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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 June-2006C, June 2006.
Process 13123 was started by mccune on cleo.thornwood,
Mon Jun 19 16:41:34 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="5">
    <literal><![CDATA[
      (x @ y) * (z @ y) = x * z @ y
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      x * y != x * z
    ]]></literal>
    <literal><![CDATA[
      y = z
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      x * y != z * y
    ]]></literal>
    <literal><![CDATA[
      x = z
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      (x * y) * z = x * y * z
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      x * y * (y @ x) = y * x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="11">
    <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="12">
    <literal><![CDATA[
      (A @ B) * C != C * (A @ B)
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      x * y * z != u * z
    ]]></literal>
    <literal><![CDATA[
      x * y = u
    ]]></literal>
    <justification jstring="[para(9(a,1),8(a,1))].">
      <j1 rule="para" parents="9 8"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      x * y * (y @ x) * z = y * x * z
    ]]></literal>
    <justification jstring="[para(10(a,1),9(a,1,1)),demod(9(2),9(5)),flip(a)].">
      <j1 rule="para" parents="10 9"/>
      <j2 rule="demod" parents="9 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      x * y * z * (y @ x) * (z @ x) = y * z * x
    ]]></literal>
    <justification jstring="[para(9(a,1),10(a,1,2)),demod(11(2),9(8))].">
      <j1 rule="para" parents="9 10"/>
      <j2 rule="demod" parents="11 9"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      x * (A @ B) * C != x * C * (A @ B)
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[ur(7,b,12,a)].">
      <j1 rule="ur" parents="7 12"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      x * y * (A @ B) * C != x * y * C * (A @ B)
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[ur(7,b,24,a)].">
      <j1 rule="ur" parents="7 24"/>
    </justification>
  </clause>

  <clause id="163">
    <literal><![CDATA[
      B * A * C * (A @ B) != A * B * C
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[para(19(a,1),30(a,1)),flip(a)].">
      <j1 rule="para" parents="19 30"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="167">
    <literal><![CDATA[
      B * A * C * (A @ B) * x != A * B * C * x
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[ur(14,b,163,a),demod(9(9),9(8),9(16),9(15))].">
      <j1 rule="ur" parents="14 163"/>
      <j2 rule="demod" parents="9 9 9 9"/>
    </justification>
  </clause>

  <clause id="219">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[para(10(a,1),167(a,2,2)),demod(21(13)),xx(a)].">
      <j1 rule="para" parents="10 167"/>
      <j2 rule="demod" parents="21"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
