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

<b>let </b><font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></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">c<sub>5</sub></font> be    <a href="scmpds_2.html#M1">Int_position</a> ;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>









<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="scmpds_2.html#K3">DataLoc</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>;<br/>
<b>assume </b><a NAME="E1:42"/><i><font color="Green">E27</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>assume </b><a NAME="E2:42"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="scmpds_2.html#M1">Int_position</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E3:42"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a>  st ( for <font color="Olive">b<sub>2</sub></font> being   <a href="scmpds_2.html#M1">Int_position</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="scmpds_2.html#M1">Int_position</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> ) &amp; <font color="Olive">b<sub>1</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/>( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="scmpds_6.html#R1">is_closed_on</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="scmpds_6.html#R2">is_halting_on</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="scmpds_2.html#M1">Int_position</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="scmpds_2.html#M1">Int_position</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<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> ] means ( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="scmpds_2.html#M1">Int_position</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Maroon">a<sub>1</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</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="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="scmpds_2.html#M1">Int_position</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> holds <br/><font color="Maroon">a<sub>1</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> ) );<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></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="E5:42"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a>  holds <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 implies <font color="Maroon">c<sub>11</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies <font color="Maroon">c<sub>11</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="scmpds_8.html#T5" target="_self">Th5</a></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#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = <font color="Maroon">c<sub>11</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E6:42"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a>  holds <br/>( <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" target="_self">Dstate</a> <font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R1">=</a> 0 iff <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/>

<b>thus </b><a NAME="E1:42_1"/>
( <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" target="_self">Dstate</a> <font color="Maroon">c<sub>12</sub></font>) <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">c<sub>12</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" target="_self">Dstate</a> <font color="Maroon">c<sub>12</sub></font>) <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>assume </b><a NAME="E2:42_1_1"/>
<font color="Maroon">c<sub>12</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E3:42_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" target="_self">Dstate</a> <font color="Maroon">c<sub>12</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="scmpds_8.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E4:42_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_8.html#E5:42"><i><font color="Green">E30</font></i></a>, <a class="txt" href="scmpds_8.html#E1:42_1_1"><i><font color="Green">E32</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:42_1"/>
<font color="Maroon">c<sub>12</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>

<a NAME="E3:42_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" target="_self">Dstate</a> <font color="Maroon">c<sub>12</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="ref" href="scmpds_8.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E4:42_1"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" target="_self">Dstate</a> <font color="Maroon">c<sub>12</sub></font>) <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="scmpds_8.html#E5:42"><i><font color="Green">E30</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:42"/><b>then </b><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></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" target="_self">Dstate</a> <font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" target="_self">Dstate</a> <font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R1">=</a> 0 holds <br/><font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
;<br/>
<a NAME="E8:42"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1" target="_self">Dstate</a> <font color="Maroon">c<sub>3</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="scmpds_8.html#K1" target="_self">Dstate</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="scmpds_2.html#M1">Int_position</a> ;<br/><b>assume </b><a NAME="E1:42_2_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:42_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scmpds_8.html#E2:42"><i><font color="Green">E28</font></i></a></i>;<br/><a NAME="E3:42_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" target="_self">Dstate</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="scmpds_8.html#T4" target="_self">Th4</a></i>;<br/><b>hence </b><a NAME="E4:42_2_1"/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" target="_self">Dstate</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_8.html#K1" target="_self">Dstate</a> <font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="scmpds_8.html#T4" target="_self">Th4</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:42_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="scmpds_2.html#M1">Int_position</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_8.html#K1" target="_self">Dstate</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="scmpds_8.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/><b>assume </b><a NAME="E1:42_3"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1" target="_self">Dstate</a> <font color="Maroon">c<sub>12</sub></font>] &amp; <font color="Maroon">c<sub>12</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>12</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> <b> such that </b><br/><a NAME="E3:42_3"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmpds_8.html#K1" target="_self">Dstate</a> <font color="Maroon">c<sub>12</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="scmpds_2.html#M1">Int_position</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Maroon">c<sub>13</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="scmpds_2.html#M1">Int_position</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> holds <br/><font color="Maroon">c<sub>13</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/><div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="scmpds_2.html#M1">Int_position</a> ;<br/><b>assume </b><a NAME="E1:42_3_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:42_3_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scmpds_8.html#E3:42_3"><i><font color="Green">E36</font></i></a></i>;<br/><a NAME="E3:42_3_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scmpds_8.html#E3:42_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="scmpds_8.html#T4" target="_self">Th4</a></i>;<br/><b>hence </b><a NAME="E4:42_3_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scmpds_8.html#E3:42_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="scmpds_8.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="scmpds_2.html#M1">Int_position</a> ;<br/><b>assume </b><a NAME="E1:42_3_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E2:42_3_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="scmpds_8.html#E3:42_3"><i><font color="Green">E36</font></i></a></i>;<br/><b>hence </b><a NAME="E3:42_3_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="scmpds_8.html#E3:42_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="scmpds_8.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:42_3"/>
( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="scmpds_6.html#R1">is_closed_on</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="scmpds_6.html#R2">is_halting_on</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="scmpds_8.html#E3:42"><i><font color="Green">E29</font></i></a>, <a class="txt" href="scmpds_8.html#E1:42_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="scmpds_8.html#E4:42_3"><i><font color="Green">E37</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>14</sub></font> =  <a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>15</sub></font> =  <a href="scmpds_8.html#K1" target="_self">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>16</sub></font> =  <a href="scmpds_8.html#K1" target="_self">Dstate</a> <font color="Maroon">c<sub>12</sub></font>;<br/><b>thus </b><a NAME="E7:42_3"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" target="_self">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></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" target="_self">Dstate</a> <font color="Maroon">c<sub>12</sub></font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_3_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_3_3"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" target="_self">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span>) <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" target="_self">Dstate</a> <font color="Maroon">c<sub>12</sub></font>)
 ;<br/>

<a NAME="E2:42_3_3"/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" target="_self">Dstate</a> <font color="Maroon">c<sub>12</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="scmpds_8.html#E1:42_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="scmpds_8.html#T4" target="_self">Th4</a></i>;<br/>
<b>then </b><i><font color="Green">E40</font></i>: <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" target="_self">Dstate</a> <font color="Maroon">c<sub>12</sub></font>) = 
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" target="_self">Dstate</a> <font color="Maroon">c<sub>12</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="scmpds_8.html#E5:42"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="scmpds_8.html#T4" target="_self">Th4</a></i>
;<br/>
<a NAME="E4:42_3_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="scmpds_8.html#E6:42"><i><font color="Green">E31</font></i></a>, <a class="txt" href="scmpds_8.html#E1:42_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="scmpds_8.html#E1:42_3_3"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E5:42_3_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" target="_self">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="scmpds_8.html#T4" target="_self">Th4</a></i>;<br/>
<b>then </b><font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" target="_self">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span>) = 
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" target="_self">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="scmpds_8.html#E5:42"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="scmpds_8.html#T4" target="_self">Th4</a></i>
;<br/>
<b>hence </b><a NAME="E7:42_3_3"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_8.html#E3:42"><i><font color="Green">E29</font></i></a>, <a class="txt" href="scmpds_8.html#E1:42_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="scmpds_8.html#E4:42_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="scmpds_8.html#E5:42_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="scmpds_8.html#E1:42_3_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="scmpds_8.html#E3:42_3_3"><i><font color="Green">E40</font></i></a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E8:42_3"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1" target="_self">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_3_4"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>17</sub></font> =  <a href="scmpds_8.html#K1" target="_self">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span>;<br/>
<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="scmpds_2.html#M1">Int_position</a> ;<br/><b>assume </b><a NAME="E1:42_3_4_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:42_3_4_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scmpds_8.html#E3:42"><i><font color="Green">E29</font></i></a>, <a class="txt" href="scmpds_8.html#E1:42_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="scmpds_8.html#E4:42_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="scmpds_8.html#E5:42_3"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E3:42_3_4_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" target="_self">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="scmpds_8.html#T4" target="_self">Th4</a></i>;<br/><b>hence </b><a NAME="E4:42_3_4_1"/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" target="_self">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_8.html#K1" target="_self">Dstate</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="scmpds_8.html#T4" target="_self">Th4</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="scmpds_2.html#M1">Int_position</a> ;<br/><b>assume </b><a NAME="E1:42_3_4_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E2:42_3_4_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="scmpds_8.html#E3:42"><i><font color="Green">E29</font></i></a>, <a class="txt" href="scmpds_8.html#E1:42_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="scmpds_8.html#E4:42_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="scmpds_8.html#E5:42_3"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E3:42_3_4_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" target="_self">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="scmpds_8.html#T4" target="_self">Th4</a></i>;<br/><b>hence </b><a NAME="E4:42_3_4_2"/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" target="_self">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="scmpds_8.html#E5:42_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="scmpds_8.html#E1:42_3_4_2"><i><font color="Green">E39</font></i></a></i>;<br/></div>
<b>end;</b></div>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E10:42"/>
( ( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) or <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>] ) &amp;  <a href="scmpds_8.html#K3" target="_self">while&gt;0</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="scmpds_6.html#R1">is_closed_on</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="scmpds_8.html#K3" target="_self">while&gt;0</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="scmpds_6.html#R2">is_halting_on</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>from </b><i><a class="ref" href="scmpds_8.html#S3" target="_self">SCMPDS_8:sch 3</a>(<a class="txt" href="scmpds_8.html#E1:42"><i><font color="Green">E27</font></i></a>, <a class="txt" href="scmpds_8.html#E7:42"><i><font color="Green">E32</font></i></a>, <a class="txt" href="scmpds_8.html#E8:42"><i><font color="Green">E33</font></i></a>, <a class="txt" href="scmpds_8.html#E9:42"><i><font color="Green">E34</font></i></a>);<br/></i>
<b>hence </b><a NAME="E11:42"/>
(  <a href="scmpds_8.html#K3" target="_self">while&gt;0</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="scmpds_6.html#R1">is_closed_on</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="scmpds_8.html#K3" target="_self">while&gt;0</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="scmpds_6.html#R2">is_halting_on</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E12:42"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E13:42"/>
( ( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) or <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>] ) &amp;  <a href="scmpds_4.html#K9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" target="_self">while&gt;0</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmpds_4.html#K9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" target="_self">while&gt;0</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>from </b><i><a class="ref" href="scmpds_8.html#S4" target="_self">SCMPDS_8:sch 4</a>(<a class="txt" href="scmpds_8.html#E1:42"><i><font color="Green">E27</font></i></a>, <a class="txt" href="scmpds_8.html#E12:42"><i><font color="Green">E35</font></i></a>, <a class="txt" href="scmpds_8.html#E7:42"><i><font color="Green">E32</font></i></a>, <a class="txt" href="scmpds_8.html#E8:42"><i><font color="Green">E33</font></i></a>, <a class="txt" href="scmpds_8.html#E9:42"><i><font color="Green">E34</font></i></a>);<br/></i>
<b>hence </b><a NAME="E14:42"/>
 <a href="scmpds_4.html#K9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" target="_self">while&gt;0</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmpds_4.html#K9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" target="_self">while&gt;0</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>


</div>
