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<!DOCTYPE proofs SYSTEM "proof3.dtd">

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

<proofs number_of_proofs="1">

<source>csd.out</source>

<heading><![CDATA[
Prover9 (32) version August-2006A, August 2006.
Process 11450 was started by mccune on cleo.thornwood,
Sat Aug 12 21:01:21 2006
The command was "/home/mccune/bin/prover9 -f head.in csd.in".
]]></heading>

<proof number="1" length="48" max_count="37">

<comments><![CDATA[
% Proof 1 at 0.37 (+ 0.00) seconds.
% Length of proof is 48.
% Level of proof is 14.
% Maximum clause weight is 20.
% Given clauses 243.
]]></comments>

  <clause id="4">
    <literal><![CDATA[
      (all x all y all z (CS(x,y,z) <-> (x v y) ^ (y v z) = y & (x v z) ^ y = y))
    ]]></literal>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="5">
    <literal><![CDATA[
      (all x all y all z (D(x,y,z) <-> x ^ z <= y & y <= x v z))
    ]]></literal>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6">
    <literal><![CDATA[
      (all x all y (x <= y <-> x ^ y = x))
    ]]></literal>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      (all a all x all b (CS(a,x,b) -> D(a,x,b)))
    ]]></literal>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      -CS(x,y,z)
    ]]></literal>
    <literal><![CDATA[
      (x v y) ^ (y v z) = y
    ]]></literal>
    <justification jstring="[clausify(4)].">
      <j1 rule="clausify" parents="4"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      -CS(x,y,z)
    ]]></literal>
    <literal><![CDATA[
      (x v z) ^ y = y
    ]]></literal>
    <justification jstring="[clausify(4)].">
      <j1 rule="clausify" parents="4"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      CS(c1,c2,c3)
    ]]></literal>
    <justification jstring="[deny(7)].">
      <j1 rule="deny" parents="7"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      x ^ y = y ^ x
    ]]></literal>
    <attribute><![CDATA[
      label("commutativity_meet")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <attribute><![CDATA[
      label("commutativity_join")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      (x ^ y) ^ z = x ^ (y ^ z)
    ]]></literal>
    <attribute><![CDATA[
      label("associativity_meet")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      (x v y) v z = x v (y v z)
    ]]></literal>
    <attribute><![CDATA[
      label("associativity_join")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      (x v y) ^ x = x
    ]]></literal>
    <attribute><![CDATA[
      label("absorption_1")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[copy(28),rewrite(24(2))].">
      <j1 rule="copy" parents="28"/>
      <j2 rule="rewrite" parents="24"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      (x ^ y) v x = x
    ]]></literal>
    <attribute><![CDATA[
      label("absorption_2")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[copy(30),rewrite(25(2))].">
      <j1 rule="copy" parents="30"/>
      <j2 rule="rewrite" parents="25"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      -D(x,y,z)
    ]]></literal>
    <literal><![CDATA[
      x ^ z <= y
    ]]></literal>
    <justification jstring="[clausify(5)].">
      <j1 rule="clausify" parents="5"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      D(x,y,z)
    ]]></literal>
    <literal><![CDATA[
      -(x ^ z <= y)
    ]]></literal>
    <literal><![CDATA[
      -(y <= x v z)
    ]]></literal>
    <justification jstring="[clausify(5)].">
      <j1 rule="clausify" parents="5"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      -(x <= y)
    ]]></literal>
    <literal><![CDATA[
      x ^ y = x
    ]]></literal>
    <justification jstring="[clausify(6)].">
      <j1 rule="clausify" parents="6"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      x <= y
    ]]></literal>
    <literal><![CDATA[
      x ^ y != x
    ]]></literal>
    <justification jstring="[clausify(6)].">
      <j1 rule="clausify" parents="6"/>
    </justification>
  </clause>

  <clause id="37">
    <literal><![CDATA[
      -D(c1,c2,c3)
    ]]></literal>
    <justification jstring="[deny(7)].">
      <j1 rule="deny" parents="7"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      (c1 v c2) ^ (c2 v c3) = c2
    ]]></literal>
    <justification jstring="[resolve(23,a,21,a)].">
      <j1 rule="resolve" parents="23 21"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      (c1 v c3) ^ c2 = c2
    ]]></literal>
    <justification jstring="[resolve(23,a,22,a)].">
      <j1 rule="resolve" parents="23 22"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      c2 ^ (c1 v c3) = c2
    ]]></literal>
    <justification jstring="[copy(39),rewrite(24(5))].">
      <j1 rule="copy" parents="39"/>
      <j2 rule="rewrite" parents="24"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      x ^ (y ^ z) = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(24(a,1),26(a,1,1)),rewrite(26(2))].">
      <j1 rule="para" parents="24 26"/>
      <j2 rule="rewrite" parents="26"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(25(a,1),29(a,1,2))].">
      <j1 rule="para" parents="25 29"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      x ^ ((x v y) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(29(a,1),26(a,1,1)),flip(a)].">
      <j1 rule="para" parents="29 26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(24(a,1),31(a,1,2))].">
      <j1 rule="para" parents="24 31"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      x ^ x = x
    ]]></literal>
    <justification jstring="[para(31(a,1),29(a,1,2))].">
      <j1 rule="para" parents="31 29"/>
    </justification>
  </clause>

  <clause id="57">
    <literal><![CDATA[
      D(x ^ y,z,u)
    ]]></literal>
    <literal><![CDATA[
      -(x ^ (y ^ u) <= z)
    ]]></literal>
    <literal><![CDATA[
      -(z <= (x ^ y) v u)
    ]]></literal>
    <justification jstring="[para(26(a,1),34(b,1))].">
      <j1 rule="para" parents="26 34"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      x <= x v y
    ]]></literal>
    <justification jstring="[hyper(36,b,29,a)].">
      <j1 rule="hyper" parents="36 29"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      c2 <= c1 v c3
    ]]></literal>
    <justification jstring="[hyper(36,b,40,a)].">
      <j1 rule="hyper" parents="36 40"/>
    </justification>
  </clause>

  <clause id="71">
    <literal><![CDATA[
      x ^ (x ^ y) = x ^ y
    ]]></literal>
    <justification jstring="[para(53(a,1),26(a,1,1)),flip(a)].">
      <j1 rule="para" parents="53 26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="87">
    <literal><![CDATA[
      x <= y v x
    ]]></literal>
    <justification jstring="[para(25(a,1),61(a,2))].">
      <j1 rule="para" parents="25 61"/>
    </justification>
  </clause>

  <clause id="93">
    <literal><![CDATA[
      x <= y v (z v x)
    ]]></literal>
    <justification jstring="[para(27(a,1),87(a,2))].">
      <j1 rule="para" parents="27 87"/>
    </justification>
  </clause>

  <clause id="94">
    <literal><![CDATA[
      x ^ y <= x
    ]]></literal>
    <justification jstring="[para(31(a,1),87(a,2))].">
      <j1 rule="para" parents="31 87"/>
    </justification>
  </clause>

  <clause id="98">
    <literal><![CDATA[
      x ^ y <= y
    ]]></literal>
    <justification jstring="[para(24(a,1),94(a,1))].">
      <j1 rule="para" parents="24 94"/>
    </justification>
  </clause>

  <clause id="100">
    <literal><![CDATA[
      x ^ (y ^ z) <= y
    ]]></literal>
    <justification jstring="[para(42(a,1),94(a,1))].">
      <j1 rule="para" parents="42 94"/>
    </justification>
  </clause>

  <clause id="124">
    <literal><![CDATA[
      x ^ y <= z v y
    ]]></literal>
    <justification jstring="[para(49(a,1),93(a,2,2))].">
      <j1 rule="para" parents="49 93"/>
    </justification>
  </clause>

  <clause id="129">
    <literal><![CDATA[
      c1 ^ (c2 v c3) = c1 ^ c2
    ]]></literal>
    <justification jstring="[para(38(a,1),46(a,1,2)),flip(a)].">
      <j1 rule="para" parents="38 46"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="380">
    <literal><![CDATA[
      D(x ^ y,y ^ z,z)
    ]]></literal>
    <justification jstring="[hyper(57,b,98,a,c,124,a)].">
      <j1 rule="hyper" parents="57 98 124"/>
    </justification>
  </clause>

  <clause id="445">
    <literal><![CDATA[
      D(x ^ y,z ^ y,z)
    ]]></literal>
    <justification jstring="[para(24(a,1),380(a,2))].">
      <j1 rule="para" parents="24 380"/>
    </justification>
  </clause>

  <clause id="488">
    <literal><![CDATA[
      D(x,y ^ (z v x),y)
    ]]></literal>
    <justification jstring="[para(45(a,1),445(a,1))].">
      <j1 rule="para" parents="45 445"/>
    </justification>
  </clause>

  <clause id="1371">
    <literal><![CDATA[
      D(c3,c1 ^ c2,c1)
    ]]></literal>
    <justification jstring="[para(129(a,1),488(a,2))].">
      <j1 rule="para" parents="129 488"/>
    </justification>
  </clause>

  <clause id="1433">
    <literal><![CDATA[
      c1 ^ c3 <= c1 ^ c2
    ]]></literal>
    <justification jstring="[hyper(32,a,1371,a),rewrite(24(3))].">
      <j1 rule="hyper" parents="32 1371"/>
      <j2 rule="rewrite" parents="24"/>
    </justification>
  </clause>

  <clause id="1445">
    <literal><![CDATA[
      c1 ^ (c2 ^ c3) = c1 ^ c3
    ]]></literal>
    <justification jstring="[hyper(35,a,1433,a),rewrite(24(7),42(7),24(6),42(6),24(5),71(7))].">
      <j1 rule="hyper" parents="35 1433"/>
      <j2 rule="rewrite" parents="24 42 24 42 24 71"/>
    </justification>
  </clause>

  <clause id="3975">
    <literal><![CDATA[
      c1 ^ c3 <= c2
    ]]></literal>
    <justification jstring="[para(1445(a,1),100(a,1))].">
      <j1 rule="para" parents="1445 100"/>
    </justification>
  </clause>

  <clause id="4044">
    <literal><![CDATA[
      D(c1,c2,c3)
    ]]></literal>
    <justification jstring="[hyper(34,b,3975,a,c,67,a)].">
      <j1 rule="hyper" parents="34 3975 67"/>
    </justification>
  </clause>

  <clause id="4045">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(4044,a,37,a)].">
      <j1 rule="resolve" parents="4044 37"/>
    </justification>
  </clause>

</proof>

</proofs>
