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<b>let </b><font color="Maroon" title="c2">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="c3">I</font>, <font color="Maroon" title="c4">J</font> be  <a href="scm_halt.html#V2" title="SCM_HALT:attr.2">InitHalting</a> <a href="scmfsa6a.html#NM1" title="SCMFSA6A:NM.1">Macro-Instruction</a>;<br/><b>let </b><font color="Maroon" title="c5">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/>





<a NAME="E1:74"/><i><font color="Green" title="E52">A1</font></i>: 
( <font color="Maroon" title="c3">I</font> <a href="scm_halt.html#R1" title="SCM_HALT:pred.1">is_closed_onInit</a> <font color="Maroon" title="c2">s</font> &amp; <font color="Maroon" title="c3">I</font> <a href="scm_halt.html#R2" title="SCM_HALT:pred.2">is_halting_onInit</a> <font color="Maroon" title="c2">s</font> )
 
<b>by </b><i><a class="ref" href="scm_halt.html#T35" target="_self" title="SCM_HALT:th.35">Th35</a>, <a class="ref" href="scm_halt.html#T36" target="_self" title="SCM_HALT:th.36">Th36</a></i>;<br/>
<a NAME="E2:74"/><i><font color="Green" title="E53">A2</font></i>: 
( <font color="Maroon" title="c4">J</font> <a href="scm_halt.html#R1" title="SCM_HALT:pred.1">is_closed_onInit</a> <font color="Maroon" title="c2">s</font> &amp; <font color="Maroon" title="c4">J</font> <a href="scm_halt.html#R2" title="SCM_HALT:pred.2">is_halting_onInit</a> <font color="Maroon" title="c2">s</font> )
 
<b>by </b><i><a class="ref" href="scm_halt.html#T35" target="_self" title="SCM_HALT:th.35">Th35</a>, <a class="ref" href="scm_halt.html#T36" target="_self" title="SCM_HALT:th.36">Th36</a></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" title="c6">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>assume </b><a NAME="E1:74_1"/>
 <a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p1">(<span class="default"><a href="scmfsa8b.html#K2" title="SCMFSA8B:func.2">if&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">J</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">s</font>
 ;<br/><a NAME="E2:74_1"/><b>then </b><i><font color="Green" title="E54">A3</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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="scmfsa8b.html#K2" title="SCMFSA8B:func.2">if&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">J</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#T79" title="FUNCT_4:th.79">FUNCT_4:79</a></i>;<br/><a NAME="E3:74_1"/><i><font color="Green" title="E55">A4</font></i>: 
( <font color="Maroon" title="c3">I</font> <a href="scm_halt.html#R1" title="SCM_HALT:pred.1">is_closed_onInit</a> <font color="Maroon" title="c6">s</font> &amp; <font color="Maroon" title="c3">I</font> <a href="scm_halt.html#R2" title="SCM_HALT:pred.2">is_halting_onInit</a> <font color="Maroon" title="c6">s</font> )
 <b>by </b><i><a class="ref" href="scm_halt.html#T35" target="_self" title="SCM_HALT:th.35">Th35</a>, <a class="ref" href="scm_halt.html#T36" target="_self" title="SCM_HALT:th.36">Th36</a></i>;<br/><a NAME="E4:74_1"/><i><font color="Green" title="E56">A5</font></i>: 
( <font color="Maroon" title="c4">J</font> <a href="scm_halt.html#R1" title="SCM_HALT:pred.1">is_closed_onInit</a> <font color="Maroon" title="c6">s</font> &amp; <font color="Maroon" title="c4">J</font> <a href="scm_halt.html#R2" title="SCM_HALT:pred.2">is_halting_onInit</a> <font color="Maroon" title="c6">s</font> )
 <b>by </b><i><a class="ref" href="scm_halt.html#T35" target="_self" title="SCM_HALT:th.35">Th35</a>, <a class="ref" href="scm_halt.html#T36" target="_self" title="SCM_HALT:th.36">Th36</a></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" title="c6">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Maroon" title="c6">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="c6">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:74_1_1_1"/><b>then </b>
 <a href="scmfsa8b.html#K2" title="SCMFSA8B:func.2">if&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">J</font> <a href="scm_halt.html#R2" title="SCM_HALT:pred.2">is_halting_onInit</a> <font color="Maroon" title="c6">s</font>
 <b>by </b><i><a class="txt" href="scm_halt.html#E3:74_1"><i><font color="Green" title="E55">A4</font></i></a>, <a class="ref" href="scm_halt.html#T49" target="_self" title="SCM_HALT:th.49">Th49</a></i>;<br/><b>hence </b><a NAME="E3:74_1_1_1"/>
<font color="Maroon" title="c6">s</font> is <a href="ami_1.html#V9" title="AMI_1:attr.9">halting</a>
 <b>by </b><i><a class="txt" href="scm_halt.html#E2:74_1"><i><font color="Green" title="E54">A3</font></i></a>, <a class="ref" href="scm_halt.html#D5" target="_self" title="SCM_HALT:def.5">Def5</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" title="c6">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:74_1_1_2"/><b>then </b>
 <a href="scmfsa8b.html#K2" title="SCMFSA8B:func.2">if&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">J</font> <a href="scm_halt.html#R2" title="SCM_HALT:pred.2">is_halting_onInit</a> <font color="Maroon" title="c6">s</font>
 <b>by </b><i><a class="txt" href="scm_halt.html#E4:74_1"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="scm_halt.html#T51" target="_self" title="SCM_HALT:th.51">Th51</a></i>;<br/><b>hence </b><a NAME="E3:74_1_1_2"/>
<font color="Maroon" title="c6">s</font> is <a href="ami_1.html#V9" title="AMI_1:attr.9">halting</a>
 <b>by </b><i><a class="txt" href="scm_halt.html#E2:74_1"><i><font color="Green" title="E54">A3</font></i></a>, <a class="ref" href="scm_halt.html#D5" target="_self" title="SCM_HALT:def.5">Def5</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" title="SCMFSA6A:func.3">Initialized</a> <span class="p1">(<span class="default"><a href="scmfsa8b.html#K2" title="SCMFSA8B:func.2">if&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">J</font></span>)</span> is <a href="ami_1.html#V12" title="AMI_1:attr.12">halting</a>
 
<b>by </b><i><a class="ref" href="ami_1.html#D26" title="AMI_1:def.26">AMI_1:def 26</a></i>;<br/>
<b>hence </b><a NAME="E5:74"/>
 <a href="scmfsa8b.html#K2" title="SCMFSA8B:func.2">if&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">J</font> is <a href="scm_halt.html#V2" title="SCM_HALT:attr.2">InitHalting</a>
 <b>by </b><i><a class="ref" href="scm_halt.html#D2" target="_self" title="SCM_HALT:def.2">Def2</a></i>;<br/>

<b>thus </b><a NAME="E6:74"/>
( <font color="Maroon" title="c2">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 implies  <a href="scmfsa6b.html#K3" title="SCMFSA6B:func.3">IExec</a> <span class="p1">(<span class="default"><a href="scmfsa8b.html#K2" title="SCMFSA8B:func.2">if&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">J</font></span>)</span>,<font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa6b.html#K3" title="SCMFSA6B:func.3">IExec</a> <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">s</font></span>)</span> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">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" title="CARD_1:func.4">card</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p5">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">J</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="scm_halt.html#E1:74"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="scm_halt.html#T50" target="_self" title="SCM_HALT:th.50">Th50</a></i>;<br/>

<b>thus </b><a NAME="E7:74"/>
( <font color="Maroon" title="c2">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 implies  <a href="scmfsa6b.html#K3" title="SCMFSA6B:func.3">IExec</a> <span class="p1">(<span class="default"><a href="scmfsa8b.html#K2" title="SCMFSA8B:func.2">if&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">J</font></span>)</span>,<font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa6b.html#K3" title="SCMFSA6B:func.3">IExec</a> <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c2">s</font></span>)</span> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">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" title="CARD_1:func.4">card</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p5">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">J</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="scm_halt.html#E2:74"><i><font color="Green" title="E53">A2</font></i></a>, <a class="ref" href="scm_halt.html#T52" target="_self" title="SCM_HALT:th.52">Th52</a></i>;<br/>


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