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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>eq3iii.out</source>

<heading><![CDATA[
Prover9 (32) version April-2007, April 2007.
Process 14718 was started by hoefnepe on europa,
Fri Sep 21 17:36:21 2007
The command was "../../../kleene/Prover9/bin/prover9 -f eq3iii.in".
]]></heading>

<proof number="1" length="16" max_count="33">

<comments><![CDATA[
% Proof 1 at 0.11 (+ 0.00) seconds.
% Length of proof is 16.
% Level of proof is 4.
% Maximum clause weight is 26.
% Given clauses 110.
]]></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[
      (all p (p + 1 = 1 & p ; p = p -> (p ; x) ^ + (p ; x ; p) ^ = (p ; x ; p) ^))
    ]]></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[
      c2 ; c2 = c2
    ]]></literal>
    <justification jstring="[deny(2)].">
      <j1 rule="deny" parents="2"/>
    </justification>
  </clause>

  <clause id="10" type="deny">
    <literal><![CDATA[
      (c2 ; c1 ; c2) ^ != (c2 ; c1) ^ + (c2 ; c1 ; c2) ^
    ]]></literal>
    <justification jstring="[deny(2)].">
      <j1 rule="deny" parents="2"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      (c2 ; c1) ^ + (c2 ; (c1 ; c2)) ^ != (c2 ; (c1 ; c2)) ^
    ]]></literal>
    <justification jstring="[copy(10),rewrite(4(5),4(15)),flip(a)].">
      <j1 rule="copy" parents="10"/>
      <j2 rule="rewrite" parents="4 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="12">
    <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="13">
    <literal><![CDATA[
      c2 ; (c2 ; x) = c2 ; x
    ]]></literal>
    <justification jstring="[para(9(a,1),4(a,1,1)),flip(a)].">
      <j1 rule="para" parents="9 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      (c2 ; c1) ^ + c2 ; (c1 ; (c2 ; (c2 ; c1) ^)) != c2 ; (c1 ; (c2 ; (c2 ; c1) ^))
    ]]></literal>
    <justification jstring="[ur(8,b,11,a),rewrite(4(14),4(13),4(25),4(24))].">
      <j1 rule="ur" parents="8 11"/>
      <j2 rule="rewrite" parents="4 4 4 4"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      c2 ; (c2 ; x) ^ = (c2 ; x) ^
    ]]></literal>
    <justification jstring="[para(12(a,1),13(a,1,2)),rewrite(12(11))].">
      <j1 rule="para" parents="12 13"/>
      <j2 rule="rewrite" parents="12"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[back_rewrite(14),rewrite(15(12),12(12),5(9),15(12),12(12)),xx(a)].">
      <j1 rule="back_rewrite" parents="14"/>
      <j2 rule="rewrite" parents="15 12 5 15 12"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
