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<b>let </b><font color="Maroon">s</font> be    <a href="scm_1.html#M1" target="_self">State-consisting</a> of 0,0,0,<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K5">halt</a> <a href="ami_3.html#K1">SCM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>, <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K4">INT</a> ;<br/>

<a NAME="E1:29"/><i><font color="Green">E23</font></i>: 
 <a href="ami_1.html#K6">IC</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E2:29"/>
1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K5">halt</a> <a href="ami_3.html#K1">SCM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<b>then </b><i><font color="Green">E25</font></i>: <font color="Maroon">s</font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 0</span>)</span></span>)</span> = 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K5">halt</a> <a href="ami_3.html#K1">SCM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="ami_1.html#K5">halt</a> <a href="ami_3.html#K1">SCM</a> 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/>
<b>set </b><font color="Maroon">s0</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 0;<br/>
<a NAME="E4:29"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 0
 
<b>by </b><i><a class="ref" href="ami_1.html#D19">AMI_1:def 19</a></i>;<br/>
<a NAME="E5:29"/><b>then </b>
<font color="Maroon">s</font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 0</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K5">halt</a> <a href="ami_3.html#K1">SCM</a> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:29"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">s</font> is <a href="ami_1.html#V9">halting</a>
 <b>by </b><i/>;<br/>

<a NAME="E7:29"/>
(  <a href="ami_1.html#K8">CurInstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K5">halt</a> <a href="ami_3.html#K1">SCM</a>  &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st  <a href="ami_1.html#K8">CurInstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K5">halt</a> <a href="ami_3.html#K1">SCM</a>  holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> ) )
 
<b>by </b><i><a class="ref" href="scm_1.html#T7" target="_self">Th7</a>, , </i>;<br/>
<b>hence </b><a NAME="E8:29"/>
 <a href="scm_1.html#K2" target="_self">Complexity</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="scm_1.html#E8"><i><font color="Green">Lemma29</font></i></a>, </i>;<br/>

<b>thus </b><a NAME="E9:29"/>
 <a href="ami_1.html#K12">Result</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="scm_1.html#T7" target="_self">Th7</a>, , , </i>;<br/>


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