<?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>t4_12.out</source>

<heading><![CDATA[
Prover9 (32) version 2009-02A, February 2009.
Process 11683 was started by mccune on cleo,
Wed Feb 25 09:36:39 2009
The command was "/home/mccune/bin/prover9 -f head.in t4_12.in".
]]></heading>

<proof number="1" length="78" max_count="59">

<comments><![CDATA[
% Proof 1 at 1.97 (+ 0.03) seconds.
% Length of proof is 78.
% Level of proof is 17.
% Maximum clause weight is 19.
% Given clauses 298.

]]></comments>

  <clause id="3" type="assumption">
    <literal><![CDATA[
      (all x all y all z (B(x,y,z) <-> (x ^ y) v (y ^ z) = y & (x v y) ^ (y v z) = y))
    ]]></literal>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="9" type="goal">
    <literal><![CDATA[
      B(x,y,z) & B(x,z,u) -> B(x,y,u)
    ]]></literal>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="10" type="expand_def">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z) = y & (x v y) ^ (y v z) = y) & (x ^ z) v (z ^ u) = z & (x v z) ^ (z v u) = z -> (x ^ y) v (y ^ u) = y & (x v y) ^ (y v u) = y
    ]]></literal>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[expand_def(9,3)].">
      <j1 rule="expand_def" parents="9 3"/>
    </justification>
  </clause>

  <clause id="11" type="assumption">
    <literal><![CDATA[
      x ^ (y v (x ^ z)) = (x ^ y) v (x ^ z)
    ]]></literal>
    <attribute><![CDATA[
      label("modularity_2")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      (x ^ y) v (x ^ z) = x ^ (y v (x ^ z))
    ]]></literal>
    <justification jstring="[copy(11),flip(a)].">
      <j1 rule="copy" parents="11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13" type="assumption">
    <literal><![CDATA[
      x v (y ^ (x v z)) = (x v y) ^ (x v z)
    ]]></literal>
    <attribute><![CDATA[
      label("modularity_1")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="14" type="assumption">
    <literal><![CDATA[
      (x ^ y) v x = x
    ]]></literal>
    <attribute><![CDATA[
      label("absorption_2")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="15" type="assumption">
    <literal><![CDATA[
      (x v y) ^ x = x
    ]]></literal>
    <attribute><![CDATA[
      label("absorption_1")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="16" type="assumption">
    <literal><![CDATA[
      (x v y) v z = x v (y v z)
    ]]></literal>
    <attribute><![CDATA[
      label("associativity_join")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="17" type="assumption">
    <literal><![CDATA[
      (x ^ y) ^ z = x ^ (y ^ z)
    ]]></literal>
    <attribute><![CDATA[
      label("associativity_meet")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="18" type="assumption">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <attribute><![CDATA[
      label("commutativity_join")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="19" type="assumption">
    <literal><![CDATA[
      x ^ y = y ^ x
    ]]></literal>
    <attribute><![CDATA[
      label("commutativity_meet")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="20" type="deny">
    <literal><![CDATA[
      (c1 ^ c2) v (c2 ^ c3) = c2
    ]]></literal>
    <justification jstring="[deny(10)].">
      <j1 rule="deny" parents="10"/>
    </justification>
  </clause>

  <clause id="21" type="deny">
    <literal><![CDATA[
      (c1 v c2) ^ (c2 v c3) = c2
    ]]></literal>
    <justification jstring="[deny(10)].">
      <j1 rule="deny" parents="10"/>
    </justification>
  </clause>

  <clause id="22" type="deny">
    <literal><![CDATA[
      (c1 ^ c3) v (c3 ^ c4) = c3
    ]]></literal>
    <justification jstring="[deny(10)].">
      <j1 rule="deny" parents="10"/>
    </justification>
  </clause>

  <clause id="23" type="deny">
    <literal><![CDATA[
      (c1 v c3) ^ (c3 v c4) = c3
    ]]></literal>
    <justification jstring="[deny(10)].">
      <j1 rule="deny" parents="10"/>
    </justification>
  </clause>

  <clause id="24" type="deny">
    <literal><![CDATA[
      (c1 ^ c2) v (c2 ^ c4) != c2
    ]]></literal>
    <literal><![CDATA[
      (c1 v c2) ^ (c2 v c4) != c2
    ]]></literal>
    <justification jstring="[deny(10)].">
      <j1 rule="deny" parents="10"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[back_rewrite(14),rewrite([18(2)])].">
      <j1 rule="back_rewrite" parents="14"/>
      <j2 rule="rewrite" parents="18"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[back_rewrite(15),rewrite([19(2)])].">
      <j1 rule="back_rewrite" parents="15"/>
      <j2 rule="rewrite" parents="19"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      (x ^ (y v (x ^ z))) v u = (x ^ y) v ((x ^ z) v u)
    ]]></literal>
    <justification jstring="[para(12(a,1),16(a,1,1))].">
      <j1 rule="para" parents="12 16"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      x v (y ^ (z ^ (x v u))) = (x v (y ^ z)) ^ (x v u)
    ]]></literal>
    <justification jstring="[para(17(a,1),13(a,1,2))].">
      <j1 rule="para" parents="17 13"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      x ^ (y v (x ^ z)) = x ^ (z v (x ^ y))
    ]]></literal>
    <justification jstring="[para(18(a,1),12(a,1)),rewrite([12(3)])].">
      <j1 rule="para" parents="18 12"/>
      <j2 rule="rewrite" parents="12"/>
    </justification>
  </clause>

  <clause id="37">
    <literal><![CDATA[
      x v (y ^ (z v x)) = (x v y) ^ (x v z)
    ]]></literal>
    <justification jstring="[para(18(a,1),13(a,1,2,2))].">
      <j1 rule="para" parents="18 13"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      x v (y v z) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(18(a,1),16(a,1,1)),rewrite([16(2)])].">
      <j1 rule="para" parents="18 16"/>
      <j2 rule="rewrite" parents="16"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      (x ^ y) v (y ^ z) = y ^ (x v (y ^ z))
    ]]></literal>
    <justification jstring="[para(19(a,1),12(a,1,1))].">
      <j1 rule="para" parents="19 12"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      x ^ (y ^ z) = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(19(a,1),17(a,1,1)),rewrite([17(2)])].">
      <j1 rule="para" parents="19 17"/>
      <j2 rule="rewrite" parents="17"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      c2 ^ (c1 v (c2 ^ c4)) != c2
    ]]></literal>
    <literal><![CDATA[
      (c1 v c2) ^ (c2 v c4) != c2
    ]]></literal>
    <justification jstring="[back_rewrite(24),rewrite([39(7)])].">
      <j1 rule="back_rewrite" parents="24"/>
      <j2 rule="rewrite" parents="39"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      c3 ^ (c1 v (c3 ^ c4)) = c3
    ]]></literal>
    <justification jstring="[back_rewrite(22),rewrite([39(7)])].">
      <j1 rule="back_rewrite" parents="22"/>
      <j2 rule="rewrite" parents="39"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      c2 ^ (c1 v (c2 ^ c3)) = c2
    ]]></literal>
    <justification jstring="[back_rewrite(20),rewrite([39(7)])].">
      <j1 rule="back_rewrite" parents="20"/>
      <j2 rule="rewrite" parents="39"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(25(a,1),13(a,1,2,2)),rewrite([25(5),19(4),26(4)])].">
      <j1 rule="para" parents="25 13"/>
      <j2 rule="rewrite" parents="25 19 26"/>
    </justification>
  </clause>

  <clause id="64">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(26(a,1),12(a,1,2)),rewrite([18(2),25(2),26(2)]),flip(a)].">
      <j1 rule="para" parents="26 12"/>
      <j2 rule="rewrite" parents="18 25 26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      x ^ x = x
    ]]></literal>
    <justification jstring="[para(26(a,1),12(a,2)),rewrite([12(3),25(2)])].">
      <j1 rule="para" parents="26 12"/>
      <j2 rule="rewrite" parents="12 25"/>
    </justification>
  </clause>

  <clause id="70">
    <literal><![CDATA[
      x ^ ((x v y) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(26(a,1),17(a,1,1)),flip(a)].">
      <j1 rule="para" parents="26 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="81">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(65(a,1),13(a,1,2)),rewrite([19(6),64(6)])].">
      <j1 rule="para" parents="65 13"/>
      <j2 rule="rewrite" parents="19 64"/>
    </justification>
  </clause>

  <clause id="87">
    <literal><![CDATA[
      x v (y ^ (z ^ x)) = x
    ]]></literal>
    <justification jstring="[para(17(a,1),55(a,1,2))].">
      <j1 rule="para" parents="17 55"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      x ^ (y v (z v x)) = x
    ]]></literal>
    <justification jstring="[para(16(a,1),64(a,1,2))].">
      <j1 rule="para" parents="16 64"/>
    </justification>
  </clause>

  <clause id="90">
    <literal><![CDATA[
      x ^ ((y v x) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(64(a,1),17(a,1,1)),flip(a)].">
      <j1 rule="para" parents="64 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="121">
    <literal><![CDATA[
      c1 v (c3 ^ c4) = c1 v c3
    ]]></literal>
    <justification jstring="[para(44(a,1),55(a,1,2)),rewrite([18(7),38(7),25(6)]),flip(a)].">
      <j1 rule="para" parents="44 55"/>
      <j2 rule="rewrite" parents="18 38 25"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="125">
    <literal><![CDATA[
      c1 v (c2 ^ c3) = c1 v c2
    ]]></literal>
    <justification jstring="[para(45(a,1),55(a,1,2)),rewrite([18(7),38(7),25(6)]),flip(a)].">
      <j1 rule="para" parents="45 55"/>
      <j2 rule="rewrite" parents="18 38 25"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="162">
    <literal><![CDATA[
      x v (y ^ (x ^ z)) = x
    ]]></literal>
    <justification jstring="[para(19(a,1),87(a,1,2,2))].">
      <j1 rule="para" parents="19 87"/>
    </justification>
  </clause>

  <clause id="165">
    <literal><![CDATA[
      x v (y v (z ^ x)) = x v y
    ]]></literal>
    <justification jstring="[para(26(a,1),87(a,1,2,2)),rewrite([16(3)])].">
      <j1 rule="para" parents="26 87"/>
      <j2 rule="rewrite" parents="16"/>
    </justification>
  </clause>

  <clause id="175">
    <literal><![CDATA[
      x ^ (y v (x v z)) = x
    ]]></literal>
    <justification jstring="[para(18(a,1),89(a,1,2,2))].">
      <j1 rule="para" parents="18 89"/>
    </justification>
  </clause>

  <clause id="176">
    <literal><![CDATA[
      x ^ (y ^ (z v x)) = x ^ y
    ]]></literal>
    <justification jstring="[para(25(a,1),89(a,1,2,2)),rewrite([17(3)])].">
      <j1 rule="para" parents="25 89"/>
      <j2 rule="rewrite" parents="17"/>
    </justification>
  </clause>

  <clause id="197">
    <literal><![CDATA[
      (c1 ^ c3) v ((c3 ^ c4) v x) = c3 v x
    ]]></literal>
    <justification jstring="[para(121(a,1),31(a,1,1,2)),rewrite([64(5),19(5)]),flip(a)].">
      <j1 rule="para" parents="121 31"/>
      <j2 rule="rewrite" parents="64 19"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="202">
    <literal><![CDATA[
      (c1 ^ c2) v ((c2 ^ c3) v x) = c2 v x
    ]]></literal>
    <justification jstring="[para(125(a,1),31(a,1,1,2)),rewrite([64(5),19(5)]),flip(a)].">
      <j1 rule="para" parents="125 31"/>
      <j2 rule="rewrite" parents="64 19"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="207">
    <literal><![CDATA[
      x v (y ^ (z ^ (x ^ u))) = x
    ]]></literal>
    <justification jstring="[para(17(a,1),162(a,1,2))].">
      <j1 rule="para" parents="17 162"/>
    </justification>
  </clause>

  <clause id="288">
    <literal><![CDATA[
      c1 ^ (c2 v c3) = c1 ^ c2
    ]]></literal>
    <justification jstring="[para(21(a,1),70(a,1,2)),flip(a)].">
      <j1 rule="para" parents="21 70"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="289">
    <literal><![CDATA[
      c1 ^ (c3 v c4) = c1 ^ c3
    ]]></literal>
    <justification jstring="[para(23(a,1),70(a,1,2)),flip(a)].">
      <j1 rule="para" parents="23 70"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="380">
    <literal><![CDATA[
      c3 ^ (c4 v (c1 ^ c3)) = c3
    ]]></literal>
    <justification jstring="[para(121(a,1),36(a,1,2)),rewrite([64(5),19(6)]),flip(a)].">
      <j1 rule="para" parents="121 36"/>
      <j2 rule="rewrite" parents="64 19"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="381">
    <literal><![CDATA[
      c2 ^ (c3 v (c1 ^ c2)) = c2
    ]]></literal>
    <justification jstring="[para(125(a,1),36(a,1,2)),rewrite([64(5),19(6)]),flip(a)].">
      <j1 rule="para" parents="125 36"/>
      <j2 rule="rewrite" parents="64 19"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="434">
    <literal><![CDATA[
      c4 v (c1 ^ c3) = c3 v c4
    ]]></literal>
    <justification jstring="[para(380(a,1),55(a,1,2)),rewrite([18(7),165(7)]),flip(a)].">
      <j1 rule="para" parents="380 55"/>
      <j2 rule="rewrite" parents="18 165"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="439">
    <literal><![CDATA[
      c3 v (c1 ^ c2) = c2 v c3
    ]]></literal>
    <justification jstring="[para(381(a,1),55(a,1,2)),rewrite([18(7),165(7)]),flip(a)].">
      <j1 rule="para" parents="381 55"/>
      <j2 rule="rewrite" parents="18 165"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="510">
    <literal><![CDATA[
      c1 v (c2 v c3) = c1 v c3
    ]]></literal>
    <justification jstring="[para(125(a,1),165(a,1,2)),rewrite([38(5),18(4),18(8)])].">
      <j1 rule="para" parents="125 165"/>
      <j2 rule="rewrite" parents="38 18 18"/>
    </justification>
  </clause>

  <clause id="569">
    <literal><![CDATA[
      c2 ^ (c1 v c3) = c2
    ]]></literal>
    <justification jstring="[para(510(a,1),175(a,1,2))].">
      <j1 rule="para" parents="510 175"/>
    </justification>
  </clause>

  <clause id="576">
    <literal><![CDATA[
      c2 ^ ((c1 v c3) ^ x) = c2 ^ x
    ]]></literal>
    <justification jstring="[para(569(a,1),17(a,1,1)),flip(a)].">
      <j1 rule="para" parents="569 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="616">
    <literal><![CDATA[
      c1 ^ (c2 ^ c3) = c1 ^ c3
    ]]></literal>
    <justification jstring="[para(288(a,1),176(a,1,2)),rewrite([42(5),19(4),19(8)])].">
      <j1 rule="para" parents="288 176"/>
      <j2 rule="rewrite" parents="42 19 19"/>
    </justification>
  </clause>

  <clause id="617">
    <literal><![CDATA[
      c1 ^ (c3 ^ c4) = c1 ^ c4
    ]]></literal>
    <justification jstring="[para(289(a,1),176(a,1,2)),rewrite([42(5),19(4),19(8)])].">
      <j1 rule="para" parents="289 176"/>
      <j2 rule="rewrite" parents="42 19 19"/>
    </justification>
  </clause>

  <clause id="668">
    <literal><![CDATA[
      c2 v (c1 ^ c3) = c2
    ]]></literal>
    <justification jstring="[para(616(a,1),162(a,1,2))].">
      <j1 rule="para" parents="616 162"/>
    </justification>
  </clause>

  <clause id="679">
    <literal><![CDATA[
      c2 v (x v (c1 ^ c3)) = x v c2
    ]]></literal>
    <justification jstring="[para(668(a,1),38(a,1,2)),flip(a)].">
      <j1 rule="para" parents="668 38"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="717">
    <literal><![CDATA[
      c2 v (x ^ (c1 ^ c3)) = c2
    ]]></literal>
    <justification jstring="[para(616(a,1),207(a,1,2,2))].">
      <j1 rule="para" parents="616 207"/>
    </justification>
  </clause>

  <clause id="721">
    <literal><![CDATA[
      c2 v (c1 ^ (c3 ^ x)) = c2
    ]]></literal>
    <justification jstring="[para(19(a,1),717(a,1,2)),rewrite([17(5)])].">
      <j1 rule="para" parents="19 717"/>
      <j2 rule="rewrite" parents="17"/>
    </justification>
  </clause>

  <clause id="756">
    <literal><![CDATA[
      c2 v (c1 ^ c4) = c2
    ]]></literal>
    <justification jstring="[para(617(a,1),721(a,1,2))].">
      <j1 rule="para" parents="617 721"/>
    </justification>
  </clause>

  <clause id="765">
    <literal><![CDATA[
      c1 ^ (c4 v (c1 ^ c2)) = c1 ^ c2
    ]]></literal>
    <justification jstring="[para(756(a,1),36(a,1,2)),flip(a)].">
      <j1 rule="para" parents="756 36"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1655">
    <literal><![CDATA[
      c2 ^ (c3 v c4) = c2 ^ c3
    ]]></literal>
    <justification jstring="[para(23(a,1),576(a,1,2)),flip(a)].">
      <j1 rule="para" parents="23 576"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1677">
    <literal><![CDATA[
      (c3 v c4) ^ (c3 v (x ^ c2)) = c3
    ]]></literal>
    <justification jstring="[para(1655(a,1),35(a,1,2,2)),rewrite([87(6),19(9)]),flip(a)].">
      <j1 rule="para" parents="1655 35"/>
      <j2 rule="rewrite" parents="87 19"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1678">
    <literal><![CDATA[
      c4 v (c2 ^ c3) = (c2 v c4) ^ (c3 v c4)
    ]]></literal>
    <justification jstring="[para(1655(a,1),37(a,1,2)),rewrite([18(8),18(11)])].">
      <j1 rule="para" parents="1655 37"/>
      <j2 rule="rewrite" parents="18 18"/>
    </justification>
  </clause>

  <clause id="2036">
    <literal><![CDATA[
      c2 v (c3 v c4) = c2 v c4
    ]]></literal>
    <justification jstring="[para(434(a,1),679(a,1,2)),rewrite([18(8)])].">
      <j1 rule="para" parents="434 679"/>
      <j2 rule="rewrite" parents="18"/>
    </justification>
  </clause>

  <clause id="2046">
    <literal><![CDATA[
      (c2 v c4) ^ (c3 v c4) = c3 v c4
    ]]></literal>
    <justification jstring="[para(2036(a,1),64(a,1,2)),rewrite([19(7)])].">
      <j1 rule="para" parents="2036 64"/>
      <j2 rule="rewrite" parents="19"/>
    </justification>
  </clause>

  <clause id="2059">
    <literal><![CDATA[
      c4 v (c2 ^ c3) = c3 v c4
    ]]></literal>
    <justification jstring="[back_rewrite(1678),rewrite([2046(12)])].">
      <j1 rule="back_rewrite" parents="1678"/>
      <j2 rule="rewrite" parents="2046"/>
    </justification>
  </clause>

  <clause id="4361">
    <literal><![CDATA[
      c4 ^ (c3 v (x ^ c2)) = c3 ^ c4
    ]]></literal>
    <justification jstring="[para(1677(a,1),90(a,1,2)),rewrite([19(3)]),flip(a)].">
      <j1 rule="para" parents="1677 90"/>
      <j2 rule="rewrite" parents="19"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4823">
    <literal><![CDATA[
      c3 v (c4 v (c1 ^ c2)) = c2 v c4
    ]]></literal>
    <justification jstring="[para(2036(a,1),202(a,2)),rewrite([38(10),18(9),2059(9),81(8),38(7),18(6)])].">
      <j1 rule="para" parents="2036 202"/>
      <j2 rule="rewrite" parents="38 18 2059 81 38 18"/>
    </justification>
  </clause>

  <clause id="5858">
    <literal><![CDATA[
      (x ^ c2) v (c3 ^ c4) = (c3 v (x ^ c2)) ^ (c4 v (x ^ c2))
    ]]></literal>
    <justification jstring="[para(4361(a,1),37(a,1,2)),rewrite([18(10),18(14),19(15)])].">
      <j1 rule="para" parents="4361 37"/>
      <j2 rule="rewrite" parents="18 18 19"/>
    </justification>
  </clause>

  <clause id="6088">
    <literal><![CDATA[
      (c2 v c4) ^ (c4 v (c1 ^ c2)) = c4 v (c1 ^ c2)
    ]]></literal>
    <justification jstring="[para(4823(a,1),64(a,1,2)),rewrite([19(9)])].">
      <j1 rule="para" parents="4823 64"/>
      <j2 rule="rewrite" parents="19"/>
    </justification>
  </clause>

  <clause id="6102">
    <literal><![CDATA[
      c4 v (c1 ^ c2) = c2 v c4
    ]]></literal>
    <justification jstring="[para(4823(a,1),197(a,2)),rewrite([38(12),18(11),5858(11),439(9),13(14),38(8),18(7),2036(8),6088(12),38(9),18(8),12(8),668(7)])].">
      <j1 rule="para" parents="4823 197"/>
      <j2 rule="rewrite" parents="38 18 5858 439 13 38 18 2036 6088 38 18 12 668"/>
    </justification>
  </clause>

  <clause id="6114">
    <literal><![CDATA[
      c1 ^ (c2 v c4) = c1 ^ c2
    ]]></literal>
    <justification jstring="[back_rewrite(765),rewrite([6102(6)])].">
      <j1 rule="back_rewrite" parents="765"/>
      <j2 rule="rewrite" parents="6102"/>
    </justification>
  </clause>

  <clause id="6131">
    <literal><![CDATA[
      (c1 v c2) ^ (c2 v c4) = c2
    ]]></literal>
    <justification jstring="[para(6114(a,1),13(a,1,2)),rewrite([55(5),18(4)]),flip(a)].">
      <j1 rule="para" parents="6114 13"/>
      <j2 rule="rewrite" parents="55 18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="6134">
    <literal><![CDATA[
      c2 ^ (c1 v (c2 ^ c4)) != c2
    ]]></literal>
    <justification jstring="[back_rewrite(43),rewrite([6131(16)]),xx(b)].">
      <j1 rule="back_rewrite" parents="43"/>
      <j2 rule="rewrite" parents="6131"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="6329">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[para(36(a,1),6134(a,1)),rewrite([19(5),6102(6),26(5)]),xx(a)].">
      <j1 rule="para" parents="36 6134"/>
      <j2 rule="rewrite" parents="19 6102 26"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
