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<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="scm_halt.html#V2" target="_self">InitHalting</a> <a href="scmfsa6a.html#NM1">Macro-Instruction</a>, <font color="Maroon">J</font> be  <a href="scm_halt.html#V2" target="_self">InitHalting</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/>





<a NAME="E1:74"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">I</font> <a href="scm_halt.html#R1" target="_self">is_closed_onInit</a> <font color="Maroon">s</font> &amp; <font color="Maroon">I</font> <a href="scm_halt.html#R2" target="_self">is_halting_onInit</a> <font color="Maroon">s</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:74"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">J</font> <a href="scm_halt.html#R1" target="_self">is_closed_onInit</a> <font color="Maroon">s</font> &amp; <font color="Maroon">J</font> <a href="scm_halt.html#R2" target="_self">is_halting_onInit</a> <font color="Maroon">s</font> )
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>now </b></a><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>assume </b><a NAME="E1:74_1"/>
 <a href="scmfsa6a.html#K3">Initialized</a> <span class="p1">(<span class="default"><a href="scmfsa8b.html#K2">if&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">J</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">s</font>
 ;<br/><a NAME="E2:74_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K3">Initialized</a> <span class="p2">(<span class="default"><a href="scmfsa8b.html#K2">if&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">J</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#T79">FUNCT_4:79</a></i>;<br/><a NAME="E3:74_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">I</font> <a href="scm_halt.html#R1" target="_self">is_closed_onInit</a> <font color="Maroon">s</font> &amp; <font color="Maroon">I</font> <a href="scm_halt.html#R2" target="_self">is_halting_onInit</a> <font color="Maroon">s</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:74_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">J</font> <a href="scm_halt.html#R1" target="_self">is_closed_onInit</a> <font color="Maroon">s</font> &amp; <font color="Maroon">J</font> <a href="scm_halt.html#R2" target="_self">is_halting_onInit</a> <font color="Maroon">s</font> )
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_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="74_1_1_1"><b>suppose </b></a><a NAME="E1:74_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="E2:74_1_1_1"/><b>then </b>
 <a href="scmfsa8b.html#K2">if&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">J</font> <a href="scm_halt.html#R2" target="_self">is_halting_onInit</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="scm_halt.html#T6" target="_self">Th6</a>, <a class="txt" href="scm_halt.html#E4:74_1"><i><font color="Green">E47</font></i></a></i>;<br/><b>hence </b><a NAME="E3:74_1_1_1"/>
<font color="Maroon">s</font> is <a href="ami_1.html#V9">halting</a>
 <b>by </b><i><a class="ref" href="scm_halt.html#T5" target="_self">Th5</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_2"><b>suppose </b></a><a NAME="E1:74_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="E2:74_1_1_2"/><b>then </b>
 <a href="scmfsa8b.html#K2">if&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">J</font> <a href="scm_halt.html#R2" target="_self">is_halting_onInit</a> <font color="Maroon">s</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:74_1_1_2"/>
<font color="Maroon">s</font> is <a href="ami_1.html#V9">halting</a>
 <b>by </b><i><a class="ref" href="scm_halt.html#T5" target="_self">Th5</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E4:74"/><b>then </b>
 <a href="scmfsa6a.html#K3">Initialized</a> <span class="p1">(<span class="default"><a href="scmfsa8b.html#K2">if&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">J</font></span>)</span> is <a href="ami_1.html#V12">halting</a>
 
<b>by </b><i><a class="ref" href="ami_1.html#D26">AMI_1:def 26</a></i>;<br/>
<b>hence </b><a NAME="E5:74"/>
 <a href="scmfsa8b.html#K2">if&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">J</font> is <a href="scm_halt.html#V2" target="_self">InitHalting</a>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E6:74"/>
( <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 implies  <a href="scmfsa6b.html#K3">IExec</a> <span class="p1">(<span class="default"><a href="scmfsa8b.html#K2">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>, </i>;<br/>

<b>thus </b><a NAME="E7:74"/>
( <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 implies  <a href="scmfsa6b.html#K3">IExec</a> <span class="p1">(<span class="default"><a href="scmfsa8b.html#K2">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>, </i>;<br/>


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