<?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>DA-gt.out</source>

<heading><![CDATA[
Prover9 (32) version August-2006A, August 2006.
Process 11352 was started by mccune on cleo.thornwood,
Sat Aug 12 20:59:51 2006
The command was "/home/mccune/bin/prover9 -f gt.in DA.in".
]]></heading>

<proof number="1" length="63" max_count="25">

<comments><![CDATA[
% Proof 1 at 0.20 (+ 0.00) seconds: A.
% Length of proof is 63.
% Level of proof is 17.
% Maximum clause weight is 25.
% Given clauses 141.
]]></comments>

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

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

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

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

  <clause id="7">
    <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="14">
    <literal><![CDATA[
      (x * y @ z) * u = (x @ z) * (y @ z) * u
    ]]></literal>
    <justification jstring="[para(6(a,1),1(a,1,1))].">
      <j1 rule="para" parents="6 1"/>
    </justification>
  </clause>

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

  <clause id="17">
    <literal><![CDATA[
      x' * y @ x = y @ x
    ]]></literal>
    <justification jstring="[para(12(a,1),6(a,1,1)),rewrite(2(3)),flip(a)].">
      <j1 rule="para" parents="12 6"/>
      <j2 rule="rewrite" parents="2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      x * y' @ y = x * e @ y
    ]]></literal>
    <justification jstring="[para(12(a,1),6(a,1,2)),rewrite(16(3)),flip(a)].">
      <j1 rule="para" parents="12 6"/>
      <j2 rule="rewrite" parents="16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      (x * y)' * x * y * z = z
    ]]></literal>
    <justification jstring="[para(1(a,1),8(a,1,2))].">
      <j1 rule="para" parents="1 8"/>
    </justification>
  </clause>

  <clause id="21">
    <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="23">
    <literal><![CDATA[
      x @ x = e
    ]]></literal>
    <justification jstring="[para(8(a,1),5(a,1,2)),rewrite(3(2)),flip(a)].">
      <j1 rule="para" parents="8 5"/>
      <j2 rule="rewrite" parents="3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="24">
    <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="25">
    <literal><![CDATA[
      (x @ y)' * (x * z @ y) = z @ y
    ]]></literal>
    <justification jstring="[para(6(a,1),8(a,1,2))].">
      <j1 rule="para" parents="6 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="27">
    <literal><![CDATA[
      x' * y * x = y * (y @ x)
    ]]></literal>
    <justification jstring="[back_rewrite(24),rewrite(26(4)),flip(a)].">
      <j1 rule="back_rewrite" parents="24"/>
      <j2 rule="rewrite" parents="26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      x * e = x
    ]]></literal>
    <justification jstring="[back_rewrite(21),rewrite(26(4))].">
      <j1 rule="back_rewrite" parents="21"/>
      <j2 rule="rewrite" parents="26"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      x * y' @ y = x @ y
    ]]></literal>
    <justification jstring="[back_rewrite(18),rewrite(29(5))].">
      <j1 rule="back_rewrite" parents="18"/>
      <j2 rule="rewrite" parents="29"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      x * y @ x = y @ x
    ]]></literal>
    <justification jstring="[para(23(a,1),6(a,1,1)),rewrite(2(3)),flip(a)].">
      <j1 rule="para" parents="23 6"/>
      <j2 rule="rewrite" parents="2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="54">
    <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="58">
    <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="62">
    <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="64">
    <literal><![CDATA[
      (x * y')' * y' * x = x @ y
    ]]></literal>
    <justification jstring="[para(3(a,1),10(a,1,2,2,2)),rewrite(29(6),31(9))].">
      <j1 rule="para" parents="3 10"/>
      <j2 rule="rewrite" parents="29 31"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      x' * y * z * x = y * z * (y * z @ x)
    ]]></literal>
    <justification jstring="[para(10(a,1),8(a,1,2)),rewrite(62(3),1(4)),flip(a)].">
      <j1 rule="para" parents="10 8"/>
      <j2 rule="rewrite" parents="62 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      (x * y)' * y * x = y @ x
    ]]></literal>
    <justification jstring="[para(8(a,1),10(a,1,2)),rewrite(32(6))].">
      <j1 rule="para" parents="8 10"/>
      <j2 rule="rewrite" parents="32"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      x' @ y = y @ x
    ]]></literal>
    <justification jstring="[back_rewrite(64),rewrite(69(6))].">
      <j1 rule="back_rewrite" parents="64"/>
      <j2 rule="rewrite" parents="69"/>
    </justification>
  </clause>

  <clause id="73">
    <literal><![CDATA[
      x @ x' = e
    ]]></literal>
    <justification jstring="[para(62(a,1),12(a,1,1))].">
      <j1 rule="para" parents="62 12"/>
    </justification>
  </clause>

  <clause id="78">
    <literal><![CDATA[
      x * y @ y' = y @ x
    ]]></literal>
    <justification jstring="[para(54(a,1),10(a,1,2,2,2)),rewrite(62(4),29(4),69(4)),flip(a)].">
      <j1 rule="para" parents="54 10"/>
      <j2 rule="rewrite" parents="62 29 69"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="80">
    <literal><![CDATA[
      x @ y' = y @ x
    ]]></literal>
    <justification jstring="[para(73(a,1),6(a,1,2)),rewrite(29(4),78(5))].">
      <j1 rule="para" parents="73 6"/>
      <j2 rule="rewrite" parents="29 78"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      x' * y' * x * z = (x @ y) * y' * z
    ]]></literal>
    <justification jstring="[para(58(a,1),9(a,1,2,2,2))].">
      <j1 rule="para" parents="58 9"/>
    </justification>
  </clause>

  <clause id="98">
    <literal><![CDATA[
      (B' @ A) @ C != D @ (F @ G)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(72(a,2),7(a,1,1))].">
      <j1 rule="para" parents="72 7"/>
    </justification>
  </clause>

  <clause id="109">
    <literal><![CDATA[
      x' @ y' = x @ y
    ]]></literal>
    <justification jstring="[para(72(a,2),72(a,1))].">
      <j1 rule="para" parents="72 72"/>
    </justification>
  </clause>

  <clause id="131">
    <literal><![CDATA[
      x' @ y = x @ y'
    ]]></literal>
    <justification jstring="[para(80(a,1),72(a,1)),flip(a)].">
      <j1 rule="para" parents="80 72"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="134">
    <literal><![CDATA[
      (F @ G) @ D' != (B @ A') @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(80(a,2),98(a,2)),rewrite(131(4)),flip(a)].">
      <j1 rule="para" parents="80 98"/>
      <j2 rule="rewrite" parents="131"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="139">
    <literal><![CDATA[
      x * y * z @ y = x * z @ y
    ]]></literal>
    <justification jstring="[para(32(a,1),6(a,1,2)),rewrite(6(3)),flip(a)].">
      <j1 rule="para" parents="32 6"/>
      <j2 rule="rewrite" parents="6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="191">
    <literal><![CDATA[
      (x @ y) @ x = e
    ]]></literal>
    <justification jstring="[para(5(a,1),17(a,1,1)),rewrite(139(6),3(4),11(4))].">
      <j1 rule="para" parents="5 17"/>
      <j2 rule="rewrite" parents="139 3 11"/>
    </justification>
  </clause>

  <clause id="202">
    <literal><![CDATA[
      x' @ (x @ y) = e
    ]]></literal>
    <justification jstring="[para(191(a,1),72(a,2))].">
      <j1 rule="para" parents="191 72"/>
    </justification>
  </clause>

  <clause id="237">
    <literal><![CDATA[
      (x * y)' * x = y'
    ]]></literal>
    <justification jstring="[para(54(a,1),19(a,1,2,2)),rewrite(29(4))].">
      <j1 rule="para" parents="54 19"/>
      <j2 rule="rewrite" parents="29"/>
    </justification>
  </clause>

  <clause id="250">
    <literal><![CDATA[
      x @ (x @ y) = e
    ]]></literal>
    <justification jstring="[para(109(a,1),202(a,1,2)),rewrite(62(2))].">
      <j1 rule="para" parents="109 202"/>
      <j2 rule="rewrite" parents="62"/>
    </justification>
  </clause>

  <clause id="267">
    <literal><![CDATA[
      (x @ y)' = x' @ y
    ]]></literal>
    <justification jstring="[para(54(a,1),25(a,1,2,1)),rewrite(11(4),29(4))].">
      <j1 rule="para" parents="54 25"/>
      <j2 rule="rewrite" parents="11 29"/>
    </justification>
  </clause>

  <clause id="305">
    <literal><![CDATA[
      (F @ G) @ D' != C @ (B @ A)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(80(a,2),134(a,2)),rewrite(267(12),109(12))].">
      <j1 rule="para" parents="80 134"/>
      <j2 rule="rewrite" parents="267 109"/>
    </justification>
  </clause>

  <clause id="307">
    <literal><![CDATA[
      D @ (F @ G) != C @ (B @ A)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(72(a,2),305(a,1)),rewrite(62(3))].">
      <j1 rule="para" parents="72 305"/>
      <j2 rule="rewrite" parents="62"/>
    </justification>
  </clause>

  <clause id="313">
    <literal><![CDATA[
      x * y @ (y @ z) = x @ (y @ z)
    ]]></literal>
    <justification jstring="[para(250(a,1),6(a,1,2)),rewrite(29(4)),flip(a)].">
      <j1 rule="para" parents="250 6"/>
      <j2 rule="rewrite" parents="29"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="331">
    <literal><![CDATA[
      (x * y)' * (y' @ z) * x * z' * y * z = x @ (y @ z)
    ]]></literal>
    <justification jstring="[para(27(a,2),10(a,1,2,2,2)),rewrite(267(4),313(13))].">
      <j1 rule="para" parents="27 10"/>
      <j2 rule="rewrite" parents="267 313"/>
    </justification>
  </clause>

  <clause id="346">
    <literal><![CDATA[
      D @ (G @ F') != C @ (B @ A)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(72(a,2),307(a,1,2)),rewrite(131(5))].">
      <j1 rule="para" parents="72 307"/>
      <j2 rule="rewrite" parents="131"/>
    </justification>
  </clause>

  <clause id="350">
    <literal><![CDATA[
      (G @ F) @ D != C @ (B @ A)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(72(a,2),346(a,1)),rewrite(267(5),109(5))].">
      <j1 rule="para" parents="72 346"/>
      <j2 rule="rewrite" parents="267 109"/>
    </justification>
  </clause>

  <clause id="372">
    <literal><![CDATA[
      D' @ (G @ F) != C @ (B @ A)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(72(a,2),350(a,1))].">
      <j1 rule="para" parents="72 350"/>
    </justification>
  </clause>

  <clause id="451">
    <literal><![CDATA[
      D' @ (G @ F) != (B @ A') @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(72(a,2),372(a,2)),rewrite(267(10),131(10))].">
      <j1 rule="para" parents="72 372"/>
      <j2 rule="rewrite" parents="267 131"/>
    </justification>
  </clause>

  <clause id="590">
    <literal><![CDATA[
      (x * y)' = y' * x'
    ]]></literal>
    <justification jstring="[para(8(a,1),237(a,1,1,1)),flip(a)].">
      <j1 rule="para" parents="8 237"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="630">
    <literal><![CDATA[
      x' * y' * (x' @ z) * y * z' * x * z = y @ (x @ z)
    ]]></literal>
    <justification jstring="[back_rewrite(331),rewrite(590(2),1(11))].">
      <j1 rule="back_rewrite" parents="331"/>
      <j2 rule="rewrite" parents="590 1"/>
    </justification>
  </clause>

  <clause id="1183">
    <literal><![CDATA[
      D' @ (G @ F) != C' @ (B @ A')
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(72(a,2),451(a,2))].">
      <j1 rule="para" parents="72 451"/>
    </justification>
  </clause>

  <clause id="1432">
    <literal><![CDATA[
      (x' @ y) * (z' @ y) * y' * z * x * y = z * x
    ]]></literal>
    <justification jstring="[para(67(a,2),67(a,1,2)),rewrite(267(3),590(2),14(9),250(14),29(12))].">
      <j1 rule="para" parents="67 67"/>
      <j2 rule="rewrite" parents="267 590 14 250 29"/>
    </justification>
  </clause>

  <clause id="1576">
    <literal><![CDATA[
      (x' @ y) * y' * x * z = x * y' * z
    ]]></literal>
    <justification jstring="[para(8(a,1),82(a,1,2,2)),rewrite(62(2)),flip(a)].">
      <j1 rule="para" parents="8 82"/>
      <j2 rule="rewrite" parents="62"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1636">
    <literal><![CDATA[
      (x' @ y) * z * y' * x * y = z * x
    ]]></literal>
    <justification jstring="[back_rewrite(1432),rewrite(1576(9))].">
      <j1 rule="back_rewrite" parents="1432"/>
      <j2 rule="rewrite" parents="1576"/>
    </justification>
  </clause>

  <clause id="1638">
    <literal><![CDATA[
      x @ (y @ z) = e
    ]]></literal>
    <justification jstring="[back_rewrite(630),rewrite(1636(9),8(4),3(2)),flip(a)].">
      <j1 rule="back_rewrite" parents="630"/>
      <j2 rule="rewrite" parents="1636 8 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1665">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[back_rewrite(1183),rewrite(1638(6),1638(8)),xx(a)].">
      <j1 rule="back_rewrite" parents="1183"/>
      <j2 rule="rewrite" parents="1638 1638"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
