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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="scmpds_4.html#V3" target="_self">shiftable</a> <a href="scmpds_4.html#NM1">Program-block</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="scmpds_2.html#M1">Int_position</a> ;<br/>





<b>assume </b><a NAME="E1:102"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="scmpds_2.html#K9">&lt;=0_goto</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="scmpds_4.html#K1" target="_self">Load</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="scmpds_2.html#K9">&lt;=0_goto</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="scmpds_4.html#K5" target="_self">';'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="scmpds_2.html#K9">&lt;=0_goto</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="ami_1.html#NM3">Instruction</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:102_1"/><i><font color="Green">E56</font></i>: 
 <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="scmpds_4.html#K5" target="_self">';'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="scmpds_2.html#K9">&lt;=0_goto</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:102_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="scmpds_4.html#K5" target="_self">';'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="scmpds_2.html#K9">&lt;=0_goto</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 ;<br/><b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="scmpds_3.html#K2">inspos</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K1" target="_self">Load</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="scmpds_2.html#K9">&lt;=0_goto</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> </span>}</span> ;<br/><a NAME="E3:102_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K6">Shift</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K1" target="_self">Load</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="scmpds_2.html#K9">&lt;=0_goto</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="scmpds_3.html#K2">inspos</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K1" target="_self">Load</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="scmpds_2.html#K9">&lt;=0_goto</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> </span>}</span> 
 <b>by </b><i><a class="ref" href="scmpds_3.html#D7">SCMPDS_3:def 7</a></i>;<br/><a NAME="E4:102_1"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="scmpds_4.html#K5" target="_self">';'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="scmpds_2.html#K9">&lt;=0_goto</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="scmpds_3.html#K2">inspos</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K1" target="_self">Load</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="scmpds_2.html#K9">&lt;=0_goto</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> </span>}</span> 
 <b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:102_1_1"/>
(  <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> or  <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="scmpds_3.html#K2">inspos</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K1" target="_self">Load</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="scmpds_2.html#K9">&lt;=0_goto</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="scmpds_4.html#E1:102_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="scmpds_4.html#E4:102_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1_1"><b>suppose </b></a><a NAME="E1:102_1_1_1"/><i><font color="Green">E59</font></i>: 
 <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:102_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="scmpds_4.html#E2:102_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="scmpds_4.html#T37" target="_self">Th37</a></i>;<br/><b>hence </b><a NAME="E3:102_1_1_1"/>
(  <a href="ami_5.html#K1">InsCode</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp;  <a href="ami_5.html#K1">InsCode</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 3 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="scmpds_4.html#R1" target="_self">valid_at</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="scmpds_4.html#E1:102_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="scmpds_4.html#D12" target="_self">Def12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1_2"><b>suppose </b></a><a NAME="E1:102_1_1_2"/>
 <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> :  <a href="scmpds_3.html#K2">inspos</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K1" target="_self">Load</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="scmpds_2.html#K9">&lt;=0_goto</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:102_1_1_2"/><i><font color="Green">E59</font></i>: 
 <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2">inspos</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:102_1_1_2"/><i><font color="Green">E60</font></i>: 
 <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K1" target="_self">Load</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="scmpds_2.html#K9">&lt;=0_goto</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E5:102_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K1" target="_self">Load</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="scmpds_2.html#K9">&lt;=0_goto</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E6:102_1_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2">inspos</a> 0
 <b>by </b><i><a class="txt" href="scmpds_4.html#E4:102_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then </b><i><font color="Green">E62</font></i>: <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="scmpds_2.html#K9">&lt;=0_goto</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K1" target="_self">Load</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="scmpds_2.html#K9">&lt;=0_goto</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="scmpds_4.html#K5" target="_self">';'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="scmpds_2.html#K9">&lt;=0_goto</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="scmpds_3.html#K3">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmpds_4.html#E4:102_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="scmpds_4.html#T38" target="_self">Th38</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="scmpds_4.html#E2:102_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="scmpds_4.html#E3:102_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="scmpds_3.html#D3">SCMPDS_3:def 3</a></i>
;<br/><b>hence </b><a NAME="E8:102_1_1_2"/>
(  <a href="ami_5.html#K1">InsCode</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp;  <a href="ami_5.html#K1">InsCode</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 3 )
 <b>by </b><i><a class="ref" href="scmpds_2.html#T26">SCMPDS_2:26</a></i>;<br/><a NAME="E9:102_1_1_2"/>
 <a href="scmpds_3.html#K2">inspos</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmpds_3.html#K2">inspos</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scmpds_4.html#E3:102_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="scmpds_4.html#E6:102_1_1_2"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E10:102_1_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="scmpds_4.html#E1:102"><i><font color="Green">E55</font></i></a></i>;<br/><a NAME="E11:102_1_1_2"/>
(  <a href="ami_5.html#K1">InsCode</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <a href="ami_5.html#K1">InsCode</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 4 &amp;  <a href="ami_5.html#K1">InsCode</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 6 )
 <b>by </b><i><a class="txt" href="scmpds_4.html#E7:102_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="scmpds_2.html#T26">SCMPDS_2:26</a></i>;<br/><b>hence </b><a NAME="E12:102_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="scmpds_4.html#R1" target="_self">valid_at</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="scmpds_4.html#E7:102_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="scmpds_4.html#E10:102_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="scmpds_4.html#D11" target="_self">Def11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:102"/>
<font color="Maroon">c<sub>1</sub></font> <a href="scmpds_4.html#K5" target="_self">';'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="scmpds_2.html#K9">&lt;=0_goto</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="scmpds_4.html#V3" target="_self">shiftable</a>
 <b>by </b><i><a class="ref" href="scmpds_4.html#D12" target="_self">Def12</a></i>;<br/>


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