<?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>na-ring-1.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 12983 was started by mccune on cleo.thornwood,
Mon Jun 19 16:39:33 2006
The command was "/home/mccune/bin/prover9 -f na-ring-1.in".
]]></heading>

<proof number="1" length="28" max_count="48">

<comments><![CDATA[
% Proof 1 at 3.32 (+ 0.01) seconds.
% Length of proof is 28.
% Level of proof is 13.
% Maximum clause weight is 48.
% Given clauses 84.
]]></comments>

  <clause id="10">
    <literal><![CDATA[
      x + 0 = x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

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

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

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

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

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

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

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

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

  <clause id="22">
    <literal><![CDATA[
      x + (- x + y) = 0 + y
    ]]></literal>
    <justification jstring="[para(11(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="11 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      0 + x = x
    ]]></literal>
    <justification jstring="[para(13(a,1),10(a,1))].">
      <j1 rule="para" parents="13 10"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      x + (y + z) = y + (x + z)
    ]]></literal>
    <justification jstring="[para(13(a,1),12(a,1,1)),demod(12(2))].">
      <j1 rule="para" parents="13 12"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      x + (- x + y) = y
    ]]></literal>
    <justification jstring="[back_demod(22),demod(23(5))].">
      <j1 rule="back_demod" parents="22"/>
      <j2 rule="demod" parents="23"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      (x * (x * y)) + ((z * (z * y)) + (((x * z) * y) + ((z * x) * y))) = (x * (x * y)) + ((z * (x * y)) + ((x * (z * y)) + (z * (z * y))))
    ]]></literal>
    <justification jstring="[para(15(a,1),16(a,1,1)),demod(14(2),14(5),12(7),15(8),16(2),15(8),15(8),16(8),13(9),24(10),15(14),14(16),15(14),15(19),12(22))].">
      <j1 rule="para" parents="15 16"/>
      <j2 rule="demod" parents="14 14 12 15 16 15 15 16 13 24 15 14 15 15 12"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      - - x = x
    ]]></literal>
    <justification jstring="[para(11(a,1),25(a,1,2)),demod(10(2)),flip(a)].">
      <j1 rule="para" parents="11 25"/>
      <j2 rule="demod" parents="10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      x + (y + - x) = y
    ]]></literal>
    <justification jstring="[para(13(a,1),25(a,1,2))].">
      <j1 rule="para" parents="13 25"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      - x + (y + x) = y
    ]]></literal>
    <justification jstring="[para(44(a,1),45(a,1,2,2))].">
      <j1 rule="para" parents="44 45"/>
    </justification>
  </clause>

  <clause id="57">
    <literal><![CDATA[
      - x + (y + (x + z)) = y + z
    ]]></literal>
    <justification jstring="[para(54(a,1),12(a,1,1)),demod(12(4)),flip(a)].">
      <j1 rule="para" parents="54 12"/>
      <j2 rule="demod" parents="12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="58">
    <literal><![CDATA[
      - x + (y + (z + x)) = y + z
    ]]></literal>
    <justification jstring="[para(12(a,1),54(a,1,2))].">
      <j1 rule="para" parents="12 54"/>
    </justification>
  </clause>

  <clause id="224">
    <literal><![CDATA[
      (x * (x * y)) + (((x * z) * y) + ((z * x) * y)) = - (z * (z * y)) + ((x * (x * y)) + ((z * (x * y)) + ((x * (z * y)) + (z * (z * y)))))
    ]]></literal>
    <justification jstring="[para(31(a,1),57(a,1,2)),flip(a)].">
      <j1 rule="para" parents="31 57"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="228">
    <literal><![CDATA[
      - x + (y + (z + (u + x))) = y + (z + u)
    ]]></literal>
    <justification jstring="[para(12(a,1),58(a,1,2,2))].">
      <j1 rule="para" parents="12 58"/>
    </justification>
  </clause>

  <clause id="229">
    <literal><![CDATA[
      (x * (x * y)) + (((x * z) * y) + ((z * x) * y)) = (x * (x * y)) + ((z * (x * y)) + (x * (z * y)))
    ]]></literal>
    <justification jstring="[back_demod(224),demod(228(23))].">
      <j1 rule="back_demod" parents="224"/>
      <j2 rule="demod" parents="228"/>
    </justification>
  </clause>

  <clause id="630">
    <literal><![CDATA[
      x + (((y * z) * u) + ((z * y) * u)) = x + ((z * (y * u)) + (y * (z * u)))
    ]]></literal>
    <justification jstring="[para(229(a,1),57(a,1,2,2)),demod(57(13)),flip(a)].">
      <j1 rule="para" parents="229 57"/>
      <j2 rule="demod" parents="57"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="707">
    <literal><![CDATA[
      ((x * y) * z) + ((y * x) * z) = (y * (x * z)) + (x * (y * z))
    ]]></literal>
    <justification jstring="[para(630(a,1),23(a,1)),demod(23(7)),flip(a)].">
      <j1 rule="para" parents="630 23"/>
      <j2 rule="demod" parents="23"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="770">
    <literal><![CDATA[
      (x * (y * x)) + ((y * x) * x) = (y * (x * x)) + (x * (y * x))
    ]]></literal>
    <justification jstring="[para(17(a,1),707(a,1,1))].">
      <j1 rule="para" parents="17 707"/>
    </justification>
  </clause>

  <clause id="821">
    <literal><![CDATA[
      x + ((y * z) * z) = x + (y * (z * z))
    ]]></literal>
    <justification jstring="[para(770(a,1),57(a,1,2,2)),demod(58(10)),flip(a)].">
      <j1 rule="para" parents="770 57"/>
      <j2 rule="demod" parents="58"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="851">
    <literal><![CDATA[
      (x * y) * y = x * (y * y)
    ]]></literal>
    <justification jstring="[para(821(a,1),23(a,1)),demod(23(4)),flip(a)].">
      <j1 rule="para" parents="821 23"/>
      <j2 rule="demod" parents="23"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="852">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(851,a,18,a)].">
      <j1 rule="resolve" parents="851 18"/>
    </justification>
  </clause>

</proof>

</proofs>
