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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="scmpds_2.html#M1">Int_position</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="ami_1.html#NM2">Instruction-Location</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="scmpds_2.html#K5">return</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 </span>}</span> ;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:19_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="amistd_1.html#K1">NIC</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K5">return</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> <b> such that </b><br/><a NAME="E3:19_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9">Following</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:19_1"/><i><font color="Green">E19</font></i>: 
 <a href="ami_1.html#K6">IC</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E5:19_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="ami_1.html#K13">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K5">return</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><i><font color="Green">E21</font></i>: <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K4">Exec</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K5">return</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span>
<b>by </b><i><a class="txt" href="scmpds_9.html#E3:19_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="scmpds_9.html#E4:19_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="scmpds_9.html#E5:19_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="scmpds_9.html#E8"><i><font color="Green">Lemma8</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K3">div</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="scmpds_1.html#D23">SCMPDS_1:def 23</a>, <a class="ref" href="scmpds_2.html#T70">SCMPDS_2:70</a></i>
;<br/><a NAME="E7:19_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> 
 <b>by </b><i><a class="txt" href="scmpds_9.html#E3:19_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="ami_2.html#D3">AMI_2:def 3</a>, <a class="ref" href="scmpds_2.html#D1">SCMPDS_2:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:19_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E10:19_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E11:19_1"/><i><font color="Green">E23</font></i>: 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K3">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="scmpds_9.html#E6:19_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="scmpds_9.html#E9:19_1"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E12:19_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K3">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E13:19_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="scmpds_9.html#E11:19_1"><i><font color="Green">E23</font></i></a></i>;<br/><a NAME="E14:19_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E15:19_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 </span>}</span> 
 <b>by </b><i><a class="txt" href="scmpds_9.html#E9:19_1"><i><font color="Green">E22</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:19"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:19"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E5:19"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K6">-</a> 2 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="scmpds_9.html#E5:19"><i><font color="Green">E19</font></i></a>, <a class="txt" href="scmpds_9.html#E17"><i><font color="Green">Lemma16</font></i></a></i>;<br/>
<a NAME="E7:19"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_2.html#K2">SCM-Data-Loc</a> 
 
<b>by </b><i><a class="ref" href="scmpds_2.html#D2">SCMPDS_2:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:19"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="ami_2.html#D2">AMI_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ami_1.html#K3">ObjectKind</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span> <b>by </b><i><a class="ref" href="ami_1.html#D11">AMI_1:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="scmpds_2.html#K5">return</a> <font color="Maroon">c<sub>1</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ami_1.html#K3">ObjectKind</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="ami_1.html#D14">AMI_1:def 14</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="ami_1.html#K16">--&gt;</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2;<br/>
<a NAME="E13:19"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E14:19"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R2">in</a>  <a href="ami_2.html#K2">SCM-Data-Loc</a> 
 
<b>by </b><i><a class="ref" href="ami_2.html#D2">AMI_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2 as    <a href="scmpds_2.html#M1">Int_position</a>  <b>by </b><i><a class="ref" href="scmpds_2.html#T9">SCMPDS_2:9</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="scmpds_9.html#K1" target="_self">--&gt;</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="ami_1.html#K16">--&gt;</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="scmpds_9.html#K1" target="_self">--&gt;</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E16:19"/><i><font color="Green">E21</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="scmpds_9.html#K1" target="_self">--&gt;</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T65">FUNCT_4:65</a></i>;<br/>
<a NAME="E17:19"/><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="ami_1.html#K16">--&gt;</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="ami_1.html#K13">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K5">return</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="ami_6.html#T6">AMI_6:6</a></i>;<br/>
<i><font color="Green">E23</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="ami_1.html#K16">--&gt;</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span> = 
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="ami_1.html#K16">--&gt;</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D15">AMI_1:def 15</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="ami_6.html#T6">AMI_6:6</a></i>
;<br/>
<a NAME="E19:19"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font> )
 
<b>by </b><i><a class="ref" href="scmpds_2.html#T53">SCMPDS_2:53</a></i>;<br/>
<a NAME="E20:19"/><b>then </b>
not <font color="Maroon">c<sub>2</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>,<font color="Maroon">c<sub>14</sub></font> <a href="scmpds_9.html#K1" target="_self">--&gt;</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="scmpds_9.html#E16:19"><i><font color="Green">E21</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E21:19"/><b>then </b><i><font color="Green">E24</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="ami_1.html#K16">--&gt;</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="scmpds_9.html#K1" target="_self">--&gt;</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K13">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K5">return</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="scmpds_9.html#E17:19"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/>
<a NAME="E22:19"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a>  &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a>  )
 
<b>by </b><i><a class="ref" href="scmpds_2.html#T52">SCMPDS_2:52</a></i>;<br/>
<a NAME="E23:19"/><b>then </b><i><font color="Green">E25</font></i>: 
not  <a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a>  <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>,<font color="Maroon">c<sub>14</sub></font> <a href="scmpds_9.html#K1" target="_self">--&gt;</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="scmpds_9.html#E16:19"><i><font color="Green">E21</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<i><font color="Green">E26</font></i>:  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="ami_1.html#K16">--&gt;</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="scmpds_9.html#K1" target="_self">--&gt;</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="ami_1.html#K16">--&gt;</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="scmpds_9.html#K1" target="_self">--&gt;</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D15">AMI_1:def 15</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="scmpds_9.html#E18:19"><i><font color="Green">E23</font></i></a>, <a class="txt" href="scmpds_9.html#E23:19"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
;<br/>
<a NAME="E25:19"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i><a class="txt" href="scmpds_9.html#E9:19"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E26:19"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2
 
;<br/>
<a NAME="E27:19"/><b>then </b><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="ami_1.html#K16">--&gt;</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="scmpds_9.html#K1" target="_self">--&gt;</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="scmpds_9.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E28:19"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="ami_1.html#K16">--&gt;</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="scmpds_9.html#K1" target="_self">--&gt;</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="scmpds_9.html#E26:19"><i><font color="Green">E27</font></i></a>, <a class="ref" href="scmpds_9.html#T7" target="_self">Th7</a></i>;<br/>
<i><font color="Green">E30</font></i>:  <a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c<sub>8</sub></font>,1 = 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="scmpds_2.html#D4">SCMPDS_2:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2

;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="ami_1.html#K16">--&gt;</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="scmpds_9.html#K1" target="_self">--&gt;</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="scmpds_9.html#E26:19"><i><font color="Green">E27</font></i></a>, <a class="ref" href="scmpds_9.html#T7" target="_self">Th7</a></i>;<br/>
<font color="Maroon">c<sub>5</sub></font> = 
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="scmpds_9.html#E4:19"><i><font color="Green">E18</font></i></a></i>
<br/>.= 
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K3">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="nat_1.html#T68">NAT_1:68</a></i>
<br/>.= 
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="nat_1.html#K3">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="scmpds_9.html#E28:19"><i><font color="Green">E29</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="ami_1.html#K16">--&gt;</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="scmpds_9.html#K1" target="_self">--&gt;</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <font color="Maroon">c<sub>8</sub></font>,1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K3">div</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="scmpds_9.html#E29:19"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4">Exec</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K5">return</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="ami_1.html#K16">--&gt;</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="scmpds_9.html#K1" target="_self">--&gt;</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span>
<b>by </b><i><a class="txt" href="scmpds_9.html#E27:19"><i><font color="Green">E28</font></i></a>, <a class="ref" href="scmpds_1.html#D23">SCMPDS_1:def 23</a>, <a class="ref" href="scmpds_2.html#T70">SCMPDS_2:70</a></i>
<br/>.= 
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9">Following</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmpds_2.html#K1">SCMPDS</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="ami_1.html#K16">--&gt;</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="scmpds_9.html#K1" target="_self">--&gt;</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmpds_9.html#E21:19"><i><font color="Green">E24</font></i></a>, <a class="txt" href="scmpds_9.html#E24:19"><i><font color="Green">E26</font></i></a>, <a class="txt" href="scmpds_9.html#E8"><i><font color="Green">Lemma8</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E32:19"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="amistd_1.html#K1">NIC</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K5">return</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="scmpds_9.html#E21:19"><i><font color="Green">E24</font></i></a>, <a class="txt" href="scmpds_9.html#E24:19"><i><font color="Green">E26</font></i></a></i>;<br/>


</div>
