<?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 June-2006C, June 2006.
Process 13584 was started by mccune on cleo.thornwood,
Mon Jun 19 16:43:06 2006
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 0.00 (+ 0.00) seconds: DEF_C.
% Length of proof is 13.
% Level of proof is 5.
% Maximum clause weight is 21.
% Given clauses 6.
]]></comments>

  <clause id="2">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="6">
    <literal><![CDATA[
      x v (y ^ (x v z)) = x v (z ^ (x v y))
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

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

  <clause id="16">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[copy(2)].">
      <j1 rule="copy" parents="2"/>
    </justification>
  </clause>

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

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

  <clause id="20">
    <literal><![CDATA[
      x v c(c(y) v c(x v z)) = x v c(c(z) v c(x v y))
    ]]></literal>
    <justification jstring="[copy(6),demod(17(2),17(8))].">
      <j1 rule="copy" parents="6"/>
      <j2 rule="demod" parents="17 17"/>
    </justification>
  </clause>

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

  <clause id="22">
    <literal><![CDATA[
      x v c(c(x) v c(y)) = x
    ]]></literal>
    <justification jstring="[back_demod(16),demod(17(1))].">
      <j1 rule="back_demod" parents="16"/>
      <j2 rule="demod" parents="17"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      c(c15) v c(c15) != c(c15)
    ]]></literal>
    <attribute><![CDATA[
      answer(DEF_C)
    ]]></attribute>
    <justification jstring="[copy(14),demod(21(3))].">
      <j1 rule="copy" parents="14"/>
      <j2 rule="demod" parents="21"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      x v c(y v c(x v x)) = x v x
    ]]></literal>
    <justification jstring="[para(22(a,1),20(a,1,2,1)),demod(18(2),18(2),18(3),18(3),18(3)),flip(a)].">
      <j1 rule="para" parents="22 20"/>
      <j2 rule="demod" parents="18 18 18 18 18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(22(a,1),20(a,1)),demod(43(6)),flip(a)].">
      <j1 rule="para" parents="22 20"/>
      <j2 rule="demod" parents="43"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(DEF_C)
    ]]></attribute>
    <justification jstring="[resolve(44,a,30,a)].">
      <j1 rule="resolve" parents="44 30"/>
    </justification>
  </clause>

</proof>

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

<comments><![CDATA[
% Proof 2 at 0.00 (+ 0.00) seconds: DEF_M.
% Length of proof is 13.
% Level of proof is 5.
% Maximum clause weight is 21.
% Given clauses 6.
]]></comments>

  <clause id="2">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="6">
    <literal><![CDATA[
      x v (y ^ (x v z)) = x v (z ^ (x v y))
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      f(f(c13,c14),f(c13,c14)) != c13 ^ c14
    ]]></literal>
    <attribute><![CDATA[
      answer(DEF_M)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[copy(2)].">
      <j1 rule="copy" parents="2"/>
    </justification>
  </clause>

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

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

  <clause id="20">
    <literal><![CDATA[
      x v c(c(y) v c(x v z)) = x v c(c(z) v c(x v y))
    ]]></literal>
    <justification jstring="[copy(6),demod(17(2),17(8))].">
      <j1 rule="copy" parents="6"/>
      <j2 rule="demod" parents="17 17"/>
    </justification>
  </clause>

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

  <clause id="22">
    <literal><![CDATA[
      x v c(c(x) v c(y)) = x
    ]]></literal>
    <justification jstring="[back_demod(16),demod(17(1))].">
      <j1 rule="back_demod" parents="16"/>
      <j2 rule="demod" parents="17"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      c(c(c13) v c(c14)) v c(c(c13) v c(c14)) != c(c(c13) v c(c14))
    ]]></literal>
    <attribute><![CDATA[
      answer(DEF_M)
    ]]></attribute>
    <justification jstring="[copy(13),demod(21(3),21(8),21(11),17(16))].">
      <j1 rule="copy" parents="13"/>
      <j2 rule="demod" parents="21 21 21 17"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      x v c(y v c(x v x)) = x v x
    ]]></literal>
    <justification jstring="[para(22(a,1),20(a,1,2,1)),demod(18(2),18(2),18(3),18(3),18(3)),flip(a)].">
      <j1 rule="para" parents="22 20"/>
      <j2 rule="demod" parents="18 18 18 18 18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(22(a,1),20(a,1)),demod(43(6)),flip(a)].">
      <j1 rule="para" parents="22 20"/>
      <j2 rule="demod" parents="43"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(DEF_M)
    ]]></attribute>
    <justification jstring="[resolve(44,a,29,a)].">
      <j1 rule="resolve" parents="44 29"/>
    </justification>
  </clause>

</proof>

<proof number="3" length="13" max_count="21">

<comments><![CDATA[
% Proof 3 at 0.00 (+ 0.00) seconds: DEF_J.
% Length of proof is 13.
% Level of proof is 5.
% Maximum clause weight is 21.
% Given clauses 6.
]]></comments>

  <clause id="2">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="6">
    <literal><![CDATA[
      x v (y ^ (x v z)) = x v (z ^ (x v y))
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      f(f(c11,c11),f(c12,c12)) != c11 v c12
    ]]></literal>
    <attribute><![CDATA[
      answer(DEF_J)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[copy(2)].">
      <j1 rule="copy" parents="2"/>
    </justification>
  </clause>

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

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

  <clause id="20">
    <literal><![CDATA[
      x v c(c(y) v c(x v z)) = x v c(c(z) v c(x v y))
    ]]></literal>
    <justification jstring="[copy(6),demod(17(2),17(8))].">
      <j1 rule="copy" parents="6"/>
      <j2 rule="demod" parents="17 17"/>
    </justification>
  </clause>

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

  <clause id="22">
    <literal><![CDATA[
      x v c(c(x) v c(y)) = x
    ]]></literal>
    <justification jstring="[back_demod(16),demod(17(1))].">
      <j1 rule="back_demod" parents="16"/>
      <j2 rule="demod" parents="17"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      c(c(c11) v c(c11)) v c(c(c12) v c(c12)) != c11 v c12
    ]]></literal>
    <attribute><![CDATA[
      answer(DEF_J)
    ]]></attribute>
    <justification jstring="[copy(12),demod(21(3),21(8),21(11))].">
      <j1 rule="copy" parents="12"/>
      <j2 rule="demod" parents="21 21 21"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      x v c(y v c(x v x)) = x v x
    ]]></literal>
    <justification jstring="[para(22(a,1),20(a,1,2,1)),demod(18(2),18(2),18(3),18(3),18(3)),flip(a)].">
      <j1 rule="para" parents="22 20"/>
      <j2 rule="demod" parents="18 18 18 18 18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(22(a,1),20(a,1)),demod(43(6)),flip(a)].">
      <j1 rule="para" parents="22 20"/>
      <j2 rule="demod" parents="43"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="48">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(DEF_J)
    ]]></attribute>
    <justification jstring="[back_demod(28),demod(44(5),18(3),44(6),18(4)),xx(a)].">
      <j1 rule="back_demod" parents="28"/>
      <j2 rule="demod" parents="44 18 44 18"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

<proof number="4" length="14" max_count="35">

<comments><![CDATA[
% Proof 4 at 0.00 (+ 0.00) seconds: MOD_SS.
% Length of proof is 14.
% Level of proof is 6.
% Maximum clause weight is 21.
% Given clauses 6.
]]></comments>

  <clause id="2">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="6">
    <literal><![CDATA[
      x v (y ^ (x v z)) = x v (z ^ (x v y))
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="11">
    <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="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[copy(2)].">
      <j1 rule="copy" parents="2"/>
    </justification>
  </clause>

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

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

  <clause id="20">
    <literal><![CDATA[
      x v c(c(y) v c(x v z)) = x v c(c(z) v c(x v y))
    ]]></literal>
    <justification jstring="[copy(6),demod(17(2),17(8))].">
      <j1 rule="copy" parents="6"/>
      <j2 rule="demod" parents="17 17"/>
    </justification>
  </clause>

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

  <clause id="22">
    <literal><![CDATA[
      x v c(c(x) v c(y)) = x
    ]]></literal>
    <justification jstring="[back_demod(16),demod(17(1))].">
      <j1 rule="back_demod" parents="16"/>
      <j2 rule="demod" parents="17"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      c(c8) v c(c(c9) v c(c(c8) v c(c(c10) v c(c10)))) != c(c8) v c(c(c10) v c(c(c8) v c(c(c9) v c(c9))))
    ]]></literal>
    <attribute><![CDATA[
      answer(MOD_SS)
    ]]></attribute>
    <justification jstring="[copy(11),demod(21(6),21(9),21(12),21(15),21(23),21(26),21(29),21(32)),flip(a)].">
      <j1 rule="copy" parents="11"/>
      <j2 rule="demod" parents="21 21 21 21 21 21 21 21"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      x v c(y v c(x v x)) = x v x
    ]]></literal>
    <justification jstring="[para(22(a,1),20(a,1,2,1)),demod(18(2),18(2),18(3),18(3),18(3)),flip(a)].">
      <j1 rule="para" parents="22 20"/>
      <j2 rule="demod" parents="18 18 18 18 18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(22(a,1),20(a,1)),demod(43(6)),flip(a)].">
      <j1 rule="para" parents="22 20"/>
      <j2 rule="demod" parents="43"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      c(c8) v c(c(c9) v c(c(c8) v c10)) != c(c8) v c(c(c10) v c(c(c8) v c9))
    ]]></literal>
    <attribute><![CDATA[
      answer(MOD_SS)
    ]]></attribute>
    <justification jstring="[back_demod(27),demod(44(11),18(9),44(23),18(21))].">
      <j1 rule="back_demod" parents="27"/>
      <j2 rule="demod" parents="44 18 44 18"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(MOD_SS)
    ]]></attribute>
    <justification jstring="[resolve(49,a,20,a)].">
      <j1 rule="resolve" parents="49 20"/>
    </justification>
  </clause>

</proof>

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

<comments><![CDATA[
% Proof 5 at 0.00 (+ 0.00) seconds: B_SS.
% Length of proof is 14.
% Level of proof is 5.
% Maximum clause weight is 21.
% Given clauses 6.
]]></comments>

  <clause id="2">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="6">
    <literal><![CDATA[
      x v (y ^ (x v z)) = x v (z ^ (x v y))
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

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

  <clause id="16">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[copy(2)].">
      <j1 rule="copy" parents="2"/>
    </justification>
  </clause>

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

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

  <clause id="20">
    <literal><![CDATA[
      x v c(c(y) v c(x v z)) = x v c(c(z) v c(x v y))
    ]]></literal>
    <justification jstring="[copy(6),demod(17(2),17(8))].">
      <j1 rule="copy" parents="6"/>
      <j2 rule="demod" parents="17 17"/>
    </justification>
  </clause>

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

  <clause id="22">
    <literal><![CDATA[
      x v c(c(x) v c(y)) = x
    ]]></literal>
    <justification jstring="[back_demod(16),demod(17(1))].">
      <j1 rule="back_demod" parents="16"/>
      <j2 rule="demod" parents="17"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      c(c(c4) v c(c4)) v c(c(c4) v c(c5)) != c4
    ]]></literal>
    <attribute><![CDATA[
      answer(B_SS)
    ]]></attribute>
    <justification jstring="[copy(9),demod(21(3),21(8),21(11))].">
      <j1 rule="copy" parents="9"/>
      <j2 rule="demod" parents="21 21 21"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      x v c(c(x) v y) = x
    ]]></literal>
    <justification jstring="[para(18(a,1),22(a,1,2,1,2))].">
      <j1 rule="para" parents="18 22"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      x v c(y v c(x v x)) = x v x
    ]]></literal>
    <justification jstring="[para(22(a,1),20(a,1,2,1)),demod(18(2),18(2),18(3),18(3),18(3)),flip(a)].">
      <j1 rule="para" parents="22 20"/>
      <j2 rule="demod" parents="18 18 18 18 18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(22(a,1),20(a,1)),demod(43(6)),flip(a)].">
      <j1 rule="para" parents="22 20"/>
      <j2 rule="demod" parents="43"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(B_SS)
    ]]></attribute>
    <justification jstring="[back_demod(25),demod(44(5),18(3),41(8)),xx(a)].">
      <j1 rule="back_demod" parents="25"/>
      <j2 rule="demod" parents="44 18 41"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

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

<comments><![CDATA[
% Proof 6 at 0.00 (+ 0.00) seconds: A_SS.
% Length of proof is 14.
% Level of proof is 5.
% Maximum clause weight is 21.
% Given clauses 6.
]]></comments>

  <clause id="2">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="6">
    <literal><![CDATA[
      x v (y ^ (x v z)) = x v (z ^ (x v y))
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="8">
    <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="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

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

  <clause id="16">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[copy(2)].">
      <j1 rule="copy" parents="2"/>
    </justification>
  </clause>

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

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

  <clause id="20">
    <literal><![CDATA[
      x v c(c(y) v c(x v z)) = x v c(c(z) v c(x v y))
    ]]></literal>
    <justification jstring="[copy(6),demod(17(2),17(8))].">
      <j1 rule="copy" parents="6"/>
      <j2 rule="demod" parents="17 17"/>
    </justification>
  </clause>

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

  <clause id="22">
    <literal><![CDATA[
      x v c(c(x) v c(y)) = x
    ]]></literal>
    <justification jstring="[back_demod(16),demod(17(1))].">
      <j1 rule="back_demod" parents="16"/>
      <j2 rule="demod" parents="17"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      c(c1) v c(c(c(c2) v c(c3)) v c(c(c2) v c(c3))) != c(c2) v c(c(c(c1) v c(c3)) v c(c(c1) v c(c3)))
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <justification jstring="[copy(8),demod(21(4),21(9),21(12),21(15),21(21),21(26),21(29),21(32)),flip(a)].">
      <j1 rule="copy" parents="8"/>
      <j2 rule="demod" parents="21 21 21 21 21 21 21 21"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      x v c(y v c(x v x)) = x v x
    ]]></literal>
    <justification jstring="[para(22(a,1),20(a,1,2,1)),demod(18(2),18(2),18(3),18(3),18(3)),flip(a)].">
      <j1 rule="para" parents="22 20"/>
      <j2 rule="demod" parents="18 18 18 18 18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(22(a,1),20(a,1)),demod(43(6)),flip(a)].">
      <j1 rule="para" parents="22 20"/>
      <j2 rule="demod" parents="43"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(A_SS)
    ]]></attribute>
    <justification jstring="[back_demod(24),demod(44(15),18(9),44(23),18(17),15(16)),xx(a)].">
      <j1 rule="back_demod" parents="24"/>
      <j2 rule="demod" parents="44 18 44 18 15"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

<proof number="7" length="18" max_count="21">

<comments><![CDATA[
% Proof 7 at 0.00 (+ 0.00) seconds: ONE_SS.
% Length of proof is 18.
% Level of proof is 6.
% Maximum clause weight is 21.
% Given clauses 7.
]]></comments>

  <clause id="2">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="5">
    <literal><![CDATA[
      x v c(x) = y v c(y)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="6">
    <literal><![CDATA[
      x v (y ^ (x v z)) = x v (z ^ (x v y))
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      f(c7,f(c7,c7)) != f(c6,f(c6,c6))
    ]]></literal>
    <attribute><![CDATA[
      answer(ONE_SS)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[copy(2)].">
      <j1 rule="copy" parents="2"/>
    </justification>
  </clause>

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

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

  <clause id="19">
    <literal><![CDATA[
      x v c(x) = y v c(y)
    ]]></literal>
    <justification jstring="[copy(5)].">
      <j1 rule="copy" parents="5"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x v c(c(y) v c(x v z)) = x v c(c(z) v c(x v y))
    ]]></literal>
    <justification jstring="[copy(6),demod(17(2),17(8))].">
      <j1 rule="copy" parents="6"/>
      <j2 rule="demod" parents="17 17"/>
    </justification>
  </clause>

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

  <clause id="22">
    <literal><![CDATA[
      x v c(c(x) v c(y)) = x
    ]]></literal>
    <justification jstring="[back_demod(16),demod(17(1))].">
      <j1 rule="back_demod" parents="16"/>
      <j2 rule="demod" parents="17"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      x v c(x) = c_0
    ]]></literal>
    <justification jstring="[new_symbol(19)].">
      <j1 rule="new_symbol" parents="19"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      c(c7) v c(c(c7) v c(c7)) != c(c6) v c(c(c6) v c(c6))
    ]]></literal>
    <attribute><![CDATA[
      answer(ONE_SS)
    ]]></attribute>
    <justification jstring="[copy(10),demod(21(4),21(7),21(13),21(16))].">
      <j1 rule="copy" parents="10"/>
      <j2 rule="demod" parents="21 21 21 21"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      x v c(y v c(x v x)) = x v x
    ]]></literal>
    <justification jstring="[para(22(a,1),20(a,1,2,1)),demod(18(2),18(2),18(3),18(3),18(3)),flip(a)].">
      <j1 rule="para" parents="22 20"/>
      <j2 rule="demod" parents="18 18 18 18 18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(22(a,1),20(a,1)),demod(43(6)),flip(a)].">
      <j1 rule="para" parents="22 20"/>
      <j2 rule="demod" parents="43"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="51">
    <literal><![CDATA[
      c(c7) v c7 != c(c6) v c6
    ]]></literal>
    <attribute><![CDATA[
      answer(ONE_SS)
    ]]></attribute>
    <justification jstring="[back_demod(26),demod(44(7),18(5),44(11),18(9))].">
      <j1 rule="back_demod" parents="26"/>
      <j2 rule="demod" parents="44 18 44 18"/>
    </justification>
  </clause>

  <clause id="57">
    <literal><![CDATA[
      c(x) v x = c_0
    ]]></literal>
    <justification jstring="[para(18(a,1),23(a,1,2))].">
      <j1 rule="para" parents="18 23"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(ONE_SS)
    ]]></attribute>
    <justification jstring="[back_demod(51),demod(57(4),57(5)),xx(a)].">
      <j1 rule="back_demod" parents="51"/>
      <j2 rule="demod" parents="57 57"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
