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

<b>let </b><font color="Maroon">N</font> be  <a href="setfam_1.html#V1">with_non-empty_elements</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="ami_1.html#V2">void</a> <a href="ami_1.html#V4">halting</a> <a href="ami_1.html#V5">IC-Ins-separated</a> <a href="ami_1.html#V8">definite</a> <a href="ami_1.html#V10">realistic</a> <a href="amistd_1.html#V4">standard</a> <a href="amistd_2.html#V6" target="_self">without_implicit_jumps</a> <a href="amistd_2.html#V8" target="_self">regular</a>  <a href="ami_1.html#L1">AMI-Struct</a> of <font color="Maroon">N</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="amistd_1.html#NM1">pre-Macro</a> of <font color="Maroon">S</font>;<br/>





<b>set </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/>
<a NAME="E1:135"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">F</font> <a href="amistd_2.html#K12" target="_self">';'</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K7" target="_self">Stop</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K11" target="_self">CutLastLoc</a> <font color="Maroon">F</font></span>)</span> <a href="scmfsa_4.html#K1">+*</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K10" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7" target="_self">Stop</a> <font color="Maroon">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="amistd_2.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E2:135"/><i><font color="Green">E45</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K11" target="_self">CutLastLoc</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><a href="amistd_1.html#K10">LastLoc</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 
<b>by </b><i/>;<br/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K10" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7" target="_self">Stop</a> <font color="Maroon">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K14">.--&gt;</a> <span class="p2">(<span class="default"><a href="ami_1.html#K5">halt</a> <font color="Maroon">S</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span><a href="realset1.html#K5">}</a></span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>
<br/>.= 
<span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><a href="amistd_1.html#K10">LastLoc</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
<b>by </b><i><a class="ref" href="amistd_1.html#T55">AMISTD_1:55</a></i>
;<br/>
<a NAME="E4:135"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="amistd_2.html#K12" target="_self">';'</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7" target="_self">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i>, , <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>
<a NAME="E5:135"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="amistd_2.html#K12" target="_self">';'</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7" target="_self">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="135_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:135_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:135_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K11" target="_self">CutLastLoc</a> <font color="Maroon">F</font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K10" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K9" target="_self">IncAddr</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7" target="_self">Stop</a> <font color="Maroon">S</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:135_2"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K11" target="_self">CutLastLoc</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K10" target="_self">Shift</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K9" target="_self">IncAddr</a> <span class="p4">(<span class="default"><a href="amistd_2.html#K7" target="_self">Stop</a> <font color="Maroon">S</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="135_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:135_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K11" target="_self">CutLastLoc</a> <font color="Maroon">F</font></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><a href="amistd_1.html#K10">LastLoc</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span> )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="135_2_1_1"><b>suppose </b></a><a NAME="E1:135_2_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K11" target="_self">CutLastLoc</a> <font color="Maroon">F</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:135_2_1_1"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K10" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K9" target="_self">IncAddr</a> <span class="p3">(<span class="default"><a href="amistd_2.html#K7" target="_self">Stop</a> <font color="Maroon">S</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="amistd_2.html#K12" target="_self">';'</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7" target="_self">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="amistd_2.html#K11" target="_self">CutLastLoc</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="135_2_1_2"><b>suppose </b></a><a NAME="E1:135_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><a href="amistd_1.html#K10">LastLoc</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 ;<br/><div class="add"><a NAME="E2:135_2_1_2"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K10">LastLoc</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:135_2_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="amistd_1.html#T55">AMISTD_1:55</a></i>;<br/><a NAME="E4:135_2_1_2"/><i><font color="Green">E64</font></i>: 
 <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K7" target="_self">Stop</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:135_2_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K10" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7" target="_self">Stop</a> <font color="Maroon">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p3">(<span class="default"><font color="Olive">m</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> where <font color="Olive">m</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K7" target="_self">Stop</a> <font color="Maroon">S</font></span>)</span> </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E6:135_2_1_2"/><b>then </b>
 <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K10" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7" target="_self">Stop</a> <font color="Maroon">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="amistd_2.html#K12" target="_self">';'</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7" target="_self">Stop</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="amistd_2.html#K10" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K7" target="_self">Stop</a> <font color="Maroon">S</font></span>)</span>,<span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">F</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="amistd_2.html#K7" target="_self">Stop</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,0</span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
 <a href="ami_1.html#K5">halt</a> <font color="Maroon">S</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="amistd_1.html#D22">AMISTD_1:def 22</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E6:135"/>
<font color="Maroon">F</font> <a href="amistd_2.html#K12" target="_self">';'</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K7" target="_self">Stop</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
