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

<b>let </b><font color="Maroon" title="c1">i</font> be   <a href="ami_1.html#NM3" title="AMI_1:NM.3">Instruction</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ;<br/>

<b>assume </b><a NAME="E1:38"/><i><font color="Green" title="E29">A1</font></i>: 
 <a href="ami_5.html#K1" title="AMI_5:func.1">InsCode</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<b>assume </b><a NAME="E2:38"/>
<font color="Maroon" title="c1">i</font> is <a href="scmpds_5.html#V2" title="SCMPDS_5:attr.2">parahalting</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c2">Li</font> =  <a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <font color="Maroon" title="c1">i</font> as  <a href="scmpds_4.html#V2" title="SCMPDS_4:attr.2">parahalting</a> <a href="scmpds_4.html#NM1" title="SCMPDS_4:NM.1">Program-block</a> <b>by </b><i><a class="ref" href="scmpds_5.html#D2" target="_self" title="SCMPDS_5:def.2">Def2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">pi</font> =  <a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c2">Li</font>;<br/>
<b>set </b><font color="Maroon" title="c4">Ii</font> =  <a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c2">Li</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c5">s</font> being   <a href="ami_1.html#NM4" title="AMI_1:NM.4">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> <b> such that </b><br/><a NAME="E5:38"/><i><font color="Green" title="E30">A2</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a>  holds  <font color="Maroon" title="c5">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <b>by </b><i><a class="ref" href="scmpds_2.html#T73" title="SCMPDS_2:th.73">SCMPDS_2:73</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">s1</font> = <font color="Maroon" title="c5">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c2">Li</font></span>)</span></span>)</span>;<br/>
<a NAME="E6:38"/><i><font color="Green" title="E31">A3</font></i>: 
 <a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c2">Li</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c2">Li</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T26" title="FUNCT_4:th.26">FUNCT_4:26</a></i>;<br/>
<a NAME="E7:38"/>
 <a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c2">Li</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="scmpds_5.html#T13" target="_self" title="SCMPDS_5:th.13">Th13</a></i>;<br/>
<b>then </b><i><font color="Green" title="E32">A4</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c2">Li</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> 0</span>)</span> = 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c2">Li</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c1">i</font>
<b>by </b><i><a class="ref" href="scmpds_5.html#T13" target="_self" title="SCMPDS_5:th.13">Th13</a></i>
;<br/>
<i><font color="Green" title="E33">A5</font></i>: <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c2">Li</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="ami_1.html#K9" title="AMI_1:func.9">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c2">Li</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D19" title="AMI_1:def.19">AMI_1:def 19</a></i>
<br/>.= 
 <a href="ami_1.html#K9" title="AMI_1:func.9">Following</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c2">Li</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D19" title="AMI_1:def.19">AMI_1:def 19</a></i>
<br/>.= 
 <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c2">Li</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmpds_5.html#E6:38"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="scmpds_5.html#E8:38"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="scmpds_5.html#T18" target="_self" title="SCMPDS_5:th.18">Th18</a></i>
;<br/>
<b>consider </b><font color="Maroon" title="c7">a</font> being    <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a> <b> such that </b><br/><a NAME="E11:38"/><i><font color="Green" title="E34">A6</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_2.html#K5" title="SCMPDS_2:func.5">return</a> <font color="Maroon" title="c7">a</font>
 <b>by </b><i><a class="txt" href="scmpds_5.html#E1:38"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="scmpds_2.html#T36" title="SCMPDS_2:th.36">SCMPDS_2:36</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c2">Li</font></span>)</span></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c2">Li</font></span>)</span></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<a href="scmpds_1.html#K23" title="SCMPDS_1:func.23">RetIC</a> </span>)</span> = 
<font color="Maroon" title="c5">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c2">Li</font></span>)</span></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<a href="scmpds_1.html#K23" title="SCMPDS_1:func.23">RetIC</a> </span>)</span>
<b>by </b><i><a class="ref" href="scmpds_5.html#T19" target="_self" title="SCMPDS_5:th.19">Th19</a></i>
<br/>.= 
2
<b>by </b><i><a class="txt" href="scmpds_5.html#E5:38"><i><font color="Green" title="E30">A2</font></i></a></i>
;<br/>
<b>then </b><i><font color="Green" title="E35">A7</font></i>: <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c2">Li</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span> = 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> 2</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4
<b>by </b><i><a class="txt" href="scmpds_5.html#E11:38"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="scmpds_2.html#T70" title="SCMPDS_2:th.70">SCMPDS_2:70</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4
<b>by </b><i><a class="ref" href="nat_2.html#T5" title="NAT_2:th.5">NAT_2:5</a></i>
<br/>.= 
 <a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> 2

;<br/>
<b>set </b><font color="Maroon" title="c8">C1</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c2">Li</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> 1;<br/>
<a NAME="E14:38"/><i><font color="Green" title="E36">A8</font></i>: 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c2">Li</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> 2
 
<b>by </b><i><a class="txt" href="scmpds_5.html#E9:38"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="scmpds_5.html#E13:38"><i><font color="Green" title="E35">A7</font></i></a></i>;<br/>
<a NAME="E15:38"/><i><font color="Green" title="E37">A9</font></i>: 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c2">Li</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c2">Li</font></span>)</span>
 
<b>by </b><i><a class="txt" href="scmpds_5.html#E6:38"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="scmpds_4.html#D9" title="SCMPDS_4:def.9">SCMPDS_4:def 9</a></i>;<br/>
<a NAME="E16:38"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c2">Li</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="ref" href="scmpds_5.html#T8" target="_self" title="SCMPDS_5:th.8">Th8</a></i>;<br/>
<b>hence </b><a NAME="E17:38"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_5.html#E14:38"><i><font color="Green" title="E36">A8</font></i></a>, <a class="txt" href="scmpds_5.html#E15:38"><i><font color="Green" title="E37">A9</font></i></a>, <a class="ref" href="scmpds_4.html#T1" title="SCMPDS_4:th.1">SCMPDS_4:1</a></i>;<br/>


</div>
