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<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> ;<br/><b>let </b><font color="Maroon">i</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <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> ;<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="E1:41"/><i><font color="Green">E32</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="E2:41"/><i><font color="Green">E37</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">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K1" target="_self">Dstate</a> <font color="Olive">t</font></span>)</span> <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/>

<b>assume </b><a NAME="E3:41"/><i><font color="Green">E38</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 ( for <font color="Olive">x</font> being   <a href="scmpds_2.html#M1">Int_position</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">x</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; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <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">I</font>,<font color="Olive">t</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K1" target="_self">Dstate</a> <font color="Olive">t</font></span>)</span> &amp; ( for <font color="Olive">x</font> being   <a href="scmpds_2.html#M1">Int_position</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</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> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">x</font> ) )
 ;<br/>

<a NAME="E4:41"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">x</font> being   <a href="scmpds_2.html#M1">Int_position</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="xcmplx_0.html#K2">+</a> <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="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>
 
;<br/>
<a NAME="E5:41"/>
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 ( for <font color="Olive">x</font> being   <a href="scmpds_2.html#M1">Int_position</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="xcmplx_0.html#K2">+</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> ) &amp; ( for <font color="Olive">x</font> being   <a href="scmpds_2.html#M1">Int_position</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">x</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; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <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">I</font>,<font color="Olive">t</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K1" target="_self">Dstate</a> <font color="Olive">t</font></span>)</span> &amp; ( for <font color="Olive">x</font> being   <a href="scmpds_2.html#M1">Int_position</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  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="Olive">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <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">I</font>,<font color="Olive">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> ) &amp; ( for <font color="Olive">x</font> being   <a href="scmpds_2.html#M1">Int_position</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</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> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">x</font> ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:41"/>
(  <a href="scmpds_8.html#K3" target="_self">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" target="_self">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>by </b><i>, , <a class="ref" href="scmpds_8.html#T1" target="_self">Th1</a>, </i>;<br/>

<b>assume </b><a NAME="E7:41"/><i><font color="Green">E42</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="E8:41"/>
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 ( for <font color="Olive">x</font> being   <a href="scmpds_2.html#M1">Int_position</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="xcmplx_0.html#K2">+</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> ) &amp; ( for <font color="Olive">x</font> being   <a href="scmpds_2.html#M1">Int_position</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">x</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; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <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">I</font>,<font color="Olive">t</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K1" target="_self">Dstate</a> <font color="Olive">t</font></span>)</span> &amp; ( for <font color="Olive">x</font> being   <a href="scmpds_2.html#M1">Int_position</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  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="Olive">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <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">I</font>,<font color="Olive">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> ) &amp; ( for <font color="Olive">x</font> being   <a href="scmpds_2.html#M1">Int_position</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</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> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">x</font> ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:41"/>
 <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">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" target="_self">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>by </b><i>, , <a class="ref" href="scmpds_8.html#T1" target="_self">Th1</a>, , </i>;<br/>


</div>
