<?xml version="1.0" encoding="ISO-8859-1"?>

<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

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

<heading><![CDATA[
Prover9 (32) version November-2006, November 2006.
Process 3864 was started by mccune on cleo.thornwood,
Wed Nov 22 11:24:17 2006
The command was "/home/mccune/bin/prover9 -f gt.in EA.in".
]]></heading>

<proof number="1" length="69" max_count="20">

<comments><![CDATA[
% Proof 1 at 0.32 (+ 0.01) seconds: A.
% Length of proof is 69.
% Level of proof is 19.
% Maximum clause weight is 20.
% Given clauses 132.
]]></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 * y * x = y * x * z * x * y
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7" type="assumption">
    <literal><![CDATA[
      (A @ B) @ C != D @ (F @ G)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></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="12">
    <literal><![CDATA[
      x' @ x = e
    ]]></literal>
    <justification jstring="[para(3(a,1),5(a,1,2,2)),rewrite(8(6)),flip(a)].">
      <j1 rule="para" parents="3 5"/>
      <j2 rule="rewrite" parents="8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      x * y * y * x = y * x * x * y
    ]]></literal>
    <justification jstring="[para(2(a,1),6(a,1,2,2)),rewrite(2(6))].">
      <j1 rule="para" parents="2 6"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x * y * (x * y)' * y * x = y * x * e
    ]]></literal>
    <justification jstring="[para(3(a,1),6(a,1,2,2)),flip(a)].">
      <j1 rule="para" parents="3 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="24">
    <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="27">
    <literal><![CDATA[
      x'' * (x @ y) = y' * x * y
    ]]></literal>
    <justification jstring="[para(5(a,1),8(a,1,2))].">
      <j1 rule="para" parents="5 8"/>
    </justification>
  </clause>

  <clause id="32">
    <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="33">
    <literal><![CDATA[
      x' * y * x = y * (y @ x)
    ]]></literal>
    <justification jstring="[back_rewrite(27),rewrite(32(4)),flip(a)].">
      <j1 rule="back_rewrite" parents="27"/>
      <j2 rule="rewrite" parents="32"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      x * e = x
    ]]></literal>
    <justification jstring="[back_rewrite(24),rewrite(32(4))].">
      <j1 rule="back_rewrite" parents="24"/>
      <j2 rule="rewrite" parents="32"/>
    </justification>
  </clause>

  <clause id="37">
    <literal><![CDATA[
      x * y * (x * y)' * y * x = y * x
    ]]></literal>
    <justification jstring="[back_rewrite(20),rewrite(35(8))].">
      <j1 rule="back_rewrite" parents="20"/>
      <j2 rule="rewrite" parents="35"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      (x @ y) * z * y * x = z * x * y
    ]]></literal>
    <justification jstring="[para(6(a,1),9(a,1,2,2)),rewrite(8(7),8(5)),flip(a)].">
      <j1 rule="para" parents="6 9"/>
      <j2 rule="rewrite" parents="8 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="56">
    <literal><![CDATA[
      e' = e
    ]]></literal>
    <justification jstring="[para(35(a,1),3(a,1))].">
      <j1 rule="para" parents="35 3"/>
    </justification>
  </clause>

  <clause id="57">
    <literal><![CDATA[
      x @ e = e
    ]]></literal>
    <justification jstring="[para(35(a,1),5(a,1,2,2)),rewrite(56(3),2(3),3(2)),flip(a)].">
      <j1 rule="para" parents="35 5"/>
      <j2 rule="rewrite" parents="56 2 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="58">
    <literal><![CDATA[
      x * x' = e
    ]]></literal>
    <justification jstring="[para(32(a,1),3(a,1))].">
      <j1 rule="para" parents="32 3"/>
    </justification>
  </clause>

  <clause id="60">
    <literal><![CDATA[
      x' * y * x * y' = x @ y'
    ]]></literal>
    <justification jstring="[para(32(a,1),5(a,1,2))].">
      <j1 rule="para" parents="32 5"/>
    </justification>
  </clause>

  <clause id="63">
    <literal><![CDATA[
      x * x' * y = y
    ]]></literal>
    <justification jstring="[para(32(a,1),8(a,1))].">
      <j1 rule="para" parents="32 8"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[para(32(a,1),35(a,1)),rewrite(35(2)),flip(a)].">
      <j1 rule="para" parents="32 35"/>
      <j2 rule="rewrite" parents="35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      (x * y')' * y' * x = x * y' @ y
    ]]></literal>
    <justification jstring="[para(3(a,1),10(a,1,2,2,2)),rewrite(35(6))].">
      <j1 rule="para" parents="3 10"/>
      <j2 rule="rewrite" parents="35"/>
    </justification>
  </clause>

  <clause id="74">
    <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="79">
    <literal><![CDATA[
      x * y' @ y = x * y' @ x
    ]]></literal>
    <justification jstring="[back_rewrite(69),rewrite(74(6)),flip(a)].">
      <j1 rule="back_rewrite" parents="69"/>
      <j2 rule="rewrite" parents="74"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="80">
    <literal><![CDATA[
      x * y * (x * y @ x) = y * x
    ]]></literal>
    <justification jstring="[back_rewrite(37),rewrite(74(4))].">
      <j1 rule="back_rewrite" parents="37"/>
      <j2 rule="rewrite" parents="74"/>
    </justification>
  </clause>

  <clause id="81">
    <literal><![CDATA[
      x @ x' = e
    ]]></literal>
    <justification jstring="[para(67(a,1),12(a,1,1))].">
      <j1 rule="para" parents="67 12"/>
    </justification>
  </clause>

  <clause id="83">
    <literal><![CDATA[
      x * y * (x * y)' = e
    ]]></literal>
    <justification jstring="[para(58(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="58 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="86">
    <literal><![CDATA[
      x * y @ y' = x * y @ x
    ]]></literal>
    <justification jstring="[para(58(a,1),10(a,1,2,2,2)),rewrite(67(4),35(4),74(4)),flip(a)].">
      <j1 rule="para" parents="58 10"/>
      <j2 rule="rewrite" parents="67 35 74"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="117">
    <literal><![CDATA[
      x * (y * x)' = y'
    ]]></literal>
    <justification jstring="[para(83(a,1),8(a,1,2)),rewrite(35(3)),flip(a)].">
      <j1 rule="para" parents="83 8"/>
      <j2 rule="rewrite" parents="35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="119">
    <literal><![CDATA[
      (x @ y) * z * (x' * y' * x * y * z)' = e
    ]]></literal>
    <justification jstring="[para(9(a,2),83(a,1,2,2,1))].">
      <j1 rule="para" parents="9 83"/>
    </justification>
  </clause>

  <clause id="130">
    <literal><![CDATA[
      (x * y)' = y' * x'
    ]]></literal>
    <justification jstring="[para(117(a,1),8(a,1,2)),flip(a)].">
      <j1 rule="para" parents="117 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="131">
    <literal><![CDATA[
      (x @ y)' = y @ x
    ]]></literal>
    <justification jstring="[para(9(a,2),117(a,1,2,1)),rewrite(130(7),130(5),130(3),130(2),1(5),67(7),1(6),1(5),67(8),1(7),1(6),1(5),5(6),63(4)),flip(a)].">
      <j1 rule="para" parents="9 117"/>
      <j2 rule="rewrite" parents="130 130 130 130 1 67 1 1 67 1 1 1 5 63"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="140">
    <literal><![CDATA[
      (x @ y) * (y @ x) = e
    ]]></literal>
    <justification jstring="[back_rewrite(119),rewrite(130(8),130(6),130(4),130(3),1(6),67(8),1(7),1(6),67(9),1(8),1(7),1(6),5(7),63(5))].">
      <j1 rule="back_rewrite" parents="119"/>
      <j2 rule="rewrite" parents="130 130 130 130 1 67 1 1 67 1 1 1 5 63"/>
    </justification>
  </clause>

  <clause id="146">
    <literal><![CDATA[
      x * y @ x = y @ x
    ]]></literal>
    <justification jstring="[back_rewrite(74),rewrite(130(2),1(5),5(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="74"/>
      <j2 rule="rewrite" parents="130 1 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="159">
    <literal><![CDATA[
      x * y @ y' = y @ x
    ]]></literal>
    <justification jstring="[back_rewrite(86),rewrite(146(5))].">
      <j1 rule="back_rewrite" parents="86"/>
      <j2 rule="rewrite" parents="146"/>
    </justification>
  </clause>

  <clause id="160">
    <literal><![CDATA[
      x * y * (y @ x) = y * x
    ]]></literal>
    <justification jstring="[back_rewrite(80),rewrite(146(2))].">
      <j1 rule="back_rewrite" parents="80"/>
      <j2 rule="rewrite" parents="146"/>
    </justification>
  </clause>

  <clause id="161">
    <literal><![CDATA[
      x * y' @ y = y' @ x
    ]]></literal>
    <justification jstring="[back_rewrite(79),rewrite(146(6))].">
      <j1 rule="back_rewrite" parents="79"/>
      <j2 rule="rewrite" parents="146"/>
    </justification>
  </clause>

  <clause id="181">
    <literal><![CDATA[
      x * y @ x' = y @ x'
    ]]></literal>
    <justification jstring="[para(8(a,1),146(a,1,1)),flip(a)].">
      <j1 rule="para" parents="8 146"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="185">
    <literal><![CDATA[
      x' * y @ x = y @ x
    ]]></literal>
    <justification jstring="[para(63(a,1),146(a,1,1)),flip(a)].">
      <j1 rule="para" parents="63 146"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="186">
    <literal><![CDATA[
      x @ x * y = x @ y
    ]]></literal>
    <justification jstring="[para(146(a,1),131(a,1,1)),rewrite(131(2)),flip(a)].">
      <j1 rule="para" parents="146 131"/>
      <j2 rule="rewrite" parents="131"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="189">
    <literal><![CDATA[
      x' @ y' * x * y = x' @ (x @ y)
    ]]></literal>
    <justification jstring="[para(5(a,1),186(a,1,2)),flip(a)].">
      <j1 rule="para" parents="5 186"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="219">
    <literal><![CDATA[
      x' @ y * x = y @ x
    ]]></literal>
    <justification jstring="[para(159(a,1),131(a,1,1)),rewrite(131(2)),flip(a)].">
      <j1 rule="para" parents="159 131"/>
      <j2 rule="rewrite" parents="131"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="240">
    <literal><![CDATA[
      (x @ y) * y * x = x * y
    ]]></literal>
    <justification jstring="[para(18(a,1),9(a,1,2,2)),rewrite(8(6),8(4)),flip(a)].">
      <j1 rule="para" parents="18 9"/>
      <j2 rule="rewrite" parents="8 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="357">
    <literal><![CDATA[
      (x @ y) @ y * x = e
    ]]></literal>
    <justification jstring="[para(240(a,1),5(a,1,2,2)),rewrite(131(2),130(3),1(6),5(6),140(3)),flip(a)].">
      <j1 rule="para" parents="240 5"/>
      <j2 rule="rewrite" parents="131 130 1 5 140"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="359">
    <literal><![CDATA[
      (x' @ (x @ y)) * (y @ x') = x @ y
    ]]></literal>
    <justification jstring="[para(5(a,1),240(a,2)),rewrite(189(5),1(8),1(7),60(8))].">
      <j1 rule="para" parents="5 240"/>
      <j2 rule="rewrite" parents="189 1 1 60"/>
    </justification>
  </clause>

  <clause id="397">
    <literal><![CDATA[
      (x @ y) @ x = e
    ]]></literal>
    <justification jstring="[para(63(a,1),357(a,1,2)),rewrite(185(3))].">
      <j1 rule="para" parents="63 357"/>
      <j2 rule="rewrite" parents="185"/>
    </justification>
  </clause>

  <clause id="415">
    <literal><![CDATA[
      x @ (x @ y) = e
    ]]></literal>
    <justification jstring="[para(397(a,1),131(a,1,1)),rewrite(56(2)),flip(a)].">
      <j1 rule="para" parents="397 131"/>
      <j2 rule="rewrite" parents="56"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="418">
    <literal><![CDATA[
      (x @ y) * x = x * (x @ y)
    ]]></literal>
    <justification jstring="[para(397(a,1),160(a,1,2,2)),rewrite(35(3)),flip(a)].">
      <j1 rule="para" parents="397 160"/>
      <j2 rule="rewrite" parents="35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="432">
    <literal><![CDATA[
      (x @ y) @ x' = e
    ]]></literal>
    <justification jstring="[para(415(a,1),159(a,2)),rewrite(418(2),181(4))].">
      <j1 rule="para" parents="415 159"/>
      <j2 rule="rewrite" parents="418 181"/>
    </justification>
  </clause>

  <clause id="471">
    <literal><![CDATA[
      x' @ (x @ y) = e
    ]]></literal>
    <justification jstring="[para(432(a,1),131(a,1,1)),rewrite(56(2)),flip(a)].">
      <j1 rule="para" parents="432 131"/>
      <j2 rule="rewrite" parents="56"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="479">
    <literal><![CDATA[
      x @ y' = y @ x
    ]]></literal>
    <justification jstring="[back_rewrite(359),rewrite(471(3),2(4))].">
      <j1 rule="back_rewrite" parents="359"/>
      <j2 rule="rewrite" parents="471 2"/>
    </justification>
  </clause>

  <clause id="579">
    <literal><![CDATA[
      (F @ G) @ D' != (A @ B) @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(479(a,2),7(a,2)),flip(a)].">
      <j1 rule="para" parents="479 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="582">
    <literal><![CDATA[
      (x @ y) @ z = z @ (y @ x)
    ]]></literal>
    <justification jstring="[para(131(a,1),479(a,1,2)),flip(a)].">
      <j1 rule="para" parents="131 479"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="593">
    <literal><![CDATA[
      x' @ y = x @ y'
    ]]></literal>
    <justification jstring="[para(479(a,2),161(a,1)),rewrite(130(3),67(2),186(3)),flip(a)].">
      <j1 rule="para" parents="479 161"/>
      <j2 rule="rewrite" parents="130 67 186"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="601">
    <literal><![CDATA[
      (F @ G) @ D' != C @ (B @ A)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[back_rewrite(579),rewrite(582(11))].">
      <j1 rule="back_rewrite" parents="579"/>
      <j2 rule="rewrite" parents="582"/>
    </justification>
  </clause>

  <clause id="653">
    <literal><![CDATA[
      D' @ (F' @ G) != C @ (B @ A)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(479(a,2),601(a,1,1)),rewrite(582(7))].">
      <j1 rule="para" parents="479 601"/>
      <j2 rule="rewrite" parents="582"/>
    </justification>
  </clause>

  <clause id="806">
    <literal><![CDATA[
      D' @ (F' @ G) != C @ (A' @ B)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(479(a,2),653(a,2,2)),rewrite(593(12,R))].">
      <j1 rule="para" parents="479 653"/>
      <j2 rule="rewrite" parents="593"/>
    </justification>
  </clause>

  <clause id="895">
    <literal><![CDATA[
      (x @ y) * z * y * (y @ x) = z * y
    ]]></literal>
    <justification jstring="[para(33(a,1),49(a,1,2,2)),rewrite(159(3),1(8),58(7),35(7))].">
      <j1 rule="para" parents="33 49"/>
      <j2 rule="rewrite" parents="159 1 58 35"/>
    </justification>
  </clause>

  <clause id="1173">
    <literal><![CDATA[
      D' @ (F' @ G) != (A' @ B) @ C'
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(479(a,2),806(a,2))].">
      <j1 rule="para" parents="479 806"/>
    </justification>
  </clause>

  <clause id="2104">
    <literal><![CDATA[
      (x @ (y @ z)) * y = y
    ]]></literal>
    <justification jstring="[para(418(a,1),9(a,1,2,2,2)),rewrite(131(3),895(6),8(3)),flip(a)].">
      <j1 rule="para" parents="418 9"/>
      <j2 rule="rewrite" parents="131 895 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2237">
    <literal><![CDATA[
      x @ ((x @ y) @ z) = (x @ y) @ z
    ]]></literal>
    <justification jstring="[para(2104(a,1),5(a,1,2,2)),rewrite(131(3),3(4),35(4),582(5)),flip(a)].">
      <j1 rule="para" parents="2104 5"/>
      <j2 rule="rewrite" parents="131 3 35 582"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2247">
    <literal><![CDATA[
      (x @ y) @ z = e
    ]]></literal>
    <justification jstring="[para(2104(a,1),219(a,1,2)),rewrite(593(2),81(2),582(4),2237(4)),flip(a)].">
      <j1 rule="para" parents="2104 219"/>
      <j2 rule="rewrite" parents="593 81 582 2237"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2256">
    <literal><![CDATA[
      x @ (y @ z) = e
    ]]></literal>
    <justification jstring="[para(2104(a,1),33(a,1,2)),rewrite(3(2),582(6),2247(5),57(5),35(5)),flip(a)].">
      <j1 rule="para" parents="2104 33"/>
      <j2 rule="rewrite" parents="3 582 2247 57 35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2284">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[back_rewrite(1173),rewrite(2256(7),2247(8)),xx(a)].">
      <j1 rule="back_rewrite" parents="1173"/>
      <j2 rule="rewrite" parents="2256 2247"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
