<?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="7">

<source>mol-ss2.out</source>

<heading><![CDATA[
Prover9 (32) version 2009-02A, February 2009.
Process 11666 was started by mccune on cleo,
Wed Feb 25 09:36:13 2009
The command was "/home/mccune/bin/prover9 -f mol-ss2.in".
]]></heading>

<proof number="1" length="13" max_count="21">

<comments><![CDATA[
% Proof 1 at 0.01 (+ 0.00) seconds: DEF_C.
% Length of proof is 13.
% Level of proof is 4.
% Maximum clause weight is 21.
% Given clauses 13.

]]></comments>

  <clause id="7" type="goal">
    <literal><![CDATA[
      x' = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      answer(DEF_C)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="9" type="assumption">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="10" type="assumption">
    <literal><![CDATA[
      x ^ y = (x' v y')'
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11" type="assumption">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="13" type="assumption">
    <literal><![CDATA[
      x v (y ^ (x v z)) = x v (z ^ (x v y))
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      x v (y' v (x v z)')' = x v (z' v (x v y)')'
    ]]></literal>
    <justification jstring="[copy(13),rewrite([10(2),10(8)])].">
      <j1 rule="copy" parents="13"/>
      <j2 rule="rewrite" parents="10 10"/>
    </justification>
  </clause>

  <clause id="15" type="assumption">
    <literal><![CDATA[
      f(x,y) = x' v y'
    ]]></literal>
    <attribute><![CDATA[
      label(definition_sheffer)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="28" type="deny">
    <literal><![CDATA[
      f(c15,c15) != c15'
    ]]></literal>
    <attribute><![CDATA[
      answer(DEF_C)
    ]]></attribute>
    <justification jstring="[deny(7)].">
      <j1 rule="deny" parents="7"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      c15' v c15' != c15'
    ]]></literal>
    <attribute><![CDATA[
      answer(DEF_C)
    ]]></attribute>
    <justification jstring="[copy(28),rewrite([15(3)])].">
      <j1 rule="copy" parents="28"/>
      <j2 rule="rewrite" parents="15"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      x v (x' v y')' = x
    ]]></literal>
    <justification jstring="[back_rewrite(9),rewrite([10(1)])].">
      <j1 rule="back_rewrite" parents="9"/>
      <j2 rule="rewrite" parents="10"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      x v (y v (x v x)')' = x v x
    ]]></literal>
    <justification jstring="[para(30(a,1),14(a,1,2,1)),rewrite([11(2),11(2),11(3),11(3),11(3)]),flip(a)].">
      <j1 rule="para" parents="30 14"/>
      <j2 rule="rewrite" parents="11 11 11 11 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(30(a,1),14(a,1)),rewrite([46(6)]),flip(a)].">
      <j1 rule="para" parents="30 14"/>
      <j2 rule="rewrite" parents="46"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="48">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(DEF_C)
    ]]></attribute>
    <justification jstring="[resolve(47,a,29,a)].">
      <j1 rule="resolve" parents="47 29"/>
    </justification>
  </clause>

</proof>

<proof number="2" length="13" max_count="21">

<comments><![CDATA[
% Proof 2 at 0.01 (+ 0.00) seconds: DEF_M.
% Length of proof is 13.
% Level of proof is 4.
% Maximum clause weight is 21.
% Given clauses 13.

]]></comments>

  <clause id="6" type="goal">
    <literal><![CDATA[
      x ^ y = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      answer(DEF_M)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="9" type="assumption">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="10" type="assumption">
    <literal><![CDATA[
      x ^ y = (x' v y')'
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11" type="assumption">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="13" type="assumption">
    <literal><![CDATA[
      x v (y ^ (x v z)) = x v (z ^ (x v y))
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      x v (y' v (x v z)')' = x v (z' v (x v y)')'
    ]]></literal>
    <justification jstring="[copy(13),rewrite([10(2),10(8)])].">
      <j1 rule="copy" parents="13"/>
      <j2 rule="rewrite" parents="10 10"/>
    </justification>
  </clause>

  <clause id="15" type="assumption">
    <literal><![CDATA[
      f(x,y) = x' v y'
    ]]></literal>
    <attribute><![CDATA[
      label(definition_sheffer)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="26" type="deny">
    <literal><![CDATA[
      f(f(c13,c14),f(c13,c14)) != c13 ^ c14
    ]]></literal>
    <attribute><![CDATA[
      answer(DEF_M)
    ]]></attribute>
    <justification jstring="[deny(6)].">
      <j1 rule="deny" parents="6"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      (c13' v c14')' v (c13' v c14')' != (c13' v c14')'
    ]]></literal>
    <attribute><![CDATA[
      answer(DEF_M)
    ]]></attribute>
    <justification jstring="[copy(26),rewrite([15(3),15(8),15(11),10(16)])].">
      <j1 rule="copy" parents="26"/>
      <j2 rule="rewrite" parents="15 15 15 10"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      x v (x' v y')' = x
    ]]></literal>
    <justification jstring="[back_rewrite(9),rewrite([10(1)])].">
      <j1 rule="back_rewrite" parents="9"/>
      <j2 rule="rewrite" parents="10"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      x v (y v (x v x)')' = x v x
    ]]></literal>
    <justification jstring="[para(30(a,1),14(a,1,2,1)),rewrite([11(2),11(2),11(3),11(3),11(3)]),flip(a)].">
      <j1 rule="para" parents="30 14"/>
      <j2 rule="rewrite" parents="11 11 11 11 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(30(a,1),14(a,1)),rewrite([46(6)]),flip(a)].">
      <j1 rule="para" parents="30 14"/>
      <j2 rule="rewrite" parents="46"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(DEF_M)
    ]]></attribute>
    <justification jstring="[resolve(47,a,27,a)].">
      <j1 rule="resolve" parents="47 27"/>
    </justification>
  </clause>

</proof>

<proof number="3" length="14" max_count="35">

<comments><![CDATA[
% Proof 3 at 0.01 (+ 0.00) seconds: MOD_SS.
% Length of proof is 14.
% Level of proof is 4.
% Maximum clause weight is 35.
% Given clauses 13.

]]></comments>

  <clause id="4" type="goal">
    <literal><![CDATA[
      f(x,f(y,f(x,f(z,z)))) = f(x,f(z,f(x,f(y,y))))
    ]]></literal>
    <attribute><![CDATA[
      answer(MOD_SS)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="9" type="assumption">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="10" type="assumption">
    <literal><![CDATA[
      x ^ y = (x' v y')'
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11" type="assumption">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="13" type="assumption">
    <literal><![CDATA[
      x v (y ^ (x v z)) = x v (z ^ (x v y))
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      x v (y' v (x v z)')' = x v (z' v (x v y)')'
    ]]></literal>
    <justification jstring="[copy(13),rewrite([10(2),10(8)])].">
      <j1 rule="copy" parents="13"/>
      <j2 rule="rewrite" parents="10 10"/>
    </justification>
  </clause>

  <clause id="15" type="assumption">
    <literal><![CDATA[
      f(x,y) = x' v y'
    ]]></literal>
    <attribute><![CDATA[
      label(definition_sheffer)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="22" type="deny">
    <literal><![CDATA[
      f(c8,f(c10,f(c8,f(c9,c9)))) != f(c8,f(c9,f(c8,f(c10,c10))))
    ]]></literal>
    <attribute><![CDATA[
      answer(MOD_SS)
    ]]></attribute>
    <justification jstring="[deny(4)].">
      <j1 rule="deny" parents="4"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      c8' v (c9' v (c8' v (c10' v c10')')')' != c8' v (c10' v (c8' v (c9' v c9')')')'
    ]]></literal>
    <attribute><![CDATA[
      answer(MOD_SS)
    ]]></attribute>
    <justification jstring="[copy(22),rewrite([15(6),15(9),15(12),15(15),15(23),15(26),15(29),15(32)]),flip(a)].">
      <j1 rule="copy" parents="22"/>
      <j2 rule="rewrite" parents="15 15 15 15 15 15 15 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      x v (x' v y')' = x
    ]]></literal>
    <justification jstring="[back_rewrite(9),rewrite([10(1)])].">
      <j1 rule="back_rewrite" parents="9"/>
      <j2 rule="rewrite" parents="10"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      c8' v (c9' v (c8' v (c10' v c10')')')' != c8' v ((c9' v c9') v (c8' v c10)')'
    ]]></literal>
    <attribute><![CDATA[
      answer(MOD_SS)
    ]]></attribute>
    <justification jstring="[para(14(a,1),23(a,2)),rewrite([11(26)])].">
      <j1 rule="para" parents="14 23"/>
      <j2 rule="rewrite" parents="11"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      x v (y v (x v x)')' = x v x
    ]]></literal>
    <justification jstring="[para(30(a,1),14(a,1,2,1)),rewrite([11(2),11(2),11(3),11(3),11(3)]),flip(a)].">
      <j1 rule="para" parents="30 14"/>
      <j2 rule="rewrite" parents="11 11 11 11 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(30(a,1),14(a,1)),rewrite([46(6)]),flip(a)].">
      <j1 rule="para" parents="30 14"/>
      <j2 rule="rewrite" parents="46"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="51">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(MOD_SS)
    ]]></attribute>
    <justification jstring="[back_rewrite(40),rewrite([47(11),11(9),47(19)]),xx(a)].">
      <j1 rule="back_rewrite" parents="40"/>
      <j2 rule="rewrite" parents="47 11 47"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

<proof number="4" length="13" max_count="21">

<comments><![CDATA[
% Proof 4 at 0.01 (+ 0.00) seconds: DEF_J.
% Length of proof is 13.
% Level of proof is 4.
% Maximum clause weight is 21.
% Given clauses 13.

]]></comments>

  <clause id="5" type="goal">
    <literal><![CDATA[
      x v y = f(f(x,x),f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      answer(DEF_J)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="9" type="assumption">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="10" type="assumption">
    <literal><![CDATA[
      x ^ y = (x' v y')'
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11" type="assumption">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="13" type="assumption">
    <literal><![CDATA[
      x v (y ^ (x v z)) = x v (z ^ (x v y))
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      x v (y' v (x v z)')' = x v (z' v (x v y)')'
    ]]></literal>
    <justification jstring="[copy(13),rewrite([10(2),10(8)])].">
      <j1 rule="copy" parents="13"/>
      <j2 rule="rewrite" parents="10 10"/>
    </justification>
  </clause>

  <clause id="15" type="assumption">
    <literal><![CDATA[
      f(x,y) = x' v y'
    ]]></literal>
    <attribute><![CDATA[
      label(definition_sheffer)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="24" type="deny">
    <literal><![CDATA[
      f(f(c11,c11),f(c12,c12)) != c11 v c12
    ]]></literal>
    <attribute><![CDATA[
      answer(DEF_J)
    ]]></attribute>
    <justification jstring="[deny(5)].">
      <j1 rule="deny" parents="5"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      (c11' v c11')' v (c12' v c12')' != c11 v c12
    ]]></literal>
    <attribute><![CDATA[
      answer(DEF_J)
    ]]></attribute>
    <justification jstring="[copy(24),rewrite([15(3),15(8),15(11)])].">
      <j1 rule="copy" parents="24"/>
      <j2 rule="rewrite" parents="15 15 15"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      x v (x' v y')' = x
    ]]></literal>
    <justification jstring="[back_rewrite(9),rewrite([10(1)])].">
      <j1 rule="back_rewrite" parents="9"/>
      <j2 rule="rewrite" parents="10"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      x v (y v (x v x)')' = x v x
    ]]></literal>
    <justification jstring="[para(30(a,1),14(a,1,2,1)),rewrite([11(2),11(2),11(3),11(3),11(3)]),flip(a)].">
      <j1 rule="para" parents="30 14"/>
      <j2 rule="rewrite" parents="11 11 11 11 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(30(a,1),14(a,1)),rewrite([46(6)]),flip(a)].">
      <j1 rule="para" parents="30 14"/>
      <j2 rule="rewrite" parents="46"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(DEF_J)
    ]]></attribute>
    <justification jstring="[back_rewrite(25),rewrite([47(5),11(3),47(6),11(4)]),xx(a)].">
      <j1 rule="back_rewrite" parents="25"/>
      <j2 rule="rewrite" parents="47 11 47 11"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

<proof number="5" length="14" max_count="21">

<comments><![CDATA[
% Proof 5 at 0.02 (+ 0.00) seconds: B_SS.
% Length of proof is 14.
% Level of proof is 4.
% Maximum clause weight is 21.
% Given clauses 13.

]]></comments>

  <clause id="2" type="goal">
    <literal><![CDATA[
      f(f(x,x),f(x,y)) = x
    ]]></literal>
    <attribute><![CDATA[
      answer(B_SS)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="9" type="assumption">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="10" type="assumption">
    <literal><![CDATA[
      x ^ y = (x' v y')'
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11" type="assumption">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="13" type="assumption">
    <literal><![CDATA[
      x v (y ^ (x v z)) = x v (z ^ (x v y))
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      x v (y' v (x v z)')' = x v (z' v (x v y)')'
    ]]></literal>
    <justification jstring="[copy(13),rewrite([10(2),10(8)])].">
      <j1 rule="copy" parents="13"/>
      <j2 rule="rewrite" parents="10 10"/>
    </justification>
  </clause>

  <clause id="15" type="assumption">
    <literal><![CDATA[
      f(x,y) = x' v y'
    ]]></literal>
    <attribute><![CDATA[
      label(definition_sheffer)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="18" type="deny">
    <literal><![CDATA[
      f(f(c4,c4),f(c4,c5)) != c4
    ]]></literal>
    <attribute><![CDATA[
      answer(B_SS)
    ]]></attribute>
    <justification jstring="[deny(2)].">
      <j1 rule="deny" parents="2"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      (c4' v c4')' v (c4' v c5')' != c4
    ]]></literal>
    <attribute><![CDATA[
      answer(B_SS)
    ]]></attribute>
    <justification jstring="[copy(18),rewrite([15(3),15(8),15(11)])].">
      <j1 rule="copy" parents="18"/>
      <j2 rule="rewrite" parents="15 15 15"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      x v (x' v y')' = x
    ]]></literal>
    <justification jstring="[back_rewrite(9),rewrite([10(1)])].">
      <j1 rule="back_rewrite" parents="9"/>
      <j2 rule="rewrite" parents="10"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      x v (x' v y)' = x
    ]]></literal>
    <justification jstring="[para(11(a,1),30(a,1,2,1,2))].">
      <j1 rule="para" parents="11 30"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      x v (y v (x v x)')' = x v x
    ]]></literal>
    <justification jstring="[para(30(a,1),14(a,1,2,1)),rewrite([11(2),11(2),11(3),11(3),11(3)]),flip(a)].">
      <j1 rule="para" parents="30 14"/>
      <j2 rule="rewrite" parents="11 11 11 11 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(30(a,1),14(a,1)),rewrite([46(6)]),flip(a)].">
      <j1 rule="para" parents="30 14"/>
      <j2 rule="rewrite" parents="46"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="60">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(B_SS)
    ]]></attribute>
    <justification jstring="[back_rewrite(19),rewrite([47(5),11(3),44(8)]),xx(a)].">
      <j1 rule="back_rewrite" parents="19"/>
      <j2 rule="rewrite" parents="47 11 44"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

<proof number="6" length="14" max_count="35">

<comments><![CDATA[
% Proof 6 at 0.02 (+ 0.00) seconds: A_SS.
% Length of proof is 14.
% Level of proof is 4.
% Maximum clause weight is 35.
% Given clauses 13.

]]></comments>

  <clause id="1" type="goal">
    <literal><![CDATA[
      f(x,f(f(y,z),f(y,z))) = f(y,f(f(x,z),f(x,z)))
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

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

  <clause id="9" type="assumption">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="10" type="assumption">
    <literal><![CDATA[
      x ^ y = (x' v y')'
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11" type="assumption">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="13" type="assumption">
    <literal><![CDATA[
      x v (y ^ (x v z)) = x v (z ^ (x v y))
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      x v (y' v (x v z)')' = x v (z' v (x v y)')'
    ]]></literal>
    <justification jstring="[copy(13),rewrite([10(2),10(8)])].">
      <j1 rule="copy" parents="13"/>
      <j2 rule="rewrite" parents="10 10"/>
    </justification>
  </clause>

  <clause id="15" type="assumption">
    <literal><![CDATA[
      f(x,y) = x' v y'
    ]]></literal>
    <attribute><![CDATA[
      label(definition_sheffer)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="16" type="deny">
    <literal><![CDATA[
      f(c2,f(f(c1,c3),f(c1,c3))) != f(c1,f(f(c2,c3),f(c2,c3)))
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      c1' v ((c2' v c3')' v (c2' v c3')')' != c2' v ((c1' v c3')' v (c1' v c3')')'
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <justification jstring="[copy(16),rewrite([15(4),15(9),15(12),15(15),15(21),15(26),15(29),15(32)]),flip(a)].">
      <j1 rule="copy" parents="16"/>
      <j2 rule="rewrite" parents="15 15 15 15 15 15 15 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      x v (x' v y')' = x
    ]]></literal>
    <justification jstring="[back_rewrite(9),rewrite([10(1)])].">
      <j1 rule="back_rewrite" parents="9"/>
      <j2 rule="rewrite" parents="10"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      x v (y v (x v x)')' = x v x
    ]]></literal>
    <justification jstring="[para(30(a,1),14(a,1,2,1)),rewrite([11(2),11(2),11(3),11(3),11(3)]),flip(a)].">
      <j1 rule="para" parents="30 14"/>
      <j2 rule="rewrite" parents="11 11 11 11 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(30(a,1),14(a,1)),rewrite([46(6)]),flip(a)].">
      <j1 rule="para" parents="30 14"/>
      <j2 rule="rewrite" parents="46"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <justification jstring="[back_rewrite(17),rewrite([47(15),11(9),47(23),11(17),8(16)]),xx(a)].">
      <j1 rule="back_rewrite" parents="17"/>
      <j2 rule="rewrite" parents="47 11 47 11 8"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

<proof number="7" length="17" max_count="21">

<comments><![CDATA[
% Proof 7 at 0.02 (+ 0.00) seconds: ONE_SS.
% Length of proof is 17.
% Level of proof is 5.
% Maximum clause weight is 21.
% Given clauses 14.

]]></comments>

  <clause id="3" type="goal">
    <literal><![CDATA[
      f(x,f(x,x)) = f(y,f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      answer(ONE_SS)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="9" type="assumption">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="10" type="assumption">
    <literal><![CDATA[
      x ^ y = (x' v y')'
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11" type="assumption">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="12" type="assumption">
    <literal><![CDATA[
      x v x' = y v y'
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="13" type="assumption">
    <literal><![CDATA[
      x v (y ^ (x v z)) = x v (z ^ (x v y))
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      x v (y' v (x v z)')' = x v (z' v (x v y)')'
    ]]></literal>
    <justification jstring="[copy(13),rewrite([10(2),10(8)])].">
      <j1 rule="copy" parents="13"/>
      <j2 rule="rewrite" parents="10 10"/>
    </justification>
  </clause>

  <clause id="15" type="assumption">
    <literal><![CDATA[
      f(x,y) = x' v y'
    ]]></literal>
    <attribute><![CDATA[
      label(definition_sheffer)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="20" type="deny">
    <literal><![CDATA[
      f(c7,f(c7,c7)) != f(c6,f(c6,c6))
    ]]></literal>
    <attribute><![CDATA[
      answer(ONE_SS)
    ]]></attribute>
    <justification jstring="[deny(3)].">
      <j1 rule="deny" parents="3"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      c7' v (c7' v c7')' != c6' v (c6' v c6')'
    ]]></literal>
    <attribute><![CDATA[
      answer(ONE_SS)
    ]]></attribute>
    <justification jstring="[copy(20),rewrite([15(4),15(7),15(13),15(16)])].">
      <j1 rule="copy" parents="20"/>
      <j2 rule="rewrite" parents="15 15 15 15"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      x v (x' v y')' = x
    ]]></literal>
    <justification jstring="[back_rewrite(9),rewrite([10(1)])].">
      <j1 rule="back_rewrite" parents="9"/>
      <j2 rule="rewrite" parents="10"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      x v x' = c_0
    ]]></literal>
    <justification jstring="[new_symbol(12)].">
      <j1 rule="new_symbol" parents="12"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      x v (y v (x v x)')' = x v x
    ]]></literal>
    <justification jstring="[para(30(a,1),14(a,1,2,1)),rewrite([11(2),11(2),11(3),11(3),11(3)]),flip(a)].">
      <j1 rule="para" parents="30 14"/>
      <j2 rule="rewrite" parents="11 11 11 11 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(30(a,1),14(a,1)),rewrite([46(6)]),flip(a)].">
      <j1 rule="para" parents="30 14"/>
      <j2 rule="rewrite" parents="46"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      c7' v c7 != c6' v c6
    ]]></literal>
    <attribute><![CDATA[
      answer(ONE_SS)
    ]]></attribute>
    <justification jstring="[back_rewrite(21),rewrite([47(7),11(5),47(11),11(9)])].">
      <j1 rule="back_rewrite" parents="21"/>
      <j2 rule="rewrite" parents="47 11 47 11"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      x' v x = c_0
    ]]></literal>
    <justification jstring="[para(11(a,1),31(a,1,2))].">
      <j1 rule="para" parents="11 31"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(ONE_SS)
    ]]></attribute>
    <justification jstring="[back_rewrite(59),rewrite([65(4),65(5)]),xx(a)].">
      <j1 rule="back_rewrite" parents="59"/>
      <j2 rule="rewrite" parents="65 65"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
