<?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>t3_12.out</source>

<heading><![CDATA[
Prover9 (32) version March-2007, March 2007.
Process 21122 was started by mccune on cleo,
Mon Mar 19 17:05:16 2007
The command was "/home/mccune/bin/prover9 -f head.in t3_12.in".
]]></heading>

<proof number="1" length="88" max_count="38">

<comments><![CDATA[
% Proof 1 at 6.72 (+ 0.08) seconds.
% Length of proof is 88.
% Level of proof is 15.
% Maximum clause weight is 38.
% Given clauses 2344.
]]></comments>

  <clause id="2" 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="24" 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="25" 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="26" 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="27" 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="28" 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="29">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[copy(28),rewrite(24(2))].">
      <j1 rule="copy" parents="28"/>
      <j2 rule="rewrite" parents="24"/>
    </justification>
  </clause>

  <clause id="30" 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="31">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[copy(30),rewrite(25(2))].">
      <j1 rule="copy" parents="30"/>
      <j2 rule="rewrite" parents="25"/>
    </justification>
  </clause>

  <clause id="32" type="clausify">
    <literal><![CDATA[
      -B(x,y,z)
    ]]></literal>
    <literal><![CDATA[
      (x ^ y) v (y ^ z) = y
    ]]></literal>
    <justification jstring="[clausify(2)].">
      <j1 rule="clausify" parents="2"/>
    </justification>
  </clause>

  <clause id="33" type="clausify">
    <literal><![CDATA[
      -B(x,y,z)
    ]]></literal>
    <literal><![CDATA[
      (x v y) ^ (y v z) = y
    ]]></literal>
    <justification jstring="[clausify(2)].">
      <j1 rule="clausify" parents="2"/>
    </justification>
  </clause>

  <clause id="34" type="clausify">
    <literal><![CDATA[
      B(x,y,z)
    ]]></literal>
    <literal><![CDATA[
      (x ^ y) v (y ^ z) != y
    ]]></literal>
    <literal><![CDATA[
      (x v y) ^ (y v z) != y
    ]]></literal>
    <justification jstring="[clausify(2)].">
      <j1 rule="clausify" parents="2"/>
    </justification>
  </clause>

  <clause id="35" type="assumption">
    <literal><![CDATA[
      x ^ (y v z) = (x ^ y) v (x ^ z)
    ]]></literal>
    <attribute><![CDATA[
      label(dist_1)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      (x ^ y) v (x ^ z) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[copy(35),flip(a)].">
      <j1 rule="copy" parents="35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="37" type="assumption">
    <literal><![CDATA[
      x v (y ^ z) = (x v y) ^ (x v z)
    ]]></literal>
    <attribute><![CDATA[
      label(dist_2)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="38" type="assumption">
    <literal><![CDATA[
      B(a,b,c)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="39" type="assumption">
    <literal><![CDATA[
      B(a,d,c)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="40" type="assumption">
    <literal><![CDATA[
      B(b,e,d)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="41" type="assumption">
    <literal><![CDATA[
      -B(a,e,c)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      x ^ ((x ^ y) v z) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[back_rewrite(36),rewrite(37(3),25(2),31(2))].">
      <j1 rule="back_rewrite" parents="36"/>
      <j2 rule="rewrite" parents="37 25 31"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      B(x,y,z)
    ]]></literal>
    <literal><![CDATA[
      (y v x) ^ ((y v y) ^ ((x ^ y) v z)) != y
    ]]></literal>
    <literal><![CDATA[
      (x v y) ^ (y v z) != y
    ]]></literal>
    <justification jstring="[back_rewrite(34),rewrite(37(4),25(3),37(3),26(7))].">
      <j1 rule="back_rewrite" parents="34"/>
      <j2 rule="rewrite" parents="37 25 37 26"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      -B(x,y,z)
    ]]></literal>
    <literal><![CDATA[
      (y v x) ^ ((y v y) ^ ((x ^ y) v z)) = y
    ]]></literal>
    <justification jstring="[back_rewrite(32),rewrite(37(4),25(3),37(3),26(7))].">
      <j1 rule="back_rewrite" parents="32"/>
      <j2 rule="rewrite" parents="37 25 37 26"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      (x v x) ^ (x v y) = x
    ]]></literal>
    <justification jstring="[back_rewrite(31),rewrite(37(2))].">
      <j1 rule="back_rewrite" parents="31"/>
      <j2 rule="rewrite" parents="37"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      x ^ (y ^ z) = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(24(a,1),26(a,1,1)),rewrite(26(2))].">
      <j1 rule="para" parents="24 26"/>
      <j2 rule="rewrite" parents="26"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      x v (y v z) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(25(a,1),27(a,1,1)),rewrite(27(2))].">
      <j1 rule="para" parents="25 27"/>
      <j2 rule="rewrite" parents="27"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(25(a,1),29(a,1,2))].">
      <j1 rule="para" parents="25 29"/>
    </justification>
  </clause>

  <clause id="51">
    <literal><![CDATA[
      x ^ ((x v y) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(29(a,1),26(a,1,1)),flip(a)].">
      <j1 rule="para" parents="29 26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      x ^ (y ^ ((x ^ y) v z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(29(a,1),26(a,1)),flip(a)].">
      <j1 rule="para" parents="29 26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      (x v y) ^ (x v (y v z)) = x v y
    ]]></literal>
    <justification jstring="[para(27(a,1),29(a,1,2))].">
      <j1 rule="para" parents="27 29"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      ((x v y) ^ (x v z)) v u = x v ((y ^ z) v u)
    ]]></literal>
    <justification jstring="[para(37(a,1),27(a,1,1))].">
      <j1 rule="para" parents="37 27"/>
    </justification>
  </clause>

  <clause id="56">
    <literal><![CDATA[
      (a v b) ^ (b v c) = b
    ]]></literal>
    <justification jstring="[hyper(33,a,38,a)].">
      <j1 rule="hyper" parents="33 38"/>
    </justification>
  </clause>

  <clause id="57">
    <literal><![CDATA[
      (a v d) ^ (c v d) = d
    ]]></literal>
    <justification jstring="[hyper(33,a,39,a),rewrite(25(6))].">
      <j1 rule="hyper" parents="33 39"/>
      <j2 rule="rewrite" parents="25"/>
    </justification>
  </clause>

  <clause id="58">
    <literal><![CDATA[
      (b v e) ^ (d v e) = e
    ]]></literal>
    <justification jstring="[hyper(33,a,40,a),rewrite(25(6))].">
      <j1 rule="hyper" parents="33 40"/>
      <j2 rule="rewrite" parents="25"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      x ^ ((y ^ x) v z) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(24(a,1),42(a,1,2,1))].">
      <j1 rule="para" parents="24 42"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      x ^ (y ^ (x v z)) = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(52),rewrite(59(3))].">
      <j1 rule="back_rewrite" parents="52"/>
      <j2 rule="rewrite" parents="59"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      B(x ^ y,z,u)
    ]]></literal>
    <literal><![CDATA[
      (z v x) ^ ((z v y) ^ ((z v z) ^ ((x ^ (y ^ z)) v u))) != z
    ]]></literal>
    <literal><![CDATA[
      ((x ^ y) v z) ^ (z v u) != z
    ]]></literal>
    <justification jstring="[para(26(a,1),43(b,1,2,2,1)),rewrite(37(4),26(11))].">
      <j1 rule="para" parents="26 43"/>
      <j2 rule="rewrite" parents="37 26"/>
    </justification>
  </clause>

  <clause id="80">
    <literal><![CDATA[
      (b v d) ^ ((b v e) ^ ((d v e) ^ (e v e))) = e
    ]]></literal>
    <justification jstring="[hyper(44,a,40,a),rewrite(25(3),25(11),37(11),25(9),24(14),26(14),47(15))].">
      <j1 rule="hyper" parents="44 40"/>
      <j2 rule="rewrite" parents="25 25 37 25 24 26 47"/>
    </justification>
  </clause>

  <clause id="81">
    <literal><![CDATA[
      (a v c) ^ ((a v d) ^ ((c v d) ^ (d v d))) = d
    ]]></literal>
    <justification jstring="[hyper(44,a,39,a),rewrite(25(3),25(11),37(11),25(9),24(14),26(14),47(15))].">
      <j1 rule="hyper" parents="44 39"/>
      <j2 rule="rewrite" parents="25 25 37 25 24 26 47"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      b ^ ((a v b) ^ (a v c)) = b
    ]]></literal>
    <justification jstring="[hyper(44,a,38,a),rewrite(25(3),25(11),37(11),25(9),25(12),47(14),45(13),24(8),47(9))].">
      <j1 rule="hyper" parents="44 38"/>
      <j2 rule="rewrite" parents="25 25 37 25 25 47 45 24 47"/>
    </justification>
  </clause>

  <clause id="85">
    <literal><![CDATA[
      (x v x) ^ ((x v y) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(45(a,1),26(a,1,1)),flip(a)].">
      <j1 rule="para" parents="45 26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="93">
    <literal><![CDATA[
      B(x,y,z)
    ]]></literal>
    <literal><![CDATA[
      y ^ (x v z) != y
    ]]></literal>
    <literal><![CDATA[
      (x v y) ^ (y v z) != y
    ]]></literal>
    <justification jstring="[para(47(a,1),43(b,1)),rewrite(85(7),59(4))].">
      <j1 rule="para" parents="47 43"/>
      <j2 rule="rewrite" parents="85 59"/>
    </justification>
  </clause>

  <clause id="95">
    <literal><![CDATA[
      x ^ ((y v x) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(50(a,1),26(a,1,1)),flip(a)].">
      <j1 rule="para" parents="50 26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="98">
    <literal><![CDATA[
      (x v y) ^ (x v (z v y)) = x v y
    ]]></literal>
    <justification jstring="[para(50(a,1),37(a,1,2)),flip(a)].">
      <j1 rule="para" parents="50 37"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="100">
    <literal><![CDATA[
      x ^ x = x
    ]]></literal>
    <justification jstring="[para(50(a,1),42(a,2)),rewrite(25(2),37(2),45(3))].">
      <j1 rule="para" parents="50 42"/>
      <j2 rule="rewrite" parents="25 37 45"/>
    </justification>
  </clause>

  <clause id="104">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(50(a,1),45(a,1))].">
      <j1 rule="para" parents="50 45"/>
    </justification>
  </clause>

  <clause id="105">
    <literal><![CDATA[
      x ^ (y ^ (z v x)) = y ^ x
    ]]></literal>
    <justification jstring="[para(50(a,1),47(a,1,2)),flip(a)].">
      <j1 rule="para" parents="50 47"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="106">
    <literal><![CDATA[
      b ^ (a v c) = b
    ]]></literal>
    <justification jstring="[back_rewrite(82),rewrite(95(9))].">
      <j1 rule="back_rewrite" parents="82"/>
      <j2 rule="rewrite" parents="95"/>
    </justification>
  </clause>

  <clause id="110">
    <literal><![CDATA[
      d ^ (a v c) = d
    ]]></literal>
    <justification jstring="[back_rewrite(81),rewrite(104(12),24(11),50(11),24(8),50(8),24(5))].">
      <j1 rule="back_rewrite" parents="81"/>
      <j2 rule="rewrite" parents="104 24 50 24 50 24"/>
    </justification>
  </clause>

  <clause id="111">
    <literal><![CDATA[
      e ^ (b v d) = e
    ]]></literal>
    <justification jstring="[back_rewrite(80),rewrite(104(12),24(11),50(11),24(8),50(8),24(5))].">
      <j1 rule="back_rewrite" parents="80"/>
      <j2 rule="rewrite" parents="104 24 50 24 50 24"/>
    </justification>
  </clause>

  <clause id="114">
    <literal><![CDATA[
      B(x ^ y,z,u)
    ]]></literal>
    <literal><![CDATA[
      z ^ ((x ^ (y ^ z)) v u) != z
    ]]></literal>
    <literal><![CDATA[
      ((x ^ y) v z) ^ (z v u) != z
    ]]></literal>
    <justification jstring="[back_rewrite(72),rewrite(104(5),47(9),51(9),47(8),51(8))].">
      <j1 rule="back_rewrite" parents="72"/>
      <j2 rule="rewrite" parents="104 47 51 47 51"/>
    </justification>
  </clause>

  <clause id="122">
    <literal><![CDATA[
      x ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(100(a,1),26(a,2,2)),rewrite(24(2))].">
      <j1 rule="para" parents="100 26"/>
      <j2 rule="rewrite" parents="24"/>
    </justification>
  </clause>

  <clause id="124">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(104(a,1),27(a,1,1)),flip(a)].">
      <j1 rule="para" parents="104 27"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="126">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(104(a,1),27(a,2,2)),rewrite(25(2))].">
      <j1 rule="para" parents="104 27"/>
      <j2 rule="rewrite" parents="25"/>
    </justification>
  </clause>

  <clause id="132">
    <literal><![CDATA[
      (x v b) ^ (x v (a v c)) = x v b
    ]]></literal>
    <justification jstring="[para(106(a,1),37(a,1,2)),flip(a)].">
      <j1 rule="para" parents="106 37"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="136">
    <literal><![CDATA[
      (x v d) ^ (x v (a v c)) = x v d
    ]]></literal>
    <justification jstring="[para(110(a,1),37(a,1,2)),flip(a)].">
      <j1 rule="para" parents="110 37"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="139">
    <literal><![CDATA[
      e ^ ((b v d) ^ x) = e ^ x
    ]]></literal>
    <justification jstring="[para(111(a,1),26(a,1,1)),flip(a)].">
      <j1 rule="para" parents="111 26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="155">
    <literal><![CDATA[
      a ^ (b v c) = a ^ b
    ]]></literal>
    <justification jstring="[para(56(a,1),51(a,1,2)),flip(a)].">
      <j1 rule="para" parents="56 51"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="160">
    <literal><![CDATA[
      a ^ (c v d) = a ^ d
    ]]></literal>
    <justification jstring="[para(57(a,1),51(a,1,2)),flip(a)].">
      <j1 rule="para" parents="57 51"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="186">
    <literal><![CDATA[
      x ^ (y ^ (z ^ x)) = y ^ (z ^ x)
    ]]></literal>
    <justification jstring="[para(26(a,1),122(a,1,2)),rewrite(26(5))].">
      <j1 rule="para" parents="26 122"/>
      <j2 rule="rewrite" parents="26"/>
    </justification>
  </clause>

  <clause id="187">
    <literal><![CDATA[
      (x v y) ^ (z ^ x) = z ^ x
    ]]></literal>
    <justification jstring="[para(122(a,1),51(a,2)),rewrite(186(4))].">
      <j1 rule="para" parents="122 51"/>
      <j2 rule="rewrite" parents="186"/>
    </justification>
  </clause>

  <clause id="218">
    <literal><![CDATA[
      x v (y v (z v x)) = y v (z v x)
    ]]></literal>
    <justification jstring="[para(27(a,1),126(a,1,2)),rewrite(27(5))].">
      <j1 rule="para" parents="27 126"/>
      <j2 rule="rewrite" parents="27"/>
    </justification>
  </clause>

  <clause id="245">
    <literal><![CDATA[
      x v ((x ^ y) v z) = x v z
    ]]></literal>
    <justification jstring="[para(104(a,1),55(a,1,1,1)),rewrite(29(2)),flip(a)].">
      <j1 rule="para" parents="104 55"/>
      <j2 rule="rewrite" parents="29"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="246">
    <literal><![CDATA[
      x v ((y ^ x) v z) = x v z
    ]]></literal>
    <justification jstring="[para(104(a,1),55(a,1,1,2)),rewrite(24(2),29(2)),flip(a)].">
      <j1 rule="para" parents="104 55"/>
      <j2 rule="rewrite" parents="24 29"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="247">
    <literal><![CDATA[
      (x ^ y) v z = (x v z) ^ (y v z)
    ]]></literal>
    <justification jstring="[para(104(a,1),55(a,2)),rewrite(25(3),245(3),25(4),246(4),25(4),37(4),126(2),126(3)),flip(a)].">
      <j1 rule="para" parents="104 55"/>
      <j2 rule="rewrite" parents="25 245 25 246 25 37 126 126"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="264">
    <literal><![CDATA[
      B(x ^ y,z,u)
    ]]></literal>
    <literal><![CDATA[
      z ^ ((x v u) ^ ((y v u) ^ (z v u))) != z
    ]]></literal>
    <literal><![CDATA[
      (x v z) ^ ((y v z) ^ (z v u)) != z
    ]]></literal>
    <justification jstring="[back_rewrite(114),rewrite(247(5),247(5),247(11),26(14))].">
      <j1 rule="back_rewrite" parents="114"/>
      <j2 rule="rewrite" parents="247 247 247 26"/>
    </justification>
  </clause>

  <clause id="281">
    <literal><![CDATA[
      x ^ (y ^ (z ^ (x v u))) = x ^ (y ^ z)
    ]]></literal>
    <justification jstring="[para(26(a,1),65(a,1,2))].">
      <j1 rule="para" parents="26 65"/>
    </justification>
  </clause>

  <clause id="283">
    <literal><![CDATA[
      d ^ (b v e) = d ^ e
    ]]></literal>
    <justification jstring="[para(58(a,1),65(a,1,2)),flip(a)].">
      <j1 rule="para" parents="58 65"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="284">
    <literal><![CDATA[
      a ^ (c ^ d) = a ^ c
    ]]></literal>
    <justification jstring="[para(160(a,1),65(a,1,2)),rewrite(47(5),24(8))].">
      <j1 rule="para" parents="160 65"/>
      <j2 rule="rewrite" parents="47 24"/>
    </justification>
  </clause>

  <clause id="286">
    <literal><![CDATA[
      B(x ^ y,z,u)
    ]]></literal>
    <literal><![CDATA[
      z ^ ((x v u) ^ (y v u)) != z
    ]]></literal>
    <literal><![CDATA[
      (x v z) ^ ((y v z) ^ (z v u)) != z
    ]]></literal>
    <justification jstring="[back_rewrite(264),rewrite(281(8))].">
      <j1 rule="back_rewrite" parents="264"/>
      <j2 rule="rewrite" parents="281"/>
    </justification>
  </clause>

  <clause id="357">
    <literal><![CDATA[
      (x v y) ^ (z ^ y) = z ^ y
    ]]></literal>
    <justification jstring="[para(122(a,1),95(a,2)),rewrite(186(4))].">
      <j1 rule="para" parents="122 95"/>
      <j2 rule="rewrite" parents="186"/>
    </justification>
  </clause>

  <clause id="373">
    <literal><![CDATA[
      a ^ (c ^ (d ^ x)) = a ^ (c ^ x)
    ]]></literal>
    <justification jstring="[para(284(a,1),26(a,1,1)),rewrite(26(4),26(9)),flip(a)].">
      <j1 rule="para" parents="284 26"/>
      <j2 rule="rewrite" parents="26 26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="433">
    <literal><![CDATA[
      a ^ (b ^ c) = a ^ c
    ]]></literal>
    <justification jstring="[para(155(a,1),105(a,1,2)),rewrite(47(5),24(4))].">
      <j1 rule="para" parents="155 105"/>
      <j2 rule="rewrite" parents="47 24"/>
    </justification>
  </clause>

  <clause id="734">
    <literal><![CDATA[
      a ^ (c ^ (b v x)) = a ^ c
    ]]></literal>
    <justification jstring="[para(433(a,1),187(a,1,2)),rewrite(247(4),24(9),26(9),65(8),433(11))].">
      <j1 rule="para" parents="433 187"/>
      <j2 rule="rewrite" parents="247 24 26 65 433"/>
    </justification>
  </clause>

  <clause id="762">
    <literal><![CDATA[
      a v (b v c) = a v c
    ]]></literal>
    <justification jstring="[para(104(a,1),132(a,1,2)),rewrite(25(5),49(5),24(9),98(9),25(8),49(8)),flip(a)].">
      <j1 rule="para" parents="104 132"/>
      <j2 rule="rewrite" parents="25 49 24 98 25 49"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="849">
    <literal><![CDATA[
      e ^ (a v c) = e
    ]]></literal>
    <justification jstring="[para(136(a,1),139(a,1,2)),rewrite(111(5),49(7),762(7)),flip(a)].">
      <j1 rule="para" parents="136 139"/>
      <j2 rule="rewrite" parents="111 49 762"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="857">
    <literal><![CDATA[
      (x v e) ^ (x v (a v c)) = x v e
    ]]></literal>
    <justification jstring="[para(849(a,1),37(a,1,2)),flip(a)].">
      <j1 rule="para" parents="849 37"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="859">
    <literal><![CDATA[
      B(a,e,c)
    ]]></literal>
    <literal><![CDATA[
      (a v e) ^ (c v e) != e
    ]]></literal>
    <justification jstring="[para(849(a,1),93(b,1)),rewrite(25(13)),xx(b)].">
      <j1 rule="para" parents="849 93"/>
      <j2 rule="rewrite" parents="25"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="4626">
    <literal><![CDATA[
      B(x ^ y,y,y v z)
    ]]></literal>
    <justification jstring="[para(53(a,1),286(c,1,2)),rewrite(124(7),24(7),50(7),29(5),104(6),24(6),50(6)),xx(b),xx(c)].">
      <j1 rule="para" parents="53 286"/>
      <j2 rule="rewrite" parents="124 24 50 29 104 24 50"/>
      <j2 rule="xx"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="4727">
    <literal><![CDATA[
      B(x ^ (y ^ z),z,z v u)
    ]]></literal>
    <justification jstring="[para(26(a,1),4626(a,1))].">
      <j1 rule="para" parents="26 4626"/>
    </justification>
  </clause>

  <clause id="9634">
    <literal><![CDATA[
      a ^ (c ^ e) = a ^ c
    ]]></literal>
    <justification jstring="[para(283(a,1),373(a,1,2,2)),rewrite(373(7),734(12))].">
      <j1 rule="para" parents="283 373"/>
      <j2 rule="rewrite" parents="373 734"/>
    </justification>
  </clause>

  <clause id="9725">
    <literal><![CDATA[
      B(a ^ c,e,e v x)
    ]]></literal>
    <justification jstring="[para(9634(a,1),4727(a,1))].">
      <j1 rule="para" parents="9634 4727"/>
    </justification>
  </clause>

  <clause id="9791">
    <literal><![CDATA[
      B(a ^ c,e,x v (y v e))
    ]]></literal>
    <justification jstring="[para(218(a,1),9725(a,3))].">
      <j1 rule="para" parents="218 9725"/>
    </justification>
  </clause>

  <clause id="23114">
    <literal><![CDATA[
      a v (c v e) = a v c
    ]]></literal>
    <justification jstring="[para(104(a,1),857(a,1,2)),rewrite(25(5),49(5),25(4),24(9),53(9),25(8),49(8),25(7)),flip(a)].">
      <j1 rule="para" parents="104 857"/>
      <j2 rule="rewrite" parents="25 49 25 24 53 25 49 25"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23166">
    <literal><![CDATA[
      (a v c) ^ (x ^ (c v e)) = x ^ (c v e)
    ]]></literal>
    <justification jstring="[para(23114(a,1),357(a,1,1))].">
      <j1 rule="para" parents="23114 357"/>
    </justification>
  </clause>

  <clause id="23222">
    <literal><![CDATA[
      B(a ^ c,e,a v c)
    ]]></literal>
    <justification jstring="[para(23114(a,1),9791(a,3))].">
      <j1 rule="para" parents="23114 9791"/>
    </justification>
  </clause>

  <clause id="23336">
    <literal><![CDATA[
      (a v e) ^ (c v e) = e
    ]]></literal>
    <justification jstring="[hyper(33,a,23222,a),rewrite(25(5),37(5),25(3),25(6),49(12),25(11),23114(12),24(11),23166(11))].">
      <j1 rule="hyper" parents="33 23222"/>
      <j2 rule="rewrite" parents="25 37 25 25 49 25 23114 24 23166"/>
    </justification>
  </clause>

  <clause id="23338">
    <literal><![CDATA[
      B(a,e,c)
    ]]></literal>
    <justification jstring="[back_rewrite(859),rewrite(23336(11)),xx(b)].">
      <j1 rule="back_rewrite" parents="859"/>
      <j2 rule="rewrite" parents="23336"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="23339">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(23338,a,41,a)].">
      <j1 rule="resolve" parents="23338 41"/>
    </justification>
  </clause>

</proof>

</proofs>
