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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 22-May-2007, May 2007.
Process 27639 was started by mccune on cleo,
Tue May 22 14:50:02 2007
The command was "/home/mccune/bin/prover9 -f lt.in uc.in H65d.in".
]]></heading>

<proof number="1" length="125" max_count="26">

<comments><![CDATA[
% Proof 1 at 52.11 (+ 0.30) seconds: distributivity.
% Length of proof is 125.
% Level of proof is 25.
% Maximum clause weight is 26.
% Given clauses 1051.
]]></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="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="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="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="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="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 (x ^ ((z ^ u) v (x ^ y)))) = x ^ (y v (z ^ u))
    ]]></literal>
    <justification jstring="[para(2(a,1),12(a,1,2,2,2))].">
      <j1 rule="para" parents="2 12"/>
    </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)))) ^ v) = x ^ ((y v (z ^ u)) ^ v)
    ]]></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="50">
    <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="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="54">
    <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="59">
    <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="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="66">
    <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="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="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="74">
    <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="75">
    <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="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="78">
    <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="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="80">
    <literal><![CDATA[
      x v (y v x') = 1
    ]]></literal>
    <justification jstring="[back_rewrite(50),rewrite([79(5)])].">
      <j1 rule="back_rewrite" parents="50"/>
      <j2 rule="rewrite" parents="79"/>
    </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="85">
    <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="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="111">
    <literal><![CDATA[
      x v (y v (z v x')) = 1
    ]]></literal>
    <justification jstring="[para(3(a,1),80(a,1,2))].">
      <j1 rule="para" parents="3 80"/>
    </justification>
  </clause>

  <clause id="116">
    <literal><![CDATA[
      x ^ (y v x)' = 0
    ]]></literal>
    <justification jstring="[para(2(a,1),103(a,1,2,1))].">
      <j1 rule="para" parents="2 103"/>
    </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="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="158">
    <literal><![CDATA[
      x v (y ^ (z ^ x))' = 1
    ]]></literal>
    <justification jstring="[para(5(a,1),152(a,1,2,1))].">
      <j1 rule="para" parents="5 152"/>
    </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="168">
    <literal><![CDATA[
      (x v y) ^ (y v x) = x v y
    ]]></literal>
    <justification jstring="[para(59(a,1),19(a,1,2))].">
      <j1 rule="para" parents="59 19"/>
    </justification>
  </clause>

  <clause id="199">
    <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="212">
    <literal><![CDATA[
      (x ^ y) v (y ^ x) = x ^ y
    ]]></literal>
    <justification jstring="[para(54(a,1),25(a,1,2))].">
      <j1 rule="para" parents="54 25"/>
    </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="244">
    <literal><![CDATA[
      x' v (y v x) = 1
    ]]></literal>
    <justification jstring="[para(228(a,1),80(a,1,2,2))].">
      <j1 rule="para" parents="228 80"/>
    </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="249">
    <literal><![CDATA[
      x' ^ (y v x) != 0
    ]]></literal>
    <literal><![CDATA[
      y v x = x
    ]]></literal>
    <justification jstring="[para(244(a,1),10(a,1)),rewrite([228(10)]),flip(c),xx(a)].">
      <j1 rule="para" parents="244 10"/>
      <j2 rule="rewrite" parents="228"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="261">
    <literal><![CDATA[
      x v y != 1
    ]]></literal>
    <literal><![CDATA[
      y ^ z != 0
    ]]></literal>
    <literal><![CDATA[
      (x v y)' = y ^ z
    ]]></literal>
    <justification jstring="[para(7(a,1),35(a,1,2)),rewrite([17(6),75(6)])].">
      <j1 rule="para" parents="7 35"/>
      <j2 rule="rewrite" parents="17 75"/>
    </justification>
  </clause>

  <clause id="289">
    <literal><![CDATA[
      x ^ (y v (z v x')) != 0
    ]]></literal>
    <literal><![CDATA[
      (y v (z v x'))' = x
    ]]></literal>
    <justification jstring="[para(111(a,1),34(a,1)),rewrite([4(7)]),xx(a)].">
      <j1 rule="para" parents="111 34"/>
      <j2 rule="rewrite" parents="4"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="305">
    <literal><![CDATA[
      x ^ (y v (x ^ (z v (x ^ y)))) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(6(a,1),37(a,1,2,2,2,1)),rewrite([6(7)])].">
      <j1 rule="para" parents="6 37"/>
      <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="723">
    <literal><![CDATA[
      ((x v y) ^ z) v (y ^ z)' = 1
    ]]></literal>
    <justification jstring="[para(75(a,1),152(a,1,2,1))].">
      <j1 rule="para" parents="75 152"/>
    </justification>
  </clause>

  <clause id="741">
    <literal><![CDATA[
      x ^ (y ^ (z ^ (u v x))) = y ^ (z ^ x)
    ]]></literal>
    <justification jstring="[para(5(a,1),78(a,1,2)),rewrite([5(6)])].">
      <j1 rule="para" parents="5 78"/>
      <j2 rule="rewrite" parents="5"/>
    </justification>
  </clause>

  <clause id="875">
    <literal><![CDATA[
      x v (y v (z ^ (y v x))') = 1
    ]]></literal>
    <justification jstring="[para(168(a,1),158(a,1,2,1,2)),rewrite([3(5)])].">
      <j1 rule="para" parents="168 158"/>
      <j2 rule="rewrite" parents="3"/>
    </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="2945">
    <literal><![CDATA[
      ((x v y) ^ (z v (u v y))) v y' = 1
    ]]></literal>
    <justification jstring="[para(74(a,1),723(a,1,2,1))].">
      <j1 rule="para" parents="74 723"/>
    </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="4779">
    <literal><![CDATA[
      x ^ ((x ^ y) v (z ^ (x ^ u))) = (x ^ y) v (z ^ (x ^ u))
    ]]></literal>
    <justification jstring="[para(66(a,1),147(a,1,1))].">
      <j1 rule="para" parents="66 147"/>
    </justification>
  </clause>

  <clause id="6939">
    <literal><![CDATA[
      x' ^ (x v y) != 0
    ]]></literal>
    <literal><![CDATA[
      y v x = x
    ]]></literal>
    <justification jstring="[para(2(a,1),249(a,1,2))].">
      <j1 rule="para" parents="2 249"/>
    </justification>
  </clause>

  <clause id="18675">
    <literal><![CDATA[
      x v y != 0
    ]]></literal>
    <literal><![CDATA[
      y v x = 0
    ]]></literal>
    <justification jstring="[para(85(a,1),289(a,1)),rewrite([199(7),68(5)]),flip(b)].">
      <j1 rule="para" parents="85 289"/>
      <j2 rule="rewrite" parents="199 68"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18678">
    <literal><![CDATA[
      x ^ y != 0
    ]]></literal>
    <literal><![CDATA[
      y ^ x = 0
    ]]></literal>
    <justification jstring="[para(212(a,1),18675(a,1)),rewrite([212(6)])].">
      <j1 rule="para" parents="212 18675"/>
      <j2 rule="rewrite" parents="212"/>
    </justification>
  </clause>

  <clause id="22906">
    <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="23062">
    <literal><![CDATA[
      x' ^ (y ^ (x v y')) = 0
    ]]></literal>
    <justification jstring="[para(6(a,1),22906(a,1,2,2,2))].">
      <j1 rule="para" parents="6 22906"/>
    </justification>
  </clause>

  <clause id="23075">
    <literal><![CDATA[
      x ^ ((y v x') ^ y') = 0
    ]]></literal>
    <justification jstring="[hyper(18678,a,23062,a),rewrite([5(5)])].">
      <j1 rule="hyper" parents="18678 23062"/>
      <j2 rule="rewrite" parents="5"/>
    </justification>
  </clause>

  <clause id="23095">
    <literal><![CDATA[
      x' ^ (y ^ (y' v x)) = 0
    ]]></literal>
    <justification jstring="[para(2(a,1),23062(a,1,2,2))].">
      <j1 rule="para" parents="2 23062"/>
    </justification>
  </clause>

  <clause id="23110">
    <literal><![CDATA[
      x' ^ ((y v x) ^ y') = 0
    ]]></literal>
    <justification jstring="[hyper(261,a,2945,a,b,23075,a),rewrite([2945(6),68(2),228(4)]),flip(a)].">
      <j1 rule="hyper" parents="261 2945 23075"/>
      <j2 rule="rewrite" parents="2945 68 228"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23562">
    <literal><![CDATA[
      x ^ (x' v y) = x ^ y
    ]]></literal>
    <justification jstring="[hyper(34,a,1402,a,b,23095,a),rewrite([228(3),657(5)]),flip(a)].">
      <j1 rule="hyper" parents="34 1402 23095"/>
      <j2 rule="rewrite" parents="228 657"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23582">
    <literal><![CDATA[
      x ^ (y v x') = x ^ y
    ]]></literal>
    <justification jstring="[para(2(a,1),23562(a,1,2))].">
      <j1 rule="para" parents="2 23562"/>
    </justification>
  </clause>

  <clause id="23589">
    <literal><![CDATA[
      x ^ (x' ^ y)' = x
    ]]></literal>
    <justification jstring="[para(143(a,1),23562(a,1,2)),rewrite([30(2)]),flip(a)].">
      <j1 rule="para" parents="143 23562"/>
      <j2 rule="rewrite" parents="30"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23590">
    <literal><![CDATA[
      x ^ (y ^ x')' = x
    ]]></literal>
    <justification jstring="[para(152(a,1),23562(a,1,2)),rewrite([30(2)]),flip(a)].">
      <j1 rule="para" parents="152 23562"/>
      <j2 rule="rewrite" parents="30"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23593">
    <literal><![CDATA[
      x' ^ (x v y) = x' ^ y
    ]]></literal>
    <justification jstring="[para(228(a,1),23562(a,1,2,1))].">
      <j1 rule="para" parents="228 23562"/>
    </justification>
  </clause>

  <clause id="23814">
    <literal><![CDATA[
      x' ^ y != 0
    ]]></literal>
    <literal><![CDATA[
      y v x = x
    ]]></literal>
    <justification jstring="[back_rewrite(6939),rewrite([23593(3)])].">
      <j1 rule="back_rewrite" parents="6939"/>
      <j2 rule="rewrite" parents="23593"/>
    </justification>
  </clause>

  <clause id="23819">
    <literal><![CDATA[
      x v (x' ^ y)' = (x' ^ y)'
    ]]></literal>
    <justification jstring="[para(23589(a,1),24(a,1,2)),rewrite([2(4)])].">
      <j1 rule="para" parents="23589 24"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="23981">
    <literal><![CDATA[
      x' ^ (y ^ x)' = x'
    ]]></literal>
    <justification jstring="[para(228(a,1),23590(a,1,2,1,2))].">
      <j1 rule="para" parents="228 23590"/>
    </justification>
  </clause>

  <clause id="24343">
    <literal><![CDATA[
      x' ^ (x v y)' = (x v y)'
    ]]></literal>
    <justification jstring="[para(6(a,1),23981(a,1,2,1)),rewrite([4(4)])].">
      <j1 rule="para" parents="6 23981"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="25293">
    <literal><![CDATA[
      x v ((y v x) ^ y') = x
    ]]></literal>
    <justification jstring="[hyper(23814,a,23110,a),rewrite([2(4)])].">
      <j1 rule="hyper" parents="23814 23110"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="25442">
    <literal><![CDATA[
      x v (y v x')' = x
    ]]></literal>
    <justification jstring="[hyper(23814,a,116,a),rewrite([2(4)])].">
      <j1 rule="hyper" parents="23814 116"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="25465">
    <literal><![CDATA[
      x' v (y v x)' = x'
    ]]></literal>
    <justification jstring="[para(228(a,1),25442(a,1,2,1,2))].">
      <j1 rule="para" parents="228 25442"/>
    </justification>
  </clause>

  <clause id="25738">
    <literal><![CDATA[
      x v ((x v y) ^ y') = x
    ]]></literal>
    <justification jstring="[para(2(a,1),25293(a,1,2,1))].">
      <j1 rule="para" parents="2 25293"/>
    </justification>
  </clause>

  <clause id="25748">
    <literal><![CDATA[
      x' v (y v (y v x)') = y v (y v x)'
    ]]></literal>
    <justification jstring="[para(199(a,1),25293(a,1,2,1)),rewrite([61(6),2(5)])].">
      <j1 rule="para" parents="199 25293"/>
      <j2 rule="rewrite" parents="61 2"/>
    </justification>
  </clause>

  <clause id="25787">
    <literal><![CDATA[
      x' v (y v (z v x)') = y v x'
    ]]></literal>
    <justification jstring="[para(25465(a,1),15(a,1,2)),flip(a)].">
      <j1 rule="para" parents="25465 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="25921">
    <literal><![CDATA[
      x v (x v y)' = x v y'
    ]]></literal>
    <justification jstring="[back_rewrite(25748),rewrite([25787(5)]),flip(a)].">
      <j1 rule="back_rewrite" parents="25748"/>
      <j2 rule="rewrite" parents="25787"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="26102">
    <literal><![CDATA[
      x v (y v (z ^ (y v x))')' = x
    ]]></literal>
    <justification jstring="[para(875(a,1),25738(a,1,2,1)),rewrite([61(7)])].">
      <j1 rule="para" parents="875 25738"/>
      <j2 rule="rewrite" parents="61"/>
    </justification>
  </clause>

  <clause id="26882">
    <literal><![CDATA[
      (x v y) ^ (x v y') = x
    ]]></literal>
    <justification jstring="[para(25921(a,1),23582(a,1,2)),rewrite([4(6),6(6)])].">
      <j1 rule="para" parents="25921 23582"/>
      <j2 rule="rewrite" parents="4 6"/>
    </justification>
  </clause>

  <clause id="26883">
    <literal><![CDATA[
      (x v y)' = x' ^ y'
    ]]></literal>
    <justification jstring="[para(25921(a,1),23593(a,1,2)),rewrite([23593(4),24343(7)]),flip(a)].">
      <j1 rule="para" parents="25921 23593"/>
      <j2 rule="rewrite" parents="23593 24343"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="26885">
    <literal><![CDATA[
      (x' ^ y')' = x v (x' ^ y)
    ]]></literal>
    <justification jstring="[para(25921(a,1),25921(a,1,2,1)),rewrite([26883(3),228(3),26883(5),23819(8)]),flip(a)].">
      <j1 rule="para" parents="25921 25921"/>
      <j2 rule="rewrite" parents="26883 228 26883 23819"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="27264">
    <literal><![CDATA[
      x v (y' ^ (z ^ (y v x))) = x
    ]]></literal>
    <justification jstring="[back_rewrite(26102),rewrite([26883(5),228(5)])].">
      <j1 rule="back_rewrite" parents="26102"/>
      <j2 rule="rewrite" parents="26883 228"/>
    </justification>
  </clause>

  <clause id="30101">
    <literal><![CDATA[
      (x v y) ^ (y v x') = y
    ]]></literal>
    <justification jstring="[para(2(a,1),26882(a,1,1))].">
      <j1 rule="para" parents="2 26882"/>
    </justification>
  </clause>

  <clause id="30178">
    <literal><![CDATA[
      x v (x' ^ y) = x v y
    ]]></literal>
    <justification jstring="[para(26883(a,1),228(a,1,1)),rewrite([26885(4)])].">
      <j1 rule="para" parents="26883 228"/>
      <j2 rule="rewrite" parents="26885"/>
    </justification>
  </clause>

  <clause id="30251">
    <literal><![CDATA[
      (x v y) ^ (x' v y) = y
    ]]></literal>
    <justification jstring="[para(2(a,1),30101(a,1,2))].">
      <j1 rule="para" parents="2 30101"/>
    </justification>
  </clause>

  <clause id="30442">
    <literal><![CDATA[
      x' v (x ^ y) = x' v y
    ]]></literal>
    <justification jstring="[para(228(a,1),30178(a,1,2,1))].">
      <j1 rule="para" parents="228 30178"/>
    </justification>
  </clause>

  <clause id="36325">
    <literal><![CDATA[
      x ^ ((y v z) ^ (y' v (x ^ ((x ^ y) v z)))) = x ^ ((x ^ y) v z)
    ]]></literal>
    <justification jstring="[para(30251(a,1),471(a,1,2)),rewrite([52(4)]),flip(a)].">
      <j1 rule="para" parents="30251 471"/>
      <j2 rule="rewrite" parents="52"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="36374">
    <literal><![CDATA[
      x' v (y ^ ((y ^ x) v z)) = x' v y
    ]]></literal>
    <justification jstring="[para(120(a,1),30442(a,1,2)),rewrite([30442(3)]),flip(a)].">
      <j1 rule="para" parents="120 30442"/>
      <j2 rule="rewrite" parents="30442"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="36479">
    <literal><![CDATA[
      x ^ ((x ^ y) v z) = (y v z) ^ x
    ]]></literal>
    <justification jstring="[back_rewrite(36325),rewrite([36374(6),78(5)]),flip(a)].">
      <j1 rule="back_rewrite" parents="36325"/>
      <j2 rule="rewrite" parents="36374 78"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="36662">
    <literal><![CDATA[
      (x ^ y) v (z ^ (x ^ u)) = x ^ (y v (z ^ (x ^ u)))
    ]]></literal>
    <justification jstring="[back_rewrite(4779),rewrite([36479(5),4(4)]),flip(a)].">
      <j1 rule="back_rewrite" parents="4779"/>
      <j2 rule="rewrite" parents="36479 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="36664">
    <literal><![CDATA[
      (x ^ y) v (x ^ z) = x ^ (y v (x ^ z))
    ]]></literal>
    <justification jstring="[back_rewrite(4772),rewrite([36479(4),4(3)]),flip(a)].">
      <j1 rule="back_rewrite" parents="4772"/>
      <j2 rule="rewrite" parents="36479 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="36949">
    <literal><![CDATA[
      c1 ^ (c2 v (c1 ^ c3)) != c1 ^ (c2 v c3)
    ]]></literal>
    <attribute><![CDATA[
      answer(distributivity)
    ]]></attribute>
    <justification jstring="[back_rewrite(13),rewrite([36664(7)])].">
      <j1 rule="back_rewrite" parents="13"/>
      <j2 rule="rewrite" parents="36664"/>
    </justification>
  </clause>

  <clause id="39377">
    <literal><![CDATA[
      x' ^ (y ^ (x v z)) = x' ^ (y ^ z)
    ]]></literal>
    <justification jstring="[para(27264(a,1),18(a,1,2)),rewrite([4(5),741(5)]),flip(a)].">
      <j1 rule="para" parents="27264 18"/>
      <j2 rule="rewrite" parents="4 741"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="39406">
    <literal><![CDATA[
      x ^ (y v (x ^ z)) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(305(a,1),27264(a,1,2,2)),rewrite([39377(7),36662(7),3(6),36662(5),30178(4),305(5)]),flip(a)].">
      <j1 rule="para" parents="305 27264"/>
      <j2 rule="rewrite" parents="39377 36662 3 36662 30178 305"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="39407">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(distributivity)
    ]]></attribute>
    <justification jstring="[resolve(39406,a,36949,a)].">
      <j1 rule="resolve" parents="39406 36949"/>
    </justification>
  </clause>

</proof>

</proofs>
