<?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="4">

<source>SET598+3.out</source>

<heading><![CDATA[
FOF-Prover9 (32) version August-2006A, August 2006.
Process 11274 was started by mccune on cleo.thornwood,
Sat Aug 12 20:59:24 2006
The command was "/home/mccune/bin/fof-prover9 -f SET598+3.in".
]]></heading>

<proof number="1" length="5" max_count="5">

<comments><![CDATA[
% Proof 1 at 0.00 (+ 0.00) seconds.
% Length of proof is 5.
% Level of proof is 2.
% Maximum clause weight is 5.
% Given clauses 0.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      subset(intersection(A,B),A)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      -subset(c1,c2)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      intersection(c2,c3) = c1
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      c1 = intersection(c2,c3)
    ]]></literal>
    <justification jstring="[copy(31),flip(a)].">
      <j1 rule="copy" parents="31"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[back_rewrite(30),rewrite(32(1)),unit_del(a,1)].">
      <j1 rule="back_rewrite" parents="30"/>
      <j2 rule="rewrite" parents="32"/>
      <j2 rule="unit_del" parents="1"/>
    </justification>
  </clause>

</proof>

<proof number="2" length="8" max_count="7">

<comments><![CDATA[
% Proof 1 at 0.00 (+ 0.00) seconds.
% Length of proof is 8.
% Level of proof is 3.
% Maximum clause weight is 7.
% Given clauses 14.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      subset(intersection(A,B),A)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      intersection(A,B) = intersection(B,A)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      intersection(c2,c3) = c1
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      c1 = intersection(c2,c3)
    ]]></literal>
    <justification jstring="[copy(30),flip(a)].">
      <j1 rule="copy" parents="30"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      -subset(c1,c3)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      -subset(intersection(c2,c3),c3)
    ]]></literal>
    <justification jstring="[copy(32),rewrite(31(1))].">
      <j1 rule="copy" parents="32"/>
      <j2 rule="rewrite" parents="31"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      subset(intersection(A,B),B)
    ]]></literal>
    <justification jstring="[para(14(a,1),1(a,1))].">
      <j1 rule="para" parents="14 1"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(49,a,33,a)].">
      <j1 rule="resolve" parents="49 33"/>
    </justification>
  </clause>

</proof>

<proof number="3" length="8" max_count="11">

<comments><![CDATA[
% Proof 1 at 0.00 (+ 0.00) seconds.
% Length of proof is 8.
% Level of proof is 3.
% Maximum clause weight is 11.
% Given clauses 33.
]]></comments>

  <clause id="2">
    <literal><![CDATA[
      -subset(A,B)
    ]]></literal>
    <literal><![CDATA[
      -subset(A,C)
    ]]></literal>
    <literal><![CDATA[
      subset(A,intersection(B,C))
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      intersection(c2,c3) = c1
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      c1 = intersection(c2,c3)
    ]]></literal>
    <justification jstring="[copy(30),flip(a)].">
      <j1 rule="copy" parents="30"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      subset(c4,c2)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      subset(c4,c3)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      -subset(c4,c1)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      -subset(c4,intersection(c2,c3))
    ]]></literal>
    <justification jstring="[copy(34),rewrite(31(2))].">
      <j1 rule="copy" parents="34"/>
      <j2 rule="rewrite" parents="31"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[ur(2,b,33,a,c,35,a),unit_del(a,32)].">
      <j1 rule="ur" parents="2 33 35"/>
      <j2 rule="unit_del" parents="32"/>
    </justification>
  </clause>

</proof>

<proof number="4" length="13" max_count="11">

<comments><![CDATA[
% Proof 1 at 0.01 (+ 0.00) seconds.
% Length of proof is 13.
% Level of proof is 4.
% Maximum clause weight is 11.
% Given clauses 67.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      subset(intersection(A,B),A)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="2">
    <literal><![CDATA[
      -subset(A,B)
    ]]></literal>
    <literal><![CDATA[
      -subset(A,C)
    ]]></literal>
    <literal><![CDATA[
      subset(A,intersection(B,C))
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      A = B
    ]]></literal>
    <literal><![CDATA[
      -subset(B,A)
    ]]></literal>
    <literal><![CDATA[
      -subset(A,B)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      intersection(A,B) = intersection(B,A)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      subset(c1,c2)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      intersection(c2,c3) != c1
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      subset(c1,c3)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      -subset(A,c2)
    ]]></literal>
    <literal><![CDATA[
      -subset(A,c3)
    ]]></literal>
    <literal><![CDATA[
      subset(A,c1)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      subset(intersection(A,B),B)
    ]]></literal>
    <justification jstring="[para(14(a,1),1(a,1))].">
      <j1 rule="para" parents="14 1"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      -subset(c1,A)
    ]]></literal>
    <literal><![CDATA[
      subset(c1,intersection(A,c2))
    ]]></literal>
    <justification jstring="[resolve(30,a,2,b)].">
      <j1 rule="resolve" parents="30 2"/>
    </justification>
  </clause>

  <clause id="129">
    <literal><![CDATA[
      subset(c1,intersection(c2,c3))
    ]]></literal>
    <justification jstring="[resolve(61,a,32,a),rewrite(14(4))].">
      <j1 rule="resolve" parents="61 32"/>
      <j2 rule="rewrite" parents="14"/>
    </justification>
  </clause>

  <clause id="134">
    <literal><![CDATA[
      -subset(intersection(c2,c3),c1)
    ]]></literal>
    <justification jstring="[resolve(129,a,11,c),flip(a),unit_del(a,31)].">
      <j1 rule="resolve" parents="129 11"/>
      <j2 rule="flip"/>
      <j2 rule="unit_del" parents="31"/>
    </justification>
  </clause>

  <clause id="156">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[ur(33,b,49,a,c,134,a),unit_del(a,1)].">
      <j1 rule="ur" parents="33 49 134"/>
      <j2 rule="unit_del" parents="1"/>
    </justification>
  </clause>

</proof>

</proofs>
