<?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="4">

<source>a1.out</source>

<heading><![CDATA[
Prover9 (32) version November-2006, November 2006.
Process 3829 was started by mccune on cleo.thornwood,
Wed Nov 22 11:23:53 2006
The command was "/home/mccune/bin/prover9 -f a1.in".
]]></heading>

<proof number="1" length="52" max_count="83">

<comments><![CDATA[
% Proof 1 at 2.26 (+ 0.01) seconds: McKenzie_2.
% Length of proof is 52.
% Level of proof is 27.
% Maximum clause weight is 83.
% Given clauses 140.
]]></comments>

  <clause id="2" type="goal">
    <literal><![CDATA[
      x ^ (y v (x v z)) = x
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie_2)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="5" type="assumption">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ (y v y)) v (u ^ y)) ^ v)) ^ (w v ((v6 v y) ^ (y v v7))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(A1)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7" type="deny">
    <literal><![CDATA[
      c4 ^ (c5 v (c4 v c6)) != c4
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie_2)
    ]]></attribute>
    <justification jstring="[deny(2)].">
      <j1 rule="deny" parents="2"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      (((x v ((y v z) ^ (z v u))) ^ ((y v z) ^ (z v u))) v ((z v (v ^ ((y v z) ^ (z v u)))) ^ w)) ^ (v6 v ((v7 v ((y v z) ^ (z v u))) ^ (((y v z) ^ (z v u)) v v8))) = (y v z) ^ (z v u)
    ]]></literal>
    <justification jstring="[para(5(a,1),5(a,1,1,2,1,1))].">
      <j1 rule="para" parents="5 5"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      (((x v y) ^ y) v (y v y)) ^ (z v ((u v y) ^ (y v v))) = y
    ]]></literal>
    <justification jstring="[para(5(a,1),5(a,1,1,2))].">
      <j1 rule="para" parents="5 5"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      (((x v (((y ^ (z v z)) v (u ^ z)) ^ v)) ^ (((y ^ (z v z)) v (u ^ z)) ^ v)) v (((w ^ ((((y ^ (z v z)) v (u ^ z)) ^ v) v (((y ^ (z v z)) v (u ^ z)) ^ v))) v (v6 ^ (((y ^ (z v z)) v (u ^ z)) ^ v))) ^ v7)) ^ (v8 v z) = ((y ^ (z v z)) v (u ^ z)) ^ v
    ]]></literal>
    <justification jstring="[para(5(a,1),5(a,1,2,2))].">
      <j1 rule="para" parents="5 5"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((y v y) v (y v y))) ^ (z v y) = y v y
    ]]></literal>
    <justification jstring="[para(12(a,1),12(a,1,2,2))].">
      <j1 rule="para" parents="12 12"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      (((x v y) ^ y) v (((y v y) v (z ^ y)) ^ u)) ^ (v v ((w v y) ^ (y v v6))) = y
    ]]></literal>
    <justification jstring="[para(16(a,1),5(a,1,1,2,1,1))].">
      <j1 rule="para" parents="16 5"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((((y v y) v (y v y)) v (z ^ (y v y))) ^ u)) ^ (v v y) = y v y
    ]]></literal>
    <justification jstring="[para(12(a,1),17(a,1,2,2))].">
      <j1 rule="para" parents="12 17"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((((y v y) v (y v y)) v (y v y)) ^ z)) ^ (u v y) = y v y
    ]]></literal>
    <justification jstring="[para(16(a,1),23(a,1,1,2,1,2))].">
      <j1 rule="para" parents="16 23"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      (((x v y) ^ y) v (((((y v y) v (z ^ y)) ^ u) v (v ^ y)) ^ w)) ^ (v6 v ((v7 v y) ^ (y v v8))) = y
    ]]></literal>
    <justification jstring="[para(17(a,1),10(a,1,1,1,1,2)),rewrite(17(17),17(22),17(26),17(27),17(31))].">
      <j1 rule="para" parents="17 10"/>
      <j2 rule="rewrite" parents="17 17 17 17 17"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ (u v ((v v y) ^ (y v w))) = y
    ]]></literal>
    <justification jstring="[para(5(a,1),33(a,1,1,2))].">
      <j1 rule="para" parents="5 33"/>
    </justification>
  </clause>

  <clause id="56">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (z ^ (y v y))) ^ (u v y) = y v y
    ]]></literal>
    <justification jstring="[para(12(a,1),43(a,1,2,2))].">
      <j1 rule="para" parents="12 43"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      (((x v (y ^ z)) ^ (y ^ z)) v (u ^ (y ^ z))) ^ (v v z) = y ^ z
    ]]></literal>
    <justification jstring="[para(43(a,1),43(a,1,2,2))].">
      <j1 rule="para" parents="43 43"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (y v y)) ^ (z v y) = y v y
    ]]></literal>
    <justification jstring="[para(16(a,1),56(a,1,1,2))].">
      <j1 rule="para" parents="16 56"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      ((x v x) v ((x v x) v (x v x))) ^ (y v ((z v (x v x)) ^ ((x v x) v u))) = x v x
    ]]></literal>
    <justification jstring="[para(68(a,1),12(a,1,1,1))].">
      <j1 rule="para" parents="68 12"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      ((x v x) v (y ^ (x v x))) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(68(a,1),56(a,1,1,1))].">
      <j1 rule="para" parents="68 56"/>
    </justification>
  </clause>

  <clause id="93">
    <literal><![CDATA[
      ((x v x) v (x v x)) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[para(68(a,1),68(a,1,1,1))].">
      <j1 rule="para" parents="68 68"/>
    </justification>
  </clause>

  <clause id="102">
    <literal><![CDATA[
      ((x v x) v (((y ^ (x v x)) v (z ^ x)) ^ u)) ^ (v v x) = x v x
    ]]></literal>
    <justification jstring="[para(13(a,1),91(a,1,1,2))].">
      <j1 rule="para" parents="13 91"/>
    </justification>
  </clause>

  <clause id="147">
    <literal><![CDATA[
      ((x v x) v (((y ^ x) v (z ^ x)) ^ u)) ^ (v v x) = x v x
    ]]></literal>
    <justification jstring="[para(66(a,1),102(a,1,1,2,1,1))].">
      <j1 rule="para" parents="66 102"/>
    </justification>
  </clause>

  <clause id="170">
    <literal><![CDATA[
      ((x v x) v x) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[para(5(a,1),147(a,1,1,2))].">
      <j1 rule="para" parents="5 147"/>
    </justification>
  </clause>

  <clause id="184">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ (y v y)) v (u ^ y)) ^ v)) ^ (w v (y v y)) = y
    ]]></literal>
    <justification jstring="[para(170(a,1),5(a,1,2,2))].">
      <j1 rule="para" parents="170 5"/>
    </justification>
  </clause>

  <clause id="604">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ y) v (u ^ y)) ^ v)) ^ (w v (y v y)) = y
    ]]></literal>
    <justification jstring="[para(66(a,1),184(a,1,1,2,1,1))].">
      <j1 rule="para" parents="66 184"/>
    </justification>
  </clause>

  <clause id="623">
    <literal><![CDATA[
      (((x v y) ^ y) v y) ^ (z v (y v y)) = y
    ]]></literal>
    <justification jstring="[para(5(a,1),604(a,1,1,2))].">
      <j1 rule="para" parents="5 604"/>
    </justification>
  </clause>

  <clause id="651">
    <literal><![CDATA[
      (((x v y) ^ y) v (y v y)) ^ (z v y) = y
    ]]></literal>
    <justification jstring="[para(623(a,1),12(a,1,2,2))].">
      <j1 rule="para" parents="623 12"/>
    </justification>
  </clause>

  <clause id="656">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ (u v y) = y
    ]]></literal>
    <justification jstring="[para(623(a,1),43(a,1,2,2))].">
      <j1 rule="para" parents="623 43"/>
    </justification>
  </clause>

  <clause id="669">
    <literal><![CDATA[
      (x v ((((x v x) v (x v x)) v (y ^ (x v x))) ^ z)) ^ (u v x) = x v x
    ]]></literal>
    <justification jstring="[para(651(a,1),23(a,1,1,1))].">
      <j1 rule="para" parents="651 23"/>
    </justification>
  </clause>

  <clause id="671">
    <literal><![CDATA[
      (x v ((((x v x) v (x v x)) v (x v x)) ^ y)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(651(a,1),27(a,1,1,1))].">
      <j1 rule="para" parents="651 27"/>
    </justification>
  </clause>

  <clause id="681">
    <literal><![CDATA[
      (x v (x v x)) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[para(651(a,1),68(a,1,1,1))].">
      <j1 rule="para" parents="651 68"/>
    </justification>
  </clause>

  <clause id="682">
    <literal><![CDATA[
      (x v x) v (x v x) = x v x
    ]]></literal>
    <justification jstring="[para(68(a,1),651(a,1,1,1)),rewrite(681(8))].">
      <j1 rule="para" parents="68 651"/>
      <j2 rule="rewrite" parents="681"/>
    </justification>
  </clause>

  <clause id="745">
    <literal><![CDATA[
      (x v ((x v x) ^ y)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(671),rewrite(682(3),682(3))].">
      <j1 rule="back_rewrite" parents="671"/>
      <j2 rule="rewrite" parents="682 682"/>
    </justification>
  </clause>

  <clause id="747">
    <literal><![CDATA[
      (x v (((x v x) v (y ^ (x v x))) ^ z)) ^ (u v x) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(669),rewrite(682(3))].">
      <j1 rule="back_rewrite" parents="669"/>
      <j2 rule="rewrite" parents="682"/>
    </justification>
  </clause>

  <clause id="791">
    <literal><![CDATA[
      (x v x) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(93),rewrite(682(3))].">
      <j1 rule="back_rewrite" parents="93"/>
      <j2 rule="rewrite" parents="682"/>
    </justification>
  </clause>

  <clause id="796">
    <literal><![CDATA[
      (x v x) ^ (y v ((z v (x v x)) ^ ((x v x) v u))) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(79),rewrite(682(4),682(3))].">
      <j1 rule="back_rewrite" parents="79"/>
      <j2 rule="rewrite" parents="682 682"/>
    </justification>
  </clause>

  <clause id="889">
    <literal><![CDATA[
      ((x v x) v ((y ^ x) v (y ^ x))) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(791(a,1),147(a,1,1,2))].">
      <j1 rule="para" parents="791 147"/>
    </justification>
  </clause>

  <clause id="1015">
    <literal><![CDATA[
      ((x v x) v ((x v x) ^ y)) ^ (z v (x v x)) = x v x
    ]]></literal>
    <justification jstring="[para(682(a,1),745(a,1,1,2,1)),rewrite(682(10))].">
      <j1 rule="para" parents="682 745"/>
      <j2 rule="rewrite" parents="682"/>
    </justification>
  </clause>

  <clause id="1453">
    <literal><![CDATA[
      ((((x v y) ^ y) v ((x v y) ^ y)) v y) ^ (z v (((x v y) ^ y) v ((x v y) ^ y))) = ((x v y) ^ y) v ((x v y) ^ y)
    ]]></literal>
    <justification jstring="[para(43(a,1),1015(a,1,1,2))].">
      <j1 rule="para" parents="43 1015"/>
    </justification>
  </clause>

  <clause id="1648">
    <literal><![CDATA[
      (x v (((x v x) v x) ^ y)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(651(a,1),747(a,1,1,2,1,2))].">
      <j1 rule="para" parents="651 747"/>
    </justification>
  </clause>

  <clause id="1704">
    <literal><![CDATA[
      (x v x) ^ (y v ((x v x) ^ ((x v x) v z))) = x v x
    ]]></literal>
    <justification jstring="[para(682(a,1),796(a,1,2,2,1))].">
      <j1 rule="para" parents="682 796"/>
    </justification>
  </clause>

  <clause id="1707">
    <literal><![CDATA[
      ((x ^ y) v (x ^ y)) ^ (z v (y v y)) = (x ^ y) v (x ^ y)
    ]]></literal>
    <justification jstring="[para(889(a,1),796(a,1,2,2))].">
      <j1 rule="para" parents="889 796"/>
    </justification>
  </clause>

  <clause id="1744">
    <literal><![CDATA[
      ((x v y) ^ y) v ((x v y) ^ y) = y
    ]]></literal>
    <justification jstring="[para(43(a,1),1704(a,1,2,2)),rewrite(656(7)),flip(a)].">
      <j1 rule="para" parents="43 1704"/>
      <j2 rule="rewrite" parents="656"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1793">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[back_rewrite(1453),rewrite(1744(5),1744(6),791(3),1744(6))].">
      <j1 rule="back_rewrite" parents="1453"/>
      <j2 rule="rewrite" parents="1744 1744 791 1744"/>
    </justification>
  </clause>

  <clause id="1821">
    <literal><![CDATA[
      (x v y) ^ y = y
    ]]></literal>
    <justification jstring="[back_rewrite(1744),rewrite(1793(5))].">
      <j1 rule="back_rewrite" parents="1744"/>
      <j2 rule="rewrite" parents="1793"/>
    </justification>
  </clause>

  <clause id="1838">
    <literal><![CDATA[
      (x ^ y) ^ (z v y) = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(1707),rewrite(1793(3),1793(2),1793(6))].">
      <j1 rule="back_rewrite" parents="1707"/>
      <j2 rule="rewrite" parents="1793 1793 1793"/>
    </justification>
  </clause>

  <clause id="1840">
    <literal><![CDATA[
      x ^ (y v (x ^ (x v z))) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1704),rewrite(1793(1),1793(1),1793(1),1793(5))].">
      <j1 rule="back_rewrite" parents="1704"/>
      <j2 rule="rewrite" parents="1793 1793 1793 1793"/>
    </justification>
  </clause>

  <clause id="1882">
    <literal><![CDATA[
      (x v (x ^ y)) ^ (z v x) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1648),rewrite(1793(1),1793(1),1793(5))].">
      <j1 rule="back_rewrite" parents="1648"/>
      <j2 rule="rewrite" parents="1793 1793 1793"/>
    </justification>
  </clause>

  <clause id="2333">
    <literal><![CDATA[
      x ^ (x ^ (x v y)) = x
    ]]></literal>
    <justification jstring="[para(1793(a,1),1840(a,1,2))].">
      <j1 rule="para" parents="1793 1840"/>
    </justification>
  </clause>

  <clause id="2339">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[para(1882(a,1),1840(a,1,2,2)),rewrite(1882(4)),flip(a)].">
      <j1 rule="para" parents="1882 1840"/>
      <j2 rule="rewrite" parents="1882"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2340">
    <literal><![CDATA[
      x ^ (x ^ y) = x ^ y
    ]]></literal>
    <justification jstring="[para(2339(a,1),1821(a,1,1))].">
      <j1 rule="para" parents="2339 1821"/>
    </justification>
  </clause>

  <clause id="2344">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[back_rewrite(2333),rewrite(2340(3))].">
      <j1 rule="back_rewrite" parents="2333"/>
      <j2 rule="rewrite" parents="2340"/>
    </justification>
  </clause>

  <clause id="2361">
    <literal><![CDATA[
      x ^ (y v (x v z)) = x
    ]]></literal>
    <justification jstring="[para(2344(a,1),1838(a,1,1)),rewrite(2344(5))].">
      <j1 rule="para" parents="2344 1838"/>
      <j2 rule="rewrite" parents="2344"/>
    </justification>
  </clause>

  <clause id="2362">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie_2)
    ]]></attribute>
    <justification jstring="[resolve(2361,a,7,a)].">
      <j1 rule="resolve" parents="2361 7"/>
    </justification>
  </clause>

</proof>

<proof number="2" length="57" max_count="83">

<comments><![CDATA[
% Proof 2 at 2.31 (+ 0.01) seconds: McKenzie_1.
% Length of proof is 57.
% Level of proof is 29.
% Maximum clause weight is 83.
% Given clauses 162.
]]></comments>

  <clause id="1" type="goal">
    <literal><![CDATA[
      x v (y ^ (x ^ z)) = x
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie_1)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="5" type="assumption">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ (y v y)) v (u ^ y)) ^ v)) ^ (w v ((v6 v y) ^ (y v v7))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(A1)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6" type="deny">
    <literal><![CDATA[
      c1 v (c2 ^ (c1 ^ c3)) != c1
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie_1)
    ]]></attribute>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      (((x v ((y v z) ^ (z v u))) ^ ((y v z) ^ (z v u))) v ((z v (v ^ ((y v z) ^ (z v u)))) ^ w)) ^ (v6 v ((v7 v ((y v z) ^ (z v u))) ^ (((y v z) ^ (z v u)) v v8))) = (y v z) ^ (z v u)
    ]]></literal>
    <justification jstring="[para(5(a,1),5(a,1,1,2,1,1))].">
      <j1 rule="para" parents="5 5"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      (((x v y) ^ y) v (y v y)) ^ (z v ((u v y) ^ (y v v))) = y
    ]]></literal>
    <justification jstring="[para(5(a,1),5(a,1,1,2))].">
      <j1 rule="para" parents="5 5"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      (((x v (((y ^ (z v z)) v (u ^ z)) ^ v)) ^ (((y ^ (z v z)) v (u ^ z)) ^ v)) v (((w ^ ((((y ^ (z v z)) v (u ^ z)) ^ v) v (((y ^ (z v z)) v (u ^ z)) ^ v))) v (v6 ^ (((y ^ (z v z)) v (u ^ z)) ^ v))) ^ v7)) ^ (v8 v z) = ((y ^ (z v z)) v (u ^ z)) ^ v
    ]]></literal>
    <justification jstring="[para(5(a,1),5(a,1,2,2))].">
      <j1 rule="para" parents="5 5"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((y v y) v (y v y))) ^ (z v y) = y v y
    ]]></literal>
    <justification jstring="[para(12(a,1),12(a,1,2,2))].">
      <j1 rule="para" parents="12 12"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      (((x v y) ^ y) v (((y v y) v (z ^ y)) ^ u)) ^ (v v ((w v y) ^ (y v v6))) = y
    ]]></literal>
    <justification jstring="[para(16(a,1),5(a,1,1,2,1,1))].">
      <j1 rule="para" parents="16 5"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((((y v y) v (y v y)) v (z ^ (y v y))) ^ u)) ^ (v v y) = y v y
    ]]></literal>
    <justification jstring="[para(12(a,1),17(a,1,2,2))].">
      <j1 rule="para" parents="12 17"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((((y v y) v (y v y)) v (y v y)) ^ z)) ^ (u v y) = y v y
    ]]></literal>
    <justification jstring="[para(16(a,1),23(a,1,1,2,1,2))].">
      <j1 rule="para" parents="16 23"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      (((x v y) ^ y) v (((((y v y) v (z ^ y)) ^ u) v (v ^ y)) ^ w)) ^ (v6 v ((v7 v y) ^ (y v v8))) = y
    ]]></literal>
    <justification jstring="[para(17(a,1),10(a,1,1,1,1,2)),rewrite(17(17),17(22),17(26),17(27),17(31))].">
      <j1 rule="para" parents="17 10"/>
      <j2 rule="rewrite" parents="17 17 17 17 17"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ (u v ((v v y) ^ (y v w))) = y
    ]]></literal>
    <justification jstring="[para(5(a,1),33(a,1,1,2))].">
      <j1 rule="para" parents="5 33"/>
    </justification>
  </clause>

  <clause id="56">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (z ^ (y v y))) ^ (u v y) = y v y
    ]]></literal>
    <justification jstring="[para(12(a,1),43(a,1,2,2))].">
      <j1 rule="para" parents="12 43"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      (((x v (y ^ z)) ^ (y ^ z)) v (u ^ (y ^ z))) ^ (v v z) = y ^ z
    ]]></literal>
    <justification jstring="[para(43(a,1),43(a,1,2,2))].">
      <j1 rule="para" parents="43 43"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (y v y)) ^ (z v y) = y v y
    ]]></literal>
    <justification jstring="[para(16(a,1),56(a,1,1,2))].">
      <j1 rule="para" parents="16 56"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      ((x v x) v ((x v x) v (x v x))) ^ (y v ((z v (x v x)) ^ ((x v x) v u))) = x v x
    ]]></literal>
    <justification jstring="[para(68(a,1),12(a,1,1,1))].">
      <j1 rule="para" parents="68 12"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      ((x v x) v (y ^ (x v x))) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(68(a,1),56(a,1,1,1))].">
      <j1 rule="para" parents="68 56"/>
    </justification>
  </clause>

  <clause id="93">
    <literal><![CDATA[
      ((x v x) v (x v x)) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[para(68(a,1),68(a,1,1,1))].">
      <j1 rule="para" parents="68 68"/>
    </justification>
  </clause>

  <clause id="102">
    <literal><![CDATA[
      ((x v x) v (((y ^ (x v x)) v (z ^ x)) ^ u)) ^ (v v x) = x v x
    ]]></literal>
    <justification jstring="[para(13(a,1),91(a,1,1,2))].">
      <j1 rule="para" parents="13 91"/>
    </justification>
  </clause>

  <clause id="111">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (z ^ y)) ^ (u v y) = y v y
    ]]></literal>
    <justification jstring="[para(66(a,1),56(a,1,1,2))].">
      <j1 rule="para" parents="66 56"/>
    </justification>
  </clause>

  <clause id="147">
    <literal><![CDATA[
      ((x v x) v (((y ^ x) v (z ^ x)) ^ u)) ^ (v v x) = x v x
    ]]></literal>
    <justification jstring="[para(66(a,1),102(a,1,1,2,1,1))].">
      <j1 rule="para" parents="66 102"/>
    </justification>
  </clause>

  <clause id="170">
    <literal><![CDATA[
      ((x v x) v x) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[para(5(a,1),147(a,1,1,2))].">
      <j1 rule="para" parents="5 147"/>
    </justification>
  </clause>

  <clause id="184">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ (y v y)) v (u ^ y)) ^ v)) ^ (w v (y v y)) = y
    ]]></literal>
    <justification jstring="[para(170(a,1),5(a,1,2,2))].">
      <j1 rule="para" parents="170 5"/>
    </justification>
  </clause>

  <clause id="604">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ y) v (u ^ y)) ^ v)) ^ (w v (y v y)) = y
    ]]></literal>
    <justification jstring="[para(66(a,1),184(a,1,1,2,1,1))].">
      <j1 rule="para" parents="66 184"/>
    </justification>
  </clause>

  <clause id="623">
    <literal><![CDATA[
      (((x v y) ^ y) v y) ^ (z v (y v y)) = y
    ]]></literal>
    <justification jstring="[para(5(a,1),604(a,1,1,2))].">
      <j1 rule="para" parents="5 604"/>
    </justification>
  </clause>

  <clause id="651">
    <literal><![CDATA[
      (((x v y) ^ y) v (y v y)) ^ (z v y) = y
    ]]></literal>
    <justification jstring="[para(623(a,1),12(a,1,2,2))].">
      <j1 rule="para" parents="623 12"/>
    </justification>
  </clause>

  <clause id="656">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ (u v y) = y
    ]]></literal>
    <justification jstring="[para(623(a,1),43(a,1,2,2))].">
      <j1 rule="para" parents="623 43"/>
    </justification>
  </clause>

  <clause id="669">
    <literal><![CDATA[
      (x v ((((x v x) v (x v x)) v (y ^ (x v x))) ^ z)) ^ (u v x) = x v x
    ]]></literal>
    <justification jstring="[para(651(a,1),23(a,1,1,1))].">
      <j1 rule="para" parents="651 23"/>
    </justification>
  </clause>

  <clause id="671">
    <literal><![CDATA[
      (x v ((((x v x) v (x v x)) v (x v x)) ^ y)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(651(a,1),27(a,1,1,1))].">
      <j1 rule="para" parents="651 27"/>
    </justification>
  </clause>

  <clause id="681">
    <literal><![CDATA[
      (x v (x v x)) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[para(651(a,1),68(a,1,1,1))].">
      <j1 rule="para" parents="651 68"/>
    </justification>
  </clause>

  <clause id="682">
    <literal><![CDATA[
      (x v x) v (x v x) = x v x
    ]]></literal>
    <justification jstring="[para(68(a,1),651(a,1,1,1)),rewrite(681(8))].">
      <j1 rule="para" parents="68 651"/>
      <j2 rule="rewrite" parents="681"/>
    </justification>
  </clause>

  <clause id="688">
    <literal><![CDATA[
      (x v (y ^ x)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(651(a,1),111(a,1,1,1))].">
      <j1 rule="para" parents="651 111"/>
    </justification>
  </clause>

  <clause id="745">
    <literal><![CDATA[
      (x v ((x v x) ^ y)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(671),rewrite(682(3),682(3))].">
      <j1 rule="back_rewrite" parents="671"/>
      <j2 rule="rewrite" parents="682 682"/>
    </justification>
  </clause>

  <clause id="747">
    <literal><![CDATA[
      (x v (((x v x) v (y ^ (x v x))) ^ z)) ^ (u v x) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(669),rewrite(682(3))].">
      <j1 rule="back_rewrite" parents="669"/>
      <j2 rule="rewrite" parents="682"/>
    </justification>
  </clause>

  <clause id="791">
    <literal><![CDATA[
      (x v x) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(93),rewrite(682(3))].">
      <j1 rule="back_rewrite" parents="93"/>
      <j2 rule="rewrite" parents="682"/>
    </justification>
  </clause>

  <clause id="796">
    <literal><![CDATA[
      (x v x) ^ (y v ((z v (x v x)) ^ ((x v x) v u))) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(79),rewrite(682(4),682(3))].">
      <j1 rule="back_rewrite" parents="79"/>
      <j2 rule="rewrite" parents="682 682"/>
    </justification>
  </clause>

  <clause id="889">
    <literal><![CDATA[
      ((x v x) v ((y ^ x) v (y ^ x))) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(791(a,1),147(a,1,1,2))].">
      <j1 rule="para" parents="791 147"/>
    </justification>
  </clause>

  <clause id="1015">
    <literal><![CDATA[
      ((x v x) v ((x v x) ^ y)) ^ (z v (x v x)) = x v x
    ]]></literal>
    <justification jstring="[para(682(a,1),745(a,1,1,2,1)),rewrite(682(10))].">
      <j1 rule="para" parents="682 745"/>
      <j2 rule="rewrite" parents="682"/>
    </justification>
  </clause>

  <clause id="1453">
    <literal><![CDATA[
      ((((x v y) ^ y) v ((x v y) ^ y)) v y) ^ (z v (((x v y) ^ y) v ((x v y) ^ y))) = ((x v y) ^ y) v ((x v y) ^ y)
    ]]></literal>
    <justification jstring="[para(43(a,1),1015(a,1,1,2))].">
      <j1 rule="para" parents="43 1015"/>
    </justification>
  </clause>

  <clause id="1648">
    <literal><![CDATA[
      (x v (((x v x) v x) ^ y)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(651(a,1),747(a,1,1,2,1,2))].">
      <j1 rule="para" parents="651 747"/>
    </justification>
  </clause>

  <clause id="1704">
    <literal><![CDATA[
      (x v x) ^ (y v ((x v x) ^ ((x v x) v z))) = x v x
    ]]></literal>
    <justification jstring="[para(682(a,1),796(a,1,2,2,1))].">
      <j1 rule="para" parents="682 796"/>
    </justification>
  </clause>

  <clause id="1707">
    <literal><![CDATA[
      ((x ^ y) v (x ^ y)) ^ (z v (y v y)) = (x ^ y) v (x ^ y)
    ]]></literal>
    <justification jstring="[para(889(a,1),796(a,1,2,2))].">
      <j1 rule="para" parents="889 796"/>
    </justification>
  </clause>

  <clause id="1744">
    <literal><![CDATA[
      ((x v y) ^ y) v ((x v y) ^ y) = y
    ]]></literal>
    <justification jstring="[para(43(a,1),1704(a,1,2,2)),rewrite(656(7)),flip(a)].">
      <j1 rule="para" parents="43 1704"/>
      <j2 rule="rewrite" parents="656"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1793">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[back_rewrite(1453),rewrite(1744(5),1744(6),791(3),1744(6))].">
      <j1 rule="back_rewrite" parents="1453"/>
      <j2 rule="rewrite" parents="1744 1744 791 1744"/>
    </justification>
  </clause>

  <clause id="1821">
    <literal><![CDATA[
      (x v y) ^ y = y
    ]]></literal>
    <justification jstring="[back_rewrite(1744),rewrite(1793(5))].">
      <j1 rule="back_rewrite" parents="1744"/>
      <j2 rule="rewrite" parents="1793"/>
    </justification>
  </clause>

  <clause id="1838">
    <literal><![CDATA[
      (x ^ y) ^ (z v y) = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(1707),rewrite(1793(3),1793(2),1793(6))].">
      <j1 rule="back_rewrite" parents="1707"/>
      <j2 rule="rewrite" parents="1793 1793 1793"/>
    </justification>
  </clause>

  <clause id="1840">
    <literal><![CDATA[
      x ^ (y v (x ^ (x v z))) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1704),rewrite(1793(1),1793(1),1793(1),1793(5))].">
      <j1 rule="back_rewrite" parents="1704"/>
      <j2 rule="rewrite" parents="1793 1793 1793 1793"/>
    </justification>
  </clause>

  <clause id="1882">
    <literal><![CDATA[
      (x v (x ^ y)) ^ (z v x) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1648),rewrite(1793(1),1793(1),1793(5))].">
      <j1 rule="back_rewrite" parents="1648"/>
      <j2 rule="rewrite" parents="1793 1793 1793"/>
    </justification>
  </clause>

  <clause id="2201">
    <literal><![CDATA[
      (x v (y ^ x)) ^ (z v x) = x
    ]]></literal>
    <justification jstring="[back_rewrite(688),rewrite(1793(5))].">
      <j1 rule="back_rewrite" parents="688"/>
      <j2 rule="rewrite" parents="1793"/>
    </justification>
  </clause>

  <clause id="2333">
    <literal><![CDATA[
      x ^ (x ^ (x v y)) = x
    ]]></literal>
    <justification jstring="[para(1793(a,1),1840(a,1,2))].">
      <j1 rule="para" parents="1793 1840"/>
    </justification>
  </clause>

  <clause id="2339">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[para(1882(a,1),1840(a,1,2,2)),rewrite(1882(4)),flip(a)].">
      <j1 rule="para" parents="1882 1840"/>
      <j2 rule="rewrite" parents="1882"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2340">
    <literal><![CDATA[
      x ^ (x ^ y) = x ^ y
    ]]></literal>
    <justification jstring="[para(2339(a,1),1821(a,1,1))].">
      <j1 rule="para" parents="2339 1821"/>
    </justification>
  </clause>

  <clause id="2344">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[back_rewrite(2333),rewrite(2340(3))].">
      <j1 rule="back_rewrite" parents="2333"/>
      <j2 rule="rewrite" parents="2340"/>
    </justification>
  </clause>

  <clause id="2376">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(2201(a,1),2344(a,1)),flip(a)].">
      <j1 rule="para" parents="2201 2344"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2446">
    <literal><![CDATA[
      (x v y) v (z ^ y) = x v y
    ]]></literal>
    <justification jstring="[para(1838(a,1),2376(a,1,2))].">
      <j1 rule="para" parents="1838 2376"/>
    </justification>
  </clause>

  <clause id="2555">
    <literal><![CDATA[
      x v (y ^ (x ^ z)) = x
    ]]></literal>
    <justification jstring="[para(2339(a,1),2446(a,1,1)),rewrite(2339(5))].">
      <j1 rule="para" parents="2339 2446"/>
      <j2 rule="rewrite" parents="2339"/>
    </justification>
  </clause>

  <clause id="2556">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie_1)
    ]]></attribute>
    <justification jstring="[resolve(2555,a,6,a)].">
      <j1 rule="resolve" parents="2555 6"/>
    </justification>
  </clause>

</proof>

<proof number="3" length="58" max_count="89">

<comments><![CDATA[
% Proof 3 at 2.41 (+ 0.01) seconds: McKenzie_4.
% Length of proof is 58.
% Level of proof is 29.
% Maximum clause weight is 89.
% Given clauses 207.
]]></comments>

  <clause id="4" type="goal">
    <literal><![CDATA[
      ((x v y) ^ (y v z)) ^ y = y
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie_4)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="5" type="assumption">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ (y v y)) v (u ^ y)) ^ v)) ^ (w v ((v6 v y) ^ (y v v7))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(A1)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="9" type="deny">
    <literal><![CDATA[
      ((c10 v c11) ^ (c11 v c12)) ^ c11 != c11
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie_4)
    ]]></attribute>
    <justification jstring="[deny(4)].">
      <j1 rule="deny" parents="4"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      (((x v ((y v z) ^ (z v u))) ^ ((y v z) ^ (z v u))) v ((z v (v ^ ((y v z) ^ (z v u)))) ^ w)) ^ (v6 v ((v7 v ((y v z) ^ (z v u))) ^ (((y v z) ^ (z v u)) v v8))) = (y v z) ^ (z v u)
    ]]></literal>
    <justification jstring="[para(5(a,1),5(a,1,1,2,1,1))].">
      <j1 rule="para" parents="5 5"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      (((x v y) ^ y) v (y v y)) ^ (z v ((u v y) ^ (y v v))) = y
    ]]></literal>
    <justification jstring="[para(5(a,1),5(a,1,1,2))].">
      <j1 rule="para" parents="5 5"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      (((x v (((y ^ (z v z)) v (u ^ z)) ^ v)) ^ (((y ^ (z v z)) v (u ^ z)) ^ v)) v (((w ^ ((((y ^ (z v z)) v (u ^ z)) ^ v) v (((y ^ (z v z)) v (u ^ z)) ^ v))) v (v6 ^ (((y ^ (z v z)) v (u ^ z)) ^ v))) ^ v7)) ^ (v8 v z) = ((y ^ (z v z)) v (u ^ z)) ^ v
    ]]></literal>
    <justification jstring="[para(5(a,1),5(a,1,2,2))].">
      <j1 rule="para" parents="5 5"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((y v y) v (y v y))) ^ (z v y) = y v y
    ]]></literal>
    <justification jstring="[para(12(a,1),12(a,1,2,2))].">
      <j1 rule="para" parents="12 12"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      (((x v y) ^ y) v (((y v y) v (z ^ y)) ^ u)) ^ (v v ((w v y) ^ (y v v6))) = y
    ]]></literal>
    <justification jstring="[para(16(a,1),5(a,1,1,2,1,1))].">
      <j1 rule="para" parents="16 5"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((((y v y) v (y v y)) v (z ^ (y v y))) ^ u)) ^ (v v y) = y v y
    ]]></literal>
    <justification jstring="[para(12(a,1),17(a,1,2,2))].">
      <j1 rule="para" parents="12 17"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((((y v y) v (y v y)) v (y v y)) ^ z)) ^ (u v y) = y v y
    ]]></literal>
    <justification jstring="[para(16(a,1),23(a,1,1,2,1,2))].">
      <j1 rule="para" parents="16 23"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      (((x v ((y v ((z v u) ^ u)) ^ (((z v u) ^ u) v v))) ^ ((y v ((z v u) ^ u)) ^ (((z v u) ^ u) v v))) v u) ^ (w v ((v6 v ((y v ((z v u) ^ u)) ^ (((z v u) ^ u) v v))) ^ (((y v ((z v u) ^ u)) ^ (((z v u) ^ u) v v)) v v7))) = (y v ((z v u) ^ u)) ^ (((z v u) ^ u) v v)
    ]]></literal>
    <justification jstring="[para(5(a,1),10(a,1,1,2))].">
      <j1 rule="para" parents="5 10"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      (((x v y) ^ y) v (((((y v y) v (z ^ y)) ^ u) v (v ^ y)) ^ w)) ^ (v6 v ((v7 v y) ^ (y v v8))) = y
    ]]></literal>
    <justification jstring="[para(17(a,1),10(a,1,1,1,1,2)),rewrite(17(17),17(22),17(26),17(27),17(31))].">
      <j1 rule="para" parents="17 10"/>
      <j2 rule="rewrite" parents="17 17 17 17 17"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ (u v ((v v y) ^ (y v w))) = y
    ]]></literal>
    <justification jstring="[para(5(a,1),33(a,1,1,2))].">
      <j1 rule="para" parents="5 33"/>
    </justification>
  </clause>

  <clause id="56">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (z ^ (y v y))) ^ (u v y) = y v y
    ]]></literal>
    <justification jstring="[para(12(a,1),43(a,1,2,2))].">
      <j1 rule="para" parents="12 43"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      (((x v (y ^ z)) ^ (y ^ z)) v (u ^ (y ^ z))) ^ (v v z) = y ^ z
    ]]></literal>
    <justification jstring="[para(43(a,1),43(a,1,2,2))].">
      <j1 rule="para" parents="43 43"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (y v y)) ^ (z v y) = y v y
    ]]></literal>
    <justification jstring="[para(16(a,1),56(a,1,1,2))].">
      <j1 rule="para" parents="16 56"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      ((x v x) v ((x v x) v (x v x))) ^ (y v ((z v (x v x)) ^ ((x v x) v u))) = x v x
    ]]></literal>
    <justification jstring="[para(68(a,1),12(a,1,1,1))].">
      <j1 rule="para" parents="68 12"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      ((x v x) v (y ^ (x v x))) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(68(a,1),56(a,1,1,1))].">
      <j1 rule="para" parents="68 56"/>
    </justification>
  </clause>

  <clause id="93">
    <literal><![CDATA[
      ((x v x) v (x v x)) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[para(68(a,1),68(a,1,1,1))].">
      <j1 rule="para" parents="68 68"/>
    </justification>
  </clause>

  <clause id="102">
    <literal><![CDATA[
      ((x v x) v (((y ^ (x v x)) v (z ^ x)) ^ u)) ^ (v v x) = x v x
    ]]></literal>
    <justification jstring="[para(13(a,1),91(a,1,1,2))].">
      <j1 rule="para" parents="13 91"/>
    </justification>
  </clause>

  <clause id="111">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (z ^ y)) ^ (u v y) = y v y
    ]]></literal>
    <justification jstring="[para(66(a,1),56(a,1,1,2))].">
      <j1 rule="para" parents="66 56"/>
    </justification>
  </clause>

  <clause id="147">
    <literal><![CDATA[
      ((x v x) v (((y ^ x) v (z ^ x)) ^ u)) ^ (v v x) = x v x
    ]]></literal>
    <justification jstring="[para(66(a,1),102(a,1,1,2,1,1))].">
      <j1 rule="para" parents="66 102"/>
    </justification>
  </clause>

  <clause id="170">
    <literal><![CDATA[
      ((x v x) v x) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[para(5(a,1),147(a,1,1,2))].">
      <j1 rule="para" parents="5 147"/>
    </justification>
  </clause>

  <clause id="184">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ (y v y)) v (u ^ y)) ^ v)) ^ (w v (y v y)) = y
    ]]></literal>
    <justification jstring="[para(170(a,1),5(a,1,2,2))].">
      <j1 rule="para" parents="170 5"/>
    </justification>
  </clause>

  <clause id="604">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ y) v (u ^ y)) ^ v)) ^ (w v (y v y)) = y
    ]]></literal>
    <justification jstring="[para(66(a,1),184(a,1,1,2,1,1))].">
      <j1 rule="para" parents="66 184"/>
    </justification>
  </clause>

  <clause id="623">
    <literal><![CDATA[
      (((x v y) ^ y) v y) ^ (z v (y v y)) = y
    ]]></literal>
    <justification jstring="[para(5(a,1),604(a,1,1,2))].">
      <j1 rule="para" parents="5 604"/>
    </justification>
  </clause>

  <clause id="651">
    <literal><![CDATA[
      (((x v y) ^ y) v (y v y)) ^ (z v y) = y
    ]]></literal>
    <justification jstring="[para(623(a,1),12(a,1,2,2))].">
      <j1 rule="para" parents="623 12"/>
    </justification>
  </clause>

  <clause id="656">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ (u v y) = y
    ]]></literal>
    <justification jstring="[para(623(a,1),43(a,1,2,2))].">
      <j1 rule="para" parents="623 43"/>
    </justification>
  </clause>

  <clause id="669">
    <literal><![CDATA[
      (x v ((((x v x) v (x v x)) v (y ^ (x v x))) ^ z)) ^ (u v x) = x v x
    ]]></literal>
    <justification jstring="[para(651(a,1),23(a,1,1,1))].">
      <j1 rule="para" parents="651 23"/>
    </justification>
  </clause>

  <clause id="671">
    <literal><![CDATA[
      (x v ((((x v x) v (x v x)) v (x v x)) ^ y)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(651(a,1),27(a,1,1,1))].">
      <j1 rule="para" parents="651 27"/>
    </justification>
  </clause>

  <clause id="681">
    <literal><![CDATA[
      (x v (x v x)) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[para(651(a,1),68(a,1,1,1))].">
      <j1 rule="para" parents="651 68"/>
    </justification>
  </clause>

  <clause id="682">
    <literal><![CDATA[
      (x v x) v (x v x) = x v x
    ]]></literal>
    <justification jstring="[para(68(a,1),651(a,1,1,1)),rewrite(681(8))].">
      <j1 rule="para" parents="68 651"/>
      <j2 rule="rewrite" parents="681"/>
    </justification>
  </clause>

  <clause id="688">
    <literal><![CDATA[
      (x v (y ^ x)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(651(a,1),111(a,1,1,1))].">
      <j1 rule="para" parents="651 111"/>
    </justification>
  </clause>

  <clause id="745">
    <literal><![CDATA[
      (x v ((x v x) ^ y)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(671),rewrite(682(3),682(3))].">
      <j1 rule="back_rewrite" parents="671"/>
      <j2 rule="rewrite" parents="682 682"/>
    </justification>
  </clause>

  <clause id="747">
    <literal><![CDATA[
      (x v (((x v x) v (y ^ (x v x))) ^ z)) ^ (u v x) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(669),rewrite(682(3))].">
      <j1 rule="back_rewrite" parents="669"/>
      <j2 rule="rewrite" parents="682"/>
    </justification>
  </clause>

  <clause id="791">
    <literal><![CDATA[
      (x v x) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(93),rewrite(682(3))].">
      <j1 rule="back_rewrite" parents="93"/>
      <j2 rule="rewrite" parents="682"/>
    </justification>
  </clause>

  <clause id="796">
    <literal><![CDATA[
      (x v x) ^ (y v ((z v (x v x)) ^ ((x v x) v u))) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(79),rewrite(682(4),682(3))].">
      <j1 rule="back_rewrite" parents="79"/>
      <j2 rule="rewrite" parents="682 682"/>
    </justification>
  </clause>

  <clause id="1015">
    <literal><![CDATA[
      ((x v x) v ((x v x) ^ y)) ^ (z v (x v x)) = x v x
    ]]></literal>
    <justification jstring="[para(682(a,1),745(a,1,1,2,1)),rewrite(682(10))].">
      <j1 rule="para" parents="682 745"/>
      <j2 rule="rewrite" parents="682"/>
    </justification>
  </clause>

  <clause id="1453">
    <literal><![CDATA[
      ((((x v y) ^ y) v ((x v y) ^ y)) v y) ^ (z v (((x v y) ^ y) v ((x v y) ^ y))) = ((x v y) ^ y) v ((x v y) ^ y)
    ]]></literal>
    <justification jstring="[para(43(a,1),1015(a,1,1,2))].">
      <j1 rule="para" parents="43 1015"/>
    </justification>
  </clause>

  <clause id="1648">
    <literal><![CDATA[
      (x v (((x v x) v x) ^ y)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(651(a,1),747(a,1,1,2,1,2))].">
      <j1 rule="para" parents="651 747"/>
    </justification>
  </clause>

  <clause id="1704">
    <literal><![CDATA[
      (x v x) ^ (y v ((x v x) ^ ((x v x) v z))) = x v x
    ]]></literal>
    <justification jstring="[para(682(a,1),796(a,1,2,2,1))].">
      <j1 rule="para" parents="682 796"/>
    </justification>
  </clause>

  <clause id="1706">
    <literal><![CDATA[
      (x v x) ^ (y v ((z v ((u v (x v x)) ^ (x v x))) ^ (((u v (x v x)) ^ (x v x)) v v))) = x v x
    ]]></literal>
    <justification jstring="[para(29(a,1),796(a,1,2,2))].">
      <j1 rule="para" parents="29 796"/>
    </justification>
  </clause>

  <clause id="1744">
    <literal><![CDATA[
      ((x v y) ^ y) v ((x v y) ^ y) = y
    ]]></literal>
    <justification jstring="[para(43(a,1),1704(a,1,2,2)),rewrite(656(7)),flip(a)].">
      <j1 rule="para" parents="43 1704"/>
      <j2 rule="rewrite" parents="656"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1793">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[back_rewrite(1453),rewrite(1744(5),1744(6),791(3),1744(6))].">
      <j1 rule="back_rewrite" parents="1453"/>
      <j2 rule="rewrite" parents="1744 1744 791 1744"/>
    </justification>
  </clause>

  <clause id="1821">
    <literal><![CDATA[
      (x v y) ^ y = y
    ]]></literal>
    <justification jstring="[back_rewrite(1744),rewrite(1793(5))].">
      <j1 rule="back_rewrite" parents="1744"/>
      <j2 rule="rewrite" parents="1793"/>
    </justification>
  </clause>

  <clause id="1839">
    <literal><![CDATA[
      x ^ (y v ((z v x) ^ (x v u))) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1706),rewrite(1793(1),1793(1),1793(2),1821(2),1793(2),1793(3),1821(3),1793(6))].">
      <j1 rule="back_rewrite" parents="1706"/>
      <j2 rule="rewrite" parents="1793 1793 1793 1821 1793 1793 1821 1793"/>
    </justification>
  </clause>

  <clause id="1840">
    <literal><![CDATA[
      x ^ (y v (x ^ (x v z))) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1704),rewrite(1793(1),1793(1),1793(1),1793(5))].">
      <j1 rule="back_rewrite" parents="1704"/>
      <j2 rule="rewrite" parents="1793 1793 1793 1793"/>
    </justification>
  </clause>

  <clause id="1882">
    <literal><![CDATA[
      (x v (x ^ y)) ^ (z v x) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1648),rewrite(1793(1),1793(1),1793(5))].">
      <j1 rule="back_rewrite" parents="1648"/>
      <j2 rule="rewrite" parents="1793 1793 1793"/>
    </justification>
  </clause>

  <clause id="2201">
    <literal><![CDATA[
      (x v (y ^ x)) ^ (z v x) = x
    ]]></literal>
    <justification jstring="[back_rewrite(688),rewrite(1793(5))].">
      <j1 rule="back_rewrite" parents="688"/>
      <j2 rule="rewrite" parents="1793"/>
    </justification>
  </clause>

  <clause id="2333">
    <literal><![CDATA[
      x ^ (x ^ (x v y)) = x
    ]]></literal>
    <justification jstring="[para(1793(a,1),1840(a,1,2))].">
      <j1 rule="para" parents="1793 1840"/>
    </justification>
  </clause>

  <clause id="2339">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[para(1882(a,1),1840(a,1,2,2)),rewrite(1882(4)),flip(a)].">
      <j1 rule="para" parents="1882 1840"/>
      <j2 rule="rewrite" parents="1882"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2340">
    <literal><![CDATA[
      x ^ (x ^ y) = x ^ y
    ]]></literal>
    <justification jstring="[para(2339(a,1),1821(a,1,1))].">
      <j1 rule="para" parents="2339 1821"/>
    </justification>
  </clause>

  <clause id="2344">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[back_rewrite(2333),rewrite(2340(3))].">
      <j1 rule="back_rewrite" parents="2333"/>
      <j2 rule="rewrite" parents="2340"/>
    </justification>
  </clause>

  <clause id="2376">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(2201(a,1),2344(a,1)),flip(a)].">
      <j1 rule="para" parents="2201 2344"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2443">
    <literal><![CDATA[
      x ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(2376(a,1),1821(a,1,1))].">
      <j1 rule="para" parents="2376 1821"/>
    </justification>
  </clause>

  <clause id="2752">
    <literal><![CDATA[
      x ^ ((y v x) ^ (x v z)) = x
    ]]></literal>
    <justification jstring="[para(1793(a,1),1839(a,1,2))].">
      <j1 rule="para" parents="1793 1839"/>
    </justification>
  </clause>

  <clause id="3224">
    <literal><![CDATA[
      ((x v y) ^ (y v z)) ^ y = y
    ]]></literal>
    <justification jstring="[para(2752(a,1),2443(a,1,2)),rewrite(2752(8))].">
      <j1 rule="para" parents="2752 2443"/>
      <j2 rule="rewrite" parents="2752"/>
    </justification>
  </clause>

  <clause id="3225">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie_4)
    ]]></attribute>
    <justification jstring="[resolve(3224,a,9,a)].">
      <j1 rule="resolve" parents="3224 9"/>
    </justification>
  </clause>

</proof>

<proof number="4" length="76" max_count="89">

<comments><![CDATA[
% Proof 4 at 4.36 (+ 0.03) seconds: McKenzie_3.
% Length of proof is 76.
% Level of proof is 31.
% Maximum clause weight is 89.
% Given clauses 353.
]]></comments>

  <clause id="3" type="goal">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v y = y
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie_3)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="5" type="assumption">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ (y v y)) v (u ^ y)) ^ v)) ^ (w v ((v6 v y) ^ (y v v7))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(A1)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="8" type="deny">
    <literal><![CDATA[
      ((c7 ^ c8) v (c8 ^ c9)) v c8 != c8
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie_3)
    ]]></attribute>
    <justification jstring="[deny(3)].">
      <j1 rule="deny" parents="3"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      (((x v ((y v z) ^ (z v u))) ^ ((y v z) ^ (z v u))) v ((z v (v ^ ((y v z) ^ (z v u)))) ^ w)) ^ (v6 v ((v7 v ((y v z) ^ (z v u))) ^ (((y v z) ^ (z v u)) v v8))) = (y v z) ^ (z v u)
    ]]></literal>
    <justification jstring="[para(5(a,1),5(a,1,1,2,1,1))].">
      <j1 rule="para" parents="5 5"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      (((x v y) ^ y) v (y v y)) ^ (z v ((u v y) ^ (y v v))) = y
    ]]></literal>
    <justification jstring="[para(5(a,1),5(a,1,1,2))].">
      <j1 rule="para" parents="5 5"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      (((x v (((y ^ (z v z)) v (u ^ z)) ^ v)) ^ (((y ^ (z v z)) v (u ^ z)) ^ v)) v (((w ^ ((((y ^ (z v z)) v (u ^ z)) ^ v) v (((y ^ (z v z)) v (u ^ z)) ^ v))) v (v6 ^ (((y ^ (z v z)) v (u ^ z)) ^ v))) ^ v7)) ^ (v8 v z) = ((y ^ (z v z)) v (u ^ z)) ^ v
    ]]></literal>
    <justification jstring="[para(5(a,1),5(a,1,2,2))].">
      <j1 rule="para" parents="5 5"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((y v y) v (y v y))) ^ (z v y) = y v y
    ]]></literal>
    <justification jstring="[para(12(a,1),12(a,1,2,2))].">
      <j1 rule="para" parents="12 12"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      (((x v y) ^ y) v (((y v y) v (z ^ y)) ^ u)) ^ (v v ((w v y) ^ (y v v6))) = y
    ]]></literal>
    <justification jstring="[para(16(a,1),5(a,1,1,2,1,1))].">
      <j1 rule="para" parents="16 5"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((((y v y) v (y v y)) v (z ^ (y v y))) ^ u)) ^ (v v y) = y v y
    ]]></literal>
    <justification jstring="[para(12(a,1),17(a,1,2,2))].">
      <j1 rule="para" parents="12 17"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((((y v y) v (y v y)) v (y v y)) ^ z)) ^ (u v y) = y v y
    ]]></literal>
    <justification jstring="[para(16(a,1),23(a,1,1,2,1,2))].">
      <j1 rule="para" parents="16 23"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      (((x v ((y v ((z v u) ^ u)) ^ (((z v u) ^ u) v v))) ^ ((y v ((z v u) ^ u)) ^ (((z v u) ^ u) v v))) v u) ^ (w v ((v6 v ((y v ((z v u) ^ u)) ^ (((z v u) ^ u) v v))) ^ (((y v ((z v u) ^ u)) ^ (((z v u) ^ u) v v)) v v7))) = (y v ((z v u) ^ u)) ^ (((z v u) ^ u) v v)
    ]]></literal>
    <justification jstring="[para(5(a,1),10(a,1,1,2))].">
      <j1 rule="para" parents="5 10"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      (((x v y) ^ y) v (((((y v y) v (z ^ y)) ^ u) v (v ^ y)) ^ w)) ^ (v6 v ((v7 v y) ^ (y v v8))) = y
    ]]></literal>
    <justification jstring="[para(17(a,1),10(a,1,1,1,1,2)),rewrite(17(17),17(22),17(26),17(27),17(31))].">
      <j1 rule="para" parents="17 10"/>
      <j2 rule="rewrite" parents="17 17 17 17 17"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ (u v ((v v y) ^ (y v w))) = y
    ]]></literal>
    <justification jstring="[para(5(a,1),33(a,1,1,2))].">
      <j1 rule="para" parents="5 33"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      (((x v (y v z)) ^ (y v z)) v ((((((y v z) v (y v z)) v (z v z)) ^ u) v (v ^ (y v z))) ^ w)) ^ (v6 v ((v7 v (y v z)) ^ ((y v z) v v8))) = y v z
    ]]></literal>
    <justification jstring="[para(16(a,1),33(a,1,1,2,1,1,1,2))].">
      <j1 rule="para" parents="16 33"/>
    </justification>
  </clause>

  <clause id="56">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (z ^ (y v y))) ^ (u v y) = y v y
    ]]></literal>
    <justification jstring="[para(12(a,1),43(a,1,2,2))].">
      <j1 rule="para" parents="12 43"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      (((x v (y ^ z)) ^ (y ^ z)) v (u ^ (y ^ z))) ^ (v v z) = y ^ z
    ]]></literal>
    <justification jstring="[para(43(a,1),43(a,1,2,2))].">
      <j1 rule="para" parents="43 43"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (y v y)) ^ (z v y) = y v y
    ]]></literal>
    <justification jstring="[para(16(a,1),56(a,1,1,2))].">
      <j1 rule="para" parents="16 56"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      ((x v x) v ((x v x) v (x v x))) ^ (y v ((z v (x v x)) ^ ((x v x) v u))) = x v x
    ]]></literal>
    <justification jstring="[para(68(a,1),12(a,1,1,1))].">
      <j1 rule="para" parents="68 12"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      ((x v x) v (y ^ (x v x))) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(68(a,1),56(a,1,1,1))].">
      <j1 rule="para" parents="68 56"/>
    </justification>
  </clause>

  <clause id="93">
    <literal><![CDATA[
      ((x v x) v (x v x)) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[para(68(a,1),68(a,1,1,1))].">
      <j1 rule="para" parents="68 68"/>
    </justification>
  </clause>

  <clause id="102">
    <literal><![CDATA[
      ((x v x) v (((y ^ (x v x)) v (z ^ x)) ^ u)) ^ (v v x) = x v x
    ]]></literal>
    <justification jstring="[para(13(a,1),91(a,1,1,2))].">
      <j1 rule="para" parents="13 91"/>
    </justification>
  </clause>

  <clause id="104">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ (u v (v v ((w v y) ^ (y v v6)))) = y
    ]]></literal>
    <justification jstring="[para(5(a,1),66(a,1,1,1,1,2)),rewrite(5(14),5(15),5(23))].">
      <j1 rule="para" parents="5 66"/>
      <j2 rule="rewrite" parents="5 5 5"/>
    </justification>
  </clause>

  <clause id="111">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (z ^ y)) ^ (u v y) = y v y
    ]]></literal>
    <justification jstring="[para(66(a,1),56(a,1,1,2))].">
      <j1 rule="para" parents="66 56"/>
    </justification>
  </clause>

  <clause id="147">
    <literal><![CDATA[
      ((x v x) v (((y ^ x) v (z ^ x)) ^ u)) ^ (v v x) = x v x
    ]]></literal>
    <justification jstring="[para(66(a,1),102(a,1,1,2,1,1))].">
      <j1 rule="para" parents="66 102"/>
    </justification>
  </clause>

  <clause id="170">
    <literal><![CDATA[
      ((x v x) v x) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[para(5(a,1),147(a,1,1,2))].">
      <j1 rule="para" parents="5 147"/>
    </justification>
  </clause>

  <clause id="184">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ (y v y)) v (u ^ y)) ^ v)) ^ (w v (y v y)) = y
    ]]></literal>
    <justification jstring="[para(170(a,1),5(a,1,2,2))].">
      <j1 rule="para" parents="170 5"/>
    </justification>
  </clause>

  <clause id="604">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ y) v (u ^ y)) ^ v)) ^ (w v (y v y)) = y
    ]]></literal>
    <justification jstring="[para(66(a,1),184(a,1,1,2,1,1))].">
      <j1 rule="para" parents="66 184"/>
    </justification>
  </clause>

  <clause id="623">
    <literal><![CDATA[
      (((x v y) ^ y) v y) ^ (z v (y v y)) = y
    ]]></literal>
    <justification jstring="[para(5(a,1),604(a,1,1,2))].">
      <j1 rule="para" parents="5 604"/>
    </justification>
  </clause>

  <clause id="651">
    <literal><![CDATA[
      (((x v y) ^ y) v (y v y)) ^ (z v y) = y
    ]]></literal>
    <justification jstring="[para(623(a,1),12(a,1,2,2))].">
      <j1 rule="para" parents="623 12"/>
    </justification>
  </clause>

  <clause id="656">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ (u v y) = y
    ]]></literal>
    <justification jstring="[para(623(a,1),43(a,1,2,2))].">
      <j1 rule="para" parents="623 43"/>
    </justification>
  </clause>

  <clause id="669">
    <literal><![CDATA[
      (x v ((((x v x) v (x v x)) v (y ^ (x v x))) ^ z)) ^ (u v x) = x v x
    ]]></literal>
    <justification jstring="[para(651(a,1),23(a,1,1,1))].">
      <j1 rule="para" parents="651 23"/>
    </justification>
  </clause>

  <clause id="671">
    <literal><![CDATA[
      (x v ((((x v x) v (x v x)) v (x v x)) ^ y)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(651(a,1),27(a,1,1,1))].">
      <j1 rule="para" parents="651 27"/>
    </justification>
  </clause>

  <clause id="681">
    <literal><![CDATA[
      (x v (x v x)) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[para(651(a,1),68(a,1,1,1))].">
      <j1 rule="para" parents="651 68"/>
    </justification>
  </clause>

  <clause id="682">
    <literal><![CDATA[
      (x v x) v (x v x) = x v x
    ]]></literal>
    <justification jstring="[para(68(a,1),651(a,1,1,1)),rewrite(681(8))].">
      <j1 rule="para" parents="68 651"/>
      <j2 rule="rewrite" parents="681"/>
    </justification>
  </clause>

  <clause id="688">
    <literal><![CDATA[
      (x v (y ^ x)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(651(a,1),111(a,1,1,1))].">
      <j1 rule="para" parents="651 111"/>
    </justification>
  </clause>

  <clause id="738">
    <literal><![CDATA[
      (((x v y) ^ y) v ((y v (z ^ y)) ^ u)) ^ (v v (y v y)) = y
    ]]></literal>
    <justification jstring="[para(651(a,1),184(a,1,1,2,1,1))].">
      <j1 rule="para" parents="651 184"/>
    </justification>
  </clause>

  <clause id="745">
    <literal><![CDATA[
      (x v ((x v x) ^ y)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(671),rewrite(682(3),682(3))].">
      <j1 rule="back_rewrite" parents="671"/>
      <j2 rule="rewrite" parents="682 682"/>
    </justification>
  </clause>

  <clause id="747">
    <literal><![CDATA[
      (x v (((x v x) v (y ^ (x v x))) ^ z)) ^ (u v x) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(669),rewrite(682(3))].">
      <j1 rule="back_rewrite" parents="669"/>
      <j2 rule="rewrite" parents="682"/>
    </justification>
  </clause>

  <clause id="791">
    <literal><![CDATA[
      (x v x) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(93),rewrite(682(3))].">
      <j1 rule="back_rewrite" parents="93"/>
      <j2 rule="rewrite" parents="682"/>
    </justification>
  </clause>

  <clause id="796">
    <literal><![CDATA[
      (x v x) ^ (y v ((z v (x v x)) ^ ((x v x) v u))) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(79),rewrite(682(4),682(3))].">
      <j1 rule="back_rewrite" parents="79"/>
      <j2 rule="rewrite" parents="682 682"/>
    </justification>
  </clause>

  <clause id="919">
    <literal><![CDATA[
      (((x v y) ^ y) v y) ^ (z v ((u v y) ^ (y v v))) = y
    ]]></literal>
    <justification jstring="[para(43(a,1),29(a,1,1,1,1,2)),rewrite(43(13),43(15),43(16),43(20))].">
      <j1 rule="para" parents="43 29"/>
      <j2 rule="rewrite" parents="43 43 43 43"/>
    </justification>
  </clause>

  <clause id="1015">
    <literal><![CDATA[
      ((x v x) v ((x v x) ^ y)) ^ (z v (x v x)) = x v x
    ]]></literal>
    <justification jstring="[para(682(a,1),745(a,1,1,2,1)),rewrite(682(10))].">
      <j1 rule="para" parents="682 745"/>
      <j2 rule="rewrite" parents="682"/>
    </justification>
  </clause>

  <clause id="1453">
    <literal><![CDATA[
      ((((x v y) ^ y) v ((x v y) ^ y)) v y) ^ (z v (((x v y) ^ y) v ((x v y) ^ y))) = ((x v y) ^ y) v ((x v y) ^ y)
    ]]></literal>
    <justification jstring="[para(43(a,1),1015(a,1,1,2))].">
      <j1 rule="para" parents="43 1015"/>
    </justification>
  </clause>

  <clause id="1604">
    <literal><![CDATA[
      (((x v y) ^ y) v ((y v (z ^ y)) ^ u)) ^ (y v y) = y
    ]]></literal>
    <justification jstring="[para(682(a,1),738(a,1,2))].">
      <j1 rule="para" parents="682 738"/>
    </justification>
  </clause>

  <clause id="1648">
    <literal><![CDATA[
      (x v (((x v x) v x) ^ y)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(651(a,1),747(a,1,1,2,1,2))].">
      <j1 rule="para" parents="651 747"/>
    </justification>
  </clause>

  <clause id="1662">
    <literal><![CDATA[
      (((x v (((y v y) v y) ^ z)) ^ (((y v y) v y) ^ z)) v (((y v y) v y) ^ z)) ^ (u v (y v y)) = ((y v y) v y) ^ z
    ]]></literal>
    <justification jstring="[para(1648(a,1),919(a,1,2,2))].">
      <j1 rule="para" parents="1648 919"/>
    </justification>
  </clause>

  <clause id="1685">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((((y v y) ^ z) v (u ^ (y v y))) ^ v)) ^ (w v ((v6 v (y v y)) ^ (y v y))) = y v y
    ]]></literal>
    <justification jstring="[para(682(a,1),47(a,1,2,2,2)),rewrite(682(7),682(7))].">
      <j1 rule="para" parents="682 47"/>
      <j2 rule="rewrite" parents="682 682"/>
    </justification>
  </clause>

  <clause id="1702">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((((y v y) ^ z) v y) ^ u)) ^ (v v ((w v (y v y)) ^ ((y v y) v v6))) = y v y
    ]]></literal>
    <justification jstring="[para(1604(a,1),47(a,1,1,2,1,2)),rewrite(682(7),682(7))].">
      <j1 rule="para" parents="1604 47"/>
      <j2 rule="rewrite" parents="682 682"/>
    </justification>
  </clause>

  <clause id="1704">
    <literal><![CDATA[
      (x v x) ^ (y v ((x v x) ^ ((x v x) v z))) = x v x
    ]]></literal>
    <justification jstring="[para(682(a,1),796(a,1,2,2,1))].">
      <j1 rule="para" parents="682 796"/>
    </justification>
  </clause>

  <clause id="1744">
    <literal><![CDATA[
      ((x v y) ^ y) v ((x v y) ^ y) = y
    ]]></literal>
    <justification jstring="[para(43(a,1),1704(a,1,2,2)),rewrite(656(7)),flip(a)].">
      <j1 rule="para" parents="43 1704"/>
      <j2 rule="rewrite" parents="656"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1756">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(104(a,1),1704(a,1,2,2)),rewrite(1744(5),1744(7))].">
      <j1 rule="para" parents="104 1704"/>
      <j2 rule="rewrite" parents="1744 1744"/>
    </justification>
  </clause>

  <clause id="1793">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[back_rewrite(1453),rewrite(1744(5),1744(6),791(3),1744(6))].">
      <j1 rule="back_rewrite" parents="1453"/>
      <j2 rule="rewrite" parents="1744 1744 791 1744"/>
    </justification>
  </clause>

  <clause id="1821">
    <literal><![CDATA[
      (x v y) ^ y = y
    ]]></literal>
    <justification jstring="[back_rewrite(1744),rewrite(1793(5))].">
      <j1 rule="back_rewrite" parents="1744"/>
      <j2 rule="rewrite" parents="1793"/>
    </justification>
  </clause>

  <clause id="1840">
    <literal><![CDATA[
      x ^ (y v (x ^ (x v z))) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1704),rewrite(1793(1),1793(1),1793(1),1793(5))].">
      <j1 rule="back_rewrite" parents="1704"/>
      <j2 rule="rewrite" parents="1793 1793 1793 1793"/>
    </justification>
  </clause>

  <clause id="1842">
    <literal><![CDATA[
      (x v (((x ^ y) v x) ^ z)) ^ (u v ((v v x) ^ (x v w))) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1702),rewrite(1793(1),1793(2),1821(2),1793(1),1793(5),1793(6),1793(10))].">
      <j1 rule="back_rewrite" parents="1702"/>
      <j2 rule="rewrite" parents="1793 1793 1821 1793 1793 1793 1793"/>
    </justification>
  </clause>

  <clause id="1855">
    <literal><![CDATA[
      (x v (((x ^ y) v (z ^ x)) ^ u)) ^ (v v x) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1685),rewrite(1793(1),1793(2),1821(2),1793(1),1793(2),1793(6),1793(7),1821(7),1793(8))].">
      <j1 rule="back_rewrite" parents="1685"/>
      <j2 rule="rewrite" parents="1793 1793 1821 1793 1793 1793 1793 1821 1793"/>
    </justification>
  </clause>

  <clause id="1870">
    <literal><![CDATA[
      (x ^ y) ^ (z v x) = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(1662),rewrite(1793(1),1793(1),1793(3),1793(3),1821(4),1793(2),1793(2),1793(3),1793(2),1793(4),1793(4))].">
      <j1 rule="back_rewrite" parents="1662"/>
      <j2 rule="rewrite" parents="1793 1793 1793 1793 1821 1793 1793 1793 1793 1793 1793"/>
    </justification>
  </clause>

  <clause id="1882">
    <literal><![CDATA[
      (x v (x ^ y)) ^ (z v x) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1648),rewrite(1793(1),1793(1),1793(5))].">
      <j1 rule="back_rewrite" parents="1648"/>
      <j2 rule="rewrite" parents="1793 1793 1793"/>
    </justification>
  </clause>

  <clause id="2201">
    <literal><![CDATA[
      (x v (y ^ x)) ^ (z v x) = x
    ]]></literal>
    <justification jstring="[back_rewrite(688),rewrite(1793(5))].">
      <j1 rule="back_rewrite" parents="688"/>
      <j2 rule="rewrite" parents="1793"/>
    </justification>
  </clause>

  <clause id="2333">
    <literal><![CDATA[
      x ^ (x ^ (x v y)) = x
    ]]></literal>
    <justification jstring="[para(1793(a,1),1840(a,1,2))].">
      <j1 rule="para" parents="1793 1840"/>
    </justification>
  </clause>

  <clause id="2335">
    <literal><![CDATA[
      (x ^ y) ^ x = x ^ y
    ]]></literal>
    <justification jstring="[para(1793(a,1),1870(a,1,2))].">
      <j1 rule="para" parents="1793 1870"/>
    </justification>
  </clause>

  <clause id="2339">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[para(1882(a,1),1840(a,1,2,2)),rewrite(1882(4)),flip(a)].">
      <j1 rule="para" parents="1882 1840"/>
      <j2 rule="rewrite" parents="1882"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2340">
    <literal><![CDATA[
      x ^ (x ^ y) = x ^ y
    ]]></literal>
    <justification jstring="[para(2339(a,1),1821(a,1,1))].">
      <j1 rule="para" parents="2339 1821"/>
    </justification>
  </clause>

  <clause id="2344">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[back_rewrite(2333),rewrite(2340(3))].">
      <j1 rule="back_rewrite" parents="2333"/>
      <j2 rule="rewrite" parents="2340"/>
    </justification>
  </clause>

  <clause id="2376">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(2201(a,1),2344(a,1)),flip(a)].">
      <j1 rule="para" parents="2201 2344"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2443">
    <literal><![CDATA[
      x ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(2376(a,1),1821(a,1,1))].">
      <j1 rule="para" parents="2376 1821"/>
    </justification>
  </clause>

  <clause id="2825">
    <literal><![CDATA[
      x v (((x ^ y) v x) ^ z) = x
    ]]></literal>
    <justification jstring="[para(1842(a,1),2344(a,1)),flip(a)].">
      <j1 rule="para" parents="1842 2344"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3103">
    <literal><![CDATA[
      x v (((x ^ y) v (z ^ x)) ^ u) = x
    ]]></literal>
    <justification jstring="[para(1855(a,1),2344(a,1)),flip(a)].">
      <j1 rule="para" parents="1855 2344"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3254">
    <literal><![CDATA[
      x v ((x ^ y) v x) = x
    ]]></literal>
    <justification jstring="[para(1756(a,1),2825(a,1,2))].">
      <j1 rule="para" parents="1756 2825"/>
    </justification>
  </clause>

  <clause id="3294">
    <literal><![CDATA[
      (x ^ y) v x = x
    ]]></literal>
    <justification jstring="[para(3254(a,1),1756(a,1,2)),rewrite(1821(3)),flip(a)].">
      <j1 rule="para" parents="3254 1756"/>
      <j2 rule="rewrite" parents="1821"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3314">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(1821(a,1),3294(a,1,1))].">
      <j1 rule="para" parents="1821 3294"/>
    </justification>
  </clause>

  <clause id="9112">
    <literal><![CDATA[
      x v ((x ^ y) v (z ^ x)) = x
    ]]></literal>
    <justification jstring="[para(1756(a,1),3103(a,1,2))].">
      <j1 rule="para" parents="1756 3103"/>
    </justification>
  </clause>

  <clause id="9271">
    <literal><![CDATA[
      x v ((x ^ y) v (x ^ z)) = x
    ]]></literal>
    <justification jstring="[para(2335(a,1),9112(a,1,2,2))].">
      <j1 rule="para" parents="2335 9112"/>
    </justification>
  </clause>

  <clause id="9432">
    <literal><![CDATA[
      x v ((y ^ x) v (x ^ z)) = x
    ]]></literal>
    <justification jstring="[para(2443(a,1),9271(a,1,2,1))].">
      <j1 rule="para" parents="2443 9271"/>
    </justification>
  </clause>

  <clause id="10040">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v y = y
    ]]></literal>
    <justification jstring="[para(9432(a,1),3314(a,1,2)),rewrite(9432(8))].">
      <j1 rule="para" parents="9432 3314"/>
      <j2 rule="rewrite" parents="9432"/>
    </justification>
  </clause>

  <clause id="10041">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie_3)
    ]]></attribute>
    <justification jstring="[resolve(10040,a,8,a)].">
      <j1 rule="resolve" parents="10040 8"/>
    </justification>
  </clause>

</proof>

</proofs>
