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

<b>set </b><font color="Maroon">c<sub>17</sub></font> =  <a href="scmp_gcd.html#K2">GBP</a> ;<br/>
<b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/><b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:17"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>18</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>18</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>18</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 3</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>20</sub></font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/><b>let </b><font color="Maroon">c<sub>21</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:17_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 3</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>20</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 3</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><b>hence </b><a NAME="E2:17_1"/>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmp_gcd.html#K2">GBP</a> ,4 <a href="scmpds_2.html#K17">:=</a> <a href="scmp_gcd.html#K2">GBP</a> ,2</span>)</span> <a href="scmpds_4.html#K6">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K13">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,<a href="scmp_gcd.html#K2">GBP</a> ,1</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K2">GBP</a> ,1 <a href="scmpds_2.html#K17">:=</a> <a href="scmp_gcd.html#K2">GBP</a> ,4</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a>  <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="scpinvar.html#E17"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>thus </b><a NAME="E3:17_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K2">GBP</a> ,4 <a href="scmpds_2.html#K17">:=</a> <a href="scmp_gcd.html#K2">GBP</a> ,2</span>)</span> <a href="scmpds_4.html#K6">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K13">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,<a href="scmp_gcd.html#K2">GBP</a> ,1</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K2">GBP</a> ,1 <a href="scmpds_2.html#K17">:=</a> <a href="scmp_gcd.html#K2">GBP</a> ,4</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_6.html#R1">is_closed_on</a> <font color="Maroon">c<sub>20</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K2">GBP</a> ,4 <a href="scmpds_2.html#K17">:=</a> <a href="scmp_gcd.html#K2">GBP</a> ,2</span>)</span> <a href="scmpds_4.html#K6">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K13">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,<a href="scmp_gcd.html#K2">GBP</a> ,1</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K2">GBP</a> ,1 <a href="scmpds_2.html#K17">:=</a> <a href="scmp_gcd.html#K2">GBP</a> ,4</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_6.html#R2">is_halting_on</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="ref" href="scmpds_6.html#T34">SCMPDS_6:34</a>, <a class="ref" href="scmpds_6.html#T35">SCMPDS_6:35</a></i>;<br/><b>thus </b><a NAME="E4:17_1"/>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmp_gcd.html#K2">GBP</a> ,4 <a href="scmpds_2.html#K17">:=</a> <a href="scmp_gcd.html#K2">GBP</a> ,2</span>)</span> <a href="scmpds_4.html#K6">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K13">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,<a href="scmp_gcd.html#K2">GBP</a> ,1</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K2">GBP</a> ,1 <a href="scmpds_2.html#K17">:=</a> <a href="scmp_gcd.html#K2">GBP</a> ,4</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 3</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 3</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="scpinvar.html#E1:17_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="scpinvar.html#E17"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>thus </b><a NAME="E5:17_1"/>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmp_gcd.html#K2">GBP</a> ,4 <a href="scmpds_2.html#K17">:=</a> <a href="scmp_gcd.html#K2">GBP</a> ,2</span>)</span> <a href="scmpds_4.html#K6">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K13">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,<a href="scmp_gcd.html#K2">GBP</a> ,1</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K2">GBP</a> ,1 <a href="scmpds_2.html#K17">:=</a> <a href="scmp_gcd.html#K2">GBP</a> ,4</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="scpinvar.html#E1:17_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="scpinvar.html#E17"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmp_gcd.html#K2">GBP</a> ,4 <a href="scmpds_2.html#K17">:=</a> <a href="scmp_gcd.html#K2">GBP</a> ,2</span>)</span> <a href="scmpds_4.html#K6">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K13">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,<a href="scmp_gcd.html#K2">GBP</a> ,1</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K2">GBP</a> ,1 <a href="scmpds_2.html#K17">:=</a> <a href="scmp_gcd.html#K2">GBP</a> ,4</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpinvar.html#E1:17_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="scpinvar.html#E17"><i><font color="Green">Lemma11</font></i></a></i>
<br/>.= 
 <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="scpinvar.html#E1:17_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="pre_ff.html#T1">PRE_FF:1</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:17"/>
( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="scmp_gcd.html#K2">GBP</a> ,4 <a href="scmpds_2.html#K17">:=</a> <a href="scmp_gcd.html#K2">GBP</a> ,2</span>)</span> <a href="scmpds_4.html#K6">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K13">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,<a href="scmp_gcd.html#K2">GBP</a> ,1</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2">GBP</a> ,1 <a href="scmpds_2.html#K17">:=</a> <a href="scmp_gcd.html#K2">GBP</a> ,4</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <font color="Maroon">c<sub>19</sub></font> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3">while&gt;0</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="scmp_gcd.html#K2">GBP</a> ,4 <a href="scmpds_2.html#K17">:=</a> <a href="scmp_gcd.html#K2">GBP</a> ,2</span>)</span> <a href="scmpds_4.html#K6">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K13">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,<a href="scmp_gcd.html#K2">GBP</a> ,1</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2">GBP</a> ,1 <a href="scmpds_2.html#K17">:=</a> <a href="scmp_gcd.html#K2">GBP</a> ,4</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="scmpds_8.html#K3">while&gt;0</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2">GBP</a> ,4 <a href="scmpds_2.html#K17">:=</a> <a href="scmp_gcd.html#K2">GBP</a> ,2</span>)</span> <a href="scmpds_4.html#K6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K13">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,<a href="scmp_gcd.html#K2">GBP</a> ,1</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K2">GBP</a> ,1 <a href="scmpds_2.html#K17">:=</a> <a href="scmp_gcd.html#K2">GBP</a> ,4</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_6.html#R1">is_closed_on</a> <font color="Maroon">c<sub>18</sub></font> &amp;  <a href="scmpds_8.html#K3">while&gt;0</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2">GBP</a> ,4 <a href="scmpds_2.html#K17">:=</a> <a href="scmp_gcd.html#K2">GBP</a> ,2</span>)</span> <a href="scmpds_4.html#K6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K13">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,<a href="scmp_gcd.html#K2">GBP</a> ,1</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K2">GBP</a> ,1 <a href="scmpds_2.html#K17">:=</a> <a href="scmp_gcd.html#K2">GBP</a> ,4</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_6.html#R2">is_halting_on</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="txt" href="scpinvar.html#E1:17"><i><font color="Green">E14</font></i></a>, <a class="txt" href="scpinvar.html#E18"><i><font color="Green">Lemma12</font></i></a>, <a class="ref" href="scpinvar.html#T10" target="_self">Th10</a></i>;<br/>


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