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

<span class="kw">set </span><font color="Maroon" title="c7">i</font> = <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">G</font> =  <a href="scmpds_6.html#K1" title="SCMPDS_6:func.1">Goto</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">IF</font> =  <a href="scmpds_6.html#K2" title="SCMPDS_6:func.2">if=0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">k1</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">J</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">IsIF</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> <span class="p2">(<span class="default"><a href="scmpds_6.html#K2" title="SCMPDS_6:func.2">if=0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">k1</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">J</font></span>)</span></span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">IJ</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p2">(<span class="default"><a href="scmpds_6.html#K1" title="SCMPDS_6:func.1">Goto</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <font color="Maroon" title="c4">J</font> as   <a href="scmpds_4.html#V3" title="SCMPDS_4:attr.3">shiftable</a>  <a href="scmnorm.html#NM2" title="SCMNORM:NM.2">Program</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>  ;<br/>
 <a href="scmpds_6.html#K2" title="SCMPDS_6:func.2">if=0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">k1</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">J</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_4.html#K4" title="SCMPDS_4:func.4">';'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K1" title="SCMPDS_6:func.1">Goto</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <font color="Maroon" title="c4">J</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_4.html#T50" title="SCMPDS_4:th.50">SCMPDS_4:50</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_4.html#K4" title="SCMPDS_4:func.4">';'</a> <font color="Maroon" title="c11">IJ</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_4.html#T50" title="SCMPDS_4:th.50">SCMPDS_4:50</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <font color="Maroon" title="c11">IJ</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_4.html#D4" title="SCMPDS_4:def.4">SCMPDS_4:def 4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:64_1_1"/>
 <a href="scmpds_6.html#K2" title="SCMPDS_6:func.2">if=0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">k1</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">J</font> is  <a href="scmpds_4.html#V3" title="SCMPDS_4:attr.3">shiftable</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="scmpds_6.html#K2" title="SCMPDS_6:func.2">if=0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">k1</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">J</font> is  <a href="scmpds_4.html#V2" title="SCMPDS_4:attr.2">parahalting</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">s</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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> <span class="p2">(<span class="default"><a href="scmpds_6.html#K2" title="SCMPDS_6:func.2">if=0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">k1</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">J</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">s</font> implies  <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Olive">b<sub>1</sub></font> <a href="ami_1.html#R1" title="AMI_1:pred.1">halts_on</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:64_1_1_2"/>
 <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> <span class="p2">(<span class="default"><a href="scmpds_6.html#K2" title="SCMPDS_6:func.2">if=0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">k1</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">J</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Olive">b<sub>1</sub></font> <a href="ami_1.html#R1" title="AMI_1:pred.1">halts_on</a> <font color="Olive">b<sub>1</sub></font></span><br/><a NAME="E2:64_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A1</font></span>: 
<font color="Maroon" title="c12">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</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> <span class="p3">(<span class="default"><a href="scmpds_6.html#K2" title="SCMPDS_6:func.2">if=0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">k1</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">J</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T79" title="FUNCT_4:th.79">FUNCT_4:79</a></span>;<br/><a NAME="E3:64_1_1_2"/><span class="lab"><font color="Green" title="E54">A2</font></span>: 
( <font color="Maroon" title="c4">J</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c12">s</font> &amp; <font color="Maroon" title="c4">J</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c12">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_6.html#T34" target="_self" title="SCMPDS_6:th.34">Th34</a>, <a class="ref" href="scmpds_6.html#T35" target="_self" title="SCMPDS_6:th.35">Th35</a></span>;<br/><a NAME="E4:64_1_1_2"/><span class="lab"><font color="Green" title="E55">A3</font></span>: 
( <font color="Maroon" title="c3">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c12">s</font> &amp; <font color="Maroon" title="c3">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c12">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_6.html#T34" target="_self" title="SCMPDS_6:th.34">Th34</a>, <a class="ref" href="scmpds_6.html#T35" target="_self" title="SCMPDS_6:th.35">Th35</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:64_1_1_2_1"/>
( <font color="Maroon" title="c12">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"><font color="Maroon" title="c12">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<font color="Maroon" title="c6">k1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c12">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"><font color="Maroon" title="c12">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<font color="Maroon" title="c6">k1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_2_1_1"/>
<font color="Maroon" title="c12">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"><font color="Maroon" title="c12">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<font color="Maroon" title="c6">k1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Olive">b<sub>1</sub></font> <a href="ami_1.html#R1" title="AMI_1:pred.1">halts_on</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:64_1_1_2_1_1"/><span class="kw">then </span>
 <a href="scmpds_6.html#K2" title="SCMPDS_6:func.2">if=0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">k1</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">J</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c12">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_1_2"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="ref" href="scmpds_6.html#T52" target="_self" title="SCMPDS_6:th.52">Th52</a></span>;<br/><span class="kw">hence </span><a NAME="E3:64_1_1_2_1_1"/>
 <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c12">s</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">halts_on</a> <font color="Maroon" title="c12">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1_1_2"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="ref" href="scmpds_6.html#D3" target="_self" title="SCMPDS_6:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_2_1_2"/>
<font color="Maroon" title="c12">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"><font color="Maroon" title="c12">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<font color="Maroon" title="c6">k1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Olive">b<sub>1</sub></font> <a href="ami_1.html#R1" title="AMI_1:pred.1">halts_on</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:64_1_1_2_1_2"/><span class="kw">then </span>
 <a href="scmpds_6.html#K2" title="SCMPDS_6:func.2">if=0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">k1</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">J</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c12">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_1_2"><span class="lab"><font color="Green" title="E54">A2</font></span></a>, <a class="ref" href="scmpds_6.html#T53" target="_self" title="SCMPDS_6:th.53">Th53</a></span>;<br/><span class="kw">hence </span><a NAME="E3:64_1_1_2_1_2"/>
 <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c12">s</font> <a href="ami_1.html#R1" title="AMI_1:pred.1">halts_on</a> <font color="Maroon" title="c12">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1_1_2"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="ref" href="scmpds_6.html#D3" target="_self" title="SCMPDS_6:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E5:64_1_1"/><span class="kw">then </span>
 <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> <span class="p2">(<span class="default"><a href="scmpds_6.html#K2" title="SCMPDS_6:func.2">if=0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">k1</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">J</font></span>)</span></span>)</span> is  <a href="ami_1.html#V11" title="AMI_1:attr.11">halting</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#D26" title="AMI_1:def.26">AMI_1:def 26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:64_1_1"/>
 <a href="scmpds_6.html#K2" title="SCMPDS_6:func.2">if=0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">k1</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">J</font> is  <a href="scmpds_4.html#V2" title="SCMPDS_4:attr.2">parahalting</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_4.html#D10" title="SCMPDS_4:def.10">SCMPDS_4:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
