<?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="2">

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

<heading><![CDATA[
Prover9 (32) version November-2006, November 2006.
Process 3929 was started by mccune on cleo.thornwood,
Wed Nov 22 11:25:20 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 11.
% Maximum clause weight is 19.
% Given clauses 17.
]]></comments>

  <clause id="1" type="goal">
    <literal><![CDATA[
      c(c(x)) = x
    ]]></literal>
    <attribute><![CDATA[
      answer(CC)
    ]]></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="18">
    <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="24">
    <literal><![CDATA[
      x v c(y v c(x)) = x
    ]]></literal>
    <justification jstring="[para(18(a,1),16(a,1,2,1))].">
      <j1 rule="para" parents="18 16"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      c(x) v c(x) = c(x)
    ]]></literal>
    <justification jstring="[para(17(a,1),24(a,1,2,1))].">
      <j1 rule="para" parents="17 24"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      c(c(x)) v x = x
    ]]></literal>
    <justification jstring="[para(44(a,1),18(a,1,2)),rewrite(17(3)),flip(a)].">
      <j1 rule="para" parents="44 18"/>
      <j2 rule="rewrite" parents="17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      c(c(c(c(x)))) v x = x
    ]]></literal>
    <justification jstring="[para(49(a,1),18(a,1,2)),rewrite(17(3)),flip(a)].">
      <j1 rule="para" parents="49 18"/>
      <j2 rule="rewrite" parents="17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      c(c(c(c(x)))) v c(c(c(c(c(c(c(x)))))) v c(x)) = x
    ]]></literal>
    <justification jstring="[para(54(a,1),7(a,1,2,1,2,1)),rewrite(54(19))].">
      <j1 rule="para" parents="54 7"/>
      <j2 rule="rewrite" parents="54"/>
    </justification>
  </clause>

  <clause id="63">
    <literal><![CDATA[
      c(c(c(x))) = c(x)
    ]]></literal>
    <justification jstring="[para(54(a,1),16(a,1,2,1)),rewrite(49(5)),flip(a)].">
      <j1 rule="para" parents="54 16"/>
      <j2 rule="rewrite" parents="49"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="64">
    <literal><![CDATA[
      c(c(x)) v c(c(c(x)) v c(x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(62),rewrite(63(3),63(5),63(5))].">
      <j1 rule="back_rewrite" parents="62"/>
      <j2 rule="rewrite" parents="63 63 63"/>
    </justification>
  </clause>

  <clause id="78">
    <literal><![CDATA[
      c(c(x)) v c(y v c(x)) = c(c(x))
    ]]></literal>
    <justification jstring="[para(63(a,1),24(a,1,2,1,2))].">
      <j1 rule="para" parents="63 24"/>
    </justification>
  </clause>

  <clause id="81">
    <literal><![CDATA[
      c(c(x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(64),rewrite(78(8))].">
      <j1 rule="back_rewrite" parents="64"/>
      <j2 rule="rewrite" parents="78"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(CC)
    ]]></attribute>
    <justification jstring="[resolve(81,a,8,a)].">
      <j1 rule="resolve" parents="81 8"/>
    </justification>
  </clause>

</proof>

<proof number="2" length="35" max_count="19">

<comments><![CDATA[
% Proof 2 at 0.02 (+ 0.00) seconds: ONE.
% Length of proof is 35.
% Level of proof is 16.
% Maximum clause weight is 19.
% Given clauses 39.
]]></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(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="18">
    <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="19">
    <literal><![CDATA[
      x v (y v c(c(x) v z)) = y v x
    ]]></literal>
    <justification jstring="[para(16(a,1),3(a,1,2)),flip(a)].">
      <j1 rule="para" parents="16 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20">
    <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="24">
    <literal><![CDATA[
      x v c(y v c(x)) = x
    ]]></literal>
    <justification jstring="[para(18(a,1),16(a,1,2,1))].">
      <j1 rule="para" parents="18 16"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      c(x) v c(x) = c(x)
    ]]></literal>
    <justification jstring="[para(17(a,1),24(a,1,2,1))].">
      <j1 rule="para" parents="17 24"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      c(c(x)) v x = x
    ]]></literal>
    <justification jstring="[para(44(a,1),18(a,1,2)),rewrite(17(3)),flip(a)].">
      <j1 rule="para" parents="44 18"/>
      <j2 rule="rewrite" parents="17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      c(c(c(c(x)))) v x = x
    ]]></literal>
    <justification jstring="[para(49(a,1),18(a,1,2)),rewrite(17(3)),flip(a)].">
      <j1 rule="para" parents="49 18"/>
      <j2 rule="rewrite" parents="17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      c(c(c(c(x)))) v c(c(c(c(c(c(c(x)))))) v c(x)) = x
    ]]></literal>
    <justification jstring="[para(54(a,1),7(a,1,2,1,2,1)),rewrite(54(19))].">
      <j1 rule="para" parents="54 7"/>
      <j2 rule="rewrite" parents="54"/>
    </justification>
  </clause>

  <clause id="63">
    <literal><![CDATA[
      c(c(c(x))) = c(x)
    ]]></literal>
    <justification jstring="[para(54(a,1),16(a,1,2,1)),rewrite(49(5)),flip(a)].">
      <j1 rule="para" parents="54 16"/>
      <j2 rule="rewrite" parents="49"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="64">
    <literal><![CDATA[
      c(c(x)) v c(c(c(x)) v c(x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(62),rewrite(63(3),63(5),63(5))].">
      <j1 rule="back_rewrite" parents="62"/>
      <j2 rule="rewrite" parents="63 63 63"/>
    </justification>
  </clause>

  <clause id="66">
    <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="78">
    <literal><![CDATA[
      c(c(x)) v c(y v c(x)) = c(c(x))
    ]]></literal>
    <justification jstring="[para(63(a,1),24(a,1,2,1,2))].">
      <j1 rule="para" parents="63 24"/>
    </justification>
  </clause>

  <clause id="81">
    <literal><![CDATA[
      c(c(x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(64),rewrite(78(8))].">
      <j1 rule="back_rewrite" parents="64"/>
      <j2 rule="rewrite" parents="78"/>
    </justification>
  </clause>

  <clause id="112">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[para(24(a,1),19(a,1,2))].">
      <j1 rule="para" parents="24 19"/>
    </justification>
  </clause>

  <clause id="114">
    <literal><![CDATA[
      x v (y v z) = z v (x v y)
    ]]></literal>
    <justification jstring="[para(112(a,1),3(a,1,2))].">
      <j1 rule="para" parents="112 3"/>
    </justification>
  </clause>

  <clause id="138">
    <literal><![CDATA[
      x v c(y v (z v c(x))) = x
    ]]></literal>
    <justification jstring="[para(19(a,1),20(a,1,2,1,2))].">
      <j1 rule="para" parents="19 20"/>
    </justification>
  </clause>

  <clause id="171">
    <literal><![CDATA[
      (x v y) v z = x v (y v z)
    ]]></literal>
    <justification jstring="[para(114(a,2),112(a,1)),flip(a)].">
      <j1 rule="para" parents="114 112"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="181">
    <literal><![CDATA[
      x v c(y v c(y)) = x
    ]]></literal>
    <justification jstring="[para(66(a,1),138(a,1,2,1))].">
      <j1 rule="para" parents="66 138"/>
    </justification>
  </clause>

  <clause id="182">
    <literal><![CDATA[
      c(x v c(x)) v y = y
    ]]></literal>
    <justification jstring="[para(181(a,1),19(a,2)),rewrite(81(7),171(6),66(6),181(7))].">
      <j1 rule="para" parents="181 19"/>
      <j2 rule="rewrite" parents="81 171 66 181"/>
    </justification>
  </clause>

  <clause id="183">
    <literal><![CDATA[
      c(x v c(x)) = c(y v c(y))
    ]]></literal>
    <justification jstring="[para(182(a,1),181(a,1))].">
      <j1 rule="para" parents="182 181"/>
    </justification>
  </clause>

  <clause id="184">
    <literal><![CDATA[
      c(x v c(x)) = c_0
    ]]></literal>
    <justification jstring="[new_symbol(183)].">
      <j1 rule="new_symbol" parents="183"/>
    </justification>
  </clause>

  <clause id="193">
    <literal><![CDATA[
      c(c_0) = x v c(x)
    ]]></literal>
    <justification jstring="[para(184(a,1),81(a,1,1))].">
      <j1 rule="para" parents="184 81"/>
    </justification>
  </clause>

  <clause id="207">
    <literal><![CDATA[
      c(c_0) != c2 v c(c2)
    ]]></literal>
    <attribute><![CDATA[
      answer(ONE)
    ]]></attribute>
    <justification jstring="[para(193(a,2),9(a,1))].">
      <j1 rule="para" parents="193 9"/>
    </justification>
  </clause>

  <clause id="208">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(ONE)
    ]]></attribute>
    <justification jstring="[resolve(207,a,193,a)].">
      <j1 rule="resolve" parents="207 193"/>
    </justification>
  </clause>

</proof>

</proofs>
