<?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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="scmfsa6b.html#D4" title="SCMFSA6B:def.4">SCMFSA6B:def 4</a>,<a class="ref" href="scmfsa6c.html#D2" target="_self" title="SCMFSA6C:def.2">SCMFSA6C:def 2</a><br/>

<b>assume </b><a NAME="E1:15_1_1"/><i><font color="Green" title="E5">A1</font></i>: 
<span class="p1">(<span class="default"><a href="scmfsa6a.html#K2" title="SCMFSA6A:func.2">Macro</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">:=len</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</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> 0</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">s</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>set </b><font color="Maroon" title="c6">Ma</font> =  <a href="scmfsa6a.html#K2" title="SCMFSA6A:func.2">Macro</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">:=len</a> <font color="Maroon" title="c2">f</font></span>)</span>;<br/>
<a NAME="E2:15_1_1"/>
 <a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K2" title="SCMFSA6A:func.2">Macro</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">:=len</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</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> 0</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T26" title="FUNCT_4:th.26">FUNCT_4:26</a></i>;<br/>
<a NAME="E3:15_1_1"/><b>then </b><i><font color="Green" title="E6">A2</font></i>: 
 <a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">s</font>
 
<b>by </b><i><a class="txt" href="scmfsa6c.html#E1:15_1_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E4:15_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</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> 0</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E5:15_1_1"/><b>then </b><i><font color="Green" title="E7">A3</font></i>: 
 <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  <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="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> 0</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<i><font color="Green" title="E8">A4</font></i>:  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c4">s</font> = 
<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> 0</span>)</span></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#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span>
<b>by </b><i><a class="txt" href="scmfsa6c.html#E3:15_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="scmfsa6c.html#E5:15_1_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
 <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0
<b>by </b><i><a class="ref" href="cqc_lang.html#T6" title="CQC_LANG:th.6">CQC_LANG:6</a></i>
;<br/>
<a NAME="E7:15_1_1"/><i><font color="Green" title="E9">A5</font></i>: 
(  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0 <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="scmfsa6a.html#K2" title="SCMFSA6A:func.2">Macro</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">:=len</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> &amp;  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 1 <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="scmfsa6a.html#K2" title="SCMFSA6A:func.2">Macro</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">:=len</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="scmfsa6b.html#T32" title="SCMFSA6B:th.32">SCMFSA6B:32</a></i>;<br/>
<a NAME="E8:15_1_1"/>
 <a href="scmfsa6a.html#K2" title="SCMFSA6A:func.2">Macro</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">:=len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">s</font>
 
<b>by </b><i><a class="txt" href="scmfsa6c.html#E1:15_1_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="scmfsa6b.html#T5" title="SCMFSA6B:th.5">SCMFSA6B:5</a></i>;<br/>
<a NAME="E9:15_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="scmfsa6a.html#K2" title="SCMFSA6A:func.2">Macro</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">:=len</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmfsa6a.html#K2" title="SCMFSA6A:func.2">Macro</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">:=len</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="scmfsa6c.html#E7:15_1_1"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/>
<a NAME="E10:15_1_1"/><b>then </b><i><font color="Green" title="E10">A6</font></i>: 
( <font color="Maroon" title="c4">s</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">:=len</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c4">s</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="ref" href="scmfsa6b.html#T33" title="SCMFSA6B:th.33">SCMFSA6B:33</a></i>;<br/>
<i><font color="Green" title="E11">A7</font></i>:  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">:=len</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c4">s</font></span>)</span> = 
 <a href="scmfsa_2.html#K7" title="SCMFSA_2:func.7">Next</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="scmfsa6c.html#E6:15_1_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="scmfsa_2.html#T100" title="SCMFSA_2:th.100">SCMFSA_2:100</a></i>
<br/>.= 
 <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="scmfsa_2.html#T32" title="SCMFSA_2:th.32">SCMFSA_2:32</a></i>
;<br/>
<i><font color="Green" title="E12">A8</font></i>: <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="ami_1.html#K9" title="AMI_1:func.9">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D19" title="AMI_1:def.19">AMI_1:def 19</a></i>
<br/>.= 
 <a href="ami_1.html#K9" title="AMI_1:func.9">Following</a> <font color="Maroon" title="c4">s</font>
<b>by </b><i><a class="ref" href="ami_1.html#D19" title="AMI_1:def.19">AMI_1:def 19</a></i>
<br/>.= 
 <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">:=len</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c4">s</font>
<b>by </b><i><a class="txt" href="scmfsa6c.html#E6:15_1_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="scmfsa6c.html#E10:15_1_1"><i><font color="Green" title="E10">A6</font></i></a></i>
;<br/>
<a NAME="E13:15_1_1"/><b>then </b><i><font color="Green" title="E13">A9</font></i>: 
 <a href="ami_1.html#K8" title="AMI_1:func.8">CurInstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="scmfsa6c.html#E10:15_1_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="scmfsa6c.html#E11:15_1_1"><i><font color="Green" title="E11">A7</font></i></a>, <a class="ref" href="ami_1.html#D13" title="AMI_1:def.13">AMI_1:def 13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_4"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_1_4"/>
( <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_4_1"><b>suppose </b></a><a NAME="E1:15_1_1_4_1"/>
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:15_1_1_4_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c5">k</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> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <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> 0</span>)</span>
 <b>by </b><i><a class="ref" href="ami_1.html#D19" title="AMI_1:def.19">AMI_1:def 19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_4_2"><b>suppose </b></a><a NAME="E1:15_1_1_4_2"/><i><font color="Green" title="E14">A10</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 ;<br/><div class="add"><a NAME="E2:15_1_1_4_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> 1</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> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <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> 0</span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa6c.html#E12:15_1_1"><i><font color="Green" title="E12">A8</font></i></a>, <a class="ref" href="scmfsa_2.html#T100" title="SCMFSA_2:th.100">SCMFSA_2:100</a></i>;<br/><b>hence </b><a NAME="E3:15_1_1_4_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c5">k</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> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <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> 0</span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa6c.html#E13:15_1_1"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="scmfsa6c.html#E1:15_1_1_4_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="ref" href="ami_1.html#T52" title="AMI_1:th.52">AMI_1:52</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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