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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

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

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13112 was started by mccune on cleo.thornwood,
Mon Jun 19 16:41:24 2006
The command was "/home/mccune/bin/prover9 -f cs.in AD.in".
]]></heading>

<proof number="1" length="24" max_count="25">

<comments><![CDATA[
% Proof 1 at 0.04 (+ 0.00) seconds: D.
% Length of proof is 24.
% Level of proof is 7.
% Maximum clause weight is 25.
% Given clauses 52.
]]></comments>

  <clause id="6">
    <literal><![CDATA[
      (A @ C) * (B @ C) != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <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 = u @ (v @ w)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      A * B @ C != (A @ C) * (B @ C)
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[copy(6),flip(a)].">
      <j1 rule="copy" parents="6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      x * y * z != x * y * u
    ]]></literal>
    <literal><![CDATA[
      z = u
    ]]></literal>
    <justification jstring="[para(9(a,1),7(a,1)),demod(9(4))].">
      <j1 rule="para" parents="9 7"/>
      <j2 rule="demod" parents="9"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      x * (x @ x) = x
    ]]></literal>
    <justification jstring="[hyper(7,a,10,a)].">
      <j1 rule="hyper" parents="7 10"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      x * y != z * x
    ]]></literal>
    <literal><![CDATA[
      z * (z @ x) = y
    ]]></literal>
    <justification jstring="[para(10(a,1),7(a,1)),flip(a)].">
      <j1 rule="para" parents="10 7"/>
      <j2 rule="flip"/>
    </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="22">
    <literal><![CDATA[
      (x @ y) * z = z * (x @ y) * (u @ (v @ w))
    ]]></literal>
    <justification jstring="[para(11(a,1),10(a,1,2,2)),flip(a)].">
      <j1 rule="para" parents="11 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      (A * B @ C) * x != (A @ C) * (B @ C) * x
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[ur(8,b,12,a),demod(9(14))].">
      <j1 rule="ur" parents="8 12"/>
      <j2 rule="demod" parents="9"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      (A @ C) * (B @ C) * x * (x @ (A * B @ C)) != x * (A * B @ C)
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(10(a,1),26(a,1)),flip(a)].">
      <j1 rule="para" parents="10 26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      x * (x @ x) * y = x * y
    ]]></literal>
    <justification jstring="[para(15(a,1),9(a,1,1)),flip(a)].">
      <j1 rule="para" parents="15 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      (x @ y) * (z @ (u @ v)) = x @ y
    ]]></literal>
    <justification jstring="[para(11(a,1),15(a,1,2))].">
      <j1 rule="para" parents="11 15"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      (x @ y) * z = z * (x @ y)
    ]]></literal>
    <justification jstring="[back_demod(22),demod(38(6))].">
      <j1 rule="back_demod" parents="22"/>
      <j2 rule="demod" parents="38"/>
    </justification>
  </clause>

  <clause id="95">
    <literal><![CDATA[
      (x @ x) * y = y
    ]]></literal>
    <justification jstring="[hyper(13,a,36,a)].">
      <j1 rule="hyper" parents="13 36"/>
    </justification>
  </clause>

  <clause id="114">
    <literal><![CDATA[
      x * ((y @ y) @ x) = x
    ]]></literal>
    <justification jstring="[para(95(a,1),10(a,1,2)),demod(95(5))].">
      <j1 rule="para" parents="95 10"/>
      <j2 rule="demod" parents="95"/>
    </justification>
  </clause>

  <clause id="116">
    <literal><![CDATA[
      ((x @ y) @ z) * u = u
    ]]></literal>
    <justification jstring="[para(11(a,2),95(a,1,1))].">
      <j1 rule="para" parents="11 95"/>
    </justification>
  </clause>

  <clause id="155">
    <literal><![CDATA[
      x * (y @ x) * y * z = y * x * z
    ]]></literal>
    <justification jstring="[para(19(a,1),19(a,1,2)),demod(116(7))].">
      <j1 rule="para" parents="19 19"/>
      <j2 rule="demod" parents="116"/>
    </justification>
  </clause>

  <clause id="165">
    <literal><![CDATA[
      x * (y @ (z @ u)) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),114(a,1,2))].">
      <j1 rule="para" parents="11 114"/>
    </justification>
  </clause>

  <clause id="180">
    <literal><![CDATA[
      (A @ C) * (B @ C) * x != x * (A * B @ C)
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[back_demod(29),demod(165(13))].">
      <j1 rule="back_demod" parents="29"/>
      <j2 rule="demod" parents="165"/>
    </justification>
  </clause>

  <clause id="285">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[ur(16,b,180,a(flip)),demod(41(11),9(11),155(13),10(8),9(10)),xx(a)].">
      <j1 rule="ur" parents="16 180"/>
      <j2 rule="demod" parents="41 9 155 10 9"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
