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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">sp</font>, <font color="Olive" title="b2">cv</font>, <font color="Olive" title="b3">result</font>, <font color="Olive" title="b4">pp</font>, <font color="Olive" title="b5">pD</font> being   <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>   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="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Olive" title="b1">sp</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">sp</font> &amp; <font color="Olive" title="b2">cv</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">result</font> &amp; <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="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Olive" title="b4">pp</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">pD</font> &amp; <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> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Olive" title="b1">sp</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">result</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">pp</font> &amp; <font color="Olive" title="b4">pp</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">pD</font> &amp; <font color="Olive" title="b5">pD</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <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="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Olive" title="b5">pD</font></span>)</span> holds <br/>(  <a href="scmpds_7.html#K2" title="SCMPDS_7:func.2">for-down</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Olive" title="b1">sp</font></span>)</span>,<font color="Olive" title="b2">cv</font>,1,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Olive" title="b1">sp</font></span>)</span>,<font color="Olive" title="b3">result</font>,<span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Olive" title="b5">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Olive" title="b4">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span> <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_7.html#K2" title="SCMPDS_7:func.2">for-down</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Olive" title="b1">sp</font></span>)</span>,<font color="Olive" title="b2">cv</font>,1,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Olive" title="b1">sp</font></span>)</span>,<font color="Olive" title="b3">result</font>,<span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Olive" title="b5">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Olive" title="b4">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c3">s</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">sp</font>, <font color="Maroon" title="c5">cv</font>, <font color="Maroon" title="c6">fr</font>, <font color="Maroon" title="c7">pp</font>, <font color="Maroon" title="c8">pD</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">sp</font> &amp; <font color="Maroon" title="c5">cv</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">fr</font> &amp; <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="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">pD</font> &amp; <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> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">fr</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">pp</font> &amp; <font color="Maroon" title="c7">pp</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">pD</font> &amp; <font color="Maroon" title="c8">pD</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <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="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span> implies (  <a href="scmpds_7.html#K2" title="SCMPDS_7:func.2">for-down</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c5">cv</font>,1,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span> <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_7.html#K2" title="SCMPDS_7:func.2">for-down</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c5">cv</font>,1,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c3">s</font> ) )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c9">BP</font> =  <a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">PD</font> =  <a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">PP</font> =  <a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:82"/><span class="lab"><font color="Green" title="E58">A1</font></span>: 
<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="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">sp</font>
 <span class="kw">and </span><br/><a NAME="E2:82"/><span class="lab"><font color="Green" title="E59">A2</font></span>: 
<font color="Maroon" title="c5">cv</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">fr</font>
 <span class="kw">and </span><br/><a NAME="E3:82"/><span class="lab"><font color="Green" title="E60">A3</font></span>: 
<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="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">pD</font>
 <span class="kw">and </span><br/><a NAME="E4:82"/><span class="lab"><font color="Green" title="E61">A4</font></span>: 
<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> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">fr</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">pp</font>
 <span class="kw">and </span><br/><a NAME="E5:82"/><span class="lab"><font color="Green" title="E62">A5</font></span>: 
<font color="Maroon" title="c7">pp</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">pD</font>
 <span class="kw">and </span><br/><a NAME="E6:82"/><span class="lab"><font color="Green" title="E63">A6</font></span>: 
<font color="Maroon" title="c8">pD</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <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="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="scmpds_7.html#K2" title="SCMPDS_7:func.2">for-down</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c5">cv</font>,1,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span> <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_7.html#K2" title="SCMPDS_7:func.2">for-down</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c5">cv</font>,1,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c3">s</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c12">i2</font> =  <a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">i3</font> =  <a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">I</font> = <span class="p1">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:82_1"/>
( <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> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">cv</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>  or <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> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">cv</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1"><span class="kw">suppose </span></a><a NAME="E1:82_1_1"/>
<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> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">cv</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="scmpds_7.html#K2" title="SCMPDS_7:func.2">for-down</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c5">cv</font>,1,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span> <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_7.html#K2" title="SCMPDS_7:func.2">for-down</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c5">cv</font>,1,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c3">s</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:82_1_1"/>
(  <a href="scmpds_7.html#K2" title="SCMPDS_7:func.2">for-down</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c5">cv</font>,1,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span> <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_7.html#K2" title="SCMPDS_7:func.2">for-down</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c5">cv</font>,1,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c3">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_7.html#T63" target="_self" title="SCMPDS_7:th.63">Th63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_2"><span class="kw">suppose </span></a><a NAME="E1:82_1_2"/><span class="lab"><font color="Green" title="E64">A7</font></span>: 
<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> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">cv</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="scmpds_7.html#K2" title="SCMPDS_7:func.2">for-down</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c5">cv</font>,1,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span> <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_7.html#K2" title="SCMPDS_7:func.2">for-down</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c5">cv</font>,1,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c3">s</font> )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c15">n</font> = <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="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span> as    <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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/><a NAME="E3:82_1_2"/>
<font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">cv</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">sp</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E4:82_1_2"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">cv</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">sp</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E5:82_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A9</font></span>: 
 <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> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">cv</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E6:82_1_2"/><span class="lab"><font color="Green" title="E66">A10</font></span>: 
<font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">fr</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">cv</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><div><span class="lab"><font color="Green" title="E67">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c16">Dv</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> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">cv</font>;<br/><span class="kw">let </span><font color="Maroon" title="c17">t</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">x</font> being   <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> holds <br/><font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">x</font> ) &amp; <font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span> implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span>,<font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span>,<font color="Maroon" title="c17">t</font></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> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">cv</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t</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> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">cv</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c17">t</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c17">t</font> &amp; ( for <font color="Olive" title="b1">y</font> being   <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span>,<font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">y</font> ) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:82_1_2_1"/><span class="lab"><font color="Green" title="E68">A12</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> holds <br/><font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">x</font>
 <span class="kw">and </span><br/><a NAME="E2:82_1_2_1"/><span class="lab"><font color="Green" title="E69">A13</font></span>: 
<font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span>,<font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span>,<font color="Maroon" title="c17">t</font></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> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">cv</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t</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> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">cv</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c17">t</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c17">t</font> &amp; ( for <font color="Olive" title="b1">y</font> being   <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span>,<font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">y</font> ) )</span><br/><span class="kw">set </span><font color="Maroon" title="c18">t0</font> =  <a href="scmpds_5.html#K1" title="SCMPDS_5:func.1">Initialized</a> <font color="Maroon" title="c17">t</font>;<br/><span class="kw">set </span><font color="Maroon" title="c19">t1</font> =  <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_5.html#K1" title="SCMPDS_5:func.1">Initialized</a> <font color="Maroon" title="c17">t</font></span>)</span>;<br/><span class="lab"><font color="Green" title="E70">A14</font></span>:  <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_5.html#K1" title="SCMPDS_5:func.1">Initialized</a> <font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">fr</font> = 
 <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c15">n</font>,<font color="Maroon" title="c6">fr</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1_2_1"><span class="lab"><font color="Green" title="E69">A13</font></span></a>, <a class="ref" href="scmpds_5.html#T40" title="SCMPDS_5:th.40">SCMPDS_5:40</a></span>
<br/>.= 
 <a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">fr</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmp_gcd.html#T5" title="SCMP_GCD:th.5">SCMP_GCD:5</a></span>
;<br/><a NAME="E4:82_1_2_1"/><span class="kw">then </span>
 <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_5.html#K1" title="SCMPDS_5:func.1">Initialized</a> <font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">fr</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A15</font></span>: <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><a href="scmpds_5.html#K1" title="SCMPDS_5:func.1">Initialized</a> <font color="Maroon" title="c17">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="scmpds_5.html#K1" title="SCMPDS_5:func.1">Initialized</a> <font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_2.html#T61" title="SCMPDS_2:th.61">SCMPDS_2:61</a></span>
<br/>.= 
<font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_5.html#T40" title="SCMPDS_5:th.40">SCMPDS_5:40</a></span>
;<br/><a NAME="E6:82_1_2_1"/>
<font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">fr</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">sp</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E7:82_1_2_1"/><span class="kw">then </span>
 <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_5.html#K1" title="SCMPDS_5:func.1">Initialized</a> <font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">fr</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82_1_2_1"><span class="lab"><font color="Green" title="E70">A14</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A16</font></span>: <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><a href="scmpds_5.html#K1" title="SCMPDS_5:func.1">Initialized</a> <font color="Maroon" title="c17">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="scmpds_5.html#K1" title="SCMPDS_5:func.1">Initialized</a> <font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_2.html#T61" title="SCMPDS_2:th.61">SCMPDS_2:61</a></span>
<br/>.= 
<font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_5.html#T40" title="SCMPDS_5:th.40">SCMPDS_5:40</a></span>
;<br/><a NAME="E9:82_1_2_1"/>
 <a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E10:82_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><a href="scmpds_5.html#K1" title="SCMPDS_5:func.1">Initialized</a> <font color="Maroon" title="c17">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">pD</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="txt" href="#E1:82_1_2_1"><span class="lab"><font color="Green" title="E68">A12</font></span></a>, <a class="txt" href="#E5:82_1_2_1"><span class="lab"><font color="Green" title="E71">A15</font></span></a></span>;<br/><a NAME="E11:82_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A17</font></span>: 
 <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p3">(<span class="default"><a href="scmpds_5.html#K1" title="SCMPDS_5:func.1">Initialized</a> <font color="Maroon" title="c17">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">pD</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmp_gcd.html#T5" title="SCMP_GCD:th.5">SCMP_GCD:5</a></span>;<br/><a NAME="E12:82_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A18</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_5.html#K1" title="SCMPDS_5:func.1">Initialized</a> <font color="Maroon" title="c17">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">pD</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E13:82_1_2_1"/>
<font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">fr</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E14:82_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">sp</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">fr</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E15:82_1_2_1"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_5.html#K1" title="SCMPDS_5:func.1">Initialized</a> <font color="Maroon" title="c17">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">sp</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="txt" href="#E5:82"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="txt" href="#E12:82_1_2_1"><span class="lab"><font color="Green" title="E74">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E16:82_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A19</font></span>: 
 <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p3">(<span class="default"><a href="scmpds_5.html#K1" title="SCMPDS_5:func.1">Initialized</a> <font color="Maroon" title="c17">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E17:82_1_2_1"/>
 <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> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">cv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">cv</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmp_gcd.html#T5" title="SCMP_GCD:th.5">SCMP_GCD:5</a></span>;<br/><a NAME="E18:82_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A20</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><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> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">cv</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">cv</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E19:82_1_2_1"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_5.html#K1" title="SCMPDS_5:func.1">Initialized</a> <font color="Maroon" title="c17">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><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> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">cv</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="txt" href="#E5:82"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="txt" href="#E6:82_1_2"><span class="lab"><font color="Green" title="E66">A10</font></span></a>, <a class="txt" href="#E12:82_1_2_1"><span class="lab"><font color="Green" title="E74">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E20:82_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A21</font></span>: 
 <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p3">(<span class="default"><a href="scmpds_5.html#K1" title="SCMPDS_5:func.1">Initialized</a> <font color="Maroon" title="c17">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">cv</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E21:82_1_2_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_5.html#K1" title="SCMPDS_5:func.1">Initialized</a> <font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">fr</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">fr</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82_1_2_1"><span class="lab"><font color="Green" title="E70">A14</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E22:82_1_2_1"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_5.html#K1" title="SCMPDS_5:func.1">Initialized</a> <font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">fr</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><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> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">cv</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E18:82_1_2_1"><span class="lab"><font color="Green" title="E76">A20</font></span></a></span>;<br/><a NAME="E23:82_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A22</font></span>: 
 <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_5.html#K1" title="SCMPDS_5:func.1">Initialized</a> <font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">fr</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">cv</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><span class="kw">thus </span><span class="lab"><font color="Green" title="E79">A23</font></span>: <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span>,<font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p3">(<span class="default"><a href="scmpds_5.html#K1" title="SCMPDS_5:func.1">Initialized</a> <font color="Maroon" title="c17">t</font></span>)</span></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_5.html#T47" title="SCMPDS_5:th.47">SCMPDS_5:47</a></span>
<br/>.= 
<font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:82_1_2_1"><span class="lab"><font color="Green" title="E72">A16</font></span></a>, <a class="txt" href="#E16:82_1_2_1"><span class="lab"><font color="Green" title="E75">A19</font></span></a>, <a class="ref" href="scmpds_2.html#T60" title="SCMPDS_2:th.60">SCMPDS_2:60</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span>,<font color="Maroon" title="c17">t</font></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> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">cv</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t</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> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">cv</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c17">t</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c17">t</font> &amp; ( for <font color="Olive" title="b1">y</font> being   <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span>,<font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">y</font> ) )</span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span>,<font color="Maroon" title="c17">t</font></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> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">cv</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p3">(<span class="default"><a href="scmpds_5.html#K1" title="SCMPDS_5:func.1">Initialized</a> <font color="Maroon" title="c17">t</font></span>)</span></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">cv</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_5.html#T47" title="SCMPDS_5:th.47">SCMPDS_5:47</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><a href="scmpds_5.html#K1" title="SCMPDS_5:func.1">Initialized</a> <font color="Maroon" title="c17">t</font></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> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">cv</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:82_1_2_1"><span class="lab"><font color="Green" title="E77">A21</font></span></a>, <a class="ref" href="scmpds_2.html#T60" title="SCMPDS_2:th.60">SCMPDS_2:60</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_5.html#K1" title="SCMPDS_5:func.1">Initialized</a> <font color="Maroon" title="c17">t</font></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> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">cv</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:82_1_2_1"><span class="lab"><font color="Green" title="E78">A22</font></span></a>, <a class="ref" href="scmpds_2.html#T61" title="SCMPDS_2:th.61">SCMPDS_2:61</a></span>
<br/>.= 
<font color="Maroon" title="c17">t</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> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">cv</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_5.html#T40" title="SCMPDS_5:th.40">SCMPDS_5:40</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c17">t</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c17">t</font> &amp; ( for <font color="Olive" title="b1">y</font> being   <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span>,<font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">y</font> ) )</span><br/><span class="kw">thus </span><a NAME="E26:82_1_2_1"/>
( <span class="p1">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c17">t</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c17">t</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_6.html#T34" title="SCMPDS_6:th.34">SCMPDS_6:34</a>, <a class="ref" href="scmpds_6.html#T35" title="SCMPDS_6:th.35">SCMPDS_6:35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">y</font> being   <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span>,<font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">y</font></span><br/><span class="lab"><font color="Green" title="E80">A24</font></span>: <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span>,<font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p2">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p3">(<span class="default"><a href="scmpds_5.html#K1" title="SCMPDS_5:func.1">Initialized</a> <font color="Maroon" title="c17">t</font></span>)</span></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_5.html#T47" title="SCMPDS_5:th.47">SCMPDS_5:47</a></span>
<br/>.= 
<font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="txt" href="#E6:82"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="txt" href="#E5:82_1_2_1"><span class="lab"><font color="Green" title="E71">A15</font></span></a>, <a class="txt" href="#E11:82_1_2_1"><span class="lab"><font color="Green" title="E73">A17</font></span></a>, <a class="ref" href="scmpds_2.html#T60" title="SCMPDS_2:th.60">SCMPDS_2:60</a></span>
;<br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c20">y</font> be    <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> implies <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span>,<font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:82_1_2_1_7"/><span class="lab"><font color="Green" title="E81">A25</font></span>: 
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span>,<font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_2_1_7_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:82_1_2_1_7_1"/>
( <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font> or <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82_1_2_1_7"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_2_1_7_1_1"><span class="kw">suppose </span></a><a NAME="E1:82_1_2_1_7_1_1"/>
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span>,<font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:82_1_2_1_7_1_1"/>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span>,<font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:82_1_2_1"><span class="lab"><font color="Green" title="E79">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_2_1_7_1_2"><span class="kw">suppose </span></a><a NAME="E1:82_1_2_1_7_1_2"/>
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span>,<font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:82_1_2_1_7_1_2"/>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span>,<font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:82_1_2_1"><span class="lab"><font color="Green" title="E80">A24</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E8:82_1_2"/>
<font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">cv</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">pp</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E4:82"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E9:82_1_2"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">cv</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">pp</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E10:82_1_2"/><span class="kw">then </span>
 <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> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">cv</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E11:82_1_2"/><span class="kw">then </span>
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> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">cv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82_1_2"><span class="lab"><font color="Green" title="E65">A9</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E12:82_1_2"/>
(  <a href="scmpds_7.html#K2" title="SCMPDS_7:func.2">for-down</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c5">cv</font>,1,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span> <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_7.html#K2" title="SCMPDS_7:func.2">for-down</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c5">cv</font>,1,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_2.html#K13" title="SCMPDS_2:func.13">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c4">sp</font></span>)</span>,<font color="Maroon" title="c6">fr</font>,<span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c8">pD</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c7">pp</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c3">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82_1_2"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="txt" href="#E5:82_1_2"><span class="lab"><font color="Green" title="E65">A9</font></span></a>, <a class="txt" href="#E7:82_1_2"><span class="lab"><font color="Green" title="E67">A11</font></span></a>, <a class="ref" href="scmpds_7.html#T67" target="_self" title="SCMPDS_7:th.67">Th67</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
