<?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 2009-02A, February 2009.
Process 11556 was started by mccune on cleo,
Wed Feb 25 09:32:32 2009
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 0.00 (+ 0.01) seconds.
% Length of proof is 15.
% Level of proof is 4.
% Maximum clause weight is 11.
% Given clauses 43.

]]></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="55">
    <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="56">
    <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="57">
    <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="89">
    <literal><![CDATA[
      -member(f1(difference(c1,c2),c3),c2)
    ]]></literal>
    <justification jstring="[resolve(56,a,6,b)].">
      <j1 rule="resolve" parents="56 6"/>
    </justification>
  </clause>

  <clause id="90">
    <literal><![CDATA[
      member(f1(difference(c1,c2),c3),c1)
    ]]></literal>
    <justification jstring="[resolve(56,a,5,b)].">
      <j1 rule="resolve" parents="56 5"/>
    </justification>
  </clause>

  <clause id="106">
    <literal><![CDATA[
      member(f1(difference(c1,c2),c3),union(c2,c3))
    ]]></literal>
    <justification jstring="[resolve(90,a,55,a)].">
      <j1 rule="resolve" parents="90 55"/>
    </justification>
  </clause>

  <clause id="148">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(106,a,3,c),unit_del(a,89),unit_del(b,57)].">
      <j1 rule="resolve" parents="106 3"/>
      <j2 rule="unit_del" parents="89"/>
      <j2 rule="unit_del" parents="57"/>
    </justification>
  </clause>

</proof>

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

<comments><![CDATA[
% Proof 1 at 0.01 (+ 0.00) seconds.
% Length of proof is 15.
% Level of proof is 4.
% Maximum clause weight is 11.
% Given clauses 43.

]]></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="55">
    <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="56">
    <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="57">
    <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="89">
    <literal><![CDATA[
      -member(f1(difference(c1,c3),c2),c3)
    ]]></literal>
    <justification jstring="[resolve(56,a,6,b)].">
      <j1 rule="resolve" parents="56 6"/>
    </justification>
  </clause>

  <clause id="90">
    <literal><![CDATA[
      member(f1(difference(c1,c3),c2),c1)
    ]]></literal>
    <justification jstring="[resolve(56,a,5,b)].">
      <j1 rule="resolve" parents="56 5"/>
    </justification>
  </clause>

  <clause id="106">
    <literal><![CDATA[
      member(f1(difference(c1,c3),c2),union(c2,c3))
    ]]></literal>
    <justification jstring="[resolve(90,a,55,a)].">
      <j1 rule="resolve" parents="90 55"/>
    </justification>
  </clause>

  <clause id="148">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(106,a,3,c),unit_del(a,57),unit_del(b,89)].">
      <j1 rule="resolve" parents="106 3"/>
      <j2 rule="unit_del" parents="57"/>
      <j2 rule="unit_del" parents="89"/>
    </justification>
  </clause>

</proof>

</proofs>
