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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>t2_12.out</source>

<heading><![CDATA[
Prover9 (32) version August-2006A, August 2006.
Process 11451 was started by mccune on cleo.thornwood,
Sat Aug 12 21:01:21 2006
The command was "/home/mccune/bin/prover9 -f head.in t2_12.in".
]]></heading>

<proof number="1" length="29" max_count="35">

<comments><![CDATA[
% Proof 1 at 0.07 (+ 0.02) seconds.
% Length of proof is 29.
% Level of proof is 8.
% Maximum clause weight is 10.
% Given clauses 175.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      (all x all y all z (A(x,y,z) <-> x <= y & y <= z | z <= y & y <= x))
    ]]></literal>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="5">
    <literal><![CDATA[
      (all x all y all z (D(x,y,z) <-> x ^ z <= y & y <= x v z))
    ]]></literal>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6">
    <literal><![CDATA[
      (all x all y (x <= y <-> x ^ y = x))
    ]]></literal>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      -D(x,y,z)
    ]]></literal>
    <literal><![CDATA[
      x ^ z <= y
    ]]></literal>
    <justification jstring="[clausify(5)].">
      <j1 rule="clausify" parents="5"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      -D(x,y,z)
    ]]></literal>
    <literal><![CDATA[
      y <= x v z
    ]]></literal>
    <justification jstring="[clausify(5)].">
      <j1 rule="clausify" parents="5"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      D(ca,cx,cb)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x ^ y = y ^ x
    ]]></literal>
    <attribute><![CDATA[
      label("commutativity_meet")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <attribute><![CDATA[
      label("commutativity_join")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      (x ^ y) v x = x
    ]]></literal>
    <attribute><![CDATA[
      label("absorption_2")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[copy(26),rewrite(21(2))].">
      <j1 rule="copy" parents="26"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      A(x,y,z)
    ]]></literal>
    <literal><![CDATA[
      -(x <= y)
    ]]></literal>
    <literal><![CDATA[
      -(y <= z)
    ]]></literal>
    <justification jstring="[clausify(1)].">
      <j1 rule="clausify" parents="1"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      A(x,y,z)
    ]]></literal>
    <literal><![CDATA[
      -(z <= y)
    ]]></literal>
    <literal><![CDATA[
      -(y <= x)
    ]]></literal>
    <justification jstring="[clausify(1)].">
      <j1 rule="clausify" parents="1"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      -(x <= y)
    ]]></literal>
    <literal><![CDATA[
      x ^ y = x
    ]]></literal>
    <justification jstring="[clausify(6)].">
      <j1 rule="clausify" parents="6"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      x <= y
    ]]></literal>
    <literal><![CDATA[
      y <= x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="37">
    <literal><![CDATA[
      -A(ca,cx,cb)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      ca ^ cb <= cx
    ]]></literal>
    <justification jstring="[resolve(19,a,17,a)].">
      <j1 rule="resolve" parents="19 17"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      cx <= ca v cb
    ]]></literal>
    <justification jstring="[resolve(19,a,18,a)].">
      <j1 rule="resolve" parents="19 18"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(20(a,1),27(a,1,2))].">
      <j1 rule="para" parents="20 27"/>
    </justification>
  </clause>

  <clause id="64">
    <literal><![CDATA[
      x ^ y = x
    ]]></literal>
    <literal><![CDATA[
      y <= x
    ]]></literal>
    <justification jstring="[hyper(34,a,36,a)].">
      <j1 rule="hyper" parents="34 36"/>
    </justification>
  </clause>

  <clause id="76">
    <literal><![CDATA[
      A(ca ^ cb,cx,ca v cb)
    ]]></literal>
    <justification jstring="[hyper(32,b,38,a,c,39,a)].">
      <j1 rule="hyper" parents="32 38 39"/>
    </justification>
  </clause>

  <clause id="667">
    <literal><![CDATA[
      x <= y
    ]]></literal>
    <literal><![CDATA[
      x ^ y = y
    ]]></literal>
    <justification jstring="[para(64(a,1),20(a,1)),flip(b)].">
      <j1 rule="para" parents="64 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="679">
    <literal><![CDATA[
      x <= y
    ]]></literal>
    <literal><![CDATA[
      x v y = x
    ]]></literal>
    <justification jstring="[para(64(a,1),55(a,1,2))].">
      <j1 rule="para" parents="64 55"/>
    </justification>
  </clause>

  <clause id="1100">
    <literal><![CDATA[
      ca <= cb
    ]]></literal>
    <literal><![CDATA[
      cb <= cx
    ]]></literal>
    <justification jstring="[para(667(b,1),38(a,1))].">
      <j1 rule="para" parents="667 38"/>
    </justification>
  </clause>

  <clause id="1168">
    <literal><![CDATA[
      ca <= cb
    ]]></literal>
    <literal><![CDATA[
      cx <= ca
    ]]></literal>
    <justification jstring="[para(679(b,1),39(a,2))].">
      <j1 rule="para" parents="679 39"/>
    </justification>
  </clause>

  <clause id="1219">
    <literal><![CDATA[
      ca <= cb
    ]]></literal>
    <justification jstring="[hyper(33,b,1100,b,c,1168,b),merge(c),unit_del(a,37)].">
      <j1 rule="hyper" parents="33 1100 1168"/>
      <j2 rule="merge"/>
      <j2 rule="unit_del" parents="37"/>
    </justification>
  </clause>

  <clause id="1220">
    <literal><![CDATA[
      ca ^ cb = ca
    ]]></literal>
    <justification jstring="[hyper(34,a,1219,a)].">
      <j1 rule="hyper" parents="34 1219"/>
    </justification>
  </clause>

  <clause id="1331">
    <literal><![CDATA[
      A(ca,cx,ca v cb)
    ]]></literal>
    <justification jstring="[back_rewrite(76),rewrite(1220(3))].">
      <j1 rule="back_rewrite" parents="76"/>
      <j2 rule="rewrite" parents="1220"/>
    </justification>
  </clause>

  <clause id="1360">
    <literal><![CDATA[
      ca v cb = cb
    ]]></literal>
    <justification jstring="[para(1220(a,1),55(a,1,2)),rewrite(21(3))].">
      <j1 rule="para" parents="1220 55"/>
      <j2 rule="rewrite" parents="21"/>
    </justification>
  </clause>

  <clause id="1375">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[back_rewrite(1331),rewrite(1360(5)),unit_del(a,37)].">
      <j1 rule="back_rewrite" parents="1331"/>
      <j2 rule="rewrite" parents="1360"/>
      <j2 rule="unit_del" parents="37"/>
    </justification>
  </clause>

</proof>

</proofs>
