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<b>let </b><font color="Maroon" title="c6">s</font> be   <a href="ami_1.html#NM4" title="AMI_1:NM.4">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ;<br/><b>let </b><font color="Maroon" title="c7">I</font> be  <a href="scmpds_4.html#V3" title="SCMPDS_4:attr.3">shiftable</a> <a href="scmpds_5.html#V3" title="SCMPDS_5:attr.3">No-StopCode</a> <a href="scmpds_4.html#NM1" title="SCMPDS_4:NM.1">Program-block</a>;<br/><b>let </b><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font>, <font color="Maroon" title="c10">c</font> be    <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a> ;<br/><b>let </b><font color="Maroon" title="c11">n</font>, <font color="Maroon" title="c12">i</font>, <font color="Maroon" title="c13">p0</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c14">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ;<br/>









<b>assume </b><a NAME="E1:6"/><i><font color="Green" title="E5">A1</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">I</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<b>assume </b><a NAME="E2:6"/><i><font color="Green" title="E6">A2</font></i>: 
( <font color="Maroon" title="c14">f</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c13">p0</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n</font> &amp; <font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c6">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="c12">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c11">n</font> &amp; <font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/>

<b>assume </b><a NAME="E3:6"/><i><font color="Green" title="E7">A3</font></i>: 
for <font color="Olive" title="b1">t</font> being  <a href="ami_1.html#NM4" title="AMI_1:NM.4">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>   st  ex <font color="Olive" title="b2">g</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  st <br/>( <font color="Olive" title="b2">g</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c13">p0</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</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="c12">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">n</font> &amp; <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <font color="Olive" title="b2">g</font> &amp; <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">g</font></span>)</span> ) &amp; <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">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="c12">i</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">p0</font> holds <br/><font color="Olive" title="b1">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="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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="b2">i</font></span>)</span> ) holds <br/>( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c7">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Olive" title="b1">t</font> &amp; <font color="Maroon" title="c7">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Olive" title="b1">t</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<font color="Olive" title="b1">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="c12">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</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="c12">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp;  ex <font color="Olive" title="b2">g</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  st <br/>( <font color="Olive" title="b2">g</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c13">p0</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">t</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="c12">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">g</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <font color="Olive" title="b2">g</font> ) &amp; ( for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">p0</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<font color="Olive" title="b1">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="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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="b2">i</font></span>)</span> ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ami_1.html#NM4" title="AMI_1:NM.4">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ] <b>means </b>( ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">p0</font> holds <br/>$1 <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">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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">i</font></span>)</span> ) &amp;  ex <font color="Olive" title="b1">g</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  st <br/>( <font color="Olive" title="b1">g</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c13">p0</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">$1 <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="c12">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">n</font> &amp; $1 <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <font color="Olive" title="b1">g</font> &amp; $1 <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="c12">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; $1 <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> ) );<br/>
<b>set </b><font color="Maroon" title="c15">da</font> =  <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<font color="Maroon" title="c12">i</font>;<br/>
<b>consider </b><font color="Maroon" title="c16">ff</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> the <a href="ami_1.html#U4" title="AMI_1:sel.4">Object-Kind</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:6"/><i><font color="Green" title="E8">A4</font></i>: 
for <font color="Olive" title="b1">t</font> being  <a href="ami_1.html#NM4" title="AMI_1:NM.4">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>  holds <br/> ( ( <font color="Olive" title="b1">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="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<font color="Maroon" title="c12">i</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c16">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b1">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="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<font color="Maroon" title="c12">i</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c16">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<font color="Maroon" title="c12">i</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="scpinvar.html#T7" target="_self" title="SCPINVAR:th.7">Th7</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ami_1.html#NM4" title="AMI_1:NM.4">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  = <font color="Maroon" title="c16">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1;<br/>
<div><i><font color="Green" title="E9">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">t</font> be   <a href="ami_1.html#NM4" title="AMI_1:NM.4">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ;<br/><b>set </b><font color="Maroon" title="c18">dt</font> =  <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c17">t</font>;<br/><b>assume </b><a NAME="E1:6_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c17">t</font>]
 ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:6_1_1"/><i><font color="Green" title="E10">A6</font></i>: 
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c17">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>assume </b><a NAME="E2:6_1_1"/><i><font color="Green" title="E11">A7</font></i>: 
<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="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<font color="Maroon" title="c12">i</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E3:6_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</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="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<font color="Maroon" title="c12">i</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></i>;<br/><a NAME="E4:6_1_1"/><b>then </b>
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c17">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</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="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<font color="Maroon" title="c12">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scpinvar.html#E5:6"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E5:6_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="scpinvar.html#E1:6_1_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="scpinvar.html#E2:6_1_1"><i><font color="Green" title="E11">A7</font></i></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E3:6_1"/>
<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="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<font color="Maroon" title="c12">i</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E4:6_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</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="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<font color="Maroon" title="c12">i</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></i>;<br/><b>hence </b><a NAME="E5:6_1"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c17">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scpinvar.html#E5:6"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:6"/><i><font color="Green" title="E10">A8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c6">s</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c17">Ds</font> =  <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c6">s</font>;<br/>
<b>thus </b><a NAME="E1:6_2"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">p0</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c6">s</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="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></i>;<br/>

<b>consider </b><font color="Maroon" title="c18">h</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E3:6_2"/><i><font color="Green" title="E11">A9</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c18">h</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c13">p0</font> )
 <b>by </b><i><a class="ref" href="scpisort.html#T3" title="SCPISORT:th.3">SCPISORT:3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c18">h</font>
;<br/>

<b>thus </b><a NAME="E4:6_2"/>
<font color="Maroon" title="c18">h</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c13">p0</font>
 <b>by </b><i><a class="txt" href="scpinvar.html#E3:6_2"><i><font color="Green" title="E11">A9</font></i></a></i>;<br/>

<b>thus </b> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">h</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c12">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">n</font>
<b>by </b><i><a class="txt" href="scpinvar.html#E2:6"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="scpinvar.html#E3:6_2"><i><font color="Green" title="E11">A9</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c6">s</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="c12">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">n</font>
<b>by </b><i><a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></i>
;<br/>

<a NAME="E6:6_2"/>
<font color="Maroon" title="c18">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="txt" href="scpinvar.html#E3:6_2"><i><font color="Green" title="E11">A9</font></i></a>, <a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></i>;<br/>
<b>hence </b><a NAME="E7:6_2"/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <font color="Maroon" title="c18">h</font>
 <b>by </b><i><a class="txt" href="scpinvar.html#E2:6"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="rvsum_1.html#T102" title="RVSUM_1:th.102">RVSUM_1:102</a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></i>;<br/>

<b>thus </b><a NAME="E8:6_2"/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c6">s</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="c12">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scpinvar.html#E2:6"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></i>;<br/>

<b>thus </b><a NAME="E9:6_2"/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">h</font></span>)</span>
 <b>by </b><i><a class="txt" href="scpinvar.html#E2:6"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="scpinvar.html#E3:6_2"><i><font color="Green" title="E11">A9</font></i></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E11">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">t</font> be   <a href="ami_1.html#NM4" title="AMI_1:NM.4">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ;<br/><b>set </b><font color="Maroon" title="c18">Dt</font> =  <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c17">t</font>;<br/><b>assume </b><a NAME="E1:6_3"/><i><font color="Green" title="E12">A11</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c17">t</font>] &amp; <font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c8">a</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="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<font color="Maroon" title="c12">i</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><div><i><font color="Green" title="E13">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:6_3_1"/><i><font color="Green" title="E14">A13</font></i>: 
<font color="Maroon" title="c19">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">p0</font>
 ;<br/><b>thus </b><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="c19">i</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</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="c19">i</font></span>)</span>
<b>by </b><i><a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></i>
<br/>.= 
<font color="Maroon" title="c6">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="c19">i</font></span>)</span>
<b>by </b><i><a class="txt" href="scpinvar.html#E1:6_3"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="scpinvar.html#E1:6_3_1"><i><font color="Green" title="E14">A13</font></i></a></i>
;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c19">h</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E4:6_3"/><i><font color="Green" title="E14">A14</font></i>: 
( <font color="Maroon" title="c19">h</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c13">p0</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</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="c12">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">n</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <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="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <font color="Maroon" title="c19">h</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <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="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">h</font></span>)</span> )
 <b>by </b><i><a class="txt" href="scpinvar.html#E1:6_3"><i><font color="Green" title="E12">A11</font></i></a></i>;<br/><a NAME="E5:6_3"/><i><font color="Green" title="E15">A15</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">t</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="c12">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">n</font> &amp; <font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <font color="Maroon" title="c19">h</font> &amp; <font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">h</font></span>)</span> )
 <b>by </b><i><a class="txt" href="scpinvar.html#E4:6_3"><i><font color="Green" title="E14">A14</font></i></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></i>;<br/><a NAME="E6:6_3"/><i><font color="Green" title="E16">A16</font></i>: 
 <a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<font color="Maroon" title="c12">i</font>
 <b>by </b><i><a class="txt" href="scpinvar.html#E2:6"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="scmp_gcd.html#T5" title="SCMP_GCD:th.5">SCMP_GCD:5</a></i>;<br/><b>hence </b><a NAME="E7:6_3"/>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<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="c8">a</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="c8">a</font>
 <b>by </b><i><a class="txt" href="scpinvar.html#E2:6"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="scpinvar.html#E3:6"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="scpinvar.html#E1:6_3"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="scpinvar.html#E2:6_3"><i><font color="Green" title="E13">A12</font></i></a>, <a class="txt" href="scpinvar.html#E4:6_3"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="scpinvar.html#E5:6_3"><i><font color="Green" title="E15">A15</font></i></a></i>;<br/><b>thus </b><a NAME="E8:6_3"/>
( <font color="Maroon" title="c7">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c17">t</font> &amp; <font color="Maroon" title="c7">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c17">t</font> )
 <b>by </b><i><a class="txt" href="scpinvar.html#E2:6"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="scpinvar.html#E3:6"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="scpinvar.html#E1:6_3"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="scpinvar.html#E2:6_3"><i><font color="Green" title="E13">A12</font></i></a>, <a class="txt" href="scpinvar.html#E4:6_3"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="scpinvar.html#E5:6_3"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="scpinvar.html#E6:6_3"><i><font color="Green" title="E16">A16</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c20">It</font> =  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c17">t</font>;<br/><a NAME="E9:6_3"/><i><font color="Green" title="E17">A17</font></i>: 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<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="c12">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">t</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="c12">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="scpinvar.html#E2:6"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="scpinvar.html#E3:6"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="scpinvar.html#E1:6_3"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="scpinvar.html#E2:6_3"><i><font color="Green" title="E13">A12</font></i></a>, <a class="txt" href="scpinvar.html#E4:6_3"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="scpinvar.html#E5:6_3"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="scpinvar.html#E6:6_3"><i><font color="Green" title="E16">A16</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c21">Dit</font> =  <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c17">t</font></span>)</span>;<br/><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_3_2_1"/>
( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<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="c12">i</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<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="c12">i</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="6_3_2_1_1"><b>suppose </b></a><a NAME="E1:6_3_2_1_1"/>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<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="c12">i</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:6_3_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<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="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<font color="Maroon" title="c12">i</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scpinvar.html#E6:6_3"><i><font color="Green" title="E16">A16</font></i></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></i>;<br/><a NAME="E3:6_3_2_1_1"/><b>then </b><i><font color="Green" title="E18">A18</font></i>: 
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c17">t</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scpinvar.html#E5:6"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/><a NAME="E4:6_3_2_1_1"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c17">t</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scpinvar.html#E6:6"><i><font color="Green" title="E9">A5</font></i></a>, <a class="txt" href="scpinvar.html#E1:6_3"><i><font color="Green" title="E12">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E5:6_3_2_1_1"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c17">t</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c17">t</font>)
 <b>by </b><i><a class="txt" href="scpinvar.html#E3:6_3_2_1_1"><i><font color="Green" title="E18">A18</font></i></a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_3_2_1_2"><b>suppose </b></a><a NAME="E1:6_3_2_1_2"/>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<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="c12">i</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:6_3_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<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="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<font color="Maroon" title="c12">i</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scpinvar.html#E6:6_3"><i><font color="Green" title="E16">A16</font></i></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></i>;<br/><b>then </b><i><font color="Green" title="E18">A19</font></i>: <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c17">t</font></span>)</span>) = 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<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="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<font color="Maroon" title="c12">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpinvar.html#E5:6"><i><font color="Green" title="E8">A4</font></i></a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">t</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="c12">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="scpinvar.html#E6:6_3"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="scpinvar.html#E9:6_3"><i><font color="Green" title="E17">A17</font></i></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">t</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="c12">i</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1

;<br/><a NAME="E4:6_3_2_1_2"/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</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="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<font color="Maroon" title="c12">i</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scpinvar.html#E1:6_3"><i><font color="Green" title="E12">A11</font></i></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></i>;<br/><b>then </b><font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c17">t</font>) = 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</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="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<font color="Maroon" title="c12">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpinvar.html#E5:6"><i><font color="Green" title="E8">A4</font></i></a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">t</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="c12">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpinvar.html#E6:6_3"><i><font color="Green" title="E16">A16</font></i></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></i>
;<br/><b>hence </b><a NAME="E6:6_3_2_1_2"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c17">t</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c17">t</font>)
 <b>by </b><i><a class="txt" href="scpinvar.html#E3:6_3_2_1_2"><i><font color="Green" title="E18">A19</font></i></a>, <a class="ref" href="xreal_1.html#T148" title="XREAL_1:th.148">XREAL_1:148</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div><b>consider </b><font color="Maroon" title="c22">g</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E12:6_3"/><i><font color="Green" title="E18">A20</font></i>: 
( <font color="Maroon" title="c22">g</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c13">p0</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">t</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="c12">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<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="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<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="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <font color="Maroon" title="c22">g</font> )
 <b>by </b><i><a class="txt" href="scpinvar.html#E2:6"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="scpinvar.html#E3:6"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="scpinvar.html#E1:6_3"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="scpinvar.html#E2:6_3"><i><font color="Green" title="E13">A12</font></i></a>, <a class="txt" href="scpinvar.html#E4:6_3"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="scpinvar.html#E5:6_3"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="scpinvar.html#E6:6_3"><i><font color="Green" title="E16">A16</font></i></a></i>;<br/><b>thus </b><a NAME="E13:6_3"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c17">t</font></span>)</span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_3"><b>proof </b></a><div class="add">

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c23">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:6_3_3_1"/><i><font color="Green" title="E19">A21</font></i>: 
<font color="Maroon" title="c23">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">p0</font>
 ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<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="c23">i</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<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="c23">i</font></span>)</span>
<b>by </b><i><a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></i>
<br/>.= 
<font color="Maroon" title="c6">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="c23">i</font></span>)</span>
<b>by </b><i><a class="txt" href="scpinvar.html#E2:6"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="scpinvar.html#E3:6"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="scpinvar.html#E1:6_3"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="scpinvar.html#E2:6_3"><i><font color="Green" title="E13">A12</font></i></a>, <a class="txt" href="scpinvar.html#E4:6_3"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="scpinvar.html#E5:6_3"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="scpinvar.html#E6:6_3"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="scpinvar.html#E1:6_3_3_1"><i><font color="Green" title="E19">A21</font></i></a></i>
;<br/></div>
<b>end;</b></div>

<b>take </b>
<font color="Maroon" title="c22">g</font>
;<br/>

<b>thus </b><a NAME="E2:6_3_3"/>
<font color="Maroon" title="c22">g</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c13">p0</font>
 <b>by </b><i><a class="txt" href="scpinvar.html#E12:6_3"><i><font color="Green" title="E18">A20</font></i></a></i>;<br/>

<b>thus </b> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<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="c12">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">n</font>
<b>by </b><i><a class="txt" href="scpinvar.html#E9:6_3"><i><font color="Green" title="E17">A17</font></i></a>, <a class="txt" href="scpinvar.html#E12:6_3"><i><font color="Green" title="E18">A20</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<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="c12">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">n</font>
<b>by </b><i><a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></i>
;<br/>

<b>thus </b><a NAME="E4:6_3_3"/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c17">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <font color="Maroon" title="c22">g</font>
 <b>by </b><i><a class="txt" href="scpinvar.html#E12:6_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></i>;<br/>

<a NAME="E5:6_3_3"/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<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="c12">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">t</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="c12">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="scpinvar.html#E9:6_3"><i><font color="Green" title="E17">A17</font></i></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></i>;<br/>
<b>hence </b><a NAME="E6:6_3_3"/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<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="c12">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scpinvar.html#E1:6_3"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="scpinvar.html#E6:6_3"><i><font color="Green" title="E16">A16</font></i></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/>

<b>thus </b><a NAME="E7:6_3_3"/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c17">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="scpinvar.html#E12:6_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c17">Iw</font> =  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;0</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c7">I</font></span>)</span>,<font color="Maroon" title="c6">s</font>;<br/>
<b>set </b><font color="Maroon" title="c18">Dw</font> =  <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <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"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;0</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c7">I</font></span>)</span>,<font color="Maroon" title="c6">s</font></span>)</span>;<br/>
<a NAME="E9:6"/><i><font color="Green" title="E12">A22</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <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"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;0</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c7">I</font></span>)</span>,<font color="Maroon" title="c6">s</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <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"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;0</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c7">I</font></span>)</span>,<font color="Maroon" title="c6">s</font></span>)</span>] )
 
<b>from </b><i><a class="ref" href="scpinvar.html#S1" target="_self" title="SCPINVAR:sch.1">SCPINVAR:sch 1</a>(<a class="txt" href="scpinvar.html#E1:6"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="scpinvar.html#E6:6"><i><font color="Green" title="E9">A5</font></i></a>, <a class="txt" href="scpinvar.html#E7:6"><i><font color="Green" title="E10">A8</font></i></a>, <a class="txt" href="scpinvar.html#E8:6"><i><font color="Green" title="E11">A10</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon" title="c19">g</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E11:6"/><i><font color="Green" title="E13">A23</font></i>: 
( <font color="Maroon" title="c19">g</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c13">p0</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;0</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c7">I</font></span>)</span>,<font color="Maroon" title="c6">s</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="c12">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">n</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;0</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c7">I</font></span>)</span>,<font color="Maroon" title="c6">s</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <font color="Maroon" title="c19">g</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;0</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c7">I</font></span>)</span>,<font color="Maroon" title="c6">s</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="c12">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;0</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c7">I</font></span>)</span>,<font color="Maroon" title="c6">s</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="c12">i</font></span>)</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"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;0</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c7">I</font></span>)</span>,<font color="Maroon" title="c6">s</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> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></i>
<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"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;0</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c7">I</font></span>)</span>,<font color="Maroon" title="c6">s</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="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<font color="Maroon" title="c12">i</font></span>)</span>
<b>by </b><i><a class="txt" href="scpinvar.html#E2:6"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="scmp_gcd.html#T5" title="SCMP_GCD:th.5">SCMP_GCD:5</a></i>
;<br/>
<a NAME="E13:6"/><b>then </b>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;0</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c7">I</font></span>)</span>,<font color="Maroon" title="c6">s</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="c12">i</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="scpinvar.html#E6:6"><i><font color="Green" title="E9">A5</font></i></a>, <a class="txt" href="scpinvar.html#E9:6"><i><font color="Green" title="E12">A22</font></i></a></i>;<br/>
<a NAME="E14:6"/><b>then </b><i><font color="Green" title="E14">A24</font></i>: 
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;0</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c7">I</font></span>)</span>,<font color="Maroon" title="c6">s</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="c12">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="scpinvar.html#E11:6"><i><font color="Green" title="E13">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>reconsider </b><font color="Maroon" title="c21">a</font> = <font color="Maroon" title="c20">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>assume </b><a NAME="E2:6_5"/>
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">n</font>
 ;<br/><a NAME="E3:6_5"/><b>then </b><i><font color="Green" title="E15">A25</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">i</font> &amp; <font color="Maroon" title="c20">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>hence </b><font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> = 
<font color="Maroon" title="c6">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> <span class="p2">(<span class="default"><font color="Maroon" title="c13">p0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpinvar.html#E2:6"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="scpisort.html#D1" title="SCPISORT:def.1">SCPISORT:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c19">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
<b>by </b><i><a class="txt" href="scpinvar.html#E11:6"><i><font color="Green" title="E13">A23</font></i></a>, <a class="txt" href="scpinvar.html#E14:6"><i><font color="Green" title="E14">A24</font></i></a>, <a class="txt" href="scpinvar.html#E3:6_5"><i><font color="Green" title="E15">A25</font></i></a>, <a class="ref" href="scpisort.html#D1" title="SCPISORT:def.1">SCPISORT:def 1</a></i>
;<br/><br/></div><b>end;</b></div>
<a NAME="E16:6"/><b>then </b>
<font color="Maroon" title="c14">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">g</font>
 
<b>by </b><i><a class="txt" href="scpinvar.html#E2:6"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="scpinvar.html#E11:6"><i><font color="Green" title="E13">A23</font></i></a>, <a class="txt" href="scpinvar.html#E14:6"><i><font color="Green" title="E14">A24</font></i></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></i>;<br/>
<b>hence </b><a NAME="E17:6"/>
<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"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;0</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c7">I</font></span>)</span>,<font color="Maroon" title="c6">s</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <font color="Maroon" title="c14">f</font>
 <b>by </b><i><a class="txt" href="scpinvar.html#E11:6"><i><font color="Green" title="E13">A23</font></i></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></i>;<br/>

<a NAME="E18:6"/><i><font color="Green" title="E15">A26</font></i>: 
for <font color="Olive" title="b1">t</font> being  <a href="ami_1.html#NM4" title="AMI_1:NM.4">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>   st <font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Olive" title="b1">t</font>] &amp; <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Olive" title="b1">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="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<font color="Maroon" title="c12">i</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="scpinvar.html#E6:6"><i><font color="Green" title="E9">A5</font></i></a></i>;<br/>
<a NAME="E19:6"/>
( ( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">s</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">s</font>) or <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">s</font>] ) &amp;  <a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;0</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c7">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c6">s</font> &amp;  <a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;0</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c7">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c6">s</font> )
 
<b>from </b><i><a class="ref" href="scmpds_8.html#S1" title="SCMPDS_8:sch.1">SCMPDS_8:sch 1</a>(<a class="txt" href="scpinvar.html#E1:6"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="scpinvar.html#E18:6"><i><font color="Green" title="E15">A26</font></i></a>, <a class="txt" href="scpinvar.html#E7:6"><i><font color="Green" title="E10">A8</font></i></a>, <a class="txt" href="scpinvar.html#E8:6"><i><font color="Green" title="E11">A10</font></i></a>);<br/></i>
<b>hence </b><a NAME="E20:6"/>
(  <a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;0</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c7">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c6">s</font> &amp;  <a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&lt;0</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c7">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c6">s</font> )
 ;<br/>


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