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

<b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="scmpds_2.html#K3">DataLoc</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span>,<font color="Maroon">F<sub>5</sub></font>();<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>3</sub></font>();<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means 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">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <font color="Olive">b<sub>1</sub></font></span>)</span>) <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>4</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <font color="Olive">b<sub>1</sub></font>] holds <br/>( <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>) <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Olive">b<sub>1</sub></font></span>)</span>] );<br/>
<a NAME="E1:12_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</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="E1:12_1_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>) <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">c<sub>13</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>4</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">c<sub>13</sub></font>] )
 ;<br/>

<a NAME="E2:12_1_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>) <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E3:12_1_1"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>) <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="scpinvar.html#E1:12_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/>
<a NAME="E4:12_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</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">F<sub>2</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span>,<font color="Maroon">F<sub>5</sub></font>()</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="txt" href="scpinvar.html#E2:12"><i><font color="Green">E11</font></i></a>, <a class="txt" href="scpinvar.html#E1:12_1_1"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E5:12_1_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="scmpds_2.html#M1">Int_position</a>  holds <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">c<sub>13</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>13</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="scpinvar.html#E1:12_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="scmpds_8.html#T23">SCMPDS_8:23</a></i>;<br/>
<b>hence </b><a NAME="E6:12_1_1"/>
( <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>) <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">c<sub>13</sub></font></span>)</span>] )
 <b>by </b><i><a class="txt" href="scpinvar.html#E1:12_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="scpinvar.html#E3:12_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="scpisort.html#T5">SCPISORT:5</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:12_1_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>13</sub></font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</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="E1:12_1_2_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>) <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>14</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>4</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">c<sub>14</sub></font>] )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1_2_1_1"/>
( <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>) <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>) <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:12_1_2_1_1_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>) <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_1_2_1_1_1"/>
( <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>) <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">c<sub>14</sub></font></span>)</span>] )
 <b>by </b><i><a class="txt" href="scpinvar.html#E1:12_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="scpinvar.html#E1:12_1_2_1"><i><font color="Green">E17</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:12_1_2_1_1_2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>) <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:12_1_2_1_1_2"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>14</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">F<sub>2</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span>,<font color="Maroon">F<sub>5</sub></font>()</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="scpinvar.html#E2:12"><i><font color="Green">E11</font></i></a>, <a class="txt" href="scpinvar.html#E1:12_1_2_1"><i><font color="Green">E17</font></i></a></i>;<br/><a NAME="E3:12_1_2_1_1_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>14</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>14</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span>,<font color="Maroon">F<sub>5</sub></font>()</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="scpinvar.html#E2:12"><i><font color="Green">E11</font></i></a>, <a class="txt" href="scpinvar.html#E1:12_1_2_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="scpinvar.html#E1:12_1_2_1_1_2"><i><font color="Green">E18</font></i></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="nat_1.html#NM1">Nat</a> = <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>);<br/><a NAME="E4:12_1_2_1_1_2"/><i><font color="Green">E21</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">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1">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">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>14</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span>,<font color="Maroon">F<sub>5</sub></font>()</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="txt" href="scpinvar.html#E2:12"><i><font color="Green">E11</font></i></a>, <a class="txt" href="scpinvar.html#E1:12_1_2_1"><i><font color="Green">E17</font></i></a></i>;<br/><a NAME="E5:12_1_2_1_1_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">c<sub>14</sub></font>]
 <b>by </b><i><a class="txt" href="scpinvar.html#E1:12_1_2_1"><i><font color="Green">E17</font></i></a></i>;<br/><a NAME="E6:12_1_2_1_1_2"/><i><font color="Green">E23</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">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <font color="Olive">b<sub>1</sub></font>] &amp; <font color="Olive">b<sub>1</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>4</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>4</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>14</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span>,<font color="Maroon">F<sub>5</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">F<sub>3</sub></font>(),<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>4</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">F<sub>4</sub></font>() &amp; <font color="Maroon">F<sub>3</sub></font>() <a href="scmpds_6.html#R1">is_closed_on</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">F<sub>3</sub></font>() <a href="scmpds_6.html#R2">is_halting_on</a> <font color="Olive">b<sub>1</sub></font> &amp; <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">F<sub>3</sub></font>(),<font color="Olive">b<sub>1</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">Dstate</a> <font color="Olive">b<sub>1</sub></font>) &amp; <font color="Maroon">P<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">F<sub>3</sub></font>(),<font color="Olive">b<sub>1</sub></font></span>)</span>] )
 <b>by </b><i><a class="txt" href="scpinvar.html#E4:12"><i><font color="Green">E13</font></i></a>, <a class="txt" href="scpinvar.html#E1:12_1_2_1"><i><font color="Green">E17</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>15</sub></font> =  <a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">c<sub>14</sub></font>;<br/><a NAME="E7:12_1_2_1_1_2"/><i><font color="Green">E24</font></i>: 
( ( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>14</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>14</sub></font>) or <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>14</sub></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">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">c<sub>14</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">while&gt;0</a> <font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">c<sub>14</sub></font></span>)</span> )
 <b>from </b><i><a class="ref" href="scmpds_8.html#S4">SCMPDS_8:sch 4</a>(<a class="txt" href="scpinvar.html#E1:12"><i><font color="Green">E10</font></i></a>, <a class="txt" href="scpinvar.html#E3:12_1_2_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="scpinvar.html#E4:12_1_2_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="scpinvar.html#E5:12_1_2_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="scpinvar.html#E6:12_1_2_1_1_2"><i><font color="Green">E23</font></i></a>);<br/></i><a NAME="E8:12_1_2_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">c<sub>14</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">Dstate</a> <font color="Maroon">c<sub>14</sub></font>)
 <b>by </b><i><a class="txt" href="scpinvar.html#E4:12"><i><font color="Green">E13</font></i></a>, <a class="txt" href="scpinvar.html#E1:12_1_2_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="scpinvar.html#E2:12_1_2_1_1_2"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E9:12_1_2_1_1_2"/><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">F<sub>3</sub></font>(),<font color="Maroon">c<sub>14</sub></font></span>)</span>) <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">c<sub>14</sub></font>)
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E10:12_1_2_1_1_2"/><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">F<sub>3</sub></font>(),<font color="Maroon">c<sub>14</sub></font></span>)</span>) <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="scpinvar.html#E1:12_1_2_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E11:12_1_2_1_1_2"/><b>then </b><i><font color="Green">E25</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">F<sub>3</sub></font>(),<font color="Maroon">c<sub>14</sub></font></span>)</span>) <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E12:12_1_2_1_1_2"/><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>4</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>4</sub></font>()
 <b>by </b><i><a class="txt" href="scpinvar.html#E4:12"><i><font color="Green">E13</font></i></a>, <a class="txt" href="scpinvar.html#E1:12_1_2_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="scpinvar.html#E2:12_1_2_1_1_2"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E13:12_1_2_1_1_2"/>
<font color="Maroon">P<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">F<sub>3</sub></font>(),<font color="Maroon">c<sub>14</sub></font></span>)</span>]
 <b>by </b><i><a class="txt" href="scpinvar.html#E4:12"><i><font color="Green">E13</font></i></a>, <a class="txt" href="scpinvar.html#E1:12_1_2_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="scpinvar.html#E2:12_1_2_1_1_2"><i><font color="Green">E19</font></i></a></i>;<br/><b>hence </b><a NAME="E14:12_1_2_1_1_2"/>
( <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">c<sub>14</sub></font></span>)</span>) <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">c<sub>14</sub></font></span>)</span>] )
 <b>by </b><i><a class="txt" href="scpinvar.html#E1:12_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="scpinvar.html#E7:12_1_2_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="scpinvar.html#E11:12_1_2_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="scpinvar.html#E12:12_1_2_1_1_2"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:12_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E3:12_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="scpinvar.html#E1:12_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="scpinvar.html#E2:12_1"><i><font color="Green">E15</font></i></a>);<br/></i>
<a NAME="E4:12_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>)]
 
;<br/>
<b>hence </b><a NAME="E5:12_1"/>
( <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span>) <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>2</sub></font>()</span>)</span>] )
 <b>by </b><i><a class="txt" href="scpinvar.html#E3:12"><i><font color="Green">E12</font></i></a></i>;<br/>


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