<?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.out</source>

<heading><![CDATA[
Prover9 (32) version November-2006, November 2006.
Process 4385 was started by mccune on cleo.thornwood,
Wed Nov 22 12:03:35 2006
The command was "/home/mccune/bin/prover9 -f LT.in LT-interp.outx".
]]></heading>

<proof number="1" length="58" max_count="29">

<comments><![CDATA[
% Proof 1 at 0.92 (+ 0.00) seconds: H2.
% Length of proof is 58.
% Level of proof is 20.
% Maximum clause weight is 29.
% Given clauses 100.
]]></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(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>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <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[
      c7 ^ (c8 v (c9 ^ ((c7 ^ (c8 v c9)) v (c8 ^ c9)))) != c7 ^ (c8 v (c7 ^ c9))
    ]]></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[
      c7 ^ (c8 v (c9 ^ ((c8 ^ c9) v (c7 ^ (c8 v c9))))) != c7 ^ (c8 v (c7 ^ c9))
    ]]></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>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <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 ^ ((y ^ (z v x)) v (z ^ (y v x))) = (z ^ x) v (y ^ x)
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(18(a,1),5(a,2)),rewrite(2(3),2(5),5(6),5(8),2(9))].">
      <j1 rule="para" parents="18 5"/>
      <j2 rule="rewrite" parents="2 2 5 5 2"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      (x v y) ^ (x v (z ^ (x v (x v y)))) = x v (z ^ (x v y))
    ]]></literal>
    <justification jstring="[para(8(a,1),18(a,2,1)),rewrite(2(6),14(6),2(5),8(7),2(5),5(8))].">
      <j1 rule="para" parents="8 18"/>
      <j2 rule="rewrite" parents="2 14 2 8 2 5"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      x ^ (x ^ y) = x ^ y
    ]]></literal>
    <justification jstring="[para(9(a,1),22(a,2)),rewrite(5(1),2(2),5(4),2(5),9(5),5(4),2(5),9(5),5(3))].">
      <j1 rule="para" parents="9 22"/>
      <j2 rule="rewrite" parents="5 2 5 2 9 5 2 9 5"/>
    </justification>
  </clause>

  <clause id="28">
    <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="30">
    <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="31">
    <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="32">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(14(a,1),17(a,1)),rewrite(2(2),14(2),17(1))].">
      <j1 rule="para" parents="14 17"/>
      <j2 rule="rewrite" parents="2 14 17"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      x ^ ((x v y) v (y ^ (x v y))) = x
    ]]></literal>
    <justification jstring="[back_rewrite(28),rewrite(32(3))].">
      <j1 rule="back_rewrite" parents="28"/>
      <j2 rule="rewrite" parents="32"/>
    </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="[back_rewrite(23),rewrite(32(3))].">
      <j1 rule="back_rewrite" parents="23"/>
      <j2 rule="rewrite" parents="32"/>
    </justification>
  </clause>

  <clause id="38">
    <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="39">
    <literal><![CDATA[
      x ^ (y ^ ((z ^ (x v u)) v (u ^ (x v z)))) = y ^ ((x ^ u) v (x ^ z))
    ]]></literal>
    <justification jstring="[para(11(a,1),15(a,2,2)),rewrite(2(5),2(10))].">
      <j1 rule="para" parents="11 15"/>
      <j2 rule="rewrite" parents="2 2"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      (x ^ (y v z)) ^ ((x ^ y) v (y ^ z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(11(a,1),38(a,1,2)),rewrite(5(4),2(5),5(9),15(9),8(8))].">
      <j1 rule="para" parents="11 38"/>
      <j2 rule="rewrite" parents="5 2 5 15 8"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      (x v y) ^ ((x ^ z) v (x ^ u)) = (x ^ z) v (x ^ u)
    ]]></literal>
    <justification jstring="[para(11(a,1),38(a,2)),rewrite(2(5),5(7),39(8),2(8))].">
      <j1 rule="para" parents="11 38"/>
      <j2 rule="rewrite" parents="2 5 39 2"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      x ^ (y ^ (z ^ (x v u))) = x ^ (y ^ z)
    ]]></literal>
    <justification jstring="[para(38(a,1),15(a,2,2)),rewrite(15(6))].">
      <j1 rule="para" parents="38 15"/>
      <j2 rule="rewrite" parents="15"/>
    </justification>
  </clause>

  <clause id="70">
    <literal><![CDATA[
      x ^ ((y ^ z) ^ (y v u)) = x ^ (y ^ z)
    ]]></literal>
    <justification jstring="[para(11(a,1),47(a,1,2,2)),rewrite(5(4),2(5),15(7),44(6),15(2),5(1),5(5),5(6),15(6)),flip(a)].">
      <j1 rule="para" parents="11 47"/>
      <j2 rule="rewrite" parents="5 2 15 44 15 5 5 5 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="87">
    <literal><![CDATA[
      (x ^ y) ^ (x v z) = x ^ y
    ]]></literal>
    <justification jstring="[para(16(a,1),70(a,2)),rewrite(25(5))].">
      <j1 rule="para" parents="16 70"/>
      <j2 rule="rewrite" parents="25"/>
    </justification>
  </clause>

  <clause id="92">
    <literal><![CDATA[
      (x ^ y) ^ (y v z) = x ^ y
    ]]></literal>
    <justification jstring="[para(5(a,1),87(a,1,1)),rewrite(5(4))].">
      <j1 rule="para" parents="5 87"/>
      <j2 rule="rewrite" parents="5"/>
    </justification>
  </clause>

  <clause id="99">
    <literal><![CDATA[
      (x ^ y) ^ (z v (x ^ y)) = x ^ y
    ]]></literal>
    <justification jstring="[para(87(a,1),87(a,1,1)),rewrite(2(3),87(7))].">
      <j1 rule="para" parents="87 87"/>
      <j2 rule="rewrite" parents="2 87"/>
    </justification>
  </clause>

  <clause id="109">
    <literal><![CDATA[
      x ^ (y v ((x v z) v (z ^ (x v z)))) = x
    ]]></literal>
    <justification jstring="[para(34(a,1),92(a,1,1)),rewrite(2(5),34(11))].">
      <j1 rule="para" parents="34 92"/>
      <j2 rule="rewrite" parents="2 34"/>
    </justification>
  </clause>

  <clause id="126">
    <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),31(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 31"/>
      <j2 rule="rewrite" parents="2 14 2 5 17 2 5 2"/>
    </justification>
  </clause>

  <clause id="180">
    <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="193">
    <literal><![CDATA[
      (x v y) ^ (x v ((y v z) v (z ^ (y v z)))) = x v y
    ]]></literal>
    <justification jstring="[para(109(a,1),35(a,1,2,2)),rewrite(5(7),109(13))].">
      <j1 rule="para" parents="109 35"/>
      <j2 rule="rewrite" parents="5 109"/>
    </justification>
  </clause>

  <clause id="223">
    <literal><![CDATA[
      (x v y) v (y ^ (x v y)) = x v y
    ]]></literal>
    <justification jstring="[para(34(a,1),180(a,1,2,2)),rewrite(2(6),180(6),17(3)),flip(a)].">
      <j1 rule="para" parents="34 180"/>
      <j2 rule="rewrite" parents="2 180 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="228">
    <literal><![CDATA[
      (x v y) ^ (x v (y v z)) = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(193),rewrite(223(5))].">
      <j1 rule="back_rewrite" parents="193"/>
      <j2 rule="rewrite" parents="223"/>
    </justification>
  </clause>

  <clause id="229">
    <literal><![CDATA[
      (x v y) v (z ^ (x v y)) = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(126),rewrite(228(7),2(5),8(6)),flip(a)].">
      <j1 rule="back_rewrite" parents="126"/>
      <j2 rule="rewrite" parents="228 2 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="236">
    <literal><![CDATA[
      x ^ ((x v y) ^ (z v u)) = x ^ (z v u)
    ]]></literal>
    <justification jstring="[para(228(a,1),47(a,2,2)),rewrite(5(5),15(6),228(5))].">
      <j1 rule="para" parents="228 47"/>
      <j2 rule="rewrite" parents="5 15 228"/>
    </justification>
  </clause>

  <clause id="244">
    <literal><![CDATA[
      (x ^ y) ^ (z v u) = x ^ (y ^ (z v u))
    ]]></literal>
    <justification jstring="[para(229(a,1),47(a,1,2,2,2)),rewrite(15(5),15(4),16(3),5(6)),flip(a)].">
      <j1 rule="para" parents="229 47"/>
      <j2 rule="rewrite" parents="15 15 16 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="285">
    <literal><![CDATA[
      x ^ (y ^ (z v (x ^ y))) = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(99),rewrite(244(4))].">
      <j1 rule="back_rewrite" parents="99"/>
      <j2 rule="rewrite" parents="244"/>
    </justification>
  </clause>

  <clause id="296">
    <literal><![CDATA[
      x ^ (y v (x ^ (y v z))) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(35(a,1),285(a,1,2))].">
      <j1 rule="para" parents="35 285"/>
    </justification>
  </clause>

  <clause id="318">
    <literal><![CDATA[
      x ^ ((y ^ x) v (y ^ z)) = y ^ x
    ]]></literal>
    <justification jstring="[para(8(a,1),39(a,1,2)),rewrite(38(3),2(6),244(7),45(6)),flip(a)].">
      <j1 rule="para" parents="8 39"/>
      <j2 rule="rewrite" parents="38 2 244 45"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="349">
    <literal><![CDATA[
      (x v y) ^ ((z v u) ^ ((x ^ v) v (x ^ (z v u)))) = x ^ (z v u)
    ]]></literal>
    <justification jstring="[para(236(a,1),318(a,1,2,1)),rewrite(2(7),244(8),236(12))].">
      <j1 rule="para" parents="236 318"/>
      <j2 rule="rewrite" parents="2 244 236"/>
    </justification>
  </clause>

  <clause id="434">
    <literal><![CDATA[
      (x v y) ^ ((z v u) ^ ((x ^ v) v (x ^ w))) = (z v u) ^ ((x ^ v) v (x ^ w))
    ]]></literal>
    <justification jstring="[para(45(a,1),244(a,1,1)),rewrite(5(5),5(11)),flip(a)].">
      <j1 rule="para" parents="45 244"/>
      <j2 rule="rewrite" parents="5 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="438">
    <literal><![CDATA[
      (x v y) ^ ((z ^ u) v (z ^ (x v y))) = z ^ (x v y)
    ]]></literal>
    <justification jstring="[back_rewrite(349),rewrite(434(8))].">
      <j1 rule="back_rewrite" parents="349"/>
      <j2 rule="rewrite" parents="434"/>
    </justification>
  </clause>

  <clause id="470">
    <literal><![CDATA[
      x ^ ((x ^ y) v ((x ^ z) v (y ^ (x v z)))) = (x ^ y) v (x ^ z)
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(11(a,1),296(a,1,2,2)),rewrite(2(5),14(6),2(5),2(12),11(13))].">
      <j1 rule="para" parents="11 296"/>
      <j2 rule="rewrite" parents="2 14 2 2 11"/>
    </justification>
  </clause>

  <clause id="471">
    <literal><![CDATA[
      (x ^ y) v (x ^ (y v z)) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(296(a,1),18(a,1,2,1)),rewrite(2(7),9(7),5(5),2(6),244(7),438(6),25(3),15(5),8(4),5(6),25(6)),flip(a)].">
      <j1 rule="para" parents="296 18"/>
      <j2 rule="rewrite" parents="2 9 5 2 244 438 25 15 8 5 25"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="488">
    <literal><![CDATA[
      x ^ ((x ^ y) v ((x ^ z) v (z ^ (x v y)))) = (x ^ y) v (x ^ z)
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(2(a,1),470(a,1,2)),rewrite(2(6),14(6),2(10))].">
      <j1 rule="para" parents="2 470"/>
      <j2 rule="rewrite" parents="2 14 2"/>
    </justification>
  </clause>

  <clause id="631">
    <literal><![CDATA[
      (x ^ y) v (y ^ (x v z)) = y ^ (x v z)
    ]]></literal>
    <justification jstring="[para(5(a,1),471(a,1,1))].">
      <j1 rule="para" parents="5 471"/>
    </justification>
  </clause>

  <clause id="656">
    <literal><![CDATA[
      (x ^ y) v (x ^ z) = x ^ ((x ^ y) v (z ^ (x v y)))
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(488(a,1),471(a,1,2)),rewrite(25(2),32(5),631(8))].">
      <j1 rule="para" parents="488 471"/>
      <j2 rule="rewrite" parents="25 32 631"/>
    </justification>
  </clause>

  <clause id="705">
    <literal><![CDATA[
      (x ^ y) v (y ^ z) = y ^ ((x ^ y) v (z ^ (x v y)))
    ]]></literal>
    <attribute><![CDATA[
      label(false)
    ]]></attribute>
    <justification jstring="[para(5(a,1),656(a,1,1)),rewrite(5(4),2(5))].">
      <j1 rule="para" parents="5 656"/>
      <j2 rule="rewrite" parents="5 2"/>
    </justification>
  </clause>

  <clause id="792">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(H2)
    ]]></attribute>
    <justification jstring="[para(705(a,2),13(a,1,2,2)),rewrite(5(8),2(9),30(10)),xx(a)].">
      <j1 rule="para" parents="705 13"/>
      <j2 rule="rewrite" parents="5 2 30"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
