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

<b>let </b><font color="Maroon">s</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> ;<br/><b>let </b><font color="Maroon">I</font> be  <a href="scmfsa6b.html#V2">parahalting</a> <a href="scmfsa6a.html#NM1">Macro-Instruction</a>, <font color="Maroon">J</font> be  <a href="scmfsa6b.html#V2">parahalting</a> <a href="scmfsa6a.html#NM1">Macro-Instruction</a>;<br/><b>let </b><font color="Maroon">a</font> be  <a href="sf_mastr.html#V1">read-write</a>  <a href="scmfsa_2.html#M1">Int-Location</a> ;<br/>





<div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1_1"/>
( <font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1"><b>suppose </b></a><a NAME="E1:32_1_1_1"/>
<font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:32_1_1_1"/><b>then </b>
 <a href="scmfsa6b.html#K3">IExec</a> <span class="p1">(<span class="default"><a href="scmfsa8b.html#K2" target="_self">if&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">J</font></span>)</span>,<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa6b.html#K3">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:32_1_1_1"/>
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><a href="scmfsa6b.html#K3">IExec</a> <span class="p2">(<span class="default"><a href="scmfsa8b.html#K2" target="_self">if&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">J</font></span>)</span>,<font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K5">insloc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span>
 <b>by </b><i><a class="ref" href="ami_5.html#T79">AMI_5:79</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_2"><b>suppose </b></a><a NAME="E1:32_1_1_2"/>
<font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:32_1_1_2"/><b>then </b>
 <a href="scmfsa6b.html#K3">IExec</a> <span class="p1">(<span class="default"><a href="scmfsa8b.html#K2" target="_self">if&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">J</font></span>)</span>,<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa6b.html#K3">IExec</a> <font color="Maroon">J</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:32_1_1_2"/>
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><a href="scmfsa6b.html#K3">IExec</a> <span class="p2">(<span class="default"><a href="scmfsa8b.html#K2" target="_self">if&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">J</font></span>)</span>,<font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K5">insloc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span>
 <b>by </b><i><a class="ref" href="ami_5.html#T79">AMI_5:79</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:32_2"/>
<font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E2:32_2"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="scmfsa6b.html#K3">IExec</a> <span class="p1">(<span class="default"><a href="scmfsa8b.html#K2" target="_self">if&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">J</font></span>)</span>,<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa6b.html#K3">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">d</font> be    <a href="scmfsa_2.html#M1">Int-Location</a> ;<br/><a NAME="E1:32_2_1"/>
not <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="scmfsa6b.html#T9">SCMFSA6B:9</a></i>;<br/><b>hence </b><a NAME="E2:32_2_1"/>
<span class="p1">(<span class="default"><a href="scmfsa6b.html#K3">IExec</a> <span class="p2">(<span class="default"><a href="scmfsa8b.html#K2" target="_self">if&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">J</font></span>)</span>,<font color="Maroon">s</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa6b.html#K3">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="scmfsa8b.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/></div>
<b>end;</b></div><b>let </b><font color="Maroon">f</font> be    <a href="scmfsa_2.html#M2">FinSeq-Location</a> ;<br/><a NAME="E4:32_2"/>
not <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="scmfsa6b.html#T10">SCMFSA6B:10</a></i>;<br/><b>hence </b><a NAME="E5:32_2"/>
<span class="p1">(<span class="default"><a href="scmfsa6b.html#K3">IExec</a> <span class="p2">(<span class="default"><a href="scmfsa8b.html#K2" target="_self">if&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">J</font></span>)</span>,<font color="Maroon">s</font></span>)</span> <a href="scmfsa_2.html#K21">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa6b.html#K3">IExec</a> <font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="scmfsa_2.html#K21">.</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="scmfsa8b.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:32"/>
<font color="Maroon">s</font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>

<a NAME="E4:32"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="scmfsa6b.html#K3">IExec</a> <span class="p1">(<span class="default"><a href="scmfsa8b.html#K2" target="_self">if&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">J</font></span>)</span>,<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa6b.html#K3">IExec</a> <font color="Maroon">J</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">d</font> be    <a href="scmfsa_2.html#M1">Int-Location</a> ;<br/><a NAME="E1:32_3"/>
not <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="scmfsa6b.html#T9">SCMFSA6B:9</a></i>;<br/><b>hence </b><a NAME="E2:32_3"/>
<span class="p1">(<span class="default"><a href="scmfsa6b.html#K3">IExec</a> <span class="p2">(<span class="default"><a href="scmfsa8b.html#K2" target="_self">if&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">J</font></span>)</span>,<font color="Maroon">s</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa6b.html#K3">IExec</a> <font color="Maroon">J</font>,<font color="Maroon">s</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="scmfsa8b.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">f</font> be    <a href="scmfsa_2.html#M2">FinSeq-Location</a> ;<br/>

<a NAME="E6:32"/>
not <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">J</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="scmfsa6b.html#T10">SCMFSA6B:10</a></i>;<br/>
<b>hence </b><a NAME="E7:32"/>
<span class="p1">(<span class="default"><a href="scmfsa6b.html#K3">IExec</a> <span class="p2">(<span class="default"><a href="scmfsa8b.html#K2" target="_self">if&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">J</font></span>)</span>,<font color="Maroon">s</font></span>)</span> <a href="scmfsa_2.html#K21">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa6b.html#K3">IExec</a> <font color="Maroon">J</font>,<font color="Maroon">s</font></span>)</span> <a href="scmfsa_2.html#K21">.</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="scmfsa8b.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/>


</div>
