<?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>BA3.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13064 was started by mccune on cleo.thornwood,
Mon Jun 19 16:40:38 2006
The command was "/home/mccune/bin/prover9 -f BA3.in".
]]></heading>

<proof number="1" length="152" max_count="29">

<comments><![CDATA[
% Proof 1 at 6.61 (+ 0.08) seconds: distributivity_1.
% Length of proof is 152.
% Level of proof is 49.
% Maximum clause weight is 25.
% Given clauses 360.
]]></comments>

  <clause id="3">
    <literal><![CDATA[
      x v x ' = y v y '
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      (x v y ') ^ (x v y) = x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="5">
    <literal><![CDATA[
      (c1 ^ c2) v (c1 ^ c3) != c1 ^ (c2 v c3)
    ]]></literal>
    <attribute><![CDATA[
      answer(distributivity_1)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

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

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

  <clause id="8">
    <literal><![CDATA[
      x v x ' = y v y '
    ]]></literal>
    <justification jstring="[copy(3)].">
      <j1 rule="copy" parents="3"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      ((x v y ') ' v (x v y) ') ' = x
    ]]></literal>
    <justification jstring="[copy(4),demod(7(4))].">
      <j1 rule="copy" parents="4"/>
      <j2 rule="demod" parents="7"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      x v x ' = c_0
    ]]></literal>
    <justification jstring="[new_symbol(8)].">
      <j1 rule="new_symbol" parents="8"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      (c1 ' v (c2 v c3) ') ' != (c1 ' v c2 ') ' v (c1 ' v c3 ') '
    ]]></literal>
    <attribute><![CDATA[
      answer(distributivity_1)
    ]]></attribute>
    <justification jstring="[copy(5),demod(7(3),7(9),7(18)),flip(a)].">
      <j1 rule="copy" parents="5"/>
      <j2 rule="demod" parents="7 7 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

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

  <clause id="13">
    <literal><![CDATA[
      ((x v (y v z) ') ' v (y v (x v z)) ') ' = x
    ]]></literal>
    <justification jstring="[para(6(a,1),9(a,1,1,2,1))].">
      <j1 rule="para" parents="6 9"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      ((x v y) ' v (x v ((y v z ') ' v (y v z) ')) ') ' = x
    ]]></literal>
    <justification jstring="[para(9(a,1),9(a,1,1,1,1,2))].">
      <j1 rule="para" parents="9 9"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      (x v (x v ((x v y ') ' v y)) ') ' = (x v y ') '
    ]]></literal>
    <justification jstring="[para(9(a,1),9(a,1,1,1)),demod(6(5))].">
      <j1 rule="para" parents="9 9"/>
      <j2 rule="demod" parents="6"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      (((x v y ') ' v (x v y) ' ') ' v x) ' = (x v y ') '
    ]]></literal>
    <justification jstring="[para(9(a,1),9(a,1,1,2))].">
      <j1 rule="para" parents="9 9"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      x v (y v x ') = y v c_0
    ]]></literal>
    <justification jstring="[para(10(a,1),6(a,1,2)),flip(a)].">
      <j1 rule="para" parents="10 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      (c_0 ' v (x v x) ') ' = x
    ]]></literal>
    <justification jstring="[para(10(a,1),9(a,1,1,1,1))].">
      <j1 rule="para" parents="10 9"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      ((x v x ' ') ' v c_0 ') ' = x
    ]]></literal>
    <justification jstring="[para(10(a,1),9(a,1,1,2,1))].">
      <j1 rule="para" parents="10 9"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      ((x v y ') ' v (x v y) ') v x = c_0
    ]]></literal>
    <justification jstring="[para(9(a,1),10(a,1,2))].">
      <j1 rule="para" parents="9 10"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      x v (y v (z v x ')) = y v (z v c_0)
    ]]></literal>
    <justification jstring="[para(17(a,1),6(a,1,2)),flip(a)].">
      <j1 rule="para" parents="17 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      ((x v (y v x ') ') ' v (y v c_0) ') ' = x
    ]]></literal>
    <justification jstring="[para(17(a,1),9(a,1,1,2,1))].">
      <j1 rule="para" parents="17 9"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      x v (y v (z v (u v v))) = u v (x v (y v (z v v)))
    ]]></literal>
    <justification jstring="[para(12(a,2),6(a,1,2))].">
      <j1 rule="para" parents="12 6"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      (c_0 ' v (x v ((x v y) v y)) ') ' = x v y
    ]]></literal>
    <justification jstring="[para(6(a,1),18(a,1,1,2,1))].">
      <j1 rule="para" parents="6 18"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      (x v (c_0 ' v (x v x)) ') ' = c_0 '
    ]]></literal>
    <justification jstring="[para(18(a,1),9(a,1,1,1))].">
      <j1 rule="para" parents="18 9"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      (c_0 ' v (x v x) ') v x = c_0
    ]]></literal>
    <justification jstring="[para(18(a,1),10(a,1,2))].">
      <j1 rule="para" parents="18 10"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      (c_0 ' v (x v x) ') v (y v x) = y v c_0
    ]]></literal>
    <justification jstring="[para(18(a,1),17(a,1,2,2))].">
      <j1 rule="para" parents="18 17"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      (c_0 ' v (x v (y v ((y v (x v z)) v z))) ') ' = y v (x v z)
    ]]></literal>
    <justification jstring="[para(12(a,1),18(a,1,1,2,1)),demod(6(6))].">
      <j1 rule="para" parents="12 18"/>
      <j2 rule="demod" parents="6"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      x v ((c_0 ' v (x v ((x v y) v y)) ') v y) = c_0
    ]]></literal>
    <justification jstring="[para(43(a,1),6(a,1)),demod(6(6)),flip(a)].">
      <j1 rule="para" parents="43 6"/>
      <j2 rule="demod" parents="6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      (c_0 ' v (x ' v x ') ') v c_0 = x v c_0
    ]]></literal>
    <justification jstring="[para(43(a,1),17(a,1,2)),flip(a)].">
      <j1 rule="para" parents="43 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      (((x v x ' ') ' v c_0 ' ') ' v x) ' = (x v x ' ') '
    ]]></literal>
    <justification jstring="[para(19(a,1),9(a,1,1,2))].">
      <j1 rule="para" parents="19 9"/>
    </justification>
  </clause>

  <clause id="63">
    <literal><![CDATA[
      (x v (x v (c_0 ' v x)) ') ' = c_0 '
    ]]></literal>
    <justification jstring="[para(6(a,1),41(a,1,1,2,1))].">
      <j1 rule="para" parents="6 41"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      (x v (c_0 ' v (x v x)) ') v c_0 ' = c_0
    ]]></literal>
    <justification jstring="[para(41(a,1),10(a,1,2))].">
      <j1 rule="para" parents="41 10"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      (c_0 ' ' v (c_0 ' ' v c_0) ') ' = c_0 '
    ]]></literal>
    <justification jstring="[para(17(a,1),41(a,1,1,2,1))].">
      <j1 rule="para" parents="17 41"/>
    </justification>
  </clause>

  <clause id="105">
    <literal><![CDATA[
      (c_0 ' v (x v (x v (c_0 ' v x))) ') ' = x
    ]]></literal>
    <justification jstring="[para(63(a,1),9(a,1,1,1))].">
      <j1 rule="para" parents="63 9"/>
    </justification>
  </clause>

  <clause id="117">
    <literal><![CDATA[
      ((x ' v y ') ' v (x ' v y) ') v c_0 = x v c_0
    ]]></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="131">
    <literal><![CDATA[
      (c_0 ' v (x v (c_0 ' v (x v x))) ') v x = c_0
    ]]></literal>
    <justification jstring="[para(41(a,1),20(a,1,1,1))].">
      <j1 rule="para" parents="41 20"/>
    </justification>
  </clause>

  <clause id="135">
    <literal><![CDATA[
      (c_0 ' v (x v (x v (c_0 ' v x))) ') v x = c_0
    ]]></literal>
    <justification jstring="[para(63(a,1),20(a,1,1,1))].">
      <j1 rule="para" parents="63 20"/>
    </justification>
  </clause>

  <clause id="154">
    <literal><![CDATA[
      (x v (c_0 ' v (x v x)) ') v c_0 = c_0 v c_0
    ]]></literal>
    <justification jstring="[para(67(a,1),17(a,1,2)),flip(a)].">
      <j1 rule="para" parents="67 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="161">
    <literal><![CDATA[
      (c_0 ' v (c_0 ' ' v (c_0 ' ' v c_0)) ') ' = c_0 ' '
    ]]></literal>
    <justification jstring="[para(69(a,1),9(a,1,1,1))].">
      <j1 rule="para" parents="69 9"/>
    </justification>
  </clause>

  <clause id="190">
    <literal><![CDATA[
      ((x v x ' ') ' v (x v ((x v x ' ') ' v c_0)) ') ' = x
    ]]></literal>
    <justification jstring="[para(19(a,1),15(a,1,1,2,1,2,1)),demod(6(11),19(22))].">
      <j1 rule="para" parents="19 15"/>
      <j2 rule="demod" parents="6 19"/>
    </justification>
  </clause>

  <clause id="195">
    <literal><![CDATA[
      (x v (y v ((x v (y v x ') ') ' v c_0)) ') ' = (x v (y v x ') ') '
    ]]></literal>
    <justification jstring="[para(21(a,1),15(a,1,1,2,1)),demod(6(8))].">
      <j1 rule="para" parents="21 15"/>
      <j2 rule="demod" parents="6"/>
    </justification>
  </clause>

  <clause id="228">
    <literal><![CDATA[
      ((x v (y v c_0) ') ' v (y v ((c_0 ' v (z v z) ') v (x v z))) ') ' = x
    ]]></literal>
    <justification jstring="[para(44(a,2),13(a,1,1,2,1,2))].">
      <j1 rule="para" parents="44 13"/>
    </justification>
  </clause>

  <clause id="251">
    <literal><![CDATA[
      ((x v ((x v x ' ') ' v c_0) ' ') ' v (x v x ' ') ') ' = x
    ]]></literal>
    <justification jstring="[para(19(a,1),16(a,1,1,1,1,1)),demod(19(24))].">
      <j1 rule="para" parents="19 16"/>
      <j2 rule="demod" parents="19"/>
    </justification>
  </clause>

  <clause id="297">
    <literal><![CDATA[
      x v (c_0 ' v (x v x)) ' = c_0
    ]]></literal>
    <justification jstring="[para(154(a,1),9(a,1,1,2,1)),demod(67(9),18(8)),flip(a)].">
      <j1 rule="para" parents="154 9"/>
      <j2 rule="demod" parents="67 18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="302">
    <literal><![CDATA[
      ((x v (y v x ') ') ' v (y v c_0) ') v (z v x) = z v c_0
    ]]></literal>
    <justification jstring="[para(22(a,1),17(a,1,2,2))].">
      <j1 rule="para" parents="22 17"/>
    </justification>
  </clause>

  <clause id="306">
    <literal><![CDATA[
      ((x v y) ' v ((y v (z v y ') ') ' v (x v (z v c_0) ')) ') ' = x
    ]]></literal>
    <justification jstring="[para(22(a,1),13(a,1,1,1,1,2))].">
      <j1 rule="para" parents="22 13"/>
    </justification>
  </clause>

  <clause id="319">
    <literal><![CDATA[
      x v (y v (c_0 ' v (x v x)) ') = y v c_0
    ]]></literal>
    <justification jstring="[para(297(a,1),6(a,1,2)),flip(a)].">
      <j1 rule="para" parents="297 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="321">
    <literal><![CDATA[
      x v (x v (c_0 ' v x)) ' = c_0
    ]]></literal>
    <justification jstring="[para(6(a,1),297(a,1,2,1))].">
      <j1 rule="para" parents="6 297"/>
    </justification>
  </clause>

  <clause id="322">
    <literal><![CDATA[
      (c_0 ' v (x v x)) v c_0 = x v c_0
    ]]></literal>
    <justification jstring="[para(297(a,1),17(a,1,2))].">
      <j1 rule="para" parents="297 17"/>
    </justification>
  </clause>

  <clause id="323">
    <literal><![CDATA[
      c_0 ' ' v (c_0 ' ' v c_0) ' = c_0
    ]]></literal>
    <justification jstring="[para(17(a,1),297(a,1,2,1))].">
      <j1 rule="para" parents="17 297"/>
    </justification>
  </clause>

  <clause id="329">
    <literal><![CDATA[
      ((x v (y v (c_0 ' v (x v x)) ') ') ' v (y v c_0) ') ' = x
    ]]></literal>
    <justification jstring="[para(297(a,1),13(a,1,1,2,1,2))].">
      <j1 rule="para" parents="297 13"/>
    </justification>
  </clause>

  <clause id="333">
    <literal><![CDATA[
      x v (y v (x v (c_0 ' v x)) ') = y v c_0
    ]]></literal>
    <justification jstring="[para(321(a,1),6(a,1,2)),flip(a)].">
      <j1 rule="para" parents="321 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="334">
    <literal><![CDATA[
      (x v (c_0 ' v x)) v c_0 = x v c_0
    ]]></literal>
    <justification jstring="[para(321(a,1),17(a,1,2))].">
      <j1 rule="para" parents="321 17"/>
    </justification>
  </clause>

  <clause id="345">
    <literal><![CDATA[
      (c_0 ' ' v c_0) v c_0 = c_0 ' ' v c_0
    ]]></literal>
    <justification jstring="[para(17(a,1),322(a,1,1))].">
      <j1 rule="para" parents="17 322"/>
    </justification>
  </clause>

  <clause id="362">
    <literal><![CDATA[
      (c_0 ' ' v c_0) v (x v c_0) = x v (c_0 ' ' v c_0)
    ]]></literal>
    <justification jstring="[para(323(a,1),21(a,1,2,2))].">
      <j1 rule="para" parents="323 21"/>
    </justification>
  </clause>

  <clause id="428">
    <literal><![CDATA[
      (c_0 ' v (x v x)) v (y v c_0) = x v (y v c_0)
    ]]></literal>
    <justification jstring="[para(319(a,1),21(a,1,2))].">
      <j1 rule="para" parents="319 21"/>
    </justification>
  </clause>

  <clause id="462">
    <literal><![CDATA[
      ((x v x) ' v ((x v (c_0 ' v x) ') ' v c_0) ') ' = x
    ]]></literal>
    <justification jstring="[para(333(a,1),14(a,1,1,2,1))].">
      <j1 rule="para" parents="333 14"/>
    </justification>
  </clause>

  <clause id="472">
    <literal><![CDATA[
      c_0 ' ' v c_0 = c_0 ' '
    ]]></literal>
    <justification jstring="[para(345(a,1),39(a,1,1,2,1,2,1)),demod(345(14),6(13),345(12),161(14),345(10)),flip(a)].">
      <j1 rule="para" parents="345 39"/>
      <j2 rule="demod" parents="345 6 345 161 345"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="484">
    <literal><![CDATA[
      c_0 ' ' v (x v c_0) = x v c_0 ' '
    ]]></literal>
    <justification jstring="[back_demod(362),demod(472(5),472(11))].">
      <j1 rule="back_demod" parents="362"/>
      <j2 rule="demod" parents="472 472"/>
    </justification>
  </clause>

  <clause id="494">
    <literal><![CDATA[
      ((c_0 ' ' v (x v c_0)) ' v (x v c_0 ') ') ' = x
    ]]></literal>
    <justification jstring="[para(484(a,2),9(a,1,1,1,1))].">
      <j1 rule="para" parents="484 9"/>
    </justification>
  </clause>

  <clause id="495">
    <literal><![CDATA[
      c_0 ' v (c_0 ' ' v (x v c_0)) = x v c_0
    ]]></literal>
    <justification jstring="[para(484(a,2),17(a,1,2))].">
      <j1 rule="para" parents="484 17"/>
    </justification>
  </clause>

  <clause id="525">
    <literal><![CDATA[
      c_0 ' ' v (x v c_0) = (x v (c_0 ' v x)) v c_0 ' '
    ]]></literal>
    <justification jstring="[para(334(a,1),484(a,1,2))].">
      <j1 rule="para" parents="334 484"/>
    </justification>
  </clause>

  <clause id="568">
    <literal><![CDATA[
      c_0 ' ' v (x v (c_0 ' v c_0)) = x v c_0
    ]]></literal>
    <justification jstring="[para(495(a,1),12(a,2))].">
      <j1 rule="para" parents="495 12"/>
    </justification>
  </clause>

  <clause id="581">
    <literal><![CDATA[
      (c_0 ' v (c_0 ' v ((c_0 ' v c_0) v c_0)) ') v c_0 = c_0 ' '
    ]]></literal>
    <justification jstring="[para(43(a,1),568(a,1,2)),demod(472(5),6(14)),flip(a)].">
      <j1 rule="para" parents="43 568"/>
      <j2 rule="demod" parents="472 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="739">
    <literal><![CDATA[
      c_0 v (c_0 ' v c_0) = c_0
    ]]></literal>
    <justification jstring="[para(428(a,1),39(a,1,1,2,1,2)),demod(6(10),6(9),105(13),6(7)),flip(a)].">
      <j1 rule="para" parents="428 39"/>
      <j2 rule="demod" parents="6 6 105 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="751">
    <literal><![CDATA[
      c_0 ' ' = c_0 v c_0
    ]]></literal>
    <justification jstring="[para(739(a,1),44(a,1,2)),demod(6(11),581(15))].">
      <j1 rule="para" parents="739 44"/>
      <j2 rule="demod" parents="6 581"/>
    </justification>
  </clause>

  <clause id="798">
    <literal><![CDATA[
      (c_0 ' v (c_0 ' v ((c_0 ' v c_0) v c_0)) ') v c_0 = c_0 v c_0
    ]]></literal>
    <justification jstring="[back_demod(581),demod(751(18))].">
      <j1 rule="back_demod" parents="581"/>
      <j2 rule="demod" parents="751"/>
    </justification>
  </clause>

  <clause id="811">
    <literal><![CDATA[
      (c_0 v c_0) v (x v (c_0 ' v c_0)) = x v c_0
    ]]></literal>
    <justification jstring="[back_demod(568),demod(751(3))].">
      <j1 rule="back_demod" parents="568"/>
      <j2 rule="demod" parents="751"/>
    </justification>
  </clause>

  <clause id="827">
    <literal><![CDATA[
      (c_0 v c_0) v (x v c_0) = c_0 v (x v c_0)
    ]]></literal>
    <justification jstring="[back_demod(525),demod(751(3),751(13),6(14),334(13))].">
      <j1 rule="back_demod" parents="525"/>
      <j2 rule="demod" parents="751 751 6 334"/>
    </justification>
  </clause>

  <clause id="845">
    <literal><![CDATA[
      ((c_0 v (x v c_0)) ' v (x v c_0 ') ') ' = x
    ]]></literal>
    <justification jstring="[back_demod(494),demod(751(3),827(6))].">
      <j1 rule="back_demod" parents="494"/>
      <j2 rule="demod" parents="751 827"/>
    </justification>
  </clause>

  <clause id="865">
    <literal><![CDATA[
      ((c_0 v ((x v x ' ') ' v c_0)) ' v x) ' = (x v x ' ') '
    ]]></literal>
    <justification jstring="[back_demod(59),demod(751(7),6(8))].">
      <j1 rule="back_demod" parents="59"/>
      <j2 rule="demod" parents="751 6"/>
    </justification>
  </clause>

  <clause id="876">
    <literal><![CDATA[
      ((c_0 v (c_0 v c_0)) ' v c_0 ') ' = c_0
    ]]></literal>
    <justification jstring="[para(751(a,1),19(a,1,1,1,1,2))].">
      <j1 rule="para" parents="751 19"/>
    </justification>
  </clause>

  <clause id="1015">
    <literal><![CDATA[
      c_0 ' v (c_0 ' v c_0 ') ' = c_0
    ]]></literal>
    <justification jstring="[para(43(a,1),845(a,1,1,2,1)),demod(49(12),876(10)),flip(a)].">
      <j1 rule="para" parents="43 845"/>
      <j2 rule="demod" parents="49 876"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1020">
    <literal><![CDATA[
      (c_0 ' v x ') ' v (c_0 ' v x) ' = c_0
    ]]></literal>
    <justification jstring="[para(20(a,1),845(a,1,1,2,1)),demod(117(13),876(10)),flip(a)].">
      <j1 rule="para" parents="20 845"/>
      <j2 rule="demod" parents="117 876"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1047">
    <literal><![CDATA[
      c_0 ' v (x v (c_0 ' v c_0 ') ') = x v c_0
    ]]></literal>
    <justification jstring="[para(1015(a,1),6(a,1,2)),flip(a)].">
      <j1 rule="para" parents="1015 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1085">
    <literal><![CDATA[
      (c_0 ' v x ') ' v c_0 = (c_0 ' v x) v c_0
    ]]></literal>
    <justification jstring="[para(1020(a,1),17(a,1,2)),flip(a)].">
      <j1 rule="para" parents="1020 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1167">
    <literal><![CDATA[
      (c_0 ' v ((c_0 ' v c_0) v ((c_0 ' v c_0) v c_0)) ') v (c_0 ' v c_0 ') ' = c_0
    ]]></literal>
    <justification jstring="[para(1047(a,1),131(a,1,1,2,1,2)),demod(1085(16),6(15),1085(14))].">
      <j1 rule="para" parents="1047 131"/>
      <j2 rule="demod" parents="1085 6 1085"/>
    </justification>
  </clause>

  <clause id="1168">
    <literal><![CDATA[
      (c_0 ' v ((c_0 ' v c_0) v ((c_0 ' v c_0) v c_0)) ') v c_0 = c_0 ' v c_0
    ]]></literal>
    <justification jstring="[para(131(a,1),1047(a,1,2)),demod(1047(28),1085(20),6(19),1085(18)),flip(a)].">
      <j1 rule="para" parents="131 1047"/>
      <j2 rule="demod" parents="1047 1085 6 1085"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1197">
    <literal><![CDATA[
      (c_0 ' v c_0 ') ' = c_0 ' v c_0
    ]]></literal>
    <justification jstring="[para(1085(a,1),15(a,1,1,2,1,2)),demod(39(14)),flip(a)].">
      <j1 rule="para" parents="1085 15"/>
      <j2 rule="demod" parents="39"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1232">
    <literal><![CDATA[
      c_0 ' v (c_0 ' v c_0) = c_0
    ]]></literal>
    <justification jstring="[back_demod(1167),demod(1197(21),6(20),1168(19))].">
      <j1 rule="back_demod" parents="1167"/>
      <j2 rule="demod" parents="1197 6 1168"/>
    </justification>
  </clause>

  <clause id="1250">
    <literal><![CDATA[
      c_0 ' v (x v (c_0 ' v c_0)) = x v c_0
    ]]></literal>
    <justification jstring="[back_demod(1047),demod(1197(8))].">
      <j1 rule="back_demod" parents="1047"/>
      <j2 rule="demod" parents="1197"/>
    </justification>
  </clause>

  <clause id="1253">
    <literal><![CDATA[
      c_0 ' v c_0 = c_0 v c_0
    ]]></literal>
    <justification jstring="[para(1232(a,1),44(a,1,2)),demod(6(11),798(15)),flip(a)].">
      <j1 rule="para" parents="1232 44"/>
      <j2 rule="demod" parents="6 798"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1254">
    <literal><![CDATA[
      c_0 v (c_0 v c_0) = c_0
    ]]></literal>
    <justification jstring="[para(44(a,2),1232(a,1,2)),demod(6(11),43(10),1253(6),6(6),1253(5))].">
      <j1 rule="para" parents="44 1232"/>
      <j2 rule="demod" parents="6 43 1253 6 1253"/>
    </justification>
  </clause>

  <clause id="1255">
    <literal><![CDATA[
      c_0 v c_0 = c_0
    ]]></literal>
    <justification jstring="[para(1232(a,1),16(a,1,1,1,1,2,1,1)),demod(1253(6),18(8),751(4),1254(5),1197(6),1253(4),1253(9),18(11))].">
      <j1 rule="para" parents="1232 16"/>
      <j2 rule="demod" parents="1253 18 751 1254 1197 1253 1253 18"/>
    </justification>
  </clause>

  <clause id="1261">
    <literal><![CDATA[
      c_0 ' v (x v (y v c_0)) = x v (y v c_0)
    ]]></literal>
    <justification jstring="[para(1232(a,1),26(a,1,2,2)),demod(1253(9),1255(8)),flip(a)].">
      <j1 rule="para" parents="1232 26"/>
      <j2 rule="demod" parents="1253 1255"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1273">
    <literal><![CDATA[
      c_0 ' v (x v c_0) = x v c_0
    ]]></literal>
    <justification jstring="[back_demod(1250),demod(1253(6),1255(5))].">
      <j1 rule="back_demod" parents="1250"/>
      <j2 rule="demod" parents="1253 1255"/>
    </justification>
  </clause>

  <clause id="1294">
    <literal><![CDATA[
      c_0 v (x v c_0) = x v c_0
    ]]></literal>
    <justification jstring="[back_demod(811),demod(1255(3),1253(5),1255(4))].">
      <j1 rule="back_demod" parents="811"/>
      <j2 rule="demod" parents="1255 1253 1255"/>
    </justification>
  </clause>

  <clause id="1303">
    <literal><![CDATA[
      c_0 ' v c_0 = c_0
    ]]></literal>
    <justification jstring="[back_demod(1253),demod(1255(7))].">
      <j1 rule="back_demod" parents="1253"/>
      <j2 rule="demod" parents="1255"/>
    </justification>
  </clause>

  <clause id="1339">
    <literal><![CDATA[
      c_0 ' ' = c_0
    ]]></literal>
    <justification jstring="[back_demod(751),demod(1255(6))].">
      <j1 rule="back_demod" parents="751"/>
      <j2 rule="demod" parents="1255"/>
    </justification>
  </clause>

  <clause id="1344">
    <literal><![CDATA[
      (((x v x ' ') ' v c_0) ' v x) ' = (x v x ' ') '
    ]]></literal>
    <justification jstring="[back_demod(865),demod(1294(8))].">
      <j1 rule="back_demod" parents="865"/>
      <j2 rule="demod" parents="1294"/>
    </justification>
  </clause>

  <clause id="1368">
    <literal><![CDATA[
      (c_0 ' v (x v ((x v c_0) v ((x v c_0) v c_0))) ') ' = x v ((x v c_0) v c_0)
    ]]></literal>
    <justification jstring="[para(428(a,1),45(a,1,1,2,1,2,2)),demod(6(11),1261(12),6(10),6(20),1261(21))].">
      <j1 rule="para" parents="428 45"/>
      <j2 rule="demod" parents="6 1261 6 6 1261"/>
    </justification>
  </clause>

  <clause id="1370">
    <literal><![CDATA[
      x v ((x v c_0) v c_0) = x v c_0
    ]]></literal>
    <justification jstring="[para(1273(a,1),105(a,1,1,2,1,2,2)),demod(6(9),6(10),1368(13))].">
      <j1 rule="para" parents="1273 105"/>
      <j2 rule="demod" parents="6 6 1368"/>
    </justification>
  </clause>

  <clause id="1372">
    <literal><![CDATA[
      (c_0 ' v (x v c_0) ') ' = x v c_0
    ]]></literal>
    <justification jstring="[para(1273(a,1),39(a,1,1,2,1,2,1)),demod(6(9),1370(9),1273(7),1273(12))].">
      <j1 rule="para" parents="1273 39"/>
      <j2 rule="demod" parents="6 1370 1273 1273"/>
    </justification>
  </clause>

  <clause id="1373">
    <literal><![CDATA[
      x v ((c_0 ' v (x v c_0) ') v c_0) = c_0
    ]]></literal>
    <justification jstring="[para(1273(a,1),135(a,1,1,2,1,2,2)),demod(6(9),1370(9),6(7),1370(7),6(9))].">
      <j1 rule="para" parents="1273 135"/>
      <j2 rule="demod" parents="6 1370 6 1370 6"/>
    </justification>
  </clause>

  <clause id="1377">
    <literal><![CDATA[
      x v (y v ((x v c_0) v c_0)) = y v (x v c_0)
    ]]></literal>
    <justification jstring="[para(1370(a,1),6(a,1,2)),flip(a)].">
      <j1 rule="para" parents="1370 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1416">
    <literal><![CDATA[
      ((x v c_0) ' v c_0 ') ' = x v c_0
    ]]></literal>
    <justification jstring="[para(1372(a,1),22(a,1,1,1,1,2)),demod(6(5),1370(5),1303(7))].">
      <j1 rule="para" parents="1372 22"/>
      <j2 rule="demod" parents="6 1370 1303"/>
    </justification>
  </clause>

  <clause id="1429">
    <literal><![CDATA[
      ((x v (y v c_0)) ' v ((y v c_0) ' v (x v c_0 ')) ') ' = x
    ]]></literal>
    <justification jstring="[para(1416(a,1),13(a,1,1,1,1,2))].">
      <j1 rule="para" parents="1416 13"/>
    </justification>
  </clause>

  <clause id="1436">
    <literal><![CDATA[
      ((x v c_0) ' v ((x v c_0) v ((x v c_0) ' v c_0)) ') ' = x v c_0
    ]]></literal>
    <justification jstring="[para(1416(a,1),15(a,1,1,2,1,2,1)),demod(6(11),1416(21))].">
      <j1 rule="para" parents="1416 15"/>
      <j2 rule="demod" parents="6 1416"/>
    </justification>
  </clause>

  <clause id="1441">
    <literal><![CDATA[
      ((x v c_0) ' v ((x v c_0) ' v c_0) ') ' = c_0
    ]]></literal>
    <justification jstring="[para(1416(a,1),22(a,1,1,1,1,2)),demod(1294(4))].">
      <j1 rule="para" parents="1416 22"/>
      <j2 rule="demod" parents="1294"/>
    </justification>
  </clause>

  <clause id="1462">
    <literal><![CDATA[
      x v (y v ((c_0 ' v (x v c_0) ') v c_0)) = y v c_0
    ]]></literal>
    <justification jstring="[para(1373(a,1),6(a,1,2)),flip(a)].">
      <j1 rule="para" parents="1373 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1486">
    <literal><![CDATA[
      (x v c_0) v c_0 = c_0
    ]]></literal>
    <justification jstring="[para(1370(a,1),47(a,1,2,1,2,1,2,1)),demod(1377(10),6(7),1370(7),6(11),1462(12))].">
      <j1 rule="para" parents="1370 47"/>
      <j2 rule="demod" parents="1377 6 1370 6 1462"/>
    </justification>
  </clause>

  <clause id="1491">
    <literal><![CDATA[
      (x v c_0) v (y v c_0) = y v c_0
    ]]></literal>
    <justification jstring="[para(1486(a,1),6(a,1,2)),flip(a)].">
      <j1 rule="para" parents="1486 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1509">
    <literal><![CDATA[
      x v c_0 = c_0
    ]]></literal>
    <justification jstring="[back_demod(1436),demod(1491(11),1441(11)),flip(a)].">
      <j1 rule="back_demod" parents="1436"/>
      <j2 rule="demod" parents="1491 1441"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1525">
    <literal><![CDATA[
      (c_0 ' v (c_0 ' v (x v c_0 ')) ') ' = x
    ]]></literal>
    <justification jstring="[back_demod(1429),demod(1509(2),1509(2),1509(4))].">
      <j1 rule="back_demod" parents="1429"/>
      <j2 rule="demod" parents="1509 1509 1509"/>
    </justification>
  </clause>

  <clause id="1536">
    <literal><![CDATA[
      (x v x ' ') ' = (c_0 ' v x) '
    ]]></literal>
    <justification jstring="[back_demod(1344),demod(1509(6)),flip(a)].">
      <j1 rule="back_demod" parents="1344"/>
      <j2 rule="demod" parents="1509"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1594">
    <literal><![CDATA[
      ((x v x) ' v c_0 ') ' = x
    ]]></literal>
    <justification jstring="[back_demod(462),demod(1509(10))].">
      <j1 rule="back_demod" parents="462"/>
      <j2 rule="demod" parents="1509"/>
    </justification>
  </clause>

  <clause id="1619">
    <literal><![CDATA[
      ((x v (y v (c_0 ' v (x v x)) ') ') ' v c_0 ') ' = x
    ]]></literal>
    <justification jstring="[back_demod(329),demod(1509(11))].">
      <j1 rule="back_demod" parents="329"/>
      <j2 rule="demod" parents="1509"/>
    </justification>
  </clause>

  <clause id="1626">
    <literal><![CDATA[
      ((x v y) ' v ((y v (z v y ') ') ' v (x v c_0 ')) ') ' = x
    ]]></literal>
    <justification jstring="[back_demod(306),demod(1509(9))].">
      <j1 rule="back_demod" parents="306"/>
      <j2 rule="demod" parents="1509"/>
    </justification>
  </clause>

  <clause id="1627">
    <literal><![CDATA[
      ((x v (y v x ') ') ' v c_0 ') v (z v x) = c_0
    ]]></literal>
    <justification jstring="[back_demod(302),demod(1509(7),1509(12))].">
      <j1 rule="back_demod" parents="302"/>
      <j2 rule="demod" parents="1509 1509"/>
    </justification>
  </clause>

  <clause id="1631">
    <literal><![CDATA[
      (c_0 ' v (c_0 ' v x) ') ' = x
    ]]></literal>
    <justification jstring="[back_demod(251),demod(1536(4),1509(6),1339(3),1509(2),1536(6))].">
      <j1 rule="back_demod" parents="251"/>
      <j2 rule="demod" parents="1536 1509 1339 1509 1536"/>
    </justification>
  </clause>

  <clause id="1633">
    <literal><![CDATA[
      ((x v c_0 ') ' v (y v ((c_0 ' v (z v z) ') v (x v z))) ') ' = x
    ]]></literal>
    <justification jstring="[back_demod(228),demod(1509(2))].">
      <j1 rule="back_demod" parents="228"/>
      <j2 rule="demod" parents="1509"/>
    </justification>
  </clause>

  <clause id="1640">
    <literal><![CDATA[
      (x v (y v x ') ') ' = (x v c_0 ') '
    ]]></literal>
    <justification jstring="[back_demod(195),demod(1509(7),1509(2)),flip(a)].">
      <j1 rule="back_demod" parents="195"/>
      <j2 rule="demod" parents="1509 1509"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1641">
    <literal><![CDATA[
      ((c_0 ' v x) ' v c_0 ') ' = x
    ]]></literal>
    <justification jstring="[back_demod(190),demod(1536(4),1536(8),1509(10),1509(6))].">
      <j1 rule="back_demod" parents="190"/>
      <j2 rule="demod" parents="1536 1536 1509 1509"/>
    </justification>
  </clause>

  <clause id="1660">
    <literal><![CDATA[
      (c_0 ' v (x v x) ') v (y v x) = c_0
    ]]></literal>
    <justification jstring="[back_demod(44),demod(1509(9))].">
      <j1 rule="back_demod" parents="44"/>
      <j2 rule="demod" parents="1509"/>
    </justification>
  </clause>

  <clause id="1675">
    <literal><![CDATA[
      x v c_0 ' = x
    ]]></literal>
    <justification jstring="[back_demod(1525),demod(1631(11))].">
      <j1 rule="back_demod" parents="1525"/>
      <j2 rule="demod" parents="1631"/>
    </justification>
  </clause>

  <clause id="1683">
    <literal><![CDATA[
      x ' ' = x
    ]]></literal>
    <justification jstring="[back_demod(1633),demod(1675(3),1660(8),1509(3),1675(4))].">
      <j1 rule="back_demod" parents="1633"/>
      <j2 rule="demod" parents="1675 1660 1509 1675"/>
    </justification>
  </clause>

  <clause id="1684">
    <literal><![CDATA[
      x ' v (y v x) = c_0
    ]]></literal>
    <justification jstring="[back_demod(1627),demod(1640(5),1675(3),1675(4))].">
      <j1 rule="back_demod" parents="1627"/>
      <j2 rule="demod" parents="1640 1675 1675"/>
    </justification>
  </clause>

  <clause id="1685">
    <literal><![CDATA[
      ((x v y) ' v (y ' v x) ') ' = x
    ]]></literal>
    <justification jstring="[back_demod(1626),demod(1640(7),1675(5),1675(6))].">
      <j1 rule="back_demod" parents="1626"/>
      <j2 rule="demod" parents="1640 1675 1675"/>
    </justification>
  </clause>

  <clause id="1692">
    <literal><![CDATA[
      c_0 ' v x = x
    ]]></literal>
    <justification jstring="[back_demod(1641),demod(1675(7),1683(5))].">
      <j1 rule="back_demod" parents="1641"/>
      <j2 rule="demod" parents="1675 1683"/>
    </justification>
  </clause>

  <clause id="1695">
    <literal><![CDATA[
      x v (y v (x v x) ') ' = x
    ]]></literal>
    <justification jstring="[back_demod(1619),demod(1692(4),1675(9),1683(7))].">
      <j1 rule="back_demod" parents="1619"/>
      <j2 rule="demod" parents="1692 1675 1683"/>
    </justification>
  </clause>

  <clause id="1697">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[back_demod(1594),demod(1675(5),1683(3))].">
      <j1 rule="back_demod" parents="1594"/>
      <j2 rule="demod" parents="1675 1683"/>
    </justification>
  </clause>

  <clause id="1747">
    <literal><![CDATA[
      x v (y v x ') ' = x
    ]]></literal>
    <justification jstring="[back_demod(1695),demod(1697(1))].">
      <j1 rule="back_demod" parents="1695"/>
      <j2 rule="demod" parents="1697"/>
    </justification>
  </clause>

  <clause id="1757">
    <literal><![CDATA[
      x v (x v ((x v y ') ' v y)) ' = x v y '
    ]]></literal>
    <justification jstring="[para(15(a,1),1683(a,1,1)),demod(1683(4)),flip(a)].">
      <j1 rule="para" parents="15 1683"/>
      <j2 rule="demod" parents="1683"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1797">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[para(1675(a,1),6(a,1,2)),demod(1675(4))].">
      <j1 rule="para" parents="1675 6"/>
      <j2 rule="demod" parents="1675"/>
    </justification>
  </clause>

  <clause id="1812">
    <literal><![CDATA[
      x v (x v y) ' = x v y '
    ]]></literal>
    <justification jstring="[back_demod(1757),demod(1797(4),1747(4))].">
      <j1 rule="back_demod" parents="1757"/>
      <j2 rule="demod" parents="1797 1747"/>
    </justification>
  </clause>

  <clause id="1883">
    <literal><![CDATA[
      (x v y) v y ' = c_0
    ]]></literal>
    <justification jstring="[para(1684(a,1),1797(a,1)),flip(a)].">
      <j1 rule="para" parents="1684 1797"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1892">
    <literal><![CDATA[
      (x v (y v z)) v (x v z) ' = c_0
    ]]></literal>
    <justification jstring="[para(6(a,1),1883(a,1,1))].">
      <j1 rule="para" parents="6 1883"/>
    </justification>
  </clause>

  <clause id="1918">
    <literal><![CDATA[
      (x v y) v x ' = c_0
    ]]></literal>
    <justification jstring="[para(1797(a,1),1883(a,1,1))].">
      <j1 rule="para" parents="1797 1883"/>
    </justification>
  </clause>

  <clause id="1937">
    <literal><![CDATA[
      (x v y) v (z v x ') = c_0
    ]]></literal>
    <justification jstring="[para(1918(a,1),6(a,1,2)),demod(1509(2)),flip(a)].">
      <j1 rule="para" parents="1918 6"/>
      <j2 rule="demod" parents="1509"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1938">
    <literal><![CDATA[
      (x v (y v z)) v y ' = c_0
    ]]></literal>
    <justification jstring="[para(6(a,1),1918(a,1,1))].">
      <j1 rule="para" parents="6 1918"/>
    </justification>
  </clause>

  <clause id="1945">
    <literal><![CDATA[
      x v (y v (z v x ') ') = y v x
    ]]></literal>
    <justification jstring="[para(1747(a,1),6(a,1,2)),flip(a)].">
      <j1 rule="para" parents="1747 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1950">
    <literal><![CDATA[
      x ' v (y v x) ' = x '
    ]]></literal>
    <justification jstring="[para(1683(a,1),1747(a,1,2,1,2))].">
      <j1 rule="para" parents="1683 1747"/>
    </justification>
  </clause>

  <clause id="1951">
    <literal><![CDATA[
      x v (x ' v y) ' = x
    ]]></literal>
    <justification jstring="[para(1797(a,1),1747(a,1,2,1))].">
      <j1 rule="para" parents="1797 1747"/>
    </justification>
  </clause>

  <clause id="1989">
    <literal><![CDATA[
      x v (y v (x ' v z) ') = y v x
    ]]></literal>
    <justification jstring="[para(1951(a,1),6(a,1,2)),flip(a)].">
      <j1 rule="para" parents="1951 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2123">
    <literal><![CDATA[
      ((x v y) v z) v x ' = c_0
    ]]></literal>
    <justification jstring="[para(1797(a,1),1938(a,1,1))].">
      <j1 rule="para" parents="1797 1938"/>
    </justification>
  </clause>

  <clause id="2125">
    <literal><![CDATA[
      (x v y) ' v (x v (z v y)) ' = (x v y) '
    ]]></literal>
    <justification jstring="[para(6(a,1),1950(a,1,2,1))].">
      <j1 rule="para" parents="6 1950"/>
    </justification>
  </clause>

  <clause id="2132">
    <literal><![CDATA[
      ((x v y) v z) v y ' = c_0
    ]]></literal>
    <justification jstring="[para(1950(a,1),1937(a,1,2))].">
      <j1 rule="para" parents="1950 1937"/>
    </justification>
  </clause>

  <clause id="2460">
    <literal><![CDATA[
      x v (y v x) ' = x v y '
    ]]></literal>
    <justification jstring="[para(1797(a,1),1812(a,1,2,1))].">
      <j1 rule="para" parents="1797 1812"/>
    </justification>
  </clause>

  <clause id="2466">
    <literal><![CDATA[
      x v ((x v y) v z) = (x v y) v z
    ]]></literal>
    <justification jstring="[para(2123(a,1),1812(a,1,2,1)),demod(1675(5),1683(6),1797(5)),flip(a)].">
      <j1 rule="para" parents="2123 1812"/>
      <j2 rule="demod" parents="1675 1683 1797"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2467">
    <literal><![CDATA[
      x v ((y v x) v z) = (y v x) v z
    ]]></literal>
    <justification jstring="[para(2132(a,1),1812(a,1,2,1)),demod(1675(5),1683(6),1797(5)),flip(a)].">
      <j1 rule="para" parents="2132 1812"/>
      <j2 rule="demod" parents="1675 1683 1797"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2515">
    <literal><![CDATA[
      x v (y v (z v x) ') = y v (x v z ')
    ]]></literal>
    <justification jstring="[para(2460(a,1),6(a,1,2)),flip(a)].">
      <j1 rule="para" parents="2460 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2677">
    <literal><![CDATA[
      (x v (y v z)) v (x v y) ' = c_0
    ]]></literal>
    <justification jstring="[para(1797(a,1),1892(a,1,1,2))].">
      <j1 rule="para" parents="1797 1892"/>
    </justification>
  </clause>

  <clause id="3985">
    <literal><![CDATA[
      (x v y) v z = x v (y v z)
    ]]></literal>
    <justification jstring="[para(2677(a,1),1812(a,1,2,1)),demod(1675(5),1683(7),1797(6),6(6),6(5),2467(5),2466(5)),flip(a)].">
      <j1 rule="para" parents="2677 1812"/>
      <j2 rule="demod" parents="1675 1683 1797 6 6 2467 2466"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4412">
    <literal><![CDATA[
      x ' v ((y v x) ' v z) = x ' v z
    ]]></literal>
    <justification jstring="[para(1950(a,1),3985(a,1,1)),flip(a)].">
      <j1 rule="para" parents="1950 3985"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4505">
    <literal><![CDATA[
      (x v y) ' v (y ' v x) ' = x '
    ]]></literal>
    <justification jstring="[para(1685(a,1),1683(a,1,1)),flip(a)].">
      <j1 rule="para" parents="1685 1683"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4636">
    <literal><![CDATA[
      (x v y) ' v (z v (y ' v x) ') = z v x '
    ]]></literal>
    <justification jstring="[para(4505(a,1),6(a,1,2)),flip(a)].">
      <j1 rule="para" parents="4505 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4667">
    <literal><![CDATA[
      (x v (y v z) ') ' = (x v (z v y ')) ' v (x v z ') '
    ]]></literal>
    <justification jstring="[para(1945(a,1),4505(a,1,2,1)),demod(1683(2),1797(4),2515(4),1683(10)),flip(a)].">
      <j1 rule="para" parents="1945 4505"/>
      <j2 rule="demod" parents="1683 1797 2515 1683"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4676">
    <literal><![CDATA[
      (c3 v (c1 ' v c2 ')) ' v (c1 ' v c3 ') ' != (c1 ' v c2 ') ' v (c1 ' v c3 ') '
    ]]></literal>
    <attribute><![CDATA[
      answer(distributivity_1)
    ]]></attribute>
    <justification jstring="[back_demod(11),demod(4667(8),6(7))].">
      <j1 rule="back_demod" parents="11"/>
      <j2 rule="demod" parents="4667 6"/>
    </justification>
  </clause>

  <clause id="4926">
    <literal><![CDATA[
      (x v y) ' v ((x v (z v y)) ' v u) = (x v y) ' v u
    ]]></literal>
    <justification jstring="[para(6(a,1),4412(a,1,2,1,1))].">
      <j1 rule="para" parents="6 4412"/>
    </justification>
  </clause>

  <clause id="5240">
    <literal><![CDATA[
      (x v (y v z)) ' v (x v y ') ' = (x v z) ' v (x v y ') '
    ]]></literal>
    <justification jstring="[para(1951(a,1),2125(a,1,2,1,2)),demod(4667(5),1683(2),1797(10),4926(10),4667(11),1683(8)),flip(a)].">
      <j1 rule="para" parents="1951 2125"/>
      <j2 rule="demod" parents="4667 1683 1797 4926 4667 1683"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="5245">
    <literal><![CDATA[
      (x v (y v z) ') ' = (x v y ') ' v (x v z ') '
    ]]></literal>
    <justification jstring="[back_demod(4667),demod(5240(12))].">
      <j1 rule="back_demod" parents="4667"/>
      <j2 rule="demod" parents="5240"/>
    </justification>
  </clause>

  <clause id="5626">
    <literal><![CDATA[
      (x v (y v z)) ' v (y v x ') ' = (y v z) ' v (y v x ') '
    ]]></literal>
    <justification jstring="[para(4636(a,1),4505(a,1,2,1)),demod(1797(6),3985(6),1989(5),5245(12),1683(9))].">
      <j1 rule="para" parents="4636 4505"/>
      <j2 rule="demod" parents="1797 3985 1989 5245 1683"/>
    </justification>
  </clause>

  <clause id="5627">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(distributivity_1)
    ]]></attribute>
    <justification jstring="[resolve(5626,a,4676,a)].">
      <j1 rule="resolve" parents="5626 4676"/>
    </justification>
  </clause>

</proof>

</proofs>
