<?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>H27d.out</source>

<heading><![CDATA[
Prover9 (32) version April-2007, April 2007.
Process 27120 was started by mccune on cleo,
Fri Apr 13 09:21:27 2007
The command was "/home/mccune/bin/prover9 -f lt.in uc.in H27d.in".
]]></heading>

<proof number="1" length="113" max_count="25">

<comments><![CDATA[
% Proof 1 at 14.84 (+ 0.14) seconds: distributivity.
% Length of proof is 113.
% Level of proof is 24.
% Maximum clause weight is 23.
% Given clauses 673.
]]></comments>

  <clause id="1" type="goal">
    <literal><![CDATA[
      x ^ (y v z) = (x ^ y) v (x ^ z)
    ]]></literal>
    <attribute><![CDATA[
      label(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 ^ (x v u))) = x ^ (y v ((x ^ z) v (z ^ (y v u))))
    ]]></literal>
    <attribute><![CDATA[
      label(H49)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      x ^ (y v ((x ^ z) v (z ^ (y v u)))) = x ^ (y v (z ^ (x v u)))
    ]]></literal>
    <justification jstring="[copy(11),flip(a)].">
      <j1 rule="copy" parents="11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13" type="deny">
    <literal><![CDATA[
      (c1 ^ c2) v (c1 ^ c3) != c1 ^ (c2 v c3)
    ]]></literal>
    <attribute><![CDATA[
      label(distributivity)
    ]]></attribute>
    <attribute><![CDATA[
      answer(distributivity)
    ]]></attribute>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

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

  <clause id="17">
    <literal><![CDATA[
      x ^ (y ^ z) = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(4(a,1),5(a,1,1)),rewrite(5(2))].">
      <j1 rule="para" parents="4 5"/>
      <j2 rule="rewrite" parents="5"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(2(a,1),6(a,1,2))].">
      <j1 rule="para" parents="2 6"/>
    </justification>
  </clause>

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

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

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

  <clause id="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="27">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(6(a,1),7(a,1,2))].">
      <j1 rule="para" parents="6 7"/>
    </justification>
  </clause>

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

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

  <clause id="30">
    <literal><![CDATA[
      x ^ 1 = x
    ]]></literal>
    <justification jstring="[para(8(a,1),6(a,1,2))].">
      <j1 rule="para" parents="8 6"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      x ^ (x' ^ y) = 0 ^ y
    ]]></literal>
    <justification jstring="[para(9(a,1),5(a,1,1)),flip(a)].">
      <j1 rule="para" parents="9 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="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="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="42">
    <literal><![CDATA[
      x ^ (y v ((z ^ x) v (z ^ (y v u)))) = x ^ (y v (z ^ (x v u)))
    ]]></literal>
    <justification jstring="[para(4(a,1),12(a,1,2,2,1))].">
      <j1 rule="para" parents="4 12"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      x ^ (y v (z ^ (x v y'))) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(8(a,1),12(a,1,2,2,2,2)),rewrite(30(3),2(2),24(2)),flip(a)].">
      <j1 rule="para" parents="8 12"/>
      <j2 rule="rewrite" parents="30 2 24"/>
      <j2 rule="flip"/>
    </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="60">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(27(a,1),3(a,1,1)),flip(a)].">
      <j1 rule="para" parents="27 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="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="75">
    <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="77">
    <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="80">
    <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="81">
    <literal><![CDATA[
      x v (x' v y) = 1
    ]]></literal>
    <justification jstring="[back_rewrite(28),rewrite(80(5))].">
      <j1 rule="back_rewrite" parents="28"/>
      <j2 rule="rewrite" parents="80"/>
    </justification>
  </clause>

  <clause id="83">
    <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="85">
    <literal><![CDATA[
      x ^ (x' ^ y) = 0
    ]]></literal>
    <justification jstring="[back_rewrite(31),rewrite(83(5))].">
      <j1 rule="back_rewrite" parents="31"/>
      <j2 rule="rewrite" parents="83"/>
    </justification>
  </clause>

  <clause id="87">
    <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 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="95">
    <literal><![CDATA[
      x v (y v x') = 1
    ]]></literal>
    <justification jstring="[back_rewrite(50),rewrite(93(5))].">
      <j1 rule="back_rewrite" parents="50"/>
      <j2 rule="rewrite" parents="93"/>
    </justification>
  </clause>

  <clause id="96">
    <literal><![CDATA[
      x ^ 0 = 0
    ]]></literal>
    <justification jstring="[para(83(a,1),4(a,1)),flip(a)].">
      <j1 rule="para" parents="83 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="97">
    <literal><![CDATA[
      x ^ (y ^ x') = 0
    ]]></literal>
    <justification jstring="[back_rewrite(77),rewrite(96(5))].">
      <j1 rule="back_rewrite" parents="77"/>
      <j2 rule="rewrite" parents="96"/>
    </justification>
  </clause>

  <clause id="108">
    <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="119">
    <literal><![CDATA[
      x ^ (y v (z ^ (x v (y' v u)))) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(81(a,1),12(a,1,2,2,2,2)),rewrite(30(3),2(2),24(2)),flip(a)].">
      <j1 rule="para" parents="81 12"/>
      <j2 rule="rewrite" parents="30 2 24"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="122">
    <literal><![CDATA[
      x ^ (x v y)' = 0
    ]]></literal>
    <justification jstring="[para(9(a,1),20(a,1,2)),rewrite(96(2)),flip(a)].">
      <j1 rule="para" parents="9 20"/>
      <j2 rule="rewrite" parents="96"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="131">
    <literal><![CDATA[
      x ^ ((x v y)' ^ z) = 0
    ]]></literal>
    <justification jstring="[para(85(a,1),20(a,1,2)),rewrite(96(2)),flip(a)].">
      <j1 rule="para" parents="85 20"/>
      <j2 rule="rewrite" parents="96"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="141">
    <literal><![CDATA[
      x ^ (y v x)' = 0
    ]]></literal>
    <justification jstring="[para(2(a,1),122(a,1,2,1))].">
      <j1 rule="para" parents="2 122"/>
    </justification>
  </clause>

  <clause id="145">
    <literal><![CDATA[
      x ^ (y v ((x v z)' ^ (y v u))) = x ^ (y v ((x v z)' ^ (x v u)))
    ]]></literal>
    <justification jstring="[para(122(a,1),12(a,1,2,2,1)),rewrite(69(6))].">
      <j1 rule="para" parents="122 12"/>
      <j2 rule="rewrite" parents="69"/>
    </justification>
  </clause>

  <clause id="157">
    <literal><![CDATA[
      x ^ (y ^ (z v (x ^ y))') = 0
    ]]></literal>
    <justification jstring="[para(141(a,1),5(a,1)),flip(a)].">
      <j1 rule="para" parents="141 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="162">
    <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="170">
    <literal><![CDATA[
      x v (y ^ (x ^ y)') != 1
    ]]></literal>
    <literal><![CDATA[
      y ^ (x ^ y)' = x'
    ]]></literal>
    <justification jstring="[para(32(a,1),10(b,1)),flip(c),xx(b)].">
      <j1 rule="para" parents="32 10"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="184">
    <literal><![CDATA[
      x v (x ^ y)' = 1
    ]]></literal>
    <justification jstring="[para(8(a,1),22(a,1,2)),rewrite(93(2)),flip(a)].">
      <j1 rule="para" parents="8 22"/>
      <j2 rule="rewrite" parents="93"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="195">
    <literal><![CDATA[
      x v (y ^ x)' = 1
    ]]></literal>
    <justification jstring="[para(4(a,1),184(a,1,2,1))].">
      <j1 rule="para" parents="4 184"/>
    </justification>
  </clause>

  <clause id="201">
    <literal><![CDATA[
      x v (y v (z ^ (x v y))') = 1
    ]]></literal>
    <justification jstring="[para(195(a,1),3(a,1)),flip(a)].">
      <j1 rule="para" parents="195 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="208">
    <literal><![CDATA[
      ((x v y) ^ z) v (x ^ z)' = 1
    ]]></literal>
    <justification jstring="[para(20(a,1),195(a,1,2,1))].">
      <j1 rule="para" parents="20 195"/>
    </justification>
  </clause>

  <clause id="276">
    <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="279">
    <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="296">
    <literal><![CDATA[
      x' v (y v x) = 1
    ]]></literal>
    <justification jstring="[para(279(a,1),95(a,1,2,2))].">
      <j1 rule="para" parents="279 95"/>
    </justification>
  </clause>

  <clause id="297">
    <literal><![CDATA[
      x' ^ (y ^ x) = 0
    ]]></literal>
    <justification jstring="[para(279(a,1),97(a,1,2,2))].">
      <j1 rule="para" parents="279 97"/>
    </justification>
  </clause>

  <clause id="301">
    <literal><![CDATA[
      x' ^ (y v x) != 0
    ]]></literal>
    <literal><![CDATA[
      y v x = x
    ]]></literal>
    <justification jstring="[para(296(a,1),10(a,1)),rewrite(279(10)),flip(c),xx(a)].">
      <j1 rule="para" parents="296 10"/>
      <j2 rule="rewrite" parents="279"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="311">
    <literal><![CDATA[
      x ^ (y ^ (y ^ ((x ^ y) v z))') = 0
    ]]></literal>
    <justification jstring="[para(21(a,1),297(a,1,2)),rewrite(4(6),5(6))].">
      <j1 rule="para" parents="21 297"/>
      <j2 rule="rewrite" parents="4 5"/>
    </justification>
  </clause>

  <clause id="430">
    <literal><![CDATA[
      x v (y v (y v (x ^ z))') = 1
    ]]></literal>
    <justification jstring="[para(162(a,1),22(a,1,2)),rewrite(93(2)),flip(a)].">
      <j1 rule="para" parents="162 22"/>
      <j2 rule="rewrite" parents="93"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="689">
    <literal><![CDATA[
      x ^ ((y ^ x) v (y ^ z)) = y ^ x
    ]]></literal>
    <justification jstring="[para(69(a,1),42(a,1,2,2,2,2)),rewrite(69(5),69(8),75(7))].">
      <j1 rule="para" parents="69 42"/>
      <j2 rule="rewrite" parents="69 69 75"/>
    </justification>
  </clause>

  <clause id="792">
    <literal><![CDATA[
      x' ^ (x v (y ^ x')) = x' ^ (x v y)
    ]]></literal>
    <justification jstring="[para(27(a,1),46(a,1,2,2,2))].">
      <j1 rule="para" parents="27 46"/>
    </justification>
  </clause>

  <clause id="878">
    <literal><![CDATA[
      ((x v y) ^ z) v (y ^ z)' = 1
    ]]></literal>
    <justification jstring="[para(87(a,1),195(a,1,2,1))].">
      <j1 rule="para" parents="87 195"/>
    </justification>
  </clause>

  <clause id="1553">
    <literal><![CDATA[
      (x ^ (y v z)) v (y ^ x)' = 1
    ]]></literal>
    <justification jstring="[para(4(a,1),208(a,1,1))].">
      <j1 rule="para" parents="4 208"/>
    </justification>
  </clause>

  <clause id="2247">
    <literal><![CDATA[
      x v ((y ^ x) v (x ^ z))' = 1
    ]]></literal>
    <justification jstring="[para(430(a,1),108(a,1)),flip(a)].">
      <j1 rule="para" parents="430 108"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4011">
    <literal><![CDATA[
      x' ^ (x v (y ^ (x' v z))) = x' ^ (x v y)
    ]]></literal>
    <justification jstring="[para(60(a,1),119(a,1,2,2,2))].">
      <j1 rule="para" parents="60 119"/>
    </justification>
  </clause>

  <clause id="7601">
    <literal><![CDATA[
      x' ^ (x v y) != 0
    ]]></literal>
    <literal><![CDATA[
      y v x = x
    ]]></literal>
    <justification jstring="[para(2(a,1),301(a,1,2))].">
      <j1 rule="para" parents="2 301"/>
    </justification>
  </clause>

  <clause id="13234">
    <literal><![CDATA[
      x' ^ (x v (x' v y)') = 0
    ]]></literal>
    <justification jstring="[para(145(a,1),792(a,2)),rewrite(4(8),131(8),33(3),4(2),9(2),4011(10)),flip(a)].">
      <j1 rule="para" parents="145 792"/>
      <j2 rule="rewrite" parents="4 131 33 4 9 4011"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13301">
    <literal><![CDATA[
      x v (x' v y)' = x
    ]]></literal>
    <justification jstring="[hyper(7601,a,13234,a),rewrite(2(4))].">
      <j1 rule="hyper" parents="7601 13234"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="13319">
    <literal><![CDATA[
      x v (y v x')' = x
    ]]></literal>
    <justification jstring="[para(2(a,1),13301(a,1,2,1))].">
      <j1 rule="para" parents="2 13301"/>
    </justification>
  </clause>

  <clause id="13326">
    <literal><![CDATA[
      x ^ (x' v y)' = (x' v y)'
    ]]></literal>
    <justification jstring="[para(13301(a,1),18(a,1,2)),rewrite(4(4))].">
      <j1 rule="para" parents="13301 18"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="13330">
    <literal><![CDATA[
      x' v (x v y)' = x'
    ]]></literal>
    <justification jstring="[para(279(a,1),13301(a,1,2,1,1))].">
      <j1 rule="para" parents="279 13301"/>
    </justification>
  </clause>

  <clause id="13428">
    <literal><![CDATA[
      x' v (y v x)' = x'
    ]]></literal>
    <justification jstring="[para(279(a,1),13319(a,1,2,1,2))].">
      <j1 rule="para" parents="279 13319"/>
    </justification>
  </clause>

  <clause id="13525">
    <literal><![CDATA[
      x' v (y v (x v z)') = y v x'
    ]]></literal>
    <justification jstring="[para(13330(a,1),15(a,1,2)),flip(a)].">
      <j1 rule="para" parents="13330 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13527">
    <literal><![CDATA[
      x' ^ (x v y)' = (x v y)'
    ]]></literal>
    <justification jstring="[para(13330(a,1),18(a,1,2)),rewrite(4(4))].">
      <j1 rule="para" parents="13330 18"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="13677">
    <literal><![CDATA[
      x' v (x ^ y)' = (x ^ y)'
    ]]></literal>
    <justification jstring="[para(7(a,1),13428(a,1,2,1)),rewrite(2(4))].">
      <j1 rule="para" parents="7 13428"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="13910">
    <literal><![CDATA[
      x' v (y v ((x' v y)' ^ (x v z))) = 1
    ]]></literal>
    <justification jstring="[para(13326(a,1),1553(a,1,2,1)),rewrite(279(9),2(8),3(8))].">
      <j1 rule="para" parents="13326 1553"/>
      <j2 rule="rewrite" parents="279 2 3"/>
    </justification>
  </clause>

  <clause id="15379">
    <literal><![CDATA[
      x ^ (y v (x ^ y'))' = 0
    ]]></literal>
    <justification jstring="[para(13527(a,1),157(a,1,2))].">
      <j1 rule="para" parents="13527 157"/>
    </justification>
  </clause>

  <clause id="15857">
    <literal><![CDATA[
      x' ^ (x ^ y)' = x'
    ]]></literal>
    <justification jstring="[para(13677(a,1),6(a,1,2))].">
      <j1 rule="para" parents="13677 6"/>
    </justification>
  </clause>

  <clause id="15862">
    <literal><![CDATA[
      x v (x' ^ y)' = (x' ^ y)'
    ]]></literal>
    <justification jstring="[para(279(a,1),13677(a,1,1))].">
      <j1 rule="para" parents="279 13677"/>
    </justification>
  </clause>

  <clause id="15960">
    <literal><![CDATA[
      x' ^ (y ^ x)' = x'
    ]]></literal>
    <justification jstring="[para(4(a,1),15857(a,1,2,1))].">
      <j1 rule="para" parents="4 15857"/>
    </justification>
  </clause>

  <clause id="16012">
    <literal><![CDATA[
      x ^ (y ^ x')' = x
    ]]></literal>
    <justification jstring="[para(279(a,1),15960(a,1,1)),rewrite(279(6))].">
      <j1 rule="para" parents="279 15960"/>
      <j2 rule="rewrite" parents="279"/>
    </justification>
  </clause>

  <clause id="16024">
    <literal><![CDATA[
      x v ((x' v y) ^ (z ^ x)') = 1
    ]]></literal>
    <justification jstring="[para(15960(a,1),208(a,1,2,1)),rewrite(279(7),2(6))].">
      <j1 rule="para" parents="15960 208"/>
      <j2 rule="rewrite" parents="279 2"/>
    </justification>
  </clause>

  <clause id="16261">
    <literal><![CDATA[
      x' v ((y v x) ^ (z ^ x')') = 1
    ]]></literal>
    <justification jstring="[para(16012(a,1),878(a,1,2,1)),rewrite(2(7))].">
      <j1 rule="para" parents="16012 878"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="17497">
    <literal><![CDATA[
      (x ^ y) v (y ^ (x ^ y)') = y
    ]]></literal>
    <justification jstring="[hyper(276,a,2247,a,b,15379,a),rewrite(279(7))].">
      <j1 rule="hyper" parents="276 2247 15379"/>
      <j2 rule="rewrite" parents="279"/>
    </justification>
  </clause>

  <clause id="17619">
    <literal><![CDATA[
      x v (y' ^ (x v y))' = 1
    ]]></literal>
    <justification jstring="[para(15862(a,1),201(a,1,2))].">
      <j1 rule="para" parents="15862 201"/>
    </justification>
  </clause>

  <clause id="18973">
    <literal><![CDATA[
      (x' ^ (x v y))' = (y ^ x')'
    ]]></literal>
    <justification jstring="[hyper(37,a,17619,a,b,311,a),rewrite(2(7),792(8)),flip(a)].">
      <j1 rule="hyper" parents="37 17619 311"/>
      <j2 rule="rewrite" parents="2 792"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="19099">
    <literal><![CDATA[
      (x' ^ (y v x))' = (y ^ x')'
    ]]></literal>
    <justification jstring="[para(2(a,1),18973(a,1,1,2))].">
      <j1 rule="para" parents="2 18973"/>
    </justification>
  </clause>

  <clause id="19101">
    <literal><![CDATA[
      ((x v y) ^ x')' = (y ^ x')'
    ]]></literal>
    <justification jstring="[para(4(a,1),18973(a,1,1))].">
      <j1 rule="para" parents="4 18973"/>
    </justification>
  </clause>

  <clause id="19107">
    <literal><![CDATA[
      (x' ^ (y v (x v y)'))' = x
    ]]></literal>
    <justification jstring="[para(29(a,1),18973(a,1,1,2)),rewrite(4(3),67(3),279(2),4(5)),flip(a)].">
      <j1 rule="para" parents="29 18973"/>
      <j2 rule="rewrite" parents="4 67 279 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="19112">
    <literal><![CDATA[
      (x ^ (x' v y))' = (y ^ x)'
    ]]></literal>
    <justification jstring="[para(279(a,1),18973(a,1,1,1)),rewrite(279(6))].">
      <j1 rule="para" parents="279 18973"/>
      <j2 rule="rewrite" parents="279"/>
    </justification>
  </clause>

  <clause id="19480">
    <literal><![CDATA[
      (x' ^ (y v x))' = (x' ^ y)'
    ]]></literal>
    <justification jstring="[para(4(a,1),19099(a,2,1))].">
      <j1 rule="para" parents="4 19099"/>
    </justification>
  </clause>

  <clause id="19575">
    <literal><![CDATA[
      (x v y) ^ (y' ^ x)' = y
    ]]></literal>
    <justification jstring="[para(19099(a,1),170(a,1,2,2)),rewrite(16261(7),19480(8),279(10)),xx(a)].">
      <j1 rule="para" parents="19099 170"/>
      <j2 rule="rewrite" parents="16261 19480 279"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="19697">
    <literal><![CDATA[
      (x v y) ^ x' = y ^ x'
    ]]></literal>
    <justification jstring="[para(19101(a,1),279(a,1,1)),rewrite(279(4)),flip(a)].">
      <j1 rule="para" parents="19101 279"/>
      <j2 rule="rewrite" parents="279"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="19983">
    <literal><![CDATA[
      x' ^ (y v (x v y)') = x'
    ]]></literal>
    <justification jstring="[para(19107(a,1),279(a,1,1)),flip(a)].">
      <j1 rule="para" parents="19107 279"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20014">
    <literal><![CDATA[
      x ^ (x' v y) = y ^ x
    ]]></literal>
    <justification jstring="[para(19112(a,1),279(a,1,1)),rewrite(279(3)),flip(a)].">
      <j1 rule="para" parents="19112 279"/>
      <j2 rule="rewrite" parents="279"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20068">
    <literal><![CDATA[
      (x' v y) ^ (y ^ x)' = x'
    ]]></literal>
    <justification jstring="[para(19112(a,1),170(a,1,2,2)),rewrite(16024(6),20014(8)),xx(a)].">
      <j1 rule="para" parents="19112 170"/>
      <j2 rule="rewrite" parents="16024 20014"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="20601">
    <literal><![CDATA[
      (x' v y)' = x ^ (y ^ x)'
    ]]></literal>
    <justification jstring="[para(13301(a,1),19575(a,1,1)),rewrite(279(4),4(3),20014(3)),flip(a)].">
      <j1 rule="para" parents="13301 19575"/>
      <j2 rule="rewrite" parents="279 4 20014"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20733">
    <literal><![CDATA[
      x' v (y v (x ^ (y ^ x)')) = 1
    ]]></literal>
    <justification jstring="[back_rewrite(13910),rewrite(20601(4),5(6),75(6),4(4))].">
      <j1 rule="back_rewrite" parents="13910"/>
      <j2 rule="rewrite" parents="20601 5 75 4"/>
    </justification>
  </clause>

  <clause id="20881">
    <literal><![CDATA[
      ((x ^ y) v z) ^ y' = z ^ y'
    ]]></literal>
    <justification jstring="[para(108(a,1),19697(a,1,1)),rewrite(19697(3)),flip(a)].">
      <j1 rule="para" parents="108 19697"/>
      <j2 rule="rewrite" parents="19697"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21224">
    <literal><![CDATA[
      x v (y v x)' = x v y'
    ]]></literal>
    <justification jstring="[para(19983(a,1),24(a,1,2)),rewrite(2(5),13525(5)),flip(a)].">
      <j1 rule="para" parents="19983 24"/>
      <j2 rule="rewrite" parents="2 13525"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21486">
    <literal><![CDATA[
      (x v y)' = x' ^ y'
    ]]></literal>
    <justification jstring="[para(18(a,1),20068(a,1,2,1)),rewrite(2(3),21224(3),19697(4)),flip(a)].">
      <j1 rule="para" parents="18 20068"/>
      <j2 rule="rewrite" parents="2 21224 19697"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21550">
    <literal><![CDATA[
      x ^ (y ^ x)' = x ^ y'
    ]]></literal>
    <justification jstring="[para(20068(a,1),170(a,1,2,2,1)),rewrite(279(6),4(5),3(6),20733(6),20068(10),279(7),4(6),21486(9),279(8)),xx(a)].">
      <j1 rule="para" parents="20068 170"/>
      <j2 rule="rewrite" parents="279 4 3 20733 20068 279 4 21486 279"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="25747">
    <literal><![CDATA[
      (x ^ y) v (y ^ x') = y
    ]]></literal>
    <justification jstring="[back_rewrite(17497),rewrite(21550(4))].">
      <j1 rule="back_rewrite" parents="17497"/>
      <j2 rule="rewrite" parents="21550"/>
    </justification>
  </clause>

  <clause id="30738">
    <literal><![CDATA[
      (x ^ y) v (x ^ (z ^ y')) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(75(a,1),25747(a,1,1)),rewrite(5(5),19697(4))].">
      <j1 rule="para" parents="75 25747"/>
      <j2 rule="rewrite" parents="5 19697"/>
    </justification>
  </clause>

  <clause id="30759">
    <literal><![CDATA[
      (x ^ y) v (x ^ z) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(689(a,1),25747(a,1,1)),rewrite(20881(6),5(4),30738(5)),flip(a)].">
      <j1 rule="para" parents="689 25747"/>
      <j2 rule="rewrite" parents="20881 5 30738"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="30760">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(distributivity)
    ]]></attribute>
    <justification jstring="[resolve(30759,a,13,a)].">
      <j1 rule="resolve" parents="30759 13"/>
    </justification>
  </clause>

</proof>

</proofs>
