<?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 November-2006, November 2006.
Process 4386 was started by mccune on cleo.thornwood,
Wed Nov 22 12:03:36 2006
The command was "/home/mccune/bin/prover9 -t 7200 -f LT.in".
]]></heading>

<proof number="1" length="82" max_count="29">

<comments><![CDATA[
% Proof 1 at 24.37 (+ 0.20) seconds: H2.
% Length of proof is 82.
% Level of proof is 19.
% Maximum clause weight is 29.
% Given clauses 550.
]]></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>
    <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="24">
    <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="25">
    <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="26">
    <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="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="31">
    <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="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="37">
    <literal><![CDATA[
      x v (y v (z v (x ^ u))) = x v (y v z)
    ]]></literal>
    <justification jstring="[para(24(a,1),14(a,2,2)),rewrite(14(6))].">
      <j1 rule="para" parents="24 14"/>
      <j2 rule="rewrite" parents="14"/>
    </justification>
  </clause>

  <clause id="39">
    <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="40">
    <literal><![CDATA[
      (x v y) ^ ((x ^ z) v (x ^ u)) = (x ^ z) v (x ^ u)
    ]]></literal>
    <justification jstring="[para(11(a,1),30(a,2)),rewrite(2(5),5(7),31(8),2(8))].">
      <j1 rule="para" parents="11 30"/>
      <j2 rule="rewrite" parents="2 5 31 2"/>
    </justification>
  </clause>

  <clause id="41">
    <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="48">
    <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="51">
    <literal><![CDATA[
      x ^ ((y ^ z) ^ (y v u)) = x ^ (y ^ z)
    ]]></literal>
    <justification jstring="[para(11(a,1),41(a,1,2,2)),rewrite(5(4),2(5),15(7),39(6),15(2),5(1),5(5),5(6),15(6)),flip(a)].">
      <j1 rule="para" parents="11 41"/>
      <j2 rule="rewrite" parents="5 2 15 39 15 5 5 5 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      x ^ (y ^ (z ^ (u ^ (x v v)))) = x ^ (y ^ (z ^ u))
    ]]></literal>
    <justification jstring="[para(41(a,1),15(a,2,2)),rewrite(15(8))].">
      <j1 rule="para" parents="41 15"/>
      <j2 rule="rewrite" parents="15"/>
    </justification>
  </clause>

  <clause id="57">
    <literal><![CDATA[
      (x ^ y) ^ (x v z) = x ^ y
    ]]></literal>
    <justification jstring="[para(16(a,1),51(a,2)),rewrite(32(5))].">
      <j1 rule="para" parents="16 51"/>
      <j2 rule="rewrite" parents="32"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      (x ^ y) ^ (y v z) = x ^ y
    ]]></literal>
    <justification jstring="[para(5(a,1),57(a,1,1)),rewrite(5(4))].">
      <j1 rule="para" parents="5 57"/>
      <j2 rule="rewrite" parents="5"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      (x ^ y) ^ ((x v z) ^ (u v (x ^ y))) = x ^ y
    ]]></literal>
    <justification jstring="[para(57(a,1),30(a,2)),rewrite(2(4))].">
      <j1 rule="para" parents="57 30"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      (x v y) ^ ((x ^ z) v (u ^ (x v y))) = (x ^ z) v (u ^ (x v y))
    ]]></literal>
    <justification jstring="[para(57(a,1),18(a,2,1)),rewrite(14(4),2(3),24(4),2(6),14(6),2(5),57(7),2(5),5(9))].">
      <j1 rule="para" parents="57 18"/>
      <j2 rule="rewrite" parents="14 2 24 2 14 2 57 2 5"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      (x ^ y) ^ (z v (x ^ y)) = x ^ y
    ]]></literal>
    <justification jstring="[para(57(a,1),57(a,1,1)),rewrite(2(3),57(7))].">
      <j1 rule="para" parents="57 57"/>
      <j2 rule="rewrite" parents="2 57"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      (x ^ y) ^ (z v (y ^ (x v u))) = x ^ y
    ]]></literal>
    <justification jstring="[para(30(a,1),62(a,1,1)),rewrite(2(4),30(8))].">
      <j1 rule="para" parents="30 62"/>
      <j2 rule="rewrite" parents="2 30"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) ^ (u v ((z ^ (x v y)) v (x ^ (y v z)))) = (x ^ y) v (y ^ z)
    ]]></literal>
    <justification jstring="[para(18(a,1),62(a,1,1)),rewrite(2(9),18(16))].">
      <j1 rule="para" parents="18 62"/>
      <j2 rule="rewrite" parents="2 18"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      (x ^ y) v (x ^ (y ^ z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(69(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="69 18"/>
      <j2 rule="rewrite" parents="17 15 5 8 15 5 16 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="93">
    <literal><![CDATA[
      ((x v y) ^ (z v u)) ^ ((x ^ y) v (x ^ (z v u))) = x ^ ((x v y) ^ (z v u))
    ]]></literal>
    <justification jstring="[para(25(a,1),30(a,1,2)),rewrite(5(12))].">
      <j1 rule="para" parents="25 30"/>
      <j2 rule="rewrite" parents="5"/>
    </justification>
  </clause>

  <clause id="94">
    <literal><![CDATA[
      (x ^ y) ^ ((x v z) ^ (u v v)) = x ^ (y ^ ((x v z) ^ (u v v)))
    ]]></literal>
    <justification jstring="[para(25(a,1),41(a,1,2,2)),rewrite(15(9),93(8),15(5),5(9),5(10)),flip(a)].">
      <j1 rule="para" parents="25 41"/>
      <j2 rule="rewrite" parents="15 93 15 5 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="98">
    <literal><![CDATA[
      x ^ (y ^ ((x v z) ^ (u v (x ^ y)))) = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(65),rewrite(94(6))].">
      <j1 rule="back_rewrite" parents="65"/>
      <j2 rule="rewrite" parents="94"/>
    </justification>
  </clause>

  <clause id="103">
    <literal><![CDATA[
      x v ((x ^ y) v (z ^ u)) = x v (z ^ u)
    ]]></literal>
    <justification jstring="[para(82(a,1),37(a,2,2)),rewrite(2(5),14(6),82(5))].">
      <j1 rule="para" parents="82 37"/>
      <j2 rule="rewrite" parents="2 14 82"/>
    </justification>
  </clause>

  <clause id="111">
    <literal><![CDATA[
      x ^ (y v ((x v z) ^ (x v u))) = x
    ]]></literal>
    <justification jstring="[para(8(a,1),74(a,1,1)),rewrite(8(7))].">
      <j1 rule="para" parents="8 74"/>
      <j2 rule="rewrite" parents="8"/>
    </justification>
  </clause>

  <clause id="147">
    <literal><![CDATA[
      (x ^ y) v (y ^ ((x ^ z) v (x ^ u))) = x ^ y
    ]]></literal>
    <justification jstring="[para(31(a,1),82(a,1,2))].">
      <j1 rule="para" parents="31 82"/>
    </justification>
  </clause>

  <clause id="215">
    <literal><![CDATA[
      (x ^ y) v (x ^ (y v z)) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(39(a,1),9(a,1,2)),rewrite(2(4))].">
      <j1 rule="para" parents="39 9"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="242">
    <literal><![CDATA[
      (x ^ y) v ((z ^ u) v (x ^ (y v v))) = (z ^ u) v (x ^ (y v v))
    ]]></literal>
    <justification jstring="[para(39(a,1),103(a,1,2,1)),rewrite(14(6),2(5),2(10))].">
      <j1 rule="para" parents="39 103"/>
      <j2 rule="rewrite" parents="14 2 2"/>
    </justification>
  </clause>

  <clause id="300">
    <literal><![CDATA[
      (x ^ y) v (y ^ (x v z)) = y ^ (x v z)
    ]]></literal>
    <justification jstring="[para(5(a,1),215(a,1,1))].">
      <j1 rule="para" parents="5 215"/>
    </justification>
  </clause>

  <clause id="362">
    <literal><![CDATA[
      x v ((x v y) ^ (x v z)) = (x v y) ^ (x v z)
    ]]></literal>
    <justification jstring="[para(8(a,1),300(a,1,1))].">
      <j1 rule="para" parents="8 300"/>
    </justification>
  </clause>

  <clause id="376">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(111(a,1),300(a,1,2)),rewrite(2(2),111(7))].">
      <j1 rule="para" parents="111 300"/>
      <j2 rule="rewrite" parents="2 111"/>
    </justification>
  </clause>

  <clause id="414">
    <literal><![CDATA[
      x ^ ((x v y) ^ (z v u)) = x ^ (z v u)
    ]]></literal>
    <justification jstring="[para(215(a,1),40(a,1,2)),rewrite(15(4),215(8))].">
      <j1 rule="para" parents="215 40"/>
      <j2 rule="rewrite" parents="15 215"/>
    </justification>
  </clause>

  <clause id="416">
    <literal><![CDATA[
      ((x v y) ^ (z v u)) ^ ((x ^ y) v (x ^ (z v u))) = x ^ (z v u)
    ]]></literal>
    <justification jstring="[back_rewrite(93),rewrite(414(12))].">
      <j1 rule="back_rewrite" parents="93"/>
      <j2 rule="rewrite" parents="414"/>
    </justification>
  </clause>

  <clause id="440">
    <literal><![CDATA[
      (x v y) ^ (x v (y v z)) = x v y
    ]]></literal>
    <justification jstring="[para(16(a,1),414(a,2)),rewrite(2(3),14(3),2(2),5(5),32(6))].">
      <j1 rule="para" parents="16 414"/>
      <j2 rule="rewrite" parents="2 14 2 5 32"/>
    </justification>
  </clause>

  <clause id="441">
    <literal><![CDATA[
      x ^ (y ^ ((x v z) ^ (u v v))) = x ^ (y ^ (u v v))
    ]]></literal>
    <justification jstring="[para(414(a,1),15(a,2,2)),rewrite(15(8))].">
      <j1 rule="para" parents="414 15"/>
      <j2 rule="rewrite" parents="15"/>
    </justification>
  </clause>

  <clause id="454">
    <literal><![CDATA[
      (x ^ y) ^ ((y ^ (x v z)) ^ (u v v)) = (x ^ y) ^ (u v v)
    ]]></literal>
    <justification jstring="[para(300(a,1),414(a,1,2,1))].">
      <j1 rule="para" parents="300 414"/>
    </justification>
  </clause>

  <clause id="458">
    <literal><![CDATA[
      x ^ (y ^ (z v (x ^ y))) = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(98),rewrite(441(6))].">
      <j1 rule="back_rewrite" parents="98"/>
      <j2 rule="rewrite" parents="441"/>
    </justification>
  </clause>

  <clause id="459">
    <literal><![CDATA[
      (x ^ y) ^ ((x v z) ^ (u v v)) = x ^ (y ^ (u v v))
    ]]></literal>
    <justification jstring="[back_rewrite(94),rewrite(441(10))].">
      <j1 rule="back_rewrite" parents="94"/>
      <j2 rule="rewrite" parents="441"/>
    </justification>
  </clause>

  <clause id="678">
    <literal><![CDATA[
      x ^ (y v (x ^ (y v z))) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(48(a,1),458(a,1,2))].">
      <j1 rule="para" parents="48 458"/>
    </justification>
  </clause>

  <clause id="683">
    <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),678(a,1,2,2)),rewrite(2(5),14(6),2(5),2(12),11(13))].">
      <j1 rule="para" parents="11 678"/>
      <j2 rule="rewrite" parents="2 14 2 2 11"/>
    </justification>
  </clause>

  <clause id="696">
    <literal><![CDATA[
      x ^ ((x ^ y) v (y ^ z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(147(a,1),678(a,1,2)),rewrite(5(1),32(2),5(2)),flip(a)].">
      <j1 rule="para" parents="147 678"/>
      <j2 rule="rewrite" parents="5 32 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="711">
    <literal><![CDATA[
      x ^ ((y ^ z) v (y ^ (x v u))) = x ^ y
    ]]></literal>
    <justification jstring="[para(696(a,1),414(a,1,2)),rewrite(5(2),30(3),5(3),2(5)),flip(a)].">
      <j1 rule="para" parents="696 414"/>
      <j2 rule="rewrite" parents="5 30 5 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="720">
    <literal><![CDATA[
      x ^ ((y ^ z) v (z ^ (x v u))) = x ^ z
    ]]></literal>
    <justification jstring="[para(5(a,1),711(a,1,2,1))].">
      <j1 rule="para" parents="5 711"/>
    </justification>
  </clause>

  <clause id="760">
    <literal><![CDATA[
      x ^ ((x ^ y) v (z ^ y)) = x ^ y
    ]]></literal>
    <justification jstring="[para(9(a,1),720(a,1,2,2,2)),rewrite(5(2),2(3))].">
      <j1 rule="para" parents="9 720"/>
      <j2 rule="rewrite" parents="5 2"/>
    </justification>
  </clause>

  <clause id="794">
    <literal><![CDATA[
      x ^ ((y ^ z) v (x ^ (y v u))) = x ^ (y v u)
    ]]></literal>
    <justification jstring="[para(57(a,1),760(a,1,2,2)),rewrite(2(4))].">
      <j1 rule="para" parents="57 760"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="814">
    <literal><![CDATA[
      x ^ ((y ^ z) ^ (u v v)) = x ^ (y ^ (z ^ (u v v)))
    ]]></literal>
    <justification jstring="[para(25(a,1),54(a,1,2,2,2)),rewrite(15(10),15(9),416(8),15(3),15(4),5(8),15(9),5(10),459(10)),flip(a)].">
      <j1 rule="para" parents="25 54"/>
      <j2 rule="rewrite" parents="15 15 416 15 15 5 15 5 459"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="834">
    <literal><![CDATA[
      (x ^ y) ^ (z v u) = x ^ (y ^ (z v u))
    ]]></literal>
    <justification jstring="[back_rewrite(454),rewrite(814(6),15(6),459(5),15(4),32(3)),flip(a)].">
      <j1 rule="back_rewrite" parents="454"/>
      <j2 rule="rewrite" parents="814 15 459 15 32"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1033">
    <literal><![CDATA[
      (x v y) v (z ^ (u ^ (x v y))) = x v y
    ]]></literal>
    <justification jstring="[para(834(a,1),376(a,1,2))].">
      <j1 rule="para" parents="834 376"/>
    </justification>
  </clause>

  <clause id="1137">
    <literal><![CDATA[
      x v (y v (z v (u ^ (x v y)))) = x v (y v z)
    ]]></literal>
    <justification jstring="[para(440(a,1),1033(a,1,2,2)),rewrite(2(5),14(5),14(4),2(3))].">
      <j1 rule="para" parents="440 1033"/>
      <j2 rule="rewrite" parents="2 14 14 2"/>
    </justification>
  </clause>

  <clause id="1201">
    <literal><![CDATA[
      x v (y v ((x v z) ^ (x v u))) = y v ((x v z) ^ (x v u))
    ]]></literal>
    <justification jstring="[para(362(a,1),14(a,2,2))].">
      <j1 rule="para" parents="362 14"/>
    </justification>
  </clause>

  <clause id="1346">
    <literal><![CDATA[
      x ^ (y v (x ^ (z v x))) = x
    ]]></literal>
    <justification jstring="[para(16(a,1),77(a,1,1,2)),rewrite(2(2),376(2),17(3),2(4),300(4),16(6),2(6),376(6))].">
      <j1 rule="para" parents="16 77"/>
      <j2 rule="rewrite" parents="2 376 17 2 300 16 2 376"/>
    </justification>
  </clause>

  <clause id="1363">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(1346(a,1),5(a,2)),rewrite(2(1),8(2),5(2))].">
      <j1 rule="para" parents="1346 5"/>
      <j2 rule="rewrite" parents="2 8 5"/>
    </justification>
  </clause>

  <clause id="1367">
    <literal><![CDATA[
      (x v y) ^ (y v (x ^ z)) = y v (x ^ z)
    ]]></literal>
    <justification jstring="[para(1346(a,1),67(a,1,2,2)),rewrite(2(1),8(2),2(3),2(6),8(7),1363(7),2(6))].">
      <j1 rule="para" parents="1346 67"/>
      <j2 rule="rewrite" parents="2 8 2 2 8 1363 2"/>
    </justification>
  </clause>

  <clause id="1683">
    <literal><![CDATA[
      x ^ ((y ^ z) v (x ^ (z v u))) = x ^ (z v u)
    ]]></literal>
    <justification jstring="[para(5(a,1),794(a,1,2,1))].">
      <j1 rule="para" parents="5 794"/>
    </justification>
  </clause>

  <clause id="1974">
    <literal><![CDATA[
      x v ((y v x) ^ (y v z)) = y v x
    ]]></literal>
    <justification jstring="[para(362(a,1),1137(a,1,2,2)),rewrite(5(3),1201(5),2(5),26(6))].">
      <j1 rule="para" parents="362 1137"/>
      <j2 rule="rewrite" parents="5 1201 2 26"/>
    </justification>
  </clause>

  <clause id="2006">
    <literal><![CDATA[
      x v ((y v z) ^ (y v (x ^ u))) = x v y
    ]]></literal>
    <justification jstring="[para(1974(a,1),103(a,1,2)),rewrite(24(3),5(5)),flip(a)].">
      <j1 rule="para" parents="1974 103"/>
      <j2 rule="rewrite" parents="24 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2093">
    <literal><![CDATA[
      (x v y) ^ (x v (z v y)) = x v y
    ]]></literal>
    <justification jstring="[para(2006(a,1),1367(a,1,2)),rewrite(2(2),14(2),2(1),2(3),5(4),2006(9),2(5))].">
      <j1 rule="para" parents="2006 1367"/>
      <j2 rule="rewrite" parents="2 14 2 2 5 2006 2"/>
    </justification>
  </clause>

  <clause id="2111">
    <literal><![CDATA[
      (x v y) ^ (x v (z ^ y)) = x v (z ^ y)
    ]]></literal>
    <justification jstring="[para(376(a,1),2093(a,1,2,2)),rewrite(5(4))].">
      <j1 rule="para" parents="376 2093"/>
      <j2 rule="rewrite" parents="5"/>
    </justification>
  </clause>

  <clause id="3428">
    <literal><![CDATA[
      x v (y v (z ^ x)) = x v y
    ]]></literal>
    <justification jstring="[para(1683(a,1),2111(a,1)),rewrite(14(3),2(2),5(5),440(5),14(5),2(4),5(7),440(7),14(4),2(3)),flip(a)].">
      <j1 rule="para" parents="1683 2111"/>
      <j2 rule="rewrite" parents="14 2 5 440 14 2 5 440 14 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4174">
    <literal><![CDATA[
      (x ^ y) v ((z ^ u) v (y ^ (x v v))) = (z ^ u) v (y ^ (x v v))
    ]]></literal>
    <justification jstring="[para(5(a,1),242(a,1,1))].">
      <j1 rule="para" parents="5 242"/>
    </justification>
  </clause>

  <clause id="4221">
    <literal><![CDATA[
      (x ^ y) v (x ^ z) = x ^ ((x ^ z) v (y ^ (x v z)))
    ]]></literal>
    <justification jstring="[back_rewrite(683),rewrite(4174(6)),flip(a)].">
      <j1 rule="back_rewrite" parents="683"/>
      <j2 rule="rewrite" parents="4174"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13496">
    <literal><![CDATA[
      x v (y ^ ((x ^ y) v (z ^ (x v y)))) = x v (y ^ z)
    ]]></literal>
    <justification jstring="[para(4221(a,1),3428(a,1,2)),rewrite(5(1),2(2))].">
      <j1 rule="para" parents="4221 3428"/>
      <j2 rule="rewrite" parents="5 2"/>
    </justification>
  </clause>

  <clause id="13559">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(H2)
    ]]></attribute>
    <justification jstring="[back_rewrite(13),rewrite(13496(14),5(5)),xx(a)].">
      <j1 rule="back_rewrite" parents="13"/>
      <j2 rule="rewrite" parents="13496 5"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
