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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="scm_1.html#M1">State-consisting</a> of 3,0,0, <a href="fib_fusc.html#K1" target="_self">Fib_Program</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="scm_1.html#K1">&lt;*</a><span class="default">1</span><a href="scm_1.html#K1">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="scm_1.html#K1">&lt;*</a><span class="default"><font color="Maroon">a<sub>1</sub></font></span><a href="scm_1.html#K1">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="scm_1.html#K1">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="scm_1.html#K1">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="scm_1.html#K1">&lt;*</a><span class="default"><font color="Olive">b<sub>3</sub></font></span><a href="scm_1.html#K1">*&gt;</a></span> st <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/>( <font color="Olive">b<sub>4</sub></font> is <a href="ami_1.html#V9">halting</a> &amp;  <a href="scm_1.html#K2">Complexity</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">6 <a href="nat_1.html#K2">*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 4 &amp; ex <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K12">Result</a> <font color="Olive">b<sub>4</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> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <font color="Olive">b<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K12">Result</a> <font color="Olive">b<sub>4</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> 3</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) );<br/>
<a NAME="E1:12"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E2:12"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:12_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>]
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="scm_1.html#M1">State-consisting</a> of 3,0,0, <a href="fib_fusc.html#K1" target="_self">Fib_Program</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="scm_1.html#K1">&lt;*</a><span class="default">1</span><a href="scm_1.html#K1">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="scm_1.html#K1">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="scm_1.html#K1">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="scm_1.html#K1">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="scm_1.html#K1">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="scm_1.html#K1">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="scm_1.html#K1">*&gt;</a></span>;<br/>



<b>assume </b><a NAME="E2:12_1"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="ami_3.html#K15">dl.</a> 0;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="ami_3.html#K15">dl.</a> 1;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="ami_3.html#K15">dl.</a> 2;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="ami_3.html#K15">dl.</a> 3;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="ami_3.html#K15">dl.</a> 4;<br/>
<a NAME="E3:12_1"/><i><font color="Green">E14</font></i>: 
 <a href="ami_1.html#K6">IC</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> 3
 
<b>by </b><i><a class="ref" href="scm_1.html#D1">SCM_1:def 1</a></i>;<br/>
<a NAME="E4:12_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="ami_3.html#K10">&gt;0_goto</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="scm_1.html#T14">SCM_1:14</a></i>;<br/>
<a NAME="E5:12_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K5">halt</a> <a href="ami_3.html#K1">SCM</a> 
 
<b>by </b><i><a class="ref" href="scm_1.html#T14">SCM_1:14</a></i>;<br/>
<a NAME="E6:12_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 3</span>)</span> <a href="ami_3.html#K3">:=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span>
 
<b>by </b><i><a class="ref" href="scm_1.html#T14">SCM_1:14</a></i>;<br/>
<a NAME="E7:12_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> 3</span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K5">SubFrom</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span>
 
<b>by </b><i><a class="ref" href="scm_1.html#T14">SCM_1:14</a></i>;<br/>
<a NAME="E8:12_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> 4</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="ami_3.html#K9">=0_goto</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="scm_1.html#T14">SCM_1:14</a></i>;<br/>
<a NAME="E9:12_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> 5</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 4</span>)</span> <a href="ami_3.html#K3">:=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="scm_1.html#T14">SCM_1:14</a></i>;<br/>
<a NAME="E10:12_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> 6</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 2</span>)</span> <a href="ami_3.html#K3">:=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 3</span>)</span>
 
<b>by </b><i><a class="ref" href="scm_1.html#T14">SCM_1:14</a></i>;<br/>
<a NAME="E11:12_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> 7</span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K4">AddTo</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 4</span>)</span>
 
<b>by </b><i><a class="ref" href="scm_1.html#T14">SCM_1:14</a></i>;<br/>
<a NAME="E12:12_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> 8</span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K8">goto</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> 3</span>)</span>
 
<b>by </b><i><a class="ref" href="scm_1.html#T14">SCM_1:14</a></i>;<br/>
<a NAME="E13:12_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 3</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="scm_1.html#T14">SCM_1:14</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> 0;<br/>
<a NAME="E14:12_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>5</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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> 0
 
<b>by </b><i><a class="ref" href="ami_1.html#D19">AMI_1:def 19</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<i><font color="Green">E26</font></i>: <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i><a class="txt" href="fib_fusc.html#E3:12_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="fib_fusc.html#E7:12_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="fib_fusc.html#E13:12_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="fib_fusc.html#E14:12_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="scm_1.html#T20">SCM_1:20</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font>

;<br/>
<a NAME="E16:12_1"/><i><font color="Green">E27</font></i>: 
( <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 3</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="fib_fusc.html#E3:12_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="fib_fusc.html#E7:12_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="fib_fusc.html#E13:12_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="fib_fusc.html#E14:12_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="fib_fusc.html#E7"><i><font color="Green">Lemma7</font></i></a>, <a class="ref" href="scm_1.html#T20">SCM_1:20</a></i>;<br/>
<i><font color="Green">E28</font></i>:  <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
 <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="fib_fusc.html#E3:12_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="fib_fusc.html#E7:12_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="fib_fusc.html#E14:12_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="scm_1.html#T20">SCM_1:20</a></i>
<br/>.= 
 <a href="ami_3.html#K16">il.</a> 4

;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<a NAME="E18:12_1"/><i><font color="Green">E29</font></i>: 
( <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 3</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="fib_fusc.html#E8:12_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="fib_fusc.html#E15:12_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="fib_fusc.html#E16:12_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="fib_fusc.html#E17:12_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="scm_1.html#T24">SCM_1:24</a></i>;<br/>
<a NAME="E19:12_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1_3"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span><b>by </b><i><a class="txt" href="fib_fusc.html#E19:12_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_3_1"><b>suppose </b></a><a NAME="E1:12_1_3_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:12_1_3_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K5">halt</a> <a href="ami_3.html#K1">SCM</a> 
 <b>by </b><i><a class="txt" href="fib_fusc.html#E5:12_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="fib_fusc.html#E8:12_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="fib_fusc.html#E15:12_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="fib_fusc.html#E17:12_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="scm_1.html#T24">SCM_1:24</a></i>;<br/><b>hence </b><a NAME="E3:12_1_3_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="ami_1.html#V9">halting</a>
 <b>by </b><i><a class="ref" href="scm_1.html#T3">SCM_1:3</a></i>;<br/><a NAME="E4:12_1_3_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ami_1.html#K5">halt</a> <a href="ami_3.html#K1">SCM</a> 
 <b>by </b><i><a class="txt" href="fib_fusc.html#E8:12_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="fib_fusc.html#E17:12_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="scm_1.html#T26">SCM_1:26</a></i>;<br/><b>hence </b><a NAME="E5:12_1_3_1"/>
 <a href="scm_1.html#K2">Complexity</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">6 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 4
 <b>by </b><i><a class="txt" href="fib_fusc.html#E1:12_1_3_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="fib_fusc.html#E2:12_1_3_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="scm_1.html#T16">SCM_1:16</a></i>;<br/><b>take </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>2</sub></font>;<br/><b>thus </b><a NAME="E6:12_1_3_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="fib_fusc.html#E1:12_1_3_1"><i><font color="Green">E31</font></i></a></i>;<br/><b>thus </b><a NAME="E7:12_1_3_1"/>
( <span class="p1">(<span class="default"><a href="ami_1.html#K12">Result</a> <font color="Maroon">c<sub>5</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> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <font color="Maroon">c<sub>14</sub></font> &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K12">Result</a> <font color="Maroon">c<sub>5</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> 3</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="fib_fusc.html#E2:12_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="fib_fusc.html#E18:12_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="fib_fusc.html#E2:12_1_3_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="scm_1.html#T4">SCM_1:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_3_2"><b>suppose </b></a><a NAME="E1:12_1_3_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:12_1_3_2"/><b>then </b><i><font color="Green">E32</font></i>: 
 <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="fib_fusc.html#E8:12_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="fib_fusc.html#E15:12_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="fib_fusc.html#E17:12_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="scm_1.html#T24">SCM_1:24</a></i>;<br/><b>set </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E3:12_1_3_2"/><i><font color="Green">E33</font></i>: 
 <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="fib_fusc.html#E9:12_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="fib_fusc.html#E2:12_1_3_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="scm_1.html#T18">SCM_1:18</a></i>;<br/><a NAME="E4:12_1_3_2"/><i><font color="Green">E34</font></i>: 
<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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 4</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="fib_fusc.html#E2:12_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="fib_fusc.html#E9:12_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="fib_fusc.html#E18:12_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="fib_fusc.html#E2:12_1_3_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="scm_1.html#T18">SCM_1:18</a></i>;<br/><a NAME="E5:12_1_3_2"/><i><font color="Green">E35</font></i>: 
( <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <font color="Maroon">c<sub>2</sub></font> &amp; <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 3</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="fib_fusc.html#E2:12_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="fib_fusc.html#E9:12_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="fib_fusc.html#E18:12_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="fib_fusc.html#E2:12_1_3_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="fib_fusc.html#E8"><i><font color="Green">Lemma8</font></i></a>, <a class="ref" href="scm_1.html#T18">SCM_1:18</a></i>;<br/><b>set </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E6:12_1_3_2"/><i><font color="Green">E36</font></i>: 
 <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default">6 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="fib_fusc.html#E10:12_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="fib_fusc.html#E3:12_1_3_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="scm_1.html#T18">SCM_1:18</a></i>;<br/><a NAME="E7:12_1_3_2"/><i><font color="Green">E37</font></i>: 
<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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="fib_fusc.html#E10:12_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="fib_fusc.html#E3:12_1_3_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="fib_fusc.html#E5:12_1_3_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="scm_1.html#T18">SCM_1:18</a></i>;<br/><a NAME="E8:12_1_3_2"/><i><font color="Green">E38</font></i>: 
( <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 3</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 4</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="fib_fusc.html#E10:12_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="fib_fusc.html#E3:12_1_3_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="fib_fusc.html#E4:12_1_3_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="fib_fusc.html#E5:12_1_3_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="fib_fusc.html#E7"><i><font color="Green">Lemma7</font></i></a>, <a class="txt" href="fib_fusc.html#E8"><i><font color="Green">Lemma8</font></i></a>, <a class="ref" href="scm_1.html#T18">SCM_1:18</a></i>;<br/><b>set </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E9:12_1_3_2"/><i><font color="Green">E39</font></i>: 
 <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default">7 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="fib_fusc.html#E11:12_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="fib_fusc.html#E6:12_1_3_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="scm_1.html#T19">SCM_1:19</a></i>;<br/><i><font color="Green">E40</font></i>: <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 3</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K15">dl.</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K15">dl.</a> 3</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fib_fusc.html#E11:12_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="fib_fusc.html#E6:12_1_3_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="scm_1.html#T19">SCM_1:19</a></i>
<br/>.= 
 <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</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="txt" href="fib_fusc.html#E8:12_1_3_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="pre_ff.html#T1">PRE_FF:1</a></i>
;<br/><a NAME="E11:12_1_3_2"/><i><font color="Green">E41</font></i>: 
( <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="fib_fusc.html#E11:12_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="fib_fusc.html#E6:12_1_3_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="fib_fusc.html#E7:12_1_3_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="fib_fusc.html#E8:12_1_3_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="fib_fusc.html#E7"><i><font color="Green">Lemma7</font></i></a>, <a class="ref" href="scm_1.html#T19">SCM_1:19</a></i>;<br/><b>set </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E12:12_1_3_2"/><i><font color="Green">E42</font></i>: 
 <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">5 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> 3
 <b>by </b><i><a class="txt" href="fib_fusc.html#E12:12_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="fib_fusc.html#E9:12_1_3_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="scm_1.html#T23">SCM_1:23</a></i>;<br/><a NAME="E13:12_1_3_2"/><i><font color="Green">E43</font></i>: 
( <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">5 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">5 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">5 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">5 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 3</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</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="txt" href="fib_fusc.html#E12:12_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="fib_fusc.html#E9:12_1_3_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="fib_fusc.html#E10:12_1_3_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="fib_fusc.html#E11:12_1_3_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="scm_1.html#T23">SCM_1:23</a></i>;<br/><a NAME="E14:12_1_3_2"/><i><font color="Green">E44</font></i>: 
( <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">5 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="ami_3.html#K10">&gt;0_goto</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> 2</span>)</span> &amp; <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">5 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K5">halt</a> <a href="ami_3.html#K1">SCM</a>  &amp; <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">5 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 3</span>)</span> <a href="ami_3.html#K3">:=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> &amp; <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">5 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> 3</span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K5">SubFrom</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> &amp; <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">5 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> 4</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="ami_3.html#K9">=0_goto</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> 1</span>)</span> &amp; <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">5 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> 5</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 4</span>)</span> <a href="ami_3.html#K3">:=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 2</span>)</span> &amp; <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">5 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> 6</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 2</span>)</span> <a href="ami_3.html#K3">:=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 3</span>)</span> &amp; <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">5 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> 7</span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K4">AddTo</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 4</span>)</span> &amp; <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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">5 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> 8</span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K8">goto</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> 3</span>)</span> )
 <b>by </b><i><a class="txt" href="fib_fusc.html#E4:12_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="fib_fusc.html#E5:12_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="fib_fusc.html#E6:12_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="fib_fusc.html#E7:12_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="fib_fusc.html#E8:12_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="fib_fusc.html#E9:12_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="fib_fusc.html#E10:12_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="fib_fusc.html#E11:12_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="fib_fusc.html#E12:12_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="ami_1.html#T54">AMI_1:54</a></i>;<br/><b>set </b><font color="Maroon">c<sub>18</sub></font> =  <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E15:12_1_3_2"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K1">+</a> 1</span>)</span> is    <a href="scm_1.html#M1">State-consisting</a> of 3,0,0, <a href="fib_fusc.html#K1" target="_self">Fib_Program</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="scm_1.html#K1">&lt;*</a><span class="default">1</span><a href="scm_1.html#K1">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="scm_1.html#K1">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="scm_1.html#K1">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="scm_1.html#K1">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="scm_1.html#K1">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="scm_1.html#K1">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="scm_1.html#K1">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="fib_fusc.html#E2:12_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="fib_fusc.html#E12:12_1_3_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="fib_fusc.html#E13:12_1_3_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="fib_fusc.html#E14:12_1_3_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="scm_1.html#T30">SCM_1:30</a></i>;<br/><a NAME="E16:12_1_3_2"/><b>then </b><i><font color="Green">E46</font></i>: 
( <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="ami_1.html#V9">halting</a> &amp;  <a href="scm_1.html#K2">Complexity</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>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default">5 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">6 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 4 &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K12">Result</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p3">(<span class="default">5 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K12">Result</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p3">(<span class="default">5 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 3</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="fib_fusc.html#E1:12_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="fib_fusc.html#E2:12_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="fib_fusc.html#E1:12_1_3_2"><i><font color="Green">E31</font></i></a></i>;<br/><b>hence </b><a NAME="E17:12_1_3_2"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="ami_1.html#V9">halting</a>
 <b>by </b><i><a class="ref" href="scm_1.html#T27">SCM_1:27</a></i>;<br/><a NAME="E18:12_1_3_2"/>
<span class="p1">(<span class="default">6 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 4 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="fib_fusc.html#E1:12_1_3_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="fib_fusc.html#E6"><i><font color="Green">Lemma6</font></i></a></i>;<br/><b>hence </b> <a href="scm_1.html#K2">Complexity</a> <font color="Maroon">c<sub>5</sub></font> = 
6 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">6 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 4</span>)</span>
<b>by </b><i><a class="txt" href="fib_fusc.html#E16:12_1_3_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="scm_1.html#T28">SCM_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default">6 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 4

;<br/><br/><b>consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E21:12_1_3_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K12">Result</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p3">(<span class="default">5 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <font color="Maroon">c<sub>19</sub></font> &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K12">Result</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p3">(<span class="default">5 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 3</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="fib_fusc.html#E1:12_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="fib_fusc.html#E2:12_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="fib_fusc.html#E1:12_1_3_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="fib_fusc.html#E15:12_1_3_2"><i><font color="Green">E45</font></i></a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>19</sub></font>
;<br/><b>thus </b><a NAME="E22:12_1_3_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="fib_fusc.html#E21:12_1_3_2"><i><font color="Green">E47</font></i></a></i>;<br/><b>thus </b><a NAME="E23:12_1_3_2"/>
( <span class="p1">(<span class="default"><a href="ami_1.html#K12">Result</a> <font color="Maroon">c<sub>5</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> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <font color="Maroon">c<sub>19</sub></font> &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K12">Result</a> <font color="Maroon">c<sub>5</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> 3</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="fib_fusc.html#E16:12_1_3_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="fib_fusc.html#E21:12_1_3_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="scm_1.html#T29">SCM_1:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>thus </b><a NAME="E3:12"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="fib_fusc.html#E1:12"><i><font color="Green">E10</font></i></a>, <a class="txt" href="fib_fusc.html#E2:12"><i><font color="Green">E11</font></i></a>);</i><br/>


</div>
