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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 June-2006C, June 2006.
Process 13585 was started by mccune on cleo.thornwood,
Mon Jun 19 16:43:06 2006
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 0.00 (+ 0.00) seconds: CC.
% Length of proof is 21.
% Level of proof is 12.
% Maximum clause weight is 19.
% Given clauses 15.
]]></comments>

  <clause id="2">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <attribute><![CDATA[
      label(B1)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

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

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

  <clause id="8">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <attribute><![CDATA[
      label(B1)
    ]]></attribute>
    <justification jstring="[copy(2)].">
      <j1 rule="copy" parents="2"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      x ^ y = c(c(x) v c(y))
    ]]></literal>
    <attribute><![CDATA[
      label(DM)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      x v c(c(c(x)) v c(x v y)) = x v y
    ]]></literal>
    <attribute><![CDATA[
      label(OM)
    ]]></attribute>
    <justification jstring="[copy(4),demod(9(3))].">
      <j1 rule="copy" parents="4"/>
      <j2 rule="demod" parents="9"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      x v c(c(x) v c(y)) = x
    ]]></literal>
    <justification jstring="[back_demod(8),demod(9(1))].">
      <j1 rule="back_demod" parents="8"/>
      <j2 rule="demod" parents="9"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      c(c(c1)) != c1
    ]]></literal>
    <attribute><![CDATA[
      answer(CC)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      x v c(c(x) v y) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),11(a,1,2,1))].">
      <j1 rule="para" parents="10 11"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x v c(c(x)) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),11(a,1,2,1))].">
      <j1 rule="para" parents="11 11"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      x v (y v c(c(x))) = y v x
    ]]></literal>
    <justification jstring="[para(20(a,1),7(a,1,2)),flip(a)].">
      <j1 rule="para" parents="20 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      x v c(y v c(x)) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),19(a,1,2,1))].">
      <j1 rule="para" parents="21 19"/>
    </justification>
  </clause>

  <clause id="48">
    <literal><![CDATA[
      c(x) v c(x) = c(x)
    ]]></literal>
    <justification jstring="[para(20(a,1),45(a,1,2,1))].">
      <j1 rule="para" parents="20 45"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      c(c(x)) v x = x
    ]]></literal>
    <justification jstring="[para(48(a,1),21(a,1,2)),demod(20(3)),flip(a)].">
      <j1 rule="para" parents="48 21"/>
      <j2 rule="demod" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      c(c(c(c(x)))) v x = x
    ]]></literal>
    <justification jstring="[para(62(a,1),21(a,1,2)),demod(20(3)),flip(a)].">
      <j1 rule="para" parents="62 21"/>
      <j2 rule="demod" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      c(c(c(c(x)))) v c(c(c(c(c(c(c(x)))))) v c(x)) = x
    ]]></literal>
    <justification jstring="[para(72(a,1),10(a,1,2,1,2,1)),demod(72(19))].">
      <j1 rule="para" parents="72 10"/>
      <j2 rule="demod" parents="72"/>
    </justification>
  </clause>

  <clause id="83">
    <literal><![CDATA[
      c(c(c(x))) = c(x)
    ]]></literal>
    <justification jstring="[para(72(a,1),19(a,1,2,1)),demod(62(5)),flip(a)].">
      <j1 rule="para" parents="72 19"/>
      <j2 rule="demod" parents="62"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="85">
    <literal><![CDATA[
      c(c(x)) v c(c(c(x)) v c(x)) = x
    ]]></literal>
    <justification jstring="[back_demod(77),demod(83(3),83(5),83(5))].">
      <j1 rule="back_demod" parents="77"/>
      <j2 rule="demod" parents="83 83 83"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      c(c(x)) v c(y v c(x)) = c(c(x))
    ]]></literal>
    <justification jstring="[para(83(a,1),45(a,1,2,1,2))].">
      <j1 rule="para" parents="83 45"/>
    </justification>
  </clause>

  <clause id="92">
    <literal><![CDATA[
      c(c(x)) = x
    ]]></literal>
    <justification jstring="[back_demod(85),demod(89(8))].">
      <j1 rule="back_demod" parents="85"/>
      <j2 rule="demod" parents="89"/>
    </justification>
  </clause>

  <clause id="93">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(CC)
    ]]></attribute>
    <justification jstring="[resolve(92,a,12,a)].">
      <j1 rule="resolve" parents="92 12"/>
    </justification>
  </clause>

</proof>

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

<comments><![CDATA[
% Proof 2 at 0.11 (+ 0.01) seconds: ONE.
% Length of proof is 33.
% Level of proof is 20.
% Maximum clause weight is 19.
% Given clauses 48.
]]></comments>

  <clause id="2">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <attribute><![CDATA[
      label(B1)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

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

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

  <clause id="8">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <attribute><![CDATA[
      label(B1)
    ]]></attribute>
    <justification jstring="[copy(2)].">
      <j1 rule="copy" parents="2"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      x ^ y = c(c(x) v c(y))
    ]]></literal>
    <attribute><![CDATA[
      label(DM)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      x v c(c(c(x)) v c(x v y)) = x v y
    ]]></literal>
    <attribute><![CDATA[
      label(OM)
    ]]></attribute>
    <justification jstring="[copy(4),demod(9(3))].">
      <j1 rule="copy" parents="4"/>
      <j2 rule="demod" parents="9"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      x v c(c(x) v c(y)) = x
    ]]></literal>
    <justification jstring="[back_demod(8),demod(9(1))].">
      <j1 rule="back_demod" parents="8"/>
      <j2 rule="demod" parents="9"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      c3 v c(c3) != c2 v c(c2)
    ]]></literal>
    <attribute><![CDATA[
      answer(ONE)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      x v (y v c(c(c(x)) v c(x v z))) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(10(a,1),7(a,1,2)),flip(a)].">
      <j1 rule="para" parents="10 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      x v c(c(x) v y) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),11(a,1,2,1))].">
      <j1 rule="para" parents="10 11"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x v c(c(x)) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),11(a,1,2,1))].">
      <j1 rule="para" parents="11 11"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      x v (y v c(c(x))) = y v x
    ]]></literal>
    <justification jstring="[para(20(a,1),7(a,1,2)),flip(a)].">
      <j1 rule="para" parents="20 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      x v c(y v (c(x) v z)) = x
    ]]></literal>
    <justification jstring="[para(7(a,1),19(a,1,2,1))].">
      <j1 rule="para" parents="7 19"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      x v c(y v c(x)) = x
    ]]></literal>
    <justification jstring="[para(21(a,1),19(a,1,2,1))].">
      <j1 rule="para" parents="21 19"/>
    </justification>
  </clause>

  <clause id="48">
    <literal><![CDATA[
      c(x) v c(x) = c(x)
    ]]></literal>
    <justification jstring="[para(20(a,1),45(a,1,2,1))].">
      <j1 rule="para" parents="20 45"/>
    </justification>
  </clause>

  <clause id="57">
    <literal><![CDATA[
      x v (c(x) v y) = x v c(x)
    ]]></literal>
    <justification jstring="[para(19(a,1),15(a,1,2)),demod(7(5)),flip(a)].">
      <j1 rule="para" parents="19 15"/>
      <j2 rule="demod" parents="7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      c(c(x)) v x = x
    ]]></literal>
    <justification jstring="[para(48(a,1),21(a,1,2)),demod(20(3)),flip(a)].">
      <j1 rule="para" parents="48 21"/>
      <j2 rule="demod" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="64">
    <literal><![CDATA[
      c(c(x)) v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(62(a,1),7(a,1,2)),flip(a)].">
      <j1 rule="para" parents="62 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      c(c(c(c(x)))) v x = x
    ]]></literal>
    <justification jstring="[para(62(a,1),21(a,1,2)),demod(20(3)),flip(a)].">
      <j1 rule="para" parents="62 21"/>
      <j2 rule="demod" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      c(c(c(c(x)))) v c(c(c(c(c(c(c(x)))))) v c(x)) = x
    ]]></literal>
    <justification jstring="[para(72(a,1),10(a,1,2,1,2,1)),demod(72(19))].">
      <j1 rule="para" parents="72 10"/>
      <j2 rule="demod" parents="72"/>
    </justification>
  </clause>

  <clause id="83">
    <literal><![CDATA[
      c(c(c(x))) = c(x)
    ]]></literal>
    <justification jstring="[para(72(a,1),19(a,1,2,1)),demod(62(5)),flip(a)].">
      <j1 rule="para" parents="72 19"/>
      <j2 rule="demod" parents="62"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="85">
    <literal><![CDATA[
      c(c(x)) v c(c(c(x)) v c(x)) = x
    ]]></literal>
    <justification jstring="[back_demod(77),demod(83(3),83(5),83(5))].">
      <j1 rule="back_demod" parents="77"/>
      <j2 rule="demod" parents="83 83 83"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      c(c(x)) v c(y v c(x)) = c(c(x))
    ]]></literal>
    <justification jstring="[para(83(a,1),45(a,1,2,1,2))].">
      <j1 rule="para" parents="83 45"/>
    </justification>
  </clause>

  <clause id="92">
    <literal><![CDATA[
      c(c(x)) = x
    ]]></literal>
    <justification jstring="[back_demod(85),demod(89(8))].">
      <j1 rule="back_demod" parents="85"/>
      <j2 rule="demod" parents="89"/>
    </justification>
  </clause>

  <clause id="100">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[back_demod(64),demod(92(2))].">
      <j1 rule="back_demod" parents="64"/>
      <j2 rule="demod" parents="92"/>
    </justification>
  </clause>

  <clause id="374">
    <literal><![CDATA[
      x v c(y v (z v c(x))) = x
    ]]></literal>
    <justification jstring="[para(100(a,1),38(a,1,2,1,2))].">
      <j1 rule="para" parents="100 38"/>
    </justification>
  </clause>

  <clause id="596">
    <literal><![CDATA[
      x v c(y v c(y)) = x
    ]]></literal>
    <justification jstring="[para(57(a,1),374(a,1,2,1))].">
      <j1 rule="para" parents="57 374"/>
    </justification>
  </clause>

  <clause id="600">
    <literal><![CDATA[
      c(x v c(x)) v y = y
    ]]></literal>
    <justification jstring="[para(596(a,1),100(a,1,2)),demod(596(8))].">
      <j1 rule="para" parents="596 100"/>
      <j2 rule="demod" parents="596"/>
    </justification>
  </clause>

  <clause id="685">
    <literal><![CDATA[
      c(x v c(x)) = c(y v c(y))
    ]]></literal>
    <justification jstring="[para(600(a,1),596(a,1))].">
      <j1 rule="para" parents="600 596"/>
    </justification>
  </clause>

  <clause id="686">
    <literal><![CDATA[
      c(x v c(x)) = c_0
    ]]></literal>
    <justification jstring="[new_symbol(685)].">
      <j1 rule="new_symbol" parents="685"/>
    </justification>
  </clause>

  <clause id="701">
    <literal><![CDATA[
      c(c_0) = x v c(x)
    ]]></literal>
    <justification jstring="[para(686(a,1),92(a,1,1))].">
      <j1 rule="para" parents="686 92"/>
    </justification>
  </clause>

  <clause id="733">
    <literal><![CDATA[
      x v c(x) = y v c(y)
    ]]></literal>
    <justification jstring="[para(701(a,1),701(a,1))].">
      <j1 rule="para" parents="701 701"/>
    </justification>
  </clause>

  <clause id="734">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(ONE)
    ]]></attribute>
    <justification jstring="[resolve(733,a,13,a)].">
      <j1 rule="resolve" parents="733 13"/>
    </justification>
  </clause>

</proof>

</proofs>
