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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 August-2006A, August 2006.
Process 11507 was started by mccune on cleo.thornwood,
Sat Aug 12 21:03:00 2006
The command was "/home/mccune/bin/prover9 -f lt.in uc.in H65d.in".
]]></heading>

<proof number="1" length="119" max_count="27">

<comments><![CDATA[
% Proof 1 at 154.65 (+ 1.84) seconds: distributivity.
% Length of proof is 119.
% Level of proof is 25.
% Maximum clause weight is 27.
% Given clauses 1707.
]]></comments>

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

  <clause id="2">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="3">
    <literal><![CDATA[
      (x v y) v z = x v (y v z)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      x ^ y = y ^ x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

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

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

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

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

  <clause id="10">
    <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">
    <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">
    <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="19">
    <literal><![CDATA[
      (x v y) ^ (x v (y v z)) = x v y
    ]]></literal>
    <justification jstring="[para(3(a,1),6(a,1,2))].">
      <j1 rule="para" parents="3 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="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="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="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="32">
    <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="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="35">
    <literal><![CDATA[
      x v (y v z) != 1
    ]]></literal>
    <literal><![CDATA[
      (x v y) ^ z != 0
    ]]></literal>
    <literal><![CDATA[
      (x v y)' = z
    ]]></literal>
    <justification jstring="[para(3(a,1),10(a,1))].">
      <j1 rule="para" parents="3 10"/>
    </justification>
  </clause>

  <clause id="45">
    <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="62">
    <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="64">
    <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="68">
    <literal><![CDATA[
      x v (y v (x ^ z)) = y v x
    ]]></literal>
    <justification jstring="[para(7(a,1),15(a,1,2)),flip(a)].">
      <j1 rule="para" parents="7 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      x v (y v x') = y v 1
    ]]></literal>
    <justification jstring="[para(8(a,1),15(a,1,2)),flip(a)].">
      <j1 rule="para" parents="8 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="73">
    <literal><![CDATA[
      1 v x = 1
    ]]></literal>
    <justification jstring="[para(62(a,1),6(a,1))].">
      <j1 rule="para" parents="62 6"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      x v (x' v y) = 1
    ]]></literal>
    <justification jstring="[back_rewrite(28),rewrite(73(5))].">
      <j1 rule="back_rewrite" parents="28"/>
      <j2 rule="rewrite" parents="73"/>
    </justification>
  </clause>

  <clause id="75">
    <literal><![CDATA[
      0' = 1
    ]]></literal>
    <justification jstring="[hyper(10,a,64,a,b,30,a)].">
      <j1 rule="hyper" parents="10 64 30"/>
    </justification>
  </clause>

  <clause id="76">
    <literal><![CDATA[
      0 ^ x = 0
    ]]></literal>
    <justification jstring="[para(64(a,1),6(a,1,2))].">
      <j1 rule="para" parents="64 6"/>
    </justification>
  </clause>

  <clause id="78">
    <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="79">
    <literal><![CDATA[
      x v (y ^ (x ^ z)) = x
    ]]></literal>
    <justification jstring="[para(17(a,1),7(a,1,2))].">
      <j1 rule="para" parents="17 7"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      x v 1 = 1
    ]]></literal>
    <justification jstring="[para(73(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="73 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="83">
    <literal><![CDATA[
      x v (y v x') = 1
    ]]></literal>
    <justification jstring="[back_rewrite(69),rewrite(82(5))].">
      <j1 rule="back_rewrite" parents="69"/>
      <j2 rule="rewrite" parents="82"/>
    </justification>
  </clause>

  <clause id="85">
    <literal><![CDATA[
      x ^ 0 = 0
    ]]></literal>
    <justification jstring="[para(76(a,1),4(a,1)),flip(a)].">
      <j1 rule="para" parents="76 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      x ^ ((y v x) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(18(a,1),5(a,1,1)),flip(a)].">
      <j1 rule="para" parents="18 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="93">
    <literal><![CDATA[
      x ^ (y ^ (z v x)) = y ^ x
    ]]></literal>
    <justification jstring="[para(18(a,1),17(a,1,2)),flip(a)].">
      <j1 rule="para" parents="18 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="94">
    <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="100">
    <literal><![CDATA[
      x v (y v ((x v y)' v z)) = 1
    ]]></literal>
    <justification jstring="[para(74(a,1),3(a,1)),flip(a)].">
      <j1 rule="para" parents="74 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="103">
    <literal><![CDATA[
      (x v y) ^ (y v (x v z)) = y v x
    ]]></literal>
    <justification jstring="[para(2(a,1),19(a,1,1))].">
      <j1 rule="para" parents="2 19"/>
    </justification>
  </clause>

  <clause id="125">
    <literal><![CDATA[
      x ^ (x v y)' = 0
    ]]></literal>
    <justification jstring="[para(9(a,1),20(a,1,2)),rewrite(85(2)),flip(a)].">
      <j1 rule="para" parents="9 20"/>
      <j2 rule="rewrite" parents="85"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="129">
    <literal><![CDATA[
      (x ^ y) v ((x v z) ^ y) = (x v z) ^ y
    ]]></literal>
    <justification jstring="[para(20(a,1),24(a,1,2)),rewrite(2(4))].">
      <j1 rule="para" parents="20 24"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="132">
    <literal><![CDATA[
      x ^ (y v x)' = 0
    ]]></literal>
    <justification jstring="[para(2(a,1),125(a,1,2,1))].">
      <j1 rule="para" parents="2 125"/>
    </justification>
  </clause>

  <clause id="139">
    <literal><![CDATA[
      x ^ (y ^ (z v (x ^ y))') = 0
    ]]></literal>
    <justification jstring="[para(132(a,1),5(a,1)),flip(a)].">
      <j1 rule="para" parents="132 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="167">
    <literal><![CDATA[
      x v (y v (z ^ (x ^ u))) = y v x
    ]]></literal>
    <justification jstring="[para(79(a,1),15(a,1,2)),flip(a)].">
      <j1 rule="para" parents="79 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="176">
    <literal><![CDATA[
      x v (x ^ y)' = 1
    ]]></literal>
    <justification jstring="[para(8(a,1),22(a,1,2)),rewrite(82(2)),flip(a)].">
      <j1 rule="para" parents="8 22"/>
      <j2 rule="rewrite" parents="82"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="186">
    <literal><![CDATA[
      x v (y ^ x)' = 1
    ]]></literal>
    <justification jstring="[para(4(a,1),176(a,1,2,1))].">
      <j1 rule="para" parents="4 176"/>
    </justification>
  </clause>

  <clause id="191">
    <literal><![CDATA[
      x v (y v (z ^ (x v y))') = 1
    ]]></literal>
    <justification jstring="[para(186(a,1),3(a,1)),flip(a)].">
      <j1 rule="para" parents="186 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="207">
    <literal><![CDATA[
      x v (y v (y v x)') = 1
    ]]></literal>
    <justification jstring="[para(2(a,1),29(a,1,2,2,1))].">
      <j1 rule="para" parents="2 29"/>
    </justification>
  </clause>

  <clause id="222">
    <literal><![CDATA[
      x ^ (y ^ (y ^ x)') = 0
    ]]></literal>
    <justification jstring="[para(4(a,1),32(a,1,2,2,1))].">
      <j1 rule="para" parents="4 32"/>
    </justification>
  </clause>

  <clause id="236">
    <literal><![CDATA[
      (x ^ y) v (y ^ (x ^ z)) = y ^ x
    ]]></literal>
    <justification jstring="[para(4(a,1),25(a,1,1))].">
      <j1 rule="para" parents="4 25"/>
    </justification>
  </clause>

  <clause id="262">
    <literal><![CDATA[
      x v y != 1
    ]]></literal>
    <literal><![CDATA[
      x ^ y != 0
    ]]></literal>
    <literal><![CDATA[
      y' = x
    ]]></literal>
    <justification jstring="[para(4(a,1),34(b,1))].">
      <j1 rule="para" parents="4 34"/>
    </justification>
  </clause>

  <clause id="265">
    <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="310">
    <literal><![CDATA[
      x v (y v z) != 1
    ]]></literal>
    <literal><![CDATA[
      (y v x) ^ z != 0
    ]]></literal>
    <literal><![CDATA[
      (y v x)' = z
    ]]></literal>
    <justification jstring="[para(15(a,1),35(a,1))].">
      <j1 rule="para" parents="15 35"/>
    </justification>
  </clause>

  <clause id="338">
    <literal><![CDATA[
      x v (y v (z ^ ((x v y) ^ z)')) != 1
    ]]></literal>
    <literal><![CDATA[
      z ^ ((x v y) ^ z)' = (x v y)'
    ]]></literal>
    <justification jstring="[para(32(a,1),35(b,1)),flip(c),xx(b)].">
      <j1 rule="para" parents="32 35"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="507">
    <literal><![CDATA[
      x v (y v (y v (x ^ z))') = 1
    ]]></literal>
    <justification jstring="[para(207(a,1),22(a,1,2)),rewrite(82(2)),flip(a)].">
      <j1 rule="para" parents="207 22"/>
      <j2 rule="rewrite" parents="82"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="508">
    <literal><![CDATA[
      x ^ (y v (y v x)') != 0
    ]]></literal>
    <literal><![CDATA[
      (y v (y v x)')' = x
    ]]></literal>
    <justification jstring="[para(207(a,1),34(a,1)),rewrite(4(7)),xx(a)].">
      <j1 rule="para" parents="207 34"/>
      <j2 rule="rewrite" parents="4"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="517">
    <literal><![CDATA[
      x ^ (y ^ (y ^ (x v z))') = 0
    ]]></literal>
    <justification jstring="[para(222(a,1),20(a,1,2)),rewrite(85(2)),flip(a)].">
      <j1 rule="para" parents="222 20"/>
      <j2 rule="rewrite" parents="85"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="518">
    <literal><![CDATA[
      x v (y v (z ^ (z ^ (x v y))')) != 1
    ]]></literal>
    <literal><![CDATA[
      z ^ (z ^ (x v y))' = (x v y)'
    ]]></literal>
    <justification jstring="[para(222(a,1),35(b,1)),flip(c),xx(b)].">
      <j1 rule="para" parents="222 35"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="787">
    <literal><![CDATA[
      (x ^ y) v (x ^ (z v y)) = x ^ (z v y)
    ]]></literal>
    <justification jstring="[para(93(a,1),24(a,1,2)),rewrite(2(4))].">
      <j1 rule="para" parents="93 24"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="1123">
    <literal><![CDATA[
      x ^ (x' v (x ^ y)) = x ^ (x' v y)
    ]]></literal>
    <justification jstring="[para(9(a,1),45(a,1,2,2,2,1)),rewrite(64(3))].">
      <j1 rule="para" parents="9 45"/>
      <j2 rule="rewrite" parents="64"/>
    </justification>
  </clause>

  <clause id="2450">
    <literal><![CDATA[
      x v ((y ^ x) v (x ^ z))' = 1
    ]]></literal>
    <justification jstring="[para(507(a,1),94(a,1)),flip(a)].">
      <j1 rule="para" parents="507 94"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2546">
    <literal><![CDATA[
      x ^ ((y v x) ^ (x v z))' = 0
    ]]></literal>
    <justification jstring="[para(517(a,1),89(a,1)),flip(a)].">
      <j1 rule="para" parents="517 89"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3674">
    <literal><![CDATA[
      x v ((x v y) ^ (x v z)) = (x v y) ^ (x v z)
    ]]></literal>
    <justification jstring="[para(6(a,1),129(a,1,1))].">
      <j1 rule="para" parents="6 129"/>
    </justification>
  </clause>

  <clause id="7097">
    <literal><![CDATA[
      (x ^ y) v (z v (y ^ (x ^ u))) = z v (y ^ x)
    ]]></literal>
    <justification jstring="[para(236(a,1),15(a,1,2)),flip(a)].">
      <j1 rule="para" parents="236 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13548">
    <literal><![CDATA[
      x v ((y v x) ^ (y v z)) = x v y
    ]]></literal>
    <justification jstring="[para(103(a,1),787(a,1,2)),rewrite(15(5),2(4),3674(4),103(8))].">
      <j1 rule="para" parents="103 787"/>
      <j2 rule="rewrite" parents="15 2 3674 103"/>
    </justification>
  </clause>

  <clause id="13552">
    <literal><![CDATA[
      x v ((x v y) ^ (y v z)) = x v y
    ]]></literal>
    <justification jstring="[para(2(a,1),13548(a,1,2,1))].">
      <j1 rule="para" parents="2 13548"/>
    </justification>
  </clause>

  <clause id="13570">
    <literal><![CDATA[
      x v ((x v y) ^ (z v y)) = x v y
    ]]></literal>
    <justification jstring="[para(2(a,1),13552(a,1,2,2))].">
      <j1 rule="para" parents="2 13552"/>
    </justification>
  </clause>

  <clause id="13763">
    <literal><![CDATA[
      x ^ (x' v y) = x ^ ((x ^ y) v x')
    ]]></literal>
    <justification jstring="[para(2(a,1),1123(a,1,2)),flip(a)].">
      <j1 rule="para" parents="2 1123"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13767">
    <literal><![CDATA[
      x ^ (x' v (x v y)') = 0
    ]]></literal>
    <justification jstring="[para(125(a,1),1123(a,1,2,2)),rewrite(2(3),64(3),9(2)),flip(a)].">
      <j1 rule="para" parents="125 1123"/>
      <j2 rule="rewrite" parents="2 64 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14000">
    <literal><![CDATA[
      x' v (x v y)' = x'
    ]]></literal>
    <justification jstring="[hyper(310,a,100,a,b,13767,a),rewrite(27(1),27(2),27(3)),flip(a)].">
      <j1 rule="hyper" parents="310 100 13767"/>
      <j2 rule="rewrite" parents="27 27 27"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14012">
    <literal><![CDATA[
      x' v (y v x)' = x'
    ]]></literal>
    <justification jstring="[para(2(a,1),14000(a,1,2,1))].">
      <j1 rule="para" parents="2 14000"/>
    </justification>
  </clause>

  <clause id="14016">
    <literal><![CDATA[
      x' ^ (x v y)' = (x v y)'
    ]]></literal>
    <justification jstring="[para(14000(a,1),18(a,1,2)),rewrite(4(4))].">
      <j1 rule="para" parents="14000 18"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="14743">
    <literal><![CDATA[
      x' v (x ^ y)' = (x ^ y)'
    ]]></literal>
    <justification jstring="[para(7(a,1),14012(a,1,2,1)),rewrite(2(4))].">
      <j1 rule="para" parents="7 14012"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="15144">
    <literal><![CDATA[
      x ^ (y v (x ^ y'))' = 0
    ]]></literal>
    <justification jstring="[para(14016(a,1),139(a,1,2))].">
      <j1 rule="para" parents="14016 139"/>
    </justification>
  </clause>

  <clause id="15187">
    <literal><![CDATA[
      (x ^ y) v (y ^ (x ^ y)') = y
    ]]></literal>
    <justification jstring="[hyper(262,a,2450,a,b,15144,a),rewrite(265(7))].">
      <j1 rule="hyper" parents="262 2450 15144"/>
      <j2 rule="rewrite" parents="265"/>
    </justification>
  </clause>

  <clause id="15404">
    <literal><![CDATA[
      x' ^ (x ^ y)' = x'
    ]]></literal>
    <justification jstring="[para(14743(a,1),6(a,1,2))].">
      <j1 rule="para" parents="14743 6"/>
    </justification>
  </clause>

  <clause id="15408">
    <literal><![CDATA[
      x' ^ (y ^ (x ^ z)') = y ^ x'
    ]]></literal>
    <justification jstring="[para(14743(a,1),78(a,1,2,2))].">
      <j1 rule="para" parents="14743 78"/>
    </justification>
  </clause>

  <clause id="15415">
    <literal><![CDATA[
      x v (y ^ (x v y'))' = 1
    ]]></literal>
    <justification jstring="[para(14743(a,1),191(a,1,2))].">
      <j1 rule="para" parents="14743 191"/>
    </justification>
  </clause>

  <clause id="15660">
    <literal><![CDATA[
      (x v y) ^ (y v (x v y)') = y
    ]]></literal>
    <justification jstring="[hyper(262,a,15415,a,b,2546,a),rewrite(265(7))].">
      <j1 rule="hyper" parents="262 15415 2546"/>
      <j2 rule="rewrite" parents="265"/>
    </justification>
  </clause>

  <clause id="15678">
    <literal><![CDATA[
      x v (y' ^ (x v y))' = 1
    ]]></literal>
    <justification jstring="[para(265(a,1),15415(a,1,2,1,2,2))].">
      <j1 rule="para" parents="265 15415"/>
    </justification>
  </clause>

  <clause id="15809">
    <literal><![CDATA[
      x v ((x v y) ^ y')' = 1
    ]]></literal>
    <justification jstring="[para(4(a,1),15678(a,1,2,1))].">
      <j1 rule="para" parents="4 15678"/>
    </justification>
  </clause>

  <clause id="15843">
    <literal><![CDATA[
      x v ((y v x) ^ y')' = 1
    ]]></literal>
    <justification jstring="[para(2(a,1),15809(a,1,2,1,1))].">
      <j1 rule="para" parents="2 15809"/>
    </justification>
  </clause>

  <clause id="15941">
    <literal><![CDATA[
      x v ((x v y) ^ x') = x v y
    ]]></literal>
    <justification jstring="[para(6(a,1),15187(a,1,1)),rewrite(6(3))].">
      <j1 rule="para" parents="6 15187"/>
      <j2 rule="rewrite" parents="6"/>
    </justification>
  </clause>

  <clause id="15960">
    <literal><![CDATA[
      x' v (x ^ (x ^ y)') = (x ^ y)'
    ]]></literal>
    <justification jstring="[para(15404(a,1),15187(a,1,1)),rewrite(15404(7),265(5),4(4))].">
      <j1 rule="para" parents="15404 15187"/>
      <j2 rule="rewrite" parents="15404 265 4"/>
    </justification>
  </clause>

  <clause id="16040">
    <literal><![CDATA[
      (x v y) ^ (y v (y v x)') = y
    ]]></literal>
    <justification jstring="[para(2(a,1),15660(a,1,2,2,1))].">
      <j1 rule="para" parents="2 15660"/>
    </justification>
  </clause>

  <clause id="16041">
    <literal><![CDATA[
      x ^ ((x ^ y) v x') = x ^ y
    ]]></literal>
    <justification jstring="[para(7(a,1),15660(a,1,1)),rewrite(7(3))].">
      <j1 rule="para" parents="7 15660"/>
      <j2 rule="rewrite" parents="7"/>
    </justification>
  </clause>

  <clause id="16062">
    <literal><![CDATA[
      x ^ (x' v y) = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(13763),rewrite(16041(7))].">
      <j1 rule="back_rewrite" parents="13763"/>
      <j2 rule="rewrite" parents="16041"/>
    </justification>
  </clause>

  <clause id="16070">
    <literal><![CDATA[
      x ^ (y v x') = x ^ y
    ]]></literal>
    <justification jstring="[para(2(a,1),16062(a,1,2))].">
      <j1 rule="para" parents="2 16062"/>
    </justification>
  </clause>

  <clause id="16280">
    <literal><![CDATA[
      x ^ (y v (z v x)') = x ^ y
    ]]></literal>
    <justification jstring="[para(16070(a,1),89(a,1,2)),rewrite(89(3)),flip(a)].">
      <j1 rule="para" parents="16070 89"/>
      <j2 rule="rewrite" parents="89"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16291">
    <literal><![CDATA[
      (x v y')' = y ^ (y ^ x)'
    ]]></literal>
    <justification jstring="[para(16070(a,1),518(a,1,2,2,2,1)),rewrite(15960(5),186(3),16070(6)),flip(b),xx(a)].">
      <j1 rule="para" parents="16070 518"/>
      <j2 rule="rewrite" parents="15960 186 16070"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="16293">
    <literal><![CDATA[
      (x v y) ^ x' = y ^ x'
    ]]></literal>
    <justification jstring="[para(15843(a,1),16070(a,1,2)),rewrite(4(5),62(5),4(7),89(7))].">
      <j1 rule="para" parents="15843 16070"/>
      <j2 rule="rewrite" parents="4 62 4 89"/>
    </justification>
  </clause>

  <clause id="16296">
    <literal><![CDATA[
      x ^ y != 0
    ]]></literal>
    <literal><![CDATA[
      x ^ y' = x
    ]]></literal>
    <justification jstring="[back_rewrite(508),rewrite(16280(4),16291(7),4(6),6(6),16293(6))].">
      <j1 rule="back_rewrite" parents="508"/>
      <j2 rule="rewrite" parents="16280 16291 4 6 16293"/>
    </justification>
  </clause>

  <clause id="16305">
    <literal><![CDATA[
      x v (y ^ x') = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(15941),rewrite(16293(3))].">
      <j1 rule="back_rewrite" parents="15941"/>
      <j2 rule="rewrite" parents="16293"/>
    </justification>
  </clause>

  <clause id="16313">
    <literal><![CDATA[
      x v (x' ^ y) = x v y
    ]]></literal>
    <justification jstring="[para(4(a,1),16305(a,1,2))].">
      <j1 rule="para" parents="4 16305"/>
    </justification>
  </clause>

  <clause id="16316">
    <literal><![CDATA[
      x v (y ^ (x ^ z)') = x v y
    ]]></literal>
    <justification jstring="[para(16305(a,1),22(a,1,2)),rewrite(22(3)),flip(a)].">
      <j1 rule="para" parents="16305 22"/>
      <j2 rule="rewrite" parents="22"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16317">
    <literal><![CDATA[
      x' v (y ^ x) = x' v y
    ]]></literal>
    <justification jstring="[para(265(a,1),16305(a,1,2,2))].">
      <j1 rule="para" parents="265 16305"/>
    </justification>
  </clause>

  <clause id="16360">
    <literal><![CDATA[
      x' v (x ^ y) = x' v y
    ]]></literal>
    <justification jstring="[para(265(a,1),16313(a,1,2,1))].">
      <j1 rule="para" parents="265 16313"/>
    </justification>
  </clause>

  <clause id="16898">
    <literal><![CDATA[
      (x v y)' = x' ^ (y ^ x')'
    ]]></literal>
    <justification jstring="[para(16293(a,1),338(a,1,2,2,2,1)),rewrite(16316(6),83(3),16293(7)),flip(b),xx(a)].">
      <j1 rule="para" parents="16293 338"/>
      <j2 rule="rewrite" parents="16316 83 16293"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="17053">
    <literal><![CDATA[
      x ^ (y ^ (y ^ x)')' = x
    ]]></literal>
    <justification jstring="[hyper(16296,a,222,a)].">
      <j1 rule="hyper" parents="16296 222"/>
    </justification>
  </clause>

  <clause id="17078">
    <literal><![CDATA[
      x' v ((y ^ x) v z) = x' v (y v z)
    ]]></literal>
    <justification jstring="[para(16317(a,1),3(a,1,1)),rewrite(3(3)),flip(a)].">
      <j1 rule="para" parents="16317 3"/>
      <j2 rule="rewrite" parents="3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="17130">
    <literal><![CDATA[
      x' v ((x v y) ^ z) = x' v z
    ]]></literal>
    <justification jstring="[para(20(a,1),16360(a,1,2)),rewrite(16360(3)),flip(a)].">
      <j1 rule="para" parents="20 16360"/>
      <j2 rule="rewrite" parents="16360"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="17235">
    <literal><![CDATA[
      x ^ (y ^ x)' = x ^ y'
    ]]></literal>
    <justification jstring="[para(32(a,1),17053(a,1,2,1,2,1)),rewrite(75(5),30(5),4(5),15408(5)),flip(a)].">
      <j1 rule="para" parents="32 17053"/>
      <j2 rule="rewrite" parents="75 30 4 15408"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="17269">
    <literal><![CDATA[
      (x v y)' = x' ^ y'
    ]]></literal>
    <justification jstring="[back_rewrite(16898),rewrite(17235(7))].">
      <j1 rule="back_rewrite" parents="16898"/>
      <j2 rule="rewrite" parents="17235"/>
    </justification>
  </clause>

  <clause id="17358">
    <literal><![CDATA[
      (x v y) ^ (y v x') = y
    ]]></literal>
    <justification jstring="[back_rewrite(16040),rewrite(17269(3),16313(5))].">
      <j1 rule="back_rewrite" parents="16040"/>
      <j2 rule="rewrite" parents="17269 16313"/>
    </justification>
  </clause>

  <clause id="21597">
    <literal><![CDATA[
      (x v y) ^ (x' v y) = y
    ]]></literal>
    <justification jstring="[para(2(a,1),17358(a,1,2))].">
      <j1 rule="para" parents="2 17358"/>
    </justification>
  </clause>

  <clause id="22264">
    <literal><![CDATA[
      (x ^ y) v z = (y v z) ^ (y' v (x v z))
    ]]></literal>
    <justification jstring="[para(94(a,1),21597(a,1,1)),rewrite(17078(5)),flip(a)].">
      <j1 rule="para" parents="94 21597"/>
      <j2 rule="rewrite" parents="17078"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22340">
    <literal><![CDATA[
      (x v y) ^ (x' v (z v y)) = (x v y) ^ (z v y)
    ]]></literal>
    <justification jstring="[para(13570(a,1),21597(a,1,1)),rewrite(17130(6))].">
      <j1 rule="para" parents="13570 21597"/>
      <j2 rule="rewrite" parents="17130"/>
    </justification>
  </clause>

  <clause id="22745">
    <literal><![CDATA[
      (x ^ y) v z = (y v z) ^ (x v z)
    ]]></literal>
    <justification jstring="[back_rewrite(22264),rewrite(22340(7))].">
      <j1 rule="back_rewrite" parents="22264"/>
      <j2 rule="rewrite" parents="22340"/>
    </justification>
  </clause>

  <clause id="23244">
    <literal><![CDATA[
      x v (y ^ z) = (x v y) ^ (x v z)
    ]]></literal>
    <justification jstring="[back_rewrite(7097),rewrite(22745(5),68(4),167(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="7097"/>
      <j2 rule="rewrite" parents="22745 68 167"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23366">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(distributivity)
    ]]></attribute>
    <justification jstring="[back_rewrite(13),rewrite(23244(7),2(5),7(5),2(6),23244(6),2(4),2(7),20(9)),xx(a)].">
      <j1 rule="back_rewrite" parents="13"/>
      <j2 rule="rewrite" parents="23244 2 7 2 23244 2 2 20"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
