<?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 April-2007, April 2007.
Process 26924 was started by mccune on cleo,
Fri Apr 13 09:16:22 2007
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 0.17 (+ 0.00) seconds: commute_meet.
% Length of proof is 17.
% Level of proof is 7.
% Maximum clause weight is 15.
% Given clauses 83.
]]></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="356">
    <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="1261">
    <literal><![CDATA[
      (x ^ y) v (y ^ x) = x ^ y
    ]]></literal>
    <justification jstring="[para(356(a,1),30(a,1,2))].">
      <j1 rule="para" parents="356 30"/>
    </justification>
  </clause>

  <clause id="1262">
    <literal><![CDATA[
      x ^ y = y ^ x
    ]]></literal>
    <justification jstring="[para(356(a,1),81(a,1,1)),rewrite(1261(3))].">
      <j1 rule="para" parents="356 81"/>
      <j2 rule="rewrite" parents="1261"/>
    </justification>
  </clause>

  <clause id="1263">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(commute_meet)
    ]]></attribute>
    <justification jstring="[resolve(1262,a,14,a)].">
      <j1 rule="resolve" parents="1262 14"/>
    </justification>
  </clause>

</proof>

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

<comments><![CDATA[
% Proof 4 at 0.31 (+ 0.00) 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="61">
    <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(61(a,1),34(a,1,1))].">
      <j1 rule="para" parents="61 34"/>
    </justification>
  </clause>

  <clause id="564">
    <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="1933">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[para(564(a,1),61(a,1,1)),rewrite(564(3),564(4))].">
      <j1 rule="para" parents="564 61"/>
      <j2 rule="rewrite" parents="564 564"/>
    </justification>
  </clause>

  <clause id="1934">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(commute_join)
    ]]></attribute>
    <justification jstring="[resolve(1933,a,12,a)].">
      <j1 rule="resolve" parents="1933 12"/>
    </justification>
  </clause>

</proof>

<proof number="5" length="40" max_count="21">

<comments><![CDATA[
% Proof 5 at 1.84 (+ 0.02) seconds: assoc_join.
% Length of proof is 40.
% Level of proof is 13.
% Maximum clause weight is 21.
% Given clauses 198.
]]></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="61">
    <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="62">
    <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(61(a,1),9(a,1,2))].">
      <j1 rule="para" parents="61 9"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      (x v y) ^ x = x
    ]]></literal>
    <justification jstring="[para(61(a,1),34(a,1,1))].">
      <j1 rule="para" parents="61 34"/>
    </justification>
  </clause>

  <clause id="117">
    <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="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="564">
    <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="572">
    <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="839">
    <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="1931">
    <literal><![CDATA[
      (x v y) ^ (y v x) = x v y
    ]]></literal>
    <justification jstring="[para(564(a,1),23(a,1,2))].">
      <j1 rule="para" parents="564 23"/>
    </justification>
  </clause>

  <clause id="1933">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[para(564(a,1),61(a,1,1)),rewrite(564(3),564(4))].">
      <j1 rule="para" parents="564 61"/>
      <j2 rule="rewrite" parents="564 564"/>
    </justification>
  </clause>

  <clause id="2249">
    <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(1933(5))].">
      <j1 rule="back_rewrite" parents="13"/>
      <j2 rule="rewrite" parents="1933"/>
    </justification>
  </clause>

  <clause id="6927">
    <literal><![CDATA[
      (x v y) v (x v (y v z)) = x v (y v z)
    ]]></literal>
    <justification jstring="[para(47(a,1),117(a,1,2)),rewrite(47(7))].">
      <j1 rule="para" parents="47 117"/>
      <j2 rule="rewrite" parents="47"/>
    </justification>
  </clause>

  <clause id="6945">
    <literal><![CDATA[
      (x v y) v ((x v z) v y) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(1933(a,1),117(a,1,2))].">
      <j1 rule="para" parents="1933 117"/>
    </justification>
  </clause>

  <clause id="7006">
    <literal><![CDATA[
      x v (y v (z ^ x)) = x v y
    ]]></literal>
    <justification jstring="[para(839(a,1),117(a,1,2)),rewrite(1933(5),6927(5),839(7))].">
      <j1 rule="para" parents="839 117"/>
      <j2 rule="rewrite" parents="1933 6927 839"/>
    </justification>
  </clause>

  <clause id="7052">
    <literal><![CDATA[
      (x v y) v (z v y) = (x v y) v z
    ]]></literal>
    <justification jstring="[para(20(a,1),7006(a,1,2,2))].">
      <j1 rule="para" parents="20 7006"/>
    </justification>
  </clause>

  <clause id="7054">
    <literal><![CDATA[
      (x v y) v (z v x) = (x v y) v z
    ]]></literal>
    <justification jstring="[para(23(a,1),7006(a,1,2,2))].">
      <j1 rule="para" parents="23 7006"/>
    </justification>
  </clause>

  <clause id="7074">
    <literal><![CDATA[
      x v ((y ^ x) v z) = x v z
    ]]></literal>
    <justification jstring="[para(1933(a,1),7006(a,1,2))].">
      <j1 rule="para" parents="1933 7006"/>
    </justification>
  </clause>

  <clause id="7119">
    <literal><![CDATA[
      (x v y) v (x v z) = y v (x v z)
    ]]></literal>
    <justification jstring="[back_rewrite(6945),rewrite(7052(4))].">
      <j1 rule="back_rewrite" parents="6945"/>
      <j2 rule="rewrite" parents="7052"/>
    </justification>
  </clause>

  <clause id="7177">
    <literal><![CDATA[
      (x v y) v (y v z) = (y v z) v x
    ]]></literal>
    <justification jstring="[back_rewrite(572),rewrite(7054(4))].">
      <j1 rule="back_rewrite" parents="572"/>
      <j2 rule="rewrite" parents="7054"/>
    </justification>
  </clause>

  <clause id="7204">
    <literal><![CDATA[
      (x v y) v z = y v (x v z)
    ]]></literal>
    <justification jstring="[para(23(a,1),7074(a,1,2,1)),rewrite(7119(3)),flip(a)].">
      <j1 rule="para" parents="23 7074"/>
      <j2 rule="rewrite" parents="7119"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="7237">
    <literal><![CDATA[
      x v (y v z) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(1931(a,1),7074(a,1,2,1)),rewrite(7204(3),7204(4),62(3),47(3),7204(4))].">
      <j1 rule="para" parents="1931 7074"/>
      <j2 rule="rewrite" parents="7204 7204 62 47 7204"/>
    </justification>
  </clause>

  <clause id="7279">
    <literal><![CDATA[
      x v (y v z) = z v (x v y)
    ]]></literal>
    <justification jstring="[back_rewrite(7177),rewrite(7237(3),7204(2),62(3),7204(4))].">
      <j1 rule="back_rewrite" parents="7177"/>
      <j2 rule="rewrite" parents="7237 7204 62 7204"/>
    </justification>
  </clause>

  <clause id="7280">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc_join)
    ]]></attribute>
    <justification jstring="[resolve(7279,a,2249,a(flip))].">
      <j1 rule="resolve" parents="7279 2249"/>
    </justification>
  </clause>

</proof>

<proof number="6" length="73" max_count="55">

<comments><![CDATA[
% Proof 6 at 3.35 (+ 0.04) seconds: all_six.
% Length of proof is 73.
% Level of proof is 14.
% Maximum clause weight is 50.
% Given clauses 233.
]]></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="61">
    <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="62">
    <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(61(a,1),9(a,1,2))].">
      <j1 rule="para" parents="61 9"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      (x v y) ^ x = x
    ]]></literal>
    <justification jstring="[para(61(a,1),34(a,1,1))].">
      <j1 rule="para" parents="61 34"/>
    </justification>
  </clause>

  <clause id="117">
    <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="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="221">
    <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="356">
    <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="364">
    <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="564">
    <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(221(a,1),41(a,1,1,2))].">
      <j1 rule="para" parents="221 41"/>
    </justification>
  </clause>

  <clause id="839">
    <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="1261">
    <literal><![CDATA[
      (x ^ y) v (y ^ x) = x ^ y
    ]]></literal>
    <justification jstring="[para(356(a,1),30(a,1,2))].">
      <j1 rule="para" parents="356 30"/>
    </justification>
  </clause>

  <clause id="1262">
    <literal><![CDATA[
      x ^ y = y ^ x
    ]]></literal>
    <justification jstring="[para(356(a,1),81(a,1,1)),rewrite(1261(3))].">
      <j1 rule="para" parents="356 81"/>
      <j2 rule="rewrite" parents="1261"/>
    </justification>
  </clause>

  <clause id="1472">
    <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(1262(21),1262(30)),xx(c)].">
      <j1 rule="back_rewrite" parents="38"/>
      <j2 rule="rewrite" parents="1262 1262"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="1931">
    <literal><![CDATA[
      (x v y) ^ (y v x) = x v y
    ]]></literal>
    <justification jstring="[para(564(a,1),23(a,1,2))].">
      <j1 rule="para" parents="564 23"/>
    </justification>
  </clause>

  <clause id="1933">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[para(564(a,1),61(a,1,1)),rewrite(564(3),564(4))].">
      <j1 rule="para" parents="564 61"/>
      <j2 rule="rewrite" parents="564 564"/>
    </justification>
  </clause>

  <clause id="2070">
    <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(1472),rewrite(1933(3),1933(12)),xx(a)].">
      <j1 rule="back_rewrite" parents="1472"/>
      <j2 rule="rewrite" parents="1933 1933"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="6927">
    <literal><![CDATA[
      (x v y) v (x v (y v z)) = x v (y v z)
    ]]></literal>
    <justification jstring="[para(47(a,1),117(a,1,2)),rewrite(47(7))].">
      <j1 rule="para" parents="47 117"/>
      <j2 rule="rewrite" parents="47"/>
    </justification>
  </clause>

  <clause id="6945">
    <literal><![CDATA[
      (x v y) v ((x v z) v y) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(1933(a,1),117(a,1,2))].">
      <j1 rule="para" parents="1933 117"/>
    </justification>
  </clause>

  <clause id="7006">
    <literal><![CDATA[
      x v (y v (z ^ x)) = x v y
    ]]></literal>
    <justification jstring="[para(839(a,1),117(a,1,2)),rewrite(1933(5),6927(5),839(7))].">
      <j1 rule="para" parents="839 117"/>
      <j2 rule="rewrite" parents="1933 6927 839"/>
    </justification>
  </clause>

  <clause id="7052">
    <literal><![CDATA[
      (x v y) v (z v y) = (x v y) v z
    ]]></literal>
    <justification jstring="[para(20(a,1),7006(a,1,2,2))].">
      <j1 rule="para" parents="20 7006"/>
    </justification>
  </clause>

  <clause id="7074">
    <literal><![CDATA[
      x v ((y ^ x) v z) = x v z
    ]]></literal>
    <justification jstring="[para(1933(a,1),7006(a,1,2))].">
      <j1 rule="para" parents="1933 7006"/>
    </justification>
  </clause>

  <clause id="7119">
    <literal><![CDATA[
      (x v y) v (x v z) = y v (x v z)
    ]]></literal>
    <justification jstring="[back_rewrite(6945),rewrite(7052(4))].">
      <j1 rule="back_rewrite" parents="6945"/>
      <j2 rule="rewrite" parents="7052"/>
    </justification>
  </clause>

  <clause id="7204">
    <literal><![CDATA[
      (x v y) v z = y v (x v z)
    ]]></literal>
    <justification jstring="[para(23(a,1),7074(a,1,2,1)),rewrite(7119(3)),flip(a)].">
      <j1 rule="para" parents="23 7074"/>
      <j2 rule="rewrite" parents="7119"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="7237">
    <literal><![CDATA[
      x v (y v z) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(1931(a,1),7074(a,1,2,1)),rewrite(7204(3),7204(4),62(3),47(3),7204(4))].">
      <j1 rule="para" parents="1931 7074"/>
      <j2 rule="rewrite" parents="7204 7204 62 47 7204"/>
    </justification>
  </clause>

  <clause id="8618">
    <literal><![CDATA[
      c17 ^ (c15 ^ c16) != c15 ^ (c16 ^ c17)
    ]]></literal>
    <attribute><![CDATA[
      answer(all_six)
    ]]></attribute>
    <justification jstring="[back_rewrite(2070),rewrite(7237(5),1933(4)),xx(a)].">
      <j1 rule="back_rewrite" parents="2070"/>
      <j2 rule="rewrite" parents="7237 1933"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="10395">
    <literal><![CDATA[
      (x ^ y) ^ (x ^ (y ^ z)) = x ^ (y ^ z)
    ]]></literal>
    <justification jstring="[para(1262(a,1),136(a,1,1))].">
      <j1 rule="para" parents="1262 136"/>
    </justification>
  </clause>

  <clause id="10430">
    <literal><![CDATA[
      x ^ (y ^ (z v x)) = x ^ y
    ]]></literal>
    <justification jstring="[para(799(a,1),136(a,1,2)),rewrite(1262(5),10395(5),799(7))].">
      <j1 rule="para" parents="799 136"/>
      <j2 rule="rewrite" parents="1262 10395 799"/>
    </justification>
  </clause>

  <clause id="10431">
    <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="10432">
    <literal><![CDATA[
      x ^ (y ^ (x v z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(815(a,1),136(a,1,2)),rewrite(1262(5),10395(5),815(7))].">
      <j1 rule="para" parents="815 136"/>
      <j2 rule="rewrite" parents="1262 10395 815"/>
    </justification>
  </clause>

  <clause id="10463">
    <literal><![CDATA[
      (x ^ y) ^ (z ^ x) = (x ^ y) ^ z
    ]]></literal>
    <justification jstring="[para(30(a,1),10430(a,1,2,2))].">
      <j1 rule="para" parents="30 10430"/>
    </justification>
  </clause>

  <clause id="10468">
    <literal><![CDATA[
      x ^ ((y v x) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(1262(a,1),10430(a,1,2))].">
      <j1 rule="para" parents="1262 10430"/>
    </justification>
  </clause>

  <clause id="10625">
    <literal><![CDATA[
      (x ^ y) ^ (y ^ z) = (y ^ z) ^ x
    ]]></literal>
    <justification jstring="[back_rewrite(364),rewrite(10463(4))].">
      <j1 rule="back_rewrite" parents="364"/>
      <j2 rule="rewrite" parents="10463"/>
    </justification>
  </clause>

  <clause id="10627">
    <literal><![CDATA[
      x ^ ((x v y) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(1262(a,1),10432(a,1,2))].">
      <j1 rule="para" parents="1262 10432"/>
    </justification>
  </clause>

  <clause id="10644">
    <literal><![CDATA[
      (x ^ y) ^ (y ^ z) = (x ^ y) ^ z
    ]]></literal>
    <justification jstring="[para(19(a,1),10468(a,1,2,1))].">
      <j1 rule="para" parents="19 10468"/>
    </justification>
  </clause>

  <clause id="10645">
    <literal><![CDATA[
      (x ^ y) ^ (x ^ z) = (x ^ y) ^ z
    ]]></literal>
    <justification jstring="[para(30(a,1),10468(a,1,2,1))].">
      <j1 rule="para" parents="30 10468"/>
    </justification>
  </clause>

  <clause id="10692">
    <literal><![CDATA[
      (x ^ y) ^ z = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(10468(a,1),136(a,1,2,2)),rewrite(10644(4),10645(3),10468(5))].">
      <j1 rule="para" parents="10468 136"/>
      <j2 rule="rewrite" parents="10644 10645 10468"/>
    </justification>
  </clause>

  <clause id="10695">
    <literal><![CDATA[
      x ^ (y ^ z) = z ^ (y ^ x)
    ]]></literal>
    <justification jstring="[back_rewrite(10625),rewrite(10692(3),69(3),10692(4))].">
      <j1 rule="back_rewrite" parents="10625"/>
      <j2 rule="rewrite" parents="10692 69 10692"/>
    </justification>
  </clause>

  <clause id="10696">
    <literal><![CDATA[
      x ^ (y ^ z) = z ^ (x ^ y)
    ]]></literal>
    <justification jstring="[back_rewrite(10431),rewrite(10692(3),10695(6),200(5),10692(5),10627(4),69(3))].">
      <j1 rule="back_rewrite" parents="10431"/>
      <j2 rule="rewrite" parents="10692 10695 200 10692 10627 69"/>
    </justification>
  </clause>

  <clause id="10697">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(all_six)
    ]]></attribute>
    <justification jstring="[resolve(10696,a,8618,a(flip))].">
      <j1 rule="resolve" parents="10696 8618"/>
    </justification>
  </clause>

</proof>

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

<comments><![CDATA[
% Proof 7 at 3.35 (+ 0.04) seconds: assoc_meet.
% Length of proof is 49.
% Level of proof is 14.
% Maximum clause weight is 21.
% Given clauses 233.
]]></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="61">
    <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(61(a,1),34(a,1,1))].">
      <j1 rule="para" parents="61 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="221">
    <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="356">
    <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="364">
    <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(221(a,1),41(a,1,1,2))].">
      <j1 rule="para" parents="221 41"/>
    </justification>
  </clause>

  <clause id="1261">
    <literal><![CDATA[
      (x ^ y) v (y ^ x) = x ^ y
    ]]></literal>
    <justification jstring="[para(356(a,1),30(a,1,2))].">
      <j1 rule="para" parents="356 30"/>
    </justification>
  </clause>

  <clause id="1262">
    <literal><![CDATA[
      x ^ y = y ^ x
    ]]></literal>
    <justification jstring="[para(356(a,1),81(a,1,1)),rewrite(1261(3))].">
      <j1 rule="para" parents="356 81"/>
      <j2 rule="rewrite" parents="1261"/>
    </justification>
  </clause>

  <clause id="1473">
    <literal><![CDATA[
      c10 ^ (c8 ^ c9) != c8 ^ (c9 ^ c10)
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc_meet)
    ]]></attribute>
    <justification jstring="[back_rewrite(15),rewrite(1262(5))].">
      <j1 rule="back_rewrite" parents="15"/>
      <j2 rule="rewrite" parents="1262"/>
    </justification>
  </clause>

  <clause id="10395">
    <literal><![CDATA[
      (x ^ y) ^ (x ^ (y ^ z)) = x ^ (y ^ z)
    ]]></literal>
    <justification jstring="[para(1262(a,1),136(a,1,1))].">
      <j1 rule="para" parents="1262 136"/>
    </justification>
  </clause>

  <clause id="10430">
    <literal><![CDATA[
      x ^ (y ^ (z v x)) = x ^ y
    ]]></literal>
    <justification jstring="[para(799(a,1),136(a,1,2)),rewrite(1262(5),10395(5),799(7))].">
      <j1 rule="para" parents="799 136"/>
      <j2 rule="rewrite" parents="1262 10395 799"/>
    </justification>
  </clause>

  <clause id="10431">
    <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="10432">
    <literal><![CDATA[
      x ^ (y ^ (x v z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(815(a,1),136(a,1,2)),rewrite(1262(5),10395(5),815(7))].">
      <j1 rule="para" parents="815 136"/>
      <j2 rule="rewrite" parents="1262 10395 815"/>
    </justification>
  </clause>

  <clause id="10463">
    <literal><![CDATA[
      (x ^ y) ^ (z ^ x) = (x ^ y) ^ z
    ]]></literal>
    <justification jstring="[para(30(a,1),10430(a,1,2,2))].">
      <j1 rule="para" parents="30 10430"/>
    </justification>
  </clause>

  <clause id="10468">
    <literal><![CDATA[
      x ^ ((y v x) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(1262(a,1),10430(a,1,2))].">
      <j1 rule="para" parents="1262 10430"/>
    </justification>
  </clause>

  <clause id="10625">
    <literal><![CDATA[
      (x ^ y) ^ (y ^ z) = (y ^ z) ^ x
    ]]></literal>
    <justification jstring="[back_rewrite(364),rewrite(10463(4))].">
      <j1 rule="back_rewrite" parents="364"/>
      <j2 rule="rewrite" parents="10463"/>
    </justification>
  </clause>

  <clause id="10627">
    <literal><![CDATA[
      x ^ ((x v y) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(1262(a,1),10432(a,1,2))].">
      <j1 rule="para" parents="1262 10432"/>
    </justification>
  </clause>

  <clause id="10644">
    <literal><![CDATA[
      (x ^ y) ^ (y ^ z) = (x ^ y) ^ z
    ]]></literal>
    <justification jstring="[para(19(a,1),10468(a,1,2,1))].">
      <j1 rule="para" parents="19 10468"/>
    </justification>
  </clause>

  <clause id="10645">
    <literal><![CDATA[
      (x ^ y) ^ (x ^ z) = (x ^ y) ^ z
    ]]></literal>
    <justification jstring="[para(30(a,1),10468(a,1,2,1))].">
      <j1 rule="para" parents="30 10468"/>
    </justification>
  </clause>

  <clause id="10692">
    <literal><![CDATA[
      (x ^ y) ^ z = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(10468(a,1),136(a,1,2,2)),rewrite(10644(4),10645(3),10468(5))].">
      <j1 rule="para" parents="10468 136"/>
      <j2 rule="rewrite" parents="10644 10645 10468"/>
    </justification>
  </clause>

  <clause id="10695">
    <literal><![CDATA[
      x ^ (y ^ z) = z ^ (y ^ x)
    ]]></literal>
    <justification jstring="[back_rewrite(10625),rewrite(10692(3),69(3),10692(4))].">
      <j1 rule="back_rewrite" parents="10625"/>
      <j2 rule="rewrite" parents="10692 69 10692"/>
    </justification>
  </clause>

  <clause id="10696">
    <literal><![CDATA[
      x ^ (y ^ z) = z ^ (x ^ y)
    ]]></literal>
    <justification jstring="[back_rewrite(10431),rewrite(10692(3),10695(6),200(5),10692(5),10627(4),69(3))].">
      <j1 rule="back_rewrite" parents="10431"/>
      <j2 rule="rewrite" parents="10692 10695 200 10692 10627 69"/>
    </justification>
  </clause>

  <clause id="10698">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc_meet)
    ]]></attribute>
    <justification jstring="[resolve(10696,a,1473,a(flip))].">
      <j1 rule="resolve" parents="10696 1473"/>
    </justification>
  </clause>

</proof>

</proofs>
