<?xml version="1.0" encoding="ISO-8859-1"?>

<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>eq3iv.out</source>

<heading><![CDATA[
Prover9 (32) version April-2007, April 2007.
Process 14728 was started by hoefnepe on europa,
Fri Sep 21 17:40:10 2007
The command was "../../../kleene/Prover9/bin/prover9 -f eq3iv.in".
]]></heading>

<proof number="1" length="29" max_count="31">

<comments><![CDATA[
% Proof 1 at 1.01 (+ 0.01) seconds.
% Length of proof is 29.
% Level of proof is 7.
% Maximum clause weight is 28.
% Given clauses 270.
]]></comments>

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

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

  <clause id="4" type="goal">
    <literal><![CDATA[
      (all p (p + 1 = 1 & p ; p = p -> (p ; x) ^ + (p ; x ; p) ^ = (p ; x) ^))
    ]]></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="clausify">
    <literal><![CDATA[
      x ; 1 = x
    ]]></literal>
    <justification jstring="[clausify(1)].">
      <j1 rule="clausify" parents="1"/>
    </justification>
  </clause>

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

  <clause id="10">
    <literal><![CDATA[
      x ; y ; z = x ; (y ; z)
    ]]></literal>
    <justification jstring="[copy(9),flip(a)].">
      <j1 rule="copy" parents="9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

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

  <clause id="12" type="assumption">
    <literal><![CDATA[
      (x + y) ; z = x ; z + y ; z
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="13" type="clausify">
    <literal><![CDATA[
      x + y != y
    ]]></literal>
    <literal><![CDATA[
      z ; y = z ; x + z ; y
    ]]></literal>
    <justification jstring="[clausify(2)].">
      <j1 rule="clausify" parents="2"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      x + y != y
    ]]></literal>
    <literal><![CDATA[
      z ; x + z ; y = z ; y
    ]]></literal>
    <justification jstring="[copy(13),flip(b)].">
      <j1 rule="copy" parents="13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="15" type="assumption">
    <literal><![CDATA[
      x ; x ^ = x ^
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="16" type="clausify">
    <literal><![CDATA[
      x ; y != y + x ; y
    ]]></literal>
    <literal><![CDATA[
      x ^ = y + x ^
    ]]></literal>
    <justification jstring="[clausify(3)].">
      <j1 rule="clausify" parents="3"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      x + y ; x != y ; x
    ]]></literal>
    <literal><![CDATA[
      x + y ^ = y ^
    ]]></literal>
    <justification jstring="[copy(16),flip(a),flip(b)].">
      <j1 rule="copy" parents="16"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18" type="deny">
    <literal><![CDATA[
      c2 + 1 = 1
    ]]></literal>
    <justification jstring="[deny(4)].">
      <j1 rule="deny" parents="4"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      1 + c2 = 1
    ]]></literal>
    <justification jstring="[copy(18),rewrite(5(3))].">
      <j1 rule="copy" parents="18"/>
      <j2 rule="rewrite" parents="5"/>
    </justification>
  </clause>

  <clause id="20" type="deny">
    <literal><![CDATA[
      (c2 ; c1) ^ != (c2 ; c1) ^ + (c2 ; c1 ; c2) ^
    ]]></literal>
    <justification jstring="[deny(4)].">
      <j1 rule="deny" parents="4"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      (c2 ; c1) ^ + (c2 ; (c1 ; c2)) ^ != (c2 ; c1) ^
    ]]></literal>
    <justification jstring="[copy(20),rewrite(10(13)),flip(a)].">
      <j1 rule="copy" parents="20"/>
      <j2 rule="rewrite" parents="10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      x + (x + y) = x + y
    ]]></literal>
    <justification jstring="[para(11(a,1),7(a,1,1)),flip(a)].">
      <j1 rule="para" parents="11 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      x + y != x
    ]]></literal>
    <literal><![CDATA[
      z ; y + z ; x = z ; x
    ]]></literal>
    <justification jstring="[para(5(a,1),14(a,1))].">
      <j1 rule="para" parents="5 14"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      x + (y ; x + z ; x) != y ; x + z ; x
    ]]></literal>
    <literal><![CDATA[
      x + (y + z) ^ = (y + z) ^
    ]]></literal>
    <justification jstring="[para(12(a,1),17(a,1,2)),rewrite(12(6))].">
      <j1 rule="para" parents="12 17"/>
      <j2 rule="rewrite" parents="12"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      x + x ; c2 = x
    ]]></literal>
    <justification jstring="[hyper(23,a,19,a),rewrite(8(4),5(3),8(5))].">
      <j1 rule="hyper" parents="23 19"/>
      <j2 rule="rewrite" parents="8 5 8"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      x ^ + (y + x) ^ = (y + x) ^
    ]]></literal>
    <justification jstring="[para(15(a,1),24(a,1,2,2)),rewrite(5(5),22(6),15(8),5(8)),xx(a)].">
      <j1 rule="para" parents="15 24"/>
      <j2 rule="rewrite" parents="5 22 15 5"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      x ^ + (x ; c2) ^ = x ^
    ]]></literal>
    <justification jstring="[para(25(a,1),26(a,1,2,1)),rewrite(5(5),25(8))].">
      <j1 rule="para" parents="25 26"/>
      <j2 rule="rewrite" parents="5 25"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      (x ; y) ^ + (x ; (y ; c2)) ^ = (x ; y) ^
    ]]></literal>
    <justification jstring="[para(10(a,1),27(a,1,2,1))].">
      <j1 rule="para" parents="10 27"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(28,a,21,a)].">
      <j1 rule="resolve" parents="28 21"/>
    </justification>
  </clause>

</proof>

</proofs>
