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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>RBA-2.out</source>

<heading><![CDATA[
Prover9 (32) version 2009-02A, February 2009.
Process 11191 was started by mccune on cleo,
Wed Feb 25 09:32:17 2009
The command was "/home/mccune/bin/prover9 -f RBA-2.in".
]]></heading>

<proof number="1" length="30" max_count="23">

<comments><![CDATA[
% Proof 1 at 0.16 (+ 0.01) seconds: Huntington.
% Length of proof is 30.
% Level of proof is 15.
% Maximum clause weight is 23.
% Given clauses 48.

]]></comments>

  <clause id="1" type="assumption">
    <literal><![CDATA[
      (exists c c + c = c)
    ]]></literal>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="2" type="goal">
    <literal><![CDATA[
      (x + y')' + (x' + y')' = y
    ]]></literal>
    <attribute><![CDATA[
      answer(Huntington)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

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

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

  <clause id="5" type="assumption">
    <literal><![CDATA[
      ((x + y)' + (x + y')')' = x
    ]]></literal>
    <attribute><![CDATA[
      label(Robbins)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6" type="clausify">
    <literal><![CDATA[
      c1 + c1 = c1
    ]]></literal>
    <justification jstring="[clausify(1)].">
      <j1 rule="clausify" parents="1"/>
    </justification>
  </clause>

  <clause id="7" type="deny">
    <literal><![CDATA[
      (c2 + c3')' + (c2' + c3')' != c3
    ]]></literal>
    <attribute><![CDATA[
      answer(Huntington)
    ]]></attribute>
    <justification jstring="[deny(2)].">
      <j1 rule="deny" parents="2"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      x + (y + z) = y + (x + z)
    ]]></literal>
    <justification jstring="[para(3(a,1),4(a,1,1)),rewrite([4(2)])].">
      <j1 rule="para" parents="3 4"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      ((x + y)' + (y + x')')' = y
    ]]></literal>
    <justification jstring="[para(3(a,1),5(a,1,1,1,1))].">
      <j1 rule="para" parents="3 5"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      ((x + y)' + (y' + x)')' = x
    ]]></literal>
    <justification jstring="[para(3(a,1),5(a,1,1,2,1))].">
      <j1 rule="para" parents="3 5"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      ((x + y)' + (x + ((y + z)' + (y + z')'))')' = x
    ]]></literal>
    <justification jstring="[para(5(a,1),5(a,1,1,2,1,2)),rewrite([3(11)])].">
      <j1 rule="para" parents="5 5"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      c1 + (c1 + x) = c1 + x
    ]]></literal>
    <justification jstring="[para(6(a,1),4(a,1,1)),flip(a)].">
      <j1 rule="para" parents="6 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      (c1' + (c1 + c1')')' = c1
    ]]></literal>
    <justification jstring="[para(6(a,1),5(a,1,1,1,1))].">
      <j1 rule="para" parents="6 5"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      ((x + c1)' + (x + (c1' + (c1 + c1')'))')' = x
    ]]></literal>
    <justification jstring="[para(18(a,1),5(a,1,1,2,1,2)),rewrite([3(14)])].">
      <j1 rule="para" parents="18 5"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      (c1 + (c1 + (c1' + c1'))')' = c1'
    ]]></literal>
    <justification jstring="[para(18(a,1),5(a,1,1,2)),rewrite([9(7),3(10)])].">
      <j1 rule="para" parents="18 5"/>
      <j2 rule="rewrite" parents="9 3"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      ((x + y)' + (x' + y)')' = y
    ]]></literal>
    <justification jstring="[para(3(a,1),10(a,1,1,2,1))].">
      <j1 rule="para" parents="3 10"/>
    </justification>
  </clause>

  <clause id="37">
    <literal><![CDATA[
      ((c1 + x)' + (c1 + (x + c1'))')' = c1 + x
    ]]></literal>
    <justification jstring="[para(16(a,1),10(a,1,1,1,1)),rewrite([4(8)])].">
      <j1 rule="para" parents="16 10"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="75">
    <literal><![CDATA[
      ((c1 + x)' + (c1' + ((c1 + c1')' + x))')' = x
    ]]></literal>
    <justification jstring="[para(18(a,1),27(a,1,1,2,1,1)),rewrite([4(9),3(14)])].">
      <j1 rule="para" parents="18 27"/>
      <j2 rule="rewrite" parents="4 3"/>
    </justification>
  </clause>

  <clause id="146">
    <literal><![CDATA[
      (c1' + (c1 + (c1' + c1'))')' = c1
    ]]></literal>
    <justification jstring="[para(26(a,1),5(a,1,1,2)),rewrite([16(9),3(11)])].">
      <j1 rule="para" parents="26 5"/>
      <j2 rule="rewrite" parents="16 3"/>
    </justification>
  </clause>

  <clause id="147">
    <literal><![CDATA[
      (c1 + (c1' + c1'))' = (c1 + c1')'
    ]]></literal>
    <justification jstring="[para(26(a,1),10(a,1,1,1)),rewrite([3(13),146(14),3(4)]),flip(a)].">
      <j1 rule="para" parents="26 10"/>
      <j2 rule="rewrite" parents="3 146 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="156">
    <literal><![CDATA[
      (c1 + (c1 + c1')')' = c1'
    ]]></literal>
    <justification jstring="[back_rewrite(26),rewrite([147(9)])].">
      <j1 rule="back_rewrite" parents="26"/>
      <j2 rule="rewrite" parents="147"/>
    </justification>
  </clause>

  <clause id="535">
    <literal><![CDATA[
      (c1' + (c1 + (c1' + (c1 + c1')'))')' = c1 + (c1 + c1')'
    ]]></literal>
    <justification jstring="[para(156(a,1),37(a,1,1,1)),rewrite([3(11)])].">
      <j1 rule="para" parents="156 37"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="648">
    <literal><![CDATA[
      c1 + (c1 + c1')' = c1
    ]]></literal>
    <justification jstring="[para(6(a,1),25(a,1,1,1,1)),rewrite([535(15)])].">
      <j1 rule="para" parents="6 25"/>
      <j2 rule="rewrite" parents="535"/>
    </justification>
  </clause>

  <clause id="669">
    <literal><![CDATA[
      c1 + ((c1 + c1')' + x) = c1 + x
    ]]></literal>
    <justification jstring="[para(648(a,1),4(a,1,1)),flip(a)].">
      <j1 rule="para" parents="648 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="678">
    <literal><![CDATA[
      (c1 + c1')' + x = x
    ]]></literal>
    <justification jstring="[para(669(a,1),10(a,1,1,1,1)),rewrite([3(12),75(15)]),flip(a)].">
      <j1 rule="para" parents="669 10"/>
      <j2 rule="rewrite" parents="3 75"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="705">
    <literal><![CDATA[
      (c1 + c1')' = (x + x')'
    ]]></literal>
    <justification jstring="[para(678(a,1),14(a,1,1,1,1)),rewrite([678(13),5(8),3(2)]),flip(a)].">
      <j1 rule="para" parents="678 14"/>
      <j2 rule="rewrite" parents="678 5 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="867">
    <literal><![CDATA[
      (x + x'')'' = x
    ]]></literal>
    <justification jstring="[para(705(a,2),5(a,1,1,1)),rewrite([678(10)])].">
      <j1 rule="para" parents="705 5"/>
      <j2 rule="rewrite" parents="678"/>
    </justification>
  </clause>

  <clause id="875">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[para(705(a,2),11(a,1,1,2)),rewrite([3(3),3(10),678(10),867(5)]),flip(a)].">
      <j1 rule="para" parents="705 11"/>
      <j2 rule="rewrite" parents="3 3 678 867"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1014">
    <literal><![CDATA[
      (x + y)' + (x' + y)' = y'
    ]]></literal>
    <justification jstring="[para(27(a,1),875(a,1,1)),flip(a)].">
      <j1 rule="para" parents="27 875"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1029">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(Huntington)
    ]]></attribute>
    <justification jstring="[back_rewrite(7),rewrite([1014(12),875(3)]),xx(a)].">
      <j1 rule="back_rewrite" parents="7"/>
      <j2 rule="rewrite" parents="1014 875"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
