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

<span class="kw">set </span><font color="Maroon" title="c3">b</font> =  <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span>,<font color="Maroon">F<sub>5</sub></font>();<br/>
<span class="kw">set </span><font color="Maroon" title="c4">WHL</font> =  <a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;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/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ] <span class="kw">means </span>for <font color="Olive" title="b1">t</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>   st <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Olive" title="b1">t</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon">F<sub>4</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Olive" title="b1">t</font>] holds <br/>( <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;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" title="b1">t</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;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" title="b1">t</font></span>)</span>] );<br/>
<a NAME="E1:8_1"/><span class="lab"><font color="Green" title="E8">A5</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">t</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c5">t</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon">F<sub>4</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c5">t</font>] implies ( <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;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" title="c5">t</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;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" title="c5">t</font></span>)</span>] ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:8_1_1"/><span class="lab"><font color="Green" title="E9">A6</font></span>: 
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c5">t</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:8_1_1"/><span class="lab"><font color="Green" title="E10">A7</font></span>: 
<font color="Maroon" title="c5">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon">F<sub>4</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon">F<sub>4</sub></font>()
 <span class="kw">and </span><br/><a NAME="E3:8_1_1"/><span class="lab"><font color="Green" title="E11">A8</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c5">t</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;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" title="c5">t</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;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" title="c5">t</font></span>)</span>] )</span><br/>

<a NAME="E4:8_1_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c5">t</font></span>)</span>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/>
<a NAME="E5:8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A9</font></span>: 
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c5">t</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E6:8_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</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#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E3:8_1_1"><span class="lab"><font color="Green" title="E11">A8</font></span></a></span>;<br/>
<a NAME="E7:8_1_1"/><span class="kw">then </span>
for <font color="Olive" title="b1">b</font> being   <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a>  holds  <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;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" title="c5">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1_1"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="ref" href="scmpds_8.html#T12" title="SCMPDS_8:th.12">SCMPDS_8:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:8_1_1"/>
( <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;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" title="c5">t</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;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" title="c5">t</font></span>)</span>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8_1_1"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="txt" href="#E5:8_1_1"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="ref" href="scpisort.html#T5" title="SCPISORT:th.5">SCPISORT:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E9">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:8_1_2"/><span class="lab"><font color="Green" title="E10">A11</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">u</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c6">u</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">u</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon">F<sub>4</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c6">u</font>] implies ( <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;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" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;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><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:8_1_2_1"/><span class="lab"><font color="Green" title="E11">A12</font></span>: 
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c6">u</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">and </span><br/><a NAME="E2:8_1_2_1"/><span class="lab"><font color="Green" title="E12">A13</font></span>: 
<font color="Maroon" title="c6">u</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon">F<sub>4</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon">F<sub>4</sub></font>()
 <span class="kw">and </span><br/><a NAME="E3:8_1_2_1"/><span class="lab"><font color="Green" title="E13">A14</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c6">u</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;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" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;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><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:8_1_2_1_1"/>
( <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c6">u</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c6">u</font></span>)</span>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:8_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" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c6">u</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;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" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;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><br/><div class="add"><span class="kw">hence </span><a NAME="E2:8_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" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;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" title="c6">u</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;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" title="c6">u</font></span>)</span>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="txt" href="#E2:8_1_2_1"><span class="lab"><font color="Green" title="E12">A13</font></span></a>, <a class="txt" href="#E3:8_1_2_1"><span class="lab"><font color="Green" title="E13">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:8_1_2_1_1_2"/><span class="lab"><font color="Green" title="E14">A15</font></span>: 
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c6">u</font></span>)</span>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;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>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;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><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c7">Iu</font> =  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c6">u</font>;<br/><a NAME="E2:8_1_2_1_1_2"/><span class="lab"><font color="Green" title="E15">A16</font></span>: 
<font color="Maroon" title="c6">u</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E3:8_1_2_1"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="txt" href="#E1:8_1_2_1_1_2"><span class="lab"><font color="Green" title="E14">A15</font></span></a></span>;<br/><a NAME="E3:8_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A17</font></span>: 
( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c6">u</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon">F<sub>4</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c6">u</font></span>)</span>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E2:8_1_2_1"><span class="lab"><font color="Green" title="E12">A13</font></span></a>, <a class="txt" href="#E3:8_1_2_1"><span class="lab"><font color="Green" title="E13">A14</font></span></a></span>;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  = <font color="Maroon">F<sub>1</sub></font>($1);<br/><a NAME="E4:8_1_2_1_1_2"/><span class="lab"><font color="Green" title="E17">A18</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c6">u</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8_1_2_1"><span class="lab"><font color="Green" title="E13">A14</font></span></a></span>;<br/><a NAME="E5:8_1_2_1_1_2"/><span class="lab"><font color="Green" title="E18">A19</font></span>: 
for <font color="Olive" title="b1">t</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>   st <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Olive" title="b1">t</font>] &amp; <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</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#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E2:8_1_2_1"><span class="lab"><font color="Green" title="E12">A13</font></span></a></span>;<br/><a NAME="E6:8_1_2_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c6">u</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c6">u</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E2:8_1_2_1"><span class="lab"><font color="Green" title="E12">A13</font></span></a>, <a class="txt" href="#E3:8_1_2_1"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="txt" href="#E2:8_1_2_1_1_2"><span class="lab"><font color="Green" title="E15">A16</font></span></a></span>;<br/><a NAME="E7:8_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c6">u</font></span>)</span>) <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c6">u</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></span>;<br/><a NAME="E8:8_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c6">u</font></span>)</span>) <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2_1"><span class="lab"><font color="Green" title="E11">A12</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:8_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A20</font></span>: 
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c6">u</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E10:8_1_2_1_1_2"/><span class="lab"><font color="Green" title="E20">A21</font></span>: 
for <font color="Olive" title="b1">t</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>   st <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Olive" title="b1">t</font>] &amp; <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon">F<sub>4</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon">F<sub>4</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">F<sub>3</sub></font>() <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Olive" title="b1">t</font> &amp; <font color="Maroon">F<sub>3</sub></font>() <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Olive" title="b1">t</font> &amp; <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Olive" title="b1">t</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Olive" title="b1">t</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Olive" title="b1">t</font></span>)</span>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E2:8_1_2_1"><span class="lab"><font color="Green" title="E12">A13</font></span></a></span>;<br/><a NAME="E11:8_1_2_1_1_2"/><span class="lab"><font color="Green" title="E21">A22</font></span>: 
<font color="Maroon" title="c6">u</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E2:8_1_2_1"><span class="lab"><font color="Green" title="E12">A13</font></span></a>, <a class="txt" href="#E3:8_1_2_1"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="txt" href="#E1:8_1_2_1_1_2"><span class="lab"><font color="Green" title="E14">A15</font></span></a></span>;<br/><a NAME="E12:8_1_2_1_1_2"/>
( ( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">u</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">u</font>) or <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">u</font>] ) &amp;  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;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" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;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" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c6">u</font></span>)</span> )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="scmpds_8.html#S2" title="SCMPDS_8:sch.2">SCMPDS_8:sch 2</a>(<a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E11:8_1_2_1_1_2"><span class="lab"><font color="Green" title="E21">A22</font></span></a>, <a class="txt" href="#E5:8_1_2_1_1_2"><span class="lab"><font color="Green" title="E18">A19</font></span></a>, <a class="txt" href="#E4:8_1_2_1_1_2"><span class="lab"><font color="Green" title="E17">A18</font></span></a>, <a class="txt" href="#E10:8_1_2_1_1_2"><span class="lab"><font color="Green" title="E20">A21</font></span></a>);<br/></span><span class="kw">hence </span><a NAME="E13:8_1_2_1_1_2"/>
( <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;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" title="c6">u</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;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" title="c6">u</font></span>)</span>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_2"><span class="lab"><font color="Green" title="E10">A11</font></span></a>, <a class="txt" href="#E9:8_1_2_1_1_2"><span class="lab"><font color="Green" title="E19">A20</font></span></a>, <a class="txt" href="#E3:8_1_2_1_1_2"><span class="lab"><font color="Green" title="E16">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:8_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E3:8_1"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E1:8_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="txt" href="#E2:8_1"><span class="lab"><font color="Green" title="E9">A10</font></span></a>);<br/></span>
<a NAME="E4:8_1"/><span class="kw">then </span>
<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" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>)]
 
;<br/>
<span class="kw">hence </span><a NAME="E5:8_1"/>
( <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;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" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8"><span class="lab"><font color="Green" title="E6">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
