<?xml version="1.0" encoding="ISO-8859-1"?>

<!DOCTYPE proofs SYSTEM "proof3.dtd">

<?xml-stylesheet type="text/xsl" href="proof3.xsl"?>

<proofs number_of_proofs="1">

<source>EA-cs.out</source>

<heading><![CDATA[
Prover9 (32) version 22-May-2007, May 2007.
Process 27446 was started by mccune on cleo,
Tue May 22 14:45:51 2007
The command was "/home/mccune/bin/prover9 -f cs.in EA.in".
]]></heading>

<proof number="1" length="109" max_count="23">

<comments><![CDATA[
% Proof 1 at 8.40 (+ 0.08) seconds: A.
% Length of proof is 109.
% Level of proof is 32.
% Maximum clause weight is 23.
% Given clauses 1107.
]]></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 = y * x * z * x * y
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6" 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="7">
    <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="8">
    <literal><![CDATA[
      x * y * z != u * z
    ]]></literal>
    <literal><![CDATA[
      x * y = u
    ]]></literal>
    <justification jstring="[para(3(a,1),2(a,1))].">
      <j1 rule="para" parents="3 2"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      x * (x @ x) = x
    ]]></literal>
    <justification jstring="[hyper(1,a,4,a)].">
      <j1 rule="hyper" parents="1 4"/>
    </justification>
  </clause>

  <clause id="10">
    <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="13">
    <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="14">
    <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="15">
    <literal><![CDATA[
      x * y * z * (y * z @ x) = y * z * x
    ]]></literal>
    <justification jstring="[para(3(a,1),4(a,1,2)),rewrite([3(7)])].">
      <j1 rule="para" parents="3 4"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

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

  <clause id="23">
    <literal><![CDATA[
      ((A @ B) @ C) * x != (D @ (F @ G)) * x
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[ur(2,b,6,a)].">
      <j1 rule="ur" parents="2 6"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      x * y * z != x * u * y
    ]]></literal>
    <literal><![CDATA[
      u * (u @ y) = z
    ]]></literal>
    <justification jstring="[para(4(a,1),7(a,1,2)),flip(a)].">
      <j1 rule="para" parents="4 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      x * y * z * y * x != y * x * u
    ]]></literal>
    <literal><![CDATA[
      z * x * y = u
    ]]></literal>
    <justification jstring="[para(5(a,1),7(a,1))].">
      <j1 rule="para" parents="5 7"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      x * ((A @ B) @ C) * y != x * (D @ (F @ G)) * y
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[ur(1,b,23,a)].">
      <j1 rule="ur" parents="1 23"/>
    </justification>
  </clause>

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

  <clause id="43">
    <literal><![CDATA[
      x * (x @ x) * y = x * y
    ]]></literal>
    <justification jstring="[para(9(a,1),3(a,1,1)),flip(a)].">
      <j1 rule="para" parents="9 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      x * y * z != u * v * z
    ]]></literal>
    <literal><![CDATA[
      x * y = u * v
    ]]></literal>
    <justification jstring="[para(3(a,1),8(a,2))].">
      <j1 rule="para" parents="3 8"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      x * y * (y @ z) != u * y * z
    ]]></literal>
    <literal><![CDATA[
      u * z = x
    ]]></literal>
    <justification jstring="[para(4(a,1),8(a,1,2)),flip(a)].">
      <j1 rule="para" parents="4 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="56">
    <literal><![CDATA[
      x * (y @ z) != y * z
    ]]></literal>
    <literal><![CDATA[
      z * y = x
    ]]></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="69">
    <literal><![CDATA[
      (x @ x) * y = y
    ]]></literal>
    <justification jstring="[hyper(7,a,43,a)].">
      <j1 rule="hyper" parents="7 43"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      x * y != y
    ]]></literal>
    <literal><![CDATA[
      z @ z = x
    ]]></literal>
    <justification jstring="[para(69(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="69 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="78">
    <literal><![CDATA[
      x * ((y @ y) @ x) = x
    ]]></literal>
    <justification jstring="[para(69(a,1),4(a,1,2)),rewrite([69(5)])].">
      <j1 rule="para" parents="69 4"/>
      <j2 rule="rewrite" parents="69"/>
    </justification>
  </clause>

  <clause id="84">
    <literal><![CDATA[
      (x @ x) @ y = y @ y
    ]]></literal>
    <justification jstring="[hyper(40,a,78,a),flip(a)].">
      <j1 rule="hyper" parents="40 78"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      x * y * z != x * y
    ]]></literal>
    <literal><![CDATA[
      (u @ u) @ y = z
    ]]></literal>
    <justification jstring="[para(78(a,1),7(a,1,2)),flip(a)].">
      <j1 rule="para" parents="78 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="100">
    <literal><![CDATA[
      ((x @ x) @ y) * z = z
    ]]></literal>
    <justification jstring="[para(84(a,2),69(a,1,1))].">
      <j1 rule="para" parents="84 69"/>
    </justification>
  </clause>

  <clause id="101">
    <literal><![CDATA[
      x * (((y @ y) @ z) @ x) = x
    ]]></literal>
    <justification jstring="[para(84(a,2),78(a,1,2,1))].">
      <j1 rule="para" parents="84 78"/>
    </justification>
  </clause>

  <clause id="105">
    <literal><![CDATA[
      x * y * (x * y @ x) = y * x
    ]]></literal>
    <justification jstring="[hyper(10,a,3,a(flip)),rewrite([3(4)])].">
      <j1 rule="hyper" parents="10 3"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="106">
    <literal><![CDATA[
      x * y * z != u * x * y
    ]]></literal>
    <literal><![CDATA[
      u * (u @ x * y) = z
    ]]></literal>
    <justification jstring="[para(3(a,1),10(a,1))].">
      <j1 rule="para" parents="3 10"/>
    </justification>
  </clause>

  <clause id="121">
    <literal><![CDATA[
      x * (x @ ((y @ y) @ z)) = x * ((y @ y) @ z)
    ]]></literal>
    <justification jstring="[hyper(10,a,100,a)].">
      <j1 rule="hyper" parents="10 100"/>
    </justification>
  </clause>

  <clause id="128">
    <literal><![CDATA[
      (((x @ x) @ y) @ z) * u = u
    ]]></literal>
    <justification jstring="[para(84(a,2),100(a,1,1,1))].">
      <j1 rule="para" parents="84 100"/>
    </justification>
  </clause>

  <clause id="129">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      ((z @ z) @ u) @ x = y
    ]]></literal>
    <justification jstring="[para(100(a,1),10(a,2)),rewrite([100(8)])].">
      <j1 rule="para" parents="100 10"/>
      <j2 rule="rewrite" parents="100"/>
    </justification>
  </clause>

  <clause id="141">
    <literal><![CDATA[
      (x @ x) @ y = z @ z
    ]]></literal>
    <justification jstring="[hyper(77,a,100,a),flip(a)].">
      <j1 rule="hyper" parents="77 100"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="142">
    <literal><![CDATA[
      x @ x = y @ y
    ]]></literal>
    <justification jstring="[hyper(77,a,69,a)].">
      <j1 rule="hyper" parents="77 69"/>
    </justification>
  </clause>

  <clause id="145">
    <literal><![CDATA[
      x @ x = c_0
    ]]></literal>
    <justification jstring="[new_symbol(142)].">
      <j1 rule="new_symbol" parents="142"/>
    </justification>
  </clause>

  <clause id="148">
    <literal><![CDATA[
      c_0 @ x = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(141),rewrite([145(1),145(3)])].">
      <j1 rule="back_rewrite" parents="141"/>
      <j2 rule="rewrite" parents="145 145"/>
    </justification>
  </clause>

  <clause id="154">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      c_0 = y
    ]]></literal>
    <justification jstring="[back_rewrite(129),rewrite([145(3),148(4),148(4)])].">
      <j1 rule="back_rewrite" parents="129"/>
      <j2 rule="rewrite" parents="145 148 148"/>
    </justification>
  </clause>

  <clause id="155">
    <literal><![CDATA[
      c_0 * x = x
    ]]></literal>
    <justification jstring="[back_rewrite(128),rewrite([145(1),148(2),148(2)])].">
      <j1 rule="back_rewrite" parents="128"/>
      <j2 rule="rewrite" parents="145 148 148"/>
    </justification>
  </clause>

  <clause id="160">
    <literal><![CDATA[
      x * (x @ c_0) = x * c_0
    ]]></literal>
    <justification jstring="[back_rewrite(121),rewrite([145(1),148(2),145(4),148(5)])].">
      <j1 rule="back_rewrite" parents="121"/>
      <j2 rule="rewrite" parents="145 148 145 148"/>
    </justification>
  </clause>

  <clause id="165">
    <literal><![CDATA[
      x * c_0 = x
    ]]></literal>
    <justification jstring="[back_rewrite(101),rewrite([145(1),148(2),148(2)])].">
      <j1 rule="back_rewrite" parents="101"/>
      <j2 rule="rewrite" parents="145 148 148"/>
    </justification>
  </clause>

  <clause id="166">
    <literal><![CDATA[
      x * y * z != x * y
    ]]></literal>
    <literal><![CDATA[
      c_0 = z
    ]]></literal>
    <justification jstring="[back_rewrite(91),rewrite([145(5),148(6)])].">
      <j1 rule="back_rewrite" parents="91"/>
      <j2 rule="rewrite" parents="145 148"/>
    </justification>
  </clause>

  <clause id="168">
    <literal><![CDATA[
      x * (x @ c_0) = x
    ]]></literal>
    <justification jstring="[back_rewrite(160),rewrite([165(5)])].">
      <j1 rule="back_rewrite" parents="160"/>
      <j2 rule="rewrite" parents="165"/>
    </justification>
  </clause>

  <clause id="191">
    <literal><![CDATA[
      x @ c_0 = c_0
    ]]></literal>
    <justification jstring="[hyper(154,a,168,a),flip(a)].">
      <j1 rule="hyper" parents="154 168"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="192">
    <literal><![CDATA[
      x * y * z != y * u
    ]]></literal>
    <literal><![CDATA[
      x * (x @ y) * z = u
    ]]></literal>
    <justification jstring="[para(13(a,1),1(a,1))].">
      <j1 rule="para" parents="13 1"/>
    </justification>
  </clause>

  <clause id="202">
    <literal><![CDATA[
      x * y * z != y * x * u
    ]]></literal>
    <literal><![CDATA[
      (y @ x) * u = z
    ]]></literal>
    <justification jstring="[para(13(a,1),7(a,2)),flip(b)].">
      <j1 rule="para" parents="13 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="216">
    <literal><![CDATA[
      x * y * (y @ y * x) = y * x
    ]]></literal>
    <justification jstring="[hyper(1,a,14,a)].">
      <j1 rule="hyper" parents="1 14"/>
    </justification>
  </clause>

  <clause id="246">
    <literal><![CDATA[
      x * y != y * x
    ]]></literal>
    <literal><![CDATA[
      x @ y = c_0
    ]]></literal>
    <justification jstring="[para(4(a,1),166(a,1)),flip(b)].">
      <j1 rule="para" parents="4 166"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="278">
    <literal><![CDATA[
      x * (y * x @ y) = x * (x @ y)
    ]]></literal>
    <justification jstring="[hyper(10,a,105,a),flip(a)].">
      <j1 rule="hyper" parents="10 105"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="294">
    <literal><![CDATA[
      x * (y @ x) * x * y = x * y * x
    ]]></literal>
    <justification jstring="[hyper(16,a,13,a)].">
      <j1 rule="hyper" parents="16 13"/>
    </justification>
  </clause>

  <clause id="301">
    <literal><![CDATA[
      x * (x @ x * y) = x * (x @ y)
    ]]></literal>
    <justification jstring="[hyper(10,a,216,a),flip(a)].">
      <j1 rule="hyper" parents="10 216"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="315">
    <literal><![CDATA[
      x * y @ x = y @ x
    ]]></literal>
    <justification jstring="[hyper(1,a,278,a)].">
      <j1 rule="hyper" parents="1 278"/>
    </justification>
  </clause>

  <clause id="332">
    <literal><![CDATA[
      x * (x @ y) @ y = x * y @ y
    ]]></literal>
    <justification jstring="[para(4(a,1),315(a,1,1)),flip(a)].">
      <j1 rule="para" parents="4 315"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="341">
    <literal><![CDATA[
      x @ x * y = x @ y
    ]]></literal>
    <justification jstring="[hyper(1,a,301,a)].">
      <j1 rule="hyper" parents="1 301"/>
    </justification>
  </clause>

  <clause id="353">
    <literal><![CDATA[
      x @ y * (y @ x) = x @ y * x
    ]]></literal>
    <justification jstring="[para(4(a,1),341(a,1,2)),flip(a)].">
      <j1 rule="para" parents="4 341"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="370">
    <literal><![CDATA[
      x * (x @ y) @ x * y = x * x * y @ x * y
    ]]></literal>
    <justification jstring="[para(341(a,1),332(a,1,1,2))].">
      <j1 rule="para" parents="341 332"/>
    </justification>
  </clause>

  <clause id="373">
    <literal><![CDATA[
      x * y @ x * (x @ y) = x * y @ x * x * y
    ]]></literal>
    <justification jstring="[para(341(a,1),353(a,1,2,2))].">
      <j1 rule="para" parents="341 353"/>
    </justification>
  </clause>

  <clause id="440">
    <literal><![CDATA[
      (x @ y) * y * x = x * y
    ]]></literal>
    <justification jstring="[hyper(1,a,294,a)].">
      <j1 rule="hyper" parents="1 294"/>
    </justification>
  </clause>

  <clause id="465">
    <literal><![CDATA[
      x * y * z != z * y
    ]]></literal>
    <literal><![CDATA[
      z @ y = x
    ]]></literal>
    <justification jstring="[para(440(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="440 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="508">
    <literal><![CDATA[
      x * y @ (y @ x) = c_0
    ]]></literal>
    <justification jstring="[para(440(a,1),246(a,2)),rewrite([3(3),4(3)]),xx(a)].">
      <j1 rule="para" parents="440 246"/>
      <j2 rule="rewrite" parents="3 4"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="510">
    <literal><![CDATA[
      x * y @ (x @ y) = c_0
    ]]></literal>
    <justification jstring="[para(440(a,1),315(a,1,1)),rewrite([508(6)])].">
      <j1 rule="para" parents="440 315"/>
      <j2 rule="rewrite" parents="508"/>
    </justification>
  </clause>

  <clause id="565">
    <literal><![CDATA[
      x * y @ (y @ x * y) = c_0
    ]]></literal>
    <justification jstring="[para(4(a,1),510(a,1,1)),rewrite([353(4)])].">
      <j1 rule="para" parents="4 510"/>
      <j2 rule="rewrite" parents="353"/>
    </justification>
  </clause>

  <clause id="716">
    <literal><![CDATA[
      (x @ y * x) * y * x = y * x * (x @ y * x)
    ]]></literal>
    <justification jstring="[para(565(a,1),4(a,1,2,2)),rewrite([165(5),3(8)])].">
      <j1 rule="para" parents="565 4"/>
      <j2 rule="rewrite" parents="165 3"/>
    </justification>
  </clause>

  <clause id="770">
    <literal><![CDATA[
      x * y != x * z
    ]]></literal>
    <literal><![CDATA[
      z = y
    ]]></literal>
    <justification jstring="[para(155(a,1),27(a,1,2)),rewrite([165(3),191(5),165(5)])].">
      <j1 rule="para" parents="155 27"/>
      <j2 rule="rewrite" parents="165 191 165"/>
    </justification>
  </clause>

  <clause id="811">
    <literal><![CDATA[
      (x @ y) * z * y * x = z * x * y
    ]]></literal>
    <justification jstring="[hyper(29,a,13,a(flip)),flip(a)].">
      <j1 rule="hyper" parents="29 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="930">
    <literal><![CDATA[
      x * y * (y @ x * y) = y * x
    ]]></literal>
    <justification jstring="[hyper(54,a,440,a),rewrite([716(4)])].">
      <j1 rule="hyper" parents="54 440"/>
      <j2 rule="rewrite" parents="716"/>
    </justification>
  </clause>

  <clause id="986">
    <literal><![CDATA[
      x * y != z * y
    ]]></literal>
    <literal><![CDATA[
      z = x
    ]]></literal>
    <justification jstring="[para(165(a,1),55(a,2,2)),rewrite([191(2),165(2),165(5)])].">
      <j1 rule="para" parents="165 55"/>
      <j2 rule="rewrite" parents="191 165 165"/>
    </justification>
  </clause>

  <clause id="1161">
    <literal><![CDATA[
      x * (x @ y * x) = (x @ y) * x
    ]]></literal>
    <justification jstring="[hyper(106,a,13,a)].">
      <j1 rule="hyper" parents="106 13"/>
    </justification>
  </clause>

  <clause id="1168">
    <literal><![CDATA[
      x * (y @ x) * y = y * x
    ]]></literal>
    <justification jstring="[back_rewrite(930),rewrite([1161(3)])].">
      <j1 rule="back_rewrite" parents="930"/>
      <j2 rule="rewrite" parents="1161"/>
    </justification>
  </clause>

  <clause id="1184">
    <literal><![CDATA[
      (x @ y) * x = x * (x @ y)
    ]]></literal>
    <justification jstring="[hyper(10,a,1168,a),flip(a)].">
      <j1 rule="hyper" parents="10 1168"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1205">
    <literal><![CDATA[
      x * (x @ y) @ y * (y @ x) = y * x @ y * y * x
    ]]></literal>
    <justification jstring="[para(1168(a,1),332(a,2,1)),rewrite([1184(2),353(3),1161(3),1184(2),1184(4),1184(8),373(9)])].">
      <j1 rule="para" parents="1168 332"/>
      <j2 rule="rewrite" parents="1184 353 1161 1184 1184 1184 373"/>
    </justification>
  </clause>

  <clause id="1207">
    <literal><![CDATA[
      x * x * y @ x * y = y * x @ y * y * x
    ]]></literal>
    <justification jstring="[para(1168(a,1),353(a,2,2)),rewrite([1184(2),1184(4),353(5),1161(5),1184(4),1205(5),1184(6),370(8)]),flip(a)].">
      <j1 rule="para" parents="1168 353"/>
      <j2 rule="rewrite" parents="1184 1184 353 1161 1184 1205 1184 370"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1234">
    <literal><![CDATA[
      x * (x @ y * x) = x * (x @ y)
    ]]></literal>
    <justification jstring="[back_rewrite(1161),rewrite([1184(5)])].">
      <j1 rule="back_rewrite" parents="1161"/>
      <j2 rule="rewrite" parents="1184"/>
    </justification>
  </clause>

  <clause id="1243">
    <literal><![CDATA[
      (x @ y) * x * z = x * (x @ y) * z
    ]]></literal>
    <justification jstring="[para(1184(a,1),3(a,1,1)),rewrite([3(3)]),flip(a)].">
      <j1 rule="para" parents="1184 3"/>
      <j2 rule="rewrite" parents="3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1743">
    <literal><![CDATA[
      x * (x @ y) * (y @ x) = x
    ]]></literal>
    <justification jstring="[hyper(192,a,4,a)].">
      <j1 rule="hyper" parents="192 4"/>
    </justification>
  </clause>

  <clause id="1766">
    <literal><![CDATA[
      (x @ y) * (y @ x) = c_0
    ]]></literal>
    <justification jstring="[hyper(154,a,1743,a),flip(a)].">
      <j1 rule="hyper" parents="154 1743"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1767">
    <literal><![CDATA[
      (x @ y) * (y @ x) * x = (y @ x) * x * (x @ y)
    ]]></literal>
    <justification jstring="[para(1743(a,1),5(a,1,2,2)),rewrite([1184(3),1766(9),165(8)]),flip(a)].">
      <j1 rule="para" parents="1743 5"/>
      <j2 rule="rewrite" parents="1184 1766 165"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1811">
    <literal><![CDATA[
      (x @ y) * (y @ x) * z = z
    ]]></literal>
    <justification jstring="[para(1766(a,1),3(a,1,1)),rewrite([155(2)]),flip(a)].">
      <j1 rule="para" parents="1766 3"/>
      <j2 rule="rewrite" parents="155"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1821">
    <literal><![CDATA[
      x * (D @ (F @ G)) * (C @ (A @ B)) != x
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(1766(a,1),31(a,1,2)),rewrite([165(2)]),flip(a)].">
      <j1 rule="para" parents="1766 31"/>
      <j2 rule="rewrite" parents="165"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1847">
    <literal><![CDATA[
      (x @ y) * z != u
    ]]></literal>
    <literal><![CDATA[
      (y @ x) * u = z
    ]]></literal>
    <justification jstring="[para(1766(a,1),16(a,1,2,2,2)),rewrite([165(4),1811(4),1766(7),165(6)]),flip(a)].">
      <j1 rule="para" parents="1766 16"/>
      <j2 rule="rewrite" parents="165 1811 1766 165"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1885">
    <literal><![CDATA[
      (x @ y) * y * (y @ x) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1767),rewrite([1811(4)]),flip(a)].">
      <j1 rule="back_rewrite" parents="1767"/>
      <j2 rule="rewrite" parents="1811"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1970">
    <literal><![CDATA[
      (x @ y) * z @ (y @ x) = z @ (y @ x)
    ]]></literal>
    <justification jstring="[para(1811(a,1),315(a,1,1)),flip(a)].">
      <j1 rule="para" parents="1811 315"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2125">
    <literal><![CDATA[
      x * (x @ y) @ (y @ x) = c_0
    ]]></literal>
    <justification jstring="[para(1885(a,1),246(a,2)),rewrite([3(4),1766(3),165(2)]),xx(a)].">
      <j1 rule="para" parents="1885 246"/>
      <j2 rule="rewrite" parents="3 1766 165"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="2131">
    <literal><![CDATA[
      x @ (y @ x) = c_0
    ]]></literal>
    <justification jstring="[para(1885(a,1),315(a,1,1)),rewrite([2125(6)])].">
      <j1 rule="para" parents="1885 315"/>
      <j2 rule="rewrite" parents="2125"/>
    </justification>
  </clause>

  <clause id="2191">
    <literal><![CDATA[
      (F @ G) @ D != C @ (A @ B)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(1811(a,1),1821(a,1)),flip(a)].">
      <j1 rule="para" parents="1811 1821"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2195">
    <literal><![CDATA[
      (x @ y) * y * z = y * (x @ y) * z
    ]]></literal>
    <justification jstring="[para(2131(a,1),13(a,1,2,2,1)),rewrite([155(3)])].">
      <j1 rule="para" parents="2131 13"/>
      <j2 rule="rewrite" parents="155"/>
    </justification>
  </clause>

  <clause id="2623">
    <literal><![CDATA[
      x @ y * x = x @ y
    ]]></literal>
    <justification jstring="[hyper(770,a,1234,a),flip(a)].">
      <j1 rule="hyper" parents="770 1234"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2663">
    <literal><![CDATA[
      x * x * y @ x * y = x @ y
    ]]></literal>
    <justification jstring="[back_rewrite(1207),rewrite([2623(8),315(6)])].">
      <j1 rule="back_rewrite" parents="1207"/>
      <j2 rule="rewrite" parents="2623 315"/>
    </justification>
  </clause>

  <clause id="2671">
    <literal><![CDATA[
      x * (x @ y) @ x * y = x @ y
    ]]></literal>
    <justification jstring="[back_rewrite(370),rewrite([2663(8)])].">
      <j1 rule="back_rewrite" parents="370"/>
      <j2 rule="rewrite" parents="2663"/>
    </justification>
  </clause>

  <clause id="2692">
    <literal><![CDATA[
      x * y @ y = x @ y
    ]]></literal>
    <justification jstring="[para(4(a,1),2623(a,1,2)),rewrite([2671(4),332(4)]),flip(a)].">
      <j1 rule="para" parents="4 2623"/>
      <j2 rule="rewrite" parents="2671 332"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2716">
    <literal><![CDATA[
      (x @ y) @ z = z @ (y @ x)
    ]]></literal>
    <justification jstring="[para(1811(a,1),2623(a,1,2)),rewrite([2692(3),1970(6)])].">
      <j1 rule="para" parents="1811 2623"/>
      <j2 rule="rewrite" parents="2692 1970"/>
    </justification>
  </clause>

  <clause id="2821">
    <literal><![CDATA[
      D @ (G @ F) != C @ (A @ B)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[back_rewrite(2191),rewrite([2716(5)])].">
      <j1 rule="back_rewrite" parents="2191"/>
      <j2 rule="rewrite" parents="2716"/>
    </justification>
  </clause>

  <clause id="3767">
    <literal><![CDATA[
      (C @ (A @ B)) * (G @ F) * D != D * (G @ F)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[ur(465,b,2821,a)].">
      <j1 rule="ur" parents="465 2821"/>
    </justification>
  </clause>

  <clause id="3896">
    <literal><![CDATA[
      (C @ (A @ B)) * (G @ F) * D * x != D * (G @ F) * x
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[ur(986,b,3767,a),rewrite([3(6),3(18),3(17)]),flip(a)].">
      <j1 rule="ur" parents="986 3767"/>
      <j2 rule="rewrite" parents="3 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4929">
    <literal><![CDATA[
      (x @ y) @ z * y * x = c_0
    ]]></literal>
    <justification jstring="[para(811(a,1),246(a,1)),rewrite([3(6),3(5),4(5)]),xx(a)].">
      <j1 rule="para" parents="811 246"/>
      <j2 rule="rewrite" parents="3 3 4"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="5283">
    <literal><![CDATA[
      (x @ y) @ z * u * y * x = c_0
    ]]></literal>
    <justification jstring="[para(3(a,1),4929(a,1,2))].">
      <j1 rule="para" parents="3 4929"/>
    </justification>
  </clause>

  <clause id="5675">
    <literal><![CDATA[
      (x @ y) @ z * u * v * y * x = c_0
    ]]></literal>
    <justification jstring="[para(3(a,1),5283(a,1,2,2))].">
      <j1 rule="para" parents="3 5283"/>
    </justification>
  </clause>

  <clause id="10907">
    <literal><![CDATA[
      (x @ y) @ z * y * x * u = c_0
    ]]></literal>
    <justification jstring="[para(811(a,1),5675(a,1,2)),rewrite([3(3)])].">
      <j1 rule="para" parents="811 5675"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="10954">
    <literal><![CDATA[
      (x @ y) @ y * x * z = c_0
    ]]></literal>
    <justification jstring="[para(155(a,1),10907(a,1,2))].">
      <j1 rule="para" parents="155 10907"/>
    </justification>
  </clause>

  <clause id="11021">
    <literal><![CDATA[
      x * y * z * (y @ x) = y * x * z
    ]]></literal>
    <justification jstring="[para(10954(a,1),4(a,1,2,2)),rewrite([165(5),3(4),3(3),2195(8),1243(7),13(8)])].">
      <j1 rule="para" parents="10954 4"/>
      <j2 rule="rewrite" parents="165 3 3 2195 1243 13"/>
    </justification>
  </clause>

  <clause id="12559">
    <literal><![CDATA[
      (C @ (A @ B)) * D * G * F != D * G * F
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(811(a,1),3896(a,1,2)),rewrite([2195(19),1184(18),4(19)])].">
      <j1 rule="para" parents="811 3896"/>
      <j2 rule="rewrite" parents="2195 1184 4"/>
    </justification>
  </clause>

  <clause id="12564">
    <literal><![CDATA[
      (C @ (B @ A)) * D * G * F != D * G * F
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[ur(1847,b,12559,a),rewrite([2716(5)])].">
      <j1 rule="ur" parents="1847 12559"/>
      <j2 rule="rewrite" parents="2716"/>
    </justification>
  </clause>

  <clause id="12595">
    <literal><![CDATA[
      (C @ (B @ A)) * G * F * D != G * F * D
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[ur(56,b,12564,a(flip)),rewrite([3(17),3(16),3(15),15(16),3(16)])].">
      <j1 rule="ur" parents="56 12564"/>
      <j2 rule="rewrite" parents="3 3 3 15 3"/>
    </justification>
  </clause>

  <clause id="12911">
    <literal><![CDATA[
      (C @ (B @ A)) * F * D * G != F * D * G
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[ur(56,b,12595,a(flip)),rewrite([3(17),3(16),3(15),15(16),3(16)])].">
      <j1 rule="ur" parents="56 12595"/>
      <j2 rule="rewrite" parents="3 3 3 15 3"/>
    </justification>
  </clause>

  <clause id="12937">
    <literal><![CDATA[
      (B @ A) * C * F * D * G != C * (B @ A) * F * D * G
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[ur(202,b,12911,a)].">
      <j1 rule="ur" parents="202 12911"/>
    </justification>
  </clause>

  <clause id="16124">
    <literal><![CDATA[
      (x @ y) * z = z * (x @ y)
    ]]></literal>
    <justification jstring="[hyper(202,a,11021,a)].">
      <j1 rule="hyper" parents="202 11021"/>
    </justification>
  </clause>

  <clause id="17093">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[back_rewrite(12937),rewrite([16124(11),3(11),3(10),3(9),16124(21),3(21),3(20)]),xx(a)].">
      <j1 rule="back_rewrite" parents="12937"/>
      <j2 rule="rewrite" parents="16124 3 3 3 16124 3 3"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
