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<b>let </b><font color="Maroon">s</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">t</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">h</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">f</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">E30</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3 &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default">0,<font color="Maroon">h</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span> &amp; <font color="Maroon">t</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">h</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font> )
 ;<br/>

<b>set </b><font color="Maroon">n1</font> = <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">f</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">n2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">f</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">f</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">n3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">f</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">f</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">f</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">E31</font></i>: 
( <font color="Maroon">t</font> <a href="turing_1.html#R1" target="_self">is_1_between</a> <font color="Maroon">h</font>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">f</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">t</font> <a href="turing_1.html#R1" target="_self">is_1_between</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">f</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">h</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">f</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">f</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">t</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">h</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">f</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">f</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">h</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">f</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">f</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">f</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>, </i>;<br/>
<a NAME="E3:79"/><b>then </b><i><font color="Green">E32</font></i>: 
( <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">f</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">k</font> being  <a href="int_1.html#NM1">Integer</a>  st <font color="Maroon">h</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">f</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">t</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 1 ) )
 
<b>by </b><i/>;<br/>
<a NAME="E4:79"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">t</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">h</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">f</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">f</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">t</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">h</font> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">f</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">f</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">f</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">k</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">h</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">f</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">f</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">k</font> &amp; <font color="Olive">k</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">h</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">f</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">f</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">f</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">t</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 1 ) )
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">h1</font> = <font color="Maroon">h</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">s03</font> = <font color="Maroon">s</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">F</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/>;<br/>
<i><font color="Green">E35</font></i>:  <a href="turing_1.html#K8" target="_self">TRAN</a> <font color="Maroon">s</font> = 
<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">s</font> <a href="mcart_1.html#K5">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s03</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">s</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="turing_1.html#T7" target="_self">Th7</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">s03</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">s</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <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">t</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">s</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <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#E3:79"><i><font color="Green">E32</font></i></a>, <a class="ref" href="turing_1.html#T6" target="_self">Th6</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"><font color="Maroon">s</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">s</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">s</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">t</font>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <font color="Maroon">s</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">s</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
<b>by </b><i>, <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">t</font>,<font color="Maroon">h</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">s</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
<b>by </b><i>, <a class="txt" href="turing_1.html#E3:79"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t</font>,<font color="Maroon">h</font>,<font color="Maroon">F</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<font color="Maroon">t</font>
<b>by </b><i>, </i>
;<br/>
<b>reconsider </b><font color="Maroon">p1</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/>;<br/>
<a NAME="E11:79"/><i><font color="Green">E114</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">s</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>;<br/>
<b>set </b><font color="Maroon">s1</font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span>;<br/>
<i><font color="Green">E116</font></i>:  <a href="turing_1.html#K9" target="_self">Following</a> <font color="Maroon">s</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">s</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">s</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">s</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">t</font></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i>, , </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">s</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">s</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">t</font></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i>, <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">p1</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="turing_1.html#D7" target="_self">Def7</a>, <a class="txt" href="turing_1.html#E3:79"><i><font color="Green">E32</font></i></a></i>
;<br/>
<a NAME="E13:79"/><i><font color="Green">E117</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">p1</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">p1</font>,0,1</span><a href="domain_1.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="turing_1.html#T7" target="_self">Th7</a>, <a class="ref" href="turing_1.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E14:79"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">p1</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#T7" target="_self">Th7</a></i>;<br/>
<b>set </b><font color="Maroon">m1</font> = <span class="p1">(<span class="default"><font color="Maroon">f</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">E121</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>assume </b><a NAME="E1:79_4"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">h</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">f</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_4"/>
<font color="Maroon">h</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">h</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_4"/><b>then </b>
<font color="Maroon">h</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="turing_1.html#T12" target="_self">Th12</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E4:79_4"/>
<font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="turing_1.html#T12" target="_self">Th12</a>, </i>;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon">s2</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">p1</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</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">f</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">t1</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">E124</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">p1</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</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">f</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">p1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</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">f</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">t1</font></span><a href="domain_1.html#K4">]</a></span> &amp; ( for <font color="Olive">i</font> being  <a href="int_1.html#NM1">Integer</a>  st <font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">h</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">f</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">t1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( for <font color="Olive">i</font> being  <a href="int_1.html#NM1">Integer</a>  st ( <font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">i</font> or <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">h</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">f</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">t1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) )
 <b>by </b><i>, , , , </i>;<br/>
<a NAME="E18:79"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</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">f</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="ref" href="turing_1.html#T12" target="_self">Th12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s23</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">p1</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</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">f</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">E126</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">p1</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</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">f</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="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">p1</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</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">f</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">s23</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">p1</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</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">f</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#T7" target="_self">Th7</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">p1</font>,<span class="p2">(<span class="default"><font color="Maroon">s23</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">p1</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</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">f</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#T12" target="_self">Th12</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">t1</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">p1</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</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">f</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#T12" target="_self">Th12</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="ref" href="turing_1.html#T12" target="_self">Th12</a>, <a class="ref" href="turing_1.html#T13" target="_self">Th13</a>, <a class="txt" href="turing_1.html#E3:79"><i><font color="Green">E32</font></i></a>, <a class="ref" href="turing_1.html#T6" target="_self">Th6</a></i>
;<br/>
<i><font color="Green">E127</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">p1</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</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">f</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">p1</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</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">f</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">p1</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</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">f</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">t1</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">p1</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</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">f</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">p1</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</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">f</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="ref" href="turing_1.html#T12" target="_self">Th12</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">t1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</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">f</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">p1</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</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">f</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="ref" href="turing_1.html#T12" target="_self">Th12</a>, <a class="txt" href="turing_1.html#E3:79"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</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">f</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">F</font>
<b>by </b><i><a class="ref" href="turing_1.html#T14" target="_self">Th14</a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<font color="Maroon">t1</font>
<b>by </b><i><a class="ref" href="turing_1.html#T13" target="_self">Th13</a>, </i>
;<br/>
<b>reconsider </b><font color="Maroon">p2</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/>;<br/>
<a NAME="E23:79"/><i><font color="Green">E129</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">p1</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</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">f</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="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="turing_1.html#T14" target="_self">Th14</a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>;<br/>
<b>set </b><font color="Maroon">j1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</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">f</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">k1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</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">f</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">s3</font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">k1</font>,<font color="Maroon">t1</font></span><a href="domain_1.html#K4">]</a></span>;<br/>
<i><font color="Green">E130</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">p1</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</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">f</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">p1</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</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">f</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">p1</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</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">f</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">p1</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</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">f</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">t1</font></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="ref" href="turing_1.html#T12" target="_self">Th12</a>, <a class="txt" href="turing_1.html#E9:79"><i><font color="Green">E75</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">p1</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</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">f</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">p1</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</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">f</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">t1</font></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i><a class="ref" href="turing_1.html#T14" target="_self">Th14</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">p2</font>,<font color="Maroon">k1</font>,<font color="Maroon">t1</font></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="turing_1.html#T12" target="_self">Th12</a>, <a class="ref" href="turing_1.html#T15" target="_self">Th15</a>, <a class="txt" href="turing_1.html#E3:79"><i><font color="Green">E32</font></i></a></i>
;<br/>
<a NAME="E26:79"/><i><font color="Green">E131</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">p2</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">p2</font>,0,1</span><a href="domain_1.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="turing_1.html#T7" target="_self">Th7</a>, <a class="ref" href="turing_1.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E27:79"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">p2</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#T7" target="_self">Th7</a></i>;<br/>
<b>set </b><font color="Maroon">m2</font> = <span class="p1">(<span class="default"><font color="Maroon">f</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">E161</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>assume </b><a NAME="E1:79_8"/><i><font color="Green">E162</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</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">f</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">i</font> &amp; <font color="Maroon">i</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">h</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">f</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">f</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_8"/>
<span class="p1">(<span class="default"><font color="Maroon">h</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">f</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">h</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">f</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_8"/><b>then </b><i><font color="Green">E163</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</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">f</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">i</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="turing_1.html#T12" target="_self">Th12</a></i>
<br/>.= 
1
<b>by </b><i>, , , </i>
;<br/><br/></div><b>end;</b></div>
<b>set </b><font color="Maroon">s4</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">p2</font>,<font color="Maroon">k1</font>,<font color="Maroon">t1</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">f</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">t2</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">E164</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">p2</font>,<font color="Maroon">k1</font>,<font color="Maroon">t1</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">f</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">p2</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">h</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">f</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">f</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">t2</font></span><a href="domain_1.html#K4">]</a></span> &amp; ( for <font color="Olive">i</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">h</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">f</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">i</font> &amp; <font color="Olive">i</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">h</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">f</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">f</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">t2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( for <font color="Olive">i</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">h</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">f</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">i</font> or <font color="Olive">i</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">h</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">f</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">f</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">t2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) )
 <b>by </b><i><a class="ref" href="turing_1.html#D10" target="_self">Def10</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">f</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">E165</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">f</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">h</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">f</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">f</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">E166</font></i>: <font color="Maroon">t2</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">h</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">f</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">f</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">t1</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">h</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">f</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">f</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/>
<br/>.= 
0
<b>by </b><i>, <a class="ref" href="turing_1.html#T12" target="_self">Th12</a>, </i>
;<br/>
<b>reconsider </b><font color="Maroon">s43</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">p2</font>,<font color="Maroon">k1</font>,<font color="Maroon">t1</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">f</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">E170</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">p2</font>,<font color="Maroon">k1</font>,<font color="Maroon">t1</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">f</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="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">p2</font>,<font color="Maroon">k1</font>,<font color="Maroon">t1</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">f</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">s43</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">p2</font>,<font color="Maroon">k1</font>,<font color="Maroon">t1</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">f</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#T7" target="_self">Th7</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">p2</font>,<span class="p2">(<span class="default"><font color="Maroon">s43</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">p2</font>,<font color="Maroon">k1</font>,<font color="Maroon">t1</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">f</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="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">t2</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">p2</font>,<font color="Maroon">k1</font>,<font color="Maroon">t1</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">f</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="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="ref" href="turing_1.html#T16" target="_self">Th16</a>, <a class="txt" href="turing_1.html#E3:79"><i><font color="Green">E32</font></i></a>, <a class="ref" href="turing_1.html#T6" target="_self">Th6</a></i>
;<br/>
<i><font color="Green">E172</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">p2</font>,<font color="Maroon">k1</font>,<font color="Maroon">t1</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">f</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">p2</font>,<font color="Maroon">k1</font>,<font color="Maroon">t1</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">f</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">p2</font>,<font color="Maroon">k1</font>,<font color="Maroon">t1</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">f</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">t2</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">p2</font>,<font color="Maroon">k1</font>,<font color="Maroon">t1</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">f</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">p2</font>,<font color="Maroon">k1</font>,<font color="Maroon">t1</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">f</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="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">t2</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">h</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">f</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">f</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">p2</font>,<font color="Maroon">k1</font>,<font color="Maroon">t1</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">f</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#E3:79"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
 <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <font color="Maroon">t2</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">h</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">f</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">f</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">F</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>
<br/>.= 
<font color="Maroon">t2</font>
<b>by </b><i><a class="ref" href="turing_1.html#T16" target="_self">Th16</a>, </i>
;<br/>
<a NAME="E37:79"/><i><font color="Green">E173</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">p2</font>,<font color="Maroon">k1</font>,<font color="Maroon">t1</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">f</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="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>;<br/>
<i><font color="Green">E174</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">p2</font>,<font color="Maroon">k1</font>,<font color="Maroon">t1</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">f</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">p2</font>,<font color="Maroon">k1</font>,<font color="Maroon">t1</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">f</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">p2</font>,<font color="Maroon">k1</font>,<font color="Maroon">t1</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">f</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">p2</font>,<font color="Maroon">k1</font>,<font color="Maroon">t1</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">f</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">t2</font></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i>, , </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">p2</font>,<font color="Maroon">k1</font>,<font color="Maroon">t1</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">f</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">p2</font>,<font color="Maroon">k1</font>,<font color="Maroon">t1</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">f</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">t2</font></span><a href="mcart_1.html#K3">]</a></span>
<b>by </b><i>, <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">h</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">f</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">f</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">t2</font></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i>, , <a class="txt" href="turing_1.html#E3:79"><i><font color="Green">E32</font></i></a></i>
;<br/>
<b>set </b><font color="Maroon">Rs</font> = <span class="p1">(<span class="default"><a href="turing_1.html#K10" target="_self">Computation</a> <font color="Maroon">s</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">f</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">f</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">E175</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_13"><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">s</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">f</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">f</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">s</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">f</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">f</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/>
<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">p1</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</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">f</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">f</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#D8" target="_self">Def8</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">p1</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</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">f</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">f</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/>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E176</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_14"><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">s</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">f</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">f</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">p1</font>,<font color="Maroon">h1</font>,<font color="Maroon">t</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">f</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">f</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>, </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">p2</font>,<font color="Maroon">k1</font>,<font color="Maroon">t1</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">f</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="ref" href="turing_1.html#D9" target="_self">Def9</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">h</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">f</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">f</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">t2</font></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div>
<b>then </b><i><font color="Green">E177</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">s</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">f</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">f</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#T7" target="_self">Th7</a></i>
;<br/>
<b>hence </b><a NAME="E42:79"/>
<font color="Maroon">s</font> is <a href="turing_1.html#V2" target="_self">Accept-Halt</a>
 <b>by </b><i/>;<br/>

<a NAME="E43:79"/><b>then </b><i><font color="Green">E178</font></i>: 
 <a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">s</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">s</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">f</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">f</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>, </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">s</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">h</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">f</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">f</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="ref" href="mcart_1.html#T68">MCART_1:68</a></i>;<br/>

<a NAME="E45:79"/><i><font color="Green">E179</font></i>: 
<span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">s</font></span>)</span> <a href="mcart_1.html#K7">`3</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font>
 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>;<br/>
<b>set </b><font color="Maroon">m3</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">h</font> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">f</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">f</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">f</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">E180</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">f</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">f</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">h</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">f</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">f</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">f</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">E181</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">f</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">h</font> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">f</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">f</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">f</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="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<i><font color="Green">E184</font></i>: <font color="Maroon">t2</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">h</font> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><font color="Maroon">f</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">f</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">f</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">t1</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">h</font> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><font color="Maroon">f</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">f</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">f</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>, </i>
<br/>.= 
0
<b>by </b><i>, <a class="ref" href="turing_1.html#T12" target="_self">Th12</a>, </i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_17"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>assume </b><a NAME="E1:79_17"/><i><font color="Green">E185</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">f</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">f</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">k</font> &amp; <font color="Maroon">k</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">h</font> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">f</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">f</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">f</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_17"/><b>then </b><i><font color="Green">E186</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">f</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">k</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>thus </b><font color="Maroon">t2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="turing_1.html#T12" target="_self">Th12</a>, </i>
<br/>.= 
1
<b>by </b><i>, , </i>
;<br/></div><b>end;</b></div>
<a NAME="E50:79"/><b>then </b>
<font color="Maroon">t2</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">h</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">f</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">f</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">h</font> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">f</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">f</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">f</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="ref" href="turing_1.html#T16" target="_self">Th16</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">s</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">h</font> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">f</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">f</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">f</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>, </i>;<br/>


</div>
