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

<b>let </b><font color="Maroon" title="c1">s</font> be   <a href="ami_1.html#NM4" title="AMI_1:NM.4">State</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ;<br/><b>let </b><font color="Maroon" title="c2">I</font> be   <a href="scmfsa6a.html#NM1" title="SCMFSA6A:NM.1">Macro-Instruction</a>;<br/><b>let </b><font color="Maroon" title="c3">a</font> be  <a href="sf_mastr.html#NV2" title="SF_MASTR:NV.2">read-write</a>  <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> ;<br/>





<b>set </b><font color="Maroon" title="c4">s0</font> =  <a href="scmfsa6c.html#K1" title="SCMFSA6C:func.1">Initialize</a> <font color="Maroon" title="c1">s</font>;<br/>
<b>set </b><font color="Maroon" title="c5">sw</font> = <font color="Maroon" title="c1">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">I</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c6">sI</font> = <font color="Maroon" title="c1">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c2">I</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">s0I</font> = <span class="p1">(<span class="default"><a href="scmfsa6c.html#K1" title="SCMFSA6C:func.1">Initialize</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">s0w</font> = <span class="p1">(<span class="default"><a href="scmfsa6c.html#K1" title="SCMFSA6C:func.1">Initialize</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span>;<br/>
<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c2">I</font> <a href="scm_halt.html#R1" title="SCM_HALT:pred.1">is_closed_onInit</a> <font color="Maroon" title="c1">s</font>
 ;<br/>

<b>assume </b><a NAME="E2:20"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c2">I</font> <a href="scm_halt.html#R2" title="SCM_HALT:pred.2">is_halting_onInit</a> <font color="Maroon" title="c1">s</font>
 ;<br/>

<b>assume </b><a NAME="E3:20"/><i><font color="Green" title="E16">A3</font></i>: 
<font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<a NAME="E4:20"/><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c2">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a>  <a href="scmfsa6c.html#K1" title="SCMFSA6C:func.1">Initialize</a> <font color="Maroon" title="c1">s</font>
 
<b>by </b><i><a class="txt" href="scmisort.html#E1:20"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="scm_halt.html#T40" title="SCM_HALT:th.40">SCM_HALT:40</a></i>;<br/>
<a NAME="E5:20"/><i><font color="Green" title="E18">A5</font></i>: 
<font color="Maroon" title="c2">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a>  <a href="scmfsa6c.html#K1" title="SCMFSA6C:func.1">Initialize</a> <font color="Maroon" title="c1">s</font>
 
<b>by </b><i><a class="txt" href="scmisort.html#E2:20"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="scm_halt.html#T41" title="SCM_HALT:th.41">SCM_HALT:41</a></i>;<br/>
<a NAME="E6:20"/><i><font color="Green" title="E19">A6</font></i>: 
<span class="p1">(<span class="default"><a href="scmfsa6c.html#K1" title="SCMFSA6C:func.1">Initialize</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="scmisort.html#E3:20"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="scmfsa6c.html#T3" title="SCMFSA6C:th.3">SCMFSA6C:3</a></i>;<br/>
<a NAME="E7:20"/><i><font color="Green" title="E20">A7</font></i>: 
<font color="Maroon" title="c1">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c2">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa6c.html#K1" title="SCMFSA6C:func.1">Initialize</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="scmfsa8a.html#T13" title="SCMFSA8A:th.13">SCMFSA8A:13</a></i>;<br/>
<a NAME="E8:20"/><i><font color="Green" title="E21">A8</font></i>: 
<font color="Maroon" title="c1">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">I</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa6c.html#K1" title="SCMFSA6C:func.1">Initialize</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="scmfsa8a.html#T13" title="SCMFSA8A:th.13">SCMFSA8A:13</a></i>;<br/>
<b>hence </b><a NAME="E9:20"/>
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p5">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0
 <b>by </b><i><a class="txt" href="scmisort.html#E4:20"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="scmisort.html#E5:20"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="scmisort.html#E6:20"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="scmisort.html#E7:20"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="scmfsa_9.html#T47" title="SCMFSA_9:th.47">SCMFSA_9:47</a></i>;<br/>

<b>thus </b><a NAME="E10:20"/>
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="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3 holds <br/> <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">I</font></span>)</span>
 <b>by </b><i><a class="txt" href="scmisort.html#E4:20"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="scmisort.html#E5:20"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="scmisort.html#E6:20"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="scmisort.html#E7:20"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="scmisort.html#E8:20"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="scmfsa_9.html#T47" title="SCMFSA_9:th.47">SCMFSA_9:47</a></i>;<br/>


</div>
