<?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 June-2006C, June 2006.
Process 13118 was started by mccune on cleo.thornwood,
Mon Jun 19 16:41:29 2006
The command was "/home/mccune/bin/prover9 -f cs.in DA.in".
]]></heading>

<proof number="1" length="73" max_count="23">

<comments><![CDATA[
% Proof 1 at 0.35 (+ 0.00) seconds: A.
% Length of proof is 73.
% Level of proof is 24.
% Maximum clause weight is 23.
% Given clauses 205.
]]></comments>

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

  <clause id="7">
    <literal><![CDATA[
      x * y != x * z
    ]]></literal>
    <literal><![CDATA[
      y = z
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      x * y != z * y
    ]]></literal>
    <literal><![CDATA[
      x = z
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      (x * y) * z = x * y * z
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      x * y * (y @ x) = y * x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="11">
    <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="12">
    <literal><![CDATA[
      (A @ B) @ C != D @ (F @ G)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      x * y * z != x * y * u
    ]]></literal>
    <literal><![CDATA[
      z = u
    ]]></literal>
    <justification jstring="[para(9(a,1),7(a,1)),demod(9(4))].">
      <j1 rule="para" parents="9 7"/>
      <j2 rule="demod" parents="9"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      x * (x @ x) = x
    ]]></literal>
    <justification jstring="[hyper(7,a,10,a)].">
      <j1 rule="hyper" parents="7 10"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      x * y != z * x
    ]]></literal>
    <literal><![CDATA[
      z * (z @ x) = y
    ]]></literal>
    <justification jstring="[para(10(a,1),7(a,1)),flip(a)].">
      <j1 rule="para" parents="10 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      x * y * (y @ x) * z = y * x * z
    ]]></literal>
    <justification jstring="[para(10(a,1),9(a,1,1)),demod(9(2),9(5)),flip(a)].">
      <j1 rule="para" parents="10 9"/>
      <j2 rule="demod" parents="9 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x * y * z * (z @ x * y) = z * x * y
    ]]></literal>
    <justification jstring="[para(10(a,1),9(a,1)),flip(a)].">
      <j1 rule="para" parents="10 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      (x @ y) * (z @ y) * ((z @ x) @ y) = (z @ y) * (x @ y)
    ]]></literal>
    <justification jstring="[para(10(a,1),11(a,1,1)),demod(11(2),11(7)),flip(a)].">
      <j1 rule="para" parents="10 11"/>
      <j2 rule="demod" parents="11 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      ((A @ B) @ C) * x != (D @ (F @ G)) * x
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[ur(8,b,12,a)].">
      <j1 rule="ur" parents="8 12"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      x * ((A @ B) @ C) * y != x * (D @ (F @ G)) * y
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[ur(7,b,23,a)].">
      <j1 rule="ur" parents="7 23"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      x @ x = y
    ]]></literal>
    <justification jstring="[para(15(a,1),7(a,1)),flip(a)].">
      <j1 rule="para" parents="15 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      x * (x @ x) * y = x * y
    ]]></literal>
    <justification jstring="[para(15(a,1),9(a,1,1)),flip(a)].">
      <j1 rule="para" parents="15 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      x * y * z != x * u * y
    ]]></literal>
    <literal><![CDATA[
      u * (u @ y) = z
    ]]></literal>
    <justification jstring="[para(10(a,1),13(a,1,2)),flip(a)].">
      <j1 rule="para" parents="10 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      x * y * z != y * x
    ]]></literal>
    <literal><![CDATA[
      y @ x = z
    ]]></literal>
    <justification jstring="[para(10(a,1),13(a,1)),flip(a)].">
      <j1 rule="para" parents="10 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      x * (D @ (F @ G)) * (((A @ B) @ C) @ x) != ((A @ B) @ C) * x
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(10(a,1),25(a,1)),flip(a)].">
      <j1 rule="para" parents="10 25"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="63">
    <literal><![CDATA[
      (x @ x) * y = y
    ]]></literal>
    <justification jstring="[hyper(13,a,32,a)].">
      <j1 rule="hyper" parents="13 32"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      x * y != y
    ]]></literal>
    <literal><![CDATA[
      z @ z = x
    ]]></literal>
    <justification jstring="[para(63(a,1),8(a,1)),flip(a)].">
      <j1 rule="para" parents="63 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="70">
    <literal><![CDATA[
      x * ((y @ y) @ x) = x
    ]]></literal>
    <justification jstring="[para(63(a,1),10(a,1,2)),demod(63(5))].">
      <j1 rule="para" parents="63 10"/>
      <j2 rule="demod" parents="63"/>
    </justification>
  </clause>

  <clause id="73">
    <literal><![CDATA[
      (x @ x * y) * (y @ x * y) * z = z
    ]]></literal>
    <justification jstring="[para(11(a,1),63(a,1,1)),demod(9(6))].">
      <j1 rule="para" parents="11 63"/>
      <j2 rule="demod" parents="9"/>
    </justification>
  </clause>

  <clause id="78">
    <literal><![CDATA[
      x * y * z != u * x * y
    ]]></literal>
    <literal><![CDATA[
      u * (u @ x * y) = z
    ]]></literal>
    <justification jstring="[para(9(a,1),16(a,1))].">
      <j1 rule="para" parents="9 16"/>
    </justification>
  </clause>

  <clause id="88">
    <literal><![CDATA[
      (x @ x) @ y = y @ y
    ]]></literal>
    <justification jstring="[hyper(29,a,70,a),flip(a)].">
      <j1 rule="hyper" parents="29 70"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="117">
    <literal><![CDATA[
      (x @ x) @ y * z = (y @ y * z) * (z @ y * z)
    ]]></literal>
    <justification jstring="[para(88(a,2),11(a,1))].">
      <j1 rule="para" parents="88 11"/>
    </justification>
  </clause>

  <clause id="120">
    <literal><![CDATA[
      ((x @ x) @ y) * z = z
    ]]></literal>
    <justification jstring="[para(88(a,2),63(a,1,1))].">
      <j1 rule="para" parents="88 63"/>
    </justification>
  </clause>

  <clause id="121">
    <literal><![CDATA[
      x * (((y @ y) @ z) @ x) = x
    ]]></literal>
    <justification jstring="[para(88(a,2),70(a,1,2,1))].">
      <j1 rule="para" parents="88 70"/>
    </justification>
  </clause>

  <clause id="129">
    <literal><![CDATA[
      x * (x @ ((y @ y) @ z)) = x * ((y @ y) @ z)
    ]]></literal>
    <justification jstring="[hyper(16,a,120,a)].">
      <j1 rule="hyper" parents="16 120"/>
    </justification>
  </clause>

  <clause id="136">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      ((z @ z) @ u) @ x = y
    ]]></literal>
    <justification jstring="[para(120(a,1),16(a,2)),demod(120(8))].">
      <j1 rule="para" parents="120 16"/>
      <j2 rule="demod" parents="120"/>
    </justification>
  </clause>

  <clause id="140">
    <literal><![CDATA[
      (((x @ x) @ y) @ z) * u = u
    ]]></literal>
    <justification jstring="[para(88(a,2),120(a,1,1,1))].">
      <j1 rule="para" parents="88 120"/>
    </justification>
  </clause>

  <clause id="164">
    <literal><![CDATA[
      (x @ x) @ y = z @ z
    ]]></literal>
    <justification jstring="[hyper(69,a,120,a),flip(a)].">
      <j1 rule="hyper" parents="69 120"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="165">
    <literal><![CDATA[
      x @ x = y @ y
    ]]></literal>
    <justification jstring="[hyper(69,a,63,a)].">
      <j1 rule="hyper" parents="69 63"/>
    </justification>
  </clause>

  <clause id="168">
    <literal><![CDATA[
      x @ x = c_0
    ]]></literal>
    <justification jstring="[new_symbol(165)].">
      <j1 rule="new_symbol" parents="165"/>
    </justification>
  </clause>

  <clause id="171">
    <literal><![CDATA[
      c_0 @ x = c_0
    ]]></literal>
    <justification jstring="[back_demod(164),demod(168(1),168(3))].">
      <j1 rule="back_demod" parents="164"/>
      <j2 rule="demod" parents="168 168"/>
    </justification>
  </clause>

  <clause id="177">
    <literal><![CDATA[
      c_0 * x = x
    ]]></literal>
    <justification jstring="[back_demod(140),demod(168(1),171(2),171(2))].">
      <j1 rule="back_demod" parents="140"/>
      <j2 rule="demod" parents="168 171 171"/>
    </justification>
  </clause>

  <clause id="180">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      c_0 = y
    ]]></literal>
    <justification jstring="[back_demod(136),demod(168(3),171(4),171(4))].">
      <j1 rule="back_demod" parents="136"/>
      <j2 rule="demod" parents="168 171 171"/>
    </justification>
  </clause>

  <clause id="184">
    <literal><![CDATA[
      x * (x @ c_0) = x * c_0
    ]]></literal>
    <justification jstring="[back_demod(129),demod(168(1),171(2),168(4),171(5))].">
      <j1 rule="back_demod" parents="129"/>
      <j2 rule="demod" parents="168 171 168 171"/>
    </justification>
  </clause>

  <clause id="186">
    <literal><![CDATA[
      x * c_0 = x
    ]]></literal>
    <justification jstring="[back_demod(121),demod(168(1),171(2),171(2))].">
      <j1 rule="back_demod" parents="121"/>
      <j2 rule="demod" parents="168 171 171"/>
    </justification>
  </clause>

  <clause id="188">
    <literal><![CDATA[
      (x @ x * y) * (y @ x * y) = c_0
    ]]></literal>
    <justification jstring="[back_demod(117),demod(168(1),171(3)),flip(a)].">
      <j1 rule="back_demod" parents="117"/>
      <j2 rule="demod" parents="168 171"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="192">
    <literal><![CDATA[
      x * (x @ c_0) = x
    ]]></literal>
    <justification jstring="[back_demod(184),demod(186(5))].">
      <j1 rule="back_demod" parents="184"/>
      <j2 rule="demod" parents="186"/>
    </justification>
  </clause>

  <clause id="203">
    <literal><![CDATA[
      x * y * ((x @ y) @ y * x) = (x @ y) * y * x
    ]]></literal>
    <justification jstring="[para(20(a,1),19(a,1)),flip(a)].">
      <j1 rule="para" parents="20 19"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="222">
    <literal><![CDATA[
      (x @ y) * ((x @ y) @ y) = x @ y
    ]]></literal>
    <justification jstring="[para(168(a,1),22(a,1,1)),demod(177(6),168(6),186(7))].">
      <j1 rule="para" parents="168 22"/>
      <j2 rule="demod" parents="177 168 186"/>
    </justification>
  </clause>

  <clause id="248">
    <literal><![CDATA[
      x @ c_0 = c_0
    ]]></literal>
    <justification jstring="[hyper(180,a,192,a),flip(a)].">
      <j1 rule="hyper" parents="180 192"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="264">
    <literal><![CDATA[
      x * (x @ x * y) = x * (x @ y)
    ]]></literal>
    <justification jstring="[hyper(42,a,20,a),flip(a)].">
      <j1 rule="hyper" parents="42 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="270">
    <literal><![CDATA[
      x * y != x * z
    ]]></literal>
    <literal><![CDATA[
      z = y
    ]]></literal>
    <justification jstring="[para(177(a,1),42(a,1,2)),demod(186(3),248(5),186(5))].">
      <j1 rule="para" parents="177 42"/>
      <j2 rule="demod" parents="186 248 186"/>
    </justification>
  </clause>

  <clause id="274">
    <literal><![CDATA[
      (x @ y) @ (z @ y) = (x @ z) @ y
    ]]></literal>
    <justification jstring="[hyper(43,a,22,a)].">
      <j1 rule="hyper" parents="43 22"/>
    </justification>
  </clause>

  <clause id="281">
    <literal><![CDATA[
      x * y != y * x
    ]]></literal>
    <literal><![CDATA[
      y @ x = c_0
    ]]></literal>
    <justification jstring="[para(186(a,1),43(a,1,2))].">
      <j1 rule="para" parents="186 43"/>
    </justification>
  </clause>

  <clause id="351">
    <literal><![CDATA[
      (x @ y) @ y = c_0
    ]]></literal>
    <justification jstring="[hyper(180,a,222,a),flip(a)].">
      <j1 rule="hyper" parents="180 222"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="352">
    <literal><![CDATA[
      (x @ y) * y = y * (x @ y)
    ]]></literal>
    <justification jstring="[para(222(a,1),10(a,1,2)),flip(a)].">
      <j1 rule="para" parents="222 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="362">
    <literal><![CDATA[
      (x @ y) * y * z = y * (x @ y) * z
    ]]></literal>
    <justification jstring="[para(351(a,1),19(a,1,2,2,1)),demod(177(3)),flip(a)].">
      <j1 rule="para" parents="351 19"/>
      <j2 rule="demod" parents="177"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="370">
    <literal><![CDATA[
      x * y * ((x @ y) @ y * x) = y * (x @ y) * x
    ]]></literal>
    <justification jstring="[back_demod(203),demod(362(8))].">
      <j1 rule="back_demod" parents="203"/>
      <j2 rule="demod" parents="362"/>
    </justification>
  </clause>

  <clause id="372">
    <literal><![CDATA[
      x @ (y @ x) = c_0
    ]]></literal>
    <justification jstring="[hyper(281,a,352,a)].">
      <j1 rule="hyper" parents="281 352"/>
    </justification>
  </clause>

  <clause id="409">
    <literal><![CDATA[
      (x @ y * x) * (y @ y * x) = c_0
    ]]></literal>
    <justification jstring="[hyper(180,a,73,a),flip(a)].">
      <j1 rule="hyper" parents="180 73"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="421">
    <literal><![CDATA[
      (x @ y) @ y * x = c_0
    ]]></literal>
    <justification jstring="[para(73(a,1),22(a,1)),demod(409(8))].">
      <j1 rule="para" parents="73 22"/>
      <j2 rule="demod" parents="409"/>
    </justification>
  </clause>

  <clause id="429">
    <literal><![CDATA[
      x * (y @ x) * y = y * x
    ]]></literal>
    <justification jstring="[back_demod(370),demod(421(3),186(2)),flip(a)].">
      <j1 rule="back_demod" parents="370"/>
      <j2 rule="demod" parents="421 186"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="453">
    <literal><![CDATA[
      x * (x @ y * x) = (x @ y) * x
    ]]></literal>
    <justification jstring="[hyper(78,a,19,a)].">
      <j1 rule="hyper" parents="78 19"/>
    </justification>
  </clause>

  <clause id="471">
    <literal><![CDATA[
      (x @ y) * x = x * (x @ y)
    ]]></literal>
    <justification jstring="[hyper(16,a,429,a),flip(a)].">
      <j1 rule="hyper" parents="16 429"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="500">
    <literal><![CDATA[
      x * (x @ y * x) = x * (x @ y)
    ]]></literal>
    <justification jstring="[back_demod(453),demod(471(5))].">
      <j1 rule="back_demod" parents="453"/>
      <j2 rule="demod" parents="471"/>
    </justification>
  </clause>

  <clause id="577">
    <literal><![CDATA[
      (x @ x * y) * y * (y @ x) = y
    ]]></literal>
    <justification jstring="[para(471(a,1),73(a,1,2)),demod(500(5))].">
      <j1 rule="para" parents="471 73"/>
      <j2 rule="demod" parents="500"/>
    </justification>
  </clause>

  <clause id="625">
    <literal><![CDATA[
      x @ x * y = x @ y
    ]]></literal>
    <justification jstring="[hyper(270,a,264,a),flip(a)].">
      <j1 rule="hyper" parents="270 264"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="643">
    <literal><![CDATA[
      x * (y @ x) * (x @ y) = x
    ]]></literal>
    <justification jstring="[back_demod(577),demod(625(2),362(4))].">
      <j1 rule="back_demod" parents="577"/>
      <j2 rule="demod" parents="625 362"/>
    </justification>
  </clause>

  <clause id="679">
    <literal><![CDATA[
      (x @ y) * (y @ x * y) = c_0
    ]]></literal>
    <justification jstring="[back_demod(188),demod(625(2))].">
      <j1 rule="back_demod" parents="188"/>
      <j2 rule="demod" parents="625"/>
    </justification>
  </clause>

  <clause id="739">
    <literal><![CDATA[
      (x @ y) * (y @ x) = c_0
    ]]></literal>
    <justification jstring="[hyper(180,a,643,a),flip(a)].">
      <j1 rule="hyper" parents="180 643"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="768">
    <literal><![CDATA[
      (x @ y) * (y @ x) * z = z
    ]]></literal>
    <justification jstring="[para(739(a,1),9(a,1,1)),demod(177(2)),flip(a)].">
      <j1 rule="para" parents="739 9"/>
      <j2 rule="demod" parents="177"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="813">
    <literal><![CDATA[
      (x @ y) @ (y @ x) = c_0
    ]]></literal>
    <justification jstring="[para(739(a,1),281(a,1)),demod(739(4)),xx(a)].">
      <j1 rule="para" parents="739 281"/>
      <j2 rule="demod" parents="739"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="1085">
    <literal><![CDATA[
      x @ y * x = x @ y
    ]]></literal>
    <justification jstring="[para(679(a,1),768(a,1,2)),demod(186(3)),flip(a)].">
      <j1 rule="para" parents="679 768"/>
      <j2 rule="demod" parents="186"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1121">
    <literal><![CDATA[
      (x @ y) @ z = z @ (y @ x)
    ]]></literal>
    <justification jstring="[para(768(a,1),1085(a,1,2)),demod(11(3),168(3),186(4),11(6),813(5),177(6))].">
      <j1 rule="para" parents="768 1085"/>
      <j2 rule="demod" parents="11 168 186 11 813 177"/>
    </justification>
  </clause>

  <clause id="1237">
    <literal><![CDATA[
      x * (D @ (F @ G)) * ((C @ (B @ A)) @ x) != (C @ (B @ A)) * x
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[back_demod(45),demod(1121(10),1121(18))].">
      <j1 rule="back_demod" parents="45"/>
      <j2 rule="demod" parents="1121 1121"/>
    </justification>
  </clause>

  <clause id="1505">
    <literal><![CDATA[
      (x @ y) @ (z @ y) = c_0
    ]]></literal>
    <justification jstring="[para(372(a,1),274(a,1,2)),demod(248(4)),flip(a)].">
      <j1 rule="para" parents="372 274"/>
      <j2 rule="demod" parents="248"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1519">
    <literal><![CDATA[
      x @ (y @ z) = c_0
    ]]></literal>
    <justification jstring="[para(274(a,2),739(a,1,1)),demod(1505(3),177(4))].">
      <j1 rule="para" parents="274 739"/>
      <j2 rule="demod" parents="1505 177"/>
    </justification>
  </clause>

  <clause id="1558">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[back_demod(1237),demod(1519(5),1519(6),171(3),186(3),186(2),1519(5),177(2)),xx(a)].">
      <j1 rule="back_demod" parents="1237"/>
      <j2 rule="demod" parents="1519 1519 171 186 186 1519 177"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
