<?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">x</font> be    <a href="scmpds_2.html#M1">Int_position</a> , <font color="Maroon">y</font> be    <a href="scmpds_2.html#M1">Int_position</a> ;<br/><b>let </b><font color="Maroon">i</font> be   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">c</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>







<b>set </b><font color="Maroon">b</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>assume </b><a NAME="E1:10"/><i><font color="Green">E42</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="E2:10"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c</font>
 ;<br/>

<b>assume </b><a NAME="E3:10"/><i><font color="Green">E59</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="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c</font> &amp; <font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">y</font> &amp; <font color="Olive">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="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#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">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> <font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">a</font> &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="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> <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> &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">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c</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> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">y</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">t</font> being  <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a>  st <br/>( <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c</font> &amp; <font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">y</font> );<br/>
<b>consider </b><font color="Maroon">f</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="E5:10"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">s</font> being  <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a>  holds <br/> ( ( <font color="Olive">s</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#R1">&lt;=</a> 0 implies <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Olive">s</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#NR4" target="_self">&gt;</a> 0 implies <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</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> ) )
 <b>by </b><i><a class="ref" href="scmpds_8.html#T5">SCMPDS_8:5</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">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E6:10"/><i><font color="Green">E61</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="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 iff <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#R1">&lt;=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </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>thus </b><a NAME="E1:10_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 implies <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#R1">&lt;=</a> 0 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1_1"/><i><font color="Green">E62</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="E2:10_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#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E3:10_1_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#NR4" 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="E4:10_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


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

<b>assume </b><a NAME="E2:10_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#R1">&lt;=</a> 0
 ;<br/>

<a NAME="E3:10_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#R1">&lt;=</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="E4:10_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="E7:10"/><b>then </b><i><font color="Green">E63</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#R1">&lt;=</a> 0
 
;<br/>
<a NAME="E8:10"/><i><font color="Green">E64</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="10_2"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">t</font> =  <a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">s</font>;<br/>

<b>thus </b><a NAME="E1:10_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">s</font>
 ;<br/>

<b>thus </b><a NAME="E2:10_2"/>
<font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="scmpds_8.html#T4">SCMPDS_8:4</a></i>;<br/>

<b>thus </b><a NAME="E3:10_2"/>
<font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">y</font>
 <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">E65</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_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>assume </b><a NAME="E1:10_3"/><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">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#NR4" target="_self">&gt;</a> 0 )
 ;<br/><b>then consider </b><font color="Maroon">v</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="E3:10_3"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">t</font> &amp; <font color="Maroon">v</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">v</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">y</font> )
 ;<br/><a NAME="E4:10_3"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="scmpds_8.html#T4">SCMPDS_8:4</a></i>;<br/><a NAME="E5:10_3"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="scmpds_8.html#T4">SCMPDS_8:4</a></i>;<br/><b>hence </b><a NAME="E6:10_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> &amp; <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/><b>set </b><font color="Maroon">t2</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/><b>set </b><font color="Maroon">t1</font> =  <a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">t</font>;<br/><b>thus </b><a NAME="E7:10_3"/>
<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>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_3_1"/><i><font color="Green">E73</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="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">t</font>)
 ;<br/>

<a NAME="E2:10_3_1"/>
<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#NR4" target="_self">&gt;</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">E74</font></i>: <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">t</font>) = 
<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>
<b>by </b><i/>
<br/>.= 
<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>
<b>by </b><i><a class="ref" href="scmpds_8.html#T4">SCMPDS_8:4</a></i>
;<br/>
<a NAME="E4:10_3_1"/><b>then </b>
<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="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#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:10_3_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#NR4" target="_self">&gt;</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> <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>) = 
<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>
<b>by </b><i/>
<br/>.= 
<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="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="scmpds_8.html#T4">SCMPDS_8:4</a></i>
;<br/>
<b>hence </b><a NAME="E7:10_3_1"/>
contradiction
 <b>by </b><i>, , , , , </i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E8:10_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="10_3_2"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">v</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/>

<b>thus </b><a NAME="E1:10_3_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <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/>

<a NAME="E2:10_3_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> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E3:10_3_2"/>
<font color="Maroon">v</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="scmpds_8.html#T4">SCMPDS_8:4</a></i>;<br/>

<a NAME="E4:10_3_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> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">y</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E5:10_3_2"/>
<font color="Maroon">v</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">y</font>
 <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>
<a NAME="E10:10"/>
( ( <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#K3">while&gt;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#K3">while&gt;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#S3">SCMPDS_8:sch 3</a>(, , , );<br/></i>
<b>hence </b><a NAME="E11:10"/>
(  <a href="scmpds_8.html#K3">while&gt;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#K3">while&gt;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/>

<b>assume </b><a NAME="E12:10"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">s</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#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E13:10"/>
( ( <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_4.html#K9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">s</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">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<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">s</font></span>)</span> )
 
<b>from </b><i><a class="ref" href="scmpds_8.html#S4">SCMPDS_8:sch 4</a>(, , , , );<br/></i>
<b>hence </b><a NAME="E14:10"/>
 <a href="scmpds_4.html#K9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">s</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">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">i</font>,<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">s</font></span>)</span>
 ;<br/>


</div>
