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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>H42.out</source>

<heading><![CDATA[
Prover9 (32) version March-2007, March 2007.
Process 21246 was started by mccune on cleo,
Mon Mar 19 17:09:50 2007
The command was "/home/mccune/bin/prover9 -f lt.in uc.in H42.in".
]]></heading>

<proof number="1" length="48" max_count="23">

<comments><![CDATA[
% Proof 1 at 66.40 (+ 0.52) seconds.
% Length of proof is 48.
% Level of proof is 10.
% Maximum clause weight is 23.
% Given clauses 1537.
]]></comments>

  <clause id="1" type="goal">
    <literal><![CDATA[
      (all x all y (x ^ y = x -> x' v y' = x'))
    ]]></literal>
    <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" type="assumption">
    <literal><![CDATA[
      x ^ y = y ^ x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="5" type="assumption">
    <literal><![CDATA[
      (x ^ y) ^ z = x ^ (y ^ z)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6" type="assumption">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7" type="assumption">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="8" type="assumption">
    <literal><![CDATA[
      x v x' = 1
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="9" type="assumption">
    <literal><![CDATA[
      x ^ x' = 0
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="10" type="assumption">
    <literal><![CDATA[
      x v y != 1
    ]]></literal>
    <literal><![CDATA[
      x ^ y != 0
    ]]></literal>
    <literal><![CDATA[
      x' = y
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11" type="assumption">
    <literal><![CDATA[
      x ^ (y v (z ^ (x v u))) = x ^ (y v (z ^ (y v (u v (x ^ z)))))
    ]]></literal>
    <attribute><![CDATA[
      label(H42)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

  <clause id="13" type="deny">
    <literal><![CDATA[
      c1 ^ c2 = c1
    ]]></literal>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

  <clause id="14" type="deny">
    <literal><![CDATA[
      c1' != c1' v c2'
    ]]></literal>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      c1' v c2' != c1'
    ]]></literal>
    <justification jstring="[copy(14),flip(a)].">
      <j1 rule="copy" parents="14"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

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

  <clause id="20">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(2(a,1),6(a,1,2))].">
      <j1 rule="para" parents="2 6"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      x ^ ((x v y) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(6(a,1),5(a,1,1)),flip(a)].">
      <j1 rule="para" parents="6 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      x v ((x ^ y) v z) = x v z
    ]]></literal>
    <justification jstring="[para(7(a,1),3(a,1,1)),flip(a)].">
      <j1 rule="para" parents="7 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(4(a,1),7(a,1,2))].">
      <j1 rule="para" parents="4 7"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      x v (x' v y) = 1 v y
    ]]></literal>
    <justification jstring="[para(8(a,1),3(a,1,1)),flip(a)].">
      <j1 rule="para" parents="8 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      x ^ 1 = x
    ]]></literal>
    <justification jstring="[para(8(a,1),6(a,1,2))].">
      <j1 rule="para" parents="8 6"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      x ^ (x' ^ y) = 0 ^ y
    ]]></literal>
    <justification jstring="[para(9(a,1),5(a,1,1)),flip(a)].">
      <j1 rule="para" parents="9 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      x v 0 = x
    ]]></literal>
    <justification jstring="[para(9(a,1),7(a,1,2))].">
      <j1 rule="para" parents="9 7"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      x ^ (y v (z ^ (y v ((x ^ z) v u)))) = x ^ (y v (z ^ (x v u)))
    ]]></literal>
    <justification jstring="[para(2(a,1),12(a,1,2,2,2,2))].">
      <j1 rule="para" parents="2 12"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      c1 ^ (c2 ^ x) = c1 ^ x
    ]]></literal>
    <justification jstring="[para(13(a,1),5(a,1,1)),flip(a)].">
      <j1 rule="para" parents="13 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="75">
    <literal><![CDATA[
      1 ^ x = x
    ]]></literal>
    <justification jstring="[para(32(a,1),4(a,1)),flip(a)].">
      <j1 rule="para" parents="32 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      0 v x = x
    ]]></literal>
    <justification jstring="[para(35(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="35 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="85">
    <literal><![CDATA[
      1 v x = 1
    ]]></literal>
    <justification jstring="[para(75(a,1),6(a,1))].">
      <j1 rule="para" parents="75 6"/>
    </justification>
  </clause>

  <clause id="86">
    <literal><![CDATA[
      x v (x' v y) = 1
    ]]></literal>
    <justification jstring="[back_rewrite(30),rewrite(85(5))].">
      <j1 rule="back_rewrite" parents="30"/>
      <j2 rule="rewrite" parents="85"/>
    </justification>
  </clause>

  <clause id="88">
    <literal><![CDATA[
      0 ^ x = 0
    ]]></literal>
    <justification jstring="[para(77(a,1),6(a,1,2))].">
      <j1 rule="para" parents="77 6"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      x ^ (x' ^ y) = 0
    ]]></literal>
    <justification jstring="[back_rewrite(33),rewrite(88(5))].">
      <j1 rule="back_rewrite" parents="33"/>
      <j2 rule="rewrite" parents="88"/>
    </justification>
  </clause>

  <clause id="96">
    <literal><![CDATA[
      x v 1 = 1
    ]]></literal>
    <justification jstring="[para(20(a,1),75(a,1)),flip(a)].">
      <j1 rule="para" parents="20 75"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="99">
    <literal><![CDATA[
      x ^ 0 = 0
    ]]></literal>
    <justification jstring="[para(88(a,1),4(a,1)),flip(a)].">
      <j1 rule="para" parents="88 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="114">
    <literal><![CDATA[
      c1 v c2 = c2
    ]]></literal>
    <justification jstring="[para(13(a,1),26(a,1,2)),rewrite(2(3))].">
      <j1 rule="para" parents="13 26"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="125">
    <literal><![CDATA[
      x ^ (x v y)' = 0
    ]]></literal>
    <justification jstring="[para(9(a,1),22(a,1,2)),rewrite(99(2)),flip(a)].">
      <j1 rule="para" parents="9 22"/>
      <j2 rule="rewrite" parents="99"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="130">
    <literal><![CDATA[
      c1 ^ (c1' v c2') != 0
    ]]></literal>
    <justification jstring="[ur(10,a,86,a,c,15,a(flip))].">
      <j1 rule="ur" parents="10 86 15"/>
    </justification>
  </clause>

  <clause id="158">
    <literal><![CDATA[
      c1 ^ c2' = 0
    ]]></literal>
    <justification jstring="[para(114(a,1),125(a,1,2,1))].">
      <j1 rule="para" parents="114 125"/>
    </justification>
  </clause>

  <clause id="172">
    <literal><![CDATA[
      x v (x ^ y)' = 1
    ]]></literal>
    <justification jstring="[para(8(a,1),24(a,1,2)),rewrite(96(2)),flip(a)].">
      <j1 rule="para" parents="8 24"/>
      <j2 rule="rewrite" parents="96"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="183">
    <literal><![CDATA[
      x v (y ^ x)' = 1
    ]]></literal>
    <justification jstring="[para(4(a,1),172(a,1,2,1))].">
      <j1 rule="para" parents="4 172"/>
    </justification>
  </clause>

  <clause id="217">
    <literal><![CDATA[
      (c1 ^ x) v (c2 ^ x) = c2 ^ x
    ]]></literal>
    <justification jstring="[para(59(a,1),26(a,1,2)),rewrite(2(5))].">
      <j1 rule="para" parents="59 26"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="222">
    <literal><![CDATA[
      (c2 ^ x) v (c1 ^ x)' = 1
    ]]></literal>
    <justification jstring="[para(59(a,1),183(a,1,2,1))].">
      <j1 rule="para" parents="59 183"/>
    </justification>
  </clause>

  <clause id="567">
    <literal><![CDATA[
      c1 ^ (x v (c2' ^ (c1 v y))) = c1 ^ (x v (c2' ^ (x v y)))
    ]]></literal>
    <justification jstring="[para(158(a,1),42(a,1,2,2,2,2,1)),rewrite(77(5)),flip(a)].">
      <j1 rule="para" parents="158 42"/>
      <j2 rule="rewrite" parents="77"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1294">
    <literal><![CDATA[
      (c2 ^ (c1 v x)) v c1' = 1
    ]]></literal>
    <justification jstring="[para(6(a,1),222(a,1,2,1))].">
      <j1 rule="para" parents="6 222"/>
    </justification>
  </clause>

  <clause id="2194">
    <literal><![CDATA[
      c1' v (c2 ^ (c1 v x)) = 1
    ]]></literal>
    <justification jstring="[para(1294(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="1294 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3012">
    <literal><![CDATA[
      c1 v (c2 ^ (c1 v x)) = c2 ^ (c1 v x)
    ]]></literal>
    <justification jstring="[para(6(a,1),217(a,1,1))].">
      <j1 rule="para" parents="6 217"/>
    </justification>
  </clause>

  <clause id="27801">
    <literal><![CDATA[
      c1 ^ (c1' v c2') = 0
    ]]></literal>
    <justification jstring="[para(2194(a,1),567(a,2,2,2,2)),rewrite(3012(11),19(10),89(10),2(5),77(5),9(4),4(8),75(8)),flip(a)].">
      <j1 rule="para" parents="2194 567"/>
      <j2 rule="rewrite" parents="3012 19 89 2 77 9 4 75"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="27802">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(27801,a,130,a)].">
      <j1 rule="resolve" parents="27801 130"/>
    </justification>
  </clause>

</proof>

</proofs>
