<?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-cs.out</source>

<heading><![CDATA[
Prover9 (32) version 22-May-2007, May 2007.
Process 27438 was started by mccune on cleo,
Tue May 22 14:45:44 2007
The command was "/home/mccune/bin/prover9 -f cs.in DA.in".
]]></heading>

<proof number="1" length="86" max_count="23">

<comments><![CDATA[
% Proof 1 at 0.28 (+ 0.01) seconds: A.
% Length of proof is 86.
% Level of proof is 26.
% Maximum clause weight is 23.
% Given clauses 211.
]]></comments>

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

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

  <clause id="3" type="assumption">
    <literal><![CDATA[
      (x * y) * z = x * y * z
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

  <clause id="5" type="assumption">
    <literal><![CDATA[
      (x @ y) * (z @ y) = x * z @ y
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6">
    <literal><![CDATA[
      x * y @ z = (x @ z) * (y @ z)
    ]]></literal>
    <justification jstring="[copy(5),flip(a)].">
      <j1 rule="copy" parents="5"/>
      <j2 rule="flip"/>
    </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 * y * z != x * y * u
    ]]></literal>
    <literal><![CDATA[
      z = u
    ]]></literal>
    <justification jstring="[para(3(a,1),1(a,1)),rewrite([3(4)])].">
      <j1 rule="para" parents="3 1"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      x * (x @ x) = x
    ]]></literal>
    <justification jstring="[hyper(1,a,4,a)].">
      <j1 rule="hyper" parents="1 4"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      x * y != z * x
    ]]></literal>
    <literal><![CDATA[
      z * (z @ x) = y
    ]]></literal>
    <justification jstring="[para(4(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="4 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      x * y * (y @ x) * z = y * x * z
    ]]></literal>
    <justification jstring="[para(4(a,1),3(a,1,1)),rewrite([3(2),3(5)]),flip(a)].">
      <j1 rule="para" parents="4 3"/>
      <j2 rule="rewrite" parents="3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      x * y * z * (z @ x * y) = z * x * y
    ]]></literal>
    <justification jstring="[para(4(a,1),3(a,1)),flip(a)].">
      <j1 rule="para" parents="4 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      (x @ y) * (z @ y) * ((z @ x) @ y) = (z @ y) * (x @ y)
    ]]></literal>
    <justification jstring="[para(4(a,1),6(a,1,1)),rewrite([6(2),6(7)]),flip(a)].">
      <j1 rule="para" parents="4 6"/>
      <j2 rule="rewrite" parents="6 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x * y * ((A @ B) @ C) != x * y * (D @ (F @ G))
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[ur(8,b,7,a)].">
      <j1 rule="ur" parents="8 7"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      x * y * z != y * x
    ]]></literal>
    <literal><![CDATA[
      y @ x = z
    ]]></literal>
    <justification jstring="[para(4(a,1),8(a,1)),flip(a)].">
      <j1 rule="para" parents="4 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      (F @ G) * D * ((A @ B) @ C) != D * (F @ G)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(4(a,1),20(a,2))].">
      <j1 rule="para" parents="4 20"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      x @ x = y
    ]]></literal>
    <justification jstring="[para(10(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="10 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="37">
    <literal><![CDATA[
      x * (x @ x) * y = x * y
    ]]></literal>
    <justification jstring="[para(10(a,1),3(a,1,1)),flip(a)].">
      <j1 rule="para" parents="10 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      x * (F @ G) * D * ((A @ B) @ C) != x * D * (F @ G)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[ur(1,b,30,a)].">
      <j1 rule="ur" parents="1 30"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      (x @ x) * y = y
    ]]></literal>
    <justification jstring="[hyper(8,a,37,a)].">
      <j1 rule="hyper" parents="8 37"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      x * y != y
    ]]></literal>
    <literal><![CDATA[
      z @ z = x
    ]]></literal>
    <justification jstring="[para(62(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="62 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      x * ((y @ y) @ x) = x
    ]]></literal>
    <justification jstring="[para(62(a,1),4(a,1,2)),rewrite([62(5)])].">
      <j1 rule="para" parents="62 4"/>
      <j2 rule="rewrite" parents="62"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      (x @ x * y) * (y @ x * y) * z = z
    ]]></literal>
    <justification jstring="[para(6(a,1),62(a,1,1)),rewrite([3(6)])].">
      <j1 rule="para" parents="6 62"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="76">
    <literal><![CDATA[
      (x @ x) @ y = y @ y
    ]]></literal>
    <justification jstring="[hyper(34,a,69,a),flip(a)].">
      <j1 rule="hyper" parents="34 69"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="94">
    <literal><![CDATA[
      (x @ x) @ y * z = (y @ y * z) * (z @ y * z)
    ]]></literal>
    <justification jstring="[para(76(a,2),6(a,1))].">
      <j1 rule="para" parents="76 6"/>
    </justification>
  </clause>

  <clause id="96">
    <literal><![CDATA[
      ((x @ x) @ y) * z = z
    ]]></literal>
    <justification jstring="[para(76(a,2),62(a,1,1))].">
      <j1 rule="para" parents="76 62"/>
    </justification>
  </clause>

  <clause id="97">
    <literal><![CDATA[
      x * (((y @ y) @ z) @ x) = x
    ]]></literal>
    <justification jstring="[para(76(a,2),69(a,1,2,1))].">
      <j1 rule="para" parents="76 69"/>
    </justification>
  </clause>

  <clause id="116">
    <literal><![CDATA[
      x * (x @ ((y @ y) @ z)) = x * ((y @ y) @ z)
    ]]></literal>
    <justification jstring="[hyper(11,a,96,a)].">
      <j1 rule="hyper" parents="11 96"/>
    </justification>
  </clause>

  <clause id="124">
    <literal><![CDATA[
      (((x @ x) @ y) @ z) * u = u
    ]]></literal>
    <justification jstring="[para(76(a,2),96(a,1,1,1))].">
      <j1 rule="para" parents="76 96"/>
    </justification>
  </clause>

  <clause id="125">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      ((z @ z) @ u) @ x = y
    ]]></literal>
    <justification jstring="[para(96(a,1),11(a,2)),rewrite([96(8)])].">
      <j1 rule="para" parents="96 11"/>
      <j2 rule="rewrite" parents="96"/>
    </justification>
  </clause>

  <clause id="135">
    <literal><![CDATA[
      (x @ x) @ y = z @ z
    ]]></literal>
    <justification jstring="[hyper(68,a,96,a),flip(a)].">
      <j1 rule="hyper" parents="68 96"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="136">
    <literal><![CDATA[
      x @ x = y @ y
    ]]></literal>
    <justification jstring="[hyper(68,a,62,a)].">
      <j1 rule="hyper" parents="68 62"/>
    </justification>
  </clause>

  <clause id="138">
    <literal><![CDATA[
      x @ x = c_0
    ]]></literal>
    <justification jstring="[new_symbol(136)].">
      <j1 rule="new_symbol" parents="136"/>
    </justification>
  </clause>

  <clause id="140">
    <literal><![CDATA[
      c_0 @ x = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(135),rewrite([138(1),138(3)])].">
      <j1 rule="back_rewrite" parents="135"/>
      <j2 rule="rewrite" parents="138 138"/>
    </justification>
  </clause>

  <clause id="145">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      c_0 = y
    ]]></literal>
    <justification jstring="[back_rewrite(125),rewrite([138(3),140(4),140(4)])].">
      <j1 rule="back_rewrite" parents="125"/>
      <j2 rule="rewrite" parents="138 140 140"/>
    </justification>
  </clause>

  <clause id="146">
    <literal><![CDATA[
      c_0 * x = x
    ]]></literal>
    <justification jstring="[back_rewrite(124),rewrite([138(1),140(2),140(2)])].">
      <j1 rule="back_rewrite" parents="124"/>
      <j2 rule="rewrite" parents="138 140 140"/>
    </justification>
  </clause>

  <clause id="150">
    <literal><![CDATA[
      x * (x @ c_0) = x * c_0
    ]]></literal>
    <justification jstring="[back_rewrite(116),rewrite([138(1),140(2),138(4),140(5)])].">
      <j1 rule="back_rewrite" parents="116"/>
      <j2 rule="rewrite" parents="138 140 138 140"/>
    </justification>
  </clause>

  <clause id="155">
    <literal><![CDATA[
      x * c_0 = x
    ]]></literal>
    <justification jstring="[back_rewrite(97),rewrite([138(1),140(2),140(2)])].">
      <j1 rule="back_rewrite" parents="97"/>
      <j2 rule="rewrite" parents="138 140 140"/>
    </justification>
  </clause>

  <clause id="157">
    <literal><![CDATA[
      (x @ x * y) * (y @ x * y) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(94),rewrite([138(1),140(3)]),flip(a)].">
      <j1 rule="back_rewrite" parents="94"/>
      <j2 rule="rewrite" parents="138 140"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="159">
    <literal><![CDATA[
      x * (x @ c_0) = x
    ]]></literal>
    <justification jstring="[back_rewrite(150),rewrite([155(5)])].">
      <j1 rule="back_rewrite" parents="150"/>
      <j2 rule="rewrite" parents="155"/>
    </justification>
  </clause>

  <clause id="183">
    <literal><![CDATA[
      x @ c_0 = c_0
    ]]></literal>
    <justification jstring="[hyper(145,a,159,a),flip(a)].">
      <j1 rule="hyper" parents="145 159"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="207">
    <literal><![CDATA[
      x * y * (y @ y * x) = y * x
    ]]></literal>
    <justification jstring="[hyper(1,a,15,a)].">
      <j1 rule="hyper" parents="1 15"/>
    </justification>
  </clause>

  <clause id="260">
    <literal><![CDATA[
      x @ x * y != c_0
    ]]></literal>
    <literal><![CDATA[
      y @ x * y = c_0
    ]]></literal>
    <justification jstring="[para(157(a,1),145(a,1)),flip(a),flip(b)].">
      <j1 rule="para" parents="157 145"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="266">
    <literal><![CDATA[
      x * (x @ x * y) = x * (x @ y)
    ]]></literal>
    <justification jstring="[hyper(11,a,207,a),flip(a)].">
      <j1 rule="hyper" parents="11 207"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="277">
    <literal><![CDATA[
      (x @ y * x) * (y @ y * x) = c_0
    ]]></literal>
    <justification jstring="[para(157(a,1),207(a,1,2,2,2)),rewrite([183(8),155(6),157(10)])].">
      <j1 rule="para" parents="157 207"/>
      <j2 rule="rewrite" parents="183 155 157"/>
    </justification>
  </clause>

  <clause id="278">
    <literal><![CDATA[
      x @ x * y = x @ y
    ]]></literal>
    <justification jstring="[hyper(1,a,266,a)].">
      <j1 rule="hyper" parents="1 266"/>
    </justification>
  </clause>

  <clause id="284">
    <literal><![CDATA[
      x * (x @ y) != x
    ]]></literal>
    <literal><![CDATA[
      x @ y = c_0
    ]]></literal>
    <justification jstring="[para(266(a,1),145(a,1)),rewrite([278(6)]),flip(b)].">
      <j1 rule="para" parents="266 145"/>
      <j2 rule="rewrite" parents="278"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="293">
    <literal><![CDATA[
      (x @ y * x) * (y @ x) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(277),rewrite([278(4)])].">
      <j1 rule="back_rewrite" parents="277"/>
      <j2 rule="rewrite" parents="278"/>
    </justification>
  </clause>

  <clause id="298">
    <literal><![CDATA[
      x @ y != c_0
    ]]></literal>
    <literal><![CDATA[
      y @ x * y = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(260),rewrite([278(2)])].">
      <j1 rule="back_rewrite" parents="260"/>
      <j2 rule="rewrite" parents="278"/>
    </justification>
  </clause>

  <clause id="307">
    <literal><![CDATA[
      (x @ y) * (y @ x * y) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(157),rewrite([278(2)])].">
      <j1 rule="back_rewrite" parents="157"/>
      <j2 rule="rewrite" parents="278"/>
    </justification>
  </clause>

  <clause id="310">
    <literal><![CDATA[
      (x @ y) * (y @ x * y) * z = z
    ]]></literal>
    <justification jstring="[back_rewrite(72),rewrite([278(2)])].">
      <j1 rule="back_rewrite" parents="72"/>
      <j2 rule="rewrite" parents="278"/>
    </justification>
  </clause>

  <clause id="322">
    <literal><![CDATA[
      (x @ y) * ((x @ y) @ y) = x @ y
    ]]></literal>
    <justification jstring="[para(138(a,1),17(a,1,1)),rewrite([146(6),138(6),155(7)])].">
      <j1 rule="para" parents="138 17"/>
      <j2 rule="rewrite" parents="146 138 155"/>
    </justification>
  </clause>

  <clause id="324">
    <literal><![CDATA[
      (x @ (y @ x)) * (y @ (y @ x)) = (y @ (y @ x)) * (x @ (y @ x))
    ]]></literal>
    <justification jstring="[para(138(a,1),17(a,1,2,2)),rewrite([155(6)])].">
      <j1 rule="para" parents="138 17"/>
      <j2 rule="rewrite" parents="155"/>
    </justification>
  </clause>

  <clause id="338">
    <literal><![CDATA[
      (x @ y * z) * (y @ x * y * z) = (x @ z) * (y @ z)
    ]]></literal>
    <justification jstring="[para(3(a,1),278(a,1,2)),rewrite([6(4),278(3),6(8)])].">
      <j1 rule="para" parents="3 278"/>
      <j2 rule="rewrite" parents="6 278 6"/>
    </justification>
  </clause>

  <clause id="339">
    <literal><![CDATA[
      x @ y * (y @ x) = x @ y * x
    ]]></literal>
    <justification jstring="[para(4(a,1),278(a,1,2)),flip(a)].">
      <j1 rule="para" parents="4 278"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="351">
    <literal><![CDATA[
      (x @ y * x) * (y @ x) * z = z
    ]]></literal>
    <justification jstring="[para(293(a,1),3(a,1,1)),rewrite([146(2)]),flip(a)].">
      <j1 rule="para" parents="293 3"/>
      <j2 rule="rewrite" parents="146"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="376">
    <literal><![CDATA[
      (x @ y) @ x * y = c_0
    ]]></literal>
    <justification jstring="[para(4(a,1),307(a,1,2,2)),rewrite([339(3),6(6),278(4),351(8)])].">
      <j1 rule="para" parents="4 307"/>
      <j2 rule="rewrite" parents="339 6 278 351"/>
    </justification>
  </clause>

  <clause id="386">
    <literal><![CDATA[
      x * y * (y @ x * y) = y * x
    ]]></literal>
    <justification jstring="[para(307(a,1),14(a,1,2,2)),rewrite([155(2)]),flip(a)].">
      <j1 rule="para" parents="307 14"/>
      <j2 rule="rewrite" parents="155"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="396">
    <literal><![CDATA[
      (x @ y) * x * y = x * y * (x @ y)
    ]]></literal>
    <justification jstring="[para(376(a,1),4(a,1,2,2)),rewrite([155(4),3(3)]),flip(a)].">
      <j1 rule="para" parents="376 4"/>
      <j2 rule="rewrite" parents="155 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="397">
    <literal><![CDATA[
      (x @ y * x) @ y * x = c_0
    ]]></literal>
    <justification jstring="[para(4(a,1),376(a,1,2)),rewrite([339(3)])].">
      <j1 rule="para" parents="4 376"/>
      <j2 rule="rewrite" parents="339"/>
    </justification>
  </clause>

  <clause id="408">
    <literal><![CDATA[
      (x @ (x @ y)) * (y @ (x @ y)) = c_0
    ]]></literal>
    <justification jstring="[para(376(a,1),293(a,1,2)),rewrite([396(4),6(5),278(4),338(8),155(7)])].">
      <j1 rule="para" parents="376 293"/>
      <j2 rule="rewrite" parents="396 6 278 338 155"/>
    </justification>
  </clause>

  <clause id="409">
    <literal><![CDATA[
      (x @ (y @ x)) * (y @ (y @ x)) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(324),rewrite([408(10)])].">
      <j1 rule="back_rewrite" parents="324"/>
      <j2 rule="rewrite" parents="408"/>
    </justification>
  </clause>

  <clause id="439">
    <literal><![CDATA[
      (x @ y * x) * y * x = x * y
    ]]></literal>
    <justification jstring="[para(397(a,1),4(a,1,2,2)),rewrite([155(5),3(4),386(4)]),flip(a)].">
      <j1 rule="para" parents="397 4"/>
      <j2 rule="rewrite" parents="155 3 386"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="447">
    <literal><![CDATA[
      (x @ y * x) @ x * y = c_0
    ]]></literal>
    <justification jstring="[para(397(a,1),278(a,2)),rewrite([439(6)])].">
      <j1 rule="para" parents="397 278"/>
      <j2 rule="rewrite" parents="439"/>
    </justification>
  </clause>

  <clause id="473">
    <literal><![CDATA[
      (x @ y * x) * x * y = x * y * (x @ y * x)
    ]]></literal>
    <justification jstring="[para(447(a,1),4(a,1,2,2)),rewrite([155(5),3(4)]),flip(a)].">
      <j1 rule="para" parents="447 4"/>
      <j2 rule="rewrite" parents="155 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="485">
    <literal><![CDATA[
      (x @ (x @ y * x)) * (y @ (x @ y * x)) = c_0
    ]]></literal>
    <justification jstring="[para(447(a,1),293(a,1,2)),rewrite([473(5),6(6),278(5),338(10),155(9)])].">
      <j1 rule="para" parents="447 293"/>
      <j2 rule="rewrite" parents="473 6 278 338 155"/>
    </justification>
  </clause>

  <clause id="491">
    <literal><![CDATA[
      (x @ y) @ (z @ y) = (x @ z) @ y
    ]]></literal>
    <justification jstring="[hyper(23,a,17,a)].">
      <j1 rule="hyper" parents="23 17"/>
    </justification>
  </clause>

  <clause id="534">
    <literal><![CDATA[
      (x @ y) @ (y @ x * y) * z = (x @ y) @ z
    ]]></literal>
    <justification jstring="[para(310(a,1),278(a,1,2)),flip(a)].">
      <j1 rule="para" parents="310 278"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="589">
    <literal><![CDATA[
      (x @ y) @ y = c_0
    ]]></literal>
    <justification jstring="[hyper(284,a,322,a)].">
      <j1 rule="hyper" parents="284 322"/>
    </justification>
  </clause>

  <clause id="590">
    <literal><![CDATA[
      (x @ y) * y = y * (x @ y)
    ]]></literal>
    <justification jstring="[para(322(a,1),4(a,1,2)),flip(a)].">
      <j1 rule="para" parents="322 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="597">
    <literal><![CDATA[
      x @ (y @ x) = c_0
    ]]></literal>
    <justification jstring="[hyper(298,a,589,a),rewrite([590(2),278(3)])].">
      <j1 rule="hyper" parents="298 589"/>
      <j2 rule="rewrite" parents="590 278"/>
    </justification>
  </clause>

  <clause id="605">
    <literal><![CDATA[
      x @ (x @ y) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(409),rewrite([597(2),146(4)])].">
      <j1 rule="back_rewrite" parents="409"/>
      <j2 rule="rewrite" parents="597 146"/>
    </justification>
  </clause>

  <clause id="610">
    <literal><![CDATA[
      x @ (y @ x * y) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(485),rewrite([605(3),146(5)])].">
      <j1 rule="back_rewrite" parents="485"/>
      <j2 rule="rewrite" parents="605 146"/>
    </justification>
  </clause>

  <clause id="669">
    <literal><![CDATA[
      (x @ y) @ ((y @ x * y) @ z) = c_0
    ]]></literal>
    <justification jstring="[para(310(a,1),610(a,1,2,2)),rewrite([6(5),138(5),155(6)])].">
      <j1 rule="para" parents="310 610"/>
      <j2 rule="rewrite" parents="6 138 155"/>
    </justification>
  </clause>

  <clause id="837">
    <literal><![CDATA[
      D * ((A @ B) @ C) != (G @ F * G) * D * (F @ G)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(351(a,1),55(a,1))].">
      <j1 rule="para" parents="351 55"/>
    </justification>
  </clause>

  <clause id="854">
    <literal><![CDATA[
      (x @ y * x) @ (y @ x) * z = (x @ y * x) @ z
    ]]></literal>
    <justification jstring="[para(351(a,1),278(a,1,2)),flip(a)].">
      <j1 rule="para" parents="351 278"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="872">
    <literal><![CDATA[
      (x @ y) @ z = z @ (y @ x * y) * z
    ]]></literal>
    <justification jstring="[para(351(a,1),339(a,2,2)),rewrite([854(9),6(9),534(8),669(5),339(8),146(6),6(7),138(7),155(8)]),flip(a)].">
      <j1 rule="para" parents="351 339"/>
      <j2 rule="rewrite" parents="854 6 534 669 339 146 6 138 155"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="874">
    <literal><![CDATA[
      x @ y * x = x @ y
    ]]></literal>
    <justification jstring="[hyper(23,a,386,a),flip(a)].">
      <j1 rule="hyper" parents="23 386"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="884">
    <literal><![CDATA[
      (x @ y) * (y @ x) = c_0
    ]]></literal>
    <justification jstring="[para(386(a,1),293(a,1,1,2)),rewrite([874(2),6(4),278(2),376(4),155(3),874(3),339(4),874(3)])].">
      <j1 rule="para" parents="386 293"/>
      <j2 rule="rewrite" parents="874 6 278 376 155 874 339 874"/>
    </justification>
  </clause>

  <clause id="911">
    <literal><![CDATA[
      (x @ y) @ z = z @ (y @ x)
    ]]></literal>
    <justification jstring="[back_rewrite(872),rewrite([874(4),874(5)])].">
      <j1 rule="back_rewrite" parents="872"/>
      <j2 rule="rewrite" parents="874 874"/>
    </justification>
  </clause>

  <clause id="944">
    <literal><![CDATA[
      (G @ F) * D * (F @ G) != D * (C @ (B @ A))
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[back_rewrite(837),rewrite([911(6),874(12)]),flip(a)].">
      <j1 rule="back_rewrite" parents="837"/>
      <j2 rule="rewrite" parents="911 874"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1376">
    <literal><![CDATA[
      x * (y @ z) * ((y @ x) @ (z @ x)) = (y @ z) * x
    ]]></literal>
    <justification jstring="[para(491(a,2),4(a,1,2,2))].">
      <j1 rule="para" parents="491 4"/>
    </justification>
  </clause>

  <clause id="1397">
    <literal><![CDATA[
      (x @ y) @ (z @ y) = c_0
    ]]></literal>
    <justification jstring="[para(597(a,1),491(a,1,2)),rewrite([183(4)]),flip(a)].">
      <j1 rule="para" parents="597 491"/>
      <j2 rule="rewrite" parents="183"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1415">
    <literal><![CDATA[
      x @ (y @ z) = c_0
    ]]></literal>
    <justification jstring="[para(491(a,2),884(a,1,1)),rewrite([1397(3),146(4)])].">
      <j1 rule="para" parents="491 884"/>
      <j2 rule="rewrite" parents="1397 146"/>
    </justification>
  </clause>

  <clause id="1425">
    <literal><![CDATA[
      (x @ y) * z = z * (x @ y)
    ]]></literal>
    <justification jstring="[back_rewrite(1376),rewrite([1415(4),155(3)]),flip(a)].">
      <j1 rule="back_rewrite" parents="1376"/>
      <j2 rule="rewrite" parents="1415 155"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1426">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[back_rewrite(944),rewrite([1425(9),3(9),884(8),155(3),1415(7),155(4)]),xx(a)].">
      <j1 rule="back_rewrite" parents="944"/>
      <j2 rule="rewrite" parents="1425 3 884 155 1415 155"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
