<?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="7">

<source>mckenzie.out</source>

<heading><![CDATA[
Prover9 (32) version November-2006, November 2006.
Process 3831 was started by mccune on cleo.thornwood,
Wed Nov 22 11:24:02 2006
The command was "/home/mccune/bin/prover9 -f mckenzie.in".
]]></heading>

<proof number="1" length="6" max_count="11">

<comments><![CDATA[
% Proof 1 at 0.00 (+ 0.00) seconds: absorb_1.
% Length of proof is 6.
% Level of proof is 2.
% Maximum clause weight is 11.
% Given clauses 3.
]]></comments>

  <clause id="5" type="goal">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <attribute><![CDATA[
      answer(absorb_1)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="9" type="assumption">
    <literal><![CDATA[
      x ^ (y v (x v z)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_2)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="10" type="assumption">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v y = y
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_3)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="16" type="deny">
    <literal><![CDATA[
      c11 ^ (c11 v c12) != c11
    ]]></literal>
    <attribute><![CDATA[
      answer(absorb_1)
    ]]></attribute>
    <justification jstring="[deny(5)].">
      <j1 rule="deny" parents="5"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),9(a,1,2))].">
      <j1 rule="para" parents="10 9"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(absorb_1)
    ]]></attribute>
    <justification jstring="[resolve(23,a,16,a)].">
      <j1 rule="resolve" parents="23 16"/>
    </justification>
  </clause>

</proof>

<proof number="2" length="6" max_count="11">

<comments><![CDATA[
% Proof 2 at 0.00 (+ 0.00) seconds: absorb_2.
% Length of proof is 6.
% Level of proof is 2.
% Maximum clause weight is 11.
% Given clauses 4.
]]></comments>

  <clause id="6" type="goal">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <attribute><![CDATA[
      answer(absorb_2)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="8" type="assumption">
    <literal><![CDATA[
      x v (y ^ (x ^ z)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_1)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11" type="assumption">
    <literal><![CDATA[
      ((x v y) ^ (y v z)) ^ y = y
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_4)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="17" type="deny">
    <literal><![CDATA[
      c13 v (c13 ^ c14) != c13
    ]]></literal>
    <attribute><![CDATA[
      answer(absorb_2)
    ]]></attribute>
    <justification jstring="[deny(6)].">
      <j1 rule="deny" parents="6"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),8(a,1,2))].">
      <j1 rule="para" parents="11 8"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(absorb_2)
    ]]></attribute>
    <justification jstring="[resolve(30,a,17,a)].">
      <j1 rule="resolve" parents="30 17"/>
    </justification>
  </clause>

</proof>

<proof number="3" length="17" max_count="15">

<comments><![CDATA[
% Proof 3 at 0.18 (+ 0.01) seconds: commute_meet.
% Length of proof is 17.
% Level of proof is 7.
% Maximum clause weight is 15.
% Given clauses 88.
]]></comments>

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

  <clause id="8" type="assumption">
    <literal><![CDATA[
      x v (y ^ (x ^ z)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_1)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="9" type="assumption">
    <literal><![CDATA[
      x ^ (y v (x v z)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_2)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="10" type="assumption">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v y = y
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_3)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11" type="assumption">
    <literal><![CDATA[
      ((x v y) ^ (y v z)) ^ y = y
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_4)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="14" type="deny">
    <literal><![CDATA[
      c7 ^ c6 != c6 ^ c7
    ]]></literal>
    <attribute><![CDATA[
      answer(commute_meet)
    ]]></attribute>
    <justification jstring="[deny(3)].">
      <j1 rule="deny" parents="3"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(9(a,1),8(a,1,2,2))].">
      <j1 rule="para" parents="9 8"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(8(a,1),9(a,1,2,2))].">
      <j1 rule="para" parents="8 9"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      (x ^ y) v y = y
    ]]></literal>
    <justification jstring="[para(8(a,1),10(a,1,1))].">
      <j1 rule="para" parents="8 10"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),8(a,1,2))].">
      <j1 rule="para" parents="11 8"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      (x ^ ((y ^ x) v z)) ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(19(a,1),11(a,1,1,1))].">
      <j1 rule="para" parents="19 11"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(20(a,1),21(a,1,1))].">
      <j1 rule="para" parents="20 21"/>
    </justification>
  </clause>

  <clause id="81">
    <literal><![CDATA[
      (x ^ y) v x = x
    ]]></literal>
    <justification jstring="[para(30(a,1),49(a,1,2)),rewrite(30(4))].">
      <j1 rule="para" parents="30 49"/>
      <j2 rule="rewrite" parents="30"/>
    </justification>
  </clause>

  <clause id="371">
    <literal><![CDATA[
      (x ^ y) ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(81(a,1),41(a,1,1,2))].">
      <j1 rule="para" parents="81 41"/>
    </justification>
  </clause>

  <clause id="1414">
    <literal><![CDATA[
      (x ^ y) v (y ^ x) = x ^ y
    ]]></literal>
    <justification jstring="[para(371(a,1),30(a,1,2))].">
      <j1 rule="para" parents="371 30"/>
    </justification>
  </clause>

  <clause id="1415">
    <literal><![CDATA[
      x ^ y = y ^ x
    ]]></literal>
    <justification jstring="[para(371(a,1),81(a,1,1)),rewrite(1414(3))].">
      <j1 rule="para" parents="371 81"/>
      <j2 rule="rewrite" parents="1414"/>
    </justification>
  </clause>

  <clause id="1416">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(commute_meet)
    ]]></attribute>
    <justification jstring="[resolve(1415,a,14,a)].">
      <j1 rule="resolve" parents="1415 14"/>
    </justification>
  </clause>

</proof>

<proof number="4" length="16" max_count="15">

<comments><![CDATA[
% Proof 4 at 0.30 (+ 0.01) seconds: commute_join.
% Length of proof is 16.
% Level of proof is 6.
% Maximum clause weight is 15.
% Given clauses 103.
]]></comments>

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

  <clause id="8" type="assumption">
    <literal><![CDATA[
      x v (y ^ (x ^ z)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_1)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="9" type="assumption">
    <literal><![CDATA[
      x ^ (y v (x v z)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_2)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="10" type="assumption">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v y = y
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_3)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11" type="assumption">
    <literal><![CDATA[
      ((x v y) ^ (y v z)) ^ y = y
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_4)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

  <clause id="19">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(9(a,1),8(a,1,2,2))].">
      <j1 rule="para" parents="9 8"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(8(a,1),9(a,1,2,2))].">
      <j1 rule="para" parents="8 9"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),9(a,1,2))].">
      <j1 rule="para" parents="10 9"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      (x v y) ^ y = y
    ]]></literal>
    <justification jstring="[para(9(a,1),11(a,1,1))].">
      <j1 rule="para" parents="9 11"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      (x v ((y v x) ^ z)) v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(20(a,1),10(a,1,1,1))].">
      <j1 rule="para" parents="20 10"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      (x v y) v x = x v y
    ]]></literal>
    <justification jstring="[para(23(a,1),19(a,1,2))].">
      <j1 rule="para" parents="23 19"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      (x v y) ^ x = x
    ]]></literal>
    <justification jstring="[para(52(a,1),34(a,1,1))].">
      <j1 rule="para" parents="52 34"/>
    </justification>
  </clause>

  <clause id="589">
    <literal><![CDATA[
      (x v y) v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(91(a,1),43(a,1,1,2))].">
      <j1 rule="para" parents="91 43"/>
    </justification>
  </clause>

  <clause id="1998">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[para(589(a,1),52(a,1,1)),rewrite(589(3),589(4))].">
      <j1 rule="para" parents="589 52"/>
      <j2 rule="rewrite" parents="589 589"/>
    </justification>
  </clause>

  <clause id="1999">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(commute_join)
    ]]></attribute>
    <justification jstring="[resolve(1998,a,12,a)].">
      <j1 rule="resolve" parents="1998 12"/>
    </justification>
  </clause>

</proof>

<proof number="5" length="42" max_count="21">

<comments><![CDATA[
% Proof 5 at 1.53 (+ 0.03) seconds: assoc_join.
% Length of proof is 42.
% Level of proof is 13.
% Maximum clause weight is 21.
% Given clauses 192.
]]></comments>

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

  <clause id="8" type="assumption">
    <literal><![CDATA[
      x v (y ^ (x ^ z)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_1)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="9" type="assumption">
    <literal><![CDATA[
      x ^ (y v (x v z)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_2)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="10" type="assumption">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v y = y
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_3)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11" type="assumption">
    <literal><![CDATA[
      ((x v y) ^ (y v z)) ^ y = y
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_4)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="13" type="deny">
    <literal><![CDATA[
      (c3 v c4) v c5 != c3 v (c4 v c5)
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc_join)
    ]]></attribute>
    <justification jstring="[deny(2)].">
      <j1 rule="deny" parents="2"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(9(a,1),8(a,1,2,2))].">
      <j1 rule="para" parents="9 8"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(8(a,1),9(a,1,2,2))].">
      <j1 rule="para" parents="8 9"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      (x ^ y) v y = y
    ]]></literal>
    <justification jstring="[para(8(a,1),10(a,1,1))].">
      <j1 rule="para" parents="8 10"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),9(a,1,2))].">
      <j1 rule="para" parents="10 9"/>
    </justification>
  </clause>

  <clause id="25">
    <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(9(a,1),10(a,1,1,1))].">
      <j1 rule="para" parents="9 10"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      (x v y) ^ y = y
    ]]></literal>
    <justification jstring="[para(9(a,1),11(a,1,1))].">
      <j1 rule="para" parents="9 11"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      (x v ((y v x) ^ z)) v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(20(a,1),10(a,1,1,1))].">
      <j1 rule="para" parents="20 10"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      x v (y v (x v z)) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(9(a,1),21(a,1,1))].">
      <j1 rule="para" parents="9 21"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      (x v ((x v y) ^ z)) v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(23(a,1),10(a,1,1,1))].">
      <j1 rule="para" parents="23 10"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      (x v y) v x = x v y
    ]]></literal>
    <justification jstring="[para(23(a,1),19(a,1,2))].">
      <j1 rule="para" parents="23 19"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(23(a,1),21(a,1,1))].">
      <j1 rule="para" parents="23 21"/>
    </justification>
  </clause>

  <clause id="71">
    <literal><![CDATA[
      x ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(19(a,1),34(a,1,1))].">
      <j1 rule="para" parents="19 34"/>
    </justification>
  </clause>

  <clause id="87">
    <literal><![CDATA[
      x ^ ((x v y) v z) = x
    ]]></literal>
    <justification jstring="[para(52(a,1),9(a,1,2))].">
      <j1 rule="para" parents="52 9"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      (x v y) ^ x = x
    ]]></literal>
    <justification jstring="[para(52(a,1),34(a,1,1))].">
      <j1 rule="para" parents="52 34"/>
    </justification>
  </clause>

  <clause id="98">
    <literal><![CDATA[
      (x v y) v (y v (x v z)) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(91(a,1),25(a,1,1,2))].">
      <j1 rule="para" parents="91 25"/>
    </justification>
  </clause>

  <clause id="125">
    <literal><![CDATA[
      (x v (y v z)) ^ y = y
    ]]></literal>
    <justification jstring="[para(9(a,1),71(a,1,2)),rewrite(9(6))].">
      <j1 rule="para" parents="9 71"/>
      <j2 rule="rewrite" parents="9"/>
    </justification>
  </clause>

  <clause id="165">
    <literal><![CDATA[
      ((x v y) v z) ^ x = x
    ]]></literal>
    <justification jstring="[para(87(a,1),71(a,1,2)),rewrite(87(6))].">
      <j1 rule="para" parents="87 71"/>
      <j2 rule="rewrite" parents="87"/>
    </justification>
  </clause>

  <clause id="278">
    <literal><![CDATA[
      (x v y) ^ (z ^ x) = z ^ x
    ]]></literal>
    <justification jstring="[para(21(a,1),165(a,1,1,1))].">
      <j1 rule="para" parents="21 165"/>
    </justification>
  </clause>

  <clause id="589">
    <literal><![CDATA[
      (x v y) v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(91(a,1),43(a,1,1,2))].">
      <j1 rule="para" parents="91 43"/>
    </justification>
  </clause>

  <clause id="598">
    <literal><![CDATA[
      (x v y) v ((y v z) v x) = (y v z) v x
    ]]></literal>
    <justification jstring="[para(165(a,1),43(a,1,1,2))].">
      <j1 rule="para" parents="165 43"/>
    </justification>
  </clause>

  <clause id="840">
    <literal><![CDATA[
      (x v (y ^ z)) v (z v x) = z v x
    ]]></literal>
    <justification jstring="[para(278(a,1),43(a,1,1,2))].">
      <j1 rule="para" parents="278 43"/>
    </justification>
  </clause>

  <clause id="869">
    <literal><![CDATA[
      (x v y) v (x v (y v z)) = x v (y v z)
    ]]></literal>
    <justification jstring="[para(125(a,1),50(a,1,1,2))].">
      <j1 rule="para" parents="125 50"/>
    </justification>
  </clause>

  <clause id="1996">
    <literal><![CDATA[
      (x v y) ^ (y v x) = x v y
    ]]></literal>
    <justification jstring="[para(589(a,1),23(a,1,2))].">
      <j1 rule="para" parents="589 23"/>
    </justification>
  </clause>

  <clause id="1998">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[para(589(a,1),52(a,1,1)),rewrite(589(3),589(4))].">
      <j1 rule="para" parents="589 52"/>
      <j2 rule="rewrite" parents="589 589"/>
    </justification>
  </clause>

  <clause id="2314">
    <literal><![CDATA[
      c5 v (c3 v c4) != c3 v (c4 v c5)
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc_join)
    ]]></attribute>
    <justification jstring="[back_rewrite(13),rewrite(1998(5))].">
      <j1 rule="back_rewrite" parents="13"/>
      <j2 rule="rewrite" parents="1998"/>
    </justification>
  </clause>

  <clause id="3523">
    <literal><![CDATA[
      (x v y) v ((x v z) v y) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(1998(a,1),98(a,1,2))].">
      <j1 rule="para" parents="1998 98"/>
    </justification>
  </clause>

  <clause id="6174">
    <literal><![CDATA[
      x v (y v (z ^ x)) = x v y
    ]]></literal>
    <justification jstring="[para(840(a,1),98(a,1,2)),rewrite(1998(5),869(5),840(7))].">
      <j1 rule="para" parents="840 98"/>
      <j2 rule="rewrite" parents="1998 869 840"/>
    </justification>
  </clause>

  <clause id="6231">
    <literal><![CDATA[
      (x v y) v (z v y) = (x v y) v z
    ]]></literal>
    <justification jstring="[para(20(a,1),6174(a,1,2,2))].">
      <j1 rule="para" parents="20 6174"/>
    </justification>
  </clause>

  <clause id="6232">
    <literal><![CDATA[
      (x v y) v (z v x) = (x v y) v z
    ]]></literal>
    <justification jstring="[para(23(a,1),6174(a,1,2,2))].">
      <j1 rule="para" parents="23 6174"/>
    </justification>
  </clause>

  <clause id="6253">
    <literal><![CDATA[
      x v ((y ^ x) v z) = x v z
    ]]></literal>
    <justification jstring="[para(1998(a,1),6174(a,1,2))].">
      <j1 rule="para" parents="1998 6174"/>
    </justification>
  </clause>

  <clause id="6292">
    <literal><![CDATA[
      (x v y) v (x v z) = y v (x v z)
    ]]></literal>
    <justification jstring="[back_rewrite(3523),rewrite(6231(4))].">
      <j1 rule="back_rewrite" parents="3523"/>
      <j2 rule="rewrite" parents="6231"/>
    </justification>
  </clause>

  <clause id="6338">
    <literal><![CDATA[
      (x v y) v (y v z) = (y v z) v x
    ]]></literal>
    <justification jstring="[back_rewrite(598),rewrite(6232(4))].">
      <j1 rule="back_rewrite" parents="598"/>
      <j2 rule="rewrite" parents="6232"/>
    </justification>
  </clause>

  <clause id="6502">
    <literal><![CDATA[
      (x v y) v z = y v (x v z)
    ]]></literal>
    <justification jstring="[para(23(a,1),6253(a,1,2,1)),rewrite(6292(3)),flip(a)].">
      <j1 rule="para" parents="23 6253"/>
      <j2 rule="rewrite" parents="6292"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="6534">
    <literal><![CDATA[
      x v (y v z) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(1996(a,1),6253(a,1,2,1)),rewrite(6502(3),6502(4),53(3),47(3),6502(4))].">
      <j1 rule="para" parents="1996 6253"/>
      <j2 rule="rewrite" parents="6502 6502 53 47 6502"/>
    </justification>
  </clause>

  <clause id="6577">
    <literal><![CDATA[
      x v (y v z) = z v (x v y)
    ]]></literal>
    <justification jstring="[back_rewrite(6338),rewrite(6534(3),6502(2),53(3),6502(4))].">
      <j1 rule="back_rewrite" parents="6338"/>
      <j2 rule="rewrite" parents="6534 6502 53 6502"/>
    </justification>
  </clause>

  <clause id="6578">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc_join)
    ]]></attribute>
    <justification jstring="[resolve(6577,a,2314,a(flip))].">
      <j1 rule="resolve" parents="6577 2314"/>
    </justification>
  </clause>

</proof>

<proof number="6" length="75" max_count="55">

<comments><![CDATA[
% Proof 6 at 3.16 (+ 0.04) seconds: all_six.
% Length of proof is 75.
% Level of proof is 14.
% Maximum clause weight is 50.
% Given clauses 239.
]]></comments>

  <clause id="7" type="goal">
    <literal><![CDATA[
      x v y = y v x & (x v y) v z = x v (y v z) & x ^ y = y ^ x & (x ^ y) ^ z = x ^ (y ^ z) & x ^ (x v y) = x & x v (x ^ y) = x
    ]]></literal>
    <attribute><![CDATA[
      label(all_six)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="8" type="assumption">
    <literal><![CDATA[
      x v (y ^ (x ^ z)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_1)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="9" type="assumption">
    <literal><![CDATA[
      x ^ (y v (x v z)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_2)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="10" type="assumption">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v y = y
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_3)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11" type="assumption">
    <literal><![CDATA[
      ((x v y) ^ (y v z)) ^ y = y
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_4)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="18" type="deny">
    <literal><![CDATA[
      c16 v c15 != c15 v c16
    ]]></literal>
    <literal><![CDATA[
      (c15 v c16) v c17 != c15 v (c16 v c17)
    ]]></literal>
    <literal><![CDATA[
      c16 ^ c15 != c15 ^ c16
    ]]></literal>
    <literal><![CDATA[
      (c15 ^ c16) ^ c17 != c15 ^ (c16 ^ c17)
    ]]></literal>
    <literal><![CDATA[
      c15 ^ (c15 v c16) != c15
    ]]></literal>
    <literal><![CDATA[
      c15 v (c15 ^ c16) != c15
    ]]></literal>
    <attribute><![CDATA[
      label(all_six)
    ]]></attribute>
    <attribute><![CDATA[
      answer(all_six)
    ]]></attribute>
    <justification jstring="[deny(7)].">
      <j1 rule="deny" parents="7"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(9(a,1),8(a,1,2,2))].">
      <j1 rule="para" parents="9 8"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(8(a,1),9(a,1,2,2))].">
      <j1 rule="para" parents="8 9"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      (x ^ y) v y = y
    ]]></literal>
    <justification jstring="[para(8(a,1),10(a,1,1))].">
      <j1 rule="para" parents="8 10"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),9(a,1,2))].">
      <j1 rule="para" parents="10 9"/>
    </justification>
  </clause>

  <clause id="25">
    <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(9(a,1),10(a,1,1,1))].">
      <j1 rule="para" parents="9 10"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      c16 v c15 != c15 v c16
    ]]></literal>
    <literal><![CDATA[
      (c15 v c16) v c17 != c15 v (c16 v c17)
    ]]></literal>
    <literal><![CDATA[
      c16 ^ c15 != c15 ^ c16
    ]]></literal>
    <literal><![CDATA[
      (c15 ^ c16) ^ c17 != c15 ^ (c16 ^ c17)
    ]]></literal>
    <literal><![CDATA[
      c15 v (c15 ^ c16) != c15
    ]]></literal>
    <attribute><![CDATA[
      answer(all_six)
    ]]></attribute>
    <justification jstring="[back_rewrite(18),rewrite(23(41)),xx(e)].">
      <j1 rule="back_rewrite" parents="18"/>
      <j2 rule="rewrite" parents="23"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),8(a,1,2))].">
      <j1 rule="para" parents="11 8"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      (x ^ ((y ^ (x ^ z)) v u)) ^ (y ^ (x ^ z)) = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(8(a,1),11(a,1,1,1))].">
      <j1 rule="para" parents="8 11"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      (x v y) ^ y = y
    ]]></literal>
    <justification jstring="[para(9(a,1),11(a,1,1))].">
      <j1 rule="para" parents="9 11"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      c16 v c15 != c15 v c16
    ]]></literal>
    <literal><![CDATA[
      (c15 v c16) v c17 != c15 v (c16 v c17)
    ]]></literal>
    <literal><![CDATA[
      c16 ^ c15 != c15 ^ c16
    ]]></literal>
    <literal><![CDATA[
      (c15 ^ c16) ^ c17 != c15 ^ (c16 ^ c17)
    ]]></literal>
    <attribute><![CDATA[
      answer(all_six)
    ]]></attribute>
    <justification jstring="[back_rewrite(27),rewrite(30(41)),xx(e)].">
      <j1 rule="back_rewrite" parents="27"/>
      <j2 rule="rewrite" parents="30"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      (x ^ ((y ^ x) v z)) ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(19(a,1),11(a,1,1,1))].">
      <j1 rule="para" parents="19 11"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      (x v ((y v x) ^ z)) v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(20(a,1),10(a,1,1,1))].">
      <j1 rule="para" parents="20 10"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      x v (y v (x v z)) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(9(a,1),21(a,1,1))].">
      <j1 rule="para" parents="9 21"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(20(a,1),21(a,1,1))].">
      <j1 rule="para" parents="20 21"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      (x v ((x v y) ^ z)) v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(23(a,1),10(a,1,1,1))].">
      <j1 rule="para" parents="23 10"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      (x v y) v x = x v y
    ]]></literal>
    <justification jstring="[para(23(a,1),19(a,1,2))].">
      <j1 rule="para" parents="23 19"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(23(a,1),21(a,1,1))].">
      <j1 rule="para" parents="23 21"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      (x ^ y) ^ x = x ^ y
    ]]></literal>
    <justification jstring="[para(30(a,1),20(a,1,2))].">
      <j1 rule="para" parents="30 20"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      x ^ (y ^ (x ^ z)) = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(8(a,1),34(a,1,1))].">
      <j1 rule="para" parents="8 34"/>
    </justification>
  </clause>

  <clause id="71">
    <literal><![CDATA[
      x ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(19(a,1),34(a,1,1))].">
      <j1 rule="para" parents="19 34"/>
    </justification>
  </clause>

  <clause id="78">
    <literal><![CDATA[
      (x ^ (y ^ z)) v y = y
    ]]></literal>
    <justification jstring="[para(8(a,1),49(a,1,2)),rewrite(8(6))].">
      <j1 rule="para" parents="8 49"/>
      <j2 rule="rewrite" parents="8"/>
    </justification>
  </clause>

  <clause id="81">
    <literal><![CDATA[
      (x ^ y) v x = x
    ]]></literal>
    <justification jstring="[para(30(a,1),49(a,1,2)),rewrite(30(4))].">
      <j1 rule="para" parents="30 49"/>
      <j2 rule="rewrite" parents="30"/>
    </justification>
  </clause>

  <clause id="87">
    <literal><![CDATA[
      x ^ ((x v y) v z) = x
    ]]></literal>
    <justification jstring="[para(52(a,1),9(a,1,2))].">
      <j1 rule="para" parents="52 9"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      (x v y) ^ x = x
    ]]></literal>
    <justification jstring="[para(52(a,1),34(a,1,1))].">
      <j1 rule="para" parents="52 34"/>
    </justification>
  </clause>

  <clause id="98">
    <literal><![CDATA[
      (x v y) v (y v (x v z)) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(91(a,1),25(a,1,1,2))].">
      <j1 rule="para" parents="91 25"/>
    </justification>
  </clause>

  <clause id="124">
    <literal><![CDATA[
      x v (y ^ (z ^ x)) = x
    ]]></literal>
    <justification jstring="[para(71(a,1),8(a,1,2,2))].">
      <j1 rule="para" parents="71 8"/>
    </justification>
  </clause>

  <clause id="125">
    <literal><![CDATA[
      (x v (y v z)) ^ y = y
    ]]></literal>
    <justification jstring="[para(9(a,1),71(a,1,2)),rewrite(9(6))].">
      <j1 rule="para" parents="9 71"/>
      <j2 rule="rewrite" parents="9"/>
    </justification>
  </clause>

  <clause id="136">
    <literal><![CDATA[
      (x ^ y) ^ (y ^ (x ^ z)) = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(81(a,1),32(a,1,1,2))].">
      <j1 rule="para" parents="81 32"/>
    </justification>
  </clause>

  <clause id="148">
    <literal><![CDATA[
      ((x ^ y) ^ z) v x = x
    ]]></literal>
    <justification jstring="[para(66(a,1),78(a,1,1))].">
      <j1 rule="para" parents="66 78"/>
    </justification>
  </clause>

  <clause id="165">
    <literal><![CDATA[
      ((x v y) v z) ^ x = x
    ]]></literal>
    <justification jstring="[para(87(a,1),71(a,1,2)),rewrite(87(6))].">
      <j1 rule="para" parents="87 71"/>
      <j2 rule="rewrite" parents="87"/>
    </justification>
  </clause>

  <clause id="200">
    <literal><![CDATA[
      x ^ (y ^ (z ^ x)) = y ^ (z ^ x)
    ]]></literal>
    <justification jstring="[para(124(a,1),34(a,1,1))].">
      <j1 rule="para" parents="124 34"/>
    </justification>
  </clause>

  <clause id="218">
    <literal><![CDATA[
      (x ^ y) v (z v x) = z v x
    ]]></literal>
    <justification jstring="[para(34(a,1),148(a,1,1,1))].">
      <j1 rule="para" parents="34 148"/>
    </justification>
  </clause>

  <clause id="220">
    <literal><![CDATA[
      (x ^ y) v (x v z) = x v z
    ]]></literal>
    <justification jstring="[para(91(a,1),148(a,1,1,1))].">
      <j1 rule="para" parents="91 148"/>
    </justification>
  </clause>

  <clause id="278">
    <literal><![CDATA[
      (x v y) ^ (z ^ x) = z ^ x
    ]]></literal>
    <justification jstring="[para(21(a,1),165(a,1,1,1))].">
      <j1 rule="para" parents="21 165"/>
    </justification>
  </clause>

  <clause id="371">
    <literal><![CDATA[
      (x ^ y) ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(81(a,1),41(a,1,1,2))].">
      <j1 rule="para" parents="81 41"/>
    </justification>
  </clause>

  <clause id="379">
    <literal><![CDATA[
      (x ^ y) ^ ((y ^ z) ^ x) = (y ^ z) ^ x
    ]]></literal>
    <justification jstring="[para(148(a,1),41(a,1,1,2))].">
      <j1 rule="para" parents="148 41"/>
    </justification>
  </clause>

  <clause id="589">
    <literal><![CDATA[
      (x v y) v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(91(a,1),43(a,1,1,2))].">
      <j1 rule="para" parents="91 43"/>
    </justification>
  </clause>

  <clause id="799">
    <literal><![CDATA[
      (x ^ (y v z)) ^ (z ^ x) = z ^ x
    ]]></literal>
    <justification jstring="[para(218(a,1),41(a,1,1,2))].">
      <j1 rule="para" parents="218 41"/>
    </justification>
  </clause>

  <clause id="815">
    <literal><![CDATA[
      (x ^ (y v z)) ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(220(a,1),41(a,1,1,2))].">
      <j1 rule="para" parents="220 41"/>
    </justification>
  </clause>

  <clause id="840">
    <literal><![CDATA[
      (x v (y ^ z)) v (z v x) = z v x
    ]]></literal>
    <justification jstring="[para(278(a,1),43(a,1,1,2))].">
      <j1 rule="para" parents="278 43"/>
    </justification>
  </clause>

  <clause id="869">
    <literal><![CDATA[
      (x v y) v (x v (y v z)) = x v (y v z)
    ]]></literal>
    <justification jstring="[para(125(a,1),50(a,1,1,2))].">
      <j1 rule="para" parents="125 50"/>
    </justification>
  </clause>

  <clause id="1414">
    <literal><![CDATA[
      (x ^ y) v (y ^ x) = x ^ y
    ]]></literal>
    <justification jstring="[para(371(a,1),30(a,1,2))].">
      <j1 rule="para" parents="371 30"/>
    </justification>
  </clause>

  <clause id="1415">
    <literal><![CDATA[
      x ^ y = y ^ x
    ]]></literal>
    <justification jstring="[para(371(a,1),81(a,1,1)),rewrite(1414(3))].">
      <j1 rule="para" parents="371 81"/>
      <j2 rule="rewrite" parents="1414"/>
    </justification>
  </clause>

  <clause id="1676">
    <literal><![CDATA[
      c16 v c15 != c15 v c16
    ]]></literal>
    <literal><![CDATA[
      (c15 v c16) v c17 != c15 v (c16 v c17)
    ]]></literal>
    <literal><![CDATA[
      c17 ^ (c15 ^ c16) != c15 ^ (c16 ^ c17)
    ]]></literal>
    <attribute><![CDATA[
      answer(all_six)
    ]]></attribute>
    <justification jstring="[back_rewrite(38),rewrite(1415(21),1415(30)),xx(c)].">
      <j1 rule="back_rewrite" parents="38"/>
      <j2 rule="rewrite" parents="1415 1415"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="1996">
    <literal><![CDATA[
      (x v y) ^ (y v x) = x v y
    ]]></literal>
    <justification jstring="[para(589(a,1),23(a,1,2))].">
      <j1 rule="para" parents="589 23"/>
    </justification>
  </clause>

  <clause id="1998">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[para(589(a,1),52(a,1,1)),rewrite(589(3),589(4))].">
      <j1 rule="para" parents="589 52"/>
      <j2 rule="rewrite" parents="589 589"/>
    </justification>
  </clause>

  <clause id="2155">
    <literal><![CDATA[
      c17 v (c15 v c16) != c15 v (c16 v c17)
    ]]></literal>
    <literal><![CDATA[
      c17 ^ (c15 ^ c16) != c15 ^ (c16 ^ c17)
    ]]></literal>
    <attribute><![CDATA[
      answer(all_six)
    ]]></attribute>
    <justification jstring="[back_rewrite(1676),rewrite(1998(3),1998(12)),xx(a)].">
      <j1 rule="back_rewrite" parents="1676"/>
      <j2 rule="rewrite" parents="1998 1998"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="3523">
    <literal><![CDATA[
      (x v y) v ((x v z) v y) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(1998(a,1),98(a,1,2))].">
      <j1 rule="para" parents="1998 98"/>
    </justification>
  </clause>

  <clause id="6174">
    <literal><![CDATA[
      x v (y v (z ^ x)) = x v y
    ]]></literal>
    <justification jstring="[para(840(a,1),98(a,1,2)),rewrite(1998(5),869(5),840(7))].">
      <j1 rule="para" parents="840 98"/>
      <j2 rule="rewrite" parents="1998 869 840"/>
    </justification>
  </clause>

  <clause id="6231">
    <literal><![CDATA[
      (x v y) v (z v y) = (x v y) v z
    ]]></literal>
    <justification jstring="[para(20(a,1),6174(a,1,2,2))].">
      <j1 rule="para" parents="20 6174"/>
    </justification>
  </clause>

  <clause id="6253">
    <literal><![CDATA[
      x v ((y ^ x) v z) = x v z
    ]]></literal>
    <justification jstring="[para(1998(a,1),6174(a,1,2))].">
      <j1 rule="para" parents="1998 6174"/>
    </justification>
  </clause>

  <clause id="6292">
    <literal><![CDATA[
      (x v y) v (x v z) = y v (x v z)
    ]]></literal>
    <justification jstring="[back_rewrite(3523),rewrite(6231(4))].">
      <j1 rule="back_rewrite" parents="3523"/>
      <j2 rule="rewrite" parents="6231"/>
    </justification>
  </clause>

  <clause id="6502">
    <literal><![CDATA[
      (x v y) v z = y v (x v z)
    ]]></literal>
    <justification jstring="[para(23(a,1),6253(a,1,2,1)),rewrite(6292(3)),flip(a)].">
      <j1 rule="para" parents="23 6253"/>
      <j2 rule="rewrite" parents="6292"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="6534">
    <literal><![CDATA[
      x v (y v z) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(1996(a,1),6253(a,1,2,1)),rewrite(6502(3),6502(4),53(3),47(3),6502(4))].">
      <j1 rule="para" parents="1996 6253"/>
      <j2 rule="rewrite" parents="6502 6502 53 47 6502"/>
    </justification>
  </clause>

  <clause id="7803">
    <literal><![CDATA[
      c17 ^ (c15 ^ c16) != c15 ^ (c16 ^ c17)
    ]]></literal>
    <attribute><![CDATA[
      answer(all_six)
    ]]></attribute>
    <justification jstring="[back_rewrite(2155),rewrite(6534(5),1998(4)),xx(a)].">
      <j1 rule="back_rewrite" parents="2155"/>
      <j2 rule="rewrite" parents="6534 1998"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="10239">
    <literal><![CDATA[
      (x ^ y) ^ (x ^ (y ^ z)) = x ^ (y ^ z)
    ]]></literal>
    <justification jstring="[para(1415(a,1),136(a,1,1))].">
      <j1 rule="para" parents="1415 136"/>
    </justification>
  </clause>

  <clause id="10273">
    <literal><![CDATA[
      x ^ (y ^ (z v x)) = x ^ y
    ]]></literal>
    <justification jstring="[para(799(a,1),136(a,1,2)),rewrite(1415(5),10239(5),799(7))].">
      <j1 rule="para" parents="799 136"/>
      <j2 rule="rewrite" parents="1415 10239 799"/>
    </justification>
  </clause>

  <clause id="10276">
    <literal><![CDATA[
      ((x ^ (y v z)) ^ u) ^ (u ^ (y ^ x)) = u ^ (y ^ x)
    ]]></literal>
    <justification jstring="[para(815(a,1),136(a,1,2,2)),rewrite(815(10))].">
      <j1 rule="para" parents="815 136"/>
      <j2 rule="rewrite" parents="815"/>
    </justification>
  </clause>

  <clause id="10277">
    <literal><![CDATA[
      x ^ (y ^ (x v z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(815(a,1),136(a,1,2)),rewrite(1415(5),10239(5),815(7))].">
      <j1 rule="para" parents="815 136"/>
      <j2 rule="rewrite" parents="1415 10239 815"/>
    </justification>
  </clause>

  <clause id="10301">
    <literal><![CDATA[
      (x ^ y) ^ (z ^ x) = (x ^ y) ^ z
    ]]></literal>
    <justification jstring="[para(30(a,1),10273(a,1,2,2))].">
      <j1 rule="para" parents="30 10273"/>
    </justification>
  </clause>

  <clause id="10310">
    <literal><![CDATA[
      x ^ ((y v x) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(1415(a,1),10273(a,1,2))].">
      <j1 rule="para" parents="1415 10273"/>
    </justification>
  </clause>

  <clause id="10482">
    <literal><![CDATA[
      (x ^ y) ^ (y ^ z) = (y ^ z) ^ x
    ]]></literal>
    <justification jstring="[back_rewrite(379),rewrite(10301(4))].">
      <j1 rule="back_rewrite" parents="379"/>
      <j2 rule="rewrite" parents="10301"/>
    </justification>
  </clause>

  <clause id="10485">
    <literal><![CDATA[
      x ^ ((x v y) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(1415(a,1),10277(a,1,2))].">
      <j1 rule="para" parents="1415 10277"/>
    </justification>
  </clause>

  <clause id="10500">
    <literal><![CDATA[
      (x ^ y) ^ (y ^ z) = (x ^ y) ^ z
    ]]></literal>
    <justification jstring="[para(19(a,1),10310(a,1,2,1))].">
      <j1 rule="para" parents="19 10310"/>
    </justification>
  </clause>

  <clause id="10501">
    <literal><![CDATA[
      (x ^ y) ^ (x ^ z) = (x ^ y) ^ z
    ]]></literal>
    <justification jstring="[para(30(a,1),10310(a,1,2,1))].">
      <j1 rule="para" parents="30 10310"/>
    </justification>
  </clause>

  <clause id="10555">
    <literal><![CDATA[
      (x ^ y) ^ z = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(10310(a,1),136(a,1,2,2)),rewrite(10500(4),10501(3),10310(5))].">
      <j1 rule="para" parents="10310 136"/>
      <j2 rule="rewrite" parents="10500 10501 10310"/>
    </justification>
  </clause>

  <clause id="10558">
    <literal><![CDATA[
      x ^ (y ^ z) = z ^ (y ^ x)
    ]]></literal>
    <justification jstring="[back_rewrite(10482),rewrite(10555(3),69(3),10555(4))].">
      <j1 rule="back_rewrite" parents="10482"/>
      <j2 rule="rewrite" parents="10555 69 10555"/>
    </justification>
  </clause>

  <clause id="10559">
    <literal><![CDATA[
      x ^ (y ^ z) = z ^ (x ^ y)
    ]]></literal>
    <justification jstring="[back_rewrite(10276),rewrite(10555(3),10558(6),200(5),10555(5),10485(4),69(3))].">
      <j1 rule="back_rewrite" parents="10276"/>
      <j2 rule="rewrite" parents="10555 10558 200 10555 10485 69"/>
    </justification>
  </clause>

  <clause id="10560">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(all_six)
    ]]></attribute>
    <justification jstring="[resolve(10559,a,7803,a(flip))].">
      <j1 rule="resolve" parents="10559 7803"/>
    </justification>
  </clause>

</proof>

<proof number="7" length="49" max_count="21">

<comments><![CDATA[
% Proof 7 at 3.16 (+ 0.04) seconds: assoc_meet.
% Length of proof is 49.
% Level of proof is 14.
% Maximum clause weight is 21.
% Given clauses 239.
]]></comments>

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

  <clause id="8" type="assumption">
    <literal><![CDATA[
      x v (y ^ (x ^ z)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_1)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="9" type="assumption">
    <literal><![CDATA[
      x ^ (y v (x v z)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_2)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="10" type="assumption">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v y = y
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_3)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11" type="assumption">
    <literal><![CDATA[
      ((x v y) ^ (y v z)) ^ y = y
    ]]></literal>
    <attribute><![CDATA[
      label(McKenzie_4)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="15" type="deny">
    <literal><![CDATA[
      (c8 ^ c9) ^ c10 != c8 ^ (c9 ^ c10)
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc_meet)
    ]]></attribute>
    <justification jstring="[deny(4)].">
      <j1 rule="deny" parents="4"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(9(a,1),8(a,1,2,2))].">
      <j1 rule="para" parents="9 8"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(8(a,1),9(a,1,2,2))].">
      <j1 rule="para" parents="8 9"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      (x ^ y) v y = y
    ]]></literal>
    <justification jstring="[para(8(a,1),10(a,1,1))].">
      <j1 rule="para" parents="8 10"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),9(a,1,2))].">
      <j1 rule="para" parents="10 9"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),8(a,1,2))].">
      <j1 rule="para" parents="11 8"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      (x ^ ((y ^ (x ^ z)) v u)) ^ (y ^ (x ^ z)) = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(8(a,1),11(a,1,1,1))].">
      <j1 rule="para" parents="8 11"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      (x v y) ^ y = y
    ]]></literal>
    <justification jstring="[para(9(a,1),11(a,1,1))].">
      <j1 rule="para" parents="9 11"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      (x ^ ((y ^ x) v z)) ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(19(a,1),11(a,1,1,1))].">
      <j1 rule="para" parents="19 11"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(20(a,1),21(a,1,1))].">
      <j1 rule="para" parents="20 21"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      (x v y) v x = x v y
    ]]></literal>
    <justification jstring="[para(23(a,1),19(a,1,2))].">
      <j1 rule="para" parents="23 19"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      (x ^ y) ^ x = x ^ y
    ]]></literal>
    <justification jstring="[para(30(a,1),20(a,1,2))].">
      <j1 rule="para" parents="30 20"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      x ^ (y ^ (x ^ z)) = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(8(a,1),34(a,1,1))].">
      <j1 rule="para" parents="8 34"/>
    </justification>
  </clause>

  <clause id="71">
    <literal><![CDATA[
      x ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(19(a,1),34(a,1,1))].">
      <j1 rule="para" parents="19 34"/>
    </justification>
  </clause>

  <clause id="78">
    <literal><![CDATA[
      (x ^ (y ^ z)) v y = y
    ]]></literal>
    <justification jstring="[para(8(a,1),49(a,1,2)),rewrite(8(6))].">
      <j1 rule="para" parents="8 49"/>
      <j2 rule="rewrite" parents="8"/>
    </justification>
  </clause>

  <clause id="81">
    <literal><![CDATA[
      (x ^ y) v x = x
    ]]></literal>
    <justification jstring="[para(30(a,1),49(a,1,2)),rewrite(30(4))].">
      <j1 rule="para" parents="30 49"/>
      <j2 rule="rewrite" parents="30"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      (x v y) ^ x = x
    ]]></literal>
    <justification jstring="[para(52(a,1),34(a,1,1))].">
      <j1 rule="para" parents="52 34"/>
    </justification>
  </clause>

  <clause id="124">
    <literal><![CDATA[
      x v (y ^ (z ^ x)) = x
    ]]></literal>
    <justification jstring="[para(71(a,1),8(a,1,2,2))].">
      <j1 rule="para" parents="71 8"/>
    </justification>
  </clause>

  <clause id="136">
    <literal><![CDATA[
      (x ^ y) ^ (y ^ (x ^ z)) = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(81(a,1),32(a,1,1,2))].">
      <j1 rule="para" parents="81 32"/>
    </justification>
  </clause>

  <clause id="148">
    <literal><![CDATA[
      ((x ^ y) ^ z) v x = x
    ]]></literal>
    <justification jstring="[para(66(a,1),78(a,1,1))].">
      <j1 rule="para" parents="66 78"/>
    </justification>
  </clause>

  <clause id="200">
    <literal><![CDATA[
      x ^ (y ^ (z ^ x)) = y ^ (z ^ x)
    ]]></literal>
    <justification jstring="[para(124(a,1),34(a,1,1))].">
      <j1 rule="para" parents="124 34"/>
    </justification>
  </clause>

  <clause id="218">
    <literal><![CDATA[
      (x ^ y) v (z v x) = z v x
    ]]></literal>
    <justification jstring="[para(34(a,1),148(a,1,1,1))].">
      <j1 rule="para" parents="34 148"/>
    </justification>
  </clause>

  <clause id="220">
    <literal><![CDATA[
      (x ^ y) v (x v z) = x v z
    ]]></literal>
    <justification jstring="[para(91(a,1),148(a,1,1,1))].">
      <j1 rule="para" parents="91 148"/>
    </justification>
  </clause>

  <clause id="371">
    <literal><![CDATA[
      (x ^ y) ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(81(a,1),41(a,1,1,2))].">
      <j1 rule="para" parents="81 41"/>
    </justification>
  </clause>

  <clause id="379">
    <literal><![CDATA[
      (x ^ y) ^ ((y ^ z) ^ x) = (y ^ z) ^ x
    ]]></literal>
    <justification jstring="[para(148(a,1),41(a,1,1,2))].">
      <j1 rule="para" parents="148 41"/>
    </justification>
  </clause>

  <clause id="799">
    <literal><![CDATA[
      (x ^ (y v z)) ^ (z ^ x) = z ^ x
    ]]></literal>
    <justification jstring="[para(218(a,1),41(a,1,1,2))].">
      <j1 rule="para" parents="218 41"/>
    </justification>
  </clause>

  <clause id="815">
    <literal><![CDATA[
      (x ^ (y v z)) ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(220(a,1),41(a,1,1,2))].">
      <j1 rule="para" parents="220 41"/>
    </justification>
  </clause>

  <clause id="1414">
    <literal><![CDATA[
      (x ^ y) v (y ^ x) = x ^ y
    ]]></literal>
    <justification jstring="[para(371(a,1),30(a,1,2))].">
      <j1 rule="para" parents="371 30"/>
    </justification>
  </clause>

  <clause id="1415">
    <literal><![CDATA[
      x ^ y = y ^ x
    ]]></literal>
    <justification jstring="[para(371(a,1),81(a,1,1)),rewrite(1414(3))].">
      <j1 rule="para" parents="371 81"/>
      <j2 rule="rewrite" parents="1414"/>
    </justification>
  </clause>

  <clause id="1677">
    <literal><![CDATA[
      c10 ^ (c8 ^ c9) != c8 ^ (c9 ^ c10)
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc_meet)
    ]]></attribute>
    <justification jstring="[back_rewrite(15),rewrite(1415(5))].">
      <j1 rule="back_rewrite" parents="15"/>
      <j2 rule="rewrite" parents="1415"/>
    </justification>
  </clause>

  <clause id="10239">
    <literal><![CDATA[
      (x ^ y) ^ (x ^ (y ^ z)) = x ^ (y ^ z)
    ]]></literal>
    <justification jstring="[para(1415(a,1),136(a,1,1))].">
      <j1 rule="para" parents="1415 136"/>
    </justification>
  </clause>

  <clause id="10273">
    <literal><![CDATA[
      x ^ (y ^ (z v x)) = x ^ y
    ]]></literal>
    <justification jstring="[para(799(a,1),136(a,1,2)),rewrite(1415(5),10239(5),799(7))].">
      <j1 rule="para" parents="799 136"/>
      <j2 rule="rewrite" parents="1415 10239 799"/>
    </justification>
  </clause>

  <clause id="10276">
    <literal><![CDATA[
      ((x ^ (y v z)) ^ u) ^ (u ^ (y ^ x)) = u ^ (y ^ x)
    ]]></literal>
    <justification jstring="[para(815(a,1),136(a,1,2,2)),rewrite(815(10))].">
      <j1 rule="para" parents="815 136"/>
      <j2 rule="rewrite" parents="815"/>
    </justification>
  </clause>

  <clause id="10277">
    <literal><![CDATA[
      x ^ (y ^ (x v z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(815(a,1),136(a,1,2)),rewrite(1415(5),10239(5),815(7))].">
      <j1 rule="para" parents="815 136"/>
      <j2 rule="rewrite" parents="1415 10239 815"/>
    </justification>
  </clause>

  <clause id="10301">
    <literal><![CDATA[
      (x ^ y) ^ (z ^ x) = (x ^ y) ^ z
    ]]></literal>
    <justification jstring="[para(30(a,1),10273(a,1,2,2))].">
      <j1 rule="para" parents="30 10273"/>
    </justification>
  </clause>

  <clause id="10310">
    <literal><![CDATA[
      x ^ ((y v x) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(1415(a,1),10273(a,1,2))].">
      <j1 rule="para" parents="1415 10273"/>
    </justification>
  </clause>

  <clause id="10482">
    <literal><![CDATA[
      (x ^ y) ^ (y ^ z) = (y ^ z) ^ x
    ]]></literal>
    <justification jstring="[back_rewrite(379),rewrite(10301(4))].">
      <j1 rule="back_rewrite" parents="379"/>
      <j2 rule="rewrite" parents="10301"/>
    </justification>
  </clause>

  <clause id="10485">
    <literal><![CDATA[
      x ^ ((x v y) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(1415(a,1),10277(a,1,2))].">
      <j1 rule="para" parents="1415 10277"/>
    </justification>
  </clause>

  <clause id="10500">
    <literal><![CDATA[
      (x ^ y) ^ (y ^ z) = (x ^ y) ^ z
    ]]></literal>
    <justification jstring="[para(19(a,1),10310(a,1,2,1))].">
      <j1 rule="para" parents="19 10310"/>
    </justification>
  </clause>

  <clause id="10501">
    <literal><![CDATA[
      (x ^ y) ^ (x ^ z) = (x ^ y) ^ z
    ]]></literal>
    <justification jstring="[para(30(a,1),10310(a,1,2,1))].">
      <j1 rule="para" parents="30 10310"/>
    </justification>
  </clause>

  <clause id="10555">
    <literal><![CDATA[
      (x ^ y) ^ z = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(10310(a,1),136(a,1,2,2)),rewrite(10500(4),10501(3),10310(5))].">
      <j1 rule="para" parents="10310 136"/>
      <j2 rule="rewrite" parents="10500 10501 10310"/>
    </justification>
  </clause>

  <clause id="10558">
    <literal><![CDATA[
      x ^ (y ^ z) = z ^ (y ^ x)
    ]]></literal>
    <justification jstring="[back_rewrite(10482),rewrite(10555(3),69(3),10555(4))].">
      <j1 rule="back_rewrite" parents="10482"/>
      <j2 rule="rewrite" parents="10555 69 10555"/>
    </justification>
  </clause>

  <clause id="10559">
    <literal><![CDATA[
      x ^ (y ^ z) = z ^ (x ^ y)
    ]]></literal>
    <justification jstring="[back_rewrite(10276),rewrite(10555(3),10558(6),200(5),10555(5),10485(4),69(3))].">
      <j1 rule="back_rewrite" parents="10276"/>
      <j2 rule="rewrite" parents="10555 10558 200 10555 10485 69"/>
    </justification>
  </clause>

  <clause id="10561">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc_meet)
    ]]></attribute>
    <justification jstring="[resolve(10559,a,1677,a(flip))].">
      <j1 rule="resolve" parents="10559 1677"/>
    </justification>
  </clause>

</proof>

</proofs>
