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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 August-2006A, August 2006.
Process 10607 was started by mccune on cleo.thornwood,
Sat Aug 12 20:57:15 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="1">
    <literal><![CDATA[
      x + 0 = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="2">
    <literal><![CDATA[
      x + -x = 0
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="3">
    <literal><![CDATA[
      (x + y) + z = x + (y + z)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      x + y = y + x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="5">
    <literal><![CDATA[
      x * (y + z) = (x * y) + (x * z)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6">
    <literal><![CDATA[
      (x + y) * z = (x * z) + (y * z)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

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

  <clause id="9">
    <literal><![CDATA[
      (B * A) * A != B * (A * A)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

  <clause id="11">
    <literal><![CDATA[
      -(x * (y * z)) + ((x * y) * z) = associator(x,y,z)
    ]]></literal>
    <justification jstring="[copy(10),rewrite(4(7)),flip(a)].">
      <j1 rule="copy" parents="10"/>
      <j2 rule="rewrite" parents="4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      x + (y + -(x + y)) = 0
    ]]></literal>
    <justification jstring="[para(3(a,1),2(a,1))].">
      <j1 rule="para" parents="3 2"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      x + (-x + y) = 0 + y
    ]]></literal>
    <justification jstring="[para(2(a,1),3(a,1,1)),flip(a)].">
      <j1 rule="para" parents="2 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      0 + x = x
    ]]></literal>
    <justification jstring="[para(4(a,1),1(a,1))].">
      <j1 rule="para" parents="4 1"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      x + (y + z) = y + (x + z)
    ]]></literal>
    <justification jstring="[para(4(a,1),3(a,1,1)),rewrite(3(2))].">
      <j1 rule="para" parents="4 3"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      x + (-x + y) = y
    ]]></literal>
    <justification jstring="[back_rewrite(15),rewrite(16(5))].">
      <j1 rule="back_rewrite" parents="15"/>
      <j2 rule="rewrite" parents="16"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      -((x * (y * z)) + (x * (u * z))) + (((x * y) * z) + ((x * u) * z)) = associator(x,y + u,z)
    ]]></literal>
    <justification jstring="[para(5(a,1),11(a,1,2,1)),rewrite(6(2),5(4),6(10))].">
      <j1 rule="para" parents="5 11"/>
      <j2 rule="rewrite" parents="6 5 6"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      -((x * (y * z)) + (u * (y * z))) + (((x * y) * z) + ((u * y) * z)) = associator(x + u,y,z)
    ]]></literal>
    <justification jstring="[para(6(a,1),11(a,1,1,1)),rewrite(6(8),6(10))].">
      <j1 rule="para" parents="6 11"/>
      <j2 rule="rewrite" parents="6 6"/>
    </justification>
  </clause>

  <clause id="37">
    <literal><![CDATA[
      associator(x,x,y) = 0
    ]]></literal>
    <justification jstring="[para(7(a,1),11(a,1,2)),rewrite(4(6),2(6)),flip(a)].">
      <j1 rule="para" parents="7 11"/>
      <j2 rule="rewrite" parents="4 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      associator(x,y,x) = 0
    ]]></literal>
    <justification jstring="[para(8(a,1),11(a,1,2)),rewrite(4(6),2(6)),flip(a)].">
      <j1 rule="para" parents="8 11"/>
      <j2 rule="rewrite" parents="4 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      (0 * x) + (y * x) = y * x
    ]]></literal>
    <justification jstring="[para(16(a,1),6(a,1,1)),flip(a)].">
      <j1 rule="para" parents="16 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      x + (y + -x) = y
    ]]></literal>
    <justification jstring="[para(2(a,1),17(a,1,2)),rewrite(1(2)),flip(a)].">
      <j1 rule="para" parents="2 17"/>
      <j2 rule="rewrite" parents="1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      x + (y + (z + -(x + z))) = y
    ]]></literal>
    <justification jstring="[para(14(a,1),17(a,1,2)),rewrite(1(2)),flip(a)].">
      <j1 rule="para" parents="14 17"/>
      <j2 rule="rewrite" parents="1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      --x = x
    ]]></literal>
    <justification jstring="[para(2(a,1),18(a,1,2)),rewrite(1(2)),flip(a)].">
      <j1 rule="para" parents="2 18"/>
      <j2 rule="rewrite" parents="1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="57">
    <literal><![CDATA[
      (x * y) * z = associator(x,y,z) + (x * (y * z))
    ]]></literal>
    <justification jstring="[para(11(a,1),18(a,1,2)),rewrite(4(4)),flip(a)].">
      <j1 rule="para" parents="11 18"/>
      <j2 rule="rewrite" parents="4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="64">
    <literal><![CDATA[
      associator(x,y,z) + associator(u,y,z) = associator(x + u,y,z)
    ]]></literal>
    <justification jstring="[back_rewrite(34),rewrite(57(8),57(12),3(15),4(16),3(16),3(15),3(14),53(15))].">
      <j1 rule="back_rewrite" parents="34"/>
      <j2 rule="rewrite" parents="57 57 3 4 3 3 3 53"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      associator(x,y,z) + associator(x,u,z) = associator(x,y + u,z)
    ]]></literal>
    <justification jstring="[back_rewrite(32),rewrite(57(8),57(12),3(15),4(16),3(16),3(15),3(14),53(15))].">
      <j1 rule="back_rewrite" parents="32"/>
      <j2 rule="rewrite" parents="57 57 3 4 3 3 3 53"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      associator(B,A,A) + (B * (A * A)) != B * (A * A)
    ]]></literal>
    <justification jstring="[back_rewrite(9),rewrite(57(5))].">
      <j1 rule="back_rewrite" parents="9"/>
      <j2 rule="rewrite" parents="57"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      -x + (y + x) = y
    ]]></literal>
    <justification jstring="[para(54(a,1),50(a,1,2,2))].">
      <j1 rule="para" parents="54 50"/>
    </justification>
  </clause>

  <clause id="143">
    <literal><![CDATA[
      0 * x = 0
    ]]></literal>
    <justification jstring="[para(49(a,1),79(a,1,2)),rewrite(4(4),2(4)),flip(a)].">
      <j1 rule="para" parents="49 79"/>
      <j2 rule="rewrite" parents="4 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="158">
    <literal><![CDATA[
      associator(0,x,y) = 0
    ]]></literal>
    <justification jstring="[para(143(a,1),57(a,1,1)),rewrite(143(2),143(6),4(5),16(5)),flip(a)].">
      <j1 rule="para" parents="143 57"/>
      <j2 rule="rewrite" parents="143 143 4 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="191">
    <literal><![CDATA[
      associator(x + y,x,z) = associator(y,x,z)
    ]]></literal>
    <justification jstring="[para(37(a,1),64(a,1,1)),rewrite(16(3)),flip(a)].">
      <j1 rule="para" parents="37 64"/>
      <j2 rule="rewrite" parents="16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="204">
    <literal><![CDATA[
      associator(-x,x,y) = 0
    ]]></literal>
    <justification jstring="[para(2(a,1),191(a,1,1)),rewrite(158(2)),flip(a)].">
      <j1 rule="para" parents="2 191"/>
      <j2 rule="rewrite" parents="158"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="206">
    <literal><![CDATA[
      associator(x,y,y + x) = 0
    ]]></literal>
    <justification jstring="[para(191(a,1),41(a,1))].">
      <j1 rule="para" parents="191 41"/>
    </justification>
  </clause>

  <clause id="217">
    <literal><![CDATA[
      associator(x,-x,y) = 0
    ]]></literal>
    <justification jstring="[para(54(a,1),204(a,1,1))].">
      <j1 rule="para" parents="54 204"/>
    </justification>
  </clause>

  <clause id="224">
    <literal><![CDATA[
      associator(x,y,x + y) = 0
    ]]></literal>
    <justification jstring="[para(4(a,1),206(a,1,3))].">
      <j1 rule="para" parents="4 206"/>
    </justification>
  </clause>

  <clause id="250">
    <literal><![CDATA[
      associator(x,-x + y,z) = associator(x,y,z)
    ]]></literal>
    <justification jstring="[para(217(a,1),66(a,1,1)),rewrite(16(3)),flip(a)].">
      <j1 rule="para" parents="217 66"/>
      <j2 rule="rewrite" parents="16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="260">
    <literal><![CDATA[
      associator(x,y,y) = 0
    ]]></literal>
    <justification jstring="[para(18(a,1),224(a,1,3)),rewrite(250(3))].">
      <j1 rule="para" parents="18 224"/>
      <j2 rule="rewrite" parents="250"/>
    </justification>
  </clause>

  <clause id="272">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[back_rewrite(74),rewrite(260(4),16(7)),xx(a)].">
      <j1 rule="back_rewrite" parents="74"/>
      <j2 rule="rewrite" parents="260 16"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
