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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="turing_1.html#NM4">All-State</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="turing_1.html#NM2">Tape</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:55_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <a href="comput_1.html#K17">[+]</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default">the <a href="turing_1.html#U4" target="_self">InitS</a> of <a href="turing_1.html#K14" target="_self">SumTuring</a> ,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="turing_1.html#R2" target="_self">storeData</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:55_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="turing_1.html#T32" target="_self">Th32</a>, <a class="ref" href="finseq_2.html#T120">FINSEQ_2:120</a></i>;<br/><a NAME="E4:55_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="turing_1.html#E1:55_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="turing_1.html#D15" target="_self">Def15</a></i>;<br/><a NAME="E5:55_1"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="finseq_4.html#K3">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="turing_1.html#E3:55_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T60">FINSEQ_1:60</a></i>;<br/><a NAME="E6:55_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="turing_1.html#R2" target="_self">storeData</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="finseq_4.html#K3">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="turing_1.html#E1:55_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="turing_1.html#E3:55_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T60">FINSEQ_1:60</a></i>;<br/><a NAME="E7:55_1"/><i><font color="Green">E47</font></i>: 
( <span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a>  <a href="turing_1.html#R2" target="_self">storeData</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="turing_1.html#E1:55_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="turing_1.html#E4:55_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="turing_1.html#E5:55_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="turing_1.html#T31" target="_self">Th31</a></i>;<br/><b>thus </b><a NAME="E8:55_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="turing_1.html#V2" target="_self">Accept-Halt</a>
 <b>by </b><i><a class="txt" href="turing_1.html#E1:55_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="turing_1.html#E4:55_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="turing_1.html#E5:55_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="turing_1.html#T31" target="_self">Th31</a></i>;<br/><b>take </b><font color="Maroon">c<sub>7</sub></font> = 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>3</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font>;<br/><b>thus </b><a NAME="E9:55_1"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="turing_1.html#E4:55_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="turing_1.html#E6:55_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="turing_1.html#T31" target="_self">Th31</a></i>;<br/><b>thus </b><a NAME="E10:55_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <a href="comput_1.html#K17">[+]</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="turing_1.html#E3:55_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="comput_1.html#T89">COMPUT_1:89</a></i>;<br/><b>thus </b><a NAME="E11:55_1"/>
<span class="p1">(<span class="default"><a href="turing_1.html#K11" target="_self">Result</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a>  <a href="turing_1.html#R2" target="_self">storeData</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="turing_1.html#E7:55_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_1.html#D9">FINSEQ_1:def 9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:55"/>
 <a href="turing_1.html#K14" target="_self">SumTuring</a>  <a href="turing_1.html#R3" target="_self">computes</a>  <a href="comput_1.html#K17">[+]</a> 
 <b>by </b><i><a class="ref" href="turing_1.html#D16" target="_self">Def16</a></i>;<br/>


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