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

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13131 was started by mccune on cleo.thornwood,
Mon Jun 19 16:41:41 2006
The command was "/home/mccune/bin/prover9 -f cs.in NA.in".
]]></heading>

<proof number="1" length="20" max_count="15">

<comments><![CDATA[
% Proof 1 at 0.02 (+ 0.00) seconds: A.
% Length of proof is 20.
% Level of proof is 6.
% Maximum clause weight is 15.
% Given clauses 36.
]]></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="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 = z * (x @ y)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </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="29">
    <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="34">
    <literal><![CDATA[
      (D @ (F @ G)) * x != x * ((A @ B) @ C)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(11(a,1),29(a,1)),flip(a)].">
      <j1 rule="para" parents="11 29"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="37">
    <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="93">
    <literal><![CDATA[
      x * (x @ (y @ z)) = x
    ]]></literal>
    <justification jstring="[hyper(16,a,11,a)].">
      <j1 rule="hyper" parents="16 11"/>
    </justification>
  </clause>

  <clause id="95">
    <literal><![CDATA[
      (x @ y) * ((x @ y) @ z) = x @ y
    ]]></literal>
    <justification jstring="[hyper(16,a,11,a(flip))].">
      <j1 rule="hyper" parents="16 11"/>
    </justification>
  </clause>

  <clause id="110">
    <literal><![CDATA[
      x @ (y @ z) = x @ x
    ]]></literal>
    <justification jstring="[hyper(37,a,93,a),flip(a)].">
      <j1 rule="hyper" parents="37 93"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="153">
    <literal><![CDATA[
      (D @ D) * x != x * ((A @ B) @ C)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(110(a,1),34(a,1,1))].">
      <j1 rule="para" parents="110 34"/>
    </justification>
  </clause>

  <clause id="177">
    <literal><![CDATA[
      (x @ x) * y = y
    ]]></literal>
    <justification jstring="[hyper(13,a,19,a)].">
      <j1 rule="hyper" parents="13 19"/>
    </justification>
  </clause>

  <clause id="202">
    <literal><![CDATA[
      x * ((A @ B) @ C) != x
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[back_demod(153),demod(177(4)),flip(a)].">
      <j1 rule="back_demod" parents="153"/>
      <j2 rule="demod" parents="177"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="203">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[resolve(202,a,95,a)].">
      <j1 rule="resolve" parents="202 95"/>
    </justification>
  </clause>

</proof>

</proofs>
