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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>na-ring-3.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 12985 was started by mccune on cleo.thornwood,
Mon Jun 19 16:39:37 2006
The command was "/home/mccune/bin/prover9 -f na-ring-3.in".
]]></heading>

<proof number="1" length="39" max_count="31">

<comments><![CDATA[
% Proof 1 at 0.03 (+ 0.00) seconds.
% Length of proof is 39.
% Level of proof is 11.
% Maximum clause weight is 31.
% Given clauses 52.
]]></comments>

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

  <clause id="11">
    <literal><![CDATA[
      x + 0 = x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      x + - x = 0
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

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

  <clause id="14">
    <literal><![CDATA[
      x + y = y + x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      x * (y + z) = (x * y) + (x * z)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      (x + y) * z = (x * z) + (y * z)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      (x * x) * y = x * (x * y)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      (x * y) * x = x * (y * x)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      (B * A) * A != B * (A * A)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      - (x * (y * z)) + ((x * y) * z) = associator(x,y,z)
    ]]></literal>
    <justification jstring="[copy(10),demod(14(7)),flip(a)].">
      <j1 rule="copy" parents="10"/>
      <j2 rule="demod" parents="14"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      x + (y + - (x + y)) = 0
    ]]></literal>
    <justification jstring="[para(13(a,1),12(a,1))].">
      <j1 rule="para" parents="13 12"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      x + (- x + y) = 0 + y
    ]]></literal>
    <justification jstring="[para(12(a,1),13(a,1,1)),flip(a)].">
      <j1 rule="para" parents="12 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      0 + x = x
    ]]></literal>
    <justification jstring="[para(14(a,1),11(a,1))].">
      <j1 rule="para" parents="14 11"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      x + (y + z) = y + (x + z)
    ]]></literal>
    <justification jstring="[para(14(a,1),13(a,1,1)),demod(13(2))].">
      <j1 rule="para" parents="14 13"/>
      <j2 rule="demod" parents="13"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      x + (- x + y) = y
    ]]></literal>
    <justification jstring="[back_demod(24),demod(25(5))].">
      <j1 rule="back_demod" parents="24"/>
      <j2 rule="demod" parents="25"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      - ((x * (y * z)) + (x * (u * z))) + (((x * y) * z) + ((x * u) * z)) = associator(x,y + u,z)
    ]]></literal>
    <justification jstring="[para(15(a,1),20(a,1,2,1)),demod(16(2),15(4),16(10))].">
      <j1 rule="para" parents="15 20"/>
      <j2 rule="demod" parents="16 15 16"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      - ((x * (y * z)) + (u * (y * z))) + (((x * y) * z) + ((u * y) * z)) = associator(x + u,y,z)
    ]]></literal>
    <justification jstring="[para(16(a,1),20(a,1,1,1)),demod(16(8),16(10))].">
      <j1 rule="para" parents="16 20"/>
      <j2 rule="demod" parents="16 16"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      associator(x,x,y) = 0
    ]]></literal>
    <justification jstring="[para(17(a,1),20(a,1,2)),demod(14(6),12(6)),flip(a)].">
      <j1 rule="para" parents="17 20"/>
      <j2 rule="demod" parents="14 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      associator(x,y,x) = 0
    ]]></literal>
    <justification jstring="[para(18(a,1),20(a,1,2)),demod(14(6),12(6)),flip(a)].">
      <j1 rule="para" parents="18 20"/>
      <j2 rule="demod" parents="14 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="58">
    <literal><![CDATA[
      (0 * x) + (y * x) = y * x
    ]]></literal>
    <justification jstring="[para(25(a,1),16(a,1,1)),flip(a)].">
      <j1 rule="para" parents="25 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      x + (y + - x) = y
    ]]></literal>
    <justification jstring="[para(12(a,1),26(a,1,2)),demod(11(2)),flip(a)].">
      <j1 rule="para" parents="12 26"/>
      <j2 rule="demod" parents="11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      x + (y + (z + - (x + z))) = y
    ]]></literal>
    <justification jstring="[para(23(a,1),26(a,1,2)),demod(11(2)),flip(a)].">
      <j1 rule="para" parents="23 26"/>
      <j2 rule="demod" parents="11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="63">
    <literal><![CDATA[
      - - x = x
    ]]></literal>
    <justification jstring="[para(12(a,1),27(a,1,2)),demod(11(2)),flip(a)].">
      <j1 rule="para" parents="12 27"/>
      <j2 rule="demod" parents="11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      (x * y) * z = associator(x,y,z) + (x * (y * z))
    ]]></literal>
    <justification jstring="[para(20(a,1),27(a,1,2)),demod(14(4)),flip(a)].">
      <j1 rule="para" parents="20 27"/>
      <j2 rule="demod" parents="14"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="73">
    <literal><![CDATA[
      associator(x,y,z) + associator(u,y,z) = associator(x + u,y,z)
    ]]></literal>
    <justification jstring="[back_demod(43),demod(66(8),66(12),13(15),14(16),13(16),13(15),13(14),62(15))].">
      <j1 rule="back_demod" parents="43"/>
      <j2 rule="demod" parents="66 66 13 14 13 13 13 62"/>
    </justification>
  </clause>

  <clause id="75">
    <literal><![CDATA[
      associator(x,y,z) + associator(x,u,z) = associator(x,y + u,z)
    ]]></literal>
    <justification jstring="[back_demod(41),demod(66(8),66(12),13(15),14(16),13(16),13(15),13(14),62(15))].">
      <j1 rule="back_demod" parents="41"/>
      <j2 rule="demod" parents="66 66 13 14 13 13 13 62"/>
    </justification>
  </clause>

  <clause id="83">
    <literal><![CDATA[
      associator(B,A,A) + (B * (A * A)) != B * (A * A)
    ]]></literal>
    <justification jstring="[back_demod(19),demod(66(5))].">
      <j1 rule="back_demod" parents="19"/>
      <j2 rule="demod" parents="66"/>
    </justification>
  </clause>

  <clause id="88">
    <literal><![CDATA[
      - x + (y + x) = y
    ]]></literal>
    <justification jstring="[para(63(a,1),59(a,1,2,2))].">
      <j1 rule="para" parents="63 59"/>
    </justification>
  </clause>

  <clause id="152">
    <literal><![CDATA[
      0 * x = 0
    ]]></literal>
    <justification jstring="[para(58(a,1),88(a,1,2)),demod(14(4),12(4)),flip(a)].">
      <j1 rule="para" parents="58 88"/>
      <j2 rule="demod" parents="14 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="167">
    <literal><![CDATA[
      associator(0,x,y) = 0
    ]]></literal>
    <justification jstring="[para(152(a,1),66(a,1,1)),demod(152(2),152(6),14(5),25(5)),flip(a)].">
      <j1 rule="para" parents="152 66"/>
      <j2 rule="demod" parents="152 152 14 25"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="200">
    <literal><![CDATA[
      associator(x + y,x,z) = associator(y,x,z)
    ]]></literal>
    <justification jstring="[para(46(a,1),73(a,1,1)),demod(25(3)),flip(a)].">
      <j1 rule="para" parents="46 73"/>
      <j2 rule="demod" parents="25"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="213">
    <literal><![CDATA[
      associator(- x,x,y) = 0
    ]]></literal>
    <justification jstring="[para(12(a,1),200(a,1,1)),demod(167(2)),flip(a)].">
      <j1 rule="para" parents="12 200"/>
      <j2 rule="demod" parents="167"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="215">
    <literal><![CDATA[
      associator(x,y,y + x) = 0
    ]]></literal>
    <justification jstring="[para(200(a,1),50(a,1))].">
      <j1 rule="para" parents="200 50"/>
    </justification>
  </clause>

  <clause id="226">
    <literal><![CDATA[
      associator(x,- x,y) = 0
    ]]></literal>
    <justification jstring="[para(63(a,1),213(a,1,1))].">
      <j1 rule="para" parents="63 213"/>
    </justification>
  </clause>

  <clause id="233">
    <literal><![CDATA[
      associator(x,y,x + y) = 0
    ]]></literal>
    <justification jstring="[para(14(a,1),215(a,1,3))].">
      <j1 rule="para" parents="14 215"/>
    </justification>
  </clause>

  <clause id="259">
    <literal><![CDATA[
      associator(x,- x + y,z) = associator(x,y,z)
    ]]></literal>
    <justification jstring="[para(226(a,1),75(a,1,1)),demod(25(3)),flip(a)].">
      <j1 rule="para" parents="226 75"/>
      <j2 rule="demod" parents="25"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="269">
    <literal><![CDATA[
      associator(x,y,y) = 0
    ]]></literal>
    <justification jstring="[para(27(a,1),233(a,1,3)),demod(259(3))].">
      <j1 rule="para" parents="27 233"/>
      <j2 rule="demod" parents="259"/>
    </justification>
  </clause>

  <clause id="281">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[back_demod(83),demod(269(4),25(7)),xx(a)].">
      <j1 rule="back_demod" parents="83"/>
      <j2 rule="demod" parents="269 25"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
