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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">i</font> be   <a href="compos_1.html#NM1" title="COMPOS_1:NM.1">Instruction</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="compos_0.html#K5" title="COMPOS_0:func.5">InsCode</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  not <font color="Maroon" title="c2">i</font> is  <a href="scmpds_5.html#V1" title="SCMPDS_5:attr.1">parahalting</a>  )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">s</font> being   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a><span class="kw"> such that </span><br/><a NAME="E2:36"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a> holds  <font color="Maroon" title="c3">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_2.html#T61" onmouseover="rs('scmpds_2/T61')" onmouseout="rh()">SCMPDS_2:61</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">P</font> =  the   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>;<br/>
<span class="kw">assume </span><a NAME="E3:36"/>
 <a href="compos_0.html#K5" title="COMPOS_0:func.5">InsCode</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c2">i</font> is  <a href="scmpds_5.html#V1" title="SCMPDS_5:attr.1">parahalting</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a><span class="kw"> such that </span><br/><a NAME="E5:36"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">return</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_2.html#T27" onmouseover="rs('scmpds_2/T27')" onmouseout="rh()">SCMPDS_2:27</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:36"/>
<font color="Maroon" title="c2">i</font> is  <a href="scmpds_5.html#V1" title="SCMPDS_5:attr.1">parahalting</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">Li</font> =  <a href="compos_1.html#K9" title="COMPOS_1:func.9">Load</a> <font color="Maroon" title="c2">i</font> as   <a href="scmpds_4.html#V2" title="SCMPDS_4:attr.2">parahalting</a>  <a href="compos_1.html#NM4" title="COMPOS_1:NM.4">Program</a> of  ;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">pi</font> =  <a href="compos_1.html#K11" title="COMPOS_1:func.11">Macro</a> <font color="Maroon" title="c2">i</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">s1</font> =  <a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <font color="Maroon" title="c3">s</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">P1</font> =  the   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="compos_1.html#K11" title="COMPOS_1:func.11">Macro</a> <font color="Maroon" title="c2">i</font></span>)</span>;<br/>
<span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<a href="scmpds_i.html#K15" title="SCMPDS_I:func.15">RetIC</a>)</span>)</span> = 
<font color="Maroon" title="c3">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<a href="scmpds_i.html#K15" title="SCMPDS_I:func.15">RetIC</a>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_5.html#T15" target="_self" onmouseover="rs('scmpds_5/T15')" onmouseout="rh()">Th4</a></span>
<br/>.= 
2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E14">A1</font></span></a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E16">A3</font></span>: <span class="p1">(<span class="default"><a href="extpro_1.html#K2" title="EXTPRO_1:func.2">Exec</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <font color="Maroon" title="c3">s</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="memstr_0.html#NK2" title="MEMSTR_0:NK.2">IC</a> </span>)</span> = 
<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default">2</span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="scmpds_2.html#T58" onmouseover="rs('scmpds_2/T58')" onmouseout="rh()">SCMPDS_2:58</a></span>
<br/>.= 
2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>
<br/>.= 
4

;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">C1</font> =  <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p1">(<span class="default"> the   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="compos_1.html#K11" title="COMPOS_1:func.11">Macro</a> <font color="Maroon" title="c2">i</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <font color="Maroon" title="c3">s</font></span>)</span>,1);<br/>
<a NAME="E10:36"/>
 <a href="compos_1.html#K10" title="COMPOS_1:func.10">stop</a> <font color="Maroon" title="c6">Li</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="compos_1.html#K11" title="COMPOS_1:func.11">Macro</a> <font color="Maroon" title="c2">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T25" onmouseover="rs('funct_4/T25')" onmouseout="rh()">FUNCT_4:25</a></span>;<br/>
<a NAME="E11:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A4</font></span>: 
 <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p2">(<span class="default"> the   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="compos_1.html#K11" title="COMPOS_1:func.11">Macro</a> <font color="Maroon" title="c2">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <font color="Maroon" title="c3">s</font></span>)</span>,1)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="compos_1.html#K11" title="COMPOS_1:func.11">Macro</a> <font color="Maroon" title="c2">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_4.html#D6" onmouseover="rs('scmpds_4/D6')" onmouseout="rh()">SCMPDS_4:def 6</a></span>;<br/>
<a NAME="E12:36"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="compos_1.html#K11" title="COMPOS_1:func.11">Macro</a> <font color="Maroon" title="c2">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="compos_1.html#T57" onmouseover="rs('compos_1/T57')" onmouseout="rh()">COMPOS_1:57</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E18">A5</font></span>: <span class="p1">(<span class="default"> the   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="compos_1.html#K11" title="COMPOS_1:func.11">Macro</a> <font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="compos_1.html#K3" title="COMPOS_1:func.3">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> = 
<span class="p1">(<span class="default"><a href="compos_1.html#K11" title="COMPOS_1:func.11">Macro</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
<font color="Maroon" title="c2">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="compos_1.html#T58" onmouseover="rs('compos_1/T58')" onmouseout="rh()">COMPOS_1:58</a></span>
;<br/>
<a NAME="E14:36"/><span class="lab"><font color="Green" title="E19">A6</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="compos_1.html#K11" title="COMPOS_1:func.11">Macro</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="compos_1.html#T56" onmouseover="rs('compos_1/T56')" onmouseout="rh()">COMPOS_1:56</a></span>;<br/>
<a NAME="E15:36"/><span class="lab"><font color="Green" title="E20">A7</font></span>: 
<span class="p1">(<span class="default"> the   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="compos_1.html#K11" title="COMPOS_1:func.11">Macro</a> <font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <span class="p2">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <font color="Maroon" title="c3">s</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="compos_1.html#K11" title="COMPOS_1:func.11">Macro</a> <font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="compos_1.html#K3" title="COMPOS_1:func.3">.</a> <span class="p1">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <span class="p2">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <font color="Maroon" title="c3">s</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T143" onmouseover="rs('pboole/T143')" onmouseout="rh()">PBOOLE:143</a></span>;<br/>
 <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p1">(<span class="default"> the   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="compos_1.html#K11" title="COMPOS_1:func.11">Macro</a> <font color="Maroon" title="c2">i</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <font color="Maroon" title="c3">s</font></span>)</span>,<span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) = 
 <a href="extpro_1.html#K4" title="EXTPRO_1:func.4">Following</a> (<span class="p1">(<span class="default"> the   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="compos_1.html#K11" title="COMPOS_1:func.11">Macro</a> <font color="Maroon" title="c2">i</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p2">(<span class="default"> the   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="compos_1.html#K11" title="COMPOS_1:func.11">Macro</a> <font color="Maroon" title="c2">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <font color="Maroon" title="c3">s</font></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="extpro_1.html#T3" onmouseover="rs('extpro_1/T3')" onmouseout="rh()">EXTPRO_1:3</a></span>
<br/>.= 
 <a href="extpro_1.html#K4" title="EXTPRO_1:func.4">Following</a> (<span class="p1">(<span class="default"> the   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="compos_1.html#K11" title="COMPOS_1:func.11">Macro</a> <font color="Maroon" title="c2">i</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <font color="Maroon" title="c3">s</font></span>)</span>)

<br/>.= 
 <a href="extpro_1.html#K2" title="EXTPRO_1:func.2">Exec</a> (<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <font color="Maroon" title="c3">s</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:36"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E15:36"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="memstr_0.html#T47" onmouseover="rs('memstr_0/T47')" onmouseout="rh()">MEMSTR_0:47</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E17:36"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:36"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E11:36"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E14:36"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="ref" href="afinsq_1.html#T66" onmouseover="rs('afinsq_1/T66')" onmouseout="rh()">AFINSQ_1:66</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
