<?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>SET593+3.out</source>

<heading><![CDATA[
FOF-Prover9 (32) version November-2006, November 2006.
Process 3776 was started by mccune on cleo.thornwood,
Wed Nov 22 11:23:39 2006
The command was "/home/mccune/bin/fof-prover9 -f SET593+3.in".
]]></heading>

<proof number="1" length="15" max_count="11">

<comments><![CDATA[
% Proof 1 at 0.02 (+ 0.00) seconds.
% Length of proof is 15.
% Level of proof is 4.
% Maximum clause weight is 11.
% Given clauses 70.
]]></comments>

  <clause id="3" type="assumption">
    <literal><![CDATA[
      member(A,B)
    ]]></literal>
    <literal><![CDATA[
      member(A,C)
    ]]></literal>
    <literal><![CDATA[
      -member(A,union(B,C))
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="5" type="assumption">
    <literal><![CDATA[
      member(A,B)
    ]]></literal>
    <literal><![CDATA[
      -member(A,difference(B,C))
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6" type="assumption">
    <literal><![CDATA[
      -member(A,B)
    ]]></literal>
    <literal><![CDATA[
      -member(A,difference(C,B))
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7" type="assumption">
    <literal><![CDATA[
      subset(A,B)
    ]]></literal>
    <literal><![CDATA[
      member(f1(A,B),A)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="8" type="assumption">
    <literal><![CDATA[
      subset(A,B)
    ]]></literal>
    <literal><![CDATA[
      -member(f1(A,B),B)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="9" type="assumption">
    <literal><![CDATA[
      -member(A,B)
    ]]></literal>
    <literal><![CDATA[
      member(A,C)
    ]]></literal>
    <literal><![CDATA[
      -subset(B,C)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="28" type="assumption">
    <literal><![CDATA[
      subset(c1,union(c2,c3))
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="29" type="assumption">
    <literal><![CDATA[
      -subset(difference(c1,c2),c3)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      -member(A,c1)
    ]]></literal>
    <literal><![CDATA[
      member(A,union(c2,c3))
    ]]></literal>
    <justification jstring="[resolve(28,a,9,c)].">
      <j1 rule="resolve" parents="28 9"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      member(f1(difference(c1,c2),c3),difference(c1,c2))
    ]]></literal>
    <justification jstring="[resolve(29,a,7,a)].">
      <j1 rule="resolve" parents="29 7"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      -member(f1(difference(c1,c2),c3),c3)
    ]]></literal>
    <justification jstring="[ur(8,a,29,a)].">
      <j1 rule="ur" parents="8 29"/>
    </justification>
  </clause>

  <clause id="60">
    <literal><![CDATA[
      -member(f1(difference(c1,c2),c3),c2)
    ]]></literal>
    <justification jstring="[resolve(42,a,6,b)].">
      <j1 rule="resolve" parents="42 6"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      member(f1(difference(c1,c2),c3),c1)
    ]]></literal>
    <justification jstring="[resolve(42,a,5,b)].">
      <j1 rule="resolve" parents="42 5"/>
    </justification>
  </clause>

  <clause id="71">
    <literal><![CDATA[
      member(f1(difference(c1,c2),c3),union(c2,c3))
    ]]></literal>
    <justification jstring="[resolve(61,a,41,a)].">
      <j1 rule="resolve" parents="61 41"/>
    </justification>
  </clause>

  <clause id="350">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(71,a,3,c),unit_del(a,60),unit_del(b,43)].">
      <j1 rule="resolve" parents="71 3"/>
      <j2 rule="unit_del" parents="60"/>
      <j2 rule="unit_del" parents="43"/>
    </justification>
  </clause>

</proof>

<proof number="2" length="15" max_count="11">

<comments><![CDATA[
% Proof 1 at 0.02 (+ 0.00) seconds.
% Length of proof is 15.
% Level of proof is 4.
% Maximum clause weight is 11.
% Given clauses 70.
]]></comments>

  <clause id="3" type="assumption">
    <literal><![CDATA[
      member(A,B)
    ]]></literal>
    <literal><![CDATA[
      member(A,C)
    ]]></literal>
    <literal><![CDATA[
      -member(A,union(B,C))
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="5" type="assumption">
    <literal><![CDATA[
      member(A,B)
    ]]></literal>
    <literal><![CDATA[
      -member(A,difference(B,C))
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6" type="assumption">
    <literal><![CDATA[
      -member(A,B)
    ]]></literal>
    <literal><![CDATA[
      -member(A,difference(C,B))
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7" type="assumption">
    <literal><![CDATA[
      subset(A,B)
    ]]></literal>
    <literal><![CDATA[
      member(f1(A,B),A)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="8" type="assumption">
    <literal><![CDATA[
      subset(A,B)
    ]]></literal>
    <literal><![CDATA[
      -member(f1(A,B),B)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="9" type="assumption">
    <literal><![CDATA[
      -member(A,B)
    ]]></literal>
    <literal><![CDATA[
      member(A,C)
    ]]></literal>
    <literal><![CDATA[
      -subset(B,C)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="28" type="assumption">
    <literal><![CDATA[
      subset(c1,union(c2,c3))
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="29" type="assumption">
    <literal><![CDATA[
      -subset(difference(c1,c3),c2)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      -member(A,c1)
    ]]></literal>
    <literal><![CDATA[
      member(A,union(c2,c3))
    ]]></literal>
    <justification jstring="[resolve(28,a,9,c)].">
      <j1 rule="resolve" parents="28 9"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      member(f1(difference(c1,c3),c2),difference(c1,c3))
    ]]></literal>
    <justification jstring="[resolve(29,a,7,a)].">
      <j1 rule="resolve" parents="29 7"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      -member(f1(difference(c1,c3),c2),c2)
    ]]></literal>
    <justification jstring="[ur(8,a,29,a)].">
      <j1 rule="ur" parents="8 29"/>
    </justification>
  </clause>

  <clause id="60">
    <literal><![CDATA[
      -member(f1(difference(c1,c3),c2),c3)
    ]]></literal>
    <justification jstring="[resolve(42,a,6,b)].">
      <j1 rule="resolve" parents="42 6"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      member(f1(difference(c1,c3),c2),c1)
    ]]></literal>
    <justification jstring="[resolve(42,a,5,b)].">
      <j1 rule="resolve" parents="42 5"/>
    </justification>
  </clause>

  <clause id="71">
    <literal><![CDATA[
      member(f1(difference(c1,c3),c2),union(c2,c3))
    ]]></literal>
    <justification jstring="[resolve(61,a,41,a)].">
      <j1 rule="resolve" parents="61 41"/>
    </justification>
  </clause>

  <clause id="350">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(71,a,3,c),unit_del(a,43),unit_del(b,60)].">
      <j1 rule="resolve" parents="71 3"/>
      <j2 rule="unit_del" parents="43"/>
      <j2 rule="unit_del" parents="60"/>
    </justification>
  </clause>

</proof>

</proofs>
