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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 March-2007, March 2007.
Process 21008 was started by mccune on cleo,
Mon Mar 19 17:02:00 2007
The command was "/home/mccune/bin/prover9 -f cs.in DN.in".
]]></heading>

<proof number="1" length="14" max_count="23">

<comments><![CDATA[
% Proof 1 at 0.03 (+ 0.00) seconds: N.
% Length of proof is 14.
% Level of proof is 4.
% Maximum clause weight is 23.
% Given clauses 54.
]]></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="2" type="assumption">
    <literal><![CDATA[
      x * y != z * y
    ]]></literal>
    <literal><![CDATA[
      x = 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 @ 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" type="assumption">
    <literal><![CDATA[
      (A @ B) * C != C * (A @ B)
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="8">
    <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="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="18">
    <literal><![CDATA[
      (A @ B) * C * x != C * (A @ B) * x
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[ur(2,b,7,a),rewrite(3(6),3(12))].">
      <j1 rule="ur" parents="2 7"/>
      <j2 rule="rewrite" parents="3 3"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      x * y * (A @ B) * C * z != x * y * C * (A @ B) * z
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[ur(8,b,18,a)].">
      <j1 rule="ur" parents="8 18"/>
    </justification>
  </clause>

  <clause id="209">
    <literal><![CDATA[
      B * A * C * (A @ B) * x != A * B * C * x
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[para(14(a,1),24(a,1)),flip(a)].">
      <j1 rule="para" parents="14 24"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="219">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[para(4(a,1),209(a,2,2)),rewrite(16(13)),xx(a)].">
      <j1 rule="para" parents="4 209"/>
      <j2 rule="rewrite" parents="16"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
