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<!DOCTYPE proofs SYSTEM "proof3.dtd">

<?xml-stylesheet type="text/xsl" href="proof3.xsl"?>

<proofs number_of_proofs="2">

<source>oml-4basis.out</source>

<heading><![CDATA[
Prover9 (32) version 2009-02A, February 2009.
Process 11667 was started by mccune on cleo,
Wed Feb 25 09:36:13 2009
The command was "/home/mccune/bin/prover9 -f oml-4basis.in".
]]></heading>

<proof number="1" length="21" max_count="19">

<comments><![CDATA[
% Proof 1 at 0.02 (+ 0.00) seconds: CC.
% Length of proof is 21.
% Level of proof is 11.
% Maximum clause weight is 19.
% Given clauses 18.

]]></comments>

  <clause id="1" type="goal">
    <literal><![CDATA[
      c(c(x)) = x
    ]]></literal>
    <attribute><![CDATA[
      answer(CC)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="3" type="assumption">
    <literal><![CDATA[
      x v (y v z) = y v (x v z)
    ]]></literal>
    <attribute><![CDATA[
      label(AJ)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="4" type="assumption">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <attribute><![CDATA[
      label(B1)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="5" type="assumption">
    <literal><![CDATA[
      x ^ y = c(c(x) v c(y))
    ]]></literal>
    <attribute><![CDATA[
      label(DM)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6" type="assumption">
    <literal><![CDATA[
      x v (c(x) ^ (x v y)) = x v y
    ]]></literal>
    <attribute><![CDATA[
      label(OM)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      x v c(c(c(x)) v c(x v y)) = x v y
    ]]></literal>
    <justification jstring="[copy(6),rewrite([5(3)])].">
      <j1 rule="copy" parents="6"/>
      <j2 rule="rewrite" parents="5"/>
    </justification>
  </clause>

  <clause id="8" type="deny">
    <literal><![CDATA[
      c(c(c1)) != c1
    ]]></literal>
    <attribute><![CDATA[
      answer(CC)
    ]]></attribute>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      x v c(c(x) v c(y)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(4),rewrite([5(1)])].">
      <j1 rule="back_rewrite" parents="4"/>
      <j2 rule="rewrite" parents="5"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      x v c(c(x) v y) = x
    ]]></literal>
    <justification jstring="[para(7(a,1),10(a,1,2,1))].">
      <j1 rule="para" parents="7 10"/>
    </justification>
  </clause>

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

  <clause id="32">
    <literal><![CDATA[
      x v (y v c(c(x))) = y v x
    ]]></literal>
    <justification jstring="[para(17(a,1),3(a,1,2)),flip(a)].">
      <j1 rule="para" parents="17 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      x v c(y v c(x)) = x
    ]]></literal>
    <justification jstring="[para(32(a,1),16(a,1,2,1))].">
      <j1 rule="para" parents="32 16"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      c(x) v c(x) = c(x)
    ]]></literal>
    <justification jstring="[para(17(a,1),42(a,1,2,1))].">
      <j1 rule="para" parents="17 42"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      c(c(x)) v x = x
    ]]></literal>
    <justification jstring="[para(53(a,1),32(a,1,2)),rewrite([17(3)]),flip(a)].">
      <j1 rule="para" parents="53 32"/>
      <j2 rule="rewrite" parents="17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      c(c(c(c(x)))) v x = x
    ]]></literal>
    <justification jstring="[para(59(a,1),32(a,1,2)),rewrite([17(3)]),flip(a)].">
      <j1 rule="para" parents="59 32"/>
      <j2 rule="rewrite" parents="17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      c(c(c(c(x)))) v c(c(c(c(c(c(c(x)))))) v c(x)) = x
    ]]></literal>
    <justification jstring="[para(69(a,1),7(a,1,2,1,2,1)),rewrite([69(19)])].">
      <j1 rule="para" parents="69 7"/>
      <j2 rule="rewrite" parents="69"/>
    </justification>
  </clause>

  <clause id="80">
    <literal><![CDATA[
      c(c(c(x))) = c(x)
    ]]></literal>
    <justification jstring="[para(69(a,1),16(a,1,2,1)),rewrite([59(5)]),flip(a)].">
      <j1 rule="para" parents="69 16"/>
      <j2 rule="rewrite" parents="59"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      c(c(x)) v c(c(c(x)) v c(x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(74),rewrite([80(3),80(5),80(5)])].">
      <j1 rule="back_rewrite" parents="74"/>
      <j2 rule="rewrite" parents="80 80 80"/>
    </justification>
  </clause>

  <clause id="96">
    <literal><![CDATA[
      c(c(x)) v c(y v c(x)) = c(c(x))
    ]]></literal>
    <justification jstring="[para(80(a,1),42(a,1,2,1,2))].">
      <j1 rule="para" parents="80 42"/>
    </justification>
  </clause>

  <clause id="99">
    <literal><![CDATA[
      c(c(x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(82),rewrite([96(8)])].">
      <j1 rule="back_rewrite" parents="82"/>
      <j2 rule="rewrite" parents="96"/>
    </justification>
  </clause>

  <clause id="100">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(CC)
    ]]></attribute>
    <justification jstring="[resolve(99,a,8,a)].">
      <j1 rule="resolve" parents="99 8"/>
    </justification>
  </clause>

</proof>

<proof number="2" length="33" max_count="19">

<comments><![CDATA[
% Proof 2 at 0.14 (+ 0.00) seconds: ONE.
% Length of proof is 33.
% Level of proof is 19.
% Maximum clause weight is 19.
% Given clauses 50.

]]></comments>

  <clause id="2" type="goal">
    <literal><![CDATA[
      x v c(x) = y v c(y)
    ]]></literal>
    <attribute><![CDATA[
      answer(ONE)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="3" type="assumption">
    <literal><![CDATA[
      x v (y v z) = y v (x v z)
    ]]></literal>
    <attribute><![CDATA[
      label(AJ)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="4" type="assumption">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <attribute><![CDATA[
      label(B1)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="5" type="assumption">
    <literal><![CDATA[
      x ^ y = c(c(x) v c(y))
    ]]></literal>
    <attribute><![CDATA[
      label(DM)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6" type="assumption">
    <literal><![CDATA[
      x v (c(x) ^ (x v y)) = x v y
    ]]></literal>
    <attribute><![CDATA[
      label(OM)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      x v c(c(c(x)) v c(x v y)) = x v y
    ]]></literal>
    <justification jstring="[copy(6),rewrite([5(3)])].">
      <j1 rule="copy" parents="6"/>
      <j2 rule="rewrite" parents="5"/>
    </justification>
  </clause>

  <clause id="9" type="deny">
    <literal><![CDATA[
      c3 v c(c3) != c2 v c(c2)
    ]]></literal>
    <attribute><![CDATA[
      answer(ONE)
    ]]></attribute>
    <justification jstring="[deny(2)].">
      <j1 rule="deny" parents="2"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      x v c(c(x) v c(y)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(4),rewrite([5(1)])].">
      <j1 rule="back_rewrite" parents="4"/>
      <j2 rule="rewrite" parents="5"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      x v (y v c(c(c(x)) v c(x v z))) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(7(a,1),3(a,1,2)),flip(a)].">
      <j1 rule="para" parents="7 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      x v c(c(x) v y) = x
    ]]></literal>
    <justification jstring="[para(7(a,1),10(a,1,2,1))].">
      <j1 rule="para" parents="7 10"/>
    </justification>
  </clause>

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

  <clause id="32">
    <literal><![CDATA[
      x v (y v c(c(x))) = y v x
    ]]></literal>
    <justification jstring="[para(17(a,1),3(a,1,2)),flip(a)].">
      <j1 rule="para" parents="17 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      x v c(y v (c(x) v z)) = x
    ]]></literal>
    <justification jstring="[para(3(a,1),16(a,1,2,1))].">
      <j1 rule="para" parents="3 16"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      x v c(y v c(x)) = x
    ]]></literal>
    <justification jstring="[para(32(a,1),16(a,1,2,1))].">
      <j1 rule="para" parents="32 16"/>
    </justification>
  </clause>

  <clause id="48">
    <literal><![CDATA[
      x v (c(x) v y) = x v c(x)
    ]]></literal>
    <justification jstring="[para(16(a,1),12(a,1,2)),rewrite([3(5)]),flip(a)].">
      <j1 rule="para" parents="16 12"/>
      <j2 rule="rewrite" parents="3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      c(x) v c(x) = c(x)
    ]]></literal>
    <justification jstring="[para(17(a,1),42(a,1,2,1))].">
      <j1 rule="para" parents="17 42"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      c(c(x)) v x = x
    ]]></literal>
    <justification jstring="[para(53(a,1),32(a,1,2)),rewrite([17(3)]),flip(a)].">
      <j1 rule="para" parents="53 32"/>
      <j2 rule="rewrite" parents="17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      c(c(x)) v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(59(a,1),3(a,1,2)),flip(a)].">
      <j1 rule="para" parents="59 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      c(c(c(c(x)))) v x = x
    ]]></literal>
    <justification jstring="[para(59(a,1),32(a,1,2)),rewrite([17(3)]),flip(a)].">
      <j1 rule="para" parents="59 32"/>
      <j2 rule="rewrite" parents="17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      c(c(c(c(x)))) v c(c(c(c(c(c(c(x)))))) v c(x)) = x
    ]]></literal>
    <justification jstring="[para(69(a,1),7(a,1,2,1,2,1)),rewrite([69(19)])].">
      <j1 rule="para" parents="69 7"/>
      <j2 rule="rewrite" parents="69"/>
    </justification>
  </clause>

  <clause id="80">
    <literal><![CDATA[
      c(c(c(x))) = c(x)
    ]]></literal>
    <justification jstring="[para(69(a,1),16(a,1,2,1)),rewrite([59(5)]),flip(a)].">
      <j1 rule="para" parents="69 16"/>
      <j2 rule="rewrite" parents="59"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      c(c(x)) v c(c(c(x)) v c(x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(74),rewrite([80(3),80(5),80(5)])].">
      <j1 rule="back_rewrite" parents="74"/>
      <j2 rule="rewrite" parents="80 80 80"/>
    </justification>
  </clause>

  <clause id="96">
    <literal><![CDATA[
      c(c(x)) v c(y v c(x)) = c(c(x))
    ]]></literal>
    <justification jstring="[para(80(a,1),42(a,1,2,1,2))].">
      <j1 rule="para" parents="80 42"/>
    </justification>
  </clause>

  <clause id="99">
    <literal><![CDATA[
      c(c(x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(82),rewrite([96(8)])].">
      <j1 rule="back_rewrite" parents="82"/>
      <j2 rule="rewrite" parents="96"/>
    </justification>
  </clause>

  <clause id="113">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[back_rewrite(61),rewrite([99(2)])].">
      <j1 rule="back_rewrite" parents="61"/>
      <j2 rule="rewrite" parents="99"/>
    </justification>
  </clause>

  <clause id="367">
    <literal><![CDATA[
      x v c(y v (z v c(x))) = x
    ]]></literal>
    <justification jstring="[para(113(a,1),35(a,1,2,1,2))].">
      <j1 rule="para" parents="113 35"/>
    </justification>
  </clause>

  <clause id="462">
    <literal><![CDATA[
      x v c(y v c(y)) = x
    ]]></literal>
    <justification jstring="[para(48(a,1),367(a,1,2,1))].">
      <j1 rule="para" parents="48 367"/>
    </justification>
  </clause>

  <clause id="468">
    <literal><![CDATA[
      c(x v c(x)) v y = y
    ]]></literal>
    <justification jstring="[para(462(a,1),113(a,1,2)),rewrite([462(8)])].">
      <j1 rule="para" parents="462 113"/>
      <j2 rule="rewrite" parents="462"/>
    </justification>
  </clause>

  <clause id="701">
    <literal><![CDATA[
      c(x v c(x)) = c(y v c(y))
    ]]></literal>
    <justification jstring="[para(468(a,1),462(a,1))].">
      <j1 rule="para" parents="468 462"/>
    </justification>
  </clause>

  <clause id="702">
    <literal><![CDATA[
      c(x v c(x)) = c_0
    ]]></literal>
    <justification jstring="[new_symbol(701)].">
      <j1 rule="new_symbol" parents="701"/>
    </justification>
  </clause>

  <clause id="711">
    <literal><![CDATA[
      c(c_0) = x v c(x)
    ]]></literal>
    <justification jstring="[para(702(a,1),99(a,1,1))].">
      <j1 rule="para" parents="702 99"/>
    </justification>
  </clause>

  <clause id="777">
    <literal><![CDATA[
      c(c_0) != c2 v c(c2)
    ]]></literal>
    <attribute><![CDATA[
      answer(ONE)
    ]]></attribute>
    <justification jstring="[para(711(a,2),9(a,1))].">
      <j1 rule="para" parents="711 9"/>
    </justification>
  </clause>

  <clause id="778">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(ONE)
    ]]></attribute>
    <justification jstring="[resolve(777,a,711,a)].">
      <j1 rule="resolve" parents="777 711"/>
    </justification>
  </clause>

</proof>

</proofs>
