<?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 March-2007, March 2007.
Process 20999 was started by mccune on cleo,
Mon Mar 19 17:01:53 2007
The command was "/home/mccune/bin/prover9 -f gt.in DA.in".
]]></heading>

<proof number="1" length="61" max_count="24">

<comments><![CDATA[
% Proof 1 at 0.28 (+ 0.00) seconds: A.
% Length of proof is 61.
% Level of proof is 16.
% Maximum clause weight is 24.
% Given clauses 167.
]]></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 * z @ 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="15">
    <literal><![CDATA[
      (x @ y)' * (z @ y)' * (x * z @ y) = (x @ y) @ (z @ y)
    ]]></literal>
    <justification jstring="[para(6(a,1),5(a,1,2,2))].">
      <j1 rule="para" parents="6 5"/>
    </justification>
  </clause>

  <clause id="16">
    <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="18">
    <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="20">
    <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="22">
    <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="23">
    <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="26">
    <literal><![CDATA[
      x * e = x
    ]]></literal>
    <justification jstring="[back_rewrite(18),rewrite(23(4))].">
      <j1 rule="back_rewrite" parents="18"/>
      <j2 rule="rewrite" parents="23"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      x * y @ x = y @ x
    ]]></literal>
    <justification jstring="[para(20(a,1),6(a,1,1)),rewrite(2(3)),flip(a)].">
      <j1 rule="para" parents="20 6"/>
      <j2 rule="rewrite" parents="2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      (x' @ y) * x = x * (y @ x)
    ]]></literal>
    <justification jstring="[para(5(a,1),9(a,1,2)),rewrite(23(4)),flip(a)].">
      <j1 rule="para" parents="5 9"/>
      <j2 rule="rewrite" parents="23"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      x' * y' * x * y * (z @ y) = x * z @ y
    ]]></literal>
    <justification jstring="[para(9(a,2),6(a,1))].">
      <j1 rule="para" parents="9 6"/>
    </justification>
  </clause>

  <clause id="49">
    <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="50">
    <literal><![CDATA[
      x * y' @ y = x @ y
    ]]></literal>
    <justification jstring="[para(12(a,1),6(a,1,2)),rewrite(26(3)),flip(a)].">
      <j1 rule="para" parents="12 6"/>
      <j2 rule="rewrite" parents="26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="51">
    <literal><![CDATA[
      x * x' = e
    ]]></literal>
    <justification jstring="[para(23(a,1),3(a,1))].">
      <j1 rule="para" parents="23 3"/>
    </justification>
  </clause>

  <clause id="56">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[para(23(a,1),26(a,1)),rewrite(26(2)),flip(a)].">
      <j1 rule="para" parents="23 26"/>
      <j2 rule="rewrite" parents="26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      x @ x' = e
    ]]></literal>
    <justification jstring="[para(56(a,1),12(a,1,1))].">
      <j1 rule="para" parents="56 12"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      (x * y')' * y' * x = x @ y
    ]]></literal>
    <justification jstring="[para(3(a,1),10(a,1,2,2,2)),rewrite(26(6),50(9))].">
      <j1 rule="para" parents="3 10"/>
      <j2 rule="rewrite" parents="26 50"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      (x * y)' * y * x = y @ x
    ]]></literal>
    <justification jstring="[para(8(a,1),10(a,1,2)),rewrite(30(6))].">
      <j1 rule="para" parents="8 10"/>
      <j2 rule="rewrite" parents="30"/>
    </justification>
  </clause>

  <clause id="70">
    <literal><![CDATA[
      x' @ y = y @ x
    ]]></literal>
    <justification jstring="[back_rewrite(61),rewrite(66(6))].">
      <j1 rule="back_rewrite" parents="61"/>
      <j2 rule="rewrite" parents="66"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      x * y @ y' = y @ x
    ]]></literal>
    <justification jstring="[para(51(a,1),10(a,1,2,2,2)),rewrite(56(4),26(4),66(4)),flip(a)].">
      <j1 rule="para" parents="51 10"/>
      <j2 rule="rewrite" parents="56 26 66"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="76">
    <literal><![CDATA[
      x @ y' = y @ x
    ]]></literal>
    <justification jstring="[para(59(a,1),6(a,1,2)),rewrite(26(4),74(5))].">
      <j1 rule="para" parents="59 6"/>
      <j2 rule="rewrite" parents="26 74"/>
    </justification>
  </clause>

  <clause id="87">
    <literal><![CDATA[
      D @ (G' @ F) != (A @ B) @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(70(a,2),7(a,2,2)),flip(a)].">
      <j1 rule="para" parents="70 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      x' @ y' = x @ y
    ]]></literal>
    <justification jstring="[para(70(a,2),70(a,1))].">
      <j1 rule="para" parents="70 70"/>
    </justification>
  </clause>

  <clause id="118">
    <literal><![CDATA[
      (x @ y) * (y @ z') = x * z @ y
    ]]></literal>
    <justification jstring="[para(76(a,2),6(a,1,2))].">
      <j1 rule="para" parents="76 6"/>
    </justification>
  </clause>

  <clause id="126">
    <literal><![CDATA[
      x' @ y = x @ y'
    ]]></literal>
    <justification jstring="[para(76(a,1),70(a,1)),flip(a)].">
      <j1 rule="para" parents="76 70"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="136">
    <literal><![CDATA[
      (G @ F') @ D' != (A @ B) @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(76(a,2),87(a,1)),rewrite(126(4))].">
      <j1 rule="para" parents="76 87"/>
      <j2 rule="rewrite" parents="126"/>
    </justification>
  </clause>

  <clause id="141">
    <literal><![CDATA[
      x * y * z @ y = x * z @ y
    ]]></literal>
    <justification jstring="[para(30(a,1),6(a,1,2)),rewrite(6(3)),flip(a)].">
      <j1 rule="para" parents="30 6"/>
      <j2 rule="rewrite" parents="6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="175">
    <literal><![CDATA[
      (x' @ y)' * (z @ x)' * (y * z @ x) = (y @ x) @ (z @ x)
    ]]></literal>
    <justification jstring="[para(70(a,2),15(a,1,1,1))].">
      <j1 rule="para" parents="70 15"/>
    </justification>
  </clause>

  <clause id="184">
    <literal><![CDATA[
      (x @ y)' * (z @ y)' * (y @ (x * z)') = (x @ y) @ (z @ y)
    ]]></literal>
    <justification jstring="[para(76(a,2),15(a,1,2,2))].">
      <j1 rule="para" parents="76 15"/>
    </justification>
  </clause>

  <clause id="216">
    <literal><![CDATA[
      (x @ y) @ x = e
    ]]></literal>
    <justification jstring="[para(5(a,1),49(a,1,1)),rewrite(141(6),3(4),11(4))].">
      <j1 rule="para" parents="5 49"/>
      <j2 rule="rewrite" parents="141 3 11"/>
    </justification>
  </clause>

  <clause id="229">
    <literal><![CDATA[
      (x @ y) @ y = e
    ]]></literal>
    <justification jstring="[para(76(a,1),216(a,1,1))].">
      <j1 rule="para" parents="76 216"/>
    </justification>
  </clause>

  <clause id="257">
    <literal><![CDATA[
      (x * y)' * x = y'
    ]]></literal>
    <justification jstring="[para(51(a,1),16(a,1,2,2)),rewrite(26(4))].">
      <j1 rule="para" parents="51 16"/>
      <j2 rule="rewrite" parents="26"/>
    </justification>
  </clause>

  <clause id="265">
    <literal><![CDATA[
      x * (y @ z) @ z = x @ z
    ]]></literal>
    <justification jstring="[para(229(a,1),6(a,1,2)),rewrite(26(3)),flip(a)].">
      <j1 rule="para" parents="229 6"/>
      <j2 rule="rewrite" parents="26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="296">
    <literal><![CDATA[
      (x @ y)' = x' @ y
    ]]></literal>
    <justification jstring="[para(51(a,1),22(a,1,2,1)),rewrite(11(4),26(4))].">
      <j1 rule="para" parents="51 22"/>
      <j2 rule="rewrite" parents="11 26"/>
    </justification>
  </clause>

  <clause id="297">
    <literal><![CDATA[
      (x @ y) * (y @ x) = e
    ]]></literal>
    <justification jstring="[para(59(a,1),22(a,2)),rewrite(296(3),91(3),74(4))].">
      <j1 rule="para" parents="59 22"/>
      <j2 rule="rewrite" parents="296 91 74"/>
    </justification>
  </clause>

  <clause id="312">
    <literal><![CDATA[
      (x @ y) * (z' * y * z @ x) = (y @ x) @ (z @ x)
    ]]></literal>
    <justification jstring="[back_rewrite(175),rewrite(296(3),56(2),296(3),6(6))].">
      <j1 rule="back_rewrite" parents="175"/>
      <j2 rule="rewrite" parents="296 56 296 6"/>
    </justification>
  </clause>

  <clause id="326">
    <literal><![CDATA[
      (x @ y) @ (z @ y) = (x @ z) @ y
    ]]></literal>
    <justification jstring="[back_rewrite(184),rewrite(296(2),296(4),118(8),6(7),5(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="184"/>
      <j2 rule="rewrite" parents="296 296 118 6 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="338">
    <literal><![CDATA[
      (G @ F) @ D != (A @ B) @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(126(a,2),136(a,1)),rewrite(296(5),91(5))].">
      <j1 rule="para" parents="126 136"/>
      <j2 rule="rewrite" parents="296 91"/>
    </justification>
  </clause>

  <clause id="340">
    <literal><![CDATA[
      (B @ A') @ C != (G @ F) @ D
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(70(a,2),338(a,2,1)),rewrite(126(9)),flip(a)].">
      <j1 rule="para" parents="70 338"/>
      <j2 rule="rewrite" parents="126"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="408">
    <literal><![CDATA[
      (G @ F) @ D != C @ (B @ A)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(76(a,2),340(a,1)),rewrite(296(6),91(6)),flip(a)].">
      <j1 rule="para" parents="76 340"/>
      <j2 rule="rewrite" parents="296 91"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="409">
    <literal><![CDATA[
      (F' @ G) @ D != C @ (B @ A)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(70(a,2),408(a,1,1))].">
      <j1 rule="para" parents="70 408"/>
    </justification>
  </clause>

  <clause id="476">
    <literal><![CDATA[
      D @ (F @ G) != C @ (B @ A)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(76(a,2),409(a,1)),rewrite(296(6),56(4))].">
      <j1 rule="para" parents="76 409"/>
      <j2 rule="rewrite" parents="296 56"/>
    </justification>
  </clause>

  <clause id="479">
    <literal><![CDATA[
      (F @ G) @ D' != C @ (B @ A)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(76(a,2),476(a,1))].">
      <j1 rule="para" parents="76 476"/>
    </justification>
  </clause>

  <clause id="504">
    <literal><![CDATA[
      (x * y)' = y' * x'
    ]]></literal>
    <justification jstring="[para(8(a,1),257(a,1,1,1)),flip(a)].">
      <j1 rule="para" parents="8 257"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="598">
    <literal><![CDATA[
      (F @ G) @ D' != (B @ A) @ C'
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(76(a,2),479(a,2))].">
      <j1 rule="para" parents="76 479"/>
    </justification>
  </clause>

  <clause id="764">
    <literal><![CDATA[
      (x @ y) * y = y * (x @ y)
    ]]></literal>
    <justification jstring="[para(70(a,1),40(a,1,1))].">
      <j1 rule="para" parents="70 40"/>
    </justification>
  </clause>

  <clause id="2051">
    <literal><![CDATA[
      x' * y * x @ z = y @ z
    ]]></literal>
    <justification jstring="[para(265(a,1),5(a,2)),rewrite(504(3),296(2),1(8),764(7),1(10),41(9),6(5))].">
      <j1 rule="para" parents="265 5"/>
      <j2 rule="rewrite" parents="504 296 1 764 1 41 6"/>
    </justification>
  </clause>

  <clause id="2120">
    <literal><![CDATA[
      (x @ y) @ (z @ y) = e
    ]]></literal>
    <justification jstring="[back_rewrite(312),rewrite(2051(5),297(3)),flip(a)].">
      <j1 rule="back_rewrite" parents="312"/>
      <j2 rule="rewrite" parents="2051 297"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2130">
    <literal><![CDATA[
      (x @ y) @ z = e
    ]]></literal>
    <justification jstring="[back_rewrite(326),rewrite(2120(3)),flip(a)].">
      <j1 rule="back_rewrite" parents="326"/>
      <j2 rule="rewrite" parents="2120"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2193">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[back_rewrite(598),rewrite(2130(6),2130(7)),xx(a)].">
      <j1 rule="back_rewrite" parents="598"/>
      <j2 rule="rewrite" parents="2130 2130"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
