<?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 August-2006A, August 2006.
Process 11366 was started by mccune on cleo.thornwood,
Sat Aug 12 21:00:03 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="1">
    <literal><![CDATA[
      x * y != x * z
    ]]></literal>
    <literal><![CDATA[
      y = z
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="2">
    <literal><![CDATA[
      x * y != z * y
    ]]></literal>
    <literal><![CDATA[
      x = z
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="3">
    <literal><![CDATA[
      (x * y) * z = x * y * z
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      x * y * (y @ x) = y * x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

  <clause id="6">
    <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="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="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="28">
    <literal><![CDATA[
      (D @ (F @ G)) * x != x * ((A @ B) @ C)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(5(a,1),23(a,1)),flip(a)].">
      <j1 rule="para" parents="5 23"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="31">
    <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="87">
    <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="89">
    <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="104">
    <literal><![CDATA[
      x @ (y @ z) = x @ x
    ]]></literal>
    <justification jstring="[hyper(31,a,87,a),flip(a)].">
      <j1 rule="hyper" parents="31 87"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="147">
    <literal><![CDATA[
      (D @ D) * x != x * ((A @ B) @ C)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(104(a,1),28(a,1,1))].">
      <j1 rule="para" parents="104 28"/>
    </justification>
  </clause>

  <clause id="171">
    <literal><![CDATA[
      (x @ x) * y = y
    ]]></literal>
    <justification jstring="[hyper(7,a,13,a)].">
      <j1 rule="hyper" parents="7 13"/>
    </justification>
  </clause>

  <clause id="196">
    <literal><![CDATA[
      x * ((A @ B) @ C) != x
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[back_rewrite(147),rewrite(171(4)),flip(a)].">
      <j1 rule="back_rewrite" parents="147"/>
      <j2 rule="rewrite" parents="171"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="197">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[resolve(196,a,89,a)].">
      <j1 rule="resolve" parents="196 89"/>
    </justification>
  </clause>

</proof>

</proofs>
