<?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="1">

<source>wang2-eq.out</source>

<heading><![CDATA[
Prover9 (32) version 2009-02A, February 2009.
Process 11695 was started by mccune on cleo,
Wed Feb 25 09:37:09 2009
The command was "/home/mccune/bin/prover9 -f wang-eq.in wang2-eq.in".
]]></heading>

<proof number="1" length="64" max_count="19">

<comments><![CDATA[
% Proof 1 at 0.02 (+ 0.00) seconds.
% Length of proof is 64.
% Level of proof is 23.
% Maximum clause weight is 19.
% Given clauses 89.

]]></comments>

  <clause id="1" type="assumption">
    <literal><![CDATA[
      x = m
    ]]></literal>
    <literal><![CDATA[
      p(x,m)
    ]]></literal>
    <literal><![CDATA[
      y = m
    ]]></literal>
    <literal><![CDATA[
      y = x
    ]]></literal>
    <literal><![CDATA[
      -p(x,y)
    ]]></literal>
    <literal><![CDATA[
      -p(y,x)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="2">
    <literal><![CDATA[
      m = x
    ]]></literal>
    <literal><![CDATA[
      p(x,m)
    ]]></literal>
    <literal><![CDATA[
      m = y
    ]]></literal>
    <literal><![CDATA[
      y = x
    ]]></literal>
    <literal><![CDATA[
      -p(x,y)
    ]]></literal>
    <literal><![CDATA[
      -p(y,x)
    ]]></literal>
    <justification jstring="[copy(1),flip(a),flip(c)].">
      <j1 rule="copy" parents="1"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="5" type="assumption">
    <literal><![CDATA[
      x = k
    ]]></literal>
    <literal><![CDATA[
      x = m
    ]]></literal>
    <literal><![CDATA[
      x = b
    ]]></literal>
    <literal><![CDATA[
      -p(x,k)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6">
    <literal><![CDATA[
      k = x
    ]]></literal>
    <literal><![CDATA[
      m = x
    ]]></literal>
    <literal><![CDATA[
      b = x
    ]]></literal>
    <literal><![CDATA[
      -p(x,k)
    ]]></literal>
    <justification jstring="[copy(5),flip(a),flip(b),flip(c)].">
      <j1 rule="copy" parents="5"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="7" type="assumption">
    <literal><![CDATA[
      x = m
    ]]></literal>
    <literal><![CDATA[
      -p(x,m)
    ]]></literal>
    <literal><![CDATA[
      f(x) != m
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      m = x
    ]]></literal>
    <literal><![CDATA[
      -p(x,m)
    ]]></literal>
    <literal><![CDATA[
      f(x) != m
    ]]></literal>
    <justification jstring="[copy(7),flip(a)].">
      <j1 rule="copy" parents="7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9" type="assumption">
    <literal><![CDATA[
      x = m
    ]]></literal>
    <literal><![CDATA[
      -p(x,m)
    ]]></literal>
    <literal><![CDATA[
      f(x) != x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

  <clause id="11" type="assumption">
    <literal><![CDATA[
      x = m
    ]]></literal>
    <literal><![CDATA[
      -p(x,m)
    ]]></literal>
    <literal><![CDATA[
      p(x,f(x))
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      m = x
    ]]></literal>
    <literal><![CDATA[
      -p(x,m)
    ]]></literal>
    <literal><![CDATA[
      p(x,f(x))
    ]]></literal>
    <justification jstring="[copy(11),flip(a)].">
      <j1 rule="copy" parents="11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13" type="assumption">
    <literal><![CDATA[
      x = m
    ]]></literal>
    <literal><![CDATA[
      -p(x,m)
    ]]></literal>
    <literal><![CDATA[
      p(f(x),x)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      m = x
    ]]></literal>
    <literal><![CDATA[
      -p(x,m)
    ]]></literal>
    <literal><![CDATA[
      p(f(x),x)
    ]]></literal>
    <justification jstring="[copy(13),flip(a)].">
      <j1 rule="copy" parents="13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="15" type="assumption">
    <literal><![CDATA[
      x = b
    ]]></literal>
    <literal><![CDATA[
      p(x,b)
    ]]></literal>
    <literal><![CDATA[
      g(x) != b
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      b = x
    ]]></literal>
    <literal><![CDATA[
      p(x,b)
    ]]></literal>
    <literal><![CDATA[
      g(x) != b
    ]]></literal>
    <justification jstring="[copy(15),flip(a)].">
      <j1 rule="copy" parents="15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="17" type="assumption">
    <literal><![CDATA[
      x = b
    ]]></literal>
    <literal><![CDATA[
      p(x,b)
    ]]></literal>
    <literal><![CDATA[
      g(x) != x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      b = x
    ]]></literal>
    <literal><![CDATA[
      p(x,b)
    ]]></literal>
    <literal><![CDATA[
      g(x) != x
    ]]></literal>
    <justification jstring="[copy(17),flip(a)].">
      <j1 rule="copy" parents="17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21" type="assumption">
    <literal><![CDATA[
      x = b
    ]]></literal>
    <literal><![CDATA[
      p(x,b)
    ]]></literal>
    <literal><![CDATA[
      p(g(x),x)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      b = x
    ]]></literal>
    <literal><![CDATA[
      p(x,b)
    ]]></literal>
    <literal><![CDATA[
      p(g(x),x)
    ]]></literal>
    <justification jstring="[copy(21),flip(a)].">
      <j1 rule="copy" parents="21"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23" type="assumption">
    <literal><![CDATA[
      x = k
    ]]></literal>
    <literal><![CDATA[
      x != m
    ]]></literal>
    <literal><![CDATA[
      p(x,k)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      k = x
    ]]></literal>
    <literal><![CDATA[
      m != x
    ]]></literal>
    <literal><![CDATA[
      p(x,k)
    ]]></literal>
    <justification jstring="[copy(23),flip(a),flip(b)].">
      <j1 rule="copy" parents="23"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="25" type="assumption">
    <literal><![CDATA[
      x = k
    ]]></literal>
    <literal><![CDATA[
      x != b
    ]]></literal>
    <literal><![CDATA[
      p(x,k)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      k = x
    ]]></literal>
    <literal><![CDATA[
      b != x
    ]]></literal>
    <literal><![CDATA[
      p(x,k)
    ]]></literal>
    <justification jstring="[copy(25),flip(a),flip(b)].">
      <j1 rule="copy" parents="25"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="27" type="assumption">
    <literal><![CDATA[
      m != b
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="28" type="assumption">
    <literal><![CDATA[
      b = k
    ]]></literal>
    <literal><![CDATA[
      k = m
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      k = b
    ]]></literal>
    <literal><![CDATA[
      k = m
    ]]></literal>
    <justification jstring="[copy(28),flip(a)].">
      <j1 rule="copy" parents="28"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="30" type="assumption">
    <literal><![CDATA[
      x = j
    ]]></literal>
    <literal><![CDATA[
      -p(x,j)
    ]]></literal>
    <literal><![CDATA[
      x = k
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      j = x
    ]]></literal>
    <literal><![CDATA[
      -p(x,j)
    ]]></literal>
    <literal><![CDATA[
      k = x
    ]]></literal>
    <justification jstring="[copy(30),flip(a),flip(c)].">
      <j1 rule="copy" parents="30"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="32" type="assumption">
    <literal><![CDATA[
      x = j
    ]]></literal>
    <literal><![CDATA[
      p(x,j)
    ]]></literal>
    <literal><![CDATA[
      x != k
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      j = x
    ]]></literal>
    <literal><![CDATA[
      p(x,j)
    ]]></literal>
    <literal><![CDATA[
      k != x
    ]]></literal>
    <justification jstring="[copy(32),flip(a),flip(c)].">
      <j1 rule="copy" parents="32"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      k = m
    ]]></literal>
    <literal><![CDATA[
      p(m,k)
    ]]></literal>
    <justification jstring="[xx_res(24,b)].">
      <j1 rule="xx_res" parents="24"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      k = b
    ]]></literal>
    <literal><![CDATA[
      p(b,k)
    ]]></literal>
    <justification jstring="[xx_res(26,b)].">
      <j1 rule="xx_res" parents="26"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      k = b
    ]]></literal>
    <literal><![CDATA[
      k = x
    ]]></literal>
    <literal><![CDATA[
      m = x
    ]]></literal>
    <literal><![CDATA[
      b = x
    ]]></literal>
    <literal><![CDATA[
      -p(x,m)
    ]]></literal>
    <justification jstring="[para(29(b,1),6(d,2))].">
      <j1 rule="para" parents="29 6"/>
    </justification>
  </clause>

  <clause id="48">
    <literal><![CDATA[
      g(j) = j
    ]]></literal>
    <literal><![CDATA[
      g(j) = k
    ]]></literal>
    <literal><![CDATA[
      j = b
    ]]></literal>
    <literal><![CDATA[
      p(j,b)
    ]]></literal>
    <justification jstring="[resolve(31,b,22,c),flip(a),flip(b),flip(c)].">
      <j1 rule="resolve" parents="31 22"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      j = k
    ]]></literal>
    <literal><![CDATA[
      p(k,j)
    ]]></literal>
    <justification jstring="[xx_res(33,c)].">
      <j1 rule="xx_res" parents="33"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      k = b
    ]]></literal>
    <literal><![CDATA[
      p(b,m)
    ]]></literal>
    <justification jstring="[para(29(b,1),46(b,2)),merge(b)].">
      <j1 rule="para" parents="29 46"/>
      <j2 rule="merge"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      k = b
    ]]></literal>
    <literal><![CDATA[
      p(f(b),b)
    ]]></literal>
    <justification jstring="[resolve(53,b,14,b),unit_del(b,27)].">
      <j1 rule="resolve" parents="53 14"/>
      <j2 rule="unit_del" parents="27"/>
    </justification>
  </clause>

  <clause id="60">
    <literal><![CDATA[
      k = b
    ]]></literal>
    <literal><![CDATA[
      p(b,f(b))
    ]]></literal>
    <justification jstring="[resolve(53,b,12,b),unit_del(b,27)].">
      <j1 rule="resolve" parents="53 12"/>
      <j2 rule="unit_del" parents="27"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      k = b
    ]]></literal>
    <literal><![CDATA[
      f(b) != b
    ]]></literal>
    <justification jstring="[resolve(53,b,10,b),unit_del(b,27)].">
      <j1 rule="resolve" parents="53 10"/>
      <j2 rule="unit_del" parents="27"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      k = b
    ]]></literal>
    <literal><![CDATA[
      f(b) != m
    ]]></literal>
    <justification jstring="[resolve(53,b,8,b),unit_del(b,27)].">
      <j1 rule="resolve" parents="53 8"/>
      <j2 rule="unit_del" parents="27"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      k = b
    ]]></literal>
    <literal><![CDATA[
      f(b) = m
    ]]></literal>
    <literal><![CDATA[
      p(f(b),m)
    ]]></literal>
    <literal><![CDATA[
      f(b) = b
    ]]></literal>
    <literal><![CDATA[
      -p(b,f(b))
    ]]></literal>
    <justification jstring="[resolve(59,b,2,e),flip(b),flip(e),unit_del(d,27)].">
      <j1 rule="resolve" parents="59 2"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
      <j2 rule="unit_del" parents="27"/>
    </justification>
  </clause>

  <clause id="94">
    <literal><![CDATA[
      k = b
    ]]></literal>
    <literal><![CDATA[
      f(b) = m
    ]]></literal>
    <literal><![CDATA[
      p(f(b),m)
    ]]></literal>
    <literal><![CDATA[
      f(b) = b
    ]]></literal>
    <justification jstring="[resolve(65,e,60,b),merge(e)].">
      <j1 rule="resolve" parents="65 60"/>
      <j2 rule="merge"/>
    </justification>
  </clause>

  <clause id="95">
    <literal><![CDATA[
      k = b
    ]]></literal>
    <literal><![CDATA[
      f(b) = m
    ]]></literal>
    <literal><![CDATA[
      f(b) = b
    ]]></literal>
    <literal><![CDATA[
      f(b) = k
    ]]></literal>
    <justification jstring="[resolve(94,c,47,e),flip(e),flip(f),flip(g),merge(d),merge(f),merge(g)].">
      <j1 rule="resolve" parents="94 47"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
      <j2 rule="merge"/>
      <j2 rule="merge"/>
      <j2 rule="merge"/>
    </justification>
  </clause>

  <clause id="104">
    <literal><![CDATA[
      k = b
    ]]></literal>
    <literal><![CDATA[
      f(b) = m
    ]]></literal>
    <literal><![CDATA[
      f(b) = b
    ]]></literal>
    <justification jstring="[para(29(b,1),95(d,2)),merge(b),merge(e)].">
      <j1 rule="para" parents="29 95"/>
      <j2 rule="merge"/>
      <j2 rule="merge"/>
    </justification>
  </clause>

  <clause id="105">
    <literal><![CDATA[
      k = b
    ]]></literal>
    <literal><![CDATA[
      f(b) = b
    ]]></literal>
    <justification jstring="[resolve(104,b,62,b),merge(c)].">
      <j1 rule="resolve" parents="104 62"/>
      <j2 rule="merge"/>
    </justification>
  </clause>

  <clause id="106">
    <literal><![CDATA[
      k = b
    ]]></literal>
    <justification jstring="[resolve(105,b,61,b),merge(b)].">
      <j1 rule="resolve" parents="105 61"/>
      <j2 rule="merge"/>
    </justification>
  </clause>

  <clause id="112">
    <literal><![CDATA[
      j = b
    ]]></literal>
    <literal><![CDATA[
      p(b,j)
    ]]></literal>
    <justification jstring="[back_rewrite(50),rewrite([106(2),106(4)])].">
      <j1 rule="back_rewrite" parents="50"/>
      <j2 rule="rewrite" parents="106 106"/>
    </justification>
  </clause>

  <clause id="113">
    <literal><![CDATA[
      g(j) = j
    ]]></literal>
    <literal><![CDATA[
      g(j) = b
    ]]></literal>
    <literal><![CDATA[
      j = b
    ]]></literal>
    <literal><![CDATA[
      p(j,b)
    ]]></literal>
    <justification jstring="[back_rewrite(48),rewrite([106(7)])].">
      <j1 rule="back_rewrite" parents="48"/>
      <j2 rule="rewrite" parents="106"/>
    </justification>
  </clause>

  <clause id="114">
    <literal><![CDATA[
      p(m,b)
    ]]></literal>
    <justification jstring="[back_rewrite(45),rewrite([106(1),106(5)]),flip(a),unit_del(a,27)].">
      <j1 rule="back_rewrite" parents="45"/>
      <j2 rule="rewrite" parents="106 106"/>
      <j2 rule="flip"/>
      <j2 rule="unit_del" parents="27"/>
    </justification>
  </clause>

  <clause id="116">
    <literal><![CDATA[
      j = x
    ]]></literal>
    <literal><![CDATA[
      -p(x,j)
    ]]></literal>
    <literal><![CDATA[
      b = x
    ]]></literal>
    <justification jstring="[back_rewrite(31),rewrite([106(5)])].">
      <j1 rule="back_rewrite" parents="31"/>
      <j2 rule="rewrite" parents="106"/>
    </justification>
  </clause>

  <clause id="118">
    <literal><![CDATA[
      b = x
    ]]></literal>
    <literal><![CDATA[
      m = x
    ]]></literal>
    <literal><![CDATA[
      -p(x,b)
    ]]></literal>
    <justification jstring="[back_rewrite(6),rewrite([106(1),106(7)]),merge(c)].">
      <j1 rule="back_rewrite" parents="6"/>
      <j2 rule="rewrite" parents="106 106"/>
      <j2 rule="merge"/>
    </justification>
  </clause>

  <clause id="130">
    <literal><![CDATA[
      g(j) = j
    ]]></literal>
    <literal><![CDATA[
      g(j) = b
    ]]></literal>
    <literal><![CDATA[
      j = b
    ]]></literal>
    <literal><![CDATA[
      j = m
    ]]></literal>
    <justification jstring="[resolve(113,d,118,c),flip(d),flip(e),merge(d)].">
      <j1 rule="resolve" parents="113 118"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
      <j2 rule="merge"/>
    </justification>
  </clause>

  <clause id="131">
    <literal><![CDATA[
      g(j) = b
    ]]></literal>
    <literal><![CDATA[
      j = b
    ]]></literal>
    <literal><![CDATA[
      j = m
    ]]></literal>
    <literal><![CDATA[
      p(j,b)
    ]]></literal>
    <justification jstring="[resolve(130,a,18,c),flip(d),merge(d)].">
      <j1 rule="resolve" parents="130 18"/>
      <j2 rule="flip"/>
      <j2 rule="merge"/>
    </justification>
  </clause>

  <clause id="132">
    <literal><![CDATA[
      g(j) = b
    ]]></literal>
    <literal><![CDATA[
      j = b
    ]]></literal>
    <literal><![CDATA[
      j = m
    ]]></literal>
    <justification jstring="[resolve(131,d,118,c),flip(d),flip(e),merge(d),merge(e)].">
      <j1 rule="resolve" parents="131 118"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
      <j2 rule="merge"/>
      <j2 rule="merge"/>
    </justification>
  </clause>

  <clause id="133">
    <literal><![CDATA[
      j = b
    ]]></literal>
    <literal><![CDATA[
      j = m
    ]]></literal>
    <literal><![CDATA[
      p(j,b)
    ]]></literal>
    <justification jstring="[resolve(132,a,16,c),flip(c),merge(c)].">
      <j1 rule="resolve" parents="132 16"/>
      <j2 rule="flip"/>
      <j2 rule="merge"/>
    </justification>
  </clause>

  <clause id="135">
    <literal><![CDATA[
      j = b
    ]]></literal>
    <literal><![CDATA[
      j = m
    ]]></literal>
    <justification jstring="[resolve(133,c,118,c),flip(c),flip(d),merge(c),merge(d)].">
      <j1 rule="resolve" parents="133 118"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
      <j2 rule="merge"/>
      <j2 rule="merge"/>
    </justification>
  </clause>

  <clause id="137">
    <literal><![CDATA[
      j = b
    ]]></literal>
    <literal><![CDATA[
      p(b,m)
    ]]></literal>
    <justification jstring="[para(135(b,1),112(b,2)),merge(b)].">
      <j1 rule="para" parents="135 112"/>
      <j2 rule="merge"/>
    </justification>
  </clause>

  <clause id="139">
    <literal><![CDATA[
      j = b
    ]]></literal>
    <literal><![CDATA[
      p(f(b),b)
    ]]></literal>
    <justification jstring="[resolve(137,b,14,b),unit_del(b,27)].">
      <j1 rule="resolve" parents="137 14"/>
      <j2 rule="unit_del" parents="27"/>
    </justification>
  </clause>

  <clause id="141">
    <literal><![CDATA[
      j = b
    ]]></literal>
    <literal><![CDATA[
      f(b) != b
    ]]></literal>
    <justification jstring="[resolve(137,b,10,b),unit_del(b,27)].">
      <j1 rule="resolve" parents="137 10"/>
      <j2 rule="unit_del" parents="27"/>
    </justification>
  </clause>

  <clause id="142">
    <literal><![CDATA[
      j = b
    ]]></literal>
    <literal><![CDATA[
      f(b) != m
    ]]></literal>
    <justification jstring="[resolve(137,b,8,b),unit_del(b,27)].">
      <j1 rule="resolve" parents="137 8"/>
      <j2 rule="unit_del" parents="27"/>
    </justification>
  </clause>

  <clause id="143">
    <literal><![CDATA[
      j = b
    ]]></literal>
    <literal><![CDATA[
      f(b) = b
    ]]></literal>
    <literal><![CDATA[
      f(b) = m
    ]]></literal>
    <justification jstring="[resolve(139,b,118,c),flip(b),flip(c)].">
      <j1 rule="resolve" parents="139 118"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="145">
    <literal><![CDATA[
      j = b
    ]]></literal>
    <literal><![CDATA[
      f(b) = b
    ]]></literal>
    <justification jstring="[resolve(143,c,142,b),merge(c)].">
      <j1 rule="resolve" parents="143 142"/>
      <j2 rule="merge"/>
    </justification>
  </clause>

  <clause id="146">
    <literal><![CDATA[
      j = b
    ]]></literal>
    <justification jstring="[resolve(145,b,141,b),merge(b)].">
      <j1 rule="resolve" parents="145 141"/>
      <j2 rule="merge"/>
    </justification>
  </clause>

  <clause id="147">
    <literal><![CDATA[
      b = x
    ]]></literal>
    <literal><![CDATA[
      -p(x,b)
    ]]></literal>
    <justification jstring="[back_rewrite(116),rewrite([146(1),146(3)]),merge(c)].">
      <j1 rule="back_rewrite" parents="116"/>
      <j2 rule="rewrite" parents="146 146"/>
      <j2 rule="merge"/>
    </justification>
  </clause>

  <clause id="149">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(147,b,114,a),flip(a),unit_del(a,27)].">
      <j1 rule="resolve" parents="147 114"/>
      <j2 rule="flip"/>
      <j2 rule="unit_del" parents="27"/>
    </justification>
  </clause>

</proof>

</proofs>
