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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">g</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>let </b><font color="Maroon">f</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>let </b><font color="Maroon">h</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">RF</font> = <span class="p1">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">RT</font> =  <a href="graphsp.html#K6" target="_self">repeat</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">cn</font> =  <a href="graphsp.html#K9" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">n</font>;<br/>
<b>assume </b><a NAME="E1:82"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <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">n</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">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 1</span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <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">n</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">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graphsp.html#K9" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">n</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">g</font>,<font color="Maroon">n</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graphsp.html#K9" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">n</font> implies <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</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">f</font></span>)</span>,<font color="Maroon">n</font> );<br/>
<a NAME="E2:82"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E3:82"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:82_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:82_1_1"/><i><font color="Green">E62</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graphsp.html#K9" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">n</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:82_1_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <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="82_1_1_1_1"><b>suppose </b></a><a NAME="E1:82_1_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:82_1_1_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">n</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1_2"><b>suppose </b></a><a NAME="E1:82_1_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:82_1_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E3:82_1_1_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E4:82_1_1_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="graphsp.html#K9" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:82_1_1_1_2"/><b>then </b>
 <a href="graphsp.html#K8" target="_self">OuterVx</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><a NAME="E6:82_1_1_1_2"/><b>then </b>
 <a href="graphsp.html#K15" target="_self">UsedVx</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" target="_self">UsedVx</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p0">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" target="_self">Argmin</a> <span class="p3">(<span class="default"><a href="graphsp.html#K8" target="_self">OuterVx</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p2">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="graphsp.html#E7"><i><font color="Green">Lemma57</font></i></a></i>;<br/><a NAME="E7:82_1_1_1_2"/><b>then </b>
 <a href="graphsp.html#K15" target="_self">UsedVx</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E8:82_1_1_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K6" target="_self">repeat</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="graphsp.html#K20" target="_self">Relax</a> <font color="Maroon">n</font></span>)</span> <a href="graphsp.html#K4" target="_self">*</a> <span class="p5">(<span class="default"><a href="graphsp.html#K17" target="_self">findmin</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="graphsp.html#K7" target="_self">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="graphsp.html#T7" target="_self">Th7</a>, , <a class="ref" href="graphsp.html#T8" target="_self">Th8</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E4:82"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E5:82"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="graphsp.html#K15" target="_self">UsedVx</a> <font color="Maroon">h</font>,<font color="Maroon">n</font>
 <b>by </b><i/>;<br/>


</div>
