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

<b>let </b><font color="Maroon" title="c4">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="c5">I</font> be   <a href="scmfsa6a.html#NM1" title="SCMFSA6A:NM.1">Macro-Instruction</a>;<br/>



<b>set </b><font color="Maroon" title="c6">s1</font> = <font color="Maroon" title="c4">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c5">I</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">s2</font> = <font color="Maroon" title="c4">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</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="scmfsa8c.html#K1" title="SCMFSA8C:func.1">loop</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">C1</font> =  <a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p2">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">C2</font> =  <a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p2">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p3">(<span class="default"><a href="scmfsa8c.html#K1" title="SCMFSA8C:func.1">loop</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span></span>)</span>;<br/>
<b>assume </b><a NAME="E1:91"/><i><font color="Green" title="E66">A1</font></i>: 
( <font color="Maroon" title="c5">I</font> <a href="scm_halt.html#R1" title="SCM_HALT:pred.1">is_closed_onInit</a> <font color="Maroon" title="c4">s</font> &amp; <font color="Maroon" title="c5">I</font> <a href="scm_halt.html#R2" title="SCM_HALT:pred.2">is_halting_onInit</a> <font color="Maroon" title="c4">s</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c10">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E2:91"/><i><font color="Green" title="E67">A2</font></i>: 
<font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="ami_1.html#K29" title="AMI_1:func.29">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p2">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E3:91"/><b>then </b>
<span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font>,<span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p4">(<span class="default"><a href="scmfsa8c.html#K1" title="SCMFSA8C:func.1">loop</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> 
 
<b>by </b><i><a class="txt" href="scm_halt.html#E1:91"><i><font color="Green" title="E66">A1</font></i></a>, <a class="ref" href="scm_halt.html#T68" target="_self" title="SCM_HALT:th.68">Th68</a></i>;<br/>
<a NAME="E4:91"/><b>then </b><i><font color="Green" title="E68">A3</font></i>: 
 <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#K9" title="AMI_1:func.9">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p4">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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#K9" title="AMI_1:func.9">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</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="scmfsa8c.html#K1" title="SCMFSA8C:func.1">loop</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ami_1.html#T121" title="AMI_1:th.121">AMI_1:121</a></i>;<br/>
<a NAME="E5:91"/>
<font color="Maroon" title="c5">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c5">I</font></span>)</span>
 
<b>by </b><i><a class="ref" href="scm_halt.html#T67" target="_self" title="SCM_HALT:th.67">Th67</a></i>;<br/>
<a NAME="E6:91"/><b>then </b><i><font color="Green" title="E69">A4</font></i>: 
<font color="Maroon" title="c5">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font>
 
<b>by </b><i><a class="ref" href="ami_1.html#T81" title="AMI_1:th.81">AMI_1:81</a></i>;<br/>
<a NAME="E7:91"/>
 <a href="scmfsa8c.html#K1" title="SCMFSA8C:func.1">loop</a> <font color="Maroon" title="c5">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</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="scmfsa8c.html#K1" title="SCMFSA8C:func.1">loop</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="scm_halt.html#T67" target="_self" title="SCM_HALT:th.67">Th67</a></i>;<br/>
<a NAME="E8:91"/><b>then </b><i><font color="Green" title="E70">A5</font></i>: 
 <a href="scmfsa8c.html#K1" title="SCMFSA8C:func.1">loop</a> <font color="Maroon" title="c5">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p4">(<span class="default"><a href="scmfsa8c.html#K1" title="SCMFSA8C:func.1">loop</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font>
 
<b>by </b><i><a class="ref" href="ami_1.html#T81" title="AMI_1:th.81">AMI_1:81</a></i>;<br/>
<a NAME="E9:91"/><i><font color="Green" title="E71">A6</font></i>: 
 <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#K9" title="AMI_1:func.9">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p4">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</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> <font color="Maroon" title="c5">I</font>
 
<b>by </b><i><a class="txt" href="scm_halt.html#E1:91"><i><font color="Green" title="E66">A1</font></i></a>, <a class="ref" href="scm_halt.html#D4" target="_self" title="SCM_HALT:def.4">Def4</a></i>;<br/>
<a NAME="E10:91"/><b>then </b><i><font color="Green" title="E72">A7</font></i>: 
 <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#K9" title="AMI_1:func.9">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p4">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</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="scmfsa8c.html#K1" title="SCMFSA8C:func.1">loop</a> <font color="Maroon" title="c5">I</font></span>)</span>
 
<b>by </b><i><a class="ref" href="scmfsa8c.html#T106" title="SCMFSA8C:th.106">SCMFSA8C:106</a></i>;<br/>
<a NAME="E11:91"/><i><font color="Green" title="E73">A8</font></i>: 
<font color="Maroon" title="c4">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c5">I</font></span>)</span> is <a href="ami_1.html#V8" title="AMI_1:attr.8">halting</a>
 
<b>by </b><i><a class="txt" href="scm_halt.html#E1:91"><i><font color="Green" title="E66">A1</font></i></a>, <a class="ref" href="scm_halt.html#D5" target="_self" title="SCM_HALT:def.5">Def5</a></i>;<br/>
<a NAME="E12:91"/><i><font color="Green" title="E74">A9</font></i>: 
 <a href="ami_1.html#K7" title="AMI_1:func.7">CurInstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p4">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p5">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="scm_halt.html#E6:91"><i><font color="Green" title="E69">A4</font></i></a>, <a class="txt" href="scm_halt.html#E9:91"><i><font color="Green" title="E71">A6</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/>
<a NAME="E13:91"/><b>then </b>
<font color="Maroon" title="c5">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p5">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> 
 
<b>by </b><i><a class="txt" href="scm_halt.html#E2:91"><i><font color="Green" title="E67">A2</font></i></a>, <a class="txt" href="scm_halt.html#E11:91"><i><font color="Green" title="E73">A8</font></i></a>, <a class="ref" href="ami_1.html#D46" title="AMI_1:def.46">AMI_1:def 46</a></i>;<br/>
<a NAME="E14:91"/><b>then </b>
<font color="Maroon" title="c5">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p5">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa8c.html#K1" title="SCMFSA8C:func.1">loop</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p5">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="scmfsa8c.html#T108" title="SCMFSA8C:th.108">SCMFSA8C:108</a></i>;<br/>
<b>hence </b><a NAME="E15:91"/>
 <a href="ami_1.html#K7" title="AMI_1:func.7">CurInstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p4">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K7" title="AMI_1:func.7">CurInstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</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="scmfsa8c.html#K1" title="SCMFSA8C:func.1">loop</a> <font color="Maroon" title="c5">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_halt.html#E4:91"><i><font color="Green" title="E68">A3</font></i></a>, <a class="txt" href="scm_halt.html#E8:91"><i><font color="Green" title="E70">A5</font></i></a>, <a class="txt" href="scm_halt.html#E10:91"><i><font color="Green" title="E72">A7</font></i></a>, <a class="txt" href="scm_halt.html#E12:91"><i><font color="Green" title="E74">A9</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/>


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