<?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="2">

<source>h1.out</source>

<heading><![CDATA[
Prover9 (32) version June-2007-, 4 June 2007.
Process 16627 was started by mccune on cleo,
Mon Jun 11 15:16:13 2007
The command was "prover9 -f h1.in".
]]></heading>

<proof number="1" length="41" max_count="35">

<comments><![CDATA[
% Proof 1 at 393.60 (+ 0.88) seconds: Winker2b.
% Length of proof is 41.
% Level of proof is 13.
% Maximum clause weight is 30.
% Given clauses 1269.
]]></comments>

  <clause id="5" type="goal">
    <literal><![CDATA[
      (exists a exists b (a + b)' = b')
    ]]></literal>
    <attribute><![CDATA[
      answer(Winker2b)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

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

  <clause id="7" type="assumption">
    <literal><![CDATA[
      (x + y) + z = x + (y + z)
    ]]></literal>
    <attribute><![CDATA[
      label(Associativity)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="8" type="assumption">
    <literal><![CDATA[
      ((x + y)' + (x + y')')' = x
    ]]></literal>
    <attribute><![CDATA[
      label(Robbins)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="9" type="assumption">
    <literal><![CDATA[
      h(x) = x + (x + x')'
    ]]></literal>
    <attribute><![CDATA[
      label(definition_h)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      x + (x + x')' = h(x)
    ]]></literal>
    <justification jstring="[copy(9),flip(a)].">
      <j1 rule="copy" parents="9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="15" type="deny">
    <literal><![CDATA[
      (x + y)' != y'
    ]]></literal>
    <attribute><![CDATA[
      answer(Winker2b)
    ]]></attribute>
    <justification jstring="[deny(5)].">
      <j1 rule="deny" parents="5"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      x + (y + z) = y + (x + z)
    ]]></literal>
    <justification jstring="[para(6(a,1),7(a,1,1)),rewrite([7(2)])].">
      <j1 rule="para" parents="6 7"/>
      <j2 rule="rewrite" parents="7"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      ((x + y)' + (y + x')')' = y
    ]]></literal>
    <justification jstring="[para(6(a,1),8(a,1,1,1,1))].">
      <j1 rule="para" parents="6 8"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      ((x + y)' + (y' + x)')' = x
    ]]></literal>
    <justification jstring="[para(6(a,1),8(a,1,1,2,1))].">
      <j1 rule="para" parents="6 8"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      ((x + (y + z))' + (x + (y + z'))')' = x + y
    ]]></literal>
    <justification jstring="[para(7(a,1),8(a,1,1,1,1)),rewrite([7(6)])].">
      <j1 rule="para" parents="7 8"/>
      <j2 rule="rewrite" parents="7"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      (x + ((x + y)' + (x + y')'')')' = (x + y)'
    ]]></literal>
    <justification jstring="[para(8(a,1),8(a,1,1,1))].">
      <j1 rule="para" parents="8 8"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      x + ((x + x')' + y) = h(x) + y
    ]]></literal>
    <justification jstring="[para(10(a,1),7(a,1,1)),flip(a)].">
      <j1 rule="para" parents="10 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      ((x + (y + z))' + (y + (x + z)')')' = y
    ]]></literal>
    <justification jstring="[para(17(a,1),8(a,1,1,1,1))].">
      <j1 rule="para" parents="17 8"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      x + (y + (x + x')') = y + h(x)
    ]]></literal>
    <justification jstring="[para(10(a,1),17(a,1,2)),flip(a)].">
      <j1 rule="para" parents="10 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      ((x + y')' + (y + x)')' = x
    ]]></literal>
    <justification jstring="[para(6(a,1),18(a,1,1))].">
      <j1 rule="para" parents="6 18"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      ((x + (y + z))' + (z + (x + y)')')' = z
    ]]></literal>
    <justification jstring="[para(7(a,1),18(a,1,1,1,1))].">
      <j1 rule="para" parents="7 18"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      ((x + y)' + (x + ((y + z)' + (z' + y)'))')' = x
    ]]></literal>
    <justification jstring="[para(19(a,1),8(a,1,1,2,1,2)),rewrite([6(11)])].">
      <j1 rule="para" parents="19 8"/>
      <j2 rule="rewrite" parents="6"/>
    </justification>
  </clause>

  <clause id="48">
    <literal><![CDATA[
      (x + (x + (y + (x + y')'))')' = (x + y')'
    ]]></literal>
    <justification jstring="[para(8(a,1),19(a,1,1,2)),rewrite([6(5),7(5),6(7)])].">
      <j1 rule="para" parents="8 19"/>
      <j2 rule="rewrite" parents="6 7 6"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      ((x + h(y))' + (y + (x + (y + y')')')')' = y
    ]]></literal>
    <justification jstring="[para(28(a,1),8(a,1,1,1,1))].">
      <j1 rule="para" parents="28 8"/>
    </justification>
  </clause>

  <clause id="122">
    <literal><![CDATA[
      ((x + (h(y) + z))' + (x + (y + ((y + y')' + z)'))')' = x + y
    ]]></literal>
    <justification jstring="[para(24(a,1),20(a,1,1,1,1,2))].">
      <j1 rule="para" parents="24 20"/>
    </justification>
  </clause>

  <clause id="159">
    <literal><![CDATA[
      ((h(x) + y)' + ((x + x')' + (x + y)')')' = (x + x')'
    ]]></literal>
    <justification jstring="[para(24(a,1),27(a,1,1,1,1))].">
      <j1 rule="para" parents="24 27"/>
    </justification>
  </clause>

  <clause id="169">
    <literal><![CDATA[
      (x + (x + ((y + z)' + (y + (z + x))'))')' = (y + (z + x))'
    ]]></literal>
    <justification jstring="[para(31(a,1),8(a,1,1,2)),rewrite([6(7),7(7),6(9)])].">
      <j1 rule="para" parents="31 8"/>
      <j2 rule="rewrite" parents="6 7 6"/>
    </justification>
  </clause>

  <clause id="846">
    <literal><![CDATA[
      (x + (x + h(x))')' = (x + x')'
    ]]></literal>
    <justification jstring="[para(10(a,1),48(a,1,1,2,1,2))].">
      <j1 rule="para" parents="10 48"/>
    </justification>
  </clause>

  <clause id="923">
    <literal><![CDATA[
      ((x + (x + h(x)))' + (x + x')')' = x
    ]]></literal>
    <justification jstring="[para(846(a,1),8(a,1,1,2))].">
      <j1 rule="para" parents="846 8"/>
    </justification>
  </clause>

  <clause id="935">
    <literal><![CDATA[
      (x + ((x + (x + h(x)))' + (x + x')'')')' = (x + (x + h(x)))'
    ]]></literal>
    <justification jstring="[para(846(a,1),21(a,1,1,2,1,2,1))].">
      <j1 rule="para" parents="846 21"/>
    </justification>
  </clause>

  <clause id="958">
    <literal><![CDATA[
      (x + (x + (h(x) + h(x)))')' = (x + x')'
    ]]></literal>
    <justification jstring="[para(846(a,1),48(a,1,1,2,1,2,2)),rewrite([7(6),28(6),846(12)])].">
      <j1 rule="para" parents="846 48"/>
      <j2 rule="rewrite" parents="7 28 846"/>
    </justification>
  </clause>

  <clause id="985">
    <literal><![CDATA[
      ((x + (x + (h(x) + h(x))))' + (x + x')')' = x
    ]]></literal>
    <justification jstring="[para(958(a,1),8(a,1,1,2))].">
      <j1 rule="para" parents="958 8"/>
    </justification>
  </clause>

  <clause id="1541">
    <literal><![CDATA[
      ((x + x)' + (h(x) + (x + (x + h(x)))')')' = x
    ]]></literal>
    <justification jstring="[para(923(a,1),68(a,1,1,2,1,2)),rewrite([6(6),6(10)])].">
      <j1 rule="para" parents="923 68"/>
      <j2 rule="rewrite" parents="6 6"/>
    </justification>
  </clause>

  <clause id="1550">
    <literal><![CDATA[
      (x + (h(x) + ((x + x)' + (x + (x + h(x)))'))')' = (x + x)'
    ]]></literal>
    <justification jstring="[para(1541(a,1),8(a,1,1,2)),rewrite([17(9),6(11)])].">
      <j1 rule="para" parents="1541 8"/>
      <j2 rule="rewrite" parents="17 6"/>
    </justification>
  </clause>

  <clause id="12225">
    <literal><![CDATA[
      ((x + (y + y))' + (x + (h(y) + (y + y)'))')' = x + y
    ]]></literal>
    <justification jstring="[para(30(a,1),122(a,1,1,2,1,2,2)),rewrite([6(10)])].">
      <j1 rule="para" parents="30 122"/>
      <j2 rule="rewrite" parents="6"/>
    </justification>
  </clause>

  <clause id="15528">
    <literal><![CDATA[
      ((x + (x + y))' + (y + (h(x) + (x + x)'))')' = y + x
    ]]></literal>
    <justification jstring="[para(6(a,1),12225(a,1,1,1,1)),rewrite([7(2)])].">
      <j1 rule="para" parents="6 12225"/>
      <j2 rule="rewrite" parents="7"/>
    </justification>
  </clause>

  <clause id="17256">
    <literal><![CDATA[
      (x + (x + (x + h(x)))')' = (x + x')'
    ]]></literal>
    <justification jstring="[para(1550(a,1),159(a,1,1,2,1,2)),rewrite([169(13),6(10),8(11),6(5)])].">
      <j1 rule="para" parents="1550 159"/>
      <j2 rule="rewrite" parents="169 6 8 6"/>
    </justification>
  </clause>

  <clause id="17258">
    <literal><![CDATA[
      ((x + (y + (y + (y + h(y)))'))' + (x + (y + y')')')' = x
    ]]></literal>
    <justification jstring="[para(17256(a,1),8(a,1,1,2,1,2))].">
      <j1 rule="para" parents="17256 8"/>
    </justification>
  </clause>

  <clause id="17259">
    <literal><![CDATA[
      ((x + (x + (x + h(x))))' + (x + x')')' = x
    ]]></literal>
    <justification jstring="[para(17256(a,1),8(a,1,1,2))].">
      <j1 rule="para" parents="17256 8"/>
    </justification>
  </clause>

  <clause id="17503">
    <literal><![CDATA[
      ((x + (y + (y + (y + (h(y) + h(y))))'))' + (x + (y + y')')')' = x
    ]]></literal>
    <justification jstring="[para(17259(a,1),45(a,1,1,2,1,2,2)),rewrite([6(13),7(13),7(12),7(11),28(11),6(12),6(15)])].">
      <j1 rule="para" parents="17259 45"/>
      <j2 rule="rewrite" parents="6 7 7 7 28 6 6"/>
    </justification>
  </clause>

  <clause id="23454">
    <literal><![CDATA[
      (x + (x + ((x + (x + h(x)))' + (x + (x + (h(x) + h(x))))'))')' = (x + (x + (h(x) + h(x))))'
    ]]></literal>
    <justification jstring="[para(985(a,1),17258(a,1,1,2)),rewrite([6(12),7(12),6(14)])].">
      <j1 rule="para" parents="985 17258"/>
      <j2 rule="rewrite" parents="6 7 6"/>
    </justification>
  </clause>

  <clause id="33336">
    <literal><![CDATA[
      (x + (x + (h(x) + h(x))))' = (x + (x + h(x)))'
    ]]></literal>
    <justification jstring="[para(923(a,1),17503(a,1,1,2)),rewrite([17(12),6(14),23454(15)])].">
      <j1 rule="para" parents="923 17503"/>
      <j2 rule="rewrite" parents="17 6 23454"/>
    </justification>
  </clause>

  <clause id="33347">
    <literal><![CDATA[
      h(x) + h(x) = x + h(x)
    ]]></literal>
    <justification jstring="[para(33336(a,1),31(a,1,1,1)),rewrite([7(10),15528(13),6(2)]),flip(a)].">
      <j1 rule="para" parents="33336 31"/>
      <j2 rule="rewrite" parents="7 15528 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="33352">
    <literal><![CDATA[
      (x + (x + (x + h(x))))' = (x + (x + h(x)))'
    ]]></literal>
    <justification jstring="[para(33336(a,1),21(a,1,1,2,1,1)),rewrite([33347(7),17256(10),935(12),33347(7)]),flip(a)].">
      <j1 rule="para" parents="33336 21"/>
      <j2 rule="rewrite" parents="33347 17256 935 33347"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="33353">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(Winker2b)
    ]]></attribute>
    <justification jstring="[resolve(33352,a,15,a)].">
      <j1 rule="resolve" parents="33352 15"/>
    </justification>
  </clause>

</proof>

<proof number="2" length="59" max_count="35">

<comments><![CDATA[
% Proof 2 at 524.26 (+ 1.06) seconds: Winker1b.
% Length of proof is 59.
% Level of proof is 22.
% Maximum clause weight is 30.
% Given clauses 1591.
]]></comments>

  <clause id="3" type="goal">
    <literal><![CDATA[
      (exists a exists b a + b = b)
    ]]></literal>
    <attribute><![CDATA[
      answer(Winker1b)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

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

  <clause id="7" type="assumption">
    <literal><![CDATA[
      (x + y) + z = x + (y + z)
    ]]></literal>
    <attribute><![CDATA[
      label(Associativity)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="8" type="assumption">
    <literal><![CDATA[
      ((x + y)' + (x + y')')' = x
    ]]></literal>
    <attribute><![CDATA[
      label(Robbins)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="9" type="assumption">
    <literal><![CDATA[
      h(x) = x + (x + x')'
    ]]></literal>
    <attribute><![CDATA[
      label(definition_h)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      x + (x + x')' = h(x)
    ]]></literal>
    <justification jstring="[copy(9),flip(a)].">
      <j1 rule="copy" parents="9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13" type="deny">
    <literal><![CDATA[
      x + y != y
    ]]></literal>
    <attribute><![CDATA[
      answer(Winker1b)
    ]]></attribute>
    <justification jstring="[deny(3)].">
      <j1 rule="deny" parents="3"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      x + (y + z) = y + (x + z)
    ]]></literal>
    <justification jstring="[para(6(a,1),7(a,1,1)),rewrite([7(2)])].">
      <j1 rule="para" parents="6 7"/>
      <j2 rule="rewrite" parents="7"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      ((x + y)' + (y + x')')' = y
    ]]></literal>
    <justification jstring="[para(6(a,1),8(a,1,1,1,1))].">
      <j1 rule="para" parents="6 8"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      ((x + y)' + (y' + x)')' = x
    ]]></literal>
    <justification jstring="[para(6(a,1),8(a,1,1,2,1))].">
      <j1 rule="para" parents="6 8"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      ((x + (y + z))' + (x + (y + z'))')' = x + y
    ]]></literal>
    <justification jstring="[para(7(a,1),8(a,1,1,1,1)),rewrite([7(6)])].">
      <j1 rule="para" parents="7 8"/>
      <j2 rule="rewrite" parents="7"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      x + ((x + x')' + y) = h(x) + y
    ]]></literal>
    <justification jstring="[para(10(a,1),7(a,1,1)),flip(a)].">
      <j1 rule="para" parents="10 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      ((x + (y + z))' + (y + (x + z)')')' = y
    ]]></literal>
    <justification jstring="[para(17(a,1),8(a,1,1,1,1))].">
      <j1 rule="para" parents="17 8"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      x + (y + (x + x')') = y + h(x)
    ]]></literal>
    <justification jstring="[para(10(a,1),17(a,1,2)),flip(a)].">
      <j1 rule="para" parents="10 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      ((x + y')' + (y + x)')' = x
    ]]></literal>
    <justification jstring="[para(6(a,1),18(a,1,1))].">
      <j1 rule="para" parents="6 18"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      ((x + (y + z))' + (z + (x + y)')')' = z
    ]]></literal>
    <justification jstring="[para(7(a,1),18(a,1,1,1,1))].">
      <j1 rule="para" parents="7 18"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      ((x + (y + z))' + (y + (z + x'))')' = y + z
    ]]></literal>
    <justification jstring="[para(7(a,1),18(a,1,1,2,1))].">
      <j1 rule="para" parents="7 18"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      ((x + (y + z))' + (x + (z + y'))')' = x + z
    ]]></literal>
    <justification jstring="[para(17(a,1),18(a,1,1,1,1)),rewrite([7(6)])].">
      <j1 rule="para" parents="17 18"/>
      <j2 rule="rewrite" parents="7"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      ((x + y)' + (x + ((y + z)' + (z' + y)'))')' = x
    ]]></literal>
    <justification jstring="[para(19(a,1),8(a,1,1,2,1,2)),rewrite([6(11)])].">
      <j1 rule="para" parents="19 8"/>
      <j2 rule="rewrite" parents="6"/>
    </justification>
  </clause>

  <clause id="48">
    <literal><![CDATA[
      (x + (x + (y + (x + y')'))')' = (x + y')'
    ]]></literal>
    <justification jstring="[para(8(a,1),19(a,1,1,2)),rewrite([6(5),7(5),6(7)])].">
      <j1 rule="para" parents="8 19"/>
      <j2 rule="rewrite" parents="6 7 6"/>
    </justification>
  </clause>

  <clause id="58">
    <literal><![CDATA[
      x + (y + ((x + x')' + z)) = y + (h(x) + z)
    ]]></literal>
    <justification jstring="[para(24(a,1),17(a,1,2)),flip(a)].">
      <j1 rule="para" parents="24 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      x + (y + (z + (x + x')')) = y + (z + h(x))
    ]]></literal>
    <justification jstring="[para(7(a,1),28(a,1,2)),rewrite([7(9)])].">
      <j1 rule="para" parents="7 28"/>
      <j2 rule="rewrite" parents="7"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      ((x + h(y))' + (y + (x + (y + y')')')')' = y
    ]]></literal>
    <justification jstring="[para(28(a,1),8(a,1,1,1,1))].">
      <j1 rule="para" parents="28 8"/>
    </justification>
  </clause>

  <clause id="122">
    <literal><![CDATA[
      ((x + (h(y) + z))' + (x + (y + ((y + y')' + z)'))')' = x + y
    ]]></literal>
    <justification jstring="[para(24(a,1),20(a,1,1,1,1,2))].">
      <j1 rule="para" parents="24 20"/>
    </justification>
  </clause>

  <clause id="159">
    <literal><![CDATA[
      ((h(x) + y)' + ((x + x')' + (x + y)')')' = (x + x')'
    ]]></literal>
    <justification jstring="[para(24(a,1),27(a,1,1,1,1))].">
      <j1 rule="para" parents="24 27"/>
    </justification>
  </clause>

  <clause id="169">
    <literal><![CDATA[
      (x + (x + ((y + z)' + (y + (z + x))'))')' = (y + (z + x))'
    ]]></literal>
    <justification jstring="[para(31(a,1),8(a,1,1,2)),rewrite([6(7),7(7),6(9)])].">
      <j1 rule="para" parents="31 8"/>
      <j2 rule="rewrite" parents="6 7 6"/>
    </justification>
  </clause>

  <clause id="483">
    <literal><![CDATA[
      ((x + (y + z))' + (z + (x' + y))')' = y + z
    ]]></literal>
    <justification jstring="[para(6(a,1),32(a,1,1,2,1)),rewrite([7(6)])].">
      <j1 rule="para" parents="6 32"/>
      <j2 rule="rewrite" parents="7"/>
    </justification>
  </clause>

  <clause id="792">
    <literal><![CDATA[
      ((x + (y + z))' + (x + (y' + z))')' = x + z
    ]]></literal>
    <justification jstring="[para(6(a,1),39(a,1,1,2,1,2))].">
      <j1 rule="para" parents="6 39"/>
    </justification>
  </clause>

  <clause id="846">
    <literal><![CDATA[
      (x + (x + h(x))')' = (x + x')'
    ]]></literal>
    <justification jstring="[para(10(a,1),48(a,1,1,2,1,2))].">
      <j1 rule="para" parents="10 48"/>
    </justification>
  </clause>

  <clause id="923">
    <literal><![CDATA[
      ((x + (x + h(x)))' + (x + x')')' = x
    ]]></literal>
    <justification jstring="[para(846(a,1),8(a,1,1,2))].">
      <j1 rule="para" parents="846 8"/>
    </justification>
  </clause>

  <clause id="958">
    <literal><![CDATA[
      (x + (x + (h(x) + h(x)))')' = (x + x')'
    ]]></literal>
    <justification jstring="[para(846(a,1),48(a,1,1,2,1,2,2)),rewrite([7(6),28(6),846(12)])].">
      <j1 rule="para" parents="846 48"/>
      <j2 rule="rewrite" parents="7 28 846"/>
    </justification>
  </clause>

  <clause id="985">
    <literal><![CDATA[
      ((x + (x + (h(x) + h(x))))' + (x + x')')' = x
    ]]></literal>
    <justification jstring="[para(958(a,1),8(a,1,1,2))].">
      <j1 rule="para" parents="958 8"/>
    </justification>
  </clause>

  <clause id="1541">
    <literal><![CDATA[
      ((x + x)' + (h(x) + (x + (x + h(x)))')')' = x
    ]]></literal>
    <justification jstring="[para(923(a,1),68(a,1,1,2,1,2)),rewrite([6(6),6(10)])].">
      <j1 rule="para" parents="923 68"/>
      <j2 rule="rewrite" parents="6 6"/>
    </justification>
  </clause>

  <clause id="1550">
    <literal><![CDATA[
      (x + (h(x) + ((x + x)' + (x + (x + h(x)))'))')' = (x + x)'
    ]]></literal>
    <justification jstring="[para(1541(a,1),8(a,1,1,2)),rewrite([17(9),6(11)])].">
      <j1 rule="para" parents="1541 8"/>
      <j2 rule="rewrite" parents="17 6"/>
    </justification>
  </clause>

  <clause id="2336">
    <literal><![CDATA[
      ((x + (h(y) + z))' + ((y + y')' + (z + (x' + y)))')' = h(y) + z
    ]]></literal>
    <justification jstring="[para(24(a,1),483(a,1,1,1,1,2)),rewrite([7(11),24(19)])].">
      <j1 rule="para" parents="24 483"/>
      <j2 rule="rewrite" parents="7 24"/>
    </justification>
  </clause>

  <clause id="12225">
    <literal><![CDATA[
      ((x + (y + y))' + (x + (h(y) + (y + y)'))')' = x + y
    ]]></literal>
    <justification jstring="[para(30(a,1),122(a,1,1,2,1,2,2)),rewrite([6(10)])].">
      <j1 rule="para" parents="30 122"/>
      <j2 rule="rewrite" parents="6"/>
    </justification>
  </clause>

  <clause id="15528">
    <literal><![CDATA[
      ((x + (x + y))' + (y + (h(x) + (x + x)'))')' = y + x
    ]]></literal>
    <justification jstring="[para(6(a,1),12225(a,1,1,1,1)),rewrite([7(2)])].">
      <j1 rule="para" parents="6 12225"/>
      <j2 rule="rewrite" parents="7"/>
    </justification>
  </clause>

  <clause id="17256">
    <literal><![CDATA[
      (x + (x + (x + h(x)))')' = (x + x')'
    ]]></literal>
    <justification jstring="[para(1550(a,1),159(a,1,1,2,1,2)),rewrite([169(13),6(10),8(11),6(5)])].">
      <j1 rule="para" parents="1550 159"/>
      <j2 rule="rewrite" parents="169 6 8 6"/>
    </justification>
  </clause>

  <clause id="17258">
    <literal><![CDATA[
      ((x + (y + (y + (y + h(y)))'))' + (x + (y + y')')')' = x
    ]]></literal>
    <justification jstring="[para(17256(a,1),8(a,1,1,2,1,2))].">
      <j1 rule="para" parents="17256 8"/>
    </justification>
  </clause>

  <clause id="17259">
    <literal><![CDATA[
      ((x + (x + (x + h(x))))' + (x + x')')' = x
    ]]></literal>
    <justification jstring="[para(17256(a,1),8(a,1,1,2))].">
      <j1 rule="para" parents="17256 8"/>
    </justification>
  </clause>

  <clause id="17503">
    <literal><![CDATA[
      ((x + (y + (y + (y + (h(y) + h(y))))'))' + (x + (y + y')')')' = x
    ]]></literal>
    <justification jstring="[para(17259(a,1),45(a,1,1,2,1,2,2)),rewrite([6(13),7(13),7(12),7(11),28(11),6(12),6(15)])].">
      <j1 rule="para" parents="17259 45"/>
      <j2 rule="rewrite" parents="6 7 7 7 28 6 6"/>
    </justification>
  </clause>

  <clause id="23454">
    <literal><![CDATA[
      (x + (x + ((x + (x + h(x)))' + (x + (x + (h(x) + h(x))))'))')' = (x + (x + (h(x) + h(x))))'
    ]]></literal>
    <justification jstring="[para(985(a,1),17258(a,1,1,2)),rewrite([6(12),7(12),6(14)])].">
      <j1 rule="para" parents="985 17258"/>
      <j2 rule="rewrite" parents="6 7 6"/>
    </justification>
  </clause>

  <clause id="31766">
    <literal><![CDATA[
      ((x + (y + h(z)))' + ((z + z')' + (y + (x' + z)))')' = h(z) + y
    ]]></literal>
    <justification jstring="[para(6(a,1),2336(a,1,1,1,1,2))].">
      <j1 rule="para" parents="6 2336"/>
    </justification>
  </clause>

  <clause id="33336">
    <literal><![CDATA[
      (x + (x + (h(x) + h(x))))' = (x + (x + h(x)))'
    ]]></literal>
    <justification jstring="[para(923(a,1),17503(a,1,1,2)),rewrite([17(12),6(14),23454(15)])].">
      <j1 rule="para" parents="923 17503"/>
      <j2 rule="rewrite" parents="17 6 23454"/>
    </justification>
  </clause>

  <clause id="33347">
    <literal><![CDATA[
      h(x) + h(x) = x + h(x)
    ]]></literal>
    <justification jstring="[para(33336(a,1),31(a,1,1,1)),rewrite([7(10),15528(13),6(2)]),flip(a)].">
      <j1 rule="para" parents="33336 31"/>
      <j2 rule="rewrite" parents="7 15528 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="34034">
    <literal><![CDATA[
      h(x) + (h(x) + y) = x + (h(x) + y)
    ]]></literal>
    <justification jstring="[para(33347(a,1),7(a,1,1)),rewrite([7(3)]),flip(a)].">
      <j1 rule="para" parents="33347 7"/>
      <j2 rule="rewrite" parents="7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="36456">
    <literal><![CDATA[
      h(x) + (y + h(x)) = x + (h(x) + y)
    ]]></literal>
    <justification jstring="[para(6(a,1),34034(a,1,2))].">
      <j1 rule="para" parents="6 34034"/>
    </justification>
  </clause>

  <clause id="36459">
    <literal><![CDATA[
      h(x) + (y + (h(x) + z)) = x + (h(x) + (y + z))
    ]]></literal>
    <justification jstring="[para(17(a,1),34034(a,1,2))].">
      <j1 rule="para" parents="17 34034"/>
    </justification>
  </clause>

  <clause id="36469">
    <literal><![CDATA[
      x + (y + (h(x) + ((x + x')' + z))) = y + (x + (h(x) + z))
    ]]></literal>
    <justification jstring="[para(34034(a,1),58(a,2,2)),rewrite([17(6)])].">
      <j1 rule="para" parents="34034 58"/>
      <j2 rule="rewrite" parents="17"/>
    </justification>
  </clause>

  <clause id="41462">
    <literal><![CDATA[
      x + (y + (h(x) + (z + (x + x')'))) = y + (x + (h(x) + z))
    ]]></literal>
    <justification jstring="[para(6(a,1),36469(a,1,2,2,2))].">
      <j1 rule="para" parents="6 36469"/>
    </justification>
  </clause>

  <clause id="41463">
    <literal><![CDATA[
      x + (h(x) + (y + ((x + x')' + z))) = y + (x + (h(x) + z))
    ]]></literal>
    <justification jstring="[para(17(a,1),36469(a,1,2))].">
      <j1 rule="para" parents="17 36469"/>
    </justification>
  </clause>

  <clause id="42432">
    <literal><![CDATA[
      x + (y + (h(y) + z)) = y + (x + (z + (h(y) + (y + y')')))
    ]]></literal>
    <justification jstring="[para(17(a,1),41462(a,1,2,2)),flip(a)].">
      <j1 rule="para" parents="17 41462"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="43117">
    <literal><![CDATA[
      x + (h(x) + (y + z)) = x + (z + (y + (h(x) + (x + x')')))
    ]]></literal>
    <justification jstring="[para(42432(a,1),6(a,1)),rewrite([7(12),7(11)]),flip(a)].">
      <j1 rule="para" parents="42432 6"/>
      <j2 rule="rewrite" parents="7 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="44016">
    <literal><![CDATA[
      h(x) + (y + (z + (h(x) + (x + x')'))) = z + (x + (h(x) + y))
    ]]></literal>
    <justification jstring="[para(43117(a,1),41463(a,1)),rewrite([7(11),24(12)])].">
      <j1 rule="para" parents="43117 41463"/>
      <j2 rule="rewrite" parents="7 24"/>
    </justification>
  </clause>

  <clause id="44684">
    <literal><![CDATA[
      (x + x')' + (y + (h(y) + x)) = y + (h(y) + h(x))
    ]]></literal>
    <justification jstring="[para(24(a,1),44016(a,1,2)),rewrite([36459(9),67(8),36456(5)]),flip(a)].">
      <j1 rule="para" parents="24 44016"/>
      <j2 rule="rewrite" parents="36459 67 36456"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="44944">
    <literal><![CDATA[
      (x + x')' + (h(y) + x) = h(x) + h(y)
    ]]></literal>
    <justification jstring="[para(44684(a,1),792(a,1,1,1,1)),rewrite([17(12),31766(16)]),flip(a)].">
      <j1 rule="para" parents="44684 792"/>
      <j2 rule="rewrite" parents="17 31766"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="45195">
    <literal><![CDATA[
      (x + x')' + (y + (h(z) + x)) = y + (h(x) + h(z))
    ]]></literal>
    <justification jstring="[para(44944(a,1),17(a,1,2)),flip(a)].">
      <j1 rule="para" parents="44944 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="46348">
    <literal><![CDATA[
      (x + x')' + (y + (x + h(x))) = y + (x + h(x))
    ]]></literal>
    <justification jstring="[para(33347(a,1),45195(a,2,2)),rewrite([6(5)])].">
      <j1 rule="para" parents="33347 45195"/>
      <j2 rule="rewrite" parents="6"/>
    </justification>
  </clause>

  <clause id="46349">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(Winker1b)
    ]]></attribute>
    <justification jstring="[resolve(46348,a,13,a)].">
      <j1 rule="resolve" parents="46348 13"/>
    </justification>
  </clause>

</proof>

</proofs>
