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<!DOCTYPE proofs SYSTEM "proof3.dtd">

<?xml-stylesheet type="text/xsl" href="proof3.xsl"?>

<proofs number_of_proofs="1">

<source>H78b.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13678 was started by mccune on cleo.thornwood,
Mon Jun 19 16:49:49 2006
The command was "/home/mccune/bin/prover9 -f lt.in uc.in H78b.in".
]]></heading>

<proof number="1" length="43" max_count="23">

<comments><![CDATA[
% Proof 1 at 74.72 (+ 0.86) seconds.
% Length of proof is 43.
% Level of proof is 11.
% Maximum clause weight is 23.
% Given clauses 1608.
]]></comments>

  <clause id="10">
    <literal><![CDATA[
      x ^ (y v (z ^ (y v u))) = x ^ (y v (z ^ (u v (y ^ (x v u)))))
    ]]></literal>
    <attribute><![CDATA[
      label(H78)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      c1 ' != c1 ' v c2 '
    ]]></literal>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      (x v y) v z = x v (y v z)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      x ^ y = y ^ x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      (x ^ y) ^ z = x ^ (y ^ z)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      x v x ' = 1
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x ^ x ' = 0
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      x v y != 1
    ]]></literal>
    <literal><![CDATA[
      x ^ y != 0
    ]]></literal>
    <literal><![CDATA[
      x ' = y
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      x ^ (y v (z ^ (u v (y ^ (x v u))))) = x ^ (y v (z ^ (y v u)))
    ]]></literal>
    <attribute><![CDATA[
      label(H78)
    ]]></attribute>
    <justification jstring="[copy(10),flip(a)].">
      <j1 rule="copy" parents="10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      c1 ^ c2 = c1
    ]]></literal>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      c1 ' v c2 ' != c1 '
    ]]></literal>
    <justification jstring="[copy(12),flip(a)].">
      <j1 rule="copy" parents="12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      x v (y v z) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(13(a,1),14(a,1,1)),demod(14(2))].">
      <j1 rule="para" parents="13 14"/>
      <j2 rule="demod" parents="14"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      x ^ ((x v y) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(17(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="17 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(15(a,1),18(a,1,2))].">
      <j1 rule="para" parents="15 18"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(17(a,1),18(a,1,2))].">
      <j1 rule="para" parents="17 18"/>
    </justification>
  </clause>

  <clause id="37">
    <literal><![CDATA[
      x v (x ' v y) = 1 v y
    ]]></literal>
    <justification jstring="[para(19(a,1),14(a,1,1)),flip(a)].">
      <j1 rule="para" parents="19 14"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      x ^ 1 = x
    ]]></literal>
    <justification jstring="[para(19(a,1),17(a,1,2))].">
      <j1 rule="para" parents="19 17"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      x ^ (y ^ (x ^ y) ') = 0
    ]]></literal>
    <justification jstring="[para(20(a,1),16(a,1)),flip(a)].">
      <j1 rule="para" parents="20 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      x v 0 = x
    ]]></literal>
    <justification jstring="[para(20(a,1),18(a,1,2))].">
      <j1 rule="para" parents="20 18"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      x v y != 1
    ]]></literal>
    <literal><![CDATA[
      y ^ x != 0
    ]]></literal>
    <literal><![CDATA[
      y ' = x
    ]]></literal>
    <justification jstring="[para(13(a,1),21(a,1))].">
      <j1 rule="para" parents="13 21"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      (x ^ y) v z != 1
    ]]></literal>
    <literal><![CDATA[
      x ^ (y ^ z) != 0
    ]]></literal>
    <literal><![CDATA[
      (x ^ y) ' = z
    ]]></literal>
    <justification jstring="[para(16(a,1),21(b,1))].">
      <j1 rule="para" parents="16 21"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      x ^ (y v ((z v (y ^ (x v z))) ^ u)) = x ^ (y v (u ^ (y v z)))
    ]]></literal>
    <justification jstring="[para(15(a,1),22(a,1,2,2))].">
      <j1 rule="para" parents="15 22"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(36(a,1),14(a,2,2)),demod(13(2))].">
      <j1 rule="para" parents="36 14"/>
      <j2 rule="demod" parents="13"/>
    </justification>
  </clause>

  <clause id="76">
    <literal><![CDATA[
      1 ^ x = x
    ]]></literal>
    <justification jstring="[para(39(a,1),15(a,1)),flip(a)].">
      <j1 rule="para" parents="39 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      0 v x = x
    ]]></literal>
    <justification jstring="[para(42(a,1),13(a,1)),flip(a)].">
      <j1 rule="para" parents="42 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="85">
    <literal><![CDATA[
      1 v x = 1
    ]]></literal>
    <justification jstring="[para(76(a,1),17(a,1))].">
      <j1 rule="para" parents="76 17"/>
    </justification>
  </clause>

  <clause id="86">
    <literal><![CDATA[
      x v (x ' v y) = 1
    ]]></literal>
    <justification jstring="[back_demod(37),demod(85(5))].">
      <j1 rule="back_demod" parents="37"/>
      <j2 rule="demod" parents="85"/>
    </justification>
  </clause>

  <clause id="88">
    <literal><![CDATA[
      0 ^ x = 0
    ]]></literal>
    <justification jstring="[para(82(a,1),17(a,1,2))].">
      <j1 rule="para" parents="82 17"/>
    </justification>
  </clause>

  <clause id="98">
    <literal><![CDATA[
      x ^ 0 = 0
    ]]></literal>
    <justification jstring="[para(88(a,1),15(a,1)),flip(a)].">
      <j1 rule="para" parents="88 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="112">
    <literal><![CDATA[
      x v (y v (z ^ x)) = y v x
    ]]></literal>
    <justification jstring="[para(33(a,1),25(a,1,2)),flip(a)].">
      <j1 rule="para" parents="33 25"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="127">
    <literal><![CDATA[
      x ^ (x v y) ' = 0
    ]]></literal>
    <justification jstring="[para(20(a,1),29(a,1,2)),demod(98(2)),flip(a)].">
      <j1 rule="para" parents="20 29"/>
      <j2 rule="demod" parents="98"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="134">
    <literal><![CDATA[
      x ^ (y v x) ' = 0
    ]]></literal>
    <justification jstring="[para(13(a,1),127(a,1,2,1))].">
      <j1 rule="para" parents="13 127"/>
    </justification>
  </clause>

  <clause id="223">
    <literal><![CDATA[
      x ' ' = x
    ]]></literal>
    <justification jstring="[para(19(a,1),43(a,1)),demod(15(5),20(5)),xx(a),xx(b)].">
      <j1 rule="para" parents="19 43"/>
      <j2 rule="demod" parents="15 20"/>
      <j2 rule="xx"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="343">
    <literal><![CDATA[
      x ^ (y ^ ((x ^ y) ' v z)) != 0
    ]]></literal>
    <literal><![CDATA[
      (x ^ y) ' v z = (x ^ y) '
    ]]></literal>
    <justification jstring="[para(86(a,1),46(a,1)),flip(c),xx(a)].">
      <j1 rule="para" parents="86 46"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="525">
    <literal><![CDATA[
      x ^ (y v ((y ' v (y ^ (x v y '))) ^ z)) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(19(a,1),55(a,2,2,2,2)),demod(39(10))].">
      <j1 rule="para" parents="19 55"/>
      <j2 rule="demod" parents="39"/>
    </justification>
  </clause>

  <clause id="1931">
    <literal><![CDATA[
      (x v (y ^ z)) ^ (x v z) ' = 0
    ]]></literal>
    <justification jstring="[para(112(a,1),134(a,1,2,1))].">
      <j1 rule="para" parents="112 134"/>
    </justification>
  </clause>

  <clause id="15256">
    <literal><![CDATA[
      x ^ (y v (x v y ') ') = x ^ y
    ]]></literal>
    <justification jstring="[para(1931(a,1),525(a,1,2,2)),demod(42(2),74(5)),flip(a)].">
      <j1 rule="para" parents="1931 525"/>
      <j2 rule="demod" parents="42 74"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="15280">
    <literal><![CDATA[
      x ' v (y ^ x) ' = (y ^ x) '
    ]]></literal>
    <justification jstring="[para(15256(a,1),343(a,1,2)),demod(41(4),223(8),33(7),13(7)),xx(a)].">
      <j1 rule="para" parents="15256 343"/>
      <j2 rule="demod" parents="41 223 33 13"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="15290">
    <literal><![CDATA[
      c1 ' v c2 ' = c1 '
    ]]></literal>
    <justification jstring="[para(23(a,1),15280(a,1,2,1)),demod(13(5),23(8))].">
      <j1 rule="para" parents="23 15280"/>
      <j2 rule="demod" parents="13 23"/>
    </justification>
  </clause>

  <clause id="15291">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(15290,a,24,a)].">
      <j1 rule="resolve" parents="15290 24"/>
    </justification>
  </clause>

</proof>

</proofs>
