<?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>NE-cs.out</source>

<heading><![CDATA[
Prover9 (32) version 22-May-2007, May 2007.
Process 27462 was started by mccune on cleo,
Tue May 22 14:46:06 2007
The command was "/home/mccune/bin/prover9 -f cs.in NE.in".
]]></heading>

<proof number="1" length="22" max_count="19">

<comments><![CDATA[
% Proof 1 at 0.10 (+ 0.00) seconds: E.
% Length of proof is 22.
% Level of proof is 10.
% Maximum clause weight is 19.
% Given clauses 100.
]]></comments>

  <clause id="1" type="assumption">
    <literal><![CDATA[
      x * y != x * z
    ]]></literal>
    <literal><![CDATA[
      y = z
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </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" type="assumption">
    <literal><![CDATA[
      x * y * (y @ x) = y * x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="5" type="assumption">
    <literal><![CDATA[
      (x @ y) * z = z * (x @ y)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6" type="assumption">
    <literal><![CDATA[
      A * B * C * B * A != B * A * C * A * B
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      x * y * z != x * y * u
    ]]></literal>
    <literal><![CDATA[
      z = u
    ]]></literal>
    <justification jstring="[para(3(a,1),1(a,1)),rewrite([3(4)])].">
      <j1 rule="para" parents="3 1"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      x * (x @ x) = x
    ]]></literal>
    <justification jstring="[hyper(1,a,4,a)].">
      <j1 rule="hyper" parents="1 4"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      x * y != z * x
    ]]></literal>
    <literal><![CDATA[
      z * (z @ x) = y
    ]]></literal>
    <justification jstring="[para(4(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="4 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      x * y * (y @ x) * z = y * x * z
    ]]></literal>
    <justification jstring="[para(4(a,1),3(a,1,1)),rewrite([3(2),3(5)]),flip(a)].">
      <j1 rule="para" parents="4 3"/>
      <j2 rule="rewrite" parents="3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      x @ x = y
    ]]></literal>
    <justification jstring="[para(9(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="9 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      x * (x @ x) * y = x * y
    ]]></literal>
    <justification jstring="[para(9(a,1),3(a,1,1)),flip(a)].">
      <j1 rule="para" parents="9 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="64">
    <literal><![CDATA[
      (x @ x) * y = y
    ]]></literal>
    <justification jstring="[hyper(7,a,38,a)].">
      <j1 rule="hyper" parents="7 38"/>
    </justification>
  </clause>

  <clause id="71">
    <literal><![CDATA[
      x * (x @ (y @ z)) = x
    ]]></literal>
    <justification jstring="[hyper(10,a,5,a)].">
      <j1 rule="hyper" parents="10 5"/>
    </justification>
  </clause>

  <clause id="83">
    <literal><![CDATA[
      x * (y @ y) = x
    ]]></literal>
    <justification jstring="[hyper(10,a,64,a),rewrite([71(3)]),flip(a)].">
      <j1 rule="hyper" parents="10 64"/>
      <j2 rule="rewrite" parents="71"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      x @ x = y @ y
    ]]></literal>
    <justification jstring="[hyper(35,a,83,a)].">
      <j1 rule="hyper" parents="35 83"/>
    </justification>
  </clause>

  <clause id="90">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      z @ z = y
    ]]></literal>
    <justification jstring="[para(83(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="83 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="93">
    <literal><![CDATA[
      x @ x = c_0
    ]]></literal>
    <justification jstring="[new_symbol(89)].">
      <j1 rule="new_symbol" parents="89"/>
    </justification>
  </clause>

  <clause id="96">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      c_0 = y
    ]]></literal>
    <justification jstring="[back_rewrite(90),rewrite([93(3)])].">
      <j1 rule="back_rewrite" parents="90"/>
      <j2 rule="rewrite" parents="93"/>
    </justification>
  </clause>

  <clause id="101">
    <literal><![CDATA[
      x * c_0 = x
    ]]></literal>
    <justification jstring="[back_rewrite(83),rewrite([93(1)])].">
      <j1 rule="back_rewrite" parents="83"/>
      <j2 rule="rewrite" parents="93"/>
    </justification>
  </clause>

  <clause id="139">
    <literal><![CDATA[
      x @ (y @ z) = c_0
    ]]></literal>
    <justification jstring="[hyper(96,a,71,a),flip(a)].">
      <j1 rule="hyper" parents="96 71"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="145">
    <literal><![CDATA[
      x * y * z * (y @ x) = y * x * z
    ]]></literal>
    <justification jstring="[para(4(a,1),13(a,1,2,2)),rewrite([139(6),101(6)])].">
      <j1 rule="para" parents="4 13"/>
      <j2 rule="rewrite" parents="139 101"/>
    </justification>
  </clause>

  <clause id="674">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[para(145(a,2),6(a,1)),rewrite([3(11),3(10),4(10)]),xx(a)].">
      <j1 rule="para" parents="145 6"/>
      <j2 rule="rewrite" parents="3 3 4"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
