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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="scmfsa6a.html#NM1">Macro-Instruction</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <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>assume </b><b>that </b><br/><a NAME="E1:26"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="funct_7.html#R1">equal_outside</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 <b>and </b><br/><a NAME="E2:26"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E3:26"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E4:26"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> holds <br/> <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</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>1</sub></font> implies <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">a<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <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> <a href="funct_7.html#R1">equal_outside</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a>  );<br/>
<a NAME="E5:26"/>
( <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="ami_1.html#D19">AMI_1:def 19</a></i>;<br/>
<a NAME="E6:26"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="scmfsa6b.html#E1:26"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E7:26"/><i><font color="Green">E26</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="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:26_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>]
 ;<br/>

<i><font color="Green">E28</font></i>: <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="ami_1.html#K9">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D19">AMI_1:def 19</a></i>
<br/>.= 
 <a href="ami_1.html#K4">Exec</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8">CurInstr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>

;<br/>
<i><font color="Green">E29</font></i>: <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="ami_1.html#K9">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D19">AMI_1:def 19</a></i>
<br/>.= 
 <a href="ami_1.html#K4">Exec</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8">CurInstr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>

;<br/>
<b>assume </b><a NAME="E4:26_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E5:26_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:26_1"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="scmfsa6b.html#E4:26"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E7:26_1"/><i><font color="Green">E32</font></i>: 
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="scmfsa6b.html#E1:26_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="scmfsa6b.html#E4:26_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="scmfsa6a.html#T29">SCMFSA6A:29</a></i>;<br/>
 <a href="ami_1.html#K8">CurInstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#T54">AMI_1:54</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa6b.html#E2:26"><i><font color="Green">E22</font></i></a>, <a class="txt" href="scmfsa6b.html#E6:26_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="scmfsa6b.html#E7:26_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa6b.html#E3:26"><i><font color="Green">E23</font></i></a>, <a class="txt" href="scmfsa6b.html#E6:26_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="scmfsa6b.html#E7:26_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#T54">AMI_1:54</a></i>
<br/>.= 
 <a href="ami_1.html#K8">CurInstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>

;<br/>
<b>hence </b><a NAME="E9:26_1"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="funct_7.html#R1">equal_outside</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 <b>by </b><i><a class="txt" href="scmfsa6b.html#E1:26_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="scmfsa6b.html#E2:26_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="scmfsa6b.html#E3:26_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="scmfsa6b.html#E4:26_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="scmfsa6a.html#T32">SCMFSA6A:32</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E8:26"/>
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="scmfsa6b.html#E6:26"><i><font color="Green">E25</font></i></a>, <a class="txt" href="scmfsa6b.html#E7:26"><i><font color="Green">E26</font></i></a>);</i><br/>


</div>
