<?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 March-2007, March 2007.
Process 21009 was started by mccune on cleo,
Mon Mar 19 17:02:00 2007
The command was "/home/mccune/bin/prover9 -f gt.in EA.in".
]]></heading>

<proof number="1" length="70" max_count="20">

<comments><![CDATA[
% Proof 1 at 0.26 (+ 0.00) seconds: A.
% Length of proof is 70.
% Level of proof is 19.
% Maximum clause weight is 20.
% Given clauses 127.
]]></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="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="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="39">
    <literal><![CDATA[
      e' = e
    ]]></literal>
    <justification jstring="[para(11(a,1),5(a,2)),rewrite(2(5),3(4),35(4))].">
      <j1 rule="para" parents="11 5"/>
      <j2 rule="rewrite" parents="2 3 35"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      x @ e = e
    ]]></literal>
    <justification jstring="[para(39(a,1),5(a,1,2,1)),rewrite(35(4),2(3),3(2)),flip(a)].">
      <j1 rule="para" parents="39 5"/>
      <j2 rule="rewrite" parents="35 2 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="50">
    <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="57">
    <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="59">
    <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="62">
    <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="63">
    <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="66">
    <literal><![CDATA[
      x @ x' = e
    ]]></literal>
    <justification jstring="[para(63(a,1),12(a,1,1))].">
      <j1 rule="para" parents="63 12"/>
    </justification>
  </clause>

  <clause id="68">
    <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="73">
    <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(68),rewrite(73(6)),flip(a)].">
      <j1 rule="back_rewrite" parents="68"/>
      <j2 rule="rewrite" parents="73"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="80">
    <literal><![CDATA[
      x * y * (x * y @ x) = y * x
    ]]></literal>
    <justification jstring="[back_rewrite(37),rewrite(73(4))].">
      <j1 rule="back_rewrite" parents="37"/>
      <j2 rule="rewrite" parents="73"/>
    </justification>
  </clause>

  <clause id="81">
    <literal><![CDATA[
      x * y * (x * y)' = e
    ]]></literal>
    <justification jstring="[para(57(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="57 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="84">
    <literal><![CDATA[
      x * y @ y' = x * y @ x
    ]]></literal>
    <justification jstring="[para(57(a,1),10(a,1,2,2,2)),rewrite(63(4),35(4),73(4)),flip(a)].">
      <j1 rule="para" parents="57 10"/>
      <j2 rule="rewrite" parents="63 35 73"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="115">
    <literal><![CDATA[
      x * (y * x)' = y'
    ]]></literal>
    <justification jstring="[para(81(a,1),8(a,1,2)),rewrite(35(3)),flip(a)].">
      <j1 rule="para" parents="81 8"/>
      <j2 rule="rewrite" parents="35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="117">
    <literal><![CDATA[
      (x @ y) * z * (x' * y' * x * y * z)' = e
    ]]></literal>
    <justification jstring="[para(9(a,2),81(a,1,2,2,1))].">
      <j1 rule="para" parents="9 81"/>
    </justification>
  </clause>

  <clause id="128">
    <literal><![CDATA[
      (x * y)' = y' * x'
    ]]></literal>
    <justification jstring="[para(115(a,1),8(a,1,2)),flip(a)].">
      <j1 rule="para" parents="115 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="129">
    <literal><![CDATA[
      (x @ y)' = y @ x
    ]]></literal>
    <justification jstring="[para(9(a,2),115(a,1,2,1)),rewrite(128(7),128(5),128(3),128(2),1(5),63(7),1(6),1(5),63(8),1(7),1(6),1(5),5(6),62(4)),flip(a)].">
      <j1 rule="para" parents="9 115"/>
      <j2 rule="rewrite" parents="128 128 128 128 1 63 1 1 63 1 1 1 5 62"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="138">
    <literal><![CDATA[
      (x @ y) * (y @ x) = e
    ]]></literal>
    <justification jstring="[back_rewrite(117),rewrite(128(8),128(6),128(4),128(3),1(6),63(8),1(7),1(6),63(9),1(8),1(7),1(6),5(7),62(5))].">
      <j1 rule="back_rewrite" parents="117"/>
      <j2 rule="rewrite" parents="128 128 128 128 1 63 1 1 63 1 1 1 5 62"/>
    </justification>
  </clause>

  <clause id="144">
    <literal><![CDATA[
      x * y @ x = y @ x
    ]]></literal>
    <justification jstring="[back_rewrite(73),rewrite(128(2),1(5),5(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="73"/>
      <j2 rule="rewrite" parents="128 1 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="157">
    <literal><![CDATA[
      x * y @ y' = y @ x
    ]]></literal>
    <justification jstring="[back_rewrite(84),rewrite(144(5))].">
      <j1 rule="back_rewrite" parents="84"/>
      <j2 rule="rewrite" parents="144"/>
    </justification>
  </clause>

  <clause id="158">
    <literal><![CDATA[
      x * y * (y @ x) = y * x
    ]]></literal>
    <justification jstring="[back_rewrite(80),rewrite(144(2))].">
      <j1 rule="back_rewrite" parents="80"/>
      <j2 rule="rewrite" parents="144"/>
    </justification>
  </clause>

  <clause id="159">
    <literal><![CDATA[
      x * y' @ y = y' @ x
    ]]></literal>
    <justification jstring="[back_rewrite(79),rewrite(144(6))].">
      <j1 rule="back_rewrite" parents="79"/>
      <j2 rule="rewrite" parents="144"/>
    </justification>
  </clause>

  <clause id="182">
    <literal><![CDATA[
      x' * y @ x = y @ x
    ]]></literal>
    <justification jstring="[para(62(a,1),144(a,1,1)),flip(a)].">
      <j1 rule="para" parents="62 144"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="184">
    <literal><![CDATA[
      x @ x * y = x @ y
    ]]></literal>
    <justification jstring="[para(144(a,1),129(a,1,1)),rewrite(129(2)),flip(a)].">
      <j1 rule="para" parents="144 129"/>
      <j2 rule="rewrite" parents="129"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="187">
    <literal><![CDATA[
      x' @ y' * x * y = x' @ (x @ y)
    ]]></literal>
    <justification jstring="[para(5(a,1),184(a,1,2)),flip(a)].">
      <j1 rule="para" parents="5 184"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="191">
    <literal><![CDATA[
      x @ x' * y = x @ y
    ]]></literal>
    <justification jstring="[para(62(a,1),184(a,1,2)),flip(a)].">
      <j1 rule="para" parents="62 184"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="217">
    <literal><![CDATA[
      x' @ y * x = y @ x
    ]]></literal>
    <justification jstring="[para(157(a,1),129(a,1,1)),rewrite(129(2)),flip(a)].">
      <j1 rule="para" parents="157 129"/>
      <j2 rule="rewrite" parents="129"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="236">
    <literal><![CDATA[
      x @ y * x' = y @ x'
    ]]></literal>
    <justification jstring="[para(63(a,1),217(a,1,1))].">
      <j1 rule="para" parents="63 217"/>
    </justification>
  </clause>

  <clause id="245">
    <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="361">
    <literal><![CDATA[
      (x @ y) @ y * x = e
    ]]></literal>
    <justification jstring="[para(245(a,1),5(a,1,2,2)),rewrite(129(2),128(3),1(6),5(6),138(3)),flip(a)].">
      <j1 rule="para" parents="245 5"/>
      <j2 rule="rewrite" parents="129 128 1 5 138"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="363">
    <literal><![CDATA[
      (x' @ (x @ y)) * (y @ x') = x @ y
    ]]></literal>
    <justification jstring="[para(5(a,1),245(a,2)),rewrite(187(5),1(8),1(7),59(8))].">
      <j1 rule="para" parents="5 245"/>
      <j2 rule="rewrite" parents="187 1 1 59"/>
    </justification>
  </clause>

  <clause id="400">
    <literal><![CDATA[
      (x @ y) @ x = e
    ]]></literal>
    <justification jstring="[para(62(a,1),361(a,1,2)),rewrite(182(3))].">
      <j1 rule="para" parents="62 361"/>
      <j2 rule="rewrite" parents="182"/>
    </justification>
  </clause>

  <clause id="422">
    <literal><![CDATA[
      (x @ y) * x = x * (x @ y)
    ]]></literal>
    <justification jstring="[para(400(a,1),158(a,1,2,2)),rewrite(35(3)),flip(a)].">
      <j1 rule="para" parents="400 158"/>
      <j2 rule="rewrite" parents="35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="424">
    <literal><![CDATA[
      x @ (x' @ y) = e
    ]]></literal>
    <justification jstring="[para(400(a,1),236(a,2)),rewrite(422(4),191(5))].">
      <j1 rule="para" parents="400 236"/>
      <j2 rule="rewrite" parents="422 191"/>
    </justification>
  </clause>

  <clause id="477">
    <literal><![CDATA[
      x' @ (x @ y) = e
    ]]></literal>
    <justification jstring="[para(63(a,1),424(a,1,2,1))].">
      <j1 rule="para" parents="63 424"/>
    </justification>
  </clause>

  <clause id="483">
    <literal><![CDATA[
      x @ y' = y @ x
    ]]></literal>
    <justification jstring="[back_rewrite(363),rewrite(477(3),2(4))].">
      <j1 rule="back_rewrite" parents="363"/>
      <j2 rule="rewrite" parents="477 2"/>
    </justification>
  </clause>

  <clause id="593">
    <literal><![CDATA[
      (F @ G) @ D' != (A @ B) @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(483(a,2),7(a,2)),flip(a)].">
      <j1 rule="para" parents="483 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="596">
    <literal><![CDATA[
      (x @ y) @ z = z @ (y @ x)
    ]]></literal>
    <justification jstring="[para(129(a,1),483(a,1,2)),flip(a)].">
      <j1 rule="para" parents="129 483"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="607">
    <literal><![CDATA[
      x' @ y = x @ y'
    ]]></literal>
    <justification jstring="[para(483(a,2),159(a,1)),rewrite(128(3),63(2),184(3)),flip(a)].">
      <j1 rule="para" parents="483 159"/>
      <j2 rule="rewrite" parents="128 63 184"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="615">
    <literal><![CDATA[
      (F @ G) @ D' != C @ (B @ A)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[back_rewrite(593),rewrite(596(11))].">
      <j1 rule="back_rewrite" parents="593"/>
      <j2 rule="rewrite" parents="596"/>
    </justification>
  </clause>

  <clause id="669">
    <literal><![CDATA[
      D' @ (F' @ G) != C @ (B @ A)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(483(a,2),615(a,1,1)),rewrite(596(7))].">
      <j1 rule="para" parents="483 615"/>
      <j2 rule="rewrite" parents="596"/>
    </justification>
  </clause>

  <clause id="803">
    <literal><![CDATA[
      D' @ (F' @ G) != C @ (A' @ B)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(483(a,2),669(a,2,2)),rewrite(607(12,R))].">
      <j1 rule="para" parents="483 669"/>
      <j2 rule="rewrite" parents="607"/>
    </justification>
  </clause>

  <clause id="909">
    <literal><![CDATA[
      D' @ (F' @ G) != (A' @ B) @ C'
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(483(a,2),803(a,2))].">
      <j1 rule="para" parents="483 803"/>
    </justification>
  </clause>

  <clause id="1040">
    <literal><![CDATA[
      (x @ y) * z * y * (y @ x) = z * y
    ]]></literal>
    <justification jstring="[para(33(a,1),50(a,1,2,2)),rewrite(157(3),1(8),57(7),35(7))].">
      <j1 rule="para" parents="33 50"/>
      <j2 rule="rewrite" parents="157 1 57 35"/>
    </justification>
  </clause>

  <clause id="1841">
    <literal><![CDATA[
      (x @ (y @ z)) * y = y
    ]]></literal>
    <justification jstring="[para(422(a,1),9(a,1,2,2,2)),rewrite(129(3),1040(6),8(3)),flip(a)].">
      <j1 rule="para" parents="422 9"/>
      <j2 rule="rewrite" parents="129 1040 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1855">
    <literal><![CDATA[
      x @ ((x @ y) @ z) = (x @ y) @ z
    ]]></literal>
    <justification jstring="[para(1841(a,1),5(a,1,2,2)),rewrite(129(3),3(4),35(4),596(5)),flip(a)].">
      <j1 rule="para" parents="1841 5"/>
      <j2 rule="rewrite" parents="129 3 35 596"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1864">
    <literal><![CDATA[
      (x @ y) @ z = e
    ]]></literal>
    <justification jstring="[para(1841(a,1),217(a,1,2)),rewrite(607(2),66(2),596(4),1855(4)),flip(a)].">
      <j1 rule="para" parents="1841 217"/>
      <j2 rule="rewrite" parents="607 66 596 1855"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1875">
    <literal><![CDATA[
      x @ (y @ z) = e
    ]]></literal>
    <justification jstring="[para(1841(a,1),33(a,1,2)),rewrite(3(2),596(6),1864(5),40(5),35(5)),flip(a)].">
      <j1 rule="para" parents="1841 33"/>
      <j2 rule="rewrite" parents="3 596 1864 40 35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1893">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[back_rewrite(909),rewrite(1875(7),1864(8)),xx(a)].">
      <j1 rule="back_rewrite" parents="909"/>
      <j2 rule="rewrite" parents="1875 1864"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
