<?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="6">

<source>mckenzie.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13095 was started by mccune on cleo.thornwood,
Mon Jun 19 16:41:20 2006
The command was "/home/mccune/bin/prover9 -f mckenzie.in".
]]></heading>

<proof number="1" length="5" max_count="11">

<comments><![CDATA[
% Proof 1 at 0.00 (+ 0.00) seconds: absorb_1.
% Length of proof is 5.
% Level of proof is 2.
% Maximum clause weight is 11.
% Given clauses 3.
]]></comments>

  <clause id="12">
    <literal><![CDATA[
      x ^ (y v (x v z)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_2)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v y = y
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_3)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      c11 ^ (c11 v c12) != c11
    ]]></literal>
    <attribute><![CDATA[
      answer(absorb_1)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[para(13(a,1),12(a,1,2))].">
      <j1 rule="para" parents="13 12"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(absorb_1)
    ]]></attribute>
    <justification jstring="[resolve(25,a,19,a)].">
      <j1 rule="resolve" parents="25 19"/>
    </justification>
  </clause>

</proof>

<proof number="2" length="5" max_count="11">

<comments><![CDATA[
% Proof 2 at 0.00 (+ 0.00) seconds: absorb_2.
% Length of proof is 5.
% Level of proof is 2.
% Maximum clause weight is 11.
% Given clauses 4.
]]></comments>

  <clause id="11">
    <literal><![CDATA[
      x v (y ^ (x ^ z)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_1)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      ((x v y) ^ (y v z)) ^ y = y
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_4)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      c13 v (c13 ^ c14) != c13
    ]]></literal>
    <attribute><![CDATA[
      answer(absorb_2)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[para(14(a,1),11(a,1,2))].">
      <j1 rule="para" parents="14 11"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(absorb_2)
    ]]></attribute>
    <justification jstring="[resolve(30,a,20,a)].">
      <j1 rule="resolve" parents="30 20"/>
    </justification>
  </clause>

</proof>

<proof number="3" length="16" max_count="15">

<comments><![CDATA[
% Proof 3 at 0.17 (+ 0.01) seconds: commute_meet.
% Length of proof is 16.
% Level of proof is 7.
% Maximum clause weight is 15.
% Given clauses 88.
]]></comments>

  <clause id="11">
    <literal><![CDATA[
      x v (y ^ (x ^ z)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_1)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      x ^ (y v (x v z)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_2)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v y = y
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_3)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      ((x v y) ^ (y v z)) ^ y = y
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_4)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      c7 ^ c6 != c6 ^ c7
    ]]></literal>
    <attribute><![CDATA[
      answer(commute_meet)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(12(a,1),11(a,1,2,2))].">
      <j1 rule="para" parents="12 11"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),12(a,1,2,2))].">
      <j1 rule="para" parents="11 12"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      (x ^ y) v y = y
    ]]></literal>
    <justification jstring="[para(11(a,1),13(a,1,1))].">
      <j1 rule="para" parents="11 13"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[para(14(a,1),11(a,1,2))].">
      <j1 rule="para" parents="14 11"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      (x ^ ((y ^ x) v z)) ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(21(a,1),14(a,1,1,1))].">
      <j1 rule="para" parents="21 14"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(22(a,1),23(a,1,1))].">
      <j1 rule="para" parents="22 23"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      (x ^ y) v x = x
    ]]></literal>
    <justification jstring="[para(30(a,1),47(a,1,2)),demod(30(4))].">
      <j1 rule="para" parents="30 47"/>
      <j2 rule="demod" parents="30"/>
    </justification>
  </clause>

  <clause id="369">
    <literal><![CDATA[
      (x ^ y) ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(79(a,1),39(a,1,1,2))].">
      <j1 rule="para" parents="79 39"/>
    </justification>
  </clause>

  <clause id="1412">
    <literal><![CDATA[
      (x ^ y) v (y ^ x) = x ^ y
    ]]></literal>
    <justification jstring="[para(369(a,1),30(a,1,2))].">
      <j1 rule="para" parents="369 30"/>
    </justification>
  </clause>

  <clause id="1413">
    <literal><![CDATA[
      x ^ y = y ^ x
    ]]></literal>
    <justification jstring="[para(369(a,1),79(a,1,1)),demod(1412(3))].">
      <j1 rule="para" parents="369 79"/>
      <j2 rule="demod" parents="1412"/>
    </justification>
  </clause>

  <clause id="1414">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(commute_meet)
    ]]></attribute>
    <justification jstring="[resolve(1413,a,17,a)].">
      <j1 rule="resolve" parents="1413 17"/>
    </justification>
  </clause>

</proof>

<proof number="4" length="15" max_count="15">

<comments><![CDATA[
% Proof 4 at 0.29 (+ 0.01) seconds: commute_join.
% Length of proof is 15.
% Level of proof is 6.
% Maximum clause weight is 15.
% Given clauses 103.
]]></comments>

  <clause id="11">
    <literal><![CDATA[
      x v (y ^ (x ^ z)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_1)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      x ^ (y v (x v z)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_2)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v y = y
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_3)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      ((x v y) ^ (y v z)) ^ y = y
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_4)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      c2 v c1 != c1 v c2
    ]]></literal>
    <attribute><![CDATA[
      answer(commute_join)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(12(a,1),11(a,1,2,2))].">
      <j1 rule="para" parents="12 11"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),12(a,1,2,2))].">
      <j1 rule="para" parents="11 12"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[para(13(a,1),12(a,1,2))].">
      <j1 rule="para" parents="13 12"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      (x v y) ^ y = y
    ]]></literal>
    <justification jstring="[para(12(a,1),14(a,1,1))].">
      <j1 rule="para" parents="12 14"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      (x v ((y v x) ^ z)) v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(22(a,1),13(a,1,1,1))].">
      <j1 rule="para" parents="22 13"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      (x v y) v x = x v y
    ]]></literal>
    <justification jstring="[para(25(a,1),21(a,1,2))].">
      <j1 rule="para" parents="25 21"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      (x v y) ^ x = x
    ]]></literal>
    <justification jstring="[para(50(a,1),34(a,1,1))].">
      <j1 rule="para" parents="50 34"/>
    </justification>
  </clause>

  <clause id="587">
    <literal><![CDATA[
      (x v y) v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(89(a,1),41(a,1,1,2))].">
      <j1 rule="para" parents="89 41"/>
    </justification>
  </clause>

  <clause id="1994">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[para(587(a,1),50(a,1,1)),demod(587(3),587(4))].">
      <j1 rule="para" parents="587 50"/>
      <j2 rule="demod" parents="587 587"/>
    </justification>
  </clause>

  <clause id="1995">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(commute_join)
    ]]></attribute>
    <justification jstring="[resolve(1994,a,15,a)].">
      <j1 rule="resolve" parents="1994 15"/>
    </justification>
  </clause>

</proof>

<proof number="5" length="37" max_count="21">

<comments><![CDATA[
% Proof 5 at 1.87 (+ 0.01) seconds: assoc_join.
% Length of proof is 37.
% Level of proof is 12.
% Maximum clause weight is 21.
% Given clauses 192.
]]></comments>

  <clause id="11">
    <literal><![CDATA[
      x v (y ^ (x ^ z)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_1)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      x ^ (y v (x v z)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_2)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v y = y
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_3)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      ((x v y) ^ (y v z)) ^ y = y
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_4)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      (c3 v c4) v c5 != c3 v (c4 v c5)
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc_join)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(12(a,1),11(a,1,2,2))].">
      <j1 rule="para" parents="12 11"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),12(a,1,2,2))].">
      <j1 rule="para" parents="11 12"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      (x ^ y) v y = y
    ]]></literal>
    <justification jstring="[para(11(a,1),13(a,1,1))].">
      <j1 rule="para" parents="11 13"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[para(13(a,1),12(a,1,2))].">
      <j1 rule="para" parents="13 12"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      (x v ((y v (x v z)) ^ u)) v (y v (x v z)) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(12(a,1),13(a,1,1,1))].">
      <j1 rule="para" parents="12 13"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      (x v y) ^ y = y
    ]]></literal>
    <justification jstring="[para(12(a,1),14(a,1,1))].">
      <j1 rule="para" parents="12 14"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      (x v ((y v x) ^ z)) v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(22(a,1),13(a,1,1,1))].">
      <j1 rule="para" parents="22 13"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      x v (y v (x v z)) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(12(a,1),23(a,1,1))].">
      <j1 rule="para" parents="12 23"/>
    </justification>
  </clause>

  <clause id="48">
    <literal><![CDATA[
      (x v ((x v y) ^ z)) v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(25(a,1),13(a,1,1,1))].">
      <j1 rule="para" parents="25 13"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      (x v y) v x = x v y
    ]]></literal>
    <justification jstring="[para(25(a,1),21(a,1,2))].">
      <j1 rule="para" parents="25 21"/>
    </justification>
  </clause>

  <clause id="51">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(25(a,1),23(a,1,1))].">
      <j1 rule="para" parents="25 23"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      x ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(21(a,1),34(a,1,1))].">
      <j1 rule="para" parents="21 34"/>
    </justification>
  </clause>

  <clause id="85">
    <literal><![CDATA[
      x ^ ((x v y) v z) = x
    ]]></literal>
    <justification jstring="[para(50(a,1),12(a,1,2))].">
      <j1 rule="para" parents="50 12"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      (x v y) ^ x = x
    ]]></literal>
    <justification jstring="[para(50(a,1),34(a,1,1))].">
      <j1 rule="para" parents="50 34"/>
    </justification>
  </clause>

  <clause id="96">
    <literal><![CDATA[
      (x v y) v (y v (x v z)) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(89(a,1),27(a,1,1,2))].">
      <j1 rule="para" parents="89 27"/>
    </justification>
  </clause>

  <clause id="123">
    <literal><![CDATA[
      (x v (y v z)) ^ y = y
    ]]></literal>
    <justification jstring="[para(12(a,1),69(a,1,2)),demod(12(6))].">
      <j1 rule="para" parents="12 69"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="163">
    <literal><![CDATA[
      ((x v y) v z) ^ x = x
    ]]></literal>
    <justification jstring="[para(85(a,1),69(a,1,2)),demod(85(6))].">
      <j1 rule="para" parents="85 69"/>
      <j2 rule="demod" parents="85"/>
    </justification>
  </clause>

  <clause id="276">
    <literal><![CDATA[
      (x v y) ^ (z ^ x) = z ^ x
    ]]></literal>
    <justification jstring="[para(23(a,1),163(a,1,1,1))].">
      <j1 rule="para" parents="23 163"/>
    </justification>
  </clause>

  <clause id="587">
    <literal><![CDATA[
      (x v y) v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(89(a,1),41(a,1,1,2))].">
      <j1 rule="para" parents="89 41"/>
    </justification>
  </clause>

  <clause id="838">
    <literal><![CDATA[
      (x v (y ^ z)) v (z v x) = z v x
    ]]></literal>
    <justification jstring="[para(276(a,1),41(a,1,1,2))].">
      <j1 rule="para" parents="276 41"/>
    </justification>
  </clause>

  <clause id="867">
    <literal><![CDATA[
      (x v y) v (x v (y v z)) = x v (y v z)
    ]]></literal>
    <justification jstring="[para(123(a,1),48(a,1,1,2))].">
      <j1 rule="para" parents="123 48"/>
    </justification>
  </clause>

  <clause id="1992">
    <literal><![CDATA[
      (x v y) ^ (y v x) = x v y
    ]]></literal>
    <justification jstring="[para(587(a,1),25(a,1,2))].">
      <j1 rule="para" parents="587 25"/>
    </justification>
  </clause>

  <clause id="1994">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[para(587(a,1),50(a,1,1)),demod(587(3),587(4))].">
      <j1 rule="para" parents="587 50"/>
      <j2 rule="demod" parents="587 587"/>
    </justification>
  </clause>

  <clause id="2309">
    <literal><![CDATA[
      c5 v (c3 v c4) != c3 v (c4 v c5)
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc_join)
    ]]></attribute>
    <justification jstring="[back_demod(16),demod(1994(5))].">
      <j1 rule="back_demod" parents="16"/>
      <j2 rule="demod" parents="1994"/>
    </justification>
  </clause>

  <clause id="3517">
    <literal><![CDATA[
      (x v y) v ((x v z) v y) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(1994(a,1),96(a,1,2))].">
      <j1 rule="para" parents="1994 96"/>
    </justification>
  </clause>

  <clause id="6168">
    <literal><![CDATA[
      x v (y v (z ^ x)) = x v y
    ]]></literal>
    <justification jstring="[para(838(a,1),96(a,1,2)),demod(1994(5),867(5),838(7))].">
      <j1 rule="para" parents="838 96"/>
      <j2 rule="demod" parents="1994 867 838"/>
    </justification>
  </clause>

  <clause id="6225">
    <literal><![CDATA[
      (x v y) v (z v y) = (x v y) v z
    ]]></literal>
    <justification jstring="[para(22(a,1),6168(a,1,2,2))].">
      <j1 rule="para" parents="22 6168"/>
    </justification>
  </clause>

  <clause id="6247">
    <literal><![CDATA[
      x v ((y ^ x) v z) = x v z
    ]]></literal>
    <justification jstring="[para(1994(a,1),6168(a,1,2))].">
      <j1 rule="para" parents="1994 6168"/>
    </justification>
  </clause>

  <clause id="6286">
    <literal><![CDATA[
      (x v y) v (x v z) = y v (x v z)
    ]]></literal>
    <justification jstring="[back_demod(3517),demod(6225(4))].">
      <j1 rule="back_demod" parents="3517"/>
      <j2 rule="demod" parents="6225"/>
    </justification>
  </clause>

  <clause id="6496">
    <literal><![CDATA[
      (x v y) v z = y v (x v z)
    ]]></literal>
    <justification jstring="[para(25(a,1),6247(a,1,2,1)),demod(6286(3)),flip(a)].">
      <j1 rule="para" parents="25 6247"/>
      <j2 rule="demod" parents="6286"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="6523">
    <literal><![CDATA[
      x v (y v z) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(1992(a,1),6247(a,1,2,1)),demod(6496(3),6496(4),51(3),45(3),6496(4))].">
      <j1 rule="para" parents="1992 6247"/>
      <j2 rule="demod" parents="6496 6496 51 45 6496"/>
    </justification>
  </clause>

  <clause id="7778">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc_join)
    ]]></attribute>
    <justification jstring="[back_demod(2309),demod(6523(5),1994(4)),xx(a)].">
      <j1 rule="back_demod" parents="2309"/>
      <j2 rule="demod" parents="6523 1994"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

<proof number="6" length="33" max_count="21">

<comments><![CDATA[
% Proof 6 at 3.70 (+ 0.01) seconds: assoc_meet.
% Length of proof is 33.
% Level of proof is 13.
% Maximum clause weight is 21.
% Given clauses 239.
]]></comments>

  <clause id="11">
    <literal><![CDATA[
      x v (y ^ (x ^ z)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_1)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      x ^ (y v (x v z)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_2)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v y = y
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_3)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      ((x v y) ^ (y v z)) ^ y = y
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_4)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      (c8 ^ c9) ^ c10 != c8 ^ (c9 ^ c10)
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc_meet)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(12(a,1),11(a,1,2,2))].">
      <j1 rule="para" parents="12 11"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),12(a,1,2,2))].">
      <j1 rule="para" parents="11 12"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      (x ^ y) v y = y
    ]]></literal>
    <justification jstring="[para(11(a,1),13(a,1,1))].">
      <j1 rule="para" parents="11 13"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[para(14(a,1),11(a,1,2))].">
      <j1 rule="para" parents="14 11"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      (x ^ ((y ^ (x ^ z)) v u)) ^ (y ^ (x ^ z)) = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(11(a,1),14(a,1,1,1))].">
      <j1 rule="para" parents="11 14"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      (x v y) ^ y = y
    ]]></literal>
    <justification jstring="[para(12(a,1),14(a,1,1))].">
      <j1 rule="para" parents="12 14"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      (x ^ ((y ^ x) v z)) ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(21(a,1),14(a,1,1,1))].">
      <j1 rule="para" parents="21 14"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(22(a,1),23(a,1,1))].">
      <j1 rule="para" parents="22 23"/>
    </justification>
  </clause>

  <clause id="64">
    <literal><![CDATA[
      (x ^ y) ^ x = x ^ y
    ]]></literal>
    <justification jstring="[para(30(a,1),22(a,1,2))].">
      <j1 rule="para" parents="30 22"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      x ^ (y ^ (x ^ z)) = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(11(a,1),34(a,1,1))].">
      <j1 rule="para" parents="11 34"/>
    </justification>
  </clause>

  <clause id="76">
    <literal><![CDATA[
      (x ^ (y ^ z)) v y = y
    ]]></literal>
    <justification jstring="[para(11(a,1),47(a,1,2)),demod(11(6))].">
      <j1 rule="para" parents="11 47"/>
      <j2 rule="demod" parents="11"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      (x ^ y) v x = x
    ]]></literal>
    <justification jstring="[para(30(a,1),47(a,1,2)),demod(30(4))].">
      <j1 rule="para" parents="30 47"/>
      <j2 rule="demod" parents="30"/>
    </justification>
  </clause>

  <clause id="134">
    <literal><![CDATA[
      (x ^ y) ^ (y ^ (x ^ z)) = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(79(a,1),32(a,1,1,2))].">
      <j1 rule="para" parents="79 32"/>
    </justification>
  </clause>

  <clause id="146">
    <literal><![CDATA[
      ((x ^ y) ^ z) v x = x
    ]]></literal>
    <justification jstring="[para(64(a,1),76(a,1,1))].">
      <j1 rule="para" parents="64 76"/>
    </justification>
  </clause>

  <clause id="216">
    <literal><![CDATA[
      (x ^ y) v (z v x) = z v x
    ]]></literal>
    <justification jstring="[para(34(a,1),146(a,1,1,1))].">
      <j1 rule="para" parents="34 146"/>
    </justification>
  </clause>

  <clause id="369">
    <literal><![CDATA[
      (x ^ y) ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(79(a,1),39(a,1,1,2))].">
      <j1 rule="para" parents="79 39"/>
    </justification>
  </clause>

  <clause id="797">
    <literal><![CDATA[
      (x ^ (y v z)) ^ (z ^ x) = z ^ x
    ]]></literal>
    <justification jstring="[para(216(a,1),39(a,1,1,2))].">
      <j1 rule="para" parents="216 39"/>
    </justification>
  </clause>

  <clause id="1412">
    <literal><![CDATA[
      (x ^ y) v (y ^ x) = x ^ y
    ]]></literal>
    <justification jstring="[para(369(a,1),30(a,1,2))].">
      <j1 rule="para" parents="369 30"/>
    </justification>
  </clause>

  <clause id="1413">
    <literal><![CDATA[
      x ^ y = y ^ x
    ]]></literal>
    <justification jstring="[para(369(a,1),79(a,1,1)),demod(1412(3))].">
      <j1 rule="para" parents="369 79"/>
      <j2 rule="demod" parents="1412"/>
    </justification>
  </clause>

  <clause id="1674">
    <literal><![CDATA[
      c10 ^ (c8 ^ c9) != c8 ^ (c9 ^ c10)
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc_meet)
    ]]></attribute>
    <justification jstring="[back_demod(18),demod(1413(5))].">
      <j1 rule="back_demod" parents="18"/>
      <j2 rule="demod" parents="1413"/>
    </justification>
  </clause>

  <clause id="10246">
    <literal><![CDATA[
      (x ^ y) ^ (x ^ (y ^ z)) = x ^ (y ^ z)
    ]]></literal>
    <justification jstring="[para(1413(a,1),134(a,1,1))].">
      <j1 rule="para" parents="1413 134"/>
    </justification>
  </clause>

  <clause id="10282">
    <literal><![CDATA[
      x ^ (y ^ (z v x)) = x ^ y
    ]]></literal>
    <justification jstring="[para(797(a,1),134(a,1,2)),demod(1413(5),10246(5),797(7))].">
      <j1 rule="para" parents="797 134"/>
      <j2 rule="demod" parents="1413 10246 797"/>
    </justification>
  </clause>

  <clause id="10320">
    <literal><![CDATA[
      x ^ ((y v x) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(1413(a,1),10282(a,1,2))].">
      <j1 rule="para" parents="1413 10282"/>
    </justification>
  </clause>

  <clause id="10518">
    <literal><![CDATA[
      (x ^ y) ^ (y ^ z) = (x ^ y) ^ z
    ]]></literal>
    <justification jstring="[para(21(a,1),10320(a,1,2,1))].">
      <j1 rule="para" parents="21 10320"/>
    </justification>
  </clause>

  <clause id="10519">
    <literal><![CDATA[
      (x ^ y) ^ (x ^ z) = (x ^ y) ^ z
    ]]></literal>
    <justification jstring="[para(30(a,1),10320(a,1,2,1))].">
      <j1 rule="para" parents="30 10320"/>
    </justification>
  </clause>

  <clause id="10572">
    <literal><![CDATA[
      (x ^ y) ^ z = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(10320(a,1),134(a,1,2,2)),demod(10518(4),10519(3),10320(5))].">
      <j1 rule="para" parents="10320 134"/>
      <j2 rule="demod" parents="10518 10519 10320"/>
    </justification>
  </clause>

  <clause id="10608">
    <literal><![CDATA[
      x ^ (y ^ z) = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[back_demod(10518),demod(10572(3),67(3),10572(4))].">
      <j1 rule="back_demod" parents="10518"/>
      <j2 rule="demod" parents="10572 67 10572"/>
    </justification>
  </clause>

  <clause id="11921">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc_meet)
    ]]></attribute>
    <justification jstring="[back_demod(1674),demod(10608(5),1413(4)),xx(a)].">
      <j1 rule="back_demod" parents="1674"/>
      <j2 rule="demod" parents="10608 1413"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
