<?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="3">

<source>IL.out</source>

<heading><![CDATA[
Prover9 (64) version 2009-02A, February 2009.
Process 1150 was started by veroff on proof,
Sat May  7 08:59:26 2011
The command was "prover9 -f IL.in".
]]></heading>

<proof number="1" length="81" max_count="22">

<comments><![CDATA[
% Proof 1 at 0.24 (+ 0.01) seconds: "HIF5".
% Length of proof is 81.
% Level of proof is 20.
% Maximum clause weight is 22.
% Given clauses 141.

]]></comments>

  <clause id="3" type="goal">
    <literal><![CDATA[
      P(k(i(n(a(x,y)),k(n(x),n(y))),i(k(n(x),n(y)),n(a(x,y)))))
    ]]></literal>
    <attribute><![CDATA[
      label("HIF5")
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="4" type="assumption">
    <literal><![CDATA[
      -P(x)
    ]]></literal>
    <literal><![CDATA[
      -P(i(x,y))
    ]]></literal>
    <literal><![CDATA[
      P(y)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="5" type="assumption">
    <literal><![CDATA[
      P(i(x,i(y,x)))
    ]]></literal>
    <attribute><![CDATA[
      label("A1")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6" type="assumption">
    <literal><![CDATA[
      P(i(i(x,i(y,z)),i(i(x,y),i(x,z))))
    ]]></literal>
    <attribute><![CDATA[
      label("A2")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7" type="assumption">
    <literal><![CDATA[
      P(i(k(x,y),x))
    ]]></literal>
    <attribute><![CDATA[
      label("A3")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="8" type="assumption">
    <literal><![CDATA[
      P(i(k(x,y),y))
    ]]></literal>
    <attribute><![CDATA[
      label("A4")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="9" type="assumption">
    <literal><![CDATA[
      P(i(i(x,y),i(i(x,z),i(x,k(y,z)))))
    ]]></literal>
    <attribute><![CDATA[
      label("A5")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="10" type="assumption">
    <literal><![CDATA[
      P(i(x,a(x,y)))
    ]]></literal>
    <attribute><![CDATA[
      label("A6")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11" type="assumption">
    <literal><![CDATA[
      P(i(x,a(y,x)))
    ]]></literal>
    <attribute><![CDATA[
      label("A7")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="12" type="assumption">
    <literal><![CDATA[
      P(i(i(x,y),i(i(z,y),i(a(x,z),y))))
    ]]></literal>
    <attribute><![CDATA[
      label("A8")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="13" type="assumption">
    <literal><![CDATA[
      P(i(i(x,n(y)),i(y,n(x))))
    ]]></literal>
    <attribute><![CDATA[
      label("A9")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="14" type="assumption">
    <literal><![CDATA[
      P(i(n(x),i(x,y)))
    ]]></literal>
    <attribute><![CDATA[
      label("A10")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="17" type="deny">
    <literal><![CDATA[
      -P(k(i(n(a(c7,c8)),k(n(c7),n(c8))),i(k(n(c7),n(c8)),n(a(c7,c8)))))
    ]]></literal>
    <attribute><![CDATA[
      label("HIF5")
    ]]></attribute>
    <attribute><![CDATA[
      answer("HIF5")
    ]]></attribute>
    <justification jstring="[deny(3)].">
      <j1 rule="deny" parents="3"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      P(i(x,i(y,i(z,y))))
    ]]></literal>
    <justification jstring="[hyper(4,a,5,a,b,5,a)].">
      <j1 rule="hyper" parents="4 5 5"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      P(i(i(i(x,i(y,z)),i(x,y)),i(i(x,i(y,z)),i(x,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,6,a,b,6,a)].">
      <j1 rule="hyper" parents="4 6 6"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      P(i(x,i(i(y,i(z,u)),i(i(y,z),i(y,u)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,6,a,b,5,a)].">
      <j1 rule="hyper" parents="4 6 5"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      P(i(x,i(k(y,z),y)))
    ]]></literal>
    <justification jstring="[hyper(4,a,7,a,b,5,a)].">
      <j1 rule="hyper" parents="4 7 5"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      P(i(i(k(x,i(y,z)),y),i(k(x,i(y,z)),z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,8,a,b,6,a)].">
      <j1 rule="hyper" parents="4 8 6"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      P(i(x,i(k(y,z),z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,8,a,b,5,a)].">
      <j1 rule="hyper" parents="4 8 5"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      P(i(i(x,y),i(x,k(i(z,x),y))))
    ]]></literal>
    <justification jstring="[hyper(4,a,5,a,b,9,a)].">
      <j1 rule="hyper" parents="4 5 9"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      P(i(x,i(y,a(y,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,10,a,b,5,a)].">
      <j1 rule="hyper" parents="4 10 5"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      P(i(x,i(y,a(z,y))))
    ]]></literal>
    <justification jstring="[hyper(4,a,11,a,b,5,a)].">
      <j1 rule="hyper" parents="4 11 5"/>
    </justification>
  </clause>

  <clause id="71">
    <literal><![CDATA[
      P(i(x,i(i(y,n(z)),i(z,n(y)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,13,a,b,5,a)].">
      <j1 rule="hyper" parents="4 13 5"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      P(i(x,i(n(y),i(y,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,14,a,b,5,a)].">
      <j1 rule="hyper" parents="4 14 5"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      P(i(i(x,y),i(x,i(z,y))))
    ]]></literal>
    <justification jstring="[hyper(4,a,18,a,b,6,a)].">
      <j1 rule="hyper" parents="4 18 6"/>
    </justification>
  </clause>

  <clause id="100">
    <literal><![CDATA[
      P(i(i(x,y),i(x,a(y,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,40,a,b,6,a)].">
      <j1 rule="hyper" parents="4 40 6"/>
    </justification>
  </clause>

  <clause id="106">
    <literal><![CDATA[
      P(i(i(x,y),i(x,a(z,y))))
    ]]></literal>
    <justification jstring="[hyper(4,a,50,a,b,6,a)].">
      <j1 rule="hyper" parents="4 50 6"/>
    </justification>
  </clause>

  <clause id="112">
    <literal><![CDATA[
      P(i(i(x,n(y)),i(x,i(y,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,77,a,b,6,a)].">
      <j1 rule="hyper" parents="4 77 6"/>
    </justification>
  </clause>

  <clause id="207">
    <literal><![CDATA[
      P(i(x,a(a(y,x),z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,11,a,b,100,a)].">
      <j1 rule="hyper" parents="4 11 100"/>
    </justification>
  </clause>

  <clause id="208">
    <literal><![CDATA[
      P(i(x,a(a(x,y),z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,10,a,b,100,a)].">
      <j1 rule="hyper" parents="4 10 100"/>
    </justification>
  </clause>

  <clause id="324">
    <literal><![CDATA[
      P(i(x,a(y,i(z,x))))
    ]]></literal>
    <justification jstring="[hyper(4,a,5,a,b,106,a)].">
      <j1 rule="hyper" parents="4 5 106"/>
    </justification>
  </clause>

  <clause id="380">
    <literal><![CDATA[
      P(i(i(x,i(y,i(z,u))),i(a(i(y,n(z)),x),i(y,i(z,u)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,112,a,b,12,a)].">
      <j1 rule="hyper" parents="4 112 12"/>
    </justification>
  </clause>

  <clause id="415">
    <literal><![CDATA[
      P(i(x,k(i(y,x),i(z,x))))
    ]]></literal>
    <justification jstring="[hyper(4,a,5,a,b,30,a)].">
      <j1 rule="hyper" parents="4 5 30"/>
    </justification>
  </clause>

  <clause id="483">
    <literal><![CDATA[
      P(i(i(x,i(y,n(z))),i(x,i(z,n(y)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,71,a,b,6,a)].">
      <j1 rule="hyper" parents="4 71 6"/>
    </justification>
  </clause>

  <clause id="711">
    <literal><![CDATA[
      P(i(i(x,i(y,i(z,u))),i(x,i(i(y,z),i(y,u)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,21,a,b,6,a)].">
      <j1 rule="hyper" parents="4 21 6"/>
    </justification>
  </clause>

  <clause id="722">
    <literal><![CDATA[
      P(i(k(x,i(x,y)),y))
    ]]></literal>
    <justification jstring="[hyper(4,a,7,a,b,24,a)].">
      <j1 rule="hyper" parents="4 7 24"/>
    </justification>
  </clause>

  <clause id="762">
    <literal><![CDATA[
      P(i(x,i(k(y,i(y,z)),z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,722,a,b,5,a)].">
      <j1 rule="hyper" parents="4 722 5"/>
    </justification>
  </clause>

  <clause id="777">
    <literal><![CDATA[
      P(i(i(x,k(y,i(y,z))),i(x,z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,762,a,b,6,a)].">
      <j1 rule="hyper" parents="4 762 6"/>
    </justification>
  </clause>

  <clause id="781">
    <literal><![CDATA[
      P(i(x,x))
    ]]></literal>
    <justification jstring="[hyper(4,a,415,a,b,777,a)].">
      <j1 rule="hyper" parents="4 415 777"/>
    </justification>
  </clause>

  <clause id="811">
    <literal><![CDATA[
      P(i(x,n(n(x))))
    ]]></literal>
    <justification jstring="[hyper(4,a,781,a,b,13,a)].">
      <j1 rule="hyper" parents="4 781 13"/>
    </justification>
  </clause>

  <clause id="815">
    <literal><![CDATA[
      P(i(i(x,y),i(x,k(x,y))))
    ]]></literal>
    <justification jstring="[hyper(4,a,781,a,b,9,a)].">
      <j1 rule="hyper" parents="4 781 9"/>
    </justification>
  </clause>

  <clause id="872">
    <literal><![CDATA[
      P(i(x,i(n(x),y)))
    ]]></literal>
    <justification jstring="[hyper(4,a,811,a,b,112,a)].">
      <j1 rule="hyper" parents="4 811 112"/>
    </justification>
  </clause>

  <clause id="961">
    <literal><![CDATA[
      P(i(i(x,i(n(y),z)),i(a(y,x),i(n(y),z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,872,a,b,12,a)].">
      <j1 rule="hyper" parents="4 872 12"/>
    </justification>
  </clause>

  <clause id="1152">
    <literal><![CDATA[
      P(i(a(x,y),i(n(x),y)))
    ]]></literal>
    <justification jstring="[hyper(4,a,5,a,b,961,a)].">
      <j1 rule="hyper" parents="4 5 961"/>
    </justification>
  </clause>

  <clause id="1190">
    <literal><![CDATA[
      P(i(x,i(a(y,z),i(n(y),z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,1152,a,b,5,a)].">
      <j1 rule="hyper" parents="4 1152 5"/>
    </justification>
  </clause>

  <clause id="1207">
    <literal><![CDATA[
      P(i(i(x,a(y,z)),i(x,i(n(y),z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,1190,a,b,6,a)].">
      <j1 rule="hyper" parents="4 1190 6"/>
    </justification>
  </clause>

  <clause id="1214">
    <literal><![CDATA[
      P(i(x,i(n(a(x,y)),z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,208,a,b,1207,a)].">
      <j1 rule="hyper" parents="4 208 1207"/>
    </justification>
  </clause>

  <clause id="1215">
    <literal><![CDATA[
      P(i(x,i(n(a(y,x)),z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,207,a,b,1207,a)].">
      <j1 rule="hyper" parents="4 207 1207"/>
    </justification>
  </clause>

  <clause id="1304">
    <literal><![CDATA[
      P(i(x,i(y,n(n(a(x,z))))))
    ]]></literal>
    <justification jstring="[hyper(4,a,1214,a,b,483,a)].">
      <j1 rule="hyper" parents="4 1214 483"/>
    </justification>
  </clause>

  <clause id="1397">
    <literal><![CDATA[
      P(i(x,i(y,i(n(a(z,x)),u))))
    ]]></literal>
    <justification jstring="[hyper(4,a,1215,a,b,82,a)].">
      <j1 rule="hyper" parents="4 1215 82"/>
    </justification>
  </clause>

  <clause id="1491">
    <literal><![CDATA[
      P(i(i(x,y),i(x,n(n(a(x,z))))))
    ]]></literal>
    <justification jstring="[hyper(4,a,1304,a,b,6,a)].">
      <j1 rule="hyper" parents="4 1304 6"/>
    </justification>
  </clause>

  <clause id="1590">
    <literal><![CDATA[
      P(i(i(x,y),i(n(a(x,z)),n(x))))
    ]]></literal>
    <justification jstring="[hyper(4,a,1491,a,b,483,a)].">
      <j1 rule="hyper" parents="4 1491 483"/>
    </justification>
  </clause>

  <clause id="1607">
    <literal><![CDATA[
      P(i(n(a(x,y)),n(x)))
    ]]></literal>
    <justification jstring="[hyper(4,a,1397,a,b,1590,a)].">
      <j1 rule="hyper" parents="4 1397 1590"/>
    </justification>
  </clause>

  <clause id="1651">
    <literal><![CDATA[
      P(i(i(n(a(x,y)),z),i(n(a(x,y)),k(n(x),z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,1607,a,b,9,a)].">
      <j1 rule="hyper" parents="4 1607 9"/>
    </justification>
  </clause>

  <clause id="2128">
    <literal><![CDATA[
      P(i(i(x,y),i(i(z,x),i(z,y))))
    ]]></literal>
    <justification jstring="[hyper(4,a,5,a,b,711,a)].">
      <j1 rule="hyper" parents="4 5 711"/>
    </justification>
  </clause>

  <clause id="2170">
    <literal><![CDATA[
      P(i(i(x,y),i(x,n(n(y)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,811,a,b,2128,a)].">
      <j1 rule="hyper" parents="4 811 2128"/>
    </justification>
  </clause>

  <clause id="2297">
    <literal><![CDATA[
      P(i(x,n(n(a(y,x)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,11,a,b,2170,a)].">
      <j1 rule="hyper" parents="4 11 2170"/>
    </justification>
  </clause>

  <clause id="2314">
    <literal><![CDATA[
      P(i(i(x,y),i(n(y),n(x))))
    ]]></literal>
    <justification jstring="[hyper(4,a,2170,a,b,483,a)].">
      <j1 rule="hyper" parents="4 2170 483"/>
    </justification>
  </clause>

  <clause id="2432">
    <literal><![CDATA[
      P(i(n(a(x,y)),n(y)))
    ]]></literal>
    <justification jstring="[hyper(4,a,2297,a,b,13,a)].">
      <j1 rule="hyper" parents="4 2297 13"/>
    </justification>
  </clause>

  <clause id="2443">
    <literal><![CDATA[
      P(i(n(a(x,y)),k(n(x),n(y))))
    ]]></literal>
    <justification jstring="[hyper(4,a,2432,a,b,1651,a)].">
      <j1 rule="hyper" parents="4 2432 1651"/>
    </justification>
  </clause>

  <clause id="2531">
    <literal><![CDATA[
      P(i(i(x,i(y,z)),i(x,i(n(z),n(y)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,2314,a,b,2128,a)].">
      <j1 rule="hyper" parents="4 2314 2128"/>
    </justification>
  </clause>

  <clause id="2927">
    <literal><![CDATA[
      P(i(i(k(x,y),i(y,z)),i(k(x,y),z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,25,a,b,19,a)].">
      <j1 rule="hyper" parents="4 25 19"/>
    </justification>
  </clause>

  <clause id="2929">
    <literal><![CDATA[
      P(i(i(k(x,y),i(x,z)),i(k(x,y),z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,23,a,b,19,a)].">
      <j1 rule="hyper" parents="4 23 19"/>
    </justification>
  </clause>

  <clause id="2931">
    <literal><![CDATA[
      P(i(i(x,i(i(y,x),z)),i(x,z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,18,a,b,19,a)].">
      <j1 rule="hyper" parents="4 18 19"/>
    </justification>
  </clause>

  <clause id="2985">
    <literal><![CDATA[
      P(a(x,i(y,i(i(z,i(i(u,z),w)),i(z,w)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,2931,a,b,324,a)].">
      <j1 rule="hyper" parents="4 2931 324"/>
    </justification>
  </clause>

  <clause id="3429">
    <literal><![CDATA[
      P(i(x,i(i(k(y,z),i(y,u)),i(k(y,z),u))))
    ]]></literal>
    <justification jstring="[hyper(4,a,2929,a,b,5,a)].">
      <j1 rule="hyper" parents="4 2929 5"/>
    </justification>
  </clause>

  <clause id="3712">
    <literal><![CDATA[
      P(i(i(x,y),i(k(x,z),y)))
    ]]></literal>
    <justification jstring="[hyper(4,a,3429,a,b,2931,a)].">
      <j1 rule="hyper" parents="4 3429 2931"/>
    </justification>
  </clause>

  <clause id="3912">
    <literal><![CDATA[
      P(i(a(i(x,n(y)),i(x,i(i(z,x),i(y,u)))),i(x,i(y,u))))
    ]]></literal>
    <justification jstring="[hyper(4,a,2931,a,b,380,a)].">
      <j1 rule="hyper" parents="4 2931 380"/>
    </justification>
  </clause>

  <clause id="3990">
    <literal><![CDATA[
      P(i(i(i(x,y),z),i(y,z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,2985,a,b,3912,a)].">
      <j1 rule="hyper" parents="4 2985 3912"/>
    </justification>
  </clause>

  <clause id="4041">
    <literal><![CDATA[
      P(i(x,i(y,k(y,x))))
    ]]></literal>
    <justification jstring="[hyper(4,a,815,a,b,3990,a)].">
      <j1 rule="hyper" parents="4 815 3990"/>
    </justification>
  </clause>

  <clause id="4063">
    <literal><![CDATA[
      P(i(i(x,i(i(y,z),u)),i(x,i(z,u))))
    ]]></literal>
    <justification jstring="[hyper(4,a,3990,a,b,2128,a)].">
      <j1 rule="hyper" parents="4 3990 2128"/>
    </justification>
  </clause>

  <clause id="4409">
    <literal><![CDATA[
      P(i(i(x,i(y,z)),i(y,i(x,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,6,a,b,4063,a)].">
      <j1 rule="hyper" parents="4 6 4063"/>
    </justification>
  </clause>

  <clause id="4472">
    <literal><![CDATA[
      P(i(i(x,y),i(i(y,z),i(x,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,2128,a,b,4409,a)].">
      <j1 rule="hyper" parents="4 2128 4409"/>
    </justification>
  </clause>

  <clause id="4477">
    <literal><![CDATA[
      P(i(n(x),i(a(x,y),y)))
    ]]></literal>
    <justification jstring="[hyper(4,a,1152,a,b,4409,a)].">
      <j1 rule="hyper" parents="4 1152 4409"/>
    </justification>
  </clause>

  <clause id="4707">
    <literal><![CDATA[
      P(i(n(x),i(n(y),n(a(x,y)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,4477,a,b,2531,a)].">
      <j1 rule="hyper" parents="4 4477 2531"/>
    </justification>
  </clause>

  <clause id="4852">
    <literal><![CDATA[
      P(i(i(i(k(x,y),z),u),i(i(x,z),u)))
    ]]></literal>
    <justification jstring="[hyper(4,a,3712,a,b,4472,a)].">
      <j1 rule="hyper" parents="4 3712 4472"/>
    </justification>
  </clause>

  <clause id="5439">
    <literal><![CDATA[
      P(i(i(x,i(y,z)),i(k(x,y),z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,2927,a,b,4852,a)].">
      <j1 rule="hyper" parents="4 2927 4852"/>
    </justification>
  </clause>

  <clause id="5533">
    <literal><![CDATA[
      P(i(k(n(x),n(y)),n(a(x,y))))
    ]]></literal>
    <justification jstring="[hyper(4,a,4707,a,b,5439,a)].">
      <j1 rule="hyper" parents="4 4707 5439"/>
    </justification>
  </clause>

  <clause id="5624">
    <literal><![CDATA[
      P(i(x,k(x,i(k(n(y),n(z)),n(a(y,z))))))
    ]]></literal>
    <justification jstring="[hyper(4,a,5533,a,b,4041,a)].">
      <j1 rule="hyper" parents="4 5533 4041"/>
    </justification>
  </clause>

  <clause id="7035">
    <literal><![CDATA[
      P(k(i(n(a(x,y)),k(n(x),n(y))),i(k(n(z),n(u)),n(a(z,u)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,2443,a,b,5624,a)].">
      <j1 rule="hyper" parents="4 2443 5624"/>
    </justification>
  </clause>

  <clause id="7036">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer("HIF5")
    ]]></attribute>
    <justification jstring="[resolve(7035,a,17,a)].">
      <j1 rule="resolve" parents="7035 17"/>
    </justification>
  </clause>

</proof>

<proof number="2" length="37" max_count="21">

<comments><![CDATA[
% Proof 2 at 0.24 (+ 0.01) seconds: "H334".
% Length of proof is 37.
% Level of proof is 12.
% Maximum clause weight is 21.
% Given clauses 142.

]]></comments>

  <clause id="1" type="goal">
    <literal><![CDATA[
      P(i(i(x,y),i(a(x,z),a(y,z))))
    ]]></literal>
    <attribute><![CDATA[
      label("H334")
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="4" type="assumption">
    <literal><![CDATA[
      -P(x)
    ]]></literal>
    <literal><![CDATA[
      -P(i(x,y))
    ]]></literal>
    <literal><![CDATA[
      P(y)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="5" type="assumption">
    <literal><![CDATA[
      P(i(x,i(y,x)))
    ]]></literal>
    <attribute><![CDATA[
      label("A1")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6" type="assumption">
    <literal><![CDATA[
      P(i(i(x,i(y,z)),i(i(x,y),i(x,z))))
    ]]></literal>
    <attribute><![CDATA[
      label("A2")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="10" type="assumption">
    <literal><![CDATA[
      P(i(x,a(x,y)))
    ]]></literal>
    <attribute><![CDATA[
      label("A6")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11" type="assumption">
    <literal><![CDATA[
      P(i(x,a(y,x)))
    ]]></literal>
    <attribute><![CDATA[
      label("A7")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="12" type="assumption">
    <literal><![CDATA[
      P(i(i(x,y),i(i(z,y),i(a(x,z),y))))
    ]]></literal>
    <attribute><![CDATA[
      label("A8")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="14" type="assumption">
    <literal><![CDATA[
      P(i(n(x),i(x,y)))
    ]]></literal>
    <attribute><![CDATA[
      label("A10")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="15" type="deny">
    <literal><![CDATA[
      -P(i(i(c1,c2),i(a(c1,c3),a(c2,c3))))
    ]]></literal>
    <attribute><![CDATA[
      label("H334")
    ]]></attribute>
    <attribute><![CDATA[
      answer("H334")
    ]]></attribute>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      P(i(x,i(y,i(z,y))))
    ]]></literal>
    <justification jstring="[hyper(4,a,5,a,b,5,a)].">
      <j1 rule="hyper" parents="4 5 5"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      P(i(i(i(x,i(y,z)),i(x,y)),i(i(x,i(y,z)),i(x,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,6,a,b,6,a)].">
      <j1 rule="hyper" parents="4 6 6"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      P(i(x,i(i(y,i(z,u)),i(i(y,z),i(y,u)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,6,a,b,5,a)].">
      <j1 rule="hyper" parents="4 6 5"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      P(i(x,i(y,a(y,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,10,a,b,5,a)].">
      <j1 rule="hyper" parents="4 10 5"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      P(i(x,i(y,a(z,y))))
    ]]></literal>
    <justification jstring="[hyper(4,a,11,a,b,5,a)].">
      <j1 rule="hyper" parents="4 11 5"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      P(i(i(x,a(y,z)),i(a(z,x),a(y,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,11,a,b,12,a)].">
      <j1 rule="hyper" parents="4 11 12"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      P(i(x,i(n(y),i(y,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,14,a,b,5,a)].">
      <j1 rule="hyper" parents="4 14 5"/>
    </justification>
  </clause>

  <clause id="100">
    <literal><![CDATA[
      P(i(i(x,y),i(x,a(y,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,40,a,b,6,a)].">
      <j1 rule="hyper" parents="4 40 6"/>
    </justification>
  </clause>

  <clause id="106">
    <literal><![CDATA[
      P(i(i(x,y),i(x,a(z,y))))
    ]]></literal>
    <justification jstring="[hyper(4,a,50,a,b,6,a)].">
      <j1 rule="hyper" parents="4 50 6"/>
    </justification>
  </clause>

  <clause id="112">
    <literal><![CDATA[
      P(i(i(x,n(y)),i(x,i(y,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,77,a,b,6,a)].">
      <j1 rule="hyper" parents="4 77 6"/>
    </justification>
  </clause>

  <clause id="324">
    <literal><![CDATA[
      P(i(x,a(y,i(z,x))))
    ]]></literal>
    <justification jstring="[hyper(4,a,5,a,b,106,a)].">
      <j1 rule="hyper" parents="4 5 106"/>
    </justification>
  </clause>

  <clause id="380">
    <literal><![CDATA[
      P(i(i(x,i(y,i(z,u))),i(a(i(y,n(z)),x),i(y,i(z,u)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,112,a,b,12,a)].">
      <j1 rule="hyper" parents="4 112 12"/>
    </justification>
  </clause>

  <clause id="593">
    <literal><![CDATA[
      P(i(a(x,y),a(y,x)))
    ]]></literal>
    <justification jstring="[hyper(4,a,10,a,b,52,a)].">
      <j1 rule="hyper" parents="4 10 52"/>
    </justification>
  </clause>

  <clause id="711">
    <literal><![CDATA[
      P(i(i(x,i(y,i(z,u))),i(x,i(i(y,z),i(y,u)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,21,a,b,6,a)].">
      <j1 rule="hyper" parents="4 21 6"/>
    </justification>
  </clause>

  <clause id="2128">
    <literal><![CDATA[
      P(i(i(x,y),i(i(z,x),i(z,y))))
    ]]></literal>
    <justification jstring="[hyper(4,a,5,a,b,711,a)].">
      <j1 rule="hyper" parents="4 5 711"/>
    </justification>
  </clause>

  <clause id="2931">
    <literal><![CDATA[
      P(i(i(x,i(i(y,x),z)),i(x,z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,18,a,b,19,a)].">
      <j1 rule="hyper" parents="4 18 19"/>
    </justification>
  </clause>

  <clause id="2985">
    <literal><![CDATA[
      P(a(x,i(y,i(i(z,i(i(u,z),w)),i(z,w)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,2931,a,b,324,a)].">
      <j1 rule="hyper" parents="4 2931 324"/>
    </justification>
  </clause>

  <clause id="3912">
    <literal><![CDATA[
      P(i(a(i(x,n(y)),i(x,i(i(z,x),i(y,u)))),i(x,i(y,u))))
    ]]></literal>
    <justification jstring="[hyper(4,a,2931,a,b,380,a)].">
      <j1 rule="hyper" parents="4 2931 380"/>
    </justification>
  </clause>

  <clause id="3990">
    <literal><![CDATA[
      P(i(i(i(x,y),z),i(y,z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,2985,a,b,3912,a)].">
      <j1 rule="hyper" parents="4 2985 3912"/>
    </justification>
  </clause>

  <clause id="4063">
    <literal><![CDATA[
      P(i(i(x,i(i(y,z),u)),i(x,i(z,u))))
    ]]></literal>
    <justification jstring="[hyper(4,a,3990,a,b,2128,a)].">
      <j1 rule="hyper" parents="4 3990 2128"/>
    </justification>
  </clause>

  <clause id="4409">
    <literal><![CDATA[
      P(i(i(x,i(y,z)),i(y,i(x,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,6,a,b,4063,a)].">
      <j1 rule="hyper" parents="4 6 4063"/>
    </justification>
  </clause>

  <clause id="4472">
    <literal><![CDATA[
      P(i(i(x,y),i(i(y,z),i(x,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,2128,a,b,4409,a)].">
      <j1 rule="hyper" parents="4 2128 4409"/>
    </justification>
  </clause>

  <clause id="4904">
    <literal><![CDATA[
      P(i(i(a(x,y),z),i(a(y,x),z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,593,a,b,4472,a)].">
      <j1 rule="hyper" parents="4 593 4472"/>
    </justification>
  </clause>

  <clause id="4920">
    <literal><![CDATA[
      P(i(i(i(x,a(y,z)),u),i(i(x,y),u)))
    ]]></literal>
    <justification jstring="[hyper(4,a,100,a,b,4472,a)].">
      <j1 rule="hyper" parents="4 100 4472"/>
    </justification>
  </clause>

  <clause id="5102">
    <literal><![CDATA[
      P(i(i(x,i(a(y,z),u)),i(x,i(a(z,y),u))))
    ]]></literal>
    <justification jstring="[hyper(4,a,4904,a,b,2128,a)].">
      <j1 rule="hyper" parents="4 4904 2128"/>
    </justification>
  </clause>

  <clause id="5950">
    <literal><![CDATA[
      P(i(i(x,y),i(a(z,x),a(y,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,52,a,b,4920,a)].">
      <j1 rule="hyper" parents="4 52 4920"/>
    </justification>
  </clause>

  <clause id="7180">
    <literal><![CDATA[
      P(i(i(x,y),i(a(x,z),a(y,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,5950,a,b,5102,a)].">
      <j1 rule="hyper" parents="4 5950 5102"/>
    </justification>
  </clause>

  <clause id="7181">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer("H334")
    ]]></attribute>
    <justification jstring="[resolve(7180,a,15,a)].">
      <j1 rule="resolve" parents="7180 15"/>
    </justification>
  </clause>

</proof>

<proof number="3" length="150" max_count="34">

<comments><![CDATA[
% Proof 3 at 0.35 (+ 0.02) seconds: "HIF4".
% Length of proof is 150.
% Level of proof is 30.
% Maximum clause weight is 34.
% Given clauses 181.

]]></comments>

  <clause id="2" type="goal">
    <literal><![CDATA[
      P(k(i(a(k(x,y),z),k(a(x,z),a(y,z))),i(k(a(x,z),a(y,z)),a(k(x,y),z))))
    ]]></literal>
    <attribute><![CDATA[
      label("HIF4")
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="4" type="assumption">
    <literal><![CDATA[
      -P(x)
    ]]></literal>
    <literal><![CDATA[
      -P(i(x,y))
    ]]></literal>
    <literal><![CDATA[
      P(y)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="5" type="assumption">
    <literal><![CDATA[
      P(i(x,i(y,x)))
    ]]></literal>
    <attribute><![CDATA[
      label("A1")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6" type="assumption">
    <literal><![CDATA[
      P(i(i(x,i(y,z)),i(i(x,y),i(x,z))))
    ]]></literal>
    <attribute><![CDATA[
      label("A2")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7" type="assumption">
    <literal><![CDATA[
      P(i(k(x,y),x))
    ]]></literal>
    <attribute><![CDATA[
      label("A3")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="8" type="assumption">
    <literal><![CDATA[
      P(i(k(x,y),y))
    ]]></literal>
    <attribute><![CDATA[
      label("A4")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="9" type="assumption">
    <literal><![CDATA[
      P(i(i(x,y),i(i(x,z),i(x,k(y,z)))))
    ]]></literal>
    <attribute><![CDATA[
      label("A5")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="10" type="assumption">
    <literal><![CDATA[
      P(i(x,a(x,y)))
    ]]></literal>
    <attribute><![CDATA[
      label("A6")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11" type="assumption">
    <literal><![CDATA[
      P(i(x,a(y,x)))
    ]]></literal>
    <attribute><![CDATA[
      label("A7")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="12" type="assumption">
    <literal><![CDATA[
      P(i(i(x,y),i(i(z,y),i(a(x,z),y))))
    ]]></literal>
    <attribute><![CDATA[
      label("A8")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="13" type="assumption">
    <literal><![CDATA[
      P(i(i(x,n(y)),i(y,n(x))))
    ]]></literal>
    <attribute><![CDATA[
      label("A9")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="14" type="assumption">
    <literal><![CDATA[
      P(i(n(x),i(x,y)))
    ]]></literal>
    <attribute><![CDATA[
      label("A10")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="16" type="deny">
    <literal><![CDATA[
      -P(k(i(a(k(c4,c5),c6),k(a(c4,c6),a(c5,c6))),i(k(a(c4,c6),a(c5,c6)),a(k(c4,c5),c6))))
    ]]></literal>
    <attribute><![CDATA[
      label("HIF4")
    ]]></attribute>
    <attribute><![CDATA[
      answer("HIF4")
    ]]></attribute>
    <justification jstring="[deny(2)].">
      <j1 rule="deny" parents="2"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      P(i(x,i(y,i(z,y))))
    ]]></literal>
    <justification jstring="[hyper(4,a,5,a,b,5,a)].">
      <j1 rule="hyper" parents="4 5 5"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      P(i(i(i(x,i(y,z)),i(x,y)),i(i(x,i(y,z)),i(x,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,6,a,b,6,a)].">
      <j1 rule="hyper" parents="4 6 6"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      P(i(x,i(i(y,i(z,u)),i(i(y,z),i(y,u)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,6,a,b,5,a)].">
      <j1 rule="hyper" parents="4 6 5"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      P(i(x,i(k(y,z),y)))
    ]]></literal>
    <justification jstring="[hyper(4,a,7,a,b,5,a)].">
      <j1 rule="hyper" parents="4 7 5"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      P(i(i(k(x,i(y,z)),y),i(k(x,i(y,z)),z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,8,a,b,6,a)].">
      <j1 rule="hyper" parents="4 8 6"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      P(i(x,i(k(y,z),z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,8,a,b,5,a)].">
      <j1 rule="hyper" parents="4 8 5"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      P(i(i(k(x,y),z),i(k(x,y),k(y,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,8,a,b,9,a)].">
      <j1 rule="hyper" parents="4 8 9"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      P(i(i(k(x,y),z),i(k(x,y),k(x,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,7,a,b,9,a)].">
      <j1 rule="hyper" parents="4 7 9"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      P(i(i(x,y),i(x,k(i(z,x),y))))
    ]]></literal>
    <justification jstring="[hyper(4,a,5,a,b,9,a)].">
      <j1 rule="hyper" parents="4 5 9"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      P(i(i(i(x,y),i(x,z)),i(i(x,y),i(x,k(y,z)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,9,a,b,6,a)].">
      <j1 rule="hyper" parents="4 9 6"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      P(i(x,i(y,a(y,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,10,a,b,5,a)].">
      <j1 rule="hyper" parents="4 10 5"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      P(i(x,i(y,a(z,y))))
    ]]></literal>
    <justification jstring="[hyper(4,a,11,a,b,5,a)].">
      <j1 rule="hyper" parents="4 11 5"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      P(i(i(x,a(y,z)),i(a(z,x),a(y,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,11,a,b,12,a)].">
      <j1 rule="hyper" parents="4 11 12"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      P(i(x,i(n(y),i(y,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,14,a,b,5,a)].">
      <j1 rule="hyper" parents="4 14 5"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      P(i(i(x,y),i(x,i(z,y))))
    ]]></literal>
    <justification jstring="[hyper(4,a,18,a,b,6,a)].">
      <j1 rule="hyper" parents="4 18 6"/>
    </justification>
  </clause>

  <clause id="88">
    <literal><![CDATA[
      P(i(i(x,k(y,z)),i(x,y)))
    ]]></literal>
    <justification jstring="[hyper(4,a,23,a,b,6,a)].">
      <j1 rule="hyper" parents="4 23 6"/>
    </justification>
  </clause>

  <clause id="94">
    <literal><![CDATA[
      P(i(i(x,k(y,z)),i(x,z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,25,a,b,6,a)].">
      <j1 rule="hyper" parents="4 25 6"/>
    </justification>
  </clause>

  <clause id="100">
    <literal><![CDATA[
      P(i(i(x,y),i(x,a(y,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,40,a,b,6,a)].">
      <j1 rule="hyper" parents="4 40 6"/>
    </justification>
  </clause>

  <clause id="106">
    <literal><![CDATA[
      P(i(i(x,y),i(x,a(z,y))))
    ]]></literal>
    <justification jstring="[hyper(4,a,50,a,b,6,a)].">
      <j1 rule="hyper" parents="4 50 6"/>
    </justification>
  </clause>

  <clause id="112">
    <literal><![CDATA[
      P(i(i(x,n(y)),i(x,i(y,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,77,a,b,6,a)].">
      <j1 rule="hyper" parents="4 77 6"/>
    </justification>
  </clause>

  <clause id="124">
    <literal><![CDATA[
      P(i(x,i(y,i(z,x))))
    ]]></literal>
    <justification jstring="[hyper(4,a,5,a,b,82,a)].">
      <j1 rule="hyper" parents="4 5 82"/>
    </justification>
  </clause>

  <clause id="149">
    <literal><![CDATA[
      P(i(k(x,k(y,z)),y))
    ]]></literal>
    <justification jstring="[hyper(4,a,8,a,b,88,a)].">
      <j1 rule="hyper" parents="4 8 88"/>
    </justification>
  </clause>

  <clause id="166">
    <literal><![CDATA[
      P(i(i(k(x,k(y,z)),u),i(k(x,k(y,z)),k(y,u))))
    ]]></literal>
    <justification jstring="[hyper(4,a,149,a,b,9,a)].">
      <j1 rule="hyper" parents="4 149 9"/>
    </justification>
  </clause>

  <clause id="170">
    <literal><![CDATA[
      P(i(k(x,k(y,z)),z))
    ]]></literal>
    <justification jstring="[hyper(4,a,8,a,b,94,a)].">
      <j1 rule="hyper" parents="4 8 94"/>
    </justification>
  </clause>

  <clause id="211">
    <literal><![CDATA[
      P(i(k(x,y),a(x,z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,7,a,b,100,a)].">
      <j1 rule="hyper" parents="4 7 100"/>
    </justification>
  </clause>

  <clause id="324">
    <literal><![CDATA[
      P(i(x,a(y,i(z,x))))
    ]]></literal>
    <justification jstring="[hyper(4,a,5,a,b,106,a)].">
      <j1 rule="hyper" parents="4 5 106"/>
    </justification>
  </clause>

  <clause id="380">
    <literal><![CDATA[
      P(i(i(x,i(y,i(z,u))),i(a(i(y,n(z)),x),i(y,i(z,u)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,112,a,b,12,a)].">
      <j1 rule="hyper" parents="4 112 12"/>
    </justification>
  </clause>

  <clause id="415">
    <literal><![CDATA[
      P(i(x,k(i(y,x),i(z,x))))
    ]]></literal>
    <justification jstring="[hyper(4,a,5,a,b,30,a)].">
      <j1 rule="hyper" parents="4 5 30"/>
    </justification>
  </clause>

  <clause id="502">
    <literal><![CDATA[
      P(i(k(x,y),k(y,x)))
    ]]></literal>
    <justification jstring="[hyper(4,a,7,a,b,27,a)].">
      <j1 rule="hyper" parents="4 7 27"/>
    </justification>
  </clause>

  <clause id="540">
    <literal><![CDATA[
      P(i(k(x,k(y,z)),k(x,z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,170,a,b,28,a)].">
      <j1 rule="hyper" parents="4 170 28"/>
    </justification>
  </clause>

  <clause id="587">
    <literal><![CDATA[
      P(i(a(x,k(y,z)),a(y,x)))
    ]]></literal>
    <justification jstring="[hyper(4,a,211,a,b,52,a)].">
      <j1 rule="hyper" parents="4 211 52"/>
    </justification>
  </clause>

  <clause id="593">
    <literal><![CDATA[
      P(i(a(x,y),a(y,x)))
    ]]></literal>
    <justification jstring="[hyper(4,a,10,a,b,52,a)].">
      <j1 rule="hyper" parents="4 10 52"/>
    </justification>
  </clause>

  <clause id="625">
    <literal><![CDATA[
      P(i(x,i(a(y,z),a(z,y))))
    ]]></literal>
    <justification jstring="[hyper(4,a,593,a,b,5,a)].">
      <j1 rule="hyper" parents="4 593 5"/>
    </justification>
  </clause>

  <clause id="653">
    <literal><![CDATA[
      P(i(i(x,a(y,z)),i(x,a(z,y))))
    ]]></literal>
    <justification jstring="[hyper(4,a,625,a,b,6,a)].">
      <j1 rule="hyper" parents="4 625 6"/>
    </justification>
  </clause>

  <clause id="711">
    <literal><![CDATA[
      P(i(i(x,i(y,i(z,u))),i(x,i(i(y,z),i(y,u)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,21,a,b,6,a)].">
      <j1 rule="hyper" parents="4 21 6"/>
    </justification>
  </clause>

  <clause id="722">
    <literal><![CDATA[
      P(i(k(x,i(x,y)),y))
    ]]></literal>
    <justification jstring="[hyper(4,a,7,a,b,24,a)].">
      <j1 rule="hyper" parents="4 7 24"/>
    </justification>
  </clause>

  <clause id="762">
    <literal><![CDATA[
      P(i(x,i(k(y,i(y,z)),z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,722,a,b,5,a)].">
      <j1 rule="hyper" parents="4 722 5"/>
    </justification>
  </clause>

  <clause id="777">
    <literal><![CDATA[
      P(i(i(x,k(y,i(y,z))),i(x,z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,762,a,b,6,a)].">
      <j1 rule="hyper" parents="4 762 6"/>
    </justification>
  </clause>

  <clause id="779">
    <literal><![CDATA[
      P(i(k(x,k(y,i(x,z))),z))
    ]]></literal>
    <justification jstring="[hyper(4,a,540,a,b,777,a)].">
      <j1 rule="hyper" parents="4 540 777"/>
    </justification>
  </clause>

  <clause id="780">
    <literal><![CDATA[
      P(i(k(i(x,y),x),y))
    ]]></literal>
    <justification jstring="[hyper(4,a,502,a,b,777,a)].">
      <j1 rule="hyper" parents="4 502 777"/>
    </justification>
  </clause>

  <clause id="781">
    <literal><![CDATA[
      P(i(x,x))
    ]]></literal>
    <justification jstring="[hyper(4,a,415,a,b,777,a)].">
      <j1 rule="hyper" parents="4 415 777"/>
    </justification>
  </clause>

  <clause id="806">
    <literal><![CDATA[
      P(i(x,i(y,i(z,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,781,a,b,124,a)].">
      <j1 rule="hyper" parents="4 781 124"/>
    </justification>
  </clause>

  <clause id="811">
    <literal><![CDATA[
      P(i(x,n(n(x))))
    ]]></literal>
    <justification jstring="[hyper(4,a,781,a,b,13,a)].">
      <j1 rule="hyper" parents="4 781 13"/>
    </justification>
  </clause>

  <clause id="815">
    <literal><![CDATA[
      P(i(i(x,y),i(x,k(x,y))))
    ]]></literal>
    <justification jstring="[hyper(4,a,781,a,b,9,a)].">
      <j1 rule="hyper" parents="4 781 9"/>
    </justification>
  </clause>

  <clause id="817">
    <literal><![CDATA[
      P(i(x,i(y,y)))
    ]]></literal>
    <justification jstring="[hyper(4,a,781,a,b,5,a)].">
      <j1 rule="hyper" parents="4 781 5"/>
    </justification>
  </clause>

  <clause id="986">
    <literal><![CDATA[
      P(i(k(i(x,y),x),k(x,y)))
    ]]></literal>
    <justification jstring="[hyper(4,a,780,a,b,27,a)].">
      <j1 rule="hyper" parents="4 780 27"/>
    </justification>
  </clause>

  <clause id="1712">
    <literal><![CDATA[
      P(i(i(x,y),i(x,k(y,x))))
    ]]></literal>
    <justification jstring="[hyper(4,a,817,a,b,31,a)].">
      <j1 rule="hyper" parents="4 817 31"/>
    </justification>
  </clause>

  <clause id="1776">
    <literal><![CDATA[
      P(i(x,k(i(y,y),x)))
    ]]></literal>
    <justification jstring="[hyper(4,a,817,a,b,1712,a)].">
      <j1 rule="hyper" parents="4 817 1712"/>
    </justification>
  </clause>

  <clause id="1850">
    <literal><![CDATA[
      P(i(x,i(i(y,z),i(y,k(z,y)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,1712,a,b,5,a)].">
      <j1 rule="hyper" parents="4 1712 5"/>
    </justification>
  </clause>

  <clause id="1930">
    <literal><![CDATA[
      P(i(i(i(x,x),y),y))
    ]]></literal>
    <justification jstring="[hyper(4,a,1776,a,b,777,a)].">
      <j1 rule="hyper" parents="4 1776 777"/>
    </justification>
  </clause>

  <clause id="2020">
    <literal><![CDATA[
      P(i(k(x,k(y,i(x,z))),k(y,z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,779,a,b,166,a)].">
      <j1 rule="hyper" parents="4 779 166"/>
    </justification>
  </clause>

  <clause id="2128">
    <literal><![CDATA[
      P(i(i(x,y),i(i(z,x),i(z,y))))
    ]]></literal>
    <justification jstring="[hyper(4,a,5,a,b,711,a)].">
      <j1 rule="hyper" parents="4 5 711"/>
    </justification>
  </clause>

  <clause id="2155">
    <literal><![CDATA[
      P(i(i(x,i(y,z)),i(x,i(i(u,y),i(u,z)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,2128,a,b,2128,a)].">
      <j1 rule="hyper" parents="4 2128 2128"/>
    </justification>
  </clause>

  <clause id="2157">
    <literal><![CDATA[
      P(i(i(x,i(i(y,y),z)),i(x,z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,1930,a,b,2128,a)].">
      <j1 rule="hyper" parents="4 1930 2128"/>
    </justification>
  </clause>

  <clause id="2166">
    <literal><![CDATA[
      P(i(i(x,k(i(y,z),y)),i(x,k(y,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,986,a,b,2128,a)].">
      <j1 rule="hyper" parents="4 986 2128"/>
    </justification>
  </clause>

  <clause id="2178">
    <literal><![CDATA[
      P(i(i(x,k(y,z)),i(x,k(z,y))))
    ]]></literal>
    <justification jstring="[hyper(4,a,502,a,b,2128,a)].">
      <j1 rule="hyper" parents="4 502 2128"/>
    </justification>
  </clause>

  <clause id="2599">
    <literal><![CDATA[
      P(i(i(x,i(x,y)),i(x,y)))
    ]]></literal>
    <justification jstring="[hyper(4,a,6,a,b,2157,a)].">
      <j1 rule="hyper" parents="4 6 2157"/>
    </justification>
  </clause>

  <clause id="2927">
    <literal><![CDATA[
      P(i(i(k(x,y),i(y,z)),i(k(x,y),z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,25,a,b,19,a)].">
      <j1 rule="hyper" parents="4 25 19"/>
    </justification>
  </clause>

  <clause id="2929">
    <literal><![CDATA[
      P(i(i(k(x,y),i(x,z)),i(k(x,y),z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,23,a,b,19,a)].">
      <j1 rule="hyper" parents="4 23 19"/>
    </justification>
  </clause>

  <clause id="2931">
    <literal><![CDATA[
      P(i(i(x,i(i(y,x),z)),i(x,z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,18,a,b,19,a)].">
      <j1 rule="hyper" parents="4 18 19"/>
    </justification>
  </clause>

  <clause id="2964">
    <literal><![CDATA[
      P(i(x,i(y,k(x,y))))
    ]]></literal>
    <justification jstring="[hyper(4,a,1850,a,b,2931,a)].">
      <j1 rule="hyper" parents="4 1850 2931"/>
    </justification>
  </clause>

  <clause id="2985">
    <literal><![CDATA[
      P(a(x,i(y,i(i(z,i(i(u,z),w)),i(z,w)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,2931,a,b,324,a)].">
      <j1 rule="hyper" parents="4 2931 324"/>
    </justification>
  </clause>

  <clause id="3080">
    <literal><![CDATA[
      P(i(x,i(i(y,z),i(y,k(x,z)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,2964,a,b,2155,a)].">
      <j1 rule="hyper" parents="4 2964 2155"/>
    </justification>
  </clause>

  <clause id="3081">
    <literal><![CDATA[
      P(i(i(x,y),i(x,i(z,k(y,z)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,2964,a,b,2128,a)].">
      <j1 rule="hyper" parents="4 2964 2128"/>
    </justification>
  </clause>

  <clause id="3144">
    <literal><![CDATA[
      P(i(i(x,y),i(x,k(i(z,i(u,i(w,w))),y))))
    ]]></literal>
    <justification jstring="[hyper(4,a,806,a,b,3080,a)].">
      <j1 rule="hyper" parents="4 806 3080"/>
    </justification>
  </clause>

  <clause id="3388">
    <literal><![CDATA[
      P(i(x,i(i(k(y,z),i(z,u)),i(k(y,z),u))))
    ]]></literal>
    <justification jstring="[hyper(4,a,2927,a,b,5,a)].">
      <j1 rule="hyper" parents="4 2927 5"/>
    </justification>
  </clause>

  <clause id="3429">
    <literal><![CDATA[
      P(i(x,i(i(k(y,z),i(y,u)),i(k(y,z),u))))
    ]]></literal>
    <justification jstring="[hyper(4,a,2929,a,b,5,a)].">
      <j1 rule="hyper" parents="4 2929 5"/>
    </justification>
  </clause>

  <clause id="3499">
    <literal><![CDATA[
      P(i(a(x,k(y,z)),k(i(u,i(w,i(v5,v5))),a(y,x))))
    ]]></literal>
    <justification jstring="[hyper(4,a,587,a,b,3144,a)].">
      <j1 rule="hyper" parents="4 587 3144"/>
    </justification>
  </clause>

  <clause id="3582">
    <literal><![CDATA[
      P(i(i(x,y),i(k(z,x),y)))
    ]]></literal>
    <justification jstring="[hyper(4,a,3388,a,b,2931,a)].">
      <j1 rule="hyper" parents="4 3388 2931"/>
    </justification>
  </clause>

  <clause id="3712">
    <literal><![CDATA[
      P(i(i(x,y),i(k(x,z),y)))
    ]]></literal>
    <justification jstring="[hyper(4,a,3429,a,b,2931,a)].">
      <j1 rule="hyper" parents="4 3429 2931"/>
    </justification>
  </clause>

  <clause id="3849">
    <literal><![CDATA[
      P(i(a(x,k(y,z)),k(a(y,x),i(u,i(w,w)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,3499,a,b,2166,a)].">
      <j1 rule="hyper" parents="4 3499 2166"/>
    </justification>
  </clause>

  <clause id="3912">
    <literal><![CDATA[
      P(i(a(i(x,n(y)),i(x,i(i(z,x),i(y,u)))),i(x,i(y,u))))
    ]]></literal>
    <justification jstring="[hyper(4,a,2931,a,b,380,a)].">
      <j1 rule="hyper" parents="4 2931 380"/>
    </justification>
  </clause>

  <clause id="3990">
    <literal><![CDATA[
      P(i(i(i(x,y),z),i(y,z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,2985,a,b,3912,a)].">
      <j1 rule="hyper" parents="4 2985 3912"/>
    </justification>
  </clause>

  <clause id="4041">
    <literal><![CDATA[
      P(i(x,i(y,k(y,x))))
    ]]></literal>
    <justification jstring="[hyper(4,a,815,a,b,3990,a)].">
      <j1 rule="hyper" parents="4 815 3990"/>
    </justification>
  </clause>

  <clause id="4063">
    <literal><![CDATA[
      P(i(i(x,i(i(y,z),u)),i(x,i(z,u))))
    ]]></literal>
    <justification jstring="[hyper(4,a,3990,a,b,2128,a)].">
      <j1 rule="hyper" parents="4 3990 2128"/>
    </justification>
  </clause>

  <clause id="4135">
    <literal><![CDATA[
      P(i(x,k(x,i(y,n(n(y))))))
    ]]></literal>
    <justification jstring="[hyper(4,a,811,a,b,4041,a)].">
      <j1 rule="hyper" parents="4 811 4041"/>
    </justification>
  </clause>

  <clause id="4409">
    <literal><![CDATA[
      P(i(i(x,i(y,z)),i(y,i(x,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,6,a,b,4063,a)].">
      <j1 rule="hyper" parents="4 6 4063"/>
    </justification>
  </clause>

  <clause id="4472">
    <literal><![CDATA[
      P(i(i(x,y),i(i(y,z),i(x,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,2128,a,b,4409,a)].">
      <j1 rule="hyper" parents="4 2128 4409"/>
    </justification>
  </clause>

  <clause id="4480">
    <literal><![CDATA[
      P(i(x,i(i(x,y),y)))
    ]]></literal>
    <justification jstring="[hyper(4,a,781,a,b,4409,a)].">
      <j1 rule="hyper" parents="4 781 4409"/>
    </justification>
  </clause>

  <clause id="4651">
    <literal><![CDATA[
      P(i(i(i(x,a(x,y)),z),z))
    ]]></literal>
    <justification jstring="[hyper(4,a,10,a,b,4480,a)].">
      <j1 rule="hyper" parents="4 10 4480"/>
    </justification>
  </clause>

  <clause id="4759">
    <literal><![CDATA[
      P(i(i(x,i(i(y,a(y,z)),u)),i(x,u)))
    ]]></literal>
    <justification jstring="[hyper(4,a,4651,a,b,2128,a)].">
      <j1 rule="hyper" parents="4 4651 2128"/>
    </justification>
  </clause>

  <clause id="4848">
    <literal><![CDATA[
      P(i(i(i(x,k(x,y)),z),i(y,z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,4041,a,b,4472,a)].">
      <j1 rule="hyper" parents="4 4041 4472"/>
    </justification>
  </clause>

  <clause id="4852">
    <literal><![CDATA[
      P(i(i(i(k(x,y),z),u),i(i(x,z),u)))
    ]]></literal>
    <justification jstring="[hyper(4,a,3712,a,b,4472,a)].">
      <j1 rule="hyper" parents="4 3712 4472"/>
    </justification>
  </clause>

  <clause id="4853">
    <literal><![CDATA[
      P(i(i(i(k(x,y),z),u),i(i(y,z),u)))
    ]]></literal>
    <justification jstring="[hyper(4,a,3582,a,b,4472,a)].">
      <j1 rule="hyper" parents="4 3582 4472"/>
    </justification>
  </clause>

  <clause id="4904">
    <literal><![CDATA[
      P(i(i(a(x,y),z),i(a(y,x),z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,593,a,b,4472,a)].">
      <j1 rule="hyper" parents="4 593 4472"/>
    </justification>
  </clause>

  <clause id="4989">
    <literal><![CDATA[
      P(i(x,i(i(k(y,x),z),i(y,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,4472,a,b,4848,a)].">
      <j1 rule="hyper" parents="4 4472 4848"/>
    </justification>
  </clause>

  <clause id="5249">
    <literal><![CDATA[
      P(i(i(k(x,y),z),i(y,i(x,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,4989,a,b,4409,a)].">
      <j1 rule="hyper" parents="4 4989 4409"/>
    </justification>
  </clause>

  <clause id="5342">
    <literal><![CDATA[
      P(i(i(k(x,y),z),i(i(y,x),i(y,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,5249,a,b,711,a)].">
      <j1 rule="hyper" parents="4 5249 711"/>
    </justification>
  </clause>

  <clause id="5376">
    <literal><![CDATA[
      P(i(i(x,a(y,z)),i(a(x,y),a(y,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,12,a,b,4759,a)].">
      <j1 rule="hyper" parents="4 12 4759"/>
    </justification>
  </clause>

  <clause id="5439">
    <literal><![CDATA[
      P(i(i(x,i(y,z)),i(k(x,y),z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,2927,a,b,4852,a)].">
      <j1 rule="hyper" parents="4 2927 4852"/>
    </justification>
  </clause>

  <clause id="5440">
    <literal><![CDATA[
      P(i(i(x,i(k(x,y),z)),i(k(x,y),z)))
    ]]></literal>
    <justification jstring="[hyper(4,a,2599,a,b,4852,a)].">
      <j1 rule="hyper" parents="4 2599 4852"/>
    </justification>
  </clause>

  <clause id="5592">
    <literal><![CDATA[
      P(i(i(x,i(y,i(z,u))),i(x,i(k(y,z),u))))
    ]]></literal>
    <justification jstring="[hyper(4,a,5439,a,b,2128,a)].">
      <j1 rule="hyper" parents="4 5439 2128"/>
    </justification>
  </clause>

  <clause id="5715">
    <literal><![CDATA[
      P(i(i(x,y),i(k(z,x),k(z,y))))
    ]]></literal>
    <justification jstring="[hyper(4,a,28,a,b,4853,a)].">
      <j1 rule="hyper" parents="4 28 4853"/>
    </justification>
  </clause>

  <clause id="5769">
    <literal><![CDATA[
      P(i(k(x,i(y,i(z,u))),k(x,i(k(y,z),u))))
    ]]></literal>
    <justification jstring="[hyper(4,a,5439,a,b,5715,a)].">
      <j1 rule="hyper" parents="4 5439 5715"/>
    </justification>
  </clause>

  <clause id="5776">
    <literal><![CDATA[
      P(i(k(x,i(y,i(i(z,a(z,u)),w))),k(x,i(y,w))))
    ]]></literal>
    <justification jstring="[hyper(4,a,4759,a,b,5715,a)].">
      <j1 rule="hyper" parents="4 4759 5715"/>
    </justification>
  </clause>

  <clause id="5782">
    <literal><![CDATA[
      P(i(k(x,i(y,i(z,u))),k(x,i(z,i(y,u)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,4409,a,b,5715,a)].">
      <j1 rule="hyper" parents="4 4409 5715"/>
    </justification>
  </clause>

  <clause id="5805">
    <literal><![CDATA[
      P(i(k(x,i(y,k(z,u))),k(x,i(y,k(u,z)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,2178,a,b,5715,a)].">
      <j1 rule="hyper" parents="4 2178 5715"/>
    </justification>
  </clause>

  <clause id="5833">
    <literal><![CDATA[
      P(i(k(x,i(y,a(z,u))),k(x,i(y,a(u,z)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,653,a,b,5715,a)].">
      <j1 rule="hyper" parents="4 653 5715"/>
    </justification>
  </clause>

  <clause id="5853">
    <literal><![CDATA[
      P(i(k(x,i(y,k(z,u))),k(x,i(y,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,88,a,b,5715,a)].">
      <j1 rule="hyper" parents="4 88 5715"/>
    </justification>
  </clause>

  <clause id="5855">
    <literal><![CDATA[
      P(i(k(x,i(y,a(z,u))),k(x,i(a(u,y),a(z,u)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,52,a,b,5715,a)].">
      <j1 rule="hyper" parents="4 52 5715"/>
    </justification>
  </clause>

  <clause id="6193">
    <literal><![CDATA[
      P(i(i(k(x,y),z),i(x,i(y,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,5342,a,b,4063,a)].">
      <j1 rule="hyper" parents="4 5342 4063"/>
    </justification>
  </clause>

  <clause id="6255">
    <literal><![CDATA[
      P(i(x,i(y,a(z,k(x,y)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,11,a,b,6193,a)].">
      <j1 rule="hyper" parents="4 11 6193"/>
    </justification>
  </clause>

  <clause id="6292">
    <literal><![CDATA[
      P(i(i(k(x,y),z),i(i(x,y),i(x,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,6193,a,b,711,a)].">
      <j1 rule="hyper" parents="4 6193 711"/>
    </justification>
  </clause>

  <clause id="6452">
    <literal><![CDATA[
      P(i(x,k(x,i(y,i(z,a(u,k(y,z)))))))
    ]]></literal>
    <justification jstring="[hyper(4,a,6255,a,b,4041,a)].">
      <j1 rule="hyper" parents="4 6255 4041"/>
    </justification>
  </clause>

  <clause id="6542">
    <literal><![CDATA[
      P(i(i(x,i(y,a(z,u))),i(x,i(a(y,z),a(z,u)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,5376,a,b,2128,a)].">
      <j1 rule="hyper" parents="4 5376 2128"/>
    </justification>
  </clause>

  <clause id="6570">
    <literal><![CDATA[
      P(i(k(x,i(y,i(k(y,z),u))),k(x,i(k(y,z),u))))
    ]]></literal>
    <justification jstring="[hyper(4,a,5440,a,b,5715,a)].">
      <j1 rule="hyper" parents="4 5440 5715"/>
    </justification>
  </clause>

  <clause id="6676">
    <literal><![CDATA[
      P(i(i(x,k(y,i(x,z))),i(x,k(y,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,2020,a,b,6292,a)].">
      <j1 rule="hyper" parents="4 2020 6292"/>
    </justification>
  </clause>

  <clause id="6903">
    <literal><![CDATA[
      P(i(i(x,y),i(x,k(y,i(z,i(u,a(w,k(z,u))))))))
    ]]></literal>
    <justification jstring="[hyper(4,a,6452,a,b,2128,a)].">
      <j1 rule="hyper" parents="4 6452 2128"/>
    </justification>
  </clause>

  <clause id="7476">
    <literal><![CDATA[
      P(i(i(x,y),i(k(x,z),k(y,z))))
    ]]></literal>
    <justification jstring="[hyper(4,a,3081,a,b,5592,a)].">
      <j1 rule="hyper" parents="4 3081 5592"/>
    </justification>
  </clause>

  <clause id="7597">
    <literal><![CDATA[
      P(i(k(i(a(x,y),z),u),k(i(a(y,x),z),u)))
    ]]></literal>
    <justification jstring="[hyper(4,a,4904,a,b,7476,a)].">
      <j1 rule="hyper" parents="4 4904 7476"/>
    </justification>
  </clause>

  <clause id="7792">
    <literal><![CDATA[
      P(i(i(x,k(y,i(z,i(u,w)))),i(x,k(y,i(k(z,u),w)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,5769,a,b,2128,a)].">
      <j1 rule="hyper" parents="4 5769 2128"/>
    </justification>
  </clause>

  <clause id="7846">
    <literal><![CDATA[
      P(i(i(x,k(y,i(z,i(u,w)))),i(x,k(y,i(u,i(z,w))))))
    ]]></literal>
    <justification jstring="[hyper(4,a,5782,a,b,2128,a)].">
      <j1 rule="hyper" parents="4 5782 2128"/>
    </justification>
  </clause>

  <clause id="8059">
    <literal><![CDATA[
      P(i(i(x,k(y,i(z,i(i(u,a(u,w)),v5)))),i(x,k(y,i(z,v5)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,5776,a,b,2128,a)].">
      <j1 rule="hyper" parents="4 5776 2128"/>
    </justification>
  </clause>

  <clause id="8112">
    <literal><![CDATA[
      P(i(i(x,k(y,i(z,a(u,w)))),i(x,k(y,i(a(w,z),a(u,w))))))
    ]]></literal>
    <justification jstring="[hyper(4,a,5855,a,b,2128,a)].">
      <j1 rule="hyper" parents="4 5855 2128"/>
    </justification>
  </clause>

  <clause id="8189">
    <literal><![CDATA[
      P(i(k(x,i(y,i(z,a(u,w)))),k(x,i(y,i(a(z,u),a(u,w))))))
    ]]></literal>
    <justification jstring="[hyper(4,a,6542,a,b,5715,a)].">
      <j1 rule="hyper" parents="4 6542 5715"/>
    </justification>
  </clause>

  <clause id="8440">
    <literal><![CDATA[
      P(i(k(x,i(y,z)),k(x,i(y,k(z,i(u,i(w,a(v5,k(u,w)))))))))
    ]]></literal>
    <justification jstring="[hyper(4,a,6903,a,b,5715,a)].">
      <j1 rule="hyper" parents="4 6903 5715"/>
    </justification>
  </clause>

  <clause id="8516">
    <literal><![CDATA[
      P(i(k(x,i(y,k(z,i(u,i(w,v5))))),k(x,i(y,k(z,i(k(u,w),v5))))))
    ]]></literal>
    <justification jstring="[hyper(4,a,7792,a,b,5715,a)].">
      <j1 rule="hyper" parents="4 7792 5715"/>
    </justification>
  </clause>

  <clause id="8591">
    <literal><![CDATA[
      P(i(k(x,i(y,k(z,i(u,i(w,v5))))),k(x,i(y,k(z,i(w,i(u,v5)))))))
    ]]></literal>
    <justification jstring="[hyper(4,a,7846,a,b,5715,a)].">
      <j1 rule="hyper" parents="4 7846 5715"/>
    </justification>
  </clause>

  <clause id="8674">
    <literal><![CDATA[
      P(i(a(x,k(y,z)),k(a(y,x),i(u,i(w,a(w,v5))))))
    ]]></literal>
    <justification jstring="[hyper(4,a,3849,a,b,8059,a)].">
      <j1 rule="hyper" parents="4 3849 8059"/>
    </justification>
  </clause>

  <clause id="8745">
    <literal><![CDATA[
      P(i(a(x,k(y,z)),k(a(y,x),i(k(u,w),a(w,v5)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,8674,a,b,7792,a)].">
      <j1 rule="hyper" parents="4 8674 7792"/>
    </justification>
  </clause>

  <clause id="8844">
    <literal><![CDATA[
      P(i(a(x,k(y,z)),k(a(y,x),i(a(u,k(w,v5)),a(v5,u)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,8745,a,b,8112,a)].">
      <j1 rule="hyper" parents="4 8745 8112"/>
    </justification>
  </clause>

  <clause id="8924">
    <literal><![CDATA[
      P(i(a(x,k(y,z)),k(a(y,x),a(z,x))))
    ]]></literal>
    <justification jstring="[hyper(4,a,8844,a,b,6676,a)].">
      <j1 rule="hyper" parents="4 8844 6676"/>
    </justification>
  </clause>

  <clause id="8984">
    <literal><![CDATA[
      P(k(i(a(x,k(y,z)),k(a(y,x),a(z,x))),i(u,n(n(u)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,8924,a,b,4135,a)].">
      <j1 rule="hyper" parents="4 8924 4135"/>
    </justification>
  </clause>

  <clause id="9077">
    <literal><![CDATA[
      P(i(i(x,k(y,i(z,i(u,a(w,v5))))),i(x,k(y,i(z,i(a(u,w),a(w,v5)))))))
    ]]></literal>
    <justification jstring="[hyper(4,a,8189,a,b,2128,a)].">
      <j1 rule="hyper" parents="4 8189 2128"/>
    </justification>
  </clause>

  <clause id="9103">
    <literal><![CDATA[
      P(k(i(a(x,k(y,z)),k(a(y,x),a(z,x))),i(u,k(n(n(u)),i(w,i(v5,a(v6,k(w,v5))))))))
    ]]></literal>
    <justification jstring="[hyper(4,a,8984,a,b,8440,a)].">
      <j1 rule="hyper" parents="4 8984 8440"/>
    </justification>
  </clause>

  <clause id="9296">
    <literal><![CDATA[
      P(i(k(x,i(y,k(z,i(u,i(w,a(v5,v6)))))),k(x,i(y,k(z,i(u,i(a(w,v5),a(v5,v6))))))))
    ]]></literal>
    <justification jstring="[hyper(4,a,9077,a,b,5715,a)].">
      <j1 rule="hyper" parents="4 9077 5715"/>
    </justification>
  </clause>

  <clause id="9346">
    <literal><![CDATA[
      P(k(i(a(x,k(y,z)),k(a(y,x),a(z,x))),i(u,k(n(n(u)),i(w,i(v5,a(v6,k(v5,w))))))))
    ]]></literal>
    <justification jstring="[hyper(4,a,9103,a,b,8591,a)].">
      <j1 rule="hyper" parents="4 9103 8591"/>
    </justification>
  </clause>

  <clause id="9433">
    <literal><![CDATA[
      P(k(i(a(x,k(y,z)),k(a(y,x),a(z,x))),i(u,k(n(n(u)),i(w,i(a(v5,v6),a(v6,k(v5,w))))))))
    ]]></literal>
    <justification jstring="[hyper(4,a,9346,a,b,9296,a)].">
      <j1 rule="hyper" parents="4 9346 9296"/>
    </justification>
  </clause>

  <clause id="9467">
    <literal><![CDATA[
      P(k(i(a(x,k(y,z)),k(a(y,x),a(z,x))),i(u,k(n(n(u)),i(a(w,v5),i(v6,a(v5,k(w,v6))))))))
    ]]></literal>
    <justification jstring="[hyper(4,a,9433,a,b,8591,a)].">
      <j1 rule="hyper" parents="4 9433 8591"/>
    </justification>
  </clause>

  <clause id="9500">
    <literal><![CDATA[
      P(k(i(a(x,k(y,z)),k(a(y,x),a(z,x))),i(u,k(n(n(u)),i(a(w,v5),i(a(v6,v5),a(v5,k(w,v6))))))))
    ]]></literal>
    <justification jstring="[hyper(4,a,9467,a,b,9296,a)].">
      <j1 rule="hyper" parents="4 9467 9296"/>
    </justification>
  </clause>

  <clause id="9535">
    <literal><![CDATA[
      P(k(i(a(x,k(y,z)),k(a(y,x),a(z,x))),i(u,k(n(n(u)),i(k(a(w,v5),a(v6,v5)),a(v5,k(w,v6)))))))
    ]]></literal>
    <justification jstring="[hyper(4,a,9500,a,b,8516,a)].">
      <j1 rule="hyper" parents="4 9500 8516"/>
    </justification>
  </clause>

  <clause id="9571">
    <literal><![CDATA[
      P(k(i(a(x,k(y,z)),k(a(y,x),a(z,x))),i(u,k(i(k(a(w,v5),a(v6,v5)),a(v5,k(w,v6))),n(n(u))))))
    ]]></literal>
    <justification jstring="[hyper(4,a,9535,a,b,5805,a)].">
      <j1 rule="hyper" parents="4 9535 5805"/>
    </justification>
  </clause>

  <clause id="9602">
    <literal><![CDATA[
      P(k(i(a(x,k(y,z)),k(a(y,x),a(z,x))),i(u,i(k(a(w,v5),a(v6,v5)),a(v5,k(w,v6))))))
    ]]></literal>
    <justification jstring="[hyper(4,a,9571,a,b,5853,a)].">
      <j1 rule="hyper" parents="4 9571 5853"/>
    </justification>
  </clause>

  <clause id="9632">
    <literal><![CDATA[
      P(k(i(a(x,k(y,z)),k(a(y,x),a(z,x))),i(k(a(u,w),a(v5,w)),a(w,k(u,v5)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,9602,a,b,6570,a)].">
      <j1 rule="hyper" parents="4 9602 6570"/>
    </justification>
  </clause>

  <clause id="9664">
    <literal><![CDATA[
      P(k(i(a(k(x,y),z),k(a(x,z),a(y,z))),i(k(a(u,w),a(v5,w)),a(w,k(u,v5)))))
    ]]></literal>
    <justification jstring="[hyper(4,a,9632,a,b,7597,a)].">
      <j1 rule="hyper" parents="4 9632 7597"/>
    </justification>
  </clause>

  <clause id="9697">
    <literal><![CDATA[
      P(k(i(a(k(x,y),z),k(a(x,z),a(y,z))),i(k(a(u,w),a(v5,w)),a(k(u,v5),w))))
    ]]></literal>
    <justification jstring="[hyper(4,a,9664,a,b,5833,a)].">
      <j1 rule="hyper" parents="4 9664 5833"/>
    </justification>
  </clause>

  <clause id="9698">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer("HIF4")
    ]]></attribute>
    <justification jstring="[resolve(9697,a,16,a)].">
      <j1 rule="resolve" parents="9697 16"/>
    </justification>
  </clause>

</proof>

</proofs>
