<?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>AE-cs.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13114 was started by mccune on cleo.thornwood,
Mon Jun 19 16:41:28 2006
The command was "/home/mccune/bin/prover9 -f cs.in AE.in".
]]></heading>

<proof number="1" length="24" max_count="19">

<comments><![CDATA[
% Proof 1 at 0.16 (+ 0.00) seconds: E.
% Length of proof is 24.
% Level of proof is 12.
% Maximum clause weight is 19.
% Given clauses 146.
]]></comments>

  <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="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 = u @ (v @ w)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      A * B * C * B * A != B * A * C * A * B
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></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="30">
    <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="33">
    <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="66">
    <literal><![CDATA[
      (x @ x) * y = y
    ]]></literal>
    <justification jstring="[hyper(13,a,33,a)].">
      <j1 rule="hyper" parents="13 33"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      x * (x @ (y @ y)) = x * (y @ y)
    ]]></literal>
    <justification jstring="[hyper(16,a,66,a)].">
      <j1 rule="hyper" parents="16 66"/>
    </justification>
  </clause>

  <clause id="76">
    <literal><![CDATA[
      x * ((y @ y) @ x) = x
    ]]></literal>
    <justification jstring="[para(66(a,1),10(a,1,2)),demod(66(5))].">
      <j1 rule="para" parents="66 10"/>
      <j2 rule="demod" parents="66"/>
    </justification>
  </clause>

  <clause id="92">
    <literal><![CDATA[
      (x @ x) @ y = y @ y
    ]]></literal>
    <justification jstring="[hyper(30,a,76,a),flip(a)].">
      <j1 rule="hyper" parents="30 76"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="96">
    <literal><![CDATA[
      x * (y @ (z @ u)) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),76(a,1,2))].">
      <j1 rule="para" parents="11 76"/>
    </justification>
  </clause>

  <clause id="108">
    <literal><![CDATA[
      x * (y @ y) = x
    ]]></literal>
    <justification jstring="[back_demod(74),demod(96(3)),flip(a)].">
      <j1 rule="back_demod" parents="74"/>
      <j2 rule="demod" parents="96"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="109">
    <literal><![CDATA[
      x @ x = y @ y
    ]]></literal>
    <justification jstring="[hyper(30,a,108,a)].">
      <j1 rule="hyper" parents="30 108"/>
    </justification>
  </clause>

  <clause id="113">
    <literal><![CDATA[
      x @ x = c_0
    ]]></literal>
    <justification jstring="[new_symbol(109)].">
      <j1 rule="new_symbol" parents="109"/>
    </justification>
  </clause>

  <clause id="117">
    <literal><![CDATA[
      x * c_0 = x
    ]]></literal>
    <justification jstring="[back_demod(108),demod(113(1))].">
      <j1 rule="back_demod" parents="108"/>
      <j2 rule="demod" parents="113"/>
    </justification>
  </clause>

  <clause id="121">
    <literal><![CDATA[
      c_0 @ x = c_0
    ]]></literal>
    <justification jstring="[back_demod(92),demod(113(1),113(3))].">
      <j1 rule="back_demod" parents="92"/>
      <j2 rule="demod" parents="113 113"/>
    </justification>
  </clause>

  <clause id="128">
    <literal><![CDATA[
      x @ (y @ z) = c_0
    ]]></literal>
    <justification jstring="[para(113(a,1),11(a,1,1)),demod(121(2)),flip(a)].">
      <j1 rule="para" parents="113 11"/>
      <j2 rule="demod" parents="121"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="138">
    <literal><![CDATA[
      x * y * z * (y @ x) = y * x * z
    ]]></literal>
    <justification jstring="[para(10(a,1),19(a,1,2,2)),demod(128(6),117(6))].">
      <j1 rule="para" parents="10 19"/>
      <j2 rule="demod" parents="128 117"/>
    </justification>
  </clause>

  <clause id="1052">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[para(138(a,2),12(a,1)),demod(9(11),9(10),10(10)),xx(a)].">
      <j1 rule="para" parents="138 12"/>
      <j2 rule="demod" parents="9 9 10"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
