<?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 22-May-2007, May 2007.
Process 27457 was started by mccune on cleo,
Tue May 22 14:46:01 2007
The command was "/home/mccune/bin/prover9 -f cs.in NA.in".
]]></heading>

<proof number="1" length="25" max_count="19">

<comments><![CDATA[
% Proof 1 at 0.03 (+ 0.00) seconds: A.
% Length of proof is 25.
% Level of proof is 10.
% Maximum clause weight is 19.
% Given clauses 51.
]]></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 = 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="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="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="35">
    <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="50">
    <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="53">
    <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="93">
    <literal><![CDATA[
      (x @ x) * y = y
    ]]></literal>
    <justification jstring="[hyper(7,a,53,a)].">
      <j1 rule="hyper" parents="7 53"/>
    </justification>
  </clause>

  <clause id="100">
    <literal><![CDATA[
      x * (x @ (y @ z)) = x
    ]]></literal>
    <justification jstring="[hyper(10,a,5,a)].">
      <j1 rule="hyper" parents="10 5"/>
    </justification>
  </clause>

  <clause id="102">
    <literal><![CDATA[
      (x @ y) * ((x @ y) @ z) = x @ y
    ]]></literal>
    <justification jstring="[hyper(10,a,5,a(flip))].">
      <j1 rule="hyper" parents="10 5"/>
    </justification>
  </clause>

  <clause id="131">
    <literal><![CDATA[
      x * (y @ y) = x
    ]]></literal>
    <justification jstring="[hyper(10,a,93,a),rewrite([100(3)]),flip(a)].">
      <j1 rule="hyper" parents="10 93"/>
      <j2 rule="rewrite" parents="100"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="137">
    <literal><![CDATA[
      x @ x = y @ y
    ]]></literal>
    <justification jstring="[hyper(50,a,131,a)].">
      <j1 rule="hyper" parents="50 131"/>
    </justification>
  </clause>

  <clause id="138">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      z @ z = y
    ]]></literal>
    <justification jstring="[para(131(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="131 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="141">
    <literal><![CDATA[
      x @ x = c_0
    ]]></literal>
    <justification jstring="[new_symbol(137)].">
      <j1 rule="new_symbol" parents="137"/>
    </justification>
  </clause>

  <clause id="144">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      c_0 = y
    ]]></literal>
    <justification jstring="[back_rewrite(138),rewrite([141(3)])].">
      <j1 rule="back_rewrite" parents="138"/>
      <j2 rule="rewrite" parents="141"/>
    </justification>
  </clause>

  <clause id="149">
    <literal><![CDATA[
      x * c_0 = x
    ]]></literal>
    <justification jstring="[back_rewrite(131),rewrite([141(1)])].">
      <j1 rule="back_rewrite" parents="131"/>
      <j2 rule="rewrite" parents="141"/>
    </justification>
  </clause>

  <clause id="229">
    <literal><![CDATA[
      x @ (y @ z) = c_0
    ]]></literal>
    <justification jstring="[hyper(144,a,100,a),flip(a)].">
      <j1 rule="hyper" parents="144 100"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="233">
    <literal><![CDATA[
      x * ((A @ B) @ C) != x
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(100(a,1),35(a,1,2)),rewrite([229(11),229(14),149(9),149(8)])].">
      <j1 rule="para" parents="100 35"/>
      <j2 rule="rewrite" parents="229 229 149 149"/>
    </justification>
  </clause>

  <clause id="234">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[resolve(233,a,102,a)].">
      <j1 rule="resolve" parents="233 102"/>
    </justification>
  </clause>

</proof>

</proofs>
