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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 November-2006, November 2006.
Process 3855 was started by mccune on cleo.thornwood,
Wed Nov 22 11:24:10 2006
The command was "/home/mccune/bin/prover9 -f cs.in AE.in".
]]></heading>

<proof number="1" length="24" max_count="19">

<comments><![CDATA[
% Proof 1 at 0.16 (+ 0.01) seconds: E.
% Length of proof is 24.
% Level of proof is 12.
% Maximum clause weight is 19.
% Given clauses 146.
]]></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="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="24">
    <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="27">
    <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="60">
    <literal><![CDATA[
      (x @ x) * y = y
    ]]></literal>
    <justification jstring="[hyper(7,a,27,a)].">
      <j1 rule="hyper" parents="7 27"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      x * (x @ (y @ y)) = x * (y @ y)
    ]]></literal>
    <justification jstring="[hyper(10,a,60,a)].">
      <j1 rule="hyper" parents="10 60"/>
    </justification>
  </clause>

  <clause id="70">
    <literal><![CDATA[
      x * ((y @ y) @ x) = x
    ]]></literal>
    <justification jstring="[para(60(a,1),4(a,1,2)),rewrite(60(5))].">
      <j1 rule="para" parents="60 4"/>
      <j2 rule="rewrite" parents="60"/>
    </justification>
  </clause>

  <clause id="86">
    <literal><![CDATA[
      (x @ x) @ y = y @ y
    ]]></literal>
    <justification jstring="[hyper(24,a,70,a),flip(a)].">
      <j1 rule="hyper" parents="24 70"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="90">
    <literal><![CDATA[
      x * (y @ (z @ u)) = x
    ]]></literal>
    <justification jstring="[para(5(a,1),70(a,1,2))].">
      <j1 rule="para" parents="5 70"/>
    </justification>
  </clause>

  <clause id="102">
    <literal><![CDATA[
      x * (y @ y) = x
    ]]></literal>
    <justification jstring="[back_rewrite(68),rewrite(90(3)),flip(a)].">
      <j1 rule="back_rewrite" parents="68"/>
      <j2 rule="rewrite" parents="90"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="103">
    <literal><![CDATA[
      x @ x = y @ y
    ]]></literal>
    <justification jstring="[hyper(24,a,102,a)].">
      <j1 rule="hyper" parents="24 102"/>
    </justification>
  </clause>

  <clause id="107">
    <literal><![CDATA[
      x @ x = c_0
    ]]></literal>
    <justification jstring="[new_symbol(103)].">
      <j1 rule="new_symbol" parents="103"/>
    </justification>
  </clause>

  <clause id="111">
    <literal><![CDATA[
      x * c_0 = x
    ]]></literal>
    <justification jstring="[back_rewrite(102),rewrite(107(1))].">
      <j1 rule="back_rewrite" parents="102"/>
      <j2 rule="rewrite" parents="107"/>
    </justification>
  </clause>

  <clause id="115">
    <literal><![CDATA[
      c_0 @ x = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(86),rewrite(107(1),107(3))].">
      <j1 rule="back_rewrite" parents="86"/>
      <j2 rule="rewrite" parents="107 107"/>
    </justification>
  </clause>

  <clause id="122">
    <literal><![CDATA[
      x @ (y @ z) = c_0
    ]]></literal>
    <justification jstring="[para(107(a,1),5(a,1,1)),rewrite(115(2)),flip(a)].">
      <j1 rule="para" parents="107 5"/>
      <j2 rule="rewrite" parents="115"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="132">
    <literal><![CDATA[
      x * y * z * (y @ x) = y * x * z
    ]]></literal>
    <justification jstring="[para(4(a,1),13(a,1,2,2)),rewrite(122(6),111(6))].">
      <j1 rule="para" parents="4 13"/>
      <j2 rule="rewrite" parents="122 111"/>
    </justification>
  </clause>

  <clause id="1046">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[para(132(a,2),6(a,1)),rewrite(3(11),3(10),4(10)),xx(a)].">
      <j1 rule="para" parents="132 6"/>
      <j2 rule="rewrite" parents="3 3 4"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
