<?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="1">

<source>LT-base.out</source>

<heading><![CDATA[
Prover9 (32) version 2009-02A, February 2009.
Process 12207 was started by mccune on cleo,
Wed Feb 25 10:10:21 2009
The command was "/home/mccune/bin/prover9 -t 7200 -f LT.in".
]]></heading>

<proof number="1" length="56" max_count="29">

<comments><![CDATA[
% Proof 1 at 26.89 (+ 0.17) seconds: H2.
% Length of proof is 56.
% Level of proof is 18.
% Maximum clause weight is 29.
% Given clauses 562.

]]></comments>

  <clause id="1" type="goal">
    <literal><![CDATA[
      x ^ (y v (x ^ z)) = x ^ (y v (z ^ ((x ^ (y v z)) v (y ^ z))))
    ]]></literal>
    <attribute><![CDATA[
      label(H2)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="2" type="assumption">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="3" type="assumption">
    <literal><![CDATA[
      (x v y) v z = x v (y v z)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      x v (y v z) = z v (x v y)
    ]]></literal>
    <justification jstring="[copy(3),rewrite([2(2)]),flip(a)].">
      <j1 rule="copy" parents="3"/>
      <j2 rule="rewrite" parents="2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="5" type="assumption">
    <literal><![CDATA[
      x ^ y = y ^ x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6" type="assumption">
    <literal><![CDATA[
      (x ^ y) ^ z = x ^ (y ^ z)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      x ^ (y ^ z) = z ^ (x ^ y)
    ]]></literal>
    <justification jstring="[copy(6),rewrite([5(2)]),flip(a)].">
      <j1 rule="copy" parents="6"/>
      <j2 rule="rewrite" parents="5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

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

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

  <clause id="10" type="assumption">
    <literal><![CDATA[
      (x ^ y) v (x ^ z) = x ^ ((y ^ (x v z)) v (z ^ (x v y)))
    ]]></literal>
    <attribute><![CDATA[
      label(H82)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      x ^ ((y ^ (x v z)) v (z ^ (x v y))) = (x ^ z) v (x ^ y)
    ]]></literal>
    <justification jstring="[copy(10),rewrite([2(8)]),flip(a),rewrite([2(5),2(9)])].">
      <j1 rule="copy" parents="10"/>
      <j2 rule="rewrite" parents="2"/>
      <j2 rule="flip"/>
      <j2 rule="rewrite" parents="2 2"/>
    </justification>
  </clause>

  <clause id="12" type="deny">
    <literal><![CDATA[
      c1 ^ (c2 v (c3 ^ ((c1 ^ (c2 v c3)) v (c2 ^ c3)))) != c1 ^ (c2 v (c1 ^ c3))
    ]]></literal>
    <attribute><![CDATA[
      label(H2)
    ]]></attribute>
    <attribute><![CDATA[
      answer(H2)
    ]]></attribute>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      c1 ^ (c2 v (c3 ^ ((c2 ^ c3) v (c1 ^ (c2 v c3))))) != c1 ^ (c2 v (c1 ^ c3))
    ]]></literal>
    <attribute><![CDATA[
      answer(H2)
    ]]></attribute>
    <justification jstring="[copy(12),rewrite([2(12)])].">
      <j1 rule="copy" parents="12"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      x v (y v z) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(4(a,1),2(a,1)),rewrite([2(3),2(4)])].">
      <j1 rule="para" parents="4 2"/>
      <j2 rule="rewrite" parents="2 2"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      x ^ (y ^ z) = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(7(a,1),5(a,1)),rewrite([5(3),5(4)])].">
      <j1 rule="para" parents="7 5"/>
      <j2 rule="rewrite" parents="5 5"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      x ^ x = x
    ]]></literal>
    <justification jstring="[para(9(a,1),8(a,1,2))].">
      <j1 rule="para" parents="9 8"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(8(a,1),9(a,1,2))].">
      <j1 rule="para" parents="8 9"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      x ^ ((y ^ (z v x)) v (z ^ (x v y))) = (z ^ x) v (x ^ y)
    ]]></literal>
    <justification jstring="[para(11(a,1),5(a,2)),rewrite([2(3),2(5),5(6),5(8),2(9)])].">
      <j1 rule="para" parents="11 5"/>
      <j2 rule="rewrite" parents="2 2 5 5 2"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      x v (y v (x ^ z)) = x v y
    ]]></literal>
    <justification jstring="[para(9(a,1),14(a,2,2)),rewrite([2(4)])].">
      <j1 rule="para" parents="9 14"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      x ^ (((x v y) ^ (z v u)) v (y ^ (x v (z v u)))) = (x ^ y) v (x ^ (z v u))
    ]]></literal>
    <justification jstring="[para(14(a,1),11(a,1,2,1,2)),rewrite([14(2),5(6),2(7),2(12)])].">
      <j1 rule="para" parents="14 11"/>
      <j2 rule="rewrite" parents="14 5 2 2"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      x ^ ((x v y) v (y ^ (x v (x v y)))) = x
    ]]></literal>
    <justification jstring="[para(16(a,1),11(a,1,2,1)),rewrite([8(9),2(8),9(8)])].">
      <j1 rule="para" parents="16 11"/>
      <j2 rule="rewrite" parents="8 2 9"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(17(a,1),14(a,1)),rewrite([2(3),14(3),17(2)]),flip(a)].">
      <j1 rule="para" parents="17 14"/>
      <j2 rule="rewrite" parents="2 14 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      x ^ ((x v y) v (y ^ (x v y))) = x
    ]]></literal>
    <justification jstring="[back_rewrite(24),rewrite([26(3)])].">
      <j1 rule="back_rewrite" parents="24"/>
      <j2 rule="rewrite" parents="26"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      x ^ (y ^ (x v z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(8(a,1),15(a,2,2)),rewrite([5(4)])].">
      <j1 rule="para" parents="8 15"/>
      <j2 rule="rewrite" parents="5"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      x ^ (x ^ y) = x ^ y
    ]]></literal>
    <justification jstring="[para(15(a,1),16(a,1)),rewrite([5(2),15(2),16(1)])].">
      <j1 rule="para" parents="15 16"/>
      <j2 rule="rewrite" parents="5 15 16"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      (x v y) ^ (x v (z ^ (x v y))) = x v (z ^ (x v y))
    ]]></literal>
    <justification jstring="[para(8(a,1),18(a,2,1)),rewrite([26(3),2(5),14(5),2(4),8(6),2(4),5(7)])].">
      <j1 rule="para" parents="8 18"/>
      <j2 rule="rewrite" parents="26 2 14 2 8 2 5"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      x v (y v (z v (x ^ u))) = x v (y v z)
    ]]></literal>
    <justification jstring="[para(22(a,1),14(a,2,2)),rewrite([14(6)])].">
      <j1 rule="para" parents="22 14"/>
      <j2 rule="rewrite" parents="14"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      (x ^ (y v z)) ^ ((x ^ y) v (y ^ z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(11(a,1),30(a,1,2)),rewrite([5(4),2(5),5(9),15(9),8(8)])].">
      <j1 rule="para" parents="11 30"/>
      <j2 rule="rewrite" parents="5 2 5 15 8"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      x ^ (y ^ (z ^ (x v u))) = x ^ (y ^ z)
    ]]></literal>
    <justification jstring="[para(30(a,1),15(a,2,2)),rewrite([15(6)])].">
      <j1 rule="para" parents="30 15"/>
      <j2 rule="rewrite" parents="15"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      (x v y) ^ ((z ^ (x v y)) v ((x v y) ^ (x v (y v z)))) = (x v y) v (z ^ (x v y))
    ]]></literal>
    <justification jstring="[para(16(a,1),23(a,2,2)),rewrite([2(3),14(3),2(2),5(5),17(8),2(8),5(11),2(13)])].">
      <j1 rule="para" parents="16 23"/>
      <j2 rule="rewrite" parents="2 14 2 5 17 2 5 2"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      x ^ ((y ^ z) ^ (y v u)) = x ^ (y ^ z)
    ]]></literal>
    <justification jstring="[para(11(a,1),44(a,1,2,2)),rewrite([5(4),2(5),15(7),42(6),15(2),5(1),5(5),5(6),15(6)]),flip(a)].">
      <j1 rule="para" parents="11 44"/>
      <j2 rule="rewrite" parents="5 2 15 42 15 5 5 5 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      (x ^ y) ^ (x v z) = x ^ y
    ]]></literal>
    <justification jstring="[para(16(a,1),54(a,2)),rewrite([32(5)])].">
      <j1 rule="para" parents="16 54"/>
      <j2 rule="rewrite" parents="32"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      (x ^ y) ^ (y v z) = x ^ y
    ]]></literal>
    <justification jstring="[para(5(a,1),65(a,1,1)),rewrite([5(4)])].">
      <j1 rule="para" parents="5 65"/>
      <j2 rule="rewrite" parents="5"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      (x ^ y) ^ (z v (x ^ y)) = x ^ y
    ]]></literal>
    <justification jstring="[para(65(a,1),65(a,1,1)),rewrite([2(3),65(7)])].">
      <j1 rule="para" parents="65 65"/>
      <j2 rule="rewrite" parents="2 65"/>
    </justification>
  </clause>

  <clause id="90">
    <literal><![CDATA[
      x ^ (y v ((x v z) v (z ^ (x v z)))) = x
    ]]></literal>
    <justification jstring="[para(27(a,1),74(a,1,1)),rewrite([2(5),27(11)])].">
      <j1 rule="para" parents="27 74"/>
      <j2 rule="rewrite" parents="2 27"/>
    </justification>
  </clause>

  <clause id="96">
    <literal><![CDATA[
      (x ^ y) v (x ^ (y ^ z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(82(a,1),18(a,1,2,1)),rewrite([17(5),15(4),5(3),8(6),15(3),5(2),16(6),2(5)]),flip(a)].">
      <j1 rule="para" parents="82 18"/>
      <j2 rule="rewrite" parents="17 15 5 8 15 5 16 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="111">
    <literal><![CDATA[
      x v ((x ^ y) v (z ^ u)) = x v (z ^ u)
    ]]></literal>
    <justification jstring="[para(96(a,1),40(a,2,2)),rewrite([2(5),14(6),96(5)])].">
      <j1 rule="para" parents="96 40"/>
      <j2 rule="rewrite" parents="2 14 96"/>
    </justification>
  </clause>

  <clause id="218">
    <literal><![CDATA[
      (x ^ y) v (x ^ (y v z)) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(42(a,1),9(a,1,2)),rewrite([2(4)])].">
      <j1 rule="para" parents="42 9"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="245">
    <literal><![CDATA[
      (x ^ y) v ((z ^ u) v (x ^ (y v w))) = (z ^ u) v (x ^ (y v w))
    ]]></literal>
    <justification jstring="[para(42(a,1),111(a,1,2,1)),rewrite([14(6),2(5),2(10)])].">
      <j1 rule="para" parents="42 111"/>
      <j2 rule="rewrite" parents="14 2 2"/>
    </justification>
  </clause>

  <clause id="256">
    <literal><![CDATA[
      x v ((x v y) v (z ^ (x v y))) = x v y
    ]]></literal>
    <justification jstring="[para(35(a,1),9(a,1,2)),rewrite([14(5)])].">
      <j1 rule="para" parents="35 9"/>
      <j2 rule="rewrite" parents="14"/>
    </justification>
  </clause>

  <clause id="270">
    <literal><![CDATA[
      (x v y) ^ (x v ((y v z) v (z ^ (y v z)))) = x v y
    ]]></literal>
    <justification jstring="[para(90(a,1),35(a,1,2,2)),rewrite([5(7),90(13)])].">
      <j1 rule="para" parents="90 35"/>
      <j2 rule="rewrite" parents="5 90"/>
    </justification>
  </clause>

  <clause id="312">
    <literal><![CDATA[
      (x ^ (y ^ z)) v (x ^ (y v u)) = x ^ (y v u)
    ]]></literal>
    <justification jstring="[para(218(a,1),22(a,2)),rewrite([5(5),15(5),5(4),2(6),245(7)])].">
      <j1 rule="para" parents="218 22"/>
      <j2 rule="rewrite" parents="5 15 5 2 245"/>
    </justification>
  </clause>

  <clause id="339">
    <literal><![CDATA[
      (x v y) v (y ^ (x v y)) = x v y
    ]]></literal>
    <justification jstring="[para(27(a,1),256(a,1,2,2)),rewrite([2(6),256(6),17(3)]),flip(a)].">
      <j1 rule="para" parents="27 256"/>
      <j2 rule="rewrite" parents="2 256 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="346">
    <literal><![CDATA[
      (x v y) ^ (x v (y v z)) = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(270),rewrite([339(5)])].">
      <j1 rule="back_rewrite" parents="270"/>
      <j2 rule="rewrite" parents="339"/>
    </justification>
  </clause>

  <clause id="347">
    <literal><![CDATA[
      (x v y) v (z ^ (x v y)) = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(49),rewrite([346(7),2(5),8(6)]),flip(a)].">
      <j1 rule="back_rewrite" parents="49"/>
      <j2 rule="rewrite" parents="346 2 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="361">
    <literal><![CDATA[
      (x ^ y) ^ (z v u) = x ^ (y ^ (z v u))
    ]]></literal>
    <justification jstring="[para(347(a,1),44(a,1,2,2,2)),rewrite([15(5),15(4),16(3),5(6)]),flip(a)].">
      <j1 rule="para" parents="347 44"/>
      <j2 rule="rewrite" parents="15 15 16 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="442">
    <literal><![CDATA[
      x ^ (y ^ (z v (x ^ y))) = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(82),rewrite([361(4)])].">
      <j1 rule="back_rewrite" parents="82"/>
      <j2 rule="rewrite" parents="361"/>
    </justification>
  </clause>

  <clause id="465">
    <literal><![CDATA[
      x ^ (y v (x ^ (y v z))) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(35(a,1),442(a,1,2))].">
      <j1 rule="para" parents="35 442"/>
    </justification>
  </clause>

  <clause id="570">
    <literal><![CDATA[
      x ^ ((x ^ y) v ((x ^ z) v (y ^ (x v z)))) = (x ^ y) v (x ^ z)
    ]]></literal>
    <justification jstring="[para(11(a,1),465(a,1,2,2)),rewrite([2(5),14(6),2(5),2(12),11(13)])].">
      <j1 rule="para" parents="11 465"/>
      <j2 rule="rewrite" parents="2 14 2 2 11"/>
    </justification>
  </clause>

  <clause id="819">
    <literal><![CDATA[
      (x ^ (y ^ z)) v (x ^ (z v u)) = x ^ (z v u)
    ]]></literal>
    <justification jstring="[para(5(a,1),312(a,1,1,2))].">
      <j1 rule="para" parents="5 312"/>
    </justification>
  </clause>

  <clause id="2012">
    <literal><![CDATA[
      (x ^ (y ^ z)) v (z ^ (y v u)) = z ^ (y v u)
    ]]></literal>
    <justification jstring="[para(5(a,1),819(a,1,1)),rewrite([5(2),15(2),5(1)])].">
      <j1 rule="para" parents="5 819"/>
      <j2 rule="rewrite" parents="5 15 5"/>
    </justification>
  </clause>

  <clause id="4246">
    <literal><![CDATA[
      x v (y v (z ^ x)) = x v y
    ]]></literal>
    <justification jstring="[para(16(a,1),2012(a,1,2)),rewrite([8(2),14(3),2(2),16(6)])].">
      <j1 rule="para" parents="16 2012"/>
      <j2 rule="rewrite" parents="8 14 2 16"/>
    </justification>
  </clause>

  <clause id="4417">
    <literal><![CDATA[
      (x ^ y) v ((z ^ u) v (y ^ (x v w))) = (z ^ u) v (y ^ (x v w))
    ]]></literal>
    <justification jstring="[para(5(a,1),245(a,1,1))].">
      <j1 rule="para" parents="5 245"/>
    </justification>
  </clause>

  <clause id="4466">
    <literal><![CDATA[
      (x ^ y) v (x ^ z) = x ^ ((x ^ z) v (y ^ (x v z)))
    ]]></literal>
    <justification jstring="[back_rewrite(570),rewrite([4417(6)]),flip(a)].">
      <j1 rule="back_rewrite" parents="570"/>
      <j2 rule="rewrite" parents="4417"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13630">
    <literal><![CDATA[
      x v (y ^ ((x ^ y) v (z ^ (x v y)))) = x v (y ^ z)
    ]]></literal>
    <justification jstring="[para(4466(a,1),4246(a,1,2)),rewrite([5(1),2(2)])].">
      <j1 rule="para" parents="4466 4246"/>
      <j2 rule="rewrite" parents="5 2"/>
    </justification>
  </clause>

  <clause id="13740">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(H2)
    ]]></attribute>
    <justification jstring="[back_rewrite(13),rewrite([13630(14),5(5)]),xx(a)].">
      <j1 rule="back_rewrite" parents="13"/>
      <j2 rule="rewrite" parents="13630 5"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
