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<!DOCTYPE proofs SYSTEM "proof3.dtd">

<?xml-stylesheet type="text/xsl" href="proof3.xsl"?>

<proofs number_of_proofs="1">

<source>NA-gt.out</source>

<heading><![CDATA[
Prover9 (32) version April-2007, April 2007.
Process 26956 was started by mccune on cleo,
Fri Apr 13 09:16:48 2007
The command was "/home/mccune/bin/prover9 -f gt.in NA.in".
]]></heading>

<proof number="1" length="12" max_count="13">

<comments><![CDATA[
% Proof 1 at 0.00 (+ 0.01) seconds: A.
% Length of proof is 12.
% Level of proof is 3.
% Maximum clause weight is 13.
% Given clauses 5.
]]></comments>

  <clause id="1" type="assumption">
    <literal><![CDATA[
      (x * y) * z = x * y * z
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

  <clause id="3" type="assumption">
    <literal><![CDATA[
      x' * x = e
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

  <clause id="5">
    <literal><![CDATA[
      x' * y' * x * y = x @ y
    ]]></literal>
    <justification jstring="[copy(4),flip(a)].">
      <j1 rule="copy" parents="4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="6" type="assumption">
    <literal><![CDATA[
      (x @ y) * z = z * (x @ y)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7" 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="8">
    <literal><![CDATA[
      x' * x * y = y
    ]]></literal>
    <justification jstring="[para(3(a,1),1(a,1,1)),rewrite(2(2)),flip(a)].">
      <j1 rule="para" parents="3 1"/>
      <j2 rule="rewrite" parents="2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      (x @ y) * (x @ y)' = e
    ]]></literal>
    <justification jstring="[para(6(a,2),3(a,1))].">
      <j1 rule="para" parents="6 3"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      (x @ y) @ z = e
    ]]></literal>
    <justification jstring="[para(6(a,1),5(a,1,2,2)),rewrite(8(6),6(4,R),17(4)),flip(a)].">
      <j1 rule="para" parents="6 5"/>
      <j2 rule="rewrite" parents="8 6 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      x @ (y @ z) = e
    ]]></literal>
    <justification jstring="[para(6(a,2),5(a,1,2,2)),rewrite(8(6),3(2)),flip(a)].">
      <j1 rule="para" parents="6 5"/>
      <j2 rule="rewrite" parents="8 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[back_rewrite(7),rewrite(18(5),19(6)),xx(a)].">
      <j1 rule="back_rewrite" parents="7"/>
      <j2 rule="rewrite" parents="18 19"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
