<?xml version="1.0" encoding="ISO-8859-1"?>

<!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 March-2007, March 2007.
Process 21175 was started by mccune on cleo,
Mon Mar 19 17:07:00 2007
The command was "/home/mccune/bin/prover9 -f lt.in uc.in H65d.in".
]]></heading>

<proof number="1" length="140" max_count="27">

<comments><![CDATA[
% Proof 1 at 169.13 (+ 1.04) seconds: distributivity.
% Length of proof is 140.
% Level of proof is 27.
% Maximum clause weight is 27.
% Given clauses 1693.
]]></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(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="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="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="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="37">
    <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="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="53">
    <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="54">
    <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="59">
    <literal><![CDATA[
      x ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(26(a,1),5(a,2,2)),rewrite(4(2))].">
      <j1 rule="para" parents="26 5"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(27(a,1),3(a,2,2)),rewrite(2(2))].">
      <j1 rule="para" parents="27 3"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="67">
    <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="69">
    <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="72">
    <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="74">
    <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="77">
    <literal><![CDATA[
      1 v x = 1
    ]]></literal>
    <justification jstring="[para(67(a,1),6(a,1))].">
      <j1 rule="para" parents="67 6"/>
    </justification>
  </clause>

  <clause id="78">
    <literal><![CDATA[
      x v (x' v y) = 1
    ]]></literal>
    <justification jstring="[back_rewrite(28),rewrite(77(5))].">
      <j1 rule="back_rewrite" parents="28"/>
      <j2 rule="rewrite" parents="77"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      0' = 1
    ]]></literal>
    <justification jstring="[hyper(10,a,69,a,b,30,a)].">
      <j1 rule="hyper" parents="10 69 30"/>
    </justification>
  </clause>

  <clause id="80">
    <literal><![CDATA[
      0 ^ x = 0
    ]]></literal>
    <justification jstring="[para(69(a,1),6(a,1,2))].">
      <j1 rule="para" parents="69 6"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      x ^ (y v (z v x)) = x
    ]]></literal>
    <justification jstring="[para(3(a,1),18(a,1,2))].">
      <j1 rule="para" parents="3 18"/>
    </justification>
  </clause>

  <clause id="83">
    <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="87">
    <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="88">
    <literal><![CDATA[
      x v 1 = 1
    ]]></literal>
    <justification jstring="[para(18(a,1),67(a,1)),flip(a)].">
      <j1 rule="para" parents="18 67"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="90">
    <literal><![CDATA[
      x v (y v x') = 1
    ]]></literal>
    <justification jstring="[back_rewrite(54),rewrite(88(5))].">
      <j1 rule="back_rewrite" parents="54"/>
      <j2 rule="rewrite" parents="88"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      x ^ 0 = 0
    ]]></literal>
    <justification jstring="[para(80(a,1),4(a,1)),flip(a)].">
      <j1 rule="para" parents="80 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="92">
    <literal><![CDATA[
      x ^ (y ^ x') = 0
    ]]></literal>
    <justification jstring="[back_rewrite(74),rewrite(91(5))].">
      <j1 rule="back_rewrite" parents="74"/>
      <j2 rule="rewrite" parents="91"/>
    </justification>
  </clause>

  <clause id="94">
    <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="102">
    <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="106">
    <literal><![CDATA[
      x v (y v (z ^ x)) = y v x
    ]]></literal>
    <justification jstring="[para(24(a,1),15(a,1,2)),flip(a)].">
      <j1 rule="para" parents="24 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="110">
    <literal><![CDATA[
      x v (y v ((x v y)' v z)) = 1
    ]]></literal>
    <justification jstring="[para(78(a,1),3(a,1)),flip(a)].">
      <j1 rule="para" parents="78 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="114">
    <literal><![CDATA[
      x ^ (x v y)' = 0
    ]]></literal>
    <justification jstring="[para(9(a,1),20(a,1,2)),rewrite(91(2)),flip(a)].">
      <j1 rule="para" parents="9 20"/>
      <j2 rule="rewrite" parents="91"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="118">
    <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="130">
    <literal><![CDATA[
      x ^ (y v x)' = 0
    ]]></literal>
    <justification jstring="[para(2(a,1),114(a,1,2,1))].">
      <j1 rule="para" parents="2 114"/>
    </justification>
  </clause>

  <clause id="145">
    <literal><![CDATA[
      x ^ (y ^ (z v (x ^ y))') = 0
    ]]></literal>
    <justification jstring="[para(130(a,1),5(a,1)),flip(a)].">
      <j1 rule="para" parents="130 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="162">
    <literal><![CDATA[
      x v (x ^ y)' = 1
    ]]></literal>
    <justification jstring="[para(8(a,1),22(a,1,2)),rewrite(88(2)),flip(a)].">
      <j1 rule="para" parents="8 22"/>
      <j2 rule="rewrite" parents="88"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="171">
    <literal><![CDATA[
      x v (y ^ x)' = 1
    ]]></literal>
    <justification jstring="[para(4(a,1),162(a,1,2,1))].">
      <j1 rule="para" parents="4 162"/>
    </justification>
  </clause>

  <clause id="176">
    <literal><![CDATA[
      x v (y v (z ^ (x v y))') = 1
    ]]></literal>
    <justification jstring="[para(171(a,1),3(a,1)),flip(a)].">
      <j1 rule="para" parents="171 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="184">
    <literal><![CDATA[
      x v (y v (z v x)) = y v (z v x)
    ]]></literal>
    <justification jstring="[para(3(a,1),65(a,1,2)),rewrite(3(5))].">
      <j1 rule="para" parents="3 65"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="217">
    <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="230">
    <literal><![CDATA[
      (x ^ y) v (y ^ x) = x ^ y
    ]]></literal>
    <justification jstring="[para(59(a,1),25(a,1,2))].">
      <j1 rule="para" parents="59 25"/>
    </justification>
  </clause>

  <clause id="232">
    <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="247">
    <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="250">
    <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="268">
    <literal><![CDATA[
      x' ^ (y ^ x) = 0
    ]]></literal>
    <justification jstring="[para(250(a,1),92(a,1,2,2))].">
      <j1 rule="para" parents="250 92"/>
    </justification>
  </clause>

  <clause id="291">
    <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="323">
    <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="549">
    <literal><![CDATA[
      x v (y v (y v (x ^ z))') = 1
    ]]></literal>
    <justification jstring="[para(217(a,1),22(a,1,2)),rewrite(88(2)),flip(a)].">
      <j1 rule="para" parents="217 22"/>
      <j2 rule="rewrite" parents="88"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="550">
    <literal><![CDATA[
      x ^ (y v (y v x)') != 0
    ]]></literal>
    <literal><![CDATA[
      (y v (y v x)')' = x
    ]]></literal>
    <justification jstring="[para(217(a,1),34(a,1)),rewrite(4(7)),xx(a)].">
      <j1 rule="para" parents="217 34"/>
      <j2 rule="rewrite" parents="4"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="552">
    <literal><![CDATA[
      x ^ (y ^ (z v (z v (x ^ y))')) != 0
    ]]></literal>
    <literal><![CDATA[
      z v (z v (x ^ y))' = (x ^ y)'
    ]]></literal>
    <justification jstring="[para(217(a,1),37(a,1)),flip(c),xx(a)].">
      <j1 rule="para" parents="217 37"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="559">
    <literal><![CDATA[
      x ^ (y ^ (y ^ (x v z))') = 0
    ]]></literal>
    <justification jstring="[para(232(a,1),20(a,1,2)),rewrite(91(2)),flip(a)].">
      <j1 rule="para" parents="232 20"/>
      <j2 rule="rewrite" parents="91"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="560">
    <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(232(a,1),35(b,1)),flip(c),xx(b)].">
      <j1 rule="para" parents="232 35"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="712">
    <literal><![CDATA[
      x v (y v (z v (x ^ u))) = y v (z v x)
    ]]></literal>
    <justification jstring="[para(3(a,1),53(a,1,2)),rewrite(3(6))].">
      <j1 rule="para" parents="3 53"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="852">
    <literal><![CDATA[
      (x ^ y) v (x ^ (z v y)) = x ^ (z v y)
    ]]></literal>
    <justification jstring="[para(87(a,1),24(a,1,2)),rewrite(2(4))].">
      <j1 rule="para" parents="87 24"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="879">
    <literal><![CDATA[
      (x ^ y) v (z v y) = z v y
    ]]></literal>
    <justification jstring="[para(65(a,1),102(a,2)),rewrite(184(4))].">
      <j1 rule="para" parents="65 102"/>
      <j2 rule="rewrite" parents="184"/>
    </justification>
  </clause>

  <clause id="976">
    <literal><![CDATA[
      x v (y v (z v (u ^ x))) = y v (z v x)
    ]]></literal>
    <justification jstring="[para(3(a,1),106(a,1,2)),rewrite(3(6))].">
      <j1 rule="para" parents="3 106"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="1046">
    <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(69(3))].">
      <j1 rule="para" parents="9 45"/>
      <j2 rule="rewrite" parents="69"/>
    </justification>
  </clause>

  <clause id="1223">
    <literal><![CDATA[
      x ^ (y ^ (z v (y ^ x))) = x ^ y
    ]]></literal>
    <justification jstring="[para(230(a,1),82(a,1,2,2)),rewrite(5(4))].">
      <j1 rule="para" parents="230 82"/>
      <j2 rule="rewrite" parents="5"/>
    </justification>
  </clause>

  <clause id="1285">
    <literal><![CDATA[
      (x ^ (y ^ z)) v (u v z) = u v z
    ]]></literal>
    <justification jstring="[para(5(a,1),879(a,1,1))].">
      <j1 rule="para" parents="5 879"/>
    </justification>
  </clause>

  <clause id="2466">
    <literal><![CDATA[
      x v ((y ^ x) v (x ^ z))' = 1
    ]]></literal>
    <justification jstring="[para(549(a,1),102(a,1)),flip(a)].">
      <j1 rule="para" parents="549 102"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2528">
    <literal><![CDATA[
      x ^ ((y v x) ^ (x v z))' = 0
    ]]></literal>
    <justification jstring="[para(559(a,1),83(a,1)),flip(a)].">
      <j1 rule="para" parents="559 83"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3163">
    <literal><![CDATA[
      x v ((x v y) ^ (x v z)) = (x v y) ^ (x v z)
    ]]></literal>
    <justification jstring="[para(6(a,1),118(a,1,1))].">
      <j1 rule="para" parents="6 118"/>
    </justification>
  </clause>

  <clause id="8802">
    <literal><![CDATA[
      (x ^ (y ^ (z ^ u))) v (v v (y ^ u)) = v v (y ^ u)
    ]]></literal>
    <justification jstring="[para(17(a,1),1285(a,1,1,2))].">
      <j1 rule="para" parents="17 1285"/>
    </justification>
  </clause>

  <clause id="8915">
    <literal><![CDATA[
      (x ^ y) v (z v (u v (y ^ x))) = z v (u v (y ^ x))
    ]]></literal>
    <justification jstring="[para(1223(a,1),1285(a,1,1))].">
      <j1 rule="para" parents="1223 1285"/>
    </justification>
  </clause>

  <clause id="25506">
    <literal><![CDATA[
      x v ((y v x) ^ (y v z)) = x v y
    ]]></literal>
    <justification jstring="[para(94(a,1),852(a,1,2)),rewrite(15(5),2(4),3163(4),94(8))].">
      <j1 rule="para" parents="94 852"/>
      <j2 rule="rewrite" parents="15 2 3163 94"/>
    </justification>
  </clause>

  <clause id="25510">
    <literal><![CDATA[
      x v ((x v y) ^ (y v z)) = x v y
    ]]></literal>
    <justification jstring="[para(2(a,1),25506(a,1,2,1))].">
      <j1 rule="para" parents="2 25506"/>
    </justification>
  </clause>

  <clause id="25528">
    <literal><![CDATA[
      x v ((x v y) ^ (z v y)) = x v y
    ]]></literal>
    <justification jstring="[para(2(a,1),25510(a,1,2,2))].">
      <j1 rule="para" parents="2 25510"/>
    </justification>
  </clause>

  <clause id="25680">
    <literal><![CDATA[
      x ^ (x' v y) = x ^ ((x ^ y) v x')
    ]]></literal>
    <justification jstring="[para(2(a,1),1046(a,1,2)),flip(a)].">
      <j1 rule="para" parents="2 1046"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="25684">
    <literal><![CDATA[
      x ^ (x' v (x v y)') = 0
    ]]></literal>
    <justification jstring="[para(114(a,1),1046(a,1,2,2)),rewrite(2(3),69(3),9(2)),flip(a)].">
      <j1 rule="para" parents="114 1046"/>
      <j2 rule="rewrite" parents="2 69 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="25919">
    <literal><![CDATA[
      x' v (x v y)' = x'
    ]]></literal>
    <justification jstring="[hyper(291,a,110,a,b,25684,a),rewrite(27(1),27(2),27(3)),flip(a)].">
      <j1 rule="hyper" parents="291 110 25684"/>
      <j2 rule="rewrite" parents="27 27 27"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="25928">
    <literal><![CDATA[
      x' v (y v x)' = x'
    ]]></literal>
    <justification jstring="[para(2(a,1),25919(a,1,2,1))].">
      <j1 rule="para" parents="2 25919"/>
    </justification>
  </clause>

  <clause id="25932">
    <literal><![CDATA[
      x' ^ (x v y)' = (x v y)'
    ]]></literal>
    <justification jstring="[para(25919(a,1),18(a,1,2)),rewrite(4(4))].">
      <j1 rule="para" parents="25919 18"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="26671">
    <literal><![CDATA[
      x' v (x ^ y)' = (x ^ y)'
    ]]></literal>
    <justification jstring="[para(7(a,1),25928(a,1,2,1)),rewrite(2(4))].">
      <j1 rule="para" parents="7 25928"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="27323">
    <literal><![CDATA[
      x ^ (y v (x ^ y'))' = 0
    ]]></literal>
    <justification jstring="[para(25932(a,1),145(a,1,2))].">
      <j1 rule="para" parents="25932 145"/>
    </justification>
  </clause>

  <clause id="27532">
    <literal><![CDATA[
      (x ^ y) v (y ^ (x ^ y)') = y
    ]]></literal>
    <justification jstring="[hyper(247,a,2466,a,b,27323,a),rewrite(250(7))].">
      <j1 rule="hyper" parents="247 2466 27323"/>
      <j2 rule="rewrite" parents="250"/>
    </justification>
  </clause>

  <clause id="28140">
    <literal><![CDATA[
      x' ^ (x ^ y)' = x'
    ]]></literal>
    <justification jstring="[para(26671(a,1),6(a,1,2))].">
      <j1 rule="para" parents="26671 6"/>
    </justification>
  </clause>

  <clause id="28147">
    <literal><![CDATA[
      x' ^ (y ^ (x ^ z)') = y ^ x'
    ]]></literal>
    <justification jstring="[para(26671(a,1),72(a,1,2,2))].">
      <j1 rule="para" parents="26671 72"/>
    </justification>
  </clause>

  <clause id="28156">
    <literal><![CDATA[
      x v (y ^ (x v y'))' = 1
    ]]></literal>
    <justification jstring="[para(26671(a,1),176(a,1,2))].">
      <j1 rule="para" parents="26671 176"/>
    </justification>
  </clause>

  <clause id="28462">
    <literal><![CDATA[
      x ^ (x' ^ y)' = x
    ]]></literal>
    <justification jstring="[para(250(a,1),28140(a,1,1)),rewrite(250(6))].">
      <j1 rule="para" parents="250 28140"/>
      <j2 rule="rewrite" parents="250"/>
    </justification>
  </clause>

  <clause id="29308">
    <literal><![CDATA[
      (x v y) ^ (y v (x v y)') = y
    ]]></literal>
    <justification jstring="[hyper(247,a,28156,a,b,2528,a),rewrite(250(7))].">
      <j1 rule="hyper" parents="247 28156 2528"/>
      <j2 rule="rewrite" parents="250"/>
    </justification>
  </clause>

  <clause id="29312">
    <literal><![CDATA[
      x v (y' ^ (x v y))' = 1
    ]]></literal>
    <justification jstring="[para(250(a,1),28156(a,1,2,1,2,2))].">
      <j1 rule="para" parents="250 28156"/>
    </justification>
  </clause>

  <clause id="29711">
    <literal><![CDATA[
      x v ((x v y) ^ y')' = 1
    ]]></literal>
    <justification jstring="[para(4(a,1),29312(a,1,2,1))].">
      <j1 rule="para" parents="4 29312"/>
    </justification>
  </clause>

  <clause id="29782">
    <literal><![CDATA[
      x v ((y v x) ^ y')' = 1
    ]]></literal>
    <justification jstring="[para(2(a,1),29711(a,1,2,1,1))].">
      <j1 rule="para" parents="2 29711"/>
    </justification>
  </clause>

  <clause id="30314">
    <literal><![CDATA[
      x v ((x v y) ^ x') = x v y
    ]]></literal>
    <justification jstring="[para(6(a,1),27532(a,1,1)),rewrite(6(3))].">
      <j1 rule="para" parents="6 27532"/>
      <j2 rule="rewrite" parents="6"/>
    </justification>
  </clause>

  <clause id="30350">
    <literal><![CDATA[
      x' v (x ^ (x ^ y)') = (x ^ y)'
    ]]></literal>
    <justification jstring="[para(28140(a,1),27532(a,1,1)),rewrite(28140(7),250(5),4(4))].">
      <j1 rule="para" parents="28140 27532"/>
      <j2 rule="rewrite" parents="28140 250 4"/>
    </justification>
  </clause>

  <clause id="30804">
    <literal><![CDATA[
      x ^ ((x ^ y) v x') = x ^ y
    ]]></literal>
    <justification jstring="[para(7(a,1),29308(a,1,1)),rewrite(7(3))].">
      <j1 rule="para" parents="7 29308"/>
      <j2 rule="rewrite" parents="7"/>
    </justification>
  </clause>

  <clause id="30843">
    <literal><![CDATA[
      x ^ (x' v y) = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(25680),rewrite(30804(7))].">
      <j1 rule="back_rewrite" parents="25680"/>
      <j2 rule="rewrite" parents="30804"/>
    </justification>
  </clause>

  <clause id="30850">
    <literal><![CDATA[
      x ^ (y v x') = x ^ y
    ]]></literal>
    <justification jstring="[para(2(a,1),30843(a,1,2))].">
      <j1 rule="para" parents="2 30843"/>
    </justification>
  </clause>

  <clause id="31255">
    <literal><![CDATA[
      x' ^ (y v x) = x' ^ y
    ]]></literal>
    <justification jstring="[para(250(a,1),30850(a,1,2,2))].">
      <j1 rule="para" parents="250 30850"/>
    </justification>
  </clause>

  <clause id="31256">
    <literal><![CDATA[
      x ^ (y v (z v x)') = x ^ y
    ]]></literal>
    <justification jstring="[para(30850(a,1),83(a,1,2)),rewrite(83(3)),flip(a)].">
      <j1 rule="para" parents="30850 83"/>
      <j2 rule="rewrite" parents="83"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="31317">
    <literal><![CDATA[
      (x v y')' = y ^ (y ^ x)'
    ]]></literal>
    <justification jstring="[para(30850(a,1),560(a,1,2,2,2,1)),rewrite(30350(5),171(3),30850(6)),flip(b),xx(a)].">
      <j1 rule="para" parents="30850 560"/>
      <j2 rule="rewrite" parents="30350 171 30850"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="31321">
    <literal><![CDATA[
      (x v y) ^ x' = y ^ x'
    ]]></literal>
    <justification jstring="[para(29782(a,1),30850(a,1,2)),rewrite(4(5),67(5),4(7),83(7))].">
      <j1 rule="para" parents="29782 30850"/>
      <j2 rule="rewrite" parents="4 67 4 83"/>
    </justification>
  </clause>

  <clause id="31330">
    <literal><![CDATA[
      x ^ y != 0
    ]]></literal>
    <literal><![CDATA[
      x ^ y' = x
    ]]></literal>
    <justification jstring="[back_rewrite(550),rewrite(31256(4),31317(7),4(6),6(6),31321(6))].">
      <j1 rule="back_rewrite" parents="550"/>
      <j2 rule="rewrite" parents="31256 31317 4 6 31321"/>
    </justification>
  </clause>

  <clause id="31355">
    <literal><![CDATA[
      x v (y ^ x') = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(30314),rewrite(31321(3))].">
      <j1 rule="back_rewrite" parents="30314"/>
      <j2 rule="rewrite" parents="31321"/>
    </justification>
  </clause>

  <clause id="31365">
    <literal><![CDATA[
      x v (x' ^ y) = x v y
    ]]></literal>
    <justification jstring="[para(4(a,1),31355(a,1,2))].">
      <j1 rule="para" parents="4 31355"/>
    </justification>
  </clause>

  <clause id="31368">
    <literal><![CDATA[
      x v (y ^ (x ^ z)') = x v y
    ]]></literal>
    <justification jstring="[para(31355(a,1),22(a,1,2)),rewrite(22(3)),flip(a)].">
      <j1 rule="para" parents="31355 22"/>
      <j2 rule="rewrite" parents="22"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="31369">
    <literal><![CDATA[
      x' v (y ^ x) = x' v y
    ]]></literal>
    <justification jstring="[para(250(a,1),31355(a,1,2,2))].">
      <j1 rule="para" parents="250 31355"/>
    </justification>
  </clause>

  <clause id="31465">
    <literal><![CDATA[
      x' v (x ^ y) = x' v y
    ]]></literal>
    <justification jstring="[para(250(a,1),31365(a,1,2,1))].">
      <j1 rule="para" parents="250 31365"/>
    </justification>
  </clause>

  <clause id="31714">
    <literal><![CDATA[
      (x' ^ y)' = x v (x v y)'
    ]]></literal>
    <justification jstring="[para(31365(a,1),552(a,1,2,2,2,1)),rewrite(31256(5),268(3),31365(6)),flip(b),xx(a)].">
      <j1 rule="para" parents="31365 552"/>
      <j2 rule="rewrite" parents="31256 268 31365"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="32024">
    <literal><![CDATA[
      (x v y) ^ (x' ^ y)' = x
    ]]></literal>
    <justification jstring="[para(31365(a,1),31255(a,1,2)),rewrite(4(5),4(9),28462(9))].">
      <j1 rule="para" parents="31365 31255"/>
      <j2 rule="rewrite" parents="4 4 28462"/>
    </justification>
  </clause>

  <clause id="32125">
    <literal><![CDATA[
      (x v y)' = x' ^ (y ^ x')'
    ]]></literal>
    <justification jstring="[para(31321(a,1),323(a,1,2,2,2,1)),rewrite(31368(6),90(3),31321(7)),flip(b),xx(a)].">
      <j1 rule="para" parents="31321 323"/>
      <j2 rule="rewrite" parents="31368 90 31321"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="32306">
    <literal><![CDATA[
      x ^ (y ^ (y ^ x)')' = x
    ]]></literal>
    <justification jstring="[hyper(31330,a,232,a)].">
      <j1 rule="hyper" parents="31330 232"/>
    </justification>
  </clause>

  <clause id="32350">
    <literal><![CDATA[
      x' v ((y ^ x) v z) = x' v (y v z)
    ]]></literal>
    <justification jstring="[para(31369(a,1),3(a,1,1)),rewrite(3(3)),flip(a)].">
      <j1 rule="para" parents="31369 3"/>
      <j2 rule="rewrite" parents="3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="32451">
    <literal><![CDATA[
      x' v ((x v y) ^ z) = x' v z
    ]]></literal>
    <justification jstring="[para(20(a,1),31465(a,1,2)),rewrite(31465(3)),flip(a)].">
      <j1 rule="para" parents="20 31465"/>
      <j2 rule="rewrite" parents="31465"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="32718">
    <literal><![CDATA[
      (x v y) ^ (y' ^ x)' = y
    ]]></literal>
    <justification jstring="[para(2(a,1),32024(a,1,1))].">
      <j1 rule="para" parents="2 32024"/>
    </justification>
  </clause>

  <clause id="32746">
    <literal><![CDATA[
      x ^ (y ^ x)' = x ^ y'
    ]]></literal>
    <justification jstring="[para(32(a,1),32306(a,1,2,1,2,1)),rewrite(79(5),30(5),4(5),28147(5)),flip(a)].">
      <j1 rule="para" parents="32 32306"/>
      <j2 rule="rewrite" parents="79 30 4 28147"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="32788">
    <literal><![CDATA[
      (x v y)' = x' ^ y'
    ]]></literal>
    <justification jstring="[back_rewrite(32125),rewrite(32746(7))].">
      <j1 rule="back_rewrite" parents="32125"/>
      <j2 rule="rewrite" parents="32746"/>
    </justification>
  </clause>

  <clause id="32972">
    <literal><![CDATA[
      (x' ^ y)' = x v y'
    ]]></literal>
    <justification jstring="[back_rewrite(31714),rewrite(32788(5),31365(7))].">
      <j1 rule="back_rewrite" parents="31714"/>
      <j2 rule="rewrite" parents="32788 31365"/>
    </justification>
  </clause>

  <clause id="35935">
    <literal><![CDATA[
      (x v y) ^ (y v x') = y
    ]]></literal>
    <justification jstring="[back_rewrite(32718),rewrite(32972(4))].">
      <j1 rule="back_rewrite" parents="32718"/>
      <j2 rule="rewrite" parents="32972"/>
    </justification>
  </clause>

  <clause id="40585">
    <literal><![CDATA[
      (x v y) ^ (x' v y) = y
    ]]></literal>
    <justification jstring="[para(2(a,1),35935(a,1,2))].">
      <j1 rule="para" parents="2 35935"/>
    </justification>
  </clause>

  <clause id="41510">
    <literal><![CDATA[
      (x ^ y) v z = (y v z) ^ (y' v (x v z))
    ]]></literal>
    <justification jstring="[para(102(a,1),40585(a,1,1)),rewrite(32350(5)),flip(a)].">
      <j1 rule="para" parents="102 40585"/>
      <j2 rule="rewrite" parents="32350"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="41589">
    <literal><![CDATA[
      (x v y) ^ (x' v (z v y)) = (x v y) ^ (z v y)
    ]]></literal>
    <justification jstring="[para(25528(a,1),40585(a,1,1)),rewrite(32451(6))].">
      <j1 rule="para" parents="25528 40585"/>
      <j2 rule="rewrite" parents="32451"/>
    </justification>
  </clause>

  <clause id="42353">
    <literal><![CDATA[
      (x ^ y) v z = (y v z) ^ (x v z)
    ]]></literal>
    <justification jstring="[back_rewrite(41510),rewrite(41589(7))].">
      <j1 rule="back_rewrite" parents="41510"/>
      <j2 rule="rewrite" parents="41589"/>
    </justification>
  </clause>

  <clause id="43752">
    <literal><![CDATA[
      x v (y v (z ^ u)) = (x v (y v z)) ^ (x v (y v u))
    ]]></literal>
    <justification jstring="[back_rewrite(8915),rewrite(42353(5),712(4),976(6)),flip(a)].">
      <j1 rule="back_rewrite" parents="8915"/>
      <j2 rule="rewrite" parents="42353 712 976"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="43767">
    <literal><![CDATA[
      x v (y ^ z) = (x v y) ^ (x v z)
    ]]></literal>
    <justification jstring="[back_rewrite(8802),rewrite(43752(6),42353(5),42353(4),42353(3),65(7),4(7),87(7),4(4),18(4),18(4),42353(6),42353(5),42353(4),65(3),18(5),18(5),18(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="8802"/>
      <j2 rule="rewrite" parents="43752 42353 42353 42353 65 4 87 4 18 18 42353 42353 42353 65 18 18 18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="43980">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(distributivity)
    ]]></attribute>
    <justification jstring="[back_rewrite(13),rewrite(43767(7),2(5),7(5),2(6),43767(6),2(4),2(7),20(9)),xx(a)].">
      <j1 rule="back_rewrite" parents="13"/>
      <j2 rule="rewrite" parents="43767 2 7 2 43767 2 2 20"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
