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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 March-2007, March 2007.
Process 21276 was started by mccune on cleo,
Mon Mar 19 17:11:02 2007
The command was "/home/mccune/bin/prover9 -f lt.in uc.in H78b.in".
]]></heading>

<proof number="1" length="44" max_count="23">

<comments><![CDATA[
% Proof 1 at 69.49 (+ 0.43) seconds.
% Length of proof is 44.
% Level of proof is 11.
% Maximum clause weight is 23.
% Given clauses 1544.
]]></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 ^ (y v u))) = x ^ (y v (z ^ (u v (y ^ (x v u)))))
    ]]></literal>
    <attribute><![CDATA[
      label(H78)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      x ^ (y v (z ^ (u v (y ^ (x v u))))) = x ^ (y v (z ^ (y 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="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="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="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="29">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(6(a,1),7(a,1,2))].">
      <j1 rule="para" parents="6 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="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="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="36">
    <literal><![CDATA[
      x v y != 1
    ]]></literal>
    <literal><![CDATA[
      y ^ x != 0
    ]]></literal>
    <literal><![CDATA[
      y' = x
    ]]></literal>
    <justification jstring="[para(2(a,1),10(a,1))].">
      <j1 rule="para" parents="2 10"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      (x ^ y) v z != 1
    ]]></literal>
    <literal><![CDATA[
      x ^ (y ^ z) != 0
    ]]></literal>
    <literal><![CDATA[
      (x ^ y)' = z
    ]]></literal>
    <justification jstring="[para(5(a,1),10(b,1))].">
      <j1 rule="para" parents="5 10"/>
    </justification>
  </clause>

  <clause id="48">
    <literal><![CDATA[
      x ^ (y v ((z v (y ^ (x v z))) ^ u)) = x ^ (y v (u ^ (y v z)))
    ]]></literal>
    <justification jstring="[para(4(a,1),12(a,1,2,2))].">
      <j1 rule="para" parents="4 12"/>
    </justification>
  </clause>

  <clause id="71">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(29(a,1),3(a,2,2)),rewrite(2(2))].">
      <j1 rule="para" parents="29 3"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="73">
    <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="75">
    <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="83">
    <literal><![CDATA[
      1 v x = 1
    ]]></literal>
    <justification jstring="[para(73(a,1),6(a,1))].">
      <j1 rule="para" parents="73 6"/>
    </justification>
  </clause>

  <clause id="84">
    <literal><![CDATA[
      x v (x' v y) = 1
    ]]></literal>
    <justification jstring="[back_rewrite(30),rewrite(83(5))].">
      <j1 rule="back_rewrite" parents="30"/>
      <j2 rule="rewrite" parents="83"/>
    </justification>
  </clause>

  <clause id="86">
    <literal><![CDATA[
      0 ^ x = 0
    ]]></literal>
    <justification jstring="[para(75(a,1),6(a,1,2))].">
      <j1 rule="para" parents="75 6"/>
    </justification>
  </clause>

  <clause id="97">
    <literal><![CDATA[
      x ^ 0 = 0
    ]]></literal>
    <justification jstring="[para(86(a,1),4(a,1)),flip(a)].">
      <j1 rule="para" parents="86 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="109">
    <literal><![CDATA[
      x v (y v (z ^ x)) = y v x
    ]]></literal>
    <justification jstring="[para(26(a,1),17(a,1,2)),flip(a)].">
      <j1 rule="para" parents="26 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="116">
    <literal><![CDATA[
      x ^ (x v y)' = 0
    ]]></literal>
    <justification jstring="[para(9(a,1),22(a,1,2)),rewrite(97(2)),flip(a)].">
      <j1 rule="para" parents="9 22"/>
      <j2 rule="rewrite" parents="97"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="132">
    <literal><![CDATA[
      x ^ (y v x)' = 0
    ]]></literal>
    <justification jstring="[para(2(a,1),116(a,1,2,1))].">
      <j1 rule="para" parents="2 116"/>
    </justification>
  </clause>

  <clause id="208">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[para(8(a,1),36(a,1)),rewrite(4(5),9(5)),xx(a),xx(b)].">
      <j1 rule="para" parents="8 36"/>
      <j2 rule="rewrite" parents="4 9"/>
      <j2 rule="xx"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="337">
    <literal><![CDATA[
      x ^ (y ^ ((x ^ y)' v z)) != 0
    ]]></literal>
    <literal><![CDATA[
      (x ^ y)' v z = (x ^ y)'
    ]]></literal>
    <justification jstring="[para(84(a,1),39(a,1)),flip(c),xx(a)].">
      <j1 rule="para" parents="84 39"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="505">
    <literal><![CDATA[
      x ^ (y v ((y' v (y ^ (x v y'))) ^ z)) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(8(a,1),48(a,2,2,2,2)),rewrite(32(10))].">
      <j1 rule="para" parents="8 48"/>
      <j2 rule="rewrite" parents="32"/>
    </justification>
  </clause>

  <clause id="2078">
    <literal><![CDATA[
      (x v (y ^ z)) ^ (x v z)' = 0
    ]]></literal>
    <justification jstring="[para(109(a,1),132(a,1,2,1))].">
      <j1 rule="para" parents="109 132"/>
    </justification>
  </clause>

  <clause id="26904">
    <literal><![CDATA[
      x ^ (y v (x v y')') = x ^ y
    ]]></literal>
    <justification jstring="[para(2078(a,1),505(a,1,2,2)),rewrite(35(2),71(5)),flip(a)].">
      <j1 rule="para" parents="2078 505"/>
      <j2 rule="rewrite" parents="35 71"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="26955">
    <literal><![CDATA[
      x' v (y ^ x)' = (y ^ x)'
    ]]></literal>
    <justification jstring="[para(26904(a,1),337(a,1,2)),rewrite(34(4),208(8),26(7),2(7)),xx(a)].">
      <j1 rule="para" parents="26904 337"/>
      <j2 rule="rewrite" parents="34 208 26 2"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="26966">
    <literal><![CDATA[
      c1' v c2' = c1'
    ]]></literal>
    <justification jstring="[para(13(a,1),26955(a,1,2,1)),rewrite(2(5),13(8))].">
      <j1 rule="para" parents="13 26955"/>
      <j2 rule="rewrite" parents="2 13"/>
    </justification>
  </clause>

  <clause id="26967">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(26966,a,15,a)].">
      <j1 rule="resolve" parents="26966 15"/>
    </justification>
  </clause>

</proof>

</proofs>
