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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="7">

<source>mol-ss1.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13583 was started by mccune on cleo.thornwood,
Mon Jun 19 16:43:06 2006
The command was "/home/mccune/bin/prover9 -f mol-ss1.in".
]]></heading>

<proof number="1" length="5" max_count="39">

<comments><![CDATA[
% Proof 1 at 0.00 (+ 0.00) seconds: AJ.
% Length of proof is 5.
% Level of proof is 2.
% Maximum clause weight is 19.
% Given clauses 0.
]]></comments>

  <clause id="8">
    <literal><![CDATA[
      c2 v (c1 v c3) != c1 v (c2 v c3)
    ]]></literal>
    <attribute><![CDATA[
      answer(AJ)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

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

  <clause id="19">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      f(f(c1,c1),f(f(f(c2,c2),f(c3,c3)),f(f(c2,c2),f(c3,c3)))) != f(f(c2,c2),f(f(f(c1,c1),f(c3,c3)),f(f(c1,c1),f(c3,c3))))
    ]]></literal>
    <attribute><![CDATA[
      answer(AJ)
    ]]></attribute>
    <justification jstring="[copy(8),demod(19(4),19(9),19(23),19(28)),flip(a)].">
      <j1 rule="copy" parents="8"/>
      <j2 rule="demod" parents="19 19 19 19"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(AJ)
    ]]></attribute>
    <justification jstring="[resolve(23,a,15,a)].">
      <j1 rule="resolve" parents="23 15"/>
    </justification>
  </clause>

</proof>

<proof number="2" length="5" max_count="11">

<comments><![CDATA[
% Proof 2 at 0.00 (+ 0.00) seconds: B1.
% Length of proof is 5.
% Level of proof is 1.
% Maximum clause weight is 11.
% Given clauses 0.
]]></comments>

  <clause id="9">
    <literal><![CDATA[
      c4 v (c4 ^ c5) != c4
    ]]></literal>
    <attribute><![CDATA[
      answer(B1)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

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

  <clause id="19">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(B1)
    ]]></attribute>
    <justification jstring="[copy(9),demod(20(4),19(9),16(18),16(7)),xx(a)].">
      <j1 rule="copy" parents="9"/>
      <j2 rule="demod" parents="20 19 16 16"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

<proof number="3" length="6" max_count="11">

<comments><![CDATA[
% Proof 3 at 0.00 (+ 0.00) seconds: DM.
% Length of proof is 6.
% Level of proof is 1.
% Maximum clause weight is 11.
% Given clauses 0.
]]></comments>

  <clause id="10">
    <literal><![CDATA[
      c(c(c6) v c(c7)) != c6 ^ c7
    ]]></literal>
    <attribute><![CDATA[
      answer(DM)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

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

  <clause id="19">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(DM)
    ]]></attribute>
    <justification jstring="[copy(10),demod(21(2),21(5),19(7),16(7),16(8),21(4),20(10)),xx(a)].">
      <j1 rule="copy" parents="10"/>
      <j2 rule="demod" parents="21 21 19 16 16 21 20"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

<proof number="4" length="4" max_count="9">

<comments><![CDATA[
% Proof 4 at 0.00 (+ 0.00) seconds: CC.
% Length of proof is 4.
% Level of proof is 1.
% Maximum clause weight is 9.
% Given clauses 0.
]]></comments>

  <clause id="11">
    <literal><![CDATA[
      c(c(c8)) != c8
    ]]></literal>
    <attribute><![CDATA[
      answer(CC)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

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

  <clause id="21">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(CC)
    ]]></attribute>
    <justification jstring="[copy(11),demod(21(2),21(4),16(7)),xx(a)].">
      <j1 rule="copy" parents="11"/>
      <j2 rule="demod" parents="21 21 16"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

<proof number="5" length="7" max_count="27">

<comments><![CDATA[
% Proof 5 at 0.00 (+ 0.00) seconds: MOD.
% Length of proof is 7.
% Level of proof is 2.
% Maximum clause weight is 19.
% Given clauses 0.
]]></comments>

  <clause id="13">
    <literal><![CDATA[
      c11 v (c13 ^ (c11 v c12)) != c11 v (c12 ^ (c11 v c13))
    ]]></literal>
    <attribute><![CDATA[
      answer(MOD)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

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

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

  <clause id="19">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      f(f(c11,c11),f(c12,f(f(c11,c11),f(c13,c13)))) != f(f(c11,c11),f(c13,f(f(c11,c11),f(c12,c12))))
    ]]></literal>
    <attribute><![CDATA[
      answer(MOD)
    ]]></attribute>
    <justification jstring="[copy(13),demod(19(5),20(10),19(21),16(42),19(18),20(23),19(34),16(55)),flip(a)].">
      <j1 rule="copy" parents="13"/>
      <j2 rule="demod" parents="19 20 19 16 19 20 19 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(MOD)
    ]]></attribute>
    <justification jstring="[resolve(29,a,18,a)].">
      <j1 rule="resolve" parents="29 18"/>
    </justification>
  </clause>

</proof>

<proof number="6" length="5" max_count="11">

<comments><![CDATA[
% Proof 6 at 0.00 (+ 0.00) seconds: DEF_SS.
% Length of proof is 5.
% Level of proof is 1.
% Maximum clause weight is 11.
% Given clauses 0.
]]></comments>

  <clause id="14">
    <literal><![CDATA[
      f(c14,c15) != c(c14) v c(c15)
    ]]></literal>
    <attribute><![CDATA[
      answer(DEF_SS)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

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

  <clause id="19">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(DEF_SS)
    ]]></attribute>
    <justification jstring="[copy(14),demod(21(5),21(8),19(10),16(10),16(11)),xx(a)].">
      <j1 rule="copy" parents="14"/>
      <j2 rule="demod" parents="21 21 19 16 16"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

<proof number="7" length="10" max_count="11">

<comments><![CDATA[
% Proof 7 at 0.01 (+ 0.00) seconds: ONE.
% Length of proof is 10.
% Level of proof is 4.
% Maximum clause weight is 11.
% Given clauses 7.
]]></comments>

  <clause id="3">
    <literal><![CDATA[
      f(x,f(x,x)) = f(y,f(y,y))
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      c10 v c(c10) != c9 v c(c9)
    ]]></literal>
    <attribute><![CDATA[
      answer(ONE)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

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

  <clause id="17">
    <literal><![CDATA[
      f(x,f(x,x)) = f(y,f(y,y))
    ]]></literal>
    <justification jstring="[copy(3)].">
      <j1 rule="copy" parents="3"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      c(x) = f(x,x)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      f(x,f(x,x)) = c_0
    ]]></literal>
    <justification jstring="[new_symbol(17)].">
      <j1 rule="new_symbol" parents="17"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      f(f(c10,c10),c10) != f(f(c9,c9),c9)
    ]]></literal>
    <attribute><![CDATA[
      answer(ONE)
    ]]></attribute>
    <justification jstring="[copy(12),demod(21(3),19(5),16(10),21(8),19(10),16(15))].">
      <j1 rule="copy" parents="12"/>
      <j2 rule="demod" parents="21 19 16 21 19 16"/>
    </justification>
  </clause>

  <clause id="57">
    <literal><![CDATA[
      f(f(x,x),x) = c_0
    ]]></literal>
    <justification jstring="[para(16(a,1),22(a,1,2))].">
      <j1 rule="para" parents="16 22"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(ONE)
    ]]></attribute>
    <justification jstring="[back_demod(28),demod(57(5),57(6)),xx(a)].">
      <j1 rule="back_demod" parents="28"/>
      <j2 rule="demod" parents="57 57"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
