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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>AE-cs.out</source>

<heading><![CDATA[
Prover9 (32) version March-2007, March 2007.
Process 20996 was started by mccune on cleo,
Mon Mar 19 17:01:53 2007
The command was "/home/mccune/bin/prover9 -f cs.in AE.in".
]]></heading>

<proof number="1" length="23" max_count="19">

<comments><![CDATA[
% Proof 1 at 0.17 (+ 0.00) seconds: E.
% Length of proof is 23.
% Level of proof is 12.
% Maximum clause weight is 19.
% Given clauses 149.
]]></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 = u @ (v @ w)
    ]]></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="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="27">
    <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="30">
    <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="50">
    <literal><![CDATA[
      (x @ x) * y = y
    ]]></literal>
    <justification jstring="[hyper(7,a,30,a)].">
      <j1 rule="hyper" parents="7 30"/>
    </justification>
  </clause>

  <clause id="58">
    <literal><![CDATA[
      x * ((y @ y) @ x) = x
    ]]></literal>
    <justification jstring="[para(50(a,1),4(a,1,2)),rewrite(50(5))].">
      <j1 rule="para" parents="50 4"/>
      <j2 rule="rewrite" parents="50"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      x * (x @ (y @ y)) = x * (y @ y)
    ]]></literal>
    <justification jstring="[para(50(a,1),4(a,1))].">
      <j1 rule="para" parents="50 4"/>
    </justification>
  </clause>

  <clause id="73">
    <literal><![CDATA[
      (x @ x) @ y = y @ y
    ]]></literal>
    <justification jstring="[hyper(27,a,58,a),flip(a)].">
      <j1 rule="hyper" parents="27 58"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="78">
    <literal><![CDATA[
      x * (y @ (z @ u)) = x
    ]]></literal>
    <justification jstring="[para(5(a,1),58(a,1,2))].">
      <j1 rule="para" parents="5 58"/>
    </justification>
  </clause>

  <clause id="88">
    <literal><![CDATA[
      x * (y @ y) = x
    ]]></literal>
    <justification jstring="[back_rewrite(59),rewrite(78(3)),flip(a)].">
      <j1 rule="back_rewrite" parents="59"/>
      <j2 rule="rewrite" parents="78"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      x @ x = y @ y
    ]]></literal>
    <justification jstring="[hyper(27,a,88,a)].">
      <j1 rule="hyper" parents="27 88"/>
    </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="97">
    <literal><![CDATA[
      x * c_0 = x
    ]]></literal>
    <justification jstring="[back_rewrite(88),rewrite(93(1))].">
      <j1 rule="back_rewrite" parents="88"/>
      <j2 rule="rewrite" parents="93"/>
    </justification>
  </clause>

  <clause id="100">
    <literal><![CDATA[
      c_0 @ x = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(73),rewrite(93(1),93(3))].">
      <j1 rule="back_rewrite" parents="73"/>
      <j2 rule="rewrite" parents="93 93"/>
    </justification>
  </clause>

  <clause id="104">
    <literal><![CDATA[
      x @ (y @ z) = c_0
    ]]></literal>
    <justification jstring="[para(93(a,1),5(a,1,1)),rewrite(100(2)),flip(a)].">
      <j1 rule="para" parents="93 5"/>
      <j2 rule="rewrite" parents="100"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="121">
    <literal><![CDATA[
      x * y * z * (y @ x) = y * x * z
    ]]></literal>
    <justification jstring="[para(4(a,1),13(a,1,2,2)),rewrite(104(6),97(6))].">
      <j1 rule="para" parents="4 13"/>
      <j2 rule="rewrite" parents="104 97"/>
    </justification>
  </clause>

  <clause id="1062">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[para(121(a,2),6(a,1)),rewrite(3(11),3(10),4(10)),xx(a)].">
      <j1 rule="para" parents="121 6"/>
      <j2 rule="rewrite" parents="3 3 4"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
