<?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="1">

<source>lm431.out</source>

<heading><![CDATA[
Prover9 (32) version April-2007, April 2007.
Process 14645 was started by hoefnepe on europa,
Fri Sep 21 17:15:45 2007
The command was "../../../kleene/Prover9/bin/prover9 -f lm431.in".
]]></heading>

<proof number="1" length="19" max_count="31">

<comments><![CDATA[
% Proof 1 at 98.20 (+ 1.67) seconds.
% Length of proof is 19.
% Level of proof is 7.
% Maximum clause weight is 31.
% Given clauses 1375.
]]></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 = y + x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="4" type="assumption">
    <literal><![CDATA[
      x ; (y ; z) = x ; y ; z
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

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

  <clause id="9" type="assumption">
    <literal><![CDATA[
      x ; x ^ = x ^
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="10" 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="11">
    <literal><![CDATA[
      x + y ; x != y ; x
    ]]></literal>
    <literal><![CDATA[
      x + y ^ = y ^
    ]]></literal>
    <justification jstring="[copy(10),flip(a),flip(b)].">
      <j1 rule="copy" parents="10"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="12" 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="13">
    <literal><![CDATA[
      (c1 ; c2) ^ + c1 ; (c2 ; c1) ^ != (c1 ; c2) ^
    ]]></literal>
    <justification jstring="[copy(12),rewrite(3(15)),flip(a)].">
      <j1 rule="copy" parents="12"/>
      <j2 rule="rewrite" parents="3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      x ; (y ; (x ; y) ^) = (x ; y) ^
    ]]></literal>
    <justification jstring="[para(9(a,1),5(a,1)),flip(a)].">
      <j1 rule="para" parents="9 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      x + y ; (z ; x) != y ; (z ; x)
    ]]></literal>
    <literal><![CDATA[
      x + (y ; z) ^ = (y ; z) ^
    ]]></literal>
    <justification jstring="[para(5(a,1),11(a,1,2)),rewrite(5(5))].">
      <j1 rule="para" parents="5 11"/>
      <j2 rule="rewrite" parents="5"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      (x + y) ; (z ; x) ^ != y ; (z ; x) ^
    ]]></literal>
    <literal><![CDATA[
      x ; (z ; x) ^ + (y ; z) ^ = (y ; z) ^
    ]]></literal>
    <justification jstring="[para(14(a,1),15(a,1,2,2)),rewrite(8(7),14(8))].">
      <j1 rule="para" parents="14 15"/>
      <j2 rule="rewrite" parents="8 14"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      x ; (y ; x) ^ + (x ; y) ^ = (x ; y) ^
    ]]></literal>
    <justification jstring="[para(6(a,1),16(a,1,1)),xx(a)].">
      <j1 rule="para" parents="6 16"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      (x ; y) ^ + x ; (y ; x) ^ = (x ; y) ^
    ]]></literal>
    <justification jstring="[para(17(a,1),3(a,1)),flip(a)].">
      <j1 rule="para" parents="17 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(18,a,13,a)].">
      <j1 rule="resolve" parents="18 13"/>
    </justification>
  </clause>

</proof>

</proofs>
