<?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>identity.out</source>

<heading><![CDATA[
Prover9 (32) version April-2007, April 2007.
Process 26439 was started by mccune on cleo,
Fri Apr 13 09:14:04 2007
The command was "/home/mccune/bin/prover9 -f identity.in".
]]></heading>

<proof number="1" length="134" max_count="67">

<comments><![CDATA[
% Proof 1 at 31.10 (+ 0.06) seconds.
% Length of proof is 134.
% Level of proof is 24.
% Maximum clause weight is 67.
% Given clauses 199.
]]></comments>

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

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

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

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

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

  <clause id="6" type="deny">
    <literal><![CDATA[
      ((c4 * c2) * c3) * c1 != ((c1 * c2) * c3) * c4
    ]]></literal>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      e * e = e
    ]]></literal>
    <justification jstring="[para(2(a,1),2(a,1,1)),rewrite(3(5)),flip(a)].">
      <j1 rule="para" parents="2 2"/>
      <j2 rule="rewrite" parents="3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      ((x * y) * x) * (e * y) = x * y
    ]]></literal>
    <justification jstring="[para(4(a,1),2(a,1))].">
      <j1 rule="para" parents="4 2"/>
    </justification>
  </clause>

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

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

  <clause id="11">
    <literal><![CDATA[
      (x * y) * ((x * z) * (y * u)) = z * u
    ]]></literal>
    <justification jstring="[para(4(a,1),3(a,1,2))].">
      <j1 rule="para" parents="4 3"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      (x * y) * ((x * z) * u) = z * (y * u)
    ]]></literal>
    <justification jstring="[para(3(a,1),4(a,1,1)),flip(a)].">
      <j1 rule="para" parents="3 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      (x * y) * (z * (y * u)) = (x * z) * u
    ]]></literal>
    <justification jstring="[para(3(a,1),4(a,1,2)),flip(a)].">
      <j1 rule="para" parents="3 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      ((x * x) * x) * e = x
    ]]></literal>
    <justification jstring="[para(5(a,1),3(a,1,2))].">
      <j1 rule="para" parents="5 3"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      (((x * y) * x) * ((x * y) * y)) * e = x * y
    ]]></literal>
    <justification jstring="[para(4(a,1),16(a,1,1))].">
      <j1 rule="para" parents="4 16"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      (x * y) * (e * (y * x)) = x
    ]]></literal>
    <justification jstring="[para(3(a,1),8(a,1,1,1)),rewrite(3(7))].">
      <j1 rule="para" parents="3 8"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      (x * y) * x = e * (y * x)
    ]]></literal>
    <justification jstring="[para(30(a,1),3(a,1,2))].">
      <j1 rule="para" parents="30 3"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      x * (e * (x * x)) = e * (x * x)
    ]]></literal>
    <justification jstring="[para(5(a,1),30(a,1,1)),rewrite(36(4),3(8),36(6))].">
      <j1 rule="para" parents="5 30"/>
      <j2 rule="rewrite" parents="36 3 36"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      e * ((x * x) * e) = x
    ]]></literal>
    <justification jstring="[para(5(a,1),30(a,1,2,2)),rewrite(36(2),41(4),7(6),36(5))].">
      <j1 rule="para" parents="5 30"/>
      <j2 rule="rewrite" parents="36 41 7 36"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      (x * x) * (e * x) = e
    ]]></literal>
    <justification jstring="[para(16(a,1),30(a,1,2,2)),rewrite(36(3),3(5))].">
      <j1 rule="para" parents="16 30"/>
      <j2 rule="rewrite" parents="36 3"/>
    </justification>
  </clause>

  <clause id="51">
    <literal><![CDATA[
      ((e * (x * y)) * ((y * x) * x)) * e = y * x
    ]]></literal>
    <justification jstring="[back_rewrite(28),rewrite(36(2))].">
      <j1 rule="back_rewrite" parents="28"/>
      <j2 rule="rewrite" parents="36"/>
    </justification>
  </clause>

  <clause id="58">
    <literal><![CDATA[
      (x * x) * e = e * x
    ]]></literal>
    <justification jstring="[para(42(a,1),3(a,1,2)),flip(a)].">
      <j1 rule="para" parents="42 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      ((x * e) * y) * z = x * (y * (x * z))
    ]]></literal>
    <justification jstring="[para(3(a,1),9(a,1,2))].">
      <j1 rule="para" parents="3 9"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      (x * (y * z)) * (e * ((x * z) * ((x * e) * y))) = (x * e) * y
    ]]></literal>
    <justification jstring="[para(9(a,1),30(a,1,1))].">
      <j1 rule="para" parents="9 30"/>
    </justification>
  </clause>

  <clause id="73">
    <literal><![CDATA[
      (x * e) * (x * x) = e
    ]]></literal>
    <justification jstring="[para(43(a,1),4(a,1)),flip(a)].">
      <j1 rule="para" parents="43 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="75">
    <literal><![CDATA[
      (e * x) * (x * x) = x * x
    ]]></literal>
    <justification jstring="[para(43(a,1),30(a,1,1)),rewrite(3(8))].">
      <j1 rule="para" parents="43 30"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="78">
    <literal><![CDATA[
      ((x * x) * y) * (e * z) = (e * x) * (y * z)
    ]]></literal>
    <justification jstring="[para(58(a,1),4(a,1,1)),flip(a)].">
      <j1 rule="para" parents="58 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      (x * e) * e = x * x
    ]]></literal>
    <justification jstring="[para(73(a,1),3(a,1,2))].">
      <j1 rule="para" parents="73 3"/>
    </justification>
  </clause>

  <clause id="99">
    <literal><![CDATA[
      ((c4 * c2) * (c1 * e)) * (c3 * c1) != ((c1 * c2) * c3) * c4
    ]]></literal>
    <justification jstring="[para(10(a,2),6(a,1))].">
      <j1 rule="para" parents="10 6"/>
    </justification>
  </clause>

  <clause id="108">
    <literal><![CDATA[
      (x * (e * (x * e))) * (x * (e * x)) = e
    ]]></literal>
    <justification jstring="[para(10(a,2),43(a,1)),rewrite(36(4))].">
      <j1 rule="para" parents="10 43"/>
      <j2 rule="rewrite" parents="36"/>
    </justification>
  </clause>

  <clause id="112">
    <literal><![CDATA[
      (x * e) * (x * e) = e * x
    ]]></literal>
    <justification jstring="[para(10(a,2),58(a,1)),rewrite(7(3))].">
      <j1 rule="para" parents="10 58"/>
      <j2 rule="rewrite" parents="7"/>
    </justification>
  </clause>

  <clause id="129">
    <literal><![CDATA[
      ((c4 * c1) * (c2 * e)) * (c3 * c1) != ((c1 * c2) * c3) * c4
    ]]></literal>
    <justification jstring="[para(4(a,1),99(a,1,1))].">
      <j1 rule="para" parents="4 99"/>
    </justification>
  </clause>

  <clause id="141">
    <literal><![CDATA[
      (x * x) * y = x * ((y * e) * (x * (e * y)))
    ]]></literal>
    <justification jstring="[para(82(a,1),10(a,2,1)),rewrite(61(8)),flip(a)].">
      <j1 rule="para" parents="82 10"/>
      <j2 rule="rewrite" parents="61"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="153">
    <literal><![CDATA[
      ((e * x) * y) * ((x * x) * z) = (x * x) * (y * z)
    ]]></literal>
    <justification jstring="[para(75(a,1),4(a,1,1)),flip(a)].">
      <j1 rule="para" parents="75 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="159">
    <literal><![CDATA[
      x * ((y * z) * ((y * x) * u)) = z * u
    ]]></literal>
    <justification jstring="[para(3(a,1),11(a,1,1))].">
      <j1 rule="para" parents="3 11"/>
    </justification>
  </clause>

  <clause id="171">
    <literal><![CDATA[
      e * (x * (y * (x * ((x * x) * z)))) = y * z
    ]]></literal>
    <justification jstring="[para(73(a,1),11(a,1,1)),rewrite(61(7))].">
      <j1 rule="para" parents="73 11"/>
      <j2 rule="rewrite" parents="61"/>
    </justification>
  </clause>

  <clause id="180">
    <literal><![CDATA[
      (x * x) * (x * (y * (x * (e * z)))) = y * z
    ]]></literal>
    <justification jstring="[para(82(a,1),11(a,1,1)),rewrite(61(7))].">
      <j1 rule="para" parents="82 11"/>
      <j2 rule="rewrite" parents="61"/>
    </justification>
  </clause>

  <clause id="191">
    <literal><![CDATA[
      (x * (y * z)) * (z * y) = (x * e) * y
    ]]></literal>
    <justification jstring="[back_rewrite(66),rewrite(159(9))].">
      <j1 rule="back_rewrite" parents="66"/>
      <j2 rule="rewrite" parents="159"/>
    </justification>
  </clause>

  <clause id="192">
    <literal><![CDATA[
      ((c4 * c1) * c3) * ((c2 * e) * c1) != ((c1 * c2) * c3) * c4
    ]]></literal>
    <justification jstring="[para(4(a,1),129(a,1))].">
      <j1 rule="para" parents="4 129"/>
    </justification>
  </clause>

  <clause id="211">
    <literal><![CDATA[
      (e * x) * x = x * e
    ]]></literal>
    <justification jstring="[para(112(a,1),30(a,1,1)),rewrite(112(8),3(6))].">
      <j1 rule="para" parents="112 30"/>
      <j2 rule="rewrite" parents="112 3"/>
    </justification>
  </clause>

  <clause id="215">
    <literal><![CDATA[
      x * (e * (x * e)) = e * x
    ]]></literal>
    <justification jstring="[para(112(a,1),10(a,2)),rewrite(82(4),3(2))].">
      <j1 rule="para" parents="112 10"/>
      <j2 rule="rewrite" parents="82 3"/>
    </justification>
  </clause>

  <clause id="218">
    <literal><![CDATA[
      (e * x) * (x * e) = x
    ]]></literal>
    <justification jstring="[para(112(a,1),36(a,1,1)),rewrite(112(11),3(9))].">
      <j1 rule="para" parents="112 36"/>
      <j2 rule="rewrite" parents="112 3"/>
    </justification>
  </clause>

  <clause id="222">
    <literal><![CDATA[
      (e * x) * (x * (e * x)) = e
    ]]></literal>
    <justification jstring="[back_rewrite(108),rewrite(215(5))].">
      <j1 rule="back_rewrite" parents="108"/>
      <j2 rule="rewrite" parents="215"/>
    </justification>
  </clause>

  <clause id="226">
    <literal><![CDATA[
      (e * x) * ((x * y) * y) = (x * y) * e
    ]]></literal>
    <justification jstring="[para(211(a,1),4(a,1)),flip(a)].">
      <j1 rule="para" parents="211 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="234">
    <literal><![CDATA[
      (x * (e * y)) * ((x * z) * (y * e)) = z * y
    ]]></literal>
    <justification jstring="[para(211(a,1),11(a,1,2,2))].">
      <j1 rule="para" parents="211 11"/>
    </justification>
  </clause>

  <clause id="235">
    <literal><![CDATA[
      (e * x) * ((x * y) * e) = (x * y) * y
    ]]></literal>
    <justification jstring="[para(211(a,1),11(a,1,2))].">
      <j1 rule="para" parents="211 11"/>
    </justification>
  </clause>

  <clause id="238">
    <literal><![CDATA[
      ((e * x) * y) * x = x * (y * x)
    ]]></literal>
    <justification jstring="[para(218(a,1),10(a,1,1)),flip(a)].">
      <j1 rule="para" parents="218 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="239">
    <literal><![CDATA[
      (x * (y * y)) * y = x * x
    ]]></literal>
    <justification jstring="[para(218(a,1),10(a,1,2)),rewrite(82(4),191(9),82(7))].">
      <j1 rule="para" parents="218 10"/>
      <j2 rule="rewrite" parents="82 191 82"/>
    </justification>
  </clause>

  <clause id="267">
    <literal><![CDATA[
      e * (x * y) = z * (x * (e * (z * y)))
    ]]></literal>
    <justification jstring="[para(30(a,1),12(a,1,2)),rewrite(36(2))].">
      <j1 rule="para" parents="30 12"/>
      <j2 rule="rewrite" parents="36"/>
    </justification>
  </clause>

  <clause id="271">
    <literal><![CDATA[
      (x * y) * e = x * (y * (e * x))
    ]]></literal>
    <justification jstring="[para(43(a,1),12(a,1,2))].">
      <j1 rule="para" parents="43 12"/>
    </justification>
  </clause>

  <clause id="274">
    <literal><![CDATA[
      (x * y) * (e * x) = x * (y * e)
    ]]></literal>
    <justification jstring="[para(58(a,1),12(a,1,2))].">
      <j1 rule="para" parents="58 12"/>
    </justification>
  </clause>

  <clause id="307">
    <literal><![CDATA[
      (x * y) * (x * x) = e * (y * e)
    ]]></literal>
    <justification jstring="[para(82(a,1),12(a,1,2))].">
      <j1 rule="para" parents="82 12"/>
    </justification>
  </clause>

  <clause id="313">
    <literal><![CDATA[
      (e * (x * y)) * (((y * x) * z) * u) = z * (y * u)
    ]]></literal>
    <justification jstring="[para(36(a,1),12(a,1,1))].">
      <j1 rule="para" parents="36 12"/>
    </justification>
  </clause>

  <clause id="343">
    <literal><![CDATA[
      e * (x * (y * e)) = y * (x * e)
    ]]></literal>
    <justification jstring="[para(112(a,1),12(a,1,2)),rewrite(274(4)),flip(a)].">
      <j1 rule="para" parents="112 12"/>
      <j2 rule="rewrite" parents="274"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="348">
    <literal><![CDATA[
      (e * x) * (y * e) = y * (x * y)
    ]]></literal>
    <justification jstring="[para(211(a,1),12(a,1,2))].">
      <j1 rule="para" parents="211 12"/>
    </justification>
  </clause>

  <clause id="369">
    <literal><![CDATA[
      (e * x) * (x * (y * (e * x))) = (x * y) * y
    ]]></literal>
    <justification jstring="[back_rewrite(235),rewrite(271(5))].">
      <j1 rule="back_rewrite" parents="235"/>
      <j2 rule="rewrite" parents="271"/>
    </justification>
  </clause>

  <clause id="370">
    <literal><![CDATA[
      (e * x) * ((x * y) * y) = x * (y * (e * x))
    ]]></literal>
    <justification jstring="[back_rewrite(226),rewrite(271(8))].">
      <j1 rule="back_rewrite" parents="226"/>
      <j2 rule="rewrite" parents="271"/>
    </justification>
  </clause>

  <clause id="394">
    <literal><![CDATA[
      x * (y * (x * y)) = y * x
    ]]></literal>
    <justification jstring="[back_rewrite(51),rewrite(271(8),3(10),313(8))].">
      <j1 rule="back_rewrite" parents="51"/>
      <j2 rule="rewrite" parents="271 3 313"/>
    </justification>
  </clause>

  <clause id="433">
    <literal><![CDATA[
      e * (x * e) = y * (x * (y * (e * y)))
    ]]></literal>
    <justification jstring="[para(222(a,1),12(a,1,2)),rewrite(36(4))].">
      <j1 rule="para" parents="222 12"/>
      <j2 rule="rewrite" parents="36"/>
    </justification>
  </clause>

  <clause id="489">
    <literal><![CDATA[
      ((x * (y * y)) * z) * ((x * x) * u) = y * (z * u)
    ]]></literal>
    <justification jstring="[para(239(a,1),12(a,1,2,1))].">
      <j1 rule="para" parents="239 12"/>
    </justification>
  </clause>

  <clause id="495">
    <literal><![CDATA[
      (x * (e * (y * e))) * (x * (e * (y * e))) = (x * x) * y
    ]]></literal>
    <justification jstring="[para(239(a,1),239(a,1,1)),rewrite(307(5),307(10)),flip(a)].">
      <j1 rule="para" parents="239 239"/>
      <j2 rule="rewrite" parents="307 307"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="497">
    <literal><![CDATA[
      x * (y * x) = y * (x * y)
    ]]></literal>
    <justification jstring="[para(394(a,1),3(a,1,2))].">
      <j1 rule="para" parents="394 3"/>
    </justification>
  </clause>

  <clause id="548">
    <literal><![CDATA[
      (e * x) * y = e * (x * (e * y))
    ]]></literal>
    <justification jstring="[para(7(a,1),13(a,1,1)),flip(a)].">
      <j1 rule="para" parents="7 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="554">
    <literal><![CDATA[
      ((x * y) * z) * u = x * (z * (e * ((y * x) * (e * u))))
    ]]></literal>
    <justification jstring="[para(30(a,1),13(a,1,1)),rewrite(548(4)),flip(a)].">
      <j1 rule="para" parents="30 13"/>
      <j2 rule="rewrite" parents="548"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="588">
    <literal><![CDATA[
      (x * y) * (z * (y * x)) = e * (z * x)
    ]]></literal>
    <justification jstring="[para(13(a,2),36(a,1))].">
      <j1 rule="para" parents="13 36"/>
    </justification>
  </clause>

  <clause id="594">
    <literal><![CDATA[
      e * (x * (e * (y * (x * (y * y))))) = y * y
    ]]></literal>
    <justification jstring="[para(13(a,2),75(a,1)),rewrite(548(6))].">
      <j1 rule="para" parents="13 75"/>
      <j2 rule="rewrite" parents="548"/>
    </justification>
  </clause>

  <clause id="627">
    <literal><![CDATA[
      e * ((x * (e * y)) * (e * z)) = (x * e) * (y * (x * z))
    ]]></literal>
    <justification jstring="[para(211(a,1),13(a,1,1)),rewrite(548(8),548(11)),flip(a)].">
      <j1 rule="para" parents="211 13"/>
      <j2 rule="rewrite" parents="548 548"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="632">
    <literal><![CDATA[
      (x * y) * y = x * (e * (y * x))
    ]]></literal>
    <justification jstring="[para(218(a,1),13(a,1,2)),rewrite(271(7),548(7),3(7))].">
      <j1 rule="para" parents="218 13"/>
      <j2 rule="rewrite" parents="271 548 3"/>
    </justification>
  </clause>

  <clause id="674">
    <literal><![CDATA[
      (x * x) * (y * (z * u)) = x * (y * (z * (e * (x * u))))
    ]]></literal>
    <justification jstring="[para(239(a,1),13(a,1,1)),rewrite(554(8),78(10),548(9),3(12))].">
      <j1 rule="para" parents="239 13"/>
      <j2 rule="rewrite" parents="554 78 548 3"/>
    </justification>
  </clause>

  <clause id="714">
    <literal><![CDATA[
      e * (x * (e * (x * (e * (y * x))))) = x * (y * (e * x))
    ]]></literal>
    <justification jstring="[back_rewrite(370),rewrite(632(4),548(7))].">
      <j1 rule="back_rewrite" parents="370"/>
      <j2 rule="rewrite" parents="632 548"/>
    </justification>
  </clause>

  <clause id="715">
    <literal><![CDATA[
      e * (x * (e * (x * (y * (e * x))))) = x * (e * (y * x))
    ]]></literal>
    <justification jstring="[back_rewrite(369),rewrite(548(7),632(11))].">
      <j1 rule="back_rewrite" parents="369"/>
      <j2 rule="rewrite" parents="548 632"/>
    </justification>
  </clause>

  <clause id="719">
    <literal><![CDATA[
      e * (x * (e * (y * e))) = y * (x * y)
    ]]></literal>
    <justification jstring="[back_rewrite(348),rewrite(548(5))].">
      <j1 rule="back_rewrite" parents="348"/>
      <j2 rule="rewrite" parents="548"/>
    </justification>
  </clause>

  <clause id="733">
    <literal><![CDATA[
      e * (x * (e * (y * e))) = x * (y * x)
    ]]></literal>
    <justification jstring="[back_rewrite(238),rewrite(548(3),548(6),274(7),36(5))].">
      <j1 rule="back_rewrite" parents="238"/>
      <j2 rule="rewrite" parents="548 548 274 36"/>
    </justification>
  </clause>

  <clause id="742">
    <literal><![CDATA[
      (x * e) * (y * (x * ((x * x) * z))) = (x * x) * (y * z)
    ]]></literal>
    <justification jstring="[back_rewrite(153),rewrite(548(3),548(8),627(10))].">
      <j1 rule="back_rewrite" parents="153"/>
      <j2 rule="rewrite" parents="548 548 627"/>
    </justification>
  </clause>

  <clause id="763">
    <literal><![CDATA[
      x * (y * (e * (z * (x * (e * ((z * z) * ((x * x) * u))))))) = z * (y * u)
    ]]></literal>
    <justification jstring="[back_rewrite(489),rewrite(554(6),554(8),3(9))].">
      <j1 rule="back_rewrite" parents="489"/>
      <j2 rule="rewrite" parents="554 554 3"/>
    </justification>
  </clause>

  <clause id="853">
    <literal><![CDATA[
      c1 * (c3 * (e * ((c2 * c1) * (e * c4)))) != c4 * (c3 * (e * ((c1 * c4) * (e * ((c2 * e) * c1)))))
    ]]></literal>
    <justification jstring="[back_rewrite(192),rewrite(554(11),554(24)),flip(a)].">
      <j1 rule="back_rewrite" parents="192"/>
      <j2 rule="rewrite" parents="554 554"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="890">
    <literal><![CDATA[
      (x * x) * y = x * (e * (x * y))
    ]]></literal>
    <justification jstring="[back_rewrite(141),rewrite(588(8))].">
      <j1 rule="back_rewrite" parents="141"/>
      <j2 rule="rewrite" parents="588"/>
    </justification>
  </clause>

  <clause id="923">
    <literal><![CDATA[
      x * (y * (e * (z * (x * (e * (z * (e * (z * (x * (e * (x * u))))))))))) = z * (y * u)
    ]]></literal>
    <justification jstring="[back_rewrite(763),rewrite(890(5),890(8))].">
      <j1 rule="back_rewrite" parents="763"/>
      <j2 rule="rewrite" parents="890 890"/>
    </justification>
  </clause>

  <clause id="925">
    <literal><![CDATA[
      (x * e) * (y * (e * (x * z))) = x * (e * (x * (y * z)))
    ]]></literal>
    <justification jstring="[back_rewrite(742),rewrite(890(4),3(7),890(10))].">
      <j1 rule="back_rewrite" parents="742"/>
      <j2 rule="rewrite" parents="890 3 890"/>
    </justification>
  </clause>

  <clause id="933">
    <literal><![CDATA[
      x * (e * (x * (y * (z * u)))) = x * (y * (z * (e * (x * u))))
    ]]></literal>
    <justification jstring="[back_rewrite(674),rewrite(890(4))].">
      <j1 rule="back_rewrite" parents="674"/>
      <j2 rule="rewrite" parents="890"/>
    </justification>
  </clause>

  <clause id="943">
    <literal><![CDATA[
      (x * (e * (y * e))) * (x * (e * (y * e))) = x * (e * (x * y))
    ]]></literal>
    <justification jstring="[back_rewrite(495),rewrite(890(13))].">
      <j1 rule="back_rewrite" parents="495"/>
      <j2 rule="rewrite" parents="890"/>
    </justification>
  </clause>

  <clause id="954">
    <literal><![CDATA[
      x * (e * (y * (x * (e * z)))) = y * z
    ]]></literal>
    <justification jstring="[back_rewrite(180),rewrite(890(7),3(7))].">
      <j1 rule="back_rewrite" parents="180"/>
      <j2 rule="rewrite" parents="890 3"/>
    </justification>
  </clause>

  <clause id="956">
    <literal><![CDATA[
      e * (x * (y * (e * (x * z)))) = y * z
    ]]></literal>
    <justification jstring="[back_rewrite(171),rewrite(890(3),3(6))].">
      <j1 rule="back_rewrite" parents="171"/>
      <j2 rule="rewrite" parents="890 3"/>
    </justification>
  </clause>

  <clause id="963">
    <literal><![CDATA[
      x * (y * (z * (x * u))) = z * (y * u)
    ]]></literal>
    <justification jstring="[back_rewrite(923),rewrite(956(14),954(8))].">
      <j1 rule="back_rewrite" parents="923"/>
      <j2 rule="rewrite" parents="956 954"/>
    </justification>
  </clause>

  <clause id="1034">
    <literal><![CDATA[
      e * (x * y) = x * (y * (e * (y * x)))
    ]]></literal>
    <justification jstring="[para(497(a,1),13(a,1)),rewrite(890(2),36(7)),flip(a)].">
      <j1 rule="para" parents="497 13"/>
      <j2 rule="rewrite" parents="890 36"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1054">
    <literal><![CDATA[
      (x * y) * z = e * (x * (e * (x * (e * (y * (e * (x * (e * (x * (e * z))))))))))
    ]]></literal>
    <justification jstring="[para(41(a,1),13(a,1,1)),rewrite(548(7),890(8),548(13),890(14)),flip(a)].">
      <j1 rule="para" parents="41 13"/>
      <j2 rule="rewrite" parents="548 890 548 890"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1141">
    <literal><![CDATA[
      e * (x * (e * (x * (e * (y * (e * (y * (e * (y * (e * (y * (e * (x * (e * (x * (e * (x * (e * (y * e))))))))))))))))))) = x * (e * (x * y))
    ]]></literal>
    <justification jstring="[back_rewrite(943),rewrite(1054(11),1054(21),3(29),3(27),3(25),1054(22),3(32),3(34),3(32),3(30))].">
      <j1 rule="back_rewrite" parents="943"/>
      <j2 rule="rewrite" parents="1054 1054 3 3 3 1054 3 3 3 3"/>
    </justification>
  </clause>

  <clause id="1151">
    <literal><![CDATA[
      e * (x * (e * (x * (e * (x * (e * (x * (e * (y * (e * (x * z))))))))))) = x * (e * (x * (y * z)))
    ]]></literal>
    <justification jstring="[back_rewrite(925),rewrite(1054(7),3(17))].">
      <j1 rule="back_rewrite" parents="925"/>
      <j2 rule="rewrite" parents="1054 3"/>
    </justification>
  </clause>

  <clause id="1171">
    <literal><![CDATA[
      e * (x * (e * (x * (e * (x * (e * (x * (e * (x * (e * y)))))))))) = x * (e * (x * y))
    ]]></literal>
    <justification jstring="[back_rewrite(890),rewrite(1054(2))].">
      <j1 rule="back_rewrite" parents="890"/>
      <j2 rule="rewrite" parents="1054"/>
    </justification>
  </clause>

  <clause id="1201">
    <literal><![CDATA[
      c4 * (c3 * (c1 * (e * (c1 * (e * (c4 * (e * (c1 * (e * (c1 * (e * (c2 * (e * (c2 * (e * (c2 * (e * (c2 * (e * c1))))))))))))))))))) != c1 * (c3 * (c2 * (e * (c2 * (e * (c1 * (e * (c2 * (e * (c2 * c4))))))))))
    ]]></literal>
    <justification jstring="[back_rewrite(853),rewrite(1054(10),3(18),3(25),1054(35),3(48),3(50),1054(47),3(66)),flip(a)].">
      <j1 rule="back_rewrite" parents="853"/>
      <j2 rule="rewrite" parents="1054 3 3 1054 3 3 1054 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1488">
    <literal><![CDATA[
      e * (x * (e * (x * (e * (y * (e * x)))))) = e * (y * e)
    ]]></literal>
    <justification jstring="[back_rewrite(307),rewrite(1054(3),41(9),3(9),3(6))].">
      <j1 rule="back_rewrite" parents="307"/>
      <j2 rule="rewrite" parents="1054 41 3 3"/>
    </justification>
  </clause>

  <clause id="1509">
    <literal><![CDATA[
      e * (x * (e * (x * (y * (e * (z * (e * (x * (e * (x * (e * (y * e)))))))))))) = z * y
    ]]></literal>
    <justification jstring="[back_rewrite(234),rewrite(1054(7),1054(23),3(28),3(25),3(24),3(21),3(20),1054(17),3(26),3(24),3(24),3(22),3(20))].">
      <j1 rule="back_rewrite" parents="234"/>
      <j2 rule="rewrite" parents="1054 1054 3 3 3 3 3 1054 3 3 3 3 3"/>
    </justification>
  </clause>

  <clause id="1563">
    <literal><![CDATA[
      e * x = x * (y * (x * (e * (y * e))))
    ]]></literal>
    <justification jstring="[para(267(a,1),394(a,1,2)),flip(a)].">
      <j1 rule="para" parents="267 394"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1571">
    <literal><![CDATA[
      x * (e * x) = y * (x * (e * (y * e)))
    ]]></literal>
    <justification jstring="[para(267(a,1),497(a,1)),flip(a)].">
      <j1 rule="para" parents="267 497"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1657">
    <literal><![CDATA[
      e * (x * (y * (x * (e * x)))) = y * e
    ]]></literal>
    <justification jstring="[para(433(a,1),3(a,1,2))].">
      <j1 rule="para" parents="433 3"/>
    </justification>
  </clause>

  <clause id="1660">
    <literal><![CDATA[
      e * (x * (e * (x * (y * (e * (y * (e * (y * (x * (e * (x * (e * y)))))))))))) = y * (e * (x * (e * (x * (y * (e * (y * (e * (y * (x * e))))))))))
    ]]></literal>
    <justification jstring="[para(433(a,2),394(a,1,2,2)),rewrite(1054(9),3(14),3(11),7(10),1054(11),3(16),1054(17),3(25),3(23),3(21),3(18),3(17),3(15),3(13),3(15),1054(21),1054(31),3(36),1054(37),3(45),3(43),3(41),3(38),3(37),3(35),3(33),3(35)),flip(a)].">
      <j1 rule="para" parents="433 394"/>
      <j2 rule="rewrite" parents="1054 3 3 7 1054 3 1054 3 3 3 3 3 3 3 3 1054 1054 3 1054 3 3 3 3 3 3 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1695">
    <literal><![CDATA[
      e * (x * (y * (e * (y * (e * (x * e)))))) = e * y
    ]]></literal>
    <justification jstring="[para(343(a,2),594(a,1,2,2,2,2,2)),rewrite(1054(9),7(14),394(15),3(13),3(11),41(10),3(10),3(7),1054(8),1488(17),3(11),3(13),1054(16),394(22),3(20),3(19),41(17),3(17),3(14))].">
      <j1 rule="para" parents="343 594"/>
      <j2 rule="rewrite" parents="1054 7 394 3 3 41 3 3 1054 1488 3 3 1054 394 3 3 41 3 3"/>
    </justification>
  </clause>

  <clause id="1742">
    <literal><![CDATA[
      x * (e * (y * (e * (x * e)))) = y * x
    ]]></literal>
    <justification jstring="[para(719(a,2),3(a,1,2))].">
      <j1 rule="para" parents="719 3"/>
    </justification>
  </clause>

  <clause id="1744">
    <literal><![CDATA[
      x * (e * (x * (e * (y * e)))) = y * x
    ]]></literal>
    <justification jstring="[para(719(a,2),394(a,1,2))].">
      <j1 rule="para" parents="719 394"/>
    </justification>
  </clause>

  <clause id="1910">
    <literal><![CDATA[
      e * (x * (e * (x * (e * (y * (e * (x * (e * (x * (e * (y * (e * (x * y))))))))))))) = x * (e * (x * (e * (y * y))))
    ]]></literal>
    <justification jstring="[para(719(a,2),714(a,1,2,2,2,2,2)),rewrite(3(13),1054(10),3(26),1054(23),1151(32),956(18),1742(12),3(6),3(11),1054(13)),flip(a)].">
      <j1 rule="para" parents="719 714"/>
      <j2 rule="rewrite" parents="3 1054 3 1054 1151 956 1742 3 3 1054"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2017">
    <literal><![CDATA[
      x * (e * (x * (e * (y * (e * x))))) = y * e
    ]]></literal>
    <justification jstring="[para(497(a,1),954(a,1,2,2)),rewrite(1054(4),3(13),3(11),3(11),3(9),1054(12),3(21),3(19),394(18),3(17),3(15))].">
      <j1 rule="para" parents="497 954"/>
      <j2 rule="rewrite" parents="1054 3 3 3 3 1054 3 3 394 3 3"/>
    </justification>
  </clause>

  <clause id="2058">
    <literal><![CDATA[
      x * (e * (y * (z * e))) = x * (z * (y * e))
    ]]></literal>
    <justification jstring="[para(343(a,1),956(a,1,2,2,2,2)),rewrite(3(10))].">
      <j1 rule="para" parents="343 956"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="2091">
    <literal><![CDATA[
      x * (y * (z * u)) = z * (y * (x * u))
    ]]></literal>
    <justification jstring="[para(963(a,1),3(a,1,2))].">
      <j1 rule="para" parents="963 3"/>
    </justification>
  </clause>

  <clause id="2092">
    <literal><![CDATA[
      x * (y * (z * (x * (y * u)))) = z * u
    ]]></literal>
    <justification jstring="[para(963(a,2),3(a,1,2))].">
      <j1 rule="para" parents="963 3"/>
    </justification>
  </clause>

  <clause id="2094">
    <literal><![CDATA[
      x * (y * (x * (z * (y * z)))) = z * x
    ]]></literal>
    <justification jstring="[para(963(a,2),394(a,1,2))].">
      <j1 rule="para" parents="963 394"/>
    </justification>
  </clause>

  <clause id="2095">
    <literal><![CDATA[
      x * (y * (z * (x * (z * y)))) = y * z
    ]]></literal>
    <justification jstring="[para(963(a,2),394(a,1))].">
      <j1 rule="para" parents="963 394"/>
    </justification>
  </clause>

  <clause id="2101">
    <literal><![CDATA[
      x * (y * (z * (x * z))) = y * (z * y)
    ]]></literal>
    <justification jstring="[para(963(a,2),497(a,1))].">
      <j1 rule="para" parents="963 497"/>
    </justification>
  </clause>

  <clause id="2103">
    <literal><![CDATA[
      x * (y * (x * (z * x))) = z * (y * z)
    ]]></literal>
    <justification jstring="[para(497(a,1),963(a,1,2,2))].">
      <j1 rule="para" parents="497 963"/>
    </justification>
  </clause>

  <clause id="2346">
    <literal><![CDATA[
      x * (y * (z * (u * (v * w)))) = v * (y * (x * (u * (z * w))))
    ]]></literal>
    <justification jstring="[para(963(a,1),963(a,1,2,2,2)),flip(a)].">
      <j1 rule="para" parents="963 963"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2350">
    <literal><![CDATA[
      x * (y * (z * (u * (y * (x * v))))) = z * (u * v)
    ]]></literal>
    <justification jstring="[para(963(a,2),963(a,1,2))].">
      <j1 rule="para" parents="963 963"/>
    </justification>
  </clause>

  <clause id="2376">
    <literal><![CDATA[
      e * (x * (e * (x * (e * (y * (e * (x * (e * (y * x))))))))) = x * (e * (x * (e * y)))
    ]]></literal>
    <justification jstring="[para(497(a,1),715(a,2,2,2)),rewrite(1054(9),1910(24),1054(12),3(17),3(14),3(13),3(10),3(9),3(6),3(10),1054(12),3(17),3(14)),flip(a)].">
      <j1 rule="para" parents="497 715"/>
      <j2 rule="rewrite" parents="1054 1910 1054 3 3 3 3 3 3 3 1054 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2389">
    <literal><![CDATA[
      e * (x * (e * (x * (y * (e * (y * (x * x))))))) = x * (e * (x * y))
    ]]></literal>
    <justification jstring="[para(733(a,1),715(a,2,2,2)),rewrite(3(21),1054(18),1054(28),3(36),3(34),3(32),1054(29),3(39),3(41),3(39),3(37),1141(38),1054(13),3(18),3(15),3(14),3(11),1054(11),3(19),3(17),3(15),1054(12),394(19),3(17),7(16),7(15),7(14),394(15),3(13),3(11),3(9),3(7),3(8),1054(14),3(19),3(16),1054(18),3(26),3(24),3(22),1054(19),3(29),1151(30),3(19),3(17),3(15)),flip(a)].">
      <j1 rule="para" parents="733 715"/>
      <j2 rule="rewrite" parents="3 1054 1054 3 3 3 1054 3 3 3 3 1141 1054 3 3 3 3 1054 3 3 3 1054 394 3 7 7 7 394 3 3 3 3 3 1054 3 3 1054 3 3 3 1054 3 1151 3 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2392">
    <literal><![CDATA[
      e * (x * (e * (x * (e * (x * (e * (x * (y * (e * (y * (e * x))))))))))) = x * (e * (x * (e * (x * (e * (x * (e * (y * (e * (y * (x * x)))))))))))
    ]]></literal>
    <justification jstring="[para(714(a,1),715(a,2,2,2)),rewrite(3(27),1054(24),1054(34),3(42),3(40),3(38),1054(35),1171(42),3(30),3(29),1054(26),3(35),3(33),1054(31),3(36),394(33),2376(40),3(33),3(31),3(29),3(30),3(28),3(26),3(28),1054(25),1171(34),3(19),1054(20),3(29),3(27),1054(25),3(30),3(27),3(26),1054(25),3(33),3(31),3(29),1054(26),2092(32),2092(28),41(24),3(24),3(25),3(23),3(21),3(23),3(21),3(19),3(21),1054(31),3(36),3(33),1054(35),3(43),3(41),3(39),1054(36),2017(42),1054(39),3(47),3(45),3(43),1054(40),1742(46),2376(48),3(41),3(39),3(37),3(38),3(36),3(34)),flip(a)].">
      <j1 rule="para" parents="714 715"/>
      <j2 rule="rewrite" parents="3 1054 1054 3 3 3 1054 1171 3 3 1054 3 3 1054 3 394 2376 3 3 3 3 3 3 3 1054 1171 3 1054 3 3 1054 3 3 3 1054 3 3 3 1054 2092 2092 41 3 3 3 3 3 3 3 3 1054 3 3 1054 3 3 3 1054 2017 1054 3 3 3 1054 1742 2376 3 3 3 3 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2488">
    <literal><![CDATA[
      x * (e * (x * (e * (x * (e * (x * (e * y))))))) = e * (x * (e * (x * (e * (x * y)))))
    ]]></literal>
    <justification jstring="[para(343(a,1),1034(a,2,2,2)),rewrite(1054(4),3(14),3(16),1054(19),2095(25),3(21),3(20))].">
      <j1 rule="para" parents="343 1034"/>
      <j2 rule="rewrite" parents="1054 3 3 1054 2095 3 3"/>
    </justification>
  </clause>

  <clause id="2538">
    <literal><![CDATA[
      e * (x * (e * (x * (e * (x * (e * (x * y))))))) = x * (e * (x * (e * (x * (e * y)))))
    ]]></literal>
    <justification jstring="[para(733(a,1),1034(a,2,2,2)),rewrite(1054(6),3(15),3(13),1054(11),3(16),3(17),3(19),3(17),3(15),3(19),1054(18),3(27),3(25),1054(23),3(28),3(25),3(27),3(29),3(27),3(25))].">
      <j1 rule="para" parents="733 1034"/>
      <j2 rule="rewrite" parents="1054 3 3 1054 3 3 3 3 3 3 1054 3 3 1054 3 3 3 3 3 3"/>
    </justification>
  </clause>

  <clause id="2618">
    <literal><![CDATA[
      x * (e * (x * (e * (x * (e * (y * (e * (y * (e * x))))))))) = x * y
    ]]></literal>
    <justification jstring="[back_rewrite(2392),rewrite(2538(18),2488(32),2389(27),3(21),3(20))].">
      <j1 rule="back_rewrite" parents="2392"/>
      <j2 rule="rewrite" parents="2538 2488 2389 3 3"/>
    </justification>
  </clause>

  <clause id="2844">
    <literal><![CDATA[
      c4 * (c3 * (c1 * (e * (c1 * (e * (c4 * (e * (c1 * (e * (c1 * (c2 * (e * (c2 * (e * (c2 * c1))))))))))))))) != c1 * (c3 * (c2 * (e * (c2 * (e * (c1 * (e * (c2 * (e * (c2 * c4))))))))))
    ]]></literal>
    <justification jstring="[back_rewrite(1201),rewrite(2488(29),3(26))].">
      <j1 rule="back_rewrite" parents="1201"/>
      <j2 rule="rewrite" parents="2488 3"/>
    </justification>
  </clause>

  <clause id="2933">
    <literal><![CDATA[
      e * (x * (e * (x * (y * (e * (y * (x * (e * (x * (e * (x * (y * (e * (y * (x * (e * (x * (e * y)))))))))))))))))) = y * (e * y)
    ]]></literal>
    <justification jstring="[para(1563(a,2),394(a,1,2,2)),rewrite(1054(9),3(14),1054(15),3(20),2095(18),3(13),1054(13),3(21),3(19),3(17),1054(14),394(21),3(19),7(18),7(17),7(16),394(17),3(15),3(13),3(11),3(9),3(10),2095(8),3(3),1054(10),1054(20),3(25),1054(26),3(34),3(32),3(30),1054(27),3(37),2488(37),3(35),3(33),3(31),3(29),3(30)),flip(a)].">
      <j1 rule="para" parents="1563 394"/>
      <j2 rule="rewrite" parents="1054 3 1054 3 2095 3 1054 3 3 3 1054 394 3 7 7 7 394 3 3 3 3 3 2095 3 1054 1054 3 1054 3 3 3 1054 3 2488 3 3 3 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2952">
    <literal><![CDATA[
      e * (x * (e * (x * (y * (e * (y * (e * (x * (e * (x * y)))))))))) = e * (y * y)
    ]]></literal>
    <justification jstring="[para(1563(a,2),714(a,1,2,2,2,2,2)),rewrite(3(18),1054(15),1054(25),3(30),1054(31),3(39),3(37),3(35),1054(32),3(42),2488(42),3(40),3(38),3(36),3(34),3(35),2933(35),394(12),1054(10),1054(20),3(25),1695(23),1054(17),3(25),3(23),3(21),1054(18),2017(26),3(21),3(23),3(21),3(19),3(21),1509(21),2092(17),3(12),1054(11),7(16),394(17),3(15),1054(12),1054(22),3(30),3(28),3(26),1054(23),3(33),2618(31),3(21),3(19),3(17),3(19)),flip(a)].">
      <j1 rule="para" parents="1563 714"/>
      <j2 rule="rewrite" parents="3 1054 1054 3 1054 3 3 3 1054 3 2488 3 3 3 3 3 2933 394 1054 1054 3 1695 1054 3 3 3 1054 2017 3 3 3 3 3 1509 2092 3 1054 7 394 3 1054 1054 3 3 3 1054 3 2618 3 3 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2975">
    <literal><![CDATA[
      x * (e * (y * (e * (y * (x * (e * (x * (e * (x * (y * e)))))))))) = e * (y * (e * (y * (e * (x * (e * (x * y)))))))
    ]]></literal>
    <justification jstring="[para(1571(a,2),394(a,1,2,2)),rewrite(1054(9),394(14),3(12),7(11),1054(12),3(20),3(18),3(16),1054(13),3(23),2488(23),3(21),3(19),3(17),3(15),1054(22),1054(32),3(40),3(38),3(36),1054(33),3(43),2618(41),3(31),3(29),3(27))].">
      <j1 rule="para" parents="1571 394"/>
      <j2 rule="rewrite" parents="1054 394 3 7 1054 3 3 3 1054 3 2488 3 3 3 3 1054 1054 3 3 3 1054 3 2618 3 3 3"/>
    </justification>
  </clause>

  <clause id="3000">
    <literal><![CDATA[
      e * (x * (e * (x * (y * (e * (y * (e * (y * (x * (e * (x * (e * y)))))))))))) = e * (x * (e * (x * (e * (y * (e * (y * x)))))))
    ]]></literal>
    <justification jstring="[back_rewrite(1660),rewrite(2975(35))].">
      <j1 rule="back_rewrite" parents="1660"/>
      <j2 rule="rewrite" parents="2975"/>
    </justification>
  </clause>

  <clause id="3225">
    <literal><![CDATA[
      x * (e * (x * (y * (e * (y * (e * (y * x))))))) = e * (y * y)
    ]]></literal>
    <justification jstring="[para(1657(a,1),394(a,1,2,2)),rewrite(1054(9),1054(19),3(24),1695(22),1054(16),3(21),715(21),1054(17),3(26),3(24),3(24),3(22),3(20),3(17),3(16),3(13),3(12),3(14),3(15),1054(18),7(23),394(24),3(22),1054(19),1054(29),3(34),1054(35),3(43),3(41),3(39),3(36),3(35),3(33),3(31),3(33),3000(33),3(27),2952(27))].">
      <j1 rule="para" parents="1657 394"/>
      <j2 rule="rewrite" parents="1054 1054 3 1695 1054 3 715 1054 3 3 3 3 3 3 3 3 3 3 3 1054 7 394 3 1054 1054 3 1054 3 3 3 3 3 3 3 3 3000 3 2952"/>
    </justification>
  </clause>

  <clause id="3242">
    <literal><![CDATA[
      c4 * (c3 * (c1 * (e * (c1 * (e * (c4 * (c2 * c2))))))) != c1 * (c3 * (c2 * (e * (c2 * (e * (c1 * (e * (c2 * (e * (c2 * c4))))))))))
    ]]></literal>
    <justification jstring="[back_rewrite(2844),rewrite(3225(25),3(14))].">
      <j1 rule="back_rewrite" parents="2844"/>
      <j2 rule="rewrite" parents="3225 3"/>
    </justification>
  </clause>

  <clause id="4062">
    <literal><![CDATA[
      c1 * (c3 * (e * (c2 * (c4 * (c2 * (c4 * (c2 * (c1 * e)))))))) != c4 * (c3 * (c1 * (e * (c1 * (e * (c4 * (c2 * c2)))))))
    ]]></literal>
    <justification jstring="[para(1744(a,2),3242(a,2,2,2,2,2,2,2)),rewrite(1054(48),3(56),3(54),3(52),1054(49),1054(62),3(70),3(68),3(66),1054(63),2488(74),3(71),2092(70),2092(64),3(58),3(56),3(54),3(56),3(54),3(52),1054(50),3(58),3(56),3(54),1054(51),2488(62),3(59),1054(56),3(65),3(63),1054(61),3(69),3(66),3(63),3(60),3(57),3(60),3(58),3(56),2488(56),3(53),3(51),3(49),3(47),3(44),3(41),3(38)),flip(a)].">
      <j1 rule="para" parents="1744 3242"/>
      <j2 rule="rewrite" parents="1054 3 3 3 1054 1054 3 3 3 1054 2488 3 2092 2092 3 3 3 3 3 3 1054 3 3 3 1054 2488 3 1054 3 3 1054 3 3 3 3 3 3 3 3 2488 3 3 3 3 3 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4217">
    <literal><![CDATA[
      x * (e * (x * (y * (z * (u * (e * (x * v))))))) = x * (y * (z * (u * v)))
    ]]></literal>
    <justification jstring="[para(267(a,2),933(a,2,2,2,2,2)),rewrite(3(15))].">
      <j1 rule="para" parents="267 933"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="4835">
    <literal><![CDATA[
      x * (y * (z * (y * x))) = z * (x * y)
    ]]></literal>
    <justification jstring="[para(394(a,1),2091(a,1,2)),flip(a)].">
      <j1 rule="para" parents="394 2091"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="5030">
    <literal><![CDATA[
      c1 * (c3 * (e * (c2 * (c4 * (c2 * (c4 * (c2 * (c1 * e)))))))) != c4 * (c3 * (c1 * (e * (c1 * (c2 * (c4 * (e * c2)))))))
    ]]></literal>
    <justification jstring="[para(2091(a,1),4062(a,2,2,2,2,2,2))].">
      <j1 rule="para" parents="2091 4062"/>
    </justification>
  </clause>

  <clause id="5282">
    <literal><![CDATA[
      x * (y * (z * (e * (z * (x * (y * u)))))) = z * (e * (z * u))
    ]]></literal>
    <justification jstring="[para(933(a,2),2092(a,1,2,2))].">
      <j1 rule="para" parents="933 2092"/>
    </justification>
  </clause>

  <clause id="6472">
    <literal><![CDATA[
      e * (x * (e * (x * (e * (x * (e * (y * (e * (y * (e * (y * (e * (x * (e * (x * (e * z)))))))))))))))) = e * (x * (e * (x * (y * (e * (y * (x * (y * (e * (y * (x * (e * (x * (e * z))))))))))))))
    ]]></literal>
    <justification jstring="[para(714(a,1),1054(a,1,1)),rewrite(1054(5),1054(15),3(20),1054(21),3(29),3(27),3(25),3(24),3(22),3(20),3(21),3(40),3(38),1054(36),3(41),1054(42),3(50),3(48),3(46),1054(43),2538(52),1054(50),3(59),3(57),1054(55),3(60),5282(61),2538(60),3(59),3(57),3(55),3(56),3(54),3(52),2538(53),3(52),3(50)),flip(a)].">
      <j1 rule="para" parents="714 1054"/>
      <j2 rule="rewrite" parents="1054 1054 3 1054 3 3 3 3 3 3 3 3 3 1054 3 1054 3 3 3 1054 2538 1054 3 3 1054 3 5282 2538 3 3 3 3 3 3 2538 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="7036">
    <literal><![CDATA[
      x * (e * (x * (e * (x * (e * (y * (e * (y * (e * (y * (e * (x * (e * z))))))))))))) = e * (x * (e * (x * (y * (e * (y * (e * (y * (x * z)))))))))
    ]]></literal>
    <justification jstring="[para(2103(a,1),4835(a,1,2,2)),rewrite(394(5),1054(4),1054(14),3(19),5282(17),2092(14),3(14),1054(17),1054(27),3(32),5282(33),2538(32)),flip(a)].">
      <j1 rule="para" parents="2103 4835"/>
      <j2 rule="rewrite" parents="394 1054 1054 3 5282 2092 3 1054 1054 3 5282 2538"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="7089">
    <literal><![CDATA[
      e * (x * (e * (x * (y * (e * (y * (x * (y * (e * (y * (x * (e * (x * (e * z)))))))))))))) = x * (e * (x * (y * (e * (y * (e * (y * (e * z))))))))
    ]]></literal>
    <justification jstring="[back_rewrite(6472),rewrite(7036(25),3(9),3(17)),flip(a)].">
      <j1 rule="back_rewrite" parents="6472"/>
      <j2 rule="rewrite" parents="7036 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10026">
    <literal><![CDATA[
      x * (e * (y * (e * (z * (y * z))))) = x * (z * e)
    ]]></literal>
    <justification jstring="[para(2101(a,2),2058(a,1,2,2)),rewrite(7(11))].">
      <j1 rule="para" parents="2101 2058"/>
      <j2 rule="rewrite" parents="7"/>
    </justification>
  </clause>

  <clause id="11355">
    <literal><![CDATA[
      c4 * (c3 * (c1 * (c2 * (e * (c2 * (c4 * (c2 * (c1 * e)))))))) != c1 * (c3 * (e * (c2 * (c4 * (c2 * (c4 * (c2 * (c1 * e))))))))
    ]]></literal>
    <justification jstring="[para(2094(a,2),5030(a,2,2,2,2,2)),rewrite(1054(42),1054(55),3(63),1054(64),3(72),3(70),3(68),3(67),3(65),3(63),3(64),7089(64),1054(54),2092(63),1054(59),3(67),2350(64),3(57),2092(55),1054(48),3(57),3(55),3(55),3(53),3(51),4217(50),10026(41),3(38),3(39)),flip(a)].">
      <j1 rule="para" parents="2094 5030"/>
      <j2 rule="rewrite" parents="1054 1054 3 1054 3 3 3 3 3 3 3 7089 1054 2092 1054 3 2350 3 2092 1054 3 3 3 3 3 4217 10026 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11356">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(11355,a,2346,a(flip))].">
      <j1 rule="resolve" parents="11355 2346"/>
    </justification>
  </clause>

</proof>

</proofs>
