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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>bw.out</source>

<heading><![CDATA[
Prover9 (32) version 2009-02A, February 2009.
Process 11192 was started by mccune on cleo,
Wed Feb 25 09:32:17 2009
The command was "/home/mccune/bin/prover9 -f bw.in".
]]></heading>

<proof number="1" length="13" max_count="60">

<comments><![CDATA[
% Proof 1 at 5.80 (+ 0.02) seconds: fixed_point_combinator.
% Length of proof is 13.
% Level of proof is 6.
% Maximum clause weight is 60.
% Given clauses 117.

]]></comments>

  <clause id="1" type="goal">
    <literal><![CDATA[
      (exists Q all x a(Q,x) = a(x,a(Q,x)))
    ]]></literal>
    <attribute><![CDATA[
      label(fixed_point_combinator)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="2" type="assumption">
    <literal><![CDATA[
      a(a(a(B,x),y),z) = a(x,a(y,z))
    ]]></literal>
    <attribute><![CDATA[
      label(B)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="3" type="assumption">
    <literal><![CDATA[
      a(a(W,x),y) = a(a(x,y),y)
    ]]></literal>
    <attribute><![CDATA[
      label(W)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="4" type="deny">
    <literal><![CDATA[
      a(x,f1(x)) != a(f1(x),a(x,f1(x)))
    ]]></literal>
    <attribute><![CDATA[
      label(fixed_point_combinator)
    ]]></attribute>
    <attribute><![CDATA[
      answer(fixed_point_combinator)
    ]]></attribute>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

  <clause id="5">
    <literal><![CDATA[
      a(f1(x),a(x,f1(x))) != a(x,f1(x))
    ]]></literal>
    <attribute><![CDATA[
      answer(fixed_point_combinator)
    ]]></attribute>
    <justification jstring="[copy(4),flip(a)].">
      <j1 rule="copy" parents="4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      a(a(W,a(B,x)),y) = a(x,a(y,y))
    ]]></literal>
    <justification jstring="[para(3(a,2),2(a,1))].">
      <j1 rule="para" parents="3 2"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      a(a(W,W),x) = a(a(x,x),x)
    ]]></literal>
    <justification jstring="[para(3(a,1),3(a,2))].">
      <j1 rule="para" parents="3 3"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      a(f1(a(a(B,x),y)),a(x,a(y,f1(a(a(B,x),y))))) != a(x,a(y,f1(a(a(B,x),y))))
    ]]></literal>
    <attribute><![CDATA[
      answer(fixed_point_combinator)
    ]]></attribute>
    <justification jstring="[para(2(a,1),5(a,1,2)),rewrite([2(19)])].">
      <j1 rule="para" parents="2 5"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      a(f1(a(a(B,x),a(a(B,y),z))),a(x,a(y,a(z,f1(a(a(B,x),a(a(B,y),z))))))) != a(x,a(y,a(z,f1(a(a(B,x),a(a(B,y),z))))))
    ]]></literal>
    <attribute><![CDATA[
      answer(fixed_point_combinator)
    ]]></attribute>
    <justification jstring="[para(2(a,1),13(a,1,2,2)),rewrite([2(29)])].">
      <j1 rule="para" parents="2 13"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="119">
    <literal><![CDATA[
      a(a(a(W,a(B,a(B,x))),y),z) = a(x,a(a(y,y),z))
    ]]></literal>
    <justification jstring="[para(8(a,2),2(a,1,1))].">
      <j1 rule="para" parents="8 2"/>
    </justification>
  </clause>

  <clause id="6487">
    <literal><![CDATA[
      a(f1(a(a(B,a(a(B,x),y)),a(a(B,z),u))),a(x,a(y,a(z,a(u,f1(a(a(B,a(a(B,x),y)),a(a(B,z),u)))))))) != a(x,a(y,a(z,a(u,f1(a(a(B,a(a(B,x),y)),a(a(B,z),u)))))))
    ]]></literal>
    <attribute><![CDATA[
      answer(fixed_point_combinator)
    ]]></attribute>
    <justification jstring="[para(2(a,1),27(a,1,2)),rewrite([2(41)])].">
      <j1 rule="para" parents="2 27"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="7180">
    <literal><![CDATA[
      a(x,a(a(W,W),a(W,a(B,a(B,x))))) = a(a(W,W),a(W,a(B,a(B,x))))
    ]]></literal>
    <justification jstring="[para(119(a,1),11(a,2)),rewrite([11(30,R)]),flip(a)].">
      <j1 rule="para" parents="119 11"/>
      <j2 rule="rewrite" parents="11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="7181">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(fixed_point_combinator)
    ]]></attribute>
    <justification jstring="[resolve(7180,a,6487,a)].">
      <j1 rule="resolve" parents="7180 6487"/>
    </justification>
  </clause>

</proof>

</proofs>
