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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="turing_1.html#NM4">All-State</a> of <a href="turing_1.html#K20" target="_self">U3(n)Turing</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="turing_1.html#NM2">Tape</a> of <a href="turing_1.html#K20" target="_self">U3(n)Turing</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>







<b>assume </b><a NAME="E1:79"/><i><font color="Green">E61</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="turing_1.html#R2" target="_self">storeData</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 4;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 6;<br/>
<a NAME="E2:79"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="turing_1.html#R1" target="_self">is_1_between</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="turing_1.html#R1" target="_self">is_1_between</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 4 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="turing_1.html#R1" target="_self">is_1_between</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 4,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 6 )
 
<b>by </b><i><a class="txt" href="turing_1.html#E1:79"><i><font color="Green">E61</font></i></a>, <a class="ref" href="turing_1.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E3:79"/><b>then </b><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) )
 
<b>by </b><i><a class="ref" href="turing_1.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E4:79"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <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"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 6</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 4 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 6 holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) )
 
<b>by </b><i><a class="txt" href="turing_1.html#E2:79"><i><font color="Green">E62</font></i></a>, <a class="ref" href="turing_1.html#D13" target="_self">Def13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="mcart_1.html#K7">`3</a>  as   <a href="turing_1.html#NM2">Tape</a> of <a href="turing_1.html#K20" target="_self">U3(n)Turing</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = 0 as   <a href="turing_1.html#NM3">Symbol</a> of <a href="turing_1.html#K20" target="_self">U3(n)Turing</a>  <b>by </b><i><a class="txt" href="turing_1.html#E75"><i><font color="Green">Lemma57</font></i></a></i>;<br/>
<i><font color="Green">E65</font></i>:  <a href="turing_1.html#K8" target="_self">TRAN</a> <font color="Maroon">c<sub>1</sub></font> = 
the <a href="turing_1.html#U3" target="_self">Tran</a> of <a href="turing_1.html#K20" target="_self">U3(n)Turing</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<a href="turing_1.html#K19" target="_self">U3(n)Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="turing_1.html#D22" target="_self">Def22</a></i>
<br/>.= 
<a href="turing_1.html#K19" target="_self">U3(n)Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:79"><i><font color="Green">E61</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<a href="turing_1.html#K19" target="_self">U3(n)Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:79"><i><font color="Green">E61</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default">1,0,1</span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:79"><i><font color="Green">E61</font></i></a>, <a class="txt" href="turing_1.html#E3:79"><i><font color="Green">E63</font></i></a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a>, <a class="ref" href="turing_1.html#T41" target="_self">Th41</a></i>
;<br/>
<i><font color="Green">E66</font></i>:  <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="mcart_1.html#K7">`3</a> </span>)</span>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> = 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:79"><i><font color="Green">E61</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:79"><i><font color="Green">E61</font></i></a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E8:79"><i><font color="Green">E65</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E3:79"><i><font color="Green">E63</font></i></a>, <a class="ref" href="turing_1.html#T28" target="_self">Th28</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = 1 as   <a href="turing_1.html#NM1">State</a> of <a href="turing_1.html#K20" target="_self">U3(n)Turing</a>  <b>by </b><i><a class="txt" href="turing_1.html#E74"><i><font color="Green">Lemma56</font></i></a></i>;<br/>
<i><font color="Green">E67</font></i>:  <a href="turing_1.html#K6" target="_self">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> 

<br/>.= 
1
<b>by </b><i><a class="txt" href="turing_1.html#E8:79"><i><font color="Green">E65</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span>;<br/>
<i><font color="Green">E68</font></i>:  <a href="turing_1.html#K9" target="_self">Following</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" target="_self">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:79"><i><font color="Green">E61</font></i></a>, <a class="txt" href="turing_1.html#E9:79"><i><font color="Green">E66</font></i></a>, <a class="txt" href="turing_1.html#E76"><i><font color="Green">Lemma58</font></i></a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default">1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" target="_self">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E8:79"><i><font color="Green">E65</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E1:79"><i><font color="Green">E61</font></i></a>, <a class="txt" href="turing_1.html#E11:79"><i><font color="Green">E67</font></i></a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a></i>
;<br/>
<a NAME="E13:79"/><i><font color="Green">E69</font></i>: 
the <a href="turing_1.html#U3" target="_self">Tran</a> of <a href="turing_1.html#K20" target="_self">U3(n)Turing</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,1</span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,0,1</span><a href="domain_1.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="turing_1.html#D22" target="_self">Def22</a>, <a class="ref" href="turing_1.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E14:79"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <a href="turing_1.html#K20" target="_self">U3(n)Turing</a> 
 
<b>by </b><i><a class="ref" href="turing_1.html#D22" target="_self">Def22</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<div><i><font color="Green">E71</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>assume </b><a NAME="E1:79_5"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/><a NAME="E2:79_5"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E3:79_5"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="turing_1.html#E1:79_5"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E4:79_5"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="turing_1.html#E2:79"><i><font color="Green">E62</font></i></a>, <a class="txt" href="turing_1.html#E1:79_5"><i><font color="Green">E72</font></i></a>, <a class="ref" href="turing_1.html#D13" target="_self">Def13</a></i>;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="turing_1.html#NM2">Tape</a> of <a href="turing_1.html#K20" target="_self">U3(n)Turing</a> <b> such that </b><br/><a NAME="E17:79"/><i><font color="Green">E72</font></i>: 
( <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st ( <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) holds <br/><font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="turing_1.html#E13:79"><i><font color="Green">E69</font></i></a>, <a class="txt" href="turing_1.html#E14:79"><i><font color="Green">E70</font></i></a>, <a class="txt" href="turing_1.html#E15:79"><i><font color="Green">E71</font></i></a>, <a class="txt" href="turing_1.html#E75"><i><font color="Green">Lemma57</font></i></a>, <a class="txt" href="turing_1.html#E77"><i><font color="Green">Lemma59</font></i></a></i>;<br/>
<a NAME="E18:79"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="turing_1.html#E3:79"><i><font color="Green">E63</font></i></a>, <a class="txt" href="turing_1.html#E17:79"><i><font color="Green">E72</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K7">`3</a>  as   <a href="turing_1.html#NM2">Tape</a> of <a href="turing_1.html#K20" target="_self">U3(n)Turing</a>  ;<br/>
<i><font color="Green">E74</font></i>:  <a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
the <a href="turing_1.html#U3" target="_self">Tran</a> of <a href="turing_1.html#K20" target="_self">U3(n)Turing</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p5"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p0"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<a href="turing_1.html#K19" target="_self">U3(n)Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p5"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p0"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="turing_1.html#D22" target="_self">Def22</a></i>
<br/>.= 
<a href="turing_1.html#K19" target="_self">U3(n)Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p0"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E17:79"><i><font color="Green">E72</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<a href="turing_1.html#K19" target="_self">U3(n)Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p0"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E17:79"><i><font color="Green">E72</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default">2,0,1</span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E17:79"><i><font color="Green">E72</font></i></a>, <a class="txt" href="turing_1.html#E18:79"><i><font color="Green">E73</font></i></a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a>, <a class="ref" href="turing_1.html#T41" target="_self">Th41</a></i>
;<br/>
<i><font color="Green">E75</font></i>:  <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <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="turing_1.html#K8" target="_self">TRAN</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p5"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> = 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <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="turing_1.html#K8" target="_self">TRAN</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p5"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E17:79"><i><font color="Green">E72</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p5"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E17:79"><i><font color="Green">E72</font></i></a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E20:79"><i><font color="Green">E74</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E18:79"><i><font color="Green">E73</font></i></a>, <a class="ref" href="turing_1.html#T28" target="_self">Th28</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = 2 as   <a href="turing_1.html#NM1">State</a> of <a href="turing_1.html#K20" target="_self">U3(n)Turing</a>  <b>by </b><i><a class="txt" href="turing_1.html#E74"><i><font color="Green">Lemma56</font></i></a></i>;<br/>
<i><font color="Green">E76</font></i>:  <a href="turing_1.html#K6" target="_self">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K7">`3</a> 

<br/>.= 
1
<b>by </b><i><a class="txt" href="turing_1.html#E20:79"><i><font color="Green">E74</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span>;<br/>
<i><font color="Green">E77</font></i>:  <a href="turing_1.html#K9" target="_self">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p0"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p0"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" target="_self">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>15</sub></font></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E17:79"><i><font color="Green">E72</font></i></a>, <a class="txt" href="turing_1.html#E21:79"><i><font color="Green">E75</font></i></a>, <a class="txt" href="turing_1.html#E76"><i><font color="Green">Lemma58</font></i></a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default">2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p0"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" target="_self">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>15</sub></font></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E20:79"><i><font color="Green">E74</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E17:79"><i><font color="Green">E72</font></i></a>, <a class="txt" href="turing_1.html#E23:79"><i><font color="Green">E76</font></i></a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a></i>
;<br/>
<a NAME="E26:79"/><i><font color="Green">E78</font></i>: 
the <a href="turing_1.html#U3" target="_self">Tran</a> of <a href="turing_1.html#K20" target="_self">U3(n)Turing</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,1</span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,0,1</span><a href="domain_1.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="turing_1.html#D22" target="_self">Def22</a>, <a class="ref" href="turing_1.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E27:79"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <a href="turing_1.html#K20" target="_self">U3(n)Turing</a> 
 
<b>by </b><i><a class="ref" href="turing_1.html#D22" target="_self">Def22</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<div><i><font color="Green">E80</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>22</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>assume </b><a NAME="E1:79_10"/><i><font color="Green">E81</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/><a NAME="E2:79_10"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <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="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E3:79_10"/><b>then </b><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="turing_1.html#E1:79_10"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E17:79"><i><font color="Green">E72</font></i></a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="turing_1.html#E2:79"><i><font color="Green">E62</font></i></a>, <a class="txt" href="turing_1.html#E1:79_10"><i><font color="Green">E81</font></i></a>, <a class="txt" href="turing_1.html#E3:79_10"><i><font color="Green">E82</font></i></a>, <a class="ref" href="turing_1.html#D13" target="_self">Def13</a></i>
;<br/><br/></div><b>end;</b></div>
<b>set </b><font color="Maroon">c<sub>22</sub></font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>23</sub></font> being   <a href="turing_1.html#NM2">Tape</a> of <a href="turing_1.html#K20" target="_self">U3(n)Turing</a> <b> such that </b><br/><a NAME="E30:79"/><i><font color="Green">E81</font></i>: 
( <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>23</sub></font></span><a href="domain_1.html#K4">]</a></span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) holds <br/><font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="turing_1.html#E26:79"><i><font color="Green">E78</font></i></a>, <a class="txt" href="turing_1.html#E27:79"><i><font color="Green">E79</font></i></a>, <a class="txt" href="turing_1.html#E28:79"><i><font color="Green">E80</font></i></a>, <a class="txt" href="turing_1.html#E75"><i><font color="Green">Lemma57</font></i></a>, <a class="txt" href="turing_1.html#E77"><i><font color="Green">Lemma59</font></i></a></i>;<br/>
<a NAME="E31:79"/>
2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 4
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E32:79"/><b>then </b><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<i><font color="Green">E83</font></i>: <font color="Maroon">c<sub>23</sub></font> <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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon">c<sub>15</sub></font> <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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E30:79"><i><font color="Green">E81</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="turing_1.html#E4:79"><i><font color="Green">E64</font></i></a>, <a class="txt" href="turing_1.html#E17:79"><i><font color="Green">E72</font></i></a>, <a class="txt" href="turing_1.html#E32:79"><i><font color="Green">E82</font></i></a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K7">`3</a>  as   <a href="turing_1.html#NM2">Tape</a> of <a href="turing_1.html#K20" target="_self">U3(n)Turing</a>  ;<br/>
<i><font color="Green">E84</font></i>:  <a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
the <a href="turing_1.html#U3" target="_self">Tran</a> of <a href="turing_1.html#K20" target="_self">U3(n)Turing</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p5"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p0"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<a href="turing_1.html#K19" target="_self">U3(n)Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p5"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p0"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="turing_1.html#D22" target="_self">Def22</a></i>
<br/>.= 
<a href="turing_1.html#K19" target="_self">U3(n)Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p0"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E30:79"><i><font color="Green">E81</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<a href="turing_1.html#K19" target="_self">U3(n)Tran</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,<span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p0"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E30:79"><i><font color="Green">E81</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default">3,0,0</span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E30:79"><i><font color="Green">E81</font></i></a>, <a class="txt" href="turing_1.html#E33:79"><i><font color="Green">E83</font></i></a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a>, <a class="ref" href="turing_1.html#T41" target="_self">Th41</a></i>
;<br/>
<i><font color="Green">E85</font></i>:  <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <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="turing_1.html#K8" target="_self">TRAN</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p5"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> = 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>23</sub></font>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <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="turing_1.html#K8" target="_self">TRAN</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p5"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E30:79"><i><font color="Green">E81</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>23</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p5"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E30:79"><i><font color="Green">E81</font></i></a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">c<sub>23</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E35:79"><i><font color="Green">E84</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<font color="Maroon">c<sub>23</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E33:79"><i><font color="Green">E83</font></i></a>, <a class="ref" href="turing_1.html#T28" target="_self">Th28</a></i>
;<br/>
<i><font color="Green">E86</font></i>:  <a href="turing_1.html#K6" target="_self">offset</a> <span class="p1">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K7">`3</a> 

<br/>.= 
0
<b>by </b><i><a class="txt" href="turing_1.html#E35:79"><i><font color="Green">E84</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
;<br/>
<i><font color="Green">E87</font></i>:  <a href="turing_1.html#K9" target="_self">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p0"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p0"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" target="_self">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>23</sub></font></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E30:79"><i><font color="Green">E81</font></i></a>, <a class="txt" href="turing_1.html#E36:79"><i><font color="Green">E85</font></i></a>, <a class="txt" href="turing_1.html#E76"><i><font color="Green">Lemma58</font></i></a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default">3,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p0"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="turing_1.html#K6" target="_self">offset</a> <span class="p4">(<span class="default"><a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>23</sub></font></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E35:79"><i><font color="Green">E84</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default">3,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 0</span>)</span>,<font color="Maroon">c<sub>23</sub></font></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E30:79"><i><font color="Green">E81</font></i></a>, <a class="txt" href="turing_1.html#E37:79"><i><font color="Green">E86</font></i></a>, <a class="ref" href="turing_1.html#T27" target="_self">Th27</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>25</sub></font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<div><i><font color="Green">E88</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_16"><b>now </b></a><div class="add"><b>thus </b><span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="turing_1.html#T13" target="_self">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E12:79"><i><font color="Green">E68</font></i></a>, <a class="ref" href="turing_1.html#T12" target="_self">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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="turing_1.html#T13" target="_self">Th13</a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E89</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_17"><b>now </b></a><div class="add"><b>thus </b><span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p0"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <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="turing_1.html#E39:79"><i><font color="Green">E88</font></i></a>, <a class="ref" href="turing_1.html#T13" target="_self">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <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="turing_1.html#E25:79"><i><font color="Green">E77</font></i></a>, <a class="ref" href="turing_1.html#T12" target="_self">Th12</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default">3,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>23</sub></font></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="turing_1.html#E38:79"><i><font color="Green">E87</font></i></a>, <a class="ref" href="turing_1.html#D8" target="_self">Def8</a></i>
;<br/></div><b>end;</b></div>
<b>then </b><i><font color="Green">E90</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K5">`1</a>  = 
3
<b>by </b><i><a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <a href="turing_1.html#K20" target="_self">U3(n)Turing</a> 
<b>by </b><i><a class="ref" href="turing_1.html#D22" target="_self">Def22</a></i>
;<br/>
<b>hence </b><a NAME="E42:79"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="turing_1.html#V2" target="_self">Accept-Halt</a>
 <b>by </b><i><a class="ref" href="turing_1.html#D9" target="_self">Def9</a></i>;<br/>

<a NAME="E43:79"/><b>then </b><i><font color="Green">E91</font></i>: 
 <a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="turing_1.html#E41:79"><i><font color="Green">E90</font></i></a>, <a class="ref" href="turing_1.html#D10" target="_self">Def10</a></i>;<br/>
<b>hence </b><a NAME="E44:79"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 4
 <b>by </b><i><a class="txt" href="turing_1.html#E40:79"><i><font color="Green">E89</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>;<br/>

<a NAME="E45:79"/><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="txt" href="turing_1.html#E40:79"><i><font color="Green">E89</font></i></a>, <a class="txt" href="turing_1.html#E43:79"><i><font color="Green">E91</font></i></a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>26</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2;<br/>
<a NAME="E46:79"/><i><font color="Green">E93</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 4 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E47:79"/><b>then </b><i><font color="Green">E94</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="txt" href="turing_1.html#E32:79"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<i><font color="Green">E95</font></i>: <font color="Maroon">c<sub>23</sub></font> <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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> = 
<font color="Maroon">c<sub>15</sub></font> <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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E30:79"><i><font color="Green">E81</font></i></a>, <a class="txt" href="turing_1.html#E46:79"><i><font color="Green">E93</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="turing_1.html#E4:79"><i><font color="Green">E64</font></i></a>, <a class="txt" href="turing_1.html#E17:79"><i><font color="Green">E72</font></i></a>, <a class="txt" href="turing_1.html#E47:79"><i><font color="Green">E94</font></i></a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_20"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>27</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>assume </b><a NAME="E1:79_20"/><i><font color="Green">E96</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 4 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 )
 ;<br/><a NAME="E2:79_20"/><b>then </b><i><font color="Green">E97</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="turing_1.html#E32:79"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> = 
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E30:79"><i><font color="Green">E81</font></i></a>, <a class="txt" href="turing_1.html#E1:79_20"><i><font color="Green">E96</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
<b>by </b><i><a class="txt" href="turing_1.html#E17:79"><i><font color="Green">E72</font></i></a>, <a class="txt" href="turing_1.html#E2:79_20"><i><font color="Green">E97</font></i></a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="turing_1.html#E2:79"><i><font color="Green">E62</font></i></a>, <a class="txt" href="turing_1.html#E1:79_20"><i><font color="Green">E96</font></i></a>, <a class="ref" href="turing_1.html#D13" target="_self">Def13</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E50:79"/><b>then </b>
<font color="Maroon">c<sub>23</sub></font> <a href="turing_1.html#R1" target="_self">is_1_between</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 4,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="txt" href="turing_1.html#E33:79"><i><font color="Green">E83</font></i></a>, <a class="txt" href="turing_1.html#E48:79"><i><font color="Green">E95</font></i></a>, <a class="ref" href="turing_1.html#D13" target="_self">Def13</a></i>;<br/>
<b>hence </b><a NAME="E51:79"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a>  <a href="turing_1.html#R2" target="_self">storeData</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="turing_1.html#E45:79"><i><font color="Green">E92</font></i></a>, <a class="ref" href="turing_1.html#T19" target="_self">Th19</a></i>;<br/>


</div>
