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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a> ;<br/>





<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="graphsp.html#R2" target="_self">equal_at</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:80"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

<a NAME="E1:80_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="graphsp.html#T22" target="_self">Th22</a></i>;<br/>
<b>hence </b><a NAME="E2:80_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="ref" href="graphsp.html#T43" target="_self">Th43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:80"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:80_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>]
 ;<br/>

<a NAME="E2:80_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="graphsp.html#R2" target="_self">equal_at</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="graphsp.html#T45" target="_self">Th45</a></i>;<br/>
<b>hence </b><a NAME="E3:80_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="graphsp.html#E1:80_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="graphsp.html#T44" target="_self">Th44</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:80"/>
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="graphsp.html#E1:80"><i><font color="Green">E67</font></i></a>, <a class="txt" href="graphsp.html#E2:80"><i><font color="Green">E68</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:80"/>
<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="graphsp.html#R2" target="_self">equal_at</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>


</div>
