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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>qg1.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13017 was started by mccune on cleo.thornwood,
Mon Jun 19 16:40:19 2006
The command was "/home/mccune/bin/prover9 -f qg1.in".
]]></heading>

<proof number="1" length="25" max_count="15">

<comments><![CDATA[
% Proof 1 at 0.01 (+ 0.00) seconds.
% Length of proof is 25.
% Level of proof is 12.
% Maximum clause weight is 15.
% Given clauses 27.
]]></comments>

  <clause id="7">
    <literal><![CDATA[
      x * (x \ y) = y
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      x \ (x * y) = y
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      (x / y) * y = x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      (x * y) / y = x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      x * (y * (y * x)) = y * x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      c2 * c1 != c1 * (c1 * c2)
    ]]></literal>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      (x / y) \ x = y
    ]]></literal>
    <justification jstring="[para(9(a,1),8(a,1,2))].">
      <j1 rule="para" parents="9 8"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      (x \ y) * (x * y) = y
    ]]></literal>
    <justification jstring="[para(7(a,1),11(a,1,2,2)),demod(7(5))].">
      <j1 rule="para" parents="7 11"/>
      <j2 rule="demod" parents="7"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      x * ((y / x) * y) = y
    ]]></literal>
    <justification jstring="[para(9(a,1),11(a,1,2,2)),demod(9(5))].">
      <j1 rule="para" parents="9 11"/>
      <j2 rule="demod" parents="9"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      (x * y) / (x * (x * y)) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),10(a,1,1))].">
      <j1 rule="para" parents="11 10"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      (x * (x * y)) * (y * (x * y)) = x * y
    ]]></literal>
    <justification jstring="[para(11(a,1),11(a,1,2,2)),demod(11(8))].">
      <j1 rule="para" parents="11 11"/>
      <j2 rule="demod" parents="11"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      x \ y = y / (x * y)
    ]]></literal>
    <justification jstring="[para(15(a,1),10(a,1,1)),flip(a)].">
      <j1 rule="para" parents="15 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      x / ((x / y) * x) = y
    ]]></literal>
    <justification jstring="[back_demod(13),demod(24(2))].">
      <j1 rule="back_demod" parents="13"/>
      <j2 rule="demod" parents="24"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      ((x / y) * x) * (y * x) = x
    ]]></literal>
    <justification jstring="[para(17(a,1),11(a,1,2,2)),demod(17(7))].">
      <j1 rule="para" parents="17 11"/>
      <j2 rule="demod" parents="17"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      x / (y * x) = (x / y) * x
    ]]></literal>
    <justification jstring="[para(17(a,1),18(a,1,1)),demod(17(3))].">
      <j1 rule="para" parents="17 18"/>
      <j2 rule="demod" parents="17"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      (x / (x / y)) * x = y
    ]]></literal>
    <justification jstring="[back_demod(29),demod(34(3))].">
      <j1 rule="back_demod" parents="29"/>
      <j2 rule="demod" parents="34"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      x / (x / y) = y / x
    ]]></literal>
    <justification jstring="[para(35(a,1),10(a,1,1)),flip(a)].">
      <j1 rule="para" parents="35 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      (x / y) * (y / x) = y
    ]]></literal>
    <justification jstring="[para(38(a,1),9(a,1,1))].">
      <j1 rule="para" parents="38 9"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      (x * y) / x = (y / x) * y
    ]]></literal>
    <justification jstring="[para(10(a,1),38(a,1,2)),demod(34(4))].">
      <j1 rule="para" parents="10 38"/>
      <j2 rule="demod" parents="34"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      ((x / y) * x) * y = y * x
    ]]></literal>
    <justification jstring="[para(10(a,1),44(a,1,2)),demod(34(2))].">
      <j1 rule="para" parents="10 44"/>
      <j2 rule="demod" parents="34"/>
    </justification>
  </clause>

  <clause id="98">
    <literal><![CDATA[
      (x / y) * x = y * (x / y)
    ]]></literal>
    <justification jstring="[para(35(a,1),53(a,1,1)),flip(a)].">
      <j1 rule="para" parents="35 53"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="101">
    <literal><![CDATA[
      (x * (y / x)) * x = x * y
    ]]></literal>
    <justification jstring="[para(53(a,1),19(a,2)),demod(98(2),98(4),98(8),19(11))].">
      <j1 rule="para" parents="53 19"/>
      <j2 rule="demod" parents="98 98 98 19"/>
    </justification>
  </clause>

  <clause id="104">
    <literal><![CDATA[
      (x * (y / x)) * (x * y) = y
    ]]></literal>
    <justification jstring="[para(32(a,1),53(a,2)),demod(98(3),98(6),98(9),101(10))].">
      <j1 rule="para" parents="32 53"/>
      <j2 rule="demod" parents="98 98 98 101"/>
    </justification>
  </clause>

  <clause id="108">
    <literal><![CDATA[
      x * (x * y) = y * x
    ]]></literal>
    <justification jstring="[para(45(a,1),53(a,1,1,1)),demod(98(2),104(4)),flip(a)].">
      <j1 rule="para" parents="45 53"/>
      <j2 rule="demod" parents="98 104"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="109">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(108,a,12,a(flip))].">
      <j1 rule="resolve" parents="108 12"/>
    </justification>
  </clause>

</proof>

</proofs>
