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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 <span class="p1">(<span class="default"><a href="turing_1.html#K18" target="_self">ZeroTuring</a>  <a href="turing_1.html#K32" target="_self">';'</a> <a href="turing_1.html#K16" target="_self">SuccTuring</a> </span>)</span>;<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#K18" target="_self">ZeroTuring</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>assume </b><a NAME="E1:102"/><i><font color="Green">E75</font></i>: 
( <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"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">0,0</span><a href="domain_1.html#K1">]</a></span>,<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_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> 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>6</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default">the <a href="turing_1.html#U4" target="_self">InitS</a> of <a href="turing_1.html#K18" target="_self">ZeroTuring</a> ,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span>;<br/>
<a NAME="E3:102"/><i><font color="Green">E76</font></i>: 
0 <a href="hidden.html#R1">=</a> the <a href="turing_1.html#U4" target="_self">InitS</a> of <a href="turing_1.html#K18" target="_self">ZeroTuring</a> 
 
<b>by </b><i><a class="ref" href="turing_1.html#D20" target="_self">Def20</a></i>;<br/>
<a NAME="E4:102"/><b>then </b><i><font color="Green">E77</font></i>: 
( <span class="p1"><a href="domain_1.html#K4">[</a><span class="default">the <a href="turing_1.html#U4" target="_self">InitS</a> of <a href="turing_1.html#K18" target="_self">ZeroTuring</a> ,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span> is <a href="turing_1.html#V2" target="_self">Accept-Halt</a> &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default">the <a href="turing_1.html#U4" target="_self">InitS</a> of <a href="turing_1.html#K18" target="_self">ZeroTuring</a> ,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default">the <a href="turing_1.html#U4" target="_self">InitS</a> of <a href="turing_1.html#K18" target="_self">ZeroTuring</a> ,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span></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"><font color="Maroon">c<sub>3</sub></font>,0</span><a href="finseq_4.html#K2">*&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="turing_1.html#E1:102"><i><font color="Green">E75</font></i></a>, <a class="txt" href="turing_1.html#E100"><i><font color="Green">Lemma74</font></i></a></i>;<br/>
the <a href="turing_1.html#U1" target="_self">Symbols</a> of <a href="turing_1.html#K18" target="_self">ZeroTuring</a>  = 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
<b>by </b><i><a class="ref" href="turing_1.html#D20" target="_self">Def20</a></i>
<br/>.= 
the <a href="turing_1.html#U1" target="_self">Symbols</a> of <a href="turing_1.html#K16" target="_self">SuccTuring</a> 
<b>by </b><i><a class="ref" href="turing_1.html#D18" target="_self">Def18</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default">the <a href="turing_1.html#U4" target="_self">InitS</a> of <a href="turing_1.html#K18" target="_self">ZeroTuring</a> ,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></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#K16" target="_self">SuccTuring</a>  ;<br/>
<a NAME="E7:102"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="turing_1.html#R2" target="_self">storeData</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,0</span><a href="finseq_4.html#K2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="turing_1.html#E4:102"><i><font color="Green">E77</font></i></a>, <a class="ref" href="turing_1.html#T53" target="_self">Th53</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default">the <a href="turing_1.html#U4" target="_self">InitS</a> of <a href="turing_1.html#K16" target="_self">SuccTuring</a> ,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K4">]</a></span>;<br/>
<a NAME="E8:102"/><i><font color="Green">E79</font></i>: 
0 <a href="hidden.html#R1">=</a> the <a href="turing_1.html#U4" target="_self">InitS</a> of <a href="turing_1.html#K16" target="_self">SuccTuring</a> 
 
<b>by </b><i><a class="ref" href="turing_1.html#D18" target="_self">Def18</a></i>;<br/>
<a NAME="E9:102"/><b>then </b><i><font color="Green">E80</font></i>: 
( <span class="p1"><a href="domain_1.html#K4">[</a><span class="default">the <a href="turing_1.html#U4" target="_self">InitS</a> of <a href="turing_1.html#K16" target="_self">SuccTuring</a> ,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K4">]</a></span> is <a href="turing_1.html#V2" target="_self">Accept-Halt</a> &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default">the <a href="turing_1.html#U4" target="_self">InitS</a> of <a href="turing_1.html#K16" target="_self">SuccTuring</a> ,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default">the <a href="turing_1.html#U4" target="_self">InitS</a> of <a href="turing_1.html#K16" target="_self">SuccTuring</a> ,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K4">]</a></span></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"><font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="turing_1.html#E7:102"><i><font color="Green">E78</font></i></a>, <a class="ref" href="turing_1.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E10:102"/><i><font color="Green">E81</font></i>: 
<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">the <a href="turing_1.html#U4" target="_self">InitS</a> of <span class="p2">(<span class="default"><a href="turing_1.html#K18" target="_self">ZeroTuring</a>  <a href="turing_1.html#K32" target="_self">';'</a> <a href="turing_1.html#K16" target="_self">SuccTuring</a> </span>)</span>,<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>
 
<b>by </b><i><a class="txt" href="turing_1.html#E1:102"><i><font color="Green">E75</font></i></a>, <a class="txt" href="turing_1.html#E3:102"><i><font color="Green">E76</font></i></a>, <a class="txt" href="turing_1.html#E8:102"><i><font color="Green">E79</font></i></a>, <a class="ref" href="turing_1.html#D32" target="_self">Def32</a></i>;<br/>
<b>hence </b><a NAME="E11:102"/>
<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="txt" href="turing_1.html#E4:102"><i><font color="Green">E77</font></i></a>, <a class="txt" href="turing_1.html#E9:102"><i><font color="Green">E80</font></i></a>, <a class="ref" href="turing_1.html#T50" target="_self">Th50</a></i>;<br/>

<a NAME="E12:102"/><i><font color="Green">E82</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#K6">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default">the <a href="turing_1.html#U4" target="_self">InitS</a> of <a href="turing_1.html#K16" target="_self">SuccTuring</a> ,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K6">`2</a>  &amp; <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> <span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default">the <a href="turing_1.html#U4" target="_self">InitS</a> of <a href="turing_1.html#K16" target="_self">SuccTuring</a> ,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K7">`3</a>  )
 
<b>by </b><i><a class="txt" href="turing_1.html#E4:102"><i><font color="Green">E77</font></i></a>, <a class="txt" href="turing_1.html#E9:102"><i><font color="Green">E80</font></i></a>, <a class="txt" href="turing_1.html#E10:102"><i><font color="Green">E81</font></i></a>, <a class="ref" href="turing_1.html#T50" target="_self">Th50</a></i>;<br/>
<b>thus </b><a NAME="E13:102"/>
<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> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="turing_1.html#E4:102"><i><font color="Green">E77</font></i></a>, <a class="txt" href="turing_1.html#E9:102"><i><font color="Green">E80</font></i></a>, <a class="txt" href="turing_1.html#E10:102"><i><font color="Green">E81</font></i></a>, <a class="ref" href="turing_1.html#T50" target="_self">Th50</a></i>;<br/>

<b>thus </b><a NAME="E14:102"/>
<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"><font color="Maroon">c<sub>3</sub></font>,1</span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="turing_1.html#E9:102"><i><font color="Green">E80</font></i></a>, <a class="txt" href="turing_1.html#E12:102"><i><font color="Green">E82</font></i></a>, <a class="ref" href="turing_1.html#T53" target="_self">Th53</a></i>;<br/>


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