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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>AN-gt.out</source>

<heading><![CDATA[
Prover9 (32) version November-2006, November 2006.
Process 3856 was started by mccune on cleo.thornwood,
Wed Nov 22 11:24:11 2006
The command was "/home/mccune/bin/prover9 -f gt.in AN.in".
]]></heading>

<proof number="1" length="43" max_count="20">

<comments><![CDATA[
% Proof 1 at 0.04 (+ 0.00) seconds: N.
% Length of proof is 43.
% Level of proof is 14.
% Maximum clause weight is 20.
% Given clauses 56.
]]></comments>

  <clause id="1" type="assumption">
    <literal><![CDATA[
      (x * y) * z = x * y * z
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="2" type="assumption">
    <literal><![CDATA[
      e * x = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="3" type="assumption">
    <literal><![CDATA[
      x' * x = e
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="4" type="assumption">
    <literal><![CDATA[
      x @ y = x' * y' * x * y
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="5">
    <literal><![CDATA[
      x' * y' * x * y = x @ y
    ]]></literal>
    <justification jstring="[copy(4),flip(a)].">
      <j1 rule="copy" parents="4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="6" type="assumption">
    <literal><![CDATA[
      (x @ y) @ z = u @ (v @ w)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </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' * x * y = y
    ]]></literal>
    <justification jstring="[para(3(a,1),1(a,1,1)),rewrite(2(2)),flip(a)].">
      <j1 rule="para" parents="3 1"/>
      <j2 rule="rewrite" parents="2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      x' * y' * x * y * z = (x @ y) * z
    ]]></literal>
    <justification jstring="[para(5(a,1),1(a,1,1)),rewrite(1(7),1(6)),flip(a)].">
      <j1 rule="para" parents="5 1"/>
      <j2 rule="rewrite" parents="1 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

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

  <clause id="11">
    <literal><![CDATA[
      e @ x = e
    ]]></literal>
    <justification jstring="[para(2(a,1),5(a,1,2,2)),rewrite(3(4),3(4)),flip(a)].">
      <j1 rule="para" parents="2 5"/>
      <j2 rule="rewrite" parents="3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      x @ (y @ z) = e
    ]]></literal>
    <justification jstring="[para(11(a,1),6(a,1,1)),rewrite(11(2)),flip(a)].">
      <j1 rule="para" parents="11 6"/>
      <j2 rule="rewrite" parents="11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      x'' * e = x
    ]]></literal>
    <justification jstring="[para(3(a,1),8(a,1,2))].">
      <j1 rule="para" parents="3 8"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      x'' * y = x * y
    ]]></literal>
    <justification jstring="[para(8(a,1),8(a,1,2))].">
      <j1 rule="para" parents="8 8"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      x * e = x
    ]]></literal>
    <justification jstring="[back_rewrite(22),rewrite(26(4))].">
      <j1 rule="back_rewrite" parents="22"/>
      <j2 rule="rewrite" parents="26"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      x' * y * x * z = y * (y @ x) * z
    ]]></literal>
    <justification jstring="[para(9(a,1),8(a,1,2)),rewrite(26(5)),flip(a)].">
      <j1 rule="para" parents="9 8"/>
      <j2 rule="rewrite" parents="26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="51">
    <literal><![CDATA[
      x * x' = e
    ]]></literal>
    <justification jstring="[para(26(a,1),3(a,1))].">
      <j1 rule="para" parents="26 3"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      x * x' * y = y
    ]]></literal>
    <justification jstring="[para(26(a,1),8(a,1))].">
      <j1 rule="para" parents="26 8"/>
    </justification>
  </clause>

  <clause id="56">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[para(26(a,1),29(a,1)),rewrite(29(2)),flip(a)].">
      <j1 rule="para" parents="26 29"/>
      <j2 rule="rewrite" parents="29"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="60">
    <literal><![CDATA[
      x * y * (x * y)' = e
    ]]></literal>
    <justification jstring="[para(51(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="51 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      (x * y)' * y * x = x * y @ x
    ]]></literal>
    <justification jstring="[para(8(a,1),10(a,1,2))].">
      <j1 rule="para" parents="8 10"/>
    </justification>
  </clause>

  <clause id="71">
    <literal><![CDATA[
      x' * (y * x @ y) * x = x @ y * x
    ]]></literal>
    <justification jstring="[para(9(a,1),10(a,1,2)),rewrite(8(3),8(8))].">
      <j1 rule="para" parents="9 10"/>
      <j2 rule="rewrite" parents="8 8"/>
    </justification>
  </clause>

  <clause id="84">
    <literal><![CDATA[
      x * (y * x)' = y'
    ]]></literal>
    <justification jstring="[para(60(a,1),8(a,1,2)),rewrite(29(3)),flip(a)].">
      <j1 rule="para" parents="60 8"/>
      <j2 rule="rewrite" parents="29"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="86">
    <literal><![CDATA[
      (x @ y) * z * (x' * y' * x * y * z)' = e
    ]]></literal>
    <justification jstring="[para(9(a,2),60(a,1,2,2,1))].">
      <j1 rule="para" parents="9 60"/>
    </justification>
  </clause>

  <clause id="113">
    <literal><![CDATA[
      (x * y)' = y' * x'
    ]]></literal>
    <justification jstring="[para(84(a,1),8(a,1,2)),flip(a)].">
      <j1 rule="para" parents="84 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="114">
    <literal><![CDATA[
      (x @ y)' = y @ x
    ]]></literal>
    <justification jstring="[para(9(a,2),84(a,1,2,1)),rewrite(113(7),113(5),113(3),113(2),1(5),56(7),1(6),1(5),56(8),1(7),1(6),1(5),5(6),55(4)),flip(a)].">
      <j1 rule="para" parents="9 84"/>
      <j2 rule="rewrite" parents="113 113 113 113 1 56 1 1 56 1 1 1 5 55"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="127">
    <literal><![CDATA[
      (x @ y) * (y @ x) = e
    ]]></literal>
    <justification jstring="[back_rewrite(86),rewrite(113(8),113(6),113(4),113(3),1(6),56(8),1(7),1(6),56(9),1(8),1(7),1(6),5(7),55(5))].">
      <j1 rule="back_rewrite" parents="86"/>
      <j2 rule="rewrite" parents="113 113 113 113 1 56 1 1 56 1 1 1 5 55"/>
    </justification>
  </clause>

  <clause id="131">
    <literal><![CDATA[
      x * y @ x = y @ x
    ]]></literal>
    <justification jstring="[back_rewrite(69),rewrite(113(2),1(5),5(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="69"/>
      <j2 rule="rewrite" parents="113 1 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="148">
    <literal><![CDATA[
      x' * (x @ y) * x = x @ y * x
    ]]></literal>
    <justification jstring="[back_rewrite(71),rewrite(131(3))].">
      <j1 rule="back_rewrite" parents="71"/>
      <j2 rule="rewrite" parents="131"/>
    </justification>
  </clause>

  <clause id="154">
    <literal><![CDATA[
      x' * (y @ z) * x = y @ z
    ]]></literal>
    <justification jstring="[para(127(a,1),9(a,1,2,2,2)),rewrite(114(3),29(4),16(6),2(7))].">
      <j1 rule="para" parents="127 9"/>
      <j2 rule="rewrite" parents="114 29 16 2"/>
    </justification>
  </clause>

  <clause id="155">
    <literal><![CDATA[
      x @ y * x = x @ y
    ]]></literal>
    <justification jstring="[back_rewrite(148),rewrite(154(4)),flip(a)].">
      <j1 rule="back_rewrite" parents="148"/>
      <j2 rule="rewrite" parents="154"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="170">
    <literal><![CDATA[
      x * y @ x' = y @ x'
    ]]></literal>
    <justification jstring="[para(8(a,1),131(a,1,1)),flip(a)].">
      <j1 rule="para" parents="8 131"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="172">
    <literal><![CDATA[
      x' * y @ x = y @ x
    ]]></literal>
    <justification jstring="[para(55(a,1),131(a,1,1)),flip(a)].">
      <j1 rule="para" parents="55 131"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="177">
    <literal><![CDATA[
      x * y @ y = y @ x'
    ]]></literal>
    <justification jstring="[para(8(a,1),155(a,1,2)),rewrite(170(5))].">
      <j1 rule="para" parents="8 155"/>
      <j2 rule="rewrite" parents="170"/>
    </justification>
  </clause>

  <clause id="180">
    <literal><![CDATA[
      x' * y @ y = y @ x
    ]]></literal>
    <justification jstring="[para(55(a,1),155(a,1,2)),rewrite(172(6))].">
      <j1 rule="para" parents="55 155"/>
      <j2 rule="rewrite" parents="172"/>
    </justification>
  </clause>

  <clause id="181">
    <literal><![CDATA[
      x * y @ y = x @ y
    ]]></literal>
    <justification jstring="[para(155(a,1),114(a,1,1)),rewrite(114(2)),flip(a)].">
      <j1 rule="para" parents="155 114"/>
      <j2 rule="rewrite" parents="114"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="184">
    <literal><![CDATA[
      x' @ y = y @ x
    ]]></literal>
    <justification jstring="[back_rewrite(180),rewrite(181(3))].">
      <j1 rule="back_rewrite" parents="180"/>
      <j2 rule="rewrite" parents="181"/>
    </justification>
  </clause>

  <clause id="185">
    <literal><![CDATA[
      x @ y' = y @ x
    ]]></literal>
    <justification jstring="[back_rewrite(177),rewrite(181(2)),flip(a)].">
      <j1 rule="back_rewrite" parents="177"/>
      <j2 rule="rewrite" parents="181"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="202">
    <literal><![CDATA[
      (B @ A') * C != C * (A @ B)
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[para(185(a,2),7(a,1,1))].">
      <j1 rule="para" parents="185 7"/>
    </justification>
  </clause>

  <clause id="211">
    <literal><![CDATA[
      x' @ y = x @ y'
    ]]></literal>
    <justification jstring="[para(185(a,1),184(a,1)),flip(a)].">
      <j1 rule="para" parents="185 184"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="227">
    <literal><![CDATA[
      (B @ A') * C != C * (B @ A')
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[para(184(a,2),202(a,2,2)),rewrite(211(11))].">
      <j1 rule="para" parents="184 202"/>
      <j2 rule="rewrite" parents="211"/>
    </justification>
  </clause>

  <clause id="367">
    <literal><![CDATA[
      (x @ y) * z = z * (x @ y)
    ]]></literal>
    <justification jstring="[para(127(a,1),43(a,1,2,2)),rewrite(114(2),29(3),16(4),2(5))].">
      <j1 rule="para" parents="127 43"/>
      <j2 rule="rewrite" parents="114 29 16 2"/>
    </justification>
  </clause>

  <clause id="368">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(N)
    ]]></attribute>
    <justification jstring="[resolve(367,a,227,a)].">
      <j1 rule="resolve" parents="367 227"/>
    </justification>
  </clause>

</proof>

</proofs>
