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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>lm431hl.out</source>

<heading><![CDATA[
Prover9 (32) version April-2007, April 2007.
Process 14655 was started by hoefnepe on europa,
Fri Sep 21 17:18:30 2007
The command was "../../../kleene/Prover9/bin/prover9 -f lm431hl.in".
]]></heading>

<proof number="1" length="12" max_count="17">

<comments><![CDATA[
% Proof 1 at 0.00 (+ 0.00) seconds.
% Length of proof is 12.
% Level of proof is 3.
% Maximum clause weight is 16.
% Given clauses 5.
]]></comments>

  <clause id="1" type="assumption">
    <literal><![CDATA[
      x + y ; x = y ; x -> x + y ^ = y ^
    ]]></literal>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="2" type="goal">
    <literal><![CDATA[
      x ; (y ; x) ^ + (x ; y) ^ = (x ; y) ^
    ]]></literal>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="3" type="assumption">
    <literal><![CDATA[
      x ; (y ; z) = x ; y ; z
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      x ; y ; z = x ; (y ; z)
    ]]></literal>
    <justification jstring="[copy(3),flip(a)].">
      <j1 rule="copy" parents="3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="5" type="assumption">
    <literal><![CDATA[
      x + x = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6" type="assumption">
    <literal><![CDATA[
      x ; x ^ = x ^
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7" type="clausify">
    <literal><![CDATA[
      x ; y != y + x ; y
    ]]></literal>
    <literal><![CDATA[
      x ^ = y + x ^
    ]]></literal>
    <justification jstring="[clausify(1)].">
      <j1 rule="clausify" parents="1"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      x + y ; x != y ; x
    ]]></literal>
    <literal><![CDATA[
      x + y ^ = y ^
    ]]></literal>
    <justification jstring="[copy(7),flip(a),flip(b)].">
      <j1 rule="copy" parents="7"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9" type="deny">
    <literal><![CDATA[
      (c1 ; c2) ^ != c1 ; (c2 ; c1) ^ + (c1 ; c2) ^
    ]]></literal>
    <justification jstring="[deny(2)].">
      <j1 rule="deny" parents="2"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      c1 ; (c2 ; c1) ^ + (c1 ; c2) ^ != (c1 ; c2) ^
    ]]></literal>
    <justification jstring="[copy(9),flip(a)].">
      <j1 rule="copy" parents="9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      x ; (y ; (x ; y) ^) = (x ; y) ^
    ]]></literal>
    <justification jstring="[para(6(a,1),4(a,1)),flip(a)].">
      <j1 rule="para" parents="6 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[ur(8,b,10,a),rewrite(4(16),11(15),5(13),4(16),11(15)),xx(a)].">
      <j1 rule="ur" parents="8 10"/>
      <j2 rule="rewrite" parents="4 11 5 4 11"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
