<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">s</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/><b>let </b><font color="Maroon">I</font> be  <a href="scmpds_4.html#V3">shiftable</a> <a href="scmpds_5.html#V3">No-StopCode</a> <a href="scmpds_4.html#NM1">Program-block</a>;<br/><b>let </b><font color="Maroon">a</font> be    <a href="scmpds_2.html#M1">Int_position</a> , <font color="Maroon">b</font> be    <a href="scmpds_2.html#M1">Int_position</a> , <font color="Maroon">c</font> be    <a href="scmpds_2.html#M1">Int_position</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">p0</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">f</font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a> ;<br/>









<b>assume </b><a NAME="E1:6"/><i><font color="Green">E45</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">I</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>assume </b><a NAME="E2:6"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">f</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <font color="Maroon">s</font>,<font color="Maroon">p0</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n</font> &amp; <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p0</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<b>assume </b><a NAME="E3:6"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">t</font> being  <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a>   st  ex <font color="Olive">g</font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a>  st <br/>( <font color="Olive">g</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <font color="Maroon">s</font>,<font color="Maroon">p0</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> &amp; <font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K6">Sum</a> <font color="Olive">g</font> &amp; <font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">g</font></span>)</span> ) &amp; <font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p0</font> holds <br/><font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Olive">i</font></span>)</span> ) holds <br/>( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Olive">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">I</font> <a href="scmpds_6.html#R1">is_closed_on</a> <font color="Olive">t</font> &amp; <font color="Maroon">I</font> <a href="scmpds_6.html#R2">is_halting_on</a> <font color="Olive">t</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Olive">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 &amp;  ex <font color="Olive">g</font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a>  st <br/>( <font color="Olive">g</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <font color="Maroon">s</font>,<font color="Maroon">p0</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Olive">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">g</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Olive">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K6">Sum</a> <font color="Olive">g</font> ) &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p0</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Olive">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Olive">i</font></span>)</span> ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ] <b>means </b>( ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p0</font> holds <br/><font color="Maroon">a<sub>1</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Olive">i</font></span>)</span> ) &amp;  ex <font color="Olive">g</font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a>  st <br/>( <font color="Olive">g</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <font color="Maroon">s</font>,<font color="Maroon">p0</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K6">Sum</a> <font color="Olive">g</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">a<sub>1</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">g</font></span>)</span> ) );<br/>
<b>set </b><font color="Maroon">da</font> =  <a href="scmpds_2.html#K3">DataLoc</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">i</font>;<br/>
<b>consider </b><font color="Maroon">ff</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="card_3.html#K4">product</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="scmpds_2.html#K1">SCMPDS</a> , <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:6"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">t</font> being  <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a>  holds <br/> ( ( <font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 implies <font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 implies <font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span> ) )
 <b>by </b><i/>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  = <font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<div><i><font color="Green">E63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/><b>set </b><font color="Maroon">dt</font> =  <a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">t</font>;<br/><b>assume </b><a NAME="E1:6_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">t</font>]
 ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:6_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">t</font>) <a href="hidden.html#R1">=</a> 0
 ;<br/><b>assume </b><a NAME="E2:6_1_1"/>
<font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><a NAME="E3:6_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="ref" href="scmpds_8.html#T4">SCMPDS_8:4</a></i>;<br/><a NAME="E4:6_1_1"/><b>then </b>
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">t</font>) <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:6_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E3:6_1"/>
<font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><a NAME="E4:6_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="scmpds_8.html#T4">SCMPDS_8:4</a></i>;<br/><b>hence </b><a NAME="E5:6_1"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">t</font>) <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E7:6"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">s</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">Ds</font> =  <a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">s</font>;<br/>
<b>thus </b><a NAME="E1:6_2"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p0</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">s</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Olive">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="scmpds_8.html#T4">SCMPDS_8:4</a></i>;<br/>

<b>consider </b><font color="Maroon">h</font> being    <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a> <b> such that </b><br/><a NAME="E3:6_2"/><i><font color="Green">E68</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">h</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <font color="Maroon">s</font>,<font color="Maroon">p0</font> )
 <b>by </b><i><a class="ref" href="scpisort.html#T3">SCPISORT:3</a></i>;<br/>
<b>take </b>
<font color="Maroon">h</font>
;<br/>

<b>thus </b><a NAME="E4:6_2"/>
<font color="Maroon">h</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <font color="Maroon">s</font>,<font color="Maroon">p0</font>
 <b>by </b><i/>;<br/>

<b>thus </b> <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> = 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">s</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="scmpds_8.html#T4">SCMPDS_8:4</a></i>
;<br/>

<a NAME="E6:6_2"/>
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/>
<b>hence </b><a NAME="E7:6_2"/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">s</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K6">Sum</a> <font color="Maroon">h</font>
 <b>by </b><i>, <a class="ref" href="rvsum_1.html#T102">RVSUM_1:102</a>, <a class="ref" href="scmpds_8.html#T4">SCMPDS_8:4</a></i>;<br/>

<b>thus </b><a NAME="E8:6_2"/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">s</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i>, <a class="ref" href="scmpds_8.html#T4">SCMPDS_8:4</a></i>;<br/>

<b>thus </b><a NAME="E9:6_2"/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">s</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="scmpds_8.html#T4">SCMPDS_8:4</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E70</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/><b>set </b><font color="Maroon">Dt</font> =  <a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">t</font>;<br/><b>assume </b><a NAME="E1:6_3"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">t</font>] &amp; <font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 ;<br/><div><i><font color="Green">E72</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:6_3_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p0</font>
 ;<br/><b>thus </b><font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="scmpds_8.html#T4">SCMPDS_8:4</a></i>
<br/>.= 
<font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">h</font> being    <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a> <b> such that </b><br/><a NAME="E4:6_3"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">h</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <font color="Maroon">s</font>,<font color="Maroon">p0</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K6">Sum</a> <font color="Maroon">h</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E5:6_3"/><i><font color="Green">E75</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> &amp; <font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K6">Sum</a> <font color="Maroon">h</font> &amp; <font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="scmpds_8.html#T4">SCMPDS_8:4</a></i>;<br/><a NAME="E6:6_3"/><i><font color="Green">E76</font></i>: 
 <a href="scmp_gcd.html#K1">intpos</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K3">DataLoc</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="scmp_gcd.html#T5">SCMP_GCD:5</a></i>;<br/><b>hence </b><a NAME="E7:6_3"/>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, , , , , </i>;<br/><b>thus </b><a NAME="E8:6_3"/>
( <font color="Maroon">I</font> <a href="scmpds_6.html#R1">is_closed_on</a> <font color="Maroon">t</font> &amp; <font color="Maroon">I</font> <a href="scmpds_6.html#R2">is_halting_on</a> <font color="Maroon">t</font> )
 <b>by </b><i>, , , , , , </i>;<br/><b>set </b><font color="Maroon">It</font> =  <a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font>;<br/><a NAME="E9:6_3"/><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, , , , , , </i>;<br/><b>set </b><font color="Maroon">Dit</font> =  <a href="scmpds_8.html#K1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span>;<br/><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_3_2_1"/>
( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_3_2_1_1"><b>suppose </b></a><a NAME="E1:6_3_2_1_1"/>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:6_3_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="scmpds_8.html#T4">SCMPDS_8:4</a></i>;<br/><a NAME="E3:6_3_2_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span>) <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><a NAME="E4:6_3_2_1_1"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">t</font>) <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:6_3_2_1_1"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span>) <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">t</font>)
 <b>by </b><i>, <a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_3_2_1_2"><b>suppose </b></a><a NAME="E1:6_3_2_1_2"/>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:6_3_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i>, <a class="ref" href="scmpds_8.html#T4">SCMPDS_8:4</a></i>;<br/><b>then </b><i><font color="Green">E80</font></i>: <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span>) = 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="scmpds_8.html#T4">SCMPDS_8:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1

;<br/><a NAME="E4:6_3_2_1_2"/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i>, <a class="ref" href="scmpds_8.html#T4">SCMPDS_8:4</a></i>;<br/><b>then </b><font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">t</font>) = 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="scmpds_8.html#T4">SCMPDS_8:4</a></i>
;<br/><b>hence </b><a NAME="E6:6_3_2_1_2"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span>) <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">t</font>)
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T148">XREAL_1:148</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div><b>consider </b><font color="Maroon">g</font> being    <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a> <b> such that </b><br/><a NAME="E12:6_3"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">g</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <font color="Maroon">s</font>,<font color="Maroon">p0</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K6">Sum</a> <font color="Maroon">g</font> )
 <b>by </b><i>, , , , , , </i>;<br/><b>thus </b><a NAME="E13:6_3"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_3"><b>proof </b></a><div class="add">

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:6_3_3_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p0</font>
 ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="scmpds_8.html#T4">SCMPDS_8:4</a></i>
<br/>.= 
<font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , , , , , , </i>
;<br/></div>
<b>end;</b></div>

<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>thus </b><a NAME="E2:6_3_3"/>
<font color="Maroon">g</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <font color="Maroon">s</font>,<font color="Maroon">p0</font>
 <b>by </b><i/>;<br/>

<b>thus </b> <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="scmpds_8.html#T4">SCMPDS_8:4</a></i>
;<br/>

<b>thus </b><a NAME="E4:6_3_3"/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K6">Sum</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="scmpds_8.html#T4">SCMPDS_8:4</a></i>;<br/>

<a NAME="E5:6_3_3"/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i>, <a class="ref" href="scmpds_8.html#T4">SCMPDS_8:4</a></i>;<br/>
<b>hence </b><a NAME="E6:6_3_3"/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i>, , <a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/>

<b>thus </b><a NAME="E7:6_3_3"/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="scmpds_8.html#T4">SCMPDS_8:4</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>set </b><font color="Maroon">Iw</font> =  <a href="scmpds_4.html#K9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K2">while&lt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">s</font>;<br/>
<b>set </b><font color="Maroon">Dw</font> =  <a href="scmpds_8.html#K1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2">while&lt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">s</font></span>)</span>;<br/>
<a NAME="E9:6"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2">while&lt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">s</font></span>)</span>) <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2">while&lt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">s</font></span>)</span>] )
 
<b>from </b><i><a class="ref" href="scpinvar.html#S1" target="_self">SCPINVAR:sch 1</a>(, , , );<br/></i>
<b>then consider </b><font color="Maroon">g</font> being    <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a> <b> such that </b><br/><a NAME="E11:6"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">g</font> <a href="scpisort.html#R1">is_FinSequence_on</a> <font color="Maroon">s</font>,<font color="Maroon">p0</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K2">while&lt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">s</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K2">while&lt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">s</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K6">Sum</a> <font color="Maroon">g</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K2">while&lt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">s</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 ;<br/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K2">while&lt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">s</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2">while&lt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">s</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="scmpds_8.html#T4">SCMPDS_8:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2">while&lt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">s</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="scmp_gcd.html#T5">SCMP_GCD:5</a></i>
;<br/>
<a NAME="E13:6"/><b>then </b>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K2">while&lt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">s</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:6"/><b>then </b><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K2">while&lt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">s</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:6_5"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:6_5"/><b>then </b><i><font color="Green">E103</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon">p0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , , <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>
;<br/><br/></div><b>end;</b></div>
<a NAME="E16:6"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 
<b>by </b><i>, , , <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>
<b>hence </b><a NAME="E17:6"/>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2">while&lt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">s</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K6">Sum</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="scmpds_8.html#T4">SCMPDS_8:4</a></i>;<br/>

<a NAME="E18:6"/><i><font color="Green">E104</font></i>: 
for <font color="Olive">t</font> being  <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a>   st <font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <font color="Olive">t</font>] &amp; <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1">Dstate</a> <font color="Olive">t</font>) <a href="hidden.html#R1">=</a> 0 holds <br/><font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E19:6"/>
( ( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">s</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">s</font>) or <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s</font>] ) &amp;  <a href="scmpds_8.html#K2">while&lt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font> <a href="scmpds_6.html#R1">is_closed_on</a> <font color="Maroon">s</font> &amp;  <a href="scmpds_8.html#K2">while&lt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font> <a href="scmpds_6.html#R2">is_halting_on</a> <font color="Maroon">s</font> )
 
<b>from </b><i><a class="ref" href="scmpds_8.html#S1">SCMPDS_8:sch 1</a>(, , , );<br/></i>
<b>hence </b><a NAME="E20:6"/>
(  <a href="scmpds_8.html#K2">while&lt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font> <a href="scmpds_6.html#R1">is_closed_on</a> <font color="Maroon">s</font> &amp;  <a href="scmpds_8.html#K2">while&lt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font> <a href="scmpds_6.html#R2">is_halting_on</a> <font color="Maroon">s</font> )
 ;<br/>


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