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

<b>let </b><font color="Maroon" title="c1">I</font> be  <a href="scmfsa6b.html#V2" title="SCMFSA6B:attr.2">parahalting</a> <a href="scmfsa7b.html#V1" title="SCMFSA7B:attr.1">good</a> <a href="scmfsa6a.html#NM1" title="SCMFSA6A:NM.1">Macro-Instruction</a>;<br/><b>let </b><font color="Maroon" title="c2">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>assume </b><a NAME="E1:116"/><i><font color="Green" title="E91">A1</font></i>: 
<font color="Maroon" title="c1">I</font> <a href="scmfsa7b.html#R4" title="SCMFSA7B:pred.4">does_not_destroy</a> <font color="Maroon" title="c2">a</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">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/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="116_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:116_1_1"/>
( <font color="Maroon" title="c3">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c3">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="116_1_1_1"><b>suppose </b></a><a NAME="E1:116_1_1_1"/>
<font color="Maroon" title="c3">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:116_1_1_1"/><b>then </b><i><font color="Green" title="E92">A2</font></i>: 
( <span class="p1">(<span class="default"><a href="scmfsa6c.html#K1" title="SCMFSA6C:func.1">Initialize</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1">(<span class="default"><a href="scmfsa6c.html#K1" title="SCMFSA6C:func.1">Initialize</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="scmfsa6c.html#T3" title="SCMFSA6C:th.3">SCMFSA6C:3</a></i>;<br/><a NAME="E3:116_1_1_1"/>
 <a href="scmfsa6a.html#K1" title="SCMFSA6A:func.1">Directed</a> <span class="p1">(<span class="default"><a href="scmfsa8c.html#K1" title="SCMFSA8C:func.1">loop</a> <span class="p2">(<span class="default"><a href="scmfsa8b.html#K1" title="SCMFSA8B:func.1">if=0</a> <font color="Maroon" title="c2">a</font>,<span class="p3">(<span class="default"><a href="scmfsa8a.html#K2" title="SCMFSA8A:func.2">Goto</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 2</span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="scmfsa6a.html#K6" title="SCMFSA6A:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K10" title="SCMFSA_2:func.10">SubFrom</a> <font color="Maroon" title="c2">a</font>,<span class="p5">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa8c.html#K1" title="SCMFSA8C:func.1">loop</a> <span class="p1">(<span class="default"><a href="scmfsa8b.html#K1" title="SCMFSA8B:func.1">if=0</a> <font color="Maroon" title="c2">a</font>,<span class="p2">(<span class="default"><a href="scmfsa8a.html#K2" title="SCMFSA8A:func.2">Goto</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="scmfsa6a.html#K6" title="SCMFSA6A:func.6">';'</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K10" title="SCMFSA_2:func.10">SubFrom</a> <font color="Maroon" title="c2">a</font>,<span class="p4">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="scmfsa8a.html#T40" title="SCMFSA8A:th.40">SCMFSA8A:40</a></i>;<br/><a NAME="E4:116_1_1_1"/><b>then </b>
 <a href="scmfsa6a.html#K1" title="SCMFSA6A:func.1">Directed</a> <span class="p1">(<span class="default"><a href="scmfsa8c.html#K1" title="SCMFSA8C:func.1">loop</a> <span class="p2">(<span class="default"><a href="scmfsa8b.html#K1" title="SCMFSA8B:func.1">if=0</a> <font color="Maroon" title="c2">a</font>,<span class="p3">(<span class="default"><a href="scmfsa8a.html#K2" title="SCMFSA8A:func.2">Goto</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 2</span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="scmfsa6a.html#K6" title="SCMFSA6A:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K10" title="SCMFSA_2:func.10">SubFrom</a> <font color="Maroon" title="c2">a</font>,<span class="p5">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmfsa8a.html#R1" title="SCMFSA8A:pred.1">is_pseudo-closed_on</a>  <a href="scmfsa6c.html#K1" title="SCMFSA6C:func.1">Initialize</a> <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="txt" href="scmfsa8c.html#E1:116"><i><font color="Green" title="E91">A1</font></i></a>, <a class="txt" href="scmfsa8c.html#E2:116_1_1_1"><i><font color="Green" title="E92">A2</font></i></a>, <a class="ref" href="scmfsa8c.html#T117" target="_self" title="SCMFSA8C:th.117">Th117</a></i>;<br/><a NAME="E5:116_1_1_1"/><b>then </b>
 <a href="scmfsa8c.html#K2" title="SCMFSA8C:func.2">Times</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">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="c3">s</font>
 <b>by </b><i><a class="txt" href="scmfsa8c.html#E2:116_1_1_1"><i><font color="Green" title="E92">A2</font></i></a>, <a class="ref" href="scmfsa8c.html#T68" target="_self" title="SCMFSA8C:th.68">Th68</a></i>;<br/><b>hence </b><a NAME="E6:116_1_1_1"/>
 <a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p1">(<span class="default"><a href="scmfsa8c.html#K2" title="SCMFSA8C:func.2">Times</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="ref" href="scmfsa8c.html#T22" target="_self" title="SCMFSA8C:th.22">Th22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="116_1_1_2"><b>suppose </b></a><a NAME="E1:116_1_1_2"/>
<font color="Maroon" title="c3">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:116_1_1_2"/><b>then </b><i><font color="Green" title="E92">A3</font></i>: 
<span class="p1">(<span class="default"><a href="scmfsa6c.html#K1" title="SCMFSA6C:func.1">Initialize</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="scmfsa6c.html#T3" title="SCMFSA6C:th.3">SCMFSA6C:3</a></i>;<br/><a NAME="E3:116_1_1_2"/>
(  <a href="scmfsa_4.html#K4" title="SCMFSA_4:func.4">SCM+FSA-Stop</a>  <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="c3">s</font> &amp;  <a href="scmfsa_4.html#K4" title="SCMFSA_4:func.4">SCM+FSA-Stop</a>  <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="c3">s</font> )
 <b>by </b><i><a class="ref" href="scmfsa7b.html#T24" title="SCMFSA7B:th.24">SCMFSA7B:24</a>, <a class="ref" href="scmfsa7b.html#T25" title="SCMFSA7B:th.25">SCMFSA7B:25</a></i>;<br/><a NAME="E4:116_1_1_2"/><b>then </b>
 <a href="scmfsa8c.html#K2" title="SCMFSA8C:func.2">Times</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">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="c3">s</font>
 <b>by </b><i><a class="txt" href="scmfsa8c.html#E2:116_1_1_2"><i><font color="Green" title="E92">A3</font></i></a>, <a class="ref" href="scmfsa8b.html#T24" title="SCMFSA8B:th.24">SCMFSA8B:24</a></i>;<br/><b>hence </b><a NAME="E5:116_1_1_2"/>
 <a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p1">(<span class="default"><a href="scmfsa8c.html#K2" title="SCMFSA8C:func.2">Times</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="ref" href="scmfsa8c.html#T22" target="_self" title="SCMFSA8C:th.22">Th22</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:116"/>
 <a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p1">(<span class="default"><a href="scmfsa8c.html#K2" title="SCMFSA8C:func.2">Times</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">I</font></span>)</span> is <a href="ami_1.html#V11" title="AMI_1:attr.11">halting</a>
 <b>by </b><i><a class="ref" href="scmfsa8c.html#T24" target="_self" title="SCMFSA8C:th.24">Th24</a></i>;<br/>


</div>
