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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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<proofs number_of_proofs="1">

<source>H49.out</source>

<heading><![CDATA[
Prover9 (32) version 2008-04A, April 2008.
Process 24358 was started by mccune on cleo,
Fri Apr  4 11:38:19 2008
The command was "/home/mccune/LADR/bin/prover9 -f lt.in uc.in H49.in".
]]></heading>

<proof number="1" length="42" max_count="23">

<comments><![CDATA[
% Proof 1 at 1.74 (+ 0.03) seconds.
% Length of proof is 42.
% Level of proof is 11.
% Maximum clause weight is 23.
% Given clauses 330.
]]></comments>

  <clause id="1" type="goal">
    <literal><![CDATA[
      (all x all y (x ^ y = x -> x' v y' = x'))
    ]]></literal>
    <attribute><![CDATA[
      label(non_clause)
    ]]></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" 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 ((x ^ z) v (z ^ (y v u))))
    ]]></literal>
    <attribute><![CDATA[
      label(H49)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      x ^ (y v ((x ^ z) v (z ^ (y v u)))) = 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="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="53">
    <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="64">
    <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="65">
    <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="71">
    <literal><![CDATA[
      1 v x = 1
    ]]></literal>
    <justification jstring="[para(64(a,1),6(a,1))].">
      <j1 rule="para" parents="64 6"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      x v (x' v y) = 1
    ]]></literal>
    <justification jstring="[back_rewrite(30),rewrite([71(5)])].">
      <j1 rule="back_rewrite" parents="30"/>
      <j2 rule="rewrite" parents="71"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      x v 1 = 1
    ]]></literal>
    <justification jstring="[para(71(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="71 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="83">
    <literal><![CDATA[
      0 ^ x = 0
    ]]></literal>
    <justification jstring="[para(65(a,1),6(a,1,2))].">
      <j1 rule="para" parents="65 6"/>
    </justification>
  </clause>

  <clause id="85">
    <literal><![CDATA[
      x ^ (x' ^ y) = 0
    ]]></literal>
    <justification jstring="[back_rewrite(33),rewrite([83(5)])].">
      <j1 rule="back_rewrite" parents="33"/>
      <j2 rule="rewrite" parents="83"/>
    </justification>
  </clause>

  <clause id="117">
    <literal><![CDATA[
      c1 ^ c2' = 0
    ]]></literal>
    <justification jstring="[para(9(a,1),53(a,1,2)),rewrite([4(3),83(3)]),flip(a)].">
      <j1 rule="para" parents="9 53"/>
      <j2 rule="rewrite" parents="4 83"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="123">
    <literal><![CDATA[
      c1 ^ (x v (c2' ^ (c1 v y))) = c1 ^ (x v (c2' ^ (x v y)))
    ]]></literal>
    <justification jstring="[para(117(a,1),12(a,1,2,2,1)),rewrite([65(7)]),flip(a)].">
      <j1 rule="para" parents="117 12"/>
      <j2 rule="rewrite" parents="65"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="175">
    <literal><![CDATA[
      x v (x ^ y)' = 1
    ]]></literal>
    <justification jstring="[para(8(a,1),24(a,1,2)),rewrite([79(2)]),flip(a)].">
      <j1 rule="para" parents="8 24"/>
      <j2 rule="rewrite" parents="79"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="231">
    <literal><![CDATA[
      (c1 ^ x) v (c2 ^ x) = c2 ^ x
    ]]></literal>
    <justification jstring="[para(53(a,1),26(a,1,2)),rewrite([2(5)])].">
      <j1 rule="para" parents="53 26"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="251">
    <literal><![CDATA[
      c1 ^ (c1' v c2') != 0
    ]]></literal>
    <justification jstring="[ur(10,a,72,a,c,15,a(flip))].">
      <j1 rule="ur" parents="10 72 15"/>
    </justification>
  </clause>

  <clause id="352">
    <literal><![CDATA[
      x v (y ^ x)' = 1
    ]]></literal>
    <justification jstring="[para(4(a,1),175(a,1,2,1))].">
      <j1 rule="para" parents="4 175"/>
    </justification>
  </clause>

  <clause id="794">
    <literal><![CDATA[
      (c2 ^ x) v (c1 ^ x)' = 1
    ]]></literal>
    <justification jstring="[para(53(a,1),352(a,1,2,1))].">
      <j1 rule="para" parents="53 352"/>
    </justification>
  </clause>

  <clause id="1447">
    <literal><![CDATA[
      (c2 ^ (c1 v x)) v c1' = 1
    ]]></literal>
    <justification jstring="[para(6(a,1),794(a,1,2,1))].">
      <j1 rule="para" parents="6 794"/>
    </justification>
  </clause>

  <clause id="1476">
    <literal><![CDATA[
      c1' v (c2 ^ (c1 v x)) = 1
    ]]></literal>
    <justification jstring="[para(1447(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="1447 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2645">
    <literal><![CDATA[
      c1 v (c2 ^ (c1 v x)) = c2 ^ (c1 v x)
    ]]></literal>
    <justification jstring="[para(6(a,1),231(a,1,1))].">
      <j1 rule="para" parents="6 231"/>
    </justification>
  </clause>

  <clause id="6686">
    <literal><![CDATA[
      c1 ^ (c1' v c2') = 0
    ]]></literal>
    <justification jstring="[para(1476(a,1),123(a,2,2,2,2)),rewrite([2645(11),19(10),85(10),2(5),65(5),9(4),4(8),64(8)]),flip(a)].">
      <j1 rule="para" parents="1476 123"/>
      <j2 rule="rewrite" parents="2645 19 85 2 65 9 4 64"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="6687">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(6686,a,251,a)].">
      <j1 rule="resolve" parents="6686 251"/>
    </justification>
  </clause>

</proof>

</proofs>
