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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>H7b.out</source>

<heading><![CDATA[
Prover9 (32) version Aug-2007, Aug 2007.
Process 16660 was started by mccune on cleo,
Tue Aug  7 09:38:12 2007
The command was "/home/mccune/LADR/bin/prover9 -f lt.in uc.in H7b.in".
]]></heading>

<proof number="1" length="33" max_count="24">

<comments><![CDATA[
% Proof 1 at 7.80 (+ 0.10) seconds.
% Length of proof is 33.
% Level of proof is 13.
% Maximum clause weight is 23.
% Given clauses 1051.
]]></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 (x ^ z)) = x ^ (y v (x ^ ((x ^ y) v (z ^ (x v y)))))
    ]]></literal>
    <attribute><![CDATA[
      label(H7)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      x ^ (y v (x ^ ((x ^ y) v (z ^ (x v y))))) = 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="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="17">
    <literal><![CDATA[
      x v (y v z) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(2(a,1),3(a,1,1)),rewrite([3(2)])].">
      <j1 rule="para" parents="2 3"/>
      <j2 rule="rewrite" parents="3"/>
    </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="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="34">
    <literal><![CDATA[
      x ^ (y ^ (x ^ y)') = 0
    ]]></literal>
    <justification jstring="[para(9(a,1),5(a,1)),flip(a)].">
      <j1 rule="para" parents="9 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      x ^ (y v (x ^ ((x ^ y) v (z ^ (y v x))))) = x ^ (y v (x ^ z))
    ]]></literal>
    <justification jstring="[para(2(a,1),12(a,1,2,2,2,2,2))].">
      <j1 rule="para" parents="2 12"/>
    </justification>
  </clause>

  <clause id="50">
    <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="56">
    <literal><![CDATA[
      1 v x = 1
    ]]></literal>
    <justification jstring="[para(50(a,1),6(a,1))].">
      <j1 rule="para" parents="50 6"/>
    </justification>
  </clause>

  <clause id="63">
    <literal><![CDATA[
      x v 1 = 1
    ]]></literal>
    <justification jstring="[para(56(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="56 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="151">
    <literal><![CDATA[
      x v (x ^ y)' = 1
    ]]></literal>
    <justification jstring="[para(8(a,1),24(a,1,2)),rewrite([63(2)]),flip(a)].">
      <j1 rule="para" parents="8 24"/>
      <j2 rule="rewrite" parents="63"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="163">
    <literal><![CDATA[
      x v (y ^ x)' = 1
    ]]></literal>
    <justification jstring="[para(4(a,1),151(a,1,2,1))].">
      <j1 rule="para" parents="4 151"/>
    </justification>
  </clause>

  <clause id="173">
    <literal><![CDATA[
      c2 v c1' = 1
    ]]></literal>
    <justification jstring="[para(13(a,1),163(a,1,2,1))].">
      <j1 rule="para" parents="13 163"/>
    </justification>
  </clause>

  <clause id="366">
    <literal><![CDATA[
      c1' ^ (c2 v (c1' ^ ((c2 ^ c1') v x))) = c1' ^ (c2 v (c1' ^ x))
    ]]></literal>
    <justification jstring="[para(173(a,1),39(a,1,2,2,2,2,2)),rewrite([4(9),32(11)])].">
      <j1 rule="para" parents="173 39"/>
      <j2 rule="rewrite" parents="4 32"/>
    </justification>
  </clause>

  <clause id="13996">
    <literal><![CDATA[
      c1' ^ (c2 v (c1' ^ (c2 ^ c1')')) = c1'
    ]]></literal>
    <justification jstring="[para(8(a,1),366(a,1,2,2,2)),rewrite([4(7),50(7),173(6),4(4),50(4)]),flip(a)].">
      <j1 rule="para" parents="8 366"/>
      <j2 rule="rewrite" parents="4 50 173 4 50"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14329">
    <literal><![CDATA[
      c2 v (c1' ^ (c2 ^ c1')') = 1
    ]]></literal>
    <justification jstring="[para(13996(a,1),26(a,1,2)),rewrite([2(13),17(13),7(12),173(4)]),flip(a)].">
      <j1 rule="para" parents="13996 26"/>
      <j2 rule="rewrite" parents="2 17 7 173"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14337">
    <literal><![CDATA[
      c1' ^ (c2 ^ c1')' = c2'
    ]]></literal>
    <justification jstring="[hyper(10,a,14329,a,b,34,a),flip(a)].">
      <j1 rule="hyper" parents="10 14329 34"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14350">
    <literal><![CDATA[
      c1' v c2' = c1'
    ]]></literal>
    <justification jstring="[para(14337(a,1),7(a,1,2))].">
      <j1 rule="para" parents="14337 7"/>
    </justification>
  </clause>

  <clause id="14351">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(14350,a,15,a)].">
      <j1 rule="resolve" parents="14350 15"/>
    </justification>
  </clause>

</proof>

</proofs>
