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<!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 November-2006, November 2006.
Process 4013 was started by mccune on cleo.thornwood,
Wed Nov 22 11:27:08 2006
The command was "/home/mccune/bin/prover9 -f lt.in uc.in H27d.in".
]]></heading>

<proof number="1" length="167" max_count="25">

<comments><![CDATA[
% Proof 1 at 16.04 (+ 0.16) seconds: distributivity.
% Length of proof is 167.
% Level of proof is 26.
% Maximum clause weight is 23.
% Given clauses 716.
]]></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="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="23">
    <literal><![CDATA[
      x v (y v ((x v y) ^ z)) = x v y
    ]]></literal>
    <justification jstring="[para(7(a,1),3(a,1)),flip(a)].">
      <j1 rule="para" parents="7 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(4(a,1),7(a,1,2))].">
      <j1 rule="para" parents="4 7"/>
    </justification>
  </clause>

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

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

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

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

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

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

  <clause id="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="43">
    <literal><![CDATA[
      x ^ (y v ((x ^ z) v ((y v u) ^ z))) = x ^ (y v (z ^ (x v u)))
    ]]></literal>
    <justification jstring="[para(4(a,1),12(a,1,2,2,2))].">
      <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="48">
    <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="50">
    <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="54">
    <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="57">
    <literal><![CDATA[
      x ^ (y v ((x ^ z) v (z ^ y))) = x ^ (y v (z ^ (x v y)))
    ]]></literal>
    <justification jstring="[para(27(a,1),12(a,1,2,2,2,2))].">
      <j1 rule="para" parents="27 12"/>
    </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="64">
    <literal><![CDATA[
      0 v x = x
    ]]></literal>
    <justification jstring="[para(33(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="33 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      x ^ (y v (z ^ (x v y))) = x ^ (y v (z ^ x))
    ]]></literal>
    <justification jstring="[para(33(a,1),12(a,1,2,2,2,2)),rewrite(57(5),33(6))].">
      <j1 rule="para" parents="33 12"/>
      <j2 rule="rewrite" parents="57 33"/>
    </justification>
  </clause>

  <clause id="71">
    <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="72">
    <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="77">
    <literal><![CDATA[
      1 v x = 1
    ]]></literal>
    <justification jstring="[para(61(a,1),6(a,1))].">
      <j1 rule="para" parents="61 6"/>
    </justification>
  </clause>

  <clause id="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="80">
    <literal><![CDATA[
      0 ^ x = 0
    ]]></literal>
    <justification jstring="[para(64(a,1),6(a,1,2))].">
      <j1 rule="para" parents="64 6"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      x ^ (x' ^ y) = 0
    ]]></literal>
    <justification jstring="[back_rewrite(31),rewrite(80(5))].">
      <j1 rule="back_rewrite" parents="31"/>
      <j2 rule="rewrite" parents="80"/>
    </justification>
  </clause>

  <clause id="83">
    <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="85">
    <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="88">
    <literal><![CDATA[
      x v 1 = 1
    ]]></literal>
    <justification jstring="[para(77(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="77 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      x v (y v x') = 1
    ]]></literal>
    <justification jstring="[back_rewrite(72),rewrite(88(5))].">
      <j1 rule="back_rewrite" parents="72"/>
      <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(85),rewrite(91(5))].">
      <j1 rule="back_rewrite" parents="85"/>
      <j2 rule="rewrite" parents="91"/>
    </justification>
  </clause>

  <clause id="95">
    <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="99">
    <literal><![CDATA[
      (x v y) ^ (x v (z v y)) = x v y
    ]]></literal>
    <justification jstring="[para(15(a,1),18(a,1,2))].">
      <j1 rule="para" parents="15 18"/>
    </justification>
  </clause>

  <clause id="101">
    <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="102">
    <literal><![CDATA[
      x v (y v (z ^ (x v y))) = x v y
    ]]></literal>
    <justification jstring="[para(24(a,1),3(a,1)),flip(a)].">
      <j1 rule="para" parents="24 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="103">
    <literal><![CDATA[
      x v (y ^ (z ^ x)) = x
    ]]></literal>
    <justification jstring="[para(5(a,1),24(a,1,2))].">
      <j1 rule="para" parents="5 24"/>
    </justification>
  </clause>

  <clause id="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 ^ (y v (z ^ (x v (y' v u)))) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(78(a,1),12(a,1,2,2,2,2)),rewrite(30(3),2(2),24(2)),flip(a)].">
      <j1 rule="para" parents="78 12"/>
      <j2 rule="rewrite" parents="30 2 24"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="112">
    <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="128">
    <literal><![CDATA[
      x v (y v (z v x')) = 1
    ]]></literal>
    <justification jstring="[para(3(a,1),89(a,1,2))].">
      <j1 rule="para" parents="3 89"/>
    </justification>
  </clause>

  <clause id="135">
    <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="142">
    <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="145">
    <literal><![CDATA[
      x ^ (y ^ ((x v z) ^ u)) = y ^ (x ^ u)
    ]]></literal>
    <justification jstring="[para(20(a,1),17(a,1,2)),flip(a)].">
      <j1 rule="para" parents="20 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="147">
    <literal><![CDATA[
      x ^ ((x v y)' ^ z) = 0
    ]]></literal>
    <justification jstring="[para(82(a,1),20(a,1,2)),rewrite(91(2)),flip(a)].">
      <j1 rule="para" parents="82 20"/>
      <j2 rule="rewrite" parents="91"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="153">
    <literal><![CDATA[
      x ^ (y v ((x v z)' ^ (y v u))) = x ^ (y v ((x v z)' ^ (x v u)))
    ]]></literal>
    <justification jstring="[para(142(a,1),12(a,1,2,2,1)),rewrite(64(6))].">
      <j1 rule="para" parents="142 12"/>
      <j2 rule="rewrite" parents="64"/>
    </justification>
  </clause>

  <clause id="164">
    <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="170">
    <literal><![CDATA[
      x ^ (y ^ (x ^ z)) = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(48(a,1),5(a,2,2)),rewrite(17(3),5(2))].">
      <j1 rule="para" parents="48 5"/>
      <j2 rule="rewrite" parents="17 5"/>
    </justification>
  </clause>

  <clause id="180">
    <literal><![CDATA[
      x ^ (y ^ (z ^ x)) = y ^ (z ^ x)
    ]]></literal>
    <justification jstring="[para(5(a,1),50(a,1,2)),rewrite(5(5))].">
      <j1 rule="para" parents="5 50"/>
      <j2 rule="rewrite" parents="5"/>
    </justification>
  </clause>

  <clause id="190">
    <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="202">
    <literal><![CDATA[
      x v (y ^ x)' = 1
    ]]></literal>
    <justification jstring="[para(4(a,1),190(a,1,2,1))].">
      <j1 rule="para" parents="4 190"/>
    </justification>
  </clause>

  <clause id="208">
    <literal><![CDATA[
      x v (y v (z ^ (x v y))') = 1
    ]]></literal>
    <justification jstring="[para(202(a,1),3(a,1)),flip(a)].">
      <j1 rule="para" parents="202 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="215">
    <literal><![CDATA[
      ((x v y) ^ z) v (x ^ z)' = 1
    ]]></literal>
    <justification jstring="[para(20(a,1),202(a,1,2,1))].">
      <j1 rule="para" parents="20 202"/>
    </justification>
  </clause>

  <clause id="228">
    <literal><![CDATA[
      x v (y v ((y v x) ^ z)) = x v y
    ]]></literal>
    <justification jstring="[para(2(a,1),23(a,1,2,2,1))].">
      <j1 rule="para" parents="2 23"/>
    </justification>
  </clause>

  <clause id="301">
    <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="320">
    <literal><![CDATA[
      x' v (y v x) = 1
    ]]></literal>
    <justification jstring="[para(301(a,1),89(a,1,2,2))].">
      <j1 rule="para" parents="301 89"/>
    </justification>
  </clause>

  <clause id="321">
    <literal><![CDATA[
      x' ^ (y ^ x) = 0
    ]]></literal>
    <justification jstring="[para(301(a,1),92(a,1,2,2))].">
      <j1 rule="para" parents="301 92"/>
    </justification>
  </clause>

  <clause id="324">
    <literal><![CDATA[
      x' v (y v (z v x)) = 1
    ]]></literal>
    <justification jstring="[para(301(a,1),128(a,1,2,2,2))].">
      <j1 rule="para" parents="301 128"/>
    </justification>
  </clause>

  <clause id="326">
    <literal><![CDATA[
      x' ^ (y v x) != 0
    ]]></literal>
    <literal><![CDATA[
      y v x = x
    ]]></literal>
    <justification jstring="[para(320(a,1),10(a,1)),rewrite(301(10)),flip(c),xx(a)].">
      <j1 rule="para" parents="320 10"/>
      <j2 rule="rewrite" parents="301"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="335">
    <literal><![CDATA[
      x ^ (y ^ (y ^ ((x ^ y) v z))') = 0
    ]]></literal>
    <justification jstring="[para(21(a,1),321(a,1,2)),rewrite(4(6),5(6))].">
      <j1 rule="para" parents="21 321"/>
      <j2 rule="rewrite" parents="4 5"/>
    </justification>
  </clause>

  <clause id="631">
    <literal><![CDATA[
      (x ^ (y ^ z))' v (u v z) = 1
    ]]></literal>
    <justification jstring="[para(103(a,1),324(a,1,2,2))].">
      <j1 rule="para" parents="103 324"/>
    </justification>
  </clause>

  <clause id="775">
    <literal><![CDATA[
      x ^ (y v ((y v z) ^ (u ^ (x v z)))) = x ^ (y v ((y v z) ^ u))
    ]]></literal>
    <justification jstring="[para(48(a,1),43(a,1,2,2,2)),rewrite(2(6),24(6),5(8)),flip(a)].">
      <j1 rule="para" parents="48 43"/>
      <j2 rule="rewrite" parents="2 24 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="807">
    <literal><![CDATA[
      (x v (y ^ z))' v (u v (x v y)) = 1
    ]]></literal>
    <justification jstring="[para(71(a,1),324(a,1,2,2))].">
      <j1 rule="para" parents="71 324"/>
    </justification>
  </clause>

  <clause id="819">
    <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="850">
    <literal><![CDATA[
      (x v y) ^ ((x v (z v y)) ^ u) = (x v y) ^ u
    ]]></literal>
    <justification jstring="[para(15(a,1),95(a,1,2,1))].">
      <j1 rule="para" parents="15 95"/>
    </justification>
  </clause>

  <clause id="857">
    <literal><![CDATA[
      ((x v y) ^ z) v (y ^ z)' = 1
    ]]></literal>
    <justification jstring="[para(95(a,1),202(a,1,2,1))].">
      <j1 rule="para" parents="95 202"/>
    </justification>
  </clause>

  <clause id="903">
    <literal><![CDATA[
      x v (y v ((z ^ x) v y)') = 1
    ]]></literal>
    <justification jstring="[para(29(a,1),101(a,1,2)),rewrite(88(2)),flip(a)].">
      <j1 rule="para" parents="29 101"/>
      <j2 rule="rewrite" parents="88"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1039">
    <literal><![CDATA[
      x ^ (x' v (y v (z ^ x))) = x ^ (x' v (y v z))
    ]]></literal>
    <justification jstring="[para(78(a,1),66(a,1,2,2,2)),rewrite(30(4),3(3),3(8)),flip(a)].">
      <j1 rule="para" parents="78 66"/>
      <j2 rule="rewrite" parents="30 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1602">
    <literal><![CDATA[
      (x ^ (y v z)) v (y ^ x)' = 1
    ]]></literal>
    <justification jstring="[para(4(a,1),215(a,1,1))].">
      <j1 rule="para" parents="4 215"/>
    </justification>
  </clause>

  <clause id="2237">
    <literal><![CDATA[
      (x v y) ^ (y v (z ^ (x v y))) = y v (z ^ (x v y))
    ]]></literal>
    <justification jstring="[para(102(a,1),18(a,1,2)),rewrite(4(5))].">
      <j1 rule="para" parents="102 18"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="2268">
    <literal><![CDATA[
      (x v y) ^ (x v (z ^ (x v y))) = x v (z ^ (x v y))
    ]]></literal>
    <justification jstring="[para(102(a,1),99(a,1,2)),rewrite(4(5))].">
      <j1 rule="para" parents="102 99"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="2817">
    <literal><![CDATA[
      x' ^ (x v (y ^ (x' v z))) = x' ^ (x v y)
    ]]></literal>
    <justification jstring="[para(54(a,1),110(a,1,2,2,2))].">
      <j1 rule="para" parents="54 110"/>
    </justification>
  </clause>

  <clause id="7278">
    <literal><![CDATA[
      x' ^ (x v y) != 0
    ]]></literal>
    <literal><![CDATA[
      y v x = x
    ]]></literal>
    <justification jstring="[para(2(a,1),326(a,1,2))].">
      <j1 rule="para" parents="2 326"/>
    </justification>
  </clause>

  <clause id="7442">
    <literal><![CDATA[
      (x v y) ^ (z ^ (y v x)) = z ^ (y v x)
    ]]></literal>
    <justification jstring="[para(112(a,1),170(a,1,2,2)),rewrite(112(8))].">
      <j1 rule="para" parents="112 170"/>
      <j2 rule="rewrite" parents="112"/>
    </justification>
  </clause>

  <clause id="8658">
    <literal><![CDATA[
      (x v ((x v y) ^ z)) ^ (y v x) = x v ((x v y) ^ z)
    ]]></literal>
    <justification jstring="[para(228(a,1),18(a,1,2))].">
      <j1 rule="para" parents="228 18"/>
    </justification>
  </clause>

  <clause id="13466">
    <literal><![CDATA[
      x' ^ (x v (x' v y)') = 0
    ]]></literal>
    <justification jstring="[para(153(a,1),819(a,2)),rewrite(4(8),147(8),33(3),4(2),9(2),2817(10)),flip(a)].">
      <j1 rule="para" parents="153 819"/>
      <j2 rule="rewrite" parents="4 147 33 4 9 2817"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13521">
    <literal><![CDATA[
      x v (x' v y)' = x
    ]]></literal>
    <justification jstring="[hyper(7278,a,13466,a),rewrite(2(4))].">
      <j1 rule="hyper" parents="7278 13466"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="13539">
    <literal><![CDATA[
      x v (y v x')' = x
    ]]></literal>
    <justification jstring="[para(2(a,1),13521(a,1,2,1))].">
      <j1 rule="para" parents="2 13521"/>
    </justification>
  </clause>

  <clause id="13546">
    <literal><![CDATA[
      x ^ (x' v y)' = (x' v y)'
    ]]></literal>
    <justification jstring="[para(13521(a,1),18(a,1,2)),rewrite(4(4))].">
      <j1 rule="para" parents="13521 18"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="13550">
    <literal><![CDATA[
      x' v (x v y)' = x'
    ]]></literal>
    <justification jstring="[para(301(a,1),13521(a,1,2,1,1))].">
      <j1 rule="para" parents="301 13521"/>
    </justification>
  </clause>

  <clause id="13759">
    <literal><![CDATA[
      x' v (y v x)' = x'
    ]]></literal>
    <justification jstring="[para(301(a,1),13539(a,1,2,1,2))].">
      <j1 rule="para" parents="301 13539"/>
    </justification>
  </clause>

  <clause id="13855">
    <literal><![CDATA[
      x' v (y v (x v z)') = y v x'
    ]]></literal>
    <justification jstring="[para(13550(a,1),15(a,1,2)),flip(a)].">
      <j1 rule="para" parents="13550 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13958">
    <literal><![CDATA[
      x' v (x ^ y)' = (x ^ y)'
    ]]></literal>
    <justification jstring="[para(7(a,1),13759(a,1,2,1)),rewrite(2(4))].">
      <j1 rule="para" parents="7 13759"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="13961">
    <literal><![CDATA[
      x' v (y v (z v x)') = y v x'
    ]]></literal>
    <justification jstring="[para(13759(a,1),15(a,1,2)),flip(a)].">
      <j1 rule="para" parents="13759 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14264">
    <literal><![CDATA[
      x' v (y v ((x' v y)' ^ (x v z))) = 1
    ]]></literal>
    <justification jstring="[para(13546(a,1),1602(a,1,2,1)),rewrite(301(9),2(8),3(8))].">
      <j1 rule="para" parents="13546 1602"/>
      <j2 rule="rewrite" parents="301 2 3"/>
    </justification>
  </clause>

  <clause id="16177">
    <literal><![CDATA[
      x' ^ (x ^ y)' = x'
    ]]></literal>
    <justification jstring="[para(13958(a,1),6(a,1,2))].">
      <j1 rule="para" parents="13958 6"/>
    </justification>
  </clause>

  <clause id="16182">
    <literal><![CDATA[
      x v (x' ^ y)' = (x' ^ y)'
    ]]></literal>
    <justification jstring="[para(301(a,1),13958(a,1,1))].">
      <j1 rule="para" parents="301 13958"/>
    </justification>
  </clause>

  <clause id="16282">
    <literal><![CDATA[
      x' ^ (y ^ x)' = x'
    ]]></literal>
    <justification jstring="[para(4(a,1),16177(a,1,2,1))].">
      <j1 rule="para" parents="4 16177"/>
    </justification>
  </clause>

  <clause id="16309">
    <literal><![CDATA[
      x' ^ (y ^ (z ^ x))' = x'
    ]]></literal>
    <justification jstring="[para(180(a,1),16177(a,1,2,1))].">
      <j1 rule="para" parents="180 16177"/>
    </justification>
  </clause>

  <clause id="16342">
    <literal><![CDATA[
      x ^ (y ^ x')' = x
    ]]></literal>
    <justification jstring="[para(301(a,1),16282(a,1,1)),rewrite(301(6))].">
      <j1 rule="para" parents="301 16282"/>
      <j2 rule="rewrite" parents="301"/>
    </justification>
  </clause>

  <clause id="16353">
    <literal><![CDATA[
      x v ((x' v y) ^ (z ^ x)') = 1
    ]]></literal>
    <justification jstring="[para(16282(a,1),215(a,1,2,1)),rewrite(301(7),2(6))].">
      <j1 rule="para" parents="16282 215"/>
      <j2 rule="rewrite" parents="301 2"/>
    </justification>
  </clause>

  <clause id="16536">
    <literal><![CDATA[
      x' v ((y v x) ^ (z ^ x')') = 1
    ]]></literal>
    <justification jstring="[para(16342(a,1),857(a,1,2,1)),rewrite(2(7))].">
      <j1 rule="para" parents="16342 857"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="17831">
    <literal><![CDATA[
      x v (y' ^ (x v y))' = 1
    ]]></literal>
    <justification jstring="[para(16182(a,1),208(a,1,2))].">
      <j1 rule="para" parents="16182 208"/>
    </justification>
  </clause>

  <clause id="18093">
    <literal><![CDATA[
      x' ^ ((y ^ (z ^ x))' ^ u) = x' ^ u
    ]]></literal>
    <justification jstring="[para(16309(a,1),5(a,1,1)),flip(a)].">
      <j1 rule="para" parents="16309 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="19214">
    <literal><![CDATA[
      (x' ^ (x v y))' = (y ^ x')'
    ]]></literal>
    <justification jstring="[hyper(37,a,17831,a,b,335,a),rewrite(2(7),819(8)),flip(a)].">
      <j1 rule="hyper" parents="37 17831 335"/>
      <j2 rule="rewrite" parents="2 819"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="19344">
    <literal><![CDATA[
      (x' ^ (y v x))' = (y ^ x')'
    ]]></literal>
    <justification jstring="[para(2(a,1),19214(a,1,1,2))].">
      <j1 rule="para" parents="2 19214"/>
    </justification>
  </clause>

  <clause id="19346">
    <literal><![CDATA[
      ((x v y) ^ x')' = (y ^ x')'
    ]]></literal>
    <justification jstring="[para(4(a,1),19214(a,1,1))].">
      <j1 rule="para" parents="4 19214"/>
    </justification>
  </clause>

  <clause id="19352">
    <literal><![CDATA[
      (x' ^ (y v (x v y)'))' = x
    ]]></literal>
    <justification jstring="[para(29(a,1),19214(a,1,1,2)),rewrite(4(3),61(3),301(2),4(5)),flip(a)].">
      <j1 rule="para" parents="29 19214"/>
      <j2 rule="rewrite" parents="4 61 301 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="19358">
    <literal><![CDATA[
      (x' ^ (y v (y v x)'))' = x
    ]]></literal>
    <justification jstring="[para(135(a,1),19214(a,1,1,2)),rewrite(4(3),61(3),301(2),4(5)),flip(a)].">
      <j1 rule="para" parents="135 19214"/>
      <j2 rule="rewrite" parents="4 61 301 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="19360">
    <literal><![CDATA[
      (x ^ (x' v y))' = (y ^ x)'
    ]]></literal>
    <justification jstring="[para(301(a,1),19214(a,1,1,1)),rewrite(301(6))].">
      <j1 rule="para" parents="301 19214"/>
      <j2 rule="rewrite" parents="301"/>
    </justification>
  </clause>

  <clause id="19727">
    <literal><![CDATA[
      ((x v y) ^ y')' = (x ^ y')'
    ]]></literal>
    <justification jstring="[para(4(a,1),19344(a,1,1))].">
      <j1 rule="para" parents="4 19344"/>
    </justification>
  </clause>

  <clause id="19728">
    <literal><![CDATA[
      (x' ^ (y v x))' = (x' ^ y)'
    ]]></literal>
    <justification jstring="[para(4(a,1),19344(a,2,1))].">
      <j1 rule="para" parents="4 19344"/>
    </justification>
  </clause>

  <clause id="19740">
    <literal><![CDATA[
      (x ^ (y v x'))' = (y ^ x)'
    ]]></literal>
    <justification jstring="[para(301(a,1),19344(a,1,1,1)),rewrite(301(6))].">
      <j1 rule="para" parents="301 19344"/>
      <j2 rule="rewrite" parents="301"/>
    </justification>
  </clause>

  <clause id="19786">
    <literal><![CDATA[
      ((x ^ (y ^ z))' ^ (u v z)')' = u v z
    ]]></literal>
    <justification jstring="[para(631(a,1),19344(a,1,1,2)),rewrite(4(4),61(4),301(3)),flip(a)].">
      <j1 rule="para" parents="631 19344"/>
      <j2 rule="rewrite" parents="4 61 301"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="19794">
    <literal><![CDATA[
      (x ^ (y v ((z ^ x) v y)')')' = y v ((z ^ x) v y)'
    ]]></literal>
    <justification jstring="[para(903(a,1),19344(a,1,1,2)),rewrite(4(7),61(7),301(6)),flip(a)].">
      <j1 rule="para" parents="903 19344"/>
      <j2 rule="rewrite" parents="4 61 301"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="19821">
    <literal><![CDATA[
      (x v y) ^ (y' ^ x)' = y
    ]]></literal>
    <justification jstring="[para(19344(a,1),164(a,1,2,2)),rewrite(16536(7),19728(8),301(10)),xx(a)].">
      <j1 rule="para" parents="19344 164"/>
      <j2 rule="rewrite" parents="16536 19728 301"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="19948">
    <literal><![CDATA[
      (x v y) ^ x' = y ^ x'
    ]]></literal>
    <justification jstring="[para(19346(a,1),301(a,1,1)),rewrite(301(4)),flip(a)].">
      <j1 rule="para" parents="19346 301"/>
      <j2 rule="rewrite" parents="301"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20237">
    <literal><![CDATA[
      x' ^ (y v (x v y)') = x'
    ]]></literal>
    <justification jstring="[para(19352(a,1),301(a,1,1)),flip(a)].">
      <j1 rule="para" parents="19352 301"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20265">
    <literal><![CDATA[
      x' ^ (y v (y v x)') = x'
    ]]></literal>
    <justification jstring="[para(19358(a,1),301(a,1,1)),flip(a)].">
      <j1 rule="para" parents="19358 301"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20308">
    <literal><![CDATA[
      x ^ (x' v y) = y ^ x
    ]]></literal>
    <justification jstring="[para(19360(a,1),301(a,1,1)),rewrite(301(3)),flip(a)].">
      <j1 rule="para" parents="19360 301"/>
      <j2 rule="rewrite" parents="301"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20359">
    <literal><![CDATA[
      (x' v y) ^ (y ^ x)' = x'
    ]]></literal>
    <justification jstring="[para(19360(a,1),164(a,1,2,2)),rewrite(16353(6),20308(8)),xx(a)].">
      <j1 rule="para" parents="19360 164"/>
      <j2 rule="rewrite" parents="16353 20308"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="20440">
    <literal><![CDATA[
      x ^ (y v (z ^ x)) = (y v z) ^ x
    ]]></literal>
    <justification jstring="[back_rewrite(1039),rewrite(20308(5),4(3),20308(7))].">
      <j1 rule="back_rewrite" parents="1039"/>
      <j2 rule="rewrite" parents="20308 4 20308"/>
    </justification>
  </clause>

  <clause id="20473">
    <literal><![CDATA[
      x v (y ^ (x v z)) = (x v y) ^ (x v z)
    ]]></literal>
    <justification jstring="[back_rewrite(2268),rewrite(20440(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="2268"/>
      <j2 rule="rewrite" parents="20440"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20474">
    <literal><![CDATA[
      x v (y ^ (z v x)) = (x v y) ^ (z v x)
    ]]></literal>
    <justification jstring="[back_rewrite(2237),rewrite(20440(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="2237"/>
      <j2 rule="rewrite" parents="20440"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20697">
    <literal><![CDATA[
      (x v y) ^ y' = x ^ y'
    ]]></literal>
    <justification jstring="[para(19727(a,1),301(a,1,1)),rewrite(301(4)),flip(a)].">
      <j1 rule="para" parents="19727 301"/>
      <j2 rule="rewrite" parents="301"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20716">
    <literal><![CDATA[
      x ^ (y v x') = y ^ x
    ]]></literal>
    <justification jstring="[para(19740(a,1),301(a,1,1)),rewrite(301(3)),flip(a)].">
      <j1 rule="para" parents="19740 301"/>
      <j2 rule="rewrite" parents="301"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20765">
    <literal><![CDATA[
      (x v y) ^ (y ^ (x v (z ^ y))')' = x v (z ^ y)
    ]]></literal>
    <justification jstring="[para(106(a,1),19821(a,1,1)),rewrite(4(5))].">
      <j1 rule="para" parents="106 19821"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="20802">
    <literal><![CDATA[
      (x' v y)' = x ^ (y ^ x)'
    ]]></literal>
    <justification jstring="[para(13521(a,1),19821(a,1,1)),rewrite(301(4),4(3),20308(3)),flip(a)].">
      <j1 rule="para" parents="13521 19821"/>
      <j2 rule="rewrite" parents="301 4 20308"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20803">
    <literal><![CDATA[
      (x v y')' = y ^ (x ^ y)'
    ]]></literal>
    <justification jstring="[para(13539(a,1),19821(a,1,1)),rewrite(301(4),4(3),20716(3)),flip(a)].">
      <j1 rule="para" parents="13539 19821"/>
      <j2 rule="rewrite" parents="301 4 20716"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20935">
    <literal><![CDATA[
      x' v (y v (x ^ (y ^ x)')) = 1
    ]]></literal>
    <justification jstring="[back_rewrite(14264),rewrite(20802(4),5(6),83(6),4(4))].">
      <j1 rule="back_rewrite" parents="14264"/>
      <j2 rule="rewrite" parents="20802 5 83 4"/>
    </justification>
  </clause>

  <clause id="20960">
    <literal><![CDATA[
      (x ^ (y ^ z'))' = z v ((x ^ y) v z)'
    ]]></literal>
    <justification jstring="[back_rewrite(19794),rewrite(20803(5),18(5),20697(4),5(3),170(4))].">
      <j1 rule="back_rewrite" parents="19794"/>
      <j2 rule="rewrite" parents="20803 18 20697 5 170"/>
    </justification>
  </clause>

  <clause id="21193">
    <literal><![CDATA[
      (x v (y v z)) ^ (y v (z ^ u)) = y v (z ^ u)
    ]]></literal>
    <justification jstring="[para(807(a,1),19948(a,1,1)),rewrite(301(5),61(4),301(8)),flip(a)].">
      <j1 rule="para" parents="807 19948"/>
      <j2 rule="rewrite" parents="301 61 301"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21454">
    <literal><![CDATA[
      x v (y v x)' = x v y'
    ]]></literal>
    <justification jstring="[para(20237(a,1),24(a,1,2)),rewrite(2(5),13855(5)),flip(a)].">
      <j1 rule="para" parents="20237 24"/>
      <j2 rule="rewrite" parents="2 13855"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21476">
    <literal><![CDATA[
      (x ^ (y ^ z'))' = z v (x ^ y)'
    ]]></literal>
    <justification jstring="[back_rewrite(20960),rewrite(21454(8))].">
      <j1 rule="back_rewrite" parents="20960"/>
      <j2 rule="rewrite" parents="21454"/>
    </justification>
  </clause>

  <clause id="21582">
    <literal><![CDATA[
      x v (x v y)' = x v y'
    ]]></literal>
    <justification jstring="[para(20265(a,1),24(a,1,2)),rewrite(2(5),13961(5)),flip(a)].">
      <j1 rule="para" parents="20265 24"/>
      <j2 rule="rewrite" parents="2 13961"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21840">
    <literal><![CDATA[
      (x v y)' = y' ^ x'
    ]]></literal>
    <justification jstring="[para(6(a,1),20359(a,1,2,1)),rewrite(2(3),21582(3),19948(4)),flip(a)].">
      <j1 rule="para" parents="6 20359"/>
      <j2 rule="rewrite" parents="2 21582 19948"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21904">
    <literal><![CDATA[
      x ^ (y ^ x)' = y' ^ x
    ]]></literal>
    <justification jstring="[para(20359(a,1),164(a,1,2,2,1)),rewrite(301(6),4(5),3(6),20935(6),20359(10),301(7),4(6),21840(9),301(9)),xx(a)].">
      <j1 rule="para" parents="20359 164"/>
      <j2 rule="rewrite" parents="301 4 3 20935 20359 301 4 21840 301"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="22013">
    <literal><![CDATA[
      x v (y' ^ x') = x v y'
    ]]></literal>
    <justification jstring="[back_rewrite(21582),rewrite(21840(2))].">
      <j1 rule="back_rewrite" parents="21582"/>
      <j2 rule="rewrite" parents="21840"/>
    </justification>
  </clause>

  <clause id="22032">
    <literal><![CDATA[
      x v (x' ^ y') = x v y'
    ]]></literal>
    <justification jstring="[back_rewrite(21454),rewrite(21840(2))].">
      <j1 rule="back_rewrite" parents="21454"/>
      <j2 rule="rewrite" parents="21840"/>
    </justification>
  </clause>

  <clause id="22170">
    <literal><![CDATA[
      (x v y) ^ (x v (z' ^ y)') = x v (z ^ y)
    ]]></literal>
    <justification jstring="[back_rewrite(20765),rewrite(21840(4),21476(7),21904(4))].">
      <j1 rule="back_rewrite" parents="20765"/>
      <j2 rule="rewrite" parents="21840 21476 21904"/>
    </justification>
  </clause>

  <clause id="22231">
    <literal><![CDATA[
      (x' ^ y')' = y v x
    ]]></literal>
    <justification jstring="[back_rewrite(19786),rewrite(21840(5),17(7),18093(7))].">
      <j1 rule="back_rewrite" parents="19786"/>
      <j2 rule="rewrite" parents="21840 17 18093"/>
    </justification>
  </clause>

  <clause id="26267">
    <literal><![CDATA[
      (x' ^ y)' = y' v x
    ]]></literal>
    <justification jstring="[para(301(a,1),22231(a,1,1,2))].">
      <j1 rule="para" parents="301 22231"/>
    </justification>
  </clause>

  <clause id="26294">
    <literal><![CDATA[
      (x v y) ^ (x' v y) = y
    ]]></literal>
    <justification jstring="[para(22231(a,1),19821(a,1,2)),rewrite(4(4))].">
      <j1 rule="para" parents="22231 19821"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="26646">
    <literal><![CDATA[
      (x v y) ^ (x v (y' v z)) = x v (z ^ y)
    ]]></literal>
    <justification jstring="[back_rewrite(22170),rewrite(26267(4))].">
      <j1 rule="back_rewrite" parents="22170"/>
      <j2 rule="rewrite" parents="26267"/>
    </justification>
  </clause>

  <clause id="31281">
    <literal><![CDATA[
      x v (y ^ x') = x v y
    ]]></literal>
    <justification jstring="[para(301(a,1),22013(a,1,2,1)),rewrite(301(5))].">
      <j1 rule="para" parents="301 22013"/>
      <j2 rule="rewrite" parents="301"/>
    </justification>
  </clause>

  <clause id="31337">
    <literal><![CDATA[
      x v (x' ^ y) = x v y
    ]]></literal>
    <justification jstring="[para(301(a,1),22032(a,1,2,2)),rewrite(301(5))].">
      <j1 rule="para" parents="301 22032"/>
      <j2 rule="rewrite" parents="301"/>
    </justification>
  </clause>

  <clause id="31523">
    <literal><![CDATA[
      (x' v y) ^ (x v ((z ^ x') v y)) = (z ^ x') v y
    ]]></literal>
    <justification jstring="[para(101(a,1),26294(a,1,2)),rewrite(4(7))].">
      <j1 rule="para" parents="101 26294"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="31730">
    <literal><![CDATA[
      x v ((y ^ x') v z) = x v (y v z)
    ]]></literal>
    <justification jstring="[para(31281(a,1),3(a,1,1)),rewrite(3(2)),flip(a)].">
      <j1 rule="para" parents="31281 3"/>
      <j2 rule="rewrite" parents="3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="31763">
    <literal><![CDATA[
      (x ^ y') v z = (y' v z) ^ (y v (x v z))
    ]]></literal>
    <justification jstring="[back_rewrite(31523),rewrite(31730(6)),flip(a)].">
      <j1 rule="back_rewrite" parents="31523"/>
      <j2 rule="rewrite" parents="31730"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="31966">
    <literal><![CDATA[
      x v (y ^ (x' ^ z)) = x v (y ^ z)
    ]]></literal>
    <justification jstring="[para(17(a,1),31337(a,1,2))].">
      <j1 rule="para" parents="17 31337"/>
    </justification>
  </clause>

  <clause id="31987">
    <literal><![CDATA[
      x v (y ^ ((x' v z) ^ u)) = x v (y ^ u)
    ]]></literal>
    <justification jstring="[para(145(a,1),31337(a,1,2)),rewrite(31966(4)),flip(a)].">
      <j1 rule="para" parents="145 31337"/>
      <j2 rule="rewrite" parents="31966"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="32647">
    <literal><![CDATA[
      x ^ (y v ((y v z) ^ u)) = x ^ ((y v u) ^ (y v z))
    ]]></literal>
    <justification jstring="[para(4(a,1),775(a,2,2,2)),rewrite(775(6),20473(7))].">
      <j1 rule="para" parents="4 775"/>
      <j2 rule="rewrite" parents="775 20473"/>
    </justification>
  </clause>

  <clause id="32649">
    <literal><![CDATA[
      x v ((x v y) ^ z) = (x v z) ^ (x v y)
    ]]></literal>
    <justification jstring="[para(26(a,1),775(a,2)),rewrite(2(8),228(8),7442(7),20474(6),17(7),8658(6),32647(5),48(5)),flip(a)].">
      <j1 rule="para" parents="26 775"/>
      <j2 rule="rewrite" parents="2 228 7442 20474 17 8658 32647 48"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="32663">
    <literal><![CDATA[
      x ^ (y v (z ^ u)) = x ^ ((y v z) ^ (y v u))
    ]]></literal>
    <justification jstring="[para(20308(a,1),775(a,2,2,2)),rewrite(21840(3),31763(5),5(9),31987(11),32649(8),32649(6),4(8),850(8),26646(5),20473(6))].">
      <j1 rule="para" parents="20308 775"/>
      <j2 rule="rewrite" parents="21840 31763 5 31987 32649 32649 4 850 26646 20473"/>
    </justification>
  </clause>

  <clause id="32906">
    <literal><![CDATA[
      x v (y ^ z) = (x v y) ^ (x v z)
    ]]></literal>
    <justification jstring="[back_rewrite(21193),rewrite(32663(5),17(6),95(6)),flip(a)].">
      <j1 rule="back_rewrite" parents="21193"/>
      <j2 rule="rewrite" parents="32663 17 95"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="33497">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(distributivity)
    ]]></attribute>
    <justification jstring="[back_rewrite(13),rewrite(32906(7),2(5),7(5),2(6),32906(6),2(4),2(7),20(9)),xx(a)].">
      <j1 rule="back_rewrite" parents="13"/>
      <j2 rule="rewrite" parents="32906 2 7 2 32906 2 2 20"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
