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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>5.1.out</source>

<heading><![CDATA[
Prover9 (64) version 2008-06A, June 2008.
Process 23416 was started by veroff on pengy,
Tue Jul  8 08:19:41 2008
The command was "prover9 -f 5.1.in".
]]></heading>

<proof number="1" length="24" max_count="16">

<comments><![CDATA[
% Proof 1 at 0.03 (+ 0.01) seconds.
% Length of proof is 24.
% Level of proof is 5.
% Maximum clause weight is 16.
% Given clauses 30.
]]></comments>

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

  <clause id="2" type="assumption">
    <literal><![CDATA[
      P((x => y) => ((y => z) => (x => z)))
    ]]></literal>
    <attribute><![CDATA[
      label("(A1)")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7" type="assumption">
    <literal><![CDATA[
      P((x ^ y) => x)
    ]]></literal>
    <attribute><![CDATA[
      label("(A6)")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="8" type="assumption">
    <literal><![CDATA[
      P((x ^ y) => y)
    ]]></literal>
    <attribute><![CDATA[
      label("(A7)")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="9" type="assumption">
    <literal><![CDATA[
      P((x => y) => ((x => z) => (x => (y ^ z))))
    ]]></literal>
    <attribute><![CDATA[
      label("(A8)")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="10" type="assumption">
    <literal><![CDATA[
      P(x => (x v y))
    ]]></literal>
    <attribute><![CDATA[
      label("(A9)")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11" type="assumption">
    <literal><![CDATA[
      P(x => (y v x))
    ]]></literal>
    <attribute><![CDATA[
      label("(A10)")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="12" type="assumption">
    <literal><![CDATA[
      P((x => y) => ((z => y) => ((x v z) => y)))
    ]]></literal>
    <attribute><![CDATA[
      label("(A11)")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="15" type="assumption">
    <literal><![CDATA[
      P(A => B)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="17" type="assumption">
    <literal><![CDATA[
      P(C => D)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="19" type="assumption">
    <literal><![CDATA[
      -P((A ^ C) => (B ^ D))
    ]]></literal>
    <literal><![CDATA[
      -P((A v C) => (B v D))
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      P((x => y) => ((x ^ z) => y))
    ]]></literal>
    <justification jstring="[hyper(1,a,2,a,b,7,a)].">
      <j1 rule="hyper" parents="1 2 7"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      P((x => y) => ((z ^ x) => y))
    ]]></literal>
    <justification jstring="[hyper(1,a,2,a,b,8,a)].">
      <j1 rule="hyper" parents="1 2 8"/>
    </justification>
  </clause>

  <clause id="132">
    <literal><![CDATA[
      P((B => x) => (A => x))
    ]]></literal>
    <justification jstring="[hyper(1,a,2,a,b,15,a)].">
      <j1 rule="hyper" parents="1 2 15"/>
    </justification>
  </clause>

  <clause id="146">
    <literal><![CDATA[
      P((D => x) => (C => x))
    ]]></literal>
    <justification jstring="[hyper(1,a,2,a,b,17,a)].">
      <j1 rule="hyper" parents="1 2 17"/>
    </justification>
  </clause>

  <clause id="165">
    <literal><![CDATA[
      P(A => (B v x))
    ]]></literal>
    <justification jstring="[hyper(1,a,132,a,b,10,a)].">
      <j1 rule="hyper" parents="1 132 10"/>
    </justification>
  </clause>

  <clause id="168">
    <literal><![CDATA[
      P((x => (B v y)) => ((A v x) => (B v y)))
    ]]></literal>
    <justification jstring="[hyper(1,a,12,a,b,165,a)].">
      <j1 rule="hyper" parents="1 12 165"/>
    </justification>
  </clause>

  <clause id="185">
    <literal><![CDATA[
      P(C => (x v D))
    ]]></literal>
    <justification jstring="[hyper(1,a,146,a,b,11,a)].">
      <j1 rule="hyper" parents="1 146 11"/>
    </justification>
  </clause>

  <clause id="213">
    <literal><![CDATA[
      P((A ^ x) => B)
    ]]></literal>
    <justification jstring="[hyper(1,a,49,a,b,15,a)].">
      <j1 rule="hyper" parents="1 49 15"/>
    </justification>
  </clause>

  <clause id="230">
    <literal><![CDATA[
      P(((A ^ x) => y) => ((A ^ x) => (B ^ y)))
    ]]></literal>
    <justification jstring="[hyper(1,a,9,a,b,213,a)].">
      <j1 rule="hyper" parents="1 9 213"/>
    </justification>
  </clause>

  <clause id="252">
    <literal><![CDATA[
      P((x ^ C) => D)
    ]]></literal>
    <justification jstring="[hyper(1,a,54,a,b,17,a)].">
      <j1 rule="hyper" parents="1 54 17"/>
    </justification>
  </clause>

  <clause id="287">
    <literal><![CDATA[
      P((A v C) => (B v D))
    ]]></literal>
    <justification jstring="[hyper(1,a,168,a,b,185,a)].">
      <j1 rule="hyper" parents="1 168 185"/>
    </justification>
  </clause>

  <clause id="315">
    <literal><![CDATA[
      P((A ^ C) => (B ^ D))
    ]]></literal>
    <justification jstring="[hyper(1,a,230,a,b,252,a)].">
      <j1 rule="hyper" parents="1 230 252"/>
    </justification>
  </clause>

  <clause id="323">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[hyper(19,a,315,a,b,287,a)].">
      <j1 rule="hyper" parents="19 315 287"/>
    </justification>
  </clause>

</proof>

</proofs>
