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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>H65d.out</source>

<heading><![CDATA[
Prover9 (32) version 2008-04A, April 2008.
Process 24353 was started by mccune on cleo,
Fri Apr  4 11:37:19 2008
The command was "/home/mccune/LADR/bin/prover9 -f lt.in uc.in H65d.in".
]]></heading>

<proof number="1" length="123" max_count="26">

<comments><![CDATA[
% Proof 1 at 49.13 (+ 0.48) seconds: distributivity.
% Length of proof is 123.
% Level of proof is 26.
% Maximum clause weight is 26.
% Given clauses 996.
]]></comments>

  <clause id="1" type="goal">
    <literal><![CDATA[
      x ^ (y v z) = (x ^ y) v (x ^ z)
    ]]></literal>
    <attribute><![CDATA[
      answer(distributivity)
    ]]></attribute>
    <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 ^ u)) = x ^ (y v (x ^ ((x ^ y) v (z ^ u))))
    ]]></literal>
    <attribute><![CDATA[
      label(H65)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      x ^ (y v (x ^ ((x ^ y) v (z ^ u)))) = x ^ (y v (z ^ 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) v (c1 ^ c3) != c1 ^ (c2 v c3)
    ]]></literal>
    <attribute><![CDATA[
      answer(distributivity)
    ]]></attribute>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

  <clause id="15">
    <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="17">
    <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="18">
    <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="20">
    <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="21">
    <literal><![CDATA[
      x ^ (y ^ ((x ^ y) v z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(6(a,1),5(a,1)),flip(a)].">
      <j1 rule="para" parents="6 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22">
    <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="23">
    <literal><![CDATA[
      x v (y v ((x v y) ^ z)) = x v y
    ]]></literal>
    <justification jstring="[para(7(a,1),3(a,1)),flip(a)].">
      <j1 rule="para" parents="7 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="24">
    <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="25">
    <literal><![CDATA[
      (x ^ y) v (x ^ (y ^ z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(5(a,1),7(a,1,2))].">
      <j1 rule="para" parents="5 7"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      x ^ x = x
    ]]></literal>
    <justification jstring="[para(7(a,1),6(a,1,2))].">
      <j1 rule="para" parents="7 6"/>
    </justification>
  </clause>

  <clause id="27">
    <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="28">
    <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="29">
    <literal><![CDATA[
      x v (y v (x v y)') = 1
    ]]></literal>
    <justification jstring="[para(8(a,1),3(a,1)),flip(a)].">
      <j1 rule="para" parents="8 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="30">
    <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="31">
    <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="33">
    <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="34">
    <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 (x ^ ((y ^ x) v (z ^ u)))) = x ^ (y v (z ^ u))
    ]]></literal>
    <justification jstring="[para(4(a,1),12(a,1,2,2,2,1))].">
      <j1 rule="para" parents="4 12"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      x ^ ((y v (x ^ ((x ^ y) v (z ^ u)))) ^ w) = x ^ ((y v (z ^ u)) ^ w)
    ]]></literal>
    <justification jstring="[para(12(a,1),5(a,1,1)),rewrite([5(4)]),flip(a)].">
      <j1 rule="para" parents="12 5"/>
      <j2 rule="rewrite" parents="5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      x ^ (y v (x ^ ((x ^ y) v z))) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(6(a,1),12(a,1,2,2,2,2)),rewrite([6(7)])].">
      <j1 rule="para" parents="6 12"/>
      <j2 rule="rewrite" parents="6"/>
    </justification>
  </clause>

  <clause id="51">
    <literal><![CDATA[
      x v (y v z) != 1
    ]]></literal>
    <literal><![CDATA[
      y ^ (x v z) != 0
    ]]></literal>
    <literal><![CDATA[
      y' = x v z
    ]]></literal>
    <justification jstring="[para(15(a,1),10(a,1))].">
      <j1 rule="para" parents="15 10"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      x ^ (x ^ y) = x ^ y
    ]]></literal>
    <justification jstring="[para(26(a,1),5(a,1,1)),flip(a)].">
      <j1 rule="para" parents="26 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="57">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(27(a,1),3(a,1,1)),flip(a)].">
      <j1 rule="para" parents="27 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      1 ^ x = x
    ]]></literal>
    <justification jstring="[para(30(a,1),4(a,1)),flip(a)].">
      <j1 rule="para" parents="30 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      0 v x = x
    ]]></literal>
    <justification jstring="[para(33(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="33 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      x ^ (y ^ (x v z)) = y ^ x
    ]]></literal>
    <justification jstring="[para(6(a,1),17(a,1,2)),flip(a)].">
      <j1 rule="para" parents="6 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      x ^ (y ^ x') = y ^ 0
    ]]></literal>
    <justification jstring="[para(9(a,1),17(a,1,2)),flip(a)].">
      <j1 rule="para" parents="9 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      1' = 0
    ]]></literal>
    <justification jstring="[hyper(10,a,33,a,b,61,a)].">
      <j1 rule="hyper" parents="10 33 61"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      1 v x = 1
    ]]></literal>
    <justification jstring="[para(61(a,1),6(a,1))].">
      <j1 rule="para" parents="61 6"/>
    </justification>
  </clause>

  <clause id="70">
    <literal><![CDATA[
      x v (x' v y) = 1
    ]]></literal>
    <justification jstring="[back_rewrite(28),rewrite([69(5)])].">
      <j1 rule="back_rewrite" parents="28"/>
      <j2 rule="rewrite" parents="69"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      0 ^ x = 0
    ]]></literal>
    <justification jstring="[para(62(a,1),6(a,1,2))].">
      <j1 rule="para" parents="62 6"/>
    </justification>
  </clause>

  <clause id="73">
    <literal><![CDATA[
      x ^ (x' ^ y) = 0
    ]]></literal>
    <justification jstring="[back_rewrite(31),rewrite([72(5)])].">
      <j1 rule="back_rewrite" parents="31"/>
      <j2 rule="rewrite" parents="72"/>
    </justification>
  </clause>

  <clause id="76">
    <literal><![CDATA[
      x ^ (y ^ (z v (x ^ y))) = x ^ y
    ]]></literal>
    <justification jstring="[para(18(a,1),5(a,1)),flip(a)].">
      <j1 rule="para" parents="18 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      x v 1 = 1
    ]]></literal>
    <justification jstring="[para(18(a,1),61(a,1)),flip(a)].">
      <j1 rule="para" parents="18 61"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="81">
    <literal><![CDATA[
      x ^ 0 = 0
    ]]></literal>
    <justification jstring="[para(72(a,1),4(a,1)),flip(a)].">
      <j1 rule="para" parents="72 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      x ^ (y ^ x') = 0
    ]]></literal>
    <justification jstring="[back_rewrite(67),rewrite([81(5)])].">
      <j1 rule="back_rewrite" parents="67"/>
      <j2 rule="rewrite" parents="81"/>
    </justification>
  </clause>

  <clause id="92">
    <literal><![CDATA[
      x v ((y ^ x) v z) = x v z
    ]]></literal>
    <justification jstring="[para(24(a,1),3(a,1,1)),flip(a)].">
      <j1 rule="para" parents="24 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="93">
    <literal><![CDATA[
      x v (y v (z ^ (x v y))) = x v y
    ]]></literal>
    <justification jstring="[para(24(a,1),3(a,1)),flip(a)].">
      <j1 rule="para" parents="24 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="94">
    <literal><![CDATA[
      x v (y ^ (z ^ x)) = x
    ]]></literal>
    <justification jstring="[para(5(a,1),24(a,1,2))].">
      <j1 rule="para" parents="5 24"/>
    </justification>
  </clause>

  <clause id="97">
    <literal><![CDATA[
      (x ^ y) v (x ^ (z ^ y)) = x ^ y
    ]]></literal>
    <justification jstring="[para(17(a,1),24(a,1,2))].">
      <j1 rule="para" parents="17 24"/>
    </justification>
  </clause>

  <clause id="103">
    <literal><![CDATA[
      x ^ (x v y)' = 0
    ]]></literal>
    <justification jstring="[para(9(a,1),20(a,1,2)),rewrite([81(2)]),flip(a)].">
      <j1 rule="para" parents="9 20"/>
      <j2 rule="rewrite" parents="81"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="120">
    <literal><![CDATA[
      x ^ (y ^ ((y ^ x) v z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(4(a,1),21(a,1,2,2,1))].">
      <j1 rule="para" parents="4 21"/>
    </justification>
  </clause>

  <clause id="125">
    <literal><![CDATA[
      (x ^ y) v (y ^ ((x ^ y) v z)) = y ^ ((x ^ y) v z)
    ]]></literal>
    <justification jstring="[para(21(a,1),24(a,1,2)),rewrite([2(5)])].">
      <j1 rule="para" parents="21 24"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="143">
    <literal><![CDATA[
      x v (x ^ y)' = 1
    ]]></literal>
    <justification jstring="[para(8(a,1),22(a,1,2)),rewrite([79(2)]),flip(a)].">
      <j1 rule="para" parents="8 22"/>
      <j2 rule="rewrite" parents="79"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="147">
    <literal><![CDATA[
      (x v y) ^ ((x ^ z) v y) = (x ^ z) v y
    ]]></literal>
    <justification jstring="[para(22(a,1),18(a,1,2)),rewrite([4(4)])].">
      <j1 rule="para" parents="22 18"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="152">
    <literal><![CDATA[
      x v (y ^ x)' = 1
    ]]></literal>
    <justification jstring="[para(4(a,1),143(a,1,2,1))].">
      <j1 rule="para" parents="4 143"/>
    </justification>
  </clause>

  <clause id="161">
    <literal><![CDATA[
      (x ^ y) v (x ^ (z ^ y))' = 1
    ]]></literal>
    <justification jstring="[para(17(a,1),152(a,1,2,1))].">
      <j1 rule="para" parents="17 152"/>
    </justification>
  </clause>

  <clause id="176">
    <literal><![CDATA[
      (x v y) ^ (y v ((x v y) ^ z)) = y v ((x v y) ^ z)
    ]]></literal>
    <justification jstring="[para(23(a,1),18(a,1,2)),rewrite([4(5)])].">
      <j1 rule="para" parents="23 18"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="228">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[para(8(a,1),34(a,1)),rewrite([4(5),9(5)]),xx(a),xx(b)].">
      <j1 rule="para" parents="8 34"/>
      <j2 rule="rewrite" parents="4 9"/>
      <j2 rule="xx"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="245">
    <literal><![CDATA[
      x' ^ (y ^ x) = 0
    ]]></literal>
    <justification jstring="[para(228(a,1),82(a,1,2,2))].">
      <j1 rule="para" parents="228 82"/>
    </justification>
  </clause>

  <clause id="408">
    <literal><![CDATA[
      x ^ (y v (x ^ ((y ^ x) v z))) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(6(a,1),39(a,1,2,2,2,2)),rewrite([6(7)])].">
      <j1 rule="para" parents="6 39"/>
      <j2 rule="rewrite" parents="6"/>
    </justification>
  </clause>

  <clause id="417">
    <literal><![CDATA[
      x' ^ (y ^ (x v (x' ^ (y ^ (z ^ u))))) = x' ^ (y ^ (x v (z ^ u)))
    ]]></literal>
    <justification jstring="[para(73(a,1),39(a,1,2,2,2,1)),rewrite([62(7),5(6),5(8),5(13)])].">
      <j1 rule="para" parents="73 39"/>
      <j2 rule="rewrite" parents="62 5 5 5"/>
    </justification>
  </clause>

  <clause id="471">
    <literal><![CDATA[
      x ^ ((y v (x ^ ((x ^ y) v z))) ^ u) = x ^ ((y v z) ^ u)
    ]]></literal>
    <justification jstring="[para(6(a,1),40(a,1,2,1,2,2,2)),rewrite([6(8)])].">
      <j1 rule="para" parents="6 40"/>
      <j2 rule="rewrite" parents="6"/>
    </justification>
  </clause>

  <clause id="657">
    <literal><![CDATA[
      x ^ (x' v (x ^ y)) = x ^ (x' v y)
    ]]></literal>
    <justification jstring="[para(9(a,1),42(a,1,2,2,2,1)),rewrite([62(3)])].">
      <j1 rule="para" parents="9 42"/>
      <j2 rule="rewrite" parents="62"/>
    </justification>
  </clause>

  <clause id="671">
    <literal><![CDATA[
      x ^ (y v (x ^ (y v z))) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(57(a,1),42(a,2,2)),rewrite([15(3),92(3)])].">
      <j1 rule="para" parents="57 42"/>
      <j2 rule="rewrite" parents="15 92"/>
    </justification>
  </clause>

  <clause id="762">
    <literal><![CDATA[
      x v (y v ((z ^ x) v u)) = y v (x v u)
    ]]></literal>
    <justification jstring="[para(92(a,1),15(a,1,2)),flip(a)].">
      <j1 rule="para" parents="92 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="793">
    <literal><![CDATA[
      x' ^ (x v y) != 0
    ]]></literal>
    <literal><![CDATA[
      x v y = x
    ]]></literal>
    <justification jstring="[para(70(a,1),51(a,1)),rewrite([228(10)]),flip(c),xx(a)].">
      <j1 rule="para" parents="70 51"/>
      <j2 rule="rewrite" parents="228"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="1402">
    <literal><![CDATA[
      (x ^ (y v (x ^ z))) v (x ^ z)' = 1
    ]]></literal>
    <justification jstring="[para(76(a,1),161(a,1,2,1))].">
      <j1 rule="para" parents="76 161"/>
    </justification>
  </clause>

  <clause id="2323">
    <literal><![CDATA[
      (x ^ y) v (x ^ ((y ^ x) v z)) = x ^ ((y ^ x) v z)
    ]]></literal>
    <justification jstring="[para(120(a,1),97(a,1,2)),rewrite([2(5)])].">
      <j1 rule="para" parents="120 97"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="4772">
    <literal><![CDATA[
      x ^ ((x ^ y) v (x ^ z)) = (x ^ y) v (x ^ z)
    ]]></literal>
    <justification jstring="[para(7(a,1),147(a,1,1))].">
      <j1 rule="para" parents="7 147"/>
    </justification>
  </clause>

  <clause id="4781">
    <literal><![CDATA[
      (x ^ (y v z)) v (x ^ ((y ^ u) v z)) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(147(a,1),25(a,1,2,2))].">
      <j1 rule="para" parents="147 25"/>
    </justification>
  </clause>

  <clause id="6957">
    <literal><![CDATA[
      x' ^ (y v x) != 0
    ]]></literal>
    <literal><![CDATA[
      x v y = x
    ]]></literal>
    <justification jstring="[para(2(a,1),793(a,1,2))].">
      <j1 rule="para" parents="2 793"/>
    </justification>
  </clause>

  <clause id="7297">
    <literal><![CDATA[
      x ^ ((y ^ x) v (x ^ z)) = (y ^ x) v (x ^ z)
    ]]></literal>
    <justification jstring="[para(24(a,1),176(a,1,1)),rewrite([24(3),24(7)])].">
      <j1 rule="para" parents="24 176"/>
      <j2 rule="rewrite" parents="24 24"/>
    </justification>
  </clause>

  <clause id="22426">
    <literal><![CDATA[
      x v (y ^ ((x ^ y) v z)) = x v (y ^ (x v z))
    ]]></literal>
    <justification jstring="[para(408(a,1),24(a,1,2)),rewrite([2(7),15(7),4781(6)]),flip(a)].">
      <j1 rule="para" parents="408 24"/>
      <j2 rule="rewrite" parents="2 15 4781"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22734">
    <literal><![CDATA[
      x' ^ (y ^ (x v (y' ^ z))) = 0
    ]]></literal>
    <justification jstring="[para(73(a,1),417(a,1,2,2,2,2)),rewrite([4(4),72(4),33(3),245(3)]),flip(a)].">
      <j1 rule="para" parents="73 417"/>
      <j2 rule="rewrite" parents="4 72 33 245"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23047">
    <literal><![CDATA[
      x' ^ (y ^ (x v y')) = 0
    ]]></literal>
    <justification jstring="[para(6(a,1),22734(a,1,2,2,2))].">
      <j1 rule="para" parents="6 22734"/>
    </justification>
  </clause>

  <clause id="23081">
    <literal><![CDATA[
      x' ^ (y ^ (y' v x)) = 0
    ]]></literal>
    <justification jstring="[para(2(a,1),23047(a,1,2,2))].">
      <j1 rule="para" parents="2 23047"/>
    </justification>
  </clause>

  <clause id="23642">
    <literal><![CDATA[
      x ^ (x' v y) = x ^ y
    ]]></literal>
    <justification jstring="[hyper(34,a,1402,a,b,23081,a),rewrite([228(3),657(5)]),flip(a)].">
      <j1 rule="hyper" parents="34 1402 23081"/>
      <j2 rule="rewrite" parents="228 657"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23665">
    <literal><![CDATA[
      x ^ (y v x') = x ^ y
    ]]></literal>
    <justification jstring="[para(2(a,1),23642(a,1,2))].">
      <j1 rule="para" parents="2 23642"/>
    </justification>
  </clause>

  <clause id="23672">
    <literal><![CDATA[
      x ^ (x' ^ y)' = x
    ]]></literal>
    <justification jstring="[para(143(a,1),23642(a,1,2)),rewrite([30(2)]),flip(a)].">
      <j1 rule="para" parents="143 23642"/>
      <j2 rule="rewrite" parents="30"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23673">
    <literal><![CDATA[
      x ^ (y ^ x')' = x
    ]]></literal>
    <justification jstring="[para(152(a,1),23642(a,1,2)),rewrite([30(2)]),flip(a)].">
      <j1 rule="para" parents="152 23642"/>
      <j2 rule="rewrite" parents="30"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23675">
    <literal><![CDATA[
      x ^ (y v (x' v y)') = x
    ]]></literal>
    <justification jstring="[para(29(a,1),23642(a,1,2)),rewrite([30(2)]),flip(a)].">
      <j1 rule="para" parents="29 23642"/>
      <j2 rule="rewrite" parents="30"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23676">
    <literal><![CDATA[
      x' ^ (x v y) = x' ^ y
    ]]></literal>
    <justification jstring="[para(228(a,1),23642(a,1,2,1))].">
      <j1 rule="para" parents="228 23642"/>
    </justification>
  </clause>

  <clause id="24127">
    <literal><![CDATA[
      (x ^ y')' v (z ^ y) = (x ^ y')'
    ]]></literal>
    <justification jstring="[para(23673(a,1),94(a,1,2,2))].">
      <j1 rule="para" parents="23673 94"/>
    </justification>
  </clause>

  <clause id="24128">
    <literal><![CDATA[
      x' ^ (y ^ x)' = x'
    ]]></literal>
    <justification jstring="[para(228(a,1),23673(a,1,2,1,2))].">
      <j1 rule="para" parents="228 23673"/>
    </justification>
  </clause>

  <clause id="24133">
    <literal><![CDATA[
      x v ((y ^ x')' v z) = (y ^ x')' v z
    ]]></literal>
    <justification jstring="[para(23673(a,1),92(a,1,2,1)),rewrite([15(5)])].">
      <j1 rule="para" parents="23673 92"/>
      <j2 rule="rewrite" parents="15"/>
    </justification>
  </clause>

  <clause id="24280">
    <literal><![CDATA[
      x ^ ((y v x') ^ z) = x ^ (y ^ z)
    ]]></literal>
    <justification jstring="[para(23665(a,1),5(a,1,1)),rewrite([5(2)]),flip(a)].">
      <j1 rule="para" parents="23665 5"/>
      <j2 rule="rewrite" parents="5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="24284">
    <literal><![CDATA[
      x' ^ (y v x) = x' ^ y
    ]]></literal>
    <justification jstring="[para(228(a,1),23665(a,1,2,2))].">
      <j1 rule="para" parents="228 23665"/>
    </justification>
  </clause>

  <clause id="24384">
    <literal><![CDATA[
      x' ^ y != 0
    ]]></literal>
    <literal><![CDATA[
      x v y = x
    ]]></literal>
    <justification jstring="[back_rewrite(6957),rewrite([24284(3)])].">
      <j1 rule="back_rewrite" parents="6957"/>
      <j2 rule="rewrite" parents="24284"/>
    </justification>
  </clause>

  <clause id="24564">
    <literal><![CDATA[
      x' ^ (y ^ (z ^ x)') = y ^ x'
    ]]></literal>
    <justification jstring="[para(24128(a,1),17(a,1,2)),flip(a)].">
      <j1 rule="para" parents="24128 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="24566">
    <literal><![CDATA[
      x' ^ (y v x)' = (y v x)'
    ]]></literal>
    <justification jstring="[para(18(a,1),24128(a,1,2,1)),rewrite([4(4)])].">
      <j1 rule="para" parents="18 24128"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="25215">
    <literal><![CDATA[
      x v (y v (y' v x)') = x v (y' v x)'
    ]]></literal>
    <justification jstring="[para(23675(a,1),93(a,1,2,2)),rewrite([2(4)])].">
      <j1 rule="para" parents="23675 93"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="25429">
    <literal><![CDATA[
      x' ^ ((x ^ y) v z) = x' ^ z
    ]]></literal>
    <justification jstring="[para(22(a,1),23676(a,1,2)),rewrite([23676(3)]),flip(a)].">
      <j1 rule="para" parents="22 23676"/>
      <j2 rule="rewrite" parents="23676"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="27413">
    <literal><![CDATA[
      x v (x' v y)' = x
    ]]></literal>
    <justification jstring="[hyper(24384,a,103,a)].">
      <j1 rule="hyper" parents="24384 103"/>
    </justification>
  </clause>

  <clause id="27444">
    <literal><![CDATA[
      x v (y' v x)' = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(25215),rewrite([27413(4)]),flip(a)].">
      <j1 rule="back_rewrite" parents="25215"/>
      <j2 rule="rewrite" parents="27413"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="28877">
    <literal><![CDATA[
      x v (x' ^ y)' = (x' ^ y)'
    ]]></literal>
    <justification jstring="[para(143(a,1),27444(a,1,2,1)),rewrite([68(5),2(5),62(5),2(7)]),flip(a)].">
      <j1 rule="para" parents="143 27444"/>
      <j2 rule="rewrite" parents="68 2 62 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="28917">
    <literal><![CDATA[
      (x' v y) ^ (y v x) = y
    ]]></literal>
    <justification jstring="[para(27444(a,1),23665(a,1,2)),rewrite([4(7),18(7)])].">
      <j1 rule="para" parents="27444 23665"/>
      <j2 rule="rewrite" parents="4 18"/>
    </justification>
  </clause>

  <clause id="28918">
    <literal><![CDATA[
      (x' v y)' = y' ^ x
    ]]></literal>
    <justification jstring="[para(27444(a,1),23676(a,1,2)),rewrite([23676(3),24566(7)]),flip(a)].">
      <j1 rule="para" parents="27444 23676"/>
      <j2 rule="rewrite" parents="23676 24566"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="28923">
    <literal><![CDATA[
      (x ^ y) v (y' ^ x) = x
    ]]></literal>
    <justification jstring="[para(27444(a,1),27444(a,1,2,1)),rewrite([28918(4),228(2),28918(4),28918(8),228(6),2(6),7(6)])].">
      <j1 rule="para" parents="27444 27444"/>
      <j2 rule="rewrite" parents="28918 228 28918 28918 228 2 7"/>
    </justification>
  </clause>

  <clause id="28924">
    <literal><![CDATA[
      x v (x' ^ y) = x v y
    ]]></literal>
    <justification jstring="[para(27444(a,1),27444(a,2)),rewrite([28918(3),2(4),28877(4),228(4)])].">
      <j1 rule="para" parents="27444 27444"/>
      <j2 rule="rewrite" parents="28918 2 28877 228"/>
    </justification>
  </clause>

  <clause id="29047">
    <literal><![CDATA[
      (x v y) ^ (y v x') = y
    ]]></literal>
    <justification jstring="[para(228(a,1),28917(a,1,1,1))].">
      <j1 rule="para" parents="228 28917"/>
    </justification>
  </clause>

  <clause id="31196">
    <literal><![CDATA[
      (x ^ y) v x' = y v x'
    ]]></literal>
    <justification jstring="[para(18(a,1),28923(a,1,2)),rewrite([4(3),23665(3)])].">
      <j1 rule="para" parents="18 28923"/>
      <j2 rule="rewrite" parents="4 23665"/>
    </justification>
  </clause>

  <clause id="31396">
    <literal><![CDATA[
      x' v (x ^ (x ^ y)') = (x ^ y)'
    ]]></literal>
    <justification jstring="[para(23672(a,1),28923(a,1,2)),rewrite([228(2),4(3),2(5),228(7)])].">
      <j1 rule="para" parents="23672 28923"/>
      <j2 rule="rewrite" parents="228 4 2 228"/>
    </justification>
  </clause>

  <clause id="31399">
    <literal><![CDATA[
      x ^ (x ^ y)' = x ^ y'
    ]]></literal>
    <justification jstring="[para(28923(a,1),23676(a,1,2)),rewrite([4(3),4(8),5(8),24564(8)])].">
      <j1 rule="para" parents="28923 23676"/>
      <j2 rule="rewrite" parents="4 4 5 24564"/>
    </justification>
  </clause>

  <clause id="31418">
    <literal><![CDATA[
      (x ^ y)' = x' v (x ^ y')
    ]]></literal>
    <justification jstring="[back_rewrite(31396),rewrite([31399(4)]),flip(a)].">
      <j1 rule="back_rewrite" parents="31396"/>
      <j2 rule="rewrite" parents="31399"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="31900">
    <literal><![CDATA[
      x' v ((x ^ y) v z) = x' v (y v z)
    ]]></literal>
    <justification jstring="[back_rewrite(24133),rewrite([31418(3),228(3),3(4),762(5),31418(6),228(6),3(7)]),flip(a)].">
      <j1 rule="back_rewrite" parents="24133"/>
      <j2 rule="rewrite" parents="31418 228 3 762 31418 228 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="31901">
    <literal><![CDATA[
      x' v (x ^ y) = x' v y
    ]]></literal>
    <justification jstring="[back_rewrite(24127),rewrite([31418(3),228(3),3(5),31900(5),24(3),31418(5),228(5)]),flip(a)].">
      <j1 rule="back_rewrite" parents="24127"/>
      <j2 rule="rewrite" parents="31418 228 3 31900 24 31418 228"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="37086">
    <literal><![CDATA[
      (x ^ y)' = x' v y'
    ]]></literal>
    <justification jstring="[back_rewrite(31418),rewrite([31901(6)])].">
      <j1 rule="back_rewrite" parents="31418"/>
      <j2 rule="rewrite" parents="31901"/>
    </justification>
  </clause>

  <clause id="37146">
    <literal><![CDATA[
      x v (y ^ (x' ^ z)) = x v (y ^ z)
    ]]></literal>
    <justification jstring="[para(17(a,1),28924(a,1,2))].">
      <j1 rule="para" parents="17 28924"/>
    </justification>
  </clause>

  <clause id="37166">
    <literal><![CDATA[
      (x ^ y) v (y ^ (x' ^ z)) = y ^ ((x ^ y) v z)
    ]]></literal>
    <justification jstring="[para(125(a,1),28924(a,2)),rewrite([37086(3),17(8),24280(8),25429(5)])].">
      <j1 rule="para" parents="125 28924"/>
      <j2 rule="rewrite" parents="37086 17 24280 25429"/>
    </justification>
  </clause>

  <clause id="37831">
    <literal><![CDATA[
      x ^ ((y v z) ^ ((x ^ ((x ^ y) v z)) v y')) = x ^ ((x ^ y) v z)
    ]]></literal>
    <justification jstring="[para(29047(a,1),471(a,1,2)),rewrite([52(4)]),flip(a)].">
      <j1 rule="para" parents="29047 471"/>
      <j2 rule="rewrite" parents="52"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="39476">
    <literal><![CDATA[
      (x ^ ((x ^ y) v z)) v y' = x v y'
    ]]></literal>
    <justification jstring="[para(120(a,1),31196(a,1,1)),rewrite([31196(3)]),flip(a)].">
      <j1 rule="para" parents="120 31196"/>
      <j2 rule="rewrite" parents="31196"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="39588">
    <literal><![CDATA[
      x ^ ((x ^ y) v z) = (y v z) ^ x
    ]]></literal>
    <justification jstring="[back_rewrite(37831),rewrite([39476(6),65(5)]),flip(a)].">
      <j1 rule="back_rewrite" parents="37831"/>
      <j2 rule="rewrite" parents="39476 65"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="40124">
    <literal><![CDATA[
      (x ^ y) v (x ^ z) = x ^ (y v (x ^ z))
    ]]></literal>
    <justification jstring="[back_rewrite(4772),rewrite([39588(4),4(3)]),flip(a)].">
      <j1 rule="back_rewrite" parents="4772"/>
      <j2 rule="rewrite" parents="39588 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="40469">
    <literal><![CDATA[
      x ^ ((y ^ x) v z) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[back_rewrite(2323),rewrite([40124(5),22426(4),671(4)]),flip(a)].">
      <j1 rule="back_rewrite" parents="2323"/>
      <j2 rule="rewrite" parents="40124 22426 671"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="40484">
    <literal><![CDATA[
      c1 ^ (c2 v (c1 ^ c3)) != c1 ^ (c2 v c3)
    ]]></literal>
    <attribute><![CDATA[
      answer(distributivity)
    ]]></attribute>
    <justification jstring="[back_rewrite(13),rewrite([40124(7)])].">
      <j1 rule="back_rewrite" parents="13"/>
      <j2 rule="rewrite" parents="40124"/>
    </justification>
  </clause>

  <clause id="40615">
    <literal><![CDATA[
      (x ^ y) v (y ^ (x' ^ z)) = y ^ (x v z)
    ]]></literal>
    <justification jstring="[back_rewrite(37166),rewrite([40469(8)])].">
      <j1 rule="back_rewrite" parents="37166"/>
      <j2 rule="rewrite" parents="40469"/>
    </justification>
  </clause>

  <clause id="40706">
    <literal><![CDATA[
      (x ^ y) v (y ^ z) = y ^ (x v (y ^ z))
    ]]></literal>
    <justification jstring="[back_rewrite(7297),rewrite([40469(4)]),flip(a)].">
      <j1 rule="back_rewrite" parents="7297"/>
      <j2 rule="rewrite" parents="40469"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="40765">
    <literal><![CDATA[
      x ^ (y v (x ^ z)) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[back_rewrite(40615),rewrite([40706(5),37146(4)])].">
      <j1 rule="back_rewrite" parents="40615"/>
      <j2 rule="rewrite" parents="40706 37146"/>
    </justification>
  </clause>

  <clause id="40766">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(distributivity)
    ]]></attribute>
    <justification jstring="[resolve(40765,a,40484,a)].">
      <j1 rule="resolve" parents="40765 40484"/>
    </justification>
  </clause>

</proof>

</proofs>
