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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="8">

<source>ALG043+1.out</source>

<heading><![CDATA[
FOF-Prover9 (32) version 22-May-2007, May 2007.
Process 27301 was started by mccune on cleo,
Tue May 22 14:45:11 2007
The command was "/home/mccune/bin/fof-prover9 -f ALG043+1.in".
]]></heading>

<proof number="1" length="4" max_count="5">

<comments><![CDATA[
% Proof 1 at 0.00 (+ 0.00) seconds.
% Length of proof is 4.
% Level of proof is 1.
% Maximum clause weight is 5.
% Given clauses 0.
]]></comments>

  <clause id="7" type="assumption">
    <literal><![CDATA[
      op(e0,e0) = e0
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="23" type="assumption">
    <literal><![CDATA[
      unit = e0
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="24" type="assumption">
    <literal><![CDATA[
      op(unit,e0) != e0
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[copy(24),rewrite([23(1),7(3)]),xx(a)].">
      <j1 rule="copy" parents="24"/>
      <j2 rule="rewrite" parents="23 7"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

<proof number="2" length="4" max_count="5">

<comments><![CDATA[
% Proof 1 at 0.00 (+ 0.00) seconds.
% Length of proof is 4.
% Level of proof is 1.
% Maximum clause weight is 5.
% Given clauses 0.
]]></comments>

  <clause id="7" type="assumption">
    <literal><![CDATA[
      op(e0,e0) = e0
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="23" type="assumption">
    <literal><![CDATA[
      unit = e0
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="24" type="assumption">
    <literal><![CDATA[
      op(e0,unit) != e0
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[copy(24),rewrite([23(2),7(3)]),xx(a)].">
      <j1 rule="copy" parents="24"/>
      <j2 rule="rewrite" parents="23 7"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

<proof number="3" length="4" max_count="5">

<comments><![CDATA[
% Proof 1 at 0.00 (+ 0.00) seconds.
% Length of proof is 4.
% Level of proof is 1.
% Maximum clause weight is 5.
% Given clauses 0.
]]></comments>

  <clause id="8" type="assumption">
    <literal><![CDATA[
      op(e0,e1) = e1
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="23" type="assumption">
    <literal><![CDATA[
      unit = e0
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="24" type="assumption">
    <literal><![CDATA[
      op(unit,e1) != e1
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[copy(24),rewrite([23(1),8(3)]),xx(a)].">
      <j1 rule="copy" parents="24"/>
      <j2 rule="rewrite" parents="23 8"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

<proof number="4" length="4" max_count="5">

<comments><![CDATA[
% Proof 1 at 0.00 (+ 0.00) seconds.
% Length of proof is 4.
% Level of proof is 1.
% Maximum clause weight is 5.
% Given clauses 0.
]]></comments>

  <clause id="11" type="assumption">
    <literal><![CDATA[
      op(e1,e0) = e1
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="23" type="assumption">
    <literal><![CDATA[
      unit = e0
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="24" type="assumption">
    <literal><![CDATA[
      op(e1,unit) != e1
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[copy(24),rewrite([23(2),11(3)]),xx(a)].">
      <j1 rule="copy" parents="24"/>
      <j2 rule="rewrite" parents="23 11"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

<proof number="5" length="4" max_count="5">

<comments><![CDATA[
% Proof 1 at 0.00 (+ 0.00) seconds.
% Length of proof is 4.
% Level of proof is 1.
% Maximum clause weight is 5.
% Given clauses 0.
]]></comments>

  <clause id="9" type="assumption">
    <literal><![CDATA[
      op(e0,e2) = e2
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="23" type="assumption">
    <literal><![CDATA[
      unit = e0
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="24" type="assumption">
    <literal><![CDATA[
      op(unit,e2) != e2
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[copy(24),rewrite([23(1),9(3)]),xx(a)].">
      <j1 rule="copy" parents="24"/>
      <j2 rule="rewrite" parents="23 9"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

<proof number="6" length="4" max_count="5">

<comments><![CDATA[
% Proof 1 at 0.00 (+ 0.00) seconds.
% Length of proof is 4.
% Level of proof is 1.
% Maximum clause weight is 5.
% Given clauses 0.
]]></comments>

  <clause id="15" type="assumption">
    <literal><![CDATA[
      op(e2,e0) = e2
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="23" type="assumption">
    <literal><![CDATA[
      unit = e0
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="24" type="assumption">
    <literal><![CDATA[
      op(e2,unit) != e2
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[copy(24),rewrite([23(2),15(3)]),xx(a)].">
      <j1 rule="copy" parents="24"/>
      <j2 rule="rewrite" parents="23 15"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

<proof number="7" length="4" max_count="5">

<comments><![CDATA[
% Proof 1 at 0.00 (+ 0.00) seconds.
% Length of proof is 4.
% Level of proof is 1.
% Maximum clause weight is 5.
% Given clauses 0.
]]></comments>

  <clause id="10" type="assumption">
    <literal><![CDATA[
      op(e0,e3) = e3
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="23" type="assumption">
    <literal><![CDATA[
      unit = e0
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="24" type="assumption">
    <literal><![CDATA[
      op(unit,e3) != e3
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[copy(24),rewrite([23(1),10(3)]),xx(a)].">
      <j1 rule="copy" parents="24"/>
      <j2 rule="rewrite" parents="23 10"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

<proof number="8" length="4" max_count="5">

<comments><![CDATA[
% Proof 1 at 0.00 (+ 0.00) seconds.
% Length of proof is 4.
% Level of proof is 1.
% Maximum clause weight is 5.
% Given clauses 0.
]]></comments>

  <clause id="19" type="assumption">
    <literal><![CDATA[
      op(e3,e0) = e3
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="23" type="assumption">
    <literal><![CDATA[
      unit = e0
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="24" type="assumption">
    <literal><![CDATA[
      op(e3,unit) != e3
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[copy(24),rewrite([23(2),19(3)]),xx(a)].">
      <j1 rule="copy" parents="24"/>
      <j2 rule="rewrite" parents="23 19"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
