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<b>let </b><font color="Maroon" title="c3">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">I</font> being  <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/> for <font color="Olive" title="b2">a</font> being   <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a> <br/> for <font color="Olive" title="b3">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon" title="c3">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="c3">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b2">a</font></span>)</span>,<font color="Olive" title="b3">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>(  <a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">i</font>,<font color="Olive" title="b1">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c3">s</font> &amp;  <a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">i</font>,<font color="Olive" title="b1">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c3">s</font> )</span><br/><b>let </b><font color="Maroon" title="c4">I</font> be   <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being   <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a> <br/> for <font color="Olive" title="b2">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon" title="c3">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="c3">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<font color="Olive" title="b2">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>(  <a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">i</font>,<font color="Maroon" title="c4">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c3">s</font> &amp;  <a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">i</font>,<font color="Maroon" title="c4">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c3">s</font> )</span><br/><b>let </b><font color="Maroon" title="c5">a</font> be    <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon" title="c3">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="c3">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="Olive" title="b1">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>(  <a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c3">s</font> &amp;  <a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c3">s</font> )</span><br/><b>let </b><font color="Maroon" title="c6">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">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="c3">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">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies (  <a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c3">s</font> &amp;  <a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c3">s</font> ) )</span><br/>







<b>set </b><font color="Maroon" title="c7">d1</font> =  <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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">i</font>;<br/>
<b>assume </b><a NAME="E1:32"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c3">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="c3">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">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c3">s</font> &amp;  <a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c3">s</font> )</span><br/>

<b>set </b><font color="Maroon" title="c8">WHL</font> =  <a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font>;<br/>
<b>set </b><font color="Maroon" title="c9">pWHL</font> =  <a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c10">iWHL</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_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c11">s3</font> = <font color="Maroon" title="c3">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</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_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c12">s4</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</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> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,1;<br/>
<b>set </b><font color="Maroon" title="c13">i1</font> = <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=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="c4">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c14">i2</font> =  <a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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="c4">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>;<br/>
<a NAME="E2:32"/><i><font color="Green" title="E21">A2</font></i>: 
 <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</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> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="scmpds_6.html#T21" title="SCMPDS_6:th.21">SCMPDS_6:21</a></i>;<br/>
<a NAME="E3:32"/><i><font color="Green" title="E22">A3</font></i>: 
 <a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=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="c4">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="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="scmpds_4.html#T51" title="SCMPDS_4:th.51">SCMPDS_4:51</a></i>;<br/>
<i><font color="Green" title="E23">A4</font></i>:  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</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> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
 <a href="ami_1.html#K9" title="AMI_1:func.9">Following</a> <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="c3">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</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> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#T14" title="AMI_1:th.14">AMI_1:14</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="c3">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</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> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#T13" title="AMI_1:th.13">AMI_1:13</a></i>
<br/>.= 
 <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=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="c4">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</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> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:32"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="scmpds_6.html#T22" title="SCMPDS_6:th.22">SCMPDS_6:22</a></i>
;<br/>
<a NAME="E5:32"/><i><font color="Green" title="E24">A5</font></i>: 
( not  <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">s</font> )
 
<b>by </b><i><a class="ref" href="scmpds_2.html#T49" title="SCMPDS_2:th.49">SCMPDS_2:49</a>, <a class="ref" href="scmpds_4.html#T31" title="SCMPDS_4:th.31">SCMPDS_4:31</a></i>;<br/>
<a NAME="E6:32"/>
( not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">s</font> )
 
<b>by </b><i><a class="ref" href="scmpds_2.html#T49" title="SCMPDS_2:th.49">SCMPDS_2:49</a>, <a class="ref" href="scmpds_4.html#T31" title="SCMPDS_4:th.31">SCMPDS_4:31</a></i>;<br/>
<b>then </b><i><font color="Green" title="E25">A6</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</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> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></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="c3">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</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> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <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">i</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</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> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></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"><font color="Maroon" title="c3">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">i</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon" title="c3">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="c3">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">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:32"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
;<br/>
<a NAME="E8:32"/>
 <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_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</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_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></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="E9:32"/><b>then </b><i><font color="Green" title="E26">A7</font></i>: 
 <a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</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> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,1
 
<b>by </b><i><a class="ref" href="ami_1.html#T81" title="AMI_1:th.81">AMI_1:81</a>, <a class="ref" href="scmpds_4.html#T57" title="SCMPDS_4:th.57">SCMPDS_4:57</a></i>;<br/>
<a NAME="E10:32"/><i><font color="Green" title="E27">A8</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<b>by </b><i><a class="ref" href="scmpds_8.html#T17" target="_self" title="SCMPDS_8:th.17">Th17</a></i>;<br/>
<a NAME="E11:32"/><b>then </b><i><font color="Green" title="E28">A9</font></i>: 
 <a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</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">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="scmpds_6.html#T25" title="SCMPDS_6:th.25">SCMPDS_6:25</a></i>;<br/>
<i><font color="Green" title="E29">A10</font></i>:  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <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="c3">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</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> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,1</span>)</span> = 
 <a href="scmpds_2.html#K18" title="SCMPDS_2:func.18">ICplusConst</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</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> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<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">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="#E1:32"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E4:32"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="#E7:32"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="scmpds_2.html#T68" title="SCMPDS_2:th.68">SCMPDS_2:68</a></i>
<br/>.= 
 <a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:32"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="scmpds_6.html#T23" title="SCMPDS_6:th.23">SCMPDS_6:23</a></i>
;<br/>
<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="c3">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</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> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,1</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> <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="c4">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> = 
<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_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</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> <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="c4">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E9:32"><i><font color="Green" title="E26">A7</font></i></a>, <a class="txt" href="#E11:32"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
 <a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> 
<b>by </b><i><a class="txt" href="#E10:32"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="scmpds_6.html#T25" title="SCMPDS_6:th.25">SCMPDS_6:25</a></i>
;<br/>
<a NAME="E14:32"/><b>then </b><i><font color="Green" title="E30">A11</font></i>: 
 <a href="ami_1.html#K8" title="AMI_1:func.8">CurInstr</a> <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="c3">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</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> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> 
 
<b>by </b><i><a class="txt" href="#E12:32"><i><font color="Green" title="E29">A10</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <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="c3">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</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> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_5_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">k</font> or <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="32_5_1_1"><b>suppose </b></a><a NAME="E1:32_5_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <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="c3">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</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> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:32_5_1_1"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font>
 <b>by </b><i><a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:32_5_1_1"/>
 <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <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="c3">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</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> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E11:32"><i><font color="Green" title="E28">A9</font></i></a>, <a class="txt" href="#E12:32"><i><font color="Green" title="E29">A10</font></i></a>, <a class="txt" href="#E14:32"><i><font color="Green" title="E30">A11</font></i></a>, <a class="ref" href="ami_1.html#T52" title="AMI_1:th.52">AMI_1:52</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_5_1_2"><b>suppose </b></a><a NAME="E1:32_5_1_2"/>
<font color="Maroon" title="c15">k</font> <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <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="c3">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</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> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:32_5_1_2"/><b>then </b>
 <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</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> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</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_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="ami_1.html#T13" title="AMI_1:th.13">AMI_1:13</a></i>;<br/><b>hence </b><a NAME="E3:32_5_1_2"/>
 <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <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="c3">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</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> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:32"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="scmpds_4.html#T75" title="SCMPDS_4:th.75">SCMPDS_4:75</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E16:32"/>
 <a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="ref" href="scmpds_6.html#D2" title="SCMPDS_6:def.2">SCMPDS_6:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c3">s</font></span><br/>

<a NAME="E17:32"/>
<font color="Maroon" title="c3">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</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_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span> is <a href="ami_1.html#V8" title="AMI_1:attr.8">halting</a>
 
<b>by </b><i><a class="txt" href="#E14:32"><i><font color="Green" title="E30">A11</font></i></a>, <a class="ref" href="ami_1.html#D20" title="AMI_1:def.20">AMI_1:def 20</a></i>;<br/>
<b>hence </b><a NAME="E18:32"/>
 <a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="ref" href="scmpds_6.html#D3" title="SCMPDS_6:def.3">SCMPDS_6:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
