
      <j1 rule="back_demod" parents="37"/>
      <j2 rule="demod" parents="2409 2425 2409 2425"/>
    </justification>
  </clause>

  <clause id="2486">
    <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_demod(2166),demod(2419(6),2425(4),2419(8),2425(9),2419(9),2425(3),2419(10))].">
      <j1 rule="back_demod" parents="2166"/>
      <j2 rule="demod" parents="2419 2425 2419 2425 2419 2425 2419"/>
    </justification>
  </clause>

  <clause id="2489">
    <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_demod(1969),demod(2419(5),2419(8),2419(13),2425(12),2419(15),2425(16),2419(16),2419(19))].">
      <j1 rule="back_demod" parents="1969"/>
      <j2 rule="demod" parents="2419 2419 2419 2425 2419 2425 2419 2419"/>
    </justification>
  </clause>

  <clause id="2490">
    <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_demod(1816),demod(2419(3),2419(6),2419(9),2419(12),2425(3),2419(13))].">
      <j1 rule="back_demod" parents="1816"/>
      <j2 rule="demod" parents="2419 2419 2419 2419 2425 2419"/>
    </justification>
  </clause>

  <clause id="2501">
    <literal><![CDATA[
      x v c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[back_demod(1711),demod(2419(4),2425(3),2419(6),2425(7),2419(7),2419(10),2425(2))].">
      <j1 rule="back_demod" parents="1711"/>
      <j2 rule="demod" parents="2419 2425 2419 2425 2419 2419 2425"/>
    </justification>
  </clause>

  <clause id="2514">
    <literal><![CDATA[
      c(x v c(c(x v c(c(x) v x)) v x)) = c(x)
    ]]></literal>
    <justification jstring="[para(2486(a,1),1757(a,1)),flip(a)].">
      <j1 rule="para" parents="2486 1757"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2516">
    <literal><![CDATA[
      c(x v y) v c(x) = c(x)
    ]]></literal>
    <justification jstring="[para(2486(a,1),2419(a,2)),demod(2419(10),2514(11))].">
      <j1 rule="para" parents="2486 2419"/>
      <j2 rule="demod" parents="2419 2514"/>
    </justification>
  </clause>

  <clause id="2520">
    <literal><![CDATA[
      (c(c(x) v y) v c(y)) v c(y) = c(y)
    ]]></literal>
    <justification jstring="[back_demod(2448),demod(2514(14))].">
      <j1 rule="back_demod" parents="2448"/>
      <j2 rule="demod" parents="2514"/>
    </justification>
  </clause>

  <clause id="2521">
    <literal><![CDATA[
      c(c(x) v y) v x = x
    ]]></literal>
    <justification jstring="[para(2425(a,1),2516(a,1,2)),demod(2425(6))].">
      <j1 rule="para" parents="2425 2516"/>
      <j2 rule="demod" parents="2425"/>
    </justification>
  </clause>

  <clause id="2522">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(2516(a,1),2516(a,1,1,1)),demod(2425(2),2425(3),2425(5))].">
      <j1 rule="para" parents="2516 2516"/>
      <j2 rule="demod" parents="2425 2425 2425"/>
    </justification>
  </clause>

  <clause id="2527">
    <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(2434(a,1),2516(a,1,1,1))].">
      <j1 rule="para" parents="2434 2516"/>
    </justification>
  </clause>

  <clause id="2535">
    <literal><![CDATA[
      (c(c(x) v c(y)) v y) v y = y
    ]]></literal>
    <justification jstring="[para(2425(a,1),2520(a,1,1,2)),demod(2425(7),2425(8))].">
      <j1 rule="para" parents="2425 2520"/>
      <j2 rule="demod" parents="2425 2425"/>
    </justification>
  </clause>

  <clause id="2542">
    <literal><![CDATA[
      (c(x v c(y)) v y) v y = y
    ]]></literal>
    <justification jstring="[para(2425(a,1),2535(a,1,1,1,1,1))].">
      <j1 rule="para" parents="2425 2535"/>
    </justification>
  </clause>

  <clause id="2546">
    <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(2425(a,1),2447(a,1,1,2,1,2)),demod(2425(6),2425(7),2425(11))].">
      <j1 rule="para" parents="2425 2447"/>
      <j2 rule="demod" parents="2425 2425 2425"/>
    </justification>
  </clause>

  <clause id="2548">
    <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(2447(a,1),2501(a,1,2,1,2,1,1,1,2,1)),demod(2425(25),2490(38),2425(13),2521(12))].">
      <j1 rule="para" parents="2447 2501"/>
      <j2 rule="demod" parents="2425 2490 2425 2521"/>
    </justification>
  </clause>

  <clause id="2552">
    <literal><![CDATA[
      c(x) v c(x v c(y)) = c(x)
    ]]></literal>
    <justification jstring="[para(2447(a,1),2522(a,1,2)),demod(2548(7),2546(23))].">
      <j1 rule="para" parents="2447 2522"/>
      <j2 rule="demod" parents="2548 2546"/>
    </justification>
  </clause>

  <clause id="2563">
    <literal><![CDATA[
      c(x) v c(x v y) = c(x)
    ]]></literal>
    <justification jstring="[para(2425(a,1),2552(a,1,2,1,2))].">
      <j1 rule="para" parents="2425 2552"/>
    </justification>
  </clause>

  <clause id="2564">
    <literal><![CDATA[
      (x v y) v x = x v y
    ]]></literal>
    <justification jstring="[para(2516(a,1),2552(a,1,2,1)),demod(2425(3),2425(3),2425(5))].">
      <j1 rule="para" parents="2516 2552"/>
      <j2 rule="demod" parents="2425 2425 2425"/>
    </justification>
  </clause>

  <clause id="2567">
    <literal><![CDATA[
      x v c(c(x) v y) = x
    ]]></literal>
    <justification jstring="[para(2521(a,1),2564(a,1,1)),demod(2521(8))].">
      <j1 rule="para" parents="2521 2564"/>
      <j2 rule="demod" parents="2521"/>
    </justification>
  </clause>

  <clause id="2586">
    <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(2516(a,1),2489(a,1,1,1,1,1)),demod(2425(4))].">
      <j1 rule="para" parents="2516 2489"/>
      <j2 rule="demod" parents="2425"/>
    </justification>
  </clause>

  <clause id="2587">
    <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(2516(a,1),2489(a,1,1,1)),demod(2425(4))].">
      <j1 rule="para" parents="2516 2489"/>
      <j2 rule="demod" parents="2425"/>
    </justification>
  </clause>

  <clause id="2594">
    <literal><![CDATA[
      c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[para(2489(a,1),2542(a,1,1)),demod(2567(10)),flip(a)].">
      <j1 rule="para" parents="2489 2542"/>
      <j2 rule="demod" parents="2567"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2608">
    <literal><![CDATA[
      c(x v c(y)) v y = y
    ]]></literal>
    <justification jstring="[back_demod(2587),demod(2594(12))].">
      <j1 rule="back_demod" parents="2587"/>
      <j2 rule="demod" parents="2594"/>
    </justification>
  </clause>

  <clause id="2609">
    <literal><![CDATA[
      c((c(x) v y) v c(z)) v x = x
    ]]></literal>
    <justification jstring="[back_demod(2586),demod(2594(14))].">
      <j1 rule="back_demod" parents="2586"/>
      <j2 rule="demod" parents="2594"/>
    </justification>
  </clause>

  <clause id="2617">
    <literal><![CDATA[
      c(x v y) v c(y) = c(y)
    ]]></literal>
    <justification jstring="[para(2425(a,1),2608(a,1,1,1,2))].">
      <j1 rule="para" parents="2425 2608"/>
    </justification>
  </clause>

  <clause id="2619">
    <literal><![CDATA[
      x v c(y v c(x)) = x
    ]]></literal>
    <justification jstring="[para(2608(a,1),2564(a,1,1)),demod(2608(8))].">
      <j1 rule="para" parents="2608 2564"/>
      <j2 rule="demod" parents="2608"/>
    </justification>
  </clause>

  <clause id="2625">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(2617(a,1),2516(a,1,1,1)),demod(2425(2),2425(3),2425(5))].">
      <j1 rule="para" parents="2617 2516"/>
      <j2 rule="demod" parents="2425 2425 2425"/>
    </justification>
  </clause>

  <clause id="2633">
    <literal><![CDATA[
      c((c(x) v y) v z) v x = x
    ]]></literal>
    <justification jstring="[para(2425(a,1),2609(a,1,1,1,2))].">
      <j1 rule="para" parents="2425 2609"/>
    </justification>
  </clause>

  <clause id="2646">
    <literal><![CDATA[
      c(c(x) v y) v (x v z) = x v z
    ]]></literal>
    <justification jstring="[para(2516(a,1),2633(a,1,1,1,1))].">
      <j1 rule="para" parents="2516 2633"/>
    </justification>
  </clause>

  <clause id="2648">
    <literal><![CDATA[
      x v c((c(x) v y) v z) = x
    ]]></literal>
    <justification jstring="[para(2633(a,1),2564(a,1,1)),demod(2633(10))].">
      <j1 rule="para" parents="2633 2564"/>
      <j2 rule="demod" parents="2633"/>
    </justification>
  </clause>

  <clause id="2652">
    <literal><![CDATA[
      c((x v c(y)) v z) v y = y
    ]]></literal>
    <justification jstring="[para(2625(a,1),2633(a,1,1,1,1))].">
      <j1 rule="para" parents="2625 2633"/>
    </justification>
  </clause>

  <clause id="2653">
    <literal><![CDATA[
      c(x v (c(y) v z)) v y = y
    ]]></literal>
    <justification jstring="[para(2625(a,1),2633(a,1,1,1))].">
      <j1 rule="para" parents="2625 2633"/>
    </justification>
  </clause>

  <clause id="2679">
    <literal><![CDATA[
      (x v y) v c(c(x) v z) = x v y
    ]]></literal>
    <justification jstring="[para(2516(a,1),2648(a,1,2,1,1))].">
      <j1 rule="para" parents="2516 2648"/>
    </justification>
  </clause>

  <clause id="2683">
    <literal><![CDATA[
      x v c((y v c(x)) v z) = x
    ]]></literal>
    <justification jstring="[para(2625(a,1),2648(a,1,2,1,1))].">
      <j1 rule="para" parents="2625 2648"/>
    </justification>
  </clause>

  <clause id="2684">
    <literal><![CDATA[
      x v c(y v (c(x) v z)) = x
    ]]></literal>
    <justification jstring="[para(2625(a,1),2648(a,1,2,1))].">
      <j1 rule="para" parents="2625 2648"/>
    </justification>
  </clause>

  <clause id="2690">
    <literal><![CDATA[
      c(x v (y v c(z))) v z = z
    ]]></literal>
    <justification jstring="[para(2625(a,1),2652(a,1,1,1))].">
      <j1 rule="para" parents="2625 2652"/>
    </justification>
  </clause>

  <clause id="2694">
    <literal><![CDATA[
      c(x v (y v z)) v c(y) = c(y)
    ]]></literal>
    <justification jstring="[para(2425(a,1),2653(a,1,1,1,2,1))].">
      <j1 rule="para" parents="2425 2653"/>
    </justification>
  </clause>

  <clause id="2708">
    <literal><![CDATA[
      x v c(y v (z v c(x))) = x
    ]]></literal>
    <justification jstring="[para(2625(a,1),2683(a,1,2,1))].">
      <j1 rule="para" parents="2625 2683"/>
    </justification>
  </clause>

  <clause id="2709">
    <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(2425(a,1),2527(a,1,1)),demod(2425(3),2425(5),2425(11),2425(13))].">
      <j1 rule="para" parents="2425 2527"/>
      <j2 rule="demod" parents="2425 2425 2425 2425"/>
    </justification>
  </clause>

  <clause id="2732">
    <literal><![CDATA[
      (x v y) v c(z v c(x)) = x v y
    ]]></literal>
    <justification jstring="[para(2516(a,1),2684(a,1,2,1,2))].">
      <j1 rule="para" parents="2516 2684"/>
    </justification>
  </clause>

  <clause id="2740">
    <literal><![CDATA[
      c(x v (y v z)) v c(z) = c(z)
    ]]></literal>
    <justification jstring="[para(2425(a,1),2690(a,1,1,1,2,2))].">
      <j1 rule="para" parents="2425 2690"/>
    </justification>
  </clause>

  <clause id="2771">
    <literal><![CDATA[
      x v (y v (x v z)) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(2694(a,1),2516(a,1,1,1)),demod(2425(2),2425(4),2425(7))].">
      <j1 rule="para" parents="2694 2516"/>
      <j2 rule="demod" parents="2425 2425 2425"/>
    </justification>
  </clause>

  <clause id="2780">
    <literal><![CDATA[
      c(c(x v y) v x) v c(x v y) = c(x)
    ]]></literal>
    <justification jstring="[para(2563(a,1),2594(a,1,1,2,1,1,1,2,1)),demod(2425(2),2425(4),2709(9),2425(9))].">
      <j1 rule="para" parents="2563 2594"/>
      <j2 rule="demod" parents="2425 2425 2709 2425"/>
    </justification>
  </clause>

  <clause id="2799">
    <literal><![CDATA[
      (x v (y v z)) v z = x v (y v z)
    ]]></literal>
    <justification jstring="[para(2740(a,1),2567(a,1,2,1)),demod(2425(4))].">
      <j1 rule="para" parents="2740 2567"/>
      <j2 rule="demod" parents="2425"/>
    </justification>
  </clause>

  <clause id="2815">
    <literal><![CDATA[
      c((x v (y v c(z))) v u) v z = z
    ]]></literal>
    <justification jstring="[para(2690(a,1),2646(a,1,2)),demod(2425(5),2690(11))].">
      <j1 rule="para" parents="2690 2646"/>
      <j2 rule="demod" parents="2425 2690"/>
    </justification>
  </clause>

  <clause id="2876">
    <literal><![CDATA[
      x v c((y v (z v c(x))) v u) = x
    ]]></literal>
    <justification jstring="[para(2690(a,1),2679(a,1,1)),demod(2425(5),2690(11))].">
      <j1 rule="para" parents="2690 2679"/>
      <j2 rule="demod" parents="2425 2690"/>
    </justification>
  </clause>

  <clause id="4078">
    <literal><![CDATA[
      c(x v c(x v y)) v x = x v y
    ]]></literal>
    <justification jstring="[para(2516(a,1),2780(a,1,1,1,1,1)),demod(2425(2),2516(8),2425(6),2425(8))].">
      <j1 rule="para" parents="2516 2780"/>
      <j2 rule="demod" parents="2425 2516 2425 2425"/>
    </justification>
  </clause>

  <clause id="4123">
    <literal><![CDATA[
      c(x) v x = x v c(x)
    ]]></literal>
    <justification jstring="[para(2619(a,1),4078(a,1,1,1))].">
      <j1 rule="para" parents="2619 4078"/>
    </justification>
  </clause>

  <clause id="4125">
    <literal><![CDATA[
      x v (y v c(x)) = x v c(x)
    ]]></literal>
    <justification jstring="[para(2708(a,1),4078(a,1,1,1)),demod(4123(2)),flip(a)].">
      <j1 rule="para" parents="2708 4078"/>
      <j2 rule="demod" parents="4123"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4128">
    <literal><![CDATA[
      (x v y) v c(x v y) = (x v y) v c(x)
    ]]></literal>
    <justification jstring="[para(2732(a,1),4078(a,1,1,1)),demod(4123(4))].">
      <j1 rule="para" parents="2732 4078"/>
      <j2 rule="demod" parents="4123"/>
    </justification>
  </clause>

  <clause id="4163">
    <literal><![CDATA[
      c(c22 v c(c22)) != c(c21 v c(c21))
    ]]></literal>
    <attribute><![CDATA[
      answer(0)
    ]]></attribute>
    <justification jstring="[back_demod(2475),demod(4123(4),4123(9))].">
      <j1 rule="back_demod" parents="2475"/>
      <j2 rule="demod" parents="4123 4123"/>
    </justification>
  </clause>

  <clause id="4164">
    <literal><![CDATA[
      (x v y) v c(x) = x v c(x)
    ]]></literal>
    <justification jstring="[para(4123(a,1),2771(a,1,2)),demod(4128(4),4125(4),4123(6),4128(6)),flip(a)].">
      <j1 rule="para" parents="4123 2771"/>
      <j2 rule="demod" parents="4128 4125 4123 4128"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4166">
    <literal><![CDATA[
      (x v c(x)) v y = x v c(x)
    ]]></literal>
    <justification jstring="[para(4123(a,1),2799(a,1,1)),demod(4128(4),4164(3),4123(7),4128(7),4164(6))].">
      <j1 rule="para" parents="4123 2799"/>
      <j2 rule="demod" parents="4128 4164 4123 4128 4164"/>
    </justification>
  </clause>

  <clause id="4167">
    <literal><![CDATA[
      c(x v c(x)) v y = y
    ]]></literal>
    <justification jstring="[para(4123(a,1),2815(a,1,1,1,1)),demod(4128(6),4164(4),4166(3))].">
      <j1 rule="para" parents="4123 2815"/>
      <j2 rule="demod" parents="4128 4164 4166"/>
    </justification>
  </clause>

  <clause id="4168">
    <literal><![CDATA[
      x v c(y v c(y)) = x
    ]]></literal>
    <justification jstring="[para(4123(a,1),2876(a,1,2,1,1)),demod(4128(6),4164(4),4166(3))].">
      <j1 rule="para" parents="4123 2876"/>
      <j2 rule="demod" parents="4128 4164 4166"/>
    </justification>
  </clause>

  <clause id="4243">
    <literal><![CDATA[
      c(x v c(x)) = c(y v c(y))
    ]]></literal>
    <justification jstring="[para(4168(a,1),4167(a,1))].">
      <j1 rule="para" parents="4168 4167"/>
    </justification>
  </clause>

  <clause id="4244">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(0)
    ]]></attribute>
    <justification jstring="[resolve(4243,a,4163,a)].">
      <j1 rule="resolve" parents="4243 4163"/>
    </justification>
  </clause>

</proof>

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

<comments><![CDATA[
% Proof 9 at 1.31 (+ 0.04) seconds: commutativity_meet.
% Length of proof is 82.
% Level of proof is 23.
% Maximum clause weight is 37.
% Given clauses 247.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="2">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="3">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(Def_meet)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      f(f(f(f(x,y),f(y,z)),u),f(y,f(f(z,f(f(y,y),z)),z))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(OML_Sh)
    ]]></attribute>
    <justification jstring="[copy(1)].">
      <j1 rule="copy" parents="1"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      f(f(x,x),f(y,y)) = x v y
    ]]></literal>
    <attribute><![CDATA[
      label(Def_join)
    ]]></attribute>
    <justification jstring="[copy(2),flip(a)].">
      <j1 rule="copy" parents="2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      f(f(x,y),f(x,y)) = x ^ y
    ]]></literal>
    <attribute><![CDATA[
      label(Def_meet)
    ]]></attribute>
    <justification jstring="[copy(3),flip(a)].">
      <j1 rule="copy" parents="3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      f(x,x) = c(x)
    ]]></literal>
    <attribute><![CDATA[
      label(Def_complement)
    ]]></attribute>
    <justification jstring="[copy(4),flip(a)].">
      <j1 rule="copy" parents="4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      c(f(x,y)) = x ^ y
    ]]></literal>
    <justification jstring="[back_demod(21),demod(22(3))].">
      <j1 rule="back_demod" parents="21"/>
      <j2 rule="demod" parents="22"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      f(c(x),c(y)) = x v y
    ]]></literal>
    <justification jstring="[back_demod(20),demod(22(1),22(2))].">
      <j1 rule="back_demod" parents="20"/>
      <j2 rule="demod" parents="22 22"/>
    </justification>
  </clause>

  <clause id="25">
    <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_demod(19),demod(22(5))].">
      <j1 rule="back_demod" parents="19"/>
      <j2 rule="demod" parents="22"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      c18 v c17 != c17 v c18
    ]]></literal>
    <attribute><![CDATA[
      answer(commutativity_meet)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      x ^ x = c(c(x))
    ]]></literal>
    <justification jstring="[para(22(a,1),23(a,1,1)),flip(a)].">
      <j1 rule="para" parents="22 23"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      c(c(x)) = x v x
    ]]></literal>
    <justification jstring="[para(24(a,1),22(a,1)),flip(a)].">
      <j1 rule="para" parents="24 22"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      c(x) ^ c(y) = c(x v y)
    ]]></literal>
    <justification jstring="[para(24(a,1),23(a,1,1)),flip(a)].">
      <j1 rule="para" parents="24 23"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      x ^ x = x v x
    ]]></literal>
    <justification jstring="[back_demod(40),demod(42(3))].">
      <j1 rule="back_demod" parents="40"/>
      <j2 rule="demod" parents="42"/>
    </justification>
  </clause>

  <clause id="49">
    <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(22(a,1),25(a,1,1,1,1))].">
      <j1 rule="para" parents="22 25"/>
    </justification>
  </clause>

  <clause id="50">
    <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(22(a,1),25(a,1,1,1,2))].">
      <j1 rule="para" parents="22 25"/>
    </justification>
  </clause>

  <clause id="51">
    <literal><![CDATA[
      f(f(x v x,y),f(x,f(f(x,f(c(x),x)),x))) = x
    ]]></literal>
    <justification jstring="[para(22(a,1),25(a,1,1,1)),demod(22(1),42(2))].">
      <j1 rule="para" parents="22 25"/>
      <j2 rule="demod" parents="22 42"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      f(f(x,y) ^ f(y,z),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(22(a,1),25(a,1,1)),demod(23(4))].">
      <j1 rule="para" parents="22 25"/>
      <j2 rule="demod" parents="23"/>
    </justification>
  </clause>

  <clause id="56">
    <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(24(a,1),25(a,1,1,1,1)),demod(42(8))].">
      <j1 rule="para" parents="24 25"/>
      <j2 rule="demod" parents="42"/>
    </justification>
  </clause>

  <clause id="58">
    <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(24(a,1),25(a,1,2,2,1,2))].">
      <j1 rule="para" parents="24 25"/>
    </justification>
  </clause>

  <clause id="59">
    <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(25(a,1),25(a,1,1,1))].">
      <j1 rule="para" parents="25 25"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      f(x v x,c(y)) = c(x) v y
    ]]></literal>
    <justification jstring="[para(42(a,1),24(a,1,1))].">
      <j1 rule="para" parents="42 24"/>
    </justification>
  </clause>

  <clause id="63">
    <literal><![CDATA[
      f(c(x),y v y) = x v c(y)
    ]]></literal>
    <justification jstring="[para(42(a,1),24(a,1,2))].">
      <j1 rule="para" parents="42 24"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      c(x v x) = c(x) v c(x)
    ]]></literal>
    <justification jstring="[para(42(a,1),42(a,1,1))].">
      <j1 rule="para" parents="42 42"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      (x v x) ^ c(y) = c(c(x) v y)
    ]]></literal>
    <justification jstring="[para(42(a,1),43(a,1,1))].">
      <j1 rule="para" parents="42 43"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      f(x v x,y v y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[para(42(a,1),62(a,1,2))].">
      <j1 rule="para" parents="42 62"/>
    </justification>
  </clause>

  <clause id="92">
    <literal><![CDATA[
      (x v x) ^ (y v y) = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[para(42(a,1),69(a,1,2))].">
      <j1 rule="para" parents="42 69"/>
    </justification>
  </clause>

  <clause id="167">
    <literal><![CDATA[
      f(x,f(c(x),f(f(y,f(x v x,y)),y))) = c(x)
    ]]></literal>
    <justification jstring="[para(50(a,1),25(a,1,1)),demod(42(3))].">
      <j1 rule="para" parents="50 25"/>
      <j2 rule="demod" parents="42"/>
    </justification>
  </clause>

  <clause id="225">
    <literal><![CDATA[
      f(c(x) v y,f(x,f(f(x,f(c(x),x)),x))) = x
    ]]></literal>
    <justification jstring="[para(62(a,1),51(a,1,1))].">
      <j1 rule="para" parents="62 51"/>
    </justification>
  </clause>

  <clause id="318">
    <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(62(a,1),52(a,1,1,2)),demod(65(9),62(12),42(9))].">
      <j1 rule="para" parents="62 52"/>
      <j2 rule="demod" parents="65 62 42"/>
    </justification>
  </clause>

  <clause id="773">
    <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(24(a,1),225(a,1,2,2,1,2)),demod(42(2))].">
      <j1 rule="para" parents="24 225"/>
      <j2 rule="demod" parents="42"/>
    </justification>
  </clause>

  <clause id="782">
    <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(225(a,1),167(a,1,2,2,1,2)),demod(42(3),42(18))].">
      <j1 rule="para" parents="225 167"/>
      <j2 rule="demod" parents="42 42"/>
    </justification>
  </clause>

  <clause id="1350">
    <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(63(a,1),58(a,1,1,1,1)),demod(62(5))].">
      <j1 rule="para" parents="63 58"/>
      <j2 rule="demod" parents="62"/>
    </justification>
  </clause>

  <clause id="1641">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,f(c(y),z)),z))) = y
    ]]></literal>
    <justification jstring="[para(59(a,1),25(a,1,1))].">
      <j1 rule="para" parents="59 25"/>
    </justification>
  </clause>

  <clause id="1643">
    <literal><![CDATA[
      f(c(x),f(x,f(f(y,f(c(x),y)),y))) = x
    ]]></literal>
    <justification jstring="[para(59(a,1),49(a,1,1))].">
      <j1 rule="para" parents="59 49"/>
    </justification>
  </clause>

  <clause id="1646">
    <literal><![CDATA[
      f(x v y,f(c(y),f(f(z,f(y v y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[para(59(a,1),56(a,1,1))].">
      <j1 rule="para" parents="59 56"/>
    </justification>
  </clause>

  <clause id="1649">
    <literal><![CDATA[
      f(c(x),f(x v x,x)) = x v x
    ]]></literal>
    <justification jstring="[back_demod(782),demod(1641(14))].">
      <j1 rule="back_demod" parents="782"/>
      <j2 rule="demod" parents="1641"/>
    </justification>
  </clause>

  <clause id="1665">
    <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(1649(a,1),52(a,1,1,1)),demod(23(10))].">
      <j1 rule="para" parents="1649 52"/>
      <j2 rule="demod" parents="23"/>
    </justification>
  </clause>

  <clause id="1666">
    <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(1649(a,1),52(a,1,1,2)),demod(42(9))].">
      <j1 rule="para" parents="1649 52"/>
      <j2 rule="demod" parents="42"/>
    </justification>
  </clause>

  <clause id="1711">
    <literal><![CDATA[
      f(c(x),f(x,f(f(c(y),x v y),c(y)))) = x
    ]]></literal>
    <justification jstring="[para(24(a,1),1643(a,1,2,2,1,2))].">
      <j1 rule="para" parents="24 1643"/>
    </justification>
  </clause>

  <clause id="1757">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(1643(a,1),1643(a,1,2,2,1,2)),demod(1641(13),22(2),22(3),42(2))].">
      <j1 rule="para" parents="1643 1643"/>
      <j2 rule="demod" parents="1641 22 22 42"/>
    </justification>
  </clause>

  <clause id="1804">
    <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_demod(1666),demod(1757(3),1757(5),22(5),1757(6),1757(6),22(6),1757(9),22(9))].">
      <j1 rule="back_demod" parents="1666"/>
      <j2 rule="demod" parents="1757 1757 22 1757 1757 22 1757 22"/>
    </justification>
  </clause>

  <clause id="1805">
    <literal><![CDATA[
      f(x ^ f(c(x),y),f(c(x),f(f(y,f(x,y)),y))) = c(x)
    ]]></literal>
    <justification jstring="[back_demod(1665),demod(1757(1),1757(1),22(1),1757(4),22(4),1757(5),47(5),1757(5),1757(10),22(10))].">
      <j1 rule="back_demod" parents="1665"/>
      <j2 rule="demod" parents="1757 1757 22 1757 22 1757 47 1757 1757 22"/>
    </justification>
  </clause>

  <clause id="1816">
    <literal><![CDATA[
      f(x v y,f(c(y),f(f(z,f(y,z)),z))) = c(y)
    ]]></literal>
    <justification jstring="[back_demod(1646),demod(1757(3))].">
      <j1 rule="back_demod" parents="1646"/>
      <j2 rule="demod" parents="1757"/>
    </justification>
  </clause>

  <clause id="1969">
    <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_demod(1350),demod(1757(7),1757(8),1757(14))].">
      <j1 rule="back_demod" parents="1350"/>
      <j2 rule="demod" parents="1757 1757 1757"/>
    </justification>
  </clause>

  <clause id="2166">
    <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_demod(773),demod(1757(1))].">
      <j1 rule="back_demod" parents="773"/>
      <j2 rule="demod" parents="1757"/>
    </justification>
  </clause>

  <clause id="2332">
    <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_demod(318),demod(1757(1),1757(5),1757(6),1757(12))].">
      <j1 rule="back_demod" parents="318"/>
      <j2 rule="demod" parents="1757 1757 1757 1757"/>
    </justification>
  </clause>

  <clause id="2409">
    <literal><![CDATA[
      x ^ y = c(c(x) v c(y))
    ]]></literal>
    <justification jstring="[back_demod(92),demod(1757(1),1757(1))].">
      <j1 rule="back_demod" parents="92"/>
      <j2 rule="demod" parents="1757 1757"/>
    </justification>
  </clause>

  <clause id="2419">
    <literal><![CDATA[
      f(x,y) = c(x) v c(y)
    ]]></literal>
    <justification jstring="[back_demod(74),demod(1757(1),1757(1))].">
      <j1 rule="back_demod" parents="74"/>
      <j2 rule="demod" parents="1757 1757"/>
    </justification>
  </clause>

  <clause id="2425">
    <literal><![CDATA[
      c(c(x)) = x
    ]]></literal>
    <justification jstring="[back_demod(42),demod(1757(3))].">
      <j1 rule="back_demod" parents="42"/>
      <j2 rule="demod" parents="1757"/>
    </justification>
  </clause>

  <clause id="2434">
    <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_demod(2332),demod(2419(1),2409(6),2419(12),2425(11),2419(14),2425(15),2419(15),2419(18),2425(10))].">
      <j1 rule="back_demod" parents="2332"/>
      <j2 rule="demod" parents="2419 2409 2419 2425 2419 2425 2419 2419 2425"/>
    </justification>
  </clause>

  <clause id="2447">
    <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_demod(1805),demod(2419(2),2425(2),2409(3),2419(8),2419(11),2419(14),2419(17),2425(8),2419(18),2425(7))].">
      <j1 rule="back_demod" parents="1805"/>
      <j2 rule="demod" parents="2419 2425 2409 2419 2419 2419 2419 2425 2419 2425"/>
    </justification>
  </clause>

  <clause id="2448">
    <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_demod(1804),demod(2419(2),2425(3),2409(3),2419(10),2425(11),2419(11),2425(9),2419(13),2425(14),2419(14),2425(8),2419(15),2425(7))].">
      <j1 rule="back_demod" parents="1804"/>
      <j2 rule="demod" parents="2419 2425 2409 2419 2425 2419 2425 2419 2425 2419 2425 2419 2425"/>
    </justification>
  </clause>

  <clause id="2486">
    <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_demod(2166),demod(2419(6),2425(4),2419(8),2425(9),2419(9),2425(3),2419(10))].">
      <j1 rule="back_demod" parents="2166"/>
      <j2 rule="demod" parents="2419 2425 2419 2425 2419 2425 2419"/>
    </justification>
  </clause>

  <clause id="2489">
    <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_demod(1969),demod(2419(5),2419(8),2419(13),2425(12),2419(15),2425(16),2419(16),2419(19))].">
      <j1 rule="back_demod" parents="1969"/>
      <j2 rule="demod" parents="2419 2419 2419 2425 2419 2425 2419 2419"/>
    </justification>
  </clause>

  <clause id="2490">
    <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_demod(1816),demod(2419(3),2419(6),2419(9),2419(12),2425(3),2419(13))].">
      <j1 rule="back_demod" parents="1816"/>
      <j2 rule="demod" parents="2419 2419 2419 2419 2425 2419"/>
    </justification>
  </clause>

  <clause id="2501">
    <literal><![CDATA[
      x v c(c(x) v c(c(y v c(x v y)) v y)) = x
    ]]></literal>
    <justification jstring="[back_demod(1711),demod(2419(4),2425(3),2419(6),2425(7),2419(7),2419(10),2425(2))].">
      <j1 rule="back_demod" parents="1711"/>
      <j2 rule="demod" parents="2419 2425 2419 2425 2419 2419 2425"/>
    </justification>
  </clause>

  <clause id="2514">
    <literal><![CDATA[
      c(x v c(c(x v c(c(x) v x)) v x)) = c(x)
    ]]></literal>
    <justification jstring="[para(2486(a,1),1757(a,1)),flip(a)].">
      <j1 rule="para" parents="2486 1757"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2516">
    <literal><![CDATA[
      c(x v y) v c(x) = c(x)
    ]]></literal>
    <justification jstring="[para(2486(a,1),2419(a,2)),demod(2419(10),2514(11))].">
      <j1 rule="para" parents="2486 2419"/>
      <j2 rule="demod" parents="2419 2514"/>
    </justification>
  </clause>

  <clause id="2520">
    <literal><![CDATA[
      (c(c(x) v y) v c(y)) v c(y) = c(y)
    ]]></literal>
    <justification jstring="[back_demod(2448),demod(2514(14))].">
      <j1 rule="back_demod" parents="2448"/>
      <j2 rule="demod" parents="2514"/>
    </justification>
  </clause>

  <clause id="2521">
    <literal><![CDATA[
      c(c(x) v y) v x = x
    ]]></literal>
    <justification jstring="[para(2425(a,1),2516(a,1,2)),demod(2425(6))].">
      <j1 rule="para" parents="2425 2516"/>
      <j2 rule="demod" parents="2425"/>
    </justification>
  </clause>

  <clause id="2522">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(2516(a,1),2516(a,1,1,1)),demod(2425(2),2425(3),2425(5))].">
      <j1 rule="para" parents="2516 2516"/>
      <j2 rule="demod" parents="2425 2425 2425"/>
    </justification>
  </clause>

  <clause id="2527">
    <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(2434(a,1),2516(a,1,1,1))].">
      <j1 rule="para" parents="2434 2516"/>
    </justification>
  </clause>

  <clause id="2535">
    <literal><![CDATA[
      (c(c(x) v c(y)) v y) v y = y
    ]]></literal>
    <justification jstring="[para(2425(a,1),2520(a,1,1,2)),demod(2425(7),2425(8))].">
      <j1 rule="para" parents="2425 2520"/>
      <j2 rule="demod" parents="2425 2425"/>
    </justification>
  </clause>

  <clause id="2542">
    <literal><![CDATA[
      (c(x v c(y)) v y) v y = y
    ]]></literal>
    <justification jstring="[para(2425(a,1),2535(a,1,1,1,1,1))].">
      <j1 rule="para" parents="2425 2535"/>
    </justification>
  </clause>

  <clause id="2546">
    <literal><![CDATA[
      (c(x) v c(x v y)) v c(x v c(c(y v c(c(x