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

<b>let </b><font color="Maroon">tm</font> be    <a href="turing_1.html#L1" target="_self">TuringStr</a> ;<br/><b>let </b><font color="Maroon">s</font> be   <a href="turing_1.html#NM4">All-State</a> of <font color="Maroon">tm</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="turing_1.html#NM1">State</a> of <font color="Maroon">tm</font>;<br/><b>let </b><font color="Maroon">h</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> ;<br/><b>let </b><font color="Maroon">t</font> be   <a href="turing_1.html#NM2">Tape</a> of <font color="Maroon">tm</font>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">d</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>











<b>assume </b><a NAME="E1:51"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> &amp; 1 is   <a href="turing_1.html#NM3">Symbol</a> of <font color="Maroon">tm</font> &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"><font color="Maroon">p</font>,<font color="Maroon">h</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span> &amp; the <a href="turing_1.html#U3" target="_self">Tran</a> of <font color="Maroon">tm</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</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">p</font>,1,1</span><a href="domain_1.html#K4">]</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="turing_1.html#U5" target="_self">AcceptS</a> of <font color="Maroon">tm</font> &amp; ( for <font color="Olive">i</font> being  <a href="int_1.html#NM1">Integer</a>  st <font color="Maroon">d</font> <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> <font color="Maroon">d</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> holds <br/><font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 1 ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> implies <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> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span> );<br/>
<a NAME="E2:51"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:51"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:51_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:51_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>set </b><font color="Maroon">dk</font> = <font color="Maroon">d</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>;<br/><b>reconsider </b><font color="Maroon">ik</font> = <font color="Maroon">d</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><b>set </b><font color="Maroon">sk</font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">ik</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span>;<br/><b>reconsider </b><font color="Maroon">tt</font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">ik</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K7">`3</a>  as   <a href="turing_1.html#NM2">Tape</a> of <font color="Maroon">tm</font> ;<br/><a NAME="E5:51_1_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E6:51_1_1"/>
<font color="Maroon">d</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">d</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E7:51_1_1"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">t</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ik</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, </i>;<br/><i><font color="Green">E117</font></i>:  <a href="turing_1.html#K8" target="_self">TRAN</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">ik</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span> = 
the <a href="turing_1.html#U3" target="_self">Tran</a> of <font color="Maroon">tm</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">tt</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"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">ik</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></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/>.= 
the <a href="turing_1.html#U3" target="_self">Tran</a> of <font color="Maroon">tm</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<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> <span class="p4"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">ik</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></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"><font color="Maroon">p</font>,1,1</span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i>, , <a class="txt" href="turing_1.html#E3:51"><i><font color="Green">E32</font></i></a></i>
;<br/><b>reconsider </b><font color="Maroon">T</font> = 1 as   <a href="turing_1.html#NM3">Symbol</a> of <font color="Maroon">tm</font> <b>by </b><i/>;<br/><i><font color="Green">E118</font></i>:  <a href="turing_1.html#K5" target="_self">Tape-Chg</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">ik</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></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"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">ik</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></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"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">ik</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></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">t</font>,<span class="p1">(<span class="default"><a href="turing_1.html#K7" target="_self">Head</a> <span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">ik</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></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"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">ik</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></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">t</font>,<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</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> <span class="p3"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">ik</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E3:51"><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>,<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">T</font>
<b>by </b><i><a class="ref" href="turing_1.html#T3" target="_self">Th3</a>, <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/><a NAME="E11:51_1_1"/><i><font color="Green">E121</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"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">ik</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="turing_1.html#T3" target="_self">Th3</a>, <a class="ref" href="mcart_1.html#T68">MCART_1:68</a></i>;<br/><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"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="turing_1.html#K9" target="_self">Following</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">ik</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#T2" target="_self">Th2</a>, , , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>
<br/>.= 
<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"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">ik</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></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"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">ik</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></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"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">ik</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span></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="txt" href="turing_1.html#E1:51_1_1"><i><font color="Green">E35</font></i></a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">p</font>,<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"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">ik</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></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"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">ik</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span></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="turing_1.html#T3" target="_self">Th3</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">p</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i>, <a class="txt" href="turing_1.html#E3:51"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span>

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:51_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:51"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="ref" href="turing_1.html#T1" target="_self">Th1</a>);<br/></i>
<b>hence </b><a NAME="E5:51"/>
<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> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">t</font></span><a href="domain_1.html#K4">]</a></span>
 ;<br/>


</div>
