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<!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 August-2006A, August 2006.
Process 11370 was started by mccune on cleo.thornwood,
Sat Aug 12 21:00:08 2006
The command was "/home/mccune/bin/prover9 -f cs.in NE.in".
]]></heading>

<proof number="1" length="25" max_count="19">

<comments><![CDATA[
% Proof 1 at 0.14 (+ 0.00) seconds: E.
% Length of proof is 25.
% Level of proof is 8.
% Maximum clause weight is 19.
% Given clauses 118.
]]></comments>

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

  <clause id="2">
    <literal><![CDATA[
      x * y != z * y
    ]]></literal>
    <literal><![CDATA[
      x = z
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="3">
    <literal><![CDATA[
      (x * y) * z = x * y * z
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

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

  <clause id="6">
    <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="11">
    <literal><![CDATA[
      x * y * (y @ z) != y * z
    ]]></literal>
    <literal><![CDATA[
      z = x
    ]]></literal>
    <justification jstring="[para(4(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="4 2"/>
      <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="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="64">
    <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="75">
    <literal><![CDATA[
      x @ (y @ z) = x @ x
    ]]></literal>
    <justification jstring="[hyper(27,a,64,a),flip(a)].">
      <j1 rule="hyper" parents="27 64"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="92">
    <literal><![CDATA[
      (x @ (y @ z)) * u = u * (x @ x)
    ]]></literal>
    <justification jstring="[para(75(a,2),5(a,1,1))].">
      <j1 rule="para" parents="75 5"/>
    </justification>
  </clause>

  <clause id="99">
    <literal><![CDATA[
      x * y != y
    ]]></literal>
    <literal><![CDATA[
      y @ y = x
    ]]></literal>
    <justification jstring="[para(9(a,1),11(a,2)),rewrite(64(3))].">
      <j1 rule="para" parents="9 11"/>
      <j2 rule="rewrite" parents="64"/>
    </justification>
  </clause>

  <clause id="107">
    <literal><![CDATA[
      (x @ x) * y = y
    ]]></literal>
    <justification jstring="[hyper(7,a,30,a)].">
      <j1 rule="hyper" parents="7 30"/>
    </justification>
  </clause>

  <clause id="116">
    <literal><![CDATA[
      x @ x = y @ y
    ]]></literal>
    <justification jstring="[hyper(99,a,107,a)].">
      <j1 rule="hyper" parents="99 107"/>
    </justification>
  </clause>

  <clause id="124">
    <literal><![CDATA[
      (x @ (y @ z)) * u = u
    ]]></literal>
    <justification jstring="[para(75(a,2),107(a,1,1))].">
      <j1 rule="para" parents="75 107"/>
    </justification>
  </clause>

  <clause id="128">
    <literal><![CDATA[
      x @ x = c_0
    ]]></literal>
    <justification jstring="[new_symbol(116)].">
      <j1 rule="new_symbol" parents="116"/>
    </justification>
  </clause>

  <clause id="130">
    <literal><![CDATA[
      x * c_0 = x
    ]]></literal>
    <justification jstring="[back_rewrite(92),rewrite(124(3),128(1)),flip(a)].">
      <j1 rule="back_rewrite" parents="92"/>
      <j2 rule="rewrite" parents="124 128"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="144">
    <literal><![CDATA[
      x @ (y @ z) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(75),rewrite(128(3))].">
      <j1 rule="back_rewrite" parents="75"/>
      <j2 rule="rewrite" parents="128"/>
    </justification>
  </clause>

  <clause id="154">
    <literal><![CDATA[
      x * y * z * (y @ x) = y * x * z
    ]]></literal>
    <justification jstring="[para(4(a,1),13(a,1,2,2)),rewrite(144(6),130(6))].">
      <j1 rule="para" parents="4 13"/>
      <j2 rule="rewrite" parents="144 130"/>
    </justification>
  </clause>

  <clause id="808">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[para(154(a,2),6(a,1)),rewrite(3(11),3(10),4(10)),xx(a)].">
      <j1 rule="para" parents="154 6"/>
      <j2 rule="rewrite" parents="3 3 4"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
