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<b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="scm_comp.html#NM1">bin-term</a>;<br/><b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="scm_1.html#M1">State-consisting</a> of <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K5">halt</a> <a href="ami_3.html#K1">SCM</a> </span>)</span></span><a href="lang1.html#K3">*&gt;</a></span>, <a href="lang1.html#K2">&lt;*&gt;</a> <a href="numbers.html#K4">INT</a> ;<br/>





<b>assume </b><a NAME="E1:31"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="scm_comp.html#K17" target="_self">max_Data-Loc_in</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<a NAME="E2:31"/>
<font color="Maroon">c<sub>17</sub></font> is    <a href="scm_1.html#M1">State-consisting</a> of <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>, <a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>, <a href="lang1.html#K2">&lt;*&gt;</a> <a href="numbers.html#K4">INT</a> 
 
<b>by </b><i><a class="ref" href="scm_comp.html#T1" target="_self">Th1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">c<sub>19</sub></font> being   <a href="ami_1.html#NM4">State</a> of <a href="ami_3.html#K1">SCM</a> <b> such that </b><br/><a NAME="E4:31"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> &amp;  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> &amp;  <a href="ami_1.html#K6">IC</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">c<sub>17</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>14</sub></font> holds <br/><font color="Maroon">c<sub>17</sub></font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="scm_comp.html#E1:31"><i><font color="Green">E27</font></i></a>, <a class="ref" href="scm_comp.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E5:31"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K5">halt</a> <a href="ami_3.html#K1">SCM</a> </span>)</span></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E6:31"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K5">halt</a> <a href="ami_3.html#K1">SCM</a> </span>)</span></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="scm_comp.html#E4:31"><i><font color="Green">E28</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E7:31"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K5">halt</a> <a href="ami_3.html#K1">SCM</a> </span>)</span></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>then </b><i><font color="Green">E29</font></i>: <font color="Maroon">c<sub>17</sub></font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K5">halt</a> <a href="ami_3.html#K1">SCM</a> </span>)</span></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="scm_1.html#D1">SCM_1:def 1</a></i>
<br/>.= 
 <a href="ami_1.html#K5">halt</a> <a href="ami_3.html#K1">SCM</a> 
<b>by </b><i><a class="txt" href="scm_comp.html#E4:31"><i><font color="Green">E28</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>
;<br/>
<b>hence </b><a NAME="E9:31"/>
<font color="Maroon">c<sub>17</sub></font> is <a href="ami_1.html#V9">halting</a>
 <b>by </b><i><a class="txt" href="scm_comp.html#E4:31"><i><font color="Green">E28</font></i></a>, <a class="ref" href="scm_1.html#T3">SCM_1:3</a></i>;<br/>

<b>thus </b><a NAME="E10:31"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K12">Result</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="scm_comp.html#E4:31"><i><font color="Green">E28</font></i></a>, <a class="txt" href="scm_comp.html#E8:31"><i><font color="Green">E29</font></i></a>, <a class="ref" href="scm_1.html#T4">SCM_1:4</a></i>;<br/>

<a NAME="E11:31"/>
<font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E12:31"/><b>then </b>
(  <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="ami_3.html#T53">AMI_3:53</a></i>;<br/>
<b>hence </b><a NAME="E13:31"/>
 <a href="scm_1.html#K2">Complexity</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E4:31"><i><font color="Green">E28</font></i></a>, <a class="txt" href="scm_comp.html#E8:31"><i><font color="Green">E29</font></i></a>, <a class="ref" href="scm_1.html#T17">SCM_1:17</a></i>;<br/>


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