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

<b>let </b><font color="Maroon">c<sub>3</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><a NAME="E1:4"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="ami_1.html#V9">halting</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E2:4"/><i><font color="Green">E6</font></i>: 
 <a href="scm_1.html#NK3" target="_self">LifeSpan</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</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 (  <a href="scm_1.html#NK3" target="_self">LifeSpan</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> implies  <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">a<sub>1</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>3</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> );<br/>
<a NAME="E3:4"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:4_1"/><i><font color="Green">E8</font></i>: 
 <a href="scm_1.html#NK3" target="_self">LifeSpan</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>

<a NAME="E2:4_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="scm_1.html#NK3" target="_self">LifeSpan</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<b>hence </b><a NAME="E3:4_1"/>
 <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> 0</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>3</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa8a.html#E1:4_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>now </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="E1:4_2"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_2_1"/><i><font color="Green">E10</font></i>: 
 <a href="scm_1.html#NK3" target="_self">LifeSpan</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_2_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="scm_1.html#NK3" target="_self">LifeSpan</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="scm_1.html#NK3" target="_self">LifeSpan</a> <font color="Maroon">c<sub>3</sub></font> )
 </span><b>by </b><i><a class="txt" href="scmfsa8a.html#E1:4_2_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1_1_1"><b>suppose </b></a><a NAME="E1:4_2_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="scm_1.html#NK3" target="_self">LifeSpan</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:4_2_1_1_1"/>
 <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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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>3</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1_1_2"><b>suppose </b></a><a NAME="E1:4_2_1_1_2"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="scm_1.html#NK3" target="_self">LifeSpan</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:4_2_1_1_2"/><b>then </b>
 <a href="scm_1.html#NK3" target="_self">LifeSpan</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:4_2_1_1_2"/><b>then </b><i><font color="Green">E12</font></i>: 
 <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> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K5">halt</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 <b>by </b><i><a class="txt" href="scmfsa8a.html#E1:4"><i><font color="Green">E5</font></i></a>, <a class="ref" href="scmfsa8a.html#T4" target="_self">Th4</a></i>;<br/><b>thus </b> <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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9">Following</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#D19">AMI_1:def 19</a></i>
<br/>.= 
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K4">Exec</a> <span class="p2">(<span class="default"><a href="ami_1.html#K5">halt</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span>,<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="scmfsa8a.html#E3:4_2_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="ami_1.html#D18">AMI_1:def 18</a></i>
<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>3</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa8a.html#E1:4_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="scmfsa8a.html#E1:4_2_1_1_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="ami_1.html#D8">AMI_1:def 8</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:4_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E5:4"/>
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="scmfsa8a.html#E3:4"><i><font color="Green">E7</font></i></a>, <a class="txt" href="scmfsa8a.html#E4:4"><i><font color="Green">E8</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:4"/>
 <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>4</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>3</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa8a.html#E2:4"><i><font color="Green">E6</font></i></a></i>;<br/>


</div>
