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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>w_sk.out</source>

<heading><![CDATA[
Prover9 (32) version August-2006A, August 2006.
Process 10884 was started by mccune on cleo.thornwood,
Sat Aug 12 20:58:58 2006
The command was "/home/mccune/bin/prover9 -f w_sk.in".
]]></heading>

<proof number="1" length="9" max_count="45">

<comments><![CDATA[
% Proof 1 at 0.00 (+ 0.00) seconds.
% Length of proof is 9.
% Level of proof is 4.
% Maximum clause weight is 45.
% Given clauses 4.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      (exists W all x all y (a(a(W,x),y) = a(a(x,y),y)))
    ]]></literal>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

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

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

  <clause id="4">
    <literal><![CDATA[
      a(a(x,f1(x)),f2(x)) != a(a(f1(x),f2(x)),f2(x))
    ]]></literal>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

  <clause id="5">
    <literal><![CDATA[
      a(a(f1(x),f2(x)),f2(x)) != a(a(x,f1(x)),f2(x))
    ]]></literal>
    <justification jstring="[copy(4),flip(a)].">
      <j1 rule="copy" parents="4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      a(a(a(S,K),x),y) = y
    ]]></literal>
    <justification jstring="[para(3(a,1),2(a,2))].">
      <j1 rule="para" parents="3 2"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      a(a(f1(a(a(S,x),y)),f2(a(a(S,x),y))),f2(a(a(S,x),y))) != a(a(a(x,f1(a(a(S,x),y))),a(y,f1(a(a(S,x),y)))),f2(a(a(S,x),y)))
    ]]></literal>
    <justification jstring="[para(2(a,1),5(a,2,1))].">
      <j1 rule="para" parents="2 5"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      a(a(a(S,x),a(a(S,K),y)),z) = a(a(x,z),z)
    ]]></literal>
    <justification jstring="[para(14(a,1),2(a,2,2))].">
      <j1 rule="para" parents="14 2"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(20,a,15,a(flip))].">
      <j1 rule="resolve" parents="20 15"/>
    </justification>
  </clause>

</proof>

</proofs>
