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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

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

<heading><![CDATA[
Prover9 (32) version August-2006A, August 2006.
Process 11357 was started by mccune on cleo.thornwood,
Sat Aug 12 20:59:57 2006
The command was "/home/mccune/bin/prover9 -f cs.in DN.in".
]]></heading>

<proof number="1" length="15" max_count="19">

<comments><![CDATA[
% Proof 1 at 0.02 (+ 0.00) seconds: N.
% Length of proof is 15.
% Level of proof is 5.
% Maximum clause weight is 19.
% Given clauses 42.
]]></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 @ y) = x * z @ y
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6">
    <literal><![CDATA[
      x * y @ z = (x @ z) * (y @ z)
    ]]></literal>
    <justification jstring="[copy(5),flip(a)].">
      <j1 rule="copy" parents="5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      (A @ B) * C != C * (A @ B)
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      x * y * z != u * z
    ]]></literal>
    <literal><![CDATA[
      x * y = u
    ]]></literal>
    <justification jstring="[para(3(a,1),2(a,1))].">
      <j1 rule="para" parents="3 2"/>
    </justification>
  </clause>

  <clause id="14">
    <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="16">
    <literal><![CDATA[
      x * y * z * (y @ x) * (z @ x) = y * z * x
    ]]></literal>
    <justification jstring="[para(3(a,1),4(a,1,2)),rewrite(6(2),3(8))].">
      <j1 rule="para" parents="3 4"/>
      <j2 rule="rewrite" parents="6 3"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      x * (A @ B) * C != x * C * (A @ B)
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[ur(1,b,7,a)].">
      <j1 rule="ur" parents="1 7"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      x * y * (A @ B) * C != x * y * C * (A @ B)
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[ur(1,b,19,a)].">
      <j1 rule="ur" parents="1 19"/>
    </justification>
  </clause>

  <clause id="158">
    <literal><![CDATA[
      B * A * C * (A @ B) != A * B * C
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[para(14(a,1),25(a,1)),flip(a)].">
      <j1 rule="para" parents="14 25"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="162">
    <literal><![CDATA[
      B * A * C * (A @ B) * x != A * B * C * x
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[ur(9,b,158,a),rewrite(3(9),3(8),3(16),3(15))].">
      <j1 rule="ur" parents="9 158"/>
      <j2 rule="rewrite" parents="3 3 3 3"/>
    </justification>
  </clause>

  <clause id="214">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[para(4(a,1),162(a,2,2)),rewrite(16(13)),xx(a)].">
      <j1 rule="para" parents="4 162"/>
      <j2 rule="rewrite" parents="16"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
