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

<b>let </b><font color="Maroon">c<sub>4</sub></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">c<sub>5</sub></font> be    <a href="scmfsa_2.html#M1">Int-Location</a> ;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be  <a href="scmfsa7b.html#V1">good</a> <a href="scmfsa6a.html#NM1">Macro-Instruction</a>;<br/>





<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font>;<br/>
<b>assume </b><a NAME="E1:21"/><i><font color="Green">E15</font></i>: 
 <a href="sfmastr2.html#R1" target="_self">ProperTimesBody</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="sfmastr2.html#K2" target="_self">StepTimes</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = 1 <a href="sfmastr1.html#NK7" target="_self">-stRWNotIn</a> <span class="p1">(<span class="default"><span class="p2"><a href="sfmastr1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="sfmastr1.html#K1">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="scmfsa6c.html#K1">Initialize</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">c<sub>5</sub></font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr2.html#K2" target="_self">StepTimes</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 1 );<br/>
<a NAME="E2:21"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr2.html#K2" target="_self">StepTimes</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> 0</span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K4">Exec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="sfmastr1.html#NK7" target="_self">-stRWNotIn</a> <span class="p4">(<span class="default"><span class="p5"><a href="sfmastr1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="sfmastr1.html#K1">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p5">(<span class="default"><a href="sf_mastr.html#K2">UsedIntLoc</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="scmfsa_2.html#K8">:=</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa6c.html#K1">Initialize</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="scmfsa_9.html#D5">SCMFSA_9:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmfsa6c.html#K1">Initialize</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="scmfsa_2.html#T89">SCMFSA_2:89</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="scmfsa6c.html#T3">SCMFSA6C:3</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E3:21"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:21_2"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">c<sub>5</sub></font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr2.html#K2" target="_self">StepTimes</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>and </b><br/><a NAME="E2:21_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">c<sub>5</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="sfmastr2.html#E2:21_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<a NAME="E4:21_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="sfmastr2.html#E2:21_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:21_2"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="scmfsa7b.html#R5">is_closed_on</a> <span class="p1">(<span class="default"><a href="sfmastr2.html#K2" target="_self">StepTimes</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="scmfsa7b.html#R6">is_halting_on</a> <span class="p1">(<span class="default"><a href="sfmastr2.html#K2" target="_self">StepTimes</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="sfmastr2.html#E1:21"><i><font color="Green">E15</font></i></a>, <a class="ref" href="sfmastr2.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E6:21_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr2.html#K2" target="_self">StepTimes</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="scmfsa_2.html#K20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="sfmastr2.html#E1:21_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="sfmastr2.html#E4:21_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="sfmastr2.html#T12" target="_self">Th12</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:21"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="sfmastr2.html#E2:21"><i><font color="Green">E16</font></i></a>, <a class="txt" href="sfmastr2.html#E3:21"><i><font color="Green">E17</font></i></a>);</i><br/>


</div>
