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<span class="kw">let </span><font color="Maroon" title="c3">P</font> be   <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 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">  for <font color="Olive" title="b1">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><br/>  for <font color="Olive" title="b2">I</font> being   <a href="compos_1.html#NM4" title="COMPOS_1:NM.4">Program</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a><br/>  for <font color="Olive" title="b3">a</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a><br/>  for <font color="Olive" title="b4">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b1">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"><font color="Olive" title="b1">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">a</font></span>)</span>,<font color="Olive" title="b4">i</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/>(  <a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">i</font>,<font color="Olive" title="b2">I</font>) <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c3">P</font> &amp;  <a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">i</font>,<font color="Olive" title="b2">I</font>) <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c3">P</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">s</font> be   <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>; <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">  for <font color="Olive" title="b1">I</font> being   <a href="compos_1.html#NM4" title="COMPOS_1:NM.4">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#NM1" title="SCMPDS_2:NM.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="c4">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"><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#K5" title="NUMBERS:func.5">0</a>  holds <br/>(  <a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">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="c4">s</font>,<font color="Maroon" title="c3">P</font> &amp;  <a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">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="c4">s</font>,<font color="Maroon" title="c3">P</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">I</font> be   <a href="compos_1.html#NM4" title="COMPOS_1:NM.4">Program</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">  for <font color="Olive" title="b1">a</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.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="c4">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"><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#K5" title="NUMBERS:func.5">0</a>  holds <br/>(  <a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">i</font>,<font color="Maroon" title="c5">I</font>) <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> &amp;  <a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">i</font>,<font color="Maroon" title="c5">I</font>) <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">a</font> be   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</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">  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="c4">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"><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">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#K5" title="NUMBERS:func.5">0</a>  holds <br/>(  <a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c5">I</font>) <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> &amp;  <a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c5">I</font>) <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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"> ( <font color="Maroon" title="c4">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"><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c7">i</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies (  <a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>) <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> &amp;  <a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>) <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> ) )</span><br/>









<span class="kw">set </span><font color="Maroon" title="c8">d1</font> =  <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c7">i</font>);<br/>
<span class="kw">assume </span><a NAME="E1:31"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<font color="Maroon" title="c4">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"><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c7">i</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>) <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> &amp;  <a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>) <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c9">i1</font> = (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;=0_goto</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">i2</font> =  <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">WHL</font> =  <a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c12">pWHL</font> =  <a href="compos_1.html#K10" title="COMPOS_1:func.10">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">s3</font> =  <a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <font color="Maroon" title="c4">s</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">P3</font> = <font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="compos_1.html#K10" title="COMPOS_1:func.10">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">s4</font> =  <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="compos_1.html#K10" title="COMPOS_1:func.10">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></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="c4">s</font></span>)</span>,1);<br/>
<span class="kw">set </span><font color="Maroon" title="c16">P4</font> = <font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="compos_1.html#K10" title="COMPOS_1:func.10">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></span>)</span>;<br/>
<a NAME="E2:31"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
 <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.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="memstr_0.html#D11" onmouseover="rs('memstr_0/D11')" onmouseout="rh()">MEMSTR_0:def 11</a></span>;<br/>
<a NAME="E3:31"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
 not  <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c7">i</font>) <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="memstr_0.html#K7" title="MEMSTR_0:func.7">Start-At</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_4.html#T18" onmouseover="rs('scmpds_4/T18')" onmouseout="rh()">SCMPDS_4:18</a></span>;<br/>
<a NAME="E4:31"/>
 not <font color="Maroon" title="c6">a</font> <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="memstr_0.html#K7" title="MEMSTR_0:func.7">Start-At</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_4.html#T18" onmouseover="rs('scmpds_4/T18')" onmouseout="rh()">SCMPDS_4:18</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E19">A4</font></span>: <span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <font color="Maroon" title="c4">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="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c7">i</font>)</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="c4">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"><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c7">i</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T11" onmouseover="rs('funct_4/T11')" onmouseout="rh()">FUNCT_4:11</a></span>
<br/>.= 
<font color="Maroon" title="c4">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"><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c7">i</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="funct_4.html#T11" onmouseover="rs('funct_4/T11')" onmouseout="rh()">FUNCT_4:11</a></span>
;<br/>
<a NAME="E6:31"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
 <a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;=0_goto</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">';'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">I</font> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_4.html#T15" onmouseover="rs('scmpds_4/T15')" onmouseout="rh()">SCMPDS_4:15</a></span>;<br/>
 <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="compos_1.html#K10" title="COMPOS_1:func.10">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></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="c4">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"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="compos_1.html#K10" title="COMPOS_1:func.10">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></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"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="compos_1.html#K10" title="COMPOS_1:func.10">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></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="c4">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"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="compos_1.html#K10" title="COMPOS_1:func.10">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></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="c4">s</font></span>)</span>)

<br/>.= 
 <a href="extpro_1.html#K2" title="EXTPRO_1:func.2">Exec</a> (<span class="p1">(<span class="default">(<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>) <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;=0_goto</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</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="c4">s</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="scmpds_6.html#T11" onmouseover="rs('scmpds_6/T11')" onmouseout="rh()">SCMPDS_6:11</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E21">A6</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"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="compos_1.html#K10" title="COMPOS_1:func.10">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></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="c4">s</font></span>)</span>,1)</span>)</span> = 
 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">ICplusConst</a> (<span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="scmpds_2.html#T56" onmouseover="rs('scmpds_2/T56')" onmouseout="rh()">SCMPDS_2:56</a></span>
<br/>.= 
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="scmpds_6.html#T12" onmouseover="rs('scmpds_6/T12')" onmouseout="rh()">SCMPDS_6:12</a></span>
;<br/>
<a NAME="E9:31"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <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="scmpds_8.html#T17" target="_self" onmouseover="rs('scmpds_8/T17')" onmouseout="rh()">Th15</a></span>;<br/>
<a NAME="E10:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A8</font></span>: 
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <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#K10" title="COMPOS_1:func.10">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="compos_1.html#T64" onmouseover="rs('compos_1/T64')" onmouseout="rh()">COMPOS_1:64</a></span>;<br/>
<a NAME="E11:31"/>
 <a href="compos_1.html#K10" title="COMPOS_1:func.10">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="compos_1.html#K10" title="COMPOS_1:func.10">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></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/>
<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="compos_1.html#K10" title="COMPOS_1:func.10">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></span>)</span></span>)</span> <a href="compos_1.html#K3" title="COMPOS_1:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><a href="compos_1.html#K10" title="COMPOS_1:func.10">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="grfunc_1.html#T2" onmouseover="rs('grfunc_1/T2')" onmouseout="rh()">GRFUNC_1:2</a></span>
<br/>.= 
 <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:31"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="compos_1.html#T64" onmouseover="rs('compos_1/T64')" onmouseout="rh()">COMPOS_1:64</a></span>
;<br/>
<a NAME="E13:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A9</font></span>: 
 <a href="extpro_1.html#K3" title="EXTPRO_1:func.3">CurInstr</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="compos_1.html#K10" title="COMPOS_1:func.10">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></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"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="compos_1.html#K10" title="COMPOS_1:func.10">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></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="c4">s</font></span>)</span>,1)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:31"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="ref" href="pboole.html#T143" onmouseover="rs('pboole/T143')" onmouseout="rh()">PBOOLE:143</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="31_5"><span class="kw">now </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">  for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds   <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"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="compos_1.html#K10" title="COMPOS_1:func.10">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></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="c4">s</font></span>)</span>,<font color="Olive" title="b1">k</font>)</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#K10" title="COMPOS_1:func.10">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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="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"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="compos_1.html#K10" title="COMPOS_1:func.10">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></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="c4">s</font></span>)</span>,<font color="Olive">b<sub>1</sub></font>)</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#K10" title="COMPOS_1:func.10">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></span>)</span></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_5_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:31_5_1"/>
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">k</font> or <font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:31_5_1_1"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">k</font>
 ; <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="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"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="compos_1.html#K10" title="COMPOS_1:func.10">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></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="c4">s</font></span>)</span>,<font color="Olive">b<sub>1</sub></font>)</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#K10" title="COMPOS_1:func.10">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:31_5_1_1"/><span class="kw">then </span>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T7" onmouseover="rs('int_1/T7')" onmouseout="rh()">INT_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:31_5_1_1"/>
 <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"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="compos_1.html#K10" title="COMPOS_1:func.10">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></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="c4">s</font></span>)</span>,<font color="Maroon" title="c17">k</font>)</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#K10" title="COMPOS_1:func.10">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E8:31"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E13:31"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="ref" href="extpro_1.html#T5" onmouseover="rs('extpro_1/T5')" onmouseout="rh()">EXTPRO_1:5</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:31_5_1_2"/>
<font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="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"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="compos_1.html#K10" title="COMPOS_1:func.10">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></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="c4">s</font></span>)</span>,<font color="Olive">b<sub>1</sub></font>)</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#K10" title="COMPOS_1:func.10">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:31_5_1_2"/><span class="kw">then </span>
 <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="compos_1.html#K10" title="COMPOS_1:func.10">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></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="c4">s</font></span>)</span>,<font color="Maroon" title="c17">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <font color="Maroon" title="c4">s</font>
 ;<br/><span class="kw">hence </span><a NAME="E3:31_5_1_2"/>
 <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"><font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="compos_1.html#K10" title="COMPOS_1:func.10">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></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="c4">s</font></span>)</span>,<font color="Maroon" title="c17">k</font>)</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#K10" title="COMPOS_1:func.10">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="compos_1.html#T36" onmouseover="rs('compos_1/T36')" onmouseout="rh()">COMPOS_1:36</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E15:31"/>
 <a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>) <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_6.html#D2" onmouseover="rs('scmpds_6/D2')" onmouseout="rh()">SCMPDS_6:def 2</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">  <a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>) <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font></span><br/>

<a NAME="E16:31"/>
<font color="Maroon" title="c3">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="compos_1.html#K10" title="COMPOS_1:func.10">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span></span>)</span> <a href="extpro_1.html#R1" title="EXTPRO_1:pred.1">halts_on</a>  <a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <font color="Maroon" title="c4">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:31"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="ref" href="extpro_1.html#T29" onmouseover="rs('extpro_1/T29')" onmouseout="rh()">EXTPRO_1:29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:31"/>
 <a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c5">I</font>) <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_6.html#D3" onmouseover="rs('scmpds_6/D3')" onmouseout="rh()">SCMPDS_6:def 3</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>
