<?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>dn1.out</source>

<heading><![CDATA[
Prover9 (32) version 2009-02A, February 2009.
Process 11743 was started by mccune on cleo,
Wed Feb 25 09:39:21 2009
The command was "/home/mccune/bin/prover9 -f dn1.in".
]]></heading>

<proof number="1" length="120" max_count="36">

<comments><![CDATA[
% Proof 1 at 3.00 (+ 0.04) seconds: robbins_basis.
% Length of proof is 120.
% Level of proof is 37.
% Maximum clause weight is 36.
% Given clauses 301.

]]></comments>

  <clause id="1" type="goal">
    <literal><![CDATA[
      x v y = y v x & (y v x) v z = y v (x v z) & ((y v x)' v (y' v x)')' = x
    ]]></literal>
    <attribute><![CDATA[
      answer(robbins_basis)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="2" type="assumption">
    <literal><![CDATA[
      (((x v y)' v z)' v (x v (z' v (z v u)')')')' = z
    ]]></literal>
    <attribute><![CDATA[
      label(DN1)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="3" type="deny">
    <literal><![CDATA[
      c1 v c2 != c2 v c1
    ]]></literal>
    <literal><![CDATA[
      (c2 v c1) v c3 != c2 v (c1 v c3)
    ]]></literal>
    <literal><![CDATA[
      ((c2 v c1)' v (c2' v c1)')' != c1
    ]]></literal>
    <attribute><![CDATA[
      answer(robbins_basis)
    ]]></attribute>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      c2 v c1 != c1 v c2
    ]]></literal>
    <literal><![CDATA[
      (c2 v c1) v c3 != c2 v (c1 v c3)
    ]]></literal>
    <literal><![CDATA[
      ((c2 v c1)' v (c2' v c1)')' != c1
    ]]></literal>
    <attribute><![CDATA[
      answer(robbins_basis)
    ]]></attribute>
    <justification jstring="[copy(3),flip(a)].">
      <j1 rule="copy" parents="3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

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

  <clause id="18">
    <literal><![CDATA[
      ((x v x')' v x)' = x'
    ]]></literal>
    <justification jstring="[para(2(a,1),5(a,1,1,2))].">
      <j1 rule="para" parents="2 5"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      (x' v (x v (x' v (x v y)')')')' = x
    ]]></literal>
    <justification jstring="[para(18(a,1),2(a,1,1,1))].">
      <j1 rule="para" parents="18 2"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      ((x v y)' v (x' v (y' v (y v z)')')')' = y
    ]]></literal>
    <justification jstring="[para(22(a,1),2(a,1,1,1,1,1))].">
      <j1 rule="para" parents="22 2"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      (((x v y)' v z)' v (x v z)')' = z
    ]]></literal>
    <justification jstring="[para(22(a,1),2(a,1,1,2,1,2))].">
      <j1 rule="para" parents="22 2"/>
    </justification>
  </clause>

  <clause id="58">
    <literal><![CDATA[
      ((x v y)' v (x' v y)')' = y
    ]]></literal>
    <justification jstring="[para(22(a,1),31(a,1,1,1,1,1))].">
      <j1 rule="para" parents="22 31"/>
    </justification>
  </clause>

  <clause id="64">
    <literal><![CDATA[
      (x v ((y v z)' v (y v x)')')' = (y v x)'
    ]]></literal>
    <justification jstring="[para(31(a,1),31(a,1,1,1))].">
      <j1 rule="para" parents="31 31"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      (((((x v y)' v z)' v u)' v (x v z)')' v z)' = (x v z)'
    ]]></literal>
    <justification jstring="[para(31(a,1),31(a,1,1,2))].">
      <j1 rule="para" parents="31 31"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      c2 v c1 != c1 v c2
    ]]></literal>
    <literal><![CDATA[
      (c2 v c1) v c3 != c2 v (c1 v c3)
    ]]></literal>
    <attribute><![CDATA[
      answer(robbins_basis)
    ]]></attribute>
    <justification jstring="[back_rewrite(4),rewrite([58(29)]),xx(c)].">
      <j1 rule="back_rewrite" parents="4"/>
      <j2 rule="rewrite" parents="58"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="94">
    <literal><![CDATA[
      ((((x v (x v y)')' v z)' v x)' v (x v y)')' = x
    ]]></literal>
    <justification jstring="[para(58(a,1),2(a,1,1,2))].">
      <j1 rule="para" parents="58 2"/>
    </justification>
  </clause>

  <clause id="101">
    <literal><![CDATA[
      (((x v x') v x)' v x')' = x
    ]]></literal>
    <justification jstring="[para(18(a,1),58(a,1,1,2))].">
      <j1 rule="para" parents="18 58"/>
    </justification>
  </clause>

  <clause id="111">
    <literal><![CDATA[
      ((x v y)' v ((z v x)' v y)')' = y
    ]]></literal>
    <justification jstring="[para(58(a,1),31(a,1,1,1,1,1))].">
      <j1 rule="para" parents="58 31"/>
    </justification>
  </clause>

  <clause id="112">
    <literal><![CDATA[
      (x v (y v (y' v x)')')' = (y' v x)'
    ]]></literal>
    <justification jstring="[para(58(a,1),31(a,1,1,1))].">
      <j1 rule="para" parents="58 31"/>
    </justification>
  </clause>

  <clause id="138">
    <literal><![CDATA[
      (x v ((x v x') v x')')' = x'
    ]]></literal>
    <justification jstring="[para(101(a,1),31(a,1,1,1))].">
      <j1 rule="para" parents="101 31"/>
    </justification>
  </clause>

  <clause id="156">
    <literal><![CDATA[
      (x' v x')' = x
    ]]></literal>
    <justification jstring="[para(138(a,1),22(a,1,1,2,1,2,1,2)),rewrite([112(9)])].">
      <j1 rule="para" parents="138 22"/>
      <j2 rule="rewrite" parents="112"/>
    </justification>
  </clause>

  <clause id="160">
    <literal><![CDATA[
      ((x' v y)' v (x v y)')' = y
    ]]></literal>
    <justification jstring="[para(138(a,1),31(a,1,1,1,1,1))].">
      <j1 rule="para" parents="138 31"/>
    </justification>
  </clause>

  <clause id="176">
    <literal><![CDATA[
      ((x v y)' v z)' v (x v (z' v (z v u)')')' = (z v z)'
    ]]></literal>
    <justification jstring="[para(2(a,1),156(a,1,1,1)),rewrite([2(13)]),flip(a)].">
      <j1 rule="para" parents="2 156"/>
      <j2 rule="rewrite" parents="2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="179">
    <literal><![CDATA[
      (x v x')' v x = x
    ]]></literal>
    <justification jstring="[para(18(a,1),156(a,1,1,1)),rewrite([18(6),156(4)]),flip(a)].">
      <j1 rule="para" parents="18 156"/>
      <j2 rule="rewrite" parents="18 156"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="186">
    <literal><![CDATA[
      (((x' v y)' v x')' v x)' = x'
    ]]></literal>
    <justification jstring="[para(156(a,1),31(a,1,1,2))].">
      <j1 rule="para" parents="156 31"/>
    </justification>
  </clause>

  <clause id="187">
    <literal><![CDATA[
      (x v x)' = ((y v z)' v x)' v (y v x)'
    ]]></literal>
    <justification jstring="[para(31(a,1),156(a,1,1,1)),rewrite([31(8)])].">
      <j1 rule="para" parents="31 156"/>
      <j2 rule="rewrite" parents="31"/>
    </justification>
  </clause>

  <clause id="193">
    <literal><![CDATA[
      (x v x)' = (y v x)' v (y' v x)'
    ]]></literal>
    <justification jstring="[para(58(a,1),156(a,1,1,1)),rewrite([58(7)])].">
      <j1 rule="para" parents="58 156"/>
      <j2 rule="rewrite" parents="58"/>
    </justification>
  </clause>

  <clause id="198">
    <literal><![CDATA[
      (x v x)' = x' v x'
    ]]></literal>
    <justification jstring="[para(156(a,1),156(a,1,1,1)),rewrite([156(4)])].">
      <j1 rule="para" parents="156 156"/>
      <j2 rule="rewrite" parents="156"/>
    </justification>
  </clause>

  <clause id="203">
    <literal><![CDATA[
      x'' v x'' = x
    ]]></literal>
    <justification jstring="[back_rewrite(2),rewrite([176(12),198(2),198(4)])].">
      <j1 rule="back_rewrite" parents="2"/>
      <j2 rule="rewrite" parents="176 198 198"/>
    </justification>
  </clause>

  <clause id="212">
    <literal><![CDATA[
      (x v y)' v (x' v y)' = y' v y'
    ]]></literal>
    <justification jstring="[back_rewrite(193),rewrite([198(2)]),flip(a)].">
      <j1 rule="back_rewrite" parents="193"/>
      <j2 rule="rewrite" parents="198"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="213">
    <literal><![CDATA[
      ((x v y)' v z)' v (x v z)' = z' v z'
    ]]></literal>
    <justification jstring="[back_rewrite(187),rewrite([198(2)]),flip(a)].">
      <j1 rule="back_rewrite" parents="187"/>
      <j2 rule="rewrite" parents="198"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="247">
    <literal><![CDATA[
      (x' v y)' v (x v y)' = y' v y'
    ]]></literal>
    <justification jstring="[para(160(a,1),203(a,1,1,1)),rewrite([160(8)]),flip(a)].">
      <j1 rule="para" parents="160 203"/>
      <j2 rule="rewrite" parents="160"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="253">
    <literal><![CDATA[
      (x v (x v x')')' = x'
    ]]></literal>
    <justification jstring="[para(198(a,1),160(a,1,1,1)),rewrite([203(5)])].">
      <j1 rule="para" parents="198 160"/>
      <j2 rule="rewrite" parents="203"/>
    </justification>
  </clause>

  <clause id="281">
    <literal><![CDATA[
      x v (x v x')' = x
    ]]></literal>
    <justification jstring="[para(253(a,1),203(a,1,1,1)),rewrite([253(7),203(5)]),flip(a)].">
      <j1 rule="para" parents="253 203"/>
      <j2 rule="rewrite" parents="253 203"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="366">
    <literal><![CDATA[
      ((x' v x)' v x')' = x
    ]]></literal>
    <justification jstring="[para(179(a,1),111(a,1,1,2,1))].">
      <j1 rule="para" parents="179 111"/>
    </justification>
  </clause>

  <clause id="367">
    <literal><![CDATA[
      (x v y)' v ((z v x)' v y)' = y' v y'
    ]]></literal>
    <justification jstring="[para(111(a,1),203(a,1,1,1)),rewrite([111(9)]),flip(a)].">
      <j1 rule="para" parents="111 203"/>
      <j2 rule="rewrite" parents="111"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="394">
    <literal><![CDATA[
      (x' v x)' v x' = x' v x'
    ]]></literal>
    <justification jstring="[para(366(a,1),203(a,1,1,1)),rewrite([366(7)]),flip(a)].">
      <j1 rule="para" parents="366 203"/>
      <j2 rule="rewrite" parents="366"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="400">
    <literal><![CDATA[
      (x v y)' v (x' v (y' v (y v z)')')' = y' v y'
    ]]></literal>
    <justification jstring="[para(27(a,1),203(a,1,1,1)),rewrite([27(13)]),flip(a)].">
      <j1 rule="para" parents="27 203"/>
      <j2 rule="rewrite" parents="27"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="405">
    <literal><![CDATA[
      ((x v (y v y))' v (x' v ((y' v y') v ((y v y) v z)')')')' = y v y
    ]]></literal>
    <justification jstring="[para(198(a,1),27(a,1,1,2,1,2,1,1))].">
      <j1 rule="para" parents="198 27"/>
    </justification>
  </clause>

  <clause id="415">
    <literal><![CDATA[
      (x v (x' v x'))' v x = x v x
    ]]></literal>
    <justification jstring="[para(27(a,1),394(a,1,1,1,1)),rewrite([400(11),400(16),198(9),203(10),400(17),198(10),203(11),400(17),198(10),203(11)])].">
      <j1 rule="para" parents="27 394"/>
      <j2 rule="rewrite" parents="400 400 198 203 400 198 203 400 198 203"/>
    </justification>
  </clause>

  <clause id="430">
    <literal><![CDATA[
      ((x' v y)' v x')' v x = x
    ]]></literal>
    <justification jstring="[para(186(a,1),203(a,1,1,1)),rewrite([186(10),203(5)]),flip(a)].">
      <j1 rule="para" parents="186 203"/>
      <j2 rule="rewrite" parents="186 203"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="432">
    <literal><![CDATA[
      x' v x' = x'
    ]]></literal>
    <justification jstring="[para(394(a,1),186(a,1,1,1,1)),rewrite([198(4),203(5),198(2)])].">
      <j1 rule="para" parents="394 186"/>
      <j2 rule="rewrite" parents="198 203 198"/>
    </justification>
  </clause>

  <clause id="433">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(186(a,1),415(a,1,1,1,2,1)),rewrite([430(7),430(8),432(3),430(10),179(4),430(7),430(7)]),flip(a)].">
      <j1 rule="para" parents="186 415"/>
      <j2 rule="rewrite" parents="430 430 432 430 179 430 430"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="436">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[back_rewrite(405),rewrite([433(1),433(6),433(5),400(11),433(3),433(3)])].">
      <j1 rule="back_rewrite" parents="405"/>
      <j2 rule="rewrite" parents="433 433 433 400 433 433"/>
    </justification>
  </clause>

  <clause id="440">
    <literal><![CDATA[
      (x v y)' v ((z v x)' v y)' = y'
    ]]></literal>
    <justification jstring="[back_rewrite(367),rewrite([433(10)])].">
      <j1 rule="back_rewrite" parents="367"/>
      <j2 rule="rewrite" parents="433"/>
    </justification>
  </clause>

  <clause id="457">
    <literal><![CDATA[
      (x' v y)' v (x v y)' = y'
    ]]></literal>
    <justification jstring="[back_rewrite(247),rewrite([433(9)])].">
      <j1 rule="back_rewrite" parents="247"/>
      <j2 rule="rewrite" parents="433"/>
    </justification>
  </clause>

  <clause id="463">
    <literal><![CDATA[
      ((x v y)' v z)' v (x v z)' = z'
    ]]></literal>
    <justification jstring="[back_rewrite(213),rewrite([433(10)])].">
      <j1 rule="back_rewrite" parents="213"/>
      <j2 rule="rewrite" parents="433"/>
    </justification>
  </clause>

  <clause id="464">
    <literal><![CDATA[
      (x v y)' v (x' v y)' = y'
    ]]></literal>
    <justification jstring="[back_rewrite(212),rewrite([433(9)])].">
      <j1 rule="back_rewrite" parents="212"/>
      <j2 rule="rewrite" parents="433"/>
    </justification>
  </clause>

  <clause id="637">
    <literal><![CDATA[
      x v ((y' v x) v (y v x)')' = y v x
    ]]></literal>
    <justification jstring="[para(457(a,1),457(a,1,1,1)),rewrite([436(2),436(10)])].">
      <j1 rule="para" parents="457 457"/>
      <j2 rule="rewrite" parents="436 436"/>
    </justification>
  </clause>

  <clause id="643">
    <literal><![CDATA[
      x v ((y v x) v (y' v x)')' = y' v x
    ]]></literal>
    <justification jstring="[para(464(a,1),457(a,1,1,1)),rewrite([436(2),436(11)])].">
      <j1 rule="para" parents="464 457"/>
      <j2 rule="rewrite" parents="436 436"/>
    </justification>
  </clause>

  <clause id="742">
    <literal><![CDATA[
      x v ((y v (z v x)')' v ((u v z)' v x)')' = (u v z)' v x
    ]]></literal>
    <justification jstring="[para(440(a,1),440(a,1,1,1)),rewrite([436(2),436(16)])].">
      <j1 rule="para" parents="440 440"/>
      <j2 rule="rewrite" parents="436 436"/>
    </justification>
  </clause>

  <clause id="755">
    <literal><![CDATA[
      (x v (y v x))' = (y v x)'
    ]]></literal>
    <justification jstring="[para(433(a,1),64(a,1,1,2,1)),rewrite([436(3)])].">
      <j1 rule="para" parents="433 64"/>
      <j2 rule="rewrite" parents="436"/>
    </justification>
  </clause>

  <clause id="756">
    <literal><![CDATA[
      x v ((y v z)' v (y v x)')' = y v x
    ]]></literal>
    <justification jstring="[para(64(a,1),436(a,1,1)),rewrite([436(3)]),flip(a)].">
      <j1 rule="para" parents="64 436"/>
      <j2 rule="rewrite" parents="436"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="796">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(755(a,1),436(a,1,1)),rewrite([436(3)]),flip(a)].">
      <j1 rule="para" parents="755 436"/>
      <j2 rule="rewrite" parents="436"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="807">
    <literal><![CDATA[
      (x v y)' v y' = y'
    ]]></literal>
    <justification jstring="[para(457(a,1),796(a,1,2)),rewrite([457(10)])].">
      <j1 rule="para" parents="457 796"/>
      <j2 rule="rewrite" parents="457"/>
    </justification>
  </clause>

  <clause id="808">
    <literal><![CDATA[
      (x v y')' v y = y
    ]]></literal>
    <justification jstring="[para(436(a,1),807(a,1,2)),rewrite([436(6)])].">
      <j1 rule="para" parents="436 807"/>
      <j2 rule="rewrite" parents="436"/>
    </justification>
  </clause>

  <clause id="819">
    <literal><![CDATA[
      (((((x v y)' v (x v x')')' v z)' v x')' v (x v x')')' = x'
    ]]></literal>
    <justification jstring="[para(281(a,1),65(a,1,1,1,1,2,1)),rewrite([281(21)])].">
      <j1 rule="para" parents="281 65"/>
      <j2 rule="rewrite" parents="281"/>
    </justification>
  </clause>

  <clause id="842">
    <literal><![CDATA[
      ((x v y) v y)' = (x v y)'
    ]]></literal>
    <justification jstring="[para(807(a,1),65(a,1,1,1,1)),rewrite([436(3)])].">
      <j1 rule="para" parents="807 65"/>
      <j2 rule="rewrite" parents="436"/>
    </justification>
  </clause>

  <clause id="860">
    <literal><![CDATA[
      (x v y) v y = x v y
    ]]></literal>
    <justification jstring="[para(842(a,1),436(a,1,1)),rewrite([436(3)]),flip(a)].">
      <j1 rule="para" parents="842 436"/>
      <j2 rule="rewrite" parents="436"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="864">
    <literal><![CDATA[
      x' v (y v x)' = x'
    ]]></literal>
    <justification jstring="[para(457(a,1),860(a,1,1)),rewrite([457(10)])].">
      <j1 rule="para" parents="457 860"/>
      <j2 rule="rewrite" parents="457"/>
    </justification>
  </clause>

  <clause id="865">
    <literal><![CDATA[
      x v (y v x')' = x
    ]]></literal>
    <justification jstring="[para(436(a,1),864(a,1,1)),rewrite([436(6)])].">
      <j1 rule="para" parents="436 864"/>
      <j2 rule="rewrite" parents="436"/>
    </justification>
  </clause>

  <clause id="866">
    <literal><![CDATA[
      x v (x v (y v x)')' = y v x
    ]]></literal>
    <justification jstring="[para(864(a,1),457(a,1,1,1)),rewrite([436(2),436(8)])].">
      <j1 rule="para" parents="864 457"/>
      <j2 rule="rewrite" parents="436 436"/>
    </justification>
  </clause>

  <clause id="911">
    <literal><![CDATA[
      x v (y' v (y v x)')' = y v x
    ]]></literal>
    <justification jstring="[para(457(a,1),463(a,1,1,1)),rewrite([436(2),436(9)])].">
      <j1 rule="para" parents="457 463"/>
      <j2 rule="rewrite" parents="436 436"/>
    </justification>
  </clause>

  <clause id="946">
    <literal><![CDATA[
      (x v y')' v (y v (x' v (x v y')')')' = x' v (x v y')'
    ]]></literal>
    <justification jstring="[para(911(a,1),457(a,1,1,1)),rewrite([436(19)])].">
      <j1 rule="para" parents="911 457"/>
      <j2 rule="rewrite" parents="436"/>
    </justification>
  </clause>

  <clause id="948">
    <literal><![CDATA[
      (x v y)' v (y' v (x' v (x v y)')')' = x' v (x v y)'
    ]]></literal>
    <justification jstring="[para(911(a,1),464(a,1,1,1)),rewrite([436(17)])].">
      <j1 rule="para" parents="911 464"/>
      <j2 rule="rewrite" parents="436"/>
    </justification>
  </clause>

  <clause id="950">
    <literal><![CDATA[
      (x v y)' v ((z v y)' v (x' v (x v y)')')' = x' v (x v y)'
    ]]></literal>
    <justification jstring="[para(911(a,1),440(a,1,1,1)),rewrite([436(18)])].">
      <j1 rule="para" parents="911 440"/>
      <j2 rule="rewrite" parents="436"/>
    </justification>
  </clause>

  <clause id="952">
    <literal><![CDATA[
      (x v (y v z)')' v (y v (x' v (x v (y v z)')')')' = x' v (x v (y v z)')'
    ]]></literal>
    <justification jstring="[para(911(a,1),463(a,1,1,1)),rewrite([436(22)])].">
      <j1 rule="para" parents="911 463"/>
      <j2 rule="rewrite" parents="436"/>
    </justification>
  </clause>

  <clause id="954">
    <literal><![CDATA[
      (x v y)' v ((x v z)' v y)' = y'
    ]]></literal>
    <justification jstring="[para(463(a,1),911(a,1,2,1,2,1)),rewrite([436(7),436(7),860(6),463(14)])].">
      <j1 rule="para" parents="463 911"/>
      <j2 rule="rewrite" parents="436 436 860 463"/>
    </justification>
  </clause>

  <clause id="999">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(94(a,1),807(a,1,1)),rewrite([436(3),436(5)])].">
      <j1 rule="para" parents="94 807"/>
      <j2 rule="rewrite" parents="436 436"/>
    </justification>
  </clause>

  <clause id="1001">
    <literal><![CDATA[
      (x' v (x v y)')' = x
    ]]></literal>
    <justification jstring="[para(808(a,1),94(a,1,1,1,1))].">
      <j1 rule="para" parents="808 94"/>
    </justification>
  </clause>

  <clause id="1059">
    <literal><![CDATA[
      (x v (y v z)')' v (y v x)' = x' v (x v (y v z)')'
    ]]></literal>
    <justification jstring="[back_rewrite(952),rewrite([1001(11)])].">
      <j1 rule="back_rewrite" parents="952"/>
      <j2 rule="rewrite" parents="1001"/>
    </justification>
  </clause>

  <clause id="1061">
    <literal><![CDATA[
      (x v y)' v ((z v y)' v x)' = x' v (x v y)'
    ]]></literal>
    <justification jstring="[back_rewrite(950),rewrite([1001(9)])].">
      <j1 rule="back_rewrite" parents="950"/>
      <j2 rule="rewrite" parents="1001"/>
    </justification>
  </clause>

  <clause id="1063">
    <literal><![CDATA[
      (x v y)' v (y' v x)' = x' v (x v y)'
    ]]></literal>
    <justification jstring="[back_rewrite(948),rewrite([1001(8)])].">
      <j1 rule="back_rewrite" parents="948"/>
      <j2 rule="rewrite" parents="1001"/>
    </justification>
  </clause>

  <clause id="1065">
    <literal><![CDATA[
      (x v y')' v (y v x)' = x' v (x v y')'
    ]]></literal>
    <justification jstring="[back_rewrite(946),rewrite([1001(9)])].">
      <j1 rule="back_rewrite" parents="946"/>
      <j2 rule="rewrite" parents="1001"/>
    </justification>
  </clause>

  <clause id="1069">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[back_rewrite(911),rewrite([1001(5)])].">
      <j1 rule="back_rewrite" parents="911"/>
      <j2 rule="rewrite" parents="1001"/>
    </justification>
  </clause>

  <clause id="1128">
    <literal><![CDATA[
      ((x v x')' v (x' v (((x v y)' v (x v x')')' v z)')')' = x'
    ]]></literal>
    <justification jstring="[back_rewrite(819),rewrite([1069(11),1069(16)])].">
      <j1 rule="back_rewrite" parents="819"/>
      <j2 rule="rewrite" parents="1069 1069"/>
    </justification>
  </clause>

  <clause id="1138">
    <literal><![CDATA[
      c3 v (c1 v c2) != c2 v (c1 v c3)
    ]]></literal>
    <attribute><![CDATA[
      answer(robbins_basis)
    ]]></attribute>
    <justification jstring="[back_rewrite(66),rewrite([1069(3),1069(10),1069(12)]),xx(a)].">
      <j1 rule="back_rewrite" parents="66"/>
      <j2 rule="rewrite" parents="1069 1069 1069"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="1139">
    <literal><![CDATA[
      x' v (x v y)' = x'
    ]]></literal>
    <justification jstring="[para(1069(a,1),464(a,1,1,1)),rewrite([1063(6)])].">
      <j1 rule="para" parents="1069 464"/>
      <j2 rule="rewrite" parents="1063"/>
    </justification>
  </clause>

  <clause id="1140">
    <literal><![CDATA[
      x v (x' v y)' = x
    ]]></literal>
    <justification jstring="[para(1069(a,1),865(a,1,2,1))].">
      <j1 rule="para" parents="1069 865"/>
    </justification>
  </clause>

  <clause id="1141">
    <literal><![CDATA[
      (x v y')' v (y v x)' = x'
    ]]></literal>
    <justification jstring="[back_rewrite(1065),rewrite([1139(11)])].">
      <j1 rule="back_rewrite" parents="1065"/>
      <j2 rule="rewrite" parents="1139"/>
    </justification>
  </clause>

  <clause id="1143">
    <literal><![CDATA[
      (x v y)' v (y' v x)' = x'
    ]]></literal>
    <justification jstring="[back_rewrite(1063),rewrite([1139(10)])].">
      <j1 rule="back_rewrite" parents="1063"/>
      <j2 rule="rewrite" parents="1139"/>
    </justification>
  </clause>

  <clause id="1145">
    <literal><![CDATA[
      (x v y)' v ((z v y)' v x)' = x'
    ]]></literal>
    <justification jstring="[back_rewrite(1061),rewrite([1139(11)])].">
      <j1 rule="back_rewrite" parents="1061"/>
      <j2 rule="rewrite" parents="1139"/>
    </justification>
  </clause>

  <clause id="1147">
    <literal><![CDATA[
      (x v (y v z)')' v (y v x)' = x'
    ]]></literal>
    <justification jstring="[back_rewrite(1059),rewrite([1139(13)])].">
      <j1 rule="back_rewrite" parents="1059"/>
      <j2 rule="rewrite" parents="1139"/>
    </justification>
  </clause>

  <clause id="1166">
    <literal><![CDATA[
      x v ((y v x)' v (z v (x v y')')')' = y v x
    ]]></literal>
    <justification jstring="[para(1141(a,1),440(a,1,1,1)),rewrite([436(2),1069(8),436(13)])].">
      <j1 rule="para" parents="1141 440"/>
      <j2 rule="rewrite" parents="436 1069 436"/>
    </justification>
  </clause>

  <clause id="1167">
    <literal><![CDATA[
      (x v y)' v (x v y')' = x'
    ]]></literal>
    <justification jstring="[para(1069(a,1),1141(a,1,2,1)),rewrite([1069(6)])].">
      <j1 rule="para" parents="1069 1141"/>
      <j2 rule="rewrite" parents="1069"/>
    </justification>
  </clause>

  <clause id="1183">
    <literal><![CDATA[
      ((x v y) v (x' v y)')' v ((x' v y)' v (y' v ((x v y) v (x' v y)')'))' = x' v y
    ]]></literal>
    <justification jstring="[para(643(a,1),637(a,1,2,1,2,1)),rewrite([1069(18),643(27)])].">
      <j1 rule="para" parents="643 637"/>
      <j2 rule="rewrite" parents="1069 643"/>
    </justification>
  </clause>

  <clause id="1192">
    <literal><![CDATA[
      x v ((x v y')' v (z v (x v y)')')' = x v y'
    ]]></literal>
    <justification jstring="[para(1167(a,1),440(a,1,1,1)),rewrite([436(2),1069(8),436(14)])].">
      <j1 rule="para" parents="1167 440"/>
      <j2 rule="rewrite" parents="436 1069 436"/>
    </justification>
  </clause>

  <clause id="1199">
    <literal><![CDATA[
      (x v y)' v ((y v z)' v x)' = x'
    ]]></literal>
    <justification jstring="[para(1069(a,1),954(a,1,1,1))].">
      <j1 rule="para" parents="1069 954"/>
    </justification>
  </clause>

  <clause id="1410">
    <literal><![CDATA[
      x v (y v (y v x)')' = x
    ]]></literal>
    <justification jstring="[para(1199(a,1),742(a,1,2,1)),rewrite([436(2),433(1),1069(3),1069(5)]),flip(a)].">
      <j1 rule="para" parents="1199 742"/>
      <j2 rule="rewrite" parents="436 433 1069 1069"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1422">
    <literal><![CDATA[
      x v (y v (x v y)')' = x
    ]]></literal>
    <justification jstring="[para(1069(a,1),1410(a,1,2,1,2,1))].">
      <j1 rule="para" parents="1069 1410"/>
    </justification>
  </clause>

  <clause id="1423">
    <literal><![CDATA[
      (x v x')' v (x' v y)' = (x' v y)'
    ]]></literal>
    <justification jstring="[para(1140(a,1),1410(a,1,2,1,2,1)),rewrite([1069(7)])].">
      <j1 rule="para" parents="1140 1410"/>
      <j2 rule="rewrite" parents="1069"/>
    </justification>
  </clause>

  <clause id="1424">
    <literal><![CDATA[
      (x v y)' v (x v x')' = (x v y)'
    ]]></literal>
    <justification jstring="[para(1139(a,1),1410(a,1,2,1,2,1)),rewrite([436(5),1069(4)])].">
      <j1 rule="para" parents="1139 1410"/>
      <j2 rule="rewrite" parents="436 1069"/>
    </justification>
  </clause>

  <clause id="1458">
    <literal><![CDATA[
      x' v ((x v y) v z)' = x'
    ]]></literal>
    <justification jstring="[back_rewrite(1128),rewrite([1424(10),436(7),1423(10),436(7)])].">
      <j1 rule="back_rewrite" parents="1128"/>
      <j2 rule="rewrite" parents="1424 436 1423 436"/>
    </justification>
  </clause>

  <clause id="1537">
    <literal><![CDATA[
      x v ((y v x)' v ((y v z)' v (y v x)')')' = x
    ]]></literal>
    <justification jstring="[para(756(a,1),1422(a,1,2,1,2,1)),rewrite([1069(9)])].">
      <j1 rule="para" parents="756 1422"/>
      <j2 rule="rewrite" parents="1069"/>
    </justification>
  </clause>

  <clause id="1538">
    <literal><![CDATA[
      x v ((x' v y) v z)' = x
    ]]></literal>
    <justification jstring="[para(436(a,1),1458(a,1,1)),rewrite([436(7)])].">
      <j1 rule="para" parents="436 1458"/>
      <j2 rule="rewrite" parents="436"/>
    </justification>
  </clause>

  <clause id="1544">
    <literal><![CDATA[
      x' v ((y v x) v z)' = x'
    ]]></literal>
    <justification jstring="[para(796(a,1),1458(a,1,2,1,1))].">
      <j1 rule="para" parents="796 1458"/>
    </justification>
  </clause>

  <clause id="1548">
    <literal><![CDATA[
      (x v y) v x' = x v x'
    ]]></literal>
    <justification jstring="[para(1458(a,1),866(a,1,2,1)),rewrite([436(3),1069(2)]),flip(a)].">
      <j1 rule="para" parents="1458 866"/>
      <j2 rule="rewrite" parents="436 1069"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1550">
    <literal><![CDATA[
      (x v y) v (x' v z)' = x v y
    ]]></literal>
    <justification jstring="[para(1143(a,1),1458(a,1,2,1,1)),rewrite([436(3),436(8)])].">
      <j1 rule="para" parents="1143 1458"/>
      <j2 rule="rewrite" parents="436 436"/>
    </justification>
  </clause>

  <clause id="1576">
    <literal><![CDATA[
      x v (y v x)' = y' v x
    ]]></literal>
    <justification jstring="[back_rewrite(1183),rewrite([1550(5),1550(11),864(9),1069(7),864(7),436(4),1069(3)])].">
      <j1 rule="back_rewrite" parents="1183"/>
      <j2 rule="rewrite" parents="1550 1550 864 1069 864 436 1069"/>
    </justification>
  </clause>

  <clause id="1592">
    <literal><![CDATA[
      x v ((y v z) v (y v x)')' = x
    ]]></literal>
    <justification jstring="[back_rewrite(1537),rewrite([1576(9),436(3)])].">
      <j1 rule="back_rewrite" parents="1537"/>
      <j2 rule="rewrite" parents="1576 436"/>
    </justification>
  </clause>

  <clause id="1626">
    <literal><![CDATA[
      x v ((y v x') v z)' = x
    ]]></literal>
    <justification jstring="[para(796(a,1),1538(a,1,2,1,1))].">
      <j1 rule="para" parents="796 1538"/>
    </justification>
  </clause>

  <clause id="1634">
    <literal><![CDATA[
      (x v y) v (x v y)' = y' v (x v y)
    ]]></literal>
    <justification jstring="[para(464(a,1),1548(a,1,1)),rewrite([436(4),436(8),1069(7)]),flip(a)].">
      <j1 rule="para" parents="464 1548"/>
      <j2 rule="rewrite" parents="436 436 1069"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1635">
    <literal><![CDATA[
      (x v y) v y' = y v y'
    ]]></literal>
    <justification jstring="[para(796(a,1),1548(a,1,1))].">
      <j1 rule="para" parents="796 1548"/>
    </justification>
  </clause>

  <clause id="1636">
    <literal><![CDATA[
      x' v (x v y) = x v x'
    ]]></literal>
    <justification jstring="[para(1548(a,1),1069(a,1)),flip(a)].">
      <j1 rule="para" parents="1548 1069"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1647">
    <literal><![CDATA[
      (x v y) v (z v (x v y)') = z v z'
    ]]></literal>
    <justification jstring="[para(1147(a,1),1548(a,1,1)),rewrite([436(6),1636(5),436(11),1069(10),1634(10),436(5)]),flip(a)].">
      <j1 rule="para" parents="1147 1548"/>
      <j2 rule="rewrite" parents="436 1636 436 1069 1634 436"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1741">
    <literal><![CDATA[
      x v (y v y')' = x
    ]]></literal>
    <justification jstring="[para(1548(a,1),1626(a,1,2,1))].">
      <j1 rule="para" parents="1548 1626"/>
    </justification>
  </clause>

  <clause id="1746">
    <literal><![CDATA[
      x v x' = y v y'
    ]]></literal>
    <justification jstring="[para(1741(a,1),464(a,1,1,1)),rewrite([1741(6),436(3),1069(2),436(6)])].">
      <j1 rule="para" parents="1741 464"/>
      <j2 rule="rewrite" parents="1741 436 1069 436"/>
    </justification>
  </clause>

  <clause id="1749">
    <literal><![CDATA[
      (x v x')' v y = y
    ]]></literal>
    <justification jstring="[para(1741(a,1),796(a,1,2)),rewrite([1741(8)])].">
      <j1 rule="para" parents="1741 796"/>
      <j2 rule="rewrite" parents="1741"/>
    </justification>
  </clause>

  <clause id="1752">
    <literal><![CDATA[
      (x v y) v (z v (x v y)') = y' v (x v y)
    ]]></literal>
    <justification jstring="[para(1741(a,1),742(a,1,2,1)),rewrite([436(3),436(4),796(3),436(6),436(10),1069(9),1634(9)])].">
      <j1 rule="para" parents="1741 742"/>
      <j2 rule="rewrite" parents="436 436 796 436 436 1069 1634"/>
    </justification>
  </clause>

  <clause id="1754">
    <literal><![CDATA[
      x v x' = c_0
    ]]></literal>
    <justification jstring="[new_symbol(1746)].">
      <j1 rule="new_symbol" parents="1746"/>
    </justification>
  </clause>

  <clause id="1761">
    <literal><![CDATA[
      x' v (y v x) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(1647),rewrite([1752(5),1754(5)])].">
      <j1 rule="back_rewrite" parents="1647"/>
      <j2 rule="rewrite" parents="1752 1754"/>
    </justification>
  </clause>

  <clause id="1764">
    <literal><![CDATA[
      c_0' v x = x
    ]]></literal>
    <justification jstring="[back_rewrite(1749),rewrite([1754(2)])].">
      <j1 rule="back_rewrite" parents="1749"/>
      <j2 rule="rewrite" parents="1754"/>
    </justification>
  </clause>

  <clause id="1765">
    <literal><![CDATA[
      x v c_0' = x
    ]]></literal>
    <justification jstring="[back_rewrite(1741),rewrite([1754(2)])].">
      <j1 rule="back_rewrite" parents="1741"/>
      <j2 rule="rewrite" parents="1754"/>
    </justification>
  </clause>

  <clause id="1769">
    <literal><![CDATA[
      (x v y) v y' = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(1635),rewrite([1754(5)])].">
      <j1 rule="back_rewrite" parents="1635"/>
      <j2 rule="rewrite" parents="1754"/>
    </justification>
  </clause>

  <clause id="1843">
    <literal><![CDATA[
      (x' v (y v (z v x))')' = x
    ]]></literal>
    <justification jstring="[para(1761(a,1),1145(a,1,1,1)),rewrite([1069(7),1764(9),436(8)])].">
      <j1 rule="para" parents="1761 1145"/>
      <j2 rule="rewrite" parents="1069 1764 436"/>
    </justification>
  </clause>

  <clause id="1860">
    <literal><![CDATA[
      x' v (y v (z v x))' = x'
    ]]></literal>
    <justification jstring="[para(1769(a,1),742(a,1,2,1,1,1,2,1)),rewrite([1765(4),1069(7),1843(8),864(5),1069(6)]),flip(a)].">
      <j1 rule="para" parents="1769 742"/>
      <j2 rule="rewrite" parents="1765 1069 1843 864 1069"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="5490">
    <literal><![CDATA[
      (x v y) v (x v z) = (x v z) v y
    ]]></literal>
    <justification jstring="[para(1592(a,1),1166(a,1,2,1,2,1)),rewrite([1069(7),1544(7),436(3)]),flip(a)].">
      <j1 rule="para" parents="1592 1166"/>
      <j2 rule="rewrite" parents="1069 1544 436"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="6140">
    <literal><![CDATA[
      (x v y) v (x v z) = (x v y) v z
    ]]></literal>
    <justification jstring="[para(1592(a,1),1192(a,1,2,1,2,1)),rewrite([436(5),1069(7),1860(7),436(3),436(6)]),flip(a)].">
      <j1 rule="para" parents="1592 1192"/>
      <j2 rule="rewrite" parents="436 1069 1860 436 436"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="6167">
    <literal><![CDATA[
      (x v y) v z = (x v z) v y
    ]]></literal>
    <justification jstring="[back_rewrite(5490),rewrite([6140(3)])].">
      <j1 rule="back_rewrite" parents="5490"/>
      <j2 rule="rewrite" parents="6140"/>
    </justification>
  </clause>

  <clause id="6179">
    <literal><![CDATA[
      (x v y) v z = y v (x v z)
    ]]></literal>
    <justification jstring="[para(6167(a,1),1069(a,1))].">
      <j1 rule="para" parents="6167 1069"/>
    </justification>
  </clause>

  <clause id="6181">
    <literal><![CDATA[
      x v (y v z) = z v (y v x)
    ]]></literal>
    <justification jstring="[para(6167(a,1),999(a,1,2)),rewrite([6179(3),796(4),6179(4)])].">
      <j1 rule="para" parents="6167 999"/>
      <j2 rule="rewrite" parents="6179 796 6179"/>
    </justification>
  </clause>

  <clause id="6182">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(robbins_basis)
    ]]></attribute>
    <justification jstring="[resolve(6181,a,1138,a)].">
      <j1 rule="resolve" parents="6181 1138"/>
    </justification>
  </clause>

</proof>

</proofs>
