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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>lmA1short.out</source>

<heading><![CDATA[
Prover9 (32) version April-2007, April 2007.
Process 6906 was started by hoefnepe on europa,
Mon Oct  1 08:01:50 2007
The command was "../../../kleene/Prover9/bin/prover9 -f lmA1short.in".
]]></heading>

<proof number="1" length="31" max_count="70">

<comments><![CDATA[
% Proof 1 at 1.07 (+ 0.02) seconds.
% Length of proof is 31.
% Level of proof is 5.
% Maximum clause weight is 31.
% Given clauses 457.
]]></comments>

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

  <clause id="2" type="assumption">
    <literal><![CDATA[
      x <= y & y <= z -> x <= z
    ]]></literal>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="3" type="assumption">
    <literal><![CDATA[
      (all p (p <= 1 & p ; p = p -> p ; (p ; x) ^ = (p ; x) ^))
    ]]></literal>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="4" type="goal">
    <literal><![CDATA[
      (all p all q all r (p <= 1 & p ; p = p & q <= 1 & q ; q = q & r <= 1 & r ; r = r -> (p ; x ; q ; y ; r ; z) ^ <= p ; (p ; x ; q + q ; y ; r + r ; z) ^))
    ]]></literal>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="5" type="assumption">
    <literal><![CDATA[
      x + y = y + x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

  <clause id="7">
    <literal><![CDATA[
      x + y + z = x + (y + z)
    ]]></literal>
    <justification jstring="[copy(6),flip(a)].">
      <j1 rule="copy" parents="6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="8" type="assumption">
    <literal><![CDATA[
      x ; (y ; z) = x ; y ; z
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      x ; y ; z = x ; (y ; z)
    ]]></literal>
    <justification jstring="[copy(8),flip(a)].">
      <j1 rule="copy" parents="8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10" type="clausify">
    <literal><![CDATA[
      -(x <= y)
    ]]></literal>
    <literal><![CDATA[
      z ; x <= z ; y
    ]]></literal>
    <justification jstring="[clausify(1)].">
      <j1 rule="clausify" parents="1"/>
    </justification>
  </clause>

  <clause id="11" type="assumption">
    <literal><![CDATA[
      x <= x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="12" type="clausify">
    <literal><![CDATA[
      -(x <= y)
    ]]></literal>
    <literal><![CDATA[
      -(y <= z)
    ]]></literal>
    <literal><![CDATA[
      x <= z
    ]]></literal>
    <justification jstring="[clausify(2)].">
      <j1 rule="clausify" parents="2"/>
    </justification>
  </clause>

  <clause id="13" type="assumption">
    <literal><![CDATA[
      (x ; y ; z) ^ <= (x + y + z) ^
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      (x ; (y ; z)) ^ <= (x + (y + z)) ^
    ]]></literal>
    <justification jstring="[copy(13),rewrite(9(2),7(5))].">
      <j1 rule="copy" parents="13"/>
      <j2 rule="rewrite" parents="9 7"/>
    </justification>
  </clause>

  <clause id="15" type="clausify">
    <literal><![CDATA[
      -(x <= 1)
    ]]></literal>
    <literal><![CDATA[
      x ; x != x
    ]]></literal>
    <literal><![CDATA[
      (x ; y) ^ = x ; (x ; y) ^
    ]]></literal>
    <justification jstring="[clausify(3)].">
      <j1 rule="clausify" parents="3"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      -(x <= 1)
    ]]></literal>
    <literal><![CDATA[
      x ; x != x
    ]]></literal>
    <literal><![CDATA[
      x ; (x ; y) ^ = (x ; y) ^
    ]]></literal>
    <justification jstring="[copy(15),flip(c)].">
      <j1 rule="copy" parents="15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="17" type="deny">
    <literal><![CDATA[
      c4 <= 1
    ]]></literal>
    <justification jstring="[deny(4)].">
      <j1 rule="deny" parents="4"/>
    </justification>
  </clause>

  <clause id="18" type="deny">
    <literal><![CDATA[
      c4 ; c4 = c4
    ]]></literal>
    <justification jstring="[deny(4)].">
      <j1 rule="deny" parents="4"/>
    </justification>
  </clause>

  <clause id="19" type="deny">
    <literal><![CDATA[
      c5 ; c5 = c5
    ]]></literal>
    <justification jstring="[deny(4)].">
      <j1 rule="deny" parents="4"/>
    </justification>
  </clause>

  <clause id="20" type="deny">
    <literal><![CDATA[
      c6 ; c6 = c6
    ]]></literal>
    <justification jstring="[deny(4)].">
      <j1 rule="deny" parents="4"/>
    </justification>
  </clause>

  <clause id="21" type="deny">
    <literal><![CDATA[
      -((c4 ; c1 ; c5 ; c2 ; c6 ; c3) ^ <= c4 ; (c4 ; c1 ; c5 + c5 ; c2 ; c6 + c6 ; c3) ^)
    ]]></literal>
    <justification jstring="[deny(4)].">
      <j1 rule="deny" parents="4"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      -((c4 ; (c1 ; (c5 ; (c2 ; (c6 ; c3))))) ^ <= c4 ; (c6 ; c3 + (c4 ; (c1 ; c5) + c5 ; (c2 ; c6))) ^)
    ]]></literal>
    <justification jstring="[copy(21),rewrite(9(5),9(7),9(6),9(9),9(8),9(7),9(11),9(10),9(9),9(8),9(18),9(23),5(28))].">
      <j1 rule="copy" parents="21"/>
      <j2 rule="rewrite" parents="9 9 9 9 9 9 9 9 9 9 9 9 5"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      x + (y + z) = z + (x + y)
    ]]></literal>
    <justification jstring="[para(7(a,1),5(a,1))].">
      <j1 rule="para" parents="7 5"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      x + (y + z) = y + (x + z)
    ]]></literal>
    <justification jstring="[para(5(a,1),7(a,1,1)),rewrite(7(2))].">
      <j1 rule="para" parents="5 7"/>
      <j2 rule="rewrite" parents="7"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      -((c4 ; (c1 ; (c5 ; (c2 ; (c6 ; c3))))) ^ <= c4 ; (c4 ; (c1 ; c5) + (c5 ; (c2 ; c6) + c6 ; c3)) ^)
    ]]></literal>
    <justification jstring="[back_rewrite(22),rewrite(23(28),5(27),24(28))].">
      <j1 rule="back_rewrite" parents="22"/>
      <j2 rule="rewrite" parents="23 5 24"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      x ; (y ; (z ; u)) ^ <= x ; (y + (z + u)) ^
    ]]></literal>
    <justification jstring="[hyper(10,a,14,a)].">
      <j1 rule="hyper" parents="10 14"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      c4 ; (c4 ; x) ^ = (c4 ; x) ^
    ]]></literal>
    <justification jstring="[hyper(16,a,17,a,b,18,a)].">
      <j1 rule="hyper" parents="16 17 18"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      c5 ; (c5 ; x) = c5 ; x
    ]]></literal>
    <justification jstring="[para(19(a,1),9(a,1,1)),flip(a)].">
      <j1 rule="para" parents="19 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      c6 ; (c6 ; x) = c6 ; x
    ]]></literal>
    <justification jstring="[para(20(a,1),9(a,1,1)),flip(a)].">
      <j1 rule="para" parents="20 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      -((c4 ; (c1 ; (c5 ; (c2 ; (c6 ; c3))))) ^ <= (c4 ; (c1 ; (c5 ; (c2 ; (c6 ; c3))))) ^)
    ]]></literal>
    <justification jstring="[ur(12,b,26,a,c,25,a),rewrite(9(27),9(26),29(25),9(26),9(25),28(24),27(26))].">
      <j1 rule="ur" parents="12 26 25"/>
      <j2 rule="rewrite" parents="9 9 29 9 9 28 27"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(30,a,11,a)].">
      <j1 rule="resolve" parents="30 11"/>
    </justification>
  </clause>

</proof>

</proofs>
