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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

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

<heading><![CDATA[
Prover9 (32) version 22-May-2007, May 2007.
Process 27460 was started by mccune on cleo,
Tue May 22 14:46:01 2007
The command was "/home/mccune/bin/prover9 -f cs.in ND.in".
]]></heading>

<proof number="1" length="31" max_count="23">

<comments><![CDATA[
% Proof 1 at 0.08 (+ 0.00) seconds: D.
% Length of proof is 31.
% Level of proof is 11.
% Maximum clause weight is 23.
% Given clauses 84.
]]></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 = z * (x @ y)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

  <clause id="7">
    <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="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="10">
    <literal><![CDATA[
      x * (x @ x) = x
    ]]></literal>
    <justification jstring="[hyper(1,a,4,a)].">
      <j1 rule="hyper" parents="1 4"/>
    </justification>
  </clause>

  <clause id="11">
    <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="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="24">
    <literal><![CDATA[
      (A * B @ C) * x != (A @ C) * (B @ C) * x
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[ur(2,b,7,a),rewrite([3(14)])].">
      <j1 rule="ur" parents="2 7"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      (A @ C) * (B @ C) * x != x * (A * B @ C)
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(5(a,1),24(a,1)),flip(a)].">
      <j1 rule="para" parents="5 24"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      (A @ C) * x * (B @ C) != x * (A * B @ C)
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(5(a,1),40(a,1,2))].">
      <j1 rule="para" parents="5 40"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      x @ x = y
    ]]></literal>
    <justification jstring="[para(10(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="10 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="58">
    <literal><![CDATA[
      x * (x @ x) * y = x * y
    ]]></literal>
    <justification jstring="[para(10(a,1),3(a,1,1)),flip(a)].">
      <j1 rule="para" parents="10 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="98">
    <literal><![CDATA[
      (x @ x) * y = y
    ]]></literal>
    <justification jstring="[hyper(8,a,58,a)].">
      <j1 rule="hyper" parents="8 58"/>
    </justification>
  </clause>

  <clause id="106">
    <literal><![CDATA[
      x * (x @ (y @ z)) = x
    ]]></literal>
    <justification jstring="[hyper(11,a,5,a)].">
      <j1 rule="hyper" parents="11 5"/>
    </justification>
  </clause>

  <clause id="111">
    <literal><![CDATA[
      C * (A @ C) * A * B * (B @ C) != A * B * C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[ur(11,b,50,a(flip)),rewrite([3(11),3(18)])].">
      <j1 rule="ur" parents="11 50"/>
      <j2 rule="rewrite" parents="3 3"/>
    </justification>
  </clause>

  <clause id="120">
    <literal><![CDATA[
      x * C * (A @ C) * A * B * (B @ C) != x * A * B * C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[ur(1,b,111,a)].">
      <j1 rule="ur" parents="1 111"/>
    </justification>
  </clause>

  <clause id="135">
    <literal><![CDATA[
      x * (y @ y) = x
    ]]></literal>
    <justification jstring="[hyper(11,a,98,a),rewrite([106(3)]),flip(a)].">
      <j1 rule="hyper" parents="11 98"/>
      <j2 rule="rewrite" parents="106"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="141">
    <literal><![CDATA[
      x @ x = y @ y
    ]]></literal>
    <justification jstring="[hyper(55,a,135,a)].">
      <j1 rule="hyper" parents="55 135"/>
    </justification>
  </clause>

  <clause id="143">
    <literal><![CDATA[
      x * y * z != x * y
    ]]></literal>
    <literal><![CDATA[
      u @ u = z
    ]]></literal>
    <justification jstring="[para(135(a,1),8(a,1,2)),flip(a)].">
      <j1 rule="para" parents="135 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="145">
    <literal><![CDATA[
      x @ x = c_0
    ]]></literal>
    <justification jstring="[new_symbol(141)].">
      <j1 rule="new_symbol" parents="141"/>
    </justification>
  </clause>

  <clause id="147">
    <literal><![CDATA[
      x * y * z != x * y
    ]]></literal>
    <literal><![CDATA[
      c_0 = z
    ]]></literal>
    <justification jstring="[back_rewrite(143),rewrite([145(5)])].">
      <j1 rule="back_rewrite" parents="143"/>
      <j2 rule="rewrite" parents="145"/>
    </justification>
  </clause>

  <clause id="154">
    <literal><![CDATA[
      c_0 * x = x
    ]]></literal>
    <justification jstring="[back_rewrite(98),rewrite([145(1)])].">
      <j1 rule="back_rewrite" parents="98"/>
      <j2 rule="rewrite" parents="145"/>
    </justification>
  </clause>

  <clause id="265">
    <literal><![CDATA[
      x * y * ((x @ y) @ x) * z = y * (x @ y) * x * z
    ]]></literal>
    <justification jstring="[para(14(a,1),14(a,1,2)),flip(a)].">
      <j1 rule="para" parents="14 14"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="320">
    <literal><![CDATA[
      x * y != y * x
    ]]></literal>
    <literal><![CDATA[
      x @ y = c_0
    ]]></literal>
    <justification jstring="[para(4(a,1),147(a,1)),flip(b)].">
      <j1 rule="para" parents="4 147"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="409">
    <literal><![CDATA[
      (x @ y) @ z = c_0
    ]]></literal>
    <justification jstring="[hyper(320,a,5,a)].">
      <j1 rule="hyper" parents="320 5"/>
    </justification>
  </clause>

  <clause id="414">
    <literal><![CDATA[
      x * (y @ x) * y * z = y * x * z
    ]]></literal>
    <justification jstring="[back_rewrite(265),rewrite([409(2),154(2)]),flip(a)].">
      <j1 rule="back_rewrite" parents="265"/>
      <j2 rule="rewrite" parents="409 154"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="415">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[back_rewrite(120),rewrite([414(13),4(8)]),xx(a)].">
      <j1 rule="back_rewrite" parents="120"/>
      <j2 rule="rewrite" parents="414 4"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
