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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>subset_trans.out</source>

<heading><![CDATA[
Prover9 (32) version August-2006A, August 2006.
Process 10891 was started by mccune on cleo.thornwood,
Sat Aug 12 20:59:02 2006
The command was "/home/mccune/bin/prover9 -f subset_trans.in".
]]></heading>

<proof number="1" length="14" max_count="29">

<comments><![CDATA[
% Proof 1 at 0.00 (+ 0.00) seconds.
% Length of proof is 14.
% Level of proof is 4.
% Maximum clause weight is 6.
% Given clauses 6.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      (all x all y (subset(x,y) <-> (all z (mem(z,x) -> mem(z,y)))))
    ]]></literal>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="2">
    <literal><![CDATA[
      (all x all y all z (subset(x,y) & subset(y,z) -> subset(x,z)))
    ]]></literal>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="3">
    <literal><![CDATA[
      subset(x,y)
    ]]></literal>
    <literal><![CDATA[
      mem(f1(x,y),x)
    ]]></literal>
    <justification jstring="[clausify(1)].">
      <j1 rule="clausify" parents="1"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      -subset(x,y)
    ]]></literal>
    <literal><![CDATA[
      -mem(z,x)
    ]]></literal>
    <literal><![CDATA[
      mem(z,y)
    ]]></literal>
    <justification jstring="[clausify(1)].">
      <j1 rule="clausify" parents="1"/>
    </justification>
  </clause>

  <clause id="5">
    <literal><![CDATA[
      subset(x,y)
    ]]></literal>
    <literal><![CDATA[
      -mem(f1(x,y),y)
    ]]></literal>
    <justification jstring="[clausify(1)].">
      <j1 rule="clausify" parents="1"/>
    </justification>
  </clause>

  <clause id="6">
    <literal><![CDATA[
      subset(c1,c2)
    ]]></literal>
    <justification jstring="[deny(2)].">
      <j1 rule="deny" parents="2"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      subset(c2,c3)
    ]]></literal>
    <justification jstring="[deny(2)].">
      <j1 rule="deny" parents="2"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      -subset(c1,c3)
    ]]></literal>
    <justification jstring="[deny(2)].">
      <j1 rule="deny" parents="2"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      -mem(x,c1)
    ]]></literal>
    <literal><![CDATA[
      mem(x,c2)
    ]]></literal>
    <justification jstring="[resolve(6,a,4,a)].">
      <j1 rule="resolve" parents="6 4"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      -mem(x,c2)
    ]]></literal>
    <literal><![CDATA[
      mem(x,c3)
    ]]></literal>
    <justification jstring="[resolve(7,a,4,a)].">
      <j1 rule="resolve" parents="7 4"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      mem(f1(c1,c3),c1)
    ]]></literal>
    <justification jstring="[resolve(8,a,3,a)].">
      <j1 rule="resolve" parents="8 3"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      -mem(f1(c1,c3),c3)
    ]]></literal>
    <justification jstring="[resolve(8,a,5,a)].">
      <j1 rule="resolve" parents="8 5"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      mem(f1(c1,c3),c2)
    ]]></literal>
    <justification jstring="[resolve(13,a,11,a)].">
      <j1 rule="resolve" parents="13 11"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[ur(12,b,14,a),unit_del(a,15)].">
      <j1 rule="ur" parents="12 14"/>
      <j2 rule="unit_del" parents="15"/>
    </justification>
  </clause>

</proof>

</proofs>
