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<span class="kw">let </span><font color="Maroon" title="c11">s</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">p0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> <br/> for <font color="Olive" title="b3">f</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 <font color="Olive" title="b2">p0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 &amp; <font color="Olive" title="b3">f</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c11">s</font>,<font color="Olive" title="b2">p0</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> holds <br/>( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scpinvar.html#K1" title="SCPINVAR:func.1">sum</a> <font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">p0</font></span>)</span>,<font color="Maroon" title="c11">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> 1</span>)</span> <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="b3">f</font> &amp;  <a href="scpinvar.html#K1" title="SCPINVAR:func.1">sum</a> <font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">p0</font> is  <a href="scmpds_4.html#V2" title="SCMPDS_4:attr.2">parahalting</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c12">n</font>, <font color="Maroon" title="c13">p0</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</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 <font color="Maroon" title="c13">p0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 &amp; <font color="Olive" title="b1">f</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c11">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">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n</font> holds <br/>( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scpinvar.html#K1" title="SCPINVAR:func.1">sum</a> <font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c13">p0</font></span>)</span>,<font color="Maroon" title="c11">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> 1</span>)</span> <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">f</font> &amp;  <a href="scpinvar.html#K1" title="SCPINVAR:func.1">sum</a> <font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c13">p0</font> is  <a href="scmpds_4.html#V2" title="SCMPDS_4:attr.2">parahalting</a>  )</span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">p0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 &amp; <font color="Maroon" title="c14">f</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c11">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="c12">n</font> implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scpinvar.html#K1" title="SCPINVAR:func.1">sum</a> <font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c13">p0</font></span>)</span>,<font color="Maroon" title="c11">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> 1</span>)</span> <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> &amp;  <a href="scpinvar.html#K1" title="SCPINVAR:func.1">sum</a> <font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c13">p0</font> is  <a href="scmpds_4.html#V2" title="SCMPDS_4:attr.2">parahalting</a>  ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:16"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c13">p0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3
 <span class="kw">and </span><br/><a NAME="E2:16"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
( <font color="Maroon" title="c14">f</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c11">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="c12">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scpinvar.html#K1" title="SCPINVAR:func.1">sum</a> <font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c13">p0</font></span>)</span>,<font color="Maroon" title="c11">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> 1</span>)</span> <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> &amp;  <a href="scpinvar.html#K1" title="SCPINVAR:func.1">sum</a> <font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c13">p0</font> is  <a href="scmpds_4.html#V2" title="SCMPDS_4:attr.2">parahalting</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E3:16"/>
<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="scpinvar.html#K1" title="SCPINVAR:func.1">sum</a> <font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c13">p0</font></span>)</span>,<font color="Maroon" title="c11">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> 1</span>)</span> <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="scpinvar.html#E13"><span class="lab"><font color="Green" title="E8">Lm4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="scpinvar.html#K1" title="SCPINVAR:func.1">sum</a> <font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c13">p0</font> is  <a href="scmpds_4.html#V2" title="SCMPDS_4:attr.2">parahalting</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">t</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="scpinvar.html#K1" title="SCPINVAR:func.1">sum</a> <font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c13">p0</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c15">t</font></span><br/><a NAME="E1:16_1"/>
 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/>(  <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> <font color="Maroon" title="c12">n</font> &amp; <font color="Olive" title="b1">g</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c15">t</font>,<font color="Maroon" title="c13">p0</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scpisort.html#T3" title="SCPISORT:th.3">SCPISORT:3</a></span>;<br/><span class="kw">hence </span><a NAME="E2:16_1"/>
 <a href="scpinvar.html#K1" title="SCPINVAR:func.1">sum</a> <font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c13">p0</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c15">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="scpinvar.html#E13"><span class="lab"><font color="Green" title="E8">Lm4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:16"/>
 <a href="scpinvar.html#K1" title="SCPINVAR:func.1">sum</a> <font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c13">p0</font> is  <a href="scmpds_4.html#V2" title="SCMPDS_4:attr.2">parahalting</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_6.html#T35" title="SCMPDS_6:th.35">SCMPDS_6:35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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