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

<a NAME="E1:17_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">4 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a></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="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T8" title="FUNCOP_1:th.8">FUNCOP_1:8</a></span>;<br/>
<a NAME="E2:17_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A1</font></span>: 
 <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">4 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E3:17_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">4 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> <a href="ami_3.html#K10" title="AMI_3:func.10">&gt;0_goto</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default">4 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T18" title="FUNCT_4:th.18">FUNCT_4:18</a></span>;<br/>
<a NAME="E4:17_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A2</font></span>: 
 <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> <a href="ami_3.html#K10" title="AMI_3:func.10">&gt;0_goto</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default">4 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1_1"><span class="lab"><font color="Green" title="E17">A1</font></span></a></span>;<br/>
<a NAME="E5:17_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> <a href="ami_3.html#K10" title="AMI_3:func.10">&gt;0_goto</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default">4 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="ami_3.html#K3" title="AMI_3:func.3">:=</a> <span class="p4">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 2</span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> <a href="ami_3.html#K10" title="AMI_3:func.10">&gt;0_goto</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default">4 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p4">(<span class="default"><a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T18" title="FUNCT_4:th.18">FUNCT_4:18</a></span>;<br/>
<a NAME="E6:17_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A3</font></span>: 
 <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="ami_3.html#K3" title="AMI_3:func.3">:=</a> <span class="p4">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 2</span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> <a href="ami_3.html#K10" title="AMI_3:func.10">&gt;0_goto</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default">4 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p4">(<span class="default"><a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a></span>;<br/>
<a NAME="E7:17_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="ami_3.html#K3" title="AMI_3:func.3">:=</a> <span class="p4">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 2</span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> <a href="ami_3.html#K10" title="AMI_3:func.10">&gt;0_goto</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default">4 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p4">(<span class="default"><a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a></span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><a href="ami_3.html#K7" title="AMI_3:func.7">Divide</a> (<span class="p4">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>,<span class="p4">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span>)</span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="ami_3.html#K3" title="AMI_3:func.3">:=</a> <span class="p5">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 2</span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">3 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> <a href="ami_3.html#K10" title="AMI_3:func.10">&gt;0_goto</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default">4 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p5">(<span class="default"><a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T18" title="FUNCT_4:th.18">FUNCT_4:18</a></span>;<br/>
<a NAME="E8:17_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A4</font></span>: 
 <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><a href="ami_3.html#K7" title="AMI_3:func.7">Divide</a> (<span class="p4">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>,<span class="p4">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span>)</span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="ami_3.html#K3" title="AMI_3:func.3">:=</a> <span class="p5">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 2</span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">3 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> <a href="ami_3.html#K10" title="AMI_3:func.10">&gt;0_goto</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default">4 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p5">(<span class="default"><a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17_1_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a></span>;<br/>
<a NAME="E9:17_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><a href="ami_3.html#K7" title="AMI_3:func.7">Divide</a> (<span class="p4">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>,<span class="p4">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span>)</span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="ami_3.html#K3" title="AMI_3:func.3">:=</a> <span class="p5">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 2</span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">3 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> <a href="ami_3.html#K10" title="AMI_3:func.10">&gt;0_goto</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default">4 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p5">(<span class="default"><a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <a href="ami_4.html#K1" title="AMI_4:func.1">Euclide-Algorithm</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T18" title="FUNCT_4:th.18">FUNCT_4:18</a></span>;<br/>
<a NAME="E10:17_1_1"/><span class="kw">then </span>
 <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <a href="ami_4.html#K1" title="AMI_4:func.1">Euclide-Algorithm</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:17_1_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:17_1_1"/>
 not  <a href="ami_4.html#K1" title="AMI_4:func.1">Euclide-Algorithm</a>  is  <a href="compos_1.html#V2" title="COMPOS_1:attr.2">halt-free</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="compos_1.html#D3" title="COMPOS_1:def.3">COMPOS_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
